,transcrib]'s ,note3 ,?is text is int5d$ = us]s ^: text read]s _c use ! real 7,unicode_/,,utf-#h7 v]sion ( ! file4 ,lett]s ) ov]l9e 7repres5t+ foll[+ n or m7 h be5 writt5 out1 )\t fur!r m>k+4 ,! ae ligature is %[n z two lett]s4 ,! pilcr[ is giv5 z ,sec44,greek--ma9ly 9 ! introduction--has be5 translit]at$ & %[n betwe5 m>ks1 ) eta & omega r5d]$ z e3 & o34 ,unless o!rwise not$1 spell+1 punctuation & capitalization are un*ang$4 ,details are giv5 at ! end ( ! e-text4 ,sup]scripts are %[n ) a c>et 1 add+ braces :]e necess>y = cl>ity4 ,t]m9ology3 "r l9e /raight l9e gem[ 7l9e7 p>allel gemew tw9 squ>e quadrilat]al 7also spell$ squire1 squyre7 also angle squ>e z describ$ "u hexagons 7siseangles7 likeiamme p>allelogram iam7me7 jamb limb1 side longsqu>e rectangle t\* l9e tang5t cantle segm5t ( a circle cantle slice 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,! pa?way to ,,kn[l$g1 ,,contai- ,,n+ ,,! ,,fir/ ,,pr9- ciples ( ,geometrie1 z !y may mo/e aptly be appli$ vn- to practice1 bo! = vse ( in/rum5tes ,geome- tricall1 & a/rono- micall & also = proiection ( plattes 9 eu]ye k9de1 & !r=e mu* ne- cess>y = all sortes ( m54 ,geometries v]dicte ,all fres%e f9e wittes by me are fil$1 ,all grosse dull wittes wi%e me exil$3 ,?\ghe no mannes witte reiect w ,i1 ,yet z !y be1 ,i wyll !m trye4 ,! argum5tes ( ! f\re bookes ,! fir/ booke decl>e? ! def9itions ( ! t]mes & names vs$ 9 ,geometry1 ) c]ta9e ( ! *iefe gr\nds :]eon ! arte is f\nd$4 ,& !n tea*e? ^? conclusions1 :i* may s]ue diu]sely 9 al workes ,geometricall4 ,! second booke do? sette =? ! ,!oremes1 7:i*e maye be call$ appr\$ tru!s7 s]u+e = ! due k & sure pro(e ( all conclusions & workes 9 ,geometrye4 ,! ?ird booke intreate? ( diu]s =mes1 & sondry protractions !reto belong+1 ) ! vse ( c]ta9 conclusions4 ,! f\r? booke tea*e? ! "r ord] ( measur+e all platte =mes1 & bodies also1 by reson ,geometricall4 ,,to ,,! ,,g5tle ,,read]4 ,excvse me1 g5tle r$] if oughte be amisse1 /raung pa?s ar n trod5 al truly at ! fir/3 ! way mu/e ne$es be comb]\s1 :] none ha! gone be=e4 ,:]e no man ha! geu5 light1 lighte is x to (f5d1 b :5 ! light is %ew$ ones1 light is x to am5de4 ,if my light may s light "s o!r1 to espie & m>ke my faultes1 ,i wi% x may s light5 !m1 t !y may voide (f5ce4 ,( /agg]+e & /ombl+e1 & vncon/aunt turmoil+e3 (t5 (f5d+1 & seldome am5d+1 su* vices to es*ewe1 & _! f9e wittes to %ew t !y may w9ne ! praise1 & ,i to hold ! c&le1 :ile/ !y _! glori\s works ) eloqu5ce sette =?1 s cunn+ly inu5t$1 s f9ely indit$1 t my bokes maie seme wor?ie to occupie no roome4 ,= nei!r is mi wit s f9elie fil$1 no!r mi le>n+ s l>gly lettr$1 no!r yet mi lais] s quiet & vncomb]$1 t ,i maie p]=m iu/lie s le>n$ a lab\re or accord+lie to accompli%e s haulte an en=cem5t1 yet maie ,i ?9ke ?us3 ,?is c&le did ,i light3 ?is light haue ,i k9del$3 t le>n$ m5 maie se1 to practise _! p5nes1 _! eloqu5ce to aduaunce1 to regi/] _! names 9 ! booke ( memorie ,i drew ! platte rudelie1 :]eon !i maie builde1 :om god ha? indu$ ) le>n+ & liuelihod4 ,= liu+ by lab\re do? le>n+ s h9d]1 t le>n+ s]ue? liu+e1 :i*e is a p]u]s trade4 ,yet z c>efull familie %all cease hir cruell call+e1 & suffre anie lais] to le>n+e to repaire1 ,i w n cease f trauaile ! pa! s to trade1 t f9] wittes maie fa%ion !m selues ) su* glims+e dull light1 a m complete woorke at lais] to f9is%e1 ) inu5cion agreable1 & aptnes ( eloqu5ce4 ,& ?is g5tle read] ,i h>telie prote/ :]e err\re ha! happ5$ ,i wis%e x r$re/4 ,,to ,,! ,,mo/ ,,no- ble & puissaunt pr9ce ,edw>de ! sixte by ! grace ( ,god1 ( ,en- gl& ,fraunce & ,irel& kynge1 de- f5d\r ( ! fai!1 & ( ! ,*ur*e ( ,engl& & ,ire- l&e 9 e>? ! su- preme head4 ,x is n vnkn[5 to y\re maie/ie1 mo/e s\]aigne lorde1 :at great disceptacion ha? be5 amonge/ ! wyttie m5 ( all nacions1 = ! exacte kn[el$ge ( true felicitie1 bo! :at x is1 & :]e9 x consi/e?3 t\*ynge :i*e ?yng1 _! op9ions almo/e 7 z _m 9 numbre1 z 7 ! p]sons ( !m1 t ei!r disput$ or wrote !re(4 ,b & if ! diu]sitie ( op9ions 9 ! vulg> sort = placyng ( _! felicitie %all be consid]$ also1 ! v>ietie %all be f\nd s great1 & ! op9ions s dissonant1 yea pla9ly mon/]\se1 t no hone/ witte w\ld v\*esafe to lose "t 9 he>yng !m1 or r 7z ,i may saie7 no witte is ( s exact remembrance1 t c consid] toge!r ! mon/]\se multitude ( !m all4 ,& yet n )/&yng ?is repugnant diu]sitie1 9 two ?ynges d !y all agree4 ,fir/ all d agre1 t felicitie is & ought to be ! /op & end ( all _! doynges1 s t he t ha? a full desire to any ?yng h[ s eu] x be e/em$ ( o!r m51 yet he e/eme? him self happie1 if he maie obta9 x3 & contr>y waies vnhappie if he c n atta9e x4 ,& !r=e d all m5 put _! :ole /udie to gette t ?yng1 :]9 !y haue p]swad$ !m self t felicitie do? consi/4 ,:]=e "s :i*e put _! felicitie 9 f$yng _! bellies1 ?9ke no pa9 to be h>d1 nor no d$e to be vnhone/1 t may be a meanes to fill t f\le pan*e4 ,o!r :i* put _! felicitie 9 play & ydle pa/imes1 iudge no "t euill sp5t1 t is employ$ !reab\t3 nor no fraude vnlawfull t may fur!r _! w9n+4 ,if ,i %\ld p>ticul>ly ou]runne b ! common sortes ( m51 :i* put _! felicitie 9 _! desires1 x wold make a great boke ( x self4 ,!r=e wyl ,i let !m al g1 & conclude z ,i began1 ,t all m5 employ _! :ole endeu\r to t ?+1 :]9 !i ?9ke felicitie to /&4 :i*e ?yng :o s li/e? to m>k exactly1 %all be able to espie & iudge ! natures ( al m51 ^: conu]sacion he do? "k1 ?\gh !i vse great dissimulacion to col\r _! desires1 especially :5 !y p]ceiue o!r m5 to mislyke t :i* !i s mu* desire3 ,= no man wold gladly haue 8 appetite impr\$4 ,& h]( comme? t seconnde ?+ :]9 al agree1 t eu]y man w\ld mo/ gladly w9 all o!r m5 to 8 sect1 & to make !m ( 8 op9ion1 & z f> z he d>e1 w dispraise all o!r m5s iudgem5tes1 & praise 8 own waies only1 onles x be :5 he dissimule?1 & t = ! fur!rance ( 8 own purpose4 ,& ?is prop]tie also do? geue great light to ! full k ( m5s natures1 :i* z all m5 ought to obs]ue1 s pr9ces ab\e o!r haue mo/ cause to m>k = sundrie occasions :i* may lye !m on1 :]( ,i %all n n$e to speke any f>!r1 consid]yng n only ! greatnes ( wit1 & exactnes ( iudgem5t :i*e god ha? l5t vnto y\r highnes p]son1 b also ye mo/ graue wisdom & pr(\nd k ( y\r maie/ies mo/ honorable c\ncel1 by :om y\r highnes may s suffici5tly vnd]/& all ?+es conu5i5t1 t lesse %al x n$e to vnd]/& by priuate ready+1 b yet n vtt]ly to refuse to read z (t5 z occasion may s]ue1 = bokes d>e speake1 :5 m5 fe>e to displease4 ,b to returne agayne to my fir/e matt]1 if none o!r good ?+ maie be l]n$ at _! man]s1 :i* s wrongfully place _! felicity1 9 s mis]able a condition 7t :ile !y ?9ke !m selfes happy1 _! felicitie mu/ n$es seme vnluckie1 to be by !m s euill plac$7 yet ?is may m5 le>n at !m1 by ^? 4ij4 spectacles to espye ! secrete natures & dispositions ( o!rs1 :i*e ?yng vnto a wise man is mu*e auailable4 ,& ?us w ,i omit ?is great tablem5t ( vnhappie hap1 & wil come to 4iij4 o!r sortes ( a bett] degre1 :]( ! "o putte? felicitie to consi/ 9 p[] & royaltie4 ,! second sorte vnto p[] annexe? worldly wisdome1 ?9kyng him full happie1 t c\ld attayn ^? two1 :]by he might n onely haue k 9 all ?ynges1 b also p[] to bryng 8 desires to ende4 ,! ?yrd sort e/eme? true felicitie to consi/ 9 wysdom annex$ ) v]tu\se man]s1 ?9kyng t !y c take h>me ( no?yng1 if !y c ) _! wys$ome ou]come all vyces4 ,( ! fir/e ( ^? ?ree sortes !re ha? be5 a great numbre 9 all ages1 yea _m mightie k+es & great g\]n\res :i* c>$ n greately h[e !y myght at*ieue _! p\rpose1 s t !y dyd preuayle3 nor did n take any greatt] c>e = g\]nance1 !n to kepe ! p 9 onely fe>e ( !m1 ,^: common s5t5ce 0 alwaies ?is3 ,od]9t dum metuant4 ,& :at good successe su*e m5ne _h1 all hi/ories doe report4 ,yet haue !y n want$ excuses3 yea ,iulius ,caes> 7:i*e 9 d$e 0 ( ! second sorte7 make? a kynde ( excuse by 8 common s5t5ce1 = !m ( t fyr/e sorte1 = he 0 eu] woonte to saie3 eip] g> adikeim *re31 turannidos p]i kalli/om adikeim1 t' alla d' eusebeim *reo3m4 ,:i*e s5t5ce ,i wys%e _h neu] be5 le>n$ out ( ,grecia4 ,b n[ to speake ( ! second sort1 ( :i*e !re ha! be5 v]ye _m also1 yet = ?is pres5t "t amonge/ !m all1 ,i wyll take ! exaumples ( kynge ,phylippe ( ,mac$onie1 & ( ,alex&] 8 sonne1 t valiaunt conqu]\r4 ,fir/ ( k+e ,phylip x appe>e? by 8 lett] s5te vnto ,ari/otle t fam\s philosoph]1 t he m delit$ 9 ! bir! ( 8 sonne1 = ! hope ( le>n+ & good education1 t might happ5 to him by ! said ,ari/otle1 !n he didde reioyse 9 ! cont9uaunce ( 8 succession1 = ^! 7 8 wordes & 8 :ole epi/le1 wor?ye to bee remembr$ & regi/r$ eu]y :]e4 ,philippos ,ari/otelei *aireim4 e/he moi gegonota huom4 polle3m oum tois !ois *>im e*o31 ou* h\to3s epi te3 g5ne3sei t\ paidos1 ho3s epi to3 kata te3m se3m he3likiam autom gegon5ai elpizo3 g> autom hupo s\ graph5ta kai paideu!nta axiom ese/hai kai heimo3m kai te3s to3m pragmato3m diado*e3s4 ,t is ?us 9 s5se1 ,philip vnto ,ari/otle s5de? gretyng4 ,y %all vnd]/&e1 t ,i haue a sonne borne1 = :i*e cause ,i yelde vnto ,god mo/e h>tie ?ankes1 n s mu*e = ! byr! ( ! *ilde1 z t x 0 8 *aunce to be borne 9 y\r tyme4 ,= my tru/ is1 t he %all be s br\ght vp & in/ruct$ by y1 t he %all become wor?ie n only to be nam$ our sonne1 b also to be ! success\r ( our affayres4 ,& 8 good desire 0 n all vayne1 = x app]$ t ,alex&] 0 neu] s busi$ ) w>res 7yet 0 he neu] out ( mo/e t]rible battaile7 b t 9 ! middes !re( he _h 9 remembraunce 8 /udies1 & caus$ 9 all c\ntreies z he w5t1 all /range bea/es1 f[les & fis%es1 to be tak5 & kept = ! ayd ( t kn[l$g1 :i* he le>n$ ( ,ari/otle3 ,& also to be _h ) him alwayes a greate numbre ( le>n$ m54 ,& 9 ! mo/e busye tyme ( all 8 w>res aga9/ ,d>ius k+e ( ,p]sia1 :5 he h>de t ,ari/otle _h putte =! c]ta9e bookes ( su*e k :]e9 he hadde be=e /udi$1 hee 0 (f5d$ ) ,ari/otle1 & wrote to hym ?is lett]4 ,alez&ros ,arisotelei eu pratteim4 ,ouk or?o3s epoie3sas ekd\s t\s akroamatik\s to3m logo3m1 t9i g> dioisome3n he3meis to3m allo3m1 ei ka?' h\s epaideu!3m5 log\s1 h\toi panto3m esontai ko9oi1 ego3 de b\loi me3m am tais p]i ta ari/a empeiriais1 e3 tais dunamesi diaph]im4 erro3s4 t is ,alex&] vnto ,ari/otle s5de? greet+4 ,y haue n doone well1 to put =! ^? bookes ( secrete phylosophy intitul$1 akroamatikoi4 ,= :]9 %all we excell o!r1 yf t k t wee haue /udi$1 %all be made comm5 to all o!r m51 namely si! our desire is to excelle o!r m5 9 exp]i5ce & k1 r !n 9 p[] & /r5g?4 ,f>ewell4 ,by :y*e lettre x appe>e? t hee e/em$ le>n+e & k ab\e p[] ( m54 ,& ! l iudgem5t did he vtt]1 :5 he beheld ! /ate ( ,diog5es ,l9icus1 adiudg+e x ! be/e /ate next to 8 owne1 s t he said3 ,if ,i 7 n ,alex&]1 ,i wolde wi%e to be ,diog5es4 ,:]eby ape>e?1 h[ he e/eem$ le>n+1 & :at felicity he putte !r91 reput+ al ! worlde saue him selfe to be inf]i\r to ,diog5es4 ,& bi al coniecturs1 ,alex&] did e/eme ,diog5es "o ( !m :i*e contemn$ ! va9e e/imation ( ! disceitfull _w1 & put 8 :ole felicity 9 kn[l$g ( v]tue1 & practise ( ! same1 ?\gh "s reporte t he knew m v]tue !n he fol[$3 ,b :atso eu] he 01 x appe>e? t ,socrates & ,plato & _m o!r did =sake _! liu+s & sel away _! patrimony1 to ! int5t to seeke & trauaile = le>n+1 :i* examples ,i %all n ne$ to repete to y\r ,maie/y1 p>tly = t y\r highnes do? (t5 reade !m & o!r lyke1 & p>tly si? y\r maie/y ha? at h& su* le>n$ s*oolemay/]s1 :i* c mu* bett] !n ,i1 decl>e !m vnto y\r highnes1 & t m l>gely also !n ! %ort5es ( ?ys epi/le w p]mit4 ,b ?ys may ,i yet adde1 t ,k+ ,salomon ^: r5\me spr$ s f>re abroad1 0 v greatlye e/em$ = 8 wond]full p[] & excead+ treasure1 b yet mu* m 0 he e/em$ = 8 wisdom4 ,& him selfe do? be> witnes1 t wis$om is bett] !n preti\s /ones 4 yea all ?+es t c be desir$ ar n to be comp>$ to x4 ,b :at ne$e? to all$ge "o s5t5ce ( him1 ^: bookes altogi!r d none o!r ?+1 !n set =? ! praise ( wis$om kn[l$g8 ,& 8 fa!r k+ ,dauid ioyne? u]tu\s conu]sacion & kn[l$g togi!r1 z ! summe ( p]fection & *ief felicity4 ,:]=e ,i maye iu/elye conclude1 t true felicity do? consi/ 9 wisdome & v]tu4 ,!n if wisdome be z ,cic]o def9e? x1 ,diu9>um atq2 human>um r]um sci5tia1 !n ought all m5 to trauail = kn[l$g 9 matt]s bo? ( religion & huma9e docr9e1 if he %all be c\nt$ wyse1 & able to atta9e true felicitie3 ,b z ! /udy ( religi\s matt]s is mo/ pr9cipall1 s ,i leue x = ?is "t to !m t bett] c write ( x !n ,i c4 ,& = huma9e k ?ys wil ,i boldly say1 t :o soeu] wyll atta9 true iudgm5t !re91 mu/ n only trauail 9 ye kn[l$g ( ! tungs1 b mu/ also be=e al o!r arts1 ta/e ( ! ma!matical sci5ces1 specially ,ari?metike & ,geometry1 )\t :i* x is n possible to attayn full kn[l$g 9 any art4 ,:i* may suffici5tly by ga!r$ by ,ari/otle n only 9 8 bookes ( demon/ration 7:i*e c n be vnd]/& )\t ,geometry7 b also 9 all 8 o!r workes4 ,& be=e him ,plato 8 mai/] wrote ?is s5t5ce on 8 s*ole h\se dore4 ,ageometre3tos oudeis eisito34 ,let no man entre "h 7sai? he7 )\t kn[l$g 9 ,geometry4 ,:]=e mo/e mighty pr9ce1 z y\r mo/ excell5t ,maie/y appe>e? to be borne vnto mo/ p]fect felicity1 n only by reason t ,god m\$ ) ! long pray]s ( ?is realme1 did s5d y\r highnes z mo/e com=table inh]it\r to ! same1 b also 9 t y\r ,maie/y 0 borne 9 ! "t ( su* skilful s*oolmai/]s le>n$ te*]s1 z y\r highnes do? n a little reioyse 91 & pr(ite by !m 9 all k9d ( v]tu kn[l$g4 ,among/ :i* is t heau5ly kn[l$g mo/ wor!ly to be prais$1 :]bi ! bl9dnes ( err\r sup]/ition is exil$1 good hope conceiu$ t al ! s$es fruts !r(1 ) all k9des ( vice iniquite1 :]by v]tu is h9d]$1 iu/ice defac$1 %al be clean extrirp$ & root$ out ( ?is realm1 :i* hope %al increase m & m1 if x may appe> t le>n+ be e/em$ flori% )9 ?is realm4 ,& al be x ! *ief le>n+ be ! diu9e scriptures1 :i* in/ruct ! m9d pr9cipally1 nexte !rto ! lawes politike1 :i* mo/ specially def5d ! "r ( goodes1 yet is x n possible t ^? two c long be wel vs$1 if t ayde want t g\]ne? heal? & expelle? sicknes1 :i* ?+ is done by ,physik1 ^! require ! help ( ! vij4 lib]all sci5ces1 b ( none m !n ( ,ari?metik & ,geometry1 by :i* n only great ?+es ar wr\ght t\*+ accomptes 9 al k9ds1 9 suruaiyng measur+ ( l&es1 b also al arts dep5d p>tly ( !m1 build+ :i* is mo/ necess>y c n be wtout !m1 :i* ?+ consid]+1 m\$ me to help to s]ue y\r maie/y 9 ?is po9t z wel z o!r wais1 to d :at mai be 9 me1 yt n only !i :i* /udi pr9cipalli = l]n+1 mai haue furd]ance bi mi poore help1 b also ^? :i*e haue no tyme to trauaile = exact] kn[lege1 may haue "s helpe to vnd]/& 9 ^? ,ma!maticall artes1 9 :i*e z ,i haue all readye set =? sum:at ( ,ari?metike1 s god will+ ,i int5d %ortly to set=? a m exact] worke !r(4 ,& 9 ! meane ceason = a ta/e ( ,geometry1 ,i haue sette =! ?is small introduction1 desir+ y\r grace n s mu*e to beholde ! simpl5es ( ! woorke1 9 comp>ison to y\r ,maie/ies excell5cye1 z to fau\r ! edition !re(1 = ! ayde ( y\r humble subiectes1 :i* %al ?9ke !m selues m & m dayly b\nd5 to y\r highnes1 if :5 !y %all p]ceaue y\r graces desyre to haue !ym pr(it$ 9 all k & v]tue4 ,& ,i = my poore ability consid]+ y\r ,maie/ies /udye = ! increase ( le>n+ g5]ally ?r\gh all y\r high5es dom9ions1 & namely 9 ! vniu]sities ( ,ox=de & ,camebridge1 z ,i haue an e>ne/ good w z f> z my simple s]uice & small kn[l$g w suffice1 to helpe t[>d ! satisfiyng ( y\r graces desire1 s if ,i %all p]ceaue t my s]uice may be to y\r maie/ies cont5tacion1 ,i wil n only put =? ! o!r two books1 :i*e %\lde haue be5e sette =? ) ^! two1 yf mis=tune _h n h9d]$ x1 b also ,i wil set =? o!r bookes ( m exact] arte1 bo! 9 ! ,lat9e tongue & also 9 ! ,engly%e1 :]e( p>te bee all readye writt51 & newe in/rum5tes to !ym deuis$1 & ! residue %all bee e&$ ) all possible spe$e4 ,i 0 bold5$ to d$icate ?is booke ( ,geometrye vnto y\r ,maie/ye1 n s mu*e bycause x is ! fir/e t eu] 0 sette =! 9 ,engli%e1 & !re=e = ! n\eltye a /raunge pres5te1 b = t ,i 0 p]swad$1 t su*e a wyse pr9ce doo! desire to haue a wise sorte ( subiectes4 ,= x is a kynges *iefe reioys+e & glory1 if 8 subiectes be ri*e 9 sub/aunce1 & wytty 9 k3 & contr>ye waies no?yng c bee m greu\se to a noble kyng1 !n t 8 realme %\ld be o!r begg]ly or full ( ignoraunce3 ,b z god ha? geu5 y\r grace a realme bo! ri*e 9 commodities & also full ( wyttie m51 s ,i tru/e by ! readyng ( wyttie artes 7:i*e be z ! :ette /ones ( witte7 !y mu/e ne$es increase m & m 9 wys$ome1 & p]adu5ture fynde "s ?ynge t[>de ! ayde ( _! sub/aunce1 :]eby y\r grace %all haue newe occasion to reioyce1 seyng y\r subiectes to increase 9 sub/ance or wisdom1 or 9 bo?4 ,& !i aga9 %al haue new & new causes to pray = y\r maie/ie1 p]ceiuyng s graci\se a m9d t[>de _! b5efite4 ,& ,i tru/e 7z ,i desire7 t a great numbre ( g5tlem51 especially ab\t ! c\rte1 :i*e vnd]/& n ! lat9 tong1 or els = ! h>dnesse ( ! mat] c\ld n away ) o!r m5s writyng1 w fall 9 trade ) ?is easie =me ( tea*yng 9 _! vulg> tong1 & s employe "s ( _! tyme 9 hone/ /udie1 :i*e 7 wont to be/[e mo/ "p ( _! "t 9 triflyng pa/ime3 ,= vnd\bt$ly if !y mean o!r y\r maie/ies s]uice1 o!r _! own wisdome1 !y w be cont5t to employ "s tyme ab\te ?is hone/ & wittie ex]cise4 ,= ^: enc\ragem5t to ! int5t !y maie p]ceiue :at %all be ! vse ( ?is sci5ce1 ,i haue n onely writt5 some:at ( ! vse ( ,geometrie1 b also ,i haue annex$ to ?is boke ! names & brefe argum5tes ( ^? o!r bokes :i*e ,i w set =? h]eaft]1 & t z %ortly z x %all appe>e vnto y\r maie/ie by coniecture ( _! dilig5t vsyng ( ?is fir/ boke1 t !y wyll vse well ! o!r bokes also4 ,9 ! meane ceason1 & at all times ,i wil be a cont9uall petition]1 t god may "w 9 all engli% h>tes an erne/ mynde to all hone/ ex]cises1 :]by !i may s]ue ! bett] y\r maie/ie & ! realm4 ,& = y\r highnes ,i bese* ! mo/ m]cifull god1 z he ha? mo/ fau\rably s5t y vnto vs1 z our *efe com=t] 9 e>!1 s t he w increase y\r maie/ie daiely 9 all v]tue & honor ) mo/e prosp]\se successe1 & augm5t 9 vs y\r mo/ humble subiectes1 true l\e to godw>d1 & iu/ ob$i5ce t[>d y\r highnes ) all reu]5ce & subiection4 ,at ,london ! 4xxviij4 daie ( ,ianu>ie4 ,m4 ,d4 ,l ,i4 ,y\r maie/ies mo/e humble s]uant & ob$i5t subiect1 ,rob]t ,recorde4 ,,! ,,preface1 decl>+ briefely ! commodi- tes ( ,geometrye1 & ! necessitye !re(4 ,geometrye may ?9ke x selfe to su/a9e great iniury1 if x %all be in=c$ o!r to %[ h] manifold commodities1 or els n to prease into ! sight ( m51 & !re=e might ?is wayes answ]e briefely3 ,o!r ,i am able to d y mu* good1 or els b litle4 ,if ,i bee able to doo y mu* good1 !n be y n y\r owne fri5des1 b greatlye y\r owne enemies to make s little ( me1 :i* maye pr(ite y s mu*e4 ,= if ,i 7 z vncurte\s z y vnk9d1 ,i %uld vtt]ly refuse to d !m any good1 :i* w s curi\sly put me to ! trial & pr(e ( my commodities1 or els to suffre exile1 & namely si! ,i %al only yeld b5efites to o!r1 & receaue none aga9e4 ,b & if y c\ld saye truely1 t my b5efites be no!r _m nor yet greate1 yet if !y bee anye1 ,i doo yelde m to y1 !n ,i doo receaue aga9e ( y1 & !re=e ,i oughte n to bee repell$ ( !m t l\e !m selfe1 al?\ghe !y l\e me n all = my selfe4 ,b z ,i am 9 nature a lib]all sci5ce1 s canne ,i n aga9/e nature cont5de ) y\r inhumanitye1 b mu/e %ewe my selfe lib]all eu5 to myne enemies4 ,yet ?is is my com=te aga9e1 t ,i haue none enemies b !m t kn[e me n1 & !re=e may hurte !mselues1 b c n noye me4 ,yf !y dispraise ! ?+e t !y "k n1 all wise m5 w blame !m & n cr$ite !m1 & yf !y ?9ke !y kn[e me1 lette !ym %ewe "o vntru! & err\re 9 me1 & ,i wyll geue ! victorye4 ,yet c no humayne sci5ce saie ?us1 b ,i onely1 t !re is no sp>ke ( vntru! 9 me3 b all my doctr9e & workes are )\t any blemi%e ( err\r t mans reason c disc]ne4 ,& nexte vnto me 9 c]ta9tie are my ?ree sy/]s1 ,ari?metike1 ,musike1 & ,a/ronomie1 :i*e are also s n]e knitte 9 amitee1 t he t l\e? ! "o1 c n despise ! o!r1 & 9 especiall ,geometrie1 ( :i*e n only ^! ?re1 b all o!r artes d bor[ great ayde1 z p>tly h]eaft] %all be %ew$4 ,b fir/ w ,i beg9ne ) ! vnle>n$ sorte1 t y maie p]ceiue h[ t no arte c /& )\t me4 ,= if ,i %\ld decl>e h[ _m wayes my helpe is vs$1 9 measuryng ( gr\nd1 = m$[1 corne1 & wodde3 9 h$gyng1 9 di*yng1 & 9 /ackes makyng1 ,i ?9ke ! poore ,husb& man w\ld be m ?ankefull vnto me1 !n he is n[e1 :yles he ?9ke? t he ha? small b5efite by me4 ,yet ?is maie he coniecture c]ta9ly1 t if he kepe n ! rules ( ,geometrie1 he c n measure any gr\nd truely4 ,& 9 di*yng1 if he kepe n a proportion ( br$? 9 ! m\!1 to ! br$! ( ! bottome1 & iu/e slop5esse 9 ! sides agreable to !m bo!1 ! di*e %all be faultie _m waies4 ,:5 he do? make /ackes = corne1 or = heye1 he practise? good ,geometrie1 els w\ld !i n long /&3 ,s t 9 "s /akes1 :i*e /& on f\re pill]s1 & yet made r\nd1 doe increase greatt] & greatt] a good height1 & !n aga9e turne small] & small] vnto ! toppe3 y maie see s good ,geometrie1 t x 7 v difficult to c\nt]faite ! lyke 9 any kynde ( buildyng4 ,z = o!r inf9ite waies t he vse? my b5efite1 ,i ou]passe = %ortnesse4 ,c>p5t]s1 ,k>u]s1 ,ioyn]s1 & ,masons1 doe will+ly ackn[l$ge t !y c worke no?yng )\t reason ( ,geometrie1 9 s mu*e t !y *al5ge me z a peculi>e sci5ce = !m4 ,b 9 t !y %\ld d wrong to all o!r m51 seyng eu]ie kynde ( m5 haue som b5efit by me1 n only 9 buildyng1 :i*e is b o!r m5nes co/es1 & ! arte ( ,c>p5t]s1 ,masons1 & ! o!r a=esayd1 b 9 _! owne priuate pr(ession1 :]e( to auoide t$i\snes ,i make ?is reh]sall4 ,si? ,m]*auntes by %ippes great ri*es d w9ne1 ,i may ) good righte at _! seate beg9ne4 ,! ,%ippes on ! sea ) ,saile & ) ,ore1 7 fir/e f\nde1 & /yll made1 by ,geometries lore4 ,_! ,compas1 _! ,c>de1 _! ,pulleis1 _! ,ank]s1 7 f\nde by ! skill ( witty ,geomet]s4 ,to sette =? ! ,cap/ocke1 & e*e o!r p>te1 wold make a greate %[e ( ,geometries arte4 ,c>p5t]s1 ,c>u]s1 ,io9]s & ,masons1 ,pa9t]s & ,limn]s ) su*e occupations1 ,brod]]s1 ,goldesmi!s1 if !y be cunn+1 ,mu/ yelde to ,geometrye ?ankes = _! le>n+4 ,! ,c>te & ! ,pl[e1 :o do? !m well m>ke1 ,are made by good ,geometrye4 ,& s 9 ! w>ke ,( ,tail]s & ,%oomak]s1 9 all %apes & fa%ion1 ,! woorke is n prais$1 if x wante proportion4 ,s weau]s by ,geometrye hade _! f\ndacion1 ,_! ,loome is a frame ( /raunge imag9acion4 ,! :eele t do? sp9ne1 ! /one t do? gr9d1 ,! ,myll t is driu5 by wat] or w9de1 ,are workes ( ,geometrye /raunge 9 _! trade1 ,fewe c\ld !m deuise1 if !y 7 vnmade4 ,& all t is wr\ght by waight or by measure1 )\t pro(e ( ,geometry c neu] be sure4 ,clockes t be made ! times to deuide1 ,! wittie/ inu5cion t eu] 0 spi$1 ,n[e t !y are common !y are n reg>d$1 ,! artes man contemn$1 ! woorke vnrew>d$4 ,b if !y 7 sc>se1 & "o = a %ewe1 ,made by ,geometrye1 !n %\lde m5 "k1 ,t neu] 0 arte s wond]full witty1 ,s ne$efull to man1 z is good ,geometry4 ,! fir/e f9d+e out ( eu]y good arte1 ,seem$ !n vnto m5 s godly a p>te1 ,t no recomp5ce might satisfye ! f9d]1 ,b to make him a god1 & hon\re him = eu]4 ,s ,c]es & ,pallas1 & ,m]cury also1 ,eolus & ,neptune1 & _m o!r mo1 ,7 hon\r$ z goddes1 bicause !y did tea*e1 ,fir/e tillage & weu+e & eloqu5t spea*e1 ,or w9des to obs]ue1 ! seas to saile ou]1 ,!y 7 call$ goddes = _! good indeu\r4 ,!n 7 m5 m ?ankefull 9 t gold5 age3 ,?is yron wolde n[e vngratefull 9 rage1 ,wyll yelde ! ?y rew>d = trauaile & pa9e1 ,) sclaund]\s repro*1 & spitefull disda9e4 ,yet ?\ghe o!r m5 vn?ankfull w be1 ,suruay]s haue cause to make mu*e ( me4 ,& s haue all ,lordes1 t l&es d possesse3 ,b ,t5naunt$ ,i fe>e w l me ! lesse4 ,yet d ,i n wrong b measure all truely1 ,all yelde ! full "r ( eu]ye man iu/ely4 ,proportion ,geometricall ha? no man oppre/1 ,yf anye bee wrong$1 ,i wi%e x r$re/4 ,b n[ to proc$e ) le>n$ pr(essions1 9 ,logike & ,rhetorike & all p>tes ( phylosophy1 !re neade? none o!r pro(e !n ,ari/otle 8 te/imony1 :i*e )\t ,geometry pr\e? almo/ no?+e4 ,9 ,logike all 8 good syllogismes & demon/rations1 hee decl>e? by ! pr9ciples ( ,geometrye4 ,9 philosophye1 ne!r motion1 nor "t1 nor ayrye impressions c\ld hee aptely decl>e1 b by ! helpe ( ,geometrye z 8 woorkes d witnes4 ,yea ! faculties ( ! m9de do! hee expresse by similitude to figures ( ,geometrye4 ,& 9 morall phylosophy he ?\ght t iu/ice c\lde n wel be taught1 nor yet well execut$ )\t proportion geometricall4 ,& ?is e/imacion ( ,geometry he maye seeme to haue le>n$ ( 8 mai/] ,plato1 :i* )\t ,geometrye wolde tea*e no?+e1 no!r wold admitte any to he>e him1 except he 7 exp]te 9 ,geometry4 ,& :at m]ualle if he s mu*e e/em$ geometrye1 se+e 8 op9ion 01 t ,godde 0 alwaies work+e by ,geometrie8 ,:i*e s5t5ce ,plut>*e decl>e? at l>ge4 ,& al?\gh ,platto d vse ! helpe ( ,geometrye 9 all ! mo/ waighte matt] ( a common weal?1 yet x is s g5]all 9 vse1 t no small ?+es almo/ c be wel done )\t x4 ,& !r=e sai? he3 t ,geometrye is to be le>n$1 if x 7 = none o!r cause1 b t all o!r artes are bo! son] & m surely vnd]/& by helpe ( x4 ,:at greate help x do! 9 physike1 ,gal5e do? s (t5 & s copi\sely decl>e1 t no man :i*e ha? r$de any booke almo/e ( 81 c be ignorant !re(1 9 s mu* t he c\lde neu] cure well a r\nde vlc]e1 tyll reason geometricall dydde tea*e x hym4 ,hippocrates is e>ne/ 9 admony%ynge t /udy ( geometrie mu/ prep>e ! way to physike1 z well z to all o!r artes4 ,i %\lde seeme some:at to t$i\s1 if ,i %\lde reck5 vp1 h[e ! diu9es also 9 all _! my/]ies ( scripture doo vse healpe ( geometrie3 & also t lawy]s c neu] vnd]/&e ! hole lawe1 no nor yet ! fir/e title !r( exactly )\t ,geometrie4 ,= if lawes c n well be e/abli%$1 nor iu/ice duelie execut$ )\t geometricall proportion1 z bo! ,plato 9 8 ,politike bokes1 & ,ari/otle 9 8 ,moralles doo l>gely decl>e4 ,yea si! ,lycurgus t *eefe lawmak] amonge/ ! ,lac$emonians1 is mo/e prais$ = t he didde *aunge ! /ate ( _! common weal! frome ! proportion ,ari?meticall to a proportion geometricall1 :i*e )\t kn[l$g ( bo! he c\lde n dooe1 ?an is x easye to p]ceaue h[e necess>ie ,geometrie is = ! lawe & /ud5tes !re(4 ,& if ,i %all saie preciselie & freelie z ,i ?9ke1 he is vtt]lie de/itute ( all abilitee to iudge 9 anie arte1 t is n somme:at exp]te 9 ! ,!oremes ( ,geometrie4 ,& t caus$ ,gal5e to say ( hym selfe1 t he c\lde neu] p]ceaue :at a demon/ration 01 no n s mu*e1 z :e!r !re 7 any or none1 tyll he _h by geometrie gott5 abilitee to vnd]/&e x1 al?\gh he he>d ! be/e tea*]s t 7 9 8 tyme4 ,x %uld be to longe & n$elesse also to decl>e :at helpe all o!r artes ,ma!maticall haue by geometrie1 si? x is ! gr\nde ( all !yr c]te9tie1 & no man /udi\s 9 !m is s d\btful !r(1 t he %all n$e any p]suasion to procure cr$ite !reto4 ,= he c n reade 4ij4 l9es almo/e 9 any ma!maticall sci5ce1 b he %all espie ! n$efulnes ( geometrie4 ,b to auoyde t$i\snesse ,i w make an ende h]e( ) t fam\s s5t5ce ( aunci5t ,py?agoras1 ,t :o s w trauayle by le>nyng to attayne wys$ome1 %all neu] appro*e to any excell5cie )\t ! artes ma!maticall1 & especially ,ari?metike & ,geometrie4 ,& yf ,i %all some:at speake ( noble m51 & g\]n\rs ( realmes1 h[e ne$efull ,geometrye maye bee vnto !m1 !n mu/ ,i repete all t ,i haue sayde be=e1 si! 9 !m ought all k to ab\nde1 namely t maye app]ta9e ei!r to good g\]naunce 9 "t ( peace1 ey!r wittye pollicies 9 "t ( w>re4 ,= m9i/ration ( good lawes 9 "t ( peace ,lycurgus example ) ! te/imonies ( ,plato & ,ari/otle may suffise4 ,& z = w>res1 ,i might ?9ke x suffici5t t ,vegetius ha? writt51 & aft] him ,valturius 9 comm5dation ( ,geometry1 = vse ( w>res1 b all _! woordes seeme to saye no?+e1 9 comp>ison to ! example ( ,ar*im$es wor?y woorkes make by geometrie1 = ! def5ce ( 8 c\ntrey1 to reade ! wond]full praise ( 8 wittie deuises1 set foor! by ! mo/ fam\s hy/ories ( ,liuius1 ,plut>*e1 & ,pl9ie1 & all o!r hy/oriographi]s1 :y*e wryte ( ! /ronge siege ( ,syracusae made by t valiant capitayne1 & noble w>ri\r ,m>cellus1 ^: p[] 0 s great1 t all m5 m]uayl$ h[ t "o citee c\lde )/&e 8 wond]full =ce s longe4 ,b mu* m w\lde !y m]uaile1 if !y vnd]/ode t "o man onely dyd )/& all ,m>cellus /r5g?1 & ) c\nt] eng9es de/roi$ 8 eng9es to ! vtt] a/ony%m5t ( ,m>cellus1 & all t 7 ) hym4 ,he _h inu5t$ su*e bala/elas t dyd %oote out a hundr$ d>tes at "o %otte1 to ! great de/ruction ( ,m>cellus s\ldi\rs1 :]by a fonde tale 0 spr$de abrode1 h[ t 9 ,syracusae !re 0 a wond]full gyant1 :i*e _h a hundr$ h&es1 & c\lde %oote a hundr$ d>tes at ones4 ,& z ?is fable 0 spr$de ( ,ar*im$es1 s _m o!r haue be5 fayn$ to bee gyantes & mon/]s1 bycause !y dyd su*e ?ynges1 :i*e f>re pass$ ! witte ( ! common p4 ,s dyd !y feyne ,argus to haue a hundr$ eies1 bicause !y h]de ( 8 wond]full circumspection1 & ?