,transcrib]'s ,note3 ,?is e-text includes *>act]s t w only display 9 ,,utf-#h 7,unicode7 text read]s1 includ+ a few words ( ,greek3 ,taktik ov]l9e or tilde to %[ foll[+ -n or -m leaf symbol2 po9t+-f+] symbol d\ble-end$ dagg]1 us$ 9 size notations 7bel[7 ,if any ( ^! *>act]s d n display prop]ly-- 9 p>ticul>1 if ! diacritic does n appe> directly above ! lett]-- or if ! quotation m>ks 9 ?is p>agraph appe> z g>bage1 make sure y\r text read]'s 8"* set0 or 8file encod+0 is set to ,unicode 7,,utf-#h74 ,y may also ne$ to *ange ! default font4 ,z a la/ resort1 use ! ,,ascii v]sion ( ?is file in/ead4 ,"s aspects ( ! orig9al book _h to be modifi$ = all v]sions ( ?is pla9-text file4 ,sup]script lett]s are %[n ) 3 ye1 yt4 ,m>g9al quotation m>ks are %[n inl9e z 8 & 01 approximat+ ! beg9n+ & end ( ! m>k$ passage4 ,9 ! orig9al text1 no quotation m>ks 7 pr9t$ inl9e4 ,p>agraphs are brok5 up = sid5otes1 ) blank l9es be=e & aft]4 ,orig9al p>agraph breaks are %[n z two blank l9es4 ,brackets )9 ! body text are 9 ! orig9al4 ,all sid5otes except ! "o beg9n+ 8,?is noble ,e>le0 7 pr9t$ 9 italics2 m>kup has be5 omitt$ to r$uce visual clutt]4 ,at lea/ f\r sizes ( text 7 us$1 (t5 9 comb9ation ) italics4 ,! v>iants are %[n "h z3 v l>ge l>g] small] ,fur!r errors & anomalies are li/$ at ! end ( ! text1 along ) ^? ,euclid citations id5tifi$ by numb]4 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,,! ,,elem5ts ,,( ,,geometrie ( ! mo/ aunci5t ,philosoph] ,,evclide ( ,meg>a4 ,fai?fully 7n[ fir/7 translat$ into ! ,engli%e t\ng1 by ,h4 ,bill+sley1 ,citiz5 ( ,london4 ,:]eunto are annex$ c]ta9e ,s*olies1 ,annotations1 & ,inu5tions1 ( ! be/ ,ma!matici5s1 bo? ( "t pa/1 & 9 ?is our age4 ,) a v fruitfull ,prface made by ,m4 ,i4 ,dee1 specify+ ! *iefe ,ma!maticall ,scices1 :at !y are1 & :]unto commodi\s3 :]e1 also1 are disclos$ c]ta9e new ,secrets ,ma!maticall & ,me*anicall1 vntill ^! our daies1 greatly miss$4 ,impr9t$ at ,london by ,iohn ,daye4 ,! ,translator to ! ,read]4 ,!re is 7g5tle ,read]7 no?+ 7! ^w ( ,god onely set ap>t7 :i* s mu* beautifie? & adorne? ! s\le & m9de ( m1 z do? ! k ( good artes & sci5ces3 z ! k ( naturall & morall ,philosophie4 ,! "o sette? be=e our eyes1 ! creatures ( ,god1 bo? 9 ! heau5s ab\e1 & 9 ! e>? b5ea?3 9 :i* z 9 a glasse1 we beholde ! exc$+ maie/ie & wis$ome ( ,god1 9 adorn+ & beautify+ !m z we see3 9 geu+ vnto !m su* wond]full & manifolde proprieties1 & naturall work+es1 & t s diu]sly & 9 su* v>ietie3 f>!r 9 ma9ta9+ & cons]u+ !m cont9ually1 :]eby to praise & adore him1 z by ,s4 ,paule we are taught4 ,! o!r tea*e? vs rules & preceptes ( v]tue1 h[1 9 common life amonge/ m51 we ought to walke vprightly3 :at dueties p]ta9e to our selues1 :at p]ta9e to ! g\]nm5t or good ord] bo? ( an h\%olde1 & also ( a citie or common weal?4 ,! read+ likewise ( hi/ories1 conduce? n a litle1 to ! adorn+ ( ! s\le m9de ( man1 a /udie ( all m5 cm5d$3 by x are se5e & kn[5 ! artes & do+es ( inf9ite wise m5 gone be=e vs4 ,9 hi/ories are conta9$ inf9ite examples ( h]oicall v]tues to be ( vs foll[$1 & horrible examples ( vices to be ( vs es*ew$4 ,_m o!r artes also !re are :i* beautifie ! m9de ( man3 b ( all o!r none d m g>ni%e beautifie x1 !n ^? artes :i* are call$ ,ma!maticall4 ,unto ! k ( :i* no man c atta9e1 )\t ! p]fecte k & in/ruction ( ! pr9ciples1 gr\ndes1 & ,elem5tes ( ,geometrie4 ,b p]fectly to be in/ruct$ 9 !m1 require? dilig5t /udie & read+ ( olde aunci5t au?ors4 ,amonge/ :i*1 none = a beg9n] is to be pref]r$ be=e ! mo/ aunci5t ,philosoph] ,euclide ( ,meg>a4 ,= ( all o!rs he ha? 9 a true me?ode & iu/e ord]1 ga!r$ toge!r :atsoeu] any be=e him _h ( ^! ,elem5tes writt53 inu5t+ also & add+ _m ?+es ( 8 owne3 :]by he ha? 9 due =me accompli%$ ! arte3 fir/ geu+ def9itions1 pr9ciples1 gr\ndes1 :]( he d$uce? 8 ,propositions or conclusions1 9 su* wond]full wise1 t t :i* goe? be=e1 is ( necessitie requir$ to ! pr\fe ( t :i* foll[e?4 ,s t )\t ! dilig5t /udie ( ,euclides ,elem5tes1 x is impossible to atta9e vnto ! p]fecte k ( ,geometrie1 & consequ5tly ( any ( ! o!r ,ma!maticall sci5ces4 ,:]e=e consid]+ ! want lacke ( su* good au?ors hi!rto 9 our ,engli%e t\nge1 lam5t+ also ! neglig5ce1 & lacke ( zeale to _! c\ntrey 9 ^? ( our nation1 to :om ,god ha? geu5 bo? k1 also abilitie to translate into our t\nge1 & to publi%e abroad su* good au?ors1 & bookes 7! *iefe in/rum5tes ( all le>n+es73 se+ more\] t _m good wittes bo? ( g5tlem5 & ( o!rs ( all degrees1 mu* desir\s & /udi\s ( ^! artes1 & seek+ = !m z mu* z !y c1 sp>+ no pa9es1 & yet fru/rate ( _! int5t1 by no meanes atta9+ to t :i* !y seeke3 ,i haue = _! sakes1 ) "s *>ge great trauaile1 fai?fully translat$ into our vulg>e toge1 set abroad 9 ,pr9t1 ?is booke ( ,euclide4 ,:]eunto ,i haue add$ easie & pla9e decl>ations & examples by figures1 ( ! def9itions4 ,9 :i* booke also ye %all 9 due place f9de manifolde additions1 ,s*olies1 ,annotations1 & ,inu5tions3 :i* ,i haue ga!r$ out ( _m ( ! mo/ fam\s *iefe ,ma!maticis1 bo? ( old "t1 & 9 our age3 z by dilig5t read+ x 9 c\rse1 ye %all well p]ceaue4 ,! fruite & ga9e :i* ,i require = ^! my pa9es & trauaile1 %all be no?+ els1 b onely t ?\ g5tle read]1 w gratefully accept ! same3 & t ?\ maye/ !reby receaue "s pr(ite3 & more\] to excite & /irre vp o!rs le>n$1 to d ! l1 to take pa9es 9 t behalfe4 ,by meanes :](1 our ,engli%e t\nge %all no lesse be enri*$ ) good ,au?ors1 !n are o!r /raunge t\nges3 z ! ,dut*1 ,fr5*1 ,italian1 & ,spani%e3 9 :i* are r$ all good au?ors 9 a man]1 f\nd amonge/ ! ,grekes or ,lat9es4 ,:i* is ! *iefe/ cause1 t amonge/ ? d flori%e s _m cunn+ & skilfull m51 9 ! inu5tions ( /raunge & wond]full ?+es1 z 9 ^! our daies we see !re d4 ,:i* fruite & ga9e if ,i atta9e vnto1 x %all enc\rage me h]eaft]1 9 su* l sort to translate1 & set abroad "s o!r good au?ors1 bo? p]ta9+ to religion 7z p>tly ,i haue already done7 & also p]ta9+ to ! ,ma!maticall ,artes4 ,?us g5tle read] f>ewell4 787 ,decoration ,,to ,,! ,,vnfa9$ ,,lov]s ( tru!1 & con/ant ,/ud5tes ( ,noble ,sci5ces1 ,,iohn ,,dee ( ,london1 h>tily wi%e? grace f heau51 & mo/ prosp]\s successe 9 all _! hone/ attemptes & ex]cises4 ,diu9e ,plato1 ! great ,ma/] ( _m wor?y ,philosoph]s1 & ! con/ant au\*]1 & pi?y p]swad] ( ,vnum1 ,bonum1 & ,ens3 9 8 ,s*ole & ,academie1 sundry times 7besides 8 ord9>y ,s*ol]s7 0 visit$ ( a c]ta9e k9de ( m51 allur$ by ! noble fame ( ,plato1 & ! great comm5dation ( hys pr(\nd & pr(itable doctr9e4 ,b :5 su* ,he>]s1 aft] long h>k5+ to him1 p]ceau$1 t ! drift ( 8 disc\rses issu$ out1 to conclude1 ?is ,vnum1 ,bonum1 & ,ens1 to be ,spirituall1 ,inf9ite1 t]nall1 ,omnipot5t1 c4 ,no?yng beyng all$g$ or express$1 ,h[1 worldly goods3 h[1 worldly dignitie3 h[1 heal?1 ,/rg? or lu/9es ( body3 nor yet ! meanes1 h[ a m]ueil\s s5sible & bodyly blysse & felicitie h]eaft]1 might be atteyn$3 ,/raightway1 ! fantasies ( ^? he>]s1 7 dampt3 _! op9ion ( ,plato1 0 cl5e *aung$3 yea 8 doctr9e 0 by !m despis$3 & 8 s*ole1 no m ( !m visit$4 ,:i* ?+1 8 ,s*ol]1 ,ari/otle1 n>r[ly csid]+1 f\nde ! cause !r(1 to be1 8,= t !y _h no =w>nyng & in=mation1 9 g5]all10 :]eto 8 doctr9e t5d$4 ,=1 s1 might !y haue _h occasion1 ei!r to haue =borne 8 s*ole hauntyng3 7if !y1 !n1 _h mislik$ 8 ,scope & purpose7 or con/antly to haue cont9u$ !r93 to _! full satisfaction3 if su* 8 f9all scope int5t1 _h b5 to _! desire4 ,:]=e1 ,ari/otle1 eu]1 aft] t1 vs$ 9 brief1 to =ew>ne 8 owne ,s*ol]s & he>]s1 8bo? ( :at matt]1 & also to :at ende1 he tooke 9 h& to speake1 or tea*40 ,:ile ,i consid] ! diu]se trades ( ^! two excell5t ,philosoph]s 7& am mo/ sure1 bo?1 t ,plato "r well1 o!rwise c\ld tea*3 & t ,ari/otle m\ght boldely1 ) 8 he>]s1 haue dealt 9 l sorte z ,plato did7 ,i am 9 no little pang ( p]plexitie3 ,bycause1 t1 :i* ,i mislike1 is mo/ easy = me to p]=me 7& to haue ,plato = my exple47 ,& t1 :i* ,i "k to be mo/ comm5dable3 & 79 ?is fir/ br+yng1 into common h&l+1 ! ,artes ,ma!maticall7 to be mo/ necess>y3 is full ( great difficultie & sundry daung]s4 ,yet1 nei!r d ,i ?9k x mete1 = s /raunge matt] 7z n[ is m5t to be publi%$7 & to s /raunge an audi5ce1 to be bluntly1 at fir/1 put =?1 )\t a peculi> ,preface3 ,nor 7,imitatyng ,ari/otle7 well c ,i hope1 t accordyng to ! ampl5es & dignitie ( ! ,/ate ,ma!maticall1 ,i am able1 ei!r playnly to prescribe ! mat]iall b\ndes3 or precisely to expresse ! *ief purposes1 & mo/ wond]full applications !r(4 ,& ?\gh ,i am sure1 t su* z did %r9ke f ,plato 8 s*ole1 aft] !y _h p]ceiu$ 8 f9all conclusion1 w\ld 9 ^! ?+es haue b5 8 mo/ dilig5t he>]s 7s inf9itely m\ght _! desires1 9 f9e & at l5g?1 by our ,artes ,ma!maticall be satisfi$7 yet1 by ?is my ,prface =ew>nyng1 ,aswell all su*1 may 7to _! great beh(e7 ! son]1 hi!r be allur$3 z also ! ,py?agoricall1 & ,platonicall p]fect s*ol]1 & ! con/ant pr(\nd ,philosoph]1 ) m ease & sp$e1 may 7l ! ,bee17 ga!r1 h]eby1 bo? wax & hony4 ,! int5t ( ?is ,preface4 ,:]=e1 seyng ,i f9de great occasion 7= ! causes alleg$1 & f>d]1 9 respect ( my ,art ,ma!matike g5]all7 to vse 8a c]ta9e =ew>nyng & ,prface1 ^: cont5t %albe1 t mighty1 mo/ plesaunt1 & frutefull ,ma!maticall ,tree1 ) 8 *ief armes & second 7grift$7 braun*es3 ,bo?1 :at eu]y "o is1 & also1 :at commodity1 9 g5]all1 is to be look$ =1 aswell ( griff z /ocke3 ,& =asmu* z ?is ent]prise is s great1 t1 to ?is our tyme1 x neu] 0 7to my k7 by any a*ieu$3 ,& also x is mo/ h>d1 9 ^! our dr]y dayes1 to su* r>e & /raunge ,artes1 to wyn due & common cr$it30 ,neu]!les1 if1 = my s9c]e endeu\r to satisfie y\r hone/ expectation1 y w b l5d me y\r ?kefull mynde a :ile3 &1 to su* matt] z1 = ?is "t1 my p5ne 7) sp$e7 is hable to deliu]1 apply y\r eye or e>e att5tifely3 p]*aunce1 at once1 & = ! fir/ salutyng1 ?is ,preface y w f9de a lesson long en\gh4 ,& ei!r y w1 = a second 7by ?is7 be made mu* ! apt]3 or %ortly become1 well hable y\r selues1 ( ! lyons claw1 to coniecture 8 royall symmetrie1 & f>d] prop]tie4 ,n[ !n1 g5tle1 my fr5des1 & c\ntrey m51 ,turne y\r eyes1 & b5d y\r myndes to t doctr9e1 :i* = our pres5t purpose1 my simple tal5t is hable to yeld y4 ,all ?+es :i* are1 haue beyng1 are f\nd vnd] a triple diu]sitie g5]all4 ,=1 ei!r1 !y are dem$ ,sup]naturall1 ,naturall1 or1 ( a ?ird be+4 ,?+es ,sup]naturall1 are immat]iall1 simple1 indiuisible1 incorruptible1 vn*angeable4 ,?+s ,naturall1 are mat]iall1 comp\nd$1 diuisible1 corruptible1 & *aungeable4 ,?+es ,sup]naturall1 are1 ( ! m9de onely1 compreh5d$3 ,?+s ,naturall1 ( ! s5se ext]ior1 ar hable to be p]ceiu$4 ,9 ?+es ,naturall1 probabilitie & coniecture ha? place3 ,b 9 ?+s ,sup]naturall1 *ief dem/ration1 mo/ sure ,sci5ce is to be _h4 ,by :i* prop]ties comp>asons ( ^! two1 m easily may be describ$1 ! /ate1 condition1 nature & prop]ty ( ^? ?+es1 :i*1 we be=e t]m$ ( a ?ird be+3 :i*1 by a peculi] "n also1 are call$ ,?ynges ,ma!maticall4 ,=1 ^!1 beyng 79 a man]7 middle1 betw5e ?+es sup]naturall & naturall3 are n s absolute & excell5t1 z ?+es sup]natural3 ,nor yet s base & grosse1 z ?+s naturall3 ,b are ?+es immat]iall3 & neu]!lesse1 by mat]iall ?+s hable some:at to be signifi$4 ,& ?\gh _! p>ticul> ,images1 by ,art1 are aggregable & diuisible3 yet ! g5]all ,=mes1 not)/&yng1 are con/ant1 vn*aungeable1 vntrs=mable1 & incorruptible4 ,nei!r ( ! s5se1 c !y1 at any tyme1 be p]ceiu$ or iudg$4 ,nor yet1 = all t1 9 ! royall mynde ( man1 fir/ conceiu$4 ,b1 surm\ntyng ! imp]fecti ( coniecture1 we5yng & op9ion3 & commyng %ort ( high intellectuall ccepti1 are ! ,m]curial fruite ( ,dianticall disc\rse1 9 p]fect imag9ation subsi/yng4 ,a m]uayl\s newtralitie haue ^! ?+es ,ma!maticall1 & also a /raunge p>ticipati betw5e ?+es sup]naturall1 immortall1 intellectual1 simple & indiuisible3 & ?ynges naturall1 mortall1 s5sible1 comp\nd$ & diuisible4 ,probabilitie & s5sible prose1 may well s]ue 9 ?+es naturall3 & is comm5dable3 ,9 ,ma!maticall reason+es1 a probable ,argum5t1 is no?yng reg>d$3 nor yet ! te/imony ( s5se1 any :it cr$it$3 ,b onely a p]fect demon/ration1 ( tru!s c]ta9e1 necess>y1 & inu9cible3 vniu]sally & necess>yly conclud$3 is all[$ z suffici5t = 8an ,argum5t exactly & purely ,ma!matical40 ,note ! worde1 ,vnit1 to expresse ! ,greke ,monas1 n ,vnitie3 z we haue all1 commonly1 till n[1 vs$4 ,( ,ma!maticall ?+es1 are two pr9cipall k9des3 namely1 ,numb]1 & ,magnitude4 ,numb]4 ,numb]1 we def9e1 to be1 a c]tayne ,ma!maticall ,sme1 ( ,vnits4 ,&1 an ,vnit1 is t ?+ ,ma!maticall1 ,indiuisible1 by p>ticipation ( "s lik5es ( ^: prop]ty1 any ?+1 :i* is 9 de$e1 or is c\nt$ ,"o1 may resonably be call$ ,"o4 ,we acc\nt an ,vnit1 a ?+ ,ma!maticall1 ?\gh x be no ,numb]1 & also indiuisible3 because1 ( x1 mat]ially1 ,numb] do? consi/3 :i*1 pr9cipally1 is a ?+ ,ma!maticall4 ,magnitude4 ,magnitude is a ?+ ,ma!maticall1 by p>ticipation ( "s lik5es ( ^: nature1 any ?+ is iudg$ long1 broade1 or ?icke4 8,a ?icke ,magnitude we call a ,solide1 or a ,body4 ,:at ,magnitude s eu]1 is ,solide or ,?icke1 is also broade1 long4 ,a broade magnitude1 we call a ,sup]ficies or a ,pla9e4 ,eu]y playne magnitude1 ha? also l5g?4 ,a long magnitude1 we t]me a ,l9e4 ,a ,l9e is nei!r ?icke nor broade1 b onely long3 ,eu]y c]tayne ,l9e1 ha? two endes3 ,a po9t4 ,! endes ( a l9e1 are ,po9tes call$4 ,a ,po9t1 is a ?+ ,ma!maticall1 indiuisible1 :i* may haue a c]tayne det]m9$ situation40 ,if a ,poynt m\e f a det]m9$ situation1 ! way :]e9 x m\$1 is also a ,l9e3 ma!matically produc$1 :]eupon1 ( ! aunci5t ,ma!matici5s1 ,a ,l9e4 a ,l9e is call$ ! race or c\rse ( a ,po9t4 ,a ,poynt we def9e1 by ! "n ( a ?+ ,ma!maticall3 ?\gh x be no ,magnitude1 & indiuisible3 because x is ! propre ende1 & b\nd ( a ,l9e3 :i* is a true ,magnitude4 ,magnitude4 ,& ,magnitude we may def9e to be t ?+ ,ma!maticall1 :i* is diuisible = eu]1 9 p>tes diuisible1 long1 broade or ?icke4 ,!re=e ?\gh a ,poynt be no ,magnitude1 yet ,t]m9atiuely1 we reck5 x a ?+ ,ma!maticall 7z ,i sayd7 by reason x is prop]ly ! end1 & b\nd ( a l9e4 ,nei!r ,numb]1 nor ,magnitude1 haue any ,mat]ialitie4 ,fir/1 we w consid] ( ,numb]1 & ( ! ,sci5ce ,ma!maticall1 to x appropriate1 call$ ,ari?metike3 & aft]w>d ( ,magnitude1 & 8 ,sci5ce1 call$ ,geometrie4 ,b t "n cont5te? me n3 :]e( a ^w or two h]eaft] %all be sayd4 ,h[ ,immat]iall & free f all matt]1 ,numb] is1 :o do? n p]ceaue8 yea1 :o do? n wond]fully wd] at x8 ,=1 nei!r pure ,elem5t1 nor ,ari/oteles1 ,qu9ta ,ess5tia1 is hable to s]ue = ,numb]1 z 8 propre matt]4 ,nor yet ! puritie & simpl5es ( ,sub/ance ,spirituall or ,angelicall1 w be f\nd propre en\gh !reto4 ,& !re=e ! great godly ,philosoph] ,anitius ,boetius1 sayd3 ,omnia qucunque a primua r]um natura con/ructa sunt1 ,num]orum vid5tur ratione =mata4 ,hoc enim fuit pr9cipale 9 animo ,conditoris ,exempl>4 ,t is3 ,all ?+es 7:i* f ! v fir/ orig9all be+ ( ?+es1 haue b5e fram$ & made7 d appe>e to be ,=m$ by ! reason ( ,numb]s4 ,= ?is 0 ! pr9cipall example or patt]ne 9 ! m9de ( ! ,creator4 ,o com=table allurem5t1 ,o raui%+ p]swasion1 to deale ) a ,sci5ce1 ^: ,subiect1 is s ,aunci5t1 s pure1 s excell5t1 s surm\nt+ all creatures1 s vs$ ( ! ,almighty & incompreh5sible wisdome ( ! ,creator1 9 ! di/9ct creation ( all creatures3 9 all _! di/9ct p>tes1 prop]ties1 natures1 & v]tues1 by ord]1 & mo/ absolute numb]1 br\ght1 f ,no?+1 to ! ,=malitie ( _! be+ & /ate4 ,by ,numb]s prop]tie !re=e1 ( vs1 by all possible meanes1 7to ! p]fection ( ! ,sci5ce7 le>n$1 we may bo? w9de & draw our selues into ! inw>d & deepe se>* & vew1 ( all creatures di/9ct v]tues1 natures1 prop]ties1 & ,=mes3 ,& also1 f>d]1 arise1 clime1 asc5d1 & m\nt vp 7) ,speculatiue w+es7 9 _s1 to behold 9 ! ,glas ( ,creation1 ! ,=me ( ,=mes1 ! ,exempl> ,numb] ( all ?+es ,num]able3 bo? visible & inuisible1 mortall & immortall1 ,corporall & ,spirituall4 ,"p ( ?is pr(\nd & diu9e ,sci5ce1 _h ,ioa*im ! ,prophesi] atteyn$ vnto3 by ,numb]s ,=mall1 ,naturall1 & ,rationall1 =seyng1 concludyng1 & =%ewyng great p>ticul> eu5ts1 long be=e _! comm+4 ,8 bookes yet rema9yng1 h]e(1 are good pr(e3 ,& ! noble ,e>le ( ,mir&ula1 7besides t17 a suffici5t witnesse3 t ,ioa*im1 9 8 prophesies1 proc$$ by no o!r way1 !n by ,numb]s ,=mall4 ,& ?is ,e>le hym selfe1 9 ,rome1 ,ano4 #adhh4 99 set vp #ijj4 ,conclusions1 9 all k9de ( ,sci5ces1 op5ly to be disput$ (3 & among ! re/1 9 8 ,conclusions ,ma!maticall1 79 ! eleu5? ,conclusion7 ha? 9 ,lat91 ?is ,engli% s5t5ce4 ,by ,numb]s1 a way is _h1 to ! se>*yng out1 & vnd]/&yng ( eu]y ?yng1 hable to be kn[54 ,= ! v]ify+ ( :i* ,conclusion1 ,i promise to aunsw]e to ! #gd4 ,que/ions1 vnd] writt51 by ! way ( ,numb]s4 ,:i* ,cclusions1 ,i omit "h to rehe>se3 aswell auoidyng sup]flu\s prolixitie3 z1 bycause ,ioannes ,picus1 workes1 are commonly _h4 ,b1 9 any case1 ,i w\ld wi% t ^? ,conclusions 7 r$ dilig5tly1 & p]ceiu$ ( su*1 z are e>ne/ ,obs]u]s & ,consid]]s ( ! con/ant law ( nb]s3 :i* is plant$ 9 ?yngs ,naturall & ,sup]naturall3 & is prescrib$ to all ,creatures1 inuiolably to be kept4 ,=1 s1 besides _m o!r ?+es1 9 ^? ,conclusions to be m>k$1 x w\ld ape>e1 h[ s9c]ely1 )9 my b\ndes1 ,i disclose ! wond]full my/]ies1 by numb]s1 to be atteyn$ vnto4 ,( my =m] wordes1 easy x is to be ga!r$1 t ,numb] ha? a treble /ate3 ,"o1 9 ! ,creator3 an o!r 9 eu]y ,creature 79 respect ( 8 complete con/itution37 & ! ?ird1 9 ,spirituall & ,angelicall ,myndes1 & 9 ! ,s\le ( m4 ,9 ! fir/ & ?ird /ate1 ,numb]1 is t]m$ ,numb] ,numbryng4 ,b 9 all ,creatures1 o!rwise1 ,numb]1 is t]m$ ,nb] ,numbr$4 ,& 9 our ,s\le1 ,nb] be>e? su* a swaye1 & ha? su* an aff9itie !r)3 t "s ( ! old ,philosoph]s taught1 ,mans ,s\le1 to be a ,numb] m\yng x selfe4 ,& 9 d$e1 9 vs1 ?\gh x be a v ,accid5t3 yet su* an ,accid5t x is1 t be=e all ,creatures x _h p]fect beyng1 9 ! ,creator1 ,sempit]nally4 ,numb] ,numbryng !r=e1 is ! discretion disc]n+1 & di/9ct+ ( ?+es4 ,b 9 ,god ! ,creator1 ,?is discretion1 9 ! beg9nyng1 produc$ ord]ly & di/9ctly all ?+es4 ,= 8 ,numbryng1 !n1 0 8 ,creatyng ( all ?+es4 ,& 8 ,cont9uall ,numbryng1 ( all ?+es1 is ! ,cons]uation ( !m 9 be+3 ,&1 :]e & :5 he w lacke an ,vnit3 !re & !n1 t p>ticul> ?yng %albe ,discreat$4 ,"h ,i /ay4 ,b our ,seu]allyng1 di/9ctyng1 & ,numbryng1 create? no?yng3 b ( ,multitude consid]$1 make? c]ta9e & di/9ct det]m9ation4 ,& albeit ^! ?ynges be waighty & tru!s ( great importance1 yet 7by ! inf9ite goodnes ( ! ,almighty ,t]n>ie17 ,artificiall ,me?ods & easy wayes are made1 by :i* ! zel\s ,philosoph]1 may wyn n]e ?is ,riu]i% ,ida1 ?is ,m\ntayne ( ,contemplation3 & m !n ,contemplation4 ,& also1 ?\gh ,numb]1 be a ?yng s ,immat]iall1 s diu9e1 & t]nall3 yet by degrees1 by litle & litle1 /ret*yng =?1 & apply+ "s lik5es ( x1 z fir/1 to ?+es ,spirituall3 & !n1 bryngyng x l[]1 to ?ynges s5sibly p]ceiu$3 z ( a mom5tanye s\nde it]at$3 !n to ! lea/ ?ynges t may be se51 num]able3 ,& at l5g?1 7mo/ grossely17 to a multitude ( any corporall ?ynges se51 or felt3 & s1 ( ^! grosse & s5sible ?ynges1 we are trayn$ to le>ne a c]ta9e ,image or lik5es ( numb]s3 & to vse ,arte 9 !m to our pleasure & pr(fit4 ,s grosse is our conu]sation1 & dull is our appreh5sion3 :ile mortall ,s5se1 9 vs1 rule? ! common weal? ( our litle _w4 ,h]eby we say1 ,?ree ,lyons1 are ?ree3 or a ,t]n>ie4 ,?ree ,egles1 are ?ree1 or a ,t]n>ie4 ,:i* 99 ,t]n>ies1 are e*e1 ! ,vnion1 knot1 & ,vni=mitie1 ( ?ree discrete & di/9ct ,vnits4 ,t is1 we may 9 e*e ,t]n>ie1 ?rise1 seu]ally po9te1 & %ew a "p1 ,"o1 ,"o1 & ,"o4 ,:]e1 9 ,numbryng1 we say ,"o1 two1 ,?ree4 ,b h[ f>re1 ^! visible ,ones1 d diff]re f our ,indiuisible ,vnits 79 pure ,ari?metike1 pr9cipally consid]$7 no man is ignorant4 ,yet f ^! grosse & mat]iall ?ynges1 may we be l$ vpw>d1 by degrees1 s1 in=myng our rude ,imag9ation1 t[>d ! cceiuyng ( ,numb]s1 absolutely 73,n suppos+1 nor admixtyng any ?yng creat$1 ,corporall or ,spirituall1 to support1 conteyne1 or repres5t ^? ,numb]s imag9$37 t at l5g?1 we may be hable1 to f9de ! numb] ( our owne "n1 glori\sly exemplifi$ & regi/r$ 9 ! booke ( ! ,tr9itie mo/ bless$ & t]nall4 ,b f>d] vnd]/&1 t vulg> ,practis]s1 haue ,numb]s1 o!rwise1 9 sundry ,consid]ations3 & ext5d _! "n f>d]1 !n to ,numb]s1 ^: lea/ "p is an ,vnit4 ,= ! common ,logi/1 ,reck5ma/]1 or ,ari?metici51 9 hys vs+ ( ,numb]s3 ( an ,vnit1 imag9e? lesse p>tes3 & calle? !m ,fractions4 ,z ( an ,vnit1 he make? an halfe1 & ?us note? x1 4 & s ( o!r1 7inf9itely diu]se7 p>tes ( an ,vnit4 ,yea & f>d]1 ha?1 ,fractions ( ,fractions4 c4 ,&1 =asmu*1 z1 ,addition1 ,sub/raction1 ,multiplication1 ,diuision & ,extraction ( ,rotes1 are ! *ief1 & suffici5t p>tes ( ,ari?metike3 ,ari?metike4 :i* is1 ! ,sci5ce t demon/rate? ! prop]ties1 ( ,numb]s1 & all op]atis1 9 numb]s to be p]=m$3 ,note4 8,h[ (t51 !r=e1 ^! fiue sundry sortes ( ,op]ations1 d1 = ! mo/ "p1 ( _! execution1 diff]re f ! fiue op]ations ( l g5]all prop]ty & "n1 9 our ,:ole numb]s practisable1 ,s (t51 7= a m di/9ct doctr9e7 we1 vulg>ly acc\nt & "n x1 an o!r kynde ( ,ari?metike40 ,& by ?is reason3 #a4 ! ,consid]ation1 doctr9e1 & work+1 9 :ole numb]s onely3 :]e1 ( an ,vnit1 is no lesse "p to be all[$3 is nam$ 7z x 77 an ,ari?metike by x selfe4 ,& s ( ! ,ari?metike ( ,fractions4 #b4 ,9 lyke sorte1 ! necess>y1 wond]full & ,secret doctr9e ( ,proportion1 & proportionalytie ha? pur*as$ vnto x selfe a peculi] man] ( h&lyng & workyng3 & s may seme an o!r =me ( ,ari?metike4 #c4 ,more\]1 ! ,a/ronom]s1 = sp$e & m commodi\s calculation1 haue deuis$ a peculi] man] ( ord]yng nb]s1 ab\t !yr circul> motions1 by ,sexag5es1 & ,sexagesmes4 ,by ,signes1 ,degrees & ,m9utes c4 :i* commonly is call$ ! ,ari?metike ( ,a/ronomical or ,phisicall ,fractions4 ,t1 haue ,i briefly not$1 by ! "n ( ,ari?metike ,circul>4 ,bycause x is also vs$ 9 circles1 n ,a/ronomicall4 c4 #d4 ,practise ha? l$ ,numb]s f>d]1 & ha? fram$ !m1 to take vpon !m1 ! %ew ( ,magnitudes prop]tie3 ,:i* is ,incomm5surabilitie & ,irrationalitie4 7,= 9 pure ,ari?metike1 an ,vnit1 is ! common ,measure ( all ,numb]s47 ,&1 "h1 ,nb]s are become1 z ,lynes1 ,playnes & ,solides3 "s tymes ,rationall1 "s tymes ,irrationall4 ,& haue propre & peculi] *>act]s1 7z 4 4 & s ( o!r4 ,:i* is to signifie ,rote ,squ>e1 ,rote ,cubik3 & s =?37 propre & peculi] fa%ions 9 ! fiue pr9cipall p>tes3 ,:]=e ! practis]1 e/eme? ?is1 a diu]se ,ari?metike f ! o!r4 ,practise brynge? 