,,transcrib]',s ,,note ,italic text is d5ot$ by und]scores4 ,bold text is d5ot$ by equal signs4 ,a sup]script is d5ot$ by x4 ,= example1 y#d indicates y to ! p[] #d4 ,a subscript is d5ot$ by x4 ,= example1 aik indicates ! ik-? elem5t ( array a4 ,algebraic expressions 9 ! orig9al text usually _h ! v>iables 9 italic4 ,! italic und]score m>kup has be5 remov$ = readability1 s = example3 dxi dxi dxi has be5 *ang$ to3 dxi dxi dxi ,obvi\s typographical errors & punctuation errors h be5 correct$ aft] c>eful comp>ison ) o!r occurr5ces )9 ! text & consultation ( ext]nal s\rces4 ,m detail c be f\nd at ! end ( ! book4 ,,! ,,f\ndations ,,( ,,geometry4 ,london3 ,c4 ,j4 ,,clay ,,& ,,sons1 ,,cambridge ,,univ]sity ,,press ,,w>eh\se1 ,,ave ,,m>ia ,,lane4 ,glasg[3 #bfc1 ,,argyle ,,/reet4 ,illu/ration ,leipzig3 ,f4 ,a4 ,,brockhaus4 ,new ,york3 ,,! ,,macmillan ,,company4 ,bombay3 ,,george ,,bell ,,& ,,sons4 ,,an ,,essay ,,on ,,! ,,f\ndations ,,( ,,geometry ,,by ,,b]tr& ,a4 ,w4 ,,russell4 ,m4,a4 ,,fell[ ,,( ,,tr9ity ,,college1 ,,cambridge4 ,,cambridge3 ,,at ,,! ,,univ]sity ,,press4 #ahig ,all ,rights res]v$4 ,cambridge3 ,,pr9t$ ,,by ,j4 ,,& ,c4 ,f4 ,,clay1 ,,at ,,! ,,univ]sity ,,press4 ,,preface4 ,! pres5t "w is bas$ on a diss]tation submitt$ at ! ,fell[%ip ,exam9ation ( ,tr9ity ,college1 ,cambridge1 9 ! ye> #ahie4 ,section ,b ( ! ?ird *apt] is 9 ! ma9 a repr9t1 ) "s s]i\s alt]ations1 ( an article 9 ,m9d 7,new ,s]ies1 ,no4 #ag74 ,! sub/ance ( ! book has be5 giv5 9 ! =m ( lectures at ! ,johns ,hopk9s ,univ]sity1 ,baltimore1 & at ,bryn ,mawr ,college1 ,p5nsylvania4 ,my *ief obligation is to ,pr(essor ,kle94 ,?r\gh\t ! fir/ *apt]1 ,i h f\nd 8 ,lectures on non-,euclidean ,geometry an invaluable guide2 ,i h accept$ f him ! division ( ,metageometry into ?ree p]iods1 & h f\nd my hi/orical "w mu* light5$ by 8 ref]5ces to previ\s writ]s4 ,9 ,logic1 ,i h le>nt mo/ f ,mr ,bradley1 & next to him1 f ,sigw>t & ,dr ,bosanquet4 ,on sev]al important po9ts1 ,i h d]iv$ useful sugge/ions f ,pr(essor ,james's ,pr9ciples ( ,psy*ology4 ,my ?anks are due to ,mr ,g4 ,f4 ,/\t & ,mr ,a4 ,n4 ,:itehead = k9dly read+ my pro(s1 & help+ me by _m useful criticisms4 ,to ,mr ,:itehead ,i owe1 also1 ! ine/imable assi/ance ( con/ant criticism & sugge/ion ?r\gh\t ! c\rse ( con/ruction1 especially z reg>ds ! philosophical importance ( projective ,geometry4 ,,haslem]e4 ,may1 #ahig4 ,,to ,,john ,mc,t,a,g,g,a,r,t ,,ellis ,mc,t,a,g,g,a,r,t ,,to ,,^: ,,disc\rse ,,& ,,fri5d%ip ,,is ,,ow+ ,,! ,,exi/5ce ,,( ,,?is ,,book4 ,,table ,,( ,,cont5ts4 ,,introduction4 ,,our ,,problem ,,def9$ ,,by ,,its ,,relations ,,to ,,logic1 ,,psy*ology ,,& ,,ma!matics4 ,,page #a4 ,! problem fir/ receiv$ a mod]n =m ?r\gh ,kant1 :o connect$ ! priori ) ! subjective #a #b4 ,a m5tal /ate is subjective1 = ,psy*ology1 :5 its imm$iate cause does n lie 9 ! out] _w #b #c4 ,a piece ( k is priori1 = ,epi/emology1 :5 )\t x k w\ld be impossible #b #d4 ,! subjective & ! priori belong respectively to ,psy*ology & to ,epi/emology4 ,! latt] alone w be inve/igat$ 9 ?is essay #c #e4 ,my te/ ( ! priori w be purely logical3 :at k is necess>y = exp]i5ce8 #c #f4 ,b s9ce ! necess>y is hypo!tical1 we mu/ include1 9 ! priori1 ! gr\nd ( necessity #d #g4 ,?is may be ! ess5tial po/ulate ( our sci5ce1 or ! elem5t1 9 ! subject-matt]1 :i* is necess>y to exp]i5ce2 #d #h4 ,:i*1 h[ev]1 are bo? at bottom ! same gr\nd #e #i4 ,=eca/ ( ! "w #e ,,*apt] ,i4 ,a ,,%ort ,,hi/ory ,,( ,,metageometry4 #aj4 ,metageometry began by reject+ ! axiom ( p>allels #g #aa4 ,its hi/ory may be divid$ into ?ree p]iods3 ! syn!tic1 ! metrical & ! projective #g #ab4 ,! fir/ p]iod 0 inaugurat$ by ,gauss1 #aj #ac4 ,^: sugge/ions 7 develop$ indep5d5tly by ,lobat*ewsky #aj #ad4 ,& ,bolyai #aa #ae4 ,! purpose ( all ?ree 0 to %[ t ! axiom ( p>allels c\ld n be d$uc$ f ! o!rs1 s9ce its d5ial did n lead to contradictions #ab #af4 ,! second p]iod _h a m philosophical aim1 & 0 inspir$ *iefly by ,gauss & ,h]b>t #ac #ag4 ,! fir/ "w ( ?is p]iod1 t ( ,riemann1 inv5t$ two new conceptions3 #ad #ah4 ,! fir/1 t ( a manifold1 is a class-conception1 conta9+ space z a species1 #ad #ai4 ,& def9$ z su* t its det]m9ations =m a collection ( magnitudes #ae #bj4 ,! second1 ! measure ( curvature ( a manifold1 grew out ( curvature 9 curves & surfaces #af #ba4 ,by means ( ,gauss's analytical =mula = ! curvature ( surfaces1 #ai #bb4 ,:i* enables u to def9e a con/ant measure ( curvature ( a ?ree-dim5sional space )\t ref]5ce to a f\r? dim5sion #bj #bc4 ,! ma9 result ( ,riemann's ma!matical "w 0 to %[ t1 if magnitudes are indep5d5t ( place1 ! measure ( curvature ( space mu/ be con/ant #ba #bd4 ,helmholtz1 :o 0 m ( a philosoph] ?an a ma!matician1 #bb #be4 ,gave a new b incorrect =mulation ( ! ess5tial axioms1 #bc #bf4 ,& d$uc$ ! quadratic =mula = ! inf9itesimal arc1 :i* ,riemann _h assum$ #bd #bg4 ,beltrami gave ,lobat*ewsky's planimetry a ,euclidean int]pretation1 #be #bh4 ,:i* is analog\s to ,cayley's !ory ( di/ance2 #bf #bi4 ,& dealt ) n-dim5sional spaces ( con/ant negative curvature #bg #cj4 ,! ?ird p]iod ab&ons ! metrical me?ods ( ! second1 & extrudes ! notion ( spatial quantity #bg #ca4 ,cayley r$uc$ metrical prop]ties to projective prop]ties1 relative to a c]ta9 conic or quadric1 ! ,absolute2 #bh #cb4 ,& ,kle9 %[$ t ! ,euclidean or non-,euclidean sy/ems result1 accord+ to ! nature ( ! ,absolute2 #bi #cc4 ,h5ce ,euclidean space appe>$ to give rise to all ! k9ds ( ,geometry1 & ! "q1 :i* is true1 appe>$ r$uc$ to "o ( conv5tion #cj #cd4 ,b ?is view is due to a confusion z to ! nature ( ! coord9ates employ$ #cj #ce4 ,projective coord9ates h be5 reg>d$ z dep5d5t on di/ance1 & ?us really metrical #ca #cf4 ,b ?is is n ! case1 s9ce anh>monic ratio c be projectively def9$ #cb #cg4 ,projective coord9ates1 be+ purely descriptive1 c give no in=mation z to metrical prop]ties1 & ! r$uction ( metrical to projective prop]ties is purely te*nical #cc #ch4 ,! true connection ( ,cayley's measure ( di/ance ) non-,euclidean ,geometry is t sugge/$ by ,beltrami's ,saggio1 & work$ out by ,sir ,r4 ,ball1 #cf #ci4 ,:i* provides a ,euclidean equival5t = e non-,euclidean proposition1 & s removes ! possibility ( contradictions 9 ,metageometry #ch #dj4 ,kle9's elliptic ,geometry has n be5 prov$ to h a correspond+ v>iety ( space #ci #da4 ,! geometrical use ( imag9>ies1 ( :i* ,cayley dem&$ a philosophical discussion1 #da #db4 ,has a m]ely te*nical validity1 #db #dc4 ,& is capable ( giv+ geometrical results only :5 x beg9s & ends ) real po9ts & figures #de #dd4 ,we h n[ se5 t projective ,geometry is logically prior to metrical ,geometry1 b _c sup]s$e x #df #de4 ,sophus ,lie has appli$ projective me?ods to ,helmholtz's =mulation ( ! axioms1 & has %[n ! axiom ( ,monodromy to be sup]flu\s #df #df4 ,metageometry has gradually gr[n indep5d5t ( philosophy1 b has gr[n cont9ually m int]e/+ to philosophy #ej #dg4 ,metrical ,geometry has ?ree indisp5sable axioms1 #ej #dh4 ,:i* we %all f9d to be n results1 b conditions1 ( measurem5t1 #ea #di4 ,& :i* are ne>ly equival5t to ! ?ree axioms ( projective ,geometry #eb #ej4 ,bo? sets ( axioms are necessitat$1 n by facts1 b by logic #eb ,,*apt] ,,ii4 ,,critical ,,acc\nt ,,( ,,"s ,,previ\s ,,philosophical ,,!ories ,,( ,,geometry4 #ea4 ,a criticism ( repres5tative mod]n !ories ne$ n beg9 be=e ,kant #ed #eb4 ,kant's doctr9e mu/ be tak51 9 an argum5t ab\t ,geometry1 on its purely logical side #ee #ec4 ,kant cont5ds t s9ce ,geometry is apodeictic1 space mu/ be priori & subjective1 :ile s9ce space is priori & subjective1 ,geometry mu/ be apodeictic #ee #ed4 ,metageometry has upset ! fir/ l9e ( argum5t1 n ! second #ef #ee4 ,! second may be attack$ by criticiz+ ei!r ! di/9ction ( syn!tic & analytic judgm5ts1 or ! fir/ two argum5ts ( ! metaphysical d$uction ( space #eg #ef4 ,mod]n ,logic reg>ds e judgm5t z bo? syn!tic & analytic1 #eg #eg4 ,b leaves ! priori1 z t :i* is presuppos$ 9 ! possibility ( exp]i5ce #ei #eh4 ,kant's fir/ two argum5ts z to space suffice to prove "s =m ( ext]nality1 b n necess>ily ,euclidean space1 a necess>y condition ( exp]i5ce #fj #ei4 ,among ! successors ( ,kant1 ,h]b>t alone advanc$ ! !ory ( ,geometry1 by influ5c+ ,riemann #fb #fj4 ,riemann reg>d$ space z a p>ticul> k9d ( manifold1 i4e4 :olly quantitatively #fc #fa4 ,he !re=e unduly neglect$ ! qualitative adjectives ( space #fd #fb4 ,8 philosophy re/s on a vici\s disjunction #fe #fc4 ,8 def9ition ( a manifold is obscure1 #ff #fd4 ,& 8 def9ition ( measurem5t applies only to space #fg #fe4 ,?\gh ma!matically invaluable1 8 view ( space z a manifold is philosophically mislead+ #fi #ff4 ,helmholtz attack$ ,kant bo? on ! ma!matical & on ! psy*ological side2 #gj #fg4 ,b 8 crit]ion ( apriority is *angeable & (t5 invalid2 #ga #fh4 ,8 pro( t non-,euclidean spaces are imag9able is inconclusive2 #gb #fi4 ,& 8 ass]tion ( ! dep5d5ce ( measurem5t on rigid bodies1 :i* may be tak5 9 ?ree s5ses1 #gd #gj4 ,is :olly false if x means t ! axiom ( ,congru5ce actually ass]ts ! exi/5ce ( rigid bodies1 #ge #ga4 ,is untrue if x means t ! necess>y ref]5ce ( geometrical propositions to matt] r5d]s pure ,geometry empirical1 #gf #gb4 ,& is inadequate to 8 conclusion if x means1 :at is true1 t actual measurem5t involves approximately rigid bodies #gh #gc4 ,geometry deals ) an ab/ract matt]1 ^: physical prop]ties are disreg>d$2 & ,physics mu/ presuppose ,geometry #hj #gd4 ,erdmann accept$ ! conclusions ( ,riemann & ,helmholtz1 #ha #ge4 ,& reg>d$ ! axioms z necess>ily successive /eps 9 classify+ space z a species ( manifold #hb #gf4 ,8 d$uction involves f\r fallaci\s assumptions1 namely3 #hb #gg4 ,t conceptions mu/ be ab/ract$ f a s]ies ( in/ances2 #hc #gh4 ,t all def9ition is classification2 #hc #gi4 ,t conceptions ( magnitude c be appli$ to space z a :ole2 #hd #hj4 ,& t if conceptions ( magnitude c\ld be s appli$1 all ! adjectives ( space w\ld result f _! application #hf #ha4 ,erdmann reg>ds ,geometry alone z incapable ( decid+ on ! tru? ( ! axiom ( ,congru5ce1 #hf #hb4 ,:i* he affirms to be empirically prov$ by ,me*anics4 #hh #hc4 ,! v>iety & inadequacy ( ,erdmann's te/s ( apriority #hi #hd4 ,invalidate 8 f9al conclusions on ! !ory ( ,geometry #ij #he4 ,lotze has discuss$ two que/ions 9 ! !ory ( ,geometry3 #ic #hf4 7#a7 ,he reg>ds ! possibility ( non-,euclidean spaces z sugge/$ by ! subjectivity ( space1 #ic #hg4 ,& rejects x ow+ to a ma!matical misund]/&+1 #if #hh4 ,hav+ miss$ ! mo/ important s5se ( _! possibility1 #if #hi4 ,:i* is t !y fulfil ! logical conditions to :i* any =m ( ext]nality mu/ con=m #ig #ij4 7#b7 ,he attacks ! ma!matical proc$ure ( ,metageometry #ih #ia4 ,! attack beg9s ) a "q-begg+ def9ition ( p>allels #ii #ib4 ,lotze ma9ta9s t all app>5t dep>tures f ,euclid c\ld be physically expla9$1 a view :i* really makes ,euclid empirical #ii #ic4 ,8 criticism ( ,helmholtz's analogies re/s :olly on ma!matical mi/akes #aja #id4 ,8 pro( t space mu/ h ?ree dim5sions re/s on neglect ( diff]5t ord]s ( inf9ity #ajd #ie4 ,he attacks non-,euclidean spaces on ! mi/ak5 gr\nd t !y are n homog5e\s #ajg #if4 ,lotze's objections fall "u f\r heads #ajh #ig4 ,two o!r semi-philosophical objections may be urg$1 #aji #ih4 ,"o ( :i*1 ! abs5ce ( simil>ity1 has be5 made ! basis ( attack by ,delbuf1 #aaj #ii4 ,b does n =m a valid gr\nd ( objection #aaa #ajj4 ,rec5t ,fr5* speculation on ! f\ndations ( ,geometry has sugge/$ few new views #aab #aja4 ,all homog5e\s spaces are priori possible1 & ! decision betwe5 !m is empirical #aad ,,*apt] ,,iii4 ,,section ,a4 ,,! ,,axioms ,,( ,,projective ,,geometry4 #ajb4 ,projective ,geometry does n deal ) magnitude1 & applies to all spaces alike #aag #ajc4 ,x w be f\nd :olly priori #aag #ajd4 ,its axioms h n yet be5 =mulat$ philosophically #aah #aje4 ,coord9ates1 9 projective ,geometry1 are n spatial magnitudes1 b conv5i5t names = po9ts #aah #ajf4 ,! possibility ( di/+ui%+ v>i\s po9ts is an axiom #aai #ajg4 ,! qualitative relations betwe5 po9ts1 dealt ) by projective ,geometry1 are presuppos$ by ! quantitative treatm5t #aai #ajh4 ,! only qualitative relation betwe5 two po9ts is ! /raight l9e1 & all /raight l9es are qualitatively simil> #abj #aji4 ,h5ce foll[s1 by ext5sion1 ! pr9ciple ( projective trans=mation #aba #aaj4 ,by :i* figures qualitatively indi/+ui%able f a giv5 figure are obta9$ #abb #aaa4 ,anh>monic ratio may & mu/ be descriptively def9$ #abb #aab4 ,! quadrilat]al con/ruction is ess5tial to ! projective def9ition ( po9ts1 #abc #aac4 ,& c be projectively def9$1 #abd #aad4 ,by ! g5]al pr9ciple ( projective trans=mation #abf #aae4 ,! pr9ciple ( duality is ! ma!matical =m ( a philosophical circle1 #abg #aaf4 ,:i* is an inevitable consequ5ce ( ! relativity ( space1 & makes any def9ition ( ! po9t contradictory #abh #aag4 ,we def9e ! po9t z t :i* is spatial1 b conta9s no space1 :5ce o!r def9itions foll[ #abh #aah4 ,:at is meant by qualitative equival5ce 9 ,geometry8 #abi #aai4 ,two pairs ( po9ts on "o /raight l9e1 or two pairs ( /raight l9es ?r\gh "o po9t1 are qualitatively equival5t #abi #abj4 ,?is expla9s :y f\r coll9e> po9ts are ne$$1 to give an intr9sic relation by :i* ! f\r? c be descriptively def9$ :5 ! fir/ ?ree are giv5 #acj #aba4 ,any two projectively relat$ figures are qualitatively equival5t1 i4e4 diff] 9 no non-quantitative conceptual prop]ty #aca #abb4 ,?ree axioms are us$ by projective ,geometry1 #acb #abc4 ,& are requir$ = qualitative spatial comp>ison1 #acb #abd4 ,:i* involves ! homog5eity1 relativity & passivity ( space #acc #abe4 ,! conception ( a =m ( ext]nality1 #acd #abf4 ,be+ a creature ( ! intellect1 c be dealt ) by pure ma!matics #acd #abg4 ,! result+ doctr9e ( ext5sion w be1 = ! mom5t1 hypo!tical #ace #abh4 ,b is r5d]$ ass]torical by ! necessity1 = exp]i5ce1 ( "s =m ( ext]nality #acf #abi4 ,any su* =m mu/ be relational #acf #acj4 ,& homog5e\s #acg #aca4 ,& ! relations con/itut+ x mu/ appe> inf9itely divisible #acg #acb4 ,x mu/ h a f9ite integral numb] ( dim5sions1 #aci #acc4 ,ow+ to its passivity & homog5eity #adj #acd4 ,& to ! sy/ematic unity ( ! _w #adj #ace4 ,a "o-dim5sional =m alone w\ld n suffice = exp]i5ce #ada #acf4 ,s9ce its elem5ts w\ld be immovably fix$ 9 a s]ies #adb #acg4 ,two positions h a relation indep5d5t ( o!r positions1 #adc #ach4 ,s9ce positions are :olly def9$ by mutually indep5d5t relations #adc #aci4 ,h5ce projective ,geometry is :olly priori1 #adf #adj4 ,?\gh metrical ,geometry conta9s an empirical elem5t #adf ,,section ,b4 ,,! ,,axioms ,,( ,,metrical ,,geometry4 #ada4 ,metrical ,geometry is di/9ct f projective1 b has ! same fundam5tal po/ulate #adg #adb4 ,x introduces ! new idea ( motion1 & has ?ree priori axioms #adh ,i4 ,! ,axiom ( ,free ,mobility4 #adc4 ,measurem5t requires a crit]ion ( spatial equality #adi #add4 ,:i* is giv5 by sup]position1 & involves ! axiom ( ,free ,mobility #aej #ade4 ,! d5ial ( ?is axiom involves an action ( empty space on ?+s #aea #adf4 ,!re is a ma!matically possible alt]native to ! axiom1 #aeb #adg4 ,:i*1 h[ev]1 is logically & philosophically unt5able #aec #adh4 ,?\gh ,free ,mobility is priori1 actual measurem5t is empirical #aed #adi4 ,"s objections rema9 to be answ]$1 conc]n+-- #aed #aej4 7#a7 ,! comp>ison ( volumes & ( ,kant's symmetrical objects #aed #aea4 7#b7 ,! measurem5t ( "t1 :]e congru5ce is impossible #aef #aeb4 7#c7 ,! imm$iate p]ception ( spatial magnitude2 & #aeg #aec4 7#d7 ,! ,geometry ( non-congru5t surfaces #aeh #aed4 ,free ,mobility includes ,helmholtz's ,monodromy #aei #aee4 ,free ,mobility involves ! relativity ( space #aei #aef4 ,f :i*1 reciprocally1 x c be d$uc$ #afj #aeg4 ,our axiom is !re=e priori 9 a d\ble s5se #afj ,,ii4 ,! ,axiom ( ,dim5sions4 #aeh4 ,space mu/ h a f9ite integral numb] ( dim5sions #afa #aei4 ,b ! re/riction to ?ree is empirical #afb #afj4 ,! g5]al axiom foll[s f ! relativity ( position #afb #afa4 ,! limitation to ?ree dim5sions1 unlike mo/ empirical k1 is accurate & c]ta9 #afc ,,iii4 ,! ,axiom ( ,di/ance4 #afb4 ,! axiom ( di/ance corresponds1 "h1 to t ( ! /raight l9e 9 projective ,geometry #afd #afc4 ,! possibility ( spatial measurem5t involves a magnitude uniquely det]m9$ by two po9ts1 #afd #afd4 ,s9ce two po9ts mu/ h "s relation1 & ! passivity ( space proves ?is to be indep5d5t ( ext]nal ref]5ce #afe #afe4 ,!re c be only "o su* relation #aff #aff4 ,?is mu/ be measur$ by a curve jo9+ ! two po9ts1 #aff #afg4 ,& ! curve mu/ be uniquely det]m9$ by ! two po9ts #afg #afh4 ,sph]ical ,geometry conta9s an exception to ?is axiom1 #afh #afi4 ,:i*1 h[ev]1 is n q equival5t to ,euclid's #afh #agj4 ,! exception is due to ! fact t two po9ts1 9 sph]ical space1 may h an ext]nal relation unalt]$ by motion1 #afi #aga4 ,:i*1 h[ev]1 be+ a relation ( l9e> magnitude1 presupposes ! possibility ( l9e> magnitude #agj #agb4 ,a relation betwe5 two po9ts mu/ be a l9e jo9+ !m #agj #agc4 ,conv]sely1 ! exi/5ce ( a unique l9e betwe5 two po9ts c be d$uc$ f ! nature ( a =m ( ext]nality1 #aga #agd4 ,& necess>ily leads to di/ance1 :5 quantity is appli$ to x #agb #age4 ,h5ce ! axiom ( di/ance1 also1 is priori 9 a d\ble s5se #agb #agf4 ,no metrical coord9ate sy/em c be set up )\t ! /raight l9e #agd #agg4 ,no axioms besides ! above ?ree are necess>y to metrical ,geometry #age #agh4 ,b ^! ?ree are necess>y to ! direct measurem5t ( any cont9uum #agf #agi4 ,two philosophical que/ions rema9 = a f9al *apt] #agg ,,*apt] ,,iv4 ,,philosophical ,,consequ5ces4 #ahj4 ,:at is ! relation to exp]i5ce ( a =m ( ext]nality 9 g5]al8 #agh #aha4 ,?is =m is ! class-conception1 conta9+ e possible intuition ( ext]nality2 & "s su* intuition is necess>y to exp]i5ce #agh #ahb4 ,:at relation does ?is view be> to ,kant's8 #agi #ahc4 ,x is less psy*ological1 s9ce x does n discuss :e!r space is giv5 9 s5sation1 #ahj #ahd4 ,& ma9ta9s t n only space1 b any =m ( ext]nality :i* r5d]s exp]i5ce possible1 mu/ be giv5 9 s5se-p]ception #aha #ahe4 ,ext]nality %\ld mean1 n ext]nality to ! ,self1 b ! mutual ext]nality ( pres5t$ ?+s #aha #ahf4 ,w\ld ?is be unkn[able )\t a giv5 =m ( ext]nality8 #ahb #ahg4 ,bradley has prov$ t space & "t preclude ! exi/5ce ( m]e p>ticul>s1 #ahb #ahh4 ,& t k requires ! ,?is to be nei!r simple nor self-subsi/5t #ahc #ahi4 ,to prove t exp]i5ce requires a =m ( ext]nality1 ,i assume t all k requires ! recognition ( id5tity 9 diff]5ce #ahd #aij4 ,su* recognition involves "t #ahd #aia4 ,& "s o!r =m giv+ simultane\s div]sity #ahe #aib4 ,! above argum5t has n d$uc$ s5se-p]ception f ! categories1 b has %[n ! =m]1 unless x conta9s a c]ta9 elem5t1 to be un9telligible to ! latt] #ahf #aic4 ,h[ to acc\nt = ! realization ( ?is elem5t1 is a "q = metaphysics #ahg #aid4 ,:at are we to d ) ! contradictions 9 space8 #ahh #aie4 ,?ree contradictions w be discuss$ 9 :at foll[s #ahh #aif4 7#a7 ,! ant9omy ( ! ,po9t proves ! relativity ( space1 #ahi #aig4 ,& %[s t ,geometry mu/ h "s ref]5ce to matt]1 #aij #aih4 ,by :i* means x is made to ref] to spatial ord]1 n to empty space #aia #aii4 ,! causal prop]ties ( matt] are irrelevant to ,geometry1 :i* mu/ reg>d x z compos$ ( unext5d$ atoms1 by :i* po9ts are replac$ #aia #bjj4 7#b7 ,! circle 9 def9+ /raight l9es & planes is ov]come by ! same ref]5ce to matt] #aib #bja4 7#c7 ,! ant9omy t space is relational & yet m ?an relational1 #aic #bjb4 ,seems to dep5d on ! confusion ( empty space ) spatial ord] #aic #bjc4 ,kant reg>d$ empty space z ! subject-matt] ( ,geometry1 #aid #bjd4 ,b ! argum5ts ( ! ,ae/hetic are inconclusive on ?is po9t1 #aie #bje4 ,& are upset by ! ma!matical ant9omies1 :i* prove t spatial ord] %\ld be ! subject-matt] ( ,geometry #aif #bjf4 ,! app>5t ?+hood ( space is a psy*ological illusion1 due to ! fact t spatial relations are imm$iately giv5 #aif #bjg4 ,! app>5t divisibility ( spatial relations is ei!r an illusion1 aris+ out ( empty space1 or ! expression ( ! possibility ( quantitatively diff]5t spatial relations #aig #bjh4 ,ext]nality is n a relation1 b an aspect ( relations4 ,spatial ord]1 ow+ to its ref]5ce to matt]1 is a real relation #aih #bji4 ,conclusion #aii ,,introduction4 ,,our ,,problem ,,def9$ ,,by ,,its ,,relations ,,to ,,logic1 ,,psy*ology ,,& ,,ma!matics4 #a4 ,geometry1 ?r\gh\t ! #ag? & #ah? c5turies1 rema9$1 9 ! w> aga9/ empiricism1 an impregnable =tress ( ! ideali/s4 ,^? :o held--z 0 g5]ally held on ! ,cont95t--t c]ta9 k1 indep5d5t ( exp]i5ce1 0 possible ab\t ! real _w1 _h only to po9t to ,geometry3 none b a madman1 !y said1 w\ld ?r[ d\bt on its validity1 & none b a fool w\ld d5y its objective ref]5ce4 ,! ,engli% ,empirici/s1 9 ?is matt]1 _h1 !re=e1 a some:at difficult task2 ei!r !y _h to ignore ! problem1 or if1 l ,hume & ,mill1 !y v5tur$ on ! assault1 !y 7 driv5 into ! app>5tly p>adoxical ass]tion t ,geometry1 at bottom1 _h no c]ta9ty ( a diff]5t k9d f t ( ,me*anics--only ! p]petual pres5ce ( spatial impressions1 !y said1 made our exp]i5ce ( ! tru? ( ! axioms s wide z to seem absolute c]ta9ty4 ,"h1 h[ev]1 z 9 _m o!r in/ances1 m]ciless logic drove ^! philosoph]s1 :e!r !y w\ld or no1 into gl>+ opposition to ! common s5se ( _! day4 ,x 0 only ?r\gh ,kant1 ! creator ( mod]n ,epi/emology1 t ! geometrical problem receiv$ a mod]n =m4 ,he r$uc$ ! "q to ! foll[+ hypo!ticals3 ,if ,geometry has apodeictic c]ta9ty1 its matt]1 i4e4 space1 mu/ be priori1 & z su* mu/ be purely subjective2 & conv]sely1 if space is purely subjective1 ,geometry mu/ h apodeictic c]ta9ty4 ,! latt] hypo!tical has m weight ) ,kant1 inde$ x is in]adicably b\nd up ) 8 :ole ,epi/emology2 nev]!less x has1 ,i ?9k1 mu* less =ce ?an ! =m]4 ,let u accept1 h[ev]1 = ! mom5t1 ! ,kantian =mulation1 & endeav\r to give precision to ! t]ms priori & subjective4 #b4 ,"o ( ! great difficulties1 ?r\gh\t ?is controv]sy1 is ! extremely v>iable use to :i* ^! words1 z well z ! ^w empirical1 are put by diff]5t au?ors4 ,to ,kant1 :o 0 no?+ ( a psy*ologi/1 priori & subjective 7 almo/ int]*angeable t]ms#a2 9 mod]n usage !re is1 on ! :ole1 a t5d5cy to conf9e ! ^w subjective to ,psy*ology1 leav+ priori to d duty = ,epi/emology4 ,if we accept ?is diff]5tiation1 we may set up1 correspond+ to ! problems ( ^! two sci5ces1 ! foll[+ provisional def9itions3 priori applies to any piece ( k :i*1 ?\gh p]haps elicit$ by exp]i5ce1 is logically presuppos$ 9 exp]i5ce3 subjective applies to any m5tal /ate ^: imm$iate cause lies1 n 9 ! ext]nal _w1 b )9 ! limits ( ! subject4 ,! latt] def9ition1 ( c\rse1 is fram$ exclusively = ,psy*ology3 f ! po9t ( view ( physical ,sci5ce all m5tal /ates are subjective4 ,b = a ,sci5ce ^: matt]1 /rictly speak+1 is only m5tal /ates1 we require1 if we are to use ! ^w to any purpose1 "s diff]5tia among m5tal /ates1 z a m>k ( a m special subjectivity on ! "p ( ^? to :i* ?is t]m is appli$4 ,n[ ! only m5tal /ates ^: imm$iate causes lie 9 ! ext]nal _w are s5sations4 ,a pure s5sation is1 ( c\rse1 an impossible ab/raction--we are nev] :olly passive "u ! action ( an ext]nal /imulus--b = ! purposes ( ,psy*ology ! ab/raction is a useful "o4 ,:atev]1 !n1 is n s5sation1 we %all1 9 ,psy*ology1 call subjective4 ,x is 9 s5sation alone t we are directly affect$ by ! ext]nal _w1 & only "h does x give u direct in=mation ab\t itself4 #c4 ,let u n[ consid] ! epi/emological "q1 z to ! sort ( k :i* c be call$ priori4 ,"h we h no?+ to d--9 ! fir/ in/ance1 at any rate--) ! cause or g5esis ( a piece ( k2 we accept k z a datum to be analys$ & classifi$4 ,su* analysis w reveal a =mal & a mat]ial elem5t 9 k4 ,! =mal elem5t w consi/ ( po/ulates :i* are requir$ to make k possible at all1 & ( all t c be d$uc$ f ^! po/ulates2 ! mat]ial elem5t1 on ! o!r h&1 w consi/ ( all t comes to fill 9 ! =m giv5 by ! =mal po/ulates--all t is cont+5t or dep5d5t on exp]i5ce1 all t might h be5 o!rwise )\t r5d]+ k impossible4 ,we %all !n call ! =mal elem5t priori1 ! mat]ial elem5t empirical4 #d4 ,n[ :at is ! connection betwe5 ! subjective & ! priori8 ,x is a connection1 obvi\sly--if x exi/s at all--f ! outside1 i4e4 n d$ucible directly f ! nature ( ei!r1 b provable--if x c be prov$--only by a g5]al view ( ! conditions ( bo?4 ,! "q1 :at k is priori1 mu/1 on ! above def9ition1 dep5d on a logical analysis ( k1 by :i* ! conditions ( possible exp]i5ce may be reveal$2 b ! "q1 :at elem5ts ( a cognitive /ate are subjective1 is to be inve/igat$ by pure ,psy*ology1 :i* has to det]m9e :at1 9 our p]ceptions1 belongs to s5sation1 & :at is ! "w ( ?\ght or ( association4 ,s9ce1 !n1 ^! two que/ions belong to diff]5t sci5ces1 & c be settl$ indep5d5tly1 w x n be wise to conduct ! two inve/igations sep>ately8 ,to decree t ! priori %all always be subjective1 seems dang]\s1 :5 we reflect t su* a view places our results1 z to ! priori1 at ! m]cy ( empirical psy*ology4 ,h[ s]i\s ?is dang] is1 ! controv]sy z to ,kant's pure intuition suffici5tly %[s4 #e4 ,i %all1 !re=e1 ?r\gh\t ! pres5t ,essay1 use ! ^w priori )\t any psy*ological implication4 ,my te/ ( apriority w be purely logical3 ,w\ld exp]i5ce be impossible1 if a c]ta9 axiom or po/ulate 7 d5i$8 ,or1 9 a m re/rict$ s5se1 :i* gives apriority only )9 a p>ticul> sci5ce3 ,w\ld exp]i5ce z to ! subject-matt] ( t sci5ce be impossible1 )\t a c]ta9 axiom or po/ulate8 ,my results also1 !re=e1 w be purely logical4 ,if ,psy*ology decl>es t "s ?+s1 :i* ,i h decl>$ priori1 are n subjective1 !n1 fail+ an error ( detail 9 my pro(s1 ! connection ( ! priori & ! subjective1 s f> z ^? ?+s are conc]n$1 mu/ be giv5 up4 ,!re w be no discussion1 accord+ly1 ?r\gh\t ?is ,essay1 ( ! relation ( ! priori to ! subjective--a relation :i* _c det]m9e :at pieces ( k are priori1 b r dep5ds on t det]m9ation1 & belongs1 9 any case1 r to ,metaphysics ?an to ,epi/emology4 #f4 ,z ,i h v5tur$ to use ! ^w priori 9 a slightly unconv5tional s5se1 ,i w give a few elucidatory rem>ks ( a g5]al nature4 ,! priori1 s9ce ,kant at any rate1 has g5]ally /ood = ! necess>y or apodeictic elem5t 9 k4 ,b mod]n logic has %[n t necess>y propositions are always1 9 "o aspect at lea/1 hypo!tical4 ,!re may be1 & usually is1 an implication t ! connection1 ( :i* necessity is pr$icat$1 has "s exi/5ce1 b /ill1 necessity always po9ts beyond itself to a gr\nd ( necessity1 & ass]ts ?is gr\nd r ?an ! actual connection4 ,z ,bradley po9ts out1 ars5ic poisons rema9s true1 ev5 if x is poison+ no "o4 ,if1 !re=e1 ! priori 9 k be prim>ily ! necess>y1 x mu/ be ! necess>y on "s hypo!sis1 & ! gr\nd ( necessity mu/ be includ$ z priori4 ,b ! gr\nd ( necessity is1 s f> z ! necess>y connection 9 "q c %[1 a m]e fact1 a m]ely categorical judgm5t4 ,h5ce necessity alone is an insuffici5t crit]ion ( apriority4 ,to supplem5t ?is crit]ion1 we mu/ supply ! hypo!sis or gr\nd1 on :i* alone ! necessity holds1 & ?is gr\nd w v>y f "o sci5ce to ano!r1 & ev51 ) ! progress ( k1 9 ! same sci5ce at diff]5t times4 ,= z k becomes m develop$ & articulate1 m & m necess>y connections are p]ceiv$1 & ! m]ely categorical tru?s1 ?\gh !y rema9 ! f\ndation ( apodeictic judgm5ts1 dim9i% 9 relative numb]4 ,nev]!less1 9 a fairly advanc$ sci5ce su* z ,geometry1 we c1 ,i ?9k1 pretty completely supply ! appropriate gr\nd1 & e/abli%1 )9 ! limits ( ! isolat$ sci5ce1 ! di/9ction betwe5 ! necess>y & ! m]ely ass]torical4 #g4 ,!re are two gr\nds1 ,i ?9k1 on :i* necessity may be s\ght )9 any sci5ce4 ,^! may be 7v r\ghly7 di/+ui%$ z ! gr\nd :i* ,kant seeks 9 ! ,prolegom5a1 & t :i* he seeks 9 ! ,pure ,reason4 ,we may />t f ! exi/5ce ( our sci5ce z a fact1 & analyse ! reason+ employ$ ) a view to discov]+ ! fundam5tal po/ulate on :i* its logical possibility dep5ds2 9 ?is case1 ! po/ulate1 & all :i* foll[s f x alone1 w be priori4 ,or we may accept ! exi/5ce ( ! subject-matt] ( our sci5ce z our basis ( fact1 & d$uce dogmatically :atev] pr9ciples we c f ! ess5tial nature ( ?is subject-matt]4 ,9 ?is latt] case1 h[ev]1 x is n ! :ole empirical nature ( ! subject-matt]1 z reveal$ by ! subsequ5t rese>*es ( our sci5ce1 :i* =ms our gr\nd2 = if x 71 ! :ole sci5ce w\ld1 ( c\rse1 be priori4 ,r x is t elem5t1 9 ! subject-matt]1 :i* makes possible ! bran* ( exp]i5ce dealt ) by ! sci5ce 9 "q#b4 ,! importance ( ?is di/9ction w appe> m cle>ly z we proce$#c4 #h4 ,^! two gr\nds ( necessity1 9 ultimate analysis1 fall toge!r4 ,! me?ods ( inve/igation 9 ! two cases diff] widely1 b ! results _c diff]4 ,= 9 ! fir/ case1 by analysis ( ! sci5ce1 we discov] ! po/ulate on :i* alone its reason+s are possible4 ,n[ if reason+ 9 ! sci5ce is impossible )\t "s po/ulate1 ?is po/ulate mu/ be ess5tial to exp]i5ce ( ! subject-matt] ( ! sci5ce1 & ?us we get ! second gr\nd4 ,nev]!less1 ! two me?ods are useful z supplem5t+ "o ano!r1 & ! fir/1 z />t+ f ! actual sci5ce1 is ! safe/ & easie/ me?od ( inve/igation1 ?\gh ! second seems ! m conv9c+ = exposition4 #i4 ,! c\rse ( my argum5t1 !re=e1 w be z foll[s3 ,9 ! fir/ *apt]1 z a prelim9>y to ! logical analysis ( ,geometry1 ,i %all give a brief hi/ory ( ! rise & developm5t ( non-,euclidean sy/ems4 ,! second *apt] w prep>e ! gr\nd = a con/ructive !ory ( ,geometry1 by a criticism ( "s previ\s philosophical views2 9 ?is *apt]1 ,i %all endeav\r to exhibit su* views z p>tly true1 p>tly false1 & s to e/abli%1 by prelim9>y polemics1 ! tru? ( su* p>ts ( my own !ory z are to be f\nd 9 =m] writ]s4 ,a l>ge "p ( ?is !ory1 h[ev]1 _c be s introduc$1 s9ce ! :ole field ( projective ,geometry1 s f> z ,i am aw>e1 has be5 hi!rto unkn[n to philosoph]s4 ,pass+1 9 ! ?ird *apt]1 f criticism to con/ruction1 ,i %all deal fir/ ) projective ,geometry4 ,?is1 ,i %all ma9ta91 is necess>ily true ( any =m ( ext]nality1 & is1 s9ce "s su* =m is necess>y to exp]i5ce1 completely priori4 ,9 metrical ,geometry1 h[ev]1 :i* ,i %all next consid]1 ! axioms w fall into two classes3 7#a7 ,^? common to ,euclidean & non-,euclidean spaces4 ,^! w be f\nd1 on ! "o h&1 ess5tial to ! possibility ( measurem5t 9 any cont9uum1 & on ! o!r h&1 necess>y prop]ties ( any =m ( ext]nality ) m ?an "o dim5sion4 ,!y w1 !re=e1 be decl>$ priori4 7#b7 ,^? axioms :i* di/+ui% ,euclidean f non-,euclidean spaces4 ,^! w be reg>d$ z :olly empirical4 ,! axiom t ! numb] ( dim5sions is ?ree1 h[ev]1 ?\gh empirical1 w be decl>$1 s9ce small errors are "h impossible1 exactly & c]ta9ly true ( our actual _w2 :ile ! two rema9+ axioms1 :i* det]m9e ! value ( ! space-con/ant1 w be reg>d$ z only approximately kn[n1 & c]ta9 only )9 ! errors ( obs]vation#d4 ,! f\r? *apt]1 f9ally1 w endeav\r to prove1 :at 0 assum$ 9 ,*apt] ,,iii41 t "s =m ( ext]nality is necess>y to exp]i5ce1 & w conclude by exhibit+ ! logical impossibility1 if k ( su* a =m is to be fre$ f contradictions1 ( :olly ab/ract+ ?is k f all ref]5ce to ! matt] conta9$ 9 ! =m4 ,i %all hope to h t\*$1 ) ?is discussion1 on all ! ma9 po9ts relat+ to ! ,f\ndations ( ,geometry4 ,,footnotes3 #a ,cf4 ,erdmann1 ,axiome d] ,geometrie1 p4 #aaa3 ,fr ,kant s9d ,aprioritt und auss*liessli*e ,subjectivitt all]d+s ,we*selbegriffe4 #b ,i use exp]i5ce "h 9 ! wide/ possible s5se1 ! s5se 9 :i* ! ^w is us$ by ,bradley4 #c ,:]e ! bran* ( exp]i5ce 9 "q is ess5tial to all exp]i5ce1 ! result+ apriority may be reg>d$ z absolute2 :]e x is necess>y only to "s special sci5ce1 z relative to t sci5ce4 #d ,i h giv5 no acc\nt ( ^! empirical pro(s1 z !y seem to be con/itut$ by ! :ole body ( physical sci5ce4 ,ev]y?