,transcrib]'s ,notes ,text pr9t$ 9 italics is repres5t$ betwe5 und]scores1 bold face text betwe5 tildes4 ,small capitals h be5 transcrib$ z ,,all ,,capitals4 ,a lett] 9 squ>e brackets prec$$ by an equal sign1 z ,x1 repres5ts a lett] ) b> above4 ,m ,transcrib]'s ,notes may be f\nd at ! end ( ?is text4 ,a ,,new ,,era ,,( ,,?\ght4 ,,sci5tific ,,romances4 ,by ,c4 ,,h[>d ,,h9ton1 ,m4,a4 ,cr[n #hvo1 clo? gilt1 #fs42 or sep>ately1 #as4 ea*4 #a4 ,:at is ! ,f\r? ,dim5sion8 #as4 ,,gho/s ,,expla9$4 8,a %ort treatise ( admirable cle>ness4 4 4 4 ,mr4 ,h9ton br+s u1 pant+ b delight$1 to at lea/ a mom5t>y fai? 9 ! ,f\r? ,dim5sion1 & ^u ! eye ( ?is fai? !re op5s a vi/a ( int]e/+ problems4 4 4 4 ,8 pamphlet exhibits a boldness ( speculation1 & a p[] ( conceiv+ & express+ ev5 ! inconceivable1 :i* r\ses "o's faculties l a tonic40--,pall ,mall4 #b4 ,! ,p]sian ,k+2 or1 ,! ,law ( ! ,valley1 #as4 ,,! ,,my/]y ,,( ,,pleasure ,,& ,,pa94 8,a v sugge/ive & well-writt5 speculation1 by ! inh]itor ( an hon\r$ "n40--,m9d4 8,w arre/ ! att5tion ( ! read] at once40--,k4 #c4 ,a ,plane ,_w1 #as4 #d4 ,a ,picture ( our ,univ]se1 #as4 #e4 ,ca/+ out ! ,self1 #as4 ,,second ,,s]ies4 #a4 ,on ! ,education ( ! ,imag9ation4 #b4 ,_m ,dim5sions1 #as4 ,,london3 ,,swan ,,sonn5s*e9 ,,co4 ,a ,new ,era ( ,?\ght4 ,,by ,,*>les ,,h[>d ,,h9ton1 ,m4,a41 ,,oxon4 ,au?or ( 8,:at is ! ,f\r? ,dim5sion10 & o!r 8,sci5tific ,romances40 ,illu/ration ,london3 ,,swan ,,sonn5s*e9 ,,co41 ,,pat]no/] ,,squ>e4 #ahhh4 ,,butl] ,,tann]1 ,,! ,,selwood ,,pr9t+ ,,works1 ,,frome1 ,,& ,,london4 ,,preface4 ,! ,,mss4 :i* =m$ ! basis ( ?is book 7 committ$ to u by ! au?or1 on 8 leav+ ,engl& = a di/ant =eign appo9tm5t4 ,x 0 8 wi% t we %\ld con/ruct ^u !m a mu* m complete treatise ?an we h effect$1 & ) t int5tion he ask$ u to make any *anges or additions we ?\ght desirable4 ,b long alliance ) him 9 ?is "w has conv9c$ u t 8 ?\ght 7especially t ( a g5]al philosophical "*7 loses mu* ( its =ce if subject$ to any extrane\s t\*4 ,?is feel+ has induc$ u to pr9t ,"p ,i4 almo/ exactly z x came f 8 h&s1 al?\gh x w\ld probably h receiv$ mu* re>rangem5t if he c\ld h wat*$ x ?r\gh ! press himself4 ,"p ,,ii4 has be5 writt5 f a hurri$ sket*1 :i* he consid]$ v inadequate1 & :i* we h consequ5tly correct$ & supplem5t$4 ,*apt] ,,xi4 ( ?is "p has be5 entirely re-writt5 by u1 & has ?us n _h ! advantage ( 8 sup]vision4 ,?is rem>k also applies to ,app5dix ,e1 :i* is an elaboration ( a !orem he sugge/$4 ,app5dix ,h1 & all ! ex]cises h1 9 accordance ) 8 wi%1 be5 writt5 solely by u4 ,x w be app>5t to ! read] t ,app5dix ,h is little m ?an a brief introduction to a v l>ge subject1 :i*1 be+ conc]n$ ) tess>acts & solids1 is really beyond treatm5t 9 writ+ & diagrams4 ,?is difficulty recalls u to ! "o great fact1 ^u :i* we feel b\nd to insi/1 t ! matt] ( ?is book mu/ receive objective treatm5t f ! read]1 :o w f9d x q useless ev5 to attempt to appreh5d x )\t actually build+ 9 squ>es & cubes all ! facts ( space :i* we ask him to impress on 8 consci\sness4 ,inde$1 we consid] t pr9t+1 z a me?od ( spread+ space-k1 is b a 8pis all]10 & we w\ld g back to t anci5t & m fruitful me?od ( ! ,greek geomet]s1 &1 :ile describ+ figures on ! s&1 or pil+ up pebbles 9 s]ies1 w\ld communicate to o!rs t _s ( le>n+ & g5]alization begott5 9 our consci\sness by cont9u\s contact ) facts1 & only by cont9u\s contact ) facts vitally ma9ta9$4 ,,alicia ,,boole1 ,h4 ,,john ,,falk4 ,n4,b4 ,models4--,x is unque/ionably a mo/ important "p ( ! process ( le>n+ space to con/ruct ^!1 & ! read] %\ld d s1 h[ev] r\ghly & ha/ily4 ,b1 if ,models are requir$ z patt]ns1 !y may be ord]$ f ,messrs4 ,swan ,sonn5s*e9 ,co41 ,pat]no/] ,squ>e1 ,london1 & w be suppli$ z soon z possible1 ! unc]ta9ty z to dem& = same n all[+ u to h a l>ge numb] made 9 advance4 ,mu* ( ! "w c be done ) pla9 cubes by us+ names )\t col\rs1 b fur!r on ! read] w f9d col\rs necess>y to enable him to grasp & reta9 ! complex s]ies ( obs]vations4 ,col\r$ models c easily be made by cov]+ ,k9d]g>t5 cubes ) :ite pap] & pa9t+ !m ) wat]-col\r1 &1 if p]man5ce be desir$1 dipp+ !m 9 size & copal v>ni%4 ,,table ,,( ,,cont5ts4 ,,"p ,i4 ,,page ,,introduction #a-g ,,*apt] ,i4 ,scepticism & ,sci5ce4 ,beg9n+ ( ,k #h-ac ,,*apt] ,,ii4 ,appreh5sion ( ,nature4 ,intellig5ce4 ,/udy ( ,arrangem5t or ,%ape #ad-bj ,,*apt] ,,iii4 ,! ,elem5ts ( ,k #ba-bc ,,*apt] ,,iv4 ,!ory & ,practice #bd-bh ,,*apt] ,v4 ,k3 ,self-,elem5ts #bi-cd ,,*apt] ,,vi4 ,function ( ,m9d4 ,space aga9/ ,metaphysics4 ,self-,limitation & its ,te/4 ,a ,plane ,_w #ce-df ,,*apt] ,,vii4 ,self ,elem5ts 9 our ,consci\sness #dg-ej ,,*apt] ,,viii4 ,relation ( ,l[] to ,high] ,space4 ,!ory ( ! !r #ea-fj ,,*apt] ,,ix4 ,ano!r ,view ( ! !r4 ,mat]ial & !rial ,bodies #fa-ff ,,*apt] ,x4 ,high] ,space & ,high] ,be+4 ,p]ception & ,inspiration #fg-hd ,,*apt] ,,xi4 ,space ! ,sci5tific ,basis ( ,altruism & ,religion #he-ii ,,"p ,,ii4 ,,*apt] ,i4 ,?ree-space4 ,g5esis ( a ,cube4 ,appe>ances ( a ,cube to a ,plane-be+ #aja-aab ,,*apt] ,,ii4 ,fur!r ,appe>ances ( a ,cube to a ,plane-be+ #aac-aag ,,*apt] ,,iii4 ,f\r-space4 ,g5esis ( a ,tess>act2 its ,repres5tation 9 ,?ree-space #aah-abi ,,*apt] ,,iv4 ,tess>act mov+ ?r\gh ,?ree-space4 ,models ( ! ,sections #acj-acd ,,*apt] ,v4 ,repres5tation ( ,?ree-space by ,names & 9 a ,plane #ace-adh ,,*apt] ,,vi4 ,! ,means by :i* a ,plane-be+ w\ld ,acquire a ,conception ( our ,figures #adi-aee ,,*apt] ,,vii4 ,f\r-space3 its ,repres5tation 9 ,?ree-space #aef-aff ,,*apt] ,,viii4 ,repres5tation ( ,f\r-space by ,"n4 ,/udy ( ,tess>acts #afg-agf ,,*apt] ,,ix4 ,fur!r ,/udy ( ,tess>acts #agg-agi ,,*apt] ,x4 ,cyclical ,projections #ahj-ahc ,,*apt] ,,xi4 ,a ,tess>actic ,figure & its ,projections #ahd-aid ,,app5dices4 ,a4 #ajj ,names us$ = ,plane ,space #aig ,b4 #baf ,names us$ = ,cubic ,space #aih ,c4 #bef ,names us$ = ,tess>actic ,space #bjj-bja ,d4 ,li/ ( ,col\rs1 ,names1 & ,symbols #bjb-bjc ,e4 ,a ,!orem 9 ,f\r-space #bjd-bje ,f4 ,ex]cises on ,%apes ( ,?ree ,dim5sions #bje-bjg ,g4 ,ex]cises on ,%apes ( ,f\r ,dim5sions #bjg-bji ,h4 ,sections ( ! ,tess>act #bji-bag ,k4 ,draw+s ( ! ,cubic ,sides & ,sections ( ! ,tess>act 7,models #a-ab7 ) ,col\rs & ,names #bai-bda ,,introductory ,,note ,,to ,,"p ,i4 ,at ! completion ( a "w1 or at ! completion ( ! fir/ "p ( a "w1 ! feel+s are necess>ily v diff]5t f ^? ) :i* ! "w 0 begun2 & ! mean+ & value ( ! "w itself be> a v diff]5t appe>ance4 ,x w !re=e be ! simple/ & %orte/ plan1 if ,i tell ! read] briefly :at ! "w is to :i* ^! pages are a guide1 & :at ,i consid] to be its value :5 done4 ,! task 0 to obta9 a s5se ( ! prop]ties ( high] space1 or space ( f\r dim5sions1 9 ! same way z t by :i* we rea* a s5se ( our ord9>y ?ree-dim5sional space4 ,i n[ pref] to call ! task t ( obta9+ a famili>ity ) high] matt]1 :i* %all be z intuitive to ! m9d z t ( ord9>y matt] has become4 ,! expression 8high] matt]0 is pref]able to 8high] space10 because x is a some:at ha/y proce$+ to split ?is concrete matt]1 :i* we t\* & feel1 into ! ab/ractions ( ext5sion & imp5etrability4 ,x seems to me t ,i _c ?9k ( space )\t matt]1 & !re=e1 z no necessity compels me to su* a c\rse1 ,i d n split up ! concrete object into subtleties1 b ,i simply ask3 8,:at is t :i* is to a cube or block or %ape ( any k9d z ! cube is to a squ>e80 ,9 ent]+ ^u ?is inquiry we f9d ! task is tw(old4 ,fir/ly1 !re is ! !oretical "p1 :i* is easy1 viz4 to set cle>ly be=e u ! relative conditions :i* w\ld obta9 if !re 7 a matt] physically high] ?an ?is matt] ( ours1 & to *oose ! be/ means ( lib]at+ our m9ds f ! limitations impos$ on x by ! p>ticul> conditions "u :i* we are plac$4 ,! second "p ( ! task is some:at labori\s1 & consi/s ( a con/ant pres5tation to ! s5ses ( ^? appe>ances :i* portions ( high] matt] w\ld pres5t1 & ( a cont9ual dwell+ on !m1 until ! high] matt] becomes famili>4 ,! read] mu/ und]take ?is task1 if he accepts x at all1 z an exp]im5t4 ,^? ( u :o h done x1 are satisfi$ t !re is t 9 ! results ( ! exp]im5t :i* make x well wor?y ( a trial4 ,& 9 a few words ,i may /ate ! g5]al be>+s ( ?is "w1 = e bran* ( "w has its g5]al be>+s4 ,x is an attempt1 9 ! mo/ elem5t>y & simple doma91 to pass f ! l[] to ! high]4 ,9 pursu+ x ! m9d passes f "o k9d ( intuition to a high] "o1 & ) t transition ! horizon ( ?\ght is alt]$4 ,x becomes cle> t !re is a physical exi/5ce transc5d+ ! ord9>y physical exi/5ce2 & "o becomes incl9$ to ?9k t ! "r direction to look is1 n away f matt] to spiritual exi/5ces1 b t[>ds ! discov]y ( conceptions ( high] matt]1 & !reby ( ^? mat]ial exi/5ces ^: def9ite relations to u are appreh5d$ z spiritual intuitions4 ,?us1 8mat]ial0 w\ld simply mean 8grasp$ by ! intellect1 become kn[n & famili>40 ,our appreh5sion ( any?+ :i* is n express$ 9 t]ms ( matt]1 is vague & indef9ite4 ,to realize & live ) t :i* we vaguely disc]n1 we ne$ to apply ! intuition ( high] matt] to ! _w ar\nd u4 ,& ?is seems to me ! great inducem5t to ?is /udy4 ,let u =m our intuition ( high] space1 & !n look out ^u ! _w4 ,secondly1 9 ?is progress f ord9>y to high] matt]1 z a g5]al type ( progress f l[] to high]1 we make ! foll[+ obs]vations4 ,fir/ly1 we become aw>e t !re are c]ta9 limitations affect+ our reg>d4 ,secondly1 we discov] by our reason :at ^? limitations are1 & !n =ce ourselves to g ?r\gh ! exp]i5ce :i* w\ld be ours if ! limitations did n affect u4 ,?irdly1 we become aw>e ( a capacity )9 u = transc5d+ ^? limitations1 & = liv+ 9 ! high] mode z we _h liv$ 9 ! previ\s "o4 ,we may rem>k t ?is progress f ! ord9>y to ! high] k9d ( matt] dem&s an absolute att5tion to details4 ,x is only 9 ! ret5tion ( details t su* progress becomes possible4 ,& z1 9 ?is "q ( matt]1 an absolute & unconv5tional exam9ation gives u ! indication ( a high]1 s1 d\btless1 9 o!r que/ions1 if we b come to facts )\t presupposition1 we beg9 to "k t !re is a high] & to discov] indications ( ! way :]eby we c approa*4 ,t way lies 9 ! fulness ( detail r ?an 9 ! g5]alization4 ,biology has %[n u t !re is a univ]sal ord] ( =ms or organisms1 pass+ f l[] to high]4 ,!re9 we f9d an indication t we ourselves take "p 9 ?is progress4 ,& 9 us+ ! little cubes we c g ?r\gh ! process ourselves1 & le>n :at x is 9 a little in/ance4 ,b ( all ! ways 9 :i* ! confid5ce ga9$ f ?is lesson c be appli$1 ! ne>e/ to u lies 9 ! sugge/ion x gives1--& m ?an ! sugge/ion1 if incl9ation to ?9k be c\nt$ = any?+1--9 ! sugge/ion ( t :i* is high] ?an ourselves4 ,we1 z individuals1 are n ! limit & end-all1 b !re is a high] be+ ?an ours4 ,:at our relation to x is1 we _c tell1 = t is unlike our relation to any?+ we "k4 ,b1 p]haps all t happ5s to u is1 c\ld we b grasp x1 our relation to x4 ,at any rate1 ! discov]y ( x is ! great object beside :i* all else is z second>y z ! r\t9e ( m]e exi/5ce is to companion%ip4 ,& ! me?od ( discov]y is full k ( ea* o!r4 ,!reby is ! high] be+ to be kn[n4 ,9 z mu* z ! lea/ ( u kn[s & is kn[n by ano!r1 9 s mu* does he "k ! high]4 ,?us1 sci5tific pray] is :5 two or ?ree meet toge!r1 &1 9 ! belief ( "o high] ?an !mselves1 mutually compreh5d t vision ( ! high]1 :i* ea* "o is1 &1 by absolute fulness ( k ( ! facts ( ea* o!r's p]sonality1 /rive to atta9 a k ( t :i* is to ea* ( _! p]sonalities z a high] figure is to its solid sides4 ,c4 ,h4 ,h4 ,a ,,new ,,era ,,( ,,?\ght4 ,,"p ,i4 ,,introduction4 ,!re are no new tru?s 9 ?is book1 b x consi/s ( an ef=t to impress ^u & br+ home to ! m9d "s ( ! m mod]n developm5ts ( ?\ght4 ,a few s5t5ces ( ,kant1 a few lead+ ideas ( ,gauss & ,lobats*ewski =m ! mat]ial out ( :i* x is built up4 ,x may be ?\ght to be unduly long2 b x mu/ be rememb]$ t 9 ^! times !re is a tw(old process go+ on--"o ( discov]y ab\t ext]nal nature1 "o ( education1 by :i* our m9ds are br\ght into h>mony ) t :i* we "k4 ,9 c]ta9 respects we f9d ourselves br\ght on by ! g5]al curr5t ( ideas--we feel t matt] is p]man5t & _c be annihilat$1 & x is almo/ an axiom 9 our m9ds t en]gy is p]si/5t1 & all its trans=mations rema9s ! same 9 am\nt4 ,b !re are o!r directions 9 :i* !re is ne$ ( def9ite tra9+ if we are to ent] into ! ?\ghts ( ! "t4 ,& x seems to me t a return to ,kant1 ! creator ( mod]n philosophy1 is ! fir/ condition4 ,n[ ( ,kant's enorm\s "w only a small "p is treat$ "h1 b ) ! diff]5ce t %\ld be f\nd betwe5 ! "w ( a ma/] & t ( a foll[]4 ,kant's /atem5ts are tak5 z lead+ ideas1 sugge/+ a field ( "w1 & x is 9 detail & manipulation m]ely t !re is an opportunity = workman%ip4 ,( ,kant's "w x is only 8 doctr9e ( space :i* is "h exp]im5t$ ^u4 ,) ,kant ! p]ception ( ?+s z be+ 9 space is n treat$ z x seems s obvi\s to d4 ,we %\ld naturally say t !re is space1 & !re are ?+s 9 x4 ,f a comp>ison ( ^? prop]ties :i* are common to all ?+s we obta9 ! prop]ties ( space4 ,b ,kant says t ?is prop]ty ( be+ 9 space is n s mu* a quality ( any def9able objects1 z ! means by :i* we obta9 an appreh5sion ( def9able objects--x is ! condition ( our m5tal "w4 ,n[ z ,kant's doctr9e is usually comm5t$ on1 ! negative side is br\ght into prom95ce1 ! positive side is neglect$4 ,x is g5]ally said t ! m9d _c p]ceive ?+s 9 !mselves1 b c only appreh5d !m subject to space conditions4 ,& 9 ?is way ! space conditions are z x 7 consid]$ some:at 9 ! light ( h9drances1 :]eby we are prev5t$ f see+ :at ! objects 9 !mselves truly are4 ,b if we take ! /atem5t simply z x is--t we appreh5d by means ( space--!n x is equally all[able to consid] our space s5se z a positive means by :i* ! m9d grasps its exp]i5ce4 ,!re is 9 s _m books 9 :i* ! subject is treat$ a c]ta9 air ( despond5cy--z if ?is space appreh5sion 7 a k9d ( veil :i* %ut u (f f nature4 ,b !re is no ne$ to adopt ?is feel+4 ,! fir/ po/ulate ( ?is book is a full recognition ( ! fact1 t x is by means ( space t we appreh5d :at is4 ,space is ! in/rum5t ( ! m9d4 ,& "h = ! purposes ( our "w we c avoid all metaphysical discussion4 ,v (t5 a /atem5t :i* seems to be v deep & ab/ruse & h>d to grasp1 is simply ! =m into :i* deep ?9k]s h ?r[n a v simple & practical obs]vation4 ,& = ! pres5t let u look on ,kant's great doctr9e ( space f a practical po9t ( view1 & x comes to ?is--x is important to develop ! space s5se1 = x is ! means by :i* we ?9k ab\t real ?+s4 ,!re is a doctr9e :i* f\nd mu* fav\r ) ! fir/ foll[]s ( ,kant1 t also af=ds u a simple & practical rule ( "w4 ,x 0 consid]$ by ,fi*te t ! :ole ext]nal _w 0 simply a projection f ! ego1 & ! manifold ( nature 0 a recognition by ! _s ( itself4 ,:at ?is comes to z a practical rule is1 t we c only und]/& nature 9 virtue ( our own activity2 t !re is no su* ?+ z m]e passive obs]vation1 b e act ( sight & ?\ght is an activity ( our own4 ,n[ accord+ to ,kant ! space s5se1 or ! intuition ( space1 is ! mo/ fundam5tal p[] ( ! m9d4 ,b ,i d n f9d any:]e a sy/ematic & ?or\ghgo+ education ( ! space s5se4 ,9 e practical pursuit x is ne$$--9 "s x is develop$4 ,9 geometry x is us$2 b ! great reason ( failure 9 education is t1 in/ead ( a sy/ematic tra9+ ( ! space s5se1 x is left to be organiz$ by accid5t & is call$ ^u to act )\t hav+ be5 =m$4 ,accord+ to ,kant & accord+ to common exp]i5ce x w be f\nd t a tra9$ ?9k] is "o 9 :om ! space s5se has be5 well develop$4 ,) reg>d to ! education ( ! space s5se1 ,i mu/ ask ! indulg5ce ( ! read]4 ,x w seem obvi\s to him t any real pursuit or real obs]vation tra9s ! space s5se1 & t x is go+ out ( ! way to und]take any special discipl9e4 ,to ?is ,i w\ld answ] t1 accord+ to my own exp]i5ce1 ,i 0 p]fectly ignorant ( space relations myself be=e ,i actually work$ at ! subject1 & t directly ,i got a true view ( space facts a :ole s]ies ( conceptions1 :i* be=e ,i _h kn[n m]ely by repute & grasp$ by an ef=t1 became p]fectly simple & cle> to me4 ,moreov]1 to take "o in/ance3 9 /udy+ ! relations ( space we always h to d ) col\r$ objects1 we always h ! s5se ( weight2 = if ! ?+s !mselves h no weight1 !re is always a direction ( up & d[n--:i* implies ! s5se ( weight1 & to get rid ( ^! elem5ts requires c>eful sift+4 ,b p]haps ! be/ po9t ( view to take is ?is--if ! read] has ! space s5se well develop$ he w h no difficulty 9 go+ ?r\gh ! "p ( ! book :i* relates to x1 & ! phraseology w s]ve him = ! consid]ations :i* come next4 ,among/ ! foll[]s ( ,kant1 ^? :o pursu$ "o ( ! l9es ( ?\ght 9 8 works h attract$ ! mo/ att5tion & h be5 consid]$ z 8 successors4 ,fi*te1 ,s*ell+1 ,hegel h develop$ c]ta9 t5d5cies & h writt5 rem>kable books4 ,b ! true successors ( ,kant are ,gauss & ,lobat*ewski4 ,= if our intuition ( space is ! means by :i* we appreh5d1 !n x foll[s t !re may be diff]5t k9ds ( intuitions ( space4 ,:o c tell :at ! absolute space intuition is8 ,?is intuition ( space mu/ be col\r$1 s to speak1 by ! conditions ( ! be+ :i* uses x4 ,n[1 aft] ,kant _h laid d[n 8 doctr9e ( space1 x 0 important to inve/igate h[ mu* 9 our space intuition is due to exp]i5ce--is a matt] ( ! physical circum/ances ( ! ?9k+ be+--& h[ mu* is ! pure act ( ! m9d4 ,! only way to inve/igate ?is is ! practical way1 & by a rem>kable analysis ! great geomet]s above m5tion$ h %[n t space is n limit$ z ord9>y exp]i5ce w\ld seem to in=m u1 b t we are q capable ( conceiv+ diff]5t k9ds ( space4 ,our space z we ord9>ily ?9k ( x is conceiv$ z limit$--n 9 ext5t1 b 9 a c]ta9 way :i* c only be realiz$ :5 we ?9k ( our ways ( measur+ space objects4 ,x is f\nd t !re are only ?ree indep5d5t directions 9 :i* a body c be measur$--x mu/ h height1 l5g? & bread?1 b x has no m ?an ^! dim5sions4 ,if any o!r measurem5t be tak5 9 x1 ?is new measurem5t w be f\nd to be comp\nd$ ( ! old measurem5ts4 ,x is impossible to f9d a po9t 9 ! body :i* c\ld n be arriv$ at by travell+ 9 comb9ations ( ! ?ree directions already tak54 ,b :y %\ld space be limit$ to ?ree indep5d5t directions8 ,geomet]s h f\nd t !re is no reason :y bodies %\ld be ?us limit$4 ,z a matt] ( fact all ! bodies :i* we c measure are ?us limit$4 ,s we come to ?is conclusion1 t ! space :i* we use = conceiv+ ord9>y objects 9 ! _w is limit$ to ?ree dim5sions4 ,b x might be possible = !re to be be+s liv+ 9 a _w su* t !y w\ld conceive a space ( f\r dim5sions4 ,all t we c say ab\t su* a supposition is1 t x is n dem&$ by our exp]i5ce4 ,x may be t 9 ! v l>ge or ! v m9ute a f\r? dim5sion ( space w h to be po/ulat$ to acc\nt = p>ts--b ) reg>d to objects ( ord9>y magnitudes we are c]ta9ly n 9 a f\r dim5sional _w4 ,& ?is 0 ! po9t at :i* ab\t t5 ye>s ago ,i took up ! inquiry4 ,x is possible to say a great deal ab\t space ( high] dim5sions ?an our own1 & to "w out analytically _m problems :i* sugge/ !mselves4 ,b c we conceive f\r-dim5sional space 9 ! same way 9 :i* we c conceive our own space8 ,c we ?9k ( a body 9 f\r dim5sions z a unit hav+ prop]ties 9 ! same way z we ?9k ( a body hav+ a def9ite %ape 9 ! space ) :i* we are famili>8 ,n[ ?is "q1 z e o!r ) :i* ,i am acqua9t$1 c only be answ]$ by exp]im5t4 ,& ,i comm5c$ a s]ies ( exp]im5ts to arrive at a conclusion "o way or ! o!r4 ,x is obvi\s t ?is is n a sci5tific inquiry--b "o = ! practical tea*]4 ,& j z 9 exp]im5tal rese>*es ! skilful manipulator w demon/rate a law ( nature1 ! less skill$ manipulator w fail2 s "h1 ev]y?+ dep5d$ on ! manipulation4 ,i 0 n sure t ?is p[] lay hidd5 9 ! m9d1 b to put ! "q fairly w\ld surely dem& e res\rce ( ! practical art ( education4 ,& s x prov$ to be2 = aft] _m ye>s ( "w1 dur+ :i* ! conception ( f\r-dim5sional bodies lay absolutely d>k1 at l5g?1 by a c]ta9 *ange ( plan1 ! :ole subject ( f\r-dim5sional exi/5ce became p]fectly cle> & easy to imp>t4 ,!re is really no m difficulty 9 conceiv+ f\r-dim5sional %apes1 :5 we g ab\t x ! "r way1 ?an 9 conceiv+ ! idea ( solid %apes1 nor is !re any my/]y at all ab\t x4 ,:5 ! faculty is acquir$--or r :5 x is br\ght into consci\sness1 = x exi/s 9 e "o 9 imp]fect =m--a new horizon op5s4 ,! m9d acquires a developm5t ( p[]1 & 9 ?is use ( ampl] space z a mode ( ?\ght1 a pa? is op5$ by us+ t v tru? :i*1 :5 fir/ /at$ by ,kant1 seem$ to close ! m9d )9 su* fa/ limits4 ,our p]ception is subject to ! condition ( be+ 9 space4 ,b space is n limit$ z we at fir/ ?9k4 ,! next /ep aft] hav+ =m$ ?is p[] ( conception 9 ampl] space1 is to inve/igate nature & see :at ph5om5a are to be expla9$ by f\r-dim5sional relations4 ,b ?is "p ( ! subject is h>dly "o = ! same work] z ! "o :o inve/igates h[ to ?9k 9 f\r-dim5sional space4 ,! "w ( build+ up ! p[] is ! "w ( ! practical educator1 ! "w ( apply+ x to nature is ! "w ( ! sci5tific man4 ,& x is n possible to accompli% bo? tasks at ! same "t4 ,consequ5tly ! cr[n is /ill to be won4 ,"h ! me?od is giv5 ( tra9+ ! m9d2 x w be an exhil>at+ mom5t :5 an inve/igator comes ^u ph5om5a :i* %[ t ext]nal nature _c be expla9$ except by ! assumption ( a f\r-dim5sion space4 ,! ?\ght ( ! pa/ ages has us$ ! conception ( a ?ree-dim5sional space1 & by t means has classifi$ _m ph5om5a & has obta9$ rules = deal+ ) matt]s ( great practical utility4 ,! pa? :i* op5s imm$iately be=e u 9 ! future is t ( apply+ ! conception ( f\r-dim5sional space to ! ph5om5a ( nature1 & ( inve/igat+ :at c be f\nd out by ?is new means ( appreh5sion4 ,9 fact1 :at has be5 pass$ ?r\gh may be call$ ! ?ree-dim5sional era2 ,gauss & ,lobat*ewski h inaugurat$ ! f\r-dim5sional era4 ,,*apt] ,i4 ,,scepticism ,,& ,,sci5ce4 ,,beg9n+ ,,( ,,k4 ,! foll[+ pages h = _! object to induce ! read] to apply himself to ! /udy1 9 ! fir/ place ( ,space1 & !n ( ,high] ,space2 &1 !re=e1 ,i h tri$ by indirect means to %[ =? ^? ?\ghts & conceptions to :i* ! practical "w leads4 ,& ,i feel t ,i h a great advantage 9 ?is project1 inasmu* z _m ( ! ?\ghts :i* spr+ up 9 ! m9d ( "o :o /udies high] space1 & _m ( ! conceptions to :i* he is driv51 turn out to be no?+ m nor less ?an old tru?s--! prop]ty ( e m9d t ?9ks & feels--tru?s :i* are n g5]ally associat$ ) ! sci5tific appreh5sion ( ! _w1 b :i* are n = t reason any ! less valuable4 ,& = my own "p ,i _c d m ?an put !m =w>d 9 a v feeble & halt+ mann]4 ,= ,i h come ^u !m1 n 9 ! way ( feel+ or direct appreh5sion1 b z ! result ( a s]ies ( works und]tak5 purely ) ! desire to "k--a desire :i* did n lift itself to ! height ( expect+ or look+ = ! beautiful or ! good1 b :i* simply ask$ = some?+ to "k4 ,= ,i f\nd myself--& _m o!rs ,i f9d d s also--,i f\nd myself 9 respect to k l a man :o is 9 ! mid/ ( pl5ty & yet :o _c f9d any?+ to eat4 ,all ar\nd me 7 ! evid5ces ( k--! arts1 ! sci5ces1 int]e/+ talk1 useful inv5tions--& yet ,i myself 0 pr(it$ no?+ at all2 = someh[1 amid/ all ?is activity1 ,i 0 left alone1 ,i c\ld get no?+ :i* ,i c\ld "k4 ,! dialect 0 =eign to me--its inn] mean+ 0 hidd54 ,if ,i w\ld1 imitat+ ! utt]ance ( my fell[s1 say a few words1 ! ef=t 0 =c$1 ! :ole result 0 an artificiality1 &1 if successful1 w\ld be b a plausible impo/ure4 ,! ^w 8sceptical0 has a c]ta9 unpleasant association atta*$ to x1 = x has be5 us$ by s _m p :o are absolutely c]ta9 9 a p>ticul> l9e1 & attack o!r p's convictions4 ,b to be sceptical 9 ! real s5se is a f> m unpleasant /ate ( m9d to ! sceptic ?an to any "o ( 8 companions4 ,= to a m9d t inquires into :at x really does "k1 x is h>dly possible to enunciate complete s5t5ces1 mu* less to put be=e x ^? complex ideas :i* h s l>ge a "p 9 true human life4 ,e ^w we use has s wide & fugitive a mean+1 & e expression t\*es or r grazes fact by s v m9ute a po9t1 t1 if we wi% to />t ) some?+ :i* we d "k1 & !nce proce$ 9 a c]ta9 mann]1 we are =c$ away f ! /udy ( reality & driv5 to an artificial sy/em1 su* z logic or ma!matics1 :i*1 />t+ f po/ulates & axioms1 develops a body ( ideal tru? :i* r comes into contact ) nature ?an is nature4 ,sci5tific a*ievem5t is res]v$ = ^? :o are cont5t to absorb into _! consci\sness1 by any means & by :atev] way !y come1 ! v>i$ appe>ances ( nature1 :5ce & 9 :i* by reflection !y f9d float+ z x 7 on ! sea ( ! unkn[n1 c]ta9 simil>ities1 c]ta9 resemblances & analogies1 by means ( :i* !y collect toge!r a body ( possible pr$ictions & inf]5ces2 & 9 nature !y f9d correspond5ces :i* are actually v]ifi$4 ,h5ce sci5ce exi/s1 al?\gh ! conceptions 9 ! m9d _c be said to h any real correspond5ce 9 nature4 ,we =m a set ( conceptions 9 ! m9d1 & ! relations betwe5 ^! conceptions give u relations :i* we f9d actually vibrat+ 9 ! _w ar\nd u4 ,b ! conceptions !mselves are ess5tially artificial4 ,we h a conception ( atoms2 b no "o supposes t atoms actually exi/4 ,we suppose a =ce v>y+ inv]sely z ! squ>e ( ! di/ance2 b no "o supposes su* a my/]i\s ?+ to really be 9 nature4 ,& :5 we come to ! region ( descriptive sci5ce1 :5 we come to simple obs]vation1 we d n f9d ourselves any bett] provid$ ) a real k ( nature4 ,if1 = in/ance1 we ?9k ( a plant1 we picture to ourselves a c]ta9 gre5 %ape1 ( a m or less def9ite "*4 ,?is gre5 %ape enables u to recognise ! plant we ?9k (1 & to describe x to a c]ta9 ext5t4 ,b if we inquire into our imag9ation ( x1 we f9d t our m5tal image v soon div]ges f ! fact4 ,if1 = in/ance1 we cut ! plant 9 half1 we f9d cells & tissues ( v>i\s k9ds4 ,if we exam9e our idea ( ! plant1 x has m]ely an ext]nal & sup]ficial resemblance to ! plant itself4 ,x is a m5tal draw+ meet+ ! real plant 9 ext]nal appe>ance2 b ! two ?+s1 ! plant & our ?\ght ( x1 come z x 7 f diff]5t sides--!y j t\* ea* o!r z f> z ! col\r & %ape are conc]n$1 b z /ructures & z liv+ organisms !y are z wide ap>t z possible4 ,( c\rse by obs]vation & /udy ! image ( a plant :i* we be> 9 our m9ds may be made to resemble a plant z f\nd 9 ! fields m & m4 ,b ! agreem5t ) nature lies 9 ! multitude ( po9ts sup]add$ on to ! notion ( gre5ness :i* we h at fir/--!re is no natural />t+-po9t :]e ! m9d meets nature1 & :5ce !y c travel h& 9 h&4 ,x almo/ seems z if1 by sympa?y & feel+1 a human be+ 0 easi] to "k ?an ! simple/ object4 ,to "k any object1 h[ev] simple1 by ! reason & obs]vation requires an endless process ( ?\ght & look+1 build+ up ! fir/ vague impression into some?+ l 9 m & m respects4 ,:ile1 on ! o!r h&1 9 deal+ ) human be+s !re is an inw>d sympa?y & capacity = kn[+ :i* is indep5d5t (1 ?\gh call$ into play by1 ! obs]vation ( ! actions & outw>d appe>ance ( ! human be+4 ,b = ! purpose ( kn[+ we mu/ leave out ^! human relation%ips4 ,!y are an affair ( in/9ct & inh]it$ unconsci\s exp]i5ce4 ,! m9d may "s day rise to ! level ( ^! inh]it$ appreh5sions1 & be able to expla9 !m2 b at pres5t x is f> m ?an ov]task$ to give an acc\nt ( ! simple/ portions ( matt]1 & is q inadequate to give an acc\nt ( ! nature ( a human be+4 ,ask+1 !n1 :at !re 0 :i* ,i c\ld "k1 ,i f\nd no po9t ( beg9n+4 ,!re 7 pl5ty ( ways ( accumulat+ obs]vations1 b none 9 :i* "o c\ld g h& 9 h& ) nature4 ,a *ild is provid$ 9 ! e>ly "p ( its life ) a provision ( food adapt$ = x4 ,b x seem$ t our m9ds are left )\t a natural subsi/5ce1 = on ! "o h& !re are arid ma!matics1 & on ! o!r !re is obs]vation1 & 9 obs]vation !re is1 out ( ! great mass ( con/ruct$ m5tal images1 b little :i* ! m9d c assimilate4 ,to ! work] at sci5ce ( c\rse ?is crude & omnivor\s obs]vation is ev]y?+2 b if we ask = some?+ :i* we c "k1 x is l a va/ mass ( indige/ible mat]ial ) e "h & !re a fibre or ?read :i* we c assimilate4 ,9 ?is p]plexity ,i 0 r$uc$ to ! la/ condition ( m5tal despair2 & 9 default ( f9d+ any?+ :i* ,i c\ld und]/& 9 nature1 ,i 0 suffici5tly humbl$ to le>n any?+ :i* seem$ to af=d a capacity ( be+ kn[n4 ,& ! objects :i* came be=e me = ?is endeav\r 7 ! simple ones :i* w be pl5tifully us$ 9 ! practical "p ( ?is book4 ,= ,i f\nd t ! only ass]tion ,i c\ld make ab\t ext]nal objects1 )\t br++ 9 unkn[n & un9telligible relations1 0 ?is3 ,i c\ld say h[ ?+s 7 arrang$4 ,if a /one lay betwe5 two o!rs1 t 0 a def9ite & intelligible fact1 & seem$ prim>y4 ,z a /one itself1 x 0 an unkn[n some:at :i* "o c\ld get m & m in=mation ab\t ! m "o /udi$ ! v>i\s sci5ces4 ,b grant+ t !re 7 "s ?+s !re :i* we call /ones1 ! way !y 7 arrang$ 0 a simple & obvi\s fact :i* c\ld be easily express$ & easily rememb]$4 ,& s 9 despair ( be+ able to obta9 any o!r k9d ( m5tal possession 9 ! way ( k1 ,i comm5c$ to le>n arrangem5ts1 & ,i took z ! objects to be arrang$ c]ta9 artificial objects ( a simple %ape4 ,i built up a block ( cubes1 & giv+ ea* a "n ,i le>nt a mass ( !m4 ,n[ ,i d n recomm5d ?is z a ?+ to be done4 ,all ,i c say is t g5u9ely !n & n[ x seem$ & seems to be ! only k9d ( m5tal possession :i* "o c call k4 ,x is p]fectly def9ite & c]ta94 ,i c\ld tell :]e ea* cube came & h[ x 0 relat$ to ea* ( ! o!rs4 ,z to ! cube itself1 ,i 0 pr(\ndly ignorant ( t2 b assum+ t z a necess>y />t+-po9t1 tak+ t z grant$1 ,i _h a def9ite mass ( k4 ,b ,i d n wi% to say t ?is is bett] ?an any k9d ( k :i* o!r p may f9d come home to !m4 ,all ,i want to d is to take ?is humble beg9n+ ( k & %[ h[ inevitably1 by devotion to x1 x leads to m>vell\s & f>-di/ant tru?s1 & h[1 by a /range pa?1 x leads directly into ! pres5ce ( "s ( ! highe/ conceptions :i* great m9ds h giv5 u4 ,i d n ?9k x ought to be any objection to an inquiry1 t x beg9s ) obvi\s & common details4 ,9 fact ,i d n ?9k t x is possible to get any?+ simpl]1 ) less ( hypo!sis ab\t x1 & m obvi\sly a simple tak+ 9 ( facts ?an ! /udy ( ! arrangem5t ( a block ( cubes4 ,_m philosoph]s h assum$ a />t+ po9t = _! ?\ght4 ,i want ! read] to accept a v humble "o & see :at comes ( x4 ,if ?is leads u to any?+1 no d\bt great] results w come f m ambiti\s beg9n+s4 ,& n[ ,i feel t ,i h c&idly expos$ myself to ! criticism ( ! read]4 ,if he w h ! pati5ce to g on1 we w beg9 & build up on our f\ndations4 ,,*apt] ,,ii4 ,,appreh5sion ,,( ,,nature4 ,,intellig5ce4 ,,/udy ,,( ,,arrangem5t ,,or ,,%ape4 ,nature is t :i* is ar\nd u4 ,b x is by no means easy to get to nature4 ,! savage liv+ we may say 9 ! bosom ( nature1 is c]ta9ly unappreh5sive ( x1 9 fact x has ne$$ ! greatness ( a ,wordswor? & ( g5]ations ( poets & pa9t]s to op5 our eyes ev5 9 a slight measure to ! wond] ( nature4 ,?us x is cle> t x is n by m]e passivity t we c compreh5d nature2 x is ! goal ( an activity1 n a free gift4 ,& !re are _m ways ( appreh5d+ nature4 ,!re are ! s\nds & sights ( nature :i* delight ! s5ses1 & ! involv$ h>monies & ! secret aff9ities :i* poetry makes u feel2 !n1 moreov]1 !re is ! def9ite k ( natural facts 9 :i* ! memory & reason are employ$4 ,?us we may divide our means ( com+ into contact ) nature into ?ree ma9 *annels3 ! s5ses1 ! imag9ation1 & ! m9d4 ,! imag9ation is p]haps ! highe/ faculty1 b we leave x out ( consid]ation "h1 & ask3 ,h[ c we br+ our m9ds into contact ) nature8 ,n[ :5 we see two p ( div]se *>act]s we sometimes say t !y _c und]/& "o ano!r--!re is no?+ 9 ! "o by :i* he c und]/& ! o!r--he is %ut out by a limitation ( 8 own faculties4 ,& ?us our p[] ( und]/&+ nature dep5ds on our own possession2 x is 9 virtue ( "s m5tal activity ( our own t we c appreh5d t outside activity :i* we call nature4 ,& ?us ! tra9+ to enable u to approa* nature ) our m9ds w be "s active process on our own "p4 ,9 ! c\rse ( my exp]i5ce z a tea*] ,i h (t5 be5 /ruck by ! want ( ! p[] ( reason display$ by pupils2 !y are n able to put two & two toge!r1 z ! say+ goes1 & ,i h be5 at "s pa9s to inve/igate :]e9 ?is curi\s defici5cy lies1 & h[ x c be suppli$4 ,& ,i h f\nd t !re is 9 ! curriculum no direct cure = x--! discipl9e :i* supplies x is n "o :i* comes into s*ool me?ods1 x is a some?+ :i* mo/ *ildr5 obta9 9 ! natural & unsup]vis$ education ( _! fir/ contact ) ! _w1 & lies be=e any recognis$ mode ( di/9ction4 ,!y c only und]/& 9 virtue ( an activity ( _! own1 & !y h n _h suffici5t ex]cise 9 ?is activity4 ,9 ! pres5t /ate ( education x is impossible to div]ge f ! ord9>y r\t9e4 ,b x is always possible to exp]im5t on *ildr5 :o are out ( ! common l9e ( education4 ,& ,i believe ,i am amply ju/ifi$ by ! result ( my exp]im5ts4 ,i h se5 t ! same activity :i* ,i h f\nd makes t habit ( m9d :i* we call intellig5ce 9 a *ild1 is ! s\rce ( our common & ev]yday rational intellectual "w1 & t j z ! faculties ( a *ild c be call$ =? by x1 s also ! p[]s ( a man are be/ prep>$ by ! same means1 b on an ampl] scale4 ,a m detail$ developm5t ( ! practical "w ( ,"p ,,ii41 w\ld be ! be/ tra9+ = ! m9d ( a *ild4 ,an ext5sion ( ! "w ( t ,"p w\ld be ! tra9+ :i*1 h& 9 h& ) obs]vation & recapitulation1 w\ld be/ develop a man's ?\ght p[]4 ,9 ord] to tell :at ! activity is by ! prosecution ( :i* we c obta9 m5tal contact ) nature we %\ld obs]ve :at x is :i* we say we 8und]/&0 9 any ph5om5on ( nature :i* has become cle> to u4 ,:5 we look at a bright object x seems v diff]5t f a dull "o4 ,a piece ( bright /eel h>dly looks l ! same sub/ance z a piece ( dull /eel4 ,b ! diff]5ce ( appe>ance 9 ! two is easily acc\nt$ = by ! diff]5t nature ( ! surface 9 ! two cases2 9 ! "o all ! irregul>ities are done away )1 & ! rays ( light :i* fall on x are s5t (f aga9 )\t be+ disp]s$ & brok5 up4 ,9 ! case ( ! dull iron ! rays ( light are brok5 up & divid$1 s t !y are n transmitt$ ) int5sity 9 any "o direction1 b flung (f 9 all sorts ( directions4 ,"h ! diff]5ce betwe5 ! bright object & ! dull object lies 9 ! arrangem5t ( ! p>ticles on its surface & _! influ5ce on ! rays ( light4 ,aga91 ) light itself ! diff]5ces ( col\r are expla9$ z be+ ! effect on u ( rays ( diff]5t rates ( vibration4 ,n[ a vibration is ess5tially ?is1 a s]ies ( arrangem5ts ( matt] :i* foll[ ea* o!r 9 a clos$ ord]1 s t :5 ! set has be5 run ?r\gh1 ! fir/ arrangem5t foll[s aga94 ,! :ole !ory ( light is an acc\nt ( arrangem5ts ( ! p>ticles 9 ! transmitt+ m$ium1 only ! arrangem5ts alt]--are n p]man5t 9 any "o *>act]i/ic1 b g ?r\gh a complete cycle ( v>ieties4 ,aga91 :5 ! movem5ts ( ! heav5ly bodies are d$uc$ f ! !ory ( univ]sal gravitation1 :at we prim>ily d is to take acc\nt ( arrangem5t2 = ! law ( gravity connects ! movem5ts :i* ! attract$ bodies t5d to make ) _! di/ances1 t is1 x %[s h[ _! movem5ts dep5d on _! arrangem5t4 ,& if gravity z a =ce is to be expla9$ itself1 ! suppositions :i* h be5 put =w>d resolve x into ! effect ( ! movem5ts ( small bodies2 t is to say1 gravity1 if expla9$ at all1 is expla9$ z ! result ( ! arrangem5t & alt]+ arrangem5ts ( small p>ticles4 ,aga91 to take ! idea :i* proce$+ f ,goe! ca/s su* an influ5ce on botanical obs]vation4 ,goe! 7& also ,wolf7 laid d[n t ! p>ts ( a fl[] 7 modifi$ leaves--& trac$ ! /ages & int]m$iate /ates betwe5 ! ord9>y gre5 leaf & ! mo/ gorge\s petal or /am5 or c>pel1 s unlike a leaf 9 =m & function4 ,n[ ! ess5tial value 9 ?is conception is1 t x enables u to look1 ^u ^! diff]5t organs ( a plant z modifications ( "o & ! same organ--x enables u to ?9k ab\t ! diff]5t v>ieties ( ! fl[] head z modifications ( "o s+le plant =m4 ,we c trace correspond5ces betwe5 !m1 & are l$ to possible explanations ( _! gr[?4 ,& all ?is is done by gett+ rid ( pi/il & /am5 z sep>ate entities1 & look+ on !m z leaves1 & _! p>ts due to diff]5t arrangem5t ( ! leaf /ructure4 ,we h r$uc$ ^! div]se objects to a common elem5t1 we h f\nd ! unit by ^: arrangem5ts ! :ole is produc$4 ,& 9 ?is dep>tm5t ( ?\ght1 z also to take ano!r in/ance1 9 *emi/ry1 to und]/& is practically ?is3 we f9d units 7leaves or atoms7 comb9ations ( :i* acc\nt = ! results :i* we see4 ,?us we see t t :i* ! m9d ess5tially appreh5ds is arrangem5t4 ,& ?is holds ov] ! :ole range ( m5tal "w1 f ! simple/ obs]vation to ! mo/ complex !ory4 ,:5 ! eye takes 9 ! =m ( an ext]nal object !re is some?+ m ?an a s5se impression1 some?+ m ?an a s5sation ( gre5ness & light & d>k4 ,! m9d works z well z ! s5se1 & ^! s5se impressions are def9itely gr\p$ 9 :at we call ! %ape ( ! object4 ,! ess5tial act ( p]ceiv+ lies 9 ! appreh5sion ( a %ape1 & a %ape is an arrangem5t ( p>ts4 ,x does n matt] :at ^! p>ts are2 if we take mean+less dots ( col\r & arrange !m we obta9 a %ape :i* repres5ts ! appe>ance ( a /one or a leaf to a c]ta9 degree4 ,if we want to make our repres5tation /ill m l1 we mu/ treat ea* ( ! dots z 9 !mselves arrangem5ts1 we mu/ compose ea* ( !m by _m /rokes & dots ( ! bru%4 ,b ev5 9 ?is case we h n got any?+ else besides arrangem5t4 ,! ultimate elem5t1 ! small items ( light & %ade or ( col\r1 are 9 !mselves mean+less2 x is 9 _! arrangem5t t ! lik5ess ( ! repres5tation consi/s4 ,?us1 f a draw+ to our notion ( ! planet>y sy/em1 all our contact ) nature lies 9 ?is1 9 an appreciation ( arrangem5t4 ,h5ce to prep>e ourselves = ! und]/&+ ( nature1 we mu/ 8arrange40 ,9 virtue ( our activity 9 mak+ arrangem5ts we prep>e ourselves to d :at is call$ und]/& nature4 ,or we may say1 t :i* we call und]/&+ nature is to disc]n some?+ simil> 9 nature to t :i* we d :5 we arrange elem5ts into comp\nd$ gr\ps4 ,n[ if we /udy arrangem5t 9 ! active way1 we mu/ h some?+ to arrange2 & ! ?+s we "w ) may be ei!r all alike1 or ea* ( !m v>y+ f e o!r4 ,if ! elem5ts are n alike !n we are n /udy+ pure arrangem5t2 b our k is affect$ by ! comp\nd nature ( t ) :i* we deal4 ,if ! elem5ts are all alike1 we h :at we call units4 ,h5ce ! discipl9e prep>atory = ! und]/&+ ( nature is ! active arrangem5t ( l units4 ,& ?is is v mu* ! case ) all educational processes2 only ! ?+s *os5 to arrange are 9 g5]al words1 :i* are s complicat$ & c>ry su* a tra9 ( association t1 unless ! m9d has already acquir$ a k ( arrangem5t1 x is puzzl$ & hamp]$1 & nev] gets a cle> appreh5sion ( :at its "w is4 ,n[ :at %all we *oose = our units8 ,any unit w\ld d2 b x ought to be a real ?+--x ought to be some?+ :i* c be t\*$ & se51 n some?+ :i* no "o has ev] t\*$ or se51 & :i* is ev5 incapable ( def9ition1 l a 8numb]40 ,i w\ld divide /udies into two classes3 ^? :i* create ! faculty ( arrangem5t1 & ^? :i* use x & ex]cise x4 ,ma!matics ex]cises x1 b ,i d n ?9k x creates x2 & un=tunately1 9 ma!matics z x is n[ (t5 taught1 ! pupil is laun*$ 1 into a va/ sy/em ( symbols--! :ole use & mean+ ( symbols 7namely1 z means to acquire a cle> grasp ( facts7 is lo/ to him4 ,( ! possible units :i* w s]ve1 ,i take ! cube2 & ,i h f\nd t :5ev] ,i took any o!r unit ,i got wrong1 puzzl$ & lo/ my way4 ,) ! cube "o does n get along v fa/1 b ev]y?+ is p]fectly obvi\s & simple1 & builds up into a :ole ( :i* e "p is evid5t4 ,& ,i mu/ ask ! read] to absolutely erase f 8 m9d all desire or wi% to be able to pr$ict or ass]t any?+ ab\t nature1 & he mu/ please look ) horror on any m5tal process by :i* he gets at a tru? 9 an +5i\s b obscure & inexplicable way4 ,let him take no?+ :i* is n p]fectly cle>1 pat5t & evid5t1 demon/rable to 8 s5ses1 a simple repetition ( obvi\s fact4 ,our "w w !n be ?is3 a /udy1 by means ( cubes1 ( ! facts ( arrangem5t4 ,& ! process ( le>n+ w be an active "o ( actually putt+ up ! cubes4 ,9 ?is way we d = ! m9d :at ,wordswor? does = ! imag9ation--we br+ x into contact ) nature4 ,,*apt] ,,iii4 ,,! ,,elem5ts ,,( ,,k4 ,!re are two elem5ts :i* ent] into our k ) respect to any ph5om5on4 ,if1 = in/ance1 we take ! sun1 & ask ourselves :at we obs]ve1 we notice t x is a bright1 mov+ body2 & ( ^! two qualities1 ! brightness & ! movem5t1 ea* seems equally pr$icable ( ! sun4 ,x does move1 & x is bright4 ,n[ fur!r /udy discloses to u t !re is a diff]5ce betwe5 ^! two affirmations4 ,! motion ( ! sun 9 its diurnal c\rse r\nd ! e>? is only app>5t2 b x is really a bright1 hot body4 ,n[ ( ^! two ass]tions :i* ! m9d naturally makes ab\t ! sun1 "o--t x is mov+--dep5ds on ! relation ( ! behold] to ! sun1 ! o!r is true ab\t ! sun itself4 ,! obs]v$ motion dep5ds on a fact affect+ oneself & hav+ no?+ to d ) ! sun1 :ile ! brightness is really a quality ( ! sun itself4 ,n[ we w call ^? qualities or appe>ances :i* we notice 9 a body :i* are due to ! p>ticul> conditions "u :i* oneself is plac$ 9 obs]v+ x1 ! self elem5ts2 ^? facts ab\t x :i* are indep5d5t ( ! obs]v]'s p>ticul> relation%ip we w call ! residual elem5t4 ,?us ! sun's motion is a self elem5t 9 our ?