91 "h1 diu]se comp\ndyng ( ,numb]s3 z "s tyme1 two1 ?ree1 f\re 7or m7 ,radicall nb]s1 diu]sly knit1 by signes1 ( ,m ,lesse3 z ?us #ab #ae4 ,or ?us #ai #ab - #b4 c4 ,& "s tyme ) :ole numb]s1 or fractions ( :ole ,numb]1 amg !m3 z #bj #bd4 #af #cc - #aj4 #dd #ab #i4 ,& s1 inf9itely1 may hap ! v>ietie4 ,aft] ?is3 ,bo? ! "o & ! o!r ha? fractions incid5t3 & s is ?is ,ari?metike greately enl>g$1 by diu]se exhibityng & vse ( ,compositions & mixtynges4 ,consid] h[1 ,i 7beyng desir\s to deliu] ! /ud5t f error & ,cauillation7 d giue to ?is ,practise1 ! "n ( ! ,ari?metike ( ,radicall numb]s3 ,n1 ( ,irrationall or ,surd ,numb]s3 :i* o!r :ile1 are ,rationall3 ?\gh !y haue ! ,signe ( a ,rote be=e !m1 :i*1 ,ari?metike ( :ole ,numb]s mo/ vsuall1 w\ld say !y _h no su* ,roote3 & s acc\nt !m ,surd ,numb]s3 :i*1 g5]ally spok1 is vntrue3 z ,euclides t5? booke may tea* y4 ,!r=e to call !m1 g5]ally1 ,radicall ,numb]s1 7by reason ( ! signe 4 prefix$17 is a sure way3 & a suffici5t g5]all di/9ction f all o!r ordryng & vs+ ( ,numb]s3 ,& yet 7beside all ?is7 ,consid]3 ! inf9ite desire ( k1 & incr$ible p[] ( mans ,se>* & ,capacitye3 h[1 !y1 ioyntly haue wad$ f>d] 7by mixtyng ( speculation & practise7 & haue f\nd out1 & atteyn$ to ! v *ief p]fection 7almo/7 ( ,numb]s ,practicall vse4 ,:i* ?+1 is well to be p]ceiu$ 9 t great ,ari?meticall ,arte ( quation3 commonly call$ ! ,rule ( ,coss4 or ,algebra4 ,! ,lat9es t]m$ x1 ,regulam ,rei ,c5sus1 t is1 ! ,rule ( ! ?yng & 8 value4 ,) an apt "n3 compreh5dyng ! fir/ & la/ po9tes ( ! worke4 ,& ! vulg> names1 bo? 9 ,italian1 ,fr5*e & ,spani%1 dep5d 79 namyng x17 vpon ! signification ( ! ,lat9 ^w1 ,res3 ,a ?+3 vnlea/ !y vse ! "n ( ,algebra4 ,& !r9 7commonly7 is a dubble error4 ,! "o1 ( !m1 :i* ?9ke x to be ( ,geb] 8 inu5tyng3 ! o!r ( su* z call x ,algebra4 ,=1 fir/1 ?\gh ,geb] = 8 great skill 9 ,numb]s1 ,geometry1 ,a/ronomy1 & o!r m>uail\s ,artes1 m\ght haue sem$ hable to haue fir/ deuis$ ! sayd ,rule3 & also ! "n c>rye? ) x a v n]e lik5es ( ,geb] 8 "n3 yet true x is1 t a ,greke ,philosoph] & ,ma!matici51 nam$ ,diophantus1 be=e ,geb] 8 tyme1 wrote #ac4 bookes !r( 7( :i*1 six are yet extant3 & ,i _h !m to 99vse1 99 ,anno4 #aeej4 ( ! fam\s ,ma!matici51 & my great fr5de1 ,petrus ,montaureus37 ,& secondly1 ! v "n1 is ,algieb>1 & n ,algebra3 z by ! ,arabi5 ,auic51 may be pr\$3 :o ha? ^! precise wordes 9 ,lat9e1 by ,&reas ,alpagus 7mo/ p]fect 9 ! ,arabik tung7 s translat$4 ,sci5tia faci5di ,algieb> ,alma*abel4 i4 ,sci5tia inu5i5di num]um ignotum1 p] additionem ,num]i1 diuisionem quationem4 ,:i* is to say3 ,! ,sci5ce ( workyng ,algieb> & ,alma*abel1 t is1 ! ,sci5ce ( f9dyng an vnkn[5 numb]1 by ,addyng ( a ,numb]1 ,diuision quation4 ,"h haue y ! "n3 & also ! pr9cipall p>tes ( ! ,rule1 t\*$4 ,to "n x1 ,! rule1 or ,art ( quation1 do? signifie ! middle "p & ! ,/ate ( ! ,rule4 ,?is ,rule1 ha? 8 peculi] ,*>act]s3 #e4 & ! pr9cipal p>tes ( ,ari?metike1 to x app]tayn+1 d diff]re f ! o!r ,ari?meticall op]ations4 ,?is ,ari?metike1 ha? ,nb]s ,simple1 ,cp\nd1 ,mixt3 & ,fractions1 accord+ly4 ,?is ,rule1 & ,ari?metike ( ,algieb>1 is s pr(\nd1 s g5]all & s 79 man]7 conteyne? ! :ole p[] ( ,numb]s ,application practicall3 t mans witt1 c deale ) no?yng1 m pr(fitable ab\t numb]s3 nor mat*1 ) a ?yng1 m mete = ! diu9e =ce ( ! ,s\le1 79 humane ,/udies1 affaires1 or ex]cises7 to be try$ 94 ,p]*aunce y look$ =1 7long ere n[17 to haue _h "s p>ticul> pr(e1 or euid5t te/imony ( ! vse1 pr(fit & ,commodity ( ,ari?metike vulg>1 9 ! ,common lyfe & trade ( m54 ,!rto1 !n1 ,i w n[ frame my selfe3 ,b h]e9 great c>e ,i haue1 lea/ l5g? ( sundry pr(es1 might make y deme1 t ei!r ,i did misd\te y\r zel\s mynde to v]tues s*ole3 or els mi/ru/ y\r hable witts1 by "s1 to gesse mu* m4 ,a pr(e !n1 f\re1 fiue1 or six1 su*1 w ,i bryng1 z any reasonable man1 !r) may be p]suad$1 to l\e honor1 yea le>ne & ex]cise ! excell5t ,sci5ce ( ,ari?metike4 ,& fir/3 :o1 n]] at h&1 c be a bett] witnesse ( ! frute receiu$ by ,ari?metike1 !n all kynde ( ,m>*ants8 ,?\gh n all1 alike1 ei!r n$e x1 or vse x4 ,h[ c\ld !y =be>e ! vse & helpe ( ! ,rule1 call$ ! ,gold5 ,rule8 ,simple & ,comp\nde3 bo? =w>d & backw>d8 ,h[ might !y misse ,ari?meticall helpe 9 ! ,rules ( ,fel[%yp3 ei!r )\t tyme1 or ) tyme8 & betw5e ! ,m>*ant 8 ,factor8 ,! ,rules ( ,b>t]+ 9 w>es onely3 or "p 9 w>es1 & "p 9 money1 w\ld !y gladly want8 ,our ,m>*ant v5tur]s1 & ,trauayl]s ou] ,sea1 h[ c\ld !y ord] _! doynges iu/ly & )\t losse1 vnlea/ c]ta9e & g5]all ,rules = ,ex*age ( money1 & ,re*aunge1 71 = _! vse1 deuis$8 ,! ,rule ( ,alligation1 9 h[ sundry cases1 do? x conclude = !m1 su* precise v]ities1 z nei!r by naturall witt1 nor o!r exp]i5ce1 !y1 7 hable1 els1 to "k8 ,& 7) ! ,m>*ant !n to make an end7 h[ ample wond]full is ! ,rule ( ,false positions8 especially z x is n[1 by two excell5t ,ma!matici5s 7( my famili] acquayntance 9 _! life "t7 enl>g$8 ,i meane ,gemma ,frisius1 & ,simon ,iacob4 ,:o c ei!r 9 brief conclude1 ! g5]all & ,capitall ,rules8 or :o c ,imag9e ! ,myriades ( sundry ,cases1 & p>ticul> examples1 9 ,act & e>ne/1 cont9ually wr\ght1 tri$ & conclud$ by ! =5am$ ,rules1 onely8 ,h[ sundry o!r ,ari?meticall practises1 are commonly 9 ,m>*antes h&es1 & k3 ,!y !m selues1 c1 at l>ge1 te/ifie4 ,! ,m9tma/]1 & ,goldsmi?1 9 _! ,mixture ( ,metals1 ei!r ( diu]se k9des1 or diu]se values3 h[ are !y1 or may !y1 exactly be direct$1 & m]uail\sly pleasur$1 if ,ari?metike be _! guide8 ,& ! honorable ,phisicis1 w gladly confesse !m selues1 mu* behold+ to ! ,sci5ce ( ,ari?metike1 & t sundry wayes3 ,b *iefly 9 _! ,art ( ,graduation1 & comp\nde ,m$ic9es4 ,& ?\gh ,gal5us1 ,au]rois1 ,arnoldus1 ,lullus1 & o!r haue publi%$ _! positions1 aswell 9 ! quantities ( ! ,degrees ab\e ,temp]am5t1 z 9 ! ,rules1 conclud+ ! new ,=me result+3 yet a m precise1 commodi\s1 & easy ,me?od1 is extant3 by a ,c\ntreyman ( ours ,r4 ,b4 7ab\e #bjj4 ye>es ago7 inu5t$4 ,& =asmu* z ,i am vnc]ta9e1 :o ha? ! same3 or :5 t litle ,lat9 treatise1 7z ! ,au?or writ x17 %all come to be ,pr9t$3 7,bo? to decl>e ! desire ,i haue to pleasure my ,c\ntrey1 :]9 ,i may3 & also1 = v good pr(e ( ,numb]s vse1 9 ?is mo/ subtile & frutefull1 ,philosophicall ,conclusion17 ,i ent5d 9 ! meane :ile1 mo/ briefly1 & ) my f>d] helpe1 to communicate ! pi? !r( vnto y4 ,fir/ describe a circle3 ^: diamet] let be an in*4 ,diuide ! ,circumf]5ce into f\re equall p>tes4 ,fr ! ,c5t]1 by ^? #d4 sections1 ext5d #d4 "r l9es3 e*e ( #d4 in*es & a halfe long3 or ( z _m z y li/e1 ab\e #d4 )\t ! circumf]5ce ( ! circle3 ,s t !y %all be ( #d4 in*es long 7at ! lea/7 )\t ! ,circle4 ,make good euid5t m>kes1 at eu]y in*es end4 ,if y li/1 y may subdiuide ! in*es aga9e into #aj4 or #ab4 small] p>tes1 equall4 ,at ! endes ( ! l9es1 write ! names ( ! #d4 pr9cipall elem5tall ,qualities4 ,hote & ,colde1 "o aga9/ ! o!r4 ,& likewise ,moy/ & ,dry1 "o aga9/ ! o!r4 ,& 9 ! ,circle write ,temp]ate4 ,:i* ,temp]ature ha? a good ,latitude3 z appe>e? by ! ,complexion ( man4 ,& !re=e we haue all[$ vnto x1 ! =esayd ,circle3 & n a po9t ,ma!maticall or ,physicall4 99 ,take "s "p ( ,lullus c\nsayle 9 8 booke de ,q4 ,ess5tia4 ,n[1 :5 y haue two ?+es ,miscible1 ^: degrees are 99 truely kn[53 ,( necessitie1 ei!r !y are ( "o ,quantitie & waight1 or ( diu]se4 ,if !y be ( "o ,quantitie & waight3 :e!r _! =mes1 be ,contr>y ,qualities1 or ( "o k9de 7b ( diu]se int5tions & degrees7 or a ,temp]ate1 & a ,contr>y1 ,! =me result+ ( _! ,mixture1 is 9 ! ,middle betw5e ! degrees ( ! =mes mixt4 ,z = example1 let ,a1 be ,moi/ 9 ! fir/ degree3 & ,b1 ,dry 9 ! ?ird degree4 ,adde #a4 & #c4 t make? #d3 ! halfe or middle ( #d4 is #b4 ,?is #b4 is ! middle1 equally di/ant f ,a & ,b 99 ,note4 7= ! 99 ,temp]am5t is c\nt$ none4 ,& = x1 y mu/ put a ,ciphre1 if at any "t1 x be 9 mixture74 ,,hote ,c ,e ,,moi/ ,a ,,temp]ate ,b ,,drye ---------------------------------------- -------- ,d ,,cold ,c\nt+ !n f ,b1 #b4 degrees1 t[>d ,a3 y f9de x to be ,dry 9 ! fir/ degree3 ,s is ! ,=me result+ ( ! ,mixture ( ,a1 & ,b1 9 our example4 ,i w geue y an o!r example4 ,suppose1 y haue two ?+es1 z ,c1 & ,d3 & ( ,c1 ! ,heate to be 9 ! #d4 degree3 & ( ,d1 ! ,colde1 to be remisse1 eu5 vnto ! ,temp]am5t4 ,n[1 = ,c1 y take #d3 & = ,d1 y take a ,ciphre3 :i*1 add$ vnto #d1 yelde? onely #d4 ,! middle1 or halfe1 :]e(1 is #b4 ,:]e=e ! ,=me result+ ( ,c1 & ,d1 is ,hote 9 ! second degree3 =1 #b4 degrees1 acc\nt$ f ,c1 t[>d ,d1 ende iu/e 9 ! #b4 degree ( heate4 ,( ! ?ird man]1 ,i w geue also an example3 :i* let be ?is3 ,note4 ,i haue a liquid ,m$ic9e ^: ,qualitie ( heate is 9 ! #d4 degree exalt$3 z 0 ,c1 9 ! example =ego+3 & an o!r liquid ,m$ic9e ,i haue3 ^: ,qualitie1 is heate1 9 ! fir/ degree4 ,( e*e ( ^!1 ,i mixt a l quantitie3 ,subtract "h1 ! lesse fr ! m3 & ! residue diuide into two equall p>tes3 :]e(1 ! "o "p1 ei!r add$ to ! lesse1 or subtract$ f ! high] degree1 do? produce ! degree ( ! ,=me result+1 by ?is mixture ( ,c1 & ,e4 ,z1 if f #d4 ye abate #a4 !re re/e? #c4 ! halfe ( #c4 is #a3 ,adde to #a4 ?is #a3 y haue #b4 ,or subtract f #d4 ?is #a3 y haue likewise #b remayn+4 ,:i* decl>e?1 ! ,=me result+1 to be ,heate1 9 ! middle ( ! ?ird degree4 ,! ,second ,rule4 8,b if ! ,quantities ( two ?+es ,commixt1 be diu]se1 & ! ,int5sions 7( _! ,=mes ,miscible7 be 9 diu]se degrees1 & heig!s4 7,:e!r ^? ,=mes be ( "o k9de1 or ( ,contr>y k9des1 or ( a ,temp]ate & a ,contr>y1 ,:at proportion is ( ! lesse quantitie to ! great]1 ! same %all be ( ! diff]5ce1 :i* is betw5e ! degree ( ! ,=me result+1 & ! degree ( ! great] quantitie ( ! ?+ miscible1 to ! diff]5ce1 :i* is betw5e ! same degree ( ! ,=me result+1 & ! degree ( ! lesse quantitie4 ,z = example4 ,let two p\nd ( ,liquor be geu51 hote 9 ! #d4 degree3 "o p\nd ( ,liquor be geu51 hote 9 ! ?ird degree40 ,i w\ld gladly "k ! ,=me result+1 9 ! ,mixture ( ^! two ,liquors4 ,set d[ne y\r nb]s 9 ord]1 ?us4 ,p4 #b4 ,hote4 #d4 ,p4 #a4 ,hote4 #c4 ,n[ by ! rule ( ,algieb>1 haue ,i deuis$ a v easie1 briefe1 & g5]all man] ( work+ 9 ?is case4 ,let vs fir/1 suppose t ,middle ,=me result+1 to be #ax3 z t ,rule tea*e?4 ,& because 7by our ,rule1 "h geu57 z ! waight ( #a4 is to #b3 ,s is ! diff]5ce betw5e #d4 7! degree ( ! great] quantitie7 & #ax3 to ! diff]5ce betw5e #ax & #c3 7! degree ( ! ?+1 9 lesse qutitie4 ,& ) all1 #ax1 be+ alwayes 9 a c]ta9e middell1 betw5e ! two heig!s or degrees74 ,= ! fir/ diff]5ce1 ,i set #d-ax3 & = ! second1 ,i set #ax-#c4 ,&1 n[ aga9e1 ,i say1 z #a4 is to #b4 s is #d-ax to #ax-#c4 ,:]=e1 ( ^! f\re proportionall numb]s1 ! fir/ & ! f\r? ,multipli$1 "o by ! o!r1 d make z mu*1 z ! second & ! ?ird ,multipli$ ! "o by ! o!r4 ,let ^! ,multiplications be made accord+ly4 ,& ( ! fir/ & ! f\r?1 we haue #ax-#c4 & ( ! second ! ?ird1 #h-bx4 ,:]=e1 our quation is betw5e #ax-#c3 & #h-bx4 ,:i* may be r$uc$1 accord+ to ! ,arte ( ,algieb>3 z1 "h1 add+ #c4 to e*e "p1 geue? ! quation1 ?us1 #ax#aa-bx4 ,& yet aga9e1 contract+1 or ,r$uc+ x3 ,adde to e*e "p1 #bx3 ,!n haue y #cx quall to #aa3 ?us repres5t$ #cx#aa4 ,:]e=e1 diuid+ #aa4 by #c3 ! ,quoti5t is #c3 ! ,valew ( our #ax1 ,coss1 or ,?+1 fir/ suppos$4 ,& t is ! heig?1 or ,int5sion ( ! ,=me result+3 :i* is1 ,heate1 9 two ?irdes ( ! f\r? degree3 ,& "h ,i set ! %ew ( ! worke 9 conclusion1 ?us4 ,! pr\fe h]e( is easie3 by subtract+ #c4 f #c1 re/e? 4 ,subtracte ! same heig? ( ! ,=me result+1 7:i* is #c7 fr #d3 !n re/e? 3 ,y see1 t is d\ble to 3 z #b4,p4 is d\ble to #a4,p4 ,s %\ld x be3 by ! rule "h geu54 ,note4 ,z y add$ to e*e "p ( ! quation1 #c3 s if ye fir/ add$ to e*e "p #bx1 x w\ld /&1 #cx-#ch4 ,& n[ add+ to e*e "p #c3 y haue 7z a=e7 #cx#aa4 ,p4 #b4 ,hote4 #d4 ,! =me #c result+4 ,p4 #a4 ,hote4 #c4 ,& ?\gh ,i1 "h1 speake onely ( two ?yngs ,miscible3 & mo/ commonly mo !n ?ree1 f\re1 fiue or six1 7c47 are to be ,mix$3 7& 9 "o ,comp\nd to be r$uc$3 ! ,=me resultyng ( ! same1 to s]ue ! turne7 yet ^! ,rules are suffici5t3 duely repeat$ & it]at$4 ,note4 ,9 proc$yng fir/1 ) any two3 & !n1 ) ! ,=me ,result+1 & an o!r3 s =?3 ,=1 ! la/ worke1 conclude? ! ,=me resultyng ( !m all3 ,i n$e no?+ to speake1 ( ! ,mixture 7"h suppos$7 :at x is4 ,common ,philosophie ha? def9$ x1 say+1 ,mixtio e/ miscibilium1 alt]atorum1 p] m9ima coniunctorum1 ,vnio4 ,eu]y ^w 9 ! def9ition1 is ( great importance4 ,i n$e n also sp5d any "t1 to %ew1 h[1 ! o!r mann] ( di/ribut+ ( degrees1 do? agree to ^! ,rules4 ,nei!r n$e ,i ( ! f>d] vse belong+ to ! ,crosse ( ,graduation 7be=e describ$7 9 ?is place decl>e1 vnto su* z are capable ( t1 :i* ,i haue all ready sayd4 ,nei!r yet ) examples specifie ! ,manifold v>ieties1 by ! =esayd two g5]all ,rules1 to be ord]$4 ,! witty & ,/udi\s1 "h1 haue suffici5t3 ,& !y :i* are n hable to atte9e to ?is1 )\t liuely tea*+1 & m 9 p>ticul>3 w\ld haue l>g] disc\rs+1 !n is mete 9 ?is place to be dealt )all3 ,& o!r 7p]*aunce7 ) a pr\de snuffe w disda9e ?is litle3 & w\ld be vn?ankefull = mu* m4 ,i1 !r=e conclude3 & wi% su* z haue mode/ & e>ne/ ,philosophicall m9des1 to laude ,god highly = ?is3 & to ,m]uayle1 t ! pr(\nde/ & subtile/ po9t1 conc]n+ ,mixture ( ,=mes & ,qualities ,naturall1 is s ,mat*t & m>y$ ) ! mo/ simple1 easie1 & %ort way ( ! noble ,rule ( ,algieb>4 ,:o c rema9e1 !r=e vnp]suad$1 to l\e1 al[1 & honor ! excell5t ,sci5ce ( ,ari?metike8 ,=1 "h1 y may p]ceiue t ! litle f+] ( ,ari?metike1 is ( m might & contriu+1 !n a hund]d ?\s& m5s wittes1 ( ! middle sorte1 are hable to p]f\rme1 or truely to conclude1 ) out helpe !re(4 ,n[ w we f>d]1 by ! wise & valiant ,capita9e1 be c]tifi$1 :at helpe he ha?1 by ! ,rules ( ,ari?metike3 9 "o ( ! ,artes to him app]ta9+3 ,& ( ! ,grekes nam$ 4 4 8,t is1 ! ,skill ( ,ordr+ ,s\ldi]s 9 ,battell ray aft] ! be/ man] to all purposes40 ,?is ,art s mu* dep5de? vppon ,numb]s vse1 & ! ,ma!maticals1 t lianus 7! be/ writ] !r(17 9 8 worke1 to ! ,emp]\r ,hadrianus1 by 8 p]fection1 9 ! ,ma!maticals1 7beyng great]1 !n o!r be=e him _h17 ?9ke? 8 booke to passe all o!r ! excell5t workes1 writt5 ( t ,art1 vnto 8 dayes4 ,=1 ( x1 _h writt5 neas3 ,cyneas ( ,!ssaly3 ,pyrrhus ,epirota3 & ,alex&] 8 sonne3 ,cle>*us3 ,pausanias3 ,euangelus3 ,polybius1 famili] fr5de to ,scipio3 ,eupolemus3 ,iphicrates1 ,possidonius3 & v _m o!r wor?y ,capita9es1 ,philosoph]s & ,pr9ces ( ,immortall fame & memory3 ,^: fayre/ fl\re ( _! g>l& 79 ?is feat7 0 ,ari?metike3 & a litle p]ceiu]ance1 9 ,geometricall ,figures4 ,b 9 _m o!r cases do? ,ari?metike /& ! ,capita9e 9 great /$e4 ,z 9 proportionyng ( vittayles1 = ! ,army1 ei!r rema9+ at a /ay3 or sudd5ly to be encreas$ ) a c]ta9e numb] ( ,s\ldi]s3 & = a c]ta9 tyme4 ,or by good ,art to dim9i% 8 company1 to make ! victuals1 long] to s]ue ! reman5t1 = a c]ta9e det]m9$ tyme3 if n$e s require4 ,& s 9 sundry 8 o!r acc\ntes1 ,reck5+es1 ,measurynges1 & proportionynges1 ! wise1 exp]t1 & ,circumspect ,capita9e w affirme ! ,sci5ce ( ,ari?metike1 to be "o ( 8 *ief ,c\nsaylors1 direct]s & aid]s4 ,:i* ?+ 7by good meanes7 0 euid5t to ! ,noble1 ! ,c\ragi\s1 ! loyall1 & ,curte\s ,iohn1 late ,e>le ( ,w>wicke4 ,:o 0 a yong ,g5tleman1 ?r\ghly kn[ne to v few4 ,albeit 8 lu/y valiantnes1 =ce1 & ,skill 9 ,*iualr\s feates & ex]cises3 8 humbl5es1 & fr5delynes to all m51 7 ?+es1 op5ly1 ( ! _w p]ceiu$4 ,b :at rotes 7o!rwise17 v]tue _h fa/5$ 9 8 bre/1 :at ,rules ( godly & honorable life he _h fram$ to him selfe3 :at vices1 79 "s !n liu+7 notable1 he tooke great c>e to es*ew3 :at manly v]tues1 9 o!r noble m51 7flori%+ be=e 8 eyes17 he ,sy?+ly aspir$ aft]3 :at pr[esses he purpos$ & m5t to a*ieue3 ) :at feats & ,artes1 he began to furni% & fraught him selfe1 = ! bett] s]uice ( 8 ,kyng & ,c\ntrey1 bo? 9 peace w>re4 ,^! 7,i say7 8 ,h]oicall ,m$itations1 =eca/+es & det]m9ations1 no twayne1 7,i ?9ke7 beside my selfe1 c s p]fectly1 & truely report4 ,& !r=e1 9 ,consci5ce1 ,i c\nt x my "p1 = ! honor1 pref]m5t1 procur+ ( v]tue 7?us1 briefly7 to haue put 8 ,"n1 9 ! ,regi/] ( ,fame ,immortall4 ,to our purpose4 ,?is ,iohn1 by "o ( 8 actes 7besides _m o!r3 bo? 9 ,engl& & ,fraunce1 by me1 9 him not$47 did disclose 8 h>ty l\e to v]tu\s ,sci5ces3 & 8 noble int5t1 to excell 9 ,m>tiall pr[esse3 ,:5 he1 ) humble reque/1 & in/ant ,sollicit+3 got ! be/ ,rules 7ei!r 9 "t pa/ by ,greke or ,roma9e1 or 9 our "t vs$3 & new ,/ratagemes !r9 deuis$7 = ordr+ ( all ,companies1 summes & ,numb]s ( m1 7,_m1 or few7 ) "o k9de ( weapon1 or mo1 appo9t$3 ) ,artill]y1 or )\t3 on horsebacke1 or on fote3 to giue1 or take onset3 to seem _m1 be+ few3 to seem few1 be+ _m4 ,to m>*e 9 battaile or ,iornay3 ) _m su* feates1 to ,f\ght5 field1 ,sk>m\%1 or ,ambu%e app>ta9+3 ,?is noble ,e>le1 dy$ ,anno4 #aeed4 sk>se ( #bd4 ye>es ( age3 hau+ no issue by 8 wife3 ,daught] to ! ,duke ( ,som]set4 ,& ( all ^!1 liuely designem5tes 7mo/ curi\sly7 to be 9 velame p>*em5t describ$3 ) ,notes peculi] m>kes1 z ! ,arte require?3 & all ^! ,rules1 & descriptions ,ari?meticall1 inclos$ 9 a ri*e ,case ( ,gold1 he vs$ to we>e ab\t 8 necke3 z 8 ,iuell mo/ preci\s1 & ,c\nsayl\r mo/ tru/y4 ,?us1 ,ari?metike1 ( him1 0 %ryn$ 9 gold3 ,( ,numb]s frute1 he _h good hope4 ,n[1 ,numb]s !r=e innum]able1 9 ,numb]s prayse1 8 %ryne %all f9de4 ,:at n$e ,i1 7= f>d] pr(e to y7 ( ! ,s*olema/]s ( ,iu/ice1 to require te/imony3 h[ n$efull1 h[ frutefull1 h[ skillfull a ?+ ,ari?metike is8 ,i meane1 ! ,lawy]s ( all sortes4 ,vnd\bt$ly1 ! ,ciuilians1 c m]uayl\sly decl>e3 h[1 nei!r ! ,aunci5t ,roma9e lawes1 )\t good k ( ,numb]s art1 c be p]ceiu$3 ,nor 7,iu/ice 9 inf9ite ,cases7 )\t due proportion1 7n>r[ly consid]$17 is hable to be execut$4 ,h[ ,iu/ly1 ) great k ( ,arte1 did ,pap9ianus in/itute a law ( p>tition1 & all[ance1 betw5e man & wife aft] a diuorce8 ,b h[ ,accursius1 ,baldus1 ,b>tolus1 ,iason1 ,alex&]1 & f9ally ,alciatus1 7be+ o!rwise1 notably well le>n$7 d iumble1 gesse1 & erre1 f ! quity1 art & ,int5t ( ! lawmak]3 ,ari?metike c detect1 & conu9ce3 & cl]ely1 make ! tru? to %9e4 ,good ,b>tolus1 tyr$ 9 ! exam9+ proportion+ ( ! matt]3 & ) ,accursius ,glosse1 mu* cumbr$3 bur/ out1 & sayd3 ,nulla e/ 9 toto libro1 hac glossa difficilior3 ,cuius computationem nec ,s*ola/ici nec ,doctores intelligunt4 c4 ,t is3 ,9 ! :ole booke1 !re is no ,glosse h>d] !n ?is3 ,^: acc\mpt or reck5yng1 nei!r ! ,s*ol]s1 nor ! ,doct\rs vnd]/&4 c4 ,:at c !y say ( ,iulianus law1 ,si ita ,scriptum4 c4 ,( ! ,te/ators w iu/ly p]=m+1 betw5e ! wife1 ,sonne & daught]8 ,h[ c !y p]ceiue ! quitie ( ,aphricanus1 ,ari?meticall ,reck5+1 :]e he treate? ( ,lex ,falcidia8 ,h[ c !y deliu] him1 f 8 ,repr\]s3 & _! ma9ta9]s3 z ,ioannes1 ,accursius ,hypolitus & ,alciatus8 ,h[ ,iu/ly & artificially1 0 ,africanus reck5+ made8 ,proportionat+ to ! ,sommes bequea!d1 ! ,contributions ( e*e "p8 ,namely1 = ! hundr$ pres5tly receiu$1 #ag-a_/g4 ,& = ! hundr$1 receiu$ aft] t5 mone!s1 #ab-f_/g3 :i* make ! #cj3 :i* 7 to be ctribut$ by ! legat>ies to ! heire4 ,=1 :at proportion1 #ajj ha? to #ge3 ! same ha? #ag-a_/g to #ab-f_/g3 ,:i* is ,sesquit]tia3 t is1 z #d1 to #c4 :i* make #g4 ,wond]full _m places1 9 ! ,ciuile law1 require an exp]t ,ari?metici51 to vnd]/& ! deepe ,iudgemt1 ,iu/ det]m9ati ( ! ,aunci5t ,roma9e ,lawmak]s4 ,b mu* m exp]t ought he to be1 :o %\ld be hable1 to decide ) quitie1 ! inf9ite v>ietie ( ,cases1 :i* d1 or may happ51 vnd] eu]y "o ( ^? lawes & ord9ances ,ciuile4 ,h]eby1 easely1 ye may n[ coniecture3 t 9 ! ,canon law3 & 9 ! lawes ( ! ,realme 7:i* ) vs1 be>e ! *ief ,au?oritie71 ,iu/ice & equity might be greately pref]r$1 & skilfully execut$1 ?r\gh due skill ( ,ari?metike1 & proportions app]ta9yng4 ,! wor?y ,philosoph]s1 & prud5t lawmak]s 7:o haue writt5 _m bookes ,de ,republica3 ,h[ ! be/ /ate ( ,common weal!s might be procur$ & ma9te9$17 haue v well det]m9$ ( ,iu/ice3 7:i*1 n onely1 is ! ,base & f\ndacion ( ,common weales3 b also ! totall p]fection ( all our workes1 words1 & ?\ghtes37 def9+ x1 ,iu/ice4 8to be t v]tue1 by :i*1 to eu]y "o1 is r5dr$1 t to him app]ta9e?40 ,god *all5ge? ?is at our h&es1 to be honor$ z ,god3 to be l\$1 z a fa!r3 to be fe>$ z a ,"l ma/]4 ,our neighb\rs proporti1 is also prescrib$ ( ! ,almighty lawmak]3 :i* is1 to d to o!r1 eu5 z we w\ld be done vnto4 ,^! proportions1 are 9 ,iu/ice necess>y3 9 duety1 comm5dable3 & ( ,common weal!s1 ! life1 /r5g?1 /ay & flori%+4 ,ari/otle 9 8 ,e?ikes 7to fat* ! s$e ( ,iu/ice1 & light ( direction1 to vse & execute ! same7 0 fayne to fly to ! p]fection1 & p[] ( ,numb]s3 = proportions ,ari?meticall & ,geometricall4 ,plato 9 8 booke call$ ,ep9omis 7:i* boke1 is ! ,?reasury ( all 8 doctr9e7 :]e1 8 purpose is1 to seke a ,sci5ce1 :i*1 :5 a man _h x1 p]fectly3 he might seme1 & s be1 9 d$e1 ,wise4 ,he1 briefly1 ( o!r ,sci5ces disc\rs+1 f9de? !m1 n hable to br+ x to passe3 ,b ( ! ,sci5ce ( ,numb]s1 he say?4 ,illa1 qu num]um mortalium g5]i d$it1 id pr(ecto efficiet4 ,deum autem aliquem1 magis quam =tunam1 ad salutem no/ram1 hoc munus nobis arbitror contulisse4 c4 ,nam ipsum bonorum omnium ,au?orem1 cur non maximi boni1 ,prud5ti dico1 causam arbitramur8 ,t ,sci5ce1 v]ely1 :i* ha? taught mankynde numb]1 %all be able to bryng x to passe4 ,&1 ,i ?9ke1 a c]ta9e ,god1 r !n =tune1 to haue giu5 vs ?is gift1 = our blisse4 ,=1 :y %\ld we n ,iudge him1 :o is ! ,au?or ( all good ?+s1 to be also ! cause ( ! greate/ good ?yng1 namely1 ,wis$ome8 ,!re1 at l5g?1 he pr\e? ,wis$ome to be atteyn$1 by good ,skill ( ,numb]s4 ,) :i* great ,te/imony1 & ! manifold pr(es1 & reasons1 be=e express$1 y may be suffici5tly & fully p]suad$3 ( ! p]fect ,sci5ce ( ,ari?metike1 to make ?is acc\nte3 ,t ( all ,sci5ces1 next to ,!ologie1 x is mo/ diu9e1 mo/ pure1 mo/ ample & g5]all1 mo/ pr(\nde1 mo/ subtile1 mo/ commodi\s & mo/ necess>y4 ,^: next ,si/]1 is ! ,absolute ,sci5ce ( ,magnitudes3 ( :i* 7by ! ,direction & aide ( him1 ^: ,magnitude is ,inf9ite1 & ( vs ,incompreh5sible7 ,i n[ ent5d1 s to write1 t bo? ) ! ,multitude1 & also ) ! ,magnitude ( ,m]uayl\s & frutefull v]ities1 y 7my fr5des & ,c\ntreym57 may be /ird vp1 & awak$1 to behold :at c]ta9e ,artes & ,sci5ces1 7to our vnspeakable beh(e7 our heau5ly fa!r1 ha? = vs prep>$1 & reueal$1 by sundry ,philosoph]s & ,ma!matici5s4 ,bo?1 ,numb] & ,magnitude1 haue a c]ta9e ,orig9all s$e1 7z x 77 ( an incr$ible prop]ty3 & ( man1 neu] hable1 ,fully1 to be decl>$4 ,( ,numb]1 an ,vnit1 & ( ,magnitude1 a ,poynte1 doo seeme to be mu* l ,orig9all causes3 ,b ! diu]sitie neu]!lesse1 is great4 ,we def9$ an ,vnit1 to be a ?