,,! ,,philosophical & ,,ma!matical ,,comm5t>ies ,,( ,,proclus1 ,,on ,,! ,,fir/ ,,book ,,( ,,euclid',s ,,elem5ts4 ,,to ,,:i* ,,are ,,add$1 ,a ,hi/ory ( ! ,re/oration ( ,,platonic ,,!ology1 ,,by ,,! ,,latt] ,,platoni/s3 ,& a ,translation f ! ,greek ( ,,proclus',s ,,!ological ,,elem5ts4 ,,9 ,,two ,,volumes4 ,illu/ration ,,vol ,i4 ,,london1 ,,pr9t$ ,,= ,,! ,,au?or3 ,& ,sold by ,t4 ,,payne & ,,son2 ,b4 ,,:ite & ,,son2 ,j4 ,,robson2 ,t4 ,,cadell2 ,,leigh & ,,co42 ,g4 ,,nicol2 ,r4 ,,fauld]2 & ,t4 & ,j4 ,,eg]ton4 #agib4 ,price ,two ,gu9eas 9 ,bo>ds4 ,extracts f ,curiosities ( ,lit]ature4 ,second ,edition4 ,pr9t$ = ,murray4 ,page #che4 ,mr4 ,t4 ,,taylor1 ! ,platonic ,philosoph] & ! mod]n ,ple?on1 consonant to t philosophy1 pr(esses ,poly!ism4#a ,! ,read] is reque/$ to correct ! foll[+ ,errors4 ,page #d4 ( ! ,diss]tation1 ,vol4 ,i4 l9e #h4 = admitt$1 read omitt$4 ,page #af4 l9e #h4 = f1 read =m4 ,page #ea4 ,vol4 ,i4 ( ! ,comm5t>ies1 l9e #af4 in/ead ( ,!y are surely n !1 c4 read ,= surely x _c be said t !re are1 c4 ,l9e #ag4 = b we1 read b t we4 ,& l9e #ai4 = is by mu* prior to1 read is by a mu* great] priority4 ,vol4 ,,ii4 page #ah4 l9e #bf4 = & "o is1 read & "o "p is4 ,& l9e #bg4 = ano!r1 read ! o!r4 ,page #aad4 l9e #ac4 = ,! angle1 read ,let ! angle2 & in/ead ( is bisect$1 9 ! same l9e1 read be bisect$4 ,page #daa4 l9e #b4 = is fill$ ) intellect1 read fills intellect4 ,& l9e #c4 = x also p>ticipates1 read also x p>ticipates4 ,,to ,,! ,,sacr$ ,,maje/y ,,( ,,tru?4 ,,preface4 ,! design ( ! pres5t "w is to br+ u acqua9t$ ) ! nature & end ( ,ma!matics 9 g5]al1 & ( ,geometry 9 p>ticul>3 & 9 ! execution ( ?is design our ,au?or has display$ an uncommon elegance ( composition1 & a mo/ valuable /ore ( recondite le>n+4 ,he is n cont5t ) e :]e unfold+ ! full1 & mo/ accurate mean+ ( ,euclid2 b he cont9ually rises 9 8 disc\rse1 & leads u into ! dep?s ( ! ,py?agoric & ,platonic philosophy4 ,we are surpris$ to f9d an use 9 ,geometry1 :i* at pres5t x is by no means suspect$ to af=d4 ,= :o w\ld conceive t x is ! g5u9e passage to true !ology1 & ! ve/ibule ( div9ity8 ,?is1 inde$1 is by no means ! case :5 x is /udi$ = lucre1 & appli$ to me*anical purposes2 = !n ! s\l is nei!r elevat$ nor enlight5$2 b degrad$ & fill$ ) mat]ial d>kness4 ,h5ce ^! ,comm5t>ies are alone valuable to ! lib]al "p ( mank9d1 :o look beyond s5se = c]ta9ty2 & :o pref] ?+s desirable = _! own sakes1 be=e su* z m9i/] to ! necessities ( life4 ,! translation ( ?is "w is att5d$ ) great difficulty & lab\r2 n only f ! s5se ( ! philosoph]1 :i* is always pr(\nd1 & frequ5tly obscure1 b f ! great incorrectness ( ! ,greek edition1 9 :i*1 exclusive ( numb]less typographical errors1 entire s5t5ces1 ess5tial to ! connection1 are frequ5tly omitt$2 & 9 "o place two pages ( ! ,lat9 translation are want+ 9 ! orig9al1 z w be %ewn 9 our foll[+ notes4 ,inde$1 ! ,lat9 translation ( ,francis ,b>ocius ! ,v5etian1 7,patav4 #aefj47 :i* 0 made f a v>iety ( manuscripts1 is inconceivably valuable2 = ! diagrams1 s necess>y to a "w ( ?is k9d1 b :i* are omitt$ 9 ! ,greek1 are "h ins]t$2 & ! v]sion is e :]e fai?ful1 & suffici5tly p]spicu\s to ^? :o are conv]sant 9 ! anci5t philosophy4 ,b>ocius ju/ly cautions ! read] n to comp>e 8 v]sion ) ! pr9t$ ,greek1 :i* he obs]ves is r lac]at$ ?an pr9t$2 z inde$1 )\t 8 translation1 x is impossible = any "o to read ! half ( ?is invaluable "w1 ev5 ?\gh he %\ld be z p]fect 9 ,greek z 9 8 native tongue4 ,if ,i _h n1 !re=e1 =tunately acquir$ ?is translation1 :i* is at pres5t v r>e1 ,i w\ld h by no means engag$ 9 ?is ardu\s und]tak+4 ,b>ocius1 inde$1 gives evid5t pro(s ( 8 possess+ ! philosophical g5ius1 by ! excell5ce ( 8 translation1 & 8 preface to ! read]2 & x is greatly to be lam5t$ t he did n adorn 8 v]sion ) explanatory notes1 :i* ?is pr(\nd "w frequ5tly dem&s1 & :i* he 0 d\btless well qualifi$ to accompli%4 ,?is defect ,i h endeav\r$1 z f> z ,i 0 able1 to supply2 & at ! same "t h be5 cauti\s nei!r to we>y ! read] by prolixity1 nor by too mu* brevity to leave him de/itute ( prop] in=mation4 ,9 ! di/ribution ( ! fir/ book ( ?is "w into *apt]s1 ,i h foll[$ ! ord] ( ,b>ocius1 because x is natural & obvi\s2 & mu/ beg leave to solicit ! read]'s indulg5ce = us+ ! words p>tible & imp>tible1 diff]5tly f _! common signification4 ,^! words ,i h g5]ally employ$ to express ! mean+ ( #b & #c 9 ! ,greek1 z ,i d n conceive t ! words divisible & indivisible always convey _! full signification4 ,i h likewise us$ quadrangle in/ead ( squ>e1 & qu9quangle = ! ^w p5tagon4 ,= if be r5d]$ triangle1 :y %\ld n be r5d]$ quadrangle8 ,&1 z ,b>ocius obs]ves1 :y1 = a simil> reason1 %\ld n & be r5d]$ qu9quangle & sexangle2 & s ( ! re/8 ,uni=mity is always desirable :5 x c be obta9$2 & is no :]e s necess>y z 9 sci5tifical disquisitions4 ,x is likewise necess>y to in=m ! read]1 t ?\gh ,i h always endeav\r$ to give ! fai?ful mean+ ( my ,au?or1 yet ,i h occasionally p>aphras$ 8 s5se1 :5 mo/ obscure1 & add$ su* elucidations ( my own1 z ,i ei!r ?\ght necess>y to ! full compreh5sion ( 8 matt]2 or :i* 7 naturally excit$ by ! fire & _s ( ! ,orig9al4 ,if x %all appe> t ,i h succe$$ 9 ! execution ( ?is "w1 & r5d]$ x intelligible to ! lov]s ( tru?1 ,i %all rejoice 9 my success1 & consid] my lab\rs suffici5tly rew>d$4 ,! applause ( ! multitude ,i am nei!r likely nor desir\s to ga92 b ,i am anxi\s to procure ! approbation ( ! disc]n+ few1 :o "k t ! age ( philosophy is pa/2 & :o e/eem ! works ( h] anci5t h]oes z ! mo/ preci\s treasures :i* h escap$ ! ravages ( "t4 99 99 99 99 99 ,"t1 inde$1 is l a deep & rapid riv]2 :atev] is trifl+ & light1 is precipitately borne on its surface1 & :at is valuable & weighty1 s9ks to its bottom4 ,h5ce1 ! sup]ficial obs]v] collects no?+ m ?an ! rubbi%1 :i* x is =ev] devolv+ into ! abyss ( oblivion2 :ile ! pr(\nd & contemplative g5ius explores ! dep?s ( ! /ream1 & acc\nts himself happy if he c ga!r any ( ! pe>l :i* its bottom conta9s4 ,?us ! discov]ies ( exp]im5tal philosophy1 float l /raws on ! surface1 :ile ! wisdom ( ,py?agoras & ,plato lies conceal$ 9 ! dep?s ( ! riv]4 ,i am well aw>e x w be said1 t ! rev]se ( ?is similitude is true2 t ! mod]n philosophy is ! pe>l1 & ! anci5t ! /ubble2 & t ! =m] w be celebrat$ by po/]ity1 & increase 9 reputation :5 ! latt] %all sc>cely be kn[n4 ,b let u att5tively exam9e ! tru? ( ?is ass]tion1 & %ut our e>s to ! unsub/antial e*oes ( popul> applause4 ,is x reasonable to suppose t m5 ( su* exalt$ abilities1 z ! ,py?agoric & ,platonic philosoph]s possess$1 ev5 9 ! e/imation ( _! oppon5ts1 accompani$ ) ! greate/ advantages ( bir? & =tune1 & ! mo/ unwe>i$ att5tion1 h discov]$ no?+ valuable1 & h left no?+ beh9d !m1 b j>gon & rev]ies8 ,is x to be suppos$1 t 9 an age :5 philosophy 0 almo/ ador$2 :5 x 0 e/eem$ by k+s1 cultivat$ by noblem51 & ev5 rev]5c$ by ! vulg>2 :5 empire 0 rel9qui%$ = its pursuit1 & e dang] enc\nt]$ = its possession3 is x to be suppos$1 t no?+ b delusion 0 ! (fspr+ ( s glori\s a p]iod1 & no?+ b folly ! rew>d ( su* g5]\s endurance8 ,or %all we say1 t ! discov]y ( tru? 0 res]v$ = ! age ( exp]im5t2 & t %e is alone to be appreh5d$ 9 ! inf9ite labyr9? ( p>ticul>s8 ,t %e is to be inve/igat$ ) ! corporeal s5ses1 & n ) ! p[]s ( intellect2 & t ! crucible1 ! alembic1 & ! air-pump1 are ! only m$iums ( detection8 ,if ?is be ! case1 tru? is mat]ial1 & may be calc9$1 di/ill$1 & r>efi$1 l any o!r corporeal sub/ance4 ,x is no long] et]nal & immutable1 b p]i%able & fluctuat+2 ! phanta/ic subject ( s5sible inspection1 & n ! /eady & real object ( ! p]man5t en]gies ( sci5ce4 ,%all we call ?is ! age ( philosophy1 9 :i* tal5ts are pro/itut$ = su/5ance1 & le>n+ submits to ! impud5ce ( weal?8 ,%all we say t we h /r5g!n$ ! cause ( philosophy1 by demoli%+ h] s*ools2 & increas$ h] indep5d5ce1 by enl>g+ ! empire ( comm]ce8 ,:]e %all we f9d ! man1 :o is at pres5t rev]5c$ = ! pr(ession ( tea*+ speculative tru?1 or inde$ :o tea*es x at all8 ,or %\ld we *ance to meet ) su* an obsolete "*1 %all we f9d him support$ by ! pr(ession8 ,x is a well kn[n fact1 t m5 =m]ly liv$ 9 ! highe/ e/eem by its propagation3 x is equally z notori\s1 t a man at pres5t w\ld />ve by su* an attempt4 ,d>e we ass]t1 t ! reason ( ?is diff]5ce mu/ be ascrib$ to ! great] lib]ality1 & m philosophical _s ( ! pres5t age8 ,%all we n r say1 t ! p]iod1 9 :i* ^! anci5t h]oes liv$1 0 ! gold5 age ( philosophy2--a p]iod s diff]5t f ! pres5t1 z to appe> fabul\s on ! comp>ison8 ,= m>k ! di/+ui%+ *>act]i/ics ( our inf]iority4 ,! great object ( anci5t philosophy1 0 an accurate speculation ( pr9ciples & causes3 b t ( ! mod]n1 is a confus$ inve/igation ( effects4 ,& if pursuits p>ticipate ( ! nature ( _! subjects1 & causes are m noble ?an effects1 ! anci5t philosophy mu/ und\bt$ly be m elevat$ ?an ! mod]n4 ,aga91 ! object ( ! ,py?agorean & ,platonic philosophy 0 to make its possessors wise & virtu\s2 & to elevate !m above ! common frailties & imp]fections ( degrad$ humanity2 & ?is end 0 happily accompli%$ 9 its vot>ies1 z _! lives abundantly ev9ce3 b ! object ( mod]n philosophy1 is a promotion ( ! conv5i5cies & ref9em5ts ( life1 by enl>g+ ! b\nd>ies ( traffic2 & ! ,ma!matical ,sci5ces are /udi$ solely ) a view to ?is enl>gem5t4 ,! design ( ! anci5t philosophy 0 to remove ! causes ( wond]1 by contemplat+ effects 9 _! causes3 ! gr& object ( ! mod]n1 is to increase admiration1 by attempt+ to inve/igate causes ?r\gh ! inf9ity ( p>ticul> effects4 ,s t philosophy1 z ,mr4 ,h>ris ju/ly obs]ves1 n[ ends :]e x =m]ly began4 ,= ei!r !re is no su* ?+ z sci5ce1 or if its exi/5ce be admitt$1 x c nev] be obta9$ by exp]im5tal enquiries2 z ^! mu/ be liable to all ! inaccuracy & imp]fection ( _! mat]ial subjects4 ,9 %ort1 ! philosophy ( ,py?agoras & ,plato w be f\nd1 :5 imp>tially consid]$1 to conta9 e ?+ :i* c enlight5 ! m9d1 improve ! morals1 & exalt ! "* ( man4 ,x is built on ! /eady basis ( tru?1 & w survive ! wreck ( ages4 ,its f\ndation is deep1 & its summit rea*es ! heav5s4 ,x is a mighty rock1 :i* mod]n sy/ems may assail1 l a rag+ sea2 b1 l /ormy waves1 !y w only be brok5 ab\t its imp5etrable sides4 ,to w> aga9/ wisdom is folly2 = opposition 9 ?is case is ! de/ruction ( its au?or4 ,! mod]ns may1 inde$1 expect1 because _! m]it is rais$ by ! pres5t age1 above t ( ! anci5ts1 to appe> z giants 9 ! eyes ( po/]ity2 b !y w only v]ify ! elegant obs]vation ( ! poet#d1 t ,pygmies are ,pygmies /ill1 ?\gh p]*'d on ,alps1 ,& ,pyramids are ,pyramids 9 vales4 ,a ,,diss]tation ,,on ,,! ,,platonic ,,doctr9e ,,( ,,ideas1 c4 ,,section ,i4 ,! ,platonic doctr9e ( ,ideas has be51 9 all ages1 ! d]ision ( ! vulg>1 & ! admiration ( ! wise4 ,inde$1 if we consid] t ideas are ! mo/ sublime objects ( speculation1 & t _! nature is no less bright 9 itself1 ?an difficult to inve/igate1 ?is opposition 9 ! conduct ( mank9d w be natural & necess>y2 =1 f our connection ) a mat]ial nature1 our intellectual eye1 previ\s to ! irradiations ( sci5ce1 is z ill adapt$ to objects ! mo/ spl5did ( all1 8z ! eyes ( bats to ! light ( day#e40 ,& yet 7z ,i presume1 x w appe> f ! foll[+ disc\rse71 unless ! exi/5ce ( ^! lucid be+s is admitt$1 !re c be no su* ?+ z sci5ce2 nor1 inde$1 any g5u9e k at all4 ,h5ce1 an enquiry conc]n+ _! nature & reality1 is highly prop]1 z an introduction to ! ensu+ ,comm5t>ies1 9 :i* !y are consid]$ z ! /able pill>s ( all tru?1 & ! prolific pr9ciples ( ! univ]se4 ,b previ\s to ?is enquiry1 x is prop] to obs]ve1 t ,plato 0 n ! inv5tor1 ?\gh he 0 a /r5u\s ass]t]1 ( ideas2 =1 9 ! ,sophi/a he affirms1 t ideas 7 ! discov]y ( m5 :o excell$ 9 wisdom & piety1 & :o cont5d$ = an invisible ess5ce4 ,diog5es ,lartius1 inde$1 ass]ts1 t ,plato receiv$ ! doctr9e ( ideas f ,epi*>mus4 ,b ,epi*>mus 0 n _! inv5tor1 because ,py?agoras1 & o!rs ( /ill high] antiquity1 7 well acqua9t$ ) ideas2 s t x may be affirm$1 ) mu* great] tru?1 t ,plato 0 in/ruct$ 9 _! nature by ,philolaus 8 preceptor1 & ! disciple ( ,py?agoras4 ,= ,py?agoras1 aft] 8 my/]i\s mann]1 signifi$ ideas by numb]s4 ,b1 prior to ,py?agoras1 ,orpheus 0 an ass]t] ( ideas1 & call$ ,jupit]1 or ! dimiurgus ( ! _w1 8! idea ( all ?+s40 ,&1 accord+ to ,syrianus1 ! mundane sph]e1 celebrat$ by ,emp$ocles1 is no o!r ?an ! ideal _w2 s t ! doctr9e ( ideas is z anci5t z t ( wisdom itself4 ,b to beg9 ) our enquiry3 9 ! fir/ place1 )\t univ]sals !re c be no sci5ce2 = ! fl[+ & p]i%+ nature ( p>ticul>s is p]fectly =eign f t /ability & duration :i* is requisite to objects ( inv>iable tru?4 ,nei!r is x possible1 t inf9ite individuals c exi/ )\t ! subsi/5ce ( "o cause endu$ ) inf9ite p[]2 = all multitude mu/ necess>ily orig9ate f "o1 & mu/ resemble its cause 9 z great a degree ( p]fection z its nature c admit2 by a diffus$ inf9ity1 %ad[+ =? t inf9ite p[] :i* subsi/s 9 indivisible union4 ,h5ce1 if ?is be ! case1 & if inf9ite m51 horses1 & a multitude ( o!r univocals1 are produc$ 9 an inf9ite "t1 an unity ( inf9ite p[] mu/ be ! s\rce ( ea*1 accord+ to :i* !y are g5]at$ 9 a t]m9at$ mann] to inf9ity 9 ! univ]se4 ,aga91 all animals are transmut$ f t :i* is 9 capacity 7i4 e4 se$71 into en]gy4 ,b if ?is be true1 x is requisite !re %\ld be "s animal 9 ! univ]se1 subsi/+ 9 ev]-vital en]gy1 :i* may call =? t :i* is conceal$ 9 dormant capacity1 into p]fect actuality4 ,?irdly1 ! cele/ial orbs w\ld n p]petually revolve 9 ! same spaces1 & aft] ! same mann]1 unless "o & ! same univ]sal numb]1 or idea1 rul$ 9 ea*4 ,s1 likewise1 !re is a natural numb] 9 e animal2 or ^? ( ! same species1 w\ld n always 7:5 p]fect7 be di/+ui%$ ) ! same inv>iable organs2 nor w\ld !y be subject to pub]ty & old age1 at ! same "t1 unless !y 7 deta9$ by ! same measure ( nature4 ,besides1 ! p>ticipation ( univ]sals1 is evid5t 9 e s5sible object4 ,?us1 ! rational nature is unit$ ) e individual man4 ,?us1 animal subsi/s 9 a lion & a horse1 9 a man & a dog4 ,& ?us ! p5tad1 or numb] five1 is p>ticipat$ 9 ! five f+]s1 & ! duad 9 ! no/rils1 eyes1 h&s1 & feet4 ,b s9ce ^! d n subsi/ )\t a cause1 b are p]fect$ by c]ta9 det]m9ate natures1 x is necess>y !re %\ld be an univ]sal animal1 9 ! :ole ( nature1 sep>ate f s5sibles1 by means ( :i* ?is s5sible animal is g5]at$4 ,& t !re %\ld subsi/ 9 nature a p5tad1 ?r\gh :i* ! h&s are always adorn$ ) t numb] ( extremities2 & a duad1 f :i* ! two eyes & no/rils are d]iv$4 ,b if nature does n possess ^! numb]s f h]self1 z %e is n ! fir/ cause ( all1 b d]ives !m f ano!r cause1 9 ! same mann] z matt] f nature1 x is necess>y !re %\ld be univ]sals & numb]s prior to nature1 subsi/+ 9 f> great] purity & p]fection4 ,aga91 we may demon/rate ! exi/5ce ( ideas z foll[s3 if ! ,deity1 9 fabricat+ ! univ]se1 op]at$ ess5tially 7& !re is no o!r way 9 :i* we c conceive him to op]ate71 he mu/ fabricate ! univ]se1 an image ( ,himself4 ,b1 if ?is be ! case1 he conta9s 9 himself1 9 ! mann] ( an exempl>1 ! causes ( ! univ]se2 & ^! causes are no o!r ?an ideas4 ,besides1 ?is consid]ation is n to be omitt$1 t ! p]fect mu/ necess>ily antec$e & preside ov] ! imp]fect2 unity ov] multitude2 ! imp>tible ov] ! p>tible2 & t :i* is p]petually ! same1 ov] t :i* admits ( v>iation & *ange4 ,f :5ce x may be inf]r$1 t ?+s d n orig9ate f bas] natures1 b t _! gradual processions end 9 ^!2 & t !y beg9 f ! mo/ p]fect1 be/1 & mo/ beautiful natures4 ,b let u pursue ?is reason+ m m9utely1 z x af=ds ! /ronge/ argum5ts = ! exi/5ce ( ideas4 ,:5 ! ,deity fabricat$ ! v>i\s species ( animals1 & be/[$ on !m ! diff]5t s5ses1 x 0 d\btless ) a view to ! b5efit ( _! possessors1 z he =esaw1 t )\t ^!1 ! animal c\ld nei!r provide = its own support1 nor def5d itself f surr\nd+ dang]s4 ,b may we n enquire f :5ce ?is previ\s p]ception orig9at$8 ,= x is n to be suppos$1 t he fir/ made animals de/itute ( s5ses1 & s1 be+ admoni%$ by _! sudd5 de/ruction1 aft]w>ds assign$ !m to _! nature4 ,%all we say1 ?is =ekn[l$ge 0 ! result ( a reason+ process8 ,b !n1 we aga9 ask1 ,:at 7 ! pr9ciples ( ?is ratioc9ation8 ,= if !y orig9at$ f o!r reason+s1 x is necess>y1 at l5g?1 to arrive at some?+ prior to ^! discursive op]ations1 on :i* !y ultimately dep5d2 s9ce all reason+ mu/ be f\nd$ on indemon/rable pr9ciples4 ,0 s5se1 !n1 or intellect ! pr9ciple ( ?is previ\s p]ception8 ,b1 s5se1 9 ! pres5t in/ance1 _h n !n a be+1 = x c\ld n exi/ prior to ! animal nature3 x 01 !re=e1 intellect4 ,b if intellect be ! repository ( c]ta9 propositions1 & ! conclusion be sci5ce1 x mu/ foll[1 t !re c\ld n !n be a consultation ( any ?+ s5sible4 ,= ! pr9ciple & ! conclusion mu/ bo? dep5d on some?+ intelligible4 ,besides1 may we n ask1 h[ su* a habit ( ?\ght arose be=e ! exi/5ce ( a s5sible nature6 ,x is absurd 9 ! extreme1 to say f *ance1 & to resolve x into a sudd5 volition ( ! ,deity1 is an ass]tion t may1 inde$1 satisfy vulg> m9ds1 b c by no means quiet ! re/less _s ( philosophical inve/igation4 ,s9ce1 to suppose ! cause ( ! univ]se1 actuat$ by sudd5 volitions1 is to place him on a level ) ! vile/ natures1 & subject him to ! irrational impulses ( ! brute4 ,h5ce we inf] t ! =mation ( animals1 & by ! same argum5ts ( ! _w1 0 n ! result ( any reason+ process4 ,=1 inde$1 argum5t & =ekn[l$ge _c ) propriety be attribut$ to ! ,deity2 b :5 !y are ascrib$ to him1 we mu/ consid] x z no?+ m ?an an indication ( 8 con/itut+ p>ticul>s1 9 a mann] some:at simil> to ! provid5ce ( a wise man1 9 inf]ior conc]ns4 ,=1 9 subord9ate natures1 ^: op]ations _c take effect prior to enquiry1 reason is necess>y1 on acc\nt ( ! inf]iority ( t p[] :i* prec$es ! reason+ en]gy4 ,9 l mann]1 =ekn[l$ge is necess>y1 because a p[] is want+ to its possessor1 :i* might r5d] him sup]ior to its use4 ,= =ekn[l$ge is direct$ to ?is end1 t "o p>ticul> circum/ance may take place 9 pref]5ce to ano!r4 ,b if x be requisite t e en]gy 9 ! ,deity %\ld be void ( defect1 & if x is n lawful t any ?+ %\ld be pres5t ) him1 :i* is n total & univ]sal1 x is necess>y t all ?+s %\ld be conta9$ 9 e ?+ ess5tial to ! nature ( ! ,deity4 ,h5ce1 s9ce ev5 futurity is ) him pres5t1 !re is no?+ 9 him po/]ior2 b :at is pres5t 9 him becomes po/]ior1 by its p>ticipation 9 ano!r4 ,if !n futurity be pres5t ) ! ,deity1 x is necess>y x %\ld be s pres5t1 z if =ekn[n 9 a po/]ior nature2 t is1 9 su* a mann] t no?+ may be want+ to any be+2 & t is1 la/ly1 s t e ?+ may be complete4 ,besides1 reason+ _c1 by any means1 belong to an et]nal ess5ce l ! deity2 = if ?is be admitt$1 he mu/ be =getful ( 8 =m] op]ations4 ,& if1 9 consequ5ce ( reason+1 he produces m p]fect natures aft]w>ds1 8 works c\ld n be p]fectly beautiful be=e3 b if !y 7 beautiful be=e1 !y mu/ be co-exi/5t ) _! cause1 i4e4 !y mu/ be et]nally beautiful1 antec$5t to ! reason+ en]gy4 ,aga91 if we suppose ! supreme intellect1 ! demiurgus ( ! _w1 to op]ate by enquiry1 8 en]gy c\ld n be spontane\s1 & truly 8 own2 b 8 ess5ce w\ld be simil> to t ( ! artific]1 :o does n d]ive 8 productions f himself1 b procures !m z some?+ adv5titi\s by le>n+ & enquiry4 ,b if ! univ]se 0 n =m$ by delib]ation1 x mu/ be co-exi/5t ) its cause1 & reside 9 8 ess5ce2 = if x be n co-exi/5t !re mu/ h be5 "s p>ticul> "t1 9 :i* its artific] det]m9$ on its production2 & ?is det]m9ation mu/ h be5 ! result ( a reason+ process1 conclud+ t x w\ld n be good to produce x be=e t p>ticul> "t1 7f :5ce1 by ! way1 we inf] ! et]nity ( ! _w47 ,& if ! univ]se be co-exi/5t ) its au?or1 x mu/ p]petually emanate f 8 nature1 & be dep5d5t on x1 l ! %ad[ on its =m+ sub/ance4 ,b 9 ?is case1 its ar*etype mu/ be conta9$ 9 ! ess5ce ( its au?or2 = e cause is t prim>ily1 :i* its effect is second>ily4 ,& h5ce we inf]1 t if ! s5sible univ]se be replete ) =ms ( e k9d1 ! exempl>s ( ^? =ms1 mu/ subsi/ 9 immat]ial p]fection1 9 ! artific] ( ! _w4 ,if ?is s5sible _w1 !n1 be =m$ accord+ to ! exempl> ( t :i* is intelligible2 may we n say1 ) ! great ,plot9us1 t x is requisite univ]sal animal %\ld !re prim>ily subsi/ 9 p]fect vital en]gy1 conta9+ all ?+s 9 its omni=m ess5ce4 8,h5ce 7says he#f7 ! heav5s are !re a div9e animal1 replete ) ideal />s4 ,e>? too does n !re subsi/ solit>y1 b is mu* m vital ?an ?is corporeal e>?1 = x is full ( intellectual life4 ,! sea too is !re1 & all wat] subsi/+ 9 life1 & an ev]-abid+ /ream4 ,= h[ is x possible t any ?+ n vital1 c be ! prog5y ( life itself8 ,he1 !re=e1 :o enquires f :5ce animals orig9ate 9 ! intelligible _w1 might z well enquire f :5ce all life1 & s\l1 & univ]sal intellect1 arose4 ,= "h !re is no?+ indig5t nor defective1 b e ?+ is p]fect & exub]ant4 ,"h !y all fl[ f "o f\nta91 n z f a c]ta9 _s1 or heat1 b z if f an univ]sal quality1 possess+ & pres]v+ 9 itself1 all qualities2 su* z sweetness1 accompani$ ) fragrance ( smell1 ! vig\r ( w9e1 & ! /r5g? ( all juices1 bright col\rs1 & :atev] is p]ceiv$ by ! ta/e40 #c4 ,su* !n are ! argum5ts :i* ! ,platonic philosophy af=ds 9 def5ce ( ideas2 ! exi/5ce ( :i* 0 s evid5t to ,plato1 t1 9 ! ,sophi/a1 he comp>es ^? :o oppose ! fri5ds ( ideas to ! giants ( old1 w>r+1 z x 71 on cele/ial s\ls1 & su* z are engag$ 9 sublime inve/igations4 ,let u n[ consid] to :at univ]sals ^! lucid be+s are conf9$2 s9ce1 accord+ to ! ,py?agoreans & ,platoni/s1 !re are n ideas ( all univ]sal conceptions4 8,=1 9 ! fir/ place 7says ,syrianus#g71 !re are no ideas ( ?+s evil & base1 because ^! subsi/ 9 nature r by a privation & abs5ce ( ideas4 ,&1 on ?is acc\nt1 !y are said to exi/ contr>y to nature4 ,nor1 secondly1 ( negations1 = ^! are de/ructive ( ! b\nd & limitation :i* is attribut$ to e ?+ f ! unify+ & compreh5d+ nature ( ideas2 & h5ce1 sep>ation is r ! result ( mat]ial inf9ity ?an ( t :i* is =mal or ideal4 ,nor aga91 are !re any ideas ( ?+s :i* at diff]5t times receive a v>iety ( conditions4 ,= ^! p>ticipate ( transmutation f a moveable cause1 b n f ! immoveable & /able illu/ration ( ideas4 ,nor aga9 ( p>ts1 su* z ! h&1 head1 f+]s1 & ! l4 ,= ! causes ( ?+s exi/+ entire1 produce :ole species & =ms2 n divid$ ab\t ! p>ts ( ^!1 l ! reasons ( nature4 ,b nei!r did ^! wise m5 place 9 intellect ! det]m9ate causes ( accid5ts 9 bodies1 su* z sweetness & :it5ess4 ,= !y consid]$ t natural reasons 7 suffici5t = ! production ( accid5ts4 ,nor aga91 ( composites1 z ( a wise man4 ,= s9ce ideas are simple1 !y preside ov] ! simple ess5ce ( e ?+4 ,b ! composition & division ( ?+s is ! bus9ess ( our intellect2 ideas1 at ! same "t1 & t intellection :i* is co-ord9ate to ideas1 be+ exempt f all ^!1 on acc\nt ( sup]lative simplicity4 ,nei!r1 !re=e1 mu/ we e/abli% ideas ( ?+s g5]at$ f dissimil>s1 su* z mules2 nor ( fruit produc$ by engraft+ f diff]5t trees4 ,= all ^! h a po/]ior & adv5titi\s g5]ation1 & are n ! "w ( nature alone1 nor ( nature proce$+ accord+ to h] own reasons1 b1 z x 71 compell$ to lab\r contr>y to h] own det]m9ations4 ,h5ce x is manife/1 t all art1 :i* imitates nature1 & alone m9i/]s to ! use ( mortal life1 is sep>at$ f ! cause ( ideas4 ,b nei!r are ! works :i*1 dep5d+ on ! purpose ( ! s\l1 are p]fect$ by a conc\rse ( _m causes1 & :i* we are accu/om$ to call ! op]ations ( =tune1 to be conjo9$ to ! cause ( ideas4 ,= ?+s :i* are !re p]fect$1 are et]nal1 & subsi/ p]petually ! same1 free f ! nature ( cont+5t ev5ts4 ,x rema9s1 !re=e1 t ideas mu/ be conf9$ to univ]sal & p]fect ess5ces1 & to :atev] conf]s to _! natural disposition2 z = in/ance1 to man1 & e ?+ p]fective ( man1 su* z wisdom & virtue4 ,= ideas exi/+ z ! g5]ative & en]getic causes ( ! p]fection ( e ?+1 di/ribute be+ to ess5ces1 & conv]t !m to ! inexhau/ible pl5itude ( _! own omni=m natures40 #d4 ,b let u n[ consid] ! nature ( numb]s2 = z e =m is a numb]1 accord+ to ! ,py?agoreans#h1 a speculation ( ?is k9d mu/ af=d no small light to ! ardu\s inve/igation ( ideas4 ,w x n1 !re=e1 be prop]1 9 ! fir/ place1 to enquire1 ) ! great ,plot9us#i1 :e!r multitude is n a dep>ture & di/ance f "o1 s t inf9ity itself is a sep>ation f unity 9 ! extreme1 because x is no o!r ?an innum]able multitude2 t on ?is acc\nt x becomes evil2 & t we contract a simil> nature :5 dep>t+ f intellectual unity1 we are divid$ by s5sible multitude8 ,= a be+ !n prop]ly becomes _m1 :5 no long] able to rema9 collect$ 9 itself1 ! same1 x is diffus$ abroad1 & ?us1 be+ disp]s$1 is v>i\sly ext5d$2 s t :51 by diffusion1 x is absolutely depriv$ ( unity1 x becomes p]fect multitude1 de/itute ( t univ]sal cem5t1 :i* unites "o "p ) ano!r4 ,b :5ev] ! conciliat+ "o is pres5t1 !n t :i* 0 scatt]$ & diffus$1 becom+ p]man5t by its b\nd+ p[]1 passes into magnitude4 ,b if any "o %\ld d5y ! subsi/5ce ( unity1 ass]t+ t "o is no :]e to be f\nd1 :i* is n "s p>ticul> "o2 & %\ld h5ce affirm1 t :at is call$ "o ab/ract$ly1 is only a c]ta9 affection ( ! s\l t[>ds any be+2 we ask1 :at prohibits ! appellation ( ess5ce1 f be+ no?+ m ?an an affection ( ! s\l1 & consequ5tly ! exi/5ce ( be+1 a delusion8 ,= we pr$icate unity ( p>ticul>s ) z great propriety z be+4 ,i am well aw>e1 t philosoph]s ( ! pres5t day w answ]1 t we h an evid5t pro( ( ! reality ( be+1 f its agitat+ ! s\l1 & becom+ app>5t 9 ! phantasy3 to :i* we reply1 t 9 l mann]1 ! s\l is agitat$1 & ! imag9ation influ5c$ ab\t ! "o4 ,= e individual z mu* excites ! p]ception ( "o1 z ( be+4 ,besides1 x is necess>y to enquire :e!r we behold ?is passion & conception ( ! s\l1 z "o or multitude4 ,& aga91 :5 we say n "o1 we d n !n possess "o f ! ?+ itself2 = we say t "o is n conta9$ 9 t individual4 ,& h5ce we mu/ possess "o 9 our own nature1 & ?is mu/ reside 9 ! s\l1 sep>ate f t :i* is d5om9at$ "s p>ticul> "o4 ,b "h x may be object$1 t ! "o we possess is receiv$ f ext]nals1 & is no?+ m ?an a conception ( ! m9d1 produc$ by ! ?+ itself4 ,= x w be said1 t z multitude is no?+ besides a numb] ( individuals1 :i* are call$ _m1 s "o is no?+ besides "o ?+2 & is =m$ by ?\ght sep>at+ t "o p>ticul> f o!rs4 ,to ?is we reply z foll[s3 ,h[ c x be consonant to reason to suppose t ! conception ( "o arises f ! s5sation ( "s "o p>ticul> subject8 ,= "o p>ticul> man1 :o is disc]n$ by s5se1 is by no means ! same ) "o itself1 s9ce1 if ?is 7 ! case1 ?\ght c\ld nev] pr$icate "o ( t :i* is n a man4 ,besides1 z cogitation1 on behold+ ! diff]5t positions ( ?+s1 affirms t ?is is "h or !re1 s :5 x p]ceives an individual1 pron\nces "o2 = t passion is n va91 nor does x ass]t "o ( a non-entity4 ,nor mu/ we ?9k x pr$icates ones1 because ?is individual is diff]5t f ano!r2 = :5 cogitation affirms su* a ?+ is ?is1 & n ano!r1 x decl>es1 9 ! mean "t1 t ! o!r is "o4 ,likewise :5 x affirms t any ?+ is ?is alone1 x !n decl>es1 t :at is alone is "o3 on :i* acc\nt1 x pr$icates "o1 prior to alone4 ,besides1 if !re be multitude1 x is necess>y t "o %\ld antec$e2 s9ce :5 x pr$icates _m1 x pron\nces m ?an "o4 ,& :5 x affirms t an army conta9s a multitude ( m51 x conceives ! soldi]s r$uc$ to "o ord]4 ,= ?\ght1 inde$1 does n p]mit multitude to rema9 p]fect multitude1 de/itute ( ! conciliat+ p[] ( unity2 9 :i* v circum/ance1 ! subsi/5ce ( "o is ev9c$2 = acutely & swiftly p]ceiv+ ! "o :i* results f ord]1 x r$uces ! nature ( ! _m into "o4 ,besides1 we affirm t a h\se & an army are ea* "o1 b t a h\se is m "o ?an an army1 on acc\nt ( ! cont9uity ( its p>ts4 ,if !re=e1 "o is conta9$ m 9 t :i* is cont9u$ ?an 9 t :i* is discrete1 & /ill m 9 :at is p]fectly indivisible1 x is evid5t t ! "o is a c]ta9 nature1 & has a real be+4 ,= x is impossible t ! m & ! less %\ld take place among ?+s :i* h no subsi/5ce4 ,if !n x be n possible to und]/& any ?+ )\t "o or two1 or "s o!r numb]1 x is by no means prop] to d5y exi/5ce to t1 )\t :i* we _c compreh5d ! exi/5ce or prop]ties ( any be+3 b x is requisite t nature %\ld antec$e all disc\rse1 & intellig5ce1 :i* is e :]e necess>y to _! exi/5ce4 ,aga91 if unity has no real subsi/5ce1 & is no?+ m ?an a "n or conception ( ! m9d1 x may be de/roy$ )\t ! de/ruction ( its subject4 ,! unity1 !re=e1 ( a h\se may be tak5 away1 )\t ! ru9 ( a h\se4 ,b if a h\se is no?+ m ?an c]ta9 mat]ials1 r$uc$ into "o =m1 ?is is impossible4 ,&1 on ! contr>y1 ! alt]ation ( t subject1 ( :i* unity is pr$icat$1 c make no real alt]ation 9 unity 7on ?is hypo!sis7 any m ?an ! dea? ( a man c affect 8 "n4 ,:51 !re=e1 a body1 ( :i* "o 0 pr$icat$1 is divid$ into a multitude ( p>ts1 !re is no real alt]ation made 9 ! unity ( ! body1 because unity is no?+ m ?an a "n4 ,x 0 9 consequ5ce ( ?is reason+1 & p]ceiv+ t unity 0 p>ticipat$ by e be+1 t ! ,py?agoreans plac$ a sup]-ess5tial "o at ! top ( ! univ]se1 intelligibly ab/ract$ f all be+s 9 simplicity & excell5ce ( nature4 ,= !y consid]$1 t unless !re 0 a self-subsi/+ "o 9 all ?+s1 !re c\ld nei!r be univ]sals nor p>ticul>s4 ,n ! fir/1 because !y are by nature "o & _m4 ,b x is requisite t ! "o itself1 %\ld preside ov] t :i* is n "o alone4 ,nor aga91 ! second1 because !y are _m & "o1 7t is1 !y p>ticipate m ( multitude ?an unity1 & _! nature is det]m9$ m by ! _m ?an ! "o47 ,& because ( ?+s 9 p>ticipation1 unless an unp>ticipat$ "o is add$1 !re c be no cause ( union to be+s2 9 ! same mann] z ! cause ( ess5ce to be+s1 is tak5 away by ^? :o d5y t be+ itself1 is ! pr9ciple ( all ess5ce4 ,= z ! good itself1 is ! "o pr9ciple ( good to ! univ]se1 & is no?+ besides good2 & z a self-motive nature1 :i* is no?+ besides self-motion1 is ! cause ( motion to all ?+s2 s all ?+s proce$ f be+ itself1 & all unit$ natures receive _! union f ! "o1 ab/ract$ f all ?+s4 ,h5ce 7su* is ! absolute dom9ion ( unity71 cont9u$ quantities w\ld h no exi/5ce )\t its p>ticipation2 = :5 !y are divid$1 s f> z !y lose unity1 !y *ange _! be+ into "s o!r =m4 ,h5ce1 ! bodies ( plants or animals1 :i* are ea* ( !m "o1 :5 !y fly f unity1 & are dissipat$ into multitude1 imm$iately lose ! ess5ce !y =m]ly possess$1 & become some?+ else2 :i* new /ate ( be+ !y likewise possess s f> z !y are "o4 ,add too1 t heal? !n fl\ri%es 9 ! corporeal frame1 :5 ! body is conciliat$ into "o2 !n beauty fl\ri%es1 :5 ! p[] ( "o connects ! memb]s into proportion & cons5t2 & !n virtue reigns 9 ! s\l1 :5 ! s\l is r$uc$ into "o similitude ) t :i* is div9e4 #e4 ,b let u n[ inve/igate ! nature ( numb]s4 ,all numb]1 accord+ to ! ,py?agoreans1 orig9ates f unity & ! indef9ite duad2 ! fir/ hav+ ! relation ( =m1 & ! second1 t ( matt] to all ! ord]s ( numb]s4 ,b !y likewise divid$ numb] into two k9ds1 ess5tial & monadic4 ,! ess5tial numb] !y consid]$ z fir/ subsi/+ 9 ! intelligible _w1 toge!r ) be+1 & f !nce di/ribut$ into all ! v>i\s gradations ( =ms4 ,b ! monadic1 or t :i* is compos$ f c]ta9 units1 !y ju/ly consid]$ z no?+ m ?an ! image ( ess5tial numb]4 ,& ) respect to ! numb]s :i* ! human s\l p>ticipates1 ^! f its imp]fect condition h a middle subsi/5ce2 i4 e4 !y exi/ 9 a vital1 gno/ic1 & speculative1 b n 9 an op]ative mann]4 ,h5ce1 :5 receiv+ "o ?+ ) ano!r1 we affirm1 t !y are two1 z a dog & a man1 or two m52 or :5 we compute m ?an two1 z t51 & say t !re is a decad ( m51 ?is numb] is n ess5tial to ! two or t5 individuals1 nor is x to be conceiv$ z subsi/+ 9 s5sible natures2 b x is purely quantity4 ,b :5 we di/ribute ?is t51 into units1 we produce ! pr9ciple ( quantity1 & g5]ate a subject 9 op9ion#aj1 capable ( p>ticipat+ ! ess5tial decad ( our s\l4 ,b :51 consid]+ man 9 himself1 we affirm t he is a c]ta9 numb]1 z ! duad1 compos$ ( animal & rational1 we d n obs]ve "o mode 9 ?is pr$ication2 b s f> z by a discursive op]ation ( ! s\l1 we num]ate1 we effect a p>ticul> quantum2 b s f> z ! subjects are two1 & at ! same "t bo? "o 7s9ce "o fills ! ess5ce ( bo?1 & 9 bo? unity is conta9$71 we pron\nce ano!r1 & an ess5tial numb]3 & ?is duad is n ( a po/]ior orig91 nor alone signifies a c]ta9 quantity1 ext]nal to ! subject1 b a duad subsi/+ 9 ! ess5ce ( man1 & conta9+ 8 nature4 ,= "h we d n produce a numb] by a discursive op]ation1 :ile we pursue ess5tial natures4 ,b :5 we numb] any t5 ?+s1 :i* are n connect$ by any conciliat+ unity1 l a *oir1 or an army1 !n ?is decad1 :i* we pr$icate ( ! t5 p>ticul>s1 subsi/s alone 9 our num]at+ s\l1 :i* r5d]s ! t5 individuals 9 op9ion1 a def9ite quantum4 ,b 9 a *oir1 or an army1 ess5tial numb] is p>ticipat$ exclusive ( t :i* subsi/s 9 our s\l4 ,& if x be enquir$ h[ numb] subsi/s 9 ! human s\l1 we mu/ say1 t ! s\l1 by h] self-mov+ en]gies1 procreates numb]1 :ile %e num]ates1 & by ?is en]gy1 causes ! exi/5ce ( quantity2 9 ! same mann] z 9 walk+1 we give rise to a c]ta9 motion4 ,?us1 monadic numb]1 or a collection ( units ( v>i\s k9ds1 subsi/s 9 op9ion1 9 a mann] correspond5t to t ( geometrical figures2 & by ?is means p>ticipates ! ess5tial numb] ( ! s\l4 ,= z a triangul> figure 9 ! phantasy1 is ! recipi5t ( a triangul> nature1 or ( triangle itself2 s e ?ree units 9 op9ion1 receive ! ess5tial triad ( ! s\l1 &1 by ?is means1 =m a def9ite quantum4 ,9 %ort1 z 9 e be+ we may disc]n ! resemblances ( matt] & =m1 s 9 ! p5tad1 or any o!r numb]1 ! five units1 :i* are ! subject ( p>ticipation1 & ! quantity ( ! numb]1 orig9ate f ! duad2 b ! =m1 t is ! p5tad itself1 f unity4 ,= e =m is an unity1 :i* unites its subject quantity1 & connects x ) its ideal species4 ,x is1 !re=e1 requisite to und]/&1 t ! two pr9ciples ( ma!matical numb]s are resid5t 9 our s\ls1 ) :i* e ma!matical numb] is co-exi/5t2 ,i mean unity1 compreh5d+ 9 itself all ! =ms ( numb]s1 & :i* corresponds to unity 9 intellectual natures2 & ! duad1 endu$ ) a g5]ative p[]1 ( a =mless nature1 & ( inf9ite virtue2 & :i* is call$ b\ndless1 on acc\nt ( its be+ ! image ( nev]-fail+ & intelligible duality4 ,h5ce1 ! unity ( ! s\l1 ) a nev]-ceas+ en]gy1 cont9ually di/+ui%es & =ms all ! ord]ly processions ( h] numb]s1 suff]s no vacuum to int]v5e1 & leaves no quantity =mless & innum]able4 ,h5ce too1 no ess5tial numb] ( ! s\l1 z = in/ance1 ! p5tad1 is compos$ f sub/ance & accid5t1 z a :ite man2 nor f g5us & diff]5ce1 z man f animal & bip$2 nor aga91 f five unities mutually t\*+ ea* o!r1 l a bundle ( wood2 nor f ?+s mixt1 l wat] & w9e1 nor f ?+s subsi/+ by position1 9 ! mann] t /ones compose a h\se2 nor la/ly1 does x subsi/ l ?+s num]able2 = x is n because !y are compos$ f indivisible units1 t !y possess any ?+ besides units4 ,= _m po9ts are indivisible1 yet quantity is n produc$ on ?is acc\nt2 b because !y p>ticipate ( two natures1 ! "o correspond+ to matt]1 & ! o!r to =m4 ,la/ly1 x is n prop] to say1 t ! numb] sev5 7& s ( any o!r numb]71 is compos$ f ! triad & ! tetrad2 = units1 inde$1 compos$ ) units1 =m a subject adapt$ to ! reception ( ! heptad1 or ! ideal & ess5tial numb] sev52 b ! def9ite num]ical quantity sev51 is =m$ f s _m units1 & ! ideal heptad4 ,h5ce1 z ! s\l ( ! %ipwright gives =m to ! timb]1 f h] inh]5t art2 s ! num]ative s\l1 f ! unity endu$ ) ! relation ( a pr9ciple :i* %e possesses1 gives =m & subsi/5ce to all h] inh]5t numb]s4 ,b !re is ?is diff]5ce betwe5 ! two1 t ! %ipwright's art is n ess5tial to our nature1 & requires manual op]ation1 because x is conv]sant ) s5sible matt]2 b ! num]ative art is ess5tially inh]5t 9 ! s\l1 & is !re=e pres5t ) all m51 & possesses an intellectual matt]1 :i* x easily =ms )\t ! assi/ance ( "t4 ,& ?is1 p]haps1 is :at deceives _m1 :o ?9k t ! heptad is no?+ m ?an sev5 units4 ,= ! imag9ation ( ! vulg>1 unless x fir/ p]ceives a ?+ de/itute ( ornam5t1 & aft]w>ds ! op]ations ( ! adorn+ artific] sup]v5+ its nature2 & la/ly1 beholds ! ?+ p]fect1 & inve/$ ) =m1 _c be p]suad$ t x possesses two natures1 ! "o =mless1 b ! o!r endu$ ) an en]getic & =m+ p[]4 ,& "h x is necess>y to obs]ve1 t ?\gh unity is ! =m ( all ari?metical =ms1 yet x is *iefly ! =m ( ! decad4 ,= :at unity is simply to all ! s]ies ( numb]s1 t ! decad is to ! foll[+ hundr$s1 ?\s&s1 & millions2 f :5ce1 accord+ to a second>y progression1 x is d5om9at$ unity4 ,z intellect1 !re=e1 is ! =m ( all ?+s1 b especially ( ! s\l1 s unity1 ?\gh x is ! idea ( all numb]s1 yet especially ( ! decad4 ,b ! reason :y ! ,py?agoreans ext5d$ ideal numb]s no f>!r ?an t51 is because ?is numb] is ! ultimate p]fection ( be+s1 conta9+ all ?+s 9 its omni=m nature4 ,= all proportion subsi/s )9 ! numb] t52 ! ari?metical 9 a natural progression ( numb]s f unity2 ! geometrical 9 ! numb]s #a1 #b1 #d1 & #a1 #c1 #i1 & ! h>monical 9 ! numb]s #b1 #c1 #f1 & #c1 #d1 #f4 ,& s9ce ! causes ( all ?+s are conta9$ 9 numb]s1 z f> z to ! decad#aa1 x is sup]flu\s to suppose exempl>s ( ! foll[+ numb]s4 ,if x %\ld be ask$ 9 :at mann] we mu/ conceive numb] z subsi/+ 9 ! intelligible _w1 we answ]1 ) ! great ,plot9us1 t we mu/ conceive x z subsi/+ 9 be+ itself1 ) a p[] ( impell+ x to ! production ( multitude4 8,h5ce 7says he1 ,ennead vi4 lib4 vi47 numb] is ei!r ! ess5ce or ! en]gy ( be+1 & animal itself1 & intellect is numb]4 ,b1 p]haps1 we mu/ call be+1 numb] unit$ 7 71 b be+s1 numb] evolv$1 or unfold$2 7 7 intellect1 numb] mov+ 9 itself2 7 7 & la/ly1 animal1 numb] compreh5d+ 7 487 ,x 0 9 consequ5ce ( ?is reason+1 t ! ,py?agoreans call$ ideas numb]s2 because ! gradual evolution ( ^! f ineffable unity1 produc$ all ! beautiful v>iety ( =ms4 ,_! exalt$ conceptions ( numb]s1 likewise1 orig9at$ f ! same sublime !ory4 ,h5ce1 #ab,py?agoras1 9 ! sacr$ disc\rse1 calls numb] 8! rul] ( =ms & ideas40 ,b #ac,philolaus1 8! comm&+ & self-begott5 conta9] ( ! et]nal duration ( mundane conc]ns40 ,& #ad;,hippasus1 & all ^? :o 7 call$ 7or su* z 7 yet "u ! probation ( ! qu9qu5nial sil5ce71 8! fir/ exempl> ( ! mundane fabric1 & ! judici>y in/rum5t ( its artific]40 #f4 ,& "h ,i _c b take notice1 ) regret1 ( ! v unphilosophical mi/ake committ$ by t great ma!matician ,dr4 ,b>r[#ae3 ,i say1 ) regret1 on acc\nt ( ! extraord9>y obligations ,i am "u to 8 writ+s1 = my pr(ici5cy 7:atev] x may be7 9 ma!matical le>n+4 ,b respect mu/ yield to ! tru?4 8,unity1 says he1 is n indivisible4 7,= h[ ex4 gr4 c #b_/f add$ to #d_/f be equal to unity1 if unity be indivisible & incompos$1 & repres5t a po9t7 b r only unity is prop]ly divisible1 & numb]s arise f ! division ( unity40 ,"h ! ,doctor evid5tly conf\nds s5sible units1 :i* are ! subjects ( vulg> practical ari?metic1 ) ^? units :i* are ! objects ( sci5ce4 ,e individual s5sible object1 is inde$ an unit1 s f> z x p>ticipates ! connect+ & conciliat+ p[] ( an immat]ial "o3 b ! unity :i* /&s at ! top ( speculative ari?metic1 is p]fectly indivisible1 or ari?metic w\ld cease to be a sci5ce4 ,! tru? ( ?is is evid5t f ,euclid's def9ition3 8,unity 7says he7 is t accord+ to :i* ea* ( ! ?+s :i* are1 is call$ "o40 ,b if unity be a composite1 ! def9ition is false2 s9ce a composite1 or a c]ta9 multitude1 c nev] be ! cause ( unity1 b ! contr>y4 ,& t ?is immat]ial "o subsi/s 9 s5sible natures1 has1 ,i hope1 be5 suffici5tly prov$ 9 ! prec$+ "p ( ?is disc\rse4 ,b ! ,platonic ,!o#af ( ,smyrna1 fully e/abli%es ! indivisibility ( unity1 z foll[s3 8,unity is t]m9at+ quantity1 ! pr9ciple & elem5t ( numb]s1 :i* rema9s undim9i%$ by ! mo/ imm5se multitude ( subtractions1 & be+ depriv$ ( all numb]1 cont9ues firm & fixt1 because x is impossible = division to proce$ beyond ! b\nd ( unity4 ,?us1 if we sep>ate any "o corporeal sub/ance into p>ts1 ! "o aga9 becomes _m2 & by subtract+ ! sev]al p>ts1 we end 9 "o "p2 & f ?is rema9+ "p1 aga9 divid$1 arises multitude2 & by tak+ away e "p1 we aga9 arrive at "o4 ,s t "o1 consid]$ z "o1 is incapable ( dim9ution1 & p]fectly indivisible4 ,on ! contr>y1 e numb] is dim9i%$ by division1 & is sep>at$ into p>ts less ?an itself2 z ! numb] #f into #c & #c1 or into #d & #b1 or into #e & #a4 ,b unity 9 s5sible p>ticul>s1 if divid$1 is dim9i%$ aft] ! mann] ( body1 & by section is di/ribut$ into p>ts less ?an itself3 b x receives increase aft] ! mann] ( numb]2 = in/ead ( ! "o1 multitude is produc$4 ,9 ?is s5se1 !re=e1 is unity indivisible2 = no?+ is divid$ into p>ts great] ?an itself4 ,b t :i* is cut into p>ts great] ?an ! :ole1 & into p>ts equal to ! :ole1 is divid$ z numb]4 ,?us1 = in/ance1 if any "o s5sible body is divid$ into six p>ts1 #a1 #a1 #a1 #a1 #a1 #a1 ^! %all be equal to ! :ole2 b by a section into #d & #b1 x is divid$ into p>ts great] ?an ! :ole1 consid]$ z "o2 = #d & #b consid]$ z numb]s1 exce$ unity1 & ! body 0 suppos$ to be "o4 ,unity1 !re=e1 z numb] is p]fectly indivisible4 ,b unity is call$ by ! ,greek ^w 1 only1 or alone1 ei!r because x rema9s immoveable1 & does n des]t itself1 nor surpass ! b\nds ( its nature 7= x rema9s ! same1 h[ev] multipli$ into itself1 ?r\gh an inf9ite progression7 or because x is plac$ sep>ate & ap>t f ! multitude ( o!r numb]s1 x is d5om9at$ ! monad1 or "o40 ,9 consequ5ce ( ?is v mi/ak5 hypo!sis1 :i* opposes n only all ! wisdom ( antiquity1 b ! sublime/ tru?s1 ! ,doctor ass]ts1 t an ari?metical cyph] is ! pr9ciple ( numb]s2 & t x is analog\s to a po9t 9 geometry4 ,j z if a cyph]1 :i* is no?+ m ?an a m>k expressive by its position ) numb]s1 ( a c]ta9 quantity1 _h a real exi/5ce1 & 0 productive ( numb]3 :51 at ! same "t1 any o!r arbitr>y "* w\ld s]ve ! same purposes1 if appli$ 9 a simil> mann]4 ,x mu/ surely afflict e ?9k+ m9d1 to see h[ dreadfully ! me*anical sy/em ( philosophy1 :i* has be5 s long 9 fa%ion1 enslaves & p]v]ts ! m9ds ( its vot>ies2 = !re _c1 ,i ?9k1 be a m egregi\s in/ance ( its fatal t5d5cy1 ?an ! pres5t1 9 :i* no?+ is consid]$ z ! f\ndation ( t noble sci5ce1 ari?metic2 :i* 0 des]v$ly plac$ by ! anci5ts1 9 ! fir/ rank ( ! ma!matical discipl9es4 ,su* a f\ndation1 inde$1 may be prop] to ! me*anical philosophy1 b is v ill adapt$ to support ! solid fabric ( ! ari?metical sci5ce4 ,b let u att5d to ! argum5ts ( ?is mo/ le>n$ man1 9 def5ce ( s /range an ass]tion1 8,a cyph]1 or ari?metical no?+ 7says he7 is really ! b\nd ( e numb] com+ betwe5 x & ! numb]s next foll[+1 b n z a "p4 ,a cyph] be+ add$ to1 or tak5 f a numb]1 does nei!r increase nor dim9i% x2 f x is tak5 ! beg9n+ ( computation1 :ile itself is n comput$2 & x be>s a manife/ relation to ! pr9cipal prop]ties ( a geometrical po9t40 ,b 9 :at mann] are we to conceive ! no?+ :i* int]v5es betwe5 any two numb]s1 to be _! t]m or b\nd>y8 ,= ,euclid def9es a t]m to be ! extremity ( any ?+2 imply+ by ! extremity1 some?+ belong+ to t ( :i* x is ! b\nd4 ,b h[ c a cyph]1 or no?+1 9 any respect belong to numb]1 or some?+8 ,= if no?+ be a b\nd>y1 m]ely f its int]v5tion1 a po9t exi/+ betwe5 any two disjo9$ l9es1 ?\gh at ! greate/ di/ance f ea*1 mu/ be _! common b\nd>y1 :i* is evid5tly absurd4 ,besides1 :at relation does x be> to a po9t1 :i* is endu$ ) a g5]ative p[]1 by its flux =m+ ! simple ext5sion ( a l9e1 &1 at ! same "t1 e :]e limit+ its progression1 & subsi/+ 9 inf9ite capacity 9 its e "p8 ,:]e are ! real & div9e prop]ties to be f\nd 9 an ari?metical no?+1 :i* ,proclus1 9 ! foll[+ ,comm5t>ies1 exhibits 9 a po9t8 ,& h[ c computation orig9ate f a m]e non-entity8 ,b a little consid]ation w conv9ce u1 t ?is ,s>ac51 or ,indian cyph]1 is no?+ m ?an an arbitr>y "*1 inv5t$ = ! purpose ( facilitat+ computation4 ,=1 suppose ! lett] 7a7 to be plac$ 9 its /ead1 & to signify1 :5 connect$ ) ! m>k = unity1 t51 or t5 multipli$ by "o2 :5 connect$ ) ! m>k = two1 t5 multipli$ by two1 & s on4 ,& aga91 :5 plac$ twice aft] unity1 let x express ! second p[] ( t51 or "o hundr$1 9 ?is mann]1 a a2 :5 ?rice connect$1 "o ?\s&1 or ! ?ird p[] ( t51 & s on3 %all we say1 9 consequ5ce ( ?is1 t 7a7 is ! b\nd ( numb]s1 & ! pr9ciple ( ari?metic8 ,or1 %all we n r say1 t x is an arbitr>y symbol1 l any o!r algebraic "*1 hav+ no real connection ) numb]s1 & dep5d+1 = its exi/5ce & application1 entirely on ! w ( its inv5tor4 ,b ?is op9ion is too absurd to ne$ any f>!r refutation4 #g4 ,x may "h1 p]haps1 be expect$1 t ,i %\ld expla9 h[1 9 ! language ( ,syrianus#ag1 8div9e numb] proce$s f ! immortal retreats ( unity1 until x arrives at ! div9e tetrad#ah20 & t ,i %\ld unfold ! prop]ties ( ! tetractys1 accord+ to ! ,py?agoreans2 b an und]tak+ ( ?is k9d1 w\ld n only f> exce$ ! limits ( ?is diss]tation1 b1 p]haps1 9 ! pres5t age1 might be ju/ly deem$1 by ! lov]s ( wisdom1 a pro/itution & pr(anation ( ! mo/ exalt$ tru?s4 ,en\gh1 ,i hope1 has be5 said to excite ! curiosity1 & r\se ! att5tion ( ! ?9k+ & lib]al "p ( mank9d2 & ^? :o und]/& :at is "h briefly deliv]$1 may apply !mselves1 ) advantage1 to ,proclus on ,plato's ,!ology1 :]e !y w f9d all ! my/]ies ( numb]s unravell$2 & to ! works ( ! great ,plot9us1 :o w lead !m into ! p5etralia ( ! mo/ recondite wisdom4 ,b1 9 p]us+ ! works ( ^! great m51 ! read] mu/ n expect to f9d ! sublime/ tru?s expla9$ 9 a famili> mann]1 & adapt$1 l _m mod]n publications1 to ! meane/ capacities4 ,= ?is1 inde$1 is impossible to be effect$4 8,mank9d 7says ,petv9#ai71 are n to be made any m truly kn[+ ?an happy by ano!r's und]/&+4--,!re is no man c at once convey light 9 ! high] subjects1 to ano!r man's und]/&+4 ,x mu/ come into ! m9d f its own motions1 )9 itself3 & ! gr& art ( philosophy1 is to set ! m9d a-go+2 &1 ev5 :5 we ?9k no?+ ( x1 to assi/ x 9 its lab\r40 ,aft] :i* he obs]ves1 t 8! anci5ts nev] attempt to lead u into k1 by a cont9u$ *a9 ( reason+2 on ! contr>y1 !y write 9 su* a mann]1 z to =ce u to ?9k = ourselves40 ,&1 previ\s to ?is1 he rem>ks1 8t !re are c]ta9 tru?s acquir$ by a long ex]cise ( reason1 bo? 9 p>ticul>1 & likewise 9 ^? subjects t are mo/ g5]al1 z mu*1 p]haps1 out ( ! rea* ( ! greate/ ma!matician1 z ,sir ,isaac ,newton's speculations are above ! capacity ( "s t are n[ call$ ma!maticians40 ,! tru? ( ?is obs]vation is suffici5tly ev9c$1 9 ,plato's def9ition ( a philosoph] 79 8 ,sophi/a71 8,! philosoph] 7says he7 is ! man :o suffici5tly sees "o idea e way ext5d$ ?r\gh _m1 e "o ( !m ly+ ap>t2 & _m ideas diff]5t f "o ano!r1 ext]nally compreh5d$ "u "o4--,& f>!r1 "o idea1 ?r\gh\t all manys1 wrapt up 9 "o2 & _m ideas1 e way sep>ate or discreet4 ,?is is to h ! k to disc]n h[ ideas1 z !y are g5]al1 agree & disagree40 ,n[1 he :o ?9ks t a p]ception ( ?is k9d may be acquir$ by b>ely read+ an accurate disc\rse on ! nature ( ideas1 compos$ 9 intelligible t]ms1 )\t1 at ! same "t1 employ+ a long c\rse ( pr(\nd m$itation1 & pati5t ?\ght1 kn[s b little ! difficulty ( ! task1 & until he *anges 8 op9ion w nev] be ! wis]4 ,b ! folly & presumption ( m51 ) respect to ?is sublime philosophy1 is really unp>donable2 = !re are v few :o conceive t mu* previ\s in/ruction is requisite to its acquisition2 b almo/ e man decides p]emptorily on ! mo/ ab/ract speculations1 & reckons himself suffici5t = ! mo/ pr(\nd inve/igations4 ,9 ! sci5ces & arts !y are will+ to proce$ to p]fection by gradual advances2 b !y consid] philosophy z easy1 ( in/ant access1 & ha/ily approa* to h] embraces ) an assur$ confid5ce ( success4 ,?\gh1 l unhappy ,ixion1 ?r\gh _! presumption1 in/ead ( a goddess1 !y grasp no?+ b an empty cl\d4 ,plato 0 s s5sible ( ?is tru?1 t1 9 8 sev5? epi/le to ,dion1 he expressly affirms1 t he nei!r has writt51 nor ev] w write explicitly conc]n+ ^! sublime speculations2 8,= a ?+ ( ?is k9d 7says he7 _c be express$ by words1 l o!r discipl9es1 b by a la/+ famili>ity1 & conjunction ( life1 ) ?is div9e object1 a bright light#bj on a sudd51 z x 7 leap+ f a fire1 w illum9ate ! s\l1 & !re pres]ve & n\ri% its spl5dor4 ,he adds1 t a publication ( su* conc]ns1 is alone useful to a few ( mank9d1 :o f "s small ve/iges previ\sly demon/rat$1 are suffici5tly sagaci\s to _! inv5tion4 ,b x w fill o!rs p>tly ) a base contempt1 & p>tly ) a ra% & va9 confid5ce1 z if !y _h n[ le>n$ "s v excell5t ?+s40 ,he !n subjo9s ! foll[+ in/ance ( ! difficulty att5d+ su* an und]tak+3 8,!re are ?ree ?+s 7says he71 f :i* sci5ce mu/ necess>ily be produc$2 b ! f\r? is sci5ce itself4 ,& x is requisite to e/abli% ! fif? z t :i* is ! object ( k1 & has a true exi/5ce4 ,"o ( ^! is ! "n ( a ?+2 ! second its def9ition2 ! ?ird ! resemblance2 ! f\r? sci5ce4 ,n[ take ea* ( ^!1 desir+ to le>n :at we h lately ass]t$1 & ?9k conc]n+ !m all1 9 a simil> mann]4 ,a circle is call$ some?+1 ^: "n we h j express$4 ,aft] ?is foll[s its def9ition1 compos$ f n\ns & v]bs4 ,= t :i* e :]e is equally di/ant f ! extremes to ! middle1 is ! def9ition ( t :i* we signify by ! "n ( a r\nd1 & a circumf]5ce1 & a circle4 ,b ! ?ird is ! circle :i* may be pa9t$1 or blott$ out1 :i* may be made by a :eel1 or de/roy$4 ,none ( :i* affections1 ! circle itself1 :i* ea* ( ^! respects1 suff]s1 z be+ ( a diff]5t nature4 ,b ! f\r? is sci5ce1 & intellect1 & true op9ion ab\t ^!4 ,& ?is aga9 mu/ be e/abli%$ z "o :ole1 :i* nei!r subsi/s 9 voice1 nor 9 corporeal figures1 b 9 intellect & intellig5ce4 ,x is !re=e manife/1 t ?is f\r? is diff]5t f ! nature itself ( ! circle1 & aga9 diff]5t f ! ?ree we h previ\sly m5tion$4 ,b among ! numb] ( ^!1 intellect1 by its relation & similitude1 proximately adh]es to ! fif?1 :ile ! re/ are m remote f its nature4 ,! same may likewise be affirm$ ( a /raight & crook$ figure1 ( col\r1 & ( ! good1 ! beautiful1 & ! j4 ,& aga91 ( e body1 :e!r fa%ion$ by ! h&1 or ! "w ( nature1 :e!r fire or wat]1 & ! re/ ( ?is k9d2 likewise ( e animal1 & ! mann]s ( animals2 & ( all actions & passions4 ,= unless1 among ^!1 "s "o1 aft] a mann]1 receives t f\r?1 he w nev] p]fectly p>ticipate ! sci5ce ab\t ! fif?40 ,he !n proce$s to %ew 9 :at respect ea* ( ! prec$+ f\r are diff]5t f ! fif?4 8,e circle 7says he7 :i* by ! h&s ( m5 is ei!r pa9t$1 or fa%ion$ by a :eel1 is pla9ly contr>y to our fif?4 ,= x e :]e p>ticipates ( ! "r-l9e4 ,b we mu/ affirm1 t ! circle itself has nei!r m nor less ( any ?+ :atev]2 t is1 x possesses 9 itself1 no?+ ( a contr>y nature4 ,besides1 none ( ^! are endu$ ) any /ability ( "n4 ,= no?+ h9d]s our apply+ ! appellation ( /raight to t :i* we n[ d5om9ate r\nd1 & call+ ! /raight by ! d5om9ation ( ! r\nd2 nor w !re be any less /ability 9 ^!1 :5 _! names are *ang$ into ! contr>y4 ,! same reason+ is likewise true ( def9ition1 s9ce x is compos$ f n\ns & v]bs1 :i* possess no /ability4 ,& 9 a v>iety ( ways1 x may be prov$1 t no "o ( ^! f\r is c]ta9 & firm40 ,n[1 ?is fif? division ( ,plato's entirely respects ideas1 consid]$ z fl\ri%+ 9 intellect2 by a conjunction ) :i*1 we acquire true intellig5ce1 & ! p]fection ( human k4 ,! fir/ ?ree ( ! prec$+ are obnoxi\s to v>i\s mutations2 ! f\r? less2 b ! la/ is p]fectly /able & inv>iable4 ,! ?ree fir/ are r conv]sant ab\t ! qualities ( ?+s1 ab\t ! image & %ad[2 ! f\r? raises u to ! p>ticipation ( tru?2 b ! fif? to tru? itself1 & p]man5t ess5ce4 ,9 ! fir/ degrees almo/ all are conv]sant2 9 ! f\r? a few2 9 ! fif?1 all ! gods1 b a v small "p ( mank9d1 z x is ass]t$ 9 ! ,timu4 ,! f\r fir/ may be kn[n1 inde$1 )\t ! fif?1 confus$ly2 b f ! k ( ! fif? !y become p]fectly manife/1 z effects f ! k ( _! cause4 ,b we _c1 by any means1 atta9 to ! appreh5sion ( ! fif?1 unless we h be5 fir/ accurately conv]sant ) ! re/2 = f our imp]fect condition we are compell$ to rise f diff]5ce to id5tity1 f multitude to unity1 & f %ad[ to sub/ance4 ,:ile we inve/igate ! k ( ?+s1 if we are alone desir\s to appreh5d _! resemblance 7:i* is ! case ) ! multitude7 we %all be plac$ 9 ! ?ird degree1 & may easily acquire ! object ( our pursuit4 ,b if we %\ld =tunately possess ! true philosophical g5ius1 :i* is r>e 9 ! extreme1 & aspir+ to ! fif? degree1 %\ld1 by a happy ev5t1 atta9 to its conjunction1 ?\gh su* a contact is cle>] & m c]ta9 ?an all k2 yet x is difficult to express x 9 words1 & to manife/ x to o!rs4 ,& ! reason ( ?is is obvi\s3 fir/1 because words are want+1 :i* exactly correspond to ! ess5ce ( a ?+1 s9ce ^! are only ! symbols ( %ad[s4 ,secondly1 because we speak ) ^?1 :o are alone conv]sant ) %ad[s1 & are on ?is acc\nt d]id$ by !m1 :5 !y f9d t our fif? does n1 by any means1 accord ) mat]ial resemblances1 :i* !y consid] z ! only realities4 #h4 ,& "h a "q v naturally pres5ts itself = our solution1 :e!r ! s\l1 :ile unit$ ) ! body1 is able to p]ceive ideas1 )\t ! assi/ance ( ! phantasy1 ,= x seems difficult to appreh5d h[ ! s\l1 ?us depress$ & weigh$ d[n ) ! t]r5e mass1 %\ld be able to raise h]self to ! sup]nal light ( ideas1 & become unit$ ) _! refulg5ce4 ,! op9ion ( ! ,p]ipatetics is well kn[n1 t "s phantasm mu/ always accompany intellig5ce2 b ?is is d5i$ by ! ,platoni/s1 & ,i ?9k ) great reason4 ,= ! op]ations ( intellect are n dep5d5t on ! phantasy1 ?\gh ! p]ceptions ( ! latt] proce$ f ! en]gies ( ! =m]4 ,besides1 z ,plot9us beautifully obs]ves1 our mo/ vigor\s en]gies are accompani$ ) ! lea/ animadv]sion2 & !re is no absurdity 9 suppos+ t by increas+ ! =ce ( intellectual en]gy1 we may speculate free f all imag9ation2 s9ce ! phantasms att5d+ our conceptions1 became weak 9 proportion z ! intellectual sight increases 9 vig\r4 ,on ?is acc\nt1 ! ,platoni/s affirm1 t ! moral virtues free u f ! vehem5ce ( p]turbations2 b ! contemplative f imag9ation1 & ! s5ses4 ,h5ce too1 ! sci5ces may be call$ liv+ wat]s2 9 :i* ! w+s ( ! s\l be+ dipt1 h] fea!rs1 :i* 7 ei!r sep>at$ or brok5 by h] lapse into body1 are repair$1 & re/or$ to a resemblance ( _! =m] p]fection4 ,= ! w+s are ! p[]s ( ! s\l1 lead+ to intelligibles3 b ! fea!rs are z well ! natural in/9cts to good & tru?1 z reasons ins]t$ 9 ! s\l2 :i* ei!r fall (f1 or are brok5 by h] desc5t into body1 & conjunction ) its ru9\s bonds4 ,b ^! are repair$ & invigorat$ by ! sci5ces1 :i*1 l liv+ /reams1 fl[+ f ! f\nta9s ( ideas1 re/ore life & p]fection to ! s\l4 ,h5ce ,plato1 9 ! ,phdrus1 ass]ts t ^! w+s ( ! s\l are increas$ by e ?+ :i* conf]s to sup]nal elevation2 z beauty1 wisdom1 & ! l2 & by a conv5i5t metaphor1 9 ! same dialogue1 he consid]s ! *>iot ( ! s\ls lives1 h] *>iote]1 & ! horses by :i* h] c> is drawn2 & la/ly1 e ?+ :i* contributes to ! elevation ( ! s\l1 & h] conjunction ) intellect & ideas4 ,we may !re=e conclude1 t ?is conjunction is possible to be effect$1 ?\gh x is r>ely obta9$2 & t x is a flight too ardu\s & sacr$ = ! grovel+ & sordid2 a spl5dor too bright = ! s5sible eye2 & a contact too ineffable to be describ$ by ! un/able composition ( words4 ,b ,i _c conclude ?is section1 )\t solicit+ ! read]'s att5tion to a comp>ison ( ! diff]5ce betwe5 ! anci5t philosophy1 & t inv5t$ by ,mr4 ,locke1 & ! mod]ns4 ,accord+ to ,mr4 ,locke's sy/em ideas are =m$ f s5sible p>ticul>s1 by a k9d ( me*anical op]ation2 s t tru? is some?+ by its nature1 po/]ior to s5sation1 & entirely dep5d5t on x = exi/5ce4 ,accord+ to ,plato1 ideas are et]nal & immat]ial be+s1 ! orig9als ( all s5sible =ms1 & ! f\nta9s ( all evid5ce & tru?2 s t on ?is sy/em tru? ranks among ! fir/1 & n 9 ! la/ ( ?+s2 & w\ld /ill reta9 its nature1 ?\gh ! corporeal s5ses 7 no m4 ,accord+ to ,mr4 ,locke1 ! s\l is a m]e rasa tabula1 an empty recipi5t1 a me*anical blank4 ,accord+ to ,plato1 %e is an ev]-writt5 tablet1 a pl5itude ( =ms1 a vital & intellectual en]gy4 ,on ! =m] sy/em1 %e is on a level ) ! mo/ degrad$ natures1 ! receptacle ( mat]ial species1 & ! spectator ( delusion & non-entity#ba4 ,h5ce1 h] en]gies are no?+ b somnol5t p]ceptions1 & encumb]$ cogitations2 = all h] k t]m9ates 9 s5se1 & h] sci5ce 9 passion4 ,l a man betwe5 sleep+ & wak+1 h] visions are turbid & confus$1 & ! phantoms ( a mat]ial night1 cont9ually glide be=e h] dr[sy eye4 ,b on ! latt] sy/em1 ! s\l is ! connect+ m$ium ( an intelligible & s5sible nature1 ! bright repository ( all middle =ms1 & ! vigilant eye ( all cogitative reasons4 ,h5ce %e is capable ( r\s+ h]self f ! sleep ( a corporeal life1 & em]g+ f ?is d>k ,cimm]ian l&1 into ! regions ( light & reality4 ,at fir/1 inde$1 be=e %e is excit$ by sci5ce1 %e is oppress$ ) le?>gy1 & cl\d$ ) oblivion2 b 9 proportion z le>n+ & enquiry /imulate h] dormant p[]s1 %e wak5s f ! dreams ( ignorance1 & op5s h] eye to ! irradiations ( wisdom4 ,on ,mr4 ,locke's sy/em1 ! pr9ciples ( sci5ce & s5se are ! same1 = ! en]gies ( bo? orig9ate f mat]ial =ms1 on :i* !y are cont9ually employ$4 ,h5ce1 sci5ce is subject to ! fl[+ & p]i%able nature ( p>ticul>s2 & if body & its attributes 7 de/roy$1 w\ld be no?+ b a "n4 ,b on ! sy/em ( ,plato1 !y diff] z mu* z delusion & reality2 = "h ! vital1 p]man5t1 & lucid nature ( ideas is ! f\nta9 ( sci5ce2 & ! in]t1 un/able1 & obscure nature ( s5sible objects1 ! s\rce ( s5sation4 ,on ,mr4 ,locke's sy/em1 body may be modifi$ into ?\ght1 & become an intellig5t creature2 x may be subtiliz$ into life1 & %r9k1 by its exility1 into intellect4 ,on t ( ,plato1 body c nev] alt] its nature by modification1 h[ev]1 x may be r>efi$ & ref9$1 v>i$ by ! transposition ( its p>ts1 or tortur$ by ! h& ( exp]im5t4 ,9 %ort1 ! two sy/ems may be aptly repres5t$ by ! two sections ( a l9e1 9 ,plato's ,republic4 ,9 ! anci5t1 y h tru? itself1 & :atev] p>ticipates ( ! brighte/ evid5ce & reality3 9 ! mod]n1 ignorance1 & :atev] belongs to obscurity & %ad[4 ,! =m] fills ! s\l ) intelligible light1 breaks h] le?>gic fett]s1 & elevates h] to ! pr9ciple ( ?+s2 ! latt] cl\ds ! intellectual eye ( ! s\l1 by increas+ h] oblivion1 /r5g!ns h] corporeal b&s1 & hurries h] d[nw>ds into ! d>k labyr9?s ( matt]4 ,nor is x wond]ful !re %\ld be s great a diff]5ce betwe5 ! two sy/ems1 & s mu* 9 fav\r ( ! anci5ts1 if we consid] ! great advantages ^! anci5ts possess$ ov] ! mod]ns 9 e ?+ :i* contributes to ! advancem5t ( philosophy4 ,=1 9 ! fir/ place1 !y liv$ 9 an age :5 ab/ract inve/igations 7 9 ! greate/ reque/1 & ! pr(essors ( su* pursuits 9 ! highe/ e/imation4 ,besides ?is1 !y unit$ ! mo/ exalt$ abilities ) ! mo/ unwe>i$ att5tion & ob/9ate p]sev]ance2 !y devot$ _! :ole lives to ! se>* ( tru?2 & rel9qui%$ e ?+ :i* might be an ob/acle to its acquisition4 ,we may add1 likewise1 ! advantages ( a language extremely philosophical2 & a fre$om f ! toil ( le>n+ any tongue b _! own4 ,n[ ! rev]se ( all ?is is ! portion ( ! mod]ns3 = 9 ! pres5t age1 ab/ract speculations are ridicul$2 & its pr(essors despis$4 ,! pursuit ( tru? is consid]$ z p]fectly consi/5t ) ord9>y avocations1 & is r prosecut$ z a relief f ! toils ( bus9ess ?an z a ?+ desirable = its own sake1 & ( ! greate/ dignity & wor?4 ,h5ce1 a few ye>s desultory application at a college1 :]e language is "o ( ! fir/ objects ( att5tion1 qualifies a mod]n = philosophy1 raises him above ,py?agoras & ,plato1 & p]suades him1 ) presumptu\s confid5ce1 to ent] ! li/s aga9/ ^! v5]able h]oes4 ,& la/ly1 all mod]n languages are b>b>\s ) respect to ! ,greek2 fall+ f> %ort ( its h>mony & en]gy1 its copi\sness & propriety4 ,if su* !n be ! true /ate ( ! case1 :at judgm5t mu/ we =m ( m5 :o1 ) all ^! disadvantages1 philosophiz$ )\t ! assi/ance ( ! anci5ts1 despis+ _! works1 & be+ ignorant ( _! cont5ts8 ,%all we call x prud5ce or presumption1 wisdom or folly8 ,tru? w c]ta9ly pron\nce ! latt]2 & ! g5]al voice ( po/]ity w confirm h] decision4 ,!re are two egregi\s in/ances 9 our own c\ntry ( ?is d>+ presumption2 ,i mean ,bacon & ,locke4 ,! =m] ( ^! is celebrat$ = hav+ de/roy$ ! j>gon ( ! s*oolm51 & br\ght exp]im5tal enquiries into repute2 & = attempt+ to inve/igate causes ?r\gh ! imm5sity ( p>ticul> effects4 ,h5ce1 he fondly expect$1 by exp]im5t pil$ on exp]im5t1 to rea* ! pr9ciple ( ! univ]se2 n consid]+ t 8 und]tak+ 0 z ridicul\s z t ( ! giants ( old1 :o attempt$ to invade ! heav5s1 by plac+ ,ossa ^u ,pelion1 & ,olympus ^u ,ossa2 & ignorant t ,heav5 /ill1 ) laught]1 ! va9 toil surveys1 ,& buries madm5 9 ! heaps !y raise4 ,! latt] ( ^!1 ,mr4 ,locke1 is applaud$ = hav+1 )\t assi/ance f ! anci5ts1 expla9$ ! nature1 & exhibit$ ! g5u9e !ory ( human und]/&+4 ,b t ?is applause is false1 ! prec$+ comp>ison betwe5 8 & ! anci5t philosophy1 may ev9ce2 & ! v>iety ( o!r self-taught sy/ems :i*1 l nocturnal meteors1 blaze = a :ile1 & !n vani% 9 obscurity1 abundantly confirms4 ,_h ^! m51 inde$1 :5 !y ju/ly d]id$ ! b>b>\s writ+s ( ! s*oolm51 explor$ ! works ( antiquity1 p5etrat$ ! wisdom !y conta91 & enri*$ _! native language ) its illu/ration1 !y _h d\btless be5 celebrat$ by ! late/ po/]ity3 b1 desir\s ( becom+ ma/]s 9 philosophy by intuition1 !y disda9$ ! in/ruction ( ! anci5ts1 & va9ly attempt$ to so> on artificial w+s to ! v summit ( sci5ce4 ,!y are1 h[ev]1 de/9$1 l ,ic>us1 to a precipitate fall2 = ! influ5ce ( "t1 :i* is cont9ually dissolv+ ! cem5t ( _! plumes1 is likewise cont9ually weak5+ _! =ce1 & w at la/ effect _! f9al sep>ation4 ,& ?us mu* conc]n+ ! doctr9e ( ideas1 & numb]s1 accord+ to ,py?agoras & ,plato4 ,,section ,,ii4#bb ,b let u n[ consid] ! prop]ties ( ! demon/rative syllogism1 & endeav\r to unravel its intricate web2 appo9t+ ,ari/otle = our guide 9 ?is ardu\s inve/igation4 ,= an enquiry ( ?is k9d is naturally connect$ ) ! doctr9e ( ideas1 z x enables u to ga9 a glimpse ( ! univ]sals p>ticipat$ 9 ma!matical =ms1 & to rise to ! pr9ciples ( sci5ce4 ,x br+s u acqua9t$ ) ! laws :i* b9d demon/ration2 & tea*es u t objects ( intellect are alone ! objects ( sci5ce1 & ! s\rces ( tru?4 ,previ\s to ! acquisition ( all le>n+ & ratioc9ative discipl9e1 x is necess>y we %\ld possess c]ta9 natural pr9ciples ( k1 z subs]vi5t to our future progress & atta9m5ts4 ,?us1 9 e sci5ce !re are "s ?+s :i* require an imm$iate ass5t z soon z propos$2 ^: c]ta9ty is too evid5t & illu/ri\s to /& 9 ne$ ( any demon/rative pro( d$uc$ f t p>ticul> sci5ce :i*1 l /ately pill>s1 !y equally support & adorn4 ,h5ce we are in=m$ by ! geometrician1 t a po9t is t :i* is de/itute ( all p>ts :atev]2 b we mu/ previ\sly und]/& ! mean+ ( ! ^w "p4 ,?us ! ari?metician def9es an odd numb]1 t :i* is divid$ accord+ to unequal p>ts2 b x is necess>y we %\ld antec$5tly "k ! mean+ ( ! ^w unequal4 ,?us1 too1 art z well z sci5ce op]ates by antec$5t k2 & h5ce ! ar*itect1 ! /atu>y1 & ! %ipwright1 le>n ! names & ! use ( _! respective implem5ts1 previ\s to ! ex]cise ( ! mat]ials !mselves4 ,?is is p>ticul>ly evid5t 9 ! discursive arts ( rhetoric & logic2 ?us ! logician reasons by syllogism1 ! rhetorician by induction1 & ! sophi/ by digressions & examples2 :ile ea* proce$s 9 an ord]ly progression f pr9ciples simple & evid5t1 to ! mo/ remote & complicat$ conclusions4 #b4 ,! antec$5t k ( ?+s may be divid$ into two p>ts3 ! "o a k ( _! exi/5ce1 or t !y exi/2 ! o!r a k ( ! t]ms expressive ( _! exi/5ce4 ,?us1 previ\s to ! enquiry :y iron is attract$ by ! magnet1 x is necess>y we %\ld le>n ! reality ( ?is attraction1 & ! g5]al mode ( its op]ation3 ?us too1 9 an enquiry conc]n+ ! nature ( motion & "t1 we mu/ be previ\sly conv9c$ ( _! exi/5ce 9 ! nature ( ?+s4 ,! second division ( antec$5t k takes place 9 subjects ^: v exi/5ce admits ( a dispute3 ?us previ\s to a solution ( ! que/ions1 ,:e!r !re are any gods or n8 ,:e!r !re is a provid5ce or n8 & ! l1 x is necess>y we %\ld fir/ und]/& ! mean+ ( ! t]ms2 s9ce we 9 va9 inve/igate ! nature ( any ?+ :ile we are ignorant ( ! mean+ ( its "n2 al?\gh1 on ! contr>y1 we may h a p]fect conception ( ! mean+ ( "s words1 & yet be totally ignorant :e!r ! ?+s !y express h a real1 or only an imag9>y exi/5ce4 ,?us1 ! mean+ ( ! ^w c5taur is well und]/ood by e "o2 b its exi/5ce is que/ion$ by mo/4 #c4 ,f h5ce x w easily appe>1 t no small diff]5ce subsi/s betwe5 le>n+ & k4 ,he :o is ab\t to und]/& ! tru? ( any proposition1 may be said to possess a previ\s conception ( its tru?2 :ile1 on ! contr>y1 x may happ5 t he :o is 9 ! capacity ( a le>n]1 has no antec$5t k ( ! sci5ce he is ab\t to le>n4 ,?us we atta9 to ! di/9ct k ( a ?+ :i* we =m]ly knew 9 a g5]al way2 & frequ5tly1 ?+s ( :i* we 7 ignorant are le>n$ & kn[n 9 ! same in/ant4 ,( ?is k9d are ! ?+s conta9$ "u "s g5]al idea1 ( :i* we possess a previ\s k3 ?us1 he :o already kn[s t ! ?ree int]ior angles ( e triangle are equal to two "r1 & is z yet ignorant t "s p>ticul> figure del9eat$ on pap] is a triangle1 is no soon] conv9c$ f inspection ( its be+ a triangle1 ?an he imm$iately le>ns & kn[s3 he le>ns x is a triangle2 he kn[s ! equality ( its angles to two "r ones4 ,t x is n[ a triangle he bo? sees & le>ns2 b ! equality ( its angles he previ\sly knew 9 t g5]al & compreh5sive idea1 :i* embraces e p>ticul> triangle4 ,inde$1 a def9ite k ( ?is triangle requires two conditions3 ! "o1 t x is a triangle2 & ! o!r1 t x has angles equal to two "r4 ,! fir/ we receive f inspection2 ! second is ! result ( a syllogi/ic process2 an op]ation too ref9$ = ! en]gies ( s5se1 & alone ! prov9ce ( intellect & demon/ration4 ,b demon/ration )\t ! k ( t :i* is univ]sal1 _c subsi/2 & s9ce ! proposition is univ]sal1 t 9 e triangle ! angles are equal to two "r2 z soon z any figure is ackn[l$g$ to be a triangle1 x mu/ necess>ily possess ?is g5]al prop]ty4 ,h5ce we inf]1 t ( ! triangle del9eat$ on pap]1 & conceal$1 we are p>tly ignorant ( ?is g5]al prop]ty1 ! equality ( its angles 7because we are ignorant ( its exi/5ce72 & we p>tly und]/& x z includ$ 9 t univ]sal idea we previ\sly possess$4 ,h5ce too1 x is evid5t t actual sci5ce arises f a m$ium betwe5 absolute ignorance & p]fect k2 & t he :o possesses ! pr9ciples ( demon/ration1 possesses 9 capacity ! conclusions also1 h[ev] complicat$ & remote2 & t by an evocation ( ^! pr9ciples f dormant p[] into en]gy1 we advance f g5]al & ab/ract$ k to t :i* is s5sible & p>ticul>4 #d4 ,two acceptations ( k may be admitt$2 ! "o common & )\t any re/riction2 ! o!r limit$ & peculi>4 ,s9ce all k1 :e!r aris+ f accid5ts1 or support$ by necess>y pr9ciples1 is call$ sci5ce4 ,k1 prop]ly s call$1 arises f a possession ( t cause f :i* a ?+ d]ives its exi/5ce1 & by :i* we inf] ! necessity ( its exi/5ce2 & ?is con/itutes simple & absolute sci5ce4 ,?us too1 ! def9itions ( ^? g5]al conceptions & suppositions1 :i* f _! prim>y nature are incapable ( demon/ration1 are call$ sci5ce4 ,b ! sci5ce :i* treats ( ! me?od ( arriv+ at k1 is call$ demon/ration2 = e demon/ration is a syllogism produc+ sci5ce4 ,h5ce1 if 9 e syllogism x is necess>y t ! propositions %\ld be ! cause ( ! conclusion2 & to "k any ?+ prop]ly1 a k ( its cause is requisite2 9 ! propositions ( demon/ration1 bo? ^! conditions are requir$3 t !y %\ld be effective ( ! conclusion2 & ! causes ( ! ?+ demon/rat$4 ,?us1 f ! ru9s ( a /ately edifice1 we may ju/ly inf]1 t ! build+ 0 beautiful :5 entire2 & f ! smoke we may collect ! exi/5ce ( ! fire1 ?\gh conceal$3 b ! ru9s ( ! edifice are n ! cause ( its beauty2 nor does fire orig9ate f smoke1 b1 on ! contr>y1 smoke is ! natural result ( fire3 ! inf]5ce1 !re=e1 is 9 nei!r case a demon/rative "o4 ,aga91 s9ce e cause is bo? prior to1 & m excell5t ?an its effect1 x is necess>y t ! propositions %\ld be m peculi>1 prim>y1 & excell5t ?an ! conclusions4 ,& because we !n "k a ?+ prop]ly :5 we believe x to h a necess>y exi/5ce1 h5ce x is requisite t ! propositions %\ld be true2 = if false1 a false conclusion may ensue1 su* z1 t ! diamet] ( a squ>e is comm5surable ) its side4 ,b if e sci5ce arises f antec$5t k1 demon/ration mu/ be f\nd$ on some?+ previ\s2 & on ?is acc\nt x is requisite t ! propositions %\ld be m kn[n ?an ! conclusions4 ,! necess>y prop]ties1 !n1 ( all demon/rative propositions1 are ^!2 t !y exi/ z causes1 are prim>y1 m excell5t1 peculi>1 true1 & kn[n1 ?an ! conclusions4 ,inde$1 e demon/ration n only consi/s ( pr9ciples prior to o!rs1 b ( su* z are em95tly fir/2 = if ! assum$ propositions may be demon/rat$ by o!r assumptions1 su* propositions may1 inde$1 appe> prior to ! conclusions1 b are by no means entitl$ to ! appellation ( fir/4 ,b o!rs1 on ! contr>y1 :i* require no demon/ration1 b are ( !mselves probable or manife/1 are des]v$ly e/eem$ ! fir/1 ! true/1 & ! be/4 ,su* indemon/rable tru?s 7 call$ by ! anci5ts1 axioms1 f _! maje/y & au?ority2 z ! assumptions :i* con/itute ! be/ syllogisms d]ive all _! =ce & efficacy f ^!4 ,& on ?is acc\nt1 above all o!rs1 !y m]it ! title ( ! pr9ciples ( demon/ration4 ,b "h x is wor? obs]v+1 t ^! prim>y propositions are n ! fir/ 9 ! ord] ( our conceptions2 b fir/ to nature1 or 9 ! nature ( ?+s4 ,to u1 t :i* is fir/ is p>ticul>1 & subject to s5sible inspection2 to nature1 t :i* is univ]sal1 & f> remote f ! appreh5sion ( s5se4 ,demon/ration does n submit itself to ! measure ( our +5uity1 b1 ) inv>iable rectitude1 t5ds to tru? z its ultimate aim2 & )\t /opp+ to consid] :at our limit$ p[]s c atta91 x alone explores & traces out ! nature ( a ?+1 ?\gh to u unp]ceiv$ & unkn[n4 ,?is demon/rative syllogism diff]s n a little f o!rs1 by ! above prop]ty2 ! re/ c z well educe a true conclusion f false premises1 :i* is frequ5t among ! rhetoricians1 z t :i* is prior f t :i* is po/]ior2 su* z1 ,is e syllogism d]iv$ f conjecture8 ,) respect to ! re/1 z we h already confess$1 !y may be =m$ f pr9ciples t are true1 b n f su* z are prop] & peculi>2 z if a physician %\ld endeav\r to prove an orbicul> w\nd ! mo/ difficult to coalesce & heal1 because its figure is ( all o!rs1 ! mo/ capaci\s2 s9ce ! demon/ration ( ?is is n ! prov9ce ( ! physician1 b ( ! geometrician alone4 #e4 ,t proposition is call$ imm$iate1 :i* has none sup]ior to itself1 & :i* no demon/ration :atev] c confirm3 su* z ^! are held toge!r by ! embraces ( univ]sals4 ,!re are "s1 inde$1 unit$ f t :i* is s5sible & p>ticul>3 ?us1 t ! g>m5t is :ite1 is an imm$iate proposition1 b n ( t k9d ^: pr9ciples require to be demon/rative ones2 ! cause ( :i* we %all h]eaft] inve/igate4 ,( imm$iate propositions subs]vi5t to ! purposes ( demon/ration1 "s are ( su* a sup]ior nature1 t all m5 possess a k ( !m )\t any previ\s in/ruction2 & ^! are call$ axioms1 or g5]al notions2 = )\t ^! all k & enquiry is va94 ,ano!r species ( imm$iate propositions is position2 incapable ( be+ /r5g!n$ by demon/ration1 yet n necess>ily =ekn[n by ! le>n]1 b receiv$ f ! tea*]4 ,) respect to ! g5us ( position1 "o ( its species is def9ition1 & ano!r hypo!sis4 ,def9ition is an oration1 9 :i* we nei!r speak ( ! exi/5ce1 nor non-exi/5ce ( a ?+2 b alone det]m9e its nature & ess5ce4 ,x is common to e hypo!sis1 n to be d]iv$ f nature1 b to be ! entire result ( ! art ( ! preceptor4 ,x likewise always affirms ! exi/5ce or non-exi/5ce ( its subject3 su* z1 t motion is1 & t f no?+ no?+ is produc$4 ,^? :i* are n s p]spicu\s are call$ po/ulates1 or petitions2 z t a circle may be describ$ f any c5tre1 & ) any radius2 & su* z ^! are prop]ly hypo!ses & po/ulates4 #f4 ,we h n[ se5 ! privilege assign$ to ! pr9ciples ( demon/ration3--:e!r or no our decision has be5 j1 ! ensu+ consid]ations w ev9ce4 ,we said t ! assumptions 9 demon/ration 7 m kn[n ?an ! conclusions1--n inde$ )\t reason1 s9ce ?r\gh ^! our k & belief ( ! conclusion arises4 ,= univ]sally1 t quality :i* is attribut$ to _m diff]5t ?+s s z to be assign$ to "o ?r\gh ! m$ium ( ano!r1 ab\nds mo/ 9 t m$ium by :i* x is transmitt$ to ! re/4 ,?us ! sun1 ?r\gh ! m$ium ( ! moon1 illum9ates ! e>? by night2 ?us ! fa!r loves ! preceptor ?r\gh ! m$ium ( 8 *ild4 ,& 9 ! fir/ in/ance ! moon is m lucid ?an any object x enlight5s3 9 ! second1 ! *ild possesses m ( ! fa!r's reg>d ?an 8 preceptor4 ,if !n we ass5t to ! conclusions ?r\gh our belief ( ! pr9ciples alone1 x is necess>y t ! pr9ciples %\ld be m kn[n1 & inh]it a great] degree ( our ass5t4 ,h5ce1 if x be true t ! pr9ciples are m kn[n ?an ! conclusions1 x foll[s1 t ei!r our k ( !m is d]iv$ f demon/ration1 or t x is sup]ior to any demon/rative pro(2 & aft] ?is mann] we mu/ conceive ( ^? g5]al self-evid5t notions :i*1 on acc\nt ( _! indemon/rable c]ta9ty1 are des]v$ly plac$ at ! top ( all human sci5ce4 ,^! propositions n only possess great] cr$ibility ?an _! conclusions1 !y likewise inh]it ?is prop]ty z an accession to _! dignity & importance2 t no contr>y propositions des]ve great] belief2 = if y give no m ass5t to any pr9ciple ?an to its contr>y1 nei!r c y give m cr$it to ! conclusion d$uc$ f t pr9ciple ?an to its opposite4 ,7 ?is ! case1 ! doctr9e ( ^! propositions w\ld imm$iately lose its inv>iable c]ta9ty4 #g4 ,!re are1 inde$1 "s :o1 f errone\sly apply+ :at we h rightly det]m9$1 endeav\r to take away ! possibility ( demon/ration4 ,f ! prec$+ doctr9e x appe>s t ! pr9ciples are m aptly kn[n ?an ! conclusions4 ,?is is n evid5t to "s1 :o ?9k no?+ c be kn[n by u )\t a demon/rative process2 & consequ5tly believe t ! mo/ simple pr9ciples mu/ d]ive all _! cr$it f ! light ( demon/ration4 ,b if x be necess>y t all assumptions %\ld be demon/rat$ by o!rs1 & ^! aga9 by o!rs2 ei!r ! enquiry mu/ be cont9u$ to inf9ity1 7b inf9ity c nev] be absolv$71 or if1 we>i$ by ! imm5se process1 y at l5g? /op1 y mu/ d\btless leave ^? propositions unkn[n1 ^: demon/ration 0 decl9$ ?r\gh ! fatigue ( inve/igation4 ,b h[ c sci5ce be d]iv$ f unkn[n pr9ciples8 ,= he :o is ignorant ( ! pr9ciples1 _c und]/& ! conclusions :i* fl[ f ^! z _! prop] s\rce1 unless f an hypo!sis or supposition ( _! reality4 ,?is argum5t ( ! sophi/s is1 inde$1 s f> true1 t he :o does n und]/& t :i* is fir/ 9 ! ord] ( demon/ration1 mu/ rema9 ignorant ( t :i* is la/3--,b 9 ?is x fails1 t all k is demon/rative2 s9ce ?is is an ass]tion no less ridicul\s ?an to ma9ta9 t no?+ c be kn[n4 ,= z x is manife/ t "s ?