\ghte t z x 0 ab\e _! capacitee1 s x c\ld n be1 onlesse he _h a hundr$ eies4 ,s imag9$ !y ,ianus to haue two faces1 "o lokyng =w>de1 & an o!r backw>de1 bycause he c\lde s wittily comp>e ?ynges pa/e ) ?ynges t 7 to come1 & s duely pondre !m1 z yf !y 7 all pres5t4 ,( l reason did !y feyn ,lynceus to haue su* %>p syght1 t he c\ld see ?r\gh walles & hylles1 bycause p]adu5ture he dyd by naturall iudgem5t decl>e :at commoditees myght be digg$ out ( ! gr\nde4 ,& an inf9ite n\mbre lyke fables are !re1 :i*e sprange all ( lyke reason4 ,= :at o!r ?yng meane? ! fable ( ! great gyant ,atlas1 :i*e 0 ymag9$ to be>e vp heau5 on 8 %uld]s8 b t he 0 a man ( s high a witte1 t x rea*$ vnto ! skye1 & 0 s skylfull 9 ,a/ronomie1 & c\lde tell be=e h&e ( ,eclipses1 & o!r l ?ynges z truely z ?\gh he dyd rule ! /]res1 & g\]ne ! planettes4 ,s 0 ,eolus accompt$ god ( ! wyndes1 & to haue !im all 9 a caue at 8 pleasure1 by reason t he 0 a wittie man 9 naturall kn[lege1 & obs]u$ well ! *ange ( we!rs1 aud 0 ! fyr/ t taught ! obs]uation ( ! wyndes4 ,& lyke reson is to be geu5 ( al ! old fables4 ,b to ret\rne agayne to ,ar*im$es1 he dyd also by arte p]spectiue 7:i*e is a p>te ( geometrie7 deuise su* glasses )9 ! t[ne ( ,syracusae1 t dyd b\rne _! ennemies %yppes a great way f ! t[ne1 :y*e 0 a m]uayl\s politike ?ynge4 ,& if ,i %ulde repete ! v>ietees ( su*e /raunge inu5tions1 z ,ar*im$es & o!rs haue wr\ght by geometrie1 ,i %\ld n onely exc$e ! ord] ( a preface1 b ,i %\ld also speake ( su*e ?ynges z c n well be vnd]/&e 9 talke1 )\t somme k 9 ! pr9ciples ( geometrie4 ,b ?is w ,i promyse1 t if ,i may p]ceaue my paynes to be ?ankfully tak51 ,i wyll n onely write ( su*e pleasant inu5tions1 decl>yng :at !y 71 b also wil tea*e h[e a great numbre ( !m 7 wr\ghte1 t !y may be practis$ 9 ?is tyme also4 ,:]by %allbe pla9ly p]ceau$1 t _m ?ynges seme impossible to be done1 :i*e by arte may v well be wr\ght4 ,& :an !y be wr\ght1 & ! reason !r( n vnd]/&e1 ?an say ! vulg>e p1 t ^? ?ynges are done by negromancy4 ,& h]e( came x t fry] ,bakon 0 accompt$ s greate a negromanci]1 :i*e neu] vs$ t arte 7by any coniecture t ,i c fynde7 b 0 9 geometrie & o!r ma!maticall sci5ces s exp]te1 t he c\lde dooe by !im su*e ?ynges z 7 wond]full 9 ! syght ( mo/ p4 ,great talke !re is ( a glasse t he made 9 ,ox=de1 9 :i*e m5 myght see ?ynges t 7 doon 9 o!r places1 & t 0 iudg$ to be done by p[] ( euyll spirites4 ,b ,i kn[e ! reason ( x to bee good & naturall1 & to be wr\ght by geometrie 7sy! p]spectiue is a p>te ( x7 & to /&e z well ) reason z to see y\r face 9 common glasse4 ,b ?is conclusion & o!r dyu]s ( lyke sorte1 are m mete = pr9ces1 = sundry causes1 ?an = o!r m51 & ought n to bee taught commonly4 ,yet to repete x1 ,i ?\ght good = ?is cause1 t ! wor?ynes ( geometry myght ! bett] be kn[51 p>tly vnd]/&+ geu51 :at wond]full ?ynges may be wr\ght by x1 & s consequ5tly h[ pleasant x is1 & h[ necess>y also4 ,& ?us = ?is tyme ,i make an end4 ,! reason ( som ?ynges done 9 ?is boke1 or omitt$ 9 ! same1 y %all fynde 9 ! preface be=e ! ,!oremes4 ,! def9itions ( ! pr9ciples ( ,,geometry4 ,geometry tea*e? ! drawyng1 ,measur+ & proporcion ( figures4 b 9 z mu*e z no figure c bee draw51 b x mu/e haue c]tayne b\ndes & inclosures ( l9es3 & eu]y lyne also is begon & end$ at "s c]ta9e prycke1 fyr/ x %al be meete to "k ^! small] p>tes ( eu]y figure1 t !rby ! :ole figures may ! bett] bee iudg$1 & di/9cte 9 sond]4 ,sid5ote3 ,a po9cte4 ,a ,poynt or a ,prycke1 is nam$ ( ,geometricians t small & vns5sible %ape1 :i*e ha? 9 x no p>tes1 t is to say3 no!r l5g?1 bread? nor dep?4 ,b z _! exactnes ( def9ition is m meet] = onlye ,!orike speculacion1 !n = practise & outw>de worke 7consid]+e t myne int5t is to applye all ^! :ole pr9ciples to woorke7 ,i ?ynke meet] = ?is purpose1 to call a poynt or prycke1 t small pr9te ( p5ne1 p5cyle1 or o!r in/rum5te1 :i*e is n m\$1 nor draw5 f 8 fyr/ t\*e1 & !r=e ha? no notable l5g? nor br$!3 z ?is example doe? decl>e4 ,illu/ration3 !re=e symbol ,:]e ,i haue set 4iij4 prickes1 e*e ( !m hauyng bo? l5g? & br$?1 ?ogh x be b smal1 & !=e n notable4 ,n[e ( a great numbre ( ^! prickes1 is made a ,lyne1 z y may p]ceiue by ?is =me ensuyng4 444444444444444444444444 :]e z ,i haue set a numbre ( prickes1 s if y ) y\r p5 w set 9 m o!r prickes betwe5e eu]ye two ( ^!1 ,sid5ote3 ,a lyne4 !n wil x be a lyne1 z "h y may see -------- & ?is lyne1 is call$ ( ,geometricians1 ,l5g! )\te bread?4 ,b z !y 9 !yr !orikes 7:i* ar only m9d workes7 d precisely vnd]/& ^! def9itions1 s x %al be suffici5t = ^? m51 :i*e seke ! vse ( ! same ?+es1 z s5se may duely iudge !m1 & applye to h&y workes if !y vnd]/& !m s to be true1 t outw>de s5se canne fynde none err\re !re94 ,( lynes !re bee two pr9cipall kyndes4 ,! "o is call$ a "r or /raight lyne1 & ! o!r a crok$ lyne4 ,sid5ote3 ,a /reghte lyne4 ,a ,/raight lyne1 is ! %orte/ t maye be draw5ne betwe5 two prickes4 ,sid5ote3 ,a crokyd lyne4 ,& all o!r l9es1 t g n "r =? f prick to prick1 b b[e? any waye1 su* are call$ ,crok$ lynes z 9 ^! examples fol[yng ye may se1 :]e ,i haue set b "o =me ( a /raight lyne1 = m =mes !re be n1 b ( crook$ lynes !re bee innum]able diu]sities1 :]e( = examples sum ,i haue sette "h4 ,illu/ration3 ,a "r lyne4 ,illu/ration3 ,crok$ lynes4 ,illu/ration3 ,crok$ l9es4 ,s n[ y mu/ vnd]/&1 t eu]y lyne is draw5 betw5e twoo prickes1 :]( ! "o is at ! beg9n+1 & ! o!r at ! ende4 ,illu/ration ,!re=e :5 soeu] y d see any =mes ( lynes to t\*e at "o notable pricke1 z 9 ?is example1 !n %all y n call x "o crok$ lyne1 b r twoo lynes3 ,sid5ote3 an ,angle4 9 z mu*e z !re is a notable & s5sible angle by 4,a4 :i*e eu]more is made by ! meetyng ( two seu]all lynes4 ,& likewayes %all y iudge ( ?is figure1 :i*e is made ( two l9es1 & n ( "o onely4 ,illu/ration ,s t :an s eu] any su*e meetyng ( l9es do? happ51 ! place ( _! metyng is call$ an ,angle or corn]4 ,( angles !re be ?ree g5]all k9des3 a %>pe angle1 a squ>e angle1 & a blunte angle4 ,sid5ote3 ,a righte angle4 ,! squ>e angle1 :i*e is commonly nam$ a "r corn]1 is made ( twoo lynes meetyng toge!r 9 f\rme ( a squire1 :i*e two l9es1 if !y be draw5 =? 9 l5g?1 w crosse "o an o!r3 z 9 ! examples fol[yng y maie see4 ,sid5ote3 ,a %>pe corn]4 ,a %>pe angle is s call$1 because x is less] ?an is a squ>e angle1 & ! l9es t make x1 d n op5 s wide 9 _! dep>tynge z 9 a squ>e corn]1 & if !i be draw5 crosse1 all f[] corn]s w n be equall4 ,sid5ote3 ,a blunte angle4 ,a blunt or brode corn]1 is great] !n is a squ>e angle1 & 8 l9es d p>te m 9 sond] !n 9 a "r angle1 ( :i*e all take ^! examples4 ,illu/ration3 ,"r angles4 ,illu/ration3 ,%>pe angles4 ,illu/ration3 ,blunte or brode angles4 ,& ^! angles 7z y see7 are made p>tly ( /reght lynes1 p>tly ( crok5 l9es1 & p>tly ( bo? toge!r4 ,h[beit 9 "r angles ,i haue put none example ( crok$ l9es1 because x w\ld mu*e tr\ble a l]n] to iudge !m3 = _! true iudgm5t do? app]ta9e to arte p]spectiue1 & z ,i may say1 r to reason !n to s5se4 ,b n[ z ( _m prickes !re is made "o l9e1 s ( diu]se l9es are !re made sundry =mes1 figures1 & %apes1 :i*e all yet be call$ by "o propre "n1 ,sid5ote3 ,a platte =me4 ,platte =mes1 & !i haue bo! l5g? & br$?1 b yet no dep5esse4 ,& ! b\ndes ( eu]ie platte =me are l9es3 z by ! examples y maie p]ceiue4 ,( platte =mes "s be pla91 & "s be crok$1 & "s p>ly pla9e1 & p>tlie crok$4 ,sid5ote3 ,a pla9e platte4 ,a pla9e platte is t1 :i*e is made al equall 9 height1 s t ! middle p>tes no!r bulke vp1 no!r %r9k d[n m !n ! bo! endes4 ,sid5ote3 ,a crook$ platte4 ,= :an ! "o p>te is high] !n ! o!r1 !n is x nam$ a ,crok$ platte4 ,& if x be p>tlie pla9e1 & p>tlie crook$1 !n is x call$ a ,myxte platte1 ( all :i*e1 ^! are exaumples4 ,illu/ration3 ,a pla9e platte4 ,illu/ration3 ,a crok$ platte4 ,illu/ration3 ,a myxte platte4 ,& z ( _m prickes is made a l9e1 & ( diu]se l9es "o platte =me1 ,sid5ote3 ,a bodie4 s ( manie plattes is made a bodie1 :i*e conteigne? ,l5g!1 br$?1 & dep5esse4 ,sid5ote3 ,dep5esse4 ,by ,dep5esse ,i vnd]/&1 n z ! common sort do?1 ! hol[nesse ( any ?+1 z ( a well1 a di*e1 a potte1 & su*e l1 b ,i meane ! massie ?icknesse ( any bodie1 z 9 exaumple ( a potte3 ! dep5esse is aft] ! common "n1 ! space f 8 brimme to 8 bottome4 ,b z ,i take x "h1 ! dep5esse ( 8 bodie is 8 ?icknesse 9 ! sides1 :i*e is an o!r ?yng cleane diff]5t f ! dep5esse ( 8 hol[nes1 t ! common p meane?4 ,n[ all bodies haue platte =mes = _! b\ndes1 ,sid5ote3 ,cubike4 s 9 a dye 7:i*e is call$ a cubike bodie7 by geomatricians1 ,sid5ote3 ,a%el]4 & an a%l] ( masons1 !re are 4vi4 sides1 :i*e are 4vi4 platte =mes1 & are ! b\ndes ( ! dye4 ,sid5ote3 ,a globe4 ,b 9 a ,globe1 7:i*e is a bodie r\nde z a b[le7 !re is b "o platte =me1 & "o b\nde1 & ^! are ! exaumples ( !m bo!4 ,illu/ration3 ,a dye or a%l]4 ,illu/ration3 ,a globe4 ,b because y %all n muse :at ,i dooe call a b\nd1 ,sid5ote3 ,a b\nde4 ,i mean !rby a g5]all "n1 betok5+ ! beg9n+1 end & side1 ( any =me4 ,sid5ote3 ,=me1 ,fygure4 ,a =me1 figure1 or %ape1 is t ?yng t is inclos$ )9 "o bond or manie bondes1 s t y vnd]/& t %ape1 t ! eye do? disc]ne1 & n ! sub/ance ( ! bodie4 ,( figures !re be manie sortes1 = ei!r !i be made ( prickes1 l9es1 or platte =mes4 ,n )/&yng to speake prop]lie1 a figure is eu] made by platte =mes1 & n ( b>e l9es vnclos$1 nei!r yet ( prickes4 ,yet = ! light] =me ( tea*yng1 x %all n be vnsemely to call all su*e %apes1 =mes & figures1 :i*e ye eye maie disc]ne di/9ctly4 ,& fir/ to beg9 ) prickes1 !re maie be made diu]se =mes ( !m1 z p>tely "h doe? fol[e4 ,illu/ration3 ,a lyne>ic numbre4 ,triangul] numbres ,longsqu>e numbre4 ,iu/ squ>e numbres a ?recorn]$ spire4 ,a squ>e spire4 ,& s maie !re be inf9ite =mes m1 :i*e ,i omitte = ?is "t1 consid]+ t _! kn[l$g app]ta9e? m to ,ari?metike figurall1 ?an to ,geometrie4 ,b yet "o "n ( a pricke1 :i*e he take? r ( 8 place !n ( 8 f\rme1 maie ,i n ou]passe4 ,& t is1 :5 a pricke /&e? 9 ! middell ( a circle 7z no circle c be made by compasse )\t x7 !n is x call$ a c5tre4 ,sid5ote3 ,a c5tre ,& !re( doe masons1 & o!r worke m5ne call t patron1 a c5tre1 :]eby !i drawe ! l9es1 = iu/ hewyng ( /ones = ar*es1 vaultes1 & *imneies1 because ! *efe vse ( t patron is wr\ght by f9dyng t pricke or c5tre1 f :i*e all ! lynes are draw51 z 9 ! ?irde booke x doe? app]e4 ,lynes make diu]se figures also1 ?\gh prop]ly !i maie n be call$ figures1 z ,i said be=e 7vnles ! l9es d close7 b onely = easie man] ( tea*yng1 all %all be call$ figures1 t ! eye c disc]ne1 ( :i*e ?is is "o1 :5 "o l9e lye? flatte 7:i*e is nam$ ,sid5ote3 ,a gr\nd l9e4 ! gr\nd l9e7 & an o!r comme? d[ne on x1 & is call$ ,sid5ote3 ,a p]p5dicul>4 ,sid5ote3 ,a plume lyne4 a p]p5dicul] or plumme lyne1 z 9 ?is example y may see4 :]e 4,a4,b4 is ! gr\nde l9e1 & ,c4,d4 ! plumbe l9e4 ,illu/ration ,& l waies 9 ?is figure !re are ?ree l9es1 ! gr\nde lyne :i*e is ,a4,b4 ! plumme l9e t is ,a4,c4 & ! bias l9e1 :i*e goe? f ! "o ( !m to ! o!r1 & lie? aga9/ ! "r corn] 9 su* a figure :i*e is "h 4,c4,b4 ,illu/ration ,b consid]yng t ,i %all haue occasion to decl>e sundry figures anon1 ,i w fir/ %ew "s c]ta9e v>ietees ( l9es t close no figures1 b are b>e lynes1 & ( ! o!r l9es w ,i make m5cion 9 ! description ( ! figures4 ,illu/ration3 tortu\se p>alleles4 ,sid5ote3 ,p>allelys ,sid5ote3 ,gem[e lynes4 ,p>alleles1 or gem[e lynes be su*e l9es z be draw5 foor? /ill 9 "o di/aunce1 & are no n]] 9 "o place !n 9 an o!r1 = & if !y be n]] at "o ende !n at ! o!r1 !n are !y no p>alleles1 b maie bee call$ b\ght lynes1 & loe "h exaumples ( !m bo!4 ,illu/ration3 p>allelis4 ,illu/ration3 b\ght l9es ,illu/ration3 p>allelis3 circul>4 ,conc5trikes4 ,i haue add$ also p>alleles tortu\se1 :i*e b[e contr>ie waies ) _! two endes3 & p>alleles circul>1 :i*e be lyke vnp]fecte compasses3 = if !y bee :ole circles1 ,sid5ote3 ,conc5trikes !n are !y call$ conc5trikes1 t is to saie1 circles draw5 on "o c5tre4 ,"h might ,i note ! error ( good ,alb]t ,dur]1 :i* affirme? t no p]p5dicul> l9es c be p>alleles4 :i* err\r doe? spr+ p>tlie ( ou]sight ( ! diff]5ce ( a /reight l9e1 & p>tlie ( mi/akyng c]ta9 pr9ciples geometrical1 :i* al ,i wil let passe vntil an o!r tyme1 & wil n blame him1 :i* ha? des]u$ wor?yly inf9ite praise4 ,& to returne to my matt]4 ,sid5ote3 ,a tw9e l9e4 an o!r fa%ion$ l9e is !re1 :i* is nam$ a tw9e or twi/ l9e1 & x goe? z a wrey? ab\t "s o!r bodie4 ,sid5ote3 ,a spirall l9e4 ,& an o!r sorte ( l9es is !re1 t is call$ a spirall l9e1 ,sid5ote3 ,a worme l9e4 or a worm l9e1 :i*e repres5te? an app>ant =me ( _m circles1 :]e !re is n "o 9 d$e3 ( ^! 4ii4 k9des ( l9es1 ^! be examples4 ,illu/ration3 ,a twi/e lyne4 ,illu/ration3 ,a spirail lyne ,illu/ration3 ,a t\*e lyne4 ,sid5ote3 ,a tu* l9e4 ,a t\*e lyne1 is a l9e t runne? a long by ! edge ( a circle1 onely t\*+ x1 b do? n crosse ! circumf]5ce ( x1 z 9 ?is exaumple y maie see4 ,sid5ote3 ,a corde1 ,& :5 t a l9e do? crosse ! edg ( ! circle1 !n is x call$ a cord1 z y %all see anon 9 ! speakynge ( circles4 ,sid5ote3 ,mat*e corn]s ,9 ! meane season mu/ ,i n omit to decl>e :at angles bee call$ mat*e corn]s1 t is to saie1 su*e z /&e directly "o aga9/ ! o!r1 :5 twoo l9es be draw5 a crosse1 z "h app]e?4 ,illu/ration3 ,mat*e corn]4 ,mat*e corn]4 ,:]e ,a4 & ,b4 are mat*e corn]s1 s are ,c4 & ,d4 b n ,a4 & ,c4 no!r ,d4 & ,a4 ,n[e w ,i beg9ne to speak ( figures1 t be prop]ly s call$1 ( :i*e all be made ( diu]se l9es1 except onely a circle1 an egge =me1 & a tunne =me1 :i* 4iij4 haue no angle & haue b "o l9e = _! b\nde1 & an eye f\rme :i*e is made ( "o lyne1 & ha? an angle onely4 ,sid5ote3 ,a circle4 ,a circle is a figure made & enclos$ ) "o l9e1 & ha? 9 ! middell ( x a pricke or c5tre1 f :i*e all ! l9es t be draw5 to ! circumf]5ce are equall all 9 l5g?1 z "h y see4 ,illu/ration ,sid5ote3 ,circumf]5ce4 ,& ! l9e t enclose? ! :ole compasse1 is call$ ! circumf]5ce4 ,sid5ote3 ,a diamet]4 ,& all ! l9es t bee draw5 crosse ! circle1 & goe by ! c5tre1 are nam$ diamet]s1 ^: halfe1 ,i meane f ! c5t] to ! circumf]5ce any waie1 ,sid5ote3 ,semidiamet]4 is call$ ! semidiamet]1 or halfe diamet]4 ,illu/ration ,b & if ! l9e goe crosse ! circle1 & passe beside ! c5tre1 ,sid5ote3 ,a cord1 or a /r+lyne4 !n is x call$ a corde1 or a /ryng l9e1 z ,i said be=e1 & z ?is exaumple %ewe?3 :]e ,a4 is ! corde4 ,& ! compass$ l9e t aunsw]e? to x1 ,sid5ote3 ,an ar*l9e is call$ an ar*e lyne1 ,sid5ote3 ,a b[l9e4 or a b[e lyne1 :i*e "h m>k$ ) ,b4 & ! diamet] ) ,c4 ,illu/ration ,b & if t "p be sep>ate f ! re/ ( ! circle 7z 9 ?is example y see7 !n ar bo? p>tes call$ cantelles1 ,sid5ote3 ,a cantle ! "o ! greatt] cantle z ,e4 & ! o!r ! less] cantle1 z ,d4 ,& if x be p>t$ iu/e by ! c5tre 7z y see 9 ,f47 ,sid5ote3 ,a semyecircle !n is x call$ a semicircle1 or halfe compasse4 ,illu/ration ,sometimes x happ5e? t a cantle is cutte out ) two lynes draw5 f ! c5tre to ! circumf]5ce 7z ,g4 is7 ,sid5ote3 ,a nooke cantle & !n maie x be call$ a nooke cantle1 & if x be n p>t$ f ! re/e ( ! circle 7z y see 9 ,h47 ,sid5ote3 ,a nooke4 !n is x call$ a nooke pla9ely )\t any addicion4 ,& ! compass$ lyne 9 x is call$ an ar*e lyne1 z ! exaumple "h doe? %ewe4 ,illu/ration3 ,an ar*e4 ,n[e haue y he>d z t\*yng circles1 meetely suffici5t in/ruction1 s t x %\ld seme n$eles to speake any m ( figures 9 t kynde1 saue t !re doe? yet rema9e ij4 =mes ( an imp]fecte circle1 = x is lyke a circle t 7 brus$1 & !reby did runne out endelong "o waie1 :i*e =me ,geometricians dooe call an ,sid5ote3 ,an egge f\rme4 egge =me1 because x doe? repres5t ! figure & %ape ( an egge duely proportion$ 7z ?is figure %ewe?7 hauyng ! "o ende greate !n ! o!r4 ,illu/ration3 ,an egge =me ,illu/ration3 ,a tunne =me4 ,sid5ote3 ,a tunne or b>rel =m ,= if x be lyke ! figure ( a circle press$ 9 l5g?1 & bo! endes lyke bygge1 !n is x call$ a tunne =me1 or b>rell =me1 ! "r makyng ( :i*e figures1 ,i wyll decl>e h]eaft] 9 ! ?irde booke4 ,an o!r =me !re is1 :i*e y maie call a nutte =me1 & is made ( "o lyne mu*e lyke an egge =me1 saue t x ha? a %>pe angle4 ,& x *aunce? sometyme t !re is a "r l9e draw5 crosse ^! figures1 ,sid5ote3 ,an axtre or axe lyne4 & t is call$ an axelyne1 or axtre4 ,h[e be x prop]ly t l9e t is call$ an axtre1 :i*e gooe? ?r\ghe ! myddell ( a ,globe1 = z a diamet] is 9 a circle1 s is an axe lyne or axtre 9 a ,globe1 t lyne t goe? f side to syde1 & passe? by ! middell ( x4 ,& ! two poyntes t su*e a lyne make? 9 ! vtt] b\nde or platte ( ! globe1 are nam$ polis1 w* y may call aptly 9 engly%1 t\rne po9tes3 ( :i*e ,i d m l>gely intreate1 9 ! booke t ,i haue writt5 ( ! vse ( ! globe4 ,illu/ration ,b to returne to ! diu]sityes ( figures t remayne vndecl>$1 ! mo/ simple ( !m ar su* ones z be made b ( two lynes1 z are ! cantle ( a circle1 & ! halfe circle1 ( :i* ,i haue spok5 allready4 ,likewyse ! halfe ( an egge =me1 ! cantle ( an egge =me1 ! halfe ( a tunne f\rme1 & ! cantle ( a tunne f\rme1 & besyde ^! a figure mo*e l to a tunne f\rne1 saue t x is %>p c\]$ at bo? ! endes1 & !r=e do? consi/ ( twoo lynes1 :]e a tunne =me is made ( "o lyne1 ,sid5ote3 ,an yey f\rme & t figure is nam$ an yey f\rme4 ,illu/ration ,sid5ote3 ,a triangle ,! nexte kynd ( figures are ^? t be made ( 4iij4 lynes o!r be all "r lynes1 all crook$ lynes1 o!r "s "r & "s crook$4 ,b :at f\rme s eu] !y be (1 !y are nam$ g5]ally triangles4 = a triangle is no?+e els to say1 b a figure ( ?ree corn]s4 ,& ?ys is a g5]all rule1 looke h[ _m lynes any figure ha?1 s mannye corn]s x ha? also1 yf x bee a platte =me1 & n a bodye4 ,= a bodye ha? dyu]s lynes metyng sometime 9 "o corn]4 ,illu/ration3 ,a ,n[ to geue y example ( triangles1 !re is "o :i*e is all ( crok$ lynes1 & may be tak5 fur a portion ( a globe z ! figur m>k$ wt ,a4 ,illu/ration3 ,b ,an o!r ha? two compass$ l9es & "o "r lyne1 & is z ! portion ( halfe a globe1 example ( ,b4 ,illu/ration3 ,c ,an o!r ha? b "o compass$ lyne1 & is ! qu>t] ( a circle1 nam$ a quadrate1 & ! ryght lynes make a "r corn]1 z y se 9 ,c4 ,o!rlesse !n x z y se ,d1 ^: "r l9es make a %>pe corn]1 or great] !n a quadrate1 z is ,f1 & !n ! "r lynes ( x d make a blunt corn]4 ,illu/ration3 ,d ,also "s triangles haue all righte lynes & !y be di/9ct$ 9 sond] by _! angles1 or corn]s4 = o!r _! corn]s bee all %>pe1 z y see 9 ! figure1 ,e4 o!r ij4 %>pe & "o blunt1 z is ! figure ,g4 o!r ij4 %>p & "o blunt z 9 ! figure ,h4 ,illu/ration3 ,e ,illu/ration3 ,f ,!re is also an o!r di/9ction ( ! names ( triangles1 accord+ to _! sides1 :i*e o!r be all equal z 9 ! figure ,e1 & t ! ,greekes doo call ,isopleuron1 ,sid5ote3 isopleurom4 & ,lat9e m5 aeequilat]um3 & 9 engli% x may be call$ a ?relike triangle1 o!r els two sydes bee equall & ! ?yrd vnequall1 :i* ! ,greekes call ,isosceles1 ,sid5ote3 isoskeles4 ! ,lat9e m5 aequicurio1 & 9 engli% tweyleke may !y be call$1 z 9 ,g1 ,h1 & ,k4 ,=1 !y may be ( iij4 k9ds t is to say1 ) "o squ>e angle1 z is ,g1 or ) a blunte corn] z ,h1 or ) all 9 %>pe korn]s1 z y see 9 ,k4 ,illu/ration3 ,g ,illu/ration3 ,h ,illu/ration3 ,k ,fur!r m x may be yt !y haue neu] a "o syde equall to an o!r1 & !y be 9 iij kyndes also di/9ct lyke ! twilekes1 z y maye p]ceaue by ^! examples 4,m4 ,n1 & ,o4 :]e ,m4 ha? a "r angle1 ,n1 a blunte angle1 & ,o1 all %>pe angles ,sid5ote3 skal5om4 ^! ! ,greekes & lat9e m5 d cal scal5a & 9 engli%e !ye may be call$ n\elekes1 = !i haue no side equall1 or l long1 to ani o!r 9 ! same figur4 ,"h x is to be not$1 t 9 a triangle al ! angles bee call$ inn]angles except ani side bee draw5ne =? 9 l5g!1 = !n is t f\r! corn] cal$ an vtt] corn]1 z 9 ?is example because ,a4,b1 is draw5 9 l5g?1 !r=e ! angle ,c1 is call$ an vtt] angle4 ,illu/ration3 ,m ,illu/ration3 ,n ,illu/ration3 ,o ,illu/ration ,illu/ration3 ,q ,sid5ote3 ,quadrangle ,& ?us haue ,i done ) triangul$ figures1 & n[e fol[e? quadrangles1 :i* are figures ( iiij4 corn]s & ( iiij4 l9es also1 ( :i*e !re be diu]s k9des1 b *iefely v4 t is to say1 ,sid5ote3 ,a squ>e quadrate4 a squ>e quadrate1 ^: sides bee all equall1 & al ! angles squ>e1 z y se "h 9 ?is figure ,q4 ,sid5ote3 ,a longe squ>e4 ,! second k9d is call$ a long squ>e1 ^: f\re corn]s be all squ>e1 b ! sides are n equall e*e to o!r1 yet is eu]y side equall to t o!r t is aga9/ x1 z y maye p]ceaue 9 ?is figure4 ,r4 ,illu/ration3 ,r ,sid5ote3 ,a los5ge ,! ?yrd k9d is call$ los5ges ,sid5ote3 ,a diamond4 or diamondes1 ^: sides bee all equall1 b x ha? neu] a squ>e corn]1 = two ( !m be %>pe1 & ! o!r two be blunt1 z appe>e? 9 4,s4 ,illu/ration3 ,s ,! iiij4 sorte are l vnto los5ges1 saue t !y are long] "o waye1 & _! sides be n equal1 yet !r corn]s are l ! corn]s ( a los+1 & !r=e ar !y nam$ ,sid5ote3 ,a los5ge lyke4 los5gelike or diamondlike1 ^: figur is not$ ) ,t4 ,"h %al y m>ke t al ^? squ>es :i* haue _! sides al equal1 may be call$ also = easy vnd]/&+e1 likesides1 z ,q4 & ,s4 & ^? t haue only ! contr>y sydes equal1 z ,r4 & ,t4 haue1 ^? wyll ,i call likeiammys1 = a diff]5ce4 ,illu/ration3 ,t ,illu/ration ,! fift sorte do? conta9e all o!r fa%ions ( f\re corn]$ figurs1 & ar call$ ( ! ,grekes trapezia1 ( ,lat9 m5 m5sulae & ( ,arabitians1 helmu>iphe1 !y may be call$ 9 engli%e borde =mes1 ,sid5ote3 ,borde =mes4 !y haue no syde equall to an o!r z ^! examples %ew1 nei!r keepe !y any rate 9 _! corn]s1 & !r=e are !y c\nt$ vnrul$ =mes1 & ! o!r f\re k9des onely are c\nt$ rul$ =mes1 9 ! kynde ( quadrangles4 ,( ^! vnrul$ =mes !r is no numbre1 !y are s mannye & s dyu]s1 yet by arte !y may be *ang$ into o!r k9des ( figures1 & !rby be br\ght to measure & proportion1 z 9 ! ?irt5e conclusion is p>tly taught1 b m pla9ly 9 my booke ( measur+ y may see x4 ,& n[e to make an e&e ( ! dyu]s kyndes ( figures1 !re do! fol[e n[ figures ( 4v4 sydes1 o!r 4v4 corn]s1 :i* we may call c9k-angles1 ^: sydes p>tlye are all equall z 9 ,a1 & ^? are c\nt$ rul$ c9keangles1 & p>tlye vnequall1 z 9 ,b1 & !y are call$ vnrul$4 ,illu/ration3 ,a ,illu/ration3 ,b ,likewyse %all y iudge ( siseangles1 :i* haue sixe corn]s1 septangles1 :i*e haue seu5 angles1 & s =?1 = z mannye numbres z !re maye be ( sydes & angles1 s manye diu]s k9des be !re ( figures1 vnto :i* y[ %all geue names accord+ to ! numbre ( _! sides & angles1 ( :i*e = ?is tyme ,i wyll make an ende1 ,sid5ote3 ,a squyre4 & wyll sette =! on example ( a syseangle1 :i*e ,i _h almo/ =gott51 & t is x1 ^: vse comme? (t5 9 ,geometry1 & is call$ a squire1 is made ( two long squ>es ioyn$ togi!r1 z ?is example %ewe?4 ,illu/ration ,& ?us ,i make an e& to speake ( platte =mes1 & w briefelye saye som:at t\*+ ! figures ( bodeis :i* p>tly haue "o platte =me = _! b\nd1 & yt iu/ r\nd z a globe ha?1 or end$ long z 9 an egge1 & a tunne f\rme1 ^: pictures are ^!4 ,illu/ration3 ,! globe z is be=e4 ,h[e be x y mu/ m>ke t ,i meane n ! v figure ( a tunne1 :5 ,i saye tunne =m1 b a figure l a tunne1 = a tune f\rme1 ha? b "o plat =me1 & !r=e mu/ ne$s be r\nd at ! endes1 :]e z a tunne ha? ?re platte =mes1 & is flatte at e*e end1 z p>tly ^! pictures d %ewe4 ,bodies ( two plattes1 are o!r cantles or halues ( ^? o!r bodies1 t haue b "o platte =me1 or els !y are lyke 9 foorme to two su* cantles ioyn$ togi!r z ?is ,a4 do? p>tly eppresse3 or els x is call$ a r\nde spire1 or /iple f\rme1 z 9 ?is figure is "s :at express$4 ,sid5ote3 ,a r\nde spi]4 ,n[e ( ?ree plattes !re are made c]ta9 figures ( bodyes1 z ! cantels & halues ( all bodyes t haue b ij4 plattys1 & also ! halues ( halfe globys & canteles ( a globe4 ,lykewyse a r\nde pill]1 & a spyre made ( a r\nde spyre1 slytte 9 ij4 p>tes long ways4 ,b z ^! =mes be h>de to be iudg$ by _! pycturs1 s ,i doe ent5de to passe !m ou] ) a great numb] ( o!r =mes ( bodyes1 :i* aft]w>de %all be set =? 9 ! boke ( ,p]spectiue1 bicause t )\t p]spectiue k1 x is n easy to iudge truly ! =mes ( !m 9 flatte protacture4 ,& ?us ,i made an ende = ?is tyme1 ( ! def9itions ,geometricall1 app]tayn+ to ?is p>te ( practise1 & ! re/ wil ,i prosecute z cause %all s]ue4 ,,! ,,practike ,,work+e ,,( sondry conclusions geometrical4 ,,! ,,fyr/ ,,conclvsion4 ,to make a ?relike triangle on any lyne measurable4 ,take ! iu/e l5g? ( ! lyne ) y\r compasse1 & /ay ! "o foot ( ! compas 9 "o ( ! endes ( t l9e1 turn+ ! o!r vp or d\n at y\r w1 drawyng ! ar*e ( a circle aga9/ ! midle ( ! l9e1 & doo l wise ) ! same compasse vnalt]$1 at ! o!r end ( ! l9e1 & :] ^! ij4 crok$ lynes do? crosse1 frome !nce drawe a lyne to e* end ( y\r fir/ l9e1 & !re %all appe> a ?relike triangle draw5 on t l9e4 ,illu/ration ,example4 ,a4,b4 is ! fir/ l9e1 on :i* ,i wold make ! ?relike triangle1 !r=e ,i op5 ! compasse z wyde z t l9e is long1 & draw two ar* l9es t mete 9 ,c1 !n f ,c1 ,i draw ij o!r l9es "o to ,a1 ano!r to ,b1 & ?an ,i haue my purpose4 ,,! 4,,ii4 ,,conclvsion ,if y wil make a twileke or a n\elike triangle on ani c]ta9e l9e4 ,illu/ration ,consid] fyr/ ! l5g? t y[ w haue ! o!r sides to conta9e1 & to t l5g? op5 y\r compasse1 & !n worke z y did 9 ! ?releke triangle1 remembryng ?is1 t 9 a n\elike triangle y mu/ take ij4 l5g!s besyde ! fyr/e lyne1 & draw an ar*e lyne ) "o ( !m at ! "o ende1 & ) ! o!r at ! o!r end1 ! example is z 9 ! o!r be=e4 ,illu/ration ,,! ,,iii4 ,,concl4 ,to diuide an angle ( "r l9es into ij4 equal p>tes4 ,fir/ op5 y\r compasse z l>gely z y c1 s t x d n exc$e ! l5g? ( ! %orte/ l9e yt inclose? ! angle4 ,!n set "o foote ( ! compasse 9 ! v]ye po9t ( ! angle1 & ) ! o!r fote draw a compass$ ar* f ! "o lyne ( ! angle to ! o!r1 t ar* %all y deuide 9 halfe1 & !n draw a l9e f ! angle to ye middle ( ye ar*1 & s ye angle is diuid$ into ij4 equall p>tes4 ,illu/ration ,example4 ,let ! triangle be ,a4,b4,c1 !n set ,i "o foot ( ye compasse 9 ,b1 & ) ! o!r ,i draw ye ar* ,d4,e1 :i* ,i "p into ij4 equall p>ts 9 ,f1 & !n draw a l9e f ,b1 to ,f1 s ,i haue m9e int5t4 ,,! ,,iiii4 ,,concl4 ,to deuide any measurable l9e into ij4 equall p>tes4 ,illu/ration ,op5 y\r compasse to ! iu/ l5g? ( ye l9e4 ,& !n set "o foote /$dely at ! "o ende ( ! l9e1 wt ! o!r fote draw an ar* ( a circle aga9/ ye midle ( ! l9e1 bo? ou] x1 & also vnd] x1 !n doo lykewaise at ! o!r ende ( ! l9e4 ,& m>ke :]e ^? ar*e l9es d meet crosse waies1 & betw5e ^? ij4 pricks draw a l9e1 & x %all cut ! fir/ l9e 9 two equall portions4 ,example4 ,! lyne is ,a4,b4 accordyng to :i* ,i op5 ! compasse & make 4iiij4 ar*e l9es1 :i*e meete 9 ,c4 & ,d1 !n drawe ,i a lyne f ,c1 s haue ,i my purpose4 ,?is conlusion s]ue? = makyng ( quadrates & squires1 beside _m o!r commodities1 h[ebeit x maye bee don m readylye by ?is conclusion t fol[e? nexte4 ,,! ,,fift ,,conclvsion4 ,to make a plumme l9e or any pricke t y w 9 any "r lyne appo9t$4 ,op5 y\re compas s t x be n wyd] !n f ! pricke appoynt$ 9 ! l9e to ! %orte/ ende ( ! l9e1 b r %ort]4 ,!n sette ! "o foote ( ! compasse 9 ! fir/ pricke appo9t$1 & ) ! o!r fote m>ke ij4 o!r prickes1 "o ( e*e syde ( t fyr/e1 aft]w>de op5 y\r compasse to ! wyd5es ( ^? ij4 new prickes1 & draw f !m ij4 ar* lynes1 z y did 9 ! fyr/ conclusion1 = mak+ ( a ?relyke triangle4 !n if y d m>k _! cross+1 & f x drawe a l9e to y\r fyr/e pricke1 x %all bee a iu/ plum lyne on t place4 ,illu/ration ,example4 ,! lyne is ,a4,b4 ! prick on :i*e ,i %\lde make ! plumme lyne1 is ,c4 !n op5 ,i ! compasse z wyde z ,a4,c1 & sette "o foot 9 ,c4 & ) ! o!r doo ,i m>ke out ,c4,a4 & ,c4,b1 !n op5 ,i ! compasse z wide z ,a4,b1 & make ij4 ar* l9es :i* d crosse 9 ,d1 & s haue ,i doone4 ,h[e bee x1 x happ5e? s sommetymes1 t ! pricke on :i*e y w\ld make ! p]p5dicul> or plum l9e1 is s n]e ! e& ( y\r l9e1 t y c n ext5de any notable l5g? f x to ?one end ( ! l9e1 & if s be x !n t y maie n drawe y\r l9e l5g] f t end1 !n do? ?is conclusion require a newe ayde1 = ! la/ deuise w n s]ue4 ,9 su*e case !r=e %all y dooe ?us3 ,if y\r l9e be ( any notable l5g?1 deuide x into fiue p>tes4 ,& if x be n s long t x maie yelde fiue notable p>tes1 !n make an o!r l9e at w1 & p>te x into fiue equall portions3 s t ?re ( ^? p>tes maie be f\nd 9 y\r l9e4 ,!n op5 y\r compas z wide z ?re ( ^! fiue measures be1 & sette ! "o foote ( ! compas 9 ! pricke1 :]e y w\ld haue ! plumme l9e to lighte 7:i*e ,i call ! fir/ pricke17 & ) ! o!r foote drawe an ar*e l9e righte ou] ! pricke1 z y c ayme x3 !n op5 y\re compas z wide z all fiue measures be1 & set ! "o foote 9 ! f\r? pricke1 & ) ! o!r foote draw an o!r ar* l9e crosse ! fir/1 & :]e !i two d crosse1 !nse draw a l9e to ! po9ct :]e y w\lde haue ! p]p5dicul> l9e to light1 & y haue doone4 ,example4 ,illu/ration ,! l9e is ,a4,b4 & ,a4 is ! prick1 on :i*e ! p]p5dicul> l9e mu/ light4 ,!r=e ,i deuide ,a4,b4 into fiue p>tes equall1 !n d ,i op5 ! compas to ! wid5esse ( ?ree p>tes 7t is ,a4,d47 & let "o foote /aie 9 ,a4 & ) ! o!r ,i make an ar*e l9e 9 ,c4 ,aft]w>de ,i op5 ! compas z wide z ,a4,b4 7t is z wide z all fiue p>tes7 & set "o foote 9 ! 