+ ,ma!maticall ,indiuisible3 ,a ,po9t1 likewise1 we sayd to be a ,ma!maticall ?+ ,indiuisible4 ,& f>d]1 t a ,po9t may haue a c]ta9e det]m9$ ,situation3 t is1 t we may assigne1 & prescribe a ,po9t1 to be "h1 !re1 yond]4 c4 ,h]e91 7behold7 our ,vnit is free1 & c abyde no bondage1 or to be ty$ to any place1 or seat3 diuisible or indiuisible4 ,agayne1 by reason1 a ,po9t may haue a ,situation limit$ to him3 a c]ta9e motion1 !r=e 7to a place1 & f a place7 is to a ,po9t incid5t & app]ta9yng4 ,b an ,vnit1 c n be imag9$ to haue any motion4 ,a ,po9t1 by 8 motion1 produce?1 ,ma!matically1 a l9e3 7z we sayd be=e7 :i* is ! fir/ k9de ( ,magnitudes1 & mo/ simple3 ,an ,vnit1 c n produce any numb]4 ,a ,l9e1 ?\gh x be produc$ ( a ,po9t m\$1 yet1 x do? n consi/ ( po9tes3 ,numb]1 ?\gh x be n produc$ ( an ,vnit1 yet do? x ,consi/ ( vnits1 z a mat]iall cause4 ,b =mally1 ,numb]4 ,numb]1 is ! ,vnion1 & ,vnitie ( ,vnits4 ,:i* vnyt+ & knitt+1 is ! workeman%ip ( our m9de3 :i*1 ( di/9ct & discrete ,vnits1 make? a ,numb]3 by vni=mitie1 result+ ( a c]ta9e multitude ( ,vnits4 ,& s1 eu]y numb]1 may haue 8 lea/ "p1 giu53 namely1 an ,vnit3 ,b n ( a ,magnitude1 7no1 n ( a ,lyne17 ! lea/ "p c be giu3 by cause1 inf9itly1 diuision !r(1 may be conceiu$4 ,all ,magnitude1 is ei!r a ,l9e1 a ,pla9e1 or a ,solid4 ,:i* ,l9e1 ,pla9e1 or ,solid1 ( no ,s5se1 c be p]ceiu$1 nor exactly by hd 7any way7 repres5t$3 nor ( ,nature produc$3 ,b1 z 7by degrees7 ,numb] did come to our p]ceiu]ance3 ,s1 by visible =mes1 we are holp5 to imag9e1 :at our ,l9e ,ma!maticall1 is4 ,:at our ,po9t1 is4 ,s precise1 are our ,magnitudes1 t "o ,l9e is no broad] !n an o!r3 = !y haue no br$?3 ,nor our ,pla9es haue any ?icknes4 ,nor yet our ,bodies1 any weight3 be !y neu] s l>ge ( dim5si4 ,our ,bodyes1 we c haue ,small]1 !n ei!r ,arte or ,nature c produce any3 & ,great] also1 !n all ! _w c compreh5d4 ,our lea/ ,magnitudes1 c be diuid$ into s _m p>tes1 z ! greate/4 ,z1 a ,l9e ( an in* long1 7) vs7 may be diuid$ into z _m p>tes1 z may ! diamet] ( ! :ole _w1 f ,ea/ to ,we/3 or any way ext5d$3 ,:at priuil$ges1 ab\e all manual ,arte1 & ,natures might1 haue our two ,sci5ces ,ma!maticall8 to exhibite1 & to deale ) ?+es ( su* p[]1 lib]ty1 simplicity1 puritie1 & p]fection8 ,& 9 !m1 s c]ta9ly1 s ord]ly1 s precisely to proc$e3 z1 excell5t is t workem ,me*anicall ,iudg$1 :o n]e/ c appro*e to ! repres5t+ ( workes1 ,ma!matically demon/rat$8 ,& our two ,sci5ces1 rema9+ pure1 & absolute1 9 _! prop] t]mes1 & 9 _! owne ,matt]3 to haue1 & all[e1 onely su* ,demon/rations1 z are pla9e1 c]ta9e1 vniu]sall1 & ( an t]nall v]itye8 ,geometrie4 ,?is ,sci5ce ( ,magnitude1 8 prop]ties1 conditions1 & app]t5ances3 commonly1 n[ is1 & f ! beg9nyng1 ha? ( all ,philosoph]s1 b5 call$ ,geometrie4 ,b1 v]yly1 ) a "n to base & scant1 = a ,sci5ce ( su* dignitie & ampl5es4 ,&1 p]*aunce1 t "n1 by cmon & secret cons5t1 ( all wisem51 hi!rto ha? b5 suffr$ to remayne3 t x might c>ry ) x a p]petuall memorye1 ( ! fir/ & notable/ b5efite1 by t ,sci5ce1 to common p %ew$3 ,:i* 01 :5 ,b\ndes & m]es ( l& & gr\nd 7 lo/1 & conf\nd$ 7z 9 ,egypt1 ye>ely1 ) ! ou]fl[yng ( ,nilus1 ! greate/ & longe/ riu] 9 ! _w7 or1 t gr\nd bequea!d1 7 to be assign$3 or1 gr\nd sold1 7 to be layd out3 or 7:5 disord] preuail$7 t ,comms 7 di/ribut$ into seu]alties4 ,=1 :]e1 vpon ^! su* l occasis1 ,"s by ignorce1 "s by negligce1 ,"s by fraude1 & "s by viol5ce1 did wrongfully limite1 measure1 encroa*1 or *all5ge 7by pret5ce ( iu/ cont5t1 & measure7 ^? l&es & gr\ndes3 great losse1 disquietnes1 murd]1 & w>re did 7full (t7 ensue3 ,till1 by ,gods m]cy1 & mans ,indu/rie1 ,! p]fect ,sci5ce ( ,l9es1 ,pla9es1 & ,solides 7l a diu9e ,iu/ici]17 gaue vnto eu]y man1 8 owne4 ,! p !n1 by ?is art pleasur$1 & greatly relieu$1 9 _! l&es iu/ measur+3 o!r ,philosoph]s1 writ+ ,rules = l& measur+3 betw5e !m bo?1 ?us1 confirm$ ! "n ( ,geometria1 t is1 7accord+ to ! v etimologie ( ! ^w7 ,l& measur+4 ,:]91 ! p knew no f>d]1 ( ,magnitudes vse1 b 9 ,pla9es3 & ! ,philosoph]s1 ( ?1 _h no feet he>]s1 or ,s*ol]s3 f>d] to disclose vnto1 !n ( flat1 pla9e ,geometrie4 ,& ?\gh1 ^! ,philosoph]s1 knew ( f>d] vse1 & be/ vnd]/ode ! etymologye ( ! worde1 yet ?is "n ,geometria1 0 ( !m apply$ g5]ally to all sortes ( ,magnitudes3 vnlea/1 o!r:ile1 ( ,plato1 & ,py?agoras3 ,:5 !y w\ld precisely decl>e _! owne doctr9e4 ,!n1 0 99 ,plato4 #g4 de ,rep4 99 ,geometria1 ) !m1 ,/udium quod circa planum v]satur4 ,b1 well y may p]ceiue by ,euclides ,elem5tes1 t m ample is our ,sci5ce1 !n to measure ,pla9es3 & no?yng lesse !r9 is t\ght 7( purpose7 !n h[ to measure ,l&4 ,an o!r "n1 !r=e1 mu/ n$es be _h1 = our ,ma!maticall ,sci5ce ( ,magnitudes3 :i* reg>de? nei!r clod1 nor turff3 nei!r hill1 nor dale3 nei!r e>? nor heau53 b is absolute ,mege?ologia3 n crep+ on gr\nd1 & dassel+ ! eye1 ) pole p]*e1 rod or lyne3 b 8liftyng ! h>t ab\e ! heau5s1 by inuisible l9es1 & immortall beames mete? ) ! reflexions1 ( ! light incompreh5sible3 & s procure? ,ioye1 & p]fection vnspeakable40 ,( :i* true vse ( our ,mege?ica1 or ,mege?ologia1 ,diu9e ,plato seem$ to haue good ta/e1 & iudgem5t3 & 7by ! "n ( ,geometrie7 s not$ x3 & w>n$ 8 ,s*ol]s !r(3 z1 9 hys seu5? ,dialog1 ( ! ,common weal?1 may euid5tly be s5e4 ,:]e 79 ,lat97 ?us x is3 "r well translat$3 ,pr(ecto1 nobis hoc non negabunt1 ,quicunque vel paululum quid ,geometri gu/runt1 qu9 hc ,sci5tia1 contr1 omn9o se habeat1 qum de ea loquuntur1 qui 9 ipsa v]santur4 ,9 ,engli%1 ?us4 ,v]ely 7say? ,plato7 :osoeu] haue1 7b eu5 v litle7 ta/$ ( ,geometrie1 w n d5ye vnto vs1 ?is3 b t ?is ,sci5ce1 is ( an o!r condicion1 q contr>y to t1 :i* !y t are ex]cis$ 9 x1 d speake ( x4 ,& !re x foll[e?1 ( our ,geometrie1 ,qud quritur cognosc5di illius gratia1 quod semp] e/1 non eius quod oritur qu&oque int]it4 ,geometria1 eius quod e/ semp]1 ,cognitio e/4 ,attollet igitur 7 ,g5]ose vir7 ad ,v]itatem1 animum3 atque ita1 ad ,philosoph&um prep>abit cogitationem1 vt ad sup]a conu]tamus3 qu1 nunc1 contra qum decet1 ad inf]iora deijcimus4 c4 ,qum maxim igitur prcipi5dum e/1 vt qui prcl>issimam hanc habitt ,civitatem1 nullo modo1 ,geometriam sp]nant4 ,nam qu prt] ipsius propositum1 quodam modo esse vid5tur1 haud exigua sunt4 c4 ,x mu/ n$es be confess$ 7sai? ,plato7 ,t ,geometrie is le>n$1 = ! kn[yng ( t1 :i* is eu]3 & n ( t1 :i*1 9 tyme1 bo? is br$ & is br\ght to an ende4 c4 ,geometrie is ! k ( t :i* is eu]la/yng4 ,x w lift vp !r=e 7,o ,g5tle ,syr7 our mynde to ! ,v]itie3 & by t meanes1 x w prep>e ! ,?\ght1 to ! ,philosophicall l\e ( wisdome3 t we may turne or conu]t1 t[>d heau5ly ?+es bo? mynde & ?\ght :i* n[1 o!rwise !n becomme? vs1 we ca/ d[n on base or inf]ior ?+s4 c4 ,*iefly1 !r=e1 ,commaundem5t mu/ be giu51 t su* z d inhabit ?is mo/ honorable ,citie1 by no meanes1 despise ,geometrie4 ,= eu5 ^? ?+es done by x :i*1 9 mann]1 seame to be1 beside ! purpose ( ,geometrie3 are ( no small importance4 c4 ,& besides ! manifold vses ( ,geometrie1 9 matt]s app]ta9yng to w>re1 he adde? m1 ( second vnpurpos$ frute1 & commoditye1 arris+ by ,geometrie3 say+3 ,scimus qu9 etiam1 ad ,discipl9as omnes facilius p] disc5das1 int]esse omn9o1 attig]it ne ,geometriam aliquis1 an non4 c4 ,hanc ergo ,doctr9am1 secundo loco disc5dam ,iuu5ibus /atuamus4 ,t is4 ,b1 also1 we "k1 t = ! m easy le>nyng ( all ,artes1 x importe? mu*1 :e!r "o haue any k 9 ,geometrie1 or no4 c4 ,let vs !r=e make an ord9ance or decree1 t ?is ,sci5ce1 ( "y m5 %all be le>n$ 9 ! second place4 ,?is 0 ,diu9e ,plato 8 ,iudgem5t1 bo? ( ! purpos$1 *ief1 & p]fect vse ( ,geometrie3 & ( 8 second1 dep5dyng1 d]iuatiue commodities4 ,& = vs1 ,*ri/5 m51 a ?\s& ?\s& mo occasions are1 to haue n$e ( ! helpe ( 99 ,i4 ,d4 99 ,h]e91 ,i w\ld gladly %ake (1 ! e>?ly "n1 ( ,geometrie4 ,mege?ologicall ,contemplations3 :]by1 to trayne our ,imag9ations & ,myndes1 by litle & litle1 to =sake & ab&on1 ! grosse & corruptible ,obiectes1 ( our vtw>d s5ses3 & to appreh5d1 by sure doctr9e demon/ratiue1 ,?+s ,ma!maticall4 ,& by !m1 readily to be holp5 & conduct$ to conceiue1 disc\rse1 & conclude ( ?+s ,intellectual1 ,spirituall1 t]nall1 & su* z conc]ne our ,blisse eu]la/+3 :i*1 o!rwise 7)\t ,speciall priuil$ge ( ,illum9ation1 or ,reuelation fr heau57 ,no mortall mans wyt 7naturally7 is hable to rea* vnto1 or to ,compasse4 ,&1 v]yly1 by my small ,tal5t 7f ab\e7 ,i am hable to pr\e & te/ifie1 t ! litt]all ,text1 & ord] ( our diu9e ,law1 ,oracles1 & ,my/]ies1 require m skill 9 ,numb]s1 & ,magnitudes3 !n 7commonly7 ! expositors haue vtt]$3 b r onely 7at ! mo/7 s w>n$3 %ew$ _! own want !r94 7,to "n any1 is n$eles3 & to note ! places1 is1 "h1 no place3 ,b if ,i be duely ask$1 my answ]e is ready47 ,& )\t ! litt]all1 ,grammaticall1 ,ma!maticall or ,naturall v]ities ( su* places1 by good & c]ta9e ,arte1 p]ceiu$1 no ,spirituall s5se 7propre to ^? places1 by ,absolute ,!ologie7 w !reon dep5d4 8,no man1 !r=e1 c d\te1 b t[>d ! atteyn+ ( k incomp>able1 & ,heau5ly ,wis$ome3 ,ma!maticall ,speculations1 bo? ( ,numb]s & ,magnitudes3 are meanes1 aydes1 & guides3 ready1 c]ta9e1 & necess>y40 ,f h5ce=?1 9 ?is my ,preface1 w ,i frame my talke1 to ,plato 8 fugitiue ,s*ol]s3 or1 r1 to su*1 :o well c1 7& also wil17 vse _! vtw>d s5ses1 to ! glory ( ,god1 ! b5efite ( _! ,c\ntrey1 & _! owne secret cont5tation1 or hone/ pref]m5t1 on ?is e>?ly ,scaffold4 ,to !m1 ,i w ord]ly recite1 describe decl>e a great ,numb] ( ,artes1 f our two ,ma!maticall f\nta9es1 d]iu$ into ! fieldes ( ,nature4 ,:]by1 su* ,s$es1 & ,rotes1 z lye depe hyd 9 ! grod ( ,nature1 are refre%$1 quick5$1 & pr\ok$ to gr[1 %ote vp1 fl\re1 & giue frute1 inf9ite1 & incr$ible4 ,& ^! ,artes1 %albe su*1 z vpon ,magnitudes prop]ties d dep5de1 m1 !n vpon ,numb]4 ,& by good reason we may call !m ,artes1 & ,artes ,ma!maticall ,d]iuatiue3 = 7at ?is tyme7 ,i ,def9e ,an ,arte4 ,an ,arte1 to be a ,me?odicall cplete ,doctr9e1 hau+ abundancy ( suffici5t1 & peculi] matt] to deale )1 by ! all[ance ( ! ,metaphisicall ,philosoph]3 ! k :]e(1 to huma9e /ate is necess>ye4 ,& t ,i acc\nt1 ,art ,ma!maticall ,d]iuatiue4 ,an ,art ,ma!maticall d]iuatiue1 :i* by ,ma!maticall demon/ratiue ,me?od1 9 ,nb]s1 or ,magnitudes1 ordre? & confirme? 8 doctr9e1 z mu* z p]fectly1 z ! matt] subiect w admit4 ,& = t1 ,i ent5d to vse ! "n & prop]tie ( a ,a ,me*aniti54 ,me*anici51 o!rwise1 !n 7hi!rto7 x ha? b5 vs$1 ,i ?9ke x good1 7= di/9ction sake7 to giue y also a brief description1 :at ,i meane !rby4 ,a ,me*anici51 or a ,me*anicall workman is he1 ^: skill is1 )\t k ( ,ma!maticall demon/ration1 p]fectly to worke & f9i%e any s5sible worke1 by ! ,ma!matici5 pr9cipall or d]iuatiue1 demon/rat$ or demon/rable4 ,full well ,i "k1 t he :i* inu5te?1 or make? ^! demon/rations1 is g5]ally call$ ,a speculatiue ,me*anici53 :i* diff]re? no?yng f a ,me*anicall ,ma!matici54 ,s1 9 respect ( diu]se actions1 "o man may haue ! "n ( sundry artes3 z1 "s tyme1 ( a ,logici51 "s tymes 79 ! same matt] o!rwise h&l$7 ( a ,re?orici54 ,( ^! trifles1 ,i make1 7z n[1 9 respect ( my ,preface17 small acc\nt3 to fyle ? = ! f9e h&lyng ( subtile curi\s disput]s4 ,9 o!r places1 !y may commaunde me1 to giue good reason3 & yet1 "h1 ,i w n be vnreasonable4 #a4 ,fir/1 !n1 f ! puritie1 absolut5es1 & ,immat]ialitie ( ,pr9cipall ,geometrie1 is t k9de ( ,geometrie d]iu$1 :i* vulg>ly is c\nt$ ,geometrie3 & is ! ,arte ( ,measur+ s5sible magnitudes1 _! iu/ qutities & cont5tes4 ,geometrie vulg>4 ,?is1 tea*e? to measure1 ei!r at h&3 & ! practis]1 to be by ! ?+ ,measur$3 & s1 by due apply+ ( ,cumpase1 ,rule1 ,squire1 ,y>de1 ,ell1 ,p]*1 ,pole1 ,l9e1 ,gag+ rod1 7or su* l in/rum5t7 to ! ,l5g?1 ,pla9e1 or ,solide measur$1 #a4 99 to be c]tifi$1 ei!r ( ! l5g?1 p]imetry1 or di/ance l9eall3 & ?is is call$1 ,mecometrie4 ,or #b4 99 to be c]tifi$ ( ! cont5t ( any pla9e ,sup]ficies3 :e!r x be 9 gr\nd ,suruey$1 ,borde1 or ,glasse measur$1 or su* l ?+3 :i* measur+1 is nam$ ,embadometrie4 #c4 99 ,or els to vnd]/& ! ,soliditie1 & cont5t ( any bodily ?+3 z ( ,tymb] & ,/one1 or ! cont5t ( ,pits1 ,pondes1 ,wells1 ,vessels1 small great1 ( all fa%ions4 ,:]e1 ( ,w9e1 ,oyle1 ,be]e1 or ,ale vessells1 c1 ! ,measur+1 commonly1 ha? a peculi] "n3 & is call$ ,gag+4 ,& ! g5]all "n ( ^! ,solide measures1 is ,/]eometrie4 #b4 ,or els1 ?is vulg> ,geometrie1 ha? consid]ation to tea* ! practis]1 h[ to measure ?+s1 ) good di/ance betw5e him & ! ?+ measur$3 & to vnd]/& !reby1 ei!r #a4 99 h[ ,f>re1 a ?+ se5e 7on l& or wat]7 is f ! measur]3 & ?is may be call$ ,apomecometrie3 #b4 ,or1 h[ ,high or depe1 ab\e or vnd] ! leuel ( ! measur]s /d+1 any ?+ is1 :i* is s5e on l& or wat]1 call$ ,hypsometrie4 #c4 99 ,or1 x in=me? ! measur]1 h[ ,broad any ?+ is1 :i* is 9 ! measur]s vew3 s x be on ,l& or ,wat]1 situat$3 & may be call$ ,platometrie4 ,?\gh ,i vse "h to condition1 ! ?+ measur$1 to be on ,l&1 or ,wat] ,situat$3 ,note4 yet1 "k = c]ta9e1 t ! sundry heig! ( ,cl\des1 blas+ ,/>res1 & ( ! ,mone1 may 7by ^! meanes7 haue _! di/ances f ! e>?3 &1 ( ! blas+ ,/>res & ,mone1 ! ,soliditie 7aswell z di/ances7 to be measur$3 ,b because1 nei!r ^! ?+s are vulg>ly taught3 nor ( a common practis] s ready to be execut$3 ,i1 r1 let su* measures be reck5$ incid5t to "s ( our o!r ,artes1 deal+ ) ?+es on high1 m purposely1 !n ?is vulg> ,l& measur+ ,geometrie do?3 z 9 ,p]spectiue & ,a/ronomie1 c4 ,( ^! ,feates 7f>!r appli$7 is ,sprong ! ,feate ( ,geodesie1 or ,l& ,measur+3 m cunn+ly to measure ,suruey ,l&1 ,woods1 & ,wat]s1 a f>re (4 ,m cunn+ly1 ,i say3 ,b ,god kn[e? 7hi!rto7 9 ^! ,realmes ( ,engl& & ,irel& 7:e!r ?r\gh ignorance or fraude1 ,i c n tell1 9 eu]y p>ticul>7 ,note4 h[ great wrong & iniurie ha? 79 my "t7 b5e committ$ by vntrue measur+ & suruey+ ( ,l& or ,woods1 any way4 ,&1 ?is ,i am sure3 t ! ,value ( ! diff]5ce1 betw5e ! tru? & su* ,surueyes1 w\ld haue b5e hable to haue fod 7= eu]7 9 e*e ( our two ,vniu]sities1 an excell5t ,ma!maticall ,read]3 to e*e1 all[+ 7ye>ly7 a hundr$ ,m>kes ( lawfull money ( ?is realme3 :i*1 9 d$e1 w\ld seme requisit1 "h1 to be _h 7?\gh by o!r wayes pr\id$ =7 z well1 z1 ! fam\s ,vniu]sitie ( ,p>is1 ha? two ,ma!maticall ,read]s3 & e*e1 two hundre? ,fr5* ,cr[nes ye>ly1 ( ! ,fr5* ,k+es magnific5t lib]alitie onely4 ,n[1 aga9e1 to our purpose return+3 ,more\]1 ( ! =m] k ,geometricall1 are gr[5 ! ,skills ( ,geographie1 ,*orographie1 ,hydrographie1 & ,/rat>i?metrie4 8,geographie tea*e? wayes1 by :i*1 9 sdry =mes1 7z ,sphrike1 ,pla9e or o!r71 ! ,situation ( ,cities1 ,t[nes1 ,villages1 ,=tes1 ,ca/ells1 ,m\nta9es1 ,woods1 ,hau5s1 ,riu]s1 ,crekes1 su* o!r ?+s1 vp ! outface ( ! e>?ly ,globe 7ei!r 9 ! :ole1 or 9 "s pr9cipall mb] & portion !r( ctayn$7 may be describ$ & design$1 9 cm5surations ,analogicall to ,nature & v]itie3 & mo/ aptly to our vew1 may be repres5t$40 ,( ?is ,arte h[ great pleasure1 & h[ manifolde commodities d come vnto vs1 daily & h\rely3 ( mo/ m51 is p]ceau$4 ,:ile1 "s1 to beautifie _! ,halls1 ,p>l]s1 ,*amb]s1 ,gal]ies1 ,/udies1 or ,libr>ies )3 o!r "s1 = ?+es pa/1 z battels f\ght1 e>?quakes1 heau5ly fyr+es1 su* occur5tes1 9 hi/ories m5tion$3 !rby liuely1 z x 71 to vewe ! place1 ! region adioyn+1 ! di/ance f vs3 & su* o!r circum/ances4 ,"s o!r1 pres5tly to vewe ! l>ge dom9ion ( ! ,turke3 ! wide ,empire ( ! ,mos*\ite3 & ! litle morsell ( gr\nd1 :]e ,*ri/5dome 7by pr(ession7 is c]ta9ly kn[54 ,litle1 ,i say1 9 respecte ( ! re/4 c4 ,"s1 ei!r = _! owne iorneyes direct+ into f>re l&es3 or to vnd]/& ( o!r m5s trauailes4 ,to conclude1 "s1 = "o purpose3 & "s1 = an o!r1 like?1 l\e?1 gette?1 & vse?1 ,mappes1 ,*>tes1 ,geographicall ,globes4 ,( ^: vse1 to speake suffici5tly1 w\ld require a booke peculi]4 ,*orographie seeme? to be an vnd]l+1 & a twig1 ( ,geographie3 & yet neu]!lesse1 is 9 practise manifolde1 & 9 vse v ample4 8,?is tea*e? ,analogically to describe a small portion or circuite ( gr\nd1 ) ! cont5tes3 n reg>d+ :at comm5suration x ha? to ! :ole1 or any p>cell1 )\t x1 conta9$4 ,b 9 ! t]ritory or p>cell ( gr\nd :i* x take? 9 h& to make description (1 x leaue? out 7or vndescrib$7 no notable1 or odde ?+1 ab\e ! gr\nd visible4 ,yea & sometimes1 ( ?+es vnd] gr\nd1 geue? "s peculi] m>ke3 or w>n+3 z ( ,mettall m9es1 ,cole pittes1 ,/one qu>ries4 c40 ,?us1 a ,duk$ome1 a ,%i]e1 a ,lord%ip1 or lesse1 may be describ$ di/9ctly4 ,b m>ueil\s pleasant1 & pr(itable x is1 9 ! exhibit+ to our eye1 & comm5suration1 ! plat ( a ,citie1 ,t[ne1 ,=te1 or ,pallace1 9 true ,symmetry3 n appro*+ to any ( !m3 & out ( ,gunne %ot4 c4 ,h]eby1 ! ,ar*itect may furni%e him selfe1 ) /ore ( :at patt]ns he like?3 to 8 great in/ruction3 eu5 9 ^? ?+es :i* outw>dly are proportion$3 ei!r simply 9 !m selues3 or respectiuely1 to ,hilles1 ,riu]s1 ,hau5s1 & ,woods adioyn+4 ,"s also1 t]me ?is p>ticul> description ( places1 ,topographie4 8,hydrographie1 deliu]e? to our k1 on ,globe or 9 ,pla9e1 ! p]fect ,analogicall description ( ! ,ocean ,sea coa/es1 ?r\gh ! :ole _w3 or 9 ! *iefe & pr9cipall p>tes !re(30 ) ! ,iles & *iefe p>ticul> places ( daung]s1 conteyn$ )9 ! b\ndes1 & ,sea coa/es describ$3 z1 ( ,quicks&es1 ,bankes1 ,pittes1 ,rockes1 ,races1 ,c\nt]tides1 ,:orlepooles4 c4 ,?is1 deale? ) ! ,elem5t ( ! wat] *iefly3 z ,geographie did pr9cipally take ! ,elem5t ( ! ,e>!s description 7) 8 app]t5ances7 to taske4 ,& besides ?ys1 ,hydrographie1 require? a p>ticul> ,regi/] ( c]ta9e ,l&m>kes 7:]e m>kes may be _h7 f ! sea1 well hable to be skri$1 9 :at po9t ( ! ,seacumpase !y appe>e1 & :at app>5t =me1 ,situation1 & bignes !y haue1 9 respecte ( any daung]\s place 9 ! sea1 or n]e vnto x1 assign$3 ,& 9 all ,coa/es1 :at ,mone1 make? full ,sea3 & :at way1 ! ,tides & ,ebbes1 come & g1 ! ,hydrograph] ought to recorde4 ,! ,s\nd+es likewise3 & ! ,*anels wayes3 _! numb]1 & dep!s ord9>ily1 at ebbe & flud1 ought ! ,hydrograph]1 by obs]uation & dilig5ce ( ,measur+1 to haue c]ta9ly kn[54 ,& _m o!r po9tes1 are belong+ to p]fecte ,hydrographie1 & = to make a ,rutt]1 by3 ( :i*1 ,i n$e n "h speake3 z ( ! describ+1 9 any place1 vpon ,globe or ,pla9e1 ! #cb4 po9tes ( ! ,compase1 truely3 7:](1 sc>sly f\re1 9 ,engl&1 haue "r k3 bycause1 ! l9es !r(1 are no /raight l9es1 nor ,circles47 ,( mak+ due proiection ( a ,sph]e 9 pla9e4 ,( ! ,v>iacion ( ! ,compas1 f true ,nor!3 ,& su* l matt]s 7( great importance1 all7 ,i leaue to speake (1 9 ?is place3 bycause1 ,i may seame 7al ready7 to haue enl>g$ ! b\ndes1 & duety ( an ,hydograph]1 mu* m1 !n any man 7to ?is day7 ha? not$1 or prescrib$4 ,yet am ,i well hable to pr\e1 all ^! ?+es1 to app]ta9e1 & also to be prop] to ! ,hydrograph]4 ,! *ief vse & ende ( ?is ,art1 is ! ,art ( ,nauigation3 b x ha? o!r diu]se vses3 eu5 by !m to be enioy$1 t neu] lacke sight ( l&4 ,/rat>i?metrie1 is ! ,skill1 7app]ta9yng to ! w>re17 by :i* a man c set 9 figure1 analogicall to any ,geometricall figure appo9t$1 any c]ta9e numb] or summe ( m53 ( su* a figure capable3 7by reason ( ! vsuall spaces betw5e ,s\ldi]s all[$3 & = t1 ( m51 c be made no ,fractions4 ,yet1 neu]!les1 he c ord] ! giu5 summe ( m51 = ! greate/ su* figure1 t ( !m1 c be ordr$7 & c]tifie1 ( ! ou]plus3 7if any be7 & ( ! next c]ta9e summe1 :i*1 ) ! ou]plus1 w admit a figure exactly proportionall to ! figure assign$4 ,by :i* ,skill1 also1 ( any army or company ( m53 7! figure sides ( ^: ord]ly /&+1 or array1 is kn[57 he is able to expresse ! iu/ numb] ( m51 )9 t figure conte9$3 or 7ord]ly7 able to be conte9$4 99 ,note4 99 ,& ?is figure1 & sides !r(1 he is hable to "k3 ei!r beyng by1 & at h&3 or a f>re (4 ,?us f>re1 /ret*e? ! description & prop]ty ( ,/rat>i?metrie3 suffici5t = ?is tyme & place4 ,! diff]5ce betw5e ,/rat>i?metrie & ,tacticie4 8,x diff]re? f ! ,feate ,tacticall1 ,de aciebus in/ru5dis4 bycause1 !re1 is necess>y ! wis$ome & =esight1 to :at purpose he s ordre? ! m53 & ,skillfull hability1 also1 = any occasion1 or purpose1 to deuise & vse ! apte/ & mo/ necess>y ord]1 array & figure ( 8 ,company & ,summe ( m540 ,by figure1 ,i meane3 z1 ei!r ( a ,p]fect ,squ>e1 ,triangle1 ,circle1 ,ouale1 long squ>e1 7( ! ,grekes x is call$ ,et]omekes7 ,rhombe1 ,rhombod1 ,lunul>1 ,ryng1 ,s]p5t9e1 & su* o!r ,geometricall figures3 ,:i*1 9 w>res1 haue b51 & are to be vs$3 = commodi\snes1 necessity1 & auauntage c4 ,& no small skill ought he to haue1 t %\ld make true report1 or n]e ! tru?1 ( ! numb]s & ,summes1 ( footem5 or horsem51 9 ! ,enemyes ordr+4 ,a f>re (1 to make an e/imate1 betw5e n]e t]mes ( ,m & ,lesse1 is n a ?yng v rife1 among ^? t gladly w\ld d x4 ,i4 ,d4 ,fr5de1 y w f9de x h>d1 to p]=me my description ( ?is ,feate4 ,b by ,*orographie1 y may helpe y\r selfe "s :at3 :]e ! ,figures kn[ne 79 ,sides & ,angles7 are n ,regul>3 ,& :]e1 ,resolution into ,triangles c s]ue4 c4 ,& yet y w f9de x /range to deale ?us g5]ally ) ,ari?meticall figures3 &1 t = ,battayle ray4 ,_! cont5tes1 diff] s mu* f l ,geometricall ,figures4 ,great pollicy may be vs$ ( ! ,capita9es1 7at tymes fete1 & 9 places conu5i5t7 z to vse ,figures1 :i* make greate/ %ew1 ( s _m z he ha?3 & vs+ ! aduauntage ( ! ?ree k9des ( vsuall spaces3 7betw5e footem5 or horsem57 to take ! l>ge/3 or :5 he w\ld seme to haue few1 7beyng _m37 contr>ywise1 9 ,figure1 & space4 ,! ,h]ald1 ,purseuant1 ,s]geant ,royall1 ,capita9e1 or :o soeu] is c>efull to come n]e ! tru? h]e91 besides ! ,iudgem5t ( 8 exp]t eye1 8 skill ( ,ord]+ ,tacticall1 ! helpe ( 8 ,geometricall in/rum5t3 ,r+1 or ,/affe ,a/ronomicall3 7commodi\sly fram$ = c>iage & vse7 ,he may wond]fully helpe him selfe1 by p]spectiue ,glasses4 ,9 :i*1 7,i tru/7 our po/]ity w pr\e m skillfull & exp]t1 & to great] purposes1 !n 9 ^! dayes1 c 7almo/7 be cr$it$ to be possible4 ,?us haue ,i lightly pass$ ou] ! ,artificiall ,feates1 *iefly dep5dyng vpon vulg> ,geometrie3 commonly & g5]ally reck5$ vnd] ! "n ( ,geometrie4 ,b !re are o!r 7v _m7 ,me?odicall ,artes1 :i*1 declyn+ f ! purity1 simplicitie1 & ,immat]iality1 ( our ,pr9cipall ,sci5ce ( ,magnitudes3 d yet neu]!les vse ! great ayde1 direction1 & ,me?od ( ! sayd pr9cipall ,sci5ce1 & haue propre names1 & di/9ct3 bo? f ! ,sci5ce ( ,geometrie1 7f :i* !y are d]iu$7 & "o f ! o!r4 ,z ,p]spectiue1 ,a/ronomie1 ,musike1 ,cosmographie1 ,a/rologie1 ,/atike1 ,an?ropographie1 ,tro*ilike1 ,helicosophie1 ,pneumati?mie1 ,m5adrie1 ,hypogeiodie1 ,hydragogie1 ,horometrie1 ,zographie1 ,ar*itecture1 ,nauigation1 ,?aumaturgike & ,ar*ema/rie4 ,i ?9ke x necess>y1 ord]ly1 ( ^! to giue "s peculi] descriptions3 & )all1 to t\* "s ( _! commodi\s vses1 & s to make ?is ,preface1 to be a little swete1 pleasant ,nosegaye = y3 to com=t y\r ,spirites1 beyng almo/ out ( c\rage1 & 9 despayre1 7?r\gh bruti% brute7 ,we5yng t ,geometrie1 _h b s]u$ = buildyng ( an h\se1 or a curi\s bridge1 or ! r\fe ( ,we/m9/] hall1 or "s witty pretty deuise1 or engyn1 appropriate to a ,c>p5t]1 or a ,ioyn] c4 ,t ! ?