+ 9 physical sci5ce1 f ! law ( gravitation to ! build+ ( bridges1 f ! spectroscope to ! art ( navigation1 w\ld be pr(\ndly modifi$ by any consid]able inaccuracy 9 ! hypo!sis t our actual space is ,euclidean4 ,! obs]v$ tru? ( physical sci5ce1 !re=e1 con/itutes ov]:elm+ empirical evid5ce t ?is hypo!sis is v approximately correct1 ev5 if n rigidly true4 ,,*apt] ,i4 ,a ,,%ort ,,hi/ory ,,( ,,metageometry4 #aj4 ,:5 a long e/abli%$ sy/em is attack$1 x usually happ5s t ! attack beg9s only at a s+le po9t1 :]e ! weakness ( ! e/abli%$ doctr9e is peculi>ly evid5t4 ,b criticism1 :5 once invit$1 is apt to ext5d mu* fur!r ?an ! mo/ d>+1 at fir/1 w\ld h wi%$4 ,fir/ cut ! liquefaction1 :at comes la/1 ,b ,fi*te's clev] cut at ,god himself8 ,s x has be5 ) ,geometry4 ,! liquefaction ( ,euclidean or?odoxy is ! axiom ( p>allels1 & x 0 by ! refusal to admit ?is axiom )\t pro( t ,metageometry began4 ,! fir/ ef=t 9 ?is direction1 t ( ,leg5dre#e1 0 inspir$ by ! hope ( d$uc+ ?is axiom f ! o!rs--a hope :i*1 z we n[ "k1 0 doom$ to inevitable failure4 ,p>allels are def9$ by ,leg5dre z l9es 9 ! same plane1 su* t1 if a ?ird l9e cut !m1 x makes ! sum ( ! int]ior & opposite angles equal to two "r angles4 ,he proves )\t difficulty t su* l9es w\ld n meet1 b is unable to prove t non-p>allel l9es 9 a plane mu/ meet4 ,simil>ly he c prove t ! sum ( ! angles ( a triangle _c exce$ two "r angles1 & t if any "o triangle has a sum equal to two "r angles1 all triangles h ! same sum2 b he is unable to prove ! exi/5ce ( ?is "o triangle4 #aa4 ,?us ,leg5dre's attempt broke d[n2 b m]e failure c\ld prove no?+4 ,a bold] me?od1 sugge/$ by ,gauss1 0 c>ri$ out by ,lobat*ewsky & ,bolyai#f4 ,if ! axiom ( p>allels is logically d$ucible f ! o!rs1 we %all1 by d5y+ x & ma9ta9+ ! re/1 be l$ to contradictions4 ,^! ?ree ma!maticians1 accord+ly1 attack$ ! problem indirectly3 !y d5i$ ! axiom ( p>allels1 & yet obta9$ a logically consi/5t ,geometry4 ,!y inf]r$ t ! axiom 0 logically indep5d5t ( ! o!rs1 & ess5tial to ! ,euclidean sy/em4 ,_! works1 be+ all inspir$ by ?is motive1 may be di/+ui%$ z =m+ ! fir/ p]iod 9 ! developm5t ( ,metageometry4 ,! second p]iod1 inaugurat$ by ,riemann1 _h a mu* deep] import3 x 0 l>gely philosophical 9 its aims & con/ructive 9 its me?ods4 ,x aim$ at no less ?an a logical analysis ( all ! ess5tial axioms ( ,geometry1 & reg>d$ space z a p>ticul> case ( ! m g5]al conception ( a manifold4 ,tak+ its /& on ! me?ods ( analytical metrical ,geometry1 x e/abli%$ two non-,euclidean sy/ems1 ! fir/ t ( ,lobat*ewsky1 ! second--9 :i* ! axiom ( ! /raight l9e1 9 ,euclid's =m1 0 also d5i$--a new v>iety1 by analogy call$ sph]ical4 ,! lead+ conception 9 ?is p]iod is ! measure ( curvature1 a t]m inv5t$ by ,gauss1 b appli$ by him only to surfaces4 ,gauss _h %[n t free mobility on surfaces 0 only possible :5 ! measure ( curvature 0 con/ant2 ,riemann & ,helmholtz ext5d$ ?is proposition to n dim5sions1 & made x ! fundam5tal prop]ty ( space4 ,9 ! ?ird p]iod1 :i* beg9s ) ,cayley1 ! philosophical motive1 :i* _h mov$ ! fir/ pione]s1 is less app>5t1 & is replac$ by a m te*nical & ma!matical _s4 ,?is p]iod is *iefly di/+ui%$ f ! second1 9 a ma!matical po9t ( view1 by its me?od1 :i* is projective in/ead ( metrical4 ,! lead+ ma!matical conception "h is ! ,absolute 7,grundgebild71 a figure by relation to :i* all metrical prop]ties become projective4 ,cayley's "w1 :i* 0 v brief1 & attract$ little att5tion1 has be5 p]fect$ & elaborat$ by ,f4 ,kle91 & ?r\gh him has f\nd g5]al acceptance4 ,kle9 has add$ to ! two k9ds ( non-,euclidean ,geometry already kn[n1 a ?ird1 :i* he calls elliptic2 ?is ?ird k9d closely resembles ,helmholtz's sph]ical ,geometry1 b is di/+ui%$ by ! important diff]5ce t1 9 x1 two /raight l9es meet 9 only "o po9t#g4 ,! di/9ctive m>k ( ! spaces repres5t$ by bo? is t1 l ! surface ( a sph]e1 !y are f9ite b unb\nd$4 ,! r$uction ( metrical to projective prop]ties1 z w be prov$ h]eaft]1 has only a te*nical importance2 at ! same "t1 projective ,geometry is able to deal directly ) ^? purely descriptive or qualitative prop]ties ( space :i* are common to ,euclid & ,metageometry alike4 ,! ?ird p]iod has1 !re=e1 great philosophical importance1 :ile its me?od has1 ma!matically1 mu* great] beauty & unity ?an t ( ! second2 x is able to treat all k9ds ( space at once1 s t e symbolic proposition is1 accord+ to ! mean+ giv5 to ! symbols1 a proposition 9 :i*ev] ,geometry we *oose4 ,?is has ! advantage ( prov+ t fur!r rese>* _c lead to contradictions 9 non-,euclidean sy/ems1 unless x at ! same mom5t reveals contradictions 9 ,euclid4 ,^! sy/ems1 !re=e1 are logically z s\nd z t ( ,euclid himself4 ,aft] ?is brief sket* ( ! *>act]i/ics ( ! ?ree p]iods1 ,i w proce$ to a m detail$ acc\nt4 ,x w be my aim to avoid1 z f> z possible1 all te*nical ma!matics1 & br+ into relief only ^? fundam5tal po9ts 9 ! ma!matical developm5t1 :i* seem ( logical or philosophical importance4 ,fir/ ,p]iod4 #ab4 ,! orig9ator ( ! :ole sy/em1 ,gauss1 does n appe>1 z reg>ds /rictly non-,euclidean ,geometry1 9 any ( 8 hi!rto publi%$ pap]s1 to h giv5 m ?an results2 8 pro(s rema9 unkn[n to u4 ,nev]!less he 0 ! fir/ to inve/igate ! consequ5ces ( d5y+ ! axiom ( p>allels#h1 & 9 8 lett]s he communicat$ ^! consequ5ces to "s ( 8 fri5ds1 among :om 0 ,wolfgang ,bolyai4 ,! fir/ m5tion ( ! subject 9 8 lett]s occurs :5 he 0 only #ah2 f\r ye>s lat]1 9 #agii1 writ+ to ,w4 ,bolyai1 he enunciates ! important !orem t1 9 hyp]bolic ,geometry1 !re is a maximum to ! area ( a triangle4 ,f lat] writ+s x appe>s t he _h work$ out a sy/em ne>ly1 if n q1 z complete z ^? ( ,lobat*ewsky & ,bolyai#i4 ,x is important to rememb]1 h[ev]1 t ,gauss's "w on curvature1 :i* 0 publi%$1 laid ! f\ndation = ! :ole me?od ( ! second p]iod1 & 0 und]tak51 accord+ to ,riemann & ,helmholtz#aj1 ) a view to an 7unpubli%$7 inve/igation ( ! f\ndations ( ,geometry4 ,8 "w 9 ?is direction w1 ow+ to its me?od1 be bett] treat$ ( "u ! second p]iod1 b x is int]e/+ to obs]ve t he /ood1 l _m pione]s1 at ! head ( two t5d5cies :i* aft]w>ds div]g$4 #ac4 ,lobat*ewsky1 a pr(essor 9 ! ,univ]sity ( ,kasan1 fir/ publi%$ 8 results1 9 _! native ,russian1 9 ! proce$+s ( t le>n$ body = ! ye>s #ahbi-ahcj4 ,ow+ to ?is d\ble obscurity ( language & place1 !y attract$ little att5tion1 until he translat$ !m into ,fr5*#aa & ,g]man#ab3 ev5 !n1 !y d n appe> to h obta9$ ! notice !y des]v$1 until1 9 #ahfh1 ,beltrami une>!d ! article 9 ,crelle1 & made x ! !me ( a brilliant int]pretation4 ,9 ! introduction to 8 little ,g]man book1 ,lobat*ewsky lam5ts ! slight int]e/ %[n 9 8 writ+s by 8 compatriots1 & ! inatt5tion ( ma!maticians1 s9ce ,leg5dre's abortive attempt1 to ! difficulties 9 ! !ory ( p>allels4 ,! body ( ! "w beg9s ) ! enunciation ( sev]al important propositions :i* hold good 9 ! sy/em propos$ z well z 9 ,euclid3 ( ^!1 "s are 9 any case indep5d5t ( ! axiom ( p>allels1 :ile o!rs are r5d]$ s by sub/itut+1 = ! ^w p>allel1 ! phrase n int]sect+1 h[ev] f> produc$4 ,!n foll[s a def9ition1 int5tionally fram$ s z to contradict ,euclid's3 ,) respect to a giv5 /raight l9e1 all o!rs 9 ! same plane may be divid$ into two classes1 ^? :i* cut ! giv5 /raight l9e1 & ^? :i* d n cut x2 a l9e :i* is ! limit betwe5 ! two classes is call$ p>allel to ! giv5 /raight l9e4 ,x foll[s t1 f any ext]nal po9t1 two p>allels c be drawn1 "o 9 ea* direction4 ,f ?is />t+-po9t1 by ! ,euclidean syn!tic me?od1 a s]ies ( propositions are d$uc$2 ! mo/ important ( ^! is1 t 9 a triangle ! sum ( ! angles is always less ?an1 or always equal to two "r angles1 :ile 9 ! latt] case ! :ole sy/em becomes or?odox4 ,a c]ta9 analogy ) sph]ical ,geometry--^: mean+ & ext5t w appe> lat]--is also prov$1 consi/+ r\ghly 9 ! sub/itution ( hyp]bolic = circul> functions4 #ad4 ,v simil> is ! sy/em ( ,johann ,bolyai1 s simil>1 inde$1 z to make ! indep5d5ce ( ! two works1 ?\gh a well-au!nticat$ fact1 seem all b incr$ible4 ,johann ,bolyai fir/ publi%$ 8 results 9 #ahcb1 9 an app5dix to a "w by 8 fa!r ,wolfgang1 entitl$2 ,app5dix1 sci5tiam spatii absolute v]am exhib5s3 a v]itate aut falsitate ,axiomatis ,,xi4 ,euclidei 7a priori haud unquam decid5da7 indep5d5tem2 adjecta ad casum falsitatis1 quadratura circuli geometrica4 ,gauss1 ^: bosom fri5d he became at college & rema9$ ?r\gh life1 01 z we h se51 ! inspir] ( ,wolfgang ,bolyai1 & us$ to say t ! latt] 0 ! only man :o appreciat$ 8 philosophical speculations on ! axioms ( ,geometry2 nev]!less1 ,wolfgang appe>s to h left to 8 son ,johann ! detail$ work+ out ( ! hyp]bolic sy/em4 ,! works ( bo? ! ,bolyai are v r>e1 & _! me?od & results are kn[n to me only ?r\gh ! r5d]+s ( ,fris*auf & ,hal/$#ac4 ,bo? z to me?od & z to results1 ! sy/em is v simil> to ,lobat*ewsky's1 s t nei!r ne$ deta9 u "h4 ,only ! initial po/ulates1 :i* are m explicit ?an ,lobat*ewsky's1 dem& a brief att5tion4 ,fris*auf's introduction1 :i* has a philosophical & ,newtonian air1 beg9s by sett+ =? t ,geometry deals ) absolute 7empty7 space1 obta9$ by ab/ract+ f ! bodies 9 x1 t two figures are call$ congru5t :5 !y diff] only 9 position1 & t ! axiom ( ,congru5ce is indisp5sable 9 all det]m9ation ( spatial magnitudes4 ,congru5ce 0 to ref] to geometrical bodies1 ) none ( ! prop]ties ( ord9>y bodies except imp5etrability 7,erdmann1 ,axiome d] ,geometrie1 p4 #bf74 ,a /raight l9e is def9$ z det]m9$ by two ( its po9ts#ad1 & a plane z det]m9$ by ?ree4 ,^! premisses1 ) a slight exception z to ! /raight l9e1 we %all h]eaft] f9d ess5tial to e ,geometry4 ,i h drawn att5tion to !m1 z x is (t5 suppos$ t non-,euclideans d5y ! axiom ( ,congru5ce1 :i*1 "h & else:]e1 is nev] ! case4 ,! /ress laid on ?is axiom by ,bolyai is probably due to ! influ5ce ( ,gauss1 ^: "w on ! curvature ( surfaces laid ! f\ndation = ! use made ( congru5ce by ,helmholtz4 #ae4 ,x is important to rememb] t1 ?r\gh\t ! p]iod we h j review$1 ! purpose ( hyp]bolic ,geometry is indirect3 n ! tru? ( ! latt]1 b ! logical indep5d5ce ( ! axiom ( p>allels f ! re/1 is ! guid+ motive ( ! "w4 ,if1 by d5y+ ! axiom ( p>allels :ile reta9+ ! re/1 we c obta9 a sy/em free f logical contradictions1 x foll[s t ! axiom ( p>allels _c be implicitly conta9$ 9 ! o!rs4 ,if ?is be s1 attempts to disp5se ) ! axiom1 l ,leg5dre's1 _c be successful2 ,euclid mu/ /& or fall ) ! suspect$ axiom4 ,( c\rse1 x rema9$ possible t1 by fur!r developm5t1 lat5t contradictions might h be5 reveal$ 9 ^! sy/ems4 ,?is possibility1 h[ev]1 0 remov$ by ! m direct & con/ructive "w ( ! second p]iod1 to :i* we mu/ n[ turn our att5tion4 ,second ,p]iod4 #af4 ,! "w ( ,lobat*ewsky & ,bolyai rema9$1 = ne>ly a qu>t] ( a c5tury1 )\t issue--inde$1 ! inve/igations ( ,riemann & ,helmholtz1 :5 !y came1 appe> to h be5 inspir$1 n by ^! m51 b r by ,gauss#ae & ,h]b>t4 ,we f9d1 accord+ly1 v great diff]5ce1 bo? ( aim & me?od1 betwe5 ! fir/ p]iod & ! second4 ,! =m]1 beg9n+ ) a criticism ( "o po9t 9 ,euclid's sy/em1 pres]v$ 8 syn!tic me?od1 :ile x ?rew ov] "o ( 8 axioms4 ,! latt]1 on ! contr>y1 be+ guid$ by a philosophical r ?an a ma!matical _s1 endeav\r$ to classify ! conception ( space z a species ( a m g5]al conception3 x treat$ space algebraically1 & ! prop]ties x gave to space 7 express$ 9 t]ms1 n ( intuition1 b ( algebra4 ,! aim ( ,riemann & ,helmholtz 0 to %[1 by ! exhibition ( logically possible alt]natives1 ! empirical nature ( ! receiv$ axioms4 ,= ?is purpose1 !y conceiv$ space z a p>ticul> case ( a manifold1 & %[$ t v>i\s relations ( magnitude 7,massv]hltnisse7 7 ma!matically possible 9 an ext5d$ manifold4 ,_! philosophy1 :i* seems to me n always irreproa*able1 w be discuss$ 9 ,*apt] ,,ii42 "h1 :ile x is important to rememb] ! philosophical motive ( ,riemann & ,helmholtz1 we %all conf9e our att5tion to ! ma!matical side ( _! "w4 ,9 s do+1 :ile we %all1 ,i fe>1 some:at maim ! sy/em ( _! ?\ghts1 we %all secure a clos] unity ( subject1 & a m compact acc\nt ( ! purely ma!matical developm5t4 ,b !re is1 9 my op9ion1 a fur!r reason = sep>at+ _! philosophy f _! ma!matics4 ,:ile _! philosophical purpose 01 to prove t all ! axioms ( ,geometry are empirical1 & t a diff]5t cont5t ( our exp]i5ce might h *ang$ !m all1 ! un9t5d$ result ( _! ma!matical "w 01 if ,i am n mi/ak51 to af=d mat]ial = an priori pro( ( c]ta9 axioms4 ,^! axioms1 ?\gh !y believ$ !m to be unnecess>y1 7 always introduc$ 9 _! ma!matical works1 be=e lay+ ! f\ndations ( non-,euclidean sy/ems4 ,i %all cont5d1 9 ,*apt] ,,iii41 t ?is ret5tion 0 logically inevitable1 & 0 n m]ely due1 z !y suppos$1 to a desire = con=mity ) exp]i5ce4 ,if ,i am "r 9 ?is1 !re is a div]g5ce betwe5 ,riemann & ,helmholtz ! philosoph]s1 & ,riemann & ,helmholtz ! ma!maticians4 ,?is div]g5ce makes x ! m desirable to trace ! ma!matical developm5t ap>t f ! accompany+ philosophy4 #ag4 ,riemann's epo*-mak+ "w1 ,ueb] die ,hypo!s51 wel*e d] ,geometrie zu ,gr&e lieg5#af1 0 writt51 & read to a small circle1 9 #ahed2 ow+1 h[ev]1 to "s *anges :i* he desir$ to make 9 x1 x rema9$ unpubli%$ till #ahfg1 :5 x 0 publi%$ by 8 executors4 ,! two fundam5tal conceptions1 on ^: inv5tion re/s ! hi/oric importance ( ?is diss]tation1 are t ( a manifold1 & t ( ! measure ( curvature ( a manifold4 ,! =m] conception s]ves a ma9ly philosophical purpose1 & is design$1 pr9cipally1 to exhibit space z an in/ance ( a m g5]al conception4 ,on ?is aspect ( ! manifold1 ,i %all h mu* to say 9 ,*apt] ,,ii42 its ma!matical aspect1 :i* alone conc]ns u "h1 is less complicat$ & less fruitful ( controv]sy4 ,! latt] conception also s]ves a d\ble purpose1 b its ma!matical use is ! m prom95t4 ,we w consid] ^! two conceptions successively4 #ah4 7#a7 ,conception ( a manifold#ag4 ,! g5]al purpose ( ,riemann's diss]tation is1 to exhibit ! axioms z successive /eps 9 ! classification ( ! species space4 ,! axioms ( ,geometry1 l ! m>ks ( a s*ola/ic def9ition1 appe> z successive det]m9ations ( class-conceptions1 end+ ) ,euclidean space4 ,we h ?us1 f ! analytical po9t ( view1 ab\t z logical & precise a =mulation z c be desir$--a =mulation 9 :i*1 f its classificatory "*1 we seem c]ta9 ( hav+ no?+ sup]flu\s or r$undant1 & obta9 ! axioms explicitly 9 ! mo/ desirable =m1 namely z adjectives ( ! conception ( space4 ,at ! same "t1 x is a pity t ,riemann1 9 accordance ) ! metrical bias ( 8 "t1 reg>d$ space z prim>ily a magnitude#ah1 or assemblage ( magnitudes1 9 :i* ! ma9 problem consi/s 9 assign+ quantities to ! diff]5t elem5ts or po9ts1 )\t reg>d to ! qualitative nature ( ! quantities assign$4 ,consid]able obscurity ?us arises z to ! :ole nature ( magnitude#ai4 ,?is view ( ,geometry und]lies ! def9ition ( ! manifold1 z ! g5]al conception ( :i* space =ms a special case4 ,?is def9ition1 :i* is n v cle>1 may be r5d]$ z foll[s4 #ai4 ,conceptions ( magnitude1 accord+ to ,riemann1 are possible !re only1 :]e we h a g5]al conception1 capable ( v>i\s det]m9ations 7,be/immungsweis574 ,! v>i\s det]m9ations ( su* a conception toge!r =m a manifold1 :i* is cont9u\s or discrete1 accord+ z ! passage f "o det]m9ation to ano!r is cont9u\s or discrete4 ,p>ticul> bits ( a manifold1 or quanta1 c be comp>$ by c\nt+ :5 discrete1 & by measurem5t :5 cont9u\s4 ,measurem5t consi/s 9 a sup]position ( ! magnitudes to be comp>$4 ,if ?is be abs5t1 magnitudes c only be comp>$ :5 "o is "p ( ano!r1 & !n only ! m or less1 n ! h[ mu*1 c be decid$ 7p4 #bef74 ,we ?us rea* ! g5]al conception ( a manifold ( sev]al dim5sions1 ( :i* space & col\rs are m5tion$ z special cases4 ,to ! abs5ce ( ?is conception ,riemann attributes ! obscurity :i*1 on ! subject ( ! axioms1 la/$ f ,euclid to ,leg5dre 7p4 #bed74 ,& ,riemann c]ta9ly has succe$$1 f an algebraic po9t ( view1 9 exhibit+1 f> m cle>ly ?an any ( 8 pr$ecessors1 ! axioms :i* di/+ui% spatial quantity f o!r quantities ) :i* ma!matics is conv]sant4 ,b by ! assumption1 f ! />t1 t space c be reg>d$ z a quantity1 he has be5 l$ to /ate ! problem z3 ,:at sort ( magnitude is space8 r ?an3 ,:at mu/ space be 9 ord] t we may be able to reg>d x z a magnitude at all8 ,he does n realise1 ei!r--inde$ 9 8 day !re 7 few :o realiz$--t an elaborate ,geometry is possible :i* does n deal ) space z a quantity at all4 ,8 def9ition ( space z a species ( manifold1 !re=e1 ?\gh = analytical purposes x def9es1 mo/ satisfactorily1 ! nature ( spatial magnitudes1 leaves obscure ! true gr\nd = ?is nature1 :i* lies 9 ! nature ( space z a sy/em ( relations1 & is ant]ior to ! possibility ( reg>d+ x z a sy/em ( magnitudes at all4 ,b to proce$ ) ! ma!matical developm5t ( ,riemann's ideas4 ,we h se5 t he decl>$ measurem5t to consi/ 9 a sup]position ( ! magnitudes to be comp>$4 ,b 9 ord] t ?is may be a possible means ( det]m9+ magnitudes1 he cont9ues1 ^! magnitudes mu/ be indep5d5t ( _! position 9 ! manifold 7p4 #bei74 ,?is c occur1 he says1 9 sev]al ways1 z ! simple/ ( :i*1 he assumes t ! l5g?s ( l9es are indep5d5t ( _! position4 ,"o w\ld be glad to "k :at o!r ways are possible3 = my "p1 ,i am unable to imag9e any o!r hypo!sis on :i* magnitude w\ld be indep5d5t ( place4 ,sett+ ?is aside1 h[ev]1 ! problem1 ow+ to ! fact t measurem5t consi/s 9 sup]position1 becomes id5tical ) ! det]m9ation ( ! mo/ g5]al manifold 9 :i* magnitudes are indep5d5t ( place4 ,?is br+s u to ,riemann's o!r fundam5tal conception1 :i* seems to me ev5 m fruitful ?an t ( a manifold4 #bj4 7#b7 ,measure ( curvature4 ,?is conception is due to ,gauss1 b 0 appli$ by him only to surfaces2 ! novelty 9 ,riemann's diss]tation 0 its ext5sion to a manifold ( n dim5sions4 ,?is ext5sion1 h[ev]1 is r briefly & obscurely express$1 & has be5 fur!r obscur$ by ,helmholtz's attempts at popul> exposition4 ,! t]m curvature1 also1 is mislead+1 s t ! phrase has be5 ! s\rce ( m misund]/&+1 ev5 among ma!maticians1 ?an any o!r 9 ,pangeometry4 ,x is (t5 =gott51 9 spite ( ,helmholtz's explicit /atem5t#bj1 t ! measure ( curvature ( an n-dim5sional manifold is a purely analytical expression1 :i* has only a symbolic aff9ity to ord9>y curvature4 ,z appli$ to ?ree-dim5sional space1 ! implication ( a f\r-dim5sional plane space is :olly mislead+2 ,i %all1 !re=e1 g5]ally use ! t]m space-con/ant in/ead#ba4 ,nev]!less1 z ! conception grew1 hi/orically1 out ( t ( curvature1 ,i w give a v brief exposition ( ! hi/orical developm5t ( !ories ( curvature4 ,j z ! notion ( l5g? 0 orig9ally d]iv$ f ! /raight l9e1 & ext5d$ to o!r curves by divid+ !m into inf9itesimal /raight l9es1 s ! notion ( curvature 0 d]iv$ f ! circle1 & ext5d$ to o!r curves by divid+ !m into inf9itesimal circul> arcs4 ,curvature may be reg>d$1 orig9ally1 z a measure ( ! am\nt by :i* a curve dep>ts f a /raight l9e2 9 a circle1 :i* is simil> ?r\gh\t1 ?is am\nt is evid5tly con/ant1 & is measur$ by ! reciprocal ( ! radius4 ,b 9 all o!r curves1 ! am\nt ( curvature v>ies f po9t to po9t1 s t x _c be measur$ )\t inf9itesimals4 ,! measure :i* at once sugge/s itself is1 ! curvature ( ! circle mo/ ne>ly co9cid+ ) ! curve at ! po9t consid]$4 ,s9ce a circle is det]m9$ by ?ree po9ts1 ?is circle w pass ?r\gh ?ree consecutive po9ts ( ! curve4 ,we h ?us def9$ ! curvature ( any curve1 plane or tortu\s2 =1 s9ce any ?ree po9ts lie 9 a plane1 su* a circle c always be describ$4 ,if we n[ pass to a surface1 :at we want is1 by analogy1 a measure ( its dep>ture f a plane4 ,! curvature1 z above def9$1 has become indet]m9ate1 = ?r\gh any po9t ( ! surface we c draw an inf9ite numb] ( arcs1 :i* w n1 9 g5]al1 all h ! same curvature4 ,let u1 !n1 draw all ! geodesics jo9+ ! po9t 9 "q to neighb\r+ po9ts ( ! surface 9 all directions4 ,s9ce ^! arcs =m a s+ly inf9ite manifold1 !re w be among !m1 if !y h n all ! same curvature1 "o arc ( maximum1 & "o ( m9imum curvature#bb4 ,! product ( ^! maximum & m9imum curvatures is call$ ! measure ( curvature ( ! surface at ! po9t "u consid]ation4 ,to illu/rate by a few simple examples3 on a sph]e1 ! curvatures ( all su* l9es are equal to ! reciprocal ( ! radius ( ! sph]e1 h5ce ! measure ( curvature ev]y:]e is ! squ>e ( ! reciprocal ( ! radius ( ! sph]e4 ,on any surface1 su* z a cone or a cyl9d]1 on :i* /raight l9es c be drawn1 ^! h no curvature1 s t ! measure ( curvature is ev]y:]e z]o--?is is ! case1 9 p>ticul>1 ) ! plane4 ,9 g5]al1 h[ev]1 ! measure ( curvature ( a surface v>ies f po9t to po9t4 ,gauss1 ! inv5tor ( ?is conception#bc1 prov$ t1 9 ord] t two surfaces may be developable ^u ea* o!r--i4e4 may be su* t "o c be b5t into ! %ape ( ! o!r )\t /ret*+ or te>+--x is necess>y t ! two surfaces %\ld h equal measures ( curvature at correspond+ po9ts4 ,:5 ?is is ! case1 e figure :i* is possible on ! "o is1 9 g5]al1 possible on ! o!r1 & ! two h practically ! same ,geometry#bd4 ,z a coroll>y1 x foll[s t a necess>y condition1 = ! free mobility ( figures on any surface1 is ! con/ancy ( ! measure ( curvature#be4 ,?is condition 0 prov$ to be suffici5t1 z well z necess>y1 by ,m9d+#bf4 #ba4 ,s f>1 all has be5 pla9 sail+--we h be5 deal+ ) purely geometrical ideas 9 a purely geometrical mann]--b we h n1 z yet1 f\nd any s5se ( ! measure ( curvature1 9 :i* x c be ext5d$ to space1 /ill less to an n-dim5sional manifold4 ,= ?is purpose1 we mu/ exam9e ,gauss's me?od1 :i* enables u to det]m9e ! measure ( curvature ( a surface at any po9t z an inh]5t prop]ty1 q indep5d5t ( any ref]5ce to ! ?ird dim5sion4 ,! me?od ( det]m9+ ! measure ( curvature f )9 is1 briefly1 z foll[s3 ,if any po9t on ! surface be det]m9$ by two coord9ates1 u1 v1 !n small arcs ( ! surface are giv5 by ! =mula ds#b ,edu#b #b;,fdu dv ,gdv#b1 :]e ,e1 ,f1 ,g are1 9 g5]al1 functions ( u1 v4#bg ,f ?is =mula alone1 )\t ref]5ce to any space outside ! surface1 we c det]m9e ! measure ( curvature at ! po9t u1 v1 z a function ( ,e1 ,f1 ,g & _! diff]5tials ) respect to u & v4 ,?us we may reg>d ! measure ( curvature ( a surface z an inh]5t prop]ty1 & ! above geometrical def9ition1 :i* involv$ a ref]5ce to ! ?ird dim5sion1 may be dropp$4 ,b at ?is po9t a caution is necess>y4 ,x w appe> 9 ,*ap4 ,,iii4 7 #agf71 t x is logically impossible to set up a precise coord9ate sy/em1 9 :i* ! coord9ates repres5t spatial magnitudes1 )\t ! axiom ( ,free ,mobility1 & ?is axiom1 z we h j se51 holds on surfaces only :5 ! measure ( curvature is con/ant4 ,h5ce our def9ition ( ! measure ( curvature w only be really free f ref]5ce to ! ?ird dim5sion1 :5 we are deal+ ) a surface ( con/ant measure ( curvature--a po9t :i* ,riemann entirely ov]looks4 ,?is caution1 h[ev]1 applies only 9 space1 & if we take ! coord9ate sy/em z presuppos$ 9 ! conception ( a manifold1 we may neglect ! caution altoge!r--:ile rememb]+ t ! possibility ( a coord9ate sy/em 9 space involves axioms to be inve/igat$ lat]4 ,we c ?us see h[ a mean+ might be f\nd1 )\t ref]5ce to any high] dim5sion1 = a con/ant measure ( curvature ( ?ree-dim5sional space1 or = any measure ( curvature ( an n-dim5sional manifold 9 g5]al4 #bb4 ,su* a mean+ is suppli$ by ,riemann's diss]tation1 to :i*1 aft] ?is long digression1 we c n[ return4 ,we may def9e a cont9u\s manifold z any cont9uum ( elem5ts1 su* t a s+le elem5t is def9$ by n cont9u\sly v>iable magnitudes4 ,?is def9ition does n really include space1 = coord9ates 9 space d n def9e a po9t1 b its relations to ! orig91 :i* is itself arbitr>y4 ,x includes1 h[ev]1 ! analytical conception ( space ) :i* ,riemann deals1 & may1 !re=e1 be all[$ to /& = ! mom5t4 ,riemann !n assumes t ! diff]5ce--or di/ance1 z x may be loosely call$--betwe5 any two elem5ts is comp>able1 z reg>ds magnitude1 to ! diff]5ce betwe5 any o!r two4 ,he assumes fur!r1 :at x is ,helmholtz's m]it to h prov$1 t ! diff]5ce ds betwe5 two consecutive elem5ts c be express$ z ! squ>e root ( a quadratic function ( ! diff]5ces ( ! coord9ates3 i4e4 ds#b #an #an aik dxi4dxk1 :]e ! coeffici5ts aik are1 9 g5]al1 functions ( ! coord9ates x#a x#b 444 xn4#bh ,! "q is3 ,h[ are we to obta9 a def9ition ( ! measure ( curvature out ( ?is =mula8 ,x is noticeable1 9 ! fir/ place1 t1 j z 9 a surface we f\nd an inf9ite numb] ( radii ( curvature at a po9t1 s 9 a manifold ( ?ree or m dim5sions we mu/ f9d an inf9ite numb] ( measures ( curvature at a po9t1 "o = e two-dim5sional manifold pass+ ?r\gh ! po9t1 & conta9$ 9 ! high] manifold4 ,:at we h fir/ to d1 !re=e1 is to def9e su* two-dim5sional manifolds4 ,!y mu/ consi/1 z we saw on ! surface1 ( a s+ly inf9ite s]ies ( geodesics ?r\gh ! po9t4 ,n[ a geodesic is completely det]m9$ by "o po9t & its direction at t po9t1 or by "o po9t & ! next consecutive po9t4 ,h5ce a geodesic ?r\gh ! po9t consid]$ is det]m9$ by ! ratios ( ! increm5ts ( coord9ates1 dx#a dx#b 444 dxn4 ,suppose we h two su* geodesics1 9 :i* ! i? increm5ts are respectively dxi & dxi4 ,!n all ! geodesics giv5 by dxi dxi dxi ,illu/ration =m a s+ly inf9ite s]ies1 s9ce !y conta9 "o p>amet]1 namely 3 4 ,su* a s]ies ( geodesics1 !re=e1 mu/ =m a two-dim5sional manifold1 ) a measure ( curvature 9 ! ord9>y ,gaussian s5se4 ,?is measure ( curvature c be det]m9$ f ! above =mula = ! elem5t>y arc1 by ! help ( ,gauss's g5]al =mula allud$ to above4 ,we ?us obta9 an inf9ite numb] ( measures ( curvature at a po9t1 b f n47n - #a7_/b ( ^!1 ! re/ c be d$uc$ 7,riemann1 ,gesammelte ,w]ke1 p4 #bfb74 ,:5 all ! measures ( curvature at a po9t are con/ant1 & equal to all ! measures ( curvature at any o!r po9t1 we get :at ,riemann calls a manifold ( con/ant curvature4 ,9 su* a manifold free mobility is possible1 & positions d n diff] intr9sically f "o ano!r4 ,if a be ! measure ( curvature1 ! =mula = ! arc becomes1 9 ?is case1 ds#b dx#b_/7a a_/#d x#b7b4 ,9 ?is case only1 z ,i po9t$ out above1 c ! t]m measure ( curvature be prop]ly appli$ to space )\t ref]5ce to a high] dim5sion1 s9ce free mobility is logically indisp5sable to ! exi/5ce ( quantitative or metrical ,geometry4 #bc4 ,! ma!matical result ( ,riemann's diss]tation may be summ$ up z foll[s4 ,assum+ x possible to apply magnitude to space1 i4e4 to det]m9e its elem5ts & figures by means ( algebraical quantities1 x foll[s t space c be br\ght "u ! conception ( a manifold1 z a sy/em ( quantitatively det]m9able elem5ts4 ,ow+1 h[ev]1 to ! peculi> nature ( spatial measurem5t1 ! quantitative det]m9ation ( space dem&s t magnitudes %all be indep5d5t ( place--9 s f> z ?is is n ! case1 our measurem5t w be necess>ily inaccurate4 ,if we n[ assume1 z ! quantitative relation ( di/ance betwe5 two elem5ts1 ! squ>e root ( a quadratic function ( ! coord9ates--a =mula subsequ5tly prov$ by ,helmholtz & ,lie--!n x foll[s1 s9ce magnitudes are to be indep5d5t ( place1 t space mu/1 )9 ! limits ( obs]vation1 h a con/ant measure ( curvature1 or mu/1 9 o!r words1 be homog5e\s 9 all its p>ts4 ,9 ! inf9itesimal1 ,riemann says 7p4 #bfg71 obs]vation c\ld n detect a dep>ture f con/ancy on ! "p ( ! measure ( curvature2 b he makes no attempt to %[ h[ ,geometry c\ld rema9 possible "u su* circum/ances1 & ! only ,geometry he has con/ruct$ is bas$ entirely on ,free ,mobility4 ,i %all endeav\r to prove1 9 ,*apt] ,,iii41 t any metrical ,geometry1 :i* %\ld endeav\r to disp5se ) ?is axiom1 w\ld be logically impossible4 ,at pres5t ,i w only po9t out t ,riemann1 9 spite ( 8 desire to prove t all ! axioms c be disp5s$ )1 has nev]!less1 9 8 ma!matical "w1 reta9$ ?ree fundam5tal axioms1 namely1 ,free ,mobility1 ! f9ite integral numb] ( dim5sions1 & ! axiom t two po9ts h a unique relation1 namely di/ance4 ,^!1 z we %all see h]eaft]1 are reta9$1 9 actual ma!matical "w1 by all metrical ,metageomet]s1 ev5 :5 !y believe1 l ,riemann & ,helmholtz1 t no axioms are philosophically indisp5sable4 #bd4 ,helmholtz1 ! hi/orically ne>e/ foll[] ( ,riemann1 0 guid$ by a simil> empirical philosophy1 & arriv$ indep5d5tly at a v simil> me?od ( =mulat+ ! axioms4 ,al?\gh ,helmholtz publi%$ no?+ on ! subject until aft] ,riemann's dea?1 he _h !n only j se5 ,riemann's diss]tation 7:i* 0 publi%$ po/hum\sly71 & _h work$ out 8 results1 s f> z !y 7 !n complet$1 9 entire indep5d5ce bo? ( ,riemann & ( ,lobat*ewsky4 ,helmholtz is by f> ! mo/ widely read ( all writ]s on ,metageometry1 & 8 writ+s1 almo/ alone1 repres5t to philosoph]s ! mod]n ma!matical /&po9t on ?is subject4 ,b 8 importance is mu* great]1 9 ?is doma91 z a philosoph] ?an z a ma!matician2 almo/ 8 only orig9al ma!matical result1 z reg>ds ,geometry1 is 8 pro( ( ,riemann's =mula = ! inf9itesimal arc1 & ev5 ?is pro( 0 f> f rigid1 until ,lie re=m$ x by 8 me?od ( cont9u\s gr\ps4 ,9 ?is *apt]1 !re=e1 only two ( 8 writ+s ne$ occupy u1 namely ! two articles 9 ! ,wiss5s*aftli*e ,abh&lung51 ,vol4 ,,ii41 entitl$ respectively ,ueb] die ?ats*li*5 ,grundlag5 d] ,geometrie1 #ahff 7p4 #faj ff471 & ,ueb] die ,?atsa*51 die d] ,geometrie zum ,grunde lieg51 #ahfh 7p4 #fah ff474 #be4 ,9 ! fir/ ( ^!1 :i* is *iefly philosophical1 ,helmholtz gives h9ts ( 8 !n uncomplet$ ma!matical "w1 b 9 ! ma9 cont5ts himself ) a /atem5t ( results4 ,he ann\nces t he w prove ,riemann's quadratic =mula = ! inf9itesimal arc2 b = ?is purpose1 he says1 we h to />t ) ,congru5ce1 s9ce )\t x spatial measurem5t is impossible4 ,nev]!less1 he ma9ta9s t ,congru5ce is prov$ by exp]i5ce4 ,h[ we c\ld1 )\t ! help ( measurem5t1 discov] lapses f ,congru5ce1 is a po9t :i* he leaves undiscuss$4 ,he !n enunciates ! f\r axioms :i* he consid]s ess5tial to ,geometry1 z foll[s3 7#a7 ,z reg>ds cont9uity & dim5sions4 ,9 a space ( n dim5sions1 a po9t is uniquely det]m9$ by ! measurem5t ( n cont9u\s v>iables 7coord9ates74 7#b7 ,z reg>ds ! exi/5ce ( moveable rigid bodies4 ,betwe5 ! #bn coord9ates ( any po9t-pair ( a rigid body1 !re exi/s an equation :i* is ! same = all congru5t po9t-pairs4 ,by consid]+ a suffici5t numb] ( po9t-pairs1 we get m equations ?an unkn[n quantities3 ?is gives u a me?od ( det]m9+ ! =m ( ^! equations1 s z to make x possible = !m all to be satisfi$4 7#c7 ,z reg>ds free mobility4 ,e po9t c pass freely & cont9u\sly f "o position to ano!r4 ,f 7#b7 & 7#c7 x foll[s1 t if two sy/ems ,a & ,b c be br\ght into congru5ce 9 any "o position1 ?is is also possible 9 e o!r position4 7#d7 ,z reg>ds indep5d5ce ( rotation 9 rigid bodies 7,monodromy74 ,if 7n - #a7 po9ts ( a body rema9 fix$1 s t e o!r po9t c only describe a c]ta9 curve1 !n t curve is clos$4 ,^! axioms1 says ,helmholtz1 suffice to give1 ) ! axiom ( ?ree dim5sions1 ! ,euclidean & non-,euclidean sy/ems z ! only alt]natives4 ,t !y suffice1 ma!matically1 _c be d5i$1 b !y seem1 9 "s respects1 to g too f>4 ,9 ! fir/ place1 !re is no necessity to make ! axiom ( ,congru5ce apply to actual rigid bodies--on ?is subject ,i h enl>g$ 9 ,*apt] ,,ii4#bi ,aga91 ,free ,mobility1 z di/9ct f ,congru5ce1 h>dly ne$s to be specially =mulat$3 :at b>ri] c\ld empty space (f] to a po9t's progress8 ,! axiom is involv$ 9 ! homog5eity ( space1 :i* is ! same ?