\ght ( ! sun1 its brightness is a residual elem5t4 ,x is n1 ( c\rse1 possible to draw a l9e di/9ctly betwe5 ! self elem5ts & ! residual elem5ts4 ,= in/ance1 "s p h d5i$ t brightness is a quality ( ?+s1 b t x dep5ds on ! capacity ( ! be+ = receiv+ s5sations2 & = brightness !y w\ld sub/itute ! ass]tion t ! sun is giv+ =? a great deal ( en]gy 9 ! =m ( heat & light4 ,b !re is no object 9 pursu+ ! discussion fur!r4 ,! ma9 di/9ction is suffici5tly obvi\s4 ,& x is important to sep>ate ! self elem5ts involv$ 9 our k z f> z possible1 s t ! residual elem5ts may be kept = our clos] att5tion4 ,by gett+ rid ( ! self elem5ts we put ourselves 9 a position 9 :i* we c prop\nd s5sible que/ions4 ,by gett+ rid ( ! notion ( its circul> motion r\nd ! e>? we prep>e our way to /udy ! sun z x really is4 ,we get ! subject cle> ( complications & extrane\s consid]ations4 ,x w\ld h>dly be wor? :ile to dwell on ?is consid]ation 7 x n ( importance 9 our /udy ( arrangem5t4 ,b ! fact is t directly a subject has be5 cle>$ ( ! self elem5ts1 x seems s absurd to h _h !m introduc$ at all t ! great difficulty !re 0 9 gett+ rid ( !m is =gott54 ,) reg>d to ! k we h at ! pres5t day ab\t sci5tific matt]s1 !re d n seem to be any self elem5ts pres5t4 ,b ! wor/ ab\t a self elem5t is1 t its pres5ce is nev] dream$ ( till x is got rid (2 to "k t x is !re is to h done away ) x4 ,& ?us our body ( k is l a fluid :i* keeps cle>1 n because !re are no sub/ances 9 solution1 b because directly !y become evid5t !y fall d[n z precipitates4 ,n[ "o ( our s]i\s pieces ( "w w be to get rid ( ! self elem5ts 9 ! k ( arrangem5t4 ,& ! k9d ( k :i* we %all try to obta9 w be some:at diff]5t f ! k9d ( k :i* we h ab\t ev5ts or natural ph5om5a4 ,9 ! l>ge subjects :i* g5]ally occupy ! m9d ! ?+s ?\ght ( are s complicat$ t e detail _c possibly be consid]$4 ,! pr9ciples ( ! :ole are realiz$1 & !n at any requir$ "t ! pr9ciples c be work$ out4 ,?us1 ) reg>d to a k ( ! planet>y sy/em1 x is said to be kn[n if ! law ( movem5t ( ea* ( ! planets is recogniz$1 & _! positions at any "o "t committ$ to memory4 ,x is n our habit to rememb] _! relative positions ) reg>d to "o ano!r at _m int]vals1 s z to h an exhau/ive catalogue ( !m 9 our m9ds4 ,b ) reg>d to ! elem5ts ( k ) :i* we %all "w1 ! subject is s simple t we may ju/ly dem& ( ourselves t we w "k e detail4 ,& ! k we %all acquire w be mu* m "o ( ! s5se & feel+ ?an ( ! reason4 ,we d n want to h a rule 9 our m9ds by :i* we c recall ! positions ( ! diff]5t cubes1 b we want to h an imm$iate appreh5sion ( !m4 ,x 0 ,kant :o fir/ po9t$ out h[ mu* ( ?\ght !re 0 embodi$ 9 ! s5se impressions2 & x is ?is embodi$ ?\ght :i* we wi% to =m4 ,,*apt] ,,iv4 ,,!ory ,,& ,,practice4 ,bo? 9 sci5ce & 9 morals !re is an important di/9ction to be drawn betwe5 !ory & practice4 ,a k ( *emi/ry does n consi/ 9 ! intellectual appreciation ( diff]5t !ories & pr9ciples1 b 9 be+ able to act 9 accordance ) ! facts ( *emical comb9ation1 s t by means ( ! appliances ( *emi/ry practical results are produc$4 ,& s 9 morals--! !oretic acqua9tance ) ! pr9ciples ( human action may consi/ ) a m>k$ degree ( ignorance ( h[ to act among/ o!r human be+s4 ,n[ ! use ( ! ^w 8le>n0 has be5 mu* re/rict$ to m]ely !oretic /udies4 ,x requires to be enl>g$ to ! sci5tific mean+4 ,& to "k1 %\ld include practice 9 actual manipulation4 ,let u take an in/ance4 ,we all "k :at ju/ice is1 & any *ild c be taught to tell ! diff]5ce betwe5 act+ ju/ly & act+ unju/ly4 ,b x is a diff]5t ?+ to tea* !m to act ) ju/ice4 ,some?+ is done :i* affects !m unpleasantly4 ,!y feel an impulse to retaliate4 ,9 ord] to see :at ju/ice dem&s !y h to put _! p]sonal feel+ on "o side4 ,!y h to get rid ( ^? conditions "u :i* !y appreh5d$ ! effects ( ! action at fir/1 & !y h to look on x f ano!r po9t ( view4 ,!n !y h to act 9 accordance ) ?is view4 ,n[ !re are two /eps--"o an intellectual "o ( und]/&+1 "o a practical "o ( c>ry+ out ! view4 ,nei!r is a moral /ep4 ,"o dem&s intellig5ce1 ! o!r ! =mation ( a habit1 & ?is habit c be inculcat$ by precept1 by rew>d & puni%m5t1 by v>i\s means4 ,b z human nature is con/itut$1 if ! habit ( ju/ice is inculcat$ x t\*es a "p ( ! be+4 ,!re is an emotional response4 ,we "k b little ( a human be+1 b we c safely say t !re are dep?s 9 x1 beyond ! feel+s ( mom5t>y res5tm5t & ! /imulus ( pleasurable or pa9ful s5sation1 to :i* ju/ice is natural4 ,h[ little adequate is our physical k ( a human be+ z a bodily frame to expla9 ! fact ( human life4 ,n[ & aga9 we see "o ( ^! isolat$ be+s b\nd up 9 ano!r1 z if !re 0 an undiscov]$ physical bond betwe5 !m4 ,& 9 all !re is ?is s5se ( ju/ice--a k9d ( indwell+ v]dict ( ! univ]sal m9d1 if we may use su* an expression1 9 virtue ( :i* a man feels n z a s+le individual b z all m54 ,) respect to ju/ice1 x is n only necess>y to take ! view ( "o o!r p]son ?an oneself1 b t ( _m4 ,!re may be ju/ice :i* is v good ju/ice f ! po9t ( view ( a p>ty1 b v bad ju/ice f ! po9t ( view ( a nation4 ,& if we suppose an ag5cy outside ! human race1 gift$ ) intellig5ce1 & affect+ ! race1 9 ! way = in/ance ( caus+ /orms or di/urbances ( ! gr\nd1 9 ord] to judge x ) ju/ice we %\ld h to take a /&po9t outside ! race ( m5 altoge!r4 ,we c\ld n say t ?is ag5cy 0 bad4 ,we %\ld h to judge x ) ref]5ce to its effect on o!r s5ti5t be+s4 ,!re are "s words :i* are (t5 us$ 9 contra/ ) ea* o!r--egoism & altruism2 & ea* seems to me unmean+ except z t]ms 9 a contra/4 ,let u take an in/ance4 ,a boy has a bag ( cakes1 & is go+ to enjoy !m by himself4 ,8 p>5t /ops him1 & makes him set up "s /umps & beg9 to le>n to play cricket ) ano!r boy4 ,! enjoym5t ( ! cakes is lo/--he has giv5 t up2 b aft] a little :ile he has a pleasure :i* is great] ?an t ( cakes 9 solitude4 ,he ent]s into ! life ( ! game4 ,he has giv5 up1 or be5 =c$ to give up1 ! pleasure he knew1 & he has f\nd a great] "o4 ,:at he ?\ght ab\t himself be=e 0 t he lik$ cakes1 n[ :at he ?9ks ab\t himself is t he likes cricket4 ,:i* ( ^! is ! true ?\ght ab\t himself8 ,nei!r1 probably1 b at any rate x is m ne> ! tru? to say t he likes ! cricket4 ,if n[ we use ! ^w self to mean t :i* a p]son kn[s ( himself1 & x is difficult to see :at o!r mean+ x c h1 8 self z he knew x at fir/ 0 ?w>t$1 0 giv5 up1 & ?r\gh t he discov]$ 8 true self4 ,& aga9 ) ! cricket2 he w make ! sacrifice ( giv+ t up1 volunt>ily or involunt>ily1 & w f9d a tru] self /ill4 ,9 g5]al !re is n mu* difficulty 9 mak+ a boy f9d out t he likes cricket2 & x is q possible = him to eat 8 cakes fir/ & le>n to play cricket aft]w>ds--! cricket w n come to him z a ?w>t+ 9 any s5se ( :at he likes bett]4 ,b ?is ease 9 ent]+ 9 to ! pursuit only %[s t ! boy's nature is already develop$ to ! level ( enjoy+ ! game4 ,! di/9ct moral advance w\ld come 9 su* a case :5 some?+ :i* at fir/ 0 h>d to him to d 0 pres5t$ to him--& ! h>dness1 ! unpleasantness is ( a d\ble k9d1 ! giv+ up ( a pursuit or indulg5ce to :i* he is accu/om$1 & ! ex]tion ( =m+ ! habits dem&$ by ! new pursuit4 ,n[ x is unimportant :e!r ! r5unciation is =c$ or will+ly tak54 ,b z a g5]al rule x may be laid d[n1 t by giv+ up 8 own desires z he feels !m at ! mom5t1 to ! ne$s & advantage ( ^? ar\nd him1 or to ! objects :i* he f9ds be=e him dem&+ accompli%m5t1 a human be+ passes to ! discov]y ( 8 true self on & on4 ,! process is limit$ by ! responsibilities :i* a man f9ds come ^u him4 ,! me?od ( moral advance is to acquire a practical k2 he mu/ fir/ see :at ! advantage ( "s "o o!r ?an himself w\ld be1 & !n he mu/ act 9 accordance ) t view ( ?+s4 ,!n hav+ act$ & =m$ a habit1 he discov]s a response 9 himself4 ,he f9ds t he really c>es1 & t 8 =m] limit$ life 0 n really himself4 ,8 body & ! ne$s ( 8 body1 s f> z he c obs]ve !m1 ext]nally are ! same z be=e2 b he has obta9$ an inn] & un9tellectual1 b none ! less real1 appreh5sion ( :at he is4 ,?us altruism1 or ! sacrifice ( egoism to o!rs1 is foll[$ by a tru] egoism1 or ass]tion ( self1 & ?is process fla%$ across by ! transc5d5t lights ( religion1 :]e91 z 9 ! s5se ( ju/ice & duty1 untold dep?s 9 ! nature ( man are reveal$ entirely unexpress$ by ! intellectual appreh5sion :i* we h ( him z an animal frame ( a v high degree ( developm5t1 is ! normal "o by :i* f *ildhood a human be+ develops into ! full responsibilities ( a man4 ,n[ bo? 9 sci5ce & 9 conduct !re are self elem5ts4 ,9 sci5ce1 gett+ rid ( ! self elem5ts means a tru] appreh5sion ( ! facts ab\t "o2 9 conduct1 gett+ rid ( ! self elem5ts means obta9+ a tru] k ( :at we are--9 ! way ( feel+ m /rongly & deeply & be+ b\nd & l9k$ 9 a l>g] scale4 ,?us )\t pret5d+ to any sci5tific accuracy 9 ! use ( t]ms1 we c assign a c]ta9 am\nt ( mean+ to ! expression--gett+ rid ( self elem5ts4 ,& all t we c d is to take ! r\gh idea ( ?is process1 & !n tak+ our special subject matt]1 apply x4 ,9 affairs ( life exp]im5ts lead to disa/]4 ,b happily sci5ce is provid$ :]e9 ! desire to put !ories into practice c be safely satisfi$--& good results sometimes foll[4 ,7 x n = ?is ! human race might be=e n[ h be5 utopiad f (f ! face ( ! e>?4 ,9 exp]im5t1 manipulation is ev]y?+2 we mu/ be c]ta9 ( all our conditions1 o!rwise we fail assur$ly & h n ev5 ! satisfaction ( kn[+ t our failure is due to ! wrongness ( our conjectures4 ,& = our purposes we use a subject matt] s simple t ! manipulation is easy4 ,,*apt] ,v4 ,,k3 ,,self-,,elem5ts4 ,i mu/ n[ g ) some:at ( detail into ! special subject 9 :i* ^! g5]al tru?s w be exhibit$4 ,ev]y?+ ,i h to say w\ld be conceiv$ mu* m cle>ly by a v little practical manipulation4 ,b "h ,i want to put ! subject 9 z g5]al a light z possible1 s t !re may be no h9drance to ! judgm5t ( ! read]4 ,& :5 ,i use ! ^w 8"k10 ,i assume some?+ else ?an ! possession ( a rule1 by :i* x c be said h[ facts are4 ,by kn[+ ,i mean t ! facts ( a subject all lie 9 ! m9d ready to come out vividly into consci\sness :5 ! att5tion is direct$ on !m4 ,mi*ael ,angelo knew ! human frame1 he c\ld tell e little fact ab\t x2 if he *ose to call up ! image1 he w\ld see m5tally h[ ea* muscle & fold ( ! sk9 lay ) reg>d to ! surr\nd+ p>ts4 ,we want to obta9 a k z good z ,mi*ael ,angelo's4 ,!re is a great diff]5ce betwe5 ,mi*ael ,angelo & u2 b let t diff]5ce be express$1 n 9 our way ( kn[+1 b 9 ! diff]5ce betwe5 ! ?+s he knew & ! ?+s we "k4 ,we take a v simple /ructure & "k x z absolutely z he knew ! complicat$ /ructure ( ! human body4 ,& let u take a block ( cubes2 any numb] w d1 b = conv5i5ce sake let u take a set ( tw5ty-sev5 cubes put toge!r s z to =m a l>ge cube ( tw5ty-sev5 p>ts4 ,& let ea* ( ^! cubes be m>k$ s z to be recogniz$1 & let ea* h a "n s t x c be ref]r$ to4 ,& let u suppose t we h le>nt ?is block ( cubes s t ea* "o is kn[n--t is to say1 its position 9 ! block is kn[n & its relation to ! o!r blocks4 ,n[ hav+ obta9$ ?is k ( ! block z x /&s 9 front ( u1 let u ask ourselves if !re is any self elem5t pres5t 9 our k ( x4 ,& !re is obvi\sly ?is self elem5t pres5t4 ,we h le>nt ! cubes z !y /& 9 accordance ) our own conv5i5ce 9 putt+ !m up4 ,we put ! l[e/ ones fir/1 & ! o!rs on ! top ( !m1 & we di/9ctly conceive ! l[] ones z support+ ! upp] ones4 ,n[ ?is fact ( support has no?+ to d ) ! block ( cubes itself1 x dep5ds on ! conditions "u :i* we come to appreh5d ! block ( cubes1 x dep5ds on our position on ! surface ( ! e>?1 :]eby gravity is an all important factor 9 our exp]i5ce4 ,9 fact our sight has got s accu/om$ to take gravity into consid]ation 9 its view ( ?+s1 t :5 we look at a l&scape or object ) our head upside d[n we d n see x inv]t$1 b we sup]9duce on ! direct s5se impressions our k ( ! action ( gravity1 & obta9 a view diff]+ v little f :at we see :5 9 an upright position4 ,x w be f\nd t e fact ab\t ! cubes has involv$ 9 x a ref]5ce to up & d[n4 ,x is by be+ above or bel[ t we *iefly rememb] :]e ! cubes are4 ,b above & bel[ is a relation :i* dep5ds simply on gravity4 ,if x 7 n = gravity above & bel[ w\ld be int]*angeable t]ms1 in/ead ( express+ a diff]5ce ( m>k$ importance to u "u our conditions ( exi/5ce4 ,n[ we put 8be+ above0 or 8be+ bel[0 into ! cubes !mselves & feel x a quality 9 !m--x def9es _! position4 ,b ?is above or bel[ really comes f ! conditions 9 :i* we are4 ,x is a self elem5t1 & z su*1 to obta9 a true k ( ! cubes we mu/ get rid ( x4 ,& n[1 = ! sake ( a process :i* w be expla9$ aft]w>ds1 let u suppose t we _c move ! block ( cubes :i* we h put up4 ,let u keep x fix$4 ,9 ord] to le>n h[ x is indep5d5t ( gravity ! be/ way w\ld be to g to a place :]e gravity has virtually ceas$ to act2 at ! c5tre ( ! e>?1 = in/ance1 or 9 a freely fall+ %ell4 ,b ?is is impossible1 s we mu/ *oose ano!r way4 ,let u1 !n1 s9ce we _c get rid ( gravity1 see :at we h done already4 ,we h le>nt ! cubes1 & h[ev] !y are le>nt1 x w be f\nd t !re is a c]ta9 set ( !m r\nd :i* ! o!rs are m5tally gr\p$1 z be+ on ! "r or left1 above or bel[4 ,n[ to get our k z p]fect z we c be=e gett+ rid ( ! self elem5t up & d[n1 we h to take z c5tral cubes 9 our m9d diff]5t sets aga9 & aga91 until !re are none :i* are prim>y to u4 ,!n !re rema9s only ! di/9ction ( "s be+ above o!rs4 ,n[ ?is c only be made to s9k out ( ! prim>y place 9 our ?\ghts by rev]s+ ! relation4 ,if we turn$ ! block upside d[n1 & le>nt x 9 ?is new position1 !n we %\ld le>n ! position ( ! cubes ) reg>d to ea* o!r ) t elem5t 9 !m1 :i* comes f ! action ( gravity1 rev]s$4 ,& ! true nature ( ! arrangem5t to :i* we add$ some?+ 9 virtue ( our s5sation ( up & d[n1 w\ld become pur] & m isolat$ 9 our m9ds4 ,we h1 h[ev]1 suppos$ t ! cubes are fix$4 ,!n1 9 ord] to le>n !m1 we mu/ put up ano!r block %[+ :at !y w\ld be l 9 ! suppos$ new position4 ,we !n take a set ( cubes1 models ( ! orig9al cubes1 & by consid]ation we c put !m 9 su* positions z to be an exact model ( :at ! block ( cubes w\ld be if turn$ upside d[n4 ,& "h is ! :ole po9t on :i* ! process dep5ds4 ,we c tell :]e ea* cube w\ld come1 b we d n "k ! block 9 ?is new position4 ,i draw a di/9ction betwe5 ! two acts1 8to tell :]e x w\ld be10 & to 8"k40 ,tell+ :]e x w\ld be is ! prep>ation = kn[+4 ,! p[] ( assign+ ! positions may be call$ ! !ory ( ! block4 ,! actual k is got by c>ry+ out ! !ory practically1 by putt+ up ! blocks & becom+ able to realize )\t ef=t :]e ea* "o is4 ,x is n en\gh to put up ! model blocks 9 ! rev]se position4 ,x is f\nd t ?is up & d[n is a v ob/9ate elem5t inde$1 & a good deal ( "w is requisite to get rid ( x completely4 ,b :5 x is got rid ( 9 "o set ( cubes1 ! faculty is =m$ ( appreciat+ %ape indep5d5tly ( ! p>ticul> p>ts :i* are above or bel[ on fir/ exam9ation4 ,we discov] 9 our own m9ds ! faculty ( appreciat+ ! facts ( position indep5d5t ( gravity & its influ5ce on u4 ,i h f\nd a v great diff]5ce 9 diff]5t m9ds 9 ?is respect4 ,to "s x is easy1 to "s x is h>d4 ,& to use our old in/ance1 ! discov]y ( ?is capacity is l ! discov]y ( a love ( ju/ice 9 ! be+ :o has =c$ himself to act ju/ly4 ,x is a capacity = be+ able to take a view indep5d5t ( ! conditions "u :i* he is plac$1 & to feel 9 accordance ) t view4 ,!re is1 s f> z ,i "k1 no means ( arriv+ imm$iately at ?is imp>tial appreciation ( %ape4 ,x c only be done by1 z x 71 ext5d+ our own body s z to include c]ta9 cubes1 & appreciat+ !n ! relation ( ! o!r cubes to ^?4 ,& aft] ?is1 by id5tify+ ourselves ) o!r cubes1 & 9 turn appreciat+ ! relation ( ! o!r cubes to ^!4 ,& ! practical putt+ up ( ! cubes is ! way 9 :i* ?is p[] is ga9$4 ,x spr+s up ) a repetition ( ! me*anical acts4 ,?us !re are ?ree processes4 #a/1 ,an appreh5sion ( :at ! position ( ! cubes w\ld be4 #bnd1 ,an actual putt+ ( !m up 9 accordance ) t appreh5sion1 #crd1 ,! spr++ up 9 ! m9d ( a direct feel+ ( :at ! block is1 indep5d5t ( any p>ticul> pres5tation4 ,?us ! self elem5t ( up & d[n c be got rid ( out ( a block ( cubes4 ,& :5 ev5 a little block is kn[n l ?is1 ! m9d has ga9$ a great deal4 ,yet 9 ! appreh5sion & k ( ! block ( cubes ) ! up & d[n relation 9 !m1 !re is m ?an 9 ! absolute appreh5sion ( !m4 ,= !re is ! appreh5sion ( _! position & also ( ! effect ( gravity on !m 9 _! position4 ,imag9e ourselves to be translat$ sudd5ly to ano!r "p ( ! univ]se1 & to f9d !re intellig5t be+s1 & to hold conv]sation ) !m4 ,if we told !m t we came f a _w1 & 7 to describe ! sun to !m1 say+ t x 0 a bright1 hot body :i* mov$ r\nd u1 !y w\ld reply3 ,y h told u some?+ ab\t ! sun1 b y h also told u some?+ ab\t y\rselves4 ,?us 9 ! appreh5sion ( ! sun z a body mov+ r\nd u !re is m ?an 9 ! appreh5sion ( x z n mov+ r\nd1 = we really 9 ?is case appreh5d two ?+s--! sun & our own conditions4 ,b = ! purpose ( fur!r k x is mo/ important t ! m ab/ract k %\ld be acquir$4 ,! self elem5t introduc$ by ! motion ( ! e>? mu/ be got rid ( be=e ! true relations ( ! sol> sy/em c be made out4 ,& 9 our block ( cubes1 x w be f\nd t feel+s ab\t arrangem5t1 & k ( space1 :i* are q unatta9able ) our ord9>y view ( position1 become simple & cle> :5 ?is discipl9e has be5 gone ?r\gh4 ,& !re c be no possible m5tal h>m 9 go+ ?r\gh ?is bit ( tra9+1 = all t x comes to is look+ at a real ?+ z x actually is--turn+ x r\nd & ov] & le>n+ x f e po9t ( view4 ,,*apt] ,,vi4 ,,function ,,( ,,m9d4 ,,space ,,aga9/ ,,metaphysics4 ,,self-,,limitation ,,& ,,its ,,te/4 ,a ,,plane ,,_w4 ,we n[ pass on to ! "q3 ,are !re any o!r self elem5ts pres5t 9 our k ( ! block ( cubes8 ,:5 we h le>nt to free x f up & d[n1 is !re any?+ else to be got rid (8 ,x seems z if1 :5 ! cubes 7 ?us le>nt1 we _h got z ab/ract & imp]sonal a bit ( k z possible4 ,b1 9 reality1 9 ! relations ( ! cubes z we ?us appreh5d !m !re is pres5t a self elem5t to :i* ! up & d[n is a m]e trifle4 ,if we ?9k we h got absolute k we are inde$ walk+ on a ?9 cru/ 9 unconsci\sness ( ! dep?s bel[4 ,we are s c]ta9 ( t :i* we are habituat$ to1 we are s sure t ! _w is made up ( ! me*anical =ces & pr9ciples :i* we famili>ly deal )1 t x is m ( a %ock ?an a welcome surprise to u to f9d h[ mi/ak5 we 74 ,& aft] all1 d we suppose t ! facts ( di/ance & size & %ape are ! ultimate facts ( ! _w--is x 9 tru? made up l a ma*9e out ( me*anical p>ts8 ,if s1 :]e is !re room = t o!r :i* we "k--m c]ta9ly1 because inw>dly--t rev]5ce & love :i* make life wor? hav+8 ,no2 ^! me*anical relations are our means ( kn[+ ab\t ! _w2 !y are n reality itself1 & _! prim>y place 9 our imag9ations is due to ! famili>ity :i* we h ) !m1 & to ! peculi> limitations "u :i* we are4 ,b ,i d n = a mom5t wi% to g 9 ?\ght beyond physical nature--,i d n suppose t 9 ?\ght we c4 ,to ! m9d x is only ! body t appe>s1 & all t ,i hope to d is to %[ mat]ial relations1 me*anism1 arrangem5ts4 ,b mu* dep5ds on :at k9d ( mat]ial relations we p]ceive outside u4 ,a human be+1 an animal & a ma*9e are to ! m9d all m]ely portions ( matt] arrang$ 9 c]ta9 ways4 ,b ! m9d c give an exhau/ive acc\nt ( ! ma*9e1 acc\nt fairly well = ! animal1 :ile ! human be+ x only def9es ext]nally1 leav+ ! real k to be suppli$ by o!r faculties4 ,b we mu/ n "u-e/imate ! "w ( ! m9d1 = x is only by ! obs]vation ( & ?\ght ab\t ! bodies ) :i* we come into contact t we "k human be+s4 ,x is ! faculty ( ?\ght t puts u 9 a position to recognize a s\l4 ,& s1 too1 ab\t ! univ]se--x is only by correct ?\ght ab\t x t we c p]ceive its true moral nature4 ,& x w be f\nd t ! deadness :i* we ascribe to ! ext]nal _w is n really !re1 b is put 9 by u because ( our own limitations4 ,x is really ! self elem5ts 9 our k :i* make u talk ( me*anical necessity1 dead matt]4 ,:5 our limitations fall1 we behold ! _s ( ! _w l we behold ! _s ( a fri5d--some?+ :i* is disc]n$ 9 & ?r\gh ! mat]ial pres5tation ( a body to u4 ,our ?\ght means are suffici5t at pres5t to %[ u human s\ls2 b all except human be+s is1 z f> z sci5ce is conc]n$1 inanimate4 ,"o self elem5t mu/ be got rid ( f our p]ception1 & ?is w be *ang$4 ,! "o ?+ necess>y is1 t 9 matt]s ( ?9k+ we w n admit any?+ t is n p]fectly cle>1 palpable & evid5t4 ,on ! m9d ! only conceivable dem& is to seek = facts4 ,! rock on :i* s _m sy/ems ( philosophy h come to grief is ! attempt to put moral pr9ciples into nature4 ,our only duty is to accept :at we f9d4 ,man is no m ! c5tre ( ! moral _w ?an he is ( ! physical _w4 ,!n relegate ! intellect to its "r position ( deal+ ) facts ( arrangem5t--x c appreciate /ructure--& let x simply look on ! _w & report on x4 ,we h to *oose betwe5 metaphysics & space ?\ght4 ,9 metaphysics we f9d l(ty ideals--pr9ciples en?ron$ high 9 our s\ls1 b :i* r$uce ! _w to a phantom1 & ourselves to ! l(ty spectators ( an arid solitude4 ,on ! o!r h&1 if we foll[ ,kant's advice1 we use our means & f9d realities l9k$ toge!r1 & 9 ! physical int]play ( =ces & connexion ( /ructure we behold ! relations betwe5 spirits--^? dwell+ 9 man & ^? above him4 ,x is difficult to expla9 ?is next self elem5t t has to be remov$ f ! block ( cubes2 x requires a little c>eful prep>ation1 9 fact our language h>dly af=ds u ! means4 ,b x is possible to approa* indirectly1 & to detect ! self-elem5t by means ( an analogy4 ,if we suspect !re be "s condition affect+ ourselves :i* make u p]ceive ?+s n z !y are1 b falsely1 !n x is possible to te/ ! matt] by mak+ ! supposition ( o!r be+s subject to c]ta9 conditions1 & !n exam9+ :at ! effect on _! exp]i5ce w\ld be ( ^! conditions4 ,?us if we make up ! appe>ances :i* w\ld pres5t !mselves to a be+ subject to a limitation or condition1 we %all f9d t ?is limitation or condition1 :5 unrecogniz$ by him1 pres5ts itself z a g5]al law ( 8 outw>d _w1 or z prop]ties & qualities ( ! objects ext]nal to him4 ,he w1 moreov]1 f9d c]ta9 op]ations possible1 o!rs impossible1 & ! b\nd>y l9e betwe5 ! possible & impossible w dep5d q z mu* on ! conditions "u :i* he is z on ! nature ( ! op]ations4 ,& if we f9d t 9 our exp]i5ce ( ! outw>d _w !re are analog\s prop]ties & qualities ( matt]1 analog\s possibilities & impossibilities1 !n x w %[ to u t we 9 our turn are "u analog\s limitations1 & t :at we p]ceive z ! ext]nal _w is bo? ! ext]nal _w & our own conditions4 ,& ! task be=e u w be to sep>ate ! two4 ,n[ ! problem we take up "h is ?is--to sep>ate ! self elem5ts f ! true fact4 ,to sep>ate !m n m]ely z an outw>d !ory & intellig5t appreh5sion1 b to sep>ate !m 9 ! consci\sness itself1 s t our p[] ( p]ception is rais$ to a high] level4 ,we f9d out t we are "u limitations4 ,our next /ep is to s famili>ize ourselves ) ! real aspect ( ?+s1 t we p]ceive l be+s n "u our limitations4 ,or m truly1 we f9d t inw>d s\l :i* itself n subject to ^! limitations1 is awak5$ to its own natural action1 :5 ! v]dicts convey$ to x ?r\gh ! s5ses are purg$ ( ! self elem5ts introduc$ by ! s5ses4 ,ev]y?+ dep5ds on ?is--,is !re a native & spontane\s p[] ( appreh5sion1 :i* spr+s into activity :5 we take ! tr\ble to pres5t to x a view f :i* ! self elem5ts are elim9at$8 ,ab\t ?is e "o mu/ judge = himself4 ,b ! process :]eby ?is inn] vision is call$ on is a def9ite "o4 ,& j z a human be+ plac$ 9 natural human relation%ips f9ds 9 himself a spontane\s motive t[>ds ! fulfilm5t ( !m1 discov]s 9 himself a be+ ^: motives transc5d ! limits ( bodily self-reg>d1 s we %\ld expect to f9d 9 our m9ds a p[] :i* is ready to appreh5d a m absolute ord] ( fact ?an t :i* comes ?r\gh ! s5ses4 ,i d n mean a !oretical p[]4 ,a !ory is always ab\t x1 & ab\t x only4 ,i mean an inn] view1 a vision :]eby ! see+ m9d z x 7 id5tifies itself ) ! ?+ se54 ,n ! tree ( k1 b ( ! inn] & vital sap :i* builds up ! tree ( k4 ,& if ?is po9t is settl$1 x w be ( "s use 9 answ]+ ! "q3 ,:at are we8 ,are we !n bodies only8 ,?is "q has be5 answ]$ 9 ! negative by our in/9cts4 ,:y %\ld we despair ( a rational answ]8 ,let u adopt our space ?\ght & develop x4 ,! supposition :i* we mu/ make is ! foll[+4 ,let u imag9e a smoo? surface--l ! surface ( a table2 b let ! solid body at :i* we are look+ be v ?91 s t our surface is m l ! surface ( a ?9 %eet ( metal ?an ! top ( a table4 ,& let u imag9e small p>ticles1 l p>ticles ( du/1 to lie on ?is surface1 & to be attract$ d[nw>ds s t !y keep on ! surface4 ,b let u suppose !m to move freely ov] ! surface4 ,let !m nev] 9 _! movem5ts rise "o ov] ! o!r2 let !m all s+ly & collectively be close to ! surface4 ,& let u suppose all sorts ( attractions & repulsions betwe5 ^! atoms1 & let !m h all k9ds ( movem5ts l ! atoms ( our matt] h4 ,!n !re may be conceiv$ a :ole _w1 & v>i\s k9ds ( be+s z =m$ out ( ?is matt]4 ,! peculi>ity ab\t ?is _w & ^! be+s w\ld be1 t nei!r ! inanimate nor ! animate memb]s ( x w\ld move away f ! surface4 ,_! movem5ts w\ld all lie 9 "o plane1 a plane p>allel to & v ne> ! surface on :i* !y are4 ,& if we suppose a va/ mass to be =m$ out ( ^! atoms1 & to lie l a great r\nd disk on ! surface1 compact & coh]+ closely toge!r1 !n ?is great disk w\ld af=d a support = ! small] %apes1 :i* we may suppose to be animate be+s4 ,! small] %apes w\ld be attract$ to ! great disk1 b w\ld be arre/$ at its rim4 ,!y w\ld t5d to ! c5tre ( ! disk1 b be unable to get ne>] to ! c5tre ?an its rim4 ,?us1 z we are attract$ to ! c5tre ( ! e>?1 b walk on its surface1 ! be+s on ?is disk w\ld be attract$ to its c5tre1 b walk on its rim4 ,! =ce ( attraction :i* !y w\ld feel w\ld be ! attraction ( ! disk4 ,! o!r =ce ( attraction1 act+ p]p5dicul>ly to ! plane :i* keeps !m & all ! matt] ( _! _w to ! surface1 !y w\ld "k no?+ ab\t4 ,= !y _c move ei!r t[>ds ?is =ce or away f x2 & ! surface is q smoo?1 s t !y feel no friction 9 _! movem5t ov] x4 ,n[ let u realize cle>ly "o ( ^! be+s z he proce$s along ! rim ( 8 _w4 ,let u imag9e him 9 ! =m ( an outl9e ( a human be+1 ) no ?ickness except t ( ! atoms ( 8 _w4 ,z to ! mode 9 :i* he walks1 we mu/ imag9e t he proce$s by spr+s or hops1 because !re w\ld be no room = 8 limbs to pass ea* o!r4 ,imag9e a l>ge disk on ! table be=e y1 & a be+1 su* z ! "o describ$1 proce$+ r\nd x4 ,let !re be small movable p>ticles surr\nd+ him1 :i* move out ( 8 way z he goes along1 & let ^! s]ve him = respiration2 let !m con/itute an atmosph]e4 ,=w>ds & backw>ds w\ld be to su* a be+ direction along ! rim--! direction 9 :i* he 0 proce$+ & its rev]se4 ,!n up & d[n w\ld evid5tly be ! direction away f ! disk's c5tre & t[>ds x4 ,?us backw>ds & =w>ds1 up & d[n1 w\ld bo? lie 9 ! plane 9 :i* he 04 ,& he w\ld h no o!r lib]ty ( movem5t except ^!4 ,?us ! words "r & left w\ld h no mean+ to him4 ,all ! directions 9 :i* he c\ld move1 or c\ld conceive movem5t possible1 w\ld be exhau/$ :5 he _h ?\ght ( ! directions along ! rim & at "r angles to x1 bo? 9 ! plane4 ,:at he w\ld call solid bodies1 w\ld be gr\ps ( ! atoms ( 8 _w coh]+ toge!r4 ,su* a mass ( atoms w\ld1 we "k1 h a slight ?ickness2 namely1 ! ?ickness ( a s+le atom4 ,b ( ?is he w\ld "k no?+4 ,he w\ld say1 8,a solid body has two dim5sions--height 7by h[ mu* x goes away f ! rim7 & ?ickness 7by h[ mu* x lies along ! rim740 ,?us a solid w\ld be a two-dim5sional body1 & a solid w\ld be b\nd$ by l9es4 ,l9es w\ld be all t he c\ld see ( a solid body4 ,?us "o ( ! results ( ! limitations "u :i* he exi/s w\ld be1 t he w\ld say1 8,!re are only two dim5sions 9 real ?+s40 ,9 ord] = 8 _w to be p]man5t1 we mu/ suppose ! surface on :i* he is to be v compact1 comp>$ to ! p>ticles ( 8 matt]2 to be v rigid2 &1 if he is n to obs]ve x by ! friction ( matt] mov+ on x1 to be v smoo?4 ,& if x is v compact ) reg>d to 8 matt]1 ! vibrations ( ! surface mu/ h ! effect ( di/urb+ ! portions ( 8 matt]1 & ( sep>at+ comp\nd bodies up into simpl] ones4 ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,ano!r consequ5ce ( ! limitation "u :i* ?is be+ lies1 w\ld be ! foll[+3--,if we cut out f ! corn]s ( a piece ( pap] two triangles1 ,a ,b ,c & ,a ,b ,c1 & suppose !m to be r$uc$ to su* a ?9ness t !y are capable ( be+ put on to ! imag9>y surface1 & ( be+ obs]v$ by ! flat be+ l o!r bodies kn[n to him2 he w1 aft] /udy+ ! b\nd+ l9es1 :i* are all t he c see or t\*1 come to ! conclusion t !y are equal & simil> 9 e respect2 & he c conceive ! "o occupy+ ! same space z ! o!r occupies1 )\t its be+ alt]$ 9 any way4 ,if1 h[ev]1 in/ead ( putt+ d[n ^! triangles into ! surface on :i* ! suppos$ be+ lives1 z %[n 9 ,fig4 #a1 we fir/ ( all turn "o ( !m ov]1 & !n put !m d[n1 !n ! plane-be+ has pres5t$ to him two triangles1 z %[n 9 ,fig4 #b4 ,& if he /udies ^!1 he f9ds t !y are equal 9 size & simil> 9 e respect4 ,b he _c make ! "o occupy ! same space z ! o!r "o2 ?is w become evid5t if ! triangles be mov$ ab\t on ! surface ( a table4 ,"o w n lie on ! same portion ( ! table t ! o!r has m>k$ out by ly+ on x4 ,h5ce ! plane-be+ by no means c\ld make ! "o triangle 9 ?is case co9cide ) ! space occupi$ by ! o!r1 nor w\ld he be able to conceive ! "o z co9cid5t ) ! o!r4 ,! reason ( ?is impossibility is1 n t ! "o _c be made to co9cide1 b t be=e hav+ be5 put d[n on 8 plane x has be5 turn$ r\nd4 ,x has be5 turn$1 us+ a direction ( motion :i* ! plane-be+ has nev] _h any exp]i5ce (1 & :i* !re=e he _c use 9 8 m5tal "w any m ?an 9 8 practical endeav\rs4 ,?us1 ow+ to 8 limitations1 !re is a c]ta9 l9e ( possibility :i* he _c ov]/ep4 ,b ?is l9e does n correspond to :at is actually possible & impossible4 ,x corresponds to a c]ta9 condition affect+ him1 n affect+ ! triangle4 ,8 say+ t x is impossible to make ! two triangles co9cide1 is an ass]tion1 n ab\t ! triangles1 b ab\t himself4 ,n[1 to return to our own _w1 no d\bt !re are _m ass]tions :i* we make ab\t ! ext]nal _w :i* are really ass]tions ab\t ourselves4 ,& we h a set ( /atem5ts :i* are precisely simil> to ^? :i* ! plane-be+ w\ld make ab\t 8 surr\nd+s4 ,?us1 he w\ld say1 !re are only two indep5d5t directions2 we say !re are only ?ree4 ,he w\ld say t solids are b\nd$ by l9es2 we say t solids are b\nd$ by planes4 ,moreov]1 !re are figures ab\t :i* we ass]t exactly ! same k9d ( impossibility z 8 plane-be+ did ab\t ! triangles 9 ,fig4 #b4 ,we "k c]ta9 %apes :i* are equal ! "o to ! o!r1 :i* are exactly simil>1 & yet :i* we _c make fit into ! same portion ( space1 ei!r practically or by imag9ation4 ,if we look at our two h&s we see ?is cle>ly1 ?\gh ! two h&s are a complicat$ case ( a v common fact ( %ape4 ,n[1 !re is "o way 9 :i* ! "r h& & ! left h& may practically be br\ght into lik5ess4 ,if we take ! "r-h& glove & ! left-h& glove1 !y w n fit any m ?an ! "r h& w co9cide ) ! left h&4 ,b if we turn "o glove inside out1 !n x w fit4 ,n[1 to suppose ! same ?+ done ) ! solid h& z is done ) ! glove :5 x is turn$ inside out1 we mu/ suppose x1 s to speak1 pull$ ?r\gh itself4 ,if ! h& 7 inside ! glove all ! "t ! glove 0 be+ turn$ inside out1 !n1 if su* an op]ation 7 possible1 ! "r h& w\ld be turn$ into an exact model ( ! left h&4 ,su* an op]ation is impossible4 ,b curi\sly en\gh !re is a precisely simil> op]ation :i*1 if x 7 possible1 w\ld1 9 a plane1 turn ! "o triangle 9 ,fig4 #b into ! exact copy ( ! o!r4 ,illu/ration ,look at ! triangle 9 ,fig4 #b1 ,a ,b ,c1 & imag9e ! po9t ,a to move into ! int]ior ( ! triangle & to pass ?r\gh x1 c>ry+ aft] x ! p>ts ( ! l9es ,a ,b & ,a ,c to :i* x is atta*$1 we %\ld h f9ally a triangle ,a ,b ,c1 :i* 0 q l ! o!r ( ! two triangles ,a ,b ,c 9 ,fig4 #b4 ,?us we "k ! op]ation :i* produces ! result ( ! 8pull+ ?r\gh0 is n an impossible "o :5 ! plane-be+ is conc]n$4 ,!n may x n be t !re is a way 9 :i* ! results ( ! impossible op]ation ( pull+ a h& ?r\gh c\ld be p]=m$8 ,! "q is an op5 "o4 ,our feel+ ( x be+ impossible to produce ?is result 9 any way1 may be because x really is impossible1 or x may be a useful bit ( in=mation ab\t ourselves4 ,n[ at ?is po9t my special "w comes 94 ,if !re be really a f\r-dim5sional _w1 & we are limit$ to a space or ?ree-dim5sional view1 !n ei!r we are absolutely ?ree-dim5sional ) no exp]i5ce at all or capacity ( appreh5d+ f\r-dim5sional facts1 or we may be1 z f> z our outw>d exp]i5ce goes1 s limit$2 b we may really be f\r-dim5sional be+s ^: consci\sness is by c]ta9 undet]m9$ conditions limit$ to a section ( ! real space4 ,?us we may really be l ! plane-be+s m5tion$ above1 or we may be 9 su* a condition t our p]ceptions1 n ourselves1 are s limit$4 ,! "q is "o :i* calls = exp]im5t4 ,we "k t if we take an animal1 su* z a dog or cat1 we c by c>eful tra9+1 & by us+ rew>ds & puni%m5t1 make !m act 9 a c]ta9 way1 9 c]ta9 def9$ cases1 9 accordance ) ju/ice2 we c produce ! me*anical action4 ,b ! feel+ ( ju/ice w n be ar\s$2 x w be b a m]e outw>d con=mity4 ,b a human be+1 if s tra9$1 & see+ o!rs s act+1 gets a feel+ ( ju/ice4 ,n[1 if we are really f\r-dim5sional1 by go+ ?r\gh ^? acts :i* correspond to a f\r-dim5sional exp]i5ce 7s f> z we c71 we %all obta9 an appreh5sion ( f\r-dim5sional exi/5ce--n ) ! outw>d eye1 b ess5tially ) ! m9d4 ,& aft] a numb] ( ye>s ( exp]im5t :i* 7 entirely nugatory1 ,i c n[ lay x d[n z a v]ifiable fact1 t by tak+ ! prop] /eps we c feel f\r-dim5sional exi/5ce1 t ! human be+ someh[1 & 9 "s way1 is n simply a ?ree-dim5sional be+--9 :at way x is ! prov9ce ( sci5ce to discov]4 ,all t ,i %all d "h is1 to put =w>d c]ta9 suppositions :i*1 9 an arbitr>y & =c$ mann]1 give an outl9e ( ! relation ( our body to f\r-dim5sional exi/5ce1 & %[ h[ 9 our m9ds we h faculties by :i* we recognise x4 ,,*apt] ,,vii4 ,,self ,,elem5ts ,,9 ,,our ,,consci\sness4 ,x is (t5 tak5 = grant$ t our consci\sness ( ourselves & ( our own feel+s has a sort ( direct & absolute value4 ,x is suppos$ to af=d a te/imony :i* does n require to be sift$ l our consci\sness ( ext]nal ev5ts4 ,b 9 reality x ne$s f> m criticism to be appli$ to x ?an any o!r mode ( appreh5sion4 ,to a c]ta9 degree we c sift our exp]i5ce ( ! ext]nal _w1 & divide x into two portions4 ,we c det]m9e ! self elem5ts & ! realities4 ,b ) reg>d to our own nature & emotions1 ! discov]y :i* makes a sci5ce possible has yet to be made4 ,!re are c]ta9 indications1 h[ev]1 spr++ f our obs]vation ( our own bodies1 :i* h a c]ta9 degree ( int]e/4 ,x is f\nd t ! processes ( ?\ght & feel+ are connect$ ) ! bra94 ,if ! bra9 is di/urb$1 ?\ghts1 sights1 & s\nds come into ! consci\sness :i* h no objective cause 9 ! ext]nal _w4 ,h5ce we may conclusively say t ! human be+1 :atev] he is1 is 9 contact ) ! bra91 & ?r\gh ! bra9 ) ! body1 & ?r\gh ! body ) ! ext]nal _w4 ,x is ! /ructures & movem5ts 9 ! bra9 :i* ! human be+ p]ceives4 ,x is by a /ructure 9 ! bra9 t he appreh5ds nature1 n imm$iately4 ,! mo/ beautiful sights & s\nds h no effect on a human be+ unless !re is ! faculty 9 ! bra9 ( tak+ !m 9 & h&+ !m on to ! consci\sness4 ,h5ce1 cle>ly1 x is ! movem5ts & /ructure ( ! m9ute portions ( matt] =m+ ! bra9 :i* ! consci\sness p]ceives4 ,& x is only by models & repres5tations made 9 ! /uff ( ! bra9 t ! m9d kn[s ext]nal *anges4 ,n[1 our bra9s are well furni%$ ) models & repres5tations ( ! facts & ev5ts ( ! ext]nal _w4 ,b a mo/ important fact /ill requires its due weight to be laid ^u x4 ,^! models & repres5tations are made on a v m9ute scale--! p>ticles ( bra9 matt] :i* =m images & repres5tations are beyond ! p[] ( ! microscope 9 _! m9ut5ess4 ,h5ce ! consci\sness prim>ily appreh5ds ! movem5ts ( matt] ( a degree ( smallness :i* is beyond ! p[] ( obs]vation 9 any o!r way4 ,h5ce we h a means ( obs]v+ ! movem5ts ( ! m9ute portions ( matt]4 ,let u call ^? portions ( ! bra9 matt] :i* are directly in/rum5tal 9 mak+ repres5tations ( ! ext]nal _w--let u call !m bra9 molecules4 ,n[1 ^! bra9 molecules are v m9ute portions ( matt] inde$2 g5]ally !y are made to g ?r\gh movem5ts & =m /ructures 9 su* a way z to repres5t ! movem5ts & /ructures ( ! ext]nal _w ( masses ar\nd u4 ,b x does n foll[ t ! /ructures & movem5ts :i* !y p]=m ( _! own nature are id5tical ) ! movem5ts ( ! portions ( matt] :i* we see ar\nd u 9 ! _w ( matt]4 ,x may be t ^! bra9 molecules h ! p[] ( f\r-dim5sional movem5t1 & t !y c g ?r\gh f\r-dim5sional movem5ts & =m f\r-dim5sional /ructures4 ,if s1 !re is a practical way ( le>n+ ! movem5ts ( ! v small p>ticles ( matt]--by obs]v+1 n :at we c see1 b :at we c ?9k4 ,=1 suppose ^! small molecules ( ! bra9 7 to build up /ructures & g ?r\gh movem5ts n 9 accordance ) ! rule ( repres5t+ :at goes on 9 ! ext]nal _w1 b 9 accordance ) _! own activity1 !n !y might g ?r\gh f\r-dim5sional movem5ts & =m f\r-dim5sional /ructures4 ,& ^! movem5ts & /ructures w\ld be appreh5d$ by ! consci\sness along ) ! o!r movem5ts & /ructures1 & w\ld seem z real z ! o!rs--b w\ld h no correspond5ce 9 ! ext]nal _w4 ,!y w\ld be ?\ghts & imag9ations1 n obs]vations ( ext]nal facts4 ,n[1 ?is field ( inve/igation is "o :i* requires to be work$ at4 ,at pres5t x is only ^? /ructures & movem5ts ( ! bra9 molecules :i* correspond to ! realities ( our ?ree-dim5sional space :i* are 9 g5]al work$ at consi/5tly4 ,b 9 ! practical "p ( ?is book x w be f\nd t by prop] /imulus ! bra9 molecules w arrange !mselves 9 /ructures repres5t+ a f\r-dim5sional exi/5ce4 ,x only requires a c]ta9 am\nt ( c>e to build up m5tal models ( high] space exi/5ces4 ,9 fact1 x is probably "p ( ! difficulty ( =m+ ?ree-dim5sional bra9 models1 t ! bra9 molecules h to be limit$ 9 _! own fre$om ( motion to ! requirem5ts ( ! limit$ space 9 :i* our practical daily life is c>ri$ on4 ,note4--,= my own "p ,i %\ld say t all ^? confusions 9 rememb]+ :i* come f an image tak+ ! place ( ! orig9al m5tal model--z1 = in/ance1 ! difficulty 9 rememb]+ :i* way to turn a screw1 & ! num]\s cases ( images 9 ?\ght transf]5ce--may be due to a toppl+ ov] 9 ! bra91 f\r-dim5sionalwise1 ( ! /ructures =m$--:i* /ructures w\ld be absolutely safe f be+ turn$ into image /ructures if ! bra9 molecules mov$ only ?ree-dim5sionalwise4 ,x is rem>kable h[ 9 sci5ce 8expla9+0 means ! ref]5ce ( ! movem5ts & t5d5cies to movem5t ( ! masses ab\t u to ! movem5ts & t5d5cies to movem5t ( ! m9ute portions ( matt]4 ,?us1 ! behavi\r ( gase\s bodies--! pressure :i* !y ex]t1 ! laws ( _! cool+ & int]mixture are expla9$ by trac+ ! movem5ts ( ! v m9ute p>ticles ( :i* !y are compos$4 ,,*apt] ,,viii4 ,,relation ,,( ,,l[] ,,to ,,high] ,,space4 ,,!ory ,,( ,,! ,,!r4 ,at ?is po9t ( our inquiries ! be/ plan is to turn to ! practical "w1 & try if ! faculty ( ?9k+ 9 high] space c be awak5$ 9 ! m9d4 ,! g5]al outl9e ( ! me?od is ! same z t :i* has be5 describ$ = gett+ rid ( ! limitation ( up & d[n f a block ( cubes4 ,we suppos$ t ! block 0 fix$2 & to get ! s5se ( :at x w\ld be :5 gravity act$ 9 a diff]5t way ) reg>d to x1 we made a model ( x z x w\ld be "u ! new circum/ances4 ,we ?\ght out ! relations :i* w\ld exi/2 & by practis+ ?is new arrangem5t we gradually =m$ ! direct appreh5sion4 ,& s ) high]-space arrangem5ts4 ,we _c put !m up actually1 b we c say h[ !y w\ld look & be to ! t\* f v>i\s sides4 ,& we c put up ! actual appe>ances ( !m1 n altoge!r1 b z models succe$+ "o ano!r2 & by contemplation & active arrangem5t ( ^! diff]5t views we call ^u our inw>d p[] to manife/ itself4 ,9 prep>+ our g5]al plan ( "w1 x is necess>y to make def9ite assumptions ) reg>d to our _w1 our univ]se1 or we may call x our space1 9 relation to ! wid] univ]se ( f\r-dim5sional space4 ,:at our relation to x may be1 is altoge!r undet]m9$4 ,! real relation%ip w require a great deal ( /udy to appreh5d1 & :5 appreh5d$ w seem z natural to u z ! position ( ! e>? among ! o!r planets does to u n[4 ,b we h n got to wait = ?is exploration 9 ord] to comm5ce our "w ( high]-space ?\ght1 = we "k def9itely t :atev] our real physical relation%ip to ?is wid] univ]se may be1 we are practically 9 exactly ! same relation%ip to x z ! creature we h suppos$ liv+ on ! surface ( a smoo? %eet is to ! _w ( ?reefold space4 ,& ?is relation%ip ( a surface to a solid or ( a solid1 z we conjecture1 to a high] solid1 is "o :i* we (t5 f9d 9 nature4 ,a surface is no?