+ is f>re o!rwise1 !n ! _w1 7commonly7 to ?is day1 ha? dem$1 by worde & worke1 good pr(e wilbe made4 ,among ^! ,artes1 by good reason1 ,p]spectiue ought to be _h1 ere ( ,a/ronomicall ,app>5ces1 p]fect k c be atteyn$4 ,& bycause ( ! pr]ogatiue ( ,light1 beyng ! fir/ ( ,gods ,creatures3 & ! eye1 ! light ( our body1 & 8 ,s5se mo/ mighty1 & 8 organ mo/ ,artificiall & ,geometricall3 ,at ,p]spectiue1 we w begyn !r=e4 ,p]spectiue1 is an ,art ,ma!maticall1 :i* demon/rate? ! man]1 & prop]ties1 ( all ,radiations ,direct1 ,brok51 & ,reflect$4 ,?is ,description1 or ,notation1 is brief3 b x rea*e? s f>re1 z ! _w is wyde4 ,x conc]ne? all ,creatures1 all ,actions1 & passions1 by ,emanation ( beames p]f\rm$4 ,beames1 or naturall l9es1 7"h7 ,i meane1 n ( light onely1 or ( col\r 7?\gh !y1 to eye1 giue %ew1 witnes1 & pr(e1 :]by to gr\nd ! ,arte vpon7 b also ( o!r ,=mes1 bo? ,sub/antiall1 & ,accid5tall1 ! c]ta9e & det]m9$ actiue ,radiall emanations4 ,by ?is ,art 7omitt+ to speake ( ! highe/ po9tes7 we may vse our eyes1 & ! light1 ) great] pleasure3 & p]fect] ,iudgem5t3 bo? ( ?+s1 9 light se51 ( o!r3 :i* by l ord] ( ,lightes ,radiations1 worke & produce _! effectes4 ,we may be a%am$ to be ignorant ( ! cause1 :y s sundry wayes our eye is deceiu$1 & abus$3 z1 :ile ! eye we5e? a rod ,globe or ,sph]e 7beyng f>re (7 to be a flat & pla9e ,circle1 & s likewise iudge? a pla9e ,squ>e1 to be rod3 suppose? walles p>allels1 to appro*e1 a f>re (3 r(e & fl\re p>allels1 ! "o to b5d d[nw>d1 ! o!r to rise vpw>d1 at a little di/ance f y4 ,aga9e1 ( ?+es be+ 9 l swiftnes ( m\+1 to ?9ke ! n]]1 to m\e fa/]3 & ! f>d]1 mu* sl[]4 ,nay1 ( two ?+es1 :]( ! "o 7incomp>ably7 do? m\e swift] !n ! o!r1 to deme ! sl[] to m\e v swift1 ! o!r to /&3 :at an error is ?is1 ( our eye8 ,( ! ,raynb[1 bo? ( 8 ,col\rs1 ( ! ord] ( ! col\rs1 ( ! bignes ( x1 ! place & hei? ( x1 7c7 to "k ! causes demon/ratiue1 is x n pleasant1 is x n necess>y8 ( two or ?ree ,sonnes appe>+3 ( ,blas+ ,/]res3 & su* l ?+es3 by naturall causes1 br\ght to passe1 7& yet neu]!les1 ( f>d] matt]1 ,significatiue7 is x n commodi\s = man to "k ! v true cause1 occasion ,naturall8 ,yea1 r1 is x n1 greatly1 aga9/ ! ,s\]a9ty ( ,mans nature1 to be s ou]%ot & abus$1 ) ?+es 7at h&7 be=e 8 eyes8 z ) a ,pecockes tayle1 & a ,d\es necke3 or a :ole ore1 9 wat]1 hold51 to seme brok54 ,?ynges1 f>re (1 to seeme n]e3 & n]e1 to seme f>re (4 ,small ?+es1 to seme great3 & great1 to seme small4 ,"o man1 to seme an ,army4 ,or a man to be cur/ly affray$ ( 8 owne %add[4 ,yea1 s mu*1 to fe>e1 t1 if y1 be+ 7alone7 n]e a c]ta9e glasse1 & pr(f]1 ) dagg] or sword1 to foyne at ! glasse1 y %all sudd5ly be m\$ to giue backe 79 man]7 by reason ( an ,image1 ,a m>ueil\s ,glasse4 appe>+ 9 ! ayre1 betw5e y ! glasse1 ) l h&1 sword or dagg]1 ) l quicknes1 foyn+ at y\r v eye1 likewise z y d at ! ,glasse4 ,/raunge1 ?is is1 to he>e (3 b m m]uail\s to behold1 !n ^! my wordes c signifie4 ,& neu]!lesse by demon/ration ,opticall1 ! ord] & cause !r(1 is c]tifi$3 eu5 s1 z ! effect is consequ5t4 ,yea1 ?us mu* m1 d>e ,i take vpon me1 t[>d ! satisfy+ ( ! noble c\rrage1 t longe? ard5tly = ! wis$ome ( ,causes ,naturall3 z to let him vnd]/&1 t1 9 ,london1 he may ) 8 owne eyes1 haue pr(e ( t1 :i* ,i haue sayd h]e94 ,a ,g5tleman1 7:i*1 = 8 good s]uice1 done to 8 ,c\ntrey1 is fam\s & honorable3 ,s4 ,w4 ,p4 & = skill 9 ! ,ma!maticall ,sci5ces1 & ,languages1 is ! ,od man ( ?is l&4 c47 eu5 he1 is hable3 & 7,i am sure7 w1 v will+ly1 let ! ,glasse1 & pr(e be s5e3 & s ,i 7"h7 reque/ him3 = ! encrease ( wis$ome1 9 ! honorable3 & = ! /opp+ ( ! m\!s malici\s3 & repress+ ! arrogancy ( ! ignorant4 ,ye may easily gesse1 :at ,i meane4 ,?is ,art ( ,p]spectiue1 is ( t excell5cy1 & may be l$1 to ! c]tify+1 & execut+ ( su* ?+es1 z no man w\ld easily beleue3 )\t ,actuall pr(e p]ceiu$4 ,i speake no?+ ( ,naturall ,philosophie1 :i*1 )\t ,p]spectiue1 c n be fully vnd]/&$1 nor p]fectly atte9$ vnto4 ,nor1 ( ,a/ronomie3 :i*1 )\t ,p]spectiue1 c n well be gr\nd$3 ,nor ,a/rologie1 naturally ,v]ifi$1 & au\*$4 ,t "p h]e(1 :i* deale? ) ,glasses 7:i* "n1 ,glasse1 is a g5]all "n1 9 ?is ,arte1 = any ?+1 f :i*1 a ,beame reb\nde?7 is call$ ,catoptrike3 & ha? s _m vses1 bo? m]ueil\s1 & pr(fitable3 t1 bo?1 x w\ld hold me to long1 to note !r9 ! pr9cipall conclusions1 all ready kn[ne3 ,& also 7p]*aunce7 "s ?+es1 might lacke due cr$ite ) y3 ,& ,i1 !rby1 to leese my labor3 & y1 to slip into light ,iudgem5t1 99 ,be=e y haue le>n$ suffici5tly ! p[re ( ,nature & ,arte4 ,n[1 to proc$e3 ,a/ronomie1 is an ,arte ,ma!maticall1 :i* demon/rate? ! di/ance1 magnitudes1 & all naturall motions1 app>5ces1 & passions propre to ! ,planets & fix$ ,/]res3 = any "t pa/1 pres5t & to come3 9 respect ( a c]ta9e ,horizon1 or )\t respect ( any ,horizon4 ,by ?is ,arte we are c]tifi$ ( ! di/ance ( ! ,/>ry ,skye1 & ( e*e ,planete f ! ,c5tre ( ! ,e>?3 & ( ! greatnes ( any ,fix$ />re s5e1 or ,planete1 9 respect ( ! ,e>!s greatnes4 ,z1 we are sure 7by ?is ,arte7 t ! ,solidity1 ,mass9es & ,body ( ! ,sonne1 conte9e? ! quantitie ( ! :ole ,e>? & ,sea1 a hundr$ ?re score & two times1 lesse by "o eight p>te ( ! e>?4 ,b ! ,body ( ! :ole e>?ly globe & ,sea1 is bigg] !n ! body ( ! ,mone1 ?ree & =ty times lesse by ( ! ,mone4 ,:]=e ! ,sonne is bigg] !n ! ,mone1 #gjjj times1 lesse1 by #ei #ci_/fd t is1 precisely #fidj #be_/fd bigg] !n ! ,mone4 ,& yet ! vnskillfull man1 w\ld iudge !m a l bigge4 ,:]=e1 ( ,necessity1 ! "o is mu* f>d] f vs1 !n ! o!r4 ,! ,sonne1 :5 he is f>de/ f ! e>? 7:i*1 n[1 9 our age1 is1 :5 he is 9 ! #h4 degree1 ( ,canc]7 is1 #aagi ,semidiamet]s ( ! ,e>?1 di/ante4 ,& ! ,mone :5 %e is f>de/ f ! e>?1 is #fh ,semidiamet]s ( ! e>? & ,! n]e/1 t ! ,mone comme? to ! e>?1 is ,semidiamet]s #eb ,! di/ance ( ! ,/>ry ,skye is1 fr vs1 9 ,semidiamet]s ( ! e>? #bjjha ,tw5ty ?\s& f\rescore1 "o1 & almo/ a halfe4 ,subtract f ?is1 ! ,mones n]e/ di/ance1 f ! ,e>?3 & !r( rema9e? ,semidiamet]s ( ! e>? #bjjbi ,tw5ty ?\s& n9e & tw5ty & a qu>t]4 ,note4 ,s ?icke is ! heau5ly ,palace1 t ! ,planetes haue all _! ex]cise 91 & mo/ m]uail\sly p]f\rme ! ,commadem5t & ,*>ge to !m giu5 by ! omnipot5t ,maie/ie ( ! k+ ( k+s4 ,?is is t1 :i* 9 ,g5esis is call$ ,ha ,rakia4 ,consid] x well4 ,! ,semidiamet] ( ! e>?1 cte9e? ( our common miles #cdcf #d_/aa ?ree ?\s&1 f\re hundr$ ?irty six & f\re eleu5? p>tes ( "o myle3 ,su* z ! :ole e>? & ,sea1 r\nd ab\t1 is #bafjj4 ,"o & tw5ty ?\s& six hundr$ ( our myles4 ,all[yng = eu]y degree ( ! greate/ circle1 ?re score myles4 ,n[ if y way well ) y\r selfe b ?is litle p>cell ( frute ,a/ronomicall1 z conc]n+ ! bignesse1 ,di/ances ( ,sonne1 ,mone1 ,/]ry ,sky1 & ! huge mass9es ( ,ha ,rakia1 w y n f9de y\r ,consci5ces m\$1 ) ! k+ly ,prophet1 to s+ ! confession ( ,gods ,glory1 & say1 ,! ,heau5s decl>e ! glory ( ,god1 & ! ,firmam5t ,ha ,rakia %ewe? =? ! workes ( 8 h&es4 ,& s =?1 = ^? fiue fir/ /aues1 ( t k+ly ,psalme4 ,well1 well1 ,x is "t = "s to lay hold on wis$ome1 & to ,iudge truly ( ?+es3 & notso to exp\nd ! ,holy ^w1 all by ,allegories3 z to ,neglect ! wis$ome1 p[re & ,goodnes ( ,god1 91 & by 8 ,creatures1 & ,creation to be se5 & le>n$4 ,by p>ables & ,analogies ( ^: natures & prop]ties1 ! c\rse ( ! ,holy ,scripture1 also1 decl>e? to vs v _m ,my/]ies4 ,! :ole ,frame ( ,gods ,creatures1 7:i* is ! :ole _w17 is to vs1 a bright glasse3 f :i*1 by reflexion1 reb\nde? to our k & p]ceiu]ance1 ,beames1 & ,radiations3 repres5t+ ! ,image ( 8 ,inf9ite goodnes1 ,omnipotcy1 & wis$ome4 ,& we !rby1 are taught & p]suad$ to ,glorifie our ,creator1 z ,god3 & be ?ankefull !r=e4 ,c\ld ! ,hea!ni/es f9de ^! vses1 ( ^! mo/ pure1 beawtifull1 & ,mighty ,corporall ,creatures3 & %all we1 aft] t ! true ,sonne ( rightwis5esse is ris5 ab\e ! ,horizon1 ( our temporall ,hemisphrie1 & ha? s abundantly /ream$ into our h>tes1 ! direct beames ( 8 goodnes1 m]cy1 & grace3 ,^: heat ,all ,creatures feele3 ,spirituall & ,corporall3 ,visible & ,inuisible4 ,%all we 7,i say7 looke vpon ! ,heau51 ,/]res1 & ,planets1 z an ,oxe & an ,asse do?3 no furd] c>efull or inquisitiue1 :at !y are3 :y 7 !y ,creat$1 ,h[ d !y execute t !y 7 ,creat$ =8 ,se+1 ,all ,creatures1 7 = our sake creat$3 & bo? we1 & !y1 ,creat$1 *iefly to glorifie ! ,almighty ,creator3 & t1 by all meanes1 to vs possible4 ,nolite ignor>e 7sai? ,plato 9 ,ep9omis7 ,a/ronomiam1 ,sapi5tissim quiddam esse4 ,be ye n ignorant1 ,a/ronomie to be a ?yng ( excell5t wis$ome4 ,a/ronomie1 0 to vs1 f ! beg9n+ comm5d$1 & 9 man] commaund$ by ,god him selfe4 ,9 asmu* z he made ! ,sonne1 ,mone1 & ,/]res1 to be to vs1 = ,signes1 & k ( ,seasons1 & = ,di/9ctions ( ,dayes1 & ye>es4 ,_m wordes n$e n4 ,b ,i wi%1 eu]y man %\ld way ?is ^w1 ,signes4 ,& besides t1 conf]re x also ) ! t5? ,*apt] ( ,hi]emie4 ,& ?\gh ,"s ?9ke1 t !re1 !y haue f\nd a rod3 ,yet ,mode/ ,reason1 w be indiff]5t ,iudge1 :o ought to be beat5 !r)1 9 respect ( our purpose4 ,leau+ t3 ,i pray y vnd]/& ?is3 t )\t great dilig5ce ( ,obs]uation1 exam9ation & ,calculation1 _! p]iods & c\rses 7:]by ,di/9ction ( ,seasons1 ye>es1 & ,new ,mones might precisely be kn[ne7 c\ld n exactely be c]tifi$4 ,:i* ?+ to p]=me1 is t ,art1 :i* we "h haue ,def9$ to be ,a/ronomie4 ,:]by1 we may haue ! di/9ct ,c\rse ( ,times1 dayes1 ye>es1 & ,ages3 aswell = ,consid]ati ( ,sacr$ ,prophesies1 accompli%$ 9 due "t1 =etold3 z = high ,my/icall ,solemnities hold+3 ,& = all o!r huma9e affaires1 ,conditions1 & c\5antes1 vpon c]ta9e "t1 betw5e man & man3 ) _m o!r great vses3 ,:]91 7v]ely71 w\ld be great inc]ta9ty1 ,confusion1 vntru?1 & bruti% ,b>b>\snes3 )\t ! wond]full dilig5ce & skill ( ?is ,arte3 cont9ually le>n+1 & det]m9+ ,times1 & p]iodes ( ,"t1 by ! ,record ( ! heau5ly booke1 :]9 all times are writt53 & to be read ) an ,a/ronomicall /affe1 9 /$e ( a fe/ue4 ,musike1 ( ,motion1 ha? 8 ,orig9all cause3 ,!r=e1 aft] ! motions mo/ swift1 & mo/ ,sl[1 :i* are 9 ! ,firmam5t1 ( ,nature p]=m$3 & vnd] ! ,a/ronom]s ,consid]ation3 n[ ,i w ,speake ( an o!r k9de ( ,motion1 produc+ s\nd1 audible1 & ( ,man num]able4 ,musike ,i call "h t ,sci5ce1 :i* ( ! ,grekes is call$ ,h>monice4 ,n m$l+ ) ! ,contr\]sie betw5e ! aunci5t ,h>moni/es1 & ,canoni/es4 ,musike is a ,ma!maticall ,sci5ce1 :i* tea*e?1 by s5se & reason1 p]fectly to iudge1 & ord] ! diu]sities ( s\ndes1 hye & l[4 ,a/ronomie & ,musike are ,si/]s1 sai? ,plato4 ,z1 = ,a/ronomie1 ! eyes3 ,s1 = ,h>moni\s ,motion1 ! e>es 7 made4 ,b z ,a/ronomie ha? a m diu9e ,contemplation1 & cmodity1 !n mortall eye c p]ceiue3 ,s1 is ,musike to be consid]$1 #a4 t ! 99 ,m9de may be pref]r$1 be=e ! e>e4 ,& f audible s\nd1 we ought to asc5de1 to ! exam9ation3 :i* numb]s are ,h>moni\s1 & :i* n4 ,& :y1 ei!r1 ! "o are3 or ! o!r are n4 ,i c\ld at l>ge1 #b4 9 ! heau5ly 99 motions & di/ances1 describe a m]uail\s ,h>monie1 ( ,py?agoras ,h>pe #c4 ) eight /r+es4 ,also1 som:at might be sayd ( ,m]curius 99 two ,h>pes1 #d4 e*e ( f\re ,/r+es ,elem5tall4 ,& v /raunge matt]1 might be all$g$ ( ! ,h>monie1 #e4 to our 99 ,spirituall "p appropriate4 ,z 9 ,ptolomaus ?ird boke1 9 ! f\r? & six? ,*apt]s may appe>e4 99 #f4 ,& :at is ! cause ( ! apt bonde1 & fr5dly fel[%ip1 ( ! ,intellectuall & ,m5tall "p ( vs1 ) our grosse corruptible body3 b a c]ta9e ,meane1 & ,h>moni\s ,spiritualitie1 ) bo? p>ticipatyng1 ( bo? 79 a man]7 resultynge ,9 #g4 ! 99 ,tune ( ,mans voyce1 & also #h4 99 ! s\nd ( ,in/rum5t1 :at might be sayd1 ( ,h>monie3 ,no common ,musici5 w\ld lightly beleue4 ,i4 ,d4 ,read 9 ,ari/otle 8 #h4 booke ( ,politikes3 ! #e1 #f1 & #g4 *apt]s4 ,:]e y %all haue "s occasion f>d] to ?9ke ( ,musike1 ?an commonly is ?\ght4 ,b ( ! sundry ,mixture 7z ,i may t]me x7 & concurse1 diu]se collation1 & ,application ( ^! ,h>monies3 z ( ?re1 f\re1 fiue1 or mo3 ,m>uail\s haue ! effectes b53 & yet may be f\nde1 & produc$ ! l3 ) "s proportionall consid]ation = our "t1 & be+3 9 respect ( ! ,/ate1 ( ! ?+es !n3 9 :i*1 & by :i*1 ! wondr\s effectes 7 wr\ght4 ,democritus & ,!ophra/us affirm$1 t1 by ,musike1 griefes & diseases ( ! ,m9de1 & body might be cur$1 or inf]r$4 ,& we f9de 9 ,recorde1 t ,t]p&]1 ,arion1 ,ism5ias1 ,orpheus1 ,amphion1 ,dauid1 ,py?agoras1 ,emp$ocles1 ,asclepiades & ,timo!us1 by ,h>monicall ,consoncy1 haue done1 & br\ght to pas1 ?+es1 m !n m]uail\s1 to "h (4 ,( !m !n1 mak+ no f>d] disc\rse1 9 ?is place3 ,sure ,i am1 t ,common ,musike1 commonly vs$1 is f\nd to ! ,musici5s & ,he>]s1 to be s ,commodi\s & pleasant1 ,t if ,i w\ld say & dispute1 b ?us mu*3 ,t x 7 to be o!rwise vs$1 !n x is1 ,i %\ld f9de m repreeu]s1 !n ,i c\ld f9de priuy1 or skilfull ( my mean+4 ,9 ?+es !r=e euid5t1 & bett] kn[51 !n ,i c expresse3 & s all[$ & lik$ (1 7z ,i w\ld wi%1 "s o!r ?+es1 _h ! l hap7 ,i w sp>e to enl>ge my l9es any f>d]1 b consequ5tly foll[ my purpose4 ,( ,cosmographie1 ,i appo9t$ briefly 9 ?is place1 to geue y "s intellig5ce4 ,cosmographie1 is ! :ole & p]fect description ( ! heau5ly1 & also elem5tall p>te ( ! _w1 & _! homologall application1 & mutuall collation necess>ie4 ,?is ,art1 require? ,a/ronomie1 ,geographie1 ,hydrographie & ,musike4 ,!r=e1 x is no small ,arte1 nor s simple1 z 9 common practise1 x is 7slightly7 consid]$4 ,?is mat*e? ,heau51 & ! ,e>?1 9 "o frame1 & aptly applie? p>ts ,correspd5t3 ,s1 z1 ! ,heau5ly ,globe1 may 79 practise7 be duely describ$ vpon ! ,geographicall1 & ,hydrographicall ,globe4 ,& !re1 = vs to consid] an quonoctiall ,circle1 an ,ecliptike l9e1 ,colures1 ,poles1 ,/]res 9 _! true ,longitudes1 ,latitudes1 ,decl9ations1 & ,v]ticalitie3 also ,climes1 & ,p>allels3 & by an ,horizon annex$1 & reuolution ( ! e>?ly ,globe 7z ! ,heau51 is1 by ! ,prim\ant1 c>i$ ab\t 9 #bd4 quall ,h\res7 to le>ne ! ,ris+es & ,sett+es ( ,/]res 7( ,virgill 9 8 ,georgikes3 ( ,hesiod3 ( ,hippocrates 9 8 ,m$ic9all ,sphre1 to ,p]dicca ,k+ ( ! ,mac$onians3 ( ,diocles1 to ,k+ ,antigonus1 & ( o!r fam\s ,philosoph]s prescrib$7 a ?+ necess>y1 = due manur+ ( ! e>?1 = ,nauigation1 = ! ,alt]ation ( mans body3 be+1 :ole1 ,sicke1 w\nd$1 or brus$4 ,by ! ,reuolution1 also1 or m\+ ( ! ,globe ,cosmographicall1 ! ,ris+ & ,sett+ ( ! ,sonne3 ! ,l5g!s1 ( dayes & nightes3 ! ,h\res & times 7bo? night & day7 are kn[ne3 ) v _m o!r pleasant & necess>y vses3 ,:](1 "s are kn[ne3 b bett] rema9e1 = su* to "k & vse3 :o ( a sp>ke ( true fire1 c make a wond]full bonfire1 by apply+ ( due matt]1 duely4 ,( ,a/rologie1 "h ,i make an ,arte1 seu]all f ,a/ronomie3 n by new deuise1 b by good reason & au?oritie3 =1 ,a/rologie1 is an ,arte ,ma!maticall1 :i* reasonably demon/rate? ! op]ations & effectes1 ( ! naturall beames1 ( light1 & secrete influ5ce3 ( ! ,/]res & ,planets3 9 eu]y elem5t & elem5tall body3 at all times1 9 any ,horizon assign$4 ,?is ,arte is furni%$ ) _m o!r great ,artes & exp]i5ces3 ,z ) p]fecte ,p]spectiue1 ,a/ronomie1 ,cosmographie1 ,naturall ,philosophie ( ! #d4 ,elem5tes1 ! ,arte ( ,graduation1 & "s good vnd]/d+ 9 ,musike3 & yet more\]1 ) an o!r great ,arte1 h]eaft] foll[+1 ?\gh ,i1 "h1 set ?is be=e1 = "s consid]ations me m\+4 ,suffici5t 7y see7 is ! /uffe1 to make ?is r>e & secrete ,arte1 (3 & h>d en\gh to frame to ! ,conclusion ,syllogi/icall4 ,yet bo? ! manifolde & cont9uall trauailes ( ! mo/ aunci5t & wise ,philosoph]s1 = ! atteyn+ ( ?is ,arte3 & by examples ( effectes1 to confirme ! same3 ha? left vnto vs suffici5t pr\fe & witnesse3 & we1 also1 daily may p]ceaue1 ,t mans body1 & all o!r ,elem5tall bodies1 are alt]$1 dispos$1 ordr$1 pleasur$1 & displeasur$1 by ! ,influ5tiall work+ ( ! ,sunne1 ,mone1 & ! o!r ,/>res & ,planets4 ,& !r=e1 say? ,ari/otle1 9 ! fir/ ( 8 ,meteorologicall bookes1 9 ! second ,*apt]3 ,e/ autem necess>i ,mundus i/e1 sup]nis lationibus f] cont9uus4 ,vt1 inde1 vis eius vniu]sa regatur4 ,ea siquidem ,caus prima put&a omnibus e/1 vnde motus pr9cipium exi/it4 ,t is3 ,?is ,elem5tall ,_w is ( necessitie1 almo/1 next adioyn+1 to ! heau5ly motions3 ,t1 f !nce1 all 8 v]tue or =ce may be g\]n$4 ,=1 t is to be ?\ght ! fir/ ,cause vnto all3 f :i*1 ! beg9n+ ( motion1 is4 ,& aga9e1 9 ! t5? ,*apt]4 ,oportet igitur horum pr9cipia sumamus1 causas omnium similit]4 ,pr9cipium igitur vt m\5s1 prcipuumque omnium primum1 ,circulus ille e/1 9 quo manife/e ,solis latio1 c4 ,& s =?4 ,8 ,meteorologicall bookes1 are full ( argum5tes1 & effectuall demon/rations1 ( ! v]tue1 op]ation1 & p[] ( ! heau5ly bodies1 9 & vpon ! f[] ,elem5tes1 & o!r bodies1 ( !m 7ei!r p]fectly1 or vnp]fectly7 compos$4 ,& 9 8 second booke1 ,de ,g5]atione ,corruptione1 9 ! t5? ,*apt]4 ,quocirca prima latio1 ,ortus ,int]itus causa non e/3 ,s$ obliqui ,circuli latio3 ea namque cont9ua e/1 duobus motibus fit3 ,9 ,engli%e1 ?us4 ,:]e=e ! vpp]mo/ motion1 is n ! cause ( ,g5]ation & ,corruption1 b ! motion ( ! ,zodiake3 =1 t1 bo?1 is cont9uall1 & is caus$ ( two m\+es4 ,& 9 8 second booke1 & second ,*apt] ( hys ,physikes4 ,homo namque g5]at hom9em1 atque ,sol4 ,= ,man 7say? he7 & ! ,sonne1 are cause ( mans g5]ation4 ,au?orities may be br\ght1 v _m3 bo? ( #ajjj4 #bjjj4 yea & #cjjj4 ye>es ,antiquitie3 ( great ,philosoph]s1 ,exp]t1 ,wise1 & godly m51 = t ,conclusion3 :i*1 daily & h\rely1 we m51 may disc]ne & p]ceaue by s5se & reason3 ,all bea/es d feele1 & simply %ew1 by _! actions & passions1 outw>d & inw>d3 ,all ,plants1 ,h]bes1 ,trees1 ,fl[]s1 & ,fruites4 ,& f9ally1 ! ,elem5tes1 & all ?+es ( ! ,elem5tes compos$1 d geue ,te/imonie 7z ,ari/otle sayd7 t !yr ,:ole ,dispositions1 v]tues1 & naturall motions1 dep5d ( ! ,actiuitie ( ! heau5ly motions & ,influ5ces4 ,:]eby1 beside ! specificall ord] & =me1 due to eu]y se$e3 & beside ! ,nature1 propre to ! ,indiuiduall ,matrix1 ( ! ?+ produc$3 ,:at %all be ! heau5ly ,impression1 ! p]fect & circumspecte ,a/rologi5 ha? to ,conclude4 ,n onely 7by ,apotelesmes7 4 b by ,naturall & ,ma!maticall demon/ration 4 ,:]eunto1 :at ,sci5ces are requisite 7)\t exception7 ,i p>tly haue "h w>n$3 ,& 9 my ,propdeumes 7besides o!r matt] !re disclos$7 ,i haue ,ma!matically furni%$ vp ! :ole ,me?od3 ,to ?is our age1 n s c>efully h&l$ by any1 t eu] ,i saw1 or he>d (4 ,i 01 99 ,anno4 #aedh & #aedi4 9 ,l\ayn4 7= 99 #ba4 ye>es ago7 by c]ta9e e>ne/ disputations1 ( ! ,le>n$ ,g]>dus ,m]cator1 & ,antonius ,gogaua1 7& o!r17 !rto s pr\ok$3 & 7by my con/ant & inu9cible zeale to ! v]itie7 9 obs]uations ( ,heau5ly ,influ5cies 7to ! ,m9ute ( "t17 ?an1 s dilig5t3 ,& *iefly by ! ,sup]naturall influ5ce1 f ! ,/>re ( ,iacob1 s direct$3 ,t any ,mode/ & ,sob] ,/ud5t1 c>efully & dilig5tly sek+ = ! ,tru?1 w bo? f9de cfesse1 !r91 to be ! ,v]itie1 ( ^! my wordes3 ,& also become a ,reasonable ,re=m]1 ( ?ree ,sortes ( p3 ab\t ^! ,influ5tiall ,op]ations1 greatly err+ f ! tru?4 ,note4 ,:](1 ! "o1 is ,light ,beleu]s1 ! o!r1 ,light ,despis]s1 & ! ?ird ,light ,practis]s4 ,! fir/1 mo/ cmon ,sort1 ?9ke ! ,heau5 & ,/]res1 to be answ]able to any _! d\tes or desires3 #a4 :i* is n s3 &1 9 d$e1 !y1 to mu*1 ou] rea*e4 ,! ,second sorte ?9ke no ,influ5tiall v]tue 7fr ! heau5ly bodies7 to be>e any ,sway 9 ,g5]ation #b4 & ,corruption1 9 ?is ,elem5tall _w4 ,& to ! ,sunne1 ,mone & ,/]res 7be+ s _m1 s pure1 s bright1 s wond]full bigge1 s f>re 9 di/ance1 s manifold 9 _! motions1 s con/ant 9 _! p]iodes4 c47 !y assigne a sleight1 simple (fice or two1 & s all[ vnto ? 7accord+ to _! capacities7 z mu* v]tue1 & p[] ,influ5tiall1 z to ! ,signe ( ! ,sunne1 ,mone1 & seu5 ,/]res1 hang$ vp 7= ,signes7 9 ,london1 = di/9ction ( h\ses1 su* grosse helpes1 9 our worldly affaires3 ,& !y vnd]/& n 7or w n vnd]/&7 ( ! o!r work+es1 & v]tues ( ! ,heau5ly ,sunne1 ,mone1 & ,/]res3 n s mu*1 z ! ,m>9]1 or ,husb& man3 no1 n s mu*1 z ! ,elephant do?1 z ! ,cynocephalus1 z ! ,porp5t9e do?3 nor w all[ ^! p]fect1 & incorruptible mighty bodies1 s mu* v]tuall ,radiation1 ,=ce1 z !y see 9 a litle peece ( a ,magnes /one3 :i*1 at great di/ance1 %ewe? 8 op]ation4 ,& p]*aunce !y ?9ke1 ! ,sea ,riu]s 7z ! ,?ames7 to be "s quicke ?+1 & s to ebbe1 & fl[1 run 9 & out1 ( !m selues1 at _! owne fantasies4 ,god helpe1 ,god helpe4 ,surely1 ^! m51 come to %ort3 & ei!r are to dull3 or willfully bl9d3 or1 p]haps1 to malici\s4 ,! ?ird man1 is ! common & vulg>e ,a/rologi51 or ,practis]3 :o1 be+ n duely1 artificially1 & p]fectly #c4 furni%$3 yet1 ei!r = va9e glory1 or gayne3 or l a simple dolt1 bl9de ,bay>d1 bo? 9 matt] & man]1 erre?3 to ! discr$it ( ! ,w>y1 & mode/ ,a/rologi53 & to ! robb+ ( ^? mo/ noble corporall ,creatures1 ( _! ,naturall ,v]tue3 be+ mo/ mighty3 mo/ b5eficiall to all elem5tall ,g5]ation1 ,corruption & ! app>t5ances3 & mo/ ,h>moni\s 9 _! ,mon>*ie3 ,= :i* ?+es1 be+ kn[51 & mode/ly vs$3 we might highly1 & cont9ually glorifie ,god1 ) ! pr9cely ,prophet1 say+4 ,! ,heau5s decl>e ! ,glorie ( ,god3 :o made ! ,heaus 9 8 wis$ome3 :o made ! ,sonne1 = to haue dom9ion ( ! day3 ! ,mone & ,/]res to haue dom9ion ( ! nyght3 :]eby1 ,day to day vtt]e? talke3 & night1 to night decl>e? k4 ,prayse him1 all ye ,/]res1 & ,light4 ,am54 ,9 ord]1 n[ fol[e?1 ( ,/atike1 some:at to say1 :at we meane by t "n3 & :at commodity1 do?1 on su* ,art1 dep5d4 ,/atike1 is an ,arte ,ma!maticall1 :i* demon/rate? ! causes ( heauynes1 & lightnes ( all ?ynges3 & ( motions & prop]ties1 to heauynes & lightnes1 belong+4 ,& = asmu* z1 by ! ,bilanx1 or ,balance 7z ! *ief s5sible ,in/rum5t17 ,exp]i5ce ( ^! demon/rations may be _h3 we call ?is ,art1 ,/atike3 t is1 ! ,exp]im5tes ( ! ,balance4 ,oh1 t m5 wi/1 :at pr(fit1 7all man] ( wayes7 by ?is ,arte might gr[1 to ! hable exam9]1 & dilig5t practis]4 8,?\ onely1 kn[e/ all ?+es precisely 7,o ,god7 :o ha/ made weight & ,balance1 ?y ,iudgem5t3 :o ha/ creat$ all ?