+ z ! axiom ( ,congru5ce4 ,monodromy1 also1 has be5 sev]ely criticiz$2 n only is x evid5t t x might h be5 includ$ 9 ,congru5ce1 b ev5 f ! purely analytical po9t ( view1 ,sophus ,lie has prov$ x to be sup]flu\s#cj4 ,?us ! axiom ( ,congru5ce1 rightly =mulat$1 includes ,helmholtz's ?ird & f\r? axioms & "p ( 8 second axiom4 ,all ! f\r1 or r1 z mu* ( !m z is relevant to ,geometry1 are consequ5ces1 z we %all see h]eaft]1 ( ! "o fundam5tal pr9ciple ( ! relativity ( position4 #bf4 ,! second article1 :i* is ma9ly ma!matical1 supplies ! promis$ pro( ( ! arc-=mula1 :i* is ,helmholtz's mo/ important contribution to ,geometry4 ,riemann _h assum$ ?is =mula1 z ! simple/ ( a numb] ( alt]natives3 ,helmholtz prov$ x to be a necess>y consequ5ce ( 8 axioms4 ,! pres5t pap] beg9s ) a %ort repetition ( ! fir/1 includ+ ! /atem5t ( ! axioms1 to :i*1 at ! end ( ! pap]1 two m are add$1 7#e7 t space has ?ree dim5sions1 & 7#f7 t space is inf9ite4 ,x is suppos$ 9 ! text1 z also 9 ! fir/ pap]1 t ! measure ( curvature _c be negative1 &1 consequ5tly1 t an inf9ite space mu/ be ,euclidean4 ,?is error 9 bo? pap]s is correct$ 9 notes1 add$ aft] ! appe>ance ( ,beltrami's pap] on negative curvature4 ,x is a sample ( ! slightly unpr(essional nature ( ,helmholtz's ma!matical "w on ?is subject1 :i* elicits f ,kle9 ! foll[+ rem>ks#ca3 ,helmholtz is n a ma!matician by pr(ession1 b a physici/ & physiologi/4444 ,f ?is non-ma!matical quality ( ,helmholtz1 x foll[s naturally t he does n treat ! ma!matical portion ( 8 "w ) ! ?or\ghness :i* "o w\ld dem& ( a ma!matician by trade 7von ,fa*74 ,he tells u himself t x 0 ! physiological /udy ( vision :i* l$ him to ! "q ( ! axioms1 & x is z a physici/ t he makes 8 axioms ref] to actual rigid bodies4 ,accord+ly1 we f9d errors 9 8 ma!matics1 su* z ! axiom ( ,monodromy1 & ! assumption t ! measure ( curvature mu/ be positive4 ,nev]!less1 ! pro( ( ,riemann's arc-=mula is extremely able1 & has1 on ! :ole1 be5 sub/antiat$ by ,lie's m ?or\gh inve/igations4 #bg4 ,helmholtz's o!r writ+s on ,geometry are almo/ :olly philosophical1 & w be discuss$ at l5g? 9 ,*apt] ,,ii4 ,= ! pres5t1 we may pass to ! only o!r important writ] ( ! second p]iod1 ,beltrami4 ,z 8 "w is purely ma!matical1 & conta9s few controv]t$ po9ts1 x ne$ n1 despite its great importance1 deta9 u long4 ,! ,saggio di ,int]pretazione della ,geometria non-,euclidea#cb1 :i* is pr9cipally conf9$ to two dim5sions1 int]prets ,lobat*ewsky's results by ! *>act]i/ic me?od ( ! second p]iod4 ,x %[s1 by a developm5t ( ! "w ( ,gauss & ,m9d+#cc1 t all ! propositions 9 plane ,geometry1 :i* ,lobat*ewsky _h set =?1 hold1 )9 ord9>y ,euclidean space1 on surfaces ( con/ant negative curvature4 ,x is /range1 z ,kle9 po9ts out#cd1 t ?is int]pretation1 :i* 0 kn[n to ,riemann & p]haps ev5 to ,gauss1 %\ld h rema9$ s long )\t explicit /atem5t4 ,?is is ! m /range1 z ,lobat*ewsky's ,gomtrie ,imag9aire _h appe>$ 9 ,crelle1 ,vol4 ,,xvii4#ce1 & ,m9d+'s article1 f :i* ! int]pretation foll[s at once1 _h appe>$ 9 ,crelle1 ,vol4 ,,xix4 ,m9d+ _h %ewn t ! ,geometry ( surfaces ( con/ant negative curvature1 9 p>ticul> z reg>ds geodesic triangles1 c\ld be d$uc$ f t ( ! sph]e by giv+ ! radius a purely imag9>y value ia#cf4 ,?is result1 z we h se51 _h also be5 obta9$ by ,lobat*ewsky = 8 ,geometry1 & yet x took ?irty ye>s = ! connection to be br\ght to g5]al notice4 #bh4 ,9 ,beltrami's ,saggio1 /raight l9es are1 ( c\rse1 replac$ by geodesics2 8 coord9ates are obta9$ ?r\gh a po9t-by-po9t correspond5ce ) an auxili>y plane1 9 :i* /raight l9es correspond to geodesics on ! surface4 ,?us geodesics h l9e> equations1 & are always uniquely det]m9$ by two po9ts4 ,di/ances on ! surface1 h[ev]1 are n equal to di/ances on ! plane2 ?us :ile ! surface is inf9ite1 ! correspond+ portion ( ! plane is conta9$ )9 a c]ta9 f9ite circle4 ,! di/ance ( two po9ts on ! surface is a c]ta9 function ( ! coord9ates1 n ! ord9>y function ( elem5t>y ,geometry4 ,^! relations ( plane & surface are important 9 connection ) ,cayley's !ory ( di/ance1 :i* we %all h to consid] next4 ,if we 7 to def9e di/ance on ! plane z t function ( ! coord9ates :i* gives ! correspond+ di/ance on ! surface1 we %\ld obta9 :at ,kle9 calls a plane ) a hyp]bolic sy/em ( measurem5t 7,massbe/immung7 9 :i* ,cayley's !ory ( di/ance w\ld hold4 ,x is evid5t1 h[ev]1 t ! ord9>y notion ( di/ance has be5 presuppos$ 9 sett+ up ! coord9ate sy/em1 s t we d n really get alt]native ,geometries on "o & ! same plane4 ,! be>+ ( ^! rem>ks w appe> m fully :5 we come to consid] ,cayley & ,kle94 #bi4 ,! value ( ,beltrami's ,saggio1 9 8 own eyes1 lies 9 ! intelligible ,euclidean s5se :i* x gives to ,lobat*ewsky's planimetry3 ! correspond+ sy/em ( ,solid ,geometry1 s9ce x has no mean+ = ,euclidean space1 is b>ely m5tion$ 9 ?is "w4 ,9 a second pap]#cg1 h[ev]1 almo/ contemporane\s ) ! fir/1 he proce$s to consid] ! g5]al !ory ( n-dim5sional manifolds ( con/ant negative curvature4 ,?is pap] is greatly influ5c$ by ,riemann's diss]tation2 x beg9s ) ! =mula = ! l9e> elem5t1 & proves f ?is fir/1 t ,congru5ce holds = su* spaces1 & next1 t !y h1 accord+ to ,riemann's def9ition1 a con/ant negative measure ( curvature4 7,x is in/ructive to obs]ve1 t bo? 9 ?is & 9 ! =m] ,essay1 great /ress is laid on ! necessity ( ! ,axiom ( ,congru5ce47 ,?is "w has less philosophical int]e/ ?an ! =m]1 s9ce x does little m ?an repeat1 9 a g5]al =m1 ! results :i* ! ,saggio _h obta9$ = two dim5sions--results :i* s9k1 :5 ext5d$ to n dim5sions1 to ! level ( m]e ma!matical con/ructions4 ,nev]!less1 ! pap] is important1 bo? z a re/oration ( negative curvature1 :i* _h be5 ov]look$ by ,helmholtz1 & z an analytical treatm5t ( ,lobat*ewsky's results--a treatm5t :i*1 toge!r ) ! ,saggio1 at la/ re/or$ to !m ! prom95ce !y des]v$4 ,?ird ,p]iod4 #cj4 ,! ?ird p]iod diff]s radically1 alike 9 its me?ods & aims1 & 9 ! und]ly+ philosophical ideas1 f ! p]iod :i* x replac$4 ,:]eas ev]y?+1 9 ! second p]iod1 turn$ on measurem5t1 ) its app>atus ( ,congru5ce1 ,free ,mobility1 ,rigid ,bodies1 & ! re/1 ^! vani% completely 9 ! ?ird p]iod1 :i*1 sw++ to ! opposite extreme1 reg>ds quantity z a p]fectly irrelevant category 9 ,geometry1 & disp5ses ) congru5ce & ! me?od ( sup]position4 ,! ideas ( ?is p]iod1 un=tunately1 h f\nd no expon5t s philosophical z ,riemann or ,helmholtz1 b h be5 set =? only by te*nical ma!maticians4 ,moreov] ! *ange ( fundam5tal ideas1 :i* is imm5se1 has n br\ght ab\t an equally great *ange 9 actual proc$ure2 = ?\gh spatial quantity is no long] a "p ( projective ,geometry1 quantity is /ill employ$1 & we /ill h equations1 algebraic trans=mations1 & s on4 ,?is is apt to give rise to confusion1 especially 9 ! m9d ( ! /ud5t1 :o fails to realise t ! quantities us$1 s f> z ! propositions are really projective1 are m]e names = po9ts1 & n1 z 9 metrical ,geometry1 actual spatial magnitudes4 ,nev]!less1 ! fundam5tal diff]5ce betwe5 ?is p]iod & ! =m] mu/ /rike any "o at once4 ,:]eas ,riemann & ,helmholtz dealt ) metrical ideas1 & took1 z _! f\ndations1 ! measure ( curvature & ! =mula = ! l9e> elem5t--bo? purely metrical--! new me?od is erect$ on ! =mulae = trans=mation ( coord9ates requir$ to express a giv5 coll9eation4 ,x beg9s by r$uc+ all s-call$ metrical notions--di/ance1 angle1 etc4--to projective =ms1 & obta9s1 f ?is r$uction1 a me?odological unity & simplicity be=e impossible4 ,?is r$uction dep5ds1 h[ev]1 except :]e ! space-con/ant is negative1 ^u imag9>y figures--9 ,euclid1 ! circul> po9ts at inf9ity2 x is moreov] purely symbolic & analytical1 & mu/ be reg>d$ z philosophically irrelevant4 ,z ! "q conc]n+ ! import ( ?is r$uction is ( fundam5tal importance to our !ory ( ,geometry1 & z ,cayley1 9 8 ,presid5tial ,address to ! ,briti% ,association 9 #ahhc1 =mally *all5g$ philosoph]s to discuss ! use ( imag9>ies1 on :i* x dep5ds1 ,i w treat ?is "q at "s l5g?4 ,b fir/ let u see h[1 z a matt] ( ma!matics1 ! r$uction is effect$4 #ca4 ,we %all f9d1 ?r\gh\t ?is p]iod1 t almo/ e important proposition1 ?\gh mislead+ 9 its obvi\s int]pretation1 has nev]!less1 :5 rightly int]pret$1 a wide philosophical be>+4 ,s x is ) ! "w ( ,cayley1 ! pione] ( ! projective me?od4 ,! projective =mula = angles1 9 ,euclidean ,geometry1 0 fir/ obta9$ by ,lagu]re1 9 #ahec4 ,?is =mula _h1 h[ev]1 a p]fectly ,euclidean "*1 & x 0 left = ,cayley to g5]alize x s z to include bo? angles & di/ances 9 ,euclidean & non-,euclidean sy/ems alike#ch4 ,cayley 01 to ! la/1 a /aun* support] ( ,euclidean space1 ?\gh he believ$ t non-,euclidean ,geometries c\ld be appli$1 )9 ,euclidean space1 by a *ange 9 ! def9ition ( di/ance#ci4 ,he has ?us1 9 spite ( 8 ,euclidean or?odoxy1 provid$ ! believ]s 9 ! possibility ( non-,euclidean spaces ) "o ( _! mo/ p[]ful weapons4 ,9 8 ,six? ,memoir ^u ,quantics 7#ahei71 he set himself ! task ( e/abli%+ ! notion ( di/ance ^u purely descriptive pr9ciples4 ,he %[$ t1 ) ! ord9>y notion ( di/ance1 x c be r5d]$ projective by ref]5ce to ! circul> po9ts & ! l9e at inf9ity1 & t ! same is true ( angles#dj4 ,n cont5t ) ?is1 he sugge/$ a new def9ition ( di/ance1 z ! inv]se s9e or cos9e ( a c]ta9 function ( ! coord9ates2 ) ?is def9ition1 ! prop]ties usually kn[n z metrical become projective prop]ties1 hav+ ref]5ce to a c]ta9 conic1 call$ by ,cayley ! ,absolute4 7,! circul> po9ts are1 analytically1 a deg5]ate conic1 s t ord9>y ,geometry =ms a p>ticul> case ( ! above47 ,he proves t1 :5 ! ,absolute is an imag9>y conic1 ! ,geometry s obta9$ = two dim5sions is sph]ical ,geometry4 ,! correspond5ce ) ,lobat*ewsky1 9 ! case :]e ! ,absolute is real1 is n work$ out3 inde$ !re is1 ?r\gh\t1 no evid5ce ( acqua9tance ) non-,euclidean sy/ems4 ,! importance ( ! memoir1 to ,cayley1 lies entirely 9 its pro( t metrical is only a bran* ( descriptive ,geometry4 #cb4 ,! connection ( ,cayley's ,!ory ( ,di/ance ) ,metageometry 0 fir/ po9t$ out by ,kle9#da4 ,kle9 %[$ 9 detail t1 if ! ,absolute be real1 we get ,lobat*ewsky's 7hyp]bolic7 sy/em2 if x be imag9>y1 we get ei!r sph]ical ,geometry or a new sy/em1 analog\s to t ( ,helmholtz1 call$ by ,kle9 elliptic2 if ! ,absolute be an imag9>y po9t-pair1 we get p>abolic ,geometry1 & if1 9 p>ticul>1 ! po9t-pair be ! circul> po9ts1 we get ord9>y ,euclid4 ,9 elliptic ,geometry1 two /raight l9es 9 ! same plane meet 9 only "o po9t1 n two z 9 ,helmholtz's sy/em4 ,! di/9ction betwe5 ! two k9ds ( ,geometry is difficult1 & w be discuss$ lat]4 #cc4 ,s9ce ^! sy/ems are all obta9$ f a ,euclidean plane1 by a m]e alt]ation 9 ! def9ition ( di/ance1 ,cayley & ,kle9 t5d to reg>d ! :ole "q z "o1 n ( ! nature ( space1 b ( ! def9ition ( di/ance4 ,s9ce ?is def9ition1 on _! view1 is p]fectly arbitr>y1 ! philosophical problem vani%es--,euclidean space is left 9 undisput$ possession1 & ! only problem rema9+ is "o ( conv5tion & ma!matical conv5i5ce#db4 ,?is view has be5 =cibly express$ by ,po9c>3 ,:at ought "o to ?9k1 he says1 ( ?is "q3 ,is ! ,euclidean ,geometry true8 ,! "q is nons5se4 ,geometrical axioms1 accord+ to him1 are m]e conv5tions3 !y are def9itions 9 disguise#dc4 ,?us ,kle9 blames ,beltrami = reg>d+ 8 auxili>y plane z m]ely auxili>y1 & rem>ks t1 if he _h kn[n ,cayley's ,memoir1 he w\ld h se5 ! relation betwe5 ! plane & ! pseudosph]e to be f> m intimate ?an he suppos$#dd4 ,a view :i* removes ! problem entirely f ! ar5a ( philosophy dem&s1 pla9ly1 a full discussion4 ,to ?is discussion we w n[ proce$4 #cd4 ,! view 9 "q has aris51 x w\ld seem1 f a natural confusion z to ! nature ( ! coord9ates employ$4 ,^? :o hold ! view h n adequately realis$1 ,i believe1 t _! coord9ates are n spatial quantities1 z 9 metrical ,geometry1 b m]e conv5tional signs1 by :i* diff]5t po9ts c be di/9ctly designat$4 ,!re is no reason1 !re=e1 until we already h metrical ,geometry1 = reg>d+ "o function ( ! coord9ates z a bett] expression ( di/ance ?an ano!r1 s long z ! fundam5tal addition-equation#de is pres]v$4 ,h5ce1 if our coord9ates are reg>d$ z adequate = all ,geometry1 an indet]m9at5ess arises 9 ! expression ( di/ance1 :i* c only be avoid$ by a conv5tion4 ,b projective coord9ates--s our argum5t w cont5d--?\gh p]fectly adequate = all projective prop]ties1 & entirely free f any metrical presupposition1 are inadequate to express metrical prop]ties1 j because !y h no metrical presupposition4 ,?us :]e metrical prop]ties are 9 "q1 ,beltrami rema9s ju/ifi$ z aga9/ ,kle92 ! r$uction ( metrical to projective prop]ties is only app>5t1 ?\gh ! indep5d5ce ( ^! la/1 z aga9/ metrical ,geometry1 is p]fectly real4 #ce4 ,b :at are projective coord9ates1 & h[ are !y introduc$8 ,?is "q 0 n t\*$ ^u 9 ,cayley's ,memoir1 & x seem$1 !re=e1 z if a logical error 7 involv$ 9 us+ coord9ates to def9e di/ance4 ,= coord9ates1 9 all previ\s sy/ems1 _h be5 d$uc$ f di/ance2 to use any exi/+ coord9ate sy/em 9 def9+ di/ance 01 accord+ly1 to incur a vici\s circle4 ,cayley m5tions ?is difficulty 9 a note1 :]e he only rem>ks1 h[ev]1 t he _h reg>d$ 8 coord9ates z numb]s arbitr>ily assign$1 on "s sy/em n fur!r inve/igat$1 to diff]5t po9ts4 ,! difficulty has be5 treat$ at l5g? by ,sir ,r4 ,ball 7,!ory ( ! ,cont5t1 ,trans4 ,r4 ,i4 ,a4 #ahhi71 :o urges t if ! values ( our coord9ates already involve ! usual measure ( di/ance1 !n to give a new def9ition1 :ile reta9+ ! usual coord9ates1 is to incur a contradiction4 ,he says 7op4 cit4 p4 #a73 ,9 ! /udy ( non-,euclidean ,geometry ,i h (t5 felt a difficulty :i* has1 ,i "k1 be5 %>$ by o!rs4 ,9 t !ory x seems z if we try to replace our ord9>y notion ( di/ance betwe5 two po9ts by ! log>i?m ( a c]ta9 anh>monic ratio#df4 ,b ?is ratio itself involves ! notion ( di/ance measur$ 9 ! ord9>y way4 ,h[1 !n1 c we sup]s$e our old notion ( di/ance by ! non-,euclidean notion1 inasmu* z ! v def9ition ( ! latt] involves ! =m]8 #cf4 ,?is objection is valid1 we mu/ admit1 s long z anh>monic ratio is def9$ 9 ! ord9>y metrical mann]4 ,x w\ld be valid1 = example1 aga9/ any attempt to f\nd a new def9ition ( di/ance on ,cremona's acc\nt ( anh>monic ratio#dg1 9 :i* x appe>s z a metrical prop]ty unalt]$ by projective trans=mation4 ,if a logical error is to be avoid$1 9 fact1 all ref]5ce to spatial magnitude ( any k9d mu/ be avoid$2 = all spatial magnitude1 z w be %[n h]eaft]#dh1 is logically dep5d5t on ! fundam5tal magnitude ( di/ance4 ,anh>monic ratio & coord9ates mu/ alike be def9$ by purely descriptive prop]ties1 if ! use aft]w>ds made ( !m is to be free f metrical presuppositions1 & !re=e f ! objections ( ,sir ,r4 ,ball4 ,su* a def9ition has be5 satisfactorily giv5 by ,kle9#di1 :o appeals1 = ! purpose1 to v4 ,/audt's quadrilat]al con/ruction#ej4 ,by ?is con/ruction1 :i* ,i h reproduc$ 9 outl9e 9 ,*apt] ,,iii4 ,section ,a1 #aab ff41 we obta9 a purely descriptive def9ition ( h>monic & anh>monic ratio1 &1 giv5 a pair ( po9ts1 we c obta9 ! h>monic conjugate to any ?ird po9t on ! same /raight l9e4 ,on ?is con/ruction1 ! introduction ( projective coord9ates is bas$4 ,/>t+ ) any ?ree po9ts on a /raight l9e1 we assign to !m arbitr>ily ! numb]s #j1 #a1 4 ,we !n f9d ! h>monic conjugate to ! fir/ ) respect to #a1 1 & assign to x ! numb] #b4 ,! object ( assign+ ?is numb] r ?an any o!r1 is to obta9 ! value -#a = ! anh>monic ratio ( ! f\r numb]s correspond+ to ! f\r po9ts#ea4 ,we !n f9d ! h>monic conjugate to ! po9t #a1 ) respect to #b1 1 & assign to x ! numb] #c2 & s on4 ,kle9 has %[n t by ?is con/ruction1 we c obta9 any numb] ( po9ts1 & c con/ruct a po9t correspond+ to any giv5 numb]1 fractional or negative4 ,moreov]1 :5 two sets ( f\r po9ts h ! same anh>monic ratio1 descriptively def9$#eb1 ! correspond+ numb]s also h ! same anh>monic ratio4 ,by introduc+ su* a num]ical sy/em on two /raight l9es1 or on ?ree1 we obta9 ! coord9ates ( any po9t 9 a plane1 or 9 space4 ,by ?is con/ruction1 :i* is ( fundam5tal importance to projective ,geometry1 ! logical error1 ^u :i* ,sir ,r4 ,ball bases 8 criticism1 is satisfactorily avoid$4 ,our coord9ates are introduc$ by a purely descriptive me?od1 & involve no presupposition :atev] z to ! measurem5t ( di/ance4 #cg4 ,) ?is coord9ate sy/em1 !n1 to def9e di/ance z a c]ta9 function ( ! coord9ates is n to be guilty ( a vici\s circle4 ,b x by no means foll[s t ! def9ition ( di/ance is arbitr>y4 ,all ref]5ce to di/ance has be5 hi!rto exclud$1 to avoid metrical ideas2 b :5 di/ance is introduc$1 metrical ideas inevitably reappe>1 & we h to rememb] t our coord9ates give no in=mation1 prim facie1 z to any ( ^! metrical ideas4 ,x is op5 to u1 ( c\rse1 if we *oose1 to cont9ue to exclude di/ance 9 ! ord9>y s5se1 z ! quantity ( a f9ite /raight l9e1 & to def9e ! ^w di/ance 9 any way we please4 ,b ! conception1 = :i* ! ^w has hi!rto /ood1 w !n require a new "n1 & ! only result w be a confusion betwe5 ! app>5t mean+ ( our propositions1 to ^? :o reta9 ! associations belong+ to ! old s5se ( ! ^w1 & ! real mean+1 result+ f ! new s5se 9 :i* ! ^w is us$4 ,?is confusion1 ,i believe1 has actually occurr$1 9 ! case ( ^? :o reg>d ! "q betwe5 ,euclid & ,metageometry z "o ( ! def9ition ( di/ance4 ,di/ance is a quantitative relation1 & z su* presupposes id5tity ( quality4 ,b projective ,geometry deals only ) quality--= :i* reason x is call$ descriptive--& _c di/+ui% betwe5 two figures :i* are qualitatively alike4 ,n[ ! mean+ ( qualitative lik5ess1 9 ,geometry1 is ! possibility ( mutual trans=mation by a coll9eation#ec4 ,any two pairs ( po9ts on ! same /raight l9e1 !re=e1 are qualitatively alike2 _! only qualitative relation is ! /raight l9e1 :i* bo? pairs h 9 common2 & x is exactly ! qualitative id5tity ( ! relations ( ! two pairs1 :i* enables ! diff]5ce ( _! relations to be exhau/ively dealt ) by quantity1 z a diff]5ce ( di/ance4 ,b :]e quantity is exclud$1 any two pairs ( po9ts on ! same /raight l9e appe> z alike1 & ev5 any two sets ( ?ree3 = any ?ree po9ts on a /raight l9e c be projectively trans=m$ into any o!r ?ree4 ,x is only ) f\r po9ts 9 a l9e t we acquire a projective prop]ty di/+ui%+ !m f o!r sets ( f\r1 & ?is prop]ty is anh>monic ratio1 descriptively def9$4 ,! projective ,geomet]1 !re=e1 sees no reason to give a "n to ! relation betwe5 two po9ts1 9 s f> z ?is relation is any?+ ov] & above ! unlimit$ /raight l9e on :i* !y lie2 & :5 he introduces ! notion ( di/ance1 he def9es x1 9 ! only way 9 :i* projective pr9ciples all[ him to def9e x1 z a relation betwe5 f\r po9ts4 ,z he nev]!less wi%es ! ^w to give him ! p[] ( di/+ui%+ betwe5 diff]5t pairs ( po9ts1 he agrees to take two out ( ! f\r po9ts z fix$4 ,9 ?is way1 ! only v>iables 9 di/ance are ! two rema9+ po9ts1 & di/ance appe>s1 !re=e1 z a function ( two v>iables1 namely ! coord9ates ( ! two v>iable po9ts4 ,:5 we h fur!r def9$ our function s t di/ance may be additive1 we h a function ) _m ( ! prop]ties ( di/ance 9 ! ord9>y s5se4 ,?is function1 !re=e1 ! projective ,geomet] reg>ds z ! only prop] def9ition ( di/ance4 ,we c see1 9 fact1 f ! mann] 9 :i* our projective coord9ates 7 introduc$1 t "s function ( ^! coord9ates mu/ express di/ance 9 ! ord9>y s5se4 ,= !y 7 introduc$ s]ially1 s t1 z we proce$$ f ! z]o-po9t t[>ds ! inf9ity-po9t1 our coord9ates cont9ually grew4 ,to e po9t1 a def9ite coord9ate correspond$3 to ! di/ance betwe5 two v>iable po9ts1 !re=e1 z a function dep5d5t on no o!r v>iables1 mu/ correspond "s def9ite function ( ! coord9ates1 s9ce ^! are !mselves functions ( _! po9ts4 ,! function discuss$ above1 !re=e1 mu/ c]ta9ly include di/ance 9 ! ord9>y s5se4 ,b ! arbitr>y & conv5tional nature ( di/ance1 z ma9ta9$ by ,po9c> & ,kle91 arises f ! fact t ! two fix$ po9ts1 requir$ to det]m9e our di/ance 9 ! projective s5se1 may be arbitr>ily *os51 & al?\gh1 :5 our *oice is once made1 any two po9ts h a def9ite di/ance1 yet1 accord+ z we make t *oice1 di/ance w become a diff]5t function ( ! two v>iable po9ts4 ,! ambiguity ?us introduc$ is unavoidable on projective pr9ciples2 b are we to conclude1 f ?is1 t x is really unavoidable8 ,mu/ we n r conclude t projective ,geometry _c adequately deal ) di/ance8 ,if ,a1 ,b1 ,c1 be ?ree diff]5t po9ts on a l9e1 !re mu/ be "s diff]5ce betwe5 ! relation ( ,a to ,b & ( ,a to ,c1 = o!rwise1 ow+ to ! qualitative id5tity ( all po9ts1 ,b & ,c c\ld n be di/+ui%$4 ,b su* a diff]5ce involves a relation1 betwe5 ,a & ,b1 :i* is indep5d5t ( o!r po9ts on ! l9e2 = unless we h su* a relation1 ! o!r po9ts _c be di/+ui%$ z diff]5t4 ,be=e we c di/+ui% ! two fix$ po9ts1 !re=e1 f :i* ! projective def9ition />ts1 we mu/ already suppose "s relation1 betwe5 any two po9ts on our l9e1 9 :i* !y are indep5d5t ( o!r po9ts2 & ?is relation is di/ance 9 ! ord9>y s5se#ed4 ,:5 we h measur$ ?is quantitative relation by ! ord9>y me?ods ( metrical ,geometry1 we c proce$ to decide :at base-po9ts mu/ be *os51 on our l9e1 9 ord] t ! projective function discuss$ above may h ! same value z ord9>y di/ance4 ,b ! *oice ( ^! base-po9ts1 :5 we are discuss+ di/ance 9 ! ord9>y s5se1 is n arbitr>y1 & _! introduction is only a te*nical device4 ,di/ance1 9 ! ord9>y s5se1 rema9s a relation betwe5 two po9ts1 n betwe5 f\r2 & x is ! failure to p]ceive t ! projective s5se diff]s f1 & _c sup]s$e1 ! ord9>y s5se1 :i* has giv5 rise to ! views ( ,kle9 & ,po9c>4 ,! "q is n "o ( conv5tion1 b ( ! irr$ucible metrical prop]ties ( space4 ,to sum up3 ,quantities1 z us$ 9 projective ,geometry1 d n /& = spatial magnitudes1 b are conv5tional symbols = purely qualitative spatial relations4 ,b di/ance1 qu quantity1 presupposes id5tity ( quality1 z ! condition ( quantitative comp>ison4 ,di/ance 9 ! ord9>y s5se is1 9 %ort1 t quantitative relation1 betwe5 two po9ts on a l9e1 by :i* _! diff]5ce f o!r po9ts c be def9$4 ,! projective def9ition1 h[ev]1 be+ unable to di/+ui% a collection ( less ?an f\r po9ts f any o!r on ! same /raight l9e1 makes di/ance dep5d on two o!r po9ts besides ^? ^: relation x def9es4 ,no "n rema9s1 !re=e1 = di/ance 9 ! ord9>y s5se1 & _m projective ,geomet]s1 hav+ aboli%$ ! "n1 believe ! ?+ to be aboli%$ also1 & are incl9$ to d5y t two po9ts h a unique relation at all4 ,?is confusion1 9 projective ,geometry1 %[s ! importance ( a "n1 & %\ld make u *>y ( all[+ new mean+s to obscure "o ( ! fundam5tal prop]ties ( space4 #ch4 ,x rema9s to discuss ! mann] 9 :i* non-,euclidean ,geometries result f ! projective def9ition ( di/ance1 z also ! true int]pretation to be giv5 to ?is view ( ,metageometry4 ,x is to be obs]v$ t ! projective me?ods :i* foll[ ,cayley deal ?r\gh\t ) a ,euclidean plane1 on :i* !y introduce diff]5t measures ( di/ance4 ,h5ce arises1 9 any int]pretation ( ^! me?ods1 an app>5t subord9ation ( ! non-,euclidean spaces1 z ?\gh ^! 7 less self-subsi/5t ?an ,euclid's4 ,?is subord9ation is n int5d$ 9 :at foll[s2 on ! contr>y1 ! correlation ) ,euclidean space is reg>d$ z valuable1 fir/1 because ,euclidean space has be5 long] /udi$ & is m famili>1 b secondly1 because ?is correlation proves1 :5 truly int]pret$1 t ! o!r spaces are self-subsi/5t4 ,we may conf9e ourselves *iefly1 9 discuss+ ?is int]pretation1 to di/ances measur$ along a s+le /raight l9e4 ,b we mu/ be c>eful to rememb] t ! metrical def9ition ( di/ance--:i*1 accord+ to ! view "h advocat$1 is ! only adequate def9ition--is ! same 9 ,euclidean & 9 non-,euclidean spaces2 to argue 9 its fav\r is n1 !re=e1 to argue 9 fav\r ( ,euclid4 ,! projective s*eme ( coord9ates consi/s ( a s]ies ( numb]s1 ( :i* ea* repres5ts a c]ta9 anh>monic ratio & d5otes "o & only "o po9t1 & :i* increase uni=mly ) ! di/ance f a fix$ orig91 until !y become inf9ite on rea*+ a c]ta9 po9t4 ,n[ ,cayley %[$ t1 9 ,euclidean ,geometry1 di/ance may be express$ z ! limit ( ! log>i?m ( ! anh>monic ratio ( ! two po9ts & ! 7co9cid5t7 po9ts at inf9ity on _! /raight l9e2 :ile1 if we assum$ t ! po9ts at inf9ity 7 di/9ct1 we obta9$ ! =mula = di/ance 9 hyp]bolic or sph]ical ,geometry1 accord+ z ^! po9ts 7 real or imag9>y4 ,h5ce x foll[s t1 ) ! projective def9ition ( di/ance1 we %all obta9 precisely ! =mulae ( hyp]bolic1 p>abolic or sph]ical ,geometry1 accord+ z we *oose ! po9t1 to :i* ! value is assign$1 at a f9ite1 inf9ite or imag9>y di/ance 79 ! ord9>y s5se7 f ! po9t to :i* we assign ! value #j4 ,our /raight l9e rema9s1 all ! :ile1 an ord9>y ,euclidean /raight l9e4 ,b we h se5 t ! projective def9ition ( di/ance fits ) ! true def9ition only :5 ! two fix$ po9ts to :i* x ref]s are suitably *os54 ,n[ ! ord9>y mean+ ( di/ance is requir$ 9 non-,euclidean z 9 ,euclidean ,geometries--inde$1 x is only 9 metrical prop]ties t ^! ,geometries diff]4 ,h5ce our ,euclidean /raight l9e1 ?\gh x may s]ve to illu/rate o!r ,geometries ?an ,euclid's1 c only be dealt ) correctly by ,euclid4 ,:]e we give a diff]5t def9ition ( di/ance f ,euclid's1 we are /ill 9 ! doma9 ( purely projective prop]ties1 & d]ive no in=mation z to ! metrical prop]ties ( our /raight l9e4 ,b ! importance1 to ,metageometry1 ( ?is new int]pretation1 lies 9 ! fact t1 hav+ indep5d5tly e/abli%$ ! metrical =mulae ( non-,euclidean spaces1 we f9d1 z 9 ,beltrami's ,saggio1 t ^! spaces c be relat$1 by a homographic correspond5ce1 ) ! po9ts ( ,euclidean space2 & t ?is c be effect$ 9 su* a mann] z to give1 = ! di/ance betwe5 two po9ts ( our non-,euclidean space1 ! hyp]bolic or sph]ical measure ( di/ance = ! correspond+ po9ts ( ,euclidean space4 #ci4 ,on ! :ole1 !n1 a modification ( ,sir ,r4 ,ball's view1 :i* is practically a g5]aliz$ /atem5t ( ,beltrami's me?od1 seems ! mo/ t5able4 ,he imag9es :at1 ) ,grassmann1 he calls a ,cont5t1 i4e4 a p]fectly g5]al ?ree-dim5sional manifold1 & !n correlates its elem5ts1 "o by "o1 ) po9ts 9 ,euclidean space4 ,?us e elem5t ( ! ,cont5t acquires1 z its coord9ates1 ! ord9>y ,euclidean coord9ates ( ! correspond+ po9t 9 ,euclidean space4 ,by means ( ?is correlation1 our calculations1 ?\gh !y ref] to ! ,cont5t1 are c>ri$ on1 z 9 ,beltrami's ,saggio1 9 ord9>y ,euclidean space4 ,?us ! confusion disappe>s1 b ) x1 ! suppos$ ,euclidean int]pretation also disappe>s4 ,sir ,r4 ,ball's ,cont5t1 if x is to be a space at all1 mu/ be a space radically diff]5t f ,euclid's#ee2 to speak1 z ,kle9 does1 ( ord9>y planes ) hyp]bolic or elliptic measures ( di/ance1 is ei!r to incur a contradiction1 or to =ego any metrical mean+ ( di/ance4 ,in/ead ( ord9>y planes1 we h surfaces l ,beltrami's1 ( con/ant measure ( curvature2 in/ead ( ,euclid's space1 we h hyp]bolic or sph]ical space4 ,at ! same "t1 x rema9s true t we c1 by ,kle9's me?od1 give a ,euclidean mean+ to e symbolic proposition 9 non-,euclidean ,geometry4 ,= by sub/itut+1 = di/ance1 ! log>i?m above allud$ to1 we obta91 f ! non-,euclidean result1 a result :i* foll[s f ! ord9>y ,euclidean axioms4 ,?is correspond5ce removes1 once = all1 ! possibility ( a lurk+ contradiction 9 ,metageometry1 s9ce1 to a proposition 9 ! "o1 corresponds "o & only "o proposition 9 ! o!r1 & contradictory results 9 "o sy/em1 !re=e1 w\ld correspond to contradictory results 9 ! o!r4 ,h5ce ,metageometry _c lead to contradictions1 unless ,euclidean ,geometry1 at ! same mom5t1 leads to correspond+ contradictions4 ,?us ! ,euclidean plane ) hyp]bolic or elliptic measure ( di/ance1 ?\gh ei!r contradictory or n metrical z an indep5d5t notion1 has1 z a help 9 ! int]pretation ( non-,euclidean results1 a v high degree ( utility4 #dj4 ,we h /ill to discuss ,kle9's ?ird k9d ( non-,euclidean ,geometry1 :i* he calls elliptic4 ,! diff]5ce betwe5 ?is & sph]ical ,geometry is difficult to grasp1 b x may be illu/rat$ by a simpl] example4 ,a plane1 z e "o kn[s1 c be wrapp$1 )\t /ret*+1 on a cyl9d]1 & /raight l9es 9 ! plane become1 by ?is op]ation1 geodesics on ! cyl9d]4 ,! ,geometries ( ! plane & ! cyl9d]1 !re=e1 h mu* 9 common4 ,b s9ce ! g5]at+ circle ( ! cyl9d]1 :i* is "o ( its geodesics1 is f9ite1 only a portion ( ! plane is us$ up 9 wrapp+ x once r\nd ! cyl9d]4 ,h5ce1 if we endeav\r to e/abli% a po9t-to-po9t correspond5ce betwe5 ! plane & ! cyl9d]1 we %all f9d an inf9ite s]ies ( po9ts on ! plane = a s+le po9t on ! cyl9d]4 ,?us x happ5s t geodesics1 ?\gh on ! plane !y h only "o po9t 9 common1 may on ! cyl9d] h an inf9ite numb] ( int]sections4 ,some:at simil> to ?is is ! relation betwe5 ! sph]ical & elliptic ,geometries4 ,to any "o po9t 9 elliptic space1 two po9ts correspond 9 sph]ical space4 ,?us geodesics1 :i* 9 sph]ical space may h two po9ts 9 common1 c nev]1 9 elliptic space1 h m ?an "o int]section4 ,b ,kle9's me?od c only prove t elliptic ,geometry holds ( ! ord9>y ,euclidean plane ) elliptic measure ( di/ance4 ,kle9 has made great endeav\rs to en=ce ! di/9ction betwe5 ! sph]ical & elliptic ,geometries#ef1 b x is n imm$iately evid5t t ! latt]1 z di/9ct f ! =m]1 is valid4 ,9 ! fir/ place1 ,kle9's elliptic ,geometry1 :i* arises z "o ( ! alt]native metrical sy/ems on a ,euclidean plane or 9 a ,euclidean space1 does n by itself suffice1 if ! above discussion has be5 correct1 to prove ! possibility ( an elliptic space1 i4e4 ( a space hav+ a po9t-to-po9t correspond5ce ) ! ,euclidean space1 & hav+ z ! ord9>y di/ance betwe5 two ( its po9ts ! elliptic def9ition ( ! di/ance betwe5 correspond+ po9ts ( ! ,euclidean space4 ,to prove ?is possibility1 we mu/ adopt ! direct me?od ( ,newcomb 7,crelle's ,j\rnal1 ,vol4 #hc74 ,n[ 9 ! fir/ place ,newcomb has n prov$ t 8 po/ulates are self-consi/5t2 he has only fail$ to prove t !y are contradictory#eg4 ,?is w\ld leave elliptic space 9 ! same position 9 :i* ,lobat*ewsky & ,bolyai left hyp]bolic space4 ,b fur!r !re seems to be1 at fir/ sight1 9 two-dim5sional elliptic space1 a positive contradiction4 ,to expla9 ?is1 h[ev]1 "s acc\nt ( ! peculi>ities ( ! elliptic plane w be necess>y4 ,illu/ration ,! elliptic plane1 reg>d$ z a figure 9 ?ree-dim5sional elliptic space1 is :at is call$ a d\ble surface#eh1 i4e4 z ,newcomb says 7loc4 cit4 p4 #bih73 ,! two sides ( a complete plane are n di/9ct1 z 9 a ,euclidean surface4444 ,if 444 a be+ %\ld travel to di/ance #b;,d1 he w\ld1 on 8 return1 f9d himself on ! opposite surface to t on :i* he />t$1 & w\ld h to repeat 8 j\rney 9 ord] to return to 8 orig9al position )\t leav+ ! surface4 ,n[ if we imag9e a two-dim5sional elliptic space1 ! di/9ction betwe5 ! sides ( a plane becomes unmean+1 s9ce x only acquires significance by ref]5ce to ! ?ird dim5sion4 ,nev]!less1 "s su* di/9ction w\ld be =c$ ^u u4 ,suppose1 = example1 t we took a small circle provid$ ) an arr[1 z 9 ! figure1 & mov$ ?is circle once r\nd ! univ]se4 ,!n ! s5se ( ! arr[ w\ld be rev]s$4 ,we %\ld ?us be =c$1 ei!r to reg>d ! new position z di/9ct f ! =m]1 :i* trans=ms our plane into a sph]ical plane1 or to attribute ! rev]sal ( ! arr[ to ! action ( a motion :i* re/ores our circle to its orig9al place4 ,x is to be obs]v$ t no?+ %ort ( mov+ r\nd ! univ]se w\ld suffice to rev]se ! s5se ( ! arr[4 ,?is rev]sal seems l an action ( empty space1 :i* w\ld =ce u to reg>d ! po9ts :i*1 f a ?ree-dim5sional po9t ( view1 are co9cid5t ?