,transcrib]'s ,notes ,obvi\s typographical errors h be5 sil5tly correct$4 ,all o!r spell+ & punctuation rema9s un*ang$4 ,italics are repres5t$ ?us italic1 bold ?us bold & sup]scripts ?us yen4 ,x %\ld be not$ t mu* ( ! text is a discussion c5tr$ on ! _m illu/rations :i* h n be5 includ$4 ,,! ,,f\r? ,,dim5sion ,,"s ,,op9ions ,,( ,,! ,,press 8,mr4 ,c4 ,h4 ,h9ton discusses ! subject ( ! high] dim5sionality ( space1 8 aim be+ to avoid ma!matical subtleties & te*nicalities1 & ?us enable 8 argum5t to be foll[$ by read]s :o are n suffici5tly conv]sant ) ma!matics to foll[ ^! processes ( reason+40--,,notts ,,gu>dian4 8,! f\r? dim5sion is a subject :i* has _h a great fasc9ation = _m tea*]s1 & ?\gh "o _c pret5d to h q grasp$ ,mr4 ,h9ton's conceptions & argum5ts1 yet x mu/ be admitt$ t he reveals ! elusive idea 9 q a fasc9at+ light4 ,q ap>t f ! ma9 !sis ( ! book _m *apt]s are ( great indep5d5t int]e/4 ,altoge!r an int]e/+1 clev] & +5i\s book40--,,dundee ,,c\ri]4 8,! book w well repay ! /udy ( m5 :o l to ex]cise _! wits ^u ! problems ( ab/ract ?\ght40--,,scotsman4 8,pr(essor ,h9ton has done well to attempt a treatise ( mod]ate size1 :i* %all at once be cle> 9 me?od & free f te*nicalities ( ! s*ools40--,,pall ,,mall ,,gazette4 8,a v int]e/+ book he has made ( x40--,,publi%]s' ,,circul>4 8,mr4 ,h9ton tries to expla9 ! !ory ( ! f\r? dim5sion s t ! ord9>y reason+ m9d c get a grasp ( :at metaphysical ma!maticians mean by x4 ,if he is n altoge!r successful x is n f want ( cle>ness on 8 "p1 b because ! :ole !ory comes z su* an absolute %ock to all "o's preconceiv$ ideas40--,,bri/ol ,,times4 8,mr4 ,h9ton's en?usiasm is only ! result ( an exhau/ive /udy1 :i* has enabl$ him to set 8 subject be=e ! read] ) f> m ?an ! am\nt ( lucidity to :i* x is accu/om$40--,,pall ,,mall ,,gazette4 8,! book ?r\gh\t is a v solid piece ( reason+ 9 ! doma9 ( high] ma!matics40--,,glasg[ ,,h]ald4 8,^? :o wi% to grasp ! mean+ ( ?is some:at difficult subject w\ld d well to read ,! ,f\r? ,dim5sion4 ,no ma!matical k is dem&$ ( ! read]1 & any "o1 :o is n afraid ( a little h>d ?9k+1 %\ld be able to foll[ ! argum5t40--,,light4 8,a spl5didly cle> re-/atem5t ( ! old problem ( ! f\r? dim5sion4 ,all :o are int]e/$ 9 ?is subject w f9d ! "w n only fasc9at+1 b lucid1 x be+ writt5 9 a /yle easily und]/&able4 ,! illu/rations make /ill m cle> ! lett]press1 & ! :ole is mo/ admirably adapt$ to ! requirem5ts ( ! novice or ! /ud5t40--,,two ,,worlds4 8,^? 9 se>* ( m5tal gymna/ics w f9d abundance ( ex]cise 9 ,mr4 ,c4 ,h4 ,h9ton's ,f\r? ,dim5sion40--,,we/m9/] ,,review4 ,,fir/ ,,edition1 ,april #aijd2 ,,second ,,edition1 ,may #aijf4 ,views ( ! ,tess>act4 ,no4 #a4 ,no4 #b4 ,no4 #c4 ,no4 #d4 ,no4 #e4 ,no4 #f4 ,no4 #g4 ,no4 #h4 ,no4 #i4 ,no4 #aj4 ,no4 #aa4 ,no4 #ab4 ,,! ,,f\r? ,,dim5sion ,,by ,c4 ,,h[>d ,,h9ton1 ,m4,a4 ,,au?or ,,( 8,,sci5tific ,,romances0 8,a ,,new ,,era ,,( ,,?\ght10 ,,etc41 ,,etc4 ,illu/ration3 ,colophon ,,london ,,swan ,,sonn5s*e9 ,,co41 ,,limit$ #be ,,high ,,/reet1 ,,bloomsbury #aijf ,,pr9t$ ,,by ,,hazell1 ,,watson ,,& ,,v9ey1 ,,ld41 ,,london ,,& ,,aylesbury4 ,,preface ,i h endeav\r$ to pres5t ! subject ( ! high] dim5sionality ( space 9 a cle> mann]1 devoid ( ma!matical subtleties & te*nicalities4 ,9 ord] to engage ! int]e/ ( ! read]1 ,i h 9 ! e>li] *apt]s dwelt on ! p]spective ! hypo!sis ( a f\r? dim5sion op5s1 & h treat$ ( ! _m connections !re are betwe5 ?is hypo!sis & ! ord9>y topics ( our ?\ghts4 ,a lack ( ma!matical k w prove ( no disadvantage to ! read]1 = ,i h us$ no ma!matical processes ( reason+4 ,i h tak5 ! view t ! space :i* we ord9>ily ?9k (1 ! space ( real ?+s 7:i* ,i w\ld call p]meable matt]71 is diff]5t f ! space treat$ ( by ma!matics4 ,ma!matics w tell u a great deal ab\t space1 j z ! atomic !ory w tell u a great deal ab\t ! *emical comb9ations ( bodies4 ,b aft] all1 a !ory is n precisely equival5t to ! subject ) reg>d to :i* x is held4 ,!re is an op5+1 !re=e1 f ! side ( our ord9>y space p]ceptions = a simple1 altoge!r rational1 me*anical1 & obs]vational way ( treat+ ?is subject ( high] space1 & ( ?is opportunity ,i h avail$ myself4 ,! details introduc$ 9 ! e>li] *apt]s1 especially 9 ,*apt]s ,,viii41 ,,ix41 ,x41 may p]haps be f\nd we>isome4 ,!y are ( no ess5tial importance 9 ! ma9 l9e ( argum5t1 & if left till ,*apt]s ,,xi4 & ,,xii4 h be5 read1 w be f\nd to af=d int]e/+ & obvi\s illu/rations ( ! prop]ties discuss$ 9 ! lat] *apt]s4 ,my ?anks are due to ! fri5ds :o h assi/$ me 9 design+ & prep>+ ! modifications ( my previ\s models1 & 9 no small degree to ! publi%] ( ?is volume1 ,mr4 ,sonn5s*e91 to ^: unique appreciation ( ! l9e ( ?\ght ( ?is1 z ( my =m] essays1 _! publication is ow+4 ,by ! provision ( a col\r$ plate1 9 addition to ! o!r illu/rations1 he has add$ greatly to ! conv5i5ce ( ! read]4 ,c4 ,,h[>d ,,h9ton4 ,,cont5ts ,,*ap4 ,,page ,i4 ,,f\r-,,dim5sional ,,space #a ,,ii4 ,,! ,,analogy ,,( ,a ,,plane ,,_w #f ,,iii4 ,,! ,,significance ,,( ,a ,,f\r-,,dim5sional ,,exi/5ce #ae ,,iv4 ,,! ,,fir/ ,,*apt] ,,9 ,,! ,,hi/ory ,,( ,,f\r ,,space #bc ,v4 ,,! ,,second ,,*apt] ,,9 ,,! ,,hi/ory ,,( ,,f\r ,,space #da ,lobat*ewsky1 ,bolyai1 & ,gauss ,metageometry ,,vi4 ,,! ,,high] ,,_w #fa ,,vii4 ,,! ,,evid5ce ,,= ,a ,,f\r? ,,dim5sion #gf ,,viii4 ,,! ,,use ,,( ,,f\r ,,dim5sions ,,9 ,,?\ght #he ,,ix4 ,,application ,,to ,,kant',s ,,!ory ,,( ,,exp]i5ce #ajg ,x4 ,a ,,f\r-,,dim5sional ,,figure #abb ,,xi4 ,,nom5clature ,,& ,,analogies #acf ,,xii4 ,,! ,,simple/ ,,f\r-,,dim5sional ,,solid #aeg ,,xiii4 ,,rem>ks ,,on ,,! ,,figures #agh ,,xiv4 ,a ,,recapitulation ,,& ,,ext5sion ,,( ,,! ,,physical ,,argum5t #bjc ,,app5dix ,i4--,,! ,,models #bca ,,ii4--,a ,,language ,,( ,,space #bdh ,,! ,,f\r? ,,dim5sion ,,*apt] ,i ,,f\r-,,dim5sional ,,space ,!re is no?+ m indef9ite1 & at ! same "t m real1 ?an t :i* we indicate :5 we speak ( ! 8high]40 ,9 our social life we see x evid5c$ 9 a great] complexity ( relations4 ,b ?is complexity is n all4 ,!re is1 at ! same "t1 a contact )1 an appreh5sion (1 some?+ m fundam5tal1 m real4 ,) ! great] developm5t ( man !re comes a consci\sness ( some?+ m ?an all ! =ms 9 :i* x %[s itself4 ,!re is a read9ess to give up all ! visible & tangible = ! sake ( ^? pr9ciples & values ( :i* ! visible & tangible are ! repres5tation4 ,! physical life ( civilis$ man & ( a m]e savage are practically ! same1 b ! civilis$ man has discov]$ a dep? 9 8 exi/5ce1 :i* makes him feel t t :i* appe>s all to ! savage is a m]e ext]nality & appurt5age to 8 true be+4 ,n[1 ?is high]--h[ %all we appreh5d x8 ,x is g5]ally embrac$ by our religi\s faculties1 by our idealis+ t5d5cy4 ,b ! high] exi/5ce has two sides4 ,x has a be+ z well z qualities4 ,& 9 try+ to realise x ?r\gh our emotions we are always tak+ ! subjective view4 ,our att5tion is always fix$ on :at we feel1 :at we ?9k4 ,is !re any way ( appreh5d+ ! high] aft] ! purely objective me?od ( a natural sci5ce8 ,i ?9k t !re is4 ,plato1 9 a wond]ful allegory1 speaks ( "s m5 liv+ 9 su* a condition t !y 7 practically r$uc$ to be ! d5iz5s ( a %ad[ _w4 ,!y 7 *a9$1 & p]ceiv$ b ! %ad[s ( !mselves & all real objects project$ on a wall1 t[>ds :i* _! faces 7 turn$4 ,all movem5ts to !m 7 b movem5ts on ! surface1 all %apes b ! %apes ( outl9es ) no sub/antiality4 ,plato uses ?is illu/ration to portray ! relation betwe5 true be+ & ! illusions ( ! s5se _w4 ,he says t j z a man lib]at$ f 8 *a9s c\ld le>n & discov] t ! _w 0 solid & real1 & c\ld g back & tell 8 b\nd companions ( ?is great] high] reality1 s ! philosoph] :o has be5 lib]at$1 :o has gone into ! ?\ght ( ! ideal _w1 into ! _w ( ideas great] & m real ?an ! ?+s ( s5se1 c come & tell 8 fell[ m5 ( t :i* is m true ?an ! visible sun--m noble ?an ,a!ns1 ! visible /ate4 ,n[1 ,i take ,plato's sugge/ion2 b lit]ally1 n metaphorically4 ,he imag9es a _w :i* is l[] ?an ?is _w1 9 t %ad[ figures & %ad[ motions are its con/itu5ts2 & to x he contra/s ! real _w4 ,z ! real _w is to ?is %ad[ _w1 s is ! high] _w to our _w4 ,i accept 8 analogy4 ,z our _w 9 ?ree dim5sions is to a %ad[ or plane _w1 s is ! high] _w to our ?ree-dim5sional _w4 ,t is1 ! high] _w is f\r-dim5sional2 ! high] be+ is1 s f> z its exi/5ce is conc]n$ ap>t f its qualities1 to be s\ght ?r\gh ! conception ( an actual exi/5ce spatially high] ?an t :i* we realise ) our s5ses4 ,"h y w obs]ve ,i necess>ily leave out all t gives its *>m & int]e/ to ,plato's writ+s4 ,all ^? conceptions ( ! beautiful & good :i* live immortally 9 8 pages4 ,all t ,i keep f 8 great /oreh\se ( weal? is ?is "o ?+ simply--a _w spatially high] ?an ?is _w1 a _w :i* c only be approa*$ ?r\gh ! /ocks & /ones ( x1 a _w :i* mu/ be appreh5d$ labori\sly1 pati5tly1 ?r\gh ! mat]ial ?+s ( x1 ! %apes1 ! movem5ts1 ! figures ( x4 ,we mu/ le>n to realise ! %apes ( objects 9 ?is _w ( ! high] man2 we mu/ become famili> ) ! movem5ts t objects make 9 8 _w1 s t we c le>n some?+ ab\t 8 daily exp]i5ce1 8 ?\ghts ( mat]ial objects1 8 ma*9]y4 ,! means = ! prosecution ( ?is enquiry are giv5 9 ! conception ( space itself4 ,x (t5 happ5s t t :i* we consid] to be unique & unrelat$ gives u1 )9 itself1 ^? relations by means ( :i* we are able to see x z relat$ to o!rs1 det]m9+ & det]m9$ by !m4 ,?us1 on ! e>? is giv5 t ph5om5on ( weight by means ( :i* ,newton br\ght ! e>? into its true relation to ! sun & o!r planets4 ,our t]re/rial globe 0 det]m9$ 9 reg>d to o!r bodies ( ! sol> sy/em by means ( a relation :i* subsi/$ on ! e>? itself4 ,& s space itself be>s )9 x relations ( :i* we c det]m9e x z relat$ to o!r space4 ,= )9 space are giv5 ! conceptions ( po9t & l9e1 l9e & plane1 :i* really involve ! relation ( space to a high] space4 ,:]e "o segm5t ( a /raight l9e leaves (f & ano!r beg9s is a po9t1 & ! /raight l9e itself c be g5]at$ by ! motion ( ! po9t4 ,"o portion ( a plane is b\nd$ f ano!r by a /raight l9e1 & ! plane itself c be g5]at$ by ! /raight l9e mov+ 9 a direction n conta9$ 9 itself4 ,aga91 two portions ( solid space are limit$ ) reg>d to ea* o!r by a plane2 & ! plane1 mov+ 9 a direction n conta9$ 9 itself1 c g5]ate solid space4 ,?us1 go+ on1 we may say t space is t :i* limits two portions ( high] space f ea* o!r1 & t our space w g5]ate ! high] space by mov+ 9 a direction n conta9$ 9 itself4 ,ano!r indication ( ! nature ( f\r-dim5sional space c be ga9$ by consid]+ ! problem ( ! arrangem5t ( objects4 ,if ,i h a numb] ( swords ( v>y+ degrees ( brightness1 ,i c repres5t !m 9 respect ( ?is quality by po9ts arrang$ along a /raight l9e4 ,if ,i place a sword at ,a1 fig4 #a1 & reg>d x z hav+ a c]ta9 brightness1 !n ! o!r swords c be arrang$ 9 a s]ies along ! l9e1 z at ,a1 ,b1 ,c1 etc41 accord+ to _! degrees ( brightness4 ,illu/ration3 ,fig4 #a4 ,if n[ ,i take acc\nt ( ano!r quality1 say l5g?1 !y c be arrang$ 9 a plane4 ,/>t+ f ,a1 ,b1 ,c1 ,i c f9d po9ts to repres5t diff]5t degrees ( l5g? along su* l9es z ,,af1 ,,bd1 ,,ce1 drawn f ,a & ,b & ,c4 ,po9ts on ^! l9es repres5t diff]5t degrees ( l5g? ) ! same degree ( brightness4 ,?us ! :ole plane is occupi$ by po9ts repres5t+ all conceivable v>ieties ( brightness & l5g?4 ,illu/ration3 ,fig4 #b4 ,br++ 9 a ?ird quality1 say %>pness1 ,i c draw1 z 9 fig4 #c1 any numb] ( upright l9es4 ,let di/ances along ^! upright l9es repres5t degrees ( %>pness1 ?us ! po9ts ,f & ,g w repres5t swords ( c]ta9 def9ite degrees ( ! ?ree qualities m5tion$1 & ! :ole ( space w s]ve to repres5t all conceivable degrees ( ^! ?ree qualities4 ,illu/ration3 ,fig4 #c4 ,if n[ ,i br+ 9 a f\r? quality1 su* z weight1 & try to f9d a means ( repres5t+ x z ,i did ! o!r ?ree qualities1 ,i f9d a difficulty4 ,e po9t 9 space is tak5 up by "s conceivable comb9ation ( ! ?ree qualities already tak54 ,to repres5t f\r qualities 9 ! same way z t 9 :i* ,i h repres5t$ ?ree1 ,i %\ld ne$ ano!r dim5sion ( space4 ,?us we may indicate ! nature ( f\r-dim5sional space by say+ t x is a k9d ( space :i* w\ld give positions repres5tative ( f\r qualities1 z ?ree-dim5sional space gives positions repres5tative ( ?ree qualities4 ,,*apt] ,,ii ,,! ,,analogy ,,( ,a ,,plane ,,_w ,at ! risk ( "s prolixity ,i w g fully into ! exp]i5ce ( a hypo!tical creature conf9$ to motion on a plane surface4 ,by s do+ ,i %all obta9 an analogy :i* w s]ve 9 our subsequ5t enquiries1 because ! *ange 9 our conception1 :i* we make 9 pass+ f ! %apes & motions 9 two dim5sions to ^? 9 ?ree1 af=ds a patt]n by :i* we c pass on /ill fur!r to ! conception ( an exi/5ce 9 f\r-dim5sional space4 ,a piece ( pap] on a smoo? table af=ds a ready image ( a two-dim5sional exi/5ce4 ,if we suppose ! be+ repres5t$ by ! piece ( pap] to h no k ( ! ?ickness by :i* he projects above ! surface ( ! table1 x is obvi\s t he c h no k ( objects ( a simil> description1 except by ! contact ) _! edges4 ,8 body & ! objects 9 8 _w h a ?ickness ( :i* h[ev]1 he has no consci\sness4 ,s9ce ! direction /ret*+ up f ! table is unkn[n to him he w ?9k ( ! objects ( 8 _w z ext5d+ 9 two dim5sions only4 ,figures are to him completely b\nd$ by _! l9es1 j z solid objects are to u by _! surfaces4 ,he _c conceive ( approa*+ ! c5tre ( a circle1 except by break+ ?r\gh ! circumf]5ce1 = ! circumf]5ce encloses ! c5tre 9 ! directions 9 :i* motion is possible to him4 ,! plane surface ov] :i* he slips & ) :i* he is always 9 contact w be unkn[n to him2 !re are no diff]5ces by :i* he c recognise its exi/5ce4 ,b = ! purposes ( our analogy ?is repres5tation is defici5t4 ,a be+ z ?us describ$ has no?+ ab\t him to pu% (f f1 ! surface ov] :i* he slips af=ds no means by :i* he c move 9 "o direction r ?an ano!r4 ,plac$ on a surface ov] :i* he slips freely1 he is 9 a condition analog\s to t 9 :i* we %\ld be if we 7 susp5d$ free 9 space4 ,!re is no?+ :i* he c pu% (f f 9 any direction kn[n to him4 ,let u !re=e modify our repres5tation4 ,let u suppose a v]tical plane aga9/ :i* p>ticles ( ?9 matt] slip1 nev] leav+ ! surface4 ,let ^! p>ticles possess an attractive =ce & coh]e toge!r into a disk2 ?is disk w repres5t ! globe ( a plane be+4 ,he mu/ be conceiv$ z exi/+ on ! rim4 ,illu/ration3 ,fig4 #d4 ,let #a repres5t ?is v]tical disk ( flat matt] & #b ! plane be+ on x1 /&+ ^u its rim z we /& on ! surface ( our e>?4 ,! direction ( ! attractive =ce ( 8 matt] w give ! creature a k ( up & d[n1 det]m9+ = him "o direction 9 8 plane space4 ,also1 s9ce he c move along ! surface ( 8 e>?1 he w h ! s5se ( a direction p>allel to its surface1 :i* we may call =w>ds & backw>ds4 ,he w h no s5se ( "r & left--t is1 ( ! direction :i* we recognise z ext5d+ out f ! plane to our "r & left4 ,! di/9ction ( "r & left is ! "o t we mu/ suppose to be abs5t1 9 ord] to project ourselves into ! condition ( a plane be+4 ,let ! read] imag9e himself1 z he looks along ! plane1 fig4 #d1 to become m & m id5tifi$ ) ! ?9 body on x1 till he f9ally looks along p>allel to ! surface ( ! plane e>?1 & up & d[n1 los+ ! s5se ( ! direction :i* /ret*es "r & left4 ,?is direction w be an unkn[n dim5sion to him4 ,our space conceptions are s intimately connect$ ) ^? :i* we d]ive f ! exi/5ce ( gravitation t x is difficult to realise ! condition ( a plane be+1 )\t pictur+ him z 9 mat]ial surr\nd+s ) a def9ite direction ( up & d[n4 ,h5ce ! necessity ( our some:at elaborate s*eme ( repres5tation1 :i*1 :5 its import has be5 grasp$1 c be disp5s$ ) = ! simpl] "o ( a ?9 object slipp+ ov] a smoo? surface1 :i* lies 9 front ( u4 ,x is obvi\s t we mu/ suppose "s means by :i* ! plane be+ is kept 9 contact ) ! surface on :i* he slips4 ,! simple/ supposition to make is t !re is a transv]se gravity1 :i* keeps him to ! plane4 ,?is gravity mu/ be ?\ght ( z diff]5t to ! attraction ex]cis$ by 8 matt]1 & z unp]ceiv$ by him4 ,at ?is /age ( our enquiry ,i d n wi% to ent] into ! "q ( h[ a plane be+ c\ld arrive at a k ( ! ?ird dim5sion1 b simply to inve/igate 8 plane consci\sness4 ,x is obvi\s t ! exi/5ce ( a plane be+ mu/ be v limit$4 ,a /raight l9e /&+ up f ! surface ( 8 e>? af=ds a b> to 8 progress4 ,an object l a :eel :i* rotates r\nd an axis w\ld be unkn[n to him1 = !re is no conceivable way 9 :i* he c get to ! c5tre )\t go+ ?r\gh ! circumf]5ce4 ,he w\ld h sp9n+ disks1 b c\ld n get to ! c5tre ( !m4 ,! plane be+ c repres5t ! motion f any "o po9t ( 8 space to any o!r1 by means ( two /raight l9es drawn at "r angles to ea* o!r4 ,let ,,ax & ,,ay be two su* axes4 ,he c accompli% ! translation f ,a to ,b by go+ along ,,ax to ,c1 & !n f ,c along ,,cb p>allel to ,,ay4 ,! same result c ( c\rse be obta9$ by mov+ to ,d along ,,ay & !n p>allel to ,,ax f ,d to ,b1 or ( c\rse by any diagonal movem5t comp\nd$ by ^! axial movem5ts4 ,illu/ration3 ,fig4 #e4 ,by means ( movem5ts p>allel to ^! two axes he c proce$ 7except = mat]ial ob/acles7 f any "o po9t ( 8 space to any o!r4 ,if n[ we suppose a ?ird l9e drawn out f ,a at "r angles to ! plane x is evid5t t no motion 9 ei!r ( ! two dim5sions he kn[s w c>ry him 9 ! lea/ degree 9 ! direction repres5t$ by ,,az4 ,illu/ration3 ,fig4 #f4 ,! l9es ,,az & ,,ax det]m9e a plane4 ,if he c\ld be tak5 (f 8 plane1 & transf]r$ to ! plane ,,axz1 he w\ld be 9 a _w exactly l 8 own4 ,f e l9e 9 8 _w !re goes (f a space _w exactly l 8 own4 ,illu/ration3 ,fig4 #g4 ,f e po9t 9 8 _w a l9e c be drawn p>allel to ,,az 9 ! direction unkn[n to him4 ,if we suppose ! squ>e 9 fig4 #g to be a geometrical squ>e f e po9t ( x1 inside z well z on ! cont\r1 a /raight l9e c be drawn p>allel to ,,az4 ,! assemblage ( ^! l9es con/itute a solid figure1 ( :i* ! squ>e 9 ! plane is ! base4 ,if we consid] ! squ>e to repres5t an object 9 ! plane be+'s _w !n we mu/ attribute to x a v small ?ickness1 = e real ?+ mu/ possess all ?ree dim5sions4 ,?is ?ickness he does n p]ceive1 b ?9ks ( ?is real object z a geometrical squ>e4 ,he ?9ks ( x z possess+ area only1 & no degree ( solidity4 ,! edges :i* project f ! plane to a v small ext5t he ?9ks ( z hav+ m]ely l5g? & no bread?--z be+1 9 fact1 geometrical l9es4 ,) ! fir/ /ep 9 ! appreh5sion ( a ?ird dim5sion !re w\ld come to a plane be+ ! conviction t he _h previ\sly =m$ a wrong conception ( ! nature ( 8 mat]ial objects4 ,he _h conceiv$ !m z geometrical figures ( two dim5sions only4 ,if a ?ird dim5sion exi/s1 su* figures are incapable ( real exi/5ce4 ,?us he w\ld admit t all 8 real objects _h a c]ta91 ?\gh v small ?ickness 9 ! unkn[n dim5sion1 & t ! conditions ( 8 exi/5ce dem&$ ! supposition ( an ext5d$ %eet ( matt]1 f contact ) :i* 9 _! motion 8 objects nev] div]ge4 ,analog\s conceptions mu/ be =m$ by u on ! supposition ( a f\r-dim5sional exi/5ce4 ,we mu/ suppose a direction 9 :i* we c nev] po9t ext5d+ f e po9t ( our space4 ,we mu/ draw a di/9ction betwe5 a geometrical cube & a cube ( real matt]4 ,! cube ( real matt] we mu/ suppose to h an ext5sion 9 an unkn[n direction1 real1 b s small z to be imp]ceptible by u4 ,f e po9t ( a cube1 int]ior z well z ext]ior1 we mu/ imag9e t x is possible to draw a l9e 9 ! unkn[n direction4 ,! assemblage ( ^! l9es w\ld con/itute a high] solid4 ,! l9es go+ (f 9 ! unkn[n direction f ! face ( a cube w\ld con/itute a cube />t+ f t face4 ,( ?is cube all t we %\ld see 9 our space w\ld be ! face4 ,aga91 j z ! plane be+ c repres5t any motion 9 8 space by two axes1 s we c repres5t any motion 9 our ?ree-dim5sional space by means ( ?ree axes4 ,!re is no po9t 9 our space to :i* we _c move by "s comb9ation ( movem5ts on ! directions m>k$ out by ^! axes4 ,on ! assumption ( a f\r? dim5sion we h to suppose a f\r? axis1 :i* we w call ,,aw4 ,x mu/ be suppos$ to be at "r angles to ea* & e "o ( ! ?ree axes ,,ax1 ,,ay1 ,,az4 ,j z ! two axes1 ,,ax1 ,,az1 det]m9e a plane :i* is simil> to ! orig9al plane on :i* we suppos$ ! plane be+ to exi/1 b :i* runs (f f x1 & only meets x 9 a l9e2 s 9 our space if we take any ?ree axes su* z ,,ax1 ,,ay1 & ,,aw1 !y det]m9e a space l our space _w4 ,?is space runs (f f our space1 & if we 7 transf]r$ to x we %\ld f9d ourselves 9 a space exactly simil> to our own4 ,we mu/ give up any attempt to picture ?is space 9 its relation to ours1 j z a plane be+ w\ld h to give up any attempt to picture a plane at "r angles to 8 plane4 ,su* a space & ours run 9 diff]5t directions f ! plane ( ,,ax & ,,ay4 ,!y meet 9 ?is plane b h no?+ else 9 common1 j z ! plane space ( ,,ax & ,,ay & t ( ,,ax & ,,az run 9 diff]5t directions & h b ! l9e ,,ax 9 common4 ,omitt+ all discussion ( ! mann] on :i* a plane be+ might be conceiv$ to =m a !ory ( a ?ree-dim5sional exi/5ce1 let u exam9e h[1 ) ! means at 8 disposal1 he c\ld repres5t ! prop]ties ( ?ree-dim5sional objects4 ,!re are two ways 9 :i* ! plane be+ c ?9k ( "o ( our solid bodies4 ,he c ?9k ( ! cube1 fig4 #h1 z compos$ ( a numb] ( sections p>allel to 8 plane1 ea* ly+ 9 ! ?ird dim5sion a little fur!r (f f 8 plane ?an ! prec$+ "o4 ,^! sections he c repres5t z a s]ies ( plane figures ly+ 9 8 plane1 b 9 s repres5t+ !m he de/roys ! coh]5ce ( !m 9 ! high] figure4 ,! set ( squ>es1 ,a1 ,b1 ,c1 ,d1 repres5ts ! section p>allel to ! plane ( ! cube %[n 9 figure1 b !y are n 9 _! prop] relative positions4 ,illu/ration3 ,fig4 #h4 ,! plane be+ c trace out a movem5t 9 ! ?ird dim5sion by assum+ discont9u\s leaps f "o section to ano!r4 ,?us1 a motion along ! edge ( ! cube f left to "r w\ld be repres5t$ 9 ! set ( sections 9 ! plane z ! succession ( ! corn]s ( ! sections ,a1 ,b1 ,c1 ,d4 ,a po9t mov+ f ,a ?r\gh ,,bcd 9 our space mu/ be repres5t$ 9 ! plane z appe>+ 9 ,a1 !n 9 ,b1 & s on1 )\t pass+ ?r\gh ! int]v5+ plane space4 ,9 ^! sections ! plane be+ leaves out1 ( c\rse1 ! ext5sion 9 ! ?ird dim5sion2 ! di/ance betwe5 any two sections is n repres5t$4 ,9 ord] to realise ?is di/ance ! conception ( motion c be employ$4 ,illu/ration3 ,fig4 #i4 ,let fig4 #i repres5t a cube pass+ transv]se to ! plane4 ,x w appe> to ! plane be+ z a squ>e object1 b ! matt] ( :i* ?is object is compos$ w be cont9ually alt]+4 ,"o mat]ial p>ticle takes ! place ( ano!r1 b x does n come f any:]e or g any:]e 9 ! space :i* ! plane be+ kn[s4 ,! analog\s mann] ( repres5t+ a high] solid 9 our case1 is to conceive x z compos$ ( a numb] ( sections1 ea* ly+ a little fur!r (f 9 ! unkn[n direction ?an ! prec$+4 ,illu/ration3 ,fig4 #aj4 ,we c repres5t ^! sections z a numb] ( solids4 ,?us ! cubes ,a1 ,b1 ,c1 ,d1 may be consid]$ z ! sections at diff]5t int]vals 9 ! unkn[n dim5sion ( a high] cube4 ,arrang$ ?us _! coh]5ce 9 ! high] figure is de/roy$1 !y are m]e repres5tations4 ,a motion 9 ! f\r? dim5sion f ,a ?r\gh ,b1 ,c1 etc41 w\ld be cont9u\s1 b we c only repres5t x z ! occupation ( ! positions ,a1 ,b1 ,c1 etc41 9 succession4 ,we c exhibit ! results ( ! motion at diff]5t /ages1 b no m4 ,9 ?is repres5tation we h left out ! di/ance betwe5 "o section & ano!r2 we h consid]$ ! high] body m]ely z a s]ies ( sections1 & s left out its cont5ts4 ,! only way to exhibit its cont5ts is to call 9 ! aid ( ! conception ( motion4 ,illu/ration3 ,fig4 #aa4 ,if a high] cube passes transv]se to our space1 x w appe> z a cube isolat$ 9 space1 ! "p t has n come into our space & ! "p t has pass$ ?r\gh w n be visible4 ,! gradual pass+ ?r\gh our space w\ld appe> z ! *ange ( ! matt] ( ! cube be=e u4 ,"o mat]ial p>ticle 9 x is succe$$ by ano!r1 nei!r com+ nor go+ 9 any direction we c po9t to4 ,9 ?is mann]1 by ! duration ( ! figure1 we c exhibit ! high] dim5sionality ( x2 a cube ( our matt]1 "u ! circum/ances suppos$1 namely1 t x has a motion transv]se to our space1 w\ld in/antly disappe>4 ,a high] cube w\ld la/ till x _h pass$ transv]se to our space by its :ole di/ance ( ext5sion 9 ! f\r? dim5sion4 ,z ! plane be+ c ?9k ( ! cube z consi/+ ( sections1 ea* l a figure he kn[s1 ext5d+ away f 8 plane1 s we c ?9k ( a high] solid z compos$ ( sections1 ea* l a solid :i* we "k1 b ext5d+ away f our space4 ,?us1 tak+ a high] cube1 we c look on x z />t+ f a cube 9 our space & ext5d+ 9 ! unkn[n dim5sion4 ,illu/ration3 ,fig4 #ab4 ,take ! face ,a & conceive x to exi/ z simply a face1 a squ>e ) no ?ickness4 ,f ?is face ! cube 9 our space ext5ds by ! occupation ( space :i* we c see4 ,b f ?is face !re ext5ds equally a cube 9 ! unkn[n dim5sion4 ,we c ?9k ( ! high] cube1 !n1 by tak+ ! set ( sections ,a1 ,b1 ,c1 ,d1 etc41 & consid]+ t f ea* ( !m !re runs a cube4 ,^! cubes h no?+ 9 common ) ea* o!r1 & ( ea* ( !m 9 its actual position all t we c h 9 our space is an isolat$ squ>e4 ,x is obvi\s t we c take our s]ies ( sections 9 any mann] we please4 ,we c take !m p>allel1 = in/ance1 to any "o ( ! ?ree isolat$ faces %[n 9 ! figure4 ,correspond+ to ! ?ree s]ies ( sections at "r angles to ea* o!r1 :i* we c make ( ! cube 9 space1 we mu/ conceive ( ! high] cube1 z compos$ ( cubes />t+ f squ>es p>allel to ! faces ( ! cube1 & ( ^! cubes all t exi/ 9 our space are ! isolat$ squ>es f :i* !y />t4 ,,*apt] ,,iii ,,! ,,significance ,,( ,a ,,f\r-,,dim5sional ,,exi/5ce ,hav+ n[ obta9$ ! conception ( a f\r-dim5sional space1 & hav+ =m$ ! analogy :i*1 )\t any fur!r geometrical difficulties1 enables u to enquire into its prop]ties1 ,i w ref] ! read]1 ^: int]e/ is pr9cipally 9 ! me*anical aspect1 to ,*apt]s ,,vi4 & ,,vii4 ,9 ! pres5t *apt] ,i w deal ) ! g5]al significance ( ! enquiry1 & 9 ! next ) ! hi/orical orig9 ( ! idea4 ,fir/1 ) reg>d to ! "q ( :e!r !re is any evid5ce t we are really 9 f\r-dim5sional space1 ,i w g back to ! analogy ( ! plane _w4 ,a be+ 9 a plane _w c\ld n h any exp]i5ce ( ?ree-dim5sional %apes1 b he c\ld h an exp]i5ce ( ?ree-dim5sional movem5ts4 ,we h se5 t 8 matt] mu/ be suppos$ to h an ext5sion1 ?\gh a v small "o1 9 ! ?ird dim5sion4 ,& ?us1 9 ! small p>ticles ( 8 matt]1 ?ree-dim5sional movem5ts may well be conceiv$ to take place4 ,( ^! movem5ts he w\ld only p]ceive ! resultants4 ,s9ce all movem5ts ( an obs]vable size 9 ! plane _w are two-dim5sional1 he w\ld only p]ceive ! resultants 9 two dim5sions ( ! small ?ree-dim5sional movem5ts4 ,?us1 !re w\ld be ph5om5a :i* he c\ld n expla9 by 8 !ory ( me*anics--motions w\ld take place :i* he c\ld n expla9 by 8 !ory ( motion4 ,h5ce1 to det]m9e if we are 9 a f\r-dim5sional _w1 we mu/ exam9e ! ph5om5a ( motion 9 our space4 ,if movem5ts occur :i* are n explicable on ! suppositions ( our ?ree-dim5sional me*anics1 we %\ld h an indication ( a possible f\r-dim5sional motion1 & if1 moreov]1 x c\ld be %[n t su* movem5ts w\ld be a consequ5ce ( a f\r-dim5sional motion 9 ! m9ute p>ticles ( bodies or ( ! e!r1 we %\ld h a /rong presumption 9 fav\r ( ! reality ( ! f\r? dim5sion4 ,by proce$+ 9 ! direction ( f9] & f9] subdivision1 we come to =ms ( matt] possess+ prop]ties diff]5t to ^? ( ! l>g] masses4 ,x is probable t at "s /age 9 ?is process we %\ld come to a =m ( matt] ( su* m9ute subdivision t its p>ticles possess a fre$om ( movem5t 9 f\r dim5sions4 ,?is =m ( matt] ,i speak ( z f\r-dim5sional e!r1 & attribute to x prop]ties approximat+ to ^? ( a p]fect liquid4 ,def]r+ ! detail$ discussion ( ?is =m ( matt] to ,*apt] ,,vi41 we w n[ exam9e ! means by :i* a plane be+ w\ld come to ! conclusion t ?ree-dim5sional movem5ts exi/$ 9 8 _w1 & po9t out ! analogy by :i* we c conclude ! exi/5ce ( f\r-dim5sional movem5ts 9 our _w4 ,s9ce ! dim5sions ( ! matt] 9 8 _w are small 9 ! ?ird direction1 ! ph5om5a 9 :i* he w\ld detect ! motion w\ld be ^? ( ! small p>ticles ( matt]4 ,suppose t !re is a r+ 9 8 plane4 ,we c imag9e curr5ts fl[+ r\nd ! r+ 9 ei!r ( two opposite directions4 ,^! w\ld produce unlike effects1 & give rise to two diff]5t fields ( influ5ce4 ,if ! r+ ) a curr5t 9 x 9 "o direction be tak5 up & turn$ ov]1 & put d[n aga9 on ! plane1 x w\ld be id5tical ) ! r+ hav+ a curr5t 9 ! opposite direction4 ,an op]ation ( ?is k9d w\ld be impossible to ! plane be+4 ,h5ce he w\ld h 9 8 space two irreconcilable objects1 namely1 ! two fields ( influ5ce due to ! two r+s ) curr5ts 9 !m 9 opposite directions4 ,by irreconcilable objects 9 ! plane ,i mean objects :i* _c be ?\ght ( z trans=m$ "o into ! o!r by any movem5t 9 ! plane4 ,in/ead ( curr5ts fl[+ 9 ! r+s we c imag9e a diff]5t k9d ( curr5t4 ,imag9e a numb] ( small r+s /rung on ! orig9al r+4 ,a curr5t r\nd ^! second>y r+s w\ld give two v>ieties ( effect1 or two diff]5t fields ( influ5ce1 accord+ to its direction4 ,^! two v>ieties ( curr5t c\ld be turn$ "o into ! o!r by tak+ "o ( ! r+s up1 turn+ x ov]1 & putt+ x d[n aga9 9 ! plane4 ,?is op]ation is impossible to ! plane be+1 h5ce 9 ?is case also !re w\ld be two irreconcilable fields 9 ! plane4 ,n[1 if ! plane be+ f\nd two su* irreconcilable fields & c\ld prove t !y c\ld n be acc\nt$ = by curr5ts 9 ! r+s1 he w\ld h to admit ! exi/5ce ( curr5ts r\nd ! r+s--t is1 9 r+s /rung on ! prim>y r+4 ,?us he w\ld come to admit ! exi/5ce ( a ?ree-dim5sional motion1 = su* a disposition ( curr5ts is 9 ?ree dim5sions4 ,n[ 9 our space !re are two fields ( diff]5t prop]ties1 :i* c be produc$ by an electric curr5t fl[+ 9 a clos$ circuit or r+4 ,^! two fields c be *ang$ "o into ! o!r by rev]s+ ! curr5ts1 b !y _c be *ang$ "o into ! o!r by any turn+ ab\t ( ! r+s 9 our space2 = ! disposition ( ! field ) reg>d to ! r+ itself is diff]5t :5 we turn ! r+1 ov] & :5 we rev]se ! direction ( ! curr5t 9 ! r+4 ,z hypo!ses to expla9 ! diff]5ces ( ^! two fields & _! effects we c suppose ! foll[+ k9ds ( space motions3--,fir/1 a curr5t along ! conductor2 second1 a curr5t r\nd ! conductor--t is1 ( r+s ( curr5ts /rung on ! conductor z an axis4 ,nei!r ( ^! suppositions acc\nts = facts ( obs]vation4 ,h5ce we h to make ! supposition ( a f\r-dim5sional motion4 ,we f9d t a f\r-dim5sional rotation ( ! nature expla9$ 9 a subsequ5t *apt]1 has ! foll[+ *>act]i/ics3--,fir/1 x w\ld give u two fields ( influ5ce1 ! "o ( :i* c\ld be turn$ into ! o!r by tak+ ! circuit up into ! f\r? dim5sion1 turn+ x ov]1 & putt+ x d[n 9 our space aga91 precisely z ! two k9ds ( fields 9 ! plane c\ld be turn$ "o into ! o!r by a rev]sal ( ! curr5t 9 our space4 ,second1 x involves a ph5om5on precisely id5tical ) t mo/ rem>kable & my/]i\s feature ( an electric curr5t1 namely t x is a field ( action1 ! rim ( :i* necess>ily abuts on a cont9u\s b\nd>y =m$ by a conductor4 ,h5ce1 on ! assumption ( a f\r-dim5sional movem5t 9 ! region ( ! m9ute p>ticles ( matt]1 we %\ld expect to f9d a motion analog\s to electricity4 ,n[1 a ph5om5on ( su* univ]sal occurr5ce z electricity _c be due to matt] & motion 9 any v complex relation1 b ought to be se5 z a simple & natural consequ5ce ( _! prop]ties4 ,i inf] t ! difficulty 9 its !ory is due to ! attempt to expla9 a f\r-dim5sional ph5om5on by a ?ree-dim5sional geometry4 ,9 view ( ?is piece ( evid5ce we _c disreg>d t af=d$ by ! exi/5ce ( symmetry4 ,9 ?is connection ,i w allude to ! simple way ( produc+ ! images ( insects1 sometimes practis$ by *ildr54 ,!y put a few blots ( ink 9 a /raight l9e on a piece ( pap]1 fold ! pap] along ! blots1 & on op5+ x ! lifelike pres5tm5t ( an insect is obta9$4 ,if we 7 to f9d a multitude ( ^! figures1 we %\ld conclude t !y _h orig9at$ f a process ( fold+ ov]2 ! *ances aga9/ ?is k9d ( r$uplication ( p>ts is too great to admit ( ! assumption t !y _h be5 =m$ 9 any o!r way4 ,! production ( ! symmetrical =ms ( organis$ be+s1 ?\gh n ( c\rse due to a turn+ ov] ( bodies ( any appreciable size 9 f\r-dim5sional space1 c well be imag9$ z due to a disposition 9 t mann] ( ! smalle/ liv+ p>ticles f :i* !y are built up4 ,?us1 n only electricity1 b life1 & ! processes by :i* we ?9k & feel1 mu/ be attribut$ to t region ( magnitude 9 :i* f\r-dim5sional movem5ts take place4 ,i d n mean1 h[ev]1 t life c be expla9$ z a f\r-dim5sional movem5t4 ,x seems to me t ! :ole bias ( ?\ght1 :i* t5ds to expla9 ! ph5om5a ( life & volition1 z due to matt] & motion 9 "s peculi> relation1 is adopt$ r 9 ! int]e/s ( ! explicability ( ?+s ?an ) any reg>d to probability4 ,( c\rse1 if we c\ld %[ t life 7 a ph5om5on ( motion1 we %\ld be able to expla9 a great deal t is at pres5t obscure4 ,b !re are two great difficulties 9 ! way4 ,x w\ld be necess>y to %[ t 9 a g]m capable ( develop+ into a liv+ be+1 !re 7 modifications ( /ructure capable ( det]m9+ 9 ! develop$ g]m all ! *>act]i/ics ( its =m1 & n only ?is1 b ( det]m9+ ^? ( all ! desc5dants ( su* a =m 9 an inf9ite s]ies4 ,su* a complexity ( me*anical relations1 und5iable ?\gh x be1 _c surely be ! be/ way ( gr\p+ ! ph5om5a & giv+ a practical acc\nt ( !m4 ,& ano!r difficulty is ?is1 t no am\nt ( me*anical adaptation w\ld give t elem5t ( consci\sness :i* we possess1 & :i* is %>$ 9 to a modifi$ degree by ! animal _w4 ,9 ^? complex /ructures :i* m5 build up & direct1 su* z a %ip or a railway tra9 7& :i*1 if se5 by an obs]v] ( su* a size t ! m5 guid+ !m 7 invisible1 w\ld seem to pres5t "s ( ! ph5om5a ( life7 ! appe>ance ( animation is n due to any diffusion ( life 9 ! mat]ial p>ts ( ! /ructure1 b to ! pres5ce ( a liv+ be+4 ,! old hypo!sis ( a s\l1 a liv+ organism )9 ! visible "o1 appe>s to me mu* m rational ?an ! attempt to expla9 life z a =m ( motion4 ,& :5 we consid] ! region ( extreme m9ut5ess *>act]is$ by f\r-dim5sional motion ! difficulty ( conceiv+ su* an organism alongside ! bodily "o disappe>s4 ,"l ,kelv9 supposes t matt] is =m$ f ! e!r4 ,we may v well suppose t ! liv+ organisms direct+ ! mat]ial ones are co-ord9ate ) !m1 n compos$ ( matt]1 b consi/+ ( e!rial bodies1 & z su* capable ( motion ?r\gh ! e!r1 & able to orig9ate mat]ial liv+ bodies ?r\gh\t ! m9]al4 ,hypo!ses su* z ^! f9d no imm$iate gr\nd = pro( or dispro( 9 ! physical _w4 ,let u1 !re=e1 turn to a diff]5t field1 &1 assum+ t ! human s\l is a f\r-dim5sional be+1 capable 9 itself ( f\r dim5sional movem5ts1 b 9 its exp]i5ces ?r\gh ! s5ses limit$ to ?ree dim5sions1 ask if ! hi/ory ( ?\ght1 ( ^! productivities :i* *>act]ise man1 correspond to our assumption4 ,let u pass 9 review ^? /eps by :i* man1 presumably a f\r-dim5sional be+1 despite 8 bodily environm5t1 has come to recognise ! fact ( f\r-dim5sional exi/5ce4 ,def]r+ ?is enquiry to ano!r *apt]1 ,i w "h recapitulate ! argum5t 9 ord] to %[ t our purpose is entirely practical & indep5d5t ( any philosophical or metaphysical consid]ations4 ,if two %ots are fir$ at a t>get1 & ! second bullet hits x at a diff]5t place to ! fir/1 we suppose t !re 0 "s diff]5ce 9 ! conditions "u :i* ! second %ot 0 fir$ f ^? affect+ ! fir/ %ot4 ,! =ce ( ! p[d]1 ! direction ( aim1 ! /r5g? ( ! w9d1 or "s condition mu/ h be5 diff]5t 9 ! second case1 if ! c\rse ( ! bullet 0 n exactly ! same z 9 ! fir/ case4 ,correspond+ to e diff]5ce 9 a result !re mu/ be "s diff]5ce 9 ! antec$5t mat]ial conditions4 ,by trac+ out ?is *a9 ( relations we expla9 nature4 ,b !re is also ano!r mode ( explanation :i* we apply4 ,if we ask :at 0 ! cause t a c]ta9 %ip 0 built1 or t a c]ta9 /ructure 0 erect$1 we might proce$ to inve/igate ! *anges 9 ! bra9 cells ( ! m5 :o design$ ! works4 ,e v>iation 9 "o %ip or build+ f ano!r %ip or build+ is accompani$ by a v>iation 9 ! processes t g on 9 ! bra9 matt] ( ! design]s4 ,b practically ?is w\ld be a v long task4 ,a m effective mode ( expla9+ ! production ( ! %ip or build+ w\ld be to enquire into ! motives1 plans1 & aims ( ! m5 :o con/ruct$ !m4 ,we obta9 a cumulative & consi/5t body ( k mu* m easily & effectively 9 ! latt] way4 ,sometimes we apply ! "o1 sometimes ! o!r mode ( explanation4 ,b x mu/ be obs]v$ t ! me?od ( explanation f\nd$ on aim1 purpose1 volition1 always presupposes a me*anical sy/em on :i* ! volition & aim works4 ,! conception ( man z will+ & act+ f motives involves t ( a numb] ( uni=m processes ( nature :i* he c modify1 & ( :i* he c make application4 ,9 ! me*anical conditions ( ! ?ree-dim5sional _w1 ! only volitional ag5cy :i* we c demon/rate is ! human ag5cy4 ,b :5 we consid] ! f\r-dim5sional _w ! conclusion rema9s p]fectly op54 ,! me?od ( explanation f\nd$ on purpose & aim does n1 surely1 sudd5ly beg9 ) man & end ) him4 ,!re is z mu* beh9d ! exhibition ( w & motive :i* we see 9 man z !re is beh9d ! ph5om5a ( movem5t2 !y are co-ord9ate1 nei!r to be resolv$ into ! o!r4 ,& ! comm5cem5t ( ! inve/igation ( t w & motive :i* lies beh9d ! w & motive manife/$ 9 ! ?ree-dim5sional me*anical field is 9 ! conception ( a s\l--a f\r-dim5sional organism1 :i* expresses its high] physical be+ 9 ! symmetry ( ! body1 & gives ! aims & motives ( human exi/5ce4 ,our prim>y task is to =m a sy/ematic k ( ! ph5om5a ( a f\r-dim5sional _w & f9d ^? po9ts 9 :i* ?is k mu/ be call$ 9 to complete our me*anical explanation ( ! univ]se4 ,b a subsidi>y contribution t[>ds ! v]ification ( ! hypo!sis may be made by pass+ 9 review ! hi/ory ( human ?\ght1 & enquir+ if x pres5ts su* features z w\ld be naturally expect$ on ?is assumption4 ,,*apt] ,,iv ,,! ,,fir/ ,,*apt] ,,9 ,,! ,,hi/ory ,,( ,,f\r ,,space ,p>m5ides1 & ! ,asiatic ?9k]s ) :om he is 9 close aff9ity1 prop\nd a !ory ( exi/5ce :i* is 9 close accord ) a conception ( a possible relation betwe5 a high] & a l[] dim5sional space4 ,?is !ory1 prior & 9 m>k$ contra/ to ! ma9 /ream ( ?\ght1 :i* we %all aft]w>ds describe1 =ms a clos$ circle by itself4 ,x is "o :i* 9 all ages has _h a /rong attraction = pure intellect1 & is ! natural mode ( ?\ght = ^? :o refra9 f project+ _! own volition into nature "u ! guise ( causality4 ,accord+ to ,p>m5ides ( ! s*ool ( ,elea ! all is "o1 unmov+ & un*ang+4 ,! p]man5t amid ! transi5t--t foo?old = ?\ght1 t solid gr\nd = feel+ on ! discov]y ( :i* dep5ds all our life--is no phantom2 x is ! image amid/ deception ( true be+1 ! et]nal1 ! unmov$1 ! "o4 ,?us says ,p>m5ides4 ,b h[ expla9 ! %ift+ sc5e1 ^! mutations ( ?+s6 8,illusion10 answ]s ,p>m5ides4 ,di/+ui%+ betwe5 tru? & error1 he tells ( ! true doctr9e ( ! "o--! false op9ion ( a *ang+ _w4 ,he is no less memorable = ! mann] ( 8 advocacy ?an = ! cause he advocates4 ,x is z if f 8 firm foo?old ( be+ he c\ld play ) ! ?\ghts "u ! burd5 ( :i* o!rs lab\r$1 = f him spr+s t flu5cy ( supposition & hypo!sis :i* =ms ! texture ( ,plato's dialectic4 ,c ! m9d conceive a m delightful intellectual picture ?an t ( ,p>m5ides1 po9t+ to ! "o1 ! true1 ! un*ang+1 & yet on ! o!r h& ready to discuss all mann] ( false op9ion1 =m+ a cosmogony too1 false 8b m9e own0 aft] ! fa%ion ( ! "t8 ,9 support ( ! true op9ion he proce$$ by ! negative way ( %[+ ! self-contradictions 9 ! ideas ( *ange & motion4 ,x is d\btful if 8 criticism1 save 9 m9or po9ts1 has ev] be5 successfully refut$4 ,to express 8 doctr9e 9 ! pond]\s mod]n way we mu/ make ! /atem5t t motion is ph5om5al1 n real4 ,let u repres5t 8 doctr9e4 ,illu/ration3 ,fig4 #ac4 ,imag9e a %eet ( /ill wat] into :i* a slant+ /ick is be+ l[]$ ) a motion v]tically d[nw>ds4 ,let #a1 #b1 #c 7,fig4 #ac71 be ?ree consecutive positions ( ! /ick4 ,a1 ,b1 ,c1 w be ?ree consecutive positions ( ! meet+ ( ! /ick1 ) ! surface ( ! wat]4 ,z ! /ick passes d[n1 ! meet+ w move f ,a on to ,b & ,c4 ,suppose n[ all ! wat] to be remov$ except a film4 ,at ! meet+ ( ! film & ! /ick !re w be an int]ruption ( ! film4 ,if we suppose ! film to h a prop]ty1 l t ( a soap bubble1 ( clos+ up r\nd any p5etrat+ object1 !n z ! /ick goes v]tically d[nw>ds ! int]ruption 9 ! film w move on4 ,illu/ration3 ,fig4 #ad4 ,if we pass a spiral ?r\gh ! film ! int]section w give a po9t mov+ 9 a circle %[n by ! dott$ l9es 9 ! figure4 ,suppose n[ ! spiral to be /ill & ! film to move v]tically upw>ds1 ! :ole spiral w be repres5t$ 9 ! film ( ! consecutive positions ( ! po9t ( int]section4 ,9 ! film ! p]man5t exi/5ce ( ! spiral is exp]i5c$ z a "t s]ies--! record ( trav]s+ ! spiral is a po9t mov+ 9 a circle4 ,if n[ we suppose a consci\sness connect$ ) ! film 9 su* a way t ! int]section ( ! spiral ) ! film gives rise to a consci\s exp]i5ce1 we see t we %all h 9 ! film a po9t mov+ 9 a circle1 consci\s ( its motion1 kn[+ no?+ ( t real spiral ! record ( ! successive int]sections ( :i* by ! film is ! motion ( ! po9t4 ,x is easy to imag9e complicat$ /ructures ( ! nature ( ! spiral1 /ructures consi/+ ( filam5ts1 & to suppose also t ^! /ructures are di/+ui%able f ea* o!r at e section4 ,if we consid] ! int]sections ( ^! filam5ts ) ! film z x passes to be ! atoms con/itut+ a film> univ]se1 we %all h 9 ! film a _w ( app>5t motion2 we %all h bodies correspond+ to ! filam5t>y /ructure1 & ! positions ( ^! /ructures ) reg>d to "o ano!r w give rise to bodies 9 ! film mov+ among/ "o ano!r4 ,?is mutual motion is app>5t m]ely4 ,! reality is ( p]man5t /ructures /ation>y1 & all ! relative motions acc\nt$ = by "o /eady movem5t ( ! film z a :ole4 ,?us we c imag9e a plane _w1 9 :i* all ! v>iety ( motion is ! ph5om5on ( /ructures consi/+ ( filam5t>y atoms trav]s$ by a plane ( consci\sness4 ,pass+ to f\r dim5sions & our space1 we c conceive t all ?+s & movem5ts 9 our _w are ! read+ (f ( a p]man5t reality by a space ( consci\sness4 ,ea* atom at e mom5t is n :at x 01 b a new "p ( t endless l9e :i* is itself4 ,& all ?is sy/em successively reveal$ 9 ! "t :i* is b ! succession ( consci\sness1 sep>ate z x is 9 p>ts1 9 its entirety is "o va/ unity4 ,repres5t+ ,p>m5ides' doctr9e ?us1 we ga9 a firm] hold on x ?an if we m]ely let 8 words re/1 gr& & massive1 9 our m9ds4 ,& we h ga9$ ! means also ( repres5t+ phases ( t ,ea/]n ?\ght to :i* ,p>m5ides 0 no /rang]4 ,modify+ 8 uncompromis+ doctr9e1 let u suppose1 to g back to ! plane ( consci\sness & ! /ructure ( filam5t>y atoms1 t ^! /ructures are !mselves mov+--are act+1 liv+4 ,!n1 9 ! transv]se motion ( ! film1 !re w\ld be two ph5om5a ( motion1 "o due to ! read+ (f 9 ! film ( ! p]man5t exi/5ces z !y are 9 !mselves1 & ano!r ph5om5on ( motion due to ! modification ( ! record ( ! ?+s !mselves1 by _! prop] motion dur+ ! process ( trav]s+ !m4 ,?us a consci\s be+ 9 ! plane w\ld h1 z x 71 a two-fold exp]i5ce4 ,9 ! complete trav]s+ ( ! /ructure1 ! int]section ( :i* ) ! film gives 8 consci\s all1 ! ma9 & pr9cipal movem5ts & actions :i* he w5t ?r\gh w\ld be ! record ( 8 high] self z x exi/$ unmov$ & unact+4 ,slight modifications & deviations f ^! movem5ts & actions w\ld repres5t ! activity & self-det]m9ation ( ! complete be+1 ( 8 high] self4 ,x is admissible to suppose t ! consci\sness 9 ! plane has a %>e 9 t volition by :i* ! complete exi/5ce det]m9es itself4 ,?us ! motive & w1 ! initiative & life1 ( ! high] be+1 w\ld be repres5t$ 9 ! case ( ! be+ 9 ! film by an initiative & a w capable1 n ( det]m9+ any great ?+s or important movem5ts 9 8 exi/5ce1 b only ( small & relatively insignificant activities4 ,9 all ! ma9 features ( 8 life 8 exp]i5ce w\ld be repres5tative ( "o /ate ( ! high] be+ ^: exi/5ce det]m9es 8 z ! film passes on4 ,b 9 8 m9ute & app>5tly unimportant actions he w\ld %>e 9 t w & det]m9ation by :i* ! :ole ( ! be+ he really is acts & lives4 ,an alt]ation ( ! high] be+ w\ld correspond to a diff]5t life hi/ory = him4 ,let u n[ make ! supposition t film aft] film trav]ses ^! high] /ructures1 t ! life ( ! real be+ is read (f aga9 & aga9 9 successive waves ( consci\sness4 ,!re w\ld be a succession ( lives 9 ! diff]5t advanc+ planes ( consci\sness1 ea* diff]+ f ! prec$+1 & diff]+ 9 virtue ( t w & activity :i* 9 ! prec$+ _h n be5 devot$ to ! great] & app>5tly mo/ significant ?+s 9 life1 b ! m9ute & app>5tly unimportant4 ,9 all great ?+s ! be+ ( ! film %>es 9 ! exi/5ce ( 8 high] self z x is at any "o "t4 ,9 ! small ?+s he %>es 9 t volition by :i* ! high] be+ alt]s & *anges1 acts & lives4 ,?us we ga9 ! conception ( a life *ang+ & develop+ z a :ole1 a life 9 :i* our sep>ation & cessation & fugitiv5ess are m]ely app>5t1 b :i* 9 its ev5ts & c\rse alt]s1 *anges1 develops2 & ! p[] ( alt]+ & *ang+ ?is :ole lies 9 ! w & p[] ! limit$ be+ has ( direct+1 guid+1 alt]+ himself 9 ! m9ute ?+s ( 8 exi/5ce4 ,transf]r+ our conceptions to ^? ( an exi/5ce 9 a high] dim5sionality trav]s$ by a space ( consci\sness1 we h an illu/ration ( a ?\ght :i* has f\nd frequ5t & v>i$ expression4 ,:51 h[ev]1 we ask ourselves :at degree ( tru? !re lies 9 x1 we mu/ admit t1 z f> z we c see1 x is m]ely symbolical4 ,! true pa? 9 ! inve/igation ( a high] dim5sionality lies 9 ano!r direction4 ,! significance ( ! ,p>m5idean doctr9e lies 9 ?is t "h1 z aga9 & aga91 we f9d t ^? conceptions :i* man introduces ( himself1 :i* he does n d]ive f ! m]e record ( 8 outw>d exp]i5ce1 h a /rik+ & significant correspond5ce to ! conception ( a physical exi/5ce 9 a _w ( a high] space4 ,h[ close we come to ,p>m5ides' ?\ght by ?is mann] ( repres5tation x is impossible to say4 ,:at ,i want to po9t out is ! adequat5ess ( ! illu/ration1 n only to give a /atic model ( 8 doctr9e1 b "o capable z x 71 ( a pla/ic modification into a correspond5ce into k9dr$ =ms ( ?\ght4 ,ei!r "o ( two ?+s mu/ be true--t f\r-dim5sional conceptions give a wond]ful p[] ( repres5t+ ! ?\ght ( ! ,ea/1 or t ! ?9k]s ( ! ,ea/ mu/ h be5 look+ at & reg>d+ f\r-dim5sional exi/5ce4 ,com+ n[ to ! ma9 /ream ( ?\ght we mu/ dwell 9 "s detail on ,py?agoras1 n because ( 8 direct relation to ! subject1 b because ( 8 relation to inve/igators :o came lat]4 ,py?agoras inv5t$ ! two-way c\nt+4 ,let u repres5t ! s+le-way c\nt+ by ! posits aa1 ab1 ac1 ad1 us+ ^! pairs ( lett]s in/ead ( ! numb]s #a1 #b1 #c1 #d4 ,i put an a 9 ea* case fir/ = a reason :i* w imm$iately appe>4 ,we h a sequ5ce & ord]4 ,!re is no conception ( di/ance necess>ily involv$4 ,! diff]5ce betwe5 ! posits is "o ( ord] n ( di/ance--only :5 id5tifi$ ) a numb] ( equal mat]ial ?