+s d]ive _! cr$it & support f o!rs1 x is equally obvi\s t _m1 by _! intr9sic excell5ce1 possess indubitable c]ta9ty & tru?2 & comm& our imm$iate ass5t z soon z propos$4 ,!y inh]it1 inde$1 a high] degree ( evid5ce ?an ^? we ass5t to by ! confirmation ( o!rs2 & ^! are ! fir/ pr9ciples ( demon/ration3 propositions indisputable1 imm$iate1 & p]spicu\s by t native lu/re !y always possess4 ,by means ( ^!1 we advance f proposition to proposition1 & f syllogism to syllogism1 till we arrive at ! mo/ complicat$ & important conclusions4 ,o!rs1 will+ to decl9e ?is inf9ite progression1 def5d ! necessity ( a circul> or reciprocal demon/ration4 ,b ?is is no?+ m ?an to build error ^u error1 9 ord] to atta9 ! tru?2 an attempt no less ridicul\s ?an t ( ! giants ( old4 ,= s9ce1 z we %all h]eaft] accurately prove1 demon/ration ought to consi/ f t :i* is fir/1 & mo/ kn[n2 & s9ce x is impossible t ! same ?+ %\ld be to itself bo? prior & po/]ior3 h5ce we inf] ! absurdity ( circul> demon/ration2 or ^? syllogisms 9 :i* ! conclusions are alt]nately sub/itut$ z pr9ciples1 & ! pr9ciples z conclusions4 ,x may1 inde$1 happ51 t ! same ?+ may be bo? prior & po/]ior to ! same2 b n at "o & ! same "t1 nor accord+ to ! same mode ( exi/5ce4 ,?us1 :at is prior 9 ! ord] ( our conceptions1 is po/]ior 9 ! ord] ( nature2 & :at is fir/ 9 ! arrangem5t ( ?+s1 is la/ 9 ! progressions ( human und]/&+4 ,b demon/ration always desires t fir/ :i* is prior 9 ! ord] & con/itution ( nature4 ,b ! folly ( su* a me?od w m pla9ly appe> f consid]+ its result3 let u suppose e a is b1 & e b is c2 h5ce we ju/ly inf]1 t e a is c4 ,9 l mann]1 if we prove t e a is b1 & by a circul> demon/ration1 t e b is a1 ! consequ5ce f ! prec$+ is no o!r ?an t e a is a2 & ?us ! conclusion t]m9ates 9 t f :i* x fir/ began2 a d$uction equally useless & ridicul\s4 ,h[ev]1 admitt+ t1 9 ! fir/ figure1 circul> demon/ration may be 9 "s cases adopt$1 yet ?is c b seldom happ5 f ! paucity ( reciprocal t]ms4 ,b t reciprocal t]ms are v few1 is pla9 f h5ce3 let any species be assum$1 z man2 :atev] is ! pr$icate ( man1 is ei!r con/itutive ( 8 ess5ce1 or expressive ( "s accid5t belong+ to 8 nature4 ,! sup]ior g5]a & diff]5ces compose 8 ess5ce1 among :i* no equal pr$icate c be assign$ reciprocable ) man1 except ! ultimate diff]5ces :i* _c be o!rwise ?an "o1 i4 e41 risibility1 :i* mutually reciprocates ) its subject2 s9ce e man is risible1 & :atev] is risible is man4 ,( accid5ts "s are common1 o!rs peculi>2 & ! common are f> m 9 numb] ?an ! peculi>2 consequ5tly ! pr$icates :i* reciprocate ) man1 are mu* few] ?an ^? :i* d n reciprocate4 #h4 ,x is n[ necess>y to enum]ate ! que/ions p]ta9+ to demon/ration2 & = ?is purpose1 we %all beg9 ) propositions1 s9ce f ^!1 syllogisms are =m$2 & s9ce e proposition consi/s ( a subject & pr$icate1 ! modes ( pr$ication mu/ be consid]$1 & ^! are ?ree :i* ,i call total1 ess5tial1 & univ]sal2 a total pr$ication takes place :5 t :i* is affirm$ or d5i$ ( "o individual is affirm$ or d5i$ ( e individual compreh5d$ "u ! same common species4 ,?us1 animal is pr$icat$ ( e man1 & x has ?is f>!r prop]ty besides1 t ( :atev] subject x is true to affirm man1 x is at ! same "t true to affirm animal4 ,^? ?+s are said to be ess5tially pr$icat$2 fir/1 :5 ! pr$icate is n only total1 b con/itutes ! ess5ce ( ! subject2 in/ances ( ?is k9d are1 animal ( man2 tree ( ! planta92 a l9e ( a triangle2 = a triangle is t :i* is conta9$ "u ?ree "r-l9es4 ,b "h we mu/ obs]ve1 t n e total pr$icate is an ess5tial "o2 ?us1 :it5ess is pr$icat$ ( e swan1 because x is inh]5t 9 e swan1 & at e in/ant ( "t2 b because :it5ess does n con/itute ! ess5ce ( a swan1 x is n ess5tially pr$icat$2 & ?is1 fir/1 is "o ( ! modes ( ess5tial pr$ication ( ! greate/ importance 9 demon/ration4 ,! second mode is ( accid5ts1 9 ! def9ition ( :i* _! common subject is appli$3 ?us1 a l9e is ess5tially inh]5t 9 rectitude1 because 9 its geometrical def9ition1 a l9e is adopt$2 = rectitude is no o!r ?an a measure1 equally ext5d$ betwe5 ! po9ts ( a l9e4 ,9 ! same mann]1 imp>ity is conta9$ 9 numb]2 = :at is t :i* is odd1 b a numb] divid$ into unequal p>ts8 ,?us1 virtues are resid5t 9 ! s\l1 because1 9 _! def9ition1 ei!r "s "p ( ! s\l1 or "s "o ( its p[]s1 is always appli$4 ,! ?ird mode ( ess5tial pr$icates p]ta9s to accid5ts :i* are insep>ably conta9$ 9 "s p>ticul> subject1 s z to exclude a prior exi/5ce 9 any o!r subject2 su* z col\r 9 sup]ficies4 ,! f\r? mode is ( ?+s nei!r conta9$ 9 ano!r1 nor pr$icat$ ( o!rs2 & su* are all individuals1 z ,callias1 ,socrates1 ,plato4 ,causes are likewise said to exi/ sub/antially1 :i* op]ate nei!r f accid5t nor =tune4 ,?us1 digg+ up ! gr\nd = ! purposes ( agriculture1 may be ! cause ( discov]+ a treasure1 b x is only an accid5tal "o4 ,b ! dea? ( ,socrates1 9 despite ( vigilance1 is n ! result ( a =tuit\s cause1 b ( an ess5tial "o1 viz4 ! op]ation ( poison4 #i4 ,^! po/]ior significations ( ess5tial pr$icates are add$ m = ! sake ( ornam5t ?an use2 b ! two =m] h a necess>y exi/5ce1 s9ce !y _c b exi/ 9 ! def9ition ( names :i* pr$icate ! ess5ce ( a ?+1 & 9 subjects :i* are s entirely ! support ( accid5ts1 t !y are always appli$ 9 _! def9ition4 ,b x is a d\bt ) "s1 :e!r ^? accid5ts are necess>y1 :i* _c be def9$ indep5d5t ( _! common subject8 ,to ?is we answ]1 t no su* accid5t c1 f its nature1 be conta9$ 9 e individual ( any species2 = curvature is n conta9$ 9 e l9e2 nor imp>ity 9 e numb]2 f :5ce we inf]1 t nei!r is curvature necess>ily exi/5t 9 a l9e1 nor p>ity 9 numb]4 ,! tru? ( ?is is evid5t f consid]+ ^! accid5ts ab/ract$ f _! subjects2 = !n we %all p]ceive t a l9e may exi/ )\t curvature1 & numb] )\t imp>ity4 ,aga91 ,i call t an univ]sal pr$icate1 :i* is pr$icat$ ( a subject totally & ess5tially1 & consid]$ z prim>ily & insep>ably inh]5t 9 t subject3 = x does n foll[ t a pr$icate1 :i* is total1 %\ld be imm$iately univ]sal2 = :it5ess is affirm$ ( e swan1 & blackness ( e cr[1 yet nei!r univ]sally4 ,9 l mann]1 a sub/antial pr$icate is n consequ5tly an univ]sal "o2 = ! ?ird mode ( ess5tial pr$icates1 & ! two foll[+ 7in/anc$ be=e7 _c be univ]sal4 ,?us1 col\r1 al?\gh inh]5t 9 sup]ficies ess5tially1 is n inh]5t 9 e sup]ficies1 & consequ5tly n univ]sally4 ,?us aga91 ,socrates1 ,callias1 & ,plato1 ?\gh !y exi/ ess5tially1 are n univ]sals1 b p>ticul>s2 & ?us1 la/ly1 ! dr9k+ ( poison 0 an ess5tial cause ( ! dea? ( ,socrates1 b n an univ]sal "o1 because ,socrates might h di$ by o!r means ?an poison4 ,if !n1 we wi% to r5d] an accurate def9ition ( an univ]sal pr$icate1 we mu/ n only say x is total & ess5tial1 b t x is prim>ily pres5t to its subject & no o!r4 ,?us1 ! possession ( angles equal to two "r1 prim>ily belongs to a triangle2 = ?is ass]tion is ess5tially pr$icat$ ( triangle1 & is inh]5t 9 e triangle4 ,?is prop]ty1 !re=e1 is n univ]sally 9 figure1 because x is n ! prop]ty ( e figure1 n ( a squ>e1 = in/ance2 nor z univ]sal 9 a scal5e triangle3 = al?\gh x is conta9$ 9 e scal5e1 & 9 e equilat]al1 & isosceles triangle1 yet x is n prim>ily conta9$ 9 !m1 b 9 triangle itself2 because ^! sev]al figures inh]it ?is prop]ty1 n f ! p>ticul> species to :i* !y belong1 b f ! common g5us triangle4 ,& ?us mu* conc]n+ total1 ess5tial1 & univ]sal pr$icates4 #aj4 ,conc]n+ t :i* is univ]sal1 we are frequ5tly liable to err2 (t5 f a belief t our demon/ration is univ]sal1 :5 x is only p>ticul>2 & frequ5tly f suppos+ x p>ticul> :5 x is1 on ! contr>y1 univ]sal4 ,!re are ?ree causes ( ?is mi/ake2 ! fir/1 :5 we demon/rate any p>ticul> prop]ty ( t :i* is s+ul> & individual1 z ! sun1 ! e>?1 or ! _w4 ,= s9ce !re is b "o sun1 "o e>?1 & "o _w1 :5 we demon/rate t ! orb ( ! e>? possesses ! middle place1 or t ! heav5s revolve1 we d n !n appe> to demon/rate t :i* is univ]sal4 ,to ?is we answ]3 :5 we demon/rate an eclipse ( ! sun to arise f ! opposition ( ! moon1 we d n consid] ! sun z "o p>ticul> lum9>y1 b we d$uce ?is consequ5ce z if _m o!r suns exi/$ besides ! pres5t4 ,j z if !re 7 b "o species ( triangles exi/$2 = in/ance1 ! isosceles2 ! equality ( its angles at ! base w\ld n be consid]$ 9 ! demon/ration ( ! equality ( all its angles to two "r ones3 b its triangul>ity w\ld be ess5tial1 suppos+ e species ( triangles b ! isosceles ext9ct1 & no o!r ! subject ( ?is affection4 ,s :5 we prove t ! sun is great] ?an ! e>?1 our pro( does n arise f consid]+ x z ?is p>ticul> sun alone1 b z sun 9 g5]al2 & by apply+ our reason+ to e sun1 if ?\s&s besides ! pres5t %\ld enlight5 ! _w4 ,?is w appe> /ill m evid5t1 if we consid] t su* conclusions mu/ be univ]sal1 z !y are ! result ( an induction ( p>ticul>s3 ?us1 he :o demon/rates t an eclipse ( ! sun arises f ! opposition ( ! moon betwe5 ! sun & e>?1 mu/ previ\sly collect1 by induction1 t :5 any lum9\s body is plac$ 9 a "r-l9e ) any two o!rs opaque1 ! lucid body %all be prev5t$1 9 a great] or less degree1 f enlight5+ ! la/ ( ^! bodies1 by ! int]v5tion ( ! second2 & by ext5d+ ?is reason+ to ! sun & e>?1 ! syllogism w run ?us3 ,e lucid body plac$ 9 a "r-l9e ) two o!rs opaque1 w be eclips$ 9 respect ( ! la/ by ! int]v5tion ( ! second2 ,! sun1 or e sun1 is a lum9\s body ) ^! conditions2 ,& consequ5tly ! sun1 & s e sun1 w be eclips$ to ! e>? by ! opposition ( ! moon4 ,h5ce1 9 cases ( ?is k9d1 we mu/ ev] rememb]1 t we demon/rate no prop]ty ( !m z s+ul>s1 b z t univ]sal conceiv$ by ! ab/raction ( ! m9d4 ,ano!r cause ( deception arises1 :5 _m diff]5t species agree 9 "o ratio or analogy1 yet t 9 :i* !y agree is nameless4 ,?us numb]1 magnitude1 & "t1 diff] by ! div]sities ( species2 b agree 9 ?is1 t z any f\r comp>able numb]s correspond 9 _! proportions to ea* o!r1 s t z ! fir/ is to ! second1 s is ! ?ird to ! f\r?2 or alt]nately1 z ! fir/ to ! ?ird2 s is ! second to ! f\r?3 9 a simil> mann]1 f\r magnitudes1 or f\r times1 accord 9 _! mutual analogies & proportions4 ,h5ce1 alt]nate proportion may be attribut$ to l9es z !y are l9es1 to numb]s z !y are numb]s1 & aft]w>ds to times & to bodies1 z ! demon/ration ( ^! is usually sep>ate & s+ul>2 :5 ! same prop]ty might be prov$ ( all ^! by "o compreh5sive demon/ration1 if ! common "n ( _! g5us c\ld be obta9$3 b s9ce ?is is want+1 & ! species are diff]5t1 we are oblig$ to consid] !m sep>ately & ap>t2 & z we are n[ speak+ ( t univ]sal demon/ration :i* is prop]ly "o1 z aris+ f "o fir/ subject2 h5ce none ( ^! obta9 an univ]sal demon/ration1 because ?is affection ( alt]nate proportion is n re/rict$ to numb]s or l9es1 consid]$ 9 !mselves1 b to t common some?+ :i* is suppos$ to embrace all ^!1 & is de/itute ( a prop] "n4 ,?us too we may happ5 to be deceiv$1 %\ld we attempt to prove ! equality ( ?ree angles to two "r1 sep>ately1 ( a scal5e1 an isosceles1 & an equilat]al triangle1 only ) ?is diff]5ce1 t 9 ! latt] case ! deception is n s easy z 9 ! =m]2 s9ce "h ! "n triangle1 expressive ( _! common g5us1 is assign$4 ,a ?ird cause ( error arises f believ+ t to demon/rate any prop]ty inh]5t aft] "s p>ticul> mann] 9 ! :ole ( a ?+1 is to demon/rate t prop]ty univ]sally inh]5t4 ,?us1 geometry proves#bc t if a "r-l9e fall+ ^u two "r-l9es makes ! outw>d angle ) ! "o l9e a "r-angle1 & ! inw>d & opposite angle ) ! o!r a "r "o1 ^? two "r-l9es %all be p>allel1 or nev] meet1 ?\gh inf9itely ext5d$4 ,?is prop]ty agrees to all l9es :i* make "r-angles3 b !y are n prim>ily equidi/ant on ?is acc\nt1 s9ce1 if !y d n ea* make a "r-angle1 b ! two conjo9tly are equal to two "r1 !y may /ill be prov$ equidi/ant4 ,?is latt] demon/ration1 !n1 is prim>ily & univ]sally conceiv$2 ! o!r1 :i* always supposes ! opposite angles "r ones1 does n conclude univ]sally2 ?\gh x concludes totally ( all l9es ) su* conditions3 ! "o may be said to conclude ( a great] all2 ! o!r ( a less]4 ,x is ?is great] all :i* ! m9d embraces :5 x ass5ts to any self-evid5t tru?2 or to any ( ! propositions ( ,euclid4 ,b by :at me?od may we discov] :e!r our demon/ration is ( ?is great] or less] all8 ,we answ]1 t g5]al affection :i* con/itutes univ]sal demon/ration is always pres5t to t subject1 :i* :5 tak5 away1 ! pr$icate is imm$iately de/roy$1 because ! fir/ ( all its inh]5t prop]ties4 ,?us1 = in/ance1 "s p>ticul> s5sible triangle possesses ^! prop]ties3--x consi/s ( brass2 x is scal5e2 x is a triangle4 ,! qu]y is1 by :i* ( ^! we h j n[ enum]at$1 ?is affection ( possess+ angles equal to two "r is pr$icat$ ( ! triangle8 ,take away ! brass1 d y by ?is means de/roy ! equality ( its angles to two "r ones8 ,c]ta9ly n3--take away its scal5ity1 yet ?is g5]al affection rema9s3 la/ly1 take away its triangul>ity1 & !n y necess>ily de/roy ! pr$icate2 = no long] c ?is prop]ty rema91 if x ceases to be a triangle4 ,b p]haps "s may object f ?is reason+1 su* a g5]al affection ext5ds to figure1 sup]ficies1 & extremities1 s9ce1 if any ( ^! are tak5 away1 ! equality ( its angles to two "r c no long] rema94 ,x is true1 inde$1 t by a sep>ation ( figure1 sup]ficies1 & t]ms1 f a body1 y de/roy all ! modes & circum/ances ( its be+2 yet n because ^! are tak5 away1 b because ! triangle1 by ! sep>ation ( ^!1 is necess>ily de/roy$2 = if ! triangle c\ld /ill be pres]v$ )\t figure1 sup]ficies1 & t]ms1 ?\gh ^! 7 tak5 away x w\ld /ill reta9 angles equal to two "r2 b ?is is impossible4 ,& if all ^! rema91 & triangle is tak5 away1 ?is affection no long] rema9s4 ,h5ce ! possession ( ?is equality ( ?ree angles to two "r1 is prim>ily & univ]sally inh]5t 9 triangle1 s9ce x is n aboli%$ by ! abolition ( ! re/3----su* z to consi/ ( brass2 to be scal5e1 or ! l4 ,nei!r does x d]ive its be+ f ! exi/5ce ( ! re/ alone2 z figure1 sup]ficies1 t]ms2 s9ce x is n e figure :i* possesses ?is prop]ty1 z is evid5t 9 su* z are quadrangul>1 or multangul>4 ,& ?us x is pres]v$ by ! pres]vation ( triangle1 x is de/roy$ by its de/ruction4 #aa4 ,f ! pr9ciples already e/abli%$1 x is pla9 t demon/ration mu/ consi/ ( su* propositions z are univ]sal & necess>y4 ,t !y mu/ be univ]sal1 is evid5t f ! prec$+2 & t !y mu/ be necess>y1 we ga!r probably f h5ce2 t 9 ! subv]sion ( any demon/ration we use no o!r argum5ts ?an ! want ( necess>y exi/5ce 9 ! pr9ciples4 ,we collect _! necessity demon/ratively1 ?us2 he :o does n "k a ?+ by ! prop] cause ( its exi/5ce1 _c possess sci5ce ( t ?+2 b he :o collects a necess>y conclusion f a m$ium n necess>y1 does n "k x by ! prop] cause ( its exi/5ce1 & !re=e he has no prop] sci5ce conc]n+ x4 ,?us1 if ! necess>y conclusion c is a1 be demon/rat$ by ! m$ium ,b1 n necess>y2 su* a m$ium is n ! cause ( ! conclusion2 = s9ce ! m$ium does n exi/ necess>ily1 x may be suppos$ n to exi/2 & at ! "t :5 x no long] exi/s1 ! conclusion rema9s 9 full =ce2 because1 s9ce necess>y1 x is et]nal4 ,b an effect _c exi/ )\t a cause ( its exi/5ce2 & h5ce su* a m$ium c nev] be ! cause ( su* a conclusion4 ,aga91 s9ce 9 all sci5ce !re are ?ree ?+s1 ) ^: pres]vation ! duration ( k is connect$2 & ^! are1 fir/1 he :o possesses sci5ce2 secondly1 ! ?+ kn[n2 & ?irdly1 ! reason by :i* x is kn[n2 :ile ^! endure1 sci5ce c nev] be blott$ f ! m9d1 b on ! contr>y1 if sci5ce be ev] lo/1 x is necess>y "s ( ^! ?ree mu/ be de/roy$4 ,if !n y inf] t ! sci5ce ( a necess>y conclusion may be obta9$ f a m$ium n necess>y1 suppose ?is m$ium1 s9ce capable ( ext9ction1 to be de/roy$2 !n ! conclusion1 s9ce necess>y1 %all rema92 b w be no long] ! object ( k1 s9ce x is suppos$ to be kn[n by t m$ium :i* is n[ ext9ct4 ,h5ce1 sci5ce is lo/1 ?\gh none ( ! prec$+ ?ree are tak5 away2 b ?is is absurd1 & contr>y to ! pr9ciples we h j e/abli%$4 ,! ?+ kn[n rema9s2 = ! conclusion1 s9ce necess>y1 _c be de/roy$2--he :o kn[s /ill rema9s1 s9ce nei!r dead1 nor =getful ( ! conclusion3--la/ly1 ! demon/ration by :i* x 0 kn[n1 /ill survives 9 ! m9d2 & h5ce we collect1 t if sci5ce be no m aft] ! corruption ( ! m$ium1 nei!r 0 x sci5ce by t m$ium be=e its corruption2 = if sci5ce 0 ev] obta9$ ?r\gh su* a m$ium1 x c\ld n be lo/ :ile ^! ?ree are pres]v$4 ,! sci5ce1 !re=e1 ( a necess>y conclusion c nev] be obta9$ by a m$ium :i* is n necess>y4 #ab4 ,f h5ce x is manife/1 t demon/rations _c emigrate f "o g5us to ano!r2 or by su* a translation be comp>$ ) "o ano!r4 ,su* z1 = in/ance1 ! demon/rations ( geometry ) ^? ( ari?metic4 ,to be conv9c$ ( ?is1 we mu/ rise a little high] 9 our speculations1 & att5tively consid] ! prop]ties ( demon/ration3 "o ( :i* is1 t pr$icate :i* is always f\nd 9 ! conclusion1 & :i* affirms or d5ies ! exi/5ce ( its subject3 ano!r is1 ^? axioms or fir/ pr9ciples by ^: univ]sal embrace demon/ration is =tifi$2 & f ^: orig9al light x d]ives all its lu/re4 ,! ?ird is ! subject g5us1 & t nature ( :i* ! affections & ess5tial prop]ties are pr$icat$2 su* z magnitude & numb]4 ,9 ^! subjects we mu/ exam9e :51 & 9 :at mann] a transition 9 demon/rations f g5us to g5us may be all[$4 ,fir/1 x is evid5t1 t :5 ! g5]a are altoge!r sep>ate & discordant1 z 9 ari?metic & geometry1 !n ! demon/rations ( ! "o _c be ref]r$ to ! o!r4 ,?us1 x is impossible t ari?metical pro(s c ev] be accommodat$ ) propriety to ! accid5ts ( magnitudes3 b :5 ! g5]a1 z x 71 communicate1 & ! "o is conta9$ "u ! o!r1 !n ! "o may transf] ! pr9ciples ( ! o!r to its own conv5i5ce4 ,?us1 optics unites 9 amicable compact ) geometry1 :i* def9es all its suppositions2 su* z l9es t are "r1 angles acute1 l9es equilat]al1 & ! l4 ,! same ord] may be p]ceiv$ betwe5 ari?metic & music3 ?us1 ! d\ble1 sesquialt]1 & ! l1 are transf]r$ f ari?metic1 f :i* !y take _! rise1 & are appli$ to ! measures ( h>mony4 ,?us1 m$ic9e frequ5tly d]ives its pro(s f nature1 because ! human body1 ) :i* x is conv]sant1 is compreh5d$ "u natural body4 ,f h5ce x foll[s1 t ! geometrician _c1 by any geometrical reasons demon/rate any tru?1 ab/ract$ f l9es1 sup]ficies1 & solids2 su* z1 t ( contr>ies !re is ! same sci5ce2 or t contr>ies foll[ ea* o!r2 nor yet su* z h an exi/5ce 9 l9es & sup]ficies1 b n an ess5tial "o1 9 ! s5se previ\sly expla9$4 ,( ?is k9d is ! "q1 :e!r a "r-l9e is ! mo/ beautiful ( l9es8 or :e!r x is m oppos$ to a l9e p]fectly orbicul>1 or to an ar* only4 ,= ! consid]ation ( beauty1 & ! opposition ( contr>ies1 does n belong to geometry1 b is alone ! prov9ce ( metaphysics1 or ! fir/ philosophy4 ,b a "q "h occurs1 ,if x be requisite t ! propositions :i* con/itute demon/ration %\ld be peculi> to ! sci5ce !y e/abli%1 aft] :at mann] are we to admit 9 demon/ration ^? axioms :i* are conceiv$ 9 ! mo/ common & g5]al t]ms2 su* z1 if f equal ?+s y take away equals1 ! rema9d]s %all be equal3----z likewise1 ( e ?+ t exi/s1 ei!r affirmation or negation is true8 ,! solution is ?is3 su* pr9ciples1 ?\gh common1 yet :5 appli$ to any p>ticul> sci5ce = ! purposes ( demon/ration1 mu/ be us$ ) a c]ta9 limitation4 ,?us ! geometrician applies t g5]al pr9ciple1 if f equal ?+s1 c4 n simply1 b ) a re/riction to magnitudes2 & ! ari?metician univ]sally to numb]s4 ,?us too1 t o!r g5]al proposition3----( e ?+1 affirmation or negation is true2 is subs]vi5t to e art1 b n )\t accommodation to ! p>ticul> sci5ce x is us$ by4 ,?us numb] is or is n1 & s ( o!rs4 ,x is n !n alone suffici5t 9 demon/ration t its propositions are true1 nor t !y are imm$iate1 or su* z inh]it an evid5ce m illu/ri\s ?an ! c]ta9ty ( pro(2 b1 besides all ^!1 x is necess>y !y %\ld be made peculi> by a limitation ( _! compreh5sive nature to "s p>ticul> subject4 ,x is on ?is acc\nt t no "o e/eems ! quadrature ( ,bryso#bd1 a geometrical demon/ration1 s9ce he uses a pr9ciple :i*1 al?\gh true1 is entirely common4 ,previ\s to 8 demon/ration he supposes two squ>es describ$1 ! "o circumscrib+ ! circle1 :i* w be consequ5tly great]2 ! o!r inscrib$1 :i* w be consequ5tly less ?an ! giv5 circle4 ,h5ce1 because ! circle is a m$ium betwe5 ! two giv5 squ>es1 let a mean squ>e be f\nd betwe5 !m1 :i* is easily done f ! pr9ciples ( geometry2 ?is mean squ>e1 ,bryso affirms1 %all be equal to ! giv5 circle4 ,9 ord] to prove ?is1 he reasons aft] ! foll[+ mann]3 ^? ?+s :i* comp>$ ) o!rs )\t any respect1 are ei!r at ! same "t great]1 or at ! same "t less1 are equal among !mselves3 ! circle & ! mean squ>e are1 at ! same "t1 great] ?an ! int]nal1 & at ! same "t less ?an ! ext]nal squ>e2 !re=e !y are equal among !mselves4 ,?is demon/ration c nev] produce sci5ce1 because x is built only on "o common pr9ciple1 :i* may ) equal propriety be appli$ to numb]s 9 ari?metic1 & to times 9 natural sci5ce4 ,x is defective1 !re=e1 because x assumes no pr9ciple peculi> to ! nature ( ! circle alone1 b su* a "o z is common to quantity 9 g5]al4 #ac4 ,x is likewise evid5t1 t if ! propositions be univ]sal1 f :i* ! demon/rative syllogism consi/s1 ! conclusion mu/ necess>ily be et]nal4 ,= necess>y propositions are et]nal2 b f ?+s necess>y & et]nal1 necess>y & et]nal tru? mu/ arise4 ,!re is no demon/ration1 !re=e1 ( corruptible natures1 nor any sci5ce absolutely1 b only by accid5t2 because x is n f\nd$ on t :i* is univ]sal4 ,= :at confirmation c !re be ( a conclusion1 ^: subject is dissoluble1 & ^: pr$icate is nei!r always1 nor simply1 b only p>tially inh]5t8 ,b z !re c be no demon/ration1 s likewise !re c be no def9ition ( corruptible natures2 because def9ition is ei!r ! pr9ciple ( demon/ration1 or demon/ration diff]+ 9 ! position ( t]ms1 or x is a c]ta9 conclusion ( demon/ration4 ,x is ! beg9n+ ( demon/ration1 :5 x is ei!r assum$ = an imm$iate proposition1 or = a t]m 9 ! proposition2 z if any "o %\ld prove t man is risible1 because he is a rational animal4 ,& x alone diff]s 9 position f demon/ration1 z (t5 z ! def9ition is su* z conta9s ! cause ( its subjects exi/5ce4 ,z ! foll[+3 an eclipse ( ! sun is a concealm5t ( its light1 ?r\gh ! int]position ( ! moon betwe5 t lum9>y & ! e>?4 ,= ! ord] ( ?is def9ition be+ a little *ang$1 passes into a demon/ration2 ?us1 ,! moon is subject$ & oppos$ to ! sun3 ,t :i* is subject$ & oppos$1 conceals3 ,! moon1 !re=e1 be+ subject$ & oppos$1 conceals ! sun4 ,b t def9ition is ! conclusion ( demon/ration1 :i* ext5ds to ! mat]ial cause2 z 9 ! prec$+ in/ance1 ! conclusion affirm+ t ! subjection & opposition ( ! moon conceals ! sun1 is a def9ition ( an eclipse includ+ ! mat]ial cause4 ,aga91 we h already prov$ t all demon/ration consi/s ( su* pr9ciples z are prior 9 ! nature ( ?+s2 & f h5ce we inf]1 t x is ! bus9ess ( no sci5ce to prove its own pr9ciples1 s9ce !y c no long] be call$ pr9ciples if !y require confirmation f any ?+ prior to !mselves2 =1 admitt+ ?is z necess>y1 an inf9ite s]ies ( pro(s mu/ ensue4 ,on ! contr>y1 if ?is be n necess>y1 b ?+s mo/ kn[n & evid5t are admitt$1 ^! mu/ be con/itut$ ! pr9ciples ( sci5ce4 ,he :o possesses a k ( ^!1 & applies !m z m$iums ( demon/ration1 is bett] skill$ 9 sci5ce1 ?an he :o kn[s only po/]ior or m$iate propositions1 & demon/rates f po/]ior pr9ciples4 ,b "h a d\bt arises :e!r ! fir/ pr9ciples ( geometry1 ari?metic1 music1 & ( o!r arts1 c ev] be demon/rat$8 ,or %all we all[ !y are capable ( pro(1 n by t p>ticul> sci5ce :i* applies !m z pr9ciples or causes ( its conclusions8 ,if s1 ?is w be ! (fice ( "s sup]ior sci5ce1--:i* c be no o!r ?an ! fir/ philosophy1 to ^: *>ge ! task is committ$2 & ^: univ]sal embrace circumscribes ! :ole circle ( sci5ce1 9 ! same mann] z ari?metic compreh5ds music1 or geometry optics4--,?is is no o!r ?an t celebrat$ wisdom :i* m]its ! appellation ( sci5ce 9 a m simple1 z well z 9 a m em95t degree ?an o!rs3 n1 inde$1 t all causes are )9 its rea*1 b su* only z are ! pr9cipal & ! be/1 because no cause sup]ior to !m c ev] be f\nd4 ,h5ce ! difficulty ( kn[+ :e!r we possess sci5ce or n1 f ! difficulty ( und]/&+ :e!r x is f\nd$ on peculi> or common pr9ciples2 s9ce x is necess>y t bo? ^! %\ld be appli$ 9 ! con/itution ( all real k & sci5ce4 #be;,aga91 axioms diff] f po/ulates 9 ?is3--!y dem& our ass5t )\t any previ\s solicitation1 f ! illu/ri\s c]ta9ty !y possess4 ,_! tru? may1 inde$1 be d5i$ by ext]nal spee*1 b nev] f int]nal connection4 ,he :o d5ies t equal ?+s %all rema9 f ! subtraction ( equal1 diss5ts1 z ,euripides says1 ) 8 tongue1 & n ) 8 he>t4 ,b demon/ration dep5ds n on ext]nal spee*1 b on intellectual & int]nal conviction2 & h5ce1 axioms d]ive all _! au?ority f intr9sic approbation1 & n f public proclaim4 ,= ! prompt decisions ( ! tongue are frequ5tly dissonant f ! s5tim5ts conceal$ 9 ! secret recesses ( ! he>t4 ,?us ! #bf;geometrician does n speculate ^? l9es :i* are ! objects ( corporeal sight1 b su* z are exhibit$ by m5tal conception1 & ( :i* ! del9eations on pap]1 or 9 ! du/1 are no m ?an imp]fect copies1 notes1 & resemblances4 ,?us1 :5 he draws a p$al l9e :i* is n p$al1 or an equilat]al triangle :i* is n equilat]al1 we mu/ pay no reg>d to ! designations ( ! p51 b solely att5d to ! intellection ( ! m9d2 = ! prop]ty demon/rat$ ( "s p>ticul> l9e1 is 9 ! conclusion appli$ to "o t is univ]sal1 & ?is true l9e c\ld be no o!rwise signifi$ to ! le>n] ?an by a mat]ial description4 ,! c]ta9ty ( axioms is1 inde$1 9 a measure obvi\s to e "o4 ,= :at m evid5t ?an t no?+ exi/s ( :i* x is possible1 at ! same "t1 to affirm & d5y any circum/ance ( be+8 ,inde$1 s illu/ri\s & indubitable is ! light ( ?is axiom1 t 9 any demon/ration we are a%am$ to assign x ! place ( an assumption4 ,x w\ld almo/ seem prolix & sup]flu\s1 s9ce !re is no?+ m manife/ & c]ta92 & yet !re are cases 9 :i* x is necess>y to rank x among assumptions4 ,& ^! take place :5ev] ! int5tion is to conclude ! exi/5ce ( some?+ z true1 & ( its opposite z false4 ,?us1 = in/ance1 9 ! demon/ration t ! _w is f9ite1 we assume ?is pr9ciple1 & !n reason z foll[s3 ,b\nd & inf9ite _c be at ! same "t affirm$ & d5i$ ( any body3 ,! _w is a body3 ,!re=e ! _w is n at ! same "t f9ite & inf9ite4 ,& 9 ?is g5us ( demon/ration1 ! major proposition ought always to assimilate ) ! conclusion4 ,b ! above axiom is n ! only "o obvi\s1 = ! foll[+ possesses equal c]ta9ty2 t ( e ?+ :i* exi/s1 ei!r affirmation or negation is true4 ,?is axiom is ( great use 9 demon/rations lead+ to an absurdity2 = he :o demon/rates ! impossibility ( any opposite ass]tion1 necess>ily e/abli%es 8 own4 ,h5ce x is we affirm t ! diamet] ( a squ>e is ei!r comm5surable or incomm5surable ) its side2 & ?is g5]al pr9ciple is accommodat$1 &1 z x 71 desc5ds into its prop] matt] z (t5 z t :i* x possesses ( univ]sal is contract$ to a c]ta9 g5us2 =1 z we h previ\sly obs]v$1 common pr9ciples are n admitt$ 9 demon/ration )\t any re/riction2 b !n only :5 _! g5]al nature is limit$ to "s p>ticul> subject1 by :i* !y become peculi> & apposite4 #ad4 #bg,wisdom1 or ! fir/ philosophy & logic1 agree 9 n us+ axioms aft] ! same mann] z o!r arts2 b on ! contr>y1 !y confirm & e/abli% _! c]ta9ty1 ?\gh ) ?is diff]5ce1 t ! logician reasons only f probabilities1 b ! metaphysician f ! highe/ c]ta9ty & evid5ce4 ,besides1 we d n rank logic 9 ! ord] ( ! sci5ces1 because x is de/itute ( "s det]m9ate g5us or subject1 z x is nei!r conv]sant ab\t l9es1 nor numb]s1 nor proportions4 ,& its *ief conc]n is ab\t app>5t prop]ties1 & n su* z are ess5tial to a subject4 ,h5ce1 9 logical disquisitions1 we confid5tly employ int]rogations1 z equally subs]vi5t to ! affirmation or negation ( an op9ion3--a me?od utt]ly impracticable1 if we only employ$ ^? pr9ciples :i* are univ]sally ackn[l$g$2 s9ce x is impossible ( ! same ?+ to prove contr>y prop]ties1--z ( ! s\l1 t x is mortal & immortal2 b he :o demon/rates1 assumes "o def9ite "p ( a "q1 because 8 purpose is n to int]rogate1 b to trace out ! lat5t pa?s ( tru?4 ,& h5ce1 if any "o affirms t ! s\l is mov$1 & imm$iately aft] d5ies x1 he is no long] a subject wor?y ! ex]cise ( our discursive & reason+ p[]s4 ,aga91 x may s happ51 t ! same sci5ce at "o "t consid]s :y a ?+ is1 at ano!r only expla9s its exi/5ce1 or t x exi/s1 )\t consid]+ ! cause4 ,?us1 ! syllogism :i* concludes by m$iate propositions1 demon/rates )\t assign+ ! prop] cause3 b t :i* det]m9es by imm$iate ones1 9 a great measure expla9s ! cause or reason ( exi/5ce4 ,?us1 he :o inf]s t trees d n brea! because !y are n animals1 reasons f a m$iate & second>y cause1 because !re are _m animals1 su* z insects1 :i* exi/ )\t brea?+3 b he :o inf]s ?is f _! want ( lungs1 demon/rates f ! imm$iate & prim>y cause4 ,?us1 ! foll[+ syllogism is a m$iate "o1 or su* z requires "o or m m$iums to e/abli% its c]ta9ty3 ,e ?+ t is n an animal does n brea!2 ,a tree is n an animal2 ,!re=e a tree does n brea!4 ,"h ! major proposition is evid5tly m$iate1 because we are /ill to seek :y t :i* is n an animal does n brea!1 :i* ! foll[+ imm$iate syllogism solves4 ,e ?+ t is n endu$ ) lungs does n brea!2 ,e ?+ t is n an animal is n endu$ ) lungs2 ergo1 ,e ?+ t is n an animal does n brea!4 ,aga91 ! same sci5ce may demon/rate ! exi/5ce ( a ?+1 or t x exi/s1 & ! cause ( su* exi/5ce z (t5 z x assigns two imm$iate reasons2 b ! "o f ! prop] cause1 ! o!r only f a sign4 ,?us1 he :o demon/rates ! increase ( ! moon1 f ! pl5itude ( h] orb1 inf]s ! cause ( su* increase2 b on ! contr>y1 he :o collects ! pl5itude ( h] orb f h] increase1 reasons only f a sign1 & c alone decl>e its exi/5ce4 ,&1 inde$1 x (t5 happ5s t ! cause & sign reciprocate1 s t z f ! sign we advance to ! cause1 demon/ration f ! cause frequ5tly recurs to ! sign4 ,?us1 f ! bread? & firmness ( ! basis1 we collect ! p]man5t duration ( ! pyramid2 & f its ext5d$ exi/5ce we inf] ! /r5g? ( its support4 ,:5ev]1 !n1 ! argum5t orig9ates f a sign1 x gives evid5ce to ! conclusion1 z f some?+ m kn[n ?an its cause4 ,:5 x beg9s f ! cause1 x proce$s f t :i* is fir/ 9 ! ord] ( nature1 to t :i* is la/1 & reasons z f ! prop] pr9ciple ( ! ?+4 ,sometimes ! cause & sign d n reciprocate4 ,?us1 al?\gh :]ev] !re is smoke1 we inf] ! exi/5ce ( fire2 yet we _c inf]1 t :]ev] !re is fire smoke exi/s4 ,?us1 f ! palace & ! picture we collect ! exi/5ce ( ! ar*itect & pa9t]2 b ! la/ may exi/ )\t ! fir/2--! liv+ ar*itect )\t ! actual palace2 & ! liv+ pa9t] )\t ! en]gies ( 8 art4 ,& ?us x is t ! cause is illu/rat$ by its sign2 b n always ! sign by its cause4 ,h5ce !n1 z all causes d n reciprocate ) _! effects2 s nei!r is x always causes & effects :i* d reciprocate3 because a multitude ( signs1 mutually inf]r+ ea* o!r1 may accompany a c]ta9 cause4 ,?us1 ! signs :i* att5d ! causes ( a fev]1 are a quick pulsation ( ! art]y1 & an int5se heat3 & ^! signs mutually ass]t ea* o!r2 b no syllogism c be compos$ f ei!r express+ ! :y1 b only simply t ! o!r exi/s4 #ae4 ,we n[ propose to consid] ! mode 9 :i* ! two prec$+ demon/rations are di/ribut$ 9 diff]5t sci5ces4 ,:5 sci5ces !n are s relat$1 t ! "o is dep5d5t on ! o!r1 z optics on geometry1 navigation on a/ronomy1 & music compos$ by ! arbitration ( ! e>1 on t :i* consi/s 9 ! k ( ma!matical proportions3 9 ?is case1 ! demon/ration ( simple exi/5ce1 or t !y exi/1 p]ta9s to ! sci5ce ( s5sibles2 b ! demon/ration :y !y exi/ to ! sci5ce :i* is speculative & ma!matical4 ,?us ! ma!matician speculates ! causes ( a c]ta9 s5sible effect1 )\t consid]+ its actual exi/5ce2 = ! contemplation ( univ]sals excludes ! k ( p>ticul>s2 & he ^: intellectual eye is fix$ on t :i* is g5]al & compreh5sive1 w ?9k b little ( t :i* is s5sible & s+ul>4 ,?us1 by ma!matics we may le>n ! responsive h>mony ( ! la/ *ord1 & its consonance ) ! mean2 b we _c p]ceive ?is concord1 if unaccu/om$ to ! practice ( ! musical art4 ,9 f9e1 ^? sci5ces :i* are m ( a ma!matical nature1 ,i mean su* z are m amply conv]sant ) ! inspection ( ?+s1 consid]+ _! =ms ab/ract$ f e mat]ial subject1 always demon/rate ! :y2 & su* is geometry 9 respect ( optics4 ,?us geometry consid]s only su* ?+s z are peculi> to "r-l9es1 indep5d5t ( e s5sible connection4 ,= ! geometrician does n inve/igate a "r-l9e z conta9$ 9 /one or brass2 b consid]s x z entirely deta*$ & unconnect$ ) any object ( s5se4 ,on ! contr>y1 optics receives a "r-l9e j z x is p]ceiv$ 9 a rule1 or engrav$ 9 brass4 ,&1 inde$1 9 treat+ ( "s p>ticul>s1 natural sci5ce has ! same relation to optics1 z optics to geometry4 ,?us1 9 consid]+ ! reason ( ! appe>ance ( ! ra9b[1 ! natural philosoph] def9es ! b[ to be an image refract$ f a c]ta9 cl\d aga9/ ! sun2 b :y x is endu$ ) su* a =m1 & se5 ) su* a col\r1 mu/ be assign$ by him :o is skill$ 9 optics4 ,!re are1 aga91 sci5ces1 "o ( :i* is n subord9ate to ! o!r1 because f\nd$ on pr9ciples totally diff]5t2 yet1 9 "s p>ticul>s !y agree ) ! prec$+4 ,?us1 to "k t an orbicul> w\nd is ! mo/ difficult ( cure1 belongs to ! physician2 b to "k :y1 to ! geometrician4 #af4 ,( all syllogi/ic figures1 ! fir/ is ! be/ adapt$ to sci5ce1 s9ce ! ari?metician1 geometrician1 & la/ly all ^? :o demon/rate any effect f its prop] cause1 fabricate _! reason+s accord+ to ?is figure4 ,= ! middle figure is seldom us$1 because only adapt$ to a few occasions3 & s9ce ! k ( ! :y is ( all o!rs ! mo/ important1 :i* is alone obta9$ by ?is figure3 h5ce1 9 ! pursuit ( sci5ce1 x is always pref]r$ be=e ! re/4 ,besides1 x is equally accommodat$ to ! k ( f9al causes2 to :i* x alone t5ds3 = x composes def9itions f words univ]sal1 & affirmative4 ,9 ! second figure1 a complex negative is conceiv$2 & 9 ! la/1 a p>ticul> "o4 ,add to ?is1 t m$iate propositions are no o!r ways r$ucible to imm$iate ones ?an by ?is figure1 9 :i* ! m$iate proposition t5ds1 by a cont9u$ s]ies1 to t :i* is imm$iate4 ,b ! second does n conclude affirmatively1 nor ! la/ univ]sally2 f :5ce x appe>s1 t a m$iate proposition c nev] become imm$iate by ^! figures3 n t all affirmative propositions are imm$iate ones1 s9ce "s negatives are ( ?is k9d2 = all propositions are equally imm$iate1 :i* _c be confirm$ by syllogism2 & su* are ^? negatives1 ( ^: t]ms x is impossible any g5us c be affirm$4 ,?us ! proposition1 no sub/ance is quality1 is an imm$iate negative ( ?is k9d1 ^: t]ms are two ( ! mo/ univ]sal g5]a ( ?+s4 ,aga91 z we h frequ5tly affirm$ t he :o demon/rates1 always assumes su* ?+s z are ess5tially pr$icat$2 b t he :o argues dialectically or topically1 n always1 b g5]ally assumes su* z are accid5tally pr$icat$1 & :i* appe> m probable & kn[n ?an su* z are ess5tially inh]5t2 x is prop] we %\ld def9e :at is meant by accid5tal pr$ication2 or some?+ pr$icat$ by means ( ano!r4 ,inde$1 ! t]m has a diffuse signification3 =1 fir/1 a body is said to be :ite by some?+ else1 because by its sup]ficies2 & 9 ?is mann] v9es are :ite1 because _! bran*es are :ite4 ,?us1 if accid5t be pr$icat$ ( accid5t1 x is by means ( ano!r2 z :5 we say ! musician is fair2 = ! be+ a musician is an accid5t ( man1 & ! be+ fair ( ! musician3 & man is ! subject ( ea*4 ,! pr$icate ( sub/ance is equally accid5tal1 :5 n includ$ 9 ! numb] ( ?+s sub/antially inh]5t2 z :5 we affirm ( any p>ticul> man t he is r$1 or black4 ,b ! pr$ication is especially accid5tal1 z (t5 z1 by p]v]t+ ! ord] ( nature1 sub/ance is pr$icat$ ( accid5t2 z :5 we say some?+ :ite is an animal3 = ?is ass]tion diff]s f t o!r1 animal is :ite4 ,9 ! latt]1 ! subject animal is nei!r inh]5t 9 ano!r1 nor subsi/s by ano!r1 b has an ess5tial exi/5ce4 ,9 ! =m]1 :at is assum$ z a subject d]ives its exi/5ce f t ( :i* x is ! accid5t4 ,x is only dialectically1 !re=e1 t we c argue f pr$icates z probable & kn[n )\t any di/9ction3 b 9 demon/ration1 all t are prepo/]\s & accid5tal mu/ be c>efully avoid$1 except+ su* accid5ts z be+ ess5tially 9 a subject1 admit ( an ess5tial pr$ication2 & "s ( ^! we h enum]at$ be=e4 #ag4 ,we are n[ ent]+ on a disquisition nei!r ignoble nor useless3 x is ?is1 :e!r ! numb] ( ?+s pr$icat$ ess5tially ( a subject is f9ite1 or :e!r ?+s 9 a cont9u$ s]ies run on to inf9ity4 ,= in/ance1 let u suppose "s ultimate subject1 :i* is n ! pr$icate ( any ?+ besides2 & let c repres5t su* a subject1 ( :i* b is ! fir/ & imm$iate pr$icate2 & 9 ! same mann] d ( b1 & e ( d3 ! qu]y is1 ,:e!r or n ?is extraction mu/ necess>ily /op1 or w admit ( an imm5se progression1 s t f may be pr$icat$ ( e1 & g ( f1 & s on inf9itely2 ! p[] ( ! pr$icates1 :i* supplies ! common id5tity1 /ill rema9+ inexhau/ible & undim9i%$8 ,! second qu]y is ?is1 ,suppos+ "s g5]al subject1 :i* we call a1 ( su* a nature z to be no long] ! subject ( any f>!r pr$ication1 b to be itself ! supreme & prim>y pr$icate2 & suppos+ t x is imm$iately inh]5t 9 f1 & f 9 e1 & e 9 g1 :e!r or n ! process mu/ /op1 or ext5d to inf9ity1 & no subject be f\nd :i* is n directly pr$icable ( ano!r8 ,!re is a rem>kable diff]5ce 9 ! two consid]ations2 =1 9 ! =m] we enquire :e!r any ultimate subject c supply an inf9ite asc5t ( pr$icates2 9 ! latt]1 :e!r any fir/ pr$icate c exi/ 9 an inf9ite desc5d+ s]ies ( subjects4 ,! ?ird "q is1 suppos+ two extremes con/itut$ f a fir/ pr$icate & la/ subject1 :e!r x is possible an inf9ite numb] ( m$iums c int]v5e8 ,& ?is is no o!r ?an to enquire :e!r demon/rations admit ( an inf9ite progression1 s t :atev] is assum$ 9 pro( ( ano!r1 mu/ be prov$ itself8 ,or :e!r x is n m agreeable to tru?1 t !re %\ld be "s imm$iate propositions & ultimate t]ms1 ^: discov]y may give respite to enquiry1 & /ay ! elaborate process ( demon/ration8 ,! same "q occurs 9 negatives4 ,b t "s ( ^! are imm$iate1 ! in/ance lately all$g$ suffici5tly ev9ces4 ,! solution ( ?is enquiry is n s difficult 9 subjects :i* mutually reciprocate2 = 9 ^!1 :5 ! ultimate subject is giv51 no "o c d\bt ! exi/5ce ( _! prim>y pr$icate2 nor :5 ! prim>y pr$icate is admitt$1 c !re be any d\bt ( ! exi/5ce ( "s ultimate subject4 ,=1 9 ?+s :i* mutually reciprocate1 :atev] is enquir$ ( ! "o1 is imm$iately que/ion$ ( ! o!r2 & :]ev] !re is a la/ subject1 !re mu/ be a fir/ pr$icate2 = by ! conv]sion ( ! ultimate subject y effect ! prim>y pr$icate4 ,previ\s to ! discussion ( ! fir/ "q1 x is necess>y to "k t inf9ite int]m$iates _c int]v5e betwe5 two f9ite t]ms 9 an asc5d+ & desc5d+ s]ies ( pr$ications4 ,i call ! s]ies asc5d+ :i* rises to univ]sals2 b desc5d+1 :i*1 by a contr>y process1 /ops at p>ticul>s4 ,?us1 if any "o admits t a is "s fir/ pr$icate1 & g "s ultimate subject1 & %\ld cont5d1 t betwe5 ^! t]ms !re may be inf9ite m$iums1 he contradicts himself2 s9ce he :o beg9s f a 9 a desc5d+ progression1 w nev]1 by ?is means1 arrive at g2 & he :o dep>ts f g 9 an asc5d+ s]ies1 c nev] f9ally rise to a4 ,s t ! extremes c be no long] f9ite1 z ! hypo!sis admitt$4 ,inde$1 ! absurdity ( su* a supposition is ! same z to cont5d t betwe5 "o & t51 an inf9ity ( numb]s may exi/2 :i* is evid5tly impossible1 because ! discrete nature ( numb]s excludes _! actual exi/5ce 9 inf9itum1 betwe5 any f9ite limits2 s9ce !y c only become inf9ite f _! actual exi/5ce & prec$5ce1 & n f any dormant p[] or capacity !y possess3 = betwe5 any two giv5 numb]s !re is no?+ simil> to numb] 9 capacity1 :i* c ev] become numb] 9 en]gy2 z 9 quantity cont9u\s betwe5 any two po9ts !re are always p>ts 9 capacity1 :i*1 :5ev] a prop] ag5t is at h&1 become imm$iately actual4 ,9 l mann]1 he :o admits ! t]ms f9ite1 b believes t ! m$iums are inf9ite1 ass]ts :at is impossible1 s9ce ^! logical pr$ications are ( ! same discrete nature ) numb]s !mselves4 ,?us all ! pr$icates :i* c exi/ betwe5 ,socrates & sub/ance1 mu/ exi/ actually1 or n at all2 = surely betwe5 ^! two t]ms1 or p]iods1 no pr$icate 9 capacity c ev] be suppos$ to subsi/4 ,if x be urg$1 t ! capacity ( receiv+ ^! pr$icates exi/s betwe5 ,socrates & sub/ance1 /ill we reply1 x is n t k9d ( capacity 9 :i* ^! pr$icates c reta9 ! mo/ %ad[y exi/5ce2 out ( :i* !y c ev] be call$ =? into en]gy1 z f "s lat5t retreat2 or into :i* !y c f9ally retire1 :5 en]gy is no m4 ,& h5ce we conclude x impossible t inf9ite m$iums c exi/ betwe5 any f9ite t]ms4 #ah4 ,x n[ rema9s t we prove1 fir/1 by probable argum5ts1 & !n by su* z are demon/rative1 t ! extremes 9 any s]ies ( pr$ications are f9ite2 & t an inf9ite progression is impossible1 n only 9 sub/antial pr$icates1 b 9 su* z are accid5tal4 ,= e ?+ pr$icat$ ( ano!r is ei!r ess5tially or accid5tally inh]5t2 & is pr$icat$ 9 a natural or prepo/]\s ord]4 ,x is pr$icat$ accord+ to nature1 :5 accid5t is decl>$ ( sub/ance2 contr>y to nature1 :5 sub/ance ( accid5t4 ,t ess5tial pr$icates are f9ite1 appe>s f h5ce1 because a contr>y hypo!sis excludes ! exi/5ce ( def9ition1 by admitt+ t all ?+s are conta9$ 9 "s sup]ior g5us1 & ackn[l$ge "s f>!r def9ition2 s9ce x is impossible t ! def9itions ( g5us c ev] be circumscrib$1 :ile !re is a cont9ual supply ( o!r g5]a1 :i* c nev] be kn[n )\t def9ition2 = ?us we %all nev] obta9 ei!r a beg9n+ or an end4 ,b to def9e all ?+s is n possible1 because inf9ity c nev] be absolv$ by ! mo/ unwe>i$ progression4 ,pr$ictions !n1 ( ?is k9d1 are always circumscrib$ by a c]ta9 numb] ( t]ms1 :i* prev5t _! inf9ite process1 & cause all ! /r5g? ( demon/ration1 & all ! c]ta9ty ( human k4 ,! same may be prov$ 9 accid5ts2 = su* z are pr$icat$ ( sub/ance1 are ei!r pr$icat$ z qualities or quantities1 z relatives1 or z actions & passions2 z expressive ( "s habit1 or significant ( "s place2 or z connect$ ) "s "t4 ,?us we say ! wood is :ite1 ! triangle is scal5e2 :it5ess be+ accid5tal to ! wood1 & scal5ity to ! triangle4 ,x is !re=e c]ta91 t e accid5t is pr$icat$ ( sub/ance2 & x is no less c]ta9 t ! pr$icates ( sub/ance are f9ite1 s9ce !y are all includ$ 9 ! t5 univ]sal g5]a ( ?+s4 #ai4 ,we h hi!rto def5d$ ! impossibility ( an inf9ite progression ( logical pr$icates & subjects1 9 a demon/rative process1 by su* argum5ts z are dialectical & common3 x n[ rema9s t we adopt su* z are peculi> & c]ta94 ,demon/rations1 !n1 are d]iv$ f affections ess5tially inh]5t 9 a subject2 & ^! are ei!r su* z take place 9 def9itions ( a subject1 z multitude & quantity1 are ess5tially pr$icat$ ( numb]2 or1 secondly1 accid5ts :i* are def9$ f _! subjects1 z imp>ity by numb]4 ,b ! pr$ication _c1 9 ei!r case1 be ext5d$ to inf9ity4 ,= x is n necess>y t 9 ! same mann] t imp>ity is pr$icat$ ( numb]1 some?+ else1 suppose c1 %\ld be pr$icat$ ( imp>ity2 & s imp>ity be conta9$ 9 its def9ition1 simil> to numb] 9 ! def9ition ( imp>ity4 ,= 9 pr$ications ( ?is k9d1 ! t]ms are always assum$ m contract$ ?an _! subject2 & at l5g?1 by a cont9u$ procession1 mu/ t]m9ate 9 an indivisible4 ,?us1 z imp>ity is m contract$ ?an numb]1 c mu/ be m contract$ ?an imp>ity4 ,h5ce1 ^! pr$ications ei!r f9ally /op1 = ! reasons we h assign$2 or because :atev] is pr$icat$ ( imp>ity1 is necess>ily pr$icat$ ( numb]2 s t "o ?+ z numb] w\ld be actually conta9$ 9 ! def9ition ( an inf9ity ( ?+s2 & s actual inf9ity mu/ ensue1 :i* is absurd4 ,la/ly1 :atev] is said to reside 9 ! t]ms1 mu/ be all[$ to reside 9 ! subject2 s numb] mu/ be appli$ 9 ! def9ition ( e affection2 & an inf9ite numb] ( prop]ties w be ess5tially inh]5t 9 numb]2 & numb] w inh]it inf9ite def9itions4 ,b affections ess5tially resid5t 9 a subject _c be inf9ite1 because x is necess>y !y %\ld exi/ 9 en]gy4 ,?us1 imp>ity _c exi/ pot5tially 9 numb]2 nor reason 9 man2 nor rotundity 9 a circle1 because :]ev] ^! subjects h an actual be+1 x is necess>y ^! ess5tial attributes %\ld be actually inh]5t4 ,aga91 9 ! def9itions ( a subject1 an inf9ite process is impossible1 because f su* an hypo!sis no?+ c\ld ev] be def9$2 & ?us x appe>s t nei!r c demon/rations be inf9itely ext5d$1 nor e ?+ admit ( demon/ration1 an op9ion we h already notic$ 9 ! beg9n+ ( ?is section3 = if nei!r univ]sally1 nor 9 e proposition a middle t]m c be assum$1 b z soon z we arrive at imm$iate propositions1 ! lab\r ( inve/igation is f9i%$1 ! possibility ( demon/rat+ e ?+ c no long] be def5d$2 s9ce x is prov$ above1 t by limit+ ! extremes1 an inf9ite numb] ( m$iums is necess>ily exclud$4 ,& ?us1 by tak+ away inf9ity f ! reason+ art1 we h giv5 a support to sci5ce1 :i* ! mo/ vigor\s ef=ts ( subtle sophi/ry c nev] f9ally subv]t4 ,we h set b\nds to t re/less _s ( enquiry :i* w&]s uncontr\l$ 9 ! m9d un5light5$ by sci5ce1 by e :]e circumscrib+ its progress )9 ! limits ( t :i* is mo/ p>ticul>1 & mo/ univ]sal1 a fir/ pr$icate1 & an ultimate subject3 & f9ally1 by ass]t+ t all ! evid5ce ( human k results f ! lu/re ( prim>y & imm$iate pr9ciples1 we h held up a /eady & p]man5t light1 ev] suffici5t to direct our /eps ?r\gh ! d>k mazes ( ignorance & error1 into ! bright pa?s ( c]ta9ty & tru?4 #bj4 ,let u next consid] :e!r univ]sal demon/ration is pref]able to p>ticul>1 or n4 ,& fir/1 9 fav\r ( p>ticul>s we may say t _! evid5ce is m exquisite & c]ta9 ?an t ( univ]sals4 ,?us1 ! k1 f inspection1 t ,callias is a rational animal1 is sup]ior to t acquir$ by a reason+ process :i* inf]s 8 rationality1 because e man is a rational animal4 ,by p>ticul> demon/ration a ?+ is kn[n z x is1 by univ]sal only 9 common4 ,besides1 p>ticul>s possess "s solidity1 univ]sals none3 & ! demon/ration ( ?+s :i* h a real exi/5ce1 is m excell5t ?an t ( ?+s :i* h none4 ,& !re are no errors m frequ5t ?an ^? ab\t univ]sals2 demon/ration consid]+ !m z ?+s entirely ab/ract$ f s+ul>s4 ,on ! contr>y1 p>ticul>s are usurp$ by ! sight1 grasp$1 z x 71 by ! h&1 & ! g5]al subject ( e s5se2 s t conc]n+ ^!1 demon/ration affirms no?+ false or incon/ant4 ,b ^! reasons1 h[ev] plausible1 are easily confus$4 ,&1 fir/1 ! t]m ess5tial is m closely connect$ ) univ]sals ?an p>ticul>s4 ,?us ! possession ( ?ree angles equal to two "r1 is an affection m ess5tial to ! triangle itself1 ?an to "o equilat]al or scal5e4 ,add too1 t 9 ! demon/ration ( univ]sals we always inf] "s prop]ty ( a subject f its simple exi/5ce1 or because x is su* a subject4 ,aga91 _m affections are conta9$ 9 s+ul>s assum$ f no p>ticul> nature1 b f t :i* is univ]sal2 z rationality 9 ,socrates1 :i* is n inf]r$ f 8 exi/5ce z ,socrates1 b f 8 exi/5ce z man4 ,f>!r1 t demon/ration is ! m excell5t :i* is d]iv$ f ! bett] cause3 b an univ]sal cause is m ext5d$ & excell5t ?an a p>ticul> "o2 s9ce ! ardu\s inve/igation ( ! :y 9 any subject is /opt by ! arrival at univ]sals4 ,?us1 if we desire to "k :y ! ext]ior angles ( a triangle are equal to f\r "r ones1 & x is answ]$1 because ! triangle is isosceles2 we aga9 ask1 ,b :y because isosceles8 ,& if x be repli$1 because x is a triangle1 we may aga9 enquire1 ,b :y because a triangle8 ,to :i* we f9ally answ]1 because a triangle is a "r-l9$ figure2 & "h our enquiry re/s at t univ]sal idea :i* embraces e prec$+ p>ticul> "o1 & is conta9$ 9 no o!r m g5]al & compreh5sive ?an itself4 ,add too1 t ! demon/ration ( p>ticul>s is almo/ ! demon/ration ( inf9ites2 ( univ]sals1 ! demon/ration ( f9ites4--,we add f>!r1 t demon/ration is ! be/1 :i* furni%es ! m9d ) ! mo/ ample k2 & ?is is alone ! prov9ce ( univ]sals4 ,aga91 ! pr9ciples ( sci5ce become imm$iate only 9 proportion z ! demon/ration becomes univ]sal2 & he :o kn[s univ]sals1 kn[s p>ticul>s 9 capacity3 b we _c inf]1 t he :o has ! be/ k ( p>ticul>s1 kn[s any ?+ ( univ]sals4 ,la/ly1 t :i* is univ]sal1 is ! prov9ce ( intellect & reason1 p>ticul>s are ! (fspr+ ( s5se2 & h5ce we conclude t univ]sal demon/ration exce$s p>ticul> bo? 9 dignity & excell5ce1 & is fir/ 9 ! nature ( ?+s1 al?\gh la/ 9 ! progressions ( ! reason+ p[]4 ,aga91 ,t affirmative demon/ration is sup]ior to negative1 appe>s f h5ce3 ! affirmative does n require ! assi/ance ( ! negative2 b ! negative _c exi/ )\t ! affirmative2 on :i* acc\nt1 ! demon/ration compos$ f negatives alone1 is incapable ( produc+ real evid5ce & conviction4 ,besides1 affirmation exce$s negation bo? 9 priority & simplicity ( exi/5ce4 ,aga91 ! demon/ration :i* concludes directly1 is bett] ?an t :i* confirms a proposition by ev9c+ ! absurdity ( its contr>y4 ,! fir/ proce$+ 9 a regul> ord]1 e/abli%es1 by a natural d$uction1 ! tru? :i* 0 fir/ advanc$4 ,! second tak+ a wid] circuit1 yet ) ! same int5tions produces a conclusion q opposite to its app>5t design4 ,! "o may be comp>$ to ! op5 attack ( a valiant & skilful soldi]1 :o expects ! conque/ ( 8 enemy f /r5g? & c\rage alone3 ! progress ( ! o!r resembles ! same soldi]1 unit+ =ce ) /ratagem1 & advanc+1 by an irregul> m>*1 :i* 8 foe mi/akes = a retreat1 b f9ds ! secret cause ( 8 de/ruction4 ,! fir/ is simple & impromiscu\s1 z compos$ f propositions alone3 ! second is comp\nd & miscellane\s1 call+ 9 hypo!sis to its assi/ance4 #ba4 ,"o sci5ce is said to be prior to1 & m c]ta9 ?an ano!r 9 _m respects2--:5 ! "o reasons f prim>y causes1 b ! o!r f su* z are second>y3--:5 ! "o may be rank$ 9 ! g5]a ( intelligibles & univ]sals2 b ! o!r 9 ! g5]a ( s5sibles & p>ticul>s4 ,& su* is ! relation ( ari?metic to music2 ( geometry to optics2 & la/ly1 ( e sup]ior to e subord9ate sci5ce4 ,aga91 ?is happ5s :5 ! "o reasons f simple pr9ciples1 ! o!r f su* z are complex & connect$2 on :i* acc\nt ari?metic seems to possess great] c]ta9ty ?an geometry4 ,= ! pr9ciple ( ari?metic is unity2 b ( geometry a po9t2 & unity is )\t position1 ) :i* a po9t is always connect$4 ,& 9 ?is mann] geometry inh]its great] evid5ce ?an a/ronomy2 = ! "o consid]s body simply1 ! o!r z connect$ ) a circul> motion4 ,! sci5ce is call$ "o :i* contemplates actions belong+ to "o g5us3 ! g5us is "o :i* possesses ! same fir/ pr9ciples2 & h5ce geometry & /]eometry =m "o sci5ce4 ,on ! contr>y1 ! sci5ces are call$ diff]5t :i* h diff]5t pr9ciples1 su* z geometry & optics2 ! latt] ( :i* does n orig9ate f ! pr9ciples ( ! =m]4 ,aga91 ! same ?+ may admit ( _m demon/rations1 & may be kn[n f _m m$iums3 at "o "t f ! application ( su* z are cong5ial3 at ano!r1 f ^? ( a diff]5t ord] or g5us4 ,f cong5ials1 z :5 we demon/rate t ! planta9 is a sub/ance1 fir/1 by ! m$ium ( a tree1 & !n by ! m$ium ( a plant1 ?us3 ,e tree is a sub/ance2 ,! planta9 is a tree3 ,!re=e ! planta9 is a sub/ance4 ,& aga91 ,e plant is a sub/ance3 ,! planta9 is a plant3 ,!re=e ! planta9 is a sub/ance4 ,we demon/rate1 f m$iums1 ( a foll[+ ord] or g5us1 z :5 we prove man to be a sub/ance1 at "o "o "t f 8 be+ rational1 at ano!r f 8 be+ a bip$2 & ^! m$iums1 9 "p1 mutually conta9 ea* o!r4 #bb4 ,=tuit\s ev5ts c nev]1 9 any sci5ce1 become ! subject ( demon/ration2 s9ce !y are nei!r limit$ by necessity1 nor admit ! arrangem5t ( syllogism4 ,inde$1 s f> f obta9+ a necess>y1 !y d n possess a frequ5t exi/5ce1 b e syllogism is compos$ f "o or o!r ( ^!4 ,aga91 sci5ce is n ! bus9ess ( s5se1 s9ce t :i* is univ]sal is ! object ( p]ception 9 p>ticul>s !mselves4 ,= ! object ( sight is col\r 9 g5]al1 & n ?is p>ticul> col\r3 ! object ( he>+ is s\nd 9 g5]al1 & n any p>ticul> s\nd2 &1 on ?is acc\nt we see or he> n only ?is or t col\r or s\nd1 b likewise e o!r :i* falls "u ! cognizance ( ^! s5ses4 ,h5ce1 if x 7 possible = any "o to disc]n by 8 sight1 ! equality ( ! ?ree angles ( "s p>ticul> triangle to two "r1 he w\ld n by ?is means possess a demon/ration ( ! conclusion :i* affirms ?is to be ! prop]ty ( e triangle2 b 8 k w\ld ext5d no f>!r ?an ! triangle he inspects4 ,?us too1 if we c\ld p]ceive an eclipse ( ! moon to arise f ! int]position ( ! e>?1 we c\ld n univ]sally conclude t ?is is ! cause ( e eclipse1 b only ( ! p>ticul> "o we behold4 ,= ! explication ( causes ext5ds to univ]sals2 & compreh5ds n only ! k ( "o p>ticul> defect ( ! moon1 b simply ( e eclipse2 s9ce ! int]position ( ! e>? is n s mu* ! cause ( any pres5t eclipse1 z ( all :i* c possibly exi/ 9 e age4 ,:5ev]1 !n1 ! cause is univ]sal1 ! k ( any effect d$uc$ f su* a cause is1 9 e respect1 sup]ior to ! evid5ce aris+ f ! p]ceptions ( s5se4 ,x is likewise m excell5t ?an ! appreh5sion :i* subsi/s indep5d5t ( ! prop] cause2 z if any "o %\ld give absolute cr$it to ! proposition1 t ! ?ree angles ( a triangle are equal to two "r1 )\t a previ\s conviction t ! ext]nal angle ( a triangle1 is equal to ! two int]ior opposite ones2 & )\t apply+ ?is la/ proposition z ! cause ( ! fir/4 ,! compreh5sion1 !n1 :i* is conjo9$ ) ! prop] cause1 f> exce$s ! /ronge/ evid5ce ( s5se4 ,b p]haps x may be said t sci5ce consi/s 9 s5se1 because ! sci5ce ( any p>ticul>1 fails f a defect ( ! s5se by :i* x is appreh5d$4 ,to ?is we reply1 t sci5ce1 inde$1 is n acquir$ )\t ! assi/ance ( s5se1 b x does n foll[ f h5ce1 t to p]ceive is to "k2 because ! object ( sci5ce is t :i* is univ]sal2 b ( s5se1 t :i* is p>ticul>4 ,?us1 if we c\ld see light p5etrat+ ! pores ( glass 7on ! atomical hypo!sis7 ! cause :y x illum9ates w\ld be manife/ f s5sible inspection z ! means1 & f ! univ]sal appreh5sion ( sci5ce1 by :i* we %\ld und]/& ?is to be univ]sally true4 ,aga91 ! pr9ciples ( all sci5ces _c be ! same nei!r consid]$ z remote or proximate4 ,n consid]$ z proximate1 because ! pr9ciples always correspond to ! demon/rat$ conclusions2 b ^! are n ! same1 s9ce !y are (t5 g5]ically diff]5t2 & consequ5tly ! propositions f :i* !y result mu/ be d]iv$ f discordant g5]a4 ,b propositions consi/ ( su* ?+s z ess5tially exi/2 & h5ce we inf]1 t ! pr9ciples ( geometry are ess5tially di/+ui%$ f ^? ( ari?metic1 t !y _c admit ( reciprocal accommodation1 s t ! "o may be pr$icat$1 or become ! subject ( ! o!r1 & t ! "o c nev] be subs]vi5t z a m$ium to ! o!r4 ,aga91 common & fir/ pr9ciples are n appli$ 9 e sci5ce2 su* z ?is1 t e ?+ mu/ ei!r be affirm$ or d5i$4 ,nor c any ?+ be prov$ by _! assi/ance alone1 b z (t5 z ^! are requir$ 9 demon/ration1 o!r pr9ciples m proximate & peculi> to ! giv5 proposition1 mu/ always be adopt$4 ,aga91 axioms univ]sally conceiv$1 _c be assum$ 9 syllogism1 b !y mu/ be contract$1 z x 71 to "s subject g5us4 ,( ?is k9d is t common axiom1 t z (t5 z any f\r quantities are proportionable1 by p]mutation1 or *ang+ ! ord] ( ! t]ms1 ! same ratio w result4 ,= ! arts apply ?is axiom 9 a re/rict$ s5se2 geometry1 by consid]+ ! relatives z f\r magnitudes1 & ari?metic z f\r numb]s2 b ! natural philosoph]1 by adapt+ ! comp>ison to f\r motions1 or f\r times4 ,besides1 if ! pr9ciples ( all sci5ces 7 ! same1 x is necess>y !y %\ld be compreh5d$ by "s c]ta9 numb]1 simil> to ! limitation ( ! elem5ts3 b e sci5ce is capable ( imm5se increase f ! _m diff]5t modes ( amplification ! conclusions w admit2 & consequ5tly x is requisite to e/abli% a correspond5t numb] ( prop] pr9ciples2 = su* z are common _c be alone suffici5t4 ,la/ly1 if ! same pr9ciples accord ) e sci5ce1 x foll[s1 t any ?+ may be demon/rat$ f su* pr9ciples3 b ! c]ta9ty ( geometrical conclusions _c be e/abli%$ f ! pr9ciples ( music2 & f h5ce x foll[s1 t al?\gh ! pr9ciples ( e sci5ce are n ! same1 !y d n possess an entire div]sity1 nor yet an absolute aff9ity ( nature4 #bc4 ,!re is a rem>kable diff]5ce betwe5 sci5ce & op9ion4 ,:atev] is ! subject ( sci5ce mu/ h a necess>y exi/5ce2 on ! contr>y1 op9ion is conv]sant ) ?+s liable to mutation & decay4 ,aga91 z sci5ce dep5ds on necess>y propositions = support2 s op9ion on su* z possess only a possibility ( exi/5ce2 & s !re is "o mode ( approbation 9 subjects ( op9ion1 & ano!r 9 ^? ( sci5ce4 ,h5ce sci5ce is di/+ui%$ f op9ion by two discrim9ations1 ! "o aris+ f _! subjects1 ! o!r f ! mode ( approbation4 ,t op9ion is conv]sant ) ?+s possible or cont+5t1 we may le>n f h5ce2 cont+5cies _c belong to sci5ce1 because _! exi/5ce is n necess>y2 nor to intellect1 or t pr9ciple ( sci5ce by :i* its t]ms are kn[n2 nor to ! appreh5sion or belief ( imm$iate propositions1 call$ indemon/rable sci5ce4 ,h5ce1 if e habit by :i* tru? is kn[n1 is ei!r sci5ce1 or intellect1 or op9ion1 x rema9s t op9ion alone consi/s ( ?+s :i* are1 inde$1 true2 b n necess>y4 ,x is1 !re=e1 incon/ant & un/able1 f ! mutable nature ( its subjects4 ,besides1 no "o ?9ks he possesses an op9ion ( ?+s :i* he believes to h a necess>y exi/5ce1 s t !y _c be o!rwise ?an !y are2 b to su* conviction he prop]ly gives ! "n ( k1 & to its contr>y ! "n ( op9ion4 ,aga91 ! same ?+ f ! same propositions may at "o "t become ! subject ( k1 at ano!r1 ( op9ion2 & ?is happ5s accord+ to ! diff]5t =mation ( ! syllogism :i* ! propositions compose2 :e!r reason+ f ! prop] cause x expla9s ! :y1 or only simply decl>es a ?+ exi/s4 ,h5ce a d\bt arises :y op9ions ( ?is k9d may n be call$ sci5ce1 s9ce bo? ! subjects & propositions are ! same8 ,! solution is obvi\s4 ,if x is believ$ t ! propositions _c be o!rwise ?an !y are1 or t !y h a necess>y exi/5ce1 su* an ass5t ( ! m9d is n op9ion1 b sci5ce2 because ?+s :i* inh]it an ess5tial exi/5ce are ! ornam5ts ( sci5ce alone4 ,on ! contr>y1 if we are conv9c$ t ! propositions are true1 b at ! same "t n necess>y1 su* conviction is n sci5ce1 b op9ion4 ,h5ce1 x is impossible t sci5ce & op9ion c be ! same1 s9ce !y v>y 9 _! def9ition & mode ( approbation1 & 9 a diff]5t mann] dem& our belief4 ,simil> to ?is1 al?\gh x may happ5 t ( ! same ?+ a true & a false op9ion may arise1 x w n !re=e foll[1 t true & false op9ions are ! same4 ,= t :i* is firm & con/ant c nev] be ! same ) t :i* is mutable & frail2 & t :i* is always true mu/ be ess5tially diff]5t f t :i* may be *ang$ into false4 ,by ! p[] ( habit inde$ 9 diff]5t m51 ! same ?+ may be compreh5d$ by op9ion & sci5ce4 ,?us x 0 op9ion 9 ,epicurus :5 he said t ! sun 0 eclips$ by ! moon pass+ "u its orb1 because he ?\ght x might o!rwise happ51 & t ! moon might be int]pos$ )\t obscur+ ! light ( ! sun4 ,x 0 sci5ce 9 ,hipp>*us1 because he knew x z a necess>y ev5t4 ,b 9 ! same m9d1 at ! same "t1 & ( ! same ?+1 x is impossible t sci5ce & op9ion c exi/4 ,& ?us mu* conc]n+ ! diff]5ce ( ! two4 #bd4 ,la/ly1 sagacity is an acute & sudd5 appreh5sion ( ! m$ium1 or prop] cause ( a c]ta9 effect3 z if any "o1 behold+ ! moon1 %\ld 9 a mom5t conjecture ! cause ( ! "p opposite to ! sun be+ lucid1 & ! o!r p>ts obscure1 because %e d]ives h] spl5dor f ! sun4 ,h5ce he is univ]sally call$ acute & sagaci\s1 :o1 f ! aspect or he>+ ( ! extremes1 c readily p]ceive ! m$ium :i* exi/s betwe5 !m3 z ! t]m imports a c]ta9 revolution ( ! conclusion into its fir/ propositions1 &1 z x 71 a swift compreh5sion & cont9uation ( ! m$ium4 ,,section ,,iii4 ,9 ! ensu+ ,comm5t>ies1 ! s\l is consid]$ z immat]ial2 & z possess+ a middle nature betwe5 intelligibles & s5sibles2 b t ?is important ass]tion may n rema9 )\t pro(1 ! foll[+ demon/rative argum5ts1 d]iv$ f ! ,platonic philosophy1 are (f]$ to ! read]'s consid]ation4 ,& fir/1 t ! s\l is an unext5d$1 & consequ5tly immat]ial ess5ce1 may be ?us prov$4 ,if1 aft] ! mann] ( magnitude1 x consi/$ ( cont9u\s p>ts1 x w\ld be impossible t any "o "p c\ld be s5ti5t f ! passion ( ano!r2 b ! s\l1 = in/ance1 :i* is situat$ 9 ! f+]1 w\ld be s5sible ( passion1 z if deta*$ f s\l 9 ! o!r p>ts ( ! body1 & exi/+ by itself2 = ! s\l1 f ?is hypo!sis1 w\ld be s5ti5t by p>ts1 & n consid]$ z a :ole4 ,besides1 !re mu/ be _m s\ls gov]n+ ea* "p ( u1 diff]5t among !mselves1 & endu$ ) _! own peculi> en]gies4 ,= :atev] may be said ( cont9uity1 is to no purpose1 unless x conduces to unity ( s5sation2 s t ! hypo!sis :i* supposes t ! s5sations gradually arrive at ! pr9cipal "p ( ! s\l by a c]ta9 cont9u$ succession1 is n to be admitt$1 s9ce x may be reasonably ask$1 ,h[ is ! pr9cipal "p to be peculi>ly di/+ui%$8 ,by :at rule ( quantity c ! p>ts be disc]n$1 by :at diff]5ce are !y to be di/+ui%$1 :]e ! quantity is "o1 & ! bulk cont9u\s8 ,besides1 is ! pr9cipal "p alone1 or are ! o!r p>ts1 s5ti5t8 ,if ! pr9cipal "p alone be p]ceptive1 ! s\l c !n alone be s5ti5t :5 a s5sible passion meets ) ?is pr9cipal "p1 situat$ 9 its peculi> seat2 b if a s5sible passion falls ^u any o!r "p ( ! s\l de/itute ( s5se1 x is impossible t su* a "p %\ld be able to transmit ! same passion to ! pr9cipal1 or be at all s5ti5t4 ,= h[ c t :i* is void ( s5se1 receive passion1 & convey x entire to a s5ti5t "p8 ,besides1 if passion acc$es to ! pr9cipal1 x ei!r falls on "o ( its p>ts1 & s ei!r "o "p alone w be s5ti5t1 & ! re/ )\t s5sation1 & consequ5tly sup]flu\s1 or !re mu/ be innum]able & dissimil> s5sations2 = if ! s5sation ( ea* ( ! p>ts s+ly1 is ! same ) ! aggregate ( !m all1 ( :at use is a multitude ( p>ts8 ,b if ! s5sations are v>i\s1 a man may say1 z x 71 ,i am prim>ily s5ti5t 9 ?is place1 & second>ily 9 ano!r2 & e s5ti5t "p besides ! fir/1 w be ignorant :]e s5sation is mo/ p[]ful3 or p]haps 7f su* an hypo!sis7 e "p ( ! s\l w be equally deceiv$1 ea* "p ?9k+ ! passion to arise 9 ! place :]e x is situat$4 ,b if n ! pr9cipal "p alone1 b e "p ( ! s\l be s5ti5t1 a pr9cipal "p is sup]flu\s = ! purpose ( s5sation2 & if ! s\l be divisible l magnitude1 h[ is x able to recognize1 z belong+ to "o subject1 ! qualities :i* fl[1 z x 71 ?r\gh _m s5ses1 z ?r\gh ! eyes & e>s8 ,= t "p ( ! s\l :i* is p]ceptive ?r\gh ! eyes1 di/+ui%es no?+ b col\rs2 t :i* en]gises ?r\gh ! e>s1 no?+ b s\nds2 & t :i* acts ?r\gh ! m$ium ( ! t\*1 no?+ b ! surfaces ( bodies3 :at is x !n :i* p]ceives all ^! prop]ties ( bodies unit$ 9 "o subject1 or :at is t :i* p]ceives any ?+ z a c]ta9 "o8 ,= unless ! int5tions ( ! s5ses1 & ( s5sible objects1 7 collect$ toge!r 9 "o1 ! s\l c\ld nev] be able to judge ( ! peculi> & diff]5t prop]ties ( bodies3 ,h5ce x is necess>y t ! s\l %\ld be1 z x 71 a g5]al c5tre2 t ! sev]al s5ses %\ld be ext5d$ on all sides to ?is1 simil> to l9es v]g+ f ! circumf]5ce ( a circle to ! c5tre2 & t a p[] ( ?is k9d1 compreh5d+ all ?+s2 %\ld be truly "o4 ,= if ! s\l 0 any ?+ divisible1 & ! int5tions ( ! s5ses rea*$ ! s\l1 & end$ 9 its ess5ce l ! extremities ( v>i\s l9es1 !y mu/ ei!r aga9 concur 9 "o & ! same z a m$ium1 or h diff]5t situations1 9 su* a mann] t ea* s5se may p]ceive diff]5t f "o ano!r3 z if1 = in/ance1 ! s5se ( sight %\ld p]ceive ! =m ( ,socrates1 & ! s5se ( he>+ recognize 8 voice2 t ess5ce :i* pron\nces ! :ole to be "o p]son1 t ( ,socrates1 mu/ be some?+ diff]5t f ea* ( ! s5ses4 ,h5ce x is necess>y t ! s\l %\ld be an indivisible ess5ce2 = if %e possess$ magnitude1 %e mu/ be divid$ along ) e s5sible object %e p]ceives2 s t "o "p ( ! s\l w\ld p]ceive a c]ta9 "p ( a s5sible object1 & we %\ld possess no s5ti5t p[] capable ( p]ceiv+ ! :ole1 or ( pron\nc+ any ?+ "o4 ,?us1 9 ! p]ception ( a man1 consid]$ z "o1 h[ is x possible t ! s\l c be divid$1 s t ! p]ception ( a "p %all be ! same ) ! p]ception ( ! :ole4 ,b if we suppose ! s\l divisible 9 all h] p]ceptions1 s9ce x is impossible %e %\ld be co-ext5d$ ) e s5ti5t object1 9 h[ _m p>ts is ! division to be made8 ,is ! s\l to be di/ribut$ into ! same numb] ( p>ts z ! s5sible object %e p]ceives1 s t e "p ( ! s\l may p]ceive ! same "p ( ! object8 ,or %all we say t ! p>ts ( ! s\l h no s5sation ( ! p>ts ( ! object8 ,b ?is w\ld be absurd & impossible4 ,if e "p !n ( ! s\l p]ceiv$ e "p ( a magnitude z a :ole1 s9ce magnitude is divisible to inf9ity1 & s9ce1 on ?is hypo!sis1 !re mu/ be innum]able s5sations ( e s5sible object1 !re mu/ be innum]able images1 z x 71 ( ! same ?+ 9 our pr9cipal "p4 ,besides1 if t :i* p]ceives is corporeal1 x w n be possible = x to p]ceive 9 any o!r mann] ?an z if c]ta9 images 7 impress$ f a seal 9 wax1 or 9 brass1 or 9 any o!r s5sible sub/ance4 ,b if ! images ( s5sation exi/ z 9 humid bodies 7:i* is mo/ probable71 !y w c]ta9ly be conf\nd$ l images 9 wat]2 nor c !re be any memory1 ! image dep>t+ ) its =m+ sub/ance4 ,& if we suppose ! figures to rema9 l impressions 9 solid bodies1 ei!r x w n be possible = o!rs to succe$ :ile ! =m] endure2 & ?us1 s5sations ( o!r ?+s _c take place2 or if o!rs succe$1 ! =m] images mu/ be imm$iately de/roy$1 & memory be no m4 ,s t if we all[ x possible to rememb]1 &1 besides ?is1 to p]ceive o!r ?+s1 )\t any h9drance f =m] impressions1 x is impossible t ! s\l %\ld be corporeal4 ,s9ce ! s\l1 !re=e1 is an unext5d$1 indivisible1 & immat]ial sub/ance1 x is consequ5tly incorruptible & immortal2 = e ?+ capable ( dissolution & disp]sion is ei!r corporeal & composite1 or exi/s 9 "s subject f :i* x is insep>able4 ,& inde$1 :atev] may be dissolv$ is corruptible1 z be+ comp\nd$ f _m4 ,b :atev] naturally subsi/s 9 some?+ diff]5t f itself1 :5 x is sep>at$ f its subject1 imm$iately vani%es into non-entity4 ,b ! s\l1 z we h prov$1 is incorporeal2 x is likewise remov$ f e subject1 & naturally rev]ts to itself1 & is !re=e immortal & incorruptible4 #b4 ,let u n[ consid] h[1 & on :at acc\nt ! s\l is said to be ( a middle nature1 & to be ! receptacle ( all middle en]gies1 bo? vital & gno/ic4 ,s9ce1 !n1 !re is a long gradation ( be+s1 proce$+ f ! fir/ be+1 ev5 to =mless matt]1 :i* is no?+ m ?an ! d>k %ad[ ( ess5ce1 x is requisite to enquire :at ! prop]ties are ( ! fir/ & la/ be+s1 & :at ! condition is ( ! middle ord]s2 = ?us we %all "k :]e ! ess5ce ( ! s\l ought to be plac$4 ,! prop]ties ( intelligible natures1 !re=e1 are z foll[s3 true be+1 et]nal1 indivisible1 immoveable1 total1 p]fect1 full ( ess5ce1 replete ) life1 free1 mov+ all ?+s1 similitude1 presid+ ov] all ?+s1 & at ! same "t sep>at$ f all2 = ea* ( ^! prop]ties appe>s 9 intelligibles1 accord+ to ! processions ( be+4 ,b ! prop]ties ( s5sible natures1 diff]5t f ^! by ! greate/ int]val1 are su* z1 n-true-be+1 temporal accord+ to ess5ce1 p>tible1 moveable1 p>ticul>1 indig5t ( ano!r1 always replete ) subsi/5ce1 liv+ by p>ticipation1 mov$ by ano!r1 dissimilitude1 & occupy+ place by p>ts4 ,b ! middle prop]ties ( ^! are1 n-true-be+1 an ess5ce bett] ?an non-be+1 & inf]ior to true-be+1 accord+ to ess5ce et]nal1 b accord+ to its en]gies ext5d$ ) "t1 indivisible accord+ to its div9e "p1 b divisible accord+ to ! v>i\s processions ( reasons1 self-motive1 gov]n+ ?+s mov$ by ano!r1 b subord9ate to su* z are immoveable1 be>+ be=e itself a p>ticul> nature1 toge!r ) its totality2 7=1 because x conta9s 9 itself all reasons1 x is aft] a mann] a :ole1 b because x is dim9i%$ & fall51 ends 9 p>ts1 & suff]s a transition ( its en]gy1 x mu/ be e/eem$ a p>ticul> nature73 & aga91 p]fect+ itself1 yet1 nev]!less1 p]fect$ by natures prior to its own2 fill+ itself ) p[] & /r5g?1 & at ! same "t fill$ by o!rs3 liv+ f itself1 & receiv+ life f o!rs1 be+ m div9e1 inde$1 ?an ?+s :i* live only by p>ticipation1 b inf]ior to ?+s prim>ily vital2 mov+ o!r ?+s1 & itself mov$ by o!rs2 at ! same "t simil> & dissimil>2 & sep>at$1 at ! same "t1 f la/ natures1 & co-ord9at$ ) !m4 ,su* !n1 be+ ! prop]ties ( ! fir/1 middle1 & la/ ord]s1 let u consid] :]e we ought to place ! s\l1 :e!r 9 ! fir/ ord]1 or 9 ^? :i* reta9 ! la/ place3 b if we e/abli% x among ! fir/1 x mu/ be true be+1 e way et]nal & immoveable1 & x mu/ consequ5tly possess e ?+ :i* we h attribut$ to ! fir/ be+s2 add too1 t on ?is hypo!sis we c no long] attribute to ! s\l a p[] ( self-motion1 nor ! discursive processions ( reason+1 nor a v>iety ( o!r p>ticul>s1 :i* manife/ly belong to ! s\l4 ,b c we place x ) propriety among ! la/ ( be+s8 ,! lea/ ( all3 = on ?is hypo!sis we %all make x alone mov$ by o!rs1 divisible1 composite1 & alone possess+ p]fection f o!rs1 ! opposite ( :i* is evid5t 9 all our s\ls2 s9ce !y move & p]fect !mselves1 & are l$ :]ev] !y please4 ,s9ce !n x is n possible to place ! s\l ei!r 9 ! fir/1 or la/ ord] ( be+s1 x is requisite to assign x a middle place1 9 imitation ( its div9e cause ,rhea 7accord+ to ! !ologi/s71 :o is ! conciliat+ b& ( ! two p>5ts ,saturn & ,jupit]1 & is report$1 f h] prolific bosom to produce ! life ( ! s\l4 ,b ?\gh ! s\l is ?us ! extremity ( intelligible1 & ! pr9ciple ( s5sible natures1 we mu/ n conceive x to be su* a pr9ciple or extremity z a po9t 9 a l9e1 = x is n 9 bo? ! natures x t]m9ates1 l a po9t 9 bo? ! sections ( a l9e2 b x is to be call$ ! extremity ( intelligibles1 because x appe>s aft] an intelligible ess5ce1 & ! pr9ciple ( s5sibles1 z be+ ab/ract$ f !m1 & ! s\rce ( _! motion4 ,& ?us x w pres]ve to u a c]ta9 proportion1 t z ! natures :i* are mov$ by o!rs1 are to ^? :i* are mov$ by !mselves1 s are ^! la/ to immoveable natures2 & h5ce x w obta9 ! condition ( a bond1 on acc\nt ( its peculi> m$iocrity1 unfold+1 inde$1 unit$ causes1 b r$uc+ ! dissipat$ p[]s ( s5sibles into "o1 & be+ conta9$ by an immoveable & p]petually abid+ cause2 b conta9+ itself ! g5]ation1 :i*1 mov$ by ano!r1 is subject to cont9ual mutation4 ,x is likewise intelligible1 if we reg>d g5]at$ natures2 b g5]at$1 if we comp>e x ) intelligibles2 & ?us x exhibits 9 its middle nature bo? extremes1 imitat+ also1 by ?is means1 7accord+ to ! ,greek !ologi/s7 its div9e cause1 = x is said to be on bo? sides refulg5t1 1 & to be endu$ ) two faces 1 & to receive 9 its bosom ! processions ( intelligible natures4 ,x is likewise said to be repl5i%$ ) intellectual life1 & to be ! f\nta9 ( ! ev]-runn+ /reams ( corporeal life1 & to conta9 9 itself ! c5tre ( ! processions ( all be+s4 ,on ?is acc\nt x is1 ) great propriety1 affirm$ to be g5]at$1 & at ! same "t )\t g5]ation4 ,= true be+1 accord+ to ! ,platoni/s1 is )\t g5]ation1 because x has an inf9ite p[] ( be+ totally pres5t at ! same "t3 & body is said to be g5]at$1 because x always possesses 9 itself an inf9ite fl[+ p[]1 :i* x _c at once totally receive4 ,! s\l1 !re=e1 because x is incorporeal1 abid+ 9 itself1 has an inf9ite p[] ( be+1 & ?is total ) respect to its ess5ce1 & immortal )\t g5]ation2 b accord+ to p>ts x may be consid]$ 9 inf9ite production4 ,= x has n ! same total inf9ity ev] pres5t1 or !re w\ld be ! same inf9ity ( ! :ole & "p1 ( ! p]fect & imp]fect1 ( ! conta9$ & conta9+1 :i* is impossible4 ,b nei!r is x possible t ! :ole ( its ess5ce %\ld be 9 ! act ( p]petual production1 any m ?an t a "p ( x %\ld be et]nal be+1 le/ ! "p %\ld be m wor?y & bett] ?an ! :ole4 ,h5ce ! 1 or subsi/5ce ( ! s\l1 is at ! same "t ( inf9ite p[]1 & is g5]at$ 9 inf9itum2 = by ?is means x p>ticipates ( be+1 & obta9s ! fir/ place among g5]at$ natures2 :ile body alone1 bo? ) respect to its :ole & p>ts1 is obnoxi\s to a p]petual g5]ation4 #c4 ,b let u n[ enquire f :at g5]a ,plato composes ?is nature ( ! s\l1 :i* conta9s 9 itself ! bond ( all be+s2 previ\s to :i* x w be requisite to expla9 :at ^! g5]a are1 & f :5ce !y orig9ate4 ,( ! species1 !n1 exi/+ 9 ! intelligible _w1 or ! div9e intellect1 :i* conta9s 9 itself ! causes ( all po/]ior natures1 "s are mo/ g5]al1 ext5d+ !mselves to ! univ]sality ( ?+s2 b o!rs are m p>ticul>1 l ! mo/ special species1 & o!rs subsi/+ betwe5 ^!1 exp& !mselves1 inde$1 to a multitude ( ?+s1 b n to all1 accord+ to ! division1 ( ! ,elean gue/ 9 ! ,sophi/a4 ,= man is produc$ f ! ideal man1 & horse f ! ideal horse1 9 ! intelligible _w2 b ! similitude :i* is f\nd 9 man & horse1 & o!r animals1 is produc$ f lik5ess itself1 or ! ideal similitude1 z dissimilitude f unlik5ess itself2 b ! sam5ess & diff]5ce :i* are f\nd 9 all be+s1 proce$ f ! sam5ess & diff]5ce :i* subsi/ 9 ev]-vital en]gy & p]fection1 9 ! supreme intellect1 or ! ideal _w4 ,n[1 z among ! sci5ces "s are especially univ]sal1 s 9 intelligible causes "s are p]fectly p>ticul>1 presid+ alone ov] ! prop] & peculi> numb] ( "o species2 b o!rs ext5d !mselves to a multitude1 su* z equality1 lik5ess1 totality1 7= ! :ole consid]$ z a :ole is n common to all ?+s1 s9ce ! "p is n a :ole72 b o!rs1 aga91 exp& !mselves to all ?+s1 z all be+s p>ticipate ( ^!1 consid]$ z be+s1 & n consid]$ z vital or animat$1 or possess+ any o!r prop]ty exclusive ( ! d5om9ation ( be+4 ,because1 !re=e1 be+ is ! fir/1 ! causes ( be+ obta9 ! mo/ univ]sal ord] among g5]a2 & ^! are five 9 numb]1 z foll[s1 ess5ce1 sam5ess1 diff]5ce1 motion1 & /ation4 ,= e be+ is endu$ ) ess5ce2 is unit$ to itself2 is by itself1 or its own sam5ess1 sep>at$ f o!rs2 proce$s f itself1 & its own /ate & pr9ciple1 & no less appe>s to p>ticipate ( a c]ta9 abid+1 9 pres]v+ its own prop] species4 ,all ?+s1 !re=e1 :e!r intelligibles or s5sibles1 or subsi/+ betwe5 bo?1 dep5d on ^! g5]a = _! exi/5ce4 ,= )\t ! be+ ( ess5ce1 no?+ c\ld subsi/2 9 l mann]1 )\t sam5ess e :ole w\ld be dissipat$1 & divid$ f itself2 & diff]5ce be+ tak5 away1 all ?+s w\ld be "o alone1 & multitude be de/roy$4 ,b )\t motion & /ation1 all ?+s w\ld ei!r be inefficaci\s & dead1 or1 los+ _! prop] /ate & /ability1 w\ld end 9 non-entity4 #d4 ,su* !n be+ ! middle nature ( ! s\l1 ,plato1 ) great propriety1 9 ! ,phdrus1 & 9 8 t5? book ( laws1 def9es x to be numb] mov+ itself2 :i* def9ition he receiv$ f ,philolaus1 & ,philolaus f ,py?agoras4 ,= s9ce ma!matical species h a middle subsi/5ce1 z is prov$ 9 ! foll[+ ,comm5t>ies1 !y are ( all ?+s mo/ accommodat$ to ! nature ( ! s\l4 ,h5ce ,plato1 9 imitation ( ,py?agoras1 sometimes expla9s ! s\l by numb]1 z 9 ! pres5t in/ance2 & sometimes by figure & magnitude1 z 9 ! ,timu2 :ile he consid]s 9 ! s\l ! int]section ( l9es1 & a tw(old circle4 ,= s9ce ma!matical =ms are sep>at$ f ! flux & incon/ancy ( matt]1 !y p>ticipate ( a c]ta91 exact1 sure1 & exquisite condition1 by means ( :i* !y em95tly conf] to ! elevation ( our +5uity1 & ! explication ( lat5t conc]ns2 &1 on ?is acc\nt1 z !y p]ta9 to numb]s1 we may say1 pres]v+ ! analogy1 t !re are five ord]s ( numb]s1 ! div9e1 ! ess5tial1 ! animative1 ! natural1 & ! ma!matic4 ,! fir/ ( ^! is uni=m1 ! second immoveable1 ! ?ird self-motive1 ! f\r? mov$ by ano!r1 & ! la/ ! image ( ! o!rs1 & _! ext]nal measure4 ,! div9e numb] is consid]$ em95tly 9 ! deity1 z 9 ! pr9ciple ( all ?+s2 ! ess5tial belongs to intellect ?r\gh ideas1 & is call$ ess5ce1 unity1 & ! fir/ be+2 ! animative numb] belongs to ! s\l1 ?r\gh ! m$ium ( h] inh]5t reasons2 ! natural to physical conc]ns1 ?r\gh ! se$s ( nature2 & la/ly1 ! ma!matical belongs to op9ion1 z x is no?