+ m nor less ?an ! relation betwe5 two ?+s4 ,two bodies t\* ea* o!r4 ,! surface is ! relation%ip ( "o to ! o!r4 ,aga91 we see ! surface ( wat]4 ,?us our solid exi/5ce may be ! contact ( two f\r-dim5sional exi/5ces ) ea* o!r2 & j z s5sation ( t\* is limit$ to ! surface ( ! body1 s s5sation on a l>g] scale may be limit$ to ?is solid surface4 ,& x is a fact wor?y ( notice1 t 9 ! surface ( a fluid diff]5t laws obta9 f ^? :i* hold ?r\gh\t ! mass4 ,!re are a :ole s]ies ( facts :i* are gr\p$ toge!r "u ! "n ( surface t5sions1 :i* are ( great importance 9 physics1 & by :i* ! behavi\r ( ! surfaces ( liquids is gov]n$4 ,& x may well be t ! laws ( our univ]se are ! surface t5sions ( a high] univ]se4 ,b ^! expressions1 x is evid5t1 af=d u no practical basis = inve/igation4 ,we mu/ assume some?+ m def9ite1 & because m def9ite 79 ! abs5ce ( details drawn f exp]i5ce71 m arbitr>y4 ,& we w assume t ! conditions "u :i* we human be+s are1 exactly resemble ^? "u :i* ! plane-be+s are plac$1 :i* h be5 describ$4 ,?is =ms ! basis ( our "w2 & ! practical "p ( x consi/s 9 do+1 ) reg>d to high] space1 t :i* a plane-be+ w\ld d ) reg>d to our space 9 ord] to enable himself to realize :at x 04 ,if we imag9e "o ( ^! limit$ creatures ^: life is cramp$ & conf9$ /udy+ ! facts ( space exi/5ce1 we f9d t he c d x 9 two ways4 ,he c assume ano!r direction 9 addition to ^? :i* he kn[s2 & he c1 by means ( ab/ract reason+1 say :at w\ld take place 9 an ampl] k9d ( space ?an 8 own4 ,all ?is w\ld be =mal "w4 ,! conclusions w\ld be ab/ract possibilities4 ,! o!r mode ( /udy is ?is4 ,he c take "s ( ^! facts ( 8 high] space & he c pond] ov] !m 9 8 m9d1 & c make up 9 8 plane _w ^? diff]5t appe>ances :i* "o & ! same solid body w\ld pres5t to him1 & !n he may try to realize inw>dly :at 8 high] exi/5ce is4 ,n[1 x is evid5t t if ! creature is absolutely conf9$ to a two-dim5sional exi/5ce1 !n any?+ m ?an su* exi/5ce w always be a m]e ab/ract & =mal consid]ation to him4 ,b if ?is high]-space ?\ght becomes real to him1 if he f9ds 9 8 m9d a possibility ( ris+ to x1 !n inde$ he kn[s t someh[ he is n limit$ to 8 app>5t _w4 ,ev]y?+ he sees & comes into contact ) may be two-dim5sional2 b ess5tially1 someh[1 himself he is n two-dim5sional m]ely4 ,& a precisely simil> piece ( "w is be=e u4 ,assum+ z we mu/ t our out] exp]i5ce is limit$ to ?ree-dim5sional space1 we %all make up ! appe>ances :i* ! v simple/ high] bodies w\ld pres5t to u1 & we %all gradually arrive at a m ?an m]ely =mal & ab/ract appreciation ( !m4 ,we %all discov] 9 ourselves a faculty ( appreh5sion ( high] space simil> to t :i* we h ( space4 ,& ?us we %all discov]1 ea* = himself1 t1 limit$ z 8 s5ses are1 he ess5tially someh[ is n limit$4 ,! mode & me?od 9 :i* ?is consci\sness w be made g5]al1 is ! same 9 :i* ! _s ( an army is =m$4 ,! individuals ent] into ! s]vice f v>i\s motives1 b ea* & all h to g ?r\gh ^? movem5ts & actions :i* correspond to ! unity ( a :ole =m$ out ( diff]5t memb]s4 ,! inn] appreh5sion :i* lies 9 ea* man ( a p>ticipation 9 a life wid] ?an t ( 8 individual body1 is awak5$ & responds2 & ! active _s ( ! army is =m$4 ,s ) reg>d to high] space1 ?is faculty ( appreh5d+ intuitively f\r-dim5sional relation%ips w be tak5 up because ( its practical use4 ,individuals w be practically employ$ to d x by society because ( ! l>g] faculty ( ?\ght :i* x gives4 ,9 fact1 ?is high]-space ?\ght means z an affair ( m5tal tra9+ simply ! p[] ( appreh5d+ ! results aris+ f f\r indep5d5t causes4 ,x means ! p[] ( deal+ ) a great] numb] ( details4 ,& :5 ?is faculty ( high]-space ?\ght has be5 =m$1 !n ! faculty ( appreh5d+ t high] exi/5ce 9 :i* m5 h "p1 w come into be+4 ,x is necess>y to gu>d "h aga9/ !re be+ ascrib$ to ?is high]-space ?\ght any o!r ?an an intellectual value4 ,x has no moral value :atev]4 ,its only connexion ) moral or e?ical consid]ations is ! possibility x w af=d ( recogniz+ m ( ! facts ( ! univ]se ?an we d n[4 ,!re is a gradual process go+ on :i* may be describ$ z ! gett+ rid ( self elem5ts4 ,?is process is "o ( k & feel+1 & ei!r may be indep5d5t ( ! o!r4 ,at pres5t1 9 respect ( feel+1 we are mu* fur!r on ?an 9 respect to und]/&+1 & ! reason is v mu* ?is3 ,:5 a self elem5t has be5 got rid ( 9 respect ( feel+1 ! new appreh5sion is put into practice1 & we live x into our organization4 ,b :5 a self elem5t has be5 got rid ( intellectually1 x is all[$ to rema9 a matt] ( !ory1 n vitally ent]+ into ! m5tal /ructure ( individuals4 ,?us up & d[n 0 discov]$ to be a self elem5t m ?an a ?\s& ye>s ago2 b1 except z a matt] ( !ory1 we are p]fect b>b>ians 9 ?is respect up to ! pres5t day4 ,we h suppos$ a be+ liv+ 9 a plane _w1 t is1 a be+ ( a v small ?ickness 9 a direction p]p5dicul> to ! surface on :i* he is4 ,n[1 if we are situat$ analog\sly ) reg>d to an ampl] space1 !re mu/ be "s elem5t 9 our exp]i5ce correspond+ to ea* elem5t 9 ! plane-be+'s exp]i5ce4 ,& x is int]e/+ to ask1 9 ! case ( ! plane-be+1 :at 8 op9ion w\ld be ) respect to ! surface on :i* he 04 ,he w\ld n recognize x z a surface ) :i* he 0 9 contact2 he w\ld h no idea ( a motion away f x or t[>ds x4 ,b he w\ld discov] its exi/5ce by ! fact t movem5ts 7 transmitt$ along x4 ,by its vibrat+ & quiv]+1 x w\ld imp>t movem5t to ! p>ticles ( matt] ly+ on x4 ,h5ce1 he w\ld consid] ?is surface to be a m$ium ly+ betwe5 bodies1 & p5etrat+ !m4 ,x w\ld appe> to him to h no weight1 b to be a p[]ful means ( transmitt+ vibrations4 ,moreov]1 x w\ld be unlike any o!r sub/ance ) :i* he 0 acqua9t$1 inasmu* z he c\ld nev] get rid ( x4 ,h[ev] p]fect a vacuum be made1 !re w\ld be 9 ?is vacuum j z mu* ( ?is unkn[n m$ium z !re 0 be=e4 ,moreov]1 ?is surface w\ld n h9d] ! movem5t ( ! p>ticles ( matt] ov] x4 ,be+ smoo?1 matt] w\ld slide freely ov] x4 ,& ?is w\ld seem to him z if matt] w5t freely ?r\gh ! m$ium4 ,!n he w\ld also notice ! fact t vibrations ( ?is m$ium w\ld te> asund] portions ( matt]4 ,! plane surface1 be+ v compact1 comp>$ to ! masses ( matt] on x1 w\ld1 by its vibrations1 %ake !m into _! compon5t p>ts4 ,h5ce he w\ld h a s]ies ( obs]vations :i* t5d$ to %[ t ?is m$ium 0 unlike any ord9>y matt] ) :i* he 0 acqua9t$4 ,al?\gh matt] pass$ freely ?r\gh x1 /ill by its %ak+ x c\ld te> matt] 9 pieces4 ,^! w\ld be v difficult prop]ties to reconcile 9 "o & ! same sub/ance4 ,!n x is weightless1 & x is ev]y:]e4 ,x might well be t he w\ld reg>d ! supposition ( !re be+ a plane surface1 on :i* he 01 z a pref]able "o to ! hypo!sis ( ?is curi\s m$ium2 & ?us he might obta9 a pro( ( 8 limitations f 8 obs]vations4 ,n[1 is !re any?+ 9 our exp]i5ce :i* corresponds to ?is m$ium :i* ! plane-be+ gets to obs]ve8 ,d we suppose ! exi/5ce ( any m$ium ?r\gh :i* matt] freely moves1 :i* yet by its vibrations de/roys ! comb9ations ( matt]--"s m$ium :i* is pres5t 9 e vacuum1 h[ev] p]fect1 :i* p5etrates all bodies1 & yet c nev] be laid hold (8 ,^! are precisely obs]vations :i* h be5 made4 ,! sub/ance :i* possesses all ^! qualities is call$ ! !r4 ,& ! prop]ties ( ! !r are a p]petual object ( inve/igation 9 sci5ce4 ,n[1 x is n ! place "h to g into details1 z all we want is a basis = "w2 & h[ev] arbitr>y x may be1 x w s]ve if x enables u to inve/igate ! prop]ties ( high] space4 ,we w suppose1 !n1 t we are n 91 b on ! !r1 only n on x 9 any kn[n direction1 b t ! new direction is t :i* comes 94 ,! !r is a smoo? body1 along :i* we slide1 be+ di/ant f x at e po9t ab\t ! ?ickness ( an atom2 or1 if we take our mean di/ance1 be+ di/ant f x by half ! ?ickness ( an atom measur$ 9 ?is new direction4 ,!n1 j z 9 space objects1 a cube1 = in/ance1 c /& on ! surface ( a table1 or on ! surface ov] :i* ! plane-be+ moves1 s on ! !r c /& a high] solid4 ,all t ! plane-be+ sees or t\*es ( a cube1 is ! squ>e on :i* x re/s4 ,s all t we c\ld see or t\* ( a high] solid w\ld be t "p by :i* x /ood on ! !r2 & ?is "p w\ld be to u exactly l any ord9>y solid body4 ,! base ( a cube w\ld be to ! plane-be+ l a squ>e :i* is to him an ord9>y solid4 ,n[1 ! two ways1 9 :i* a plane-be+ w\ld appreh5d a solid body1 w\ld be by ! successive appe>ances to him ( x z x pass$ ?r\gh 8 plane2 & also by ! diff]5t views ( "o & ! same solid body :i* he got by turn+ ! body ov]1 s t diff]5t p>ts ( its surface come into contact ) 8 plane4 ,& ! practical "w ( le>n+ to ?9k 9 f\r-dim5sional space1 is to g ?r\gh ! appe>ances :i* "o & ! same high] solid has4 ,(t51 9 ! c\rse ( inve/igation 9 nature1 we come across objects :i* h a c]ta9 simil>ity1 & yet :i* are 9 p>ts entirely diff]5t4 ,! "w ( ! m9d consi/s 9 =m+ an idea ( t :ole 9 :i* !y coh]e1 & ( :i* !y are simple pres5tations4 ,! "w ( =m+ an idea ( a high] solid is ! mo/ simple & mo/ def9ite ( all su* m5tal op]ations4 ,if we imag9e a plane _w 9 :i* !re are objects :i* correspond to our sun1 to ! planets1 &1 9 fact1 to all our visible univ]se1 we mu/ suppose a surface ( enorm\s ext5t on :i* great disks slide1 ^! disks be+ worlds ( v>i\s ord]s ( magnitude4 ,^! disks w\ld "s ( !m be c5tral1 & hot1 l our sun2 r\nd !m w\ld circulate o!r disks1 l our planets4 ,& ! sy/ems ( sun & planets mu/ be conceiv$ z mov+ ) great velocity ov] ! surface :i* be>s !m all4 ,& ! movem5ts ( ! atoms ( ^! worlds w be ! c\rse ( ev5ts 9 su* worlds4 ,z ! atoms weave toge!r1 & =m bodies alt]+1 becom+1 & ceas+1 s w bodies be =m$ & disappe>4 ,& ! plane :i* be>s !m all on its smoo? surface w simply be a support to all ^! movem5ts1 & influ5ce !m 9 no way4 ,is to be consci\s ( be+ consci\s ( be+ hot1 ! same ?+ z to be consci\s ( be+ hot8 ,x is n ! same4 ,!re is a /&+ outside1 & objectivation ( a /ate ( m9d :i* e "o w\ld say 9 ! fir/ /ate 0 v diff]5t f ! simple consci\sness4 ,b ! consci\sness mu/ d z mu* 9 ! fir/ case z 9 ! second4 ,h5ce ! feel+ hot is v diff]5t f ! consci\sness ( feel+ hot4 ,a feel+ :i* we always h1 we %\ld n be consci\s (--a s\nd always pres5t ceases to be he>d4 ,h5ce consci\sness is a concomitant ( *ange1 t is1 ( ! contact betwe5 "o /ate & ano!r4 ,if a be+ liv+ on su* a plane 7 to inve/igate ! prop]ties1 he w\ld h to suppose ! solid to pass ?r\gh 8 plane 9 ord] to see ! :ole ( its surface4 ,?us we may imag9e a cube re/+ on a table to beg9 to p5etrate ?r\gh ! table4 ,if ! cube passes ?r\gh ! surface1 mak+ a clean cut all r\nd x1 s t ! plane-be+ c come up to x & inve/igate x1 !n ! diff]5t p>ts ( ! cube z x passes ?r\gh ! plane w be to him squ>es1 :i* he appreh5ds by ! b\nd>y l9es4 ,! cut :i* !re is 9 8 plane mu/ be suppos$ n to be notic$1 he mu/ be able to g "r up to ! cube )\t h9drance1 & to t\* & see t ?9 slice ( x :i* is j above ! plane4 ,& s1 :5 we /udy a high] solid1 we mu/ suppose t x passes ?r\gh ! !r1 & t we only see t ?9 ?ree-dim5sional section ( x :i* is j ab\t to pass f "o side to ! o!r ( ! !r4 ,:5 we look on a solid z a section ( a high] solid1 we h to suppose ! !r brok5 ?r\gh1 only we mu/ suppose t x runs up to ! edge ( ! body :i* is p5etrat+ x1 s t we are aw>e ( no brea* ( cont9uity4 ,! surface ( ! !r mu/ !n be suppos$ to h ! prop]ties ( ! surface ( a fluid2 only1 ( c\rse1 x is a solid ?ree-dim5sional surface1 n a two-dim5sional surface4 ,,*apt] ,,ix4 ,,ano!r ,,view ,,( ,,! ,,!r4 ,,mat]ial ,,& ,,!rial ,,bodies4 ,we h suppos$ 9 ! case ( a plane _w t ! surface on :i* ! movem5ts take place is inactive1 except by its vibrations4 ,x is simply a smoo? support4 ,= ! sake ( simplicity let u call ?is smoo? surface 8! !r0 9 ! case ( a plane _w4 ,! !r !n we h imag9$ to be simply a smoo?1 ?9 %eet1 n possess$ ( any def9ite /ructure1 b excit$ by real di/urbances ( ! matt] on x into vibrations1 :i* c>ry ! effect ( ^! di/urbances z light & heat to o!r portions ( matt]4 ,n[1 x is possible to take an entirely diff]5t view ( ! !r 9 ! case ( a plane _w4 ,let u imag9e t1 in/ead ( ! !r be+ a smoo? %eet s]v+ simply z a support1 x is def9itely m>k$ & groov$4 ,let u imag9e ^! grooves & *annels to be v m9ute1 b to be def9ite & p]man5t4 ,!n1 let u suppose t1 in/ead ( ! matt] :i* slides 9 ! !r hav+ attractions & repulsions ( its own1 t x is q in]t1 & has only ! prop]ties ( in]tia4 ,t is to say1 tak+ a disk or a plane _w z a specim51 ! :ole disk is slid+ on ! !r 9 virtue ( a c]ta9 mom5tum :i* x has1 & c]ta9 portions ( its matt] fit into ! grooves 9 ! !r1 & move along ^? grooves4 ,! size ( ! portions is det]m9$ by ! size ( ! grooves4 ,& let u call ^? portions ( matt] :i* occupy ! bread? ( a groove1 atoms4 ,!n x is evid5t t ! disk slid+ along ov] ! !r1 its atoms w move accord+ to ! arrangem5t ( ! grooves ov] :i* ! disk slides4 ,if ! grooves at any "o p>ticul> place come close toge!r1 !re w be a cond5sation ( matt] at t place :5 ! disk passes ov] x2 & if ! grooves sep>ate1 !re w be a r>efaction ( matt]4 ,if we imag9e five p>ticles1 ea* slipp+ along 9 its own groove1 if ! p>ticles are arrang$ 9 ! =m ( a regul> p5tagon1 & ! grooves are p>allel1 !n ^! five p>ticles1 mov+ ev5ly on1 w ma9ta9 _! positions ) reg>d to "o ano!r1 & a body w\ld exi/ l a p5tagon1 la/+ z long z ! groves rema9$ p>allel4 ,b if1 aft] "s di/ance _h be5 trav]s$ by ! disk1 & ^! five p>ticles 7 br\ght into a region :]e "o ( ! grooves t5d$ away f ! o!rs1 ! %ape ( ! p5tagon w\ld be de/roy$1 x w\ld become "s irregul> figure4 ,& x is easy to see t if ! grooves sep>at$1 & o!r grooves came 9 among/ !m1 along :i* o!r portions ( matt] 7 slid+1 t ! p5tagon w\ld disappe> z an isolat$ body1 t its con/itu5t matt] w\ld be sep>at$1 & t its p>ticles w\ld ent] into o!r %apes z con/itu5ts ( !m1 & n ( ! orig9al p5tagon4 ,?us1 9 cases ( great] complication1 an elaborate /ructure may be suppos$ to be =m$1 to alt]1 & to pass away2 its orig91 gr[?1 & decay be+ due1 n to any indep5d5t motion ( ! p>ticles con/itut+ x1 b to ! movem5t ( ! disk :]eby its portions ( matt] 7 br\ght to regions :]e !re 0 a p>ticul> disposition ( ! grooves4 ,!n ! nature ( ! %ape w\ld really be det]m9$ by ! grooves1 n by ! portions ( matt] :i* pass$ ov] !m--!y w\ld become manife/ z giv+ rise to a mat]ial =m :5 a disk pass$ ov] !m1 b !y w\ld subsi/ indep5d5tly ( ! disk2 & if ano!r disk 7 to pass ov] ! same grooves1 exactly ! same mat]ial /ructures w\ld spr+ up z came into be+ be=e4 ,if we make a simil> supposition ab\t our !r along :i* our e>? slides1 we may conceive ! movem5ts ( ! p>ticles ( matt] to be det]m9$1 n by attractions or repulsions ex]t$ on "o ano!r1 b to be set 9 exi/5ce by ! alt]ations 9 ! directions ( ! grooves ( ! !r along :i* !y are proce$+4 ,if ! grooves 7 all p>allel1 ! e>? w\ld proce$ )\t any o!r motion ?an t ( its pa? 9 ! heav5s4 ,b ) an alt]ation 9 ! direction ( ! grooves1 ! p>ticles1 in/ead ( proce$+ uni=mly ) ! mass ( ! e>?1 w\ld beg9 to move among/ ea* o!r4 ,& by a suffici5tly complicat$ arrangem5t ( grooves x may be suppos$ t all ! movem5ts ( ! =ms we see ar\nd u are due to int]weav+ & v>i\sly dispos$ grooves4 ,?us ! movem5ts1 :i* any body goes ?r\gh1 w\ld dep5d on ! arrangem5t ( ! !real grooves along :i* x 0 pass+4 ,z long z ! grooves rema9 gr\p$ toge!r 9 approximately ! same way1 x w\ld ma9ta9 its exi/5ce z ! same body2 b :5 ! grooves sep>at$1 & became involv$ ) ! grooves ( o!r objects1 ?is body w\ld cease to exi/ sep>ately4 ,?us ! sep>ate exi/5ces ( ! e>? might conceivably be due to ! disposition ( ^? p>ts ( ! !r ov] :i* ! e>? pass$4 ,& ?us any object w\ld h to be sep>at$ into two p>ts1 "o ! !real =m1 or modification :i* la/$1 ! o!r ! mat]ial p>ticles :i*1 com+ on ) bl9d mom5tum1 7 direct$ into su* movem5ts z to produce ! actual objects ar\nd u4 ,9 ?is way !re w\ld be two p>ts 9 any organism1 ! mat]ial "p & ! !real "p4 ,!re w\ld be ! mat]ial body1 :i* soon passes & becomes indi/+ui%able f any o!r mat]ial body1 & ! !real body :i* rema9s4 ,n[1 if we direct our att5tion to ! mat]ial body1 we see ! ph5om5a ( gr[?1 decay1 & dea?1 ! com+ & ! pass+ away ( a liv+ be+1 isolat$ dur+ 8 exi/5ce1 absolutely m]g$ at 8 dea? into ! common /oreh\se ( matt]4 ,b if we reg>d ! !real body1 we f9d some?+ diff]5t4 ,we f9d an organism :i* is n s absolutely sep>at$ f ! surr\nd+ organisms--an organism :i* is "p ( ! !r1 & :i* is l9k$ to o!r !real organisms by its v sub/ance--an organism betwe5 :i* & o!rs !re exi/s a unity incapable ( be+ brok51 & a common life :i* is r m>k$ ?an reveal$ by ! matt] :i* passes ov] x4 ,! !real body moreov] rema9s p]man5tly :5 ! mat]ial body has pass$ away4 ,! correspond5ces betwe5 ! !real body & ! life ( an organism su* z we "k1 is r to be f\nd 9 ! emotional region ?an 9 ! "o ( outw>d obs]vation4 ,to ! !real =m1 all p>ts ( x are equally "o2 b "p ( ?is =m corresponds to ! future ( ! mat]ial be+1 "p ( x to 8 pa/4 ,?us1 c>e = ! future & reg>d = ! pa/ w\ld be ! way 9 :i* ! mat]ial be+ w\ld exhibit ! unity ( ! !real body1 :i* is bo? 8 pa/1 8 pres5t1 & 8 future4 ,t is to say1 suppose ! !real body capable ( receiv+ an injury1 an injury 9 "o "p ( x w\ld correspond to an injury 9 a man's pa/2 an injury 9 ano!r "p1--t :i* ! mat]ial body 0 trav]s+1--w\ld correspond to an injury to ! man at ! pres5t mom5t2 injury to ! !real body at ano!r "p1 w\ld correspond to injury com+ to ! man at "s future "t4 ,& ! self-pres]vation ( ! !real body1 suppos+ x to h su* a motive1 w\ld 9 ! la/ case be ! motive ( reg>d+ 8 own future to ! man4 ,& inasmu* z ! man felt ! real unity ( 8 !real body1 & did n conf9e 8 att5tion to 8 mat]ial body1 :i* is absolutely disunit$ at e mom5t f its future & its pa/--inasmu* z he appreh5d$ 8 !real unity1 insomu* w\ld he c>e = 8 future welf>e1 & consid] x z equal 9 importance to 8 pres5t com=t4 ,! correspond5ce betwe5 emotion & physical fact w\ld be1 t ! emotion ( reg>d correspond$ to an undisc]n$ !real unity4 ,& !n also1 j z ! two tips ( two f+]s put d[n on a plane1 w\ld seem to a plane-be+ to be two completely diff]5t bodies1 n connect$ toge!r1 s "o & ! same !real body might appe> z two di/9ct mat]ial bodies1 & any reg>d betwe5 ! two w\ld correspond to an appreh5sion ( _! !real unity4 ,9 ! supposition ( an !real body1 x is n necess>y to keep to ! idea ( ! rigidity & p]man5ce ( ! grooves def9+ ! motion ( ! matt] :i*1 pass+ along1 exhibits ! mat]ial body4 ,! !real body may h a life ( its own1 relations ) o!r !real bodies1 & a life z full ( vicissitudes z t ( ! mat]ial body1 :i* 9 its total orbit expresses 9 ! movem5ts ( matt] "o phase 9 ! life ( ! !real body4 ,b !re are c]ta9 obvi\s consid]ations :i* prev5t any s]i\s dwell+ on ^! speculations--!y are only introduc$ "h 9 ord] to %[ h[ ! conception ( high] space l5ds itself to ! repres5tation ( c]ta9 indef9ite appreh5sions1--su* z t ( ! ess5tial unity ( ! race1--& af=ds a possible clue to correspond5ces betwe5 ! emotional & ! physical life4 ,! :ole "q ( our relation to ! !r has to be settl$4 ,t :i* we call ! !r is f> m probably ! surface ( a liquid1 & ! ph5om5a we obs]ve due to surface t5sions4 ,inde$1 ! physical que/ions conc]n u "h no?+ at all4 ,x is easy en\gh to make "s supposition :i* gives u a /&+ gr\nd to discipl9e our high]-space p]ception2 & :5 t is tra9$1 we %all turn r\nd & look at ! facts4 ,! conception :i* we %all =m ( ! univ]se w und\bt$ly be z diff]5t f our pres5t "o1 z ! ,cop]nican view diff]s f ! m pleasant view ( a wide immovable e>? b5ea? a va/ vault4 ,inde$1 any conception ( our place 9 ! univ]se w be m agreeable ?an ! ?\ght ( be+ on a sp9n+ ball1 kick$ into space )\t any means ( communication ) any o!r inhabitants ( ! univ]se4 ,,*apt] ,x4 ,,high] ,,space ,,& ,,high] ,,be+4 ,,p]ception ,,& ,,inspiration4 ,9 ! in/9ctive & s5se p]ception ( man & nature !re is all hidd51 :i* reflection aft]w>ds br+s into consci\sness4 ,we are consci\s ( some:at high] ?an ea* individual man :5 we look at m54 ,9 "s1 ?is consci\sness rea*es an extreme pit*1 & becomes a religi\s appreh5sion4 ,b 9 none is x o!rwise ?an in/9ctive4 ,! appreh5sion is suffici5tly def9ite to be c]ta94 ,b x is n expressible to u 9 t]ms ( ! reason4 ,n[1 ,i h %[n t by us+ ! conception ( high] space x is easy en\gh to make a supposition :i* %all %[ all mank9d z physical p>ts ( "o :ole4 ,our app>5t isolation z bodies f ea* o!r is by no means s necess>y to assume z x w\ld appe>4 ,b1 ( c\rse1 a supposition ( t k9d is ( no value1 except z %[+ a possibility4 ,if we came to exam9e into ! matt] closely1 we %\ld f9d a natural relation%ip :i* acc\nt$ = our consci\sness be+ limit$ z at pres5t x is4 ,! fir/ ?+ to be done1 is to organize our high]-space p]ception1 & !n look4 ,we _c tell :at ext]nal objects w bl5d toge!r into ! unity ( a high] be+4 ,b j z ! riddle ( ! two h&s becomes cle> to u f our fir/ inspection ( high] space1 s w !re gr[ be=e our eyes great] unities & great] surprises4 ,we h be5 subject to a limitation ( ! mo/ absurd "*4 ,let u op5 our eyes & see ! facts4 ,n[1 x requires "s tra9+ to op5 ! eyes4 ,= _m ye>s ,i work$ at ! subject )\t ! slighte/ success4 ,all 0 m]e =malism4 ,b by adopt+ ! simple/ means1 & by a m ?or\gh k ( space1 ! :ole fla%$ cle>4 ,space %apes c only be symbolical ( f\r-dim5sional %apes2 & if we d n deal ) space %apes directly1 b only treat !m by symbols on ! plane--z 9 analytical geometry--we are try+ to get a p]ception ( high] space ?r\gh symbols ( symbols1 & ! task is hopeless4 ,b a direct /udy ( space leads u to ! k ( high] space4 ,& ) ! k ( high] space !re come into our k5 b\ndless possibilities4 ,all ^? ?+s may be real1 :]e( sa9ts & philosoph]s h dream$4 ,look+ on ! fact ( life1 x has become cle> to ! human m9d1 t ju/ice1 tru?1 purity1 are to be s\ght--t !y are pr9ciples :i* x is well to s]ve4 ,& m5 h inv5t$ an ab/ract devotion to ^!1 & all comes toge!r 9 ! gr& b vague conception ( ,duty4 ,b all ^! ?\ghts are to ^? :i* spr+ up be=e u z ! %ad[ on a bank ( cl\ds ( a great m\nta9 is to ! m\nta9 itself4 ,on ! pil$-up cl\ds falls ! %ad[--va/1 impos+1 b d>k1 col\rless4 ,if ! behold] b turns1 he beholds ! m\nta9 itself1 t[]+ gr&ly ) v]dant p9es1 ! sn[l9e1 & ! awful peaks4 ,s all ^! conceptions are ! way 9 :i* n[1 ) vision conf9$1 we appreh5d ! great exi/5ces ( ! univ]se4 ,in/ead ( an ab/raction1 :at we h to s]ve is a reality1 to :i* ev5 our real ?+s are b %ad[s4 ,we are p>ts ( a great be+1 9 ^: s]vice1 & ) ^: love1 ! utmo/ dem&s ( duty are satisfi$4 ,h[ c x n be a /ruggle1 :5 ! claims ( righte\sness mean dim9i%$ life1--ev5 dea?1--to ! individual :o /rives8 ,& yet to a cle> & m rational view x w be se5 t 9 8 ext9ction & loss1 t :i* he loves1--t real be+ :i* is to him %ad[$ =? 9 ! pres5t exi/5ce ( wife & *ild1--t be+ lives m truly1 & 9 its life ^? he loves are 8 = ev]4 ,b1 ( c\rse1 !re are mi/akes 9 :at we consid] to be our duty1 z 9 ev]y?+ else2 & ?is is an additional reason = pursu+ ! que/ ( ?is reality4 ,= by ! rational obs]vance ( o!r mat]ial bodies ?an our own1 we come to ! conclusion t !re are o!r be+s ar\nd u l ourselves1 :om we appreh5d 9 virtue ( two processes--! "o simply a s5se "o ( obs]vation & reflection--! o!r a process ( direct appreh5sion4 ,n[1 if we did n g ?r\gh ! s5se process ( obs]vation1 we might1 x is true1 "k t !re 7 o!r human be+s ar\nd u 9 "s subtle way--9 "s mesm]ic feel+2 b we %\ld n h t organiz$ human life :i*1 deal+ ) ! ?+s ( ! _w1 gr[s into su* complicat$ =ms4 ,we %\ld = ev] be good-hum\r$ babies--a s5su\s1 affectionate k9d ( jelly-fi%4 ,& j s n[ ) ref]5ce to ! high intellig5ces by :om we are surr\nd$4 ,we feel !m1 b we d n realize !m4 ,to realize !m1 x w be necess>y to develop our p[] ( p]ception4 ,! p[] ( see+ ) our bodily eye is limit$ to ! ?ree-dim5sional section4 ,b ,i h %[n t ! inn] eye is n ?us limit$2 t we c organize our p[] ( see+ 9 high] space1 & t we c =m conceptions ( realities 9 ?is high] space1 j z we c 9 our ord9>y space4 ,& ?is af=ds ! gr\ndwork = ! p]ception & /udy ( ^! o!r be+s ?an man4 ,j z "s me*anical means are necess>y = ! appreh5sion ( our fell[s 9 space1 s a c]ta9 am\nt ( me*anical education is necess>y = ! p]ception ( high] be+s 9 high] space4 ,let u turn ! curr5t ( our ?\ght "r r\nd2 in/ead ( seek+ aft] ab/ractions1 & connect+ our obs]vations by ideas1 let u tra9 our s5se ( high] space & build up conceptions ( great] realities1 m absolute exi/5ces4 ,x is really a wa/e ( "t to write or read m g5]alities4 ,"h is ! gramm> ( ! k ( high] be+--let u le>n x1 n sp5d "t 9 speculat+ z to :i!r x w lead u4 ,yet "o ?+ m4 ,we are1 ) ref]5ce to ! high] ?+s ( life1 l bl9d & puzzl$ *ildr54 ,we "k t we are memb]s ( "o body1 limbs ( "o v9e2 b we _c disc]n1 except by in/9ct & feel+1 :at t body is1 :at ! v9e is4 ,if to "k x w\ld take away our feel+1 !n x 7 well nev] to "k x4 ,b full] k ( o!r human be+s does n take away our love = !m2 :at reason is !re !n to suppose t a k ( ! high] exi/5ces w\ld dead5 our feel+s8 ,& !n1 aga91 we ea* ( u h a feel+ t we ourselves h a "r to exi/4 ,we dem& our own p]petuation4 ,no man1 ,i believe1 is capable ( sacrific+ 8 life to any ab/ract idea2 9 all cases x is ! consci\sness ( contact ) "s be+ t enables him to make ! la/ human sacrifice4 ,& :at we c d by ?is /udy ( high] space1 is to make ?is consci\sness1 :i* has be5 res]v$ = a few1 ! prop]ty ( all4 ,d we n all feel t !re is a limit to our devotion to ab/ractions1 none to be+s :om we love4 ,& to love !m1 we mu/ "k !m4 ,!n1 j z our own individual life is empty & mean+less )\t ^? we love1 s ! life ( ! human race is empty & mean+less )\t a k ( ^? t surr\nd x4 ,& al?\gh to "s an inn] k ( ! on5ess ( all m5 is v\*saf$1 x rema9s to be demon/rat$ to ! _m4 ,! p]petual /ruggle betwe5 individual int]e/s & ! common good c only be solv$ by m]g+ bo? impulses 9 a love t[>ds "o be+ ^: life lies 9 ! fulfilm5t ( ea*4 ,& ?is se>*1 x seems to me1 af=ds ! ne$ful supplem5t to ! inquiries ( "o ) ^: ?\ght ,i h be5 v famili>1 & to :i* ,i return aga91 aft] hav+ ab&on$ x = ! purely mat]iali/ic views :i* seem =c$ ^u u by ! facts ( sci5ce4 ,all t he said seem$ to me unsupport$ by fact1 unrelat$ to :at we "k4 ,b :5 ,i f\nd t my k 0 m]ely an empty pret5ce1 t x 0 ! vanity ( be+ able to pr$ict & =etell t /ood to me 9 ! place ( an absolute appreh5sion ( fact--:5 all my intellectual possessions turn$ to no?+ness1 !n ,i 0 =c$ into t simple que/ = fact1 :i*1 :5 p]si/$ 9 & liv$ 91 op5s out to ! ?\ghts l a fl[] to ! life-giv+ sun4 ,x is inde$ a f> saf] c\rse1 to believe t :i* appeals to u z noble1 ?an simply to ask :at is true2 to take t :i* great m9ds h giv51 ?an to dem& t our puny ones %\ld be satisfi$4 ,b ,i suppose !re is "s good to "s "o 9 ! scepticism & /ruggle ( ^? :o _c foll[ 9 ! saf] c\rse4 ,! ?\ghts ( ! inquir] to :om ,i allude may r\ghly be /at$ ?us3-- ,he saw 9 human life ! work+ out ( a great process1 9 ! toil & /ra9 ( our human hi/ory he saw ! becom+ ( man4 ,!re is a defect :]eby we fall %ort ( ! true measure ( our be+1 & t defect is made good 9 ! c\rse ( hi/ory4 ,x is ow+ to t defect t we p]ceive evil2 & 9 ! p]ception ( evil & suff]+ lies our heal+1 = we %all be =c$ into t pa? at la/1 aft] try+ e o!r1 :i* is ! true "o4 ,& ?is1 ! hi/ory ( ! r$emption ( man1 is :at he saw 9 all ! sc5es ( life2 ea* mo/ trivial occurr5ce 0 great & significant 9 relation to ?is4 ,&1 fur!r1 he put =w>d a def9ite /atem5t ) reg>d to ?is defect1 ?is lack ( true be+1 = x lay1 he said1 9 ! self-c5tr$ness ( our emotions1 9 ! limitation ( !m to our bodily selves4 ,he look$ = a "t :51 driv5 f all ?\ghts ( our own pa9 or pleasure1 good or evil1 we %\ld say1 9 view ( ! mis]ies ( our fell[-creatures1 ,let me be anyh[1 use my body & my m9d 9 any way1 s t ,i s]ve4 ,& ?is1 x seems to me1 is ! true aspiration2 =1 j z a note ( music fl+s itself into ! m>* ( ! melody1 &1 los+ itself 9 x1 is us$ = x & lo/ z a sep>ate be+1 s we %\ld ?r[ ^! lives ( ours z freely into ! s]vice (--:om8 ,"h comes ! difficulty4 ,let x be grant$ t we %\ld h no self-rights1 limit our s]vice 9 no way1 /ill ! "q comes1 ,:at %all we s]ve8 ,x is f> happi] to h "s concrete object to :i* we are devot$1 or to be b\nd up 9 ! ceaseless r\nd ( active life1 :]e9 ea* day pres5ts s _m necessities t we h no room = *oice4 ,b besides & ap>t f all ^!1 !re comes to "s ! "q1 8,:at does x all mean80 ,to o!rs1 an unlovable & gloomy aspect is pres5t$1 :]e9 _! life seems to be b us$ z a mat]ial wor?less 9 itself & ungift$ ) any dignity or hon\r2 :ile to o!rs aga91 ) ! love ( ^? !y love1 comes a cessation ( all p]sonal int]e/ 9 life1 & a disappo9tm5t & feel+ ( valuelessness4 ,& 9 all ^! cases "s answ] is ne$$4 ,& "h human duty ceases4 ,we _c make objects to love4 ,we c make ma*9es & works ( art1 b no?+ :i* directly excites our love4 ,to give u t :i* r\ses our love1 is ! duty ( "o high] ?an ourselves4 ,& yet 9 "o respect we h a duty--we mu/ look4 ,:at good w\ld x be1 to surr\nd u ) objects ( lov+ int]e/1 if we bury our reg>ds 9 ourselves & w n see8 ,& does x n seem z if ) l[]$ eyelids1 till only ! ?9ne/ slit 0 op51 we gaz$ p]si/5tly1 n on :at is1 b on ! ?9ne/ conceivable section ( x8 ,let x be grant$ t our "r attitude is1 s to devote ourselves t !re is no "q z to :at we w d or :at we w n d1 b we are p]fectly ob$i5t s]vants4 ,! "q is1 ,:om are we to s]ve8 ,x _c be ea* individual1 = _! claims are conflict+1 & z (t5 z n !re is m ne$ ( a ma/] ?an ( a s]vant4 ,moreov]1 ! aspect ( our fell[s does n always excite love1 :i* is ! only possible induc] ( ! "r attitude ( s]vice4 ,if we d n love1 we c only s]ve = a self motive1 because x is 9 "s way good = ourselves4 ,?us x seems to me t we are r$uc$ to ?is3 our only duty is to look = t :i* x is giv5 u to love4 ,b ?is look+ is n m]e gaz+4 ,to "k1 we mu/ act4 ,let any "o try x4 ,he w f9d t unless he goes ?r\gh a s]ies ( actions correspond+ to 8 k1 he gets m]ely a !oretic & outside view ( any facts4 ,! way to "k is ?is3 ,get someh[ a means ( tell+ :at y\r p]ceptions w\ld be if y knew1 & act 9 accordance ) ^? p]ceptions4 ,?us1 ) reg>d to a fell[-creature1 if we knew him we %\ld feel :at 8 feel+s are4 ,let u !n le>n 8 feel+s1 & act z if we _h !m4 ,x is by ! practical "w ( satisfy+ 8 ne$s t we get to "k him4 ,!n1 may-be1 we love him2 or p]*ance x is said we may f9d t ?r\gh him we h be5 br\ght into contact ) "o great] ?an him4 ,?is is our duty--to "k--to "k1 n m]ely !oretically1 b practically2 & !n1 :5 we "k1 we h done our "p2 if !re is no?+1 we _c supply x4 ,all we h to d is to look = realities4 ,we mu/ n take ?is view ( education--t we are horribly press$ = "t1 & mu/ le>n1 someh[1 a knack ( say+ h[ ?+s mu/ be1 )\t look+ at !m4 ,b r1 we mu/ say t we h a long "t--all our lives1 9 :i* we w press facts clos] & clos] to our m9ds2 & we beg9 by le>n+ ! simple/4 ,!re is an idea 9 t home ( our inspiration--! fact t !re are c]ta9 me*anical processes by :i* m5 c acquire m]it4 ,?is is p]fectly true4 ,x is by me*anical processes t we become diff]5t2 & ! sci5ce ( education consi/s l>gely 9 sy/ematiz+ ^! processes4 ,!n1 j z space p]ceptions are necess>y = ! k ( our fell[-m51 & enable u to ent] into human relation%ips ) !m 9 all ! organiz$ v>iety ( civiliz$ life1 s x is necess>y to develop our p]ceptions ( high] space1 s t we c appreh5d ) our m9ds ! relation%ip :i* we h to be+s high] ?an ourselves1 & br+ our in/9ctive k into cle>] consci\sness4 ,x appe>s to me self-evid5t1 t 9 ! p>ticul> disposition ( any portion ( matt]1 t is1 9 any physical action1 !re c be nei!r "r nor wrong2 ! ?+ done is p]fectly indiff]5t4 ,at ! same "t1 x is only 9 ?+s done t we come into relation%ip ) ! be+s ab\t u & high] ?an u4 ,consequ5tly1 9 ! ?+s we d lies ! :ole importance ( our lives4 ,n[1 _m ( our impulses are directly signs ( a relation%ip 9 u to a be+ ( :i* we are n imm$iately consci\s4 ,! feel+ ( love1 = in/ance1 is always direct$ t[>ds a p>ticul> individual2 b by love man t5ds t[>ds ! pres]vation & improvem5t ( 8 race2 ?us 9 ! commone/ & mo/ univ]sal impulses lie 8 relations to high] be+s ?an ! individuals by :om he is surr\nd$4 ,n[1 along ) ^! impulses are _m in/9cts ( a modify+ t5d5cy2 &1 be+ altoge!r 9 ! d>k z to ! nature ( ! high] be+s to :om we are relat$1 x is difficult to say 9 :at ! s]vice ( ! high] be+s consi/s1 9 :at x does n4 ,! only way is1 z 9 e o!r pre-rational dep>tm5t ( life1 to take ! v]dict ( ^? ) ! mo/ insight & inspiration4 ,& any /riv+ aga9/ su* v]dicts1 & discont5t ) !m1 %\ld be turn$ into en]gy t[>ds f9d+ out exactly :at relation we h t[>ds ^! high] be+s by ! /udy ( ,space4 ,human life at pres5t is an art con/ruct$ 9 its regulations & rules on ! inspirations ( ^? :o love ! undisc]n$ high] be+s1 ( :i* we are a "p4 ,!y love ^! high] be+s1 & "k _! s]vice4 ,b our p]ceptions are co>s]2 & x is only by lab\r & toil t we %all be br\ght also to see1 & !n lose ! re/ra9ts t n[ are necess>y to u 9 ! fulness ( love4 ,exactly :at relation%ip !re is t[>ds u on ! "p ( ^! high] be+s we _c say 9 ! lea/4 ,we _c ev5 say :e!r !re is m ?an humanity be=e ! highe/2 & any conception :i* we =m n[ mu/ use ! human drama z its only possible mode ( pres5tation4 ,b t !re is su* a relation seems cle>2 & ! ludicr\s mann]1 9 :i* our p]ceptions h be5 limit$1 is a suffici5t explanation ( :y !y h n be5 sci5tifically appreh5d$4 ,! mode1 9 :i* an appreh5sion ( ^! high] be+s or be+ is at pres5t secur$1 is z foll[s2 & x be>s a /rik+ analogy to ! mode by :i* ! self is cut out ( a block ( cubes4 ,:5 we /udy a block ( cubes1 we fir/ ( all le>n x1 by />t+ f a p>ticul> cube1 & le>n+ h[ all ! o!rs come ) reg>d to t4 ,all ! o!rs are "r or left1 up or d[n1 ne> or f>1 ) reg>d to t p>ticul> cube4 ,& ! l9e ( cubes />t+ f ?is fir/ "o1 :i* we take z ! direction 9 :i* we look1 is1 z x 71 an axis ab\t :i* ! re/ ( ! cubes are gr\p$4 ,we le>n ! block ) reg>d to ?is axis1 s t we c m5tally conceive ! disposition ( e cube z x comes reg>d$ f "o po9t ( view4 ,next we suppose ourselves to be 9 ano!r cube at ! extremity ( ano!r axis2 &1 look+ f ?is axis1 we le>n ! aspects ( all ! cubes1 & s on4 ,?us we impress on ! feel+ :at ! block ( cubes is l f e axis4 ,9 ?is way we get a k ( ! block ( cubes4 ,n[1 to get a k ( humanity1 we mu/ feel ) _m individuals4 ,ea* individual is an axis z x 71 & we mu/ reg>d human be+s f _m diff]5t axes4 ,& z1 9 le>n+ ! block ( cubes1 muscul> action1 z us$ 9 putt+ up ! block ( cubes1 is ! means by :i* we impress on ! feel+ ! diff]5t views ( ! block2 s1 ) reg>d to humanity1 x is by act+ ) reg>d to ! view ( ea* individual t a k is obta9$4 ,t is to say1 t1 besides sympa?iz+ ) ea* individual1 we mu/ act ) reg>d to 8 view2 & act+ s1 we %all feel 8 view1 & ?us get to "k humanity f m ?an "o axis4 ,?us !re spr+s up a feel+ ( humanity1 & ( m4 ,^? :o feel sup]ficially ) a great _m p1 are l ^? le>n]s :o h a slight acqua9tance ) a block ( cubes f _m po9ts ( view4 ,^? :o h "s deep atta*m5ts1 are l ^? :o "k !m well f "o or two po9ts ( view4 ,?us !re are two def9ite pa?s--"o by :i* ! in/9ctive feel+ is call$ out & develop$1 ! o!r by :i* we ga9 ! faculty ( rationally appreh5d+ & le>n+ ! high] be+s4 ,9 ! "o way x is by ! ex]cise ( a sympa!tic & active life2 9 ! o!r1 by ! /udy ( high] space4 ,bo? %\ld be foll[$2 b ! latt] way is m accessible to ^? :o are n good4 ,= we at any rate h ! indu/ry to g ?r\gh me*anical op]ations1 & "k t we ne$ some?+4 ,& aft] all1 p]haps1 ! diff]5ce betwe5 ! good & ! re/ ( u1 lies r 9 ! =m] be+ aw>e4 ,!re is some?+ outside !m :i* draws !m to x1 :i* !y see :ile we d n4 ,!re is no reason1 h[ev]1 :y ?is k %\ld n become demon/rable fact4 ,surely1 x is only by becom+ demon/rable fact t ! errors :i* h be5 necess>ily introduc$ into x by human weakness w fall away f x4 ,! rational k w n replace feel+1 b w =m ! vehicle by :i* ! facts w be pres5t$ to our consci\sness4 ,j z we le>n to "k our fell[s by wat*+ _! de$s1--b x is some?+ beyond ! m]e p[] ( obs]v+ !m t makes u reg>d !m1--s ! high] exi/5ces ne$ to be kn[n2 &1 :5 kn[n1 !n !re is a *ance t 9 ! dep?s ( our nature !y w awak5 feel+s t[>ds !m l ! natural response ( "o human be+ to ano!r4 ,& :5 we reflect on :at surr\nds u1 :5 we ?9k t ! beauty ( fruit & fl[]1 ! blue dep?s ( ! sky1 ! maje/y ( rock & ocean1--all ^! are b ! *ance & arbitr>y view :i* we h ( true be+1--!n we c imag9e some:at ( ! glories t await our com+4 ,h[ set out 9 exquisite lovel9ess are all ! budd+ trees & h$g][s on a spr+ day--f "h1 :]e !y almo/ s+ to u 9 _! ne>ness1 to :]e1 9 ! di/ance1 !y /& up delicately di/ant & di/9ct 9 ! ame?y/ ocean ( ! air6 ,& !re1 quiet & /ately1 revolve ! sl[ mov+ sun & ! />s ( ! night4 ,all ^! are ! fragm5t>y views :i* we h ( great be+s to :om we are relat$1 to :om we are l9k$1 did we b realize x1 by a bond ( love & s]vice 9 close connexions ( mutual helpfulness4 ,j z "h & !re on ! face ( a woman sits ! div9e _s ( beauty1 s t all _c b love :o look--s1 pres5t+ itself to u 9 all ?is m+l$ sc5e ( air & ocean1 pla9 & m\nta91 is a be+ ( su* lovel9ess t1 did we b "k ) "o accord 9 "o /ream1 all our he>ts w\ld be c>ri$ 9 a p]fect & will+ s]vice4 ,x is n t we ne$ to be made diff]5t2 we h b to look & gaze1 & see t c5tre :]eunto ) joyful love all creat$ be+s move4 ,b n ) ef=tless wond] w our days be fill$1 b 9 toil & /rong ex]tion2 =1 j z n[ we all lab\r & /rive = an object1 our s]vice is b\nd up ) ?+s :i* we d--s !n we f9d no re/ f lab\r1 b ! s5se ( solitude & isolation is gone4 ,! bonds ( bro!rhood ) our fell[-m5 gr[ /rong1 = we "k "o common purpose4 ,& ?r\gh ! exquisite face ( nature %9es ! spiritual light t gives u a great & nev]-fail+ comrade4 ,our task is a simple "o--to lift f our m9d t veil :i* someh[ has fall5 on u1 to take t curi\s limitation f our p]ception1 :i* at pres5t is only transc5d$ by inspiration4 ,& ! means to d x is by ?r[+ aside our reason--by giv+ up ! idea t :at we ?9k or are has any value4 ,we too (t5 sit z judges ( nature1 :5 all we c be are h] humble le>n]s4 ,we h b to dr9k 9 ( ! inexhau/ible fulness ( be+1 press+ x close into our m9ds1 & lett+ our pride ( be+ able to =etell vani% into du/4 ,!re is a curi\s passage 9 ! works ( ,immanuel ,kant1#a 9 :i* he %[s t space mu/ be 9 ! m9d be=e we c obs]ve ?+s 9 space4 8,=10 he says1 8s9ce ev]y?+ we conceive is conceiv$ z be+ 9 space1 !re is no?+ :i* comes be=e our m9ds f :i* ! idea ( space c be d]iv$2 x is equally pres5t 9 ! mo/ rudim5t>y p]ception & ! mo/ complete40 ,h5ce he says t space belongs to ! p]ceiv+ s\l itself4 ,)\t go+ into ?is argum5t to ab/ract regions1 x has a great am\nt ( practical tru?4 ,all our p]ceptions are ( ?+s 9 space2 we _c ?9k ( any detail1 h[ev] limit$ or isolat$1 :i* is n 9 space4 #a ,! idea ( space c 8ni*t aus d5 ,v]hltniss5 d] uss]5 ,ers*e9ung dur* ,erfahrung erborgt se91 sond]n diese uss]e ,erfahrung i/ nur dur* g$a*te ,vor/ellung all]]/ mgli*40 ,h5ce1 9 ord] to ex]cise our p]ceptive p[]s1 x is well to h prep>$ be=eh& a /rong appreh5sion ( space & space relations4 ,& s1 z we pass on1 is x n easily conceivable t1 ) our p[] ( high] space p]ception s rudim5t>y & s unorganiz$1 we %\ld f9d x impossible to p]ceive high] exi/5ces8 ,t mode ( p]ception :i* x belongs to u to ex]cise is want+4 ,:at wond]1 !n1 t we _c see ! objects :i* are ready1 7 b our own "p done8 ,?9k h[ mu* has come into human life ?r\gh ex]cis+ ! p[] ( ! ?ree-dim5sional space p]ception1 & we c =m "s measure1 9 a fa9t way1 ( :at is 9 /ore = u4 ,!re is a c]ta9 reluctance 9 u 9 br++ any?+1 :i* be=e has be5 a matt] ( feel+1 )9 ! doma9 ( consci\s reason4 ,we d n l to expla9 :y ! grass is gre51 fl[]s bright1 &1 above all1 :y we h ! feel+s :i* we pass ?r\gh4 ,b ?is objection & in/9ctive reluctance is *iefly d]iv$ f ! fact t expla9+ has got to mean expla9+ away4 ,we s (t5 ?9k t a ?+ is expla9$1 :5 x c be %[n simply to be ano!r =m ( some?