+s 9 juxtaposition does ! notion ( di/ance arise4 ,n[1 besides ! simple s]ies ,i c h1 />t+ f aa1 ba1 ca1 da1 f ab1 bb1 cb1 db1 & s on1 & =m+ a s*eme3 da db dc dd ca cb cc cd ba bb bc bd aa ab ac ad ,?is complex or manifold gives a two-way ord]4 ,i c repres5t x by a set ( po9ts1 if ,i am on my gu>d aga9/ assum+ any relation ( di/ance4 ,illu/ration3 ,fig4 #ae4 ,py?agoras /udi$ ?is two-fold way ( c\nt+ 9 ref]5ce to mat]ial bodies1 & discov]$ t mo/ rem>kable prop]ty ( ! comb9ation ( numb] & matt] t be>s 8 "n4 ,! ,py?agorean prop]ty ( an ext5d$ mat]ial sy/em c be exhibit$ 9 a mann] :i* w be ( use to u aft]w>ds1 & :i* !re=e ,i w employ n[ in/ead ( us+ ! k9d ( figure :i* he himself employ$4 ,consid] a two-fold field ( po9ts arrang$ 9 regul> r[s4 ,su* a field w be presuppos$ 9 ! foll[+ argum5t4 ,illu/ration3 ,fig4 #af4 #a & #b ,x is evid5t t 9 fig4 #af f\r ( ! po9ts det]m9e a squ>e1 :i* squ>e we may take z ! unit ( measurem5t = areas4 ,b we c also measure areas 9 ano!r way4 ,fig4 #af 7#a7 %[s f\r po9ts det]m9+ a squ>e4 ,b f\r squ>es also meet 9 a po9t1 fig4 #af 7#b74 ,h5ce a po9t at ! corn] ( a squ>e belongs equally to f\r squ>es4 ,?us we may say t ! po9t value ( ! squ>e %[n is "o po9t1 = if we take ! squ>e 9 fig4 #af 7#a7 x has f\r po9ts1 b ea* ( ^! belong equally to f\r o!r squ>es4 ,h5ce "o f\r? ( ea* ( !m belongs to ! squ>e 7#a7 9 fig4 #af4 ,?us ! po9t value ( ! squ>e is "o po9t4 ,! result ( c\nt+ ! po9ts is ! same z t arriv$ at by reckon+ ! squ>e units enclos$4 ,h5ce1 if we wi% to measure ! area ( any squ>e we c take ! numb] ( po9ts x encloses1 c\nt ^! z "o ea*1 & take "o-f\r? ( ! numb] ( po9ts at its corn]s4 ,illu/ration3 ,fig4 #ag4 ,n[ draw a diagonal squ>e z %[n 9 fig4 #ag4 ,x conta9s "o po9t & ! f\r corn]s c\nt = "o po9t m2 h5ce its po9t value is #b4 ,! value is ! measure ( its area--! size ( ?is squ>e is two ( ! unit squ>es4 ,look+ n[ at ! sides ( ?is figure we see t !re is a unit squ>e on ea* ( !m--! two squ>es conta9 no po9ts1 b h f\r corn] po9ts ea*1 :i* gives ! po9t value ( ea* z "o po9t4 ,h5ce we see t ! squ>e on ! diagonal is equal to ! squ>es on ! two sides2 or z x is g5]ally express$1 ! squ>e on ! hypo!nuse is equal to ! sum ( ! squ>es on ! sides4 ,illu/ration3 ,fig4 #ah4 ,notic+ ?is fact we c proce$ to ask if x is always true4 ,draw+ ! squ>e %[n 9 fig4 #ah1 we c c\nt ! numb] ( its po9ts4 ,!re are five altoge!r4 ,!re are f\r po9ts inside ! squ>e on ! diagonal1 & h5ce1 ) ! f\r po9ts at its corn]s ! po9t value is #e--t is1 ! area is #e4 ,n[ ! squ>es on ! sides are respectively ( ! area #d & #a4 ,h5ce 9 ?is case also ! squ>e on ! diagonal is equal to ! sum ( ! squ>e on ! sides4 ,?is prop]ty ( matt] is "o ( ! fir/ great discov]ies ( appli$ ma!matics4 ,we %all prove aft]w>ds t x is n a prop]ty ( space4 ,= ! pres5t x is en\gh to rem>k t ! positions 9 :i* ! po9ts are arrang$ is entirely exp]im5tal4 ,x is by means ( equal pieces ( "s mat]ial1 or ! same piece ( mat]ial mov$ f "o place to ano!r1 t ! po9ts are arrang$4 ,py?agoras next enquir$ :at ! relation mu/ be s t a squ>e drawn slant+-wise %\ld be equal to "o /raight-wise4 ,he f\nd t a squ>e ^: side is five c be plac$ ei!r rectangul>ly along ! l9es ( po9ts1 or 9 a slant+ position4 ,& ?is squ>e is equival5t to two squ>es ( sides #d & #c4 ,"h he came ^u a num]ical relation embodi$ 9 a prop]ty ( matt]4 ,numb]s imman5t 9 ! objects produc$ ! equality s satisfactory = intellectual appreh5sion4 ,& he f\nd t numb]s :5 imman5t 9 s\nd--:5 ! /r+s ( a musical in/rum5t 7 giv5 c]ta9 def9ite proportions ( l5g?--7 no less captivat+ to ! e> ?an ! equality ( squ>es 0 to ! reason4 ,:at wond] !n t he ascrib$ an active p[] to numb]6 ,we mu/ rememb] t1 %>+ l ourselves ! se>* = ! p]man5t 9 *ang+ ph5om5a1 ! ,greeks _h n t conception ( ! p]man5t 9 matt] t we h4 ,to !m mat]ial ?+s 7 n p]man5t4 ,9 fire solid ?+s w\ld vani%2 absolutely disappe>4 ,rock & e>? _h a m /able exi/5ce1 b !y too grew & decay$4 ,! p]man5ce ( matt]1 ! cons]vation ( en]gy1 7 unkn[n to !m4 ,& t di/9ction :i* we draw s readily betwe5 ! fleet+ & p]man5t causes ( s5sation1 betwe5 a s\nd & a mat]ial object1 = in/ance1 _h n ! same mean+ to !m :i* x has = u4 ,let u b imag9e = a mom5t t mat]ial ?+s are fleet+1 disappe>+1 & we %all ent] ) a f> bett] appreciation into t se>* = ! p]man5t :i*1 ) ! ,greeks1 z ) u1 is ! prim>y intellectual dem&4 ,:at is t :i* amid a ?\s& =ms is ev] ! same1 :i* we c recognise "u all its vicissitudes1 ( :i* ! div]se ph5om5a are ! appe>ances8 ,to ?9k t ?is is numb] is n s v wide ( ! m>k4 ,) an intellectual appreh5sion :i* f> outran ! evid5ces = its application1 ! atomi/s ass]t$ t !re 7 ev]la/+ mat]ial p>ticles1 :i*1 by _! union1 produc$ all ! v>y+ =ms & /ates ( bodies4 ,b 9 view ( ! obs]v$ facts ( nature z !n kn[n1 ,ari/otle1 ) p]fect reason1 refus$ to accept ?is hypo!sis4 ,he expressly /ates t !re is a *ange ( quality1 & t ! *ange due to motion is only "o ( ! possible modes ( *ange4 ,) no p]man5t mat]ial _w ab\t u1 ) ! fleet+1 ! unp]man5t1 all ar\nd we %\ld1 ,i ?9k1 be ready to foll[ ,py?agoras 9 8 id5tification ( numb] ) t pr9ciple :i* subsi/s amid/ all *anges1 :i* 9 multitud9\s =ms we appreh5d imman5t 9 ! *ang+ & disappe>+ sub/ance ( ?+s4 ,& f ! num]ical idealism ( ,py?agoras !re is b a /ep to ! m ri* & full idealism ( ,plato4 ,t :i* is appreh5d$ by ! s5se ( t\* we put z prim>y & real1 & ! o!r s5ses we say are m]ely conc]n$ ) appe>ances4 ,b ,plato took !m all z valid1 z giv+ qualities ( exi/5ce4 ,t ! qualities 7 n p]man5t 9 ! _w z giv5 to ! s5ses =c$ him to attribute to !m a diff]5t k9d ( p]man5ce4 ,he =m$ ! conception ( a _w ( ideas1 9 :i* all t really is1 all t affects u & gives ! ri* & wond]ful weal? ( our exp]i5ce1 is n fleet+ & transitory1 b et]nal4 ,& ( ?is real & et]nal we see 9 ! ?+s ab\t u ! fleet+ & transi5t images4 ,& ?is _w ( ideas 0 no exclusive "o1 :]e9 0 no place = ! inn]mo/ convictions ( ! s\l & its mo/ au?oritative ass]tions4 ,!re9 exi/$ ju/ice1 beauty--! "o1 ! good1 all t ! s\l dem&$ to be4 ,! _w ( ideas1 ,plato's wond]ful creation pres]v$ = man1 = 8 delib]ate inve/igation & _! sure developm5t1 all t ! rude incompreh5sible *anges ( a h>% exp]i5ce scatt]s & de/roys4 ,plato believ$ 9 ! reality ( ideas4 ,he meets u fairly & squ>ely4 ,divide a l9e into two p>ts1 he says2 "o to repres5t ! real objects 9 ! _w1 ! o!r to repres5t ! transitory appe>ances1 su* z ! image 9 /ill wat]1 ! glitt] ( ! sun on a bright surface1 ! %ad[s on ! cl\ds4 ,a ,b ---------------------------------------- ---------------------------------------- ---------------------------------------- ----- ,real ?+s3 ,appe>ances3 e4g41 ! sun4 e4g41 ! reflection ( ! sun4 ,take ano!r l9e & divide x into two p>ts1 "o repres5t+ our ideas1 ! ord9>y occupants ( our m9ds1 su* z :it5ess1 equality1 & ! o!r repres5t+ our true k1 :i* is ( et]nal pr9ciples1 su* z beauty1 goodness4 ,a#a ,b#a ---------------------------------------- ---------------------------------------- ---------------------------------------- ----- ,et]nal pr9ciples1 ,appe>ances 9 ! m9d1 z beauty z :it5ess1 equality ,!n z ,a is to ,b1 s is ,a#a to ,b#a ,t is1 ! s\l c proce$1 go+ away f real ?+s to a region ( p]fect c]ta9ty1 :]e x beholds :at is1 n ! scatt]$ reflections2 beholds ! sun1 n ! glitt] on ! s&s2 true be+1 n *ance op9ion4 ,n[1 ?is is to u1 z x 0 to ,ari/otle1 absolutely inconceivable f a sci5tific po9t ( view4 ,we c und]/& t a be+ is kn[n 9 ! fulness ( 8 relations2 x is 9 8 relations to 8 circum/ances t a man's "* is kn[n2 x is 9 8 acts "u 8 conditions t 8 "* exi/s4 ,we _c grasp or conceive any pr9ciple ( individuation ap>t f ! fulness ( ! relations to ! surr\nd+s4 ,b suppose n[ t ,plato is talk+ ab\t ! high] man--! f\r-dim5sional be+ t is limit$ 9 our ext]nal exp]i5ce to a ?ree-dim5sional _w4 ,d n 8 words beg9 to h a mean+8 ,su* a be+ w\ld h a consci\sness ( motion :i* is n z ! motion he c see ) ! eyes ( ! body4 ,he1 9 8 own be+1 kn[s a reality to :i* ! outw>d matt] ( ?is too solid e>? is flimsy sup]ficiality4 ,he too kn[s a mode ( be+1 ! fulness ( relations1 9 :i* c only be repres5t$ 9 ! limit$ _w ( s5se1 z ! pa9t] unsub/antially portrays ! dep?s ( woodl&1 pla9s1 & air4 ,?9k+ ( su* a be+ 9 man1 0 n ,plato's l9e well divid$8 ,x is notewor?y t1 if ,plato omitt$ 8 doctr9e ( ! indep5d5t orig9 ( ideas1 he w\ld pres5t exactly ! f\r-dim5sional argum5t2 a real ?+ z we ?9k x is an idea4 ,a plane be+'s idea ( a squ>e object is ! idea ( an ab/raction1 namely1 a geometrical squ>e4 ,simil>ly our idea ( a solid ?+ is an ab/raction1 = 9 our idea !re is n ! f\r-dim5sional ?ickness :i* is necess>y1 h[ev] slight1 to give reality4 ,! argum5t w\ld !n run1 z a %ad[ is to a solid object1 s is ! solid object to ! reality4 ,?us ,a & ,b w\ld be id5tifi$4 ,9 ! allegory :i* ,i h already allud$ to1 ,plato 9 almo/ z _m words %[s =? ! relation betwe5 exi/5ce 9 a sup]ficies & 9 solid space4 ,& he uses ?is relation to po9t to ! conditions ( a high] be+4 ,he imag9es a numb] ( m5 prison]s1 *a9$ s t !y look at ! wall ( a cav]n 9 :i* !y are conf9$1 ) _! backs to ! road & ! light4 ,ov] ! road pass m5 & wom51 figures & processions1 b ( all ?is pageant all t ! prison]s behold is ! %ad[ ( x on ! wall :]eon !y gaze4 ,_! own %ad[s & ! %ad[s ( ! ?+s 9 ! _w are all t !y see1 & id5tify+ !mselves ) _! %ad[s relat$ z %ad[s to a _w ( %ad[s1 !y live 9 a k9d ( dream4 ,plato imag9es "o ( _! numb] to pass out f among/ !m into ! real space _w1 & !n return+ to tell !m ( _! condition4 ,"h he pres5ts mo/ pla9ly ! relation betwe5 exi/5ce 9 a plane _w & exi/5ce 9 a ?ree-dim5sional _w4 ,& he uses ?is illu/ration z a type ( ! mann] 9 :i* we are to proce$ to a high] /ate f ! ?ree-dim5sional life we "k4 ,x mu/ h hung ^u ! weight ( a %ad[ :i* pa? he took6--:e!r ! "o we %all foll[ t[>d ! high] solid & ! f\r-dim5sional exi/5ce1 or ! "o :i* makes ideas ! high] realities1 & ! direct p]ception ( !m ! contact ) ! tru] _w4 ,pass+ on to ,ari/otle1 we w t\* on ! po9ts :i* mo/ imm$iately conc]n our enquiry4 ,j z a sci5tific man ( ! pres5t day 9 review+ ! speculations ( ! anci5t _w w\ld treat !m ) a curiosity half amus$ b :olly respectful1 ask+ ( ea* & all :]e9 lay _! relation to fact1 s ,ari/otle1 9 discuss+ ! philosophy ( ,greece z he f\nd x1 asks1 above all o!r ?+s3 8,does ?is repres5t ! _w8 ,9 ?is sy/em is !re an adequate pres5tation ( :at is80 ,he f9ds !m all defective1 "s = ! v reasons :i* we e/eem !m mo/ highly1 z :5 he criticises ! ,atomic !ory = its r$uction ( all *ange to motion4 ,b 9 ! l(ty m>* ( 8 reason he nev] loses sight ( ! :ole2 & t :]e9 our views diff] f 8 lies n s mu* 9 a sup]iority ( our po9t ( view1 z 9 ! fact :i* he himself enunciates--t x is impossible = "o pr9ciple to be valid 9 all bran*es ( enquiry4 ,! conceptions ( "o me?od ( inve/igation are n ^? ( ano!r2 & our div]g5ce lies 9 our exclusive att5tion to ! conceptions useful 9 "o way ( appreh5d+ nature r ?an 9 any possibility we f9d 9 our !ories ( giv+ a view ( ! :ole transc5d+ t ( ,ari/otle4 ,he takes acc\nt ( ev]y?+2 he does n sep>ate matt] & ! manife/ation ( matt]2 he fires all toge!r 9 a conception ( a va/ _w process 9 :i* ev]y?+ takes "p--! motion ( a gra9 ( du/1 ! unfold+ ( a leaf1 ! ord]$ motion ( ! sph]es 9 heav5--all are p>ts ( "o :ole :i* he w n sep>ate into dead matt] & adv5titi\s modifications4 ,& j z our !ories1 z repres5tative ( actuality1 fall be=e 8 unequall$ grasp ( fact1 s ! doctr9e ( ideas fell4 ,x is n an adequate acc\nt ( exi/5ce1 z ,plato himself %[s 9 8 8,p>m5ides02 x only expla9s ?+s by putt+ _! d\bles beside !m4 ,= 8 own "p ,ari/otle inv5t$ a great m>*+ def9ition :i*1 ) a k9d ( p[] ( its own1 cleaves its way ?r\gh ph5om5a to limit+ conceptions on ei!r h&1 t[>ds ^: exi/5ce all exp]i5ce po9ts4 ,9 ,ari/otle's def9ition ( matt] & =m z ! con/itu5t ( reality1 z 9 ,plato's my/ical vision ( ! k+dom ( ideas1 ! exi/5ce ( ! high] dim5sionality is implicitly involv$4 ,sub/ance accord+ to ,ari/otle is relative1 n absolute4 ,9 ev]y?+ t is !re is ! matt] ( :i* x is compos$1 ! =m :i* x exhibits2 b ^! are indissolubly connect$1 & nei!r c be ?\ght )\t ! o!r4 ,! blocks ( /one out ( :i* a h\se is built are ! mat]ial = ! build]2 b1 z reg>ds ! qu>rym51 !y are ! matt] ( ! rocks ) ! =m he has impos$ on !m4 ,words are ! f9al product ( ! gramm>ian1 b ! m]e matt] ( ! orator or poet4 ,! atom is1 ) u1 t out ( :i* *emical sub/ances are built up1 b look$ at f ano!r po9t ( view is ! result ( complex processes4 ,n[h]e d we f9d f9ality4 ,! matt] 9 "o sph]e is ! matt]1 plus =m1 ( ano!r sph]e ( ?\ght4 ,mak+ an obvi\s application to geometry1 plane figures exi/ z ! limitation ( diff]5t portions ( ! plane by "o ano!r4 ,9 ! b\nd+ l9es ! sep>at$ matt] ( ! plane %[s its det]m9ation into =m4 ,& z ! plane is ! matt] relatively to det]m9ations 9 ! plane1 s ! plane itself exi/s 9 virtue ( ! det]m9ation ( space4 ,a plane is t :]e9 =mless space has =m sup]impos$ on x1 & gives an actuality ( real relations4 ,we _c refuse to c>ry ?is process ( reason+ a /ep f>!r back1 & say t space itself is t :i* gives =m to high] space4 ,z a l9e is ! det]m9ation ( a plane1 & a plane ( a solid1 s solid space itself is ! det]m9ation ( a high] space4 ,z a l9e by itself is inconceivable )\t t plane :i* x sep>ates1 s ! plane is inconceivable )\t ! solids :i* x limits on ei!r h&4 ,& s space itself _c be positively def9$4 ,x is ! negation ( ! possibility ( movem5t 9 m ?an ?ree dim5sions4 ,! conception ( space dem&s t ( a high] space4 ,z a surface is ?9 & unsub/antial )\t ! sub/ance ( :i* x is ! surface1 s matt] itself is ?9 )\t ! high] matt]4 ,j z ,ari/otle inv5t$ t algebraical me?od ( repres5t+ unkn[n quantities by m]e symbols1 n by l9es necess>ily det]m9ate 9 l5g? z 0 ! habit ( ! ,greek geomet]s1 & s /ruck out ! pa? t[>ds ^? objectifications ( ?\ght :i*1 l indep5d5t ma*9es = reason+1 supply ! ma!matician ) 8 analytical weapons1 s 9 ! =mulation ( ! doctr9e ( matt] & =m1 ( pot5tiality & actuality1 ( ! relativity ( sub/ance1 he produc$ ano!r k9d ( objectification ( m9d--a def9ition :i* _h a vital =ce & an activity ( its own4 ,9 none ( 8 writ+s1 z f> z we "k1 did he c>ry x to its legitimate conclusion on ! side ( matt]1 b 9 ! direction ( ! =mal qualities he 0 l$ to 8 limit+ conception ( t exi/5ce ( pure =m :i* lies beyond all kn[n det]m9ation ( matt]4 ,! unmov$ mov] ( all ?+s is ,ari/otle's highe/ pr9ciple4 ,t[>ds x1 to p>take ( its p]fection all ?+s move4 ,! univ]se1 accord+ to ,ari/otle1 is an active process--he does n adopt ! illogical conception t x 0 once set 9 motion & has kept on ev] s9ce4 ,!re is room = activity1 w1 self-det]m9ation1 9 ,ari/otle's sy/em1 & = ! cont+5t & accid5tal z well4 ,we d n foll[ him1 because we are accu/om$ to f9d 9 nature inf9ite s]ies1 & d n feel oblig$ to pass on to a belief 9 ! ultimate limits to :i* !y seem to po9t4 ,b ap>t f ! pu%+ to ! limit1 z a relative pr9ciple ?is doctr9e ( ,ari/otle's z to ! relativity ( sub/ance is irrefragible 9 its logic4 ,he 0 ! fir/ to %[ ! necessity ( t pa? ( ?\ght :i* :5 foll[$ leads to a belief 9 a f\r-dim5sional space4 ,antagoni/ic z he 0 to ,plato 9 8 conception ( ! practical relation ( reason to ! _w ( ph5om5a1 yet 9 "o po9t he co9cid$ ) him4 ,& 9 ?is he %[$ ! c&\r ( 8 intellect4 ,he 0 m anxi\s to lose no?+ ?an to expla9 ev]y?+4 ,& t :]e9 s _m h detect$ an inconsi/5cy1 an inability to free himself f ! s*ool ( ,plato1 appe>s to u 9 connection ) our enquiry z an in/ance ( ! acut5ess ( 8 obs]vation4 ,= beyond all k giv5 by ! s5ses ,ari/otle held t !re is an active intellig5ce1 a m9d n ! passive recipi5t ( impressions f )\t1 b an active & orig9ative be+1 capable ( grasp+ k at fir/ h&4 ,9 ! active s\l ,ari/otle recognis$ some?+ 9 man n produc$ by 8 physical surr\nd+s1 some?+ :i* creates1 ^: activity is a k und]iv$ f s5se4 ,?is1 he says1 is ! immortal & undy+ be+ 9 man4 ,?us we see t ,ari/otle 0 n f> f ! recognition ( ! f\r-dim5sional exi/5ce1 bo? )\t & )9 man1 & ! process ( adequately realis+ ! high] dim5sional figures to :i* we %all come subsequ5tly is a simple r$uction to practice ( 8 hypo!sis ( a s\l4 ,! next /ep 9 ! unfold+ ( ! drama ( ! recognition ( ! s\l z connect$ ) our sci5tific conception ( ! _w1 &1 at ! same "t1 ! recognition ( t high] ( :i* a ?ree-dim5sional _w pres5ts ! sup]ficial appe>ance1 took place _m c5turies lat]4 ,if we pass ov] ! int]v5+ "t )\t a ^w x is because ! s\l 0 occupi$ ) ! ass]tion ( itself 9 o!r ways ?an t ( k4 ,:5 x took up ! task 9 e>ne/ ( kn[+ ?is mat]ial _w 9 :i* x f\nd itself1 & ( direct+ ! c\rse ( inanimate nature1 f t mo/ objective aim came1 reflect$ back z f a mirror1 its k ( itself4 ,,*apt] ,v ,,! ,,second ,,*apt] ,,9 ,,! ,,hi/ory ,,( ,,f\r ,,space ,,lobat*ewsky1 ,,bolyai1 ,,& ,,gauss ,be=e ent]+ on a description ( ! "w ( ,lobat*ewsky & ,bolyai x w n be out ( place to give a brief acc\nt ( !m1 ! mat]ials = :i* are to be f\nd 9 an article by ,franz ,s*midt 9 ! =ty-second volume ( ! ,ma!matis*e ,annal51 & 9 ,engel's edition ( ,lobat*ewsky4 ,lobat*ewsky 0 a man ( ! mo/ complete & wond]ful tal5ts4 ,z a y\? he 0 full ( vivacity1 c>ry+ 8 exub]ance s f> z to fall into s]i\s tr\ble = haz+ a pr(essor1 & o!r freaks4 ,sav$ by ! good (fices ( ! ma!matician ,b>tels1 :o appreciat$ 8 ability1 he manag$ to re/ra9 himself )9 ! b\nds ( prud5ce4 ,appo9t$ pr(essor at 8 own ,univ]sity1 ,kasan1 he ent]$ on 8 duties "u ! regime ( a pieti/ic reaction>y1 :o surr\nd$ himself ) sycophants & hypocrites4 ,e/eem+ probably ! int]e/s ( 8 pupils z high] ?an any attempt at a va9 resi/ance1 he made himself ! tyrant's "r-h& man1 do+ an incr$ible am\nt ( tea*+ & p]=m+ ! mo/ v>i$ (ficial duties4 ,amid/ all 8 activities he f\nd "t to make important contributions to sci5ce4 ,8 !ory ( p>allels is mo/ closely connect$ ) 8 "n1 b a /udy ( 8 writ+s %[s t he 0 a man capable ( c>ry+ on ma!matics 9 its ma9 l9es ( advance1 & ( a judgm5t equal to disc]n+ :at ^! l9es 74 ,appo9t$ rector ( 8 ,univ]sity1 he di$ at an advanc$ age1 surr\nd$ by fri5ds1 hon\r$1 ) ! results ( 8 b5efic5t activity all ar\nd him4 ,to him no subject came amiss1 f ! f\ndations ( geometry to ! improvem5t ( ! /oves by :i* ! peasants w>m$ _! h\ses4 ,he 0 born 9 #agic4 ,8 sci5tific "w 0 unnotic$ till1 9 #ahfg1 ,h\el1 ! ,fr5* ma!matician1 drew att5tion to its importance4 ,johann ,bolyai de ,bolyai 0 born 9 ,klaus5burg1 a t[n 9 ,transylvania1 ,decemb] #ae?1 #ahjb4 ,8 fa!r1 ,wolfgang ,bolyai1 a pr(essor 9 ! ,re=m$ ,college ( ,m>os ,vas>hely1 reta9$ ! ard\r 9 ma!matical /udies :i* _h made him a *os5 companion ( ,gauss 9 _! e>ly /ud5t days at ,gtt+54 ,he f\nd an eag] pupil 9 ,johann4 ,he relates t ! boy sprang be=e him l a devil4 ,z soon z he _h enunciat$ a problem ! *ild w\ld give ! solution & comm& him to g on fur!r4 ,z a ?irte5-ye>-old boy 8 fa!r sometimes s5t him to fill 8 place :5 incapacitat$ f tak+ 8 classes4 ,! pupils li/5$ to him ) m att5tion ?an to 8 fa!r = !y f\nd him cle>] to und]/&4 ,9 a lett] to ,gauss ,wolfgang ,bolyai writes3-- 8,my boy is /rongly built4 ,he has le>n$ to recognise _m con/ellations1 & ! ord9>y figures ( geometry4 ,he makes apt applications ( 8 notions1 draw+ = in/ance ! positions ( ! />s ) _! con/ellations4 ,la/ w9t] 9 ! c\ntry1 see+ ,jupit] he ask$3 #8,h[ is x t we c see him f "h z well z f ! t[n8 ,he mu/ be f> (f4' ,& z to ?ree diff]5t places to :i* he _h be5 he ask$ me to tell him ab\t !m 9 "o ^w4 ,i did n "k :at he meant1 & !n he ask$ me if "o 0 9 a l9e ) ! o!r & all 9 a r[1 or if !y 7 9 a triangle4 8,he enjoys cutt+ pap] figures ) a pair ( scissors1 & )\t my ev] hav+ told him ab\t triangles rem>k$ t a "r-angl$ triangle :i* he _h cut out 0 half ( an oblong4 ,i ex]cise 8 body ) c>e1 he c dig well 9 ! e>? ) 8 little h&s4 ,! blossom c fall & no fruit left4 ,:5 he is fifte5 ,i want to s5d him to y to be y\r pupil40 ,9 ,johann's autobiography he says3-- 8,my fa!r call$ my att5tion to ! imp]fections & gaps 9 ! !ory ( p>allels4 ,he told me he _h ga9$ m satisfactory results ?an 8 pr$ecessors1 b _h obta9$ no p]fect & satisfy+ conclusion4 ,none ( 8 assumptions _h ! necess>y degree ( geometrical c]ta9ty1 al?\gh !y suffic$ to prove ! elev5? axiom & appe>$ acceptable on fir/ sight4 8,he begg$ ( me1 anxi\s n )\t a reason1 to hold myself alo( & to %un all inve/igation on ?is subject1 if ,i did n wi% to live all my life 9 va940 ,johann1 9 ! failure ( 8 fa!r to obta9 any response f ,gauss1 9 answ] to a lett] 9 :i* he ask$ ! great ma!matician to make ( 8 son 8an apo/le ( tru? 9 a f> l&10 ent]$ ! ,eng9e]+ ,s*ool at ,vi5na4 ,he writes f ,temesv>1 :]e he 0 appo9t$ sub-lieut5ant ,septemb]1 #ahbc3-- 8,temesv>1 ,novemb] #crd1 #ahbc4 8,,de> ,,good ,,fa!r1 8,i h s ov]:elm+ly mu* to write ab\t my discov]y t ,i "k no o!r way ( *eck+ myself ?an tak+ a qu>t] ( a %eet only to write on4 ,i want an answ] to my f\r-%eet lett]4 8,i am unbrok5 9 my det]m9ation to publi% a "w on ,p>allels1 z soon z ,i h put my mat]ial 9 ord] & h ! means4 8,at pres5t ,i h n made any discov]y1 b ! way ,i h foll[$ almo/ c]ta9ly promises me ! atta9m5t ( my object if any possibility ( x exi/s4 8,i h n got my object yet1 b ,i h produc$ su* /up5d\s ?+s t ,i 0 ov]:elm$ myself1 & x w\ld be an et]nal %ame if !y 7 lo/4 ,:5 y see !m y w f9d t x is s4 ,n[ ,i c only say t ,i h made a new _w out ( no?+4 ,ev]y?+ t ,i h s5t y be=e is a h\se ( c>ds 9 comp>ison ) a t[]4 ,i am conv9c$ t x w be no less to my hon\r ?an if ,i _h already discov]$ x40 ,! discov]y ( :i* ,johann "h speaks 0 publi%$ z an app5dix to ,wolfgang ,bolyai's ,t5tam54 ,s5d+ ! book to ,gauss1 ,wolfgang writes1 aft] an int]ruption ( eighte5 ye>s 9 8 correspond5ce3-- 8,my son is fir/ lieut5ant ( ,eng9e]s & w soon be capta94 ,he is a f9e y\?1 a good viol9 play]1 a skilful f5c]1 & brave1 b has _h _m duels1 & is wild ev5 = a soldi]4 ,yet he is di/+ui%$--light 9 d>kness & d>kness 9 light4 ,he is an impassion$ ma!matician ) extraord9>y capacities4444 ,he w ?9k m ( y\r judgm5t on 8 "w ?an t ( all ,europe40 ,wolfgang receiv$ no answ] f ,gauss to ?is lett]1 b s5d+ a second copy ( ! book receiv$ ! foll[+ reply3-- 8,y h rejoic$ me1 my un=gott5 fri5d1 by y\r lett]s4 ,i delay$ answ]+ ! fir/ because ,i want$ to wait = ! arrival ( ! promis$ little book4 8,n[ some?+ ab\t y\r son's "w4 8,if ,i beg9 ) say+ t #8,i ought n to praise x1' y w be /agg]$ = a mom5t4 ,b ,i _c say any?+ else4 ,to praise x is to praise myself1 = ! pa? y\r son has brok5 9 ^u & ! results to :i* he has be5 l$ are almo/ exactly ! same z my own reflections1 "s ( :i* date f ?irty to ?irty-five ye>s ago4 8,9 fact ,i am a/oni%$ to ! utt]mo/4 ,my int5tion 0 to let no?+ be kn[n 9 my lifetime ab\t my own "w1 ( :i*1 = ! re/1 b little is committ$ to writ+4 ,mo/ p h b little p]ception ( ! problem1 & ,i h f\nd v few :o took any int]e/ 9 ! views ,i express$ to !m4 ,to be able to d t "o mu/ fir/ ( all h _h a real live feel+ ( :at is want+1 & z to t mo/ m5 are completely 9 ! d>k4 8,/ill x 0 my int5tion to commit ev]y?+ to writ+ 9 ! c\rse ( "t1 s t at lea/ x %\ld n p]i% ) me4 8,i am deeply surpris$ t ?is task c be sp>$ me1 & ,i am mo/ ( all pleas$ 9 ?is t x is ! son ( my old fri5d :o has 9 s rem>kable a mann] prec$$ me40 ,! impression :i* we receive f ,gauss's inexplicable sil5ce t[>ds 8 old fri5d is swept away by ?is lett]4 ,h5ce we brea! ! cle> air ( ! m\nta9 tops4 ,gauss w\ld n h fail$ to p]ceive ! va/ significance ( 8 ?\ghts1 sure to be all ! great] 9 _! effect on future ages f ! want ( compreh5sion ( ! pres5t4 ,yet !re is n a ^w or a sign 9 8 writ+ to claim ! ?\ght = himself4 ,he publi%$ no s+le l9e on ! subject4 ,by ! measure ( :at he ?us sil5tly rel9qui%es1 by su* a measure ( a _w-trans=m+ ?\ght1 we c appreciate 8 greatness4 ,x is a long /ep f ,gauss's s]5ity to ! di/urb$ & passionate life ( ,johann ,bolyai--he & ,galois1 ! two mo/ int]e/+ figures 9 ! hi/ory ( ma!matics4 ,= ,bolyai1 ! wild soldi]1 ! duelli/1 fell at odds ) ! _w4 ,x is relat$ ( him t he 0 *all5g$ by ?irte5 (fic]s ( 8 g>rison1 a ?+ n unlikely to happ5 consid]+ h[ diff]5tly he ?\ght f e "o else4 ,he f\ght !m all 9 succession--mak+ x 8 only condition t he %\ld be all[$ to play on 8 viol9 = an int]val betwe5 meet+ ea* oppon5t4 ,he dis>m$ or w\nd$ all 8 antagoni/s4 ,x c be easily imag9$ t a temp]am5t su* z 8 0 "o n cong5ial to 8 milit>y sup]iors4 ,he 0 retir$ 9 #ahcc4 ,8 epo*-mak+ discov]y awoke no att5tion4 ,he seems to h conceiv$ ! idea t 8 fa!r _h betray$ him 9 "s inexplicable way by 8 communications ) ,gauss1 & he *all5g$ ! excell5t ,wolfgang to a duel4 ,he pass$ 8 life 9 pov]ty1 _m a "t1 says 8 biograph]1 seek+ to snat* himself f dissipation & apply himself aga9 to ma!matics4 ,b 8 ef=ts _h no result4 ,he di$ ,janu>y #bg?1 #ahfj1 fall5 out ) ! _w & ) himself4 ,,metageometry ,! !ories :i* are g5]ally connect$ ) ! names ( ,lobat*ewsky & ,bolyai be> a s+ul> & curi\s relation to ! subject ( high] space4 ,9 ord] to %[ :at ?is relation is1 ,i mu/ ask ! read] to be at ! pa9s to c\nt c>efully ! sets ( po9ts by :i* ,i %all e/imate ! volumes ( c]ta9 figures4 ,no ma!matical processes beyond ?is simple "o ( c\nt+ w be necess>y4 ,illu/ration3 ,fig4 #ai4 ,let u suppose we h be=e u 9 fig4 #ai a plane cov]$ ) po9ts at regul> int]vals1 s plac$ t e f\r det]m9e a squ>e4 ,n[ x is evid5t t z f\r po9ts det]m9e a squ>e1 s f\r squ>es meet 9 a po9t4 ,illu/ration3 ,fig4 #bj4 ,?us1 consid]+ a po9t inside a squ>e z belong+ to x1 we may say t a po9t on ! corn] ( a squ>e belongs to x & to ?ree o!rs equally3 belongs a qu>t] ( x to ea* squ>e4 ,illu/ration3 ,fig4 #ba4 ,illu/ration3 ,fig4 #bb4 ,?us ! squ>e ,,acde 7fig4 #ba7 conta9s "o po9t1 & has f\r po9ts at ! f\r corn]s4 ,s9ce "o-f\r? ( ea* ( ^! f\r belongs to ! squ>e1 ! f\r toge!r c\nt z "o po9t1 & ! po9t value ( ! squ>e is two po9ts--! "o inside & ! f\r at ! corn] make two po9ts belong+ to x exclusively4 ,n[ ! area ( ?is squ>e is two unit squ>es1 z c be se5 by draw+ two diagonals 9 fig4 #bb4 ,we also notice t ! squ>e 9 "q is equal to ! sum ( ! squ>es on ! sides ,,ab1 ,,bc1 ( ! "r-angl$ triangle ,,abc4 ,?us we recognise ! proposition t ! squ>e on ! hypo!nuse is equal to ! sum ( ! squ>es on ! two sides ( a "r-angl$ triangle4 ,n[ suppose we set ourselves ! "q ( det]m9+ ! :]eab\ts 9 ! ord]$ sy/em ( po9ts1 ! end ( a l9e w\ld come :5 x turn$ ab\t a po9t keep+ "o extremity fix$ at ! po9t4 ,we c solve ?is problem 9 a p>ticul> case4 ,if we c f9d a squ>e ly+ slantwise among/ ! dots :i* is equal to "o :i* goes regul>ly1 we %all "k t ! two sides are equal1 & t ! slant+ side is equal to ! /raight-way side4 ,?us ! volume & %ape ( a figure rema9+ un*ang$ w be ! te/ ( its hav+ rotat$ ab\t ! po9t1 s t we c say t its side 9 its fir/ position w\ld turn into its side 9 ! second position4 ,n[1 su* a squ>e c be f\nd 9 ! "o ^: side is five units 9 l5g?4 ,illu/ration3 ,fig4 #bc4 ,9 fig4 #bc1 9 ! squ>e on ,,ab1 !re are-- #i po9ts int]ior #i #d at ! corn]s #a #d sides ) #c on ea* side1 consid]$ z #a on ea* side1 because belong+ equally to two squ>es #f ,! total is #af4 ,!re are #i po9ts 9 ! squ>e on ,,bc4 ,9 ! squ>e on ,,ac !re are-- #bd po9ts inside #bd #d at ! corn]s #a or #be altoge!r4 ,h5ce we see aga9 t ! squ>e on ! hypo!nuse is equal to ! squ>es on ! sides4 ,n[ take ! squ>e ,,afhg1 :i* is l>g] ?an ! squ>e on ,,ab4 ,x conta9s #be po9ts4 #af inside #af #af on ! sides1 c\nt+ z #h #d on ! corn]s #a mak+ #be altoge!r4 ,if two squ>es are equal we conclude ! sides are equal4 ,h5ce1 ! l9e ,,af turn+ r\nd ,a w\ld move s t x w\ld aft] a c]ta9 turn+ co9cide ) ,,ac4 ,?is is prelim9>y1 b x involves all ! ma!matical difficulties t w pres5t !mselves4 ,!re are two alt]ations ( a body by :i* its volume is n *ang$4 ,"o is ! "o we h j consid]$1 rotation1 ! o!r is :at is call$ %e>4 ,consid] a book1 or heap ( loose pages4 ,!y c be slid s t ea* "o slips ov] ! prec$+ "o1 & ! :ole assumes ! %ape b 9 fig4 #bd4 ,illu/ration3 ,fig4 #bd4 ,?is de=mation is n %e> alone1 b %e> accompani$ by rotation4 ,%e> c be consid]$ z produc$ 9 ano!r way4 ,take ! squ>e ,,abcd 7fig4 #be71 & suppose t x is pull$ out f along "o ( its diagonals bo? ways1 & proportionately compress$ along ! o!r diagonal4 ,x w assume ! %ape 9 fig4 #bf4 ,?is compression & expansion along two l9es at "r angles is :at is call$ %e>2 x is equival5t to ! slid+ illu/rat$ above1 comb9$ ) a turn+ r\nd4 ,illu/ration3 ,fig4 #be4 ,illu/ration3 ,fig4 #bf4 ,9 pure %e> a body is compress$ & ext5d$ 9 two directions at "r angles to ea* o!r1 s t its volume rema9s un*ang$4 ,n[ we "k t our mat]ial bodies resi/ %e>--%e> does viol5ce to ! int]nal arrangem5t ( _! p>ticles1 b !y turn z :oles )\t su* int]nal resi/ance4 ,b !re is an exception4 ,9 a liquid %e> & rotation take place equally easily1 !re is no m resi/ance aga9/ a %e> ?an !re is aga9/ a rotation4 ,n[1 suppose all bodies 7 to be r$uc$ to ! liquid /ate1 9 :i* !y yield to %e> & to rotation equally easily1 & !n 7 to be recon/ruct$ z solids1 b 9 su* a way t %e> & rotation _h int]*ang$ places4 ,t is to say1 let u suppose t :5 !y _h become solids aga9 !y w\ld %e> )\t (f]+ any int]nal resi/ance1 b a rotation w\ld d viol5ce to _! int]nal arrangem5t4 ,t is1 we %\ld h a _w 9 :i* %e> w\ld h tak5 ! place ( rotation4 ,a %e> does n alt] ! volume ( a body3 ?us an inhabitant liv+ 9 su* a _w w\ld look on a body %e>$ z we look on a body rotat$4 ,he w\ld say t x 0 ( ! same %ape1 b _h turn$ a bit r\nd4 ,let u imag9e a ,py?agoras 9 ?is _w go+ to "w to inve/igate1 z is 8 wont4 ,illu/ration3 ,fig4 #bg4 ,illu/ration3 ,fig4 #bh4 ,fig4 #bg repres5ts a squ>e un%e>$4 ,fig4 #bh repres5ts a squ>e %e>$4 ,x is n ! figure into :i* ! squ>e 9 fig4 #bg w\ld turn1 b ! result ( %e> on "s squ>e n drawn4 ,x is a simple slant+ plac$ figure1 tak5 n[ z we took a simple slant+ plac$ squ>e be=e4 ,n[1 s9ce bodies 9 ?is _w ( %e> (f] no int]nal resi/ance to %e>+1 & keep _! volume :5 %e>$1 an inhabitant accu/om$ to !m w\ld n consid] t !y alt]$ _! %ape "u %e>4 ,he w\ld call ,,acde z mu* a squ>e z ! squ>e 9 fig4 #bg4 ,we w call su* figures %e> squ>es4 ,c\nt+ ! dots 9 ,,acde1 we f9d-- #b inside #b #d at corn]s #a or a total ( #c4 ,n[1 ! squ>e on ! side ,,ab has #d po9ts1 t on ,,bc has #a po9t4 ,"h ! %e> squ>e on ! hypo!nuse has n #e po9ts b #c2 x is n ! sum ( ! squ>es on ! sides1 b ! diff]5ce4 ,?is relation always holds4 ,look at fig4 #bi4 ,illu/ration3 ,fig4 #bi4 ,%e> squ>e on hypo!nuse-- #g int]nal #g #d at corn]s #a -- #h ,illu/ration3 ,fig4 #bi bis4 ,squ>e on "o side--:i* ! read] c draw = himself-- #d int]nal #d #h on sides #d #d at corn]s #a -- #i & ! squ>e on ! o!r side is #a4 ,h5ce 9 ?is case aga9 ! diff]5ce is equal to ! %e> squ>e on ! hypo!nuse1 #i - #a #h4 ,?us 9 a _w ( %e> ! squ>e on ! hypo!nuse w\ld be equal to ! diff]5ce ( ! squ>es on ! sides ( a "r-angl$ triangle4 ,9 fig4 #bi bis ano!r %e> squ>e is drawn on :i* ! above relation c be te/$4 ,:at n[ w\ld be ! position a l9e on turn+ by %e> w\ld take up8 ,we mu/ settle ?is 9 ! same way z previ\sly ) our turn+4 ,s9ce a body %e>$ rema9s ! same1 we mu/ f9d two equal bodies1 "o 9 ! /raight way1 "o 9 ! slant+ way1 :i* h ! same volume4 ,!n ! side ( "o w by turn+ become ! side ( ! o!r1 = ! two figures are ea* :at ! o!r becomes by a %e> turn+4 ,we c solve ! problem 9 a p>ticul> case-- ,illu/ration3 ,fig4 #cj4 ,9 ! figure ,,acde 7fig4 #cj7 !re are-- #ae inside #ae #d at corn]s #a a total ( #af4 ,n[ 9 ! squ>e ,,abgf1 !re are #af-- #i inside #i #ab on sides #f #d at corn]s #a -- #af ,h5ce ! squ>e on ,,ab w\ld1 by ! %e> turn+1 become ! %e> squ>e ,,acde4 ,& h5ce ! inhabitant ( ?is _w w\ld say t ! l9e ,,ab turn$ into ! l9e ,,ac4 ,^! two l9es w\ld be to him two l9es ( equal l5g?1 "o turn$ a little way r\nd f ! o!r4 ,t is1 putt+ %e> 9 place ( rotation1 we get a diff]5t k9d ( figure1 z ! result ( ! %e> rotation1 f :at we got ) our ord9>y rotation4 ,& z a consequ5ce we get a position = ! end ( a l9e ( inv>iable l5g? :5 x turns by ! %e> rotation1 diff]5t f ! position :i* x w\ld assume on turn+ by our rotation4 ,a real mat]ial rod 9 ! %e> _w w\ld1 on turn+ ab\t ,a1 pass f ! position ,,ab to ! position ,,ac4 ,we say t its l5g? alt]s :5 x becomes ,,ac1 b ?is trans=mation ( ,,ab w\ld seem to an inhabitant ( ! %e> _w l a turn+ ( ,,ab )\t alt]+ 9 l5g?4 ,if n[ we suppose a communication ( ideas t takes place betwe5 "o ( ourselves & an inhabitant ( ! %e> _w1 !re w\ld evid5tly be a diff]5ce betwe5 8 views ( di/ance & ours4 ,we %\ld say t 8 l9e ,,ab increas$ 9 l5g? 9 turn+ to ,,ac4 ,he w\ld say t our l9e ,,af 7fig4 #bc7 decreas$ 9 l5g? 9 turn+ to ,,ac4 ,he w\ld ?9k t :at we call$ an equal l9e 0 9 reality a %ort] "o4 ,we %\ld say t a rod turn+ r\nd w\ld h its extremities 9 ! positions we call at equal di/ances4 ,s w\ld he--b ! positions w\ld be diff]5t4 ,he c\ld1 l u1 appeal to ! prop]ties ( matt]4 ,8 rod to him alt]s z little z ours does to u4 ,n[1 is !re any /&>d to :i* we c\ld appeal1 to say :i* ( ! two is "r 9 ?is argum5t8 ,!re is no /&>d4 ,we %\ld say t1 ) a *ange ( position1 ! configuration & %ape ( 8 objects alt]$4 ,he w\ld say t ! configuration & %ape ( our objects alt]$ 9 :at we call$ m]ely a *ange ( position4 ,h5ce di/ance indep5d5t ( position is inconceivable1 or practically di/ance is solely a prop]ty ( matt]4 ,!re is no pr9ciple to :i* ei!r p>ty 9 ?is controv]sy c\ld appeal4 ,!re is no?+ to connect ! def9ition ( di/ance ) our ideas r ?an ) 81 except ! behavi\r ( an actual piece ( matt]4 ,= ! /udy ( ! processes :i* g on 9 our _w ! def9ition ( di/ance giv5 by tak+ ! sum ( ! squ>es is ( p>am\nt importance to u4 ,b z a "q ( pure space )\t mak+ any unnecess>y assumptions ! %e> _w is j z possible & j z int]e/+ z our _w4 ,x 0 ! geometry ( su* conceivable worlds t ,lobat*ewsky & ,bolyai /udi$4 ,?is k9d ( geometry has evid5tly no?+ to d directly ) f\r-dim5sional space4 ,b a connection arises 9 ?is way4 ,x is evid5t t1 in/ead ( tak+ a simple %e> z ,i h done1 & def9+ x z t *ange ( ! arrangem5t ( ! p>ticles ( a solid :i* !y w und]go )\t (f]+ any resi/ance due to _! mutual action1 ,i might take a complex motion1 compos$ ( a %e> & a rotation toge!r1 or "s o!r k9d ( de=mation4 ,let u suppose su* an alt]ation pick$ out & def9$ z ! "o :i* means simple rotation1 !n ! type1 accord+ to :i* all bodies w alt] by ?is rotation1 is fix$4 ,look+ at ! movem5ts ( ?is k9d1 we %\ld say t ! objects 7 alt]+ _! %ape z well z rotat+4 ,b to ! inhabitants ( t _w !y w\ld seem to be unalt]$1 & our figures 9 _! motions w\ld seem to !m to alt]4 ,9 su* a _w ! features ( geometry are diff]5t4 ,we h se5 "o su* diff]5ce 9 ! case ( our illu/ration ( ! _w ( %e>1 :]e ! squ>e on ! hypo!nuse 0 equal to ! diff]5ce1 n ! sum1 ( ! squ>es on ! sides4 ,9 our illu/ration we h ! same laws ( p>allel l9es z 9 our ord9>y rotation _w1 b 9 g5]al ! laws ( p>allel l9es are diff]5t4 ,9 "o ( ^! worlds ( a diff]5t con/itution ( matt] ?r\gh "o po9t !re c be two p>allels to a giv5 l9e1 9 ano!r ( !m !re c be none1 t is1 al?\gh a l9e be drawn p>allel to ano!r x w meet x aft] a "t4 ,n[ x 0 precisely 9 ?is respect ( p>allels t ,lobat*ewsky & ,bolyai discov]$ ^! diff]5t worlds4 ,!y did n ?9k ( !m z worlds ( matt]1 b !y discov]$ t space did n necess>ily mean t our law ( p>allels is true4 ,!y made ! di/9ction betwe5 laws ( space & laws ( matt]1 al?\gh t is n ! =m 9 :i* !y /at$ _! results4 ,! way 9 :i* !y 7 l$ to ^! results 0 ! foll[+4 ,euclid _h /at$ ! exi/5ce ( p>allel l9es z a po/ulate--putt+ frankly ?is unprov$ proposition--t "o l9e & only "o p>allel to a giv5 /raight l9e c be drawn1 z a dem&1 z some?+ t mu/ be assum$4 ,! words ( 8 n9? po/ulate are ^!3 8,if a /raight l9e meet+ two o!r /raight l9es makes ! int]ior angles on ! same side ( x equal to two "r angles1 ! two /raight l9es w nev] meet40 ,! ma!maticians ( lat] ages did n l ?is bald assumption1 & n be+ able to prove ! proposition !y call$ x an axiom--! elev5? axiom4 ,_m attempts 7 made to prove ! axiom2 no "o d\bt$ ( its tru?1 b no means c\ld be f\nd to demon/rate x4 ,at la/ an ,italian1 ,sac*i]i1 unable to f9d a pro(1 said3 8,let u suppose x n true40 ,he d$uc$ ! results ( !re be+ possibly two p>allels to "o giv5 l9e ?r\gh a giv5 po9t1 b feel+ ! wat]s too deep = ! human reason1 he devot$ ! latt] half ( 8 book to disprov+ :at he _h assum$ 9 ! fir/ "p4 ,!n ,bolyai & ,lobat*ewsky ) firm /ep ent]$ on ! =bidd5 pa?4 ,!re c be no great] evid5ce ( ! indomitable nature ( ! human _s1 or ( its manife/ de/9y to conqu] all ^? limitations :i* b9d x d[n )9 ! sph]e ( s5se ?an ?is gr& ass]tion ( ,bolyai & ,lobat*ewsky4 ,c ,d ,a ,b ,take a l9e ,,ab & a po9t ,c4 ,we say & see & "k t ?r\gh ,c c only be drawn "o l9e p>allel to ,,ab4 ,b ,bolyai said3 8,i w draw two40 ,let ,,cd be p>allel to ,,ab1 t is1 n meet ,,ab h[ev] f> produc$1 & let l9es beyond ,,cd also n meet ,,ab2 let !re be a c]ta9 region betwe5 ,,cd & ,,ce1 9 :i* no l9e drawn meets ,,ab4 ,,ce & ,,cd produc$ backw>ds ?r\gh ,c w give a simil> region on ! o!r side ( ,c4 ,illu/ration3 ,fig4 #cb4 ,no?+ s triumphantly1 "o may almo/ say s insol5tly1 ignor+ ( s5se _h ev] be5 writt5 be=e4 ,m5 _h /ruggl$ aga9/ ! limitations ( ! body1 f\ght !m1 despis$ !m1 conqu]$ !m4 ,b no "o _h ev] ?\ght simply z if ! body1 ! bodily eyes1 ! organs ( vision1 all ?is va/ exp]i5ce ( space1 _h nev] exi/$4 ,! age-long conte/ ( ! s\l ) ! body1 ! /ruggle = ma/]y1 _h come to a culm9ation4 ,bolyai & ,lobat*ewsky simply ?\ght z if ! body 0 n4 ,! /ruggle = dom9ion1 ! /rife & combat ( ! s\l 7 ov]2 !y _h ma/]$1 & ! ,hung>ian drew 8 l9e4 ,c we po9t out any connection1 z 9 ! case ( ,p>m5ides1 betwe5 ^! speculations & high] space8 ,c we suppose x 0 any inn] p]ception by ! s\l ( a motion n kn[n to ! s5ses1 :i* result$ 9 ?is !ory s free f ! bonds ( s5se8 ,no su* supposition appe>s to be possible4 ,practically1 h[ev]1 metageometry _h a great influ5ce 9 br++ ! high] space to ! front z a work+ hypo!sis4 ,?is c be trac$ to ! t5d5cy ! m9d has to move 9 ! direction ( lea/ resi/ance4 ,! results ( ! new geometry c\ld n be neglect$1 ! problem ( p>allels _h occupi$ a place too prom95t 9 ! developm5t ( ma!matical ?\ght = its f9al solution to be neglect$4 ,b ?is utt] indep5d5ce ( all me*anical consid]ations1 ?is p]fect cutt+ loose f ! famili> intuitions1 0 s difficult t almo/ any o!r hypo!sis 0 m easy ( acceptance1 & :5 ,beltrami %[$ t ! geometry ( ,lobat*ewsky & ,bolyai 0 ! geometry ( %orte/ l9es drawn on c]ta9 curv$ surfaces1 ! ord9>y def9itions ( measurem5t be+ reta9$1 att5tion 0 drawn to ! !ory ( a high] space4 ,an illu/ration ( ,beltrami's !ory is furni%$ by ! simple consid]ation ( hypo!tical be+s liv+ on a sph]ical surface4 ,illu/ration3 ,fig4 #cc4 ,let ,,abcd be ! equator ( a globe1 & ,,ap1 ,,bp1 m]idian l9es drawn to ! pole1 ,p4 ,! l9es ,,ab1 ,,ap1 ,,bp w\ld seem to be p]fectly /raight to a p]son mov+ on ! surface ( ! sph]e1 & unconsci\s ( its curvature4 ,n[ ,,ap & ,,bp bo? make "r angles ) ,,ab4 ,h5ce !y satisfy ! def9ition ( p>allels4 ,yet !y meet 9 ,p4 ,h5ce a be+ liv+ on a sph]ical surface1 & unconsci\s ( its curvature1 w\ld f9d t p>allel l9es w\ld meet4 ,he w\ld also f9d t ! angles 9 a triangle 7 great] ?an two "r angles4 ,9 ! triangle ,,pab1 = in/ance1 ! angles at ,a & ,b are "r angles1 s ! ?ree angles ( ! triangle ,,pab are great] ?an two "r angles4 ,n[ 9 "o ( ! sy/ems ( metageometry 7= aft] ,lobat*ewsky _h %[n ! way x 0 f\nd t o!r sy/ems 7 possible besides 87 ! angles ( a triangle are great] ?an two "r angles4 ,?us a be+ on a sph]e w\ld =m conclusions ab\t 8 space :i* are ! same z he w\ld =m if he liv$ on a plane1 ! matt] 9 :i* _h su* prop]ties z are presuppos$ by "o ( ^! sy/ems ( geometry4 ,beltrami also discov]$ a c]ta9 surface on :i* !re c\ld be drawn m ?an "o 8/raight0 l9e ?r\gh a po9t :i* w\ld n meet ano!r giv5 l9e4 ,i use ! ^w /raight z equival5t to ! l9e hav+ ! prop]ty ( giv+ ! %orte/ pa? betwe5 any two po9ts on x4 ,h5ce1 )\t giv+ up ! ord9>y me?ods ( measurem5t1 x 0 possible to f9d conditions 9 :i* a plane be+ w\ld necess>ily h an exp]i5ce correspond+ to ,lobat*ewsky's geometry4 ,& by ! consid]ation ( a high] space1 & a solid curv$ 9 su* a high] space1 x 0 possible to acc\nt = a simil> exp]i5ce 9 a space ( ?ree dim5sions4 ,n[1 x is f> m easy to conceive ( a high] dim5sionality to space ?an to imag9e t a rod 9 rotat+ does n move s t its end describes a circle4 ,h5ce1 a logical conception hav+ be5 f\nd h>d] ?an t ( a f\r dim5sional space1 ?\ght turn$ to ! latt] z a simple explanation ( ! possibilities to :i* ,lobat*ewsky _h awak5$ x4 ,?9k]s became accu/om$ to deal ) ! geometry ( high] space--x 0 ,kant1 says ,v]onese1 :o fir/ us$ ! expression ( 8diff]5t spaces0--& ) famili>ity ! inevitabl5ess ( ! conception made itself felt4 ,f ?is po9t x is b a small /ep to adapt ! ord9>y me*anical conceptions to a high] spatial exi/5ce1 & !n ! recognition ( its objective exi/5ce c\ld be delay$ no long]4 ,"h1 too1 z 9 s _m cases1 x turns out t ! ord] & connection ( our ideas is ! ord] & connection ( ?+s4 ,:at is ! significance ( ,lobat*ewsky's & ,bolyai's "w8 ,x mu/ be recognis$ z some?+ totally diff]5t f ! conception ( a high] space2 x is applicable to spaces ( any numb] ( dim5sions4 ,by imm]s+ ! conception ( di/ance 9 matt] to :i* x prop]ly belongs1 x promises to be ( ! greate/ aid 9 analysis = ! effective di/ance ( any two p>ticles is ! product ( complex mat]ial conditions & _c be measur$ by h>d & fa/ rules4 ,its ultimate significance is altoge!r unkn[n4 ,x is a cutt+ loose f ! bonds ( s5se1 n co9cid5t ) ! recognition ( a high] dim5sionality1 b indirectly contributory !reto4 ,?us1 f9ally1 we h come to accept :at ,plato held 9 ! holl[ ( 8 h&2 :at ,ari/otle's doctr9e ( ! relativity ( sub/ance implies4 ,! va/ univ]se1 too1 has its high]1 & 9 recognis+ x we f9d t ! direct+ be+ )9 u no long] /&s inevitably outside our sy/ematic k4 ,,*apt] ,,vi ,,! ,,high] ,,_w ,x is inde$ /range1 ! mann] 9 :i* we mu/ beg9 to ?9k ab\t ! high] _w4 ,^? simple/ objects analog\s to ^? :i* are ab\t u on e side 9 our daily exp]i5ce su* z a door1 a table1 a :eel are remote & uncognisable 9 ! _w ( f\r dim5sions1 :ile ! ab/ract ideas ( rotation1 /ress & /ra91 ela/icity into :i* analysis resolves ! famili> elem5ts ( our daily exp]i5ce are transf]able & applicable ) no difficulty :atev]4 ,?us we are 9 ! unwont$ position ( be+ oblig$ to con/ruct ! daily & habitual exp]i5ce ( a f\r-dim5sional be+1 f a k ( ! ab/ract !ories ( ! space1 ! matt]1 ! motion ( x2 in/ead (1 z 9 our case1 pass+ to ! ab/ract !ories f ! ri*ness ( s5sible ?+s4 ,:at w\ld a :eel be 9 f\r dim5sions8 ,:at ! %aft+ = ! transmission ( p[] :i* a f\r-dim5sional be+ w\ld use4 ,! f\r-dim5sional :eel1 & ! f\r-dim5sional %aft+ are :at w occupy u = ^! few pages4 ,& x is no futile or insignificant enquiry4 ,= 9 ! attempt to p5etrate into ! nature ( ! high]1 to grasp )9 our k5 t :i* transc5ds all analogies1 because :at we "k are m]ely p>tial views ( x1 ! purely mat]ial & physical pa? af=ds a means ( approa* pursu+ :i* we are 9 less likelihood ( error ?an if we use ! m frequ5tly trodd5 pa? ( fram+ conceptions :i* 9 _! elevation & beauty seem to u ideally p]fect4 ,= :]e we are conc]n$ ) our own ?\ghts1 ! developm5t ( our own ideals1 we are z x 7 on a curve1 mov+ at any mom5t 9 a direction ( tang5cy4 ,:i!r we g1 :at we set up & exalt z p]fect1 repres5ts n ! true tr5d ( ! curve1 b our own direction at ! pres5t--a t5d5cy condition$ by ! pa/1 & by a vital en]gy ( motion ess5tial b only true :5 p]petually modifi$4 ,t et]nal corrector ( our aspirations & ideals1 ! mat]ial univ]se draws sublimely away f ! simple/ ?+s we c t\* or h&le to ! inf9ite dep?s ( />ry space1 9 "o & all un9flu5c$ by :at we ?9k or feel1 pres5t+ unmov$ fact to :i*1 ?9k x good or ?9k x evil1 we c b con=m1 yet out ( all t impassivity ) a ref]5ce to some?