+ m ?an ! image ( ess5tial numb]1 =m$ by ! en]gies ( ! rational s\l4 ,! s\l1 !re=e1 is numb]1 n limit$ by quantity1 & ma!matical1 b animative2 x is numb]1 n inde$ numb]+1 b numb]$1 g5]at+ & conv]t$ into itself4 ,h5ce too1 because h>mony arises f numb]1 ! s\l is call$ h>mony2 n1 inde$1 a h>mony ( ! p>ts ( ! body1 nor ! h>monic quantity :i* subsi/s 9 s\nd or 9 voice2 b a h>mony aris+ f its ess5tial numb]s1 plac$ 9 its inh]5t reasons1 & 9 ! g5]a :i* con/itute its nature4 ,x is ?is h>mony :i* produces1 z f its prop] cause1 ! h>mony ( ! corporeal p>ts1 ! rhy?m ( motions1 & ! melody ( voices & s\nds4 ,x is ?is :i* produces t delight 9 ! s\l f s5sible h>mony1 :i* suffici5tly indicates x to be some?+ famili> & dome/ic to h] nature4 ,f h5ce x may be inf]r$1 t ,plato is n inconsi/5t ) himself :51 9 ! ,phd1 he d5ies1 & 9 ! ,timu affirms1 ! s\l to be h>mony2 = he d5ies t x is a h>mony ( a def9ite quantity1 or su* z arises f ! p>ts ( ! body2 b he ass]ts x to be a h>mony 9 ! mann] already expla9$4 ,x may likewise be inf]r$1 t ,plato is ignorantly accus$ by a _m1 = affirm+ t ! s\l is h>mony1 or numb]2 = !y only reg>d vulg> ma!matical numb]1 & s5sible h>mony2 :ile ,plato1 f> m elevat$1 disc\rses ( intelligible numb]s1 & ideal h>mony1 subsi/+ 9 immat]ial en]gy & p]fection4 #e4 ,& "h x is necess>y to consid] :at numb]1 9 a p>ticul> mann] belongs to ! s\l2 = v>i\s numb]s1 diff]5tly consid]$1 accord ) h] self-motive nature4 ,9 ! fir/ place1 union & unity may be consid]$ 9 ! s\l1 z 9 h] prop] degree %e p>ticipates ( div9e unity2 & likewise ) relation to h] totality1 = %e is "o c]ta9 :ole4 ,& because a :ole may be consid]$ 9 a triple respect1 "o be=e ! p>ts1 ano!r ris+ f ! aggregate ( p>ts1 & a ?ird subsi/+ 9 ! s+le p>ts2 ! s\l is a :ole 9 ea* ( ^! respects4 ,?us %e is a :ole prior to ! p>ts1 :ile %e is consid]$ z divisible into !m1 9 an incorporeal mann]2 %e is a :ole ris+ f p>ts1 :ile assum+ ! p>ts 9 ! fir/ place1 we consid] h[ h] nature is fabricat$ f _! conjunction2 & %e is a :ole 9 ! s+le p>ts1 s9ce %e is total 9 ! :ole1 & 9 e "p4 ,besides1 ! duad belongs to ! s\l1 because %e conta9s 9 h] nature b\nd & inf9ite1 sam5ess & diff]5ce2 & la/ly1 a conv]sion to intelligibles & s5sibles4 ,&1 inde$1 ! duad conjo9$ ) unity1 v prop]ly accords ) ! s\l2 = to intellect above s\l1 unity p>ticul>ly belongs2 to body b5ea? s\l1 ! inf9ite alone2 & to s\l situat$ 9 ! middle1 duality prop]ly agrees1 be+1 z x 71 inf9ite1 connect$ ) unity4 ,aga91 ! t]n>y numb] is attribut$ to ! s\l1 z well on acc\nt ( h] beg9n+1 middle1 & end1 z because %e abides 9 h]self1 proce$s to inf]iors1 & returns to sup]nal natures4 ,besides1 %e fl[s f ! "o1 rec$es f him1 & is reflect$ into ! "o :5 %e acquires h] prop] p]fection4 ,la/ly1 z ,proclus obs]ves1 ! nature ( ! s\l is divid$ into ess5ce1 p[]1 & en]gy2 s t %e may be said to rejoice 9 ! t]n>y numb]1 & to be replete ) its p]fection4 ,b ! quat]n>y numb] belongs to ! s\l s f> z %e is connect$ ) matt]1 :i* is temp]$ ) f\r qualities1 & f\r elem5ts2 & %e is endu$ ) f\r pr9cipal faculties1 nutrition1 s5sation1 local motion1 & intellection4 ,b to omit o!r numb]s1 & _! con=mity ) ! s\l1 ! qu9>y1 & sept5>y numb]s are especially attribut$ to ! s\l4 ,! qu9>y1 because ! s\l is compos$ f ! five g5]a ( ?+s1 we h previ\sly expla9$2 & because five p>ticul>s m]it a pr9cipal consid]ation 9 ! s\l2 fir/1 h] ess5ce2 secondly1 ! h>mony ( h] reasons2 ?irdly1 ! species aris+ f ! concord ( h] p>ts or reasons2 f\r?ly1 h] virtue2 &1 la/ly1 h] en]gies3 & on ?is acc\nt1 ,proclus obs]ves1 ! consid]ation ( ! s\l ought to receive a qu9tuple di/ribution4 ,besides1 z ! s\l consi/s f a divisible & indivisible nature2 s ! qu9>y is compos$ f ! fir/ ev5 & ! fir/ odd numb]4 ,la/ly1 z ! s\l is ! connect+ m$ium ( ! univ]se2 s ! qu9>y obta9s ! middle situation 9 univ]sal numb]1 t is 9 ! decad4 ,b ! sept5>y numb] belongs to ! s\l1 because1 z ,plato %ews1 9 8 ,timu1 all h>monical reasons are conta9$ 9 ! sev5 numb]s1 #a1 #b1 #c1 #d1 #i1 #h1 #bg2 & f ^! ! s\l is compos$4 ,h5ce ,proclus elegantly obs]ves1 t ! sept5>y numb] is d$icat$ to ,apollo1 ! p>5t ( all h>mony2 because 9 "o1 two1 & f\r1 f :i* ! sept5>y results1 ! fir/ bisdiapason is f\nd4 ,besides1 ! writ]s on h>mony affirm t all ! diff]5ce ( voices proce$s z f> z to ! sev5? degree4 ,s t ,plato uses1 ) great propriety1 ! sept5>y numb] = ! composition ( ! s\l4 ,aga91 9 ^! numb]s ( ! s\l1 e m$ium is f\nd 9 a conv5i5t proportion2 9 ^! ! geometric m$ium is conta9$1 correspond+ to ! "r in/itution ( laws2 ( :i* ,plato1 9 8 ,republic1 says1 t by ?is cities are prop]ly gov]n$3 !re is f\nd too1 an h>monical m$ium1 :i* is ! similitude ( ju/ice3 & la/ly1 we may discov] an ari?metical m$ium1 :i* is call$ ! illu/ri\s symbol ( peace4 ,aft] ?is mann]1 !re=e1 ,plato1 ) a wond]ful fecundity ( significations1 affirms t ! s\l is compos$ f numb]s4 ,he likewise consid]s ! figures %e conta9s1 ,i mean ! circle & triangle2 because z ! s\l is ! fir/ nature :i* v]ges to body1 s ^! are ! fir/ ( all figures1 z well rectil9e> z curvil9e>4 ,besides1 an orbiculation agrees to ! s\l1 ?r\gh intellect2 b progression & rectitude accord+ to h] own prop] nature2 & on ?is acc\nt %e conta9s bo? a /raight & circul> figure4 ,i omit o!r correspond5cies ( a circle & triangle ) ! s\l1 z !y are exhibit$ 9 ! foll[+ ,comm5t>ies2 & p>ticul>ly by ,proclus1 9 ! ?ird book ( 8 ,comm5t>ies on ! ,timu3 x is suffici5t to ! design ( ?is ,diss]tation1 j to h m5tion$ ?is analogy1 t ! nature ( ! numb]s & figures may appe>1 :i* are consid]$ by ,plato 9 ! composition ( ! s\l4 #f4 ,let u n[ pass f contemplat+ ! nature ( ! s\l1 to a survey ( its v>i\s gradations ( k1 & ! means by :i* x acquires ! illum9ations ( sci5ce2 z ?is is a speculation p]fectly ess5tial to a full compreh5sion ( ! ensu+ ,comm5t>ies4 ,accord+ to ,plato1 !n1 9 ! six? & sev5? books ( 8 ,republic#bh1 !re are f\r degrees ( ! int]nal cognitions ( our s\l2 imag9ation1 or assimilation2 fai?1 cogitation 772 & la/ly1 sci5ce or wisdom4 ,! two fir/ degrees conjo9$ con/itute op9ion2 b ! two la/ equally jo9$ produce intellig5ce 9 its l>ge acceptation4 ,i say 9 its l>ge acceptation1 because ! ^w intellig5ce is consid]$ by ! ,platoni/s 9 a triple respect4 ,fir/1 z x rises f op9ion & sci5ce2 z ,plato ass]ts1 9 ! sev5? book ( 8 ,republic4 ,secondly1 z x passes into ! same ) sci5ce2 = ?us1 9 ! end ( ! six? book1 he consid]s intellig5ce & sci5ce z ! same4 ,la/ly1 z x is di/+ui%$ f sci5ce1 & intelligible f t :i* is ! object ( cognition3 ?us1 sci5ce reg>ds ! ess5tial reasons ( ! s\l2 b intellig5ce elevates u to ideas1 & ?is is intellig5ce 9 its prop] acceptation4 ,! di/9ction ( ^! f\r degrees1 especially dep5ds on ! di/9ction ( ?+s ) :i* ! s\l is conv]sant4 ,= ^! f\r degrees ( ?+s are1 ! image ( a s5sible object1 ! s5sible object itself1 ! image ( an intelligible object1 & ! intelligible itself4 ,imag9ation or assimilation reg>ds ! image ( ! s5sible object2 :i* image is no?+ m ?an ! %ad[ or resemblance appe>+ 9 wat]1 or o!r lucid & poli%$ bodies4 ,fai? is conv]sant ) s5sible objects2 & ^! are animals1 plants1 & e ?+ subject to ! en]gies ( s5se4 ,f ! junction ( imag9ation & fai?1 op9ion is produc$4 ,cogitation is conv]sant ) ! image ( an intelligible object2 :i* is no?+ m ?an a c]ta9 univ]sal1 collect$ f s5sibles1 relat$ to ! reasons ( ?+s exi/+ 9 ! s\l1 & con/itut$ by _! assi/ance2 b n elevat$ to ideas1 & resolv$ into _! lucid nature4 ,^? :o are conv]sant ) ?is image ( an intelligible object1 use hypo!ses2 :i* d n elevate u to pr9ciples1 :i* are reasons & ideas1 b br+ u d[n to subord9ate objects4 ,la/ly1 sci5ce1 consid]$ z ! same ) intellig5ce1 is conv]sant ) t :i* is intelligible1 or ! ess5ce ( ?+s2 & ( ?is k9d are reasons p]ta9+ to ! s\l1 & ideas to intellect4 ,h5ce1 z intelligible is to s5sible 9 spl5dor & tru?1 & =m to its image2 su* is ! relation ( ! sup]ior to ! inf]ior degrees ( cognition4 ,& z ^! f\r degrees may be resolv$ into five1 by sep>at+ intellig5ce f sci5ce2 s !y are r$uc$ by ,plato into two pr9cipal degrees2 s f> z ! two fir/ are conv]sant ab\t g5]ation2 b ! two la/ ab\t ess5ce4 ,& ?us mu* = ! fir/ p>ticul> propos$4 ,x n[ rema9s t we inve/igate ! mode 9 :i* sci5ce is produc$ 9 ! s\l1 accord+ to ! doctr9e ( ,plato4 ,= ?is purpose1 ,i ?9k x w be necess>y to consid] ! rational s\l1 9 ^: nature all reasons correspond+ to ideas1 h be5 ins]t$ f et]nity1 receiv$ into ! human body1 z into ! pla9 ( oblivion2 bord]+ on ! riv] ( neglig5ce1 t is plac$ ne> to ! flux ( hum\rs2 :i* produc+ 9 its nature v>i\s k9ds ( p]turbations1 are ! causes ( its self-oblivion & neglect4 ,h5ce ! s\l1 ?us con/itut$1 z x 71 sleep+ & intoxicat$2 sleep+ be=e x is r\s$2 intoxicat$ be=e x is purifi$2 beg9s f ext]nal s5sible objects to be mov$ & excit$1 & to seek ) avidity1 ! k cong5ial to its nature4 ,! s\l n[1 enquir+ aft] tru? & ! sci5ces1 is fir/ conv]sant ) ! ext]nal images ( ?+s1 9 :i* ! glimm]+ light ( similitude to tru?1 pres5ts itself to ! view2 !n x di/+ui%es ^! images among !mselves by its reason+ p[]2 & if !y agree 9 any p>ticul>1 collects !m into "o4 ,?us1 be+ employ$ 9 sep>at+ into _m1 ?+s unit$1 & r$uc+ _m into "o1 x advances f %ade to sub/ance1 & is elevat$ f similitude to tru? itself2 & ?us appreh5ds ! ess5ce ( a ?+ free f e =eign or contr>y quality1 %9+ 9 reason & idea4 ,aft]w>ds1 f ?is contact1 especially salut>y to itself1 x exp]i5ces an ineffable joy1 z f a return into its prop] nature1 & be/ disposition2 & s great is its exultation1 t x neglects & despises ! %ad[s :i* x =m]ly pursu$4 ,!n ! s\l truly kn[s1 t :ile a man reg>ds corporeal natures1 he is employ$ 9 resemblances2 & t ?\gh he may e/eem himself kn[+ 9 _m ?+s1 he kn[s no?+ 9 reality2 b is !n alone elevat$ to ! sublime degree ( sci5ce1 :5 he arrives at ideas4 ,h5ce x appe>s1 t !re are f\r subord9ate dispositions ( mank9d 9 ord] to sci5ce4 ,=1 9 ! fir/ place1 *ildr51 z new gue/s ( ?is _w1 are ignorant ( e ?+1 )\t be+ consci\s ( _! ignorance4 ,b z !y advance 9 ye>s1 !y are employ$ 9 ! %ad[s & images ( natural conc]ns2 & n be+ yet conv]t$ to ! ess5ces ( ?+s1 ?\gh !y possess no real k1 yet !y conceive !mselves to ab\nd 9 k4 ,b 9 ! ?ird place1 be+ elevat$ to sup]nal natures1 & judg+ ^! to be alone true1 !y affirm !mselves to be ignorant ( all !y =m]ly imag9$ !mselves to "k2 9 :i* degree ,socrates pr(ess$ to f9d himself1 :5 he said1 ?is "o ?+ ,i "k1 t ,i "k no?+2 an ignorance pref]able to all ! k ga9$ by ! mo/ unwe>i$ exp]im5tal enquiries4 ,inde$1 ?is ord] is indicat$ by ,socrates himself1 9 ! ,phd1 :5 he says1 t on 8 fir/ acqua9tance ) natural conc]ns1 he ?\ght he ab\nd$ 9 k1 b z he advanc$ 9 ^! pursuits1 he p]ceiv$ t he knew no?+4 ,la/ly1 :51 ?r\gh ! pi]c+ sight ( ! s\l1 m5 are elevat$ to ideas1 !y become illu/rat$ ) ! spl5dors ( true sci5ce1 & pass into ! regions ( p]fect reality4 ,& h5ce x appe>s h[ true sci5ce1 :i* is ! same ) intellig5ce & wisdom1 produces piety & religion3 = sci5ce elevates u to intellect & div9ity2 & copulates ! s\l ) natures ( ! greate/ purity & p]fection2 s t an union ( ?is k9d _c take place )\t piety1 sanctity1 & religion2 z dissimil>s c nev] be bl5d$ 9 amicable conjunction4 ,,section ,,iv4 ,we are in=m$ by ,proclus1 9 ! ensu+ ,comm5t>ies1 t ! end ( geometry1 &1 inde$1 ( ma!matics 9 g5]al1 is to be ref]r$ to ! en]gies ( intellect2 & t x is degrad$ :5 made subs]vi5t to ! common utilities ( a m]e animal life4 ,b z ! v opposite to ?is is ! prevail+ op9ion ( ! pres5t age1 let u exam9e ! tru? ( ?is doctr9e1 & att5d to ! argum5ts :i* ! ,platonic philosophy af=ds 9 its def5ce4 ,= if we c prove t ?is ass]tion ( ,proclus is support$ by ! /ronge/ evid5ce1 we %all v9dicate ! dignity ( true geometry1 re/ore x to its anci5t e/eem 9 ! m9ds ( ! lib]al1 & %ew h[ mu* x is p]v]t$ by apply+ x to contr>y purposes4 ,9 ord] to ?is1 ,i %all endeav\r to prove ! foll[+ position1 t ?+s valuable = _! own sakes1 are pref]able to su* z ref] to some?+ else4 ,n[1 ?is may be demon/rat$1 by consid]+ t e natural production 0 made ) ref]5ce to "s end1 z is evid5t f an induction ( p>ticul>s2 & if ?is be ! case1 x may be safely inf]r$1 t e ?+ exi/s = ! sake ( ! end4 ,b t = ! sake ( :i* any be+ subsi/s is ! be/ ( all2 & ! end1 accord+ to nature1 is t :i* is p]fect$ ! la/ ( all1 f ! bir? ( any be+4 ,h5ce ! human body receives its end or p]fection fir/1 b ! s\l la/4 ,& h5ce ! s\l is po/]ior to ! body1 9 ! accompli%m5t ( its nature2 & its ultimate p]fection is wisdom4 ,x is on ?is acc\nt t old age alone pursues & desires ! goods ( prud5ce & wisdom4 ,h5ce1 wisdom is a c]ta9 end to u accord+ to nature2 & to be wise1 is ! extreme or f9al cause = :i* we 7 produc$4 ,x 01 !re=e1 beautifully said by ,py?agoras1 t man 0 con/itut$ by div9ity1 t he might "k1 & contemplate4 ,if !n wisdom be ! end ( our nature1 to be endu$ ) wisdom mu/ be ! be/ ( all4 ,s t o!r ?+s are to be p]=m$ = ! sake ( ! good :i* ?is conta9s4 ,b to enquire 9 e sci5ce some?+ besides ?is1 & to require t x %\ld be useful1 is alone ! employm5t ( "o ignorant ( ! great diff]5ce betwe5 ! mo/ illu/ri\s goods1 & ?+s necess>y4 ,= !y diff]1 inde$1 widely2 s9ce ?+s are to be call$ necess>ies1 :i* are ! objects ( desire = ! sake ( o!rs1 & )\t :i* x is impossible to live4 ,b ^? conc]ns alone are prop]ly good1 :i* are lov$ by !mselves1 ?\gh no?+ else %\ld fall to ! lot ( _! possessor2 = "o ?+ is n to be desir$ = ! sake ( ano!r inf9itely1 b x is requisite to /op at "s limit$ object ( desire1 ( :i* x w\ld be ridicul\s to require any utility ab/ract$ f itself4 ,b y w ask1 ,:at is ! emolum5t ( contemplative wisdom1 :at ! good x conf]s on its possessor8 ,:at if we %\ld say 7= su* is ! tru? ( ! case7 t x transports u by intellect & cogitation1 to regions simil> to ! =tunate isl&s2 = utility & necessity are /rang]s to ^? happy & lib]al realms4 ,& if ?is be admitt$1 ought we n to blu%1 t hav+ x 9 our p[] to become inhabitants ( ! =tunate isl&s1 we neglect ! pursuit1 ?r\gh a sordid enquiry aft] :at is useful & pr(itable accord+ to vulg> e/imation8 ,! rew>ds ( sci5ce1 !re=e1 are n to be repreh5d$1 nor is x a trifl+ good :i* results f its acquisition4 ,besides1 z m5 travel to ! m\nta9 ,olympus = its spectacle alone1 pref]r+ a view ( its l(ty summit to mu* weal?2 & z _m o!r spectacles are desir$ = _! own sakes1 & valu$ beyond gold1 9 l mann] ! speculation ( ! univ]se is to be priz$ above e ?+ :i* appe>s useful to ! purposes ( life3 = x is surely %ameful t we %\ld eag]ly frequ5t ! !atre1 & ! race1 = ! sake ( ! delight af=d$ to our corporeal sight1 & %\ld look = no f>!r utility 9 ^! ?an ! pleasure !y produce2 & yet %\ld be s sordidly /upid z to ?9k t ! nature ( ?+s1 & tru? itself are n to be speculat$ )\t "s f>!r rew>d ?an ! s9c]e delight _! contemplation af=ds4 ,x is on ?is acc\nt t ! appreh5sion ( tru? is comp>$ to corporeal vision2 = ! sight is ! mo/ lib]al ( all ! s5ses1 z is confirm$ by ! g5]al te/imony ( mank9d4 ,h5ce1 ! sight ( ! sun & moon1 & ! glori\s spectacle ( ! />s is desir$ by ! mo/ illit]ate z well z ! mo/ kn[+1 = ! delight su* visions af=d2 :ile1 on ! contr>y1 ! desires ( ! o!r s5ses are = ! mo/ "p direct$ to some?+ f>!r ?an ! m]e objects ( _! en]gy4 ,?us1 ev5 ! s5se ( he>+1 :i* is ! next 9 dignity to ! sight1 is n always desirable = its own sake2 = light is ! g5]al object ( sight1 & s\nd t ( he>+2 b x is evid5t t light is m univ]sally desir$ ?an s\nd1 s9ce all light1 :5 n excessive1 is always pleas+1 b ?is is by no means ! case ) e k9d ( s\nd4 ,h5ce x is1 t all contemplation is s delightful1 & ?is 9 proportion z x becomes ab/ract$ f s5sible objects2 = ! mo/ beautiful =ms d n produce g5u9e delight1 until !y are /rongly repres5t$ 9 ! phantasy1 z is evid5t 9 ! passion ( love2 s9ce ! faire/ face !n alone causes love :5 x pres5ts itself cle>ly to ! inw>d eye ( ?\ght1 9 ! mirror ( imag9ation1 accompani$ ) liv+ elegance1 & a resi/less en]gy ( =m4 ,inde$1 s lib]al & s exalt$ an employm5t is contemplation1 t ,plot9us1 ) 8 usual pr(undity1 proves t ! univ]se subsi/s = its sake2 t all ! productions ( nature orig9ate f ?is2 & t ev5 actions !mselves are und]tak5 ) a view to ! enjoym5t ( aft]-speculation4 ,may we n1 !re=e1 say t ! sportsman foll[s ! *ace = ! sake ( a subsequ5t review ( 8 fav\rite pursuit8 ,t ! glutton = ?is rejoices 9 ! meal2 & ev5 ! mis] 9 8 weal?8 ,& t conv]sation is alone sollicit$1 t x may recal pa/ images to ! s\l8 ,9 %ort1 contemplation is ! fir/ spr+ ( action1 & its only end2 s9ce we are fir/ incit$ to any ext]nal object by speculat+ its image 9 ! phantasy3 & our subsequ5t conduct t5ds1 )\t ceas+1 to ! en]gy ( reflection2 = de/roy prior & po/]ior contemplation1 & action is no m4 ,n[ if ?is be ! case1 & if geometry is a speculative sci5ce 7,i mean ! geometry ( ! anci5ts71 x is bo? desirable = its own sake1 & = /ill high] contemplations1 ! visions ( intellect1 to :i* x is ultimately subs]vi5t4 ,=1 :5 /udi$ ) ?is view1 x op5s ! eye ( ! s\l to spectacles ( p]fect reality1 & purifies x f ! d>kness ( mat]ial oblivion4 ,away !n1 ye sordid vulg>1 :o are p]petually dem&+ ! utility ( ab/ract speculations1 & :o are impati5t to br+ d[n & debase ! noble/ en]gies1 to ! mo/ grovel+ purposes2 ignorant ( t mighty pr9ciple ( action1 :i* influ5ces e "p ( ! univ]se1 & ?r\gh :i* ev5 division & discord t5d z mu* z possible to union & cons5t2 ignorant t f ! depravity ( y\r nature1 & ! bl9dness ( y\r inw>d eye1 y are incapable ( speculat+ ! sub/ance ( reality1 & are !re=e eag]ly gaz+ on its %ad[3 & la/ly1 unconsci\s t ?is is ! po9t ab\t :i* y are cont9ually mak+ exc5tric revolutions1 mi/ak+ ! circumf]5ce = ! c5tre1 motion = re/1 & a dep>ture f good = a t5d5cy to felicity4 ,x 0 = ! sake ( ?is mo/ exalt$ & lib]al contemplation t ,h]aclitus yield$ 8 "r ( succession to a ?rone1 to 8 bro!r2 & t ,anaxagoras neglect$ 8 patrimony1 e/eem+ "o drop ( g5u9e wisdom pref]able to :ole tuns ( ri*es4 ,l$ by a desire ( ?is1 z by "s guid+ />1 ,py?agoras travell$ into ,egypt1 & *e]fully enc\nt]$ ! greate/ difficulties1 & ma9ta9$ ! mo/ ob/9ate p]sev]ance1 until at l5g? he happily p5etrat$ ! dep?s ( ,egyptian wisdom1 & br\ght into ,greece a treasury ( tru? = future speculation4 ,b ^! 7 happy days2 ?is 0 ! p]iod de/9$ to ! reign ( true philosophy1 & to ! advancem5t ( ! human s\l to ! greate/ p]fection its union ) ?is t]r5e body c admit4 ,= 9 our times1 ! voice ( wisdom is no long] he>d 9 ! sil5ce ( sacr$ solitude2 b folly usurp+ h] place1 has fill$ e qu>t] ) ! b>b>\s & deafn+ clam\rs ( despicable sect>ies2 :ile ! brutal h& ( comm]ce has bl9d$ ! lib]al eye ( div9e contemplation4 ,= un=tunately1 ! circle ( "t1 z x produces cont9ual v>iations1 at l5g? rev]ses ! objects ( pursuit2 & h5ce1 t :i* 0 once des]v$ly fir/1 becomes at l5g?1 by a degrad$ revolution1 ! la/ 9 ! g5]al e/eem4 #b4 ,if geometry1 !re=e1 be bo? valuable = its own sake1 & = its subs]vi5cy to ! mo/ exalt$ contemplations1 !re c be no d\bt b t ! great p]fection to :i* ?is sci5ce 0 br\ght by ! ,greeks1 0 entirely ow+ to _! deep conviction ( ?is important tru?4 ,euclid1 we are in=m$ by ,proclus1 9 ?is "w1 0 ( ! ,platonic sect2 & ,ar*im$es is report$1 by ,plut>*1 9 8 ,life ( ,m>cellus1 to h possess$ su* elevat$ s5tim5ts ( ! intr9sic dignity ( geometry1 t he consid]$ x p]v]t$ & degrad$1 :5 subs]vi5t to me*anical op]ations2 ?\gh1 at ! reque/ ( k+ ,hi]o1 he fabricat$ su* admirable eng9es = ! def5ce ( ,syracuse4 ,f ?is s\rce alone1 ! great accuracy & elegance ( _! demon/rations 0 d]iv$1 :i* h be5 s des]v$ly applaud$ by ! greate/ mod]n ma!maticians1 & ! w>me/ advocates = ! f>rago ( algebraic calculation4 ,algebra1 inde$1 or z x is call$1 speci\s analysis1 is ! mod]n sub/itute = ! p]fect me?od adopt$ by ! anci5ts 9 geometrical demon/rations2 & ?is solely1 because x is capable ( be+ appli$ ) great] facility to ! common purposes ( life4 ,h5ce1 hypo!ses h be5 eag]ly admitt$ 9 geometry1 :i* ! anci5ts w\ld h blu%$ to own3 ,i mean ! multiplications & divisions ( l9es & spaces z if !y 7 numb]s1 & consid]+ geometry & ari?metic z sci5ces p]fectly ! same4 ,b we h =tunately ! te/imony ( ! fir/ ma!maticians among ! mod]ns aga9/ ! unlawfulness ( ?is ungeometrical invasion4 ,& to beg9 ) ! great sir ,isaac ,newton1 9 8 ,univ]sal ,ari?metic#bi3 8,equations 7says he7 are expressions ( ari?metical computation1 & prop]ly h no place 9 geometry1 except s f> z quantities truly geometrical 7t is1 l9es1 surfaces1 solids1 & proportions71 may be said to be "s equal to o!rs4 ,multiplications1 divisions1 & su* sort ( computations1 are newly receiv$ into geometry1 & t unw>ily1 & contr>y to ! fir/ design ( ?is sci5ce4 ,= :oev] consid]s ! con/ruction ( problems by a "r-l9e & a circle1 f\nd out by ! fir/ geometricians1 w easily p]ceive t geometry 0 inv5t$ t we might exp$iti\sly avoid1 by draw+ l9es1 ! t$i\sness ( computation4 ,!re=e1 ^! two sci5ces ought n to be conf\nd$4 ,! anci5ts s indu/ri\sly di/+ui%$ !m f "o ano!r1 t !y nev] introduc$ ari?metical t]ms into geometry4 ,& ! mod]ns1 by conf\nd+ bo?1 h lo/ ! simplicity 9 :i* all ! elegancy ( geometry consi/s40 ,& 9 ano!r "p#cj ( ! same "w he obs]ves1 t 8! mod]n geomet]s are too fond ( ! speculation ( equations40 ,to ?is v high au?ority we may add t ( ,dr4 ,halley1 9 ! preface to 8 translation ( ,apollonius de ,sectione ,rationis2 = :i* "w he conceiv$ s great an e/eem1 t he 0 at ! pa9s to le>n ,arabic 9 ord] to accompli% its translation into ,lat9#ca3 8,?is me?od1 says he1 7( ,apollonius7 cont5ds ) speci\s algebra 9 facility1 b f> excels x 9 evid5ce & elegance ( demon/rations2 z w be abundantly manife/ if any "o comp>es ?is doctr9e ( ,apollonius de ,sectione ,rationis1 ) ! algebraic analysis ( ! same problem1 :i* ! mo/ illu/ri\s ,wallis exhibits 9 ! second volume ( 8 ma!matical works1 cap4 liv4 p4 #bbj40 ,& 9 ! conclusion ( 8 preface1 he obs]ves#cb1 8t x is "o ?+ to give ! resolution ( a problem "s h[ or o!r1 :i* may be accompli%$ by v>i\s ways1 b ano!r to effect ?is by ! mo/ elegant me?od2 by an analysis ! %orte/1 & at ! same "t p]spicu\s2 by a syn!sis elegant1 & by no means op]ose40 ,& ,dr4 ,b>r[1 not)/&+ he 0 s great an advocate = ! id5tity ( ari?metic & geometry1 expressly ass]ts#cc1 t algebra is no sci5ce4 ,to ^! au?orities we may add ,simson & ,lawson1 :o1 s5sible ( ! sup]ior skill ( ! anci5ts1 bo? 9 analysis & syn!sis1 h made laudable attempts to re/ore ! ,greek geometry to its pri/9e purity & p]fection4 ,aga91 ! greate/ m5 ( ! pres5t times h be5 ( op9ion1 t algebra 0 n unkn[n to ! anci5ts2 & if ?is be true1 _! sil5ce respect+ x is a suffici5t pro( ( _! disapprobation4 ,inde$1 if we consid] x :5 appli$ to geometry1 z an art alone subs]vi5t to ! facility ( practice1 z convey+ no evid5ce1 & possess+ no elegance ( demon/ration1 we %all n wond] at its be+ unnotic$ by ! anci5ts1 ) :om practice 0 ev] consid]$ z subs]vi5t to speculation2 & 9 ^: writ+s elegance ( !ory & accuracy ( reason+ are f\nd p]petually unit$4 #c4 ,b ! lives ( ! fir/ cultivators ( ?is sci5ce 7,i mean ! ,egyptian prie/s7 z well z ( ! ,py?agoreans & ,platoni/s1 by :om x aft]w>ds receiv$ su* improvem5ts1 suffici5tly ev9ce t ?is sci5ce advanc$ to p]fection f an intellectual !ory z its s\rce1 & f be+ ref]r$ to contemplation z its end2 & ?is w be evid5t1 by att5d+ to ! foll[+ hi/ory ( ! ,egyptian prie/s1 z pres]v$ to u by ,porphyry1 9 8 excell5t "w on ab/95ce#cd2 a translation ( :i* w n1 ,i presume1 be unacceptable to ! philosophical read]1 8,*remon1 ! ,/oic 7says he7 expla9+ ! rites ( ! ,egyptian prie/s1 :o1 he says1 are acc\nt$ philosoph]s by ! ,egyptians1 relates1 t !y *oose a place be/ adapt$ to ! /udy & p]=mance ( sacr$ rites2 s t a desire ( contemplation is excit$ by only frequ5t+ ^? recesses :i* are d$icat$ to _! use1 & :i* procure safety to ! prie/s1 on acc\nt ( t rev]5ce ( ! div9ity1 ^: sacr$ my/]ies !y p]=m2 s t all possible hon\r is paid to ^! philosoph]s1 9 ! same mann] z to "s sacr$ animals4 ,b he says !y live entirely solit>y1 except at p>ticul> times1 :5 !y mix ) o!rs 9 su* assemblies z are usually held1 & 9 public fea/s2 & t on all o!r occasions !y are sc>cely to be approa*$4 ,= he :o desires to conv]se ) !m mu/ fir/ purify himself1 & ab/a9 f a multitude ( ?+s aft] ! mann] ( ^! ,egyptian prie/s4 ,he adds1 t ^! m51 r5\nc+ e o!r occupation1 & all human affairs1 give !mselves entirety1 ?r\gh ! :ole ( life1 to ! contemplation ( div9e conc]ns1 & to enquir+ into ! div9e w3 by ! latt] ( ^! employm5ts procur+ to !mselves hon\r1 security1 & ! e/imation ( piety2 by contemplation1 trac+ out ! lat5t pa?s ( sci5ce2 & by bo? ^! occupations unit$1 accu/om+ !mselves to mann]s truly occult1 & wor?y ( antiquity4 ,= to dwell always on div9e k1 & be dispos$ = div9e inspiration1 removes a man beyond all immod]ate desires1 calms ! passions ( ! s\l1 & raises h] intellectual eye to ! p]ception ( t :i* is real & true4 ,b !y /udi$ t5uity ( alim5t1 & frugality 9 _! app>el1 & cultivat$ temp]ance & pati5ce1 toge!r ) ju/ice & equity1 9 all _! conc]ns4 ,inde$1 a solit>y life r5d]$ !m p]fectly v5]able2 = dur+ t p]iod :i* !y call ! "t ( purification1 !y sc>cely mix$ ) ! associates ( _! own ord]1 or saw any "o ( !m1 except him :o 0 conv]sant ) !m 9 t ex]cise ( purity1 on acc\nt ( necess>y uses4 ,b !y by no means conc]n$ !mselves ) ^? :o 7 unemploy$ 9 ! bus9ess ( purification4 ,! rema9+ "p ( _! "t !y conv]s$ famili>ly ) ^? simil> to !mselves2 b !y liv$ sep>ate & ap>t f ^? :o 7 e/rang$ f _! c]emonies & mann] ( liv+4 ,he adds1 !y are always se5 employ$ among ! resemblances ( ! gods1 ei!r c>ry+ _! images1 or prec$+ !m 9 _! accu/om$ processions1 & dispos+ !m ) gravity ( deportm5t1 & 9 a graceful ord]4 ,9 all :i* op]ations !y did n indicate any pride ( disposition2 b exhibit$ "s p>ticul> natural reason4 ,b _! gravity 0 conspicu\s f _! habit2 = :5 !y walk$1 _! pace 0 equable1 & _! aspect s p]fectly /eady1 t !y refra9$ f w9k+ :5ev] !y pleas$4 ,_! risibility too1 ext5d$ no f>!r ?an to a smile4 ,b _! h&s 7 always conta9$ )9 _! g>m5ts2 & z !re 7 _m ord]s ( prie/s1 e "o c>ri$ ab\t him "s rem>kable symbol ( ! ord] he 0 allott$ 9 sacr$ conc]ns4 ,_! su/5ance too 0 sl5d] & simple2 & ) respect to w9e1 "s ( !m entirely refra9$ f x2 & o!rs drank x v sp>+ly1 affirm+ t x hurt ! n]ves1 0 an imp$im5t to ! inv5tion ( ?+s1 & an inc5tive to v5]eal desires4 ,!y also ab/a9$ f _m o!r ?+s1 nev] us+ bread 9 ex]cises ( purity2 & if !y ate x at o!r times1 x 0 fir/ cut 9 pieces1 & m+l$ ) hyssop4 ,b !y ab/a9$1 = ! mo/ "p1 f oil1 & :5 !y us$ x mixt ) olives1 x 0 only 9 small quantities1 & z mu* z 0 suffici5t to mitigate ! ta/e ( ! h]bs4 ,9 ! mean "t1 x 0 n lawful = any "o to ta/e ( ! alim5t1 :e!r solid or fluid1 :i* 0 br\ght into ,egypt f =eign p>ts4 ,!y likewise ab/a9$ f ! fi% :i* ,egypt produc$2 & f all quadrup$s hav+ solid or _m fissur$ ho(s2 f su* z 7 )\t horns2 & f all c>niv]\s birds3 b _m ( !m ab/a9$ entirely f animal food4 ,& at ^? times :5 !y all r5d]$ !mselves pure1 !y did n ev5 eat an egg4 ,b :5 ! "t drew ne> 9 :i* !y 7 to celebrate "s sacr$ rites1 or fe/ival1 !y employ$ _m days 9 previ\s prep>ation1 "s ( !m sett+ ap>t =ty-two days1 o!rs a great] l5g? ( "t ?an ?is2 & o!rs aga9 a %ort]2 b nev] less ?an sev5 days2 ab/a9+1 dur+ ?is p]iod1 f all animals1 & f all legum9\s & oily nutrim5t1 b especially f v5]eal congress4 ,e day1 !y wa%$ !mselves ?ree times 9 cold wat]2 aft] ris+ f b$1 be=e d9n]1 & :5 !y betook !mselves to re/4 ,& if !y happ5$ to be pollut$ 9 _! sleep1 !y imm$iately purifi$ _! bodies 9 a ba?4 ,!y made cold wat] too subs]vi5t to ! purposes ( purification at o!r times1 b n s (t5 z ! ba?4 ,_! b$s 7 compos$ f ! bran*es ( palm1 :i* !y call$ 1 bais4 ,a piece ( wood1 ( a semi-cyl9drical =m1 & well plan$1 s]v$ !m = a pill[4 ,b ?r\gh ! :ole ( life1 !y 7 ex]cis$ 9 ! endurance ( hung] & ?ir/1 & accu/om$ to a paucity & simplicity ( nutrim5t4 ,b z a te/imony ( _! temp]ance1 ?\gh !y nei!r us$ ! ex]cise ( walk+ nor rid+1 yet !y liv$ free f disease1 & 7 mod]ately /rong4 ,=1 inde$1 !y endur$ great lab\r 9 _! sacr$ c]emonies1 & p]=m$ _m s]vices exce$+ ! common /r5g? ( m54 ,!y divid$ ! night betwe5 obs]vations ( ! cele/ial bodies1 & (fices ( purity2 b ! day 0 de/9$ by !m to ! cultivation ( ! div9ities1 :om !y wor%ip$ ) hymns ea* day ?ree or f\r times2 9 ! morn+ & ev5+1 :5 ! sun is at 8 m]idian1 & :5 he is sett+4 ,b ! re/ ( _! "t !y 7 occupi$ 9 ari?metical & geometrical speculations1 always labori\s & inv5t+1 & cont9ually employ$ 9 ! inve/igation ( ?+s4 ,9 w9t] nights also1 !y 7 dilig5t 9 ! same employm5ts1 & 7 ev] vigilant to lit]>y /udies2 s9ce !y 7 n solicit\s ab\t ext]nal conc]ns1 & 7 fre$ f ! base dom9ion ( intemp]ate desires4 ,_! unwe>i$ & assidu\s lab\r1 !re=e1 is an argum5t ( _! great pati5ce2 & _! cont95ce is suffici5tly indicat$ by _! privation ( desire4 ,besides ?is1 x 0 e/eem$ v impi\s to sail f ,egypt1 z !y 7 p>ticul>ly c>eful 9 ab/a9+ f ! mann]s & luxuries ( =eign nations2 s t to leave ,egypt 0 alone lawful to ^? :o 7 compell$ to x by /ate necessities4 ,b !y disc\rs$ mu* conc]n+ a ret5tion ( _! native mann]s2 & if any prie/ 0 judg$ to h transgress$ ! laws 9 ! lea/ p>ticul>1 he 0 expell$ f ! college4 ,besides1 ! true me?od ( philosophiz+ 0 pres]v$ 9 ,comm5t>ies & ,di>ies1 by ! prophets & m9i/]s ( sacr$ conc]ns3 b ! rema9+ multitude ( prie/s1 ,pa/ophori1 or prie/s ( ,isis & ,osiris1 gov]nors ( temples1 & s]vants ( ! gods1 /udi$ purity1 yet n s exactly1 nor ) s great cont95ce z ^? we h m5tion$4 ,& ?us mu* is relat$ ( ! ,egyptians1 by a man :o is equally a lov] ( tru?1 & ( accurate dilig5ce1 & :o is deeply skill$ 9 ! ,/oic philosophy40 #d4 ,b ! lives ( ! ,py?agoreans & ,platoni/s1 :o c>ri$ ?is div9e sci5ce to its ultimate p]fection1 no less em95tly ev9ce ! tru? ( our position4 ,=1 z ,porphyry in=ms u1 9 ! same invaluable treatise#ce1 8"s ( ! anci5t ,py?agoreans1 & wise m51 inhabit$ ! mo/ des]t places2 & o!rs retir$ into temples1 f :i* ! multitude & e tumult 7 expell$4 ,b ,plato 0 will+ to fix 8 academy 9 a place n only solit>y1 & remote f ! city1 b1 z !y report1 insalubri\s4 ,o!rs1 aga91 h n sp>$ _! eyes1 ?r\gh a desire ( m p]fectly enjoy+ t blissful contemplation1 f :i* !y wi%$ nev] to be sep>at$40 ,aft] ?is1 he pres5ts u ) a description f ,plato#cf ( ^? intellectual m51 by :om ! _w has be5 enlight5$ ) ! sublime/ wisdom & tru?3 8,= x 0 n falsly1 or 9 va9 7says he71 t a c]ta9 philosoph]1 speak+ ( contemplative m51 affirms1 t su* z ^! are ignorant1 f _! e>ly y\?1 ( ! way :i* leads to ! =um1 or 9 :at place ! c\rt or s5ate-h\se is situat$1 or any public c\ncil ( ! /ate4 ,!y nei!r see nor he> ! laws1 :e!r decre$ or promulgat$1 or writt52 & ) respect to ! factions & cont5tions ( _! companions = magi/racy1 = assemblies & spl5did ent]ta9m5ts1 luxuri\s eat+ & m9/rels1 !y d n ev5 ?9k ( ^! z 9 a dream4 ,su* an "o kn[s no m ( ! evil :i* has happ5$ to "s "o ( 8 ance/ors1 :e!r male or female1 or any ?+ belong+ to !m1 ?an h[ _m pit*]s ( wat] are conta9$ 9 ! sea4 ,nor does he ab/a9 f ?+s ( ?is nature = ! sake ( acquir+ fame2 b 9 reality1 8 body alone abides 9 ! city1 & w&]s ab\t f place to place1 b 8 intellect e/eem+ all ^! z ( small importance1 or r z non-entities1 he despises !m1 &1 accord+ to ,p9d>1 8f ^! on e side he so>s30 by no means apply+ himself to ?+s :i* are ne> him1 & to s5sible conc]ns40 ,if su* !n 7 ! lives ( ! m5 :o br\ght ?is contemplative sci5ce to its pres5t p]fection1 & :o are to ?is day our ma/]s 9 geometry2 if su* 7 ! exalt$ s5tim5ts !y ent]ta9$ ( its dignity & wor?1 :at great] pro( c we require ( its be+ valuable = its own sake1 & z subs]vi5t to ! en]gies ( intellect8 ,we h ample evid5ce too1 ( its be+ degrad$ :5 br\ght d[n to ! common purposes ( life1 9 ! example ( ^? :o1 ) ?is view1 h disguis$ x ) ! d>k & sordid involutions ( algebraic calculation2 = x 0 solely to facilitate practice1 t ?is b>b>\s invasion has be5 admitt$ by ! mod]ns4 ,let me !n be p]mitt$ to p]suade ! few :o /udy geometry 9 its anci5t purity1 & :o consid] ! ru9s ( ,grecian lit]ature on ?is1 z well z on e o!r sci5ce1 ! models ( p]fection1 to ent] ) avidity on ! /udy ( ! ensu+ ,comm5t>ies1 & endeav\r to fa?om ! dep? ( our pr(\nd & elegant philosoph]3 = by ?is means !y may happily obta9 ! end ( all true sci5ce1 ! purification ( ! s\l2 & be able to draw ! light ( p]fect wisdom1 f ! undecay+ & inexhau/ible f\nta9 ( good4 ,b if x %\ld be ask$ 9 :at ^! en]gies ( intellect consi/1 to :i* all sci5ce ultimately ref]s8 ,i answ]1 9 ! contemplation ( true be+1 or ^? ideal & div9e =ms1 ) :i* ! intelligible _w is replete4 ,n[ ?is great end is n to be accompli%$ )\t previ\s discipl9e1 a long ex]cise ( ! reason+ p[]1 & a cont9u$ s]ies ( philosophic endurance4 ,= ?is end1 :5 atta9$1 is no o!r ?an ! enjoym5t ( t felicity cong5ial to ! s\l previ\s to h] imm]sion 9 body4 ,b1 = ! fur!r in=mation ( ! lib]al read] on ?is important subject1 ! foll[+ p>aphrases f ,porphyry & ,proclus are subjo9$2 ! =m] in/ruct+ u 9 ! v>i\s purifications necess>y to ?is end2 & ! latt] exhibit+ ! gradations by :i* we may rise to ! speculation ( reality1 & 7leav+ all multitude beh9d7 asc5d to ! div9ely solit>y pr9ciple ( ?+s1 ! ineffable ,"o4 #e4 8,9 ! fir/ place1 !n 7says ,porphyry#cg7 my reasons are n address$ to ^? :o are occupi$ 9 illib]al arts1 nor to ^? engag$ 9 corporeal ex]cises1 nei!r to soldi]s nor sailors1 nei!r to rhetoricians nor to ^? :o h und]tak5 ! duties ( an active life4 ,b ,i write to ! man cont9ually employ$ 9 ?9k+ :at he is1 f :5ce he comes1 & :i!r he ought to t5d3 & :o1 ) respect to e ?+ p]ta9+ to food1 & o!r (fices ( life1 is entirely *ang$ f ^? :o propose to !mselves a diff]5t mann] ( liv+2 = to a man ( ?is k9d alone is my pres5t disc\rse address$4 ,inde$1 9 ?is common /ate ( exi/5ce1 "o & ! same mode ( p]suasion _c be address$ to ! sleep]1 :o1 if x 0 possible1 w\ld conciliate to himself p]petual sleep1 & :o1 = ?is purpose1 seeks on e side = soporif]\s inc5tives1 z to him :o /udies cont9ually to drive away sleep1 & to dispose e ?+ ab\t him to vigilance & intellectual activity4 ,b to ! =m]1 x is necess>y to advise intoxication1 surfeit+1 & satiety1 & to recomm5d a d>k h\se2 &1 z ! poets say1 a b$ luxuri\s1 broad1 & s(t4 ,su* a "o %\ld *use :atev] t5ds to produce /upor1 & give bir? to indol5ce & oblivion1 :e!r consi/+ ( od\rs1 o9tm5ts1 or m$icam5ts :i* are accu/om$ to be eat or drank4 ,b x is necess>y t ! intellectual man %\ld use sob] dr9k1 unmix$ ) ! le?>gic fumes ( w9e2 nutrim5t sl5d]1 & almo/ approa*+ to fa/+2 a lucid h\se1 receiv+ a subtle air & w9d2 t he %\ld be cont9ually agitat$ ) c>es & griefs2 & la/ly1 t he prep>es = himself a small & h>d b$1 :ile ?us employ$ 9 purify+ 8 s\l f ! /a9s contract$ by corporeal involution4 ,b :e!r we are born = ?is exalt$ purpose1 ,i mean = vigilant intellectual en]gies1 all[+ z small a "p ( our life z possible to sleep2 7s9ce we d n exi/ 9 a place :]e s\ls p]petually vigilant abide71 or :e!r we are de/9$ to a contr>y purpose1 ,i mean1 to sleep & oblivion1 w\ld be =eign f our design to expla92 & w\ld require a long] demon/ration ?an ! limits ( our "w w admit4 ,b :oev] once cauti\sly surmizes ! delusions ( our life 9 ! pres5t _w1 & ! in*antm5ts ( ?is mat]ial h\se 9 :i* we are employ$1 & :o p]ceives himself naturally adapt$ to vigilant en]gies2 la/ly1 :o appreh5ds ! soporif]\s nature ( ! place 9 :i* he acts1 to su* a "o we w\ld prescribe a diet congru\s to 8 suspicion ( ?is fallaci\s abode1 & to ! k he possesses ( himself2 9 ! mean "t1 advis+ him to bid a long f>ewel to ! sleep]1 /ret*$ on 8 c\*1 z on ! lap ( oblivion4 ,nev]!less1 we %\ld be c>eful le/1 z ^? :o behold ! ble]-ey$1 contract a simil> defect1 & z we gape :5 pres5t ) ^? :o are gap+1 s we %\ld be fill$ ) dr[s9ess & sleep1 :5 ! place 9 :i* we reside is cold1 & adapt$ to fill ! eyes ) wat]y hum\rs1 f its ab\nd+ ) m>%es & vap\rs1 :i* incl9e _! inhabitants to heav9ess & sleep4 ,if !n1 legislators _h compos$ ! laws ) a view to ! utility ( ! /ate1 & _h ref]r$ ^! to a contemplative & intellectual life z _! end1 we ought to submit to _! in/itutions1 & acquiesce 9 ! diet !y h prescrib$ = our subsi/5ce4 ,b if !y1 only reg>d+ t life :i* is accord+ to nature1 & is call$ ( ! middle k9d1 orda9 su* ?+s z ! vulg> admit1 :o only e/imate good & evil z !y respect ! body1 :y %\ld any "o1 adduc+ ^! laws1 we>y himself 9 endeav\r+ to subv]t a life :i* is f> m excell5t ?an e law writt5 & compos$ = ! sake ( ! vulg>1 & :i* foll[s a law n writt51 b div9ely deliv]$8 ,= su* is ! tru? ( ! case4 ,t contemplation :i* procures u felicity1 is n a mass ( disc\rses1 & a multitude ( discipl9es2 or1 z "s may ?9k1 consi/+ f h5ce2 nor does x receive any increase f a quantity ( words4 ,= if ?is 0 ! case1 no?+ c\ld h9d] ^? f be+ happy1 :o compreh5d all discipl9es1 & are accurately skill$ 9 a v>iety ( languages4 ,b ! :ole circle ( ! sci5ces _c by any means accompli% ?is blissful contemplation1 nor ev5 ^? discipl9es :i* are conv]sant ) true & sub/antial be+1 unless !re is also a con=mation ( our nature & life to ?is div9e end4 ,= s9ce !re are1 z !y say1 ?ree ends ( liv+1 if we reg>d ! p>ticul> objects to :i* mank9d t5d1 ! end ) u is to foll[ ! contemplation ( true be+1 promot+1 z mu* z possible1 by an acquisition ( ?is k9d1 an intimate union ( ! contemplat+ individual ) ! object ( contemplation4 ,=1 9 no?+ else besides true be+1 is x possible = ! s\l to return to its pri/9e felicity2 nor c ?is be effect$ by any o!r conjunction4 ,b intellect is true be+ itself3 s t ! prop] end is to live accord+ to intellect4 ,& on ?is acc\nt1 exot]ic disc\rses & discipl9es1 ret>d+ ! purgation ( ! s\l1 are f> f fill+ up ! measure ( our felicity4 ,if !n1 felicity 0 def9$ by ! compreh5sion ( words or sci5ces1 !y :o d n pay a prop] att5tion to ! k9d & quantity ( _! food1 nor to any ?+ else p]ta9+ to _! pres5t exi/5ce1 might obta9 ?is end3 b s9ce x is requisite to *ange our life1 & to be pure bo? 9 spee* & action1 let u consid] :at disc\rses & :at works may r5d] u p>tak]s ( ?is mo/ necess>y means ( acquir+ sub/antial felicity4 ,are1 !n1 ^? ?+s :i* sep>ate u f s5sible objects1 & f ! affections :i* !y excite1 & :i* lead to a life intellectual1 & void ( imag9ation & passion1 are ^! ! means we are 9 pursuit (8 ,s t e ?+ contr>y is =eign f our purpose1 & wor?y to be reject$8 ,& 9 su* proportion z x draws u aside f intellect8 ,inde$1 ,i ?9k x is consonant to tru?1 t we %\ld eag]ly cont5d :]e intellect leads2 = 9 ?is mat]ial abode1 we are simil> to ^? :o ent] or dep>t f a =eign region1 n only 9 ca/+ aside our native mann]s & cu/oms1 b f ! long use ( a /range c\ntry1 we are imbu$ ) affections1 mann]s1 & laws =eign f our natural & true region1 & ) a /rong prop5sity to ^! unnatural habits4 ,su* an "o1 !re=e1 %\ld n only ?9k e>ne/ly ( ! way1 h[ev] long & labori\s1 by :i* he may return to 8 own1 b t he may meet ) a m fav\rable reception f 8 prop] k9dr$1 %\ld also m$itate by :at means he may dive/ himself ( e ?+ ali5 f 8 true c\ntry1 :i* he has contract$2 & 9 :at mann] he may be/ recal to 8 memory1 ^? habits & dispositions )\t :i* he _c be admitt$ by 8 own1 & :i*1 f long disuse1 h dep>t$ f 8 s\l4 ,9 l mann]1 x is requisite1 if we wi% to return to su* ?+s z are truly our own1 & prop] to man consid]$ z a rational s\l1 to lay aside :atev] we h associat$ to ourselves f a mortal nature1 toge!r ) all t prop5sity to mat]ial connections1 by :i* ! s\l is allur$1 & desc5ds into ! obscure regions ( s5se2 b to be m9dful ( t bless$ & et]nal ess5ce intellect1 our true fa!r1 & ha/5+ our return to ! contemplation ( ! uncol\r$ light ( good1 to take especial c>e ( ^! two ?+s2 "o1 t we dive/ ourselves 7z ( =eign g>m5ts7 ( e ?+ mortal & mat]ial2 ! o!r h[ we may return ) safety1 s9ce ?us1 asc5d+ to our native l&1 we are diff]5t f ourselves be=e we desc5d$ into mortality4 ,= we 7 =m]ly intellectual natures2 & ev5 n[ we are ess5ces purifi$ f e /a9 contract$ by s5se1 & f t "p :i* is de/itute ( reason3 b we are complicat$ ) s5sible connections1 on acc\nt ( our impot5ce & infirmity1 :i* is ! cause t we _c always be conv]sant ) intellectual conc]ns2 b ) mundane affairs we c be pres5t ) frequ5cy & ease2 = all our en]getic p[]s are /upifi$ & cl\d$ ) oblivion1 ?r\gh body & s5se2 ! s\l n rema9+ 9 an intellectual /ate2 7z ! e>? :5 badly affect$1 ?\gh good fruit is deposit$ 9 its bosom1 produces no?+ b we$s72 & ?is1 ?r\gh ! improbity ( ! s\l1 :i* does n1 inde$1 de/roy its ess5ce1 :ile x acquires brutality2 b by su* an accession x becomes complicat$ ) a p]i%+ nature1 is b\nd 9 ! d>k folds ( matt]1 & is drawn aside f its prop] /ate1 into "o t is =eign & base4 ,s t x is highly requisite to /udy1 if we are solicit\s ( return+ to our pri/9e /ate ( felicity1 h[ to dep>t f s5se & imag9ation1 & h] att5dant brutality1 & f ^? passions :i* are rais$ by h] phanta/ic eye1 z mu* z ! necessity ( our nature w p]mit4 ,= ! intellect mu/ be accurately compos$2 & x is prop] x %\ld obta9 a peace & tranquility free f ! cont5tions ( t "p :i* is de/itute ( reason1 t we may n only he> ) att5tion conc]n+ intellect & intelligible objects1 b to ! utmo/ ( our ability1 may enjoy _! contemplation2 & ?us1 be+ r$uc$ into an incorporeal nature1 may truly lead an intellectual life1 & n 9 a false delusive mann]1 l ^! :o are at ! same "t entangl$ ) corporeal conc]ns4 ,we mu/1 !re=e1 dive/ ourselves ( ! v>i\s g>m5ts ( mortality by :i* our vig\r is imp$$2 z well ?is visible & fle%ly g>m5t1 z t m int]ior "o ) :i* we are inve/$ contigu\s to ! sk94 ,we mu/ ent] ! place ( conte/ nak$1 & )\t ! incumbrance ( dress1 /riv+ = ! mo/ glori\s ( all prizes1 ! ,olympiad ( ! s\l4 ,b ! fir/ requisite1 & )\t :i* x is n lawful to cont5d1 is1 t we /rip (f our g>m5ts4 ,& s9ce our ve/m5ts are "s ( !m ext]ior1 & "s int]ior1 s ) respect to ! d5udation ( ! s\l1 "o process is by ?+s m op51 ano!r by su* z are m occult4 ,= in/ance1 n to eat1 or n to accept :at is (f]$1 is among ?+s obvi\s & op52 b n to desire is m obscure2 s t x is "h requisite n only to ab/a9 f ?+s improp] 9 de$s1 b likewise 9 desire4 ,= :at does x pr(it to ab/a9 9 actions f :at is base1 9 ! mean "t adh]+ to ! causes :i* produce su* actions1 z if b\nd 9 indissoluble *a9s8 ,b ?is rec$+ f mat]ial affections is br\ght ab\t p>tly by =ce1 & p>tly by p]suasion2 & by ! assi/ance ( reason ! affections langui%1 & are1 z x 71 buri$ 9 oblivion1 or 9 a c]ta9 philosophical dea?2 :i* is1 inde$1 ! be/ mode ( des]tion1 )\t oppress+ ! t]r5e b&age f :i* ! s\l dep>ts4 ,= 9 ?+s :i* are ! objects ( s5se1 a viol5t devulsion _c take place )\t ei!r a lac]ation ( "s "p1 or at lea/ a ve/ige ( sep>ation4 ,b vice /eals 9 ^u ! s\l ?r\gh cont9ual neglig5ce3 & c>elessness is produc$ by n suffici5tly att5d+ to intelligible objects2 ! affections 9 ! mean "t be+ excit$ by ! dr[sy p]ceptions ( s5se1 among :i* mu/ be also reckon$ ! s5sations aris+ f food4 ,we mu/ !re=e ab/a91 n less ?an f o!r ?+s1 f su* food z usually excites ! passions ( our s\l4 ,let u !n 9 ?is p>ticul> enquire a little f>!r4 ,!re are two f\nta9s1 ^: noxi\s /reams deta9 ! s\l 9 matt]1 & ) :i*1 z if saturat$ ) le?>gic potions1 %e =gets h] own prop] speculations3 ,i mean pleasure & grief1 ! artific] ( :i* is s5se & its p]ceptions1 toge!r ) ! op]ations att5dant on ! s5ses1 imag9ations1 op9ions1 & memory4 ,! passions1 r\s$ by ! en]gies ( ^!1 & ! irrational "p1 n[ fatt5$ ) noxi\s nutrim5t1 draw d[n ! s\l1 & av]t h] incl9ations f h] native love ( true be+4 ,x is requisite1 !re=e1 t we revolt f ^! to ! utmo/ ( our ability4 ,b true defections c alone take place by avoid+ ! passions & ra% motions produc$ by ! s5ses4 ,b1 s5sation respects :atev] moves ! sight1 or ! he>+1 or ! ta/e1 or ! smell4 ,& s5se is1 z x 71 ! metropolis ( t =eign colony ( passions :i* reside 9 ! s\l1 & :i* mu/ be expell$ by him :o wi%es1 :ile connect$ ) body1 to become an inhabitant ( ! royal regions ( intellect4 ,let u !n enquire h[ mu* fuel ( ! passions ent]s into u ?r\gh ea* ( ! s5ses2 & ?is ei!r :5 we behold ! spectacles ( horses 9 ! race1 & ! lab\rs ( ! a?letic1 or ! conte/s ( ^? :o twi/ & b5d _! bodies 9 leap+1 or :5 we survey beautiful wom54 ,= all ^! insn>e u1 unconsci\s ( ! dang]1 & subject to _! dom9ion ! irrational appetite1 by pr(f]$ in*antm5ts ( e k9d4 ,= by all su* in*antm5ts ! s\l1 z if driv5 into fury1 compels ! comp\nd man to leap ra%ly1 & )\t reason1 & full ( ! brutal nature to bell[ & exclaim4 ,9 ! mean "t1 ! p]turbation appe>+ f )\t1 be+ inflam$ by ! int]nal1 :i* 0 fir/ ( all r\s$ by s5se4 ,b ! vehem5t motions excit$ by ! he>+1 arise f c]ta9 noises & s\nds1 f base disc\rse1 & mix$ assemblies2 s t "s1 exil$ f reason1 behave z if /ruck mad2 & o!rs1 en]vat$ by effem9ate s(tness1 agitate !mselves by a multitude ( trifl+ ge/iculations4 ,& :o is ignorant h[ mu* ! s\l is fatt5$1 & infe/$ ) mat]ial grossness1 by ! o9tm5ts & p]fumes :i* comm5d lov]s to ea* o!r8 ,b :y is x necess>y to speak ( ! passions orig9at+ f ! ta/e3 9 ?is respect especially1 b9d+ ! s\l 9 a d\ble b&2 "o ( :i* is ?ick5$ by ! passions excit$ by ! ta/e2 ! o!r becomes /rong & p[]ful by ! diff]5t bodies :i* we receive 9 food4 ,= z a c]ta9 physician obs]v$1 ^? are n ! only poisons :i* are prep>$ by ! m$ical art1 b su* ?+s z we daily receive = food1 z well liquid z solid1 are to be reckon$ among ?is numb]2 & mu* great] dang] arises to our life f ^!1 ?an to our bodies f poisons4 ,b ! t\* does all b transmute ! s\l into body1 & excites 9 x1 z 9 a dissonant body1 c]ta9 brok5 & en]vat$ s\nds4 ,! remembrance1 imag9ation1 & cogitation ( all ^! raise a collect$ sw>m ( passions1 i4 e4 ( fe>1 desire1 ang]1 love1 emulation1 c>es1 & griefs1 !y fill ! s\l ) p]turbations ( ?is k9d1 cl\d its intellectual eye ) oblivion1 & bury its div9e light 9 mat]ial d>kness4 ,on :i* acc\nt x is a great und]tak+ to be purifi$ f all ?is r\t ( pollutions2 & to be/[ mu* lab\r 9 m$itat+ day & night1 :at measures we %all adopt to be fre$ f ^! bonds1 & ?is because we are complicat$ ) s5se1 f a c]ta9 necessity4 ,f :5ce1 z mu* z our ability w p]mit1 we ought to rec$e f ^? places 9 :i* we may 7p]haps unwill+ly71 meet ) ?is ho/ile r\t2 & x is requisite we %\ld be solicit\s n to engage 9 combat ) ^! dang]\s foes1 le/1 ?r\gh too great a confid5ce ( victory & success1 in/ead ( vigor\s cont5tion1 we produce only unskilfulness & indol5ce40 ,& 9 ! conclusion ( ! fir/ book1 he adds1 8,=1 inde$1 if x be lawful to speak freely1 & )\t fe>1 we c by no o!r means obta9 ! true end ( a contemplative1 intellectual life1 b by adh]+ to ! ,deity 7if ,i may be all[$ ! expression71 z if fa/5$ by a nail1 at ! same "t be+ torn away & sep>at$ f body & corporeal delights2 hav+ procur$ safety f our de$s1 & n f ! m]e att5tion to words4 ,b if fri5d%ip is n to be conciliat$ ) a div9ity1 :o is only ! gov]nor ( "s p>ticul> region1 ) any k9d ( food1 or by ! use ( animal nutrim5t1 mu* less c a gross diet effect an union ) t ,god :o is exalt$ above all ?+s1 & :o is sup]ior to a nature simply incorporeal2 b aft] e mode ( purgation1 & ! greate/ *a/ity ( body1 & purity ( s\l1 we %all sc>cely be ?\ght wor?y to obta9 ! vision ( 8 ineffable beauty2 ?\gh ?is is sometimes p]mitt$ to him ^: s\l is well dispos$1 & :o has pass$ ?r\gh life ) ! greate/ sanctity & purity ( mann]s4 ,s t1 by h[ mu* ! ,fa!r ( all exce$s e nature 9 simplicity1 purity1 & self-suffici5cy1 z be+ inf9itely remote f all suspicion ( mat]ial contagion1 by s mu* ! m ought he :o approa*es to ! ,deity1 to be entirely pure & holy1 fir/ 9 8 body1 & aft]w>ds 9 ! mo/ secret recesses ( 8 s\l2 hav+ di/ribut$ a purgation adapt$ to e "p1 & be+ completely inve/$ ) purity1 z ) a transp>5t g>m5t1 fit = ! intimate reception ( div9e illum9ation40 ,?us f> ,porphyry1 ^: excell5t s5tim5ts on ?is subject are a la/+ monum5t ( ! elevation & purity ( s\l :i* ! ,platonic philosophy af=ds2 & at ! same "t suffici5tly prove ! arrogance & ignorance ( ^? :o depreciate ! wisdom ( ! anci5ts1 & consid] _! greate/ philosoph]s z involv$ 9 m5tal d>kness & delusion4 ,b presumption ( ?is k9d is cont9ually increas$ by indol5ce1 & /r5g!n$ by int]e/2 & x is common to f9d scribbl]s ( e k9d1 laugh+ at ,plato & 8 philosophy1 :o are too mean = criticism1 & ev5 too insignificant = contempt4 ,let u1 !re=e1 leave su* 9 _! native inanity1 & li/5 to ! in/ructions ( ! div9ely elegant ,proclus1 by :i* we may asc5d to ! contemplation ( true be+1 & ! ineffable pr9ciple ( ?+s4 #f4 #ch8,py?agoras & ,plato comm& u to fly f ! multitude1 t we may pursue ! mo/ simple tru?1 & apply ourselves :olly to ! contemplation ( real be+4 ,f ! multitude ( ext]ior p draw+ u aside 9 v>i\s ways1 & deceiv+ u by fallaci\s appe>ances4 ,b mu* m to %un ! multitude ( int]ior p2 = ?is mu* m di/racts & deceives4 ,we mu/1 !re=e1 fly f ! v>i\s multitude ( affections1 ! obscure in=mations ( s5se1 ! %ad[y objects ( imag9ation1 & ! dusky light ( op9ion4 ,= e multitude ( ?is k9d is s diff]5t 9 itself1 t its p>ts are contr>y to "o ano!r2 f :5ce x is necess>y to betake ourselves to ! sci5ces1 9 :i* multitude has no contr>iety4 ,= ?\gh affections are contr>y to affections1 "o p]ception ( s5se to ano!r1 imag9ations to imag9ations1 & op9ions to op9ions1 yet no "o sci5ce is f\nd contr>y to ano!r4 ,9 ?is multitude1 !re=e1 ( propositions & notions1 we may collect into "o ! numb] ( sci5ces b9d+ !m 9 "o accord+ bond4 ,= !y are s remote f contr>iety to ea* o!r1 t notion is subs]vi5t to notion1 & inf]ior sci5ces m9i/] to sup]ior1 dep5d+ on !m = _! orig94 ,above all1 x is "h necess>y1 f _m sci5ces :i* pre-suppose "o1 to betake ourselves to "o sci5ce itself1 no long] suppos+ ano!r1 & 9 an ord]ly s]ies to ref] !m all to ?is orig9al "o4 ,b aft] sci5ce1 & its /udy1 x w be necess>y to lay aside compositions1 divisions1 & multi=m disc\rses1 & f !nce to asc5d to intellectual life1 to its simple vision1 & intimate p]ception4 ,= sci5ce is n ! summit ( k1 b beyond x is intellect2 n t intellect only :i* is sep>at$ f s\l1 b ! illu/ration infus$ f !nce into ! s\l1 :i* ,ari/otle affirms to be ! intellect by :i* we ackn[l$ge ! pr9ciples ( sci5ce2 & ,timu says1 t ?is exi/s 9 no place b ! s\l4 ,asc5d+1 !re=e1 to ?is intellect1 we mu/ contemplate toge!r ) x intelligible ess5ce1 by indivisible & simple p]ceptions1 speculat+ ! simple g5]a ( be+s4 ,b aft] v5]able intellect itself1 x w be prop] to contemplate t summit ( ! s\l1 by :i* we are "o1 & "u ^: influ5ce our multitude is unit$4 ,= z by our intellect we t\* ! div9e intellect2 s by our unity1 & z x 7 ! fl[] ( our ess5ce1 x w be lawful to t\* t fir/ "o1 f :om all subord9ate unities proce$4 ,& by ?is our "o1 we are especially conjo9$ ) div9ity4 ,= similitude may be e :]e compreh5d$ by t :i* is simil>2 ! objects ( k by sci5ce2 ?+s intelligible by intellect2 & ! mo/ unify+ measures ( be+1 by ! unity ( ! s\l4 ,b ?is unity & its en]gy is ! summit ( our actions2 = by ?is we become div9e1 :51 fly+ f all multitude1 we retire into ! dep?s ( our unity1 &1 be+ collect$ into "o1 uni=mly en]gize4 ,?us f> we admoni% to %un ! multitude1 by /eps proce$+ f ! ord] ( k3 9 ! next place1 we %all proce$ 9 ! same design by ! s]ies ( kn[able objects4 ,fly !n e s5sible species1 = !y are heap$ toge!r1 are divisible1 & p]fectly mutable1 & incapable ( af=d+ s9c]e & g5u9e k4 ,f ^! d>k in=mations1 !re=e1 betake y\rself to incorporeal ess5ce2 s9ce e s5sible object possesses adv5titi\s unity1 is by itself scatt]$ & confus$1 & full ( =mless inf9ity4 ,h5ce its good is divisible1 & adv5titi\s1 di/ant & sep>at$ f itself1 & resid+ 9 a =eign seat4 ,:5 y h asc5d$ ?i!r1 & are plac$ among incorporeal be+s1 y w behold above ! fluctuat+ empire ( bodies1 ! sublime animal ord]1 self-mov+1 spontane\sly en]giz+ 9 itself1 & f itself possess+ its own ess5ce1 yet multipli$1 & anticipat+ 9 itself a c]ta9 app>ition or image ( ! ess5ce divisible ab\t ! un/able ord] ( bodies4 ,y w !re p]ceive _m habitudes ( reasons1 v>i\s proportions1 & accord+ bonds4 ,likewise ! :ole & p>ts1 vivid circles1 & a multi=m v>iety ( p[]s2 toge!r ) a p]fection ( s\ls n-et]nal1 n subsi/+1 toge!r z a :ole1 b1 unfold$ by "t1 gradually dep>t+ f _! integrity1 & conv]sant ) cont9ual circulations4 ,= su* is ! nature ( ! s\l4 ,b aft] ! multitude belong+ to s\ls1 betake y\rself to intellect1 & ! intellectual k+doms1 t y may possess ! unity ( ?+s4 ,!re rema9 9 contemplation ( a nature ev] abid+ 9 et]nity1 ( life ev] fl\ri%+1 intellig5ce ev] vigilant1 to :i* no p]fection ( be+ is want+1 & :i* does n desire ! *>iot ( "t1 = ! full en]gy ( its ess5ce4 ,:5 y h beheld natures ( ?is exalt$ k9d1 & h se5 by h[ great an int]val !y are sup]ior to s\ls2 9 ! next place enquire :e!r any multitude is !re1 & if intellect1 s9ce x is "o1 is also univ]sal2 & aga91 s9ce x is uni=m1 if n also multi=m3 = y w f9d x subsi/s aft] ?is mann]4 ,:51 !re=e1 y h intimately beheld ?is intellectual multitude1 ?\gh pr(\ndly indivisible & unit$1 transport y\rself aga9 to ano!r pr9ciple1 & hav+ consid]$1 z 9 a m exalt$ rank1 ! unities ( intellectual ess5ces1 9 ! la/ place proce$ to unity p]fectly sep>ate & free f all ?+s4 ,& :5 advanc$ ?us f>1 lay aside all multitude1 & y w at l5g? arrive at ! ineffable f\nta9 ( good4 ,& s9ce x appe>s1 f ^! v>i\s gradations1 t ! s\l !n prop]ly obta9s p]fection1 :5 %e flies f all ext]nal & int]nal multitude1 & ! b\ndless v>iety ( ! univ]se1 we may likewise conclude f h5ce1 t our s\ls d n alone collect _! k f ! obscure objects ( s5se1 nor f ?+s p>ticul> & divisible discov] a p]fect :ole1 & a p]fect "o1 b draw =? sci5ce f _! inmo/ recesses1 & produce accuracy & p]fection f :atev] 9 appe>ances is inaccurate & imp]fect4 ,= x is n prop] to suppose t ?+s false & obscure1 %\ld be ! pr9cipal s\rces ( k to ! s\l2 & t ?+s discordant among !mselves1 :i* require ! reason+s & argum5ts ( ! s\l1 & :i* are ambigu\s & confus$1 %\ld prec$e sci5ce :i* is immutable2 nor t ?+s v>i\sly *ang$1 %\ld g5]ate reasons abid+ 9 "o2 nor t indet]m9ate be+s %\ld exi/ z ! causes ( det]m9ate intellig5ce4 ,x is n1 !re=e1 fit to receive ! tru? ( et]nal entities f b\ndless multitude2 nor f s5sible objects ! judgm5t ( univ]sals2 nor f ?+s de/itute ( reason1 accurate discrim9ation ( t :i* is good3 b x is prop] t ! s\l1 retir+ into h] immortal ess5ce1 %\ld !re scrut9ize ! good & ! true1 & ! immutable reasons ( all ?+s3 = ! ess5ce ( ! s\l is full ( ^!1 ?\gh !y are cl\d$ by oblivion4 ,! s\l1 !re=e1 behold+ ext]iors1 enquires aft] tru?1 9 ! mean "t possess+ x 9 ! dep?s ( h] ess5ce1 & des]t+ h]self1 explores ! good 9 ! d>k regions ( matt]4 ,h5ce1 e "o 9 ! pursuit ( reality ought to beg9 ) ! k ( himself4 ,=1 if we con/antly ext5d our views among ! multitude ( m51 we %all nev] disc]n ! "o species man1 obscur$ by ! multitude1 & di/ract$ by ! division & discord1 & ! v>i\s mutations ( ^? :o p>ticipate ! species4 ,b if we turn our eye inw>ds1 !re1 remote f p]turbation1 we %all behold "o reason & nature ( m52 s9ce multitude is an imp$im5t to ! conv]sion ( ! s\l into h]self4 ,= "h v>iety d>k5s unity1 diff]5ce obscures id5tity1 & dissimilitude cl\ds similitude2 s9ce species are confus$ 9 ! folds ( matt]2 & e :]e t :i* is excell5t is mix$ ) ! base40 ,?us f> ,proclus2 & ?us mu* = our int5d$ ,diss]tation4 ,,life ,,& ,,comm5t>ies ,,( ,,proclus4 ,,! ,,life ,,( ,,proclus1 ,,by ,,m>9us#ci2 ,,or1 ,,conc]n+ ,,felicity4 ,:5 ,i consid] ! magnitude ( m9d1 & dignity ( "* belong+ to ,proclus1 a philosoph] ( our "t1 & att5d to ^? requisites1 & t p[] ( composition :i* ^? ought to possess :o und]take a description ( 8 life2 & la/ly1 :5 ,i reg>d my own pov]ty ( diction1 ,i am incl9$ to believe x m prop] to refra9 f su* an und]tak+1 n to leap ov] ! fosse 7accord+ to ! prov]b71 & to decl9e a disc\rse involv$ 9 s mu* difficulty & dang]4 ,b my scruples are some?+ dim9i%$ :5 ,i consid]1 on ! o!r h&1 t ev5 9 temples1 ^? :o approa* to ! alt>s d n all sacrifice alike2 b t "s are solicit\sly employ$ 9 prep>+ bulls1 goats1 & o!r ?+s ( a simil> k9d1 z n unwor?y ! b5efic5ce ( ! ,gods to :om ^? alt>s belong3 likewise t !y compose hymns1 "s ( :i* are m elegant 9 v]se1 b o!rs 9 prose2 :ile "s1 :o are de/itute ( all su* gifts1 & sacrifice ) no?+ m p]haps ?an a cake & a small quantity ( bread1 ) frank9c5se1 & :o f9i% _! invocations ) a %ort address to ! p>ticul> div9ity !y adore1 are n less he>d ?an o!rs4 ,:ile ,i ?us ?9k ) myself1 ,i am afraid1 accord+ to ,ibycus#dj1 le/ ,i %\ld n (f5d aga9/ ! ,gods 7= ^! are 8 words7 b aga9/ a wise man1 & ?us obta9 ! praise ( m54 ,= ,i d n ?9k x lawful1 t ,i :o 0 "o ( 8 famili>s1 %\ld be sil5t conc]n+ 8 life2 & %\ld n1 accord+ to my utmo/ ability1 relate su* p>ticul>s conc]n+ him z are true1 & :i* p]haps ought to be publi%$ 9 pref]5ce to o!rs4 ,& inde$ by su* a neglect ,i %all n p]haps obta9 ! e/eem & hon\r ( mank9d1 :o w n entirely ascribe my conduct to ! desire ( avoid+ o/5tation1 b w suppose ,i avoid$ su* a design f indol5ce1 or "s1 m dreadful disease ( ! s\l4 ,incit$1 !re=e1 by all ^! consid]ations1 ,i h tak5 ^u me to relate "s illu/ri\s p>ticul>s ( ?is philosoph]1 s9ce !y are almo/ inf9ite1 & may be dep5d$ on = _! und\bt$ reality4 ,i %all beg9 !re=e n accord+ to ! usual mann] ( writ]s1 :o are accu/om$ to di/ribute _! disc\rse into *apt]s2 b ,i consid] t ! felicity ( ?is bless$ man ought1 ) ! greate/ propriety1 to be plac$ z ! f\ndation ( ?is treatise4 ,= ,i reg>d him z ! mo/ happy ( ^? m5 :o 7 celebrat$ 9 =m] ages2 ,i d n say happy only f ! felicity ( wisdom1 ?\gh he possess$ ?is 9 ! highe/ degree ( all m52 nor because he abundantly enjoy$ ! goods ( an animal life2 nor aga9 on acc\nt ( 8 =tune1 ?\gh ?is belong$ to him 9 a mo/ em95t degree1 = he 0 suppli$ ) a great abundance ( all su* ?+s z are call$ ext]nal goods3 b ,i call him happy1 because 8 felicity 0 p]fect1 complete 9 all p>ts1 & compos$ f ea* ( ! prec$+ p>ticul>s4 ,hav+ !n 9 ! fir/ place di/ribut$#da virtues accord+ to _! k9ds into natural1 moral1 & political1 & also into ^? ( a sublim] rank1 :i* are :olly conv]sant ) purification & contemplation1 & are !re=e call$ ,ca?>tic & ,!oretic1 & also su* z are d5om9at$ ,!urgic1 by :i* we acquire a similitude ) "s p>ticul> div9ity2 b omitt+ su* z are sup]ior to ^!1 z beyond ! rea* ( man1 we %all beg9 f su* z are m natural1 & :i* are fir/ 9 ! progressions ( ! human s\l1 ?\gh n fir/ 9 ! nature ( ?+s4 ,?is bless$ man1 !n1 ^: praise is ! subject ( ?is treatise1 naturally possess$1 f ! h\r ( 8 bir?1 all ^? physical virtues :i* fall to ! lot ( mank9d2 ! traces ( :i* 7 manife/ 9 ! late/ p]iod ( 8 life1 & appe>$ to surr\nd & inve/ 8 body aft] ! mann] ( a t5aci\s %ell4 ,9 ! fir/ place1 he 0 endu$ ) a s+ul> p]fection ( s5sation1 :i* !y d5om9ate corporeal prud5ce2 & ?is 0 p>ticul>ly evid5t 9 ! nobl] s5ses ( see+ & he>+1 :i* are inde$ giv5 by ! gods to m5 = ! purpose ( philosophiz+1 & = ! great] conv5i5ce ( ! animal life2 & :i* rema9$ entire to ?is div9e man ?r\gh ! :ole ( 8 life4 ,secondly1 he possess$ a /r5g? ( body :i* 0 n affect$ by cold1 & :i* 0 nei!r weak5$ nor di/urb$ by any vici\s or neglig5t diet1 nor by any endurance ( lab\rs1 ?\gh x 0 exhau/$ day & night1 :ile he 0 employ$ 9 pray]1 9 p]us+ ! works ( o!rs1 9 writ+ books himself1 & 9 conv]s+ ) 8 famili>s2 all :i* he p]=m$ ) su* exp$ition1 t he appe>$ to /udy b "o ?+ alone4 ,b a p[] ( ?is k9d may ) propriety be call$ =titude ( body1 f ! s+ul> /r5g? employ$ 9 su* ex]tions4 ,! ?ird corporeal virtue ) :i* he 0 endu$ 0 beauty1 :i*1 :5 comp>$ ) temp]ance1 ! au?ors ( ^! appellations h v prop]ly consid]$ z possess+ a similitude ( nature4 ,= z we consid] temp]ance z consi/+ 9 a c]ta9 symphony & cons5t ( ! p[]s ( ! s\l1 s corporeal beauty is und]/ood to consi/ 9 a c]ta9 agreem5t ( ! organical p>ts4 ,he 0 inde$ ( a mo/ pleas+ aspect1 n only because he 0 endu$ ) ?is excell5t proportion ( body1 b because ! fl\ri%+ condition ( 8 s\l beam$ ?r\gh 8 corporeal frame l a liv+ light1 ) spl5dors too wond]ful = language to expla94 ,& inde$ he 0 s beautiful t no pa9t] c\ld accurately describe 8 resemblance2 & all ! pictures ( him :i* 7 circulat$1 al?\gh v beautiful1 7 f> %ort ( ! true beauty ( ! orig9al4 ,b ! f\r? corporeal virtue :i* he possess$ 0 heal?1 :i* !y affirm corresponds to ju/ice 9 ! s\l2 & t ?is is a c]ta9 ju/ice 9 ! disposition ( ! corporeal p>ts1 z ! o!r 9 ^? ( ! s\l4 ,= ju/ice is no?+ m ?an a c]ta9 habit1 conta9+ ! p>ts ( ! s\l 9 _! prop] duty4 ,h5ce1 t is call$ heal? by physicians1 :i* conciliates ! j>r+ elem5ts ( ! body into union & cons5t2 & :i* ,proclus possess$ 9 su* p]fection1 t he affirm$ he 0 n ill above twice or ?rice1 9 ! c\rse ( s long a life z sev5ty-five ye>s4 ,b a suffici5t pro( ( ?is is evid5t f h5ce1 t1 9 8 la/ illness1 he 0 entirely ignorant :at ! disord]s 7 :i* invad$ 8 body1 on acc\nt ( ! great r>ity ( _! incursions4 ,su* !n 7 ! corporeal goods :i* ,proclus possess$1 & :i* may be call$ ! =]unn]s1 & z x 7 mess5g]s1 ( ^? =ms into :i* we h di/ribut$ p]fect virtue4 ,b ! fir/ p[]s & prog5y ( 8 s\l1 :i* he naturally possess$1 previ\s to in/ruction1 & ^? p>ts ( virtue ) :i* he 0 adorn$1 & :i* ,plato reckons ! elem5ts ( a philosophic nature#db1 mu/ excite ! wond] ( any "o :o consid]s _! excell5t quality4 ,= he 0 rem>kable = 8 memory & +5uity2 he 0 ( a disposition magnific5t1 g5tle1 & fri5dly2 & a companion1 z x 71 ( tru?1 ju/ice1 =titude & temp]ance2 & 8 love ( tru? 0 s great1 t he nev] admitt$ any prud5t dissimulation1 b hat$ falsehood vehem5tly4 ,inde$ x is necess>y t he :o prosecutes tru?1 ) s mu* e>ne/ness & s9c]ity1 %\ld be extremely desir\s ( x f 8 infancy1 s9ce tru? is ! s\rce ( e good1 bo? to gods & m54 ,b t he despis$ corporeal pleasures1 & 0 an em95t lov] ( temp]ance1 is suffici5tly evid5t f 8 great prop5sity to discipl9es1 & 8 desire ( e k9d ( /udies2 = dispositions ( ?is k9d nev] suff] bea/ly & illib]al pleasure to dwell x ! m9d1 b are able to excite 9 ! s\l1 f h] own int]nal op]ations1 s9c]e pleasure & delight4 ,b x is impossible to say h[ =eign he 0 f av>ice1 s t :5 a boy he despis$ ! weal? ( 8 p>5ts1 ?\gh v ri*1 on acc\nt ( 8 incr$ible love t[>ds philosophy4 ,h5ce he 0 f> remov$ f illib]ality1 & f ! c>e ( less] conc]ns1 z he 0 mo/ /udi\s ( ! univ]se1 & ( e ?+ div9e & human4 ,b f su* a disposition ( ! rational s\l1 hav+ acquir$ true magnanimity1 he consid]$ human life z ( no acc\nt1 &1 unlike ! multitude1 view$ no?+ dreadful 9 dea?4 ,s t he by no means fe>$ all t r\t ( mole/ations :i* appe> t]rible to o!rs1 & ?is 9 consequ5ce ( t natural affection :i* x is prop] to call by no o!r "n ?an t ( ,=titude alone4 ,b1 f all ^! virtues1 ,i ?9k x mu/ be evid5t to ^? :o h n exp]i5c$ 8 be/ ( dispositions1 t he lov$ equity f a boy2 t he 0 j & mild1 & by no means difficult or unju/ 9 8 associations or contracts4 ,to u inde$ he c]ta9ly appe>$ mode/ & elegant1 nei!r av>iti\s nor illib]al1 nei!r arrogant nor timid4 ,b w x n be sup]flu\s to m5tion ! goodness & f]tility ( 8 +5uity8 ,especially among ^? :o "k & :o h he>d1 t he 0 full ( ! mo/ beautiful discipl9es1 & :o are acqua9t$ ) ! multitude he produc$ & publi%$ to ! _w1 s t he alone seem$ to h drank no?+ ( ! cup ( oblivion1 z he 0 endu$ ) a p[] ( memory :i* 0 nev] di/urb$1 & t :i* belongs to ! oblivi\s1 nev] happ5$ to him4 ,besides1 he nev] neglect$ fre% acquisitions1 z if possess+ a suffici5cy ( discipl9es2 & z "o :o is m]ely delight$ ) _! /udy4 ,b he 0 mo/ remote f a nature ru/ic & horrid1 & av]se f ! ,muses1 & p>ticul>ly prop5se to m cultivat$ end[m5ts3 = on acc\nt ( 8 s+ul> urbanity & fe/ivity 7)\t transgress+ ! b\nds ( true hone/y7 9 8 common associations1 sacr$ fea/s1 & o!r actions1 he allur$ & *>m$ 8 famili>s1 & always dismiss$ !m m *e]ful & pleas$4 ,8 "m1 !re=e1 ,m>cella1 lawfully unit$ to 8 fa!r ,patricius1 bo? ( ! ,lycian nation1 & excell+ 9 bir? & virtue1 produc$ our philosoph]1 ?us endu$ f ! beg9n+ ) all ^!1 & o!r gifts ( nature4 ,&#dc ,m9]va1 ! tutel> goddess ( ,byzantium1 receiv$ him :5 born1 & took c>e ( him z a midwife1 %e be+ ! cause ( 8 bir? 9 t city3 b aft]w>ds %e provid$ = 8 well-be+1 :5 he 0 numb]$ among boys & "y m52 = %e appe>$ once to him 9 a dream1 & exhort$ him to ! /udy ( philosophy1 f :5ce arose 8 great prop5sity to ?is goddess1 z he p>ticul>ly p]=m$ h] sacr$ rites1 & cultivat$ ) a great] fury 7z ,i may say7 h] in/itutions4 ,la/ly1 8 p>5ts br\ght him1 :5 born1 into _! native c\ntry ,zan?us1 consecrat$ to ,apollo3 & ,i _c b ?9k t ?is c\ntry happ5$ to him by a c]ta9 div9e provid5ce2 z x 0 requisite t he1 :o 0 to be ! pr9ce ( all sci5ces1 %\ld be educat$ "u ! presid+ deity ( ! ,muses4 ,"h1 be+ in/itut$ 9 ! mo/ elegant mann]s1 he pursu$ moral virtues1 & 0 accu/om$ to "r conduct1 & to a decl9ation ( its contr>y1 t :i* is base4 ,b at t "t ! love ( ! gods1 :o _h att5d$ him f 8 nativity1 manife/ly appe>$2 = be+ once deta9$ by "s disease ( body1 & x appe>+ v difficult1 & sc>cely possible to cure him1 !re /ood at 8 b$ a y\? ( a m ?an ord9>y appe>ance1 s t ev5 previ\s to ! decl>ation ( 8 "n1 he might be consid]$ z#dd ,telesphorus or ,apollo3 b ! god proclaim+ :o he 01 & pron\nc+ 8 "n1 t\*$ ! head ( ,proclus 7= he /ood recl9+ 8 head on ,proclus' pill[7 & hav+ imm$iately re/or$ him to heal?1 vani%$ f 8 sight4 ,& su* 0 !#de div9e vision1 & ! div9e b5evol5ce at t "t exhibit$ to our y\?4 ,b hav+1 = a %ort space ( "t1 9 ,lycia1 appli$ himself to gramm>1 he w5t to ,alex&ria 9 ,egypt1 br++ ) him v s+ul> moral virtues1 by :i* he excit$ t[>ds himself ! love ( ! ma/]s resid5t 9 t place4 ,h5ce ,leonas ! rhetorician1 :o d]iv$ 7z ,i ?9k7 8 l9eage f ,isaurus1 & 0 illu/ri\s among _m ( t pr(ession1 :o 7 !n at ,alex&ria1 n only made him a p>tak] ( 8 /udies1 b ?\ght him wor?y to become 8 dome/ic1 & ord]$ t he %\ld be suppli$ ) food1 toge!r ) 8 wife & *ildr51 no o!rwise ?an if he _h be5 8#df true son4 ,he likewise took c>e to procure him ! notice ( ! pr9cipal m5 9 ,egypt1 :o be+ wond]fully delight$ ) ! acum5 ( ! y\?'s +5uity1 & ) ! elegance & integrity ( 8 morals1 reckon$ him among _! greate/ fri5ds4 ,b he 0 also in/ruct$ by ,orion ! gramm>ian1 ^: ance/ors dis*>g$ ! sac]dotal (fice among ! ,egyptians1 & :o _h made su* a progress 9 ! k ( 8 art1 t he compos$ elaborate books on ?is subject1 :i* he left n )\t advantage to po/]ity4 ,he also w5t to ! s*ools ( ! ,roman preceptors1 & made a great progress 9 t language2 = he 0 at fir/ l$ to ! /udy ( 8 fa!r's pr(ession1 9 :i* he 0 illu/ri\s1 8 employm5t be+ ! /udy ( law 9 ! royal city4 ,b :5 x appe>$ h[ vehem5tly ! "y man 0 delight$ ) ! /udy ( rhetoric1 z he _h n yet t\*$ ! writ+s ( ! philosoph]s1 he bo? acquir$ great glory f 8 acquisitions1 & became ! admiration ( 8 fell[ pupils & ma/]s1 on acc\nt ( ! elegance ( 8 disc\rse1 & 8 cel]ity 9 p]ceiv+2 & f 8 exhibit+ m ! habit & indu/ry ( ! ma/]1 ?an t ( ! s*ol>4 ,b :ile he yet frequ5t$ ! rhetorical s*ool1 ! sophi/ ,leonas1 made him ! companion ( 8 j\rney to ,byzantium3 :i* he und]took = ! purpose ( gratify+ 8 fri5d ,!odorus1 :o 0 at t "t prfect ( ,alex&ria1 & :o 0 a man bo? polite & magnific5t1 & a lov] ( philosophy4 ,b ,proclus1 ?\gh a y\?1 foll[$ 8 ma/] m *e]fully 9 ?is j\rney1 le/ he %\ld be compell$ to int]rupt 8 /udies4 ,h[ev]1 t ,i may speak m truly1 a c]ta9 good =tune br\ght him back to ! s\rce ( 8 nativity4 ,=1 on 8 return1 8 tutel> goddess exhort$ him to philosophy1 & to visit ! ,a!nian s*ools4 ,b hav+ fir/ return$ to ,alex&ria1 & bid f>ewel to rhetoric1 & ! o!r arts :i* he _h =m]ly /udi$1 he gave himself up to ! disc\rses ( ! philosoph]s !n resid5t at ,alex&ria4 ,b he frequ5t$#dg ,olympiodorus1 ! mo/ illu/ri\s ( philosoph]s1 = ! sake ( imbib+ ! doctr9e ( ,ari/otle2 & 9 ma!matical discipl9es gave himself to ,h]o#dh1 a religi\s man1 & "o :o 0 em95tly skilful 9 ! prop] me?ods ( in/itution4 ,b ^! m5 7 s delight$ ) ! mann]s ( ! y\?1 t ,olympiodorus wi%$ him to esp\se 8 daught]1 :om he _h tak5 c>e to in/ruct 9 philosophy1 & ,h]o committ$ to him all 8 religion1 & made him 8 con/ant companion4 ,b hav+1 on a c]ta9 "t1 he>d ,olympiodorus1 a man :o 0 endu$ ) a great p[] ( speak+1 & on acc\nt ( ! cel]ity ( 8 spee*1 & ! gravity ( 8 subjects1 0 und]/ood by v few ( 8 auditors1 z he 0 dep>t+ ) ! dismiss$ multitude1 he repeat$ to 8 companions all t 0 said1 & almo/ v]batim1 ?\gh ! disc\rse 0 copi\s2 z ,ulpianus ,gazu1 "o ( 8 fell[-disciples in=m$ me1 :o also consum$ n ! lea/ "p ( 8 life 9 ! /udy ( philosophy4 ,b he likewise le>n$1 ) great facility1 ! writ+s ( ,ari/otle1 p]ta9+ to rational philosophy#di1 ?\gh ! b>e read+ ( !m is difficult to ^? :o are engag$ 9 ! attempt4 ,hav+ !re=e1 at ,alex&ria1 appli$ himself to ^! ma/]s1 & enjoy$ _! confid5ce 9 su* in/ruction z !y 7 able to af=d1 :5 ^u read+ toge!r ) !m a c]ta9 au?or1 !y appe>$ to him n to int]pret ! m9d ( ! philosoph] z !y ought2 conceiv+ a contempt = ^! s*ools1 & at ! same "t be+ m9dful ( ! exhortation :i* _h be5 div9ely s5t to him at ,byzantium1 he w5t to ,a!ns1 att5d$ by ! presid+ deities ( eloqu5ce & philosophy1 & by b5efic5t dmons4 ,= t he might pres]ve ! g5u9e & entire succession#ej ( ,plato1 he 0 br\ght by ! gods to ! gu>dian city ( philosophy1 z ! circum/ances :i* happ5$ on 8 fir/ entrance into ! city1 & all ! div9e excitations manife/ly ev9ce3 = !y op5ly presag$1 t ?is gift 0 s5t f ! fa!r ,apollo1 & 0 a future suffrage ( 8 succession confirm$ by div9e ev5ts4 ,= :5 8 vessel drove to ! ,pyrum1 & x 0 told to ! citiz5s1 ,nicolaus1 :o aft]w>ds fl\ri%$ 9 ! rhetorical art1 b at t "t /udi$ "u ! ma/]s ( ,a!ns1 desc5d$ to ! %ore z if to an acqua9tance1 receiv$ him = 8 gue/ z if he _h be5 a citiz51 & br\ght him to ! city2 = ,nicolaus 0 also a ,lycian4 ,b ,proclus1 :o p]ceiv$ himself we>y f 8 j\rney1 sat d[n at ! temple ( ,socrates1 ?\gh he did n yet "k1 nor _h he>d t ,socrates 0 wor%ipp$ 9 t place1 & reque/$ ,nicolaus t he w\ld /ay !re = a %ort "t1 &1 if possible1 procure him "s wat]1 z he said t he 0 exce$+ ?ir/y4 ,imm$iately ,nicolaus1 f t v consecrat$ place1 br\ght him "s wat]2 = ! f\nta9 belong+ to ! /atue ( ,socrates 0 n f> di/ant4 ,b :ile he 0 dr9k+1 ,nicolaus1 = ! fir/ "t1 consid]+ ! circum/ance3 ,?is is an om51 says he1 because y h sate 9 ! temple ( ,socrates1 & h !re fir/ drank ! ,attic wat]4 ,!n ,proclus ris+1 & hav+ paid due rev]5ce to ! place1 proce$$ to ! city4 ,b :5 he came to ! t[]1 ! port] :o 0 pres5t at 8 entrance1 & 0 ab\t to close ! gates ) b>s1 said to him1 7t ,i may repeat ! words ( ! man17 8,c]ta9ly unless y _h come1 ,i %\ld h clos$ ! gates40 ,& :at om5 c\ld be m manife/1 or c\ld less require ! int]pretations (#ea ,polletes or ,melampodes1 or su* l div9]s8 ,b ,proclus1 despis+ ! s*ools ( ! rhetoricians1 ?\gh !y v mu* desir$ 8 association1 z if he _h come = t v purpose1 met ) ! pr9ce ( philosoph]s ,syrianus#eb1 ! son ( ,philox5us4 ,at t "t too1 ,la*>es 0 9 ! same company1 a man mu* conv]sant 9 philosophy1 & ! companion ( ,syrianus 9 8 /udy1 b 9 eloqu5ce he 0 9 z great admiration z ,hom] 9 ! poetic art4 ,he !n 01 z ,i h said1 pres5t at ! same "t4 ,b x 0 n[ ! ev5+ twilight1 & :ile !y 7 engag$ 9 mutual conv]se1 ! sun sate1 & ! moon made h] fir/ appe>ance aft] h] *ange3 :]e=e1 hav+ salut$ ! /rang]1 !y endeav\r$ to dismiss him1 z be+ a "y man1 f _! company1 t !y might adore ! goddess ap>t4 ,b he n hav+ proce$$ f>1 beheld also ! moon appe>+ f ! same h\se1 & lay+ aside 8 s&als1 9 _! pres5ce salut$ ! goddess4 ,"h ,la*>es1 admir+ ! confid5ce ( ! y\?1 said1 speak+ to ,syrianus1 8,?is is :at ,plato#ec div9ely affirms ( great g5iuses2 t !y ei!r produce great good1 or its contr>y40 ,& su*1 t ,i may m5tion a few out ( _m1 7 ! signatures ( div9e orig9al1 :i* happ5$ to our philosoph]1 on 8 fir/ arrival at ,a!ns4 ,b ,syrianus br\ght him to ! great ,plut>*#ed1 ! son ( ,ne/or1 :o1 :5 he saw ! "y man1 n yet tw5ty ye>s old1 & he>d ( 8 love & desire ( a philosophic life1 he 0 mu* delight$1 & imm$iately made him a p>tak] ( 8 philosophic /udy1 ?