\gh opposite1 z really di/9ct1 & s r$uce ! elliptic to ! sph]ical plane4 ,b motion1 n space1 really causes ! *ange1 & ! elliptic plane is !re=e n prov$ to be impossible4 ,! "q is n1 h[ev]1 ( any great philosophic importance4 #da4 ,9 connection ) ! r$uction ( metrical to projective ,geometry1 we h "o m topic = discussion4 ,?is is ! geometrical use ( imag9>ies1 by means ( :i*1 except 9 ! case ( hyp]bolic space1 ! r$uction is effect$4 ,i h already cont5d$1 on o!r gr\nds1 t ?is r$uction1 9 spite ( its imm5se te*nical importance1 & 9 spite ( ! complete logical fre$om ( projective ,geometry f metrical ideas1 is purely te*nical1 & is n philosophically valid4 ,! same conclusion w appe>1 if we take up ,cayley's *all5ge at ! ,briti% ,association1 9 8 ,presid5tial ,address ( #ahhc4 ,9 ?is address1 ,pr(essor ,cayley devot$ mo/ ( 8 "t to non-,euclidean sy/ems4 ,non-,euclidean spaces1 he decl>$1 seem$ to him mi/ak5 priori#ei2 b non-,euclidean ,geometries1 "h z 9 8 ma!matical works1 7 accept$ z fl[+ f a *ange 9 ! def9ition ( di/ance4 ,?is view has be5 already discuss$1 & ne$ n1 !re=e1 be fur!r criticis$ "h4 ,:at ,i wi% to speak ab\t1 is ! "q ) :i* ,cayley himself op5$ 8 address1 namely1 ! geometrical use & mean+ ( imag9>y quantities4 ,f ! mann] 9 :i* he spoke ( ?is "q1 x becomes imp]ative to treat x some:at at l5g?4 ,= he said 7pp4 #h-i73 444 ,! notion :i* is ! really fundam5tal "o 7& ,i _c too /rongly emphasize ! ass]tion7 und]ly+ & p]vad+ ! :ole notion ( mod]n analysis & ,geometry1 is t ( imag9>y magnitude 9 analysis1 & ( imag9>y space 7or space z ! locus 9 quo ( imag9>y po9ts & figures7 9 ,geometry3 ,i use 9 ea* case ! ^w imag9>y z includ+ real4444 ,say ev5 ! conclusion 7 t ! notion belongs to m]e te*nical ma!matics1 or has ref]5ce to non5tities 9 reg>d to :i* no sci5ce is possible1 /ill x seems to me t 7z a subject ( philosophical discussion7 ! notion ought n to be ?us ignor$2 x %\ld at lea/ be %[n t !re is a "r to ignore x4 #db4 ,?is "r x is n[ my purpose to demon/rate4 ,b = fe> non-ma!maticians %\ld miss ! po9t ( ,cayley's rem>k 7:i* has sometimes be5 errone\sly suppos$ to ref] to non-,euclidean spaces71 ,i may z well expla91 at ! outset1 t ?is "q is radically di/9ct f1 & only indirectly connect$ )1 ! validity or import ( ,metageometry4 ,an imag9>y quantity is "o :i* involves -#a3 its mo/ g5]al =m is a -#a;b :]e a & b are real2 ,cayley uses ! ^w imag9>y s z to include real1 9 ord] to cov] ! special case :]e b #j4 ,x w be conv5i5t1 9 :at foll[s1 to exclude ?is wid] mean+1 & assume t b is n z]o4 ,an imag9>y po9t is "o ^: coord9ates involve -#a1 i4e4 ^: coord9ates are imag9>y quantities4 ,an imag9>y curve is "o ^: po9ts are imag9>y--or1 9 "s special uses1 "o ^: equation conta9s imag9>y coeffici5ts4 ,! ma!matical subtleties to :i* ?is notion leads ne$ n be "h discuss$2 ! read] :o is int]e/$ 9 !m w f9d an excell5t elem5t>y acc\nt ( _! geometrical uses 9 ,kle9's ,ni*t-,euklid1 ,,ii4 pp4 #ch-df4 ,b = our pres5t purpose1 we may conf9e ourselves to imag9>y po9ts4 ,if ^! are f\nd to h a m]ely te*nical import1 & to be de/itute ( any philosophical mean+1 !n ! same w hold ( any collection ( imag9>y po9ts1 i4e4 ( any imag9>y curve or surface4 ,t ! notion ( imag9>y po9ts is ( supreme importance 9 ,geometry1 w be se5 by any "o :o reflects t ! circul> po9ts are imag9>y1 & t ! r$uction ( metrical to projective ,geometry1 :i* is "o ( ,cayley's greate/ a*ievem5ts1 dep5ds on ^! po9ts4 ,b to discuss adequately _! philosophical import is difficult to me1 s9ce ,i am unacqua9t$ ) any satisfactory philosophy ( imag9>ies 9 pure ,algebra4 ,i w !re=e adopt ! mo/ fav\rable hypo!sis1 & assume t no objection c be successfully urg$ aga9/ ?is use4 ,ev5 on ?is hypo!sis1 ,i ?9k1 no case c be made out = imag9>y po9ts 9 ,geometry4 ,9 ! fir/ place1 we mu/ exclude1 f ! imag9>y po9ts consid]$1 ^? ^: coord9ates are only imag9>y ) c]ta9 special sy/ems ( coord9ates4 ,= example1 if "o ( a po9t's coord9ates be ! tang5t f x to a sph]e1 ?is coord9ate w be imag9>y = any po9t inside ! sph]e1 & yet ! po9t is p]fectly real4 ,a po9t1 !n1 is only to be call$ imag9>y1 :51 :atev] real sy/em ( coord9ates we adopt1 "o or m ( ! quantities express+ ^! coord9ates rema9s imag9>y4 ,= ?is purpose1 x is ma!matically suffici5t to suppose our coord9ates ,c>tesian--a po9t ^: ,c>tesian coord9ates are imag9>y1 is a true imag9>y po9t 9 ! above s5se4 ,to discuss ! mean+ ( su* a po9t1 x is necess>y to consid] briefly ! fundam5tal nature ( ! correspond5ce betwe5 a po9t & its coord9ates4 ,assum+ t elem5t>y ,geometry has prov$--:at ,i ?9k x does satisfactorily prove--t spatial relations are susceptible ( quantitative measurem5t1 !n a giv5 po9t w h1 ) a suitable sy/em ( coord9ates1 9 a space ( n dim5sions1 n quantitative relations to ! fix$ spatial figure =m+ ! axes ( coord9ates1 & ^! n quantitative relations w1 "u c]ta9 res]vations1 be unique--i4e41 no o!r po9t w h ! same quantities assign$ to x4 7,) _m possible coord9ate sy/ems1 ?is latt] condition is n realiz$3 b = t v reason !y are inconv5i5t1 & employ$ only 9 special problems47 ,?us giv5 a coord9ate sy/em1 & giv5 any set ( quantities1 ^! quantities1 if !y det]m9e a po9t at all1 det]m9e x uniquely4 ,b1 by a natural ext5sion ( ! me?od1 ! above res]vation is dropp$1 & x is assum$ t to e set ( quantities "s po9t mu/ correspond4 ,= ?is assumption !re seems to me no ve/ige ( evid5ce4 ,z well might a po/man assume t1 because e h\se 9 a /reet is uniquely det]m9$ by its numb]1 !re=e !re mu/ be a h\se = e imag9able numb]4 ,we mu/ "k1 9 fact1 t a giv5 set ( quantities c be ! coord9ates ( "s po9t 9 space1 be=e x is legitimate to give any spatial significance to ^! quantities3 & ?is k1 obvi\sly1 _c be d]iv$ f op]ations ) coord9ates alone1 on pa9 ( a vici\s circle4 ,we mu/1 to return to ! above analogy1 "k ! numb] ( h\ses 9 ,piccadilly1 be=e we "k :e!r a giv5 numb] has a correspond+ h\se or n2 & ari?metic alone1 h[ev] subtly employ$1 w nev] give u ?is in=mation4 ,?us ! di/9ction :i* is important is1 n ! di/9ction betwe5 real & imag9>y quantities1 b betwe5 quantities to :i* po9ts correspond & quantities to :i* no po9ts correspond4 ,we c conv5tionally agree to d5ote real po9ts by imag9>y coord9ates1 z 9 ! ,gaussian me?od ( d5ot+ by ! s+le quantity 7a -#a;b7 ! po9t ^: ord9>y coord9ates are a1 b4 ,b ?is does n t\* ,cayley's mean+4 ,cayley means t x is ( great utility 9 ma!matics to reg>d1 z po9ts ) a real exi/5ce 9 space1 ! assum$ spatial correlates ( quantities :i*1 ) ! coord9ate sy/em employ$1 h no correlates 9 e-day space2 & t ?is utility is suppos$1 by _m ma!maticians1 to indicate ! validity ( s fruitful an assumption4 ,to fix our ideas1 let u consid] ,c>tesian axes 9 ?ree-dim5sional ,euclidean space4 ,!n x appe>s1 by inspection1 t a po9t may be situat$ at any di/ance to "r or left ( any ( ! ?ree coord9ate planes2 tak+ ?is di/ance z a coord9ate1 !re=e1 x appe>s t real po9ts correspond to all quantities f - to 4 ,! same appe>s = ! o!r two coord9ates2 & s9ce elem5t>y ,geometry proves _! v>iations mutually indep5d5t1 we "k t "o & only "o real po9t corresponds to any ?ree real quantities4 ,b we also "k1 f ! exhau/ive me?od pursu$1 t all space is cov]$ by ! range ( ^! ?ree v>iable quantities3 a fre% set ( quantities1 !re=e1 su* z is introduc$ by ! use ( imag9>ies1 possesses no spatial correlate1 & c be suppos$ to possess "o only by a conv5i5t fiction4 #dc4 ,! fact t ! fiction is conv5i5t1 h[ev]1 may be ?\ght to indicate t x is m ?an a fiction4 ,b ?is presumption1 ,i ?9k1 c be easily expla9$ away4 ,= all ! fruitful uses ( imag9>ies1 9 ,geometry1 are ^? :i* beg9 & end ) real quantities1 & use imag9>ies only = ! int]m$iate /eps4 ,n[ 9 all su* cases1 we h a real spatial int]pretation at ! beg9n+ & end ( our argum5t1 :]e alone ! spatial int]pretation is important3 9 ! int]m$iate l9ks1 we are deal+ 9 a purely algebraical mann] ) purely algebraical quantities1 & may p]=m any op]ations :i* are algebraically p]missible4 ,if ! quantities ) :i* we end are capable ( spatial int]pretation1 !n1 & only !n1 our result may be reg>d$ z geometrical4 ,to use geometrical language1 9 any o!r case1 is only a conv5i5t help to ! imag9ation4 ,to speak1 = example1 ( projective prop]ties :i* ref] to ! circul> po9ts1 is a m]e memoria te*nica = purely algebraical prop]ties2 ! circul> po9ts are n to be f\nd 9 space1 b only 9 ! auxili>y quantities by :i* geometrical equations are trans=m$4 ,t no contradictions arise f ! geometrical int]pretation ( imag9>ies1 is n wond]ful3 = !y are int]pret$ solely by ! rules ( ,algebra1 :i* we may admit z valid 9 _! application to imag9>ies4 ,! p]ception ( space be+ :olly abs5t1 ,algebra rules supreme1 & no inconsi/5cy c arise4 ,:]ev]1 = a mom5t1 we all[ our ord9>y spatial notions to intrude1 ! grosse/ absurdities d arise--e "o c see t a circle1 be+ a clos$ curve1 _c get to inf9ity4 ,! metaphysician1 :o %\ld inv5t any?+ s prepo/]\s z ! circul> po9ts1 w\ld be hoot$ f ! field4 ,b ! ma!matician may /eal ! horse ) impunity4 ,f9ally1 !n1 only a k ( space1 n a k ( ,algebra1 c assure u t any giv5 set ( quantities w h a spatial correlate1 & 9 ! abs5ce ( su* a correlate1 op]ations ) ^! quantities h no geometrical import4 ,?is is ! case ) imag9>ies 9 ,cayley's s5se1 & _! use 9 ,geometry1 great z are its te*nical advantages1 & rigid z is its te*nical validity1 is :olly de/itute ( philosophical importance4 #dd4 ,we h n[1 ,i ?9k1 discuss$ mo/ ( ! que/ions conc]n+ ! scope & validity ( ! projective me?od4 ,we h se5 t x is indep5d5t ( all metrical presuppositions1 & t its use ( coord9ates does n involve ! assumption t spatial magnitudes are measur$ or express$ by !m4 ,we h se5 t x is able to deal1 by its own me?ods alone1 ) ! "q ( ! qualitative lik5ess ( geometrical figures1 :i* is logically prior to any comp>ison z to quantity1 s9ce quantity presupposes qualitative lik5ess4 ,we h se5 also t1 s f> z its legitimate use ext5ds1 x applies equally to all homog5e\s spaces1 & t its crit]ion ( an indep5d5tly possible space--! det]m9ation ( a /raight l9e by two po9ts#fj--;is n subject to ! qualifications & limitations :i* belong1 z we h se5 9 ! case ( ! cyl9d]1 to ! metrical crit]ion ( con/ant curvature4 ,b we h also se5 t1 :5 projective ,geometry endeav\rs to grapple ) spatial magnitude1 & br+ di/ance & ! measurem5t ( angles b5ea? its sway1 its success1 ?\gh te*nically valid & important1 is philosophically an app>5t success only4 ,metrical ,geometry1 !re=e1 if quantity is to be appli$ to space at all1 rema9s a sep>ate1 ?\gh logically subsequ5t bran* ( ,ma!matics4 #de4 ,x only rema9s to say a few words ab\t ,sophus ,lie4 ,z a ma!matician1 z ! inv5tor ( a new & imm5sely p[]ful me?od ( analysis1 he _c be too highly prais$4 ,geometry is only "o ( ! num]\s subjects to :i* 8 !ory ( cont9u\s gr\ps applies1 b its application to ,geometry has made a revolution 9 me?od1 & has r5d]$ possible1 9 su* problems z ,helmholtz's1 a treatm5t inf9itely m precise & exhau/ive ?an any :i* 0 possible be=e4 ,! g5]al def9ition ( a gr\p is z foll[s3 ,if we h any numb] ( indep5d5t v>iables x#ax#b444xn1 & any s]ies ( trans=mations ( ^! into new v>iables--! trans=mations be+ def9$ by equations ( specifi$ =ms1 ) p>amet]s v>y+ f "o trans=mation to ano!r--!n ! s]ies ( trans=mations =m a gr\p1 if ! successive application ( any two is equival5t to a s+le memb] ( ! orig9al s]ies ( trans=mations4 ,! gr\p is cont9u\s1 :5 we c pass1 by inf9itesimal gradations )9 ! gr\p1 f any "o ( ! trans=mations to any o!r4 ,n[1 9 ,geometry1 ! result ( two successive motions or coll9eations ( a figure c always be obta9$ by a s+le motion or coll9eation1 & any motion or coll9eation c be built up ( a s]ies ( inf9itesimal motions or coll9eations4 ,moreov] ! analytical expression ( ei!r is a c]ta9 trans=mation ( ! coord9ates ( all ! po9ts ( ! figure#fa4 ,h5ce ! trans=mations det]m9+ a motion or a coll9eation are su* z to =m a cont9u\s gr\p4 ,b ! "q ( ! projective equival5ce ( two figures1 to :i* all projective ,geometry is r$ucible1 mu/ always be dealt ) by a coll9eation2 & ! "q ( ! equality ( two figures1 to :i* all metrical ,geometry is r$ucible1 mu/ always be decid$ by a motion su* z to cause sup]position2 h5ce ! :ole subject ( ,geometry may be reg>d$ z a !ory ( ! cont9u\s gr\ps :i* def9e all possible coll9eations & motions4 ,n[ ,sophus ,lie has develop$1 at great l5g?1 ! purely analytical !ory ( gr\ps2 he has !re=e1 by ?is me?od ( =mulat+ ! problem1 a v p[]ful weapon ready = ! attack4 ,9 two pap]s ,on ! f\ndations ( ,geometry#fb1 und]tak5 at ,kle9's urg5t reque/1 he takes premisses :i* r\ghly correspond to ^? ( ,helmholtz1 omitt+ ,monodromy1 & applies ! !ory ( gr\ps to ! d$uction ( _! consequ5ces#fc4 ,helmholtz's "w1 he says1 c h>dly be look$ ^u z prov+ its conclusions1 & inde$ ! m se>*+ analysis ( ! gr\p-!ory reveals sev]al possibilities unkn[n to ,helmholtz4 ,nev]!less1 z a pione]1 devoid ( ,lie's ma*9]y1 ,helmholtz des]ves1 ,i ?9k1 m praise ?an ,lie is will+ to give him#fd4 ,lie's me?od is p]fectly exhau/ive2 omitt+ ! premiss ( ,monodromy1 ! o!rs %[ t a body has six degrees ( fre$om1 i4e4 t ! gr\p giv+ all possible motions ( a body w h six indep5d5t memb]s2 if we keep "o po9t fix$1 ! numb] ( indep5d5t memb]s is r$uc$ to ?ree4 ,he !n1 f 8 g5]al !ory1 enum]ates all ! gr\ps :i* satisfy ?is condition4 ,9 ord] t su* a gr\p %\ld give possible motions1 x is necess>y1 by ,helmholtz's second axiom1 t x %\ld leave inv>iant "s function ( ! coord9ates ( any two po9ts4 ,?is elim9ates sev]al ( ! gr\ps previ\sly enum]at$1 ea* ( :i* he discusses 9 turn4 ,he is ?us l$ to ! foll[+ results3 ,i4 ,9 two dim5sions1 if free mobility is to hold univ]sally1 !re are no gr\ps satisfy+ ,helmholtz's fir/ ?ree axioms1 except ^? :i* give ! ord9>y ,euclidean & non-,euclidean motions2 b if x is to hold only )9 a c]ta9 region1 !re is also a possible gr\p 9 :i* ! curve describ$ by any po9t 9 a rotation is n clos$1 b an equiangul> spiral4 ,to exclude ?is possibility1 ,helmholtz's axiom ( ,monodromy is requir$4 ,,ii4 ,9 ?ree dim5sions1 ! results g /ill m aga9/ ,helmholtz4 ,assum+ free mobility only )9 a c]ta9 region1 we h to di/+ui% two cases3 ,ei!r free mobility holds1 )9 t region1 absolutely )\t exception1 i4e4 :5 "o po9t is held fa/1 e o!r po9t )9 ! region c move freely ov] a surface3 9 ?is case ! axiom ( ,monodromy is unnecess>y1 & ! fir/ ?ree axioms suffice to def9e our gr\p z t ( ,euclidean & non-,euclidean motions4 ,or free mobility1 )9 ! specifi$ region1 holds only ( e po9t ( g5]al position1 :ile ! po9ts ( a c]ta9 l9e1 :5 "o po9t is fix$1 are only able to move on t l9e1 n on a surface3 :5 ?is is ! case1 o!r gr\ps are possible1 & c only be exclud$ by ,helmholtz's f\r? axiom4 ,hav+ n[ /at$ ! purely ma!matical results ( ,lie's inve/igations1 we may return to philosophical consid]ations1 by :i* ,helmholtz's "w 0 ma9ly motiv$4 ,x becomes obvi\s1 n only t exceptions )9 a c]ta9 region1 b also t limitation to a c]ta9 region1 ( ! axiom ( ,free ,mobility1 are philosophically q impossible & inconceivable4 ,h[ c a c]ta9 l9e1 or a c]ta9 surface1 =m an impassable b>ri] 9 space1 or h any mobility diff]5t 9 k9d f t ( all o!r l9es or surfaces8 ,! notion _c1 9 philosophy1 be p]mitt$ = a mom5t1 s9ce x de/roys t mo/ fundam5tal ( all ! axioms1 ! homog5eity ( space4 ,we n only may1 !re=e1 b mu/ take ,helmholtz's axiom ( ,free ,mobility 9 its v /ricte/ s5se2 ! axiom ( ,monodromy ?us becomes ma!matically1 z well z philosophically1 sup]flu\s4 ,?is is1 f a philosophical /&po9t1 ! mo/ important ( ,lie's results4 #df4 ,i h n[ come to ! end ( my hi/ory ( ,metageometry4 ,x has n be5 my aim to give an exhau/ive acc\nt ( ev5 ! important works on ! subject--9 ! ?ird p]iod1 especially1 ! names ( ,po9c>1 ,pas*1 ,cremona1 ,v]onese1 & o!rs :o might be m5tion$1 w\ld h cri$ %ame ^u me1 _h ,i _h any su* object4 ,b ,i h tri$ to set =?1 z cle>ly z ,i c\ld1 ! pr9ciples at "w 9 ! v>i\s p]iods1 ! motives & results ( successive !ories4 ,we h se5 h[ ! philosophical motive1 at fir/ pr$om9ant1 has be5 gradually extrud$ by ! purely ma!matical & te*nical _s ( mo/ rec5t ,geomet]s4 ,at fir/1 to discr$it ! ,transc5d5tal ,ae/hetic seem$1 to ,metageomet]s1 z important z to advance _! sci5ce2 b f ! works ( ,cayley1 ,kle9 or ,lie1 no read] c\ld ga!r t ,kant _h ev] liv$4 ,we h also se51 h[ev]1 t z ! int]e/ 9 philosophy wan$1 ! int]e/ = philosophy increas$3 z ! ma!matical results %ook !mselves free f philosophical controv]sies1 !y assum$ gradually a /able =m1 f :i* fur!r developm5t1 we may reasonably hope1 w take ! =m ( gr[?1 r ?an trans=mation4 ,! same gradual developm5t out ( philosophy might1 ,i believe1 be trac$ 9 ! infancy ( mo/ bran*es ( ma!matics2 :5 philosophical motives cease to op]ate1 ?is is1 9 g5]al1 a sign t ! /age ( unc]ta9ty z to premisses is pa/1 s t ! future belongs entirely to ma!matical te*nique4 ,:5 ?is /able /age has be5 atta9$1 x is "t = ,philosophy to borr[ ( ,sci5ce1 accept+ its f9al premisses z ^? impos$ by a real necessity ( fact or logic4 #dg4 ,n[ 9 discuss+ ! sy/ems ( ,metageometry1 we h f\nd two k9ds1 radically di/9ct & subject to diff]5t axioms4 ,! hi/orically prior k9d1 :i* deals ) metrical ideas1 discusses1 to beg9 )1 ! conditions ( ,free ,mobility1 :i* is ess5tial to all measurem5t ( space4 ,x f9ds ! analytical expression ( ^! conditions 9 ! exi/5ce ( a space-con/ant1 or con/ant measure ( curvature1 :i* is equival5t to ! homog5eity ( space4 ,?is is its fir/ axiom4 ,its second axiom /ates t space has a f9ite integral numb] ( dim5sions1 i4e4 9 metrical t]ms1 t ! position ( a po9t1 relative to any o!r figure 9 space1 is uniquely det]m9$ by a f9ite numb] ( spatial magnitudes1 call$ coord9ates4 ,! ?ird axiom ( metrical ,geometry may be call$1 to di/+ui% x f ! correspond+ projective axiom1 ! axiom ( di/ance4 ,!re exi/s "o relation1 x says1 betwe5 any two po9ts1 :i* c be pres]v$ unalt]$ 9 a comb9$ motion ( bo? po9ts1 & :i*1 9 any motion ( a sy/em z "o rigid body1 is always unalt]$4 ,?is relation we call di/ance4 ,! above /atem5t ( ! ?ree ess5tial axioms ( metrical ,geometry is tak5 f ,helmholtz z am5d$ by ,lie4 ,lie's own /atem5t ( ! axioms1 z quot$ above1 has be5 too mu* influ5c$ by projective me?ods to give a hi/orically correct r5d]+ ( ! _s ( ! second p]iod2 ,helmholtz's /atem5t1 on ! o!r h&1 requires1 z ,lie has %ewn1 v consid]able modifications4 ,! above compromise may1 !re=e1 ,i hope be tak5 z accept+ ,lie's corrections :ile reta9+ ,helmholtz's _s4 #dh4 ,b metrical ,geometry1 ?\gh x is hi/orically prior1 is logically subsequ5t to projective ,geometry4 ,= projective ,geometry deals directly ) t qualitative lik5ess1 :i* ! judgm5t ( quantitative comp>ison requires z its basis4 ,n[ ! above ?ree axioms ( metrical ,geometry1 z we %all see 9 ,*apt] ,,iii4 ,section ,b1 d n presuppose measurem5t1 b are1 on ! contr>y1 ! conditions presuppos$ by measurem5t4 ,)\t ^! axioms1 :i* are common to all ?ree spaces1 measurem5t w\ld be impossible2 ) !m1 s ,i %all cont5d1 measurem5t is able1 ?\gh only empirically1 to decide approximately :i* ( ! ?ree spaces is valid ( our actual _w4 ,b if ^! ?ree axioms !mselves express1 n results1 b conditions1 ( measurem5t1 mu/ !y n be equival5t to ! /atem5t ( t qualitative lik5ess on :i* quantitative comp>ison dep5ds8 ,& if s1 mu/ we n expect to f9d ! same axioms1 ?\gh p]haps "u a diff]5t =m1 9 projective ,geometry8 #di4 ,?is expectation w n be disappo9t$4 ,! above ?ree axioms1 z we %all see h]eaft]1 are "o & all philosophically equival5t to ! homog5eity ( space1 & ?is 9 turn is equival5t to ! axioms ( projective ,geometry4 ,! axioms ( projective ,geometry1 9 fact1 may be r\ghly /at$ ?us3 ,i4 ,space is cont9u\s & inf9itely divisible2 ! z]o ( ext5sion1 result+ f inf9ite division1 is call$ a ,po9t4 ,all po9ts are qualitatively simil>1 & di/+ui%$ by ! m]e fact t !y lie outside "o ano!r4 ,,ii4 ,any two po9ts det]m9e a unique figure1 ! /raight l9e2 two /raight l9es1 l two po9ts1 are qualitatively simil>1 & di/+ui%$ by ! m]e fact t !y are mutually ext]nal4 ,,iii4 ,?ree po9ts n 9 "o /raight l9e det]m9e a unique figure1 ! plane1 & f\r po9ts n 9 "o plane det]m9e a figure ( ?ree dim5sions4 ,?is process may1 s f> z c be se5 priori1 be cont9u$1 )\t 9 any way int]f]+ ) ! possibility ( projective ,geometry1 to five or to n po9ts4 ,b projective ,geometry requires1 z an axiom1 t ! process %\ld /op ) "s positive integral numb] ( po9ts1 aft] :i*1 any fre% po9t is conta9$ 9 ! figure det]m9$ by ^? already giv54 ,if ! process /ops ) 7n #a7 po9ts1 our space is said to h n dim5sions4 ,^! ?ree axioms1 x w be se51 are ! equival5ts ( ! ?ree axioms ( metrical ,geometry#fe1 express$ )\t ref]5ce to quantity4 ,we %all f9d !m to be d$ucible1 z be=e1 f ! homog5eity ( space1 or1 m g5]ally /ill1 f ! possibility ( exp]i5c+ ext]nality4 ,!y w !re=e appe> z priori1 z ess5tial to ! exi/5ce ( any ,geometry & to exp]i5ce ( an ext]nal _w z su*4 #ej4 ,t "s logical necessity is involv$ 9 ^! axioms might1 ,i ?9k1 be inf]r$ z probable1 f _! hi/orical developm5t alone4 ,= ! sy/ems ( ,metageometry h n1 9 g5]al1 be5 set up z m likely to fit facts ?an ! sy/em ( ,euclid2 ) ! exception ( ,zlln]1 = example1 ,i "k ( no "o :o has reg>d$ ! f\r? dim5sion z requir$ to expla9 ph5om5a4 ,z reg>ds ! space-con/ant aga91 ?\gh a small space-con/ant is reg>d$ z empirically possible1 x is n usually reg>d$ z probable2 & ! f9ite space-con/ants1 ) :i* ,metageometry is equally conv]sant1 are n usually ?\ght ev5 possible1 z explanations ( empirical fact#ff4 ,?us ! motive has be5 ?r\gh\t n "o ( fact1 b "o ( logic4 ,does n ?is give a /rong presumption1 t ^? axioms :i* are reta9$1 are reta9$ because !y are logically indisp5sable8 ,if ?is be s1 ! axioms common to ,euclid & ,metageometry w be priori1 :ile ^? peculi> to ,euclid w be empirical4 ,aft] a criticism ( "s diff]+ !ories ( ,geometry1 ,i %all proce$1 9 ,*apt]s ,,iii4 & ,,iv41 to ! pro( & consequ5ces ( ?is !sis1 :i* w =m ! rema9d] ( ! pres5t "w4 ,,footnotes3 #e ,v4 ,mmoires de l',acadmie royale des ,sci5ces de l'ln/itut de ,france1 ,t4 ,,xii4 #ahcc1 = a full /atem5t ( 8 results1 ) ref]5ces to =m] writ+s4 #f ,?is bold] me?od1 x appe>s1 _h be5 sugge/$1 ne>ly a c5tury e>li]1 by an ,italian1 ,sac*]i4 ,8 "w1 :i* seems to h rema9$ completely unkn[n until ,beltrami r$iscov]$ x 9 #ahhi1 is call$ ,euclides ab omni naevo v9dicatus1 etc4 ,m$iolani1 #agcc4 7,see ,v]onese1 ,grundzge d] ,geometrie1 ,g]man translation1 ,leipzig1 #ahid1 p4 #fcf47 ,8 results includ$ sph]ical z well z hyp]bolic space2 b !y al>m$ him to su* an ext5t t he devot$ ! la/ half ( 8 book to disprov+ !m4 #g ,kle9's fir/ acc\nt ( elliptic ,geometry1 z a result ( ,cayley's projective !ory ( di/ance1 appe>$ 9 two articles entitl$ ,ueb] die sog5annte ,ni*t-,euklidis*e ,geometrie1 ,i1 ,,ii1 ,ma?4 ,annal5 #d1 #f 7#ahga-b74 ,x 0 aft]w>ds indep5d5tly discov]$ by ,newcomb1 9 an article entitl$ ,elem5t>y ,!orems relat+ to ! geometry ( a space ( ?ree dim5sions1 & ( uni=m positive curvature 9 ! f\r? dim5sion1 ,crelle's ,j\rnal fr die re9e und angew&te ,ma!matik1 ,vol4 #hc 7#ahgg74 ,= an acc\nt ( ! ma!matical controv]sies conc]n+ elliptic ,geometry1 see ,kle9's ,vorlesung5 b] ,ni*t-,euklidis*e ,geometrie1 ,gtt+5 #ahic1 ,i4 p4 #bhd ff4 ,a bibliography ( ! relevant lit]ature up to ! ye> #ahgh 0 giv5 by ,hal/$ 9 ! ,am]ican ,j\rnal ( ,ma!matics1 ,vols4 #a1 #b4 #h ,v]onese 7op4 cit4 p4 #fch7 d5ies ! priority ( ,gauss 9 ! inv5tion ( a non-,euclidean sy/em1 ?\gh he admits him to h be5 ! fir/ to reg>d ! axiom ( p>allels z indemon/rable4 ,8 gr\nds = ! =m] ass]tion seem sc>cely adequate3 on ! evid5ce aga9/ x1 see ,kle91 ,ni*t-,euklid1 ,i4 pp4 #aga-agd4 #i ,v4 ,briefwe*sel mit ,s*uma*]1 ,bd4 ,,ii4 p4 #bfh4 #aj f4 ,helmholtz1 ,wiss4 ,abh4 ,,ii4 p4 #faa4 #aa ,crelle's ,j\rnal1 #ahcg4 #ab ,!orie d] ,p>allell9i51 ,b]l91 #ahdj4 ,republi%$1 ,b]l91 #ahhg4 ,translat$ by ,hal/$1 ,au/91 ,texas1 ,u4,s4,a4 #d? edition1 #ahib4 #ac ,fris*auf1 ,absolute ,geometrie1 na* ,johann ,bolyai1 ,leipzig1 #ahgb4 ,hal/$1 ,! ,sci5ce ,absolute ( ,space1 translat$ f ! ,lat91 #d? edition1 ,au/91 ,texas1 ,u4,s4,a4 #ahif4 #ad ,bo? ,lobat*ewsky & ,bolyai1 z ,v]onese rem>ks1 />t r f ! po9t-pair ?an f di/ance4 ,see ,fris*auf1 ,absolute ,geometrie1 ,anhang4 #ae ,comp>e ,/allo1 ,concepts ( ,mod]n ,physics1 p4 #bdh4 #af ,gesammelte ,w]ke1 pp4 #bee-bfh4 #ag ,on ! hi/ory ( ?is ^w1 see ,/allo1 ,concepts ( ,mod]n ,physics1 p4 #beh4 ,x 0 us$ by ,kant1 & adapt$ by ,h]b>t to almo/ ! same mean+ z x be>s 9 ,riemann4 ,h]b>t1 h[ev]1 also uses ! ^w ,reih5=m to express a simil> idea4 ,see ,psy*ologie als ,wiss5s*aft1 ,i4 #ajj & ,,ii4 #aci1 :]e ,riemann's analogy ) col\rs is also sugge/$4 #ah ,comp>e ,erdmann's ,grss5begriff vom ,raum4 #ai ,comp>e ,v]onese1 op4 cit4 p4 #fdb3 ,riemann i/ 9 se9] ,def9ition des ,begriffs ,grsse dunkel4 ,see also ,v]onese's :ole foll[+ criticism4 #bj ,vortrge und ,r$51 ,vol4 ,,ii4 p4 #ah4 #ba ,cf4 ,kle91 ,ni*t-,euklid1 ,i4 p4 #afj4 #bb ,s9ce we are consid]+ ! curvature at a po9t1 we are only conc]n$ ) ! fir/ inf9itesimal elem5ts ( ! geodesics t />t f su* a po9t4 #bc ,disquisitiones g5]ales circa sup]ficies curvas1 ,w]ke1 ,bd4 ,,iv4 ,,ss4 #bai-beh1 #ahbg4 #bd ,nev]!less1 ! ,geometries ( diff]5t surfaces ( equal curvature are liable to important diff]5ces4 ,= example1 ! cyl9d] is a surface ( z]o curvature1 b s9ce its l9es ( curvature 9 "o direction are f9ite1 its ,geometry co9cides ) t ( ! plane only = l5g?s small] ?an ! circumf]5ce ( its g5]at+ circle 7see ,v]onese1 op4 cit4 p4 #fdd74 ,two geodesics on a cyl9d] may meet 9 _m po9ts4 ,= surfaces ( z]o curvature on :i* ?is is n possible1 ! id5tity ) ! plane may be all[$ to /&4 ,o!rwise1 ! id5tity ext5ds only to ! prop]ties ( figures n exce$+ a c]ta9 size4 #be ,= we may consid] two diff]5t p>ts ( ! same surface z correspond+ p>ts ( diff]5t surfaces2 ! above proposition !n %[s t a figure c be reproduc$ 9 "o "p :5 x has be5 drawn 9 ano!r1 if ! measures ( curvature correspond 9 ! two p>ts4 #bf ,crelle1 ,vols1 ,,xix41 ,,xx41 #ahci-dj4 #bg ,9 ?is =mula1 u1 v may be ! l5g?s ( l9es1 or ! angles betwe5 l9es1 drawn on ! surface1 & hav+ ?us no necess>y ref]5ce to a ?ird dim5sion4 #bh ,9 :at foll[s1 ,i h giv5 r ,kle9's exposition ( ,riemann1 ?an ,riemann's own acc\nt4 ,! =m] is mu* cle>] & full]1 & n sub/antially diff]5t 9 any way4 ,v4 ,kle91 ,ni*t-,euklid1 ,i4 pp4 #bjf ff4 #bi ,see #fi-gc4 #cj ,grundlag5 d] ,geometrie1 ,i4 & ,,ii41 ,leipzig] ,b]i*te1 #ahij2 v4 end ( pres5t *apt]1 #de4 #ca ,ni*t-,euklid1 ,i4 pp4 #beh-i4 #cb ,giornale di ,matemati*e1 ,vol4 ,,vi41 #ahfh4 ,translat$ into ,fr5* by ,j4 ,hoel 9 ! ,annales ,sci5tifiques de l'cole ,normale ,suprieure1 ,vol4 ,,vi4 #ahfi4 #cc ,crelle's ,j\rnal1 ,vols4 ,,xix4 ,,xx41 #ahci-dj4 #cd ,ni*t-,euklid1 ,i4 p4 #aij4 #ce ,?is article is m trigonometrical & analytical ?an ! ,g]man book1 & !re=e makes ! above int]pretation peculi>ly evid5t4 #cf ,su* surfaces are by no means p>ticul>ly remote4 ,"o ( !m1 = example1 is =m$ by ! revolution ( ! common ,tractrix x as9 1 y a7log tan _/#b cos 74 #cg ,teoria fondam5tale degli spazii di curvatura co/anta1 ,annali di ,matematica1 ,,ii4 ,vol4 #b1 #ahfh-i4 ,also translat$ by ,j4 ,hoel1 loc4 cit4 #ch ,see ,kle91 ,ni*t-,euklid1 ,i4 p4 #dg ff41 & ! ref]5ces !re giv54 #ci ,see quotation bel[1 f 8 ,briti% ,association ,address4 #dj ,comp>e ! op5+ s5t5ce1 due to ,cayley1 ( ,salmon's ,high] ,plane ,curves4 #da ,v4 ,ni*t-,euklid1 ,i4 ,*aps4 ,i4 & ,,ii4 #db ,see p4 #i ( ,cayley's address to ! ,brit4 ,ass4 #ahhc4 ,also a quotation f ,kle9 9 ,erdmann's ,axiome d] ,geometrie1 p4 #abd note4 #dc ,nature1 ,vol4 ,,xlv4 p4 #djg4 #dd ,ni*t-,euklid1 ,i4 p4 #bjj4 #de ,i4e4 ! equation ,,ab ,,bc ,,ac1 = ?ree po9ts 9 "o /raight l9e4 #df ,! =mula sub/itut$ by ,kle9 = ,cayley's inv]se s9e or cos9e4 ,! two are equival5t1 b ,kle9's is ma!matically mu* ! m conv5i5t4 #dg ,elem5ts ( ,projective ,geometry1 ,second ,edition1 ,ox=d1 #ahic1 ,*ap4 ,,ix4 #dh ,*ap4 ,,iii4 ,section ,b4 #di ,see ,ni*t-,euklid1 ,i4 p4 #cch ff4 #ej ,see 8 ,geometrie d] ,lage1 #h1 ,h>monis*e ,gebilde4 #ea ,! anh>monic ratio ( f\r numb]s1 p1 q1 r1 s1 is def9$ z 7p - q747r - s7 _/ 7p - r747q - s74 #eb ,i4e4 z trans=mable into ea* o!r by a coll9eation4 ,see ,*ap4 ,,iii4 ,sec4 ,a1 #aaj4 #ec ,see ,*ap4 ,,iii4 ,sec4 ,a4 #ed ,x foll[s f ?is1 t ! r$uction ( metrical to projective prop]ties1 ev5 :51 z 9 hyp]bolic ,geometry1 ! ,absolute is real1 is only app>5t1 & has a m]ely te*nical validity4 #ee ,sir ,r4 ,ball does n reg>d 8 non-,euclidean cont5t z a possible space 7v4 op4 cit4 p4 #aea74 ,9 ?is important po9t ,i disagree ) 8 int]pretation1 hold+ su* a cont5t to be a space z possible1 priori1 z ,euclid's1 & p]haps actually true )9 ! m>g9 due to errors ( obs]vation4 #ef ,see ,ni*t-,euklid1 ,i4 p4 #ig ff4 & p4 #bib ff4 #eg ,newcomb says 7loc4 cit4 p4 #bic73 ,! sy/em "h set =? is f\nd$ on ! foll[+ ?ree po/ulates4 #a4 ,i assume t space is triply ext5d$1 unb\nd$1 )\t prop]ties dep5d5t ei!r on position or direction1 & possess+ su* plan5ess 9 its smalle/ p>ts t bo? ! po/ulates ( ! ,euclidean ,geometry1 & our common conceptions ( ! relations ( ! p>ts ( space are true = e indef9itely small region 9 space4 #b4 ,i assume t ?is space is affect$ ) su* curvature t a "r l9e %all always return into itself at ! end ( a f9ite & real di/ance #b;,d )\t los+1 9 any "p ( its c\rse1 t symmetry ) respect to space on all sides ( x :i* con/itutes ! fundam5tal prop]ty ( our conception ( x4 #c4 ,i assume t if two "r l9es emanate f ! same po9t1 mak+ ! indef9itely small angle a ) ea* o!r1 _! di/ance ap>t at ! di/ance r f ! po9t ( int]section w be giv5 by ! equation s #b;a,d_/ s9 r_/#b;,d4 ,! "r l9e ?us has ?is prop]ty 9 common ) ! ,euclidean "r l9e t two su* l9es int]sect only 9 a s+le po9t4 ,x may be t ! numb] ( po9ts 9 :i* two su* l9es c int]sect admit ( be+ det]m9$ f ! laws ( curvature1 b n be+ able s to det]m9e x1 ,i assume z a po/ulate ! fundam5tal prop]ty ( ! ,euclidean "r l9e4 ,x is pla9 t 9 ! abs5ce ( ! det]m9ation spok5 (1 ! possibility ( elliptic space is n e/abli%$4 ,x may be possible1 = example1 to prove t1 9 a space :]e !re is a maximum to di/ance1 !re mu/ be an inf9ite numb] ( /raight l9es jo9+ two po9ts ( maximum di/ance4 ,9 ?is ev5t1 elliptic space w\ld become impossible4 #eh ,= an elucidation ( ?is t]m1 see ,kle91 ,ni*t-,euklid1 ,i4 p4 #ii ff4 #ei ,cf4 p4 #i ( ,report3 ,my own view is t ,euclid's twelf? axiom1 9 ,playfair's =m ( x1 does n ne$ demon/ration1 b is "p ( our notion ( space1 ( ! physical space ( our exp]i5ce1 b :i* is ! repres5tation ly+ at ! bottom ( all ext]nal exp]i5ce4 #fj ,! exception to ?is axiom1 9 sph]ical space1 presupposes metrical ,geometry1 & does n de/roy ! validity ( ! axiom = projective ,geometry4 ,see ,*ap4 ,,iii4 ,sec4 ,b1 #aga4 #fa ,ma!maticians ( ,lie's s*ool h a habit1 at fir/ some:at confus+1 ( speak+ ( motions ( space in/ead ( motions ( bodies1 z ?\gh space z a :ole c\ld move4 ,all t is meant is1 ( c\rse1 ! equival5t motion ( ! coord9ate axes1 i4e4 a *ange ( axes 9 ! usual elem5t>y s5se4 #fb ,ueb] die ,grundlag5 d] ,geometrie1 ,leipzig] ,b]i*te1 #ahij4 ,! problem ( ^! two pap]s is really metrical1 s9ce x is conc]n$1 n ) coll9eations 9 g5]al1 b ) motions4 ,! problem1 h[ev]1 is dealt ) by ! projective me?od1 motions be+ reg>d$ z coll9eations :i* leave ! ,absolute un*ang$4 ,x seem$ impossible1 !re=e1 to discuss ,lie's "w1 until "s acc\nt _h be5 giv5 ( ! projective me?od4 #fc ,lie's premisses1 to be accurate1 are ! foll[+3 ,let x#a f7x1 y1 z1 a#a1 a#b4447 x#b 7x1 y1 z1 a#a1 a#b4447 x#c 7x1 y1 z1 a#a1 a#b4447 give an inf9ite family ( real trans=mations ( space1 z to :i* we make ! foll[+ hypo!ses3 ,a4 ,! functions f1 1 1 are analytical functions ( x1 y1 z1 a#a1 a#b4444 ,b4 ,two po9ts x#ay#az#a1 x#by#bz#b possess an inv>iant1 i4e4 7x#a1 y#a1 z#a1 x#b1 y#b1 z#b7 7x#a1 y#a1 z#a1 x#b1 y#b1 z#b7 :]e x#a4441 x#b4441 are ! trans=m$ coord9ates ( ! two po9ts4 ,c4 ,free ,mobility3 i4e41 any po9t c be mov$ into any o!r position2 :5 "o po9t is fix$1 any o!r po9t ( g5]al position c take up #b positions2 :5 two po9ts are fix$1 any o!r ( g5]al position c take up #a positions2 :5 ?ree1 no motion is possible--^! limitations be+ results ( ! equations giv5 by ! inv>iant 4 #fd ,on ?is po9t1 cf4 ,kle91 ,h"h ,geometrie1 ,gtt+51 #ahic1 ,,ii4 pp4 #bbe-bdd1 especially pp4 #bcj-a4 #fe ,axiom ,,ii4 ( ! metrical triad corresponds to ,axiom ,,iii4 ( ! projective1 & vice v]s4 #ff ,cf4 ,helmholtz1 ,wiss4 ,abh4 ,vol4 ,,ii4 p4 #fdj1 note3 ,die ,be>beit] d] ,ni*t-,euklidis*5 ,geometrie 7hab57 d]5 objective ,wahrheit nie behauptet4 ,,*apt] ,,ii4 ,,critical ,,acc\nt ,,( ,,"s ,,previ\s ,,philosophical ,,!ories ,,( ,,geometry4 #ea4 ,we h n[ trac$ ! ma!matical developm5t ( ! !ory ( geometrical axioms1 f ! fir/ revolt aga9/ ,euclid to ! pres5t day4 ,we may hope1 !re=e1 to h at our comm& ! te*nical k requir$ = ! philosophy ( ! subject4 ,! importance ( ,geometry1 9 ! !ories ( k :i* h aris5 9 ! pa/1 c sc>cely be exagg]at$4 ,9 ,desc>tes1 we f9d ! :ole !ory ( me?od dom9at$ by analytical ,geometry1 ( ^: fruitfulness he 0 ju/ly pr\d4 ,9 ,sp9oza1 ! p>am\nt influ5ce ( ,geometry is too obvi\s to require comm5t4 ,among ma!maticians1 ,newton's belief 9 absolute space 0 long supreme1 & is /ill responsible = ! curr5t =mulation ( ! laws ( motion4 ,aga9/ ?is belief on ! "o h&1 & aga9/ ,leibnitz's !ory ( space on ! o!r1 & n1 z ,caird has po9t$ out#fg1 aga9/ ,hume's empiricism1 0 direct$ t key/one ( ! ,critical ,philosophy1 ! ,kantian doctr9e ( space4 ,?us ,geometry has be51 ?r\gh\t1 ( supreme importance 9 ! !ory ( k4 ,b 9 a criticism ( repres5tative mod]n !ories ( ,geometry1 :i* is design$ to be1 n a hi/ory ( ! subject1 b an introduction to1 & def5ce (1 ! views ( ! au?or1 x w n be necess>y to discuss any m anci5t !ory ?an t ( ,kant4 ,kant's views on ?is subject1 true or false1 h s dom9at$ subsequ5t ?