+ beyond our individual hopes & fe>s support+ u & giv+ u our be+4 ,& to ?is great be+ we come ) ! "q3 8,y1 too1 :at is y\r high]80 ,or to put x 9 a =m :i* w leave our conclusions 9 ! %ape ( no b>r5 =mula1 & attack+ ! problem on its mo/ assailable side3 8,:at is ! :eel & ! %aft+ ( ! f\r-dim5sional me*anic80 ,9 ent]+ on ?is enquiry we mu/ make a plan ( proc$ure4 ,! me?od :i* ,i %all adopt is to trace out ! /eps ( reason+ by :i* a be+ conf9$ to movem5t 9 a two-dim5sional _w c\ld arrive at a conception ( our turn+ & rotation1 & !n to apply an analog\s process to ! consid]ation ( ! high] movem5ts4 ,! plane be+ mu/ be imag9$ z no ab/ract figure1 b z a real body possess+ all ?ree dim5sions4 ,8 limitation to a plane mu/ be ! result ( physical conditions4 ,we w !re=e ?9k ( him z ( a figure cut out ( pap] plac$ on a smoo? plane4 ,slid+ ov] ?is plane1 & com+ into contact ) o!r figures equally ?9 z he 9 ! ?ird dim5sion1 he w appreh5d !m only by _! edges4 ,to him !y w be completely b\nd$ by l9es4 ,a 8solid0 body w be to him a two-dim5sional ext5t1 ! int]ior ( :i* c only be rea*$ by p5etrat+ ?r\gh ! b\nd+ l9es4 ,n[ su* a plane be+ c ?9k ( our ?ree-dim5sional exi/5ce 9 two ways4 ,fir/1 he c ?9k ( x z a s]ies ( sections1 ea* l ! solid he kn[s ( ext5d+ 9 a direction unkn[n to him1 :i* /ret*es transv]se to 8 tangible univ]se1 :i* lies 9 a direction at "r angles to e motion :i* he made4 ,secondly1 rel9qui%+ ! attempt to ?9k ( ! ?ree-dim5sional solid body 9 its entirety he c reg>d x z consi/+ ( a numb] ( plane sections1 ea* ( !m 9 itself exactly l ! two-dim5sional bodies he kn[s1 b ext5d+ away f 8 two-dim5sional space4 ,a squ>e ly+ 9 8 space he reg>ds z a solid b\nd$ by f\r l9es1 ea* ( :i* lies 9 8 space4 ,a squ>e /&+ at "r angles to 8 plane appe>s to him z simply a l9e 9 8 plane1 = all ( x except ! l9e /ret*es 9 ! ?ird dim5sion4 ,he c ?9k ( a ?ree-dim5sional body z consi/+ ( a numb] ( su* sections1 ea* ( :i* />ts f a l9e 9 8 space4 ,n[1 s9ce 9 8 _w he c make any draw+ or model :i* involves only two dim5sions1 he c repres5t ea* su* upright section z x actually is1 & c repres5t a turn+ f a kn[n into ! unkn[n dim5sion z a turn+ f "o to ano!r ( 8 kn[n dim5sions4 ,to see ! :ole he mu/ rel9qui% "p ( t :i* he has1 & take ! :ole portion by portion4 ,consid] n[ a plane be+ 9 front ( a squ>e1 fig4 #cd4 ,! squ>e c turn ab\t any po9t 9 ! plane--say ! po9t ,a4 ,b x _c turn ab\t a l9e1 z ,,ab4 ,=1 9 ord] to turn ab\t ! l9e ,,ab1 ! squ>e mu/ leave ! plane & move 9 ! ?ird dim5sion4 ,?is motion is out ( 8 range ( obs]vation1 & is !re=e1 except = a process ( reason+1 inconceivable to him4 ,illu/ration3 ,fig4 #cd4 ,rotation w !re=e be to him rotation ab\t a po9t4 ,rotation ab\t a l9e w be inconceivable to him4 ,! result ( rotation ab\t a l9e he c appreh5d4 ,he c see ! fir/ & la/ positions occupi$ 9 a half-revolution ab\t ! l9e ,,ac4 ,! result ( su* a half revolution is to place ! squ>e ,,abcd on ! left h& in/ead ( on ! "r h& ( ! l9e ,,ac4 ,x w\ld correspond to a pull+ ( ! :ole body ,,abcd ?r\gh ! l9e ,,ac1 or to ! production ( a solid body :i* 0 ! exact reflection ( x 9 ! l9e ,,ac4 ,x w\ld be z if ! squ>e ,,abcd turn$ into its image1 ! l9e ,,ab act+ z a mirror4 ,su* a rev]sal ( ! positions ( ! p>ts ( ! squ>e w\ld be impossible 9 8 space4 ,! occurr5ce ( x w\ld be a pro( ( ! exi/5ce ( a high] dim5sionality4 ,let him n[1 adopt+ ! conception ( a ?ree-dim5sional body z a s]ies ( sections ly+1 ea* remov$ a little f>!r ?an ! prec$+ "o1 9 direction at "r angles to 8 plane1 reg>d a cube1 fig4 #cf1 z a s]ies ( sections1 ea* l ! squ>e :i* =ms its base1 all rigidly connect$ toge!r4 ,illu/ration3 ,fig4 #ce4 ,if n[ he turns ! squ>e ab\t ! po9t ,a 9 ! plane ( xy1 ea* p>allel section turns ) ! squ>e he moves4 ,9 ea* ( ! sections !re is a po9t at re/1 t v]tically ov] ,a4 ,h5ce he w\ld conclude t 9 ! turn+ ( a ?ree-dim5sional body !re is "o l9e :i* is at re/4 ,t is a ?ree-dim5sional turn+ 9 a turn+ ab\t a l9e4 99 99 99 99 99 ,9 a simil> way let u reg>d ourselves z limit$ to a ?ree-dim5sional _w by a physical condition4 ,let u imag9e t !re is a direction at "r angles to e direction 9 :i* we c move1 & t we are prev5t$ f pass+ 9 ?is direction by a va/ solid1 t aga9/ :i* 9 e movem5t we make we slip z ! plane be+ slips aga9/ 8 plane %eet4 ,we c !n consid] a f\r-dim5sional body z consi/+ ( a s]ies ( sections1 ea* p>allel to our space1 & ea* a little f>!r (f ?an ! prec$+ on ! unkn[n dim5sion4 ,take ! simple/ f\r-dim5sional body--"o :i* beg9s z a cube1 fig4 #cf1 9 our space1 & consi/s ( sections1 ea* a cube l fig4 #cf1 ly+ away f our space4 ,if we turn ! cube :i* is its base 9 our space ab\t a l9e1 if1 e4g41 9 fig4 #cf we turn ! cube ab\t ! l9e ,,ab1 n only x b ea* ( ! p>allel cubes moves ab\t a l9e4 ,! cube we see moves ab\t ! l9e ,,ab1 ! cube beyond x ab\t a l9e p>allel to ,,ab & s on4 ,h5ce ! :ole f\r-dim5sional body moves ab\t a plane1 = ! assemblage ( ^! l9es is our way ( ?9k+ ab\t ! plane :i*1 />t+ f ! l9e ,,ab 9 our space1 runs (f 9 ! unkn[n direction4 ,illu/ration3 ,fig4 #cf4 ,9 ?is case all t we see ( ! plane ab\t :i* ! turn+ takes place is ! l9e ,,ab4 ,b x is obvi\s t ! axis plane may lie 9 our space4 ,a po9t ne> ! plane det]m9es ) x a ?ree-dim5sional space4 ,:5 x beg9s to rotate r\nd ! plane x does n move any:]e 9 ?is ?ree-dim5sional space1 b moves out ( x4 ,a po9t c no m rotate r\nd a plane 9 ?ree-dim5sional space ?an a po9t c move r\nd a l9e 9 two-dim5sional space4 ,we w n[ apply ! second ( ! modes ( repres5tation to ?is case ( turn+ ab\t a plane1 build+ up our analogy /ep by /ep f ! turn+ 9 a plane ab\t a po9t & t 9 space ab\t a l9e1 & s on4 ,9 ord] to r$uce our consid]ations to ^? ( ! greate/ simplicity possible1 let u realise h[ ! plane be+ w\ld ?9k ( ! motion by :i* a squ>e is turn$ r\nd a l9e4 ,let1 fig4 #cd1 ,,abcd be a squ>e on 8 plane1 & repres5t ! two dim5sions ( 8 space by ! axes ,ax ,ay4 ,n[ ! motion by :i* ! squ>e is turn$ ov] ab\t ! l9e ,,ac involves ! ?ird dim5sion4 ,he _c repres5t ! motion ( ! :ole squ>e 9 its turn+1 b he c repres5t ! motions ( p>ts ( x4 ,let ! ?ird axis p]p5dicul> to ! plane ( ! pap] be call$ ! axis ( z4 ,( ! ?ree axes x1 y1 z1 ! plane be+ c repres5t any two 9 8 space4 ,let him !n draw1 9 fig4 #ce1 two axes1 x & z4 ,"h he has 9 8 plane a repres5tation ( :at exi/s 9 ! plane :i* goes (f p]p5dicul>ly to 8 space4 ,9 ?is repres5tation ! squ>e w\ld n be %[n1 = 9 ! plane ( xz simply ! l9e ,,ab ( ! squ>e is conta9$4 ,! plane be+ !n w\ld h be=e him1 9 fig4 #ce1 ! repres5tation ( "o l9e ,,ab ( 8 squ>e & two axes1 x & z1 at "r angles4 ,n[ x w\ld be obvi\s to him t1 by a turn+ su* z he kn[s1 by a rotation ab\t a po9t1 ! l9e ,,ab c turn r\nd ,a1 & occupy+ all ! int]m$iate positions1 su* z ,,ab#a1 come aft] half a revolution to lie z ,ax produc$ ?r\gh ,a4 ,aga91 j z he c repres5t ! v]tical plane ?r\gh ,,ab1 s he c repres5t ! v]tical plane ?r\gh ,a,b1 fig4 #cd1 & 9 a l mann] c see t ! l9e ,a,b c turn ab\t ! po9t ,a till x lies 9 ! opposite direction f t :i* x ran 9 at fir/4 ,n[ ^! two turn+s are n inconsi/5t4 ,9 8 plane1 if ,,ab turn$ ab\t ,a1 & ,a,b ab\t ,a1 ! consi/5cy ( ! squ>e w\ld be de/roy$1 x w\ld be an impossible motion = a rigid body to p]=m4 ,b 9 ! turn+ :i* he /udies portion by portion !re is no?+ inconsi/5t4 ,ea* l9e 9 ! squ>e c turn 9 ?is way1 h5ce he w\ld realise ! turn+ ( ! :ole squ>e z ! sum ( a numb] ( turn+s ( isolat$ p>ts4 ,su* turn+s1 if !y took place 9 8 plane1 w\ld be inconsi/5t1 b by virtue ( a ?ird dim5sion !y are consi/5t1 & ! result ( !m all is t ! squ>e turns ab\t ! l9e ,,ac & lies 9 a position 9 :i* x is ! mirror image ( :at x 0 9 its fir/ position4 ,?us he c realise a turn+ ab\t a l9e by rel9qui%+ "o ( 8 axes1 & repres5t+ 8 body "p by "p4 ,let u apply ?is me?od to ! turn+ ( a cube s z to become ! mirror image ( itself4 ,9 our space we c con/ruct ?ree indep5d5t axes1 x1 y1 z1 %[n 9 fig4 #cf4 ,suppose t !re is a f\r? axis1 w1 at "r angles to ea* & e "o ( !m4 ,we _c1 keep+ all ?ree axes1 x1 y1 z1 repres5t w 9 our space2 b if we rel9qui% "o ( our ?ree axes we c let ! f\r? axis take its place1 & we c repres5t :at lies 9 ! space1 det]m9$ by ! two axes we reta9 & ! f\r? axis4 ,illu/ration3 ,fig4 #cg4 ,let u suppose t we let ! y axis drop1 & t we repres5t ! w axis z occupy+ its direction4 ,we h 9 fig4 #cg a draw+ ( :at we %\ld !n see ( ! cube4 ,! squ>e ,,abcd1 rema9s un*ang$1 = t is 9 ! plane ( xz1 & we /ill h t plane4 ,b f ?is plane ! cube /ret*es out 9 ! direction ( ! y axis4 ,n[ ! y axis is gone1 & s we h no m ( ! cube ?an ! face ,,abcd4 ,consid]+ n[ ?is face ,,abcd1 we see t x is free to turn ab\t ! l9e ,,ab4 ,x c rotate 9 ! x to w direction ab\t ?is l9e4 ,9 fig4 #ch x is %[n on its way1 & x c evid5tly cont9ue ?is rotation till x lies on ! o!r side ( ! z axis 9 ! plane ( xz4 ,we c also take a section p>allel to ! face ,,abcd1 & !n lett+ drop all ( our space except ! plane ( t section1 introduce ! w axis1 runn+ 9 ! old y direction4 ,?is section c be repres5t$ by ! same draw+1 fig4 #ch1 & we see t x c rotate ab\t ! l9e on its left until x sw+s half way r\nd & runs 9 ! opposite direction to t :i* x ran 9 be=e4 ,^! turn+s ( ! diff]5t sections are n inconsi/5t1 & tak5 all toge!r !y w br+ ! cube f ! position %[n 9 fig4 #cf to t %[n 9 fig4 #da4 ,illu/ration3 ,fig4 #ch4 ,s9ce we h ?ree axes at our disposal 9 our space1 we are n oblig$ to repres5t ! w axis by any p>ticul> "o4 ,we may let any axis we l disappe>1 & let ! f\r? axis take its place4 ,illu/ration3 ,fig4 #ci4 ,illu/ration3 ,fig4 #dj4 ,illu/ration3 ,fig4 #da4 ,9 fig4 #cf suppose ! z axis to g4 ,we h !n simply ! plane ( xy & ! squ>e base ( ! cube ,,aceg1 fig4 #ci1 is all t c\ld be se5 ( x4 ,let n[ ! w axis take ! place ( ! z axis & we h1 9 fig4 #ci aga91 a repres5tation ( ! space ( xyw1 9 :i* all t exi/s ( ! cube is its squ>e base4 ,n[1 by a turn+ ( x to w1 ?is base c rotate ar\nd ! l9e ,,ae1 x is %[n on its way 9 fig4 #dj1 & f9ally x w1 aft] half a revolution1 lie on ! o!r side ( ! y axis4 ,9 a simil> way we may rotate sections p>allel to ! base ( ! xw rotation1 & ea* ( !m comes to run 9 ! opposite direction f t :i* !y occupi$ at fir/4 ,?us aga9 ! cube comes f ! position ( fig4 #cf4 to t ( fig4 #da4 ,9 ?is x to w turn+1 we see t x takes place by ! rotations ( sections p>allel to ! front face ab\t l9es p>allel to ,,ab1 or else we may consid] x z consi/+ ( ! rotation ( sections p>allel to ! base ab\t l9es p>allel to ,,ae4 ,x is a rotation ( ! :ole cube ab\t ! plane ,,abef4 ,two sep>ate sections c\ld n rotate ab\t two sep>ate l9es 9 our space )\t conflict+1 b _! motion is consi/5t :5 we consid] ano!r dim5sion4 ,j1 !n1 z a plane be+ c ?9k ( rotation ab\t a l9e z a rotation ab\t a numb] ( po9ts1 ^! rotations n int]f]+ z !y w\ld if !y took place 9 8 two-dim5sional space1 s we c ?9k ( a rotation ab\t a plane z ! rotation ( a numb] ( sections ( a body ab\t a numb] ( l9es 9 a plane1 ^! rotations n be+ inconsi/5t 9 a f\r-dim5sional space z !y are 9 ?ree-dim5sional space4 ,we are n limit$ to any p>ticul> direction = ! l9es 9 ! plane ab\t :i* we suppose ! rotation ( ! p>ticul> sections to take place4 ,let u draw ! section ( ! cube1 fig4 #cf1 ?r\gh ,a1 ,f1 ,c1 ,h1 =m+ a slop+ plane4 ,n[ s9ce ! f\r? dim5sion is at "r angles to e l9e 9 our space x is at "r angles to ?is section also4 ,we c repres5t our space by draw+ an axis at "r angles to ! plane ,,aceg1 our space is !n det]m9$ by ! plane ,,aceg1 & ! p]p5dicul> axis4 ,if we let ?is axis drop & suppose ! f\r? axis1 w1 to take its place1 we h a repres5tation ( ! space :i* runs (f 9 ! f\r? dim5sion f ! plane ,,aceg4 ,9 ?is space we %all see simply ! section ,,aceg ( ! cube1 & no?+ else1 = "o cube does n ext5d to any di/ance 9 ! f\r? dim5sion4 ,if1 keep+ ?is plane1 we br+ 9 ! f\r? dim5sion1 we %all h a space 9 :i* simply ?is section ( ! cube exi/s & no?+ else4 ,! section c turn ab\t ! l9e ,,af1 & p>allel sections c turn ab\t p>allel l9es4 ,?us 9 consid]+ ! rotation ab\t a plane we c draw any l9es we l & consid] ! rotation z tak+ place 9 sections ab\t !m4 ,illu/ration3 ,fig4 #db4 ,to br+ out ?is po9t m cle>ly let u take two p>allel l9es1 ,a & ,b1 9 ! space ( xyz1 & let ,,cd & ,,ef be two rods runn+ above & bel[ ! plane ( xy1 f ^! l9es4 ,if we turn ^! rods 9 our space ab\t ! l9es ,a & ,b1 z ! upp] end ( "o1 ,f1 is go+ d[n1 ! l[] end ( ! o!r1 ,c1 w be com+ up4 ,!y w meet & conflict4 ,b x is q possible = ^! two rods ea* ( !m to turn ab\t ! two l9es )\t alt]+ _! relative di/ances4 ,to see ?is suppose ! y axis to g1 & let ! w axis take its place4 ,we %all see ! l9es ,a & ,b no long]1 = !y run 9 ! y direction f ! po9ts ,g & ,h4 ,illu/ration3 ,fig4 #dc4 ,fig4 #dc is a picture ( ! two rods se5 9 ! space ( xzw4 ,if !y rotate 9 ! direction %[n by ! arr[s--9 ! z to w direction--!y move p>allel to "o ano!r1 keep+ _! relative di/ances4 ,ea* w rotate ab\t its own l9e1 b _! rotation w n be inconsi/5t ) _! =m+ "p ( a rigid body4 ,n[ we h b to suppose a c5tral plane ) rods cross+ x at e po9t1 l ,,cd & ,,ef cross ! plane ( xy1 to h an image ( a mass ( matt] ext5d+ equal di/ances on ea* side ( a diametral plane4 ,z two ( ^! rods c rotate r\nd1 s c all1 & ! :ole mass ( matt] c rotate r\nd its diametral plane4 ,?is rotation r\nd a plane corresponds1 9 f\r dim5sions1 to ! rotation r\nd an axis 9 ?ree dim5sions4 ,rotation ( a body r\nd a plane is ! analogue ( rotation ( a rod r\nd an axis4 ,9 a plane we h rotation r\nd a po9t1 9 ?ree-space rotation r\nd an axis l9e1 9 f\r-space rotation r\nd an axis plane4 ,! f\r-dim5sional be+'s %aft by :i* he transmits p[] is a disk rotat+ r\nd its c5tral plane--! :ole cont\r corresponds to ! ends ( an axis ( rotation 9 our space4 ,he c imp>t ! rotation at any po9t & take x (f at any o!r po9t on ! cont\r1 j z rotation r\nd a l9e c 9 ?ree-space be imp>t$ at "o end ( a rod & tak5 (f at ! o!r end4 ,a f\r-dim5sional :eel c easily be describ$ f ! analogy ( ! repres5tation :i* a plane be+ w\ld =m = himself ( "o ( our :eels4 ,suppose a :eel to move transv]se to a plane1 s t ! :ole disk1 :i* ,i w consid] to be solid & )\t spokes1 came at ! same "t into contact ) ! plane4 ,x w\ld appe> z a circul> portion ( plane matt] completely enclos+ ano!r & small] portion--! axle4 ,?is appe>ance w\ld la/1 suppos+ ! motion ( ! :eel to cont9ue until x _h trav]s$ ! plane by ! ext5t ( its ?ickness1 :5 !re w\ld rema9 9 ! plane only ! small disk :i* is ! section ( ! axle4 ,!re w\ld be no means obvi\s 9 ! plane at fir/ by :i* ! axle c\ld be rea*$1 except by go+ ?r\gh ! sub/ance ( ! :eel4 ,b ! possibility ( rea*+ x )\t de/roy+ ! sub/ance ( ! :eel w\ld be %[n by ! cont9u$ exi/5ce ( ! axle section aft] t ( ! :eel _h disappe>$4 ,9 a simil> way a f\r-dim5sional :eel mov+ transv]se to our space w\ld appe> fir/ z a solid sph]e1 completely surr\nd+ a small] solid sph]e4 ,! out] sph]e w\ld repres5t ! :eel1 & w\ld la/ until ! :eel has trav]s$ our space by a di/ance equal to its ?ickness4 ,!n ! small sph]e alone w\ld rema91 repres5t+ ! section ( ! axle4 ,! l>ge sph]e c\ld move r\nd ! small "o q freely4 ,any l9e 9 space c\ld be tak5 z an axis1 & r\nd ?is l9e ! out] sph]e c\ld rotate1 :ile ! inn] sph]e rema9$ /ill4 ,b 9 all ^! directions ( revolution !re w\ld be 9 reality "o l9e :i* rema9$ unalt]$1 t is ! l9e :i* /ret*es away 9 ! f\r? direction1 =m+ ! axis ( ! axle4 ,! f\r-dim5sional :eel c rotate 9 any numb] ( planes1 b all ^! planes are su* t !re is a l9e at "r angles to !m all unaffect$ by rotation 9 !m4 ,an objection is sometimes exp]i5c$ z to ?is mode ( reason+ f a plane _w to a high] dim5sionality4 ,h[ artificial1 x is argu$1 ?is conception ( a plane _w is4 ,if any real exi/5ce conf9$ to a sup]ficies c\ld be %[n to exi/1 !re w\ld be an argum5t = "o relative to :i* our ?ree-dim5sional exi/5ce is sup]ficial4 ,b1 bo? on ! "o side & ! o!r ( ! space we are famili> )1 spaces ei!r ) less or m ?an ?ree dim5sions are m]ely arbitr>y conceptions4 ,9 reply to ?is ,i w\ld rem>k t a plane be+ hav+ "o less dim5sion ?an our ?ree w\ld h "o-?ird ( our possibilities ( motion1 :ile we h only "o-f\r? less ?an ^? ( ! high] space4 ,x may v well be t !re may be a c]ta9 am\nt ( fre$om ( motion :i* is dem&$ z a condition ( an organis$ exi/5ce1 & t no mat]ial exi/5ce is possible ) a m limit$ dim5sionality ?an ours4 ,?is is well se5 if we try to con/ruct ! me*anics ( a two-dim5sional _w4 ,no tube c\ld exi/1 = unless jo9$ toge!r completely at "o end two p>allel l9es w\ld be completely sep>ate4 ,! possibility ( an organic /ructure1 subject to conditions su* z ?is1 is highly problematical2 yet1 possibly 9 ! convolutions ( ! bra9 !re may be a mode ( exi/5ce to be describ$ z two-dim5sional4 ,we h b to suppose ! increase 9 surface & ! dim9ution 9 mass c>ri$ on to a c]ta9 ext5t to f9d a region :i*1 ?\gh )\t mobility ( ! con/itu5ts1 w\ld h to be describ$ z two-dim5sional4 ,b1 h[ev] artificial ! conception ( a plane be+ may be1 x is none ! less to be us$ 9 pass+ to ! conception ( a great] dim5sionality ?an ours1 & h5ce ! validity ( ! fir/ "p ( ?is objection altoge!r disappe>s directly we f9d evid5ce = su* a /ate ( be+4 ,! second "p ( ! objection has m weight4 ,h[ is x possible to conceive t 9 a f\r-dim5sional space any creatures %\ld be conf9$ to a ?ree-dim5sional exi/5ce8 ,9 reply ,i w\ld say t we "k z a matt] ( fact t life is ess5tially a ph5om5on ( surface4 ,! amplitude ( ! movem5ts :i* we c make is mu* great] along ! surface ( ! e>? ?an x is up or d[n4 ,n[ we h b to conceive ! ext5t ( a solid surface increas$1 :ile ! motions possible tranv]se to x are dim9i%$ 9 ! same proportion1 to obta9 ! image ( a ?ree-dim5sional _w 9 f\r-dim5sional space4 ,& z our habitat is ! meet+ ( air & e>? on ! _w1 s we mu/ ?9k ( ! meet+ place ( two z af=d+ ! condition = our univ]se4 ,! meet+ ( :at two8 ,:at c t va/ness be 9 ! high] space :i* /ret*es 9 su* a p]fect level t our a/ronomical obs]vations fail to detect ! slighte/ curvature8 ,! p]fection ( ! level sugge/s a liquid--a lake amid/ :at va/ sc5]y6--:]eon ! matt] ( ! univ]se floats speck-l4 ,b ?is aspect ( ! problem is l :at are call$ 9 ma!matics b\nd>y conditions4 ,we c trace out all ! consequ5ces ( f\r-dim5sional movem5ts d[n to _! la/ detail4 ,!n1 kn[+ ! mode ( action :i* w\ld be *>act]i/ic ( ! m9ute/ p>ticles1 if !y 7 free1 we c draw conclusions f :at !y actually d ( :at ! con/ra9t on !m is4 ,( ! two ?+s1 ! mat]ial conditions & ! motion1 "o is kn[n1 & ! o!r c be inf]r$4 ,if ! place ( ?is univ]se is a meet+ ( two1 !re w\ld be a "o-sid5ess to space4 ,if x lies s t :at /ret*es away 9 "o direction 9 ! unkn[n is unlike :at /ret*es away 9 ! o!r1 !n1 z f> z ! movem5ts :i* p>ticipate 9 t dim5sion are conc]n$1 !re w\ld be a diff]5ce z to :i* way ! motion took place4 ,?is w\ld be %[n 9 ! dissimil>ity ( ph5om5a1 :i*1 s f> z all ?ree-space movem5ts are conc]n$1 7 p]fectly symmetrical4 ,to take an in/ance1 m]ely1 = ! sake ( precis+ our ideas1 n = any inh]5t probability 9 x2 if x c\ld be %[n t ! electric curr5t 9 ! positive direction 7 exactly l ! electric curr5t 9 ! negative direction1 except = a rev]sal ( ! compon5ts ( ! motion 9 ?ree-dim5sional space1 !n ! dissimil>ity ( ! dis*>ge f ! positive & negative poles w\ld be an indication ( a "o-sid5ess to our space4 ,! only cause ( diff]5ce 9 ! two dis*>ges w\ld be due to a compon5t 9 ! f\r? dim5sion1 :i* direct$ 9 "o direction transv]se to our space1 met ) a diff]5t resi/ance to t :i* x met :5 direct$ 9 ! opposite direction4 ,,*apt] ,,vii ,,! ,,evid5ces ,,= ,a ,,f\r? ,,dim5sion ,! me?od necess>ily to be employ$ 9 ! se>* = ! evid5ces ( a f\r? dim5sion1 consi/s prim>ily 9 ! =mation ( ! conceptions ( f\r-dim5sional %apes & motions4 ,:5 we are 9 possession ( ^! x is possible to call 9 ! aid ( obs]vation1 )\t !m we may h be5 all our lives 9 ! famili> pres5ce ( a f\r-dim5sional ph5om5on )\t ev] recognis+ its nature4 ,to take "o ( ! conceptions we h already =m$1 ! turn+ ( a real ?+ into its mirror image w\ld be an occurr5ce :i* x w\ld be h>d to expla91 except on ! assumption ( a f\r? dim5sion4 ,we "k ( no su* turn+4 ,b !re exi/ a multitude ( =ms :i* %[ a c]ta9 relation to a plane1 a relation ( symmetry1 :i* indicates m ?an an accid5tal juxtaposition ( p>ts4 ,9 organic life ! univ]sal type is ( "r- & left-h&$ symmetry1 !re is a plane on ea* side ( :i* ! p>ts correspond4 ,n[ we h se5 t 9 f\r dim5sions a plane takes ! place ( a l9e 9 ?ree dim5sions4 ,9 our space1 rotation ab\t an axis is ! type ( rotation1 & ! orig9 ( bodies symmetrical ab\t a l9e z ! e>? is symmetrical ab\t an axis c easily be expla9$4 ,b :]e !re is symmetry ab\t a plane no simple physical motion1 su* z we are accu/om$ to1 suffices to expla9 x4 ,9 our space a symmetrical object mu/ be built up by equal additions on ea* side ( a c5tral plane4 ,su* additions ab\t su* a plane are z little likely z any o!r increm5ts4 ,! probability aga9/ ! exi/5ce ( symmetrical =m 9 inorganic nature is ov]:elm+ 9 our space1 & 9 organic =ms !y w\ld be z difficult ( production z any o!r v>iety ( configuration4 ,to illu/rate ?is po9t we may take ! *ild's amusem5t ( mak+ f dots ( ink on a piece ( pap] a lifelike repres5tation ( an insect by simply fold+ ! pap] ov]4 ,! dots spread out on a symmetrical l9e1 & give ! impression ( a segm5t$ =m ) ant5n & legs4 ,n[ see+ a numb] ( su* figures we %\ld naturally inf] a fold+ ov]4 ,c1 !n1 a fold+ ov] 9 f\r-dim5sional space acc\nt = ! symmetry ( organic =ms8 ,! fold+ _c ( c\rse be ( ! bodies we see1 b x may be ( ^? m9ute con/itu5ts1 ! ultimate elem5ts ( liv+ matt] :i*1 turn$ 9 "o way or ! o!r1 become "r- or left-h&$1 & s produce a correspond+ /ructure4 ,!re is some?+ 9 life n includ$ 9 our conceptions ( me*anical movem5t4 ,is ?is some?+ a f\r-dim5sional movem5t8 ,if we look at x f ! broade/ po9t ( view1 !re is some?+ /rik+ 9 ! fact t :]e life comes 9 !re arises an entirely diff]5t set ( ph5om5a to ^? ( ! inorganic _w4 ,! int]e/ & values ( life z we "k x 9 ourselves1 z we "k x exi/+ ar\nd u 9 subord9ate =ms1 is entirely & completely diff]5t to any?+ :i* inorganic nature %[s4 ,& 9 liv+ be+s we h a k9d ( =m1 a disposition ( matt] :i* is entirely diff]5t f t %[n 9 inorganic matt]4 ,"r- & left-h&$ symmetry does n occur 9 ! configurations ( dead matt]4 ,we h in/ances ( symmetry ab\t an axis1 b n ab\t a plane4 ,x c be argu$ t ! occurr5ce ( symmetry 9 two dim5sions involves ! exi/5ce ( a ?ree-dim5sional process1 z :5 a /one falls into wat] & makes r+s ( ripples1 or z :5 a mass ( s(t mat]ial rotates ab\t an axis4 ,x c be argu$ t symmetry 9 any numb] ( dim5sions is ! evid5ce ( an action 9 a high] dim5sionality4 ,?us consid]+ liv+ be+s1 !re is an evid5ce bo? 9 _! /ructure1 & _! diff]5t mode ( activity1 ( a some?+ com+ 9 f )\t into ! inorganic _w4 ,& ! objections :i* w readily occur1 su* z ^? d]iv$ f ! =ms ( tw9 cry/als & ! !oretical /ructure ( *emical molecules1 d n invalidate ! argum5t2 = 9 ^! =ms too ! presumable seat ( ! activity produc+ !m lies 9 t v m9ute region 9 :i* we necess>ily place ! seat ( a f\r-dim5sional mobility4 ,9 ano!r respect also ! exi/5ce ( symmetrical =ms is notewor?y4 ,x is puzzl+ to conceive h[ two %apes exactly equal c exi/ :i* are n sup]posible4 ,su* a pair ( symmetrical figures z ! two h&s1 "r & left1 %[ ei!r a limitation 9 our p[] ( movem5t1 by :i* we _c sup]pose ! "o on ! o!r1 or a def9ite influ5ce & compulsion ( space on matt]1 inflict+ limitations :i* are additional to ^? ( ! proportions ( ! p>ts4 ,we w1 h[ev]1 put aside ! argum5ts to be drawn f ! consid]ation ( symmetry z inconclusive1 reta9+ "o valuable indication :i* !y af=d4 ,if x is 9 virtue ( a f\r-dim5sional motion t symmetry exi/s1 x is only 9 ! v m9ute p>ticles ( bodies t t motion is to be f\nd1 = !re is no su* ?+ z a b5d+ ov] 9 f\r dim5sions ( any object ( a size :i* we c obs]ve4 ,! region ( ! extremely m9ute is ! "o1 !n1 :i* we %all h to inve/igate4 ,we mu/ look = "s ph5om5on :i*1 occasion+ movem5ts ( ! k9d we "k1 /ill is itself inexplicable z any =m ( motion :i* we "k4 ,n[ 9 ! !ories ( ! actions ( ! m9ute p>ticles ( bodies on "o ano!r1 & 9 ! motions ( ! e!r1 ma!maticians h tacitly assum$ t ! me*anical pr9ciples are ! same z ^? :i* prevail 9 ! case ( bodies :i* c be obs]v$1 x has be5 assum$ )\t pro( t ! conception ( motion be+ ?ree-dim5sional1 holds beyond ! region f obs]vations 9 :i* x 0 =m$4 ,h5ce x is n f any ph5om5on expla9$ by ma!matics t we c d]ive a pro( ( f\r dim5sions4 ,e ph5om5on t has be5 expla9$ is expla9$ z ?ree-dim5sional4 ,&1 moreov]1 s9ce 9 ! region ( ! v m9ute we d n f9d rigid bodies act+ on ea* o!r at a di/ance1 b ela/ic sub/ances & cont9u\s fluids su* z e!r1 we %all h a d\ble task4 ,we mu/ =m ! conceptions ( ! possible movem5ts ( ela/ic & liquid f\r-dim5sional matt]1 be=e we c beg9 to obs]ve4 ,let u1 !re=e1 take ! f\r-dim5sional rotation ab\t a plane1 & enquire :at x becomes 9 ! case ( ext5sible fluid sub/ances4 ,if f\r-dim5sional movem5ts exi/1 ?is k9d ( rotation mu/ exi/1 & ! f9] portions ( matt] mu/ exhibit x4 ,consid] = a mom5t a rod ( flexible & ext5sible mat]ial4 ,x c turn ab\t an axis1 ev5 if n /raight2 a r+ ( india rubb] c turn inside out4 ,:at w\ld ?is be 9 ! case ( f\r dim5sions8 ,let u consid] a sph]e ( our ?ree-dim5sional matt] hav+ a def9ite ?ickness4 ,to repres5t ?is ?ickness let u suppose t f e po9t ( ! sph]e 9 fig4 #dd rods project bo? ways1 9 & out1 l ,d & ,f4 ,we c only see ! ext]nal portion1 because ! int]nal p>ts are hidd5 by ! sph]e4 ,illu/ration3 ,fig4 #dd4 ,axis ( x runn+ t[>ds ! obs]v]4 ,9 ?is sph]e ! axis ( x is suppos$ to come t[>ds ! obs]v]1 ! axis ( z to run up1 ! axis ( y to g to ! "r4 ,illu/ration3 ,fig4 #de4 ,n[ take ! section det]m9$ by ! zy plane4 ,?is w be a circle z %[n 9 fig4 #de4 ,if we let drop ! x axis1 ?is circle is all we h ( ! sph]e4 ,lett+ ! w axis n[ run 9 ! place ( ! old x axis we h ! space yzw1 & 9 ?is space all t we h ( ! sph]e is ! circle4 ,fig4 #de !n repres5ts all t !re is ( ! sph]e 9 ! space ( yzw4 ,9 ?is space x is evid5t t ! rods ,,cd & ,,ef c turn r\nd ! circumf]5ce z an axis4 ,if ! matt] ( ! sph]ical %ell is suffici5tly ext5sible to all[ ! p>ticles ,c & ,e to become z widely sep>at$ z !y w\ld be 9 ! positions ,d & ,f1 !n ! /rip ( matt] repres5t$ by ,,cd & ,,ef & a multitude ( rods l !m c turn r\nd ! circul> circumf]5ce4 ,?us ?is p>ticul> section ( ! sph]e c turn inside out1 & :at holds = any "o section holds = all4 ,h5ce 9 f\r dim5sions ! :ole sph]e c1 if ext5sible turn inside out4 ,moreov]1 any "p ( x--a b[l-%ap$ portion1 = in/ance--c turn inside out1 & s on r\nd & r\nd4 ,?is is really no m ?an we _h be=e 9 ! rotation ab\t a plane1 except t we see t ! plane c1 9 ! case ( ext5sible matt]1 be curv$1 & /ill play ! "p ( an axis4 ,if we suppose ! sph]ical %ell to be ( f\r-dim5sional matt]1 our repres5tation w be a little diff]5t4 ,let u suppose !re to be a small ?ickness to ! matt] 9 ! f\r? dim5sion4 ,?is w\ld make no diff]5ce 9 fig4 #dd1 = t m]ely %[s ! view 9 ! xyz space4 ,b :5 ! x axis is let drop1 & ! w axis comes 91 !n ! rods ,,cd & ,,ef :i* repres5t ! matt] ( ! %ell1 w h a c]ta9 ?ickness p]p5dicul> to ! plane ( ! pap] on :i* !y are drawn4 ,if !y h a ?ickness 9 ! f\r? dim5sion !y w %[ ?is ?ickness :5 look$ at f ! direction ( ! w axis4 ,suppos+ ^! rods1 !n1 to be small slabs /rung on ! circumf]5ce ( ! circle 9 fig4 #de1 we see t !re w n be 9 ?is case ei!r any ob/acle to _! turn+ r\nd ! circumf]5ce4 ,we c h a %ell ( ext5sible mat]ial or ( fluid mat]ial turn+ inside out 9 f\r dim5sions4 ,& we mu/ rememb] t 9 f\r dim5sions !re is no su* ?+ z rotation r\nd an axis4 ,if we want to inve/igate ! motion ( fluids 9 f\r dim5sions we mu/ take a movem5t ab\t an axis 9 our space1 & f9d ! correspond+ movem5t ab\t a plane 9 f\r space4 ,n[1 ( all ! movem5ts :i* take place 9 fluids1 ! mo/ important f a physical po9t ( view is vortex motion4 ,a vortex is a :irl or eddy--x is %[n 9 ! gyrat+ wrea?s ( du/ se5 on a summ] day2 x is exhibit$ on a l>g] scale 9 ! de/ructive m>* ( a cyclone4 ,a :eel :irl+ r\nd w ?r[ (f ! wat] on x4 ,b :5 ?is circl+ motion takes place 9 a liquid itself x is /rangely p]si/5t4 ,!re is1 ( c\rse1 a c]ta9 cohesion betwe5 ! p>ticles ( wat] by :i* !y mutually imp$e _! motions4 ,b 9 a liquid devoid ( friction1 su* t e p>ticle is free f lat]al cohesion on its pa? ( motion1 x c be %[n t a vortex or eddy sep>ates f ! mass ( ! fluid a c]ta9 portion1 :i* always rema9 9 t vortex4 ,! %ape ( ! vortex may alt]1 b x always consi/s ( ! same p>ticles ( ! fluid4 ,n[1 a v rem>kable fact ab\t su* a vortex is t ! ends ( ! vortex _c rema9 susp5d$ & isolat$ 9 ! fluid4 ,!y mu/ always run to ! b\nd>y ( ! fluid4 ,an eddy 9 wat] t rema9s half way d[n )\t com+ to ! top is impossible4 ,! ends ( a vortex mu/ rea* ! b\nd>y ( a fluid--! b\nd>y may be ext]nal or int]nal--a vortex may exi/ betwe5 two objects 9 ! fluid1 t]m9at+ "o end on ea* object1 ! objects be+ int]nal b\nd>ies ( ! fluid4 ,aga91 a vortex may h its ends l9k$ toge!r1 s t x =ms a r+4 ,circul> vortex r+s ( ?is description are (t5 se5 9 puffs ( smoke1 & t ! smoke travels on 9 ! r+ is a pro( t ! vortex always consi/s ( ! same p>ticles ( air4 ,let u n[ enquire :at a vortex w\ld be 9 a f\r-dim5sional fluid4 ,we mu/ replace ! l9e axis by a plane axis4 ,we %\ld h !re=e a portion ( fluid rotat+ r\nd a plane4 ,we h se5 t ! cont\r ( ?is plane corresponds ) ! ends ( ! axis l9e4 ,h5ce su* a f\r-dim5sional vortex mu/ h its rim on a b\nd>y ( ! fluid4 ,!re w\ld be a region ( vorticity ) a cont\r4 ,if su* a rotation 7 />t$ at "o "p ( a circul> b\nd>y1 its edges w\ld run r\nd ! b\nd>y 9 bo? directions till ! :ole int]ior region 0 fill$ ) ! vortex %eet4 ,a vortex 9 a ?ree-dim5sional liquid may consi/ ( a numb] ( vortex filam5ts ly+ toge!r produc+ a tube1 or rod ( vorticity4 ,9 ! same way we c h 9 f\r dim5sions a numb] ( vortex %eets alongside ea* o!r1 ea* ( :i* c be ?\ght ( z a b[l-%ap$ portion ( a sph]ical %ell turn+ inside out4 ,! rotation takes place at any po9t n 9 ! space occupi$ by ! %ell1 b f t space to ! f\r? dim5sion & r\nd back aga94 ,is !re any?+ analog\s to ?is )9 ! range ( our obs]vation8 ,an electric curr5t answ]s ?is description 9 e respect4 ,electricity does n fl[ ?r\gh a wire4 ,its effect travels bo? ways f ! />t+ po9t along ! wire4 ,! sp>k :i* %[s its pass+ midway 9 its circuit is lat] ?an t :i* occurs at po9ts ne> its />t+ po9t on ei!r side ( x4 ,moreov]1 x is kn[n t ! action ( ! curr5t is n 9 ! wire4 ,x is 9 ! region enclos$ by ! wire1 ?is is ! field ( =ce1 ! locus ( ! exhibition ( ! effects ( ! curr5t4 ,& ! necessity ( a conduct+ circuit = a curr5t is exactly t :i* we %\ld expect if x 7 a f\r-dim5sional vortex4 ,accord+ to ,maxwell e curr5t =ms a clos$ circuit1 & ?is1 f ! f\r-dim5sional po9t ( view1 is ! same z say+ a vortex mu/ h its ends on a b\nd>y ( ! fluid4 ,?us1 on ! hypo!sis ( a f\r? dim5sion1 ! rotation ( ! fluid e!r w\ld give ! ph5om5on ( an electric curr5t4 ,we mu/ suppose ! e!r to be full ( movem5t1 = ! m we exam9e into ! conditions :i* prevail 9 ! obscurity ( ! m9ute1 ! m we f9d t an unceas+ & p]petual motion reigns4 ,?us we may say t ! conception ( ! f\r? dim5sion means t !re mu/ be a ph5om5on :i* pres5ts ! *>act]i/ics ( electricity4 ,we "k n[ t light is an electro-magnetic action1 & t s f> f be+ a special & isolat$ ph5om5on ?is electric action is univ]sal 9 ! realm ( ! m9ute4 ,h5ce1 may we n conclude t1 s f> f ! f\r? dim5sion be+ remote & f> away1 be+ a ?+ ( symbolic import1 a t]m = ! explanation ( dubi\s facts by a m obscure !ory1 x is really ! mo/ important fact )9 our k4 ,our ?ree-dim5sional _w is sup]ficial4 ,^! processes1 :i* really lie at ! basis ( all ph5om5a ( matt]1 escape our obs]vation by _! m9ut5ess1 b reveal to our intellect an amplitude ( motion surpass+ any t we c see4 ,9 su* %apes & motions !re is a realm ( ! utmo/ intellectual beauty1 & "o to :i* our symbolic me?ods apply ) a bett] grace ?an !y d to ^? ( ?ree dim5sions4 ,,*apt] ,,viii ,,! ,,use ,,( ,,f\r ,,dim5sions ,,9 ,,?\ght ,hav+ held be=e ourselves ?is outl9e ( a conjecture ( ! _w z f\r-dim5sional1 hav+ r\ghly ?r[n toge!r ^? facts ( movem5t :i* we c see apply to our actual exp]i5ce1 let u pass to ano!r bran* ( our subject4 ,! eng9e] uses draw+s1 graphical con/ructions1 9 a v>iety ( mann]s4 ,he has1 = in/ance1 diagrams :i* repres5t ! expansion ( /eam1 ! effici5cy ( 8 valves4 ,^! exi/ alongside ! actual plans ( 8 ma*9es4 ,!y are n ! pictures ( any?+ really exi/+1 b enable him to ?9k ab\t ! relations :i* exi/ 9 8 me*anisms4 ,& s1 besides %[+ u ! actual exi/5ce ( t _w :i* lies b5ea? ! "o ( visible movem5ts1 f\r-dim5sional space enables u to make ideal con/ructions :i* s]ve to repres5t ! relations ( ?+s1 & ?r[ :at w\ld o!rwise be obscure into a def9ite & sugge/ive =m4 ,f amid/ ! great v>iety ( in/ances :i* lies be=e me ,i w select two1 "o deal+ ) a subject ( slight intr9sic int]e/1 :i* h[ev] gives )9 a limit$ field a /rik+ example ( ! me?od ( draw+ conclusions & ! use ( high] space figures4#a #a ,x is sugge/ive also 9 ano!r respect1 because x %[s v cle>ly t 9 our processes ( ?\ght !re are 9 play faculties o!r ?an logical2 9 x ! orig9 ( ! idea :i* proves to be ju/ifi$ is drawn f ! consid]ation ( symmetry1 a bran* ( ! beautiful4 ,! o!r in/ance is *os5 on acc\nt ( ! be>+ x has on our fundam5tal conceptions4 ,9 x ,i try to discov] ! real mean+ ( ,kant's !ory ( exp]i5ce4 ,! inve/igation ( ! prop]ties ( numb]s is mu* facilitat$ by ! fact t relations betwe5 numb]s are !mselves able to be repres5t$ z numb]s--e4g41 #ab1 & #c are bo? numb]s1 & ! relation betwe5 !m is #d1 ano!r numb]4 ,! way is ?us op5$ = a process ( con/ructive !ory1 )\t !re be+ any necessity = a rec\rse to ano!r class ( concepts besides t :i* is giv5 9 ! ph5om5a to be /udi$4 ,! discipl9e ( numb] ?us creat$ is ( great & v>i$ applicability1 b x is n solely z quantitative t we le>n to und]/& ! ph5om5a ( nature4 ,x is n possible to expla9 ! prop]ties ( matt] by numb] simply1 b all ! activities ( matt] are en]gies 9 space4 ,!y are num]ically def9ite & also1 we may say1 direct$ly def9ite1 i4e4 def9ite 9 direction4 ,is !re1 !n1 a body ( doctr9e ab\t space :i*1 l t ( numb]1 is available 9 sci5ce8 ,x is ne$less to answ]3 ,yes2 geometry4 ,b !re is a me?od ly+ alongside ! ord9>y me?ods ( geometry1 :i* tacitly us$ & pres5t+ an analogy to ! me?od ( num]ical ?\ght des]ves to be br\ght into great] prom95ce ?an x usually occupies4 ,! relation ( numb]s is a numb]4 ,c we say 9 ! same way t ! relation ( %apes is a %ape8 ,we c4 ,to take an in/ance *os5 on acc\nt ( its ready availability4 ,let u take two "r-angl$ triangles ( a giv5 hypo!nuse1 b hav+ sides ( diff]5t l5g?s 7fig4 #df74 ,^! triangles are %apes :i* h a c]ta9 relation to ea* o!r4 ,let u exhibit _! relation z a figure4 ,illu/ration3 ,fig4 #df4 ,draw two /raight l9es at "r angles to ea* o!r1 ! "o ,,hl a horizontal level1 ! o!r ,,vl a v]tical level 7fig4 #dg74 ,by means ( ^! two co-ord9at+ l9es we c repres5t a d\ble set ( magnitudes2 "o set z di/ances to ! "r ( ! v]tical level1 ! o!r z di/ances above ! horizontal level1 a suitable unit be+ *os54 ,illu/ration3 ,fig4 #dg4 ,?us ! l9e m>k$ #g w pick out ! assemblage ( po9ts ^: di/ance f ! v]tical level is #g1 & ! l9e m>k$ #a w pick out ! po9ts ^: di/ance above ! horizontal level is #a4 ,! meet+ po9t ( ^! two l9es1 #g & #a1 w def9e a po9t :i* ) reg>d to ! "o set ( magnitudes is #g1 ) reg>d to ! o!r is #a4 ,let u take ! sides ( our triangles z ! two sets ( magnitudes 9 "q4 ,!n ! po9t #g1 #a1 w repres5t ! triangle ^: sides are #g & #a4 ,simil>ly ! po9t #e1 #e--e1 t is1 to ! "r ( ! v]tical level & #e above ! horizontal level--w repres5t ! triangle ^: sides are #e & #e 7fig4 #dh74 ,illu/ration3 ,fig4 #dh4 ,?us we h obta9$ a figure consi/+ ( ! two po9ts #g1 #a1 & #e1 #e1 repres5tative ( our two triangles4 ,b we c g fur!r1 &1 draw+ an arc ( a circle ab\t ,o1 ! meet+ po9t ( ! horizontal & v]tical levels1 :i* passes ?r\gh #g1 #a1 & #e1 #e1 ass]t t all ! triangles :i* are "r-angl$ & h a hypo!nuse ^: squ>e is #ej are repres5t$ by ! po9ts on ?is arc4 ,?us1 ea* individual ( a class be+ repres5t$ by a po9t1 ! :ole class is repres5t$ by an assemblage ( po9ts =m+ a figure4 ,accept+ ?is repres5tation we c atta* a def9ite & calculable significance to ! expression1 resemblance1 or simil>ity betwe5 two individuals ( ! class repres5t$1 ! diff]5ce be+ measur$ by ! l5g? ( ! l9e betwe5 two repres5tative po9ts4 ,x is ne$less to multiply examples1 or to %[ h[1 correspond+ to diff]5t classes ( triangles1 we obta9 diff]5t curves4 ,a repres5tation ( ?is k9d 9 :i* an object1 a ?+ 9 space1 is repres5t$ z a po9t1 & all its prop]ties are left out1 _! effect rema9+ only 9 ! relative position :i* ! repres5tative po9t be>s to ! repres5tative po9ts ( ! o!r objects1 may be call$1 aft] ! analogy ( ,sir ,william ,r4 ,hamilton's hodograph1 a 8,poiograph40 ,repres5tations ?us made h ! "* ( natural objects2 !y h a det]m9ate & def9ite "* ( _! own4 ,any lack ( complet5ess 9 !m is probably due to a failure 9 po9t ( complet5ess ( ^? obs]vations :i* =m ! gr\nd ( _! con/ruction4 ,e sy/em ( classification is a poiograph4 ,9 ,m5deleff's s*eme ( ! elem5ts1 = in/ance1 ea* elem5t is repres5t$ by a po9t1 & ! relations betwe5 ! elem5ts are repres5t$ by ! relations betwe5 ! po9ts4 ,s f> ,i h simply br\ght into prom95ce processes & consid]ations ) :i* we are all famili>4 ,b x is wor? :ile to br+ into ! full light ( our att5tion our habitual assumptions & processes4 ,x (t5 happ5s t we f9d !re are two ( !m :i* h a be>+ on ea* o!r1 :i*1 )\t ?is dragg+ into ! light1 we %\ld h all[$ to rema9 )\t mutual influ5ce4 ,!re is a fact :i* x conc]ns u to take into acc\nt 9 discuss+ ! !ory ( ! poiograph4 ,) respect to our k ( ! _w we are f> f t condition :i* ,laplace imag9$ :5 he ass]t$ t an all-kn[+ m9d c\ld det]m9e ! future condition ( e object1 if he knew ! co-ord9ates ( its p>ticles 9 space1 & _! velocity at any p>ticul> mom5t4 ,on ! contr>y1 9 ! pres5ce ( any natural object1 we h a great complexity ( conditions be=e u1 :i* we _c r$uce to position 9 space & date 9 "t4 ,!re is mass1 attraction app>5tly spontane\s1 electrical & magnetic prop]ties :i* mu/ be sup]add$ to spatial configuration4 ,to cut ! li/ %ort we mu/ say t practically ! ph5om5a ( ! _w pres5t u problems involv+ _m v>iables1 :i* we mu/ take z indep5d5t4 ,f ?is x foll[s t 9 mak+ poiographs we mu/ be prep>$ to use space ( m ?an ?ree dim5sions4 ,if ! symmetry & complet5ess ( our repres5tation is to be ( use to u we mu/ be prep>$ to appreciate & criticise figures ( a complexity great] ?an ( ^? 9 ?ree dim5sions4 ,x is impossible to give an example ( su* a poiograph :i* w n be m]ely trivial1 )\t go+ into details ( "s k9d irrelevant to our subject4 ,i pref] to introduce ! irrelevant details r ?an treat ?is "p ( ! subject p]functorily4 ,to take an in/ance ( a poiograph :i* does n lead u into ! complexities incid5t on its application 9 classificatory sci5ce1 let u foll[ ,mrs4 ,alicia ,boole ,/ott 9 h] repres5tation ( ! syllogism by its means4 ,%e w be int]e/$ to f9d t ! curi\s gap %e detect$ has a significance4 ,a syllogism consi/s ( two /atem5ts1 ! major & ! m9or premiss1 ) ! conclusion t c be drawn f !m4 ,?us1 to take an in/ance1 fig4 #di4 ,x is evid5t1 f look+ at ! successive figures t1 if we "k t ! region ,m lies altoge!r )9 ! region ,p1 & also "k t ! region ,s lies altoge!r )9 ! region ,m1 we c conclude t ! region ,s lies altoge!r )9 ! region ,p4 ,m is ,p1 major premiss2 ,s is ,m1 m9or premiss2 ,s is ,p1 conclusion4 ,giv5 ! fir/ two data we mu/ conclude t ,s lies 9 ,p4 ,! conclusion ,s is ,p involves two t]ms1 ,s & ,p1 :i* are respectively call$ ! subject & ! pr$icate1 ! lett]s ,s & ,p be+ *os5 ) ref]5ce to ! p>ts ! notions !y designate play 9 ! conclusion4 ,s is ! subject ( ! conclusion1 ,p is ! pr$icate ( ! conclusion4 ,! major premiss we take to be1 t :i* does n involve ,s1 & "h we always write x fir/4 ,illu/ration3 ,fig4 #di4 ,!re are sev]al v>ieties ( /atem5t possess+ diff]5t degrees ( univ]sality & mann]s ( ass]tiv5ess4 ,^! diff]5t =ms ( /atem5t are call$ ! moods4 ,we w take ! major premiss z "o v>iable1 z a ?