\gh 8 age almo/ =bade su* an attempt2 = he 0 !n v old4 ,he !re=e read to ,proclus 8 comm5t>y on ,ari/otle's books on ! s\l1 & on ! ,phd ( ,plato3 & ?is great man exhort$ him to commit to writ+ :at he he>d1 employ+ ! ambition ( ! y\? z an in/rum5t1 by tell+ him1 t if he complet$ ^? s*olia1 !y w\ld be report$ z ! comm5t>ies ( ,proclus4 ,& z he lov$ ! y\? v mu* on acc\nt ( 8 incl9ation to ! be/ /udies1 he call$ him 8 son1 & caus$ him to reside ) him z a dome/ic4 ,b aft] he saw 8 temp]ance1 ) reg>d to animal food1 he exhort$ him n to ab/a9 f animals entirely1 b to use !m s f> z 0 necess>y to ! vital en]gies ( ! corporeal "p4 ,he likewise gave ! same advice to ,syrianus1 conc]n+ ! y\?'s diet4 ,b he answ]$ ! old man1 z t div9e head 7,proclus7 in=m$ u3 8,suff] him ) t frugality to le>n :at ,i wi%1 & !n1 if he pleases1 he may die40 ,su* 0 ! c>e ( 8 ma/]s respect+ him1 9 e conc]n4 ,b ! old man liv$ ab\t two ye>s aft] ! arrival ( ,proclus2 & dy+1 comm5d$ ! y\? to 8 successor ,syrianus1 z also 8 gr&son ,ar*iadas4 ,b ,syrianus1 :5 he _h receiv$ ,proclus z 8 pupil1 n only mu* assi/$ him 9 le>n+1 b made him 8 dome/ic z to o!r conc]ns1 & ! companion ( 8 philosophic life1 hav+ f\nd him su* an auditor & successor z he _h = a long "t s\ght =2 & "o :o 0 adapt$ = ! reception ( a multitude ( discipl9es1 & div9e dogmata4 ,9 a %ort] space1 !re=e1 ?an two ye>s1 he read1 toge!r ) ,syrianus1 all ! works ( ,ari/otle1 i4 e4 8 logic1 e?ics1 politics1 physics1 & !ological sci5ce4 ,b be+ suffici5tly in/ruct$ 9 ^!1 z 9 c]ta9#ee proteleia1 & small my/]ies1 ,syrianus l$ him to ! sacr$ discipl9e ( ,plato1 & ?is by an ord]ly progression1 & n#ef accord+ to ! oracle1 ) a transc5d5t foot4 ,& he 0 c>eful t he might survey ) him true my/]ies1 ) ! eyes ( 8 s\l1 free f mat]ial d>kness1 & ) a speculation ( intellect ref9$ & pure4 ,h5ce ,proclus 0 employ$ night & day 9 vigilant en]gies1 & 9 writ+ comp5di\sly :at he _h he>d1 employ+ 8 own judgm5t 9 ! selection & ord]4 ,& 9 consequ5ce ( ?is unwe>i$ assiduity1 he made s great a progress 9 a %ort "t1 t by !n he 0 tw5ty-eight ye>s ( age1 he compos$ a multitude ( works2 & among ! re/ 8 v le>n$ & elegant comm5t>ies on ! ,timu4 ,b f an in/itution ( ?is k9d1 8 mann]s also receiv$ a great] ornam5t1 s9ce z he advanc$ 9 sci5ce he accumulat$ virtue4 ,9 ! orig9al 1 :i* ,i wond] ,fabricius %\ld translate1 quod aiunt1 z x is usual ) ! ,platoni/s1 to cite ! ,zoroa/rian oracles exactly 9 ^! words1 in/ances ( :i* may be f\nd 9 ,proclus on ,plato's !ology2 & ! v words prove !mselves to be a "p ( an oracle1 :5 att5tively consid]$4 ,b he likewise pursu$ ! civil virtues f ,ari/otle's political writ+s1 & ,plato's books conc]n+ laws & a republic4 ,h[ev]1 le/ he %\ld be conv]sant only 9 ! contemplation ( ^!1 & %\ld attempt no?+ actual1 s9ce he 0 h9d]$ f engag+ 9 public affairs himself1 f 8 be+ employ$ 9 m important conc]ns1 he exhort$ ,ar*iadas1 :o 0 a religi\s man1 to a political life1 at ! same "t in/ruct+ him 9 its nature & virtues1 & expla9+ to him ! me?ods prop] to be obs]v$ 9 ! dis*>ge ( its duties4 ,& 9 ! same mann] z !y incite ^? :o run 9 ! race1 s he exhort$ him to preside 9 common ov] ! :ole ( 8 city1 & to conf] b5efits privately ^u ea* accord+ to all ! virtues1 b p>ticul>ly 9 con=mity ) ! laws ( ju/ice4 ,b he excit$ 9 reality ! emulation ( himself 9 ,ar*iadas1 :5 he exhibit$ to him 8 own lib]ality & magnific5ce ) respect to ri*es1 by be/[+ gifts at "o "t on 8 fri5ds1 & at ano!r "t on 8 k9dr$1 gue/s1 & citiz5s2 prov+ himself by ?is means n 9 ! lea/ affect$ ) av>ice1 & conf]r+ n a little to ! good ( public conc]ns4 ,b :5 dy+1 he bequea!d t "p ( 8 possessions :i* he did n leave to ,ar*iadas1 to 8 own c\ntry1 & to ,a!ns4 ,b ,ar*iadas became s great a lov] ( tru?1 z well f ! company ( 8 own associates z f ! fri5d%ip ( ,proclus1 t he is nev] m5tion$ by our acqua9tance1 )\t be+ at ! same "t celebrat$ z ! religi\s ,ar*iadas4 ,b ! philosoph] also employ$ himself 9 civil consultations among public assemblies1 :]e ?+s p]ta9+ to ! republic 7 discuss$2 giv+ ! mo/ prud5t c\nsels1 conf]r+ ) gov]nors conc]n+ equity1 & n only exhort+ !m to an imp>tial di/ribution ( ju/ice1 b 9 a mann] compell+ !m by philosophical au?ority4 ,= he _h a c]ta9 public c>e ( ! morals ( pr9ces1 & n only in/ruct$ !m 9 ! art ( temp]ate gov]nm5t by 8 disc\rse1 b also by 8 own example ?r\gh ! :ole ( 8 life2 s9ce he 01 z x 71 ! exempl> ( temp]ance to ! re/4 ,b he gave a specim5 ( civil =titude p]fectly ,h]culean3 = s9ce at t "t !re 01 z x 71 a sea ( tr\bles ^u him1 & mighty waves ( /ormy employm5ts 7 r\s$ by adv]se w9ds aga9/ 8 upright life1 he conduct$ himself1 ?\gh 9 dang]1 ) gravity1 & an un%ak5 con/ancy4 ,& :5 he 0 once v mu* mole/$ by ! improbity ( "s viol5t m51 :i* 0 bo? p]nici\s & dang]\s to himself1 he und]took a j\rney into ,asia1 :i* contribut$ greatly to 8 own advantage3 = z he 0 n unskilful 9 ! m anci5t rites ( t place1 :i* he yet pres]v$1 a div9e p[] af=d$ him ?is occasion ( dep>ture4 ,h5ce1 z he well knew ! :ole ( _! conc]ns1 he taught !m m accurately 9 ?+s p]ta9+ to ! gods1 if !y happ5$ to h neglect$ any ?+ ?r\gh a long int]val ( "t4 ,& :ile he 0 engag$ 9 all ^! employm5ts1 & liv$ 9 a correspond5t mann]1 he 0 s conceal$ f ! multitude t he ev5 excell$ ! ,py?agoreans1 :o keep ) un%ak5 con/ancy ?is precept ( _! f\nd]1 1 live conceal$4 ,b hav+ pass$ a ye> only 9 ! p>ts ( ,lydia1 he return$ to ,a!ns1 "u ! provid5tial protection ( ! presid+ goddess ( philosophy4 ,& ?us =titude 0 p]fect$ 9 our philosoph]1 fir/ by nature1 !n by cu/om1 & aft]w>ds by sci5ce & ! consid]ation ( causes4 ,besides ?is1 he exhibit$ 9 ano!r mann] 8 politic habit practically1 by compos+ lett]s = noblem52 & by ?is means procur+ good to entire cities4 ,b ( ?is ,i h a suffici5t te/imony f ^? on :om !y 7 be/[$1 z well ,a!nians z ,argives1 & o!rs ( diff]5t nations4 ,b he likewise mu* promot$ & increas$ lit]>y /udies1 dem&+ ( pr9ces rew>ds = ! preceptors1 accord+ to _! sev]al des]ts4 ,nor did he und]take ?is ra%ly1 nor ) any int]e/$ views1 b he compell$ !m 7z he consid]$ x a matt] ( great mom5t7 to be dilig5t 9 _! pr(ession1 int]rogat+ & disc\rs+ ) !m respect+ e p>ticul>3 = he 0 a judge suffici5tly in/ruct$ 9 ! employm5ts ( !m all4 ,& if he ev] f\nd any "o neglig5t 9 8 pr(ession1 he %>ply reprov$ him2 s t he appe>$ v vehem5t & ambiti\s1 because he 0 bo? will+ & able to give a j det]m9ation on e subject3 & he 0 inde$ a lov] ( glory4 ,b ?is 0 n a fault 9 him1 z 9 mo/1 because x alone reg>d$ virtue & goodness4 ,&1 p]haps1 )\t an en]gy ( ?is k9d1 no?+ great & excell5t w\ld ev] subsi/ 9 ! human m9d4 ,b he 0 9 ?is respect vehem5t3 ?is ,i w n d5y4 ,yet1 at ! same "t1 he 0 g5tle2 = he 0 easily pleas$1 & demon/rat$ 9 a mom5t t 8 ang] 0 z pliable z wax4 ,=1 almo/ at ! same "t1 he 0 7z ,i may say7 :olly transport$ 9 repreh5sion1 & ) a desire ( becom+ subs]vi5t to _! int]e/1 & t he might int]c$e ) pr9ces 9 _! names2 be+ mov$ ) a c]ta9 natural conjunction ( s\l1 &1 z x 71 sympa?y ( grief4 ,& "h1 ,i v opportunely recollect a peculi> example ( 8 natural sympa?y ( s\l ) o!rs3 nor d ,i ?9k ! l 0 ev] relat$ ( any o!r man4 ,=1 not)/&+ he 0 unm>ri$1 & _h no *ildr5 7because he 0 n desir\s ( su* connections1 b rema9$ free f !m all1 ?\gh _m noble & ri* alliances 7 (f]$ him71 yet 8 c>e ( all 8 famili>s & fri5ds1 & ( _! wives & *ildr51 0 z great z if he _h be5 "s common fa!r1 & ! cause ( _! bir?2 = he be/[$ a s+ul> att5tion on ! life ( ea*4 ,& :5ev] any "o ( !m 0 deta9$ by any disease1 he fir/ e>ne/ly supplicat$ ! gods on _! behalf ) sacrifices & hymns2 aft]w>ds he gave a prompt att5dance on ! sick p]son himself1 conv5$ ! physicians1 & urg$ !m to make an imm$iate trial1 if !y knew ( any ?+ 9 _! art advantage\s to ! condition ( ! diseas$2 & sometimes he produc$ "s s+ul> advice himself1 among ! physicians2 & ?us deliv]$ _m f imm95t dang]s4 ,& ! greatness ( ?is bless$ man's humanity t[>ds 8 s]vants1 may be und]/ood by ^? :o desire x1 f 8 w4 ,b ( all 8 famili>s1 he lov$ ,ar*iadas & 8 k9dr$ ! mo/2 because1 9 ! fir/ place1 _! succession 0 d]iv$ f ! g5us ( ,plut>* ! philosoph]2 & aft]w>ds on acc\nt ( t#eg ,py?agoric fri5d%ip :i* he ma9ta9$ ) ,ar*iadas1 z he 0 bo? ! companion & preceptor ( 8 /udies4 ,& ?is o!r k9d ( fri5d%ip1 diff]+ f ! two already m5tion$1 appe>s to h be5 ! mo/ firm & excell5t1 among ^! illu/ri\s m54 ,= ,ar*iadas desir$ no?+1 :i* 0 n also ! wi% ( ,proclus2 & on ! contr>y1 ! desires ( ,proclus 7 ! con/ant wi%es ( ,ar*iadas4 ,b hav+ n[ br\ght ! political virtues1 :i* are inf]ior to ! true ones1 to an end1 & t]m9at+ !m 9#eh fri5d%ip1 z _! prop] b\nd1 we %all n[ pass to ! ,ca?>tic diff]+ f ! politic virtues4 ,inde$1 ! employm5t ( ^! la/ consi/s 9 purify+ ! s\l1 t s be+ lib]at$ f ! body z mu* z !y are able to effect1 x may reg>d human conc]ns1 & possess a c]ta9 similitude ) div9ity2 :i* is ! s\l's be/ & mo/ exalt$ end4 ,yet !y d n all lib]ate aft] ! same mann]1 b "s m1 & o!rs less4 ,s9ce !re are c]ta9 political purgations :i* adorn _! possessors1 ev5 :ile connect$ ) body1 & r$uce !m to a bett] condition2 br++ "u ! dom9ion ( reason1 ang] & desire1 & entirely de/roy+ passion & e false op9ion3 b ! ,ca?>tic virtues1 :i* are sup]ior to ^!1 sep>ate entirely f ?is truly lead5 weight ( body1 & procure an easy flight f mundane conc]ns4 ,& 9 ^!1 inde$1 our philosoph] 0 /udi\sly employ$ dur+ ! :ole ( 8 life1 :i* 0 devot$ to philosophy2 s9ce he bo? taught by 8 disc\rses :at !y 71 & aft] :at mann] !y 7 prep>atory to felicity1 & 9 a p>ticul> mann] con=m$ 8 life to _! in/itutions2 p]=m+ e ?+ :i* c\ld contribute to ! sep>ation ( 8 s\l1 us+ bo? night & day pray]s1 lu/rations1 & o!r purifications1 z well accord+ to ! ,orphic z ! ,*aldaic in/itutions3 & e mon? he desc5d$1 ) great dilig5ce1 to ! sea2 & ?is sometimes twice or ?rice4 ,b he 0 ex]cis$ 9 ^!1 n only 9 ! vig\r ( 8 age1 b also t[>ds ! close ( 8 life2 & ^! cu/oms he obs]v$ p]petually1 z if !y 7 c]ta9 inv>iable /atutes4 ,b he us$ meat & dr9k1 & o!r necess>y pleasures1 only s f> z 0 necess>y to avoid ! mole/ations ( disease2 = he 0 9 ^! by mu* ! mo/ frugal1 & p>ticul>ly lov$ ab/95ce f animal food4 ,& if at any "t he 0 invit$ to eat x m vehem5tly1 he 0 s cauti\s 9 its use1 t he ate x m]ely aft] ! mann] ( a ta/]4 ,b he purifi$ himself e mon? by ! sacr$ rites1 9 hon\r ( ! "m ( ! gods1 celebrat$ by ! ,romans1 & prior to !m by ! ,phrygians3 he likewise m dilig5tly obs]v$ ! un=tunate days ( ! ,egyptians ?an !y !mselves2 &1 besides ?is1 fa/$1 on c]ta9 days 9 a peculi> mann]1 on acc\nt ( ! lun> appe>ances#ei4 ,he likewise in/itut$ a fa/ on ! la/ day ( ! mon?1 n hav+ supp$ ! day be=e4 ,b 9 :at a spl5did mann]1 & ) :at piety1 he celebrat$ ! new moon1 & prop]ly obs]v$1 ) sacrifices1 ! m illu/ri\s fea/s ( almo/ all nations1 accord+ to ! mann] ( ea* c\ntry2 & h[ f ^! he did n1 accord+ to ! cu/om ( _m1 take occasion ( becom+ idle & intemp]ate1 b employ$ himself 9 cont9ual pray]s1 hymns1 & ! l1 8 hymns suffici5tly ev9ce1 :i* n only celebrate ! div9ities ( ! ,greeks1 b likewise ,m>na ,gazu1 ,esculapius ,leonteu*us1 ,ascalonites1 & ,!&rites1 ano!r god mu* v5]at$ by ! ,arabians2 toge!r ) ,isis1 wor%ipp$ by ! ,philians2 & la/ly1 all ! re/ :i* 7 ! subjects ( 8 devotion4 ,= ?is s5tim5t 0 v famili> to ?is mo/ religi\s man1 t x 0 prop] a philosoph] %\ld n be c>eful 9 ! obs]vance ( ! rites & in/itutions ( "o p>ticul> city1 nor ( c]ta9 nations only1 b t he %\ld be ! g5]al prie/ ( ! univ]se4 ,& ?us 0 he pure & holy1 s f> z p]ta9s to ! virtue ( temp]ance4 ,b he decl9$1 z mu* z possible1 pa93 & if x ev] happ5$ to him1 he bore x ) g5tl5ess1 & dim9i%$ x ) ?is view1 t 8 be/ "p might n at ! same "t be affect$ ) its mole/ations4 ,& ! =titude ( 8 s\l 9 ?is respect1 0 suffici5tly ev9c$ 9 8 la/ illness2 = :51 at t "t1 he 0 oppress$ & torm5t$ ) ! mo/ excruciat+ pa9s1 he endeav\r$ to ! utmo/ to mitigate & expel _! afflictive invasions4 ,h5ce1 on su* occasions1 he (t5 comm&$ u to repeat c]ta9 hymns1 :i* :5 repeat$ procur$ him a remission & cessation ( pa94 ,& :at is m wond]ful1 he rememb]$ :at he he>d ( ^!1 ?\gh =getful ( almo/ all human conc]ns1 f ! dissolution ( 8 corporeal "p cont9ually increas+4 ,= :5 we began to repeat1 he suppli$ :at 0 unf9i%$ ( ! hymns1 toge!r ) _m ( ! ,orphic v]ses2 = x 0 ^! we 7 !n recit+4 ,nor 0 he only ?us con/ant 9 endur+ corporeal evils1 b mu* m s 9 ext]nal un=tunate ev5ts1 & su* z appe>$ to happ5 contr>y to expectation4 ,s t he w\ld say1 conc]n+ p>ticul>s ( ?is k9d1 ,s x is1 su* ?+s are usual2 :i* seem$ to me1 or r at t "t appe>$ to be1 wor?y ( remembrance1 & an evid5t argum5t ( our philosoph]'s magnanimity4 ,b besides ?is1 he re/ra9$ ang] z mu* z possible1 s t x might ei!r rema9 free f all excitation1 or t at lea/ reason might n cons5t to its indulg5ce1 b ! irrational faculty alone1 contr>y to 8 w1 might be mod]ately & lightly excit$4 ,& ) respect to v5]eal conc]ns1 he us$ !m 9 ! natural way1 b s z t he might n proce$ beyond a v mod]ate & light phantasy1 9 _! indulg5ce4 ,& ?us ! s\l ( ?is bless$ man1 hav+ collect$ itself f all p>ts1 & retir+ into ! dep?s ( its ess5ce1 dep>t$ aft] a mann] f body2 :ile x yet appe>$ to be conta9$ 9 its d>k receptacle4 ,= he possess$ a prud5ce1 n l t ( a civil nature1 :i* is conv]sant 9 ! adm9i/ration ( fluctuat+ p>ticul>s1 b prud5ce itself1 by itself s9c]e1 :i* is engag$ 9 contemplat+1 & conv]t+ itself into itself1 )\t any long] cons5t+ to a corporeal nature4 ,he likewise possess$ a temp]ance free f evil2 & :i* is n ev5 mod]ately influ5c$ by p]turbations1 b is ab/ract$ f all affections4 ,& la/ly1 he acquir$ a =titude1 :i* does n fe> a dep>ture f body4 ,b reason & intellect hav+ obta9$ 9 him a p]fect dom9ion1 & ! inf]ior p[]s ( 8 s\l no long] oppos+ !mselves to purify+ ju/ice1 8 :ole life 0 adorn$ ) ! div9e irradiations ( g5u9e virtue4 ,our philosoph]1 !re=e1 hav+ mo/ happily absolv$ ?is =m ( virtues1 advanc+ n[1 z x 71 by ! highe/ & mo/ my/ical /ep1 he asc5d$ to ! greate/ & mo/ consummate or tele/ic virtues2 employ+ = ?is purpose1 ! felicity ( 8 nature1 & a sci5tial in/itution4 ,h5ce1 be+ n[ purifi$1 & ! victor ( 8 nativity1 & despis+ ! va9 ,?yrsus-be>]s1 & boa/]s ( wisdom1 he happily p5etrat$ into h] pr(\nd recesses1 & enjoy$ ! contemplation ( ! truly bless$ spectacles %e conta9s4 ,no long] requir+ prolix diss]tations1 or demon/rations1 = ! purpose ( collect+ ! sci5ce ( ^!1 b1 ) a simple vision & en]gy ( intellect1 behold+ ! exempl> ( ! div9e m9d1 he obta9$ a virtue :i* _c ) suffici5t propriety be call$ prud5ce1 b is m prop]ly d5om9at$ wisdom1 or some?+1 if possible1 /ill m v5]able & div9e4 ,b ! philosoph] en]giz+1 accord+ to ?is virtue1 easily compreh5d$ all ! !ology ( ! ,greeks & ,b>b>ians1 & :atev] is %ad[$ ov] by ! figm5ts ( fables1 & plac$ x 9 a cle> light1 = ! use ( ^? :o are will+ & able to pursue its lat5t signification4 ,b hav+ int]pret$ div9ely e ?+ ( ?is k9d1 & %ew+ ! symphony betwe5 !m all2 at ! same "t1 inve/igat+ all ! writ+s ( ! anci5ts1 :atev] he f\nd 9 !m ( g5u9e wisdom1 & approv$ by g5]al cons5t1 ?is he judici\sly appli$ to use2 b if he f\nd any ?+ ( a diff]5t & dissonant nature1 ?is he entirely reject$1 z vici\s & false4 ,& :atev] he met ) contr>y to wisdom1 ?\gh endu$ ) a fri5dly appe>ance1 ?is he vigor\sly subv]t$ by a dilig5t exam9ation4 ,nor did he employ less =ce & p]spicuity 9 8 association ) o!r m54 ,= he 0 a man labori\s to a miracle2 z he (t51 9 "o day absolv$ five1 & sometimes m lectures2 & writ besides1 _m v]ses1 (t5 to ! numb] ( sev5 hundr$4 ,besides ?is1 he w5t to o!r philosoph]s1 & frequ5t$ _! company2 & aga9 celebrat$ ) !m an ev5+ association1 ceas+ f ! lab\r ( writ+4 ,& all ^! employm5ts he execut$ 9 su* a mann]1 z n to neglect 8 nocturnal & vigilant piety to ! gods1 & assidu\sly supplicat+ ! sun :5 ris+1 :5 at 8 m]idian1 & :5 he sets4 ,b he 0 ! p>5t ( _m dogmata1 :i* 7 nev] kn[n be=e1 bo? 9 physics1 & 9 intellectual & m div9e conc]ns4 ,= he fir/ taught1 t !re 0 a k9d ( s\ls#fj1 endu$ ) ! p[] ( contemplat+ _m =ms at once1 :i* he plac$1 n )\t great probability1 betwe5 intellect1 collectively1 & z x 7 ) "o intuition compreh5d+ e ?+1 & s\ls1 :i* are alone able to direct _! vision to "o p>ticul> =m4 ,& ^? :o are will+ to p]use 8 works w meet ) a great v>iety ( dogmata1 peculi> to him alone2 ! relation ( :i* ,i %all omit1 le/ ,i %\ld give a too great ext5t to my disc\rse4 ,b he :o evolves 8 writ+s1 w easily p]ceive t all we h above relat$ ( him is mo/ true1 & mu* m if he happ5s to h kn[n him1 to h se5 8 face1 & to h he>d him int]pret+ 9 ! mo/ excell5t diss]tations1 & deliv]+ ! ,platonic & ,socratic dogmata 9 8 ye>ly s*ools4 ,nor did he seem de/itute ( div9e inspiration2 = he produc$ f 8 wise m\?1 words simil> to ! mo/ :ite & ?ick fall+ sn[#fa2 s t 8 eyes emitt$ a bright radiance1 & ! re/ ( 8 c\nt5ance 0 respl5d5t ) a div9e light4 ,h5ce1 :5 on a c]ta9 "t1 "o ,ruf9us1 a man ( a great "n 9 ! republic1 :o 0 /udi\s ( tru?1 & 9 o!r respects wor?y ( v5]ation1 came to him :5 he 0 tea*+ & int]pret+1 he p]ceiv$ t ! head ( ,proclus 0 surr\nd$ ) a light2 & :5 ! philosoph] _h f9i%$ 8 int]pretation1 ,ruf9us ris+1 ador$ him2 & (f]$ to give a public te/imony1 by oa?1 ( ! div9e vision :i* he _h obs]v$4 ,& mu* gold 0 (f]$ to him1 by ! same ,ruf9us1 on 8 return f ,asia1 hav+ escap$ ! dang] ( ! w>4 ,b ,proclus likewise reject$ ?is gift1 & 0 by no means will+ to receive x4 ,b t we may return to our fir/ design1 hav+ n[ disc\rs$ conc]n+ ! contemplative wisdom ( ! philosoph]1 ?\gh 9 a mann] b little suit$ to its dignity1 x rema9s t we n[ speak ( ! ju/ice p]ta9+ to ?is k9d ( virtues4 ,= ?is1 unlike t ( :i* we spoke be=e1 is n conv]sant 9 di/ribution1 or proportion2 b mu/ be equally remov$ f ! k9d ( self-en]giz+ ju/ice1 by :i* all ?+s are alone direct$ to ! rational s\l4 ,= to t1 conc]n+ :i* we n[ treat1 x is alone prop] to ref] e en]gy to intellect & ! deity1 :i* our philosoph] p]=m$ 9 ! mo/ exalt$ mann]4 ,= he sc>cely re/$ f 8 diurnal lab\rs1 or refre%$ 8 body ) sleep1 & p]haps ev5 !n 0 n free f m$itation & contemplation4 ,?is is c]ta91 t hav+ v spe$ily r\s$ himself f sleep1 z f a c]ta9 torpor ( ! s\l1 he aspir$ aft] ! morn+1 ! "t ( pray]2 & le/ ! great] "p ( ! night1 %\ld glide f him )\t advantage1 z he 0 ly+ alone 9 8 b$1 he ei!r compos$ hymns1 or exam9$ & =tifi$ ^? dogmata :i* aft]w>ds1 9 ! day "t1 he committ$ to writ+4 ,aft] a simil> mann] he pursu$ t temp]ance :i* has an aff9ity ) ^! virtues1 & :i* consi/s 9 a conv]sion ( ! s\l to intellect1 s z n to suff] itself to be t\*$1 nor mov$ ) any o!r conc]ns4 ,la/ly1 he jo9$ =titude 9 alliance ) ^!1 by a c]ta9 p]fect me?od1 zeal\sly aspir+ aft] t lib]ty :i* is ignorant ( all passion1 & :i* he p]ceiv$ 0 natural to ! div9e object ( 8 contemplation4 ,& ?us1 ?r\gh ! :ole ( 8 conduct1 he did n lead ! life ( a man m]ely good1 to :i*1 z ,plot9us says1 ! political virtues may lead1 b leav+ ?is f> beh9d him1 he endeav\r$ to *ange x = "o mu* m p]fect & div9e1 ! life ( ! gods !mselves2 s9ce1 to become simil> to ^!1 & n to virtu\s m51 0 ! great object ( 8 cont5tion4 ,& ?us he _h r5d]$ virtues ( ?is k9d famili> to himself1 :ile he frequ5t$ ! philosoph] ,syrianus1 & evolv$ & /udi$ ! comm5t>ies ( ! anci5ts4 ,b he receiv$ f ! m\? ( 8 preceptor c]ta9 small se$s1 z x 71 ( ! ,orphic & ,*aldaic !ology2 because he 0 prev5t$ f he>+ ! complete int]pretation ( 8 ma/] on ! ,orphic v]ses4 ,= ,syrianus left to ! *oice ( ,proclus & "o ,domn9us#fb1 a philosoph] ( ! ,syrian nation1 & :o aft]w>ds succe$$ ,syrianus1 ! exposition ( ! ,orphic writ+s1 or ! oracles4 ,b !y 7 by no means unanim\s 9 _! *oice2 = ,domn9us pref]r$ ! int]pretation ( ! ,orphic v]ses1 & ,proclus t ( ! oracles4 ,b our philosoph] did n p]fect 8 und]tak+1 because ! dea? ( ! great ,syrianus happ5$ n long aft]4 ,hav+ !re=e1 z ,i h said1 receiv$ ! outl9es f ! m\? ( 8 ma/]1 he appli$ himself ) ! greate/ dilig5ce to ! writt5 comm5t>ies ( ,syrianus ^u ,orpheus2 & be+ assidu\sly n\ri%$ ) ! copi\s lucubrations ( ,porphyry & ,jambli*us on ! oracles1 & simil> writ+s ( ! ,*aldeans1 he arriv$1 z mu* z is possible to man1 to ! top ( ^? highe/ virtues1 :i* ! div9e ,jambli*us 0 accu/om$ to call aft] a truly div9e mann]1 !urgic4 ,he lab\r$ !re=e1 n )\t exquisite judgm5t1 9 collect+ ! expositions ( philosoph]s prior to 8 "t2 & contract$ into "o1 o!r ,*aldaic hypo!ses1 & ! mo/ excell5t ( ! comm5t>ies on ! div9e oracles1 complet+ ?is great "w 9 ! space ( five ye>s2 conc]n+ :i* ?is div9e vision appe>$ to him 9 8 sleep4 ,= he saw ! great ,plut>* approa* to him1 affirm+ t he %\ld live s _m ye>s z he _h compos$ tetrads or quat]nions on ! oracles4 ,aft]w>ds1 hav+ collect$ ! numb] ( ^!1 he f\nd !y am\nt$ to sev5ty4 ,b t ?is 0 a div9e dream1 0 suffici5tly ev9c$ by ! la/ "p ( 8 life2 = ?\gh he liv$ sev5ty-five ye>s1 z we h m5tion$ above1 yet he _h n ! p]fect use ( 8 p[]s1 9 ! five la/4 ,b 8 body1 ?\gh naturally ( ! be/ con/itution1 be+ debilitat$ by t h>d & sc>cely tol]able k9d ( diet :i* he us$1 & by s _m frequ5t lab\rs & fa/+s1 began to langui% exactly at 8 sev5tie? ye>1 s t he !n became mu* m remiss 9 all 8 en]gies ?an be=e4 ,yet1 ev5 at ?is p]iod1 & ?us affect$1 he compos$ orations & hymns3 he also writ "s ?+s1 & conv]s$ ) 8 fri5ds2 b 8 anci5t vig\r 0 want+ 9 ea*4 ,h5ce ! memory ( ! dream excit$ 8 wond]1 & he e :]e said1 t he _h only liv$ sev5ty ye>s4 ,b :ile he lab\r$ "u ?is infirmity ( body1 a c]ta9 y\?1 nam$ ,hegias1 r5d]$ him m al]t 9 ! bus9ess ( int]pretation4 ,?is "y man1 :o already exhibit$ f 8 t5d] ye>s su* egregi\s signs ( all ! virtues ( 8 ance/ors1 0 "o ( t gold5 *a9 ( philosoph]s1 :o _h =m]ly appe>$ to m52 & adh]$ mo/ dilig5tly to ,proclus deliv]+ ! ,platonic & o!r !ologies4 ,b ,proclus1 at ?is advanc$ p]iod1 0 n mod]ately rejoic$ 9 communicat+ ) ! "y man1 8 own writ+s1 aft] he und]/ood t he made cubital advances 9 e k9d ( discipl9es4 ,& ?us we h briefly disc\rs$ conc]n+ ! app>atus ( ! philosoph] 9 ! ,*aldean oracles4 ,b ,i1 on a c]ta9 "t1 hav+ read ) him ! ,orphic v]ses1 & he>d1 among 8 int]pretations1 n only ! recondite !ology :i* is to be f\nd 9 ,jambli*us & ,syrianus1 b also1 among _m o!r div9e m51 ,i reque/$ ! philosoph] t he w\ld n leave ^! div9e v]ses also )\t 8 explanation3 b 8 answ] 01 ,t he _h (t5 ?\ght ( writ+ comm5t>ies on ,orpheus1 b t he _h be5 /rongly prohibit$ 9 m ?an "o dream4 ,= ,syrianus appe>+ to him 9 8 sleep1 _h det]r$ him ) ?reats f ! design4 ,hav+ !re=e employ$ o!r ma*9es1 ,i intreat$ t at lea/ he w\ld m>k :at he pr9cipally approv$ ( 9 ! books ( 8 ma/]2 :i* :5 ?is be/ ( m5 _h p]=m$1 9 consequ5ce ( my p]suasions1 & _h not$ "s ?+s 9 ! front ( ea* ( ,syrianus's comm5t>ies1 we obta9$ a collection ( all ^!1 & by ?is means s*olia1 & comm5t>ies ( no small bulk2 ?\gh to accompli% ?is on ! :ole ( t div9e poetry1 & on all ! ,orphic rhapsiodies1 0 n ! int5tion ( ,proclus4 ,b s9ce1 z we h said1 8 incr$ible /udy ( ^! conc]ns1 procur$ him a great] & m p]fect degree ( !urgic virtue1 h5ce he no long] rema9$ 9 ! contemplative ord]1 nor cont5t$ ) ei!r ( ! tw(old prop]ties 9 div9e conc]ns1 ex]cis$ intellect & ?\ght alone on ! m excell5t objects ( speculation3 b he 0 c>eful to obta9 ano!r k9d ( virtues m div9e ?an ! =m]1 & sep>at$ f ! politic mode2 = he us$ ! ,*aldean assemblies & conf]5ces1 & _! div9e & ineffable concealm5ts4 ,& hav+ compreh5d$ ^!1 he le>n$ ! mann] ( pron\nc+ & express+ !m1 ) _! rema9+ use1 f ,asclepig5ia1 ! daught] ( ,plut>*3 = %e alone1 at t "t1 pres]v$ ! k ( ! great ,orgies1 & ( ! :ole !urgic discipl9e1 deliv]$ to h] by h] fa!r1 :o receiv$ x f ,ne/or4 ,besides ?is1 our philosoph] also be+ purifi$ 9 an ord]ly mann] 9 ! ,*aldean lu/rations1 0 an inspector ( ! lucid hecatic phasmai 7or visions7 ( :i* he himself makes m5tion 9 "o ( 8 comm5t>ies4 ,b by mov+ a c]ta9 hecatic sphrula#fc1 he v opportunely br\ght d[n %[]s ( ra91 & fre$ ,a!ns f an unseasonable heat4 ,besides ?is1 by c]ta9 phylact]ia1 or *>ms1 he /opt an e>?quake1 & _h ?or\ghly tri$ ! en]gies ( a div9+ tripod1 hav+ le>n$ f c]ta9 v]ses conc]n+ its defect4 ,= :5 he 0 9 8 =tie? ye>1 he ?\ght 9 a dream1 t he repeat$ to himself1 ! foll[+ v]ses3 ,high above !r !re ) radiance bright1 ,a pure immortal spl5dor w+s its flight2 ,^: beams div9e ) vivid =ce aspire1 ,& leap res\nd+ f a f\nt ( fire4 ,& 9 ! beg9n+ ( 8 =ty-second ye>1 he appe>$ to himself to pron\nce ^! v]ses1 ) a l\d voice3 ,lo6 on my s\l a sacr$ fire desc5ds1 ,^: vivid p['r ! intellect ext5ds2 ,f :5ce f>-beam+ ?r\gh dull bodies night1 ,x so>s to !r deck'd ) />ry light2 ,& ) s(t murmurs ?r\gh ! azure r\nd1 ,! lucid regions ( ! gods res\nd4 ,besides1 he cle>ly p]ceiv$ t he belong$ to ! ,m]curial s]ies2 & 0 p]suad$ f a dream t he possess$ ! s\l (#fd ,nicoma*us ! ,py?agorean4 ,inde$1 if ,i 7 will+ to be prolix1 ,i c\ld relate _m !urgical op]ations ( ?is bless$ man2 f ! inf9ite numb] ( :i*1 ,i w only relate "o 9 ?is place ( a v wond]ful nature4 ,asclepig5ia1 ! daught] ( ,ar*iades & ,plut>*1 b ! wife ( ,!ag5es1 f :om ,i h receiv$ _m b5efits1 :ile %e 0 yet a girl1 & 0 n\ri%$ by h] p>5ts1 fell into a disease1 griev\s1 & incurable by ! physicians4 ,& ,ar*iades1 :o _h repos$ 9 h] alone ! hope ( 8 race1 0 deeply afflict$ ) h] condition4 ,b ! physicians despair+ ( h] recov]y1 ! fa!r fl$ to 8 la/ an*or1 ,proclus ! philosoph]1 :i* he 0 accu/om$ to d 9 affairs ( ! greate/ mom5t1 consid]+ him z 8 be/ pres]v]1 & e>ne/ly intreat$ him to int]c$e by 8 pray]s ) ! god = 8 daught]1 ^: condition 0 well kn[n to our philosoph]4 ,proclus !re=e1 tak+ ) him ,p]icles#fe ! ,lydian1 a man well des]v+ ! "n ( a philosoph]1 w5t to ! temple ( ,esculapius#ff1 t he might pray to ! god = ! heal? ( ! sick virg93 = ! city at t "t happily possess$ ?is div9ity1 & rejoic$ 9 ! temple ( ! savi\r god1 :i* 0 z yet free f ! de/ruction ( ! ,*ri/ians4 ,z soon1 !re=e1 z ,proclus _h pray$ aft] ! anci5t mann]1 ! girl imm$iately p]ceiv$ a great *ange & alleviation ( h] disease2 = ! pres]v+ god _h easily re/or$ h] to heal?4 ,! sacr$ rites be+ f9i%$1 ,proclus w5t to ,asclepig5ia1 & f\nd h] deliv]$ f ! mole/ations ( disease1 & 9 a heal?y condition4 ,b ?is affair1 ) _m o!rs1 0 p]=m$ privately1 s t no traces ( x rema9$ to succe$+ inve/igators2 & ! h\se 9 :i* he resid$ greatly assi/$ him 9 ?is design4 ,= besides 8 o!r =tunes1 he enjoy$ a mo/ conv5i5t dwell+1 :i* 8 fa!r ,syrianus & 8 gr&-fa!r ,plut>* 7= s he did n hesitate to call him7 _h once inhabit$2 & ?is 0 situat$ ne> ! temple ( ,esculapius1 toge!r ) t ( ! celebrat$ ,sophocles1 & ( ,bac*us1 :i* is next to ! !atre1 & looks t[>ds1 or is at lea/ se5 f1 ! l(ty t[]s ( ,m9]va4 ,b h[ mu* ,proclus 0 lov$ by ! philosophic goddess is abundantly ev9c$ by 8 philosophic life1 :i* he *ose ?r\gh h] p]suasions1 & t ) ! great success we h hi!rto describ$4 ,b %e cle>ly demon/rat$ h] affection to ,proclus1 by ! foll[+ circum/ance4 ,:5 h] image1 :i* _h be5 s long d$icat$ 9 ! p>!non1 or temple1 0 tak5 away by ^?#fg :o1 )\t any hesitation1 mov$ out ( _! places ?+s ! mo/ holy1 & :i* ought to be immoveable1 !re appe>$ to ! philosoph] 9 a dream1 a woman ( a graceful =m1 :o admoni%$ him to build a temple ) great exp$ition1 =1 says %e1 x pleases ,m9]va1 ! presid+ deity ( philosophy1 to dwell ) y4 ,& h[ famili> he 0 ) ,esculapius1 besides :at we h m5tion$ above1 may be ev9c$ f 8 p]ceiv+ ! pres5ce ( ! same god 9 8 la/ illness4 ,= :ile he 0 betwe5 sleep+ & wak+1 he p]ceiv$ a dragon creep+ on 8 head1 :i* vision occasion$ a remission ( 8 disease1 & a mitigation ( 8 pa92 s t x is probable he w\ld h be5 re/or$ to p]fect heal?1 if 8 desire & vehem5t expectation ( dea? _h n prev5t$ 8 recov]y1 or 8 no long] be/[+ a dilig5t att5tion on 8 body4 ,b he also relat$ ! foll[+ circum/ances1 7wor?y ( be+ rememb]$7 b n )\t te>s1 ?r\gh ! sympa?y ( 8 m9d4 ,=1 :5 a "y man1 he 0 afraid le/ he %\ld be infe/$ ) ! g\t1 :i* 0 ! disease ( 8 fa!r1 & :i* loves to desc5d f p>5ts to _! *ildr54 ,nor 0 he afraid1 z x seems1 )\t reason2 = prior to t :i* we h already relat$ conc]n+ him1 he 0 torm5t$ ) pa9s ( ?is k9d1 :5 ano!r extraord9>y circum/ance happ5$ to ?is bless$ man4 ,9 consequ5ce ( a c]ta9 p]son's prescription1 :om he _h consult$1 he appli$ a pla/] to 8 torm5t$ foot1 :i* an unexpect$ bird flew away ) z he 0 ly+ on 8 b$4 ,& ?is 0 c]ta9ly a div9e & salut>y symbol to ,proclus1 & might h p]suad$ him n to be afraid ( ?is calamity 9 future4 ,b he1 z ,i h said1 0 not)/&+ afflict$ ) ! dread ( ?is disease2 he !re=e supplicat$ ! heal+ god conc]n+ ?is1 & intreat$ him to af=d him a m evid5t tok5 ( 8 w3 & aft] ?is he saw 9 8 sleep 7x is bold inde$ to conceive su* a circum/ance 9 ! m9d1 b we mu/ d>e1 nev]!less1 nor dread to br+ tru? to light7 ,esculapius approa*+ to him1 & accurately contemplat+ 8 leg1 nor disda9+1 ?r\gh 8 great philan?ropy to embrace 8 knee4 ,h5ce1 9 consequ5ce ( ?is vision1 ,proclus 01 ?r\gh ! :ole ( 8 life1 free f ! appreh5sions ( ?is disease1 & 0 nev] m tortur$ ) ?is k9d ( pa9s4 ,aga91 ! god :o is wor%ipp$ by ! ,adrott5i1 did n less conspicu\sly ev9ce 8 great famili>ity ) ?is fri5d ( ! gods2 =1 m ?an once1 ! god b5evol5tly pres5t$ himself to ! sight ( ,proclus1 frequ5t+ 8 temple4 ,& :5 he 0 d\btful1 & wi%$ to "k :at god or gods resid$ & 7 wor%ipp$ 9 t place1 because ! inhabitants 7 ( diff]5t op9ions 9 ?is affair1 "s believ+ t ! temple ( ,esculapius 0 !re1 p]suad$ ( ?is by v>i\s signs1 z voices are c]ta9ly said to be he>d 9 t place1 & a table is consecrat$ to ! gods1 & oracles are giv5 conc]n+ ! recov]y ( heal?1 & ^? :o approa* are miracul\sly deliv]$ f ! mo/ griev\s dang]s ( life4 ,o!rs1 on ! contr>y1 ?9k t ! ,dioscuri reside 9 t place1 because !re h be5 ^? :o saw 9 ! way t5d+ t[>ds ,adrotta1 two "y m5 ( ! mo/ beautiful =m1 & rid+ ) great cel]ity on horseback2 at ! same "t decl>+1 t !y ha/5$ to ! temple4 ,!y add besides1 t ! c\nt5ance ( ^! 0 inde$ human1 b t !y imm$iately gave evid5ce ( a m div9e pres5ce2 = :5 ! m5 7 arriv$ at ! temple1 ! above m5tion$ y\?s appe>$ to !m1 mak+ no enquiry conc]n+ ! affair1 & occupi$ 9 ! sacr$ conc]ns2 b pres5tly aft]1 )drew !mselves f _! sight4 ,z ! philosoph]1 !re=e1 0 9 d\bt conc]n+ ^! div9ities1 & did n discr$it ! relations1 hav+ reque/$ ! gods :o inhabit$ t place1 t !y w\ld condesc5d to manife/ :o !y 72 a god appe>$ to him 9 a dream1 & cle>ly spoke to him z foll[s3 ,:at6 ,ha/ ?\ n he>d ( ,jambli*us tea*+ conc]n+1 & celebrat+ ^? two1 ,ma*aon & ,podalirius8 ,& besides ?is1 ! god condesc5d$ to af=d s great an in/ance ( b5evol5ce to ?is bless$ man1 t he /ood aft] ! mann] ( ^? :o be/[ encomiums on o!rs 9 ! !atres1 & ) a cle> voice1 & compos$ habit1 ext5d+ 8 "r h&1 did n hesitate to exclaim1 7t ,i may relate ! words ( ! god17 ,proclus ! ornam5t ( ! city4 ,& :at c\ld possibly be a great] te/imony ( ?is bless$ man's fri5d%ip ) div9ity8 ,b he1 inde$1 9 consequ5ce ( a mo/ rem>kable sympa?y1 by :i* he 0 unit$ ) ! gods1 c\ld nev] refra9 f te>s :5 he m5tion$ ?is affair to u1 & relat$ ! div9e encomium ( himself4 ,inde$1 if ,i 7 will+ to pursue e p>ticul> 9 l mann]1 & to relate 8 famili>ity ) ! ,h]metic ,pan1 toge!r ) ! great b5evol5ce & manifold assi/ance :i* t div9ity condesc5d$ to af=d ,proclus at ,a!ns1 & ( t p]fectly s+ul> k9d ( felicity :i* he obta9$ f ! "m ( ! gods1 & 9 :i* he 0 accu/om$ especially to triumph & rejoice2 ,i %\ld p]haps seem to _m read]s1 to be ra% 9 my ass]tions1 & to o!rs1 ! au?or ( ?+s p]fectly incr$ible4 ,= _m & great 7 ! daily in/ances ( ?is goddess's b5evol5ce t[>ds him1 9 words & actions1 :i* are bo? innum]able & unhe>d (1 & conc]n+ :i*1 ,i d n at pres5t reta9 an accurate remembrance4 ,b if any "o is desir\s ( kn[+ h[ great he 0 9 ^!1 he mu/ evolve 8 book conc]n+ ! "m ( ! gods1 f :i* he w und]/&1 t x 0 n )\t div9e assi/ance he produc$ all ! !ology ( t goddess into light2 z likewise1 :atev] is act$ or affirm$ conc]n+ ! same 9 ! fables ( ,a?is1 :i* he has expla9$ aft] a philosophical mann]1 t vulg> e>s might aft]w>ds cease to be di/urb$1 on p]ceiv+ ! lam5tations & o!r obscure c]emonies ) :i* h] my/]ies are celebrat$4 ,aft] hav+1 !re=e1 run ?r\gh ! !urgic virtue ( ,proclus1 & its en]gies1 & ! happy circum/ances :i* befell him1 respect+ its op]ations1 & hav+ %ewn1 t he did n less excel 9 e k9d ( virtue1 & t he 0 a man su* z mortals h n beheld = a long p]iod ( "t1 x rema9s t we n[ br+ our disc\rse conc]n+ him to a conclusion3 = we are n n[ at ! beg9n+ ( our n>ration1 nor does ! half ( ! :ole rema9 to u1 accord+ to ! prov]b1 b ! :ole itself is n[ p]fect & complete4 ,s9ce hav+ begun f ! felicity ( ! philosoph]1 & proce$$ 9 its exemplification1 our disc\rse n[ returns to x aga94 ,= we h expla9$ ! great goods1 & provid5tial ex]tions1 :i* 7 grant$ to ?is mo/ excell5t man f ! gods1 & h %ewn _! prompt att5tion to 8 pray]s1 ! div9e visions :i* he enjoy$1 & ! help & solicitude :i* ! gods te/ifi$ t[>ds him4 ,we h likewise expla9$ 8 prosp]\s fate1 & propiti\s =tune1 8 c\ntry1 p>5ts1 ! good habit ( 8 body1 8 ma/]s1 fri5ds1 & o!r ext]nal advantages2 ea* ( :i* we h %ewn to h be5 f> great] & m spl5did 9 him ?an 9 o!r m52 & h likewise dilig5tly enum]at$ su* ?+s z _c be reckon$ among/ outw>d allurem5ts1 b entirely dep5d$ on 8 w1 su* z are ! upright & illu/ri\s de$s ( 8 s\l1 accord+ to univ]sal virtue4 ,& ?us we h summ>ily demon/rat$ t 8 s\l arriv$ 9 reality at ! summit ( ! mo/ consummate virtue1 & 0 happily e/abli%$ 9 a p]fect life1 by human & div9e goods ( e k9d4 ,b t ! lov]s ( m elegant /udies way be able to conjecture f ! position ( ! />s "u :i* he 0 born1 t ! condition ( 8 life1 0 by no means among ! la/ or middl+ classes1 b rank$ among ! highe/ & mo/ happy ord]s1 we h ?\ght fit to expose 9 ?is place1 ! foll[+ s*eme ( 8 nativity#fh4 #af #bf' #ag #bi #bd #bc #bd #da #bi #ej -- #bc #d #db ,horos4 #h #ai ,mid4 ,heav54 #d #db or ! head ( ! dragon4 #bd #cc ,! new moon prec$+ 8 bir?4 #h #ea ,b ,proclus dep>t$ f ?is corporeal life1 9 ! "o hundr$ & tw5ty-f\r? ye> aft] ! gov]nm5t ( ,julian1 on ! sev5te5? day ( ! ,attic ,muni*ion1 or ! ,april ( ! ,romans4 ,nicagoras1 ! junior1 be+ at t "t ! ,a!nian ar*on4 ,! dead body 0 tak5 c>e ( accord+ to ! ,a!nian rites1 z he himself :ile liv+ _h direct$4 ,& if t dilig5t att5tion :i* is due to ! deceas$1 0 ev] paid by any "o1 x 0 c]ta9ly be/[$ by ?is mo/ bless$ man4 ,= he did n neglect any p>ticul> ( ! accu/om$ c]emonies1 b e ye>1 on c]ta9 days1 visit$ ! sepul*res ( ! ,a!nian h]oes & philosoph]s1 & besides ?is sacrific$ himself1 & n ?r\gh ! m$ium ( ano!r1 to ! manes ( 8 fri5ds & famili>s4 ,& hav+ ?us exhibit$ to ea* a prop] rev]5ce & hon\r1 proce$+ into ! ,academy1 he !re1 9 a c]ta9 place1 pleas$ "o by "o ! manes ( 8 ance/ors & k9dr$2 & %ortly aft]1 9 ano!r "p ( ! same ,academy1 he supplicat$ 9 common ! s\ls ( all philosoph]s4 ,& ^! c]emonies be+ end$1 ?is mo/ excell5t man1 hav+ *os5 a ?ird place1 p]=m$ sacr$ rites to all ! s\ls ( deceas$ mank9d4 ,! dead body ( ,proclus1 !re=e1 be+ dispos$ ( accord+ to 8 appo9tm5t1 z we h said1 0 c>ri$ by 8 fri5ds & buri$ 9 ! ea/]n "p ( ! suburbs#fi1 ne> to ,lycabetus1 :]e also ! body ( 8 preceptor ,syrianus reposes1 :o1 :5 he 0 alive1 _h reque/$ ?is ( our philosoph]1 &1 9 consequ5ce ( ?is1 _h tak5 c>e to procure a tw(old receptacle 9 8 sepul*re4 ,b :5 ?is mo/ holy man 0 consult$ h[ he w\ld wi% 8 fun]al to be conduct$1 le/ !re %\ld be any ?+ indec5t1 or )\t a prop] decorum1 he desir$ flutes1 ) :i* he 0 ?reat5$ 9 a dream1 & no?+ besides4 ,an epigram1 consi/+ ( f\r v]ses1 is inscrib$ on 8 tomb1 :i* he compos$ himself1 z foll[s3 ,i ,proclus1 "h ! debt ( nature paid1 7,my c\ntry ,lycia7 9 ! du/ am laid2 ,great ,syrianus =m'd my e>ly y\?1 ,& left me 8 successor 9 ! tru?4 ,"o common tomb1 our e>?ly "p conta9s1 ,"o place our k9dr$ s\ls1--?' !rial pla9s4 ,nor 7 progno/ics want+ 9 ! ye> prior to 8 decease1 su* z an eclipse#gj ( ! sun1 s great t night 0 produc$ 9 ! day-"t3 = ! d>kness 0 s pr(\nd1 t ! />s became visible4 ,& ?is happ5$ :5 ! sun 0 9 ,capricorn1 9 ! ea/]n c5tre4 ,besides1 ! writ]s ( ,ephem]ide obs]ve t !re w be ano!r eclipse ( ! sun1 at ! conclusion ( ! next ye>4 ,b su* l affections ( ! heav5ly bodies are said to port5d ev5ts on ! e>?3 to u inde$1 ! eclipse p]spicu\sly signifi$ a privation &1 z x 71 defection ( ! illu/ri\s lum9>y ( philosophy4 ,& ?us mu* may suffice = ! life ( ! philosoph]4 ,x n[ rema9s = ^? :o are will+ to und]take ! lab\r1 to write conc]n+ 8 disciples & fri5ds4 ,= x appe>s t _m resort$ to him f all p>ts2 among :i* "s 7 only 8 auditors1 b o!rs firm adh]5ts to 8 doctr9e2 &1 on acc\nt ( philosophy1 8 famili>s4 ,i c\ld likewise wi% t "s "o1 :o is m labori\s ?an myself1 w\ld give a p>ticul> acc\nt ( 8 writ+s4 ,= ,i h alone deliv]$ ^! p>ticul>s ( 8 life1 t ,i might satisfy my consci5ce1 & might te/ify t ,i religi\sly rev]5ce ! s\l ( ,proclus1 & 8 good dmon4 ,i %all1 !re=e1 add no?+ conc]n+ ! philosoph]'s writ+s1 except t ,i h always he>d him pref] 8 comm5t>ies on ! ,timu ( ,plato beyond all 8 works4 ,he likewise vehem5tly approv$ ( 8 comm5t>ies on ! ,!tetus4 ,& he us$ frequ5tly to say1 t if he 0 endu$ ) despotic p[]1 he w\ld alone pres]ve1 ( all ! writ+s ( ! anci5ts1 ! ,oracles & ! ,timu4 ,=1 said he1 ,i w\ld aboli% all ! re/1 & remove !m f ! pres5t age1 because x happ5s t _m are (f5d$1 :o und]take to read !m ra%ly1 & )\t prop] in/itution4 ,conc]n+ ! ,publi%$ ,writ+s ( ,,proclus4 ,m>9us hav+ neglect$ to give u an acc\nt ( ! writ+s ( ,proclus1 ,i ?\ght x w\ld n be unacceptable to ! read]1 to pres5t him ) a catalogue ( 8 works :i* are /ill extant2 & ! mo/ p]fect relation ,i am able ( su* z are un=tunately lo/4 ,9 ! execution ( ?is design1 ,i %all foll[1 = ! mo/ "p1 ! accurate ,fabricius 9 ! ord]1 & critical acc\nt ( 8 works2 b %all n hesitate to diss5t f him 9 decid+ on _! philosophical m]it3 = x is v r>e t philology & philosophy are unit$ 9 ! same p]son1 & coalesce 9 amicable conjunction4 99 99 99 99 99 #a4 ,f\r ,hymns4 ,9 ! prec$+ life we are in=m$1 by ,m>9us1 t ,proclus compos$ _m hymns1 9 :i* ! div9ities1 bo? ( ,greeks & ,b>b>ians1 7 celebrat$2 b un=tunately !re are only f\r pres]v$1 ! fir/ ( :i* is to ! ,sun1 ! second & ?ird to ,v5us 7:i* may be f\nd 9 ! ,florilegium ( ,/obus1 p4 #bdi4 edit4 ,grotii17 & ! f\r? to ! ,muses4 ,!y are collect$ by ,fabricius1 9 ! eigh? volume ( 8 ,biblio!ca ,grca2 & are mo/ happy specim5s ( philosophical poetry4 ,inde$1 !y be> mo/ evid5t m>ks ( a m9d full ( div9e light1 & agitat$ ) ! fury ( ! ,muses2 & possess all t elegance ( composition = :i* ! writ+s ( ,proclus are s rem>kable4 ,s t x is v /range ,gyraldus %\ld ascribe !m to a ,hi]ophant ( ,laodica1 ( t "n2 s9ce1 z ,fabricius obs]ves1 ,suidas m5tions o!r writ+s ( ?is prie/1 b does n speak ( him z ! au?or ( any hymns4 ,& if he _h1 ^! hymns brea! too mu* ( ! _s & mann] ( ,proclus1 to be ! production ( any o!r4 #b4 ,two ,books conc]n+ ! useful ,p>ts ( ,le>n+1 conta9$ 9 ,gramm>4 ,?ree books on ?is subject are m5tion$ by ,suidas1 & f\r by ,photius2 b !re are alone extant fragm5ts ( two1 9 ! ,biblio!ca ( ,photius2 :i* 7 publi%$ sep>ate by ,&rew ,s*ottus1 at ! end ( ! syntax ( ,apollonius ,alex&r9us1 ,franc(4 #aeij4 ,hanov4 #afae4 qu>to4 ,x appe>s f ^! extracts1 t ,proclus1 9 ?is "w1 n only treats ( e k9d ( v]se1 b also ( ! m celebrat$ poets4 ,& h5ce ,fabricius conjectures1 t ! %ort life ( ,hom]1 :i* ,leo ,allatius publi%$ "u ! "n ( ,proclus1 0 tak5 f ! fir/ book ( ?is ,*re/oma?ia4 #c4 ,eighte5 ,argum5ts aga9/ ! ,*ri/ians4 ,! le>n$ ,cave1 0 egregi\sly mi/ak51 9 suppos+ t ?is "w1 z well z ! answ] to x by ,philoponus1 is lo/2 n consid]+1 probably1 t ^! argum5ts 9 def5ce ( ! _w's et]nity are 7z ,fabricius well obs]ves7 directly aga9/ "o ( ! ,*ri/ian dogmata1 ! creation ( ! _w4 ,^! argum5ts 7except ! fir/1 :i* is lo/17 are pres]v$ 9 ! answ] ( ,philoponus2 ! ,greek edition ( :i* 0 publi%$ at ,v5ice1 9 #aece1 folio2 & 9 ,lat9 f ! v]sion ( ,joh4 ,mahatius1 at ,lyons1 #aeeg4 folio4 ,simplicius1 9 8 comm5t>ies on ! fir/ book ( ,ari/otle de ,clo1 & else:]e1 opposes ?is "w ( ,philoponus2 at ! same "t1 suppress+ 8 "n4 ,! argum5ts are1 9 my op9ion1 extremely subtle1 & unansw]able2 & ,philoponus1 9 8 refutation1 e :]e discov]s1 t he has by no means fa?om$ ! dep? ( our philosoph]4 #d4 ,a ,comm5t>y on ! ,timu ( ,plato1 9 five ,books4 ,a mo/ admirable "w1 compos$ 9 ! fl[] ( 8 age1 :5 he 0 tw5ty-eight ye>s old1 accord+ to ,m>9us4 ,?is invaluable treasure1 may be said to conta9 all ! wisdom ( ! highe/ antiquity3 = ,proclus e :]e cites ! mo/ celebrat$ int]pret]s on ! ,timu1 su* z ,adra/us ! ,p]ipatetic2 lianus & ,alb9us1 ,platoni/s2 ,ari/ocles1 ,cle>*us1 ,crantor1 ,d]cyllides2 & especially ,jambli*us2 likewise ,long9us1 :om he (t5 refutes2 ,num5ius ! ,py?agorean1 & ,orig51 7diff]5t f ,adamantius17 toge!r ) ,plot9us1 ,porphyry ! ,platonic1 ,ptolemy1 ,sev]us1 & ,!odorus ,as9u2 b he always pref]s ! explications ( 8 ma/] ,syrianus1 be=e all ! re/4 ,add too1 t he e :]e conciliates ! ,orphic writ+s & ! ,*aldean oracles ) ! !ology ( ,plato4 ,& ) respect to ,orpheus 9 p>ticul>1 we are indebt$ to ^! sublime comm5t>ies = ! greate/ "p ( ! fragm5ts ( t div9e poet4 ,he is p]petually on ! w+1 & rises gradually 9 elevat$ conceptions1 until 8 m9d1 l ,hom]'s discord1 rea*es ! heav5s4 ,8 /yle is at ! same "t n]v\s & diffuse1 accurate & elegant4 ,we are a/oni%$ ) ! magnific5ce ( 8 metaphors1 delight$ ) ! copi\s v>iety ( 8 matt]1 & fill$ ) a div9e light1 by ! sacr$ tru?s he unfolds4 ,?is great "w1 h[ev]1 un=tunately1 sc>cely expla9s a ?ird "p ( ! ,timu2 f :5ce x is probable1 z ,fabricius well obs]ves1 t sev]al books h be5 lo/ ?r\gh ! injury ( "t4 ,x 0 publi%$ 9 ,greek1 at ,basil1 9 ! ye> #aeef2 & is full ( mi/akes1 z is g5]ally ! case ) ! ,basil editions ( books1 s t a m correct edition is greatly to be desir$1 ?\gh at pres5t n mu* to be expect$4 ,on conclud+ my acc\nt ( ?is ine/imable "w1 ,i f9d my indignation r\s$ by ! foll[+ words ( ,dr4 ,cudwor?1 9 8 ,intellectual ,sy/em1 p4 #cjf4 8,proclus 7says he7 _h "s peculi> fancies & :ims ( 8 own1 & 0 inde$ a conf\nd] ( ! ,platonic !ology1 & a m+l] ( mu* un9telligible /uff ) x40 ,i mu/ confess1 7& ,i am nei!r afraid1 nor a%am$ ( ! decl>ation17 t ,i nev] f\nd any ?+ 9 ,proclus1 b :at by pati5t ?\ght1 accompani$ ) a s9c]e & vehem5t ?ir/ aft] tru?1 ,i h be5 able to fa?om4 ,_h ,dr4 ,cudwor? be5 endu$ ) ^! requisites1 he w\ld d\btless h _h equal success2 b )\t !m1 ! sublime/ tru?s w c]ta9ly appe> to be un9telligible /uff4 ,besides ?is consid]ation is n to be omitt$1 t a mod]n prie/ makes a bad philosoph]4 #e4 ,on ,plato's ,republic4 ,^! comm5t>ies1 or r fragm5ts ( comm5t>ies1 are extant 9 ,greek1 at ! end ( ,proclus on ! ,timu4 ,suidas m5tions f\r books ( ,proclus on ,plato's politics2 & "s ( ,proclus's diss]tations on ^! books1 7 f\nd 7accord+ to ,fabricius7 9 ! libr>y ( ,lucas ,hol/5ius4 ,! *ief design ( ?is "w seems to be ! unveil+ ! !ological my/]ies conceal$ "u ! fables ( ,hom]1 & o!r div9e poets2 :i* ,proclus has accompli%$ 79 my op9ion7 9 a mo/ wond]ful mann]4 ,t ,hom]1 inde$1 e :]e ab\nds ) ,egyptian le>n+1 is obvi\s to e "o2 b few are acqua9t$ ) ! pr(\nd wisdom :i* 8 fables conceal4 ,! lat5t mean+ ( mo/ ( ^! is unfold$ 9 ! pres5t invaluable1 ?\gh imp]fect "w2 & he ^: m9d is suffici5tly enlight5$ by ! anci5t philosophy1 to compreh5d ! beauty ( ^! illu/rations1 w receive an additional delight f ! /udy ( ,hom]1 :i* x is impossible to express4 ,an epitome ( ?is "w 0 publi%$ 9 ,lat91 by ! le>n$ ,gesn]1 #aedb4 #hvo4 "u ! foll[+ title3 ,apologi qudam pro ,hom]o1 et ,arte ,poetica1 ,fabul>umque aliquot ,en>rationes ex comm5t>iis ,procli ,lycii ,diado*i philosophi ,platonici 9 libros ,platonis de ,rep4 9 quibus plurim de ,diis ,fabul non juxta grammaticorum vulgus hi/oric1 physic aut e?ic tractantur1 s$ ,!ologicis1 ut ,g5tiles loquntur1 ex prima ,philosophia rationibus explanantur4 #f4 ,on ! fir/ ,alcibiades ( ,plato4 ,fic9us translat$ p>ts ( ?is "w into ,lat91 & publi%$ !m "u ! title ( ,procli de ,anima ac dmone1 de ,sacrificio et ,magia1 ,v5ice #adig4 & #aeaf4 fol4 by ,aldus2 & 9 a m simple =m at ,lyons4 ,fabricius in=ms u1 t ! manuscript comm5t>y ( ,proclus 9 ,greek1 b sc>cely expla9+ ! half "p ( ! ,alcibiades1 is to be f\nd 9 v>i\s libr>ies ( ,france1 ,engl&1 & ,italy4 ,also at ,lyons1 among ! books ( ,isaac ,vossius2 & at ,hamburgh 9 ! ,johannean libr>y4 ,f ! specim5 giv5 ( ?is "w by ,fic9us1 x appe>s1 l all ,proclus's philosophical writ+s1 to be an invaluable treasury ( wisdom2 & no?+ c]ta9ly1 reflects great] disgrace on a nation ?an suff]+ su* monum5ts ( anci5t le>n+ & wisdom to lie conceal$ 9 colleges1 cov]$ ) du/1 & nev] consult$4 #g4 ,six ,books on ,plato's ,!ology4 ,a mo/ div9e "w1 9 :i* ! philosoph] collects into a sy/em ! !ology disp]s$ 9 ! writ+s ( ,plato1 & e/abli%es x by inv9cible demon/rations4 ,he d$uces1 9 a beautiful & connect$ s]ies1 all ! div9e ord]s1 f ! retreats ( ineffable unity2 e :]e connects !m by prop] m$iums1 &1 aft] lead+ u ?r\gh ! long gradation ( pr9ciples1 br+s u back aga9 to ! orig9al f :5ce !y fl[$1 & to :i* !y con/antly t5d4 ,! :ole is uncommonly pr(\nd & ab/ruse2 & x 0 n be=e ! ?ird read+1 t ,i c\ld fa?om ! dep? x conta9s4 ,fabricius obs]ves1 8t x is a subtle & le>n$ "w1 b f :i*1 y w soon] le>n ! op9ion ( ,syrianus & ,proclus1 conc]n+ ! deity & div9e conc]ns1 ?an t ( ,plato4 ,he adds1 t x is usual ) ! ,platoni/s1 ev5 f ,plot9us1 to unite to ! doctr9e ( ,plato1 a ?\s& dogmata1 =eign f 8 philosophy1 z if ,plato1 ?\gh he did n p]ceive aft] ?is mann]1 ought c]ta9ly s to p]ceive40 ,:5 m5 mi/ake _! abilities1 !y always act absurdly1 & (t5 dang]\sly4 ,z a labori\s & accurate critic on philological matt]s1 ,fabricius m]its ! highe/ comm5dation su* atta9m5ts c des]ve2 b :5 he leaves ! beat5 road 9 :i* nature design$ him to walk1 & attempts ! tractless pa?s ( philosophy1 he p]petually /umbles1 & (t5 falls on ! gr\nd4 ,! w+s ( philology1 l ^? ( ! swall[1 7 nev] de/9$ = a l(ty flight3--x mu/ be ! eagle w+ ( g5ius1 :i* c alone so> to ! sun ( philosophy4 ,! ,greek & ,lat9 edition ( ?is valuable "w1 0 publi%$ at ,hamburgh1 by milius ,portus1 #afah1 folio4 #h4 ,!ological ,in/itutions2 or1 z x may be call$1 ! ,elem5ts ( ,!ology4 ,?is admirable "w conta9s two hundr$ & t5 propositions1 dispos$ 9 a sci5tific ord]1 & support$ by ! firme/ demon/rations4 ,!y beg9 f sup]-ess5tial unity1 & proce$ gradually ?r\gh all ! beautiful & wond]ful progressions ( div9e causes1 end+ 9 ! self-mov+ en]gies ( s\l4 ,!y possess all ! accuracy ( ,euclid1 & all ! subtilty & sublimity necess>y to a k ( ! mo/ pr(\nd !ology2 & may be consid]$ z be>+ ! same relation to ! ,py?agoric & ,platonic wisdom1 z ,euclid's ,elem5ts1 to ! mo/ ab/ruse geometry4 ,patricius1 ! fir/ ,lat9 translator ( ?is div9e "w1 seems to h be5 v s5sible ( ! tru? ( ?is obs]vation3 = he e :]e c>efully di/+ui%es ! propositions f _! demon/rations2 & adds ! ^w coroll>y to su* consequ5cies z m]it t appellation4 ,8 edition 0 publi%$ at ,f]r>ia1 #aehc4 qu>to1 "u ! title ( ,!ological ,elem5ts4 ,! ,greek & ,lat9 edition1 is subjo9$ to ,proclus's six books on ,plato's ,!ology1 ,hamburgh #afah4 folio4 #i4 ,two ,books conc]n+ ,motion4 ,?is useful "w1 collect$1 z ,fabricius obs]ves1 f ! ?ird & foll[+ books ( ,ari/otle's physics1 0 publi%$ 9 ,greek at ,basil1 #aeca1 & ) ! ,lat9 v]sion ( "o ,ju/us ,velsius1 a physician1 ,basil1 #aede4 octavo4 ,x 0 likewise translat$ by ,patricius1 & is annex$ to 8 v]sion ( ! ,!ological ,in/itutions4 #aj4 ,an ,hypotyposis1 or ,in=mation conc]n+ ,a/ronomical ,hypo!ses4 ,?is "w1 :i* ,fabricius obs]ves is a comp5dium ( ,ptolemy's ,almage/1 0 publi%$ 9 ,greek1 at ,basil1 #aedj4 qu>to2 & 9 ,lat9 by ,george ,valla1 folio1 #aeda4 ,a "p ( ?is "w1 :i* treats ( ! use ( ! a/rolabe1 ,fabricius in=ms u1 is extant 9 manuscript1 9 v>i\s libr>ies4 ,! same accurate critic likewise obs]ves1 t a small treatise1 inscrib$ ,uranodromus1 is extant1 "u ! "n ( ,proclus1 9 "s libr>ies1 z 9 t ( ,v9dobona1 & ( ,ox=d1 among ! ,b>rocian volumes4 ,! compreh5sive v>iety ( ,proclus's g5ius equally dem&s our admiration & applause4 #aa4 ,a small ,treatise conc]n+ ! ,sph]e1 or ,cele/ial ,circles4 ,?is little "w is an accurate & elegant introduction to a/ronomy2 & is almo/ :olly tak5 f ! ,isagoge ( ,gem9us ,rhodius1 on ! phnom5a4 ,! be/ editions are ! ,greek & ,lat9 "o publi%$ at ,p>is 9 #aeec1 qu>to2 & t ( ,ba9bridge1 pr(essor ( a/ronomy at ,ox=d1 ,london #afbj4 qu>to4 #ab4 ,a ,p>aphrase 9 f\r ,books1 on ! ,quadrip>tite ( ,ptolemy4 ,?is elegant "w mu/1 ,i %\ld imag9e1 be an invaluable treasure to ! lov]s ( a/rology4 ,x 0 fir/ publi%$ 9 ,greek by ,melanc?on2 & aft]w>ds 9 ,greek & ,lat9 by ,leo ,allatius1 at ,lyons ,bat4 #afed4 octavo4 #ac4 ,f\r ,books1 on ! fir/ ,book ( ,euclid's ,elem5ts4 ,= an acc\nt ( ?is "w1 see ! introduction1 & ! foll[+ %eets1 9 :i* x speaks = itself1 9 an ,engli% dress4 #ad4 ,a ,comm5t>y on ,hesiod's ,works & ,days4 ,?is "w conta9s a valuable moral explanation ( ?is great poet's mean+2 & ,fabricius ju/ly obs]ves1 t he is (t5 assault$ )\t occasion1 by ! petulant je]s ( t va9 man ,joh4 ,tzetzes4 ,! be/ edition ( ?is "w is t ( ,daniel ,he9sius1 ,lugd4 ,bat4 #afjc4 qu>to4 #ae4 ,fabricius in=ms u1 t 9 "s manuscripts1 z 9 ! ,v9dobon5sian & ,b>rocian1 a small treatise is usually ascrib$ to ,proclus1 entitl$ ,epi/olic ,*>act]s2 & is prefix$ to ! ,epi/les ( ,phal>is & ,brutis1 & publi%$ "u ! "n ( ,libanius1 9 ,greek1 ) ! v]sion ( ,casp4 ,/ibil9us1 ,commel94 #aeig1 octavo4 ,b x is d\btful :e!r ,proclus is ! g5u9e au?or ( ?is "w3 f ! title1 ,i %\ld suppose ! contr>y4 ,& ?us mu* = an acc\nt ( ^? writ+s ( ,proclus :i* h escap$ ! ravages ( "t1 & h be5 =tunately expos$ to public inspection3 x n[ rema9s t we relate su* ine/imable works ( ?is philosoph]1 z are yet pres]v$ 9 %ameful concealm5t2 or are utt]ly lo/ 9 ! ru9s ( antiquity4 ,conc]n+ ! ,unpubli%$ ,writ+s ( ,,proclus4 #af4 ,on ! ,alcibiades ( ,plato4 ,see num4 #f4 #ag4 ,on ,plato's ,politics4 ,see num4 #e4 #ah4 ,on ,plato's ,p>m5ides4 ,a comm5t>y1 9 sev5 books2 ! la/ ( :i* 0 n complet$ by ,proclus1 b by ,damascius4 ,f occasional fragm5ts1 :i* h be5 publi%$ ( ?is comm5t>y1 x appe>s to be a mo/ div9e "w2 & inde$ x _c be o!rwise1 if we consid] x z ! production ( "o ( ! greate/ philosoph]s1 on ! mo/ sublime & pr(\nd ( all ,plato's ,dialogues4 ,x is d$icat$ to ,asclepiodotus1 a physician & philosoph]1 & is n only extant 9 ,greek ,,ms4 9 ! libr>y ( ! ,g]man emp]or1 accord+ to ,lambecius1 lib4 vii4 p4 #da4 b also 9 ,lat91 f ! unpubli%$ v]sion ( "o ,antonius ,h]mannus ,gogava1 z ! same ,lambecius in=ms u1 p4 #da4 ,f\r books ( ?is "w are extant 9 ,greek1 9 ! ,bodleian libr>y at ,ox=d2 & x is mu* to be lam5t$ t ,?omson did n publi% ^!1 in/ead ( 8 trifl+ edition ( ! ,p>m5ides4 ,fabricius likewise in=ms u1 t ,livius ,galantes m5tions 8 hav+ f\nd six ( ^! books 9 "s ( ! ,italian libr>ies4 ,!y are also extant 9 ! ,m$ican libr>y ( ! great ,etruscan comm&]4 #ai4 ,on ! ,cratylus ( ,plato4 ,we h already obs]v$1 9 ! diss]tation on ! ,orphic !ology1 p4 #aje4 :at a great treasure ( anci5t my?ology1 mu/ be conta9$ 9 ?is "w2 b !re is little hope ( its ev] em]g+ f ! obscurity ( public libr>ies4 ,x is extant 9 ,greek1 n only 9 ! ,italian libr>ies1 b also among ! manuscript books ( ,isaac ,vossius4 #bj4 ,uranodromus4 ,see above1 num4 #aj4 #ba4 ,t5 ,d\bts conc]n+ ,provid5ce1 9 "o ,book4 ,philoponus m5tions ?is "w1 9 8 second book aga9/ ,proclus on ! et]nity ( ! _w2 & a ,lat9 v]sion ( x is extant by "o ,william de ,morbeka1 9 ! ,johannean libr>y ( ,hamburgh4 ,extracts f ?is translation are pres]v$ by ,fabricius1 9 8 ,greek ,libr>y2 & !y are 9 e respect wor?y ( ! g5ius ( ,proclus4 #bb4 ,conc]n+ ,provid5ce & ,fate1 & t :i* is 9 our ,p[]1 "o ,book4 ,?is "w is d$icat$ to "o ,!odorus1 a me*ani/2 & is extant 9 ! ,lat9 translation ( ! same ,morbeka1 9 ! ,greek ,libr>y ( ,fabricius4 ,! translation is = ! mo/ "p b>b>\s1 b is1 h[ev]1 suffici5tly legible to discov] t x is a mo/ valuable treatise1 replete ) ! usual elegance1 subtilty1 & sublimity ( our philosoph]4 #bc4 ,conc]n+ ! ,hypo/asis1 or ,subsi/5ce ( ,evil4 ,?is book is extant 9 ,lat91 9 ! ,johannean libr>y2 & fragm5ts ( x are pres]v$ by ,fabricius1 9 8 ,greek libr>y4 ,x is to be regrett$1 t ,fabricius did n pres]ve ! :ole 9 t excell5t philological "w4 ,conc]n+ ! ,lo/ ,writ+s ( ,,proclus4 #bd4 ,on ! ,spee* ( ,diotima1 9 ,plato's ,banquet1 conc]n+ ! ,subsi/5ce ( ! ,beautiful4 ,fabricius in=ms u1 t ?is "w is di/ribut$ into _m books2 & ,hol/5ius obs]ves1 t x is m5tion$ 9 a c]ta9 s*olium ( ! ,m$icean copy ( ,proclus's comm5t>ies on ,plato's politics2 b x is un=tunately no :]e extant4 #be4 ,on ! ,philebus ( ,plato2 z may be inf]r$ f ! n>ration ( ,damascius 9 ,photius1 p4 #eej2 & ,suidas 9 ,m>9us4 ,= ,damascius relates1 t ,m>9us hav+ compos$ a comm5t>y on ?is dialogue1 on %ew+ x to ,isidorus = 8 approbation1 t philosoph] obs]v$1 t ^? ( 8 ma/] 7 suffici5t2 :i* words ,fabricius1 ) great propriety1 applies to ! comm5t>ies ( ,proclus on ! ,philebus4 #bf4 ,on ! ,!tetus ( ,plato4 ,?is "w is prais$ by ,m>9us1 9 ! la/ *apt] ( ! prec$+ life2 & no d\bt ) great propriety3 = ?is ab/ruse & sublime dialogue w\ld naturally call =? all ! div9e fire & elegance ( our philosoph]4 #bg4 ,comm5t>ies on ! ,enneads ( ,plot9us4 ,?is "w is m5tion$ by ,gyraldus1 9 8 second dialogue on anci5t poets2 by ,fic9us on ,plot9us2 by ,philip ,labbeus1 9 8 acc\nt ( ,,ms4 books1 p4 #bhf2 & 9 ! notes ( ,bullialdus to ,!o ( ,smyrna1 p4 #bbd4 ,b also 9 a c]ta9 note prefix$ to an anci5t manuscript ( ,jambli*us1 on ! ,egyptian my/]ies1 to ?is effect3 8,! philosoph] ,proclus1 comm5t+ on ! ,enneads ( ! great ,plot9us1 says1 t x is ! div9e ,jambli*us :o answ]s ! epi/le ( ,porphyry40 ,?is note is 9 ,greek1 9 ! orig9al1 & is 79 my op9ion7 ( itself suffici5t to prove t su* a "w 0 once extant1 ?\gh n[ un=tunately lo/4 ,h[ mu* ! want ( ^! comm5t>ies is to be regrett$1 mu/ be deeply felt by e lov] ( ! ,platonic philosophy4 ,= ! unequall$ pr(undity1 & div9e my/]ies1 conta9$ 9 ! writ+s ( ,plot9us1 c\ld nev] be m happily illu/rat$ ?an by ! irradiations ( su* a g5ius z ,proclus4 #bh4 ,lectures on ,ari/otle's ,book 1 or conc]n+ ,int]pretation4 ,?is "w1 x seems1 0 nev] publi%$2 b ,ammonius ,h]meas1 ! disciple ( ,proclus1 has ins]t$ 9 8 valuable comm5t>y on ?is book all t he c\ld reta9 9 8 memory ( ,proclus's lectures4 #bi4 ,hymns1 n a few1 see num4 #a4 #cj4 ,*re/oma!a4 ,see num4 #c4 #ca4 ,on ! ,"m ( ! ,gods1 "o book1 m5tion$ by ,m>9us1 9 ! prec$+ ,life4 #cb4 ,on ! ,!ology ( ,orpheus4 ,?is "w is m5tion$ by ,m>9us1 9 ! prec$+ ,life1 & by ,suidas2 & its loss mu/ be p>ticul>ly regrett$ by all ! lov]s ( recondite !ology4 #cc4 ,t5 ,books1 on ! ,*aldean ,oracles4 ,?is mo/ valuable "w is m5tion$ by ,m>9us1 9 ! prec$+ ,life1 & by ,proclus himself on ,plato's ,politics1 p4 #cei4 ,x 0 d\btless n extant at ! "t :5 ,psellus & ,ple?o und]took ! illu/ration ( a few ( ^! oracles3 at lea/ ! inconsid]able m]it ( _! comm5t>ies1 /rongly fav\rs ?is supposition4 #cd4 ,a ,comm5t>y on ! :ole ( ,hom]4 ,suidas4 ,a specim5 ( ! great value ( ?is "w may be se5 9 our philosoph]'s comm5t>ies on ,plato's republic4 ,! works ( ,hom] are n only ! great f\nta9 ( poetry1 b likewise ( philosophy2 & are no less admirable = inspir+ ! fury ( ! ,muses ?an = conta9+ ! my/]ies ( ! mo/ recondite !ology4 #ce4 ,conc]n+ ! ,gods1 accord+ to ,hom]4 ,_h ?is "w be5 pres]v$1 we %\ld d\btless h be5 furni%$ ) a def5ce ( ! hea!n religion1 :i* w\ld h sil5c$ ! ignorant clam\rs ( its oppon5ts4 #cf4 ,! ,symphony or ,concord ( ,orpheus1 ,py?agoras1 & ,plato4 ,suidas4 ,proclus1 9 8 publi%$ writ+s1 is e :]e /udi\s ( reconcil+ ! doctr9es ( ^! great m51 & is always successful 9 ?is und]tak+4 ,inde$1 ! same div9e g5ius seems to h irradiat$ & inspir$ ^! wond]ful h]oes1 b 9 diff]5t ways3 9 ,orpheus x 0 accompani$ ) ! fire ( ! ,muses2 9 ,py?agoras x %one ?r\gh ! my/]i\s veil ( numb]s2 & 9 ,plato1 comb9+ ! prec$+ modes1 x 0 se5 en%r9$ 9 awful maje/y ( ?\ght1 clo!d ) ! graces ( poetical diction1 & respl5d5t ) ineffable light4 #cg4 ,two ,books on ! ,!urgic ,discipl9e4 ,suidas4 ,h[ mu* ,proclus excell$ 9 ?is art1 may be se5 9 ! prec$+ ,life4 #ch4 ,conc]n+ ! ,oppositions ( ,ari/otle to ,plato's ,timu4 ,?is "w is m5tion$ by ,proclus 9 ! #c;d book ( 8 comm5t>y on ! ,timu1 p4 #bbf4 & seems to h escap$ ! notice ( ! accurate ,fabricius4 ,ari/otle may1 no d\bt1 9 _m p>ticul>s be reconcil$ ) ,plato2 b x is also c]ta91 t 9 "s he is p]fectly dissonant4 ,& ?us mu* = ! ,life & ,writ+s ( ,proclus4 ,,comm5t>ies ,,( ,,proclus4 ,,book ,i4 ,,*ap4 ,i4 ,on ! ,middle ,nature ( ! ,ma!matical ,ess5ce4 ,x is necess>y t ! ma!matical ess5ce %\ld nei!r be sep>at$ f ! fir/ nor la/ g5]a ( ?+s1 nor f t :i* obta9s a simplicity ( ess5ce2 b t x %\ld obta9 a middle situation betwe5 sub/ances de/itute ( p>ts1 simple1 incomposite & indivisible1 & su* z are subject to p>tition1 & are t]m9at$ 9 manifold compositions & v>i\s divisions4 ,= s9ce t :i* subsi/s 9 its inh]5t reasons rema9s p]petually ! same1 is firm & durable1 & _c be confut$1 x evid5tly decl>es x is sup]ior to ! =ms exi/+ 9 matt]4 ,b t p[] ( progression :i* appreh5ds1 & :i* besides uses ! dim5sions ( subjects1 & prep>es diff]5t conclusions f diff]5t pr9ciples1 gives x an ord] inf]ior to t nature :i* is allott$ an indivisible ess5ce1 p]fectly con/itut$ 9 itself4 ,h5ce 7z x appe>s to me7#ga ,plato also divides ! k ( ?+s :i* are1 into fir/1 middle1 & la/ sub/ances4 ,& to indivisible natures1 inde$1 he attributes an intellig5ce1 :i*1 9 a collective mann]1 & by a c]ta9 simple p[]1 divides ! objects ( intellectual p]ception2 s t be+ dive/$ ( matt]1 & endu$ ) ! greate/ purity1 x appreh5ds ?+s !mselves1 by a c]ta9 unify+ p]ception1 & excels ! o!r k9ds ( k4 ,b to divisible ess5ces1 & su* z are allott$ ! l[e/ nature1 & to all s5sible be+s1 he attributes op9ion1 :i* obta9s an obscure & imp]fect tru?4 ,b to middle ess5ces 7& su* are ma!matical =ms71 & to ?+s inf]ior to an indivisible & sup]ior to a divisible nature1 he attributes cogitation4 ,= ?is1 inde$1 is inf]ior to intellect1 & ! supreme sci5ce dialectic2 b is m p]fect ?an op9ion1 & m c]ta9 & pure4 ,= x advances by a discursive procession1 exp&s ! indivisibility ( intellect1 & unfolds t :i* 0 involv$ 9 ! unity ( intellectual appreh5sion3 b x collects ?+s :i* are divid$1 & br+s !m back to m9d4 ,h5ce1 z kn[l$ges diff] among !mselves1 s ! objects ( k are di/+ui%$ by nature4 ,s t intelligible ess5ces hav+ an uni=m subsi/5ce1 evid5tly excel all o!rs4 ,b s5sibles are entirely excell$ by prim>y ess5ces3 & ma!matical natures1 & :atev] falls "u cogitation1 are allott$ a middle ord]3 = !y are excell$ by ! division ( intelligibles2 b because de/itute ( matt]1 !y are sup]ior to s5sible natures2 & by a c]ta9 simple p[]1 !y are excell$ by ! fir/2 b by a c]ta9 reason are m exalt$ ?an ! la/4 ,h5ce !y possess notions ( an intellectual ess5ce1 :i* are m manife/ ?an s5sibles1 b :i* are1 at ! same "t1 only ! images ( an intellectual nature2 & !y imitate divisibly ! indivisible1 &1 9 a multi=m mann]1 ! uni=m exempl>s ( ?+s4 ,&1 t ,i may sum up ! :ole 9 a few words1 !y are plac$ 9 ! ve/ibules or entrances ( prim>y =ms1 & disclose _! indivisible & prolific subsi/5ce collect$ into "o1 b !y d n yet excel ! division & composition ( reasons1 & an ess5ce accommodat$ to ! obscurity ( images2 nor are !y capable ( pass+ beyond ! v>i\s notions ( ! s\l1 endu$ ) a discursive p[]1 & ( adh]+ to intellections p]fectly simple1 & purifi$ f all mat]ial imp]fection4 ,aft] ?is mann] !n1 is ! middle nature ( ma!matical g5]a & =ms to be und]/ood2 z fill+ up ! m$ium betwe5 ess5ces entirely indivisible1 & su* z are divisible ab\t matt]4 ,,*ap4 ,,ii4 ,conc]n+ ! common ,pr9ciples ( ,be+s1 & ( ! ,ma!matical ,ess5ce1#gb b\nd & inf9ite4 ,b x is necess>y t1 consid]+ ! pr9ciples ( ! :ole ma!matical ess5ce1 we %\ld return to ^? g5]al pr9ciples1 :i* p]vade ?r\gh & produce all ?+s f !mselves1 ,i mean b\nd & inf9ite4 ,= f ^! two aft] t cause ( "o1 :i* c nei!r be expla9$1 nor entirely compreh5d$1 e o!r ?+1 z well z ! nature ( ! ma!matical discipl9es1 is con/itut$4 ,9 ! =m]1 inde$1 produc+ all ?+s collectively & sep>ately2 b 9 ^! proce$+ 9 a conv5i5t measure1 & receiv+ a progression 9 a becom+ ord]2 & 9 "s1 subsi/+ among prim>y1 b 9 o!rs among middle1 & 9 o!rs aga9 among po/]ior natures4 ,= intelligible g5]a1 by _! simplicity ( p[]1 are ! fir/ p>ticipants ( b\nd & inf9ite3 because1 on acc\nt ( _! union & id5tity1 & _! firm & /able exi/5ce1 !y are p]fect$ by b\nd3 b on acc\nt ( _! division into multitude1 _! copi\s p[] ( g5]ation1 & _! div9e div]sity & progression1 !y obta9 ! nature ( inf9ite4 ,b ma!matical g5]a orig9ate1 inde$1 f b\nd & inf9ite1 yet n f prim>y1 intelligible1 & occult pr9ciples only2 b also f ^? pr9ciples :i* proce$ f ! fir/ to a second>y ord]1 & :i* are suffici5t to produce ! middle ornam5ts ( be+s1 & ! v>iety :i* is alt]nately f\nd 9 _! natures4 ,h5ce1 9 ^! also1 ! reasons & proportions advance to inf9ity1 b are re/ra9$ & conf9$ by t :i* is ! cause ( b\nd4 ,= numb] ris+ f ! retreats ( unity1 receives an incessant increase1 b t :i* is receiv$ z x /ops 9 its progression1 is always f9ite4 ,magnitude also suff]s an inf9ite division1 yet all ! p>ts :i* are divid$ are b\nd$1 & ! p>ticles ( ! :ole exi/ f9ite 9 en]gy4 ,s t )\t ! be+ ( inf9ity1 all magnitudes w\ld be comm5surable1 & no "o w\ld be f\nd b :at might ei!r be expla9$ by words1 or compreh5d$ by reason 79 :i* inde$ geometrical subjects appe> to diff] f su* z are ari?metical27 & numb]s w\ld be v little able to ev9ce ! prolific p[] ( unity1 & all ! multiplex & sup]-p>ticul> proportions :i* !y conta94 ,= e numb] *anges its proportion1 look+ back ^u1 & dilig5tly enquir+ aft] unity1 & a reason prior to itself4 ,b b\nd be+ tak5 away1 ! comm5surability & communication ( reasons1 & "o & ! same p]petual ess5ce ( =ms1 toge!r ) equality1 & :atev] reg>ds a bett] co-ord9ation1 w\ld nev] appe> 9 ma!matical anticipations3 nor w\ld !re be any sci5ce ( ^!2 nor any firm & c]ta9 compreh5sions4 ,h5ce !n1 z all o!r g5]a ( be+s require ^! two pr9ciples1 s likewise ! ma!matical ess5ces4 ,b su* ?+s z are la/ 9 ! ord] ( be+s1 :i* subsi/ 9 matt]1 & are =m$ by ! pla/ic h& ( nature1 are manife/ly se5 to enjoy ^! two pr9ciples ess5tially4 ,inf9ite z ! subject seat ( _! =ms2 b b\nd z t :i* inve/s !m ) reasons1 figures1 & =ms4 ,& h5ce x is manife/ t ma!matical ess5ces h ! same pre-exi/5t pr9ciples ) all ! o!r g5]a ( be+s4 ,,*ap4 ,,iii4 ,:at ! common ,!orems are ( ! ,ma!matical ,ess5ces4 ,b z we h contemplat$ ! common pr9ciples ( ?+s1 :i* are diffus$ ?r\gh all ! ma!matical g5]a1 aft] ! same mann] we mu/ consid] ^? common & simple !orems1 orig9at+ f "o sci5ce1 :i* conta9s all ma!matical k 9 "o4 ,& we mu/ inve/igate h[ !y are capable ( accord+ ) all numb]s1 magnitudes & motions4 ,b ( ?is k9d are all consid]ations respect+ proportions1 compositions1 divisions1 conv]sions1 & alt]nate *anges3 also ! speculation ( e k9d ( reasons1 multiplex1 sup]-p>ticul>1 sup]-p>ti5t1 & ! opposite to ^!3 toge!r ) ! common & univ]sal consid]ations respect+ equal & unequal1 n z conv]sant 9 figures1 or numb]s1 or motions1 b s f> z ea* ( ^! possesses a common nature ess5tially1 & af=ds a m simple k ( itself4 ,b beauty & ord] are also common to all ! ma!matical discipl9es1 toge!r ) a passage f ?+s m kn[n1 to su* z are s\ght =1 & a transition f ^! to ^? :i* are call$ resolutions & compositions4 ,besides1 a similitude & dissimilitude ( reasons are by no means abs5t f ! ma!matical g5]a3 = we call "s figures simil>1 & o!rs dissimil>2 & ! same ) respect to numb]s4 ,& aga91 all ! consid]ations :i* reg>d p[]s1 agree 9 l mann] to all ! ma!matical discipl9es1 z well ! p[]s !mselves1 z ?+s subject to _! dom9ion3 :i*1 inde$1 ,socrates1 9 ! ,republic1 d$icates to ! ,muses1 speak+ ?+s ardu\s & sublime1 because he _h embrac$ ?+s common to all ma!matical reasons1 9 t]m9at$ limits1 & _h det]m9$ !m 9 giv5 numb]s1 9 :i* ! measures bo? ( abundance & /]ility appe>4 ,,*ap4 ,,iv4 ,h[ ^! ,common ,prop]ties subsi/1 & by :at ,sci5ce !y are consid]$4 ,b x is requisite to believe1 t ^! common prop]ties d n prim>ily subsi/ 9 _m & divid$ =ms1 nor orig9ate f ?+s _m & ! la/3 b we ought to place !m z ?+s prec$+ 9 a c]ta9 simplicity & excell5ce4 ,= ! k ( ^! antec$es _m kn[l$ges1 & supplies !m ) pr9ciples2 & ! multitude ( sci5ces subsi/ ab\t ?is1 & are ref]r$ to x z _! s\rce4 ,?us ! geometrician affirms1 t :5 f\r magnitudes are proportional1 !y %all be alt]nately proportional2 & he demon/rates ?is f pr9ciples peculi> to 8 sci5ce1 & :i* ! ari?metician nev] uses4 ,9 l mann]1 ! ari?metician affirms1 t :5 f\r numb]s are proportional1 !y %all be s alt]nately3 & ?is he ev9ces f ! prop] pr9ciples ( 8 sci5ce4 ,= :o is he t kn[s alt]nate ratio consid]$ by itself1 :e!r x subsi/s 9 magnitudes or 9 numb]s8 ,& ! division ( composite magnitudes or numb]s1 & 9 l mann]1 ! composition ( su* z are divid$8 ,= surely x _c be said t !re are sci5ces & cognitions ( ?+s divisible3 b t we h no sci5ce ( ?+s de/itute ( matt]1 & :i* are assign$ a m intellectual contemplation2 = ! k ( ^! is by a mu* great] priority sci5ce1 & f ^! ! common reasons ( _m sci5ces are d]iv$4 ,& !re is a gradual asc5t 9 cognitions f ?+s m p>ticul> to m univ]sal1 till we rev]t to ! sci5ce ( t :i* is1 consid]$ z x is1 ab/ract$ f all second>y prop]ties4 ,= ?is sublime sci5ce does n ?9k x suitable to its dignity1 to contemplate ! common prop]ties :i* are ess5tially inh]5t 9 numb]s1 & are common to all quantities2 b x contemplates ! "o1 & firm ess5ce ( all ! ?+s :i* are4 ,h5ce1 x is ! mo/ capaci\s ( all sci5ces1 & f ?is all ! re/ assume _! own peculi> pr9ciples4 ,= ! sup]ior sci5ces always af=d ! fir/ suppositions ( demon/rations to su* z are subord9ate4 ,b t :i* is ! mo/ p]fect ( all ! sci5ces1 di/ributes f itself pr9ciples to all ! re/1 to "s inde$1 su* z are m univ]sal1 b to o!rs1 su* z are m p>ticul>4 ,h5ce1 ,socrates1 9 ! ,!tetus1 m+l+ ! jocose ) ! s]i\s1 comp>es ! sci5ces :i* reside 9 u to doves3 b he says !y fly away1 "s 9 flocks1 b o!rs sep>ate f "o ano!r4 ,= su*1 inde$1 z are m common & m capaci\s1 compreh5d 9 !mselves _m su* z are m p>ticul>3 b su* z be+ di/ribut$ into =ms1 t\* ?+s subject to k1 are di/ant f "o ano!r1 & c by no means be copulat$ toge!r1 s9ce !y are excit$ by diff]5t prim>y pr9ciples4 ,"o sci5ce1 !re=e1 prec$es all sci5ces & discipl9es1 s9ce x kn[s ! common prop]ties :i* p]vade ?r\gh all ! g5]a ( be+s1 & supplies pr9ciples to all ! ma!matical sci5ces4 ,& ?us f> our doctr9e conc]n+ dialectic#gc is t]m9at$4 ,,*ap4 ,v4 ,:at ! ,in/rum5t is1 :i* judges ( ! ,ma!matical ,g5]a & ,species4 ,let u n[ consid] :at t in/rum5t is#gd1 adapt$ to ! judgm5t ( ma!matical conc]ns2 & let u appo9t ,plato z our guide 9 ?is affair1 :o1 9 8 ,republic1 divides cognitions sep>ately f su* ?+s z are ! objects ( k2 & di/ributes cognitions 9 conjunction ) ?+s subject to k4 ,= ( ! ?+s :i* are1 "s he ranks among intelligibles1 & o!rs among s5sibles4 ,& ( intelligibles1 "s are aga9 pure intelligibles1 & o!rs subject to cogitation4 ,& ( s5sibles1 "s are purely s5sibles1 b o!rs conjectural4 ,to intelligibles1 inde$1 :i* are ! fir/ ( ! f\r g5]a1 he assigns an intelligible k2 b to ^? :i* are subject to cogitation1 he attributes ?\ght3 to s5sibles1 fai?2 b to conjecturals1 a conjectural or assimilatory p[]4 ,& he %ews1 t ! assimilatory p[] has ! same proportion to s5se z ?\ght to intellig5ce4 ,= ! conjectural p[] kn[s ! spectres ( s5sible =ms1 :ile !y are beheld 9 wat] & o!r bodies1 :i* p]spicu\sly repres5t _! image3 s9ce1 by _! situation 9 wat]1 !y are aft] a mann]1 allott$ ! la/ seat 9 ! gradations ( =ms1 & truly become ! resemblances ( resemblances4 ,9 l mann]1 ?\ght beholds ! images ( intelligibles 9 a degrad$ /ate1 fall5 f prim>y simple & indivisible =ms1 into multitude & division4 ,h5ce1 a k ( ?is k9d1 dep5ds on o!r m anci5t hypo!ses2 b intellig5ce arrives at t pr9ciple :i* is no long] suppos$4 ,if !n1 ma!matical conc]ns are nei!r allott$ an ess5ce sep>ate f all division & v>iety1 nor t nature :i* is appreh5d$ by s5se1 :i* is obnoxi\s to _m mutations1 & is 9 e proportion divisible1 x mu/ be manife/ to e "o1 t !y are ess5tially subject to cogitation3 b cogitation presides ov] ^! z an in/rum5t adapt$ to judgm5t1 9 ! same mann] z s5se to s5sibles1 & ! assimilatory p[] to conjecturals4 ,f :5ce1 inde$1 ,socrates det]m9es t ! k ( ^! is m obscure ?an ! fir/ sci5ce1 b is m evid5t ?an ! impulsive appreh5sion ( op9ion4 ,= 9 ?is ! ma!matical sci5ces are inf]ior to intellig5ce1 because !y contemplate t :i* is evolv$1 & is endu$ ) a p[] ( progression2 b !y are sup]ior to op9ion1 by t /ability ( reasons :i* !y conta91 & :i* _c be confut$4 ,& !y orig9ate f supposition1 ?r\gh a dim9ution ( ! fir/ sci5ce2 b !y conta9 =ms indep5d5t ( matt]1 f _! possess+ a k m p]fect ?an t ( s5sibles4 ,we h !re=e det]m9$ an in/rum5t adapt$ to ! judgm5t ( all ma!matical conc]ns1 i4 e4 cogitation1 accord+ to ! m9d ( ,plato2 :i* places itself inde$ above op9ion1 b is excell$ by intellig5ce4 ,,*ap4 ,,vi4 ,conc]n+ ! ,ess5ce ( ,ma!matical ,g5]a & ,species#ge4 ,x n[ rema9s1 t we consid] :at subsi/5ce or ess5ce ought to be assign$ to ma!matical g5]a & species8 ,:e!r we mu/ d$uce _! orig9 & subsi/5ce f s5sible objects1 or f ab/raction1 or f a collection ( su* ?+s z are disp]s$ by p>ts into "o common def9ition2 or mu/ all[ !m an exi/5ce prior to t ( s5sibles1 z ,plato affirms1 & z ! progression ( univ]sal be+ demon/rates8 ,fir/ !n1 if we affirm t ma!matical species are compos$ f s5sibles2 :il/ ! s\l f mat]ial triangles or circles1 =ms 9 h]self ! trigonic1 or circul> species1 by a k9d ( second>y g5]ation2 ,i w\ld ask f :5ce is d]iv$ ! great c]ta9ty & accuracy ( def9itions8 ,= x mu/ ei!r proce$ f s5sibles1 or f ! s\l h]self4 ,b f s5sibles is impossible1 = ^!1 9 a cont9ual fl[ ( g5]ation & decay1 d n = a mom5t reta9 an exact sam5ess ( be+2 & consequ5tly fall f> %ort ( ! exactness conta9$ 9 ! def9itions !mselves4 ,x mu/ !re=e proce$ f ! s\l1 :i*1 by h] immat]ial nature1 procures p]fection f ! imp]fect1 accurate subtilty f t :i* is nei!r accurate nor subtle1 & rek9dles ! light ( ideas f ! obscure & unreal objects ( s5se4 ,= :]e %all we f9d1 among/ s5sible objects1 an indivisible nature1 su* z t ( a po9t1 or a l9e )\t ! dim5sion ( bread?1 or a sup]ficies )\t dep?1 or ! ev] con/ant proportion ( sides1 & exact rectitude ( angles8 ,= my "p1 ,i _c see :]e1 s9ce all divisible natures are ?us mix$ & confus$ toge!r1 no?+ s9c]e1 no?+ free f its contr>y1 b ?+s e :]e yield+ to sep>ation1 z well su* z are remov$ by di/ance ( place1 z ^? :i* are unit$ toge!r4 ,h[ !n %all we obta9 ?is durable ess5ce = ^! immoveable natures f ! ev] fluctuat+ =ms ( s5se8 ,= :atev] d]ives its exi/5ce f moveable be+s1 mu/ ( necessity be mutable & frail4 ,& h[ %all we ga9 ?is p]fect accuracy = ! /able species1 f ! inaccurate & imp]fect8 ,= :atev] is ! cause ( a conception1 always immutable1 is itself mu* m /able ?an its effect4 ,we mu/ !re=e admit ! s\l to be ! g5]ator ( ^! ma!matical species & reasons4 ,b if %e conta9s !m 9 h]self1 z fir/ exempl>s1 %e gives !m an ess5tial be+1 s t ! g5]ations are no?+ else ?an propagations ( species1 :i* _h a prior subsi/5ce 9 h]self3 & ?us we %all speak agreeably to ! s5tim5ts ( ,plato1 & discov] ! true ess5ce ( ma!matical entities4 ,b if ! s\l1 ?\gh %e nei!r possesses nor receiv$ ! ma!matical reasons prior to ! en]gies ( s5se1 yet fabricates ?is admirable immat]ial build+1 & g5]ates ?is fair s]ies ( speculations2 h[ c %e disc]n :e!r h] productions are /able & con/ant1 or ?+s :i* ! w9ds may dissipate1 & phantoms r ?an realities8 ,:at /&>d c %e apply z ! measure ( _! tru?8 ,or h[1 s9ce %e is de/itute ( _! ess5ce1 c %e g5]ate su* a v>iety ( reasons8 ,= f su* an hypo!sis1 we make _! subsi/5ce =tuit\s1 n t5d+ to any sci5tific b\nd4 ,ma!matical species are !re=e ! g5u9e (fspr+ ( ! s\l3 nor does %e d]ive f s5sible objects ! def9itions %e frames1 b r ! fir/ are propagat$ f ! second2 !y are ! en]gies ( s\l1 :i*1 z x 71 pregnant ) =ms1 deliv]s h] immat]ial prog5y into ! d>k & fluctuat+ regions ( matt]1 z evid5ces ( ! p]man5t duration ( h] species4 ,aga91 if we collect ma!matical reasons f ext]nals1 :y are n demon/rations compos$ f s5sibles1 bett] ?an ! demon/rations ( univ]sal & simple species8 ,= we say1 9 ord] to ! inve/igation ( any ?+ s\ght1 t ! pr9ciples & propositions1 %\ld be alli$ to ! conclusions4 ,if !n1 p>ticul>s are ! causes ( univ]sals1 & s5sibles ! s\rces ( reason+1 :y does ! b\nd>y ( demon/ration always ref] to t :i* is m univ]sal1 & n to t :i* is p>tial & p>ticul>8 ,& h[ c we prove t ! ess5ce ( intelligibles is m alli$ to demon/ration ?an ! ess5ce ( s5sibles8 ,= ?us !y speak#gf3 8 k is n legitimate1 :o demon/rates t ! isosceles1 ! equilat]al1 or ! scal5e triangle1 h angles equal to two "r2 b he possesses sci5ce1 prop]ly s call$1 :o demon/rates ?is ( e triangle simply1 or ( triangle itself4 ,& aga91 t univ]sals1 = ! purpose ( demon/ration1 are sup]ior to p>ticul>s2 t demon/rations conc]n ?+s m univ]sal2 b t ! pr9ciples f :i* demon/rations are compos$1 h a priority ( exi/5ce1 & a prec$5cy 9 nature to s+ul>s1 & are ! causes ( ! propositions !y prove4 ,x is v remote1 !re=e1 f ! nature ( ,apodictical sci5ces1 t f conv]se ) ?