4iiij4 pricke1 :i* is ,e1 drawyng an ar* l9e ) ! o!r foote 9 ,c4 also4 ,!n d ,i draw !nce a l9e vnto ,a1 & s haue ,i doone4 ,b & if ! l9e be to %orte to be p>t$ into fiue p>tes1 ,i %all deuide x into iij4 p>tes only1 z y see ! liue ,f4,g1 & !n make ,d4 an o!r l9e 7z is ,k4,l47 :i*e ,i deuide into 4v4 su*e diuisions1 z ,f4,g4 conta9e? 4iij1 !n op5 ,i ! compass z wide z 4iiij4 p>tes 7:i*e is ,k4,m47 & s set ,i "o foote ( ! compas 9 ,f1 & ) ! o!r ,i drawe an ar* lyne t[>d ,h1 !n op5 ,i ! compas z wide z ,k4,l4 7t is all 4v4 p>tes7 & set "o foote 9 ,g1 7t is ! iij4 pricke7 & ) ! o!r ,i draw an ar* l9e t[>d ,h4 also3 & :]e ^? 4ij4 ar* l9es d crosse 7:i*e is by ,h47 !nce draw ,i a l9e vnto ,f1 & t make? a v plumbe l9e to ,f4,g1 z my desire 04 ,! man] ( workyng ( ?is conclusion1 is l to ! second conlusion1 b ! reason ( x do? dep5d ( ! 4xlvi4 proposicion ( ye fir/ boke ( ,euclide4 ,an o!r waie yet4 set "o foote ( ! compas 9 ! prick1 on :i*e y w\ld haue ! plumbe l9e to light1 & /ret*e =? ?o!r foote t[>d ! longe/ end ( ! l9e1 z wide z y c = ! l5g? ( ! l9e1 & s draw a qu>t] ( a compas or m1 !n )\t /irryng ( ! compas1 set "o foote ( x 9 ! same l9e1 :]e z ! circul> l9e did beg91 & ext5d ?o!r 9 ! circul> l9e1 settyng a m>ke :]e x do? light1 !n take half t quantitie m !re vnto1 & by t prick t ende? ! la/ "p1 draw a l9e to ! pricke assign$1 & x %all be a p]p5dicul>4 ,illu/ration ,example4 ,a4,b4 is ! l9e appo9t$1 to :i*e ,i mu/ make a p]p5dicul> l9e to light 9 ! pricke assign$1 :i* is ,a4 ,!r=e doo ,i set "o foote ( ! compas 9 ,a1 & ext5d ! o!r vnto ,d4 makyng a "p ( a circle1 m !n a qu>t]1 t is ,d4,e4 ,!n d ,i set "o foote ( ! compas vnalt]$ 9 ,d1 & /ret* ! o!r 9 ! circul> l9e1 & x do? light 9 ,f1 ?is space betw5e ,d4 & ,f4 ,i deuide into halfe 9 ! pricke ,g1 :i*e halfe ,i take ) ! compas1 & set x beyond ,f4 vnto ,h1 & !=e is ,h4 ! po9t1 by :i*e ! p]p5dicul> l9e mu/ be drawn1 s say ,i t ! l9e ,h4,a1 is a plumbe l9e to ,a4,b1 z ! conclusion w\ld4 ,,! 4,,vi4 ,,conclvsion4 ,to drawe a /reight l9e f any pricke t is n 9 a l9e1 & to make x p]p5dicul> to an o!r l9e4 ,op5 y\r compas z s wide t x may ext5d some:at f>!r1 !n f ! prick to ! l9e1 !n sette ! "o foote ( ! compas 9 ! pricke1 & ) ! o!r %all y draw a compass$ l9e1 t %all crosse t o!r fir/ l9e 9 4ij4 places4 ,n[ if y deuide t ar* l9e into 4ij4 equall p>tes1 & f ! middell pricke !r( vnto ! prick )\t ! l9e y drawe a /reight l9e1 x %albe a plumbe l9e to t fir/e lyne1 accordyng to ! conclusion4 ,illu/ration ,example4 ,c4 is ! appo9t$ pricke1 f :i*e vnto ! l9e ,a4,b4 ,i mu/ draw a p]p5dicul>4 ,!=e ,i op5 ! compas s wide1 t x may haue "o foote 9 ,c1 & ?o!r to rea* ou] ! l9e1 & ) yt foote ,i draw an ar* l9e z y see1 betw5e ,a4 & ,b1 :i* ar* l9e ,i deuide 9 ! middell 9 ! po9t ,d4 ,!n drawe ,i a l9e f ,c4 to ,d1 & x is p]p5dicul> to ! l9e ,a4,b1 accordyng z my desire 04 ,,! 4,,vii4 ,,conclvsion4 ,to make a plumbe lyne on any porcion ( a circle1 & t on ! vtt] or inn] bughte4 ,m>k fir/ ! prick :]e ye plumbe l9e %al lyght3 & prick out ( e* side ( x 4ij4 o!r po9ctes equally di/ant f t fir/ pricke4 ,!n set ! "o foote ( ! compas 9 "o ( ^? side prickes1 & ! o!r foote 9 ! o!r side pricke1 & fir/ m\e "o ( ! feete & drawe an ar*e l9e ou] ! middell pricke1 !n set ! compas /$die ) ! "o foote 9 ! o!r side pricke1 & ) ! o!r foote drawe an o!r ar*e l9e1 t %all cut t fir/ ar*e1 & f ! v po9cte ( _! meetyng1 drawe a "r l9e vnto ! fir/e pricke1 :]e y d m9de t ! plumbe l9e %all lyghte4 ,& s haue y p]=m$ ?9t5t ( ?is conclusion4 ,example4 ,illu/ration ,! ar*e ( ! circle on :i*e ,i w\ld erect a plumbe l9e1 is ,a4,b4,c4 & ,b4 is ! pricke :]e ,i w\ld haue ! plumbe l9e to light4 ,!r=e ,i meate out two equall di/aunces on e*e side ( t pricke ,b4 & !y are ,a4,c4 ,!n op5 ,i ! compas z wide z ,a4,c4 & settyng "o ( ! feete 9 ,a4 ) ! o!r ,i drawe an ar*e l9e :i* goe? by ,g4 ,l waies ,i set "o foote ( ! compas /$dily 9 ,c4 & ) ! o!r ,i drawe an ar*e l9e1 goyng by ,g4 also4 ,n[ consid]yng t ,g4 is ! pricke ( _! meetyng1 x %all be also ! po9ct fro :i*e ,i mu/ drawe ! plumbe l9e4 ,!n draw ,i a "r l9e f ,g4 to ,b4 & s haue m9e int5t4 ,n[ z ,a4,b4,c4 ha? a plumbe l9e erect$ on 8 vtt] b\ght1 s may ,i erect a plumbe l9e on ! inn] bught ( ,d4,e4,f1 doynge ) x z ,i did ) ! o!r1 t is to saye1 fyr/e settyng =! ! pricke :]e ! plumbe l9e %all light1 :i* is ,e1 & !n m>kyng "o o!r on e*e syde1 z are ,d4 & ,f4 ,& !n proc$+ z ,i dyd 9 ! example be=e4 ,,! ,,viii4 ,,conclvsyon4 ,h[ to deuide ! ar*e ( a circle into two equall p>tes1 )\t measur+ ! ar*e4 ,deuide ! corde ( t l9e info ij4 equall portions1 & !n f ! middle prycke erecte a plumbe l9e1 & x %al p>te t ar*e 9 ! middle4 ,illu/ration ,example4 ,! ar* to be diuid$ ys ,a4,d4,c1 ! corde is ,a4,b4,c1 ?is corde is diuid$ 9 ! middle ) ,b1 f :i* prick if ,i erect a plum l9e z ,b4,d1 !n w x diuide ! ar* 9 ! middle1 t is to say1 9 ,d4 ,,! ,,ix4 ,,conclvsion4 ,to d ! same ?ynge o!r wise4 ,& = %ort5es ( worke1 if y wyl make a plumbe l9e )\t mu* lab\r1 y may d x ) y\r squyre1 s t x be iu/ly made1 = yf y applye ! edge ( ! squyre to ! l9e 9 :i* ! prick is1 & =esee ! v corn] ( ! squyre doo t\*e ! pricke4 ,& ?an frome t corn] if y drawe a lyne by ! o!r edge ( ! squyre1 yt w be p]p5dicul> to ! =m] l9e4 ,illu/ration ,example4 ,a4,b4 is ! l9e1 on :i* ,i wold make ! plumme l9e1 or p]p5dicul>4 ,& !re=e ,i m>ke ! prick1 f :i* ! plumbe lyne mu/e rise1 :i* "h is ,c4 ,!n d ,i sette "o edg ( my squyre 7t is ,b4,c47 to ! l9e ,a4,b1 s at ! corn] ( ! squyre d t\*e ,c4 iu/ly4 ,& f ,c4 ,i drawe a l9e by ! o!r edge ( ! squire1 7:i* is ,c4,d47 ,& s haue ,i made ! plumme l9e ,d4,c1 :i* ,i s\ght =4 ,,! ,x4 ,,conclvsion4 ,h[ to d ! same ?+e an o!r way yet ,if s be x t y haue an ar*e ( su*e greatnes1 t y\r squyre wyll n suffice !rto1 z ! ar*e ( a brydge or ( a h\se or w9d[1 !n may y d ?is4 ,mete vnd]ne? ! ar* :]e ye midle ( 8 cord wyl be1 & !r set a m>k4 ,!n take a long l9e ) a plummet1 & holde ! l9e 9 su*e a place ( ! ar*1 t ! plummet d hang iu/ely ou] ! middle ( ! corde1 t y didde diuide be=e1 & !n ! l9e do? %ewe y ! middle ( ! ar*e4 ,illu/ration ,example4 ,! ar* is ,a4,d4,b1 ( :i* ,i trye ! midle ?us4 ,i draw a corde f "o syde to ! o!r 7z "h is ,a4,b17 :i* ,i diuide 9 ! middle 9 ,c4 ,!n take ,i a l9e ) a plummet 7t is ,d4,e17 & s hold ,i ! l9e t ! plummet ,e1 doo? hange ou] ,c1 ,& !n ,i say t ,d4 is ! middle ( ! ar*e4 ,& to !nt5t t my plummet %all po9t ! m iu/ely1 ,i doo make x %>pe at ! ne!r ende1 & s may ,i tru/ ?is woorke = c]ta9e4 ,,! ,,xi4 ,,conclvsion4 ,:5 any l9e is appo9t$ & )\t x a pricke1 :]eby a p>allel mu/ be draw5 h[e y %all doo x4 ,take ! iu/e measure beetw5e ! l9e & ! pricke1 accord+e to :i* y %al op5 y\r compasse4 ,!n pit* "o foote ( y\r compasse at ! "o ende ( ! l9e1 & ) ! o!r foote draw a b[e l9e "r ou] ! pyt*e ( ! compasse1 lyke-wise doo at ! o!r ende ( ! lyne1 !n draw a l9e t %all t\*e ! vtt]mo/e edge ( bo! ^? b[e l9es1 & x w bee a true p>allele to ! fyr/e lyne appo9t$4 ,example4 ,illu/ration ,a4,b1 is ! l9e vnto :i* ,i mu/ draw an o!r gem[ l9e1 :i* mu/e passe by ! prick ,c1 fir/ ,i meate ) my compasse ! smalle/ di/ance t is f ,c4 to ! l9e1 & t is ,c4,f1 :]=e /ay+ ! compasse at t di/aunce1 ,i seete ! "o foote 9 ,a1 & ) ! o!r foot ,i make a b[e lyne1 :i* is ,d1 !n l wise set ,i ! "o foote ( ! compasse 9 ,b1 & ) ! o!r ,i make ! second b[ l9e1 :i* is ,e4 ,& !n draw ,i a l9e1 s t x t\*e? ! vtt]mo/ edge ( bo! ^! b[e l9es1 & t lyne passe? by ! pricke ,c1 end is a gem[e l9e to ,a4,b1 z my sekyng 04 ,,! 4,,xii4 ,,conclvsion4 ,to make a triangle ( any 4iij4 l9es1 s t ! l9es be su*e1 t any 4ij4 ( !m be long] !n ! ?irde4 ,= ?is rule is g5]all1 t any two sides ( eu]ie triangle tak5 toge!r1 are long] !n ! o!r side t rema9e?4 ,if y d rememb] ! fir/ & seconde conclusions1 !n is !re no difficultie 9 ?is1 = x is 9 man] ! same woorke4 ,fir/ consid] ! 4iij4 l9es t y mu/ take1 & set "o ( !m = ! gr\nd l9e1 !n worke ) ! o!r 4ij4 l9es z y did 9 ! fir/ & second conclusions4 ,example4 ,illu/ration ,i haue 4iij4 ,a4,b4 & ,c4,d4 & ,e4,f4 ( :i*e ,i put 4,c4,d4 = my gr\nd l9e1 !n ) my compas ,i take ! l5g? ( 4,a4,b4 & set ! "o foote ( my compas 9 ,c1 & draw an ar* l9e ) ! o!r foote4 ,likewaies ,i take ! l5g? ( ,e4,f1 & set "o foote 9 ,d1 & ) ! o!r foote ,i make an ar* l9e crosse ! o!r ar*e1 & ! pricke ( _! metyng 7:i*e is ,g47 %all be ! ?irde corn] ( ! triangle1 = 9 all su*e kyndes ( woorkynge to make a tryangle1 if y haue "o l9e draw51 !re remayne? no?yng els b to fynde :]e ! pit*e ( ! ?irde corn] %all bee1 = two ( !m mu/ ne$es be at ! two e&es ( ! lyne t is draw54 ,,! ,,xiii4 ,,conclvsion4 ,if y haue a l9e appo9t$1 & a po9te 9 x limit$1 h[e y maye make on x a righte l9$ angle1 equall to an o!r "r l9$ angle1 all ready assign$4 ,fyr/e draw a l9e aga9/ ! corn] assign$1 & s is x a triangle1 !n take he$e to ! l9e & ! po9te 9 x assign$1 & consid] if t l9e f ! pricke to ?is end bee z long z any ( ! sides t make ! triangle assign$1 & if x bee longe en\ghe1 !n prick out !re ! l5g? ( "o ( ! l9es1 & !n woorke ) ! o!r two l9es1 accord+e to ! la/e conlusion1 makynge a triangle ( ?re l lynes to t assign$ triangle4 ,if x bee n longe in\ghe1 !nn l5g!n x fyr/e1 & aft]w>de doo z ,i haue sayde bee=e4 ,illu/ration ,example4 ,lette ! angle appoynt$ bee ,a4,b4,c1 & ! corn] assign$1 ,b4 ,f>!rmore let ! lymit$ l9e bee ,d4,g1 & ! pricke assign$ ,d4 ,fyr/e !re=e by draw+e ! l9e ,a4,c1 ,i make ! triangle ,a4,b4,c4 ,illu/ration ,!n consid]+e t ,d4,g1 is long] ?anne ,a4,b1 y %all cut out a l9e f ,d4 t[>d ,g1 equal to ,a4,b1 z = example ,d4,f4 ,!n measure oute ! o!r ij4 l9es & worke ) !m accord+ z ! conclusion ) ! fyr/e also & ! second tea*e? y[1 & !n haue y done4 ,,! ,,xiiii4 ,,conclvsion4 ,to make a squ>e quadrate ( any righte lyne appo9ct$4 ,fir/ make a plumbe l9e vnto y\r l9e appo9t$1 :i*e %all light at "o ( ! endes ( x1 accordyng to ! fif? conclusion1 & let x be ( l l5g? z y\r fir/ l9e is1 !n op5 y\r compasse to ! iu/e l5g? ( "o ( !m1 & sette "o foote ( ! compasse 9 ! ende ( ! "o l9e1 & ) ! o!r foote draw an ar*e l9e1 !re z y ?9ke t ! f[]? corn] %all be1 aft] t set ! "o foote ( ! same compasse vn/urr$1 9 ! e&e ( ! o!r l9e1 & drawe an o!r ar*e l9e crosse ! fir/ ar*el9e1 & ! po9cte t !y d crosse 91 is ! pricke ( ! f\r? corn] ( ! squ>e quadrate :i* y seke =1 !r=e draw a l9e f t pricke to ! e&e ( e*e l9e1 & y %all !rby haue made a squ>e quadrate4 ,example4 ,illu/ration ,a4,b4 is ! l9e propos$1 ( :i*e ,i %all make a squ>e quadrate1 !re=e fir/e ,i make a plumbe l9e vnto x1 :i*e %all lighte 9 ,a1 & t plumb l9e is ,a4,c1 !n op5 ,i my compasse z wide z ! l5g? ( ,a4,b1 or ,a4,c1 7= !y mu/ be bo! equall7 & ,i set ! "o foote ( !nd 9 ,c1 & ) ! o!r ,i make an ar*e l9e nigh vnto ,d1 aft]w>d ,i set ! compas aga9 ) "o foote 9 ,b1 & ) ! o!r foote ,i make an ar*e l9e crosse ! fir/ ar*e l9e 9 ,d1 & f ! prick ( _! crossyng ,i draw 4ij4 l9es1 "o to ,b1 & an o!r to ,c1 & s haue ,i made ! squ>e quadrate t ,i ent5d$4 ,,! 4,,xv4 ,,conclvsion4 ,to make a likeiamme equall to a triangle appo9t$1 & t 9 a "r l9$ angle limit$4 ,fir/ f "o ( ! angles ( ! triangle1 y %all drawe a gem[e l9e1 :i*e %all be a p>allele to t syde ( ! triangle1 on :i*e y w make t likeiamme4 ,!n on "o end ( ! side ( ! triangle1 :i*e lie? aga9/ ! gem[e lyne1 y %all draw =? a l9e vnto ! gem[ l9e1 s t "o angle t comme? ( ^? 4ij4 l9es be l to ! angle :i* is limit$ vnto y4 ,!n %all y deuide into ij4 equall p>tes t side ( ! triangle :i*e be>e? t l9e1 & f ! pricke ( t deuision1 y %all raise an o!r l9e p>allele to t =m] l9e1 & cont9ewe x vnto ! fir/ gem[e l9e1 & !n ( ^? 4ij4 la/ gem[e lynes1 & ! fir/ gem[e l9e1 ) ! halfe side ( ! triangle1 is made a lykeiamme equall to ! triangle appo9t$1 & ha? an angle lyke to an angle limit$1 accordyng to ! conclusion4 ,illu/ration ,example4 ,b4,c4,g1 is ! triangle appo9ct$ vnto1 :i*e ,i mu/e make an equall likeiamme4 ,& ,d1 is ! angle t ! likeiamme mu/ haue4 ,!r=e fir/ ent5dyng to erecte ! likeiamme on ! "o side1 t ! gr\nd l9e ( ! triangle 7:i*e is ,b4,g47 ,i d draw a gem[ l9e by ,c1 & make x p>allele to ! gr\nd l9e ,b4,g1 & t new gem[ l9e is ,a4,h4 ,!n d ,i raise a l9e f ,b4 vnto ! gem[e l9e1 7:i*e l9e is ,a4,b7 & make an angle equall to ,d1 t is ! appo9t$ angle 7accordyng z ! 4viij4 conclusion tea*e?7 & t angle is ,b4,a4,e4 ,!n to proc$e1 ,i doo p>te 9 ye middle ! said gr\nd l9e ,b4,g1 9 ! prick ,f1 f :i* prick ,i draw to ! fir/ gem[e l9e 7,a4,h47 an o!r l9e t is p>allele to ,a4,b1 & t l9e is ,e4,f4 ,n[ saie ,i t ! likeiamme ,b4,a4,e4,f1 is equall to ! triangle ,b4,c4,g4 ,& also t x ha? "o angle 7t is ,b4,a4,e47 l to ,d4 ! angle t 0 limitt$4 ,& s haue ,i m9e int5t4 ,! pr(e ( ! equalnes ( ^? two figures doe? dep5d ( ! 4xli4 proposition ( ,euclides fir/ boke1 & is ! 4xxxi4 proposition ( ?is second boke ( ,!oremis1 :i*e saie?1 t :an a tryangle & a likeiamme be made betw5e 4ij4 selfe same gem[ l9es1 & haue _! gr\nd l9e ( "o l5g?1 !n is ! likeiamme d\ble to ! triangle1 :]( x fol[e?1 t if 4ij4 su*e figures s draw5 diff] 9 _! gr\nd l9e onely1 s t ! gr\nd l9e ( ! likeiamme be b halfe ! gr\nd l9e ( ! triangle1 !n be ^? 4ij4 figures equall1 z y %all m at l>ge p]ceiue by ! boke ( ,!oremis1 9 ye 4xxxi4 !oreme4 ,,! 4,,xvi4 ,,conclvsion4 ,to make a likeiamme equall to a triangle appo9ct$1 accordyng to an angle limitt$1 & on a l9e also assign$4 ,9 ! la/ conclusion ! sides ( y\r likeiamme w] left to y\r lib]tie1 ?\gh y _h an angle appo9ct$4 ,n[e 9 ?is conclusion y are som:at m re/ra9$ ( lib]tie si? ! l9e is limitt$1 :i* mu/ be ! side ( ! likeiamme4 ,!r=e ?us %all y proc$e4 ,fir/e accordyng to ! la/e conclusion1 make a likeiamme 9 ! angle appo9ct$1 equall to ! triangle t is assign$4 ,!n ) y\r compasse take ! l5g? ( y\r l9e appo9t$1 & set out two l9es ( ! same l5g? 9 ! second gem[e l9es1 beg9nyng at ! "o side ( ! likeiamme1 & by ^? two prickes %all y draw an o!r gem[e l9e1 :i*e %all be p>allele to two sides ( ! likeiamme4 ,aft]w>d %all y draw 4ij4 l9es m = ! accompli%em5t ( y\r worke1 :i* bett] %all be p]ceau$ by a %orte exaumple1 !n by a greate numbre ( wordes1 only )\t example1 !re=e ,i wyl by example sette =? ! :ole worke4 ,example4 ,illu/ration ,fyr/1 accord+ to ! la/ conclusion1 ,i make ! likeiamme ,e4,f4,c4,g1 equal to ! triangle ,d1 9 ! appoynt$ angle :i*e is ,e4 ,!n take ,i ! l5g! ( ! assign$ l9e 7:i* is ,a4,b17 & ) my compas ,i sette =! ! same l5g? 9 ! ij4 gem[ l9es ,n4,f4 & ,h4,g1 sett+ "o foot 9 ,e1 & ! o!r 9 ,n1 & aga9e settyng "o foote 9 ,c1 & ! o!r 9 ,h4 ,aft]w>d ,i draw a l9e f ,n4 to ,h1 :i*e is a gem[ lyne1 to ij4 sydes ( ! likeiamme4 !nne drawe ,i a l9e also f ,n4 vnto ,c4 & ext5d x vntyll x crosse ! l9es1 ,e4,l4 & ,f4,g1 :i* bo? mu/ be draw5 =? long] !n ! sides ( ! likeiamme4 & :]e t lyne doe? crosse ,f4,g1 !re ,i sette ,m4 ,n[e to make an ende1 ,i make an o!r gem[e l9e1 :i*e is p>allel to ,n4,f4 & ,h4,g1 & t gem[e l9e do? passe by ! pricke ,m1 & !n haue ,i done4 ,n[ say ,i t ,h4,c4,k4,l1 is a likeiamme equall to ! triangle appo9t$1 :i*e 0 ,d1 & is made ( a l9e assign$ t is ,a4,b1 = ,h4,c1 is equall vnto ,a4,b1 & s is ,k4,l4 ,! pr(e ( ye equalnes ( ?is likeiam vnto ! triangle1 dep5de? ( ! ?irty & two ,!oreme3 z 9 ! boke ( ,!oremes do? appe>1 :]e x is decl>$1 t 9 al likeiammes1 :5 !re are m !n "o made ab\t "o bias l9e1 ! filsqu>es ( eu]y ( !m mu/e ne$es be equall4 ,,! ,,xvii4 ,,conclvsion4 ,to make a likeiamme equal to any "r l9$ figure1 & t on an angle appo9t$4 ,! readie/ waye to worke ?is conclusion1 is to t\rn t rightl9$ figure into triangles1 & !n = eu]y triangle toge!r an equal likeiamme1 accord+ vnto ! eleu5 conclusion1 & !n to io9e al ^? likeiammes into "o1 if _! sides happ5 to be equal1 :i* ?+ is eu] c]ta91 :5 al ! triangles happ5 iu/ly betw5e "o pair ( gem[ l9es4 b & if !y w n frame s1 !n aft] t y haue = ! fir/e triangle made 8 likeiamme1 y %all take ! l5g? ( "o ( 8 sides1 & set t z a l9e assign$1 on :i*e y %al make ! o!r likeiams1 accord+ to ! twelft conclusion1 & s %all y haue al y\r likeiammes ) ij4 sides equal1 & ij4 l angles1 s yt y mai easily ioyne !m into "o figure4 ,illu/ration ,example4 ,if ! "r l9$ figure be l vnto ,a1 !n may x be turn$ into triangles t wil /& betw5e ij4 p>allels anye ways1 z y mai se by ,c4 & ,d1 = ij4 sides ( bo? ! trianngles ar p>allels4 ,also if ! "r l9$ figure be l vnto ,e1 !n wil x be turn$ into triangles1 liyng betw5e two p>allels also1 z ye o!r did be=e1 z 9 ! example ( ,f4,g4 ,b & if ye "r l9$ figure be l vnto ,h1 & s turn$ into triangles z y se 9 ,k4,l4,m1 :] x is p>t$ into iij triangles1 !n wil n all ^? triangles lye betw5 "o pair ( p>allels or gem[ l9es1 b mu/ haue _m1 = eu]y triangle mu/ haue "o paire ( p>allels seu]all1 yet x maye happ5 t :5 !re bee ?ree or f[] triangles1 ij4 ( !ym maye happ5 to agre to "o pair ( p>allels1 :i*e ?+e ,i remit to eu]y hone/ witte to s]*e1 = ! mann] ( _! draught wil decl>e1 h[ _m paires ( p>allels !y %all ne$e1 ( :i* v>ietee bicause ! examples ar inf9ite1 ,i haue set =? ^! few1 t by !m y may coniecture duly ( all o!r l4 ,illu/ration ,fur!r explicacion y %al n greatly ne$e1 if y remembre :at ha? b5 taught be=e1 & !n dilig5tly behold h[ ^! sundry figures be turn$ into triangles4 ,9 ! fyr/ y se ,i haue made v4 triangles1 & f\r p>alleles4 9 ! seconde vij4 triangles & f\re p>alleles4 9 ! ?irde ?re triangles1 & fiue p>allels1 9 ! iiij4 y se fiue triangles f\r p>allels4 9 ! fift1 iiij4 triangles & 4iiij4 p>allels1 9 ye sixt !r ar fiue triangles iiij4 p>alels4 ,h[beit a man maye at lib]ty alt] !m into diu]s =mes ( triangles !re=e ,i leue x to ! discretion ( ! woorkmai/]1 to d 9 al su*e cases z he %al ?9ke be/1 = by ^! examples 7if !y bee well m>k$7 may all o!r l conclusions be wr\ght4 ,,! ,,xviii4 ,,conclvsion4 ,to p>te a l9e assign$ aft] su*e a sorte1 t ! squ>e t is made ( ! :ole l9e & "o ( 8 p>ts1 %al be equal to ! squ> t come? ( ! o!r p>te alone4 ,fir/ deuide y\r lyne into ij4 equal p>ts1 & ( ! l5g? ( "o "p make a p]p5dicul> to light at "o end ( y\r l9e assign$4 !n adde a bias l9e1 & make !re( a triangle1 ?is done if y take f ?is bias l9e ! halfe l5g! ( y\r l9e appo9t$1 :i* is ! iu/e l5g? ( y\r p]p5dicul>1 t "p ( ! bias l9e :i*e do! remayne1 is ! great] portion ( ! deuision t y seke =1 !re=e if y cut y\r l9e accord+ to ! l5g! ( x1 !n w ! squ>e ( t great] portion be equall to ! squ>e t is made ( ! :ole l9e & 8 less] portion4 ,& contr>y wise1 ! squ>e ( ! :ole l9e & 8 less] p>te1 wyll be equall to ! squ>e ( ! great] p>te4 ,illu/ration ,example4 ,a4,b1 is ! lyne assign$4 ,e4 is ! middle pricke ( ,a4,b1 ,b4,c4 is ! plumb l9e or p]p5dicul>1 made ( ! halfe ( ,a4,b1 equall to ,a4,e1 o!r ,b4,e1 ! byas l9e is ,c4,a1 f :i*e ,i cut a peece1 t is ,c4,d1 equall to ,c4,b1 & accordyng to ! l5g! lo ! peece t rema9e? 7:i*e is ,d4,a17 ,i doo deuide ! l9e ,a4,b1 at :i*e diuision ,i set ,f4 ,n[ say ,i1 t ?is l9e ,a4,b1 7w* 0 assign$ vnto me7 is s diuid$ 9 ?is po9t ,f1 yt ye squ>e ( ye hole l9e ,a4,b1 ( ! "o portion 7yt is ,f4,b1 ! less] "p7 is equall to ! squ>e ( ! o!r p>te1 :i*e is ,f4,a1 & is ! great] "p ( ! fir/ l9e4 ,! pr(e ( ?is equalitie %all y le>ne by ! 4xl4 ,!oreme4 ,transcrib]'s ,note3 ,!re are two ways to make ?is ,example "w3 --transpose ,e & ,f 9 ! illu/ration1 & *ange "o occurr5ce ( ,e to ,f 9 ! text 7at :i*e diuision ,i set44471 or3 --keep ! illu/ration z pr9t$1 & transpose all o!r occurr5ces ( ,e & ,f 9 ! text4 ,,! 4,,xix4 ,,conclvsion4 ,to make a squ>e quadrate equall to any "r l9$ figure appo9ct$4 ,fir/ make a likeiamme equall to t "r l9$ figure1 ) a "r angle1 accordyng to ! 4xi4 conclusion1 !n consid] ! likeiamme1 :e!r x haue all 8 sides equall1 or n3 = yf !y be all equall1 !n haue y doone y\r conclusion4 b & if ! sides be n all equall1 !n %all y make "o "r l9e iu/e z long z two ( ^? vnequall sides1 t l9e %all y deuide 9 ! middle1 & on t pricke drawe half a circle1 !n cutte f t diamet] ( ! halfe circle a c]tayne portion equall to ! "o side ( ! likeiamme1 & f t po9te ( diuision %all y erecte a p]p5dicul>1 :i* %all t\*e ! edge ( ! circle4 ,& t p]p5dicul> %all be ! iu/e side ( ! squ>e quadrate1 equall bo? to ! lykeiamme1 & also to ! "r l9$ figure appo9t$1 z ! conclusion will$4 ,example4 ,illu/ration ,k1 is ! "r l9$ figure appo9t$1 & ,b4,c4,d4,e1 is ! likeiamme1 ) "r angles equall vnto ,k1 b because t ?is likeiamme is n a squ>e quadrate1 ,i mu/ turne x into su* "o aft] ?is sort1 ,i %all make "o "r l9e1 z long z 4ij4 vnequall sides ( ! likeiamme1 t l9e "h is ,f4,g1 :i*e is equall to ,b4,c1 & ,c4,e4 ,!n "p ,i t l9e 9 ! middle 9 ! pricke ,m1 & on t pricke ,i make halfe a circle1 accordyng to ! l5g? ( ! diamet] ,f4,g4 ,aft]w>d ,i cut awaie a peece f ,f4,g1 equall to ,c4,e1 m>kyng t po9t ) ,h4 ,& on t pricke ,i erecte a p]p5dicul> ,h4,k1 :i*e is ! iu/ side to ! squ>e quadrate t ,i seke =1 !r=e accordyng to ! doctr9e ( ! 4x4 conclusion1 ( ! lyne ,i doe make a squ>e quadrate1 & s haue ,i atta9$ ! practise ( ?is conclusion4 ,,! 4,,xx4 ,,conclvsion4 ,:5 any 4ij4 squ>e quadrates are set =?1 h[ y maie make "o equall to !m bo!4 ,fir/ drawe a "r l9e equall to ! side ( "o ( ! quadrates3 & on ! ende ( x make a p]p5dicul>1 equall 9 l5g? to ! side ( ! o!r quadrate1 !n drawe a byas l9e betw5e ^? 4ij4 o!r l9es1 makyng !re( a "r angel$ triangle4 ,& t byas lyne wyll make a squ>e quadrate1 equall to ! o!r 4ij4 quadrates appo9t$4 ,illu/ration ,example4 ,a4,b4 & ,c4,d1 are ! two squ>e quadrates appo9t$1 vnto :i* ,i mu/ make "o equall squ>e quadrate4 ,fir/ !r=e ,i dooe make a righte l9e ,e4,f1 equall to "o ( ! sides ( ! squ>e quadrate ,a4,b4 ,& on ! "o end ( x ,i make a plumbe l9e ,e4,g1 equall to ! side ( ! o!r quadrate ,d4,c4 ,!n drawe ,i a byas l9e ,g4,f1 :i* beyng made ! side ( a quadrate 7accordyng to ! t5? conclusion7 w accompli%e ! worke ( ?is practise3 = ! quadrate ,h4 is mu*e iu/ z ! o!r two4 ,i meane ,a4,b4 & ,d4,c4 ,,! 4,,xxi4 ,,conclvsion4 ,:5 any two quadrates be set =?1 h[e to make a squire ab\t ! "o quadrate1 :i*e %all be equall to ! o!r quadrate4 ,det]m9e ) y\r selfe ab\t :i*e quadrate y wil make ! squire1 & drawe "o side ( t quadrate =? 9 l5gte1 accordyng to ! measure ( ! side ( ! o!r quadrate1 :i*e l9e y maie call ! gr\nde l9e1 & !n haue y a "r angle made on ?is l9e by an o!r side ( ! same quadrate3 ,!r=e turne t into a "r corn]$ triangle1 accordyng to ! worke 9 ! la/e conclusion1 by makyng ( a byas l9e1 & t byas lyne w p]=me ! worke ( y\r desire4 ,= if y take ! l5g? ( t byas l9e ) y\r compasse1 & !n set "o foote ( ! compas 9 ! f>!/ angle ( ! fir/ quadrate 7:i*e is ! "o ende ( ! gr\ndl9e7 & ext5d ! o!r foote on ! same l9e1 accordyng to ! measure ( ! byas l9e1 & ( t l9e make a quadrate1 enclosyng ye fir/ quadrate1 !n w !re app]e ! =me ( a squire ab\t ! fir/ quadrate1 :i* squire is equall to ! second quadrate4 ,illu/ration ,example4 ,! fir/ squ>e quadrate is ,a4,b4,c4,d1 & ! seconde is ,e4 ,n[ w\ld ,i make a squire ab\t ! quadrate ,a4,b4,c4,d1 :i*e %all bee equall vnto ! quadrate ,e4 ,!r=e fir/ ,i draw ! l9e ,a4,d1 m 9 l5g?1 accordyng to ! measure ( ! side ( ,e1 z y see1 f ,d4 vnto ,f1 & s ! hole l9e ( bo! ^! seu]all sides is ,a4,f1 !n make ,i a byas l9e f ,c1 to ,f1 :i*e byas l9e is ! measure ( ?is woorke4 :]e=e ,i op5 my compas accordyng to ! l5g? ( t byas l9e ,c4,f1 & set ! "o compas foote 9 ,a1 & ext5d ?o!r foote ( ! compas t[>d ,f1 makyng ?is pricke ,g1 f :i*e ,i erect a plumbel9e ,g4,h1 & s make out ! squ>e quadrate ,a4,g4,h4,k1 ^: sides are equall e*e ( !m to ,a4,g4 ,& ?is squ>e do? conta9 ! fir/ quadrate ,a4,b4,c4,d1 & also a squire ,g4,h4,k1 :i*e is equall to ! second quadrate ,e1 = z ! la/ conclusion decl>e?1 ! quadrate ,a4,g4,h4,k1 is equall to bo! ! o!r quadrates propos$1 t is ,a4,b4,c4,d1 & ,e4 ,!n mu/e ! squire ,g4,h4,k1 ne$es be equall to ,e1 consid]yng t all ! re/ ( t great quadrate is no?yng els b ! quadrate self1 ,a4,b4,c4,d1 & s haue ,i ?9t5t ( ?is conclusion4 ,,! 4,,xxii4 ,,conclvsion4 ,to f9d out ! c5tre ( any circle assign$4 ,draw a corde or /ryngl9e crosse ! circle1 !n deuide into 4ij4 equall p>tes1 bo? t corde1 & also ! b[e l9e1 or ar*e l9e1 t s]ue? to t corde1 & f ! prickes ( ^? diuisions1 if y drawe an o!r l9e crosse ! circle1 x mu/ n$es passe by ! c5tre4 ,!r=e deuide t l9e 9 ! middle1 & t middle pricke is ! c5tre ( ! circle propos$4 ,example4 ,illu/ration ,let ! circle be ,a4,b4,c4,d1 ^: c5tre ,i %all seke4 ,fir/ !r=e ,i draw a corde crosse ! circle1 t is ,a4,c4 ,!n d ,i deuide t corde 9 ! middle1 9 ,e1 & likewaies also d ,i deuide 8 ar*e l9e ,a4,b4,c1 9 ! middle1 9 ! po9te ,b4 ,aft]w>d ,i drawe a l9e f ,b4 to ,e1 & s crosse ! circle1 :i*e l9e is ,b4,d1 9 :i* l9e is ! c5tre t ,i seeke =4 ,!re=e if ,i p>te t l9e ,b4,d1 9 ! middle 9 to two equall portions1 t middle pricke 7:i* "h is ,f7 is ! v]ye c5tre ( ! sayde circle t ,i seke4 ,?is conclusion may o!r waies be wr\ght1 z ! mo/e "p ( conclusions haue sondry =mes ( practise1 & t is1 by mak+e ?re prickes 9 ! circumf]5ce ( ! circle1 at lib]ty :]e y wyll1 & !n f9d+e ! c5tre to ^? ?re pricks1 ,:i* worke bicause x s]ue? = sondry vses1 ,i ?9k meet to make x a seu]all conclusion by x selfe4 ,,! ,,xxiii4 ,,conclvsion4 ,to f9d ! comm5 c5tre belongyng to anye ?ree prickes appo9t$1 if !y be n 9 an exacte "r l9e4 ,x is to be not$1 t ?\gh eu]y small ar*e ( a greate circle d seeme to be a "r lyne1 yet 9 v d$e x is n s1 = eu]y "p ( ! circumf]5ce ( al circles is compass$1 ?