+ capable ( diff]5t modifications ( ! same k9d1 ! m9or premiss z ano!r1 & ! diff]5t moods we w consid] z def9+ ! v>iations :i* ^! v>iables und]go4 ,!re are f\r moods3-- #a4 ,! univ]sal affirmative2 all ,m is ,p1 call$ mood ,a4 #b4 ,! univ]sal negative2 no ,m is ,p1 mood ,e4 #c4 ,! p>ticul> affirmative2 "s ,m is ,p1 mood ,i4 #d4 ,! p>ticul> negative2 "s ,m is n ,p1 mood ,o4 ,illu/ration3 #a4 #b4 #c4 #d4 ,mood ,a4 ,mood ,e4 ,mood ,i4 ,mood ,o4 ,fig4 #ej4 ,! dott$ l9es 9 #c & #d1 fig4 #ej1 d5ote t x is n kn[n :e!r or no any objects exi/1 correspond+ to ! space ( :i* ! dott$ l9e =ms "o delimit+ b\nd>y2 ?us1 9 mood ,i we d n "k if !re are any ,m',s :i* are n ,p1 we only "k "s ,m',s are ,p4 ,illu/ration3 ,fig4 #ea4 ,repres5t+ ! fir/ premiss 9 its v>i\s moods by regions m>k$ by v]tical l9es to ! "r ( ,,pq1 we h 9 fig4 #ea1 runn+ up f ! f\r lett]s ,,aeio1 f\r columns1 ea* ( :i* indicates t ! major premiss is 9 ! mood d5ot$ by ! respective lett]4 ,9 ! fir/ column to ! "r ( ,,pq is ! mood ,a4 ,n[ above ! l9e ,,rs let !re be m>k$ (f f\r regions correspond+ to ! f\r moods ( ! m9or premiss4 ,?us1 9 ! fir/ r[ above ,,rs all ! region betwe5 ,,rs & ! fir/ horizontal l9e above x d5otes t ! m9or premiss is 9 ! mood ,a4 ,! lett]s ,e1 ,i1 ,o1 9 ! same way %[ ! mood *>act]is+ ! m9or premiss 9 ! r[s opposite ^! lett]s4 ,we h /ill to exhibit ! conclusion4 ,to d ?is we mu/ consid] ! conclusion z a ?ird v>iable1 *>act]is$ 9 its diff]5t v>ieties by f\r moods--?is be+ ! syllogi/ic classification4 ,! introduction ( a ?ird v>iable involves a *ange 9 our sy/em ( repres5tation4 ,be=e we />t$ ) ! regions to ! "r ( a c]ta9 l9e z repres5t+ successively ! major premiss 9 its moods2 n[ we mu/ />t ) ! regions to ! "r ( a c]ta9 plane4 ,let ,,lmnr be ! plane face ( a cube1 fig4 #eb1 & let ! cube be divid$ into f\r p>ts by v]tical sections p>allel to ,,lmnr4 ,! v>iable1 ! major premiss1 is repres5t$ by ! successive regions :i* occur to ! "r ( ! plane ,,lmnr--t region to :i* ,a /&s opposite1 t slice ( ! cube1 is significative ( ! mood ,a4 ,?is :ole qu>t]-"p ( ! cube repres5ts t = e "p ( x ! major premiss is 9 ! mood ,a4 ,illu/ration3 ,fig4 #eb4 ,9 a simil> mann] ! next section1 ! second ) ! lett] ,e opposite x1 repres5ts t = e "o ( ! sixte5 small cubic spaces 9 x1 ! major premiss is 9 ! mood ,e4 ,! ?ird & f\r? comp>tm5ts made by ! v]tical sections d5ote ! major premiss 9 ! moods ,i & ,o4 ,b ! cube c be divid$ 9 o!r ways by o!r planes4 ,let ! divisions1 ( :i* f\r /ret* f ! front face1 correspond to ! m9or premiss4 ,! fir/ wall ( sixte5 cubes1 fac+ ! obs]v]1 has z its *>act]i/ic t 9 ea* ( ! small cubes1 :atev] else may be ! case1 ! m9or premiss is 9 ! mood ,a4 ,! v>iable--! m9or premiss--v>ies ?r\gh ! phases ,a1 ,e1 ,i1 ,o1 away f ! front face ( ! cube1 or ! front plane ( :i* ! front face is a "p4 ,& n[ we c repres5t ! ?ird v>iable 9 a precisely simil> way4 ,we c take ! conclusion z ! ?ird v>iable1 go+ ?r\gh its f\r phases f ! gr\nd plane upw>ds4 ,ea* ( ! small cubes at ! base ( ! :ole cube has ?is true ab\t x1 :atev] else may be ! case1 t ! conclusion is1 9 x1 9 ! mood ,a4 ,?us1 to recapitulate1 ! fir/ wall ( sixte5 small cubes1 ! fir/ ( ! f\r walls :i*1 proce$+ f left to "r1 build up ! :ole cube1 is *>act]is$ 9 ea* "p ( x by ?is1 t ! major premiss is 9 ! mood ,a4 ,! next wall d5otes t ! major premiss is 9 ! mood ,e1 & s on4 ,proce$+ f ! front to ! back ! fir/ wall pres5ts a region 9 e "p ( :i* ! m9or premiss is 9 ! mood ,a4 ,! second wall is a region ?r\gh\t :i* ! m9or premiss is 9 ! mood ,e1 & s on4 ,9 ! lay]s1 f ! bottom upw>ds1 ! conclusion goes ?r\gh its v>i\s moods beg9n+ ) ,a 9 ! l[e/1 ,e 9 ! second1 ,i 9 ! ?ird1 ,o 9 ! f\r?4 ,9 ! g5]al case1 9 :i* ! v>iables repres5t$ 9 ! poiograph pass ?r\gh a wide range ( values1 ! planes f :i* we measure _! degrees ( v>iation 9 our repres5tation are tak5 to be indef9itely ext5d$4 ,9 ?is case1 h[ev]1 all we are conc]n$ ) is ! f9ite region4 ,we h n[ to repres5t1 by "s limitation ( ! complex we h obta9$1 ! fact t n e comb9ation ( premisses ju/ifies any k9d ( conclusion4 ,?is c be simply effect$ by m>k+ ! regions 9 :i* ! premisses1 be+ su* z are def9$ by ! positions1 a conclusion :i* is valid is f\nd4 ,tak+ ! conjunction ( ! major premiss1 all ,m is ,p1 & ! m9or1 all ,s is ,m1 we conclude t all ,s is ,p4 ,h5ce1 t region mu/ be m>k$ 9 :i* we h ! conjunction ( major premiss 9 mood ,a2 m9or premiss1 mood ,a2 conclusion1 mood ,a4 ,?is is ! cube occupy+ ! l[e/ left-h& corn] ( ! l>ge cube4 ,illu/ration3 ,fig4 #ec4 ,proce$+ 9 ?is way1 we f9d t ! regions :i* mu/ be m>k$ are ^? %[n 9 fig4 #ec4 ,to discuss ! case %[n 9 ! m>k$ cube :i* appe>s at ! top ( fig4 #ec4 ,"h ! major premiss is 9 ! second wall to ! "r--x is 9 ! mood ,e & is ( ! type no ,m is ,p4 ,! m9or premiss is 9 ! mood *>act]is$ by ! ?ird wall f ! front4 ,x is ( ! type "s ,s is ,m4 ,f ^! premisses we draw ! conclusion t "s ,s is n ,p1 a conclusion 9 ! mood ,o4 ,n[ ! mood ,o ( ! conclusion is repres5t$ 9 ! top lay]4 ,h5ce we see t ! m>k+ is correct 9 ?is respect4 ,illu/ration3 ,fig4 #ed4 ,x w\ld1 ( c\rse1 be possible to repres5t ! cube on a plane by means ( f\r squ>es1 z 9 fig4 #ed1 if we consid] ea* squ>e to repres5t m]ely ! beg9n+ ( ! region x /&s =4 ,?us ! :ole cube c be repres5t$ by f\r v]tical squ>es1 ea* /&+ = a k9d ( v]tical tray1 & ! m>k+s w\ld be z %[n4 ,9 ,no4 #a ! major premiss is 9 mood ,a = ! :ole ( ! region indicat$ by ! v]tical squ>e ( sixte5 divisions2 9 ,no4 #b x is 9 ! mood ,e1 & s on4 ,a creature conf9$ to a plane w\ld h to adopt "s su* disjunctive way ( repres5t+ ! :ole cube4 ,he w\ld be oblig$ to repres5t t :i* we see z a :ole 9 sep>ate p>ts1 & ea* "p w\ld m]ely repres5t1 w\ld n be1 t solid cont5t :i* we see4 ,! view ( ^! f\r squ>es :i* ! plane creature w\ld h w\ld n be su* z ours4 ,he w\ld n see ! int]ior ( ! f\r squ>es repres5t$ above1 b ea* w\ld be entirely conta9$ )9 its outl9e1 ! int]nal b\nd>ies ( ! sep>ate small squ>es he c\ld n see except by remov+ ! out] squ>es4 ,we are n[ ready to introduce ! f\r? v>iable involv$ 9 ! syllogism4 ,9 assign+ lett]s to d5ote ! t]ms ( ! syllogism we h tak5 ,s & ,p to repres5t ! subject & pr$icate 9 ! conclusion1 & ?us 9 ! conclusion _! ord] is inv>iable4 ,b 9 ! premisses we h tak5 arbitr>ily ! ord] all ,m is ,p1 & all ,s is ,m4 ,!re is no reason :y ,m in/ead ( ,p %\ld n be ! pr$icate ( ! major premiss1 & s on4 ,accord+ly we take ! ord] ( ! t]ms 9 ! premisses z ! f\r? v>iable4 ,( ?is ord] !re are f\r v>ieties1 & ^! v>ieties are call$ figures4 ,us+ ! ord] 9 :i* ! lett]s are writt5 to d5ote t ! lett] fir/ writt5 is subject1 ! "o writt5 second is pr$icate1 we h ! foll[+ possibilities3-- #a/ ,figure4 #bnd ,figure4 #crd ,figure4 #d? ,figure4 ,major ,m ,p ,p ,m ,m ,p ,p ,m ,m9or ,s ,m ,s ,m ,m ,s ,m ,s ,!re are !re=e f\r possibilities ) reg>d to ?is f\r? v>iable z ) reg>d to ! premisses4 ,we h us$ up our dim5sions ( space 9 repres5t+ ! phases ( ! premisses & ! conclusion 9 respect ( mood1 & to repres5t 9 an analog\s mann] ! v>iations 9 figure we require a f\r? dim5sion4 ,n[ 9 br++ 9 ?is f\r? dim5sion we mu/ make a *ange 9 our orig9s ( measurem5t analog\s to t :i* we made 9 pass+ f ! plane to ! solid4 ,?is f\r? dim5sion is suppos$ to run at "r angles to any ( ! ?ree space dim5sions1 z ! ?ird space dim5sion runs at "r angles to ! two dim5sions ( a plane1 & ?us x gives u ! opportunity ( g5]at+ a new k9d ( volume4 ,if ! :ole cube moves 9 ?is dim5sion1 ! solid itself traces out a pa?1 ea* section ( :i*1 made at "r angles to ! direction 9 :i* x moves1 is a solid1 an exact repetition ( ! cube itself4 ,! cube z we see x is ! beg9n+ ( a solid ( su* a k9d4 ,x repres5ts a k9d ( tray1 z ! squ>e face ( ! cube is a k9d ( tray aga9/ :i* ! cube re/s4 ,suppose ! cube to move 9 ?is f\r? dim5sion 9 f\r /ages1 & let ! hyp]-solid region trac$ out 9 ! fir/ /age ( its progress be *>act]is$ by ?is1 t ! t]ms ( ! syllogism are 9 ! fir/ figure1 !n we c repres5t 9 ea* ( ! ?ree subsequ5t /ages ! rema9+ ?ree figures4 ,?us ! :ole cube =ms ! basis f :i* we measure ! v>iation 9 figure4 ,! fir/ figure holds good = ! cube z we see x1 & = t hyp]-solid :i* lies )9 ! fir/ /age2 ! second figure holds good 9 ! second /age1 & s on4 ,?us we measure f ! :ole cube z f> z figures are conc]n$4 ,b we saw t :5 we measur$ 9 ! cube itself hav+ ?ree v>iables1 namely1 ! two premisses & ! conclusion1 we measur$ f ?ree planes4 ,! base f :i* we measur$ 0 9 e case ! same4 ,h5ce1 9 measur+ 9 ?is high] space we %\ld h bases ( ! same k9d to measure f1 we %\ld h solid bases4 ,! fir/ solid base is easily se51 x is ! cube itself4 ,! o!r c be f\nd f ?is consid]ation4 ,t solid f :i* we measure figure is t 9 :i* ! rema9+ v>iables run ?r\gh _! full range ( v>ieties4 ,n[1 if we want to measure 9 respect ( ! moods ( ! major premiss1 we mu/ let ! m9or premiss1 ! conclusion1 run ?r\gh _! range1 & also ! ord] ( ! t]ms4 ,t is we mu/ take z basis ( measurem5t 9 respect to ! moods ( ! major t :i* repres5ts ! v>iation ( ! moods ( ! m9or1 ! conclusion & ! v>iation ( ! figures4 ,n[ ! v>iation ( ! moods ( ! m9or & ( ! conclusion are repres5t$ 9 ! squ>e face on ! left ( ! cube4 ,"h are all v>ieties ( ! m9or premiss & ! conclusion4 ,! v>ieties ( ! figures are repres5t$ by /ages 9 a motion proce$+ at "r angles to all space directions1 at "r angles consequ5tly to ! face 9 "q1 ! left-h& face ( ! cube4 ,consequ5tly lett+ ! left-h& face move 9 ?is direction we get a cube1 & 9 ?is cube all ! v>ieties ( ! m9or premiss1 ! conclusion1 & ! figure are repres5t$4 ,?us ano!r cubic base ( measurem5t is giv5 to ! cube1 g5]at$ by movem5t ( ! left-h& squ>e 9 ! f\r? dim5sion4 ,we f9d ! o!r bases 9 a simil> mann]1 "o is ! cube g5]at$ by ! front squ>e mov$ 9 ! f\r? dim5sion s z to g5]ate a cube4 ,f ?is cube v>iations 9 ! mood ( ! m9or are measur$4 ,! f\r? base is t f\nd by mov+ ! bottom squ>e ( ! cube 9 ! f\r? dim5sion4 ,9 ?is cube ! v>iations ( ! major1 ! m9or1 & ! figure are giv54 ,consid]+ ?is z a basis 9 ! f\r /ages proce$+ f x1 ! v>iation 9 ! moods ( ! conclusion are giv54 ,any "o ( ^! cubic bases c be repres5t$ 9 space1 & !n ! high] solid g5]at$ f !m lies out ( our space4 ,x c only be repres5t$ by a device analog\s to t by :i* ! plane be+ repres5ts a cube4 ,he repres5ts ! cube %[n above1 by tak+ f\r squ>e sections & plac+ !m arbitr>ily at conv5i5t di/ances ! "o f ! o!r4 ,s we mu/ repres5t ?is high] solid by f\r cubes3 ea* cube repres5ts only ! beg9n+ ( ! correspond+ high] volume4 ,x is suffici5t = u1 !n1 if we draw f\r cubes1 ! fir/ repres5t+ t region 9 :i* ! figure is ( ! fir/ k9d1 ! second t region 9 :i* ! figure is ( ! second k9d1 & s on4 ,^! cubes are ! beg9n+s m]ely ( ! respective regions--!y are ! trays1 z x 71 aga9/ :i* ! real solids mu/ be conceiv$ z re/+1 f :i* !y />t4 ,! fir/ "o1 z x is ! beg9n+ ( ! region ( ! fir/ figure1 is *>act]is$ by ! ord] ( ! t]ms 9 ! premisses be+ t ( ! fir/ figure4 ,! second simil>ly has ! t]ms ( ! premisses 9 ! ord] ( ! second figure1 & s on4 ,^! cubes are %[n bel[4 ,= ! sake ( %[+ ! prop]ties ( ! me?od ( repres5tation1 n = ! logical problem1 ,i w make a digression4 ,i w repres5t 9 space ! moods ( ! m9or & ( ! conclusion & ! diff]5t figures1 keep+ ! major always 9 mood ,a4 ,"h we h ?ree v>iables 9 diff]5t /ages1 ! m9or1 ! conclusion1 & ! figure4 ,let ! squ>e ( ! left-h& side ( ! orig9al cube be imag9$ to be /&+ by itself1 )\t ! solid "p ( ! cube1 repres5t$ by 7#b7 fig4 #ee4 ,! ,a1 ,e1 ,i1 ,o1 :i* run away repres5t ! moods ( ! m9or1 ! ,a1 ,e1 ,i1 ,o1 :i* run up repres5t ! moods ( ! conclusion4 ,! :ole squ>e1 s9ce x is ! beg9n+ ( ! region 9 ! major premiss1 mood ,a1 is to be consid]$ z 9 major premiss1 mood ,a4 ,f ?is squ>e1 let x be suppos$ t t direction 9 :i* ! figures are repres5t$ runs to ! left h&4 ,?us we h a cube 7#a7 runn+ f ! squ>e above1 9 :i* ! squ>e itself is hidd51 b ! lett]s ,a1 ,e1 ,i1 ,o1 ( ! conclusion are se54 ,9 ?is cube we h ! m9or premiss & ! conclusion 9 all _! moods1 & all ! figures repres5t$4 ,) reg>d to ! major premiss1 s9ce ! face 7#b7 belongs to ! fir/ wall f ! left 9 ! orig9al arrangem5t1 & 9 ?is arrangem5t 0 *>act]is$ by ! major premiss 9 ! mood ,a1 we may say t ! :ole ( ! cube we n[ h put up repres5ts ! mood ,a ( ! major premiss4 ,illu/ration3 ,fig4 #ee4 ,h5ce ! small cube at ! bottom to ! "r 9 #a1 ne>e/ to ! spectator1 is major premiss1 mood ,a2 m9or premiss1 mood ,a2 conclusion1 mood ,a2 & figure ! fir/4 ,! cube next to x1 runn+ to ! left1 is major premiss1 mood ,a2 m9or premiss1 mood ,a2 conclusion1 mood ,a2 figure #b4 ,s 9 ?is cube we h ! repres5tations ( all ! comb9ations :i* c occur :5 ! major premiss1 rema9+ 9 ! mood ,a1 ! m9or premiss1 ! conclusion1 & ! figures pass ?r\gh _! v>ieties4 ,9 ?is case !re is no room 9 space = a natural repres5tation ( ! moods ( ! major premiss4 ,to repres5t !m we mu/ suppose z be=e t !re is a f\r? dim5sion1 & />t+ f ?is cube z base 9 ! f\r? direction 9 f\r equal /ages1 all ! fir/ volume corresponds to major premiss ,a1 ! second to major premiss1 mood ,e1 ! next to ! mood ,i1 & ! la/ to mood ,o4 ,! cube we see is z x 7 m]ely a tray aga9/ :i* ! f\r-dim5sional figure re/s4 ,its section at any /age is a cube4 ,b a transition 9 ?is direction be+ transv]se to ! :ole ( our space is repres5t$ by no space motion4 ,we c exhibit successive /ages ( ! result ( transf]5ce ( ! cube 9 t direction1 b _c exhibit ! product ( a transf]5ce1 h[ev] small1 9 t direction4 ,illu/ration3 ,fig4 #ef4 ,to return to ! orig9al me?od ( repres5t+ our v>iables1 consid] fig4 #ef4 ,^! f\r cubes repres5t f\r sections ( ! figure d]iv$ f ! fir/ ( !m by mov+ x 9 ! f\r? dim5sion4 ,! fir/ portion ( ! motion1 :i* beg9s ) #a1 traces out a m ?an solid body1 :i* is all 9 ! fir/ figure4 ,! beg9n+ ( ?is body is %[n 9 #a4 ,! next portion ( ! motion traces out a m ?an solid body1 all ( :i* is 9 ! second figure2 ! beg9n+ ( ?is body is %[n 9 #b2 #c & #d foll[ on 9 l mann]4 ,"h1 !n1 9 "o f\r-dim5sional figure we h all ! comb9ations ( ! f\r v>iables1 major premiss1 m9or premiss1 figure1 conclusion1 repres5t$1 ea* v>iable go+ ?r\gh its f\r v>ieties4 ,! disconnect$ cubes drawn are our repres5tation 9 space by means ( disconnect$ sections ( ?is high] body4 ,n[ x is only a limit$ numb] ( conclusions :i* are true--_! tru? dep5ds on ! p>ticul> comb9ations ( ! premisses & figures :i* !y accompany4 ,! total figure ?us repres5t$ may be call$ ! univ]se ( ?\ght 9 respect to ^! f\r con/itu5ts1 & out ( ! univ]se ( possibly exi/+ comb9ations x is ! prov9ce ( logic to select ^? :i* correspond to ! results ( our reason+ faculties4 ,we c g ov] ea* ( ! premisses 9 ea* ( ! moods1 & f9d out :at conclusion logically foll[s4 ,b ?is is done 9 ! works on logic2 mo/ simply & cle>ly ,i believe 9 8,jevon's ,logic40 ,z we are only conc]n$ ) a =mal pres5tation ( ! results we w make use ( ! mnemonic l9es pr9t$ bel[1 9 :i* ! words enclos$ 9 brackets ref] to ! figures1 & are n significative3-- ,b>b>a cel>5t ,d>ii f]ioque prioris4 ,caes>e ,came/ris ,fe/9o ,b>oko secundae4 ,t]tia d>apti disamis datisi felapton4 ,bok>do f]isson habet ,qu>ta insup] addit4 ,bramantip cam5es dim>is f]apton fresison4 ,9 ^! l9es ea* significative ^w has ?ree v[els1 ! fir/ v[el ref]s to ! major premiss1 & gives ! mood ( t premiss1 8a0 signify+1 = in/ance1 t ! major mood is 9 mood a4 ,! second v[el ref]s to ! m9or premiss1 & gives its mood4 ,! ?ird v[el ref]s to ! conclusion1 & gives its mood4 ,?us 7prioris7--( ! fir/ figure--! fir/ mnemonic ^w is 8b>b>a10 & ?is gives major premiss1 mood ,a2 m9or premiss1 mood ,a2 conclusion1 mood ,a4 ,accord+ly 9 ! fir/ ( our f\r cubes we m>k ! l[e/ left-h& front cube4 ,to take ano!r in/ance 9 ! ?ird figure 8,t]tia10 ! ^w 8f]isson0 gives u major premiss mood ,e--e4g41 no ,m is ,p1 m9or premiss mood ,i2 "s ,m is ,s1 conclusion1 mood ,o2 "s ,s is n ,p4 ,! region to be m>k$ !n 9 ! ?ird repres5tative cube is ! "o 9 ! second wall to ! "r = ! major premiss1 ! ?ird wall f ! front = ! m9or premiss1 & ! top lay] = ! conclusion4 ,x is easily se5 t 9 ! diagram ?is cube is m>k$1 & s ) all ! valid conclusions4 ,! regions m>k$ 9 ! total region %[ :i* comb9ations ( ! f\r v>iables1 major premiss1 m9or premiss1 figure1 & conclusion exi/4 ,t is to say1 we objectify all possible conclusions1 & build up an ideal manifold1 conta9+ all possible comb9ations ( !m ) ! premisses1 & !n out ( ?is we elim9ate all t d n satisfy ! laws ( logic4 ,! residue is ! syllogism1 consid]$ z a canon ( reason+4 ,look+ at ! %ape :i* repres5ts ! totality ( ! valid conclusions1 x does n pres5t any obvi\s symmetry1 or easily *>act]isable nature4 ,a /rik+ configuration1 h[ev]1 is obta9$1 if we project ! f\r-dim5sional figure obta9$ into a ?ree-dim5sional "o2 t is1 if we take 9 ! base cube all ^? cubes :i* h a m>k$ space any:]e 9 ! s]ies ( f\r regions :i* />t f t cube4 ,?is corresponds to mak+ ab/raction ( ! figures1 giv+ all ! conclusions :i* are valid :atev] ! figure may be4 ,illu/ration3 ,fig4 #eg4 ,proce$+ 9 ?is way we obta9 ! arrangem5t ( m>k$ cubes %[n 9 fig4 #eg4 ,we see t ! valid conclusions are arrang$ almo/ symmetrically r\nd "o cube--! "o on ! top ( ! column />t+ f ,,aaa4 ,!re is "o brea* ( cont9uity h[ev] 9 ?is s*eme4 ,"o cube is unm>k$1 :i* if m>k$ w\ld give symmetry4 ,x is ! "o :i* w\ld be d5ot$ by ! lett]s ,i1 ,e1 ,o1 9 ! ?ird wall to ! "r1 ! second wall away1 ! topmo/ lay]4 ,n[ ?is comb9ation ( premisses 9 ! mood ,,ie1 ) a conclusion 9 ! mood ,o1 is n notic$ 9 any book on logic ) :i* ,i am famili>4 ,let u look at x = ourselves1 z x seems t !re mu/ be some?+ curi\s 9 connection ) ?is break ( cont9uity 9 ! poiograph4 ,illu/ration3 ,fig4 #eh4 ,! propositions ,i1 ,e1 9 ! v>i\s figures are ! foll[+1 z %[n 9 ! accompany+ s*eme1 fig4 #eh3--;,fir/ figure3 "s ,m is ,p2 no ,s is ,m4 ,second figure3 "s ,p is ,m2 no ,s is ,m4 ,?ird figure3 "s ,m is ,p2 no ,m is ,s4 ,f\r? figure3 "s ,p is ,m2 no ,m is ,s4 ,exam9+ ^! figures1 we see1 tak+ ! fir/1 t if "s ,m is ,p & no ,s is ,m1 we h no conclusion ( ! =m ,s is ,p 9 ! v>i\s moods4 ,x is q indet]m9ate h[ ! circle repres5t+ ,s lies ) reg>d to ! circle repres5t+ ,p4 ,x may lie inside1 outside1 or p>tly inside ,p4 ,! same is true 9 ! o!r figures #b & #c4 ,b :5 we come to ! f\r? figure1 s9ce ,m & ,s lie completely outside ea* o!r1 !re _c lie inside ,s t "p ( ,p :i* lies inside ,m4 ,n[ we "k by ! major premiss t "s ( ,p does lie 9 ,m4 ,h5ce ,s _c conta9 ! :ole ( ,p4 ,9 words1 "s ,p is ,m1 no ,m is ,s1 !re=e ,s does n conta9 ! :ole ( ,p4 ,if we take ,p z ! subject1 ?is gives u a conclusion 9 ! mood ,o ab\t ,p4 ,"s ,p is n ,s4 ,b x does n give u conclusion ab\t ,s 9 any "o ( ! f\r =ms recognis$ 9 ! syllogism & call$ its moods4 ,h5ce ! brea* ( ! cont9uity 9 ! poiograph has enabl$ u to detect a lack ( complet5ess 9 ! relations :i* are consid]$ 9 ! syllogism4 ,to take an in/ance3--,"s ,am]icans 7,p7 are ( ,african /ock 7,m72 ,no ,aryans 7,s7 are ( ,african /ock 7,m72 ,aryans 7,s7 d n include all ( ,am]icans 7,p74 ,9 ord] to draw a conclusion ab\t ,s we h to admit ! /atem5t1 8,s does n conta9 ! :ole ( ,p10 z a valid logical =m--x is a /atem5t ab\t ,s :i* c be made4 ,! logic :i* gives u ! =m1 8"s ,p is n ,s10 & :i* does n all[ u to give ! exactly equival5t & equally prim>y =m1 8,s does n conta9 ! :ole ( ,p10 is artificial4 ,& ,i wi% to po9t out t ?is artificiality leads to an error4 ,if "o tru/$ to ! mnemonic l9es giv5 above1 "o w\ld conclude t no logical conclusion ab\t ,s c be drawn f ! /atem5t1 8"s ,p are ,m1 no ,m are ,s40 ,b a conclusion c be drawn3 ,s does n conta9 ! :ole ( ,p4 ,x is n t ! result is giv5 express$ 9 ano!r =m4 ,! mnemonic l9es d5y t any conclusion c be drawn f premisses 9 ! moods ,i1 ,e1 respectively4 ,?us a simple f\r-dim5sional poiograph has enabl$ u to detect a mi/ake 9 ! mnemonic l9es :i* h be5 h&$ d[n un*all5g$ f m$ival times4 ,to discuss ! subject ( ^! l9es m fully a logician def5d+ !m w\ld probably say t a p>ticul> /atem5t _c be a major premiss2 & s d5y ! exi/5ce ( ! f\r? figure 9 ! comb9ation ( moods4 ,to take our in/ance3 "s ,am]icans are ( ,african /ock2 no ,aryans are ( ,african /ock4 ,he w\ld say t ! conclusion is "s ,am]icans are n ,aryans2 & t ! second /atem5t is ! major4 ,he w\ld refuse to say any?+ ab\t ,aryans1 condemn+ u to an et]nal sil5ce ab\t !m1 z f> z ^! premisses are conc]n$6 ,b1 if !re is a /atem5t involv+ ! relation ( two classes1 x mu/ be expressible z a /atem5t ab\t ei!r ( !m4 ,to b> ! conclusion1 8,aryans d n include ! :ole ( ,am]icans10 is purely a make%ift 9 fav\r ( a false classification4 ,& ! argum5t drawn f ! univ]sality ( ! major premiss _c be consi/5tly ma9ta9$4 ,x w\ld preclude su* comb9ations z major ,o1 m9or ,a1 conclusion ,o--i4e41 su* z "s m\nta9s 7,m7 are n p]man5t 7,p72 all m\nta9s 7,m7 are sc5]y 7,s72 "s sc5]y 7,s7 is n p]man5t 7,p74 ,?is is all[$ 9 8,jevon's ,logic10 & 8 omission to discuss ,i1 ,e1 ,o1 9 ! f\r? figure1 is inexplicable4 ,a satisfactory poiograph ( ! logical s*eme c be made by admitt+ ! use ( ! words "s1 none1 or all1 ab\t ! pr$icate z well z ab\t ! subject4 ,!n we c express ! /atem5t1 8,aryans d n include ! :ole ( ,am]icans10 clumsily1 b1 :5 its obscurity is fa?om$1 correctly1 z 8,"s ,aryans are n all ,am]icans40 ,& ?is me?od is :at is call$ ! 8quantification ( ! pr$icate40 ,! laws ( =mal logic are co9cid5t ) ! conclusions :i* c be drawn ab\t regions ( space1 :i* ov]lap "o ano!r 9 ! v>i\s possible ways4 ,x is n difficult s to /ate ! relations or to obta9 a symmetrical poiograph4 ,b to ent] into ?is bran* ( geometry is beside our pres5t purpose1 :i* is to %[ ! application ( ! poiograph 9 a f9ite & limit$ region1 )\t any ( ^? complexities :i* att5d its use 9 reg>d to natural objects4 ,if we take ! latt]--plants1 = in/ance--&1 )\t assum+ fix$ directions 9 space z repres5tative ( def9ite v>iations1 arrange ! repres5tative po9ts 9 su* a mann] z to correspond to ! simil>ities ( ! objects1 we obta9 configuration ( s+ul> int]e/2 & p]haps 9 ?is way1 9 ! mak+ ( %apes ( %apes1 bodies ) bodies omitt$1 "s insight into ! /ructure ( ! species & g5]a might be obta9$4 ,,*apt] ,,ix ,,application ,,to ,,kant',s ,,!ory ,,( ,,exp]i5ce ,:5 we obs]ve ! heav5ly bodies we become aw>e t !y all p>ticipate 9 "o univ]sal motion--a diurnal revolution r\nd ! pol> axis4 ,9 ! case ( fix$ />s ?is is mo/ unqualifi$ly true1 b 9 ! case ( ! sun1 & ! planets also1 ! s+le motion ( revolution c be disc]n$1 modifi$1 & slightly alt]$ by o!r & second>y motions4 ,h5ce ! univ]sal *>act]i/ic ( ! cele/ial bodies is t !y move 9 a diurnal circle4 ,b we "k t ?is "o great fact :i* is true ( !m all has 9 reality no?+ to d ) !m4 ,! diurnal revolution :i* !y visibly p]=m is ! result ( ! condition ( ! obs]v]4 ,x is because ! obs]v] is on a rotat+ e>? t a univ]sal /atem5t c be made ab\t all ! cele/ial bodies4 ,! univ]sal /atem5t :i* is valid ab\t e "o ( ! cele/ial bodies is t :i* does n conc]n !m at all1 & is b a /atem5t ( ! condition ( ! obs]v]4 ,n[ !re are univ]sal /atem5ts ( o!r k9ds :i* we c make4 ,we c say t all objects ( exp]i5ce are 9 space & subject to ! laws ( geometry4 ,does ?is mean t space & all t x means is due to a condition ( ! obs]v]8 ,if a univ]sal law 9 "o case means no?+ affect+ ! objects !mselves1 b only a condition ( obs]vation1 is ?is true 9 e case8 ,!re is %[n u 9 a/ronomy a v]a causa = ! ass]tion ( a univ]sal4 ,is ! same cause to be trac$ ev]y:]e8 ,su* is a fir/ approximation to ! doctr9e ( ,kant's critique4 ,x is ! appreh5sion ( a relation into :i*1 on ! "o side & ! o!r1 p]fectly def9ite con/itu5ts ent]--! human obs]v] & ! />s--& a transf]5ce ( ?is relation to a region 9 :i* ! con/itu5ts on ei!r side are p]fectly unkn[n4 ,if spatiality is due to a condition ( ! obs]v]1 ! obs]v] _c be ?is bodily self ( ours--! body1 l ! objects ar\nd x1 are equally 9 space4 ,?is conception ,kant appli$1 n only to ! intuitions ( s5se1 b to ! concepts ( reason--:]ev] a univ]sal /atem5t is made !re is af=d$ him an opportunity = ! application ( 8 pr9ciple4 ,he con/ruct$ a sy/em 9 :i* "o h>dly kn[s :i* ! mo/ to admire1 ! ar*itectonic skill1 or ! retic5ce ) reg>d to ?+s 9 !mselves1 & ! obs]v] 9 himself4 ,8 sy/em c be comp>$ to a g>d51 some:at =mal p]haps1 b ) ! *>m ( a quality m ?an intellectual1 a besonn5heit1 an exquisite mod]ation ov] all4 ,& f ! gr\nd he s c>efully prep>$ ) t buri$ 9 obscurity1 :i* x is fitt+ %\ld be obscure1 sci5ce blossoms & ! tree ( real k gr[s4 ,! critique is a /oreh\se ( ideas ( pr(\nd int]e/4 ,! "o ( :i* ,i h giv5 a p>tial /atem5t leads1 z we %all see on /udy+ x 9 detail1 to a !ory ( ma!matics sugge/ive ( enquiries 9 _m directions4 ,! ju/ification = my treatm5t w be f\nd among/ o!r passages 9 t "p ( ! transc5d5tal analytic1 9 :i* ,kant speaks ( objects ( exp]i5ce subject to ! =ms ( s5sibility1 n subject to ! concepts ( reason4 ,kant ass]ts t :5ev] we ?9k we ?9k ( objects 9 space & "t1 b he d5ies t ! space & "t exi/ z indep5d5t entities4 ,he goes ab\t to expla9 !m1 & _! univ]sality1 n by assum+ !m1 z mo/ o!r philosoph]s d1 b by po/ulat+ _! abs5ce4 ,h[ !n does x come to pass t ! _w is 9 space & "t to u8 ,kant takes ! same position ) reg>d to :at we call nature--a great sy/em subject to law & ord]4 8,h[ d y expla9 ! law & ord] 9 nature80 we ask ! philosoph]s4 ,all except ,kant reply by assum+ law & ord] some:]e1 & !n %[+ h[ we c recognise x4 ,9 expla9+ our notions1 philosoph]s f o!r ?an ! ,kantian /&po9t1 assume ! notions z exi/+ outside u1 & !n x is no difficult task to %[ h[ !y come to u1 ei!r by inspiration or by obs]vation4 ,we ask 8,:y d we h an idea ( law 9 nature80 8,because natural processes g accord+ to law10 we are answ]$1 8& exp]i5ce inh]it$ or acquir$1 gives u ?is notion40 ,b :5 we speak ab\t ! law 9 nature we are speak+ ab\t a notion ( our own4 ,s all t ^! expositors d is to expla9 our notion by an assumption ( x4 ,kant is v diff]5t4 ,he supposes no?+4 ,an exp]i5ce su* z ours is v diff]5t f exp]i5ce 9 ! ab/ract4 ,imag9e j simply exp]i5ce1 succession ( /ates1 ( consci\sness6 ,:y1 !re w\ld be no connect+ any two toge!r1 !re w\ld be no p]sonal id5tity1 no memory4 ,x is out ( a g5]al exp]i5ce su* z ?is1 :i*1 9 respect to any?+ we call real1 is less ?an a dream1 t ,kant %[s ! g5esis ( an exp]i5ce su* z ours4 ,kant takes up ! problem ( ! explanation ( space1 "t1 ord]1 & s q logically does n presuppose !m4 ,b h[1 :5 e act ( ?\ght is ( ?+s 9 space1 & "t1 & ord]$1 %all we repres5t to ourselves t p]fectly indef9ite some:at :i* is ,kant's necess>y hypo!sis--t :i* is n 9 space or "t & is n ord]$4 ,t is our problem1 to repres5t t :i* ,kant assumes n subject to any ( our =ms ( ?\ght1 & !n %[ "s function :i* work+ on t makes x into a 8nature0 subject to law & ord]1 9 space & "t4 ,su* a function ,kant calls ! 8,unity ( ,app]ception02 i4e41 t :i* makes our /ate ( consci\sness capable ( be+ wov5 into a sy/em ) a self1 an out] _w1 memory1 law1 cause1 & ord]4 ,! difficulty t meets u 9 discuss+ ,kant's hypo!sis is t ev]y?+ we ?9k ( is 9 space & "t--h[ !n %all we repres5t 9 space an exi/5ce n 9 space1 & 9 "t an exi/5ce n 9 "t8 ,?is difficulty is /ill m evid5t :5 we come to con/ruct a poiograph1 = a poiograph is ess5tially a space /ructure4 ,b because m evid5t ! difficulty is ne>] a solution4 ,if we always ?9k 9 space1 i4e4 us+ space concepts1 ! fir/ condition requisite = adapt+ !m to ! repres5tation ( non-spatial exi/5ce1 is to be aw>e ( ! limitation ( our ?\ght1 & s be able to take ! prop] /eps to ov]come x4 ,! problem be=e u1 !n1 is to repres5t 9 space an exi/5ce n 9 space4 ,! solution is an easy "o4 ,x is provid$ by ! conception ( alt]nativity4 ,to get our ideas cle> let u g "r back beh9d ! di/9ctions ( an inn] & an out] _w4 ,bo? ( ^!1 ,kant says1 are products4 ,let u take m]ely /ates ( consci\sness1 & n ask ! "q :e!r !y are produc$ or sup]9duc$--to ask su* a "q is to h got too f> on1 to h assum$ some?+ ( :i* we h n trac$ ! orig94 ,( ^! /ates let u simply say t !y occur4 ,let u n[ use ! ^w a 8posit0 = a phase ( consci\sness r$uc$ to its la/ possible /age ( evanesc5ce2 let a posit be t phase ( consci\sness ( :i* all t c be said is t x occurs4 ,let a1 b1 c1 be ?ree su* posits4 ,we _c repres5t !m 9 space )\t plac+ !m 9 a c]ta9 ord]1 z a1 b1 c4 ,b ,kant di/+ui%es betwe5 ! =ms ( s5sibility & ! concepts ( reason4 ,a dream 9 :i* ev]y?+ happ5s at haphaz>d w\ld be an exp]i5ce subject to ! =m ( s5sibility & only p>tially subject to ! concepts ( reason4 ,x is p>tially subject to ! concepts ( reason because1 al?\gh !re is no ord] ( sequ5ce1 /ill at any giv5 "t !re is ord]4 ,p]ception ( a ?+ z 9 space is a =m ( s5sibility1 ! p]ception ( an ord] is a concept ( reason4 ,we mu/1 !re=e1 9 ord] to get at t process :i* ,kant supposes to be con/itutive ( an ord]$ exp]i5ce imag9e ! posits z 9 space )\t ord]4 ,z we "k !m !y mu/ be 9 "s ord]1 abc1 bca1 cab1 acb1 cba1 bac1 "o or ano!r4 ,to repres5t !m z hav+ no ord] conceive all ^! diff]5t ord]s z equally exi/+4 ,introduce ! conception ( alt]nativity--let u suppose t ! ord] abc1 & bac1 = example1 exi/ equally1 s t we _c say ab\t a t x comes be=e or aft] b4 ,?is w\ld correspond to a sudd5 & arbitr>y *ange ( a into b & b into a1 s t1 to use ,kant's words1 x w\ld be possible to call "o ?+ by "o "n at "o "t & at ano!r "t by ano!r "n4 ,9 an exp]i5ce ( ?is k9d we h a k9d ( *aos1 9 :i* no ord] exi/s2 x is a manifold n subject to ! concepts ( reason4 ,n[ is !re any process by :i* ord] c be introduc$ into su* a manifold--is !re any function ( consci\sness 9 virtue ( :i* an ord]$ exp]i5ce c\ld arise8 ,9 ! precise condition 9 :i* ! posits are1 z describ$ above1 x does n seem to be possible4 ,b if we imag9e a duality to exi/ 9 ! manifold1 a function ( consci\sness c be easily discov]$ :i* w produce ord] out ( no ord]4 ,let u imag9e ea* posit1 !n1 z hav+1 a dual aspect4 ,let a be #a;a 9 :i* ! dual aspect is repres5t$ by ! comb9ation ( symbols4 ,& simil>ly let b be #b;b1 c be #c;c1 9 :i* #b & b repres5t ! dual aspects ( b1 #c & c ^? ( c4 ,s9ce a c arbitr>ily *ange into b1 or into c1 & s on1 ! p>ticul> comb9ations writt5 above _c be kept4 ,we h to assume ! equally possible occurr5ce ( =m su* z #b;a1 #b;b1 & s on2 & 9 ord] to get a repres5tation ( all ^? comb9ations out ( :i* any set is alt]natively possible1 we mu/ take e aspect ) e aspect4 ,we mu/1 t is1 h e lett] ) e numb]4 ,let u n[ apply ! me?od ( space repres5tation4 ,note4--,at ! beg9n+ ( ! next *apt] ! same /ructures z ^? :i* foll[ are exhibit$ 9 m detail & a ref]5ce to !m w remove any obscurity :i* may be f\nd 9 ! imm$iately foll[+ passages4 ,!y are !re c>ri$ on to a great] multiplicity ( dim5sions1 & ! significance ( ! process "h briefly expla9$ becomes m app>5t4 ,illu/ration3 ,fig4 #ei4 ,take ?ree mutually rectangul> axes 9 space #a1 #b1 #c 7fig4 #ei71 & on ea* m>k ?ree po9ts1 ! common meet+ po9t be+ ! fir/ on ea* axis4 ,!n by means ( ^! ?ree po9ts on ea* axis we def9e #bg positions1 #bg po9ts 9 a cubical clu/]1 %[n 9 fig4 #fj1 ! same me?od ( co-ord9ation be+ us$ z has be5 describ$ be=e4 ,ea* ( ^! positions c be nam$ by means ( ! axes & ! po9ts comb9$4 ,illu/ration3 ,fig4 #fj4 ,?us1 = in/ance1 ! "o m>k$ by an a/]isk c be call$ #a;c1 #b;b1 #c;c1 because x is opposite to c on #a1 to b on #b1 to c on #c4 ,let u n[ treat ( ! /ates ( consci\sness correspond+ to ^! positions4 ,ea* po9t repres5ts a composite ( posits1 & ! manifold ( consci\sness correspond+ to !m is ( a c]ta9 complexity4 ,suppose n[ ! con/itu5ts1 ! po9ts on ! axes1 to int]*ange arbitr>ily1 any "o to become any o!r1 & also ! axes #a1 #b1 & #c1 to int]*ange among/ !mselves1 any "o to become any o!r1 & to be subject to no sy/em or law1 t is to say1 t ord] does n exi/1 & t ! po9ts :i* run abc on ea* axis may run bac1 & s on4 ,!n any "o ( ! /ates ( consci\sness repres5t$ by ! po9ts 9 ! clu/] c become any o!r4 ,we h a repres5tation ( a r&om consci\sness ( a c]ta9 degree ( complexity4 ,n[ let u exam9e c>efully "o p>ticul> case ( arbitr>y int]*ange ( ! po9ts1 a1 b1 c2 z "o su* case1 c>efully consid]$1 makes ! :ole cle>4 ,illu/ration3 ,fig4 #fa4 ,consid] ! po9ts nam$ 9 ! figure #a;c1 #b;a1 #c;c2 #a;c1 #b;c1 #c;a2 #a;a1 #b;c1 #c;c1 & exam9e ! effect on !m :5 a *ange ( ord] takes place4 ,let u suppose1 = in/ance1 t a *anges into b1 & let u call ! two sets ( po9ts we get1 ! "o be=e & ! "o aft]1 _! *ange conjugates4 ,be=e ! *ange #a;c #b;a #c;c #a;c #b;c #c;a #a;a #b;c #c;c,conjug- ,aft] ! *ange #a;c #b;b #c;c #a;c #b;c #c;b #a;b #b;c #c;c ates4 ,! po9ts surr\nd$ by r+s repres5t ! conjugate po9ts4 ,x is evid5t t z consci\sness1 repres5t$ fir/ by ! fir/ set ( po9ts & aft]w>ds by ! second set ( po9ts1 w\ld h no?+ 9 common 9 its two phases4 ,x w\ld n be capable ( giv+ an acc\nt ( itself4 ,!re w\ld be no id5tity4 ,if1 h[ev]1 we c f9d any set ( po9ts 9 ! cubical clu/]1 :i*1 :5 any arbitr>y *ange takes place 9 ! po9ts on ! axes1 or 9 ! axes !mselves1 repeats itself1 is reproduc$1 !n a consci\sness repres5t$ by ^? po9ts w\ld h a p]man5ce4 ,x w\ld h a pr9ciple ( id5tity4 ,despite ! no law1 ! no ord]1 ( ! ultimate con/itu5ts1 x w\ld h an ord]1 x w\ld =m a sy/em1 ! condition ( a p]sonal id5tity w\ld be fulfill$4 ,! "q comes to ?is1 !n4 ,c we f9d a sy/em ( po9ts :i* is self-conjugate :i* is su* t :5 any posit on ! axes becomes any o!r1 or :5 any axis becomes any o!r1 su* a set is trans=m$ into itself1 its id5tity is n subm]g$1 b rises sup]ior to ! *aos ( its con/itu5ts8 ,illu/ration3 ,fig4 #fb4 ,su* a set c be f\nd4 ,consid] ! set repres5t$ 9 fig4 #fb1 & writt5 d[n 9 ! fir/ ( ! two l9es-- ,self- #a;a #b;b #c;c #a;b #b;a #c;c #a;c #b;a #c;b conjugate4 #a;c #b;b #c;a #a;b #b;c #c;a #a;a #b;c #c;b ,self- #a;c #b;b #c;a #a;b #b;c #c;a #a;a #b;c #c;b conjugate4 #a;a #b;b #c;c #a;b #b;a #c;c #a;c #b;a #c;b ,if n[ a *ange into c & c into a1 we get ! set 9 ! second l9e1 :i* has ! same memb]s z are 9 ! upp] l9e4 ,look+ at ! diagram we see t x w\ld correspond simply to ! turn+ ( ! figures z a :ole4#b ,any arbitr>y *ange ( ! po9ts on ! axes1 or ( ! axes !mselves1 reproduces ! same set4 #b ,^! figures are describ$ m fully1 & ext5d$1 9 ! next *apt]4 ,?us1 a function1 by :i* a r&om1 an unord]$1 consci\sness c\ld give an ord]$ & sy/ematic "o1 c be repres5t$4 ,x is notewor?y t x is a sy/em ( selection4 ,if out ( all ! alt]native =ms t only is att5d$ to :i* is self-conjugate1 an ord]$ consci\sness is =m$4 ,a selection gives a feature ( p]man5ce4 ,c we say t ! p]man5t consci\sness is ?is selection8 ,an analogy betwe5 ,kant & ,d>w9 comes into light4 ,t :i* is sw+s cle> ( ! fleet+1 9 virtue ( its pres5t+ a feature ( p]man5ce4 ,!re is no ne$ to suppose any function ( 8att5d+ to40 ,a consci\sness capable ( giv+ an acc\nt ( itself is "o :i* is *>act]is$ by ?is comb9ation4 ,all comb9ations exi/--( ?is k9d is ! consci\sness :i* c give an acc\nt ( itself4 ,& ! v duality :i* we h presuppos$ may be reg>d$ z orig9at$ by a process ( selection4 ,d>w9 set himself to expla9 ! orig9 ( ! fauna & flora ( ! _w4 ,he d5i$ specific t5d5cies4 ,he assum$ an indef9ite v>iability--t is1 *ance--b a *ance conf9$ )9 n>r[ limits z reg>ds ! magnitude ( any consecutive v>iations4 ,he %[$ t organisms possess+ features ( p]man5ce1 if !y occurr$ w\ld be pres]v$4 ,s 8 acc\nt ( any /ructure or organis$ be+ 0 t x possess$ features ( p]man5ce4 ,kant1 und]tak+ n ! explanation ( any p>ticul> ph5om5a b ( t :i* we call nature z a :ole1 _h an orig9 ( species ( 8 own1 an acc\nt ( ! flora & fauna ( consci\sness4 ,he d5i$ any specific t5d5cy ( ! elem5ts ( consci\sness1 b tak+ our own consci\sness1 po9t$ out t 9 :i* x resembl$ any consci\sness :i* c\ld survive1 :i* c\ld give an acc\nt ( itself4 ,he assumes a *ance or r&om _w1 & z great & small 7 n to him any giv5 notions ( :i* he c\ld make use1 he did n limit ! *ance1 ! r&omness1 9 any way4 ,b any consci\sness :i* is p]man5t mu/ possess c]ta9 features--^? attributes namely :i* give x p]man5ce4 ,any consci\sness l our own is simply a consci\sness :i* possesses ^? attributes4 ,! ma9 ?+ is t :i* he calls ! unity ( app]ception1 :i* we h se5 above is simply ! /atem5t t a p>ticul> set ( phases ( consci\sness on ! basis ( complete r&omness w be self-conjugate1 & s p]man5t4 ,z ) ,d>w9 s ) ,kant1 ! reason = exi/5ce ( any feature comes to ?is--%[ t x t5ds to ! p]man5ce ( t :i* possesses x4 ,we c ?us reg>d ,kant z ! creator ( ! fir/ ( ! mod]n evolution !ories4 ,&1 z is s (t5 ! case1 ! fir/ ef=t 0 ! mo/ /up5d\s 9 its scope4 ,kant does n inve/igate ! orig9 ( any special "p ( ! _w1 su* z its organisms1 its *emical elem5ts1 its social communities ( m54 ,he simply inve/igates ! orig9 ( ! :ole--( all t is includ$ 9 consci\sness1 ! orig9 ( t 8?\ght ?+0 ^: progressive realisation is ! kn[able univ]se4 ,?is po9t ( view is v diff]5t f ! ord9>y "o1 9 :i* a man is suppos$ to be plac$ 9 a _w l t :i* he has come to ?9k ( x1 & !n to le>n :at he has f\nd out f ?is model :i* he himself has plac$ on ! sc5e4 ,we all "k t !re are a numb] ( que/ions 9 attempt+ an answ] to :i* su* an assumption is n all[able4 ,mill1 = in/ance1 expla9s our notion ( 8law0 by an inv>iable sequ5ce 9 nature4 ,b :at we call nature is some?+ giv5 9 ?\ght4 ,s he expla9s a ?\ght ( law & ord] by a ?\ght ( an inv>iable sequ5ce4 ,he leaves ! problem :]e he f\nd x4 ,kant's !ory is n unique & alone4 ,x is "o ( a numb] ( evolution !ories4 ,a notion ( its import & significance c be obta9$ by a comp>ison ( x ) o!r !ories4 ,?us 9 ,d>w9's !oretical _w ( natural selection a c]ta9 assumption is made1 ! assumption ( indef9ite v>iability--slight v>iability x is true1 ov] any appreciable lapse ( "t1 b indef9ite 9 ! po/ulat$ epo*s ( trans=mation--& a :ole *a9 ( results is %[n to foll[4 ,?is elem5t ( *ance v>iation is n1 h[ev]1 an ultimate re/+ place4 ,x is a prelim9>y /age4 ,?is suppos+ ! all is a prelim9>y /ep t[>ds f9d+ out :at is4 ,if e k9d ( organism c come into be+1 ^? t d survive w pres5t su* & su* *>act]i/ics4 ,?is is ! necess>y beg9n+ = asc]ta9+ :at k9ds ( organisms d come into exi/5ce4 ,& s ,kant's hypo!sis ( a r&om consci\sness is ! necess>y beg9n+ = ! rational inve/igation ( consci\sness z x is4 ,8 assumption supplies1 z x 71 ! space 9 :i* we c obs]ve ! ph5om5a4 ,x gives ! g5]al laws con/itutive ( any exp]i5ce4 ,if1 on ! assumption ( absolute r&omness 9 ! con/itu5ts1 su* & su* w\ld be *>act]i/ic ( ! exp]i5ce1 !n1 :atev] ! con/itu5ts1 ^! *>act]i/ics mu/ be univ]sally valid4 ,we w n[ proce$ to exam9e m c>efully ! poiograph1 con/ruct$ = ! purpose ( exhibit+ an illu/ration ( ,kant's unity ( app]ception4 ,9 ord] to %[ ! d]ivation ord] out ( non-ord] x has be5 necess>y to assume a pr9ciple ( duality--we h _h ! axes & ! posits on ! axes--!re are two sets ( elem5ts1 ea* non-ord]$1 & x is 9 ! reciprocal relation ( !m t ! ord]1 ! def9ite sy/em1 orig9ates4 ,is !re any?+ 9 our exp]i5ce ( ! nature ( a duality8 ,!re c]ta9ly are objects 9 our exp]i5ce :i* h ord] & ^? :i* are incapable ( ord]4 ,! two roots ( a quadratic equation h no ord]4 ,no "o c tell :i* comes fir/4 ,if a body rises v]tically & !n goes at "r angles to its =m] c\rse1 no "o c assign any priority to ! direction ( ! nor? or to ! ea/4 ,!re is no priority 9 directions ( turn+4 ,we associate turn+s ) no ord] progressions 9 a l9e ) ord]4 ,b 9 ! axes & po9ts we h assum$ above !re is no su* di/9ction4 ,x is ! same1 :e!r we assume an ord] among ! turn+s1 & no ord] among ! po9ts on ! axes1 or1 vice v]sa1 an ord] 9 ! po9ts & no ord] 9 ! turn+s4 ,a be+ ) an inf9ite numb] ( axes mutually at "r angles1 ) a def9ite sequ5ce betwe5 !m & no sequ5ce betwe5 ! po9ts on ! axes1 w\ld be 9 a condition =mally indi/+ui%able f t ( a creature :o1 accord+ to an assumption m natural to u1 _h on ea* axis an inf9ite numb] ( ord]$ po9ts & no ord] ( priority among/ ! axes4 ,a be+ 9 su* a con/itut$ _w w\ld n be able to tell :i* 0 turn+ & :i* 0 l5g? along an axis1 9 ord] to di/+ui% betwe5 !m4 ,?us to take a p]t95t illu/ration1 we may be 9 a _w ( an inf9ite numb] ( dim5sions1 ) ?ree arbitr>y po9ts on ea*--?ree po9ts ^: ord] is indiff]5t1 or 9 a _w ( ?ree axes ( arbitr>y sequ5ce ) an inf9ite numb] ( ord]$ po9ts on ea*4 ,we c't tell :i* is :i*1 to di/+ui% x f ! o!r4 ,?us x appe>s ! mode ( illu/ration :i* we h us$ is n an artificial "o4 ,!re really exi/s 9 nature a duality ( ! k9d :i* is necess>y to expla9 ! orig9 ( ord] out ( no ord]--! duality1 namely1 ( dim5sion & position4 ,let u use ! t]m gr\p = t sy/em ( po9ts :i* rema9s un*ang$1 :atev] arbitr>y *ange ( its con/itu5ts takes place4 ,we notice t a gr\p involves a duality1 is inconceivable )\t a duality4 ,?us1 accord+ to ,kant1 ! prim>y elem5t ( exp]i5ce is ! gr\p1 & ! !ory ( gr\ps w\ld be ! mo/ fundam5tal bran* ( sci5ce4 ,ow+ to an expression 9 ! critique ! au?ority ( ,kant is sometimes adduc$ aga9/ ! assumption ( m ?an ?ree dim5sions to space4 ,x seems to me1 h[ev]1 t ! :ole t5d5cy ( 8 !ory lies 9 ! opposite direction1 & po9ts to a p]fect duality betwe5 dim5sion & position 9 a dim5sion4 ,if ! ord] & ! law we see is due to ! conditions ( consci\s exp]i5ce1 we mu/ conceive nature z spontane\s1 free1 subject to no pr$ication t we c devise1 b1 h[ev] appreh5d$1 subject to our logic4 ,& our logic is simply spatiality 9 ! g5]al s5se--t resultant ( a selection ( ! p]man5t f ! unp]man5t1 ! ord]$ f ! unord]$1 by ! means ( ! gr\p & its und]ly+ duality4 ,we c pr$icate no?+ ab\t nature1 only ab\t ! way 9 :i* we c appreh5d nature4 ,all t we c say is t all t :i* exp]i5ce gives u w be condition$ z spatial1 subject to our logic4 ,?us1 9 explor+ ! facts ( geometry f ! simple/ logical relations to ! prop]ties ( space ( any numb] ( dim5sions1 we are m]ely obs]v+ ourselves1 becom+ aw>e ( ! conditions "u :i* we mu/ p]ceive4 ,d any ph5om5a pres5t !mselves incapable ( explanation "u ! assumption ( ! space we are deal+ )1 !n we mu/ habituate ourselves to ! conception ( a high] space1 9 ord] t our logic may be equal to ! task be=e u4 ,we ga9 a repetition ( ! ?\ght t came be=e1 exp]im5tally sugge/$4 ,if ! laws ( ! intellectual compreh5sion ( nature are ^? d]iv$ f consid]+ h] z absolute *ance1 subject to no law save t d]iv$ f a process ( selection1 !n1 p]haps1 ! ord] ( nature requires diff]5t faculties f ! intellectual to appreh5d x4 ,! s\rce & orig9 ( ideas may h to be s\ght else:]e ?an 9 reason+4 ,! total outcome ( ! critique is to leave ! ord9>y man j :]e he is1 ju/ifi$ 9 8 practical attitude t[>ds nature1 lib]at$ f ! fett]s ( 8 own m5tal repres5tations4 ,! tru? ( a picture lies 9 its total effect4 ,x is va9 to seek in=mation ab\t ! l&scape f an exam9ation ( ! pigm5ts4 ,& 9 any me?od ( ?\ght x is ! complexity ( ! :ole t br+s u to a k ( nature4 ,dim5sions are artificial en\gh1 b 9 ! multiplicity ( !m we cat* "s brea? ( nature4 ,we mu/ !re=e1 & ?is seems to me ! practical conclusion ( ! :ole matt]1 proce$ to =m means ( intellectual appreh5sion ( a great] & great] degree ( complexity1 bo? dim5sionally & 9 ext5t 9 any dim5sion4 ,su* means ( repres5tation mu/ always be artificial1 b 9 ! multiplicity ( ! elem5ts ) :i* we deal1 h[ev] incipi5tly arbitr>y1 lies our *ance ( appreh5d+ nature4 ,& z a conclud+ *apt] to ?is "p ( ! book1 ,i w ext5d ! figures1 :i* h be5 us$ to repres5t ,kant's !ory1 two /eps1 s t ! read] may h ! opportunity ( look+ at a f\r-dim5sional figure :i* c be del9eat$ )\t any ( ! special app>atus1 to ! consid]ation ( :i* ,i %all subsequ5tly pass on4 ,,*apt] ,x ,a ,,f\r-,,dim5sional ,,figure ,! me?od us$ 9 ! prec$+ *apt] to illu/rate ! problem ( ,kant's critique1 gives a s+ul>ly easy & direct mode ( con/ruct+ a s]ies ( important figures 9 any numb] ( dim5sions4 ,we h se5 t to repres5t our space a plane be+ mu/ give up "o ( 8 axes1 & simil>ly to repres5t ! high] %apes we mu/ give up "o among/ our ?ree axes4 ,b !re is ano!r k9d ( giv+ up :i* r$uces ! con/ruction ( high] %apes to a matt] ( ! utmo/ simplicity4 ,ord9>ily we h on a /raight l9e any numb] ( positions4 ,! weal? ( space 9 position is illimitable1 :ile !re are only ?ree dim5sions4 ,i propose to give up ?is weal? ( positions1 & to consid] ! figures obta9$ by tak+ j z _m positions z dim5sions4 ,9 ?is way ,i consid] dim5sions & positions z two 8k9ds10 & apply+ ! simple rule ( select+ e "o ( "o k9d ) e o!r ( e o!r k9d1 get a s]ies ( figures :i* are notewor?y because !y exactly fill space ( any numb] ( dim5sions 7z ! hexagon fills a plane7 by equal repetitions ( !mselves4 ,! rule w be made m evid5t by a simple application4 ,let u consid] "o dim5sion & "o position4 ,i w call ! axis i1 & ! position o4 -------------------------------i o ,"h ! figure is ! position o on ! l9e i4 ,take n[ two dim5sions & two positions on ea*4 ,illu/ration3 ,fig4 #fc4 ,we h ! two positions o2 #a on i1 & ! two positions o1 #a on j1 fig4 #fc4 ,^! give rise to a c]ta9 complexity4 ,i w let ! two l9es i & j meet 9 ! position ,i call o on ea*1 & ,i w consid] i z a direction />t+ equally f e position on j1 & j z />t+ equally f e position on i4 ,we ?us obta9 ! foll[+ figure3--,a is bo? oi & oj1 ,b is #a;i & oj1 & s on z %[n 9 fig4 #fc;b4 ,! positions on ,,ac are all oi positions4 ,!y are1 if we l to consid] x 9 t way1 po9ts at no di/ance 9 ! i direction f ! l9e ,,ac4 ,we c call ! l9e ,,ac ! oi l9e4 ,simil>ly ! po9ts on ,,ab are ^? no di/ance f ,,ab 9 ! j direction1 & we c call !m oj po9ts & ! l9e ,,ab ! oj l9e4 ,aga91 ! l9e ,,cd c be call$ ! #a;j l9e because ! po9ts on x are at a di/ance1 #a 9 ! j direction4 ,illu/ration3 ,fig4 #fc;b4 ,we h !n f\r positions or po9ts nam$ z %[n1 &1 consid]+ directions & positions z 8k9ds10 we h ! comb9ation ( two k9ds ) two k9ds4 ,n[1 select+ e "o ( "o k9d ) e o!r ( e o!r k9d w mean t we take #a ( ! k9d i & ) x o ( ! k9d j2 & !n1 t we take o ( ! k9d i & ) x #a ( ! k9d j4 ,?us we get a pair ( positions ly+ 9 ! /raight l9e ,,bc1 fig4 #fd4 ,we c call ?is pair #aj & #ja if we adopt ! plan ( m5tally1 add+ an i to ! fir/ & a j to ! second ( ! symbols writt5 ?us--#ja is a %ort expression = ,oi1 #a;j4 ,illu/ration3 ,fig4 #fd4 ,com+ n[ to our space1 we h ?ree dim5sions1 s we take ?ree positions on ea*4 ,^! positions ,i w suppose to be at equal di/ances along ea* axis4 ,! ?ree axes & ! ?ree positions on ea* are %[n 9 ! accompany+ diagrams1 fig4 #fe1 ( :i* ! fir/ repres5ts a cube ) ! front faces visible1 ! second ! re> faces ( ! same cube2 ! positions ,i w call #j1 #a1 #b2 ! axes1 i1 j1 k4 ,i take ! base ,,abc z ! />t+ place1 f :i* to det]m9e di/ances 9 ! k direction1 & h5ce e po9t 9 ! base ,,abc w be an ok position1 & ! base ,,abc c be call$ an ok plane4 ,illu/ration3 ,fig4 #fe4 ,9 ! same way1 measur+ ! di/ances f ! face ,,adc1 we see t e position 9 ! face ,,adc is an oi position1 & ! :ole plane ( ! face may be call$ an oi plane4 ,?us we see t ) ! introduction ( a new dim5sion ! signification ( a comp\nd symbol1 su* z 8oi10 alt]s4 ,9 ! plane x meant ! l9e ,,ac4 ,9 space x means ! :ole plane ,,acd4 ,n[1 x is evid5t t we h tw5ty-sev5 positions1 ea* ( !m nam$4 ,if ! read] w foll[ ?is nom5clature 9 respect ( ! positions m>k$ 9 ! figures he w h no difficulty 9 assign+ names to ea* "o ( ! tw5ty-sev5 positions4 ,a is oi1 oj1 ok4 ,x is at ! di/ance #j along i1 #j along j1 #j along k1 & io c be writt5 9 %ort #jjj1 :]e ! ijk symbols are omitt$4 ,! po9t imm$iately above is #jja1 = x is no di/ance 9 ! i direction1 & a di/ance ( #a 9 ! k direction4 ,aga91 look+ at ,b1 x is at a di/ance ( #b f ,a1 or f ! plane ,,adc1 9 ! i direction1 #j 9 ! j direction f ! plane ,,abd1 & #j 9 ! k direction1 measur$ f ! plane ,,abc4 ,h5ce x is #bjj writt5 = #b;i1 #j;j1 #jk4 ,n[1 out ( ^! tw5ty-sev5 8?