+es 9 ,numb]1 ,waight1 & ,measure3 & ha/ way$ ! m\nta9es & hils 9 a ,balance3 :o ha/ peys$ 9 ?y h&1 bo? ,heau5 & e>?4 ,we !r=e w>n$ by ! ,sacr$ ^w1 to ,consid] ?y ,creatures3 & by t consid]ation1 to wynne a glyms 7z x 717 or %add[ ( p]ceiu]ance1 t ?y wis$ome1 might1 & goodnes is inf9ite1 & vnspeakable1 9 ?y ,creatures decl>$3 ,& be+ f>d] adu]tis$1 by ?y m]cifull goodnes1 t1 ?ree pr9cipall wayes1 71 ( !1 vs$ 9 ,creation ( all ?y ,creatures1 namely1 ,numb]1 ,waight & ,measure1 ,& = z mu* z1 ( ,numb] & ,measure1 ! two ,artes 7aunci5t1 fam\s1 & to huma9e vses mo/ necess>y17 are1 all ready1 suffici5tly kn[5 & extant3 ,?is ?ird key1 we bese*e !e 7?r\gh ?y accu/om$ goodnes17 t x may come to ! n$efull & suffici5t k1 ( su* ?y ,s]uauntes1 z 9 ?y workeman%ip1 w\ld gladly f9de1 ?y true occasions 7purposely ( ! vs$7 :]eby we %\ld glorifie ?y "n1 & %ew =? 7to ! weakl+es 9 fai?7 ?y wondr\s wis$ome & ,goodnes4 ,am540 ,m]uaile no?+ at ?is pang 7godly fr5d1 y ,g5tle & zel\s ,/ud5t47 ,an o!r day1 p]*aunce1 y w p]ceiue1 :at occasion m\$ me4 ,"h1 z n[1 ,i w giue y "s gr\nd1 & )all "s %ew1 ( c]ta9e commodities1 by ?is ,arte aris+4 ,& bycause ?is ,arte is r>e1 my wordes & practises might be to d>ke3 vnlea/ y _h "s light1 hold5 be=e ! matt]3 & t1 be/ w be1 9 giu+ y1 out ( ,ar*im$es demon/rations1 a few pr9cipal ,conclusions1 z fol[e?4 #a4 ,! ,sup]ficies ( eu]y ,liquor1 by x selfe consi/yng1 & 9 quyet1 is ,sphricall3 ! c5tre :]e(1 is ! same1 :i* is ! c5tre ( ! ,e>?4 #b4 ,if ,solide ,magnitudes1 be+ ( ! same bignes1 or qutitie1 t any ,liquor is1 & hauyng also ! same ,waight3 be let d[ne into ! same ,liquor1 !y w settle d[new>d1 s1 t no p>te ( !m1 %all be ab\e ! ,sup]ficies ( ! ,liquor3 & yet neu]!les1 !y w n s9ke vtt]ly d[ne1 or dr[ne4 #c4 ,if any ,solide ,magnitude beyng ,light] !n a ,liquor1 be let d[ne into ! same ,liquor1 x w settle d[ne1 s f>re into ! same ,liquor1 t s great a quantitie ( t ,liquor1 z is ! p>te ( ! ,solid ,magnitude1 settl$ d[ne into ! same ,liquor3 is 9 ,waight1 quall1 to ! waight ( ! :ole ,solid ,magnitude4 #d4 ,any ,solide ,magnitude1 ,light] !n a ,liquor1 =c$ d[ne into ! same ,liquor1 w m\e vpw>d1 ) s great a p[]1 by h[ mu*1 ! ,liquor hauyng quall quantitie to ! :ole ,magnitude1 is heauy] !n ! same ,magnitude4 #e4 ,any ,solid ,magnitude1 heauy] !n a ,liquor1 beyng let d[ne into ! same ,liquor1 w s9ke d[ne vtt]ly3 ,& wilbe 9 t ,liquor1 ,light] by s mu*1 z is ! waight or heauynes ( ! ,liquor1 hau+ bygnes or quantitie1 quall to ! ,solid ,magnitude4 #f4 ,i4 ,d4 ,! ,cutt+ ( a ,sphre accord+ to any proportion assign$ may by ?is proposition be done ,me*anically by temp]+ ,liquor to a c]tayne waight 9 respect ( ! waight ( ! ,sphre !re9 ,swymm+4 ,if any ,solide ,magnitude1 ,light] !n a ,liquor1 be let d[ne into ! same ,liquor1 ! waight ( ! same ,magnitude1 w be1 to ! ,waight ( ! ,liquor4 7,:i* is quall 9 quantitie to ! :ole ,magnitude17 9 t proportion1 t ! p>te1 ( ! ,magnitude settl$ d[ne1 is to ! :ole ,magnitude4 ,by ^! v]ities1 great ,errors may be re=m$1 9 ,op9ion ( ! ,naturall ,motion ( ?+es1 ,light & ,heauy4 ,:i* errors1 are 9 ,naturall ,philosophie 7almo/7 ( all m all[$3 to mu* tru/+ to ,au?ority3 & false ,suppositions4 ,z1 ,( any two bodyes1 ! heauy]1 to m\e d[nw>d fa/] !n ! light]4 ,a common error1 not$4 ,?is error1 is n fir/ by me1 ,not$3 b by "o ,iohn ,bapti/ de ,b5$ictis4 ,! *ief ( 8 propositions1 is ?is3 :i* seeme? a ,p>adox4 ,if !re be two bodyes ( "o =me1 & ( "o kynde1 quall 9 quantitie or vnquall1 ,a p>adox4 !y w m\e by quall space1 9 quall tyme3 ,s t bo? !yr m\ynges be 9 ayre1 or bo? 9 wat]3 or 9 any "o ,middle4 ,h]eupon1 9 ! feate ( ,gunnyng1 ,n4 ,t4 c]ta9e good disc\rses 7o!rwise7 may receiue great am5dem5t1 & furd]ance4 ,! wond]full vse ( ^! ,propositions4 ,9 ! ent5d$ purpose1 also1 all[+ som:at to ! imp]fection ( ,nature3 n aunsw]able to ! precis5es ( demon/ration4 ,more\]1 by ! =esaid propositions 7wisely vs$47 ,! ,ayre1 ! wat]1 ! ,e>?1 ! ,fire1 may be n]ely1 kn[51 h[ light or heauy !y are 7,naturally7 9 _! assign$ p>tes3 or 9 ! :ole4 ,& !n1 to ?+es ,elem5tall1 turn+ y\r practise3 y may deale = ! proportion ( ! ,elem5tes1 9 ! ?+es ,comp\nd$4 ,!n1 to ! proportions ( ! ,hum\rs 9 ,man3 _! waightes3 & ! waight ( 8 bones1 & fle%4 c4 ,?an1 by waight1 to haue consid]ation ( ! ,=ce ( man1 any man] ( way3 9 :ole or 9 "p4 ,!n1 may y1 ( ,%ips wat] draw+1 diu]sly1 9 ! ,sea & 9 fre% wat]1 haue pleasant consid]ation3 & ( way+ vp ( any ?+1 sonk5 9 ,sea or 9 fre% wat] c4 ,& 7to lift vp y\r head a l(t37 by waight1 y may1 z precisely1 z by any in/rum5t els1 measure ! ,diamet]s ( ,sonne & ,mone4 c4 ,fr5de1 ,i pray y1 way ^! ?+es1 ) ! iu/ ,balance ( ,reason4 ,& y w f9de ,m]uailes vpon ,m]uailes3 ,& e/eme "o ,drop ( ,tru? 7yea 9 ,naturall ,philosophie7 m wor?1 !n :ole ,libr>ies ( ,op9ions1 vndemon/rat$3 or n aunsw]+ to ,natures ,law1 & y\r exp]i5ce4 ,leau+ ^! ?+es1 ?us3 ,i w giue y two or ?ree1 light practises1 to great purpose3 & s f9i% my ,annotation ,/aticall4 ,9 ,ma!maticall matt]s1 by ! ,me*anici5s ayde1 we w behold1 "h1 ! ,commodity ( waight4 ,! practise ,/aticall1 to "k ! proportion1 betw5e ! ,cube1 & ! ,sphre4 ,make a ,cube1 ( any "o ,vni=me3 & ?r\gh l heauy /uffe3 ( ! same ,/uffe1 make a ,sphre or ,globe1 precisely1 ( a ,diamet] quall to ! ,radicall side ( ! ,cube4 ,y\r /uffe1 may be wood1 ,copp]1 ,t9ne1 ,lead1 ,silu]4 c4 7be+1 z ,i sayd1 ( l nature1 condition1 & l waight ?r\gh\t47 ,& y may1 by ,say ,balance1 haue prep>$ a great numb] ( ! smalle/ waightes3 :i*1 by ^? ,balance c be disc]n$ or try$3 & s1 haue proc$$ to make y a p]fect ,pyle1 company ,numb] ( waightes3 to ! waight ( six1 eight1 or twelue p\nd waight3 mo/ dilig5tly try$1 all4 ,& ( eu]y "o1 ! ,cont5t kn[51 9 y\r lea/ waight1 t is wayable4 ,!y t c n haue ^! waightes ( precis5es3 may1 by ,s&1 ,vni=me1 & well du/$1 make !m a numb] ( waightes1 some:at n]e precis5es3 by half+ eu] ! ,s&3 !y %all1 at l5g?1 come to a lea/ common waight4 ,!re91 ,i leaue ! f>d] matt]1 to _! discretion1 :om n$e %all p9*e4 ,! ,v5etians consid]ation ( waight1 may seme precise en\gh3 by eight desc5tes progressionall1 99 half+1 f a grayne4 ,i4 ,d4 99 ,=1 s1 haue y 4#bef4 p>tes ( a ,gra9e4 ,y\r ,cube1 ,sphre1 apt ,balance1 & conu5i5t waightes1 be+ ready3 fall to worke44 ,fir/1 way y\r ,cube4 ,note ! ,numb] ( ! waight4 ,way1 aft] t1 y\r ,sphre4 ,note likewise1 ! ,nb] ( ! waight4 ,if y n[ f9d ! waight ( y\r ,cube1 to be to ! waight ( ! ,sphre1 z #ba4 is to #aa3 ,!n y see1 h[ ! ,me*anici5 & ,exp]im5t]1 )\t ,geometrie & ,demon/ration1 are 7z n]ely 9 effect7 t\ght ! proportion ( ! ,cube to ! ,sph]e3 z ,i haue demon/rat$ x1 9 ! end ( ! twelf? boke ( ,euclide4 ,(t51 try ) ! same ,cube & ,sphre4 ,!n1 *aunge1 y\r ,sphre & ,cube1 to an o!r matt]3 or to an o!r bignes3 till y haue made a p]fect vniu]sall ,exp]i5ce ( x4 ,possible x is1 t y %all wynne to n]] t]mes1 9 ! proportion4 ,:5 y haue f\nd ?is "o c]ta9e ,drop ( ,naturall v]itie1 proc$e on1 to ,inf]re1 & duely to make assay1 ( matt] dep5d+4 ,z1 bycause x is well demon/rat$1 t a ,cyl9d]1 ^: hei?1 & ,diamet] ( 8 base1 is quall to ! ,diamet] ( ! ,sphre1 is ,sesquialt] to ! same ,sphre 7t is1 z #c4 to #b37 ,to ! numb] ( ! waight ( ! ,sphre1 adde halfe s mu*1 z x is3 & s haue y ! numb] ( ! waight ( t ,cyl9d]4 ,:i* is also ,compreh5d$ ( our =m] ,cube3 ,s1 t ! base ( t ,cyl9d]1 is a ,circle describ$ 9 ! ,squ>e1 :i* is ! base ( our ,cube4 ,b ! ,cube & ! ,cyl9d]1 be+ bo? ( "o hei?1 haue _! ,bases 9 ! same proportion1 9 ! :i*1 !y are1 "o to an o!r1 9 _! ,mass9es or ,soliditie4 ,b1 be=e1 we haue two numb]s1 express+ _! ,mass9es1 ,solidities1 & ,quantities1 by waight3 :]=e1 99 ,! proportion ( ! ,squ>e to ! ,circle inscrib$4 we haue 99 ! proportion ( ! ,squ>e1 to ! ,circle1 inscrib$ 9 ! same ,squ>e4 ,& s are we fall5 into ! k s5sible1 & ,exp]im5tall ( ,ar*im$es great ,secret3 ( him1 by great trauaile ( m9de1 s\ght & f\nd4 ,:]=e1 to any ,circle giu51 y c giue a ,squ>e quall3 99 ,! ,squ>+ ( ! ,circle1 ,me*anically4 99 z ,i haue taught1 9 my ,annotation1 vpon ! fir/ proposition ( ! twelf? boke1 ,& likewise1 to any ,squ>e giu51 y may giue a ,circle quall3 99 ,to any ,squ>e geu51 to geue a ,circle1 equall4 99 ,if y describe a ,circle1 :i* %all be 9 t proportion1 to y\r ,circle inscrib$1 z ! ,squ>e is to ! same ,circle3 ,?is1 y may d1 by my ,annotations1 vpon ! second proposition ( ! twelf? boke ( ,euclide1 9 my ?ird ,probleme !re4 ,y\r dilig5ce may come to a proportion1 ( ! ,squ>e to ! ,circle inscrib$1 n]] ! tru?1 !n is ! proportion ( #ad4 to #aa4 ,& consid]1 t y may begyn at ! ,circle & ,squ>e1 & s come to conclude ( ! ,sphre1 ! ,cube1 :at _! proportion is3 z n[1 y came f ! ,sphre to ! ,circle4 ,=1 ( ,silu]1 or ,gold1 or ,latton ,lamyns or plates 7?or\gh "o hole draw1 z ! man] is7 if y make a ,squ>e figure way x3 & !n1 describ+ !ron1 ! ,circle inscrib$3 cut (1 file away1 precisely 7to ! ,circle7 ! ou]plus ( ! ,squ>e3 y %all !n1 way+ y\r ,circle1 see1 :e!r ! waight ( ! ,squ>e1 be to y\r ,circle1 z #ad4 to #aa4 ,z ,i haue ,not$1 9 ! beg9n+ ( ,euclides twelf? boke4 c4 aft] ?is resort to my la/ proposition1 vpon ! la/ ( ! twelf?4 ,& !re1 helpe y\r selfe1 to ! end4 ,&1 "h1 ,note ?is1 by ! way4 ,note ,squ>+ ( ! ,circle )\t k ( ! proportion betw5e ,circumf]5ce & ,diamet]4 ,t we may ,squ>e ! ,circle1 )\t hau+ k ( ! proportion1 ( ! ,circumf]5ce to ! ,diamet]3 z y haue "h p]ceiu$4 ,& o!rwayes also1 ,i c demon/rate x4 ,s t1 _m haue cumb]d !m selues sup]flu\sly1 by trauail+ 9 t po9t fir/1 :i* 0 n ( necessitie1 fir/3 & also v intricate4 ,& easily1 y may1 7& t diu]sly7 come to ! k ( ! ,circumf]5ce3 ! ,circles ,quantitie1 be+ fir/ kn[54 ,:i* ?+1 ,i leaue to y\r consid]ation3 mak+ ha/ to despat* an o!r ,magi/rall ,probleme3 & to br+ x1 n]] to y\r k1 & readi] deal+ )1 !n ! _w 7be=e ?is day17 _h x = y1 t ,i c tell (4 ,& t is1 ,a ,me*anicall ,dubblyng ( ! ,cube3 c4 ,:i* may1 ?us1 be done3 ,to ,dubble ! ,cube r$ily3 by ,art ,me*anicall3 dep5d+ vppon ,demon/ration ,ma!maticall4 ,make ( ,copp] plates1 or ,tyn plates1 a f\rsqu>e vpright ,pyramis1 or a ,cone3 p]fectly fa%ion$ 9 ! hol[1 )94 ,:]91 let great dilig5ce be vs$1 to appro*e 7z n]e z may be7 to ! ,ma!maticall p]fection ( ^? figures4 ,at _! bases1 let !m be all op53 eu]y :]e1 els1 mo/ close1 & iu/ to4 ,f ! v]tex1 to ! ,circumf]5ce ( ! base ( ! ,cone3 to ! sides ( ! base ( ! ,pyramis3 ,i4 ,d4 ,! #d4 sides ( ?is ,pyramis mu/ be #d4 ,isosceles ,triangles alike & quall4 ,let #d4 /raight l9es be draw51 9 ! inside ( ! ,cone & ,pyramis3 makyng at _! fall1 on ! p]imet]s ( ! bases1 equall angles on bo? sides !m selues1 ) ! sayd p]imet]s4 ,^! #d4 l9es 79 ! ,pyramis3 & z _m1 9 ! ,cone7 diuide3 "o1 9 #ab4 quall p>tes3 & an o!r1 9 #bd4 an o!r1 9 #fj1 & an o!r1 9 #ajj4 7reck5yng vp f ! v]tex47 ,or vse o!r numb]s ( diuision1 z exp]i5ce %all tea* y4 ,i4 ,d4 99 ,9 all work+es ) ?is ,pyramis or ,cone1 ,let _! ,situations be 9 all ,po9tes & ,conditions1 alike1 or all "o3 :ile y are ab\t "o ,worke4 ,els y w erre4 ,!n1 99 set y\r ,cone or ,pyramis1 ) ! v]tex d[nw>d1 p]p5dicul>ly1 9 respect ( ! ,base4 7,?\gh x be o!rwayes1 x h9dre? no?yng47 ,s let ? mo/ /$ily be /ay$4 ,n[1 if !re be a ,cube1 :i* y wold haue ,dubbl$4 ,make y a prety ,cube ( ,copp]1 ,silu]1 ,lead1 ,tynne1 ,wood1 ,/one1 or ,bone4 ,or els make a holl[ ,cube1 or ,cubik c(f51 ( ,copp]1 ,silu]1 ,tynne1 or ,wood c4 ,^!1 y may s proporti 9 respect ( y\r ,pyramis or ,cone1 t ! ,pyramis or ,cone1 w be hable to conte9e ! waight ( !m1 9 wat]1 #c4 or #d4 times3 at ! lea/3 :at /uff s eu] !y be made (4 ,let n y\r ,solid angle1 at ! v]tex1 be to %>pe3 b t ! wat] may come ) ease1 to ! v v]tex1 ( y\r holl[ ,cone or ,pyramis4 ,put "o ( y\r ,solid ,cubes 9 a ,balance apt3 take ! waight !r( exactly 9 wat]4 ,p[re t wat]1 7)\t losse7 into ! holl[ ,pyramis or ,cone1 quietly4 ,m>ke 9 y\r l9es1 :at numb]s ! wat] ,cutte?3 ,take ! waight ( ! same ,cube aga9e3 9 ! same k9de ( wat]1 :i* y _h be=e3 ,i4 ,d4 99 ,consid] well :an y mu/ put y\r wat]s togy!r3 & :an1 y mu/ empty y\r fir/ wat]1 out ( y\r ,pyramis or ,cone4 ,els y w erre4 put t99 also1 into ! ,pyramis or ,cone1 :]e y did put ! fir/4 ,m>ke n[ aga9e1 9 :at numb] or place ( ! l9es1 ! wat] ,cutte? !m4 ,two wayes y may conclude y\r purpose3 x is to wete1 ei!r by numb]s or l9es4 ,by numb]s3 z1 if y diuide ! side ( y\r ,fundam5tall ,cube into s _m quall p>tes1 z x is capable (1 conu5i5tly1 ) y\r ease1 & precis5es ( ! diuision4 ,=1 z ! numb] ( y\r fir/ & lesse l9e 79 y\r holl[ ,pyramis or ,cone17 is to ! second or great] 7bo? be+ c\nt$ f ! v]tex7 s %all ! numb] ( ! side ( y\r ,fundam5tall ,cube1 be to ! nb] belong+ to ! ,radicall side1 ( ! ,cube1 dubble to y\r ,fundam5tall ,cube3 ,:i* be+ multipli$ ,cubik wise1 w sone %ew x selfe1 :e!r x be dubble or no1 to ! ,cubik numb] ( y\r ,fundam5tall ,cube4 ,by l9es1 ?us3 ,z y\r lesse & fir/ l9e1 79 y\r holl[ ,pyramis or ,cone17 is to ! second or great]1 s let ! ,radical side ( y\r ,fundamtall ,cube1 be to a f\r? proportionall l9e1 by ! #ab4 proposition1 ( ! six? boke ( ,euclide4 ,:i* f\r? l9e1 %all be ! ,rote ,cubik1 or ,radicall side ( ! ,cube1 dubble to y\r ,fundam5tall ,cube3 :i* is ! ?+ we desir$4 ,god be ?ank$ = ?is ,inu5tion1 ! fruite ensu+4 ,= ?is1 may ,i 7) ioy7 say1 1 1 3 ?ank+ ! holy & glori\s ,tr9ity3 hau+ great] cause !rto1 !n 99 ,vitruuius4 ,lib4 #i4 ,cap4 #c4 99 ,ar*im$es _h 7= f9d+ ! fraude vs$ 9 ! ,k+es ,cr[ne1 ( ,gold73 z all m5 may easily ,iudge3 by ! diu]sitie ( ! frute foll[+ ( ! "o1 & ! o!r4 ,:]e ,i spake be=e1 ( a holl[ ,cubik ,c(f53 ! l vse1 is ( x3 & )\t waight4 ,?us4 ,fill x ) wat]1 precisely full1 & p\re t wat] into y\r ,pyramis or ,cone4 ,& "h note ! l9es cutt+ 9 y\r ,pyramis or ,cone4 ,aga9e1 fill y\r c(f51 l z y did be=e4 ,put t ,wat]1 also1 to ! fir/4 ,m>ke ! second cutt+ ( y\r l9es4 ,n[1 z y proc$$ be=e1 s mu/ y "h proc$e4 99 ,note4 99 ,& if ! ,cube1 :i* y %\ld ,d\ble1 be neu] s great3 y haue1 ?us1 ! proportion 79 small7 betw5e y\r two litle ,cubes3 ,& !n1 ! side1 ( t great ,cube 7to be d\bl$7 be+ ! ?ird1 w haue ! f\r?1 f\nd1 to x proportionall3 by ! #ab4 ( ! six? ( ,euclide4 ,note1 z conc]n+ ! ,sphricall ,sup]ficies ( ! ,wat]4 ,note1 t all ?is :ile1 ,i =get n my fir/ ,proposition ,/aticall1 "h rehe>s$3 t1 ! ,sup]ficies ( ! wat]1 is ,sphricall4 ,:]e91 vse y\r discretion3 to ! fir/ l9e1 add+ a small he>e bread?1 m3 & to ! second1 halfe a he>e bread? m1 to 8 l5g?4 ,=1 y w easily p]ceaue1 t ! diff]5ce c be no great]1 9 any ,pyramis or ,cone1 ( y to be h&l$4 ,:i* y %all ?us trye4 ,= f9d+ ! swell+ ( ! wat] ab\e leuell4 8,squ>e ! ,semidiamet]1 f ! ,c5tre ( ! e>?1 to y\r fir/ ,wat]s ,sup]ficies4 ,squ>e !n1 halfe ! ,subt5d5t ( t watry ,sup]ficies 7:i* ,subt5d5t mu/ haue ! equall p>tes ( 8 measure1 all "o1 ) ^? ( ! ,semidiamet] ( ! e>? to y\r watry ,sup]ficies73 ,subtracte ?is squ>e1 f ! fir/3 ,( ! residue1 take ! ,rote ,squ>e4 ,t ,rote1 ,subtracte f y\r fir/ ,semidiamet] ( ! e>? to y\r watry ,sup]ficies3 t1 :i* rema9e?1 is ! hei? ( ! wat]1 9 ! middle1 ab\e ! leuell40 ,:i*1 y w f9de1 to be a ?+ ins5sible4 ,& ?\gh x 7 greatly s5sible1 99 99 ,note4 yet1 by helpe ( my sixt ,!oreme vpon ! la/ ,proposition ( ,euclides twelf? booke1 not$3 y may r$uce all1 to a true ,leuell4 ,b1 f>!r dilig5ce1 ( y is to be vs$1 aga9/ accid5tall causes ( ! wat]s swell+3 z by hau+ 7som:at7 ) a moy/ ,sponge1 be=e1 made moy/ y\r holl[ ,pyramis or ,cone1 w preu5t an accid5tall cause ( ,swell+1 c4 ,exp]i5ce w tea* y abundantly3 ) great ease1 pleasure1 & cmoditie4 ,?us1 may y ,d\ble ! ,cube ,me*anically1 ,treble x1 & s =?1 9 any proportion4 ,note ?is ,abridgem5t ( ,dubbl+ ! ,cube4 c4 ,n[ w ,i ,abridge y\r pa9e1 co/1 & ,c>e h]e94 ,)\t all prep>+ ( y\r ,fundam5tall ,cubes3 y may 7alike7 worke ?is ,conclusion4 ,=1 t1 0 r a k9de ( ,exp]im5tall dem/ration1 !n ! %orte/ way3 & all1 vpon "o ,ma!maticall ,demon/ration dep5d+4 8,take wat] 7z mu* z conu5i5tly w s]ue y\r turne3 z ,i w>n$ be=e ( y\r ,fundam5tall ,cubes bignes7 ,way x precisely4 ,put t wat]1 into y\r ,pyramis or ,cone4 ,( ! same k9de ( wat]1 !n take aga9e1 ! same waight y _h be=e3 put t likewise into ! ,pyramis or ,cone4 ,=1 9 e*e "t1 y\r m>k+ ( ! l9es1 h[ ! ,wat] do? cut !m1 %all geue y ! proportion betw5 ! ,radicall sides1 ( any two ,cubes1 :]( ! "o is ,d\ble to ! o!r3 work+ z be=e ,i haue taught y3 99 ,note4 99 sau+ t = y ,fundam5tall ,cube 8 ,radicall side3 "h1 y may take a "r l9e1 at pleasure40 ,yet f>!r proc$+ ) our droppe ( ,naturall tru?3 ,to giue ,cubes "o to ! o!r 9 any proportion1 ,rationall or ,irrationall4 y may 7n[7 geue ,cubes1 "o to ! o!r1 9 any proporti geu3 ,rationall or ,irrationall3 on ?is man]4 ,make a holl[ ,p>allelipip$on ( ,copp] or ,t9ne3 ) "o ,base wt+1 or op53 z 9 our ,cubike ,c(f54 ,fr ! bottome ( t ,p>allelipip$on1 raise vp1 _m p]p5dicul>s1 9 eu]y ( 8 f[] sides4 ,n[ if any proportion be assign$ y1 9 "r l9es3 ,cut "o ( y\r p]p5dicul>s 7or a l9e equall to x1 or lesse !n x7 likewise3 by ! #aj4 ( ! six? ( ,euclide4 ,& ^? two p>tes1 set 9 two sundry l9es ( ^? p]p5dicul>s 7or y may set !m bo?1 9 "o l9e7 mak+ _! beg9n+es1 to be1 at ! base3 & s _! l5g!s to ext5d vpw>d4 ,n[1 set y\r holl[ ,p>allelipip$on1 vpright1 p]p5dicul>ly1 /eadie4 ,p\re 9 wat]1 h&somly1 to ! hei? ( y\r %ort] l9e4 ,p\re t wat]1 into ! holl[ ,pyramis or ,cone4 ,m>ke ! place ( ! ris+4 ,settle y\r holl[ ,p>allelipip$on aga9e4 ,p\re wat] into x3 vnto ! hei? ( ! second l9e1 exactly4 99 ,empty+ ! fir/4 ,p\re t wat] 99 duely into ! holl[ ,pyramis or ,cone3 ,m>ke n[ aga9e1 :]e ! wat] cutte? ! same l9e :i* y m>k$ be=e4 ,=1 !re1 z ! fir/ m>k$ l9e1 is to ! second3 ,s %all ! two ,radicall sides be1 "o to ! o!r1 ( any two ,cubes3 :i*1 9 _! ,soliditie1 %all haue ! same proportion1 :i* 0 at ! fir/ assign$3 7 x ,rationall or ,irrationall4 ,?us1 9 sundry waies y may furni%e y\r selfe ) su* /raunge & pr(itable matt]3 :i*1 long ha? b5e wi%$ =4 ,& ?\gh x be ,naturally done & ,me*anically3 yet ha? x a good ,demon/ration ,ma!maticall4 ,! demon/rations ( ?is ,dubbl+ ( ! ,cube1 & ( ! re/4 ,:i* is ?is3 ,alwaies1 y haue two ,l ,pyramids3 or two ,l ,cones1 9 ! proportions assign$3 & l ,pyramids or ,cones1 are 9 proportion1 "o to ! o!r1 9 ! proportion ( _! ,homologall sides 7or l9es7 tripl$4 ,:]e=e1 if to ! fir/1 & second l9es1 f\nd 9 y\r holl[ ,pyramis or ,cone1 y ioyne a ?ird & a f\r?1 9 cont9uall proportion3 t f\r? l9e1 %all be to ! fir/1 z ! great] ,pyramis or ,cone1 is to ! lesse3 by ! #cc4 ( ! eleu5? ( ,euclide4 ,if ,pyramis to ,pyramis1 or ,cone to ,cone1 be d\ble1 ,i4 ,d4 99 ,h]eby1 helpe y\r self to become a prcise practis]4 ,& s consid]1 h[1 no?+ at all1 y are h9dr$ 7s5sibly7 by ! ,conuexitie ( ! wat]4 !n %all 99 ,l9e to ,l9e1 be also d\ble1 c4 ,b1 z our fir/ l9e1 is to ! second1 s is ! ,radicall side ( our ,fundam5tall ,cube1 to ! ,radicall side ( ! ,cube to be made1 or to be d\bl$3 & !re=e1 to ^? twa9e also1 a ?ird & a f\r? l9e1 9 cont9uall proportion1 ioyn$3 w geue ! f\r? l9e 9 t proportion to ! fir/1 z our f\r? ,pyramidall1 or ,conike l9e1 0 to 8 fir/3 b t 0 d\ble1 or treble1 c4 z ! ,pyramids or ,cones 71 "o to an o!r 7z we haue pr\$7 !r=e1 ?is f\r?1 %albe also d\ble or treble to ! fir/1 z ! ,pyramids or ,cones 7 "o to an o!r3 ,b our made ,cube1 is describ$ ( ! second 9 proportion1 ( ! f[] proportionall l9es3 99 ,by ! #cc4 ( ! eleu5? booke ( ,euclide4 !r=e 99 z ! f\r? l9e1 is to ! fir/1 s is t ,cube1 to ! fir/ ,cube3 & we haue pr\$ ! f\r? l9e1 to be to ! fir/1 z ! ,pyramis or ,cone1 is to ! ,pyramis or ,cone3 ,:]e=e ! ,cube is to ! ,cube1 z ,pyramis is to ,pyramis1 or ,cone is to ,cone4 ,i4 ,d4 99 ,& y\r dilig5ce 9 practise1 c s 79 waight ( wat]7 p]=me x3 ,!re=e1 n[1 y are able to geue good reason ( y\r :ole do+4 ,b we 99 ,suppose ,pyramis to ,pyramis1 or ,cone to ,cone1 to be d\ble or treble4 c4 ,!r=e ,cube1 is to ,cube1 d\ble1 or treble1 c4 ,:i* 0 to be demon/rat$4 ,& ( ! ,p>allelipip$1 x is euidt1 t ! wat] ,solide ,p>allelipip$ons1 are "o to ! o!r1 z _! hei!s are1 se+ !y haue "o base4 ,:]=e ! ,pyramids or ,cones1 made ( ^? wat] ,p>allelipip$ons1 are "o to ! o!r1 z ! l9es are 7"o to ! o!r7 betw5e :i*1 our proportion 0 assign$4 ,b ! ,cubes made ( l9es1 aft] ! proporti ( ! ,pyramidal or ,conik homologall l9es1 are "o to ! o!r1 z ! ,pyramides or ,cones are1 "o to ! o!r 7z we be=e did pr\e7 !r=e1 ! ,cubes made1 %albe "o to ! o!r1 z ! l9es assign$1 are "o to ! o!r3 ,:i* 0 to be demon/rat$4 ,note4 99 ,note ?is ,coroll>y4 99 ,?is1 my ,demon/rati is m g5]all1 !n onely 9 ,squ>e ,pyramis or ,cone3 ,consid] well4 ,?us1 haue ,i1 bo? ,ma!matically & ,me*anically1 b5 v long 9 wordes3 yet 7,i tru/7 no?+ t$i\s to !m1 :o1 to ^! ?+es1 are well affect$4 ,& v]ily ,i am =c$ 7auoid+ prolixitie7 to omit sundry su* ?+s1 easie to be practis$3 :i* to ! ,ma!matici51 w\ld be a great ,?reasure3 & to ! ,me*anici51 no small ga9e4 99 ,! great ,commodities foll[+ ( ^! new ,inu5tions4 99 ,n[ may y1 ,betw5e two l9es giu51 f9de two middle proportionals1 9 ,cont9uall proportion3 by ! holl[ ,p>allelipip$on1 & ! holl[ ,pyramis1 or ,cone4 ,n[1 any ,p>allelipip$on rectangle be+ giu53 ?re "r l9es may be f\nd1 proportionall 9 any proportion assign$1 ( :i*1 %al be produc$ a ,p>allelipip$on1 quall to ! ,p>allelipip$on giu54 ,h]e(1 ,i not$ som:at1 vpon ! #cf4 proposition1 ( ! #aa4 boke ( ,euclide4 ,n[1 all ^? ?+es1 :i* ,vitruuius 9 8 ,ar*itecture1 specifi$ hable to be done1 by dubbl+ ( ! ,cube3 ,or1 by f9d+ ( two middle proportionall l9es1 betw5e two l9es giu51 may easely be p]=m$4 ,n[1 t ,probleme1 :i* ,i not$ vnto y1 9 ! end ( my ,addition1 vpon ! #cd4 ( ! #aa4 boke ( ,euclide1 is pr\$ possible4 ,n[1 may any regul> body1 be ,trans=m$ into an o!r1 c4 ,n[1 any regul> body3 any ,sph]e1 yea any ,mixt ,solid3 & 7t m is7 ,irregul> ,solides1 may be made 79 any proporti assign$7 l vnto ! body1 fir/ giu54 ,?us1 ( a ,mannek51 7z ! ,dut* ,pa9t]s t]me x7 9 ! same ,symmetrie1 may a ,giant be made3 & t1 ) any ge/ure1 by ! ,mannek5 vs$3 & contr>ywise4 ,n[1 may y1 ( any ,m\ld1 or ,modell ( a ,%ip1 make "o1 ( ! same ,m\ld 79 any assign$ proportion7 bigg] or less]4 99 ,n[1 may y1 ( any 99 ,gunne1 or little peece ( ord9ace1 make an o!r1 ) ! same ,symmetrie 79 all po9tes7 z great1 & z little1 z y w4 ,m>ke t3 & ?9ke on x4 ,inf9itely1 may y apply ?is1 s long s\ght =1 & n[ s easily conclud$3 & )all1 s will+ly & frankly communicat$ to su*1 z fai?fully deale ) v]tu\s /udies4 ,su* is ! ,fruite ( ! ,ma!maticall ,sci5ces & ,artes4 ,?us1 c ! ,ma!maticall m9de1 deale ,speculatiuely 9 8 own ,arte3 & by good meanes1 ,m\nt ab\e ! cl\des & /]res3 ,& ?irdly1 he c1 by ord]1 ,desc5d1 to frame ,naturall ?