\gh 9 litle ar*es ( great circles ! eye _c disc]ne ! crok$nes1 yet reason doe? alwais decl>e x1 !r=e iij4 prickes 9 an exact "r l9e c n bee br\ght into ! circumf]5ce ( a circle4 ,b & if !y be n 9 a "r l9e h[ s eu] !y /&e1 ?us %all y f9d _! common c5tre4 ,op5 y\r compas s wide1 t x be some:at m !n ! halfe di/ance ( two ( ^? prickes4 ,!n sette ! "o foote ( ! compas 9 ! "o pricke1 & ) ! o!r foot draw an ar*e lyne t[>d ! o!r pricke1 ,!n aga9e putte ! foot ( y\r compas 9 ! second pricke1 & ) ! o!r foot make an ar*e l9e1 t may crosse ! fir/e ar* l9e 9 ij4 places4 ,n[ z y haue done ) ^? two pricks1 s d ) ! middle pricke1 & ! ?irde t remayne?4 ,!n draw ij4 l9es by ! poyntes :]e ^? ar*e l9es d crosse1 & :]e ^? two l9es d meete1 !re is ! c5tre t y seeke =4 ,example ,illu/ration ,! iij4 prickes ,i haue set to be ,a4,b1 & ,c1 :i*e ,i wold br+ into ! edg ( "o common circle1 by f9d+ a c5tre comm5 to !m all1 fyr/ !re=e ,i op5 my compas1 s t !i occupye m !n ye halfe di/ance betw5e ij4 pricks 7z are ,a4,b47 & s sett+e "o foote 9 ,a4 & ext5d+e ! o!r t[>d ,b1 ,i make ! ar*e l9e ,d4,e4 ,likewise sett+ "o foot 9 ,b1 & turn+e ! o!r t[>d ,a1 ,i draw an o!r ar*e l9e t crosse? ! fir/ 9 ,d4 & ,e4 ,!n f ,d4 to ,e1 ,i draw a "r lyne ,d4,h4 ,aft] ?is ,i op5 my compasse to a new di/ance1 & make ij4 ar*e l9es betw5e ,b4 & ,c1 :i*e crosse "o ! o!r 9 ,f4 & ,g1 by :i*e two po9tes ,i draw an o!r l9e1 t is ,f4,h4 ,& bycause t ! lyne ,d4,h4 & ! lyne ,f4,h4 doo meete 9 ,h1 ,i saye t ,h4 is ! c5tre t s]ue? to ^? iij4 prickes4 ,n[ !r=e if y set "o foot ( y\r compas 9 ,h1 & ext5d ! o!r to any ( ! iij4 pricks1 y may draw a circle w* %al enclose ^? iij4 pricks 9 ! edg ( 8 circumf]5ce ?us haue y atta9$ ye vse ( ?is conclusion4 ,,! ,,xxiiii4 ,,conclvsion4 ,to drawe a t\*e l9e onto a circle1 f any po9cte assign$4 ,"h mu/ y vnd]/& t ! pricke mu/ be )\t ! circle1 els ! conclusion is n possible4 ,b ! pricke or po9ct beyng )\t ! circle1 ?us %all y proc$e3 ,op5 y\r compas1 s t ! "o foote ( x maie be set 9 ! c5tre ( ! circle1 & ! o!r foote on ! pricke appo9ct$1 & s draw an o!r circle ( t l>g5esse ab\t ! same c5tre3 & x %all g\]ne y c]ta9ly 9 makyng ! said t\*e l9e4 ,= if y draw a l9e f ! pricke appo9t$ vnto ! c5tre ( ! circle1 & m>ke ! place :]e x doe? crosse ! less] circle1 & f t po9cte erect a plumbe l9e t %all t\*e ! edge ( ! vtt] circle1 & m>ke also ! place :]e t plumbe l9e crosse? t vtt] circle1 & f t place drawe an o!r l9e to ! c5tre1 takyng he$e :]e x crosse? ! less] circle1 if y drawe a plumbe l9e f t pricke vnto ! edge ( ! greatt] circle1 t l9e ,i say is a t\*e l9e1 draw5 f ! po9t assign$1 accord+ to ! mean+ ( ?is conclusion4 ,illu/ration ,example4 ,let ! circle be call$ ,b4,c4,d1 & 8 c5tre ,e1 & ye prick assign$ ,a1 op5 y\r compas n[ ( su* wid5es1 yt ! "o foote may be set 9 ,e1 w* is ye c5tre ( ye circle1 ye o!r 9 ,a1 w* is ye po9te assign$1 s make an o!r gret] circle 7z "h is ,a4,f4,g7 !n draw a l9e f ,a4 vnto ,e1 & :] t l9e do? cross ye inn] circle 7w* he]e is 9 ! prick ,b47 !re erect a plumb l9e vnto ! l9e4 ,a4,e4 & let t plumb l9e t\* ! vtt] circle1 z x do? "h 9 ! po9t ,f1 s %all ,b4,f4 bee t plumbe lyne4 ,!n f ,f4 vnto ,e4 drawe an o!r l9e :i*e %al be ,f4,e1 & x w cutte ! inn] circle1 z x do? "h 9 ! po9t ,c1 f :i* po9te ,c4 if y erect a plumb l9e vnto ,a1 !n is t l9e ,a4,c1 ! t\*e l9e1 :i*e y %\lde f9de4 ,n )/&+e t ?is is a c]ta9e waye to fynde any t\*e l9e1 & a demon/rable =me1 yet m easyly by _m folde may y fynde & make any su*e l9e ) a true rul]1 lay+e ! edge ( ! rul] to ! edge ( ! circle & to ! pricke1 & s draw+ a "r l9e1 z ?is example %ewe?1 :]e ! circle is ,e1 ! pricke assign$ is ,a4 & ! rul] ,c4,d4 by :i* ! t\* l9e is draw51 & t is ,a4,b1 & z ?is way is light to doo1 s is x c]ta9e in\ghe = any k9de ( work+e4 ,illu/ration ,,! ,,xxv4 ,,conclvsion4 ,:5 y haue any peece ( ! circumf]5ce ( a circle assign$1 h[e y may make oute ! :ole circle agreynge !revnto4 ,fir/ seeke out ( ! c5tre ( t ar*e1 accord+ to ! doctr9e ( ! seu5t5? conclusion1 & !n sett+ "o foote ( y\r compas 9 ! c5tre1 & ext5d+ ! o!r foot vnto ! edge ( ! ar*e or peece ( ! circumf]5ce1 x is easy to drawe ! :ole circle4 ,example4 ,a peece ( an olde pill> 0 f\nd1 l 9 =me to ?ys figure ,a4,d4,b4 ,n[ to kn[e h[e mu*e ! compasse ( ! hole pill] 01 se+ by ?is p>te x app]e? t x 0 r\nd1 ?us %al y d4 ,make 9 a table ! l draught ( yt circumf]5ce by ! self patron1 vs+ x z x w] a crok$ rul]4 ,!n make 4iij4 prickes 9 t ar*e l9e1 z ,i haue made1 ,c4 ,d4 & ,e4 ,& !n f9de out ! common c5tre to !m all1 z ! 4xvij4 conclusion tea*e?4 ,& t c5tre is "h ,f1 n[e settyng "o foote ( y\r compas 9 ,f1 & ! o!r 9 ,c4 ,d1 o!r 9 ,e1 & s makyng a compasse1 y haue y\re :ole int5t4 ,illu/ration ,,! ,,xxvi4 ,,conclvsion4 ,to f9de ! c5tre to any ar*e ( a circle4 ,if s be x t y desire to f9d ! c5tre by any o!r way !n by ^? 4iij4 prickes1 consid]yng t sometimes y c n haue s mu* space 9 ! ?yng :]e ! ar*e is draw51 z %\ld s]ue to make ^? 4iiij4 b[e l9es1 !n %all y d ?us3 ,p>te t ar*e l9e into two p>tes1 equall o!r vnequall1 x make? no =ce1 & vnto e* portion draw a corde1 o!r a /r+l9e4 ,& !n accordyng z y dyd 9 "o ar*e 9 ! 4xvi4 conclusion1 s doe 9 bo! ^? ar*es "h1 t is to saie1 deuide ! ar*e 9 ! middle1 & also ! corde1 & drawe !n a l9e by ^? two deuisions1 s !n are y sure t t l9e goe? by ! c5tre4 ,aft]w>d d lykewaies ) ! o!r ar*e & 8 corde1 & :]e ^? 4ij4 l9es d crosse1 !re is ! c5tre1 t y seke =4 ,example4 ,illu/ration ,! ar*e ( ! circle is ,a4,b4,c1 vnto :i*e ,i mu/ seke a c5tre1 !r=e fir/e ,i d deuide x into 4ij4 p>tes1 ! "o ( !m is ,a4,b1 & ! o!r is ,b4,c4 ,!n doe ,i cut eu]y ar*e 9 ! middle1 s is ,e4 ! middle ( ,a4,b1 & ,g4 is ! middle ( ,b4,c4 ,likewaies1 ,i take ! middle ( _! cordes1 :i*e ,i m>k ) ,f4 & ,h1 settyng ,f4 by ,e1 & ,h4 by ,g4 ,!n drawe ,i a l9e f ,e4 to ,f1 & f ,g4 to ,h1 & !y d crosse 9 ,d1 :]e=e saie ,i1 t ,d4 is ! c5tre1 t ,i seke =4 ,,! ,,xxvii4 ,,conclvsion4 ,to drawe a circle )9 a triangle appo9ct$4 ,= ?is conclusion & all o!r lyke1 y mu/e vnd]/&e1 t :5 "o figure is nam$ to be )9 an o!r1 t x is n o!r waies to be vnd]/&e1 b t ey!r eu]y syde ( ! inn] figure dooe? t\*e eu]ie corn] ( ! o!r1 o!r els eu]y corn] ( ! "o dooe? t\*e eu]ie side ( ! o!r4 ,s ,i call t triangle draw5 9 a circle1 ^: corn]s d t\*e ! circumf]5ce ( ! circle4 ,& t circle is conta9$ 9 a triangle1 ^: circumf]5ce doe? t\*e iu/ely eu]y side ( ! triangle1 & yet dooe? n crosse ou] any side ( x4 ,& s t quadrate is call$ prop]ly to be draw5 9 a circle1 :5 all 8 f[] angles doe? t\*e ! edge ( ! circle1 ,& t circle is draw5 9 a quadrate1 ^: circumf]5ce doe? t\*e eu]y side ( ! quadrate1 & lykewaies ( o!r figures4 ,examples are ^!4 ,a4,b4,c4,d4,e4,f4 ,illu/ration3 ,a4 is a circle 9 a triangle4 ,b4 a triangle 9 a circle4 ,c4 a quadrate 9 a circle4 ,d4 a circle 9 a quadrate4 ,9 ^! 4ij4 la/ figures ,e4 & ,f1 ! circle is n nam$ to be draw5 9 a triangle1 because x do? n t\*e ! sides ( ! triangle1 nei!r is ! triangle c\nt$ to be draw5 9 ! circle1 because "o ( 8 corn]s do? n t\*e ! circumf]5ce ( ! circle1 yet 7z y see7 ! circle is )9 ! triangle1 & ! triangle )9 ! circle1 b no!r ( !m is prop]ly nam$ to be 9 ! o!r4 ,n[ to come to ! conclusion4 ,if ! triangle haue all 4iij4 sides lyke1 !n %all y take ! middle ( eu]y side1 & f ! contr>y corn] drawe a "r l9e vnto t poynte1 & :]e ^? l9es d crosse "o an o!r1 !re is ! c5tre4 ,!n set "o foote ( ! compas 9 ! c5tre & /ret*e out ! o!r to ! middle pricke ( any ( ! sides1 & s drawe a compas1 :i*e %all t\*e eu]y side ( ! triangle1 b %all n passe ) out any ( !m4 ,example4 ,! triangle is ,a4,b4,c1 ^: sides ,i d "p into 4ij4 equall p>tes1 e*e by x selfe 9 ^! po9tes ,d4,e4,f1 puttyng ,f4 betw5e ,a4,b1 & ,d4 betw5e ,b4,c1 & ,e4 betw5e ,a4,c4 ,!n draw ,i a l9e f ,c4 to ,f1 & an o!r f ,a4 to ,d1 & ! ?ird f ,b4 to ,e4 ,illu/ration ,& :]e all ^? l9es d mete 7t is to saie ,m4 ,g17 ,i set ! "o foote ( my compasse1 because x is ! common c5tre1 & s drawe a circle accordyng to ! di/aunce ( any ( ! sides ( ! triangle4 ,& !n f9d ,i t circle to agree iu/ely to all ! sides ( ! triangle1 s t ! circle is iu/ely made 9 ! triangle1 z ! conclusion did purporte4 ,& ?is is eu] true1 :5 ! triangle ha? all ?re sides equall1 o!r at ! lea/ 4ij4 sides lyke long4 ,b 9 ! o!r k9des ( triangles y mu/ deuide eu]y angle 9 ! middle1 z ! ?ird conclusion tea*es y4 ,& s drawe l9es f e*e angle to _! middle pricke4 ,& :]e ^? l9es d crosse1 !re is ! common c5tre1 f :i* y %all draw a p]p5dicul> to "o ( ! sides4 ,!n sette "o foote ( ! compas 9 t c5tre1 & /ret*e ! o!r foote accordyng to ! l5g? ( ! p]p5dicul>1 & s drawe y\r circle4 ,illu/ration ,example4 ,! triangle is ,a4,b4,c1 ^: corn]s ,i haue diuid$ 9 ! middle ) ,d4,e4,f1 & haue draw5 ! l9es ( diuision ,a4,d4 ,b4,e1 & ,c4,f1 :i* crosse 9 ,g1 !r=e %all ,g4 be ! common c5tre4 ,!n make ,i "o p]p5dicul> f ,g4 vnto ! side ,b4,c1 & t is ,g4,h4 ,n[ sette ,i "o fote ( ! compas 9 ,g1 & ext5d ! o!r foote vnto ,h4 & s drawe a compas1 :i*e wyll iu/ly answ]e to t triangle accord+ to ! mean+ ( ! conclusion4 ,,! ,,xxviii4 ,,conclvsion4 ,to drawe a circle ab\t any triangle assign$4 ,fyr/e deuide two sides ( ! triangle equally 9 half & f ^? ij4 prickes erect two p]p5dicul>s1 :i* mu/e ne$es meet 9 crosse1 & t po9t ( _! met+ is ! c5tre ( ! circle t mu/ be draw51 !re=e sette "o foote ( ! compasse 9 t po9te1 & ext5d ! o!r foote to "o corn] ( ! triangle1 & s make a circle1 & x %all t\*e all iij4 corn]s ( ! triangle4 ,example4 ,illu/ration ,a4,b4,c4 is ! triangle1 ^: two sides ,a4,c4 & ,b4,c4 are diuid$ into two equall p>tes 9 ,d4 & ,e1 settyng ,d4 betw5e ,b4 & ,c1 & ,e4 betw5e ,a4 & ,c4 ,& f e*e ( ^? two po9tes is !r erect$ a p]p5dicul> 7z y se ,d4,f1 & ,e4,f47 :i* mete1 & crosse 9 ,f1 & /ret*e =? ! o!r foot ( any corn] ( ! triangle1 & s make a circle1 t circle %al t\* eu]y corn] ( ! triangle1 & %al enclose ! :ole triangle1 accord+e1 z ! conclusion wille?4 ,an o!r way to d ! same4 ,& yet an o!r waye may y doo x1 accord+e z y le>n$ 9 ! seu5t5? conclusion1 = if y call ! ?ree corn]s ( ! triangle iij4 prickes1 & !n 7z y le>n$ !re7 yf y seeke out ! c5tre to ^? ?ree prickes1 & s make x a circle to include ^? ?re prickes 9 8 circumf]5ce1 y %all p]ceaue t ! same circle %all iu/elye include ! triangle propos$4 ,example4 ,illu/ration ,a4,b4,c4 is ! triangle1 ^: iij4 corn]s ,i c\nt to be iij4 po9tes4 ,!n 7z ! seu5t5e conclusion do? tea*e7 ,i seeke a common c5tre1 on :i* ,i may make a circle1 t %all enclose ^? iij prickes4 t c5tre z y se is ,d1 = 9 ,d4 do? ! "r l9es1 t passe by ! angles ( ! ar*e l9es1 meete & crosse4 ,& on t c5tre z y se1 haue ,i made a circle1 :i* do? inclose ! iij4 angles ( ! triangle1 & consequ5tlye ! triangle itselfe1 z ! conclusion dydde int5de4 ,,! ,,xxix4 ,,conclvsion4 ,to make a triangle 9 a circle appoynt$ ^: corn]s %al be equall to ! corn]s ( any triangle assign$4 ,:5 ,i w draw a triangle 9 a circle appo9t$1 s t ! corn]s ( t triangle %all be equall to ! corn]s ( any triangle assign$1 !n mu/ ,i fir/ draw a tu*e lyne vnto t circle1 z ! tw5ty conclusion do? tea*1 & 9 ! v poynte ( ! t\*e mu/e ,i make an angle1 equall to "o angle ( ! triangle1 & t inw>de t[>d ! circle3 likewise 9 ! same pricke mu/ ,i make an o!r angle wt ! o!r halfe ( ! t\*e l9e1 equall to an o!r corn] ( ! triangle appo9t$1 & !n betw5 ^? two corn]s w !re resulte a ?ird angle1 equall to ! ?ird corn] ( t triangle4 ,n[e :]e ^? two l9es t entre into ! circle1 doo t\*e ! circumf]5ce 7beside ! t\*e l9e7 !re set ,i two prickes1 & betw5e !m ,i drawe a ?yrde l9e4 ,& s haue ,i made a triangle 9 a circle appo9t$1 ^: corn]s bee equall to ! corn]s ( ! triangle assign$4 ,example4 ,illu/ration ,a4,b4,c1 is ! triangle appo9t$1 & ,f4,g4,h4 is ! circle1 9 :i* ,i mu/e make an o!r triangle1 ) lyke angles to ! angles ( ,a4,b4,c4 ! triangle appo9t$4 ,!re=e fyr/ ,i make ! t\* lyne ,d4,f4,e4 ,& !n make ,i an angle 9 ,f1 equall to ,a1 :i*e is "o ( ! angles ( ! triangle4 ,& ! lyne t make? t angle ) ! t\*e l9e1 is ,f4,h1 :i*e ,i drawe 9 l5g! vntill x t\*e ! edge ( ! circle4 ,!n aga9e 9 ! same po9t ,f1 ,i make an o!r corn] equall to ! angle ,c4 & ! l9e t make? t corn] ) ! t\*e l9e1 is ,f4,g4 :i*e also ,i drawe foor! vntill x t\*e ! edge ( ! circle4 ,& !n haue ,i made ?ree angles vpon t "o t\* l9e1 & 9 yt "o po9t ,f1 & ^? iij4 angles be equall to ! iij4 angles ( ! triangle assign$1 :i*e ?+e do? pla9ely appe>e1 9 s mu*e z !y bee equall to ij4 "r angles1 z y may gesse by ! fixt !oreme4 ,& ! ?re angles ( eu]ye triangle are equill also to ij4 righte angles1 z ! two & tw5ty !oreme do! %[1 s t bicause !y be equall to "o ?irde ?+e1 !y mu/ ne$es be equal togi!r1 z ! common s5t5ce sai?4 ,!n d ,i draw a l9e frome ,g4 to ,h1 & t l9e make? a triangle ,f4,g4,h1 :ole angles be equall to ! angles ( ! triangle appo9t$4 ,& ?is triangle is drawn 9 a circle1 z ! conclusion didde wyll4 ,! pro(e ( ?is conclusion do? appe>e 9 ! seu5ty & iiij4 ,!oreme4 ,,! ,,xxx4 ,,conclvsion4 ,to make a triangle ab\t a circle assign$ :i* %all haue corn]s1 equall to ! corn]s ( any triangle appo9t$4 ,fir/ draw =? 9 l5g? ! "o side ( ! triangle assign$ s t !rby y may haue ij4 vtt] angles1 vnto :i* two vtt] angles y %all make ij4 o!r equall on ! c5tre ( ! circle propos$1 draw+ ?re halfe diamet]s frome ! circumf]5ce1 :i*e %al enclose ^? ij4 angles1 !n draw iij4 t\*e l9es :i* %all make ij4 "r angles1 e*e ( !m ) "o ( ^? semidiamet]s4 ,^? iij4 l9es w make a triangle equally corn]$ to ! triangle assign$1 & t triangle is draw5 ab\t a circle apo9t$1 z ! conclusion did wil4 ,example4 ,a4,b4,c1 is ! triangle assign$1 & ,g4,h4,k1 is ! circle appo9t$1 ab\t :i* ,i mu/e make a triangle hau+ equall angles to ! angles ( t triangle ,a4,b4,c4 ,fyr/ !re=e ,i draw ,a4,c4 7:i* is "o ( ! sides ( ! triangle7 9 l5g? t !re may appe>e two vtt] angles 9 t triangle1 z y se ,b4,a4,d1 & ,b4,c4,e4 ,illu/ration ,!n drawe ,i 9 ! circle appo9t$ a semidiamet]1 :i* is "h ,h4,f1 = ,f4 is ! c5tre ( ! circle ,g4,h4,k4 ,!n make ,i on t c5tre an angle equall to ! vtt] angle ,b4,a4,d1 & t angle is ,h4,f4,k4 ,l waies on ! same c5tre by drawyng an o!r semidiamet]1 ,i make an o!r angle ,h4,f4,g1 equall to ! second vtt] angle ( ! triangle1 :i*e is ,b4,c4,e4 ,& ?us haue ,i made 4iij4 semidiamet]s 9 ! circle appo9t$4 ,!n at ! ende ( e*e semidiamet]1 ,i draw a t\*e l9e1 :i*e %all make righte angles ) ! semidiamet]4 ,& ^? 4iij4 t\* l9es mete1 z y see1 & make ! trianagle ,l4,m4,n1 :i*e is ! triangle t ,i %\ld make1 = x is draw5 ab\t a circle assign$1 & ha? corn]s equall to ! corn]s ( ! triangle appo9t$1 = ! corn] ,m4 is equall to ,c4 ,likewaies ,l4 to ,a1 & ,n4 to ,b1 :i*e ?yng y %all bett] p]ceiue by ! vi4 ,!oreme1 z ,i w decl>e 9 ! booke ( pro(es4 ,,! ,,xxxi4 ,,conclvsion4 ,to make a portion ( a circle on any "r l9e assign$1 :i*e %all conte9e an angle equall to a "r l9$ angle appo9t$4 ,! angle appo9t$1 maie be a %>pe angle1 a "r angle1 o!r a blunte angle1 s t ! worke mu/ be diu]sely h&el$ accord+ to ! diu]sities ( ! angles1 b consid]+e ! h>d5es ( ^? seu]all woorkes1 ,i wyll omitte !m = a m met] "t1 & at ?is tyme wyll %ewe y "o light waye :i* s]ue? = all k9des ( angles1 & t is ?is4 ,:5 ! l9e is propos$1 & ! angle assign$1 y %all ioyne t l9e propos$ s to ! o!r twoo l9es contayn+e ! angle assign$1 t y %all make a triangle ( !ym1 = ! easy doo+e :]e(1 y may enl>ge or %ort5 z y see cause1 anye ( ! two lynes contayn+e ! angle appo9t$4 ,& :5 y haue made a triangle ( ^? iij4 l9es1 !n accord+e to ! doctr9e ( ! seu5 & tw5ty coclusion1 make a circle ab\t t triangle4 ,& s haue y wr\ghte ! reque/ ( ?is conclusion4 ,:y*e yet y maye woorke by ! tw5ty & eight conclusion also1 s t ( y\r l9e appo9t$1 y make "o side ( ! triangle be equal to ye angle assign$ z y\re selfe mai easily gesse4 ,illu/ration ,example4 ,fir/ = example ( a %>pe angle let ,a4 /& ,b4,c %al be ye lyne assign$4 ,!n d ,i make a triangle1 by add+ ,b4,c1 z a ?irde side to ^? o!r ij4 :i* doo include ! angle assign$1 & t triangle is ,d4,e4,f1 s yt ,e4,f4 is ! l9e appo9t$1 & ,d4 is ! angle assign$4 ,!n doo ,i drawe a portion ( a circle ab\t t triangle1 f ! "o ende ( t l9e assign$ vnto ! o!r1 t is to saie1 f ,e4 a long by ,d4 vnto ,f1 :i*e portion is eu]more greatt] !n ! halfe ( ! circle1 by reason t ! angle is a %>pe angle4 ,b if ! angle be "r 7z 9 ! second exaumple y see x7 !n %all ! portion ( ! circle t conta9e? t angle1 eu] m be ! iu/e halfe ( a circle4 ,& :5 ! angle is a blunte angle1 z ! ?irde exaumple dooe? prop\nde1 !n %all ! portion ( ! circle eu]more be lesse !n ! halfe circle4 ,s 9 ! seconde example1 ,g4 is ! "r angle assign$1 & ,h4,k4 is ! lyne appo9t$1 & ,l4,m4,n4 ! portion ( ! circle aunsw]yng !reto4 ,9 ! ?ird exaumple1 ,o4 is ! blunte corn] assign$1 ,p4,q4 is ! l9e1 & ,r4,s4,t4 is ! portion ( ! circle1 t conta9e? t blunt corn]1 & is draw5 on ,r4,t4 ! l9e appo9t$4 ,,! ,,xxxii4 ,,conclvsion4 ,to cutte ( f a circle appo9t$1 a portion conta9yng an angle equall to a "r lyn$ angle assign$4 ,:5 ! angle & ! circle are assign$1 fir/ draw a t\* l9e vnto t circle1 & !n drawe an o!r l9e f ! pricke ( ! t\*yng to "o side ( ! circle1 s t !reby ^? two lynes d make an angle equall to ! angle assign$4 ,!n saie ,i t ! portion ( ! circle ( ! contr>ie side to ! angle draw51 is ! p>te t y seke =4 ,example4 ,illu/ration ,a4 is ! angle appo9t$1 & ,d4,e4,f4 is ! circle assign$1 f :i* ,i mu/ cut away a portion t do? conta9 an angle equall to ?is angle ,a4 ,!r=e fir/ ,i d draw a t\*e l9e to ! circle assign$1 & t t\* l9e is ,b4,c1 ! v pricke ( ! t\*e is ,d1 f :i*e ,d4 ,i drawe a lyne ,d4,e1 s t ! angle made ( ^? two l9es be equall to ! angle appo9t$4 ,!n say ,i1 t ! ar* ( ! circle ,d4,f4,e1 is ! ar*e t ,i seke aft]4 ,= if ,i doo deuide t ar*e 9 ! middle 7z "h is done 9 ,f47 & s draw !nce two l9es1 "o to ,d1 & ! o!r to ,e1 !n w ! angle ,f1 be equall to ! angle assign$4 ,,! ,,xxxiii4 ,,conclvsion4 ,to make a squ>e quadrate 9 a circle assign$4 ,draw 4ij4 diamet]s 9 ! circle1 s t !y runne a crosse1 & t !y make 4iiij4 "r angles4 ,!n drawe 4iiij4 l9es1 t may ioyne ! 4iiij4 ends ( ^? diamet]s1 "o to an o!r1 & !n haue y made a squ>e quadrate 9 ! circle appo9t$4 ,example4 ,illu/ration ,a4,b4,c4,d4 is ! circle assign$1 & ,a4,c4 & ,b4,d4 are ! two diamet]s :i* crosse 9 ! c5tre ,e1 & make 4iiij4 "r corn]s4 ,!n d ,i make f[re o!r l9es1 t is ,a4,b1 ,b4,c1 ,c4,d1 & ,d4,a1 :i* d ioyne toge!r ! f[re endes ( ! ij4 diamet]s4 ,& s is ! squ>e quadrate made 9 ! circle assign$1 z ! conclusion wille?4 ,,! ,,xxxiiii4 ,,conclvsion4 ,to make a squ>e quadrate ab\te annye circle assign$4 ,drawe two diamet]s 9 crosse waies1 s t !y make f\re righte angles 9 ! c5tre4 ,!n ) y\r compasse take ! l5g? ( ! halfe diamet]1 & set "o foote ( ! compas 9 e*e end ( ! compas1 s %all y haue viij4 ar*el9es4 ,!n yf y m>ke ! prickes :]9 ^? ar* l9es d crosse1 & draw betw5e ^? iiij4 prickes iiij "r l9es1 !n haue y made ! squ>e quadrate accord+e to ! reque/ ( ! conclusion4 ,example4 ,illu/ration ,a4,b4,c4 is ! circle assign$ 9 :i* fir/ ,i draw two diamet]s1 9 crosse waies1 mak+ iiij4 righte angles1 & ^? ij4 diamet]s are ,a4,c4 & ,b4,d4 ,!n sette ,i my compasse 7:i*e is op5$ accord+ to ! semidiamet] ( ! said circle7 fix+ "o foote 9 ! end ( eu]y semidiamet]1 & drawe ) ! o!r foote twoo ar*e l9es1 "o on eu]y side4 ,z fir/e1 :5 ,i sette ! "o foote 9 ,a1 !n ) ! o!r foote ,i doo make twoo ar*e l9es1 "o 9 ,e1 & an o!r 9 ,f4 ,!n sette ,i ! "o foote ( ! compasse 9 ,b1 & drawe twoo ar*e l9es ,f4 & ,g4 ,l wise sett+ ! compasse foote 9 ,c1 ,i drawe twoo o!r ar*e l9es1 ,g4 & ,h1 & on ,d4 ,i make twoo o!r1 ,h4 & ,e4 ,!n frome ! cross+es ( ^? eighte ar*e l9es ,i drawe iiij4 /raighte lynes1 t is to saye1 ,e4,f1 & ,f4,g1 also ,g4,h1 & ,h4,e1 :i*e iiij4 /raighte lynes d make ! squ>e quadrate t ,i %\ld draw ab\t ! circle assign$4 ,,! ,,xxxv4 ,,conclvsion4 ,to draw a circle 9 any squ>e quadrate appo9t$4 ,fyr/e deuide eu]y side ( ! quadrate into twoo equall p>tes1 & s drawe two lynes betw5e e*e two contr>y po9ctes1 & :]e ^? twoo l9es doo crosse1 !re is ! c5tre ( ! circle4 ,!n sette ! foote ( ! compasse 9 t po9t1 & /ret* =? ! o!r foot1 accord+ to ! l5g? ( halfe "o ( ^? l9es1 & s make a compas 9 ! squ>e quadrate assign$4 ,example4 ,illu/ration ,a4,b4,c4,d4 is ! quadrate appo9t$1 9 :i*e ,i mu/e make a circle4 ,!re=e fir/ ,i d deuide eu]y side 9 ij4 equal p>tes1 & draw ij4 l9es acrosse1 betw5e e*e ij4 contr>y prickes1 z y se ,e4,g1 & ,f4,h1 :i*e mete 9 ,k1 & !r=e %al ,k1 be ! c5tre ( ! circle4 ,!n d ,i set "o foote ( ! compas 9 ,k4 & op5 ! o!r z wide z ,k4,e1 & s draw a circle1 :i* is made accord+e to ! conclusion4 ,,! ,,xxxvi4 ,,conclvsion4 ,to draw a circle ab\t a squ>e quadrate4 ,draw ij4 l9es betw5e ! iiij4 corn]s ( ! quadrate1 & :]e !y mete 9 crosse1 !r is ! c5tre ( ! circle t y seeke =4 ,!n set "o foot ( ! compas 9 t c5tre1 & ext5d ! o!r foote vnto "o corn] ( ! quadrate1 & s may y draw a circle :i* %all iu/ely inclose ! quadrate propos$4 ,example4 ,illu/ration ,a4,b4,c4,d4 is ! squ>e quadrate propos$1 ab\t :i* ,i mu/ make a circle4 ,!r=e d ,i draw ij4 l9es crosse ! squ>e quadrate f angle to angle1 z y se ,a4,c4 ,b4,d4 ,& :]e !y ij4 d crosse 7t is to say 9 ,e47 !re set ,i ! "o foote ( ! compas z 9 ! c5tre1 & ! o!r foote ,i d ext5d vnto "o angle ( ! quadrate1 z = example to ,a1 & s make a compas1 :i*e do? iu/ly inclose ! quadrate1 accord+ to ! m9de ( ! conclusion4 ,,! ,,xxxvii4 ,,conclvsion4 ,to make a twileke triangle1 :i*e %all haue eu]y ( ! ij4 angles t lye ab\t ! gr\nd l9e1 d\ble to ! o!r corn]4 ,fyr/e make a circle1 & deuide ! circumf]5ce ( x into fyue equall p>tes4 ,& !nne drawe frome "o pricke 7:i* y w7 two l9es to ij4 o!r prickes1 t is to say to ! iij4 & iiij4 pricke1 c\nt+ t = ! fir/1 :]h5ce y drewe bo? ^? l9es1 ,!n drawe ! ?yrde lyne to make a triangle ) ^? o!r twoo1 & y haue doone accord+ to ! conclusion1 & haue made a twelike triangle1 ^: ij4 corn]s ab\t ! gr\nde l9e1 are e*e ( !ym d\ble to ! o!r corn]4 ,example4 ,illu/ration ,a4,b4,c4 is ! circle1 :i*e ,i haue deuid$ into fiue equal portions4 ,& f "o ( ! prickes 7:i* is ,a17 ,i haue draw5 ij4 l9es1 ,a4,b4 & ,b4,c1 :i*e are draw5 to ! ?ird & iiij4 prickes4 ,!n draw ,i ! ?ird l9e ,c4,b1 :i* is ! gr\nde l9e1 & make? ! triangle1 t ,i w\ld haue1 = ! angle ,c4 is d\ble to ! angle ,a1 & s is ! angle ,b4 also4 ,,! ,,xxxviii4 ,,conclvsion4 ,to make a c9kangle ( equall sides1 & equall corn]s 9 any circle appo9t$4 ,deuide ! circle appo9t$ into fiue equall p>tes1 z y didde 9 ! la/e conclusion1 & drawe ij4 l9es f eu]y pricke to ! o!r ij4 t are nexte vnto x4 ,& s %all y make a c9kangle aft] ! meanynge ( ! conclusion4 ,example4 ,y[ se "h ?is circle ,a4,b4,c4,d4,e4 deuid$ into fiue equall portions4 ,& f e*e pricke ij4 l9es draw5 to ! o!r ij4 nexte prickes1 s f ,a4 are draw5 ij4 l9es1 "o to ,b1 & ! o!r to ,e1 & s f ,c4 "o to ,b4 & an o!r to ,d1 & likewise ( ! re/e4 ,s t y haue n only le>n$ h]eby h[ to make a s9kangle 9 anye circle1 b also h[ y %al make a l figure sp$ely1 :anne & :]e y w1 onlye draw+e ! circle = ! int5te1 readylye to make ! o!r figure 7,i meane ! c9kangle7 !reby4 ,illu/ration ,,! ,,xxxix4 ,,conclvsion4 ,h[ to make a c9kangle ( equall sides & equall angles ab\t any circle appo9t$4 ,deuide fir/e ! circle z y did 9 ! la/ conclusion into fiue equall portions1 & draw fiue semidiamet]s 9 ! circle4 ,!n make fiue t\*e l9es1 9 su*e sorte t eu]y t\*e l9e make two "r angles ) "o ( ! semidiamet]s4 ,& ^? fiue t\*e l9es w make a c9kangle ( equall sides & equall angles4 ,illu/ration ,example4 ,a4,b4,c4,d4,e4 is ! circle appo9t$1 :i* is deuid$ into fiue equal p>tes4 ,& vnto eu]y prycke is draw5 a semidiamet]1 z y see4 ,!n doo ,i make a t\*e l9e 9 ! pricke ,b1 :i*e is ,f4,g1 mak+ ij4 "r angles ) ! semidiamet] ,b1 & lyke waies on ,c4 is made ,g4,h1 on ,d4 /&e? ,h4,k1 & on ,e1 is set ,k4,l1 s t ( ^? 4v4 t\*e lynes are made ! 4v4 sides ( a c9keangle1 accordyng to ! conclusion4 ,an o!r waie4 ,ano!r waie also maie y drawe a c9keangle ab\te a circle1 drawyng fir/ a c9keangle 9 ! circle 7:i*e is an easie ?yng to doe1 by ! doctr9e ( ! 4xxxvij4 conclusion7 & !n draw+ 4v4 t\*e l9es :i*e %all be iu/e p>alleles to ! 4v4 sides ( ! c9keangle 9 ! circle1 =seeyng t "o ( !m d n crosse ou]?w>te an o!r & !n haue y done4 ,! exaumple ( ?is 7because x is easie7 ,i leaue to y\r owne ex]cise4 ,,! ,,xl4 ,,conclvsion4 ,to make a circle 9 any appo9t$ c9keangle ( equall sides & equall corn]s4 ,drawe a plumbe l9e f any "o corn] ( ! c9keangle1 vnto ! middle ( ! side t lie? iu/e aga9/ t angle4 ,& d likewaies 9 drawyng an o!r l9e f "s o!r corn]1 to ! middle ( ! side t lie? aga9/ t corn] also4 ,& ^? two l9es wyll meete 9 crosse 9 ! pricke ( _! crossyng1 %all y iudge ! c5tre ( ! circle to be4 ,!r=e set "o foote ( ! compas 9 t pricke1 & ext5d ! o!r to ! end ( ! l9e t t\*e? ! middle ( "o side1 :i*e y li/e1 & s drawe a circle4 ,& x %all be iu/ly made 9 ! c9keangle1 accord+ to ! conclusion4 ,example4 ,! c9keangle assign$ is ,a4,b4,c4,d4,e1 9 :i*e ,i mu/e make a circle1 :]e=e ,i draw a "r l9e f ! "o angle 7z f ,b17 to ! middle ( ! contr>y side 7:i*e is ,e4 ,d17 & t middle pricke is ,f4 ,!n lykewaies f an o!r corn] 7z f ,e7 ,i drawe a "r l9e to ! middle ( ! side t lie? aga9/ x 7:i*e is ,b4,c47 & t pricke is ,g4 ,n[e because t ^! two l9es d crosse 9 ,h1 ,i saie t ,h4 is ! c5tre ( ! circle1 :i*e ,i w\ld make4 ,!r=e ,i set "o foote ( ! compasse 9 ,h1 & ext5d ! o!r foote vnto ,g1 or ,f4 7:i*e are ! endes ( ! lynes t lighte 9 ! middle ( ! side ( t c9keangle7 & s make ,i ! circle 9 ! c9kangle1 "r z ! conclusion meane?4 ,illu/ration ,,! ,,xli4 ,,conclvsion ,to make a circle ab\t any assign$ c9keangle ( equall sides1 & equall corn]s4 ,drawe 4ij4 l9es )9 ! c9keangle1 f 4ij4 corn]s to ! middle on tbe 4ij4 contr>y sides 7z ! la/ conclusion tea*e?7 & ! po9te ( _! crossyng %all be ! c5tre ( ! circle t ,i seke =4 ,!n sette ,i "o foote ( ! compas 9 t c5tre1 & ! o!r foote ,i ext5d to "o ( ! angles ( ! c9kangle1 & s draw ,i a circle ab\t ! c9kangle assign$4 ,example4 ,a4,b4,c4,d4,e1 is ! c9kangle assign$1 ab\t :i* ,i w\ld make a circle4 ,!r=e ,i drawe fir/e ( all two lynes 7z y see7 "o f ,e4 to ,g1 & ! o!r f ,c4 to ,f1 & because !i d meete 9 ,h1 ,i saye t ,h4 is ! c5tre ( ! circle t ,i w\lde haue1 :]=e ,i sette "o foote ( ! compasse 9 ,h4 & ext5de ! o!r to "o corn] 7:i*e happ5e? fyr/e1 = all are l di/aunte f ,h47 & s make ,i a circle ab\te ! c9keangle assign$4 ,illu/ration ,an o!r waye also4 ,ano!r waye maye ,i d x1 ?us presuppos+ any ?ree corn]s ( ! c9kangle to be ?ree prickes appo9t$1 vnto :i*e ,i %\lde f9de ! c5tre1 & !n draw+e a circle t\*+e !m all ?re1 accord+e to ! doctr9e ( ! seu5t5e1 "o & tw5ty1 & two & tw5ty conclusions4 ,& :5 ,i haue f\nde ! c5tre1 !n doo ,i drawe ! circle z ! same conclusions d tea*e1 & ?is =ty conclusion also4 ,,! ,,xlii4 ,,conclvsion4 ,to make a siseangle ( equall sides1 & equall angles1 9 any circle assign$4 ,yf ! c5tre ( ! circle be n kn[51 !n seeke oute ! c5tre accord+ to ! doctr9e ( ! sixt5? conclusion4 ,& ) y\r compas take ! quantitee ( ! semidiamet] iu/ly4 ,& !n sette "o foote 9 "o pricke ( ! circumf]5ce ( ! circle1 & ) ! o!r make a m>ke 9 ! circumf]5ce also t[>de bo? sides4 ,!n sette "o foote ( ! compas /$ily 9 e*e ( ^? new prickes1 & po9t out two o!r prickes4 ,& if y haue done well1 y %al p]ceaue t !re w be b eu5 sixe su* diuisions 9 ! circumf]5ce4 ,:]eby x do! well appe>e1 t ! side ( anye sisangle made 9 a circle1 is equalle to ! semidiamet] ( ! same circle4 ,example4 ,illu/ration ,! circle is ,b4,c4,d4,e4,f4,g1 ^: c5tre ,i f9de to bee ,a4 ,!re=e ,i sette "o foote ( ! compas 9 ,a1 & d ext5d ! o!r foote to ,b1 !reby tak+e ! semidiamet]4 ,!n sette ,i "o foote ( ! compas vnrem\$ 9 ,b1 & m>ke ) ! o!r foote on e*e side ,c4 & ,g4 ,!n f ,c4 ,i m>ke ,d1 & f ,d1 ,e3 f ,e4 m>ke ,i ,f4 ,& !n haue ,i b "o space iu/e vnto ,g4 & s haue ,i made a iu/e siseangle ( equall sides & equall angles1 9 a circle appo9t$4 ,,! ,,xliii4 ,,conclvsion4 ,to make a siseangle ( equall sides1 & equall angles ab\t any circle assign$4 ,,! ,,xliiii4 ,,conclvsion4 ,to make a circle 9 any siseangle appo9t$1 ( equall sides & equal angles4 ,,! ,,xlv4 ,,conclvsion4 ,to make a circle ab\t any sise angle limit$ ( equall sides & equall angles4 ,bicause y maye easily coniecture ! mak+e ( ^! figures by t t is saide be=e ( c9kangles1 only consid]+e t !re is a diff]5ce 9 ! numbre ( sides1 ,i ?\ght be/e to leue ^! vnto y\r owne deuice1 t y %\ld /udy 9 "s ?