\ght1 t :e!r !y 7 accept$ or reject$1 !y seem$ equally pot5t 9 =m+ ! op9ions1 & ! mann] ( exposition1 ( almo/ all lat] writ]s4 ,kant4 #eb4 ,x is n my purpose1 9 ?is *apt]1 to add to ! volum9\s lit]ature ( ,kantian criticism1 b only to discuss ! be>+ ( ,metageometry on ! argum5t ( ! ,transc5d5tal ,ae/hetic1 & ! aspect "u :i* ?is argum5t mu/ be view$ 9 a discussion ( ,geometry#fh4 ,on ?is po9t sev]al misund]/&+s seem to me to h _h wide preval5ce1 bo? among fri5ds & foes1 & ^! misund]/&+s ,i %all endeav\r1 if ,i c1 to remove4 ,9 ! fir/ place1 :at does ,kant's doctr9e mean = ,geometry8 ,obvi\sly n ! aspect ( ! doctr9e :i* has be5 attack$ by psy*ologi/s1 ! ,kantian ma*9e-%op z ,james calls x--at any rate1 if ?is c be cle>ly sep>at$ f ! logical aspect4 ,! "q :e!r space is giv5 9 s5sation1 or :e!r1 z ,kant ma9ta9$1 x is giv5 by an intuition to :i* no ext]nal matt] corresponds1 may = ! pres5t be disreg>d$4 ,if1 inde$1 we held ! view :i* seems crudely to sum up ! /&po9t ( ! ,critique1 ! view t all c]ta9 k is self-k1 !n we %\ld be committ$1 if we _h decid$ t ,geometry 0 apodeictic1 to ! view t space is subjective4 ,b ev5 !n1 ! psy*ological "q c\ld only arise :5 ! epi/emological "q _h be5 solv$1 & c\ld n1 !re=e1 be tak5 into acc\nt 9 our fir/ inve/igation4 ,! "q be=e u is precisely ! "q :e!r1 or h[ f>1 ,geometry is apodeictic1 & = ! mom5t we h only to inve/igate ?is "q1 )\t fe> ( psy*ological consequ5ces4 #ec4 ,n[ on ?is "q1 z on almo/ all que/ions 9 ! ,ae/hetic or ! ,analytic1 ,kant's argum5t is tw(old4 ,on ! "o h&1 he says1 ,geometry is kn[n to h apodeictic c]ta9ty3 !re=e space mu/ be priori & subjective4 ,on ! o!r h&1 x foll[s1 f gr\nds indep5d5t ( ,geometry1 t space is subjective & priori2 !re=e ,geometry mu/ h apodeictic c]ta9ty4 ,^! two argum5ts are n cle>ly di/+ui%$ 9 ! ,ae/hetic1 b a little analysis1 ,i ?9k1 w dis5tangle !m4 ,?us 9 ! fir/ edition1 ! fir/ two argum5ts d$uce1 f non-geometrical gr\nds1 ! apriority ( space2 ! ?ird d$uces ! apodeictic c]ta9ty ( ,geometry1 & ma9ta9s1 conv]sely1 t no o!r view c acc\nt = ?is c]ta9ty#fi2 ! la/ two argum5ts only ma9ta9 t space is an intuition1 n a concept4 ,9 ! second edition1 ! d\ble argum5t is cle>]1 ! apriority ( space be+ prov$ indep5d5tly ( ,geometry 9 ! metaphysical d$uction1 & d$uc$ f ! c]ta9ty ( ,geometry1 z ! only possible explanation ( ?is1 9 ! transc5d5tal d$uction4 ,9 ! ,prolegom5a1 ! latt] argum5t alone is us$1 b 9 ! ,critique bo? are employ$4 #ed4 ,n[ x mu/ be admitt$1 ,i ?9k1 t ,metageometry has de/roy$ ! legitimacy ( ! argum5t f ,geometry to space2 we c no long] affirm1 on purely geometrical gr\nds1 ! apodeictic c]ta9ty ( ,euclid4 ,b unless ,metageometry has done m ?an ?is--unless x has prov$1 :at ,i believe x alone _c prove1 t ,euclid has n apodeictic c]ta9ty--!n ,kant's o!r l9e ( argum5t reta9s :at =ce x may ev] h _h4 ,! actual space we "k1 x may say1 is admitt$ly ,euclidean1 & is prov$1 )\t any ref]5ce to ,geometry1 to be priori2 h5ce ,euclid has apodeictic c]ta9ty1 & non-,euclid /&s condemn$4 ,to ?is x is no answ] to urge1 ) ! ,metageomet]s1 t non-,euclidean sy/ems are logically self-consi/5t2 = ,kant is c>eful to argue t geometrical reason+1 by virtue ( our intuition ( space1 is syn!tic1 & _c1 ?\gh priori1 be upheld by ! pr9ciple ( contradiction alone#gj4 ,unless non-,euclideans c prove1 :at !y h c]ta9ly fail$ to prove up to ! pres5t1 t we c frame an intuition ( non-,euclidean spaces1 ,kant's position _c be upset by ,metageometry alone1 b mu/ also be attack$1 if x is to be successfully attack$1 on its purely philosophical side4 #ee4 ,= su* an attack1 two roads lie op53 ei!r we may disprove ! fir/ two argum5ts ( ! ,ae/hetic1 or we may criticize1 f ! /&po9t ( g5]al logic1 ! ,kantian doctr9e ( syn!tic priori judgm5ts & _! connection ) subjectivity4 ,bo? ^! attacks1 ,i believe1 c\ld be conduct$ ) "s success2 b if we are to disprove ! apodeictic c]ta9ty ( ,geometry1 "o or o!r is ess5tial1 & bo?1 ,i believe1 w be f\nd only p>tially successful4 ,x w be my aim to prove1 9 discuss+ ^! two l9es ( attack1 7#a7 t ! di/9ction ( syn!tic & analytic judgm5ts is unt5able1 & fur!r1 t ! pr9ciple ( contradiction c only give fruitful results on ! assumption t exp]i5ce 9 g5]al1 or1 9 a p>ticul> sci5ce1 "s special bran* ( exp]i5ce1 is to be =mally possible2 7#b7 t ! fir/ two argum5ts ( ! ,transc5d5tal ,ae/hetic suffice to prove1 n ,euclidean space1 b "s =m ( ext]nality--:i* may be s5sational or intuitional1 b n m]ely conceptual--a necess>y pr]equisite ( exp]i5ce ( an ext]nal _w4 ,9 ! ?ird & f\r? *apt]s1 ,i %all cont5d1 z a result ( ^! conclusions1 t ^? axioms1 :i* ,euclid & ,metageometry h 9 common1 co9cide ) ^? prop]ties ( any =m ( ext]nality :i* are d$ucible1 by ! pr9ciple ( contradiction1 f ! possibility ( exp]i5ce ( an ext]nal _w4 ,^! prop]ties1 !n1 may be said1 ?\gh n q 9 ! ,kantian s5se1 to be priori prop]ties ( space1 & z to ^!1 ,i ?9k1 a modifi$ ,kantian position may be ma9ta9$4 ,b ! "q ( ! subjective or objective nature ( space may be left :olly out ( acc\nt dur+ ! c\rse ( ?is discussion1 :i* w ga9 by deal+ exclusively ) logical1 z oppos$ to psy*ological po9ts ( view4 #ef4 7#a7 ,kant's logical position4 ,! doctr9e ( syn!tic & analytic judgm5ts--at any rate if ?is is tak5 z ! corn]-/one ( ,epi/emology--has be5 s completely reject$ by mo/ mod]n logicians#ga1 t x w\ld dem& little att5tion "h1 b = ! fact t an en?usia/ic ,fr5* ,kantian1 ,m4 ,r5\vi]1 has rec5tly appeal$ to x1 ) p]fect confid5ce1 on ! v "q ( ,geometry#gb4 ,& x mu/ be own$1 ) ,m4 ,r5\vi]1 t if su* judgm5ts exi/$1 9 ! ,kantian s5se1 non-,euclidean ,geometry1 :i* makes no appeal to intuition1 c\ld h no?+ to say aga9/ !m4 ,m4 ,r5\vi]'s cont5tion1 !re=e1 =ces u briefly to review ! argum5ts aga9/ ,kant's doctr9e1 & briefly to discuss :at logical canon is to replace x4 ,e judgm5t--s mod]n logic cont5ds--is bo? syn!tic & analytic2 x comb9es p>ts into a :ole1 & analyses a :ole into p>ts#gc4 ,if ?is be s1 ! di/9ction ( analysis & syn!sis1 :atev] may be its importance 9 pure ,logic1 c h no value 9 ,epi/emology4 ,b su* a doctr9e1 x mu/ be obs]v$1 all[s full scope to ! pr9ciple ( contradiction3 ?is crit]ion1 s9ce all judgm5ts1 9 "o aspect at lea/1 are analytic1 is applicable to all judgm5ts alike4 ,on ! o!r h&1 ! :ole :i* is analys$ mu/ be suppos$ already giv51 be=e ! p>ts c be mutually contradictory3 = only by connection 9 a giv5 :ole c two p>ts or adjectives be incompatible4 ,?us ! pr9ciple ( contradiction rema9s b>r5 until we already h "s judgm5ts1 & ev5 "s inf]5ce3 = ! p>ts may be reg>d$1 to "s ext5t1 z an inf]5ce f ! :ole1 or vice v]s4 ,:5 once ! ar* ( k is con/ruct$1 ! p>ts support "o ano!r1 & ! pr9ciple ( contradiction is ! key/one3 b until ! ar* is built1 ! key/one rema9s susp5d$1 unsupport$ & unsupport+1 9 ! empty air4 ,9 o!r words1 k once exi/5t c be analys$1 b k :i* %\ld h to w9 e in* ( ! way aga9/ a critical scepticism1 c\ld nev] beg91 & c\ld nev] atta9 t circul> condition 9 :i* alone x c /&4 ,b ,kant's doctr9e1 if true1 is design$ to re/ra9 a critical scepticism ev5 :]e x might be effective4 ,c]ta9 fundam5tal propositions1 he says1 are n d$ucible f logic1 i4e4 _! contradictories are n self-contradictory2 !y comb9e a subject & pr$icate :i* _c1 9 any purely logical way1 be %ewn to h any connection1 & yet ^! judgm5ts h apodeictic c]ta9ty4 ,b conc]n+ su* judgm5ts1 ,kant is g5]ally c>eful n to rely ^u ! m]e subjective conviction t !y are und5iable3 he proves1 ) e precaution1 t )\t !m exp]i5ce w\ld be impossible4 ,exp]i5ce consi/s 9 ! comb9ation ( t]ms :i* =mal logic leaves ap>t1 & presupposes1 !re=e1 c]ta9 judgm5ts by :i* a framework is made = br++ su* t]ms toge!r4 ,)\t ^! judgm5ts--s ,kant cont5ds--all syn!sis & all exp]i5ce w\ld be impossible4 ,if1 !re=e1 ! detail ( ! ,kantian reason+ be s\nd1 8 results may be obta9$ by ! pr9ciple ( contradiction plus ! possibility ( exp]i5ce1 z well z by 8 di/9ction ( syn!tic & analytic judgm5ts4 ,logic1 at ! pres5t day1 arrogates to itself at once a wid] & a n>r[] sph]e ?an ,kant all[$ to x4 ,wid]1 because x believes itself capable ( condemn+ any false pr9ciple or po/ulate2 n>r[]1 because x believes t its law ( contradiction1 )\t a giv5 :ole or a giv5 hypo!sis1 is p[]less1 & t two t]ms1 p] se1 ?\gh !y may be diff]5t1 _c be contradictories1 b acquire ?is relation only by comb9ation 9 a :ole ab\t :i* some?+ is kn[n1 or by connection ) a po/ulate :i*1 = "s reason1 mu/ be pres]v$4 ,?us no judgm5t1 p] se1 is ei!r analytic or syn!tic1 = ! sev]ance ( a judgm5t f its context robs x ( its vitality1 & makes x n truly a judgm5t at all4 ,b 9 its prop] context x is nei!r purely syn!tic nor purely analytic2 = :ile x is ! fur!r det]m9ation ( a giv5 :ole1 & ?us 9 s f> analytic1 x also involves ! em]g5ce ( new relations )9 ?is :ole1 & is s f> syn!tic4 #eg4 ,we may reta91 h[ev]1 a di/9ction r\ghly correspond+ to ! ,kantian priori & po/]iori1 ?\gh less rigid1 & m liable to *ange ) ! degree ( organisation ( k4 ,kant usually endeav\r$ to prove1 z obs]v$ above1 t 8 syn!tic priori propositions 7 necess>y pr]equisites ( exp]i5ce2 n[ al?\gh we _c reta9 ! t]m syn!tic1 we c reta9 ! t]m priori1 = ^? assumptions1 or ^? po/ulates1 f :i* alone ! possibility ( exp]i5ce foll[s4 ,:atev] c be d$uc$ f ^! po/ulates1 )\t ! aid ( ! matt] ( exp]i5ce1 w also1 ( c\rse1 be priori4 ,f ! /&po9t ( g5]al logic1 ! laws ( ?\ght & ! categories1 ) ! indisp5sable conditions ( _! applicability1 w be alone priori2 b f ! /&po9t ( any special sci5ce1 we may call priori :atev] r5d]s possible ! exp]i5ce :i* =ms ! subject-matt] ( our sci5ce4 ,9 ,geometry1 to p>ticul>ize1 we may call priori :atev] r5d]s possible exp]i5ce ( ext]nality z su*4 ,x is to be obs]v$ t ?is use ( ! t]m is at once m rationali/ic & less precise ?an t ( ,kant4 ,kant w\ld seem to h suppos$ himself imm$iately aw>e1 by inspection1 t "s k 0 apodeictic1 & its subject-matt]1 !re=e1 priori3 b he did n always d$uce its apriority f any fur!r pr9ciple4 ,"h1 h[ev]1 x is to be %[n1 be=e admitt+ apriority1 t ! falsehood ( ! judgm5t 9 "q w\ld n be effect$ by a m]e *ange 9 ! matt] ( exp]i5ce1 b only by a *ange :i* %\ld r5d] "s bran* ( exp]i5ce =mally impossible1 i4e4 inaccessible to our me?ods ( cognition4 ,! above use is also less precise1 = x v>ies accord+ to ! specialization ( ! exp]i5ce we are assum+ possible1 & ) e progress ( k "s new connection is p]ceiv$1 two previ\sly isolat$ judgm5ts are br\ght into logical relation1 & ! priori may ?us1 at any mom5t1 enl>ge its sph]e1 z m is f\nd d$ucible f fundam5tal po/ulates4 #eh4 7#b7 ,kant's argum5ts = ! apriority ( space4 ,hav+ n[ discuss$ ! logical canon to be us$ z reg>ds ! priori1 we may proce$ to te/ ,kant's argum5ts z reg>ds space4 ,! argum5t f ,geometry1 z rem>k$ above1 is upset by ,metageometry1 at lea/ s f> z ^? prop]ties are conc]n$1 :i* belong to ,euclid b n to non-,euclidean spaces2 z reg>ds ! common prop]ties ( bo? k9ds ( space1 we _c decide on _! apriority till we h discuss$ ! consequ5ces ( d5y+ !m1 :i* w be done 9 ,*apt] ,,iii4 ,z reg>ds ! two argum5ts :i* prove t space is an intuition1 n a concept1 !y w\ld call = mu* discussion 9 a special criticism ( ,kant1 b "h !y may be pass$ by ) ! obvi\s comm5t t inf9ite homog5e\s ,euclidean space is a concept1 n an intuition--a concept inv5t$ to expla9 an intuition1 x is true1 b /ill a pure concept#gd4 ,& x is ?is pure concept :i*1 9 all discussions ( ,geometry1 is prim>ily to be dealt )2 ! intuition ne$ only be ref]r$ to :]e x ?r[s light on ! functions or ! nature ( ! concept4 ,! second ( ,kant's argum5ts1 t we c imag9e empty space1 ?\gh n ! abs5ce ( space1 is false if x means a space )\t matt] any:]e1 & irrelevant if x m]ely means a space betwe5 matt]s & reg>d$ z empty#ge4 ,! only argum5t ( importance1 !n1 is ! fir/ argum5t4 ,b ,i mu/ insi/1 at ! outset1 t our problem is purely logical1 & t all psy*ological implications mu/ be exclud$ to ! utmo/ possible ext5t4 ,moreov]1 z w be prov$ 9 ,*apt] ,,iv41 ! prop] function ( space is to di/+ui% betwe5 diff]5t pres5t$ ?+s1 n betwe5 ! ,self & ! object ( s5sation or p]ception4 ,! argum5t !n becomes ! foll[+3 consci\sness ( a _w ( mutually ext]nal ?+s dem&s1 9 pres5tations1 a cognitive b non-inf]5tial elem5t lead+ to ! discrim9ation ( ! objects pres5t$4 ,?is elem5t mu/ be non-inf]5tial1 = f :atev] numb] or comb9ation ( pres5tations1 :i* did n ( !mselves dem& div]sity 9 _! objects1 ,i c\ld nev] be l$ to inf] ! mutual ext]nality ( _! objects4 ,kant says3 ,9 ord] t s5sations may be ascrib$ to some?+ ext]nal to me 444 & simil>ly 9 ord] t ,i may be able to pres5t !m z outside & beside "o ano!r1 444 ! pres5tation ( space mu/ be already pres5t4 ,b ?is goes r too f>3 9 ! fir/ place1 ! "q %\ld be only z to ! mutual ext]nality ( pres5t$ ?+s1 n z to _! ext]nality to ! ,self#gf2 & 9 ! second place1 ?+s w appe> mutually ext]nal if ,i h ! pres5tation ( any =m ( ext]nality1 :e!r ,euclidean or non-,euclidean4 ,:atev] may be true ( ! psy*ological scope ( ?is argum5t--^: validity is "h irrelevant--! logical scope ext5ds1 n to ,euclidean space1 b only to any =m ( ext]nality :i* c\ld exi/ intuitively1 & p]mit k1 9 be+s ) our laws ( ?\ght1 ( a _w ( div]se b int]relat$ ?+s4 ,moreov] ext]nality1 to r5d] ! scope ( ! argum5t :olly logical1 mu/ n be left ) a s5sational or intuitional mean+1 ?\gh x mu/ be suppos$ giv5 9 s5sation or intuition4 ,x mu/ mean1 9 ?is argum5t1 ! fact ( ,o!rness#gg1 ! fact ( be+ diff]5t f "s o!r ?+3 x mu/ involve ! di/9ction betwe5 diff]5t ?+s1 & mu/ be t elem5t1 9 a cognitive /ate1 :i* leads u to discrim9ate con/itu5t p>ts 9 its object4 ,s mu*1 !n1 w\ld appe> to result f ,kant's argum5t1 t exp]i5ce ( div]se b int]relat$ ?+s dem&s1 z a necess>y pr]equisite1 "s s5sational or intuitional elem5t1 9 p]ception1 by :i* we are l$ to attribute complexity to objects ( p]ception#gh2 t ?is elem5t1 9 its isolation may be call$ a =m ( ext]nality2 & t ^? prop]ties ( ?is =m1 if any su* be f\nd1 :i* c be d$uc$ f its m]e function ( r5d]+ exp]i5ce ( int]relat$ div]sity possible1 are to be reg>d$ z priori4 ,:at ^! prop]ties are1 & h[ ! v>i\s l9es ( argum5t "h sugge/$ conv]ge to a s+le result1 we %all see 9 ,*apt]s ,,iii4 & ,,iv4 #ei4 ,9 ! philosoph]s :o foll[$ ,kant1 ,metaphysics1 = ! mo/ "p1 s pr$om9at$ ov] ,epi/emology1 t little 0 add$ to ! !ory ( ,geometry4 ,:at 0 add$1 came indirectly f ! "o philosoph] :o /ood out aga9/ ! purely ontological speculations ( 8 "t1 namely ,h]b>t4 ,h]b>t's actual views on ,geometry1 :i* are to be f\nd *iefly 9 ! fir/ section ( 8 ,syne*ologie1 are n ( any great value1 & h borne no great fruit 9 ! developm5t ( ! subject4 ,b 8 psy*ological !ory ( space1 8 con/ruction ( ext5sion out ( s]ies ( po9ts1 8 comp>ison ( space ) ! tone & col\r-s]ies1 8 g5]al pref]5ce = ! discrete above ! cont9u\s1 & f9ally 8 belief 9 ! great importance ( classify+ space ) o!r =ms ( s]ies 7,reih5=m5#gi71 gave rise to _m ( ,riemann's epo*-mak+ speculations1 & enc\rag$ ! attempt to expla9 ! nature ( space by its analytical & quantitative aspect alone#hj4 ,?r\gh 8 influ5ce on ,riemann1 he acquir$1 indirectly1 a great importance 9 geometrical philosophy4 ,to ,riemann's diss]tation1 :i* we h already discuss$ 9 its ma!matical aspect1 we mu/ n[ return1 consid]+1 ?is "t1 only its philosophical views4 ,riemann4 #fj4 ,! aim ( ,riemann's diss]tation1 z we saw 9 ,*apt] ,i41 0 to def9e space z a species ( manifold1 i4e4 z a p>ticul> k9d ( collection ( magnitudes4 ,x 0 ?us assum$1 to beg9 )1 t spatial figures c\ld be reg>d$ z magnitudes1 & ! axioms :i* em]g$1 accord+ly1 det]m9$ only ! p>ticul> place ( ^! among ! _m algebraically possible v>ieties ( magnitudes4 ,! result+ =mulation ( ! axioms--:ile1 f ! ma!matical /&po9t ( metrical ,geometry1 x 0 almo/ :olly laudable--mu/1 f ! /&po9t ( philosophy1 be reg>d$1 9 my op9ion1 z a petitio pr9cipii4 ,= :5 we h arriv$ at reg>d+ spatial figures z magnitudes1 we h already trav]s$ ! mo/ difficult "p ( ! gr\nd4 ,! axioms ( metrical ,geometry--& x is metrical ,geometry1 exclusively1 :i* is consid]$ 9 ,riemann's ,essay--w appe>1 9 ,*apt] ,,iii41 to be divisible into two classes4 ,( ^!1 ! fir/ class--:i* conta9s ! axioms common to ,euclid & ,metageometry1 ! only axioms s]i\sly discuss$ by ,riemann--are n ! results ( measurem5t1 nor ( any conception ( magnitude1 b are conditions to be fulfill$ be=e measurem5t becomes possible4 ,! second class only--^? :i* express ! diff]5ce betwe5 ,euclidean & non-,euclidean spaces--c be d$uc$ z results ( measurem5t or ( conceptions ( magnitude4 ,z reg>ds ! fir/ class1 on ! contr>y1 we %all see t ! relativity ( position--by :i* space is di/+ui%$ f all o!r kn[n manifolds1 except "t--leads logically to ! necessity ( ?ree ( ! mo/ di/9ctive axioms ( ,geometry1 & yet ?is relativity _c be call$ a d$uction f conceptions ( magnitude4 ,9 analytical ,geometry1 ow+ to ! fact t coord9ate sy/ems />t f po9ts1 & h5ce build up l9es & surfaces1 x is easy to suppose t po9ts c be giv5 indep5d5tly ( l9es & ( ea* o!r1 & ?us ! relativity ( position is lo/ sight (4 ,! error ?us sugge/$ by ma!matics 0 probably re9=c$ by ,h]b>t's !ory ( space1 :i*1 by its s]ial "*1 z we h se51 appe>$ to him to facilitate a con/ruction out ( successive po9ts1 & to :i* ,riemann ackn[l$ges 8 indebt$ness bo? 9 8 ,diss]tation & else:]e4 ,! same error reappe>s 9 ,helmholtz1 9 :om x is probably due :olly to ! me?ods ( analytical ,geometry4 ,x is a /rik+ fact t1 ?r\gh\t ! writ+s ( ^! two m51 !re is n1 s f> z ,i "k1 "o allusion to ! relativity ( position1 t prop]ty ( space f :i*1 z our next *apt] w %ew1 ! ri*e/ qu>ry ( consequ5ces c be extract$4 ,?is is n a result ( any conception ( magnitude1 b foll[s f ! nature ( our space-intuition2 yet no "o1 surely1 c\ld call x empirical1 s9ce x is b\nd up 9 ! v possibility ( locat+ ?+s !re z oppos$ to "h4 #fa4 ,inde$ we c see1 f a purely logical consid]ation ( ! judgm5t ( quantity1 t ,riemann's mann] ( approa*+ ! problem c nev]1 by legitimate me?ods1 atta9 to a philosophically s\nd =mulation ( ! axioms4 ,= quantity is a result ( comp>ison ( two qualitatively simil> objects1 & ! judgm5t ( quantity neglects altoge!r ! qualitative aspect ( ! objects comp>$4 ,h5ce a k ( ! ess5tial prop]ties ( space c nev] be obta9$ f judgm5ts ( quantity1 :i* neglect ^! prop]ties1 :ile !y yet presuppose !m4 ,z well might "o hope to le>n ! nature ( man f a c5sus4 ,moreov]1 ! judgm5t ( quantity is ! result ( comp>ison1 & !re=e presupposes ! possibility ( comp>ison4 ,to "k :e!r1 or by :at means1 comp>ison is possible1 we mu/ "k ! qualities ( ! ?+s comp>$ & ( ! m$ium 9 :i* comp>ison is effect$2 :ile to "k t quantitative comp>ison is possible1 we mu/ "k t !re is a qualitative id5tity betwe5 ! ?+s comp>$1 :i* aga9 involves a previ\s qualitative k4 ,:5 spatial figures h once be5 r$uc$ to quantity1 _! quality has already be5 neglect$1 z kn[n & simil> to ! quality ( o!r figures4 ,to hope1 !re=e1 = ! qualities ( space1 f a comp>ison ( its expression z pure quantity ) o!r pure quantities1 is an error natural to an analytical geomet]1 b an error1 none ! less1 f :i* !re is no return to ! qualitative basis ( spatial quantity4 #fb4 ,we mu/ entirely diss5t1 !re=e1 f ! disjunction :i* und]lies ,riemann's philosophy ( space4 ,ei!r ! axioms mu/ be consequ5ces ( g5]al conceptions ( magnitude1 he ?9ks1 or else !y c only be prov$ by exp]i5ce 7p4 #bee74 ,:atev] c be d]iv$ f g5]al conceptions ( magnitude1 we may retort1 _c be an priori adjective ( space3 = all ! necess>y adjectives ( space are presuppos$ 9 any judgm5t ( spatial quantity1 & _c1 !re=e1 be consequ5ces ( su* a judgm5t4 ,riemann's disjunction1 accord+ly1 s9ce "o ( its alt]natives is obvi\sly impossible1 really begs ! "q4 ,9 =mulat+ ! axioms ( metrical ,geometry1 our "q %\ld be3 ,:at axioms1 i4e4 :at adjectives ( space1 mu/ be presuppos$1 9 ord] t quantitative comp>ison ( ! p>ts ( space may be possible at all8 ,& only :5 we h det]m9$ ^! conditions1 :i* are priori necess>y to any quantitative sci5ce ( space1 does ! second "q arise3 :at inf]5ces c we draw1 z to space1 f ! obs]v$ results ( ?is quantitative sci5ce1 i4e4 ( ?is measurem5t ( spatial figures8 ,! conditions ( measurem5t !mselves1 ?\gh n results ( any conception ( magnitude1 w be priori1 if x c be %[n t1 )\t !m1 exp]i5ce ( ext]nality w\ld be impossible4 ,aft] ?is initial prote/ aga9/ ,riemann's g5]al philosophical position1 let u proce$ to exam9e1 9 detail1 8 use ( ! notion ( a manifold4 #fc4 ,9 ! fir/ place !re is1 if ,i am n mi/ak51 consid]able obscurity 9 ! def9ition ( a manifold1 ( :i* an almo/ v]bal r5d]+ 0 giv5 9 ,*apt] ,i4 ,:at is meant1 to beg9 )1 by a g5]al conception capable ( v>i\s det]m9ations8 ,does n ?is prop]ty belong to all conceptions8 ,x af=ds1 c]ta9ly1 a basis = c\nt+1 b if cont9u\s quantity is to arise1 we mu/1 surely1 h "s less discrete =mulation4 ,x might af=d a basis1 = example1 = ! di/9ction ( po9ts 9 projective ,geometry1 b projective ,geometry has no?+ to d ) quantity4 ,some?+ m fluid & flexible ?an a conception1 "o w\ld ?9k1 is necess>y z ! basis ( cont9ua4 ,!n1 aga91 :at is meant by a quantum ( a manifold8 ,9 space1 ! answ] is obvi\s3 :at is meant is a piece ( volume4 ,b h[ ab\t ,riemann's o!r cont9u\s manifold1 col\r8 ,does a quantum ( col\r mean a s+le l9e 9 ! spectrum1 or a b& ( f9ite ?ickness8 ,9 ei!r case1 :at are ! magnitudes to be comp>$8 ,& h[ is sup]position necess>y1 or ev5 possible8 ,a col\r is fix$ by its position 9 ! spectrum3 two l9es 9 ! same spectrum _c be sup]pos$1 & two l9es 9 diff]5t spectra ne$ n be--_! positions 9 _! respective spectra suffice1 or ev51 r\ghly1 _! imm$iate s5se-quality4 ,! fact is1 ,riemann _h space 9 8 m9d f ! />t1 & _m ( ! prop]ties1 :i* he enunciates z belong+ to all manifolds1 belong1 z a matt] ( fact1 only to space4 ,x is f> f cle> :at ! magnitudes are :i* ! v>i\s det]m9ations make possible4 ,d ^! magnitudes measure ! elem5ts ( ! manifold1 or ! relations betwe5 elem5ts8 ,?is is surely a v fundam5tal po9t1 b x is "o :i* ,riemann nev] t\*es on4 ,9 ! =m] case1 ! sup]position :i* he speaks ( becomes unnecess>y1 s9ce ! magnitude is inh]5t 9 ! elem5t consid]$4 ,we d n require sup]position to measure quantities correspond+ to diff]5t tones or col\rs2 ^! c be discov]$ by analysis ( s+le tones or col\rs4 ,) space1 on ! o!r h&1 if we seek = elem5ts1 we c f9d none except po9ts1 & no analysis ( a po9t w f9d magnitudes inh]5t 9 x--su* magnitudes are a fiction ( coord9ate ,geometry4 ,! magnitudes :i* space deals )1 z we %all see 9 ,*apt] ,,iii41 are relations betwe5 po9ts1 & x is = ?is reason t sup]position is ess5tial to space-measurem5t4 ,!re is no inh]5t quality 9 a s+le po9t1 z !re is 9 a s+le col\r1 by :i* x c be quantitatively di/+ui%$ f ano!r4 ,?us ! conception ( a manifold1 z def9$ by ,riemann1 ei!r does n include col\rs1 or does n involve sup]position z ! only means ( measurem5t4 ,f ?is dilemma !re is no escape4 #fd4 ,b if measurem5t consi/s 9 a sup]position ( ! magnitudes comp>$ 7p4 #bef71 does x n foll[ imm$iately t measurem5t is logically possible only :]e su* sup]position leaves ! magnitudes un*ang$8 ,& !re=e t measurem5t1 z above def9$1 involves1 z an priori condition1 t magnitudes are un*ang$ by motion8 ,?is consequ5ce is n drawn by ,riemann2 inde$ he proce$s imm$iately 7pp4 #bef-g7 to consid] :at he calls a g5]al portion ( ! doctr9e ( magnitude 7,grss5lehre71 indep5d5t ( measurem5t4 ,b h[ is any doctr9e ( magnitude possible1 9 :i* ! magnitudes _c be measur$8 ,! reason ( ! confusion is1 t ,riemann's def9ition ( measurem5t is applicable to no s+le manifold except space1 s9ce x dep5ds on ! notewor?y prop]ty t :at we measure 9 ,geometry is n po9ts1 b relations betwe5 po9ts1 & ! latt]1 ?\gh n ! =m]1 may ( c\rse be unalt]$ by motion4 ,let u try1 9 illu/ration1 to apply ,riemann's def9ition ( measurem5t to col\rs4 ,we mu/ rememb] t motion1 9 deal+ ) ! col\r manifold1 means--n motion 9 space b--motion 9 ! col\r manifold itself4 ,n[ s9ce e po9t ( ! col\r manifold is completely det]m9$ by ?ree magnitudes1 :i* are giv5 9 fact1 & _c be arbitr>ily *os51 x is pla9 t measurem5t by sup]position--involv+1 z x does1 motion1 & !re=e *ange 9 ^! det]m9+ magnitudes--is totally out ( ! "q4 ,! sup]position ( "o col\r on ano!r1 z a means ( measurem5t1 is %e] nons5se4 ,& yet measurem5t is possible 9 ! col\r-manifold1 by means ( ,helmholtz's law ( mixture 7,mis*ungsgesetz72 b ! measurem5t is ( e sep>ate elem5t1 n ( ! relations betwe5 elem5ts1 & is ?us radically diff]5t f space-measurem5t#ha4 ,! elem5ts are n1 l po9ts 9 space1 qualitatively alike1 & di/+ui%$ by ! m]e fact ( _! mutual ext]nality4 ,:at we h1 9 col\rs1 is ?ree fundam5tal qualitatively di/9ct elem5ts1 out ( c]ta9 proportions ( :i* we c build up all ! o!r elem5ts ( ! manifold--ea* ( ! result+ elem5ts hav+ ! same comb9ation ( qualitative div]sity & simil>ity z ! ?ree orig9al elem5ts4 ,b 9 space1 :at c\ld we make ( su* a proc$ure8 ,giv5 ?ree po9ts1 h[ are we to comb9e !m 9 c]ta9 proportions8 ,! phrase is mean+less4 ,if "s "o makes ! obvi\s retort1 t we h to comb9e l9es1 n po9ts1 my rejo9d] is equally obvi\s4 ,to beg9 )1 l9es are n elem5ts4 ,metaphysically1 space has no elem5ts1 be+1 z ! sequel w %[1 m]e relations betwe5 non-spatial elem5ts4 ,ma!matically1 ?is fact exhibits itself 9 ! self-contradictory notion ( ! po9t1 or z]o magnitude 9 space1 z ! limit 9 our va9 se>* = spatial elem5ts4 ,b ev5 if we all[ ! l9e to pass z ! spatial elem5t1 :at does ! comb9ation ( ?ree l9es 9 def9ite proportions give u8 ,x gives u1 simply1 ! coord9ates ( a po9t4 ,"h aga9 we see a great diff]5ce betwe5 ! col\r & space-manifolds4 ,9 col\rs1 ! comb9ation ( magnitudes gives a new magnitude ( ! same k9d2 9 space1 x def9es1 n a magnitude at all1 b a w\ld-be elem5t ( a diff]5t k9d f ! def9+ magnitudes4 ,9 ! tone-manifold1 we %\ld f9d /ill diff]5t conditions4 ,"h1 no "o ( ! measur+ magnitudes c vani% )\t ! tone vani%+ too1 & all ?ree are s b\nd up toge!r1 9 ! s+le result+ s5sation1 t none c exi/ )\t a f9ite quantity ( ! o!rs4 ,!y are all qualitatively diff]5t1 bo? f ea* o!r1 & f any possible tone1 be+ con/itu5ts ( x1 z mass & velocity are con/itu5ts ( mom5tum4 ,all ^! diff]5t conditions require to be exam9$1 be=e a manifold c be completely def9$2 & until we h conduct$ su* an exam9ation 9 detail1 we _c pron\nce z to ! priori or empirical nature ( ! laws ( ! manifold4 ,z reg>ds space1 ,i h attempt$ su* an exam9ation 9 ! ?ird & f\r? *apt]s ( ?is ,essay4 #fe4 ,i d n wi% to d5y1 h[ev]1 ! great value ( ! conception ( space z a manifold4 ,on ! contr>y1 ?is conception seems to h become ess5tial to any treatm5t ( ! "q4 ,i only wi% to urge t ! purely algebraical treatm5t ( any manifold1 important z x may be 9 d$uc+ fre% consequ5ces f kn[n premisses1 t5ds r to conceal ?an to make cle> ! basis ( ! premisses !mselves1 & is !re=e mislead+ 9 a philosophical inve/igation4 ,= ma!matics1 :]e quantity reigns supreme1 ,riemann's conception has prov$ itself abundantly fruitful2 = philosophy1 on ! contr>y1 :]e quantity appe>s r z a cloak to conceal ! qualities x ab/racts f1 ! conception seems to me m productive ( error & confusion ?an ( s\nd doctr9e4 ,we are ?us br\ght back to ! po9t f :i* we />t$1 namely1 ! falsity ( ,riemann's initial disjunction1 & ! consequ5t fallacy 9 8 pro( ( ! empirical nature ( ! axioms4 ,8 philosophy is *iefly vitiat$1 to my m9d1 by ?is fallacy1 & by ! uncritical assumption t a metrical coord9ate sy/em c be set up indep5d5tly ( any axioms z to space-measurem5t#hb4 ,riemann has fail$ to obs]ve1 :at ,i h endeav\r$ to prove 9 ! next *apt]1 t1 unless space _h a /rictly con/ant measure ( curvature1 ,geometry w\ld become impossible2 also t ! abs5ce ( con/ant measure ( curvature involves absolute position1 :i* is an absurdity4 ,h5ce he is l$ to ! conclusion t all geometrical axioms are empirical1 & may n hold 9 ! inf9itesimal1 :]e obs]vation is impossible4 ,?us he says 7p4 #bfg73 ,n[ ! empirical conceptions1 on :i* spatial measurem5ts are bas$1 ! conceptions ( ! rigid body & ! light-ray1 appe> to lose _! validity 9 ! inf9itesimal3 x is !re=e q conceivable t ! relations ( spatial magnitudes 9 ! inf9itesimal d n correspond to ! presuppositions ( ,geometry1 & ?is w\ld1 9 fact1 h to be assum$1 z soon z x w\ld enable u to expla9 ! ph5om5a m simply4 ,f ?is conclusion ,i mu/ entirely diss5t4 ,9 v l>ge spaces1 !re might be a dep>ture f ,euclid2 = !y dep5d ^u ! axiom ( p>allels1 :i* is n conta9$ 9 ! axiom ( ,free ,mobility2 b 9 ! inf9itesimal1 dep>tures f ,euclid c\ld only be due to ! abs5ce ( ,free ,mobility1 :i*1 z ,i hope my ?ird *apt] w %[1 is once = all impossible4 ,helmholtz4 #ff4 ,helmholtz1 l ,riemann1 0 important bo? 9 ! ma!matics & 9 ! philosophy ( ,geometry4 ,f ! ma!matical po9t ( view1 8 "w has be5 already consid]$ 9 ,*apt] ,i42 ! consid]ation ( 8 philosophy1 :i* mu/ occupy u "h1 w be a m s]i\s task4 ,l ,riemann1 he endeav\r$ to prove t all ! axioms are empirical1 & l ,riemann1 he bas$ 8 pro( *iefly on ,metageometry4 ,he _h an additional res\rce1 h[ev]1 9 ! physiology ( ! s5ses1 :i* fir/ l$ him to reject ! ,transc5d5tal ,ae/hetic1 & enabl$ him to attack ,kant f ! psy*ological z well z ! ma!matical side#hc4 ,! pr9cipal topics1 = a criticism ( ,helmholtz1 are ?ree3 ,fir/1 8 crit]ion ( ! priori2 second1 8 discussion ) ,l& z to ! imag9ability ( non-,euclidean spaces2 ?ird--& ?is is by f> ! mo/ important ( ! ?ree--8 !ory ( ! dep5d5ce ( ,geometry on ,me*anics4 ,let u discuss ^! ?ree po9ts successively4 #fg4 ,helmholtz's crit]ion ( apriority is difficult to discov]1 z he nev]1 to my k1 gives a precise /atem5t ( x4 ,f 8 discussion ( physical & transc5d5tal ,geometry#hd1 h[ev]1 x w\ld appe> t he reg>ds z empirical :atev] applies to empirical matt]4 ,= he !re ma9ta9s1 t ev5 if space 7 an priori =m1 yet any ,geometry1 :i* aim$ at an application to ,physics1 w\ld1 s9ce ! actual places ( bodies are n kn[n priori1 be necess>ily empirical#he4 ,x seems ! m probable t he reg>ds ?is z a possible crit]ion1 z x is adopt$1 9 sev]al passages1 by 8 disciple ,erdmann#hf1 & s /range a te/ c\ld h>dly be accept$ by a philosoph]1 unless he _h f\nd x 9 8 ma/]4 ,i h call$ ?is a /range te/1 because x seems to me completely to ignore ! "w ( ! ,critical ,philosophy4 ,= if !re is "o ?+ :i*1 "o might h hop$1 _h be5 made suffici5tly cle> by ,kant's ,critique1 x is ?is1 t k :i* is priori1 be+ itself ! condition ( possible exp]i5ce1 applies--& 9 ,kant's view1 applies only--to empirical matt]4 ,helmholtz & ,erdmann1 !re=e1 9 sett+ up ?is te/ )\t discussion1 simply ignore ! exi/5ce ( ,kant & ! possibility ( a transc5d5tal argum5t4 ,helmholtz assumes always t empirical k mu/ be :olly empirical1 t !re c be no priori conditions ( ! exp]i5ce 9 "q1 t exp]i5ce w always be possible1 & may give any k9d ( result4 ,?us 9 discuss+ physical ,geometry1 he assumes t ! possibility ( empirical measurem5t involves no priori axioms1 & t no priori elem5t c be conta9$ 9 ! process4 ,?is assumption1 z we %all see 9 ,*apt] ,,iii41 is q unw>rantable3 c]ta9 prop]ties ( space1 9 fact1 are involv$ 9 ! possibility ( measur+ matt]4 ,9 spite ( ! fact1 !re=e1 t we apply measurem5t to empirical matt]1 & t our results are !re=e empirical1 !re may well be an priori elem5t 9 measurem5t1 :i* is presuppos$ 9 its possibility4 ,su* a crit]ion1 !re=e1 mu/ pron\nce ev]y?