+es1 to wond]full vses3 & :5 he li/1 retire home into 8 owne ,c5tre3 & !re1 prep>e m ,meanes1 to ,asc5d or ,desc5d by3 &1 all1 to ! glory ( ,god1 & our hone/ delectation 9 e>?4 ,al?\gh1 ! ,pr9t]1 ha? look$ = ?is ,prface1 a day or two1 yet c\ld ,i n br+ my p5 f ! pap]1 be=e ,i _h giu5 y com=table w>n+1 & brief in/ructions1 ( "s ( ! ,commodities1 by ,/atike1 hable to be reap$3 ,9 ! re/1 ,i w !r=e1 be z brief1 z x is possible3 & ) all1 describ+ !m1 som:at accord+ly4 ,& t1 y %all p]ceiue1 by ?is1 :i* 9 ord] comme? next4 ,=1 :]as1 x is s ample & wond]full1 t1 an :ole ye>e long1 "o might f9de fruitfull matt] !r91 to speake (3 & also 9 practise1 is a ,?reasure endeles3 yet w ,i glanse ou] x1 ) wordes v few4 ,?is d ,i call ,an?ropographie4 ,:i* is an ,art re/or$1 & ( my pref]m5t to y\r ,s]uice4 ,i pray y1 ?9ke ( x1 z ( "o ( ! *ief po9tes1 ( ,humane k4 ,al?\gh x be1 b n[1 fir/ ,cfirm$1 ) ?is new "n3 yet ! matt]1 ha? f ! beg9n+1 b5 9 consid]ation ( all p]fect ,philosoph]s4 ,an?ropographie1 is ! description ( ! ,numb]1 ,measure1 ,waight1 figure1 ,situation1 & col\r ( eu]y diu]se ?+1 conteyn$ 9 ! p]fect body ( ,,man3 ) c]ta9 k ( ! ,symmetrie1 figure1 waight1 ,*>act]ization1 & due locall motion1 ( any p>cell ( ! sayd body1 assign$3 & ( ,nb]s1 to ! sayd p>cell app]ta9yng4 ,?is1 is ! "o "p ( ! ,def9ition1 mete = ?is place3 ,suffici5t to notifie1 ! p>ticul>itie1 & excell5cy ( ! ,arte3 & :y x is1 "h1 ascrib$ to ! ,ma!maticals4 ,yf ! description ( ! heau5ly "p ( ! _w1 _h a peculi] ,art1 call$ ,a/ronomie3 ,if ! description ( ! e>?ly ,globe1 ha? 8 peculi] arte1 call$ ,geographie4 ,if ! ,mat*+ ( bo?1 ha? 8 peculi] ,arte1 call$ ,cosmographie3 ,:i* is ! ,descripti ( ! :ole1 & vniu]sall frame ( ! _w3 ,:y %\ld n ! description ( ,,man is ! ,lesse ,_w4 him1 :o is ! ,lesse _w3 &1 fr ! beg9n+1 call$ ,microcosmus 7t is4 ,! ,lesse ,_w47 ,& = ^: sake1 & s]uice1 all bodily creatures els1 7 creat$3 ,:o1 also1 p>ticipate? ) ,spirites1 & ,angels3 & is made to ! ,image & similitude ( ,god3 haue 8 peculi] ,art8 & be call$ ! ,arte ( ,artes3 r1 !n1 ei!r to want a "n1 or to haue to base & impropre a "n8 ,y mu/ ( sundry pr(essions1 bor[ or *all5ge home1 peculi] p>tes h]e(3 & f>d] proc$e3 z1 ,god1 ,nature1 ,reason & ,exp]i5ce %all in=me y4 ,! ,anatomi/es w re/ore to y1 "s "p3 ,! ,physiognomi/es1 "s3 ,! ,*yromanti/es "s4 ,! ,metaposcopi/es1 "s3 ,! excell5t1 ,alb]t ,dur]1 a good "p3 ! ,arte ( ,p]spectiue1 w som:at1 = ! ,eye1 helpe =w>d3 ,py?agoras1 ,hipocrates1 ,plato1 ,gal5us1 ,meletius1 _m o!r 79 c]ta9e ?+es7 w be ,contribut>ies4 ,& f>d]1 ! ,heau51 ! ,e>?1 & all o!r ,creatures1 w e*e %ew1 & (f] _! ,h>moni\s s]uice1 to fill vp1 t1 :i* wante? h]e(3 & ) y\r own ,exp]i5ce1 conclud+3 y may ,me?odically regi/] ! :ole1 = ! po/]itie3 ,:]eby1 good pr(e w be _h1 ( our ,h>moni\s1 & ,micro ,cosmus4 ,microcosmicall con/itution4 99 ,! outw>d ,image1 & vew h]e(3 to ! ,art ( ,zographie & ,pa9t+1 to ,sculpture1 & ,ar*itecture3 7= ,*ur*1 ,h\se1 ,=t1 or ,%ip7 is mo/ necess>y & pr(itable3 = t1 x is ! *iefe base & f\ndation ( !m4 99 ,lib4 #c4 ,cap4 #a4 ,looke 9 99 ,vitruuius1 :e!r ,i deale s9c]ely = y\r beh\fe1 or no4 ,looke 9 ,alb]tus ,dur]us1 ,de ,symmetria humani ,corporis4 ,looke 9 ! #bg4 & #bh4 ,*apt]s1 ( ! second booke1 ,de occulta ,philosophia4 ,consid] ! ,arke ( ,noe4 ,& by t1 wade f>!r4 ,rememb] ! ,delphicall ,oracle ,,nosce ,,teipsvm 7,kn[e ?y selfe7 s long agoe pron\nc$3 ( s _m a ,philosoph] repeat$3 & ( ! ,wise/ attempt$3 ,& !n1 y w p]ceaue1 h[ long agoe1 y haue b5e call$ to ! ,s*ole1 :]e ?is ,arte might be le>n$4 ,well4 ,i am no?+ affrayde1 ( ! disdayne ( "s su*1 z ?9ke ,sci5ces & ,artes1 to be b ,seu54 ,p]haps1 ^? ,su*1 may1 ) ignorance1 & %ame en\gh1 come %ort ( !m ,seu5 also3 & yet neu]!lesse !y c n prescribe a c]ta9e numb] ( ,artes3 & 9 e*e1 c]ta9e vnpassable b\ndes1 to ,god1 ,nature1 & mans ,indu/rie4 ,new ,artes1 dayly rise vp3 & !re 0 no su* ord] tak51 t1 ,all ,artes1 %\ld 9 "o age1 or 9 "o l&1 or ( "o man1 be made kn[5 to ! _w4 ,let vs embrace ! giftes ( ,god1 & wayes to wis$ome1 9 ?is "t ( grace1 f ab\e1 cont9ually be/[$ on !m1 :o ?ankefully w receiue !m3 ,et bonis ,omnia ,coop]abuntur 9 bonum4 ,tro*ilike1 is t ,art ,ma!maticall1 :i* demon/rate? ! prop]ties ( all ,circul> motions1 ,simple & ,comp\nde4 ,& bycause ! frute h]e(1 vulg>ly receiu$1 is 9 ,:eles1 x ha? ! "n ( ,tro*ilike3 z a man w\ld say1 ,:ele ,art4 ,by ?is art1 a ,:ele may be geu5 :i* %all m\e ones ab\t1 9 any tyme assign$4 ,two ,:eles may be giu51 ^: turnynges ab\t 9 "o & ! same tyme1 7or equall tymes71 %all haue1 "o to ! o!r1 any proportion appo9t$4 ,by ,:eles1 may a /raight l9e be describ$3 ,likewise1 a ,spirall l9e 9 pla9e1 ,conicall ,section l9es1 & o!r ,irregul> l9es1 at pleasure1 may be draw54 ,^!1 & su* l1 are pr9cipall ,conclusions ( ?is ,arte3 & helpe =w>d _m pleasant & pr(itable ,me*anicall workes3 ,saw ,milles4 ,z ,milles1 to ,saw great & v long ,deale bordes1 no man be+ by4 ,su* haue ,i se5e 9 ,g]many3 & 9 ! ,citie ( ,prage3 9 ! k+dome ( ,bohemia3 ,coyn+ ,milles1 ,h& ,milles = ,corne gr9d+3 ,& all man] ( ,milles1 & ,:ele worke3 ,by ,w9de1 ,smoke1 ,wat]1 ,waight1 ,spr+1 ,man or ,bea/1 m\$4 ,take 9 y\r h&1 ,agricola ,de re ,metallica3 & !n %all y 79 all ,m9es7 p]ceaue1 h[ great n$e is1 ( ,:ele worke4 ,by ,:eles1 /raunge workes & incr$ible1 are done3 z w1 9 o!r ,artes h]eaft]1 appe>e4 ,a wond]full example ( f>!r possibilitie1 & pres5t commoditie1 0 s5e 9 my "t1 9 a c]ta9e ,in/rum5t3 :i* by ! ,inu5t] & ,artific] 7be=e7 0 solde = xx4 ,tal5tes ( ,golde3 & !n _h 7by mis=tune7 receau$ "s iniurie & hurt3 ,& "o ,ianellus ( ,cremona did m5d ! same1 & pres5t$ x vnto ! ,emp]\r ,*>les ! fif?4 ,hi]onymus ,c>danus1 c be my witnesse1 t !re91 0 "o ,:ele1 :i* m\$1 & t1 9 su* rate1 t1 9 #gjjj4 ye>es onely1 8 owne p]iode %\ld be f9i%$4 ,a ?+ almo/ incr$ible3 ,b h[ f>re1 ,i keepe me )9 my b\ndes3 v _m m5 7yet aliue7 c tell4 ,helicosophie1 is n]e ,si/] to ,tro*ilike3 & is1 ,an ,arte ,ma!maticall1 :i* demon/rate? ! design+ ( all ,spirall l9es 9 ,pla9e1 on ,cyl9d]1 ,cone1 ,sphre1 ,conoid1 & ,sphroid1 & _! prop]ties app]tayn+4 ,! vse h]e(1 9 ,ar*itecture1 & diu]se ,in/rum5tes & ,eng9es1 is mo/ necess>y4 ,=1 9 _m ?+es1 ! ,skrue worke? ! feate1 :i*1 els1 c\ld n be p]=m$4 ,by helpe h]e(1 99 ,a!neus ,lib4 #e4 cap4 #h4 x is 99 record$1 t1 :]e all ! p[] ( ! ,citie ( ,syracusa1 0 n hable to m\e a c]ta9e ,%ip 7be+ on gr\nd7 mightie ,ar*im$es1 sett+ to1 8 ,skrui% ,eng9e1 caus$ ,hi]o ! k+1 by him self1 at ease1 to rem\e h]1 z he w\ld4 ,proclus4 ,pag4 #ah4 ,:]at1 ! ,k+ wondr+3 1 1 4 ,f ?is day1 =w>d 7said ! ,k+7 ,cr$it ought to be giu5 to ,ar*im$es1 :at soeu] he say?4 ,pneumati?mie demon/rate? by close holl[ ,geometricall ,figures1 7regul> & irregul>7 ! /raunge prop]ties 79 motion or /ay7 ( ! ,wat]1 ,ayre1 ,smoke1 & ,fire1 9 !yr ct9uitie1 & z !y are ioyn$ to ! ,elem5tes next !m4 ,?is ,arte1 to ! ,naturall ,philosoph]1 is v pr(fitable3 to pr\e1 t ,vacuum1 or ,empt9es is n 9 ! _w4 ,& t1 all ,nature1 abhorre? x s mu*3 t1 contr>y to ord9>y law1 ! ,elem5tes w m\e or /&4 ,z1 ,wat] to asc5d3 r !n betw5e him & ,ayre1 ,space or place %\ld be left1 m !n 7naturally7 t qutitie ( ,ayre require?1 or c fill4 ,aga9e1 ,wat] to hang1 & n desc5d3 r !n by desc5d+1 to leaue ,empt9es at 8 backe4 ,! l1 is ( ,fire & ,ayre3 !y w desc5d3 :51 ei!r1 _! ,ct9uitie %\ld be dissolu$3 or _! next ,elem5t =c$ f !m4 ,& z !y w n be ext5d$1 to discont9uitie3 ,s1 w !y n1 nor yet ( mans =ce1 c be pre/ or p5t1 9 space1 n suffici5t & aunsw]able to _! bodily sub/ance4 ,great =ce & viol5ce w !y vse1 to enioy _! naturall "r & lib]tie4 ,to g to ! bottom ( ! ,sea )\t daung]4 ,h]eupon1 two or ?ree m5 toge!r1 by kep+ ,ayre vnd] a great ,cauldron1 & =cyng ! same d[ne1 ord]ly1 may )\t h>me desc5d to ! ,sea bottome3 & cont9ue !re a tyme c4 ,:]e1 ,note1 h[ ! ?ick] ,elem5t 7z ! ,wat]7 giue? place to ! ?ynn] 7z1 is ! ayre37 & receiue? viol5ce ( ! ?9n]1 9 man]4 c4 ,pumps & all man] ( ,bell[es1 haue _! gr\nd ( ?is ,art3 & _m o!r /raunge deuises4 ,z1 ,hydraulica1 ,organes goyng by wat]4 c4 ,( ?is ,feat1 7call$ commonly ,pneumatica17 goodly workes are extant1 bo? 9 ,greke1 & ,lat94 ,) old & le>n$ ,s*ole m51 x is call$ ,sci5tia de pl5o vacuo4 ,m5adrie1 is an ,arte ,ma!maticall1 :i* demon/rate?1 h[1 ab\e ,natures v]tue & p[] simple3 ,v]tue & =ce may be multipli$3 & s1 to direct1 to lift1 to pull to1 & to put or ca/ fro1 any multipli$ or simple1 det]m9$ ,v]tue1 ,waight or ,=ce3 naturally1 n1 s1 directible or m\eable4 ,v mu* is ?is ,art furdr$ by o!r ,artes3 z1 9 "s po9tes1 by ,p]spectiue3 9 "s1 by ,/atike3 9 "s1 by ,tro*ilike3 & 9 o!r1 by ,helicosophie3 & ,pneumati?mie4 ,by ?is ,art1 all ,cranes1 ,gybbettes1 ,+9es to lift vp1 or to =ce any ?+1 any man] way1 are ordr$3 & ! c]ta9e cause ( _! =ce1 is kn[ne3 ,z1 ! =ce :i* "o man ha? ) ! ,du*e wagh5 ,racke3 !r)1 to set vp agayne1 a mighty wagh5 lad51 be+ ou]?r[ne4 ,! =ce ( ! ,crosseb[ ,racke1 is c]ta9ly1 "h1 demon/rat$4 ,! reason1 :y "o m1 do? ) a leau]1 lift t1 :i* ,sixe m51 ) _! h&es onely1 c\ld n1 s easily d4 ,by ?is ,arte1 9 our common ,cranes 9 ,london1 :]e p[re is to ,crane vp1 ! waight ( #bjjj4 p\nd3 by two ,:eles m 7by good ord] add$7 ,arte conclude?1 t !re may be ,cran$ vp #bjjjjj4 p\nd waight c4 ,s well knew ,ar*im$es ?is ,arte3 t he alone1 ) 8 deuises & engynes1 7twise or ?rise7 spoyl$ & discomfit$ ! :ole ,army & ,ho/e ( ! ,roma9es1 besieg+ ,syracusa1 ,plut>*us 9 ,m>co ,m>cello4 ,m>cus ,m>cellus ! ,consul1 be+ _! ,g5]all ,capita9e4 ,synesius 9 ,epi/olis4 ,su* huge ,/ones1 s _m1 ) su* =ce1 & s f>re1 did he ) 8 engynes hayle among !m1 out ( ! ,citie4 ,polybius4 ,pl9ius4 ,qu9tilianus4 ,t4 ,liuius4 ,& by ,sea likewise3 ?\gh _! ,%ips might come to ! walls ( ,syracusa1 yet hee vtt]ly conf\nd$ ! ,roma9e ,nauye4 ,:at ) 8 mighty ,/ones hurlyng3 99 ,a!nu4 :at ) ,pikes ( 99 #ah fote long1 made l %aftes3 :i* he =c$ almo/ a qu>t] ( a myle4 ,:at1 ) 8 cat*yng hold ( _! ,%yps1 & hoys+ !m vp ab\e ! wat]1 & sudd5ly lett+ !m fall into ! ,sea aga9e3 99 ,gal5us4 ,an!mius4 :at ) 8 99 ,burn+ ,glasses3 by :i* he fir$ _! o!r ,%ippes a f>-(3 :at1 ) 8 o!r pollicies1 deuises1 & eng9es1 he s manfully acquit him selfe3 t all ! ,=ce1 c\rage1 & pollicie ( ! ,roma9es 7= a great season7 c\ld no?+ preuaile1 = ! w9n+ ( ,syracusa4 ,:]upon1 ! ,romanes nam$ ,ar*im$es1 ,bri>eus1 & ,c5timanus4 ,zon>as make? m5tion ( "o ,proclus1 :o s well _h p]ceiu$ ,ar*im$es ,arte ( ,m5adrie1 & _h s well inu5t$ ( 8 owne1 t ) 8 ,burn+ ,glasses1 ,burn+ ,glasses4 be+ plac$ vpon ! walles ( ,bysance1 he multipli$ s ! heate ( ! ,sunne1 & direct$ ! beames ( ! same aga9/ 8 enemies ,nauie ) su* =ce1 & s sode9ly 7l light5+7 t he burn$ & de/roy$ bo? man & %ip4 ,& ,dionspecifie? ( ,priscus1 a ,geometrici5 9 ,bysance1 :o inu5t$ & vs$ sondry ,eng9s1 ( ,=ce multipli$3 ,:i* 0 cause1 t ! ,emp]\r ,seu]us p>don$ him1 8 life1 aft] he _h wonne ,bysance3 ,bycause he honor$ ! ,arte1 wytt1 & r>e indu/rie ( ,priscus4 ,b no?+ inf]ior to ! inu5tion ( ^! eng9es ( ,=ce1 0 ! inu5tion ( ,gunnes4 ,gunnes4 ,:i*1 f an ,engli% man1 _h ! occasion & ord] ( fir/ inu5t+3 ?\gh 9 an o!r l&1 & by o!r m51 x 0 fir/ execut$4 ,& !y t %\ld see ! record1 :]e ! occasion & ord] g5]all1 ( ,gunn+1 is fir/ disc\rs$ (1 w\ld ?9ke3 t1 8small ?+es1 slight1 & cmon3 comm+ to wise m5s consid]ation1 & indu/ri\s m5s h&l+1 may gr[ to be ( =ce incr$ible40 ,hypogeiodie1 is an ,arte ,ma!maticall1 demon/ratyng1 h[1 vnd] ! ,sphricall ,sup]ficies ( ! e>?1 at any dep?1 to any p]p5dicul> l9e assign$ 7^: di/ance f ! p]p5dicul> ( ! entrance3 & ! ,azimu?1 likewise1 9 respect ( ! said entrance1 is kn[57 c]ta9e way may be prscrib$ & gone3 ,& h[1 any way ab\e ! ,sup]ficies ( ! e>? design$1 may vnd] e>?1 at any dep? limit$1 be kept3 goyng alwayes1 p]p5dicul>ly1 vnd] ! way1 on e>? design$3 ,&1 contr>ywise1 ,any way1 7/raight or crok$17 vnd] ! e>?1 beyng giu53 vppon ! vtface1 or ,sup]ficies ( ! e>?1 to ,lyne out ! same3 ,s1 z1 f ! ,c5tre ( ! e>?1 p]p5dicul>s draw5 to ! ,sphricall ,sup]ficies ( ! e>?1 %all precisely fall 9 ! ,correspond5t po9tes ( ^? two wayes4 ,?is1 ) all o!r ,cases & circum/ances h]e91 & app]t5ances1 ?is ,arte demon/rate?4 ,?is ,arte1 is v ample 9 v>ietie ( ,conclusions3 & v pr(itable sundry wayes to ! ,common ,weal?4 ,! occasion ( my ,inu5t+ ?is ,arte1 0 at ! reque/ ( two ,g5tlem51 :o _h a c]ta9e worke 7( ga9e7 vnd] gr\nd3 & _! gr\ndes did ioyne ou] ! worke3 & by reason ( ! crok$nes1 diu]s dep!s1 & hei!s ( ! way vnd] gr\nd1 !y 7 9 d\bt1 & at contr\]sie1 vnd] ^: gr\nd1 z !n1 ! worke 04 ,! "n onely 7be=e ?is7 0 ( me publi%$1 ,de ,it9]e ,subt]raneo3 ,! re/1 be at ,gods w4 ,= ,pion]s1 ,m9]s1 ,digg]s = ,mettalls1 ,/one1 ,cole1 & = secrete passages vnd] gr\nd1 betw5e place & place 7z ?is l& ha? diu]se7 & = o!r purposes1 any man may easily p]ceaue1 bo? ! great fruite ( ?is ,arte1 & also 9 ?is ,arte1 ! great aide ( ,geometrie4 ,hydragogie1 demon/rate? ! possible lead+ ( ,wat]1 by ,natures lawe1 & by artificiall helpe1 f any head 7be+ a ,spr+1 /&+1 or runn+ ,wat]7 to any o!r place assign$4 ,long1 ha? ?is ,arte b5e 9 vse3 & mu* !re( writt53 & v m>ueil\s workes !re91 p]=m$3 z may yet appe>e1 9 ,italy3 by ! ,ruynes rema9+ ( ! ,aqu$uctes4 ,9 o!r places1 ( ,riu]s lead+ ?r\gh ! ,ma9e l&1 ,nauigable _m a ,mile4 ,& 9 o!r places1 ( ! m>ueil\s =c+es ( ,wat] to ,asc5d4 :i* all1 decl>e ! great skill1 to be requir$ ( him1 :o %\ld 9 ?is ,arte be p]fecte1 = all occasions ( wat]s possible lead+4 ,to speake ( ! all[ance ( ! ,fall1 = eu]y hundr$ foote3 or ( ! ,v5tills 7if ! wat]s lab\r be f>re1 & great7 ,i ne$e n3 ,se+1 at h& 7ab\t vs7 _m exp]t m5 c suffici5tly te/ifie1 9 effecte1 ! ord]3 ?\gh ! ,demon/ration ( ! ,necessitie !re(1 !y "k n3 ,nor yet1 if !y %\ld be l$1 vp & d[ne1 & ab\t ,m\nta9es1 f ! head ( ! ,spr+3 & !n1 a place be+ assign$3 & ( !m1 to be demaund$1 h[ l[ or high1 t la/ place is1 9 respecte ( ! head1 f :i* 7s crok$ly1 & vp & d[ne7 !y be come3 ,p]haps1 !y w\ld n1 or c\ld n1 v r$ily1 or n]ely assoyle t "q4 ,geometrie !re=e1 is necess>y to ,hydragogie4 ,( ! sundry wayes to =ce wat] to asc5d1 ey!r by ,tympane1 ,kettell mills1 ,skrue1 ,ctesibike1 or su* l3 9 ,vitruuius1 ,agricola1 7& o!r17 fully1 ! man] may appe>e4 ,& s1 !reby1 also be mo/ euid5t1 h[ ! ,artes1 ( ,pneumati?mie1 ,helicosophie1 ,/atike1 ,tro*ilike1 & ,m5adrie1 come to ! furniture ( ?is1 9 ,speculation1 & to ! ,commoditie ( ! ,common ,weal?1 9 practise4 ,horometrie1 is an ,arte ,ma!maticall1 :i* dem/rate?1 h[1 at all times appo9t$1 ! precise vsuall d5om9ati ( "t1 may be kn[51 = any place assign$4 ,^! wordes1 are smo? & pla9e easie ,engli%e1 b ! rea* ( _! mean+1 is f>!r1 !n y w\lde lightly imag9e4 ,"s "p ( ?is ,arte1 0 call$ 9 olde "t1 ,gnomonice3 & ( late1 ,horologiographia3 & 9 ,engli%e1 may be t]m$1 ,diall+4 ,aunci5t is ! vse1 & m aunci5t1 is ! ,inu5tion4 ,! vse1 do? well appe>e to haue b5e 7at ! lea/7 ab\e two ?\s& & ?ree hundr$ ye>e agoe3 #d4 ,reg4 #bj4 9 99 ,k+ ,a*az ,diall1 !n1 by ! ,sunne1 %ew+ ! di/9ction ( "t4 ,by ,sunne1 ,mone1 & ,/]res1 ?is ,diall+ may be p]=m$1 & ! precise ,"t ( day or night kn[54 ,b ! demon/ratiue del9eation ( ^! ,dialls1 ( all sortes1 require? good skill1 bo? ( ,a/ronomie1 & ,geometrie ,elem5tall1 ,sphricall1 ,phnom5all1 & ,conikall4 ,!n1 to vse ! gr\ndes ( ! ,arte1 = any regul> ,sup]ficies1 9 any place (fr$3 & 79 any possible apt position !r(7 !ron1 to describe 7all man] ( wayes7 h[1 vsuall h[]s1 may be 7by ! ,sunnes %ad[7 truely det]m9$3 w be f\nd no sleight ,pa9t]s worke4 ,s to ,pa9t1 & prescribe ! ,sunnes ,motion1 to ! bread? ( a he>e4 ,9 ?is ,feate 79 my y\?7 ,i ,inu5t$ a way1 ,h[ 9 any ,horizontall1 ,murall1 or qu9octiall ,diall1 c4 ,at all h[]s 7! ,sunne %9+7 ! ,signe & ,degree asc5d5t1 may be kn[54 ,:i* is a ?+ v necess>y = ! ,ris+ ( ^? fix$ ,/]res3 ^: ,op]ation 9 ! ,ayre1 is ( great might1 euid5tly4 ,i speake no fur!r1 ( ! vse h]e(4 ,bur =asmu* z1 ,mans affaires require k ( ,times ,mom5tes1 :51 nei!r ,sunne1 ,mone1 or ,/]re1 c be s5e3 ,!re=e1 by ,indu/rie ,me*anicall1 0 inu5t$1 fir/1 h[1 by ,wat]1 runn+ ord]ly1 ! ,"t & h[]s might be kn[53 :]e(1 ! fam\s ,ctesibius1 0 ,inu5tor3 a man1 ( ,vitruuius1 to ! ,skie 7iu/ly7 extoll$4 ,!n1 aft] t1 by ,s& runn+1 7 h[]s measur$3 ,!n1 by ,tro*ilike ) waight3 ,& ( late "t1 by ,tro*ilike ) ,spr+3 )\t waight4 ,all ^!1 by ,sunne or ,/]res direction 79 c]ta9e "t7 require ou]sight & re=mation1 accord+ to ! heau5ly qu9octiall ,motion3 besides ! 9qualitie ( _! owne ,op]ation4 ,!re remayne? 7)\t p>abolicall mean+ h]e97 among ! ,philosoph]s1 ,a p]petuall ,motion4 a m excell5t1 m commodi\s1 & m m>ueil\s way1 !n all ^!3 ( hau+ ! motion ( ! ,prim\ant 7or fir/ qu9octiall motion17 by ,nature & ,arte1 ,imitat$3 :i* y %all 7by furd] se>* 9 waighti] /udyes7 h]eaft]1 vnd]/& m (4 ,& s1 x is tyme to f9i% ?is ,annotation1 ( ,tymes di/9ction1 vs$ 9 our common1 & priuate affaires3 ,! commoditie :](1 no man w\ld want1 t c tell1 h[ to be/[ 8 tyme4 ,zographie1 is an ,arte ,ma!maticall1 :i* tea*e? & demon/rate?1 h[1 ! ,int]section ( all visuall ,pyramides1 made by any playne assign$1 7! ,c5tre1 di/ance1 & lightes1 beyng det]m9$7 may be1 by lynes1 & due propre col\rs1 repres5t$4 ,a notable ,arte1 is ?is3 & w\ld require a :ole ,volume1 to decl>e ! prop]ty !re(3 & ! ,commodities ensuyng4 ,great skill ( ,geometrie1 ,ari?metike1 ,p]spectiue1 & ,an?ropographie1 ) _m o!r p>ticul> ,artes1 ha? ! ,zograph]1 n$e (1 = 8 p]fection4 ,=1 ! mo/ excell5t ,pa9t]1 7:o is b ! propre ,me*anici51 ,imitator s5sible1 ( ! ,zograph]7 ha? atte9$ to su* p]fection1 t ,s5se ( ,man & bea/1 haue iudg$ ?+es pa9t$1 to be ?+s naturall1 & n artificiall3 aliue1 & n dead4 ,?is ,me*anicall ,zograph] 7commonly call$ ! ,pa9t]7 is m]uail\s 9 8 skill3 & seeme? to haue a c]ta9e diu9e p[]3 ,z1 ( fr5des abs5t1 to make a fr5dly1 pres5t com=t3 yea1 & ( fr5des dead1 to giue a cont9uall1 sil5t pres5ce3 n onely ) vs1 b ) our po/]itie1 = _m ,ages4 ,& s proc$yng1 ,consid]1 ,h[1 9 ,w9t]1 he c %ew y1 ! liuely vew ( ,somm]s ,ioy1 & ri*es3 & 9 ,somm]1 exhibite ! c\nt5ance ( ,w9t]s dolefull ,/ate1 & nak$nes4 ,cities1 ,t[nes1 ,=tes1 ,woodes1 ,armyes1 yea :ole ,k+domes 7be !y neu] s f>re1 or greate7 c he1 ) ease1 br+ ) him1 home 7to any mans ,iudgem5t7 z ,pat]nes liuely1 ( ! ?+es rehe>s$4 ,9 "o little h\se1 c he1 enclose 7) great pleasure ( ! behold]s17 ! portrayture liuely1 ( all visible ,creatures1 ei!r on e>?1 or 9 ! e>?1 liu+3 or 9 ! wat]s ly+1 ,crep+1 slyd+1 or swimm+3 or ( any f\le1 or fly1 9 ! ayre fly+4 ,nay1 9 respect ( ! ,/>res1 ! ,skie1 ! ,cl\des3 yea1 9 ! %ew ( ! v light x selfe 7t ,diu9e ,creature7 c he mat* our eyes ,iudgem5t1 mo/ n]ely4 ,:at a ?+ is ?is8 ?+es n yet be+1 he c repres5t s1 z1 at _! be+1 ! ,picture %all seame 79 man]7 to haue ,creat$ !m4 ,to :at ,artific]1 is n ,picture1 a great pleasure & ,commoditie8 ,:i* ( !m all1 w refuse ! ,direction & ayde ( ,picture8 ,! ,ar*itect1 ! ,goldsmi?1 & ! ,arras ,weau]3 ( ,picture1 make great acc\nt4 ,our liuely ,h]bals1 our portraitures ( birdes1 bea/es1 & fi%es3 & our curi\s ,anatomies1 :i* way1 are !y mo/ p]fectly made1 or ) mo/ pleasure1 ( vs behold58 ,is x n1 by ,picture onely8 ,& if ,picture1 by ! ,indu/ry ( ! ,pa9t]1 be ?us commodi\s & m]uail\s3 :at %all be ?\ght ( ,zographie1 ! ,s*olema/] ( ,picture1 & *ief g\]nor8 ,?\gh ,i m5cion n ,sculpture1 9 my ,table ( ,artes ,ma!maticall3 yet may all m5 p]ceiue1 ,h[1 t ,picture & ,sculpture1 are ,si/]s g]ma9e3 & bo?1 "r pr(itable1 9 a ,comm weal?4 & ( ,sculpture1 aswell z ( ,picture1 excell5t ,artific]s haue writt5 great bokes 9 comm5dation4 ,witnesse ,i take1 ( ,georgio ,vas>i1 ,pittore ,aret9o3 ( ,pomponius ,gauricus3 & o!r4 ,to ^! two ,artes1 7) o!r17 is a c]ta9e od ,arte1 call$ ,al?almasat1 mu* beholdyng3 m1 !n ! common ,sculptor1 ,entayl]1 ,k]u]1 ,cutt]1 ,grau]1 ,f\nd]1 or ,paynt] 7c7 "k _! ,arte1 to be commodi\s4 ,an objection4 ,ar*itecture1 to _m may seme n wor?y1 or n mete1 to be reckn$ among ! ,artes ,ma!maticall4 ,to :om1 ,i ?9ke good1 to giue "s acc\nt ( my s doyng4 ,n wor?y1 7w !y say17 bycause x is b = build+1 ( a h\se1 ,pallace1 ,*ur*1 ,=te1 or su* l1 grosse workes4 ,& y1 also1 def9$ ! ,artes ,ma!maticall1 to be su*1 z deal$ ) no ,mat]iall or corruptible ?+3 & also did demon/ratiuely proc$e 9 _! faculty1 by ,numb] or ,magnitude4 ,fir/1 ,! ,answ]4 y see1 t ,i c\nt1 "h1 ,ar*itecture1 among ^? ,artes ,ma!maticall1 :i* are ,d]iu$ f ! ,pr9cipals3 & y "k1 t su*1 may deale ) ,naturall ?+es1 & s5sible matt]4 ,( :i*1 8"s draw n]]1 to ! ,simple & absolute ,ma!maticall ,speculation1 !n o!r d4 ,& ?\gh1 ! ,ar*itect procure?1 en=me?1 directe?1 ! ,me*anici51 to h&worke1 ! build+ actuall1 ( h\se1 ,ca/ell1 or ,pallace1 & is *ief ,iudge ( ! same3 yet1 ) him selfe 7z *ief ,ma/] & ,ar*itect17 rema9e? ! ,demon/ratiue reason & cause1 ( ! ,me*anici5s worke3 9 ,lyne1 pla9e1 & ,solid3 by ,geometricall1 ,ari?meticall1 ,opticall1 ,musicall1 ,a/ronomicall1 ,cosmographicall0 7 to be brief7 by all ! =m] ,d]iu$ ,artes ,ma!maticall1 & o!r ,naturall ,artes1 hable to be confirm$ & /abli%$4 ,if ?is be s3 !n1 may y ?9ke1 t ,ar*itecture1 ha? good & due all[ance1 9 ?is hone/ ,company ( ,artes ,ma!maticall ,d]iuatiue4 ,i w1 h]e91 craue ,iudgem5t ( two mo/ p]fect ,ar*itectes3 ! "o1 be+ ,vitruuius1 ! ,roma9e3 :o did write t5 bookes !re(1 to ! ,emp]\r ,augu/us 79 ^: daies our ,heau5ly ,ar*ema/]1 0 borne73 & ! o!r1 ,leo ,bapti/a ,alb]tus1 a ,flor5t9e3 :o also publi%$ t5 bookes !r(4 ,ar*itectura 7say? ,vitruuius7 e/ ,sci5tia pluribus discipl9is v>ijs eruditionibus ornata3 cuius ,iudicio probantur omnia1 qu ab ct]is ,artificibus p]ficiuntur op]a4 ,t is4 ,ar*itecture1 is a ,sci5ce g>ni%$ ) _m doctr9es diu]se in/ructions3 by ^: ,iudgem5t1 all workes1 by o!r workm5 f9i%$1 are ,iudg$4 ,x foll[e?4 ,ea nascitur ex ,fabrica1 ,ratioc9atione4 c4 ,ratioc9atio autem e/1 qu1 res fabricatas1 ,sol]tia ac ratione proportionis1 demon/r>e atque explic>e pote/4 ,ar*itecture1 gr[e? ( ,fram+1 & ,reason+4 c4 ,reason+1 is t1 :i* ( ?