+es to ex]cise y\r witte )all & t y mighte haue ! bett] occasion to p]ceaue :at diff]5ce !re is betw5e e*e twoo ( ^? conclusions4 ,= ?\ghe x seeme "o ?+ to make a siseangle 9 a circle1 & to make a circle ab\t a siseangle1 yet %all y p]ceaue1 t is n "o ?+e1 no!r are ^? twoo conclusions wr\ght "o way4 ,likewaise %all y ?9ke ( ^? o!r two conclusions4 ,to make a siseangle ab\t a circle1 & to make a circle 9 a siseangle1 ?\ghe ! figures be "o 9 fa%ion1 :5 !y are made1 yet are !y n "o 9 work+1 z y may well p]ceaue by ! xxxvij4 xxxviij4 xxxix4 & xl4 conclusions1 9 :i*e ! same workes are taught1 t\*+ a circle & a c9kangle1 yet ?is mu*e wyll ,i saye1 = y\r helpe 9 work+1 t :5 y %all seeke ! c5tre 9 a siseangle 7:e!r x be to make a circle 9 x o!r ab\t x7 y %all drawe ! two crossel9es1 f "o angle to ! o!r angle t lie? aga9/e x1 & n to ! middle ( any side1 z y did 9 ! c9kangle4 ,,! ,,xlvi4 ,,conclvsion4 ,to make a figure ( fifte5e equall sides & angles 9 any circle appo9t$4 ,?is rule is g5]all1 t h[ _m sides ! figure %all haue1 t %all be draw5 9 any circle1 into s _m p>tes iu/ely mu/e ! circles bee deuid$4 ,& !re=e x is ! m easi] woorke commonly1 to drawe a figure 9 a circle1 !n to make a circle 9 an o!r figure4 ,n[ !re=e to end ?is conclusion1 deuide ! circle fir/e into fiue p>tes1 & !n e*e ( !m into ?ree p>tes aga9e3 ,or els fir/ deuide x into ?ree p>tes1 & !n e* ( !m into fiue o!r p>tes1 z y li/1 & canne mo/ readilye4 ,!n draw l9es betw5e eu]y two prickes t be nighe/ togi!r1 & !r wil appe> rightly draw5 ! figure1 ( fift5e sides1 & angles equall4 ,& s d ) any o!r figure ( :at numbre ( sides s eu] x bee4 ,,f9is4 ,,! ,,second ,,booke ,,( ,,! ,,pr9ciples ( ,geometry1 conta9+ c]ta9e ,!oremes1 :i*e may be cal- l$ ,appr\$ tru!s4 ,& be z x 7 ! mo/e c]ta9e gr\ndes1 :]on ! practike conclusions ( ,geometry ar f\nd$4 ,leaf ,:]vnto are annex$ c]ta9e decl>ations by examples1 = ! "r vnd]/&+ ( ! same1 to ! ende t ! simple read] might n iu/ly compla9 ( h>dnes or obscuritee1 & = ! same cause ar ! demon/ra- tions & iu/ pr(es omitt$1 vntill a m conu5i- ent "t4 #aeea4 ,if tru! maie trie x selfe1 ,by ,reasons prud5t skyll1 ,if reason maie preuayle by "r1 ,& rule ! rage ( w1 ,i d>e ! triall byde1 ,= tru! t ,i pret5de4 ,& ?\gh "s ly/ at me rep9e1 ,iu/e tru! %all me def5de4 ,,! ,,preface ,,vnto ! ,!oremes4 ,i ,d\bt n g5tle read]1 b z my argum5t is /raunge & vnacqua9t$ ) ! vulg>e t\ngue1 s %all ,i ( _m m5 be /raungly talk$ (1 & z /raungly iudg$4 ,"s m5 w saye p]adu5ture1 ,i mighte haue bett] imploy$ my tyme 9 "s pleasaunte hi/orye1 compris+e matt] ( *iualrye4 ,"s o!r wolde m haue preis$ my trauaile1 if ,i hadde sp5te ! l "t 9 "s morall matt]1 o!r 9 decid+ "s contr\]sy ( religion4 ,& yet "s m5 7z ,i iudg7 w n mislike ?is k9d ( mat]1 b !n w !y wi%e t ,i _h vs$ a m c]ta9e ord] 9 plac+e bo! ! ,propositions & ,!oremes1 & also a m exact] pro(e ( e*e ( !im bo!1 by demon/rations ma!maticall4 ,"s also w mislike my %ort5es & simple pla9esse1 z o!r ( o!r affections diu]sely %all espye som:at t !y %all ?9ke blame wor?y1 & %al misse some:at1 t !i wold ) to haue b5e "h vs$1 s t eu]ie manne %all giue 8 v]dicte ( me accord+ to 8 phantasie1 vnto :ome io9ctly1 ,i make ?is my fir/e answ]e3 t z !y ar _m & 9 op9ions v]ie diu]s1 s 7 x sc>se possible to please !m all ) anie "o argum5te1 ( :at k9de s eu] x 74 ,& = my seconde aunsw]e1 ,i saye ?us4 ,t if annye "o argum5te mighte please !m all1 !n %\ld !i be ?ankfull vnto me = ?is k9d ( matt]4 ,= no!r is !re anie matt] m /raunge 9 ! engli%e tungue1 !n ?is :]e( neu] booke 0 writt5 be=e n[1 9 t tungue1 & !re=e oughte to delite all !m1 t desire to vnd]/& /range matt]s1 z mo/ m5 commonlie doo4 ,& aga9e ! practise is s pleasaunt 9 vs+e1 & s pr(itable 9 appliynge1 t :o s eu] do! delite 9 anie ( bo!1 ought n ( "r to mislike ?is arte4 ,& if any manne %all l ! arte welle = x selfe1 b %all mislyke ! f\rme t ,i haue vs$ 9 tea*yng ( x1 to hym ,i %all saie1 ,fir/e1 t ,i dooe wi%e ) hym t "s o!r man1 :i*e c\lde bett] haue doone x1 hadde %ew$ 8 good w1 & vs$ 8 dilig5ce 9 su*e sorte1 t ,i myght haue b5e !rby occasion$ iu/ely to haue left ( my lab\re1 or aft] my trauaile to haue suppress$ my bookes4 ,b si! no manne ha? yet attempt$ ! l1 z f> z ,i canne le>ne1 ,i tru/e all su*e z bee n ex]cis$ 9 ! /udie ( ,geometrye1 %all f9de greate ease & fur!raunce by ?is simple1 pla9e1 & easie =me ( writ+e4 ,& %all p]ceaue ! exacte woorkes ( ,!on1 & o!rs t write on ,euclide1 a great deale ! son]1 by ?is blunte del9eacion a=e h&e to !m taughte4 ,= ,i d>e presuppose ( !m1 t ?+ :i* ,i haue sette 9 my selfe1 & haue m>k$ 9 o!rs1 t is to saye1 t x is n easie = a man t %all trauaile 9 a /raunge arte1 to vnd]/& at ! beg9n+e bo! ! ?+ t is taught & also ! iu/e reason :ie x is s4 ,& by exp]i5ce ( tea*+e ,i haue tri$ x to bee true1 = :5ne ,i haue taughte ! proposition1 z x is import$ 9 mean+e1 & annex$ ! demon/ration ) all1 ,i didde p]ceaue t x 0 a greate tr\ble & a pa9efull vexacion ( mynde to ! le>n]1 to compreh5d bo! ^? ?+es at ones4 ,& !r=e did ,i pr\e fir/e to make !m to vnd]/&e ! s5ce ( ! propositions1 & !n aft]w>d did !y conceaue ! demon/rations mu*e son]1 :5 !y hadde ! s5t5ce ( ! propositions fir/ +raft$ 9 _! m9des4 ,?is ?+e caus$ me 9 bo! ^! bookes to omitte ! demon/rations1 & to vse onlye a pla9e =me ( decl>ation1 :i* might be/ s]ue = ! fir/e introduction4 ,:i*e example ha? be5e vs$ by o!r le>n$ m5ne be=e n[e1 = n only ,georgius ,ioa*imus ,rheticus1 b also ,boetius t wittye cl>ke did set =? "s :ole books ( ,euclide1 )\t any demon/ration or any o!r decl>ation at al4 ,b if ,i %al h]eaft] p]ceaue t x maie be a ?ankefull trauaile to sette foor? ! propositions ( geometrie ) demon/rations1 ,i w n refuse to dooe x1 & t ) sundry v>ietees ( demon/rations1 bo! pleasaunt & pr(itable also4 ,& !n w ,i 9 l man] prep>e to sette foor? ! o!r bookes1 :i*e n[ are lefte vnpr9t$1 by occasion n s mu*e ( ! *>ges 9 cuttyng ( ! figures1 z = o!r iu/e hynd]ances1 :i*e ,i tru/e h]eaft] %all bee rem$i$4 ,9 ! meane season if any man muse :y ,i haue sette ! ,conclusions bee=e ! ,teoremes1 seynge _m ( ! ,!oremes seeme to include ! cause ( "s ( ! conclusions1 & !r=e oughte to haue gone be=e !m1 z ! cause goe? be=e ! effecte4 ,"h vnto ,i saie1 t al?\gh ! cause doo g bee=e ! effect 9 ord] ( nature1 yet 9 ord] ( tea*yng ! effect mu/ be fyr/ decl>$1 & ?an ! cause !r( %ew$1 = s t m5 be/ vnd]/& ?+s ,fir/ to l]ne t su* ?+es ar to be wr\ght1 & second>ily :at !i ar1 & :at !i d import1 & ?an ?irdly :at is ! cause !r(4 ,an o!r cause :y yt ! !oremes be put aft] ! conclusions is ?is1 :an ,i wrote ^! fir/ connclusions 7:i* 0 4iiiij4 y]es pass$7 ,i ?\ght n !n to haue add$ any !oremes1 b next vnto ye conclusions to haue taught ! ord] h[ to haue appli$ !m to "w1 = draw+ ( plottes su* l vses4 ,b aft]w>d consid]+ ! great commoditie yt !i s]ue =1 & ! light t !i d geue to all sortes ( practise geometricall1 besyde o!r m notable b5efites1 :i*e %all be decl>$ m specially 9 a place conu5i5t1 ,i ?\ghte be/e to geue y "s ta/e ( !ym1 & ! pleasaunt contemplation ( su*e geometrical propositions1 :i* might s]ue diu]selye 9 o!r bookes = ! demon/rations & pro(es ( all ,geometricall woorkes4 ,& 9 !im1 z well z 9 ! propositions1 ,i haue draw5 9 ! ,l9e>ie examples _m tymes m lynes1 ?an be spok5 ( 9 ! explication ( !m1 :i*e is doone to ?is int5t1 t yf any manne ly/ to le>ne ! demon/rations by h>te1 7z somme le>n$ m5 haue iudg$ be/e to doo7 ^? same m5 %\ld f9d ! ,l9e>ye exaumples to s]ue = ?is purpose1 & to wante no ?yng ne$efull to ! iu/e pro(e1 :]eby ?is booke may bee wel appr\$ to be m complete !n _m m5 wolde suppose x4 ,& ?us = ?is tyme ,i wyll make an ende )\t any l>g] decl>ation ( ! commoditiees ( ?is arte1 or any f>!r answ]yng to t may bee obiect$ agayn/ my h&elyng ( x1 wyllyng !m t myslike x1 n to m$le ) x3 & vnto ^? t w n disda9e ! /udie ( x1 ,i promise all su*e aide z ,i %all be able to %ewe = _! f>!r proc$yng bo? 9 ! same1 & 9 all o!r commoditees t !re( maie ensue4 ,& = _! inc\ragem5t ,i haue "h annex$ ! names & brefe argum5tes ( su*e bookes1 z ,i int5de 7,god willyng7 %ortly to sette =?1 if ,i %all p]ceaue t my paynes maie pr(yte o!r1 z my desyre is4 ,! brefe argum5tes ( su*e bokes z ar appoynt$ %ortly to be set =? by ! au?or h](4 ,,! seconde "p ( ,ari?metike1 tea*yng ! workyng by fractions1 ) extraction ( rootes bo? squ>e & cubike3 ,& decl>yng ! rule ( allegation1 ) sundrye plesaunt exaumples 9 metalles & o!r ?ynges4 ,also ! rule ( false position1 ) dyu]s examples n onely vulg>1 b "s app]taynyng to ! rule ( ,algeb]1 appli$ vnto quantitees p>tly rationall1 & p>tly surde4 ,,! arte ( ,measuryng by ! quadrate geometricall1 & ! disord]s committ$ by vsyng ! same1 n only reuel$ b re=m$ also 7z mu*e z to ! in/rum5t p]tayne?7 by ! deuise ( a new quadrate newely inu5t$ by ! au?or h]e(4 ,,! arte ( measuryng by ! a/ronom]s /affe1 & by ! a/ronom]s ryng1 & ! =m ( makyng !m bo?4 ,,! arte ( makyng ( ,dials1 bo! = ! daie & ! nyght1 ) c]tayn new =mes ( fix$ dialles = ! moon & o!r = ! /]res1 :i*e may bee sette 9 glasse w9d[es to s]ue by daie & by night4 ,& h[e y may by ^? dialles kn[e 9 :at degree ( ! ,zodiake n only ! sonne1 b also ! moone is4 ,& h[ _m h[rs old %e is4 ,& also by ! same dial to "k :e!r any eclipse %all be t mone?1 ( ! sonne or ( ! moone4 ,! makyng & vse ( an in/rum5t1 :]by y maye n onely measure ! di/ance at ones ( all places t y c see togy!r1 h[e mu*e e*e "o is f y1 & eu]y "o f o!r1 b also !rby to drawe ! plotte ( any c\ntreie t y %all come 91 z iu/ely z maie be1 by mannes dilig5ce & lab\r4 ,,! vse bo! ( ! ,globe & ! ,sph]e1 & !r9 also ( ! arte ( ,nauigation1 & :at in/rum5tes s]ue be/e !rvnto1 & ( ! trew latitude & longitude ( regions & t[nes4 ,euclides woorkes 9 foore p>tes1 ) diu]s demon/rations ,ari?meticall & ,geometricall or ,l9e>ie4 ,! fyr/ p>te ( platte =mes4 ,! second ( numbres & quantitees surde or irrationall4 ,! ?ird ( bodies & solide =mes4 ,! f\r! ( p]spectiue1 & o!r ?ynges !reto annex$4 ,,beside ^! ,i haue o!r sundrye woorkes p>tely end$1 & p>tely to bee end$1 ,( ! p]egr9ation ( man1 & ! orig9all ( al nations1 ,! /ate ( tymes1 & mutations ( realmes1 ,! image ( a p]fect common wel?1 ) diu]s o!r woorkes 9 naturall sci5ces1 ,( ! wond]full workes & effectes 9 bea/es1 plantes1 & m9]als1 ( :i*e at ?is tyme1 ,i w omitte ! argum5tes1 beecause !i doo app]ta9e littel to ?is arte1 & h&le o!r matt]s 9 an o!r sorte4 ,to haue1 or leaue1 ,n[e maie y *use1 ,no pa9e to please1 ,w ,i refuse4 ,! ,!oremes ( ,geometry1 be=e ,,:i*e ,,are ,,set ,,=! c]ta9e grauntable reque/es :i* s]ue = demon/rations ,ma!maticall4 ,sid5ote3 ,i4 ,t f any pricke to "o o!r1 !re may be draw5 a "r l9e4 ,z = example ,a--------,b4 ,a4 be+ ! "o pricke1 & ,b4 ! o!r1 y maye drawe betw5e !m f ! "o to ! o!r1 t is to say1 frome ,a4 vnto ,b1 & f ,b4 to ,a4 ,sid5ote3 ,,ii4 ,t any "r l9e ( measurable l5g? may be draw5 =? long]1 & /raight4 ,illu/ration ,example ( ,a4,b1 :i* z x is a l9e ( measurable l5g!1 s may x be draw5 =? f>!r1 z = example vnto ,c1 & t 9 true /reight5es )\t crok+e4 ,sid5ote3 ,,iii4 ,illu/ration ,t vpon any c5tre1 !re may be made a circle ( anye quantitee t a man wyll4 ,let ! c5tre be set to be ,a1 :at %al h9d] a man to drawe a circle ab\te x1 ( :at quantitee t he lu/e?1 z y se ! =me "h3 o!r bygg] or lesse1 z x %all lyke him to doo3 ,t all "r angles be equall e*e to o!r4 ,illu/ration ,set = an example ,a4 & ,b1 ( :i* two ?\gh ,a4 seme ! greatt] angle to "s m5 ( small exp]i5ce1 x happ5e? only bicause t ! l9es ab\te ,a1 are long] !n ! l9es ab\t ,b1 z y may pr\e by draw+ !m long]1 = s t ,b4 seme ! great] angle yf y make 8 l9es long] !n ! l9es t make ! angle ,a4 ,& to pr\e x by demon/ration1 ,i say ?us4 ,if any ij4 "r corn]s be n equal1 !n "o "r corn] is great] !n an o!r1 b t corn] :i* is greatt] !n a "r angle1 is a blunt corn] 7by 8 def9ition7 s mu/ "o corn] be bo? a "r corn] & a blunt corn] also1 :i* is n possible3 ,& aga9e3 ! less] "r corn] mu/ be a %>pe corn]1 by 8 def9ition1 bicause x is lesse !n a "r angle4 :i* ?+ is impossible4 ,!re=e ,i conclude t all "r angles be equall4 ,yf "o "r l9e d crosse two o!r "r l9es1 & make ij4 inn] corn]s ( "o side less] !n ij4 righte corn]s1 x is c]ta9e1 t if ^? two l9es be draw5 =? "r on t side t ! %>pe inn] corn]s be1 !y wil at l5g? mete togi!r1 & crosse on an o!r4 ,illu/ration ,! ij4 l9es be+e z ,a4,b4 & ,c4,d1 & ! ?ird l9e cross+ !m z doo? he]e ,e4,f1 mak+ ij inn] cornes 7z ar ,g4,h47 less] !n two "r corn]s1 si? e* ( !m is lesse !n a "r corn]1 z y\r eyes maye iudge1 !n say ,i1 if ^? ij4 l9es ,a4,b4 & ,c4,d4 be draw5 9 l5g! on t side t ,g4 & ,h4 are1 ! w at l5g? meet & crosse "o an o!r4 ,two "r l9es make no platte =me4 ,illu/ration ,a platte =me1 z y h>de be=e1 ha? bo! l5g? & br$!1 & is inclos$ ) l9es z ) 8 b\ndes1 b ij4 "r l9es _c inclose al ! bondes ( any platte =me4 ,take = an example fir/e ^! two "r l9es ,a4,b4 & ,a4,c4 :i*e meete togi!r 9 ,a1 b yet _c be call$ a platte =me1 bicause !re is no bond f ,b4 to ,c1 b if y w drawe a l9e betw5e !m twoo1 t is frome ,b4 to ,c1 !n w x be a platte =me1 t is to say1 a triangle1 b !n are !re iij4 l9es1 & n only ij4 ,likewise may y say ( ,d4,e4 & ,f4,g1 :i*e doo make a platte =me1 no!r yet c !y make any )\t helpe ( two l9es m1 :]e( ! "o mu/ be draw5 f ,d4 to ,f1 & ! o!r frome ,e4 to ,g1 & !n w x be a longe rqu>e4 ,s !n ( two "r l9es c bee made no platte =me4 ,b ( ij4 crok$ l9es be made a platte =me1 z y se 9 ! eye =m4 ,& also ( "o "r l9e1 "o crok$ l9e1 maye a platte f\rme bee made1 z ! semicircle ,f4 doo! sette =?4 ,c]tayn common s5t5ces manife/ to s5ce1 & ackn[l$g$ ( all m54 ,! fir/e common s5t5ce4 ,:at s eu] ?+s be equal to "o o!r ?+e1 ^? same bee equall betw5e !m selues4 ,illu/ration ,examples !r( y may take bo? 9 greatnes & also 9 numbre4 ,fir/ 7?\gh x p]ta9e n proprely to geometry1 b to helpe ! vnd]/&+e ( ! rules1 :i*e may bee wr\ght by bo! artes7 ?us may y p]ceaue4 ,if ! summe ( monnye 9 my purse1 & ! mony 9 y\r purse be equall e*e ( !m to ! mony t any o!r man ha!1 !n mu/ ne$es y\r mony & m9e be equall togy!r4 ,likewise1 if anye ij4 quantities1 z ,a4 & ,b1 be equal to an o!r1 z vnto ,c1 !n mu/e n$es ,a4 & ,b4 be equall e*e to o!r1 z ,a4 equall to ,b1 & ,b4 equall to ,a1 :i*e ?+e ! bett] to p]ceaue1 t\rne ^! quantities into numbre1 s %all ,a4 & ,b4 make sixte5e1 & ,c4 z _m4 ,z y may p]ceaue by multipliyng ! numbre ( _! sides togi!r4 ,! seconde common s5t5ce4 ,& if y adde equall portions to ?+es t be equall1 :at s am\nte? ( !m %all be equall4 ,example1 ,yf y & ,i haue l summes ( mony1 & !n receaue e*e ( vs l summes m1 !n our summes wil be l /yll4 ,also if ,a4 & ,b4 7z 9 ! =m] example7 bee equall1 !n by add+ an equal portion to !m bo?1 z to e* ( !m1 ! qu>t] ( ,a4 7t is f\re7 !y w be equall /ill4 ,! ?irde common s5t5ce4 ,& if y abate eu5 portions f ?+s t are equal1 ^? p>tes t rema9 %all be equall also4 ,?is y may p]ceaue by ! la/ example4 ,= t t 0 add$ !re1 is subtract$ he]e4 & s ! "o doo! appr\e ! o!r4 ,! f\r? common s5t5ce4 ,if y abate equalle p>tes f vnequal ?+es1 ! rema9]s %all be vnequall4 ,z bicause t a hundre? & eight & =ty be vnequal if ,i take t5ne f !m bo?1 !re w rema9e nynetye & eight & ?irty1 :i* are also vnequall4 & likewise 9 quantities x is to be iudg$4 ,! fifte common s5t5ce4 ,:5 eu5 portions are add$ to vnequalle ?+es1 ^? t am\nte %albe vnequall4 ,s if y adde tw5ty to fifty1 & lyke ways to nynty1 y %all make seu5ty & a hundr$ & t5 :i*e are no lesse vnequall1 ?an 7 fifty & nynty4 ,! syxt common s5t5ce4 ,if two ?+es be d\ble to any o!r1 ^? same two ?+es are equal togi!r4 ,illu/ration ,bicause ,a4 & ,b4 are e*e ( !m d\ble to ,c1 !re=e mu/ ,a4 & ,b4 n$es be equall togi!r4 ,= z v4 times viij4 make? xl4 :i* is d\ble to iiij4 times v1 t is xx s iiij4 times x1 likewise is d\ble to xx4 7= x make? =tie7 & !re=e mu/e neades be equall to =ty4 ,! seu5? common s5t5ce4 ,if any two ?+es be ! halfes ( "o o!r ?+1 !n are !i 4ij4 equall togi!r4 ,s are ,d4 & ,c4 9 ! la/e example equal togy!r1 bicause !y are e*e ( !m ! halfe ( ,a4 o!r ( ,b1 z _! numbre decl>e?4 ,! eyght common s5t5ce4 ,if any "o quantitee be laide on an o!r1 & !i agree1 s t ! "o exc$e? n ! o!r1 !n are !y equall togi!r4 ,illu/ration ,z if ?is figure ,a4,b4,c1 be lay$ on t o!r ,d4,e4,f1 s t ,a4 be lay$ to ,d1 ,b4 to ,e1 & ,c4 to ,f1 y %all see !m agre 9 sides exactlye & ! "o n to exc$e ! o!r1 = ! l9e ,a4,b4 is equall to ,d4,e1 & ! ?ird lyne ,c4,a1 is equall to ,f4,d s t eu]yside 9 ! "o is equall to "s "o side ( ! o!r4 ,:]=e x is playne1 t ! two triangles are equall togi!r4 ,! nyn? common s5t5ce4 ,eu]y :ole ?+ is great] ?an any ( 8 p>tes4 ,?is s5t5ce n$e? none example4 ,= ! ?yng is m playn] !n any decl>ation1 yet consid]+ t o!r common s5t5ce t fol[e? nexte t4 ,! t5! common s5t5ce4 ,eu]y :ole ?+e is equall to all 8 p>tes tak5 togi!r4 ,illu/ration ,illu/ration ,x %all be mete to expresse bo? wt "o example1 = ( ?ys la/ s5t5ce _m m5 at ! fir/ he>+ d make a d\bt4 ,!r=e z 9 ?is example ( ! circle deuid$ into sundry p>tes x doe? app]e t no p>te c be s great z ! :ole circle1 7accordyng to ! meanyng ( ! eight s5t5ce7 s yet x is c]ta91 t all ^? eight p>tes toge!r be equall vnto ! :ole circle4 ,& ?is is ! meanyng ( t common s5t5ce 7:i*e _m vse1 & fewe d rightly vnd]/&7 t is1 t ,all ! p>tes ( any ?+ are no?+ els1 b ! :ole4 ,& contr>y waies3 ,! :ole is no?+ els1 b all 8 p>tes tak5 togi!r4 :i*e saiynges "s haue vnd]/& to meane ?us3 t all ! p>tes are ( ! same k9d t ! :ole ?yng is3 b t t meanyng is false1 x do? pla9ly app]e by ?is figure ,a4,b1 ^: p>tes ,a4 & ,b1 are triangles1 & ! :ole figure is a squ>e1 & s are !y n ( "o k9d4 ,b & if !y applie x to ! matt] or sub/ance ( ?+es 7z "s d7 !n x is mo/ false1 = eu]y comp\nd ?yng is made ( p>tes ( diu]se matt] & sub/ance4 ,take = example a man1 a h\se1 a boke1 & all o!r comp\nd ?+es4 ,"s vnd]/& x ?us1 t ! p>tes all toge!r c make none o!r =me1 b t t ! :ole do? %ewe1 :i*e is also false1 = ,i maie make fiue hundr$ diu]se figures ( ! p>tes ( "s "o figure1 z y %all bett] p]ceiue 9 ! ?ird boke4 ,& 9 ! meane season take = an example ?is squ>e figure foll[+ ,a4,b4,c4,d1 w* is deuid$ b 9 two p>ts1 & yet 7z y se7 ,i haue made fiue figures m beside ! fir/e1 ) onely diu]se ioynyng ( ^? two p>tes4 ,b ( ?is %all ,i speake m l>gely 9 an o!r place4 ,9 ! mean season cont5t y\r self ) ^! pr9ciples1 :i*e are c]ta9 ( ! *iefe gr\ndes :]on all demon/rations ma!matical are f\rm$1 ( :i* ?\gh ! mo/e p>te seeme s pla9e1 t no *ilde do? d\bte ( !m1 ?9ke n !r=e t ! art vnto :i*e !y s]ue1 is simple1 o!r *ildi%e1 b r consid]1 h[e c]tayne ! pr(es ( t arte is1 yt ha? = 8 gr\ndes so*e playne tru!s1 z ,i may say1 su*e vnd[btfull & s5sible pr9ciples1 ,& ?is is ! cause :y all le>n$ m5ne doo? appr\e ! c]t5ty ( geometry1 & consequ5tly ( ! o!r artes ma!matical1 :i* haue ! gr\nds 7z ,ari?meticke1 musike & a/ronomy7 ab\e all o!r artes & sci5ces1 t be vs$ amonge/ m54 ,?us mu*e haue ,i sayd ( ! fir/ pr9ciples1 & n[ w ,i g on ) ! !oremes1 :i*e ,i d only by examples decl>e1 m9d+ to res]ue ! pro(es to a peculi> boke :i* ,i w !n set =?1 :5 ,i p]ceaue ?is to be ?ankfully tak5 ( ! read]s ( x4 ,illu/ration ,! !oremes ( ,geometry brieflye decl>$ by %orte examples4 ,! fir/e ,!oreme4 ,:5 4ij4 triangles be s draw51 t ! "o ( !m ha? ij4 sides equal to ij sides ( ! o!r triangle1 & t ! angles enclos$ ) ^? sides1 bee equal also 9 bo! triangles1 !n is ! ?irde side likewise equall 9 !m4 ,& ! :ole triangles be ( "o greatnes1 & eu]y angle 9 ! "o equall to 8 mat*e angle 9 ! o!r1 ,i meane ^? angles t be inclos$ ) l sides4 ,example4 ,illu/ration ,?is triangle ,a4,b4,c4 ha? ij4 sides 7t is to say7 ,c4,a4 & ,c4,b1 equal to ij4 sides ( ! o!r triangle ,f4,g4,h1 = ,a4,c4 is equall to ,f4,g1 & ,b4,c4 is equall to ,g4,h4 ,& also ! angle ,c4 contayn$ beetwe5e ,f4,g1 & ,g4,h1 = bo? ( !m answ]e to ! eight p>te ( a circle4 ,!r=e do? x remayne t ,a4,b4 :i*e is ! ?irde lyne 9 ! fir/e triangle1 do? agre 9 l5g! ) ,f4,h1 w* is ! ?ird l9e 9 ye second triangle ye hole triangle4 ,a4,b4,c4 mu/ n$es be equal to ye hole triangle ,f4,g4,h4 ,& eu]y corn] equall to 8 mat*1 t is to say1 ,a4 equall to ,f1 ,b4 to ,h1 & ,c4 to ,g1 = ^? bee call$ mat* corn]s1 :i* are inclos$ ) l sides1 o!r els d lye aga9/ l sides4 ,! second ,!oreme4 ,9 twileke triangles ! ij4 corn]s t be ab\t ! gr\nd l9e1 are equal togi!r4 ,& if ! sides t be equal1 be draw5 out 9 l5g? !n wil ! corn]s t are vnd] ! gr\nd l9e1 be equal also togi!r4 ,example ,illu/ration ,a4,b4,c4 is a twileke triangle1 = ! "o side ,a4,c1 is equal to ! o!r side ,b4,c4 ,& !r=e ,i saye t ! inn] corn]s ,a4 & ,b1 :i* are ab\t ! gr\nd l9es1 7t is ,a4,b47 be equall togi!r4 ,& f>!r if ,c4,a4 & ,c4,b4 bee draw5 =! vnto ,d4 & ,e4 z y se t ,i haue draw5 !m1 !n saye ,i t ! two vtt] angles vnd] ,a4 & ,b1 are equal also togi!r3 z ! !orem said4 ,! pr(e :](1 z ( al ! re/1 %al ape>e 9 ,euclide1 :ome ,i int5de to set foor? 9 engli% ) sondry new additions1 if ,i may p]ceaue t x wilbe ?ankfully tak54 ,! ?irde ,!oreme4 ,if 9 annye triangle !re bee twoo angles equall togi!r1 !n %all ! sides1 t lie aga9/ ^? angles1 be equal also4 ,illu/ration ,example4 ,?is triangle ,a4,b4,c4 ha? two corn]s equal e*e to o!r1 t is ,a4 & ,b1 z ,i d by supposition limite1 :]=e x fol[e? t ! side ,a4,c1 is equal to t o!r side ,b4,c1 = ! side ,a4,c1 lie? aga9/e ! angle ,b1 & ! side ,b4,c1 lie? aga9/ ! angle ,a4 ,! f\r? ,!oreme4 ,:5 two l9es are draw5 f ! endes ( anie "o l9e1 & meet 9 anie po9te1 x is n possible to draw two o!r l9es ( l l5g! e* to 8 mat* t %al beg9 at ! same po9tes1 & end 9 anie o!r po9te !n ! twoo fir/ did4 ,example4 ,illu/ration ,! fir/ l9e is ,a4,b1 on :i* ,i haue erect$ two o!r l9es ,a4,c1 & ,b4,c1 t meete 9 ! pricke ,c1 :]e=e ,i say1 x is n possible to draw ij4 o!r l9es f ,a4 & ,b4 :i* %al mete 9 "o po9t 7z y se ,a4,d4 & ,b4,d4 mete 9 ,d47 b t ! mat* l9es %albe vnequal1 ,i mean by mat* l9es1 ! two l9es on "o side1 t is ! ij4 on ! "r h&1 or ! ij4 on ! lefte h&1 = z y se 9 ?is example ,a4,d4 is long] !n ,a4,c1 & ,b4,c4 is long] !n ,b4,d4 ,& x is n possible1 t ,a4,c4 & ,a4,d4 %all bee ( "o l5g!1 if ,b4,d4 & ,b4,c4 bee l longe4 ,= if "o c\ple ( mat*e l9es be equall 7z ! same example ,a4,e4 is equall to ,a4,c4 9 l5g?7 !n mu/ ,b4,e4 ne$es be vnequall to ,b4,c4 z y see1 x is "h %ort]4 ,! fifte ,!oreme4 ,if two triangles haue !re ij4 sides equal "o to an o!r1 & _! gr\nd l9es equal also1 !n %all _! corn]s1 :i*e are conta9$ betw5e l sides1 be equall "o to ! o!r4 ,example4 ,illu/ration ,because ^! two triangles ,a4,b4,c1 & ,d4,e4,f4 haue two sides equall "o to an o!r4 ,= ,a4,c4 is equall to ,d4,f1 & ,b4,c4 is equall to ,e4,f1 & aga9 _! gr\nd l9es ,a4,b4 & ,d4,e4 are lyke 9 l5g?1 !r=e is e*e angle ( ! "o triangle equall to e* angle ( ! o!r1 comp>yng toge!r ^? angles t are conta9$ )9 lyke sides1 s is ,a4 equall to ,d1 ,b4 to ,e1 & ,c4 to ,f1 = !y are contayn$ )9 l sides1 z be=e is said4 ,! sixt ,!oreme4 ,:5 any "r l9e /&e? on an o!r1 ! ij4 angles t !i make1 o!r are bo? "r angles1 or els equall to 4ij4 righte angles4 ,example4 ,illu/ration ,a4,b4 is a "r l9e1 & on x !re do? light ano!r "r l9e1 draw5 f ,c4 p]p5dicul>ly on x1 !re=e saie ,i1 t ! 4ij4 angles t !i d make1 are 4ij4 "r angles z maie be iudg$ by ! def9ition ( a "r angle4 ,b 9 ! second "p ( ! example1 :]e ,a4,b4 beyng /ill ! "r l9e1 on :i* ,d4 /&e? 9 slope wayes1 ! two angles t be made ( !m are n righte angles1 b yet !y are equall to two righte angles1 = s mu*e z ! "o is to greate1 m !n a righte angle1 s mu*e iu/e is ! o!r to little1 s t bo! togi!r are equall to two "r angles1 z y maye p]ceiue4 ,! seu5? ,!oreme4 ,if 4ij4 l9es be draw5 to any "o pricke 9 an o!r lyne1 & ^? 4ij4 l9es d make ) ! fyr/ lyne1 two "r angles1 o!r su*e z be equall to two "r angles1 & t t[>de "o h&e1 ?an ^? two l9es doo make "o /reyght lyne4 ,example4 ,illu/ration ,a4,b4 is a /reyght lyne1 on :i* !re do? lyght two o!r l9es "o frome ,d1 & ! o!r frome ,c1 b consid]ynge t !y meete 9 "o pricke ,e1 & t ! angles on "o h& be equal to two "r corn]s 7z ! la/e !oreme do! decl>e7 !r=e maye ,d4,e4 & ,e4,c4 be c\nt$ = "o ryght lyne4 ,! eight ,!oreme4 ,:5 two l9es d cut "o an o!r crosseways !y d make _! mat*e angles equall4 ,illu/ration ,example4 ,:at mat*e angles are1 ,i haue tolde y 9 ! def9itions ( ! t]mes4 ,& "h ,a1 & ,b4 are mat*e corn]s 9 ?is example1 z are also ,c4 & ,d1 s t ! corn] ,a1 is equall to ,b1 & ! angle ,c1 is equall to ,d4 ,! nyn? ,!oreme4 ,:an s eu] 9 any triangle ! l9e ( "o side is draw5 =! 9 l5g!1 t vtt] angle is great] ?an any ( ! two inn] corn]s1 t ioyne n ) x4 ,example4 ,illu/ration ,! triangle ,a4,d4,c ha! hys gr\nde lyne ,a4,c4 draw5 =! 9 l5g! vnto ,b1 s t ! vtt] corn] t x make? at ,c1 is great] !n any ( ! two inn] corn]s t lye aga9/e x1 & ioyne n wy? x1 :y*e are ,a4 & ,d1 = !y bo? are less] !n a ryght angle1 & be %>pe angles1 b ,c4 is a blonte angle1 & !r=e great] !n a ryght angle4 ,! t5? ,!oreme4 ,9 eu]y triangle any 4ij4 corn]s1 h[ s eu] y take !m1 ar lesse !n ij4 "r corn]s4 ,example4 ,illu/ration ,9 ! fir/e triangle ,e1 :i*e is a ?relyke1 & !r=e ha? all 8 angles %>pe1 take anie twoo corn]s t y w1 & y %all p]ceiue t !y be less] !n ij4 "r corn]s1 = 9 eu]y triangle t ha? all %>pe corn]s 7z y see x to be 9 ?is example7 eu]y corn] is lesse !n a "r corn]4 ,& !r=e also eu]y two corn]s mu/ n$es be lesse !n two "r corn]s4 ,fur!rmore 9 t o!r triangle m>k$ ) ,m1 :i*e ha? 4ij4 %>pe corn]s & "o "r1 any 4ij4 ( !m also are lesse !n two "r angles4 ,= ?\gh y take ! "r corn] = "o1 yet ! o!r :i*e is a %>pe corn]1 is lesse !n a "r corn]4 ,& s x is true 9 all k9des ( triangles1 z y maie p]ceiue m pla9ly by ! 4xxij4 ,!oreme4 ,! 4xi4 ,!oreme4 ,9 eu]y triangle1 ! greatte/ side lie? aga9/ ! greatte/ angle4 ,example4 ,illu/ration ,z 9 ?is triangle ,a4,b4,c1 ! greatte/ angle is ,c4 ,& ,a4,b4 7:i*e is ! side t lie? aga9/ x7 is ! greate/ & longe/ side4 ,& contr>y waies1 z ,a4,c4 is ! %orte/ side1 s ,b4 7:i*e is ! angle liyng aga9/ x7 is ! smalle/ & %>pe/ angle1 = ?is do? fol[ also1 t is ! longe/ side lye? aga9/ ! greate/ angle1 s x t fol[e? ,! twelft ,!oreme4 ,9 eu]y triangle ! greatte/ angle lie? aga9/ ! longe/ side4 ,= ^! ij4 !oremes are "o 9 tru!4 ,! ?irt5? !oreme4 ,9 eu]ie triangle anie ij4 sides togi!r h[ s eu] y take !m1 are long] !n ! ?irde4 ,illu/ration ,= example y %al take ?is triangle ,a4,b4,c4 :i* ha? a v blunt corn]1 & !r=e "o ( 8 sides great] a good deale !n any ( ! o!r1 & yet ! ij4 less] sides togi!r ar great] !n x4 ,& if x bee s 9 a blunte angel$ triangle1 x mu/ n$es be true 9 all o!r1 = !re is no o!r k9de ( triangles t ha! ! "o side s greate ab\e ! o!r sids1 z !i yt haue blunt corn]s4 ,! f\rt5? !oreme4 ,if !re be draw5 f ! endes ( anie side ( a triangle 4ij4 l9es met+e )9 ! triangle1 ^? two l9es %all be lesse !n ! o!r twoo sides ( ! triangle1 b yet ! corn] t !i make1 %all bee great] !n t corn] ( ! triangle1 :i*e /&e? ou] x4 ,example4 ,illu/ration ,a4,b4,c4 is a triangle4 on ^: gr\nd l9e ,a4,b4 !re is draw5 ij4 l9es1 f ! ij4 endes ( x1 ,i say f ,a4 & ,b1 & !y meete )9 ! triangle 9 ! po9te ,d1 :]=e ,i say1 t z ^? two lynes ,a4,d4 & ,b4,d1 are less] !n ,a4,c4 & ,b4,c1 s ! angle ,d1 is greatt] !n ! angle ,c1 :i* is ! angle aga9/ x4 ,! fift5? ,!oreme4 ,if a triangle haue two sides equall to ! two sides ( an o!r triangle1 b yet ! angle t is conta9$ betw5e ^? sides1 great] !n ! l angle 9 ! o!r triangle1 !n is 8 gr\nde l9e great] !n ! gr\nde l9e ( ! o!r triangle4 ,illu/ration ,example4 ,a4,b4,c4 is a triangle1 ^: sides ,a4,c4 & ,b4,c1 are equall to ,e4,d4 & ,d4,f1 ! two sides ( ! triangle ,d4,e4,f1 b bicause ! angle 9 ,d1 is greatt] !n ! angle ,c4 7:i*e are ! ij4 angles contayn$ betw5e ! equal lynes7 !r=e mu/e ! gr\nd l9e ,e4,f4 n$es bee greatt] !nne ! gr\nde l9e ,a4,b1 z y se pla9ely4 ,illu/ration ,! xvi4 ,!oreme4 ,if a triangle haue twoo sides equalle to ! two sides ( an o!r triangle1 b yet ha! a long] gr\nd l9e !n t o!r triangle1 !n is 8 angle t lie? betw5e ! equall sides1 great] !n ! l corn] 9 ! o!r triangle4 ,example4 ,?is ,!oreme is no?