+ empirical1 b mu/ itself be pron\nc$ wor?less4 ,ano!r & a bett] crit]ion1 x is true1 is also to be f\nd 9 ,helmholtz1 & has also be5 adopt$ by ,erdmann4 ,:atev] might1 by a diff]5t exp]i5ce1 h be5 r5d]$ diff]5t--s ?is crit]ion cont5ds--mu/ itself be dep5d5t on exp]i5ce1 & s empirical4 ,?is crit]ion seems p]fectly s\nd1 b ,helmholtz's use ( x is usually vitiat$ by 8 neglect+ to prove ! possibility ( ! diff]5t exp]i5ce 9 "q4 ,he says1 = example1 t if our exp]i5ce %[$ u only bodies :i* *ang$ _! %apes 9 motion1 we %\ld n arrive at ! axiom ( ,congru5ce1 :i* he pron\nces accord+ly to be empirical4 ,b ,i %all endeav\r to prove1 9 ,*apt] ,,iii41 t )\t ! axiom ( ,congru5ce1 exp]i5ce ( spatial magnitude w\ld be impossible4 ,if my pro( be correct1 x foll[s t no exp]i5ce c ev] reveal spatial magnitudes :i* contradict ?is axiom--a possibility :i* ,helmholtz n[h]e discusses1 9 sett+ up 8 hypo!tical exp]i5ce4 ,?us ?is second crit]ion1 ?\gh p]fectly s\nd1 requires always an accompany+ transc5d5tal argum5t1 z to ! conditions ( possible exp]i5ce4 ,b ?is accompanim5t is seldom to be f\nd 9 ,helmholtz4 #fh4 ,"o ( ! few cases1 9 :i* ,helmholtz has attempt$ su* an accompanim5t1 occurs 9 connection ) our second po9t1 ! imag9ability ( non-,euclidean spaces4 ,! argum5t on ?is po9t 0 elicit$ by ,helmholtz's ,kantian oppon5ts1 :o ma9ta9$ t ! m]ely logical possibility ( ^! spaces 0 irrelevant1 s9ce ! basis ( ,geometry 0 n logic1 b intuition4 ,! axioms1 !y said1 are syn!tic propositions1 & _! contr>ies are1 !re=e1 n self-contradictory2 !y are nev]!less apodeictic propositions1 s9ce no o!r intuition ?an ! ,euclidean is possible to u#hg4 ,i h already criticiz$ ?is l9e ( argum5t 9 ! beg9n+ ( ! pres5t *apt]4 ,helmholtz's criticism1 h[ev]1 0 diff]5t3 admitt+ ! int]nal consi/5cy ( ! argum5t1 he d5i$ "o ( its premisses4 ,we c imag9e non-,euclidean spaces1 he said1 ?\gh _! unfamili>ity makes ?is difficult4 ,f ?is view x foll[$1 ( c\rse1 t ,kant's argum5t1 ev5 if x 7 =mally valid1 c\ld n prove ! apriority ( ,euclidean space 9 p>ticul>1 b only ( t g5]al space :i* includ$ ,euclid & non-,euclid alike#hh4 ,al?\gh ,i agree ) ,helmholtz 9 ?9k+ ! di/9ction betwe5 ,euclidean & non-,euclidean spaces empirical1 ,i _c ?9k 8 argum5t on ! imag9ability ( ! latt] a v happy "o4 ,! validity ( any pro( mu/ turn1 obvi\sly1 on ! def9ition ( imag9ability4 ,! def9ition :i* ,helmholtz gives 9 8 answ] to ,l& is z foll[s3 ,imag9ability requires die voll/ndige ,vor/ellb>keit d]j5ig5 ,s9nese9drcke1 wel*e das betreff5de ,object 9 uns na* d5 bekannt5 ,gesetz5 uns]] ,s9nesorgane unt] all5 d5kb>5 ,b$+ung5 d] ,beoba*tung erreg51 und wodur* es si* von &]5 hnli*5 ,object5 unt]s*eid5 wrde 7,wiss4 ,abh4 ,,ii4 p4 #fdd74 ,?is def9ition is n v cle>1 ow+ to ! ambiguity ( ! ^w ,vor/ellb>keit4 ,! foll[+ def9ition seems less ambigu\s3 ,w5n die ,reihe d] ,s9nese9drcke voll/ndig und e9deutig angegeb5 w]d5 kann1 muss man m4 ,e4 die ,sa*e fr ans*auli* vor/ellb> erklr5 7,vortrge und ,r$51 ,,ii4 p4 #bcd74 ,?is makes cle>1 :at also appe>s f 8 mann] ( pro(1 t he reg>ds ?+s z imag9able :i* c be describ$ 9 conceptual t]ms4 ,su*1 z ,l& rem>ks 7,m9d1 ,vol4 ,,ii4 p4 #de71 is n ! s5se requir$ = argum5tation 9 ?is case4 ,t ,l&'s criticism is j1 is %[n by ,helmholtz's pro( = non-,euclidean spaces1 = x consi/s only 9 an analogy to ! volume inside a sph]e1 :i* is ma!matically obta9$ ?us3 ,we take ! symbols repres5t+ magnitudes 9 pseudo-sph]ical 7hyp]bolic7 space1 & give !m a new ,euclidean mean+2 ?us all our symbolic propositions become capable ( two int]pretations1 "o = pseudo-sph]ical space1 & "o = ! volume inside a sph]e4 ,x is1 h[ev]1 suffici5tly obvi\s t ?is proc$ure1 ?\gh x enables u to describe our new space1 does n enable u to imag9e x1 9 ! s5se ( call+ up images ( ! way ?+s w\ld look 9 x4 ,we really d]ive1 f ?is analogy1 no m k ?an a man born bl9d may d]ive1 z to light1 f an analogy ) heat4 ,! dictum ,nihil e/ 9 intellectu quod non fu]it ante 9 s5su1 w\ld unque/ionably be true1 if = intellect we 7 to sub/itute imag9ation2 x is va91 !re=e1 if our actual space be ,euclidean1 to hope = a p[] ( imag9+ a non-,euclidean space4 ,:at ,helmholtz might1 ,i believe ) p]fect tru?1 h urg$ aga9/ ,l&1 is t ! image we actually h ( space is n suffici5tly accurate to exclude1 9 ! actual space we "k1 all possibility ( a slight dep>ture f ! ,euclidean type4 ,b 9 ma9ta9+ t we _c imag9e1 ?\gh we c conceive & describe1 a space diff]5t f t we actually h1 ,l& is1 9 my op9ion1 unque/ionably 9 ! "r4 ,= a pure ,kantian1 :o ma9ta9s1 ) ,l&1 t none ( ! axioms c be prov$1 ?is "q is ( great importance4 ,b if1 z ,i h ma9ta9$1 "s ( ! axioms are susceptible ( a transc5d5tal pro(1 :ile ! o!rs c be v]ifi$ empirically1 ! "q is fre$ f psy*ological implications1 & ! imag9ability or non-imag9ability ( metageometrical spaces becomes unimportant4 #fi4 ,we come n[ to ! ?ird & mo/ important "q1 ! relation ( ,geometry to ,me*anics4 ,!re are ?ree s5ses 9 :i* ,helmholtz's appeal to rigid bodies may be tak53 ! fir/1 ,i ?9k1 is ! s5se 9 :i* he orig9ally int5d$ x2 ! second seems to be ! s5se :i* he adopt$ 9 8 def5ce aga9/ ,l&2 :ile ! ?ird is admitt$ by ,l&1 & w be admitt$ 9 ! foll[+ argum5t4 ,^! ?ree s5ses are z foll[s3 7#a7 ,x may be ass]t$ t ! actual mean+ ( ! axiom ( ,free ,mobility lies 9 ! ass]tion ( empirical rigid bodies1 & t ! two propositions are equival5t to "o ano!r4 ,?is is c]ta9ly false4 7#b7 ,! axiom ( ,free ,mobility1 x may be said1 is logically di/+ui%able f ! ass]tion ( rigid bodies1 & may ev5 be n empirical2 b x is b>r51 ev5 = pure ,geometry1 )\t ! aid ( measures1 :i* mu/ !mselves be empirical rigid bodies4 ,?is s5se is m plausible ?an ! fir/1 b ,i believe we c %[ t1 9 ?is s5se also1 ! proposition is false4 7#c7 ,= pure ,geometry & ! ab/ract /udy ( space1 x may be said1 ,free ,mobility1 z appli$ to an ab/ract geometrical matt]1 gives a suffici5t possibility ( quantitative comp>ison2 b ! mom5t we ext5d our results to mix$ ma!matics1 & apply !m to empirically giv5 matt]1 we require also1 z measures1 empirically giv5 rigid bodies1 or bodies1 at lea/1 ^: dep>tures f rigidity are empirically kn[n4 ,9 ?is s5se1 ,i admit1 ! proposition is correct#hi4 ,9 discuss+ ^! ?ree mean+s1 ,i %all n conf9e myself /rictly to ! text ( ,helmholtz or ,l&3 if ,i endeav\r$ to d s1 ,i %\ld be met by ! difficulty t nei!r ( !m def9es ! priori1 & t ea* is too mu* incl9$1 9 my op9ion1 to te/ x by psy*ological crit]ia4 ,i %all1 !re=e1 take ! ?ree mean+s 9 turn1 )\t lay+ /ress on _! hi/orical adequacy to ! views ( ,l& or ,helmholtz4 #gj4 7#a7 ,congru5ce may be tak5 to mean--z ,helmholtz w\ld c]ta9ly seem to desire--t we f9d actual bodies1 9 our me*anical exp]i5ce1 to pres]ve _! %apes ) approximate con/ancy1 & t we inf]1 f ?is exp]i5ce1 ! homog5eity ( space4 ,?is view1 9 my op9ion1 radically misconceives ! nature ( measurem5t1 & ( ! axioms involv$ 9 x4 ,= :at is meant by ! non-rigidity ( a body8 ,we mean1 simply1 t x has *ang$ its %ape4 ,b ?is involves ! possibility ( comp>ison ) its =m] %ape1 9 o!r words1 ( measurem5t4 ,9 ord]1 !re=e1 t !re may be any "q ( rigidity or non-rigidity1 ! measurem5t ( spatial magnitudes mu/ be already possible4 ,x foll[s t measurem5t _c1 )\t a vici\s circle1 be itself d]iv$ f exp]i5ce ( rigid bodies4 ,geometrical measurem5t1 9 fact1 is ! comp>ison ( spatial magnitudes1 & su* comp>ison involves1 z w be prov$ at l5g? 9 ,*apt] ,,iii41 ! homog5eity ( space4 ,?is is1 !re=e1 ! logical pr]equisite ( all exp]i5ce ( rigid bodies1 & _c be ! result ( su* exp]i5ce4 ,)\t ! homog5eity ( space1 ! v notion ( rigidity or non-rigidity c\ld n exi/1 s9ce ^! mean1 respectively1 ! con/ancy or incon/ancy ( spatial magnitude 9 pieces ( matt]1 & bo? alike1 !re=e1 presuppose ! possibility ( spatial measurem5t4 ,f ! homog5eity ( space1 we le>n t a body1 :5 x moves1 w n *ange its %ape )\t "s physical cause2 t x actually does n *ange its %ape1 is nev] ass]t$1 & is inde$ kn[n to be false4 ,z soon z measurem5t is possible1 actual *anges ( %ape c be e/imat$1 & _! empirical causes c be s\ght4 ,b if space 7 n homog5e\s1 measurem5t w\ld be impossible1 con/ant %ape w\ld be a mean+less phrase1 & rigidity c\ld nev] be exp]i5c$4 ,congru5ce ass]ts1 9 %ort1 t a body c1 s f> z m]e space is conc]n$1 move )\t *ange ( %ape2 rigidity ass]ts t x actually does s move--a v diff]5t proposition1 involv+ obvi\sly1 z its logical prius1 ! =m] geometrical proposition4 ,?is argum5t may be summ$ up by ! foll[+ disjunction3 ,if bodies *ange _! %apes 9 motion--& to "s ext5t1 s9ce no body is p]fectly rigid1 !y mu/ all d s--!n "o ( two cases mu/ occur4 ,ei!r ! *anges ( %ape1 z bodies move f place to place1 foll[ no geometrical law1 are n1 = in/ance1 functions ( ! am\nt or direction ( motion2 9 :i* case ! law ( causation requires t !y %\ld n be effects ( ! *ange ( place1 b ( "s simultane\s non-geometrical *ange1 su* z temp]ature4 ,or ! *anges are regul>1 & ! %ape ,s becomes1 9 a new position p1 ,sf7p74 ,9 ?is case1 ! law ( concomitant v>iations leads u to attribute ! *ange ( %ape to ! m]e motion1 & %ape ?us becomes a function ( absolute position4 ,b ?is is absurd1 = position means m]ely a relation or set ( relations2 x is impossible1 !re=e1 t m]e position %\ld be able to effect *anges 9 a body4 ,position is "o t]m 9 a relation1 n a ?+ p] se2 x _c1 !re=e1 act on a ?+1 nor exi/ by itself1 ap>t f ! o!r t]ms ( ! relation4 ,?us ,helmholtz's view1 t ,congru5ce dep5ds on ! exi/5ce ( rigid bodies1 mu/1 s9ce x involves absolute position1 be condemn$ z a logical fallacy4 ,congru5ce1 9 fact1 z ,i %all prove m fully 9 ,*apt] ,,iii41 is an priori d$uction f ! relativity ( position4 #ga4 7#b7 ,! above argum5t seems to me to answ] satisfactorily ,helmholtz's cont5tion 9 ! precise =m :i* he fir/ gave x4 ,! axiom ( ,congru5ce1 we mu/ agree1 is logically di/+ui%able f ! exi/5ce ( rigid bodies4 ,nev]!less "s ref]5ce to matt] is logically involv$ 9 ,geometry#ij1 b :e!r ?is ref]5ce makes ,geometry empirical1 or does n1 r1 %[ an priori elem5t 9 dynamics1 is a fur!r "q4 ,! ref]5ce to matt] is necessitat$ by ! homog5eity ( empty space4 ,= s long z we leave matt] out ( acc\nt1 "o position is p]fectly indi/+ui%able f ano!r1 & a sci5ce ( ! relations ( positions is impossible4 ,inde$1 be=e spatial relations c arise at all1 ! homog5eity ( empty space mu/ be de/roy$1 & ?is de/ruction mu/ be effect$ by matt]4 ,! blank page is useless to ! geomet] until he defaces its homog5eity by l9es 9 ink or p5cil4 ,no spatial figures1 9 %ort1 are conceivable1 )\t a ref]5ce to a n purely spatial matt]4 ,aga91 if ,congru5ce is ev] to be us$1 !re mu/ be motion3 b a purely geometrical po9t1 be+ def9$ solely by its spatial attributes1 _c be suppos$ to move )\t a contradiction 9 t]ms4 ,:at moves1 !re=e1 mu/ be matt]4 ,h5ce1 9 ord] t motion may af=d a te/ ( equality1 we mu/ h "s matt] :i* is kn[n to be unaffect$ ?r\gh\t ! motion1 t is1 we mu/ h "s rigid bodies4 ,& ! difficulty is1 t ^! bodies mu/ n only und]go no *ange due solely to ! nature ( space1 b mu/1 fur!r1 be un*ang$ by _! *ang+ relation to o!r bodies4 ,& "h we h a requisite :i* c no long] be fulfill$ priori3 :i*1 inde$1 we "k to be1 9 /rictness1 untrue4 ,= ! =ces act+ on a body dep5d ^u its spatial relations to o!r bodies1 & *ang+ =ces are liable to produce *ang+ configuration4 ,h5ce1 x w\ld seem1 actual measurem5t mu/ be purely empirical1 & mu/ dep5d on ! degree ( rigidity to be obta9$1 dur+ ! process ( measurem5t1 9 ! bodies ) :i* we are conv]sant4 ,?is conclusion1 ,i believe1 is valid ( all actual measurem5t4 ,b ! possibility ( su* empirical & approximate rigidity1 ,i mu/ insi/1 dep5ds on ! priori law t m]e motion1 ap>t f ! action ( o!r matt]1 _c effect a *ange ( %ape4 ,= )\t ?is law1 ! effect ( o!r matt] w\ld n be discov]able2 ! laws ( motion w\ld be absurd1 & ,physics w\ld be impossible4 ,consid] ! second law1 = example3 ,h[ c\ld we measure ! *ange ( motion1 if motion itself produc$ a *ange 9 our measures8 ,or consid] ! law ( gravitation3 ,h[ c\ld we e/abli% ! inv]se squ>e1 unless we 7 able1 indep5d5tly ( ,dynamics1 to measure di/ances8 ,! :ole sci5ce ( ,dynamics1 9 %ort1 is fundam5tally dep5d5t on ,geometry1 & b = ! indep5d5t possibility ( measur+ spatial magnitudes1 none ( ! magnitudes ( ,dynamics c\ld be measur$4 ,"t1 =ce1 & mass are alike measur$ by spatial correlates3 ^! correlates are giv51 = "t1 by ! fir/ law1 = =ce & mass1 by ! second & ?ird4 ,x is true1 !n1 t an empirical elem5t appe>s unavoidably 9 all actual measurem5t1 inasmu* z we c only "k empirically t a giv5 piece ( matt] pres]ves its %ape ?r\gh\t ! necess>y *ange ( dynamical relations to o!r matt] involv$ 9 motion2 b x is fur!r true t1 = ,geometry--:i* reg>ds matt] simply z supply+ ! necess>y brea* 9 ! homog5eity ( space1 & ! necess>y t]m = spatial relations1 n z ! be>] ( =ces :i* *ange ! configuration ( o!r mat]ial sy/ems--= ,geometry1 :i* deals ) ?is ab/ract & m]ely k9ematical matt]1 rigidity is priori1 9 s f> z ! only *anges ) :i* x is cognizant--*anges ( m]e position1 namely--are incapable ( affect+ ! %apes ( ! imag9>y & ab/ract bodies ) :i* x deals4 ,to use a s*ola/ic di/9ction1 we may say t matt] is ! causa ess5di ( space1 b ,geometry is ! causa cognosc5di ( ,physics4 ,)\t a ,geometry indep5d5t ( ,physics1 ,physics itself1 :i* necess>ily assumes ! results ( ,geometry1 c\ld nev] arise2 b :5 ,geometry is us$ 9 ,physics1 x loses "s ( its priori c]ta9ty1 & acquires ! empirical & approximate "* :i* belongs to all acc\nts ( actual ph5om5a4 #gb4 7#c7 ,?is argum5t leads u to ,l&'s di/9ction ( physical & geometrical rigidity4 ,! di/9ction may be express$--& ,i ?9k x is bett] express$--by di/+ui%+ betwe5 ! conceptions ( matt] prop] to ,dynamics & to ,geometry respectively4 ,9 ,dynamics1 we are conc]n$ ) matt] z subject to & z caus+ motion1 z affect$ by & z ex]t+ =ce4 ,we are !re=e conc]n$ ) ! *anges ( spatial configuration to :i* mat]ial sy/ems are liable3 ! description & explanation ( ^! *anges is ! prop] subject-matt] ( all ,dynamics4 ,b 9 ord] t su* a sci5ce may exi/1 x is obvi\sly necess>y t spatial configuration %\ld be already measurable4 ,if ?is 7 n ! case1 motion1 accel]ation & =ce w\ld rema9 p]fectly indet]m9ate4 ,geometry1 !re=e1 mu/ already exi/ be=e ,dynamics becomes possible3 to make ,geometry dep5d5t = its possibility on ! laws ( motion or any ( _! consequ5ces1 is a gross 4 ,nev]!less1 z we h se51 "s sort ( matt] is ess5tial to ,geometry4 ,b ?is geometrical matt] is a m ab/ract & :olly diff]5t matt] f t ( ,dynamics4 ,9 ord] to /udy space by itself1 we r$uce ! prop]ties ( matt] to a b>e m9imum3 we avoid entirely ! category ( causation1 s ess5tial to ,dynamics1 & reta9 no?+1 9 our matt]1 b its spatial adjectives#ia4 ,! k9d ( rigidity affirm$ ( ?is ab/ract matt]--a k9d :i* suffices = ! !ory ( our sci5ce1 ?\gh n = its application to ! objects ( daily life--is purely geometrical1 & ass]ts no m ?an ?is3 ,t s9ce our matt] is devoid1 ex hypo!si1 ( causal prop]ties1 !re rema9s no?+1 9 m]e empty space1 :i* is capable ( *ang+ ! configuration ( any geometrical sy/em4 ,a *ange ( absolute position1 x ass]ts1 is no?+2 !re=e ! only real *ange involv$ 9 motion is a *ange ( relation to o!r matt]2 b su* o!r matt]1 = ! purposes ( our sci5ce1 is reg>d$ z de/itute ( causal p[]s2 h5ce no *ange c occur1 9 ! configuration ( our sy/em1 by ! m]e effect ( motion ?r\gh empty space4 ,! necessity ( su* a pr9ciple may be %[n by a simple r$uctio ad absurdum1 z foll[s4 ,a motion ( translation ( ! univ]se z a :ole1 ) con/ant direction & velocity1 is dynamically negligeable2 inde$ x is1 philosophically1 no motion at all1 = x involves no *ange 9 ! condition or mutual relations ( ! ?+s 9 ! univ]se4 ,b if our geometrical rigidity 7 d5i$1 ! *ange 9 ! p>amet] ( space might cause all bodies to *ange _! %apes ow+ to ! m]e *ange ( absolute position1 :i* is obvi\sly absurd4 ,to make q pla9 ! function ( rigid bodies 9 ,geometry1 let u suppose a liquid geomet] 9 a liquid _w4 ,we _c suppose ! liquid p]fectly homog5e\s & undiff]5tiat$1 9 ! fir/ place because su* a liquid w\ld be indi/+ui%able f empty space1 9 ! second place because our geomet]'s body--unless he be a disembodi$ _s--w itself con/itute a diff]5tiation = him4 ,we may !re=e assume dim beams1 ,:i* amid ! /reams ,weave a network ( col\r$ light1 & we may suppose ?is network to =m ! occasion = our geomet]'s reflections4 ,!n he w be able to imag9e a network 9 :i* ! l9es are /raight1 or circul>1 or p>abolic1 or any o!r %ape1 & he w be able to inf] t su* a network1 if x c be wov5 9 "o "p ( ! fluid1 c be wov5 9 ano!r4 ,?is w =m suffici5t basis = 8 d$uctions4 ,! sup]position he is conc]n$ )--s9ce n actual equality1 b only ! =mal conditions ( equality1 are ! subject-matt] ( ,geometry--is purely ideal1 & is unaffect$ by ! impossibility ( congeal+ any actual network4 ,b 9 ord] to apply 8 ,geometry to ! exig5cies ( life1 he w\ld ne$ "s /&>d ( comp>ison betwe5 actual networks1 & "h1 x is true1 he w\ld ne$ ei!r a rigid body1 or a k ( ! conditions "u :i* simil> networks arose4 ,moreov] ^! conditions1 be+ necess>ily empirical1 c\ld h>dly be kn[n ap>t f previ\s measurem5t4 ,h5ce = appli$1 ?\gh n = pure ,geometry1 "o rigid body at lea/ seems ess5tial4 #gc4 ,! utility1 = ,dynamics1 ( our ab/ract geometrical matt]1 is suffici5tly evid5t4 ,= hav+1 by its means1 a p[] ( det]m9+ ! configurations ( mat]ial sy/ems 9 :atev] "p ( space1 & kn[+ t *anges ( configuration are n due to m]e *ange ( place1 we are able to attribute ^! *anges to ! action ( o!r matt]1 & ?us to e/abli% ! notion ( =ce1 :i* w\ld be impossible if *ange ( %ape might be due to empty space4 ,?us1 to conclude3 ,geometry requires1 if x is to be practically possible1 "s body or bodies :i* are ei!r rigid 79 ! dynamical s5se71 or kn[n to und]go "s def9ite *anges ( %ape accord+ to "s def9ite law4 7,^! *anges1 we may suppose1 are kn[n by ! laws ( ,physics1 :i* h be5 exp]im5tally e/abli%$1 & :i* ?r\gh\t assume ! tru? ( ,geometry47 ,"o or m su* bodies are necess>y to appli$ ,geometry--b only 9 ! s5se 9 :i* rul]s & compasses are necess>y4 ,!y are necess>y z1 9 mak+ ! ,ordnance ,survey1 an elaborate app>atus 0 necess>y = measur+ ! base l9e on ,salisbury ,pla94 ,b = ! !ory ( ,geometry1 geometrical rigidity suffices1 & geometrical rigidity means only t a %ape1 :i* is possible 9 "o "p ( space1 is possible 9 any o!r4 ,! empirical elem5t 9 practice1 aris+ f ! purely empirical nature ( physical rigidity1 is comp>able to ! empirical inaccuracies aris+ f ! failure to f9d /raight l9es or circles 9 ! _w--:i* no "o b ,mill has reg>d$ z r5d]+ ,geometry itself empirical or inaccurate4 ,b to make ,geometry await ! p]fection ( ,physics1 is to make ,physics1 :i* dep5ds ?r\gh\t on ,geometry1 =ev] impossible4 ,z well might we leave ! =mation ( numb]s until we _h c\nt$ ! h\ses 9 ,piccadilly4 ,erdmann4 #gd4 ,9 connection ) ,riemann & ,helmholtz1 x is natural to consid] ,erdmann's philosophical "w on _! !ories#ib4 ,?is is c]ta9ly ! mo/ important book on ! subject :i* has appe>$ f ! philosophical side1 & 9 spite ( ! fact t1 l ! :ole !ory ( ,riemann & ,helmholtz1 x is inapplicable to projective ,geometry1 x /ill des]ves a v full discussion4 ,erdmann agrees ?r\gh\t ) ! conclusions ( ,riemann & ,helmholtz1 except on a few po9ts ( m9or importance2 & 8 views1 z ?is agreem5t w\ld lead "o to expect1 are ultra-empirical4 ,inde$ 8 logic seems--?\gh ,i say ?is ) hesitation--to be incompatible ) any sy/em b t ( ,mill3 !re is app>5tly no di/9ction1 to him1 betwe5 ! g5]al & ! univ]sal1 & consequ5tly no concept n embodi$ 9 a s]ies ( in/ances4 ,su* a !ory ( logic1 to my m9d1 vitiates mo/ ( 8 "w1 z x vitiat$ ,riemann's philosophy#ic4 ,?is g5]al criticism w f9d abundant illu/ration 9 ! c\rse ( our acc\nt ( ,erdmann's views4 #ge4 ,aft] a g5]al introduction1 & a %ort hi/ory ( ! developm5t ( ,metageometry1 ,erdmann proce$s1 9 8 second *apt]1 to discuss :at are ! axioms ( ,euclidean ,geometry4 ,! ari?metical axioms1 z !y are call$1 he leaves aside1 z apply+ to magnitude 9 g5]al2 :at we want "h1 he says1 is a def9ition ( space1 = :i* ! geometrical axioms are alone relevant4 ,b a def9ition ( space1 he says--foll[+ ,riemann--dem&s a g5us ( :i* space %all be a species1 & ?is1 s9ce our space is psy*ologically unique1 c only be furni%$ by analytical ma!matics 7p4 #cf74 ,n[ ! space-=ms dealt ) by ,geometry are magnitudes1 & conceptions ( magnitude are ev]y:]e appli$ 9 ,geometry4 ,b be=e ,riemann1 only p>ticul> det]m9ations ( space c\ld be exhibit$ z magnitudes1 & ?us ! desir$ def9ition 0 impossible to obta94 ,n[1 h[ev]1 we c subsume space z a :ole "u a g5]al conception ( magnitude1 & ?us obta91 besides ! space-intuition & ! space-conception1 a ?ird =m1 namely1 ! conception ( space z a magnitude 7,grss5begriff vom ,raum1 pp4 #ch-ci74 ,! def9ition ( ?is w give u ! complete1 b n r$undant1 sy/em ( axioms1 :i* c\ld n be obta9$ by trans=m+ ! g5]al intuition ( space into ! space-conception1 = want ( a plurality ( in/ances 7p4 #dj74 #gf4 ,be=e consid]+ ! subsequ5t me?od ( def9ition1 let u reflect on ! !ories involv$ 9 ! above acc\nt ( ! conception ( space z a magnitude4 ,9 ! fir/ place1 x is assum$ t conceptions _c be =m$ unless we h a s]ies ( sep>ate objects f :i* to ab/ract a common prop]ty--9 o!r words1 t ! univ]sal is always ! g5]al4 ,9 ! second place1 x is assum$ t all def9ition is classification "u a g5us4 ,9 ! ?ird place1 ! conception ( magnitude1 if ,i am n mi/ak51 is fundam5tally misund]/ood :5 x is suppos$ applicable to space z a :ole4 ,b 9 ! f\r? place1 ev5 if su* a conception exi/$1 x c\ld give none ( ! ess5tial prop]ties ( space4 ,let u consid] ^! f\r po9ts successively4 #gg4 ,z reg>ds ! fir/ po9t1 x is to be obs]v$ t p c]ta9ly _h "s conception ( space be=e ,riemann inv5t$ ! notion ( a manifold1 & t ?is conception 0 c]ta9ly some?+ o!r ?an ! common qualities ( all ! po9ts1 l9es or figures 9 space4 ,9 ! second place1 ,erdmann's view w\ld make x impossible to conceive ,god1 unless "o 7 a poly!i/1 or ! univ]se--unless1 l ,leibnitz1 "o imag9$ a s]ies ( possible worlds1 set ov] aga9/ ,god1 & none ( !m1 !re=e1 a true ,univ]se--or1 to take an in/ance m likely to appeal to an empirici/1 ! necess>ily unique c5tre ( mass ( ! mat]ial univ]se4 ,any univ]sal1 9 %ort1 :i* is a bond or unity betwe5 ?+s1 & n m]ely a common prop]ty among indep5d5t objects1 becomes impossible on ,erdmann's view4 ,we _c1 !re=e1 unless we adopt ,mill's philosophy intact1 reg>d ! conception ( space z dem&+ a s]ies ( in/ances f :i* to ab/ract4 ,b ev5 if we did s reg>d x1 ,riemann's manifolds w\ld leave u )\t res\rces4 ,= ,euclidean space /ill appe>s z unique1 at ! end ( 8 s]ies ( det]m9ations4 ,we h in/ances ( manifolds1 b n in/ances ( ,euclidean space4 ,?us if ,erdmann's !ory ( conceptions 7 correct1 he w\ld /ill be left se>*+ 9 va9 = ! conception ( ,euclidean space4 #gh4 ,! second po9t1 ! view t all def9ition is classification1 is closely alli$ to ! fir/1 & ! two toge!r plunge u into ! dep?s ( s*ola/ic =mal logic4 ,! same in/ances ( ?+s :i* c\ld n1 on ,erdmann's view1 be conceiv$1 may n[ be adduc$ z ?+s :i* _c be def9$4 ,:atev] 0 said above applies "h also1 & ! po9t ne$ n1 !re=e1 be fur!r discuss$#id4 #gi4 ,z reg>ds ! ?ird po9t1 ! impossibility ( apply+ conceptions ( magnitude to space z a :ole1 a long] argum5t w be necess>y1 = we are conc]n$1 "h1 ) ! :ole "q ( ! logical nature ( judgm5ts ( magnitude4 ,z we _h be=e too mu* comp>ison = our ne$s1 s we h n[ too little4 ,i w endeav\r to expla9 ?is po9t1 :i* is ( great importance1 & und]lies1 ,i ?9k1 mo/ ( ! philosophical fallacies ( ,riemann's s*ool4 ,a judgm5t ( magnitude is always a judgm5t ( comp>ison1 & :at is m1 ! comp>ison is nev] conc]n$ ) quality1 b only ) quantity4 ,quality1 9 ! judgm5t ( magnitude1 is suppos$ id5tical1 9 ! object ^: magnitude is /at$1 & 9 ! unit ) :i* x is comp>$4 ,b quality1 except 9 pure numb]1 & 9 pure quantity z dealt ) by ! ,calculus1 is always pres5t1 & is p>tly absorb$ into quantity1 p>tly unt\*$ by ! judgm5t ( magnitude4 ,z ,bosanquet says 7,logic1 ,vol4 ,i4 p4 #abd72 ,quantitative comp>ison is n prima facie coord9ate ) qualitative1 b r /&s 9 its place z ! effect ( comp>ison on quality1 :i* s f> z comp>able becomes quantity1 & s f> z incomp>able furni%es ! di/9ction ( p>ts ess5tial to ! quantitative :ole 7italics 9 ! orig9al74 ,?us1 if we are to reg>d space z a magnitude1 we mu/ be able to adduce all ^? s]ies ( in/ances ( :i* ,erdmann speaks1 & :i*1 = ! conception ( space1 seem$ irrelevant4 ,b x rema9s to be prov$ t ! comp>ison1 :i* we c in/itute betwe5 v>i\s spaces1 is capable ( expression 9 a quantitative =m4 ,r x w\ld seem t ! diff]5ce ( quality is su* z to preclude quantitative comp>ison betwe5 diff]5t spaces1 & !re=e also to preclude all judgm5ts ( magnitude ab\t space z a :ole4 ,"h an exception might seem to be dem&$ by non-,euclidean spaces1 ^: space-con/ants give a def9ite magnitude1 inh]5t 9 space z a :ole1 & !re=e1 "o might ?9k1 *>act]iz+ space z a magnitude4 ,b ?is is a mi/ake4 ,= ! space-con/ant1 9 su* spaces1 is ! ultimate unit1 ! fix$ t]m 9 all quantitative comp>ison2 x is itself1 !re=e1 de/itute ( quantity1 s9ce !re is no indep5d5tly giv5 magnitude ) :i* to comp>e x4 ,a non-,euclidean _w1 9 :i* ! space-con/ant & all l9es & figures 7 sudd5ly multipli$ 9 a con/ant ratio1 w\ld be :olly un*ang$2 ! l9es1 z measur$ aga9/ ! space-con/ant1 w\ld h ! same magnitude z be=e1 & ! space-con/ant itself1 hav+ no outside /&>d ( comp>ison1 w\ld be de/itute ( quantity1 & !re=e n subject to *ange ( quantity4 ,su* an enl>gem5t ( a non-,euclidean _w1 9 o!r words1 is unmean+2 & ?is proves h[ inapplicable is ! notion ( quantity to space z a :ole4 ,x might be object$ t ?is only proves ! abs5ce ( quantitative diff]5ce betwe5 diff]5t spaces ( positive space-con/ant1 or betwe5 ^? ( negative space-con/ant3 ! quantitative diff]5ce p]si/s1 x might be said1 betwe5 ^? ( positive curvature 9 g5]al & ^? ( negative curvature 9 g5]al1 or betwe5 bo? toge!r & ,euclidean space4 ,?is ,i entirely d5y4 ,!re is no qualitatively simil> unit1 9 ! ?ree k9ds ( space1 by :i* quantitative comp>ison c\ld be effect$4 ,! /raight l9es ( "o space _c be put into ! o!r3 ! two /raight l9es1 9 "o space1 ^: product is ! reciprocal ( ! measure ( curvature1 h no correspond+ curves 9 ! o!r space1 & ! measures ( curvature _c1 !re=e1 be quantitatively comp>$ ) ea* o!r4 ,t ! "o may be reg>d$ z positive1 ! o!r negative1 ,i admit1 b _! values are indet]m9ate1 & ! units 9 ! two cases are qualitatively diff]5t4 ,a debt ( #cjj may be repres5t$ z ! asset ( -#cjj1 & ! height ( ! ,eiffel ,t[] is #cjj metres2 b x does n foll[ t ! two are quantitatively comp>able4 ,s ) space-con/ants3 ! space-con/ant is itself ! unit = magnitudes 9 its own space1 & diff]s qualitatively f ! space-con/ant ( ano!r k9d ( space4 ,aga91 to proce$ to a m philosophical argum5t1 two diff]5t spaces _c co-exi/ 9 ! same _w3 we may be unable to decide betwe5 ! alt]natives ( ! disjunction1 b !y rema91 none ! less1 absolutely incompatible alt]natives4 ,h5ce we _c get t coexi/5ce ( two spaces :i* is ess5tial to comp>ison4 ,! fact seems to be t ,erdmann1 9 8 admiration = ,riemann & ,helmholtz1 has fall5 9 ) _! ma!matical bias1 & assum$1 z ma!maticians naturally t5d to assume1 t quantity is ev]y:]e & always applicable & adequate1 & c deal ) m ?an ! m]e comp>ison ( ?+s ^: qualities are already kn[n z simil>#ie4 #hj4 ,?is sugge/s ! f\r? & la/ ( ! above po9ts1 t ! qualities ( space1 ev5 if space c\ld be successfully reg>d$ z a magnitude1 w\ld h to be entirely omitt$ 9 su* a mann] ( reg>d+ x1 & t1 !re=e1 none ( its important or ess5tial prop]ties w\ld em]ge f su* treatm5t4 ,= to reg>d space z a magnitude involves1 z we saw1 a comp>ison ) some?+ qualitatively simil>1 & an ab/raction f ! simil> qualities4 ,to "s ext5t & by ! help ( c]ta9 d\btful argum5ts1 su* a comp>ison is in/itut$ by ,riemann & ,erdmann2 b :5 !y h in/itut$ x1 !y =get all ab\t ! common qualities on :i* its possibility dep5ds4 ,b ^! are precisely ! fundam5tal prop]ties ( space1 & ^? f :i*1 z ,i %all endeav\r to prove 9 ,*apt] ,,iii41 ! axioms common to ,euclid & ,metageometry foll[ priori4 ,su* are ! dang]s ( ! quantitative bias4 #ha4 ,aft] ?is prote/ aga9/ ! initial assumptions 9 ,erdmann's d$uction ( space1 let u return to consid] ! mann]1 9 :i* ?is d$uction is c>ri$ out4 ,"h !re w be less gr\nd = criticism1 z ! d$uction1 giv5 its presuppositions1 is1 ,i ?9k1 z good z su* a d$uction c be4 ,to def9e space z a magnitude1 he says1 let u />t ) two ( its mo/ obvi\s prop]ties1 cont9uity & ! ?ree dim5sions4 ,tones & col\rs af=d o!r in/ances ( a manifold ) ^! two prop]ties1 b diff] f space 9 t _! dim5sions are n homog5e\s & int]*angeable4 ,to designate ?is diff]5ce1 ,erdmann introduces a useful pair ( t]ms3 9 ! g5]al case1 he calls a manifold n-det]m9$ 7n-be/immt72 9 ! case :]e1 z 9 space1 ! dim5sions are homog5e\s1 he calls ! manifold n-ext5d$ 7n-ausg$ehnt74 ,manifolds ( ! latt] sort he calls ext5ts 7,ausg$ehn!it574 ,t ! diff]5ce betwe5 ! two k9ds is "o ( quality1 n ( quantity1 he seems n to p]ceive2 he also ov]looks ! fact t1 9 ! second k9d1 f its v def9ition1 ! axiom ( ,congru5ce mu/ hold1 on acc\nt ( ! qualitative simil>ity ( diff]5t p>ts4 ,9 spite ( ?is fact1 he def9es space z an ext5t1 & !n reg>ds ,congru5ce z empirical1 & z possibly false 9 ! inf9itesimal4 ,?is is ! m /range1 z he actually proves 7p4 #ej7 t measurem5t is impossible1 9 an ext5t1 unless ! p>ts are indep5d5t ( _! place1 & c be c>ri$ ab\t unalt]$ z measures4 ,9 spite ( ?is1 he proce$s imm$iately to discuss :e!r ! measure ( curvature is con/ant or v>iable1 )\t inve/igat+ h[1 9 ! latt] case1 ,geometry c\ld exi/4 ,we _c "k1 he says1 f geometrical sup]position1 t geometrical bodies are indep5d5t ( place1 = if _! dim5sions alt]$ 9 motion accord+ to any fix$ law1 two bodies :i* c\ld be sup]pos$ 9 "o place c\ld be sup]pos$ 9 any o!r4 ,t su* a hypo!sis involves absolute position1 & d5ies ! qualitative simil>ity ( ! p>ts ( space1 :i* he decl>es 7p4 #aga7 to be ! pr9ciple ( 8 !ory ( ,geometry1 is n[h]e p]ceiv$4 ,b :at is m1 8 notion t magnitude is some?+ absolute1 indep5d5t ( comp>ison1 has prev5t$ him f see+ t su* a hypo!sis is unmean+4 ,he says himself t1 ev5 on ?is hypo!sis1 a geometrical body c be def9$ z "o ^: po9ts reta9 con/ant di/ances f ea* o!r1 =1 s9ce we h no absolute measure1 measurem5t c\ld n reveal to u ! *ange ( absolute magnitude 7p4 #fj74 ,b is n ?is a r$uctio ad absurdum8 ,= magnitude is no?+ ap>t f comp>ison1 & ! comp>ison "h c only be effect$ by sup]position2 if1 !n1 z on ! above hypo!sis1 sup]position always gives ! same result1 by :atev] motion x is effect$1 !re is no s5se 9 speak+ ( magnitudes z no long] equal :5 sep>at$3 absolute magnitude is an absurdity1 & ! magnitude result+ f comp>ison does n diff] f t :i* w\ld result if ! dim5sions ( bodies 7 un*ang$ 9 motion4 ,!re=e1 s9ce magnitude is only intelligible z ! result ( comp>ison1 ! dim5sions ( bodies are un*ang$ 9 motion1 & ! sugge/$ hypo!sis is unmean+4 ,on ?is subject ,i %all h m to say 9 ,*apt] ,,iii4#if #hb4 ,?is hypo!sis1 h[ev]1 is n introduc$ = its own sake1 b only to u%] 9 ! ,helmholtzian deus ex ma*9a1 ,me*anics4 ,= ,me*anics proves--s ,erdmann confid5tly cont9ues--t rigidity mu/ hold1 n m]ely z to ratios1 9 ! above re/rict$ geometrical s5se1 b z to absolute magnitudes 7p4 #fb74 ,h5ce we get at la/ true ,congru5ce1 empirical z ,me*anics is empirical1 & impossible to prove ap>t f ,me*anics4 ,i h already criticiz$ ,helmholtz's view ( ! dep5d5ce ( ,geometry on ,me*anics1 & ne$ n "h speak ( x at l5g?4 ,x is a pity t ,erdmann has 9 no way specifi$ ! proc$ure by :i* ,me*anics decides ! geometrical alt]natives--inde$ he seems to rely on ! ipse dixit ( ,helmholtz4 ,h[1 if ,geometry w\ld be totally unable to discov] a *ange 9 dim5sions ( ! k9d sugge/$1 ! ,laws ( ,motion1 :i* ?r\gh\t dep5d on ,geometry1 %\ld be able to discov] x if x exi/$1 ,i am :olly at a loss to und]/&4 ,uni=m motion 9 a /raight l9e1 = example1 presupposes geometrical measurem5t2 if ?is measurem5t is mi/ak51 :at ,me*anics imag9es to be uni=m motion is n really su*1 b ,me*anics c nev] discov] ! discrepancy4 ,if ! ,laws ( ,motion _h be5 reg>d$ z priori1 ,geometry might possibly h be5 re9=c$ by !m2 b s long z !y are empirical1 !y presuppose geometrical measurem5t1 & _c !re=e condition or affect x4 ,erdmann's conclusion1 9 ! second *apt]1 is t ,congru5ce is probable1 b _c be v]ifi$ 9 ! inf9itesimal2 t its tru? involves ! actual exi/5ce ( rigid bodies 7?