+s0 or comp\nds ( position & dim5sion1 select ^? :i* are giv5 by ! rule1 e "o ( "o k9d ) e o!r ( e o!r k9d4 ,take #b ( ! i k9d4 ,) ?is we mu/ h a #a ( ! j k9d1 & !n by ! rule we c only h a #j ( ! k k9d1 = if we _h any o!r ( ! k k9d we %\ld repeat "o ( ! k9ds we already _h4 ,9 #b;i1 #a;j1 #ak1 = in/ance1 #a is repeat$4 ,! po9t we obta9 is t m>k$ #baj1 fig4 #ff4 ,illu/ration3 ,fig4 #ff4 ,proce$+ 9 ?is way1 we pick out ! foll[+ clu/] ( po9ts1 fig4 #fg4 ,!y are jo9$ by l9es1 dott$ :]e !y are hidd5 by ! body ( ! cube1 & we see t !y =m a figure--a hexagon :i* c\ld be tak5 out ( ! cube & plac$ on a plane4 ,x is a figure :i* w fill a plane by equal repetitions ( itself4 ,! plane be+ repres5t+ ?is con/ruction 9 8 plane w\ld take ?ree squ>es to repres5t ! cube4 ,let u suppose t he takes ! ij axes 9 8 space & k repres5ts ! axis runn+ out ( 8 space1 fig4 #fh4 ,9 ea* ( ! ?ree squ>es %[n "h z drawn sep>ately he c\ld select ! po9ts giv5 by ! rule1 & he w\ld !n h to try to discov] ! figure det]m9$ by ! ?ree l9es drawn4 ,! l9e f #baj to #abj is giv5 9 ! figure1 b ! l9e f #bja to #ajb or ,,gk is n giv54 ,he c det]m9e ,,gk by mak+ ano!r set ( draw+s & discov]+ 9 !m :at ! relation betwe5 ^! two extremities is4 ,illu/ration3 ,fig4 #fg4 ,illu/ration3 ,fig4 #fh4 ,illu/ration3 ,fig4 #fi4 ,let him draw ! i & k axes 9 8 plane1 fig4 #fi4 ,! j axis !n runs out & he has ! accompany+ figure4 ,9 ! fir/ ( ^! ?ree squ>es1 fig4 #fi1 he c pick out by ! rule ! two po9ts #bja1 #ajb--;,g1 & ,k4 ,"h !y occur 9 "o plane & he c measure ! di/ance betwe5 !m4 ,9 8 fir/ repres5tation !y occur at ,g & ,k 9 sep>ate figures4 ,?us ! plane be+ w\ld f9d t ! ends ( ea* ( ! l9es 0 di/ant by ! diagonal ( a unit squ>e f ! correspond+ end ( ! la/ & he c\ld !n place ! ?ree l9es 9 _! "r relative position4 ,jo9+ !m he w\ld h ! figure ( a hexagon4 ,illu/ration3 ,fig4 #gj4 ,we may also notice t ! plane be+ c\ld make a repres5tation ( ! :ole cube simultane\sly4 ,! ?ree squ>es1 %[n 9 p]spective 9 fig4 #gj1 all lie 9 "o plane1 & on ^! ! plane be+ c\ld pick out any selection ( po9ts j z well z on ?ree sep>ate squ>es4 ,he w\ld obta9 a hexagon by jo9+ ! po9ts m>k$4 ,?is hexagon1 z drawn1 is ( ! "r %ape1 b x w\ld n be s if actual squ>es 7 us$ in/ead ( p]spective1 because ! relation betwe5 ! sep>ate squ>es z !y lie 9 ! plane figure is n _! real relation4 ,! figure1 h[ev]1 z ?us con/ruct$1 w\ld give him an idea ( ! correct figure1 & he c\ld det]m9e x accurately by rememb]+ t di/ances 9 ea* squ>e 7 correct1 b 9 pass+ f "o squ>e to ano!r _! di/ance 9 ! ?ird dim5sion _h to be tak5 into acc\nt4 ,com+ n[ to ! figure made by select+ accord+ to our rule f ! :ole mass ( po9ts giv5 by f\r axes & f\r positions 9 ea*1 we mu/ fir/ draw a catalogue figure 9 :i* ! :ole assemblage is %[n4 ,we c repres5t ?is assemblage ( po9ts by f\r solid figures4 ,! fir/ giv+ all ^? positions :i* are at a di/ance ,o f our space 9 ! f\r? dim5sion1 ! second %[+ all ^? t are at a di/ance #a1 & s on4 ,^! figures w ea* be cubes4 ,! fir/ two are drawn %[+ ! front faces1 ! second two ! re> faces4 ,we w m>k ! po9ts #j1 #a1 #b1 #c1 putt+ po9ts at ^? di/ances along ea* ( ^! axes1 & suppose all ! po9ts ?us det]m9$ to be conta9$ 9 solid models ( :i* our draw+s 9 fig4 #ga are repres5tatives4 ,"h we notice t z on ! plane #j;i meant ! :ole l9e f :i* ! di/ances 9 ! i direction 0 measur$1 & z 9 space #j;i means ! :ole plane f :i* di/ances 9 ! i direction are measur$1 s n[ #j;h means ! :ole space 9 :i* ! fir/ cube /&s--measur+ away f t space by a di/ance ( "o we come to ! second cube repres5t$4 ,illu/ration3 ,fig4 #ga4 ,n[ select+ accord+ to ! rule e "o ( "o k9d ) e o!r ( e o!r k9d1 we mu/ take1 = in/ance1 #c;i1 #b;j1 #ak1 #j;h4 ,?is po9t is m>k$ #cbaj at ! l[] /> 9 ! figure4 ,x is #c 9 ! i direction1 #b 9 ! j direction1 #a 9 ! k direction1 #j 9 ! h direction4 ,) #c;i we mu/ also take #a;j1 #bk1 #j;h4 ,?is po9t is %[n by ! second /> 9 ! cube #j;h4 ,illu/ration3 ,fig4 #gb4 ,9 ! fir/ cube1 s9ce all ! po9ts are #j;h po9ts1 we c only h v>ieties 9 :i* i1 j1 k1 are accompani$ by #c1 #b1 #a4 ,! po9ts det]m9$ are m>k$ (f 9 ! diagram fig4 #gb1 & l9es are drawn jo9+ ! adjac5t pairs 9 ea* figure1 ! l9es be+ dott$ :5 !y pass )9 ! sub/ance ( ! cube 9 ! fir/ two diagrams4 ,opposite ea* po9t1 on "o side or ! o!r ( ea* cube1 is writt5 its "n4 ,x w be notic$ t ! figures are symmetrical "r & left2 & "r & left ! fir/ two numb]s are simply int]*ang$4 ,n[ ?is be+ our selection ( po9ts1 :at figure d !y make :5 all are put toge!r 9 _! prop] relative positions8 ,to det]m9e ?is we mu/ f9d ! di/ance betwe5 correspond+ corn]s ( ! sep>ate hexagons4 ,illu/ration3 ,fig4 #gc4 ,to d ?is let u keep ! axes i1 j1 9 our space1 & draw h in/ead ( k1 lett+ k run out 9 ! f\r? dim5sion1 fig4 #gc4 ,"h we h f\r cubes aga91 9 ! fir/ ( :i* all ! po9ts are #jk po9ts2 t is1 po9ts at a di/ance z]o 9 ! k direction f ! space ( ! ?ree dim5sions ijh4 ,we h all ! po9ts select$ be=e1 & "s ( ! di/ances1 :i* 9 ! la/ diagram l$ f figure to figure are %[n "h 9 ! same figure1 & s capable ( measurem5t4 ,take = in/ance ! po9ts #cabj to #cjba1 :i* 9 ! fir/ diagram 7fig4 #gb7 lie 9 ! fir/ & second figures4 ,_! actual relation is %[n 9 fig4 #gc 9 ! cube m>k$ #b,k1 :]e ! po9ts 9 "q are m>k$ ) a 994 ,we see t ! di/ance 9 "q is ! diagonal ( a unit squ>e4 ,9 l mann] we f9d t ! di/ance betwe5 correspond+ po9ts ( any two hexagonal figures is ! diagonal ( a unit squ>e4 ,! total figure is n[ easily con/ruct$4 ,an idea ( x may be ga9$ by draw+ all ! f\r cubes 9 ! catalogue figure 9 "o 7fig4 #gd74 ,^! cubes are exact repetitions ( "o ano!r1 s "o draw+ w s]ve z a repres5tation ( ! :ole s]ies1 if we take c>e to rememb] :]e we are1 :e!r 9 a #j;h1 a #a;h1 a #b;h1 or a #c;h figure1 :5 we pick out ! po9ts requir$4 ,fig4 #gd is a repres5tation ( all ! catalogue cubes put 9 "o4 ,= ! sake ( cle>ness ! front faces & ! back faces ( ?is cube are repres5t$ sep>ately4 ,illu/ration3 ,fig4 #gd4 ,! figure det]m9$ by ! select$ po9ts is %[n bel[4 ,9 putt+ ! sections toge!r "s ( ! outl9es 9 !m disappe>4 ,! l9e ,,tw = in/ance is n want$4 ,we notice t ,,pqtw & ,,twrs are ea* ! half ( a hexagon4 ,n[ ,,qv & ,,vr lie 9 "o /raight l9e4 ,h5ce ^! two hexagons fit toge!r1 =m+ "o hexagon1 & ! l9e ,,tw is only want$ :5 we consid] a section ( ! :ole figure1 we ?us obta9 ! solid repres5t$ 9 ! l[] "p ( fig4 #gd4 ,equal repetitions ( ?is figure1 call$ a tetrakaidecagon1 w fill up ?ree-dim5sional space4 ,to make ! correspond+ f\r-dim5sional figure we h to take five axes mutually at "r angles ) five po9ts on ea*4 ,a catalogue ( ! positions det]m9$ 9 five-dim5sional space c be f\nd ?us4 ,take a cube ) five po9ts on ea* ( its axes1 ! fif? po9t is at a di/ance ( f\r units ( l5g? f ! fir/ on any "o ( ! axes4 ,& s9ce ! f\r? dim5sion also /ret*es to a di/ance ( f\r we %all ne$ to repres5t ! successive sets ( po9ts at di/ances #j1 #a1 #b1 #c1 #d1 9 ! f\r? dim5sions1 five cubes4 ,n[ all ( ^! ext5d to no di/ance at all 9 ! fif? dim5sion4 ,to repres5t :at lies 9 ! fif? dim5sion we %all h to draw1 />t+ f ea* ( our cubes1 five simil> cubes to repres5t ! f\r /eps on 9 ! fif? dim5sion4 ,by ?is assemblage we get a catalogue ( all ! po9ts %[n 9 fig4 #ge1 9 :i* ,l repres5ts ! fif? dim5sion4 ,illu/ration3 ,fig4 #ge4 ,n[1 z we saw be=e1 !re is no?+ to prev5t u f putt+ all ! cubes repres5t+ ! diff]5t /ages 9 ! f\r? dim5sion 9 "o figure1 if we take note :5 we look at x1 :e!r we consid] x z a #j;h1 a #a;h1 a #b;h1 etc41 cube4 ,putt+ !n ! #j;h1 #a;h1 #b;h1 #c;h1 #d;h cubes ( ea* r[ 9 "o1 we h five cubes ) ! sides ( ea* conta9+ five positions1 ! fir/ ( ^! five cubes repres5ts ! #jl po9ts1 & has 9 x ! i po9ts f #j to #d1 ! j po9ts f #j to #d1 ! k po9ts f #j to #d1 :ile we h to specify ) reg>d to any selection we make f x1 :e!r we reg>d x z a #j;h1 a #a;h1 a #b;h1 a #c;h1 or a #d;h figure4 ,9 fig4 #gf ea* cube is repres5t$ by two draw+s1 "o ( ! front "p1 ! o!r ( ! re> "p4 ,let !n our five cubes be arrang$ be=e u & our selection be made accord+ to ! rule4 ,take ! fir/ figure 9 :i* all po9ts are #jl po9ts4 ,we _c h #j ) any o!r lett]4 ,!n1 keep+ 9 ! fir/ figure1 :i* is t ( ! #jl positions1 take fir/ ( all t selection :i* always conta9s #a;h4 ,we suppose1 !re=e1 t ! cube is a #a;h cube1 & 9 x we take i1 j1 k 9 comb9ation ) #d1 #c1 #b accord+ to ! rule4 ,! figure we obta9 is a hexagon1 z %[n1 ! "o 9 front4 ,! po9ts on ! "r h& h ! same figures z ^? on ! left1 ) ! fir/ two num]als int]*ang$4 ,next keep+ /ill to ! #jl figure let u suppose t ! cube be=e u repres5ts a section at a di/ance ( #b 9 ! h direction4 ,let all ! po9ts 9 x be consid]$ z #b;h po9ts4 ,we !n h a #jl1 #b;h region1 & h ! sets ijk & #dca left ov]4 ,we mu/ !n pick out 9 accordance ) our rule all su* po9ts z #d;i1 #c;j1 #ak4 ,^! are %[n 9 ! figure & we f9d t we c draw !m )\t confusion1 =m+ ! second hexagon f ! front4 ,go+ on 9 ?is way x w be se5 t 9 ea* ( ! five figures a set ( hexagons is pick$ out1 :i* put toge!r =m a ?ree-space figure some?+ l ! tetrakaidecagon4 ,illu/ration3 ,fig4 #gf4 ,^! sep>ate figures are ! successive /ages 9 :i* ! :ole f\r-dim5sional figure 9 :i* !y coh]e c be appreh5d$4 ,! fir/ figure & ! la/ are tetrakaidecagons4 ,^! are two ( ! solid b\nd>ies ( ! figure4 ,! o!r solid b\nd>ies c be trac$ easily4 ,"s ( !m are complete f "o face 9 ! figure to ! correspond+ face 9 ! next1 z = in/ance ! solid :i* ext5ds f ! hexagonal base ( ! fir/ figure to ! equal hexagonal base ( ! second figure4 ,?is k9d ( b\nd>y is a hexagonal prism4 ,! hexagonal prism also occurs 9 ano!r sectional s]ies1 z = in/ance1 9 ! squ>e at ! bottom ( ! fir/ figure1 ! oblong at ! base ( ! second & ! squ>e at ! base ( ! ?ird figure4 ,o!r solid b\nd>ies c be trac$ ?r\gh f\r ( ! five sectional figures4 ,?us tak+ ! hexagon at ! top ( ! fir/ figure we f9d 9 ! next a hexagon also1 ( :i* "s alt]nate sides are elongat$4 ,! top ( ! ?ird figure is also a hexagon ) ! o!r set ( alt]nate rules elongat$1 & f9ally we come 9 ! f\r? figure to a regul> hexagon4 ,^! f\r sections are ! sections ( a tetrakaidecagon z c be recognis$ f ! sections ( ?is figure :i* we h _h previ\sly4 ,h5ce ! b\nd>ies are ( two k9ds1 hexagonal prisms & tetrakaidecagons4 ,^! f\r-dim5sional figures exactly fill f\r-dim5sional space by equal repetitions ( !mselves4 ,,*apt] ,,xi ,,nom5clature ,,& ,,analogies ,,prelim9>y ,,to ,,! ,,/udy ,,( ,,f\r-,,dim5sional ,,figures ,9 ! foll[+ pages a me?od ( designat+ diff]5t regions ( space by a sy/ematic col\r s*eme has be5 adopt$4 ,! explanations h be5 giv5 9 su* a mann] z to involve no ref]5ce to models1 ! diagrams w be f\nd suffici5t4 ,b to facilitate ! /udy a description ( a set ( models is giv5 9 an app5dix :i* ! read] c ei!r make = himself or obta94 ,if models are us$ ! diagrams 9 ,*apt]s ,,xi4 & ,,xii4 w =m a guide suffici5t to indicate _! use4 ,cubes ( ! col\rs designat$ by ! diagrams %\ld be pick$ out & us$ to re9=ce ! diagrams4 ,! read]1 9 ! foll[+ description1 %\ld suppose t a bo>d or wall /ret*es away f him1 aga9/ :i* ! figures are plac$4 ,illu/ration3 ,fig4 #gg4 ,take a squ>e1 "o ( ^? %[n 9 ,fig4 #gg & give x a neutral col\r1 let ?is col\r be call$ 8null10 & be su* t x makes no appreciable diff]5ce to any col\r ) :i* x mix$4 ,if !re is no su* real col\r let u imag9e su* a col\r1 & assign to x ! prop]ties ( ! numb] z]o1 :i* makes no diff]5ce 9 any numb] to :i* x is add$4 ,above ?is squ>e place a r$ squ>e4 ,?us we symbolise ! go+ up by add+ r$ to null4 ,away f ?is null squ>e place a yell[ squ>e1 & repres5t go+ away by add+ yell[ to null4 ,to complete ! figure we ne$ a f\r? squ>e4 ,col\r ?is orange1 :i* is a mixture ( r$ & yell[1 & s appropriately repres5ts a go+ 9 a direction comp\nd$ ( up & away4 ,we h ?us a col\r s*eme :i* w s]ve to "n ! set ( squ>es drawn4 ,we h two axes ( col\rs--r$ & yell[--& !y may occupy z 9 ! figure ! direction up & away1 or !y may be turn$ ab\t2 9 any case !y enable u to "n ! f\r squ>es drawn 9 _! relation to "o ano!r4 ,n[ take1 9 ,fig4 #gh1 n9e squ>es1 & suppose t at ! end ( ! go+ 9 any direction ! col\r />t$ ) repeats itself4 ,illu/ration3 ,fig4 #gh4 ,we obta9 a squ>e nam$ z %[n4 ,let u n[1 9 fig4 #gi1 suppose ! numb] ( squ>es to be increas$1 keep+ /ill to ! pr9ciple ( col\r+ already us$4 ,"h ! nulls rema9 f\r 9 numb]4 ,!re are ?ree r$s betwe5 ! fir/ null & ! null above x1 ?ree yell[s betwe5 ! fir/ null & ! null beyond x1 :ile ! oranges increase 9 a d\ble way4 ,illu/ration3 ,fig4 #gi4 ,suppose ?is process ( enl>g+ ! numb] ( ! squ>es to be indef9itely pursu$ & ! total figure obta9$ to be r$uc$ 9 size1 we %\ld obta9 a squ>e ( :i* ! int]ior 0 all orange1 :ile ! l9es r\nd x 7 r$ & yell[1 & m]ely ! po9ts null col\r1 z 9 fig4 #hj4 ,?us all ! po9ts1 l9es1 & ! area w\ld h a col\r4 ,illu/ration3 ,fig4 #hj4 ,we c consid] ?is s*eme to orig9ate ?us3--,let a null po9t move 9 a yell[ direction & trace out a yell[ l9e & end 9 a null po9t4 ,!n let ! :ole l9e ?us trac$ move 9 a r$ direction4 ,! null po9ts at ! ends ( ! l9e w produce r$ l9es1 & end 9 null po9ts4 ,! yell[ l9e w trace out a yell[ & r$1 or orange squ>e4 ,n[1 turn+ back to fig4 #gh1 we see t ^! two ways ( nam+1 ! "o we />t$ ) & ! "o we arriv$ at1 c be comb9$4 ,by its position 9 ! gr\p ( f\r squ>es1 9 fig4 #gg1 ! null squ>e has a relation to ! yell[ & to ! r$ directions4 ,we c speak !re=e ( ! r$ l9e ( ! null squ>e )\t confusion1 mean+ !reby ! l9e ,,ab1 fig4 #ha1 :i* runs up f ! initial null po9t ,a 9 ! figure z drawn4 ,! yell[ l9e ( ! null squ>e is its l[] horizontal l9e ,,ac z x is situat$ 9 ! figure4 ,illu/ration3 ,fig4 #ha4 ,if we wi% to d5ote ! upp] yell[ l9e ,,bd1 fig4 #ha1 we c speak ( x z ! yell[ l9e1 mean+ ! yell[ l9e :i* is sep>at$ f ! prim>y yell[ l9e by ! r$ movem5t4 ,9 a simil> way ea* ( ! o!r squ>es has null po9ts1 r$ & yell[ l9es4 ,al?\gh ! yell[ squ>e is all yell[1 its l9e ,,cd1 = in/ance1 c be ref]r$ to z its r$ l9e4 ,?is nom5clature c be ext5d$4 ,if ! eight cubes drawn1 9 fig4 #hb1 are put close toge!r1 z on ! "r h& ( ! diagram1 !y =m a cube1 & 9 !m1 z ?us arrang$1 a go+ up is repres5t$ by add+ r$ to ! z]o1 or null col\r1 a go+ away by add+ yell[1 a go+ to ! "r by add+ :ite4 ,:ite is us$ z a col\r1 z a pigm5t1 :i* produces a col\r *ange 9 ! pigm5ts ) :i* x is mix$4 ,f :atev] cube ( ! l[] set we />t1 a motion up br+s u to a cube %[+ a *ange to r$1 ?us light yell[ becomes light yell[ r$1 or light orange1 :i* is call$ o*re4 ,& go+ to ! "r f ! null on ! left we h a *ange involv+ ! introduction ( :ite1 :ile ! yell[ *ange runs f front to back4 ,!re are ?ree col\r axes--! r$1 ! :ite1 ! yell[--& ^! run 9 ! position ! cubes occupy 9 ! draw+--up1 to ! "r1 away--b !y c\ld be turn$ ab\t to occupy any positions 9 space4 ,illu/ration3 ,fig4 #hb4 ,illu/ration3 ,fig4 #hc4 ,! ?ree lay]s4 ,we c conv5i5tly repres5t a block ( cubes by ?ree sets ( squ>es1 repres5t+ ea* ! base ( a cube4 ,?us ! block1 fig4 #hc1 c be repres5t$ by ! lay]s on ! "r4 ,"h1 z 9 ! case ( ! plane1 ! initial col\rs repeat !mselves at ! end ( ! s]ies4 ,proce$+ n[ to increase ! numb] ( ! cubes we obta9 fig4 #hd1 9 :i* ! initial lett]s ( ! col\rs are giv5 in/ead ( _! full names4 ,"h we see t !re are f\r null cubes z be=e1 b ! s]ies :i* spr+ f ! initial corn] w t5d to become l9es ( cubes1 z also ! sets ( cubes p>allel to !m1 />t+ f o!r corn]s4 ,?us1 f ! initial null spr+s a l9e ( r$ cubes1 a l9e ( :ite cubes1 & a l9e ( yell[ cubes4 ,if ! numb] ( ! cubes is l>gely increas$1 & ! size ( ! :ole cube is dim9i%$1 we get a cube ) null po9ts1 & ! edges col\r$ ) ^! ?ree col\rs4 ,illu/ration3 ,fig4 #hd4 ,! light yell[ cubes increase 9 two ways1 =m+ ultimately a %eet ( cubes1 & ! same is true ( ! orange & p9k sets4 ,h5ce1 ultimately ! cube ?us =m$ w\ld h r$1 :ite1 & yell[ l9es surr\nd+ p9k1 orange1 & light yell[ faces4 ,! o*re cubes increase 9 ?ree ways1 & h5ce ultimately ! :ole int]ior ( ! cube w\ld be col\r$ o*re4 ,we h ?us a nom5clature = ! po9ts1 l9es1 faces1 & solid cont5t ( a cube1 & x c be nam$ z exhibit$ 9 fig4 #he4 ,illu/ration3 ,fig4 #he4 ,we c consid] ! cube to be produc$ 9 ! foll[+ way4 ,a null po9t moves 9 a direction to :i* we atta* ! col\r indication yell[2 x g5]ates a yell[ l9e & ends 9 a null po9t4 ,! yell[ l9e ?us g5]at$ moves 9 a direction to :i* we give ! col\r indication r$4 ,?is lies up 9 ! figure4 ,! yell[ l9e traces out a yell[1 r$1 or orange squ>e1 & ea* ( its null po9ts trace out a r$ l9e1 & ends 9 a null po9t4 ,?is orange squ>e moves 9 a direction to :i* we attribute ! col\r indication :ite1 9 ?is case ! direction is ! "r4 ,! squ>e traces out a cube col\r$ orange1 r$1 or o*re1 ! r$ l9es trace out r$ to :ite or p9k squ>es1 & ! yell[ l9es trace out light yell[ squ>es1 ea* l9e end+ 9 a l9e ( its own col\r4 ,:ile ! po9ts ea* trace out a null :ite1 or :ite l9e to end 9 a null po9t4 ,n[ return+ to ! fir/ block ( eight cubes we c "n ea* po9t1 l9e1 & squ>e 9 !m by ref]5ce to ! col\r s*eme1 :i* !y det]m9e by _! relation to ea* o!r4 ,?us1 9 fig4 #hf1 ! null cube t\*es ! r$ cube by a light yell[ squ>e2 x t\*es ! yell[ cube by a p9k squ>e1 & t\*es ! :ite cube by an orange squ>e4 ,!re are ?ree axes to :i* ! col\rs r$1 yell[1 & :ite are assign$1 ! faces ( ea* cube are designat$ by tak+ ^! col\rs 9 pairs4 ,tak+ all ! col\rs toge!r we get a col\r "n = ! solidity ( a cube4 ,illu/ration3 ,fig4 #hf4 ,let u n[ ask ourselves h[ ! cube c\ld be pres5t$ to ! plane be+4 ,)\t go+ into ! "q ( h[ he c\ld h a real exp]i5ce ( x1 let u see h[1 if we c\ld turn x ab\t & %[ x to him1 he1 "u 8 limitations1 c\ld get in=mation ab\t x4 ,if ! cube 7 plac$ ) its r$ & yell[ axes aga9/ a plane1 t is re/+ aga9/ x by its orange face1 ! plane be+ w\ld obs]ve a squ>e surr\nd$ by r$ & yell[ l9es1 & hav+ null po9ts4 ,see ! dott$ squ>e1 fig4 #hg4 ,illu/ration3 ,fig4 #hg4 ,we c\ld turn ! cube ab\t ! r$ l9e s t a diff]5t face comes into juxtaposition ) ! plane4 ,suppose ! cube turn$ ab\t ! r$ l9e4 ,z x is turn+ f its fir/ position all ( x except ! r$ l9e leaves ! plane--goes absolutely out ( ! range ( ! plane be+'s appreh5sion4 ,b :5 ! yell[ l9e po9ts /raight out f ! plane !n ! p9k face comes into contact ) x4 ,?us ! same r$ l9e rema9+ z he saw x at fir/1 n[ t[>ds him comes a face surr\nd$ by :ite & r$ l9es4 ,if we call ! direction to ! "r ! unkn[n direction1 !n ! l9e he saw be=e1 ! yell[ l9e1 goes out into ?is unkn[n direction1 & ! l9e :i* be=e w5t into ! unkn[n direction1 comes 94 ,x comes 9 9 ! opposite direction to t 9 :i* ! yell[ l9e ran be=e2 ! int]ior ( ! face n[ aga9/ ! plane is p9k4 ,x is a prop]ty ( two l9es at "r angles t1 if "o turns out ( a giv5 direction & /&s at "r angles to x1 !n ! o!r ( ! two l9es comes 91 b runs ! opposite way 9 t giv5 direction1 z 9 fig4 #hh4 ,illu/ration3 ,fig4 #hh4 ,n[ ^! two pres5tations ( ! cube w\ld seem1 to ! plane creature l p]fectly diff]5t mat]ial bodies1 ) only t l9e 9 common 9 :i* !y bo? meet4 ,aga9 our cube c be turn$ ab\t ! yell[ l9e4 ,9 ?is case ! yell[ squ>e w\ld disappe> z be=e1 b a new squ>e w\ld come into ! plane aft] ! cube _h rotat$ by an angle ( #ij ab\t ?is l9e4 ,! bottom squ>e ( ! cube w\ld come 9 ?us 9 figure #hi4 ,! cube suppos$ 9 contact ) ! plane is rotat$ ab\t ! l[] yell[ l9e & !n ! bottom face is 9 contact ) ! plane4 ,"h1 z be=e1 ! r$ l9e go+ out into ! unkn[n dim5sion1 ! :ite l9e :i* be=e ran 9 ! unkn[n dim5sion w\ld come 9 d[nw>ds 9 ! opposite s5se to t 9 :i* ! r$ l9e ran be=e4 ,illu/ration3 ,fig4 #hi4 ,n[ if we use i1 j1 k1 = ! ?ree space directions1 i left to "r1 j f ne> away1 k f bel[ up2 !n1 us+ ! col\r names = ! axes1 we h t fir/ ( all :ite runs i1 yell[ runs j1 r$ runs k2 !n aft] ! fir/ turn+ r\nd ! k axis1 :ite runs negative j1 yell[ runs i1 r$ runs k2 ?us we h ! table3-- i j k #a/ position :ite yell[ r$ #bnd position yell[ :ite-- r$ #crd position r$ yell[ :ite-- ,"h :ite ) a negative sign aft] x 9 ! column "u j means t :ite runs 9 ! negative s5se ( ! j direction4 ,we may express ! fact 9 ! foll[+ way3-- ,9 ! plane !re is room = two axes :ile ! body has ?ree4 ,!re=e 9 ! plane we c repres5t any two4 ,if we want to keep ! axis t goes 9 ! unkn[n dim5sion always runn+ 9 ! positive s5se1 !n ! axis :i* orig9ally ran 9 ! unkn[n dim5sion 7! :ite axis7 mu/ come 9 9 ! negative s5se ( t axis :i* goes out ( ! plane into ! unkn[n dim5sion4 ,x is obvi\s t ! unkn[n direction1 ! direction 9 :i* ! :ite l9e runs at fir/1 is q di/9ct f any direction :i* ! plane creature kn[s4 ,! :ite l9e may come 9 t[>ds him1 or runn+ d[n4 ,if he is look+ at a squ>e1 :i* is ! face ( a cube 7look+ at x by a l9e71 !n any "o ( ! b\nd+ l9es rema9+ unmov$1 ano!r face ( ! cube may come 91 any "o ( ! faces1 namely1 :i* h ! :ite l9e 9 !m4 ,& ! :ite l9e comes sometimes 9 "o ( ! space directions he kn[s1 sometimes 9 ano!r4 ,n[ ?is turn+ :i* leaves a l9e un*ang$ is some?+ q unlike any turn+ he kn[s 9 ! plane4 ,9 ! plane a figure turns r\nd a po9t4 ,! squ>e c turn r\nd ! null po9t 9 8 plane1 & ! r$ & yell[ l9es *ange places1 only ( c\rse1 z ) e rotation ( l9es at "r angles1 if r$ goes :]e yell[ w5t1 yell[ comes 9 negative ( r$'s old direction4 ,?is turn+1 z ! plane creature conceives x1 we %\ld call turn+ ab\t an axis p]p5dicul> to ! plane4 ,:at he calls turn+ ab\t ! null po9t we call turn+ ab\t ! :ite l9e z x /&s out f 8 plane4 ,!re is no su* ?+ z turn+ ab\t a po9t1 !re is always an axis1 & really mu* m turns ?an ! plane be+ is aw>e (4 ,tak+ n[ a diff]5t po9t ( view1 let u suppose ! cubes to be pres5t$ to ! plane be+ by be+ pass$ transv]se to 8 plane4 ,let u suppose ! %eet ( matt] ov] :i* ! plane be+ & all objects 9 8 _w slide1 to be ( su* a nature t objects c pass ?r\gh x )\t break+ x4 ,let u suppose x to be ( ! same nature z ! film ( a soap bubble1 s t x closes ar\nd objects pu%$ ?r\gh x1 &1 h[ev] ! object alt]s its %ape z x passes ?r\gh x1 let u suppose ?is film to run up to ! cont\r ( ! object 9 e "p1 ma9ta9+ its plane surface unbrok54 ,!n we c pu% a cube or any object ?r\gh ! film & ! plane be+ :o slips ab\t 9 ! film w "k ! cont\r ( ! cube j & exactly :]e ! film meets x4 ,illu/ration3 ,fig4 #ij4 ,fig4 #ij repres5ts a cube pass+ ?r\gh a plane film4 ,! plane be+ n[ comes into contact ) a v ?9 slice ( ! cube some:]e betwe5 ! left & "r h& faces4 ,?is v ?9 slice he ?9ks ( z hav+ no ?ickness1 & consequ5tly 8 idea ( x is :at we call a section4 ,x is b\nd$ by him by p9k l9es front & back1 com+ f ! "p ( ! p9k face he is 9 contact )1 & above & bel[1 by light yell[ l9es4 ,its corn]s are n null-col\r$ po9ts1 b :ite po9ts1 & its int]ior is o*re1 ! col\r ( ! int]ior ( ! cube4 ,if n[ we suppose ! cube to be an in* 9 ea* dim5sion1 & to pass across1 f "r to left1 ?r\gh ! plane1 !n we %\ld expla9 ! appe>ances pres5t$ to ! plane be+ by say+3 ,fir/ ( all y h ! face ( a cube1 ?is la/s only a mom5t2 !n y h a figure ( ! same %ape b diff]5tly col\r$4 ,?is1 :i* appe>s n to move to y 9 any direction :i* y "k (1 is really mov+ transv]se to y\r plane _w4 ,its appe>ance is unalt]$1 b ea* mom5t x is some?+ diff]5t--a section fur!r on1 9 ! :ite1 ! unkn[n dim5sion4 ,f9ally1 at ! end ( ! m9ute1 a face comes 9 exactly l ! face y fir/ saw4 ,?is f9i%es up ! cube--x is ! fur!r face 9 ! unkn[n dim5sion4 ,! :ite l9e1 :i* ext5ds 9 l5g? j l ! r$ or ! yell[1 y d n see z ext5sive2 y appreh5d x simply z an endur+ :ite po9t4 ,! null po9t1 "u ! condition ( movem5t ( ! cube1 vani%es 9 a mom5t1 ! la/+ :ite po9t is really y\r appreh5sion ( a :ite l9e1 runn+ 9 ! unkn[n dim5sion4 ,9 ! same way ! r$ l9e ( ! face by :i* ! cube is fir/ 9 contact ) ! plane la/s only a mom5t1 x is succe$$ by ! p9k l9e1 & ?is p9k l9e la/s = ! inside ( a m9ute4 ,?is la/+ p9k l9e 9 y\r appreh5sion ( a surface1 :i* ext5ds 9 two dim5sions j l ! orange surface ext5ds1 z y "k x1 :5 ! cube is at re/4 ,b ! plane creature might answ]1 8,?is orange object is sub/ance1 solid sub/ance1 b\nd$ completely & on e side40 ,"h1 ( c\rse1 ! difficulty comes 94 ,8 solid is our surface--8 notion ( a solid is our notion ( an ab/ract surface ) no ?ickness at all4 ,we %\ld h to expla9 to him t1 f e po9t ( :at he call$ a solid1 a new dim5sion runs away4 ,f e po9t a l9e c be drawn 9 a direction unkn[n to him1 & !re is a solidity ( a k9d great] ?an t :i* he kn[s4 ,?is solidity c only be realis$ by him by 8 suppos+ an unkn[n direction1 by motion 9 :i* :at he conceives to be solid matt] in/antly disappe>s4 ,! high] solid1 h[ev]1 :i* ext5ds 9 ?is dim5sion z well z 9 ^? :i* he kn[s1 la/s :5 a motion ( t k9d takes place1 diff]5t sections ( x come consecutively 9 ! plane ( 8 appreh5sion1 & take ! place ( ! solid :i* he at fir/ conceives to be all4 ,?us1 ! high] solid--our solid 9 contradi/9ction to 8 area solid1 8 two-dim5sional solid1 mu/ be conceiv$ by him z some?+ :i* has duration 9 x1 "u circum/ances 9 :i* 8 matt] disappe>s out ( 8 _w4 ,we may put ! matt] ?us1 us+ ! conception ( motion4 ,a null po9t mov+ 9 a direction away g5]ates a yell[ l9e1 & ! yell[ l9e ends 9 a null po9t4 ,we suppose1 t is1 a po9t to move & m>k out ! products ( ?is motion 9 su* a mann]4 ,n[ suppose ?is :ole l9e z ?us produc$ to move 9 an upw>d direction2 x traces out ! two-dim5sional solid1 & ! plane be+ gets an orange squ>e4 ,! null po9t moves 9 a r$ l9e & ends 9 a null po9t1 ! yell[ l9e moves & g5]ates an orange squ>e & ends 9 a yell[ l9e1 ! f>!r null po9t g5]ates a r$ l9e & ends 9 a null po9t4 ,?us1 by movem5t 9 two successive directions kn[n to him1 he c imag9e 8 two-dim5sional solid produc$ ) all its b\nd>ies4 ,n[ we tell him3 8,?is :ole two-dim5sional solid c move 9 a ?ird or unkn[n dim5sion to y4 ,! null po9t mov+ 9 ?is dim5sion out ( y\r _w g5]ates a :ite l9e & ends 9 a null po9t4 ,! yell[ l9e mov+ g5]ates a light yell[ two-dim5sional solid & ends 9 a yell[ l9e1 & ?is two-dim5sional solid1 ly+ end on to y\r plane _w1 is b\nd$ on ! f> side by ! o!r yell[ l9e4 ,9 ! same way ea* ( ! l9es surr\nd+ y\r squ>e traces out an area1 j l ! orange area y "k4 ,b !re is some?+ new produc$1 some?+ :i* y _h no idea ( be=e2 x is t :i* is produc$ by ! movem5t ( ! orange squ>e4 ,t1 ?an :i* y c imag9e no?+ m solid1 itself moves 9 a direction op5 to x & produces a ?ree-dim5sional solid4 ,us+ ! addition ( :ite to symbolise ! products ( ?is motion ?is new k9d ( solid w be light orange or o*re1 & x w be b\nd$ on ! f> side by ! f9al position ( ! orange squ>e :i* trac$ x out1 & ?is f9al position we suppose to be col\r$ l ! squ>e 9 its fir/ position1 orange ) yell[ & r$ b\nd>ies & null corn]s40 ,?is product ( movem5t1 :i* x is s easy = u to describe1 w\ld be difficult = him to conceive4 ,b ?is difficulty is connect$ r ) its totality ?an ) any p>ticul> "p ( x4 ,any l9e1 or plane ( ?is1 to him high]1 solid we c\ld %[ to him1 & put 9 8 s5sible _w4 ,we h already se5 h[ ! p9k squ>e c\ld be put 9 8 _w by a turn+ ( ! cube ab\t ! r$ l9e4 ,& any section :i* we c conceive made ( ! cube c\ld be exhibit$ to him4 ,y h simply to turn ! cube & pu% x ?r\gh1 s t ! plane ( 8 exi/5ce is ! plane :i* cuts out ! giv5 section ( ! cube1 !n ! section w\ld appe> to him z a solid4 ,9 8 _w he w\ld see ! cont\r1 get to any "p ( x by digg+ d[n into x4 ,,! ,,process ,,by ,,:i* ,a ,,plane ,,be+ ,,w\ld ,,ga9 ,a ,,notion ,,( ,a ,,solid4 ,if we suppose ! plane be+ to h a g5]al idea ( ! exi/5ce ( a high] solid--our solid--we mu/ next trace out 9 detail ! me?od1 ! discipl9e1 by :i* he w\ld acquire a work+ famili>ity ) our space exi/5ce4 ,! process beg9s ) an adequate realisation ( a simple solid figure4 ,= ?is purpose we w suppose eight cubes =m+ a l>g] cube1 & fir/ we w suppose ea* cube to be col\r$ ?r\gh\t uni=mly4 ,let ! cubes 9 fig4 #ia be ! eight mak+ a l>g] cube4 ,illu/ration3 ,fig4 #ia4 ,n[1 al?\gh ea* cube is suppos$ to be col\r$ entirely ?r\gh ) ! col\r1 ! "n ( :i* is writt5 on x1 /ill we c speak ( ! faces1 edges1 & corn]s ( ea* cube z if ! col\r s*eme we h inve/igat$ held = x4 ,?us1 on ! null cube we c speak ( a null po9t1 a r$ l9e1 a :ite l9e1 a p9k face1 & s on4 ,^! col\r designations are %[n on ,no4 #a ( ! views ( ! tess]act 9 ! plate4 ,"h ^! col\r names are us$ simply 9 _! geometrical significance4 ,!y d5ote :at ! p>ticul> l9e1 etc41 ref]r$ to w\ld h z its col\r1 if 9 ref]5ce to ! p>ticul> cube ! col\r s*eme describ$ previ\sly 7 c>ri$ out4 ,if su* a block ( cubes 7 put aga9/ ! plane & !n pass$ ?r\gh x f "r to left1 at ! rate ( an in* a m9ute1 ea* cube be+ an in* ea* way1 ! plane be+ w\ld h ! foll[+ appe>ances3-- ,fir/ ( all1 f\r squ>es null1 yell[1 r$1 orange1 la/+ ea* a m9ute2 & secondly1 tak+ ! exact places ( ^! f\r squ>es1 f\r o!rs1 col\r$ :ite1 light yell[1 p9k1 o*re4 ,?us1 to make a catalogue ( ! solid body1 he w\ld h to put side by side 9 8 _w two sets ( f\r squ>es ea*1 z 9 fig4 #ib4 ,! fir/ are suppos$ to la/ a m9ute1 & !n ! o!rs to come 9 9 place ( !m1 & also la/ a m9ute4 ,illu/ration3 ,fig4 #ib4 ,9 speak+ ( !m he w\ld h to d5ote :at "p ( ! respective cube ea* squ>e repres5ts4 ,?us1 at ! beg9n+ he w\ld h null cube orange face1 & aft] ! motion _h begun he w\ld h null cube o*re section4 ,z he c\ld get ! same col\r$ section :i*ev] way ! cube pass$ ?r\gh1 x w\ld be be/ = him to call ?is section :ite section1 mean+ t x is transv]se to ! :ite axis4 ,^! col\r-names1 ( c\rse1 are m]ely us$ z names1 & d n imply 9 ?is case t ! object is really col\r$4 ,f9ally1 aft] a m9ute1 z ! fir/ cube 0 pass+ beyond 8 plane he w\ld h null cube orange face aga94 ,! same names w hold = ea* ( ! o!r cubes1 describ+ :at face or section ( !m ! plane be+ has be=e him2 & ! second wall ( cubes w come on1 cont9ue1 & g out 9 ! same mann]4 ,9 ! area he ?us has he c repres5t any movem5t :i* we c>ry out 9 ! cubes1 z long z x does n involve a motion 9 ! direction ( ! :ite axis4 ,! relation ( p>ts t succe$ "o ano!r 9 ! direction ( ! :ite axis is realis$ by him z a consecution ( /ates4 ,n[1 8 means ( develop+ 8 space appreh5sion lies 9 ?is1 t t :i* is repres5t$ z a "t sequ5ce 9 "o position ( ! cubes1 c become a real co-exi/5ce1 if some?+ t has a real co-exi/5ce becomes a "t sequ5ce4 ,we mu/ suppose ! cubes turn$ r\nd ea* ( ! axes1 ! r$ l9e1 & ! yell[ l9e1 !n some?+1 :i* 0 giv5 z "t be=e1 w n[ be giv5 z ! plane creature's space2 some?+1 :i* 0 giv5 z space be=e1 w n[ be giv5 z a "t s]ies z ! cube is pass$ ?r\gh ! plane4 ,! ?ree positions 9 :i* ! cubes mu/ be /udi$ are ! "o giv5 above & ! two foll[+ ones4 ,9 ea* case ! orig9al null po9t :i* 0 ne>e/ to u at fir/ is m>k$ by an a/]isk4 ,9 figs4 #ic & #id ! po9t m>k$ ) a /> is ! same 9 ! cubes & 9 ! plane view4 ,illu/ration3 ,fig4 #ic4 ,! cube swung r\nd ! r$ l9e1 s t ! :ite l9e po9ts t[>ds u4 ,9 fig4 #ic ! cube is swung r\nd ! r$ l9e s z to po9t t[>ds u1 & consequ5tly ! p9k face comes next to ! plane4 ,z x passes ?r\gh !re are two v>ieties ( appe>ance designat$ by ! figures #a & #b 9 ! plane4 ,^! appe>ances are nam$ 9 ! figure1 & are det]m9$ by ! ord] 9 :i* ! cubes come 9 ! motion ( ! :ole block ?r\gh ! plane4 ,) reg>d to ^! squ>es sev]ally1 h[ev]1 diff]5t names mu/ be us$1 det]m9$ by _! relations 9 ! block4 ,?us1 9 fig4 #ic1 :5 ! cube fir/ re/s aga9/ ! plane ! null cube is 9 contact by its p9k face2 z ! block passes ?r\gh we get an o*re section ( ! null cube1 b ?is is bett] call$ a yell[ section1 z x is made by a plane p]p5dicul> to ! yell[ l9e4 ,:5 ! null cube has pass$ ?r\gh ! plane1 z x is leav+ x1 we get aga9 a p9k face4 ,illu/ration3 ,fig4 #id4 ,! cube swung r\nd yell[ l9e1 ) r$ l9e runn+ f left to "r1 & :ite l9e runn+ d[n4 ,! same s]ies ( *anges take place ) ! cube appe>ances :i* foll[ on ^? ( ! null cube4 ,9 ?is motion ! yell[ cube foll[s on ! null cube1 & ! squ>e m>k$ yell[ 9 #b 9 ! plane w be fir/ 8yell[ p9k face10 !n 8yell[ yell[ section10 !n 8yell[ p9k face40 ,9 fig4 #id1 9 :i* ! cube is turn$ ab\t ! yell[ l9e1 we h a c]ta9 difficulty1 = ! plane be+ w f9d t ! position 8 squ>es are to be plac$ 9 w lie bel[ t :i* !y fir/ occupi$4 ,!y w come :]e ! support 0 on :i* he /ood 8 fir/ set ( squ>es4 ,he w get ov] ?is difficulty by mov+ 8 support4 ,!n1 s9ce ! cubes come ^u 8 plane by ! light yell[ face1 he w h1 tak+ ! null cube z be=e = an example1 null1 light yell[ face2 null1 r$ section1 because ! section is p]p5dicul> to ! r$ l9e2 & f9ally1 z ! null cube leaves ! plane1 null1 light yell[ face4 ,!n1 9 ?is case r$ foll[+ on null1 he w h ! same s]ies ( views ( ! r$ z he _h ( ! null cube4 ,illu/ration3 ,fig4 #ie4 ,!re is ano!r set ( consid]ations :i* we w briefly allude to4 ,suppose !re is a holl[ cube1 & a /r+ is /ret*$ across x f null to null1 r1 y1 :1 z we may call ! f> diagonal po9t1 h[ w ?is /r+ appe> to ! plane be+ z ! cube moves transv]se to 8 plane8 ,let u repres5t ! cube z a numb] ( sections1 say #e1 correspond+ to #d equal divisions made along ! :ite l9e p]p5dicul> to x4 ,we numb] ^! sections #j1 #a1 #b1 #c1 #d1 correspond+ to ! di/ances along ! :ite l9e at :i* !y are tak51 & imag9e ea* section to come 9 successively1 tak+ ! place ( ! prec$+ "o4 ,^! sections appe> to ! plane be+1 c\nt+ f ! fir/1 to exactly co9cide ea* ) ! prec$+ "o4 ,b ! section ( ! /r+ occupies a diff]5t place 9 ea* to t :i* x does 9 ! prec$+ section4 ,! section ( ! /r+ appe>s 9 ! position m>k$ by ! dots4 ,h5ce ! slant ( ! /r+ appe>s z a motion 9 ! frame "w m>k$ out by ! cube sides4 ,if we suppose ! motion ( ! cube n to be recognis$1 !n ! /r+ appe>s to ! plane be+ z a mov+ po9t4 ,h5ce ext5sion on ! unkn[n dim5sion appe>s z duration4 ,ext5sion slop+ 9 ! unkn[n direction appe>s z cont9u\s movem5t4 ,,*apt] ,,xii ,,! ,,simple/ ,,f\r-,,dim5sional ,,solid ,a plane be+1 9 le>n+ to appreh5d solid exi/5ce1 mu/ fir/ ( all realise t !re is a s5se ( direction altoge!r want+ to him4 ,t :i* we call "r & left does n exi/ 9 8 p]ception4 ,he mu/ assume a movem5t 9 a direction1 & a di/9ction ( positive & negative 9 t direction1 :i* has no reality correspond+ to x 9 ! movem5ts he c make4 ,?is direction1 ?is new dim5sion1 he c only make s5sible to himself by br++ 9 "t1 & suppos+ t *anges1 :i* take place 9 "t1 are due to objects ( a def9ite configuration 9 ?ree dim5sions pass+ transv]se to 8 plane1 & ! diff]5t sections ( x be+ appreh5d$ z *anges ( "o & ! same plane figure4 ,he mu/ also acquire a di/9ct notion ab\t 8 plane _w1 he mu/ no long] believe t x is ! all ( space1 b t space ext5ds on bo? sides ( x4 ,9 ord]1 !n1 to prev5t 8 mov+ (f 9 ?is unkn[n direction1 he mu/ assume a %eet1 an ext5d$ solid %eet1 9 two dim5sions1 aga9/ :i*1 9 contact ) :i*1 all 8 movem5ts take place4 ,:5 we come to ?9k ( a f\r-dim5sional solid1 :at are ! correspond+ assumptions :i* we mu/ make8 ,we mu/ suppose a s5se :i* we h n1 a s5se ( direction want+ 9 u1 some?+ :i* a be+ 9 a f\r-dim5sional _w has1 & :i* we h n4 ,x is a s5se correspond+ to a new space direction1 a direction :i* ext5ds positively & negatively f e po9t ( our space1 & :i* goes "r away f any space direction we "k (4 ,! p]p5dicul> to a plane is p]p5dicul>1 n only to two l9es 9 x1 b to e l9e1 & s we mu/ conceive ?is f\r? dim5sion z runn+ p]p5dicul>ly to ea* & e l9e we c draw 9 our space4 ,& z ! plane be+ _h to suppose some?+ :i* prev5t$ 8 mov+ (f 9 ! ?ird1 ! unkn[n dim5sion to him1 s we h to suppose some?+ :i* prev5ts u mov+ (f 9 ! direction unkn[n to u4 ,?is some?+1 s9ce we mu/ be 9 contact ) x 9 e "o ( our movem5ts1 mu/ n be a plane surface1 b a solid2 x mu/ be a solid1 :i* 9 e "o ( our movem5ts we are aga9/1 n 94 ,x mu/ be suppos$ z /ret*+ out 9 e space dim5sion t we "k2 b we are n 9 x1 we are aga9/ x1 we are next to x1 9 ! f\r? dim5sion4 ,t is1 z ! plane be+ conceives himself z hav+ a v small ?ickness 9 ! ?ird dim5sion1 ( :i* he is n aw>e 9 8 s5se exp]i5ce1 s we mu/ suppose ourselves z hav+ a v small ?ickness 9 ! f\r? dim5sion1 &1 be+ ?us f\r-dim5sional be+s1 to be prev5t$ f realis+ t we are su* be+s by a con/ra9t :i* keeps u always 9 contact ) a va/ solid %eet1 :i* /ret*es on 9 e direction4 ,we are aga9/ t %eet1 s t1 if we _h ! p[] ( f\r-dim5sional movem5t1 we %\ld ei!r g away f x or ?r\gh x2 all our space movem5ts z we "k !m be+ su* t1 p]=m+ !m1 we keep 9 contact ) ?is solid %eet4 ,n[ consid] ! exposition a plane be+ w\ld make = himself z to ! "q ( ! enclosure ( a squ>e1 & ( a cube4 ,he w\ld say ! squ>e ,a1 9 ,fig4 #if1 is completely enclos$ by ! f\r squ>es1 ,a f>1 ,a ne>1 ,a above1 ,a bel[1 or z !y are writt5 ,an1 ,af1 ,aa1 ,ab4 ,illu/ration3 ,fig4 #if4 ,if n[ he conceives ! squ>e ,a to move 9 !1 to him1 unkn[n dim5sion x w trace out a cube1 & ! b\nd+ squ>es w =m cubes4 ,w ^! completely surr\nd ! cube g5]at$ by ,a8 ,no2 !re w be two faces ( ! cube made by ,a left uncov]$2 ! fir/1 t face :i* co9cides ) ! squ>e ,a 9 its fir/ position2 ! next1 t :i* co9cides ) ! squ>e ,a 9 its f9al position4 ,aga9/ ^! two faces cubes mu/ be plac$ 9 ord] to completely enclose ! cube ,a4 ,^! may be call$ ! cubes left & "r or ,al & ,ar4 ,?us ea* ( ! enclos+ squ>es ( ! squ>e ,a becomes a cube & two m cubes are want$ to enclose ! cube =m$ by ! movem5t ( ,a 9 ! ?ird dim5sion4 ,illu/ration3 ,fig4 #ig4 ,! plane be+ c\ld n see ! squ>e ,a ) ! squ>es ,an1 ,af1 etc41 plac$ ab\t x1 because !y completely hide x f view2 & s we1 9 ! analog\s case 9 our ?ree-dim5sional _w1 _c see a cube ,a surr\nd$ by six o!r cubes4 ,^! cubes we w call ,a ne> ,an1 ,a f> ,af1 ,a above ,aa1 ,a bel[ ,ab1 ,a left ,al1 ,a "r ,ar1 %[n 9 fig4 #ig4 ,if n[ ! cube ,a moves 9 ! f\r? dim5sion "r out ( space1 x traces out a high] cube--a tess]act1 z x may be call$4 ,ea* ( ! six surr\nd+ cubes c>ri$ on 9 ! same motion w make a tess]act also1 & ^! w be gr\p$ ar\nd ! tess]act =m$ by ,a4 ,b w !y enclose x completely8 ,all ! cubes ,an1 ,af1 etc41 lie 9 our space4 ,b !re is no?+ betwe5 ! cube ,a & t solid %eet 9 contact ) :i* e p>ticle ( matt] is4 ,:5 ! cube ,a moves 9 ! f\r? direction x />ts f its position1 say ,ak1 & ends 9 a f9al position ,an 7us+ ! words 8ana0 & 8kata0 = up & d[n 9 ! f\r? dim5sion74 ,n[ ! movem5t 9 ?is f\r? dim5sion is n b\nd$ by any ( ! cubes ,an1 ,af1 nor by :at !y =m :5 ?us mov$4 ,! tess]act :i* ,a becomes is b\nd$ 9 ! positive & negative ways 9 ?is new direction by ! fir/ position ( ,a & ! la/ position ( ,a4 ,or1 if we ask h[ _m tess]acts lie ar\nd ! tess]act :i* ,a =ms1 !re are eight1 ( :i* "o meets x by ! cube ,a1 & ano!r meets x by a cube l ,a at ! end ( its motion4 ,we come "h to a v curi\s ?+4 ,! :ole solid cube ,a is to be look$ on m]ely z a b\nd>y ( ! tess]act4 ,yet ?is is exactly analog\s to :at ! plane be+ w\ld come to 9 8 /udy ( ! solid _w4 ,! squ>e ,a 7fig4 #if71 :i* ! plane be+ looks on z a solid exi/5ce 9 8 plane _w1 is m]ely ! b\nd>y ( ! cube :i* he supposes g5]at$ by its motion4 ,! fact is t we h to recognise t1 if !re is ano!r dim5sion ( space1 our pres5t idea ( a solid body1 z "o :i* has ?ree dim5sions only1 does n correspond to any?+ real1 b is ! ab/ract idea ( a ?ree-dim5sional b\nd>y limit+ a f\r-dim5sional solid1 :i* a f\r-dim5sional be+ w\ld =m4 ,! plane be+'s ?\ght ( a squ>e is n ! ?\ght ( :at we %\ld call a possibly exi/+ real squ>e1 b ! ?\ght ( an ab/ract b\nd>y1 ! face ( a cube4 ,let u n[ take our eight col\r$ cubes1 :i* =m a cube 9 space1 & ask :at additions we mu/ make to !m to repres5t ! simple/ collection ( f\r-dim5sional bodies--namely1 a gr\p ( !m ( ! same ext5t 9 e direction4 ,9 plane space we h f\r squ>es4 ,9 solid space we h eight cubes4 ,s we %\ld expect 9 f\r-dim5sional space to h sixte5 f\r-dim5sional bodies-bodies :i* 9 f\r-dim5sional space correspond to cubes 9 ?ree-dim5sional space1 & ^! bodies we call tess]acts4 ,giv5 !n ! null1 :ite1 r$1 yell[ cubes1 & ^? :i* make up ! block1 we notice t we repres5t p]fectly well ! ext5sion 9 ?ree directions 7fig4 #ih74 ,f ! null po9t ( ! null cube1 travell+ "o in*1 we come to ! :ite cube2 travell+ "o in* away we come to ! yell[ cube2 travell+ "o in* up we come to ! r$ cube4 ,n[1 if !re is a f\r? dim5sion1 !n travell+ f ! same null po9t = "o in* 9 t direction1 we mu/ come to ! body ly+ beyond ! null region4 ,illu/ration3 ,fig4 #ih4 ,i say null region1 n cube2 = ) ! introduction ( ! f\r? dim5sion ea* ( our cubes mu/ become some?+ diff]5t f cubes4 ,if !y are to h exi/5ce 9 ! f\r? dim5sion1 !y mu/ be 8fill$ up f0 9 ?is f\r? dim5sion4 ,n[ we w assume t z we get a transf]5ce f null to :ite go+ 9 "o way1 f null to yell[ go+ 9 ano!r1 s go+ f null 9 ! f\r? direction we h a transf]5ce f null to blue1 us+ ?us ! col\rs :ite1 yell[1 r$1 blue1 to d5ote transf]5ces 9 ea* ( ! f\r directions--"r1 away1 up1 unkn[n or f\r? dim5sion4 ,illu/ration3 ,fig4 #ii4 ,a plane be+'s repres5tation ( a block ( eight cubes by two sets ( f\r squ>es4 ,h5ce1 z ! plane be+ mu/ repres5t ! solid regions1 he w\ld come to by go+ "r1 z f\r squ>es ly+ 9 "s position 9 8 plane1 arbitr>ily *os51 side by side ) 8 orig9al f\r squ>es1 s we mu/ repres5t ^? eight f\r-dim5sional regions1 :i* we %\ld come to by go+ 9 ! f\r? dim5sion f ea* ( our eight cubes1 by eight cubes plac$ 9 "s arbitr>y position relative to our fir/ eight cubes4 ,illu/ration3 ,fig4 #ajj4 ,our repres5tation ( a block ( sixte5 tess]acts by two blocks ( eight cubes4#c #c ,! eight cubes us$ "h 9 #b c be f\nd 9 ! second ( ! model blocks4 ,!y c be tak5 out & us$4 ,h5ce1 ( ! two sets ( eight cubes1 ea* "o w s]ve u z a repres5tation ( "o ( ! sixte5 tess]acts :i* =m "o s+le block 9 f\r-dim5sional space4 ,ea* cube1 z we h x1 is a tray1 z x 71 aga9/ :i* ! real f\r-dim5sional figure re/s--j z ea* ( ! squ>es :i* ! plane be+ has is a tray1 s to speak1 aga9/ :i* ! cube x repres5ts c\ld re/4 ,if we suppose ! cubes to be "o in* ea* way1 !n ! orig9al eight cubes w give eight tess]acts ( ! same col\rs1 or ! cubes1 ext5d+ ea* "o in* 9 ! f\r? dim5sion4 ,b aft] ^! !re come1 go+ on 9 ! f\r? dim5sion1 eight o!r bodies1 eight o!r tess]acts4 ,^! mu/ be !re1 if we suppose ! f\r-dim5sional body we make up to h two divisions1 "o in* ea* 9 ea* ( f\r directions4 ,! col\r we *oose to designate ! transf]5ce to ?is second region 9 ! f\r? dim5sion is blue4 ,?us1 />t+ f ! null cube & go+ 9 ! f\r? dim5sion1 we fir/ g ?r\gh "o in* ( ! null tess]act1 !n we come to a blue cube1 :i* is ! beg9n+ ( a blue tess]act4 ,?is blue tess]act /ret*es "o in* f>!r on 9 ! f\r? dim5sion4 ,?us1 beyond ea* ( ! eight tess]acts1 :i* are ( ! same col\r z ! cubes :i* are _! bases1 lie eight tess]acts ^: col\rs are d]iv$ f ! col\rs ( ! fir/ eight by add+ blue4 ,?us-- ,null gives blue ,yell[ 0 gre5 ,r$ 0 purple ,orange 0 br[n ,:ite 0 light blue ,p9k 0 light purple ,light yell[ 0 light gre5 ,o*re 0 light br[n ,! addition ( blue to yell[ gives gre5--?is is a natural supposition to make4 ,x is also natural to suppose t blue add$ to r$ makes purple4 ,orange & blue c be made to give a br[n1 by us+ c]ta9 %ades & proportions4 ,& o*re & blue c be made to give a light br[n4 ,b ! s*eme ( col\rs is m]ely us$ = gett+ a def9ite & realisable set ( names & di/9ctions visible to ! eye4 ,_! naturalness is app>5t to any "o 9 ! habit ( us+ col\rs1 & may be assum$ to be ju/ifiable1 z ! sole purpose is to devise a set ( names :i* are easy to rememb]1 & :i* w give u a set ( col\rs by :i* diagrams may be made easy ( compreh5sion4 ,no sci5tific classification ( col\rs has be5 attempt$4 ,/>t+1 !n1 ) ^! sixte5 col\r names1 we h a catalogue ( ! sixte5 tess]acts1 :i* =m a f\r-dim5sional block analog\s to ! cubic block4 ,b ! cube :i* we c put 9 space & look at is n "o ( ! con/itu5t tess]acts2 x is m]ely ! beg9n+1 ! solid face1 ! side1 ! aspect1 ( a tess]act4 ,we w n[ proce$ to d]ive a "n = ea* region1 po9t1 edge1 plane face1 solid & a face ( ! tess]act4 ,! sy/em w be cle>1 if we look at a repres5tation 9 ! plane ( a tess]act ) ?ree1 & "o ) f\r divisions 9 its side4 ,! tess]act made up ( ?ree tess]acts ea* way corresponds to ! cube made up ( ?ree cubes ea* way1 & w give u a complete nom5clature4 ,9 ?is diagram1 fig4 #aja1 #a repres5ts a cube ( #bg cubes1 ea* ( :i* is ! beg9n+ ( a tess]act4 ,^! cubes are repres5t$ simply by _! l[e/ squ>es1 ! solid cont5t mu/ be und]/ood4 #b repres5ts ! #bg cubes :i* are ! beg9n+s ( ! #bg tess]acts "o in* on 9 ! f\r? dim5sion4 ,^! tess]acts are repres5t$ z a block ( cubes put side by side ) ! fir/ block1 b 9 _! prop] positions !y c\ld n be 9 space ) ! fir/ set4 #c repres5ts #bg cubes 7=m+ a l>g] cube7 :i* are ! beg9n+s ( ! tess]acts1 :i* beg9 two in*es 9 ! f\r? direction f our space & cont9ue ano!r in*4 ,illu/ration3 ,fig4 #aja4 ,illu/ration3 ,fig4 #ajbd #d ,! col\r$ plate1 figs4 #a1 #b1 #c1 %[s ^! relations m conspicu\sly4 ,9 fig4 #ajb1 we h ! repres5tation ( a block ( #d #d #d #d or #bef tess]acts4 ,!y are giv5 9 f\r consecutive sections1 ea* suppos$ to be tak5 "o in* ap>t 9 ! f\r? dim5sion1 & s giv+ f\r blocks ( cubes1 #fd 9 ea* block4 ,"h we see1 comp>+ x ) ! figure ( #ha tess]acts1 t ! numb] ( ! diff]5t regions %[ a diff]5t t5d5cy ( increase4 ,by tak+ five blocks ( five divisions ea* way ?is w\ld become ev5 m cle>4 ,we see1 fig4 #ajb1 t />t+ f ! po9t at any corn]1 ! :ite col\r$ regions only ext5d out 9 a l9e4 ,! same is true = ! yell[1 r$1 & blue4 ,) reg>d to ! latt] x %\ld be notic$ t ! l9e ( blues does n consi/ 9 regions next to ea* o!r 9 ! draw+1 b 9 portions :i* come 9 9 diff]5t cubes4 ,! portions :i* lie next to "o ano!r 9 ! f\r? dim5sion mu/ always be repres5t$ s1 :5 we h a ?ree-dim5sional repres5tation4 ,aga91 ^? regions su* z ! p9k "o1 g on increas+ 9 two dim5sions4 ,ab\t ! p9k region ?is is se5 )\t go+ out ( ! cube itself1 ! p9k regions increase 9 l5g? & height1 b 9 no o!r dim5sion4 ,9 exam9+ ^! regions x is suffici5t to take "o z a sample4 ,! purple increases 9 ! same mann]1 = x comes 9 9 a succession f bel[ to above 9 block #b1 & 9 a succession f block to block 9 #b & #c4 ,n[1 a succession f bel[ to above repres5ts a cont9u\s ext5sion upw>ds1 & a succession f block to block repres5ts a cont9u\s ext5sion 9 ! f\r? dim5sion4 ,?us ! purple regions increase 9 two dim5sions1 ! upw>d & ! f\r?1 s :5 we take a v great _m divisions1 & let ea* become v small1 ! purple region =ms a two-dim5sional ext5sion4 ,9 ! same way1 look+ at ! regions m>k$ l4 b4 or light blue1 :i* />ts ne>e/ a corn]1 we see t ! tess]acts occupy+ x increase 9 l5g? f left to "r1 =m+ a l9e1 & t !re are z _m l9es ( light blue tess]acts z !re are sections betwe5 ! fir/ & la/ section4 ,h5ce ! light blue tess]acts increase 9 numb] 9 two ways--9 ! "r & left1 & 9 ! f\r? dim5sion4 ,!y ultimately =m :at we may call a plane surface4 ,n[ all ^? regions :i* conta9 a mixture ( two simple col\rs1 :ite1 yell[1 r$1 blue1 increase 9 two ways4 ,on ! o!r h&1 ^? :i* conta9 a mixture ( ?ree col\rs increase 9 ?ree ways4 ,take1 = in/ance1 ! o*re region2 ?is has ?ree col\rs1 :ite1 yell[1 r$2 & 9 ! cube itself x increases 9 ?ree ways4 ,n[ reg>d ! orange region2 if we add blue to ?is we get a br[n4 ,! region ( ! br[n tess]acts ext5ds 9 two ways on ! left ( ! second block1 ,no4 #b 9 ! figure4 ,x ext5ds also f left to "r 9 succession f "o section to ano!r1 f section #b to section #c 9 our figure4 ,h5ce ! br[n tess]acts increase 9 numb] 9 ?ree dim5sions upw>ds1 to & fro1 f\r? dim5sion4 ,h5ce !y =m a cubic1 a ?ree-dim5sional region2 ?is region ext5ds up & d[n1 ne> & f>1 & 9 ! f\r? direction1 b is ?9 9 ! direction f left to "r4 ,x is a cube :i*1 :5 ! complete tess]act is repres5t$ 9 our space1 appe>s z a s]ies ( faces on ! successive cubic sections ( ! tess]act4 ,comp>e fig4 #ajc 9 :i* ! middle block1 #b1 /&s z repres5t+ a great numb] ( sections int]m$iate betwe5 #a & #c4 ,9 a simil> way f ! p9k region by addition ( blue we h ! light purple region1 :i* c be se5 to increase 9 ?ree ways z ! numb] ( divisions becomes great]4 ,! ?ree ways 9 :i* ?is region ( tess]acts ext5ds is up & d[n1 "r & left1 f\r? dim5sion4 ,f9ally1 !re=e1 x =ms a cubic mass ( v small tess]acts1 & :5 ! tess]act is giv5 9 space sections x appe>s on ! faces conta9+ ! upw>d & ! "r & left dim5sions4 ,we get !n altoge!r1 z ?ree-dim5sional regions1 o*re1 br[n1 light purple1 light gre54 ,f9ally1 !re is ! region :i* corresponds to a mixture ( all ! col\rs2 !re is only "o region su* z ?is4 ,x is ! "o t spr+s f o*re by ! addition ( blue--?is col\r we call light br[n4 ,look+ at ! light br[n region we see t x increases 9 f\r ways4 ,h5ce1 ! tess]acts ( :i* x is compos$ increase 9 numb] 9 ea* ( f\r dim5sions1 & ! %ape !y =m does n rema9 ?9 9 any ( ! f\r dim5sions4 ,consequ5tly ?is region becomes ! solid cont5t ( ! block ( tess]acts1 itself2 x is ! real f\r-dim5sional solid4 ,all ! o!r regions are !n b\nd>ies ( ?is light br[n region4 ,if we suppose ! process ( increas+ ! numb] ( tess]acts & dim9i%+ _! size c>ri$ on indef9itely1 !n ! light br[n col\r$ tess]acts become ! :ole int]ior mass1 ! ?ree-col\r$ tess]acts become ?ree-dim5sional b\nd>ies1 ?9 9 "o dim5sion1 & =m ! o*re1 ! br[n1 ! light purple1 ! light gre54 ,! two-col\r$ tess]acts become two-dim5sional b\nd>ies1 ?9 9 two dim5sions1 e4g41 ! p9k1 ! gre51 ! purple1 ! orange1 ! light blue1 ! light yell[4 ,! "o-col\r$ tess]acts become b\nd+ l9es1 ?9 9 ?ree dim5sions1 & ! null po9ts become b\nd+ corn]s1 ?9 9 f\r dim5sions4 ,f ^! ?9 real b\nd>ies we c pass 9 ?