+ els1 b ! s5t5ce ( ! la/ ,!oreme turn$ backw>d1 & !r=e n$e? none o!r pr(e no!r decl>ation1 !n ! o!r example4 ,! seu5t5? ,!oreme4 ,if two triangles be su* sort1 t two angles ( ! "o be equal to ij4 angles ( ! o!r1 & t "o side ( ! "o be equal to on side ( ! o!r1 :e!r t side d adioyne to "o ( ! equall corn]s1 or els lye aga9/e "o ( !m1 !n %all ! o!r twoo sides ( ^? triangles bee equalle togi!r1 & ! ?irde corn] also %all be equall 9 ^? two triangles4 ,example4 ,illu/ration ,bicause t ,a4,b4,c1 ! "o triangle ha? two corn]s ,a4 & ,b1 equal to ,d4,e1 t are twoo corn]s ( ! o!r triangle4 ,d4,e4,f4 & t !y haue "o side 9 !ym bo! equall1 t is ,a4,b1 :i* is equall to ,d4,e1 !re=e %all bo? ! o!r ij4 sides be equall "o to an o!r1 z ,a4,c4 & ,b4,c4 equall to ,d4,f4 & ,e4,f1 & also ! ?irde angle 9 !m bo? %al be equall1 t is1 ! angle ,c4 %al be equall to ! angle ,f4 ,! eight5? ,!oreme4 ,:5 on ij4 "r l9es !r is draw5 a ?ird "r l9e crosse waies1 & make? 4ij4 mat*e corn]s ( ! "o l9e equall to ! l twoo mat*e corn]s ( ! o!r l9e1 !n ar ^? two l9es gemm[ l9es1 or p>alleles4 ,example4 ,illu/ration ,! 4ij4 fyr/ lynes are ,a4,b4 & ,c4,d1 ! ?yrd lyne t crosse? !m is ,e4,f4 ,& bycause t ,e4,f4 make? ij4 mat*e angles ) ,a4,b1 equall to 4ij4 o!r lyke mat*e angles on ,c4,d1 7t is to say ,e4,g1 equall to ,k4,f1 & ,m4,n4 equall also to ,h4,l47 !r=e are ^? ij4 lynes ,a4,b4 & ,c4,d4 gem[ lynes1 vnd]/& "h by lyke mat*e corn]s1 ^? t g "o way z do? ,e4,g1 & ,k4,f1 lyke ways ,n4,m1 & ,h4,l1 = z ,e4,g4 & ,h4,l1 o!r ,n4,m4 & ,k4,f4 g n "o waie1 s be n !y lyke mat* corn]s4 ,! nynt5? ,!oreme4 ,:5 on two "r l9es !re is draw5 a ?irde "r l9e crossewaies1 & make? ! ij4 ou] corn]s t[>de "o h&e equall togi!r1 !n ar ^? 4ij4 l9es p>alleles4 ,& 9 l man] if two inn] corn]s t[>d "o h&e1 be equall to 4ii4 "r angles4 ,example4 ,z ! ,!oreme do! speake ( 4ij4 ou] angles1 s mu/e y vnd]/&e also ( 4ij4 ne!r angles1 = ! iudgem5t is lyke 9 bo!4 ,take = example ! figure ( ! la/ !oreme1 :]e ,a4,b1 & ,c4,d1 be call$ p>alleles also1 bicause ,e4 & ,k1 7:i*e are 4ij4 ou] corn]s7 are equall1 & lykewaies ,l4 & ,m4 ,& s are 9 lyke man] ! ne!r corn]s ,n4 & ,h1 & ,g4 & ,f4 ,n[e to ! seconde p>te ( ! !oreme1 ^? 4ij4 lynes ,a4,b4 & ,c4,d1 %all be call$ p>alleles1 because ! ij4 inn] corn]s4 ,z = example ^? two t bee t[>d ! "r h&e 7t is ,g4 & ,l47 are equall 7by ! fyr/ p>te ( ?is nynt5? !oreme7 !r=e mu/e ,g4 & ,l4 be equall to two ryght angles4 ,! xx4 ,!oreme4 ,:5 a "r l9e is draw5 crosse ou] 4ij4 "r gem[ l9es1 x make? 4ij4 mat*e corn]s ( ! "o l9e1 equall to two mat*e corn]s ( ! o!r l9e1 & also bo! ou] corn]s ( "o h&e equall togi!r1 & bo! ne!r corn]s l waies1 & m ou] two inn] corn]s1 & two vtt] corn]s also t[>de "o h&e1 equall to two "r angles4 ,example4 ,bycause ,a4,b4 & ,c4,d1 79 ! la/e figure7 are p>alleles1 !re=e ! two mat*e corn]s ( ! "o lyne1 z ,e4,g4 be equall vnto ! 4ij4 mat*e corn]s ( ! o!r l9e1 t is ,k4,f1 & lykewaies ,m4,n1 equall to ,h4,l4 ,& also ,e4 & ,k4 bo! ou] corn]s ( ! lefte h&e equall togy!r1 & s are ,m4 & ,l1 ! two ou] corn]s on ! ryghte h&e1 9 lyke man] ,n4 & ,h1 ! two ne!r corn]s on ! lefte h&e1 equall e*e to o!r1 & ,g4 & ,f4 ! two ne!r angles on ! "r h&e equall togi!r4 ,sec4 ,f>!rmore yet ,g4 & ,l4 ! 4ij4 inn] angles on ! "r h&e bee equall to two "r angles1 & s are ,m4 & ,f4 ! 4ij4 vtt] angles on ! same h&e1 9 lyke mann] %all y say ( ,n4 & ,k4 ! two inn] corn]s on ! left h&4 & ( ,e4 & ,h4 ! two vtt] corn]s on ! same h&e4 ,& ?us y see ! agreable s5t5ce ( ^! 4iii4 !oremes to t5de to ?is purpose1 to decl>e by ! angles h[ to iudge p>alleles1 & contr>y waies h[e y may by p>alleles iudge ! proportion ( ! angles4 ,! xxi4 ,!oreme4 ,:at s eu] l9es be p>alleles to any o!r l9e1 ^? same be p>alleles togi!r4 ,example4 ,illu/ration ,a4,b4 is a gem[ l9e1 or a p>allele vnto ,c4,d4 ,& ,e4,f1 lykewaies is a p>allele vnto ,c4,d4 ,:]=e x fol[e?1 t ,a4,b4 mu/ n$es bee a p>allele vnto ,e4,f4 ,! 4xxij4 !oreme4 ,9 eu]y triangle1 :5 any side is draw5 =? 9 l5g?1 ! vtt] angle is equall to ! ij4 inn] angles t lie aga9/e x4 ,& all iij4 inn] angles ( any triangle are equall to ij4 "r angles4 ,illu/ration ,example4 ,! triangle beeyng ,a4,d4,e4 & ! syde ,a4,e4 draw5 foor! vnto ,b1 !re is made an vtt] corn]1 :i*e is ,c1 & ?is vtt] corn] ,c1 is equall to bo!r ! inn] corn]s t lye agayn/ x1 :y*e are ,a4 & ,d4 ,& all ?re inn] corn]s1 t is to say1 ,a4,d4 & ,e1 are equall to two ryght corn]s1 :]e( x fol[e?1 t all ! ?ree corn]s ( any "o triangle are equall to all ! ?ree corn]s ( eu]ye o!r triangle4 ,= :at s eu] ?ynges are equalle to anny "o ?yrde ?ynge1 ^? same are equalle togit!r1 by ! fyr/e common s5t5ce1 s t bycause all ! 4iij4 angles ( eu]y triangle are equall to two ryghte angles1 & all ryghte angles bee equall togy!r 7by ! f\r? reque/7 !r=e mu/ x n$es fol[1 t all ! ?re corn]s ( eu]y triangle 7accomptyng !m togy!r7 are equall to iij4 corn]s ( any triangle1 tak5 all togy!r4 ,! 4xxiii4 !oreme4 ,:5 any ij4 "r l9es do? t\*e & c\ple 4ij4 o!r righte l9es1 :i*e are equall 9 l5g? & p>alleles1 & if ^? 4ij4 l9es bee draw5 t[>de "o h&e1 !n are !i also equall toge!r1 & p>alleles4 ,example4 ,illu/ration ,a4,b4 & ,c4,d4 are ij4 ryght lynes & p>alleles & equall 9 l5g?1 & !y ar t\*$ & ioyn$ togi!r by ij4 o!r lynes ,a4,c4 & ,b4,d1 ?is beyng s1 & ,a4,c4 & ,b4,d4 beyng draw5 t[>de "o syde 7t is to saye1 bo! t[>de ! lefte h&e7 !re=e are ,a4,c4 & ,b4,d4 bo! equall & also p>alleles4 ,! 4xxiiij4 !oreme4 ,9 any likeiamme ! two contr>y sides ar equall togi!r1 & s are e*e 4ij4 contr>y angles1 & ! bias l9e t is draw5 9 x1 do! diuide x into two equall portions4 ,example4 ,illu/ration ,"h ar two likeiammes ioyn$ togi!r1 ! "o is a longe squ>e ,a4,b4,e1 & ! o!r is a los5gelike ,d4,c4,e4,f4 :i* ij4 likeiammes ar pr\$ equall togi!r1 bycause !y haue "o gr\nd l9e1 t is1 ,f4,e1 ,& are made betw5e "o payre ( gem[ l9es1 ,i meane ,a4,d4 & ,e4,h4 ,by ?is ,!oreme may y "k ! arte ( ! righte measur+e ( likeiammes1 z 9 my booke ( measur+ ,i wil m pla9ly decl>e4 ,! xxvi4 ,!oreme4 ,all likeiammes t haue equal gr\nde l9es & are draw5 betw5e "o paire ( p>alleles1 are equal togi!r4 ,example4 ,fyr/e y mu/e m>ke ! diff]5ce betw5e ?is ,!oreme & ! la/e1 = ! la/e ,!oreme presuppos$ to ! diu]s likeiammes "o gr\nd l9e common to !m1 b ?is !oreme do? presuppose a diu]s gr\nd l9e = eu]y likeiamme1 only mean+ !m to be equal 9 l5g?1 ?\gh !y be diu]s 9 numbre4 ,z = example4 ,9 ! la/ figure !r are two p>allels1 ,a4,d4 & ,e4,h1 & betw5e !m are draw5 ?re likeiammes1 ! fir/e is1 ,a4,b4,e4,f1 ! second is ,e4,c4,d4,f1 & ! ?irde is ,c4,g4,h4,d4 ,! fir/e & ! seconde haue "o gr\nd l9e1 7t is ,e4,f47 & !r=e 9 s mu*e z !y are betw5e "o paire ( p>alleles1 !y are equall accord+e to ! fiue & tw5tye ,!oreme1 b ! ?irde likeiamme t is ,c4,g4,h4,d4 ha! 8 gr\nde l9e ,g4,h1 seu]all frome ! o!r1 b yet equall vnto x4 :]e=e ! ?ird likeiam is equall to ! o!r two fir/e likeiammes4 ,& = a pro(e t ,g4,h4 be+ ! gr\nd or gr\nd l9e ( ! ?ird likeiamme1 is equal to ,e4,f1 :i*e is ! gr\nd l9e to bo? ! o!r likeiams1 t may be ?us decl>$1 ,g4,h4 is equall to ,c4,d1 seynge !y are ! contr>y sides ( "o likeiamme 7by ! f\re & tw5ty !oreme7 & s are ,c4,d4 & ,e4,f4 by ! same !oreme4 ,!r=e seynge bo? ^? gr\nd l9es ,e4,f4 & ,g4,h1 are equall to "o ?irde l9e 7t is ,c4,d47 !y mu/ n$es bee equall togy!r by ! fir/e common s5t5ce4 ,! xxvii4 ,!oreme4 ,all triangles hau+e "o gr\nde lyne1 & /&+ betw5e "o paire ( p>allels1 ar equall togi!r4 ,example4 ,illu/ration ,a4,b4 & ,c4,f4 are twoo gem[e l9es1 betwe5e :i* !re be made two triangles1 ,a4,d4,e4 & ,d4,e4,b1 s t ,d4,e1 is ! common gr\nd l9e to !m bo!4 :]=e x do? fol[1 t ^? two triangles ,a4,d4,e4 & ,d4,e4,b4 are equall e*e to o!r4 ,! xxviij4 ,!oreme4 ,all triangles t haue l long gr\nd l9es1 & bee made betwe5e "o paire ( gem[ l9es1 are equall togi!r4 ,example4 ,example ( ?is ,!oreme y may see 9 ! la/ figure1 :]e z sixe triangles made betw5e ^? two gem[e l9es ,a4,b4 & ,c4,f1 ! fir/ triangle is ,a4,c4,d1 ! seconde is ,a4,d4,e1 ! ?irde is ,a4,d4,b1 ! f\r? is ,a4,b4,e1 ! fifte is ,d4,e4,b1 & ! sixte is ,b4,e4,f1 ( :i* sixe triangles1 ,a4,d4,e4 & ,d4,e4,b4 are equall1 bicause !y haue "o common gr\nde l9e4 ,& s likewise ,a4,b4,e4 & ,a4,b4,d1 ^: comm5 gr\nde l9e is ,a4,b1 b ,a4,c4,d4 is equal to ,b4,e4,f1 be+ bo? betw5e "o c\ple ( p>allels1 n bicause !i haue "o gr\nd l9e1 b bicause !y haue _! gr\nd l9es equall1 = ,c4,d4 is equall to ,e4,f1 z y may decl>e ?us4 ,c4,d1 is equall to ,a4,b4 7by ! f\re & tw5ty ,!oreme7 = !i are two contr>y sides ( "o lykeiamme4 ,a4,c4,d4,b1 & ,e4,f by ! same !oreme1 is equall to ,a4,b1 = !i ar ! two ye contr>y sides ( ! likeiamme1 ,a4,e4,f4,b1 :]=e ,c4,d4 mu/ ne$es be equall to ,e4,f4 l wise ! triangle ,a4,c4,d1 is equal to ,a4,b4,e1 bicause !y ar made betw5e "o paire ( p>allels & haue _! gr\ndl9es l1 ,i meane ,c4,d4 & ,a4,b4 ,aga9e ,a4,d4,e1 is equal to e*e ( !m bo?1 = 8 gr\nd l9e ,d4,e1 is equall to ,a4,b1 inso mu*e z !y are ! contr>y sides ( "o likeiamme1 t is ! long squ>e ,a4,b4,d4,e4 ,& ?us may y pr\e ! equalnes ( all ! re/e4 ,! xxix4 ,!oreme4 ,al equal triangles t are made on "o gr\nde l9e1 & rise "o waye1 mu/ ne$es be betw5e "o paire ( p>allels4 ,example4 ,take = example ,a4,d4,e1 & ,d4,e4,b1 :i* 7z ! xxvij4 conclusion doo? pr\e7 are equall togi!r1 & z y see1 !y haue "o gr\nd l9e ,d4,e4 ,& aga9e !y rise t[>de "o side1 t is to say1 vpw>de t[>d ! l9e ,a4,b1 :]=e !y mu/ ne$es be inclos$ betwe5e "o paire ( p>allels1 :i* are he]e 9 ?is example ,a4,b4 & ,d4,e4 ,! ?irty ,!oreme4 ,equal triangles t haue _! gr\nd l9es equal1 & be draw5 t[>d "o side1 ar made betw5e "o paire ( p>alleles4 ,example4 ,! example t decl>$ ! la/ !oreme1 maye well s]ue to ! decl>acion ( ?is also4 ,= ^? ij4 !oremes d diffre b 9 ?is "o po9te1 t ! la/e !oreme meane? ( triangles1 t haue "o gr\nd l9e common to !m bo?1 & ?is !oreme do! presuppose ! gr\nde l9es to bee diu]s1 b yet ( "o l5g?1 z ,a4,c4,d1 & ,b4,e4,f1 z !y are ij4 equall triangles appr\$1 by ! eighte & tw5tye ,!orem1 s 9 ! same ,!orem x is decl>$1 yt _! gr\nd l9es are equall togi!r1 t is ,c4,d1 & ,e4,f1 n[ ?is beeynge true1 & consid]+ t !y are made t[>de "o side1 x fol[e?1 t !y are made betw5e "o paire ( p>allels :5 ,i saye1 draw5 t[>de "o side1 ,i meane t ! triangles mu/ be draw5 o!r bo? vpw>d frome "o p>allel1 o!r els bo? d[nw>d1 = if ! "o be draw5 vpw>d & ! o!r d[nw>d1 !n are !y draw5 betw5e two paire ( p>allels1 presuppos+e "o to bee draw5 by _! gr\nd l9e1 & !n d !y ryse t[>d contr>y sides4 ,! xxxi4 !oreme4 ,if a likeiamme haue "o gr\nd l9e ) a triangle1 & be draw5 betw5e "o paire ( p>alleles1 !n %all ! likeiamme be d\ble to ! triangle4 ,example4 ,illu/ration ,a4,h4 & ,b4,g4 are 4ij4 gem[ l9es1 betw5e :i* !re is made a triangle ,b4,c4,g1 & a lykeiamme1 ,a4,b4,g4,c1 :i*e haue a gr\nde lyne1 t is to saye1 ,b4,g4 ,!r=e do? x fol[ t ! lyke iamme ,a4,b4,g4,c4 is d\ble to ! triangle ,b4,c4,g4 ,= eu]y halfe ( t lykeiamme is equall to ! triangle1 ,i meane ,a4,b4,f4,e4 o!r ,f4,e4,c4,g4 z y may coniecture by ! 4xi4 conclusion geometrical4 ,& z ?is ,!oreme do! speake ( a triangle & likeiamme t haue "o gr\ndelyne1 s is x true also1 yf !yr gr\ndelynes bee equall1 ?\gh !y bee dyu]s1 s t !i be made betw5e "o payre ( p>alleles4 ,& h]e( may y p]ceaue ! reason1 :y 9 measuryng ! platte ( a triangle1 y mu/ multiply ! p]p5dicul> lyne by halfe ! gr\nde lyne1 or els ! hole gr\nde lyne by halfe ! p]p5dicul>1 = by any ( ^! bo! waies is !re made a lykeiamme equall to halfe su*e a "o z %ulde be made on ! same hole gr\nde lyne ) ! triangle1 & betwe5e "o payre ( p>alleles4 ,!r=e z t lykeiamme is d\ble to ! triangle1 s ! halfe ( x1 mu/ ne$es be equall to ! triangle4 ,comp>e ! 4xi4 conclusion ) ?is !oreme4 ,! 4xxxij4 ,!oreme4 ,9 all likeiammes :]e !re are m ?an "o made ab\te "o bias l9e1 ! fill squ>es ( eu]y ( !m mu/ n$es be equall4 ,illu/ration ,example4 ,fyr/ be=e ,i decl>e ! examples1 x %al be mete to %ew ! true vnd]/&yng ( ?is !orem4 ,sid5ote3 ,bias lyne4 ,!r=e by ! ,bias l9e1 ,i meane t lyne1 :i*e 9 any squ>e figure doo? runne f corn] to corn]4 ,& eu]y squ>e :i* is diuid$ by t bias l9e into equall halues f corn] to corn] 7t is to say1 into 4ij4 equall triangles7 ^? be c\nt$ to /&e ab\te "o bias l9e1 & ! o!r squ>es1 :i*e t\*e t bias l9e1 ) "o ( _! corn]s onely1 ^? doo ,i call ,fyll squ>es1 ,sid5ote3 ,fyll squ>es4 accordyng to ! greke "n1 :i* is anapl]omata1 ,sid5ote3 anaple3ro3mata & call$ 9 lat9 supplem5ta1 bycause t !y make "o g5]all squ>e1 includyng & enclosyng ! o!r diu]s squ>es1 z 9 ?is example ,h4,c4,e4,n4 is "o squ>e likeiamme1 & ,l4,m4,g4,c4 is an o!r1 :i*e bo! are made ab\te "o bias l9e1 t is ,n4,m1 ?an ,k4,l4,h4,c4 & ,c4,e4,f4,g4 are 4ij4 fyll squ>es1 = !y doo fyll vp ! sydes ( ! 4ij4 fyr/e squ>e lykeiammes1 9 su*e sorte1 t all !m f\re is made "o greate g5]all squ>e ,k4,m4,f4,n4 ,n[e to ! s5t5ce ( ! !oreme1 ,i say1 t ! 4ij4 fill squ>es1 ,h4,k4,l4,c4 & ,c4,e4,f4,g4 are bo? equall togi!r1 7z x %all bee decl>$ 9 ! booke ( pro(es7 bicause !y are ! fill squ>es ( two likeiammes made ab\te "o bias l9e1 z ! exaumple %ewe?4 ,conf]re ! twelf! conclusion ) ?is !oreme4 ,! xxxiij4 ,!oreme4 ,9 all "r angul$ triangles1 ! squ>e ( t side :i*e lie? aga9/ ! "r angle1 is equall to ! 4ij4 squ>es ( bo? ! o!r sides4 ,example4 ,illu/ration ,a4,b4,c4 is a triangle1 hau+ a ryght angle 9 ,b4 ,:]=e x fol[e?1 t ! squ>e ( ,a4,c1 7:i*e is ! side t lye? agayn/ ! "r angle7 %all be z mu*e z ! two squ>es ( ,a4,b4 & ,b4,c4 :i* are ! o!r 4ij4 sides4 ,sec4 ,by ! squ>e ( any lyne1 y mu/e vnd]/&e a figure made iu/e squ>e1 hauyng all 8 iiij4 sydes equall to t l9e1 :]e( x is ! squ>e1 s is ,a4,c4,f1 ! squ>e ( ,a4,c4 ,lykewais ,a4,b4,d4 is ! squ>e ( ,a4,b4 ,& ,b4,c4,e4 is ! squ>e ( ,b4,c4 ,n[ by ! numbre ( ! diuisions 9 e*e ( ^! squ>es1 may y p]ceaue n onely :at ! squ>e ( any l9e is call$1 b also t ! !oreme is true1 & express$ playnly bo! by l9es & numbre4 ,= z y see1 ! greatt] squ>e 7t is ,a4,c4,f47 ha? fiue diuisions on e*e syde1 all equall togy!r1 & ^? 9 ! :ole squ>e are tw5ty & fiue4 ,n[e 9 ! left squ>e1 :i*e is ,a4,b4,d4 !re are b 4iij4 ( ^? diuisions 9 "o syde1 & t yelde? nyne 9 ! :ole4 ,s lykeways y see 9 ! meane squ>e ,a4,c4,e4 9 eu]y syde 4iiij4 p>tes1 :i*e 9 ! :ole am\nt vnto sixt5e4 ,n[e adde togy!r all ! p>tes ( ! two less] squ>es1 t is to saye1 sixt5e & nyne1 & y p]ceyue t !y make tw5ty & fiue1 :y*e is an equall numbre to ! summe ( ! greatt] squ>e4 ,by ?is !oreme y may vnd]/& a r$y way to "k ! syde ( any ryght angul$ triangle t is vnkn[51 s t y kn[e ! l5g! ( any two sydes ( x4 ,= by t\rnynge ! two sydes c]tayne into !yr squ>es1 & s addynge !m togy!r1 o!r subtractynge ! "o f ! o!r 7accordyng z 9 ! vse ( ^! !oremes ,i haue sette foor!7 & !n fyndynge ! roote ( ! squ>e t remayne?1 :i* roote 7,i meane ! syde ( ! squ>e7 is ! iu/e l5g? ( ! vnkn[5 syde1 :y*e is s\ght =4 ,b ?is app]ta9e? to ! ?yrde booke1 & !re=e ,i wyll speake no m ( x at ?is tyme4 ,! xxxiiij4 ,!oreme4 ,if s be x1 t 9 any triangle1 ! squ>e ( ! "o syde be equall to ! 4ij4 squ>es ( ! o!r 4ij4 sides1 ?an mu/ n$es t corn] be a "r corn]1 :i* is conte9$ betw5e ^? two less] sydes4 ,example4 ,z 9 ! figure ( ! la/e ,!oreme1 bicause ,a4,c1 made 9 squ>e1 is asmu* z ! squ>e ( ,a4,b1 & also z ! squ>e ( ,b4,c4 ioyn$ bo! togy!r1 !re=e ! angle t is inclos$ betw5e ^? 4ij4 less] lynes1 ,a4,b4 & ,b4,c4 7t is to say7 ! angle ,b4 :i*e lie? aga9/ ! l9e ,a4,c1 mu/ n$es be a ryght angle4 ,?is !oreme do! s dep5de ( ! tru! ( ! la/e1 t :an y p]ceaue ! tru! ( ! "o1 y c n iu/ly d\bt ( ! o!rs tru!1 = !y conte9e "o s5t5ce1 contr>y waies pron\nc$4 ,! 4xxxv4 !oreme4 ,if !re be set =? 4ij4 "r l9es1 & "o ( !m p>t$ into sundry p>tes1 h[ _m or few s eu] !y be1 ! squ>e t is made ( ^? ij4 "r l9es propos$1 is equal to all ! squ>es1 t are made ( ! vndiuid$ l9e1 & eu]y p>te ( ! diuid$ l9e4 ,illu/ration ,example4 ,! ij4 l9es propos$ ar ,a4,b4 & ,c4,d1 & ! lyne ,a4,b4 is deuid$ into ?re p>tes by ,e4 & ,f4 ,n[ sai? ?is !oreme1 t ! squ>e t is made ( ^? two :ole l9es ,a4,b4 & ,c4,d1 s t ! l9e ,a4,b4 /&e? = ! l5g? ( ! squ>e1 & ! o!r l9e ,c4,d4 = ! br$? ( ! same4 ,t squ>e 7,i say7 wil be equall to all ! squ>es t be made1 ( ! vndiu$$ lyne 7:i* is ,c4,d47 & eu]y portion ( ! diuid$ l9e4 ,& to decl>e t p>ticul>ly1 ,fyr/ ,i make an o!r l9e ,g4,k1 equall to ! l9e 4,c4,d1 & ! l9e ,g4,h4 to be equal to ! l9e ,a4,b1 & to bee diuid$ into iij4 l p>tes1 s t ,g4,m4 is equall to ,a4,e1 & ,m4,n4 equal to ,e4,f1 & !n mu/e ,n4,h4 n$es rema9e equall to ,f4,b4 ,!n ( ^? ij4 l9es ,g4,k1 vndeuid$1 & ,g4,h4 :i* is deuid$1 ,i make a squ>e1 t is ,g4,h4,k4,l1 ,9 :i* squ>e if ,i drawe crosse l9es frome "o side to ! o!r1 accord+ to ! diuisions ( ! l9e ,g4,h1 !n w x appe> pla9e1 t ! !oreme do? affirme4 ,= ! fir/ squ>e ,g4,m4,o4,k1 mu/ ne$es be equal to ! squ>e ( ! l9e ,c4,d1 & ! fir/ portion ( ! diuid$ l9e1 :i* is ,a4,e1 = bicause _! sides are equall4 ,& s ! seconde squ>e t is ,m4,n4,p4,o1 %all be equall to ! squ>e ( ,c4,d1 & ! second "p ( ,a4,b1 t is ,e4,f4 ,also ! ?ird squ>e :i* is ,n4,h4,l4,p1 mu/ ( necessitee be equal to ! squ>e ( ,c4,d1 & ,f4,b1 bicause ^? l9es be s c\pel$ t eu]y c\ple are equall 9 ! seu]all figures4 ,& s %al y n only 9 ?is example1 b 9 all o!r f9de x true1 t if "o l9e be deuid$ into sondry p>tes1 & an o!r l9e :ole & vndeuid$1 mat*$ ) him 9 a squ>e1 t squ>e :i* is made ( ^! two :ole l9es1 is z mu*e iu/e & equally1 z all ! seu]all squ>es1 :i*e bee made ( ! :ole l9e vndiuid$1 & eu]y "p seu]ally ( ! diuid$ l9e4 ,! xxxvi4 ,!oreme4 ,if a "r l9e be p>t$ into ij4 p>tes1 z *aunce may happe1 ! squ>e t is made ( ! :ole l9e1 is equall to bo! ! squ>es t are made ( ! same l9e1 & ! twoo p>tes ( x seu]ally4 ,example4 ,illu/ration ,! l9e prop\nd$ beyng ,a4,b4 & deuid$1 z *aunce happ5e?1 9 ,c4 into ij4 vnequall p>tes1 ,i say t ! squ>e made ( ! hole l9e ,a4,b1 is equal to ! two squ>es made ( ! same l9e ) ! twoo p>tes ( itselfe1 z ) ,a4,c1 & ) ,c4,b1 = ! squ>e ,d4,e4,f4,g4 is equal to ! two o!r p>tial squ>es ( ,d4,h4,k4,g & ,h4,e4,f4,k1 b t ! great] squ>e is equall to ! squ>e ( ! :ole l9e ,a4,b1 & ! p>tiall squ>es equall to ! squ>es ( ! second p>tes ( ! same l9e ioyn$ ) ! :ole l9e1 y\r eye may iudg )\t mu*e decl>acion1 s t ,i %all n ne$e to make m exposition !r(1 b t y may exam9e x1 z y did 9 ! la/e ,!oreme4 ,! xxxvij4 ,!oreme4 ,if a "r l9e be deuid$ by *aunce1 z x maye happ51 ! squ>e t is made ( ! :ole l9e1 & "o ( ! p>tes ( x :i* soeu] x be1 %al be equall to t squ>e t is made ( ! ij4 p>tes ioyn$ togi!r1 & to an o!r squ>e made ( t "p1 :i* 0 be=e ioyn$ ) ! :ole l9e4 ,example4 ,illu/ration ,! l9e ,a4,b4 is deuid$ 9 ,c4 into twoo p>tes1 ?\gh n equally1 ( :i* two p>tes = an example ,i take ! fir/1 t is ,a4,c1 & ( x ,i make "o side ( a squ>e1 z = example ,d4,g4 accompt+e ^? two l9es to be equall1 ! o!r side ( ! squ>e is ,d4,e1 :i*e is equall to ! :ole l9e ,a4,b4 ,n[ may x appe>e1 to y\r eye1 t ! great squ>e made ( ! :ole l9e ,a4,b1 & ( "o ( 8 p>tes t is ,a4,c1 7:i* is equall ) ,d4,g47 is equal to two p>tiall squ>es1 :]e( ! "o is made ( ! saide greatt] portion ,a4,c1 9 z mu*e z n only ,d4,g1 beynge "o ( 8 sides1 b also ,d4,h4 be+e ! o!r side1 are e*e ( !m equall to ,a4,c4 ,! second squ>e is ,h4,e4,f4,k1 9 :i* ! "o side ,h4,e1 is equal to ,c4,b1 be+ ! less] p>te ( ! l9e1 ,a4,b1 & ,e4,f4 is equall to ,a4,c4 :i* is ! great] p>te ( ! same l9e4 ,s t ^? two squ>es ,d4,h4,k4,g & ,h4,e4,f4,k1 bee bo! ( !m no m !n ! greate squ>e ,d4,e4,f4,g1 accord+e to ! wordes ( ! ,!oreme a=e saide4 ,! xxxviij4 ,!oreme4 ,if a righte l9e be deuid$ by *aunce1 into p>tes1 ! squ>e t is made ( t :ole l9e1 is equall to bo? ! squ>es t ar made ( e*e p>te ( ! l9e1 & more\] to two squ>es made ( ! "o portion ( ! diuid$ l9e ioyn$ ) ! o!r 9 squ>e4 ,illu/ration ,example4 ,lette ! diuid$ l9e bee ,a4,b1 & p>t$ 9 ,c1 into twoo p>tes3 ,n[e sai! ! ,!oreme1 t ! squ>e ( ! :ole lyne ,a4,b1 is z m\*e iu/e z ! squ>e ( ,a4,c1 & ! squ>e ( ,c4,b1 e*e by x selfe1 & m ou] by z mu*e twise1 z ,a4,c4 & ,c4,b4 ioyn$ 9 "o squ>e w make4 ,= z y se1 ! great squ>e ,d4,e4,f4,g1 conteyne? 9 hym f\re less] squ>es1 ( :i*e ! fir/ & ! greate/ is ,n4,m4,f4,k1 & is equall to ! squ>e ( ! lyne ,a4,c4 ,! second squ>e is ! le/ ( !m all1 t is ,d4,h4,l4,n1 & x is equall to ! squ>e ( ! l9e ,c4,b4 ,!n are !re two o!r longe squ>es bo? ( "o bygnes1 t is ,h4,e4,n4,m4 & ,l4,n4,g4,k1 e*e ( !m bo? hauyng 4ij4 sides equall to ,a4,c1 ! long] p>te ( ! diuid$ l9e1 & !re o!r two sides equall to ,c4,b1 beeyng he %ort] p>te ( ! said l9e ,a4,b4 ,s is t greate/ squ>e1 beeyng made ( ! hole lyne ,a4,b1 equal to ! ij4 squ>es ( e*e ( 8 p>tes seu]ally1 & m by z mu*e iu/ z 4ij4 longe squ>es1 made ( ! long] portion ( ! diuid$ lyne ioyn$ 9 squ>e ) ! %ort] p>te ( ! same diuid$ l9e1 z ! !oreme wold4 ,& z "h ,i haue put an example ( a lyne diuid$ into 4ij4 p>tes1 s ! !oreme is true ( all diuid$ l9es1 ( :at numb] s eu] ! p>tes be1 f\re1 fyue1 or syxe4 etc4 ,?is !oreme ha? great vse1 n only 9 geometrie1 b also 9 ari?metike1 z h]aft] ,i w decl>e 9 conu5i5t place4 ,! 4xxxix4 !oreme4 ,if a "r l9e be deuid$ into two equall p>tes1 & "o ( ^! 4ij4 p>tes diuid$ agayn into two o!r p>tes1 z happ5e? ! longe squ>e t is made ( ! ?yrd or lat] "p ( t diuid$ l9e1 ) ! residue ( ! same l9e1 & ! squ>e ( ! mydlemo/e p>te1 are bo! togi!r equall to ! squ>e ( halfe ! fir/e l9e4 ,example4 ,illu/ration ,! l9e ,a4,b4 is diuid$ into ij4 equal p>tes 9 ,c1 & t p>te ,c4,b4 is diuid$ agayne z hapne? 9 ,d4 ,:]=e sai? ! ,!orem t ! long squ>e made ( ,d4,b4 & ,a4,d1 ) ! squ>e ( ,c4,d4 7:i* is ! mydle portion7 %all bo! be equall to ! squ>e ( half ! lyne ,a4,b1 t is to saye1 to ! squ>e ( ,a4,c1 or els ( ,c4,d1 :i* make all "o4 ,! long squ>e ,f4,g4,n4,o4 :i*e is ! longe squ>e t ! !oreme speake? (1 is made ( 4ij4 long squ>es1 :]( ! fyr/ is ,f4,g4,m4,k1 & ! seconde is ,k4,n4,o4,m4 ,! squ>e ( ! myddle portion is ,l4,m4,o4,p4 & ! squ>e ( ! halfe ( ! fyr/e lyne is ,e4,k4,q4,l4 ,n[e by ! !oreme1 t longe squ>e ,f4,g4,m4,o1 ) ! iu/e squ>e ,l4,m4,o4,p1 mu/e bee equall to ! greate squ>e ,e4,k4,q4,l1 :y*e ?ynge bycause x seeme? some:at difficult to vnd]/&e1 al?\ghe ,i int5de n "h to make demon/rations ( ! ,!oremes1 bycause x is appoynt$ to be done 9 ! newe edition ( ,euclide1 yet ,i wyll %ew y brefely h[ ! equalitee ( ! p>tes do? /&e4 ,& fyr/ ,i say1 t :]e ! comp>yson ( equalitee is made betwe5e ! greate squ>e 7:i*e is made ( halfe ! l9e ,a4,b47 & two o!r1 :]e ( ! fyr/ is ! longe squ>e ,f4,g4,n4,o1 & ! second is ! full squ>e ,l4,m4,o4,p1 :i* is "o portion ( ! great squ>e all r$ye1 & s is t longe squ>e ,k4,n4,m4,o1 beynge a p>cell also ( ! longe squ>e ,f4,g4,n4,o1 ,:]=e z ^? two p>tes are common to bo! p>tes comp>$ 9 equalitee1 & !r=e beynge bo! abat$ f e*e p>te1 if ! re/e ( bo! ! o!r p>tes bee equall1 ?an 7 ^? :ole p>tes equall be=e3 ,n[e ! re/e ( ! great squ>e1 ^? two less] squ>es beyng tak5 away1 is t longe squ>e ,e4,n4,p4,q1 :y*e is equall to ! long squ>e ,f4,g4,k4,m1 beyng ! re/ ( ! o!r p>te4 ,& t !y two be equall1 !yr sydes doo decl>e4 ,= ! longe/ lynes t is ,f4,k & ,e4,q are equall1 & s are ! %ort] lynes1 ,f4,g1 & ,e4,n1 & s app]e? ! tru! ( ! ,!oreme4 ,! 4xl4 !oreme4 ,if a "r l9e be diuid$ into 4ij4 eu5 p>tes1 & an o!r "r l9e annex$ to "o ende ( t l9e1 s t x make "o righte l9e ) ! fir/e4 ,! longe squ>e t is made ( ?is :ole l9e s augm5t$1 & ! portion t is add$1 ) ! squ>e ( halfe ! "r l9e1 %all be equall to ! squ>e ( t l9e1 :i*e is comp\nd$ ( halfe ! fir/e l9e1 & ! p>te newly add$4 ,example4 ,illu/ration ,! fyr/ lyne prop\nd$ is ,a4,b1 & x is diuid$ into ij4 equall p>tes 9 ,c1 & an o!r ryght lyne1 ,i meane ,b4,d annex$ to "o ende ( ! fyr/e lyne4 ,n[e say ,i1 t ! long squ>e ,a4,d4,m4,k1 is made ( ! :ole lyne s augm5t$1 t is ,a4,d1 & ! portion annex$1 yt is ,d4,m1 = ,d4,m is equall to ,b4,d1 :]=e yt long squ>e ,a4,d4,m4,k1 ) ! squ>e ( halfe ! fir/ l9e1 t is ,e4,g4,h4,l1 is equall to ! great squ>e ,e4,f4,d4,c4 :i*e squ>e is made ( ! l9e ,c4,d4 t is to saie1 ( a l9e comp\nd$ ( halfe ! fir/ l9e1 beyng ,c4,b1 & ! portion annex$1 t is ,b4,d4 ,& x is easyly p]ceau$1 if y consyd] t ! longe squ>e ,a4,c4,l4,k4 7:i*e onely is lefte out ( ! great squ>e7 ha? ano!r longe squ>e equall to hym1 & to supply 8 /e$e 9 ! great squ>e1 & t is ,g4,f4,m4,h4 ,= _! sydes be ( lyke l9es 9 l5g?4 ,! xli4 ,!oreme4 ,if a "r l9e bee diuid$ by *aunce1 ! squ>e ( ! same :ole l9e1 & ! squ>e ( "o ( 8 p>tes are iu/e equall to ! long squ>e ( ! :ole l9e1 & ! sayde p>te twise tak51 & m ou] to ! squ>e ( ! o!r p>te ( ! sayd l9e4 ,example4 ,illu/ration ,a4,b4 is ! l9e diuid$ 9 ,c4 ,& ,d4,e4,f4,g1 is ! squ>e ( ! :ole l9e1 ,d4,h4,k4,m4 is ! squ>e ( ! less] portion 7:y*e ,i take = an example7 & !r=e mu/ bee twise reck5$4 ,n[e ,i saye t ^? ij4 squ>es are equall to two longe squ>es ( ! :ole l9e ,a4,b1 & 8 sayd portion ,a4,c1 & also to ! squ>e ( ! o!r portion ( ! sayd fir/ l9e1 :i*e portion is ,c4,b1 & 8 squ>e ,k4,n4,f4,l4 ,9 ?is !oreme !re is no difficultie1 if y consyd] t ! litle squ>e ,d4,h4,k4,m4 is 4iiij4 tymes reck5$1 t is to say1 fyr/ ( all z a p>te ( ! greate/ squ>e1 :i*e is ,d4,e4,f4,g4 ,secondly he is rekn$ by him selfe4 ,?irdely he is accompt$ z p>cell ( ! long squ>e ,d4,e4,n4,m1 ,& f\r?ly he is tak5 z a "p ( ! o!r long squ>e ,d4,h4,l4,g1 s t 9 z mu*e z he is twise reck5$ 9 "o "p ( ! comp>ison ( equalitee1 & twise also 9 ! second p>te1 !re c rise none occasion ( err\r or d\btfulnes !rby4 ,! xlij4 ,!oreme4 ,if a "r l9e be deuid$ z *ance happ5e? ! iiij4 long squ>es1 t may be made ( t :ole l9e & "o ( 8 p>tes ) ! squ>e ( ! o!r "p1 %all be equall to ! squ>e t is made ( ! :ole l9e & ! saide fir/ portion ioyn$ to him 9 l5g! z "o :ole l9e4 ,example4 ,illu/ration ,! fir/e l9e is ,a4,b1 & is deuid$ by ,c4 into two vnequall p>tes z happ5e?4 ,! long squ>e ( yt1 & 8 less] portion ,a4,c1 is f\re times draw51 ! fir/ is ,e4,g4,m4,k1 ! seconde is ,k4,m4,q4,o1 ! ?ird is ,h4,k4,r4,s1 & ! f\r! is ,k4,l4,s4,t4 ,& :]e z x appe>e? t "o ( ! little squ>es 7,i meane ,k4,l4,p4,o7 is reck5$ twise1 ones z p>cell ( ! second long squ>e & agayne z p>te ( ! ?irde long squ>e1 to auoide ambiguite1 y may place "o in/e$e ( x1 an o!r squ>e ( equalitee1 ) x4 t is to saye1 ,d4,e4,k4,h1 :i* 0 at no tyme accompt+ z p>cell ( any "o ( !m1 & !n haue y iiij4 long squ>es di/9ctly made ( ! :ole l9e ,a4,b1 & 8 less] portion ,a4,c4 ,& )9 !m is !re a greate full squ>e ,p4,q4,t4,v4 :i*e is ! iu/ squ>e ( ,b4,c1 beynge ! greatt] portion ( ! l9e ,a4,b4 ,& t ^? fiue squ>es doo make iu/e z mu*e z ! :ole squ>e ( t long] l9e ,d4,g1 7:i*e is z longe z ,a4,b1 & ,a4,c4 ioyn$ togi!r7 x may be iudg$ easyly by ! eye1 si? t "o greate squ>e do? compreh5d 9 x all ! o!r fiue squ>es1 t is to say1 f\re long squ>es 7z is be=e m5cion$7 & "o full squ>e4 :i* is ! int5t ( ! ,!oreme4 ,! xliij4 ,!oreme4 ,if a "r l9e be deuid$ into ij4 equal p>tes fir/1 & "o ( ^? p>ts aga9 into o!r ij4 p>ts1 z *aunce hap5e?1 ! squ>e t is made ( ! la/ "p ( ! l9e s diuid$1 & ! squ>e ( ! residue ( t :ole l9e1 are d\ble to ! squ>e ( halfe t l9e1 & to ! squ>e ( ! middle portion ( ! same l9e4 ,example4 ,illu/ration ,! l9e to be deuid$ is ,a4,b1 & is p>t$ 9 ,c4 into two equall p>tes1 & !n ,c4,b1 is deuid$ aga9e into two p>tes 9 ,d1 s t ! mean+e ( ! ,!oreme1 is t ! squ>e ( ,d4,b4 :i* is ! latt] p>te ( ! l9e1 & ! squ>e ( ,a4,d1 :i* is ! residue ( ! :ole l9e4 ,^? two squ>es1 ,i say1 ar d\ble to ! squ>e ( "o halfe ( ! l9e1 & to ! squ>e ( ,c4,d1 :i* is ! middle portion ( ^? ?re diuisions4 ,:i* ?