\gh1 by ! way1 we "k ^! to be1 /rictly speak+1 non-exi/5t71 t rigid bodies are freely moveable1 & d n alt] _! size 9 rotation 7,helmholtz's ,monodromy72 t ! axiom ( ?ree dim5sions is c]ta91 s9ce small errors are impossible2 & t ! rema9+ axioms ( ,euclid--^? ( ! /raight l9e & ( p>allels--are approximately1 if n accurately1 true ( our actual space 7pp4 #gh1 #hc74 ,he does n discuss h[ ,congru5ce1 on ! above view1 is compatible ) ! atomic !ory1 or ev5 ) ! obs]v$ de=mations ( approximately rigid bodies2 nor h[1 if space1 z he assumes1 is homog5e\s1 rigid bodies c fail to be freely moveable ?r\gh space4 ,! axioms are all lump$ toge!r z empirical1 & x appe>s1 9 ! foll[+ *apt]s1 t ,erdmann reg>ds _! empirical nature z suffici5tly prov$ by _! applicability to empirical mat]ial 7cf4 pp4 #aei1 #afe7--;a /range crit]ion1 :i* w\ld prove ! same conclusion1 ) equal facility1 ( ,ari?metic & ( ! laws ( ?\ght4 #hc4 ,! ?ird *apt]1 on ! philosophical consequ5ces ( ,metageometry1 ne$ n be discuss$ at l5g?1 s9ce x deals r ) space ?an ) ,geometry4 ,at ! same "t1 x w be wor? :ile to treat briefly ( ,erdmann's crit]ion ( apriority4 ,on ?is subject x is v difficult to discov] 8 mean+1 s9ce x seems to v>y ) ! topic he is discuss+4 ,?us at "o "t 7p4 #adg7 he rejects mo/ emphatically ! ,kantian connection ( ! priori & ! subjective#ig1 & yet at ano!r "t 7p4 #if7 he reg>ds e pres5tation ( ext]nal ?+s z p>tly priori1 p>tly empirical1 m]ely because su* a pres5tation is due to an int]action betwe5 ourselves & ?+s1 & is !re=e p>tly due to subjective activity1 p>tly due to outside objects4 ,h5ce1 he says1 ! di/9ction is n betwe5 diff]5t pres5tations1 b betwe5 diff]5t aspects ( "o & ! same pres5tation4 ,?is seems to return :olly to ! ,kantian psy*ological crit]ion ( subjectivity1 ) ! add$ disadvantage t x makes ! di/9ction1 l t ( analytic & syn!tic1 epi/emologically wor?less4 ,& yet he nev] hesitates to pron\nce e piece ( k 9 turn empirical4 ,! fact seems to be1 t :]e he wants a m logical crit]ion1 he adopts a modification ( ,helmholtz's crit]ion = s5sations4 ,if space be an priori =m1 he says1 no exp]i5ce c\ld possibly *ange x 7p4 #ajh72 b ?is ,metageometry has prov$ n to be ! case1 s9ce we c intuit ! p]ceptions :i* non-,euclidean space w\ld give u 7p4 #aae74 ,i h criticis$ ?is argum5t 9 discuss+ ,helmholtz2 at pres5t we are conc]n$ ) ,erdmann's crit]ion ( apriority4 ,! subjectivity-crit]ion--?\gh he c]ta9ly uses x 9 discuss+ ! apriority ( space1 & solemnly decides1 by its means1 t space is bo? priori & empirical s9ce a *ange ei!r 9 u or 9 ! out] _w c\ld *ange x 7p4 #ig7--w\ld seem1 l sev]al ( 8 o!r te/s1 to be a lapse on 8 "p3 ! crit]ion :i* he means to use is ,helmholtz's4 ,?is crit]ion1 ,i ?9k1 ) a slight *ange ( word+1 might be accept$2 x seems to me a necess>y1 b n a suffici5t condition4 ,! priori1 we may say1 is n only t :i* no exp]i5ce c *ange1 b t )\t :i* exp]i5ce w\ld become impossible4 ,x is ! omission to discuss ! conditions :i* r5d] geometrical 7& me*anical7 exp]i5ce possible1 to my m9d1 :i* vitiates ! empirical conclusions ( ,helmholtz & ,erdmann4 ,:y c]ta9 conditions %\ld be necess>y = exp]i5ce--:e!r on acc\nt ( ! con/itution ( ! m9d1 or = "s o!r reason--is a fur!r "q1 :i* introduces ! relation ( ! priori to ! subjective4 ,b 9 discuss+ ! "q z to :at k is priori1 z oppos$ to ! "q conc]n+ ! fur!r consequ5ces ( apriority1 x is well to keep to ! purely logical crit]ion1 & s pres]ve our indep5d5ce ( psy*ological controv]sies4 ,! fact1 if x be a fact1 t ! _w might be su* z to defy our attempts to "k x1 w n1 ) ! above crit]ion1 invalidate ! conclusion t c]ta9 elem5ts 9 k are priori2 = :e!r fulfill$ or n1 !y rema9 necess>y conditions = ! exi/5ce ( any k at all4 #hd4 ,) ?is caution z to ! mean+ ( apriority1 we %all f9d1 ,i ?9k1 t ! conclusions ( ,erdmann's f9al *apt]1 on ! pr9ciples ( a !ory ( ,geometry1 are l>gely invalidat$ by ! div]sity & inadequacy ( 8 te/s ( ! priori4 ,he beg9s by ass]t+1 9 con=mity ) ! quantitative bias notic$ above1 t ! "q z to ! nature ( geometrical axioms is completely analog\s to ! correspond+ "q ( ! f\ndations ( pure ma!matics 7p4 #ach74 ,?is is1 ,i ?9k1 a radical error3 = ! function ( ! axioms seems to be1 to e/abli% t qualitative basis on :i*1 z we saw1 all qualitative comp>ison mu/ re/4 ,b 9 pure ma!matics1 ?is qualitative basis is irrelevant1 = we deal !re ) pure quantity1 i4e4 ) ! m]ely quantitative result ( quantitative comp>ison1 :]ev] x is possible1 indep5d5tly ( ! qualities und]ly+ ! comp>ison4 ,geometry1 z ,grassmann insi/s#ih1 ought n to be class$ ) pure ma!matics1 = x deals ) a matt] :i* is giv5 to ! intellect1 n creat$ by x4 ,! axioms give ! means by :i* ?is matt] is made am5able to quantity1 & _c1 !re=e1 be !mselves d$uc$ f purely quantitative consid]ations4 ,leav+ ?is po9t aside1 h[ev]1 let u return to ,erdmann4 ,he di/+ui%es1 )9 space1 a =m & a matt]3 ! =m is to conta9 ! prop]ties common to all ext5ts1 ! matt] ! prop]ties :i* di/+ui% space f o!r ext5ts4 ,?is di/9ction1 he says1 is purely logical1 & does n correspond ) ,kant's3 matt] & =m1 = ,erdmann1 are alike empirical4 ,! axioms & def9itions ( ,geometry1 he says1 deal exclusively ) ! matt] ( space4 ,x seems a pity1 hav+ made ?is di/9ction1 to put x to s little use3 aft] a few pages1 x is dropp$1 & no epi/emological consequ5ces are drawn f x4 ,! reason is1 ,i ?9k1 t ,erdmann has n p]ceiv$ h[ mu* c be d$uc$ f 8 def9ition ( an ext5t1 z a manifold 9 :i* ! dim5sions are homog5e\s & int]*angeable4 ,= ?is prop]ty suffices to prove ! complete homog5eity ( an ext5t1 & h5ce--f ! abs5ce ( qualitative diff]5ces among elem5ts--! relativity ( position & ! axiom ( ,congru5ce4 ,?is d$uction w be made at l5g? 9 ! sequel#ii2 at pres5t1 ,i h only to obs]ve t e ext5t1 on ?is view1 possesses all ! prop]ties 7except ! ?ree dim5sions7 common to ,euclidean & non-,euclidean spaces4 ,! axioms :i* express ^! prop]ties1 !re=e1 apply to ! =m ( space1 & foll[ f homog5eity alone1 :i* ,erdmann all[s 7p4 #aga7 z ! pr9ciple ( any !ory ( space4 ,! above di/9ction ( =m & matt]1 !re=e1 corresponds1 :5 its full consequ5ces are d$uc$1 to ! di/9ction betwe5 ! axioms :i* foll[ f ! homog5eity ( space & ^? :i* d n4 ,s9ce1 !n1 homog5eity is equival5t to ! relativity ( position1 & ! relativity ( position is ( ! v ess5ce ( a =m ( ext]nality1 x w\ld seem t 8 di/9ction ( =m & matt] c also be made coext5sive ) ! di/9ction ( ! priori & empirical 9 ,geometry4 ,on ?is subject1 ,i %all h m to say 9 ,*apt] ,,iii4 ,9 ! rema9d] ( ! *apt]1 ,erdmann insi/s t ! /raight l9e1 etc41 ?\gh n ab/ract$ f exp]i5ce1 :i* n[h]e pres5ts /raight l9es1 mu/ yet1 z applicable to admitt$ly empirical sci5ces1 be empirical 7p4 #aei7--;a crit]ion :i* he appe>s to employ only :5 all o!r gr\nds = an empirical op9ion fail1 & "o :i*1 obvi\sly1 c nev] refuse to d its "w1 s9ce all elem5ts ( k are susceptible ( employm5t on "s empirical mat]ial4 ,he also def9es ! /raight l9e 7p4 #aee7 z a l9e ( con/ant curvature z]o1 z ?\gh curvature c\ld be measur$ indep5d5tly ( ! /raight l9e4 ,ev5 ! ari?metical axioms are decl>$ empirical 7p4 #afe71 s9ce 9 a _w :]e ?+s 7 all hopelessly diff]5t f "o ano!r1 ^! axioms c\ld n be appli$4 ,aft] ?is rem9d] ( ,mill1 we are n surpris$1 a few pages lat] 7p4 #agb71 at a vague appeal to ,engli% logicians z hav+ prov$ ,geometry to be an inductive sci5ce4 ,nev]!less1 ,erdmann decl>es1 almo/ on ! la/ page ( 8 book 7p4 #agc71 t ,geometry is di/+ui%$ f all o!r sci5ces by ! homog5eity ( its mat]ial3 a pr9ciple ( :i* no s+le application occurs ?r\gh\t 8 book1 & :i*1 z we %all see 9 ,*apt] ,,iii41 flatly contradicts ! philosophical !ories advocat$ ?r\gh\t 8 prec$+ pages4 ,on ! :ole1 !n1 x _c be said t ,erdmann has done mu* to /r5g!n ! philosophical position ( ,riemann & ,helmholtz4 ,i h criticiz$ him at l5g?1 because 8 book has ! appe>ance ( great ?or\ghness1 & because x is und\bt$ly ! be/ def5ce extant ( ! position :i* x takes up4 ,we %all n[ h ! opposite task to p]=m1 9 def5d+ ,metageometry1 on its ma!matical side1 f ! attacks ( ,lotze & o!rs1 & 9 v9dicat+ = x t measure ( philosophical importance--f> inf]ior1 inde$1 to ! hopes ( ,erdmann--:i* x seems really to possess4 ,lotze4 #he4 ,lotze's argum5t z reg>ds ,geometry#ajj--:i* foll[s a metaphysical argum5t z to ! ontological nature ( space1 & assumes ! results ( ?is argum5t--consi/s ( two p>ts3 ! fir/ discusses ! v>i\s mean+s logically assignable 7pp4 #bcc-bdg7 to ! proposition t o!r spaces ?an ,euclid's are possible1 & ! second criticizes1 9 detail1 ! proc$ure ( ,metageometry4 ,! fir/ ( ^! que/ions is v important1 & dem&s consid]able c>e z to ! logical import ( a judgm5t ( possibility4 ,al?\gh ,lotze's discussion is excell5t 9 _m respects1 ,i _c p]suade myself t he has hit on ! only true s5se 9 :i* non-,euclidean spaces are possible4 ,i %all endeav\r to make good ?is /atem5t 9 ! foll[+ pages4 #hf4 ,lotze op5s ) a some:at />tl+ /atem5t1 :i*1 ?\gh philosophically wor?y to be true1 does n appe> to be hi/orically borne out4 ,euclidean ,geometry has be5 *iefly %ak51 he says1 by ! ,kantian notion ( ! exclusive subjectivity ( space--if space is only our private =m ( intuition1 to :i* !re exi/s no analogue 9 ! objective _w1 !n o!r be+s may h o!r spaces1 )\t suppos+ any diff]5ce 9 ! _w :i* !y arrange 9 ^! spaces 7p4 #bcc74 ,?is c]ta9ly seems a legitimate d$uction f ! subjectivity ( space1 :i*1 s f> f e/abli%+ ! univ]sal validity ( ,euclid1 e/abli%es 8 validity only aft] an empirical inve/igation ( ! nature ( space z intuit$ by ,tom1 ,dick or ,h>ry4 ,b z a matt] ( fact1 ^? :o h done mo/ to fur!r non-,euclidean ,geometry--) ! exception ( ,riemann1 :o 0 a disciple ( ,h]b>t--h usually inh]it$ f ,newton a nave realism z reg>ds absolute space4 ,i might in/ance ! passage quot$ f ,bolyai 9 ,*apt] ,i41 or ,clif=d1 :o seems to h ?\ght t we actually see ! images ( ?+s on ! ret9a#aja1 or aga9 ,helmholtz's belief 9 ! dep5d5ce ( ,geometry on ! behavi\r ( rigid bodies4 ,?is belief l$ to ! view t ,geometry1 l ,physics1 is an exp]im5tal sci5ce1 9 :i* objective tru? c be atta9$1 x is true1 b only by empirical me?ods4 ,h[ev]1 ,lotze's gr\nd = unc]ta9ty ab\t ,euclid is a philosophically t5able gr\nd1 & x w be in/ructive to obs]ve ! v>i\s possibilities :i* arise f x4 ,if space is only a subjective =m--s ,lotze op5s 8 argum5t--o!r be+s may h a diff]5t =m4 ,if ?is corresponds to a diff]5t _w1 ! diff]5ce1 he says1 is un9t]e/+3 = our _w alone is relevant to any metaphysical discussion4 ,b if ?is diff]5t space corresponds to ! same _w :i* we "k "u ! ,euclidean =m1 !n1 9 8 op9ion1 we get a "q ( g5u9e philosophic int]e/4 ,& "h he di/+ui%es two cases3 ei!r ! relations betwe5 ?+s1 :i* are pres5t$ to ^! hypo!tical be+s "u ! =m ( "s diff]5t space1 are relations :i* d n appe> to u1 or at any rate d n appe> spatial2 or !y are ! same relations :i* appe> to u z figures 9 ,euclidean space 7p4 #bce74 ,! fir/ possibility w\ld be illu/rat$1 he says1 by be+s to :om ! tone or col\r-manifolds appe>$ ext5d$2 b we _c1 9 8 op9ion1 imag9e a manifold1 su* z is requir$ = ?is case1 to h its dim5sions homog5e\s & comp>able int] se1 & !re=e ! cont5ts ( ! v>i\s pres5tations con/itut+ su* a manifold c\ld n be comb9$ into a s+le cont5t conta9+ !m all4 ,b ! possibility ( su* a comb9ation is ( ! ess5ce ( any?+ wor? call+ a space3 !re=e ! fir/ ( ! above possibilities is unmotiv$ & un9t]e/+4 ,lotze's conclusion on ?is po9t1 ,i ?9k1 is und5iable1 b ,i d\bt :e!r 8 argum5t is v cog5t4 ,h[ev]1 z ?is possibility has no connection ) t contemplat$ by non-,euclideans1 x is n wor? :ile to discuss x fur!r4 ,! second possibility also1 ,lotze ?9ks1 is n t ( ,metageometry1 b 9 tru? x comes ne>] to x ?an any ( ! o!r possibilities discuss$4 ,if a non-,euclidean 7 at ! same "t a believ] 9 ! subjectivity ( space1 he w\ld h to be an adh]5t ( ?is view4 ,let u see m precisely :at ! view is4 ,9 ,book ,,ii41 ,*apt] ,i41 ,lotze has accept$ ! argum5t ( ! ,transc5d5tal ,ae/hetic1 b reject$ t ( ! ma!matical ant9omies3 he has decid$ t space is1 z ,kant believ$1 subjective1 b possesses nev]!less1 :at ,kant d5i$ x1 an objective c\nt]p>t4 ,! relation ( pres5t$ space to its objective c\nt]p>t1 z conceiv$ by ,lotze1 is r h>d to und]/&4 ,x seems sc>cely to resemble ! relation ( s5sation to its object--e4g4 ( light to e!r-vibrations--= if x did1 space w\ld n be 9 any peculi> s5se subjective4 ,x seems r to resemble ! relation ( a p]ceiv$ bodily motion to ! /ate ( m9d ( ! p]son will+ ! motion4 ,h[ev] ?is may be1 ! objective c\nt]p>t ( space is suppos$ to consi/ ( c]ta9 imm$iate int]actions ( monads1 :o exp]i5ce ! int]actions z modifications ( _! int]nal /ates4 ,su* int]actions1 x is pla91 d n =m ! subject-matt] ( ,geometry1 :i* deals only ) our result+ p]ceptions ( spatial figures4 ,n[ if ,lotze's con/ruction ( space be correct1 !re seems c]ta9ly no reason :y ^! result+ p]ceptions %\ld n1 = "o & ! same int]action betwe5 monads1 be v diff]5t 9 be+s diff]5tly con/itut$ f ourselves4 ,b if !y 7 diff]5t1 says ,lotze1 !y w\ld h to be utt]ly diff]5t--z diff]5t1 = example1 z ! int]val betwe5 two notes is f a /raight l9e4 ,! possibility is1 !re=e1 9 8 op9ion1 "o ab\t :i* we c "k no?+1 & "o :i* mu/ rema9 always a m]e empty idea4 ,?is seems to me to g too f>3 = :atev] ! objective c\nt]p>t may be1 any argum5t :i* gives u in=mation ab\t x mu/1 :5 rev]s$1 give u in=mation ab\t any possible =m ( intuition 9 :i* ?is c\nt]p>t is pres5t$4 ,! argum5t :i* ,lotze has us$ 9 8 =m] *apt]1 = example1 d$uc+1 f ! relativity ( position1 ! m]ely relational nature ( ! objective c\nt]p>t1 all[s u1 conv]sely1 to inf]1 f ?is relational nature1 ! complete relativity ( position 9 any possible space-intuition--unless1 inde$1 x bore a :olly deceitful relation to ^? int]actions ( monads :i* =m its objective c\nt]p>t4 ,b ! complete relativity ( position1 z ,i %all endeav\r to e/abli% 9 ,*apt] ,,iii41 suffices to prove t our ,geometry mu/ be ,euclidean1 elliptic1 sph]ical or pseudo-sph]ical4 ,we h1 !re=e1 x w\ld seem1 v consid]able k1 on ,lotze's !ory ( space1 ( ! mann] 9 :i* :at appe>s to u z space mu/ appe> to any be+s ) our laws ( ?\ght4 ,we _c "k1 x is true1 :at psy*ological !ory ( space-p]ception w\ld apply to su* be+s3 !y might h a s5se diff]5t f any ( ours1 & !y might h no s5se 9 any way resembl+ ours1 b yet _! ,geometry w\ld h po9ts ( resemblance to ours1 z t ( ! bl9d co9cides ) t ( ! see+4 ,if space has any objective c\nt]p>t :atev]1 9 %ort1 & if any inf]5ce is possible1 z ,lotze holds x to be1 f space to its c\nt]p>t1 !n a conv]se argum5t is also possible1 ?\gh x may give "s only ( ! qualities ( ,euclidean space1 s9ce "s only ( ^! qualities may be f\nd to h a necess>y analogue 9 ! c\nt]p>t4 #hg4 ,admitt+1 !n1 9 ,lotze's s5se1 ! subjectivity ( space1 ! above possibility does n seem s empty z he imag9es4 ,he discusses x briefly1 h[ev]1 9 ord] to pass on to :at he reg>ds z ! real mean+ ( ,metageometry4 ,9 ?is he is guilty ( a ma!matical mi/ake1 :i* causes mu* irrelevant reason+4 ,= he believes t ,metageometry con/ructs its spaces out ( /raight l9es & angles 9 all respects simil> to ,euclid's1 :5ce he d]ives an easy victory 9 prov+ t ^! elem5ts c lead only to ! "o space4 ,9 ?is he has be5 misl$ by ! phraseology ( non-,euclideans1 z well z by ,euclid's sep>ation ( def9itions & axioms4 ,= ! fact is1 ( c\rse1 t /raight l9es are only fully def9$ :5 we add to ! =mal def9ition ! axioms ( ! /raight l9e & ( p>allels4 ,)9 ,euclidean space1 ,euclid's def9ition suffices to di/+ui% ! /raight l9e f all o!r curves2 ! two axioms ref]r$ to are !n absorb$ into ! def9ition ( space4 ,b ap>t f ! re/riction to ,euclidean space1 ! def9ition has to be supplem5t$ by ! two axioms1 9 ord] to def9e completely ! ,euclidean /raight l9e4 ,?us ,lotze has misconceiv$ ! be>+ ( non-,euclidean con/ructions1 & has simply miss$ ! po9t 9 argu+ z he does4 ,! possibility contemplat$ by a non-,euclidean1 if x fell "u any ( ,lotze's cases1 w\ld fall "u ! second case discuss$ above4 #hh4 ,b ! be>+ ( ,metageometry is really1 ,i ?9k1 diff]5t f any?+ imag9$ by ,lotze2 & z few writ]s seem cle> on ?is po9t1 ,i w ent] some:at fully into :at ,i conceive to be its purpose4 ,9 ! fir/ place1 !re are "s writ]s--notably ,clif=d--:o1 be+ nave reali/s z reg>ds space1 hold t our evid5ce is :olly insuffici5t1 z yet1 to decide z to its nature 9 ! inf9ite or 9 ! inf9itesimal 7cf4 ,essays1 ,vol4 ,i4 p4 #cbj73 ^! writ]s are n conc]n$ ) any possibility ( be+s diff]5t f ourselves1 b simply ) ! ev]yday space we "k1 :i* !y inve/igate 9 ! _s ( a *emi/ discuss+ :e!r hydrog5 is a metal1 or an a/ronom] discuss+ ! nebul> hypo!sis4 ,b ^! are a m9ority3 mo/1 m cauti\s1 admit t our space1 s f> z obs]vation ext5ds1 is ,euclidean1 & if n accurately ,euclidean1 mu/ be only slightly sph]ical or pseudo-sph]ical4 ,"h aga91 x is ! space ( daily life :i* is "u discussion1 & "h fur!r ! discussion is1 ,i ?9k1 indep5d5t ( any philosophical assumption z to ! nature ( our space-intuition4 ,= ev5 if ?is be purely subjective1 ! translation ( an intuition into a conception c only be accompli%$ approximately1 )9 ! errors ( obs]vation incid5t to self-analysis2 & until ! intuition ( space has become a conception1 we get no sci5tific ,geometry4 ,! apodeictic c]ta9ty ( ! axiom ( p>allels %r9ks to an unmotiv$ subjective conviction1 & vani%es altoge!r 9 ^? :o ent]ta9 non-,euclidean d\bts4 ,to re9=ce ! ,euclidean fai?1 reason mu/ n[ be br\ght to ! aid ( intuition2 b reason1 un=tunately1 ab&ons u1 & we are left to ! m]cy ( approximate obs]vations ( /ell> triangles--a meagre support1 inde$1 = ! *]i%$ religion ( our *ildhood4 #hi4 ,b ! possibility ( an inaccuracy s slight1 t our f9e/ in/rum5ts & our mo/ di/ant p>allaxes %[ no trace ( x1 w\ld tr\ble m5's m9ds no m ?an ! analog\s *ance ( inaccuracy 9 ! law ( gravitation1 7 x n = ! philosophical import ( ev5 ! sl5d]e/ possibility 9 ?is sph]e4 ,& x is ! philosophical be>+ ( ,metageometry alone1 ,i ?9k1 :i* con/itutes its real importance4 ,ev5 if1 z we w suppose = ! mom5t1 obs]vation _h e/abli%$1 beyond ! possibility ( d\bt1 t our space might be safely reg>d$ z ,euclidean1 /ill ,metageometry w\ld h %[n a philosophical possibility1 & on ?is gr\nd alone x c\ld claim1 ,i ?9k1 v ne>ly all ! att5tion :i* x at pres5t des]ves4 ,b :at is ?is possibility8 ,a ?+ is possible1 accord+ to ,bradley 7,logic1 p4 #ahg71 :5 x w\ld foll[ f a c]ta9 numb] ( conditions1 "s ( :i* are kn[n to be realiz$4 ,n[ ! conditions to :i* a =m ( ext]nality mu/ con=m1 9 ord] to be affirm$1 are3 fir/1 ( c\rse1 t x %\ld be exp]i5c$1 or legitimately inf]r$ f some?+ exp]i5c$2 b secondly1 t x %\ld con=m to c]ta9 logical conditions1 detail$ 9 ,*apt] ,,iii41 :i* may be summ$ up 9 ! relativity ( position4 ,n[ :at ,metageometry has done1 9 any case1 is to sugge/ ! pro( t ! second ( ^! conditions is fulfill$ by non-,euclidean spaces4 ,euclid is affirm$1 !re=e1 on ! gr\nd ( imm$iate exp]i5ce alone1 & 8 tru?1 z unm$iat$ by logical necessity1 is m]ely ass]torical1 or1 if we pref] x1 empirical4 ,?is is ! mo/ important s5se1 x seems to me1 9 :i* non-,euclidean spaces are possible4 ,!y are1 9 %ort1 a /ep 9 a philosophical argum5t1 r ?an 9 ! inve/igation ( fact3 !y ?r[ light on ! nature ( ! gr\nds = ,euclid1 r ?an on ! actual con=mation ( space#ajb4 ,?is import ( ,metageometry is d5i$ by ,lotze1 on ! gr\nd t non-,euclidean logic is faulty1 a gr\nd :i* he endeav\rs1 by mu* detail & ?r\gh _m pages1 to make good--) :at success1 we w n[ proce$ to exam9e4 #ij4 ,lotze's attack on ,metageometry--al?\gh x rema9s1 s f> z ,i "k1 ! be/ ho/ile criticism extant1 & al?\gh its argum5ts h become "p ( ! regul> /ock-9-trade ( ,euclidean philosoph]s--conta9s1 if ,i am n mi/ak51 sev]al misund]/&+s due to insuffici5t ma!matical k ( ! subject4 ,z ^! misund]/&+s h be5 widely spread among philosoph]s1 & _c be easily remov$ except by a critic :o has gone into non-,euclidean ,geometry ) "s c>e1 x seems desirable to discuss ,lotze's /rictures po9t by po9t4 ,illu/ration #ia4 ,! ma!matical criticism beg9s 7 #aca7 ) a some:at "q-begg+ def9ition ( p>allel /raight l9es4 ,two /raight l9es a1 b1 accord+ to ?is def9ition1 are p>allel :5--a & b be+ arbitr>y po9ts on ! two l9es--if a b1 !n ab 1 :]e 1 are two o!r po9ts on ! two /raight l9es respectively4 ,?is def9ition--:i* conta9s ,euclid's axiom & def9ition comb9$ 9 a v conv5i5t & entic+ =m--is ( c\rse ?or\ghly suitable to ,euclidean ,geometry1 & leads imm$iately to all ! ,euclidean propositions ab\t p>allels4 ,b x is p]haps m hone/ to foll[ ,euclid's c\rse2 :5 an axiom is ?us buri$ 9 a def9ition1 x is apt to seem1 s9ce def9itions are suppos$ to be arbitr>y1 z ?\gh ! difficulty _h be5 ov]come1 :ile 9 reality1 ! possibility ( p>allels1 z above def9$1 involves ! v po9t 9 "q1 namely1 ! disput$ axiom ( p>allels4 ,= :at ?is axiom ass]ts is simply ! exi/5ce ( l9es con=m+ to ,lotze's def9ition4 ,! d$uction ( ! pr9cipal propositions on p>allels1 ) :i* ,lotze foll[s up 8 def9ition1 is ( c\rse a v simple proce$+--a proce$+1 h[ev]1 9 :i* ! fir/ /ep begs ! "q4 #ib4 ,! next argum5t = ! apriority ( ,euclidean ,geometry has1 oddly en\gh1 an exactly opposite be>+1 al?\gh x is a great fav\rite ) oppon5ts ( ,metageometry4 ,measurem5ts ( /ell> triangles1 & all simil> attempts at an empirical det]m9ation ( ! space-con/ant are1 accord+ to ,lotze1 beside ! m>k2 = any obs]v$ dep>ture f two "r angles1 or any f9ite annual p>allax = di/ant />s1 w\ld be attribut$ to "s new k9d ( refraction1 or1 z 9 ! case ( ab]ration1 to "s o!r physical cause1 & nev] to ! geometrical nature ( space4 ,?is is a /rong argum5t = ! empirical validity ( ,euclid1 b z an argum5t = ! apodeictic c]ta9ty ( ! or?odox sy/em1 x has an opposite t5d5cy4 ,= obs]vations ( ! k9d contemplat$ w\ld h to be due to dep>tures f ,euclidean /raightness1 hi!rto unkn[n1 on ! "p ( /ell> light-rays4 ,su* dep>ture c\ld1 9 c]ta9 cases1 be acc\nt$ = by a f9ite space-con/ant1 b x c\ld also1 probably1 be acc\nt$ = by a *ange 9 ,optics1 = example1 by attribut+ refractive prop]ties to ! e!r4 ,su* prop]ties c\ld only exi/ if e!r 7 ( v>y+ d5sity1 if 7say7 x 7 d5s] 9 ! neighb\rhood ( any ( ! heav5ly bodies4 ,b su* an assumption w\ld1 ,i believe1 de/roy ! utility ( e!r = ,physics2 a slight alt]ation 9 our ,geometry1 s slight z n appreciably to affect di/ances )9 ! ,sol> ,sy/em1 w\ld probably be 9 ! end1 !re=e1 %\ld su* errors ev] be discov]$1 a simpl] explanation ?an any t ,physics c\ld (f]4 ,b ?is is n ! po9t ( my cont5tion4 ,! po9t is t1 if ! physical explanation1 z ,lotze holds1 be possible 9 ! above case1 ! conv]se mu/ also hold3 x mu/ be possible to expla9 ! pres5t ph5om5a by suppos+ e!r refractive & space non-,euclidean4 ,f ?is conclusion !re is no escape4 ,if e conceivable behavi\r ( light-rays c be expla9$1 )9 ,euclid1 by physical causes1 x mu/ also be possible1 by a suitable *oice ( hypo!tical physical causes1 to expla9 ! actual ph5om5a z belong+ to a non-,euclidean space4 ,su* a hypo!sis w\ld be rightly reject$ by ,sci5ce1 = ! pres5t1 on acc\nt ( its unnecess>y complexity4 ,nev]!less x w\ld rema91 = philosophy1 a possibility to be reckon$ )1 & ! *oice c\ld only be decid$ ^u empirical gr\nds ( simplicity4 ,x may well be d\bt$ :e!r1 9 ! _w we "k1 ! ph5om5a c\ld be attribut$ to a di/9ctly non-,euclidean space1 b ?is conclusion foll[s inevitably f ! cont5tion t no ph5om5a c\ld =ce u to assume su* a space4 ,lotze's argum5t1 !re=e1 if pu%$ home1 disproves 8 own view1 & puts ,euclidean space1 z an empirical explanation ( ph5om5a1 on a level ) lum9if]\s e!r#ajc4 #ic4 ,lotze n[ proce$s 7 #acb7 to a detail$ criticism ( ,helmholtz1 :om he reg>ds z a typical expon5t ( ,metageometry4 ,x is possible t1 at ! "t :5 he wrote1 ,helmholtz really did occupy ?is position2 b x is un=tunate t1 9 ! m9ds ( philosoph]s1 he %\ld /ill cont9ue to d s1 aft] ! v mat]ial advances br\ght ab\t by ! projective treatm5t ( ! subject4 ,x is also un=tunate t 8 some:at c>eless attempts to popul>ise ma!matical results h s (t5 be5 dispos$ (1 )\t due att5tion to 8 m te*nical & solid contributions4 ,?us 8 romances ab\t ,flatl& & ,sph]el&--at be/ only fairy-tale analogies ( d\btful value--h be5 attack$ z if !y =m$ an ess5tial feature ( ,metageometry4 ,b to proce$ to p>ticul>s3 ,lotze readily all[s t ! ,flatl&]s w\ld set up ,plane ,geometry1 z we "k x1 b refuses to admit t ! ,sph]el&]s c\ld1 )\t inf]r+ ! ?ird dim5sion1 set up a two-dim5sional sph]ical ,geometry :i* %\ld be free f contradictions4 ,i w endeav\r to give a free r5d]+ ( ,lotze's argum5t on ?is po9t4 ,illu/ration ,suppose1 he says1 a nor? & s\? pole1 ,n & ,s1 arbitr>ily fix$1 & an equator ,,ew4 ,suppose a be+1 ,b1 capable ( impressions only f ?+s on ! surface ( ! sph]e1 to move 9 a m]idian ,,nbs4 ,let ,b />t f "s po9t a1 & f9ally1 aft] describ+ a great circle1 return to ! same po9t a4 ,if a is kn[n only by ! quality ( ! impression x makes on ,b1 ,b may imag9e he has n rea*$ ! same po9t a1 b ano!r simil> po9t a1 be>+ a relation to a simil> to t ( ! octave 9 s++3 he might ev5 n arrange 8 impressions spatially at all4 ,9 ord] t ?is may occur1 we require ! fur!r assumption1 t e diff]5ce 9 ! above-m5tion$ feel+s 7z he describes ! m]idian7 may be pres5t$ z a spatial di/ance betwe5 two places4 ,ev5 n[1 ,b may ?9k he is describ+ a ,euclidean /raight l9e1 conta9+ simil> po9ts at c]ta9 int]vals4 ,all[+1 h[ev]1 t he realizes ! id5tity ( a ) 8 initial position1 he w n[ seem1 by motion 9 a /raight l9e1 to h return$ to ! po9t f :i* he />t$1 = 8 motion _c1 )\t ! ?ird dim5sion1 seem to him o!r ?an rectil9e>4 ,up to ?is po9t1 !re seems little gr\nd = objection1 except1 p]haps1 to ! idea ( a /raight l9e ) p]iodical simil> po9ts--if ,b 7 z philosophical z1 9 ^! discussions1 we usually suppose him to be1 he w\ld probably object to ?is int]pretation ( 8 exp]i5ces1 on ! gr\nd t x reg>ds empty space z some?+ indep5d5t ( ! objects 9 x4 ,x is wor? po9t+ out1 also1 t ,b w\ld n ne$ to describe ! :ole circle1 9 ord] sudd5ly to f9d himself home aga9 ) 8 old fri5ds4 ,accurate measurem5ts ( small triangles w\ld suffice to det]m9e 8 space-con/ant1 & %[ him ! l5g? ( a great circle 7or /raight l9e1 z he w\ld call x74 ,we mu/ admit1 also1 t s hypo!tical a be+ z ,b might =m no space-intuition at all1 b z he is introduc$ solely = ! purposes ( ! analogy1 x is conv5i5t to all[ him all possible qualifications = 8 po/4 ,b ^! po9ts d n t\* ! k]nel ( ! argum5t1 :i* lies 9 ! /atem5t t su* a /raight l9e1 return+ into itself aft] a f9ite "t1 w\ld appe> to ,b z an un5durable contradiction1 & ?us =ce him1 = logical ?\gh n = s5sational purposes1 into ! assumption ( a ?ird dim5sion4 ,?is ass]tion seems to me q unw>rant$3 ! :ole ( ,metageometry is a solid array 9 dispro( ( x4 ,helmholtz's argum5t is1 x mu/ be rememb]$1 only an analogy1 & ! contradiction w\ld exi/ only = a ,euclidean4 ,a complete ?ree-dim5sional ,geometry has1 we h se5 9 ,*apt] ,i41 be5 develop$ on ! assumption t /raight l9es are ( f9ite l5g?4 ,a con/ant value = ! measure ( curvature1 z our discussion ( ,riemann %[$1 involves nei!r ref]5ce to ! f\r? dim5sion1 nor any k9d ( int]nal contradiction4 ,?is fact disproves ,lotze's cont5tion1 :i* arises solely f inability to dive/ 8 imag9ation ( ,euclidean ideas4 ,lotze next attacks ,helmholtz = ! ass]tion t ,b w\ld "k no?+ ( p>allel l9es--p>allel /raight l9es1 z ! context %[s1 he meant to say#ajd4 ,lotze1 h[ev]1 takes him z mean+1 app>5tly1 m]e curves ( con/ant di/ance f a giv5 /raight l9e1 :i* are "p ( ! regul> /ock-9-trade ( ,metageometry4 ,p>allels ( latitude1 9 ! geographical s5se1 w\ld n--) ! exception ( ! equator--appe> to ,b z /raight l9es1 b z circles4 ,great circles he w\ld call /raight1 & ?is fact seems to h misl$ ,lotze into ?9k+ all circles 7 to be treat$ z /raight l9es4 ,p>allels ( latitude1 !re=e1 ?\gh ,b might call !m p>allels1 w\ld n invalidate ,helmholtz's cont5tion1 :i* applies only to /raight l9es4 ,! argum5t t su* small circles w\ld be p>allel1 :i* we h j dispos$ (1 is only ! preface to ano!r pro( t ,b w\ld ne$ a ?ird dim5sion4 ,let u call two ( ^! p>allels ( latitude ln & ls1 & let !m be equidi/ant f ! equator1 "o 9 ! nor!rn1 "o 9 ! s\!rn hemisph]e4 ,consecutive tang5t planes1 along ^! p>allels1 conv]ge1 9 ! "o case nor?w>ds1 9 ! o!r s\?w>ds4 ,ei!r ,b c\ld become aw>e ( _! diff]5ce1 says ,lotze1 or he c\ld n4 ,9 ! =m] case1 :i* he reg>ds z ! m probable1 he easily proves t ,b w\ld inf] a ?ird dim5sion4 ,b ?is alt]native is1 ,i ?9k1 :olly inadmissible4 ,tang5t planes1 l ,euclidean planes 9 g5]al1 w\ld h no mean+ to ,b2 unless1 inde$1 he 7 a metageometrician1 :i*1 ) all 8 metaphysical & ma!matical subtlety1 ! argum5t supposes him n to be--& to su* a supposition ,lotze1 surely1 is ! la/ p]son :o has a "r to object4 ,lotze's attempt$ pro( t ?is is ! "r alt]native re/s1 if ,i und]/& him aright1 on a %e] error 9 ord9>y sph]ical ,geometry4 ,b w\ld obs]ve1 he says1 t ! m]idians made small] angles ) 8 pa? t[>ds ! ne>] ?an t[>ds ! fur!r pole--z a matt] ( fact1 !y w\ld be simply p]p5dicul> to 8 pa? 9 bo? directions4 ,:at ,lotze means is1 p]haps1 t all ! m]idians w\ld meet soon] 9 "o direction ?an 9 ! o!r1 & ?is1 ( c\rse1 is true4 ,b ! poles1 9 :i* ! m]idians meet1 w\ld appe> to ,b z ! c5tres ( ! respective p>allels1 :ile ! p>allels !mselves w\ld appe> to be circles4 ,n[ ,i am at a loss to see :at difficulty w\ld arise1 to ,b1 9 suppos+ two diff]5t circles to h diff]5t c5tres#aje4 ,we mu/1 !re=e1 take ! fir/ alt]native1 t ,b w\ld h no sort ( k z to ! direction 9 :i* ! tang5t planes conv]g$4 ,"h ,lotze attempts1 if ,i h n misund]/ood him1 to prove a r$uctio ad absurdum3 ,b w\ld ?9k1 he says1 t he 0 describ+ two pa?s :olly ! same 9 direction1 & !n he might reg>d bo? pa?s z circles 9 a plane4 ,x may be obs]v$ t direction1 :5 appli$ to a circle z a :ole1 is mean+less2 inde$ direction1 9 all ,metageometry1 c only mean1 ev5 :5 appli$ to /raight l9es1 direction t[>ds a po9t4 ,to speak ( two l9es1 :i* d n meet1 z hav+ ! same direction1 is a surreptiti\s introduction ( ! axiom ( p>allels4 ,ap>t f ?is1 ,i _c conceive any objection1 on ,b's "p1 to su* a view--"o %\ld say mu/1 n might4 ,! :ole argum5tation1 !re=e1 unless its obscurity has l$ me a/ray1 mu/ be pron\nc$ fruitless & inconclusive4 #id4 ,aft] ?is prelim9>y discussion ( ,sph]el&1 ,lotze proce$s to ! "q ( a f\r? dim5sion1 & !nce to sph]ical & pseudo-sph]ical space4 ,z be=e1 he appe>s to "k only ! m c>eless & popul> utt]ances ( ,helmholtz & ,riemann1 & to h tak5 no tr\ble to und]/& ev5 ! f\ndations ( ma!matical ,metageometry4 ,by ?is neglect1 mu* ( :at he says is r5d]$ :olly wor?less4 ,to beg9 )1 he reg>ds1 z ! purpose ( ,helmholtz's fairy tale1 ! sugge/ion ( a possible f\r? dim5sion1 :]eas ! real purpose 0 q ! opposite--to make intelligible a purely ?ree-dim5sional non-,euclidean space4 ,helmholtz introduc$ ,flatl& only because its relation to ,sph]el& is analog\s to ! relation ( ours to sph]ical space#ajf4 ,b ,lotze says3 ,! ,flatl&]s w\ld f9d no difficulty 9 a ?ird dim5sion1 s9ce x w\ld 9 no way contradict _! own ,geometry1 :ile ! p 9 ,sph]el&1 f ! contradictions 9 _! two-dim5sional sy/em1 w\ld already h be5 l$ to x4 ,! latt] cont5tion ,i h already tri$ to answ]2 ! =m] has an odd s\nd1 9 view ( ! attempt1 a few pages lat]1 to prove priori t all =ms ( intuition1 9 any way analog\s to space1 mu/ h ?ree dim5sions4 ,"o _c help suspect+ t ! ,flatl&]s1 ) two in/ead ( ?ree dim5sions1 w\ld make a simil> attempt4 ,b to return to ,lotze's argum5t3 ,nei!r analogy c be us$1 he says1 to prove t we ought p]haps to set up a f\r? dim5sion1 s9ce1 = u1 no contradictions or o!rwise inexplicable ph5om5a exi/4 ,! only p1 s f> z ,i "k1 :o h us$ ?is analogy1 are ,dr ,abbot & a few ,spirituali/s--! =m] 9 joke1 ! latt] to expla9 c]ta9 ph5om5a m simply expla9$1 p]haps1 by ,maskelyne & ,cooke4 ,b al?