+ :i* we "k already4 ,&1 9 fact1 ! we>i$ m9d (t5 does long to h a ph5om5on %[n to be m]ely a d$uction f c]ta9 kn[n laws4 ,b explanation prop] is n ( ?is k9d2 x is introduc+ into ! m9d ! new conception :i* is indicat$ by ! ph5om5on already pres5t4 ,nature consi/s ( _m entities t[>ds ! appreh5sion ( :i* we /rive4 ,if = a "t we break d[n ! b\nds :i* we h set up1 & unify va/ fields ( obs]vation "u "o common law1 x is t ! conceptions we =m$ at fir/ are inadequate1 & mu/ be replac$ by great] ones4 ,b x is always ! case1 t1 to und]/& nature1 a conception mu/ be =m$ 9 ! m9d4 ,?is process ( gr[? 9 ! m5tal hi/ory is hidd52 b x is ! really important "o4 ,! new conception satisfies m facts ?an ! old ones1 is tru] ph5om5ally2 & ! argum5ts = x are its simplicity1 its p[] ( acc\nt+ = _m facts4 ,b ! conception has to be =m$ fir/4 ,& ! real hi/ory ( advance lies 9 ! gr[? ( ! new conceptions :i* e n[ & !n come to light4 ,:5 ! wea!r-wise savage look$ at ! sky at night1 he saw _m specks ( yell[ light1 l fire-flies1 spr9kl$ amid/ :iti% fleece2 & sometimes ! fleece rema9$1 ! fire-spots w5t1 & ra9 came2 sometimes ! fire-spots rema9$1 & ! night 0 f9e4 ,he did n see t ! fire-po9ts 7 ev] ! same1 ! cl\ds diff]5t2 b by feel+ dimly1 he knew en\gh = 8 purpose4 ,b :5 ! ?9k+ m9d turn$ itself on ^! appe>ances1 !re sprang up1--n all at once1 b gradually1--! k ( ! sublime exi/5ces ( ! di/ant heav5s1 & all ! lore ( ! m>vell\s =ms ( wat]1 ( air1 & ! movem5ts ( ! e>?4 ,surely ^! realities1 9 :i* lies a weal? ( embodi$ poetry1 are well wor? ! delight$ s5su\s appreh5sion ( ! savage z he gaz$4 ,p]haps some?+ is lo/1 b 9 ! realities1 ( :i* we "k1 !re is comp5sation4 ,& s1 :5 we le>n to und]/& ! mean+ ( ^! my/]i\s *anges1 ?is c\rse ( natural ev5ts1 we %all f9d 9 ! great] realities among/ :i* we move a fair ex*ange = ! in/9ctive rev]5ce1 :i* !y n[ awak5 9 u4 ,9 ?is book ! task is tak5 up ( =m+ ! mo/ simple & elem5t>y ( ! great conceptions t are ab\t u4 ,9 ! works ( ! poets1 & /ill m 9 ! pages ( religi\s ?9k]s1 lies an untold weal? ( conception1 ! organization ( :i* 9 our e-day intellectual life is ! "w ( ! practical educator4 ,b none is capable ( su* simple demon/ration & absolute pres5tation z ?is ( high] space1 & none s imm$iately op5s our eyes to see ! _w z a diff]5t place4 ,&1 inde$1 x is v in/ructive2 = :5 ! new conception is =m$1 x is f\nd to be q simple & natural4 ,we ask ourselves :at we h ga9$2 & we answ]3 ,no?+2 we h simply remov$ an obvi\s limitation4 ,& ?is is univ]sally true2 x is n t we mu/ rise to ! high] by a long & labori\s process4 ,we may h a long & labori\s process to g ?r\gh1 b1 :5 we f9d ! high]1 x is ?is3 we discov] our true selves1 our ess5tial be+1 ! fact ( our lives4 ,9 ?is case1 we pass f ! ridicul\s limitation1 to :i* our eyes & h&s seem to be subject1 ( act+ 9 a m]e section ( space1 to ! full] k & feel+ ( space z x is4 ,h[ d we pass to ?is tru] intellectual life8 ,simply by obs]v+1 by lay+ aside our intellectual p[]s1 & by look+ at :at is4 ,we take t :i* is easie/ to obs]ve1 n t :i* is easie/ to def9e2 we take t :i* is ! mo/ def9itely limit$ real ?+1 & use x z our t\*/one :]eby to explore nature4 ,z x seems to me1 ,kant made ! great & fundam5tal /atem5t 9 philosophy :5 he explod$ all previ\s sy/ems1 & all physics 7 reft f (f ! p]ceiv+ s\l4 ,b :at he did once & = all1 0 too great to be a practical means ( intellectual "w4 ,! dynamic =m ( 8 absolute insight _h to be f\nd2 & x is 9 o!r works t ! practical in/ances ( ! ,kantian me?od are to be f\nd4 ,=1 in/ead ( look+ at ! l>ge f\ndations ( k1 ! ultimate pr9ciples ( exp]i5ce1 late writ]s turn$ to ! details ( exp]i5ce1 & te/$ e ph5om5on1 n ) ! "q1 ,:at is ?is8 b ) ! "q1 8,:at makes me p]ceive ?us80 ,& surely ! "q1 z s put1 is m capable ( an answ]2 = x is only ! p]cipi5t1 z a subject ( ?\ght1 ab\t :i* we c speak4 ,! absolute s\l1 s9ce x is ! ?9k]1 c nev] be ! subject ( ?\ght2 b1 z physically condition$1 x c be ?\ght ab\t4 ,?us we c nev]1 )\t committ+ a ludicr\s error1 ?9k ( ! m9d ( man except z a mat]ial organ ( "s k9d2 & ! pa? ( discov]y lies 9 inve/igat+ :at ! devi\s l9e ( 8 ?\ght hi/ory is due to1 :i* w9ds betwe5 two doma9s ( physics--! unkn[n conditions :i* affect ! p]ceiv]1 ! p>tially kn[n physics :i* con/itute :at we call ! ext]nal _w4 ,x is a pity to sp5d "t ov] ^! reflections2 if !y d n seem tame & poor comp>$ to ! practical appreh5sion :i* comes ( work+ ) ! models1 !n !re is no?+ 9 ! :ole subject4 ,if 9 ! little real objects :i* ! read] has to h&le & obs]ve does n lie to him a poetry ( a high] k9d ?an any express$ ?\ght1 !n all ^! words are n only useless1 b false4 ,if1 on ! o!r h&1 !re is true "w to be done ) !m1 !n ^! sugge/ions w be felt to be b mean & insuffici5t appreh5sions4 ,=1 9 ! simple/ appreh5sion ( a high] space lies a k ( a reality :i* is1 to ! realities we "k1 z _s is to matt]2 & yet to ?is new vision all our solid facts & mat]ial conditions are b z a %ad[ is to t :i* ca/s x4 ,9 ! awak5+ light ( ?is new appreh5sion1 ! flimsy _w quiv]s & %akes1 rigid solids fl[ & m+le1 all our mat]ial limitations turn into graci\sness1 & ! new field ( possibility waits = u to look & behold4 ,,*apt] ,,xi4 ,,space ,,! ,,sci5tific ,,basis ,,( ,,altruism ,,& ,,religion4 ,! read] w d\btless ask = "s def9ite result correspond+ to ^! words--some?+ n ( ! nature ( an hypo!sis or a might-be4 ,& 9 t ,i c only satisfy him aft] my own p[]s4 ,my only /r5g? is 9 detail & pati5ce2 & if he w g ?r\gh ! practical "p ( ! book1 x w assur$ly dawn ^u him t "h is ! beg9n+ ( an answ] to 8 reque/4 ,i only /udy ! blocks & /ones ( ! high] life4 ,b "h !y are def9ite en\gh4 ,& ! m eag] he is = p]sonal & spiritual tru?1 ! m eag]ly d ,i urge him to take up ! practical "w1 = ! true good comes to u ?r\gh ^? :o1 aspir+ greatly1 /ill submit _! aspirations to fact1 & :o1 desir+ to appreh5d _s1 /ill are will+ to manipulate matt]4 ,! p>ticul> problem at :i* ,i h work$ = m ?an t5 ye>s1 has be5 completely solv$4 ,x is possible = ! m9d to acquire a conception ( high] space z adequate z t ( our ?ree-dim5sional space1 & to use x 9 ! same mann]4 ,!re are two di/9ct ways ( /udy+ space--our famili> space at pres5t 9 use4 ,"o is t ( ! analy/1 :o treats space relations by 8 algebra1 & discov]s m>vell\s relations4 ,! o!r is t ( ! obs]v] or me*anician1 :o /udies ! %apes ( ?+s 9 space directly4 ,a practical design] ( ma*9es w\ld n f9d ! k ( geometrical analysis ( imm$iate help to him2 & an arti/ or draughtsman /ill less s4 ,n[1 my inquiry 01 :e!r x 0 possible to get ! same p[] ( conception ( f\r-dim5sional space1 z ! design] & draughtsman h ( ?ree-dim5sional space4 ,x is possible4 ,& ) ?is p[] x is possible = u to design ma*9es 9 high] space1 & to conceive objects 9 ?is space1 j z a draughtsman or arti/ does4 ,analytical skill is n ( mu* use 9 design+ a /atue or inv5t+ a ma*9e1 or 9 appreciat+ ! detail ( ei!r a "w ( art or a me*anical contrivance4 ,& hi!rto ! /udy ( f\r-dim5sional space has be5 conduct$ by analysis4 ,"h1 = ! fir/ "t1 ! fact ( ! p[] ( conception ( f\r-dim5sional space is demon/rat$1 & ! means ( educat+ x are giv54 ,& ,i propose a complete sy/em ( "w1 ( :i* ! volume on f\r space#b is ! fir/ in/alm5t4 #b 8,sci5ce ,romance10 ,no4 ,i41 by ,c4 ,h4 ,h9ton4 ,publi%$ by ,swan ,sonn5s*e9 ,co4 ,i %all br+ =w>d a complete sy/em ( f\r-dim5sional ?\ght--me*anics1 sci5ce1 & art4 ,! necess>y condition is1 t ! m9d acquire ! p[] ( us+ f\r-dim5sional space z x n[ does ?ree-dim5sional4 ,& !re is ano!r condition :i* is no less important4 ,we c nev] see1 = in/ance1 f\r-dim5sional pictures ) our bodily eyes1 b we c ) our m5tal & inn] eye4 ,! condition is1 t we %\ld acquire ! p[] ( m5tally c>ry+ a great numb] ( details4 ,if1 = in/ance1 we c\ld ?9k ( ! human body "r d[n to e m9ute "p 9 its "r position1 & conceive its aspect1 we %\ld h a f\r-dim5sional picture :i* is a solid /ructure4 ,n[1 to d ?is1 we mu/ =m ! habit ( m5tal pa9t+1 t is1 ( putt+ def9ite col\rs 9 def9ite positions1 n ) our h&s on pap]1 b ) our m9ds 9 ?\ght1 s t we c recall1 alt]1 & view complicat$ arrangem5ts ( col\r exi/+ 9 ?\ght ) ! same ease ) :i* we c pa9t on canvas4 ,?is is simply an affair ( indu/ry2 & ! m5tal p[] lat5t 9 u 9 ?is direction is simply m>vell\s4 ,9 any picture1 a /roke ( ! bru% put on )\t ?\ght is valueless4 ,! arti/ is n consci\s ( ! ?\ght process he goes ?r\gh4 ,= our purpose x is necess>y t ! manipulation ( col\r & =m :i* ! arti/ goes ?r\gh unconsci\sly1 %\ld become a consci\s p[]1 & t1 at :atev] sacrifice ( imm$iate beauty1 ! art ( m5tal pa9t+ %\ld exi/ beside our m unconsci\s art4 ,all t ,i mean is ?is--t 9 ! c\rse ( our campaign x is necess>y to take up ! task ( le>n+ pictures by he>t1 s t1 j z an arti/ ?9ks ov] ! outl9es ( a figure he wants to draw1 s we ?9k ov] ea* /roke 9 our pictures4 ,! means by :i* ?is c be done w be giv5 9 a future volume4 ,we ?r[ ourselves on an ent]prise 9 :i* we h to leave altoge!r ! direct pres5tation to ! s5ses4 ,we mu/ acquire a s5se-p]ception & memory ( s ke5 & accurate a k9d t we c build up m5tal pictures ( great] complexity ?an any :i* we c see4 ,we h a va/ "w ( organization1 b x is m]ely organization4 ,! p[] really exi/s & %[s itself :5 x is look$ =4 ,mu* fault may be f\nd ) ! sy/em ( organization :i* ,i h adopt$1 b x is ! survivor ( _m attempts2 & al?\gh ,i c\ld bett] x 9 p>ts1 /ill ,i ?9k x is be/ to use x until1 ! full importance ( ! subject be+ realiz$1 x w be ! lifework ( m5 ( sci5ce to reorganize ! me?ods4 ,! "o ?+ on :i* ,i mu/ insi/ is ?is--t k is ( no value1 x does n exi/ unless x comes into ! m9d4 ,to "k t a ?+ mu/ be is no use at all4 ,x mu/ be cle>ly realiz$1 & 9 detail z x is1 be=e x c be us$4 ,a :ole _w swims be=e u1 ! appreh5sion ( :i* simply dem&s a pati5t cultivation ( our p[]s2 & !n1 :5 ! faculty is =m$1 we %all recognize :at ! univ]se 9 :i* we are is l4 ,we %all le>n ab\t ourselves & pass into a new doma94 ,& ,i w\ld speak to "s m9ds :o1 l myself1 %>e to a l>ge ext5t ! feel+ ( unsettl$ness & unfix$ness ( our pres5t k4 ,religion has suff]$ 9 "s respects f ! inaccuracy ( its /atem5ts2 & x is n always se5 t x consi/s ( two p>ts--"o a set ( rules z to ! managem5t ( our relations to ! physical _w ab\t u1 & to our own bodies2 ano!r1 a set ( rules z to our relation%ip to be+s high] ?an ourselves4 ,n[1 on ! =m] ( ^! subjects1 on physical facts1 on ! laws ( heal?1 sci5ce has a fair /&+ gr\nd ( criticism1 & c correct ! religi\s doctr9es 9 _m important respects4 ,b on ! o!r "p ( ! subject matt]1 z to our relation%ip to be+s high] ?an ourselves1 sci5ce has n yet ! mat]ials = judg+4 ,! proposition :i* und]lies ?is book is1 t we %\ld beg9 to acquire ! faculties = judg+4 ,to judge1 we mu/ fir/ appreciate2 & h[ f> we are f appreciat+ ) sci5ce ! fundam5tal religi\s doctr9es ,i leave to any "o to judge4 ,!re is absolutely no sci5tific basis = morality1 us+ morality 9 ! high] s5se ( o!r ?an a code ( rules to promote ! greate/ physical & m5tal heal? & gr[? ( a human be+4 ,sci5ce does n give u any in=mation :i* is n equally acceptable to ! mo/ selfi% & mo/ g5]\s man2 x simply tells him ( means by :i* he may atta9 8 own ends1 x does n %[ him ends4 ,! prosecution ( sci5ce is an ennobl+ pursuit2 b x is ( sci5tific k t ,i am n[ speak+ 9 itself4 ,we h no sci5tific k ( any exi/5ces high] ?an ourselves--at lea/1 n recogniz$ z high]4 ,b we h abundant k ( ! actions ( be+s less develop$ ?an ourselves1 f ! /rik+ unanimity ) :i* all inorganic be+s t5d to move t[>ds ! e>?'s c5tre1 to ! almo/ equally uni=m modes ( response 9 elem5t>y organiz$ matt] to diff]5t /imuli4 ,! "q may be put3 ,9 :at way d we come into contact ) ^! high] be+s at pres5t8 ,& evid5tly ! answ] is1 ,9 ^? ways 9 :i* we t5d to =m organic unions--unions 9 :i* ! activities ( individuals coalesce 9 a liv+ way4 ,! coh]5ce ( a milit>y empire or ( a subjugat$ population1 pres5t+ no natural nucleus ( gr[?1 is n "o ?r\gh :i* we %\ld hope to gr[ into direct contact ) our high] de/9ies4 ,b 9 fri5d%ip1 9 volunt>y associations1 & above all1 9 ! family1 we t5d t[>ds our great] life4 ,& x seems t ! in/9cts ( wom5 are mu* m relative to ?is1 ! mo/ fundam5tal & important side ( life1 ?an are ^? ( m54 ,9 fact1 until we "k1 ! l9e ( advance _h bett] be left to ! feel+ ( wom51 z !y organize ! home & ! social life spread+ out !refrom4 ,x is difficult1 p]haps1 = a man to be /ill & p]ceive2 b if he is s1 he f9ds t :at1 :5 ?w>t$1 are mean+less caprices & empty emotionalities1 are1 on ! "p ( woman1 :5 all[$ to gr[ freely & un*eck$1 ! fir/ beg9n+s ( a new life--! %ad[y filam5ts1 z x 71 by :i* an organism beg9s to coagulate toge!r f ! m$ium 9 :i* x makes its appe>ance4 ,9 v _m respects m5 h to make ! conditions1 & !n le>n to recognize4 ,h[ c we see ! high] be+s ab\t u1 :5 we _c ev5 conceive ! simple/ high] %apes8 ,we may talk ab\t space1 & use big words1 b1 aft] all1 ! pref]able way ( putt+ our ef=ts is ?is3 let u look fir/ at ! simple/ facts ( high] exi/5ce1 & !n1 :5 we h le>nt to realize ^!1 ,we %all be able to see :at ! _w pres5ts4 ,& !n1 also1 light w be ?r[n on ! con/itu5t organisms ( our own bodies1 :5 we see 9 ! ?or\gh developm5t ( our social life a relation betwe5 ourselves & a l>g] organism simil> to t :i* exi/s betwe5 u & ! m9ute con/itu5ts ( our frame4 ,! problem1 z x comes to me1 is ?is3 x is cle>ly demon/rat$ t self-reg>d is to be put on "o side--& self-reg>d 9 e respect--n only %\ld ?+s pa9ful & ardu\s be done1 b ?+s degrad+ & vile1 s t !y s]ve4 ,i am to sign any li/ ( any numb] ( de$s :i* ! mo/ f\l imag9ation c sugge/1 z ?+s :i* ,i w\ld d did ! occasion come :5 ,i c\ld b5efit ano!r by do+ !m2 &1 9 fact1 !re is to be no *>act]i/ic 9 any action :i* ,i w\ld %r9k f did ! occasion come :5 x pres5t$ itself to be done = ano!r's sake4 ,& ,i believe t ! s\l is absolutely un/a9$ by ! action1 provid$ ! reg>d is = ano!r4 ,b ?is is1 9 tru?1 a dang]\s doctr9e2 at "o ,sweep x puts away all absolute comm&m5ts1 all absolute v]dicts ( "r ab\t ?+s1 & leaves ! ag5t to 8 own judgm5t4 ,x is a k9d ( rule ( life :i* requires mo/ absolute op5ness1 & dem&s t society %\ld frame sev]e & insup]able regulations2 = o!rwise1 ) ! motives ( ! individual ?us lib]at$ f absolute law1 endless v>ieties ( conduct w\ld spr+ =?1 & ! wisdom ( individual m5 is h>dly en\gh to ju/ify _! irresponsible action4 ,/ill1 x does seem t1 z an ideal1 ! absolute abs5ce ( self-reg>d is to be aim$ at4 ,) a /rong religi\s basis1 ?is w\ld "w no h>m1 = ! rules ( life1 z laid d[n by religions1 w\ld suffice4 ,b !re are _m :o d n accept ^! rules z any absolute indication ( ! w ( ,god1 b only z ! regulations ( good m51 :i* h a claim to respect & no?+ m4 ,& ?us x seems to me t altruism--?or\ghgo+ altruism--h&s ov] ^? :o reg>d x z an ideal1 & :o are also ( a sceptical turn ( m9d1 to ! mo/ absolute unfix$ness ( !ory1 &1 v possibly1 to ! greate/ errors 9 life4 ,& "h we come to ! po9t :]e ! /udy ( space becomes s important4 ,= if ?is rule ( altruism is ! "r "o1 if x appeals ) a great invitation to u1 we ne$ n !re=e try x ) less precaution ?an we %\ld use 9 o!r affairs ( inf9itely less importance4 ,:5 we want to "k if a plank w be>1 we entru/ x ) a diff]5t load f t ( a human body4 ,& if ?is law ( altruism is ! true "o1 let u try x :]e failure w n mean ! ru9 ( human be+s4 ,n[1 9 k1 pure altruism means s to bury ! m9d 9 ! ?+ kn[n t all p>ticul> relations ( "o's self pass away4 ,! altrui/ic k ( ! heav5s w\ld be1 to feel t ! />s 7 va/ bodies1 & t ,i am mov+ rapidly4 ,x w\ld be1 to "k ?is1 n z a matt] ( !ory1 b z a matt] ( habitual feel+4 ,:e!r ?is is possible1 ,i d n "k2 b a some:at simil> attempt c be made ) mu* simpl] means4 ,9 a diff]5t place ,i h describ$ ! process ( acquir+ an altrui/ic k ( a block ( cubes2 & ! results ( ! labori\s processes involv$ are well wor? ! tr\ble4 ,= z a cle>ly demon/rable fact ?is comes be=e "o4 ,to acquire an absolute k ( a block ( cubes1 s t all self relations are ca/ out1 means t "o has to take ! view ( a high] be+4 ,x sudd5ly comes be=e "o1 t ! p>ticul> relations :i* are s fix$ & important1 & seem s absolutely sure :5 "o beg9s ! process ( le>n+1 are by no means absolute facts1 b m>ks ( a s+ul> limitation1 almo/ a degradation1 on "o's own "p4 ,9 ! det]m9$ attempt to "k ! mo/ insignificant object p]fectly & ?or\ghly1 !re fla%es be=e "o's eyes an exi/5ce inf9itely high] ?an "o's own4 ,& ) t vision !re comes1--,i d n speak f my own exp]i5ce only1--a conviction t our exi/5ce also is n :at we suppose--t ?is bodily self ( ours is b a limit too4 ,& ! "q ( altruism1 z aga9/ self-reg>d1 seems almo/ to vani%1 = by altruism we come to "k :at we truly are4 8,:at we truly are10 ,i d n mean ap>t f space & matt]1 b :at we really are z be+s hav+ a space exi/5ce2 = our way ( ?9k+ ab\t exi/5ce is to conceive x z ! relations ( bodies 9 space4 ,to ?9k is to conceive realities 9 space4 ,j z1 to explore ! di/ant />s ( ! heav5s1 a p>ticul> mat]ial arrangem5t is necess>y :i* we call a telescope1 s to explore ! nature ( ! be+s :o are high] ?an u1 a m5tal arrangem5t is necess>y4 ,we mu/ prep>e our p[] ( ?9k+ z we prep>e a m ext5d$ p[] ( look+4 ,we want a /ructure develop$ inside ! skull = ! "o purpose1 :ile an ext]ior telescope w d = ! o!r4 ,& ?us x seems t ! difficulties :i* we fir/ appreh5d$ fall away4 ,to u1 look+ ) half-bl9d$ eyes at m]ely our own little slice ( exi/5ce1 our filmy all1 x seem$ t altruism meant disord]1 vag>y1 dang]4 ,b :5 we put x into practice 9 k1 we f9d t x means ! direct revelation ( a high] be+ & a call to u to p>ticipate ourselves too 9 a high] life--nay1 a consci\sness comes t we are high] ?an we "k4 ,& s ) our moral life z ) our intellectual life4 ,is x n ! case t ^?1 :o truly accept ! rule ( altruism1 le>n life 9 new dang]\s ways8 ,x is true t we mu/ give up ! precepts ( religion z be+ ! w ( ,god2 b !n we %all le>n t ! w ( ,god %[s itself p>tly 9 ! religi\s precepts1 & comes to be m fully & m pla9ly kn[n z an inw>d _s4 ,& t difficulty1 too1 ab\t :at we may d & :at we may n1 vani%es also4 ,=1 if x is ! same ab\t our fell[-creatures z x is ab\t ! block ( cubes1 :5 we h ?r[n out ! self-reg>d f our relation%ip to !m1 we %all feel t[>ds !m z a high] be+ ?an man feels t[>ds !m1 we %all feel t[>ds !m z !y are 9 _! true selves1 n 9 _! outw>d =ms1 b z et]nal lov+ spirits4 ,& !n ^? in/9cts :i* humanity feels ) a secret impulse to be sacr$ & high] ?an any tempor>y good w be ju/ifi$--or fulfill$4 ,!re are two t5d5cies--"o t[>ds ! direct cultivation ( ! religi\s p]ceptions1 ! o!r to r$uc+ ev]y?+ to reason4 ,x w be b j = ! expon5ts ( ! latt] t5d5cy to look at ! :ole univ]se1 n ! m]e section ( x :i* we "k1 be=e !y deal au?oritatively ) ! high] p>ts ( religion4 ,& ^? :o feel ! imman5ce ( a high] life 9 u w be ne$$ 9 ?is outlook on ! wid] field ( reality1 s t !y1 be+ fitt$ to recognize1 may tell u :at lies ready = u to "k4 ,! true pa? ( wisdom consi/s 9 see+ t our intellect is fooli%ness--t our conclusions are absurd & mi/ak51 n 9 speculat+ on ! _w z a =m ( ?\ght project$ f ! ?9k+ pr9ciple )9 u--r to be amaz$ t our ?\ght has s limit$ ! _w & hidd5 f u its real exi/5ces4 ,to ?9k ( ourselves z any o!r ?an ?+s 9 space & subject to mat]ial conditions1 is absurd1 x is absurd on ei!r ( two hypo!ses4 ,if we are really ?+s 9 space1 !n ( c\rse x is absurd to ?9k ( ourselves z if we 7 n s4 ,on ! o!r h&1 if we are n ?+s 9 space1 !n conceiv+ 9 space is ! mode 9 :i* t unkn[n :i* we are exi/s z a m9d4 ,its m5tal action is space-conception1 & !n to give up ! idea ( ourselves z 9 space1 is n to get a tru] idea1 b to lose ! only p[] ( appreh5sion ( ourselves :i* we possess4 ,& yet !re is1 x mu/ be confess$1 "o way 9 :i* x may be possible = u to ?9k )\t ?9k+ ( ?+s 9 space4 ,t way is1 n to ab&on ! use ( space-?\ght1 b to pass ?r\gh x4 ,:5 we ?9k ( space1 we h to ?9k ( x z inf9ity ext5d$1 & we h to ?9k ( x z ( inf9ite dim5sions4 ,n[1 z ,i h %[n 9 8,! ,law ( ! ,valley10#c :5 we come ^u inf9ity 9 any mode ( our ?\ght1 x is a sign t t mode ( ?\ght is deal+ ) a high] reality ?an x is adapt$ =1 & 9 /ruggl+ to repres5t x1 c only d s by an inf9ite numb] ( t]ms4 ,n[1 space has an inf9ite numb] ( positions & turns1 & ?is may be due to ! attempt =c$ ^u u to ?9k ( ?+s high] ?an space z 9 space4 ,if s1 !n ! way to get rid ( space f our ?\ghts1 is1 n to g away f x1 b to pass ?r\gh x--to ?9k ab\t l>g] & l>g] sy/ems ( space1 & space ( m & m dim5sions1 till at la/ we get to su* a repres5tation 9 space ( :at is high] ?an space1 t we c pass f ! space-?\ght to ! m absolute ?\ght )\t t leap :i* w\ld be necess>y if we 7 to try to pass beyond space ) our pres5t v inadequate repres5tation 9 x ( :at really is4 #c 8,sci5ce ,romances10 ,no4 ,,ii4 ,aga9 & aga9 has human nature aspir$ & fall54 ,! vision has pres5t$ itself ( a law :i* 0 love1 a duty :i* c>ri$ away ! en?usiasm1 & 9 :i* ! conflict ( ! high] & l[] natures ceas$ because all 0 enli/$ 9 "o lov+ s]vice4 ,b aga9 & aga9 h su* attempts fail$4 ,! common-s5se view1 t man is subject to law1 ext]nal law1 rema9s--t !re are fates :om he mu/ propitiate & obey4 ,& !re is a /rong %>p curb1 :i*1 if x be n br\ght to be> by ! w1 is soon pull$ tight by ! _w1 & "o m trag$y is enact$1 & ! ov]-confid5t s\l is br\ght l[4 ,& ! rock on :i* su* attempts always split1 is 9 ! indulg5ce ( "s limit$ passion4 ,"s "o object fills ! s\l ) its image1 & 9 devotion to t1 o!r ?+s are sacrific$1 until at la/ all comes to ru94 ,b :at does ?is mean8 ,surely x is simply ?is1 t :]e !re %\ld be k !re is ignorance4 ,x is n t !re is too mu* devotion1 too mu* passion1 b t we are ignorant & bl9d1 & w&] 9 error4 ,we d n "k :at x is we c>e =1 & wa/e our ef=t on ! appe>ance4 ,!re is no su* ?+ z wrong love2 !re is good love & bad k1 & m5 :o err1 clasp phantoms to !mselves4 ,religion is b ! se>* = realities2 & ?\ght1 consci\s ( its own limitations1 is its be/ aid4 ,let a man c>e = any "o object--let 8 reg>d = x be z conc5trat$ & exclusive z y w1 !re w be no dang] if he truly appreh5ds t :i* he c>es =4 ,its true be+ is b\nd up ) all ! re/ ( exi/5ce1 &1 if 8 reg>d is true to "o1 !n1 if t "o is really kn[n1 8 reg>d is true to all4 ,!re is a "q sometimes ask$1 :i* %[s ! m]e =malism into :i* we h fall54 ,we ask3 ,:at is ! end ( exi/5ce8 ,a m]e play on words6 ,= to conceive exi/5ce is to feel ends4 ,! k ( exi/5ce is ! c>+ = objects1 ! fe> ( dang]s1 ! anxieties ( love4 ,imm]s$ 9 ^!1 ! triviality ( ! "q1 :at is ! end ( exi/5ce8 becomes obvi\s4 ,if1 h[ev]1 lett+ reality fade away1 we play ) words1 "s que/ions ( ?is k9d are possible2 b !y are m]e que/ions ( words1 & all cont5t & mean+ has pass$ out ( !m4 ,! task be=e u is ?is3 we /rive to f9d out t physical unity1 t body :i* m5 are p>ts (1 & 9 ! life ( :i* _! true unity lies4 ,! exi/5ce ( ?is "o body we "k f ! utt]ances ( ^? :om we _c b feel to be inspir$2 we feel c]ta9 t5d5cies 9 ourselves :i* _c be expla9$ except by a supposition ( ?is k9d4 ,&1 n[1 we set to "w delib]ately to =m 9 our m9ds ! means ( inve/igation1 ! faculty ( high]-space conception4 ,to our ord9>y space-?\ght1 m5 are isolat$1 di/9ct1 9 great measure antagoni/ic4 ,b ) ! fir/ use ( ! weapon ( high] ?\ght1 x is easily se5 t all m5 may really be memb]s ( "o body1 _! isolation may be b an affair ( limit$ consci\sness4 ,!re is1 ( c\rse1 no value z sci5ce 9 su* a supposition4 ,b x sugge/s to u _m possibilities2 x reveals to u ! conf9$ nature ( our pres5t physical views1 & /imulates u to und]take ! "w necess>y to enable u to deal adequately ) ! subject4 ,! "w is entirely practical & detail$2 x is ! elaboration1 beg9n+ f ! simple/ objects ( an exp]i5ce 9 ?\ght1 ( a high]-space _w4 ,to beg9 x1 we take up ^? details ( position & relation :i* are g5]ally relegat$ to symbolism or unconsci\s appreh5sion1 & br+ ^! wa/e products ( ?\ght into ! c5tral position ( ! laboratory ( ! m9d4 ,we turn all our att5tion on ! mo/ simple & obvi\s details ( our e-day exp]i5ce1 & !nce we build up a conception ( ! fundam5tal facts ( position & arrangem5t 9 a high] _w4 ,we next /udy m complicat$ high] %apes1 & get our space p]ception drill$ & discipl9$4 ,!n we proce$ to put a cont5t into our framework4 ,! means ( do+ ?is are tw(old--obs]vation & inspiration4 ,z to obs]vation1 x is h>dly possible to describe ! feel+s ( t inve/igator :o %all di/9ctly trace 9 ! physical _w1 & exp]im5tally demon/rate ! exi/5ce ( ! high]-space facts :i* are s curi\sly hidd5 f u4 ,he w lay ! fir/ /one = ! obs]vation & k ( ! high] be+s to :om we are relat$4 ,z to ! o!r means1 x is obvi\s1 surely1 t if !re has ev] be5 inspiration1 !re is inspiration n[4 ,inspiration is n a unique ph5om5on4 ,x has exi/$ 9 absolutely m>vell\s degree 9 "s ( ! tea*]s ( ! anci5t _w2 b t1 :atev] x 01 :i* !y possess$1 mu/ be pres5t n[1 &1 if we c\ld isolate x1 be a demon/rable fact4 ,& ,i w\ld propose to def9e inspiration z ! faculty1 :i*1 to take a p>ticul> in/ance1 does ! foll[+3-- ,if a squ>e p5etrates a l9e corn]wise1 x m>ks out on ! l9e a segm5t b\nd$ by two po9ts--t is1 we suppose a l9e drawn on a piece ( pap]1 & a squ>e ly+ on ! pap] to be pu%$ s t its corn] passes ov] ! l9e4 ,!n1 suppos+ ! pap] & ! l9e to be 9 ! same plane1 ! l9e is int]rupt$ by ! squ>e2 &1 ( ! squ>e1 all t is obs]vable 9 ! l9e1 is a segm5t b\nd$ by two po9ts4 ,next1 suppose a cube to be pu%$ corn]wise ?r\gh a plane1 & let ! plane make a section ( ! cube4 ,! section w be a plane figure1 & x w be a triangle4 ,n[1 fir/1 ! section ( a squ>e by a l9e is a segm5t b\nd$ by two po9ts2 second1 ! section ( a cube by a plane is a triangle b\nd$ by ?ree l9es4 ,h5ce1 we inf] t ! section ( a figure 9 f\r dim5sions analog\s to a cube1 by ?ree-dim5sional space1 w be a tetrah$ron--a figure b\nd$ by f\r planes4 ,?is is f\nd to be true2 ) a little famili>ity ) f\r-dim5sional movem5ts ?is is se5 to be obvi\s4 ,b ,i w\ld def9e inspiration z ! faculty by :i* )\t actual exp]i5ce ?is conclusion is =m$4 ,h[ x is we come to ?is conclusion ,i am p]fectly unable to say4 ,someh[1 look+ at m]e =mal consid]ations1 !re comes into ! m9d a conclusion ab\t some?+ beyond ! range ( actual exp]i5ce4 ,we may call ?is reason+ f analogy2 b us+ ?is phrase does n expla9 ! process4 ,x seems to me j z rational to say t ! facts ( ! l9e & plane rem9d u ( facts :i* we "k already ab\t f\r-dim5sional figures--t !y t5d to br+ ^! facts out into consci\sness1 z ,plato %[s ) ! boy's k ( ! cube4 ,we mu/ be really f\r-dim5sional creatures1 or we c\ld n ?9k ab\t f\r dim5sions4 ,b :atev] "n we give to ?is peculi> & inexplicable faculty1 t we d possess x is c]ta92 & 9 our inve/igations x w be ( s]vice to u4 ,we mu/ c>efully inve/igate exi/5ce 9 a plane _w1 & !n1 mak+ sure1 & impress+ on our inw>d s5se1 z we g1 e /ep we take ) reg>d to a high] _w1 we %all be rem9d$ cont9ually ( fre% possibilities ( our high] exi/5ce4 ,,"p ,,ii4 ,,*apt] ,i4 ,,?ree-,,space4 ,,g5esis ,,( ,a ,,cube4 ,,appe>ances ,,( ,a ,,cube ,,to ,a ,,plane-,,be+4 ,! models consi/ ( a set ( eight & a set ( f\r cubes4 ,!y are m>k$ ) diff]5t col\rs1 s z to %[ ! prop]ties ( ! figure 9 ,high] ,space1 to :i* !y belong4 ,! simple/ figure 9 "o-dim5sional space1 t is1 9 a /raight l9e1 is a /raight l9e b\nd$ at ! two extremities4 ,! figure 9 ?is case consi/s ( a l5g? b\nd$ by two po9ts4 ,look+ at ,cube #a1 & plac+ x s t ! figure #a is upp]mo/1 we notice a /raight l9e 9 contact ) ! table1 :i* is col\r$ ,orange4 ,x beg9s 9 a ,gold po9t & ends 9 a ,fawn po9t4 ,! ,orange ext5ds to "s di/ance on two faces ( ! ,cube2 b = our pres5t purpose we suppose x to be simply a ?9 l9e4 ,?is l9e we conceive to be g5]at$ 9 ! foll[+ way4 ,let a po9t move & trace out a l9e4 ,let ! po9t be ! ,gold po9t1 & let x1 mov+1 trace out ! ,orange l9e & t]m9ate 9 ! ,fawn po9t4 ,?us ! figure consi/s ( ! po9t at :i* x beg9s1 ! po9t at :i* x ends1 & ! portion betwe54 ,we may suppose ! po9t to />t z a ,gold po9t1 to *ange its col\r to ,orange dur+ ! motion1 & :5 x /ops to become ,fawn4 ,! motion we suppose f left to "r1 & its direction we call ,x4 ,if1 n[1 ?is ,orange l9e move away f u at "r angles1 x w trace out a squ>e4 ,let ?is be ! ,black squ>e1 :i* is se5 und]nea? ,model #a4 ,! po9ts1 :i* b\nd ! l9e1 w dur+ ?is motion trace out l9es1 & to ^! l9es !re w be t]m9al po9ts4 ,also1 ! ,squ>e w be t]m9at$ by a l9e on ! opposite side4 ,let ! ,gold po9t 9 mov+ away trace out a ,blue l9e & end 9 a ,buff po9t2 ! ,fawn po9t a ,crimson l9e end+ 9 a ,t]racotta po9t4 ,! ,orange l9e1 hav+ trac$ a ,black squ>e1 ends 9 a ,gre5-grey l9e4 ,?is direction1 away f ! obs]v]1 we call ,y4 ,n[1 let ! :ole ,black squ>e trac$ out by ! ,orange l9e move upw>ds at "r angles4 ,x w trace out a new figure1 a ,cube4 ,& ! edges ( ! squ>e1 :ile mov+ upw>ds1 w trace out squ>es4 ,b\nd+ ! cube1 & opposite to ! ,black squ>e1 w be ano!r squ>e4 ,let ! ,orange l9e mov+ upw>ds trace a ,d>k ,blue squ>e & end 9 a ,r$di% l9e4 ,! ,gold po9t traces a ,br[n l9e2 ! ,fawn po9t traces a ,fr5*-grey l9e1 & ^! l9es end 9 a ,light-blue & a ,dull-purple po9t4 ,let ! ,blue l9e trace a ,v]milion squ>e & end 9 a ,deep-yell[ l9e4 ,let ! ,buff po9t trace a ,gre5 l9e1 & end 9 a ,r$ po9t4 ,! ,gre5-grey l9e traces a ,light-yell[ squ>e & ends 9 a ,lead5 l9e2 ! ,t]racotta po9t traces a ,d>k-slate l9e & ends 9 a ,deep-blue po9t4 ,! ,crimson l9e traces a ,blue-gre5 squ>e & ends 9 a ,bright-blue l9e4 ,f9ally1 ! ,black squ>e traces a ,cube1 ! col\r ( :i* is invisible1 & ends 9 a :ite squ>e4 ,we suppose ! col\r ( ! cube to be a ,light-buff4 ,! upw>d direction we call ,z4 ,?us we say3 ,! ,gold po9t mov$ ,z1 traces a ,br[n l9e1 & ends 9 a ,light-blue po9t4 ,we c n[ cle>ly realize & ref] to ea* region ( ! cube by a col\r4 ,at ! ,gold po9t1 l9es f ?ree directions meet1 ! ,x l9e ,orange1 ! ,y l9e ,blue1 ! ,z l9e ,br[n4 ,?us we began ) a figure ( "o dim5sion1 a l9e1 we pass$ on to a figure ( two dim5sions1 a squ>e1 & end$ ) a figure ( ?ree dim5sions1 a cube4 99 99 99 99 99 ,! squ>e repres5ts a figure 9 two dim5sions2 b if we want to realize :at x is to a be+ 9 two dim5sions1 we mu/ n look d[n on x4 ,su* a view c\ld n be tak5 by a plane-be+4 ,let u suppose a be+ mov+ on ! surface ( ! table & unable to rise f x4 ,let x n "k t x is ^u any?+1 b let x believe t ! two directions & comp\nds ( ^? two directions are all possible directions4 ,moreov]1 let x n ask ! "q3 8,on :at am ,i support$80 ,let x see no reason = any su* "q1 b simply call ! smoo? surface1 along :i* x moves1 ,space4 ,su* a be+ c\ld n tell ! col\r ( ! squ>e trac$ by ! ,orange l9e4 ,! squ>e w\ld be b\nd$ by ! l9es :i* surr\nd x1 & only by break+ ?r\gh "o ( ^? l9es c\ld ! plane-be+ discov] ! col\r ( ! squ>e4 ,9 try+ to realize ! exp]i5ce ( a plane-be+ x is be/ to suppose t its two dim5sions are upw>ds & sideways1 i4e41 ,z & ,x1 because1 if !re be any matt] 9 ! plane-_w1 x w1 l matt] 9 ! solid _w1 ex]t attractions & repulsions4 ,! matt]1 l ! be+s1 mu/ be suppos$ v ?91 t is1 ( s slight ?ickness t x is q unnotic$ by ! be+4 ,n[1 if !re be a v l>ge mass ( su* matt] ly+ on ! table1 & a plane-be+ be free to move ab\t x1 he w be attract$ to x 9 e direction4 8,t[>ds ?is huge mass0 w\ld be 8,d[n10 & 8,away f x0 w\ld be 8,up10 j z 8,t[>ds ! e>?0 is to solid be+s 8,d[n10 & 8,away f x0 is 8,up10 at :atev] "p ( ! globe !y may be4 ,h5ce1 if we want to realize a plane-be+'s feel+s1 we mu/ keep ! s5se ( up & d[n4 ,!re=e we mu/ use ! ,z direction1 & x is m conv5i5t to take ,z & ,x ?an ,z & ,y4 ,any direction ly+ betwe5 ^! is said to be comp\nd$ ( ! two2 =1 if we move slantwise = "s di/ance1 ! po9t rea*$ might h be5 also rea*$ by go+ a c]ta9 di/ance ,x1 & !n a c]ta9 di/ance ,z1 or vice v]s4 ,let u suppose ! ,orange l9e has mov$ ,z1 & trac$ ! ,d>k-blue squ>e end+ 9 ! ,r$di% l9e4 ,if we n[ place a piece ( /iff pap] aga9/ ! ,d>k-blue squ>e1 & suppose ! plane-be+s to move to & fro on t surface ( ! pap]1 :i* t\*es ! squ>e1 we %all h means ( repres5t+ _! exp]i5ce4 ,to obta9 a m consi/5t view ( _! exi/5ce1 let u suppose ! piece ( pap] ext5d$1 s t x cuts ?r\gh our e>? & comes out at ! antipodes1 ?us cutt+ ! e>? 9 two4 ,!n suppose all ! e>? remov$ away1 bo? hemisph]es vani%+1 & only a v ?9 lay] ( matt] left ^u ! pap] on t side :i* t\*es ! ,d>k-blue squ>e4 ,?is repres5ts :at ! _w w\ld be to a plane-be+4 ,x is ( "s importance to get ! notion ( ! directions 9 a plane-_w1 z great difficulty arises f our notions ( up & d[n4 ,we miss ! "r analogy if we conceive ( a plane-_w )\t ! conception ( up & d[n4 ,a good plan is1 to use a slant+ surface1 a /iff c>d or book cov]1 s plac$ t x slopes upw>ds to ! eye4 ,!n gravity acts z two =ces4 ,x acts 7#a7 z a =ce press+ all p>ticles ^u ! slant+ surface into x1 & 7#b7 z a =ce ( gravity along ! plane1 mak+ p>ticles t5d to slip d[n its incl9e4 ,we may suppose t 9 a plane-_w !re are two su* =ces1 "o keep+ ! be+s !reon to ! plane1 ! o!r act+ betwe5 bodies 9 x1 & ( su* a nature t by virtue ( x any l>ge mass ( plane-matt] produces on small p>ticles ar\nd x ! same effects z ! l>ge mass ( solid matt] call$ our e>? produces on small objects l our bodies situat$ ar\nd x4 ,9 bo? cases ! l>g] draws ! small] to itself1 & creates ! s5sations ( up & d[n4 ,if we hold ! cube s t its ,d>k-blue side t\*es a %eet ( pap] held upw>ds to ! eye1 & if we !n look /raight d[n along ! pap]1 conf9+ our view to t :i* is 9 actual contact ) ! pap]1 we see ! same view ( ! cube z a plane-be+ w\ld get4 ,we see a ,light-blue po9t1 a ,r$di% l9e1 & a ,dull-purple po9t4 ,! plane-be+ only sees a l9e1 j z we only see a squ>e ( ! cube4 ,! l9e :]e ! pap] re/s on ! table may be tak5 z repres5tative ( ! surface ( ! plane-be+'s e>?4 ,x w\ld be m]ely a l9e to him1 b x w\ld h ! same prop]ty 9 relation to ! plane-_w1 z a squ>e has 9 relation to a solid _w2 9 nei!r case c ! notion ( :at 9 ! latt] is t]m$ solidity be q exclud$4 ,if ! plane-be+ broke ?r\gh ! l9e b\nd+ 8 e>?1 he w\ld f9d m matt] beyond x4 ,let u n[ leave out ( consid]ation ! "q ( 8up & d[n0 9 a plane-_w4 ,let u no long] consid] x 9 ! v]tical1 or ,,zx1 position1 b simply take ! surface 7,,xy7 ( ! table z t :i* supports a plane-_w4 ,let u repres5t its inhabitants by ?9 pieces ( pap]1 :i* are free to move ov] ! surface ( ! table1 b _c rise f x4 ,also1 let ! ?ickness 7i4e41 height above ! surface7 ( ^! be+s be s small t !y _c disc]n x4 ,la/ly let u premise !re is no attraction 9 _! _w1 s t !y h n any up & d[n4 ,plac+ ,cube #a 9 front ( u1 let u n[ ask h[ a plane-be+ c\ld appreh5d su* a cube4 ,! ,black face he c\ld easily /udy4 ,he w\ld f9d x b\nd$ by ,gold po9t1 ,orange l9e1 ,fawn po9t1 ,crimson l9e1 & s on4 ,& he w\ld discov] x 0 ,black by cutt+ ?r\gh any ( ^! l9es & ent]+ x4 7,?is op]ation w\ld be equival5t to ! m9+ ( a solid be+74 ,b ( :at came above ! ,black squ>e he w\ld be completely ignorant4 ,let u n[ suppose a squ>e hole to be made 9 ! table1 s t ! cube c\ld pass ?r\gh1 & let ! cube fit ! op5+ s exactly t no trace ( ! cutt+ ( ! table be visible to ! plane-be+4 ,if ! cube began to pass ?r\gh1 x w\ld seem to him simply to *ange1 = ( its motion he c\ld n be aw>e1 z he w\ld n "k ! direction 9 :i* x mov$4 ,let x pass d[n till ! ,:ite squ>e be j on a level ) ! surface ( ! table4 ,! plane-be+ w\ld !n p]ceive a ,light-blue po9t1 a ,r$di% l9e1 a ,dull-purple po9t1 a ,bright-blue l9e1 & s on4 ,^! w\ld surr\nd a ,:ite squ>e1 :i* belong$ to ! same body z t to :i* ! ,black squ>e belong$4 ,b 9 ?is body !re w\ld be a dim5sion1 :i* 0 n 9 ! squ>e4 ,our upw>d direction w\ld n be appreh5d$ by him directly4 ,motion f above d[nw>ds w\ld only be appreh5d$ z a *ange 9 ! figure be=e him4 ,he w\ld n say t he _h be=e him diff]5t sections ( a cube1 b only a *ang+ squ>e4 ,if he want$ to look at ! upp] squ>e1 he c\ld only d s :5 ! ,black squ>e _h gone an in* bel[ 8 plane4 ,to /udy ! upp] squ>e simultane\sly ) ! l[]1 he w\ld h to make a model ( x1 & !n he c\ld place x beside ! l[] "o4 ,look+ at ! cube1 we see t ! ,r$di% l9e corresponds precisely to ! ,orange l9e1 & ! ,deep-yell[ to ! ,blue l9e4 ,b if ! plane-be+ _h a model ( ! upp] squ>e1 & plac$ x on ! "r-h& side ( ! ,black squ>e1 ! ,deep-yell[ l9e w\ld come next to ! ,crimson l9e ( ! ,black squ>e4 ,!re w\ld be a discont9uity ab\t x4 ,all t he c\ld d w\ld be to obs]ve :i* "p 9 ! "o squ>e correspond$ to :i* "p 9 ! o!r4 ,obvi\sly too !re lies some?