+s ( po/]ior orig91 & f ! d>k p]ceptions ( s5se1 !y %\ld falsely collect _! indubitable propositions4 ,i add f>!r1 t !y :o affirm ?is1 make ! s\l ( a bas] nature ?an ! mat]ial species !mselves4 ,= if matt] d]ives f nature be+s ess5tial1 & p>ticipat+ a high degree ( entity & evid5ce2 b ! s\l1 by a po/]ior en]gy1 receives ^! f s5sible objects1 & fa%ions 9 h]self resemblances & images ( po/]ior orig91 contemplat+ vile ess5ces1 & ab/ract+ f matt]1 ! =ms insep>able f its nature2 d !y n make ! s\l m obscure & indig5t ?an matt] itself8 ,= matt] is ! receptacle ( =ms mat]ializ$1 z ! s\l is ( species immat]ializ$4 ,b 9 ?is case1 matt] w\ld be ! place ( prim>y be+s1 & ! s\l ( su* z are second>y & subord9ate3 matt] & its =ms obta9+ ! lead 9 be+1 & exi/+ z ! s\rces ( ! subsi/5ce ( immat]ial =ms4 ,la/ly1 ! mat]ial =ms w\ld h an ess5tial exi/5ce1 ! o!rs only an int5tional d5om9ation4 ,h[ !n c ! s\l1 :i* is ! fir/ p>ticipant ( intellect1 & an intellective ess5ce1 & :i* d]ives f !nce consummate k1 & a pl5itude ( life1 become ! receptacle ( ! mo/ obscure species1 ! l[e/ 9 ! ord] ( ?+s1 & p>ticipat+ ! mo/ imp]fect exi/5ce4 ,b ?is op9ion1 :i* has be5 suffici5tly explod$ by o!rs1 ne$s no f>!r confutation4 ,if !n1 ma!matical species d n subsi/ by mat]ial ab/raction1 nor by a collection ( ^? common prop]ties inh]5t 9 individuals2 nor are at all1 9 _! orig91 po/]ior to s5sibles1 nor d]iv$ 9 any mann] f !m3 x is necess>y t ! s\l %\ld ei!r d$uce !m f h]self1 or f intellect2 or la/ly1 f h]self & intellect unit$4 ,b if f h]self alone1 ,:5ce d ! images ( intellectual species arise2 :5ce d !y d]ive _! middle nature1 l9k+1 z x 71 ! divisible & indivisible ess5ce toge!r2 if !y d n p>ticipate ! fullness ( entity f prim>y ess5ces8 ,la/ly1 h[1 ^u ?is hypo!sis1 are ! fir/ exempl>s1 p>adigms1 or ideas1 :i* subsi/ 9 intellect1 ! pr9ciples ( univ]sals8 ,b if !y are d]iv$ f intellect alone into ! s\l1 h[ c ! s\l rema9 self-op]ative1 & self-motive1 if h] inh]5t reasons fl[ f an ext]nal s\rce1 & are regulat$ by its op]ations8 ,& 9 :at respect does ! s\l diff] f matt]1 :i* is all ?+s 9 m]e dormant capacity1 b g5]ates no?+ app]ta9+ to mat]ial species8 ,x rema9s1 !re=e1 t ! s\l d$uces ^! species f h]self1 & intellect2 & t %e is ! absolute consummation ( ! =ms :i* orig9ate f intellectual exempl>s1 b :i* are allott$ f !mselves a transition to p]man5t be+4 ,! s\l1 !re=e1 is by no means to be comp>$ to a smoo? tablet1 void ( all reasons2 b %e is an ev]-writt5 tablet1 h]self inscrib+ ! *>act]s 9 h]self1 ( :i* %e d]ives an et]nal pl5itude f intellect4 ,= s\l is a c]ta9 subord9ate intellect1 revolv+ r\nd an intellect prior to h]self1 =m$ to its image1 & p>ticipat+ its div9e irradiations4 ,if !n1 ?is sup]ior intellect is all ?+s intellectually1 s\l w all ?+s animally2 if ! fir/ exi/s z ! exempl>1 s\l w be z its image2 if z contract$ & unit$ 9 itself1 s\l z divisible & exp&$4 ,& ?is is :at ,plato und]/ood1 :5 9 8 ,timu1 he composes ! s\l ( ! _w f all ?+s1 divid+ h] accord+ to h>monical reasons1 & analogies2 assign+ to h] ! fir/ pr9ciples effective ( figures1 ,i mean ! "r & circul> l9e1 & giv+ an intellectual motion to h] inh]5t circles4 ,all ma!matical species1 !re=e1 h a prim>y subsi/5ce 9 ! s\l3 s t1 be=e s5sible numb]s1 !re are to be f\nd 9 h] inmo/ recesses1 self-mov+ numb]s2 vital figures1 prior to ! app>5t1 ideal proportions ( h>mony previ\s to concordant s\nds2 & invisible orbs1 prior to ! bodies :i* revolve 9 a circle4 ,s t s\l is ! prolific abundance ( all ^!1 & is ano!r ornam5t produc+ h]self1 & produc$ f a prop] pr9ciple1 fill+ h]self ) life1 & at ! same "t fill$ f ! demiurgus ( ! univ]se1 is an incorporeal & indi/ant mann]4 ,:51 !re=e1 %e produces & unfolds h] lat5t reasons1 %e !n detects e sci5ce & virtue4 ,! ess5ce ( s\l !n consi/s 9 ^! species1 nor mu/ we suppose h] inh]5t numb]s to be a multitude ( units1 nor h] ar*ytipal ideas ( divisible =ms to be corporeal3 b we mu/ conceive all ^! z subsi/+ ev] vitally1 & intellectually1 z ! exempl>s ( app>5t numb]s1 figures1 reasons & motions4 ,& "h we mu/ foll[ ! doctr9e ( ,timu1 :o d]ives ! orig91 & consummates ! fabric ( ! s\l1 f ma!matical =ms1 & reposes 9 h] nature ! causes ( e ?+ :i* exi/s4 ,= ! sev5 b\nd+ t]ms#gg1 compreh5d+ ! pr9ciples ( all numb]s1 l9es1 planes & solids1 pre-exi/ 9 s\l accord+ to cause4 ,& aga91 ! pr9ciples ( figures are plac$ 9 h] ess5ce1 accord+ to a demiurgical p[]4 ,& la/ly1 ! fir/ ( all motions1 :i* embraces e o!r motion 9 its compreh5sive ambit1 is co-exi/5t ) s\l4 ,= ! pr9ciple ( e ?+ :i* is mov$ is a circle1 & ! circul> motion4 ,! ma!matical reasons1 !re=e1 :i* fully consummate ! s\l1 are ess5tial1 & self-mov+3 & ! s\l1 by h] cogitative p[]1 diffus+1 propagat+1 & evolv+ ^!1 f h] pr(\nd recesses1 con/itutes all ! fair v>iety ( ma!matical sci5ces4 ,nor w %e ev] cease to g5]ate1 & wak5 into en]gy1 succe$+ species1 :ile %e dive/s h] indivisible reasons ( _! intellectual simplicity4 ,= %e previ\sly receiv$ all ?+s1 aft] a prim>y mann]2 & accord+ to h] inf9ite p[]1 f pre-exi/5t pr9ciples1 d$uces a beautiful s]ies ( v>i\s speculations4 ,,*ap4 ,,vii4 ,:at ! ,employm5ts & ,p[]s are ( ! ,ma!matical ,sci5ce1 & h[ f> !y ext5d !mselves 9 _! ,en]gies4 ,b1 aft] contemplat+ ! ess5ce ( ma!matical =ms1 x is necess>y we %\ld recur to t "o ma/]-sci5ce ( ^!1 :i* we h %ewn is prior to a multitude ( o!rs1 & t we %\ld contemplate :at its employm5t is1 :at are its p[]s1 & h[ f> x advances 9 its en]gies4 ,! employm5t1 !re=e1 ( ! :ole ma!matical sci5ce1 possess+1 z we h be=e said1 ! p[] ( cogitation1 mu/ n be plac$ s high z t ( intellig5ce2 :i* is firmly seat$ 9 its own /able ess5ce1 is p]fect1 is conta9$ by itself1 & 9 itself cont9ually v]ges4 ,nor mu/ x be situat$ s l[ z t ( op9ion & s5se1 s9ce ^! cognitions dwell ^u ext]nal conc]ns1 en]gize ^u !m1 & d n possess ! causes ( ! objects ( _! k4 ,b ! ma!matical sci5ce1 receives its comm5cem5t1 inde$1 extr9sically f#gh recollection1 b ends 9 ! mo/ intimate reasons1 resid+ 9 ! dep?s ( ! s\l2 & is excit$1 inde$1 f ?+s po/]ior1 b arrives by gradual advances at ! pr9cipal ess5ce ( =ms4 ,nor is its en]gy immoveable1 l t ( intellig5ce1 nor is x affect$ ) local motion & alt]ation1 l s5se1 b x revolves ) a vital en]gy1 & runs ?r\gh ! ornam5t ( incorporeal reasons1 sometimes advanc+ f pr9ciples to su* ?+s z are p]fect$ by pr9ciples1 b at o!r times yield+ 9 a retrograde progression f conclusions to _! =m+ pr9ciples3 & sometimes proce$+ f ?+s previ\sly kn[n1 to su* z are ! subject ( inve/igation3 b at o!r times1 f ?+s plac$ 9 ! "q1 to su* z prec$e 9 cognition4 ,besides1 x does n excel all inquisition1 z if x 7 p]fect f itself1 l intellect1 nor is x p]fect$ f o!rs1 l s5se1 b x proce$s by enquiry to inv5tion1 & asc5ds f ! imp]fect to p]fection4 ,b x likewise possesses tw(old p[]s1 "o k9d ( ^! d$uc+ pr9ciples into multitude1 & g5]at+ ! diff]5t pa?s ( contemplation3 b ! o!r endu$ ) a p[] ( collect+ _m transitions into prop] suppositions4 ,= s9ce x proposes to itself z pr9ciples1 z well unity1 & multitude1 z b\nd & inf9ite1 & su* ?+s z are subject to its compreh5sion1 are allott$ a middle ord]1 betwe5 =ms indivisible & e way divisible2 ) great propriety 7,i ?9k7 ! gno/ic p[]s ( ! :ole sci5ce ( ^! are ess5tially tw(old4 ,"o species inde$1 ha/5s to union1 & contracts ! expansion ( multitude3 b ! o!r possesses a p[] ( di/+ui%+ ?+s simple into su* z are v>i\s1 m univ]sals into m p>ticul>s1 & reasons dige/$ 9 _! pr9ciple1 into ?+s second>y & multif>i\sly multipli$ f ^! pr9ciples4 ,= ris+ high] f its comm5cem5t x p5etrates ev5 to su* ?+s z are ! p]fections ( s5sible conc]ns1 is jo9$ ) nature1 & demon/rates _m ?+s toge!r ) natural sci5ce4 ,s9ce asc5d+ f inf]iors1 x acc$es 9 a c]ta9 respect proximate to intellectual k1 & t\*es ! contemplation ( ?+s prim>y & div9e4 ,& h5ce1 9 ! limits :i* fl[ f its ess5ce1 x produces ! :ole me*anic1 optic1 & catoptric speculation1 toge!r ) _m o!r sci5ces :i* are inwov5 & entangl$ ) s5sible conc]ns1 & :i* op]ate ?r\gh _! assi/ance4 ,besides1 9 its asc5sions f corporeal natures1 x d]ives intellig5ces indivisible & de/itute ( matt]3 & ) ^! x p]fects its divisible appreh5sions1 ^? cognitions :i* subsi/ 9 progressions1 & its own g5]a & =ms3 x likewise indicates ! tru? respect+ ! gods !mselves1 & 9 its peculi> treatises exhibits a contemplation ( ! ?+s :i* are4 ,& ?us mu* conc]n+ ! employm5t & p[]s ( ! ,ma!matical ,sci5ce4 ,,*ap4 ,,viii4 ,conc]n+ ! ,utility ( ! ,ma!matical ,sci5ce4 ,b let u n[ consid] ! utility ( ?is ,sci5ce1 :i* ext5ds itself f ! mo/ pr9cipal to ! la/ cognitions4 ,timu1 !re=e1 calls ! k ( ! ma!matical discipl9es ! pa? ( erudition1 because1 inde$1 x has ! same proportion to univ]sal sci5ce1 & ! fir/ philosophy1 :i* le>n+ has to virtue4 ,= ?is la/ frames our s\l to a p]fect life1 by ! possession ( wor?y mann]s2 b ! =m] prep>es our cogitation1 & ! div9e eye ( our s\l to an elevation f ! obscurity ( s5sible in=mation4 ,h5ce1 ,socrates 9 ! ,republic1 says1 8,t ! eye ( ! s\l1 :i* is d>k5$ & buri$ by o!r /udies1 c by ! ma!matical discipl9es alone be invigorat$1 & aga9 excit$ to ! contemplation ( t :i* is1 & transf]r$ f resemblances to real be+s1 f an obscure light to t light :i* has ! p[] ( intellig5ce1 & f a cave1 & ^? bonds :i* exi/ 9 x z ! au?ors ( g5]ation1 & f mat]ial imp$im5ts be able to rise to an incorporeal & indivisible ess5ce4 ,= ! beauty & ord] ( ma!matical reasons1 & ! firmness & /ability ( ! contemplations !y af=d1 conjo9s u ) intelligible objects1 & p]fectly det]m9es u 9 _! ess5ces2 :i* p]petually rema9 ! same1 ev] %9+ ) div9e beauty1 & pres]v+ a mutual ord] )\t end4 ,b ,socrates1 9 ! ,phdrus1 deliv]s to u ?ree *>act]s :o are elevat$ f s5se1 because !y fill up & accompli% ! prim>y life ( ! s\l1 i4 e4 ! philosoph]1 ! lov]1 & ! musician4 ,b ! beg9n+ & pa? ( elevation to ! lov]1 is a progression f app>5t beauty1 us+ z excitations ! middle =ms ( beautiful objects4 ,b to ! musician1 :o is allott$ ! ?ird seat1 ! way consi/s 9 a transition f s5sible to invisible h>monies1 & to ! reasons exi/+ 9 ^!4 ,s t to ! "o1 sight is ! in/rum5t ( rem9isc5ce1 & to ! o!r1 he>+4 ,b to him :o is by nature a philosoph]1 f :5ce & by :at means is rem9isc5ce ! prelude ( intellectual k1 & an excitation to t :i* truly is1 & to tru? itself8 ,= ?is "* also1 on acc\nt ( its imp]fection1 requires a prop] pr9ciple3 = x is allott$ a natural virtue1 an imp]fect eye1 & a degrad$ mann]4 ,x mu/ !re=e be excit$ f itself2 & he :o is ( su* a nature1 rejoices 9 t :i* is4 ,b to ! philosoph]1 says ,plot9us1 ! ma!matical discipl9es mu/ be exhibit$1 t !y may accu/om him to an incorporeal nature1 & t aft]w>ds us+ ^! z figures1 he may be l$ to dialectic reasons1 & to ! contemplation ( all ! ?+s :i* are4 ,& ?us x is manife/1 f h5ce1 t ! ma!matics are ( ! greate/ utility to philosophy4 ,b x is requisite t we %\ld be m explicit1 & m5tion ! sev]al p>ticul>s to :i* !y conduce1 & ev9ce t !y prep>e ! intellectual appreh5sions ( !ology4 ,= :atev] to imp]fect natures appe>s difficult & ardu\s 9 obta9+ ! true k ( ! gods1 ! ma!matical reasons r5d]1 by _! images1 cr$ible1 manife/1 & c]ta94 ,?us1 9 numb]s1 !y indicate ! significations ( sup]-ess5tial prop]ties1 b !y ev9ce ! p[]s ( intellectual figures1 9 ^? figures :i* fall "u cogitation4 ,h5ce x is1 t ,plato1 by ma!matical =ms tea*es u _m & admirable s5t5ces conc]n+ ! gods1 & ! philosophy ( ! ,py?agoreans1 us+ ^! z veils1 conceals f vulg> inspection ! discipl9e ( div9e s5t5ces4 ,= su* is ! :ole ( ! ,sacr$ & ,div9e ,disc\rse#gi1 t ( ,philolaus 9 8 ,bac*ics1 & ! univ]sal me?od ( ! ,py?agoric n>ration conc]n+ ! ,gods4 ,b x especially ref]s to ! contemplation ( nature1 s9ce x discloses ! ord] ( ^? reasons by :i* ! univ]se is fabricat$1 & t proportion :i* b9ds1 z ,timu says1 :atev] ! _w conta9s1 9 union & cons5t2 besides1 x conciliates 9 amity ?+s mutually oppos+ ea* o!r1 & gives conv5i5ce & cons5t to ?+s mutually disagree+1 & exhibits to our view simple & prim>y elem5ts1 f :i* ! univ]se is compos$1 on e side compreh5d$ by comm5surability & equality1 because x receives conv5i5t figures 9 its proportions1 & numb]s prop] to e production1 & f9ds out _! revolutions & r5ovations1 by :i* we are enabl$ to reason conc]n+ ! be/ orig91 & ! contr>y dissolution ( p>ticul>s4 ,9 consequ5ce ( ?is1 z x appe>s to me1 ,timu discloses ! contemplation conc]n+ ! nature ( ! univ]se1 by ma!matical names1 adorns ! orig9 ( ! elem5ts ) numb]s & figures1 ref]r+ to ^! _! p[]s1 passions1 & en]gies2 & e/eem+ z well ! acut5ess z ! obtus5ess ( angles1 ! levity ( sides1 or contr>y p[]s1 & _! multitude & paucity to be ! cause ( ! all-v>i\s mutation ( ! elem5ts4 ,b :y may we n say1 t x pr(its mu*1 & 9 an admirable mann]1 to t philosophy :i* is call$ ,politic1 z well by measur+ ! times ( actions z af=d+ ! v>i\s revolutions ( ! univ]se1 & numb]s conv5i5t to ?+s ris+ into be+2 ,i mean ! assimilat+1 & au?ors ( dissimilitude1 ! prolific too & ! p]fect1 & ! contr>ies to ^!2 toge!r ) ord]ly & elegant m9i/]s ( life1 & inelegance2 & f9ally1 su* numb]s z procure f]tility & /]ility4 ,:i*1 inde$1 ! spee* ( ! ,muses 9 ! ,republic#hj ev9ces1 plac+ ! univ]sal ,geometric ,numb] z ! au?or ( bett] & m debas$ g5]ations1 & z ! cause ( ! indissoluble p]sev]ance ( good mann]s1 & ( ! mutation ( ! be/ ,republics into su* z are remote f reason1 & are giv5 to affections4 ,= x is suffici5tly evid5t1 t x belongs to ! :ole ma!matical discipl9e to deliv] ! sci5ce ( ?is numb] :i* is call$ geometrical1 & n to "o p>ticul> sci5ce1 su* z ari?metic1 or geometry3 s9ce ! reasons or proportions ( abundance & /]ility1 p]meate ?r\gh all ! ma!matical discipl9es4 ,aga91 x is ! means ( our in/itution 9 moral philosophy :i* x br+s to its ultimate p]fection1 & gives ord] & an elegant life to our mann]s4 ,besides ?is1 x deliv]s to u figures1 & modulations & motions conv5i5t to virtue1 by :i* ! ,a!nian gue/ wi%es ^? to be in/itut$ & p]fect$1 :o are de/9$ to pursue moral virtue f _! e>ly y\?4 ,add too1 t x places be=e our view ! reasons ( virtues1 9 "o mann]1 inde$1 9 numb]s1 9 ano!r 9 figures1 b diff]5tly 9 musical symphonies2 & la/ly1 x indicates ! excess & defect ( vices1 by :i* we are enabl$ to mod]ate & adorn our mann]s4 ,h5ce x is1 t ,socrates1 9 ! ,gorgias1 accus+ ,calicles ( an inord9ate & intemp]ate life1 says to him1 8,y neglect geometry & geometric equality30 b1 9 ! ,republic1 he f9ds out ! proportion ( tyrannic pleasure to a royal int]val1 accord+ to a plane & solid g5]ation4 ,b we %all le>n :at great utility is d]iv$ to o!r sci5ces & arts f ! ma!matical sci5ce1 :5 we consid] t x adds ord] & p]fection to contemplative arts2 ,i mean rhetoric1 & all su* z consi/ 9 disc\rse4 ,b x proposes to ! poetic arts1 ! reasons ( poems 9 ! place ( an example1 because x presides ov] ! measures exi/+ 9 ^!4 ,b to ! active arts x det]m9es action & motion1 by its own abid+ & immoveable =ms4 ,= all arts1 z ,socrates says1 9 ! ,philebus1 require ari?metic1 m5suration1 & /atics1 ei!r 9 all1 or 9 "s ( _! op]ations4 ,b all ^! are conta9$ 9 ! disc\rses ( ! ma!matical sci5ce1 & are t]m9at$ accord+ to _! div]sity4 ,= f ?is sci5ce ! divisions ( numb]s1 & ! v>iety ( dim5sions1 & ! diff]5ce ( weights are kn[n4 ,! utility1 !re=e1 ( ! :ole ma!matical sci5ce to philosophy itself1 & to o!r sci5ces & arts1 may be f h5ce kn[n to intellig5t he>]s40 ,,*ap4 ,,ix4 ,a ,solution ( an ,objection rais$ by "s aga9/ ! ,utility ( ! ,ma!matical ,sci5ces4 ,b "s1 :o are prone to contradiction ?r\gh ^? :o wi% to subv]t geometry1 endeav\r to de/roy ! dignity ( ?is sci5ce4 ,"o "p1 inde$1 depriv+ x ( ornam5t & good1 because x does n disc\rse on ^!4 ,b ano!r "p#ha affirm+ t s5sible exp]im5ts are m useful ?an ! univ]sal objects ( its speculation2 ,i mean1 t ,geodesia 7= in/ance17 or ! m5suration ( ! e>?1 is pref]able to geometry1 & vulg> ari?metic to t ari?metic :i* is conv]sant ) !orems alone3 & t nautical a/rology is m useful ?an t :i* tea*es univ]sally1 ab/ract$ f any application to s5sible conc]ns4 ,= we are n1 say !y1 made ri* by our k ( ri*es1 b by us+ !m2 nor are we happy by ! m]ely und]/&+ felicity1 b by liv+ happily4 ,h5ce we mu/ confess t ^? ma!matical sci5ces1 :i* are conv]sant ) cognition1 d n pr(it human life1 & conf] to action1 b ^? only :i* are engag$ 9 ex]cise4 ,= ^? :o are ignorant ( ! reasons ( ?+s1 b are ex]cis$ 9 p>ticul> & s5sible exp]im5ts1 are 9 e respect m excell5t1 = ! purposes ( human life1 ?an ^? :o are employ$ 9 contemplation alone4 ,aga9/ objections !n1 ( ?is k9d1 we %all reply1 by %ew+ ! beauty ( ! ma!matical discipl9es f ^? argum5ts by :i* ,ari/otle endeav\rs to p]suade u4 ,we mu/ !re=e confess t !re are ?ree ?+s :i* especially cause beauty1 bo? 9 bodies & s\ls2 ,i mean1 ord]1 conv5i5ce1 & det]m9ation4 ,s9ce corporeal bas5ess1 inde$1 arises f mat]ial inord9ation1 de=mity1 & inconv5i5ce1 & f ! dom9ion ( ! indef9ite 9 ! composite body4 ,b ! bas5ess ( ! s\l orig9ates f its irrationality1 & inord9ate motion1 & f its be+ 9 a /ate ( discord ) reason1 & n receiv+ f !nce its prop] limitation4 ,h5ce1 beauty exi/s ev5 9 contr>ies1 by means ( ord]1 conv5i5ce & det]m9ation4 ,b we may behold ^! 9 a m em95t degree 9 ! ma!matical sci5ce2 ord]1 inde$1 9 ! p]petual exhibition ( ?+s po/]ior & m v>i\s1 f su* z are prim>y & m simple2 = ?+s subsequ5t are always annex$ to _! prec$5ts1 ! latt] rank+ z pr9ciples1 & ! =m] z ! fir/ suppositions ( ?+s consequ5t3 b conv5i5ce is ev9c$ 9 ! mutual consonance ( ?+s demon/rat$1 & 9 ! relation ( all ( !m to intellect1 s9ce intellect is ! common measure ( all sci5ce1 f :i* x receives its pr9ciples1 & to :i* x conv]ts ! le>n]3 b det]m9ation is p]ceiv$ 9 its p]petually abid+ & immoveable reasons1 = ! objects ( its k are n1 at times1 subject to v>iation1 l ^? ( op9ion & s5se1 b pres5t !mselves = ev] ! same1 & are b\nd$ by intellectual =ms4 ,if su* !n1 are ! pr9cipal requisites ( beauty1 x is evid5t1 t 9 ^! sci5ces t illu/ri\s ornam5t & gracefulness is f\nd4 ,= h[ is x possible ?is %\ld n be ! case ) a sci5ce receiv+ a sup]nal illum9ation f intellect1 to :i* x cont9ually advances1 ha/5+ to transf] u f ! obscure light ( s5sible in=mation8 ,) respect to ! second objection1 we ?9k x prop] to judge ( its utility1 )\t reg>d+ ! conv5i5cies & necessities ( human life4 ,= o!rwise1 we mu/ confess t contemplative virtue is also useless1 :i* sep>ates itself f human conc]ns1 :i* x is v little desir\s to look d[n ^u & und]/&4 ,inde$ ,socrates1 9 ! ,!tetus1 affirm+ ?is conc]n+ noblem5 endu$ ) ! prophetic p[]1 says1 8t x )draws !m f all reg>d to human life1 & raises _! ?\ghts1 prop]ly lib]at$1 f all necessity & use1 to ! v summit ( all true be+40 ,! ma!matical sci5ce1 !re=e1 mu/ be consid]$ z desirable = its own sake1 & = ! contemplation x af=ds1 & n on acc\nt ( ! utility x adm9i/]s to human conc]ns4 ,b if x is necess>y to ref] ! utility x produces to some?+ diff]5t f itself1 x mu/ be ref]r$ to intellectual k4 ,= x leads u to ?is1 & prep>es ! eye ( ! s\l = ! k ( univ]sals1 remov+ & oblit]at+ ! imp$im5ts aris+ f ! s5ses1 & f corporeal involution4 ,z !re=e we call ! :ole ( purgative virtue useful1 or ! contr>y1 n reg>d+ ! use ( ! s5sible life1 b ( t :i* is contemplative1 s inde$ x is requisite to ref] ! end ( ma!matics to intellect1 & univ]sal wisdom4 ,h5ce its en]gy is wor?y our /udy1 bo? on its own acc\nt1 & on acc\nt ( an intellectual life4 ,b x appe>s1 z ,ari/otle#hb says1 t ?is sci5ce is desirable ( itself to its vot>ies1 because ?\gh no rew>d is propos$ to its enquir]s1 yet ! ma!matical contemplation receives1 9 a small "t1 an abundant increase4 ,besides1 ?is is f>!r evid5t f h5ce1 t all m5 are will+ly employ$ 9 its pursuit1 & wi% to dwell on its speculations1 omitt+ e o!r conc]n2 ev5 ^? :o h1 ) _! lips1 z x 71 b j t\*$ its utility4 ,& h5ce x foll[s1 t !y :o despise ! k ( ! ma!matical discipl9es1 h v little ta/$ ( ! pleasures !y conta94 ,! ma!matics1 !re=e1 are n to be despis$ because _! speculative p>ts d n imm$iately conf] to human utility1 7= ! ultimate limits ( its progressions1 & :atev] op]ates ) matt]1 consid] a use ( ?is k9d27 b on ! contr>y we %\ld admire its immat]iality1 & ! good :i* x conta9s1 consid]$ by itself alone4 ,= :5 mank9d 7 entirely dis5gag$ f ! c>e ( necess>y conc]ns1 !y conv]t$ !mselves to ! inve/igation ( ! ma!matical discipl9es2 & ?is1 inde$1 ) ! greate/ propriety4 ,s9ce affairs famili> to human life 9 its mo/ imp]fect /ate1 & :i* are imm$iately connect$ ) its orig91 fir/ ( all employ$ ! /udies ( mank9d3 b1 9 ! second place1 ^? conc]ns succe$$ :i* sep>ate ! s\l f g5]ation1 & re/ore its memory ( t :i* ,,is4 ,aft] ?is mann]1 !n1 we are engag$ 9 necess>ies1 be=e ?+s hon\rable = _! own sakes1 on acc\nt ( _! intr9sic dignity & wor?2 & 9 ?+s relat$ to s5se1 be=e su* z are appreh5d$ by ! nobl] en]gies ( m9d4 ,= e orig9 & life ( ! s\l :i* is conv]t$ into h]self1 is naturally adapt$ to proce$ f ! imp]fect to ! p]fect4 ,& ?us mu* aga9/ ^? :o despise ! ma!matical sci5ce40 ,,*ap4 ,x4 ,a ,solution ( ano!r ,objection ( c]ta9 ,platoni/s1 aga9/ ! ,utility ( ! ,ma!matical ,sci5ces4 ,b1 p]haps1 "s ( our own family w "h rise up aga9/ u1 &1 propos+ ,plato z a witness1 w endeav\r to provoke rud] und]/&+s into a contemptu\s disreg>d ( ! ma!matical discipl9es4 ,= !y w say1 t ?is philosoph] entirely excludes 79 8 ,republic7 ! ma!matical k f ! *oir ( ! sci5ces1 & t he accuses x z be+ ignorant ( its own pr9ciples1 t its v pr9ciple is to itself unkn[n1 & its ends & m$iums compos$ f ?+s ( :i* x is ignorant4 ,to ^! objections !y may likewise add :atev] o!r reproa*es are !re urg$ by ,socrates aga9/ ?is contemplation4 ,9 answ] !n1 to ! objections ( our fri5ds1 we %all recall into _! memory1 t ,plato himself p]spicu\sly ass]ts ! ma!matical sci5ce to be ! purgation ( ! s\l1 & t x is endu$ ) a p[] ( lead+ x on high2 because1 l ! ,hom]ic ,m9]va1 x removes ! d>kness ( a s5sible nature f ! intellectual light ( ?\ght1 :i* is bett] wor? sav+ ?an t5 ?\s& corporeal eyes1 & :i* n only p>ticipates ( a m]curial gift1 7pres]v+ u f ! incantations & delusions ( ?is mat]ial abode1 :i* is simil> to ! fasc9at+ realms ( ,circe17 b also ( ! m div9e arts ( ,m9]va4 ,he likewise e :]e calls x by ! "n ( sci5ce1 & ass]ts t x is ! cause ( ! greate/ felicity to ^? :o are ex]cis$ 9 its contemplation4 ,b ,i w briefly expla9 :y1 9 ! ,republic he takes f x ! surname ( sci5ce3 = my pres5t disc\rse is address$ to ! le>n$4 ,plato1 inde$1 9 mo/ places1 calls all ! k 7z ,i may say7 ( univ]sals by ! "n ( sci5ce1 oppos+ x 9 a division to s5se :i* appreh5ds only p>ticul>s1 :e!r su* a mode ( cognition is accompli%$ by art or exp]i5ce4 ,& 9 ?is s5se1 z x appe>s to me 9 ! ,civil ,dialogue1 & 9 ! ,sophi/a1 he seems to use ! "n ( sci5ce2 plac+ likewise ! illu/ri\s ,sophi/ic sci5ce1 :i* ,socrates 9 ! ,gorgias1 says1 is a c]ta9 exp]i5ce3 also1 ! adulatory1 & _m o!rs1 :i* are exp]i5ces1 b n true sci5ces4 ,b aga91 divid+ ?is k ( univ]sals into t :i* kn[s causes1 & into t :i* und]/&s )\t a cause1 he ?9ks t ! "o %\ld be call$ sci5ce1 b ! o!r exp]i5ce4 ,& h5ce1 to arts he sometimes attributes ! "n ( sci5ce1 b to exp]i5ce nev]4 ,= h[ 7says he 9 ! ,banquet7 c a ?+ :i* possesses no reason be sci5ce8 ,all k1 !re=e1 :i* conta9s ! reason & cause ( ! ?+s kn[n1 is a c]ta9 sci5ce4 ,aga91 !re=e1 he divides ?is sci5ce :i* is endu$ ) a p[] f ! cause ( kn[+1 by ! peculi>ity ( its subjects1 & he places "o1 conjectural ( ?+s divisible2 b ! o!r ( su* z subsi/ by !mselves1 & are ev] kn[able aft] ! same mann]4 ,& accord+ to ?is division he sep>ates f sci5ce1 m$ic9e1 & e faculty :i* is conv]sant ) mat]ial conc]ns4 ,b he calls ma!matical k1 & :atev] possesses a p[] ( contemplat+ et]nal objects1 by ! "n ( sci5ce4 ,la/ly1 divid+ ?is sci5ce1 :i* we di/+ui%$ f arts1 he consid]s "o "p z void ( supposition2 b ! o!r z fl[+ f supposition4 ,& t ! "o :i* is void ( supposition1 has a p[] ( kn[+ univ]sals3 t x rises to good1 & ! supreme cause ( all2 & t x consid]s good z ! end ( its elevation3 b t ! o!r1 :i* previ\sly fabricates = itself def9ite & det]m9ate pr9ciples1 f :i* x ev9ces ?+s consequ5t to su* pr9ciples1 does n t5d to ! pr9ciple1 b to ! conclusion4 ,& h5ce he ass]ts1 t ma!matical k1 because x makes use ( supposition1 falls %ort ( t sci5ce :i* is )\t supposition1 & is p]fect4 ,= !re is "o true sci5ce1 by means ( :i* we are dispos$ to "k all ! ?+s :i* are1 & f :i* also pr9ciples em]ge to all sci5ces2 to "s1 inde$1 con/itut$ m proximately1 b to o!rs m remotely4 ,we mu/ n say1 !re=e1 t ,plato expels ma!matical k f ! numb] ( ! sci5ces1 b t he ass]ts x to be ! second f t "o sci5ce1 :i* possesses ! supreme seat ( all3 nor mu/ we affirm1 t he accuses x z ignorant ( its own pr9ciples1 b t receiv+ ^! f ! ma/] sci5ce dialectic1 & possess+ !m )\t any demon/ration1 x demon/rates f ^! its consequ5t propositions4 ,=1 inde$1 he sometimes all[s ! s\l1 :i* is con/itut$ f ma!matical reasons1 to be ! pr9ciple ( motion3 & sometimes he affirms1 t x receives its motion f g5]a :i* are subject to intellig5ce4 ,& ^! v>iations accord among !mselves4 ,= to su* ?+s z are mov$ by ano!r1 ! s\l is a c]ta9 cause ( motion1 b x is n ! cause ( e motion4 ,aft] ! same mann]1 ! ma!matical sci5ce is inde$ ! second f ! fir/ ( all sci5ces1 &1 ) ref]5ce to x1 imp]fect3 b x is1 nev]!less1 a sci5ce1 n z be+ free f supposition1 b z kn[+ ! peculi> reasons resid5t 9 ! s\l1 & z br++ ! causes ( conclusions1 & conta9+ ! reason ( su* ?+s z are subject to its k4 ,& ?us mu* = ! op9ion ( ,plato respect+ ma!matics4 ,,*ap4 ,,xi4 ,b let u n[ consid] :at are ! ?+s :i* may be requir$ ( a ma!matician1 & h[ any "o may rightly judge conc]n+ 8 di/+ui%+ peculi>ities4 ,=#hc ,ari/otle inde$1 says1 t he :o is simply le>n$ 9 all discipl9es1 is adapt$ to judge ( all3 b t he :o is alone skill$ 9 ! ma!matical sci5ces1 c alone det]m9e conc]n+ ! magnitude ( reasons inh]5t 9 ^!4 ,x is requisite1 !re=e1 t we %\ld previ\sly assume ! t]ms ( judg+1 & t we %\ld "k1 9 ! fir/ place1 9 :at ?+s x is prop] to demon/rate g5]ally1 & 9 :at to reg>d ! peculi>ities ( s+ul>s4 ,= _m ( ! same prop]ties reside 9 ?+s diff]+ 9 species1 z two "r angles 9 all triangles3 b _m h inde$ ! same pr$icam5t1 yet diff] 9 _! individuals 9 a common species1 z similitude 9 figures & numb]s4 ,b "o demon/ration is n to be s\ght = by ! ma!matician 9 ^!1 = ! pr9ciples ( figures & numb]s are n ! same1 b diff] 9 _! subject g5us4 ,& if ! ess5tial accid5t is "o1 ! demon/ration w also be "o#hd3 = ! possession ( two "r angles is ! same 9 all triangles1 & t g5]al some?+ to :i* ?is p]ta9s is ! same 9 all1 ,i mean triangle1 & a triangul> reason4 ,9 ! same mann]1 likewise1 ! possession ( ext]nal angles to f\r "r ones1 n only p]ta9s to triangles1 b also to all "r-l9$ figures2 & ! demon/ration1 s f> z !y are "r-l9$1 agrees 9 all4 ,= e reason br+s ) x1 at ! same "t1 a c]ta9 prop]ty & passion1 ( :i* all p>ticipate ?r\gh t reason1 :e!r triangul>1 or rectil9e>1 or univ]sally figure4 ,b ! second limit by :i* a ma!matician is to be judg$1 is1 if he demon/rates accord+ to 8 subject-matt]1 & r5d]s necess>y reasons1 & su* z _c be confut$1 b are at ! same "t nei!r probable1 nor repl5i%$ ) a similitude ( tru?4 ,=1 says ,ari/otle1 x is j ! same to require demon/rations f a rhetorician1 & to ass5t to a ma!matician disput+ probably2 s9ce e "o1 endu$ ) sci5ce & art1 ought to r5d] reasons adapt$ to ! subjects ( 8 inve/igation4 ,9 l mann] also1 ,plato 9 ! ,timu1 requires cr$ible reasons ( ! natural philosoph]1 z "o :o is employ$ 9 ! resemblances ( tru?3 b ( him :o disc\rses conc]n+ intelligibles1 & a /able ess5ce1 he dem&s reasons :i* c nei!r be confut$ nor mov$4 ,= subjects e :]e cause a diff]5ce 9 sci5ces & arts1 s9ce1 if "s ( !m are immoveable1 o!rs are conv]sant ) motion2 & "s are m simple1 b o!rs m composite2 & "s are intelligibles1 b o!rs s5sibles4 ,h5ce we mu/ n require ! same c]ta9ty f e "p ( ! ma!matical sci5ce4 ,= if "o "p1 aft] a mann]1 bord]s ^u s5sibles1 b ano!r "p is ! k ( intelligible subjects1 !y _c bo? be equally c]ta91 b "o mu/ inh]it a high] degree ( evid5ce ?an ! o!r4 ,& h5ce x is1 t we call ari?metic m c]ta9 ?an ! sci5ce ( h>mony4 ,nor mu/ we ?9k x j t ma!matics & o!r sci5ces %\ld use ! same demon/rations2 = _! subjects af=d !m no small v>iety4 ,9 ! ?ird place1 we mu/ affirm1 t he :o rightly judges ma!matical reasons1 mu/ consid] sam5ess & diff]5ce1 :at subsi/s by itself1 & :at is accid5tal1 :at proportion is1 & e consid]ation ( a simil> k9d4 ,= almo/ all errors ( ?is sort happ5 to ^? :o ?9k !y demon/rate ma!matically1 :5 at ! same "t !y by no means demon/rate1 s9ce !y ei!r demon/rate ! same ?+ z if diff]5t 9 ea* species1 or t :i* is diff]5t z if x 7 ! same3 or :5 !y reg>d t :i* is accid5tal1 z if x 7 an ess5tial prop]ty2 or t :i* subsi/s by itself1 z if x 7 accid5tal4 ,= in/ance1 :5 !y endeav\r to demon/rate t ! circumf]5ce ( a circle is m beautiful ?an a "r l9e1 or an equilat]al ?an an isosceles triangle4 ,= ! det]m9ation ( ^! does n belong to ! ma!matician1 b to ! fir/ philosoph] alone4 ,la/ly1 9 ! f\r? place1 we mu/ affirm1 t s9ce ! ma!matical sci5ce obta9s a middle situation betwe5 intelligibles & s5sibles1 & exhibits 9 itself _m images ( div9e conc]ns1 & _m exempl>s ( natural reasons1 we may behold 9 x ?ree k9ds ( demon/ration#he1 "o approa*+ ne>] to intellect1 ! second m accommodat$ to cogitation1 & ! ?ird bord]+ on op9ion4 ,= x is requisite t demon/rations %\ld diff] accord+ to ! v>ieties ( problems1 & receive a division correspond5t to ! g5]a ( be+s1 s9ce ! ma!matical sci5ce is connect$ ) all ^!1 & adapts its reasons to ! univ]sality ( ?+s4 ,& ?us mu* = a discussion ( ! subject propos$4 ,,*ap4 ,,xii4 ,:at & h[ _m ! ,species ( ! :ole ,ma!matical ,sci5ce are1 accord+ to ! ,op9ion ( ! ,py?agoreans4 ,b aft] ^! consid]ations1 x is requisite to det]m9e conc]n+ ! p>ts ( ! ma!matical sci5ce1 :at1 & h[ _m !y are4 ,= x is j1 aft] speculat+ its :ole & entire g5us1 to consid] ! diff]5ces ( its m p>ticul> sci5ces1 accord+ to _! species4 ,! ,py?agoreans#hf1 !re=e1 ?\ght t ! :ole ma!matical sci5ce %\ld receive a f\rfold di/ribution1 attribut+ "o ( its p>ts to ! h[-_m1 b ! o!r to ! h[-mu*2 & !y assign$ to ea* ( ^! p>ts a tw(old division4 ,= !y said1 t discrete quantity1 or ! h[-_m1 ei!r subsi/s by itself1 or mu/ be consid]$ ) relation to "s o!r2 b t cont9u$ quantity1 or ! h[-mu*1 is ei!r /able or 9 motion4 ,h5ce !y affirm$1 t ari?metic contemplates t discrete quantity :i* subsi/s by itself1 b music t :i* is relat$ to ano!r2 & t geometry consid]s cont9u$ quantity s f> z x is immoveable2 b sph]ics contemplates cont9u$ quantity z mov+ f itself1 9 consequ5ce ( its union ) a self-motive nature4 ,!y affirm$ besides1 t ^! two sci5ces1 discrete & cont9u$ quantity1 did n consid] ei!r magnitude or multitude absolutely1 b t alone :i* 9 ea* ( ^! is def9ite f ! p>ticipation ( b\nd4 ,= sci5ces alone speculate ! def9ite1 reject+ z va9 ! compreh5sion ( inf9ite quantity4 ,b :5 ^! wise m5 assign$ ?is di/ribution1 we mu/ n suppose !y und]/ood t discrete quantity :i* is f\nd 9 s5sible natures1 nor t cont9u$ quantity :i* subsi/s ab\t ! fluctuat+ ord] ( bodies4 ,=1 ,i ?9k1 ! contemplation ( ^! p]ta9s to ! natural & n to ! ma!matical sci5ce4 ,b because ! demiurgus ( ! univ]se1 employ$ ! union1 division1 & id5tity ( g5]al natures1 toge!r ) diff]5ce1 /ation1 & motion1 = ! purpose ( complet+ ! ess5ce ( ! s\l1 & compos$ x f ^! g5]a1 z ,timu in=ms u1 we mu/ affirm1 t cogitation1 abid+ accord+ to its div]sity1 its division ( reasons1 & its multitude1 & und]/&+ itself to be bo? "o & _m1 proposes inde$ to itself1 & produces numb]s1 toge!r ) an ari?metical k ( ^!3 b x provides = itself music accord+ to an union ( its multitude1 & a communication & junction ) itself2 & h5ce x is t ari?metic excels music 9 antiquity2 s9ce1 accord+ to ! n>ration ( ,plato1 ! demiurgus fir/ divid$ ! s\l1 & aft]w>ds collect$ x 9 h>monical proportions4 ,aga91 ?\ght e/abli%+ its en]gy accord+ to ! /ability :i* x conta9s1 draws f its inmo/ retreats geometry1 toge!r ) "o ess5tial figure1 & ! demiurgical pr9ciples ( all figures#hg3 b1 accord+ to its inh]5t motion1 x produces ! sph]ical sci5ce4 ,= x is mov$ also by circles1 b abides p]petually ! same f ! causes ( circles4 ,h5ce1 likewise1 geometry prec$es sph]ics1 9 ! same mann] z /ation is prior to motion4 ,b because cogitation itself produces ^! sci5ces1 n by look+ back ^u its convolution ( =ms1 endu$ ) an inf9ite p[]1 b ^u ! inclosure ( b\nd accord+ to its def9ite g5]a2 h5ce !y say1 t ! ma!matical sci5ces take away inf9ite f multitude & magnitude1 & are only conv]sant ab\t f9ite quantity4 ,inde$1 intellect has plac$ 9 cogitation all ! pr9ciples bo? ( multitude & magnitude4 ,= s9ce x :olly consi/s1 ) ref]5ce to itself1 ( simil> p>ts1 & is "o & indivisible1 & aga9 divisible1 educ+ ! ornam5t ( =ms1 x p>ticipates ( b\nd & inf9ite1 f intelligible ess5ces !mselves4 ,b x und]/&s1 inde$1 f its p>ticipation ( b\nd1 & g5]ates vital en]gies1 & v>i\s reasons f ! nature ( inf9ite4 ,! intellections1 !re=e1 ( ?\ght1 con/itute ^! sci5ces accord+ to ! b\nd :i* !y conta91 & n accord+ to an inf9ity ( life2 s9ce !y br+ ) !m an image ( intellect1 b n ( life4 ,su* !n is ! op9ion ( ! ,py?agoreans1 & ! division ( ! f\r ma!matical sci5ces4 ,,*ap4 ,,xiii4 ,ano!r ,division ( ! ,ma!matical ,sci5ce1 accord+ to ,gem9us4 ,aga91 "s ?9k 7among :om is ,gem9us7 t ! ma!matical sci5ce is to be divid$ 9 a diff]5t mann] f ! prec$+4 ,& !y consid] t "o ( its p>ts is conv]sant ) intelligibles only1 b ! o!r ) s5sibles1 ^u :i* x bord]s2 d5om9at+ z intelligibles :atev] inspections ! s\l r\ses into en]gy by h]self1 :5 sep>at+ h]self f mat]ial =ms4 ,& ( t :i* is conv]sant ) intelligibles !y e/abli% two1 by f> ! fir/ & mo/ pr9cipal p>ts1 ari?metic & geometry3 b ( t :i* unfolds its (fice & employm5t 9 s5sibles1 !y appo9t six p>ts1 me*anics1 a/rology1 optics1 geodsia1 canonics1 & logi/ics1 or ! art ( reckon+4 ,b !y d n ?9k t ! milit>y art1 or tactics1 %\ld be call$ any "o "p ( ma!matics1 accord+ to ! op9ion ( "s#hh2 b !y consid] x z us+ at "o "t ! art ( reckon+1 z 9 ! numb]+ ( legions2 b at ano!r "t geodsia1 z 9 divid+ & measur+ ! spaces fill$ by a field ( camps4 ,z1 say !y1 nei!r ! art ( writ+1 nor ! art ( heal+1 are any "p ( ma!matics1 ?\gh frequ5tly bo? ! hi/orian & physician use ma!matical !orems4 ,?is is ! case ) hi/orians inde$1 :5 relat+ ! situation ( climates1 or collect+ ! magnitudes & dim5sions ( cities1 or _! compass & circuit3 b ) physicians1 :5 elucidat+ by ways ( ?is k9d1 _m ?+s 9 _! art4 ,= ,hippocrates himself %ews ! utility d]iv$ to m$ic9e f a/rology1 & almo/ all :o speak ( opportune times & places4 ,by ! same reason he also1 :o accommodates 8 "w to tactics1 uses inde$ ma!matical !orems1 yet is n on ?is acc\nt a ma!matician1 al?\gh he is sometimes will+ t a num]\s camp %\ld exhibit a v small multitude1 & =ms 8 army accord+ to a circul> figure2 b sometimes 9 a quadrangul>1 qu9quangul>1 or "s o!r multangul> figure1 :5 he desires x to appe> num]\s4 ,b s9ce ^! are ! species ( ! :ole ma!matical sci5ce1 geometry is aga9 divid$ into ! contemplation ( planes1 & ! dim5sion ( solids1 :i* is call$ /]eometry4 ,= !re is n any peculi> treatise ab\t po9ts & l9es1 because no figure c be produc$ f ^! )\t planes or solids4 ,= geometry treats ( no?+ else 9 e "o ( its p>ts1 ?an t x may con/itute ei!r planes or solids3 or t :5 con/itut$1 x may comp>e & divide !m among !mselves4 ,9 l mann]1 ari?metic is di/ribut$ into ! contemplation ( l9e>1 plane1 & solid numb]s4 ,= x consid]s ! species ( numb]s sep>ate f s5sible connections1 proce$+ f unity1 & ! orig9 ( plane numb]s2 ,i mean ( ! simil>1 dissimil>1 & solid1 ev5 to ! ?ird increase4 ,b geodsia1 & ! art ( reckon+1 are divid$ simil>ly to ari?metic & geometry1 z !y d n disc\rse conc]n+ intelligible numb]s or figures1 b ( su* z are s5sible alone4 ,= nei!r is x ! (fice ( geodesia to measure ! cyl9d] or ! cone1 b mat]ial masses z if !y 7 cones1 & wells z if !y 7 cyl9d]s4 ,nei!r does x accompli% ?is purpose by intelligible "r l9es1 b by su* z are s5sible1 sometimes inde$ by a m c]ta9 means1 z by ! sol> rays3 b at o!r times by gross] ones1 z by a l9e & p]p5dicul>4 ,9 l mann]1 ! reckon] does n survey ! passions ( numb]s by !mselves1 b z !y are resid5t 9 s5sible objects4 ,f :5ce he also imposes a "n ^u ^! d]iv$ f ! ?+s :i* he reckons1 call+ !m 1 4 ,besides ?is1 he does n1 admit ( any lea/1 l ! ari?metician1 :o receives t m9imum1 z a g5us ( relation4 ,= "s "o man is consid]$ by him z ! measure ( ! :ole multitude ( m51 z unity also is ! common measure ( all numb]s4 ,aga91 optics & canonics are produc$ f geometry & ari?metic4 ,& optics uses ! visual rays :i* are con/itut$ by ! rays ( ! eyes1 z l9es & angles4 ,b x is divid$ into t :i* is prop]ly call$ optics 7because x r5d]s ! cause ( ^! appe>ances1 :i* are accu/om$ to pres5t !mselves to u diff]5t f _! reality1 on acc\nt ( ! diff]5t situations & di/ances ( visible objects1 z ! co9cid5ce ( p>allel l9es1 or ! appe>ance ( quadrangles z if !y 7 circles72 & into univ]sal catoptrics1 :i* is conv]sant ab\t v>i\s & manifold refractions1 & is connect$ ) imag9ative or conjectural k3 z also into t :i* is call$ sciography#hi1 or ! del9eation ( %ad[s1 :i* %ews h[ appe>ances 9 images may seem nei!r inelegant nor de=m$1 on acc\nt ( ! di/ances & altitudes ( ! ?+s design$4 ,b canonics 7music7 or ! regul> art1 consid]s ! app>5t reasons ( h>monies1 f9d+ out ! sections ( rules1 e :]e us+ ! assi/ance ( s5se1 &1 z ,plato says1 seem+ to pref] ! te/imony ( ! e>s to intellect itself4 ,b to ! p>ts we h hi!rto enum]at$1 me*anics mu/ he add$1 z x is a c]ta9 "p ( ! :ole sci5ce1 & ( ! k ( s5sible objects1 & ( ?+s unit$ ) matt]4 ,b "u ?is exi/s ! art effective ( in/rum5ts1 :i* is call$ 7o7 ,i mean ( ^? in/rum5ts prop] = ! purposes ( w>3 su*1 inde$1 z ,ar*im$es is report$ to h con/ruct$1 resi/+ ! besieg]s ( sea & l&2 & t :i* is effective ( miracles1 & :i* is call$ 747 ,"o "p ( ?is con/ructs ) ! greate/ artifice pneumatic eng9es1 su* z ,ctesibius & ,h]on fabricat$3 b ano!r op]ates ) weights1 ! motion ( :i* is reckon$ to be ! cause ( inequilibrity2 b _! /ation ( equilibrity1 z ,timu also has det]m9$3 & aga91 ano!r "p imitates animate fold+s & motions by /r+s & ropes4 ,aga91 "u me*anics is plac$ ! k ( equilibriums1 & ( su* in/rum5ts z are call$ c5tropond]ants3 also 77 or ! art effective ( sph]es1 imitat+ ! cele/ial revolutions1 su* z ,ar*im$es fabricat$2 & la/ly1 e ?+ endu$ ) a p[] ( mov+ matt]4 ,b ! la/ ( all is a/rology1 :i* treats ( ! mundane motions1 ( ! magnitudes ( ! cele/ial bodies1 _! figures & illum9ations1 _! di/ances f ! e>?1 & e ?+ ( ?is k9d2 assum+ _m ?+s inde$ to itself f s5se1 b communicat+ mu* ) ! natural speculation4 ,"o "p ( ?is is gnomonics1 :i* is ex]cis$ 9 settl+ ! dim5sion ( horor>y gnomons3 b ! o!r is me!oroscopics1 :i* f9ds out ! diff]5ces ( elevations1 & ! di/ances ( ! />s1 & also tea*es _m o!r & v>i\s a/rological !orems4 ,! ?ird "p is dioptrics1 :i* asc]ta9s by dioptric in/rum5ts ( ?is k9d ! di/ances ( ! sun & moon1 & ( ! five o!r />s4 ,& su* is ! acc\nt ( ! p>ts ( ! ma!matical sci5ce1 deliv]$ by ! anci5ts1 & transmitt$ to our memory by ! in=m+ h& ( "t4 ,,*ap4 ,,xiv4 ,h[ ,dialectic is ! ,top ( ! ,ma!matical ,sci5ces1 & :at _! ,conjunction is1 accord+ to ,plato4 ,let u aga9 consid] aft] :at mann] ,plato1 9 8 ,republic1 calls dialectic ! top ( ! ma!matical discipl9es2 & :at _! conjunction is1 accord+ to ! tradition ( ! au?or ( ! ,ep9omis#ij4 ,& 9 ord] to ?is we mu/ ass]t1 t z intellect is sup]ior to cogitation1 supply+ x ) sup]nal pr9ciples1 & f itself giv+ p]fection to cogitation2 9 ! same mann] dialectic also1 be+ ! pure/ "p ( philosophy1 excels 9 simplicity ! ma!matical discipl9es1 to :i* x is proximate1 & ) :i* x is conjo9$4 ,inde$ x embraces ! complete circle ( ^! sci5ces1 to :i* x elevates f itself v>i\s en]gies1 endu$ ) a p[] ( caus+ p]fection1 judgm5t1 & intellig5ce4 ,& ^! en]gies consi/ 9 resolv+1 divid+1 def9+1 & demon/rat+2 by :i* ma!matics itself1 receiv+ assi/ance & p]fection1 inv5ts "s ?+s by resolution1 b o!rs by composition3 & "s ?+s x expla9s by division1 o!rs by def9ition3 b collects o!r subjects ( its inve/igation by demon/ration2 accommodat+1 inde$1 ^! ways to its subjects1 b us+ ea* ( !m = ! purpose ( behold+ its middle enquiries4 ,f :5ce inde$1 bo? ! resolutions1 def9itions1 divisions1 & demon/rations :i* x conta9s1 are peculi>1 & adapt$ to its nature1 & revolve accord+ to ! mode ( ma!matical cognition4 ,n undes]v$ly1 !re=e1 is dialectic ! v]tex z x 71 & summit ( ma!matics4 ,s9ce x p]fects all :i* ma!matics conta9s ( intellig5ce2 r5d]s its c]ta9ty free f repreh5sion1 pres]ves ! /ability ( its immovable ess5ce1 & ref]s :at x conta9s de/itute ( matt] & pure to ! simplicity ( intellect1 & a nature sep>at$ f mat]ial connections4 ,besides1 x di/+ui%es ! fir/ pr9ciples ( ^! sci5ces1 by def9itions3 exhibits ! sep>ations ( g5]a & =ms conta9$ "u ! g5]a !mselves3 & besides ?is1 tea*es ! compositions1 :i*1 f pr9ciples1 produce ?+s consequ5t to pr9ciples3 & ! resolutions :i* rise & m\nt up to ?+s fir/1 & to pr9ciples !mselves4 ,b ) respect to :at rema9s1 proportion itself is n to be consid]$ 7z ,erato/h5es ?\ght x 07 z ! conjunction ( ! ma!matical discipl9es4 ,s9ce proportion is said to be1 & inde$ is "o ( ^? ?+s common to ! ma!matics4 ,b 9 %ort1 _m o!r ?+s besides proportion reg>d all ! ma!matical discipl9es1 :i* are ess5tially inh]5t 9 ! common nature ( ! ma!matics4 ,b z x appe>s to me1 we %\ld say1 t !re is "o proximate conjunction ( ^!1 & ( ! :ole ma!matical sci5ce1 :i* especially embraces 9 itself1 9 a m simple mann]1 ! pr9ciples ( all sci5ces2 :i* consid]s _! community & diff]5ce2 tea*es :atev] is f\nd 9 ^! ! same2 toge!r ) :at ?+s are inh]5t 9 a _m1 & :at 9 a few4 ,s t to ^? :o aptly le>n !re is a rev]sion f _m o!r sci5ces to ?is alone#ia4 ,b1 dialectic is a conjunction ( ! ma!matical discipl9es sup]ior to ! prec$+2 :i* ,plato1 z ,i h already obs]v$1 calls 9 8 ,republic _! v]tex3 =1 inde$1 x p]fects ! :ole ( ma!matics1 br+s x back to intellect by its p[]s1 %ews x to be a true sci5ce1 & causes x to be c]ta9 & obnoxi\s to no repro(4 ,b1 intellect obta9s ! ?ird ord] betwe5 ^! conjunctions1 :i* compreh5ds 9 itself uni=mly all ! dialectic p[]s1 contracts _! v>iety by its simplicity1 _! p>tition by its indivisible k1 & _! multitude by its occult union4 ,h5ce1 intellect itself congregates inde$ ! involutions & deviations ( ! dialectic pa?s1 into an intelligible ess5ce1 b x collects sup]nally all ! progression ( ma!matical disc\rses3 & x is ! be/ end bo? ( ! elevat+ p[] ( ! s\l1 & ( ! en]gy consi/+ 9 cognition4 ,& su* are ! s5tim5ts decl>$ by me on ! pres5t enquiry4 ,,*ap4 ,,xv4 ,f :5ce ! ,"n ,ma!matics orig9at$4 ,aga91 f :5ce %all we say ?is "n ( ma!matics1 & ma!matical discipl9es1 0 assign$ by ! anci5ts1 & :at apt reason c we r5d] ( its position8 ,inde$1 x appe>s to me1 t su* an appellation ( a sci5ce :i* respects cogitative reasons1 0 n1 l mo/ names1 inv5t$ by indiff]5t p]sons1 b 7z ! tru? ( ! case is1 & accord+ to report7 by ! ,py?agoreans alone4 ,& ?is1 :5 !y p]ceiv$1 t :atev] is call$ ma!sis or discipl9e1 is no?+ m ?an rem9isc5ce2 :i* does n approa* ! s\l extr9sically1 l ! images :i* ris+ f s5sible objects are =m$ 9 ! phantasy3 nor is x adv5titi\s & =eign1 l ! k consi/+ 9 op9ion1 b x is excit$1 inde$1 f app>5t objects1 & is p]fect$ )91 by ?\ght intimately conv]t$ to itself4 ,& :5 !y likewise p]ceiv$ t ?\gh rem9isc5ce might be %ewn f _m p>ticul>s1 yet x 0 ev9c$ 9 a m em95t mann] 7z ,plato also says#ib7 f ! ma!matical discipl9es4 ,= if any "o1 says he1 is l$ into ! descriptions1 he w !re easily prove t discipl9e is rem9isc5ce4 ,f :5ce ,socrates also1 9 ! ,m5o1 %ews by ?is me?od ( argu+1 t le>n+ is no?+ else ?an ! s\l's recollection ( h] inh]5t reasons4 ,& ?is1 because t :i* recollects1 is alone ! cogitative "p ( ! s\l2 b ?is p]fects h] ess5ce 9 ! reasons ( ! ma!matical discipl9es1 ! sci5ces ( :i* %e previ\sly receiv$ into h]self1 ?\gh %e does n always en]gize on _! fair v>iety4 ,inde$1 %e conta9s !m all ess5tially & occultly2 b %e produces ea* ( !m :5 %e is fre$ f ! imp$im5ts orig9at+ f s5se4 ,= s5se unites h] ) divisible objects3 ! phantasy fills h] ) =m+ motions1 & appetite b5ds h] to an indulg5t & luxuri\s life4 ,b e ?+ divisible is an ob/acle to our self-conv]sion4 ,& :atev] inve/s ) =m1 di/urbs & (f5ds t k :i* is de/itute ( =m4 ,& :atev] is obnoxi\s to p]turbations is an imp$im5t to t en]gy :i* is unimpair$ by affections4 ,:51 !re=e1 we h mov$ all ^! f ! cogitative p[]1 !n %all we be able to und]/& by ?\ght itself1 ! reasons :i* ?\ght conta9s3 !n %all we become sci5tific 9 en]gy2 & unfold our ess5tial k4 ,b :il/ we are captive & b\nd1 & w9k+ ) ! eye ( ! s\l1 we _c by any means atta9 to a p]fection conv5i5t to our nature4 ,su* !n is ma!sis or discipl9e3 a rem9isc5ce ( ! et]nal reasons conta9$ 9 ! s\l4 ,& ! ma!matical or discipl9ative sci5ce is on ?is acc\nt p>ticul>ly d5om9at$ t k :i* especially conf]s to our rem9isc5ce ( ^! ess5tial reasons4 ,h5ce1 ! bus9ess & (fice ( ?is sci5ce#ic1 is app>5t f its "n4 ,= its duty is to move ! inh]5t k ( ! s\l2 to awak5 its intellig5ce2 to purify its cogitation2 to call =? its ess5tial =ms f _! dormant retreats2 to remove t oblivion & ignorance1 :i* are cong5ial ) our bir?2 & to dissolve ! bonds aris+ f our union ) an irrational nature4 ,x pla9ly leads u to a similitude ( t div9ity :o presides ov] ?is sci5ce1 :o manife/s intellectual gifts1 & fills ! univ]se ) div9e reasons2 :o elevates s\ls to intellect1 wak5s !m z f a pr(\nd sleep1 conv]ts !m by enquiry to !mselves2 & by a c]ta9 ob/etric art1 & inv5tion ( pure intellect1 br+s !m to a bless$ life4 ,to :om inde$1 d$icat+ ! pres5t "w1 we "h conclude our contemplation ( ! ma!matical sci5ce4 ,,book ,,ii4 ,,*ap4 ,i4 ,:at ,"p ,geometry is ( ,ma!matics1 & :at ! ,matt] is ( :i* x consi/s4 ,9 ! prec$+ disc\rses we h consid]$ ^? common prop]ties :i* respect ! :ole ( ! ma!matical sci5ce2 & ?is we h done agreeable to ! doctr9e ( ,plato2 at ! same "t collect+ su* p>ticul>s z p]ta9 to our pres5t design4 ,b consequ5t to ?is x is requisite t we %\ld disc\rse on geometry itself1 & on ! propos$ in/itution ( ! elem5ts1 = ! sake ( :i* we h und]tak5 ! :ole ( ! pres5t "w4 ,t geometry !n1 is a "p ( ! :ole ( ma!matics1 & t x obta9s ! second place aft] ari?metic1 s9ce x is p]fect$ & b\nd$ by ?is1 7= :atev] 9 geometry may be express$ & kn[n1 is det]m9$ by ari?metical reasons7 has be5 ass]t$ by ! anci5ts1 & requires no long discussion 9 ! pres5t enquiry4 ,b we also may be able to relate our op9ion on ?is p>ticul>1 if we consid] :at place1 & :at ess5ce its subject matt]#id is allott$ among ! univ]sality ( ?+s4 ,= f a prop] survey ( ?is1 ! p[] ( ! sci5ce :i* kn[s ?is subject matt]1 ! utility aris+ f x1 & ! good acquir$ by its le>n]s1 w imm$iately appe>4 ,inde$1 "s "o may d\bt 9 :at g5us ( ?+s he ought to place geometrical matt]1 s z n to deviate f ! tru? x conta9s4 ,= if ! figures conc]n+ :i* geometry disc\rses1 exi/ 9 s5sible natures1 & _c be sep>at$ f ! d>k receptacle ( matt]2 h[ c we ass]t t geometry frees u f s5sible objects1 t x br+s u to an incorporeal ess5ce1 t x accu/oms u to an inspection ( intelligibles1 & prep>es u = intellectual en]gy8 ,:]e %all we ev] survey among s5sible objects a po9t )\t p>ts1 or a l9e de/itute ( bread?1 or a sup]ficies )\t pr(undity1 or ! equality ( l9es f ! c5tre to ! circumf]5ce2 or ! multangles1 & all ! figures ( _m bases1 conc]n+ :i* geometry in=ms u8 ,la/ly1 aft] :at mann] c ! reasons ( su* a sci5ce rema9 free f all possible confutation2 s9ce1 inde$1 s5sible =ms & figures are susceptive ( ! m & ! less1 are all moveable & mutable1 & are full ( mat]ial v>iety2 among :i* equality subsi/s mixt & confus$ ) its contr>y inequality1 & into :i* ?+s )\t p>ts h proce$$ into p>tition1 & int]val1 d>k5$ ) ! %ades ( matt]1 & lo/ 9 its inf9ite folds8 ,b if ! subjects ( geometry are remov$ f matt]1 are pure =ms1 & are sep>at$ f s5sible objects3 !y w be all ( !m1 )\t d\bt1 void ( p>ts1 incorporeal1 & de/itute ( magnitude4 ,= ext5sion1 tumor1 & int]val1 approa* to =ms1 on acc\nt ( ! mat]ial receptacle 9 :i* !y are involv$1 & :i* receives ?+s de/itute ( p>ts1 di/ribut$ into p>ts2 ?+s void ( dim5sion1 ext5d$ into dim5sion2 & immoveable natures accompani$ ) motion4 ,h[ !n1 if ?is is ! case1 %all we cut a "r l9e1 triangle1 & circle8 ,h[ c we speak ( ! div]sities ( angles1 & ! increm5ts & decrem5ts ( triangul> & quadrangul> figures8 ,or h[ exhibit ! contacts ( circles or "r l9es8 ,= all ^! ev9ce t ! geometric matt] consi/s ( p>ts1 & does n reside among indivisible reasons4 ,su* !n are ! d\bts conc]n+ ! matt] ( geometry1 to :i* we may add1 t ,plato consid]s ! =ms ( geometry z plac$ 9 cogitation2 & grants1 t we advance f s5sibles to =ms ( ?is k9d1 & t we rise f s5sibles to intellect1 ?\gh 7z we h previ\sly obs]v$7 ! reasons subsi/+ 9 cogitation are indivisible1 are sep>at$ by no int]val1 & subsi/ accord+ to ! peculi>ity ( ! s\l4 ,b if reasons are to be r5d]$ agreeable to ?+s !mselves1 & to ! doctr9e ( ,plato1 ! foll[+ division mu/ be adopt$4 #ie;,e univ]sal1 & "o ?+ conta9+ _m1 is ei!r naturally dispos$ to be ?\ght ( 9 p>ticul>s1 or to appe> su*1 because x possesses its exi/5ce 9 ^!2 is insep>able f !m2 is dispos$ & di/ribut$ 9 !m2 & toge!r ) ^! is ei!r mov$1 or firmly & immoveably abides4 ,or x is adapt$ to subsi/ prior to _m1 & to possess a p[] ( g5]at+ multitude1 af=d+ to _m ?+s images f itself1 be+ furni%$ ) a nature de/itute ( p>ts1 f ! ess5ces :i* x p>ticipates1 & rais+ v>i\s p>ticipations to second>y natures3 or x is dispos$ to be =m$ by ?\ght1 f ! _m1 to possess a g5]at+ exi/5ce1 & to reside 9 ! la/ place 9 ! _m4 ,=1 accord+ to ^! ?ree modes ( subsi/5ce1 we %all f9d1 ,i ?9k1 t "s subsi/ be=e ! _m1 o!rs 9 ! _m1 & o!rs f ! relation & pr$ication :i* !y possess to ^!4 ,b1 t ,i may absolve all 9 "o ^w1 univ]sal =ms be+ ?reefold1 we %all consid] ! diff]5ces ( t =m :i* _m p>ticipate1 :i* exi/s 9 _m1 & fills p>ticul> natures accord+ to its subject matt]4 ,besides ?is1 e/abli%+ a tw(old ord] ( p>ticipants1 "o subsi/+ 9 s5sible objects1 b ! o!r 9 ! phantasy1 7s9ce matt] is tw(old2 "o inde$1 ( ?+s unit$ ) s5se1 b ! o!r ( su* z fall "u ! inspection ( phantasy1 z ,ari/otle ass]ts1 9 a c]ta9 place#if7 we mu/ all[ t ! univ]sal1 :i* is di/ribut$ 9 ! _m1 is likewise tw(old4 ,! "o1 inde$1 s5sible1 z be+ t :i* s5sible objects p>ticipate2 b ! o!r imag9ative1 z t :i* subsi/s 9 ! _m ( ! phantasy4 ,= ! phantasy1 on acc\nt ( its =m+ motion1 & because x subsi/s )1 & 9 body1 always receives impressions :i* are bo? divid$ & figur$4 ,s t :atev] is kn[n by x1 is allott$ a correspond5t exi/5ce3 on :i* acc\nt1 ,ari/otle#ig does n hesitate to call x passive intellect4 ,b if x is intellect1 :y is x n impassive1 & de/itute ( matt]8 ,& if x op]ates ) passion1 h[ c x ) propriety be call$ intellect8 ,= impassivity1 inde$1 prop]ly belongs to intellect & an intellig5t nature3 b passivity is v remote f su* an ess5ce4 ,b 7unless ,i am deceiv$7 ,ari/otle be+ will+ to expla9 its middle nature betwe5 cognitions ! mo/ prim>y1 & su* z are ! la/1 calls x at ! same "t intellect1 because simil> to prim>y cognitions1 & passive f t alliance :i* x possesses ) su* z are po/]ior4 ,= fir/ cognitions are inde$ de/itute ( figures & =ms2 compreh5d+ 9 !mselves1 intelligible natures1 en]giz+ ab\t !mselves1 unit$ ) ! objects ( k1 & free f all extr9sical impression & passion4 ,b la/ cognitions ex]cise !mselves ?r\gh ! m$ium ( in/rum5ts1 are r passions ?an en]gies1 admit extr9sical k1 & move !mselves toge!r ) _! v>i\s subjects4 ,= su* 7says ,plato7 are ! s5sations :i* arise f viol5t passions4 ,b ! phantasy1 obta9+ a middle c5tre 9 ! ord] ( cognitions1 is excit$1 inde$1 by itself1 & produces t :i* falls "u cogitation3 b because x is n sep>ate f body1 x d$uces into p>tition1 int]val1 & figure1 ! objects ( its k1 f ! indivisibility ( an intellectual life4 ,h5ce1 :atev] x kn[s1 is a c]ta9 impression & =m ( intellig5ce4 ,= x und]/&s ! circle1 toge!r ) its int]val1 void1 inde$1 ( ext]nal matt]1 b possess+ intelligible matt]4 ,on ?is acc\nt1 l s5sible matt]1 x does n conta9 "o circle only3 = we behold 9 its receptacle1 di/ance1 toge!r ) ! m & ! less1 & a multitude ( circles & triangles4 ,if !n an univ]sal nature is di/ribut$ 9 s5sible circles1 s9ce ea* ( ^! completes a circul> figure1 & !y are all mutually simil>1 subsi/+ 9 "o reason1 b diff]+ 9 magnitudes or subjects3 9 l mann]1 !re is a common some?+ 9 ! circles1 :i* subsi/ 9 ! receptacle ( ! phantasy1 ( :i* all its circles p>ticipate1 & accord+ to :i* !y all possess ! same =m2 b 9 ! phantasy !y possess b "o diff]5ce only1 t ( magnitude4 ,= :5 y imag9e _m circles ab\t ! same c5tre1 !y all ( !m exi/ 9 "o immat]ial subject & life1 :i* is insep>able f a simple body1 :i*1 by ! possession ( int]val1 exce$s an ess5ce de/itute ( p>ts2 b !y diff] 9 magnitude & p>vitude1 & because !y are conta9$ & conta94 ,h5ce1 t univ]sal is tw(old1 :i* is und]/ood z subsi/+ 9 ! _m3 "o1 inde$1 9 s5sible =ms2 b ! o!r 9 su* z are imag9ative4 ,& ! reason ( a circul> & triangul> figure1 & ( figure univ]sal1 is tw(old4 ,! "o subsi/+ 9 intelligible1 b ! o!r 9 s5sible matt]4 ,b prior to ^! is ! reason :i* resides 9 cogitation1 & t :i* is seat$ 9 nature h]self4 ,! =m] be+ ! au?or ( imag9ative circles1 & ( ! "o =m :i* !y conta92 b ! o!r1 ( su* z are s5sible4 ,= !re are circles exi/+ 9 ! heav5s1 & univ]sally ^? produc$ by nature1 ! reason ( :i* does n fall "u a cogitative di/ribution4 ,= 9 incorporeal causes1 ?+s possess+ int]val1 are di/+ui%$ by no int]vals3 su* z are endu$ ) p>ts1 subsi/ )\t p>ts3 & magnitudes )\t ! diffusion ( magnitude1 z on ! contr>y 9 corporeal causes1 ?+s )\t p>ts subsi/ divisibly1 & su* z are void ( magnitude ) ! ext5sion ( magnitude4 ,h5ce1 ! circle resid5t 9 cogitation1 is "o1 simple & free f int]val3 & magnitude itself is !re de/itute ( magnitude2 & figure express$ by no figure3 = su* are reasons sep>ate f matt]4 ,b ! circle subsi/+ 9 ! phantasy1 is divisible1 figur$1 endu$ ) int]val1 n "o only1 b "o & _m1 nor =m alone1 b di/ribut$ ) =m4 ,& ! circle1 9 s5sible objects1 is composite1 di/ant ) magnitude1 dim9i%$ by a c]ta9 reason1 full ( ineptitude1 & v remote f ! purity ( immat]ial natures4 ,we mu/ !re=e say1 t geometry1 :5 x ass]ts any ?+ ( circle & diamet]1 & ( ! passions & affections :i* reg>d ! circle2 z ( contacts1 divisions1 & ! l3 nei!r tea*es nor disc\rses conc]n+ s5sible =ms1 7s9ce x endeav\rs to sep>ate u f ^!71 nor yet conc]n+ ! =m resid5t 9 cogitation1 7= "h ! circle is "o1 b geometry disc\rses ( _m1 propos+ some?+ ( ea*1 & contemplat+ ! same ( all3 & "h x is indivisible1 b ! geometric circle is divisible72 b we mu/ confess1 t x consid]s univ]sal itself2 yet z di/ribut$ 9 imag9ative circles4 ,& t x beholds1 inde$1 "o circle#ih3 & by ! m$ium ( ano!r1 contemplates ! circle resid5t 9 ! dep?s ( cogitation3 b by ano!r1 diff]5t f ! prec$+1 fabricates ! fair v>iety ( its demon/rations4 ,= s9ce cogitation is endu$ ) reasons1 b _c behold !m contract$ly1 sep>at$ f mat]ial figure2 x di/ributes & removes !m1 & draws !m =? seat$ 9 ! %ad[y bosom ( ! phantasy1 & plac$ 9 ! ve/ibules ( prim>y =ms2 revolv+ 9 x1 or toge!r ) x1 ! k ( ^!3 lov+1 inde$1 a sep>ation f s5sibles1 b f9d+ imag9ative matt] prop] = ! reception ( its univ]sal =ms4 ,h5ce1 its intellection does n subsi/ )\t ! phantasy4 ,& ! compositions & divisions ( figures are imag9ative2 & _! k is ! way :i* leads u to t ess5ce pursu$ by cogitation3 b cogitation itself1 does n yet arrive at ?is /able ess5ce1 :ile x looks abroad to ext]nals1 contemplates its int]nal =ms accord+ to ^!1 uses ! impressions ( reasons1 & is mov$ f itself to ext]nal & mat]ial =ms4 ,b if x %\ld ev] be able to return to itself1 :5 x has contract$ int]vals & impressions1 & beholds multitude )\t impression1 & subsi/+ uni=mly2 !n x w excell5tly p]ceive geometrical reasons1 void ( division & int]val1 ess5tial & vital1 ( :i* !re is a copi\s v>iety4 ,& ?is en]gy w be ! be/ end ( ! geometric /udy2 & truly ! employm5t ( a ,m]curial gift1 br++ x back z f a c]ta9 ,calypso1 & h] deta9+ *>ms1 to a m intellectual k2 & free+ x f ^? =m+ appreh5sions ) :i* ! mirror ( ! phantasy is replete4 ,inde$1 x is requisite t a true geometrician %\ld be employ$ 9 ?is m$itation1 & %\ld e/abli%1 z 8 prop] end1 ! excitation & transition f ! phantasy to cogitation alone2 & t he %\ld accompli% ?is by sep>at+ himself f int]vals1 & ! passive intellect to t en]gy :i* cogitation conta9s4 ,= by ?is means he w p]ceive all ?+s )\t an int]val1 ! circle & diamet] )\t a "p1 ! polygons 9 ! circle1 all 9 all1 & yet e "o sep>ate & ap>t4 ,s9ce1 on ?is acc\nt1 we exhibit also 9 ! phantasy1 bo? circles inscrib$ 9 polygons1 & polygons 9 circles2 imitat+ ! alt]nate exhibition ( reasons de/itute ( p>ts4 ,h5ce1 !re=e1 we describe ! con/itutions1 ! orig91 divisions1 positions1 & applications ( figures3 because we use ! phantasy1 & di/ances ( ?is k9d proce$+ f its mat]ial nature2 s9ce =m itself is immoveable1 )\t g5]ation1 indivisible1 & free f e subject4 ,b :atev] =m conta9s occultly1 & 9 an indi/ant mann]1 is produc$ into ! phantasy subsi/+ ) int]vals1 divisibly & exp&$4 ,& t :i*1 inde$1 produces ! =ms ( geometric speculation1 is cogitation3 b t f :i* !y are produc$1 is ! =m resid5t 9 cogitation3 & t 9 :i* ! produc$ figure resides is :at is call$ ! passive intellect4 ,:i* folds itself ab\t ! imp>tibility ( true intellect1 sep>ates f itself ! p[] ( pure intellig5ce free f int]val2 con=ms itself accord+ to all =mless species1 & becomes p]fectly e ?+ f :i* cogitation itself1 & our indivisible reason consi/s4 ,& ?us mu* conc]n+ ! geometric matt]1 z we are n ignorant ( :atev] ,porphyry ! ,philosoph] has obs]v$ 9 8 miscellanies1 & :atev] _m ( ! ,platoni/s describe4 ,b we ?9k t ! pres5t discussions are m agreeable to geometric diss]tations1 & to ,plato himself1 :o subjects to geometry ! objects ( cogitation4 ,= ^! mutually agree among !mselves2 because ! causes1 inde$1 ( geometrical =ms1 by :i* cogitation produces demon/rations1 pre-exi/ 9 demon/ration itself3 b ! p>ticul> figures :i* are divid$ & comp\nd$1 are situat$ 9 ! receptacle ( ! phantasy4 ,,*ap4 ,,ii4 ,:at k9d ( ,sci5ce ,geometry is4 ,b let u n[ speak ( t sci5ce :i* possesses a p[] ( contemplat+ ! univ]sal =ms p>ticipat$ by imag9ative matt]4 ,geometry1 !re=e1 is endu$ ) ! k ( magnitudes & figures1 & ( ! t]ms & reasons subsi/+ 9 ^!2 toge!r ) ! passions1 v>i\s positions & motions :i* are cont+5t ab\t ^!4 ,= x proce$s1 inde$1 f an imp>tible po9t1 b desc5ds ev5 to solids1 & f9ds out _! multi=m div]sities4 ,& aga91 runs back f ?+s m composite1 to ?+s m simple1 & to ! pr9ciples ( ^!3 s9ce x uses compositions & resolutions1 always beg9n+ f suppositions1 & assum+ its pr9ciples f a previ\s sci5ce2 b employ+ all ! dialectic ways4 ,9 pr9ciples1 by ! divisions ( =ms f _! g5]a1 & by def9+ its orations4 ,b 9 ?+s po/]ior to pr9ciples1 by demon/rations & resolutions4 ,z likewise1 x exhibits ?+s m v>i\s1 proce$+ f su* z are m simple1 & return+ to !m aga94 ,besides ?is1 x sep>ately disc\rses ( its subjects2 sep>ately ( its axioms2 f :i* x rises to demon/rations2 & sep>ately ( ess5tial accid5ts1 :i* x %ews likewise are resid5t 9 its subjects4 ,= e sci5ce has1 inde$1 a g5us1 ab\t :i* x is conv]sant1 & ^: passions x proposes to consid]3 & besides ?is1 pr9ciples1 :i* x uses 9 demon/rations2 & ess5tial accid5ts4 ,axioms1 inde$1 are common to all sci5ces 7?\gh ea* employs !m 9 its peculi> subject matt]71 b g5us & ess5tial accid5t v>y accord+ to ! sci5tial v>iety4 ,! subjects ( geometry are !re=e1 inde$1 triangles1 quadrangles1 circles1 & univ]sally figures & magnitudes1 & ! b\nd>ies ( ^!4 ,b its ess5tial accid5ts are divisions1 ratios1 contacts1 equalities1 applications1 excesses1 defects1 & ! l4 ,b its petitions & axioms1 by :i* x demon/rates e p>ticul> are1 ?is1 to draw a "r l9e f any po9t to any po9t2 & t1 if f equals y take away equals1 ! rema9d]s w be equal2 toge!r ) ! petitions & axioms consequ5t to ^!4 ,h5ce1 n e problem nor ?+ s\ght is geometrical1 b su* only z fl[ f geometric pr9ciples4 ,& he :o is reprov$ & convict$ f ^!1 is conv9c$ z a geometrician4 ,b :oev] is conv9c$ f pr9ciples diff]5t f ^!1 is n a geometrician1 b is =eign f ! geometric contemplation4 ,b ! objects ( ! non-geometric inve/igation1 are ( two k9ds4 ,= ! ?+ s\ght =1 is ei!r f entirely diff]5t pr9ciples1 z we say t a musical enquiry is =eign f geometry1 because x emanates f o!r suppositions1 & n f ! pr9ciples ( geometry3 or x is su* z uses1 inde$1 geometrical pr9ciples1 b at ! same "t p]v]sely1 z if any "o %\ld say1 t p>allels co9cide4 ,& on ?is acc\nt1 geometry also exhibits to u in/rum5ts ( judg+1 by :i* we may "k :at ?+s are consequ5t to its pr9ciples1 & :at ^? are :i* fall f ! tru? ( its pr9ciples3 = "s ?+s att5d geometrical1 b o!rs ari?metical pr9ciples4 ,& :y %\ld we speak ( o!rs1 s9ce !y are f> di/ant f ^!8 ,= "o sci5ce is m c]ta9 ?an ano!r 7z ,ari/otle says#ii7 t1 inde$1 :i* emanates f m simple suppositions1 ?an t :i* uses m v>i\s pr9ciples2 & t :i* tells ! :y1 ?an t :i* kn[s only ! simple exi/5ce ( a ?+2 & t :i* is conv]sant ab\t intelligibles1 ?an t :i* t\*es & is employ$ ab\t s5sibles4 ,& accord+ to ^! def9itions ( c]ta9ty1 ari?metic is1 inde$1 m c]ta9 ?an geometry1 s9ce its pr9ciples excel by _! simplicity4 ,= unity is void ( position1 ) :i* a po9t is endu$4 ,& a po9t1 inde$1 :5 x receives position1 is ! pr9ciple ( geometry3 b unity1 ( ari?metic4 ,b geometry is m c]ta9 ?an sph]ics2 & ari?metic1 ?an music4 ,= ^! r5d] univ]sally ! causes ( ^? !orems1 :i* are conta9$ "u !m4 ,aga91 geometry is m c]ta9 ?an me*anics1 optics1 & catoptrics4 ,because ^! disc\rse only on s5sible objects4 ,! pr9ciples1 !re=e1 ( geometry & ari?metic1 diff]1 inde$1 f ! pr9ciples ( o!r sci5ces2 b ! hypo!ses ( ^! two1 alt]nately diff] & agree accord+ to ! diff]5ce we h already describ$4 ,h5ce1 also1 ) respect to ! !orems :i* are demon/rat$ 9 ^! sci5ces1 "s are1 inde$1 common to !m1 b o!rs peculi>4 ,= ! !orem :i* says1 e proportion may be express$1 alone belongs to ari?metic2 b by no means to geometry3 s9ce ?is la/ sci5ce conta9s ?+s :i* _c be express$#ajj4 ,t !orem also1 :i* affirms1 t ! gnomons ( quadrangles are t]m9at$ accord+ to ! lea/#aja1 is ! prop]ty ( ari?metic3 = 9 geometry1 a m9imum _c be giv54 ,b ^? ?+s are peculi> to geometry1 :i* are conv]sant ab\t positions2 = numb]s h no position3 :i* respect contacts2 = contact is f\nd 9 cont9u$ quantities3 & :i* are conv]sant ab\t ineffable proportions2 = :]e division proce$s to inf9ity1 !re also t :i* is ineffable is f\nd#ajb4 ,b ?+s common to bo? ^! sci5ces1 are su* z respect divisions1 :i* ,euclid treats ( 9 ! second book2 except t proposition :i* divides a "r l9e into extreme & mean proportion#ajc4 ,aga91 ( ^! common !orems1 "s1 inde$1 are transf]r$ f geometry into ari?metic2 b o!rs1 on ! contr>y1 f ari?metic into geometry3 & o!rs simil>ly accord ) bo?1 :i* are d]iv$ into !m f ! :ole ma!matical sci5ce4 ,= ! p]mutation1 inde$1 conv]sions1 compositions1 & divisions ( ratios are1 aft] ?is mann]1 common to bo?4 ,b su* ?+s z are comm5surable1 ari?metic fir/ beholds2 b aft]w>ds geometry1 imitat+ ari?metic4 ,f :5ce1 also1 x det]m9es su* ?+s to be comm5surables ( ?is k9d1 :i* h ! same mutual ratio to "o ano!r1 z numb] to numb]2 because comm5surability pr9cipally subsi/s 9 numb]s4 ,= :]e numb] is1 !re also t :i* is comm5surable is f\nd2 & :]e comm5surable is1 !re also numb]4 ,la/ly1 geometry fir/ inspects triangles & quadrangles3 b1 ari?metic1 receiv+ ^! f geometry1 consid]s !m accord+ to proportion4 ,= 9 numb]s1 figures reside 9 a causal mann]4 ,be+ excit$1 !re=e1 f effects1 we pass to _! causes1 :i* are conta9$ 9 numb]s4 ,& at "o "t1 we indiff]5tly behold ! same accid5ts1 z :5 e polygon is resolv$ by u into triangles#ajd3 b1 at ano!r "t1 we are cont5t ) :at is ne>e/ to ! tru?1 z :5 we f9d 9 geometry "o quadrangle ! d\ble ( ano!r1 b n f9d+ ?is 9 numb]s1 we say t "o squ>e is d\ble ( ano!r1 except by a defici5ce ( unity4 ,z = in/ance1 ! squ>e f #g1 is d\ble ! squ>e f #e1 want+ "o4 ,b we h produc$ our discussion to ?is l5g?1 = ! purpose ( ev9c+ ! communion & diff]5ce 9 ! pr9ciples ( ^! two sci5ces4 ,s9ce x belongs to a geometrician to survey f :at common pr9ciples common !orems are divid$2 & f :at pr9ciples su* z are peculi> proce$2 & ?us to di/+ui% betwe5 ! geometrical1 & non-geometrical1 ref]r+ ea* ( !m to diff]5t sci5ces4 ,,*ap4 ,,iii4 ,f :5ce ! :ole ( ,geometry orig9at$1 h[ f> x proce$s1 & 9 :at its ,utility consi/s4 ,b1 beg9n+ /ill high]1 let u contemplate ! :ole ( geometry1 f :5ce x orig9at$1 & h[ f> x proce$s 9 its en]gies3 = ?us we %all prop]ly p]ceive ! ornam5t :i* x conta9s4 ,inde$1 x is necess>y to und]/& t x is ext5d$ ?r\gh ! univ]sality ( ?+s3 t x accommodates its animadv]sions#aje to all be+s2 & conta9s 9 itself ! =ms ( all ?+s3 t1 accord+ to its supreme "p1 & :i* is endu$ ) ! highe/ p[] ( intellig5ce1 x surveys true be+s2 & tea*es by images ! prop]ties ( div9e ornam5ts1 & ! p[]s ( intellectual =ms3 = x conta9s ! reasons ( ^! also 9 its peculi> contemplations4 ,& x exhibits :at figures are conv5i5t to ! god1 to prim>y ess5ces1 & to ! natures ( s\ls4 ,b1 accord+ to its middle cognitions1 x evolves cogitative reasons2 expla9s & beholds ! v>iety :i* !y conta92 exhibits _! exi/5ce1 & inh]5t passions2 z also1 _! communities & div]sities4 ,f :i*1 inde$1 x compreh5ds1 9 t]m9at$ b\nds1 ! imag9ative =mations ( figures1 & r$uces !m to ! ess5tial sub/ance ( reasons4 ,b1 accord+ to ! ?ird propagations ( cogitative intellig5ce1 x consid]s nature1 & deliv]s ! mann] 9 :i* ! =ms ( s5sible elem5ts1 & ! p[]s :i* !y conta91 are previ\sly receiv$ accord+ to cause1 9 ! reasons !mselves4 ,= x possesses1 inde$1 ! images ( univ]sal intelligible g5]a2 b ! exempl>s ( su* z are s5sible3 & completes its own ess5ce1 accord+ to su* ?+s z are subject to cogitation4 ,& ?r\gh ^!1 z ?r\gh prop] m$iums1 x asc5ds & desc5ds to ^? univ]sals :i* truly are1 & to s5sible =ms :i* are 9 a /ate ( p]petual =mation4 ,b always geometrically philosophis+ conc]n+ ! ?+s :i* are1 x compreh5ds 9 all ! proportions ( virtues1 ! images ( intellectual1 animal1 & natural conc]ns4 ,& x deliv]s1 9 an ord]ly mann]1 all ! ornam5ts ( republics3 & exhibits 9 itself _! v>i\s mutations4 ,su* !n are its en]gies aris+ f a c]ta9 immat]ial p[] ( cognition3 b :5 x t\*es ^u matt]1 x produces f itself a multitude ( sci5ces2 su* z geodsia1 me*anics1 & p]spective3 by :i* x procures ! greate/ b5efit to ! life ( mortals4 ,= x con/ructs by ^! sci5ces1 w>-in/rum5ts1 & ! bulw>ks ( cities2 & makes kn[n ! circuits ( m\nta9s1 & ! situations ( places4 ,la/ly1 x in/ructs u 9 measures3 at "o "t ( ! div]sifi$ ways ( ! e>?2 & at ano!r1 ( ! re/less pa?s ( ! deep4 ,add too1 t x con/ructs balances & scales1 by :i* x r5d]s to cities a sure equality accord+ to ! inv>iable /&>d ( numb]4 ,likewise1 x cle>ly expresses1 by images1 ! ord] ( ! :ole orb ( ! e>?2 & by ^!1 manife/s _m ?+s incr$ible to mank9d1 & r5d]s !m cr$ible to all4 ,su*1 inde$1 z ,hi]o ( ,syracuse is report$ to h said ( ,ar*im$es#ajf1 :5 he _h fabricat$ a %ip furni%$ ) ?ree sails1 :i* he _h prep>$ to s5d to ,ptolemy k+ ( ,egypt4 ,= :5 all ! ,syracusians toge!r1 7 unable to draw ?is %ip1 ,ar*im$es enabl$ ,hi]o to draw x himself1 )\t any assi/ance f o!rs4 ,b he1 be+ a/oni%$1 said1 ,f ?is day1 ,ar*im$es %all be believ$ 9 :atev] he %all affirm4 ,!y also report1 t ,gelo said ! same1 :5 ,ar*im$es discov]$ ! weight ( ! sev]al mat]ials f :i* 8 cr[n 0 compos$1 )\t dissolv+ _! union4 ,& su* are ! n>rations :i* _m ( ! anci5ts h deliv]$ to our memory1 :o 7 will+ to speak 9 praise ( ! ma!matics3 &1 on ?is acc\nt1 we h plac$ be=e ! read]1 = ! pres5t1 a few out ( ! _m1 z n =eign f our design ( exhibit+ ! k & utility ( geometry4 ,,*ap4 ,,iv4 ,on ! ,orig9 ( ,geometry1 & its ,inv5tors4 ,b let u n[ expla9 ! orig9 ( geometry1 z exi/+ 9 ! pres5t age ( ! _w4 ,= ! demoniacal ,ari/otle#ajg obs]ves1 t ! same op9ions (t5 subsi/ among m51 accord+ to c]ta9 ord]ly revolutions ( ! _w3 & t sci5ces did n receive _! fir/ con/itution 9 our times1 nor 9 ^? p]iods :i* are kn[n to u f hi/orical tradition1 b h appe>$ & vani%$ aga9 9 o!r revolutions ( ! univ]se2 nor is x possible to say h[ (t5 ?is has happ5$ 9 pa/ ages1 & w aga9 take place 9 ! future circulations ( "t4 ,b1 because ! orig9 ( arts & sci5ces is to be consid]$ accord+ to ! pres5t revolution ( ! univ]se1 we mu/ affirm1 9 con=mity ) ! mo/ g5]al tradition1 t geometry 0 fir/ inv5t$ by ! ,egyptians1 d]iv+ its orig9 f ! m5suration ( _! fields3 s9ce ?is1 inde$1 0 necess>y to !m1 on acc\nt ( ! inundation ( ! ,nile wa%+ away ! b\nd>ies ( l& belong+ to ea*4 ,nor ought x to seem wond]ful t ! inv5tion ( ?is z well z ( o!r sci5ces1 %\ld receive its comm5cem5t f conv5i5ce & opportunity4 ,s9ce :atev] is c>ri$ 9 ! circle ( g5]ation1 proce$s f ! imp]fect to ! p]fect4 ,a transition1 !re=e1 is n undes]v$ly made f s5se to consid]ation1 & f ?is to ! nobl] en]gies ( intellect#ajh4 ,h5ce1 z ! c]ta9 k ( numb]s receiv$ its orig9 among ! ,phnicians1 on acc\nt ( m]*&ise & comm]ce1 s geometry 0 f\nd out among ! ,egyptians f ! di/ribution ( l&4 ,:5 ,?ales1 !re=e1 fir/ w5t into ,egypt1 he transf]r$ ?is k f !nce into ,greece3 & he inv5t$ _m ?+s himself1 & communicat$ to 8 successors ! pr9ciples ( _m4 ,"s ( :i* 71 inde$1 m univ]sal1 b o!rs ext5d$ to s5sibles4 ,aft] him ,am]i/us1 ! bro!r ( ,/esi*orus ! poet1 is celebrat$ z "o :o t\*$ ^u1 & ta/$ ! /udy ( geometry1 & :o is m5tion$ by ,hippias ! ,elean1 z re/or+ ! glory ( geometry4 ,b aft] ^!1 ,py?agoras *ang$ t philosophy1 :i* is conv]sant ab\t geometry itself1 into ! =m ( a lib]al doctr9e1 consid]+ its pr9ciples 9 a m exalt$ mann]2 & inve/igat+ its !orems immat]ially & intellectually2 :o likewise inv5t$ a treatise ( su* ?+s z _c be expla9$#aji 9 geometry1 & discov]$ ! con/itution ( ! mundane figures4 ,aft] him1 ,anaxagoras ! ,clazom5ian succe$$1 :o und]took _m ?+s p]ta9+ to geometry4 ,& ,o5opides ! ,*ian1 0 some:at junior to ,anaxagoras1 & :om ,plato m5tions 9 8 ,rivals1 z "o :o obta9$ ma!matical glory4 ,to ^!1 succe$$ ,hippocrates1 ! ,*ian1 :o inv5t$ ! quadrature ( ! lunula#aaj1 & ,!odorus ! ,cyr5ean1 bo? ( !m em95t 9 geometrical k4 ,= ! fir/ ( ^!1 ,hippocrates compos$ geometrical elem5ts3 b ,plato1 :o 0 po/]ior to ^!1 caus$ z well geometry itself1 z ! o!r ma!matical discipl9es1 to receive a rem>kable addition1 on acc\nt ( ! great /udy he be/[$ 9 _! inve/igation4 ,?is he himself manife/s1 & 8 books1 replete ) ma!matical disc\rses1 ev9ce3 to :i* we may add1 t he e :]e excites :atev] 9 !m is wond]ful1 & ext5ds to philosophy4 ,b 9 8 "t also liv$ ,leodamas ! ,?asian1 ,ar*ytas ! ,t>ant9e1 & ,!tetus ! ,a!nian2 by :om !orems 7 increas$1 & advanc$ to a m skilful con/itution4 ,b ,neoclides 0 junior to ,leodamas1 & 8 disciple 0 ,leon2 :o add$ _m ?+s to ^? ?\ght ( by =m] geometricians4 ,s t ,leon also con/ruct$ elem5ts m accurate1 bo? on acc\nt ( _! multitude1 & on acc\nt ( ! use :i* !y exhibit3 & besides ?is1 he discov]$ a me?od ( det]m9+ :5 a problem1 ^: inve/igation is s\ght =1 is possible1 & :5 x is impossible4 ,b ,eudoxus ! ,cnidian1 :o 0 some:at junior to ,leon1 & ! companion ( ,plato1 fir/ ( all r5d]$ ! multitude ( ^? !orems :i* are call$ univ]sals m abundant2 & to ?ree proportions add$ ?ree o!rs2 & ?+s relative to a section1 :i* receiv$ _! comm5cem5t f ,plato1 he diffus$ into a ri*] multitude1 employ+ also resolutions 9 ! prosecution ( ^!4 ,aga91 ,amyclas ! ,h]acleotean1 "o ( ,plato's famili>s1 & ,m5*mus1 ! disciple1 inde$1 ( ,eudoxus1 b conv]sant ) ,plato1 & 8 bro!r ,d9o/ratus1 r5d]$ ! :ole ( geometry z yet m p]fect4 ,b ,!udius1 ! ,magnian1 appe>s to h excell$1 z well 9 ma!matical discipl9es1 z 9 ! re/ ( philosophy4 ,= he con/ruct$ elem5ts egregi\sly1 & r5d]$ _m p>ticul>s m univ]sal4 ,besides1 ,cyzic9us ! ,a!nian1 fl\ri%$ at ! same p]iod1 & became illu/ri\s 9 o!r ma!matical discipl9es1 b especially 9 geometry4 ,^!1 !re=e1 resort$ by turns to ! ,academy1 & employ$ !mselves 9 propos+ common que/ions4 ,b ,h]motimus1 ! ,colophonian1 r5d]$ m abundant :at 0 =m]ly publi%$ by ,eudoxus & ,!tetus1 & inv5t$ a multitude ( elem5ts1 & wrote conc]n+ "s geometrical places4 ,b ,philippus ! ,m5dan#aaa1 a disciple ( ,plato1 & by him inflam$ 9 ! ma!matical discipl9es1 bo? compos$ que/ions1 accord+ to ! in/itutions ( ,plato1 & propos$ z ! object ( 8 enquiry :atev] he ?