+es fram$1 ) =eca/1 & proportion3 c make demon/ration1 & manife/ decl>ation4 ,aga9e4 ,cm1 9 omnibus enim rebus1 tm maxim etiam 9 ,ar*itectura1 hc duo insunt3 quod significatur1 quod significat4 ,significatur proposita res1 de qua dicitur3 hanc autem ,significat ,demon/ratio1 rationibus doctr9>um explicata4 ,=asmu* z1 9 all ?+es3 !re=e *iefly 9 ,ar*itecture1 ^! two ?+es are3 ! ?+ signifi$3 & t :i* signifie?4 ,! ?+ prop\nd$1 :]e( we speake1 is ! ?+ ,signifi$4 ,b ,demon/ration1 express$ ) ! reasons ( diu]se doctr9es1 do? signifie ! same ?+4 ,aft] t4 ,vt lit]atus sit1 p]itus ,graphidos1 eruditus ,geometri1 ,optices non ign>us3 in/ructus ,ari?metica3 hi/orias complures n\]it1 ,philosophos dilig5t] audiu]it3 ,musicam sciu]it3 ,m$ic9 non sit ign>us1 responsa ,iurisp]itor n\]it3 ,a/rologiam1 ,clique rationes cognitas habeat4 ,an ,ar*itect 7say? he7 ought to vnd]/& ,languages1 to be skilfull ( ,pa9t+1 well in/ruct$ 9 ,geometrie1 n ignorant ( ,p]spectiue1 furni%$ ) ,ari?metike1 haue k ( _m hi/ories1 & dilig5tly haue he>d ,philosoph]s1 haue skill ( ,musike1 n ignorant ( ,physike1 "k ! aunsw]es ( ,lawy]s1 & haue ,a/ronomie1 & ! c\rses ,cle/iall1 9 good k4 ,he geue? reason1 ord]ly1 :]e=e all ^! ,artes1 ,doctr9es1 & ,in/ructions1 are requisite 9 an excell5t ,ar*itect4 ,& 7= breuitie7 omitt+ ! ,lat9 text1 ?us he ha?4 ,secondly1 x is beh(efull = an ,ar*itect to haue ! k ( ,pa9t+3 t he may ! m easilie fa%ion out1 9 patt]nes pa9t$1 ! =me ( :at worke he like?4 ,& ,geometrie1 geue? to ,ar*itecture _m helpes3 & fir/ tea*e? ! ,vse ( ! ,rule1 & ! ,cumpasse3 :]by 7*iefly & easilie7 ! descriptions ( ,build+es1 are despat*$ 9 ,gr\ndplats3 & ! directions ( ,squires1 ,leuells1 & ,l9es4 ,likewise1 by ,p]spectiue1 ! ,lightes ( ! heau51 are well l$1 9 ! build+es3 f c]ta9e qu>t]s ( ! _w4 ,by ,ari?metike1 ! *>ges ( ,build+es are summ$ toge!r3 ! measures are express$1 & ! h>d que/ions ( ,symmetries1 are by ,geometricall ,meanes & ,me?ods disc\rs$ on4 c4 ,besides ?is1 ( ! ,nature ( ?+es 7:i* 9 ,greke is call$ 7 ,philosophie do? make decl>ation4 ,:i*1 x is necess>y1 = an ,ar*itect1 ) dilig5ce to haue le>n$3 because x ha? _m & diu]s naturall que/ions3 z specially1 9 ,aqu$uctes4 ,= 9 _! c\rses1 lead+es ab\t1 9 ! leuell gr\nd1 & 9 ! m\nt+es1 ! naturall ,spirites or brea!s are +5dr$ diu]s wayes3 ,! h9drances1 :i* !y cause1 no man c helpe1 b he1 :i* out ( ,philosophie1 ha? le>n$ ! orig9all causes ( ?+es4 ,likewise1 :o soeu] %all read ,ctesibius1 or ,ar*im$es bookes1 7& ( o!rs1 :o haue writt5 su* ,rules7 c n ?9ke1 z !y d3 vnlesse he %all haue receau$ ( ,philosoph]s1 in/ructions 9 ^! ?+es4 ,& ,musike he mu/ n$es "k3 t he may haue vnd]/&+1 bo? ( ,regul> & ,ma!maticall ,musike3 t he may temp] well 8 ,bali/es1 ,catapultes1 & ,scorpions4 c4 ,more\]1 ! ,bras5 ,vessels1 :i* 9 ,!atres1 are plac$ by ,ma!maticall ord]1 9 ambries1 vnd] ! /eppes3 & ! diu]sities ( ! s\ndes 7:i* ye ,grecians call 7 are ordr$ accord+ to ,musicall ,symphonies ,h>monies3 be+ di/ribut$ 9 ye ,circuites1 by ,diatess>on1 ,diap5te1 & ,diapason4 ,t ! conu5i5t voyce1 ( ! play]s s\nd1 : x came to ^! prep>ations1 made 9 ord]1 !re be+ increas$3 ) yt increas+1 might come m cle>e pleasant1 to ye e>es ( ! lok]s on4 c4 ,& ( ,a/ronomie1 is "k ye ,ea/1 ,we/1 ,s\?1 & ,nor?4 ,! fa%ion ( ! heau51 ! qu9ox1 ! ,sol/icie1 & ! c\rse ( ! /]res4 ,:i* ?+es1 vnlea/ "o "k3 he c n p]ceiue1 any ?yng at all1 ! reason ( ,horologies4 ,seyng !r=e ?is ample ,sci5ce1 is g>ni%$1 beautifi$ & /or$1 ) s _m & sundry skils & kn[l$ges3 ,i ?9ke1 t none c iu/ly acc\nt !m selues ,ar*itectes1 ( ! suddeyne4 ,b !y onely1 :o f _! *ildes ye>es1 asc5dyng by ^! degrees ( kn[l$ges1 beyng fo/]$ vp ) ! atteynyng ( _m ,languages & ,artes1 haue wonne to ! high ,tab]nacle ( ,ar*itecture4 c4 ,& to :om ,nature ha? giu5 su* quicke ,circumspection1 %>pnes ( witt1 & ,memorie1 t !y may be v absolutely skillfull 9 ,geometrie1 ,a/ronomie1 ,musike1 & ! re/ ( ! ,artes ,ma!maticall3 ,su*1 surm\nt & passe ! callyng1 & /ate1 ( ,ar*itectes3 ,a ,ma!matici54 & are become ,ma!matici5s4 c4 ,& !y are f\nd1 seldome4 ,z1 9 tymes pa/1 0 ,ari/>*us ,samius3 ,philolaus1 & ,ar*ytas1 ,t>5tynes3 ,apollonius ,p]gu3 ,erato/h5es ,cyr5eus3 ,ar*im$es1 & ,scopas1 ,syracusians4 ,:o also1 left to !yr po/]itie1 _m ,eng9es & ,gnomonicall workes3 by numb]s & naturall meanes1 inu5t$ & decl>$4 ,?us mu*1 & ! same wordes 79 s5se7 9 "o onely ,*apt] ( ?is ,incp>able ,ar*itect ,vitruuius1 %all y f9de4 ,& if y %\ld1 b take 8 boke 9 y\r h&1 & slightly loke ?or\gh x1 y w\ld say /raight way3 ,vitruuius4 ,?is is ,geometrie1 ,ari?metike1 ,a/ronomie1 ,musike1 ,an?ropographie1 ,hydragogie1 ,horometrie4 c4 & 7to cclude7 ! ,/oreh\se ( all workm%ip4 ,n[1 let vs li/5 to our o!r ,iudge1 our ,flor5t9e1 ,leo ,bapti/a3 & n>r[ly consid]1 h[ he do? det]m9e ( ,ar*itecture4 ,s$ anteque vltra progr$i>4 c4 ,b be=e ,i proc$e any fur!r 7say? he7 ,i ?9ke1 t ,i ought to expresse1 :at man ,i w\ld haue to bee all[$ an ,ar*itect4 ,=1 ,i w n bryng 9 place a ,c>p5t]3 z ?\gh y might ,comp>e him to ! ,*ief ,ma/]s ( o!r ,artes4 ,= ! h& ( ! ,c>p5t]1 is ! ,ar*itectes ,in/rum5t4 ,v,vho is an ,ar*itect4 ,b ,i w appo9t ! ,ar*itect to be 8t man1 :o ha? ! skill1 7by a c]ta9e & m]uail\s meanes & way17 bo? 9 m9de & ,imag9ation to det]m9e & also 9 worke to f9i%3 :at workes s eu]1 by motion ( waight1 & cuppl+ & framyng toge!r ( bodyes1 may mo/ aptly be ,commodi\s = ! wor?ie/ ,vses ( ,man40 ,& t he may be able to p]=me ^! ?+es1 he ha? n$e ( atteynyng & k ( ! be/1 & mo/ wor?y ?ynges4 c4 ,! :ole ,feate ( ,ar*itecture 9 buildyng1 consi/e? 9 ,l9eam5tes1 & 9 ,framyng4 ,& ! :ole p[] & skill ( ,l9eam5tes1 t5de? to ?is3 t ! "r & absolute way may be _h1 ( ,coaptyng & ioyn+ ,l9es & angles3 by :i*1 ! face ( ! buildyng or frame1 may be compreh5d$ & conclud$4 ,& x is ! prop]ty ( ,l9eam5tes1 to prescribe vnto buildynges1 & eu]y "p ( !m1 an apt place1 c]ta9e nb]3 a wor?y man]1 & a semely ord]3 t1 s1 ye :ole =me & figure ( ! buildyng1 may re/ 9 ! v ,l9eam5tes4 c4 ,& we may prescribe 9 mynde & imag9ation ! :ole =mes1 99 99 ,! ,immat]ialitie ( p]fect ,ar*itecture4 all mat]ial /uffe beyng seclud$4 ,:i* po9t we %all atteyne1 by ,notyng & =epo9tyng ! angles1 & l9es1 by a sure & c]ta9e direction & connexion4 ,seyng !n1 ^! ?+es1 are ?us3 ,:at1 ,l9eam5t is4 ,l9eam5te1 %albe ! c]ta9e & con/ant prescribyng1 conceiu$ 9 mynde3 made 9 l9es & angles3 & f9i%$ ) a le>n$ m9de & wyt4 8,we ?anke y ,ma/] ,bapti/1 t y haue s aptly br\ght y\r ,arte1 & phrase !r(1 to haue "s ,ma!maticall p]fection3 ,note4 by c]ta9e ord]1 nb]1 =me1 figure1 & ,symmetrie m5tall30 all naturall s5sible /uffe set a "p4 ,n[1 !n1 x is euid5t1 7,g5tle read]7 h[ aptely & wor!ly1 ,i haue pref]r$ ,ar*itecture1 to be br$ & fo/]$ vp 9 ! ,dom9ion ( ! p]eles ,pr9cesse1 ,ma!matica3 & to be a naturall ,subiect ( h]s4 ,& ! "n ( ,ar*itecture1 is ( ! pr9cipalitie1 :i* ?is ,sci5ce ha?1 ab\e all o!r ,artes4 ,& ,plato affirme?1 ! ,ar*itect to be ,ma/] ou] all1 t make any worke4 ,:]upon1 he is nei!r ,smi?1 nor ,build]3 nor1 sep>ately1 any ,artific]3 b ! ,h$1 ! ,pr\o/1 ! ,direct]1 & ,iudge ( all ,artificiall workes1 & all ,artific]s4 ,=1 ! true ,ar*itect1 is hable to tea*1 ,demon/rate1 di/ribute1 describe1 & ,iudge all workes wr\ght4 ,& he1 onely1 se>*e? out ! causes & reasons ( all ,artificiall ?ynges4 ,?us excell5t1 is ,ar*itecture3 ?\gh few 79 our dayes7 atteyne !reto3 yet may n ! ,arte1 be o!rwise ?\ght on1 !n 9 v d$e x is wor?y4 ,nor we may n1 ( aunci5t ,artes1 make new & imp]fect ,def9itions 9 our dayes3 = sc>sitie ( ,artific]s3 ,no m1 ?an we may pyn*e 91 ! ,def9itions ( ,wis$ome1 or ,hone/ie1 or ( ,fr5de%yp or ( ,iu/ice4 ,no m w ,i cons5t1 to ,dim9i% any :it1 ( ! p]fection & dignitie1 7by iu/ cause7 all[$ to absolute ,ar*itecture4 ,vnd] ! ,direction ( ?is ,arte1 are ?re pr9cipall1 necess>y ,me*anicall ,artes4 ,namely1 ,h[s+1 ,=tification1 & ,naupegie4 ,h[s+1 ,i vnd]/&1 bo? = ,diu9e ,s]uice1 & ,mans common vsage3 publike1 & priuate4 ,( ,=tification & ,naupegie1 /raunge matt] might be told y3 ,b p]*aunce1 "s w be tyr$1 ) ?is ,b$]oll1 all ready rehe>s$3 & o!r "s1 w nycely nip my grosse & homely disc\rs+ ) y3 made 9 po/ ha/3 = fe>e y %\ld wante ?is true & fr5dly w>nyng1 & ta/ giuyng1 ( ! ,p[] ,ma!maticall4 ,lyfe is %ort1 & vnc]ta9e3 ,tymes are p]il\se3 c4 ,& /ill ! ,pr9t] awayt+1 = my p5 /ay+3 ,all ^! ?+es1 ) f>d] matt] ( ,+ratefulnes1 giue me occasion to passe away1 to ! o!r ,artes rema9yng1 ) all sp$e possible4 ,! ,arte ( ,nauigation1 demon/rate? h[1 by ! %orte/ good way1 by ! apte/ ,directi1 9 ! %orte/ "t1 a suffici5t ,%ip1 betw5e any two places 79 passage ,nauigable17 assign$3 may be cduct$3 & 9 all /ormes1 naturall di/urbances *auncyng1 h[1 to vse ! be/ possible meanes1 :]eby to rec\] ! place fir/ assign$4 ,:at n$e1 ! ,ma/] ,pilote1 ha? ( o!r ,artes1 "h be=e recit$1 x is easie to "k3 z1 ( ,hydrographie1 ,a/ronomie1 ,a/rologie1 & ,horometrie4 ,presuppos+ cont9ually1 ! common ,base1 & f\ndacion ( all3 namely ,ari?metike & ,geometrie4 ,s t1 he be hable to vnd]/&1 & ,iudge 8 own necess>y ,in/rum5tes1 & furniture ,necess>y3 ,:e!r !y be p]fectly made or no3 & also c1 7if n$e be7 make !m1 hym selfe4 ,z ,quadrantes1 ,! ,a/ronom]s ,ryng1 ,! ,a/ronom]s /affe1 ,! ,a/rolabe vniu]sall4 ,an ,hydrographicall ,globe4 ,*>ts ,hydrographicall1 true1 7n ) p>allell ,m]idians74 ,! ,common ,sea ,compas3 ,! ,compas ( v>iacion3 ,! ,proportionall1 & ,p>adoxall ,compasses ,anno4 #aeei4 7( me ,inu5t$1 = our two ,mosc\y ,ma/] ,pilotes1 at ! reque/ ( ! ,company7 ,clockes ) spryng3 h\re1 halfe h\re1 & ?ree h\re ,s&glasses3 sundry o!r ,in/rumtes3 ,& also1 be hable1 on ,globe1 or ,playne to describe ! ,p>adoxall ,compasse3 & duely to vse ! same1 to all man] ( purposes1 :]eto x 0 inu5t$4 ,& also1 be hable to ,calculate ! ,planetes places = all tymes4 ,more\]1 ) ,sonne ,mone or ,/]re 7or )\t7 be hable to def9e ! ,longitude ,latitude ( ! place1 :i* he is 93 ,s t1 ! ,longitude ,latitude ( ! place1 f :i* he sayl$1 be giu53 or by him1 be kn[ne4 :]eto1 app]tayne? exp]t meanes1 to be c]tifi$ eu]1 ( ! ,%ips way4 c4 ,& by =ese+ ! ,ris+1 ,settyng1 ,none/$yng1 or ,midnightyng ( c]ta9e tempe/u\s fix$ ,/]res3 or _! ,coniunctions1 & ,anglynges ) ! ,planetes1 c4 he ought to haue exp]t coniecture ( ,/ormes1 ,tempe/es1 & ,sp\tes3 & su* lyke ,meteorologicall effectes1 daung]\s on ,sea4 ,= 7z ,plato say?17 ,mutationes1 opportunitatesque temporum pres5tire1 non m9us rei milit>i1 qum ,agricultur1 ,nauigationique conu5it4 ,to =esee ! alt]ations & opportunities ( tymes1 is conu5i5t1 no lesse to ! ,art ( ,w>re1 !n to ,husb&ry & ,nauigation4 ,& besides su* cunnyng meanes1 m euid5t tok5s 9 ,sonne & ,mone1 ought ( hym to be kn[53 su* z 7! ,philosophicall ,pote7 ,virgilius tea*e?1 9 hys ,georgikes4 ,:]e he say?1 ,sid5ote3 ,georgic4 #a4 ,sol quoque exori5s quum se condet 9 vndas1 ,signa dabit1 ,solem c]tissima signa sequuntur4 c4 -------- ,nam spe videmus1 ,ipsius 9 vultu v>ios err>e colores4 ,cruleus1 pluuiam d5unciat1 igneus ,euros4 ,s9 macul incipi5t rutilo immisc]i] igni1 ,omnia tum p>it] v5to1 nimbisque videbis ,f]u]e3 non illa quisquam me nocte p] altum ,ire1 neque a t]ra m\eat conuell]e funem4 c4 ,sol tibi signa dabit4 ,solem quis dic]e falsum ,audeat8 -------- c4 ,& s ( ,mone1 ,/]res1 ,wat]1 ,ayre1 ,fire1 ,wood1 ,/ones1 ,birdes1 & ,bea/es1 & ( _m ?ynges els1 a c]ta9e ,sympa?icall =ew>nyng may be _h3 sometymes to great pleasure & pr(fit1 bo? on ,sea & ,l&4 ,suffici5tly1 = my pres5t purpose1 x do? appe>e1 by ! premisses1 h[ ,ma!maticall1 ! ,arte ( ,nauigation1 is3 & h[ x n$e? & also vse? o!r ,ma!maticall ,artes3 ,& n[1 if ,i w\ld g ab\t to speake ( ! manifold ,commodities1 commyng to ?is ,l&1 & o!rs1 by ,%ypps & ,nauigation1 y might ?9ke1 t ,i cat* at occasions1 to vse _m wordes1 :]e no n$e is4 ,yet1 ?is "o ?yng may ,i1 7iu/ly7 say4 ,9 ,nauigation1 none ought to haue great] c>e1 to be skillfull1 !n our ,engli% ,pylotes4 ,& p]*aunce1 ,"s1 w\ld m attempt3 ,& o!r ,"s1 m will+ly w\ld be aydyng1 x !y wi/ c]ta9ely1 ,:at ,priuil$ge1 ,god _h endu$ ?is ,il& )1 by reason ( ,situation1 mo/ commodi\s = ,nauigation1 to ,places mo/ ,fam\s ,ri*e4 ,& ?\gh1 99 ,anno4 #aefg ,s4 ,h4 ,g4 7( 99 ,late7 a "y ,g5tleman1 a ,c\rragi\s ,capita9e1 0 9 a great readynes1 ) good hope1 & great causes ( p]suasion1 to haue v5tur$1 = a ,disc\]ye1 7ei!r ,we/]ly1 by ,cape de ,p>amantia3 or ,e/]ly1 ab\e ,n\a ,zemla1 & ! ,cyremisses7 & 01 at ! v n]e tyme ( ,attemptyng1 call$ & employ$ o!rwise 7bo? !n1 & s9ce17 9 great good s]uice to 8 ,c\ntrey1 z ! ,iri% ,rebels haue 99 ta/$3 99 ,anno4 #aefi ,yet1 ,i say1 7?\gh ! same ,g5tleman1 doo n h]eaft]1 deale !re)7 ,"s "o1 or o!r1 %\ld li/5 to ! ,matt]3 & by good aduise1 & discrete ,circumspection1 by little1 & little1 wynne to ! suffici5t k ( t ,trade & ,voyage3 ,:i*1 n[1 ,i w\ld be sory1 7?r\gh ,c>elesnesse1 want ( ,skill1 & ,c\rrage17 %\ld remayne ,vnkn[ne & vnhe>d (4 ,seyng1 also1 we are h]e91 halfe ,*all5g$1 by ! le>n$1 by halfe reque/1 publi%$4 ,!r(1 v]ely1 might gr[ ,commoditye1 to ?is ,l& *iefly1 & to ! re/ ( ! ,*ri/5 ,common weal?1 f>re pass+ all ri*es & worldly ,?reasure4 ,?aumaturgike1 is t ,art ,ma!maticall1 :i* giue? c]ta9e ord] to make /raunge workes1 ( ! s5se to be p]ceiu$1 & ( m5 greatly to be wondr$ at4 ,by sundry meanes1 ?is ,wond]-worke is wr\ght4 ,"s1 by ,pneumati?mie4 ,z ! workes ( ,ctesibius & ,h]o1 ,"s by waight4 :]( ,timu speake?4 ,"s1 by ,/r+es /rayn$1 or ,spr+s1 !r) ,imitat+ liuely ,motions4 ,"s1 by o!r meanes1 z ! ,images ( ,m]curie3 & ! bras5 h$1 made by ,alb]tus ,magnus1 :i* dyd seme to speake4 ,boe?ius 0 excell5t 9 ^! feates4 ,to :om1 ,cassiodorus writyng1 say?4 ,y\r purpose is to "k pr(\nd ?ynges3 & to %ew m]uayles4 ,by ! disposition ( y\r ,arte1 ,metals d l[3 ,diom$es ( brasse1 do? bl[ a ,trumpet l\de3 a bras5 ,s]p5t hisse?3 byrdes made1 s+ swetely4 ,small ?ynges we rehe>se ( y1 :o c ,imitate ! heau54 c4 ,( ! /raunge ,selfm\yng1 :i*1 at ,sa9t ,d5ys1 by ,p>is1 99 ,anno4 #aeea 99 ,i saw1 ones or twise 7,orontius beyng !n ) me1 9 ,company7 x 7 to /raunge to tell4 ,b "s haue writt5 x4 ,& yet1 7,i hope7 x is !re1 ( o!r to be s5e4 ,& by ,p]spectiue also /raunge ?+es1 are done4 ,z p>tly 7be=e7 ,i gaue y to vnd]/& 9 ,p]spectiue4 ,z1 to see 9 ! ,ayre1 a l(t1 ! lyuely ,image ( an o!r man1 ei!r walkyng to & fro3 or /&yng /ill4 ,likewise1 to come into an h\se1 & !re to see ! liuely %ew ( ,gold1 ,silu] or preci\s /ones3 & commyng to take !m 9 y\r h&1 to f9de n\ght b ,ayre4 ,h]eby1 haue "s m5 79 all o!r matt]s c\nt$ wise7 f\ly ou]%ot ? selues3 misdeam+ ( ! meanes4 ,!r=e sayd ,claudius ,cle/9us4 ,de 8 qu ,mundo mirabilit] eu5iunt4 cap4 #h4 ,hodie magn lit]atur viros magna reputationis videmus1 op]a qu$am quasi mir&a1 supra ,natur put>e3 de quibus 9 ,p]spectiua doctus causam facilit] r$didisset4 ,t is4 ,n[ a dayes1 we see "s m51 yea ( great le>nyng & reputation1 to ,iudge c]ta9 workes z m]uayl\s1 ab\e ! p[] ( ,nature3 ,( :i* workes1 "o t 7 skillfull 9 ,p]spectiue might easely haue giu5 ! ,cause4 ,( ,ar*im$es ,sphre1 ,cic]o witnesse?4 ,tusc4 #a4 ,:i* is v /raunge to ?9ke on4 ,= :5 ,ar*im$es 7say? he7 did fa/5 9 a ,sphre1 ! m\ynges ( ! ,sonne1 ,mone1 & ( ! fiue o!r ,planets1 he did1 z ! ,god1 :i* 79 ,timu ( ,plato7 did make ! _w4 ,t1 "o turnyng1 %\ld rule motions mo/ vnlike 9 sl[nes1 & swiftnes4 ,b a great] cause ( m]uayl+ we haue by ,claudianus report h]e(4 ,:o affirme? ?is ,ar*im$es worke1 to haue b5 ( ,glasse4 ,& disc\rse? ( x m at l>ge3 :i* ,i omit4 ,! ,d\e ( wood1 :i* ! ,ma!matici5 ,ar*ytas did make to flye1 is by ,agellius spok5 (4 ,( ,ddalus /raunge ,images1 ,plato reporte?4 ,hom]e ( ,vulcans ,selfm\]s1 7by secret :eles7 leaue? 9 writyng4 ,ari/otle1 9 hys ,politikes1 ( bo?1 make? m5tion4 ,m]uayl\s 0 ! workeman%yp1 ( late dayes1 p]=m$ by good skill ( ,tro*ilike4 c4 ,= 9 ,noremb]ge1 ,a flye ( ,i]n1 beyng let out ( ! ,artific]s h&1 did 7z x 77 fly ab\t by ! ge/es1 at ! table1 & at l5g?1 z ?\gh x 7 we>y1 ret\rne to 8 ma/]s h& agayne4 ,more\]1 an ,artificiall ,egle1 0 ordr$1 to fly out ( ! same ,t[ne1 a mighty way1 & t a l(t 9 ! ,ayre1 t[>d ! ,emp]\r comm+ !!r3 & foll[$ hym1 beyng come to ! gate ( ! t[ne4 99 99 ,?us1 y see1 :at1 ,arte ,ma!maticall c p]=me1 :5 ,skill1 w1 ,indu/ry1 & ,hability1 are duely apply$ to pr(e4 ,a ,digression4 ,& = ^!1 & su* l m>ueil\s ,actes & ,feates1 ,naturally1 ,ma!matically1 & ,me*anically1 wr\ght & contriu$3 ,apologeticall4 ought any hone/ ,/ud5t1 & ,mode/ ,*ri/ian ,philosoph]1 be c\nt$1 call$ a ,coniur]8 ,%all ! folly ( ,idiotes1 & ! ,mallice ( ! ,scornfull1 s mu* preuaile1 t ,he1 :o seeke? no worldly ga9e or glory at _! h&es3 ,b onely1 ( ,god1 ! ?reasor ( heau5ly wis$ome1 k ( pure v]itie3 ,%all he 7,i say7 9 ! meane space1 be robb$ & spoil$ ( 8 hone/ "n & fame8 ,he t seke? 7by ,s4 ,paules adu]tisem5t7 9 ! ,creatures ,prop]ties1 & wond]full v]tues1 to f9de iu/e cause1 to glorifie ! t]nall1 & ,almightie ,creator by3 ,%all t man1 be 79 hugg] mugg]7 condemn$1 z a ,companion ( ! ,helh\ndes1 & a ,call]1 & ,coniur] ( wick$ & damn$ ,spirites8 ,he t bewaile? 8 great want ( "t1 suffici5t 7to 8 cont5tation7 = le>n+ ( ,godly wisdome1 & ,godly ,v]ities 93 & onely !r9 sette? all 8 delight3 ,w t m leese & abuse 8 "t1 9 deal+ ) ! ,*iefe enemie ( ,*ri/ our ,r$em]3 ! deadly foe ( all mank9de3 ! subtile & impud5t p]u]t] ( ,godly ,v]itie3 ! ,hypocriticall ,crocodile3 ! ,enui\s ,basiliske1 cont9ually desir\s1 9 ! tw9ke ( an eye1 to de/roy all ,mank9de1 bo? 9 ,body & ,s\le1 t]nally8 ,surely 7= my "p1 some:at to say h]e97 ,i haue n le>n$ to make s bruti%1 & s wick$ a ,b>ga9e4 ,%\ld ,i1 = my xx4 or xxv4 ye>es ,/udie3 = two or ?ree ?\s& ,m>kes sp5d+3 seu5 or eight ?\s& ,miles go+ & trauail+1 onely = good le>n+es sake3 ,& t1 9 all man] ( we!rs3 9 all man] ( waies & passages3 bo? e>ly & late3 9 daung] ( viol5ce by man3 9 daung] ( de/ruction by wilde bea/es3 9 hung]3 9 ?ir/3 9 p]il\s heates by day1 ) toyle on foote3 9 daung]\s dampes ( colde1 by night1 almo/ b]eu+ life3 7z ,god kn[e?73 ) lodg+es1 (t times1 to small ease3 & somtime to lesse securitie4 ,& = mu* m 7!n all ?is7 done suffr$1 = ,le>n+ & atta9+ ( ,wis$ome3 ,%\ld ,i 7,i pray y7 = all ?is1 no o!rwise1 nor m w>ily3 or 7by ,gods m]cifulnes7 no m luckily1 haue fi%$1 ) s l>ge1 & co/ly1 a ,nette1 s long "t 9 draw+ 7& t ) ! helpe & aduise ( ,lady ,philosophie1 ,que5e ,!ologie73 b at l5g?1 to haue cat*$1 & draw5 vp1 99 a ,frog8 99 ,a pr\]b4 ,fayre fi%t1 & caught a ,frog4 ,nay1 a ,deuill8 ,=1 s1 do? ! ,common peui% ,pratl] ,imag9e & ,iangle3 ,&1 s1 do? ! ,malici\s skorn]1 secretly wi%e1 brauely & boldly face d[n1 beh9de my backe4 ,ah1 :at a mis]able ?+1 is ?is k9de ( ,m58 ,h[ great is ! bl9dnes boldnes1 ( ! ,multitude1 9 ?+es ab\e _! ,capacitie8 ,:at a ,l&3 :at a ,p3 :at ,man]s3 :at ,times are ^!8 ,are !y become ,deuils1 !m selues3 &1 by false witnesse be>+ aga9/ _! ,neighb\r1 w\ld !y also1 become ,murd]]s8 ,do? ,god1 s long geue !m respite1 to reclaime !m selues 91 f ?is horrible slaund]+ ( ! giltlesse3 contr>y to _! owne ,consci5ces3 & yet w !y n cease8 ,do? ! ,innoc5t1 =be>e ! call+ ( !m1 ,iuridically to aunsw]e him1 accord+ to ! rig\r ( ! ,lawes3 & w !y despise 8 ,*>itable paci5ce8 ,z !y1 aga9/ him1 by "n1 d =ge1 fable1 rage1 & raise slaund]1 by ,worde ,pr9t3 ,w !y pr\oke him1 by worde & ,pr9t1 likewise1 to ,note _! ,names to ! ,_w3 ) _! p>ticul> deuises1 fables1 bea/ly ,imag9ations1 & vn*ri/5-l slaund]s8 ,well3 ,well4 ,o 7y su*7 my vnk9de ,c\ntrey m54 ,o vnnaturall ,c\ntrey m54 ,o vn?ankfull ,c\ntrey m54 ,o ,bra9sicke1 ,ra%e1 ,spitefull1 & ,disda9full ,c\ntrey m54 ,:y oppresse y me1 ?us viol5tly1 ) y\r slaund]+ ( me3 ,contr>y to ,v]itie3 & contr>y to y\r owne ,consci5ces8 ,& ,i1 to ?is h[]1 nei!r by worde1 de$e1 or ?\ght1 haue b5e1 any way1 hurtfull1 damageable1 or iniuri\s to y1 or y\rs8 ,haue ,i1 s long1 s de>ly1 s f>re1 s c>efully1 s pa9fully1 s daung]\sly s\ght trauail$ = ! le>n+ ( ,wis$ome1 atteyn+ ( ,v]tue3 ,& 9 ! end 79 y\r iudgemt7 am ,i become1 worse1 !n :5 ,i beg8 ,worse1 ? a ,mad man8 ,a dang]\s ,memb] 9 ! ,common ,weal?3 & no ,memb] ( ! ,*ur* ( ,*ri/8 ,call y ?is1 to be ,le>n$8 ,call y ?is1 to be a ,philosoph]8 & a l\] ( ,wis$ome8 ,to =sake ! /raight heau5ly way3 & to wall[ 9 ! broad way ( damnation8 ,to =sake ! light ( heau5ly ,wis$ome3 & to lurke 9 ! dungeon ( ! ,pr9ce ( d>k5esse8 ,to =sake ! ,v]itie ( ,god1 8 ,creatures3 & to fawne vpon ! ,impud5t1 ,craftie1 ,ob/9ate ,li]1 & cont9uall disgrac] ( ,gods ,v]itie1 to ! vtt]mo/ ( 8 p[]8 ,to =sake ! ,life ,blisse t]nall3 & to cleaue vnto ! ,au?or ( ,dea? eu]la/+8 t ,murd]\s ,tyrant1 mo/ gr$ily await+ ! ,pray ( ,mans ,s\le8 ,well3 ,i ?anke ,god & our ,lorde ,iesus ,*ri/1 = ! ,com=t :i* ,i haue by ! ,examples ( o!r m51 be=e my "t3 ,to :om1 nei!r 9 godl9es ( life1 nor 9 p]fection ( le>n+1 ,i am wor?y to be comp>$3 & yet1 !y su/a9$ ! v l ,iniuries1 t ,i d3 or r1 great]4 ,paci5t ,socrates1 8 ,apologie w te/ifie3 ,apuleius 8 ,apologies1 w decl>e ! ,bruti%nesse ( ! ,multitude4 ,ioannes ,picus1 ,e>le ( ,mir&ula1 8 ,apologie w tea* y1 ( ! ,rag+ slaund] ( ! ,malici\s ,ignorant aga9/ him4 ,ioannes ,tri!mius1 8 ,apologie w specifie1 h[ he _h occasion to make publike ,prote/ation3 z well by reason ( ! ,rude ,simple3 z also1 9 respect ( su*1 z 7 c\nt$ to be ( ! wise/ sort ( m54 8,_m c\ld ,i recite3 ,b ,i def]re ! precise & det]m9$ h&l+ ( ?is matt]3 be+ lo? to detect ! ,folly ,mallice ( my ,natiue ,c\ntrey m54 99 99 ,:o1 s h>dly1 c disge/ or l any extraord9>y c\rse ( ,philosophicall ,/udies3 n fall+ )9 ! ,cumpasse ( _! ,capacitie3 or :]e !y are n made priuie ( ! true & secrete cause1 ( su* wond]full ,philosophicall ,feates40 ,^! m51 are ( f[] sortes1 *iefly4 ,! fir/1 ,i may "n1 ,va9e pratl+ busie bodies3 ,! second1 ,fond ,fr5des3 ,! ?ird1 ,imp]fectly zel\s3 & ! f\r?1 ,malici\s ,ignorant4 ,to e*e ( ^! 