+ betwe5 ! ,black squ>e & ! ,:ite4 ,! plane-be+ w\ld notice t :5 a l9e moves 9 a direction n its own1 x traces out a squ>e4 ,:5 ! ,orange l9e is mov$ away1 x traces out ! ,black squ>e4 ,! conception ( a new direction ?us obta9$1 he w\ld und]/& t ! ,orange l9e mov+ s w\ld trace out a squ>e1 & ! ,blue l9e mov+ s w\ld d ! same4 ,to u ^! squ>es are visible z :oles1 ! ,d>k-blue1 & ! ,v]milion4 ,to him !y w\ld be matt]s ( v]bal def9ition r ?an asc]ta9$ facts4 ,h[ev]1 giv5 t he _h ! exp]i5ce ( a cube be+ pu%$ ?r\gh 8 plane1 he w\ld "k !re 0 "s figure1 :]e( 8 squ>e 0 "p1 :i* 0 b\nd$ by 8 squ>e on "o side1 & by a ,:ite squ>e on ano!r side4 ,we h suppos$ him to make models ( ^! b\nd>ies1 a ,black squ>e & a ,:ite squ>e4 ,! ,black squ>e1 :i* is 8 solid matt]1 is only "o b\nd>y ( a figure 9 ,high] ,space4 ,b we c suppose ! cube to be pres5t$ to him o!rwise ?an by pass+ ?r\gh 8 plane4 ,x c be turn$ r\nd ! ,orange l9e1 9 :i* case ! ,blue l9e goes out1 &1 aft] a "t1 ! ,br[n l9e comes 94 ,x mu/ be notic$ t ! ,br[n l9e comes into a direction opposite to t 9 :i* ! ,blue l9e ran4 ,^! two l9es are at "r angles to ea* o!r1 &1 if "o be mov$ upw>ds till x is at "r angles to ! surface ( ! table1 ! o!r comes on to ! surface1 b runs 9 a direction opposite to t 9 :i* ! fir/ ran4 ,?us1 by turn+ ! cube ab\t ! ,orange l9e & ! ,blue l9e1 diff]5t sides ( x c be %[n to a plane-be+4 ,by comb9+ ! two processes ( turn+ & pu%+ ?r\gh ! plane1 all ! sides c be %[n to ! plane-be+4 ,= in/ance1 if ! cube be turn$ s t ! ,d>k-blue squ>e be on ! plane1 & x be !n pass$ ?r\gh1 ! ,light-yell[ squ>e w come 94 ,n[1 if ! plane-be+ made a set ( models ( ^! diff]5t appe>ances & /udi$ !m1 he c\ld =m "s rational idea ( ! ,high] ,solid :i* produc$ !m4 ,he w\ld become able to give "s consi/5t acc\nt ( ! prop]ties ( ?is new k9d ( exi/5ce2 he c\ld say :at came into 8 plane space1 if ! o!r space p5etrat$ ! plane edge-wise or corn]-wise1 & c\ld describe all t w\ld come 9 z x turn$ ab\t 9 any way4 ,he w\ld h six models4 ,let u consid] two ( !m--! ,black & ! ,:ite squ>es4 ,we c obs]ve !m on ! cube4 ,e col\r on ! "o is diff]5t f e col\r on ! o!r4 ,if we n[ ask :at lies betwe5 ! ,orange l9e & ! ,r$di% l9e1 we "k x is a squ>e1 = ! ,orange l9e mov+ 9 any direction gives a squ>e4 ,&1 if ! six models 7 be=e ! plane-be+1 he c\ld easily select t :i* %[$ :at he want$4 ,= t :i* lies betwe5 ,orange l9e & ,r$di% l9e mu/ be b\nd$ by ,orange & ,r$di% l9es4 ,he w\ld se>* among ! six models ly+ beside ea* o!r on 8 plane1 till he f\nd ! ,d>k-blue squ>e4 ,x is evid5t t only "o o!r squ>e diff]s 9 all its col\rs f ! ,black squ>e1 viz41 ! ,:ite squ>e4 ,= x is entirely sep>ate4 ,! o!rs meet x 9 "o ( _! l9es4 ,?is total diff]5ce exi/s 9 all ! pairs ( opposite surfaces on ! cube4 ,n[1 suppose ! plane-be+ ask$ himself :at w\ld appe> if ! cube turn$ r\nd ! ,blue l9e4 ,! cube w\ld beg9 to pass ?r\gh 8 space4 ,! ,crimson l9e w\ld disappe> b5ea? ! plane & ! ,blue-gre5 squ>e w\ld cut x1 s t opposite to ! ,blue l9e 9 ! plane !re w\ld be a ,blue-gre5 l9e4 ,! ,fr5*-grey l9e & ! ,d>k-slate l9e w\ld be cut 9 po9ts1 & f ! ,gold po9t to ! ,fr5*-grey po9t w\ld be a ,d>k-blue l9e2 & opposite to x w\ld be a ,light-yell[ l9e1 f ! ,buff po9t to ! ,d>k-slate po9t4 ,?us ! figure 9 ! plane _w w\ld be an oblong in/ead ( a squ>e1 & ! int]ior ( x w\ld be ( ! same ,light-buff col\r z ! int]ior ( ! cube4 ,x is assum$ t ! plane closes up r\nd ! pass+ cube1 z ! surface ( a liquid does r\nd any object imm]s$4 ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,illu/ration3 ,fig4 #c4 ,illu/ration3 ,fig4 #d4 ,illu/ration3 ,fig4 #e4 ,b1 9 ord] to appreh5d :at w\ld take place :5 ?is twi/+ r\nd ! ,blue l9e began1 ! plane-be+ w\ld h to set to "w by p>ts4 ,he has no conception ( :at a solid w\ld d 9 twi/+1 b he kn[s :at a plane does4 ,let him1 !n1 in/ead ( ?9k+ ( ! :ole ,black squ>e1 ?9k only ( ! ,orange l9e4 ,! ,d>k-blue squ>e /&s on x4 ,z f> z ?is squ>e is conc]n$1 twi/+ r\nd ! ,blue l9e is ! same z twi/+ r\nd ! ,gold po9t4 ,let him imag9e himself 9 t plane at "r angles to 8 plane-_w1 :i* conta9s ! ,d>k-blue squ>e4 ,let him keep 8 att5tion fix$ on ! l9e :]e ! two planes meet1 viz41 t :i* is at fir/ m>k$ by ! ,orange l9e4 ,we w call ?is l9e ! l9e ( 8 plane1 = all t he kn[s ( 8 own plane is ?is l9e4 ,n[1 let ! ,d>k-blue squ>e turn r\nd ! ,gold po9t4 ,! ,orange l9e at once dips bel[ ! l9e ( 8 plane1 & ! ,d>k-blue squ>e passes ?r\gh x4 ,!re=e1 9 8 plane he w see a ,d>k-blue l9e 9 place ( ! ,orange "o4 ,& 9 place ( ! ,fawn po9t1 only fur!r (f f ! ,gold po9t1 w be a ,fr5*-grey po9t4 ,! ,diagrams 7#a71 7#b7 %[ h[ ! cube appe>s z x is be=e & aft] ! turn+4 ,g is ! ,gold1 ,f ! ,fawn po9t4 ,9 7#b7 ,g is unmov$1 & ! plane is cut by ! ,fr5*-grey l9e1 ,gr4 ,in/ead ( imag9+ a direction he did n "k1 ! plane-be+ c\ld ?9k ( ! ,d>k-blue squ>e z ly+ 9 8 plane4 ,b 9 ?is case ! ,black squ>e w\ld be out (f 8 plane1 & only ! ,orange l9e w\ld rema9 9 x4 ,diagram 7#c7 %[s ! ,d>k-blue squ>e ly+ 9 8 plane1 & ,diagram 7#d7 %[s x turn+ r\nd ! ,gold po9t4 ,"h1 in/ead ( ?9k+ ab\t 8 plane & also t at "r angles to x1 he has only to ?9k h[ ! squ>e turn+ r\nd ! ,gold po9t w cut ! l9e1 :i* runs left to "r f ,g1 viz41 ! dott$ l9e4 ,! ,fr5*-grey l9e is cut by ! dott$ l9e 9 a po9t4 ,to f9d out :at w\ld come 9 at o!r p>ts1 he ne$ only treat a numb] ( ! plane sections ( ! cube p]p5dicul> to ! ,black squ>e 9 ! same mann] z he _h treat$ ! ,d>k-blue squ>e4 ,e su* section w\ld turn r\nd a po9t1 z ! :ole cube turn$ r\nd ! ,blue l9e4 ,?us he w\ld treat ! cube z a numb] ( squ>es by tak+ p>allel sections f ! ,d>k-blue to ! ,light-yell[ squ>e1 & he w\ld turn ea* ( ^! r\nd a corn] ( ! same col\r z ! ,blue l9e4 ,comb9+ ^! s]ies ( appe>ances1 he w\ld discov] :at came into 8 plane z ! cube turn$ r\nd ! ,blue l9e4 ,?us1 ! problem ( ! turn+ ( ! cube c\ld be settl$ by ! consid]ation ( ! turn+s ( a numb] ( squ>es4 ,z ! cube turn$1 a numb] ( diff]5t appe>ances w\ld be pres5t$ to ! plane-be+4 ,! ,black squ>e w\ld *ange into a ,light-buff oblong1 ) ,d>k-blue1 ,blue-gre51 ,light-yell[1 & ,blue sides1 & w\ld gradually elongate itself until x became z long z ! diagonal ( ! squ>e side ( ! cube2 & !n ! b\nd+ l9e opposite to ! ,blue l9e w\ld *ange f ,blue-gre5 to ,bright-blue1 ! o!r l9es rema9+ ! same col\r4 ,if ! cube !n turn$ /ill fur!r1 ! ,bright-blue l9e w\ld become ,:ite1 & ! oblong w\ld dim9i% 9 l5g?4 ,x w\ld 9 "t become a ,v]milion squ>e1 ) a ,deep-yell[ l9e opposite to ! ,blue l9e4 ,x w\ld !n pass :olly bel[ ! plane1 & only ! ,blue l9e w\ld rema94 ,if ! turn+ 7 cont9u$ till half a revolution _h be5 accompli%$1 ! ,black squ>e w\ld come 9 aga94 ,b n[ x w\ld come up into ! plane f und]nea?4 ,x w\ld appe> z a ,black squ>e exactly simil> to ! fir/2 b ! ,orange l9e1 in/ead ( runn+ left to "r f ,gold po9t1 w\ld run "r to left4 ,! squ>e w\ld be ! same1 only diff]5tly dispos$ ) reg>d to ! ,blue l9e4 ,x w\ld be ! look+-glass image ( ! fir/ squ>e4 ,!re w\ld be a diff]5ce 9 respect ( ! lie ( ! p>ticles ( :i* x 0 compos$4 ,if ! plane-be+ c\ld exam9e its ?ickness1 he w\ld f9d t p>ticles :i*1 9 ! fir/ case1 lay above o!rs1 n[ lay bel[ !m4 ,b1 if he 7 really a plane-be+1 he w\ld h no idea ( ?ickness 9 8 squ>es1 & he w\ld f9d !m bo? q id5tical4 ,only ! "o w\ld be to ! o!r z if x _h be5 pull$ ?r\gh itself4 ,9 ?is ph5om5on ( symmetry he w\ld appreh5d ! diff]5ce ( ! lie ( ! l9e1 :i* w5t 9 !1 to him1 unkn[n direction ( up-&-d[n4 ,,*apt] ,,ii4 ,,fur!r ,,appe>ances ,,( ,a ,,cube ,,to ,a ,,plane-,,be+4 ,be=e leav+ ! obs]vation ( ! cube1 x is well to look at x = a mom5t z x w\ld appe> to a plane-be+1 9 ^: _w !re 0 su* a fact z attraction4 ,to d ?is1 let ! cube re/ on ! table1 s t its ,d>k-blue face is p]p5dicul> 9 front ( u4 ,n[1 let a %eet ( pap] be plac$ 9 contact ) ! ,d>k-blue squ>e4 ,let up & sideways be ! two dim5sions ( ! plane-be+1 & away ! unkn[n direction4 ,let ! l9e :]e ! pap] meets ! table1 repres5t ! surface ( 8 e>?4 ,!n1 !re is to him1 z all t he c appreh5d ( ! cube1 a ,d>k-blue squ>e /&+ upright2 &1 :5 we look ov] ! edge ( ! pap]1 & reg>d m]ely ! "p 9 contact ) ! pap]1 we see :at ! plane-be+ w\ld see4 ,if ! cube be turn$ r\nd ! up l9e1 ! ,br[n l9e1 ! ,orange l9e w pass to ! ne> side ( ! pap]1 & ! section made by ! cube 9 ! pap] w be an oblong4 ,su* an oblong c be cut out2 & :5 ! cube is fitt$ into x1 x c be se5 t x is b\nd$ by a ,br[n l9e & a ,blue-gre5 l9e opposite !reto1 :ile ! o!r b\nd>ies are ,black & ,:ite l9es4 ,next1 if we take a section half-way betwe5 ! ,black & ,:ite squ>es1 we %all h a squ>e cutt+ ! plane ( ! a=esaid pap] 9 a s+le l9e4 ,) reg>d to ?is section1 all we h to inquire is1 ,:at w take ! place ( ?is l9e z ! cube turns8 ,obvi\sly1 ! l9e w elongate4 ,f a ,d>k-blue l9e x w *ange to a ,light-buff l9e1 ! col\r ( ! inside ( ! section1 & w t]m9ate 9 a ,blue-gre5 po9t in/ead ( a ,fr5*-grey4 ,aga91 x is obvi\s t1 if ! cube turns r\nd ! ,orange l9e1 x w give rise to a s]ies ( oblongs1 /ret*+ upw>ds4 ,?is turn+ c be cont9u$ till ! cube is :olly on ! ne> side ( ! pap]1 & only ! ,orange l9e rema9s4 ,&1 :5 ! cube has made half a revolution1 ! ,d>k-blue squ>e w return into ! plane2 b x w run d[nw>ds in/ead ( upw>ds z at fir/4 ,!reaft]1 if ! cube turn fur!r1 a s]ies ( oblongs w appe>1 all runn+ d[nw>ds f ! ,orange l9e4 ,h5ce1 if all ! appe>ances produc$ by ! revolution ( ! cube h to be %[n1 x mu/ be suppos$ to be rais$ "s di/ance above ! plane-be+'s e>?1 s t ^? appe>ances may be %[n :i* occur :5 x is turn$ r\nd ! ,orange l9e d[nw>ds1 z well z :5 x is turn$ upw>ds4 ,! unkn[n direction comes into ! plane ei!r upw>ds or d[nw>ds1 b !re is no special connection betwe5 x & ei!r ( ^! directions4 ,if x come 9 upw>ds1 ! ,br[n l9e goes ne>w>ds or -,y2 if x come 9 d[nw>ds1 or -,z1 ! ,br[n l9e goes away1 or ,y4 ,let u consid] m closely ! directions :i* ! plane-be+ w\ld h4 ,fir/ly1 he w\ld h up-&-d[n1 t is1 away f 8 e>? & t[>ds x on ! plane ( ! pap]1 ! surface ( 8 e>? be+ ! l9e :]e ! pap] meets ! table4 ,!n1 if he mov$ along ! surface ( 8 e>?1 !re w\ld only be a l9e = him to move 91 ! l9e runn+ "r & left4 ,b1 be+ ! direction ( 8 movem5t1 he w\ld say x ran =w>ds & backw>ds4 ,?us he w\ld simply h ! words up & d[n1 =w>ds & backw>ds1 & ! expressions "r & left w\ld h no mean+ = him4 ,if he 7 to frame a notion ( a _w 9 high] dim5sions1 he mu/ inv5t new words = di/9ctions n )9 8 exp]i5ce4 ,to repeat ! obs]vations already made1 let ! cube be held 9 front ( ! obs]v]1 & suppose ! ,d>k-blue squ>e ext5d$ on e side s z to =m a plane4 ,!n let ?is plane be consid]$ z indep5d5t ( ! ,d>k-blue squ>e4 ,n[1 hold+ ! ,br[n l9e betwe5 f+] & ?umb1 & t\*+ its extremities1 ! ,gold & ,light-blue po9ts1 turn ! cube r\nd ! ,br[n l9e4 ,! ,d>k-blue squ>e w leave ! plane1 ! ,orange l9e w t5d t[>ds ! -,y direction1 & ! ,blue l9e w f9ally come into ! plane po9t+ 9 ! ,x direction4 ,if we move ! cube s t ! l9e :i* leaves ! plane runs ,y1 !n ! l9e :i* be=e ran ,y w come into ! plane 9 ! direction opposite to t ( ! l9e :i* has left ! plane4 ,! ,blue l9e1 :i* runs 9 ! unkn[n direction c come into ei!r ( ! two kn[n directions ( ! plane4 ,x c take ! place ( ! ,orange l9e by turn+ ! cube r\nd ! ,br[n l9e1 or ! place ( ! ,br[n l9e by turn+ x r\nd ! ,orange l9e4 ,if ! plane-be+ made models to repres5t ^! two appe>ances ( ! cube1 he w\ld h id5tically ! same l9e1 ! ,blue l9e1 runn+ 9 "o ( 8 kn[n directions 9 ! fir/ model1 & 9 ! o!r ( 8 kn[n directions 9 ! second4 ,9 /udy+ ! cube he w\ld f9d x be/ to turn x s t ! l9e ( unkn[n direction ran 9 t direction 9 ! positive s5se4 ,9 t case1 x w\ld come into ! plane 9 ! negative s5se ( ! kn[n directions4 ,/>t+ ) ! cube 9 front ( ! obs]v]1 !re are two ways 9 :i* ! ,v]milion squ>e c be br\ght into ! imag9>y plane1 t is ! ext5sion ( ! ,d>k-blue squ>e4 ,if ! cube turn r\nd ! ,br[n l9e s t ! ,orange l9e goes away1 7i4e4 ,y71 ! ,v]milion squ>e comes 9 on ! left ( ! ,br[n l9e4 ,if x turn 9 ! opposite direction1 ! ,v]milion squ>e comes 9 on ! "r ( ! ,br[n l9e4 ,?us1 if we id5tify ! plane-be+ ) ! ,br[n l9e1 ! ,v]milion squ>e w\ld appe> ei!r beh9d or be=e him4 ,^! two appe>ances ( ! ,v]milion squ>e w\ld seem id5tical1 b !y c\ld n be made to co9cide by any movem5t 9 ! plane4 ,! diagram 7,fig4 #e47 %[s ! diff]5ce 9 !m4 ,x is obvi\s t no turn 9 ! plane c\ld put "o 9 ! place ( ! o!r1 "p = "p4 ,?us ! plane-be+ appreh5ds ! rev]sal ( ! unkn[n direction by ! disposition ( 8 figures4 ,if a figure1 :i* lay on "o side ( a l9e1 *ang$ into an id5tical figure on ! o!r side ( x1 he c\ld be sure t a l9e ( ! figure1 :i* at fir/ ran 9 ! positive unkn[n direction1 n[ ran 9 ! negative unkn[n direction4 ,we h dwelt at great l5g? on ! appe>ances1 :i* a cube w\ld pres5t to a plane-be+1 & x w be f\nd t all ! po9ts :i* w\ld be likely to cause difficulty h]eaft]1 h be5 expla9$ 9 ?is obvi\s case4 ,!re is1 h[ev]1 "o o!r way1 op5 to a plane-be+ ( /udy+ a cube1 to :i* we mu/ att5d4 ,?is is1 by /eady motion4 ,let ! cube come into ! imag9>y plane1 :i* is ! ext5sion ( ! ,d>k-blue squ>e1 i4e4 let x t\* ! piece ( pap] :i* is /&+ v]tical on ! table4 ,!n let x travel ?r\gh ?is plane at "r angles to x at ! rate ( an in* a m9ute4 ,! plane-be+ w\ld fir/ p]ceive a ,d>k-blue squ>e1 t is1 he w\ld see ! col\r$ l9es b\nd+ t squ>e1 & enclos$ !re9 w\ld be :at he w\ld call a ,d>k-blue solid4 ,9 ! movem5t ( ! cube1 h[ev]1 ?is ,d>k-blue squ>e w\ld n la/ = m ?an a fla% ( "t4 7,! edges & po9ts on ! models are made v l>ge2 9 reality !y mu/ be suppos$ v m9ute47 ,?is ,d>k-blue squ>e w\ld be succe$$ by "o ( ! col\r ( ! cube's int]ior1 i4e4 by a ,light-buff squ>e4 ,b ?is col\r ( ! int]ior w\ld n be visible to ! plane-be+4 ,he w\ld g r\nd ! squ>e on 8 plane1 & w\ld see ! b\nd+ l9es1 viz4 ,v]milion1 ,:ite1 ,blue-gre51 ,black4 ,& at ! corn]s he w\ld see ,deep-yell[1 ,bright-blue1 ,crimson1 & ,blue po9ts4 ,^! l9es & po9ts w\ld really be ^? p>ts ( ! faces & l9es ( ! cube1 :i* 7 on ! po9t ( pass+ ?r\gh 8 plane4 ,n[1 !re w\ld be "o diff]5ce betwe5 ! ,d>k-blue squ>e & ! ,light-buff ) _! respective b\nd>ies4 ,! fir/ only la/$ = a fla%2 ! second w\ld la/ = a m9ute or all b a m9ute4 ,consid] ! ,v]milion squ>e4 ,x appe>s to ! plane-be+ z a l9e4 ,! ,br[n l9e also appe>s to him z a l9e4 ,b !re is a diff]5ce betwe5 !m4 ,! ,br[n l9e only la/s = a fla%1 :]eas ! ,v]milion l9e la/s = a m9ute4 ,h5ce1 9 ?is mode ( pres5tation1 we may say t = a plane-be+ a la/+ l9e is ! mode ( appreh5d+ a plane1 & a la/+ plane 7:i* is a plane-be+'s solid7 is ! mode ( appreh5d+ our solids4 ,9 ! same way1 ! ,blue l9e1 z x passes ?r\gh 8 plane1 gives rise to a po9t4 ,?is po9t la/s = a m9ute1 :]eas ! ,gold po9t only la/$ = a fla%4 ,,*apt] ,,iii4 ,,f\r-,,space4 ,,g5esis ,,( ,a ,,tess>act4 ,,its ,,repres5tation ,,9 ,,?ree-,,space4 ,hi!rto we h only look$ at ,model #a4 ,?is1 ) ! next sev51 repres5t :at we c obs]ve ( ! simple/ body 9 ,high] ,space4 ,a few words w expla9 _! con/ruction4 ,a po9t by its motion traces a l9e4 ,a l9e by its motion traces ei!r a long] l9e or an area2 if x moves at "r angles to its own direction1 x traces a rectangle4 ,= ! sake ( simplicity1 we w suppose all movem5ts to be an in* 9 l5g? & at "r angles to ea* o!r4 ,?us1 a po9t mov+ traces a l9e an in* long2 a l9e mov+ traces a squ>e in*2 a squ>e mov+ traces a cubic in*4 ,9 ^! cases ea* ( ^! movem5ts produces some?+ intr9sically diff]5t f :at we _h be=e4 ,a squ>e is n a long] l9e1 nor a cube a l>g] squ>e4 ,:5 ! cube moves1 we are unable to see any new direction 9 :i* x c move1 & are compell$ to make x move 9 a direction :i* has previ\sly be5 us$4 ,let u suppose !re is an unkn[n direction at "r angles to all our kn[n directions1 j z a ?ird direction w\ld be unkn[n to a be+ conf9$ to ! surface ( ! table4 ,& let ! cube move 9 ?is unkn[n direction = an in*4 ,we call ! figure x traces a ,tess>act4 ,! models are repres5tations ( ! appe>ances a ,tess>act w\ld pres5t to u if %[n 9 v>i\s ways4 ,consid] = a mom5t :at happ5s to a squ>e :5 mov$ to =m a cube4 ,ea* ( its l9es1 mov$ 9 ! new direction1 traces a squ>e2 ! squ>e itself traces a new figure1 a cube1 :i* ends 9 ano!r squ>e4 ,n[1 our cube1 mov$ 9 a new direction1 w trace a tess>act1 :]e( ! cube itself is ! beg9n+1 & ano!r cube ! end4 ,^! two cubes are to ! tess>act z ! ,black squ>e & ,:ite squ>e are to ! cube4 ,a plane-be+ c\ld n see bo? ^? squ>es at once1 b he c\ld make models ( !m & let !m bo? re/ 9 8 plane at once4 ,s also we c make models ( ! beg9n+ & end ( ! tess>act4 ,model #a is ! cube1 :i* is its beg9n+2 ,model #b is ! cube :i* is its end4 ,x w be notic$ t !re are no two col\rs alike 9 ! two models4 ,! ,silv] po9t corresponds to ! ,gold po9t1 t is1 ! ,silv] po9t is ! t]m9ation ( ! l9e trac$ by ! ,gold po9t mov+ 9 ! new direction4 ,! sides correspond 9 ! foll[+ mann]3-- ,,sides4 ,model #a4 ,model #b4 ,black corresponds to ,bright-gre5 ,:ite ,light-grey ,v]milion ,indian-r$ ,blue-gre5 ,yell[-o*re ,d>k-blue ,burnt-si5na ,light-yell[ ,dun ,! two cubes %\ld be look$ at & comp>$ long en\gh to ensure t ! correspond+ sides c be f\nd quickly4 ,!n !re are ! foll[+ correspond5cies 9 po9ts & l9es4 ,,po9ts4 ,model #a4 ,model #b4 ,gold corresponds to ,silv] ,fawn ,turquoise ,t]ra-cotta ,e>!n ,buff ,blue t9t ,light-blue ,quak]-gre5 ,dull-purple ,peacock-blue ,deep-blue ,orange-v]milion ,r$ ,purple ,,l9es4 ,model #a4 ,model #b4 ,orange corresponds to ,leaf-gre5 ,crimson ,dull-gre5 ,gre5-grey ,d>k-purple ,blue ,purple-br[n ,br[n ,dull-blue ,fr5*-grey ,d>k-p9k ,d>k-slate ,pale-p9k ,gre5 ,indigo ,r$di% ,br[n-gre5 ,bright-blue ,d>k-gre5 ,lead5 ,pale-yell[ ,deep-yell[ ,d>k ,! col\r ( ! cube itself is invisible1 z x is cov]$ by its b\nd>ies4 ,we suppose x to be ,sage-gre54 ,^! two cubes are j z disconnect$ :5 look$ at by u z ! black & :ite squ>es w\ld be to a plane-be+ if plac$ side by side on 8 plane4 ,he _c see ! squ>es 9 _! "r position ) reg>d to ea* o!r1 nor c we see ! cubes 9 !irs4 ,let u n[ consid] ! v]milion side ( ,model #a4 ,if x move 9 ! ,x direction1 x traces ! cube ( ,model #a4 ,its ,gold po9t travels along ! ,orange l9e1 & itself1 aft] trac+ ! cube1 ends 9 ! ,blue-gre5 squ>e4 ,b if x moves 9 ! new direction1 x w also trace a cube1 = ! new direction is at "r angles to ! up & away directions1 9 :i* ! ,br[n & ,blue l9es run4 ,let ?is squ>e1 !n1 move 9 ! unkn[n direction1 & trace a cube4 ,?is cube we _c see1 because ! unkn[n direction runs out ( our space at once1 j z ! up direction runs out ( ! plane ( ! table4 ,b a plane-be+ c\ld see ! squ>e1 :i* ! ,blue l9e traces :5 mov$ upw>ds1 by ! cube be+ turn$ r\nd ! ,blue l9e1 ! ,orange l9e go+ upw>ds2 !n ! ,br[n l9e comes into ! plane ( ! table 9 ! -,x direction4 ,s also ) our cube4 ,z treat$ above1 x runs f ! ,v]milion squ>e out ( our space4 ,b if ! tess>act 7 turn$ s t ! l9e :i* runs f ! ,gold po9t 9 ! unkn[n direction lay 9 our space1 & ! ,orange l9e lay 9 ! unkn[n direction1 we c\ld !n see ! cube :i* is =m$ by ! movem5t ( ! ,v]milion squ>e 9 ! new direction4 ,take ,model #e4 ,!re is on x a ,v]milion squ>e4 ,place ?is s t x t\*es ! ,v]milion squ>e on ,model #a4 ,all ! m>ks ( ! two squ>es are id5tical4 ,?is ,cube #e1 is ! "o trac$ by ! ,v]milion squ>e mov+ 9 ! unkn[n direction4 ,9 ,model #e1 ! :ole figure1 ! tess>act1 produc$ by ! movem5t ( ! cube 9 ! unkn[n direction1 is suppos$ to be s turn$ t ! ,orange l9e passes into ! unkn[n direction1 & t ! l9e :i* w5t 9 ! unkn[n direction1 runs opposite to ! old direction ( ! ,orange l9e4 ,look+ at ?is new cube1 we see t !re is a ,/one l9e runn+ to ! left f ! ,gold po9t4 ,?is l9e is t :i* ! ,gold po9t traces :5 mov+ 9 ! unkn[n direction4 ,x is obvi\s t1 if ! ,tess>act turns s z to %[ u ! side1 ( :i* ,cube #e is a model1 !n ,cube #a w no long] be visible4 ,! ,orange l9e w run 9 ! unkn[n or f\r? direction1 & be out ( our sight1 toge!r ) ! :ole cube :i* ! ,v]milion squ>e g5]ates1 :5 ! ,gold po9t moves along ! ,orange l9e4 ,h5ce1 if we consid] ^! models z real portions ( ! tess>act1 we mu/ n h m ?an "o be=e u at once4 ,:5 we look at "o1 ! o!rs mu/ necess>ily be beyond our sight & t\*4 ,b we may consid] !m simply z models1 &1 z su*1 we may let !m lie alongside ( ea* o!r4 ,9 ?is case1 we mu/ rememb] t _! real relation%ips are n ^? 9 :i* we see !m4 ,we n[ enum]ate ! sides ( ! new ,cube #e1 s t1 :5 we ref] to x1 any col\r may be recognis$ by "n4 ,! squ>e ,v]milion traces a ,pale-gre5 cube1 & ends 9 an ,indian-r$ squ>e4 7,! col\r ,pale-gre5 ( ?is cube is n se51 z x is entirely surr\nd$ by squ>es & l9es ( col\r47 ,ea* ,l9e traces a ,squ>e & ends 9 a ,l9e4 ,! ,blue l9e ,light-br[n squ>e &,purple-br[n l9e ,br[n traces,yell[ ends,dull-blue ,deep-yell[ a ,light-r$ 9 ,d>k ,gre5 ,deep-crimson a ,indigo 4 ,ea* ,po9t traces a ,l9e & ends 9 a ,po9t4 ,! ,gold po9t ,/one l9e &,silv] po9t ,buff traces,light-gre5 ends,blue-t9t ,light-blue a ,ri*-r$ 9 ,quak]-gre5 ,r$ ,em]ald a ,purple 4 ,x w be notic$ t besides ! ,v]milion squ>e ( ?is cube ano!r squ>e ( x has be5 se5 be=e4 ,a mom5t's comp>ison ) ! exp]i5ce ( a plane-be+ w make ?is m cle>4 ,if a plane-be+ has be=e him models ( ! ,black & ,:ite squ>es ( ! ,cube1 he sees t all ! col\rs ( ! "o are diff]5t f all ! col\rs ( ! o!r4 ,next1 if he looks at a model ( ! ,v]milion squ>e1 he sees t x />ts f ! ,blue l9e & ends 9 a l9e ( ! ,:ite squ>e1 ! ,deep-yell[ l9e4 ,9 ?is squ>e he has two l9es :i* he _h be=e1 ! ,blue l9e ) ,gold & ,buff po9ts1 ! ,deep-yell[ l9e ) ,light-blue & ,r$ po9ts4 ,to him ! ,black & ,:ite squ>es are 8 ,models #a & #b1 & ! ,v]milion squ>e is to him z our ,model #e is to u4 ,! left-h& squ>e ( ,model #e is ,indian-r$1 & is id5tical ) t ( ! same col\r on ! left-h& side ( ,model #b4 ,9 fact1 ,model #e %[s u :at lies betwe5 ! ,v]milion face ( #a1 & ! ,indian-r$ face ( #b4 ,f ! ,gold po9t we suppose f\r p]fectly indep5d5t l9es to spr+ =?1 ea* ( !m at "r angles to all ! o!rs4 ,9 our space !re is only room = ?ree l9es mutually at "r angles4 ,x w be f\nd1 if we try to introduce a f\r? at "r angles to ea* ( ?ree1 t we fail2 h5ce1 ( ^! f\r l9es "o mu/ g out ( ! space we "k4 ,! col\rs ( ^! f\r l9es are ,br[n1 ,orange1 ,blue1 ,/one4 ,9 ,model #a are %[n ! ,br[n1 ,orange1 & ,blue4 ,9 ,model #e are %[n ! ,br[n1 ,blue1 & ,/one4 ,^! l9es might h _h any directions at fir/1 b we *ose to beg9 ) ! ,br[n l9e go+ up1 or ,z1 ! ,orange go+ ,x1 ! ,blue go+ ,y1 & ! ,/one l9e go+ 9 ! unkn[n direction1 :i* we w call ,w4 ,consid] = a mom5t ! ,/one & ! ,orange l9es4 ,!y c be se5 toge!r on ,model #g by look+ at ! l[] face ( x4 ,!y are at "r angles to ea* o!r1 & if ! ,orange l9e be turn$ to take ! place ( ! ,/one l9e1 ! latt] w run into ! negative "p ( ! direction previ\sly occupi$ by ! =m]4 ,?is is ! reason t ! ,models #c1 #e1 & #g are made ) ! ,/one l9e always runn+ 9 ! rev]se direction ( t l9e ( ,model #a1 :i* is want+ 9 ea* respectively4 ,x w n[ be easy to f9d out ,models #c & #g4 ,all t has to be done is1 to discov] :at faces !y h 9 common ) #a & #b1 & ^! faces w %[ f :i* planes ( #a !y are g5]at$ by motion 9 ! unkn[n direction4 ,take ,model #g4 ,on "o side ( x !re is a ,d>k-blue squ>e1 :i* is id5tical ) ! ,d>k-blue squ>e ( ,model #a4 ,plac+ x s t x co9cides ) #a by ?is squ>e l9e = l9e1 we see t ! squ>e ne>e/ to u is ,burnt-si5na1 ! same z ! ne> squ>e on ,model #b4 ,h5ce ?is cube is a model ( :at ! ,d>k-blue squ>e traces on mov+ 9 ! unkn[n direction4 ,"h ! unkn[n direction co9cides ) ! negative away direction4 ,9 fact1 to see ?is cube1 we h be5 oblig$ to suppose ! ,blue l9e turn$ into ! unkn[n direction1 = we _c look at m ?an ?ree ( ^! rectangul> l9es at once 9 our space1 & 9 ?is ,model #g we h ! ,br[n1 ,orange1 & ,/one l9es4 ,! faces1 l9es1 & po9ts ( ,cube #g c be id5tifi$ by ! foll[+ li/4 ,! ,d>k-blue squ>e traces a ,d>k-/one cube 7^: int]ior is r5d]$ invisible by ! b\nd+ squ>es71 & ends 9 a ,burnt-si5na squ>e4 ,ea* ,l9e traces a ,squ>e & ends 9 a ,l9e4 ,! ,orange l9e ,azure squ>e &,leaf-gre5 l9e ,br[n traces,yell[ ends,dull-blue ,fr5*-grey an ,yell[-gre5 9 ,d>k-p9k ,r$di% ,o*re a ,br[n-gre5 4 ,ea* ,po9t traces a ,l9e & ends 9 a ,po9t4 ,! ,gold po9t ,/one l9e &,silv] po9t ,fawn traces,smoke ends,turquoise ,light-blue a ,ri*-r$ 9 ,quak]-gre5 ,dull-purple ,gre5-blue a ,peacock-blue 4 ,if we n[ take ,model #c1 we see t x has a ,black squ>e upp]mo/1 & has ,blue & ,orange l9es4 ,h5ce1 x evid5tly proce$s f ! ,black squ>e 9 ,model #a2 & x has 9 x ,blue & ,orange l9es1 :i* proce$ f ! ,gold po9t4 ,b besides ^!1 x has runn+ d[nw>ds a ,/one l9e4 ,! l9e want+ is ! ,br[n l9e1 &1 z 9 ! o!r cases1 :5 "o ( ! ?ree l9es ( ,model #a turns out into ! unkn[n direction1 ! ,/one l9e turns into ! direction opposite to t f :i* ! l9e has turn$4 ,take ?is ,model #c & place x und]nea? ,model #a1 rais+ ! latt] s t ! ,black squ>es on ! two co9cide l9e = l9e4 ,!n we see :at w\ld come into our view if ! ,br[n l9e 7 to turn into ! unkn[n direction1 & ! ,/one l9e come into our space d[nw>ds4 ,look+ at ?is cube1 we see t ! foll[+ p>ts ( ! tess>act h be5 g5]at$4 ,! ,black squ>e traces a ,brick-r$ cube 7invisible because cov]$ by its own sides & edges71 & ends 9 a ,bright-gre5 squ>e4 ,ea* ,l9e traces a ,squ>e & ends 9 a ,l9e4 ,! ,orange l9e ,azure squ>e &,leaf-gre5 l9e ,crimson traces,rose ends,dull-gre5 ,gre5-grey an ,sea-blue 9 ,d>k-purple ,blue ,light-br[n a ,purple-br[n 4 ,ea* ,po9t traces a ,l9e & ends 9 a ,po9t4 ,! ,gold po9t ,/one l9e &,silv] po9t ,fawn traces,smoke ends,turquoise ,t]ra-cotta a ,mag5ta 9 ,e>!n ,buff ,light-gre5 a ,blue-t9t 4 ,?is completes ! enum]ation ( ! regions ( ,cube #c4 ,x may seem a little unnatural t x %\ld come 9 d[nw>ds2 b x mu/ be rememb]$ t ! new f\r? direction has no m relation to up-&-d[n ?an to "r-&-left or to ne>-&-f>4 ,& if1 in/ead ( ?9k+ ( a plane-be+ z liv+ on ! surface ( a table1 we suppose 8 _w to be ! surface ( ! %eet ( pap] t\*+ ! ,d>k-blue squ>e ( ,cube #a1 !n we see t a turn r\nd ! ,orange l9e1 :i* makes ! ,br[n l9e g into ! plane-be+'s unkn[n direction1 br+s ! ,blue l9e into 8 d[nw>ds direction4 ,!re /ill rema9 to be describ$ ,models #d1 #f1 & #h4 ,x w be %[n t ,model #d is to ,model #c :at ,model #b is to ,model #a4 ,t is1 if1 :5 #c is 9 our space1 x be mov$ s z to trace a tess>act1 #d w be ! opposite cube 9 :i* ! tess>act ends4 ,!re is no col\r common to #c & #d4 ,simil>ly1 #f is ! opposite b\nd>y ( ! tess>act g5]at$ by #e1 & #h ( t by #g4 ,a little clos] consid]ation w reveal sev]al po9ts4 ,look+ at ,cube #e1 we see proce$+ f ! ,gold po9t a ,br[n1 a ,blue1 & a ,/one l9e4 ,! ,orange l9e is want+2 !re=e1 x goes 9 ! unkn[n direction4 ,if we want to discov] :at exi/s 9 ! unkn[n direction f ,cube #e1 we c get help f ,cube #a4 ,=1 s9ce ! ,orange l9e lies 9 ! unkn[n direction f ,cube #e1 ! ,gold po9t w1 if mov$ along ! ,orange l9e1 pass 9 ! unkn[n direction4 ,s also1 ! ,v]milion squ>e1 if mov$ along 9 ! direction ( ! ,orange l9e1 w proce$ 9 ! unkn[n direction4 ,look+ at ,cube #a we see t ! ,v]milion squ>e ?us mov$ ends 9 a ,blue-gre5 squ>e4 ,!n1 look+ at ,model #f1 on x1 correspond+ to ! ,v]milion squ>e on ,cube #e1 is a ,blue-gre5 squ>e4 ,cube #f ?us %[s :at exi/s an in* beyond #e 9 ! unkn[n direction4 ,betwe5 ! "r-h& face on #e & ! "r-h& face on #f lies a cube1 ! "o :i* is %[n 9 ,model #a4 ,model #a is trac$ by ! ,v]milion squ>e mov+ an in* along ! direction ( ! ,orange l9e4 ,9 ,model #e1 ! ,orange l9e goes 9 ! unkn[n direction2 & look+ at ,model #f we see :at we %\ld get at ! end ( a movem5t ( "o in* 9 t direction4 ,we h /ill to enum]ate ! col\rs ( ,cubes #d1 #f1 & #h1 & we d s 9 ! foll[+ li/4 ,9 ! fir/ column is designat$ ! "p ( ! cube2 9 ! columns "u #d1 #f1 #h1 come ! col\rs :i* #d1 #f1 #h1 respectively h 9 ! p>ts designat$ 9 ! correspond+ l9e 9 ! fir/ column4 ,cube itself3-- #d #f #h ,*ocolate ,oak-yell[ ,salmon ,squ>es3-- ,l[] face ,light-grey ,rose ,sea-blue ,upp] ,:ite ,deep-br[n ,deep-gre5 ,left-h& ,light-r$ ,yell[-o*re ,deep-crimson ,"r-h& ,deep-br[n ,blue-gre5 ,d>k-grey ,ne> ,o*re ,yell[-gre5 ,dun ,f> ,deep-gre5 ,d>k-grey ,light-yell[ ,l9es3-- ,on gr\nd1 go+ r\nd ! squ>e f left to "r3-- #d #f #h #a4 ,br[n-gre5 ,smoke ,d>k-purple #b4 ,d>k-gre5 ,crimson ,mag5ta #c4 ,pale-yell[ ,mag5ta ,gre5-grey #d4 ,d>k ,dull-gre5 ,light-gre5 ,v]tical1 go+ r\nd ! sides f left to "r3-- #a4 ,ri*-r$ ,d>k-p9k ,indigo #b4 ,gre5-blue ,fr5*-grey ,pale-p9k #c4 ,sea-gre5 ,d>k-slate ,d>k-slate #d4 ,em]ald ,pale-p9k ,gre5 ,r\nd upp] face 9 same ord]3-- #a4 ,r$di% ,gre5-blue ,pale-yell[ #b4 ,bright-blue ,bright-blue ,sea-gre5 #c4 ,lead5 ,sea-gre5 ,lead5 #d4 ,deep-yell[ ,d>k-gre5 ,em]ald ,po9ts3-- ,on l[] face1 go+ f left to "r3-- #a4 ,quak]-gre5 ,turquoise ,blue-t9t #b4 ,peacock-blue ,fawn ,e>!n #c4 ,orange-v]milion ,t]ra-cotta ,t]ra-cotta #d4 ,purple ,e>!n ,buff ,on upp] face3-- #a4 ,light-blue ,peacock-blue ,purple #b4 ,dull-purple ,dull-purple ,orange-v]milion #c4 ,deep-blue ,deep-blue ,deep-blue #d4 ,r$ ,orange-v]milion ,r$ ,if any "o ( ^! cubes be tak5 at r&om1 x is easy en\gh to f9d out to :at "p ( ! ,tess>act x belongs4 ,9 all ( !m1 except #b1 !re w be "o face1 :i* is a copy ( a face on #a2 ?is face is1 9 fact1 id5tical ) ! face on #a :i* x resembles4 ,& ! model %[s :at lies 9 ! unkn[n direction f t face4 ,?is unkn[n direction is turn$ into our space1 s t we c see & t\* ! result ( mov+ a squ>e 9 x4 ,& we h sacrific$ "o ( ! ?ree orig9al directions 9 ord] to d ?is4 ,x w be f\nd t ! l9e1 :i* 9 #a goes 9 ! #d? direction1 9 ! o!r models always runs 9 a negative direction4 ,let u take ,model #h1 = in/ance4 ,se>*+ x = a face we "k1 we come to a ,light-yell[ face away f u4 ,we place ?is face p>allel ) ! ,light-yell[ face on ,cube #a1 & we see t x has a ,gre5 l9e go+ up1 & a ,gre5-grey l9e go+ to ! "r f ! ,buff po9t4 ,9 ^! respects x is id5tical ) ! ,light-yell[ face on ,cube #a4 ,b in/ead ( a ,blue l9e com+ t[>ds u f ! ,buff po9t1 !re is a ,light-gre5 l9e4 ,?is ,light-gre5 l9e1 !n1 is t :i* proce$s 9 ! unkn[n direction f ! ,buff po9t4 ,! l9e is turn$ t[>ds u 9 ?is ,model #h 9 ! negative ,y direction2 & look+ at ! model1 we see exactly :at is =m$ :5 9 ! motion ( ! :ole cube 9 ! unkn[n direction1 ! ,light-yell[ face is mov$ an in* 9 t direction4 ,x traces out a ,salmon cube 7v4 ,table on p4 #abg71 & x has ,sea-blue & ,deep-gre5 sides bel[ & above1 & ,deep-crimson & ,d>k-grey sides left & "r1 & ,dun & ,light-yell[ sides ne> & f>4 ,if we want to v]ify ! correctness ( any ( ^! details1 we mu/ turn to ,models #a & #b4 ,:at lies an in* f ! ,light-yell[ squ>e 9 ! unkn[n direction8 ,model #b tells u1 a ,dun squ>e4 ,n[1 look+ at #h1 we see t t[>ds u lies a ,dun squ>e4 ,?is is :at lies an in* 9 ! unkn[n direction f ! ,light-yell[ squ>e4 ,x is "h turn$ to face u1 & we c see :at lies betwe5 x & ! ,light-yell[ squ>e4 ,,*apt] ,,iv4 ,,tess>act ,,mov+ ,,?r\gh ,,?ree-,,space4 ,,models ,,( ,,! ,,sections4 ,9 ord] to obta9 a cle> conception ( ! high] solid1 a c]ta9 am\nt ( famili>ity ) ! facts %[n 9 ^! models is necess>y4 ,b ! be/ way ( obta9+ a sy/ematic k is %[n h]eaft]4 ,:at ^! models enable u to d1 is to take a g5]al review ( ! subject4 ,9 all ( !m we see simply ! b\nd>ies ( ! tess>act 9 our space2 we c no m see or t\* ! tess>act's solidity ?an a plane-be+ c t\* ! cube's solidity4 ,!re rema9 ! f\r models #i1 #aj1 #aa1 #ab4 ,model #i repres5ts :at lies betwe5 #a & #b4 ,if #a be mov$ an in* 9 ! unkn[n direction1 x traces out ! tess>act & ends 9 #b4 ,b1 obvi\sly1 betwe5 #a & #b !re mu/ be an inf9ite numb] ( exactly simil> solid sections2 ^! are all l ,model #i4 ,take ! case ( a plane-be+ on ! table4 ,he sees ! ,black squ>e1--t is1 he sees ! l9es r\nd x1--& he kn[s t1 if x moves an in* 9 "s my/]i\s direction1 x traces a new k9d ( figure1 ! opposite b\nd>y :]e( is ! ,:ite squ>e4 ,if1 !n1 he has models ( ! ,:ite & ,black squ>es1 he has be=e him ! end & beg9n+ ( our cube4 ,b betwe5 ^! squ>es are any numb] ( o!rs1 ! plane sections ( ! cube4 ,we c see :at !y are4 ,! int]ior ( ea* is a ,light-buff 7! col\r ( ! sub/ance ( ! cube71 ! sides are ( ! col\rs ( ! v]tical faces ( ! cube1 & ! po9ts ( ! col\rs ( ! v]tical l9es ( ! cube1 viz41 ,d>k-blue1 ,blue-gre51 ,light-yell[1 ,v]milion l9es1 & ,br[n1 ,fr5*-grey1 ,d>k-slate1 ,gre5 po9ts4 ,?us1 ! squ>e1 9 mov+ 9 ! unkn[n direction1 traces out a succession ( squ>es1 ! assemblage ( :i* makes ! cube 9 lay]s4 ,s also ! cube1 mov+ 9 ! unkn[n direction1 w at any po9t ( its motion1 /ill be a cube2 & ! assemblage ( cubes ?us plac$ con/itutes ! tess>act 9 lay]s4 ,we suppose ! cube to *ange its col\r directly x beg9s to move4 ,its col\r betwe5 #a & #b we c easily det]m9e by f9d+ :at col\rs its diff]5t p>ts assume1 z !y move 9 ! unkn[n direction4 ,! ,gold po9t imm$iately beg9s to trace a ,/one-l9e4 ,we w look at ,cube #e to see :at ! ,v]milion face becomes2 we "k ! int]ior ( t cube is ,pale-gre5 7v4 ,table1 p4 #abb74 ,h5ce1 z x moves 9 ! unkn[n direction1 ! ,v]milion squ>e =ms 9 its c\rse a s]ies ( ,pale-gre5 squ>es4 ,! ,br[n l9e gives rise to a ,yell[ squ>e2 h5ce1 at e po9t ( its c\rse 9 ! f\r? direction1 x is a ,yell[ l9e1 until1 on tak+ its f9al position1 x becomes a ,dull-blue l9e4 ,look+ at ,cube #e1 we see t ! ,deep yell[ l9e becomes a ,light-r$ l9e1 ! ,gre5 l9e a ,deep ,crimson "o1 ! ,gold po9t a ,/one "o1 ! ,light-blue po9t a ,ri*-r$ "o1 ! ,r$ po9t an ,em]ald "o1 & ! ,buff po9t a ,light-gre5 "o4 ,n[1 take ! ,model #i4 ,look+ at ! left side ( x1 we see exactly t into :i* ! ,v]milion squ>e is trans=m$1 z x moves 9 ! unkn[n direction4 ,! left side is an exact copy ( a section ( ,cube #e1 p>allel to ! ,v]milion face4 ,b we h only acc\nt$ = "o side ( our ,model #i4 ,!re are five o!r sides4 ,take ! ne> side correspond+ to ! ,d>k-blue squ>e on ,cube #a4 ,:5 ! ,d>k-blue squ>e moves1 x traces a ,d>k-/one cube1 ( :i* we h a copy 9 ,cube #g4 ,look+ at #g 7v4 ,table1 p4 #abd71 we see t1 z soon z ! ,d>k-blue squ>e beg9s to move1 x becomes ( a ,d>k-/one col\r1 & has ,yell[1 ,o*re1 ,yell[-gre51 & ,azure sides1 & ,/one1 ,ri*-r$1 ,gre5-blue1 ,smoke l9es runn+ 9 ! unkn[n direction f x4 ,n[1 ! side ( ,model #i1 :i* faces u1 has ^! col\rs ! squ>es be+ se5 z l9es1 & ! l9es z po9ts4 ,h5ce ,model #i is a copy ( :at ! cube becomes1 s f> z ! ,v]milion & ,d>k-blue sides are conc]n$1 :51 mov+ 9 ! unkn[n direction1 x traces ! tess>act4 ,we w n[ look at ! l[] squ>e ( our model4 ,x is a ,brick-r$ squ>e1 ) ,azure1 ,rose1 ,sea-blue1 & ,light-br[n l9es1 & ) ,/one1 ,smoke1 ,mag5ta1 & ,light-gre5 po9ts4 ,?is1 !n1 is :at ! ,black squ>e %\ld *ange into1 z x moves 9 ! unkn[n direction4 ,let u look at ,model #c4 ,"h ! ,/one l9e1 :i* is ! l9e 9 ! unkn[n direction1 runs d[nw>ds4 ,x is turn$ into ! d[nw>ds direction1 s t ! cube trac$ by ! ,black squ>e may be 9 our space4 ,! col\r ( ?is cube is ,brick-r$2 ! ,orange l9e has trac$ an ,azure1 ! ,blue l9e a ,light-br[n1 ! ,crimson l9e a ,rose1 & ! ,gre5-grey l9e a ,sea-blue squ>e4 ,h5ce1 ! l[] squ>e ( ,model #i %[s :at ! ,black squ>e becomes1 z x traces ! tess>act2 or1 9 o!r words1 ! section ( ,model #c betwe5 ! ,black & ,bright-gre5 squ>es exactly corresponds to ! l[] face ( ,model #i4 ,!re=e1 x appe>s t ,model #i is a model ( a section ( ! tess>act1 t x is to ! tess>act :at a squ>e betwe5 ! ,black & ,:ite squ>es is to ! cube4 ,to prove ! o!r sides correct1 we h to see :at ! ,:ite1 ,blue-gre51 & ,light-yell[ squ>es ( ,cube #a become1 z ! cube moves 9 ! unkn[n direction4 ,?is c be effect$ by means ( ! ,models #d1 #f1 #h4 ,ea* cube c be us$ z an index = %[+ ! *anges ?r\gh :i* any side ( ! fir/ model passes1 z x moves 9 ! unkn[n direction till x becomes ,cube #b4 ,?us1 :at becomes ( ! ,:ite squ>e8 ,look at ,cube #d4 ,f ! ,light-blue corn] ( its ,:ite squ>e runs d[nw>ds ! ,ri*-r$ l9e 9 ! unkn[n direction4 ,if we take a p>allel section bel[ ! ,:ite squ>e1 we h a squ>e b\nd$ by ,o*re1 ,deep-br[n1 ,deep-gre51 & ,light-r$ l9es2 & by ,ri*-r$1 ,gre5-blue1 ,sea-gre51 & ,em]ald po9ts4 ,! col\r ( ! cube is ,*ocolate1 & !re=e its section is ,*ocolate4 ,?is description is exactly true ( ! upp] surface ( ,model #i4 ,!re /ill rema9 two sides1 ^? correspond+ to ! ,light-yell[ & ,blue-gre5 ( ,cube #a4 ,:at ! ,blue-gre5 squ>e becomes midway betwe5 ,cubes #a & #b c be se5 on ,model #f4 ,! col\r ( ! la/-nam$ is ,oak-yell[1 & a section p>allel to its ,blue-gre5 side is surr\nd$ by ,yell[-gre51 ,deep-br[n1 ,d>k-grey & ,rose l9es & by ,gre5-blue1 ,smoke1 ,mag5ta1 & ,sea-gre5 po9ts4 ,?is is exactly simil> to ! "r side ( ,model #i4 ,la/ly1 t :i* becomes ( ! ,light-yell[ side c be se5 on ,model #h4 ,! section ( ! cube is a ,salmon squ>e b\nd$ by ,deep-crimson1 ,deep-gre51 ,d>k-grey & ,sea-blue l9es & by ,em]ald1 ,sea-gre51 ,mag5ta1 & ,light-gre5 po9ts4 ,?us ! models c be us$ to answ] any "q ab\t sections4 ,= we h simply to take1 in/ead ( ! :ole cube1 a plane1 & ! relation ( ! :ole tess>act to t plane c be told by look+ at ! model1 :i*1 />t+ ) t plane1 /ret*es f x 9 ! unkn[n direction4 ,we h n z yet settl$ ! col\r ( ! int]ior ( ,model #i4 ,x is t "p ( ! tess>act :i* is trac$ out by ! int]ior ( ,cube #a4 ,! unkn[n direction />ts equally & simultane\sly f e po9t ( e "p ( ,cube #a1 j z ! up direction />ts equally & simultane\sly f e po9t ( a squ>e4 ,let u suppose t ! cube1 :i* is ,light-buff1 *anges to a ,wood-col\r directly x beg9s to trace ! tess>act4 ,!n ! int]nal "p ( ! section betwe5 #a & #b w be a ,wood-col\r4 ,! sides ( ! ,model #i are ( ! greate/ importance4 ,!y are ! col\r ( ! six cubes1 #c1 #d1 #e1 #f1 #g1 & #h4 ,! col\rs ( #a & #b are want+1 viz4 ,light-buff & ,sage-gre54 ,?us ! section betwe5 #a & #b c be f\nd by its want+ ! col\rs ( ! ,cubes #a & #b4 ,look+ at ,models #aj1 #aa1 & #ab 9 a simil> mann]1 ! read] w f9d !y repres5t ! sections betwe5 ,cubes #c & #d1 ,cubes #e & #f1 & ,cubes #g & #h respectively4 ,,*apt] ,v4 ,,repres5tation ,,( ,,?ree-,,space ,,by ,,names1 ,,& ,,9 ,a ,,plane4 ,we may n[ ask ourselves ! be/ way ( pass+ on to a cle> compreh5sion ( ! facts ( high] space4 ,some?+ c be effect$ by look+ at ^! models2 b x is improbable t m ?an a slight s5se ( analogy w be obta9$ ?us4 ,inde$1 we h be5 tru/+ hi!rto to a me?od :i* has some?+ vici\s ab\t x--we h be5 tru/+ to our s5se ( :at mu/ be4 ,! plan adopt$1 z ! s]i\s ef=t t[>ds ! compreh5sion ( ?is subject1 is to le>n a small portion ( high] space4 ,if any read] feel a difficulty 9 ! =ego+ *apt]s1 or if ! subject is to be taught to "y m9ds1 x is f> bett] to ab&on all attempt to see :at high] space mu/ be1 & to le>n :at x is f ! foll[+ *apt]s4 ,,nam+ ,a ,,piece ,,( ,,space4 ,! diagram 7,fig4 #f7 repres5ts a block ( #bg cubes1 :i* =m ,set #a ( ! #ha cubes4 ,! cubes are col\r$1 & x w be se5 t ! col\rs are arrang$ aft] ! patt]n ( ,model #a ( previ\s *apt]s1 :i* w s]ve z a key to ! block4 ,9 ! diagram1 ,g4 d5otes ,gold1 ,o4 ,orange1 ,f4 ,fawn1 ,br4 ,br[n1 & s on4 ,we w give names to ! cubes ( ?is block4 ,!y %\ld n be le>nt1 b kept = ref]5ce4 ,we w write ^! names 9 ?ree sets1 ! l[e/ consi/+ ( ! cubes :i* t\* ! table1 ! next ( ^? imm$iately above !m1 & ! ?ird ( ^? at ! top4 ,?us ! ,gold cube is call$ ,corvus1 ! ,orange1 ,cuspis1 ! ,fawn1 ,nug1 & ! c5tral "o bel[1 ,syce4 ,! correspond+ col\rs ( ! foll[+ set c easily be trac$4 ,olus ,semita ,lama ,via ,mel ,it] ,ilex ,callis ,sors ,buc9a ,murex ,daps ,alvus ,mala ,proes ,arctos ,mna ,f> ,ci/a ,cadus ,crus ,dos ,syce ,bolus ,corvus ,cuspis ,nug ,?us ! c5tral or ,light-buff cube is call$ ,mala2 ! middle "o ( ! l[] face is ,syce2 ( ! upp] face ,mel2 ( ! "r face1 ,proes2 ( ! left1 ,alvus2 ( ! front1 ,mna 7! ,d>k-blue squ>e ( ,model #a72 & ( ! back1 ,murex 7! ,light-yell[ squ>e74 ,n[1 if ,model #a be tak51 & consid]$ z repres5t+ a block ( #fd cubes1 ! ,gold corn] z "o cube1 ! ,orange l9e z two cubes1 ! ,fawn po9t z "o cube1 ! ,d>k-blue squ>e z f\r cubes1 ! ,light-buff int]ior z eight cubes1 & s on1 x w correspond to ! diagram 7,fig4 #g74 ,?is block diff]s f ! la/ 9 ! numb] ( cubes1 b ! arrangem5t ( ! col\rs is ! same4 ,! foll[+ table gives ! names :i* we w use = ^! cubes4 ,!re are no new names2 !y are only appli$ m ?an once to all cubes ( ! same col\r4 ,olus ,semita ,semita ,lama ,f\r?,via ,mel ,mel ,it] ,floor4,via ,mel ,mel ,it] ,ilex ,callis ,callis ,sors ,buc9a ,murex ,murex ,daps ,?ird ,alvus ,mala ,mala ,proes ,floor4,alvus ,mala ,mala ,proes ,arctos ,mna ,mna ,f> ,buc9a ,murex ,murex ,daps ,second,alvus ,mala ,mala ,proes ,floor4,alvus ,mala ,mala ,proes ,arctos ,mna ,mna ,f> ,ci/a ,cadus ,cadus ,crus ,fir/ ,dos ,syce ,syce ,bolus ,floor4,dos ,syce ,syce ,bolus ,corvus ,cuspis ,cuspis ,nug ,illu/ration3 ,fig4 #f4 ,illu/ration3 ,fig4 #g4 ,illu/ration3 ,fig4 #h4 ,if we n[ consid] ,model #a to repres5t a block1 five cubes ea* way1 built up ( in* cubes1 & col\r x 9 ! same way1 t is1 ) simil> col\rs = ! corn]-cubes1 edge-cubes1 face-cubes1 & int]ior-cubes1 we obta9 :at is repres5t$ 9 ! diagram 7,fig4 #h74 ,"h we h n9e ,d>k-blue cubes call$ ,mna2 t is1 ,mna d5otes ! n9e ,d>k-blue cubes1 =m+ a lay] on ! front ( ! cube1 & fill+ up ! :ole front except ! edges & po9ts4 ,cuspis d5otes ?ree ,orange1 ,dos ?ree ,blue1 & ,arctos ?ree ,br[n cubes4 ,n[1 ! block ( cubes c be simil>ly increas$ to any size we please4 ,! corn]s w always consi/ ( s+le cubes2 t is1 ,corvus w rema9 a s+le cubic in*1 ev5 ?\gh ! block be a hundr$ in*es ea* way4 ,cuspis1 9 t case1 w be #ih in*es long1 & consi/ ( a r[ ( #ih cubes2 ,arctos1 also1 w be a long ?9 l9e ( cubes /&+ up4 ,mna w be a ?9 lay] ( cubes almo/ cov]+ ! :ole front ( ! block2 ! numb] ( !m w be #ih times #ih4 ,syce w be a simil> squ>e lay] ( cubes on ! gr\nd1 s also ,mel1 ,alvus1 ,proes1 & ,murex 9 _! respective places4 ,mala1 ! int]ior ( ! cube1 w consi/ ( #ih times #ih times #ih in* cubes4 ,illu/ration3 ,fig4 #i ,n[1 if we cont9u$ 9 ?is mann] till we _h a v l>ge block ( ?