\ght to ! ab/ractions--po9ts1 l9es1 faces1 solids--b\nd+ ! f\r-dim5sional solid1 :i* 9 ?is case is light br[n col\r$1 & "u ?is supposition ! light br[n col\r$ region is ! only real "o1 is ! only "o :i* is n an ab/raction4 ,x %\ld be obs]v$ t1 9 tak+ a squ>e z ! repres5tation ( a cube on a plane1 we only repres5t "o face1 or ! section betwe5 two faces4 ,! squ>es1 z drawn by a plane be+1 are n ! cubes !mselves1 b repres5t ! faces or ! sections ( a cube4 ,?us 9 ! plane be+'s diagram a cube ( tw5ty-sev5 cubes 8null0 repres5ts a cube1 b is really1 9 ! normal position1 ! orange squ>e ( a null cube1 & may be call$ null1 orange squ>e4 ,a plane be+ w\ld save himself confusion if he nam$ 8 repres5tative squ>es1 n by us+ ! names ( ! cubes simply1 b by add+ to ! names ( ! cubes a ^w to %[ :at "p ( a cube 8 repres5tative squ>e 04 ,?us a cube null /&+ aga9/ 8 plane t\*es x by null orange face1 pass+ ?r\gh 8 plane x has 9 ! plane a squ>e z trace1 :i* is null :ite section1 if we use ! phrase :ite section to mean a section drawn p]p5dicul> to ! :ite l9e4 ,9 ! same way ! cubes :i* we take z repres5tative ( ! tess]act are n ! tess]act itself1 b def9ite faces or sections ( x4 ,9 ! prec$+ figures we %\ld say !n1 n null1 b 8null tess]act o*re cube10 because ! cube we actually h is ! "o det]m9$ by ! ?ree axes1 :ite1 r$1 yell[4 ,!re is ano!r way 9 :i* we c reg>d ! col\r nom5clature ( ! b\nd>ies ( a tess]act4 ,consid] a null po9t to move trac+ out a :ite l9e "o in* 9 l5g?1 & t]m9at+ 9 a null po9t1 see fig4 #ajc or 9 ! col\r$ plate4 ,!n consid] ?is :ite l9e ) its t]m9al po9ts itself to move 9 a second dim5sion1 ea* ( ! po9ts traces out a l9e1 ! l9e itself traces out an area1 & gives two l9es z well1 its initial & its f9al position4 ,?us1 if we call 8a region0 any elem5t ( ! figure1 su* z a po9t1 or a l9e1 etc41 e 8region0 9 mov+ traces out a new k9d ( region1 8a high] region10 & gives two regions ( its own k9d1 an initial & a f9al position4 ,! 8high] region0 means a region ) ano!r dim5sion 9 x4 ,n[ ! squ>e c move & g5]ate a cube4 ,! squ>e light yell[ moves & traces out ! mass ( ! cube4 ,lett+ ! addition ( r$ d5ote ! region made by ! motion 9 ! upw>d direction we get an o*re solid4 ,! light yell[ face 9 its initial & t]m9al positions give ! two squ>e b\nd>ies ( ! cube above & bel[4 ,!n ea* ( ! f\r l9es ( ! light yell[ squ>e--:ite1 yell[1 & ! :ite1 yell[ opposite !m--trace out a b\nd+ squ>e4 ,s !re are 9 all six b\nd+ squ>es1 f\r ( ^! squ>es be+ designat$ 9 col\r by add+ r$ to ! col\r ( ! g5]at+ l9es4 ,f9ally1 ea* po9t mov+ 9 ! up direction gives rise to a l9e col\r$ null r$1 or r$1 & !n !re are ! initial & t]m9al positions ( ! po9ts giv+ eight po9ts4 ,! numb] ( ! l9es is evid5tly twelve1 = ! f\r l9es ( ?is light yell[ squ>e give f\r l9es 9 _! initial1 f\r l9es 9 _! f9al position1 :ile ! f\r po9ts trace out f\r l9es1 t is altoge!r twelve l9es4 ,n[ ! squ>es are ea* ( !m sep>ate b\nd>ies ( ! cube1 :ile ! l9es belong1 ea* ( !m1 to two squ>es1 ?us ! r$ l9e is t :i* is common to ! orange & p9k squ>es4 ,n[ suppose t !re is a direction1 ! f\r? dim5sion1 :i* is p]p5dicul> alike to e "o ( ! space dim5sions already us$--a dim5sion p]p5dicul>1 = in/ance1 to up & to "r h&1 s t ! p9k squ>e mov+ 9 ?is direction traces out a cube4 ,a dim5sion1 moreov]1 p]p5dicul> to ! up & away directions1 s t ! orange squ>e mov+ 9 ?is direction also traces out a cube1 & ! light yell[ squ>e1 too1 mov+ 9 ?is direction traces out a cube4 ,"u ?is supposition1 ! :ole cube mov+ 9 ! unkn[n dim5sion1 traces out some?+ new--a new k9d ( volume1 a high] volume4 ,?is high] volume is a f\r-dim5sional volume1 & we designate x 9 col\r by add+ blue to ! col\r ( t :i* by mov+ g5]ates x4 ,x is g5]at$ by ! motion ( ! o*re solid1 & h5ce x is ( ! col\r we call light br[n 7:ite1 yell[1 r$1 blue1 mix$ toge!r74 ,x is repres5t$ by a numb] ( sections l #b 9 fig4 #ajc4 ,n[ ?is light br[n high] solid has = b\nd>ies3 fir/1 ! o*re cube 9 its initial position1 second1 ! same cube 9 its f9al position1 #a & #c1 fig4 #ajc4 ,ea* ( ! squ>es :i* b\nd ! cube1 moreov]1 by movem5t 9 ?is new direction traces out a cube1 s we h f ! front p9k faces ( ! cube1 ?ird1 a p9k blue or light purple cube1 %[n z a light purple face on cube #b 9 fig4 #ajc1 ?is cube /&+ = any numb] ( int]m$iate sections2 f\r?1 a simil> cube f ! opposite p9k face2 fif?1 a cube trac$ out by ! orange face--?is is col\r$ br[n & is repres5t$ by ! br[n face ( ! section cube 9 fig4 #ajc2 six?1 a correspond+ br[n cube on ! "r h&2 sev5?1 a cube />t+ f ! light yell[ squ>e bel[2 ! unkn[n dim5sion is at "r angles to ?is also4 ,?is cube is col\r$ light yell[ & blue or light gre52 &1 f9ally1 eigh?1 a correspond+ cube f ! upp] light yell[ face1 %[n z ! light gre5 squ>e at ! top ( ! section cube4 ,! tess]act has ?us eight cubic b\nd>ies4 ,^! completely enclose x1 s t x w\ld be invisible to a f\r-dim5sional be+4 ,n[1 z to ! o!r b\nd>ies1 j z ! cube has squ>es1 l9es1 po9ts1 z b\nd>ies1 s ! tess]act has cubes1 squ>es1 l9es1 po9ts1 z b\nd>ies4 ,! numb] ( squ>es is f\nd ?us--r\nd ! cube are six squ>es1 ^! w give six squ>es 9 _! initial & six 9 _! f9al positions4 ,!n ea* ( ! twelve l9es ( ! cube trace out a squ>e 9 ! motion 9 ! f\r? dim5sion4 ,h5ce !re w be altoge!r #ab #ab #bd squ>es4 ,if we look at any "o ( ^! squ>es we see t x is ! meet+ surface ( two ( ! cubic sides4 ,?us1 ! r$ l9e by its movem5t 9 ! f\r? dim5sion1 traces out a purple squ>e--?is is common to two cubes1 "o ( :i* is trac$ out by ! p9k squ>e mov+ 9 ! f\r? dim5sion1 & ! o!r is trac$ out by ! orange squ>e mov+ 9 ! same way4 ,to take ano!r squ>e1 ! light yell[ "o1 ?is is common to ! o*re cube & ! light gre5 cube4 ,! o*re cube comes f ! light yell[ squ>e by mov+ x 9 ! up direction1 ! light gre5 cube is made f ! light yell[ squ>e by mov+ x 9 ! f\r? dim5sion4 ,! numb] ( l9es is ?irty-two1 = ! twelve l9es ( ! cube give twelve l9es ( ! tess]act 9 _! initial position1 & twelve 9 _! f9al position1 mak+ tw5ty-f\r1 :ile ea* ( ! eight po9ts traces out a l9e1 ?us =m+ ?irty-two l9es altoge!r4 ,! l9es are ea* ( !m common to ?ree cubes1 or to ?ree squ>e faces2 take1 = in/ance1 ! r$ l9e4 ,?is is common to ! orange face1 ! p9k face1 & t face :i* is =m$ by mov+ ! r$ l9e 9 ! six? dim5sion1 namely1 ! purple face4 ,x is also common to ! o*re cube1 ! pale purple cube1 & ! br[n cube4 ,! po9ts are common to six squ>e faces & to f\r cubes2 ?us1 ! null po9t f :i* we />t is common to ! ?ree squ>e faces--p9k1 light yell[1 orange1 & to ! ?ree squ>e faces made by mov+ ! ?ree l9es :ite1 yell[1 r$1 9 ! f\r? dim5sion1 namely1 ! light blue1 ! light gre51 ! purple faces--t is1 to six faces 9 all4 ,! f\r cubes :i* meet 9 x are ! o*re cube1 ! light purple cube1 ! br[n cube1 & ! light gre5 cube4 ,illu/ration3 ,fig4 #ajc4 ,! tess]act1 r$1 :ite1 yell[ axes 9 space4 ,9 ! l[] l9e ! ?ree re> faces are %[n1 ! int]ior be+ remov$4 ,illu/ration3 ,fig4 #ajd4 ,! tess]act1 r$1 yell[1 blue axes 9 space1 ! blue axis runn+ to ! left1 opposite faces are col\r$ id5tically4 ,a complete view ( ! tess]act 9 its v>i\s space pres5tations is giv5 9 ! foll[+ figures or catalogue cubes1 figs4 #ajc-ajf4 ,! fir/ cube 9 ea* figure repres5ts ! view ( a tess]act col\r$ z describ$ z x beg9s to pass transv]se to our space4 ,! int]m$iate figure repres5ts a sectional view :5 x is p>tly ?r\gh1 & ! f9al figure repres5ts ! f> end z x is j pass+ out4 ,^! figures w be expla9$ 9 detail 9 ! next *apt]4 ,illu/ration3 ,fig4 #aje4 ,! tess]act1 ) r$1 :ite1 blue axes 9 space4 ,opposite faces are col\r$ id5tically4 ,illu/ration3 ,fig4 #ajf4 ,! tess]act1 ) blue1 :ite1 yell[ axes 9 space4 ,! blue axis runs d[nw>d f ! base ( ! o*re cube z x /&s orig9ally4 ,opposite faces are col\r$ id5tically4 ,we h ?us obta9$ a nom5clature = ea* ( ! regions ( a tess]act2 we c speak ( any "o ( ! eight b\nd+ cubes1 ! tw5ty squ>e faces1 ! ?irty-two l9es1 ! sixte5 po9ts4 ,,*apt] ,,xiii ,,rem>ks ,,on ,,! ,,figures ,an inspection ( above figures w give an answ] to _m que/ions ab\t ! tess]act4 ,if we h a tess]act "o in* ea* way1 !n x c be repres5t$ z a cube--a cube hav+ :ite1 yell[1 r$ axes1 & f ?is cube z a beg9n+1 a volume ext5d+ into ! f\r? dim5sion4 ,n[ suppose ! tess]act to pass transv]se to our space1 ! cube ( ! r$1 yell[1 :ite axes disappe>s at once1 x is indef9itely ?9 9 ! f\r? dim5sion4 ,its place is occupi$ by ^? p>ts ( ! tess]act :i* lie fur!r away f our space 9 ! f\r? dim5sion4 ,ea* "o ( ^! sections w la/ only = "o mom5t1 b ! :ole ( !m w take up "s appreciable "t 9 pass+4 ,if we take ! rate ( "o in* a m9ute ! sections w take ! :ole ( ! m9ute 9 _! passage across our space1 !y w take ! :ole ( ! m9ute except ! mom5t :i* ! beg9n+ cube & ! end cube occupy 9 _! cross+ our space4 ,9 ea* "o ( ! cubes1 ! section cubes1 we c draw l9es 9 all directions except 9 ! direction occupi$ by ! blue l9e1 ! f\r? dim5sion2 l9es 9 t direction are repres5t$ by ! transition f "o section cube to ano!r4 ,?us to give ourselves an adequate repres5tation ( ! tess]act we ought to h a limitless numb] ( section cubes int]m$iate betwe5 ! fir/ b\nd+ cube1 ! o*re cube1 & ! la/ b\nd+ cube1 ! o!r o*re cube4 ,practically ?ree int]m$iate sectional cubes w be f\nd suffici5t = mo/ purposes4 ,we w take !n a s]ies ( five figures--two t]m9al cubes1 & ?ree int]m$iate sections--& %[ h[ ! diff]5t regions appe> 9 our space :5 we take ea* set ( ?ree out ( ! f\r axes ( ! tess]act z ly+ 9 our space4 ,9 fig4 #ajg initial lett]s are us$ = ! col\rs4 ,a ref]5ce to fig4 #ajc w %[ ! complete nom5clature1 :i* is m]ely indicat$ "h4 ,illu/ration3 ,fig4 #ajg4 ,9 ?is figure ! tess]act is %[n 9 five /ages di/ant f our space3 fir/1 z]o2 second1 #a_/d 942 ?ird1 #b_/d 942 f\r?1 #c_/d 942 fif?1 #a 942 :i* are call$ b#j1 b#a1 b#b1 b#c1 b#d1 because !y are sections tak5 at di/ances #j1 #a1 #b1 #c1 #d qu>t] in*es along ! blue l9e4 ,all ! regions c be nam$ f ! fir/ cube1 ! b#j cube1 z be=e1 simply by rememb]+ t transf]5ce along ! b axis gives ! addition ( blue to ! col\r ( ! region 9 ! o*re1 ! b#j cube4 ,9 ! f9al cube b#d1 ! col\r+ ( ! orig9al b#j cube is repeat$4 ,?us ! r$ l9e mov$ along ! blue axis gives a r$ & blue or purple squ>e4 ,?is purple squ>e appe>s z ! ?ree purple l9es 9 ! sections b#a1 b#b1 b#c1 tak5 at #a_/d1 #b_/d1 #c_/d ( an in* 9 ! f\r? dim5sion4 ,if ! tess]act moves transv]se to our space we h !n 9 ?is p>ticul> region1 fir/ ( all a r$ l9e :i* la/s = a mom5t1 secondly a purple l9e :i* takes its place4 ,?is purple l9e la/s = a m9ute--t is1 all ( a m9ute1 except ! mom5t tak5 by ! cross+ our space ( ! initial & f9al r$ l9e4 ,! purple l9e hav+ la/$ = ?is p]iod is succe$$ by a r$ l9e1 :i* la/s = a mom5t2 !n ?is goes & ! tess]act has pass$ across our space4 ,! f9al r$ l9e we call r$ bl41 because x is sep>at$ f ! initial r$ l9e by a di/ance along ! axis = :i* we use ! col\r blue4 ,?us a l9e t la/s repres5ts an area duration2 is 9 ?is mode ( pres5tation equival5t to a dim5sion ( space4 ,9 ! same way ! :ite l9e1 dur+ ! cross+ our space by ! tess]act1 is succe$$ by a light blue l9e :i* la/s = ! inside ( a m9ute1 & z ! tess]act leaves our space1 hav+ cross$ x1 ! :ite bl4 l9e appe>s z ! f9al t]m9ation4 ,take n[ ! p9k face4 ,mov$ 9 ! blue direction x traces out a light purple cube4 ,?is light purple cube is %[n 9 sections 9 b#a1 b#b1 b#c1 & ! f>!r face ( ?is cube 9 ! blue direction is %[n 9 b#d--;a p9k face1 call$ p9k b because x is di/ant f ! p9k face we began ) 9 ! blue direction4 ,?us ! cube :i* we col\r light purple appe>s z a la/+ squ>e4 ,! squ>e face itself1 ! p9k face1 vani%es in/antly ! tess]act beg9s to move1 b ! light purple cube appe>s z a la/+ squ>e4 ,"h also duration is ! equival5t ( a dim5sion ( space--a la/+ squ>e is a cube4 ,x is useful to connect ^! diagrams ) ! views giv5 9 ! col\r$ plate4 ,take aga9 ! orange face1 t det]m9$ by ! r$ & yell[ axes2 f x goes a br[n cube 9 ! blue direction1 = r$ & yell[ & blue are suppos$ to make br[n4 ,?is br[n cube is %[n 9 ?ree sections 9 ! faces b#a1 b#b1 b#c4 ,9 b#d is ! opposite orange face ( ! br[n cube1 ! face call$ orange b1 = x is di/ant 9 ! blue direction f ! orange face4 ,z ! tess]act passes transv]se to our space1 we h !n 9 ?is region an in/antly vani%+ orange squ>e1 foll[$ by a la/+ br[n squ>e1 & f9ally an orange face :i* vani%es in/antly4 ,n[1 z any ?ree axes w be 9 our space1 let u s5d ! :ite axis out into ! unkn[n1 ! f\r? dim5sion1 & take ! blue axis into our kn[n space dim5sion4 ,s9ce ! :ite & blue axes are p]p5dicul> to ea* o!r1 if ! :ite axis goes out into ! f\r? dim5sion 9 ! positive s5se1 ! blue axis w come into ! direction ! :ite axis occupi$1 9 ! negative s5se4 ,illu/ration3 ,fig4 #ajh4 ,h5ce1 n to complicate matt]s by hav+ to ?9k ( two s5ses 9 ! unkn[n direction1 let u s5d ! :ite l9e into ! positive s5se ( ! f\r? dim5sion1 & take ! blue "o z runn+ 9 ! negative s5se ( t direction :i* ! :ite l9e has left2 let ! blue l9e1 t is1 run to ! left4 ,we h n[ ! r[ ( figures 9 fig4 #ajh4 ,! dott$ cube %[s :]e we _h a cube :5 ! :ite l9e ran 9 our space--n[ x has turn$ out ( our space1 & ano!r solid b\nd>y1 ano!r cubic face ( ! tess]act comes into our space4 ,?is cube has r$ & yell[ axes z be=e2 b n[1 in/ead ( a :ite axis runn+ to ! "r1 !re is a blue axis runn+ to ! left4 ,"h we c di/+ui% ! regions by col\rs 9 a p]fectly sy/ematic way4 ,! r$ l9e traces out a purple squ>e 9 ! transf]5ce along ! blue axis by :i* ?is cube is g5]at$ f ! orange face4 ,?is purple squ>e made by ! motion ( ! r$ l9e is ! same purple face t we saw be=e z a s]ies ( l9es 9 ! sections b#a1 b#b1 b#c4 ,"h1 s9ce bo? r$ & blue axes are 9 our space1 we h no ne$ ( duration to repres5t ! area !y det]m9e4 ,9 ! motion ( ! tess]act across space ?is purple face w\ld in/antly disappe>4 ,f ! orange face1 :i* is common to ! initial cubes 9 fig4 #ajg & fig4 #ajh1 !re goes 9 ! blue direction a cube col\r$ br[n4 ,?is br[n cube is n[ all 9 our space1 because ea* ( its ?ree axes run 9 space directions1 up1 away1 to ! left4 ,x is ! same br[n cube :i* appe>$ z ! successive faces on ! sections b#a1 b#b1 b#c4 ,hav+ all its ?ree axes 9 our space1 x is giv5 9 ext5sion2 no "p ( x ne$s to be repres5t$ z a succession4 ,! tess]act is n[ 9 a new position ) reg>d to our space1 & :5 x moves across our space ! br[n cube in/antly disappe>s4 ,9 ord] to exhibit ! o!r regions ( ! tess]act we mu/ rememb] t n[ ! :ite l9e runs 9 ! unkn[n dim5sion4 ,:]e %all we put ! sections at di/ances along ! l9e8 ,any arbitr>y position 9 our space w d3 !re is no way by :i* we c repres5t _! real position4 ,h[ev]1 z ! br[n cube comes (f f ! orange face to ! left1 let u put ^! successive sections to ! left4 ,we c call !m :#j1 :#a1 :#b1 :#c1 :#d1 because !y are sections along ! :ite axis1 :i* n[ runs 9 ! unkn[n dim5sion4 ,runn+ f ! purple squ>e 9 ! :ite direction we f9d ! light purple cube4 ,?is is repres5t$ 9 ! sections :#a1 :#b1 :#c1 :#d1 fig4 #ajh4 ,x is ! same cube t is repres5t$ 9 ! sections b#a1 b#b1 b#c3 9 fig4 #ajg ! r$ & :ite axes are 9 our space1 ! blue out ( x2 9 ! o!r case1 ! r$ & blue are 9 our space1 ! :ite out ( x4 ,x is evid5t t ! face p9k y1 opposite ! p9k face 9 fig4 #ajg1 makes a cube %[n 9 squ>es 9 b#a1 b#b1 b#c1 b#d1 on ! opposite side to ! l purple squ>es4 ,also ! light yell[ face at ! base ( ! cube b#j1 makes a light gre5 cube1 %[n z a s]ies ( base squ>es4 ,! same light gre5 cube c be f\nd 9 fig4 #ajg4 ,! base squ>e 9 :#j is a gre5 squ>e1 = x is enclos$ by blue & yell[ axes4 ,f x goes a cube 9 ! :ite direction1 ?is is !n a light gre5 cube & ! same z ! "o j m5tion$ z exi/+ 9 ! sections b#j1 b#a1 b#b1 b#c1 b#d4 ,! case is1 h[ev]1 a little diff]5t ) ! br[n cube4 ,?is cube we h altoge!r 9 space 9 ! section :#j1 fig4 #ajh1 :ile x exi/s z a s]ies ( squ>es1 ! left-h& ones1 9 ! sections b#j1 b#a1 b#b1 b#c1 b#d4 ,! br[n cube exi/s z a solid 9 our space1 z %[n 9 fig4 #ajh4 ,9 ! mode ( repres5tation ( ! tess]act exhibit$ 9 fig4 #ajg1 ! same br[n cube appe>s z a succession ( squ>es4 ,t is1 z ! tess]act moves across space1 ! br[n cube w\ld actually be to u a squ>e--x w\ld be m]ely ! la/+ b\nd>y ( ano!r solid4 ,x w\ld h no ?ickness at all1 only ext5sion 9 two dim5sions1 & its duration w\ld %[ its solidity 9 ?ree dim5sions4 ,x is obvi\s t1 if !re is a f\r-dim5sional space1 matt] 9 ?ree dim5sions only is a m]e ab/raction2 all mat]ial objects mu/ !n h a slight f\r-dim5sional ?ickness4 ,9 ?is case ! above /atem5t w und]go modification4 ,! mat]ial cube :i* is us$ z ! model ( ! b\nd>y ( a tess]act w h a slight ?ickness 9 ! f\r? dim5sion1 & :5 ! cube is pres5t$ to u 9 ano!r aspect1 x w\ld n be a m]e surface4 ,b x is mo/ conv5i5t to reg>d ! cubes we use z hav+ no ext5sion at all 9 ! f\r? dim5sion4 ,?is consid]ation s]ves to br+ out a po9t allud$ to be=e1 t1 if !re is a f\r? dim5sion1 our conception ( a solid is ! conception ( a m]e ab/raction1 & our talk+ ab\t real ?ree-dim5sional objects w\ld seem to a f\r-dim5sional be+ z incorrect z a two-dim5sional be+'s tell+ ab\t real squ>es1 real triangles1 etc41 w\ld seem to u4 ,! consid]ation ( ! two views ( ! br[n cube %[s t any section ( a cube c be look$ at by a pres5tation ( ! cube 9 a diff]5t position 9 f\r-dim5sional space4 ,! br[n faces 9 b#a1 b#b1 b#c1 are ! v same br[n sections t w\ld be obta9$ by cutt+ ! br[n cube1 :#j1 across at ! "r di/ances along ! blue l9e1 z %[n 9 fig4 #ajh4 ,b z ^! sections are plac$ 9 ! br[n cube1 :#j1 !y come beh9d "o ano!r 9 ! blue direction4 ,n[1 9 ! sections :#a1 :#b1 :#c1 we are look+ at ^! sections f ! :ite direction--! blue direction does n exi/ 9 ^! figures4 ,s we see !m 9 a direction at "r angles to t 9 :i* !y occur beh9d "o ano!r 9 :#j4 ,!re are int]m$iate views1 :i* w\ld come 9 ! rotation ( a tess]act4 ,^! br[n squ>es c be look$ at f directions int]m$iate betwe5 ! :ite & blue axes4 ,x mu/ be rememb]$ t ! f\r? dim5sion is p]p5dicul> equally to all ?ree space axes4 ,h5ce we mu/ take ! comb9ations ( ! blue axis1 ) ea* two ( our ?ree axes1 :ite1 r$1 yell[1 9 turn4 ,9 fig4 #aji we take r$1 :ite1 & blue axes 9 space1 s5d+ yell[ into ! f\r? dim5sion4 ,if x goes into ! positive s5se ( ! f\r? dim5sion ! blue l9e w come 9 ! opposite direction to t 9 :i* ! yell[ l9e ran be=e4 ,h5ce1 ! cube det]m9$ by ! :ite1 r$1 blue axes1 w />t f ! p9k plane & run t[>ds u4 ,! dott$ cube %[s :]e ! o*re cube 04 ,:5 x is turn$ out ( space1 ! cube com+ t[>ds f its front face is ! "o :i* comes into our space 9 ?is turn+4 ,s9ce ! yell[ l9e n[ runs 9 ! unkn[n dim5sion we call ! sections y#j1 y#a1 y#b1 y#c1 y#d1 z !y are made at di/ances #j1 #a1 #b1 #c1 #d1 qu>t] in*es along ! yell[ l9e4 ,we suppose ^! cubes arrang$ 9 a l9e com+ t[>ds u--n t t is any m natural ?an any o!r arbitr>y s]ies ( positions1 b x agrees ) ! plan previ\sly adopt$4 ,illu/ration3 ,fig4 #aji4 ,! int]ior ( ! fir/ cube1 y#j1 is t d]iv$ f p9k by add+ blue1 or1 z we call x1 light purple4 ,! faces ( ! cube are light blue1 purple1 p9k4 ,z drawn1 we c only see ! face ne>e/ to u1 :i* is n ! "o f :i* ! cube />ts--b ! face on ! opposite side has ! same col\r "n z ! face t[>ds u4 ,! successive sections ( ! s]ies1 y#j1 y#a1 y#b1 etc41 c be consid]$ z d]iv$ f sections ( ! b#j cube made at di/ances along ! yell[ axis4 ,:at is di/ant a qu>t] in* f ! p9k face 9 ! yell[ direction8 ,?is "q is answ]$ by tak+ a section f a po9t a qu>t] in* along ! yell[ axis 9 ! cube b#j1 fig4 #ajg4 ,x is an o*re section ) l9es orange & light yell[4 ,?is section w !re=e take ! place ( ! p9k face 9 y#a :5 we g on 9 ! yell[ direction4 ,?us1 ! fir/ section1 y#a1 w beg9 f an o*re face ) light yell[ & orange l9es4 ,! col\r ( ! axis :i* lies 9 space t[>ds u is blue1 h5ce ! regions ( ?is section-cube are det]m9$ 9 nom5clature1 !y w be f\nd 9 full 9 fig4 #aje4 ,!re rema9s only "o figure to be drawn1 & t is ! "o 9 :i* ! r$ axis is replac$ by ! blue4 ,"h1 z be=e1 if ! r$ axis goes out into ! positive s5se ( ! f\r? dim5sion1 ! blue l9e mu/ come into our space 9 ! negative s5se ( ! direction :i* ! r$ l9e has left4 ,accord+ly1 ! fir/ cube w come 9 b5ea? ! position ( our o*re cube1 ! "o we h be5 9 ! habit ( />t+ )4 ,illu/ration3 ,fig4 #aaj4 ,to %[ ^! figures we mu/ suppose ! o*re cube to be on a movable /&4 ,:5 ! r$ l9e sw+s out into ! unkn[n dim5sion1 & ! blue l9e comes 9 d[nw>ds1 a cube appe>s bel[ ! place occupi$ by ! o*re cube4 ,! dott$ cube %[s :]e ! o*re cube 04 ,t cube has gone & a diff]5t cube runs d[nw>ds f its base4 ,?is cube has :ite1 yell[1 & blue axes4 ,its top is a light yell[ squ>e1 & h5ce its int]ior is light yell[ blue or light gre54 ,its front face is =m$ by ! :ite l9e mov+ along ! blue axis1 & is !re=e light blue1 ! left-h& side is =m$ by ! yell[ l9e mov+ along ! blue axis1 & !re=e gre54 ,z ! r$ l9e n[ runs 9 ! f\r? dim5sion1 ! successive sections c he call$ r#j1 r#a1 r#b1 r#c1 r#d1 ^! lett]s indicat+ t at di/ances #j1 #a_/d1 #b_/d1 #c_/d1 #a in* along ! r$ axis we take all ( ! tess]act t c be f\nd 9 a ?ree-dim5sional space1 ?is ?ree-dim5sional space ext5d+ n at all 9 ! f\r? dim5sion1 b up & d[n1 "r & left1 f> & ne>4 ,we c see :at %\ld replace ! light yell[ face ( r#j1 :5 ! section r#a comes 91 by look+ at ! cube b#j1 fig4 #ajg4 ,:at is di/ant 9 x "o-qu>t] ( an in* f ! light yell[ face 9 ! r$ direction8 ,x is an o*re section ) orange & p9k l9es & r$ po9ts2 see also fig4 #ajc4 ,?is squ>e !n =ms ! top squ>e ( r#a4 ,n[ we c det]m9e ! nom5clature ( all ! regions ( r#a by consid]+ :at w\ld be =m$ by ! motion ( ?is squ>e along a blue axis4 ,b we c adopt ano!r plan4 ,let u take a horizontal section ( r#j1 & f9d+ t section 9 ! figures1 ( fig4 #ajg or fig4 #ajc1 f !m det]m9e :at w replace x1 go+ on 9 ! r$ direction4 ,a section ( ! r#j cube has gre51 light blue1 gre51 light blue sides & blue po9ts4 ,n[ ?is squ>e occurs on ! base ( ea* ( ! section figures1 b#a1 b#b1 etc4 ,9 !m we see t #a_/d in* 9 ! r$ direction f x lies a section ) br[n & light purple l9es & purple corn]s1 ! int]ior be+ ( light br[n4 ,h5ce ?is is ! nom5clature ( ! section :i* 9 r#a replaces ! section ( r#j made f a po9t along ! blue axis4 ,h5ce ! col\r+ z giv5 c be d]iv$4 ,we h ?us obta9$ a p]fectly nam$ gr\p ( tess]acts4 ,we c take a gr\p ( eighty-"o ( !m #c #c #c #c1 9 f\r dim5sions1 & ea* tess]act w h its "n null1 r$1 :ite1 yell[1 blue1 etc41 & :atev] cubic view we take ( !m we c say exactly :at sides ( ! tess]acts we are h&l+1 & h[ !y t\* ea* o!r4#e #e ,at ?is po9t ! read] w f9d x advantage\s1 if he has ! models1 to g ?r\gh ! manipulations describ$ 9 ! app5dix4 ,?us1 = in/ance1 if we h ! sixte5 tess]acts %[n bel[1 we c ask h[ does null t\* blue4 ,illu/ration3 ,fig4 #aaa4 ,9 ! arrangem5t giv5 9 fig4 #aaa we h ! axes :ite1 r$1 yell[1 9 space1 blue runn+ 9 ! f\r? dim5sion4 ,h5ce we h ! o*re cubes z bases4 ,imag9e n[ ! tess]actic gr\p to pass transv]se to our space--we h fir/ ( all null o*re cube1 :ite o*re cube1 etc42 ^! in/antly vani%1 & we get ! section %[n 9 ! middle cube 9 fig4 #ajc1 & f9ally1 j :5 ! tess]act block has mov$ "o in* transv]se to our space1 we h null o*re cube1 & !n imm$iately aft]w>ds ! o*re cube ( blue comes 94 ,h5ce ! tess]act null t\*es ! tess]act blue by its o*re cube1 :i* is 9 contact1 ea* & e po9t ( x1 ) ! o*re cube ( blue4 ,h[ does null t\* :ite1 we may ask8 ,look+ at ! beg9n+ ,a1 fig4 #aaa1 :]e we h ! o*re cubes1 we see t null o*re t\*es :ite o*re by an orange face4 ,n[ let u g5]ate ! null & :ite tess]acts by a motion 9 ! blue direction ( ea* ( ^! cubes4 ,ea* ( !m g5]ates ! correspond+ tess]act1 & ! plane ( contact ( ! cubes g5]ates ! cube by :i* ! tess]acts are 9 contact4 ,n[ an orange plane c>ri$ along a blue axis g5]ates a br[n cube4 ,h5ce null t\*es :ite by a br[n cube4 ,illu/ration3 ,fig4 #aab4 ,if we ask aga9 h[ r$ t\*es light blue tess]act1 let u re>range our gr\p1 fig4 #aab1 or r turn x ab\t s t we h a diff]5t space view ( x2 let ! r$ axis & ! :ite axis run up & "r1 & let ! blue axis come 9 space t[>ds u1 !n ! yell[ axis runs 9 ! f\r? dim5sion4 ,we h !n two blocks 9 :i* ! b\nd+ cubes ( ! tess]acts are giv51 diff]5tly arrang$ ) reg>d to u--! arrangem5t is really ! same1 b x appe>s diff]5t to u4 ,/>t+ f ! plane ( ! r$ & :ite axes we h ! f\r squ>es ( ! null1 :ite1 r$1 p9k tess]acts z %[n 9 ,a1 on ! r$1 :ite plane1 unalt]$1 only f !m n[ comes out t[>ds u ! blue axis4 ,h5ce we h null1 :ite1 r$1 p9k tess]acts 9 contact ) our space by _! cubes :i* h ! r$1 :ite1 blue axis 9 !m1 t is by ! light purple cubes4 ,foll[+ on ^! f\r tess]acts we h t :i* comes next to !m 9 ! blue direction1 t is ! f\r blue1 light blue1 purple1 light purple4 ,^! are likewise 9 contact ) our space by _! light purple cubes1 s we see a block z nam$ 9 ! figure1 ( :i* ea* cube is ! "o det]m9$ by ! r$1 :ite1 blue1 axes4 ,! yell[ l9e n[ runs out ( space2 accord+ly "o in* on 9 ! f\r? dim5sion we come to ! tess]acts :i* foll[ on ! eight nam$ 9 ,c1 fig4 #aab1 9 ! yell[ direction4 ,^! are %[n 9 ,c4y#a1 fig4 #aab4 ,betwe5 figure ,c & ,c4y#a is t f\r-dim5sional mass :i* is =m$ by mov+ ea* ( ! cubes 9 ,c "o in* 9 ! f\r? dim5sion--t is1 along a yell[ axis2 = ! yell[ axis n[ runs 9 ! f\r? dim5sion4 ,9 ! block ,c we obs]ve t r$ 7light purple cube7 t\*es light blue 7light purple cube7 by a po9t4 ,n[ ^! two cubes mov+ toge!r rema9 9 contact dur+ ! p]iod 9 :i* !y trace out ! tess]acts r$ & light blue4 ,?is motion is along ! yell[ axis1 consequ5tly r$ & light blue t\* by a yell[ l9e4 ,we h se5 t ! p9k face mov$ 9 a yell[ direction traces out a cube2 mov$ 9 ! blue direction x also traces out a cube4 ,let u ask :at ! p9k face w trace out if x is mov$ 9 a direction )9 ! tess]act ly+ equally betwe5 ! yell[ & blue directions4 ,:at section ( ! tess]act w x make8 ,we w fir/ consid] ! r$ l9e alone4 ,let u take a cube ) ! r$ l9e 9 x & ! yell[ & blue axes4 ,! cube ) ! yell[1 r$1 blue axes is %[n 9 fig4 #aac4 ,if ! r$ l9e is mov$ equally 9 ! yell[ & 9 ! blue direction by f\r equal motions ( in* ea*1 x takes ! positions #aa1 #bb1 #cc1 & ends z a r$ l9e4 ,illu/ration3 ,fig4 #aac4 ,n[1 ! :ole ( ?is r$1 yell[1 blue1 or br[n cube appe>s z a s]ies ( faces on ! successive sections ( ! tess]act />t+ f ! o*re cube & lett+ ! blue axis run 9 ! f\r? dim5sion4 ,h5ce ! plane trac$ out by ! r$ l9e appe>s z a s]ies ( l9es 9 ! successive sections1 9 our ord9>y way ( repres5t+ ! tess]act2 ^! l9es are 9 diff]5t places 9 ea* successive section4 ,illu/ration3 ,fig4 #aad4 ,?us draw+ our initial cube & ! successive sections1 call+ !m b#j1 b#a1 b#b1 b#c1 b#d1 fig4 #aae1 we h ! r$ l9e subject to ?is movem5t appe>+ 9 ! positions indicat$4 ,we w n[ inve/igate :at positions 9 ! tess]act ano!r l9e 9 ! p9k face assumes :5 x is mov$ 9 a simil> mann]4 ,take a section ( ! orig9al cube conta9+ a v]tical l9e1 #d1 9 ! p9k plane1 fig4 #aae4 ,we h1 9 ! section1 ! yell[ direction1 b n ! blue4 ,f ?is section a cube goes (f 9 ! f\r? dim5sion1 :i* is =m$ by mov+ ea* po9t ( ! section 9 ! blue direction4 ,illu/ration3 ,fig4 #aae4 ,illu/ration3 ,fig4 #aaf4 ,draw+ ?is cube we h fig4 #aaf4 ,n[ ?is cube occurs z a s]ies ( sections 9 our orig9al repres5tation ( ! tess]act4 ,tak+ f\r /eps z be=e ?is cube appe>s z ! sections drawn 9 b#j1 b#a1 b#b1 b#c1 b#d1 fig4 #aag1 & if ! l9e #d is subject$ to a movem5t equal 9 ! blue & yell[ directions1 x w occupy ! positions designat$ by #d1 #da1 #db1 #dc1 #dd4 ,illu/ration3 ,fig4 #aag4 ,h5ce1 reason+ 9 a simil> mann] ab\t e l9e1 x is evid5t t1 mov$ equally 9 ! blue & yell[ directions1 ! p9k plane w trace out a space :i* is %[n by ! s]ies ( section planes repres5t$ 9 ! diagram4 ,?us ! space trac$ out by ! p9k face1 if x is mov$ equally 9 ! yell[ & blue directions1 is repres5t$ by ! set ( planes del9eat$ 9 ,fig4 #aah1 p9k face or #j1 !n #a1 #b1 #c1 & f9ally p9k face or #d4 ,?is solid is a diagonal solid ( ! tess]act1 runn+ f a p9k face to a p9k face4 ,its l5g? is ! l5g? ( ! diagonal ( a squ>e1 its side is a squ>e4 ,let u n[ consid] ! unlimit$ space :i* spr+s f ! p9k face ext5d$4 ,?is space1 if x goes (f 9 ! yell[ direction1 gives u 9 x ! o*re cube ( ! tess]act4 ,?us1 if we h ! p9k face giv5 & a po9t 9 ! o*re cube1 we h det]m9$ ?is p>ticul> space4 ,simil>ly go+ (f f ! p9k face 9 ! blue direction is ano!r space1 :i* gives u ! light purple cube ( ! tess]act 9 x4 ,& any po9t be+ tak5 9 ! light purple cube1 ?is space go+ (f f ! p9k face is fix$4 ,illu/ration3 ,fig4 #aah4 ,! space we are speak+ ( c be conceiv$ z sw++ r\nd ! p9k face1 & 9 ea* ( its positions x cuts out a solid figure f ! tess]act1 "o ( :i* we h se5 repres5t$ 9 fig4 #aah4 ,ea* ( ^! solid figures is giv5 by "o position ( ! sw++ space1 & by "o only4 ,h5ce 9 ea* ( !m1 if "o po9t is tak51 ! p>ticul> "o ( ! slant+ spaces is fix$4 ,?us we see t giv5 a plane & a po9t out ( x a space is det]m9$4 ,n[1 two po9ts det]m9e a l9e4 ,aga91 ?9k ( a l9e & a po9t outside x4 ,imag9e a plane rotat+ r\nd ! l9e4 ,at "s "t 9 its rotation x passes ?r\gh ! po9t4 ,?us a l9e & a po9t1 or ?ree po9ts1 det]m9e a plane4 ,& f9ally f\r po9ts det]m9e a space4 ,we h se5 t a plane & a po9t det]m9e a space1 & t ?ree po9ts det]m9e a plane2 s f\r po9ts w det]m9e a space4 ,^! f\r po9ts may be any po9ts1 & we c take1 = in/ance1 ! f\r po9ts at ! extremities ( ! r$1 :ite1 yell[1 blue axes1 9 ! tess]act4 ,^! w det]m9e a space slant+ ) reg>d to ! section spaces we h be5 previ\sly consid]+4 ,?is space w cut ! tess]act 9 a c]ta9 figure4 ,"o ( ! simple/ sections ( a cube by a plane is t 9 :i* ! plane passes ?r\gh ! extremities ( ! ?ree edges :i* meet 9 a po9t4 ,we see at once t ?is plane w\ld cut ! cube 9 a triangle1 b we w g ?r\gh ! process by :i* a plane be+ w\ld mo/ conv5i5tly treat ! problem ( ! det]m9ation ( ?is %ape1 9 ord] t we may apply ! me?od to ! det]m9ation ( ! figure 9 :i* a space cuts a tess]act :5 x passes ?r\gh ! #d po9ts at unit di/ance f a corn]4 ,we "k t two po9ts det]m9e a l9e1 ?ree po9ts det]m9e a plane1 & giv5 any two po9ts 9 a plane ! l9e betwe5 !m lies :olly 9 ! plane4 ,illu/ration3 ,fig4 #aai4 ,let n[ ! plane be+ /udy ! section made by a plane pass+ ?r\gh ! null r1 null :1 & null y po9ts1 fig4 #aai4 ,look+ at ! orange squ>e1 :i*1 z usual1 we suppose to be initially 9 8 plane1 he sees t ! l9e f null r to null y1 :i* is a l9e 9 ! section plane1 ! plane1 namely1 ?r\gh ! ?ree extremities ( ! edges meet+ 9 null1 cuts ! orange face 9 an orange l9e ) null po9ts4 ,?is !n is "o ( ! b\nd>ies ( ! section figure4 ,let n[ ! cube be s turn$ t ! p9k face comes 9 8 plane4 ,! po9ts null r & null : are n[ visible4 ,! l9e betwe5 !m is p9k ) null po9ts1 & s9ce ?is l9e is common to ! surface ( ! cube & ! cutt+ plane1 x is a b\nd>y ( ! figure 9 :i* ! plane cuts ! cube4 ,aga91 suppose ! cube turn$ s t ! light yell[ face is 9 contact ) ! plane be+'s plane4 ,he sees two po9ts1 ! null : & ! null y4 ,! l9e betwe5 ^! lies 9 ! cutt+ plane4 ,h5ce1 s9ce ! ?ree cutt+ l9es meet & enclose a portion ( ! cube betwe5 !m1 he has det]m9$ ! figure he s\ght4 ,x is a triangle ) orange1 p9k1 & light yell[ sides1 all equal1 & enclos+ an o*re area4 ,let u n[ det]m9e 9 :at figure ! space1 det]m9$ by ! f\r po9ts1 null r1 null y1 null :1 null b1 cuts ! tess]act4 ,we c see ?ree ( ^! po9ts 9 ! prim>y position ( ! tess]act re/+ aga9/ our solid %eet by ! o*re cube4 ,^! ?ree po9ts det]m9e a plane :i* lies 9 ! space we are consid]+1 & ?is plane cuts ! o*re cube 9 a triangle1 ! int]ior ( :i* is o*re 7fig4 #aai w s]ve = ?is view71 ) p9k1 light yell[ & orange sides1 & null po9ts4 ,go+ 9 ! f\r? direction1 9 "o s5se1 f ?is plane we pass into ! tess]act1 9 ! o!r s5se we pass away f x4 ,! :ole area inside ! triangle is common to ! cutt+ plane we see1 & a b\nd>y ( ! tess]act4 ,h5ce we conclude t ! triangle drawn is common to ! tess]act & ! cutt+ space4 ,n[ let ! o*re cube turn out & ! br[n cube come 94 ,! dott$ l9es %[ ! position ! o*re cube has left 7fig4 #abj74 ,illu/ration3 ,fig4 #abj4 ,"h we see ?ree out ( ! f\r po9ts ?r\gh :i* ! cutt+ plane passes1 null r1 null y1 & null b4 ,! plane !y det]m9e lies 9 ! cutt+ space1 & ?is plane cuts out ( ! br[n cube a triangle ) orange1 purple & gre5 sides1 & null po9ts4 ,! orange l9e ( ?is figure is ! same z ! orange l9e 9 ! la/ figure4 ,n[ let ! light purple cube sw+ into our space1 t[>ds u1 fig4 #aba4 ,illu/ration3 ,fig4 #aba4 ,! cutt+ space :i* passes ?r\gh ! f\r po9ts1 null r1 y1 :1 b1 passes ?r\gh ! null r1 :1 b1 & !re=e ! plane ^! det]m9e lies 9 ! cutt+ space4 ,?is triangle lies be=e u4 ,x has a light purple int]ior & p9k1 light blue1 & purple edges ) null po9ts4 ,?is1 s9ce x is all ( ! plane t is common to x1 & ?is b\nd+ ( ! tess]act1 gives u "o ( ! b\nd+ faces ( our sectional figure4 ,! p9k l9e 9 x is ! same z ! p9k l9e we f\nd 9 ! fir/ figure--t ( ! o*re cube4 ,f9ally1 let ! tess]act sw+ ab\t ! light yell[ plane1 s t ! light gre5 cube comes into our space4 ,x w po9t d[nw>ds4 ,! ?ree po9ts1 n4y1 n4:1 n4b1 are 9 ! cutt+ space1 & ! triangle !y det]m9e is common to ! tess]act & ! cutt+ space4 ,h5ce ?is b\nd>y is a triangle hav+ a light yell[ l9e1 :i* is ! same z ! light yell[ l9e ( ! fir/ figure1 a light blue l9e & a gre5 l9e4 ,illu/ration3 ,fig4 #abb4 ,we h n[ trac$ ! cutt+ space betwe5 e set ( ?ree t c be made out ( ! f\r po9ts 9 :i* x cuts ! tess]act1 & h got f\r faces :i* all jo9 on to ea* o!r by l9es4 ,illu/ration3 ,fig4 #abc4 ,! triangles are %[n 9 fig4 #abc z !y jo9 on to ! triangle 9 ! o*re cube4 ,b !y jo9 on ea* to ! o!r 9 an exactly simil> mann]2 _! edges are all id5tical two & two4 ,!y =m a clos$ figure1 a tetrah$ron1 enclos+ a light br[n portion :i* is ! portion ( ! cutt+ space :i* lies inside ! tess]act4 ,we _c expect to see ?is light br[n portion1 any m ?an a plane be+ c\ld expect to see ! inside ( a cube if an angle ( x 7 pu%$ ?r\gh 8 plane4 ,all he c d is to come ^u ! b\nd>ies ( x 9 a diff]5t way to t 9 :i* he w\ld if x pass$ /raight ?r\gh 8 plane4 ,?us 9 ?is solid section2 ! :ole int]ior lies p]fectly op5 9 ! f\r? dim5sion4 ,g r\nd x z we may we are simply look+ at ! b\nd>ies ( ! tess]act :i* p5etrates ?r\gh our solid %eet4 ,if ! tess]act 7 n to pass across s f>1 ! triangle w\ld be small]2 if x 7 to pass f>!r1 we %\ld h a diff]5t figure1 ! outl9es ( :i* c be det]m9$ 9 a simil> mann]4 ,! prec$+ me?od is op5 to ! objection t x dep5ds r on our inf]r+ :at mu/ be1 ?an our see+ :at is4 ,let u !re=e consid] our sectional space z consi/+ ( a numb] ( planes1 ea* v close to ! la/1 & obs]ve :at is to be f\nd 9 ea* plane4 ,! correspond+ me?od 9 ! case ( two dim5sions is z foll[s3--,! plane be+ c see t l9e ( ! sectional plane ?r\gh null y1 null :1 null r1 :i* lies 9 ! orange plane4 ,let him n[ suppose ! cube & ! section plane to pass half way ?r\gh 8 plane4 ,replac+ ! r$ & yell[ axes are l9es p>allel to !m1 sections ( ! p9k & light yell[ faces4 ,illu/ration3 ,fig4 #abd4 ,:]e w ! section plane cut ^! p>allels to ! r$ & yell[ axes8 ,let him suppose ! cube1 9 ! position ( ! draw+1 fig4 #abd1 turn$ s t ! p9k face lies aga9/ 8 plane4 ,he c see ! l9e f ! null r po9t to ! null : po9t1 & c see 7comp>e fig4 #aai7 t x cuts ,,ab a p>allel to 8 r$ axis1 drawn at a po9t half way along ! :ite l9e1 9 a po9t ,b1 half way up4 ,i %all speak ( ! axis z hav+ ! l5g? ( an edge ( ! cube4 ,simil>ly1 by lett+ ! cube turn s t ! light yell[ squ>e sw+s aga9/ 8 plane1 he c see 7comp>e fig4 #aai7 t a p>allel to 8 yell[ axis drawn f a po9t half-way along ! :ite axis1 is cut at half its l5g? by ! trace ( ! section plane 9 ! light yell[ face4 ,h5ce :5 ! cube _h pass$ half-way ?r\gh he w\ld h--in/ead ( ! orange l9e ) null po9ts1 :i* he _h at fir/--an o*re l9e ( half its l5g?1 ) p9k & light yell[ po9ts4 ,?us1 z ! cube pass$ sl[ly ?r\gh 8 plane1 he w\ld h a succession ( l9es gradually dim9i%+ 9 l5g? & =m+ an equilat]al triangle4 ,! :ole int]ior w\ld be o*re1 ! l9e f :i* x />t$ w\ld be orange4 ,! succession ( po9ts at ! ends ( ! succe$+ l9es w\ld =m p9k & light yell[ l9es & ! f9al po9t w\ld be null4 ,?us look+ at ! successive l9es 9 ! section plane z x & ! cube pass$ across 8 plane he w\ld det]m9e ! figure cut out bit by bit4 ,com+ n[ to ! section ( ! tess]act1 let u imag9e t ! tess]act & its cutt+ space pass sl[ly across our space2 we c exam9e portions ( x1 & _! relation to portions ( ! cutt+ space4 ,take ! section space :i* passes ?r\gh ! f\r po9ts1 null r1 :1 y1 b2 we c see 9 ! o*re cube 7fig4 #aai7 ! plane belong+ to ?is section space1 :i* passes ?r\gh ! ?ree extremities ( ! r$1 :ite1 yell[ axes4 ,n[ let ! tess]act pass half way ?r\gh our space4 ,in/ead ( our orig9al axes we h p>allels to !m1 purple1 light blue1 & gre51 ea* ( ! same l5g? z ! fir/ axes1 = ! section ( ! tess]act is ( exactly ! same %ape z its o*re cube4 ,b ! sectional space se5 at ?is /age ( ! transf]5ce w\ld n cut ! section ( ! tess]act 9 a plane dispos$ z at fir/4 ,to see :]e ! sectional space w\ld cut ^! p>allels to ! orig9al axes let ! tess]act sw+ s t1 ! orange face rema9+ /ation>y1 ! blue l9e comes 9 to ! left4 ,"h 7fig4 #abe7 we h ! null r1 y1 b po9ts1 & ( ! sectional space all we see is ! plane ?r\gh ^! ?ree po9ts 9 x4 ,illu/ration3 ,fig4 #abe4 ,9 ?is figure we c draw ! p>allels to ! r$ & yell[ axes & see t1 if !y />t$ at a po9t half way along ! blue axis1 !y w\ld ea* be cut at a po9t s z to be half ( _! previ\s l5g?4 ,sw++ ! tess]act into our space ab\t ! p9k face ( ! o*re cube we likewise f9d t ! p>allel to ! :ite axis is cut at half its l5g? by ! sectional space4 ,h5ce 9 a section made :5 ! tess]act _h pass$ half across our space ! p>allels to ! r$1 :ite1 yell[ axes1 :i* are n[ 9 our space1 are cut by ! section space1 ea* ( !m half way along1 & = ?is /age ( ! trav]s+ motion we %\ld h fig4 #abf4 ,! section made ( ?is cube by ! plane 9 :i* ! sectional space cuts x1 is an equilat]al triangle ) purple1 l4 blue1 gre5 po9ts1 & l4 purple1 br[n1 l4 gre5 l9es4 ,illu/ration3 ,fig4 #abf4 ,?us ! orig9al o*re triangle1 ) null po9ts & p9k1 orange1 light yell[ l9es1 w\ld be succe$$ by a triangle col\r$ 9 mann] j describ$4 ,?is triangle w\ld initially be only a v little small] ?an ! orig9al triangle1 x w\ld gradually dim9i%1 until x end$ 9 a po9t1 a null po9t4 ,ea* ( its edges w\ld be ( ! same l5g?4 ,?us ! successive sections ( ! successive planes into :i* we analyse ! cutt+ space w\ld be a tetrah$ron ( ! description %[n 7fig4 #abc71 & ! :ole int]ior ( ! tetrah$ron w\ld be light br[n4 ,illu/ration3 ,fig4 #abg4 ,front view4 ,! re> faces4 ,9 fig4 #abg ! tetrah$ron is repres5t$ by means ( its faces z two triangles :i* meet 9 ! p4 l9e1 & two re> triangles :i* jo9 on to !m1 ! diagonal ( ! p9k face be+ suppos$ to run v]tically upw>d4 ,we h n[ rea*$ a natural t]m9ation4 ,! read] may pursue ! subject 9 fur!r detail1 b w f9d no ess5tial novelty4 ,i conclude ) an indication z to ! mann] 9 :i* figures previ\sly giv5 may be us$ 9 det]m9+ sections by ! me?od develop$ above4 ,apply+ ?is me?od to ! tess]act1 z repres5t$ 9 ,*apt] ,,ix41 sections made by a space cutt+ ! axes equidi/antly at any di/ance c be drawn1 & also ! sections ( tess]acts arrang$ 9 a block4 ,if we draw a plane1 cutt+ all f\r axes at a po9t six units di/ance f null1 we h a slant+ space4 ,?is space cuts ! r$1 :ite1 yell[ axes 9 ! po9ts ,,lmn 7fig4 #abh71 & s 9 ! region ( our space be=e we g (f into ! f\r? dim5sion1 we h ! plane repres5t$ by ,,lmn ext5d$4 ,?is is :at is common to ! slant+ space & our space4 ,illu/ration3 ,fig4 #abh4 ,?is plane cuts ! o*re cube 9 ! triangle ,,efg4 ,comp>+ ?is ) 7fig4 #gb7 oh1 we see t ! hexagon !re drawn is "p ( ! triangle ,,efg4 ,let u n[ imag9e ! tess]act & ! slant+ space bo? toge!r to pass transv]se to our space1 a di/ance ( "o unit1 we h 9 #a;h a section ( ! tess]act1 ^: axes are p>allels to ! previ\s axes4 ,! slant+ space cuts !m at a di/ance ( five units along ea*4 ,draw+ ! plane ?r\gh ^! po9ts 9 #a;h x w be f\nd to cut ! cubical section ( ! tess]act 9 ! hexagonal figure drawn4 ,9 #b;h 7fig4 #gb7 ! slant+ space cuts ! p>allels to ! axes at a di/ance ( f\r along ea*1 & ! hexagonal figure is ! section ( ?is section ( ! tess]act by x4 ,f9ally :5 #c;h comes 9 ! slant+ space cuts ! axes at a di/ance ( ?ree along ea*1 & ! section is a triangle1 ( :i* ! hexagon drawn is a truncat$ portion4 ,aft] ?is ! tess]act1 :i* ext5ds only ?ree units 9 ea* ( ! f\r dim5sions1 has completely pass$ transv]se ( our space1 & !re is no m ( x to be cut4 ,h5ce1 putt+ ! plane sections toge!r 9 ! "r relations1 we h ! section det]m9$ by ! p>ticul> slant+ space3 namely an octah$ron4 ,,*apt] ,,xiv4#f ,a ,,recapitulation ,,& ,,ext5sion ,,( ,,! ,,physical ,,argum5t ,!re are two directions ( inquiry 9 :i* ! rese>* = ! physical reality ( a f\r? dim5sion c be prosecut$4 ,"o is ! inve/igation ( ! inf9itely great1 ! o!r is ! inve/igation ( ! inf9itely small4 #f ,! cont5ts ( ?is *apt] are tak5 f a pap] read be=e ! ,philosophical ,society ( ,wa%+ton4 ,! ma!matical portion ( ! pap] has appe>$ 9 "p 9 ! ,proce$+s ( ! ,royal ,iri% ,academy "u ! title1 8,cayley's =mul ( or?ogonal trans=mation10 ,nov4 #bi?1 #aijc4 ,by ! measurem5t ( ! angles ( va/ triangles1 ^: sides are ! di/ances betwe5 ! />s1 a/ronom]s h s\ght to det]m9e if !re is any deviation f ! values giv5 by geometrical d$uction4 ,if ! angles ( a cele/ial triangle d n toge!r equal two "r angles1 !re w\ld be an evid5ce = ! physical reality ( a f\r? dim5sion4 ,?is conclusion des]ves a ^w ( explanation4 ,if space is really f\r-dim5sional1 c]ta9 conclusions foll[ :i* mu/ be br\ght cle>ly into evid5ce if we are to frame ! que/ions def9itely :i* we put to ,nature4 ,to acc\nt = our limitation let u assume a solid mat]ial %eet aga9/ :i* we move4 ,?is %eet mu/ /ret* alongside e object 9 e direction 9 :i* x visibly moves4 ,e mat]ial body mu/ slip or slide along ?is %eet1 n deviat+ f contact ) x 9 any motion :i* we c obs]ve4 ,! necessity = ?is assumption is cle>ly app>5t1 if we consid] ! analog\s case ( a supposition>y plane _w4 ,if !re 7 any creatures ^: exp]i5ces 7 conf9$ to a plane1 we mu/ acc\nt = _! limitation4 ,if !y 7 free to move 9 e space direction1 !y w\ld h a ?ree-dim5sional motion2 h5ce !y mu/ be physically limit$1 & ! only way 9 :i* we c conceive su* a limitation to exi/ is by means ( a mat]ial surface aga9/ :i* !y slide4 ,! exi/5ce ( ?is surface c\ld only be kn[n to !m indirectly4 ,x does n lie 9 any direction f !m 9 :i* ! k9ds ( motion !y "k ( leads !m4 ,if x 7 p]fectly smoo? & always 9 contact ) e mat]ial object1 !re w\ld be no diff]5ce 9 _! relations to x :i* w\ld direct _! att5tion to x4 ,b if ?is surface 7 curv$--if x 71 say1 9 ! =m ( a va/ sph]e--! triangles !y drew w\ld really be triangles ( a sph]e1 & :5 ^! triangles are l>ge en\gh ! angles div]ge f ! magnitudes !y w\ld h = ! same l5g?s ( sides if ! surface 7 plane4 ,h5ce by ! measurem5t ( triangles ( v great magnitude a plane be+ might detect a diff]5ce f ! laws ( a plane _w 9 8 physical _w1 & s be l$ to ! conclusion t !re 0 9 reality ano!r dim5sion to space--a ?ird dim5sion--z well z ! two :i* 8 ord9>y exp]i5ce made him famili> )4 ,n[1 a/ronom]s h ?\ght x wor? :ile to exam9e ! measurem5ts ( va/ triangles drawn f "o cele/ial body to ano!r ) a view to det]m9e if !re is any?+ l a curvature 9 our space--t is to say1 !y h tri$ a/ronomical measurem5ts to f9d out if ! va/ solid %eet aga9/ :i*1 on ! supposition ( a f\r? dim5sion1 ev]y?