+ t y maye m easilye p]ceaue1 ,i haue draw5 f\re squ>es1 :]e( ! greate/ be+ m>k$ ) ,e4 is ! squ>e ( ,a4,d4 ,! next1 :i* is m>k$ ) ,g1 is ! squ>e ( halfe ! l9e1 t is1 ( ,a4,c1 ,& ! o!r two little squ>es m>k$ ) ,f4 & ,h1 be bo? ( "o bignes1 by reason t ,i did diuide ,c4,b4 into two equall p>tes1 s t y amy take ! squ>e ,f1 = ! squ>e ( ,d4,b1 & ! squ>e ,h1 = ! squ>e ( ,c4,d4 ,n[ ,i ?9ke y d\bt n1 b t ! squ>e ,e4 & ! squ>e ,f1 ar d\ble s mu* z ! squ>e ,g4 & ! squ>e ,h1 :i* ?+ ! easy] is to be vnd]/&e1 bicause t ! greate squ>e ha? 9 8 side iij4 qu>t]s ( ! fir/e l9e1 :i* multipli$ by itselfe make? nyne qu>t]s1 & ! squ>e ,f4 conta9e? b "o qu>t]1 s t bo! doo make t5ne qu>t]s4 ,!n ,g4 contayne? iiij4 qu>t]s1 seynge 8 side conta9e? twoo1 & ,h4 conta9e? b "o qu>t]1 :i*e bo? make b fiue qu>t]s1 & t is b halfe ( t5ne4 ,:]eby y may easylye coniecture1 t ! meanynge ( ! !oreme is v]ifi$ 9 ! figures ( ?is example4 ,! xliiij4 ,!oreme4 ,if a "r l9e be deuid$ into ij4 p>tes equally1 & an o!r portion ( a righte lyne annex$ to t fir/e l9e1 ! squ>e ( ?is :ole l9e s comp\nd$1 & ! squ>e ( ! portion t is annex$1 ar d\le z mu* z ! squ>e ( ! halfe ( ! fir/e l9e1 & ! squ>e ( ! o!r halfe ioyn$ 9 "o ) ! annex$ portion1 z "o :ole l9e4 ,example4 ,illu/ration ,! l9e is ,a4,b1 & is diuid$ fir/e into twoo equal p>tes 9 ,c1 & !n is !re annex$ to x an o!r portion :i*e is ,b4,d4 ,n[ sai? ! ,!oreme1 t ! squ>e ( ,a4,d1 & ! squ>e ( ,b4,d1 ar d\ble to ! squ>e ( ,a4,c1 & to ! squ>e ( ,c4,d4 ,! l9e ,a4,b4 conta9+ f\r p>tes1 !n mu/ ne$es 8 halfe conta9e ij4 p>tes ( su* p>tes ,i suppose ,b4,d4 7:i* is ! annex$ l9e7 to conta9e ?re1 s %al ! hole l9e compreh5d vij4 p>ts1 & 8 squ>e xlix4 p>ts1 :]e vnto if y ad ye squ>e ( ! annex$ lyne1 :i*e make? nyne1 ?an ^? bo! doo yelde1 lviij4 :y*e mu/ be d\ble to ! squ>e ( ! halfe lyne ) ! annex$ portion4 ,! halfe lyne by x selfe conteyne? b 4ij4 p>tes1 & !r=e 8 squ>e doo? make f\re4 ,! halfe lyne ) ! annex$ portion conteyne? fiue1 & ! squ>e ( x is 4xxv1 n[ put f\re to 4xxv1 & x make? iu/ 4xxix1 ! eu5 halfe ( fifty & eight1 :]by app]e? ! tru! ( ! !oreme4 ,! 4xlv4 !oreme4 ,9 all triangles t haue a blunt angle1 ! squ>e ( ! side t lie? aga9/ ! blunt angle1 is great] ?an ! two squ>es ( ! o!r twoo sydes1 by twise z mu*e z is compreh5d$ ( ! "o ( ^? 4ij4 sides 7inclosyng ! blunt corn]7 & ! portion ( ! same l9e1 beyng draw5 foor? 9 l5g!1 :i* lie? betw5e ! said blunt corn] & a p]p5dicul> l9e lightyng on x1 & draw5 f "o ( ! %>pe angles ( ! =esayd triangle4 ,example4 ,= ! decl>ation ( ?is !oreme & ! next also1 ^: vse are wond]full 9 ! practise ( ,geometrie1 & 9 measuryng especially1 x %all be n$efull to decl>e t eu]y triangle t ha? no ryght angle z ^? :y*e are call$ 7z 9 ! boke ( practise is decl>$7 %>p corn]$ triangles1 & blunt corn]$ triangles1 yet may !y be br\ght to haue a ryght angle1 ey!r by p>tyng !m into two less] triangles1 or els by addyng an o!r triangle vnto !m1 :i*e may be a great helpe = ! ayde ( measuryng1 z m l>gely %all be sette foor! 9 ! boke ( measuryng4 ,b = ?is pres5t place1 ?is =me wyll ,i vse1 7:i*e ,!on also vse?7 to adde "o triangle vnto an o!r1 to bryng ! blunt corn]$ triangle into a ryght angl$ triangle1 :]eby ! proportion ( ! squ>es ( ! sides 9 su*e a blunt corn]$ triangle may ! bett] bee kn[54 ,illu/ration ,fyr/ !r=e ,i sette foor? ! triangle ,a4,b4,c1 ^: corn] by ,c4 is a blunt corn] z y maye well iudge1 ?an to make an o!r triangle ( yt ) a ryght angle1 ,i mu/ drawe =? ! side ,b4,c4 vnto ,d1 & f ! %>p corn] by ,a4 ,i brynge a plumbe lyne or p]p5dicul> on ,d4 ,& s is !re n[e a newe triangle ,a4,b4,d4 ^: angle by ,d4 is a "r angle4 ,n[e accordyng to ! meanyng ( ! ,!oreme1 ,i saie1 t 9 ! fir/ triangle ,a4,b4,c1 because x ha? a blunt corn] at ,c1 ! squ>e ( ! l9e ,a4,b4 :i*e lie? aga9/ ! said blunte corn]1 is m !n ! squ>e ( ! l9e ,a4,c1 & also ( ! lyne ,b4,c1 7:i*e inclose ! blunte corn]7 by z mu*e z w am\nt twise ( ! l9e ,b4,c1 & t portion ,d4,c4 :i*e lie? betw5e ! blunt angle by ,c1 & ! p]p5dicul> l9e ,a4,d4 ,! squ>e ( ! l9e ,a4,b1 is ! great squ>e m>k$ ) ,e4 ,! squ>e ( ,a4,c1 is ! meane squ>e m>k$ ) ,f4 ,! squ>e ( ,b4,c1 is ! lea/ squ>e m>k$ ) ,g4 ,& ! long squ>e m>k$ ) ,k1 is sette 9 /e$e ( two squ>es made ( ,b4,c1 & ,c4,d4 ,= z ! %ort] side is ! iu/e l5g! ( ,c4,d1 s ! o!r long] side is iu/ twise s longe z ,b4,c1 ,:]=e ,i saie n[ accordyng to ! ,!oreme1 t ! greatte squ>e ,e1 is m !n ! o!r two squ>es ,f4 & ,g1 by ! quantitee ( ! longe squ>e ,k1 :]( ,i res]ue ! pr(e to a m conu5i5t place1 :]e ,i w also tea*e ! reason h[e to fynde ! l5g! ( all su*e p]p5dicul> lynes1 & also ( ! l9e t is draw5 betwe5e ! blunte angle & ! p]p5dicul> l9e1 ) sundrie o!r v pleasant conclusions4 ,! 4xlvi4 ,!oreme4 ,9 %>pe corn]$ triangles1 ! squ>e ( anie side t lie? aga9/ a %>pe corn]1 is less] !n ! two squ>es ( ! o!r two sides1 by z mu*e z is compris$ twise 9 ! long squ>e ( t side1 on :i*e ! p]p5dicul> l9e falle?1 & ! portion ( t same l9e1 liyng betwe5e ! p]p5dicul>1 & ! =esaid %>pe corn]4 ,example4 ,illu/ration ,fyr/ ,i sette foor? ! triangle ,a4,b4,c1 & 9 yt ,i draw a plumbe l9e f ! angle ,c4 vnto ! l9e ,a4,b1 & x lighte? 9 ,d4 ,n[e by ! !oreme ! squ>e ( ,b4,c4 is n s mu*e z ! squ>e ( ! o!r two sydes1 t ( ,b4,a4 & ( ,a4,c4 by z mu*e z is twise conteyn$ 9 ! long squ>e made ( ,a4,b1 & ,a4,d1 ,a4,b4 beyng ! l9e or syde on :i* ! p]p5dicul> l9e falle?1 & ,a4,d4 beeyng t portion ( ! same l9e :i*e do? lye betw5e ! p]p5dicul> l9e1 & ! sayd %>pe angle limitt$1 :i*e angle is by ,a4 ,= decl>ation ( ! figures1 ! squ>e m>k$ ) ,e4 is ! squ>e ( ,b4,c1 :i*e is ! syde t lie? agayn/ ! %>pe angle1 ! squ>e m>k$ ) ,g4 is ! squ>e ( ,a4,b1 & ! squ>e m>k$ ) ,f4 is ! squ>e ( ,a4,c1 & ! two longe squ>es m>k$ ) ,h4,k1 are made ( ! hole l9e ,a4,b1 & "o ( 8 portions ,a4,d4 ,& tru! x is t ! squ>e ,e4 is less] ?an ! o!r two squ>es ,c4 & ,f4 by ! quantitee ( ^? two long squ>es ,h4 & ,k4 ,:]by y may consyd] agayn1 an o!r proportion ( equalitee1 t is to saye1 t ! squ>e ,e4 ) ! twoo longsqu>es ,h4,k1 are iu/e equall to ! o!r twoo squ>es ,c4 & ,f4 ,& s maye y make1 z x 7 an o!r !oreme4 ,t 9 al %>pe corn]$ triangles1 :]e a p]p5dicul> l9e is draw5 frome "o angle to ! side t lye? aga9/e x1 ! squ>e ( anye "o side1 ) ! ij4 longesqu>es made at t hole l9e1 :]eon ! p]p5dicul> l9e do? lighte1 & ( t portion ( x1 :i* ioyne? to t side ^: squ>e is all ready tak51 ^? ?re figures1 ,i say1 are equall to ! ij4 squ>es1 ( ! o!r ij4 sides ( ! triangle4 ,9 :i*e y mu/e vnd]/&1 t ! side on :i* ! p]p5dicul>e falle?1 is ?rise vs$1 yet is 8 squ>e b ones m5cion$1 = twise he is tak5 = "o side ( ! two long squ>es4 ,& z ,i haue ?us made z x 7 an o!r !oreme out ( ?is f\rty & sixe !oreme1 s mighte ,i out ( x1 & ! o!r t goe? nexte be=e1 make z manny z w\lde suffice = a :ole booke1 s t :5 !y %all bee apply$ to practise1 & consequ5tly to expresse _! b5efite1 no manne t ha! n well wayde _! wond]full commoditee1 w\ld cr$ite ! possibilitie ( _! wond]full vse1 & l>ge ayde 9 k4 ,b all ?is wyll ,i remitte to a place conu5i5t4 ,! xlvij4 ,!oreme4 ,if ij4 po9ts be m>k$ 9 ! circumf]5ce ( a circle1 & a "r l9e draw5 frome ! "o to ! o!r1 t l9e mu/ ne$es fal )9 ! circle4 ,example4 ,illu/ration ,! circle is ,a4,b4,c4,d1 ! ij4 po9ctes are ,a4,b1 ! righte l9e t is draw5ne frome ! "o to ! o!r1 is ! l9e ,a4,b1 :i* z y see1 mu/ ne$es lyghte )9 ! circle4 ,s if y putte ! po9tes to be ,a4,d1 or ,d4,c1 or ,a4,c1 o!r ,b4,c1 or ,b4,d1 9 any ( ^! cases y see1 t ! l9e t is draw5 f ! "o pricke to ! o!r do! eu]more run )9 ! edge ( ! circle1 els canne x be no "r l9e4 ,h[ be x1 t a crok$ l9e1 especially be+ m crok$ !n ! portion ( ! circumf]5ce1 maye bee draw5 f po9te to po9te )\te ! circle4 ,b ! !oreme speake? only ( "r l9es1 & n ( crok$ l9es4 ,! xlviij4 ,!oreme4 ,if a righte l9e pass+e by ! c5tre ( a circle1 doo crosse an o!r "r l9e )9 ! same circle1 pass+e beside ! c5tre1 if he deuide ! saide l9e into twoo equal p>tes1 !n doo !y make all _! angles righte4 ,& contr>ie waies1 if !y make all _! angles righte1 !n do? ! long] l9e cutte ! %ort] 9 twoo p>tes4 ,example4 ,illu/ration ,! circle is ,a4,b4,c4,d1 ! l9e t passe? by ! c5tre1 is ,a4,e4,c1 ! l9e t goe? beside ! c5tre is ,d4,b4 ,n[e saye ,i1 t ! l9e ,a4,e4,c1 do! cutte t o!r l9e ,d4,b4 into twoo iu/e p>tes1 & !re=e all _! f\r angles ar righte angles4 ,& contr>ye wayes1 bicause all _! angles are righte angles1 !r=e x mu/e be true1 t ! great] cutte? ! less] into two equal p>tes1 accord+e z ! ,!oreme w\ld4 ,! xlix4 ,!oreme4 ,if twoo "r l9es draw5 9 a circle doo crosse "o an o!r1 & doo n passe by ! c5tre1 eu]y ( !m do! n deuide ! o!r into equall p>tions4 ,example4 ,illu/ration ,! circle is ,a4,b4,c4,d1 & ! c5tre is ,e1 ! "o l9e ,a4,c1 & ! o!r is ,b4,d1 :i* two l9es crosse "o an o!r1 b yet !y g n by ! c5tre1 :]e=e accord+e to ! woordes ( ! !oreme1 e*e ( !im do? cuytte ! o!r into equall portions4 ,= z y may easily iudge1 ,a4,c4 ha? "o portion long] & an o!r %ort]1 & s l wise ,b4,d4 ,h[beit1 x is n s to be vnd]/&1 b "o ( !m may be deuid$ into ij4 eu5 p>ts1 b bo! to bee cutte equally 9 ! middle1 is n possible1 onles bo? passe ?r\gh ! c5tre1 !r=e mu* r :5 bo! g beside ! c5tre1 x c n be t e*e ( !ym %\lde be iu/ely p>t$ into ij4 eu5 p>tes4 ,! ,l4 ,!oreme4 ,if two circles crosse & cut "o an o!r1 !n haue n !y bo? "o c5tre4 ,example4 ,illu/ration ,?is !oreme seeme? ( x selfe s manife/1 t x neade? no!r demon/ration no!r decl>acion4 ,yet = ! pla9e vnd]/&+ ( x1 ,i haue sette =! a figure "h1 :]e ij4 circles be draw51 s t "o ( !m do? crosse ! o!r 7z y see7 9 ! po9tes ,b4 & ,g1 & _! c5tres appe> at ! fir/e sighte to bee diu]s4 ,= ! c5tre ( ! "o is ,f1 & ! c5tre ( ! o!r is ,e1 :i* diffre z f>re asondre z ! edges ( ! circles1 :]e !y bee mo/ di/aunte 9 sond]4 ,! ,li4 ,!oreme4 ,if two circles be s draw51 t "o ( !m d t\*e ! o!r1 !n haue !y n "o c5tre4 ,example4 ,illu/ration ,!re are two circles made1 z y see1 ! "o is ,a4,b4,c1 & ha? 8 c5tre by ,g1 ! o!r is ,b4,d4,e1 & 8 c5tre is by ,f1 s t x is easy en\gh to p]ceaue t _! c5tres doe dyff] z mu*e a sond]1 z ! halfe diamet] ( ! great] circle is long] !n ! half diamet] ( ! less] circle4 ,& s mu/ x ne$es be ?\ght & said ( all o!r circles 9 lyke k9de4 ,! 4lij4 !oreme4 ,if a c]ta9e po9te be assign$ 9 ! diamet] ( a circle1 di/ant f ! c5tre ( ! said circle1 & f t po9te diu]se lynes draw5 to ! edge & circumf]5ce ( ! same circle1 ! longe/ l9e is t :i*e passe? by ! c5tre1 & ! %orte/ is ! residew ( ! same l9e4 ,& ( al ! o!r l9es t is eu] ! greate/1 t is nighe/ to ! l9e1 :i* passe? by ! c5tre4 ,& contr>y waies1 t is ! %orte/1 t is f>!/ f x4 ,& amonge/ !m all !re c be b onely 4ij4 equall toge!r1 & !y mu/ n$es be s plac$1 t ! %orte/ l9e %all be 9 ! iu/ middle betwixte !m4 ,example4 ,illu/ration ,! circle is ,a4,b4,c4,d4,e4,h1 & 8 c5tre is ,f1 ! diamet] is ,a4,e1 9 :i*e diamet] ,i haue tak5 a c]ta9 po9t di/aunt f ! c5tre1 & t po9te is ,g1 f :i* ,i haue draw5 4iiij4 l9es to ! circumf]5ce1 beside ! two p>tes ( ! diamet]1 :i*e make? vp vi4 lynes 9 all4 ,n[e = ! diu]sitee 9 quantitie ( ^! lynes1 ,i saie accordyng to ! ,!oreme1 t ! l9e :i*e goe? by ! c5tre is ! longe/ l9e1 t is to saie1 ,a4,g1 & ! residewe ( ! same diamet] beeyng ,g4,e1 is ! %orte/ lyne4 ,& ( all ! o!r t lyne is longe/1 t is ne]e/ vnto t p>te ( ! diamet] :i*e gooe? by ! c5tre1 & t is %orte/1 t is f>!/ di/ant f x1 :]e=e ,i saie1 t ,g4,b1 is long] !n ,g4,c1 & !r=e mu*e m long] !n ,g4,d1 si? ,g4,c1 also is long] !n ,g4,d1 & by ?is maie y soone p]ceiue1 t x is n possible to drawe 4ij4 lynes on any "o side ( ! diamet]1 :i*e might be equall 9 l5g! toge!r1 b on ! "o side ( ! diamet] maie y easylie make "o lyne equall to an o!r1 on ! o!r side ( ! same diamet]1 z y see 9 ?is example ,g4,h1 to bee equall to ,g4,d1 betwe5e :i*e ! lyne ,g4,e1 7z ! %orte/ 9 all ! circle7 doo! /&e eu5 di/aunte f e*e ( !m1 & x is ! precise kn[el$ge ( _! equalitee1 if !y be equally di/aunt f "o halfe ( ! diamet]4 ,:]e z contr>y waies if ! "o be ne]] to any "o halfe ( ! diamet] !n ! o!r is1 x is n possible t !y two may be equall 9 l5g!1 namely if !y dooe ende bo! 9 ! circumf]5ce ( ! circle1 & be bo! draw5 f "o poynte 9 ! diamet]1 s t ! saide poynte be 7z ! ,!oreme doe? suppose7 some:at di/aunt f ! c5tre ( ! said circle4 ,= if !y be draw5 f ! c5tre1 !n mu/ !y ( necessitee be all equall1 h[e _m s eu] !y bee1 z ! def9ition ( a circle dooe? importe1 )\te any reg>de h[ ne]e s eu] !y be to ! diamet]1 or h[ di/ante f x4 ,& "h is to be not$1 t 9 ?is ,!oreme1 by ne]5esse & di/aunce is vnd]/& ! n]5ess & di/aunce ( ! extreeme p>tes ( ^? lynes :]e !y t\*e ! circumf]5ce4 ,= at ! o!r end !y d all meete & t\*e4 ,! 4liij4 ,!oreme4 ,if a po9te bee m>k$ )\t a circle1 & f x diu]se l9es draw5 crosse ! circle1 to ! circumf]5ce on ! o!r side1 s t "o ( !m passe by ! c5tre1 !n t l9e :i*e passe? by ! c5tre %all be ! loonge/ ( !m all t crosse ! circle4 ,& ( ! o!r l9es ^? are longe/1 t be nexte vnto x t passe? by ! c5tre4 ,& ^? ar %orte/1 t be f>!/ di/ant f x4 ,b among ^? p>tes ( ^? l9es1 :i*e ende 9 ! outew>de circumf]5ce1 t is mo/ %orte/1 :i*e is p>te ( ! l9e t passe? by ! c5tre1 & amonge/e ! o!re e*e1 ( !m1 ! n]] !y are vnto x1 ! %ort] !y are1 & ! f>!r f x1 ! long] !y be4 ,& amonge/ !m all !re c n be m !n 4ij4 ( any "o l5g?1 & !y two mu/e be on ! two contr>ie sides ( ! %orte/ l9e4 ,example4 ,illu/ration ,take ! circle to be ,a4,b4,c1 & ! po9t assign$ )\t x to be ,d4 ,n[ say ,i1 t if !re be draw5 sundrie l9es f ,d1 & crosse ! circle1 endyng 9 ! circumf]5ce on ! contr>y side1 z "h y see1 ,d4,a1 ,d4,e1 ,d4,f1 & ,d4,b1 !n ( all ^! l9es ! longe/ mu/ ne$es be ,d4,a1 :i* goe? by ! c5tre ( ! circle1 & ! nexte vnto x1 t is ,d4,e1 is ! longe/ amonge/ ! re/4 ,& contr>ie waies1 ,d4,b1 is ! %orte/e1 because x is f>!/ di/aunt f ,d4,a4 ,& s maie y iudge ( ,d4,f1 because x is n]] vnto ,d4,a1 !n is ,d4,b1 !re=e is x long] !n ,d4,b4 ,& likewaies because x is f>!r ( f ,d4,a1 !n is ,d4,e1 !r=e is x %ort] !n ,d4,e4 ,n[ = ^? p>tes ( ! l9es :i*e bee )\te ! circle 7z y see7 ,d4,c1 is ! %orte/4 because x is ! p>te ( t l9e :i* passe? by ! c5tre1 ,& ,d4,k1 is next to x 9 di/ance1 & !re=e also 9 %ortnes1 s ,d4,g1 is f>!/ f x 9 di/ance1 & !r=e is ! longe/ ( !m4 ,n[ ,d4,h1 beyng n]] !n ,d4,g1 is also %ort] !n x1 & beynge f>!r (1 !n ,d4,k1 is long] !n x4 ,s t = ?is p>te ( ! !oreme 7z ,i ?9k7 y d pla9ly p]ceaue ! tru! !re(1 s ! residue ha! no difficulte4 ,= se+ t ! ne>] any l9e is to ,d4,c1 7:i* ioyne? ) ! diamet]7 ! %ort] x is & ! f>!r ( f x1 ! long] x is4 ,& seyng two lynes c n be ( l di/aunce be+e bo! on "o side1 !re=e if !y %al be ( "o l5g!1 & consequ5tly ( "o di/aunce1 !y mu/ ne$es bee on contr>y sides ( ! saide l9e ,d4,c4 ,& s appe>e? ! mean+ ( ! :ole ,!oreme4 ,& ( ?is ,!oreme do! !re fol[e an o!r lyke4 :i*e y maye calle o!r a !oreme by x selfe1 or else a ,coroll>y vnto ?is la/e !oreme1 ,i passe n s mu*e = ! "n4 ,b 8 s5t5ce is ?is3 :5 s eu] any lynes be draw5 frome any po9te1 )\te a circle1 :e!r !y crosse ! circle1 or e&e 9 ! utt] edge ( 8 circumf]5ce1 ^? two l9es t bee equally di/aunt f ! lea/ l9e are equal togi!r1 & contr>y waies1 if !y be equall togi!r1 !y ar also equally di/ant f t lea/ l9e4 ,= ! decl>acion ( ?is proposition1 x %all n ne$ to vse any o!r example1 !n t :i* is br\ght = ! explication ( ?is la/e !oreme1 by :i*e y may )\t any tea*+e easyly p]ceaue bo? ! meanyng & also ! tru? ( ?is proposition4 ,! ,liiij4 ,!oreme4 ,if a po9t be set =! 9 a circle1 & f t po9te vnto ! circumf]5ce _m l9es draw51 ( :i* m !n two are equal togi!r1 !n is t po9t ! c5tre ( t circle4 ,example4 ,illu/ration ,! circle is ,a4,b4,c1 & )9 x ,i haue sette f\r? = an example ?ree prickes1 :i* are ,d4,e4 & ,f1 f eu]y "o ( !m ,i haue draw5 7at ! lea/e7 iiij4 l9es vnto ! circumf]5ce ( ! circle b frome ,d1 ,i haue draw5 m1 yet maye x appe> readily vnto y\r eye1 t ( all ! l9es :i*e be draw5 f ,e4 & ,f1 vnto ! circumf]5ce1 !re are b twoo equall1 & m c n bee1 = ,g4,e4 nor ,e4,h4 ha? none o!r equal to !im1 nor canne n haue any be+e draw5 f ! same po9t ,e4 ,no m c ,l4,f1 or ,f4,k1 haue anye l9e equall to ei!r ( !im1 be+e draw5 f ! same po9te ,f4 ,& yet f ei!r ( ^? two po9ctes are !re draw5 twoo l9es equall togi!r1 z ,a4,e1 is equall to ,e4,b1 & ,b4,f1 is equall to ,f4,c1 b !re c no ?ird l9e be draw5 equall to ei!r ( ^! two c\ples1 & t is by reason t !y be draw5 f a po9te di/aunte f ! c5tre ( ! circle4 ,b f ,d1 al?\ghe !re be seu5 l9es draw51 to ! circumf]5ce1 yet all bee equall1 bicause x is ! c5tre ( ! circle4 ,& !re=e if y drawe neu] s mannye m f x vnto ! circumf]5ce1 all %all be equal1 s t ?is is ! priuilege 7z x 7 ( ! c5tre7 & !r=e no o!r po9t c haue ab\e two equal l9es draw5 f x vnto ! circumf]5ce4 ,& f all po9tes y maye drawe ij4 equall l9es to ! circumf]5ce ( ! circle1 :e!r t po9te be )9 ! circle or )\t x4 ,! lv4 ,!oreme4 ,no circle canne cut an o!r circle 9 m po9tes !n two4 ,example4 ,illu/ration ,! fir/ circle is ,a4,b4,f4,e1 ! second circle is ,b4,c4,d4,e1 & !y crosse "o an o!r 9 ,b4 & 9 ,e1 & 9 no m po9tes4 ,no!r is x possible t !y %\ld1 b o!r figures !r be1 :i* maye cutte a circle 9 f\re p>tes1 z y se 9 ?is example4 ,:]e ,i haue set =! "o tunne =me1 & "o eye =me1 & e*e ( !m cutte? eu]y ( _! two circles into f\re p>tes4 ,b z !y be irregul>e =mes1 t is to saye1 su*e =mes z haue no precise measure no!r proportion 9 _! draughte1 s c !re sc>cely be made any c]ta9e !orem ( !m4 ,b circles are regul>e =mes1 t is to say1 su* =mes z haue 9 _! protracture a iu/e & c]ta9e proportion1 s t c]ta9 & det]m9ate tru?s may be affirm$ ( !m1 si? !y ar vni=me & vn*aungable4 ,! lvi4 ,!oreme4 ,if two circles be s draw51 t ! "o be )9 ! o!r1 & t !y t\*e "o an o!r3 ,if a l9e bee draw5 by bo! _! c5tres1 & s =! 9 l5g!1 t l9e %all runne to t po9te1 :]e ! circles d t\*e4 ,example4 ,illu/ration ,! "o circle1 :i* is ! greatte/ & vtt]mo/ is ,a4,b4,c1 ! o!r circle t is ye less]1 & is draw5 )9 ! fir/e1 is ,a4,d4,e4 ,! c5tre ( ! great] circle is ,f1 & ! c5tre ( ! less] circle is ,g1 ! po9te :]e !y t\*e is ,a4 ,& n[ y may see ! tru! ( ! !oreme s pla9ely1 t x ne$e? no f>!r decl>acion4 ,= y maye see1 t draw+e a l9e f ,f4 to ,g1 & s =? 9 l5g!1 vntill x come to ! circumf]5ce1 x wyll lighte 9 ! v po9cte ,a1 :]e ! circles t\*e "o an o!r4 ,! ,lvij4 ,!oreme4 ,if two circles bee draw5 s "o )\te an o!r1 t _! edges doo t\*e & a "r l9e bee drawn5ne frome ! c5tre ( ! "o to ! c5tre ( ! o!r1 t l9e %all passe by ! place ( _! t\*+4 ,example4 ,illu/ration ,! fir/e circle is ,a4,b4,e1 & 8 c5tre is ,k1 ,! second circle is ,d4,b4,c1 & 8 c5tre is ,h1 ! po9t :] !y d t\* is ,b4 ,n[e doo y se t ! l9e ,k4,h1 :i*e is draw5 f ,k1 t is c5tre ( ! fir/e circle1 vnto ,h1 beyng c5tre ( ! second circle1 do? passe 7z x mu/ n$es by ! po9te ,b17 :i*e is ! v]ye poynte :] !y d to t\*e toge!r4 ,! 4lviij4 !oreme4 ,"o circle c n t\*e an o!r 9 m po9tes !n "o1 :e!r !y t\*e )9 or )\t4 ,example4 ,illu/ration ,= ! decl>ation ( ?is ,!oreme1 ,i haue draw5 iiij4 circles1 ! fir/ is ,a4,b4,c1 & 8 c5tre ,h4 ! second is ,a4,d4,g1 & 8 c5tre ,f4 ! ?ird is ,l4,m1 & 8 c5tre ,k4 ! 4iiij4 is ,d4,g4,l4,m1 & 8 c5tre ,e4 ,n[e z y p]ceiue ! second circle ,a4,d4,g1 t\*e? ! fir/ 9 ! inn] side1 9 s mu* z x is draw5 )9 ! o!r1 & yet x t\*e? him b 9 "o po9t1 t is to say 9 ,a1 s lykewaies ! ?ird circle ,l4,m1 is draw5 )\t ! fir/e circle & t\*e? hym1 z y maie see1 b 9 "o place4 ,& n[ z = ! 4iiij4 circle1 x is draw5 to decl>e ! diu]sitie betw5e t\*yng & cuttyng1 or crossyng4 ,= "o circle maie crosse & cutte a great _m o!r circles1 yet c be n cutte any "o 9 m places !n two1 z ! fiue & fiftie ,!oreme affirme?4 ,! 4lix4 ,!oreme4 ,9 eu]ie circle ^? l9es are to be c\nt$ equall1 :i*e are 9 lyke di/aunce f ! c5tre1 ,& contr>ie waies !y are 9 lyke di/ance f ! c5tre1 :i*e be equall4 ,example4 ,illu/ration ,9 ?is figure y see fir/e ! circle draw51 :i*e is ,a4,b4,c4,d1 & 8 c5tre is ,e4 ,9 ?is circle also !re are draw5 two l9es equally di/aunt f ! c5tre1 = ! l9e ,a4,b1 & ! l9e ,d4,c1 are iu/e ( "o di/aunce f ! c5tre1 :i*e is ,e1 & !r=e are !y ( "o l5g?4 ,aga9 !i are ( "o l5g! 7z %all be pr\$ 9 ! boke ( pr(es7 & !re=e _! di/aunce f ! c5tre is all "o4 ,! lx4 ,!oreme4 ,9 eu]ie circle ! longe/ l9e is ! diamet]1 & ( all ! o!r l9es1 !i are /ill longe/ t be nexte vnto ! c5tre1 & !y be ! %orte/1 t be f>!/ di/aunt f x4 ,example4 ,illu/ration ,9 ?is circle ,a4,b4,c4,d1 ,i haue draw5 fir/ ! diamet]1 :i*e is ,a4,d1 :i*e passe? 7z x mu/7 by ! c5tre ,e1 ,!n haue ,i draw5 ij4 o!r l9es z ,m4,n1 :i*e is ne]] ! c5tre1 & ,f4,g1 t is f>!r f ! c5tre4 ,! f\r? l9e also on ! o!r side ( ! diamet]1 t is ,b4,c1 is ne]] to ! c5tre !n ! l9e ,f4,g1 = x is ( lyke di/ance z is ! lyne ,m4,n4 ,n[e saie ,i1 t ,a4,d1 beyng ! diamet]1 is ! longe/ ( all ^? lynes1 & also ( any o!r t maie be draw5 )9 t circle1 ,& ! o!r l9e ,m4,n1 is long] !n ,f4,g4 ,also ! l9e ,f4,g1 is %ort] !n ! l9e ,b4,c1 = because x is f>!r f ! c5tre !n is ! lyne ,b4,c4 ,& ?us maie y iudge ( al l9es draw5 9 any circle1 h[ to "k ! proportion ( _! l5g?1 by ! proportion ( _! di/ance1 & contr>y waies1 h[e to disc]ne ! proportion ( _! di/ance by _! l5g!s1 if y kn[e ! proportion ( _! l5g?4 ,& to speake ( x by ! waie1 x is a m>uayl\se ?yng to consid]1 t a man maie kn[e an exacte proportion betw5e two ?ynges1 & yet c n "n nor attayne ! precise quantitee ( ^? two ?ynges1 ,z = exaumple1 ,if two squ>es be sette foor!1 :]e( ! "o conta9e? 9 x fiue squ>e feete1 & ! o!r contayne? fiue & =tie foote1 ( l squ>e feete1 ,i am n able to tell1 no nor yet anye manne liuyng1 :at is ! precyse measure ( ! sides ( any ( ^? 4ij4 squ>es1 & yet ,i c pr\e by vnfallible reason1 t _! sides be 9 a triple proportion1 t is to saie1 t ! side ( ! great] squ>e 7:i*e conta9e? 4xlv4 foote7 is ?ree tymes s long iu/e z ! side ( ! less] squ>e1 t include? b fiue foote4 ,b ?is seeme? to be spok5 out ( ceason 9 ?is place1 !r=e ,i w omitte x n[1 res]uyng ! exact] decl>ation !r( to a m conu5i5t place & "t1 & w proc$e ) ! residew ( ! ,!oremes appo9t$ = ?is boke4 ,! 4lxi4 ,!oreme4 ,if a "r l9e be draw5 at any end ( a diamet] 9 p]p5dicul> =me1 & d make a "r angle ) ! diamet]1 t "r l9e %all light )\t ! circle1 & yet s io9tly knitte to x1 t x is n possible to draw any o!r "r l9e betw5e t saide l9e & ! circumf]5ce ( ! circle4 ,& ! angle t is made 9 ! semicircle is great] !n any %>pe angle t may be made ( "r l9es1 b ! o!r angle )\t1 is less] !n any t c be made ( "r l9es4 ,example4 ,illu/ration ,9 ?is circle ,a4,b4,c1 ! diamet] is ,a4,c1 ! p]p5dicul> l9e1 :i* make? a "r angle ) ! diamet]1 is ,c4,a1 :i*e l9e falle? )\t ! circle1 & yet ioyne? s exactly vnto x1 t x is n possible to draw an o!r "r l9e betw5e ! circumf]5ce ( ! circle & x1 :i*e ?yng is s pla9ly se5e ( ! eye1 t x ne$e? no f>!r decl>acion4 ,= eu]y man wil easily cons5t1 t betw5e ! crok$ l9e ,a4,f1 7:i*e is a p>te ( ! circumf]5ce ( ! circle7 & ,a4,e 7:i* is ! said p]p5dicul> l9e7 !re c none o!r l9e bee draw5 9 t place :]e !y make ! angle4 ,n[e = ! residue ( ! !oreme4 ,! angle ,d4,a4,b1 :i* is made 9 ! semicircle1 is great] !n anye %>pe angle t may bee made ( ryghte l9es4 & yet is x a %>pe angle also1 9 z mu* z x is less] !n a "r angle1 :i* is ! angle ,e4,a4,d1 & ! residue ( t "r angle1 :i* lie? )\t ! circle1 t is to saye1 ,e4,a4,b1 is less] !n any %>pe angle t c be made ( "r l9es also4 ,= z x 0 be=e reh]s$1 !re canne no "r l9e be draw5 to ! angle1 betw5e ! circumf]5ce & ! "r l9e ,e4,a4 ,!n mu/ x ne$es fol[1 t !re c be made no less] angle ( righte l9es4 ,& aga9e1 if !r canne be no less] !n ! "o1 !n do? x sone appe>1 t !re canne be no great] !n ! o!r1 = !y twoo doo make ! :ole "r angle1 s t if anye corn] c\lde be made great] !n ! "o p>te1 !n %\lde ! residue bee less] !n ! o!r p>te1 s t o!r bo! p>tes mu/e be false1 or els bo! graunt$ to be true4 ,! lxij4 ,!oreme4 ,if a "r l9e doo t\*e a circle1 & an o!r "r l9e draw5 frome ! c5tre ( ! circle to ! po9te :]e !y t\*e1 t l9e :i*e is draw5ne frome ! c5tre1 %all be a p]p5dicul> l9e to ! t\* l9e4 ,example4 ,illu/ration ,! circle is ,a4,b4,c1 & 8 c5tre is ,f4 ,! t\*e l9e is ,d4,e1 & ! po9t :] !y t\* is ,c4 ,n[ by reason t a "r l9e is draw5 frome ! c5tre ,f4 vnto ,c1 :i* is ! po9t ( ! t\*e1 !re=e sai? ! !oreme1 t ! sayde l9e ,f4,c1 mu/e ne$es bee a p]p5dicul> l9e vnto ! t\*e l9e ,d4,e4 ,! lxiij4 ,!oreme4 ,if a righte l9e doo t\*e a circle1 & an o!r "r l9e be draw5 f ! po9te ( _! t\*+e1 s t x doo make righte corn]s ) ! t\*e l9e1 !n %al ! c5tre ( ! circle bee 9 t same l9e1 s draw54 ,example4 ,illu/ration ,! circle is ,a4,b4,c1 & ! c5tre ( x is ,g4 ,! t\*e l9e is ,d4,c4,e1 & ! po9te :]e x t\*e?1 is ,c4 ,n[e x appe>e? manife/1 t if a righte l9e be draw5 f ! po9te :]e ! t\* l9e do? io9e ) ! circle1 & t ! said lyne doo make righte corn]s ) ! t\*e l9e1 !n mu/e x ne$es g by ! c5tre ( ! circle1 & !n consequ5tly x mu/ haue ! sayde c5tre 9 him4 ,= if ! saide l9e %\lde g beside ! c5tre1 z ,f4,c4 do?1 !n do! x n make righte angles ) ! t\*e l9e1 :i* 9 ! !oreme is suppos$4 ,! lxiiij4 ,!oreme4 ,if an angle be made on ! c5tre ( a circle1 & an o!r angle made on ! circumf]5ce ( ! same circle1 & _! gr\nde l9e be "o common portion ( ! circumf]5ce1 !n is ! angle on ! c5tre twise s great z ! o!r angle on ! circumf]5ce4 ,example4 ,illu/ration ,! circle is ,a4,b4,c4,d1 & 8 c5tre is ,e3 ! angle on ! c5tre is ,c4,e4,d1 & ! angle on ! circumf]5ce is ,c4,a4,d t _! comm5 gr\nd l9e1 is ,c4,f4,d4 ,n[ say ,i t ! angle ,c4,e4,d1 :i*e is on ! c5tre1 is twise s greate z ! angle ,c4,a4,d1 :i* is on ! circumf]5ce4 ,! lxv4 ,!oreme4 ,^? angles :i*e be made 9 "o cantle ( a circle1 mu/ ne$es be equal togi!r4 ,example4 ,be=e ,i decl>e ?is !oreme by example1 x %all bee ne$efull to decl>e1 :at is x to be vnd]/&e by ! wordes 9 ?is !oreme4 ,= ! s5t5ce canne n be kn[51 onles ! u]y mean+ ( ! wordes be fir/e vnd]/&4 ,!re=e :5 x speake? ( angles made 9 "o cantle ( a circle1 x is ?is to be vnd]/&1 t ! angle mu/e t\* ! circumf]5ce3 & ! l9es t doo inclose t angle1 mu/e be draw5 to ! extremities ( t l9e1 :i* make? ! cantle ( ! circle4 ,s t if any angle d n t\* ! circumf]5ce1 or if ! l9es t inclose t angle1 doo n ende 9 ! extremities ( ! corde l9e1 b ende o!r 9 "s o!r "p ( ! said corde1 or 9 ! circumf]5ce1 or t any "o ( !m d s e&e1 !n is n t angle accompt$ to be draw5 9 ! said cantle ( ! circle4 ,& ?is promis$1 n[e w ,i cumme to ! mean+e ( ! !oreme4 ,i sette =! a circle :i*e is ,a4,b4,c4,d1 & 8 c5tre ,e1 9 ?is circle ,i drawe a l9e ,d4,c1 :]eby !re ar made two cantels1 a m & a less]4 ,! less] is ,d4,e4,c1 & ! geat] is ,d4,a4,b4,c4 ,9 ?is great] cantle ,i drawe two angles1 ! fir/e is ,d4,a4,c1 & ! second is ,d4,b4,c :i* two angles by reason !y are made bo! 9 "o cantle ( a circle 7t is ! cantle ,d4,a4,b4,c7 !re=e are !y bo? equall4 ,n[ do? !re app]e an o!r triangle1 ^: angle lighte? on ! c5tre ( ! circle1 & t triangle is ,d4,e4,c1 ^: angle is d\ble to ! o!r angles1 z is decl>$ 9 ! lxiiij4 ,!oreme1 :i*e maie /&e well en\gh ) ?is ,!oreme1 = x is n made 9 ?is cantle ( ! circle1 z ! o!r are1 by reason t 8 angle do? n light 9 ! circumf]5ce ( ! circle1 b on ! c5tre ( x4 ,illu/ration ,! 4lxvi4 !oreme4 ,eu]ie figure ( f\re sides1 draw5 9 a circle1 ha? 8 two contr>ie angles equall vnto two "r angles4 ,example4 ,illu/ration ,! circle is ,a4,b4,c4,d1 & ! figure ( f\re sides 9 x1 is made ( ! sides ,b4,c1 & ,c4,d1 & ,d4,a1 & ,a4,b4 ,n[ if y take any two angles t be contr>y1 z ! angle by ,a1 & ! angle by ,c1 ,i saie t ^? 