\ght conduc$ to ! ,platonic philosophy4 ,& ?us f> hi/orians produce ! p]fection ( ?is sci5ce4 ,b ,euclid 0 n mu* junior to ^!1 :o collect$ elem5ts1 & con/ruct$ _m ( ^? ?+s :i* 7 inv5t$ by ,eudoxus2 & p]fect$ _m :i* 7 discov]$ by ,!tetus4 ,besides1 he r$uc$ to inv9cible demon/rations1 su* ?+s z 7 exhibit$ by o!rs ) a weak] arm4 ,b he liv$ 9 ! times ( ! fir/ ,ptolemy3 = ,ar*im$es m5tions ,euclid1 9 8 fir/ book1 & also 9 o!rs4 ,besides1 !y relate t ,euclid 0 ask$ by ,ptolemy1 :e!r !re 0 any %ort] way to ! atta9m5t ( geometry ?an by 8 elem5t>y in/itution1 & t he answ]$1 !re 0 no o!r royal pa? :i* l$ to geometry4 ,euclid1 !re=e1 0 junior to ! famili>s ( ,plato1 b m anci5t ?an ,erato/h5es & ,ar*im$es 7= ^! liv$ at "o & ! same "t1 accord+ to ! tradition ( ,erato/h5es7 b he 0 ( ! ,platonic sect1 & famili> ) its philosophy3 & f h5ce he appo9t$ ! con/itution ( ^? figures :i* are call$ ,platonic#aab1 z ! end ( 8 elem5t>y in/itutions4 ,,*ap4 ,v4 ,:at ,ma!matical ,volumes ,euclid compos$4 ,!re are1 !re=e1 _m o!r ma!matical volumes ( ?is man1 full ( admirable dilig5ce1 & skilful consid]ation3 = su* are 8 ,optics#aac1 & ,catoptrics3 & su* also1 are 8 elem5t>y in/itutions1 :i* conduce to ! atta9m5t ( music#aad2 & 8 book conc]n+ divisions#aae4 ,b 8 geometrical in/itution ( ! ,elem5ts is especially admirable1 on acc\nt ( ! ord] & election ( ^? !orems & problems1 :i* are di/ribut$ ?r\gh ! ,elem5ts4 ,= he does n assume all :i* might be said1 b t only :i* c\ld be deliv]$ 9 an elem5t>y ord]4 ,besides ?is1 he exhibits modes ( syllogisms ( e k9d2 "s1 inde$1 receiv+ cr$ibility f causes1 b o!rs proce$+ f c]ta9 signs2 b all ( !m inv9cible & sure1 & accommodat$ to sci5ce4 ,b1 besides ^!1 he employs all ! dialectic ways1 divid+1 inde$1 9 ! inv5tions ( =ms2 b def9+ 9 ess5tial reasons3 & aga91 demon/rat+ 9 ! progressions f pr9ciples to ?+s s\ght1 b resolv+ 9 ! rev]sions f ?+s s\ght to pr9ciples4 ,besides ?is1 we may view 9 8 geometrical elem5ts1 ! v>i\s species ( conv]sions1 z well ( su* z are simple z ( su* z are m composite4 ,& aga91 :at :oles may be conv]t$ ) :oles3 :at :oles ) p>ts2 & on ! o!r h&1 :at z p>ts ) p>ts#aaf4 ,besides ?is1 we mu/ say1 t 9 ! cont9uation ( inv5tions1 ! dispositions & ord] ( ?+s prec$+ & foll[+1 & 9 ! p[] ) :i* he treats e p>ticul>1 he is n deceiv$1 z if fall+ f sci5ce1 & c>ri$ to its contr>y1 falsehood & ignorance4 ,b because we may imag9e _m ?+s z adh]+ to tru?1 & :i* are consequ5t to pr9ciples produc+ sci5ce1 :i* nev]!less t5d to t error :i* fl[s f ! pr9ciples1 & :i* deceives rud] m9ds1 he has also deliv]$ me?ods ( ! p]spicaci\s prud5ce belong+ to ^!4 ,f ! possession ( :i*1 we may ex]cise ^? 9 ! inv5tion ( fallacies1 :o und]take ?is inspection1 & may pres]ve ourselves f all deception4 ,& ?is book1 by :i* he procures u ?is prep>ation1 is inscrib$ 1 or1 conc]n+ fallacies#aag4 ,because he enum]ates 9 ord] _! v>i\s modes1 & 9 ea* ex]cises our cogitation ) v>i\s !orems4 ,& he comp>es tru? ) falsehood1 & adapts ! confutation ( deception to exp]i5ce itself4 ,?is book1 !re=e1 conta9s a purgative & ex]cis+ p[]4 ,b ! in/itution ( 8 elem5t>y1 skilful contemplation ( geometrical conc]ns1 possesses an inv9cible & p]fect n>ration4 ,,*ap4 ,,vi4 ,conc]n+ ! ,purport ( ,geometry4 ,b1 p]haps1 "s "o may enquire 9 :at ! design ( ?is treatise consi/s8 ,to ?is ,i answ]1 t its design is to be di/+ui%$ z well accord+ to ! objects ( enquiry1 z accord+ to ! le>n]4 ,&1 inde$1 reg>d+ ! subject1 we mu/ affirm1 t all ! disc\rse ( geometry is conc]n+ ! mundane figures4 ,because x beg9s f su* ?+s z are simple1 b ends 9 ! v>iety ( _! con/itution4 ,&1 inde$1 x con/itutes ea* ( !m sep>ately1 b at ! same "t deliv]s _! inscriptions 9 a sph]e1 & ! proportions :i* !y conta94 ,on :i* acc\nt "s h ?\ght1 t ! design ( ea* ( ! books is to be ref]r$ to ! _w2 & !y h deliv]$ to our memory1 ! utility :i* !y af=d u 9 ! contemplation ( ! univ]se4 ,b di/+ui%+ ! design ) respect to ! le>n]1 we mu/ affirm1 t its purpose is ! in/itution ( elem5ts2 & ! p]fection ( ! le>n]s cogitative p[]s 9 univ]sal geometry4 ,= beg9n+ f ^!1 we are enabl$ to und]/& ! o!r p>ts ( ?is sci5ce1 & to compreh5d ! v>iety :i* !y conta94 ,&1 inde$1 )\t ^!1 ! discipl9e ( ! re/1 is to u impossible & incompreh5sible4 ,= su* !orems z are mo/ pr9cipal & simple1 & are mo/ alli$ to fir/ suppositions1 are "h collect$ 9 a becom+ ord]4 ,& ! demon/rations ( o!r ma!maticians1 use ^! z mo/ kn[n1 & advance f ^! 9 _! mo/ complicat$ progressions4 ,= ?us ,ar*im$es1 9 :at he has writ conc]n+ ! sph]e & cyl9d]1 & ,apollonius1 & ! re/ ( ma!maticians1 use1 z evid5t pr9ciples1 ! ?+s exhibit$ 9 ?is treatise4 ,its purpose1 !re=e1 is ! in/itution ( le>n]s 9 ! :ole geometric sci5ce1 & to deliv] ! det]m9ate con/itutions ( ! mundane figures4 ,,*ap4 ,,vii4 ,f :5ce ! ,"n ( ,elem5t>y ,in/itution orig9at$1 & :y ,euclid is call$ ! ,in/itutor ( ,elem5ts4 ,b :at gave rise to ! "n ( elem5t>y in/itution1 & ( elem5t itself1 f :i* elem5t>y in/itution 0 d]iv$8 ,to ?is we %all reply1 by obs]v+1 t ( !orems "s are usually call$ elem5ts1 b o!rs elem5t>y1 & o!rs aga9 are det]m9$ beyond ! p[] ( ^!4 ,h5ce1 an elem5t is t ^: consid]ation passes to ! sci5ce ( o!r ?+s1 & f :i* we d]ive a solution ( ! d\bts incid5t to ! p>ticul> sci5ce we inve/igate4 ,= z !re are c]ta9 fir/ pr9ciples ( spee*1 mo/ simple & indivisible1 :i* we d5om9ate elem5ts1 & f :i* all disc\rse is compos$2 s !re are c]ta9 pr9cipal !orems ( ! :ole ( geometry1 d5om9at$ elem5ts1 :i* h ! respect ( pr9ciples to ! foll[+ !orems2 :i* reg>d all ! subsequ5t propositions1 & af=d ! demon/rations ( _m accid5ts ess5tial to ! subjects ( geometric speculation4 ,b ?+s elem5t>y are su* z ext5d !mselves to a multitude ( propositions1 & possess a c]ta9 simplicity & sweetness1 yet are n ( ! same dignity ) elem5ts2 because _! contemplation is n common to all ! sci5ce to :i* !y belong1 z is ! case 9 ! foll[+ !orem1 t 9 triangles1 p]p5dicul>s1 drawn f _! angles to _! sides1 co9cide 9 "o po9t#aah4 ,la/ly1 :atev] nei!r possesses a k ext5d$ into multitude1 nor exhibits any ?+ skilful & elegant1 falls beyond ! elem5t>y p[]4 ,aga91 an elem5t1 z ,m5*mus says1 may h a tw(old def9ition4 ,= t :i* confirms1 is an elem5t ( t :i* is confirm$2 z ! fir/ proposition ( ,euclid ) respect to ! second1 & ! f\r? ) reg>d to ! fif?4 ,& ?us1 inde$1 _m ?+s may be mutually call$ elem5ts "o ( ano!r2 = !y are mutually confirm$4 ,?us1 because ! ext]nal angles ( "r-l9$ figures1 are equal to f\r "r angles1 ! multitude ( int]nal ones equal to "r angles2 &1 on ! contr>y1 t f ?is is exhibit$#aai4 ,besides1 an elem5t is o!rwise call$ t into :i*1 because x is m simple1 a composite is dissolv$4 ,b x mu/ be obs]v$1 t e elem5t _c be call$ ! elem5t ( e ?+3 b su* z are m pr9cipal are ! elem5ts ( su* z are con/itut$ 9 ! reason ( ! ?+ effect$2 z petitions are ! elem5ts ( !orems4 ,&1 accord+ to ?is signification ( an elem5t1 ,euclid's elem5ts are con/ruct$4 ,"s1 inde$1 ( t geometry :i* is conv]sant ab\t planes2 b o!rs ( /]eometry4 ,9 ! same mann]1 likewise1 9 ari?metic & a/ronomy1 _m h compos$ elem5t>y in/itutions4 ,b x is difficult1 9 ea* sci5ce1 to *use & conv5i5tly orda9 elem5ts1 f :i* all ! peculi>ities ( t sci5ce orig9ate1 & into :i* !y may be resolv$4 ,& among ^? :o h und]tak5 ?is employm5t1 "s h be5 able to collect m1 b o!rs few] elem5ts4 ,& "s1 inde$1 h us$ %ort] demon/rations2 b o!rs h ext5d$ _! treatise to an inf9ite l5g?4 ,& "s h omitt$ ! me?od by an impossibility2 b o!rs t by proportion2 & o!rs1 aga91 h attempt$ prep>ations aga9/ argum5ts de/roy+ pr9ciples4 ,s t _m me?ods ( elem5t>y in/itution h be5 inv5t$ by p>ticul> writ]s on ?is subject4 ,b x is requisite t ?is treatise %\ld entirely remove e ?+ sup]flu\s1 because x is an imp$im5t to sci5ce4 ,b e ?+ %\ld be *os51 :i* conta9s & concludes ! ?+ propos$2 = ?is is mo/ conv5i5t & useful 9 sci5ce4 ,! greate/ c>e1 likewise1 %\ld be paid to cle>ness & brevity2 = ! contr>ies to ^!1 di/urb our cogitation4 ,la/ly1 x %\ld v9dicate to itself1 ! univ]sal compreh5sion ( !orems1 9 _! prop] b\nds3 = su* ?+s z divide le>n+ into p>ticul> fragm5ts1 produce an incompreh5sible k4 ,b 9 all ^! modes1 any "o may easily f9d1 t ! elem5t>y in/itution ( ,euclid excels ! in/itutions ( o!rs4 ,= its utility1 inde$1 especially conf]s to ! contemplation ( prim>y figures3 b ! transition f ?+s m simple to su* z are m v>i\s1 & also t p]ception1 :i* f axioms possesses ! beg9n+ ( k1 produces cle>ness1 & an ord]ly tradition3 & ! migration f fir/ & pr9cipal !orems to ! objects ( enquiry1 effects ! univ]sality ( demon/ration4 ,= :atev] he seems to omit1 may ei!r be kn[n by ! same ways1 z ! con/ruction ( a scal5e & isosceles triangle#abj3 or because !y are difficult1 & capable ( inf9ite v>iety1 !y are f> remote f ! election ( elem5ts1 su* z ! doctr9e ( p]turbate proportions1 :i* ,apollonius has copi\sly h&l$3 or1 la/ly1 because !y may be easily con/ruct$ f ! ?+s deliv]$1 z f causes1 su* z _m species ( angles & l9es4 ,= ^!1 inde$1 7 omitt$ by ,euclid1 & are l>gely disc\rs$ ( by o!rs1 & are kn[n f simple propositions4 ,& ?us mu* conc]n+ ! univ]sal elem5t>y in/itution ( geometry4 ,,*ap4 ,,viii4 ,conc]n+ ! ,ord] ( ,geometrical ,disc\rses4 ,b let u n[ expla9 ! univ]sal ord] ( ! disc\rses conta9$ 9 geometry4 ,because !n1 we ass]t t ?is sci5ce consi/s f hypo!sis#aba1 & demon/rates its consequ5t propositions f def9ite pr9ciples 7= "o sci5ce only1 ,i mean ! fir/ philosophy1 is )\t supposition1 b all ! re/ assume _! pr9ciples f ?is7 x is necess>y t he :o con/ructs ! geometrical in/itution ( elem5ts1 %\ld sep>ately deliv] ! pr9ciples ( ! sci5ce1 & sep>ately ! conclusions :i* fl[ f ^? pr9ciples2 & t he %\ld r5d] no reason conc]n+ ! nature or tru? ( ! pr9ciples1 b %\ld confirm by reasons1 ! ?+s consequ5t to ^! geometric pr9ciples4 ,= no sci5ce demon/rates its own pr9ciples1 nor disc\rses conc]n+ !m2 b procures to itself a belief ( _! reality1 & !y become m evid5t to ! p>ticul> sci5ce to :i* !y belong ?an ! ?+s d]iv$ f !m z _! s\rce4 ,& ^!1 inde$1 sci5ce kn[s by !mselves2 b _! consequ5ts1 ?r\gh ! m$ium ( ^!4 ,= ?us1 also1 ! natural philosoph] propagates 8 reasons f a def9ite pr9ciple1 suppos+ ! exi/5ce ( motion4 ,?us too1 ! physician1 & he :o is skill$ 9 any ( ! o!r sci5ces & arts4 ,= if any "o m+les pr9ciples1 & ?+s fl[+ f pr9ciples into "o & ! same1 he di/urbs ! :ole ord] ( k1 & conglut9ates ?+s :i* c nev] mutually agree2 s9ce a pr9ciple1 & its emanat+ consequ5t1 are naturally di/9ct f ea* o!r4 ,9 ! fir/ place1 !re=e 7z ,i h said71 pr9ciples 9 ! geometric in/itution are to be di/+ui%$ f _! consequ5ts1 :i* is p]=m$ by ,euclid 9 ea* ( 8 books2 :o1 be=e e treatise1 exhibits ! common pr9ciples ( ?is sci5ce2 & aft]w>ds divides ^! common pr9ciples into hypo!ses1 petitions1 & axioms4 ,= all ^! mutually diff]2 nor is an axiom1 petition1 & hypo!sis ! same1 accord+ to ! demoniacal ,ari/otle2 b :5 t :i* is assum$ 9 ! ord] ( a pr9ciple1 is inde$ kn[n to ! le>n]1 & cr$ible by itself1 x is an axiom3 su* z1 t ?+s equal to ! same1 are mutually equal to ea* o!r4 ,b :5 any "o1 he>+ ano!r speak conc]n+ t ( :i* he has no self-evid5t k1 gives ?is ass5t to its assumption1 ?is is hypo!sis4 ,= t a circle is a figure ( su* a p>ticul> k9d1 we presume 7n accord+ to any common conception7 )\t any prec$+ doctr9e4 ,b :51 aga91 t :i* is ass]t$ 0 nei!r kn[n1 nor admitt$ by ! le>n]1 yet is assum$1 !n 7says he7 we call x petition2 z ! assumption t all "r angles are equal4 ,b ! tru? ( ?is is ev9c$ by ^? :o /udy to treat ( "s petition1 z ( t :i* _c by itself be admitt$ by any "o4 ,& ?us1 accord+ to ! doctr9e ( ,ari/otle#abb1 are axiom1 petition1 & supposition di/+ui%$4 ,b (t5times1 "s d5om9ate all ^! hypo!ses1 9 ! same mann] z ! ,/oics call e simple enunciation an axiom4 ,s t1 accord+ to _! op9ion1 hypo!ses also w be axioms2 b1 accord+ to ! op9ion ( o!rs1 axioms w be call$ suppositions4 ,aga91 su* ?+s z fl[ f pr9ciples are divid$ into problems & !orems4 ,! fir/1 inde$1 conta9+ ! orig91 sections1 ablations1 or additions ( figures1 & all ! affections ) :i* !y are conv]sant2 b ! o!r exhibit+ ! accid5ts ess5tial to ea* figure4 ,=1 z ?+s effective ( sci5ce1 p>ticipate ( contemplation1 9 ! same mann] ?+s contemplative previ\sly assume problems 9 ! place ( op]ations4 ,b =m]ly "s ( ! anci5t ma!maticians ?\ght t all geometrical propositions %\ld be call$ !orems1 z ! foll[]s ( ,speusippus & ,amph9omus1 believ+1 t to contemplative sci5ces1 ! appellation ( !orems is m prop] ?an t ( problems2 especially s9ce !y disc\rse conc]n+ et]nal & immutable objects4 ,= orig9 does n subsi/ among ?+s et]nal3 on :i* acc\nt1 problems _c h any place 9 ^! sci5ces2 s9ce !y enunciate orig91 & ! production ( t :i* =m]ly _h no exi/5ce1 z ! con/ruction ( an equilat]al triangle1 or ! description ( a squ>e on a giv5 "r l9e1 or ! position ( a "r l9e at a giv5 po9t4 ,x is bett]1 !re=e 7say !y71 to ass]t t all propositions are ( ! speculative k9d2 b t we p]ceive _! orig91 n by production1 b by k1 receiv+ ?+s et]nal z if !y 7 g5]at$2 & on ?is acc\nt we ought to conceive all ^? !orematically1 b n problematically4 ,b o!rs1 on ! contr>y1 ?9k t all %\ld be call$ problems2 z ^? ma!maticians :o h foll[$ ,m5*mus4 ,b t ! (fice ( problems is tw(old1 sometimes1 inde$1 to procure ! ?+ s\ght2 b at o!r times :5 !y h receiv$ ! det]m9ate object ( enquiry1 to see1 ei!r :at x is1 or ( :at k9d x is1 or :at affection x possesses1 or :at its relation is to ano!r4 ,&1 inde$1 ! ass]tions ( ea* are "r2 = ! foll[]s ( ,speusippus well p]ceive4 ,s9ce ! problems ( geometry are n ( ! same k9d1 ) su* z are me*anical4 ,= ^! are s5sibles1 & are endu$ ) orig91 & mutation ( e k9d4 ,&1 on ! o!r h&1 ^? :o foll[ ,m5*mus d n diss5t f tru?3 s9ce ! inv5tions ( !orems _c by any means take place )\t an approa* into matt]2 ,i mean intelligible matt]4 ,reasons1 !re=e1 proce$+ into ?is1 & giv+ =m to its =mless nature1 are n undes]v$ly said to be assimilat$ to g5]ations4 ,= we say t ! motion ( our cogitation1 & ! production ( its inh]5t reasons1 is ! orig9 ( ! figures situat$ 9 ! phantasy1 & ( ! affections ) :i* !y are conv]sant3 = !re con/ructions & sections1 positions & applications1 additions & ablations1 exi/3 b e ?+ resid5t 9 cogitation1 subsi/s )\t orig9 & mutation4 ,!re are1 !re=e1 bo? geometrical problems & !orems4 ,b1 because contemplation ab\nds 9 geometry1 z production 9 me*anics1 all problems p>ticipate ( contemplation2 b e ?+ contemplative is n problematical4 ,= demon/rations are entirely ! "w ( contemplation2 b e ?+ 9 geometry po/]ior to ! pr9ciples1 is assum$ by demon/ration4 ,h5ce1 a !orem is m common3 b all !orems d n require problems2 = !re are "s :i* possess f !mselves ! demon/ration ( ! ?+ s\ght4 ,b o!rs1 di/+ui%+ a !orem f a problem1 say1 t inde$ e problem receives :atev] is pr$icat$ ( its matt]1 toge!r ) its own opposite3 b t e !orem receives1 inde$1 its symptom pr$icate1 b n its opposite4 ,b ,i call ! matt] ( ^!1 t g5us :i* is ! subject ( enquiry2 z = in/ance1 a triangle1 quadrangle1 or a circle3 b ! symptom pr$icate1 t :i* is d5om9at$ an ess5tial accid5t1 z equality1 or section1 or position1 or "s o!r affection ( ?is k9d4 ,:51 !re=e1 any "o proposes to inscribe an equilat]al triangle 9 a circle1 he proposes a problem3 = x is possible to inscribe "o t is n equilat]al4 ,b :5 any "o ass]ts t ! angles at ! base ( an isosceles triangle are equal1 we mu/ affirm t he proposes a !orem2 = x is n possible t ! angles at ! base ( an isosceles triangle %\ld be unequal to ea* o!r4 ,on :i* acc\nt1 if any "o =m+ problematically1 %\ld say t he wi%es to inscribe a "r angle 9 a semi-circle1 he mu/ be consid]$ z ignorant ( geometry2 s9ce e angle 9 a semi-circle is necess>ily a "r "o4 ,h5ce1 propositions :i* h an univ]sal symptom1 att5d+ ! :ole matt]1 mu/ be call$ !orems2 b ^? 9 :i* ! symptom is n univ]sal1 & does n att5d its subject1 mu/ be consid]$ z problems4 ,z to bisect a giv5 t]m9at$ "r l9e1 or to cut x into equal p>ts3 = x is possible to cut x into unequal p>ts4 ,to bisect e rectil9e> angle1 or divide x into equal p>ts2 = a division may be giv5 into unequal p>ts4 ,on a giv5 "r l9e to describe a quadrangle2 = a figure t is n quadrangul> may be describ$4 ,&1 9 %ort1 all ( ?is k9d belong to ! problematical ord]4 ,b ! foll[]s ( ,z5odotus1 :o 0 famili> ) ! doctr9e ( ,o5opides1 b ! disciple ( ,&ron1 di/+ui% a !orem f a problem1 s f> z a !orem enquires :at ! symptom is :i* is pr$icat$ ( ! matt] x conta9s2 b a problem enquires :at t is1 ! exi/5ce ( :i* is grant$4 ,f :5ce ! foll[]s ( ,possidonius def9e a !orem a proposition1 by :i* x is enquir$ :e!r a ?+ exi/s or n2 b a problem1 a proposition1 9 :i* x is enquir$ :at a ?+ is1 or ! mann] ( its exi/5ce4 ,& !y say t we ought to =m ! contemplat+ proposition by enunciat+1 z t e triangle has two sides great] ?an ! rema9+ "o1 & t ! angles at ! base ( e isosceles triangle are equal3 b we mu/ =m ! problematical proposition1 z if enquir+ :e!r a triangle is to be con/ruct$ ^u ?is "r l9e4 ,= !re is a diff]5ce1 say !y1 absolutely & indef9itely1 to enquire :e!r ! ?+ propos$ is f a giv5 po9t to erect a "r l9e at "r angles to a giv5 l9e1 & to behold :at ! p]p5dicul> is4 ,& ?us1 f :at has be5 said1 x is manife/ !re is "s diff]5ce betwe5 a problem & a !orem4 ,b t ! elem5t>y in/itution ( ,euclid1 also1 consi/s p>tly ( problems1 & p>tly ( !orems1 w be manife/ f consid]+ ! sev]al propositions4 ,s9ce1 9 ! conclusion ( 8 demon/rations1 he sometimes adds 7:i* 0 to be %ewn7 sometimes 7:i* 0 to be done7 ! latt] s5t5ce be+ ! m>k or symbol ( problems1 & ! =m] ( !orems4 ,= al?\gh1 z we h said1 demon/ration takes place 9 problems1 yet x is (t5 = ! sake ( g5]ation2 = we assume demon/ration 9 ord] to %ew1 t :at 0 comm&$ is accompli%$3 b sometimes x is wor?y by itself1 s9ce ! nature ( ! ?+ s\ght aft] may be br\ght into ! mid/4 ,b y w f9d ,euclid sometimes comb9+ !orems ) problems1 & us+ !m alt]nately1 z 9 ! fir/ book2 b sometimes ab\nd+ ) ! "o & n ! o!r4 ,= ! f\r? book is :olly problematical2 b ! fif? is entirely compos$ f !orems4 ,& ?us mu* conc]n+ ! ord] ( geometrical propositions4 ,,*ap4 ,,ix4 ,conc]n+ ! ,design ( ! fir/ ,book1--its ,division1--& a previ\s ,admonition to ! ,read]4 ,b1 aft] ^! consid]ations1 :5 we h det]m9$ ! design ( ! fir/ book1 & h exhibit$ its division1 we %all ent] ^u ! treatise ( ! def9itions4 ,! design1 !n1 ( ?is book1 is to deliv] ! pr9ciples ( ! contemplation ( "r l9es4 ,= ?\gh a circle1 & its consid]ation1 is m excell5t ?an ! ess5ce & k ( "r l9es1 yet ! doctr9e conc]n+ ^! is m adapt$ to u1 :o are ha/5+ to transf] our cogitation f m imp]fect & s5sible natures1 to su* z are intelligible4 ,=1 inde$1 "r l9$ figures are prop] to s5sibles1 b a circle to intelligibles4 ,because t :i* is simple1 uni=m1 & def9ite1 is prop] to ! nature ( ! ?+s :i* are3 b t :i* is v>i\s1 & :i* increases indef9itely f ! numb] ( its conta9+ sides1 reg>ds ! fluctuat+ ess5ce ( s5sible p>ticul>s4 ,h5ce1 9 ?is book1 ! fir/ & mo/ pr9cipal ( "r l9$ figures are deliv]$2 ,i mean ! triangle & p>allelogram4 ,= 9 ^!1 z "u _! prop] g5us1 ! causes ( ! elem5ts are conta9$3 viz4 ! isosceles & scal5e1 & ^? :i* are =m$ f ^!1 ! equilat]al triangle1 & ! quadrangle1 f :i* ! f\r figures ( ! elem5ts are compos$4 ,we %all f9d1 !re=e1 z well ! orig9 ( ! equilat]al triangle z ( ! quadrangle2 ( ! la/1 inde$1 ^u1 b ( ! fir/ f a giv5 "r l9e4 #abc;,an equilat]al triangle1 !re=e1 is ! proximate cause ( ! ?ree elem5ts1 fire1 air1 & wat]3 b a quadrangle is annex$ to e>?4 ,& la/ly1 ! design ( ! fir/ book is adapt$ to ! :ole treatise1 & conf]s to ! univ]sal k ( ! mundane elem5ts4 ,besides1 x in/ructs le>n]s 9 ! sci5ce conc]n+ "r-l9$ figures2 s9ce x rightly inv5ts1 & accurately collects1 ! fir/ pr9ciples ( ^!4 ,b ?is book is divid$ into ?ree greate/ p>ts1 ( :i* ! fir/ decl>es ! orig9 & prop]ties ( triangles1 z well accord+ to angles1 z also accord+ to sides4 ,besides1 x makes mutual comp>isons ( ^!1 & beholds e "o by itself4 ,= receiv+ "o triangle1 sometimes x consid]s ! angles f ! sides2 b sometimes ! sides f ! angles3 & ?is accord+ to equality & inequality4 ,& suppos+ two triangles1 x discov]s ! same prop]ty aga91 by v>i\s me?ods4 ,b ! second "p comb9es ! contemplation ( p>allelograms1 describ+ _! prop]ties & g5]ations4 ,& ! ?ird "p %ews ! communication ( triangles & p>allelograms1 bo? 9 symptoms & mutual comp>isons4 ,= x %ews t triangles & p>allelograms con/itut$ on ! same & on equal bases1 are affect$ ) ! same passions2 & by complication1 :5 bo? /& ^u "o base3 & aga91 aft] :at mann] a p>allelogram may be made equal to a triangle2 & la/ly1 conc]n+ ! proportion :i* 9 "r angl$ triangles1 ! squ>e made f ! side subt5d+1 has to ! squ>es conta9+ ! "r angle4 ,& su* is ! division ( ! fir/ ,book4 ,b1 previ\s to our enquiry into ea* ( ^! p>ts1 we ?9k x requisite to admoni% ! read]1 t he mu/ n require ( u1 ^? small assumptions1 & cases1 & :atev] else !re may be ( t k9d1 :i* has be5 divulg$ by our pr$ecessors4 ,= we are satiat$ ) ^!1 & %all1 !re=e1 b r>ely adopt !m 9 our disc\rse4 ,b :atev] has a m difficult contemplation1 & reg>ds univ]sal philosophy1 ( ?is we %all make a p>ticul> relation3 imitat+ ! ,py?agoreans1 ) :om ?is nigma 0 common1 8a#abd figure & a /ep3 b n a figure & ?ree oboli10 %ew+ by ?is1 t x is requisite to pursue t philosophy :i* asc5ds e !orem by a /ep1 & raises ! s\l on high2 b does n suff] x to rema9 among s5sibles1 to fill up ! use att5dant on mortals1 &1 consult+ = ?is1 to neglect ! elevation :i* rises f h5ce to an intelligible ess5ce4 ,,def9itions4 ,,def9ition ,i4 ,a ,,po9t is t :i* has no ,,p>ts4 ,t geometry1 accord+ to ! transition :i* takes place f ?+s m composite to su* z are m simple1 runs f body1 :i* is diffus$ into di/ance by ?ree dim5sions1 to a sup]ficies by :i* x is b\nd$2 b f sup]ficies to a l9e1 ! b\nd>y ( sup]ficies2 & f a l9e to a po9t de/itute ( all dim5sion1 has be5 (t5 said1 & is p]fectly manife/4 ,b because ^! t]ms1 9 _m places1 on acc\nt ( _! simplicity1 appe> to be m excell5t ?an ! nature ( composites2 b 9 _m1 z :5 !y subsi/ 9 ?+s :i* !y t]m9ate1 !y are simil> to accid5ts1 x is necess>y to det]m9e 9 :at g5]a ( be+s ea* ( ^! may be beheld#abe4 ,i say !n1 t su* ?+s z are de/itute ( matt]1 :i* subsi/ 9 sep>ate reasons1 & 9 ^? =ms :i* are plac$ "u !mselves1 are always allott$ a subsi/5ce ( m simple ess5ces1 sup]ior to ! subsi/5ce ( su* z are m composite4 ,on ?is acc\nt1 bo? 9 intellect1 & 9 ! ornam5ts1 z well ( ! middle k9d z among ^? peculi> to ! s\l1 & 9 natures !mselves1 ! t]ms :i* proximately vivify bodies1 excel accord+ to ess5ce ! ?+s :i* are t]m9at$2 & are m imp>tible1 m uni=m1 & m prim>y ?an ^!4 ,= 9 immat]ial =ms1 unity is m p]fect ?an multitude2 t :i* is imp>tible1 ?an t :i* is endu$ ) unb\nd$ progression2 & t :i* t]m9ates1 ?an t :i* receives b\nd f ano!r4 ,b su* ?+s z are indig5t ( matt]1 & abide 9 o!rs1 & deg5]ate f ! p]fection ( _! ess5ce1 :i* are scatt]$ ab\t subjects1 & h an unnatural union1 are allott$ m composite reasons1 prior to su* z are m simple4 ,h5ce1 ?+s :i* appe> 9 ! phantasy inve/$ ) =m1 & ! matt] ( ! figures :i* ! phantasy conta9s1 & :atev] 9 s5sibles is g5]at$ by nature1 h1 9 a prec$+ ord]1 ! reasons ( ! ?+s t]m9at$2 b ! reasons :i* t]m9ate1 9 a foll[+ & adv5titi\s rank#abf4 ,= le/ t :i* is di/ribut$ into ?ree dim5sions1 %\ld be ext5d$ into inf9ite magnitude1 ei!r accord+ to intellig5ce or s5se1 x 0 e way t]m9at$ by sup]ficies4 ,& le/ a plane sup]ficies %\ld conceal itself 9 an inf9ite progression1 a l9e approa*+ oppos$ its diffusion1 & gave b\nd to its indef9ite ext5sion4 ,&1 9 l mann]1 a po9t limit$ ! progressions ( a l9e2 composite natures d]iv+ _! subsi/5ce f su* z are simple4 ,= ?is also is aga9 manife/1 t 9 sep>ate =ms ! reasons ( t]ms subsi/ 9 !mselves1 b n 9 ^? :i* are t]m9at$2 & abid+ su* z !y are 9 reality1 possess a p[] ( con/itut+ second>y natures4 ,b1 9 insep>able =ms !y give !mselves up to ?+s :i* are t]m9at$1 reside 9 !m1 become1 z x 71 _! p>ts1 & are repl5i%$ ) bas] natures4 ,on :i* acc\nt1 t :i* is imp>tible is !re endu$ ) a p>tible ess5ce1 & t :i* is void ( latitude is diffus$ into bread?4 ,& t]ms are no long] able to pres]ve _! simplicity & purity4 ,= s9ce !y abide 9 ano!r1 !y necess>ily *ange _! own nature into ! matt] ( _! conta9+ subject4 ,matt]1 inde$1 di/urbs ! p]fection ( ^!1 & causes ! reason ( a plane to become a pr(\nd plane2 b obscur+ ! "o dim5sion ( a l9e1 causes x to be e way p>tible2 & gives corporeity to ! indivisibility ( a po9t1 & sep>ates x toge!r ) ! natures :i* x t]m9ates4 ,= all ^! reasons fall+ into matt]1 ! "o k9d f cogitation into intelligible matt]1 b ! o!r f nature into t :i* is s5sible1 are repl5i%$ ) _! conta9+ subjects2 & dep>t f _! own simplicity1 into =eign compositions & int]vals4 ,b "h a d\bt arises h[ all ^!1 exi/+ 9 intellect & s\l 9 an imp>tible mann]1 & )\t any dim5sion1 are di/ribut$ into matt]1 "s inde$1 pr9cipally1 b o!rs on acc\nt ( its nature8 ,%all we say t !re is a c]ta9 ord] 9 immat]ial =ms1 s t "s are allott$ ! fir/1 "s ! middle1 & o!rs ! la/ place2 & t ( =ms "s are m uni=m1 b t o!rs are m multipli$2 & t "s h _! p[]s collect$ toge!r1 b o!rs t5d+ into int]val2 & t "s1 aga91 bord] ^u b\nd1 b t o!rs are proximate to inf9ity8 ,= ?\gh all p>ticipate ( ^! two pr9ciples1 yet "s orig9ate f b\nd1 b o!rs f inf9ity1 ( :i* !y m l>gely p>ticipate4 ,h5ce1 a po9t is entirely imp>tible1 s9ce x subsi/s accord+ to b\nd1 yet x occultly conta9s an inf9ite p[]1 by :i* x produces e int]val1 & ! progression ( all int]vals1 unfolds its inf9ite p[]4 ,b body1 & ! reason ( body1 p>ticipates m ( an inf9ite nature2 on :i* acc\nt x is among ! numb] ( ?+s t]m9at$ by ano!r1 & divisible 9 inf9itum1 accord+ to all dim5sions4 ,b ! m$iums betwe5 ^!1 accord+ to ! di/ance ( ! extremes1 are ei!r among ! numb] ( ?+s :i* h an abundance ( b\nd2 or among su* z h an afflu5ce ( inf9ity3 on :i* acc\nt !y bo? t]m9ate & are t]m9at$4 ,=1 inde$1 s f> z !y consi/ f b\nd1 !y are able to t]m9ate o!rs2 b s f> z !y p>ticipate ( inf9ity1 !y are indig5t ( t]m9ation f o!rs1 ,h5ce1 s9ce a po9t is also a b\nd1 x pres]ves its prop] p[] 9 p>ticipation3 b s9ce x likewise conta9s inf9ity occultly1 & is compell$ to be e :]e pres5t ) ! natures :i* x t]m9ates1 x resides ) !m inf9itely4 ,&1 because among immat]ial =ms !re 0 a c]ta9 inf9ite p[] capable ( produc+ ?+s di/ant f ea* o!r by int]vals1 a po9t is pres5t ) its p>ticipants 9 capacity4 ,= inf9ity 9 intelligibles is ! prim>y cause & prolific p[] ( ! univ]se2 b 9 mat]ial natures x is imp]fect1 & is alone all ?+s 9 dormant capacity4 ,& 9 %ort1 ^? =ms :i*1 on acc\nt ( _! simplicity & imp>tibility1 hold a sup]ior rank among pr9ciples1 pres]ve1 inde$1 79 con=mity to _! nature17 _! own prop]ty 9 _! p>ticipations1 b become worse ?an m composite reasons4 ,= matt] is able to p>ticipate ^! m cle>ly1 & to be prep>$ = _! reception1 r ?an t ( ! mo/ simple causes ( be+s4 ,on :i* acc\nt1 ! ve/igies ( sep>ate pr9ciples desc5d into matt]2 b ! p>ticipations ( ^? 9 a second & ?ird ord]1 become m conspicu\s4 ,h5ce1 matt] p>ticipates m ( ! cause ( body1 ?an ( a plane2 & ( ?is m ?an ! =m ( a l9e2 & ( ?is /ill m ?an t ( a po9t1 :i* conta9s all ^!1 & is ! b\nd>y ( !m all4 ,= ! reason ( a po9t presides ov] ?is :ole s]ies1 unites & conta9s all p>tible natures1 t]m9ates _! progressions1 produces !m all by its inf9ite p[]1 & compreh5ds !m 9 its indivisible b\nd4 ,on :i* acc\nt also1 9 ! images ( immat]ial =ms1 "s are ! b\nd>ies ( o!rs2 b a po9t is ! limit ( !m all4 ,b t we mu/ n ?9k ) ! ,/oics1 t ^! b\nd>ies ( bodies alone subsi/ f cogitation2 b t !re are c]ta9 natures ( ?is k9d among be+s1 :i* previ\sly conta9 ! demiurgical reasons ( ?+s1 we %all be enabl$ to rememb]1 if we reg>d ! :ole _w1 ! convolutions ( its p>ts1 ! c5tres ( ^? convolutions1 & ! axes :i* p5etrate ?r\gh ! :ole ( ^! revolv+ circles4 ,= ! c5tres subsi/ 9 en]gy1 s9ce !y conta9 ! sph]es1 pres]ve !m 9 _! prop] /ate1 unite _! int]vals1 & b9d & e/abli% to !mselves ! p[]s :i* !y possess4 ,b ! axes !mselves be+ 9 an immoveable position1 evolve ! sph]es1 give !m a circul> motion1 & a revolution r\nd _! own abid+ nature4 ,& ! poles ( ! sph]es1 :i* bo? t]m9ate ! axes1 & b9d 9 !mselves ! o!r convolutions1 d !y n p]spicu\sly ev9ce1 t po9ts are endu$ ) demiurgical & capaci\s p[]s1 t !y are p]fective ( e ?+ di/ant by int]vals1 & are ! s\rces ( union1 & an unceas+ motion8 ,f :5ce1 inde$1 ,plato#abg also says1 t !y h an adamant9e subsi/5ce2 %ew+ by ?is1 ! immutable1 et]nal1 & /able p[] ( _! ess5ce1 ev] pres]v+ itself 9 ! same uni=m mode ( exi/5ce4 ,he adds too1 t ! :ole sp9dle ( ! ,fates1 is turn$ ab\t ^!1 & leaps r\nd _! co]cive union4 ,b o!r m recondite & ab/ruse disc\rses affirm1 t ! demiurgus presides ov] ! _w1 seat$ 9 ! poles1 &1 by 8 div9e love1 conv]t+ ! univ]se to himself4 ,b ! ,py?agoreans ?\ght t ! pole %\ld be call$ ! ,seal ( ,rhea#abh2 because ! zoogonic1 or vivific goddess1 p\rs ?r\gh ^! into ! univ]se1 an inexplicable & efficaci\s p[]4 ,& ! c5tre !y call$ ! prison ( ,jupit]2 because1 s9ce ,jupit] has plac$ a demiurgical gu>d 9 ! bosom ( ! _w1 he has firmly e/abli%$ x 9 ! mid/4 ,=1 inde$1 ! c5tre abid+1 ! univ]se possesses its immoveable ornam5t1 & unceas+ convolution3 & ! gods :o preside ov] ! poles1 obta9 a p[] collective ( divisible natures1 & unific ( su* z are multipli$3 & ^? :o are allott$ ! gov]nm5t ( ! axes1 re/ra9 & et]nally evolve _! p]petual convolutions4 ,&1 if x is lawful to (f] our own op9ion on ?is subject1 ! c5tres & poles ( all ! sph]es are ! symbols ( ! conciliat+ gods1 %ad[+ =? _! imp]ceptible & unify+ composition4 ,b ! axes express ! coh]5cies ( ! univ]sal ornam5ts2 & are endu$ ) a p[] ( compreh5d+ ! mundane integrities & p]iods1 9 ! same mann] z _! presid+ deities1 ( su* z are intellectual4 ,b ! sph]es !mselves are images ( ! gods1 call$ p]fectors ( works1 copulat+ ! pr9ciple ) ! end1 & excell+ all figures 9 simplicity1 similitude1 & p]fection4 ,b we h be5 ?us prolix1 t we might ev9ce ! p[] ( imp>tibles1 & ( ! t]ms :i* ! _w conta9s1 & t s f> z !y be> an image ( prim>y & mo/ pr9cipal causes1 !y are allott$ ! mo/ excell5t ord] 9 ! univ]se4 ,= c5tres & poles are n ( ! same k9d ) ?+s :i* are t]m9at$2 b !y subsi/ 9 en]gy1 & possess an ess5ce1 & p]fect p[]1 :i* p]vades ?r\gh all p>tible natures4 ,b _m behold+ ^? t]ms :i* imp]fectly subsi/ 9 t]m9at$ ess5ces1 consid] !m z endu$ ) a sl5d] subsi/5ce2 & "s inde$ say1 t !y are alone sep>at$ f s5sibles by ?\ght2 b o!rs1 t !y h an ess5ce no :]e b 9 our ?\ghts4 ,h[ev]1 s9ce ! =ms ( all ^! are f\nd bo? 9 ! nature ( intellect1 9 ! ornam5ts ( s\l1 9 ! nature ( ?+s1 & 9 inf]ior bodies1 let u consid] h[1 accord+ to ! ord] !y conta91 !y subsi/ 9 ! g5]a ( be+s4 ,& inde$1 all ( !m pre-exi/ 9 intellect1 b 9 an imp>tible & uni=m mann]3 s t !y all subsi/ accord+ to "o =m1 ! reason ( a po9t1 :i* exi/s occultly & imp>tibly4 ,b !y all subsi/ 9 s\l accord+ to ! =m ( a l9e3 on :i* acc\nt ,timu also composes ! s\l f "r & circul> l9es3 = e circle is a l9e alone#abi4 ,b !y all subsi/ 9 natures1 accord+ to ! reason ( a plane2 & on ?is acc\nt1 ,plato comm&s u to manife/ ^? natural reasons1 :i* are endu$ ) a p[] ( con/itut+ bodies by a plane4 ,& ! resolution ( bodies into planes leads u to ! proximate cause ( appe>ances4 ,la/ly1 !y all subsi/ 9 bodies1 b 9 a corporeal mann]2 s9ce all =ms h _! be+ 9 ^!1 accord+ to ! p>tible nature ( bodies4 ,h5ce1 all ( !m appe> e :]e1 & ea* accord+ to its prop] ord]2 & div]sity arises f pre-dom9at+ p[]4 ,! po9t1 inde$1 is e :]e imp>tible1 & :5 t :i* is divisible into p>ts1 excels accord+ to ! dim9ution ( be+s1 x v9dicates to itself1 an illu/ri\s subsi/5ce ( p>tible natures4 ,& sometimes ! po9t is entirely sup]ior1 accord+ to ! excell5ce ( cause2 b sometimes x is connect$ ) divisibles1 & sometimes x is allott$ 9 !m an adv5titi\s exi/5ce2 &1 z if swall[$ up by ! p>tition ( ! l[e/ natures1 loses its own prop] imp>tibility4 ,z1 !re=e1 ) respect to ! monad1 "o#acj is ! "m ( numb]1 b ! o!r is z matt] spread "u1 & ! receptacle ( numb]s2 & ea* ( !m a pr9ciple1 7yet nei!r ( !m is numb]71 b 9 a diff]5t respect3 9 ! same mann] a po9t also1 is p>tly ! p>5t & au?or ( magnitudes2 b is p>tly a pr9ciple 9 ano!r respect1 & n accord+ to a g5]ative cause4 ,b is a po9t1 !n1 ! only imp>tible8 ,or may we affirm ?is ( ! n[ 9 "t1 & ( unity 9 numb]s8 ,%all we n say1 t to ! philosoph]1 inde$1 disc\rs+ conc]n+ ! univ]sality ( ?+s1 x is prop] to behold e ?+1 h[ev] fall+ "u di/ribution2 b t to him :o is endu$ ) ! sci5ce ( p>ticul>s1 :o produces 8 contemplation f c]ta9 def9ite pr9ciples1 & runs back ev5 to ^!1 b v little scrut9izes ! progressions ( be+s1 x is requisite to attempt1 consid]1 & treat conc]n+ t imp>tible nature alone1 :i* reg>ds 8 fir/ pr9ciples2 & to behold t simplicity :i* presides ov] all ! p>ticul> subjects ( 8 k8 ,9 consequ5ce ( ?is reason+1 !re=e1 a po9t alone1 accord+ to ! geometric matt]1 is de/itute ( p>tition2 b unity accord+ to t :i* is ari?metical4 ,& ! reason ( a po9t1 h[ev] 9 "s o!r respects x may be imp]fect1 yet is p]fect 9 ! pres5t sci5ce4 ,=1 inde$1 ! physician also says1 t ! elem5ts ( bodies are fire & wat]1 & ?+s simil> to ^!2 & z f> z to ^! ! resolution ( bodies proce$s4 ,b ! natural philosoph] passes on to m simple elem5ts2 & ! "o def9es an elem5t simple z to s5se1 b ! o!r simple z to reason2 & bo? ( !m prop]ly z to _! peculi> sci5ce4 ,we mu/ n1 !re=e1 ?9k t ! def9ition ( a po9t is faulty1 nor det]m9e x z imp]fect2 = s f> z p]ta9s to ! geometric matt]1 & its pr9ciples1 x is suffici5tly deliv]$4 ,?is alone1 inde$1 is want+ to its completion1 t ! def9ition does n cle>ly say1 t :i* is imp>tible ) me is a po9t2 & my pr9ciple1 & t :i* ,i conta9 z mo/ simple1 is no?+ else ?an ?is4 ,& aft] ?is mann] x is prop] to he> ! geometrician address+ u4 ,euclid1 !re=e1 f a negation ( p>ts1 decl>es to u a pr9ciple1 lead+ to ! !ory ( its :ole subject nature4 ,= negative disc\rses are prop] to pr9ciples1 z ,p>m5ides tea*es u1 :o deliv]s ! doctr9e conc]n+ ! fir/ & la/ cause1 by negations alone4 ,s9ce e pr9ciple consi/s ( an ess5ce diff]5t f its fl[+ consequ5ts2 & ! negations ( ^! exhibit to u ! prop]ty ( _! s\rce4 ,= t x is1 inde$1 ! cause ( ^!1 yet at ! same "t has no?+ 9 common ) ^!1 becomes p]spicu\s f a doctr9e ( ?is k9d4 ,b "h a d\bt may arise1 h[1 s9ce ! phantasy receives all ?+s inve/$ ) =ms1 & 9 a p>tible mann]1 ! geometrician beholds 9 x ! po9t de/itute ( p>ts8 ,= x is n because !y are reasons exi/+ 9 cogitation1 b ! phantasy receives ! resemblances ( intellectual & div9e =ms accord+ to its own prop] nature1 exhibit+ 9 its %ad[y bosom ! =ms ( =mless natures1 & clo?+ ) figure ?+s entirely free f ! affections ( figure4 ,to ?is ambiguity we mu/ say1 t ! species ( imag9ative motion is nei!r alone p>tible1 nor imp>tible2 b t x proce$s f ! imp>tible to ! p>tible1 & f ! =mless nature to t :i* is express$ by =m4 ,= if x 0 p>tible alone1 x c\ld n pres]ve 9 itself _m impressions ( =ms1 s9ce ! subsequ5t w\ld obscure ! pre-exi/5t figures3 = no body c conta9 at once1 & accord+ to ! same situation1 a multitude ( figures2 b ! =m] w be blott$ out by ! succession ( ! latt]4 ,b if x 0 alone imp>tible1 x w\ld n be inf]ior to cogitation1 & to s\l1 :i* surveys all ?+s 9 an imp>tible mann]4 ,h5ce1 x is necess>y t x %\ld inde$ beg9 f an imp>tible accord+ to its motion1 & f !nce draw =? ! fold$ & scatt]$ =m ( e ?+ fall+ "u cogitation1 & p5etrat+ to its %ad[y receptacle3 b1 t x %\ld at l5g? end 9 =m1 figure1 & int]val4 ,& if x be allott$ a nature ( ?is k9d1 x w1 aft] a c]ta9 mann]1 conta9 an imp>tible ess5ce3 & a po9t1 accord+ to ?is1 mu/ be said to h its pr9cipal subsi/5ce3 = ! =m ( a l9e is contract$ 9 ! phantasy accord+ to ?is4 ,h5ce1 because x possesses a tw(old p[]1 imp>tible & p>tible1 x w inde$ conta9 a po9t 9 an imp>tible1 & int]vals 9 a p>tible mann]4 ,b z ! ,py?agoreans def9e a po9t to be unity hav+ position1 let u consid] :at !y mean4 ,t numb]s1 inde$1 are m immat]ial & m pure ?an magnitudes1 & t ! pr9ciple ( numb]s is m simple ?an ! pr9ciple ( magnitudes1 is manife/ to e "o3 b :5 !y say t a po9t is unity endu$ ) position1 !y appe> to me to ev9ce t unity & numb] subsi/ 9 op9ion3 ,i mean monadic numb]#aca4 ,on :i* acc\nt1 e numb]1 z ! p5tad & ! heptad1 is "o 9 e s\l1 & n _m2 & !y are de/itute ( figure & adv5titi\s =m4 ,b a po9t op5ly pres5ts itself 9 ! phantasy1 subsi/s1 z x 71 9 place1 & is mat]ial accord+ to intelligible matt]4 ,unity1 !re=e1 has no position1 s f> z x is immat]ial1 & free f all int]val & place3 b a po9t has position1 s f> z x appe>s seat$ 9 ! bosom ( ! phantasy1 & has a mat]ial subsi/5ce4 ,b unity is /ill m simple ?an a po9t1 on acc\nt ( ! community ( pr9ciples4 ,s9ce a po9t exce$s unity accord+ to position2 b appositions 9 incorporeals produce dim9utions ( ^? natures1 by :i* ! appositions are receiv$4 ,,def9ition ,,ii4 ,a ,l9e is a ,l5g? )\t ,bread?4 ,a ,l9e obta9s ! second place 9 ! ,def9itions1 z x is by f> ! fir/ & mo/ simple int]val1 :i* ! geometrician calls a l5g?1 add+ also )\t bread?2 s9ce a l9e1 9 respect ( a sup]ficies1 ranks z a pr9ciple4 ,= he def9es a po9t1 z x is ! pr9ciple ( all magnitudes1 by negation alone2 b a l9e1 z well by affirmation z by negation4 ,h5ce x is a l5g?1 & by ?is exce$s ! imp>tibility ( a po9t2 b x is )\t bread?1 because x is sep>at$ f o!r dim5sions4 ,=1 inde$1 e ?+ :i* is void ( bread?1 is also de/itute ( bulk1 b ! contr>y is n true1 t e ?+ void ( bulk is also de/itute ( bread?4 ,s9ce1 !re=e1 he has remov$ bread? f a l9e1 he has also remov$ at ! same "t bulk4 ,on :i* acc\nt he does n add1 t a l9e also has no ?ickness1 because ?is prop]ty is consequ5t to ! notion ( be+ )\t bread?4 ,b x is def9$ by o!rs 9 v>i\s ways3 = "s call x ! flux ( a po9t1 b o!rs a magnitude conta9$ by "o int]val4 ,& ?is def9ition1 inde$2 is p]fect1 & suffici5tly expla9s ! ess5ce ( a l9e2 b t :i* calls x ! flux ( a po9t1 appe>s to manife/ its nature f its produc+ cause2 & does n express e l9e1 b alone t :i* is immat]ial4 ,= ?is is produc$ by a po9t1 :i* ?\gh imp>tible itself1 is ! cause ( be+ to p>tible natures4 ,b ! flux ( a po9t1 %ews its progression & prolific p[]1 approa*+ to e int]val1 receiv+ no detrim5t1 p]petually abid+ ! same1 & af=d+ ess5ce to all p>tible magnitudes4 ,h[ev]1 ^! obs]vations are kn[n1 & manife/ to e "o4 ,b we %all recall into our memory1 disc\rses m ,py?agorical1 :i* det]m9e a po9t z analog\s to unity1 a l9e to ! duad1 a sup]ficies to ! triad1 & body to ! tetrad4 #acb,yet :5 we comp>e ^? :i* receive int]val toge!r1 we %all f9d a l9e monadic2 b a sup]ficies dyadic1 & a solid body triadic4 ,f :5ce also1 ,ari/otle#acc says2 t body is p]fect$ by ! t]n>y numb]4 ,&1 inde$1 ?is is n wond]ful1 t a po9t1 on acc\nt ( its imp>tibility1 %\ld be assimilat$ to unity2 b t ?+s subsequ5t to a po9t1 %\ld subsi/ accord+ to numb]s proce$+ f unity1 & %\ld pres]ve ! same proportion to a po9t1 z numb]s to unity2 & t e "o %\ld p>ticipate ( its proximate sup]ior1 & h ! same proportion to its k9dr$1 & foll[+ degree1 z ! sup]ior to ?is1 :i* is ! imm$iate consequ5t4 #acd;,= example1 t a l9e has ! ord] ( ! duad ) respect to ! po9t1 b ( unity to a sup]ficies2 & t ?is la/ has ! relation ( a triad to ! po9t1 b ( ! duad to a solid4 ,& on ?is acc\nt1 body is tetradic1 ) respect to a po9t1 b triadic z to a l9e4 ,ea* ord]1 !re=e1 has its proportion2 b ! ord] ( ! ,py?agoreans is ! m pr9cipal1 :i* receives its comm5cem5t f an exalt$ s\rce1 & foll[s ! nature ( be+s4 ,= a po9t is inde$ tw(old2 s9ce x ei!r subsi/s by itself1 or 9 a l9e2 9 :i* la/ respect also1 s9ce z a b\nd>y x is alone & "o1 nei!r hav+ a :ole nor p>ts1 x imitates ! supreme nature ( be+s4 ,on :i* acc\nt too1 x 0 plac$ 9 a correspond5t proportion to unity4 #ace;,= z ! oracle says1 ,unity is !re fir/1 :]e ! pat]nal unity abides4 ,b a l9e is ! fir/ endu$ ) p>ts & a :ole1 & x is monadic because x is di/ant by "o int]val only2 & dyadic on acc\nt ( its progression3 = if x be inf9ite1 x p>ticipates ( ! indef9ite duad2 b if f9ite1 x requires two t]ms1 f :5ce & to :at place2 s9ce1 on acc\nt ( ^! x imitates totality1 & is allott$ an ord] among totals4 ,= unity1 accord+ to ! oracle1 is ext5d$#acf1 & g5]ates two2 & ?is produces a progression into longitude1 toge!r ) t :i* is di/ant ext5d$ly1 & ) "o int]val1 & ! matt] ( ! duad4 ,b sup]ficies1 s9ce x is bo? a triad & duad1 z also ! receptacle ( ! prim>y figures1 & t :i* receives ! fir/ =m & species1 is 9 a c]ta9 respect simil> to ! triadic nature1 :i* fir/ t]m9ates be+s2 & to ! duad1 by :i* !y are divid$ & disp]s$4 ,b a solid1 s9ce x has a triple di/ance1 & is di/+ui%$ by ! tetrad1 :i* is endu$ ) a p[] ( compreh5d+ all reasons1 is r$uc$ to t ord] 9 :i* ! di/9ction ( corporeal ornam5ts appe>s2 z also ! division ( ! univ]se into ?ree p>ts1 toge!r ) ! tetradic prop]ty1 :i* is g5]ative & female4 ,& ^! obs]vations1 inde$1 might be m l>gely discuss$1 b = ! pres5t1 mu/ be omitt$4 ,aga91 ! disc\rse ( ! ,py?agoreans1 n undes]v$ly1 calls a l9e1 :i* is ! second 9 ord]1 & is con/itut$ accord+ to ! fir/ motion f an imp>tible nature1 dyadic4 ,& t a po9t is po/]ior to unity1 a l9e to ! duad1 & a sup]ficies to ! triad1 ,p>m5ides himself %ews1 by fir/ ( all tak+ away multitude f "o by negation1 & aft]w>ds ! :ole4 ,because1 if multitude is be=e t :i* is a :ole1 numb] also w be prior to t :i* is cont9u\s1 & ! duad to ! l9e1 & unity to ! po9t3 s9ce ! epi!t n _m1 belongs to unity :i* g5]ates multitude1 b to ! po9t1 ! t]m n a :ole1 is prop]1 because x produces a :ole2 = ?is is said to h no "p4 ,& ^! ?+s are affirm$ ( a l9e1 :ile we m accurately contemplate its nature4 ,b we %\ld also admit ! foll[]s ( ,apollonius1 :o say1 t we obta9 a notion ( a l9e1 :5 we are ord]$ to measure ! l5g?s alone1 ei!r ( ways or walls2 = we d n !n subjo9 ei!r bread? or bulk1 b only make "o di/ance ! object ( our consid]ation4 ,9 ! same mann] we p]ceive sup]ficies1 :5 we measure fields2 & a solid1 :5 we take ! dim5sions ( wells4 ,= !n1 collect+ all ! di/ances toge!r1 we say1 t ! space ( ! well is s mu*1 accord+ to l5g?1 bread?1 & dep?4 ,b a l9e may become ! object ( our s5sation1 if we behold ! divisions ( lucid places f ^? :i* are d>k1 & survey ! moon :5 di*otomiz$3 = ?is m$ium has no di/ance ) respect to latitude2 b is endu$ ) longitude1 :i* is ext5d$ toge!r ) ! light & %ad[4 ,,def9ition ,,iii4 ,b ! ,extremities ( a ,l9e are ,po9ts4 ,e composite receives its b\nd f t :i* is simple1 & e ?+ p>tible f t :i* is imp>tible2 & ! images ( ^! op5ly pres5t !mselves 9 ma!matical pr9ciples4 ,= :5 x is said t a l9e is t]m9at$ by po9ts1 x seems manife/ly to make x ( itself inf9ite1 because1 on acc\nt ( its prop] progression1 x has no extremity4 ,z1 !re=e1 ! duad is t]m9at$ by unity1 & r$uces its own intol]able boldness "u b\nd1 :5 x is re/ra9$ 9 its compreh5sive embrace3 s a l9e also is limit$ by ! po9ts :i* x conta9s4 ,=1 s9ce x is simil> to ! duad1 x p>ticipates ( a po9t hav+ ! relation ( unity1 accord+ to ! nature ( ! duad4 ,inde$1 9 imag9ative1 z well z 9 s5sible =ms1 ! po9ts !mselves t]m9ate ! l9es 9 :i* !y reside4 ,b 9 immat]ial =ms1 ! reason ( ! imp>tible po9t pre-exi/s sep>ate & ap>t2 b :5 proce$+ f !nce by f> ! fir/ ( all1 by det]m9+ itself ) int]val1 mov+ itself1 & fl[+ 9 inf9ite progression1 & imitat+ ! indef9ite duad1 x is re/ra9$ inde$1 by its prop] pr9ciple1 is unit$ by its p[]1 & on e side seiz$ by its co]cive b\nd4 ,h5ce x is1 at ! same "t1 bo? inf9ite & f9ite3 inf9ite1 inde$1 accord+ to its progression2 b f9ite accord+ to its p>ticipation ( a t]m9at+ cause4 ,s t1 :5 x approa*es to ?is cause1 x is deta9$ 9 its compreh5sion1 & is t]m9at$ accord+ to its union4 ,h5ce too1 9 ! images ( incorporeal =ms1 a po9t is said to t]m9ate a l9e1 by occupy+ its beg9n+ & end4 ,b\nd1 !re=e1 9 immat]ials1 is sep>at$ f t :i* is b\nd$3 b "h x is tw(old2 = x subsi/s 9 t :i* is t]m9at$4 ,& ?is af=ds a wond]ful symptom1 t =ms2 inde$1 abid+ 9 !mselves1 prec$e _! p>ticipants accord+ to cause2 b :5 giv+ !mselves up to _! subord9ate natures1 subsi/ accord+ to _! div]sifi$ prop]ties3 s9ce !y are multipli$ & di/ribut$ toge!r ) ^!1 & receive ! division ( _! subjects4 ,besides1 ?is also mu/ be previ\sly receiv$ conc]n+ a l9e1 t our geometrician uses x 9 a ?reefold acceptation4 ,z t]m9at$ on bo? sides1 & f9ite2 z 9 ! problem#acg :i* says1 ,^u a giv5 t]m9at$ "r l9e to con/ruct an equilat]al triangle4 ,& z p>tly inf9ite & p>tly f9ite2 z 9 ! problem :i* comm&s u f ?ree "r l9es1 :i* are equal to ?ree giv5 "r l9es1 to con/ruct a triangle2 = 9 ! con/ruction ( ! problem1 he says1 ,let !re be plac$ a c]ta9 "r l9e1 on "o "p f9ite1 b on ! o!r "p inf9ite4 ,& aga91 a l9e is receiv$ by ,euclid z on bo? sides inf9ite2 z 9 ! problem :i* says1 ,^u a giv5 inf9ite "r l9e1 f a giv5 po9t1 :i* is n 9 t l9e1 to let fall a p]p5dicul>4 ,b1 besides ?is1 ! foll[+ d\bts1 s9ce !y are wor?y ( solution1 mu/ n be omitt$4 ,h[ are po9ts call$ ! extremities ( a l9e8 & ( :at l9e1 s9ce !y c nei!r be ! b\nds ( "o t is inf9ite1 nor ( e f9ite8 ,= !re is a c]ta9 l9e1 :i* is bo? f9ite1 & has n po9ts = its extremities4 ,& su* is a circul> l9e1 :i* returns into itself1 & is n b\nd$ by po9ts1 l a "r l9e4 ,& su* also is ! ellipsis1 or l9e l a %ield4 ,is x !re=e requisite to behold a l9e1 consid]$ z a l9e8 = we mu/ receive a c]ta9 circumf]5ce1 :i* is t]m9at$ by po9ts1 & a "p ( ! elliptic l9e2 hav+1 9 l mann]1 its extremities b\nd$ by po9ts4 ,b e circul> & elliptic l9e1 assumes to itself ano!r c]ta9 prop]ty1 by :i* x is n l9e alone1 b is also endu$ ) a p[] ( p]fect+ figure#ach4 ,l9es1 !mselves1 !re=e1 h _! extremities t]m9at$ by po9ts2 b ^? :i* are effective ( su* l figures1 return into !mselves4 ,&1 inde$1 if y conceive !m to be describ$1 y w also f9d h[ !y are b\nd$ by po9ts2 b if y receive !m already describ$1 & connect ! end ) ! beg9n+1 y c no long] behold _! extremes4 ,,def9ition ,,iv4 ,a ,"r ,l9e1 is t :i* is equally situat$ betwe5 its b\nd+ ,po9ts4 ,plato e/abli%+ two mo/ simple & pr9cipal species ( l9es1 ! "r & ! circul>1 composes all ! re/ f ! mixture ( ^!2 ,i mean su* z are call$ curve l9es1 "s ( :i* are =m$ f planes2 b o!rs subsi/ ab\t solids2 & :atev] species ( curve l9es are produc$ by ! sections ( solids4 ,& x seems1 inde$1 t a po9t 7if x be lawful s to speak7 be>s an image ( ! "o itself1 accord+ to ,plato3 = unity has no "p1 z he also %ews 9 ! ,p>m5ides4 ,b1 because aft] unity itself !re are ?ree hypo/ases1 or sub/ances1 b\nd1 inf9ite1 & t :i* is mix$ f ^!1 ! species ( l9es1 angles1 & figures1 :i* subsi/ 9 ! nature ( ?+s orig9ate f !nce4 ,&1 inde$1 a circumf]5ce & a circul> angle1 & a circle among plane figures1 & a sph]e among solids1 are analog\s to b\nd4 ,b a "r l9e corresponds to inf9ity1 accord+ to all ^!2 = x prop]ly belongs to all1 if x is beheld z exi/+ 9 ea*4 ,b t :i* is mix$ 9 all ^!1 is analog\s to ! mixt :i* subsi/s among intelligibles4 ,= l9es are mix$1 z ^? :i* are call$ spirals4 ,& angles1 z ! semi-circul> & cornicul>#aci4 ,& plane figures1 z segm5ts & apsides2 b solids1 z cones & cyl9d]s1 & o!rs ( t k9d4 ,b\nd1 !re=e1 inf9ite1 & t :i* is mix$1 are p>ticipat$ by all ^!4 ,b ,ari/otle#adj likewise ass5ts to ,plato2 = e species ( l9es1 says he1 is ei!r "r or circul>1 or mix$ f ^! two4 ,f :5ce also !re are ?ree motions1 "o accord+ to a "r l9e2 ! o!r circul>2 & ! ?ird mix$4 ,b "s oppose ?is division1 & say t !re are n two simple l9es alone1 b t !re is a c]ta9 ?ird l9e giv51 i4 e4 a helix or spiral1 :i* is describ$ ab\t a cyl9d]#ada1 :51 :il/ a "r l9e is mov$ r\nd ! sup]ficies ( ! cyl9d]1 a po9t 9 ! l9e is c>ri$ along ) an equal cel]ity4 ,= by ?is means1 a helix1 or circumvolute l9e1 is produc$1 :i* adapts all ! p>ts ( itself to all1 accord+ to a similitude ( p>ts1 z ,apollonius %ews 9 8 book conc]n+ ! ,co*lea2 :i* passion1 among all spirals1 agrees to ?is alone4 ,= ! p>ts ( a plane helix are dissimil> among !mselves2 z also ( ^? :i* are describ$ ab\t a cone & sph]e4 ,b ! cyl9dric spiral alone1 consi/s ( simil> p>ts 9 ! same mann] z a "r & circul> l9e4 ,are !re1 !n1 ?ree simple l9es1 & n two only8 ,to :i* d\bt we reply1 t a helix ( ?is k9d is1 inde$1 ( simil> p>ts1 z ,apollonius tea*es1 b is by no means simple2 s9ce among natural productions1 gold & silv] are compos$ ( simil> p>ts1 b are n simple bodies4 ,b ! g5]ation ( ! cyl9dric helix ev9ces t its mixture is f ?+s simple2 = x orig9ates :ile a "r l9e is circul>ly mov$ r\nd ! axis ( ! cyl9d]1 a po9t at ! same "t fl[+ along 9 ! "r l9e4 ,two simple motions1 !re=e1 compose its nature2 &1 on ?is acc\nt1 x is among ! numb] ( mixt l9es1 & n among su* z are simple3 = t :i* is compos$ f dissimil>s is n simple1 b mixt4 ,h5ce1 ,gem9us1 ) great propriety1 :5 he admits t "s simple l9es may be produc$ f _m motions1 does n grant t e su* l9e is mixt2 b t alone1 :i* arises f dissimil> motions4 ,= if y conceive a squ>e1 & two motions :i* are p]=m$ ) an equal cel]ity1 "o accord+ to ! l5g?1 b ! o!r accord+ to ! bread?1 a "r l9e or ! diamet] w be produc$2 b ! "r l9e w n1 on ?is acc\nt1 be mix$3 = no o!r l9e prec$es x1 =m$ by a simple motion1 z we ass]t$ ( ! cyl9dric helix4 ,nor yet1 if y suppose a "r l9e1 mov+ 9 a "r angle1 & by a bisection to describe a circle#adb1 is ! circul> l9e1 on ?is acc\nt1 produc$ ) mixture3 = ! extremities ( t :i* is mov$ aft] ?is mann]1 s9ce !y are equally mov$1 w describe a "r l9e2 & ! bisection1 s9ce x is unequally devolv$1 w del9eate a circle2 b ! o!r po9ts w describe an ellipsis4 ,on :i* acc\nt1 ! g5]ation ( a circul> l9e is ! consequ5ce ( t inequality ( lation aris+ f ! bisection2 because a "r l9e 0 suppos$ to be mov$ 9 a "r angle1 b n 9 a natural mann]4 ,& ?us mu* conc]n+ ! g5]ation ( l9es4 ,b x seems1 t ( ! two simple l9es1 ! "r & ! circul>1 ! "r l9e is ! m simple2 = 9 ?is1 dissimilitude _c be conceiv$1 ev5 9 op9ion4 ,b 9 ! circul> l9e1 ! concave & ! convex1 indicate dissimilitude4 ,& a "r l9e1 inde$1 does n inf] a circumf]5ce accord+ to ?\ght2 b a circumf]5ce br+s ) x a "r l9e1 ?\gh n accord+ to its g5]ation1 yet ) respect to its c5tre4 ,b :at if x %\ld be said t a circumf]5ce requires a "r l9e to its con/ruction6 ,= if ei!r extreme ( a "r l9e rema9s fixt1 b ! o!r is mov$1 x w d\btless describe a circle1 ^: c5tre w be ! abid+ extreme ( ! "r l9e4 ,%all we say t ! g5]ator ( ! circle is ! po9t :i* is c>ri$ ab\t ! abid+ po9t1 b n ! "r l9e itself8 ,= ! l9e only det]m9es ! di/ance1 b ! po9t composes ! circul> l9e1 :ile x is mov$ 9 a circul> mann]3 b ( ?is en\gh4 ,aga91 a circumf]5ce appe>s to be proximate to b\nd1 & to h ! same proportion to o!r l9es1 z b\nd to ! univ]sality ( ?+s4 ,= x is f9ite1 & is alone among simple l9es p]fective ( figure4 ,b a "r l9e is proximate to inf9ity2 = its capacity ( inf9ite ext5sion nev] fails3 & z all ! re/ are produc$ f b\nd & inf9ite1 9 ! same mann] f ! circul> & "r l9e1 e mixt g5us ( l9es is compos$1 z well ( planes z ( ^? :i* consi/ 9 solid bodies4 ,& on ?is acc\nt1 ! s\l also#adc previ\sly assum$ into h]self ! "r & circul> accord+ to h] ess5ce1 t %e might mod]ate all ! co-ord9ation ( inf9ite1 & all ! nature ( b\nd1 :i* ! _w conta9s4 ,by a "r l9e1 inde$1 con/itut+ ! progression ( ^! pr9ciples into ! univ]se2 b by a circul> l9e1 _! return to _! orig9al s\rce3 & by ! "o1 produc+ all ?+s into multitude2 b by ! o!r1 collect+ !m into "o4 ,& n only ! s\l1 b he also :o produc$ ! s\l1 & endu$ h] ) ^! p[]s1 conta9s 9 himself bo? ^! prim>y causes4 ,= :5 he previ\sly assum$ ! beg9n+1 middle1 & end ( all ?+s1 he t]m9at$ "r l9es 7says ,plato#add71 by a circul> progression accord+ to nature4 ,& proce$+ to all ?+s by provid5t en]gies1 & return+ to himself1 he e/abli%$ himself1 says ,timu1 aft] 8 own peculi> mann]4 ,b a "r l9e is ! m>k or symbol ( a provid5ce1 indecl9able1 incapable ( p]v]sion1 immaculate1 nev]-fail+1 omnipot5t1 & pres5t to all be+s1 & to e "p ( ! univ]se4 ,b a circumf]5ce1 & t :i* environs1 is ! symbol ( an en]gy retir+ into union ) itself1 & :i* rules ov] all ?+s accord+ to "o intellectual b\nd4 ,:51 !re=e1 ! demiurgus ( ! univ]se _h e/abli%$ 9 himself ^! two pr9ciples1 ! "r & ! circul> l9e1 & _h giv5 !m dom9ion1 he produc$ f himself two unities2 ! "o1 inde$1 en]giz+ accord+ to ! circul> l9e1 & be+ effective ( intellectual ess5ces2 b ! o!r accord+ to ! "r l9e1 & af=d+ an orig9 to s5sible natures4 ,b because ! s\l is allott$ a middle situation betwe5 intellectuals & s5sibles1 s f>1 inde$1 z %e adh]es to an intellectual nature1 %e en]gizes accord+ to ! circle2 b s f> z %e presides ov] s5sibles1 %e provides = _! welf>e accord+ to ! "r l9e3 & ?us mu* conc]n+ ! similitude ( ^! =ms to ! univ]sality ( ?+s4 ,b ,euclid1 inde$1 has prop]ly deliv]$ ! pres5t def9ition ( a l9e2 by :i* he %ews t a "r l9e alone occupies a space equal to t :i* is situat$ betwe5 its po9ts3 = z mu* z is ! di/ance ( "o po9t f ano!r1 s great is ! magnitude ( ! l9es t]m9at$ by ! po9ts4 ,& ?is is ! mean+ ( be+ equally situat$ betwe5 its extremes4 ,= if y take two po9ts 9 a circumf]5ce1 or 9 any o!r c]ta9 l9e1 ! space ( l9e :i* is includ$ betwe5 ^!1 exce$s _! di/ance f ea* o!r2 & e l9e1 besides a "r "o1 appe>s to suff] ?is prop]ty4 ,h5ce1 accord+ to a common conception1 ! vulg> also say1 t he :o walks by a "r l9e1 p]=ms only a necess>y j\rney3 b t !y necess>ily w&] mu*1 :o d n proce$ 9 a "r l9e4 ,b ,plato ?us def9es x2 a "r l9e is t ^: middle p>ts d>k5 its extremes4 ,= ?is passion necess>ily att5ds ?+s :i* h a direct position2 b x is n necess>y t ?+s situat$ 9 ! circumf]5ce ( a circle1 or 9 ano!r int]val1 %\ld be endu$ ) ?is prop]ty4 ,h5ce1 ! a/rolog]s also say1 t ! sun !n suff]s an eclipse :5 t lum9>y1 ! moon1 & our eye are 9 "o "r l9e2 = x is !n d>k5$ ?r\gh ! middle position ( ! moon betwe5 u & its orb4 ,& p]haps1 ! passion ( a "r l9e w ev9ce1 t 9 ! ?+s :i* are1 accord+ to processions emanat+ f causes1 ! m$iums are endu$ ) a p[] ( divid+ ! di/ance ( ! extremes1 & _! mutual communication ) ea* o!r4 ,z also1 accord+ to regressions1 su* ?+s z are di/ant f ! extremes1 are conv]t$ by m$iums to _! prim>y causes4 ,b ,ar*im$es def9es a "r l9e ! lea/ ( ?+s hav+ ! same b\nds4 ,= s9ce1 accord+ to ,euclid1 a "r l9e is equally situat$ betwe5 its po9ts1 x is on ?is acc\nt1 ! lea/ ( ?+s hav+ ! same b\nds3 = if a less l9e c\ld be giv51 x w\ld n lie equally betwe5 its extremes3 b all ! o!r def9itions ( a "r l9e1 fall into ! same conclusions2 z = in/ance1 t x is con/itut$ 9 its extremities1 & t "o "p ( x is n 9 its subject plane1 b ano!r1 9 "o m sublime3 & t all its p>ts simil>ly agree to all3 & t its extremes abid+1 x also abides4 ,la/ly1 t x does n p]fect figure1 ) "o l9e simil> 9 species to itself3 = all ^! def9itions express ! prop]ty ( a "r l9e1 :i* x possesses f ! simplicity ( its ess5ce1 & f its hav+ "o progression ! %orte/ ( all f "o extremity to ano!r4 ,& ?us mu* conc]n+ ! def9itions ( a "r l9e4 ,b aga91 ,gem9us divides a l9e fir/ into an incomposite & composite2 call+ a composite1 t :i* is refract$1 & =ms an angle2 b all ! re/ ( !m1 he d5om9ates incomposites4 ,aft]w>ds1 he divides a composite l9e into t :i* produces figure1 & t :i* may be inf9itely ext5d$4 ,& he calls t :i* produces figure1 a circul> l9e1 & ! l9e ( a %ield#ade1 & t :i* is simil> to an ivy leaf#adf2 b t :i* is n effective ( figure1 ! section ( a rectangul> & obtuse angul> cone1 ! l9e simil> to a %ell#adg1 ! "r l9e1 & all ( t k9d4 ,& aga91 aft] ano!r mann]1 ( ! incomposite l9e1 "o sort is simple1 b ! o!r mixt4 ,& ( ! simple1 "o produces figure1 z ! circul>2 b ! o!r is indef9ite1 z ! "r l9e4 ,b ( ! mixt1 "o subsi/s 9 planes1 b ! o!r 9 solids4 ,& ( t :i* is 9 planes1 "o co9cides 9 itself1 z ! figure ( ! ivy leaf1 :i* is call$ ! cissoid2 b ! o!r may be produc$ 9 inf9itum1 z ! helix4 ,b ( t :i* is 9 solids1 "o may be consid]$ 9 ! sections ( solids2 b ! o!r z consi/+ ab\t ! solids !mselves4 ,= ! helix1 inde$1 :i* is describ$ ab\t a sph]e or a cone1 consi/s ab\t solids2 b conic1 or spirical sections are g5]at$ f a p>ticul> section ( solids4 ,b1 ) respect to ^! sections1 ! conic 7 inv5t$ by ,mne*mus1 :i* also ,erato/h5es relat+1 says1 8,nor 9 a cone ,mne*mian t]n>ies divide40 ,b ! spiric by ,p]seus1 :o compos$ an epigram on _! inv5tion1 to ?is purpose1 8,:5 ,p]seus _h inv5t$ ?ree spiral l9es 9 five sections1 he sacrific$ to ! gods on ! occasion40 ,& ! ?ree sections ( a cone1 are ! p>abola1 hyp]bola1 & ellipsis3 b ( spiral sections1 "o k9d is twi/$ & involv$1 l ! fetlock ( a horse2 b ano!r is dilat$ 9 ! middle1 & defici5t 9 ea* extremity3 & ano!r :i* is oblong1 has less space 9 ! middle1 b is dilat$ on ea* side4 ,b ! multitude ( ! o!r mixt l9es is inf9ite4 ,= !re is an innum]able multitude ( solid figures1 f :i* !re are con/itut$ multi=m sections4 ,= a "r l9e1 :ile x is circul>ly mov$#adh1 does n make a c]ta9 det]m9ate sup]ficies1 nor yet conical1 nor con*oidal l9es1 nor circumf]5ces !mselves4 ,h5ce1 if ^! solids are multif>i\sly cut1 !y w exhibit v>i\s species ( l9es4 ,la/ly1 ( ^? l9es :i* consi/ ab\t solids1 "s are ( simil> p>ts1 z ! helixes ab\t a cyl9d]2 b o!rs ( dissimil> p>ts1 z all ! re/4 ,f ^! divisions1 !re=e1 we may collect1 t !re are only ?ree l9es ( simil> p>ts1 ! "r1 ! circul>1 & ! cyl9dric helix4 ,! two simple ones1 inde$1 exi/+ 9 a plane1 b ! "o mixt1 ab\t a solid4 ,& ?is ,gem9us evid5tly demon/rates1 :5 he %ews1 t if two "r l9es are ext5d$ f "o po9t1 to a l9e ( simil> p>ts1 s z to make equal angles ^u t l9e1 !y %all be equal to ea* o!r4 ,& ! demon/rations ( ?is may be receiv$ by ! /udi\s1 f 8 volumes2 s9ce 9 ^! he deliv]s ! orig9 ( spiral1 con*oidal1 & cissoidal l9es4 ,b we h b>ely relat$ ! names & divisions ( ^! l9es1 = ! purpose ( excit+ ! +5i\s to _! inve/igation2 z we ?9k1 t an accurate enquiry aft] ! me?od ( detect+ ! prop]ties ( ea*1 w\ld be sup]flu\s 9 ! pres5t und]tak+3 s9ce ! geometrician only unfolds to u 9 ?is "w1 simple & prim>y l9es1 i4e4 ! "r l9e1 9 ! pres5t def9ition2 b a circul> l9e1 9 ! tradition ( a circle4 ,= he !n says1 t ! l9e t]m9at+ ! circle1 is ! circumf]5ce4 ,b he makes no m5tion ( mixt l9es1 ?\gh he 0 well acqua9t$ ) mixt angles1 ,i mean1 ! semi-circul> & cornicul>3 z also ) plane mixt figures1 i4e4 segm5ts & sectors2 & ) solids1 viz4 cones & cyl9d]s4 ,( ea* ( ! re/1 !re=e1 he deliv]s ?ree species2 b ( l9es only two1 i4 e4 ! "r & circul>3 = he ?\ght x requisite 9 disc\rses conc]n+ ?+s simple1 to assume simple species2 & all ! re/ are m composite ?an l9es4 ,h5ce1 9 imitation ( ! geometrician1 we also %all t]m9ate _! explanation ) simple l9es4 ,,def9ition ,v4 ,a ,,sup]ficies is t :i* has only ,l5g? & ,bread?4 ,aft] a po9t & a l9e1 a sup]ficies is plac$1 :i* is di/ant by a tw(old int]val1 l5g? & bread?4 ,b ?is also rema9+ de/itute ( ?ickness or bulk1 possesses a nature m simple ?an body1 :i* is di/ant by a triple dim5sion4 ,on :i* acc\nt ! geometrician adds to ! two int]vals ! p>ticle only1 because ! ?ird int]val does n exi/ 9 sup]ficies4 ,& ?is is equival5t to a negation ( bulk1 z "h also he %ews ! excell5cy ( sup]ficies comp>$ to a solid ) respect to simplicity1 by negation1 or by an addition equival5t to negation3 b ! dim9ution :i* x possesses1 if comp>$ ) ! prec$+ t]ms1 by ! affirmations !mselves4 ,b o!rs def9e a sup]ficies to be ! b\nd>y ( body1 :i* is almo/ affirm+ ! same z ! def9ition ( ,euclid2 s9ce t :i* t]m9ates is exce$$ 9 "o dim5sion1 by t :i* is t]m9at$4 ,& o!rs1 a magnitude diff]5t by two int]vals4 ,la/ly1 o!rs decl>+ ! same affection1 =m its assignation 9 a some:at diff]5t mann]4 ,b !y say we h a k ( sup]ficies :5 we measure fields1 & di/+ui% _! extremities accord+ to l5g? & bread?2 b t we receive a c]ta9 s5sation ( x1 :5 we behold %ad[s4 ,= z !y are )\t bulk1 because !y _c p5etrate into ! int]ior "p ( ! e>?1 !y h only l5g? & bread?4 ,b ! ,py?agoreans say1 t x is assimilat$ to ! triad2 because ! t]n>y is by f> ! fir/ cause to all ! figures2 :i* a sup]ficies conta9s4 ,= a circle1 :i* is ! pr9ciple ( orbicul> figures1 occultly possesses ! t]n>y1 by its c5tre1 int]val1 & circumf]5ce4 ,b a triangle1 :i* ranks z ! fir/ among all "r-l9$ figures1 on e side ev9ces t x is enclos$ by ! triad1 & receives its =m f its p]fect nature4 ,,def9ition ,,vi4 ,! ,extremities ( a ,,sup]ficies are ,l9es4 ,f ^! also1 z images1 we may und]/&1 t ?+s m simple procure b\nd & an end to e "o ( _! proximate natures3 = s\l p]fects & det]m9es ! op]ations ( nature2 & nature ! motion ( bodies4 ,& prior to ^!1 intellect measures ! convolutions ( s\l2 & unity ! life ( intellect2 = t is ! measure ( all4 ,j z 9 ^! also1 a solid is t]m9at$ by a sup]ficies2 b a sup]ficies by a l9e2 & a l9e by a po9t2 = t is ! b\nd>y ( !m all4 ,h5ce1 ! l9e exi/+ uni=mly 9 immat]ial =ms & imp>tible reasons1 t]m9ates & re/ra9s ! v>i\s motion ( a sup]ficies 9 its progression1 & proximately unites its inf9ity4 ,b 9 ! images ( ^!1 :5 t :i* b\nds sup]v5es t :i* is b\nd$1 x causes1 by ?is means1 its limitation & b\nd4 ,b if x %\ld be enquir$ h[ l9es are ! extremities ( e sup]ficies1 s9ce !y are n ! extremes ( e f9ite figure2 = ! sup]ficies ( a sph]e is t]m9at$ inde$1 yet n by l9es1 b by itself8 ,9 answ] to ?is1 we mu/ say1 t by receiv+ a sup]ficies s f> z x is di/ant by a tw(old int]val1 we %all f9d x t]m9at$ by l9es accord+ to l5g? & bread?4 ,b if we behold a sph]ical sup]ficies1 we mu/ receive x z t :i* is endu$ ) figure2 :i* possesses ano!r quality1 & conjo9s ! end ) ! beg9n+2 & loses its two extremities 9 ! compreh5sive embraces ( "o3 & ?is "o extremity subsi/s 9 capacity only1 & n 9 en]gy4 ,,def9ition ,,vii4 ,a ,plane ,,sup]ficies is t :i* is equally situat$ betwe5 its b\nd+ ,l9es4 ,x 0 n agreeable to ! anci5t philosoph]s to e/abli% a plane species ( sup]ficies2 b !y consid]$ sup]ficies 9 g5]al1 z ! repres5tative ( magnitude1 :i* is di/ant by a tw(old int]val4 ,= ?us ! div9e ,plato#adi says1 t geometry is contemplative ( planes1 oppos+ x 9 division to /]eometry1 z if a plane & a sup]ficies 7 ! same4 ,& ?is 0 likewise ! op9ion ( ! demoniacal ,ari/otle#aej4 ,b ,euclid & 8 foll[]s consid] sup]ficies z a g5us1 b a plane z its species1 9 ! same mann] z rectitude ( a l9e4 ,& on ?is acc\nt he def9es a plane sep>ate f a sup]ficies1 aft] ! similitude ( a "r l9e4 ,= he def9es ?is la/ z equal to ! space1 plac$ betwe5 its po9ts4 ,& 9 l mann]1 he says1 t two "r l9es be+ giv51 a plane sup]ficies occupies a place equal to ! space situat$ betwe5 ^? two l9es4 ,= ?is is equally situat$ betwe5 its l9es2 & o!rs also expla9+ ! same b\nd>y1 ass]t t x is con/itut$ 9 its extremities4 ,b o!rs def9e x z t to all ! p>ts ( :i* a "r l9e may be adapt$#aea4 ,b p]haps o!rs w say1 t x is ! %orte/ ( sup]ficies1 hav+ ! same b\nd>ies2 & t its middle p>ts d>k5 its extremities2 & t all ! def9itions ( a "r l9e may be transf]r$ into a plane sup]ficies1 by only *ang+ ! g5us3 s9ce a "r1 circul>1 & mixt l9e1 comm5c+ f l9es1 arrive ev5 at solids1 z we h ass]t$ above2 = !y are proportionally1 bo? 9 sup]ficies & solids4 ,h5ce also1 ,p>m5ides says1 t e figure is ei!r "r1 or circul>1 or mixt4 ,b if y wi% to consid] ! "r 9 sup]ficies1 take a plane1 to :i* a "r l9e agrees 9 v>i\s ways2 b if a circul> receive a sph]ical sup]ficies2 & if a mixt1 a conic or cyl9dric1 or "s "o ( t g5us4 ,b x is requisite 7says ,gem9us7 s9ce a l9e1 & also a sup]ficies is call$ mixt1 to "k ! measure ( mixture1 because x is v>i\s4 ,= mixture 9 l9es1 is nei!r by composition1 nor by temp]am5t only3 s9ce1 inde$1 a helix is mix$1 yet "o "p ( x is n /raight1 & ano!r "p circul>1 l ^? ?+s :i* are mix$ by composition3 nor if a helix is cut aft] any mann]1 does x exhibit an image ( ?+s simple1 su* z ^? :i* are mix$ ?r\gh temp]am5t2 b 9 ^! ! extremes are1 at ! same "t1 corrupt$ & confus$4 ,h5ce1 ,!odorus ! ma!matician1 does n rightly p]ceive1 9 ?9k+ t ?is mixture is 9 l9es4 ,b mixture 9 sup]ficies1 is nei!r by composition1 nor by confusion2 b subsi/s r by a c]ta9 temp]am5t4 ,= conceiv+ a circle 9 a subject plane1 & a po9t on high1 & produc+ a "r l9e f ! po9t to ! circumf]5ce ( ! circle1 ! revolution ( ?is l9e w produce a conical sup]ficies :i* is mixt4 ,& we aga9 resolve x into its simple elem5ts1 by a p>allel section3 = by draw+ a section betwe5 ! v]tex & ! base1 :i* %all cut ! plane ( ! g5]ative "r l9e1 we effect a circul> l9e4 ,b ! idea ( l9es1 %ews t ! mode ( mixture is n by temp]am5t2 = nei!r does x s5d u back to ! simple nature ( elem5ts3 on ! contr>y1 :5 sup]ficies are cut1 !y imm$iately exhibit to u _! produc+ l9es4 ,! mode ( mixture1 !re=e1 is n ! same 9 l9es & sup]ficies4 ,b z among l9es !re 7 "s simple1 t is1 ! "r & circul>1 ( :i* ! vulg> also possess an anticipat$ k )\t any previ\s in/ruction2 b ! species ( mixt l9es require a m artificial appreh5sion3 s among sup]ficies1 we possess an innate notion ( ^? :i* are especially elem5t>y1 ! plane & sph]ical2 b sci5ce & its reason inve/igates ! v>iety ( ^? :i* are compos$ ?r\gh mixture4 ,b ?is is an admirable prop]ty ( sup]ficies1 t _! mixture 9 g5]ation is (t5times produc$ f a circul> l9e2 & ?is also happ5s to a spiral sup]ficies4 ,= ?is is und]/ood by ! revolution ( a circle rema9+ erect1 & turn+ itself ab\t ! same po9t :i* is n its c5tre4 ,& on ?is acc\nt1 a spiral also is ?reefold2 = its c5tre is ei!r 9 a circumf]5ce1 or )91 or ext]nal to a circumf]5ce4 ,if ! c5tre is 9 ! circumf]5ce1 a cont9u$ spiral is produc$3 if )9 ! circumf]5ce1 an intangl$ "o2 if )\t1 a divid$ "o4 ,& !re are ?ree spiral sections correspond+ to ^! ?ree diff]5ces4 ,b e spiral l9e is mixt1 al?\gh ! motion f :i* x is produc$ is "o & circul>4 ,& mixt sup]ficies are produc$ z well f simple l9es1 7z we h said17 :ile !y are mov$ ) a motion ( ?is k9d1 z f mixt l9es4 ,s9ce1 !re=e1 !re are ?ree conic l9es1 !y produce f\r mixt sup]ficies1 :i* !y call conoids4 ,= a rectangul> conoid1 is produc$ f ! revolution ( ! p>abola ab\t its axis3 b t :i* is =m$ by ! ellipsis1 is call$ a sph]oid2 & is ! revolution is made ab\t ! great] axis1 x is an oblong2 b if ab\t ! less] a broad sph]oid4 ,la/ly1 an obtuse-angl$ conoid is g5]at$ f ! revolution ( ! hyp]bola4 ,b x is requisite to "k1 t sometimes we arrive at ! k ( sup]ficies f l9es1 & sometimes ! contr>y2 = f conical & spiral sup]ficies1 we appreh5d conical & spiral l9es4 ,besides1 ?is also mu/ be previ\sly receiv$ conc]n+ ! diff]5ce ( l9es & sup]ficies1 t !re are ?ree l9es ( simil> p>ts 7z we h already obs]v$71 b only two sup]ficies1 ! plane & ! sph]ical4 ,= ?is is n true ( ! cyl9dric1 s9ce all p>ts ( ! cyl9dric sup]ficies _c agree to all4 ,& ?us mu* conc]n+ ! diff]5ces ( sup]ficies1 "o ( :i* ! geometrician hav+ *os5 7,i mean ! plane71 ?is also he has def9$2 & 9 ?is1 z a subject1 he contemplates figures1 & _! att5dant passions3 = 8 disc\rse is m copi\s 9 ?is ?an 9 o!r sup]ficies3 s9ce1 inde$1 we may und]/& "r l9es1 & circles1 & helixes 9 a plane2 also ! sections ( circles & "r l9es1 contacts1 & applications1 & ! con/ructions ( angles ( e k9d4 ,b 9 o!r sup]ficies1 all ^! _c be beheld4 ,= h[ 9 "o t is sph]ical1 c we appreh5d a "r l9e1 or a "r-l9$ angle8 ,h[1 la/ly1 9 a conic or cyl9dric sup]ficies1 c we behold sections ( circles or "r l9es8 ,n undes]v$ly1 !re=e1 does he bo? def9e ?is sup]ficies1 & discuss 8 geometrical conc]ns1 by exhibit+ e ?+ 9 ?is z 9 a subject2 = f h5ce he calls ! pres5t treatise plane4 ,&1 aft] ?is mann]1 x is requisite to und]/& t :i* is plane1 z project$ & con/itut$ be=e ! eyes3 b cogitation z describ+ all ?+s 9 ?is1 ! phantasy correspond+ to a plane mirror1 & ! reasons resid5t 9 cogitation z dropp+ _! images#aeb into its %ad[y receptacle4 ,,def9ition ,,viii4 #aec;,a ,,plane ,,angle1 is ! incl9ation ( two ,l9es to ea* o!r 9 a ,plane1 :i* meet toge!r1 b are n 9 ! same direction4 ,"s ( ! anci5t philosoph]s1 plac+ an angle 9 ! pr$icam5t ( relation1 h said1 t x is ! mutual incl9ation ( l9es or planes to ea* o!r4 ,b o!rs1 includ+ ?is 9 quality1 z well z rectitude & obliquity1 say1 t x is a c]ta9 passion ( a sup]ficies or a solid4 ,& o!rs1 ref]r+ x to quantity1 confess t x is sup]ficies or a solid4 ,= ! angle :i* subsi/s 9 sup]ficies is divid$ by a l9e2 b t :i* is 9 solids1 by a sup]ficies4 ,b 7say !y7 t :i* is divid$ by ^!1 is no o!r ?an magnitude1 & ?is is n l9e>1 s9ce a l9e is divid$ by a po9t2 & !re=e x foll[s t x mu/ be ei!r a sup]ficies or a solid4 ,b if x is magnitude1 & all f9ite magnitudes ( ! same k9d h a mutual proportion2 all angles ( ! same k9d1 i4 e4 :i* subsi/ 9 sup]ficies1 w h a mutual proportion4 ,& h5ce1 ! cornicul> w be proportionable to a "r-l9$ angle4 ,b ?+s :i* h a mutual proportion1 may1 by multiplication1 exce$ ea* o!r2 & !re=e x may be possible = ! cornicul> to exce$ a "r-l9$ angle1 :i*1 x is well kn[n1 is impossible1 s9ce x is %ewn to be less ?an e "r-l9$ angle4 ,b if x is quality alone1 l heat & cold1 h[ is x divisible into equal p>ts8 ,= equality1 inequality1 & divisibility1 are n less resid5t 9 angles ?an 9 magnitudes2 b !y are1 9 l mann]1 ess5tial4 ,b if ! ?+s 9 :i* ^! are ess5tially inh]5t1 are quantities1 & n qualities1 x is manife/ t angles also are n qualities4 ,s9ce ! m & ! less are ! prop] passions ( quality#aed1 b n equal & unequal4 ,on ?is hypo!sis1 !re=e1 angles ought n to be call$ unequal1 & ?is great]1 b ! o!r less2 b !y ought to be d5om9at$ dissimil>s1 & "o m an angle1 b ! o!r less4 ,b t ^! appellations are =eign f ! ess5ce ( ma!matical conc]ns1 is obvi\s to e "o3 = e angle receives ! same def9ition1 nor is ?is m an angle1 b t less4 ,?irdly1 if an angle is incl9ation1 & belongs to ! category ( relation1 x mu/ foll[1 t f ! exi/5ce ( "o incl9ation1 !re w also be "o angle1 & n m ?an "o4 ,= if x is no?+ else ?an ! relation ( l9es or planes1 h[ is x possible !re c be "o relation ( l9es or planes1 b _m angles8 ,if1 !re=e1 we conceive a cone cut by a triangle f ! v]tex to ! base1 we %all behold "o incl9ation ( ! triangul> l9es 9 ! semicone to ! v]tex2 b two di/9ct angles3 "o ( :i* is plane1 ,i mean t ( ! triangle2 b ! o!r subsi/s 9 ! mixt sup]ficies ( ! cone1 & bo? are compreh5d$ by ! two triangul> l9es4 ,! relation1 !re=e1 ( ^!1 d n make ! angle4 ,aga91 if is necess>y to call an angle ei!r quality or quantity1 or relation2 = figures1 inde$1 are qualities1 b _! mutual proportions belong to relation4 ,x is necess>y1 !re=e1 t an angle %\ld be r$uc$ "u "o ( ^! ?ree g5]a4 ,su* d\bts1 !n1 aris+ conc]n+ an angle1 & ,euclid call+ x incl9ation1 b ,apollonius ! collection ( a sup]ficies1 or a solid 9 "o po9t1 "u a refract$ l9e or sup]ficies 7= he seems to def9e e angle univ]sally71 we %all affirm1 agreeable to ! s5tim5ts ( our preceptor ,syrianus1 t an angle is ( itself none ( ! a=esaid2 b is con/itut$ f ! concurr5ce ( !m all4 ,& t1 on ?is acc\nt1 a d\bt arises among ^? :o reg>d "o category alone4 ,b ?is is n peculi> to an angle1 b is likewise ! prop]ty ( a triangle4 ,= ?is1 too1 p>ticipates ( quantity1 & is call$ equal & unequal2 because x has to quantity ! proportion ( matt]4 ,b quality also1 is pres5t ) ?is1 9 consequ5ce ( its figure 7s9ce triangles are call$ z well simil> z equal72 b x possesses ?is f "o category1 & t f ano!r4 ,h5ce1 an angle is p]fectly indig5t ( quantity1 ! subject ( magnitude4 ,b x is also indig5t ( quality1 by :i* x possesses1 z x 71 its prop] =m & figure1 ,la/ly1 x is indig5t ( ! relation ( l9es t]m9at+1 or ( sup]ficies compreh5d+ its =m4 ,s t an angle consi/s f all ^!1 yet is n any "o ( !m 9 p>ticul>4 ,& x is inde$ divisible1 & capable ( receiv+ equality & inequality1 accord+ to ! quantity :i* x conta9s4 ,b x is n compell$ to admit ! proportion ( magnitudes ( ! same k9d1 s9ce x has also a peculi> quantity1 by :i* angles are also incapable ( a comp>ison ) ea* o!r4 ,nor c "o incl9ation p]fect "o angle3 s9ce ! quantity also1 :i* is plac$ betwe5 ! incl9$ l9es1 completes its ess5ce4 ,if !n we reg>d ^! di/9ctions1 we %all dissolve all absurdities1 & discov] t ! prop]ty ( an angle is n ! collection ( a sup]ficies or solid1 accord+ to ,apollonius 7s9ce ^! also complete its ess5ce17 b t x is no?+ else ?an a sup]ficies itself1 collect$ into "o po9t1 & compreh5d$ by incl9$ l9es1 or by "o l9e incl9$ to itself3 & t a solid angle is ! collection ( sup]ficies mutually incl9$ to ea* o!r4 ,h5ce1 we %all f9d t a =m$ quantum1 con/itut$ 9 a c]ta9 relation1 supplies its p]fect def9ition4 ,& ?us mu* we h ?