+ slides is curv$ or n4 ,^! results h be5 negative4 ,! solid %eet1 if x exi/s1 is n curv$ or1 be+ curv$1 has n a suffici5t curvature to cause any obs]vable deviation f ! !oretical value ( ! angles calculat$4 ,h5ce ! exam9ation ( ! inf9itely great leads to no decisive crit]ion4 ,if x did we %\ld h to decide betwe5 ! pres5t !ory & t ( metageometry4 ,com+ n[ to ! prosecution ( ! inquiry 9 ! direction ( ! inf9itely small1 we h to /ate ! "q ?us3 ,our laws ( movem5t are d]iv$ f ! exam9ation ( bodies :i* move 9 ?ree-dim5sional space4 ,all our conceptions are f\nd$ on ! supposition ( a space :i* is repres5t$ analytically by ?ree indep5d5t axes & v>iations along !m--t is1 x is a space 9 :i* !re are ?ree indep5d5t movem5ts4 ,any motion possible 9 x c be comp\nd$ out ( ^! ?ree movem5ts1 :i* we may call3 up1 "r1 away4 ,to exam9e ! actions ( ! v small portions ( matt] ) ! view ( asc]ta9+ if !re is any evid5ce 9 ! ph5om5a = ! supposition ( a f\r? dim5sion ( space1 we mu/ comm5ce by cle>ly def9+ :at ! laws ( me*anics w\ld be on ! supposition ( a f\r? dim5sion4 ,x is ( no use ask+ if ! ph5om5a ( ! smalle/ p>ticles ( matt] are l--we d n "k :at4 ,we mu/ h a def9ite conception ( :at ! laws ( motion w\ld be on ! supposition ( ! f\r? dim5sion1 & !n inquire if ! ph5om5a ( ! activity ( ! small] p>ticles ( matt] resemble ! conceptions :i* we h elaborat$4 ,n[1 ! task ( =m+ ^! conceptions is by no means "o to be lightly dismiss$4 ,movem5t 9 space has _m features :i* diff] entirely f movem5t on a plane2 & :5 we set ab\t to =m ! conception ( motion 9 f\r dim5sions1 we f9d t !re is at lea/ z great a /ep z f ! plane to ?ree-dim5sional space4 ,i d n say t ! /ep is difficult1 b ,i want to po9t out t x mu/ be tak54 ,:5 we h =m$ ! conception ( f\r-dim5sional motion1 we c ask a rational "q ( ,nature4 ,be=e we h elaborat$ our conceptions we are ask+ if an unkn[n is l an unkn[n--a futile inquiry4 ,z a matt] ( fact1 f\r-dim5sional movem5ts are 9 e way simple & m easy to calculate ?an ?ree-dim5sional movem5ts1 = f\r-dim5sional movem5ts are simply two sets ( plane movem5ts put toge!r4 ,)\t ! =mation ( an exp]i5ce ( f\r-dim5sional bodies1 _! %apes & motions1 ! subject c be b =mal--logically conclusive1 n intuitively evid5t4 ,x is to ?is logical appreh5sion t ,i mu/ appeal4 ,x is p]fectly simple to =m an exp]i5tial famili>ity ) ! facts ( f\r-dim5sional movem5t4 ,! me?od is analog\s to t :i* a plane be+ w\ld h to adopt to =m an exp]i5tial famili>ity ) ?ree-dim5sional movem5ts1 & may be briefly summ$ up z ! =mation ( a comp\nd s5se by means ( :i* duration is reg>d$ z equival5t to ext5sion4 ,consid] a be+ conf9$ to a plane4 ,a squ>e enclos$ by f\r l9es w be to him a solid1 ! int]ior ( :i* c only be exam9$ by break+ ?r\gh ! l9es4 ,if su* a squ>e 7 to pass transv]se to 8 plane1 x w\ld imm$iately disappe>4 ,x w\ld vani%1 go+ 9 no direction to :i* he c\ld po9t4 ,if1 n[1 a cube be plac$ 9 contact ) 8 plane1 its surface ( contact w\ld appe> l ! squ>e :i* we h j m5tion$4 ,b if x 7 to pass transv]se to 8 plane1 break+ ?r\gh x1 x w\ld appe> z a la/+ squ>e4 ,! ?ree-dim5sional matt] w give a la/+ appe>ance 9 circum/ances "u :i* two-dim5sional matt] w at once disappe>4 ,simil>ly1 a f\r-dim5sional cube1 or1 z we may call x1 a tess]act1 :i* is g5]at$ f a cube by a movem5t ( e "p ( ! cube 9 a f\r? direction at "r angles to ea* ( ! ?ree visible directions 9 ! cube1 if x mov$ transv]se to our space1 w\ld appe> z a la/+ cube4 ,a cube ( ?ree-dim5sional matt]1 s9ce x ext5ds to no di/ance at all 9 ! f\r? dim5sion1 w\ld in/antly disappe>1 if subject$ to a motion transv]se to our space4 ,x w\ld disappe> & be gone1 )\t x be+ possible to po9t to any direction 9 :i* x _h mov$4 ,all attempts to visualise a f\r? dim5sion are futile4 ,x mu/ be connect$ ) a "t exp]i5ce 9 ?ree space4 ,! mo/ difficult notion = a plane be+ to acquire w\ld be t ( rotation ab\t a l9e4 ,consid] a plane be+ fac+ a squ>e4 ,if he 7 told t rotation ab\t a l9e 7 possible1 he w\ld move 8 squ>e ?is way & t4 ,a squ>e 9 a plane c rotate ab\t a po9t1 b to rotate ab\t a l9e w\ld seem to ! plane be+ p]fectly impossible4 ,h[ c\ld ^? p>ts ( 8 squ>e :i* 7 on "o side ( an edge come to ! o!r side )\t ! edge mov+8 ,he c\ld und]/& _! reflection 9 ! edge4 ,he c\ld =m an idea ( ! look+-glass image ( 8 squ>e ly+ on ! opposite side ( ! l9e ( an edge1 b by no motion t he kn[s ( c he make ! actual squ>e assume t position4 ,! result ( ! rotation w\ld be l reflection 9 ! edge1 b x w\ld be a physical impossibility to produce x 9 ! plane4 ,! demon/ration ( rotation ab\t a l9e mu/ be to him purely =mal4 ,if he conceiv$ ! notion ( a cube /ret*+ out 9 an unkn[n direction away f 8 plane1 !n he c see ! base ( x1 8 squ>e 9 ! plane1 rotat+ r\nd a po9t4 ,he c likewise appreh5d t e p>allel section tak5 at successive int]vals 9 ! unkn[n direction rotates 9 l mann] r\nd a po9t4 ,?us he w\ld come to conclude t ! :ole body rotates r\nd a l9e--! l9e consi/+ ( ! succession ( po9ts r\nd :i* ! plane sections rotate4 ,?us1 giv5 ?ree axes1 x1 y1 z1 if x rotates to take ! place ( y1 & y turns s z to po9t to negative x1 !n ! ?ird axis rema9+ unaffect$ by ?is turn+ is ! axis ab\t :i* ! rotation takes place4 ,?is1 !n1 w\ld h to be 8 crit]ion ( ! axis ( a rotation--t :i* rema9s un*ang$ :5 a rotation ( e plane section ( a body takes place4 ,!re is ano!r way 9 :i* a plane be+ c ?9k ab\t ?ree-dim5sional movem5ts2 &1 z x af=ds ! type by :i* we c mo/ conv5i5tly ?9k ab\t f\r-dim5sional movem5ts1 x w be no loss ( "t to consid] x 9 detail4 ,illu/ration3 ,fig4 #a 7#abi74 ,we c repres5t ! plane be+ & 8 object by figures cut out ( pap]1 :i* slip on a smoo? surface4 ,! ?ickness ( ^! bodies mu/ be tak5 z s m9ute t _! ext5sion 9 ! ?ird dim5sion escapes ! obs]vation ( ! plane be+1 & he ?9ks ab\t !m z if !y 7 ma!matical plane figures 9 a plane in/ead ( be+ mat]ial bodies capable ( mov+ on a plane surface4 ,let ,ax1 ,ay be two axes & ,,abcd a squ>e4 ,z f> z movem5ts 9 ! plane are conc]n$1 ! squ>e c rotate ab\t a po9t ,a1 = example4 ,x _c rotate ab\t a side1 su* z ,,ac4 ,b if ! plane be+ is aw>e ( ! exi/5ce ( a ?ird dim5sion he c /udy ! movem5ts possible 9 ! ample space1 tak+ 8 figure portion by portion4 ,8 plane c only hold two axes4 ,b1 s9ce x c hold two1 he is able to repres5t a turn+ into ! ?ird dim5sion if he neglects "o ( 8 axes & repres5ts ! ?ird axis z ly+ 9 8 plane4 ,he c make a draw+ 9 8 plane ( :at /&s up p]p5dicul>ly f 8 plane4 ,let ,az be ! axis1 :i* /&s p]p5dicul> to 8 plane at ,a4 ,he c draw 9 8 plane two l9es to repres5t ! two axes1 ,ax & ,az4 ,let ,fig4 #b be ?is draw+4 ,"h ! z axis has tak5 ! place ( ! y axis1 & ! plane ( ,ax ,az is repres5t$ 9 8 plane4 ,9 ?is figure all t exi/s ( ! squ>e ,,abcd w be ! l9e ,,ab4 ,illu/ration3 ,fig4 #b 7#acj74 ,! squ>e ext5ds f ?is l9e 9 ! y direction1 b m ( t direction is repres5t$ 9 ,fig4 #b4 ,! plane be+ c /udy ! turn+ ( ! l9e ,,ab 9 ?is diagram4 ,x is simply a case ( plane turn+ ar\nd ! po9t ,a4 ,! l9e ,,ab occupies int]m$iate portions l ,,ab#a & aft] half a revolution w lie on ,ax produc$ ?r\gh ,a4 ,n[1 9 ! same way1 ! plane be+ c take ano!r po9t1 ,a1 & ano!r l9e1 ,a,b1 9 8 squ>e4 ,he c make ! draw+ ( ! two directions at ,a1 "o along ,a,b1 ! o!r p]p5dicul> to 8 plane4 ,he w obta9 a figure precisely simil> to ,fig4 #b1 & w see t1 z ,,ab c turn ar\nd ,a1 s ,a,c ar\nd ,a4 ,9 ?is turn+ ,,ab & ,a,b w\ld n int]f]e ) ea* o!r1 z !y w\ld if !y mov$ 9 ! plane ar\nd ! sep>ate po9ts ,a & ,a4 ,h5ce ! plane be+ w\ld conclude t a rotation r\nd a l9e 0 possible4 ,he c\ld see 8 squ>e z x began to make ?is turn+4 ,he c\ld see x half way r\nd :5 x came to lie on ! opposite side ( ! l9e ,,ac4 ,b 9 int]m$iate portions he c\ld n see x1 = x runs out ( ! plane4 ,com+ n[ to ! "q ( a f\r-dim5sional body1 let u conceive ( x z a s]ies ( cubic sections1 ! fir/ 9 our space1 ! re/ at int]vals1 /ret*+ away f our space 9 ! unkn[n direction4 ,we mu/ n ?9k ( a f\r-dim5sional body z =m$ by mov+ a ?ree-dim5sional body 9 any direction :i* we c see4 ,ref] = a mom5t to ,fig4 #c4 ,! po9t ,a1 mov+ to ! "r1 traces out ! l9e ,,ac4 ,! l9e ,,ac1 mov+ away 9 a new direction1 traces out ! squ>e ,,aceg at ! base ( ! cube4 ,! squ>e ,,aegc1 mov+ 9 a new direction1 w trace out ! cube ,,acegbdhf4 ,! v]tical direction ( ?is la/ motion is n id5tical ) any motion possible 9 ! plane ( ! base ( ! cube4 ,x is an entirely new direction1 at "r angles to e l9e t c be drawn 9 ! base4 ,to trace out a tess]act ! cube mu/ move 9 a new direction--a direction at "r angles to any & e l9e t c be drawn 9 ! space ( ! cube4 ,! cubic sections ( ! tess]act are relat$ to ! cube we see1 z ! squ>e sections ( ! cube are relat$ to ! squ>e ( its base :i* a plane be+ sees4 ,let u imag9e ! cube 9 our space1 :i* is ! base ( a tess]act1 to turn ab\t "o ( its edges4 ,! rotation w c>ry ! :ole body ) x1 & ea* ( ! cubic sections w rotate4 ,! axis we see 9 our space w rema9 un*ang$1 & likewise ! s]ies ( axes p>allel to x ab\t :i* ea* ( ! p>allel cubic sections rotates4 ,! assemblage ( all ( ^! is a plane4 ,h5ce 9 f\r dim5sions a body rotates ab\t a plane4 ,!re is no su* ?+ z rotation r\nd an axis4 ,we may reg>d ! rotation f a diff]5t po9t ( view4 ,consid] f\r indep5d5t axes ea* at "r angles to all ! o!rs1 drawn 9 a f\r-dim5sional body4 ,( ^! f\r axes we c see any ?ree4 ,! f\r? ext5ds normal to our space4 ,rotation is ! turn+ ( "o axis into a second1 & ! second turn+ to take ! place ( ! negative ( ! fir/4 ,x involves two axes4 ,?us1 9 ?is rotation ( a f\r-dim5sional body1 two axes *ange & two rema9 at re/4 ,f\r-dim5sional rotation is !re=e a turn+ ab\t a plane4 ,z 9 ! case ( a plane be+1 ! result ( rotation ab\t a l9e w\ld appe> z ! production ( a look+-glass image ( ! orig9al object on ! o!r side ( ! l9e1 s to u ! result ( a f\r-dim5sional rotation w\ld appe> l ! production ( a look+-glass image ( a body on ! o!r side ( a plane4 ,! plane w\ld be ! axis ( ! rotation1 & ! pa? ( ! body betwe5 its two appe>ances w\ld be unimag9able 9 ?ree-dim5sional space4 ,illu/ration3 ,fig4 #c 7#aca74 ,let u n[ apply ! me?od by :i* a plane be+ c\ld exam9e ! nature ( rotation ab\t a l9e 9 our exam9ation ( rotation ab\t a plane4 ,fig4 #c repres5ts a cube 9 our space1 ! ?ree axes x1 y1 z d5ot+ its ?ree dim5sions4 ,let w repres5t ! f\r? dim5sion4 ,n[1 s9ce 9 our space we c repres5t any ?ree dim5sions1 we c1 if we *oose1 make a repres5tation ( :at is 9 ! space det]m9$ by ! ?ree axes x1 z1 w4 ,?is is a ?ree-dim5sional space det]m9$ by two ( ! axes we h drawn1 x & z1 & 9 place ( y ! f\r? axis1 w4 ,we _c1 keep+ x & z1 h bo? y & w 9 our space2 s we w let y g & draw w 9 its place4 ,:at w be our view ( ! cube8 ,evid5tly we %all h simply ! squ>e t is 9 ! plane ( xz1 ! squ>e ,,acdb4 ,! re/ ( ! cube /ret*es 9 ! y direction1 &1 z we h none ( ! space s det]m9$1 we h only ! face ( ! cube4 ,?is is repres5t$ 9 fig4 #d4 ,illu/ration3 ,fig4 #d 7#acb74 ,n[1 suppose ! :ole cube to be turn$ f ! x to ! w direction4 ,con=mably ) our me?od1 we w n take ! :ole ( ! cube into consid]ation at once1 b w beg9 ) ! face ,,abcd4 ,let ?is face beg9 to turn4 ,fig4 #e repres5ts "o ( ! positions x w occupy2 ! l9e ,,ab rema9s on ! z axis4 ,! re/ ( ! face ext5ds betwe5 ! x & ! w direction4 ,illu/ration3 ,fig4 #e 7#acc74 ,n[1 s9ce we c take any ?ree axes1 let u look at :at lies 9 ! space ( zyw1 & exam9e ! turn+ !re4 ,we mu/ n[ let ! z axis disappe> & let ! w axis run 9 ! direction 9 :i* ! z ran4 ,mak+ ?is repres5tation1 :at d we see ( ! cube8 ,obvi\sly we see only ! l[] face4 ,! re/ ( ! cube lies 9 ! space ( xyz4 ,9 ! space ( xyz we h m]ely ! base ( ! cube ly+ 9 ! plane ( xy1 z %[n 9 fig4 #f4 ,illu/ration3 ,fig4 #f 7#acd74 ,n[ let ! x to w turn+ take place4 ,! squ>e ,,aceg w turn ab\t ! l9e ,,ae4 ,?is edge w rema9 along ! y axis & w be /ation>y1 h[ev] f> ! squ>e turns4 ,?us1 if ! cube be turn$ by an x to w turn+1 bo? ! edge ,,ab & ! edge ,,ac rema9 /ation>y2 h5ce ! :ole face ,,abef 9 ! yz plane rema9s fix$4 ,! turn+ has tak5 place ab\t ! face ,,abef4 ,illu/ration3 ,fig4 #g 7#ace74 ,suppose ?is turn+ to cont9ue till ,,ac runs to ! left f ,a4 ,! cube w occupy ! position %[n 9 fig4 #h4 ,?is is ! look+-glass image ( ! cube 9 fig4 #c4 ,by no rotation 9 ?ree-dim5sional space c ! cube be br\ght f ! position 9 fig4 #c to t %[n 9 fig4 #h4 ,illu/ration3 ,fig4 #h 7#acf74 ,we c ?9k ( ?is turn+ z a turn+ ( ! face ,,abcd ab\t ,,ab1 & a turn+ ( ea* section p>allel to ,,abcd r\nd ! v]tical l9e 9 :i* x int]sects ! face ,,abef1 ! space 9 :i* ! turn+ takes place be+ a diff]5t "o f t 9 :i* ! cube lies4 ,"o ( ! conditions1 !n1 ( our inquiry 9 ! direction ( ! inf9itely small is t we =m ! conception ( a rotation ab\t a plane4 ,! production ( a body 9 a /ate 9 :i* x pres5ts ! appe>ance ( a look+-glass image ( its =m] /ate is ! crit]ion = a f\r-dim5sional rotation4 ,!re is "s evid5ce = ! occurr5ce ( su* trans=mations ( bodies 9 ! *ange ( bodies f ^? :i* produce a "r-h&$ pol>isation ( light to ^? :i* produce a left-h&$ pol>isation2 b ?is is n a po9t to :i* any v great importance c be atta*$4 ,/ill1 9 ?is connection1 let me quote a rem>k f ,pr(4 ,john ,g4 ,mc,kendrick's address on ,physiology be=e ! ,briti% ,association at ,glasg[4 ,discuss+ ! possibility ( ! h]$it>y production ( *>act]i/ics ?r\gh ! mat]ial /ructure ( ! ovum1 he e/imates t 9 x !re exi/ #ab1jjj1jjj1jjj biophors1 or ultimate p>ticles ( liv+ matt]1 a suffici5t numb] to acc\nt = h]$it>y transmission1 & obs]ves3 8,?us x is conceivable t vital activities may also be det]m9$ by ! k9d ( motion t takes place 9 ! molecules ( t :i* we speak ( z liv+ matt]4 ,x may be diff]5t 9 k9d f "s ( ! motions kn[n to physici/s1 & x is conceivable t life may be ! transmission to dead matt]1 ! molecules ( :i* h already a special k9d ( motion1 ( a =m ( motion sui g5]is40 ,n[1 9 ! realm ( organic be+s symmetrical /ructures--^? ) a "r & left symmetry--are ev]y:]e 9 evid5ce4 ,grant$ t f\r dim5sions exi/1 ! simple/ turn+ produces ! image =m1 & by a fold+-ov] /ructures c\ld be produc$1 duplicat$ "r & left1 j z is ! case ( symmetry 9 a plane4 ,?us "o v g5]al *>act]i/ic ( ! =ms ( organisms c\ld be acc\nt$ = by ! supposition t a f\r-dim5sional motion 0 involv$ 9 ! process ( life4 ,b :e!r f\r-dim5sional motions correspond 9 o!r respects to ! physiologi/'s dem& = a special k9d ( motion1 or n1 ,i d n "k4 ,our bus9ess is ) ! evid5ce = _! exi/5ce 9 physics4 ,= ?is purpose x is necess>y to exam9e into ! significance ( rotation r\nd a plane 9 ! case ( ext5sible & ( fluid matt]4 ,let u dwell a mom5t long] on ! rotation ( a rigid body4 ,look+ at ! cube 9 fig4 #c1 :i* turns ab\t ! face ( ,,abfe1 we see t any l9e 9 ! face c take ! place ( ! v]tical & horizontal l9es we h exam9$4 ,take ! diagonal l9e ,,af & ! section ?r\gh x to ,,gh4 ,! portions ( matt] :i* 7 on "o side ( ,,af 9 ?is section 9 fig4 #c are on ! opposite side ( x 9 fig4 #h4 ,!y h gone r\nd ! l9e ,,af4 ,?us ! rotation r\nd a face c be consid]$ z a numb] ( rotations ( sections r\nd p>allel l9es 9 x4 ,! turn+ ab\t two diff]5t l9es is impossible 9 ?ree-dim5sional space4 ,to take ano!r illu/ration1 suppose ,a & ,b are two p>allel l9es 9 ! xy plane1 & let ,,cd & ,,ef be two rods cross+ !m4 ,n[1 9 ! space ( xyz if ! rods turn r\nd ! l9es ,a & ,b 9 ! same direction !y w make two indep5d5t circles4 ,:5 ! end ,f is go+ d[n ! end ,c w be com+ up4 ,!y w meet & conflict4 ,illu/ration3 ,fig4 #i 7#acg74 ,b if we rotate ! rods ab\t ! plane ( ,,ab by ! z to w rotation ^! movem5ts w n conflict4 ,suppose all ! figure remov$ ) ! exception ( ! plane xz1 & f ?is plane draw ! axis ( w1 s t we are look+ at ! space ( xzw4 ,"h1 fig4 #aj1 we _c see ! l9es ,a & ,b4 ,we see ! po9ts ,g & ,h1 9 :i* ,a & ,b int]cept ! x axis1 b we _c see ! l9es !mselves1 = !y run 9 ! y direction1 & t is n 9 our draw+4 ,n[1 if ! rods move ) ! z to w rotation !y w turn 9 p>allel planes1 keep+ _! relative positions4 ,! po9t ,d1 = in/ance1 w describe a circle4 ,at "o "t x w be above ! l9e ,a1 at ano!r "t bel[ x4 ,h5ce x rotates r\nd ,a4 ,illu/ration3 ,fig4 #aj 7#ach74 ,n only two rods b any numb] ( rods cross+ ! plane w move r\nd x h>moni\sly4 ,we c ?9k ( ?is rotation by suppos+ ! rods /&+ up f "o l9e to move r\nd t l9e & rememb]+ t x is n inconsi/5t ) ?is rotation = ! rods /&+ up along ano!r l9e also to move r\nd x1 ! relative positions ( all ! rods be+ pres]v$4 ,n[1 if ! rods are ?ick toge!r1 !y may repres5t a disk ( matt]1 & we see t a disk ( matt] c rotate r\nd a c5tral plane4 ,rotation r\nd a plane is exactly analog\s to rotation r\nd an axis 9 ?ree dim5sions4 ,if we want a rod to turn r\nd1 ! ends mu/ be free2 s if we want a disk ( matt] to turn r\nd its c5tral plane by a f\r-dim5sional turn+1 all ! cont\r mu/ be free4 ,! :ole cont\r corresponds to ! ends ( ! rod4 ,ea* po9t ( ! cont\r c be look$ on z ! extremity ( an axis 9 ! body1 r\nd ea* po9t ( :i* !re is a rotation ( ! matt] 9 ! disk4 ,if ! "o end ( a rod be clamp$1 we c twi/ ! rod1 b n turn x r\nd2 s if any "p ( ! cont\r ( a disk is clamp$ we c imp>t a twi/ to ! disk1 b n turn x r\nd its c5tral plane4 ,9 ! case ( ext5sible mat]ials a long1 ?9 rod w twi/ r\nd its axis1 ev5 :5 ! axis is curv$1 z1 = in/ance1 9 ! case ( a r+ ( ,india rubb]4 ,9 an analog\s mann]1 9 f\r dim5sions we c h rotation r\nd a curv$ plane1 if ,i may use ! expression4 ,a sph]e c be turn$ inside out 9 f\r dim5sions4 ,illu/ration3 ,fig4 #aa 7#aci74 ,let fig4 #aa repres5t a sph]ical surface1 on ea* side ( :i* a lay] ( matt] exi/s4 ,! ?ickness ( ! matt] is repres5t$ by ! rods ,,cd & ,,ef1 ext5d+ equally )\t & )94 ,illu/ration3 ,fig4 #ab 7#adj74 ,n[1 take ! section ( ! sph]e by ! yz plane we h a circle--fig4 #ab4 ,n[1 let ! w axis be drawn 9 place ( ! x axis s t we h ! space ( yzw repres5t$4 ,9 ?is space all t !re w be se5 ( ! sph]e is ! circle drawn4 ,"h we see t !re is no ob/acle to prev5t ! rods turn+ r\nd4 ,if ! matt] is s ela/ic t x w give en\gh = ! p>ticles at ,e & ,c to be sep>at$ z !y are at ,f & ,d1 !y c rotate r\nd to ! position ,d & ,f1 & a simil> motion is possible = all o!r p>ticles4 ,!re is no matt] or ob/acle to prev5t !m f mov+ out 9 ! w direction1 & !n on r\nd ! circumf]5ce z an axis4 ,n[1 :at w hold = "o section w hold = all1 z ! f\r? dim5sion is at "r angles to all ! sections :i* c be made ( ! sph]e4 ,we h suppos$ ! matt] ( :i* ! sph]e is compos$ to be ?ree-dim5sional4 ,if ! matt] _h a small ?ickness 9 ! f\r? dim5sion1 !re w\ld be a slight ?ickness 9 fig4 #ab above ! plane ( ! pap]--a ?ickness equal to ! ?ickness ( ! matt] 9 ! f\r? dim5sion4 ,! rods w\ld h to be replac$ by ?9 slabs4 ,b ?is w\ld make no diff]5ce z to ! possibility ( ! rotation4 ,?is motion is discuss$ by ,newcomb 9 ! fir/ volume ( ! ,am]ican ,j\rnal ( ,ma!matics4 ,let u n[ consid]1 n a m]ely ext5sible body1 b a liquid "o4 ,a mass ( rotat+ liquid1 a :irl1 eddy1 or vortex1 has _m rem>kable prop]ties4 ,on fir/ consid]ation we %\ld expect ! rotat+ mass ( liquid imm$iately to spread (f & lose itself 9 ! surr\nd+ liquid4 ,! wat] flies (f a :eel :irl$ r\nd1 & we %\ld expect ! rotat+ liquid to be disp]s$4 ,b see ! eddies 9 a riv] /rangely p]si/5t4 ,! r+s t occur 9 puffs ( smoke & la/ s long are :irls or vortices curv$ r\nd s t _! opposite ends jo9 toge!r4 ,a cyclone w travel ov] great di/ances4 ,helmholtz 0 ! fir/ to inve/igate ! prop]ties ( vortices4 ,he /udi$ !m z !y w\ld occur 9 a p]fect fluid--t is1 "o )\t friction ( "o mov+ portion or ano!r4 ,9 su* a m$ium vortices w\ld be inde/ructible4 ,!y w\ld g on = ev]1 alt]+ _! %ape1 b consi/+ always ( ! same portion ( ! fluid4 ,b a /raight vortex c\ld n exi/ surr\nd$ entirely by ! fluid4 ,! ends ( a vortex mu/ rea* to "s b\nd>y inside or outside ! fluid4 ,a vortex :i* is b5t r\nd s t its opposite ends jo9 is capable ( exi/+1 b no vortex has a free end 9 ! fluid4 ,! fluid r\nd ! vortex is always 9 motion1 & "o produces a def9ite movem5t 9 ano!r4 ,"l ,kelv9 has propos$ ! hypo!sis t portions ( a fluid segregat$ 9 vortices acc\nt = ! orig9 ( matt]4 ,! prop]ties ( ! e!r 9 respect ( its capacity ( propagat+ di/urbances c be expla9$ by ! assumption ( vortices 9 x in/ead ( by a prop]ty ( rigidity4 ,x is difficult to conceive1 h[ev]1 ( any arrangem5t ( ! vortex r+s & endless vortex filam5ts 9 ! e!r4 ,n[1 ! fur!r consid]ation ( f\r-dim5sional rotations %[s ! exi/5ce ( a k9d ( vortex :i* w\ld make an e!r fill$ ) a homog5e\s vortex motion easily ?9kable4 ,to und]/& ! nature ( ?is vortex1 we mu/ g on & take a /ep by :i* we accept ! full significance ( ! f\r-dim5sional hypo!sis4 ,grant$ f\r-dim5sional axes1 we h se5 t a rotation ( "o into ano!r leaves two unalt]$1 & ^! two =m ! axial plane ab\t :i* ! rotation takes place4 ,b :at ab\t ^! two8 ,d !y necess>ily rema9 motionless8 ,!re is no?+ to prev5t a rotation ( ^! two1 "o into ! o!r1 tak+ place concurr5tly ) ! fir/ rotation4 ,?is possibility ( a d\ble rotation des]ves ! mo/ c>eful att5tion1 = x is ! k9d ( movem5t :i* is di/9ctly typical ( f\r dim5sions4 ,rotation r\nd a plane is analog\s to rotation r\nd an axis4 ,b 9 ?ree-dim5sional space !re is no motion analog\s to ! d\ble rotation1 9 :i*1 :ile axis #a *anges into axis #b1 axis #c *anges into axis #d4 ,consid] a f\r-dim5sional body1 ) f\r indep5d5t axes1 x1 y1 z1 w4 ,a po9t 9 x c move 9 only "o direction at a giv5 mom5t4 ,if ! body has a velocity ( rotation by :i* ! x axis *anges into ! y axis & all p>allel sections move 9 a simil> mann]1 !n ! po9t w describe a circle4 ,if1 n[1 9 addition to ! rotation by :i* ! x axis *anges into ! y axis ! body has a rotation by :i* ! z axis turns into ! w axis1 ! po9t 9 "q w h a d\ble motion 9 consequ5ce ( ! two turn+s4 ,! motions w comp\nd1 & ! po9t w describe a circle1 b n ! same circle :i* x w\ld describe 9 virtue ( ei!r rotation sep>ately4 ,we "k t if a body 9 ?ree-dim5sional space is giv5 two movem5ts ( rotation !y w comb9e into a s+le movem5t ( rotation r\nd a def9ite axis4 ,x is 9 no diff]5t condition f t 9 :i* x is subject$ to "o movem5t ( rotation4 ,! direction ( ! axis *anges2 t is all4 ,! same is n true ab\t a f\r-dim5sional body4 ,! two rotations1 x to y & z to w1 are indep5d5t4 ,a body subject to ! two is 9 a totally diff]5t condition to t :i* x is 9 :5 subject to "o only4 ,:5 subject to a rotation su* z t ( x to y1 a :ole plane 9 ! body1 z we h se51 is /ation>y4 ,:5 subject to ! d\ble rotation no "p ( ! body is /ation>y except ! po9t common to ! two planes ( rotation4 ,if ! two rotations are equal 9 velocity1 e po9t 9 ! body describes a circle4 ,all po9ts equally di/ant f ! /ation>y po9t describe circles ( equal size4 ,we c repres5t a f\r-dim5sional sph]e by means ( two diagrams1 9 "o ( :i* we take ! ?ree axes1 x1 y1 z2 9 ! o!r ! axes x1 w1 & z4 ,9 fig4 #ac we h ! view ( a f\r-dim5sional sph]e 9 ! space ( xyz4 ,fig4 #ac %[s all t we c see ( ! f\r sph]e 9 ! space ( xyz1 = x repres5ts all ! po9ts 9 t space1 :i* are at an equal di/ance f ! c5tre4 ,let u n[ take ! xz section1 & let ! axis ( w take ! place ( ! y axis4 ,"h1 9 fig4 #ad1 we h ! space ( xzw4 ,9 ?is space we h to take all ! po9ts :i* are at ! same di/ance f ! c5tre1 consequ5tly we h ano!r sph]e4 ,if we _h a ?ree-dim5sional sph]e1 z has be5 %[n be=e1 we %\ld h m]ely a circle 9 ! xzw space1 ! xz circle se5 9 ! space ( xzw4 ,b n[1 tak+ ! view 9 ! space ( xzw1 we h a sph]e 9 t space also4 ,9 a simil> mann]1 :i*ev] set ( ?ree axes we take1 we obta9 a sph]e4 ,illu/ration3 ,%[+ axes xyz ,fig4 #ac 7#ada74 ,illu/ration3 ,%[+ axes xwz ,fig4 #ad 7#adb74 ,9 fig4 #ac1 let u imag9e ! rotation 9 ! direction xy to be tak+ place4 ,! po9t x w turn to y1 & p to p4 ,! axis zz rema9s /ation>y1 & ?is axis is all ( ! plane zw :i* we c see 9 ! space section exhibit$ 9 ! figure4 ,9 fig4 #ad1 imag9e ! rotation f z to w to be tak+ place4 ,! w axis n[ occupies ! position previ\sly occupi$ by ! y axis4 ,?is does n mean t ! w axis c co9cide ) ! y axis4 ,x indicates t we are look+ at ! f\r-dim5sional sph]e f a diff]5t po9t ( view4 ,any ?ree-space view w %[ u ?ree axes1 & 9 fig4 #ad we are look+ at xzw4 ,! only "p t is id5tical 9 ! two diagrams is ! circle ( ! x & z axes1 :i* axes are conta9$ 9 bo? diagrams4 ,?us ! plane zxz is ! same 9 bo?1 & ! po9t p repres5ts ! same po9t 9 bo? diagrams4 ,n[1 9 fig4 #ad let ! zw rotation take place1 ! z axis w turn t[>d ! po9t w ( ! w axis1 & ! po9t p w move 9 a circle ab\t ! po9t x4 ,?us 9 fig4 #ac ! po9t p moves 9 a circle p>allel to ! xy plane2 9 fig4 #ad x moves 9 a circle p>allel to ! zw plane1 indicat$ by ! arr[4 ,n[1 suppose bo? ( ^! indep5d5t rotations comp\nd$1 ! po9t p w move 9 a circle1 b ?is circle w co9cide ) nei!r ( ! circles 9 :i* ei!r "o ( ! rotations w take x4 ,! circle ! po9t p w move 9 w dep5d on its position on ! surface ( ! f\r sph]e4 ,9 ?is d\ble rotation1 possible 9 f\r-dim5sional space1 !re is a k9d ( movem5t totally unlike any ) :i* we are famili> 9 ?ree-dim5sional space4 ,x is a requisite prelim9>y to ! discussion ( ! behavi\r ( ! small p>ticles ( matt]1 ) a view to det]m9+ :e!r !y %[ ! *>act]i/ics ( f\r-dim5sional movem5ts1 to become famili> ) ! ma9 *>act]i/ics ( ?is d\ble rotation4 ,& "h ,i mu/ rely on a =mal & logical ass5t r ?an on ! intuitive appreh5sion1 :i* c only be obta9$ by a m detail$ /udy4 ,9 ! fir/ place ?is d\ble rotation consi/s 9 two v>ieties or k9ds1 :i* we w call ! ,a & ,b k9ds4 ,consid] f\r axes1 x1 y1 z1 w4 ,! rotation ( x to y c be accompani$ ) ! rotation ( z to w4 ,call ?is ! ,a k9d4 ,b also ! rotation ( x to y c be accompani$ by ! rotation1 ( n z to w1 b w to z4 ,call ?is ! ,b k9d4 ,!y diff] 9 only "o ( ! compon5t rotations4 ,"o is n ! negative ( ! o!r4 ,x is ! semi-negative4 ,! opposite ( an x to y1 z to w rotation w\ld be y to x1 w to z4 ,! semi-negative is x to y & w to z4 ,if f\r dim5sions exi/ & we _c p]ceive !m1 because ! ext5sion ( matt] is s small 9 ! f\r? dim5sion t all movem5ts are )held f direct obs]vation except ^? :i* are ?ree-dim5sional1 we %\ld n obs]ve ^! d\ble rotations1 b only ! effects ( !m 9 ?ree-dim5sional movem5ts ( ! type ) :i* we are famili>4 ,if matt] 9 its small p>ticles is f\r-dim5sional1 we %\ld expect ?is d\ble rotation to be a univ]sal *>act]i/ic ( ! atoms & molecules1 = no portion ( matt] is at re/4 ,! consequ5ces ( ?is corpuscul> motion c be p]ceiv$1 b only "u ! =m ( ord9>y rotation or displacem5t4 ,?us1 if ! !ory ( f\r dim5sions is true1 we h 9 ! corpuscles ( matt] a :ole _w ( movem5t1 :i* we c nev] /udy directly1 b only by means ( inf]5ce4 ,! rotation ,a1 z ,i h def9$ x1 consi/s ( two equal rotations--"o ab\t ! plane ( zw1 ! o!r ab\t ! plane ( xy4 ,x is evid5t t ^! rotations are n necess>ily equal4 ,a body may be mov+ ) a d\ble rotation1 9 :i* ^! two indep5d5t compon5ts are n equal2 b 9 su* a case we c consid] ! body to be mov+ ) a composite rotation--a rotation ( ! ,a or ,b k9d &1 9 addition1 a rotation ab\t a plane4 ,if we comb9e an ,a & a ,b movem5t1 we obta9 a rotation ab\t a plane2 =1 ! fir/ be+ x to y & z to w1 & ! second be+ x to y & w to z1 :5 !y are put toge!r ! z to w & w to z rotations neutralise ea* o!r1 & we obta9 an x to y rotation only1 :i* is a rotation ab\t ! plane ( zw4 ,simil>ly1 if we take a ,b rotation1 y to x & z to w1 we get1 on comb9+ ?is ) ! ,a rotation1 a rotation ( z to w ab\t ! xy plane4 ,9 ?is case ! plane ( rotation is 9 ! ?ree-dim5sional space ( xyz1 & we h--:at has be5 describ$ be=e--a twi/+ ab\t a plane 9 our space4 ,consid] n[ a portion ( a p]fect liquid hav+ an ,a motion4 ,x c be prov$ t x possesses ! prop]ties ( a vortex4 ,x =ms a p]man5t individuality--a sep>at$-out portion ( ! liquid--accompani$ by a motion ( ! surr\nd+ liquid4 ,x has prop]ties analog\s to ^? ( a vortex filam5t4 ,b x is n necess>y = its exi/5ce t its ends %\ld rea* ! b\nd>y ( ! liquid4 ,x is self-conta9$ &1 unless di/urb$1 is circul> 9 e section4 ,illu/ration3 ,fig4 #ae 7#adc74 ,if we suppose ! e!r to h its prop]ties ( transmitt+ vibration giv5 x by su* vortices1 we mu/ inquire h[ !y lie toge!r 9 f\r-dim5sional space4 ,plac+ a circul> disk on a plane & surr\nd+ x by six o!rs1 we f9d t if ! c5tral "o is giv5 a motion ( rotation1 x imp>ts to ! o!rs a rotation :i* is antagoni/ic 9 e two adjac5t ones4 ,if ,a goes r\nd1 z %[n by ! arr[1 ,b & ,c w be mov+ 9 opposite ways1 & ea* t5ds to de/roy ! motion ( ! o!r4 ,n[1 if we suppose sph]es to be arrang$ 9 a correspond+ mann] 9 ?ree-dim5sional space1 !y w be gr\p$ 9 figures :i* are = ?ree-dim5sional space :at hexagons are = plane space4 ,if a numb] ( sph]es ( s(t clay be press$ toge!r1 s z to fill up ! int]/ices1 ea* w assume ! =m ( a f\rte5-sid$ figure call$ a tetrakaidecagon4 ,n[1 assum+ space to be fill$ ) su* tetrakaidecagons1 & plac+ a sph]e 9 ea*1 x w be f\nd t "o sph]e is t\*$ by eight o!rs4 ,! rema9+ six sph]es ( ! f\rte5 :i* surr\nd ! c5tral "o w n t\* x1 b w t\* ?ree ( ^? 9 contact ) x4 ,h5ce1 if ! c5tral sph]e rotates1 x w n necess>ily drive ^? ar\nd x s t _! motions w be antagoni/ic to ea* o!r1 b ! velocities w n arrange !mselves 9 a sy/ematic mann]4 ,9 f\r-dim5sional space ! figure :i* =ms ! next t]m ( ! s]ies hexagon1 tetrakaidecagon1 is a ?irty-sid$ figure4 ,x has = its faces t5 solid tetrakaidecagons & tw5ty hexagonal prisms4 ,su* figures w exactly fill f\r-dim5sional space1 five ( !m meet+ at e po9t4 ,if1 n[1 9 ea* ( ^! figures we suppose a solid f\r-dim5sional sph]e to be plac$1 any "o sph]e is surr\nd$ by ?irty o!rs4 ,( ^! x t\*es t51 &1 if x rotates1 x drives ! re/ by means ( ^!4 ,n[1 if we imag9e ! c5tral sph]e to be giv5 an ,a or a ,b rotation1 x w turn ! :ole mass ( sph]e r\nd 9 a sy/ematic mann]4 ,suppose f\r-dim5sional space to be fill$ ) su* sph]es1 ea* rotat+ ) a d\ble rotation1 ! :ole mass w\ld =m "o consi/5t sy/em ( motion1 9 :i* ea* "o drove e o!r "o1 ) no friction or lagg+ beh9d4 ,e sph]e w\ld h ! same k9d ( rotation4 ,9 ?ree-dim5sional space1 if "o body drives ano!r r\nd ! second body rotates ) ! opposite k9d ( rotation2 b 9 f\r-dim5sional space ^! f\r-dim5sional sph]es w\ld ea* h ! d\ble negative ( ! rotation ( ! "o next x1 & we h se5 t ! d\ble negative ( an ,a or ,b rotation is /ill an ,a or ,b rotation4 ,?us f\r-dim5sional space c\ld be fill$ ) a sy/em ( self-pres]vative liv+ en]gy4 ,if we imag9e ! f\r-dim5sional sph]es to be ( liquid & n ( solid matt]1 !n1 ev5 if ! liquid 7 n q p]fect & !re 7 a slight ret>d+ effect ( "o vortex on ano!r1 ! sy/em w\ld /ill ma9ta9 itself4 ,9 ?is hypo!sis we mu/ look on ! e!r z possess+ en]gy1 & its transmission ( vibrations1 n z ! convey+ ( a motion imp>t$ f )\t1 b z a modification ( its own motion4 ,we are n[ 9 possession ( "s ( ! conceptions ( f\r-dim5sional me*anics1 & w turn aside f ! l9e ( _! developm5t to inquire if !re is any evid5ce ( _! applicability to ! processes ( nature4 ,is !re any mode ( motion 9 ! region ( ! m9ute :i*1 giv+ ?ree-dim5sional movem5ts = its effect1 /ill 9 itself escapes ! grasp ( our me*anical !ories8 ,i w\ld po9t to electricity4 ,?r\gh ! lab\rs ( ,f>aday & ,maxwell we are conv9c$ t ! ph5om5a ( electricity are ( ! nature ( ! /ress & /ra9 ( a m$ium2 b !re is /ill a gap to be bridg$ ov] 9 _! explanation--! laws ( ela/icity1 :i* ,maxwell assumes1 are n ^? ( ord9>y matt]4 ,&1 to take ano!r in/ance3 a magnetic pole 9 ! neighb\rhood ( a curr5t t5ds to move4 ,maxwell has %[n t ! pressures on x are analog\s to ! velocities 9 a liquid :i* w\ld exi/ if a vortex took ! place ( ! electric curr5t3 b we _c po9t out ! def9ite me*anical explanation ( ^! pressures4 ,!re mu/ be "s mode ( motion ( a body or ( ! m$ium 9 virtue ( :i* a body is said to be electrifi$4 ,take ! ions :i* convey *>ges ( electricity #ejj times great] 9 proportion to _! mass ?an are c>ri$ by ! molecules ( hydrog5 9 electrolysis4 ,9 respect ( :at motion c ^! ions be said to be electrifi$8 ,x c be %[n t ! en]gy !y possess is n en]gy ( rotation4 ,?9k ( a %ort rod rotat+4 ,if x is turn$ ov] x is f\nd to be rotat+ 9 ! opposite direction4 ,n[1 if rotation 9 "o direction corresponds to positive electricity1 rotation 9 ! opposite direction corresponds to negative electricity1 & ! smalle/ electrifi$ p>ticles w\ld h _! *>ges rev]s$ by be+ turn$ ov]--an absurd supposition4 ,if we fix on a mode ( motion z a def9ition ( electricity1 we mu/ h two v>ieties ( x1 "o = positive & "o = negative2 & a body possess+ ! "o k9d mu/ n become possess$ ( ! o!r by any *ange 9 its position4 ,all ?ree-dim5sional motions are comp\nd$ ( rotations & translations1 & none ( !m satisfy ?is fir/ condition = s]v+ z a def9ition ( electricity4 ,b consid] ! d\ble rotation ( ! ,a & ,b k9ds4 ,a body rotat+ ) ! ,a motion _c h its motion trans=m$ into ! ,b k9d by be+ turn$ ov] 9 any way4 ,suppose a body has ! rotation x to y & z to w4 ,turn+ x ab\t ! xy plane1 we rev]se ! direction ( ! motion x to y4 ,b we also rev]se ! z to w motion1 = ! po9t at ! extremity ( ! positive z axis is n[ at ! extremity ( ! negative z axis1 & s9ce we h n int]f]$ ) its motion x goes 9 ! direction ( position w4 ,h5ce we h y to x & w to z1 :i* is ! same z x to y & z to w4 ,?us bo? compon5ts are rev]s$1 & !re is ! ,a motion ov] aga94 ,! ,b k9d is ! semi-negative1 ) only "o compon5t rev]s$4 ,h5ce a sy/em ( molecules ) ! ,a motion w\ld n de/roy x 9 "o ano!r1 & w\ld imp>t x to a body 9 contact ) !m4 ,?us ,a & ,b motions possess ! fir/ requisite :i* mu/ be dem&$ 9 any mode ( motion repres5tative ( electricity4 ,let u trace out ! consequ5ces ( def9+ positive electricity z an ,a motion & negative electricity z a ,b motion4 ,! comb9ation ( positive & negative electricity produces a curr5t4 ,imag9e a vortex 9 ! e!r ( ! ,a k9d & unite ) ?is "o ( ! ,b k9d4 ,an ,a motion & ,b motion produce rotation r\nd a plane1 :i* is 9 ! e!r a vortex r\nd an axial surface4 ,x is a vortex ( ! k9d we repres5t z a "p ( a sph]e turn+ inside out4 ,n[ su* a vortex mu/ h its rim on a b\nd>y ( ! e!r--on a body 9 ! e!r4 ,let u suppose t a conductor is a body :i* has ! prop]ty ( s]v+ z ! t]m9al abutm5t ( su* a vortex4 ,!n ! conception we mu/ =m ( a clos$ curr5t is ( a vortex %eet hav+ its edge along ! circuit ( ! conduct+ wire4 ,! :ole wire w !n be l ! c5tres on :i* a sp9dle turns 9 ?ree-dim5sional space1 & any int]ruption ( ! cont9uity ( ! wire w produce a t5sion 9 place ( a cont9u\s revolution4 ,z ! direction ( ! rotation ( ! vortex is f a ?ree-space direction into ! f\r? dim5sion & back aga91 !re w be no direction ( fl[ to ! curr5t2 b x w h two sides1 accord+ to :e!r z goes to w or z goes to negative w4 ,we c draw any l9e f "o "p ( ! circuit to ano!r2 !n ! e!r along t l9e is rotat+ r\nd its po9ts4 ,?is geometric image corresponds to ! def9ition ( an electric circuit4 ,x is kn[n t ! action does n lie 9 ! wire1 b 9 ! m$ium1 & x is kn[n t !re is no direction ( fl[ 9 ! wire4 ,no explanation has be5 (f]$ 9 ?ree-dim5sional me*anics ( h[ an action c be impress$ ?r\gh\t a region & yet necess>ily run itself out along a clos$ b\nd>y1 z is ! case 9 an electric curr5t4 ,b ?is ph5om5on corresponds exactly to ! def9ition ( a f\r-dim5sional vortex4 ,if we take a v long magnet1 s long t "o ( its poles is practically isolat$1 & put ?is pole 9 ! vic9ity ( an electric circuit1 we f9d t x moves4 ,n[1 assum+ = ! sake ( simplicity t ! wire :i* det]m9es ! curr5t is 9 ! =m ( a circle1 if we take a numb] ( small magnets & place !m all po9t+ 9 ! same direction normal to ! plane ( ! circle1 s t !y fill x & ! wire b9ds !m r\nd1 we f9d t ?is %eet ( magnets has ! same effect on ! magnetic pole t ! curr5t has4 ,! %eet ( magnets may be curv$1 b ! edge ( x mu/ co9cide ) ! wire4 ,! collection ( magnets is !n equival5t to ! vortex %eet1 & an elem5t>y magnet to a "p ( x4 ,?us1 we mu/ ?9k ( a magnet z condition+ a rotation 9 ! e!r r\nd ! plane :i* bisects at "r angles ! l9e jo9+ its poles4 ,if a curr5t is />t$ 9 a circuit1 we mu/ imag9e vortices l b[ls turn+ !mselves inside out1 />t+ f ! cont\r4 ,9 rea*+ a p>allel circuit1 if ! vortex %eet 7 int]rupt$ & jo9$ mom5t>ily to ! second circuit by a free rim1 ! axis plane w\ld lie betwe5 ! two circuits1 & a po9t on ! second circuit opposite a po9t on ! fir/ w\ld correspond to a po9t opposite to x on ! fir/2 h5ce we %\ld expect a curr5t 9 ! opposite direction 9 ! second circuit4 ,?us ! ph5om5a ( induction are n inconsi/5t ) ! hypo!sis ( a vortex ab\t an axial plane4 ,9 f\r-dim5sional space1 9 :i* all f\r dim5sions 7 comm5surable1 ! int5sity ( ! action transmitt$ by ! m$ium w\ld v>y inv]sely z ! cube ( ! di/ance4 ,n[1 ! action ( a curr5t on a magnetic pole v>ies inv]sely z ! squ>e ( ! di/ance2 h5ce1 ov] measurable di/ances ! ext5sion ( ! e!r 9 ! f\r? dim5sion _c be assum$ z o!r ?an small 9 comp>ison ) ^? di/ances4 ,if we suppose ! e!r to be fill$ ) vortices 9 ! %ape ( f\r-dim5sional sph]es rotat+ ) ! ,a motion1 ! ,b motion w\ld correspond to electricity 9 ! "o-fluid !ory4 ,!re w\ld ?us be a possibility ( electricity exi/+ 9 two =ms1 /atically1 by itself1 &1 comb9$ ) ! univ]sal motion1 9 ! =m ( a curr5t4 ,to arrive at a def9ite conclusion x w be necess>y to inve/igate ! resultant pressures :i* accompany ! collocation ( solid vortices ) surface ones4 ,to recapitulate3 ,! movem5ts & me*anics ( f\r-dim5sional space are def9ite & intelligible4 ,a vortex ) a surface z its axis af=ds a geometric image ( a clos$ circuit1 & !re are rotations :i* by _! pol>ity af=d a possible def9ition ( /atical electricity4#g #g ,^! d\ble rotations ( ! ,a & ,b k9ds ,i %\ld l to call ,hamiltons & co-,hamiltons1 = x is a s+ul> fact t 9 8 8,quat]nions0 ,sir ,wm4 ,r[an ,hamilton has giv5 ! !ory ( ei!r ! ,a or ! ,b k9d4 ,!y foll[ ! laws ( 8 symbols1 ,i1 ,j1 ,k4 ,hamiltons & co-,hamiltons seem to be natural units ( geometrical expression4 ,9 ! pap] 9 ! 8,proce$+s ( ! ,royal ,iri% ,academy10 ,nov4 #aijc1 already allud$ to1 ,i h %[n some?+ ( ! rem>kable facility :i* is ga9$ 9 deal+ ) ! composition ( ?ree- & f\r-dim5sional rotations by an alt]ation 9 ,hamilton's notation1 :i* enables 8 sy/em to be appli$ to bo? ! ,a & ,b k9ds ( rotations4 ,! objection :i* has be5 (t5 made to ,hamilton's sy/em1 namely1 t x is only "u special conditions ( application t 8 processes give geometrically int]pretable results1 c be remov$1 if we assume t he 0 really deal+ ) a f\r-dim5sional motion1 & alt] 8 notation to br+ ?is circum/ance into explicit recognition4 ,,app5dix ,i ,,! ,,models ,9 ,*apt] ,,xi4 a description has be5 giv5 :i* w enable any "o to make a set ( models illu/rative ( ! tess]act & its prop]ties4 ,! set "h suppos$ to be employ$ consi/s (3-- #a4 ,?ree sets ( tw5ty-sev5 cubes ea*4 #b4 ,tw5ty-sev5 slabs4 #c4 ,twelve cubes ) po9ts1 l9es1 faces1 di/+ui%$ by col\rs1 :i* w be call$ ! catalogue cubes4 ,! prep>ation ( ! twelve catalogue cubes involves ! exp5diture ( a consid]able am\nt ( "t4 ,x is advantage\s to use !m1 b !y c be replac$ by ! draw+ ( ! views ( ! tess]act or by a ref]5ce to figs4 #ajc1 #ajd1 #aje1 #ajf ( ! text4 ,! slabs are col\r$ l ! tw5ty-sev5 cubes ( ! fir/ cubic block 9 fig4 #aja1 ! "o ) r$1 :ite1 yell[ axes4 ,! col\rs ( ! ?ree sets ( tw5ty-sev5 cubes are ^? ( ! cubes %[n 9 fig4 #aja4 ,! slabs are us$ to =m ! repres5tation ( a cube 9 a plane1 & c well be disp5s$ ) by any "o :o is accu/om$ to deal ) solid figures4 ,b ! :ole !ory dep5ds on a c>eful obs]vation ( h[ ! cube w\ld be repres5t$ by ^! slabs4 ,9 ! fir/ /ep1 t ( =m+ a cle> idea h[ a plane be+ w\ld repres5t ?ree-dim5sional space1 only "o ( ! catalogue cubes & "o ( ! ?ree blocks is ne$$4 ,,application ,,to ,,! ,,/ep ,,f ,,plane ,,to ,,solid4 ,look at fig4 #a ( ! views ( ! tess]act1 or1 :at comes to ! same ?+1 take catalogue cube ,no4 #a & place x be=e y ) ! r$ l9e runn+ up1 ! :ite l9e runn+ to ! "r1 ! yell[ l9e runn+ away4 ,! ?ree dim5sions ( space are !n m>k$ out by ^! l9es or axes4 ,n[ take a piece ( c>dbo>d1 or a book1 & place x s t x =ms a wall ext5d+ up & d[n n opposite to y1 b runn+ away p>allel to ! wall ( ! room on y\r left h&4 ,plac+ ! catalogue cube aga9/ ?is wall we see t x comes into contact ) x by ! r$ & yell[ l9es1 & by ! includ$ orange face4 ,9 ! plane be+'s _w ! aspect he has ( ! cube w\ld be a squ>e surr\nd$ by r$ & yell[ l9es ) grey po9ts4 ,n[1 keep+ ! r$ l9e fix$1 turn ! cube ab\t x s t ! yell[ l9e goes out to ! "r1 & ! :ite l9e comes into contact ) ! plane4 ,9 ?is case a diff]5t aspect is pres5t$ to ! plane be+1 a squ>e1 namely1 surr\nd$ by r$ & :ite l9es & grey po9ts4 ,y %\ld p>ticul>ly notice t :5 ! yell[ l9e goes out1 at "r angles to ! plane1 & ! :ite comes 91 ! latt] does n run 9 ! same s5se t ! yell[ did4 ,f ! fix$ grey po9t at ! base ( ! r$ l9e ! yell[ l9e ran away f y4 ,! :ite l9e n[ runs t[>ds y4 ,?is turn+ at "r angles makes ! l9e :i* 0 out ( ! plane be=e1 come into x 9 an opposite s5se to t 9 :i* ! l9e ran :i* has j left ! plane4 ,if ! cube does n break ?r\gh ! plane ?is is always ! rule4 ,aga9 turn ! cube back to ! normal position ) r$ runn+ up1 :ite to ! "r1 & yell[ away1 & try ano!r turn+4 ,y c keep ! yell[ l9e fix$1 & turn ! cube ab\t x4 ,9 ?is case ! r$ l9e go+ out to ! "r ! :ite l9e w come 9 po9t+ d[nw>ds4 ,y w be oblig$ to elevate ! cube f ! table 9 ord] to c>ry out ?is turn+4 ,x is always necess>y :5 a v]tical axis goes out ( a space to imag9e a movable support :i* w all[ ! l9e :i* ran out be=e to come 9 bel[4 ,hav+ look$ at ! ?ree ways ( turn+ ! cube s z to pres5t diff]5t faces to ! plane1 exam9e :at w\ld be ! appe>ance if a squ>e hole 7 cut 9 ! piece ( c>dbo>d1 & ! cube 7 to pass ?r\gh x4 ,a hole c be actually cut1 & x w be se5 t 9 ! normal position1 ) r$ axis runn+ up1 yell[ away1 & :ite to ! "r1 ! squ>e fir/ p]ceiv$ by ! plane be+--! "o conta9$ by r$ & yell[ l9es--w\ld be replac$ by ano!r squ>e ( :i* ! l9e t[>ds y is p9k--! section l9e ( ! p9k face4 ,! l9e above is light yell[1 bel[ is light yell[ & on ! opposite side away f y is p9k4 ,9 ! same way ! cube c be pu%$ ?r\gh a squ>e op5+ 9 ! plane f any ( ! positions :i* y h already turn$ x into4 ,9 ea* case ! plane be+ w p]ceive a diff]5t set ( cont\r l9es4 ,hav+ obs]v$ ^! facts ab\t ! catalogue cube1 turn n[ to ! fir/ block ( tw5ty-sev5 cubes4 ,y notice t ! col\r s*eme on ! catalogue cube & t ( ?is set ( blocks is ! same4 ,place !m be=e y1 a grey or null cube on ! table1 above x a r$ cube1 & on ! top a null cube aga94 ,!n away f y place a yell[ cube1 & beyond x a null cube4 ,!n to ! "r place a :ite cube & beyond x ano!r null4 ,!n complete ! block1 accord+ to ! s*eme ( ! catalogue cube1 putt+ 9 ! c5tre ( all an o*re cube4 ,y h n[ a cube l t :i* is describ$ 9 ! text4 ,= ! sake ( simplicity1 9 "s cases1 ?is cubic block c be r$uc$ to "o ( eight cubes1 by leav+ out ! t]m9ations 9 ea* direction4 ,?us1 in/ead ( null1 r$1 null1 ?ree cubes1 y c take null1 r$1 two cubes1 & s on4 ,x is useful1 h[ev]1 to practise ! repres5tation 9 a plane ( a block ( tw5ty-sev5 cubes4 ,= ?is purpose take ! slabs1 & build !m up aga9/ ! piece ( c>dbo>d1 or ! book 9 su* a way z to repres5t ! diff]5t aspects ( ! cube4 ,proce$ z foll[s3-- ,fir/1 cube 9 normal position4 ,place n9e slabs aga9/ ! c>dbo>d to repres5t ! n9e cubes 9 ! wall ( ! r$ & yell[ axes1 fac+ ! c>dbo>d2 ^! repres5t ! aspect ( ! cube z x t\*es ! plane4 ,n[ pu% ^! along ! c>dbo>d & make a diff]5t set ( n9e slabs to repres5t ! appe>ance :i* ! cube w\ld pres5t to a plane be+1 if x 7 to pass half way ?r\gh ! plane4 ,!re w\ld be a :ite slab1 above x a p9k "o1 above t ano!r :ite "o1 & six o!rs1 repres5t+ :at w\ld be ! nature ( a section across ! middle ( ! block ( cubes4 ,! section c be ?\ght ( z a ?9 slice cut out by two p>allel cuts across ! cube4 ,hav+ arrang$ ^! n9e slabs1 pu% !m along ! plane1 & make ano!r set ( n9e to repres5t :at w\ld be ! appe>ance ( ! cube :5 x _h almo/ completely gone ?r\gh4 ,?is set ( n9e w be ! same z ! fir/ set ( n9e4 ,n[ we h 9 ! plane ?ree sets ( n9e slabs ea*1 :i* repres5t ?ree sections ( ! tw5ty-sev5 block4 ,!y are put alongside "o ano!r4 ,we see t x does n matt] 9 :at ord] ! sets ( n9e are put4 ,z ! cube passes ?r\gh ! plane !y repres5t appe>ances :i* foll[ ! "o aft] ! o!r4 ,if !y 7 :at !y repres5t$1 !y c\ld n exi/ 9 ! same plane toge!r4 ,?is is a r important po9t1 namely1 to notice t !y %\ld n co-exi/ on ! plane1 & t ! ord] 9 :i* !y are plac$ is indiff]5t4 ,:5 we repres5t a f\r-dim5sional body our solid cubes are to u 9 ! same position t ! slabs are to ! plane be+4 ,y %\ld also notice t ea* ( ^! slabs repres5ts only ! v ?9ne/ slice ( a cube4 ,! set ( n9e slabs fir/ set up repres5ts ! side surface ( ! block4 ,x is1 z x 71 a k9d ( tray--a beg9n+ f :i* ! solid cube goes (f4 ,! slabs z we use !m h ?ickness1 b ?is ?ickness is a necessity ( con/ruction4 ,!y are to be ?