7briefly1 & 9 *>itie7 ,i w say a ^w or two1 & s returne to my ,prface4 #a4 ,va9e pratl+ busie bodies1 vse y\r idle assemblies1 & conf]5ces1 o!rwise1 !n 9 talke ( matt]1 ei!r ab\e y\r ,capacities1 = h>dnesse3 or contr>y to y\r ,consci5ces1 9 ,v]itie4 #b4 ,fonde ,fr5des1 leaue (1 s to comm5d y\r vnacqua9t$ fr5d1 vpon bl9de affection3 ,z1 because he kn[e? m1 !n ! common ,/ud5t3 t1 !r=e1 he mu/ ne$es be skilfull1 & a do]1 9 su* matt] & man]1 z y t]me ,coniur+4 ,we5+1 !reby1 y aduaunce 8 fame3 & t y make o!r m51 great m>ueil]s ( y\r hap1 to haue su* a le>n$ fr5d4 ,cease to ascribe ,impietie1 :]e y pret5d ,amitie4 ,=1 if y\r t\nges 7 true1 !n 7 t y\r fr5d1 ,vntrue1 bo? to ,god1 & 8 ,s\]aigne4 ,su* ,fr5des & ,fondl+es1 ,i %ake (1 & r5\nce y3 ,%ake y (1 y\r ,folly4 #c4 ,imp]fectly zel\s1 to y1 d ,i say3 t 7p]haps7 well1 d y ,meane3 ,b f>re y misse ! ,m>ke3 ,if a ,lambe y w kill1 to fe$e ! flocke ) 8 bl\d4 ,%eepe1 ) ,lambes bl\d1 haue no naturall su/5aunce3 ,no m1 is ,*ri/es flocke1 ) horrible slaund]s1 duely difi$4 ,nor y\r faire pret5se1 by su* ra%e ragg$ ,rhetorike1 any :it1 well grac$4 ,b su*1 z s vse me1 w f9de a f[le ,cracke 9 _! ,cr$ite4 ,speake t y "k3 ,& "k1 z y ought3 ,"k n1 by ,he>e say1 :5 life lie? 9 daung]4 ,se>* to ! quicke1 let ,*>itie be y\r guide4 #d4 ,malici\s ,ignorant1 :at %all ,i say to !e8 ,prohibe l+uam tuam a malo4 ,a detractione p>cite l+u4 ,cause ?y t\ng to refra9e fr euill4 ,refra9e y\r t\ng f slaund]4 ,?\gh y\r t\nges be %>pn$1 ,s]p5t l1 ,add]s poyson lye 9 y\r lippes3 ,psal4 #adj4 yet take he$e1 & ?9ke1 betimes1 ) y\r selfe1 ,vir l+uosus non /abilietur 9 t]ra4 ,virum viol5tum v5abitur malum1 donec prcipitetur4 ,=1 sure ,i am1 ,quia faciet ,dom9us ,iudicium afflicti3 v9dictam paup]um4 ,?us1 ,i require y1 my assur$ fr5des1 & ,c\ntrey m5 7y ,ma!matici5s1 ,me*anici5s1 & ,philosoph]s1 ,*>itable & discrete7 to deale 9 my behalf1 ) ! light vntrue t\ng$1 my enui\s ,adu]s>ies1 or ,fond fr5ds4 ,& f>!r1 ,i w\ld wi%e1 t at leysor1 y w\ld consid]1 h[ ,basilius ,magnus1 laye? ,moses & ,daniel1 be=e ! eyes ( ^?1 :i* c\nt all su* ,/udies ,philosophicall 7z m9e ha? b5e7 to be vngodly1 or vnpr(itable4 ,waye well ,s4 ,/eph5 8 witnesse ( ,moses4 ,act4 #g4 ,c4 ,eruditus e/ ,moses omni ,sapi5tia gyptior3 erat pot5s 9 v]bis op]ibus suis4 ,moses 0 in/ruct$ 9 all man] ( wis$ome ( ! gyptians3 & he 0 ( p[] bo? 9 8 wordes1 & workes4 ,y see ?is ,philosophicall ,p[] ,wis$ome1 :i* ,moses _h1 to be no?+ mislik$ ( ! ,holy ,gho/4 ,yet ,pl9ius ha? record$1 ,moses to be a wick$ ,magici54 ,& t 7( =ce7 mu/ be1 ei!r = ?is ,philosophicall wis$ome1 le>n$1 be=e 8 call+ to ! lead+ ( ! ,*ildr5 ( ,israel3 or = ^? 8 wond]s1 wr\ght be=e ,k+ ,ph>ao1 aft] he _h ! conduct+ ( ! ,israelites4 ,z conc]n+ ! fir/1 y p]ceaue1 h[ ,s4 ,/eph51 at 8 ,m>tyrdome 7be+ full ( ! ,holy ,gho/7 9 8 ,recapitulation ( ! olde ,te/am5t1 ha? made m5tion ( ,moses ,philosophie3 ) good lik+ ( x3 ,& ,basilius ,magnus also1 au\*e? x1 to haue b5e to ,moses pr(itable 7& !re=e1 ,i say1 to ! ,*ur* ( ,god1 necess>y74 ,b z cc]n+ ,moses wond]s1 done be=e ,k+ ,ph>ao3 ,god1 him selfe1 sayd3 ,vide vt omnia o/5ta1 qu posui 9 manu tua1 facias coram ,ph>aone4 ,see t ?\ d all ^? wond]s be=e ,ph>ao1 :i* ,i haue put 9 ?y h&4 ,?us1 y euid5tly p]ceaue1 h[ ra%ly1 ,pl9ius ha? slaund]$ ,moses1 ,lib4 #cj4 ,cap4 #a4 ( vayne fraudul5t ,magike1 say+3 ,e/ alia ,magices ,factio1 a ,mose1 ,iamne1 ,iotape1 ,iudis p5d5s3 s$ multis millibus annorum po/ ,zoroa/rem4 c4 #a4 ,let all su*1 !re=e1 :o1 9 ,iudgem5t & ,skill ( ,philosophie1 are f>re ,inf]ior to ,pl9ie1 8take good he$e1 lea/ !y ou]%oote !m selues ra%ly10 9 ,iudg+ ( ,philosoph]s /raunge ,actes3 & ! ,meanes1 h[ !y are done4 #b4 ,b1 mu* m1 ought !y to bew>e ( =g+1 deuis+1 & imag9+ mon/r\s feates1 & wond]full workes1 :5 & :]e1 no su* 7 done3 no1 n any sp>ke or likelihode1 ( su*1 z !y1 )\t all %ame1 d report4 #c4 ,& 7to conclude7 mo/ ( all1 let !m be a%am$ ( ,man1 & afraide ( ! dreadfull & ,iu/e ,iudge3 bo? ,foli%ly or ,malici\sly to deuise3 & !n1 deuili%ly to fa!r _! new fond ,mon/]s on me3 ,innoc5t1 9 h& & h>t3 = trespac+ ei!r aga9/ ! lawe ( ,god1 or ,man1 9 any my ,/udies or ,ex]cises1 ,philosophicall1 or ,ma!maticall3 ,z 9 due "t1 ,i hope1 w be m manife/4 ,n[ end ,i1 ) ,ar*ema/rie4 ,:i* "n1 is n s new1 z ?is ,arte is r>e4 ,= an o!r ,arte1 vnd] ?is1 a degree 7= skill & p[]7 ha? b5e indu$ ) ?is ,engli% "n be=e4 ,& yet1 ?is1 may s]ue = our purpose1 suffici5tly1 at ?is pres5t4 ,?is ,arte1 tea*e? to bryng to actuall exp]i5ce s5sible1 all wor?y conclusions by all ! ,artes ,ma!maticall purpos$1 by true ,naturall ,philosophie conclud$3 bo? adde? to !m a f>d] scope1 9 ! t]mes ( ! same ,artes1 also by hys propre ,me?od1 & 9 peculi] t]mes1 proc$e?1 ) helpe ( ! =esayd ,artes1 to ! p]=mance ( complet ,exp]ices1 :i* ( no p>ticul> ,art1 are hable 7,=mally7 to be *all5g$4 ,if y rememb]1 h[ we consid]$ ,ar*itecture1 9 respect ( all common h&workes3 "s light may y haue1 !rby1 to vnd]/& ! ,s\]a9ty & prop]tie ( ?is ,sci5ce4 ,sci5ce ,i may call x1 r1 !n an ,arte3 = ! excell5cy & ,ma/]%yp x ha?1 ou] s _m1 & s mighty ,artes & ,sci5ces4 ,& bycause x proc$e? by ,exp]i5ces1 & se>*e? =? ! causes ( ,conclusions1 by ,exp]i5ces3 & also putte? ! ,conclusions !m selues1 9 ,exp]i5ce1 x is nam$ ( "s1 ,sci5tia ,exp]im5talis4 ,! ,exp]im5tall ,sci5ce4 ,nicolaus ,cusanus t]me? x s1 9 hys ,exp]im5tes ,/atikall1 ,& an o!r ,philosoph]1 ,r4 ,b4 ( ?is l& ,natiue 7! fl\re ( ^: wor?y fame1 c neu] dye nor )]7 did write !r( l>gely1 at ! reque/ ( ,clem5t ! sixt4 ,! ,arte c>rie? ) x1 a wond]full ,cr$it3 ,by reason1 x c]tefie?1 s5sibly1 fully1 & completely to ! vtmo/ p[] ( ,nature1 & ,arte4 ,?is ,arte1 c]tifie? by ,exp]i5ce complete & absolute3 & o!r ,artes1 ) _! ,argum5tes1 & ,demon/rations1 p]suade3 & 9 wordes1 pr\e v well _! ,conclusions4 99 ,b wordes1 & ,argum5tes1 are no s5sible c]tify+3 nor ! full & f9all frute ( ,sci5ces practisable4 ,& ?\gh "s ,artes1 haue 9 !m1 ,exp]i5ces1 yet !y are n complete1 & br\ght to ! vtt]mo/1 !y may be /ret*$ vnto1 & apply$ s5sibly4 ,z = example3 ! ,naturall ,philosoph] dispute? & make? goodly %ew ( reason3 ,& ! ,a/ronom]1 & ! ,opticall ,me*anici51 put "s ?ynges 9 ,exp]i5ce3 b nei!r1 all1 t !y may3 nor yet suffici5tly1 & to ! vtmo/1 ^?1 :i* !y d1 ,!re1 !n1 ! ,ar*ema/] /eppe? 91 & leade? =? on1 ! ,exp]i5ces1 by ord] ( 8 doctr9e ,exp]im5tall1 to ! *ief & f9all p[] ( ,naturall & ,ma!maticall ,artes4 ,( two or ?ree m51 9 :om1 ?is ,description ( ,ar*ema/ry 0 ,exp]im5tally1 v]ifi$1 ,i haue read & h>d3 & good record1 is ( _! su* p]fection4 ,s t1 ?is ,art1 is no fanta/icall ,imag9ation3 z "s ,sophi/]1 might1 ,cum suis ,insolubilibus1 make a flori%3 & dassell y\r ,imag9ation3 & da% y\r hone/ desire & ,c\rage1 f beleu+ ^! ?+es1 s vnhe>d (1 s m]uayl\s1 ( su* ,importance4 ,well3 z y w4 ,i haue =ew>n$ y4 ,i haue done ! "p ( a fr5de3 ,i haue dis*>g$ my ,duety t[>d ,god3 = my small ,tal5t1 at hys mo/ m]cyfull h&es receiu$4 ,to ?is ,sci5ce1 do? ! ,sci5ce ,alnirangiat1 great ,s]uice4 ,muse no?yng ( ?is "n4 ,i *aunge n ! "n1 s vs$1 & 9 ,pr9t publi%$ by o!r3 beyng a "n1 propre to ! ,sci5ce4 ,vnd] ?is1 comme? ,ars ,s9trillia1 by ,artephius1 briefly writt54 ,b ! *ief ,sci5ce1 ( ! ,ar*ema/]1 79 ?is _w7 z yet kn[51 is an o!r 7z x 77 ,,optical ,sci5ce3 :](1 ! "n %all be told 7,god willyng7 :5 ,i %all haue "s1 7m iu/7 occasion1 !r(1 to ,disc\rse4 ,"h1 ,i mu/ end1 ?us abruptly 7,g5tle fr5de1 & vnfayn$ l\] ( hone/ & necess>y v]ities47 ,=1 !y1 :o haue 7= y\r sake1 & v]tues cause7 reque/$ me1 7an old =worne ,ma!matici57 to take p5 9 h&3 7?r\gh ! confid5ce !y repos$ 9 my long exp]i5ce3 & try$ s9c]ity7 = ! decl>yng & reportyng some:at1 ( ! frute & commodity1 by ! ,artes ,ma!maticall1 to be atteyn$ vnto3 eu5 !y1 ,sore agayn/ _! willes1 are =c$1 = sundry causes1 to satisfie ! workemans reque/1 9 endyng =?)3 ,he1 s fe>e? ?is1 s new an attempt1 s co/ly3 ,& 9 matt] s sl5d]ly 7he!rto7 among ! common ,sorte ( ,/ud5tes1 consid]$ or e/em$4 ,& :]e ,i 0 will$1 some:at to all$ge1 :y1 9 our vulg>e ,spe*e1 ?is "p ( ! ,pr9cipall ,sci5ce ( ,geometrie1 call$ ,euclides ,geometricall ,elem5tes1 is publi%$1 to y\r h&lyng3 be+ vnlat9$ p1 & n ,vniu]sitie ,s*ol]s3 ,v]ily1 ,i ?9ke x n$elesse4 #a4 ,=1 ! ,hon\r1 & ,e/imation ( ! ,vniu]sities1 & ,graduates1 is1 h]eby1 no?+ dim9i%$4 ,se+1 f1 & by _! ,nurse ,*ildr51 y receaue all ?is ,b5efite3 h[ great soeu] x be4 #b4 ,nei!r are _! ,/udies1 h]eby1 any :it h9dr$4 ,no m1 !n ! ,italian ,vniu]sities1 z ,academia ,bononi5sis1 ,f]r>i5sis1 ,flor5t9a1 ,m$iolan5sis1 ,patau9a1 ,papi5sis1 ,p]us9a1 ,pisana1 ,romana1 ,s55sis1 or any "o ( !m1 f9de !m selues1 any deale1 disgrac$1 or _! ,/udies any ?+ h9dr$1 by ,frat] ,lucas de ,burgo1 or by ,nicolaus ,t>talea1 :o 9 vulg> ,italian language1 haue publi%$1 n onely ,euclides ,geometrie1 b ( ,ar*im$es some:at3 & 9 ,ari?metike & ,practicall ,geometrie1 v l>ge volumes1 all 9 _! vulg> spe*e4 ,nor 9 ,g]many haue ! fam\s ,vniu]sities1 any ?+ b5e discont5t ) ,alb]tus ,dur]us1 8 ,geometricall ,in/itutions 9 ,dut*3 or ) ,gulielmus ,xyl&]1 8 le>n$ translation ( ! fir/ sixe bookes ( ,euclide1 out ( ! ,greke into ! high ,dut*4 ,nor ) ,gualt]us ,h4 ,riffius1 8 ,geometricall ,volume3 v dilig5tly translat$ into ! high ,dut* t\nge1 & publi%$4 ,nor yet ! ,vniu]sities ( ,spa9e1 or ,portugall1 ?9ke _! reputation to be decay$3 or suppose any _! ,/udies to be h9dr$ by ! ,excell5t ,p4 ,nonnius1 8 ,ma!maticall workes1 9 vulg>e spe*e by him put =?4 ,haue y n1 likewise1 9 ! ,fr5* t\nge1 ! :ole ,ma!maticall ,quadriuie8 & yet nei!r ,p>is1 ,orleance1 or any ( ! o!r ,vniu]sities ( ,fraunce1 at any "t1 ) ! ,translat]s1 or ,publi%]s (f5d$3 or any mans ,/udie !reby h9dr$8 #c4 ,& surely1 ! ,common & ,vulg> ,s*ol] 7mu* m1 ! ,gram>ian7 be=e 8 comm+ to ! ,vniu]sitie1 %all 7or may7 be1 n[ 7accord+ to ,plato 8 ,c\nsell7 suffici5tly in/ruct$ 9 ,ari?metike & ,geometrie1 = ! bett] & easi] le>n+ ( all man] ( ,philosophie1 ,academicall1 or ,p]ipateticall4 ,& by t meanes1 goe m *]efully1 m skilfully1 & sp$ily =w>de1 9 8 ,/udies1 !re to be le>n$4 ,&1 s1 9 lesse "t1 pr(ite m1 !n 7o!rwise7 he %\ld1 or c\ld d4 #d4 ,also _m good & pregnant ,engli%e wittes1 ( "y ,g5tlem51 & ( o!r1 :o neu] int5d to m$dle ) ! pr(\nd se>* & ,/udie ( ,philosophie 79 ! ,vniu]sities to be le>n$7 may neu]!lesse1 n[1 ) m ease & lib]tie1 haue good occasion1 v]tu\sly to occupie ! %>pnesse ( _! wittes3 :]e1 els 7p]*ance7 o!rwise1 !y w\ld 9 fond ex]cises1 sp5d 7or r leese7 _! "t3 nei!r s]u+ ,god3 nor furd]+ ! ,weale1 common or priuate4 #e4 ,& great ,com=t1 ) good hope1 may ! ,vniu]sities haue1 by reason ( ?is ,engli%e ,geometrie1 & ,ma!maticall ,prface1 t !y 7h]eaft]7 %all be ! m reg>d$1 e/eem$1 & resort$ vnto4 ,=1 :5 x %all be kn[5 & report$1 t ( ! ,ma!maticall ,sci5ces onely1 su* great ,commodities are ensu+ 7z ,i haue specifi$73 & t 9 d$e1 "s ( y vnlat9$ ,/ud5tes1 c be good witnesse1 ( su* r>e fruite by y enioy$ 7!reby73 z ei!r1 be=e ?is1 0 n he>d (3 or els1 n s fully cr$it$3 8,well1 may all m5 coniecture1 t f>re great] ayde1 & bett] furniture1 to w9ne to ! ,p]fection ( all ,philosophie1 ,vniu]sities4 may 9 ! ,vniu]sities be _h3 be+ ! ,/oreh\ses ,?reasory ( all ,sci5ces1 & all ,artes1 necess>y = ! be/1 & mo/ noble ,/ate ( ,common ,weal!s40 #f4 ,besides ?is1 h[ _m a ,common ,artific]1 is !re1 9 ^! ,realmes ( ,engl& & ,irel&1 t deale? ) ,numb]s1 ,rule1 ,cumpasse3 ,:o1 ) _! owne ,skill & exp]i5ce1 already _h1 w be hable 7by ^! good helpes & in=mations7 to f9de out1 & deuise1 new workes1 /raunge ,eng9es1 & ,in/rum5tes3 = sundry purposes 9 ! ,common ,weal?8 or = priuate pleasure8 & = ! bett] ma9tayn+ ( _! owne e/ate8 ,i w n 7!re=e7 fight aga9/ myne owne %ad[e4 ,=1 no man 7,i am sure7 w op5 8 m\? aga9/ ?is ,ent]prise4 ,no m 7,i say7 :o ei!r ha? ,*>itie t[>d 8 bro!r 7& w\ld be glad ( 8 fur!rance 9 v]tu\s k73 or t ha? any c>e zeale = ! bett]+ ( ! ,cmon /ate ( ?is ,realme4 ,nei!r any1 t make accompt1 :at ! wis] sort ( m5 7,sage & ,/ay$7 d ?9ke ( !m4 ,to none 7!re=e7 w ,i make any ,apologie1 = a v]tu\s acte do+3 & = cm5d+1 or sett+ =?1 ,pr(itable ,artes to ,engli% m51 9 ! ,engli% t\ng4 8,b1 vnto ,god our ,creator1 let vs all be ?ankefull3 = t1 ,z he1 ( 8 ,goodnes1 by 8 ,p[re1 & 9 8 wis$ome1 ha? ,creat$ all ?ynges1 9 ,numb]1 ,waight1 & ,measure3 ,s1 to vs1 ( hys great ,m]cy1 he ha? reueal$ ,meanes1 :]eby1 to atteyne ! suffici5t & necess>y k ( ! =esayd hys ?ree pr9cipall ,in/rum5tes3 ,:i* ,meanes1 ,i haue abundantly pr\$ vnto y1 to be ! ,sci5ces & ,artes ,ma!maticall40 ,& ?\gh ,i haue b5 p9*$ ) /raightnes ( tyme3 t1 no way1 ,i c\ld s p5 d[ne ! matt] 79 my ,mynde7 z ,i det]m9$3 hopyng ( conu5i5t laysure3 ,yet4 if v]tu\s zeale1 & hone/ ,int5t pr\oke & bryng y to ! readyng & exam9yng ( ?is ,comp5di\s treatise1 ,i d n d\te1 b1 z ! v]itie !r( 7accordyng to our purpose7 w be euid5t vnto y3 ,s ! pi? & =ce !r(1 w p]suade y3 & ! wond]full frute !r(1 highly pleasure y4 ,& t y may ! easi] p]ceiue1 & bett] rememb]1 ! pr9cipall po9tes1 :]e( my ,preface treate?1 ,! ,gr\nd platt ( ?is ,prface 9 a ,table4 ,i w giue y ! ,gr\ndplatt ( my :ole disc\rse1 9 a ,table annex$3 f ! fir/ to ! la/1 some:at ,me?odically contriu$4 ,if ,ha/1 ha? caus$ my poore p51 any :]e1 to /umble3 ,y w1 7,i am sure7 9 "p ( recomp5ce1 7= my e>ne/ & s9c]e good w to pleasure y71 ,consid] ! rocki% huge m\nta9es1 & ! p]il\s vnbeat5 wayes1 :i* 7bo? night & day1 = ! :ile7 x ha? toyl$ & labor$ ?r\gh1 to bryng y ?is good ,newes1 & ,com=table pr(e1 ( ,v]tues frute4 ,s1 ,i ,commit y vnto ,gods ,m]cyfull direction1 = ! re/3 h>tely bese*yng hym1 to prosp] y\r ,/udyes1 & hone/ ,int5tes3 to 8 ,glory1 ! ,commodity ( our ,c\ntrey4 ,am54 ,writt5 at my poore ,h\se ,at ,mortlake4 ,anno4 #aegj4 ,febru>y4 #i4 ,decoration ,transcrib]'s ,note3 ,! 8,gr\ndplat0 0 pr9t$ 9 ! =m ( a /emma1 or tree1 on an ov]siz$ fold-out page4 ,! lay\t 0 impossible to reproduce = ?is e-text1 s ! in=mation has be5 re>rang$ 9 ne/$-li/ =m4 ,size m>k+s 7see note at beg9n+ ( e-text7 are relative )9 ea* p>agraph4 ,j4 ,,dee ,"h haue y 7accord+ to my promisse7 ! ,gr\ndplat ( my ,,ma!maticall ,prface3 annex$ to ,euclide 7n[ fir/7 publi%$ 9 our ,engli%e t\nge4 ,an4 #aegj4 ,febr4 #c4 ,sci5ces1 & ,artes ,ma!maticall1 are1 ei!r ,pr9cipall1 :i* are two1 onely1 ,ari?metike4 ,simple1 ,:i* deale? ) ,numb]s onely3 & demon/rate? all _! prop]ties & app]t5ances3 :]e1 an ,vnit1 is ,indiuisible4 ,mixt1 ,:i* ) aide ( ,geometrie pr9cipall1 demon/rate? "s ,ari?meticall ,conclusion1 or ,purpose4 ,geometrie4 ,simple1 ,:i* deale? ) ,magnitudes1 onely3 & demon/rate? all _! prop]ties1 passions1 & app]t5ances3 ^: ,po9t1 is ,indiuisible4 ,mixt1 ,:i* ) aide ( ,ari?metike pr9cipall1 demon/rate? "s ,geometricall purpose1 z ,,evclides ,,elem5tes4 ,! vse :]e(1 is ei!r1 ,9 ?+es ,sup]naturall1 t]nall1 ,diu9e3 ,by ,application1 ,asc5d+4 ,9 ?+es ,ma!maticall3 )\t f>!r ,application4 ,9 ?+es ,naturall3 bo? ,sub/tiall1 ,accid5tall1 ,visible1 ,inuisible4 c4 ,by ,application3 ,desc5d+4 ,! l ,vses & ,applications are1 7?\gh 9 a degree l[]7 9 ! ,artes ,ma!maticall ,d]iuatiue4 ,d]iuatiue fr ! ,pr9cipalls3 ( :i*1 "s haue ,! names ( ! ,pr9cipalls3 z1 ,ari?metike1 vulg>3 :i* consid]e? --,ari?metike ( mo/ vsuall :ole numb]s3 ,& ( ,fractions to !m app]ta9+4 --,ari?metike ( ,proportions4 --,ari?metike ,circul>4 --,ari?metike ( ,radicall ,nb]s3 ,simple1 ,comp\nd1 ,mixt3 ,& ( _! ,fractions4 --,ari?metike ( ,cossike ,nb]s3 ) _! ,fractions3 ,& ! great ,arte ( ,algieb>4 ,geometrie1 vulg>3 :i* tea*e? ,measur+ ,at h& ,all ,l5g!s4--,mecometrie4 ,all ,pla9es3 ,z1 ,l&1 ,borde1 ,glasse1 c4--,embadometrie4 ,all ,solids3 ,z1 ,timb]1 ,/one1 ,vessels1 c4--,/]eometrie4 ,) di/ce f ! ?+ ,measur$1 z1 ,h[ f>re1 f ! ,measur]1 any ?+ is3 ( him s5e1 on ,l& or ,wat]3 call$ ,apomecometrie4 ,h[ high or deepe1 f ! leuell ( ! ,measur]s /&+1 any ?+ is3 ,se5e ( hym1 on ,l& or ,wat]3 call$ ,hypsometrie4 ,h[ broad1 a ?+ is1 :i* is 9 ! ,measur]s view3 s x be situat$ on ,l& or ,wat]3 call$ ,platometrie4 ,( :i* are gr[5 ! ,feates ,artes ( ,geodesie3 m cunn+ly to ,measure & ,suruey ,l&es1 ,woods1 ,wat]s4 c4 ,geographie4 ,*orographie4 ,hydrographie4 ,/rat>i?metrie4 ,propre names z1 ,p]spectiue1--,:i* demon/rate? ! man]s & prop]ties ( all ,radiations3 ,directe1 ,brok51 & ,reflect$4 ,a/ronomie1--,:i* demon/rate? ! ,di/ances1 ,magnitudes1 & all ,naturall motions1 ,app>5ces1 & ,passions1 prop] to ! ,planets & fix$ ,/>res3 = any "t1 pa/1 pres5t1 & to come3 9 respecte ( a c]ta9e ,horizon1 or )\t respecte ( any ,horizon4 ,musike1--,:i* demon/rate? by reason1 & tea*e? by s5se1 p]fectly to iudge & ord] ! diu]sitie ( ,s\ndes1 hie or l[4 ,cosmographie1--,:i*1 :oly & p]fectly make? description ( ! ,heau5lym & also ,elem5tall "p ( ! ,_w3 & ( ^! p>tes1 make? homologall application1 & mutuall collation necess>y4 ,a/rologie1--,:i* reasonably demon/rate? ! op]ations & effectes ( ! naturall beames ( light1 & secrete ,influ5ce ( ! ,planets1 & fix$ ,/>res1 9 eu]y ,elem5t & ,elem5tall body3 at all times1 9 any ,horizon assign$4 ,/atike1--,:i* demon/rate? ! causes ( heau9es & lightnes ( all ?+es3 & ( ! motions & prop]ties to heau9es & lightnes belong+4 ,an?ropographie1 ,:i* describe? ! ,nb]1 ,measure1 ,waight1 ,figure1 ,situation1 & col\r ( eu]y diu]s ?+ conta9$ 9 ! p]fecte body ( ,,man3 & geue? c]ta9e k ( ! ,figure1 ,symmetrie1 ,waight1 ,*>act]ization1 due ,locall motion ( any p]cell ( ! said body assign$3 & ( numb]s to ! said p]cell app]ta9+4 ,tro*ilike1--,:i* demon/rate? ! prop]ties ( all ,circul> motions3 ,simple & ,comp\nd4 ,helicosophie1--,:i* demon/rate? ! design+ ( all ,spirall l9es3 9 ,pla9e1 on ,cyl9d]1 ,cone1 ,sphre1 ,conod1 & ,sphroid3 & _! prop]ties4 ,pneumati?mie1--,:i* demon/rate? by close holl[ ,geometricall figures 7,regul> & ,irregul>7 ! /raunge prop]ties 79 motion or /ay7 ( ! ,wat]1 ,ayre1 ,smoke1 & ,fire1 9 _! ,cont9uitie1 & z !y are ioyn$ to ! ,elem5tes next !m4 ,m5adrie1--,:i* demon/rate?1 h[1 ab\e ,natures ,v]tue1 & p[] simple3 ,v]tue & =ce1 may be multipli$3 & s to directe1 to lift1 to pull to1 & to put or ca/ fro1 any multipli$1 or simple det]m9$ ,v]tue1 ,waight1 or ,=ce3 naturally1 n1 s1 directible1 or m\eable4 ,hypogeiodie1--,:i* demon/rate?1 h[1 vnd] ! ,sphricall ,sup]ficies ( ! ,e>?1 at any dep?1 to any p]p5dicul> l9e assign$ 7^: di/ance f ! p]p5dicul> ( ! entrance3 & ! ,azimu? likewise1 9 respecte ( ! sayd entrance1 is kn[57 c]ta9e way1 may be prescrib$ & gone1 c4 ,hydragogie1--,:i* demon/rate? ! possible lead+ ( wat] by ,natures law1 & by artificiall helpe1 f any head 7be+ ,spr+1 /&+1 or runn+ wat]7 to any o!r place assign$4 ,horometrie1--,:i* demon/rate?1 h[1 at all times appo9t$1 ! precise1 vsuall d5om9ation ( "t1 may be kn[51 = any place assign$4 ,zographie1--,:i* demon/rate? & tea*e?1 h[1 ! ,int]section ( all visuall ,pyramids1 made by any pla9e assign$ 7! ,c5t]1 di/ance1 & lightes be+ det]m9$7 may be1 by l9es1 & prop] col\rs repres5t$4 ,ar*itecture1--,:i* is a ,sci5ce g>ni%$ ) _m doctr9es1 & diu]s ,in/ructions3 by ^: iudgem5t1 all workes by o!r workm5 f9i%$1 are iudg$4 ,nauigation1--,:i* demon/rate?1 h[1 by ! ,%orte/ good way1 by ! apte/ direction1 & 9 ! %orte/ "t3 a suffici5t ,%ippe1 betw5e any two places 79 passage nauigable7 assign$1 may be conduct$3 & 9 all /ormes & naturall di/urbances *aunc+1 h[ to vse ! be/ possible meanes1 to rec\] ! place fir/ assign$4 ,?aumaturgike1--,:i* geue? c]ta9e ord] to make /raunge workes1 ( ! s5se to be p]ceiu$3 & ( m5 greatly to be wondr$ at4 ,ar*ema/rie1--,:i* tea*e? to br+ to actuall exp]i5ce s5sible1 all wor?y conclusions1 by all ! ,artes ,ma!maticall purpos$3 & by true ,naturall philosophie1 conclud$3 ,& bo? adde? to !m a f>d] ,scope1 9 ! t]mes ( ! same ,artes3 & also1 by 8 prop] ,me?od1 & 9 peculi> t]mes1 proc$e?1 ) helpe ( ! =sayd ,artes1 to ! p]=mance ( complete ,exp]i5ces3 :i*1 ( no p>ticul> ,arte1 are hable 7,=mally7 to be *all5g$4 ,impr9t$ by ,iohn ,day4 ,an4 #aegj4 ,feb4 #be4 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,errors & ,anomalies3 ,unless o!rwise not$1 spell+ & punctuation are un*ang$4 ,errors are li/$ bel[1 ) ! orig9al =m1 if *ang$1 %[n 9 brackets4 ,unusual words include 8fat*0 7probably us$ z a v>iant ( 8fet*07 & ! ma!matical t]ms 8sexag5e0 & 8sexagesme04 ,h[1 worldly goods3 h[1 worldly dignitie 8o0 9 second 8worldly0 invisible 8 mo/ dilig5t he>]s 7s inf9itely m\ght he>]s7 s ! b\ndes1 & duety ( an ,hydrograph] ,hydograph] ( ! ,grekes x is call$ ,et]omekes text un*ang$3 correct =m is 8,het]omekes0 acc5t un*ang$ 9 our worldly affaires wordly fall to worke44 ,"s text read]s may n display ! ov]siz$-a/]isk symbol4 ,empty+ ! fir/4 ,emptyt+ 1 1 444 ( ! suddeyne snddeyne t ! "r & absolute way may be _h he _h ,georgic ,i3 ,! quot$ segm5ts1 ea* end+ 9 8c401 are #dch-dci2 #dea-deg2 #dfc-dfd4 ,additional ,notes3 ,! ,greek lett] 7eta7 0 consi/5tly pr9t$ z if x 7 ! ou-ligature 4 ,! ,lat9 8-que0 0 writt5 z an abbreviation resembl+ 8-q204 ,x is %[n "h z que4 ,ma!matical symbols se5 9 ! section accompany+ ! diagrams c\ld n be reproduc$4 ,! foll[+ sub/itutions 7 made3 --,! curly 8,p0 us$ = 8,p\nds0 is %[n z ,p4 --,! 8pote/as0 symbol1 us$ to repres5t 8x0 7! unkn[n71 is %[n z x4 --,all roots 7 express$ z ! 8root0 sign comb9$ ) symbols = ! p[] ( #b 7d\bl$ = p[] ( #d1 or f\r? root7 & #c4 ,!y are %[n z 4 ,euclid3 ,! foll[+ ,propositions 7 id5tifi$ by numb]4 #f4ab3 7,h[7 to f9d a f\r? 7l9e7 proportional to ?ree giv5 /raight l9es4 #aa4cd3 ,9 equal p>allelepip$al solids ! bases are reciprocally proportional to ! heights2 & ^? p>allelepip$al solids 9 :i* ! bases are reciprocally proportional to ! heights are equal4 #aa4cf3 ,if ?ree /raight l9es are proportional1 !n ! p>allelepip$al solid =m$ out ( ! ?ree equals ! p>allelepip$al solid on ! mean :i* is equilat]al1 b equiangul> ) ! a=esaid solid4 #ab4a3 ,simil> polygons inscrib$ 9 circles are to "o ano!r z ! squ>es on _! diamet]s4 #ab4b3 ,circles are to "o ano!r z ! squ>es on _! diamet]s4 #ab4ah 78la/073 ,sph]es are to "o ano!r 9 triplicate ratio ( _! respective diamet]s4