\ght requisite to ass]t conc]n+ ! sub/ance ( angles1 previ\sly contemplat+ ! common ess5ce ( e triangle1 be=e we divide x into species4 ,b s9ce !re are ?ree op9ions ( an angle1 ,eudemus ! ,p]ipatetic1 :o compos$ a book conc]n+ an angle1 affirms t x is quality4 ,=1 consid]+ ! orig9 ( an angle1 he says t x is no?+ else ?an ! fraction ( l9es3 because1 if rectitude is quality1 fraction also w be quality4 ,& h5ce1 s9ce its g5]ation is 9 quality1 an angle w be entirely quality4 ,b ,euclid1 & ^? :o call x incl9ation1 place x 9 ! category ( relation4 ,b !y call x quantity1 :o say t x is ! fir/ int]val "u a po9t1 t is imm$iately subsi/+ aft] a po9t4 ,9 ! numb] ( :i* is ,plut>*1 :o con/ra9s ,apollonius also into ! same op9ion4 ,= x is requisite 7says he7 !re %\ld be "s fir/ int]val1 "u ! incl9ation ( conta9+ l9es or sup]ficies4 ,b s9ce ! int]val1 :i* is "u a po9t1 is cont9u\s1 x is n possible t a fir/ int]val c be assum$2 s9ce e int]val is divisible 9 inf9itum4 ,besides1 if we any h[ di/+ui% a fir/ int]val1 & ?r\gh x draw a "r l9e1 a triangle is produc$1 & n "o angle4 ,b ,c>pus ,antio*5us says1 t an angle is quantity1 & is ! di/ance ( its compreh5d+ l9es1 or sup]ficies2 & t ?is is di/ant by "o int]val1 & yet an angle is n on t acc\nt a l9e3 s9ce x is n true t e ?+ :i* is di/ant by only "o int]val1 is a l9e4 ,b ?is surely is ! mo/ absurd ( all1 t !re %\ld be any magnitude except a l9e1 :i* is di/ant only by "o int]val4 ,& ?us mu* conc]n+ ! nature ( an angle4 ,b ) respect to ! division ( angles1 "s consi/ 9 sup]ficies1 b o!rs 9 solids4 ,& ( ^? :i* are 9 sup]ficies1 "s are 9 simple ones1 b o!rs 9 su* z are mixt4 ,= an angle may be produc$ 9 a cyl9dric1 conic1 sph]ical1 & plane sup]ficies4 ,b ( ^? :i* consi/ 9 simple sup]ficies1 "s are con/itut$ 9 ! sph]ical2 b o!rs 9 ! plane4 ,= ! zodiac itself =ms angles1 divid+ ! equ9octial 9 two p>ts1 at ! v]tex ( ! cutt+ sup]ficies4 ,& angles ( ?is k9d subsi/ 9 a sph]ical sup]ficies4 ,b ( ^? :i* are 9 planes1 "s are compreh5d$ by simple l9es1 o!rs by mixt ones2 & o!rs1 aga91 by bo?4 ,= 9 ! %ield-l figure#aee1 an angle is compreh5d$ by ! axis1 & ! l9e ( ! %ield3 b "o ( ^! l9es is mixt1 & ! o!r simple4 ,b if a circle cuts ! %ield1 ! angle w be compreh5d$ by ! circumf]5ce1 & ! ellipsis4 ,& :5 cissoids1 or l9es simil> to an ivy leaf1 clos+ 9 "o po9t l ! leaves ( ivy 7f :5ce !y d]ive _! appellation7 make an angle1 su* an angle is compreh5d$ by mixt l9es4 ,also1 :5 ! hippop$e1 or l9e famili> to ! foot ( a m>e1 :i* is "o ( ! spirals1 incl9+ to ano!r l9e1 =ms an angle1 x is compreh5d$ by mixt l9es4 ,la/ly1 ! angles conta9$ by a circumf]5ce & a "r l9e1 are compreh5d$ by simple l9es4 ,b ( ^! aga91 "s are conta9$ by su* z are simil> 9 species1 b o!rs by su* z are dissimil>4 ,= two circumf]5ces1 mutually cutt+1 or t\*+ ea* o!r1 produce angles3 & ^! triple1 = !y are ei!r on bo? sides convex1 :5 ! convexities ( ! circumf]5ces are ext]nal3 or on bo? sides concave1 :5 bo? ! concavities are ext]nal2 :i* !y call si/roides2 or mixt f convex & concave l9es1 z ! l9es call$ lunulas4 ,b besides ?is1 angles are conta9$ 9 a tw(old mann]1 by a "r l9e & a circumf]5ce3 = !y are ei!r conta9$ by a "r l9e1 & a concave circumf]5ce1 z ! semi-circul> angle2 or by a "r l9e & a convex circumf]5ce1 z ! cornicul> angle4 ,b all ^? :i* are compreh5d$ by two "r l9es1 are call$ rectil9e> angles1 :i* h likewise a triple diff]5ce#aef4 ,! geometrician1 !re=e1 9 ! pres5t hypo!sis1 def9es all ^? angles :i* are con/itut$ 9 plane sup]ficies1 & gives !m ! common "n ( a plane angle4 ,& ! g5us ( ^! he d5om9ates incl9ation3 b ! place1 ! plane itself1 = angles h position3 b _! orig9 su*1 t x is requisite !re %\ld be two l9es at lea/1 & n ?ree z 9 a solid4 ,& t ^! %\ld t\* ea* o!r1 & by t\*+1 mu/ n lie 9 a "r l9e1 z an angle is ! incl9ation & compreh5sion ( l9es3 b is n di/ance only1 accord+ to "o int]val4 ,b if we exam9e ?is def9ition1 9 ! fir/ place x appe>s t x does n admit1 an angle c be p]fect$ by "o l9e2 ?\gh a cissoid1 :i* is b "o1 p]fects an angle4 ,&1 9 l mann]1 ! hippop$e4 ,= we call ! :ole a cissoid1 & n its portions 7le/ any "o %\ld say1 t ! conjunction ( ^! =ms an angle7 & ! :ole a spiral1 b n its p>ts4 ,ea*1 !re=e1 s9ce x is "o1 =ms an angle to itself1 & n to ano!r4 ,b aft] ?is1 he is faulty1 9 def9+ an angle to be incl9ation4 ,= h[1 on ?is hypo!sis1 w !re be two angles1 f "o incl9ation8 ,h[ c we call angles equal & unequal8 ,& :atev] else is usually object$ aga9/ ?is op9ion4 ,?irdly1 & la/ly1 t "p ( ! def9ition1 :i* says1 & n plac$ 9 a "r l9e1 is sup]flu\s 9 c]ta9 angles1 z 9 ^? :i* are =m$ f orbicul> l9es4 ,= )\t ! assi/ance ( ?is "p1 ! def9ition is p]fect2 s9ce ! incl9ation ( "o ( ! l9es to ! o!r1 =ms ! angle4 ,& x is n possible t orbicul> angles %\ld be plac$ 9 a "r l9e4 ,& ?us mu* we h ?\ght prop] to say conc]n+ ! def9ition ( ,euclid2 p>tly1 inde$1 int]pret+1 & p>tly d\bt+ its tru?4 ,,def9ition ,,ix4 ,b :5 ! ,l9es conta9+ ! ,angle1 are "r1 ! ,angle is call$ ,,rectil9e>4 ,an angle is ! symbol & image ( ! connection & compression1 :i* subsi/s 9 ! div9e g5]a1 & ( t ord] :i* collects divisibles into "o1 p>ticles into an imp>tible nature1 & ! _m into conciliat+ community4 ,= x is ! bond ( a multitude ( l9es & sup]ficies1 ! collector ( magnitude into ! imp>tibility ( po9ts1 & ! compreh5d] ( e figure :i* is compos$ by its conf9+ nature4 ,on :i* acc\nt1 ! oracles#aeg call ! angul> junctions ( figures1 knots1 s f> z !y br+ ) !m an image ( connect+ union1 & div9e conjunctions1 by :i* discrete natures mutually coh]e ) ea* o!r4 ,! angles1 !re=e1 subsi/+ 9 sup]ficies1 express ! m immat]ial1 simple1 & p]fect unions :i* sup]ficies conta93 b ^? :i* are 9 solids1 repres5t ! unions1 :i* proce$ ev5 to inf]iors1 & supply a community to ?+s disjunct1 & a con/ruction ( ! same nature1 to ?+s :i* on e side receive a p]fect p>tition4 ,b ( ! angles 9 sup]ficies1 "s %ad[ =? prim>y & unmixt unions2 b o!rs1 su* z compreh5d 9 !mselves1 an inf9ity ( progressions4 ,& "s1 inde$1 are ! s\rces ( union to intellectual =ms2 b o!rs1 to s5sible reasons2 & o!rs1 aga91 are copulative ( ^? =ms :i* obta9 betwe5 ^!1 a middle situation4 ,h5ce1 ! angles :i* are made f circumf]5ces1 imitate ^? causes :i* envelop intellectual v>iety 9 co]cive union2 = circumf]5ces1 ha/5+ to coalesce ) ea* o!r1 are images ( intellect1 & intellectual =ms4 ,on ! contr>y1 rectil9eal angles1 are ! symbols ( ^? unions :i* preside ov] s5sibles1 & af=d a conjunction ( ! reasons subsi/+ 9 ^!3 b mixt angles repres5t ! pres]v]s ( ! communion1 z well ( s5sible1 z ( intellectual =ms1 accord+ to "o immoveable union4 ,x is requisite1 !re=e1 by reg>d+ ^! p>adigms1 or exempl>s1 to r5d] ! causes ( ea*4 ,= among ! ,py?agoreans we %all f9d v>i\s angles d$icat$ to v>i\s gods4 ,?us1 ,philolaus1 consecrates to "s a triangul>1 b to o!rs a quadrangul> angle2 & to o!rs1 aga91 diff]5t angles4 ,likewise1 he p]mits ! same to _m gods1 & _m to ! same god1 accord+ to ! diff]5t p[]s :i* !y conta94 ,& ) a view to ?is1 & to ! demiurgic triangle1 :i* is ! prim>y cause ( all ! ornam5t ( ! elem5ts1 x appe>s to me1 t ,!odorus ,as9u ! philosoph]1 con/itutes "s ( ! gods1 accord+ to sides2 b o!rs1 accord+ to angles4 ,! fir/1 inde$1 supply+ progression & p[]2 b ! second1 ! conjunction ( ! univ]se1 & ! collection ( progressive natures aga9 into "o4 ,b ^!1 inde$1 direct u to ! k ( ! ?+s :i* are4 ,& we mu/ n wond] t l9es are "h said to conta9 an angle4 ,= ! "o & imp>tible nature :i* is f\nd 9 ^!1 is adv5titi\s3 b 9 ! gods !mselves1 & 9 true be+s1 ! :ole1 & imp>tible good1 prec$es ?+s _m1 & divid$4 ,,def9ition ,x4 ,:5 a ,"r ,l9e /&+ on a ,"r ,l9e1 makes ! successive ,angles on ea* side equal to "o ano!r1 ea* ( ! equal ,angles1 is a ,"r ,angle2 & ! insi/+ ,"r ,l9e1 is call$ a ,,p]p5dicul> to t ^u :i* x /&s4 ,,def9ition ,,xi4 ,an ,,obtuse ,,angle is t :i* is great] ?an a ,,"r ,,angle4 ,,def9ition ,,xii4 ,b an ,,acute ,,angle1 is t :i* is less ?an a ,,"r ,,angle4 ,^! are ! triple species ( angles1 :i* ,socrates speaks ( 9 ! ,republic1 & :i* are receiv$ by geometricians f hypo!sis2 a "r-l9e con/itut+ ^! angles1 accord+ to a division into species2 ,i mean1 ! "r1 ! obtuse1 & ! acute4 ,! fir/ ( ^! be+ def9$ by equality1 id5tity & similitude2 b ! o!rs be+ compos$ ?r\gh ! nature ( ! great] & ! less]2 & la/ly1 ?r\gh inequality & div]sity1 & ?r\gh ! m & ! less1 indet]m9ately assum$4 ,b _m geometricians1 are unable to r5d] a reason ( ?is division1 & use ! ass]tion1 t !re are ?ree angles1 z an hypo!sis#aeh4 ,s t1 :5 we int]rogate !m conc]n+ its cause1 !y answ]1 ?is is n to be requir$ ( !m z geometricians4 ,h[ev]1 ! ,py?agoreans1 ref]r+ ! solution ( ?is triple di/ribution to pr9ciples1 are n want+ 9 r5d]+ ! causes ( ?is diff]5ce ( "r-l9$ angles4 ,=1 s9ce "o ( ! pr9ciples subsi/s accord+ to b\nd1 & is ! cause ( limitation1 id5tity1 & equality1 & la/ly1 ( ! :ole ( a bett] co-ord9ation3 b ! o!r is ( an inf9ite nature1 & conf]s on its prog5y1 a progression to inf9ity1 increase1 & decrease1 inequality1 & div]sity ( e k9d1 & entirely presides ov] ! worse s]ies2 h5ce1 ) great propriety1 s9ce ! pr9ciples ( a "r-l9$ angle are con/itut$ by ^!1 ! reason proce$+ f b\nd1 produces a "r angle1 "o1 ) respect to ! equality ( e "r angle1 endu$ ) similitude1 always f9ite & det]m9ate1 ev] abid+ ! same1 & nei!r receiv+ increm5t nor decrease4 ,b ! reason proce$+ f inf9ity1 s9ce x is ! second 9 ord]1 & ( a dyadic nature1 produces tw(old angles ab\t ! "r angle1 di/+ui%$ by inequality1 accord+ to ! nature ( ! great] & ! less]1 & possess+ an inf9ite motion1 accord+ to ! m & ! less1 s9ce ! "o becomes m or less obtuse2 b ! o!r m or less acute4 ,h5ce1 9 consequ5ce ( ?is reason1 !y ascribe "r angles to ! pure & immaculate gods ( ! div9e ornam5ts1 & div9e p[]s :i* proce$ into ! univ]se1 z ! au?ors ( ! inv>iable provid5ce ( inf]iors2 = rectitude1 & an inflexibility & immutability to subord9ate natures accords ) ^! gods3 b !y affirm1 t ! obtuse & acute angles %\ld be ascrib$ to ! gods1 :o af=d progression1 & motion1 & a v>iety ( p[]s4 ,s9ce obtus5ess is ! image ( an exp&$ progression ( =ms2 b acut5ess possesses a similitude to ! cause divid+ & mov+ ! univ]se4 ,b likewise1 among ! ?+s :i* are1 rectitude is1 inde$1 simil> to ess5ce1 pres]v+ ! same b\nd ( its be+2 b ! obtuse & acute1 %ad[ =? ! nature ( accid5ts4 ,= ^! receive ! m & ! less1 & are indef9itely *ang$ )\t ceas+4 ,h5ce1 ) great propriety1 !y exhort ! s\l to make h] desc5t into g5]ation1 accord+ to ?is inv>iable species ( ! "r angle1 by n v]g+ to ?is "p m ?an to t2 & by n affect+ "s ?+s m1 & o!rs less4 ,= ! di/ribution ( a c]ta9 conv5i5ce & sympa?y ( nature1 draws x d[n into mat]ial error1 & indef9ite v>iety#aei4 ,a p]p5dicul> l9e is1 !re=e1 ! symbol ( inflexibility1 purity1 immaculate1 & inv>iable p[]1 & e ?+ ( ?is k9d4 ,b x is likewise ! symbol ( div9e & intellectual measure3 s9ce we measure ! altitudes ( figures by a p]p5dicul>1 & def9e o!r rectil9eal angles by _! relation to a "r angle1 z by !mselves !y are indef9ite & indet]m9ate4 ,= !y are beheld subsi/+ 9 excess & defect1 ea* ( :i* is1 by itself1 indef9ite4 ,h5ce !y say1 t virtue also /&s accord+ to rectitude2 b t vice subsi/s accord+ to ! inf9ity ( ! obtuse & acute1 t x produces excesses & defects1 & t ! m & ! less exhibit its immod]ation1 & inord9ate nature4 ,( rectil9eal angles1 !re=e1 we mu/ e/abli% ! "r angle1 z ! image ( p]fection1 & inv>iable en]gy1 ( limitation1 intellectual b\nd1 & ! l2 b ! obtuse & acute1 z %ad[+ =? inf9ite motion1 unceas+ progression1 division1 p>tition & inf9ity4 ,& ?us mu* = ! !ological speculation ( angles4 ,b "h we mu/ take notice1 t ! g5us is to be add$ to ! def9itions ( an obtuse & acute angle2 = ea* is "r-l9$1 & ! "o is great]1 b ! o!r less ?an a "r-angle4 ,b x is n absolutely true1 t e angle :i* is less ?an a "r "o1 is acute4 ,= ! cornicul> is less ?an e "r-angle1 because less ?an an acute "o1 yet is n on ?is acc\nt an acute angle4 ,also1 a semi-circul> is less ?an any "r-angle1 yet is n acute4 ,& ! cause ( ?is prop]ty is because !y are mixt1 & n rectil9eal angles4 ,besides1 _m curve-l9$ angles appe> great] ?an "r-l9$ angles1 yet are n on ?is acc\nt obtuse2 because x is requisite t an obtuse %\ld be a "r-l9$ angle4 ,secondly1 z x 0 ! int5tion ( ,euclid1 to def9e a "r-angle1 he consid]s a "r-l9e /&+ ^u ano!r "r-l9e1 & mak+ ! angles on ea* side equal4 ,b he def9es an obtuse & acute angle1 n f ! incl9ation ( a "r l9e to ei!r "p1 b f _! relation to a "r-angle4 ,= ?is is ! measure ( angles deviat+ f ! "r1 9 ! same mann] z equality ( ?+s unequal4 ,b l9es incl9$ to ei!r "p1 are innum]able1 & n "o alone1 l a p]p5dicul>4 ,b aft] ?is1 :5 he says1 7! angles equal to "o ano!r7 he exhibits to u a specim5 ( ! greate/ geometrical dilig5ce2 s9ce x is possible t angles may be equal to o!rs1 )\t be+ "r4 ,b :5 !y are equal to "o ano!r1 x is necess>y !y %\ld be "r4 ,besides1 ! ^w successive appe>s to me n to be add$ sup]flu\sly1 z "s h improp]ly consid]$ x2 s9ce x exhibits ! reason ( rectitude4 ,= x is on ?is acc\nt t ea* ( ! angles is "r2 because1 :5 !y are successive1 !y are equal4 ,&1 inde$1 ! insi/+ "r-l9e1 on acc\nt ( its inflexibility to ei!r "p1 is ! cause ( equality to bo?1 & ( rectitude to ea*4 ,! cause1 !re=e1 ( ! rectitude ( angles1 is n absolutely mutual equality1 b position 9 a consequ5t ord]1 toge!r ) equality4 ,b1 besides all ?is1 ,i ?9k x "h necess>y to call to m9d1 ! purpose ( our au?or2 ,i mean1 t he disc\rses 9 ?is place1 conc]n+ ! angles consi/+ 9 "o plane4 ,& h5ce1 ?is def9ition is n ( e p]p5dicul>2 b ( t :i* is 9 "o & ! same plane4 ,= x is n 8 pres5t design to def9e a solid angle4 ,z1 !re=e1 he def9es1 9 ?is place1 a plane angle1 s likewise a p]p5dicul> ( ?is k9d4 ,because a solid p]p5dicul> ought n to make "r angles to "o "r-l9e only2 b to all :i* t\* x1 & are conta9$ 9 its subject plane3 = ?is is its necess>y peculi>ity4 ,,def9ition ,,xiii4 ,a ,,b\nd is t :i* is ! ,extremity ( any ?+#afj4 ,a ,b\nd1 9 ?is place1 is n to be ref]r$ to all magnitudes1 = !re is a b\nd & extremity ( a l9e2 b to ! spaces :i* are conta9$ 9 sup]ficies1 & to solid bodies4 ,= he n[ calls a b\nd1 ! ambit :i* t]m9ates & di/+ui%es e space4 ,& a b\nd ( ?is k9d1 he def9es to be an extremity3 b n aft] ! mann] 9 :i* a po9t is call$ ! extremity ( a l9e1 b accord+ to its prop]ty ( includ+ & exclud+ f circumjac5t figures4 ,b ?is "n is prop] to geometry 9 its infant /ate1 by :i* !y measur$ fields1 & pres]v$ _! b\nd>ies di/9ct & )\t confusion1 & f :i* !y arriv$ at ! k ( ! pres5t sci5ce4 ,s9ce1 !re=e1 ,euclid calls ! ext]nal ambit1 a b\nd1 x is n )\t propriety t he1 by ?is means1 def9es ! extremity ( spaces4 ,= by ?is1 e ?+ compreh5d$ is circumscrib$4 ,i say1 = example1 9 a circle1 its b\nd & extremity is ! circumf]5ce2 b itself1 a c]ta9 plane space3 & s ( ! re/4 ,,def9ition ,,xiv4 ,a ,,figure is t :i* is compreh5d$ by "o or m ,b\nd>ies4 ,because figure is pr$icat$ 9 v>i\s ways1 & is divid$ into diff]5t species1 x is requisite1 9 ! fir/ place1 to behold its diff]5ces2 & aft]w>ds to disc\rse conc]n+ t figure :i* is propos$ 9 ?is ,def9ition4 ,!re is1 !n a c]ta9 figure :i* is con/itut$ by mutation1 & is produc$ f passion1 :ile ! recipi5ts ( ! figure are di/urb$1 divid$1 or tak5 away2 :ile !y receive additions1 or are alt]$1 or suff] o!r v>i\s affections4 ,!re is also a figure1 :i* is produc$ by ! pott]'s1 or /atu>y's art1 accord+ to ! pre-exi/5t reason1 :i* art itself conta9s3 art1 inde$1 produc+ ! =m1 b matt] receiv+ f !nce1 =m1 & beauty1 & elegance4 ,b !re are /ill m noble & m illu/ri\s figures ?an ^!1 ! skilful op]ations ( nature4 ,"s1 inde$1 exi/+ 9 ! elem5ts "u ! moon#afa1 & hav+ a p[] ( compreh5d+ ! reasons ^? elem5ts conta93 b o!rs are situat$ 9 ! cele/ial regions1 di/+ui%+ _! p[]s1 & endless revolutions4 ,= ! heav5ly bodies1 bo? :5 consid]$ by !mselves1 & ) relation to ea* o!r1 exhibit an abundant & admirable v>iety ( figures2 & at diff]5t times !y pres5t to our view diff]5t =ms1 br++ ) !m a spl5did image ( intellectual species2 &1 by _! elegant & h>moni\s revolutions1 describ+ ! incorporeal & immat]ial p[]s ( figures4 ,b !re are1 aga91 besides all ^!1 mo/ pure & p]fect beauties1 ! figures ( s\ls1 :i*1 because !y are full ( life1 & self-motive1 h an exi/5ce prior to ?+s mov$ by ano!r2 & :i*1 because !y subsi/ immat]ially1 & )\t any dim5sion1 excel ! =ms :i* are endu$ ) dim5sion & matt]4 ,9 ! nature ( :i* we are in/ruct$ by ,timu1 :o has expla9$ to u ! demiurgic1 & ess5tial figure ( s\ls#afb4 ,b aga91 ! figures ( intellects are by f> m div9e ?an ! figures ( s\ls2 = ^!1 on e side1 excel p>tible ess5ces2 are e :]e respl5d5t ) imp>tible & intellectual light2 are prolific1 effective1 & p]fective ( ! univ]se2 are equally pres5t1 & firmly abide 9 all ?+s2 & procure union to ! figures ( s\ls2 b recall ! mutation ( s5sible figures to ! limitation ( _! prop] b\nd4 ,la/ly1 !re are1 sep>ate f all ^!1 ^? p]fect1 uni=m1 unkn[n1 & ineffable figures ( ! gods1 :i* are resid5t1 inde$1 9 ! figures ( intellects2 b jo9tly t]m9ate all figures1 & compreh5d all ?+s 9 _! unify+ b\nd>ies4 ,! prop]ties ( :i* ! !urgic art1 also express+1 surr\nds v>i\s resemblances ( ! gods1 ) v>i\s figures4 ,& "s1 inde$1 x fa%ions by *>act]s1 9 an ineffable mann]2 = *>act]s ( ?is k9d1 manife/ ! unkn[n p[]s ( ! gods3 b o!rs x imitates by =ms & images2 fa%ion+ "s ( !m erect1 & o!rs fitt+2 & "s simil> to a he>t1 b o!rs sph]ical1 & o!rs express$ by diff]5t figures4 ,& aga91 "s x fabricates ( a simple =m2 b o!rs x composes f a multitude ( =ms2 & "s are sacr$ & v5]able2 b o!rs are dome/ic1 exhibit+ ! peculi> g5tl5ess ( ! gods4 ,& "s x con/ructs ( a sev]e aspect2 & la/ly1 attributes to o!rs1 diff]5t symbols1 accord+ to ! similitude & sympa?y p]ta9+ to ! gods#afc4 ,s9ce1 !re=e1 figure d]ives its orig9 f ! gods !mselves1 x arrives1 by a gradual progression1 ev5 to inf]iors1 9 ^! also appe>+ f prim>y causes4 ,s9ce x is requisite to suppose ! p]fect be=e ! imp]fect1 & ?+s situat$ 9 ! /ability ( _! own ess5ce1 prior to ^? :i* subsi/ 9 o!rs1 & previ\s to ?+s full ( _! own privation1 su* z pres]ve _! prop] nature s9c]e4 ,su* figures1 !re=e1 z are mat]ial1 p>ticipate ( mat]ial inelegance1 & d n possess a purity conv5i5t to _! nature4 ,b ! cele/ial figures are divisible1 & subsi/ 9 o!rs4 ,& ! figures ( s\ls are endu$ ) division1 & v>iety1 & involution ( e k9d2 b ! figures ( intellects1 toge!r ) immat]ial union1 possess a progression into multitude4 ,& la/ly1 ! figures ( ! gods are free1 uni=m1 simple1 & g5]ative2 !y subsi/ be=e all ?+s1 conta9+ all p]fection 9 !mselves1 & ext5d+ f !mselves to all ?+s1 ! completion ( =ms4 ,we mu/ n1 !re=e1 li/5 to1 & endure ! op9ions ( _m1 :o affirm1 t c]ta9 additions1 ablations1 & alt]ations1 produce s5sible figures1 7= motions1 s9ce !y are imp]fect1 _c possess ! pr9ciple & prim>y cause ( effects2 nor c\ld ! same figures (t5 be produc$ f contr>y motions2 = ! same =m is sometimes g5]at$ f addition & detraction17 b we mu/ consid] op]ations ( ?is k9d z subs]vi5t to o!r purposes 9 g5]ation1 & d]ive ! p]fection ( figure f o!r primog5ial causes4 ,nor mu/ we subscribe to _! op9ion1 :o ass]t t figures de/itute ( matt] c h no subsi/5ce2 b ^? only :i* appe> 9 matt]4 ,nor to !irs1 :o ackn[l$ge1 inde$1 t !y are ext]nal to matt]1 b consid] !m z subsi/+ alone1 accord+ to ?\ght & ab/raction4 ,= :]e %all we pres]ve 9 safety1 ! c]ta9ty1 beauty1 & ord] ( figures1 among ?+s :i* subsi/ by ab/raction8 ,=1 s9ce !y are ( ! same k9d ) s5sibles1 !y are f> di/ant f indubitable & pure c]ta9ty4 ,b f :5ce d !y d]ive ! c]ta9ty1 ord]1 & p]fection :i* !y receive8 ,= !y ei!r d]ive x f s5sibles 7b !y h no subsi/5ce 9 ^!71 or f intelligibles 7b 9 ^! !y are m p]fect71 s9ce1 to say f t :i* is n1 is ! mo/ absurd ( all4 ,= nature does n produce imp]fect figures1 & leave ! p]fect )\t any subsi/5ce4 ,nor is x lawful1 t our s\l %\ld fabricate m c]ta91 p]fect1 & ord]ly figures1 ?an intellect & ! gods !mselves4 ,!re are1 !re=e1 prior to s5sible figures1 self-mov+1 intellectual1 & div9e reasons ( figures4 ,& we are excit$1 inde$1 f ! obscurity ( s5sible =ms1 b we produce int]nal reasons1 :i* are ! lucid images ( o!rs4 ,& we possess a k ( s5sible figures1 by _! exempl>s resid5t 9 s\l 771 b we compreh5d by images 77 su* z are intellectual & div9e4 ,= ! reasons we conta91 em]g+ f ! d>k night ( oblivion1 & propagat+ !mselves 9 sci5tial v>iety1 exhibit ! =ms ( ! gods1 & ! uni=m b\nds ( ! univ]se1 by :i* !y ineffably conv]t all ?+s into !mselves4 ,9 ! gods1 !re=e1 !re is bo? an egregi\s k ( univ]sal figures1 & a p[] ( g5]at+ & con/itut+ all inf]iors4 ,b 9 natures1 figures are endu$ ) a p[] g5]ative ( app>5t =ms2 b are de/itute ( cognition & intellectual p]ception4 ,&1 9 p>ticul> s\ls1 !re is1 inde$1 an immat]ial intellection1 & a self-en]giz+ k2 b !re is want+ a prolific1 & efficaci\s cause4 ,z1 !re=e1 nature1 by h] =m+ p[] presides ov] s5sible figures1 9 ! same mann]1 s\l1 by h] gno/ic en]gy1 drops 9 ! phantasy z 9 a mirror1 ! reasons ( figures4 ,b ! phantasy receiv+ ^! 9 h] %ad[y =ms1 & possess+ images ( ! inh]5t reasons ( ! s\l1 af=ds by ^! ! means ( inw>d conv]sion to ! s\l1 & ( an en]gy direct$ to h]self1 f ! spectres ( imag9ation4 ,j z if any "o behold+ 8 image 9 a mirror1 & admir+ ! p[] ( nature1 & 8 own beauty1 %\ld desire to see himself 9 p]fection1 & %\ld receive a p[] ( becom+1 at ! same "t1 ! p]ceiv]1 & ! ?+ p]ceiv$4 ,= ! s\l1 aft] ?is mann]1 look+ abroad into ! bright mirror ( ! phantasy1 & survey+ ! %ad[y figures x conta9s1 & admir+ _! beauty & ord]1 pursues1 9 consequ5ce ( h] admiration1 ! reasons f :i* ^! images proce$2 & be+ wond]fully delight$1 dismisses _! beauty1 z conv]sant ab\t spectres alone2 b aft]w>ds seeks h] own pur] beauty1 & desires to pass into h] own pr(\nd retreats1 & !re to p]ceive ! circle & ! triangle1 & all ?+s subsi/+ toge!r1 9 an imp>tible mann]1 & to ins]t h]self 9 ! objects1 to contract h] multitude into "o2 & la/ly1 to behold ! occult & ineffable figures ( ! gods1 seat$ 9 ! mo/ sacr$ & div9e recesses ( h] nature4 ,%e is likewise desir\s ( br++ into light1 f its awful concealm5t1 ! solit>y beauty1 ( ! gods1 & ( p]ceiv+ ! circle1 subsi/+ 9 its true p]fection1 m imp>tible ?an any c5tre1 & ! triangle )\t int]val2 & la/ly1 by asc5d+ into an union ) h]self1 ( survey+ e object :i* is subject to ! p[] ( cognition4 ,! figure1 !re=e1 :i* is self-motive1 prec$es t :i* is mov$ by ano!r2 & ! imp>tible t :i* is self-motive3 b t :i* is ! same ) "o1 prec$es ! imp>tible itself4 ,= all ?+s are b\nd$1 :5 !y return to ! unities ( _! nature2 s9ce all ?+s pass ?r\gh ^! z a div9e entrance into be+4 ,& ?us mu* = ?is long digression1 :i* we h deliv]$ accord+ to ! s5tim5ts ( ! ,py?agoreans4 ,b ! geometrician1 contemplat+ t figure :i* is seat$ 9 ! phantasy1 & def9+ ?is1 9 ! fir/ place1 7s9ce ?is def9ition agrees ) s5sibles1 9 ! second place7 says1 t figure is t :i* is compreh5d$ by "o or m b\nd>ies4 ,=1 s9ce he receives x toge!r ) matt]1 & conceives ( x z di/ant ) int]vals1 he does n improp]ly call x f9ite & t]m9at$#afd4 ,s9ce e ?+ :i* conta9s ei!r intelligible or feasible matt]1 is allott$ an adv5titi\s b\nd2 & is n itself b\nd1 b t :i* is b\nd$4 ,nor is x ! b\nd ( itself2 b "o ( its p[]s is t]m9at+1 & ! o!r t]m9at$4 ,nor does x subsi/ 9 b\nd itself1 b is conta9$ by b\nd4 ,= figure is jo9$ to quantity1 & subsi/s toge!r ) x2 &1 at ! same "t1 quantity is subject$ to figure2 b ! reason & aspect ( t quantity is no?+ else ?an figure & =m4 ,s9ce1 inde$1 reason t]m9ates quantity1 & adds to x a p>ticul> "* & b\nd1 ei!r simple or composite4 ,=1 s9ce ?is also exhibits ! tw(old progression ( b\nd & inf9ite 9 its prop] =ms1 79 ! same mann] z ! reason ( an angle17 x inve/s ! objects ( its compreh5sion ) "o b\nd>y & simple =m1 accord+ to b\nd1 b ) _m1 accord+ to inf9ity#afe4 ,h5ce1 e ?+ figur$1 v9dicates to itself ei!r "o b\nd>y1 or a _m4 ,euclid1 !re=e1 d5om9at+ t :i* is figur$ & mat]ial1 & annex$ to quantity figure1 does n improp]ly say1 t x is conta9$ by "o or m t]ms4 ,b ,possidonius def9es figure to be conclud+ b\nd1 sep>at+ ! reason ( figure f quantity2 & consid]+ x z ! cause ( t]m9at+1 def9+1 & compreh5d+ quantity4 ,= t :i* encloses1 is diff]5t f t :i* is enclos$2 & b\nd f t :i* is b\nd$4 ,& ,possidonius1 inde$1 seems to reg>d ! ext]nal surr\nd+ b\nd2 b ,euclid1 ! :ole subject4 ,h5ce1 ! "o calls a circle a figure1 ) relation to its :ole plane1 & ext]ior ambit2 b ! o!r ) relation to its circumf]5ce only4 ,& ! "o def9es t :i* is figur$1 & :i* is beheld toge!r ) its subject3 b ! o!r desires to def9e ! reason ( ! circle2 ,i mean t :i* t]m9ates & concludes its quantity4 ,b if any logician1 & capti\s p]son1 %\ld blame ! def9ition ( ,euclid1 because he def9es g5us f species 7= ?+s conta9$ by "o or m t]ms1 are ! species ( figure17 we %all ass]t1 9 opposition to su* an objection1 t g5]a also pre-occupy 9 !mselves ! p[]s ( species4 ,& :5 m5 ( anci5t au?ority1 7 will+ to manife/ g5]a !mselves1 f ^? p[]s :i* g5]a conta91 !y appe>$1 inde$1 to ent] on _! design f species1 b1 9 reality1 !y expla9$ g5]a f !mselves1 & f ! p[]s :i* !y conta94 ,! reason ( figure1 !re=e1 s9ce x is "o1 compreh5ds ! diff]5ces ( _m figures1 accord+ to ! b\nd & inf9ity resid+ 9 its nature4 ,& he :o def9$ ?is reason1 0 n void ( und]/&+1 :il/ he compreh5d$ 9 a def9ition1 ! diff]5ces ( ! p[]s x conta9$4 ,b y w ask1 ,f :5ce does ! reason ( figure orig9ate1 & by :at causes is x p]fect$8 ,i answ]1 t x fir/ arises f b\nd & inf9ite1 & t :i* is mix$ f ^!4 ,h5ce x produces "s species f b\nd1 o!rs f inf9ite1 & o!rs f ! mixt4 ,& ?is x accompli%es by br++ ! =m ( b\nd to circles2 b t ( inf9ite1 to "r l9es3 & t ( ! mixt to figures compos$ f "r & circul> l9es4 ,b1 9 ! second place1 ?is reason is p]fect$ f t totality1 :i* is sep>at$ into dissimil> p>ts4 ,f :5ce1 inde$1 x occasions a :ole to e =m1 & ea* figure is cut into diff]5t species4 ,= a circle1 & e "r-l9$ figure may be divid$1 by reason or proportion1 into dissimil> figures2 :i* is ! bus9ess ( ,euclid 9 8 book ( divisions1 :]e he divides "o figure into figures simil> to ea* z are giv52 b ano!r into su* z are dissimil>4 ,9 ! ?ird place1 x is invigorat$ f accumulat$ multitude1 &1 on acc\nt ( ?is1 ext5ds =ms ( e k9d1 & produces ! multi=m reasons ( figures4 ,h5ce1 9 propagat+ itself1 x does n cease till x arrives at some?+ la/1 & has unfold$ all ! v>iety ( =ms4 ,&1 z 9 ! intelligible _w1 "o is %ewn to abide 9 t :i* is2 &1 at ! same "t1 t :i* is 9 "o1 s likewise1 reason exhibits circul> 9 "r-l9$ figures2 & on ! contr>y1 rectil9e> compreh5d$ 9 circul> figures4 ,& x peculi>ly manife/s its :ole nature 9 ea*1 & all ^! 9 all4 ,s9ce ! :ole subsi/s 9 all collectively1 & 9 ea* sep>ate & ap>t4 ,f t ord]1 !re=e1 x is endu$ ) ?is p[]4 ,9 ! f\r? place1 x receives f ! fir/ ( numb]s#aff1 ! measures ( ! progression ( =ms4 ,f :5ce x con/itutes all figures accord+ to numb]s2 "s1 inde$1 accord+ to ! m simple1 b o!rs accord+ to ! m composite4 ,= triangles1 quadrangles1 qu9quangles1 & all multangles1 proce$ 9 inf9itum1 toge!r ) ! mutations ( numb]s4 ,b ! cause ( ?is is1 inde$1 unkn[n to ! vulg>1 ?\gh1 to ^? :o und]/& :]e numb] & figure subsi/1 ! reason is manife/4 ,fif?ly1 x is replete ) t division ( =ms1 :i* divides =ms into o!r simil> =ms1 f ano!r second totality1 :i* is also di/ribut$ into simil> p>ts4 ,& by ?is1 a triangul> reason is divid$ into triangles1 & a quadrangul> reason into quadrangles4 ,& h5ce1 ex]cis+ our inw>d p[]s1 we effect :at ,i h said 9 images1 s9ce x pre-exi/$ by f> ! fir/ 9 its pr9ciples4 ,b by reg>d+ ^! di/ributions1 we may r5d] _m causes ( figures1 r$uc+ !m to _! fir/ pr9ciples4 ,& ! m common1 or geometrical figure1 is allott$ an ord] ( ?is k9d1 & f s _m causes1 receives ! p]fection ( its nature4 ,b1 f h5ce x advances to ! g5]a ( ! gods1 & is v>i\sly attribut$ accord+ to its v>i\s =ms1 & en]gizes diff]5tly 9 diff]5t gods4 ,to "s1 inde$1 af=d+ m simple figures2 b to o!rs1 su* z are m composite4 ,& to "s1 aga91 assign+ prim>y figures1 & ^? :i* are produc$ 9 sup]ficies2 b to o!rs 7ent]+ ! tumor ( solid bodies7 su* figures1 z 9 solids are conv5i5t to !mselves4 ,= all figures1 inde$1 subsi/ 9 all1 s9ce ! =ms ( ! gods are accumulat$1 & full ( univ]sal p[]s3 b1 by _! peculi>ity1 !y produce "o ?+ accord+ to ano!r4 ,= "o possesses all ?+s circul>ly1 ano!r 9 a triangul> mann]1 b ano!r accord+ to a quadrangul> reason4 ,& 9 a simil> mann] 9 solids4 ,,def9ition ,,xv4 ,a ,,circle is a ,plane ,figure1 compreh5d$ by "o ,l9e1 :i* is call$ ! ,circumf]5ce1 to :i* all ,"r ,l9es fall+ f a c]ta9 ,po9t )9 ! ,figure1 are equal to ea* o!r4 ,,def9ition ,,xvi4 ,& t ,po9t is call$ ! ,,c5tre ( ! ,,circle4 ,a ,circle is ! fir/1 mo/ simple1 & mo/ p]fect ( figures4 ,= x excels all solids1 because x exi/s 9 a m simple place2 b x is sup]ior to ! figures subsi/+ 9 planes1 on acc\nt ( its similitude & id5tity4 ,& x has a correspond+ proportion to b\nd1 & unity1 & a bett] co-ord9ation ( be+4 ,h5ce1 9 a di/ribution ( mundane & sup]-mundane figures1 y w always f9d t ! circle is ( a div9] nature4 ,= if y make a division into ! heav5s1 & ! univ]sal regions ( g5]ation1 y mu/ assign to ! heav5s a circul> =m2 b to g5]ation1 t ( a "r l9e1 ,= :atev] among g5]able natures is circul>1 desc5ds f ! heav5s2 s9ce g5]ation revolves into itself1 ?r\gh _! circumvolutions1 & r$uces its un/able mutation to a regul> & ord]ly cont9uance4 ,b if y di/ribute incorporeal natures into s\l & intellect1 y w say1 t ! circle belongs to intellect1 & ! "r l9e to ! s\l4 ,& on ?is acc\nt1 ! s\l1 by its conv]sion to intellect1 is said to be circul>ly mov$2 & x possesses ! same proportion to intellect1 z g5]ation to ! heav5s4 ,= x is circul>ly mov$1 7says ,socrates#afg17 because x imitates intellect4 ,b ! g5]ation & progression ( s\l is made accord+ to a "r-l9e4 ,= x is ! prop]ty1 ( ! s\l to apply h]self at diff]5t times to diff]5t =ms4 ,b if y wi% to divide into body & s\l1 y mu/ con/itute e ?+ corporeal1 accord+ to ! "r l9e2 b y mu/ assign to e animal a p>ticipation ( ! id5tity & similitude ( ! circle4 ,= body is a composite1 & is endu$ ) v>i\s p[]s1 simil> to "r-l9$ figures3 b s\l is simple & intellig5t2 self-motive1 & self-op]ative2 conv]t$ into1 & en]giz+ 9 h]self4 ,f :5ce1 inde$1 ,timu also1 :5 he _h compos$ ! elem5ts ( ! univ]se f "r-l9$ figures1 assign$ to !m a circul> motion & =mation1 f t div9e s\l :i* is seat$ 9 ! bosom ( ! _w4 ,& ?us1 t ! circle e :]e holds ! fir/ rank1 9 respect ( o!r figures1 is suffici5tly evid5t f ! prec$+ obs]vations4 ,b x is requisite to survey its :ole s]ies1 beg9n+ sup]nally1 end+ 9 inf]iors1 & p]fect+ all ?+s1 accord+ to ! aptitude ( ! natures :i* receive its alliance4 ,to ! gods1 !re=e1 x af=ds a conv]sion to _! causes1 & ineffable union3 x occasions _! abid+ 9 !mselves1 prev5ts _! dep>t+ f _! own beatitude1 /r5g!ns _! highe/ unions1 z c5tres desirable to inf]ior natures2 & /ably places ab\t ^! ! multitude ( ! p[]s :i* ! gods possess1 conta9+ !m 9 ! simplicity ( _! ess5ces4 ,b ! circle af=ds to intellectual natures1 a p]petual en]gy 9 !mselves1 is ! cause ( _! be+ fill$ ) k f !mselves1 & ( possess+ 9 _! ess5ces1 intelligibles contract$ly2 & ( p]fect+ intellections 9 !mselves4 ,= e intellect1 proposes to itself t :i* is intelligible2 & ?is is z a c5tre to intellect1 ab\t :i* x cont9ually revolves3 = intellect folds itself1 & op]ates ab\t ?is1 & is unit$ )9 itself on all sides1 by univ]sal intellectual en]gies4 ,b x ext5ds to s\ls by illum9ation1 a self-vital1 & self-motive p[]1 & an ability ( turn+1 & leap+ r\nd intellect1 & ( return+ accord+ to prop] convolutions1 unfold+ ! imp>tibility ( intellect4 ,aga91 ! intellectual ord]s excel s\ls aft] ! mann] ( c5tres1 b s\ls en]gize circul>ly ab\t _! nature4 ,= e s\l1 accord+ to its intellectual "p1 & ! supreme "o1 :i* is ! v fl[] ( its ess5ce1 receives a c5tre3 b1 accord+ to its multitude1 x has a circul> revolution1 desir+1 by ?is means1 to embrace ! intellect :i* x p>ticipates4 ,b1 to ! cele/ial bodies1 ! circle af=ds an assimilation to intellect1 equality1 a compreh5sion ( ! univ]se1 9 prop] limits1 revolutions :i* take place 9 det]m9ate measures1 a p]petual subsi/5ce1 a nature )\t beg9n+ & end1 & e ?+ ( ?is k9d4 ,& to ! elem5ts "u ! concave ( ! moon's orb1 x is ! cause ( a p]iod1 conv]sant ) mutations2 an assimilation to ! heav5s2 t :i* is )\t g5]ation1 9 g5]at$ natures2 t :i* abides 9 ?+s :i* are mov$2 & :atev] is b\nd$ 9 p>tible ess5ces4 ,= all ?+s are p]petual1 ?r\gh ! circle ( g5]ation2 & equability is e :]e pres]v$ on acc\nt ( ! reciprocation ( corruption4 ,s9ce1 if g5]ation did n return1 9 a circul> revolution1 9 a %ort space ( "t1 ! ord]1 & all ! ornam5t ( ! elem5ts w\ld vani%4 ,b aga91 ! circle procures to animals & plants1 t similitude :i* is f\nd 9 g5]ations2 = ^! are produc$ f se$s1 & se$s f ^!4 ,h5ce1 g5]ation "h1 & a circumvolution1 alt]nately takes place1 f ! imp]fect to ! p]fect1 & ! contr>y2 s t corruption subsi/s toge!r ) g5]ation4 ,b1 besides ?is1 to unnatural productions x imposes ord]1 & r$uces _! indet]m9ate v>iety to ! limitation ( b\nd2 &1 ?r\gh ?is1 nature h]self is gracefully ornam5t$ 9 ! la/ ve/iges ( h] p[]s4 ,h5ce1 ?+s contr>y to nature h a revolution accord+ to det]m9ate numb]s1 & n only f]tility1 b also /]ility1 subsi/s accord+ to ! alt]nate convolutions ( circles 7z ! disc\rse ( ! ,muses ev9ces71 & all evils ?\gh !y are dismiss$ f ! pres5ce ( ! gods1 into ! place ( mortals1 yet ^! roll r\nd1 says ,socrates1 & to ^! !re is pres5t a circul> revolution1 & a circul> ord]2 s t no?+ immod]ate & evil is des]t$ by ! gods2 b t provid5ce1 :i* is p]fective ( ! univ]se1 r$uces also ! inf9ite v>iety ( evils1 to b\nd1 & an ord] conv5i5t to _! nature4 ,! circle1 !re=e1 is ! cause ( ornam5t to all ?+s1 ev5 to ! la/ p>ticipations1 & leaves no?+ de/itute ( itself1 s9ce x supplies beauty1 similitude1 =mation1 & p]fection to ! univ]se4 ,h5ce too1 9 numb]s x conta9s ! middle c5tres ( ! :ole progression ( numb]s1 :i* revolves f unity to ! decad 7or t574 ,= five & six exhibit a circul> p[]1 because1 9 ! progressions f !mselves1 !y return aga9 into !mselves1 z is evid5t 9 ! multiplication ( ^! numb]s4 ,multiplication1 !re=e1 is an image ( progression1 s9ce x is ext5d$ into multitude2 b an end+ 9 ! same species1 is an image ( regression into !mselves4 ,b a circul> p[] af=ds ea* ( ^!1 excit+1 inde$1 z f an abid+ c5tre1 ^? causes :i* are productive ( multitude2 b conv]t+ multitude aft] ! productions to _! causes4 ,two numb]s1 !re=e1 hav+ ! prop]ties ( a circle1 possess ! middle place betwe5 all numb]s3 ( :i* "o1 inde$1 prec$es e conv]tible g5us ( males & an odd nature2 b ! o!r1 recalls e ?+ fem99e & ev51 & all prolific s]ies1 to _! prop] pr9ciples1 accord+ to a circul> p[]4 ,& ?us mu* conc]n+ ! p]fection ( ! circle4 ,let u n[ contemplate ! ma!matical def9ition ( ! circle1 :i* is e way p]fect4 ,9 ! fir/ place1 !re=e1 he def9es x a figure1 because1 inde$1 x is f9ite1 & e :]e compreh5d$ by "o limit1 & is n ( an inf9ite nature1 b associat$ to b\nd4 ,likewise plane1 because1 s9ce figures are ei!r beheld 9 sup]ficies1 or 9 solid bodies1 a circle is ! fir/ ( plane figures1 excell+ solids 9 simplicity1 b possess+ ! proportion ( unity to planes4 ,b compreh5d$ by "o l9e1 because x is simil> to "o1 by :i* x is def9$1 & because x does n extr9sically receive a v>iety ( surr\nd+ t]ms4 ,& aga91 t ?is l9e makes all ! l9es drawn to x f a c]ta9 po9t )9 equal1 because ( ! figures :i* are b\nd$ by "o l9e1 "s h all ! l9es proce$+ f ! middle equal2 b o!rs n at all4 ,= ! ellipsis is compreh5d$ by "o l9e1 yet all ! l9es issu+ f ! c5tre1 & b\nd$ by is curvature1 are n equal1 b only two4 ,also ! plane1 :i* is includ$ by ! l9e call$ a cissoid1 has "o conta9+ l9e1 yet x does n conta9 a c5tre1 f :i* all ! l9es are equal4 ,b1 because ! c5tre 9 a circle is entirely "o po9t 7= !re are n _m c5tres ( "o circle71 on ?is acc\nt1 ! geometrician adds1 t l9es fall+ f "o po9t to ! b\nd ( ! circle1 are equal4 ,= !re are inf9ite po9ts )9 x1 b ( all ^!1 "o only has ! p[] ( a c5tre4 ,& because ?is "o po9t1 f :i* all ! l9es drawn to ! circumf]5ce ( ! circle are equal1 is ei!r )9 ! circle1 or )\t 7= e circle has a pole1 f :i* all ! l9es drawn to its circumf]5ce are equal71 on ?is acc\nt he adds1 ( ! po9ts )9 ! figure1 because1 "h he receives ! c5tre alone1 & n ! pole4 ,= he wi%es to behold all its prop]ties 9 "o plane1 b ! pole is m elevat$ ?an ! subject plane4 ,h5ce1 he necess>ily adds1 9 ! end ( ! def9ition1 t ?is po9t1 :i* is plac$ )9 ! circle1 & to :i* all "r l9es drawn f x to ! circumf]5ce1 are equal1 is ! c5tre ( ! circle4 ,= !re are only two po9ts ( ?is k9d1 ! pole & ! c5tre4 ,b ! =m] is )\t1 & ! o!r )9 ! plane4 ,?us1 = in/ance1 if y conceive a p]p5dicul> /&+ on ! c5tre ( a circle1 its sup]ior extremity is ! pole3 = all l9es drawn f x to ! circumf]5ce ( ! circle1 are demon/rat$ to be equal4 ,&1 9 l mann]1 9 a cone1 ! v]tex ( ! :ole cone1 is ! pole ( ! circle at ! base4 ,& ?us f> we h det]m9$ :at a circle is1 & its c5tre1 & :at ! nature is ( its circumf]5ce1 & ! :ole circul> figure4 ,aga91 !re=e1 f ^!1 let u return to ! speculation ( _! exempl>s1 contemplat+ 9 !m ! c5tre1 accord+ to "o imp>tible & /able excell5ce4 ,b ! di/ances f ! c5tre1 accord+ to ! progressions :i* are made f "o1 to multitude inf9ite 9 capacity4 ,& ! circumf]5ce ( ! circle1 accord+ to ! regression ( ! progressions to ! c5tre1 by means ( :i* ! multitude ( p[]s are roll$ r\nd _! union1 & all ( !m ha/5 to its compreh5sion1 & desire to en]gize ab\t its indivisible embrace4 ,&1 z 9 ! circle itself1 all ?+s subsi/ toge!r1 ! c5tre1 int]vals1 & ext]nal circumf]5ce2 s 9 ^! :i* are its image1 "o ?+ has n an ess5ce pre-exi/5t1 & ano!r consequ5t 9 "t2 b all ?+s are1 inde$1 toge!r1 p]man5cy1 progression1 & regression4 ,b ^! diff] f ^?1 because ! =m] subsi/ indivisibly1 & )\t any dim5sion2 b ! latt] ) dim5sion2 & 9 a divisible mann]2 ! c5tre exi/+ 9 "o place1 ! l9es emanat+ f ! c5tre1 9 ano!r2 & ! ext]nal circumf]5ce t]m9at+ ! circle1 hav+ a /ill diff]5t situation4 ,b !re all ?+s abide 9 "o3 = if y reg>d t :i* p]=ms ! (fice ( a c5tre1 y w f9d x ! receptacle ( all ?+s4 ,if ! progression di/ant f ! c5tre1 9 ?is1 likewise1 y w f9d all ?+s conta9$4 ,&1 9 a simil> mann]1 if y reg>d its regression4 ,:51 !re=e1 y are able to p]ceive all ?+s subsi/+ toge!r1 & h tak5 away ! defect proce$+ f dim5sion1 & h remov$ f y\r inw>d vision1 ! position ab\t :i* p>tition subsi/s1 y w f9d ! true circle1 advanc+ to itself1 b\nd+1 & en]giz+ 9 itself1 exi/+ bo? "o & _m1 & abid+1 proce$+1 & return+2 likewise firmly e/abli%+ t "p ( its ess5ce :i* is mo/ imp>tible1 & especially s+ul>2 b advanc+ f ?is accord+ to rectitude1 & ! inf9ity :i* x conta9s2 & roll+ itself f itself to "o1 & excit+ itself by similitude & id5tity to ! imp>tible c5tre ( its nature1 & to ! occult p[] ( ! "o :i* x conta9s4 ,b ?is "o1 :i* ! circle conta9s1 & environs 9 its bosom1 x emulates accord+ to ! multitude ( its own nature4 ,= t :i* is convolv$1 imitates t :i* abides1 & ! p]iph]y is z a c5tre :i* is di/ant ) int]val1 & nods to itself1 ha/5+ to receive1 & to become "o ) ! c5tre1 & to t]m9ate its regress :]e x receiv$ ! pr9ciple ( its progression4 ,= ! c5tre is e :]e 9 ! place ( t :i* is lovely1 & ! object ( desire1 presid+ ov] all ?+s :i* subsi/ ab\t its nature1 & exi/+ z ! beg9n+ & au?or ( all progressions4 ,& ?is ! ma!matical c5tre also expresses1 by t]m9at+ all ! l9es fall+ f itself to ! circumf]5ce1 & by af=d+ to !m equality1 z an image ( prop] union4 ,b ! oracles likewise def9e ! c5tre1 aft] ?is mann]3 ,! c5tre is t f :i* &#afh to :i* all ! l9es to ! circumf]5ce are equal4 ,indicat+ ! beg9n+ ( ! di/ance ( ! l9es1 by ! p>ticle f :i*2 b ! middle ( ! circumf]5ce by ! p>ticle to :i*3 = ?is1 9 e "p1 is jo9$ ) ! c5tre4 ,b if x be necess>y to decl>e ! fir/ cause1 ?r\gh :i* a circul> figure appe>s & receives its p]fection1 ,i affirm1 t x is ! supreme ord] ( intelligibles4 ,= ! c5tre1 inde$1 is assimilat$ to ! cause ( b\nd2 b ! l9es emanat+ f ?is1 & :i* are inf9ite1 ) respect to !mselves1 bo? 9 multitude & magnitude1 repres5t inf9ity2 & ! l9e :i* t]m9ates _! ext5sion1 & conjo9s ! circul> figure ) ! c5tre1 is simil> to t occult ornam5t1 consi/+ f ! intelligible ord]s2 :i* ,orpheus also says1 is circul>ly borne1 9 ! foll[+ words1 ,b x is c>ri$ ) an unwe>i$ en]gy1 accord+ to an inf9ite circle4 ,=1 s9ce x is mov$ intelligibly1 ab\t t :i* is intelligible1 hav+ x = ! c5tre ( its motion1 x is1 ) great propriety1 said to en]gize 9 a circul> mann]4 ,h5ce1 f ^! also1 ! triadic god#afi proce$s1 :o conta9s 9 himself ! cause ( ! progression ( "r-l9$ figures4 ,= on ?is acc\nt1 wise m51 & ! mo/ my/ic ( !ologi/s1 h fabricat$ 8 "n4 #agj;,h5ce too1 x is manife/1 t a circle is ! fir/ ( all figures3 b a triangle is ! fir/ ( su* z are "r-l9$4 ,figures1 !re=e1 appe> fir/ 9 ! regul> ornam5ts ( ! gods2 b !y h a lat5t subsi/5ce1 accord+ to pre-exi/5t causes1 9 intelligible ess5ces4 ,,def9ition ,,xvii4 ,a ,,diamet] ( a ,circle is a c]ta9 /raight ,l9e1 drawn ?r\gh ! ,c5tre1 :i* is t]m9at$ bo? ways by ! ,circumf]5ce ( ! ,circle1 &1 divides ! ,circle into two equal ,p>ts4 ,euclid "h p]spicu\sly %ews1 t he does n def9e e diamet]1 b t :i* belongs to a circle only4 ,because !re is a diamet] ( quadrangles & all p>allelograms1 & likewise ( a sph]e among solid figures4 ,b 9 ! fir/ ( ^!1 x is d5om9at$ a diagonal3 b 9 a sph]e1 ! axis2 & 9 circles ! diamet] only4 ,inde$1 we are accu/om$ to speak ( ! axis ( an ellipsis1 cyl9d]1 & cone2 b ( a circle1 ) propriety1 ! diamet]4 ,?is1 !re=e1 9 its g5us1 is a "r-l9e2 b z !re are _m "r-l9es 9 a circle1 z likewise inf9ite po9ts1 "o ( :i* is a c5tre1 s ?is only is call$ a diamet]1 :i* passes ?r\gh ! c5tre1 & nei!r falls )9 ! circumf]5ce1 nor transc5ds its b\nd>y2 b is bo? ways t]m9at$ by its compreh5sive b\nd4 ,& ^! obs]vations exhibit its orig94 ,b t :i* is add$ 9 ! end1 t x also divides ! circle into two equal p>ts1 indicates its prop] en]gy 9 ! circle1 exclusive ( all o!r l9es drawn ?r\gh ! c5tre1 :i* are n t]m9at$ bo? ways by ! circumf]5ce4 ,b !y report1 t ,?ales fir/ demon/rat$1 t ! circle 0 bisect$ by ! diamet]4 ,& ! cause ( ?is bisection1 is ! indecl9eable transit ( ! "r l9e1 ?r\gh ! c5tre4 ,=1 s9ce x is drawn ?r\gh ! middle1 & always pres]ves ! same inflexible motion1 accord+ to all its p>ts1 x cuts (f equal portions on bo? sides to ! circumf]5ce ( ! circle4 ,b if y desire to exhibit ! same ma!matically1 conceive ! diamet] drawn1 & "o "p ( ! circle plac$ on ! o!r#aga4 ,!n1 if x is n equal1 x ei!r falls )91 or )\t2 b ! consequ5ce ei!r ( ^! ways mu/ be1 t a less "r-l9e w be equal to a great]4 ,s9ce all l9es f ! c5tre to ! circumf]5ce are equal4 ,! l9e1 !re=e1 :i* t5ds to ! ext]ior circumf]5ce1 w be equal to t :i* t5ds to ! int]ior4 ,b ?is is impossible4 ,^! p>ts ( ! circle1 !n1 agree1 & are on ?is acc\nt equal4 ,b "h a d\bt arises1 if two semi-circles are produc$ by "o diamet]1 & inf9ite diamet]s may be drawn ?r\gh ! c5tre1 a d\ble ( inf9ities w take place1 accord+ to numb]4 ,= ?is is object$#agb by "s aga9/ ! section ( magnitudes to inf9ity4 ,b ?is we may solve by affirm+1 t magnitude may1 inde$1 be divid$ inf9itely1 b n into inf9ites4 ,= ?is latt] mode produces inf9ites 9 en]gy1 b ! =m] 9 capacity only4 ,& ! "o af=ds ess5ce to inf9ite1 b ! o!r is ! s\rce ( its orig9 alone4 ,two semi-circles1 !re=e1 subsi/ toge!r ) "o diamet]1 yet !re w nev] be inf9ite diamet]s1 al?\gh !y may be inf9itely assum$4 ,h5ce1 !re c nev] be d\bles ( inf9ites2 b ! d\bles :i* are cont9ually produc$1 are ! d\bles ( f9ites2 = ! diamet]s :i* are always assum$1 are f9ite 9 numb]4 ,& :at reason c be assign$ :y e magnitude %\ld n h f9ite divisions1 s9ce numb] is prior to magnitudes1 def9es all _! sections1 pre-occupies inf9ity1 & always det]m9es ! p>ts :i* rise into en]gy1 f dormant capacity8 ,,def9ition ,,xviii4 ,a ,,semi-,,circle is ! ,figure conta9$ by ! ,diamet]1 & t ,"p ( ! ,circumf]5ce :i* is cut (f by ! ,diamet]4 ,,def9ition ,,xix4#agc ,b ! ,,c5tre ( ! ,semi-circle1 is ! same ) t ( ! ,circle4 ,f ! def9ition ( a circle ,euclid f9ds out ! nature ( ! c5tre1 diff]+ f all ! o!r po9ts :i* ! circle conta9s4 ,b f ! c5tre he def9es ! diamet]1 & sep>ates x f ! o!r "r l9es1 :i* are describ$ )9 ! circle4 ,& f ! diamet]1 he tea*es ! nature ( ! semi-circle2 & in=ms u1 t x is conta9$ by two t]ms1 always diff]+ f ea* o!r1 viz4 a "r-l9e & a circumf]5ce3 & t ?is "r-l9e is n any "o indiff]5tly1 b ! diamet] ( ! circle4 ,= bo? a less & a great] segm5t ( a circle1 are conta9$ by a "r-l9e & circumf]5ce2 yet ^! are n semi-circles1 because ! division ( ! circle is n made ?r\gh ! c5tre4 ,all ^! figures1 !re=e1 are bi=m$1 z a circle 0 monadic1 & are compos$ f dissimil>s4 ,= e figure :i* is compreh5d$ by two t]ms1 is ei!r conta9$ by two circumf]5ces1 z ! lunul>3 or by a "r-l9e & circumf]5ce1 z ! above m5tion$ figures2 or by two mixt l9es1 z if two ellipses int]sect ea* o!r 7s9ce !y enclose a figure1 :i* is int]cept$ betwe5 !m71 or by a mixt l9e & circumf]5ce1 z :5 a circle cuts an ellipsis2 or by a mixt & "r-l9e1 z ! half ( an ellipsis4 ,b a semi-circle is compos$ f dissimil> l9es1 yet su* z are1 at ! same "t1 simple1 & t\*+ ea* o!r by apposition4 ,h5ce1 be=e he def9es triadic figures1 he1 ) great propriety1 passes f ! circle to a bi=m$ figure4 ,= two "r-l9es c1 inde$1 nev] compreh5d space4 ,b ?is may be effect$ by a "r-l9e & circumf]5ce4 ,likewise by two circumf]5ces1 ei!r mak+ angles1 z 9 ! lunul> figure2 or =m+ a figure )\t angles1 z t :i* is compreh5d$ by conc5tric circles4 ,= ! middle space int]cept$ betwe5 bo?1 is compreh5d$ by two circumf]5ces2 "o int]ior1 b ! o!r ext]ior1 & no angle is produc$4 ,= !y d n mutually int]sect1 z 9 ! lunul> figure1 & t :i* is on bo? sides convex4 ,b t ! c5tre ( ! semi-circle is ! same ) t ( ! circle1 is manife/4 ,= ! diamet]1 conta9+ 9 itself ! c5tre1 completes ! semi-circle1 & f ?is all l9es drawn to ! semi-circumf]5ce are equal4 ,= ?is is a "p ( ! circumf]5ce ( ! circle4 ,b equal "r l9es proce$ f ! c5tre to all p>ts ( ! circumf]5ce4 ,! c5tre1 !re=e1 ( ! circle & semi-circle is "o & ! same4 ,& x mu/ be obs]v$1 t among all figures1 ?is alone conta9s ! c5tre 9 its own p]imet]1 ,i say1 among all plane figures4 ,h5ce y may collect1 t ! c5tre has ?ree places4 ,= x is ei!r )9 a figure1 z 9 ! circle2 or 9 its p]imet]1 z 9 ! semi-circle2 or )\t ! figure1 z 9 c]ta9 conic l9es#agd4 ,:at !n is indicat$ by ! semi-circles1 hav+ ! same c5tre ) ! circle1 or ( :at ?+s does x be> an image1 unless t all figures :i* d n entirely dep>t f su* z are fir/1 b p>ticipate !m aft] a mann]1 may be conc5tric ) !m1 & p>ticipate ( ! same causes8 ,= ! semi-circle communicates ) ! circle d\bly1 z well accord+ to ! diamet]1 z accord+ to ! circumf]5ce4 ,on ?is acc\nt1 !y possess a c5tre also 9 common4 ,& p]haps1 aft] ! mo/ simple pr9ciples1 ! semi-circle is assimilat$ to ! second co-ord9ations1 :i* p>ticipate ^? pr9ciples2 & by _! relation to !m1 al?\gh imp]fectly1 & by halves1 !y are1 nev]!less1 r$uc$ to t :i* is1 & to _! fir/ orig9al cause4 ,,def9ition ,,xx4 ,,rectil9e> ,,figures are ^? :i* are compreh5d$ by ,/raight ,l9es4 ,,def9ition ,,xxi4 ,,trilat]al ,,figures1 or ,,triangles1 by ?ree ,/raight ,l9es4 ,,def9ition ,,xxii4 ,,quadrilat]al1 by f\r ,/raight ,l9es4 ,,def9ition ,,xxiii4 ,,multilat]al ,,figures1 or ,,polygons1 by m ?an f\r ,/raight ,l9es4 ,aft] ! monadic figure hav+ ! relation ( a pr9ciple to all figures1 & ! bi=m$ semi-circle1 ! progression ( "r-l9$ figures 9 inf9itum1 accord+ to numb]s1 is deliv]$4 ,= on ?is acc\nt also1 m5tion 0 made ( ! semi-circle1 z communicat+ accord+ to t]ms or b\nd>ies2 p>tly1 inde$1 ) ! circle1 b p>tly ) "r-l9es3 j z ! duad is ! m$ium betwe5 unity & numb]4 ,= unity1 by composition1 produces m ?an by multiplication2 b numb]1 on ! contr>y1 is m increas$ by multiplication ?an composition3 & ! duad1 :e!r multipli$ into1 or comp\nd$ ) itself1 produces an equal quantity4 ,z1 !re=e1 ! duad is ! middle ( unity & numb]1 s likewise1 a semi-circle communicates1 accord+ to its base1 ) "r-l9es2 b accord+ to its circumf]5ce1 ) ! circle4 ,b "r-l9$ figures proce$ ord]ly to inf9ity1 att5d$ by numb] & its b\nd+ p[]1 :i* beg9s f ! triad4 ,on ?is acc\nt1 ,euclid also beg9s f h5ce#age4 ,= he says1 trilat]al & quadrilat]al1 & ! foll[+ figures1 call$ by ! common "n ( multilat]al3 s9ce trilat]al figures are also multilat]als2 b !y h likewise a prop]1 besides a common d5om9ation4 ,b1 z we are b little able to pursue ! re/1 on acc\nt ( ! inf9ite progression ( numb]s1 we mu/ be cont5t ) a common d5om9ation4 ,b he only makes m5tion ( trilat]als & quadrilat]als1 because ! triad & tetrad are ! fir/ 9 ! ord] ( numb]s2 ! =m] be+ a pure odd among ! odd2 b ! latt]1 an entire ev5 among ev5 numb]s4 ,euclid1 !re=e1 assumes bo? 9 ! orig9 ( "r-l9$ figures1 = ! purpose ( exhibit+ _! subsi/5ce1 accord+ to all ev5 & odd numb]s4 ,besides1 s9ce he is ab\t to tea* conc]n+ ^! 9 ! fir/ book1 z especially elem5t>y 7,i mean triangles & p>allelograms7 he does n undes]v$ly1 z f> z to ^!1 e/abli% a prop] enum]ation3 b he embraces all o!r "r-l9$ figures by a common "n1 call+ !m multilat]als3 b ( ^! en\gh4 ,aga91 assum+ a m elevat$ exordium1 we mu/ say1 t ( plane figures1 "s are conta9$ by simple l9es1 o!rs by su* z are mixt1 b o!rs aga9 by bo?4 ,& ( ^? :i* are compreh5d$ by simple l9es1 "s are conta9$ by simil>s 9 species1 z "r-l9es2 b o!rs by dissimil>s 9 species1 z semi-circles1 & segm5ts1 & apsides1 :i* are less ?an semi-circles4 ,likewise ( ^? :i* are conta9$ by simil>s 9 species1 "s are compreh5d$ by a circul> l9e2 b o!rs by a "r-l9e4 ,& ( ^? compreh5d$ by a circul> l9e1 "s are conta9$ by "o1 o!rs by two1 b o!rs by m ?an two4 ,by "o1 inde$1 ! circle itself4 ,b by two1 "s )\t angles1 z ! cr[ns#agf t]m9at$ by conc5tric circles2 b o!rs angul> 77 z ! lunula4 ,& ( ^? compreh5d$ by m ?an two1 !re is an inf9ite procession4 ,= !re are c]ta9 figures conta9$ by ?ree & f\r & succe$+ circumf]5ces4 ,?us1 if ?ree circles t\* ea* o!r1 !y w int]cept a c]ta9 trilat]al space2 b if f\r1 "o t]m9at$ by f\r circumf]5ces1 & 9 l mann]1 by a successive progression4 ,b ( ^? conta9$ by "r l9es1 "s are compreh5d$ by ?ree1 o!rs by f\r1 & o!rs by a multitude ( l9es4 ,= nei!r is space compreh5d$ by two "r-l9es1 nor mu* m by "o "r-l9e4 ,h5ce1 e space compreh5d$ by "o b\nd>y1 or by two1 is ei!r mixt or circul>4 ,& x is mixt 9 a tw(old mann]1 ei!r because ! mixt l9es compreh5d x1 z ! space int]cept$ by ! cissoidal l9e2 or because x is conta9$ by l9es dissimil> 9 species1 z ! apsis3 s9ce m+l+ is tw(old1 ei!r by apposition or confusion4 ,e "r-l9$ figure1 !re=e1 is ei!r trilat]al1 or quadrilat]al1 or gradually multilat]al2 b e trilat]al1 or quadrilat]al1 or multilat]al figure1 is n "r-l9$2 s9ce s great a numb] ( sides is also produc$ f circumf]5ces4 ,& ?us mu* conc]n+ ! division ( plane figures4 ,b we h already ass]t$#agg1 t rectitude ( progression is bo? a symbol ( motion & inf9ity1 & t x is peculi> to ! g5]ative co-ord9ations ( ! gods1 & to ! produc]s ( diff]5ce1 & to ! au?ors ( mutation & motion4 ,"r-l9$ figures1 !re=e1 are peculi> to ^! gods1 :o are ! pr9ciples ( ! prolific en]gy ( ! :ole progression ( =ms4 ,on :i* acc\nt1 g5]ation also1 0 pr9cipally adorn$ by ^! figures1 & is allott$ its ess5ce f ^!1 s f> z x subsi/s 9 cont9ual motion & mutation )\t end4 ,,def9ition ,,xxiv4 ,( ?ree-sid$ ,,figures3 an ,,equilat]al ,,triangle is t :i* has ?ree equal ,sides4 ,,def9ition ,,xxv4 ,an ,,isosceles ,,triangle1 is t :i* has only two ,sides equal4 ,,def9ition ,,xxvi4 ,a ,,scal5e ,,triangle1 is t :i* has ?ree unequal ,sides4 ,,def9ition ,,xxvii4 ,a ,,"r-,,angl$ ,,triangle is t :i* has a ,"r ,angle4 ,,def9ition ,,xxviii4 ,an ,,obtuse-,,angl$ ,,triangle is t :i* has an ,obtuse ,angle4 ,,def9ition ,,xxix4 ,an ,,acute-,,angl$ ,,triangle is t :i* has ?ree ,acute ,angles4 ,! division ( triangles sometimes comm5ces f angles1 b sometimes f sides4 ,& t1 inde$1 :i* orig9ates f sides1 prec$es z kn[n2 b t f angles foll[s z a prop] di/ribution4 ,= ^! ?ree angles alone belong to "r-l9$ figures1 viz4 ! "r1 ! obtuse1 & ! acute3 b ! equality & inequality ( sides subsi/ also 9 non-rectil9e> figures4 ,euclid says1 !re=e1 t ( triangles1 "s are equilat]al1 o!rs isosceles1 & o!rs scal5e3 = !y h ei!r all _! sides equal1 or all unequal1 or only two equal4 ,& aga91 t ( triangles "s are "r-angl$1 o!rs obtuse-angl$1 & o!rs acute-angl$4 ,& he def9es a "r-angl$ triangle1 t :i* has "o "r angle1 z likewise an obtuse-angl$ triangle1 t :i* has "o obtuse angle3 = x is impossible t a triangle c h m ?an "o "r1 or obtuse angle#agh4 ,b he def9es an acute-angl$ triangle1 t :i* has all its angles acute4 ,= "h x is n suffici5t t x %\ld h only "o acute2 s9ce1 9 ?is case1 all triangles w\ld be acute-angl$1 z e triangle has necess>ily two acute angles#agi4 ,b1 to possess ?ree acute angles1 is ! prop]ty ( an acute-angl$ triangle alone4 ,b ,euclid appe>s to me to h made a sep>ate division into angles & sides1 f consid]+ ?is alone1 t e triangle is n also trilat]al4 ,= !re are quadrilat]al triangles1 :i* are call$ by ma!maticians !mselves 77 t is1 simil> to ! po9t ( a spe>#ahj3 b by ,z5odorus 77 t is1 hav+ an holl[ angle4 ,= on "o ( ! sides ( a trilat]al figure1 con/itute two "r-l9es inw>dly2 by ?is means a c]ta9 space w be enclos$1 :i* is compreh5d$ by ext]nal & int]nal "r-l9es1 & :i* has ?ree angles2 "o1 inde$1 conta9$ by ! ext]nal l9es2 b two compreh5d$ by ^! & ! int]nal l9es1 at ! extremities 9 :i* ^! l9es are conjo9$4 ,a figure ( ?is k9d1 !re=e1 is a quadrilat]al triangle4 ,& h5ce1 x does n imm$iately foll[1 t because a figure has ?ree angles 7:e!r !y are all acute1 or "o "r1 or "o obtuse71 we %all f9d x trilat]al2 = x may be1 p]haps1 quadrilat]al4 ,9 l mann]1 y may also f9d quadrangles hav+ m ?an f\r sides4 ,& !re=e1 we mu/ n ra%ly det]m9e ! numb] ( sides f ! multitude ( angles4 ,b ( ?is en\gh4 ,b ! ,py?agoreans affirm t ! triangle is simply ! pr9ciple ( g5]ation1 & ( ! =mation ( g5]able natures4 ,on :i* acc\nt1 ,timu says1 t natural reasons1 z well z ^? ( ! con/ruction ( ! elem5ts1 are triangul>4 ,= !y are di/ant by a triple int]val1 are on all sides collective ( p>tible1 & v>i\sly mutable natures1 are replete ) mat]ial inf9ity1 & be> be=e !mselves ! conjunctions ( mat]ial bodies1 loos5$ & free3 z1 inde$1 triangles also are compreh5d$ by ?ree "r-l9es1 b !y possess angles :i* collect ! multitude ( l9es1 & af=d to !m an adv5titi\s angle & conjunction4 ,) great propriety1 !re=e1 ,philolaus has consecrat$ ! angle ( a triangle to f\r gods1 ,saturn1 ,pluto1 ,m>s1 & ,bac*us1 compreh5d+ 9 ^! ! :ole quadrip>tite ornam5t ( ! elem5ts desc5d+ f ! heav5s1 or f ! f\r segm5ts ( ! zodiac4 ,= ,saturn con/itutes an ess5ce :olly humid & frigid2 b ,m>s a nature totally fi]y2 & ,pluto conta9s ! :ole t]re/rial life2 b ,bac*us gov]ns a humid & hot g5]ation2 ( :i* w9e also is a symbol1 = ?is is humid & hot4 ,h5ce1 all ^! gods diff] accord+ to _! op]ations 9 inf]ior conc]ns3 b !y are mutually unit$ accord+ to _! prop] natures4 ,& on ?is acc\nt1 ,philolaus collects _! union accord+ to "o angle4 ,b if ! diff]5ces ( triangles contribute to g5]ation1 we %all v prop]ly confess t a triangle is ! pr9ciple & au?or ( ! con/itution ( sublun>y natures4 ,= a "r angle1 inde$1 af=ds !m ess5ce1 & det]m9es ! measure ( be+2 & ! reason ( a "r-angl$ triangle produces ! ess5ce ( ! elem5ts ( g5]able natures2 b an obtuse angle assigns to !m univ]sal di/ance2 & ! reason ( an obtuse-angl$ triangle increases mat]ial =ms 9 magnitude1 & 9 mutation ( e k9d4 ,b an acute angle effects _! divisible nature2 & ! reason ( an acute-angl$ triangle prep>es !m to receive inf9ite division4 ,b simply1 a triangul> reason con/itutes ! ess5ce ( mat]ial bodies di/ant ) int]val1 & on all sides divisible4 ,& ?us mu* %\ld we speculate conc]n+ ! nature ( triangles4 ,b f ^! divisions y may und]/&1 t all ! species ( triangles are nei!r m nor less ?an sev54 ,= ! equilat]al triangle is "o1 s9ce x is acute-angl$ only2 b ea* ( ! re/ is triple4 ,= ! isosceles is ei!r "r-angl$1 or obtuse-angl$1 or acute-angl$2 &1 9 l mann]1 ! scal5e triangle possesses ?is triple diff]5ce4 ,if !n1 ^! h a triple di/9ction1 b ! equilat]al has b "o mode ( exi/5ce1 all ! species ( triangles w be sev54 ,b aga91 y w und]/& ! proportion ( triangles to ! ?+s :i* are1 accord+ to ! division ( sides2 = ! equilat]al1 entirely excell+ 9 equality & simplicity1 is alli$ to div9e s\ls2 s9ce x is ! measure & equality ( ?+s unequal1 9 ! same mann] z div9ity ( all inf]ior conc]ns4 ,b ! isosceles triangle is alli$ to ! bett] g5]a1 :i* gov]n a mat]ial nature1 ! great] "p ( :i* g5]a is held by ! limitation ( measure2 b _! extremes ext5d to inequality & mat]ial immod]ation2 = ! two sides ( an isosceles triangle are equal1 b ! base is unequal4 ,b a scal5e triangle symbolizes ) p>tible lives1 :i* are on all sides lame & defective1 :i* prep>e !mselves = g5]ation1 & are replete ) matt] & mat]ial imp]fection4 ,,def9ition ,,xxx4 ,( ,quadrilat]al ,figures1 a ,,quadrangle or ,,squ>e is t :i* has all its ,sides equal1 & all its ,angles ,"r ,angles4 ,,def9ition ,,xxxi4 ,an ,,oblong is t :i* has all its ,angles "r ,angles1 b has n all its ,sides equal4 ,,def9ition ,,xxxii4 ,a ,,rhombus1 is t :i* has all its ,sides equal1 b its ,angles are n "r ,angles4 ,,def9ition ,,xxxiii4 ,a ,,rhomboid is t :i* has its opposite ,sides equal to "o ano!r1 b all its ,sides are n equal1 nor its ,angles ,"r ,angles4 ,,def9ition ,,xxxiv4 ,all o!r ,quadrilat]al ,figures besides ^!1 are call$ ,,trapeziums4 ,x is requisite t ! fir/ division ( quadrilat]al figures %\ld take place 9 two numb]s2 & t "s ( !m %\ld be call$ p>allelograms1 b o!rs non-p>allelograms4 ,b ( p>allelograms "s are rectangul> & equilat]al1 z quadrangles2 b o!rs nei!r ( ^!1 z rhomboids3 o!rs aga91 are rectangul>1 b n equilat]al1 z oblongs3 b o!rs1 on ! contr>y1 are equilat]al1 b n rectangul>1 z ! rhombuses4 ,= x is requisite ei!r to possess bo?1 viz4 equality ( sides & rectitude ( angles1 or nei!r2 or "o ( ^!1 & ?is 9 a tw(old respect4 ,h5ce a p>allelogram has a quadruple subsi/5ce4 ,b ( non-p>allelograms1 "s h only two p>allel sides1 & n ! re/2 b o!rs h none ( _! sides p>allel4 ,& ^? are call$ ,trapeziums1 b ^! ,trapezoids4 ,b ( ,trapeziums1 "s1 inde$1 h ! sides equal1 by :i* ! p>allel sides ( ?is k9d are conjo9$2 b o!rs unequal2 & ! =m] ( ^! are call$ isosceles trapeziums2 b ! latt] scal5e trapeziums4 ,a quadrilat]al figure1 !re=e1 is con/itut$ by u accord+ to a sev5-fold di/ribution4 ,= "o is a quadrangle2 b ! o!r an oblong2 ! ?ird a rhombus2 ! f\r? a rhomboides2 ! fif? an isosceles trapezium2 ! six? a scal5e trapezium2 ! sev5? a trapezoid4 ,b ,possidonius makes a p]fect division ( "r-l9$ quadrilat]al figures into s _m memb]s2 = he e/abli%es sev5 species ( ^!2 z likewise ( triangles4 ,b ,euclid c\ld n divide into p>allelograms & non p>allelograms1 because he nei!r m5tions p>allels1 nor tea*es u conc]n+ ! p>allelogram itself4 ,b trapeziums1 & all trapezoids1 he calls by a common "n1 describ+ trapeziums !mselves1 accord+ to ! diff]5ce ( ^? f\r figures#aha1 9 :i* ! prop]ty ( p>allelograms is v]ifi$4 ,& ?is is to h ! opposite sides & angles equal4 ,= a quadrangle & an oblong1 & a rhombus1 h _! opposite sides & angles equal4 ,b 9 a rhomboides he only adds ?is1 t its opposite sides are equal1 le/ he %\ld def9e x by negations alone1 s9ce he nei!r calls x equilat]al1 nor rectangul>4 ,= :]e we want prop] appellations1 x is necess>y to use su* z are common4 ,b we %\ld he> ,euclid %ew+ t ?is is common to all p>allelograms4 ,b a rhombus appe>s to be a quadrangle hav+ its sides mov$1 & a rhomboides a mov$ oblong4 ,h5ce1 accord+ to sides1 ^! d n diff] f ^?2 b !y v>y only accord+ to ! obtus5ess & acut5ess ( angles2 s9ce ! quadrangle & ! oblong are rectangul>4 ,= if y conceive a quadrangle or an oblong1 hav+ its sides drawn 9 su* a mann]1 t :ile two ( its opposite angles are dilat$1 ! o!r two are contract$2 !n ! dilat$ angles w appe> obtuse1 & ! contract$1 acute4 ,& ! appellation ( rhombus#ahb seems to h be5 impos$ f motion4 ,= if y conceive a quadrangle mov+ aft] ! mann] ( a rhombus1 x w appe> to y *ang$ 9 ord]1 accord+ to its angles3 j z if a circle is mov$ aft] ! mann] ( a sl+1 x w imm$iately exhibit ! appe>ance ( an ellipsis4 ,b "h y may p]haps enquire conc]n+ ! quadrangle1 :y x has ?is d5om9ation8 & :y ! appellation ( quadrangle may n be appli$ to o!r quadrilat]al figures1 z ! "n ( triangle is common to all ^? :i* are nei!r equiangul> nor equilat]al1 & 9 l mann] ( qu9quangles or p5tagons2 = ! geometrician1 9 ^!1 adds only ! p>ticle an equilat]al triangle1 or a qu9quangle1 :i* is equilat]al & equiangul>1 z if ^! c\ld n be o!rwise ?an su* z !y are8 ,b :5 he m5tions a quadrangle1 he imm$iately indicates t x mu/ be equilat]al & rectangul>4 ,b ! reason ( ?is is z foll[s3 a quadrangle alone has ! be/ space1 bo? accord+ to its sides & angles4 ,= ea* ( ! latt] is "r1 int]cept+ a measure ( angles1 :i* nei!r receives int5tion nor remission4 ,z x excels1 !re=e1 9 bo? respects1 x des]v$ly obta9s a common appellation4 ,b a triangle1 ?\gh x may h equal sides1 yet w 9 ?is case h all its angles acute1 & a qu9quangle all its angles obtuse4 ,s9ce1 !re=e1 ( all quadrilat]al figures1 a quadrangle alone is replete ) equality ( sides1 & rectitude ( angles1 x 0 n undes]v$ly allott$ ?is appellation3 =1 to excell5t =ms1 we (t5 d$icate ! "n ( ! :ole4 ,b x appe>$ also to ! ,py?agoreans1 t ?is prop]ty ( quadrilat]al figures1 pr9cipally convey$ an image ( a div9e ess5ce4 ,= !y p>ticul>ly signifi$ by ?is1 a pure & immaculate ord]4 ,s9ce rectitude imitates inflexibility1 b equality a firm & p]man5t p[]3 = motion emanates f inequality1 b quiet f equality itself4 ,! gods1 !re=e1 :o are ! au?ors to all ?+s ( /able disposition1 ( pure & uncontam9at$ ord]1 & ( indecl9able p[]1 are des]v$ly manife/$ z f an image1 by a quadrangul> figure4 ,b1 besides ^!1 ,philolaus also1 accord+ to ano!r appreh5sion1 calls a quadrangul> angle1 ! angle ( ,rhea1 ,c]es & ,ve/a4 ,=1 s9ce a quadrangle con/itutes ! e>?1 & is its proximate elem5t1 z we le>n f ,timu1 b ! e>? h]self receives f all ^! div9ities1 g5ital se$s1 & prolific p[]s1 he does n unju/ly consecrate ! angle ( a quadrangle to ^! goddesses1 ! be/[]s ( life4 ,= "s call bo? ! e>? & ,c]es1 ,ve/a#ahc1 & !y say t ,rhea totally p>ticipates h] nature1 & t all g5]ative causes are conta9$ 9 h] ess5ce4 ,philolaus1 !re=e1 says t a quadrangul> angle compreh5ds1 by a c]ta9 t]re/rial p[]1 "o union ( ! div9e g5]a4 ,b "s assimilate a quadrangle to univ]sal virtue1 s f> z e quadrangle f its p]fection has f\r "r angles4 ,j z we say t ea* ( ! virtues is p]fect1 cont5t ) itself1 ! measure & b\nd ( life1 & ! middle ( e ?+ :i*1 9 morals1 corresponds to ! obtuse & acute4 ,b x is by no means prop] to conceal1 t ,philolaus attributes a triangul> angle to f\r1 b a quadrangul> angle to ?ree gods1 exhibit+ _! alt]nate transition1 & ! community ( all ?+s 9 all1 ( odd natures 9 ! ev51 & ( ev5 9 ! odd4 ,h5ce1 ! tetradic t]n>y1 & ! triadic quat]n>y1 p>ticipat+ ( prolific & efficaci\s goods1 conta9 ! :ole ornam5t ( g5]able natures1 & pres]ve !m 9 _! prop] /ate4 ,f :i* ! duod5>y1 or ! numb] twelve1 is excit$ to a s+ul> unity1 viz4 ! gov]nm5t ( ,jupit]4 ,= ,philolaus says1 t ! angle ( a dodecagon 7or twelve-sid$ figure7 belongs to ,jove1 s f> z ,jupit] conta9s & pres]ves1 by 8 s+ul> union1 ! :ole numb] ( ! duod5>y4 ,= also1 accord+ to ,plato1 ,jupit] presides ov] ! duod5>y#ahd1 & gov]ns & mod]ates ! univ]se ) absolute sway4 ,& ?us mu* we h ?\ght prop] to disc\rse conc]n+ quadrilat]al figures1 z well decl>+ ! s5se ( our au?or1 z likewise af=d+ an occasion ( m pr(\nd inspections to su* z desire ! k ( intelligible & occult ess5ces4 ,,def9ition ,,xxxv4 ,,p>allel ,,"r ,,l9es are su* z be+ 9 ! same ,plane1 & produc$ bo? ways inf9itely1 w 9 no "p mutually co9cide4 ,:at ! elem5ts ( p>allels are1 & by :at accid5ts 9 ^! !y may be kn[n1 we %all aft]w>ds le>n3 b :at p>allel "r l9es are1 he def9es 9 ^! words3 8,x is requisite1 !re=e 7says he71 t !y %\ld be 9 "o plane1 & :ile !y are produc$ bo? ways h no co9cid5ce1 b be ext5d$ 9 inf9itum40 ,= non-p>allel l9es also1 if !y are produc$ to a c]ta9 di/ance1 w n co9cide4 ,b to be produc$ inf9itely1 )\t co9cid5ce1 expresses ! prop]ty ( p>allels4 ,nor yet ?is absolutely1 b to be ext5d$ bo? ways inf9itely1 & n co9cide4 ,= x is possible t non-p>allel l9es may also be produc$ "o way inf9itely1 b n ! o!r2 s9ce1 v]g+ 9 ?is "p1 !y are f> di/ant f mutual co9cid5ce 9 ! o!r4 ,b ! reason ( ?is is1 because two "r-l9es _c compreh5d space2 = if !y v]ge to ea* o!r bo? ways1 ?is _c happ54 ,besides ?is1 he v prop]ly consid]s ! "r-l9es z subsi/+ 9 ! same plane4 ,= if ! "o %\ld be 9 a subject plane1 b ! o!r 9 "o elevat$1 !y w n mutually co9cide accord+ to e position1 yet !y are n on ?is acc\nt p>allel4 ,! plane1 !re=e1 %\ld be "o1 & !y %\ld be produc$ bo? ways inf9itely1 & n co9cide 9 ei!r "p4 ,= ) ^! conditions1 ! "r-l9es w be p>allel4 ,& agreeable to ?is1 ,euclid def9es p>allel "r-l9es4 ,b ,posidonius says1 p>allel l9es are su* z nei!r incl9e nor div]ge 9 "o plane2 b h all ! p]p5dicul>s equal :i* are drawn f ! po9ts ( ! "o to ! o!r4 ,b su* l9es z make _! p]p5dicul>s always great] & less1 w "s "t or o!r co9cide1 because !y mutually v]ge to ea* o!r4 ,= a p]p5dicul> is capable ( b\nd+ ! altitudes ( spaces1 & ! di/ances ( l9es4 ,on :i* acc\nt1 :5 ! p]p5dicul>s are equal1 ! di/ances ( ! "r l9es are also equal2 b :5 !y are great] & less1 ! di/ance also becomes great] & less1 & !y mutually v]ge 9 ^? p>ts1 9 :i* ! less] p]p5dicul>s are f\nd4 ,b x is requisite to "k1 t non-co9cid5ce does n entirely =m p>allel l9es4 ,= ! circumf]5ces ( conc5tric circles d n co9cide3 b x is likewise requisite t !y %\ld be inf9itely produc$4 ,b ?is prop]ty is n only inh]5t 9 "r1 b also 9 o!r l9es3 = x is possible to conceive spirals describ$ 9 ord] ab\t "r l9es1 :i* if produc$ inf9itely toge!r ) ! "r l9es1 w nev] co9cide#ahe4 ,gem9us1 !re=e1 makes a v prop] division 9 ?is place1 affirm+ f ! beg9n+1 t ( l9es "s are b\nd$1 & conta9 figure1 z ! circle & ellipsis1 likewise ! cissoid1 & _m o!rs2 b o!rs are indet]m9ate1 :i* may be produc$ inf9itely1 z ! "r-l9e1 & ! section ( a "r-angl$1 & obtuse angl$ cone2 likewise ! con*oid itself4 ,b aga91 ( ^? :i* may be produc$ 9 inf9itum1 "s compreh5d no figure1 z ! "r-l9e & ! conic sections2 b o!rs1 return+ into !mselves1 & =m+ figure1 may aft]w>ds be inf9itely produc$4 ,& ( ^! "s w n h]eaft] co9cide1 :i* resi/ co9cid5ce1 h[ f> soev] !y may be produc$2 b o!rs are co9cid5t1 :i* w "s "t or o!r co9cide4 ,b ( non-co9cid5t l9es1 "s are mutually 9 "o plane2 & o!rs n4 ,& ( non-co9cid5ts subsi/+ 9 "o plane1 "s are always mutually di/ant by an equal int]val2 b o!rs always dim9i% ! int]val1 z an hyp]bola 9 its incl9ation to a "r-l9e1 & likewise ! con*oid#ahf4 ,= ^!1 ?\gh !y always dim9i% ! int]val1 nev] co9cide4 ,& !y mutually conv]ge1 inde$1 b nev] p]fectly nod to ea* o!r2 :i* is inde$ a !orem 9 geometry especially admirable1 exhibit+ c]ta9 l9es endu$ ) a non-ass5t+ nod4 ,b ! "r-l9es1 :i* are always di/ant by an equal int]val1 & :i* nev] dim9i% ! space plac$ betwe5 !m 9 "o plane1 are p>allel l9es4 ,& ?us mu* we h extract$ f ! /udies ( ! elegant ,gem9us1 = ! purpose ( expla9+ ! pres5t def9ition4 ,,end ,,( ,,! ,,fir/ ,,volume4 ,,footnotes3 #a ,! ,grecian lit]ature ( ?is writ] w n[ prove ( real utility2 & ! graces & ! sublimities ( ,,plato w soon be famili>is$ to ! ,engli% read]1 by a h& t ,i am p]suad$ w n appe> inf]ior to 8 great orig9al4 ,let me also be p]mitt$ to recomm5d 8 v]sion ( ,,plot9us on ,,! ,,beautiful4 #b i4e4 ,capable ( p>ts4 #c i4e4 ,n capable ( p>ts4 #d ,dr4 ,"y1 9 8 ,night ,?\ghts4 #e ,see book ! second1 ( ,ari/otle's ,metaphysics4 #f ,ennead vi4 lib4 vii4 #g ,9 8 comm5t>y on ! #b;d1 #ab?1 & #ac? books ( ,ari/otle's ,metaphysics1 page #fj4 ,a ,lat9 translation only ( ?is invaluable "w is extant2 b ,i h =tunately a copy 9 my possession1 ) ! v]sion e :]e correct$ by ! le>n$ ,?omas ,gale1 & ) l>ge extracts f ! ,greek4 #h ,see ,proclus on ,plato's ,!ology1 p4 #bbf4 #i ,ennead vi4 lib4 #f4 #aj ,9 giv+ monadic numb] a subsi/5ce 9 op9ion1 ,i h foll[$ ! di/ribution ( ,proclus1 9 ! conclusion ( 8 comm5t on a po9t2 &1 ,i ?9k1 n )\t suffici5t reason4 ,= s9ce monadic numb]s are m immat]ial ?an geometrical l9es & figures1 !y mu/ h a m immat]ial subsi/5ce4 ,b z !y are correspond5t to matt]1 !y _c reside 9 ! ess5tial reasons ( ! s\l2 nor c !y subsi/ 9 ! phantasy1 because !y are sup]ior to geometrical figures4 ,x rema9s1 !re=e1 t we mu/ place !m betwe5 or cogitation1 & ! phantasy2 & ?is middle situation is t ( op9ion4 ,= cogitation1 :i* ,plato def9es1 9 8 ,sophi/a1 to be an inw>d disc\rse1 )\t voice1 is an en]gy ( ! rational s\l1 ext5d+ itself f propositions to conclusions4 ,&1 accord+ to ,plato1 9 ! same place1 op9ion is ! sil5t affirmation1 or negation ( 1 or ?\ght4 ,h5ce1 says he1 8op9ion is ! conclusion ( cogitation2 b imag9ation1 ! mutual mixture ( s5se & op9ion40 ,s t op9ion may1 ) great propriety1 be said to conta9 monadic numb]1 to :i* x be>s ! proportion ( matt]4 ,& h5ce ! reason is obvi\s1 :y ! ,py?agoreans call$ ! duad op9ion4 #aa 1 1 4 ,syrian4 9 ,meta4 ,ari/ot4 p4 #aac4 ,gr4 i4e4 7,accord+ to ! ,py?agoreans7 8! immortal gods & e>?-born m51 call ! v5]able decad1 immutable & unwe>i$40 #ab 4 ,vid4 ,syrian4 9 ,ari/4 ,meta4 p4 #he4 ,gr4 #ac 1 4 ,syrian4 9 eodem loco4 #ad 1 4 4 ,jamb4 9 ,nicoma*4 ,ari?4 p4 #aa4 #ae ,9 8 ,ma!matical ,lectures1 page #dh4 #af ,9 ,ari?met4 p4 #bc4 #ag ,9 ,ari/ot4 ,meta4 p4 #aac4 ,gr4 vel #ei4 b4 ,lat4 #ah ,= ! tetrad conta9s all numb]s )9 its nature1 9 ! mann] ( an exempl>2 & h5ce x is1 t 9 monadic numb]s1 #a1 #b1 #c1 #d1 are equal to t54 #ai ,notes to ,lett]s on ,m9d1 page #hc4 #bj ,?is bright light is no o!r ?an t ( ideas !mselves2 :i*1 :5 x is once enk9dl$1 or r re-k9dl$ 9 ! s\l1 becomes ! g5]al /&>d1 & crit]ion ( tru?4 ,he :o possesses ?is1 is no long] ! slave ( op9ion2 puzzl$ ) d\bts1 & lo/ 9 ! unc]ta9ties ( conjecture4 ,"h ! f\nta9 ( evid5ce is alone to be f\nd4--,?is is ! true light1 ^: spl5dors c alone dispel ! d>kness ( ignorance1 & procure = ! s\l undecay+ good1 & sub/antial felicity4 ,( ?is ,i am c]ta91 f my own exp]i5ce2 & happy is he :o acquires ?is invaluable treasure4 ,b let ! read] bew>e ( mix+ ! extravagancies ( mod]n en?usiasm ) ?is exalt$ illum9ation4 ,= ?is light is alone br\ght into ! m9d by sci5ce1 pati5t reflection1 & unwe>i$ m$itation3 x is n produc$ by any viol5t agitation ( spirits1 or extasy ( imag9ation2 = x is f> sup]ior to ! en]gies ( ^!3 b x is tranquil & /eady1 intellectual & div9e4 ,avic5na1 ! ,arabian1 0 well acqua9t$ ) ?is light1 z is evid5t f ! beautiful description he gives ( x1 9 ! elegant introduction ( ,ebn ,tophail1 to ! ,life ( ,hai ,ebn ,yokdhan4 8,:5 a man's desires 7says he7 are consid]ably elevat$1 & he is compet5tly well ex]cis$ 9 ^! speculations1 !re w appe> to him "s small glimm]+s ( ! tru?1 z x 7 fla%es ( lightn+1 v delightful1 :i* j %9e ^u him1 & !n become ext9ct4 ,!n ! m he ex]cises himself1 ! (t5] w he p]ceive !m1 till at la/ he w become s well acqua9t$ ) !m1 t !y w occur to him spontane\sly1 )\t any ex]cise at all2 & !n z soon z he p]ceives any ?+1 he applies himself to ! div9e ess5ce1 s z to reta9 "s impression ( x2 !n some?+ occurs to him on a sudd51 :]eby he beg9s to disc]n ! tru? 9 e ?+2 till ?r\gh frequ5t ex]cise he at la/ atta9s to a p]fect tranquillity2 & t :i* us$ to appe> to him only by fits & />ts1 becomes habitual1 & t :i* 0 only a glimm]+ be=e1 a con/ant light2 & he obta9s a con/ant & /eady k40 ,he :o desires to "k m conc]n+ ?is1 & a /ill bright] light1 t aris+ f an union ) ! supreme1 mu/ consult ! eigh? book ( ,plot9us' fif? ,ennead1 & ! #g? & #i? ( ! six?1 & 8 book on ! ,beautiful1 ( :i* ,i h publi%$ a translation4 #ba ,le/ ! sup]ficial read] %\ld ?9k ?is is no?+ m ?an declamation1 let him att5d to ! foll[+ argum5t4 ,if ! s\l possesses ano!r eye diff]5t f t ( s5se 7& t %e does s1 ! sci5ces suffici5tly ev9ce71 !re mu/ be1 9 ! nature ( ?+s1 species accommodat$ to h] p]ception1 diff]5t f feasible =ms4 ,= if our intellect speculates ?+s :i* h no real subsi/5ce1 su* z ,mr4 ,locke's ideas1 its condition mu/ be mu* m unhappy ?an t ( ! s5sitive eye1 s9ce ?is is co-ord9at$ to be+s2 b intellect w\ld speculate no?