\s&s ( cubes 9 ea* side ,corvus w\ld1 9 comp>ison to ! :ole block1 be a m9ute po9t ( a cubic %ape1 & ,cuspis w\ld be a m]e l9e ( m9ute cubes1 :i* w\ld h l5g?1 b v small dep? or height4 ,next1 if we suppose ?is mu* sub-divid$ block to be r$uc$ 9 size till x becomes "o measur+ an in* ea* way1 ! cubes ( :i* x consi/s mu/ ea* ( !m become extremely m9ute1 & ! corn] cubes & l9e cubes w\ld be sc>cely disc]nible4 ,b ! cubes on ! faces w\ld be j z visible z be=e4 ,= in/ance1 ! cubes compos+ ,mna w\ld /ret* out on ! face ( ! cube s z to fill x up4 ,!y w\ld =m a lay] ( extreme ?9ness1 b w\ld cov] ! face ( ! cube 7all ( x except ! m9ute l9es & po9ts74 ,?us we may use ! words ,corvus & ,nug1 etc41 to d5ote ! corn]-po9ts ( ! cube1 ! words ,mna1 ,syce1 ,mel1 ,alvus1 ,proes1 ,murex1 to d5ote ! faces4 ,x mu/ be rememb]$ t ^! faces h a ?ickness1 b x is extremely m9ute comp>$ ) ! cube4 ,mala w\ld d5ote all ! cubes ( ! int]ior except ^?1 :i* compose ! faces1 edges1 & po9ts4 ,?us1 ,mala w\ld practically mean ! :ole cube except ! col\r+ on x4 ,& x is 9 ?is s5se t ^! words w be us$4 ,9 ! models1 ! ,gold po9t is int5d$ to be a ,corvus1 only x is made l>ge to be visible2 s too ! ,orange l9e is meant = ,cuspis1 b magnifi$ = ! same reason4 ,f9ally1 ! #bg names ( cubes1 ) :i* we began1 come to be ! names ( ! po9ts1 l9es1 & faces ( a cube1 z %[n 9 ! diagram 7,fig4 #i74 ,) ^! names x is easy to express :at a plane-be+ w\ld see ( any cube4 ,let u suppose t ,mna is only ( ! ?ickness ( 8 matt]4 ,we suppose 8 matt] to be compos$ ( p>ticles1 :i* slip ab\t on 8 plane1 & are s ?9 t he _c by any means disc]n any ?ickness 9 !m4 ,s he has no idea ( ?ickness4 ,b we "k t 8 matt] mu/ h "s ?ickness1 & we suppose ,mna to be ( t degree ( ?ickness4 ,if ! cube be plac$ s t ,mna is 9 8 plane1 ,corvus1 ,cuspis1 ,nug1 ,f>1 ,sors1 ,callis1 ,ilex & ,arctos w j come into 8 appreh5sion2 !y w be l bits ( 8 matt]1 :ile all t is beyond !m 9 ! direction he does n "k1 w be hidd5 f him4 ,?us a plane-be+ c only p]ceive ! ,mna or ,syce or "s "o o!r face ( a cube2 t is1 he w\ld take ! ,mna ( a cube to be a solid 9 8 plane-space1 & he w\ld see ! l9es ,cuspis1 ,f>1 ,callis1 ,arctos4 ,to him !y w\ld b\nd x4 ,! po9ts ,corvus1 ,nug1 ,sors1 & ,ilex1 he w\ld n see1 = !y are only z long z ! ?ickness ( 8 matt]1 & t is s slight z to be indisc]nible to him4 ,we mu/ n[ g ) great c>e ?r\gh ! exact processes by :i* a plane-be+ w\ld /udy a cube4 ,= ?is purpose we use squ>e slabs :i* h a c]ta9 ?ickness1 b are suppos$ to be z ?9 z a plane-be+'s matt]4 ,n[1 let u take ! fir/ set ( #ha cubes aga91 & build !m f #a to #bg4 ,we mu/ realize cle>ly t two k9ds ( blocks c be built4 ,x may be built ( #bg cubes1 ea* simil> to ,model #a1 9 :i* case ea* cube has its regions col\r$1 b all ! cubes are alike4 ,or x may be built ( #bg diff]5tly col\r$ cubes l ,set #a1 9 :i* case ea* cube is col\r$ :olly ) "o col\r 9 all its regions4 ,if ! latt] set be us$1 we c /ill use ! names ,mna1 ,alvus1 etc4 to d5ote ! front1 side1 etc41 ( any "o ( ! cubes1 :atev] be its col\r4 ,:5 !y are built up1 place a piece ( c>d aga9/ ! front to repres5t ! plane on :i* ! plane-be+ lives4 ,! front ( ea* ( ! cubes 9 ! front ( ! block t\*es ! plane4 ,9 previ\s *apt]s we h suppos$ ,mna to be a ,blue squ>e4 ,b we c apply ! "n to ! front ( a cube ( any col\r4 ,let u say ! ,mna ( ea* front cube is 9 ! plane2 ! ,mna ( ! ,gold cube is ,gold1 & s on4 ,to repres5t ?is1 take n9e slabs ( ! same col\rs z ! cubes4 ,place a /iff piece ( c>dbo>d 7or a book-cov]7 slant+ f y1 & put ! slabs on x4 ,!y c be support$ on ! incl9e s z to prev5t _! slipp+ d[n away f y by a ?9 book1 or ano!r %eet ( c>dbo>d1 :i* /&s = ! surface ( ! plane-be+'s e>?4 ,we w n[ give names to ! cubes ( ,block #a ( ! #ha ,set4 ,we call ea* "o ,mala1 to d5ote t x is a cube4 ,!y are writt5 9 ! foll[+ li/ 9 floors or lay]s1 & are suppos$ to run backw>ds or away f ! read]4 ,?us1 9 ! fir/ lay]1 ,fr5um ,mala is beh9d or f>!r away ?an ,urna ,mala2 9 ! second lay]1 ,o/rum is 9 front1 ,uncus beh9d x1 & ,ala beh9d ,uncus4 ,?ird1 or ,m>s ,mala ,m]ces ,mala ,tyro ,mala ,top ,spicula ,mala ,mora ,mala ,oliva ,mala ,floor4 ,comes ,mala ,tibic5 ,mala ,ve/is ,mala ,second1 or,ala ,mala ,cortis ,mala ,a] ,mala ,middle ,uncus ,mala ,pallor ,mala ,t]gum ,mala ,floor4 ,o/rum ,mala ,bid5s ,mala ,sc5a ,mala ,fir/1 or ,sector ,mala ,hama ,mala ,remus ,mala ,bottom ,fr5um ,mala ,plebs ,mala ,sypho ,mala ,floor4 ,urna ,mala ,moles ,mala ,saltus ,mala ,^! names %\ld be le>nt s t ! diff]5t cubes 9 ! block c be ref]r$ to q easily & imm$iately by "n4 ,!y mu/ be le>nt 9 e ord]1 t is1 9 ea* ( ! ?ree directions backw>ds & =w>ds1 e4g4 ,urna to ,saltus1 ,urna to ,sector1 ,urna to ,comes2 & ! same rev]s$1 viz41 ,comes to ,urna1 ,sector to ,urna1 etc4 ,only by s le>n+ !m c ! m9d id5tify any "o individually )\t ev5 a mom5t>y ref]5ce to ! o!rs ar\nd x4 ,x is well to make x a rule n to proce$ f "o cube to a di/ant "o )\t nam+ ! int]m$iate cubes4 ,=1 9 ,space we _c pass f "o "p to ano!r )\t go+ ?r\gh ! int]m$iate portions4 ,&1 9 ?9k+ ( ,space1 x is well to accu/om our m9ds to ! same limitations4 ,urna ,mala is suppos$ to be solid ,gold an in* ea* way2 s too all ! cubes are suppos$ to be entirely ( ! col\r :i* !y %[ on _! faces4 ,?us any section ( ,moles ,mala w be ,orange1 ( ,plebs ,mala ,black1 & s on4 ,illu/ration3 ,fig4 #aj4 ,let u n[ draw a pair ( l9es on a piece ( pap] or c>dbo>d l ^? 9 ! diagram 7,fig4 #aj74 ,9 ?is diagram ! top ( ! page is suppos$ to re/ on ! table1 & ! bottom ( ! page to be rais$ & br\ght ne> ! eye1 s t ! plane ( ! diagram slopes upw>ds to ! read]4 ,let ,z d5ote ! upw>d direction1 & ,x ! direction f left to "r4 ,let u turn ! ,block ( cubes ) its front ^u ?is slope i4e4 s t ,urna fits ^u ! squ>e m>k$ ,urna4 ,moles w be to ! "r & ,o/rum above ,urna1 i4e4 ne>] ! eye4 ,we might leave ! block z x /&s & put ! piece ( c>dbo>d aga9/ x2 9 ?is case our plane-_w w\ld be v]tical4 ,x is difficult to fix ! cubes 9 ?is position on ! plane1 & !re=e m conv5i5t if ! c>dbo>d be s incl9$ t !y w n slip (f4 ,b ! upw>d direction mu/ be id5tifi$ ) ,z4 ,n[1 tak+ ! slabs1 let u compose :at a plane-be+ w\ld see ( ! ,block4 ,he w\ld p]ceive j ! front faces ( ! cubes ( ! ,block1 z x comes into 8 plane2 ^! front faces we may call ! ,mo5as ( ! cubes4 ,let ea* ( ! slabs repres5t ! ,mo5a ( its correspond+ cube1 ! ,gold slab ( ! ,gold cube & s on4 ,!y are ?ick] ?an !y %\ld be2 b we mu/ ov]look ?is & suppose we simply see ! ?ickness z a l9e4 ,we ?us build a squ>e ( n9e slabs to repres5t ! appe>ance to a plane-be+ ( ! front face ( ! ,block4 ,! middle "o1 ,bid5s ,mo5a1 w\ld be completely hidd5 f him by ! o!rs on all its sides1 & he w\ld see ! edges ( ! eight out] squ>es4 ,if ! ,block n[ beg9 to move ?r\gh ! plane1 t is1 to cut ?r\gh ! piece ( pap] at "r angles to x1 x w n = "s "t appe> any diff]5t4 ,= ! sections ( ,urna are all ,gold l ! front face ,mo5a1 s t ! appe>ance ( ,urna at any po9t 9 its passage w be a ,gold squ>e exactly l ,urna ,mo5a1 se5 by ! plane-be+ z a l9e4 ,?us1 if ! spe$ ( ! ,block's passage be "o in* a m9ute1 ! plane-be+ w see no *ange = a m9ute4 ,9 o!r words1 ?is set ( slabs la/+ "o m9ute w repres5t :at he sees4 ,:5 ! ,block has pass$ "o in*1 a diff]5t set ( cubes appe>s4 ,remove ! front lay] ( cubes4 ,!re w n[ be 9 contact ) ! pap] n9e new cubes1 ^: names we write 9 ! ord] 9 :i* we %\ld see !m ?r\gh a piece ( glass /&+ upright 9 front ( ! ,block3 ,spicula ,mala ,mora ,mala ,oliva ,mala ,uncus ,mala ,pallor ,mala ,t]gum ,mala ,fr5um ,mala ,plebs ,mala ,sypho ,mala ,we pick out n9e slabs to repres5t ! ,mo5as ( ^! cubes1 & plac$ 9 ord] !y %[ :at ! plane-be+ sees ( ! second set ( cubes z !y pass ?r\gh4 ,simil>ly ! ?ird wall ( ! ,block w come into ! plane1 & look+ at !m simil>ly1 z x 7 ?r\gh an upright piece ( glass1 we write _! names3 ,m>s ,mala ,m]ces ,mala ,tyro ,mala ,ala ,mala ,cortis ,mala ,a] ,mala ,sector ,mala ,hama ,mala ,remus ,mala ,n[1 x is evid5t t ^! slabs /& at diff]5t times = diff]5t p>ts ( ! cubes4 ,we c imag9e !m to /& = ! ,mo5a ( ea* cube z x passes ?r\gh4 ,9 t case1 ! fir/ set ( slabs1 :i* we put up1 repres5ts ! ,mo5as ( ! front wall ( cubes2 ! next set1 ! ,mo5as ( ! second wall4 ,?us1 if all ! ?ree sets ( slabs be toge!r on ! table1 we h a repres5tation ( ! sections ( ! cube4 ,= "s purposes x w\ld be bett] to h f\r sets ( slabs1 ! f\r? set repres5t+ ! ,murex ( ! ?ird wall2 = ! ?ree sets only %[ ! front faces ( ! cubes1 & !re=e w\ld n indicate any?+ ab\t ! back faces ( ! ,block4 ,= in/ance1 if a l9e pass$ ?r\gh ! ,block diagonally f ! po9t ,corvus 7,gold7 to ! po9t ,lama 7,deep-blue71 x w\ld be repres5t$ on ! slabs by a po9t at ! bottom left-h& corn] ( ! ,gold slab1 a second po9t at ! same corn] ( ! ,light-buff slab1 & a ?ird at ! same corn] ( ! ,deep-blue slab4 ,?us1 we %\ld h ! po9ts mapp$ at :i* ! l9e ent]$ ! fronts ( ! walls ( cubes1 b n ! po9t 9 ,lama at :i* x w\ld leave ! ,block4 ,let ! ,diagrams #a1 #b1 #c 7,fig4 #aa71 be ! ?ree sets ( slabs4 ,to see ! diagrams prop]ly1 ! read] mu/ set ! top ( ! page on ! table1 & look along ! page f ! bottom ( x4 ,! l9e 9 "q1 :i* runs f ! bottom left-h& ne> corn] to ! top "r-h& f> corn] ( ! ,block w be repres5t$ 9 ! ?ree sets ( slabs by ! po9ts ,a1 ,b1 ,c4 ,to complete ! diagram ( its c\rse1 we ne$ a f\r? set ( slabs = ! ,murex ( ! ?ird wall2 ! same object might be atta9$1 if we _h ano!r ,block ( #bg cubes beh9d ! fir/ ,block & repres5t$ its front or ,mo5as by a set ( slabs4 ,= ! po9t1 at :i* ! l9e leaves ! fir/ ,block is id5tical ) t at :i* x ent]s ! second ,block4 ,illu/ration3 ,fig4 #aa4 ,if we suppose a %eet ( glass to be ! plane-_w1 ! ,diagrams #a1 #b1 #c 7,fig4 #aa71 may be drawn m naturally to u z ,diagrams 1 1 7,fig4 #ab74 ,"h repres5ts ! ,mo5as ( ! fir/ wall1 ^? ( ! second1 ^? ( ! ?ird4 ,b to get ! plane-be+'s view we mu/ look ov] ! edge ( ! glass d[n ! ,z axis4 ,illu/ration3 ,fig4 #ab4 ,set #b ( slabs repres5t ! ,mo5as ( ,wall #b4 ,^! ,mo5as are 9 contact ) ! ,murex ( ,wall #a4 ,?us ,set #b w %[ :]e ! l9e issues f ,wall #a z well z :]e x ent]s ,wall #b4 ,! plane-be+1 !re=e1 c\ld get an idea ( ! ,block ( cubes by le>n+ ^! slabs4 ,he ought n to call ! ,gold slab ,urna ,mala1 b ,urna ,mo5a1 & s on1 because all t he le>ns are ,mo5as1 m]ely ! ?9 faces ( ! cubes4 ,by introduc+ ! c\rse ( "t1 he c repres5t ! ,block m ne>ly4 ,=1 if he supposes x to be pass+ an in* a m9ute1 he may give ! "n ,urna ,mala to ! ,gold slab endur+ = a m9ute4 ,b1 :5 he has le>nt ! slabs 9 ?is position & sequ5ce1 he has only a v p>tial view ( ! ,block4 ,let ! ,block turn r\nd ! ,z axis1 z ,model #a turns r\nd ! ,br[n l9e4 ,a diff]5t set ( cubes comes into 8 plane1 & n[ !y come 9 on ! ,alvus faces4 7,alvus is "h us$ to d5ote ! left-h& faces ( ! cubes1 & is n suppos$ to be ,v]milion2 x is simply ! ?9ne/ slice on ! left h& ( ! cube & ( ! same col\r z ! cube47 ,to repres5t ?is1 ! plane-be+ %\ld employ a fre% set ( slabs1 = !re is no?+ common to ! ,mo5a & ,alvus faces except an edge4 ,b1 s9ce ea* cube is ( ! same col\r ?r\gh\t1 ! same slab may be us$ = its diff]5t faces4 ,?us ! ,alvus ( ,urna ,mala c be repres5t$ by a ,gold slab4 ,only x mu/ nev] be =gott5 t x is meant to be a new slab1 & is n id5tical ) ! same slab us$ = ,mo5a4 ,illu/ration3 ,fig4 #ac4 ,illu/ration3 ,fig4 #ad4 ,n[1 :5 ! ,block ( cubes has turn$ r\nd ! ,br[n l9e into ! plane1 x is cle> t !y w be on ! side ( ! ,z axis opposite to t on :i* 7 ! ,mo5a slabs4 ,! l9e1 :i* ran ,y1 n[ runs -,x4 ,?us ! slabs w occupy ! second quadrant m>k$ by ! axes1 z %[n 9 ! diagram 7,fig4 #ac74 ,ea* ( ^! slabs we w "n ,alvus4 ,9 ?is view1 z be=e1 ! book is suppos$ to be tilt$ up t[>ds ! read]1 s t ! ,z axis runs f ,o to 8 eye4 ,!n1 if ! ,block be pass$ at "r angles ?r\gh ! plane1 !re w come into view ! two sets ( slabs repres5t$ 9 ! ,diagrams 7,fig4 #ac74 ,9 copy+ ?is arrangem5t ) ! slabs1 ! c>dbo>d on :i* !y are arrang$ mu/ slant upw>ds to ! eye1 i4e41 ,,oz mu/ run up to ! eye1 & ! sides ( ! slabs se5 are 9 ,diagram #b 7,fig4 #ac71 ! upp] edges ( ,tibic51 ,mora1 ,m]ces2 9 ,diagram #c1 ! upp] edges ( ,ve/is1 ,oliva1 ,tyro4 ,!re is ano!r view ( ! ,block possible to a plane-be+4 ,if ! ,block be turn$ r\nd ! ,x axis1 ! l[] face comes into ! v]tical plane4 ,?is corresponds to turn+ ,model #a r\nd ! ,orange l9e4 ,on ref]r+ to ! diagram 7,fig4 #ad71 we n[ see t ! "n ( ! faces ( ! cubes com+ into ! plane is ,syce4 ,"h ! plane-be+ looks f ! extremity ( ! ,z axis & ! squ>es1 :i* he sees run f him 9 ! -,z direction4 7,z ?is turn ( ! ,block br+s its ,syce into ! v]tical plane s t x ext5ds ?ree in*es bel[ ! base l9e ( its ,mo5a1 x is evid5t t ! turn is only possible if ! ,mo5a be orig9ally at a height ( at lea/ ?ree in*es above ! plane-be+'s e>? l9e 9 ! v]tical plane47 ,next1 if ! ,block be pass$ ?r\gh ! plane1 ! sections %[n 9 ! ,diagrams #b & #c 7,fig4 #ad7 are br\ght into view4 ,?us1 !re are ?ree di/9ct ways ( reg>d+ ! cubic ,block1 ea* ( !m equally prim>y2 & if ! plane-be+ is to h a correct idea ( ! ,block1 he mu/ be equally famili> ) ea* view4 ,by means ( ! slabs ea* aspect c be repres5t$2 b we mu/ rememb] 9 ea* ( ! ?ree cases1 t ! slabs repres5t diff]5t p>ts ( ! cube4 ,:5 we look at ! cube ,pallor ,mala 9 space1 we see t x t\*es six o!r cubes by its six faces4 ,b ! plane-be+ c\ld only arrive at ?is fact by comp>+ diff]5t views4 ,tak+ ! ?ree ,mo5a sections ( ! ,block1 he c see t ,pallor ,mala ,mo5a t\*es ,plebs ,mo5a1 ,mora ,mo5a1 ,uncus ,mo5a1 & ,t]gum ,mo5a by l9es4 ,& x takes ! place ( ,bid5s ,mo5a1 & is itself displac$ by ,cortis ,mo5a z ! ,block passes ?r\gh ! plane4 ,next1 ?is same ,pallor ,mala c appe> to him z an ,alvus4 ,9 ?is case1 x t\*es ,plebs ,alvus1 ,mora ,alvus1 ,bid5s ,alvus1 & ,cortis ,alvus by l9es1 takes ! place ( ,uncus ,alvus1 & is itself displac$ by ,t]gum ,alvus z ! ,block moves4 ,simil>ly he c obs]ve ! relations1 if ! ,syce ( ! ,block be 9 8 plane4 ,h5ce1 ?is unkn[n body ,pallor ,mala appe>s to him n[ z "o plane-figure n[ z ano!r1 & comes be=e him 9 diff]5t connections4 ,pallor ,mala is t :i* satisfies all ^! relations4 ,ea* ( !m he c 9 turn pres5t to s5se2 b ! total conception ( ,pallor ,mala itself c only1 if at all1 gr[ up 9 8 m9d4 ,! way = him to =m ?is m5tal conception1 is to g ?r\gh all ! practical possibilities :i* ,pallor ,mala w\ld af=d him by its v>i\s movem5ts & turns4 ,9 our _w ^! v>i\s relations are f\nd by ! mo/ simple obs]vations2 b a plane-be+ c\ld only acquire !m by consid]able lab\r4 ,& if he det]m9$ to obta9 a k ( ! physical exi/5ce ( a high] _w ?an 8 own1 he mu/ pass ?r\gh su* discipl9e4 99 99 99 99 99 ,illu/ration3 ,fig4 #ae4 ,illu/ration3 ,fig4 #af4 ,we w see :at *ange c\ld be introduc$ into ! %apes he builds by ! movem5ts1 :i* he does n "k 9 8 _w4 ,let u build up ?is %ape ) ! cubes ( ! ,block3 ,urna ,mala1 ,moles ,mala1 ,bid5s ,mala1 ,tibic5 ,mala4 ,to ! plane-be+ ?is %ape w\ld be ! slabs1 ,urna ,mo5a1 ,moles ,mo5a1 ,bid5s ,mo5a1 ,tibic5 ,mo5a 7,fig4 #ae74 ,n[ let ! ,block be turn$ r\nd ! ,z axis1 s t x goes pa/ ! position1 9 :i* ! ,alvus sides ent] ! v]tical plane4 ,let x move until1 pass+ ?r\gh ! plane1 ! same ,mo5a sides come 9 aga94 ,! mass ( ! ,block w n[ h cut ?r\gh ! plane & be on ! ne> side ( x t[>ds u2 b ! ,mo5a faces only w be on ! plane-be+'s side ( x4 ,! diagram 7,fig4 #af7 %[s :at he w see1 & x w seem to him simil> to ! fir/ %ape 7,fig4 #ae7 9 e respect except its disposition ) reg>d to ! ,z axis4 ,x lies 9 ! direction -,x1 opposite to t ( ! fir/ figure4 ,h[ev] mu* he turn ^! two figures ab\t 9 ! plane1 he _c make "o occupy ! place ( ! o!r1 "p = "p4 ,h5ce x appe>s t1 if we turn ! plane-be+'s figure ab\t a l9e1 x und]goes an op]ation :i* is to him q my/]i\s4 ,he _c by any turn 9 8 plane produce ! *ange 9 ! figure produc$ by u4 ,a little obs]vation w %[ t a plane-be+ c only turn r\nd a po9t4 ,turn+ r\nd a l9e is a process unkn[n to him4 ,!re=e "o ( ! elem5ts 9 a plane-be+'s k ( a space high] ?an 8 own1 w be ! conception ( a k9d ( turn+ :i* w *ange 8 solid bodies into _! own images4 ,,*apt] ,,vi4 ,,! ,,means ,,by ,,:i* ,a ,,plane-,,be+ ,,w\ld ,,acquire ,a ,,conception ,,( ,,our ,,figures4 ,take ! ,block ( tw5ty-sev5 ,mala cubes1 & build up ! foll[+ %ape 7,fig4 #ah73-- ,urna ,mala1 ,moles ,mala1 ,plebs ,mala1 ,pallor ,mala1 ,mora ,mala4 ,if ?is %ape1 pass$ ?r\gh ! v]tical plane1 ! plane-be+ w\ld p]ceive3-- 7#a7 ,! squ>es ,urna ,mo5a & ,moles ,mo5a4 7#b7 ,! ?ree squ>es ,plebs ,mo5a1 ,pallor ,mo5a1 ,mora ,mo5a1 & !n ! :ole figure w\ld h pass$ ?r\gh 8 plane4 ,if ! :ole ,block 7 turn$ r\nd ! ,z axis till ! ,alvus sides ent]$1 & ! figure built up z x w\ld be 9 t disposition ( ! cubes1 ! plane-be+ w\ld p]ceive dur+ its passage ?r\gh ! plane3-- 7#a7 ,urna ,alvus2 7#b7 ,moles ,alvus1 ,plebs ,alvus1 ,pallor ,alvus1 ,mora ,alvus1 :i* w\ld all ent] on ! left side ( ! ,z axis4 ,aga91 if ! ,block 7 turn$ r\nd ! ,x axis1 ! ,syce side w\ld ent]1 & ! cubes appe> 9 ! foll[+ ord]3-- 7#a7 ,urna ,syce1 ,moles ,syce1 ,plebs ,syce2 7#b7 ,pallor ,syce2 7#c7 ,mora ,syce4 ,illu/ration3 ,fig4 #ag4 ,illu/ration3 ,fig4 #ah4 ,a comp>ison ( ^! ?ree sets ( appe>ances w\ld give ! plane-be+ a full acc\nt ( ! figure4 ,x is t :i* c produce ^! v>i\s appe>ances4 ,let u n[ suppose a glass plate plac$ 9 front ( ! ,block 9 its fir/ position4 ,on ?is plate let ! axes ,x & ,z be drawn4 ,!y divide ! surface into f\r p>ts1 to :i* we give ! foll[+ names 7,fig4 #ag73-- ,z ,x t qu>t] def9$ by ! positive ,z & positive ,x axis4 ,z ,x t qu>t] def9$ by ! positive ,z & negative ,x axis 7:i* is call$ 8,z negative ,x074 ,z ,x t qu>t] def9$ by ! negative ,z & negative ,x axis4 ,z ,x t qu>t] def9$ by ! negative ,z & positive ,x axis4 ,! ,block appe>s 9 ^! diff]5t qu>t]s or quadrants1 z x is turn$ r\nd ! diff]5t axes4 ,9 ,z ,x ! ,mo5as appe>1 9 ,z ,x ! ,alvus faces1 9 ,z ,x ! ,syces4 ,9 ea* quadrant are drawn n9e squ>es1 to receive ! faces ( ! cubes :5 !y ent]4 ,= in/ance1 9 ,z ,x we h ! ,mo5as (3-- ,z ,comes ,tibic5 ,ve/is ,o/rum ,bid5s ,sc5a ,urna ,moles ,saltus --------------------------------------,x ,& 9 ,z ,x we h ! ,alvus (3-- ,z ,m>s ,spicula ,comes ,ala ,uncus ,o/rum ,sector ,fr5um ,urna -,x------------------------------------- ,& 9 ! ,z ,x we h ! ,syces (3-- -----------------------------------,x ,urna ,moles ,saltus ,fr5um ,plebs ,sypho ,sector ,hama ,remus -,z ,n[1 if ! %ape tak5 at ! beg9n+ ( ?is *apt] be look$ at ?r\gh ! glass1 & ! di/ance ( ! second & ?ird walls ( ! %ape beh9d ! glass be consid]$ ( no acc\nt--t is1 if !y be treat$ z close up to ! glass--we get a plane outl9e1 :i* occupies ! squ>es ,urna ,mo5a1 ,moles ,mo5a1 ,bid5s ,mo5a1 ,tibic5 ,mo5a4 ,?is outl9e is call$ a projection ( ! figure4 ,to see x l a plane-be+1 we %\ld h to look d[n on x along ! ,z axis4 ,x is obvi\s t "o projection does n give ! %ape4 ,= in/ance1 ! squ>e ,bid5s ,mo5a might be fill$ by ei!r ,pallor or ,cortis4 ,all t a squ>e 9 ! room ( ,bid5s ,mo5a d5otes1 is t !re is a cube some:]e 9 ! ,y1 or unkn[n1 direction f ,bid5s ,mo5a4 ,?is view1 j tak51 we %\ld call ! front view 9 our space2 we are !n look+ at x along ! negative ,y axis4 ,:5 ! same %ape is turn$ r\nd on ! ,z axis1 s z to be project$ on ! ,z ,x quadrant1 we h ! squ>es--,urna ,alvus1 ,fr5um ,alvus1 ,uncus ,alvus1 ,spicula ,alvus4 ,:5 x is turn$ r\nd ! ,x axis1 & project$ on ,z ,x1 we h ! squ>es1 ,urna ,syce1 ,moles ,syce1 ,plebs ,syce1 & no m4 ,?is is :at is ord9>ily call$ ! gr\nd plan2 b we h set x 9 a diff]5t position f t 9 :i* x is usually drawn4 ,illu/ration3 ,fig4 #ai4 ,n[1 ! be/ me?od = a plane-be+ ( famili>iz+ himself ) %apes 9 our space1 w\ld be to practise ! realization ( !m f _! diff]5t projections 9 8 plane4 ,?us1 giv5 ! ?ree projections j m5tion$1 he %\ld be able to con/ruct ! figure f :i* !y are d]iv$4 ,! projections 7,fig4 #ai7 are drawn bel[ ! p]spective pictures ( ! %ape 7,fig4 #ah74 ,f ! front1 or ,mo5a view1 he w\ld conclude t ! %ape 0 ,urna ,mala1 ,moles ,mala1 ,bid5s ,mala1 ,tibic5 ,mala2 or in/ead ( ^!1 or also 9 addition to !m1 any ( ! cubes runn+ 9 ! ,y direction f ! plane4 ,t is1 f ! ,mo5a projection he might inf] ! pres5ce ( all ! foll[+ cubes 7! ^w ,mala is omitt$ = brevity73 ,urna1 ,fr5um1 ,sector1 ,moles1 ,plebs1 ,hama1 ,bid5s1 ,pallor1 ,cortis1 ,tibic51 ,mora1 ,m]ces4 ,next1 ! ,alvus view or projection might be giv5 by ! cubes 7! ^w ,mala be+ aga9 omitt$73 ,urna1 ,moles1 ,saltus1 ,fr5um1 ,plebs1 ,sypho1 ,uncus1 ,pallor1 ,t]gum1 ,spicula1 ,mora1 ,oliva4 ,la/ly1 look+ at ! gr\nd plan or ,syce view1 he w\ld inf] ! possible pres5ce ( ,urna1 ,o/rum1 ,comes1 ,moles1 ,bid5s1 ,tibic51 ,plebs1 ,pallor1 ,mora4 ,n[1 ! %ape 9 high] space1 :i* is usually !re1 is t :i* is common to all ^! ?ree appe>ances4 ,x c be det]m9$1 !re=e1 by reject+ ^? cubes :i* are n pres5t 9 all ?ree li/s ( cubes possible f ! projections4 ,& by ?is process ! plane-be+ c\ld arrive at ! enum]ation ( ! cubes :i* belong to ! %ape ( :i* he _h ! projections4 ,aft] a "t1 :5 he _h exp]i5ce ( ! cubes 7:i*1 ?\gh invisible to him z :oles1 he c\ld see "p by "p 9 turn ent]+ 8 space71 ! projections w\ld h m mean+ to him1 & he might compreh5d ! %ape !y express$ fragm5t>ily 9 8 space4 ,to practise ! realization f projections1 we %\ld proce$ 9 ?is way4 ,fir/1 we %\ld ?9k ( ! possibilities involv$ 9 ! ,mo5a view1 & build !m up 9 cubes be=e u4 ,secondly1 we %\ld build up ! cubes possible f ! ,alvus view4 ,aga91 tak+ ! %ape at ! beg9n+ ( ! *apt]1 we %\ld f9d t ! %ape ( ! ,alvus possibilities int]sect$ t ( ! ,mo5a possibilities 9 ,urna1 ,moles1 ,fr5um1 ,plebs1 ,pallor1 ,mora2 or1 9 o!r words1 ^! cubes are common to bo?4 ,?irdly1 we %\ld build up ! ,syce possibilities1 &1 comp>+ _! %ape ) ^? ( ! ,mo5a & ,alvus projections1 we %\ld f9d ,urna1 ,moles1 ,plebs1 ,pallor1 ,mora1 ( ! ,syce view co9cid+ ) ! same cubes ( ! o!r views1 ! only cube pres5t 9 ! int]section ( ! ,mo5a & ,alvus possibilities1 & n pres5t 9 ! ,syce view1 be+ ,fr5um4 ,! det]m9ation ( ! figure d5ot$ by ! ?ree projections1 may be m easily effect$ by treat+ ea* projection z an indication ( :at cubes are to be cut away4 ,tak+ ! same %ape z be=e1 we h 9 ! ,mo5a projection ,urna1 ,moles1 ,bid5s1 ,tibic52 & ! possibilities f !m are ,urna1 ,fr5um1 ,sector1 ,moles1 ,plebs1 ,hama1 ,bid5s1 ,pallor1 ,cortis1 ,tibic51 ,mora1 ,m]ces4 ,?is may aptly be call$ ! ,mo5a solution4 ,n[1 f ! ,syce projection1 we le>n at once t ^? cubes1 :i* 9 x w\ld produce ,fr5um1 ,sector1 ,hama1 ,remus1 ,sypho1 ,saltus1 are n 9 ! %ape4 ,?is abs5ce ( ,fr5um & ,sector 9 ! ,syce view proves t _! pres5ce 9 ! ,mo5a solution is sup]flu\s4 ,! abs5ce ( ,hama removes ! possibility ( ,hama1 ,cortis1 ,m]ces4 ,! abs5ce ( ,remus1 ,sypho1 ,saltus1 makes no diff]5ce1 z nei!r !y nor any ( _! ,syce possibilities are pres5t 9 ! ,mo5a solution4 ,h5ce1 ! result ( comp>ison ( ! ,mo5a & ,syce projections & possibilities is ! %ape3 ,urna1 ,moles1 ,plebs1 ,bid5s1 ,pallor1 ,tibic51 ,mora4 ,?is may be aptly call$ ! ,mo5a-,syce solution4 ,n[1 9 ! ,alvus projection we see t ,o/rum1 ,comes1 ,sector1 ,ala1 & ,m>s are abs5t4 ,! abs5ce ( ,sector1 ,ala1 & ,m>s has no effect on our ,mo5a-,syce solution2 z x does n conta9 any ( _! ,alvus possibilities4 ,b ! abs5ce ( ,o/rum & ,comes proves t 9 ! ,mo5a-,syce solution ,bid5s & ,tibic5 are sup]flu\s1 s9ce _! pres5ce 9 ! orig9al %ape w\ld give ,o/rum & ,comes 9 ! ,alvus projection4 ,?us we arrive at ! ,mo5a-,alvus-,syce solution1 :i* gives u ! %ape3 ,urna1 ,moles1 ,plebs1 ,pallor1 ,mora4 ,x w be obvi\s on trial t a %ape c be in/antly recognis$ f its ?ree projections1 if ! ,block be ?or\ghly well kn[n 9 all ?ree positions4 ,any difficulty 9 ! realization ( ! %apes comes f ! arbitr>y habit ( associat+ ! cubes ) "s "o direction 9 :i* !y happ5 to g ) reg>d to u4 ,if we rememb] ,o/rum z above ,urna1 we are n rememb]+ ! ,block1 b only "o p>ticul> relation ( ! ,block to u4 ,t position ( ,o/rum is a fact z mu* relat$ to ourselves z to ! ,block4 ,!re is1 ( c\rse1 "s in=mation ab\t ! ,block impli$ 9 t position2 b x is s mix$ ) in=mation ab\t ourselves z to be ineffectual k ( ! ,block4 ,x is ( ! highe/ importance to ent] m9utely into all ! details ( solution writt5 above4 ,=1 correspond+ to e op]ation necess>y to a plane-be+ = ! compreh5sion ( our _w1 !re is an op]ation1 ) :i* we h to become famili>1 if 9 our turn we w\ld ent] into "s compreh5sion ( a _w high] ?an our own4 ,e cube ( ! ,block ought to be ?or\ghly kn[n 9 all its relations4 ,& ! ,block mu/ be reg>d$1 n z a =mless mass out ( :i* %apes c be made1 b z ! sum ( all possible %apes1 f :i* any "o we may *oose is a selection4 ,9 fact1 to be famili> ) ! ,block1 we ought to "k e %ape t c\ld be made by any selection ( its cubes2 or1 9 o!r words1 we ought to make an exhau/ive /udy ( x4 ,9 ! ,app5dix is giv5 a set ( ex]cises 9 ! use ( ^! names 7:i* =m a language ( %ape71 & 9 v>i\s k9ds ( projections4 ,! projections /udi$ 9 ?is *apt] are n ! only1 nor ! mo/ natural1 projections by :i* a plane-be+ w\ld /udy high] space4 ,b !y suffice z an illu/ration ( our pres5t purpose4 ,if ! read] w g ?r\gh ! ex]cises 9 ! ,app5dix1 & =m o!rs = himself1 he w f9d e bit ( manipulation done w be ( s]vice to him 9 ! compreh5sion ( high] space4 ,!re is "o po9t ( view 9 ! /udy ( ! ,block1 by means ( slabs1 :i* is ( "s int]e/4 ,! cubes ( ! ,block1 & !re=e also ! repres5tative slabs ( _! faces1 c be reg>d$ z =m+ r[s & columns4 ,!re are ?ree sets ( !m4 ,if we take ! ,mo5a view1 !y repres5t ! views ( ! ?ree walls ( ! ,block1 z !y pass ?r\gh ! plane4 ,to =m ! ,alvus view1 we only h to re>range ! slabs1 & =m new sets4 ,! fir/ new set is =m$ by tak+ ! fir/1 or left-h&1 column ( ea* ( ! ,mo5a sets4 ,! second ,alvus set is =m$ by tak+ ! second or middle columns ( ! ?ree ,mo5a sets4 ,! ?ird w consi/ ( ! rema9+ or "r-h& columns ( ! ,mo5as4 ,simil>ly1 ! ?ree ,syce sets may be =m$ f ! ?ree horizontal r[s or floors ( ! ,mo5a sets4 ,h5ce1 x appe>s t ! plane-be+ w\ld /udy our space by tak+ all ! possible comb9ations ( ! correspond+ r[s & columns4 ,he w\ld break up ! fir/ ?ree sets into o!r sets1 & ! /udy ( ! ,block w\ld practically become to him ! /udy ( ^! v>i\s arrangem5ts4 ,,*apt] ,,vii4 ,,f\r-,,space3 ,,its ,,repres5tation ,,9 ,,?ree-,,space4 ,we n[ come to ! ess5tial difficulty ( our task4 ,all t has gone be=e is prelim9>y4 ,we h n[ to frame ! me?od by :i* we %all introduce ?r\gh our space-figures ! figures ( a high] space4 ,:5 a plane-be+ /udies our %apes ( cubes1 he has to use squ>es4 ,he is limit$ at ! outset4 ,a cube appe>s to him z a squ>e4 ,on ,model #a we see ! v>i\s squ>es z :i* ! cube c appe> to him4 ,we suppose ! plane-be+ to look f ! extremity ( ! ,z axis d[n a v]tical plane4 ,fir/1 !re is ! ,mo5a squ>e4 ,!n !re is ! squ>e giv5 by a section p>allel to ,mo5a1 :i* he recognises by ! v>iation ( ! b\nd+ l9es z soon z ! cube beg9s to pass ?r\gh 8 plane4 ,!n comes ! ,murex squ>e4 ,next1 if ! cube be turn$ r\nd ! ,z axis & pass$ ?r\gh1 he sees ! ,alvus & ,proes squ>es & ! int]m$iate section4 ,s too ) ! ,syce & ,mel squ>es & ! section betwe5 !m4 ,n[1 deal+ ) figures 9 high] space1 we are 9 an analog\s position4 ,we _c grasp ! elem5t ( :i* !y are compos$4 ,we c conceive a cube2 b t :i* corresponds to a cube 9 high] space is beyond our grasp4 ,b ! plane-be+ 0 oblig$ to use two-dim5sional figures1 squ>es1 9 arriv+ at a notion ( a ?ree-dim5sional figure2 s also mu/ we use ?ree-dim5sional figures to arrive at ! notion ( a f\r-dim5sional4 ,let u call ! figure :i* corresponds to a squ>e 9 a plane & a cube 9 our space1 a tess>act4 ,model #a is a cube4 ,let u assume a tess>act g5]at$ f x4 ,let u call ! tess>act ,urna4 ,! g5]at+ cube may !n be aptly call$ ,urna ,mala4 ,we may use cubes to repres5t p>ts ( f\r-space1 b we mu/ always rememb] t !y are to u1 9 our /udy1 only :at squ>es are to a plane-be+ ) respect to a cube4 ,let u aga9 exam9e ! mode 9 :i* a plane-be+ repres5ts a ,block ( cubes ) slabs4 ,take ,block #a ( ! #ha ,set ( cubes4 ,! plane-be+ repres5ts ?is by n9e slabs1 :i* repres5t ! ,mo5a face ( ! block4 ,!n1 omitt+ ! solidity ( ^! fir/ n9e cubes1 he takes ano!r set ( n9e slabs to repres5t ! next wall ( cubes4 ,la/ly1 he repres5ts ! ?ird wall by a ?ird set1 omitt+ ! solidity ( bo? second & ?ird walls4 ,9 ?is mann]1 he evid5tly repres5ts ! ext5sion ( ! ,block upw>ds & sideways1 9 ! ,z & ,x directions2 b 9 ! ,y direction he is p[]less1 & is compell$ to repres5t ext5sion 9 t direction by sett+ ! ?ree sets ( slabs alongside 9 8 plane4 ,! second & ?ird sets d5ote ! height & bread? ( ! respective walls1 b n _! dep? or ?ickness4 ,n[1 note t ! ,block ext5ds ?ree in*es 9 ea* ( ! ?ree directions4 ,! plane-be+ c repres5t two ( ^! dim5sions on 8 plane2 b ! unkn[n direction he has to repres5t by a repetition ( 8 plane figures4 ,! cube ext5ds ?ree in*es 9 ! ,y direction4 ,he has to use #c sets ( slabs4 ,! ,block is built up arbitr>ily 9 ?is mann]3 ,/>t+ f ,urna ,mala & go+ up1 we come to a ,br[n cube1 & !n to a ,light-blue cube4 ,/>t+ f ,urna ,mala & go+ "r1 we come to an ,orange & a ,fawn cube4 ,/>t+ f ,urna ,mala & go+ away f u1 we come to a ,blue & a ,buff cube4 ,n[1 ! plane-be+ repres5ts ! ,br[n & ,orange cubes by squ>es ly+ next to ! squ>e :i* repres5ts ,urna ,mala4 ,! ,blue cube is z close z ! ,br[n cube to ,urna ,mala1 b he c f9d no place 9 ! plane :]e he c place a ,blue squ>e s z to %[ ?is co-equal proximity ( bo? cubes to ! fir/4 ,s he is =c$ to put a ,blue squ>e any:]e 9 8 plane & say ( x3 8,?is ,blue squ>e repres5ts :at ,i %\ld arrive at1 if ,i />t$ f ,urna ,mala & mov$ away1 t is 9 ! ,y or unkn[n direction40 ,n[1 j z !re are ?ree cubes go+ up1 s !re are ?ree go+ away4 ,h5ce1 besides ! ,blue squ>e plac$ any:]e on ! plane1 he mu/ also place a ,buff squ>e beyond x1 to %[ t ! ,block ext5ds z f> away z x does upw>ds & sideways4 7,ea* cube be+ a diff]5t col\r1 !re w be z _m diff]5t col\rs ( squ>es z ( cubes47 ,x w easily be se5 t n only ! ,gold squ>e1 b also ! ,orange & e o!r squ>e 9 ! fir/ set ( slabs mu/ h two o!r squ>es set some:]e beyond x on ! plane to repres5t ! ext5sion ( ! ,block away1 or 9 ! unkn[n ,y direction4 ,com+ n[ to ! repres5tation ( a f\r-dim5sional block1 we see t we c %[ only ?ree dim5sions by cubic blocks1 & t ! f\r? c only be repres5t$ by repetitions ( su* blocks4 ,!re mu/ be a c]ta9 am\nt ( arbitr>y nam+ & col\r+4 ,! col\rs h be5 *os5 z n[ /at$4 ,take ! fir/ ,block ( ! #ha ,set4 ,we are famili> ) its col\rs1 & !y c be f\nd at any "t by ref]5ce to ,model #a4 ,n[1 suppose ! ,gold cube to repres5t :at we c see 9 our space ( a ,gold tess>act2 ! o!r cubes ( ,block #a give ! col\rs ( ! tess>acts :i* lie 9 ! ?ree directions ,x1 ,y1 & ,z f ! ,gold "o4 ,b :at is ! col\r ( ! tess>act :i* lies next to ! ,gold 9 ! unkn[n direction1 ,w8 ,let u suppose x to be ,/one 9 col\r4 ,tak+ out ,block #b ( ! #ha ,set & arrang+ x on ! patt]n ( ,model #i1 we f9d 9 x a ,/one cube4 ,b1 j z !re are ?ree tess>acts 9 ! ,x1 ,y1 & ,z directions1 z %[n by ! cubes 9 ,block #a1 s also mu/ !re be ?ree tess>acts 9 ! unkn[n direction1 ,w4 ,take ,block #c ( ! #ha ,set4 ,?is ,block c be arrang$ on ! patt]n ( ,model #b4 ,9 x !re is a ,silv] cube :]e ! ,gold cube lies 9 ,block #a4 ,h5ce1 we may say1 ! tess>act :i* comes next to ! ,/one "o 9 ! unkn[n direction f ! ,gold1 is ( a ,silv] col\r4 ,n[1 a cube 9 all ^! cases repres5ts a tess>act4 ,betwe5 ! ,gold & ,/one cubes !re is an in* 9 ! unkn[n direction4 ,! ,gold tess>act is suppos$ to be ,gold ?r\gh\t 9 all f\r directions1 & s also is ! ,/one4 ,we may imag9e x 9 ?is way4 ,suppose ! set ( ?ree tess>acts1 ! ,gold1 ! ,/one1 & ! ,silv] to move ?r\gh our space at ! rate ( an in* a m9ute4 ,we %\ld fir/ see ! ,gold cube :i* w\ld la/ a m9ute1 !n ! ,/one cube = a m9ute1 & la/ly ! ,silv] cube a m9ute4 7,?is is precisely analog\s to ! appe>ance ( pass+ cubes to ! plane-be+ z successive squ>es la/+ a m9ute47 ,aft] t1 no?+ w\ld be visible4 ,n[1 j z we mu/ suppose t !re are ?ree tess>acts proce$+ f ! ,gold cube 9 ! unkn[n direction1 s !re mu/ be ?ree tess>acts ext5d+ 9 ! unkn[n direction f e "o ( ! #bg cubes ( ! fir/ ,block4 ,! ,block ( #bg cubes is n a ,block ( #bg tess>acts1 b x repres5ts z mu* ( !m z we c see at once 9 our space2 & !y =m ! fir/ portion or lay] 7l ! fir/ wall ( cubes to ! plane-be+7 ( a set ( eighty-"o tess>acts1 ext5d+ to equal di/ances 9 all f\r directions4 ,?us1 to repres5t ! :ole ,block ( tess>acts !re are #ha cubes1 or ?ree ,blocks ( #bg ea*4 ,n[1 x is obvi\s t1 j z a cube has v>i\s plane b\nd>ies1 s a tess>act has v>i\s cube b\nd>ies4 ,! cubes ( ! tess>act1 :i* we h be5 reg>d+1 h be5 ^? conta9+ ! ,x1 ,y1 & ,z directions1 j z ! plane-be+ reg>d$ ! ,mo5a faces conta9+ ! ,x & ,z directions4 ,&1 z long z ! tess>act is un*ang$ 9 its position ) reg>d to our space1 we c nev] see any portion ( x :i* is 9 ! unkn[n direction4 ,simil>ly1 we saw t a plane-be+ c\ld n see ! p>ts ( a cube :i* w5t 9 ! ?ird direction1 until ! cube 0 turn$ r\nd "o ( its edges4 ,9 ord] to make x q cle> :at p>ts ( a cube came into ! plane1 we gave di/9ct names to !m4 ,?us1 ! squ>es conta9+ ! ,z & ,x directions 7 call$ ,mo5a & ,murex2 ^? conta9+ ! ,z & ,y1 ,alvus & ,proes2 & ^? ! ,x & ,y1 ,syce & ,mel4 ,n[1 simil>ly ) our f\r axes1 any ?ree w det]m9e a cube4 ,let ! tess>act 9 its normal position h ! cube ,mala det]m9$ by ! axes ,z1 ,x1 ,y4 ,let ! cube ,l> be t :i* is det]m9$ by ,x1 ,y1 ,w1 t is1 ! cube :i*1 />t+ f ! ,x ,y plane1 /ret*es "o in* 9 ! unkn[n or ,w direction4 ,let ,vesp] be ! cube det]m9$ by ,z1 ,y1 ,w1 & ,pluvium by ,z1 ,x1 ,w4 ,& let ^! cubes h opposite cubes ( ! foll[+ names3 ,mala has ,m>go ,l> ,velum ,vesp] ,idus ,pluvium ,tela ,ano!r way ( look+ at ! matt] is ?is4 ,:5 a cube is g5]at$ f a squ>e1 ea* ( ! l9es b\nd+ ! squ>e becomes a squ>e1 & ! squ>e itself becomes a cube1 giv+ two squ>es 9 its initial & f9al positions4 ,:5 a cube moves 9 ! new & unkn[n direction1 ea* ( its planes traces a cube & x g5]ates a tess>act1 giv+ 9 its initial & f9al positions two cubes4 ,?us !re are eight cubes b\nd+ ! tess>act1 six ( !m f ! six plane sides & two f ! cube itself4 ,^! latt] two are ,mala & ,m>go4 ,! cubes f ! six sides are3 ,l> f ,syce1 ,velum f ,mel1 ,vesp] f ,alvus1 ,idus f ,proes1 ,pluvium f ,mo5a1 ,tela f ,murex4 ,& j z a plane-be+ c only see ! squ>es ( a cube1 s we c only see ! cubes ( a tess>act4 ,x may be said t ! cube c be pu%$ p>tly ?r\gh ! plane1 s t ! plane-be+ sees a section betwe5 ,mo5a & ,murex4 ,simil>ly1 ! tess>act c be pu%$ ?r\gh our space s t we c see a section betwe5 ,mala & ,m>go4 ,!re is a me?od ( approa*+ ! matt]1 :i* settles all difficulties1 & provides u ) a nom5clature = e "p ( ! tess>act4 ,we h se5 h[ by writ+ d[n ! names ( ! cubes ( a block1 & !n suppos+ t _! numb] increases1 c]ta9 sets ( ! names come to d5ote po9ts1 l9es1 planes1 & solid4 ,simil>ly1 if we write d[n a set ( names ( tess>acts 9 a block1 x w be f\nd t1 :5 _! numb] is increas$1 c]ta9 sets ( ! names come to d5ote ! v>i\s p>ts ( a tess>act4 ,= ?is purpose1 let u take ! #ha ,set1 & use ! cubes to repres5t tess>acts4 ,! :ole ( ! #ha cubes make "o s+le tess>actic set ext5d+ ?ree in*es 9 ea* ( ! f\r directions4 ,! names mu/ be rememb]$ to d5ote tess>acts4 ,?us1 ,corvus is a tess>act :i* has ! tess>acts ,cuspis & ,nug to ! "r1 ,arctos & ,ilex above x1 ,dos & ,ci/a away f x1 & ,ops & ,spira 9 ! f\r? or unkn[n direction f x4 ,x w be evid5t at once1 t to write ^! names 9 any repres5tative ord] we mu/ adopt an arbitr>y sy/em4 ,we require !m runn+ 9 f\r directions2 we h only two on pap]4 ,! ,x direction 7f left to "r7 & ! ,y 7f ! bottom t[>ds ! top ( ! page7 w be assum$ to be truly repres5t$4 ,! ,z direction w be symboliz$ by writ+ ! names 9 floors1 ! upp] floors always prec$+ ! l[]4 ,la/ly1 ! f\r?1 or ,w1 direction 7:i* has to be symboliz$ 9 ?ree-dim5sional space by sett+ ! solids 9 an arbitr>y position7 w be signifi$ by writ+ ! names 9 blocks1 ! "n :i* /&s 9 any "o place 9 any block be+ next 9 ! ,w direction to t :i* occupies ! same position 9 ! block be=e or aft] x4 ,?us1 ,ops is writt5 9 ! same place 9 ! ,second ,block1 ,spira 9 ! ,?ird ,block1 z ,corvus occupies 9 ! ,fir/ ,block4 ,s9ce !re are an equal numb] ( tess>acts 9 ea* ( ! f\r directions1 !re w be ?ree floors ,z :5 !re are ?ree ,x & ,y4 ,also1 !re w be ?ree ,blocks ,w4 ,if !re be f\r tess>acts 9 ea* direction1 !re w be f\r floors ,z1 & f\r blocks ,w4 ,?us1 :5 ! numb] 9 ea* direction is enl>g$1 ! numb] ( blocks ,w is equal to ! numb] ( tess>acts 9 ea* kn[n direction4 ,on pp4 #acf1 #acg 7 giv5 ! names z us$ = a cubic block ( #bg or #fd4 ,us+ ! same & m names = a tess>actic ,set1 & rememb]+ t ea* "n n[ repres5ts1 n a cube1 b a tess>act1 we obta9 ! foll[+ nom5clature1 ! ord] 9 :i* ! names are writt5 be+ t /at$ above3 ,,?ird ,,block4 ,upp] ,solia ,livor ,talus ,floor4 ,l5sa ,l>es ,calor ,felis ,?olus ,pass] ----------------- ,middle ,lixa ,portica ,v5a ,floor4 ,crux ,m>go ,sal ,pagus ,silex ,onag] ----------------- ,l[] ,panax ,m5sura ,mugil ,floor4 ,opex ,lappa ,mappa ,spira ,luca ,ancilla ,,second ,,block4 ,upp] ,orsa ,mango ,lib]a ,floor4 ,creta ,velum ,meatus ,lucta ,limbus ,pator ----------------- ,middle ,camo5a ,tela ,orca ,floor4 ,vesp] ,tess>act ,idus ,pag9a ,pluvium ,pactum ----------------- ,l[] ,lis ,lorica ,(fex ,floor4 ,lua ,l> ,olla ,ops ,lotus ,limus ,,fir/ ,,block4 ,upp] ,olus ,semita ,lama ,floor4 ,via ,mel ,it] ,ilex ,callis ,sors ----------------- ,middle ,buc9a ,murex ,daps ,floor4 ,alvus ,mala ,proes ,arctos ,mo5a ,f> ----------------- ,l[] ,ci/a ,cadus ,crus ,floor4 ,dos ,syce ,bolus ,corvus ,cuspis ,nug ,x is evid5t t ?is set ( tess>acts c\ld be increas$ to ! numb] ( f\r 9 ea* direction1 ! names be+ us$ z be=e = ! cubic blocks on pp4 #acf1 #acg1 & 9 t case ! ,second ,block w\ld be duplicat$ to make ! f\r blocks requir$ 9 ! unkn[n direction4 ,comp>+ su* an #ha ,set & #bef ,set1 we %\ld f9d t ,cuspis1 :i* 0 a s+le tess>act 9 ! #ha ,set became two tess>acts 9 ! #bef ,set4 ,&1 if we introduc$ a l>g] numb]1 x w\ld simply become long]1 & n increase 9 any o!r dim5sion4 ,?us1 ,cuspis w\ld become ! "n ( an edge4 ,simil>ly1 ,dos w\ld become ! "n ( an edge1 & also ,arctos4 ,ops1 :i* is f\nd 9 ! ,middle ,block ( ! #ha ,set1 occurs bo? 9 ! ,second & ,?ird ,blocks ( ! #bef ,set2 t is1 x also t5ds to elongate & n ext5d 9 any o!r direction1 & may !re=e be us$ z ! "n ( an edge ( a tess>act4 ,look+ at ! cubes :i* repres5t ! ,syce tess>acts1 we f9d t1 ?