4ij4 be equall to 4ij4 "r angles4 ,also if y take ! angle by ,b1 & ! angle by ,d1 :i*e two are also contray1 ^? two angles are l waies equall to two "r angles4 ,b if any man w take ! angle by ,a1 ) ! angle by ,b1 or ,d1 !y c n be accompt$ contr>y1 no m is n ! angle by ,c4 e/em$ contray to ! angle by ,b1 or yet to ! angle by ,d1 = !y onely be accompt$ contr>y angles1 :i*e haue no "o l9e common to !m bo!4 ,su*e is ! angle by ,a1 9 respect ( ! angle by ,c1 = !re bo? lynes be di/9ct1 :]e z ! angle by ,a1 & ! angle by ,d1 haue "o common l9e ,a4,d1 & !r=e c n be accompt$ contr>y angles1 ,s ! angle by ,d1 & ! angle by ,c1 haue ,d4,c1 z a common l9e1 & !re=e be n contr>y angles4 ,& ?is maie y iudge ( ! residewe1 by l reason4 ,! lxvij4 ,!oreme4 ,vpon "o "r lyne !re c n be made two cantles ( circles1 l & vnequall1 & draw5 t[>de "o p>te4 ,example4 ,illu/ration ,cantles ( circles be !n call$ l1 :5 ! angles t are made 9 !m be equall4 ,b n[ = ! ,!oreme1 let ! "r l9e be ,a4,e4,c1 on :i*e ,i draw a cantle ( a circle1 :i*e is ,a4,b4,c4 ,n[ saie? ! ,!oreme1 t x is n possible to draw an o!r cantle ( a circle1 :i*e %all be vnequall vnto ?is fir/ cantle1 t is to say1 o!r greatt] or less] !n x1 & yet be lyke x also1 t is to say1 t ! angle 9 ! "o %all be equall to ! angle 9 ! o!r4 ,= z 9 ?is example y see a less] cantle draw5 also1 t is ,a4,d4,c1 s if an angle 7 made 9 x1 t angle w\ld be greatt] !n ! angle made 9 ! cantle ,a4,b4,c1 & !r=e c n !y be call$ lyke cantels1 b & if any o!r cantle 7 made great] !n ! fir/1 !n w\ld ! angle 9 x be less] !n t 9 ! fir/e1 & s no!r a less] no!r a great] cantle c be made vpon "o l9e ) an o!r1 b x w be vnlike to x also4 ,! 4lxviij4 ,!oreme4 ,lyke cantelles ( circles made on equal righte lynes1 are equall toge!r4 ,example4 ,:at is m5t by l cantles y haue he>d be=e4 & x is easie to vnd]/&1 t su*e figures a call$ equall1 t be ( "o bygnesse1 s t ! "o is no!r great] no!r less] !n ! o!r4 ,& 9 ?is k9de ( comp>ison1 !y mu/ s agree1 t if ! "o be lay$ on ! o!r1 !y %all exactly agree 9 all _! b\ndes1 s t no!r %all exc$e o!r4 ,illu/ration ,n[e = ! example ( ! ,!oreme1 ,i haue set =! diu]s v>ieties ( cantles ( circles1 amonge/ :i* ! fir/ & seconde are made vpon equall l9es1 & ar also bo? equall & l4 ,! ?ird c\ple ar ioyn$ 9 "o1 & be no!r equall1 no!r l1 b expressyng an absurde de=mitee1 :i*e w\ld fol[e if ?is ,!oreme w] n true4 ,& s 9 ! f\r? c\ple y maie see1 t because !y are n equall cantles1 !r=e c n !y be l cantles1 = necess>ily x goe? toge!r1 t all cantles ( circles made vpon equall "r l9es1 if !y be l !y mu/ be equall also4 ,! lxix4 ,!oreme4 ,9 equall circles1 su*e angles z be equall are made vpon equall ar* l9es ( ! circumf]5ce1 :e!r ! angle light on ! circumf]5ce1 or on ! c5tre4 ,example4 ,illu/ration ,fir/e ,i haue sette = an exaumple twoo equall circles1 t is ,a4,b4,c4,d1 ^: c5tre is ,k1 & ! second circle ,e4,f4,g4,h1 & 8 c5tre ,l1 & 9 e*e ( !m is !re made two angles1 "o on ! circumf]5ce1 & ! o!r on ! c5tre ( e*e circle1 & !y be all made on two equall ar*e l9es1 t is ,b4,c4,d4 ! "o1 & ,f4,g4,h4 ! o!r4 ,n[ saie? ! ,!oreme1 t if ! angle ,b4,a4,d1 be equall to ! angle ,f4,e4,h1 !n are !y made 9 equall circles1 & on equall ar* l9es ( _! circumf]5ce4 ,also if ! angle ,b4,k4,d1 be equal to ! angle ,f4,l4,h1 !n be !y made on ! c5tres ( equall circles1 & on equall ar*e l9es1 s t y mu/e comp>e ^? angles toge!r1 :i*e are made bo? on ! c5tres1 or bo? on ! circumf]5ce1 & maie n conf]re ^? angles1 :]( "o is draw5 on ! circumf]5ce1 & ! o!r on ! c5tre4 ,= eu]more ! angle on ! c5tre 9 su*e sorte %all be d\ble to ! angle on ! circumf]5ce1 z is decl>$ 9 ! ?ree score & f\re ,!oreme4 ,! 4lxx4 ,!oreme4 ,9 equall circles1 ^? angles :i*e bee made on equall ar*e lynes1 are eu] equall toge!r1 :e!r !y be made on ! c5tre1 or on ! circumf]5ce4 ,example4 ,?is ,!oreme do? b conu]t ! s5t5ce ( ! la/ ,!oreme be=e1 & !r=e is to be vnd]/&e by ! same examples1 = z t sai?1 t equall angles occupie equall ar*elynes1 s ?is sai?1 t equal ar*e l9es cause? equal angles1 consid]+e all o!r circum/ances1 z 0 taughte 9 ! la/e !oreme be=e1 s t ?is !oreme dooe? affirm+ speake ( ! equalitie ( ^? angles1 ( :i* ! la/e !oreme spake conditionally4 ,& :]e ! la/e !oreme spake affirmatiuely ( ! ar*e l9es1 ?is !oreme speake? conditionally ( !m1 z ?us3 ,if ! ar*e l9e ,b4,c4,d4 be equall to ! o!r ar*e l9e ,f4,g4,h1 !n is t angle ,b4,a4,d4 equall to ! o!r angle ,f4,e4,h4 ,or els ?us may y decl>e x causally3 ,bicause ! ar*e l9e ,b4,c4,d1 is equal to ! o!r ar*e l9e ,f4,g4,h1 !re=e is ! angle ,b4,k4,d4 equall to ! angle ,f4,l4,h1 consid]+e t !y are made on ! c5tres ( equall circles4 ,& s ( ! o!r angles1 bicause ^? two ar*e l9es a=esaid ar equal1 !r=e ! angle ,d4,a4,b1 is equall to ! angle ,f4,e4,h1 = z mu*e z !y are made on ^? equall ar*e l9es1 & also on ! circumf]5ce ( equall circles4 ,& ?us ^! !oremes doo "o decl>e an o!r1 & "o v]ifie ! o!r4 ,! lxxi4 ,!oreme4 ,9 equal circles1 equall "r l9es be+e draw51 doo cutte awaye equalle ar*e l9es frome _! circumf]5ces1 s t ! great] ar*e l9e ( ! "o is equall to ! great] ar*e l9e ( ! o!r1 & ! less] to ! less]4 ,example4 ,illu/ration ,! circle ,a4,b4,c4,d1 is made equall to ! circle ,e4,f4,g4,h1 & ! "r l9e ,b4,d4 is equal to ! righte l9e ,f4,h1 :]=e x fol[e?1 t ! ij4 ar*e l9es ( ! circle ,a4,b4,d1 :i*e are cut f 8 circumf]5ce by ! "r l9e ,b4,d1 are equall to two o!r ar*e l9es ( ! circle ,e4,f4,h1 be+ cutte frome 8 circumf]5ce1 by ! "r l9e ,f4,h4 t is to saye1 t ! ar*e l9e ,b4,a4,d1 be+e ! great] ar* l9e ( ! fir/e circle1 is equall to ! ar*e l9e ,f4,e4,h1 beynge ! great] ar*e l9e ( ! o!r circle4 ,& s 9 l mann] ! less] ar*e l9e ( ! fir/e circle1 beynge ,b4,c4,d1 is equal to ! less] ar*e l9e ( ! seconde circle1 t is ,f4,g4,h4 ,! lxxij4 ,!oreme4 ,9 equall circles1 vnd] equall ar*e l9es ! "r l9es t bee draw5 are equall togi!r4 ,example4 ,?is ,!oreme is none o!r1 b ! conu]sion ( ! la/e ,!oreme bee=e1 & !re=e ne$e? none o!r example4 ,= z t did decl>e ! equalitie ( ! ar*e l9es1 by ! equalitie ( ! righte l9es1 s do! ?is ,!oreme decl>e ! equalnes ( ! "r l9es to ensue ( ! equalnes ( ! ar*e l9es1 & !re=e decl>e? t "r lyne ,b4,d1 to be equal to ! o!r "r l9e ,f4,h1 bicause !y bo? are draw5 vnd] equall ar*e l9es1 t is to saye1 ! "o vnd] ,b4,a4,d1 & ?o!r vnd] ,f4,e4,h1 & ^? two ar* l9es are e/im$ equall by ! !oreme la/e be=e1 & %al be pr\$ 9 ! booke ( pro(es4 ,! lxxiij4 ,!oreme4 ,9 eu]y circle1 ! angle t is made 9 ! halfe circle1 is a iu/e righte angle1 & ! angle t is made 9 a cantle great] !n ! halfe circle1 is less] ?anne a righte angle1 b t angle t is made 9 a cantle1 less] !n ! halfe circle1 is greatt] !n a "r angle4 ,& more\] ! angle ( ! great] cantle is great] !n a righte angle & ! angle ( ! less] cantle is less] !n a "r angle4 ,example4 ,9 ?is proposition1 x %al be meete to note1 t !re is a greate diu]site betw5e an angle ( a cantle1 & an angle made 9 a cantle1 & also betw5e ! angle ( a semicircle1 & ye angle made 9 a semicircle4 ,also x is meet to note yt al angles t be made 9 ye "p ( a circle1 ar made o!r 9 a semicircle1 7:i* is ! iu/e half circle7 or els 9 a cantle ( ! circle1 :i* cantle is o!r great] or less] !n ! semicircle is1 z 9 ?is figure annex$ y maye p]ceaue eu]ye "o ( ! ?+es seu]allye4 ,illu/ration ,fir/e ! circle is1 z y see1 ,a4,b4,c4,d1 & 8 c5tre ,e1 8 diamet] is ,a4,d1 ,!n is !r a l9e draw5 f ,a4 to ,b1 & s =? vnto ,f1 :i* is )\t ! circle3 & an o!r l9e also frome ,b4 to ,d1 :i*e make? two cantles ( ! :ole circle4 ,! great] cantle is ,d4,a4,b1 & ! less] cantle is ,b4,c4,d1 ,9 :i*e less] cantle also !re are two l9es t make an angle1 ! "o l9e is ,b4,c1 & ! o!r l9e is ,c4,d4 ,n[ to %[e ! diff]5ce ( an angle 9 a cantle1 & an angle ( a cantle1 fir/ = an example ,i take ! gret] cantle ,b4,a4,d1 9 :i* is b "o angle made1 & t is ! angle by ,a1 :i* is made ( a l9e ,a4,b1 & ! l9e ,a4,d1 ,& ?is angle is !r=e call$ an angle 9 a cantle4 ,b n[ ! same cantle ha! two o!r angles1 :i* be call$ ! angles ( t cantle1 s ! twoo angles made ( ! righte l9e ,d4,b1 & ! ar*e l9e ,d4,a4,b1 are ! twoo angles ( ?is cantle1 :]e( ! "o is by ,d1 & ! o!r is by ,b4 ,:] y mu/ remembre1 t ! angle by ,d4 is made ( ! "r l9e ,b4,d1 & ! ar*e l9e ,d4,a4 ,& ?is angle is diuid$ by an o!r "r l9e ,a4,e4,d1 :i* 9 ?is case mu/ be omitt$ z no l9e4 ,also ! angle by ,b4 is made ( ! "r l9e ,d4,b1 & ( ! ar* l9e 4,b4,a1 al?\gh x be deuid$ ) ij4 o!r "r l9es1 ( w* ! "o is ! "r l9e ,b4,a1 ?o!r ! "r l9e ,b4,e1 yet 9 ?is case !y ar n to be consid]$4 ,& by ?is may y p]ceaue also :i* be ! angles ( ! less] cantle1 ! fir/ ( !m is made ( ye "r l9e ,b4,d1 ( ye ar* l9e ,b4,c1 ! second is made ( ! "r l9e 4,d4,b1 ( ! ar* l9e ,d4,c4 ,!n ar !r ij4 o!r l9es1 w* deuide ^? ij4 corn]s1 yt is ! l9e ,b4,c1 ! l9e ,c4,d1 w* ij4 l9es d meet 9 ! poynte ,c1 & !re make an angle1 :i*e is call$ an angle made 9 t less] cantle1 b yet is n any angle ( t cantle4 ,& s haue y he>d ! diff]5ce betwe5e an angle 9 a cantle1 & an angle ( a cantle4 ,& 9 lyke sorte %all y iudg ( ! angle made 9 a semicircle1 :i*e is di/9ct f ! angles ( ! semicircle4 ,= 9 ?is figure1 ! angles ( ! semicircle are ^? angles :i* be by ,a4 & ,d1 & be made ( ! "r l9e ,a4,d1 beeyng ! diamet]1 & ( ! halfe circumf]5ce ( ! circle1 b by ! angle made 9 ! semicircle is t angle by ,b1 :i*e is made ( ! righte l9e ,a4,b1 & t o!r "r l9e ,b4,d1 :i*e z !y mete 9 ! circumf]5ce1 & make an angle1 s !y ende ) _! o!r extremities at ! endes ( ! diamet]4 ,^! ?ynges premis$1 n[ saie ,i t\*yng ! ,!oreme1 t eu]ye angle t is made 9 a semicircle1 is a "r angle1 & if x be made 9 any cantle ( a circle1 !n mu/ x n$s be o!r a blunt angle1 or els a %>pe angle1 & 9 no wise a righte angle4 ,= if ! cantle :]e9 ! angle is made1 be great] !n ! halfe circle1 !n is t angle a %>pe angle4 ,& g5]ally ! great] ! cantle is1 ! less] is ! angle compris$ 9 t cantle3 & contr>y waies1 ! less] any cantle is1 ! great] is ! angle t is made 9 x4 ,:]=e x mu/ n$es fol[e1 t ! angle made 9 a cantle lesse !n a semicircle1 mu/ n$es be great] !n a "r angle4 ,s ! angle by ,b1 beyng made at ! "r l9e ,a4,b1 & ! righte l9e ,b4,d1 is a iu/e righte angle1 because x is made 9 a semicircle4 ,b ! angle made by ,a1 :i* is made ( ! "r l9e ,a4,b1 & ( ! "r l9e ,a4,d1 is less] !n a righte angle1 & is nam$ a %>pe angle1 = z mu*e z x is made 9 a cantle ( a circle1 great] !n a semicircle4 ,& contr>y waies1 ! angle by ,c1 beyng made ( ! righte l9e ,b4,c1 & ( ! "r l9e ,c4,d1 is great] !n a "r angle1 & is nam$ a blunte angle1 because x is made 9 a cantle ( a circle1 less] !n a semicircle4 ,b n[ t\*yng ! o!r angles ( ! cantles1 ,i saie accordyng to ! ,!oreme1 t ! 4ij4 angles ( ! great] cantle1 :i* are by ,b4 & ,d1 z is be=e decl>$1 are greatt] e*e ( !m !n a "r angle4 ,& ! angles ( ! less] cantle1 :i*e are by ! same lett]s ,b1 & ,d1 b be on ! o!r side ( ! corde1 are less] e*e ( !m !n a "r angle1 & be !r=e %>pe corn]s4 ,! lxxiiij4 ,!oreme4 ,if a "r l9e d t\*e a circle1 & f ! po9te :]e !y t\*e1 a righte lyne be draw5 crosse ! circle1 & deuide x1 ! angles t ! sai$ lyne dooe? make ) ! t\*e l9e1 are equall to ! angles :i*e are made 9 ! cantles ( ! same circle1 on ! contr>ie sides ( ! lyne a=esaid4 ,example4 ,illu/ration ,! circle is ,a4,b4,c4,d1 & ! t\*e l9e is ,e4,f4 ,! po9te ( ! t\*yng is ,d1 f :i* po9t ,i suppose ! l9e ,d4,b1 to be draw5 crosse ! circle1 & to diuide x into 4ij4 cantles1 :]( ! great] is ,b4,a4,d1 & ! less] is ,b4,c4,d1 & 9 e* ( !m an angle is draw51 = 9 ! great] cantle ! angle is by ,a1 & is made ( ! "r l9es ,b4,a1 & ,a4,d1 9 ! less] cantle ! angle is by ,c1 & is made ( ye "r l9es ,b4,c1 & ,c4,d4 ,n[ sai? ! ,!oreme t ! angle ,b4,d4,f1 is equall to ! angle made 9 ! cantle on ! o!r side ( ! said l9e1 t is to saie1 9 ! cantle ,b4,a4,d1 s t ! angle ,b4,d4,f1 is equall to ! angle ,b4,a4,d1 because ! angle ,b4,d4,f1 is on ! "o side ( ! l9e ,b4,d1 7:i*e is accord+ to ! supposition ( ! ,!oreme draw5 crosse ! circle7 & ! angle ,b4,a4,d1 is 9 ! cantle on ! o!r side4 ,likewaies ! angle ,b4,d4,e1 beyng on ! "o side ( ! l9e ,b4,d1 mu/ be equall to ! angle ,b4,c4,d1 7t is ! angle by ,c17 :i*e is made 9 ! cantle on ! o!r side ( ! "r l9e ,b4,d4 ,! pr(e ( all ^! ,i d res]ue1 z ,i haue (t5 saide1 to a conu5i5t boke1 :]e9 !y %all be all set at l>ge4 ,! 4lxxv4 ,!oreme4 ,9 any circle :5 4ij4 "r l9es d crosse "o an o!r1 ! likeiamme t is made ( ! portions ( ! "o l9e1 %all be equall to ! lykeiamme made ( ! p>tes ( ! o!r lyne4 ,illu/ration ,because ?is ,!oreme do? s]ue to _m vses1 & wold be wel vnd]/&e1 ,i haue set =? 4ij4 examples ( x4 ,9 ! fir/e1 ! l9es by _! crossyng d make _! portions some:at t[>d an equalitie4 ,9 ! second ! portions ( ! lynes be v f> f an equalitie1 & yet 9 bo! ^! & 9 all o!r ye ,!oreme is true4 ,9 ! fir/ example ! circle is ,a4,b4,c4,d1 9 :i* ?one l9e ,a4,c1 do? crosse ?o!r l9e ,b4,d1 9 ye po9t ,e4 ,n[ if y d make "o likeiamme or longsqu>e ( ,d4,e1 ,e4,b1 be+ ye 4ij4 portions ( ! l9e ,d4,b1 t longsqu>e %all be equall to ! o!r longsqu>e made ( ,a4,e1 & ,e4,c1 beyng ! portions ( ! o!r l9e ,a4,c4 ,lykewaies 9 ! second example1 ! circle is ,f4,g4,h4,k1 9 :i*e ! l9e ,f4,h1 do? crosse ! o!r l9e ,g4,k1 9 ! po9te ,l4 ,:]=e if y make a lykeiamme or longsqu>e ( ! two p>tes ( ! l9e ,f4,h1 t is to saye1 ( ,f4,l1 & ,l4,h1 t longsqu>e w be equall to an o!r longsqu>e made ( ! two p>tes ( ! l9e ,g4,k4 :i* p>tes are ,g4,l1 & ,l4,k4 ,^? longsqu>es haue ,i set foor? vnd] ! circles conta9yng _! sides1 t y maie some:at :et y\r own wit 9 practisyng ?is ,!oreme1 accordyng to ! doctr9e ( ! n9ete5? conclusion4 ,! 4lxxvi4 ,!oreme4 ,if a po9te be m>k$ )\t a circle1 & f t po9te two "r l9es draw5 to ! circle1 s t ! "o ( !m doe runne crosse ! circle1 & ! o!r doe t\*e ! circle onely1 ! long squ>e t is made ( t :ole lyne :i* crosse? ! circle1 & ! portion ( x1 t lye? betw5e ! vtt] circumf]5ce ( ! circle & ! po9te1 %all be equall to ! full squ>e ( ! o!r lyne1 t onely t\*e? ! circle4 ,example4 ,! circle is ,d4,b4,c1 & ! po9te )\t ! circle is ,a1 f :i*e po9te !re is draw5 "o l9e crosse ! circle1 & t is ,a4,d4,c1 & an o!r lyne is drawn f ! said pricke to ! m>ge or edge ( ! circumf]5ce ( ! circle1 & doe? only t\*e x1 t is ! l9e ,a4,b4 ,& ( t fir/ l9e ,a4,d4,c1 y maie p]ceiue "o "p ( x1 :i*e is ,a4,d1 to lie )\t ! circle1 betwe5e ! vtt] circumf]5ce ( x1 & ! po9te assign$1 :i*e 0 ,a4 ,n[e conc]nyng ! meanyng ( ! ,!oreme1 if y make a longsqu>e ( ! :ole l9e ,a4,c1 & ( t p>te ( x t lye? betw5e ! circumf]5ce & ! po9t1 7:i*e is ,a4,d17 t longesqu>e %all be equall to ! full squ>e ( ! t\*e l9e ,a4,b1 accordyng n onely z ?is figure %ewe?1 b also ! sai$ nynete5? conclusion dooe? pr\e1 if y ly/e to examyne ! "o by ! o!r4 ,illu/ration ,! lxxvii4 ,!oreme4 ,if a po9te be assign$ )\t a circle1 & f t po9te 4ij4 "r lynes be draw5 to ! circle1 s t ! "o doe crosse ! circle1 & ! o!r dooe ende at ! circumf]5ce1 & t ! longsqu>e ( ! l9e :i* crosse? ! circle made ) ! portion ( ! same l9e beyng )\t ! circle betwe5e ! vtt] circumf]5ce & ! po9te assign$1 doe equally agree ) ! iu/e squ>e ( t l9e t ende? at ! circumf]5ce1 !n is t lyne s endyng on ! circumf]5ce a t\*e l9e vnto t circle4 ,example4 ,9 z mu*e z ?is ,!oreme is no?yng els b ! s5t5ce ( ! la/ ,!oreme be=e conu]t$1 !r=e x %all n be n$efull to vse any o!r example !n ! same1 = z 9 t o!r ,!oreme because ! "o l9e is a t\*e lyne1 !r=e x make? a squ>e iu/ equal ) ! longsqu>e made ( t :ole l9e1 :i*e crosse? ! circle1 & 8 portion liyng )\t ! same circle4 ,s sai? ?is ,!oreme3 t if ! iu/ squ>e ( ! l9e t ende? on ! circumf]5ce1 be equall to t longsqu>e :i*e is made z = 8 long] sides ( ! :ole l9e1 :i* comme? f ! po9tt assign$1 & crosse? ! circle1 & = 8 o!r %ort] sides is made ( ! portion ( ! same l9e1 liyng betw5e ! circumf]5ce ( ! circle & ! po9te assign$1 !n is t l9e :i*e ende? on ! circumf]5ce a "r t\*e l9e1 t is to saie1 yf ! full squ>e ( ! "r l9e ,a4,b1 be equall to ! longsqu>e made ( ! :ole l9e ,a4,c1 z "o ( 8 l9es1 & ( 8 portion ,a4,d1 z 8 o!r l9e1 !n mu/ x n$es be1 t ! lyne ,a4,b1 is a "r t\*e lyne vnto ! circle ,d4,b4,c4 ,& ?us = ?is tyme ,i make an ende ( ! ,!oremes4 ,,f9is1 ,,impr9t$ at ,london 9 ,p\les *ur*ey>de1 at ! signe ( ! ,bra- s5s]p5t1 by ,reynold ,wolfe4 ,cum priuilegio ad imprim5- dum solum4 ,,anno ,,dom9i 4,m4,d4,l4,i4 ,illu/ration3 ,publi%]'s ,device 7braz5 s]p5t1 ,,nvm4 ,,xxi47 99 99 99 99 99 99 99 99 99 ,errors & ,inconsi/5cies3 ,unless o!rwise not$1 spell+1 punctuation & capitalization are un*ang$4 ,=ms 7 regul>iz$ only :]e !re 0 a v l>ge disp>ity betwe5 ! expect$ =m & ! app>5t errors 7= example1 a ?\s& ,a4,b aga9/ a doz5 ,a1,b71 or a flagrant mispr9t su* z cnt = cut4 ,! lett]s u & v foll[ ! conv5tional initial v1 non-initial u patt]n except 9 numb]s 7xv1 iv74 ,! l[]-case j =m occurs only z ! la/ digit ( a numb] 7ij1 xxj72 upp]-case ,i & ,j %>e a =m1 always read z ,i4 ,capital & l[]-case w 7 (t5 us$ int]*angeably4 ,words split across l9e breaks may or may n h a hyph54 ,language3 ,! ^w o!r is us$ int]*angeably ) bo? or & ei!r2 simil>ly1 no!r is us$ 9 place ( nor & nei!r4 ,! expression an o!r is almo/ always writt5 z two words4 ,! spell+s !n7ne7 & ?an7ne7 are us$ int]*angeably2 ?an7ne7 is r>e4 ,! spell+ liyng1 bo? by itself & z ! end ( a long] ^w1 is us$ consi/5tly4 ,illu/rations3 ,a numb] ( illu/rations conta9 errors su* z unm>k$ or mislabel$ po9ts 7circle ,b4,c4,d :]e only ,c & ,d are label$74 ,errors ( ?is type are id5tifi$ 9 ! illu/rat$ ,,html v]sion1 b n 9 ! pres5t text-only file4 ,greek3 ,all ,greek is %[n & translit]at$ z pr9t$4 ,errors or anomalies include miss+1 misplac$ or incorrect diacritics2 ! non-f9al =m ( sigma us$ at ! end ( words2 & ^w-f9al m = n4 ,mean+ful ,errors & ,anomalies3 %al be clean extrirp$ & root$ out spell+ un*ang$3 int5d$ =m n c]ta9 new & new causes to pray = y\r maie/ie1 p]ceiuyng text new & new causes probably int5tional ,preface ,& ?us = ?is tyme ,i make an end444 ,body text ,! def9itions ( ! pr9ciples ( ,,geometry4 ,pag9ation z %[n by signature numb]s dem&s ano!r leaf 7two pages7 betwe5 ! end ( ! ,preface & ! beg9n+ ( ! body text4 ,b no text is miss+1 & ! facsimile has no blank pages4 ,o!rlesse !n x z y se ,d text un*ang$3 int5d$ word+ unc]ta9 ,,! 4,,xxii4 ,,conclvsion4 4,,xxi4 ,,! ,,xxxviii4 ,,conclvsion4 ,,xxxvii4 ,! xxvi4 ,!oreme4 ,!re is no xxv4 7#be7 !orem4 ,! 4xxxix4 !oreme4 ,! text ( ! ,example is g>bl$1 & does n fit ! illu/ration4 ,among o!r problems1 po9ts ,c & ,d seem to h be5 swit*$1 ei!r 9 ! text or 9 ! illu/ration4 ,! circle is ,a4,b4,c4,d1 & 8 c5tre is ,e3 ! angle on ! c5tre is ,c4,e4,d1 & ! angle on ! circumf]5ce is ,c4,a4,d t _! comm5 gr\nd l9e1 is ,c4,f4,d4 cirle pr9t+ ( ,c4,e4,d uncle>3 looks l ,c4,f4,d b c5t] ( circle is ,e lone t may be error = amp]s& or o!r punctuation ,c4,f4,d1 ,mispr9ts3 ,ari/otle _h putte =! c]ta9e bookes kookes ,& 8 fa!r k+ ,dauid ioyne? ,danid ^? ?ynges are done by negromancy4 ,& h]e( came x t fry] ,bakon 0 accompt$ s greate a negromanci] spell+ un*ang$ ,= vnd\bt$ly if !y mean vnd\dt$ly if ! t ,godde 0 alwaies work+e alwaaies all ! l9es t be draw5 to ! circumf]5ce circumf]nece ,an o!r ha? b "o compass$ ha?t ^: sydes p>tlye are all equall z 9 ,a eqnall z ?is ,a4 do? p>tly expresse ,a do? p>tly eppresse ,to make a ?relike triangle on any lyne measurable4 or any & x %all cut ! fir/ l9e 9 two equall portions4 cnt !n op5 ,i ! compass z wide z 4iiij4 p>tes compaas ,to make a plumbe lyne on any porcion ( a circle or any = eu]y triangle toge!r an equal likeiamme text has to_/ge!r at l9e break3 may be meant = two words z y mai se by ,c & ,d1 = ij4 sides ( bo? ! trianngles ar p>allels4 y maise 444 triangls !n w ! squ>e ( t great] portion portior t l9e ,i say is a t\*e l9e t\! 7t is to saie ,m4 ,g17 error = 9 ,g8 "o p]p5dicul> f ,g4 vnto ! side side ,b4,c ,a4,c !n mu/ ,i fir/ draw a tu*e lyne wu/ anye ( ! two lynes contayn+e ! angle appo9t$4 nye ( draw !nce two l9es1 "o to ,d "o to ,a :i* is made accord+e to ! conclusion4 ancord+e y %al p]ceaue t !re w be peceaue a _/ m conu5i_/ent "t4 text has conui_/ent at l9e break o!r 9 decid+ "s contr\]sy decis+ a great deale ! son] somn] ,:i*e example ha? be5e vs$e hat :an ,i wrote ^! fir/ connclusions cunclusions su*e bokes z ar appoynt$ z at ,w ,i refuse4 ,willl all "r angles be equall eqnall :i*e ?+e ! bett] to p]ceaue peceaue :i*e ,i d only by examples decl>e declae ab\t ! gr\nd l9e1 are equal togi!r gr\d y %al take ?is triangle ,a4,b4,c4 :i* ha? a v blunt corn] ^w ?is illegible ve]y blunt ! "o is a longe squ>e ,a4,b4,e pr9t$ z %[n3 expect$ =m is ,a4,b4,f4,e ?\gh !y be diu]s 9 numbre4 numbhe pro(e t ,g4,h4 be+ ! gr\nd l9e gr\d & /&+ betw5e "o paire ( p>allels an = !i ar ! two ye contr>y sides ^w ye sup]flu\s8 !y haue "o gr\nd l9e ,d4,e4 on ,by ! squ>e ( any lyne sqn>e squ>es t are made ( ! same l9e sane ,! fyr/ lyne prop\nd$ is ,a4,b prop\n$ ha? ano!r longe squ>e equall to hym text has anomal\s a _/ no!r at l9e break "o ( ^? p>ts aga9 into o!r ij4 p>ts iuto :i* ?+ ! easy] is to be vnd]/&e eays] & blunt corn]$ triangles c\]$ ! squ>e m>k$ ) ,g4 is ! squ>e ( ,a4,b ) ,c4 ,& f all po9tes y maye drawe ij4 equall l9es poittes ,if two circles bee draw5 s "o )\te an o!r & o!r draw5 frome ! c5tre ( ! circle to ! po9te tge :i* 9 ! !oreme is suppos$4 ! 4heoreme & !r=e c n !y be call$ lyke cantels ban !n w\ld ! angle 9 x be less] x be+ cutte frome 8 circumf]5ce1 by ! "r l9e ,f4,h4 circum=5ce ,& 9 lyke sorte %all y iudg ,& n lyke ,punctuation1 ,spac+1 ,capitalization3 ,phrase breaks :]e a comma is foll[$ by a capital lett]1 or a p]iod by a l[]-case lett]1 are n individually not$4 ,numb] =ms su* z ^?4 ij4 la/ or l9e4 ,a4,b4 7 sil5tly regul>iz$ to ^? 4ij4 la/ & l9e 4,a4,b4 ,miss+ s5t5ce-f9al p]iods at ! end ( a pr9t$ l9e 7 sil5tly suppli$4 ,initial u or m$ial v un*ang$3 ioyne? u]tu\s conu]sacion ,,xxv4 ,,conclvsion4444 ! :ole circle agreynge !revnto4 or e&e 9 ! utt] edge ( 8 circumf]5ce onles ! u]y mean+ ( ! wordes be fir/e vnd]/& ,geometry tea*e? ! drawyng1 ,measur+ & proporcion capital ,m 9 orig9al ,& h]( comme? t seconnde ?+ :]9 al agree t seconnde 4 ?+ if !y c ) _! wys$ome ou]come all vyces4 ,( ! fir/e ( ^? ?ree sortes text reads 444 _! rwys$ome 444 _/ 444 ? ee sortes 444 on consecutive l9es4 ,! extrane\s r is directly above ! miss+ or invisible r akroamatikoi4 f9al 4 miss+ 0 he e/em$ = 8 wisdom4 f9al 4 miss+ wis$om is bett] !n preti\s /ones 4 yea punctuation un*ang$ y\r ,maie/ies excell5cye1 excell5cye11 if 8 subiectes be ri*e 9 sub/aunce1 sub /aunce new & new causes to pray = y\r maie/ie1 p]ceiuyng maie/ie1p]ceiuyng c be ignorant !re(1 9 s mu* t !re(4 9 ^? ?ynges are done by negromancy4 spell+ un*ang$ ?ogh x be b smal1 & !=e n notable4 f9al 4 miss+ b b[e? any waye1 su* are call$ waye4 su* n call x "o crok$ lyne1 b r cr ok$ ,n[ to geue y example ( triangles1 triangles11 a portion ( a globe z ! figur m>k$ wt ,a4 f9al 4 miss+ z is ! figure ,g4 o!r ij4 %>p & "o blunt ^! examples 4,m4 ,n1 & ,o4 :]e ,m4 ha? a "r angle1 ,n1 a blunte angle 4,m4,n1 & ,o :]e 444 ,n1 ,a1 blunte z 9 ?is example because ,a4,b1 is draw5 9 l5g? ,a1,b1 ! angle ,c1 is call$ an vtt] angle4 f9al 4 miss+ diamondlike1 ^: figur is not$ ) ,t4 f9al 4 miss+ & p>tlye vnequall1 z 9 ,b1 & !y 91 ,b & z 9 ?is figure is "s :at express$4 f9al 4 miss+ !n draw a l9e f ,b1 to ,f1 s ,i haue m9e int5t4 f9al 4 miss+ !n op5 ,i ! compasse z wyde z ,a4,c1 ,a1,c1 ,f1 ?is space betw5e ,d4 & ,f4 betw5e ,d & ,f4 crosse t o!r fir/ l9e 9 4ij4 places4 f9al 4 miss+ ,! ar* to be diuid$ ys ,a4,d4,c1 ! corde is ,a4,b4,c1 ,a1,b4,c1 & !r set a m>k4 ,!n take a long l9e m>k ,!n a p>allel mu/ be draw5 h[e y %all doo x4 doo x1 equal to ,a4,b1 z = example ,d4,f4 ,d1,f4 accordyng to ! fif? conclusion accordyngt o 7accordyng z ! 4viij4 conclusion tea*e?7 comma = close p>5!sis x ha? "o angle 7t is ,b4,a4,e47 l to ,d close p>5!sis miss+ !re=e ,i wyl by example sette =? !r e=e !nne drawe ,i a l9e also f ,n4 vnto ,c4 vnto ,c is equall vnto ,a4,b1 & s is ,k4,l4 ,k4,l1 y mai se by ,c4 & ,d1 = ij4 sides by ,c & ,d1 accordyng to ! l5g! lo ! peece t rema9e? error = ( !8 ? e peece ?is l9e ,a4,b1 7w* 0 assign$ vnto me7 ,a1,b1 ?us haue y atta9$ ye vse ( ?is conclusion4 f9al 4 miss+ ,make 9 a table ! l draught ,a table vs+ x z x w] a crok$ rul]4 f9al 4 miss+ ! l9es ( diuision ,a4,d1 ,b4,e ,a4 ,d4 ,b4 ,e ^: two sides ,a4,c4 & ,b4,c4 are diuid$ ,b1,c4 enclose ^? iij prickes4 t c5tre z y se is ,d prickes t c5tre 4 z z y se ,b4,a4,d1 & ,b4,c4,e4 ,b4,c1,e4 at ?is tyme wyll %ewe y %e1we z y\re selfe mai easily gesse4 f9al 4 miss+ ,a4,c4 & ,b4,d4 are ! two diamet]s ,a4,c4 & ,b ,d4 ,a4,b4,c4,d4 is ! quadrate appo9t$ ,a1,b4,c4,d4 quadrate f angle to angle1 z y se ,a4,c4 ,b ,d4 ,a4,c4 ,b ,d4 ,a4,b4,c4 is ! circle1 :i*e ,i haue deuid$ ,a1,b4,c4 ,& f e*e pricke ij4 l9es draw5 l9e sdraw5 ,i w\ld make a circle4 ,!r=e ,i drawe circle1 ,!r=e ( equall sides & equall angles4 angles1 p]ceaue by ! xxxvij4 xxxviij4 xxxix4 & xl4 conclusions xxxviij xxxix4 t be nighe/ t be 4 nighe/ le>ne ! demon/rations by h>te1 7z somme miss+ op5 p>5!sis sundrye woorkes p>tely end$1 & p>tely to bee end$ ,& ,z = example ,a--------,b4 ,a4 be+ ! "o pricke ,a be+ t corn] :i* is greatt] !n a "r angle s 9 is invisible fir/e ^! two "r l9es ,a4,b4 & ,a4,c4 ,a ,b4 & ,a4,c4 ij4 quantities1 z ,a4 & ,b1 be equal to an o!r z ,a & ,b1 9 an o!r place4 ,9 ! mean season 9 draw5 =! vnto ,d4 & ,e4 vnto ,d & ,e4 ,! ?irde ,!oreme4 ,example4 f9al 4 miss+ & yet ! ij4 less] sides togi!r ar great] !n x4 text has 444 yet ?r 444 ar greate at consecutive l9e-ends ! angle ,c4 7:i*e are ! ij4 angles contayn$ miss+ op5 p>5!sis ,m4,n4 equall also to ,h4,l4 ,h1,l4 !re=e are ,a4,c4 & ,b4,d4 bo! equall ,a1,c4 ,a4,d4,e1 & ,d4,e4,b1 :i* 7z ! xxvij4 miss+ op5 p>5!sis ,! xxxi4 !oreme4 xxxi1 !oreme4 !re is made a triangle ,b4,c4,g1 & a lykeiamme ,b4,c ,g1 fyll vp ! sydes ( ! 4ij4 fyr/e squ>e lykeiammes 4ij fyr/e equall to ! 4ij4 squ>es ( bo? ! o!r sides4 f9al 4 miss+ ,! ij4 l9es propos$ ar ,a4,b4 & ,c4,d ,a ,b4 & ,c4,d = ! squ>e ,d4,e4,f4,g4 is equal to ! two o!r p>tial squ>es ( ,d4,h4,k4,g & ,h4,e4,f4,k1 ,d1,e4,f4,g4 444 ,d4,h4,k ,g ,! xxxvij4 ,!oreme4 xxxvij ,!oreme4 ! squ>e t is made ( ! :ole l9e fir/ t 9 t invisible 7:i* is equall ) ,d4,g47 ,d1,g4 ,lette ! diuid$ l9e bee ,a4,b1 & p>t$ 9 ,c ,a1,b1 ! squ>e ( ! :ole lyne ,a4,b1 ,a1,b1 z h]aft] ,i w decl>e 9 conu5i5t place4 f9al 4 miss+ two less] squ>es beyng tak5 away1 close p>5!sis = comma ! great squ>e1 & t is ,g4,f4,m4,h4 ,g1,f4,m4,h4 two vnequall p>tes z happ5e?4 ,! long squ>e ! ,! 4xlvi4 ,!oreme4 ,!4xlvi4 ,! xlvij4 ,!oreme4 xlvij ,!oreme4 & doo n passe by ! c5tre passeby ,= z y may easily iudge1 ,a4,c4 ha? "o portion ,a ,c4 if !y be equally di/aunt f "o halfe ( ! diamet] second l 9 equally invisible !n x1 & beynge f>!r ( x4 & ! second circle is ,b4,c4,d4,e1 & !y crosse ,b4,c4,d1,e1 ,! second circle is ,d4,b4,c1 & 8 c5tre is ,h ,d1,b4,c1 t is f>!r f ! c5tre4 ,! f\r? c5tre1 ,! twise s great z ! o!r angle on ! circumf]5ce4 f9al 4 miss+ ,! less] is ,d4,e4,c1 & ! geat] is ,d4,a4,b4,c4 ,d4,f4,c1 !re=e are !y bo? equall4 f9al 4 miss+ ,:at is m5t by l cantles y haue he>d be=e m5tby by ! equalitie ( ! righte l9es1 s do! ?is ,!oreme s d ! ?is ,also x is meet to note yt al angles t be made ytal :i*e make? two cantles ( ! :ole circle4 circle1 :i* is made ( a l9e ,a4,b1 & ! l9e ,a4,d1 ,a1,b1 ,i saie accordyng to ! ,!oreme1 t ! 4ij4 angles 4ij angles s t ! angle ,b4,d4,f1 is equall to ! angle ,b4,a4,d ,b ,d4,f1 make _! portions some:at t[>d an equalitie4 f9al 4 miss+ do? crosse ?o!r l9e ,b4,d1 9 ye po9t ,e4 ,b ,d1 :i*e 0 ,a4 ,n[e conc]nyng ! meanyng ,no we