+ b illusions4 ,n[1 if ?is be absurd1 & if we possess an intellectual eye1 :i* is endu$ ) a visive p[]1 !re mu/ be =ms correspond5t & conjo9$ ) its vision2 =ms immoveable1 inde$1 by a corporeal motion1 b mov$ by an intellectual en]gy4 #bb ,! pres5t section conta9s an illu/ration ( almo/ all ! fir/ book ( ,ari/otle's la/ ,analytics4 ,i h = ! mo/ "p foll[$ ! accurate & elegant p>aphrase ( ,!mi/ius1 9 ! execution ( ?is design1 z ! le>n$ read] w p]ceive3 b ,i h likewise ev]y:]e add$ elucidations ( my own1 & endeav\r$ to r5d] ?is valuable "w intelligible to ! ?9k+ ma!matical read]4 #bc ,see ! tw5ty-eigh? proposition ( ! fir/ book ( ,euclid's ,elem5ts4 #bd ,we are in=m$ by ,simplicius1 9 8 ,comm5t>y on ,ari/otle's ?ird ,category ( ,relation1 8t ?\gh ! quadrature ( ! circle seems to h be5 unkn[n to ,ari/otle1 yet1 accord+ to ,jambli*us1 x 0 kn[n to ! ,py?agoreans1 z appe>s f ! say+s & demon/rations ( ,sextus ,py?agoricus1 :o receiv$ 7says he7 by succession1 ! art ( demon/ration2 & aft] him ,ar*im$es succe$$1 :o inv5t$ ! quadrature by a l9e1 :i* is call$ ! l9e ( ,nicom$es4 ,likewise1 ,nicom$es attempt$ to squ>e ! circle by a l9e1 :i* is prop]ly call$ 1 or ! quadrature4 ,& ,apollonius1 by a c]ta9 l9e1 :i* he calls ! si/] ( ! curve l9e1 simil> to a cockle1 or tortoise1 & :i* is ! same ) ! quadratix ( ,nicom$es4 ,also ,c>pus wi%$ to squ>e ! circle1 by a c]ta9 l9e1 :i* he calls simply =m$ f a tw(old motion4 ,& _m o!rs1 accord+ to ,jambli*us1 h accompli%$ ?is und]tak+ 9 v>i\s ways40 ,?us f> ,simplicius4 ,9 l mann]1 ,boe?ius1 9 8 ,comm5t>y on ! same "p ( ,ari/otle's ,categories 7p4 #aff47 obs]ves1 t ! quadrature ( ! circle 0 n discov]$ 9 ,ari/otle's "t1 b 0 f\nd out aft]w>ds2 ! demon/ration ( :i* 7says he7 because x is long1 mu/ be omitt$ 9 ?is place4 ,f h5ce x seems v probable1 t ! anci5t ma!maticians appli$ !mselves solely to squ>+ ! circle geometrically1 )\t attempt+ to accompli% ?is by an ari?metical calculation4 ,inde$1 no?+ c be m ungeometrical ?an to expect1 t if ev] ! circle be squ>$1 ! squ>e to :i* x is equal mu/ be comm5surable ) o!r kn[n rectil9eal spaces2 = ^? :o are skill$ 9 geometry "k t _m l9es & spaces may be exhibit$ ) ! greate/ accuracy1 geometrically1 ?\gh !y are incapable ( be+ express$ ari?metically1 )\t an inf9ite s]ies4 ,agreeable to ?is1 ,tacquet well obs]ves 79 lib4 ii4 ,geom4 ,pract4 p4 #hg47 8,d5ique admon5di hic sunt1 qui geometri1 non satis p]iti1 sibi p]suad5t ad quadraturam necess>ium esse1 ut ratio l9e circul>is ad rectam1 aut circuli ad quadratum 9 num]is exhibeatur4 ,is sane error valde crassus e/1 et indignus geometr1 quamvis enim irrationalis esset ea proportio1 modo 9 rectis l9eis exhibeatur1 rep]ta erat quadratura40 ,& t ?is quadrature is possible geometrically1 0 n only ! op9ion ( ! above m5tion$ le>n$ & acute geometrician1 b likewise ( ,wallis & ,b>r[2 z may be se5 9 ! ,me*anics ( ! =m]1 p4 #eag & 9 ! ,ma!matical ,lectures ( ! latt]1 p4 #aid4 ,b ! foll[+ discov]y w1 ,i hope1 conv9ce ! lib]al geometrical read]1 t ! quadrature ( ! circle may be obta9$ by means ( a circle & "r-l9e only1 :i* we h no me?od ( accompli%+ by any inv5tion ( ! anci5ts or mod]ns4 ,at lea/ ?is me?od1 if kn[n to ! anci5ts1 is n[ lo/1 & ?\gh x has be5 attempt$ by _m ( ! mod]ns1 x has n be5 att5d$ ) success4 ,9 ! circle g o e f1 let g o be ! quadrantal ar*1 & ! "r-l9e g x its tang5t4 ,!n conceive t ! c5tral po9t a fl[s uni=mly along ! radius a e1 inf9itely produc$2 & t x is endu$ ) an uni=m impulsive p[]4 ,let x likewise be suppos$1 t dur+ its flux1 radii emanate f x on all sides1 :i* enl>ge !mselves 9 proportion to ! di/ance ( ! po9t a f its fir/ situation4 ,?is be+ admitt$1 conceive t ! po9t a by its impulsive p[]1 ?r\gh ! radii a n1 a m1 c4 act+ e :]e equally on ! ar* g o1 impells x into its equal tang5t ar* g r4 ,& :51 by its uni=m motion along ! inf9ite l9e a 1 x has at ! same "t arriv$ at b1 ! c5tre ( ! ar* g r1 let x impel 9 a simil> mann] ! ar* g r1 into its equal tang5t ar* g s1 by act+ e :]e equally ?r\gh radii equal to b r4 ,n[1 if ?is be conceiv$ to take place inf9itely 7s9ce a circul> l9e is capable ( inf9ite remission7 ! ar* g o w at l5g? be unb5t into its equal1 ! tang5t l9e g x2 & ! extreme po9t o1 w describe by su* a motion ( unb5d+ a circul> l9e o x4 ,= s9ce ! same cause1 act+ e :]e simil>ly & equally1 produces e :]e simil> & equal effects2 & ! ar* g o1 is e :]e equally remitt$ or unb5t1 x w describe a l9e simil> 9 e "p4 ,n[1 on acc\nt ( ! simplicity ( ! impulsive motion1 su* a l9e mu/ ei!r be /raight or circul>2 = !re are only ?ree l9es e :]e simil>1 i4 e4 ! "r & circul> l9e1 & ! cyl9dric helix2 b ?is la/1 z ,proclus well obs]ves 9 8 foll[+ ,comm5t>y on ! f\r? def9ition1 is n a simple l9e1 because x is g5]at$ by two simple motions1 ! rectil9eal & circul>4 ,b ! l9e :i* b\nds m ?an two equal tang5t ar*es _c be a "r l9e1 z is well kn[n to all geometricians2 x is !re=e a circul> l9e4 ,x is likewise evid5t1 t ?is ar* o x is concave t[>ds ! po9t g3 = if n1 x w\ld pass beyond ! *ord o x1 :i* is absurd4 ,& aga91 no ar* great] ?an ! quadrant c be unb5t by ?is motion3 = any "o ( ! radii1 z a p beyond g o1 has a t5d5cy f1 & n to ! tang5t g x1 :i* la/ is necess>y to our hypo!sis4 ,n[ if we conceive ano!r quadrantal ar* ( ! circle g o e f1 t is g y1 t\*+ ! =m] 9 g to be unb5t 9 ! same mann]1 ! ar* x y %all be a cont9uation ( ! ar* x o2 = if x be drawn p]p5dicul> to x g1 z 9 ! figure1 x %all be a tang5t 9 x to ! equal ar*es y x1 x o2 because x _c fall )9 ei!r1 )\t mak+ ! s9e ( "s "o ( ! equal ar*es1 equal to ! "r-l9e x g1 :i* w\ld be absurd4 ,& h5ce we may easily inf]1 t ! c5tre ( ! ar* y x o1 is 9 ! tang5t l9e x g4 ,h5ce too1 we h an easy me?od ( f9d+ a tang5t "r-l9e equal to a quadrantal ar*3 = hav+ ! po9ts y1 o giv51 x is easy to f9d a ?ird po9t1 z s2 & !n ! circle pass+ ?r\gh ! ?ree po9ts o1 s1 y1 %all cut (f ! tang5t x g1 equal to ! quadrantal ar* g o4 ,& ! po9t s may be spe$ily obta9$1 by describ+ ! ar* g s ) a radius1 hav+ to ! radius a g ! proportion ( #f to #d2 = !n g s is ! six? "p ( its :ole circle1 & is equal to ! ar* g o4 ,& ?us1 f ?is hypo!sis1 :i*1 ,i presume1 may be z readily admitt$ z ! increm5ts & decrem5ts ( l9es 9 fluxions1 ! quadrature ( ! circle may be geometrically obta9$2 = ?is is easily f\nd1 :5 a "r-l9e is discov]$ equal to ! p]iph]y ( a circle4 ,i am well aw>e ! algebrai/s w consid] x z useless1 because x _c be accommodat$ to ! f>rago ( an ari?metical calculation2 b ,i hope ! lov]s ( ! anci5t geometry w deem x des]v+ an accurate inve/igation2 & if !y c f9d no p>alogism 9 ! reason+1 w consid] x z a legitimate demon/ration4 ,illu/ration ,illu/ration #be ,axioms h a subsi/5ce prior to t ( magnitudes & ma!matical numb]s1 b subord9ate to t ( ideas2 or1 9 o!r words1 !y h a middle situation betwe5 ess5tial & ma!matical magnitude4 ,= ( ! reasons subsi/+ 9 s\l1 "s are m simple & univ]sal1 & h a great] ambit ?an o!rs1 & on ?is acc\nt approa* ne>] to intellect1 & are m manife/ & kn[n ?an su* z are m p>ticul>4 ,b o!rs are de/itute ( all ^!1 & receive _! completion f m anci5t reasons4 ,h5ce x is necess>y 7s9ce conceptions are !n true1 :5 !y are consonant ) ?+s !mselves7 t !re %\ld be "s reason1 9 :i* ! axiom ass]t+1 if f equals y take away equals1 c4 is prim>ily inh]5t2 & :i* is nei!r ! reason ( magnitude1 nor numb]1 nor "t1 b conta9s all ^!1 & e ?+ 9 :i* ?is axiom is naturally inh]5t4 ,vide ,syrian4 9 ,ari?4 ,meta4 p4 #dh4 #bf ,geometry1 inde$1 wi%es to speculate ! imp>tible reasons ( ! s\l1 b s9ce %e _c use intellections de/itute ( imag9ation1 %e ext5ds h] disc\rses to imag9ative =ms1 & to figures endu$ ) dim5sion1 & by ?is means speculates immat]ial reasons 9 ^!2 & :5 imag9ation is n suffici5t = ?is purpose1 %e proce$s ev5 to ext]nal matt]1 9 :i* %e describes ! fair v>iety ( h] propositions4 ,b1 inde$1 ev5 !n ! pr9cipal design ( geometry is n to appreh5d s5sible & ext]nal =m1 b t int]ior vital "o1 resid5t 9 ! mirror ( imag9ation1 :i* ! ext]ior inanimate =m imitates1 z f> z its imp]fect nature w admit4 ,nor yet is x h] pr9cipal design to be conv]sant ) ! imag9ative =m2 b :51 on acc\nt ( ! imbecility ( h] intellection1 %e _c receive a =m de/itute ( imag9ation1 %e speculates ! immat]ial reason 9 ! pur] =m ( ! phantasy2 s t h] pr9cipal employm5t is ab\t univ]sal & immat]ial =ms4 ,syrian4 9 ,ari/4 ,meta4 p4 #di4 #bg ,syrianus1 9 8 excell5t ,comm5t>y on ,ari/otle's ,metaphysics1 7:i* does n s mu* expla9 ,ari/otle1 z def5d ! doctr9e ( ideas1 accord+ to ,plato1 f ! app>5t if n real opposition ( ,ari/otle to _! exi/5ce71 in=ms u t x is ! bus9ess ( wisdom1 prop]ly s call$1 to consid] immat]ial =ms or ess5ces1 & _! ess5tial accid5ts4 ,by ! me?od ( resolution receiv+ ! pr9ciples ( be+2 by a divisive & & def9itive me?od1 consid]+ ! ess5ces ( all ?+s2 b by a demon/rative process1 conclud+ conc]n+ ! ess5tial prop]ties :i* sub/ances conta94 ,h5ce 7says he7 because intelligible ess5ces are ( ! mo/ simple nature1 !y are nei!r capable ( def9ition nor demon/ration1 b are p]ceiv$ by a simple vision & en]gy ( intellect alone4 ,b middle ess5ces1 :i* are demon/rable1 exi/ accord+ to _! inh]5t prop]ties3 s9ce1 9 ! mo/ simple be+s1 no?+ is inh]5t besides _! be+4 ,on :i* acc\nt we _c say t ?is is _! ess5ce1 & t some?+ else2 & h5ce !y are bett] ?an def9ition & demon/ration4 ,b 9 univ]sal reasons1 consid]$ by !mselves1 & adorn+ a s5sible nature1 ess5tial accid5ts sup]v5e2 & h5ce demon/ration is conv]sant ) ^!4 ,b 9 mat]ial species1 individuals1 & s5sibles1 su* ?+s z are prop]ly accid5ts are p]ceiv$ by ! imag9ation1 & are pres5t & abs5t )\t ! corruption ( _! subjects4 ,& ^! aga9 be+ worse ?an demon/rable accid5ts1 are appreh5d$ by signs1 n inde$ by a wise man1 consid]$ z wise1 b p]haps by physicians1 natural philosoph]s1 & all ( ?is k9d4 #bh ,see ,note to ,*ap4 i4 ,book i4 ( ! ensu+ ,comm5t>ies4 #bi ,page #bbg4 #cj ,page #bej4 #ca ,me?odus hc cum algebr specios facilitate cont5dit1 evid5ti v]o et demon/rationum eleganti eam longe sup]>e videtur3 ut abunde con/abit1 si quis conf]at hanc ,apollonii doctr9am de ,sectione ,rationis cum ejusdem ,problematis ,analysi ,algebraic1 quam exhibuit cl>issimus ,wallisius1 tom4 ii4 ,op]um ,ma?4 cap4 liv4 p4 #bbj4 #cb ,v]um p]p5d5dum e/1 aliud esse problema aliqualit] resolutum d>e1 quod modis v>iis1 pl]umque fi]i pote/1 aliud me?odo elegantissim ipsum effic]e2 ,analysi brevissim et simul p]spicu1 ,syn!si conc9n et m9ime op]os4 #cc ,9 8 ,ma!matical ,lectures1 p4 #dd4 #cd ,lib4 iv4 #ce ,lib4 i4 p4 #cj4 #cf ,9 ,!teto4 #cg ,9 8 mo/ excell5t "w on ,ab/95ce1 lib4 i4 p4 #bb1 c4 #ch ,see ! ,exc]pta ( ,fic9us f ,proclus1 on ! fir/ ,alcibiades ( ,plato2 8 ,lat9 v]sion only ( :i* is extant4 ,fic9i ,op]a1 tom4 ii4 #ci ,m>9us1 ! au?or ( ! ensu+ life1 0 ! disciple ( ,proclus2 & 8 successor 9 ! ,a!nian s*ool4 ,8 philosophical writ+s 7 n v num]\s1 & h n be5 pres]v$4 ,a comm5t>y ascrib$ to him1 on ,euclid's data1 is /ill extant2 b 8 mo/ celebrat$ "w1 appe>s to h be51 ! pres5t life ( 8 ma/]4 ,x is inde$ 9 ! orig9al elegant & concise2 & may be consid]$ z a v happy specim5 ( philosophical biography4 ,e lib]al m9d mu/ be *>m$ & elevat$ ) ! gr&eur & sublimity ( "*1 ) :i* ,proclus is pres5t$ to our view4 ,if comp>$ ) mod]n philosophical h]oes1 he appe>s to be a be+ ( a sup]ior ord]2 & we look back ) regret on ! glori\s p]iod1 s well calculat$ = ! gr[? ( ! philosophical g5ius1 & ! enc\ragem5t ( exalt$ m]it4 ,we f9d 9 8 life1 no traces ( ! common frailties ( deprav$ humanity2 no in/ances ( meanness1 or in/ability ( conduct3 b he is uni=mly magnific5t1 & con/antly good4 ,i am well aw>e t ?is acc\nt ( him w be consid]$ by _m z highly exagg]at$2 z ! result ( weak en?usiasm1 bl9d sup]/ition1 or gross deception3 b ?is w nev] be ! p]suasion ( ^?1 :o "k by exp]i5ce :at elevation ( m9d & purity ( life ! ,platonic philosophy is capable ( procur+2 & :o truly und]/& ! div9e tru?s conta9$ 9 8 works4 ,& ! te/imony ( ! multitude1 :o measure ! m]it ( o!r m5's *>act]s by ! bas5ess ( _! own1 is surely n to be reg>d$4 ,i only add1 t our ,philosoph] fl\ri%$ #dab ye>s aft] ,*ri/1 accord+ to ! accurate *ronology ( ,fabricius2 & ,i w\ld recomm5d ^? :o desire a v>iety ( critical in=mation conc]n+ ,proclus1 to ! ,prolegom5a prefix$ by t mo/ le>n$ man to 8 excell5t ,greek & ,lat9 edition ( ?is "w1 pr9t$ at ,london 9 #agjc4 #dj ,plato 9 ,phdro4 ,mem9it et ,plut>*4 ,,viii4 ,sympos4 ,suidas 9 4 ,fabricius4 #da ,= a full acc\nt ( ! di/ribution ( ! virtues accord+ to ! ,platoni/s1 consult ! s5t5ces ( ,porphyry1 & ! ,prolegom5a ( ,fabricius to ?is "w4 #db ,see ! six? book ( 8 ,republic1 & ! ,ep9omis4 #dc ,we are in=m$ by ,fabricius1 t ! ,platonic ,olympiodorus 9 8 ,,ms4 ,comm5t>y on ! ,alcibiades ( ,plato1 divides ! ord]s ( ! ,gods1 into 1 or sup]-mundane1 :i* are sep>ate f all connection ) body2 & into 1 or mundane4 ,& t ( ^!1 "s are 1 or cele/ial1 o!rs 1 or1 or e!rial1 or 1 fi]y1 o!rs 1 or a]ial1 o!rs 1 or watry1 o!rs 1 or e>?ly2 & o!rs 1 or subt]ranean4 ,b among ! t]re/rial1 "s are 1 or gov]nors ( climates1 o!rs 1 or rul]s ov] cities1 & o!rs la/ly 1 or gov]nors ( h\ses4 #dd ,?is epi!t is likewise ascrib$ by ,onomacritus to ! ,moon1 z may be se5 9 8 hymn to t deity2 & ! reason ( :i* we h giv5 9 our notes to t hymn4 #de ,div9e visions1 & extraord9>y circum/ances1 may be fairly all[$ to happ5 to su* exalt$ g5iuses z ,proclus2 b des]ve ridicule :5 ascrib$ to ! vulg>4 #df ,:at glori\s times6 :5 x 0 consid]$ z an extraord9>y circum/ance = a tea*] ( rhetoric to treat a noble & weal?y pupil z 8 dome/ic4 ,:5 we comp>e !m ) ! pres5t1 we c only exclaim1 ,o tempora6 ,o mores6 ,philosophy sunk 9 ! ru9s ( anci5t ,greece & ,rome4 #dg ,fabricius rightly obs]ves1 t ?is ,olympiodorus is n ! same ) ! ,philosoph] ( t "n1 ^: le>n$ comm5t>ies1 on c]ta9 books ( ,plato1 are extant 9 manuscript1 9 v>i\s libr>ies4 ,z 9 ^!1 n only ,proclus himself1 b ,damascius1 :o fl\ri%$ long aft] ,proclus1 is celebrat$4 #dh ,conc]n+ ! v>i\s ma!maticians ( ?is "n1 see ,fabricius 9 ,biblio!ca ,grca4 #di ,! ^w 9 ! orig9al is 1 :i* ,fabricius rightly conjectures has 9 ?is place a m ext5sive signification ?an ei!r ,logic1 or ,rhetoric3 b ,i mu/ beg leave to diff] f t great critic1 9 n translat+ x simply philosophical1 z ,i %\ld r imag9e1 ,m>9us int5d$ to conf9e x to t "p ( ,ari/otle's works1 :i* compreh5ds only logic1 rhetoric1 & poetry4 ,= ! v]b 1 or to le>n1 :i* ,m>9us uses on ?is occasion1 _c ) propriety be appli$ to ! m ab/ruse writ+s ( ,ari/otle4 #ej ,h5ce ,proclus 0 call$1 by way ( em95ce1 1 or ! ,platonic ,successor4 #ea ,conc]n+ ,polletes1 see ,suidas2 & = ,melampodes1 consult ,fabricius 9 ,biblio!ca ,grca4 #eb ,?is ,syrianus 0 inde$ a mo/ excell5t philosoph]1 z we may be conv9c$ f 8 comm5t>y on ! metaphysics ( ,ari/otle1 a ,lat9 translation only ( :i*1 by "o ,hi]onimus ,bagol9us1 0 publi%$ at ,v5ice 9 #aeeh4 ,! ,greek is extant1 accord+ to ,fabricius1 9 _m ( ! ,italian libr>ies1 & 9 ! ,johannean libr>y at ,hamburg4 ,accord+ to ,suidas1 he writ a comm5t>y on ! :ole ( ,hom] 9 six books2 on ,plato's politics1 9 f\r books2 & on ! cons5t ( ,orpheus1 ,py?agoras1 & ,plato1 ) ! ,*aldean ,oracles1 9 t5 books4 ,all ^! are un=tunately lo/2 & ! lib]al few1 are by ?is means depriv$ ( treasures ( wisdom1 :i* ano!r philosophical age1 9 "s di/ant revolution1 is alone likely to produce4 #ec ,socrates1 9 ! #f? book ( ,plato's ,republic1 says1 t f great g5iuses no?+ ( a middle k9d mu/ be expect$2 b ei!r great good1 or great evil4 #ed ,! read] w please to take notice1 t ?is great man is n ! same ) ,plut>* ! biograph]1 ^: works are s well kn[n2 b an ,a!nian philosoph] ( a mu* lat] p]iod4 #ee ,ari/otle's philosophy1 :5 comp>$ ) ! discipl9e ( ,plato is1 ,i ?9k1 des]v$ly consid]$ 9 ?is place z be>+ ! relation ( ! proteleia to ! epopteia 9 sacr$ my/]ies4 ,n[ ! proteleia1 or ?+s previ\s to p]fection1 belong to ! initiat$1 & ! my/ics2 ! =m] ( :om 7 introduc$ into "s light] c]emonies only3 b ! my/ics1 7 p]mitt$ to be pres5t ) c]ta9 prelim9>y & less] sacr$ conc]ns4 ,on ! o!r h&1 ! epopt 7 admitt$ into ! sanctu>y ( ! great] sacr$ rites2 & became spectators ( ! symbols1 & m int]ior c]emonies4 ,ari/otle inde$ appe>s to be e :]e an enemy to ! doctr9e ( ideas1 z und]/ood by ,plato2 ?\gh !y are d\btless ! lead+ />s ( all true philosophy4 ,h[ev]1 ! great excell5ce ( 8 works1 consid]$ z an introduction to ! div9e !ology ( ,plato1 des]ves ! mo/ unb\nd$ comm5dation4 ,agreeable to ?is1 ,damascius in=ms u t ,isidorus ! philosoph]1 8:5 he appli$ himself to ! m holy philosophy ( ,ari/otle1 & saw t he tru/$ m to necess>y reasons ?an to 8 own prop] s5se1 yet did n entirely employ a div9e intellection1 0 b little solicit\s ab\t 8 doctr9e3 b t :5 he _h ta/$ ( ,plato's conceptions1 he no long] deign$ to behold him 9 ! language ( ,p9d>4 ,b hop+ he %\ld obta9 8 desir$ end1 if he c\ld p5etrate into ! sanctu>y ( ,plato's m9d1 he direct$ to ?is purpose ! :ole c\rse ( 8 application40 ,photii ,biblio!ca4 p4 #ajcd4 #ef accord+ to ! oracle4 #eg ,no?+ is m celebrat$ by ! anci5ts ?an t /rict fri5d%ip :i* subsi/$ among ! ,py?agoreans2 to ! ex]cise ( :i* !y 7 accu/om$ to admoni% ea* o!r1 n to divide ! god :i* !y conta9$1 z ,jambli*us relates1 lib4 i4 c4 #cc4 ,de ,vita ,py?agor4 ,inde$1 true fri5d%ip c alone subsi/ 9 s\ls1 prop]ly enlight5$ ) g5u9e wisdom & virtue2 = x !n becomes an union ( intellects1 & mu/ consequ5tly be immortal & div9e4 #eh ,py?agoras1 accord+ to ,damascius1 said1 t fri5d%ip 0 ! "m ( all ! political virtues4 #ei ,a g5u9e mod]n w d\btless consid] ! :ole ( ,proclus' religi\s conduct z ridicul\sly sup]/iti\s4 ,& s1 inde$1 at fir/ sight1 x appe>s2 b he :o has p5etrat$ ! dep?s ( anci5t wisdom1 w f9d 9 x m ?an meets ! vulg> e>4 ,! religion ( ! ,hea!ns1 has inde$1 = _m c5turies1 be5 ! object ( ridicule & contempt3 yet ! au?or ( ! pres5t "w is n a%am$ to own1 t he is a p]fect conv]t to x 9 e p>ticul>1 s f> z x 0 und]/ood & illu/rat$ by ! ,py?agoric & ,platonic philosoph]s4 ,inde$ ! !ology ( ! anci5t1 z well z ( ! mod]n vulg>1 0 no d\bt full ( absurdity2 b t ( ! anci5t philosoph]s1 appe>s to be wor?y ( ! highe/ comm5dations1 & ! mo/ assidu\s cultivation4 ,h[ev]1 ! pres5t prevail+ op9ions1 =bid ! def5ce ( su* a sy/em2 = ?is mu/ be ! bus9ess ( a m enlight5$ & philosophic age4 ,besides1 ! au?or is n =getful ( ,porphyry's de/9y1 ^: polemical writ+s 7 suppress$ by ! decrees ( emp]ors2 & ^: argum5ts 9 def5ce ( 8 religion 7 s v futile & easy ( solution1 t1 z ,/4 ,hi]om in=ms u1 9 8 preface on ,daniel1 ,eusebius answ]$ him 9 tw5ty-five1 & ,apoll9>is 9 ?irty volumes6 #fj ,see ,proclus on ,plato's ,politics1 p4 #cii4 ,in/it4 ,!olog4 num4 #aif2 & ! extracts ( ,fic9us f ,proclus's comm5t>y on ! fir/ ,alcibiades1 p4 #bdf4 c4 #fa ,allud+ to ! beautiful description giv5 ( ,ulysses1 9 ! #c;d book ( ! ,iliad1 v4 #bbb4 4 ,:i* is ?us elegantly p>aphras$ by ,mr4 ,pope4 ,b :5 he speaks1 :at elocution fl[s6 ,s(t z ! fleeces ( desc5d+ sn[s ,! copi\s acc5ts fall1 ) easy art2 ,melt+ !y fall1 & s9k into ! he>t6 c4 #fb ,conc]n+ ,domn9us1 see ,photius & ,suidas f ,damascius 9 8 ,life ( ,isidorus4 #fc ,nicephorus1 9 8 comm5t>y on ,synesius de ,insomniis1 p4 #efb4 in=ms u1 t ! hecatic orb1 is a gold5 sph]e1 :i* has a sapphire /one includ$ 9 its middle "p1 & ?r\gh its :ole extremity1 *>act]s & v>i\s figures4 ,he adds1 t turn+ ?is sph]e r\nd1 !y p]=m invocations1 :i* !y call ,jyng4 ,?us too1 accord+ to ,suidas1 ! magician ,julian ( ,*aldea1 & ,arnuphis ! ,egyptian1 br\ght d[n %[]s ( ra91 by a magical p[]4 ,& by an artifice ( ?is k9d1 ,emp$ocles 0 accu/om$ to re/ra9 ! fury ( ! w9ds2 on :i* acc\nt he 0 call$ 1 or a *as] ( w9ds4 #fd ,no op9ion is m celebrat$1 ?an t ( ! metempsy*osis ( ,py?agoras3 b p]haps1 no doctr9e is m g5]ally mi/ak54 ,by mo/ ( ! pres5t day x is explod$ z ridicul\s2 & ! few :o reta9 "s v5]ation = its f\nd]1 endeav\r to de/roy ! lit]al1 & to conf9e x to an allegorical mean+4 ,by "s ( ! anci5ts ?is mutation 0 limit$ to simil> bodies3 s t !y conceiv$ ! human s\l might transmigrate into v>i\s human bodies1 b n into ^? ( brutes2 & ?is 0 ! op9ion ( ,hi]ocles1 z may be se5 9 8 comm5t on ! ,gold5 ,v]ses4 ,b :y may n ! human s\l become connect$ ) subord9ate z well z ) sup]ior lives1 by a t5d5cy ( incl9ation8 ,d n simil>s love to be unit$2 & is !re n 9 all k9ds ( life1 some?+ simil> & common8 ,h5ce1 :5 ! affections ( ! s\l v]ge to a bas] nature1 :ile connect$ ) a human body1 ^! affections1 on ! dissolution ( su* a body1 become envelop$ z x 71 9 a brutal nature1 & ! rational eye1 9 ?is case1 cl\d$ ) p]turbations1 is oppress$ by ! irrational en]gies ( ! brute1 & surveys no?+ b ! d>k phantasms ( a degrad$ imag9ation4 ,b ?is doctr9e is v9dicat$ by ,proclus ) 8 usual subtilty1 9 8 admirable comm5t>y on ! ,timu1 lib4 v4 p4 #cbi1 z foll[s1 8,x is usual1 says he1 to enquire h[ s\ls c desc5d into brute animals4 ,& "s1 inde$1 ?9k t !re are c]ta9 similitudes ( m5 to brutes1 :i* !y call savage lives3 = !y by no means ?9k x possible t ! rational ess5ce c become ! s\l ( a savage animal4 ,on ! contr>y1 o!rs all[ x may be s5t into brutes1 because all s\ls are ( "o & ! same k9d2 s t !y may become wolves & pan!rs1 & i*neumons4 ,b true reason1 inde$1 ass]ts t ! human s\l way be lodg$ 9 brutes1 yet 9 su* a mann]1 z t x may obta9 its own prop] life1 & t ! degrad$ s\l may1 z x 71 be c>ri$ above x1 & be b\nd to ! bas] nature1 by a prop5sity & similitude ( affection4 ,& t ?is is ! only mode ( ins9uation1 we h prov$ by a multitude ( reasons1 9 our comm5t>ies on ! ,phdrus4 ,b if x is requisite to take notice1 t ?is is ! op9ion ( ,plato1 we add1 t 9 8 politics1 he says1 t ! s\l ( ,!rsites assum$ an ape1 b n ! body ( an ape3 & 9 ! ,phdrus1 t ! s\l desc5ds into a savage life1 b n into a savage body2 = life is conjo9$ ) its prop] s\l4 ,& 9 ?is place he says x is *ang$ into a brutal nature3 = a brutal nature is n a brutal body1 b a brutal life40 #fe ,p]icles ,lydus1 a ,/oic philosoph]4 #ff ,vide ,pausan4 lib4 i4 ,atticorum1 cap4 #ba4 et #bj4 #fg ,he means ! ,*ri/ians4 #fh ,proclus 0 born 9 ! ye> ( ,*ri/ #dab1 on ! #f? ( ! ,ides ( ,febru>y4 ,b1 = ! sake ( ! a/rolog]s1 ,i h subjo9$ ! foll[+ figure f ! ,prolegom5a ( ,fabricius to ?is life3 & ?\gh ,i am n skill$ 9 ! art myself1 ,i am p]suad$1 f ! argum5ts ( ,plot9us1 t x conta9s _m g5]al tru?s2 b :5 made subs]vi5t to p>ticul>s1 is liable to great inaccuracy & error4 ,9 %ort1 its evid5ce is :olly ( a physiognomic nature2 = su* is ! admirable ord] & connection ( ?+s1 t ?r\gh\t ! univ]se1 "o ?+ is signifi$ by ano!r1 & :oles are aft] a mann] conta9$ 9 _! p>ts4 ,s t ! language ( ! obscure & pr(\nd ,h]aclitus is p]fectly j1 :5 he says1 8,y mu/ connect ! p]fect & ! imp]fect1 ! agree+ & ! disagree+1 ! consonant & ! dissonant1 & out ( "o all ?+s1 & out ( all ?+s "o40 ,illu/ration3 ,a ,s*eme ( ! situation ( ! ,/>s1 su* z x 0 at ,byzantium1 :5 ! philosoph] ,proclus 0 born4 #fi ,x 0 =m]ly ! cu/om ( almo/ all nations1 to h _! burial places 9 ! suburbs1 & n 9 ! city itself4 #gj ,?is eclipse happ5$1 accord+ to ,fabricius1 9 ,a4 ,c4 #dhd4 #ai ,cal4 ,feb4 at sun-rise4 #ga ,all ! anci5t !ologi/s1 among :om ,plato holds a di/+ui%$ rank1 affirm$ t ! s\l 0 ( a c]ta9 middle nature & condition betwe5 intelligibles & s5sibles3 agreeable to :i* doctr9e1 ,plot9us div9ely ass]ts t %e is plac$ 9 ! horizon1 or 9 ! b\nd>y & i/hmus1 z x 71 ( et]nal & mortal natures2 & h5ce1 accord+ to ! ,magi1 %e is simil> to ! moon1 "o ( ^: p>ts is lucid1 b ! o!r d>k4 ,n[ ! s\l1 9 consequ5ce ( ?is middle condition1 mu/ necess>ily be ! receptacle ( all middle en]gies1 bo? vital & gno/ic2 s t h] k is inf]ior to ! indivisible simplicity ( intellectual compreh5sion1 b sup]ior to ! impulsive p]ceptions ( s5se4 ,h5ce ! ma!matical g5]a & species reside 9 h] ess5ce1 z 9 _! prop] & natural region2 = !y are entirely ( a middle nature1 z ,proclus proves 9 ?is & ! six? foll[+ *apt]4 ,b ?is doctr9e ( ,plato's1 orig9ally d]iv$ f ,bront9us & ,ar*ytas1 is ?us elegantly expla9$ by t philosoph]1 9 ! conclud+ "p ( ! six? book ( 8 ,republic4 8,socrates1 "k !n1 !y are1 z we say1 two 7! ,good itself1 & ! ,sun17 & t ! "o reigns ov] ! intelligible _w1 b ! o!r ov] ! visible1 n to say ! heav5s1 le/ ,i %\ld deceive y by ! "n4 ,y compreh5d !n1 ^! two ord]s ( ?+s1 ,i mean ! visible & ! intelligible8--,glauco4 ,i d4--,socrates4 ,cont9ue ?is division !n1 z if x 7 a l9e divid$ into two unequal segm5ts2 & ea* "p aga91 i4 e4 ! s5sible & intelligible1 divid$ aft] a simil> mann]1 & y w h evid5ce & obscurity plac$ by ea* o!r4 ,9 ! visible segm5t1 inde$1 "o "p w conta9 images4 ,b ,i call images1 9 ! fir/ place1 %ad[s2 aft]w>ds1 ! resemblances ( ?+s appe>+ 9 wat]1 & 9 d5se1 smoo?1 & lucid bodies1 & e ?+ ( ?is k9d1 if y appreh5d me8--,glauco4 ,i appreh5d y4--,socrates4 ,n[ conceive t ! o!r section compreh5ds ! ?+s1 ( :i* ^! images are no?+ m ?an similitudes1 su* z ! animals ar\nd u1 toge!r ) plants1 & :atev] is ! "w ( nature & art4--,glauco4 ,i conceive x4--,socrates4 ,d y consid] ?is section !n1 z divid$ into true & false8 ,& t ! hypo!sis ( op9ion is to ! k ( sci5ce1 z a resemblance to its orig9al8--,glauco4 ,i d1 v readily4--,socrates4 ,n[ !n1 consid] h[ ! section ( ! intelligible is to be divid$4--,glauco4 ,h[8--,socrates4 ,?us3 "o segm5t is t :i* ! s\l enquires aft]1 us+ ! =m] divisions z images1 & compell$ to proce$ f hypo!ses1 n to ! pr9ciple1 b to ! conclusion4 ,! o!r is t :i* employs ! cogitative p[] ( ! s\l1 z %e proce$s f an hypo!sis to a pr9ciple no long] suppos$1 &1 neglect+ images1 advances ?r\gh _! obscurity into ! light ( ideas !mselves4--,glauco4 ,i d n1 9 ?is1 suffici5tly und]/& y4--,socrates4 ,b aga91 = y w m easily und]/& me f :at has be5 already premis$4 ,i ?9k y are n ignorant1 t ^? :o are conv]sant 9 geometry1 ari?metic1 & ! l1 suppose ev5 & odd1 toge!r ) v>i\s figures1 & ! ?ree species ( angles1 & o!r ?+s simil> to ^!1 accord+ to ea* me?od ( proce$+4 ,n[1 hav+ e/abli%$ ^!1 z hypo!ses suffici5tly kn[n1 !y conceive t no reason is to be requir$ = _! position3 b beg9n+ f ^!1 !y desc5d ?r\gh ! re/1 & arrive at la/1 at ! object ( _! inve/igation4--,glauco4 ,?is ,i "k p]fectly well4--,socrates4 ,?is also y "k1 t !y use visible =ms1 & make !m ! subject ( _! disc\rse1 at ! same "t n direct+ _! intellect to ! p]ception ( ^!1 b to ! orig9als !y resemble2 ,i mean ! squ>e itself1 & ! diamet] itself2 & n to ! figures !y del9eate4 ,& ?us1 o!r =ms1 :i* are repres5t$ by %ad[s & images 9 wat]1 are employ$ by !m1 m]ely z resemblances1 :ile !y /rive to behold t :i* c be se5 by cogitation alone4--,glauco4 ,y speak ! tru?4--,socrates4 ,?is is :at ,i call$ above a species ( ! intelligible1 9 ! inve/igation ( :i*1 ! s\l 0 compell$ to use hypo!ses2 n asc5d+ to ! pr9ciple1 z incapable ( ris+ above hypo!ses1 b us+ ! images =m$ f inf]ior objects1 to a similitude ( su* z are sup]ior1 & :i* are s conceiv$ & di/+ui%$ by op9ion1 z if !y p]spicu\sly contribut$ to ! k ( ?+s !mselves4--,glauco4 ,i und]/& inde$1 t y are speak+ ( ! circum/ances :i* take place 9 geometry1 & h] k9dr$ arts4--,socrates4 ,und]/& n[1 t by ! o!r section ( ! intelligible1 ,i mean t :i* reason h]self rea*es1 by h] p[] ( demon/rat+1 :5 no long] e/eem+ hypo!ses = pr9ciples1 b receiv+ !m 9 reality = hypo!ses1 %e uses !m z s _m /eps & h&les 9 h] asc5t1 until %e arrives at t :i* is no long] hypo!tical1 ! pr9ciple ( ! univ]se2 & aft]w>ds desc5d+1 hold+ by ideas :i* adh]e to ! pr9ciple1 %e arrives at ! conclusion1 employ+ no?+ s5sible 9 h] progress1 b proce$+ ?r\gh ideas1 & 9 ^! at la/ t]m9at+ h] desc5t4--,glauco4 ,i und]/& y1 b n s well z ,i desire3 = y seem to me to propose a great und]tak+4 ,y endeav\r1 inde$1 to det]m9e t ! portion ( true be+ & intelligible1 :i* we speculate by ! sci5ce ( demon/ration1 is m evid5t ?an ! discov]ies made by ! sci5ces call$ arts2 because 9 ! fir/ hypo!ses are pr9ciples1 & _! ma/]s are compell$ to employ ! eye ( cogitation1 & n ! p]ceptions ( ! s5ses4 ,yet1 because !y d n asc5d to ! pr9ciple1 b inve/igate f hypo!ses1 !y seem to y n to h intellig5ce conc]n+ ^!1 ?\gh !y are intelligible1 ?r\gh ! light ( ! pr9ciple4 ,b y seem to me to call ! habit ( reason+ on geometrical & ! l conc]ns1 cogitation1 r ?an intellig5ce1 z if cogitation held ! middle situation betwe5 op9ion & intellect4--,socrates4 ,y und]/& me suffici5tly well4 ,& aga93 ) ^! f\r proportions take ^! f\r correspond+ affections ( ! s\l3 ) ! highe/ intellig5ce2 ) ! second cogitation2 aga9/ ! ?ird set op9ion2 & aga9/ ! f\r? assimilation1 or imag9ation4 ,besides ?is1 e/abli% !m 9 ! ord] ( alt]nate proportion1 s t !y may p>take ( evid5ce1 9 ! same mann] z _! correspond+ objects p>ticipate ( reality40 ,i h tak5 ! lib]ty ( translat+ ?is f9e passage diff]5tly f bo? ,petv9 & ,sp5s2 because !y h neglect$ to give ! prop] mean+ ( ! ^w 1 or cogitation1 ! =m] translat+ x m9d1 & ! eye ( ! m9d1 & by ?is means conf\nd+ x ) intellect2 & ! latt] call+ x und]/&+4 ,b x is c]ta9 t ,plato1 9 ?is place1 ranks intellect z ! fir/1 on acc\nt ( ! sup]ior evid5ce ( its p]ceptions2 9 ! next place1 cogitation2 9 ! ?ird1 op9ion2 & 9 ! f\r?1 imag9ation4 ,h[ev]1 ! read] w please to rememb]1 t by 1 or cogitation1 9 ! pres5t "w1 is und]/ood t p[] ( ! s\l :i* reasons f premises to conclusions1 & ^: syllogi/ic en]gy1 on active subjects1 is call$ prud5ce2 & on su* z are speculative1 sci5ce4 ,b = f>!r in=mation conc]n+ its nature1 see ! diss]tation prefix$ to ?is "w1 & ! foll[+ fif? *apt]4 #gb ,^! two pr9ciples1 b\nd & inf9ite1 w d\btless be consid]$ by ! un?9k+ "p ( mank9d1 z no?+ m ?an g5]al t]ms1 & n z ! mo/ real ( be+s4 ,h[ev]1 an accurate contemplation ( ! univ]se1 w conv9ce e truly philosophic m9d ( _! reality4 ,= ! heav5s !mselves1 by ! coh]5ce & ord] ( _! p>ts1 ev9ce _! p>ticipation ( b\nd4 ,b by _! prolific p[]s1 & ! unceas+ revolutions ( ! orbs !y conta91 !y demon/rate _! p>ticipation ( inf9ity4 ,& ! f9ite & p]petually abid+ =ms ) :i* ! _w is replete1 be> a similitude to b\nd3 :ile1 on ! contr>y1 ! v>iety ( p>ticul>s1 _! nev]-ceas+ mutation1 & ! connection ( m & less 9 ! communion ( =ms1 repres5ts an image ( inf9ity4 ,add too1 t e natural species1 by its =m is simil> to b\nd2 b by its matt]1 to inf9ity4 ,= ^! two1 =m & matt]1 dep5d on b\nd & inf9ity1 & are _! ultimate progressions4 ,& ea* ( ^!1 inde$1 p>ticipates ( unity2 b =m is ! measure & b\nd ( matt]1 & is m "o4 ,b matt] is 9 capacity all ?+s1 because x subsi/s by an emanation f ! fir/ capacity1 or ! inf9ite itself4 #gc ,( human discipl9es1 ^? alone des]ve to be call$ sci5ces :i* use no hypo!ses1 :i* resolve ?+s into _! pr9ciples1 :i* are conv]sant ) true be+1 & elevate u to ideas !mselves4 ,dialectic is :olly ( ?is k9d 7,i mean ! dialectic ( ,plato72 = ?is alone uses no suppositions1 b1 neglect+ %ad[s & images1 raises u1 by a sublime inve/igation1 to ! pr9ciple ( ! univ]se2 & on ?is acc\nt1 des]ves to be call$ ! v apex ( discipl9es4 ,b we mu/ n imag9e1 t by ! ^w dialectic "h1 is meant logic1 or any "p ( logic1 or t me?od ( disputation1 by :i* we fabricate probable reasons2 b we mu/ conceive x z signify+ a discipl9e1 endu$ ) ! greate/ acut5ess2 neglect+ all hypo!ses1 truly so>+ to prim>y causes1 & ultimately repos+ 9 _! contemplation4 ,plot9us has giv5 u mo/ happy specim5s ( ?is me?od1 9 8 books on ! g5]a ( be+4 #gd ,see note to ! fir/ *apt]4 #ge ,i w\ld p>ticul>ly recomm5d ?is *apt] to mod]n ma!maticians1 mo/ ( :om1 ,i am afraid1 h nev] consid]$ :e!r or n ! subjects ( _! speculation h any real subsi/5ce3 ?\gh x is surely an enquiry wor?y ! e>ne/ att5tion ( e lib]al m9d4 ,= if ! objects ( ma!matical inve/igation are m]ely imag9>y1 ,i mean ! po9t )\t p>ts1 ! l9e )\t bread?1 c4 ! sci5ce1 f\nd$ on ^! false pr9ciples1 mu/ ( c\rse be entirely delusive4 ,inde$1 an absolutely true conclusion1 c nev] fl[ f an errone\s pr9ciple1 z f its cause3 z ! /ream mu/ always p>ticipate ( its s\rce4 ,i mean su* a conclusion z is demon/rat$ by ! prop] cause1 1 ' 1 says ,ari/otle1 9 8 fir/ ,analytics2 t is1 a syllogism f false pr9ciples w n prove ! :y1 b only simply t x is3 inde$ x c only simply prove t x is1 to him :o admits ! false propositions2 because he :o all[s ! premises1 _c d5y ! conclusion1 :5 ! syllogism is prop]ly con/ruct$4 ,?us we way syllogize 9 ! fir/ figure1 ,e ?+ :ite1 is an animal3 ,e bird is :ite3 ,!re=e1 ,e bird is an animal4 ,& ! conclusion w be true1 ?\gh ! major & m9or t]ms are false2 b !n ^! t]ms are n ! causes ( ! conclusion1 & we h an inf]5ce )\t a pro(4 ,9 l mann]1 if ma!matical species are delusive & fictiti\s1 ! conclusions d$uc$ f !m z pr9ciples1 are m]ely hypo!tical1 & n demon/rative4 #gf ,ari/otle1 9 8 la/ ,analytics4 ,! read] w please to obs]ve1 t ! :ole =ce ( ?is n]v\s1 accurate1 & elegant reason+1 is direct$ aga9/ ,ari/otle2 :o seems un=tunately to h consid]$1 ) ! mod]ns1 t ma!matical species subsi/ 9 ! s\l1 by an ab/raction f s5sibles4 ,see ! prec$+ ,diss]tation4 #gg ,viz4 #a1 #b1 #d1 #h1 #c1 #i1 #bg4 ,conc]n+ :i*1 see lib4 iii4 ( ,proclus's excell5t ,comm5t>y on ! ,timu4 #gh ,plato frequ5tly1 bo? 9 ! ,m5o & else:]e1 %ews t sci5ce is ,rem9isc5ce2 & ,i ?9k n )\t ! /ronge/ reason4 ,= s9ce ! s\l is immat]ial1 z we h demon/rat$ 9 ! diss]tation to ?is "w1 %e mu/ be truly immortal1 i4 e4 bo? a p>te ante1 a p>te po/4 ,t %e mu/ be et]nal1 inde$1 ) respect to futurity1 if immat]ial1 is admitt$ by all2 & we may prove1 ) ,ari/otle1 9 8 fir/ book de ,clo1 t %e is immortal1 likewise a p>te ante1 z foll[s4 ,e ?+ )\t g5]ation1 is incorruptible1 & e ?+ incorruptible1 is )\t g5]ation3 = t :i* is )\t g5]ation1 has a necessity ( exi/+ inf9itely a p>te ante 7f ! hypo!sis72 & !re=e1 if x possesses a capacity ( be+ de/roy$1 s9ce !re is no great] reason :y x %\ld be corrupt$ n[1 r ?an 9 "s =m] p]iod1 x is endu$ ) a capacity ( be+ de/roy$ & ceas+ to be1 9 e in/ant ( inf9ite "t1 9 :i* x necess>ily is4 ,9 l mann]1 t :i* is incorruptible1 has a necessity ( exi/+ inf9itely a p>te po/2 !re=e1 if x possesses a capacity ( be+ g5]at$1 s9ce !re is no great] reason :y x %\ld be g5]at$ n[ r ?an aft]w>ds1 x possesses a capacity ( be+ g5]at$1 9 e in/ant ( "t1 9 :i* x necess>ily is4 ,if !n ! s\l is ess5tially immortal1 ) respect to ! pa/ & future circulations ( "t2 & if %e is replete ) =ms or ideas ( e k9d1 z we h prov$ 9 ! diss]tation1 %e mu/1 f h] circulat+ nature1 h be5 = ev] conv]sant 9 alt]nately possess+ & los+ ! k ( ^!4 ,n[1 ! recov]y ( ?is k by sci5ce1 is call$ by ,plato1 rem9isc5ce2 & is no?+ m ?an a r5ew$ contemplation ( ^? div9e =ms1 s famili> to ! s\l1 be=e %e became involv$ 9 ! d>k ve/m5t ( an e>?ly body4 ,s t we may say1 ) ! elegant ,maximus ,tyrus1 7,diss]4 #bh47 8,rem9isc5ce is simil> to t :i* happ5s to ! corporeal eye1 :i*1 ?\gh always endu$ ) a p[] ( vision1 yet d>kness sometimes ob/ructs its passage1 & av]ts x f ! p]ception ( ?+s4 ,art !re=e1 approa*es1 :i* ?\gh x does n give to ! eye ! p[] ( vision1 yet removes its imp$im5ts1 & af=ds a free egress to its rays4 ,conceive n[1 t our rational s\l is su* a p[] ( p]ceiv+1 :i* sees & kn[s ! nature ( be+s4 ,to ?is ! common calamity ( bodies happ5s1 t d>kness spread+ r\nd x1 hurries away its aspect1 blunts its %>pness1 & ext+ui%es its prop] light4 ,aft]w>ds1 ! art ( reason approa*es1 :i*1 l a physician1 does n br+ or af=d x a new sci5ce1 b r\ses t :i* x possesses1 ?\gh v sl5d]1 confus$1 & un/eady40 ,h5ce1 s9ce ! s\l1 by h] imm]sion 9 body1 is 9 a dormant /ate1 until %e is r\s$ by sci5ce to an ex]tion ( h] lat5t en]gies2 & yet ev5 previ\s to ?is awak5+1 s9ce %e conta9s ! vivid sp>ks1 z x 71 ( all k1 :i* only require to be v5tilat$ by ! w+s ( le>n+1 9 ord] to rek9dle ! light ( ideas1 %e may be said 9 ?is case to "k all ?+s z 9 a dream1 & to be ignorant ( !m ) respect to vigilant p]ceptions4 ,h5ce too1 we may inf] t "t does n antec$e our ess5tial k ( =ms1 because we possess x f et]nity3 b x prec$es our k ) respect to a production ( ^! reasons into p]fect en]gy4 ,i only add1 t ,i w\ld recomm5d ! lib]al ,engli% read]1 to ,mr4 ,syd5ham's excell5t translation ( ,plato's ,m5o1 :]e he w f9d a famili> & elegant demon/ration ( ! doctr9e ( ,rem9isc5ce4 #gi ,conc]n+ ?is valuable "w1 entitl$ #8 '1 see ! ,biblio!ca ,grca ( ,fabricius1 vol4 i4 p4 #aah & #dfb1 & 9 ! comm5t>y ( ,syrianus on ,ari/otle's metaphysics1 p4 #g1 #ga1 #hc1 & #ajh1 ! read] w f9d "s curi\s extracts f ?is celebrat$ disc\rse2 p>ticul>ly 9 p4 #hc4 ,syrianus in=ms u1 8t he :o consults ?is "w w f9d all ! ord]s bo? ( ,monads & ,numb]s1 )\t neglect+ "o1 fully celebrat$ 7470 ,!re is no d\bt1 b t ,py?agoras & 8 disciples conceal$ ! sublime/ tru?s1 "u ! symbols ( numb]s2 ( :i* he :o reads & und]/&s ! writ+s ( ! ,platoni/s w be fully conv9c$4 ,h5ce ,proclus1 9 ! ?ird book ( 8 excell5t comm5t>y on ! ,timu1 obs]ves1 8t ,plato employ$ ma!matical t]ms = ! sake ( my/]y & concealm5t1 z c]ta9 veils1 by :i* ! p5etralia ( tru? might be seclud$ f vulg> inspection1 j z ! !ologi/s made fables1 b ! ,py?agoreans symbols1 subs]vi5t to ! same purpose3 = 9 images we may speculate _! exempl>s1 & ! =m] af=d u ! means ( access to ! latt]40 #hj ,conc]n+ ?is ,geometric ,numb]1 9 ! #h? book ( ,plato's ,republic1 ?an :i* ,cic]o affirms !re is no?+ m obscure1 see ! notes ( ,bullialdus to ,!o4 p4 #bib4 #ha ,i am sorry to say1 t ?is "p ( ! enemies to pure geometry & ari?metic1 are at ! pres5t "t v num]\s2 conceptions ( utility 9 ^! sci5ces1 ext5d+ no f>!r ?an ! sordid purposes ( a m]e animal life4 ,b surely1 if intellect is a "p ( our composition1 & ! noble/ "p too1 !re mu/ be an object ( its contemplation2 & ?is1 :i* is no o!r ?an tru? 9 ! mo/ exalt$ s5se1 mu/ be ! mo/ noble & useful subject ( speculation to e rational be+4 #hb ,9 ! #ac? book ( 8 ,metaphysics1 cap4 iii4 #hc ,94 ,i4 ,de ,p>tib4 ,animalium1 et 9 primo ,e?ic4 cap4 iii4 #hd ,see m conc]n+ ?is 9 ! ,diss]tation4 #he ,s9ce numb] is prior to magnitude1 ! demon/rations ( ari?metic mu/ be m intellectual1 b ^? ( geometry m accommodat$ to ! rational p[]4 ,& :5 ei!r ari?metic or geometry is appli$ to s5sible conc]ns1 ! demon/rations1 f ! nature ( ! subjects1 mu/ p>ticipate ( ! obscurity ( op9ion4 ,if ?is is ! case1 a true ma!matician w value ^? p>ts ( 8 sci5ce mo/1 :i* p>ticipate mo/ ( evid5ce2 & w consid] !m z degrad$1 :5 appli$ to ! common purposes ( life4 #hf ,?is division ( ! ma!matical sci5ce1 accord+ to ! ,py?agoreans1 :i* is ne>ly co9cid5t ) t ( ,plato1 is blam$ by ,dr4 ,b>r[ 9 8 ,ma!matical ,lectures1 p4 #ae4 z be+ conf9$ )9 too n>r[ limits3 & ! reason he assigns = s p>tial a division1 is1 8because1 9 ,plato's "t1 o!rs 7 ei!r n yet inv5t$1 or n suffici5tly cultivat$1 or at lea/ 7 n yet receiv$ into ! numb] ( ! ma!matical sci5ces40 ,b ,i mu/ beg leave to diff] f ?is mo/ illu/ri\s ma!matician 9 ?is affair2 & to ass]t t ! reason ( s conf9$ a di/ribution 7z x is conceiv$ by ! mod]ns7 arose f ! exalt$ conceptions ^! wise m5 ent]ta9$ ( ! ma!matical sci5ces1 :i* !y consid]$ z s _m preludes to ! k ( div9ity1 :5 prop]ly pursu$2 b !y reckon$ !m degrad$ & p]v]t$1 :5 !y became mix$ ) s5sible objects1 & 7 appli$ to ! common purposes ( life4 #hg ,t is1 a "r & circul> l9e4 #hh ,i am afraid !re are few 9 ! pres5t day1 :o d n consid] tactics z "o ( ! mo/ pr9cipal p>ts ( ma!matics2 & :o w\ld n fail to cite1 9 def5ce ( _! op9ions1 t great re=m] ( philosophy1 z he is call$1 ,"l ,bacon1 comm5d+ pursuits :i* come home to m5's bus9esses & bosoms4 ,inde$1 if :at is l[e/ 9 ! true ord] ( ?+s1 & be/ adm9i/]s to ! vile/ "p ( human nature1 is to h ! pref]5ce1 _! op9ion is "r1 & ,"l ,bacon is a philosoph]6 #hi ,by ?is is to be und]/ood ! art new call$ ,p]spective3 f :5ce x is evid5t t ?is art 0 n unkn[n to ! anci5ts1 ?\gh x is que/ion$ by ! mod]ns4 #ij ,f h5ce x appe>s1 t x is d\btful :e!r ,plato is ! au?or ( ! dialogue call$ ,ep9omis2 & ,i ?9k x may ) great propriety be que/ion$4 ,= ?\gh x be>s evid5t m>ks ( high antiquity1 & is replete ) g5u9e wisdom1 x does n seem to be p]fectly aft] ,plato's mann]2 nor to conta9 t great dep? ( ?\ght ) :i* ! writ+s ( ?is philosoph] ab\nd4 ,fabricius 79 8 ,biblio!ca ,grca1 lib4 i4 p4 #bg47 wond]s t ,suidas %\ld ascribe ?is "w to a philosoph] :o di/ribut$ ,plato's laws into twelve books1 because x 0 an usual op9ion2 f :5ce x seems1 t accurate critic _h n att5d$ to ! pres5t passage4 #ia ,?is proximate conjunction ( ! ma!matical sci5ces1 :i* ,proclus consid]s z subord9ate to dialectic1 seems to diff] f t v]tex ( sci5ce 9 ?is1 t ! =m] m]ely embraces ! pr9ciples ( all sci5ce1 b ! latt] compreh5ds ! univ]sal g5]a ( be+1 & speculates ! pr9ciple ( all4 #ib ,9 ! ,m5o4 #ic ,?is is c]ta9ly ! true or philosophical employm5t ( ! ma!matical sci5ce2 = by ?is means we %all be enabl$ to asc5d f s5se to intellect1 & rek9dle 9 ! s\l t div9e light ( tru?1 :i*1 previ\s to su* an en]gy1 0 buri$ 9 ! obscurity ( a corporeal nature4 ,b by a contr>y process1 ,i mean1 by apply+ ma!matical speculations1 to exp]im5tal purposes1 we %all bl9d ! lib]al eye ( ! s\l1 & leave no?+ 9 its /ead b ! d>kness ( corporeal vision1 & ! phantoms ( a degrad$ imag9ation4 #id ,! design ( ! pres5t *apt] is to prove t ! figures :i* are ! subjects ( geometric speculation1 d n subsi/ 9 ext]nal & s5sible matt]1 b 9 ! receptacle ( imag9ation1 or ! matt] ( ! phantasy4 ,& ?is our philosoph] proves ) 8 usual elegance1 subtilty1 & dep?4 ,inde$1 x mu/ be evid5t to e att5tive obs]v]1 t s5sible figures fall f> %ort ( t accuracy & p]fection :i* are requir$ 9 geometrical def9itions3 = !re is no s5sible circle p]fectly r\nd1 s9ce ! po9t f :i* x is describ$ is n )\t p>ts2 &1 z ,vossius well obs]ves1 7de ,ma!m4 p4 #d47 !re is n any sph]e 9 ! nature ( ?+s1 t only t\*es 9 a po9t1 = ) "s "p ( its sup]ficies x always t\*es ! subject$ plane 9 a l9e1 z ,ari/otle %ews ,protagoras to h object$ aga9/ ! geometricians4 ,nor mu/ we say1 ) t great ma!matician ,dr4 ,b>r[1 9 8 ,ma!matical ,lectures1 page #gf1 8t all imag9able geometrical figures1 are really inh]5t 9 e p>ticle ( matt]1 9 ! utmo/ p]fection1 ?\gh n app>5t to s5se2 j z ! effigies ( ,cs> lies hid 9 ! unhewn m>ble1 & is no new ?+ made by ! /atu>y1 b only is discov]$ & br\ght to sight by 8 workman%ip1 i4 e4 by remov+ ! p>ts ( matt] by :i* x is ov]%ad[$ & involv$4 ,:i* made ,mi*ael ,angelus1 ! mo/ fam\s c>v]1 say1 t sculpture 0 no?+ b a purgation f ?+s sup]flu\s4 ,= take all t is sup]flu\s1 7says he7 f ! wood or /one1 & ! re/ w be ! figure y int5d4 ,s1 if ! h& ( an angel 7at lea/ ! p[] ( ,god7 %\ld ?9k fit to poli% any p>ticle ( matt]1 )\t vacuity1 a sph]ical sup]ficies w\ld appe> to ! eyes1 ( a figure exactly r\nd2 n z creat$ anew1 b z unveil$ & laid op5 f ! disguises & cov]s ( its circumjac5t matt]40 ,= ?is w\ld be giv+ a p]fection to s5sible matt]1 :i* x is naturally incapable ( receiv+3 s9ce ext]nal body is ess5tially full ( pores & irregul>ities1 :i* mu/ et]nally prev5t its receiv+ ! accuracy ( geometrical body1 ?\gh poli%$ by ! h& ( an angel4 ,besides1 :at poli%+ w\ld ev] produce a po9t )\t p>ts1 & a l9e )\t bread?8 ,= ?\gh body may be r$uc$ to ! greate/ exility1 x w n by ?is means ev] pass into an incorporeal nature1 & des]t its triple dim5sion4 ,s9ce ext]nal matt]1 !re=e1 is by no means ! receptacle ( geometrical figures1 !y mu/ necess>ily reside 9 ! catoptric matt] ( ! phantasy1 :]e !y subsi/ ) an accuracy suffici5t = ! en]gies ( ?is sci5ce4 ,x is true1 inde$1 t ev5 9 ! pur] matt] ( imag9ation1 ! po9t does n appe> p]fectly imp>tible1 nor ! l9e )\t latitude3 b !n ! magnitude ( ! po9t1 & ! bread? ( ! l9e is indef9ite1 & !y are1 at ! same "t1 unatt5d$ ) ! qualities ( body1 & exhibit to ! eye ( ?\ght1 magnitude alone4 ,h5ce1 ! figures 9 ! phantasy1 are ! prop] recipi5ts ( t univ]sal1 :i* is ! object ( geometrical speculation1 & repres5t1 z 9 a mirror1 ! p>ticipat$ subsi/5ce ( ^? vital & immat]ial =ms :i* ess5tially reside 9 ! s\l4 #ie ,?is division is elegantly expla9$ by ,ammonius1 79 ,porphyr4 p4 #ab47 z foll[s1 8,conceive a seal-r+1 :i* has ! image ( "s p>ticul> p]son1 = in/ance1 ( ,a*illes1 engrav$ 9 its seal1 & let !re be _m portions ( wax1 :i* are impress$ by ! r+4 ,aft]w>ds conceive t "s "o approa*es1 & p]ceives all ! portions ( wax1 /amp$ ) ! impression ( ?is "o r+1 & keeps ! impression ( ! r+ 9 8 m9d3 ! seal engrav$ 9 ! r+1 repres5ts ! univ]sal1 prior to ! _m3 ! impression 9 ! portions ( wax1 ! univ]sal 9 ! _m3 b t :i* rema9s 9 ! intellig5ce ( ! behold]1 may be call$ ! univ]sal1 aft] & po/]ior to ! _m4 ,! same mu/ we conceive 9 g5]a & species4 ,= t be/ & mo/ excell5t artific] ( ! _w1 possesses )9 himself ! =ms & exempl>s ( all ?+s3 s t 9 ! fabrication ( man1 he looks back ^u ! =m ( man resid5t 9 8 ess5ce1 & fa%ions all ! re/ accord+ to its exempl>4 ,b if any "o %\ld oppose ?is doctr9e1 & ass]t t ! =ms ( ?+s d n reside ) _! artific]1 let him att5d to ! foll[+ argum5ts4 ,! artific] ei!r kn[s1 or is ignorant ( t :i* he produces3 b he :o is ignorant w nev] produce any ?+4 ,= :o w attempt to d t1 :i* he is ignorant h[ to p]=m8 s9ce he _c act f an irrational p[] l nature1 ^: op]ations are n att5d$ ) animadv]sion4 ,b if he produces any ?+ by a c]ta9 reason1 he mu/ possess a k ( e ?+ :i* he produces4 ,if1 !re=e1 x is n impi\s to ass]t1 t ! op]ations ( ! ,deity1 l ^? ( m51 are att5d$ ) k1 x is evid5t t ! =ms ( ?+s mu/ reside 9 8 ess5ce3 b =ms are 9 ! demiurgus1 l ! seal 9 ! r+2 & ^! =ms are said to be prior to ! _m1 & sep>at$ f matt]4 ,b ! species man1 is conta9$ 9 ea* p>ticul> man1 l ! impression ( ! seal 9 ! wax1 & is said to subsi/ 9 ! _m1 )\t a sep>ation f matt]4 ,& :5 we behold p>ticul> m51 & p]ceive ! same =m & effigy 9 ea*1 t =m se>$ 9 our s\l1 is said to be aft] ! _m1 & to h a po/]ior g5]ation3 j z we obs]v$ 9 him1 :o beheld _m seals impress$ 9 ! wax f "o & ! same r+4 ,& ?is "o1 po/]ior to ! _m1 may be sep>at$ f body1 :5 x is conceiv$ z n inh]5t 9 body1 b 9 ! s\l3 b is incapable ( a real sep>ation f its subject40 ,we mu/ "h1 h[ev]1 obs]ve1 t :5 ,ammonius speaks ( ! k ( ! ,deity1 x mu/ be conceiv$ z f> sup]ior to ours4 ,= he possesses a nature m true ?an all ess5ce1 & a p]ception cle>] ?an all k4 ,& z he produc$ all ?+s by 8 unity1 s by an ineffable unity ( appreh5sion1 he kn[s ! univ]sality ( ?+s4 #if ,9 lib4 vii4 ,metaphys4 #ce #ci4 #ig ,9 lib4 iii4 de ,anima1 tex4 #bj4 #ih ,t is1 geometry fir/ speculates ! circle del9eat$ on pap]1 or 9 ! du/3 b by ! m$ium ( ! circul> figure 9 ! phantasy1 contemplates ! circle resid5t 9 cogitation2 & by t univ]sal1 or circul> reason1 p>ticipat$ 9 ! circle ( ! phantasy1 frames its demon/rations4 #ii ,9 8 fir/ ,analytics1 t4 #db4 ,see ! ,diss]tation to ?is "w4 #ajj ,su* z ! proportion ( ! diagonal ( a squ>e to its side2 & t ( ! diamet] ( a circle1 to ! p]iph]y4 #aja ,! gnomons1 f :i* squ>e numb]s are produc$1 are odd numb]s 9 a natural s]ies f unity1 i4 e4 #a1 #c1 #e1 #g1 #i1 #aa1 c4 = ^!1 add$ to ea* o!r cont9ually1 produce squ>e numb]s ad inf9itum4 ,b ^! gnomons cont9ually decrease f ! highe/1 & are at l5g? t]m9at$ by indivisible unity4 #ajb ,?is doctr9e ( ineffable quantities1 or su* ^: proportion _c be express$1 is l>gely & accurately discuss$ by ,euclid1 9 ! t5? book ( 8 ,elem5ts3 b its /udy is neglect$ by mod]n ma!maticians1 because x is ( no use1 t is1 because x contributes to no?+ me*anical4 #ajc ,?is proposition is ! #aa? ( ! second book3 at lea/1 ! me?od ( divid+ a l9e into extreme & mean proportion1 is imm$iately d$uc$ f x2 :i* is done by ,euclid1 9 ! #cj?1 ( ! six? book4 ,?us1 ,euclid %ews 7#aa4 #b47 h[ to divide ! l9e 7,a ,g ,b7diagram ,a ,b1 s t ! rectangle "u ! :ole ,a ,b1 & ! segm5t ,g ,b1 may be equal to ! squ>e made f ,a ,g3 = :5 ?is is done1 x foll[s1 t z ,a ,b is to ,a ,g1 s is ,a ,g to ,g ,b2 z is well kn[n4 ,b ?is proposition1 z ,dr4 ,b>r[ obs]ves1 _c be expla9$ by numb]s2 because !re is n any numb] :i* c be s divid$1 t ! product f ! :ole into "o "p1 may be equal to ! squ>e f ! o!r "p4 #ajd ,all polygon\s figures1 may1 x is well kn[n1 be resolv$ into triangles2 & ?is is no less true ( polygon\s numb]s1 z ! foll[+ obs]vations ev9ce4 ,all numb] orig9ates f indivisible unity1 :i* corresponds to a po9t3 & x is ei!r l9e>1 correspond+ to a l9e2 or sup]ficial1 :i* corresponds to a sup]ficies2 or solid1 :i* imitates a geometrical solid4 ,aft] unity1 !re=e1 ! fir/ ( l9e> numb]s is ! duad2 j z e f9ite l9e is allott$ two extremities4 ,! triad is ! fir/ ( sup]ficial numb]s2 z ! triangle ( geometrical figures4 ,& ! tetrad1 is ! fir/ ( solids2 because a triangul> pyramid1 is ! fir/ among solid numb]s1 z well z among solid figures4 ,z1 !re=e1 ! monad is assimilat$ to ! po9t1 s ! duad to ! l9e1 ! triad to ! sup]ficies1 & ! tetrad to ! solid4 ,n[1 ( sup]ficial numb]s1 "s are triangles1 o!rs squ>es1 o!rs p5tagons1 hexagons1 heptagons1 c4 ,triangul> numb]s are g5]at$ f ! cont9ual addition ( numb]s 9 a natural s]ies1 beg9n+ f unity4 ,?us1 if ! numb]s #a1 #b1 #c1 #d1 #e1 c4 be add$ to ea* o!r cont9ually1 !y w produce ! triangul> numb]s #a1 #c1 #f1 #aj1 #ae1 c4 & if e triangul> numb] be add$ to its prec$+ numb]1 x w produce a squ>e numb]4 ,?us #c add$ to #a makes #d2 #f add$ to #c is equals #i2 #aj add$ to #f is equal to #af2 & s ( ! re/4 ,p5tagons1 are produc$ f ! junction ( triangul> & squ>e numb]s1 z foll[s4 ,let !re be a s]ies ( triangul> numb]s #a1 #c1 #f1 #aj1 #ae1 c4 ,& ( squ>es #a1 #d1 #i1 #af1 #be1 c4 ,!n ! second squ>e numb]1 add$ to ! fir/ triangle1 w produce ! fir/ p5tagon f unity1 i4e4 #e4 ,! ?ird squ>e add$ to ! second triangle1 w produce ! second p5tagon1 i4e4 #ab2 & s ( ! re/1 by a simil> addition4 ,9 l mann]1 ! second p5tagon1 add$ to ! fir/ triangle1 w =m ! fir/ hexagon f unity2 ! ?ird p5tagon & ! second triangle1 w =m ! second hexagon1 c4 ,&1 by a simil> proce$+1 all ! o!r polygons may be obta9$4 #aje ,intellections are univ]sally correspond5t to _! objects1 & p>ticipate ( evid5ce or ! contr>y1 9 proportion z _! subjects are lucid or obscure4 ,h5ce1 ,porphyry1 9 8 s5t5ces1 ju/ly obs]ves1 t 8we d n und]/& 9 a simil> mann] ) all ! p[]s ( ! s\l1 b accord+ to ! p>ticul> ess5ce ( ea*4 ,= ) ! intellect we und]/& intellectually2 & ) ! s\l1 rationally3 our k ( plants is accord+ to a sem9al conception2 our und]/&+ ( bodies is imag9ative2 & our intellection ( ! div9ely solit>y pr9ciple ( ! univ]se1 :o is above all ?+s1 is 9 a mann] sup]ior to intellectual p]ception1 & by a sup]-ess5tial en]gy40 1 7#aj47 ,s t1 9 consequ5ce ( ?is reason+1 ! speculations ( geometry are !n mo/ true1 :5 mo/ ab/ract$ f s5sible & mat]ial natures4 #ajf ,see ,plut>*1 9 ! life ( ,m>cellus4 #ajg ,9 lib4 i4 de ,clo1 tex4 #bb4 et lib4 i4 ,meteo4 cap4 #c4 ,ari/otle 0 call$ demoniacal by ! ,platonic philosoph]s1 9 consequ5ce ( ! encomium be/[$ on him by 8 ma/]1 ,plato1 8,t he 0 ! dmon ( nature40 ,inde$1 8 great k 9 ?+s subject to ! dom9ion ( nature1 well des]v$ ?is encomium2 & ! epi!t div9e1 has be5 univ]sally ascrib$ to ,plato1 f 8 pr(\nd k ( ! intelligible _w4 #ajh 1 is want+ 9 ! orig9al1 b is suppli$ by ! excell5t translation ( ,b>ocius4 #aji 1 9 ! pr9t$ ,greek1 :i* ,fabricius1 9 8 ,biblio!ca ,grca1 vol4 i4 page #che1 is ( op9ion1 %\ld be read 2 b ,i h r5d]$ ! ^w accord+ to ! translation ( ,b>ocius1 :o is likely to h obta9$ ! true read+1 f ! v>iety ( manuscripts :i* he consult$4 #aaj ,! quadrature ( ! ,lunula is z foll[s4 ,illu/ration ,let ,a ,b ,c be a "r-angl$ triangle1 & ,b ,a ,c a semi-circle on ! diamet] ,b ,c3 ,b ,n ,a a semi-circle describ$ on ! diamet] ,a ,b2 ,a ,m ,c a semi-circle describ$ on ! diamet] ,a ,c4 ,!n ! semi-circle ,b ,a ,c is equal to ! semi-circle ,b ,n ,a1 & ,a ,m ,c toge!r3 7because circles are to ea* o!r z ! squ>es ( _! diamet]s1 #ca1 #f47 ,if1 !re=e1 y take away ! two spaces ,b ,a1 ,a ,c common on bo? sides1 !re w rema9 ! two lunulas ,b ,n ,a1 ,a ,m ,c1 b\nd$ on bo? sides ) circul> l9es1 equal to ! "r-angl$ triangle ,b ,a ,c4 ,& if ! l9e ,b ,a1 be equal to ! l9e ,a ,c1 & y let fall a p]p5dicul> to ! hypot5use ,b ,c1 ! triangle ,b ,a ,o w be equal to ! lunul> space ,b ,n ,a1 & ! triangle ,c ,o ,a w be equal to ! lunula ,c ,m ,a4 ,^? :o are curi\s1 may see a long acc\nt ( an attempt ( ,hippocrates to squ>e ! circle1 by ! inv5tion ( ! lunulas1 9 ,simplicius on ,ari/otle's ,physics1 lib4 i4 #aaa ,s ,b>ocius reads1 b ,fabricius 4 #aab i4 e4 ,! five regul> bodies1 ! pyramid1 cube1 octa$ron1 dodeca$ron & icosa$ron2 conc]n+ :i*1 & _! application to ! !ory ( ! univ]se1 see ,kepl]'s admirable "w1 ,de ,h>monia ,mundi4 #aac ,x may be d\bt$ :e!r ! optics & catoptrics1 ascrib$ to ,euclid 9 ! editions ( 8 works are g5u9e3 = ,savil1 & ,dr4 ,gregory1 ?9k !m sc>cely wor?y s great a man4 #aad ,!re are two excell5t editions ( ?is "w1 "o by ,meibomius1 9 8 collection ( anci5t au?ors on h>mony2 & ! o!r by ,dr4 ,gregory1 9 8 collection ( ,euclid's works4 #aae ,?is "w is mo/ probably lo/4 ,see ,dr4 ,gregory's ,euclid4 #aaf ,all ?is is %ewn by ,proclus 9 ! foll[+ ,comm5t>ies2 & is surely mo/ admirable & wor?y ! inve/igation ( e lib]al m9d2 b ,i am afraid mod]n ma!maticians v little reg>d su* k1 because x _c be appli$ to practical & me*anical purposes4 #aag ,?is "w is un=tunately lo/4 #aah ,because ?is is true only 9 isosceles & equilat]al triangles4 #aai ,?is foll[s f ! #cb;d proposition ( ! fir/ book ( ,euclid2 & is demon/rat$ by ,dr4 ,b>r[1 9 8 s*olium to t proposition4 #abj ,! me?od ( con/ruct+ ^! is %ewn by our philosoph]1 9 8 comm5t on ! fir/ proposition1 z w appe> 9 ! second volume ( ?is "w4 #aba ,! read] w please to obs]ve1 t ! def9itions are1 inde$1 hypo!ses1 accord+ to ! doctr9e ( ,plato1 z may be se5 9 ! note to *ap1 i4 book ,i4 ( ?is "w4 #abb ,9 8 la/ ,analytics4 ,see ! prec$+ ,diss]tation4 #abc ,t "p ( ?is "w enclos$ )9 ! brackets1 is want+ 9 ! orig9al2 :i* ,i h re/or$ f ! excell5t v]sion ( ,b>ocius4 ,! philosophical read]1 !re=e1 ( ! orig9al1 :o may n h ,b>ocius 9 8 possession1 w1 ,i hope1 be pleas$1 to see s great a vacancy suppli$2 especially1 z x conta9s ! beg9n+ ( ! comm5t>y on ! def9ition ( a po9t4 #abd ,i d n f9d ?is nigma among ! ,py?agoric symbols :i* are extant2 s t x is probably no :]e m5tion$ b 9 ! pres5t "w4 ,& ,i am sorry to add1 t a figure & ?ree oboli1 9 too mu* ! g5]al cry ( ! pres5t times4 #abe ,! pres5t ,comm5t1 & inde$ mo/ ( ! foll[+1 em95tly ev9ces ! tru? ( ,kepl]'s obs]vation1 9 8 excell5t "w1 ,de ,h>monia ,mundi1 p4 #aah4 ,=1 speak+ ( our au?or's composition 9 ! pres5t "w1 :i* he e :]e admires & def5ds1 he rem>ks z foll[s1 8oratio fluit ipsi torr5tis in/>1 ripas inundans1 et cca dubitationum vada gurgitesque occultans1 dum m5s pl5a maje/atis tant>um r]um1 luctatur 9 angu/iis l+u1 et conclusio nunquam sibi ipsi v]borum copi satisfaci5s1 propositionum simplicitatem exc$it40 ,b ,kepl] 0 skill$ 9 ! ,platonic philosophy1 & appe>s to h be5 no less acqua9t$ ) ! great dep? ( our au?or's m9d ?an ) ! magnific5ce & sublimity ( 8 language4 ,p]haps ,kepl] is ! only in/ance among ! mod]ns1 ( ! philosophical & ma!matical g5ius be+ unit$ 9 ! same p]son4 #abf ,t is1 ! reason ( a triangul> figure 7= in/ance7 9 ! phantasy1 or triangle itself1 is sup]ior to ! triangul> nature p>ticipat$ 9 t figure4 #abg ,9 ! t5? book ( 8 ,republic4 #abh ,see ! ,hymn to ! ,"m ( ! ,gods1 9 my translation ( ! ,orphic ,initiations4 #abi ,! philosoph] "h seems to contradict :at he ass]ts 9 ! end ( 8 comm5t on ! #ac? ,def9ition3 = !re he ass]ts1 t ! circle is a c]ta9 plane space4 ,p]haps he may be reconcil$1 by consid]+1 t z ! circle subsi/s mo/ accord+ to b\nd1 :5 we speculate its ess5ce 9 ?is respect we may def9e x accord+ to ! circumf]5ce1 :i* is ! cause ( its b\nd4 ,b :5 we consid] x z p>ticipat+ ( inf9ity also1 ?\gh n 9 s em95t a degree1 & view x f its emanations f ! c5tre z well z 9 its regressions1 we may def9e x a plane space4 #acj ,t is1 ! ess5tial "o ( ! s\l is ! "m ( numb]2 b t :i* subsi/s 9 op9ion is no?+ m ?an ! receptacle ( ! =m]2 j z matt] is ! seat ( all =ms4 ,= a f>!r acc\nt ( ! subsi/5ce ( numb]s1 see ! fir/ section ( ! prec$+ ,diss]tation4 #aca ,t is1 numb] compos$ f units4 #acb ,?is s5t5ce )9 ! brackets1 is :olly omitt$ 9 ! pr9t$ ,greek4 #acc ,9 i4 ,de ,clo4 #acd ,?is s5t5ce )9 ! brackets1 :i* is v imp]fect 9 ! ,greek1 ,i h suppli$ f ! excell5t translation ( ,b>ocius4 ,9 ! ,greek !re is no?+ m ?an 4 #ace ,9 ! ,greek1 1 4 ,! latt] "p only ( ?is oracle1 is to be f\nd 9 all ! pr9t$ editions ( ! ,zoroa/rian oracles2 ?\gh x is wond]ful h[ ?is omission c\ld escape ! notice ( s may able critics1 & le>n$ m54 ,x seems probable1 f h5ce1 t x is only to be f\nd p]fect 9 ! pres5t "w4 #acf ,! ^w 1 is omitt$ 9 ! ,greek4 #acg ,?is & ! foll[+ problems1 are ! #a/1 #bb;d1 & #ab? propositions ( ! fir/ book4 ,b 9 ! two la/1 in/ead ( ! ^w or inf9ite1 :i* is ! t]m employ$ by ,euclid1 ,mr4 ,simson1 9 8 edition ( ! ,elem5ts1 uses ! ^w unlimit$4 ,b x is no unusual ?+ ) ?is great geometrician1 to alt] ! words ( ,euclid1 :5 !y convey a philosophical mean+2 z we %all pla9ly ev9ce 9 ! c\rse ( ^! ,comm5t>ies4 ,he c]ta9ly des]ves ! greate/ praise = 8 zeal\s atta*m5t to ! anci5t geometry3 b he w\ld 79 my op9ion7 h des]v$ /ill m1 _h he be5 acqua9t$ ) ! ,greek philosophy2 & fa?om$ ! dep? ( ,proclus2 = !n he w\ld nev] h attempt$ to re/ore ,euclid's ,elem5ts1 by depriv+ !m ( "s v consid]able beauties4 #ach ,?is is d\btless ! reason :y ! proportion betwe5 a "r & circul> l9e1 _c be exactly obta9$ 9 numb]s2 = on ?is hypo!sis1 !y mu/ be incomm5surable quantities2 because ! "o conta9s prop]ty ess5tially diff]5t f ! o!r4 #aci ,illu/ration ,! cornicul> angle is t :i* is made f ! p]iph]y ( a circle & its tang5t2 t is1 ! angle compreh5d$ by ! ar* ,l ,a1 & ! "r l9e ,f ,a1 :i* ,euclid 9 7#af4 #c47 proves to be less ?an any "r-l9$ angle4 ,& f ?is admirable proposition x foll[s1 by a legitimate consequ5ce1 t any quantity may be cont9ually & inf9itely increas$1 b ano!r inf9itely dim9i%$2 & yet ! augm5t ( ! fir/1 h[ great soev] x may be1 %all always be less ?an ! decrem5t ( ! second3 :i* ,c>dan demon/rates z foll[s4 ,let !re be propos$ an angle ( contact ,b ,a ,e1 & an acute angle ,h ,g ,i4 ,n[ if !re be o!r less] circles describ$ ,a ,c1 ,a ,d1 ! angle ( contact w be evid5tly increas$4 ,& if betwe5 ! "r l9es ,g ,h1 ,g ,i1 !re fall o!r "r l9es ,g ,k1 ,g ,l1 ! acute angle %all be cont9ually dim9i%$3 yet ! angle ( contact1 h[ev] increas$1 is always less ?an ! acute angle1 h[ev] dim9i%$4 ,sir ,isaac ,newton likewise obs]ves1 9 8 ,treatise on ,fluxions1 t !re are angles ( contact made by o!r curve l9es1 & _! tang5ts inf9itely less ?an ^? made by a circle & "r l9e2 all :i* is demon/rably c]ta93 yet1 su* is ! =ce ( prejudice1 t ,mr4 ,simson is ( op9ion1 ) ,vieta1 t ?is "p ( ! #af? proposition is adult]at$2 & t ! space made by a circul> l9e & its tang5t1 is no angle4 ,at lea/ 8 words1 9 ! note ^u ?is proposition1 w be> su* a con/ruction4 ,pelet>ius 0 likewise ( ! same op9ion2 b is elaborately confut$ by ! excell5t ,clavius1 z may be se5 9 8 comm5t on ?is proposition4 ,b all ! difficulties & p>adoxes 9 ?is affair1 may be easily solv$ & admitt$1 if we consid]1 ) our philosoph]1 t ! ess5ce ( an angle does n subsi/ 9 e!r quantity1 quality1 or incl9ation1 tak5 s+ly1 b 9 ! aggregate ( !m all4 ,= if we reg>d ! incl9ation ( a circul> l9e to its tang5t1 we %all f9d x possess ! prop]ty1 by :i* ,euclid def9es an angle3 if we respect its p>ticipation ( quantity1 we %all f9d x capable ( be+ augm5t$ & dim9i%$2 & if we reg>d x z possess+ a peculi> quality1 we %all acc\nt = its be+ incomm5surable ) e "r-l9$ angle4 ,see ! ,comm5t on ! #h? ,def9ition4 #adj ,9 i4 ,de ,clo4 #ada ,x is f ?is cyl9dric spiral t ! screw is =m$4 #adb ,illu/ration ,! pres5t v obscure passage1 may be expla9$ by ! foll[+ figure4 ,let ,a ,b ,c1 be a "r angle1 & ,d ,e ! l9e to be mov$1 :i* is bisect$ 9 ,g4 ,n[1 conceive x to be mov$ along ! l9es ,a ,b1 ,b ,c1 9 su* a mann]1 t ! po9t ,d may always rema9 9 ,a ,b1 & ! po9t ,e 9 ,b ,c4 ,!n1 :5 ! l9e ,d ,e1 is 9 ! situations d e1 1 ! po9t ,g1 %all be 9 g1 1 & ^! po9ts ,g1 g1 1 %all be 9 a circle4 ,& any o!r po9t ,f 9 ! l9e ,d ,e1 w1 at ! same "t1 describe an ellipsis2 ! great] axis be+ 9 ! l9e ,a ,b1 :5 ! po9t ,f is betwe5 ,d & ,g2 & 9 ! l9e ,b ,c1 :5 ! po9t ,f is betwe5 ,g & ,e4 #adc ,t is1 ! s\l ( ! _w4 #add ,9 ,timo4 #ade ,! ellipsis4 #adf ,! cissoid4 ,= ! prop]ties ( ?is curve1 see ,dr4 ,wallis's treatise on ! cycloid1 p4 #ha4 #adg ,! con*oid4 #adh ,?us1 a "r l9e1 :5 consid]$ z ! side ( a p>allelogram1 mov+ circul>ly1 g5]ates a cyl9drical sup]ficies3 :5 mov+ circul>ly1 z ! side ( a triangle1 a conical surface2 & s 9 o!r l9es1 ! produc$ sup]ficies v>y+ accord+ to ! diff]5t positions ( _! g5]ative l9es4 #adi ,inv ii4 ,de ,rep4 #aej ,9 multis locis4 #aea ,?is def9ition is ! same ) t :i* ,mr4 ,simson has adopt$ in/ead ( ,euclid's1 express$ 9 diff]5t words3 = he says1 8a plane sup]ficies is t 9 :i* any two po9ts be+ tak51 ! /raight l9e betwe5 !m lies :olly 9 t sup]ficies40 ,b he does n m5tion to :om he 0 indebt$ = ! def9ition2 & ?is1 d\btless1 because he consid]$ x 0 n wor? :ile to relate ! trifles ( ,proclus at full l5g?3 = ^! are 8 own words1 9 8 note to proposition #g1 book i4 ,nor has he in=m$ u 9 :at respect ,euclid's def9ition is indi/9ct4 #aeb ,9 ! ,greek 1 b x %\ld d\btless be read 1 images1 z 9 ! translation ( ,b>ocius4 #aec ,mr4 ,simson1 9 8 note on ?is def9ition1 supposes x to be ! addition ( "s less skilful editor2 on :i* acc\nt1 & because x is q useless 79 8 op9ion7 he di/+ui%es x f ! re/ by inv]t$ d\ble commas4 ,b x is surely /range t ! def9ition ( angle 9 g5]al %\ld be acc\nt$ useless1 & ! "w ( an unskilful geometrician4 ,su* an ass]tion may1 inde$1 be v suitable to a pr(essor ( exp]im5tal philosophy1 :o consid]s ! useful z insep>able f practice2 b is by no means becom+ a re/or] ( ! lib]al geometry ( ! anci5ts4 ,besides1 ,mr4 ,simson seems cont9ually to =get t ,euclid 0 ( ! ,platonic sect2 & consequ5tly 0 a philosoph] z well z a ma!matician4 ,i only add1 t ! comm5t>y on ! pres5t def9ition is1 9 my op9ion1 rem>kably subtle & accurate1 & well des]ves ! pr(\nd att5tion ( ! greate/ geometricians4 #aed ,= a philosophical discussion ( ! nature ( quality & quantity1 consult ! ,comm5t>ies ( ,ammonius1 & ,simplicius on ,ari/otle's ,categories1 ,plot9us on ! g5]a ( be+s1 & ,mr4 ,h>ris's ,philosophical ,arrangem5ts4 #aee ,t is1 ! ellipsis4 #aef ,t is1 !y are ei!r "r1 acute1 or obtuse4 #aeg ,?is oracle is n m5tion$ by any ( ! collectors ( ! ,zoroa/rian oracles4 #aeh ,?is1 inde$1 mu/ always be ! case ) ^? geometricians1 :o are n at ! same "t1 philosoph]s2 a conjunction no less valuable ?an r>e4 ,h5ce1 f _! ignorance ( pr9ciples & intellectual conc]ns1 :5 any contemplative enquiry is propos$1 !y imm$iately ask1 9 :at its utility consi/s2 consid]+ e ?+ z sup]flu\s1 :i* does n contribute to ! solution ( "s practical problem4 #aei ,conc]n+ ! s\l's desc5t into body1 see lib4 ix41 ,ennead iv4 ( ,plot9us2 & = ! me?od by :i* %e may aga9 return to h] pri/9e felicity1 /udy ! fir/ book ( ,porphyry's ,treatise on ,ab/95ce4 #afj ,?is ,def9ition too1 is m>k$ by ,mr4 ,simson ) inv]t$ commas1 z a symbol ( its be+ int]polat$4 ,b = :at reason ,i "k n1 unless because x is useless1 t is1 because x is philosophical6 #afa ,t is1 ! v>i\s species ( =ms1 ) :i* ! f\r elem5ts are replete4 #afb ,t is1 ! circle4 #afc ,an admir] ( ! mod]ns1 & _! pursuits1 w d\btless consid] all ?is z ! relics ( hea!n sup]/ition & ignorance2 & w ?9k1 p]haps1 he makes a great concession 9 admitt+ ! exi/5ce ( "o supreme god1 )\t ackn[l$g+ a multitude ( deities subord9ate to ! fir/4 ,= :at ! anci5ts c urge 9 def5ce ( ?is obsolete op9ion1 ,i mu/ beg leave to ref] ! read] to ! diss]tation prefix$ to my translation ( ,orpheus2 9 addition to :i* let him att5d to ! foll[+ consid]ations4 ,is x possible t ! ma*9]y ( ! gods 9 ,hom] c\ld be s beautiful1 if su* be+s _h no exi/5ce8 ,or c any ?+ be beautiful :i* is de/itute ( all reality8 ,d n ?+s univ]sally please 9 proportion z !y resemble reality8 ,p]haps x w be answ]$1 t ! rev]se ( ?is is true1 & t fiction m g5]ally pleases ?an tru?1 z is evid5t f ! great avidity ) :i* romances are p]us$4 ,to ?is ,i reply1 t fiction itself ceases to be pleas+1 :5 x supposes absolute impossibilities3 = ! exi/5ce ( g5ii & fairies _c be prov$ impossible2 & ^! compose all ! m>vell\s ( romance4 ,?is obs]vation is v]ifi$ 9 ,sp5c]'s ,fairy ,que53 = 8 allegories1 9 :i* ! passions are p]sonifi$1 are t$i\s & unpleasant1 because !y are n disguis$ "u ! appe>ance ( reality3 :ile ! magic ( ,circe1 ! b[] ( ,calypso1 ! rocks ( ,scylla & ,*>ybdis1 & ! melody ( ! ,syr5s1 9 ! ,odyssey ( ,hom]1 ?\gh no?+ b allegories1 univ]sally en*ant & delight1 because !y are cov]$ ) ! semblance ( tru?4 ,x is on ?is acc\nt t ,mikon's battles 9 heav5 are b>b>\s & ridicul\s 9 ! extreme2 = e "o sees ! impossibility ( suppos+ gun-p[d] & cannons 9 ! cele/ial regions3 ! ma*9]y is =c$ & unnatural1 conta9s no elegance ( fancy1 & is n replete ) any my/ical in=mation4 ,on ! contr>y1 ,hom]'s ma*9]y is natural & possible1 is full ( dignity & elegance1 & is pregnant ) ! sublime/ tru?s2 x delights & enobles ! m9d ( ! read]1 a/oni%es him ) its magnific5ce & propriety1 & animates him ) ! fury ( poetic inspiration4 ,& ?is1 because x is possible & true4 #afd ,! s5t5ce )9 ! brackets is omitt$ 9 ! ,greek4 #afe ,t is1 ! circul> =m proce$s f b\nd1 b "r-l9$ figures f inf9ity4 #aff ,t is1 ! numb] ?ree4 #afg ,9 ,timo4 #afh 1 or1 to :i*1 is want+ 9 ! orig9al1 & 9 all ! publi%$ collections ( ! ,zoroa/rian oracles4 #afi ,t is ,jupit]1 :o is call$ triadic1 because he proce$s f ,saturn & ,rhea2 & because 8 gov]nm5t is p>ticipat$ by ,neptune & ,pluto1 = ea* ( ^! is call$ ,jupit] by ,orpheus4 #agj ,?is s5t5ce1 )9 ! brackets1 is omitt$ 9 ! pr9t$ ,greek4 #aga ,illu/ration3 ,fig4 ,i4 ,illu/ration3 ,fig4 ,,ii4 ,?us let a "p ,a ,e ,b cut (f by ! diamet] ,a ,b 7fig4 ,i47 ( ! circle ,a ,e ,b ,d be plac$ on ! o!r "p ,a ,d ,b1 z 9 fig4 ,,ii44 ,!n1 if x is n equal to ! o!r "p1 ei!r ,a ,e ,b w fall )9 ,a ,d ,b1 or ,a ,d ,b )9 ,a ,e ,b3 b 9 ei!r case1 ,c ,e w be equal to ,c ,d1 :i* is absurd4 #agb ,?is objection is urg$ by ,philoponus1 9 8 book aga9/ ,proclus on ! et]nity ( ! _w2 b n1 9 my op9ion1 ) any success4 ,see also ,simplicius1 9 8 ?ird digression aga9/ ,philoponus1 9 8 comm5t>y on ! #h? book ( ,ari/otle's ,physics4 #agc ,?is def9ition is no :]e extant b 9 ! comm5t>ies ( ,proclus4 ,in/ead ( x1 9 almo/ all ! pr9t$ editions ( ,euclid1 ! foll[+ is sub/itut$4 ,a segm5t ( a circle is ! figure conta9$ by a diamet]1 & ! "p ( ! circumf]5ce cut (f by ! diamet]4 ,?is ,mr4 ,simson has m>k$ ) commas1 z a symbol ( its be+ int]polat$3 b he has tak5 no notice ( ! diff]5t read+ 9 ! comm5t>ies ( ,proclus4 ,& :at is /ill m rem>kable1 ?is v>iation is n notic$ by any editor ( ,euclid's ,elem5ts1 ei!r anci5t or mod]n4 #agd ,z 9 e hyp]bola4 #age ,! ,platonic read] mu/ d\btless be pleas$ to f9d t ,euclid 0 deeply skill$ 9 ! philosophy ( ,plato1 z ,proclus e :]e ev9ces4 ,inde$1 ! great accuracy1 & elegant di/ribution ( ^! ,elem5ts1 suffici5tly prove ! tru? ( ?is ass]tion4 ,& x is no inconsid]able te/imony 9 fav\r ( ! ,platonic philosophy1 t its assi/ance enabl$ ,euclid to produce su* an admirable "w4 #agf ,conc]n+ ^! cr[ns1 or annul> spaces1 consult ! great "w ( t v subtle & elegant ma!matician ,tacquet1 entitl$ ,cyl9drica et ,annul>ia4 #agg ,9 ! prec$+ t5? comm5t>y4 #agh ,?is 9 consequ5ce ( e triangle possess+ angles alone equal to two "r4 #agi ,?is too1 foll[s f ! same cause z above4 #ahj ,?us ! foll[+ figure ,a ,b ,d ,c has f\r sides1 & b ?ree angles4 ,illu/ration #aha ,! ,greek 9 ?is place is v errone\s1 :i* ,i h re/or$ f ! v]sion ( ,b>ocius4 #ahb ,= ! ,greek ^w is d]iv$ f ! v]b 1 :i* signifies to h a circumvolute motion4 #ahc ,see ! ,orphic ,hymns ( ,onomacritus to ^! deities2 my translation ( :i* ,i mu/ recomm5d to ! ,engli% read]1 because !re is no o!r4 #ahd ,^! twelve div9ities1 ( :i* ,jupit] is ! head1 are1 ,jupit]1 ,neptune1 ,vulcan1 ,ve/a1 ,m9]va1 ,m>s1 ,c]es1 ,juno1 ,diana1 ,m]cury1 ,v5us1 & ,apollo4 ,! fir/ triad ( ^! is demiurgic1 ! second compreh5ds gu>dian deities1 ! ?ird is vivific1 or zoogonic1 & ! f\r? conta9s elevat+ gods4 ,b1 = a p>ticul> !ological acc\nt ( ^! div9ities1 /udy ,proclus on ,plato's ,!ology1 & y w f9d _! nature unfold$1 9 page #djc1 ( t admirable "w4 #ahe ,= x is easy to conceive a cyl9dric spiral describ$ ab\t a "r-l9e1 s z to pres]ve an equal di/ance f x 9 e "p2 & 9 ?is case ! spiral & "r-l9e w nev] co9cide ?\gh inf9itely produc$4 #ahf ,illu/ration ,z ! con*oid is a curve b little kn[n1 ,i h subjo9$ ! foll[+ acc\nt ( its g5]ation & pr9cipal prop]ty4 ,9 any giv5 "r l9e ,a ,p1 call ,p ! pole1 ,a ! v]tex1 & any int]m$iate po9t ,c ! c5tre ( ! con*oid3 likewise1 conceive an inf9ite "r l9e ,c ,h1 :i* is call$ a rule1 p]p5dicul> to ,a ,p4 ,!n1 if ! "r l9e ,a p cont9u$ at p z mu* z is necess>y1 is conceiv$ to be s turn$ ab\t ! abid+ pole p1 t ! po9t ,c may p]petually rema9 9 ! "r l9e ,c ,h1 ! po9t ,a w describe ! curve ,a o1 :i* ! anci5ts call$ a con*oid4 ,9 ?is curve x is manife/ 7on acc\nt ( ! "r l9e ,p ,o1 cutt+ ! rule 9 ,h t ! po9t o w nev] arrive at rule ,c ,h2 b because h ,o is p]petually equal to ,c ,a1 & ! angle ( section is cont9ually m acute1 ! di/ance ( ! po9t ,o f ,c ,h w at l5g? be less ?an any giv5 di/ance1 & consequ5tly ! "r l9e ,c ,h w be an asymptote to ! curve ,a ,o4 ,:5 ! pole is at ,p1 s t ,p ,c is equal to ,c ,a1 ! con*oid ,a ,o describ$ by ! revolution ( ,p ,a1 is call$ a prim>y con*oid1 & ^? describ$ f ! poles p1 & 1 or ! curves ,a o1 ,a 1 second>y con*oids2 & ^! are ei!r contract$ or protract$1 z ! ecc5tricity ,p ,c1 is great] or less ?an ! g5]ative radius ,c ,a1 :i* is call$ ! altitude ( ! curve4 ,n[1 f ! nature ( ! con*oid1 x may be easily inf]r$1 t n only ! ext]ior con*oid ,a w nev] co9cide ) ! "r l9e ,c ,h1 b ?is is likewise true ( ! con*oids ,a ,o1 ,a o2 & by inf9itely ext5d+ ! "r-l9e ,a 1 an inf9ite numb] ( con*oids may be describ$ betwe5 ! ext]ior con*oid ,a 1 & ! l9e ,c ,h1 no "o ( :i* %all ev] co9cide ) ! asymptote ,c ,h4 ,& ?is p>adoxical prop]ty ( ! con*oid :i* has n be5 obs]v$ by any ma!matician1 is a legitimate consequ5ce ( ! inf9ite divisibility ( quantity4 ,n1 inde$1 t quantity admits ( an actual division 9 inf9itum1 = ?is is absurd & impossible2 b x is endu$ ) an unwe>i$ capacity ( division1 & a p[] ( be+ diffus$ into multitude1 :i* c nev] be exhau/$4 ,& ?is inf9ite capacity :i* x possesses arises f its p>ticipation ( ! indef9ite duad2 ! s\rce ( b\ndless diffusion1 & innum]able multitude4 ,illu/ration ,b ?is s+ul> prop]ty is n conf9$ to ! con*oid1 b is f\nd 9 ! foll[+ curve4 ,conceive t ! "r l9e ,a ,c :i* is p]p5dicul> to ! indef9ite l9e ,x ,y1 is equal to ! quadrantal ar* ,h ,d1 describ$ f ! c5tre ,c1 ) ! radius ,c ,d3 !n f ! same c5tre ,c1 ) ! sev]al di/ances ,c ,e1 ,c ,f1 ,c ,g1 describe ! ar*es ,e l1 ,f n1 ,g p1 ea* ( :i* mu/ be conceiv$ equal to ! fir/ ar* ,h ,d1 & s on inf9itely4 ,n[1 if ! po9ts ,h1 k1 l1 n1 p1 be jo9$1 !y w =m a curve l9e1 approa*+ cont9ually ne>] to ! "r-l9e ,a ,b 7p>allel to ,c ,y7 b nev] effect+ a p]fect co9cid5ce4 ,?is w be evid5t f consid]+ t ea* ( ! s9es ( ! ar*es ,h ,d1 l ,e1 n ,f1 c4 be+ less ?an its respective ar*1 mu/ also be less ?an ! "r-l9e ,a ,c1 & consequ5tly c nev] co9cide ) ! "r-l9e ,a ,b4 ,b if o!r ar*es ,d i1 ,e m1 ,f o1 c4 ea* ( !m equal to ! "r-l9e ,a ,c1 & describ$ f "o c5tre1 tang5ts to ! =m] ar*es ,h ,d1 l ,e1 n ,f1 c4 be suppos$2 x is evid5t t ! po9ts ,h1 i1 m1 o1 c4 be+ jo9$1 w =m a curve l9e1 :i* %all pass beyond ! =m] curve1 & conv]ge /ill ne>] to ! l9e ,a ,b1 )\t a possibility ( ev] becom+ co9cid5t3 = s9ce ! ar*es ,d i1 ,e m1 ,f o1 c4 h less curvature ?an ! =m] ar*es1 b are equal to !m 9 l5g?1 x is evid5t t !y w be subt5d$ by long] l9es1 & yet c nev] t\* ! "r-l9e ,a ,b4 ,9 l mann]1 if o!r tang5t ar*es be drawn to ! =m]1 & s on inf9itely1 ) ! same conditions1 an inf9ite numb] ( curve-l9es w be =m$1 ea* ( !m pass+ betwe5 ,h p & ,a ,b1 & cont9ually div]g+ f ! latt]1 )\t a possibility ( ev] co9cid+ ) ! =m]4 ,?is curve1 :i* ,i inv5t$ "s ye>s s9ce1 ,i suspect to be a p>abola2 b ,i h n yet _h opportunity to det]m9e x ) c]ta9ty4 ,transcrib]'s ,notes3 #a4 ,obvi\s pr9t]s'1 punctuation & spell+ errors h be5 correct$ sil5tly4 #b4 ,:]e hyph5ation is 9 d\bt1 x has be5 reta9$ z 9 ! orig9al4 #c4 ,"s hyph5at$ & non-hyph5at$ v]sions ( ! same words h be5 reta9$ z 9 ! orig9al4 #d4 ,! errata h be5 soil5tly correct$4 #e4 ,italics are %[n z xxx4