\ght ( z m]ely ( ! ?ickness ( a l9e4 ,if n[ ! block ( cubes pass$ ?r\gh ! plane at ! rate ( an in* a m9ute ! appe>ance to a plane be+ w\ld be repres5t$ by3-- #a4 ,! fir/ set ( n9e slabs la/+ = "o m9ute4 #b4 ,! second set ( n9e slabs la/+ = "o m9ute4 #c4 ,! ?ird set ( n9e slabs la/+ = "o m9ute4 ,n[ ! appe>ances :i* ! cube w\ld pres5t to ! plane be+ 9 o!r positions c be %[n by means ( ^! slabs4 ,! use ( su* slabs w\ld be ! means by :i* a plane be+ c\ld acquire a famili>ity ) our cube4 ,turn ! catalogue cube 7or imag9e ! col\r$ figure turn$7 s t ! r$ l9e runs up1 ! yell[ l9e out to ! "r1 & ! :ite l9e t[>ds y4 ,!n turn ! block ( cubes to occupy a simil> position4 ,! block has n[ a diff]5t wall 9 contact ) ! plane4 ,its appe>ance to a plane be+ w n be ! same z be=e4 ,he has1 h[ev]1 en\gh slabs to repres5t ?is new set ( appe>ances4 ,b he mu/ remodel 8 =m] arrangem5t ( !m4 ,he mu/ take a null1 a r$1 & a null slab f ! fir/ ( 8 sets ( slabs1 !n a :ite1 a p9k1 & a :ite f ! second1 & !n a null1 a r$1 & a null f ! ?ird set ( slabs4 ,he takes ! fir/ column f ! fir/ set1 ! fir/ column f ! second set1 & ! fir/ column f ! ?ird set4 ,to repres5t ! half-way-?r\gh appe>ance1 :i* is z if a v ?9 slice 7 cut out half way ?r\gh ! block1 he mu/ take ! second column ( ea* ( 8 sets ( slabs1 & to repres5t ! f9al appe>ance1 ! ?ird column ( ea* set4 ,n[ turn ! catalogue cube back to ! normal position1 & also ! block ( cubes4 ,!re is ano!r turn+--a turn+ ab\t ! yell[ l9e1 9 :i* ! :ite axis comes bel[ ! support4 ,y _c break ?r\gh ! surface ( ! table1 s y mu/ imag9e ! old support to be rais$4 ,!n ! top ( ! block ( cubes 9 its new position is at ! level at :i* ! base ( x 0 be=e4 ,n[ repres5t+ ! appe>ance on ! plane1 we mu/ draw a horizontal l9e to repres5t ! old base4 ,! l9e %\ld be drawn ?ree in*es high on ! c>dbo>d4 ,bel[ ?is ! repres5tative slabs c be arrang$4 ,x is easy to see :at !y are4 ,! old arrangem5ts h to be brok5 up1 & ! lay]s tak5 9 ord]1 ! fir/ lay] ( ea* = ! repres5tation ( ! aspect ( ! block z x t\*es ! plane4 ,!n ! second lay]s w repres5t ! appe>ance half way ?r\gh1 & ! ?ird lay]s w repres5t ! f9al appe>ance4 ,x is evid5t t ! slabs individually d n repres5t ! same portion ( ! cube 9 ^! diff]5t pres5tations4 ,9 ! fir/ case ea* slab repres5ts a section or a face p]p5dicul> to ! :ite axis1 9 ! second case a face or a section :i* runs p]p5dicul>ly to ! yell[ axis1 & 9 ! ?ird case a section or a face p]p5dicul> to ! r$ axis4 ,b by means ( ^! n9e slabs ! plane be+ c repres5t ! :ole ( ! cubic block4 ,he c t\* & h&le ea* portion ( ! cubic block1 !re is no "p ( x :i* he _c obs]ve4 ,tak+ x bit by bit1 two axes at a "t1 he c exam9e ! :ole ( x4 ,,our ,,repres5tation ,,( ,a ,,block ,,( ,,tess]acts4 ,look at ! views ( ! tess]act #a1 #b1 #c1 or take ! catalogue cubes #a1 #b1 #c1 & place !m 9 front ( y1 9 any ord]1 say runn+ f left to "r1 plac+ #a 9 ! normal position1 ! r$ axis runn+ up1 ! :ite to ! "r1 & yell[ away4 ,n[ notice t 9 catalogue cube #b ! col\rs ( ea* region are d]iv$ f ^? ( ! correspond+ region ( cube #a by ! addition ( blue4 ,?us null blue blue1 & ! corn]s ( numb] #b are blue4 ,aga91 r$ blue purple1 & ! v]tical l9es ( #b are purple4 ,blue yell[ gre51 & ! l9e :i* runs away is col\r$ gre54 ,by means ( ^! obs]vations y may be sure t catalogue cube #b is rightly plac$4 ,catalogue cube #c is j l numb] #a4 ,hav+ ^! cubes 9 :at we may call _! normal position1 proce$ to build up ! ?ree sets ( blocks4 ,?is is easily done 9 accordance ) ! col\r s*eme on ! catalogue cubes4 ,! fir/ block we already "k4 ,build up ! second block1 beg9n+ ) a blue corn] cube1 plac+ a purple on x1 & s on4 ,hav+ ^! ?ree blocks we h ! means ( repres5t+ ! appe>ances ( a gr\p ( eighty-"o tess]acts4 ,let u consid] a mom5t :at ! analogy 9 ! case ( ! plane be+ is4 ,he has 8 ?ree sets ( n9e slabs ea*4 ,we h our ?ree sets ( tw5ty-sev5 cubes ea*4 ,our cubes are l 8 slabs4 ,z 8 slabs are n ! ?+s :i* !y repres5t to him1 s our cubes are n ! ?+s !y repres5t to u4 ,! plane be+'s slabs are to him ! faces ( cubes4 ,our cubes !n are ! faces ( tess]acts1 ! cubes by :i* !y are 9 contact ) our space4 ,z ea* set ( slabs 9 ! case ( ! plane be+ might be consid]$ z a sort ( tray f :i* ! solid cont5ts ( ! cubes came out1 s our ?ree blocks ( cubes may be consid]$ z ?ree-space trays1 ea* ( :i* is ! beg9n+ ( an in* ( ! solid cont5ts ( ! f\r-dim5sional solids />t+ f !m4 ,we want n[ to use ! names null1 r$1 :ite1 etc41 = tess]acts4 ,! cubes we use are only tess]act faces4 ,let u d5ote t fact by call+ ! cube ( null col\r1 null face2 or1 %ortly1 null f41 mean+ t x is ! face ( a tess]act4 ,to det]m9e :i* face x is let u look at ! catalogue cube #a or ! fir/ ( ! views ( ! tess]act1 :i* c be us$ in/ead ( ! models4 ,x has ?ree axes1 r$1 :ite1 yell[1 9 our space4 ,h5ce ! cube det]m9$ by ^! axes is ! face ( ! tess]act :i* we n[ h be=e u4 ,x is ! o*re face4 ,x is en\gh1 h[ev]1 simply to say null f41 r$ f4 = ! cubes :i* we use4 ,to impress ?is 9 y\r m9d1 imag9e t tess]acts d actually run f ea* cube4 ,!n1 :5 y move ! cubes ab\t1 y move ! tess]acts ab\t ) !m4 ,y move ! face b ! tess]act foll[s ) x1 z ! cube foll[s :5 its face is %ift$ 9 a plane4 ,! cube null 9 ! normal position is ! cube :i* has 9 x ! r$1 yell[1 :ite axes4 ,x is ! face hav+ ^!1 b want+ ! blue4 ,9 ?is way y c def9e :i* face x is y are h&l+4 ,i w write an 8f40 aft] ! "n ( ea* tess]act j z ! plane be+ might call ea* ( 8 slabs null slab1 yell[ slab1 etc41 to d5ote t !y 7 repres5tations4 ,we h !n 9 ! fir/ block ( tw5ty-sev5 cubes1 ! foll[+--null f41 r$ f41 null f41 go+ up2 :ite f41 null f41 ly+ to ! "r1 & s on4 ,/>t+ f ! null po9t & travell+ up "o in* we are 9 ! null region1 ! same = ! away & ! "r-h& directions4 ,& if we 7 to travel 9 ! f\r? dim5sion = an in* we %\ld /ill be 9 a null region4 ,! tess]act /ret*es equally all f\r ways4 ,h5ce ! appe>ance we h 9 ?is fir/ block w\ld d equally well if ! tess]act block 7 to move across our space = a c]ta9 di/ance4 ,= any?+ less ?an an in* ( _! transv]se motion we %\ld /ill h ! same appe>ance4 ,y mu/ notice1 h[ev]1 t we %\ld n h null face aft] ! motion _h begun4 ,:5 ! tess]act1 null = in/ance1 _h mov$ ev] s little we %\ld n h a face ( null b a section ( null 9 our space4 ,h5ce1 :5 we ?9k ( ! motion across our space we mu/ call our cubes tess]act sections4 ,?us on null pass+ across we %\ld see fir/ null f41 !n null s41 & !n1 f9ally1 null f4 aga94 ,imag9e n[ ! :ole fir/ block ( tw5ty-sev5 tess]acts to h mov$ tranv]se to our space a di/ance ( "o in*4 ,!n ! second set ( tess]acts1 :i* orig9ally 7 an in* di/ant f our space1 w\ld be ready to come 94 ,_! col\rs are %[n 9 ! second block ( tw5ty-sev5 cubes :i* y h be=e y4 ,^! repres5t ! tess]act faces ( ! set ( tess]acts t lay be=e an in* away f our space4 ,!y are ready n[ to come 91 & we c obs]ve _! col\rs4 ,9 ! place :i* null f4 occupi$ be=e we h blue f41 9 place ( r$ f4 we h purple f41 & s on4 ,ea* tess]act is col\r$ l ! "o ^: place x takes 9 ?is motion ) ! addition ( blue4 ,n[ if ! tess]act block goes on mov+ at ! rate ( an in* a m9ute1 ?is next set ( tess]acts w occupy a m9ute 9 pass+ across4 ,we %all see1 to take ! null "o = in/ance1 fir/ ( all null face1 !n null section1 !n null face aga94 ,at ! end ( ! second m9ute ! second set ( tess]acts has gone ?r\gh1 & ! ?ird set comes 94 ,?is1 z y see1 is col\r$ j l ! fir/4 ,altoge!r1 ^! ?ree sets ext5d ?ree in*es 9 ! f\r? dim5sion1 mak+ ! tess]act block ( equal magnitude 9 all dim5sions4 ,we h n[ be=e u a complete catalogue ( all ! tess]acts 9 our gr\p4 ,we h se5 !m all1 & we %all ref] to ?is arrangem5t ( ! blocks z ! 8normal position40 ,we h se5 z mu* ( ea* tess]act at a "t z c\ld be done 9 a ?ree-dim5sional space4 ,ea* "p ( ea* tess]act has be5 9 our space1 & we c\ld h t\*$ x4 ,! f\r? dim5sion appe>$ to u z ! duration ( ! block4 ,if a bit ( our matt] 7 to be subject$ to ! same motion x w\ld be in/antly remov$ out ( our space4 ,be+ ?9 9 ! f\r? dim5sion x is at once tak5 out ( our space by a motion 9 ! f\r? dim5sion4 ,b ! tess]act block we repres5t hav+ l5g? 9 ! f\r? dim5sion rema9s /eadily be=e our eyes = ?ree m9utes1 :5 x is subject$ to ?is transv]se motion4 ,we h n[ to =m repres5tations ( ! o!r views ( ! same tess]act gr\p :i* are possible 9 our space4 ,let u !n turn ! block ( tess]acts s t ano!r face ( x comes into contact ) our space1 & !n by obs]v+ :at we h1 & :at *anges come :5 ! block trav]ses our space1 we %all h ano!r view ( x4 ,! dim5sion :i* appe>$ z duration be=e w become ext5sion 9 "o ( our kn[n dim5sions1 & a dim5sion :i* co9cid$ ) "o ( our space dim5sions w appe> z duration4 ,leav+ catalogue cube #a 9 ! normal position1 remove ! o!r two1 or suppose !m remov$4 ,we h 9 space ! r$1 ! yell[1 & ! :ite axes4 ,let ! :ite axis g out into ! unkn[n1 & occupy ! position ! blue axis holds4 ,!n ! blue axis1 :i* runs 9 t direction n[ w come into space4 ,b x w n come 9 po9t+ 9 ! same way t ! :ite axis does n[4 ,x w po9t 9 ! opposite s5se4 ,x w come 9 runn+ to ! left in/ead ( runn+ to ! "r z ! :ite axis does n[4 ,:5 ?is turn+ takes place e "p ( ! cube #a w disappe> except ! left-h& face--! orange face4 ,& ! new cube t appe>s 9 our space w run to ! left f ?is orange face1 hav+ axes1 r$1 yell[1 blue4 ,take models #d1 #e1 #f4 ,place #d1 or suppose ,no4 #d ( ! tess]act views plac$1 ) its orange face co9cid5t ) ! orange face ( #a1 r$ l9e to r$ l9e1 & yell[ l9e to yell[ l9e1 ) ! blue l9e po9t+ to ! left4 ,!n remove cube #a & we h ! tess]act face :i* comes 9 :5 ! :ite axis runs 9 ! positive unkn[n1 & ! blue axis comes into our space4 ,n[ place catalogue cube #e 9 "s position1 x does n matt] :i*1 say to ! left2 & place x s t !re is a correspond5ce ( col\r correspond+ to ! col\r ( ! l9e t runs out ( space4 ,! l9e t runs out ( space is :ite1 h5ce1 e "p ( ?is cube #e %\ld diff] f ! correspond+ "p ( #d by an alt]ation 9 ! direction ( :ite4 ,?us we h :ite po9ts 9 #e correspond+ to ! null po9ts 9 #d4 ,we h a p9k l9e correspond+ to a r$ l9e1 a light yell[ l9e correspond+ to a yell[ l9e1 an o*re face correspond+ to an orange face4 ,?is cube section is completely nam$ 9 ,*apt] ,,xi4 ,f9ally cube #f is a replica ( #a4 ,^! catalogue cubes w enable u to set up our models ( ! block ( tess]acts4 ,fir/ ( all = ! set ( tess]acts1 :i* beg9n+ 9 our space rea* out "o in* 9 ! unkn[n1 we h ! patt]n ( catalogue cube #d4 ,we see t we c build up a block ( tw5ty-sev5 tess]act faces aft] ! col\r s*eme ( cube #d1 by tak+ ! left-h& wall ( block #a1 !n ! left-h& wall ( block #b1 & f9ally t ( block #c4 ,we take1 t is1 ! ?ree fir/ walls ( our previ\s arrangem5t to =m ! fir/ cubic block ( ?is new "o4 ,?is w repres5t ! cubic faces by :i* ! gr\p ( tess]acts 9 its new position t\*es our space4 ,we h runn+ up1 null f41 r$ f41 null f4 ,9 ! next v]tical l9e1 on ! side remote f u1 we h yell[ f41 orange f41 yell[ f41 & !n ! fir/ col\rs ov] aga94 ,!n ! ?ree foll[+ columns are1 blue f41 purple f41 blue f42 gre5 f41 br[n f41 gre5 f42 blue f41 purple f41 blue f4 ,! la/ ?ree columns are l ! fir/4 ,^! tess]acts t\* our space1 & none ( !m are by any "p ( !m di/ant m ?an an in* f x4 ,:at lies beyond !m 9 ! unkn[n8 ,?is c be told by look+ at catalogue cube #e4 ,accord+ to its s*eme ( col\r we see t ! second wall ( ea* ( our old arrangem5ts mu/ be tak54 ,putt+ !m toge!r we h1 z ! corn]1 :ite f4 above x1 p9k f4 above x1 :ite f4 ,! column next to ?is remote f u is z foll[s3--light yell[ f41 o*re f41 light yell[ f41 & beyond ?is a column l ! fir/4 ,!n = ! middle ( ! block1 light blue f41 above x light purple1 !n light blue4 ,! c5tre column has1 at ! bottom1 light gre5 f41 light br[n f4 9 ! c5tre & at ! top light gre5 f4 ,! la/ wall is l ! fir/4 ,! ?ird block is made by tak+ ! ?ird walls ( our previ\s arrangem5t1 :i* we call$ ! normal "o4 ,y may ask :at faces & :at sections our cubes repres5t4 ,to answ] ?is "q look at :at axes y h 9 our space4 ,y h r$1 yell[1 blue4 ,n[ ^! det]m9e br[n4 ,! col\rs r$1 yell[1 blue are suppos$ by u :5 mix$ to produce a br[n col\r4 ,& t cube :i* is det]m9$ by ! r$1 yell[1 blue axes we call ! br[n cube4 ,:5 ! tess]act block 9 its new position beg9s to move across our space ea* tess]act 9 x gives a section 9 our space4 ,?is section is transv]se to ! :ite axis1 :i* n[ runs 9 ! unkn[n4 ,z ! tess]act 9 its pres5t position passes across our space1 we %\ld see fir/ ( all ! fir/ ( ! blocks ( cubic faces we h put up--^! w\ld la/ = a m9ute1 !n w\ld come ! second block & !n ! ?ird4 ,at fir/ we %\ld h a cube ( tess]act faces1 ea* ( :i* w\ld be br[n4 ,directly ! movem5t began1 we %\ld h tess]act sections transv]se to ! :ite l9e4 ,!re are two m analog\s positions 9 :i* ! block ( tess]acts c be plac$4 ,to f9d ! ?ird position1 re/ore ! blocks to ! normal arrangem5t4 ,let u make ! yell[ axis g out into ! positive unkn[n1 & let ! blue axis1 consequ5tly1 come 9 runn+ t[>ds u4 ,! yell[ ran away1 s ! blue w come 9 runn+ t[>ds u4 ,put catalogue cube #a 9 its normal position4 ,take catalogue cube #g & place x s t its p9k face co9cides ) ! p9k face ( cube #a1 mak+ also its r$ axis co9cide ) ! r$ axis ( #a & its :ite ) ! :ite4 ,moreov]1 make cube #g come t[>ds u f cube #a4 ,look+ at x we see 9 our space1 r$1 :ite1 & blue axes4 ,! yell[ runs out4 ,place catalogue cube #h 9 ! neighb\rhood ( #g--obs]ve t e region 9 #h has a *ange 9 ! direction ( yell[ f ! correspond+ region 9 #g4 ,?is is because x repres5ts :at y come to n[ 9 go+ 9 ! unkn[n1 :5 ! yell[ axis runs out ( our space4 ,f9ally catalogue cube #i1 :i* is l numb] #g1 %[s ! col\rs ( ! ?ird set ( tess]acts4 ,n[ evid5tly1 />t+ f ! normal position1 to make up our ?ree blocks ( tess]act faces we h to take ! ne> wall f ! fir/ block1 ! ne> wall f ! second1 & !n ! ne> wall f ! ?ird block4 ,?is gives u ! cubic block =m$ by ! faces ( ! tw5ty-sev5 tess]acts :i* are n[ imm$iately t\*+ our space4 ,foll[+ ! col\r s*eme ( catalogue cube #h1 we make ! next set ( tw5ty-sev5 tess]act faces1 repres5t+ ! tess]acts1 ea* ( :i* beg9s "o in* (f f our space1 by putt+ ! second walls ( our previ\s arrangem5t toge!r1 & ! repres5tation ( ! ?ird set ( tess]acts is ! cubic block =m$ ( ! rema9+ ?ree walls4 ,s9ce we h r$1 :ite1 blue axes 9 our space to beg9 )1 ! cubes we see at fir/ are light purple tess]act faces1 & aft] ! transv]se motion beg9s we h cubic sections transv]se to ! yell[ l9e4 ,re/ore ! blocks to ! normal position1 !re rema9s ! case 9 :i* ! r$ axis turns out ( space4 ,9 ?is case ! blue axis w come 9 d[nw>ds1 opposite to ! s5se 9 :i* ! r$ axis ran4 ,9 ?is case take catalogue cubes #aj1 #aa1 #ab4 ,lift up catalogue cube #a & put #aj und]nea? x1 imag9+ t x goes d[n f ! previ\s position ( #a4 ,we h to keep 9 space ! :ite & ! yell[ axes1 & let ! r$ g out1 ! blue come 94 ,n[1 y w f9d on cube #aj a light yell[ face2 ?is %\ld co9cide ) ! base ( #a1 & ! :ite & yell[ l9es on ! two cubes %\ld co9cide4 ,!n ! blue axis runn+ d[n y h ! catalogue cube correctly plac$1 & x =ms a guide = putt+ up ! fir/ repres5tative block4 ,catalogue cube #aa w repres5t :at lies 9 ! f\r? dim5sion--n[ ! r$ l9e runs 9 ! f\r? dim5sion4 ,?us ! *ange f #aj to #aa %\ld be t[>ds r$1 correspond+ to a null po9t is a r$ po9t1 to a :ite l9e is a p9k l9e1 to a yell[ l9e an orange l9e1 & s on4 ,catalogue cube #ab is l #aj4 ,h5ce we see t to build up our blocks ( tess]act faces we mu/ take ! bottom lay] ( ! fir/ block1 hold t up 9 ! air1 und]nea? x place ! bottom lay] ( ! second block1 & f9ally und]nea? ?is la/ ! bottom lay] ( ! la/ ( our normal blocks4 ,simil>ly we make ! second repres5tative gr\p by tak+ ! middle c\rses ( our ?ree blocks4 ,! la/ is made by tak+ ! ?ree topmo/ lay]s4 ,! ?ree axes 9 our space be=e ! transv]se motion beg9s are blue1 :ite1 yell[1 s we h light gre5 tess]act faces1 & aft] ! motion beg9s sections transv]se to ! r$ light4 ,^! ?ree blocks repres5t ! appe>ances z ! tess]act gr\p 9 its new position passes across our space4 ,! cubes ( contact 9 ?is case are ^? det]m9al by ! ?ree axes 9 our space1 namely1 ! :ite1 ! yell[1 ! blue4 ,h5ce !y are light gre54 ,x foll[s f ?is t light gre5 is ! int]ior cube ( ! fir/ block ( repres5tative cubic faces4 ,practice 9 ! manipulations describ$1 ) a realization 9 ea* case ( ! face or section :i* is 9 our space1 is "o ( ! be/ means ( a ?or\gh compreh5sion ( ! subject4 ,we h to le>n h[ to get any "p ( ^! f\r-dim5sional figures into space1 s t we c look at !m4 ,we mu/ fir/ le>n to sw+ a tess]act1 & a gr\p ( tess]acts ab\t 9 any way4 ,:5 ^! op]ations h be5 repeat$ & ! me?od ( arrangem5t ( ! set ( blocks has become famili>1 x is a good plan to rotate ! axes ( ! normal cube #a ab\t a diagonal1 & !n repeat ! :ole s]ies ( turn+s4 ,?us1 9 ! normal position1 r$ goes up1 :ite to ! "r1 yell[ away4 ,make :ite g up1 yell[ to ! "r1 & r$ away4 ,le>n ! cube 9 ?is position by putt+ up ! set ( blocks ( ! normal cube1 ov] & ov] aga9 till x becomes z famili> to y z 9 ! normal position4 ,!n :5 ?is is le>n$1 & ! correspond+ *anges 9 ! arrangem5ts ( ! tess]act gr\ps are made1 ano!r *ange %\ld be made3 let1 9 ! normal cube1 yell[ g up1 r$ to ! "r1 & :ite away4 ,le>n ! normal block ( cubes 9 ?is new position by arrang+ !m & re-arrang+ !m till y "k )\t ?\ght :]e ea* "o goes4 ,!n c>ry out all ! tess]act arrangem5ts & turn+s4 ,if y want to und]/& ! subject1 b d n see y\r way cle>ly1 if x does n seem natural & easy to y1 practise ^! turn+s4 ,practise1 fir/ ( all1 ! turn+ ( a block ( cubes r\nd1 s t y "k x 9 e position z well z 9 ! normal "o4 ,practise by gradually putt+ up ! set ( cubes 9 _! new arrangem5ts4 ,!n put up ! tess]act blocks 9 _! arrangem5ts4 ,?is w give y a work+ conception ( high] space1 y w ga9 ! feel+ ( x1 :e!r y take up ! ma!matical treatm5t ( x or n4 ,,app5dix ,,ii ,a ,,language ,,( ,,space ,! m]e nam+ ! p>ts ( ! figures we consid] involves a c]ta9 am\nt ( "t & att5tion4 ,?is "t & att5tion leads to no result1 = ) ea* new figure ! nom5clature appli$ is completely *ang$1 e lett] or symbol is us$ 9 a diff]5t significance4 ,surely x mu/ be possible 9 "s way to utilise ! lab\r ?us at pres5t wa/$6 ,:y %\ld we n make a language = space itself1 s t e position we want to ref] to w\ld h its own "n8 ,!n e "t we nam$ a figure 9 ord] to demon/rate its prop]ties we %\ld be ex]cis+ ourselves 9 ! vocabul>y ( place4 ,if we use a def9ite sy/em ( names1 & always ref] to ! same space position by ! same "n1 we create z x 7 a multitude ( little h&s1 ea* prep>$ to grasp a special po9t1 position1 or elem5t1 & hold x = u 9 its prop] relations4 ,we make1 to use ano!r analogy1 a k9d ( m5tal pap]1 :i* has some:at ( ! prop]ties ( a s5sitive plate1 9 t x w regi/]1 )\t ef=t1 complex1 visual1 or tactual impressions4 ,b ( f> m importance ?an ! applications ( a space language to ! plane & to solid space is ! facilitation x br+s ) x to ! /udy ( f\r-dim5sional %apes4 ,i h delay$ introduc+ a space language because all ! sy/ems ,i made turn$ out1 aft] giv+ !m a fair trial1 to be intol]able4 ,i h n[ come ^u "o :i* seems to pres5t features ( p]man5ce1 & ,i w "h give an outl9e ( x1 s t x c be appli$ to ! subject ( ! text1 & 9 ord] t x may be subject$ to criticism4 ,! pr9ciple on :i* ! language is con/ruct$ is to sacrifice e o!r consid]ation = brevity4 ,x is inde$ curi\s t we are able to talk & conv]se on e subject ( ?\ght except ! fundam5tal "o ( space4 ,! only way ( speak+ ab\t ! spatial configurations t und]lie e subject ( discursive ?\ght is a co-ord9ate sy/em ( numb]s4 ,?is is s awkw>d & incommodi\s t x is nev] us$4 ,9 ?9k+ also1 9 realis+ %apes1 we d n use x2 we conf9e ourselves to a direct visualisation4 ,n[1 ! use ( words corresponds to ! /or+ up ( our exp]i5ce 9 a def9ite bra9 /ructure4 ,a *ild1 9 ! endless tactual1 visual1 m5tal manipulations x makes = itself1 is be/ left to itself1 b 9 ! c\rse ( in/ruction ! introduction ( space names w\ld make ! tea*]s "w m cumulative1 & ! *ild's k m social4 ,_! full use c only be appreciat$1 if !y are introduc$ e>ly 9 ! c\rse ( education2 b 9 a m9or degree any "o c conv9ce himself ( _! utility1 especially 9 our imm$iate subject ( h&l+ f\r-dim5sional %apes4 ,! sum total ( ! results obta9$ 9 ! prec$+ pages c be comp5di\sly & accurately express$ 9 n9e words ( ! ,space ,language4 ,9 "o ( ,plato's dialogues ,socrates makes an exp]im5t on a slave boy /&+ by4 ,he makes c]ta9 p]ceptions ( space awake 9 ! m9d ( ,m5o's slave by direct+ 8 close att5tion on "s simple facts ( geometry4 ,by means ( a few words & "s simple =ms we c repeat ,plato's exp]im5t on new gr\nd4 ,d we by direct+ our close att5tion on ! facts ( f\r dim5sions awak5 a lat5t faculty 9 ourselves8 ,! old exp]im5t ( ,plato's1 x seems to me1 has come d[n to u z novel z on ! day he incept$ x1 & its significance n bett] und]/ood ?r\gh all ! discussion ( :i* x has be5 ! subject4 ,imag9e a voiceless p liv+ 9 a region :]e ev]y?+ _h a velvety surface1 & :o 7 ?us depriv$ ( all opportunity ( exp]i5c+ :at s\nd is4 ,!y c\ld obs]ve ! sl[ pulsations ( ! air caus$ by _! movem5ts1 & argu+ f analogy1 !y w\ld no d\bt inf] t m rapid vibrations 7 possible4 ,f ! !oretical side !y c\ld det]m9e all ab\t ^! m rapid vibrations4 ,!y m]ely diff]1 !y w\ld say1 f sl[] ones1 by ! numb] t occur 9 a giv5 "t2 !re is a m]ely =mal diff]5ce4 ,b suppose !y 7 to take ! tr\ble1 g to ! pa9s ( produc+ ^! m rapid vibrations1 !n a totally new s5sation w\ld fall on _! rudim5t>y e>s4 ,probably at fir/ !y w\ld only be dimly consci\s ( ,s\nd1 b ev5 f ! fir/ !y w\ld become aw>e t a m]ely =mal diff]5ce1 a m]e diff]5ce 9 po9t ( numb] 9 ?is p>ticul> respect1 made a great diff]5ce practically1 z relat$ to !m4 ,& to u ! diff]5ce betwe5 ?ree & f\r dim5sions is m]ely =mal1 num]ical4 ,we c tell =mally all ab\t f\r dim5sions1 calculate ! relations t w\ld exi/4 ,b t ! diff]5ce is m]ely =mal does n prove t x is a futile & empty task1 to pres5t to ourselves z closely z we c ! ph5om5a ( f\r dim5sions4 ,9 our =mal k ( x1 ! :ole "q ( its actual relation to u1 z we are1 is left 9 abeyance4 ,possibly a new appreh5sion ( nature may come to u ?r\gh ! practical1 z di/+ui%$ f ! ma!matical & =mal1 /udy ( f\r dim5sions4 ,z a *ild h&les & exam9es ! objects ) :i* he comes 9 contact1 s we c m5tally h&le & exam9e f\r-dim5sional objects4 ,! po9t to be det]m9$ is ?is4 ,d we f9d some?+ cognate & natural to our faculties1 or are we m]ely build+ up an artificial pres5tation ( a s*eme only =mally possible1 conceivable1 b :i* has no real connection ) any exi/+ or possible exp]i5ce8 ,?is1 x seems to me1 is a "q :i* c only be settl$ by actually try+4 ,?is practical attempt is ! logical & direct cont9uation ( ! exp]im5t ,plato devis$ 9 ! 8,m5o40 ,:y d we ?9k true8 ,:y1 by our processes ( ?\ght1 c we pr$ict :at w happ51 & correctly conjecture ! con/itution ( ! ?+s ar\nd u8 ,?is is a problem :i* e mod]n philosoph] has consid]$1 & ( :i* ,desc>tes1 ,leibnitz1 ,kant1 to "n a few1 h giv5 memorable solutions4 ,plato 0 ! fir/ to sugge/ x4 ,& z he _h ! unique position ( be+ ! fir/ devisor ( ! problem1 s 8 solution is ! mo/ unique4 ,lat] philosoph]s h talk$ ab\t consci\sness & its laws1 s5sations1 categories4 ,b ,plato nev] us$ su* words4 ,consci\sness ap>t f a consci\s be+ meant no?+ to him4 ,8 0 always an objective se>*4 ,he made man's intuitions ! basis ( a new k9d ( natural hi/ory4 ,9 a few simple words ,plato puts u 9 an attitude ) reg>d to psy*ic ph5om5a--! m9d--! ego--8:at we are10 :i* is analog\s to ! attitude sci5tific m5 ( ! pres5t day h ) reg>d to ! ph5om5a ( outw>d nature4 ,beh9d ?is fir/ appreh5sion ( ours ( nature1 !re is an inf9ite dep? to be le>n$ & kn[n4 ,plato said t beh9d ! ph5om5a ( m9d t ,m5o's slave boy exhibit$1 !re 0 a va/1 an inf9ite p]spective4 ,& 8 s+ul>ity1 8 orig9ality1 comes out mo/ /rongly m>k$ 9 ?is1 t ! p]spective1 ! complex ph5om5a beyond 71 accord+ to him1 ph5om5a ( p]sonal exp]i5ce4 ,a footpr9t 9 ! s& means a man to a be+ t has ! conception ( a man4 ,b to a creature t has no su* conception1 x means a curi\s m>k1 someh[ result+ f ! concat5ation ( ord9>y occurr5ces4 ,su* a be+ w\ld attempt m]ely to expla9 h[ causes kn[n to him c\ld s co9cide z to produce su* a result2 he w\ld n recognise its significance4 ,plato introduc$ ! conception :i* made a new k9d ( natural hi/ory possible4 ,he said t ,m5o's slave boy ?\ght true ab\t ?+s he _h nev] le>n$1 because 8 8s\l0 _h exp]i5ce4 ,i "k ?is w s\nd absurd to "s p1 & x flies /raight 9 ! face ( ! maxim1 t explanation consi/s 9 %[+ h[ an effect dep5ds on simple causes4 ,b :at a mi/ak5 maxim t is6 ,c any s+le in/ance be %[n ( a simple cause8 ,take ! behavi\r ( sph]es = in/ance2 say ^? ivory sph]es1 billi>d balls1 = example4 ,we c expla9 _! behavi\r by suppos+ !y are homog5e\s ela/ic solids4 ,we c give =mul :i* w acc\nt = _! movem5ts 9 e v>iety4 ,b are !y homog5e\s ela/ic solids8 ,no1 c]ta9ly n4 ,!y are complex 9 physical & molecul> /ructure1 & atoms & ions beyond op5 an endless vi/a4 ,our simple explanation is false1 false z x c be4 ,! balls act z if !y 7 homog5e\s ela/ic sph]es4 ,!re is a /ati/ical simplicity 9 ! resultant ( v complex conditions1 :i* makes t artificial conception useful4 ,b its usefulness mu/ n bl9d u to ! fact t x is artificial4 ,if we really look deep into nature1 we f9d a mu* great] complexity ?an we at fir/ suspect4 ,& s beh9d ?is simple 8,i10 ?is myself1 is !re n a p>allel complexity8 ,plato's 8s\l0 w\ld be q acceptable to a l>ge class ( ?9k]s1 if by 8s\l0 & ! complexity he attributes to x1 he meant ! product ( a long c\rse ( evolution>y *anges1 :]eby simple =ms ( liv+ matt] end[$ ) rudim5t>y s5sation _h gradually develop$ into fully consci\s be+s4 ,b ,plato does n mean by 8s\l0 a be+ ( su* a k9d4 ,8 s\l is a be+ ^: faculties are clogg$ by its bodily environm5t1 or at lea/ hamp]$ by ! difficulty ( direct+ its bodily frame--a be+ :i* is ess5tially high] ?an ! acc\nt x gives ( itself ?r\gh its organs4 ,at ! same "t ,plato's s\l is n incorporeal4 ,x is a real be+ ) a real exp]i5ce4 ,! "q ( :e!r ,plato _h ! conception ( non-spatial exi/5ce has be5 mu* discuss$4 ,! v]dict is1 ,i believe1 t ev5 8 8ideas0 7 conceiv$ by him z be+s 9 space1 or1 z we %\ld say1 real4 ,plato's attitude is t ( ,sci5ce1 inasmu* z he ?9ks ( a _w 9 ,space4 ,b1 grant+ ?is1 x _c be d5i$ t !re is a fundam5tal div]g5ce betwe5 ,plato's conception & ! evolution>y !ory1 & also an absolute div]g5ce betwe5 8 conception & ! g5etic acc\nt ( ! orig9 ( ! human faculties4 ,! functions & capacities ( ,plato's 8s\l0 are n d]iv$ by ! int]action ( ! body & its environm5t4 ,plato 0 engag$ on a v>iety ( problems1 & 8 religi\s & e?ical ?\ghts 7 s ke5 & f]tile t ! exp]im5tal inve/igation ( 8 s\l appe>s involv$ ) _m o!r motives4 ,9 "o passage ,plato w comb9e matt] ( ?\ght ( all k9ds & f all s\rces1 ov]lapp+1 int]runn+4 ,& 9 no case is he m involv$ & ri* ?an 9 ?is "q ( ! s\l4 ,9 fact1 ,i wi% !re 7 two words1 "o d5ot+ t be+1 corporeal & real1 b ) high] faculties ?an we manife/ 9 our bodily actions1 :i* is to be tak5 z ! subject ( exp]im5tal inve/igation2 & ! o!r ^w d5ot+ 8s\l0 9 ! s5se 9 :i* x is made ! recipi5t & ! promise ( s mu* t m5 desire4 ,x is ! s\l 9 ! =m] s5se t ,i wi% to inve/igate1 & 9 a limit$ sph]e only4 ,i wi% to f9d out1 9 cont9uation ( ! exp]im5t 9 ! ,m5o1 :at ! 8s\l0 9 u ?9ks ab\t ext5sion1 exp]im5t+ on ! gr\nds laid d[n by ,plato4 ,he made1 to /ate ! matt] briefly1 ! hypo!sis ) reg>d to ! ?9k+ p[] ( a be+ 9 u1 a 8s\l40 ,?is s\l is n accessible to obs]vation by sight or t\*1 b x c be obs]v$ by its functions2 x is ! object ( a new k9d ( natural hi/ory1 ! mat]ials = con/ruct+ :i* lie 9 :at x is natural to u to ?9k4 ,) ,plato 8?\ght0 0 a v wide-rea*+ t]m1 b /ill ,i w\ld claim 9 8 g5]al plan ( proc$ure a place = ! p>ticul> "q ( ext5sion4 ,! problem comes to be1 8,:at is x natural to u to ?9k ab\t matt] qua ext5d$80 ,fir/ ( all1 ,i f9d t ! ord9>y intuition ( any simple object is extremely imp]fect4 ,take a block ( diff]5tly m>k$ cubes1 = in/ance1 & become acqua9t$ ) !m 9 _! positions4 ,y may ?9k y "k !m q well1 b :5 y turn !m r\nd--rotate ! block r\nd a diagonal1 = in/ance--y w f9d t y h lo/ track ( ! individuals 9 _! new positions4 ,y c m5tally con/ruct ! block 9 its new position1 by a rule1 by tak+ ! rememb]$ sequ5ces1 b y don't "k x intuitively4 ,by obs]vation ( a block ( cubes 9 v>i\s positions1 & v exp$iti\sly by a use ( ,space names appli$ to ! cubes 9 _! diff]5t pres5tations1 x is possible to get an intuitive k ( ! block ( cubes1 :i* is n di/urb$ by any displacem5t4 ,n[1 ) reg>d to ?is intuition1 we mod]ns w\ld say t ,i _h =m$ x by my tactual visual exp]i5ces 7aid$ by h]$it>y pre-disposition74 ,plato w\ld say t ! s\l _h be5 /imulat$ to recognise an in/ance ( %ape :i* x knew4 ,plato w\ld consid] ! op]ation ( le>n+ m]ely z a /imulus2 we z completely acc\nt+ = ! result4 ,! latt] is ! m common-s5se view4 ,b1 on ! o!r h&1 x presupposes ! g5]ation ( exp]i5ce f physical *anges4 ,! _w ( s5ti5t exp]i5ce1 accord+ to ! mod]n view1 is clos$ & limit$2 only ! physical _w is ample & l>ge & ( ev]-to-be-discov]$ complexity4 ,plato's _w ( s\l1 on ! o!r h&1 is at lea/ z l>ge & ample z ! _w ( ?+s4 ,let u n[ try a crucial exp]im5t4 ,c ,i =m an intuition ( a f\r-dim5sional object8 ,su* an object is n giv5 9 ! physical range ( my s5se contacts4 ,all ,i c d is to pres5t to myself ! sequ5ces ( solids1 :i* w\ld mean ! pres5tation to me "u my conditions ( a f\r-dim5sional object4 ,all ,i c d is to visualise & tactualise diff]5t s]ies ( solids :i* are alt]native sets ( sectional views ( a f\r-dim5sional %ape4 ,if n[1 on pres5t+ ^! sequ5ces1 ,i f9d a p[] 9 me ( intuitively pass+ f "o ( ^! sets ( sequ5ces to ano!r1 (1 be+ giv5 "o1 intuitively con/ruct+ ano!r1 n us+ a rule1 b directly appreh5d+ x1 !n ,i h f\nd a new fact ab\t my s\l1 t x has a f\r-dim5sional exp]i5ce2 ,i h obs]v$ x by a function x has4 ,i d n l to speak positively1 = ,i might occasion a loss ( "t on ! "p ( o!rs1 if1 z may v well be1 ,i am mi/ak54 ,b = my own "p1 ,i ?9k !re are indications ( su* an intuition2 f ! results ( my exp]im5ts1 ,i adopt ! hypo!sis t t :i* ?9ks 9 u has an ample exp]i5ce1 ( :i* ! intuitions we use 9 deal+ ) ! _w ( real objects are a "p2 ( :i* exp]i5ce1 ! intuition ( f\r-dim5sional =ms & motions is also a "p4 ,! process we are engag$ 9 intellectually is ! read+ ! obscure signals ( our n]ves into a _w ( reality1 by means ( intuitions d]iv$ f ! inn] exp]i5ce4 ,! image ,i =m is z foll[s4 ,imag9e ! capta9 ( a mod]n battle-%ip direct+ its c\rse4 ,he has 8 *>ts be=e him2 he is 9 communication ) 8 associates & subord9ates2 c convey 8 messages & comm&s to e "p ( ! %ip1 & receive in=mation f ! conn+-t[] & ! eng9e-room4 ,n[ suppose ! capta9 imm]s$ 9 ! problem ( ! navigation ( 8 %ip ov] ! ocean1 to h s absorb$ himself 9 ! problem ( ! direction ( 8 craft ov] ! plane surface ( ! sea t he =gets himself4 ,all t occupies 8 att5tion is ! k9d ( movem5t t 8 %ip makes4 ,! op]ations by :i* t movem5t is produc$ h sunk bel[ ! ?re%old ( 8 consci\sness1 8 own actions1 by :i* he pu%es ! buttons1 gives ! ord]s1 are s famili> z to be automatic1 8 m9d is on ! motion ( ! %ip z a :ole4 ,9 su* a case we c imag9e t he id5tifies himself ) 8 %ip2 all t ent]s 8 consci\s ?\ght is ! direction ( its movem5t ov] ! plane surface ( ! ocean4 ,su* is ! relation1 z ,i imag9e x1 ( ! s\l to ! body4 ,a relation :i* we c imag9e z exi/+ mom5t>ily 9 ! case ( ! capta9 is ! normal "o 9 ! case ( ! s\l ) its craft4 ,z ! capta9 is capable ( a k9d ( movem5t1 an amplitude ( motion1 :i* does n ent] into 8 ?\ghts ) reg>d to ! direct+ ! %ip ov] ! plane surface ( ! ocean1 s ! s\l is capable ( a k9d ( movem5t1 has an amplitude ( motion1 :i* is n us$ 9 its task ( direct+ ! body 9 ! ?ree-dim5sional region 9 :i* ! body's activity lies4 ,if = any reason x became necess>y = ! capta9 to consid] ?ree-dim5sional motions ) reg>d to 8 %ip1 x w\ld n be difficult = him to ga9 ! mat]ials = ?9k+ ab\t su* motions2 all he has to d is to call 8 own intimate exp]i5ce into play4 ,z f> z ! navigation ( ! %ip1 h[ev]1 is conc]n$1 he is n oblig$ to call on su* exp]i5ce4 ,! %ip z a :ole simply moves on a surface4 ,! problem ( ?ree-dim5sional movem5t does n ord9>ily conc]n its /e]+4 ,& ?us ) reg>d to ourselves all ^? movem5ts & activities :i* *>act]ise our bodily organs are ?ree-dim5sional2 we nev] ne$ to consid] ! ampl] movem5ts4 ,b we d m ?an use ! movem5ts ( our body to effect our aims by direct means2 we h n[ come to ! pass :5 we act indirectly on nature1 :5 we call processes into play :i* lie beyond ! rea* ( any explanation we c give by ! k9d ( ?\ght :i* has be5 suffici5t = ! /e]+ ( our craft z a :ole4 ,:5 we come to ! problem ( :at goes on 9 ! m9ute1 & apply ourselves to ! me*anism ( ! m9ute1 we f9d our habitual conceptions inadequate4 ,! capta9 9 u mu/ wake up to 8 own intimate nature1 realise ^? functions ( movem5t :i* are 8 own1 & 9 virtue ( 8 k ( !m appreh5d h[ to deal ) ! problems he has come to4 ,?9k ( ! hi/ory ( man4 ,:5 has !re be5 a "t1 9 :i* 8 ?\ghts ( =m & movem5t 7 n exclusively ( su* v>ieties z 7 adapt$ = 8 bodily p]=mance8 ,we h nev] _h a dem& to conceive :at our own mo/ intimate p[]s are4 ,b1 j z little z by imm]s+ himself 9 ! /e]+ ( 8 %ip ov] ! plane surface ( ! ocean1 a capta9 c lose ! faculty ( ?9k+ ab\t :at he actually does1 s little c ! s\l lose its own nature4 ,x c be r\s$ to an intuition t is n d]iv$ f ! exp]i5ce :i* ! s5ses give4 ,all t is necess>y is to pres5t "s few ( ^? appe>ances :i*1 :ile inconsi/5t ) ?ree-dim5sional matt]1 are yet consi/5t ) our =mal k ( f\r-dim5sional matt]1 9 ord] = ! s\l to wake up & n beg9 to le>n1 b ( its own intimate feel+ fill up ! gaps 9 ! pres5tim5t1 grasp ! full orb ( possibilities f ! isolat$ po9ts pres5t$ to x4 ,9 relation to ?is "q ( our p]ceptions1 let me sugge/ ano!r illu/ration1 n tak+ x too s]i\sly1 only prop\nd+ x to exhibit ! possibilities 9 a broad & g5]al way4 ,9 ! heav5s1 among/ ! multitude ( />s1 !re are "s :i*1 :5 ! telescope is direct$ on !m1 seem n to be s+le />s1 b to be split up into two4 ,reg>d+ ^! tw9 />s ?r\gh a spectroscope1 an a/ronom] sees 9 ea* a spectrum ( b&s ( col\r & black l9es4 ,comp>+ ^! spectrums ) "o ano!r1 he f9ds t !re is a slight relative %ift+ ( ! d>k l9es1 & f t %ift+ he kn[s t ! />s are rotat+ r\nd "o ano!r1 & c tell _! relative velocity ) reg>d to ! e>?4 ,by means ( 8 t]re/rial physics he reads ?is signal ( ! skies4 ,?is %ift+ ( l9es1 ! m]e slight v>iation ( a black l9e 9 a spectrum1 is v unlike t :i* ! a/ronom] kn[s x means4 ,b x is probably mu* m l :at x means ?an ! signals :i* ! n]ves deliv] are l ! ph5om5a ( ! out] _w4 ,no picture ( an object is convey$ ?r\gh ! n]ves4 ,no picture ( motion1 9 ! s5se 9 :i* we po/ulate its exi/5ce1 is convey$ ?r\gh ! n]ves4 ,! actual deliv]ances ( :i* our consci\sness takes acc\nt are probably id5tical = eye & e>1 sight & t\*4 ,if = a mom5t ,i take ! :ole e>? toge!r & reg>d x z a s5ti5t be+1 ,i f9d t ! problem ( its appreh5sion is a v complex "o1 & involves a long s]ies ( p]sonal & physical ev5ts4 ,simil>ly ! problem ( our appreh5sion is a v complex "o4 ,i only use ?is illu/ration to exhibit my mean+4 ,x has ?is especial m]it1 t1 z ! process ( consci\s appreh5sion takes place 9 our case 9 ! m9ute1 s1 ) reg>d to ?is e>? be+1 ! correspond+ process takes place 9 :at is relatively to x v m9ute4 ,n[1 ,plato's view ( a s\l leads u to ! hypo!sis t t :i* we designate z an act ( appreh5sion may be a v complex ev5t1 bo? physically & p]sonally4 ,he does n seek to expla9 :at an intuition is2 he makes x a basis f :5ce he sets out on a voyage ( discov]y4 ,k means k2 he puts consci\s be+ to acc\nt = consci\s be+4 ,he makes an hypo!sis ( ! k9d t is s f]tile 9 physical sci5ce--an hypo!sis mak+ no claim to f9ality1 :i* m>ks out a vi/a ( possible det]m9ation beh9d det]m9ation1 l ! hypo!sis ( space itself1 ! type ( s]viceable hypo!ses4 ,&1 above all1 ,plato's hypo!sis is conducive to exp]im5t4 ,he gives ! p]spective 9 :i* real objects c be det]m9$2 &1 9 our pres5t enquiry1 we are mak+ ! simple/ ( all possible exp]im5ts--we are enquir+ :at x is natural to ! s\l to ?9k ( matt] z ext5d$4 ,ari/otle says we always use a 8phantasm0 9 ?9k+1 a phantasm ( our corporeal s5ses a visualisation or a tactualisation4 ,b we c s modify t visualisation or tactualisation t x repres5ts some?+ n kn[n by ! s5ses4 ,d we by t repres5tation wake up an intuition ( ! s\l8 ,c we by ! pres5tation ( ^! hypo!tical =ms1 t are ! subject ( our pres5t discussion1 wake ourselves up to high] intuitions8 ,& c we expla9 ! _w ar\nd by a motion t we only "k by our s\ls8 ,ap>t f all speculation1 h[ev]1 x seems to me t ! int]e/ ( ^! f\r-dim5sional %apes & motions is suffici5t reason = /udy+ !m1 & t !y are ! way by :i* we c gr[ into a full] appreh5sion ( ! _w z a concrete :ole4 ,,space ,,names4 ,if ! words writt5 9 ! squ>es drawn 9 fig4 #a are us$ z ! names ( ! squ>es 9 ! positions 9 :i* !y are plac$1 x is evid5t t a comb9ation ( ^! names w d5ote a figure compos$ ( ! designat$ squ>es4 ,x is f\nd to be mo/ conv5i5t to take z ! initial squ>e t m>k$ ) an a/]isk1 s t ! directions ( progression are t[>ds ! obs]v] & to 8 "r4 ,! directions ( progression1 h[ev]1 are arbitr>y1 & c be *os5 at w4 ,illu/ration3 ,fig4 #a4 ,?us et1 at1 x1 an1 al w d5ote a figure 9 ! =m ( a cross compos$ ( five squ>es4 ,"h1 by means ( ! d\ble sequ5ce1 e1 a1 i & n1 t1 l1 x is possible to "n a limit$ collection ( space elem5ts4 ,! sy/em c obvi\sly be ext5d$ by us+ lett] sequ5ces ( m memb]s4 ,b1 )\t introduc+ su* a complexity1 ! pr9ciples ( a space language c be exhibit$1 & a nom5clature obta9$ adequate to all ! consid]ations ( ! prec$+ pages4 #a4 ,ext5sion4 ,call ! l>ge squ>es 9 fig4 #b by ! "n writt5 9 !m4 ,x is evid5t t ea* c be divid$ z %[n 9 fig4 #a4 ,!n ! small squ>e m>k$ #a w be 8en0 9 8,en10 or 8,en540 ,! squ>e m>k$ #b w be 8et0 9 8,en0 or 8,enet10 :ile ! squ>e m>k$ #d w be 8en0 9 8,et0 or 8,et540 ,?us ! squ>e #e w be call$ 8,ilil40 ,illu/ration3 ,fig4 #b4 ,?is pr9ciple ( ext5sion c be appli$ 9 any numb] ( dim5sions4 #b4 ,application to ,?ree-,dim5sional ,space4 ,to "n a ?ree-dim5sional collocation ( cubes take ! upw>d direction fir/1 secondly ! direction t[>ds ! obs]v]1 ?irdly ! direction to 8 "r h&4 ,illu/ration ,^! =m a ^w 9 :i* ! fir/ lett] gives ! place ( ! cube upw>ds1 ! second lett] its place t[>ds ! obs]v]1 ! ?ird lett] its place to ! "r4 ,we h ?us ! foll[+ s*eme1 :i* repres5ts ! set ( cubes ( column #a1 fig4 #aja1 page #afe4 ,we beg9 ) ! remote l[e/ cube at ! left h&1 :]e ! a/]isk is plac$ 7?is proves to be by f> ! mo/ conv5i5t orig9 to take = ! normal sy/em74 ,?us 8n50 is a 8null0 cube1 8t50 a r$ cube on x1 & 8l50 a 8null0 cube above 8t540 ,by us+ a m ext5d$ sequ5ce ( consonants & v[els a l>g] set ( cubes c be nam$4 ,to "n a f\r-dim5sional block ( tess]acts x is simply necess>y to prefix an 8e10 an 8a10 or an 8i0 to ! cube names4 ,?us ! tess]act blocks s*ematically repres5t$ on page #afe1 fig4 #aja are nam$ z foll[s3-- ,illu/ration3 #a #b #c #b4 ,,d]ivation ,,( ,,po9t1 ,,l9e1 ,,face1 ,,etc41 ,,names4 ,illu/ration ,! pr9ciple ( d]ivation c be %[n z foll[s3 ,tak+ ! squ>e ( squ>es ! numb] ( squ>es 9 x c be enl>g$ & ! :ole kept ! same size4 ,illu/ration ,comp>e fig4 #gi1 p4 #ach1 = in/ance1 or ! bottom lay] ( fig4 #hd4 ,n[ use an initial 8s0 to d5ote ! result ( c>ry+ ?is process on to a great ext5t1 & we obta9 ! limit names1 t is ! po9t1 l9e1 area names = a squ>e4 8,sat0 is ! :ole int]ior4 ,! corn]s are 8s510 8sel10 8s910 8sil10 :ile ! l9es are 8san10 8sal10 8set10 8sit40 ,illu/ration ,i f9d t by ! use ( ! initial 8s0 ^! names come to be practically entirely disconnect$ ) ! sy/ematic names = ! squ>e f :i* !y are d]iv$4 ,!y are easy to le>n1 & :5 le>n$ c be us$ readily ) ! axes runn+ 9 any direction4 ,to d]ive ! limit names = a f\r-dim5sional rectangul> figure1 l ! tess]act1 is a simple ext5sion ( ?is process4 ,^! po9t1 l9e1 etc41 names include ^? :i* apply to a cube1 z w be evid5t on inspection ( ! fir/ cube ( ! diagrams :i* foll[4 ,all t is necess>y is to place an 8s0 be=e ea* ( ! names giv5 = a tess]act block4 ,we !n obta9 apellatives :i*1 l ! col\r names on page #agd1 fig4 #ajc1 apply to all ! po9ts1 l9es1 faces1 solids1 & to ! hyp]-solid ( ! tess]act4 ,^! names h ! advantage ov] ! col\r m>ks t ea* po9t1 l9e1 etc41 has its own individual "n4 ,9 ! diagrams ,i give ! names correspond+ to ! positions %[n 9 ! col\r$ plate or describ$ on p4 #agd4 ,by comp>+ cubes #a1 #b1 #c ) ! fir/ r[ ( cubes 9 ! col\r$ plate1 ! sy/ematic names ( ea* ( ! po9ts1 l9es1 faces1 etc41 c be det]m9$4 ,! a/]isk %[s ! orig9 f :i* ! names run4 ,^! po9t1 l9e1 face1 etc41 names %\ld be us$ 9 connection ) ! correspond+ col\rs4 ,! names %\ld call up col\r$ images ( ! p>ts nam$ 9 _! "r connection4 ,illu/ration ,x is f\nd t a c]ta9 abbreviation adds vividness ( di/9ction to ^! names4 ,if ! f9al 8en0 be dropp$ :]ev] x occurs ! sy/em is improv$4 ,?us in/ead ( 8s5510 8set510 8sel510 x is pref]able to abbreviate to 8s510 8set10 8sel10 & also use 8san10 8s90 = 8san510 8s9540 ,illu/ration ,illu/ration ,we c n[ "n any section4 ,take e4g4 ! l9e 9 ! fir/ cube f s59 to s5el1 we %\ld call ! l9e runn+ f s59 to s5el1 s59 s5at s5el1 a l9e light yell[ 9 col\r ) null po9ts4 ,illu/ration ,"h s5at is ! "n = all ( ! l9e except its ends4 ,us+ 8s5at0 9 ?is way does n mean t ! l9e is ! :ole ( s5at1 b :at !re is ( x is s5at4 ,x is a "p ( ! s5at region4 ,?us also ! triangle1 :i* has its ?ree v]tices 9 s591 s5el1 sel51 is nam$ ?us3 ,area3 setat4 ,sides3 setan1 s5at1 setet4 ,v]tices3 s591 s5el1 sel4 ,! tetrah$ron section ( ! tess]act c be ?\ght ( z a s]ies ( plane sections 9 ! successive sections ( ! tess]act %[n 9 fig4 #aad1 p4 #aia4 ,9 b#j ! section is ! "o writt5 above4 ,9 b#a ! section is made by a plane :i* cuts ! ?ree edges f san5 int]m$iate ( _! l5g?s & ?us w be3 ,area3 satat4 ,sides3 satan1 sanat1 satet4 ,v]tices3 sanan1 sanet1 sat4 ,! sections 9 b#b1 b#c w be l ! section 9 b#a b small]4 ,f9ally 9 b#d ! section plane simply passes ?r\gh ! corn] nam$ s94 ,h5ce1 putt+ ^! sections toge!r 9 _! "r relation1 f ! face setat1 surr\nd$ by ! l9es & po9ts m5tion$ above1 !re run3 #c faces3 satan1 sanat1 satet #c l9es3 sanan1 sanet1 sat & ^! faces & l9es run to ! po9t s94 ,?us ! tetrah$ron is completely nam$4 ,! octah$ron section ( ! tess]act1 :i* c be trac$ f fig4 #gb1 p4 #abi by ext5d+ ! l9es !re drawn1 is nam$3 ,front triangle sel91 selat1 selel1 setal1 s5il1 setit1 sel9 ) area setat4 ,! sections betwe5 ! front & re> triangle1 ( :i* "o is %[n 9 #a;b1 ano!r 9 #b;b1 are ?us nam$1 po9ts & l9es1 salan1 salat1 salet1 satet1 satel1 satal1 sanal1 sanat1 sanit1 satit1 sat91 satan1 salan4 ,! re> triangle f\nd 9 #c;b by produc+ l9es is sil1 sitet1 s9el1 s9at1 s991 sitan1 sil4 ,! assemblage ( sections con/itute ! solid body ( ! octah$ron satat ) triangul> faces4 ,! "o f ! l9e selat to ! po9t sil1 = in/ance1 is nam$ sel91 selat1 selel1 salet1 salat1 salan1 sil4 ,! :ole int]ior is salat4 ,%apes c easily be cut out ( c>dbo>d :i*1 :5 fold$ toge!r1 =m n only ! tetrah$ron & ! octah$ron1 b also samples ( all ! sections ( ! tess]act tak5 z x passes corn]wise ?r\gh our space4 ,to "n & visualise ) appropriate col\rs a s]ies ( ^! sections is an admirable ex]cise = obta9+ famili>ity ) ! subject4 ,,ext5sion ,,& ,,connection ,,) ,,numb]s4 ,by ext5d+ ! lett] sequ5ce x is ( c\rse possible to "n a l>g] field4 ,by us+ ! limit names ! corn]s ( ea* squ>e c be nam$4 ,?us 8en s510 8an s510 etc41 w be ! names ( ! po9ts ne>e/ ! orig9 9 8en0 & 9 8an40 ,a field ( po9ts ( :i* ea* "o is indef9itely small is giv5 by ! names writt5 bel[4 ,illu/ration ,! squ>es are %[n 9 dott$ l9es1 ! names d5ote ! po9ts4 ,^! po9ts are n ma!matical po9ts1 b really m9ute areas4 ,in/ead ( />t+ ) a set ( squ>es & nam+ !m1 we c />t ) a set ( po9ts4 ,by an easily rememb]$ conv5tion we c give names to su* a region ( po9ts4 ,let ! space names ) a f9al 8e0 add$ d5ote ! ma!matical po9ts at ! corn] ( ea* squ>e ne>e/ ! orig94 ,we h !n = ! set ( ma!matical po9ts indicat$4 ,?is sy/em is really completely indep5d5t ( ! area sy/em & is connect$ ) x m]ely = ! purpose ( facilitat+ ! memory processes4 ,! ^w 8ene0 is pron\nc$ l 8eny10 ) j suffici5t att5tion to ! f9al v[el to di/+ui% x f ! ^w 8en40 ,illu/ration ,n[1 connect+ ! numb]s #j1 #a1 #b ) ! sequ5ce e1 a1 i1 & also ) ! sequ5ce n1 t1 l1 we h a set ( po9ts nam$ z ) numb]s 9 a co-ord9ate sy/em4 ,?us 8ene0 is 7#j1 #j7 8ate0 is 7#a1 #a7 8ite0 is 7#b1 #a74 ,to pass to ! area sy/em ! rule is t ! "n ( ! squ>e is =m$ f ! "n ( its po9t ne>e/ to ! orig9 by dropp+ ! f9al e4 ,by us+ a notation analog\s to ! decimal sy/em a l>g] field ( po9ts c be nam$4 ,x rema9s to assign a lett] sequ5ce to ! numb]s f positive #j to positive #i1 & f negative #j to negative #i1 to obta9 a sy/em :i* c be us$ to d5ote bo? ! usual co-ord9ate sy/em ( mapp+ & a sy/em ( nam$ squ>es4 ,! names d5ot+ ! po9ts all end ) e4 ,^? t d5ote squ>es end ) a consonant4 ,!re are _m consid]ations :i* mu/ be att5d$ to 9 ext5d+ ! sequ5ces to be us$1 su* z uniqu5ess 9 ! mean+ ( ! words =m$1 ease ( pronunciation1 avoidance ( awkw>d comb9ations4 ,i drop 8s0 altoge!r f ! consonant s]ies & %ort 8u0 f ! v[el s]ies4 ,x is conv5i5t to h unsignificant lett]s at disposal4 ,a d\ble consonant l 8/0 = in/ance c be ref]r$ to )\t giv+ x a local significance by call+ x 8u/40 ,i increase ! numb] ( v[els by consid]+ a s\nd l 8ra0 to be a v[el1 us+1 t is1 ! lett] 8r0 z =m+ a comp\nd v[el4 ,! s]ies is z foll[s3-- ,,consonants4 #j #a #b #c #d #e #f #g #h #i positive n t l p f % k * nt / negative z d ? b v m g j nd sp ,,v[els4 #j #a #b #c #d #e #f #g #h #i positive e a i ee ae ai ar ra ri ree negative er o oo io oe iu or ro roo rio ,pronunciation4--e z 9 m52 a z 9 man2 i z 9 92 ee z 9 betwe52 ae z ay 9 may2 ai z i 9 m9e2 ar z 9 art2 er z e> 9 e>?2 o z 9 on2 oo z oo 9 soon2 io z 9 cl>ion2 oe z oa 9 oat2 iu pron\nc$ l yew4 ,to "n a po9t su* z 7#bc1 #da7 x is consid]$ z 7#c1 #a7 on f 7#bj1 #dj7 & is call$ 8ifeete40 ,x is ! initial po9t ( ! squ>e ifeet ( ! area sy/em4 ,! prec$+ amplification ( a space language has be5 introduc$ m]ely = ! sake ( complet5ess4 ,z has already be5 said n9e words & _! comb9ations1 appli$ to a few simple models suffice = ! purposes ( our pres5t enquiry4 ,pr9t$ by ,hazell1 ,watson ,v9ey1 ,ld41 ,london & ,aylesbury4