\gh !y increase 9 numb]1 !y increase only 9 two directions2 !re=e1 ,syce may be tak5 to signify a squ>e4 ,b1 look+ at :at comes f ,syce 9 ! ,w direction1 we f9d 9 ! ,middle ,block ( ! #ha ,set "o ,l>1 & 9 ! ,second & ,?ird ,blocks ( ! #bef ,set f\r ,l>s ea*4 ,h5ce1 ,l> ext5ds 9 ?ree directions1 ,x1 ,y1 ,w1 & becomes a cube4 ,simil>ly1 ,mo5a is a plane2 b ,pluvium1 :i* proce$s f x1 ext5ds n only sideways & upw>ds l ,mo5a1 b 9 ! unkn[n direction also4 ,x occurs 9 bo? ,middle ,blocks ( ! #bef ,set4 ,h5ce1 x also is a cube4 ,we h n[ consid]$ su* p>ts ( ! ,sets z conta9 "o1 two1 & ?ree dim5sions4 ,b !re is "o "p :i* conta9s f\r4 ,x is t nam$ ,tess>act4 ,9 ! #bef ,set !re are eight su* cubes 9 ! ,second1 & eight 9 ! ,?ird ,block2 t is1 !y ext5d ,z1 ,x1 ,y1 & also ,w4 ,!y may1 !re=e1 be consid]$ to repres5t t "p ( a tess>act or tess>actic ,set1 :i* is analog\s to ! int]ior ( a cube4 ,! arrangem5t ( col\rs correspond+ to ^! names is se5 on ,model #a correspond+ to ,mala1 ,model #b to ,m>go1 & ,model #i to ! int]m$iate block4 ,:5 we take ! view ( ! tess>act ) :i* we comm5c$1 & 9 :i* ,arctos goes ,z1 ,cuspis ,x1 ,dos ,y1 & ,ops ,w1 we see ,mala 9 our space4 ,b :5 ! tess>act is turn$ s t ! ,ops l9e goes -,x1 :ile ,cuspis is turn$ ,w1 ! o!r two rema9+ z !y 71 !n we d n see ,mala1 b t cube :i*1 9 ! orig9al position ( ! tess>act1 conta9s ! ,z1 ,y1 ,w1 directions1 t is1 ! ,vesp] cube4 ,a plane-be+ may beg9 to /udy a block ( cubes by _! ,syce squ>es2 b if ! block be turn$ r\nd ,dos1 he w h ,alvus squ>es 9 8 space1 & he mu/ !n use !m to repres5t ! cubic ,block4 ,s1 :5 ! tess>actic ,set is turn$ r\nd1 ,mala cubes leave our space1 & ,vesp]s ent]4 ,!re are two ways :i* c be foll[$ 9 /udy+ ! ,set ( tess>acts4 ,i4 ,ea* tess>act ( "o in* e way c be suppos$ to be ( ! same col\r ?r\gh\t1 s t1 :i*ev] way x be turn$1 :i*ev] ( its edges co9cide ) our kn[n axes1 x appe>s to u z a cube ( "o uni=m col\r4 ,?us1 if ,urna be ! tess>act1 ,urna ,mala w\ld be a ,gold cube1 ,urna ,vesp] a ,gold cube1 & s on4 ,?is me?od is1 = ! mo/ "p1 adopt$ 9 ! foll[+ pages4 ,9 ?is case1 a :ole ,set ( #d #d #d #d tess>acts w\ld 9 col\rs resemble a set compos$ ( f\r cubes l ,models #a1 #i1 #i1 & #b4 ,b1 :5 any "q ab\t a p>ticul> tess>act has to be settl$1 x is advantage\s1 = ! sake ( di/9ctness1 to suppose x col\r$ 9 its diff]5t regions z ! :ole set is col\r$4 ,,ii4 ,! o!r plan is1 to />t ) ! cubic sides ( ! in* tess>act1 ea* col\r$ accord+ to ! s*eme ( ! ,models #a to #h4 ,9 ?is case1 ! l9es1 if %[n at all1 %\ld be v ?94 ,=1 9 fact1 only ! faces w\ld be se51 z ! wid? ( ! l9es w\ld only be equal to ! ?ickness ( our matt] 9 ! f\r? dim5sion1 :i* is indi/+ui%able to ! s5ses4 ,if su* completely col\r$ cubes be us$1 less error is likely to creep 92 b x is a disadvantage t ea* cube 9 ! sev]al blocks is exactly l ! o!rs 9 t block4 ,if ! read] make su* a set to "w ) = a "t1 he w ga9 greatly1 = ! real way ( acquir+ a s5se ( high] space is to obta9 ^? exp]i5ces ( ! s5ses exactly1 :i* ! obs]vation ( a f\r-dim5sional body w\ld give4 ,^! ,models #a-h are call$ sides ( ! tess>act4 ,to make ! matt] p]fectly cle>1 x is be/ to suppose t any tess>act or set ( tess>acts :i* we exam9e1 has a duplicate exactly simil> 9 %ape & arrangem5t ( p>ts1 b diff]5t 9 _! col\r+4 ,9 ! fir/1 let ea* tess>act h "o col\r ?r\gh\t1 s t all its cubes1 appreh5d$ 9 turn 9 our space1 w be ( "o & ! same col\r4 ,9 ! duplicate1 let ea* tess>act be s col\r$ z to %[ its diff]5t cubic sides by _! diff]5t col\rs4 ,!n1 :5 we h x turn$ to u 9 diff]5t aspects1 we %all see diff]5t cubes1 & :5 we try to trace ! contacts ( ! tess>acts ) ea* o!r1 we %all be help$ by realiz+ ea* "p ( e tess>act 9 its own col\r4 ,,*apt] ,,viii4 ,,repres5tation ,,( ,,f\r-,,space ,,by ,,"n4 ,,/udy ,,( ,,tess>acts4 ,we h n[ survey$ all ! prelim9>y gr\nd1 & c /udy ! masses ( tess>acts )\t obscurity4 ,we require a scaffold or framework = ?is purpose1 :i* 9 ?ree dim5sions w consi/ ( eight cubic spaces or octants assembl$ r\nd "o po9t1 z 9 two dim5sions x consi/$ ( f\r squ>es or quadrants r\nd a po9t4 ,^! eight octants lie betwe5 ! ?ree axes ,z1 ,x1 ,y1 :i* int]sect at ! giv5 po9t1 & c be nam$ accord+ to _! positions betwe5 ! positive & negative directions ( ^? axes4 ,?us ! octant ,z1 ,x1 ,y1 is t :i* is conta9$ by ! positive portions ( all ?ree axes2 ! octant ,z1 ,x1 ,y1 t :i* is to ! left ( ,z1 ,x1 ,y1 & betwe5 ! positive p>ts ( ,z & ,y & ! negative ( ,x4 ,to illu/rate ?is q cle>ly1 let u take ! eight cubes--,urna1 ,moles1 ,plebs1 ,fr5um1 ,uncus1 ,pallor1 ,bid5s1 ,o/rum--& place !m 9 ! eight octants4 ,let !m be plac$ r\nd ! po9t ( int]section ( ! axes2 ,pallor ,corvus1 ,plebs ,ilex1 etc41 w be at t po9t4 ,_! positions w !n be3-- ,urna 9 ! ,octant ,z ,x ,y ,moles ,z ,x ,y ,plebs ,z ,x ,y ,fr5um ,z ,x ,y ,uncus ,z ,x ,y ,pallor ,z ,x ,y ,bid5s ,z ,x ,y ,o/rum ,z ,x ,y ,! names us$ = ! cubes1 z !y are be=e u1 are z foll[s3-- ,,?ird ,,block4 ,?ird ,arcus ,mala ,ovis ,mala ,portio ,mala ,floor4 ,laurus ,mala ,tigris ,mala ,segm5 ,mala ,axis ,mala ,troja ,mala ,aries ,mala ,second ,po/is ,mala ,clipeus ,mala ,tabula ,mala ,floor4 ,orcus ,mala ,lac]ta ,mala ,te/udo ,mala ,v]bum ,mala ,luctus ,mala ,anguis ,mala ,fir/ ,telum ,mala ,nepos ,mala ,angu/a ,mala ,floor4 ,polus ,mala ,p5ates ,mala ,vulcan ,mala ,c]vix ,mala ,securis ,mala ,v9culum ,mala ,,second ,,block4 ,?ird ,ara ,mala ,vom] ,mala ,pluma ,mala ,floor4 ,pra$a ,mala ,sac]dos ,mala ,hydra ,mala ,cortex ,mala ,mica ,mala ,flagellum ,mala ,second ,pilum ,mala ,glans ,mala ,colus ,mala ,floor4 ,ocrea ,mala ,tess]a ,mala ,domitor ,mala ,c>do ,mala ,cudo ,mala ,malleus ,mala ,fir/ ,agm5 ,mala ,lacus ,mala ,arvus ,mala ,floor4 ,crates ,mala ,cura ,mala ,lim5 ,mala ,?yrsus ,mala ,vitta ,mala ,sceptrum ,mala ,,fir/ ,,block4 ,?ird ,m>s ,mala ,m]ces ,mala ,tyro ,mala ,floor4 ,spicula ,mala ,mora ,mala ,oliva ,mala ,comes ,mala ,tibic5 ,mala ,ve/is ,mala ,second ,ala ,mala ,cortis ,mala ,a] ,mala ,floor4 ,uncus ,mala ,pallor ,mala ,t]gum ,mala ,o/rum ,mala ,bid5s ,mala ,sc5a ,mala ,fir/ ,sector ,mala ,hama ,mala ,remus ,mala ,floor4 ,fr5um ,mala ,plebs ,mala ,sypho ,mala ,urna ,mala ,moles ,mala ,saltus ,mala ,_! col\rs c be f\nd by ref]5ce to ! ,models #a1 #i1 #b1 :i* correspond respectively to ! ,fir/1 ,second1 & ,?ird ,blocks4 ,?us1 ,urna ,mala is ,gold2 ,moles1 ,orange2 ,saltus1 ,fawn2 ,?yrsus1 ,/one2 ,c]vix1 ,silv]4 ,! cubes ^: col\rs are n %[n 9 ! ,models1 are ,pallor ,mala1 ,tess]a ,mala1 & ,lac]ta ,mala1 :i* are equival5t to ! int]iors ( ! ,model cubes1 & are respectively ,light-buff1 ,wood51 & ,sage-gre54 ,^! #ha cubes are ! cubic sides & sections ( ! tess>acts ( an #ha tess>actic ,set1 :i* measures ?ree in*es 9 e direction4 ,we suppose x to pass ?r\gh our space4 ,let u call ! positive unkn[n direction ,ana 7i4e41 ,w7 & ! negative unkn[n direction ,kata 7-,w74 ,!n1 z ! :ole tess>act moves ,kata at ! rate ( an in* a m9ute1 we see fir/ ! ,fir/ ,block ( #bg cubes = "o m9ute1 !n ! ,second1 & la/ly ! ,?ird1 ea* la/+ "o m9ute4 ,n[1 :5 ! ,fir/ ,block /&s 9 ! normal position1 ! edges ( ! tess>act t run f ! ,corvus corn] ( ,urna ,mala1 are3 ,arctos 9 ,z1 ,cuspis 9 ,x1 ,dos 9 ,y1 ,ops 9 ,w4 ,h5ce1 we d5ote ?is position by ! foll[+ symbol3-- ,z ,x ,y ,w a c d o :]e a /&s = ,arctos1 c = ,cuspis1 d = ,dos1 & o = ,ops1 & ! o!r lett]s = ! f\r axes 9 space4 a1 c1 d1 o are ! axes ( ! tess>act1 & c take up diff]5t directions 9 space ) reg>d to u4 99 99 99 99 99 ,let u n[ take a small] f\r-dim5sional set4 ,( ! #ha ,set let u take ! foll[+3-- ,z ,x ,y ,w a c d o ,,second ,,block4 ,second ,floor4 ,ocrea ,mala ,tess]a ,mala ,c>do ,mala ,cudo ,mala ,fir/ ,floor4 ,crates ,mala ,cura ,mala ,?yrsus ,mala ,vitta ,mala ,,fir/ ,,block4 ,second ,floor4 ,uncus ,mala ,pallor ,mala ,o/rum ,mala ,bid5s ,mala ,fir/ ,floor4 ,fr5um ,mala ,plebs ,mala ,urna ,mala ,moles ,mala ,let ! ,fir/ ,block be put up be=e u 9 ,z ,x ,y1 7,urna ,corvus is at ! junction ( our axes ,z ,x ,y74 ,! ,second ,block is n[ "o in* di/ant 9 ! unkn[n direction2 &1 if we suppose ! tess>actic ,set to move ?r\gh our space at ! rate ( "o in* a m9ute1 ! ,second w ent] 9 "o m9ute1 & replace ! fir/4 ,b1 in/ead ( ?is1 let u suppose ! tess>acts to turn s t ,ops1 :i* n[ goes ,w1 %all g -,x4 ,!n we c see 9 our space t cubic side ( ea* tess>act :i* is conta9$ by ! l9es ,arctos1 ,dos1 & ,ops1 ! cube ,vesp]2 & we %all no long] h ! ,mala sides b ! ,vesp] sides ( ! tess>actic ,set 9 our space4 ,we w n[ build x up 9 its ,vesp] view 7z we built up ! cubic ,block 9 its ,alvus view74 ,take ! ,gold cube1 :i* n[ means ,urna ,vesp]1 & place x on ! left h& ( its =m] position z ,urna ,mala1 t is1 9 ! octant ,z ,x ,y4 ,?yrsus ,vesp]1 :i* previ\sly lay j beyond ,urna ,vesp] 9 ! unkn[n direction1 w n[ lie j beyond x 9 ! -,x direction1 t is1 to ! left ( x4 ,! tess>actic ,set is n[ 9 ! position ,z ,x ,y ,w a d c 7! m9us sign ov] ! o mean+ t ,ops runs 9 ! negative direction71 & its ,vesp]s lie 9 ! foll[+ ord]3-- ,,second ,,block4 ,second ,floor4 ,tess>a ,pallor ,cudo ,bid5s ,fir/ ,floor4 ,cura ,plebs ,vitta ,moles ,,fir/ ,,block4 ,second ,floor4 ,ocrea ,uncus ,c>do ,o/rum ,fir/ ,floor4 ,crates ,fr5um ,?yrsus ,urna ,! "n ,vesp] is left out 9 ! above li/ = ! sake ( brevity1 b %\ld be us$ 9 /udy+ ! positions4 ,illu/ration3 ,fig4 #bj4 ,on comp>+ ! two li/s ( ! ,mala view & ,vesp] view1 x w be se5 t ! cubes pres5t$ 9 ! ,vesp] view are new sides ( ! tess>act1 & t ! arrangem5t ( !m is diff]5t f t 9 ! ,mala view4 7,?is is analog\s to ! *anges 9 ! slabs f ! ,mo5a to ,alvus view ( ! cubic ,block47 ,( c\rse1 ! ,vesp]s ( all ^! tess>acts are n visible at once 9 our space1 any m ?an are ! ,mo5as ( all ?ree walls ( a cubic ,block to a plane-be+4 ,b if ! tess>actic ,set be suppos$ to move ?r\gh space 9 ! unkn[n direction at ! rate ( an in* a m9ute1 ! ,second ,block w pres5t its ,vesp]s aft] ! ,fir/ ,block has la/$ a m9ute4 ,! relative position ( ! ,mala ,block & ! ,vesp] ,block may be repres5t$ 9 our space z 9 ! diagram1 ,fig4 #bj4 ,b x mu/ be di/9ctly rememb]$ t ?is arrangem5t is q conv5tional1 no m real ?an a plane-be+'s symbolization ( ! ,mo5a ,wall & ! ,alvus ,wall ( ! cubic ,block by ! arrangem5t ( _! ,mo5a & ,alvus faces1 ) ! solidity omitt$1 along "o ( 8 kn[n directions4 ,! ,vesp]s ( ! ,fir/ & ,second ,blocks _c be 9 our space simultane\sly1 any m ?an ! ,mo5as ( all ?ree walls 9 plane space4 ,to r5d] _! simultane\s pres5ce possible1 ! cubic or tess>actic ,block or ,set mu/ be brok5 up1 & its p>ts no long] reta9 _! relations4 ,?is fact is ( supreme importance 9 consid]+ high] space4 ,endless fallacies creep 9 z soon z x is =gott5 t ! cubes are m]ely repres5tative z ! slabs 71 & ! positions 9 our space m]ely conv5tional & symbolical1 l ^? ( ! slabs along ! plane4 ,& ^! fallacies are s mu* fo/]$ by aga9 symboliz+ ! cubic symbols & _! symbolical positions 9 p]spective draw+s or diagrams1 t ! read] %\ld surr5d] all hope ( le>n+ space f ?is book or ! draw+s alone1 & "w e ?\ght out ) ! cubes !mselves4 ,if we want to see :at ea* individual cube ( ! tess>actic faces pres5t$ to u 9 ! la/ example is l1 we h only to consid] ea* ( ! ,malas simil> 9 its p>ts to ,model #a1 & ea* ( ! ,vesp]s to ,model #e4 ,& x mu/ always be rememb]$ t ! cubes1 ?\gh us$ to repres5t bo? ,mala & ,vesp] faces ( ! tess>act1 mean z great a diff]5ce z ! slabs us$ = ! ,mo5a & ,alvus faces ( ! cube4 ,if ! tess>actic ,set move ,kata ?r\gh our space1 :5 ! ,vesp] faces are pres5t$ to u1 we see ! foll[+ p>ts ( ! tess>act ,urna 7&1 !re=e1 also ! same p>ts ( ! o!r tess>acts73 7#a7 ,urna ,vesp]1 :i* is ,model #e4 7#b7 ,a p>allel section betwe5 ,urna ,vesp] & ,urna ,idus1 :i* is ,model #aa4 7#c7 ,urna ,idus1 :i* is ,model #f4 ,:5 ,urna ,idus has pass$ ,kata our space1 ,moles ,vesp] ent]s x2 !n a section betwe5 ,moles ,vesp] & ,moles ,idus1 & !n ,moles ,idus4 ,"h we h evid5tly obs]v$ ! tess>act m m9utely2 z x passes ,kata ?r\gh our space1 />t+ on its ,vesp] side1 we h se5 ! p>ts :i* w\ld be g5]at$ by ,vesp] mov+ along ,cuspis--t is ,ana4 ,two o!r arrangem5ts ( ! tess>acts h to be le>nt besides ^? f ! ,mala & ,vesp] aspect4 ,"o ( !m is ! ,pluvium aspect4 ,build up ! ,set 9 ,z ,x ,y1 lett+ ,arctos run ,z1 ,cuspis ,x1 & ,ops ,y4 ,9 ! common plane ,mo5a1 ,urna ,pluvium co9cides ) ,urna ,mala1 ?\gh !y _c be 9 our space toge!r2 s too ,moles ,pluvium ) ,moles ,mala1 ,o/rum ,pluvium ) ,o/rum ,mala4 ,& ly+ t[>ds u1 or ,y1 is n[ t tess>act :i* be=e lay 9 ! ,w direction f ,urna1 viz41 ,?yrsus4 ,! ord] w !re=e be ! foll[+ 7a /> d5otes ! cube ^: corn] is at po9t ( int]section ( ! axes1 & ! "n ,pluvium mu/ be und]/ood to foll[ ea* ( ! names73 ,z ,x ,y ,w a c d ,,second ,,block4 ,second ,floor4 ,uncus ,pallor ,ocrea ,tess]a ,fir/ ,floor4 ,fr5um ,plebs ,crates ,cura ,,fir/ ,,block4 ,second ,floor4 ,o/rum ,bid5s ,c>do ,cudo ,fir/ ,floor4 99,urna ,moles ,?yrsus ,vitta ,?us ! wall ( cubes 9 contact ) t wall ( ! ,mala position :i* conta9s ! ,urna1 ,moles1 ,o/rum1 & ,bid5s ,malas1 is a wall compos$ ( ! ,pluviums ( ,urna1 ,moles1 ,o/rum1 & ,bid5s4 ,! wall next to ?is1 & ne>] to u1 is ( ,?yrsus1 ,vitta1 ,c>do1 ,cudo1 ,pluviums4 ,! ,second ,block is "o in* out ( our ,space1 & only ent]s x if ! ,block moves ,kata4 ,model #g %[s ! ,pluvium cube2 & ea* ( ! cubes ( ! tess>acts se5 9 ! ,pluvium position is a ,pluvium4 ,if ! tess>actic ,set mov$ ,kata1 we w\ld see ! ,section betwe5 ,pluvium & ,tela = all b a m9ute2 & !n ,tela w\ld ent] our space1 & ! ,tela ( ea* tess>act w\ld be se54 ,model #ab %[s ! ,section f ,pluvium to ,tela4 ,model #h is ,tela4 ,tela only la/s = a fla%1 z x has only ! m9ute/ magnitude 9 ! unkn[n or ,ana direction4 ,!n1 ,fr5um ,pluvium takes ! place ( ,urna ,tela2 &1 :5 x passes ?r\gh1 we see a simil> section betwe5 ,fr5um ,pluvium & ,fr5um ,tela1 & la/ly ,fr5um ,tela4 ,!n ! tess>actic ,set passes out1 or ,kata1 our space4 ,a simil> process takes place ) e o!r tess>act1 :5 ! ,set ( tess>acts moves ?r\gh our space4 ,!re is /ill "o m arrangem5t to be le>nt4 ,if ! l9e ( ! tess>act1 :i* 9 ! ,mala position goes ,ana1 or ,w1 be *ang$ into ! ,z or d[nw>ds direction1 ! tess>act w !n appe> 9 our space bel[ ! ,mala position2 & ! side pres5t$ to u w n be ,mala1 b t :i* conta9s ! l9es ,dos1 ,cuspis1 & ,ops4 ,?is side is ,model #c1 & is call$ ,l>4 ,und]nea? ! place :i* 0 occupi$ by ,urna ,mala1 w come ,urna ,l>2 "u ! place ( ,moles ,mala1 ,moles ,l>2 "u ! place ( ,fr5um ,mala1 ,fr5um ,l>4 ,! tess>act1 :i* 9 ! ,mala position 0 an in* out ( our space ,ana1 or ,w1 f ,urna ,mala1 w n[ come into x an in* d[nw>ds1 or ,z1 bel[ ,urna ,mala1 ) its ,l> pres5t$ to u2 t is1 ,?yrsus ,l> w be bel[ ,urna ,l>4 ,9 ! :ole arrangem5t ( !m writt5 bel[1 ! highe/ floors are writt5 fir/1 = n[ !y /ret* d[nw>ds in/ead ( upw>ds4 ,! "n ,l> is und]/ood aft] ea*4 ,z ,x ,y ,w c d a ,,second ,,block4 ,second ,floor4 ,uncus ,pallor ,o/rum ,bid5s ,fir/ ,floor4 ,ocrea ,tess]a ,c>do ,cudo ,,fir/ ,,block4 ,second ,floor4 ,fr5um ,plebs 99,urna ,moles ,fir/ ,floor4 ,crates ,cura ,?yrsus ,vitta ,"h x is evid5t t :at 0 ! l[] floor ( ,malas1 ,urna1 ,moles1 ,plebs1 ,fr5um1 n[ appe>s z if c>ri$ d[nw>ds in/ead ( upw>ds1 ,l>s be+ pres5t$ 9 our space in/ead ( ,malas4 ,?is ,block ( ,l>s is :at we see ( ! tess>act ,set :5 ! ,arctos l9e1 :i* 9 ! ,mala position goes up1 is turn$ into ! ,ana1 or ,w1 direction1 & ! ,ops l9e comes 9 d[nw>ds4 ,! re/ ( ! tess>acts1 :i* consi/s ( ! cubes opposite to ! f\r treat$ above1 & ( ! tess>actic space betwe5 !m1 is all ,ana 9 our space4 ,if ! tess>act be mov$ ?r\gh our space--= in/ance1 :5 ! ,l>s are pres5t 9 x--we see1 tak+ ,urna alone1 fir/ ! section betwe5 ,urna ,l> & ,urna ,velum 7,model #aj71 & !n ,urna ,velum 7,model #d71 & simil>ly ! sections & ,velums ( ea* tess>act 9 ! ,set4 ,:5 ! ,fir/ ,block has pass$ ,kata our space1 ,o/rum ,l> ent]s2 & ! ,l>s ( ! ,second ,block ( tess>acts occupy ! places j vacat$ by ! ,velums ( ! ,fir/ ,block4 ,!n1 z ! tess>actic ,set moves on ,kata1 ! sections betwe5 ,velums & ,l>s ( ! ,second ,block ( tess>acts ent] our space1 & f9ally _! ,velums4 ,!n ! :ole tess>actic ,set disappe>s f our space4 ,:5 we h le>nt all ^! aspects & passages1 we h exp]i5c$ "s ( ! pr9cipal features ( ?is small ,set ( tess>acts4 ,,*apt] ,,ix4 ,,fur!r ,,/udy ,,( ,,tess>acts4 ,:5 ! arrangem5t ( a small set has be5 ma/]$1 ! diff]5t views ( ! :ole #ha ,set %\ld be le>nt4 ,x is n[ cle> to u t1 9 ! li/ ( ! names ( ! eighty-"o tess>acts giv5 above1 ^? :i* lie 9 ! ,w direction appe> 9 diff]5t blocks1 :ile ^? t lie 9 ! ,z1 ,x1 or ,y directions c be f\nd 9 ! same block4 ,!re=e1 f ! arrangem5t giv51 :i* is d5ot$ by ,z ,x ,y ,w 1 a c d o or m briefly by a c d o1 we c =m any o!r arrangem5t4 ,to confirm ! mean+ ( ! symbol a c d o = position1 let u rememb] t ! ord] ( ! axes kn[n 9 our space w inv>iably be ,z ,x ,y1 & ! unkn[n direction w be /at$ la/1 ?us3 ,z ,x ,y ,w4 ,h5ce1 if we write a d c1 we "k t ! position or aspect int5d$ is t 9 :i* ,arctos 7a7 goes ,z1 ,ops 77 negative ,x1 ,dos 7d7 ,y1 & ,cuspis 7c7 ,w4 ,& su* an arrangem5t c be made by %ift+ ! n9e cubes on ! left side ( ! ,fir/ ,block ( ! eighty-"o tess>acts1 & putt+ !m into ! ,z ,x ,y octant1 s t !y j t\* _! =m] position4 ,next to !m1 to _! left1 we set ! n9e ( ! left side ( ! ,second ,block ( ! #ha ,set2 & next to ^! aga91 on _! left1 ! n9e ( ! left side ( ! ,?ird ,block4 ,?is ,block ( tw5ty-sev5 n[ repres5ts ,vesp] ,cubes1 :i* h only "o squ>e id5tical ) ! ,mala cubes ( ! previ\s blocks1 f :i* !y 7 tak54 ,simil>ly ! ,block :i* is "o in* ,ana1 c be made by tak+ ! n9e cubes :i* come v]tically 9 ! middle ( ea* ( ! ,blocks 9 ! fir/ position1 & arrang+ !m 9 a simil> mann]4 ,la/ly1 ! ,block :i* lies two in*es ,ana1 c be made by tak+ ! "r sides ( n9e cubes ea* f ea* ( ! ?ree orig9al ,blocks1 & arrang+ !m s t ^? 9 ! ,second orig9al ,block g to ! left ( ^? 9 ! ,fir/1 & ^? 9 ! ,?ird to _! left4 ,9 ?is mann] we %\ld obta9 ?ree new ,blocks1 :i* repres5t :at we c see ( ! tess>acts1 :5 ! direction 9 :i* ,urna1 ,moles1 ,saltus lie 9 ! orig9al ,set1 is turn$ ,w4 ,! ,pluvium ,block we c make by tak+ ! front wall ( ea* orig9al ,block1 & sett+ ea* an in* ne>] to u1 t is -,y4 ,! f> sides ( ^! cubes are ,mo5as ( ,pluviums4 ,by cont9u+ ?is treatm5t ( ! o!r walls ( ! ?ree orig9al ,blocks p>allel to ! front wall1 we obta9 two o!r ,blocks ( tess>acts4 ,! ?ree toge!r are ! tess>actic position a c d1 = 9 all ( !m ,ops lies 9 ! -,y direction1 & ,dos has be5 turn$ ,w4 ,! ,l> position is m difficult to con/ruct4 ,to put ! ,l>s ( ! ,blocks 9 _! natural position 9 our space1 we mu/ />t ) ! orig9al ,mala ,blocks1 at lea/ ?ree in*es above ! table4 ,! ,fir/ ,l> ,block is made by tak+ ! l[e/ floors ( ! ?ree ,mala ,blocks1 & plac+ !m s t t ( ! ,second is bel[ t ( ! ,fir/1 & t ( ! ,?ird bel[ t ( ! ,second4 ,! floor ( cubes ^: diagonal runs f ,urna ,l> to ,remus ,l>1 w be at ! top ( ! ,block ( ,l>s2 & t ^: diagonal goes f ,c]vix ,l> to ,angu/a ,l>1 w be at ! bottom4 ,! next ,block ( ,l>s w be made by tak+ ! middle horizontal floors ( ! ?ree orig9al ,blocks1 & plac+ !m 9 a simil> succession--! floor f ,o/rum ,l> to ,a] ,l> be+ at ! top1 t f ,c>do ,l> to ,colus ,l> 9 ! middle1 & ,v]bum ,l> to ,tabula ,l> at ! bottom4 ,! ,?ird ,l> ,block is compos$ ( ! top floor ( ! ,fir/ ,block on ! top--t is1 ( ,comes ,l> to ,tyro ,l>1 ( ,cortex ,l> to ,pluma ,l> 9 ! middle1 & ,axis ,l> to ,portio ,l> at ! bottom4 ,,*apt] ,x4 ,,cyclical ,,projections4 ,let u d5ote ! orig9al position ( ! cube1 t :]e9 ,arctos goes ,z1 ,cuspis ,x1 & ,dos ,y1 by ! expression1 ,z ,x ,y 7#a7 a c d ,if ! cube be turn$ r\nd ,cuspis1 ,dos goes ,z1 ,cuspis rema9s un*ang$1 & ,arctos goes ,y1 & we h ! position1 ,z ,x ,y d c a :]e ,z d means t ,dos runs 9 ! negative direction ( ! ,z axis f ! po9t :]e ! axes int]sect4 ,we might write ,z d b x is pref]able to write ,z d4 ,if we next turn ! cube r\nd ! l9e1 :i* runs ,y1 t is1 r\nd ,arctos1 we obta9 ! position1 ,z ,x ,y 7#b7 c d a & by means ( ?is d\ble turn we h put c & d 9 ! places ( a & c4 ,moreov]1 we h no negative directions4 ,?is position we call simply c d a4 ,if f x we turn ! cube r\nd a1 :i* runs ,y1 we get ,z ,x ,y d c a1 & if1 !n1 we turn x r\nd ,dos we get ,z ,x ,y d a c or simply d a c4 ,?is la/ is ano!r position 9 :i* all ! l9es are positive1 & ! projections1 in/ead ( ly+ 9 diff]5t quadrants1 w be conta9$ 9 "o4 ,! arrangem5t ( cubes 9 a c d we "k4 ,t 9 c d a is3 ,ve/is ,oliva ,tyro ,?ird ,sc5a ,t]gum ,a] ,floor4 ,saltus ,sypho ,remus ,tibic5 ,mora ,m]ces ,second ,bid5s ,pallor ,cortis ,floor4 ,moles ,plebs ,hama ,comes ,spicula ,m>s ,fir/ ,o/rum ,uncus ,ala ,floor4 ,urna ,fr5um ,sector ,x w be f\nd t le>n+ ! cubes 9 ?is position gives a great advantage1 = !reby ! axes ( ! cube become dissociat$ ) p>ticul> directions 9 space4 ,! d a c position gives ! foll[+ arrangem5t3 ,remus ,a] ,tyro ,hama ,cortis ,m]ces ,sector ,ala ,m>s ,sypho ,t]gum ,oliva ,plebs ,pallor ,mora ,fr5um ,uncus ,spicula ,saltus ,sc5a ,ve/is ,moles ,bid5s ,tibic5 ,urna ,o/rum ,comes ,! sides1 :i* t\* ! v]tical plane 9 ! fir/ position1 are respectively1 9 a c d ,mo5a1 9 c d a ,syce1 9 d a c ,alvus4 ,take ! %ape ,urna1 ,o/rum1 ,moles1 ,saltus1 ,sc5a1 ,sypho1 ,remus1 ,a]1 ,tyro4 ,?is gives 9 a c d ! projection3 ,urna ,mo5a1 ,o/rum ,mo5a1 ,moles ,mo5a1 ,saltus ,mo5a1 ,sc5a ,mo5a1 ,ve/is ,mo5a4 7,if ! diff]5t positions ( ! cube are n well kn[n1 x is be/ to h a li/ ( ! names be=e "o1 b 9 e case ! block %\ld also be built1 z well z ! names us$47 ,! same %ape 9 ! position c d a is1 ( c\rse1 express$ 9 ! same words1 b x has a diff]5t appe>ance4 ,! front face consi/s ( ! ,syces ( ,saltus ,sypho ,remus ,moles ,plebs ,hama ,urna ,fr5um ,sector ,& tak+ ! %ape we f9d we h ,urna1 & we "k t ,o/rum lies beh9d ,urna1 & does n come 92 next we h ,moles1 ,saltus1 & we "k t ,sc5a lies beh9d ,saltus & does n come 92 la/ly1 we h ,sypho & ,remus1 & we "k t ,a] & ,tyro are 9 ! ,y direction f ,remus1 & s d n come 94 ,h5ce1 altoge!r ! projection w consi/ only ( ! ,syces ( ,urna1 ,moles1 ,saltus1 ,sypho1 & ,remus4 ,next1 tak+ ! position d a c1 ! cubes 9 ! front face h _! ,alvus sides aga9/ ! plane1 & are3 ,sector ,ala ,m>s ,fr5um ,uncus ,spicula ,urna ,o/rum ,comes ,&1 tak+ ! %ape1 we f9d ,urna1 ,o/rum2 ,moles & ,saltus are hidd5 by ,urna1 ,sc5a is beh9d ,o/rum1 ,sypho gives ,fr5um1 ,remus gives ,sector1 ,a] gives ,ala1 & ,tyro gives ,m>s4 ,all ^! are ,alvus sides4 ,let u n[ take ! rev]se problem1 &1 giv5 ! ?ree cyclical projections1 det]m9e ! %ape4 ,let ! a c d projection be ! ,mo5as ( ,urna1 ,o/rum1 ,bid5s1 ,sc5a1 ,ve/is4 ,let ! c d a be ! ,syces ( ,urna1 ,fr5um1 ,plebs1 ,sypho1 & ! d a c be ! ,alvus ( ,urna1 ,fr5um1 ,uncus1 ,spicula4 ,n[1 f a c d we h ,urna1 ,fr5um1 ,sector1 ,o/rum1 ,uncus1 ,ala1 ,bid5s1 ,pallor1 ,cortis1 ,sc5a1 ,t]gum1 ,a]1 ,ve/is1 ,oliva1 ,tyro4 ,f c d a we h ,urna1 ,o/rum1 ,comes1 ,fr5um1 ,uncus1 ,spicula1 ,plebs1 ,pallor1 ,mora1 ,sypho1 ,t]gum1 ,oliva4 ,9 ord] to see h[ ^! w modify ea* o!r1 let u consid] ! a c d solution z if x 7 a set ( cubes 9 ! c d a arrangem5t4 ,"h1 ^? t g 9 ! ,arctos direction1 g away f ! plane ( projection1 & mu/ be repres5t$ by ! ,syce ( ! cube 9 contact ) ! plane4 ,look+ at ! a c d solution we write d[n 7keep+ ^? toge!r :i* g away f ! plane ( projection73 ,urna & ,o/rum1 ,fr5um & ,uncus1 ,sector & ,ala1 ,bid5s1 ,pallor1 ,cortis1 ,sc5a & ,ve/is1 ,t]gum & ,oliva1 ,a] & ,tyro4 ,"h we see t ! :ole c d a face is fill$ up 9 ! projection1 z f> z ?is solution is conc]n$4 ,b 9 ! c d a solution we h only ,syces ( ,urna1 ,fr5um1 ,plebs1 ,sypho4 ,^! ,syces only indicate ! pres5ce ( a c]ta9 numb] ( ! cubes /at$ above z possible f ! ,mo5a projection1 & ^? are ,urna1 ,o/rum1 ,fr5um1 ,uncus1 ,pallor1 ,t]gum1 ,oliva4 ,?is is ! result ( a comp>ison ( ! ,mo5a projection ) ! ,syce projection4 ,n[1 writ+ ^! la/ nam$ z !y come 9 ! d a c projection1 we h ,urna1 ,o/rum1 ,fr5um1 ,uncus & ,pallor & ,t]gum1 ,oliva4 ,& ( ^! ,o/rum ,alvus is want+ 9 ! d a c projection z giv5 above4 ,h5ce ,o/rum w be want+ 9 ! f9al %ape1 & we h z ! f9al solution3 ,urna1 ,fr5um1 ,uncus1 ,pallor1 ,t]gum1 ,oliva4 ,,*apt] ,,xi4 ,a ,,tess>actic ,,figure ,,& ,,its ,,projections4 ,we w n[ consid] a f\r?-dim5sional %ape compos$ ( tess>acts1 & ! mann] 9 :i* we c obta9 a conception ( x4 ,! op]ation is precisely analog\s to t describ$ 9 *apt] ,,vi41 by :i* a plane be+ c\ld obta9 a conception ( solid %apes4 ,x is only a little m difficult 9 t we h to deal ) "o dim5sion or direction m1 & c only d s symbolically4 ,we w assume ! %ape to consi/ ( a c]ta9 numb] ( ! #ha tess>acts1 ^: names we h giv5 on p4 #afh4 ,let x consi/ ( ! ?irte5 tess>acts3 ,urna1 ,moles1 ,plebs1 ,fr5um1 ,pallor1 ,tess]a1 ,cudo1 ,vitta1 ,cura1 ,p5ates1 ,polus1 ,orcus1 ,lac]ta4 ,fir/ly1 we w consid] :at appe>ances or projections ^! tess>acts w pres5t to u accord+ z ! tess>actic set t\*es our space ) its 7a7 ,mala cubes1 7b7 ,vesp] cubes1 7c7 ,pluvium cubes1 or 7d7 ,l> cubes4 ,secondly1 we w treat ! conv]se "q1 h[ ! %ape c be det]m9$ :5 ! projections 9 ea* ( ^? views are giv54 ,let u build up 9 cubes ! f\r diff]5t arrangem5ts ( ! tess>acts accord+ z !y ent] our space on _! ,mala1 ,vesp]1 ,pluvium or ,l> sides4 ,!y c only be pr9t$ by symboliz+ two ( ! directions4 ,9 ! foll[+ tabulations ! directions ,y1 ,x w at once be und]/ood4 ,! direction ,z 7express$ by ! wavy l9e7 indicates t ! floors ( n9e1 ea* pr9t$ ne>] ! top ( ! page1 lie above ^? pr9t$ ne>] ! bottom ( x4 ,! direction ,w is indicat$ by ! dott$ l9e1 :i* %[s t ! floors ( n9e ly+ to ! left or "r are 9 ! ,w direction 7,ana7 or ! -,w direction 7,kata7 f ^? :i* lie to ! "r or left4 ,= in/ance1 9 ! arrangem5t ( ! tess>acts1 z ,malas 7,table ,a7 ! tess>act ,tess>a1 :i* is exactly 9 ! middle ( ! eighty-"o tess>acts has ,domitor on its "r side or 9 ! ,x direction4 ,ocrea on its left -,x ,glans away f u ,y ,cudo ne>] to u -,y ,sac]dos above x ,z ,cura bel[ x -,z ,lac]ta 9 ! ,ana or ,w ,pallor 9 ! ,kata or -,w ,simil>ly ,c]vix lies 9 ! ,ana or ,w direction f ,urna1 ) ,?yrsus betwe5 !m4 ,& to take "o m in/ance1 a j\rney f ,saltus to ,arcus w\ld be made by travell+ ,y to ,remus1 !nce -,x to ,sector1 !nce ,z to ,m>s1 & f9ally ,w to ,arcus4 ,a l9e f ,saltus to ,arcus is !re=e a diagonal ( ! set ( #ha tess>acts1 because ! full l5g? ( its side has be5 trav]s$ 9 ea* ( ! f\r directions to rea* "o f ! o!r1 i4e4 ,saltus to ,remus1 ,remus to ,sector1 ,sector to ,m>s1 ,m>s to ,arcus4 ,,table ,a4 ,mala pres5tation ( #ha ,tess>acts4 ,z ,w-------------------------------------- ----------------- -,w ,y ,y ,y ,block ,a ,block ,b ,block ,c ---------------,x ---------------,x ---------------,x ,y ,y ,y ,block ,d ,block ,e ,block ,f ---------------,x ---------------,x ---------------,x ,y ,y ,y ,block ,g ,block ,h ,block ,i ---------------,x ---------------,x ---------------,x -,z ,block ,a3 ,arcus ,ovis ,portio ,laurus ,tigris ,segm5 ,axis ,troja ,aries ,block ,b3 ,ara ,vom] ,pluma ,pra$a ,sac]dos ,hydra ,cortex ,mica ,flagellum ,block ,c3 ,m>s ,m]ces ,tyro ,spicula ,mora ,oliva ,comes ,tibic5 ,ve/is ,block ,d3 ,po/is ,clipeus ,tabula ,orcus ,lac]ta ,te/udo ,v]bum ,luctus ,anguis ,block ,e3 ,pilum ,glans ,co9s ,ocrea ,tess]a ,domitor ,c>do ,cudo ,malleus ,block ,f3 ,ala ,cortis ,a] ,uncus ,pallor ,t]gum ,o/rum ,bid5s ,sc5a ,block ,g3 ,telum ,nepos ,angu/a ,polus ,p5ates ,vulcan ,c]vix ,securis ,v9culum ,block ,h3 ,agm5 ,lacus ,arvus ,crates ,cura ,lim5 ,?yrsus ,vitta ,sceptrum ,block ,i3 ,sector ,hama ,remus ,fr5um ,plebs ,sypho ,urna ,moles ,saltus ,,table ,b4 ,vesp] pres5tation ( #ha ,tess>acts4 ,z ,w-------------------------------------- ---------------- -,w ,y ,y ,y ,block ,a ,block ,b ,block ,c -,x--------------- -,x--------------- -,x--------------- ,y ,y ,y ,block ,d ,block ,e ,block ,f -,x--------------- -,x--------------- -,x--------------- ,y ,y ,y ,block ,g ,block ,h ,block ,i -,x--------------- -,x--------------- -,x--------------- -,z ,block ,a3 ,portio ,pluma ,tyro ,segm5 ,hydra ,oliva ,aries ,flagellum ,ve/is ,block ,b3 ,ovis ,vom] ,m]ces ,tigris ,sac]dos ,mora ,troja ,mica ,tibic5 ,block ,c3 ,arcus ,ara ,m>s ,laurus ,pra$a ,spicula ,axis ,cortex ,comes ,block ,d3 ,tabula ,colus ,a] ,te/udo ,domitor ,t]gum ,anguis ,malleus ,sc5a ,block ,e3 ,clipeus ,glans ,cortis ,lac]ta99 ,tess]a99 ,pallor99 ,luctus99 ,cudo99 ,bid5s99 ,block ,f3 ,po/is ,pilum ,ala ,orcus99 ,ocrea99 ,uncus99 ,v]bum ,c>do ,o/rum ,block ,g3 ,angu/a ,arvus ,remus ,vulcan ,lim5 ,sypho ,v9culum ,sceptrum ,saltus ,block ,h3 ,nepos ,lacus ,hama ,p5ates99 ,cura99 ,plebs99 ,securis99 ,vitta99 ,moles99 ,block ,i3 ,telum ,agm5 ,sector ,polus99 ,crates99 ,fr5um99 ,c]vix99 ,?yrsus99 ,urna99 ,,table ,c4 ,pluvium pres5tation ( #ha ,tess>acts4 ,z ,w-------------------------------------- ----------------- -,w ----------------,x ----------------,x ---------------,x ,block ,a ,block ,b ,block ,c -,y -,y -,y ----------------,x ----------------,x ---------------,x ,block ,d ,block ,e ,block ,f -,y -,y -,y ----------------,x ----------------,x ---------------,x ,block ,g ,block ,h ,block ,i -,y -,y -,y -,z ,block ,a3 ,m>s ,m]ces ,tyro ,ara ,vom] ,pluma ,arcus ,ovis ,portio ,block ,b3 ,spicula ,mora ,oliva ,pra$a ,sac]dos ,hydra ,laurus ,tigris ,segm5 ,block ,c3 ,comes ,tibic5 ,ve/is ,cortex ,mica ,flagellum ,axis ,troja ,aries ,block ,d3 ,ala ,cortis ,a] ,pilum ,glans ,colus ,po/is ,clipeus ,tabula ,block ,e3 ,uncus99 ,pallor99 ,t]gum ,ocrea99 ,tess]a99 ,domitor ,orcus99 ,lac]ta99 ,te/udo ,block ,f3 ,o/rum ,bid5s ,sc5a ,c>do ,cudo99 ,malleus ,v]bum ,luctus ,anguis ,block ,g3 ,sector ,hama ,remus ,agm5 ,lacus ,arvus ,telum ,nepos ,angu/a ,block ,h3 ,fr5um99 ,plebs99 ,sypho ,crates99 ,cura99 ,lim5 ,polus99 ,p5ates99 ,vulcan ,block ,i3 ,urna99 ,moles99 ,saltus ,?yrsus99 ,vitta99 ,sceptrum ,c]vix ,securis ,v9culum ,,table ,d4 ,l> pres5tation ( #ha ,tess>acts4 ,z ,w-------------------------------------- ----------------- -,w ,y ,y ,y ,block ,a ,block ,a ,block ,a ---------------,x ---------------,x ---------------,x ,y ,y ,y ,block ,a ,block ,a ,block ,a ---------------,x ---------------,x ---------------,x ,y ,y ,y ,block ,a ,block ,a ,block ,a ---------------,x ---------------,x ---------------,x -,z ,block ,a3 ,m>s ,m]ces ,tyro ,spicula ,mora ,oliva ,comes ,tibic5 ,ve/is ,block ,b3 ,ala ,cortis ,a] ,uncus ,pallor99 ,t]gum ,o/rum ,bid5s ,sc5a ,block ,c3 ,sector ,hama ,remus ,fr5um99 ,plebs99 ,sypho ,urna99 ,moles99 ,saltus ,block ,d3 ,ara ,vom] ,pluma ,pro$a ,sac]dos ,hydra ,cortex ,mica ,flagellum ,block ,e3 ,pilum ,glans ,colus ,ocrea ,tess]a99 ,domitor ,c>do ,cudo99 ,malleus ,block ,f3 ,agm5 ,laurus ,arvus ,crates ,cura99 ,lim5 ,?yrsus ,vitta99 ,sceptrum ,block ,g3 ,arcus ,ovis ,portio ,laurus ,tigris ,segm5 ,axis ,troja ,aries ,block ,h3 ,po/is ,clipeus ,tabula ,orcus99 ,lac]ta99 ,te/udo ,v]bum ,luctus ,anguis ,block ,i3 ,telum ,nepos ,angu/a ,polus99 ,p5ates99 ,vulcan ,c]vix ,securis ,v9culum ,! relation betwe5 ! f\r diff]5t arrangem5ts %[n 9 ! tables ,a1 ,b1 ,c1 & ,d1 w be und]/ood f :at has be5 said 9 *apt] ,,viii4 ab\t a small set ( sixte5 tess>acts4 ,a glance at ! l9es1 :i* indicate ! directions 9 ea*1 w %[ ! *anges effect$ by turn+ ! tess>acts f ! ,mala pres5tation4 ,9 ! ,vesp] pres5tation3 ,! tess>acts-- 7e4g4 ,urna1 ,o/rum1 ,comes71 :i* ran ,z /ill run ,z4 7e4g4 ,urna1 ,fr5um1 ,sector71 ,y ,y4 7e4g4 ,urna1 ,moles1 ,saltus71 ,x n[ run ,w4 7e4g4 ,urna1 ,?yrsus1 ,c]vix71 ,w -,x4 ,9 ! ,pluvium pres5tation3 ,! tess>acts-- 7e4g4 ,urna1 ,o/rum1 ,comes71 :i* ran ,z /ill run ,z4 7e4g4 ,urna1 ,moles1 ,saltus71 ,x ,x4 7e4g4 ,urna1 ,fr5um1 ,sector71 ,y n[ run ,w4 7e4g4 ,urna1 ,?yrsus1 ,c]vix71 ,w -,y4 ,9 ! ,l> pres5tation3 ,! tess>acts-- 7e4g4 ,urna1 ,moles1 ,saltus71 :i* ran ,x /ill run ,x4 7e4g4 ,urna1 ,fr5um1 ,sector71 ,y ,y4 7e4g4 ,urna1 ,o/rum1 ,comes71 ,z n[ run ,w4 7e4g4 ,urna1 ,?yrsus1 ,c]vix71 ,w -,z4 ,?is relation 0 already treat$ 9 *apt] ,,ix41 b x is well to h x v cle> = our pres5t purpose4 ,= x is ! app>5t *ange ( ! relative positions ( ! tess>acts 9 ea* pres5tation1 :i* enables u to det]m9e any body ( !m4 ,9 consid]+ ! projections1 we always suppose ourselves to be situat$ ,ana or ,w t[>ds ! tess>acts1 & any movem5t to be ,kata or -,w ?r\gh our space4 ,= in/ance1 9 ! ,mala pres5tation we h fir/ 9 our space ! ,malas ( t block ( tess>acts1 :i* is ! la/ 9 ! -,w direction4 ,?us1 ! ,mala projection ( any giv5 tess>act ( ! set is t ,mala 9 ! extreme -,w block1 ^: place its 7! giv5 tess>act's7 ,mala w\ld occupy1 if ! tess>actic set mov$ ,kata until ! giv5 tess>act rea*$ our space4 ,or1 9 o!r words1 if all ! tess>acts 7 transp>5t except ^? :i* con/itute ! body "u consid]ation1 & if a light %one ?r\gh ,f\r-space f ! ,ana 7,w7 side to ! ,kata 7-,w7 side1 !re w\ld be d>kness 9 ea* ( ^? ,malas1 :i* w\ld be occupi$ by ! ,mala ( any opaque tess>act1 if ! tess>actic set mov$ ,kata4 ,let u look at ! set ( #ha tess>acts we h built up 9 ! ,mala arrangem5ts1 & trace ! projections 9 ! extreme -,w block ( ! ?irte5 ( our %ape4 ,! latt] are pr9t$ 9 italics 9 ,table ,a1 & _! projections are m>k$ 4 ,?us ! cube ,uncus ,mala is ! projection ( ! tess>act ,orcus1 ,pallor ,mala ( ,pallor & ,tess]a & ,tac]ta1 ,bid5s ,mala ( ,cudo1 ,fr5um ,mala ( ,fr5um & ,polus1 ,plebs ,mala ( ,plebs & ,cura & ,p5ates1 ,moles ,mala ( ,moles & ,vitta1 ,urna ,mala ( ,urna4 ,simil>ly1 we c trace ! ,vesp] projections 7,table ,b74 ,orcus ,vesp] is ! projection ( ! tess>acts ,orcus & ,lac]ta1 ,ocrea ,vesp] ( ,tess]a1 ,uncus ,vesp] ( ,pallor1 ,c>do ,vesp] ( ,cudo1 ,polus ,vesp] ( ,polus & ,p5ates1 ,crates ,vesp] ( ,cura1 ,fr5um ,vesp] ( ,fr5um & ,plebs1 ,urna ,vesp] ( ,urna & ,moles1 ,?yrsus ,vesp] ( ,vitta4 ,next 9 ! ,pluvium pres5tation 7,table ,c7 we f9d t ,bid5s ,pluvium is ! projection ( ! tess>act ,pallor1 ,cudo ,pluvium ( ,cudo & ,tess]a1 ,luctus ,pluvium ( ,lac]ta1 ,v]bum ,pluvium ( ,orcus1 ,urna ,pluvium ( ,urna & ,fr5um1 ,moles ,pluvium ( ,moles & ,plebs1 ,vitta ,pluvium ( ,vitta & ,cura1 ,securis ,pluvium ( ,p5ates1 ,c]vix ,pluvium ( ,polus4 ,la/ly1 9 ! ,l> pres5tation 7,table ,d7 we obs]ve t ,fr5um ,l> is ! projection ( ,fr5um1 ,plebs ,l> ( ,plebs & ,pallor1 ,moles ,l> ( ,moles1 ,urna ,l> ( ,urna1 ,cura ,l> ( ,cura & ,tess>a1 ,vitta ,l> ( ,vitta & ,cudo1 ,p5ates ,l> ( ,p5ates & ,lac]ta1 ,polur ,l> ( ,polus & ,orcus4 ,secondly1 we w treat ! conv]se problem1 h[ to det]m9e ! %ape :5 ! projections 9 ea* pres5tation are giv54 ,look+ back at ! li/ j giv5 above1 let u write d[n 9 ea* pres5tation ! projections only4 ,mala projections3 ,uncus1 ,pallor1 ,bid5s1 ,fr5um1 ,plebs1 ,moles1 ,urna4 ,vesp] projections3 ,orcus1 ,ocrea1 ,uncus1 ,c>do1 ,polus1 ,crates1 ,fr5um1 ,urna1 ,?yrsus4 ,pluvium projections3 ,bid5s1 ,cudo1 ,luctus1 ,v]bum1 ,urna1 ,moles1 ,vitta1 ,securis1 ,c]vix4 ,l> projections3 ,fr5um1 ,plebs1 ,moles1 ,urna1 ,cura1 ,vitta1 ,polus1 ,p5ates4 ,n[ let u det]m9e ! %ape indicat$ by ^! projections4 ,9 n[ us+ ! same tables we mu/ n notice ! italics1 z ! %ape is suppos$ to be unkn[n4 ,x is assum$ t ! read] is build+ ! problem 9 cubes4 ,f ! ,mala projections we might inf] ! pres5ce ( all or any ( ! tess>acts writt5 9 ! brackets 9 ! foll[+ li/ ( ! ,mala pres5tation4 7,uncus1 ,ocrea1 ,orcus72 7,pallor1 ,tess]a1 ,lac]ta72 7,bid5s1 ,cudo1 ,luctus72 7,fr5um1 ,crates1 ,polus72 7,plebs1 ,cura1 ,p5ates72 7,moles1 ,vitta1 ,securis72 7,urna1 ,?yrsus1 ,c]vix74 ,let u suppose !m all to be pres5t 9 our %ape1 & obs]ve :at _! appe>ance w\ld be 9 ! ,vesp] pres5tation4 ,we m>k !m all ) an a/]isk 9 ,table ,b4 ,9 addition to ^? already m>k$ we mu/ m>k 77 ,v]bum1 ,c>do1 ,o/rum1 & !n we see all ! ,vesp] projections1 :i* w\ld be =m$ by all ! tess>acts possible f ! ,mala projections4 ,let u comp>e ^! ,vesp] projections1 viz4 ,orcus1 ,ocrea1 ,uncus1 ,v]bum1 ,c>do1 ,o/rum1 ,polus1 ,crates1 ,fr5um1 ,c]vix1 ,?yrsus1 ,urna1 ) ! giv5 ,vesp] projections4 ,we see at once t ,v]bum1 ,o/rum1 & ,c]vix are abs5t4 ,!re=e1 we may conclude t all ! tess>acts1 :i* w\ld be impli$ z possible by _! pres5ce1 are abs5t1 & ( ! ,mala possibilities may exclude ! tess>acts ,bid5s1 ,luctus1 ,securis1 & ,c]vix itself4 ,?us1 ( ! #ba tess>acts possible 9 ! ,mala view1 !re rema9 only #ag possible1 bo? 9 ! ,mala & ,vesp] views1 viz4 ,uncus1 ,ocrea1 ,orcus1 ,pallor1 ,tess]a1 ,lac]ta1 ,cudo1 ,fr5um1 ,crates1 ,polus1 ,plebs1 ,cura1 ,p5ates1 ,moles1 ,vitta1 ,urna1 ,?yrsus4 ,?is we call ! ,mala-,vesp] solution4 ,next let u take ! ,pluvium pres5tation4 ,we aga9 m>k ) an a/]isk 9 ,table ,c ! possibilities inf]r$ f ! ,mala-,vesp] solution1 & take ! projections ^? possibilities w\ld produce4 ,! additional projections are aga9 m>k$ 774 ,!re are twelve ,pluvium projections altoge!r1 viz4 ,bid5s1 ,o/rum1 ,cudo1 ,c>do1 ,luctus1 ,v]bum1 ,urna1 ,moles1 ,vitta1 ,?yrsus1 ,securis1 ,c]vix4 ,aga9 we comp>e ^! ) ! giv5 ,pluvium projections1 & f9d ?ree are abs5t1 viz4 ,o/rum1 ,c>do1 ,?yrsus4 ,h5ce ! tess>acts impli$ by ,o/rum & ,c>do & ,?yrsus _c be 9 our %ape1 viz4 ,uncus1 ,ocrea1 ,crates1 nor ,?yrsus itself4 ,exclud+ ^! f\r f ! sev5te5 possibilities ( ! ,mala-,vesp] solution we h left ! ?irte5 tess>acts3 ,orcus1 ,pallor1 ,tess]a1 ,lac]ta1 ,cudo1 ,fr5um1 ,polus1 ,plebs1 ,cura1 ,p5ates1 ,moles1 ,vitta1 ,urna4 ,?is we call ! ,mala-,vesp]-,pluvium solution4 ,la/ly1 we h to consid] :e!r ^! ?irte5 tess>acts are consi/5t ) ! giv5 ,l> projections4 ,we m>k !m aga9 on ,table ,d ) an a/]isk1 & we f9d t ! projections are exactly ^? giv51 viz4 ,fr5um1 ,plebs1 ,moles1 ,urna1 ,cura1 ,vitta1 ,polus1 ,p5ates4 ,!re=e1 we h n to exclude any ( ! ?irte51 & c inf] t !y con/itute ! %ape1 :i* produces ! f\r diff]5t giv5 views or projections4 ,9 f9e1 any %ape 9 space consi/s ( ! possibilities common to ! projections ( its p>ts ^u ! b\nd>ies ( t space1 :atev] be ! numb] ( its dim5sions4 ,h5ce ! simple rule = ! det]m9ation ( ! %ape w\ld be to write d[n all ! possibilities ( ! sets ( projections1 & !n cancel all ^? possibilities :i* are n common to all4 ,b ! process adopt$ above is mu* pref]able1 z ?r\gh x we may realize ! gradual delimitation ( ! %ape view by view4 ,= once m we mu/ rem9d ourselves t our great object is1 n to arrive at results by symbolical op]ations1 b to realize ^? results piece by piece ?r\gh realiz$ processes4 ,,app5dices4 ,,app5dix ,a4 ,?is set ( #ajj names is useful = /udy+ ,plane ,space1 & =ms a squ>e #aj #aj4 ,ain ,bios ,hup> ,neas ,kairos ,enos ,?lipsis ,*eimas ,!ion ,epei ,itea ,hagios ,pha9o ,g]as ,?olos ,ergon ,pa*s ,kin ,eris ,cleos ,loma ,ets ,tro*os ,klazo ,lutron ,hds ,is*s ,paigma ,h$na ,demas ,numphe ,ba?us ,pauo ,eu?u ,holos ,p>a ,?uos ,k> ,pyl ,sp>eis ,ania ,en ,s]anx ,mesoi ,dramo ,?allos ,akt ,ozo ,onos ,magos ,notos ,mnis ,lampas ,ornis ,?ama ,eni ,pholis ,mala ,/rizo ,rudon ,labo ,helor ,rupa ,rabdos ,doru ,epos ,!os ,idris d ,hepo ,sophos ,i*or ,kanen ,eph?ra ,oxis ,luk ,blue ,helos ,p]i ,!lus ,eunis ,limos ,ke$o ,igde ,mat ,lukos ,pt]is ,holmos ,oulo ,dokos ,aeido ,ias ,assa ,muzo ,hippeus ,es ,at ,akme r ,gua ,,app5dix ,b4 ,! foll[+ li/ ( names is us$ to d5ote cubic spaces4 ,x makes a cubic block ( six floors1 ! highe/ be+ ! six?4 ,f ,fons ,plectrum ,vulnus ,ar5a ,m5sa ,t]m9us ,s l ,te/a ,plausus ,uva ,collis ,coma ,nebula i o ,copia ,cornu ,solum ,munus ,rixum ,vitrum x o ,ars ,f]vor ,?yma ,colubra ,seges ,cor t r4 ,lupus ,classis ,modus ,flamma ,m5s ,incola h ,?alamus ,ha/a ,calamus ,cr9is ,auriga ,vallum ,f ,l9teum ,p9nis ,puppis ,nuptia ,aegis ,ci?>a ,f l ,triumphus ,curris ,lux ,portus ,latus ,funis i o ,regnum ,fascis ,bellum ,capellus ,arbor ,cu/os f o ,sagitta ,pu] ,/ella ,saxum ,humor ,pontus t r4 ,nom5 ,imago ,lapsus ,qu]cus ,mundus ,proelium h ,palae/ra ,nuncius ,bos ,ph>etra ,pumex ,tibia ,f ,f ,lignum ,focus ,ornus ,lucrum ,alea ,vox o l ,cat]va ,facies ,onus ,silva ,gelu ,flum5 u o ,tellus ,sol ,os ,arma ,bra*ium ,jaculum r o ,m]um ,signum ,umbra ,tempus ,corona ,socius t r4 ,mo5a ,opus ,honor ,campus ,rivus ,imb] h ,victor ,equus ,miles ,cursus ,lyra ,tunica ,f ,ha$us ,tab]na ,turris ,nox ,domus ,v9um ,t l ,pru9us ,*orus ,luna ,flos ,lucus ,agna h o ,fulm5 ,hiems ,v] ,c>9a ,arator ,pratum i o ,oculus ,ignis ,ae!r ,cohors ,p5na ,labor r r4 ,aes ,pectus ,pelagus ,notus ,fretum ,gradus d ,pr9ceps ,dux ,v5tus ,navis ,f9is ,robur ,s ,f ,vultus ,ho/is ,figura ,ales ,coelum ,aura e l ,hum]us ,augur ,ludus ,clamor ,galea ,pes c o ,civis ,f]rum ,pugna ,res ,c>m5 ,nubes o o ,litus ,unda ,rex ,templum ,ripa ,amnis n r4 ,pannus ,ulmus ,s$es ,columba ,aequor ,dama d ,dext] ,urbs ,g5s ,mon/rum ,pecus ,mons ,f ,nemus ,sidus ,v]tex ,nix ,gr&o ,arx ,f l ,v5ator ,c]va ,ap] ,plagua ,h$]a ,frons i o ,membrum ,aqua ,caput ,ca/rum ,lituus ,tuba r o ,fluctus ,rus ,ratis ,amphora ,p>s ,dies s r4 ,turba ,ag] ,trabs ,myrtus ,fibra ,nauta t ,decus ,pulvis ,meta ,rota ,palma ,t]ra ,,app5dix ,c4 ,! foll[+ names are us$ = a set ( #bef ,tess>acts4 ,,f\r? ,,block4 ,,?ird ,,block4 ,f\r? ,floor4 ,f\r? ,floor4 ,dolium ,caballus ,py?on ,circaea ,*>ta ,cures ,quae/or ,cli5s ,cussis ,pulsus ,dra*ma ,cordax ,frux ,pyra ,l5a ,procella ,porrum ,consul ,diota ,dyka ,h]a ,esca ,secta ,rug ,colum5 ,ravis ,corbis ,rap9a ,eurus ,gloria ,soc] ,sequela ,?ird ,floor4 ,?ird ,floor4 ,alexis ,planta ,corymbus ,lectrum ,ar*e ,agg] ,cumulus ,cassis ,ae/us ,labellum ,cala?us ,nux ,arcus ,ovis ,portio ,mimus ,septum ,sepes ,turtur ,ordo ,laurus ,tigris ,segm5 ,obolus ,morsus ,ae/as ,capella ,rh$a ,axis ,troja ,aries ,fuga ,second ,floor4 ,second ,floor4 ,corydon ,jugum ,tornus ,labrum ,ru9a ,culm5 ,f5e/ra ,a$es ,lac ,hibiscus ,donum ,cal?a ,po/is ,clipeus ,tabula ,l+ua ,s5ex ,palus ,salix ,cespes ,orcus ,lac]ta ,te/udo ,scala ,amictus ,gurges ,otium ,pomum ,v]bum ,luctus ,anguis ,dolus ,fir/ ,floor4 ,fir/ ,floor4 ,odor ,aprum ,pignus ,messor ,additus ,salus ,clades ,rana ,color ,casa ,c]a ,papav] ,telum ,nepos ,angu/a ,mucro ,spes ,lapis ,apis ,afrus ,polus ,p5ates ,vulcan ,ira ,vitula ,clavis ,fagus ,cornix ,c]vix ,securis ,v9culum ,furor ,,second ,,block4 ,,fir/ ,,block4 ,f\r? ,floor4 ,f\r? ,floor4 ,actus ,spadix ,sic]a ,ans] ,horreum ,fumus ,hircus ,erisma ,auspex ,praetor ,atta ,sonus ,anulus ,pluor ,acies ,naxos ,fulgor ,ardea ,prex ,aevum ,etna ,gemma ,alpis ,arbit] ,sp9a ,birrus ,ac]ra ,ramus ,alauda ,furca ,g5a ,alnus ,?ird ,floor4 ,?ird ,floor4 ,ma*9a ,lex ,om5 ,artus ,fax ,v55um ,syrma ,ursa ,ara ,vom] ,pluma ,odium ,m>s ,m]ces ,tyro ,fama ,pro$a ,sac]dos ,hydra ,luxus ,spicula ,mora ,oliva ,conjux ,cortex ,mica ,flagellum ,mas ,comes ,tibic5 ,ve/is ,pl5um ,second ,floor4 ,second ,floor4 ,ardor ,rupes ,pallas ,ari/a ,ro/rum ,armig] ,premium ,tribus ,pilum ,glans ,colus ,pellis ,ala ,cortis ,a] ,fragor ,ocrea ,tess>a ,domitor ,f]a ,uncus ,pallor ,t]gum ,reus ,c>do ,cudo ,malleus ,?orax ,o/rum ,bid5s ,sc5a ,torus ,fir/ ,floor4 ,fir/ ,floor4 ,reg9a ,canis ,m>mor ,tectum ,p>dus ,rubor ,nurus ,hospes ,agm5 ,lacus ,arvus ,rumor ,sector ,hama ,remus ,=tuna ,crates ,cura ,lim5 ,vita ,fr5um ,plebs ,sypho ,myrrha ,?yrsus ,vitta ,sceptrum ,pax ,urna ,moles ,saltus ,acus ,,app5dix ,d4 ,! foll[+ li/ gives ! col\rs1 & ! v>i\s uses = !m4 ,!y h already be5 us$ 9 ! =ego+ pages to di/+ui% ! v>i\s regions ( ! ,tess>act1 & ! diff]5t individual cubes or ,tess>acts 9 a block4 ,! o!r use sugge/$ 9 ! la/ column ( ! li/ has n be5 discuss$2 b x is believ$ t x may af=d great aid to ! m9d 9 amass+1 h&l+1 & reta9+ ! quantities ( =mulae requisite 9 sci5tific tra9+ & "w4 ,region ( ,tess>act ,col\r4 ,tess>act4 9 #ha ,set4 ,symbol4 ,black ,syce ,plebs #j ,:ite ,mel ,mora #a ,v]milion ,alvus ,uncus #b ,orange ,cuspis ,moles #c ,light-yell[ ,murex ,cortis #d ,bright-gre5 ,lappa ,p5ates #e ,bright-blue ,it] ,oliva #f ,light-grey ,l>es ,tigris #g ,indian-r$ ,crux ,orcus #h ,yell[-o*re ,sal ,te/udo #i ,buff ,ci/a ,sector 7plus7 ,wood ,tess>act ,tess>a - 7m9us7 ,br[n-gre5 ,?olus ,troja 7plus or m9us7 ,sage-gre5 ,m>go ,lac]ta 7multipli$ by7 ,r$di% ,callis ,tibic5 7divid$ by7 ,*ocolate ,velum ,sac]dos 7equal to7 ,fr5*-grey ,f> ,sc5a 7n equal to7 ,br[n ,arctos ,o/rum 7great] ?an7 ,d>k-slate ,daps ,a] 7less ?an7 ,dun ,portica ,clipeus signs ,orange-v]milion ,talus ,portio ( proportion ,/one ,ops ,?yrsus 7decimal po9t7 ,quak]-gre5 ,felis ,axis 7factorial7 ,lead5 ,semita ,m]ces 7p>allel7 ,dull-gre5 ,mappa ,vulcan 7n p>allel7 ,indigo ,lixa ,po/is #b 7#ij7 7at "r angles7 ,dull-blue ,pagus ,v]bum log4 base #aj ,d>k-purple ,m5sura ,nepos s94 7s9e7 ,pale-p9k ,v5a ,tabula cos4 7cos9e7 ,d>k-blue ,mo5a ,bid5s tan4 7tang5t7 ,e>!n ,mugil ,angu/a 7inf9ity7 ,blue ,dos ,fr5um a ,t]racotta ,crus ,remus b ,oak ,idus ,domitor c ,yell[ ,pag9a ,c>do d ,gre5 ,buc9a ,ala e ,rose ,olla ,lim5 f ,em]ald ,orsa ,ara g ,r$ ,olus ,m>s h ,sea-gre5 ,lib]a ,pluma i ,salmon ,tela ,glans j ,pale-yell[ ,livor ,ovis k ,purple-br[n ,opex ,polus l ,deep-crimson ,camo5a ,pilum m ,blue-gre5 ,proes ,t]gum n ,light-br[n ,lua ,crates o ,deep-blue ,lama ,tyro p ,brick-r$ ,l> ,cura q ,mag5ta ,(fex ,arvus r ,gre5-grey ,cadus ,hama s ,light-r$ ,croeta ,pra$a t ,azure ,lotus ,vitta u ,pale-gre5 ,vesp] ,ocrea v ,blue-t9t ,panax ,telum w ,yell[-gre5 ,pactum ,malleus x ,deep-gre5 ,mango ,vom] y ,light-gre5 ,lis ,agm5 z ,light-blue ,ilex ,comes ,crimson ,bolus ,sypho ,o*re ,limbus ,mica ,purple ,solia ,arcus ,leaf-gre5 ,luca ,securis ,turquoise ,ancilla ,v9culum ,d>k-grey ,orca ,colus ,fawn ,nug ,saltus ,smoke ,limus ,sceptrum ,light-buff ,mala ,pallor ,dull-purple ,sors ,ve/is ,ri*-r$ ,lucta ,cortex ,gre5-blue ,pator ,flagellum ,burnt-si5na ,silex ,luctus ,sea-blue ,lorica ,lacus ,peacock-blue ,pass] ,aries ,deep-br[n ,meatus ,hydra ,d>k-p9k ,onag] ,anguis ,d>k ,l5sa ,laurus ,d>k-/one ,pluvium ,cudo ,silv] ,spira ,c]vix ,gold ,corvus ,urna ,deep-yell[ ,via ,spicula ,d>k-gre5 ,calor ,segm5 ,,app5dix ,e4 ,a ,,!orem ,,9 ,,f\r-,,space4 ,if a pyramid on a triangul> base be cut by a plane :i* passes ?r\gh ! ?ree sides ( ! pyramid 9 su* mann] t ! sides ( ! sectional triangle are n p>allel to ! correspond+ sides ( ! triangle ( ! base2 !n ! sides ( ^! two triangles1 if produc$ 9 pairs1 w meet 9 ?ree po9ts :i* are 9 a /raight l9e1 namely1 ! l9e ( int]section ( ! sectional plane & ! plane ( ! base4 ,let ,a ,b ,c ,d be a pyramid on a triangul> base ,a ,b ,c1 & let a b c be a section su* t ,a ,b1 ,b ,c1 ,a ,c1 are respectively n p>allel to a b1 b c1 a c4 ,x mu/ be und]/ood t a is a po9t on ,a ,d1 b is a po9t on ,b ,d1 & c a po9t on ,c ,d4 ,let1 ,a ,b & a b1 produc$1 meet 9 m4 ,b ,c & b c1 produc$1 meet 9 n2 & ,a ,c & a c1 produc$1 meet 9 o4 ,^! ?ree po9ts1 m1 n1 o1 are 9 ! l9e ( int]section ( ! two planes ,a ,b ,c & a b c4 ,n[1 let ! l9e a b be project$ on to ! plane ( ! base1 by draw+ l9es f a & b at "r angles to ! base1 & meet+ x 9 a b2 ! l9e a b1 produc$1 w meet ,a ,b produc$ 9 m4 ,if ! l9es b c & a c be project$ 9 ! same way on to ! base1 to ! po9ts b c & a c2 !n ,b ,c & b c produc$1 w meet 9 n1 & ,a ,c & a c produc$1 w meet 9 o4 ,! two triangles ,a ,b ,c & a b c are su*1 t ! l9es jo9+ ,a to a1 ,b to b1 & ,c to c1 w1 if produc$1 meet 9 a po9t1 namely1 ! po9t on ! base ,a ,b ,c :i* is ! projection ( ,d4 ,any two triangles :i* fulfil ?is condition are ! possible base & projection ( ! section ( a pyramid2 !re=e ! sides ( su* triangles1 if produc$ 9 pairs1 w meet 7if !y are n p>allel7 9 ?ree po9ts :i* lie 9 "o /raight l9e4 ,a f\r-dim5sional pyramid may be def9$ z a figure b\nd$ by a polyh$ron ( any numb] ( sides1 & ! same numb] ( pyramids ^: bases are ! sides ( ! polyh$ron1 & ^: apices meet 9 a po9t n 9 ! space ( ! base4 ,if a f\r-dim5sional pyramid on a tetrah$ral base be cut by a space :i* passes ?r\gh ! f\r sides ( ! pyramid 9 su* a way t ! sides ( ! sectional figure be n p>allel to ! sides ( ! base2 !n ! sides ( ^! two tetrah$ra1 if produc$ 9 pairs1 w meet 9 l9es :i* all lie 9 "o plane1 namely1 ! plane ( int]section ( ! space ( ! base & ! space ( ! section4 ,if n[ ! sectional tetrah$ron be project$ on to ! base 7by draw+ l9es f ea* po9t ( ! section to ! base at "r angles to x71 !re w be two tetrah$ra fulfill+ ! condition t ! l9e jo9+ ! angles ( ! "o to ! angles ( ! o!r w1 if produc$1 meet 9 a po9t1 :i* po9t is ! projection ( ! apex ( ! f\r-dim5sional pyramid4 ,any two tetrah$ra :i* fulfil ?is condition1 are ! possible base & projection ( a section ( a f\r-dim5sional pyramid4 ,!re=e1 9 any two su* tetrah$ra1 :]e ! sides ( ! "o are n p>allel to ! sides ( ! o!r1 ! sides1 if produc$ 9 pairs 7"o side ( ! "o ) "o side ( ! o!r71 w meet 9 f\r /raight l9es :i* are all 9 "o plane4 ,,app5dix ,f4 ,,ex]cises ,,on ,,%apes ,,( ,,?ree ,,dim5sions4 ,! names us$ are ^? giv5 9 ,app5dix ,b4 ,f9d ! %apes f ! foll[+ projections3 #a4 ,syce projections3 ,ratis1 ,caput1 ,ca/rum1 ,plagua4 ,alvus projections3 ,m]um1 ,oculus1 ,fulm51 ,pru9us4 ,mo5a projections3 ,miles1 ,v5tus1 ,navis4 #b4 ,syce3 ,dies1 ,tuba1 ,lituus1 ,frons4 ,alvus3 ,sagitta1 ,regnum1 ,tellus1 ,fulm51 ,pru9us4 ,mo5a3 ,tibia1 ,tunica1 ,robur1 ,f9is4 #c4 ,syce3 ,nemus1 ,sidus1 ,v]tex1 ,nix1 ,c]va4 ,alvus3 ,lignum1 ,ha$us1 ,vultus1 ,nemus1 ,hum]us4 ,mo5a3 ,dext]1 ,pr9ceps1 ,equus1 ,dux1 ,urbs1 ,pullis1 ,g5s1 ,mon/rum1 ,miles4 #d4 ,syce3 ,amphora1 ,ca/rum1 ,myrtus1 ,rota1 ,palma1 ,meta1 ,trabs1 ,ratis4 ,alvus3 ,dext]1 ,pr9ceps1 ,mo5a1 ,aes1 ,m]um1 ,oculus1 ,littus1 ,civis1 ,fulm54 ,mo5a3 ,g5s1 ,v5tus1 ,navis1 ,f9is1 ,mon/rum1 ,cursus4 #e4 ,syce3 ,ca/rum1 ,plagua1 ,nix1 ,v]tex1 ,ap]1 ,caput1 ,c]va1 ,v5ator4 ,alvus3 ,triumphus1 ,tellus1 ,cat]va1 ,lignum1 ,ha$us1 ,pru9us1 ,fulm51 ,civis1 ,hum]us1 ,vultus4 ,mo5a3 ,ph>etra1 ,cursus1 ,miles1 ,equus1 ,dux1 ,navis1 ,mon/rum1 ,g5s1 ,urbs1 ,dext]4 ,,answ]s4 ,! %apes are3 #a4 ,umbra1 ,ae!r1 ,v]1 ,c>9a1 ,flos4 #b4 ,pontus1 ,cu/os1 ,jaculum1 ,pratum1 ,arator1 ,agna4 #c4 ,focus1 ,omus1 ,ha$us1 ,tabema1 ,vultus1 ,ho/is1 ,figura1 ,ales1 ,sidus1 ,augur4 #d4 ,tempus1 ,campus1 ,f9is1 ,navis1 ,v5tus1 ,pelagus1 ,notus1 ,cohors1 ,ae!r1 ,c>9a1 ,res1 ,templum1 ,rex1 ,g5s1 ,mon/rum4 #e4 ,portus1 ,arma1 ,sylva1 ,lucrum1 ,ornus1 ,onus1 ,os1 ,facies1 ,*orus1 ,c>9a1 ,flos1 ,nox1 ,ales1 ,clamor1 ,res1 ,pugna1 ,ludus1 ,figura1 ,augur1 ,hum]us4 ,,fur!r ,,ex]cises ,,9 ,,%apes ,,( ,,?ree ,,dim5sions4 ,! ,names us$ are ^? giv5 9 ,app5dix ,c2 & ?is set ( ex]cises =ms a prep>ation = _! use 9 space ( f\r dim5sions4 ,all are 9 ! #bg ,block 7,urna to ,syrma74 #a4 ,syce3 ,moles1 ,fr5um1 ,plebs1 ,sypho4 ,alvus3 ,urna1 ,fr5um1 ,uncus1 ,spicula1 ,comes4 ,mo5a3 ,moles1 ,bid5s1 ,tibic51 ,comes1 ,saltus4 #b4 ,syce3 ,urna1 ,moles1 ,plebs1 ,hama1 ,remus4 ,alvus3 ,urna1 ,fr5um1 ,sector1 ,ala1 ,m>s4 ,mo5a3 ,urna1 ,moles1 ,saltus1 ,bid5s1 ,tibic54 #c4 ,syce3 ,moles1 ,plebs1 ,hama1 ,remus4 ,alvus3 ,uma1 ,o/rum1 ,comes1 ,spicula1 ,fr5um1 ,sector4 ,mo5a3 ,moles1 ,saltus1 ,bid5s1 ,tibic54 #d4 ,syce3 ,fr5um1 ,plebs1 ,sypho1 ,moles1 ,hama4 ,alvus3 ,urna1 ,fr5um1 ,uncus1 ,sector1 ,spicula4 ,mo5a3 ,urna1 ,moles1 ,saltus1 ,sc5a1 ,ve/is4 #e4 ,syce3 ,urna1 ,moles1 ,plebs1 ,hama1 ,remus1 ,sector4 ,alvus3 ,urna1 ,fr5um1 ,sector1 ,uncus1 ,spicula1 ,comes1 ,m>s4 ,mo5a3 ,urna1 ,moles1 ,saltus1 ,bid5s1 ,tibic51 ,comes4 #f4 ,syce3 ,uma1 ,moles1 ,saltus1 ,sypho1 ,remus1 ,hama1 ,sector4 ,alvus3 ,comes1 ,o/rum1 ,uncus1 ,spicula1 ,m>s1 ,ala1 ,sector4 ,mo5a3 ,urna1 ,moles1 ,saltus1 ,sc5a1 ,ve/is1 ,tibic51 ,comes1 ,o/rum4 #g4 ,syce3 ,sypho1 ,saltus1 ,moles1 ,urna1 ,fr5um1 ,sector4 ,alvus3 ,urna1 ,fr5um1 ,uncus1 ,spicula1 ,m>s4 ,mo5a3 ,saltus1 ,moles1 ,urna1 ,o/rum1 ,comes4 #h4 ,syce3 ,moles1 ,plebs1 ,hama1 ,sector4 ,alvus3 ,o/rum1 ,fr5um1 ,uncus1 ,spicula1 ,m>s1 ,ala4 ,mo5a3 ,moles1 ,bid5s1 ,tibic51 ,o/rum4 #i4 ,syce3 ,moles1 ,saltus1 ,sypho1 ,plebs1 ,fr5um1 ,sector4 ,alvus3 ,o/rum1 ,comes1 ,spicula1 ,m>s1 ,ala4 ,mo5a3 ,o/rum1 ,comes1 ,tibic51 ,bid5s1 ,sc5a1 ,ve/is4 #aj4 ,syce3 ,urna1 ,moles1 ,saltus1 ,sypho1 ,remus1 ,sector1 ,fr5um4 ,alvus3 ,urna1 ,o/rum1 ,comes1 ,spicula1 ,m>s1 ,ala1 ,sector4 ,mo5a3 ,urna1 ,o/rum1 ,comes1 ,tibic51 ,ve/is1 ,sc5a1 ,saltus4 #aa4 ,syce3 ,fr5um1 ,plebs1 ,sypho1 ,hama4 ,alvus3 ,fr5um1 ,sector1 ,ala1 ,m>s1 ,spicula4 ,mo5a3 ,urna1 ,moles1 ,saltus1 ,bid5s1 ,tibic54 ,,answ]s4 ,! %apes are3 #a4 ,moles1 ,plebs1 ,sypho1 ,pallor1 ,mora1 ,tibic51 ,spicula4 #b4 ,urna1 ,moles1 ,plebs1 ,hama1 ,cortis1 ,m]ces1 ,remus4 #c4 ,moles1 ,bid5s1 ,tibic51 ,mora1 ,plebs1 ,hama1 ,remus4 #d4 ,fr5um1 ,plebs1 ,sypho1 ,t]gum1 ,oliva1 ,moles1 ,hama4 #e4 ,urna1 ,moles1 ,plebs1 ,hama1 ,remus1 ,pallor1 ,mora1 ,tibic51 ,m>s1 ,m]ces1 ,comes1 ,sector4 #f4 ,o/rum1 ,comes1 ,tibic51 ,ve/is1 ,sc5a1 ,t]gum1 ,oliva1 ,tyro1 ,a]1 ,remus1 ,hama1 ,sector1 ,m]ces1 ,m>s1 ,ala4 #g4 ,sypho1 ,saltus1 ,moles1 ,urna1 ,fr5um1 ,uncus1 ,spicula1 ,m>s4 #h4 ,plebs1 ,pallor1 ,mora1 ,bid5s1 ,m]ces1 ,cortis1 ,ala4 #i4 ,bid5s1 ,tibic51 ,ve/is1 ,sc5a1 ,oliva1 ,mora1 ,spicula1 ,m>s1 ,ala4 #aj4 ,urna1 ,o/rum1 ,comes1 ,spicula1 ,m>s1 ,tibic51 ,ve/is1 ,oliva1 ,tyro1 ,a]1 ,remus1 ,sector1 ,ala1 ,saltus1 ,sc5a4 #aa4 ,fr5um1 ,plebs1 ,sypho1 ,hama1 ,cortis1 ,m]ces1 ,mora4 ,,app5dix ,g4 ,,ex]cises ,,on ,,%apes ,,( ,,f\r ,,dim5sions4 ,! ,names us$ are ^? giv5 9 ,app5dix ,c4 ,! fir/ six ex]cises are 9 ! #ha ,set1 & ! re/ 9 ! #bef ,set4 #a4 ,mala projection3 ,urna1 ,moles1 ,plebs1 ,pallor1 ,cortis1 ,m]ces4 ,l> projection3 ,urna1 ,moles1 ,plebs1 ,cura1 ,p5ates1 ,nepos4 ,pluvium projection3 ,urna1 ,moles1 ,vitta1 ,cudo1 ,luctus1 ,troja4 ,vesp] projection3 ,urna1 ,fr5um1 ,crates1 ,ocrea1 ,orcus1 ,po/is1 ,arcus4 #b4 ,mala3 ,urna1 ,fr5um1 ,uncus1 ,pallor1 ,cortis1 ,a]4 ,l>3 ,urna1 ,fr5um1 ,crates1 ,cura1 ,lacus1 ,arvus1 ,angu/a4 ,pluvium3 ,urna1 ,?yrsus1 ,c>do1 ,cudo1 ,malleus1 ,anguis4 ,vesp]3 ,urna1 ,fr5um1 ,crates1 ,ocrea1 ,pilum1 ,po/is4 #c4 ,mala3 ,comes1 ,tibic51 ,mora1 ,pallor4 ,l>3 ,urna1 ,moles1 ,vitta1 ,cura1 ,p5ates4 ,pluvium3 ,comes1 ,tibic51 ,mica1 ,troja1 ,luctus4 ,vesp]3 ,comes1 ,cortex1 ,pra$a1 ,laurus1 ,orcus4 #d4 ,mala3 ,ve/is1 ,oliva1 ,tyro4 ,l>3 ,saltus1 ,sypho1 ,remus1 ,arvus1 ,angu/a4 ,pluvium3 ,ve/is1 ,flagellum1 ,aries4 ,vesp]3 ,comes1 ,spicula1 ,m>s1 ,ara1 ,arcus4 #e4 ,mala3 ,m>s1 ,m]ces1 ,tyro1 ,a]1 ,t]gum1 ,pallor1 ,plebs4 ,l>3 ,sector1 ,hama1 ,lacus1 ,nepos1 ,angu/a1 ,vulcan1 ,p5ates4 ,pluvium3 ,comes1 ,tibic51 ,mica1 ,troja1 ,aries1 ,anguis1 ,luctus1 ,securis4 ,vesp]3 ,m>s1 ,ara1 ,arcus1 ,po/is1 ,orcus1 ,polus4 #f4 ,mala3 ,pallor1 ,mora1 ,oliva1 ,tyro1 ,m]ces1 ,m>s1 ,spicula1 ,comes1 ,tibic51 ,ve/is4 ,l>3 ,plebs1 ,cura1 ,p5ates1 ,vulcan1 ,angu/a1 ,nepos1 ,telum1 ,polus1 ,c]vix1 ,securis1 ,v9culum4 ,pluvium3 ,bid5s1 ,cudo1 ,luctus1 ,troja1 ,axis1 ,aries4 ,vesp]3 ,uncus1 ,ocrea1 ,orcus1 ,laurus1 ,arcus1 ,axis4 #g4 ,mala3 ,hospes1 ,tribus1 ,fragor1 ,a]1 ,tyro1 ,mora1 ,oliva4 ,l>3 ,hospes1 ,tectum1 ,rumor1 ,arvus1 ,angu/a1 ,c]a1 ,apis1 ,lapis4 ,pluvium3 ,acus1 ,torus1 ,malleus1 ,flagellum1 ,?orax1 ,aries1 ,ae/as1 ,capella4 ,vesp]3 ,p>dus1 ,ro/rum1 ,ardor1 ,pilum1 ,ara1 ,arcus1 ,ae/us1 ,septum4 #h4 ,mala3 ,pallor1 ,t]gum1 ,a]1 ,tyro1 ,cortis1 ,syrma1 ,ursa1 ,fama1 ,naxos1 ,erisma4 ,l>3 ,plebs1 ,cura1 ,lim51 ,vulcan1 ,angu/a1 ,nepos1 ,c]a1 ,papav]1 ,pignus1 ,messor4 ,pluvium3 ,bid5s1 ,cudo1 ,malleus1 ,anguis1 ,aries1 ,luctus1 ,capella1 ,rh$a1 ,rap9a4 ,vesp]3 ,uncus1 ,ocrea1 ,orcus1 ,po/is1 ,arcus1 ,ae/us1 ,cussis1 ,dolium1 ,alexis4 #i4 ,mala3 ,fama1 ,conjux1 ,reus1 ,torus1 ,acus1 ,myrrha1 ,sypho1 ,plebs1 ,pallor1 ,mora1 ,oliva1 ,alpis1 ,acies1 ,hircus4 ,l>3 ,missale1 ,=tuna1 ,vita1 ,pax1 ,furor1 ,ira1 ,vulcan1 ,p5ates1 ,lapis1 ,apis1 ,c]a1 ,pignus4 ,pluvium3 ,torus1 ,pl5um1 ,pax1 ,?orax1 ,dolus1 ,furor1 ,v9culum1 ,securis1 ,clavis1 ,gurges1 ,ae/as1 ,capella1 ,corbis4 ,vesp]3 ,uncus1 ,spicula1 ,m>s1 ,ocrea1 ,c>do1 ,?yrsus1 ,c]vix1 ,v]bum1 ,orcus1 ,polus1 ,spes1 ,s5ex1 ,septum1 ,porrum1 ,cussis1 ,dolium4 ,,answ]s4 ,! %apes are3 #a4 ,urna1 ,moles1 ,plebs1 ,cura1 ,tess>a1 ,lac]ta1 ,clipeus1 ,ovis4 #b4 ,urna1 ,fr5um1 ,crates1 ,ocrea1 ,tess>a1 ,glans1 ,colus1 ,tabula4 #c4 ,comes1 ,tibic51 ,mica1 ,sac]dos1 ,tigris1 ,lac]ta4 #d4 ,ve/is1 ,oliva1 ,tyro1 ,pluma1 ,portio4 #e4 ,m>s1 ,m]ces1 ,vom]1 ,ovis1 ,portio1 ,tabula1 ,te/udo1 ,lac]ta1 ,p5ates4 #f4 ,pallor1 ,tess>a1 ,lac]ta1 ,tigris1 ,segm51 ,portio1 ,ovis1 ,arcus1 ,laurus1 ,axis1 ,troja1 ,aries4 #g4 ,hospes1 ,tribus1 ,ari/a1 ,pellis1 ,colus1 ,pluma1 ,portio1 ,cala?us1 ,turtur1 ,sepes4 #h4 ,pallor1 ,tess>a1 ,domitor1 ,te/udo1 ,tabula1 ,clipeus1 ,portio1 ,cala?us1 ,nux1 ,lectrum1 ,corymbus1 ,circaea1 ,cordax4 #i4 ,fama1 ,conjux1 ,reus1 ,f]a1 ,?orax1 ,pax1 ,furor1 ,dolus1 ,scala1 ,ira1 ,vulcan1 ,p5ates1 ,lapis1 ,palus1 ,sepes1 ,turtur1 ,diota1 ,dra*ma1 ,py?on4 ,,app5dix ,h4 ,,sections ,,( ,,cube ,,& ,,tess>act4 ,!re are ?ree k9ds ( sections ( a cube4 #a4 ,! sectional plane1 :i* is 9 all cases suppos$ to be inf9ite1 c be tak5 p>allel to two ( ! opposite faces ( ! cube2 t is1 p>allel to two ( ! l9es meet+ 9 ,corvus1 & cutt+ ! ?ird4 #b4 ,! sectional plane c be tak5 p>allel to "o ( ! l9es meet+ 9 ,corvus & cutt+ ! o!r two1 or "o or bo? ( !m produc$4 #c4 ,! sectional plane c be tak5 cutt+ all ?ree l9es1 or any or all ( !m produc$4 ,take ! fir/ case1 & suppose ! plane cuts ,dos half-way betwe5 ,corvus & ,ci/a4 ,s9ce x does n cut ,arctos or ,cuspis1 or ei!r ( !m produc$1 x w cut ,via1 ,it]1 & ,bolus at ! middle po9t ( ea*2 & ! figure1 det]m9$ by ! int]section ( ! ,plane & ,mala1 is a squ>e4 ,if ! l5g? ( ! edge ( ! cube be tak5 z ! unit1 ?is figure may be express$ ?us3 ,z ,x ,y #j 4 #j 4 %[+ t ! ,z & ,x l9es f ,corvus are n cut at all1 & t ! ,y l9e is cut at half-a-unit f ,corvus4 ,sections tak5 ,z ,x ,y #j 4 #j 4 & ,z ,x ,y #j 4 #j 4 #a w\ld also be squ>es4 ,take ! second case4 ,let ! plane cut ,cuspis & ,dos1 ea* at half-a-unit f ,corvus1 & n cut ,arctos or ,arctos produc$2 x w also cut ?r\gh ! middle po9ts ( ,via & ,callis4 ,! figure produc$1 is a rectangle :i* has two sides ( "o unit1 & ! o!r two are ea* ! diagonal ( a half-unit squ>$4 ,if ! plane cuts ,cuspis & ,dos1 ea* at "o unit f ,corvus1 & is p>allel to ,arctos1 ! figure w be a rectangle :i* has two sides ( "o unit 9 l5g?2 & ! o!r two ! diagonal ( "o unit squ>$4 ,if ! plane passes ?r\gh ,mala1 cutt+ ,dos produc$ & ,cuspis produc$1 ea* at "o-&-a-half unit f ,corvus1 & is p>allel to ,arctos1 ! figure w be a p>allelogram l ! "o obta9$ by ! section ,z ,x ,y #j 4 4 4 ,?is set ( figures w be express$ ,z ,x ,y ,z ,x ,y ,z ,x ,y #j 4 4 #j 4 #a 4 #a #j 4 #a 4 #a ,x w be se5 t ^! sections are p>allel to ea* o!r2 & t 9 ea* figure ,cuspis & ,dos are cut at equal di/ances f ,corvus4 ,we may express ! :ole set ?us3-- ,z ,x ,y ,o 4 ,i 4 ,i x be+ und]/ood t :]e ,roman figures are us$1 ! numb]s d n ref] to ! l5g? ( unit cut (f any giv5 l9e f ,corvus1 b to ! proportion betwe5 ! l5g?s4 ,?us ,z ,x ,y ,o 4 ,i 4 ,,ii means t ,arctos is n cut at all1 & t ,cuspis & ,dos are cut1 ,dos be+ cut twice z f> f ,corvus z is ,cuspis4 ,^! figures w also be rectangles4 ,take ! ?ird case4 ,suppose ,arctos1 ,cuspis1 & ,dos are ea* cut half-way4 ,?is figure is an equilat]al triangle1 ^: sides are ! diagonal ( a half-unit squ>$4 ,! figure ,z ,x ,y #a 4 #a 4 #a is also an equilat]al triangle1 & ! figure ,z ,x ,y #a 4 #a 4 #a is an equilat]al hexagon4 ,x is easy = u to see :at ^! %apes are1 & also1 :at ! figures ( any o!r set w\ld be1 z ,z ,x ,y ,i 4 ,,ii 4 ,,ii or ,z ,x ,y ,i 4 ,,ii 4 ,,iii b we mu/ le>n !m z a two-dim5sional be+ w\ld1 s t we may see h[ to le>n ! ?ree-dim5sional sections ( a tess>act4 ,x is evid5t t ! result+ figures are ! same :e!r we fix ! cube1 & !n turn ! sectional plane to ! requir$ position1 or :e!r we fix ! sectional plane1 & !n turn ! cube4 ,?us1 9 ! fir/ case we might h fix$ ! plane1 & !n s plac$ ! cube t "o plane side co9cid$ ) ! sectional plane1 & !n h drawn ! cube half-way ?r\gh1 9 a direction at "r angles to ! plane1 :5 we %\ld h se5 ! squ>e fir/ m5tion$4 ,9 ! second case 7,z ,x ,y7 7,o 4 ,i 4 ,i7 we might h put ! cube ) ,arctos co9cid+ ) ! plane & ) ,cuspis & ,dos equally incl9$ to x1 & !n h drawn ! cube ?r\gh ! plane at "r angles to x until ! l9es 7,cuspis & ,dos7 7 cut at ! requir$ di/ances f ,corvus4 ,9 ! ?ird case we might h put ! cube ) only ,corvus co9cid+ ) ! plane & ) ,cuspis1 ,dos1 & ,arctos equally incl9$ to x 7= any ( ! %apes 9 ! set ,z ,x ,y7 ,i 4 ,i 4 ,i7 & !n h drawn x ?r\gh z be=e4 ,! result+ figures are exactly ! same z ^? we got be=e2 b ?is way is ! be/ to use1 z x w\ld probably be easi] = a two-dim5sional be+ to ?9k ( a cube pass+ ?r\gh 8 space ?an to imag9e 8 :ole space turn$ r\nd1 ) reg>d to ! cube4 ,we h already se5 7p4 #aag7 h[ a two-dim5sional be+ w\ld obs]ve ! sections ( a cube :5 x is put ) "o plane side co9cid+ ) 8 space1 & is !n drawn p>tly ?r\gh4 ,n[1 suppose ! cube put ) ! l9e ,arctos co9cid+ ) 8 space1 & ! l9es ,cuspis & ,dos equally incl9$ to x4 ,at fir/ he w\ld only see ,arctos4 ,if ! cube 7 mov$ until ,dos & ,cuspis 7 ea* cut half-way1 ,arctos /ill be+ p>allel to ! plane1 ,arctos w\ld disappe> at once2 & to f9d out :at he w\ld see he w\ld h to take ! squ>e sections ( ! cube1 & f9d on ea* ( !m :at l9es are giv5 by ! new set ( sections4 ,?us he w\ld take ,mo5a itself1 :i* may be reg>d$ z ! fir/ section ( ! squ>e set4 ,"o po9t ( ! figure w\ld be ! middle po9t ( ,cuspis1 & s9ce ! sectional plane is p>allel to ,arctos1 ! l9e ( int]section ( ,mo5a ) ! sectional plane w be p>allel to ,arctos4 ,! requir$ l9e !n cuts ,cuspis half-way1 & is p>allel to ,arctos1 !re=e x cuts ,callis half-way4 ,illu/ration3 ,fig4 #ba4 ,next he w\ld take ! squ>e section half-way betwe5 ,mo5a & ,murex4 ,he kn[s t ! l9e ,alvus ( ?is section is p>allel to ,arctos1 & t ! po9t ,dos at "o ( its ends is half-way betwe5 ,corvus & ,ci/a1 s t ?is l9e itself is ! "o he wants 7because ! sectional plane cuts ,dos half-way betwe5 ,corvus & ,ci/a1 & is p>allel to ,arctos74 ,9 ,fig4 #ba ! two l9es ?us f\nd are %[n4 a b is ! l9e 9 ,mo5a1 & c d ! l9e 9 ! section4 ,he mu/ n[ f9d out h[ f> ap>t !y are4 ,he kn[s t f ! middle po9t ( ,cuspis to ,corvus is half-a-unit1 & f ! middle po9t ( ,dos to ,corvus is half-a-unit1 & ,cuspis & ,dos are at "r angles to ea* o!r2 !re=e f ! middle po9t ( ,cuspis to ! middle po9t ( ,dos is ! diagonal ( a squ>e ^: sides are half-a-unit 9 l5g?4 ,?is diagonal may be writt5 d 774 ,he w\ld also see t f ! middle po9t ( ,callis to ! middle po9t ( ,via is ! same l5g?2 !re=e ! figure is a p>allelogram1 hav+ two ( its sides1 ea* "o unit 9 l5g?1 & ! o!r two ea* d 774 ,he c\ld also see t ! angles are "r1 because ! l9es a c & b d are made up ( ! ,x & ,y directions1 & ! o!r two1 a b & d1 are purely ,z1 & s9ce !y h no t5d5cy 9 common1 !y are at "r angles to ea* o!r4 ,illu/ration3 ,fig4 #bb4 ,if he want$ ! figure made by ,z ,x ,y #j 4 #a 4 #a x w\ld be a little m difficult4 ,he w\ld h to take ,mo5a1 a section halfway betwe5 ,mo5a & ,murex1 ,murex & ano!r squ>e :i* he w\ld h to reg>d z an imag9>y section half-a-unit fur!r ,y ?an ,murex 7,fig4 #bb74 ,he might n[ draw a gr\nd plan ( ! sections2 t is1 he w\ld draw ,syce1 & produce ,cuspis & ,dos half-a-unit beyond ,nug & ,ci/a4 ,he w\ld see t ,cadus & ,bolus w\ld be cut half-way1 s t 9 ! half-way section he w\ld h ! po9t a 7,fig4 #bc71 & 9 ,murex ! po9t c4 ,9 ! imag9>y section he w\ld h g2 b ?is he might disreg>d1 s9ce ! cube goes no fur!r ?an ,murex4 ,f ! po9ts c & a !re w\ld be l9es go+ ,z1 s t ,it] & ,semita w\ld be cut half-way4 ,illu/ration3 ,gr\ndplan ( ,sections %[n 9 ,fig4 #bb4 ,fig4 #bc4 ,he c\ld f9d out h[ f> ! two l9es a b & c d 7,fig4 #bb7 are ap>t by ref]r+ d & b to ,lama1 & a & c to ,crus4 ,9 tak+ ! ?ird ord] ( sections1 a simil> me?od may be foll[$4 ,illu/ration3 ,fig4 #bd4 ,suppose ! sectional plane to cut ,cuspis1 ,dos1 & ,arctos1 ea* at "o unit f ,corvus4 ,he w\ld fir/ take ,mo5a1 & z ! sectional plane passes ?r\gh ,ilex & ,nug1 ! l9e on ,mo5a w\ld be ! diagonal pass+ ?r\gh ^! two po9ts4 ,!n he w\ld take ,murex1 & he w\ld see t z ! plane cuts ,dos at "o unit f ,corvus1 all he w\ld h is ! po9t ,ci/a4 ,s ! :ole figure is ! ,ilex to ,nug diagonal1 & ! po9t ,ci/a4 ,n[ ,ci/a & ,ilex are ea* "o in* f ,corvus1 & measur$ along l9es at "r angles to ea* o!r2 !re=e1 !y are d 7#a7 f ea* o!r4 ,by ref]r+ ,nug & ,ci/a to ,corvus he w\ld f9d t !y are also d 7#a7 ap>t2 !re=e ! figure is an equilat]al triangle1 ^: sides are ea* d 7#a74 ,suppose ! sectional plane to pass ?r\gh ,mala1 cutt+ ,cuspis1 ,dos1 & ,arctos ea* at unit f ,corvus4 ,to f9d ! figure1 ! plane-be+ w\ld h to take ,mo5a1 a section half-way betwe5 ,mo5a & ,murex1 ,murex1 & an imag9>y section half-a-unit beyond ,murex 7,fig4 #bd74 ,he w\ld produce ,arctos & ,cuspis to po9ts half-a-unit f ,ilex & ,nug1 & by jo9+ ^! po9ts1 he w\ld see t ! l9e passes ?r\gh ! middle po9ts ( ,callis & ,f> 7a1 b1 ,fig4 #bd74 ,9 ! la/ squ>e1 ! imag9>y section1 !re w\ld be ! po9t m2 = ?is is #a unit f ,corvus measur$ along ,dos produc$4 ,!re w\ld also be l9es 9 ! o!r two squ>es1 ! section & ,murex1 & to f9d ^! he w\ld h to make _m obs]vations4 ,he f\nd ! po9ts a & b 7,fig4 #bd7 by draw+ a l9e f r to s1 r & s be+ ea* #a unit f ,corvus1 & simply see+ t x cut ,callis & ,f> at ! middle po9t ( ea*4 ,he might n[ imag9e a cube ,mala turn$ ab\t ,arctos1 s t ,alvus came into 8 plane2 he might !n produce ,arctos & ,dos until !y 7 ea* unit long1 & jo9 _! extremities1 :5 he w\ld see t ,via & ,buc9a are ea* cut half-way4 ,aga91 by turn+ ,syce into 8 plane1 & produc+ ,dos & ,cuspis to po9ts #a unit f ,corvus & jo9+ ! po9ts1 he w\ld see t ,bolus & ,cadus are cut half-way4 ,he has n[ det]m9$ six po9ts on ,mala1 ?r\gh :i* ! plane passes1 & by ref]r+ !m 9 pairs to ,ilex1 ,olus1 ,ci/a1 ,crus1 ,nug1 ,sors1 he w\ld f9d t ea* 0 d 77 f ! next2 s he w\ld "k t ! figure is an equilat]al hexagon4 ,! angles he w\ld n h got 9 ?is obs]vation1 & !y might be a s]i\s difficulty to him4 ,x %\ld be obs]v$ t a simil> difficulty does n come to u 9 our obs]vation ( ! sections ( a tess>act3 =1 if ! angles ( ea* side ( a solid figure are det]m9$1 ! solid angles are also det]m9$4 ,!re is ano!r1 & 9 "s respects a bett]1 way by :i* he might h f\nd ! sides ( ?is figure4 ,if he _h notic$ 8 plane-space mu*1 he w\ld h f\nd out t1 if a l9e be drawn to cut two o!r l9es :i* meet1 ! ratio ( ! p>ts ( ! two l9es cut (f by ! fir/ l9e1 on ! side ( ! angle1 is ! same = ^? l9es1 & any o!r two t are p>allel to !m4 ,?us1 if a b & a c 7,fig4 #be7 meet1 mak+ an angle at a1 & b c crosses !m1 & also crosses a b & a c1 ^! la/ two be+ p>allel to a b & a c1 !n a b a c a b a c4 ,illu/ration3 ,fig4 #be ,if ! plane-be+ knew ?is1 he w\ld rightly assume t if ?ree l9es meet1 mak+ a solid angle1 & a plane passes ?r\gh !m1 ! ratio ( ! p>ts betwe5 ! plane & ! angle is ! same = ^? ?ree l9es1 & = any o!r ?ree p>allel to !m4 ,9 ! case we are deal+ ) he kn[s t f ,ilex to ! po9t on ,arctos produc$ :]e ! plane cuts1 x is half-a-unit2 & z ! ,z1 ,x1 & ,y l9es are cut equally f ,corvus1 he w\ld conclude t ! ,x & ,y l9es are cut ! same di/ance f ,ilex z ! ,z l9e1 t is half-a-unit4 ,he kn[s t ! ,x l9e is cut at #a units f ,corvus2 t is1 half-a-unit f ,nug3 s he w\ld conclude t ! ,z & ,y l9es are cut half-a-unit f ,nug4 ,he w\ld also see t ! ,z & ,x l9es f ,ci/a are cut at half-a-unit4 ,he has n[ six po9ts on ! cube1 ! middle po9ts ( ,callis1 ,via1 ,buc9a1 ,cadus1 ,bolus1 & ,f>4 ,n[1 look+ at 8 squ>e sections1 he w\ld see on ,mo5a a l9e go+ f middle ( ,f> to middle ( ,callis1 t is1 a l9e d 77 long4 ,on ! section he w\ld see a l9e f middle ( ,via to middle ( ,bolus d 7#a7 long1 & on ,murex he w\ld see a l9e f middle ( ,cadus to middle ( ,buc9a1 d 77 long4 ,( ^! ?ree l9es a b1 c d1 e f1 7,fig4 #bd7--;a b & e f are sides1 & c d is a section ( ! requir$ figure4 ,he c f9d ! di/ances betwe5 a & c by ref]5ce to ,ilex1 betwe5 b & d by ref]5ce to ,nug1 betwe5 c & e by ref]5ce to ,olus1 & betwe5 d & f by ref]5ce to ,crus2 & he w f9d t ^! di/ances are ea* d 774 ,?us1 he w\ld "k t ! figure is an equilat]al hexagon ) its sides d 77 long1 ( :i* two ( ! opposite po9ts 7c & d7 are d 7#a7 ap>t1 & ! only figure fulfill+ all ^! conditions is an equilat]al & equiangul> hexagon4 ,en\gh has be5 said ab\t sections ( a cube1 to %[ h[ a plane-be+ w\ld f9d ! %apes 9 any set z 9 ,z ,x ,y ,i 4 ,,ii 4 ,,ii or ,z ,x ,y ,i 4 ,i 4 ,,ii4 ,he w\ld always h to be> 9 m9d t ! ratio ( ! l5g?s ( ! ,z1 ,x1 & ,y l9es is ! same f ,corvus to ! sectional plane z f any o!r po9t to ! sectional plane4 ,?us1 if he 7 tak+ a section :]e ! plane cuts ,arctos & ,cuspis at "o unit f ,corvus & ,dos at "o-&-a-half1 t is :]e ! ratio ( ,z & ( ,x to ,y is z two to ?ree1 he w\ld see t ,dos itself is n cut at all2 b f ,ci/a to ! po9t on ,dos produc$ is half-a-unit2 !re=e f ,ci/a1 ! ,z & ,x l9es w be cut at ( unit f ,ci/a4 ,x is impossible 9 writ+ to %[ h[ to make ! v>i\s sections ( a tess>act2 & ev5 if x 7 n s1 x w\ld be unadvisable2 = ! value ( do+ x is n 9 see+ ! %apes !mselves1 s mu* z 9 ! conc5tration ( ! m9d on ! tess>act involv$ 9 ! process ( f9d+ !m out4 ,any "o :o wi%es to make !m %\ld g c>efully ov] ! sections ( a cube1 n look+ at !m z he himself c see !m1 or det]m9+ !m z he1 ) 8 ?ree-dim5sional conceptions1 c2 b he mu/ limit 8 imag9ation to two dim5sions1 & "w ?r\gh ! problems :i* a plane-be+ w\ld h to "w ?r\gh1 al?\gh to 8 high] m9d !y may be self-evid5t4 ,?us a ?ree-dim5sional be+ c see at a glance1 t if a sectional plane passes ?r\gh a cube at "o unit ea* way f ,corvus1 ! result+ figure is an equilat]al triangle4 ,if he wi%$ to prove x1 he w\ld %[ t ! ?ree b\nd+ l9es are ! diagonals ( equal squ>es4 ,?is is all a two-dim5sional be+ w\ld h to d2 b x is n s evid5t to him t two ( ! l9es are ! diagonals ( squ>es4 ,moreov]1 :5 ! figure is drawn1 we c look at x f a po9t outside ! plane ( ! figure1 & c ?us see x all at once2 b he :o has to look at x f a po9t 9 ! plane c only see an edge at a "t1 or he might see two edges 9 p]spective toge!r4 ,!n !re are c]ta9 suppositions he has to make4 ,= in/ance1 he kn[s t two po9ts det]m9e a l9e1 & he assumes t ?ree po9ts det]m9e a plane1 al?\gh he _c conceive any o!r plane ?an ! "o 9 :i* he exi/s4 ,we assume t f\r po9ts det]m9e a solid space4 ,or r1 we say t if ?is supposition1 toge!r ) c]ta9 o!rs ( a l nature1 are true1 we c f9d all ! sections ( a tess>act1 & ( o!r f\r-dim5sional figures by an inf9ite solid4 ,:5 any difficulty arises 9 tak+ ! sections ( a tess>act1 ! sure/ way ( ov]com+ x is to suppose a simil> difficulty occurr+ to a two-dim5sional be+ 9 tak+ ! sections ( a cube1 &1 /ep by /ep1 to foll[ ! solution he might obta91 & !n to apply ! same or simil> pr9ciples to ! case 9 po9t4 ,a few figures are giv51 :i*1 if cut out & fold$ along ! l9es1 w %[ "s ( ! sections ( a tess>act4 ,b ! read] is e>ne/ly begg$ n to be cont5t ) look+ at ! %apes only4 ,t w tea* him no?+ ab\t a tess>act1 or f\r-dim5sional space1 & w only t5d to produce 9 8 m9d a feel+ t 8! f\r? dim5sion0 is an unkn[n & un?9kable region1 9 :i* any %apes may be "r1 z giv5 sections ( its figures1 & ( :i* any /atem5t may be true4 ,:ile1 9 fact1 if x is ! case t ! laws ( spaces ( two & ?ree dim5sions may1 ) tru?1 be c>ri$ on into space ( f\r dim5sions2 !n ! little our solidity 7l ! flatness ( a plane-be+7 w all[ u to le>n ( ^! %apes & relations1 is no m a matt] ( d\bt to u ?an :at we le>n ( two- & ?ree-dim5sional %apes & relations4 ,!re are giv5 also sections ( an octa-tess>act1 & ( a tetra-tess>act1 ! equival5ts 9 f\r-space ( an octah$ron & tetrah$ron4 ,a tetrah$ron may be reg>d$ z a cube ) e alt]nate corn] cut (f4 ,?us1 if ,mala h ! corn] t[>ds ,corvus cut (f z f> z ! po9ts ,ilex1 ,nug1 ,ci/a1 & ! corn] t[>ds ,sors cut (f z f> z ,ilex1 ,nug1 ,lama1 & ! corn] t[>ds ,crus cut (f z f> z ,lama1 ,nug1 ,ci/a1 & ! corn] t[>ds ,olus cut (f z f> z ,ilex1 ,lama1 ,ci/a1 :at is left ( ! cube is a tetrah$ron1 ^: angles are at ! po9ts ,ilex1 ,nug1 ,ci/a1 ,lama4 ,9 a simil> mann]1 if e alt]nate corn] ( a tess>act be cut (f1 ! figure t is left is a tetra-tess>act1 :i* is a figure b\nd$ by sixte5 regul> tetrah$rons4 ,illu/ration3 7i7 ,fig4 #bf4 ,fig4 #bg4 ,fig4 #bg4 ,fig4 #bf4 ,illu/ration3 7ii7 ,fig4 #bh4 ,fig4 #bi4 ,fig4 #cj4 ,illu/ration3 7iii7 ,fig4 #ca4 ,fig4 #cb4 ,illu/ration3 7iv7 ,fig4 #cc4 ,fig4 #cd4 ,fig4 #ce4 ,illu/ration3 7v7 ,fig4 #cf4 ,fig4 #cg4 ,fig4 #ch4 ,illu/ration3 7vi7 ,fig4 #ci4 ,fig4 #dj4 ,fig4 #da4 ,! octa-tess>act is got by cutt+ (f e corn] ( ! tess>act4 ,if e corn] ( a cube is cut (f1 ! figure left is an octa-h$ron1 ^: angles are at ! middle po9ts ( ! sides4 ,! angles ( ! octa-tess>act are at ! middle po9ts ( its plane sides4 ,a c>eful /udy ( a tetra-h$ron & an octa-h$ron z !y are cut out ( a cube w be ! be/ prep>ation = ! /udy ( ^! f\r-dim5sional figures4 ,x w be se5 t !re is mu* to le>n ( !m1 z--,h[ _m planes & l9es !re are 9 ea*1 ,h[ _m solid sides !re are r\nd a po9t 9 ea*4 ,a ,,description ,,( ,,figures #bf ,,to #da4 ,z ,x ,y ,w ,z ,x ,y ,w #bf is a section tak5 #a 4 #a 4 #a 4 #a ,i 4 ,i 4 ,i 4 ,i #bg #a 4 #a 4 #a 4 #a #bh #b 4 #b 4 #b 4 #b ,z ,x ,y ,w ,z ,x ,y ,w #bi is a section tak5 #a 4 #a 4 #a 4 ,,ii 4 ,,ii 4 ,,ii 4 ,i #cj #a 4 #a 4 #a 4 #ca #b 4 #b 4 #b 4 #a #cb #b 4 #b 4 #b 4 #a ,! above are sections ( a tess>act4 ,figures #cc to #ce are ( a tetra-tess>act4 ,! tetra-tess>act is suppos$ to be imb$d$ 9 a tess>act1 & ! sections are tak5 ?r\gh x1 cutt+ ! ,z1 ,x & ,y l9es equally1 & correspond+ to ! figures giv5 ( ! sections ( ! tess>act4 ,figures #cf1 #cg1 & #ch are simil> sections ( an octa-tess>act4 ,figures #ci1 #dj1 & #da are ! foll[+ sections ( a tess>act4 ,z ,x ,y ,w ,z ,x ,y ,w #ci is a section tak5 #j 4 4 4 ,o 4 ,i 4 ,i 4 ,i #dj #j 4 #a 4 #a 4 #a #da #j 4 #a 4 #a 4 #a ,x is cle> t !re are f\r ord]s ( sections ( e f\r-dim5sional figure2 namely1 ^? beg9n+ ) a solid1 ^? beg9n+ ) a plane1 ^? beg9n+ ) a l9e1 & ^? beg9n+ ) a po9t4 ,!re %\ld be little difficulty 9 f9d+ !m1 if ! sections ( a cube ) a tetra-h$ron1 or an octa-h$ron enclos$ 9 x1 are c>efully exam9$4 ,,app5dix ,k4 ,illu/ration3 ,,model #a4 ,,mala4 ,,col\rs3 ,,mala1 ,,light-,,buff4 ,po9ts3 ,corvus1 ,gold4 ,nug1 ,fawn4 ,crus1 ,t]ra-cotta4 ,ci/a1 ,buff4 ,ilex1 ,light-blue4 ,sors1 ,dull-purple4 ,lama1 ,deep-blue4 ,olus1 ,r$4 ,l9es3 ,cuspis1 ,orange4 ,bolus1 ,crimson4 ,cadus1 ,gre5-grey4 ,dos1 ,blue4 ,arctos1 ,br[n4 ,f>1 ,fr5*-grey4 ,daps1 ,d>k-slate4 ,buc9a1 ,gre54 ,callis1 ,r$di%4 ,it]1 ,bright-blue4 ,semita1 ,lead54 ,via1 ,deep-yell[4 ,surfaces3 ,mo5a1 ,d>k-blue4 ,proes1 ,blue-gre54 ,murex1 ,light-yell[4 ,alvus1 ,v]milion4 ,mel1 ,:ite4 ,syce1 ,black4 ,illu/ration3 ,,model #b4 ,,m>go4 ,,col\rs3 ,,m>go1 ,,sage-,,gre54 ,po9ts3 ,spira1 ,silv]4 ,ancilla1 ,turquoise4 ,mugil1 ,e>!n4 ,panax1 ,blue-t9t4 ,felis1 ,quak]-gre54 ,pass]1 ,peacock-blue4 ,talus1 ,orange-v]milion4 ,solia1 ,purple4 ,l9es3 ,luca1 ,leaf-gre54 ,mappa1 ,dull-gre54 ,m5sura1 ,d>k-purple4 ,opex1 ,purple-br[n4 ,pagus1 ,dull-blue4 ,onag]1 ,d>k-p9k4 ,v5a1 ,pale-p9k4 ,lixa1 ,indigo4 ,?olus1 ,br[n-gre54 ,calor1 ,d>k-gre54 ,livor1 ,pale-yell[4 ,l5sa1 ,d>k4 ,surfaces3 ,silex1 ,burnt-si5na4 ,sal1 ,yell[-o*re4 ,portica1 ,dun4 ,crux1 ,indian-r$4 ,l>es1 ,light-grey4 ,lappa1 ,bright-gre54 ,illu/ration3 ,,model #c4 ,,l>4 ,,col\rs3 ,,l>1 ,,brick-,,r$4 ,po9ts3 ,spira1 ,silv]4 ,ancilla1 ,turquoise4 ,mugil1 ,e>!n4 ,panax1 ,blue-t9t4 ,corvus1 ,gold4 ,nug1 ,fawn4 ,crus1 ,t]ra-cotta4 ,ci/a1 ,buff4 ,l9es3 ,luca1 ,leaf-gre54 ,mappa1 ,dull-gre54 ,m5sura1 ,d>k-purple4 ,opex1 ,purple-br[n4 ,ops1 ,/one4 ,limus1 ,smoke4 ,(fex1 ,mag5ta4 ,lis1 ,light-gre54 ,cuspis1 ,orange4 ,bolus1 ,crimson4 ,cadus1 ,gre5-grey4 ,dos1 ,blue4 ,surfaces3 ,lotus1 ,azure4 ,olla1 ,rose4 ,lorica1 ,sea-blue4 ,lua1 ,bright-br[n4 ,syce1 ,black4 ,lappa1 ,bright-gre54 ,illu/ration3 ,,model #d4 ,,velum4 ,,col\rs3 ,,velum1 ,,*ocolate4 ,po9ts3 ,felis1 ,quak]-gre54 ,pass]1 ,peacock-blue4 ,talus1 ,orange-v]milion4 ,solia1 ,purple4 ,ilex1 ,light-blue4 ,sors1 ,dull-purple4 ,lama1 ,deep-blue4 ,olus1 ,r$4 ,l9es3 ,?olus1 ,br[n-gre54 ,calor1 ,d>k-gre54 ,livor1 ,pale-yell[4 ,l5sa1 ,d>k4 ,lucta1 ,ri*-r$4 ,pator1 ,gre5-blue4 ,lib]a1 ,sea-gre54 ,orsa1 ,em]ald4 ,callis1 ,r$di%4 ,it]1 ,bright-blue4 ,semita1 ,lead54 ,via1 ,deep-yell[4 ,surfaces3 ,limbus1 ,o*re4 ,meatus1 ,deep-br[n4 ,mango1 ,deep-gre54 ,croeta1 ,light-r$4 ,mel1 ,:ite4 ,l>es1 ,light-grey4 ,illu/ration3 ,,model #e4 ,,vesp]4 ,,col\rs3 ,,vesp]1 ,,pale-,,gre54 ,po9ts3 ,spira1 ,silv]4 ,corvus1 ,gold4 ,ci/a1 ,buff4 ,panax1 ,blue-t9t4 ,felis1 ,quak]-gre54 ,ilex1 ,light-blue4 ,olus1 ,r$4 ,solia1 ,purple4 ,l9es3 ,ops1 ,/one4 ,dos1 ,blue4 ,lis1 ,light-gre54 ,opex1 ,purple-br[n4 ,pagus1 ,dull-blue4 ,arctos1 ,br[n4 ,buc9a1 ,gre54 ,lixa1 ,indigo4 ,lucta1 ,ri*-r$4 ,via1 ,deep-yell[4 ,orsa1 ,em]ald4 ,l5sa1 ,d>k4 ,surfaces3 ,pag9a1 ,yell[4 ,alvus1 ,v]milion4 ,camo5a1 ,deep-crimson4 ,crux1 ,indian-r$4 ,croeta1 ,light-r$4 ,lua1 ,light-br[n4 ,illu/ration3 ,,model #f4 ,,idus4 ,,col\rs3 ,,idus1 ,,oak4 ,po9ts3 ,ancilla1 ,turquoise4 ,nug1 ,fawn4 ,crus1 ,t]ra-cotta4 ,mugil1 ,e>!n4 ,pass]1 ,peacock-blue4 ,sors1 ,dull-purple4 ,lama1 ,deep-blue4 ,talus1 ,orange-v]milion4 ,l9es3 ,limus1 ,smoke4 ,bolus1 ,crimson4 ,(fex1 ,mag5ta4 ,mappa1 ,dull-gre54 ,onag]1 ,d>k-p9k4 ,f>1 ,fr5*-grey4 ,daps1 ,d>k-slate4 ,v5a1 ,pale-p9k4 ,pator1 ,gre5-blue4 ,it]1 ,bright-blue4 ,lib]a1 ,sea-gre54 ,calor1 ,d>k-gre54 ,surfaces3 ,pactum1 ,yell[-gre54 ,proes1 ,blue-gre54 ,orca1 ,d>k-grey4 ,sal1 ,yell[-o*re4 ,meatus1 ,deep-br[n4 ,olla1 ,rose4 ,illu/ration3 ,,model #g4 ,,pluvium4 ,,col\rs3 ,,pluvium1 ,,d>k-,,/one4 ,po9ts3 ,spira1 ,silv]4 ,ancilla1 ,turquoise4 ,nug1 ,fawn4 ,corvus1 ,gold4 ,felis1 ,quak]-gre54 ,pass]1 ,peacock-blue4 ,sors1 ,dull-purple4 ,ilex1 ,light-blue4 ,l9es3 ,luca1 ,leaf-gre54 ,limus1 ,smoke4 ,cuspis1 ,orange4 ,ops1 ,/one4 ,pagus1 ,dull-blue4 ,onag]1 ,d>k-p9k4 ,f>1 ,fr5*-grey4 ,arctos1 ,br[n4 ,?olos1 ,br[n-gre54 ,pator1 ,gre5-blue4 ,callis1 ,r$di%4 ,lucta1 ,ri*-r$4 ,surfaces3 ,silex1 ,burnt-,si5na4 ,pactum1 ,yell[-gre54 ,mo5a1 ,d>k-blue4 ,pag9a1 ,yell[4 ,limbus1 ,o*re4 ,lotus1 ,azure4 ,illu/ration3 ,,model #h4 ,,tela4 ,,col\rs3 ,,tela1 ,,salmon4 ,po9ts3 ,panax1 ,blue-t9t4 ,mugil1 ,e>!n4 ,crus1 ,t]ra-cotta4 ,ci/a1 ,buff4 ,solia1 ,purple4 ,talus1 ,orange-v]milion4 ,lama1 ,deep-blue4 ,olus1 ,r$4 ,l9es3 ,m5sura1 ,d>k-purple4 ,(fex1 ,mag5ta4 ,cadus1 ,gre5-grey4 ,lis1 ,light-gre54 ,lixa1 ,indigo4 ,v5a1 ,pale-p9k4 ,daps1 ,d>k-slate4 ,buc9a1 ,gre54 ,livor1 ,pale-yell[4 ,lib]a1 ,sea-gre54 ,semita1 ,lead54 ,orsa1 ,em]ald4 ,surfaces3 ,portica1 ,dun4 ,orca1 ,d>k-grey4 ,murex1 ,light-yell[4 ,camo5a1 ,deep-crimson4 ,mango1 ,deep-gre54 ,lorica1 ,sea-blue4 ,illu/ration3 ,,model #i4 ,,section ,,betwe5 ,,mala ,,& ,,m>go4 ,,col\rs3 ,,int]ior ,,or ,,tess>act1 ,,wood4 ,po9ts 7,l9es73 ,ops1 ,/one4 ,limus1 ,smoke4 ,(fex1 ,mag5ta4 ,lis1 ,light-gre54 ,lucta1 ,ri*-r$4 ,pator1 ,gre5-blue4 ,lib]a1 ,sea-gre54 ,orsa1 ,em]ald4 ,l9es 7,surfaces73 ,lotus1 ,azure4 ,olla1 ,rose4 ,lorica1 ,sea-blue4 ,lua ,bright-br[n4 ,pag9a1 ,yell[4 ,pactum1 ,yell[-gre54 ,orca1 ,d>k-grey4 ,camo5a1 ,deep-crimson4 ,limbus1 ,o*re4 ,meatus1 ,deep-br[n4 ,mango1 ,deep-gre54 ,croeta1 ,light r$4 ,surfaces 7,solids73 ,pluvium1 ,d>k-/one4 ,idus1 ,oak4 ,tela1 ,salmon4 ,vesp]1 ,pale-gre54 ,velum1 ,*ocolate4 ,l>1 ,brick-r$4 ,illu/ration3 ,,model #aj4 ,,section ,,betwe5 ,,l> ,,& ,,velum4 ,,col\rs3 ,,int]ior ,,or ,,tess>act1 ,,wood4 ,po9ts 7,l9es73 ,pagus1 ,dull-blue4 ,onag]1 ,d>k-p9k4 ,v5a1 ,pale-p9k4 ,lixa1 ,indigo4 ,arctos1 ,br[n4 ,f>1 ,fr5*-grey4 ,daps1 ,d>k-slate4 ,buc9a1 ,gre54 ,l9es 7,surfaces73 ,silex1 ,burnt-si5na4 ,sal1 ,yell[-o*re4 ,portica1 ,dun4 ,crux1 ,indian-r$4 ,pag9a1 ,yell[4 ,pactum1 ,yell[-gre54 ,orca1 ,d>k-grey4 ,camo5a1 ,deep-crimson4 ,mo5a1 ,d>k-blue4 ,proes1 ,blue-gre54 ,murex1 ,light-yell[4 ,alvus1 ,v]milion4 ,surfaces 7,solids73 ,pluvium1 ,d>k-/one4 ,idus1 ,oak4 ,tela1 ,salmon4 ,vesp]1 ,pale-gre54 ,mala1 ,light-buff4 ,m>go1 ,sage-gre54 ,illu/ration3 ,,model #aa4 ,,section ,,betwe5 ,,vesp] ,,& ,,idus4 ,,col\rs3 ,,int]ior ,,or ,,tess>act1 ,,wood4 ,po9ts 7,l9es73 ,luca1 ,leaf-gre54 ,cuspis1 ,orange4 ,cadus1 ,gre5-grey4 ,m5sura1 ,d>k-purple4 ,?olus1 ,br[n-gre54 ,callis1 ,r$di%4 ,semita1 ,lead54 ,livor1 ,pale-yell[4 ,l9es 7,surfaces73 ,lotus1 ,azure4 ,syce1 ,black4 ,lorica1 ,sea-blue4 ,lappa1 ,bright-gre54 ,silex1 ,burnt-si5na4 ,mo5a1 ,d>k-blue4 ,murex1 ,light-yell[4 ,portica1 ,dun4 ,limbus1 ,o*re4 ,mel1 ,:ite4 ,mango1 ,deep-gre54 ,l>es1 ,light-grey4 ,surfaces 7,solids73 ,pluvium1 ,d>k-/one4 ,mala1 ,light-buff4 ,tela1 ,salmon4 ,m>go1 ,sage-gre54 ,velum1 ,*ocolate4 ,l>1 ,brick-r$4 ,illu/ration3 ,,model #ab4 ,,section ,,betwe5 ,,pluvium ,,& ,,tela4 ,,col\rs3 ,,int]ior ,,or ,,tess>act1 ,,wood4 ,po9ts 7,l9es73 ,opex1 ,purple-br[n4 ,mappa1 ,dull-gre54 ,bolus1 ,crimson4 ,dos1 ,blue4 ,l5sa1 ,d>k4 ,calor1 ,d>k-gre54 ,it]1 ,bright-blue4 ,via1 ,deep-yell[4 ,l9es 7,surfaces73 ,lappa1 ,bright-gre54 ,olla1 ,rose4 ,syce1 ,black4 ,lua1 ,bright-br[n4 ,crux1 ,indian-r$4 ,sal1 ,yell[-o*re4 ,proes1 ,blue-gre54 ,alvus1 ,v]milion4 ,l>es1 ,light-grey4 ,meatus1 ,deep-br[n4 ,mel1 ,:ite4 ,croeta1 ,light-r$4 ,surfaces 7,solids73 ,m>go1 ,sage-gre54 ,idus1 ,oak4 ,mala1 ,light-buff4 ,vesp]1 ,pale-gre54 ,velum1 ,*ocolate4 ,l>1 ,brick-r$4 ,transcrib]'s ,notes 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