\gh ,lotze's conclusion 9 ?is matt] is s\nd1 & "o ) :i* ,helmholtz might h agre$1 8 argum5ts1 to my m9d1 are irrelevant & unconv9c+4 ,!re is ?is diff]5ce1 he says1 betwe5 u & ! ,sph]el&]s3 ! latt] 7 logically =c$ to a new dim5sion1 & f\nd x possible2 we are n =c$ to x1 & f9d x1 9 our space1 impossible4 ,i h cont5d$ t1 on ! contr>y1 no?+ w\ld =ce ! ,sph]el&]s to assume a ?ird dim5sion1 :ile !y w\ld f9d x impossible exactly z we f9d a f\r? impossible--n logically1 t is to say1 b only z a pres5table con/ruction 9 giv5 space4 ,aft] a some:at elephant9e piece ( hum\r1 ab\t sociali/ic :ales 9 a f\r-dim5sional sea ( ,f\rri]'s eau sucre1 ,lotze proce$s to a pro(1 by logic1 t e =m ( intuition1 :i* embraces ! :ole sy/em ( ord]$ relations ( a coexi/+ manifold1 mu/ h ?ree dim5sions4 ,"o might object1 on priori gr\nds1 to any su* attempt3 :at belongs to pure intuition c\ld h>dly1 "o w\ld h ?\ght1 be det]m9$ by priori reason+#ajg4 ,i w n1 h[ev]1 develop ?is argum5t "h1 b endeav\r to po9t out1 z f> z its obscurity w all[1 ! p>ticul> fallacy ( ! pro( 9 "q4 ,illu/ration ,lotze's argum5t is z foll[s4 ,9 ?is discussion1 ?\gh our t]m9ology is necess>ily tak5 f space1 we are really conc]n$ ) a mu* m g5]al conception4 ,we assume1 9 ord] to pres]ve ! homog5eity ( dim5sions1 t ! diff]5ce 7di/ance7 betwe5 any two elem5ts 7po9ts7 ( our manifold--to borr[ ,riemann's ^w--is ( ! same k9d z1 & comm5surable )1 ! diff]5ce betwe5 any o!r two elem5ts4 ,let u take a s]ies ( elem5ts at successive di/ances x su* t ! di/ance betwe5 any two is ! sum ( ! di/ances betwe5 int]m$iate elem5ts4 ,su* a s]ies corresponds to a /raight l9e1 :i* is tak5 z ! x-axis4 ,!n a s]ies ,,oy is call$ p]p5dicul> to ! x-axis ,,ox1 :5 ! di/ances ( any elem5t y1 on ,,oy1 f mx & -mx are equal4 ,by our hypo!sis1 ^! di/ances are comp>able )1 & qualitatively simil> to1 x & y4 ,s long z ,,oy is def9$ only by relation to ,,ox1 x is conceptually unique4 ,b n[ let u suppose ! same relation z t betwe5 ,,ox & ,,oy1 to be possible betwe5 ,,oy & a new s]ies ,,oz2 we !n get a ?ird s]ies ,,oz p]p5dicul> to ,,oy1 & aga9 conceptually unique1 s long z x is def9$ by relation to ,,oy alone4 ,we might proce$1 9 ! same way1 to a f\r? l9e ,,ou p]p5dicul> to ,,oz4 ,b x is necess>y1 = our purposes1 t ,,oz %\ld be p]p5dicul> to ,,ox z well z ,,oy4 ,)\t ?is condition1 ,,oz might ext5d into ano!r _w1 & h no correspond+ relation to ,,ox--?is is a possibility only exclud$ by our unavoidable spatial images4 ,at ?is po9t comes ! crux ( ! argum5t4 ,t ,,oz1 says ,lotze1 :i*1 besides be+ p]p5dicul> to ,,oy1 is also p]p5dicul> to ,,ox1 mu/ be among ! s]ies ( ,,oy's1 = ^! 7 def9$ only by p]p5dicul>ity to ,,ox4 ,h5ce1 he concludes1 !re c only be ev5 a ?ird dim5sion if ,,oz co9cides ) "o1 &--z soon z ,,ox is consid]$ fix$--) only "o1 ( ! _m memb]s ( ! ,,oy s]ies4 ,9 ?is argum5t x is difficult--to me at any rate--to see any =ce at all4 ,! only way ,i c acc\nt = x is1 to suppose t ,lotze has neglect$ ! possibility ( any b s+le inf9ities4 ,on ?is int]pretation1 ! argum5t might be /at$ ?us3 ,!re is an inf9ite s]ies ( cont9u\sly v>y+ ,,oy's2 to ! common prop]ty ( ^!1 we add ano!r prop]ty1 :i* w divide _! total numb] by inf9ity4 ,! rema9+ ,,oz1 !re=e1 mu/ be uniquely det]m9$4 ,! same =m ( argum5t1 h[ev]1 w\ld prove t two surfaces c only cut "o ano!r 9 a s+le po9t1 & numb]less o!r absurdities4 ,! fact is1 t inf9ities may be ( diff]5t ord]s4 ,= example1 ! numb] ( po9ts 9 a l9e may be tak5 z a s+le inf9ity1 & s may ! numb] ( l9es 9 a plane ?r\gh any po9t2 h5ce1 by multiplication1 ! numb] ( po9ts 9 a plane is a d\ble inf9ity1 #b1 & if we divide ?is numb] by a s+le inf9ity1 we get /ill an inf9ite numb] left4 ,?us ,lotze's argum5t assumes :at he has to prove1 t ! numb] ( l9es p]p5dicul> to a giv5 l9e1 ?r\gh any po9t1 is a s+le inf9ity1 :i* is equival5t to ! axiom ( ?ree dim5sions4 ,! :ole passage is s obscure1 t its mean+ may h escap$ me4 ,x is obvi\s priori1 h[ev]1 z ,i po9t$ out 9 ! beg9n+1 t any pro( ( ! axiom mu/ be fallaci\s some:]e1 & ! above int]pretation ( ! argum5t is ! only "o ,i h be5 able to f9d4 #ie4 ,! re/ ( ! ,*apt] is devot$ to an attack on sph]ical & pseudo-sph]ical space1 on ! gr\nd t !y int]f]e ) ! homog5eity ( ! ?ree dim5sions1 & ) ! simil>ity ( all p>ts ( space4 ,?is is simply false4 ,su* spaces1 l ! surface ( a sph]e1 are exactly alike ?r\gh\t4 ,lotze %[s1 "h & else:]e1 t he has n tak5 ! pa9s to f9d out :at ,metageometry really is4 ,i hold myself1 & h tri$ to prove 9 ?is ,essay1 t ,congru5ce is an priori axiom1 )\t :i* ,geometry w\ld be impossible2 b ! wi% to uphold ?is axiom is1 z ,lotze ought to h kn[n1 ! precise motive :i* l$ ,metageometry to limit itself to spaces ( con/ant measure ( curvature4 ,we see "h ! importance ( di/+ui%+ betwe5 ,helmholtz ! philosoph] & ,helmholtz ! ma!matician4 ,?\gh ! philosoph] wi%$ to disp5se ) ,congru5ce1 ! ma!matician1 z we saw 9 ,*apt] ,i41 reta9$ & /rongly emphasiz$ x4 ,a little lat] ,lotze %[s1 aga91 h[ he has be5 misl$ by ! un=tunate analogy ( ,sph]el&4 ,a sph]ical surface1 he says1 he c und]/&2 b h[ are we to pass f ?is to a sph]ical space8 ,ei!r ?is surface is ! :ole ( our space1 z 9 ,sph]el&1 or x g5]ates space by a gradually gr[+ radius4 ,su* conc5tric sph]es1 z ,lotze triumphantly po9ts out1 ( c\rse g5]ate ,euclidean space4 ,8 disjunction1 h[ev]1 is utt]ly & entirely false1 & c\ld nev] h be5 sugge/$ by any "o ) ev5 a sup]ficial k ( ,metageometry4 ,?is po9t is less lab\r$ ?an ! =m]1 :i*1 9 all its nak$ness1 is ?us re-/at$ 9 ! la/ s5t5ce ( ! ,*apt]3 ,i _c p]suade myself t "o c\ld1 )\t ! elem5ts ( homog5e\s space1 ev5 =m or def9e ! pres5tation ( het]og5e\s spaces1 or ( su* z _h v>iable measures ( curvature4 ,z ?\gh su* spaces 7 ev] set up by non-,euclidean ma!matics6 ,9 conclusion1 ,lotze expresses a hope t ,philosophy1 on ?is po9t1 w n all[ itself to be impos$ ^u by ,ma!matics4 ,i mu/1 in/ead1 rejoice t ,ma!matics has n be5 impos$ ^u by ,philosophy1 b has develop$ freely an important & self-consi/5t sy/em1 :i* des]ves1 = its subtle analysis into logical & factual elem5ts1 ! gratitude ( all :o seek = a philosophy ( space4 #if4 ,! objections to non-,euclidean ,geometry :i* h j be5 discuss$ fall "u f\r heads3 ,i4 ,non-,euclidean spaces are n homog5e\s2 ,metageometry !re=e unduly reifies space4 ,,ii4 ,!y involve a ref]5ce to a f\r? dim5sion4 ,,iii4 ,!y _c be set up )\t an implicit ref]5ce to ,euclidean space1 or to ! ,euclidean /raight l9e1 on :i* !y are !re=e dep5d5t4 ,,iv4 ,!y are self-contradictory 9 "o or m ways4 ,! read] :o has foll[$ me 9 reg>d+ ^! f\r objections z fallaci\s1 w h no difficulty 9 dispos+ ( any o!r critic ( ,metageometry1 z ^! are ! only ma!matical argum5ts1 s f> z ,i "k1 ev] urg$ aga9/ non-,euclideans#ajh4 ,! logical validity ( ,metageometry1 & ! ma!matical possibility ( ?ree-dim5sional non-,euclidean spaces1 w !re=e be reg>d$1 ?r\gh\t ! rema9d] ( ! "w1 z suffici5tly e/abli%$4 #ig4 ,two o!r objections may1 inde$1 be urg$ aga9/ ,metageometry1 b ^! are r ( a philosophical ?an ( a /rictly ma!matical import4 ,! fir/ ( ^!1 :i* has be5 made ! base ( op]ations by ,delbuf1 applies equally to all non-,euclidean spaces4 ,! second1 :i* has n1 s f> z ,i "k1 be5 mu* employ$1 b yet seems to me des]v+ ( notice1 be>s directly aga9/ spaces ( positive curvature alone2 b if x c\ld discr$it ^!1 x might ?r[ d\bt on ! me?od by :i* all alike are obta9$4 ,! two objections are3 ,i4 ,space mu/ be su* z to all[ ( simil>ity1 i4e4 ( ! increase or dim9ution1 9 a con/ant ratio1 ( all ! l9es 9 a figure1 )\t *ange ( angles2 :]eas 9 non-,euclid1 l9es1 l angles1 h absolute magnitude4 ,,ii4 ,space mu/ be inf9ite1 :]eas sph]ical & elliptic spaces are f9ite4 ,i w discuss ! fir/ objection 9 connection ) ,delbuf's articles ref]r$ to above4 ,! second1 :i* has n1 to my k1 be5 widely us$ 9 criticism1 w be bett] def]r$ to ,*apt] ,,iii4 ,delbuf4 #ih4 ,m4 ,delbuf's f\r articles 9 ! ,revue ,philosophique conta9 mu* matt] t has already be5 dealt ) 9 ! criticism ( ,lotze1 & mu* t is irrelevant = our pres5t purpose4 ,! only po9t1 :i* ,i wi% to discuss "h1 is ! "q ( absolute magnitude1 z x is call$--! "q1 t is1 :e!r ! possibility ( simil> b unequal geometrical figures c be kn[n priori#aji4 ,9 discuss+ ?is "q1 x is important1 to beg9 )1 to di/+ui% cle>ly ! s5se 9 :i* absolute magnitude is requir$ 9 non-,euclidean ,geometry1 f ano!r s5se1 9 :i* x w\ld be absurd to reg>d any magnitude z absolute4 ,judgm5ts ( magnitude c only result f comp>ison1 & if ,metageometry requir$ magnitudes :i* c\ld be det]m9$ )\t comp>ison1 x w\ld c]ta9ly des]ve condemnation4 ,b ?is is n requir$4 ,all we require is1 t x %all be impossible1 :ile ! re/ ( space is unaffect$1 to alt] ! magnitude ( any figure1 z comp>$ ) o!r figures1 :ile leav+ ! relative int]nal magnitudes ( its p>ts un*ang$4 ,?is con/ruction1 :i* is possible 9 ,euclid1 is impossible 9 ,metageometry4 ,we h to discuss :e!r su* an impossibility r5d]s non-,euclidean spaces logically faulty4 ,m4 ,delbuf's position on ?is axiom--:i* he calls ! po/ulate ( homog5eity#aaj--;is1 t all ,geometry mu/ presuppose x1 & t ,metageometry1 consequ5tly1 ?\gh logically s\nd1 is logically subsequ5t to ,euclid1 & c only make its con/ructions )9 a ,euclidean homog5e\s space 7,rev4 ,phil4 ,vol4 ,,xxxvii41 pp4 #chj-a74 ,he w\ld appe> to ?9k1 nev]!less1 t homog5eity 79 8 s5se7 is le>nt f exp]i5ce1 ?\gh on ?is po9t he is n v explicit4 7,see ,vol4 ,,xxxviii41 p4 #abi47 ,no priori pro(1 at any rate1 is (f]$ 9 8 articles4 ,z a result ( exp]i5ce1 e "o w\ld admit1 simil>ity is kn[n to be possible )9 ! limits ( obs]vation2 b ! fact t ?is possibility ext5ds to ,ordnance maps1 :i* deal ) a sph]ical surface1 %\ld make u *>y ( inf]r+1 f su* a datum1 ! c]ta9ty ( ,euclid = l>ge spaces4 ,moreov] if homog5eity be empirical1 ,metageometry1 :i* disp5ses ) x1 is n necess>ily 9 logical dep5d5ce ^u ,euclid1 s9ce homog5eity & isog5eity are logically sep>able4 ,i %all assume1 !re=e1 z ! only cont5tion :i* c be int]e/+ to our argum5t1 t homog5eity is reg>d$ z priori1 & z logically ess5tial to ,geometry4 #ii4 ,n[ we saw1 9 discuss+ ,erdmann's views ( ! judgm5t ( quantity1 t 9 non-,euclidean space1 z 9 ,euclidean1 a *ange ( all spatial magnitudes1 9 ! same ratio1 w\ld be no *ange at all2 ! ratios ( all magnitudes to ! space-con/ant w\ld be un*ang$1 & ! space-con/ant1 z ! ultimate /&>d ( comp>ison1 _c1 9 any intelligible s5se1 be said to h any p>ticul> magnitude4 ,! absolute magnitudes ( ,metageometry1 !re=e1 are absolute only z aga9/ any o!r p>ticul> magnitude1 n z aga9/ o!r magnitudes 9 g5]al4 ,if ?is 7 n ! case1 ! comp>ative nature ( ! judgm5t ( magnitude w\ld be contradict$1 & metrical ,metageometry w\ld become absurd4 ,b z x is1 ! diff]5ce f ,euclid consi/s only 9 ?is3 t 9 ,metageometry we h1 :ile 9 ,euclid we h n1 a /&>d ( comp>ison involv$ 9 ! nature ( our space z a :ole1 :i* we call ! space-con/ant4 ,we h to discuss :e!r ! ass]tion ( su* a /&>d involves an undue reification ( space4 ,i d n believe t ?is is ! case4 ,= an undue reification ( space w\ld only arise1 if we 7 no long] able to reg>d position z :olly relative1 & z geometrically def9able only by dep>ture f o!r positions4 ,b ! relativity ( position1 z we h abundantly se51 is pres]v$ by all spaces ( con/ant curvature--9 all ( ^!1 positions c only be def9$1 geometrically1 by relations to fre% positions#aaa4 ,?is s]ies ( def9itions may lead to an inf9ite regress1 b x may also1 z 9 sph]ical space1 =m a vici\s circle1 & return aga9 to ! position f :i* x />t$4 ,no reification ( space1 no indep5d5t exi/5ce ( m]e relations1 seems involv$ 9 su* a proc$ure4 ,! :ole ( ,metageometry1 9 %ort1 is a pro( t ! relativity ( position is compatible ) absolute magnitude1 9 ! only s5se requir$ by non-,euclidean spaces4 ,we mu/ conclude1 !re=e1 t !re is no?+ incompatible1 9 a d5ial ( homog5eity 79 ,delbuf's s5se71 ei!r ) ! relational nature ( space1 or ) ! comp>ative nature ( magnitude4 ,?is la/ priori objection to ,metageometry1 !re=e1 _c be ma9ta9$1 & ! issue mu/ be decid$ on empirical gr\nds alone4 #ajj4 ,! f\ndations ( ,geometry h be5 ! subject ( mu* rec5t speculation 9 ,france1 & ?is seems to dem& "s notice4 ,b 9 spite ( ! spl5did "w :i* ! ,fr5* h done on ! alli$ "q ( numb] & cont9u\s quantity1 ,i _c p]suade myself t !y h succe$$ 9 greatly advanc+ ! subject ( geometrical philosophy4 ,! *ief writ]s h be51 f ! ma!matical side1 ,cal9on & ,po9c>1 f ! philosophical1 ,r5\vi] & ,delbuf2 z a m$iator betwe5 ma!matics & philosophy1 ,le*alas4 ,cal9on1 9 an int]e/+ article on ! geometrical indet]m9at5ess ( ! univ]se1 ma9ta9s t any ,geometry may be appli$ to ! actual _w by a suitable hypo!sis z to ! c\rse ( light-rays4 ,= ! e>? only is kn[n to u o!rwise ?an by ,optics1 & ! e>? is an inf9itesimal "p ( ! univ]se4 ,?is l9e ( argum5t has be5 already discuss$ 9 connection ) ,lotze1 b ,cal9on adds a new sugge/ion1 t ! space-con/ant may p]haps v>y ) ! "t4 ,?is w\ld involve a causal connection betwe5 space & o!r ?+s1 :i* seems h>dly conceivable1 & :i*1 if reg>d$ z possible1 mu/ surely de/roy ,geometry1 s9ce ,geometry dep5ds ?r\gh\t on ! irrelevance ( ,causation#aab4 ,moreov]1 9 all op]ations ( measurem5t1 "s "t is sp5t2 unless we knew t space 0 un*ang+ ?r\gh\t ! op]ation1 x is h>d to see h[ our results c\ld be tru/wor?y1 & h[1 consequ5tly1 a *ange 9 ! p>amet] c\ld be discov]$4 ,! same difficulties w\ld arise1 9 fact1 z ^? :i* result f suppos+ space n homog5e\s4 ,po9c> ma9ta9s t ! "q1 :e!r ,euclid or ,metageometry %\ld be accept$1 is "o ( conv5i5ce & conv5tion1 n ( tru?2 axioms are def9itions 9 disguise1 & ! *oice betwe5 def9itions is arbitr>y4 ,?is view has be5 discuss$ 9 ,*apt] ,i41 9 connection ) ,cayley's !ory ( di/ance1 on :i* x dep5ds4 ,le*alas is a philosophical disciple ( ,cal9on4 ,he is a rationali/ ( ! pre-,kantian type1 b a believ] 9 ! validity ( ,metageometry4 ,he holds t ,geometry c disp5se ) all purely spatial po/ulates1 & "w ) axioms ( magnitude alone#aac1 :i*1 9 8 op9ion1 are purely analytic4 ,! pr9ciple ( contradiction1 to him1 is ! sole & only te/ ( tru?2 we make long *a9s ( reason+ f our premisses to see if contradictions w em]ge4 ,x might be object$ t ?is view1 ?\gh x saves g5]al ,geometry f be+ logically empirical1 leaves x only empirically logical2 ?is mu/1 9 fact1 be ! fate ( e piece ( priori k1 if ,m4 ,le*alas's 7 ! only te/ ( tru?4 ,h[ev]1 he concludes t g5]al ,geometry is apodeictic1 :ile ! space ( our actual _w1 l all o!r ph5om5a1 is cont+5t4 ,delbuf criticizes non-,euclidean space f an ultra-reali/ /&po9t3 he holds t real space is nei!r homog5e\s nor isog5e\s1 b t conceiv$ space1 z ab/ract$ f real space1 has bo? ^! prop]ties4 ,he (f]s no ju/ification = 8 real space1 :i* seems to be ma9ta9$ 9 ! _s ( nave realism1 nor does he %[ h[ he has acquir$ 8 intimate k ( its con/itution#aad4 ,8 argum5ts aga9/ ,metageometry1 9 s f> z !y are n repetitions ( ,lotze1 h be5 discuss$ above4 ,r5\vi]1 f9ally1 is a pure ,kantian1 ( ! mo/ or?odox type4 ,8 views z to ! importance1 = ,geometry1 ( ! di/9ction betwe5 syn!tic & analytic judgm5ts1 h be5 discuss$1 9 connection ) ,kant1 at ! beg9n+ ( ! pres5t ,*apt]#aae4 #aja4 ,be=e beg9n+ ! con/ructive argum5t ( ! next ,*apt]1 let u endeav\r briefly to sum up ! !ories :i* h be5 polemically advocat$ ?r\gh\t ! criticisms we h j conclud$4 ,we agre$ to accept1 ) ,kant1 necessity = any possible exp]i5ce z ! te/ ( ! priori1 b we refus$1 = ! pres5t1 to discuss ! connection ( ! priori ) ! subjective1 reg>d+ ! purely logical te/ z suffici5t = our imm$iate purpose4 ,we also refus$ to atta* importance to ! di/9ction ( analytic & syn!tic1 s9ce x seem$ to apply1 n to diff]5t judgm5ts1 b only to diff]5t aspects ( any judgm5t4 ,we !n discuss$ ,riemann's attempt to id5tify ! empirical elem5t 9 ,geometry ) ! elem5t n d$ucible f ideas ( magnitude1 & we decid$ t ?is id5tification 0 due to a confusion z to ! nature ( magnitude4 ,= judgm5ts ( magnitude1 we said1 require always "s qualitative basis1 :i* is n quantitatively expressible4 ,9 criticiz+ ,helmholtz1 we decid$ t ,me*anics logically presupposes ,geometry1 ?\gh space presupposes matt]2 b t ! matt] :i* space presupposes1 & to :i* ,geometry indirectly ref]s1 is a m ab/ract matt] ?an t ( ,me*anics1 a matt] de/itute ( =ce & ( causal attributes1 & possess$ only ( ! purely spatial attributes requir$ = ! possibility ( spatial figures4 ,b we conc$$ t ,geometry1 :5 appli$ to mix$ ma!matics or to daily life1 dem&s m ?an ?is1 dem&s1 9 fact1 "s means ( discov]+1 9 ! m concrete matt] ( ,me*anics1 ei!r a rigid body1 or a body ^: dep>ture f rigidity foll[s "s empirically discov]able law4 ,actual measurem5t1 !re=e1 we agre$ to reg>d z empirical4 ,our conclusions1 z reg>ds ! empiricism ( ,riemann & ,helmholtz1 7 re9=c$ by a criticism ( ,erdmann4 ,we !n _h an opposite task to p]=m1 9 def5d+ ,metageometry aga9/ ,lotze4 ,"h we saw t !re are two s5ses 9 :i* ,metageometry is possible4 ,! fir/ conc]ns our actual space1 & ass]ts t x may h a v small space-con/ant2 ! second conc]ns philosophical !ories ( space1 & ass]ts a purely logical possibility1 :i* leaves ! decision to exp]i5ce4 ,we saw also t ,lotze's ma!matical /rictures arose f insuffici5t k ( ! subject1 & c\ld all be refut$ by a bett] acqua9tance ) ,metageometry4 ,f9ally1 we discuss$ ! "q ( absolute magnitude1 & f\nd 9 x no logical ob/acle to non-,euclidean spaces4 ,our conclusion1 !n1 9 s f> z we are z yet entitl$ to a conclusion1 is t all spaces ) a space-con/ant are priori ju/ifiable1 & t ! decision betwe5 !m mu/ be ! "w ( exp]i5ce4 ,spaces )\t a space-con/ant1 on ! o!r h&1 spaces1 t is1 :i* are n homog5e\s ?r\gh\t1 we f\nd logically uns\nd & impossible to "k1 & !re=e to be condemn$ priori4 ,! con/ructive pro( ( ?is !sis w =m ! argum5t ( ! foll[+ *apt]4 ,,footnotes3 #fg ,! ,critical ,philosophy ( ,kant1 ,vol4 ,i4 p4 #bhg4 #fh ,= a discussion ( ,kant f a less purely ma!matical /&po9t1 see ,*ap4 ,,iv4 #fi ,cf4 ,vaih+]'s ,comm5t>1 ,,ii4 pp4 #bjb1 #bfe4 ,also p4 #ccf ff4 #gj ,e4g4 second edition1 p4 #ci3 ,s w]d5 au* alle geometris*5 ,grundstze1 z4 ,b4 dass 9 e9em ,triangel zwei ,seit5 zusamm5 grss] s9d als die dritte1 niemals aus allgeme95 ,begriff5 von ,l9ie und ,triangel1 sond]n aus d] ,ans*auung1 und zw> priori mit apodiktis*] ,gewis%eit abgeleitet4 #ga ,cf4 ,bradley's ,logic1 ,bk4 ,,iii4 ,pt4 ,i4 ,*ap4 ,,vi42 ,bosanquet's ,logic1 ,bk4 ,i4 ,*ap4 ,i4 pp4 #ig-ajc4 #gb ,philosophie de la ,rgle et du ,compas1 ,anne ,philosophique1 ,,ii4 pp4 #a-ff4 #gc ,i h /at$ ?is doctr9e dogmatically1 z a pro( w\ld require a :ole treatise on ,logic4 ,i accept ! pro(s (f]$ by ,bradley & ,bosanquet1 to :i* ! read] is ref]r$4 #gd ,= a fur!r discussion ( ?is po9t1 see ,*aps4 ,,iii4 & ,,iv4 #ge ,see ,*ap4 ,,iv4 = a discussion ( ?is argum5t4 #gf ,see ,*ap4 ,,iv4 #ahe4 #gg ,an ,o!rness ( sub/ance1 r ?an ( attribute1 is "h int5d$2 an ,o!rness :i* may p]haps be call$ real z oppos$ to logical div]sity4 #gh ,?is proposition w be argu$ at l5g? 9 ,*ap4 ,,iv4 #gi ,see ,psy*ologie als ,wiss5s*aft1 ,i4 ,section ,,iii4 ,*ap4 ,,vii42 ,,ii4 ,section ,i4 ,*ap4 ,,iii4 & ,section ,,ii4 ,*ap4 ,,iii4 ,comp>e also ,syne*ologie1 ,section ,i4 ,*aps4 ,,ii4 & ,,iii4 #hj ,on ! influ5ce ( ,h]b>t on ,riemann1 comp>e ,erdmann1 ,die ,axiome d] ,geometrie1 p4 #cj4 #ha ,i d n mean t measurem5t ( col\rs is effect$ )\t ref]5ce to _! relations1 s9ce all measurem5t is ess5tially comp>ison4 ,b 9 col\rs1 x is ! elem5ts :i* are comp>$1 :ile 9 space1 x is ! relations betwe5 elem5ts4 #hb ,= a discussion ( ?is po9t1 see ,*ap4 ,,iii4 ,sec4 ,b1 #agf4 #hc ,! works ( ,helmholtz on geometrical philosophy comprise1 9 addition to ! articles quot$ 9 ,*ap4 ,i41 ! foll[+ articles3 ,ursprung und ,s9n d] geometris*5 ,axiome1 geg5 ,l&1 ,wiss4 ,abh4 ,vol4 ,,ii4 p4 #fdj1 #ahgh4 7,also ,m9d1 ,vol4 ,,iii43 an answ] to ,l& 9 ,m9d1 ,vol4 ,,ii47 ,ursprung und ,b$eutung d] geometris*5 ,axiome1 #ahgj1 ,vortrge und ,r$51 ,vol4 ,,ii4 p4 #a4 7,also ,m9d1 ,vol4 ,i47 ,two ,app5dices to ,die ,?atsa*5 9 d] ,wahrnehmung1 entitl$3 ,,ii4 ,d] ,raum kann transc5d5tal se91 ohne dass es die ,axiome s9d2 & ,,iii4 ,die ,anw5db>keit d] ,axiome auf die physis*e ,welt1 #ahgh1 ,vortrge und ,r$51 ,vol4 ,,ii4 p4 #bef ff4 ,! two ,app5dices la/ m5tion$ are popul>iz+s & expansions ( ! article 9 ,m9d1 ,vol4 ,,iii4 ,! mo/ widely read1 ?\gh also1 to my m9d1 ! lea/ valuable1 ( all ,helmholtz's writ+s on ,geometry1 is ! article 9 ,m9d1 ,vol4 ,i4 ,?is conta9s ! fam\s & mu* misund]/ood analogies ( ,flatl& & ,sph]el&1 :i* w be discuss$1 & z f> z possible def5d$1 9 answ]+ ,lotze's attack on ,metageometry--an attack bas$1 app>5tly1 almo/ entirely on ?is "o popul> article4 ,! pres5t discussion1 !re=e1 may be conf9$ almo/ entirely to ,m9d1 ,vol4 ,,iii41 & ! philosophical portions ( ! two pap]s quot$ 9 ,*ap4 ,i41 i4e4 to ! articles 9 ,wiss4 ,abh4 ,vol4 ,,ii4 pp4 #faj-ffj4 ,8 o!r works are popul>1 & important only because ( ! l>ge public to :i* !y appeal4 #hd ,9 ! answ] to ,l&1 ,m9d1 ,vol4 ,,iii4 & ,wiss4 ,abh4 ,,ii4 p4 #fdj4 #he ,see also ,die ,?atsa*5 9 d] ,wahrnehmung1 ,zusatz ,,ii41 ,d] ,raum kann transc5d5tal se91 ohne dass es die ,axiome s9d4 ,vortrge und ,r$51 ,vol4 ,,ii4 #hf ,see bel[1 criticism ( ,erdmann1 #hd4 #hg ,see ,pr(4 ,l&1 9 ,m9d1 ,vol4 ,,ii4 #hh ,see conclud+ p>agraph ( ,helmholtz's article 9 ,m9d1 ,vol4 ,,iii4 #hi ,cf4 ,v]onese1 ,grundzge d] ,geometrie 7,g]man translation71 p4 ix4 ,also pp4 xxxiv1 #cjd1 & ,note ,,ii4 pp4 #fib-d4 #ij ,see ,*ap4 ,,iv4 #aig ff4 #ia ,cf4 ! op9ion ( ,bolyai1 quot$ by ,erdmann1 ,axiome1 p4 #bf2 cf4 also ib4 p4 #fj4 #ib ,die ,axiome d] ,geometrie3 ,e9e philosophis*e ,unt]su*ung d] ,riemann-,helmholtz's*5 ,raum!orie1 ,leipzig1 #ahgg4 #ic ,on ! influ5ce ( ,mill1 cf4 ,/allo1 ,concepts ( ,mod]n ,physics1 p4 #baf4 #id ,?is view seems to be d]iv$1 ?r\gh ,riemann1 f ,h]b>t4 ,see ,psy*4 als ,wiss4 ed4 ,h>t4 ,vol4 ,v4 p4 #bfb4 #ie ,! same irr$ucibility ( space to m]e magnitude is prov$ by ,kant's h&s & sph]ical triangles1 9 :i* a diff]5ce p]si/s 9 spite ( complete quantitative equality4 #if ,see #adf-g4 #ig ,j$] ,v]su*1 ,kant's ,lehre von d] ,aprioritt als des subjectiv51 von all] ,erfahrung absolut unabhngig5 ,erk5ntnissfactors1 trotzdem zu halt51 i/ de%alb von voruh]e9 aussi*tslos4 #ih ,ausdehnungslehre von #ahdd1 #bnd edition1 pp4 xxii4 xxiii4 #ii ,see #abi ff4 #ajj ,metaphysik1 ,book ,,ii4 ,*ap4 ,,ii4 ,my ref]5ces are to ! orig9al4 #aja ,see ,lectures & ,essays1 ,vol4 ,i4 p4 #bfa4 #ajb ,on ! mean+ ( geometrical possibility1 cf4 ,v]onese1 ,grundzge d] ,geometrie 7,g]man translation71 pp4 xi4-xiii4 #ajc ,comp>e ,cal9on1 ,sur l',indt]m9ation gomtrique de l',univ]s1 ,revue ,philosophique1 #ahic1 ,vol4 ,,xxxvi4 pp4 #eie-fjg4 #ajd ,vortrge und ,r$51 ,vol4 ,,ii4 p4 #i3 ,p>allele ,l9i5 wrd5 die ,bewohn] d] ,kugel g> ni*t k5n54 ,sie wrd5 behaupt51 dass j$e beliebige zwei g]ade/e ,l9i51 gehrig v]lang]t1 si* s*liessli* ni*t nur 9 e9em1 sond]n 9 zwei ,punkt5 s*neid5 ms/54 7,! italics are m9e47 ,! omission ( /raight 9 su* phrases is a frequ5t laxity ( ma!maticians4 #aje ,x has be5 sugge/$ to me t ,lotze reg>ds ! m]idians z project$ on to a plane1 z 9 a map4 ,if ?is be s1 !re is an obvi\sly illegitimate introduction ( ! ?ird dim5sion4 #ajf ,?is is prov$ by ,helmholtz's rem>k at ! end ( a detail$ attempt to make sph]ical & pseudo-sph]ical spaces imag9able 7l4c4 p4 #bh73 ,&]s i/ es mit d5 drei ,dim5sion5 des ,raumes4 ,da alle uns]e ,mittel s9nli*] ,ans*auung si* nur auf e95 ,raum von drei ,dim5sion5 er/reck51 und die vi]te ,dim5sion ni*t bloss e9e ,abnd]ung von ,vorh&5em1 sond]n etwas vollkomm5 ,neues wre1 s bef9d5 wir uns s*on weg5 uns]] krp]li*5 ,organisation 9 d] absolut5 ,unmgli*keit1 uns e9e ,ans*auungsweise e9] vi]t5 ,dim5sion vorzu/ell54 #ajg ,cf4 ,grassmann1 ,ausdehnungslehre von #ahdd1 #bnd ,edition1 p4 xxiii4 #ajh ,see especially ,/allo1 ,concepts ( ,mod]n ,physics1 ,int]national ,sci5ce ,s]ies1 ,vol4 ,,xlii4 ,*aps4 ,,xiii4 & ,,xiv42 ,r5\vi]1 ,philosophie de la rgle et du compas1 ,anne ,philosophique1 ,,ii42 ,delbuf1 ,l'anci5ne et les n\velles gomtries1 ,revue ,philosophique1 ,vols4 ,,xxxvi4-,,xxxix4 #aji ,m4 ,delbuf des]ves cr$it = hav+ bas$ ,euclid1 already 9 #ahfj1 9 8 ,prolgomnes ,philosophiques de la ,gomtrie1 on ?is axiom--c]ta9ly a bett] basis1 at fir/ sight1 ?an ! axiom ( p>allels4 #aaj ,?is mean+ ( homog5eity mu/ n be conf\nd$ ) ! s5se 9 :i* ,i h us$ ! ^w4 ,9 ,delbuf's s5se1 x means t figures may be simil> ?\gh ( diff]5t sizes2 9 my s5se x means t figures may be simil> ?\gh 9 diff]5t places4 ,?is prop]ty ( space is call$ by ,delbuf isog5eity4 #aaa ,= a full pro( ( ?is proposition1 see ,*ap4 ,,iii4 #aab ,see ,*ap4 ,,iii41 especially #acc4 #aac ,= a criticism ( ?is view1 see ! above discussions on ,riemann & ,erdmann4 #aad ,cf4 ,c\turat1 ,de l',inf9i ,ma?matique1 ,p>is1 ,flix ,alcan1 #ahif1 p4 #edd4 #aae ,! foll[+ is a li/ ( ! mo/ important rec5t ,fr5* philosophical writ+s on ,geometry1 s f> z ,i am acqua9t$ ) !m4 ,&rade3 ,les bases exprim5tales de la gomtrie euclidi5ne2 ,rev4 ,phil4 #ahij1 ,,ii41 & #ahia1 ,i4 ,bonnel3 ,les hypo?ses dans la gomtrie2 ,gau?i]-,vill>s1 #ahig4 ,l',abb de ,broglie3 ,la gomtrie non-euclidi5ne1 two articles2 ,annales de ,phil4 ,*rt4 #ahij4 ,cal9on3 ,les espaces gomtriques2 ,rev4 ,phil4 #ahhi1 ,i41 & #ahia1 ,,ii4 ,sur l'indt]m9ation gomtrique de l'univ]s2 ib4 #ahic1 ,,ii4 ,c\turat3 ,l',anne ,philosophique de ,f4 ,pillon1 ,rev4 de ,mt4 et de ,morale1 ,jan4 #ahic4 ,note sur la gomtrie non-euclidi5ne et la relativit de l'espace2 ib41 ,may1 #ahic4 tudes sur l'espace et le temps1 ib4 ,sep4 #ahif4 ,delbuf3 ,l'anci5ne et les n\velles gomtries1 f\r articles2 ,rev4 ,phil4 #ahic-e4 ,le*alas3 ,la gomtrie gnrale2 ,crit4 ,phil4 #ahhi4 ,la gomtrie gnrale et les jugem5ts syn?tiques priori & ,les bases exprim5tales de la gomtrie2 ,rev4 ,phil4 #ahij1 ,,ii4 ,m4 ,delbuf et ,le problme des mondes semblables2 ib4 #ahid1 ,i4 ,note sur la gomtrie non-euclidi5ne et le pr9cipe de similitude2 ,rev4 de ,mt4 et de ,morale1 ,m>*1 #ahic4 ,la c\rbure et la di/ance en gomtrie gnrale2 ib41 ,m>*1 #ahif4 ,la gomtrie gnrale et l'intuition2 ,annales de ,phil4 ,*rt41 #ahij4 ,etude sur l'espace et le temps2 ,p>is1 ,alcan1 #ahif4 ,li>d3 ,des df9itions gomtrie et des df9itions empiriques1 #bnd ed42 ,p>is1 ,alcan1 #ahhh4 ,mansion3 ,premi]s pr9cipes de la mtagomtrie2 two articles 9 ,rev4 ,no-,s*ola/ique1 #ahif4 ,sep>ately publi%$1 ,gau?i]-,vill>s1 #ahif4 ,milhaud3 ,la gomtrie non-euclidi5ne et la ?orie de la connaissance2 ,rev4 ,phil4 #ahhh1 ,i4 ,po9c>3 ,non-,euclidian ,geometry2 ,nature1 ,vol4 ,,xlv41 #ahia-b4 ,l'espace et la gomtrie2 ,rev4 de ,mt4 et de ,morale1 ,nov4 #ahie4 ,rsponse quelques critiques1 ib4 ,jan4 #ahig4 ,r5\vi]3 ,philosophie de la rgle et du compas2 ,crit4 ,phil41 #ahhi1 & ,l',anne ,phil41 ,,iime anne1 #ahia4 ,sorel3 ,sur la gomtrie non-euclidi5ne2 ,rev4 ,phil41 #ahia1 ,i4 ,tann]y3 ,?orie de la connaissance ma?matique2 ,rev4 ,phil41 #ahid1 ,,ii4 ,,*apt] ,,iii4 ,section ,a4 ,,! ,,axioms ,,( ,,projective ,,geometry4 #ajb4 ,projective ,geometry prop]1 z we saw 9 ,*apt] ,i41 does n employ ! conception ( magnitude1 & does n1 !re=e1 require ^? axioms :i*1 9 ! sy/ems ( ! second or metrical p]iod1 7 requir$ solely to r5d] possible ! application ( magnitude to space4 ,b we saw1 also1 t ,cayley's r$uction ( metrical to projective prop]ties 0 purely te*nical & philosophically irrelevant4 ,n[ x is 9 metrical prop]ties alone--ap>t f ! exception to ! axiom ( ! /raight l9e1 :i* itself1 h[ev]1 presupposes metrical prop]ties#aaf--t non-,euclidean & ,euclidean spaces diff]4 ,! prop]ties dealt ) by projective ,geometry1 !re=e1 9 s f> z ^! are obta9$ )\t ! use ( imag9>ies1 are prop]ties common to all spaces4 ,f9ally1 ! diff]5ces :i* appe> betwe5 ! ,geometries ( diff]5t spaces ( ! same curvature--e4g4 betwe5 ! ,geometries ( ! plane & ! cyl9d]--are diff]5ces 9 projective prop]ties#aag4 ,?us ! necessity :i* arises1 9 metrical ,geometry1 = fur!r qualifications besides ^? ( con/ant curvature1 disappe>s :5 our g5]al space is def9$ by purely projective prop]ties4 #ajc4 ,we h good gr\nd = expect+1 !re=e1 t ! axioms ( projective ,geometry w be ! simple/ & mo/ complete expression ( ! indisp5sable requisites ( any geometrical reason+3 & ?is expectation1 ,i hope1 w n be disappo9t$4 ,projective ,geometry1 9 s f> z x deals only ) ! prop]ties common to all spaces1 w be f\nd1 if ,i am n mi/ak51 to be :olly priori1 to take no?+ f exp]i5ce1 & to h1 l ,ari?metic1 a creature ( ! pure intellect = its object4 ,if ?is be s1 x is t bran* ( pure ma!matics :i* ,grassmann1 9 8 ,ausdehnungslehre ( #ahdd1 felt to be possible1 & endeav\r$1 9 a brilliant failure1 to con/ruct )\t any appeal to ! space ( intuition4 #ajd4 ,b un=tunately1 ! task ( discov]+ ! axioms ( projective ,geometry is f> f easy4 ,!y h1 z yet1 f\nd no ,riemann or ,helmholtz to =mulate !m philosophically4 ,_m geomet]s h con/ruct$ sy/ems1 :i* !y int5d$ to be1 & :i*1 ) suffici5t c>e 9 int]pretation1 really are1 free f metrical presuppositions4 ,b ^! presuppositions are s root$ 9 all ! v elem5ts ( ,geometry1 t ! task ( elim9at+ !m dem&s a recon/ruction ( ! :ole geometrical edifice4 ,?us ,euclid1 = example1 deals1 f ! />t1 ) spatial equality--he employs ! circle1 :i* is necess>ily def9$ by means ( equality1 & he bases all 8 lat] propositions on ! congru5ce ( triangles z discuss$ 9 ,book ,i4#aah ,be=e we c use any elem5t>y proposition ( ,euclid1 !re=e1 ev5 if ?is expresses a projective prop]ty1 we h to prove t ! prop]ty 9 "q c be d$uc$ by projective me?ods4 ,?is has n1 9 g5]al1 be5 done by projective geomet]s1 :o h too (t5 assum$1 = example1 t ! quadrilat]al con/ruction--by :i*1 z we saw 9 ,*ap4 ,i41 !y introduce projective coord9ates--or anh>monic ratio1 :i* is prim facie metrical1 c\ld be satisfactorily e/abli%$ on _! pr9ciples4 ,bo? ^! assumptions1 h[ev]1 c be ju/ifi$1 & we may admit1 !re=e1 t ! claims ( projective ,geometry to logical indep5d5ce ( measurem5t or congru5ce are valid4 ,let u see1 !n1 h[ x proce$s4 #aje4 ,9 ! fir/ place1 x is important to realize t :5 coord9ates are us$1 9 projective ,geometry1 !y are n coord9ates 9 ! ord9>y metrical s5se1 i4e4 ! num]ical measures ( c]ta9 spatial magnitudes4 ,on ! contr>y1 !y are a set ( numb]s1 arbitr>ily b sy/ematically assign$ to diff]5t po9ts1 l ! numb]s ( h\ses 9 a /reet1 & s]v+ only1 f a philosophical /&po9t1 z conv5i5t designations = po9ts :i* ! inve/igation wi%es to di/+ui%4 ,b = ! brevity ( ! alphabet1 9 fact1 !y might1 z 9 ,euclid1 be replac$ by lett]s4 ,h[ !y are introduc$1 & :at !y mean1 has be5 discuss$ 9 ,*apt] ,i4 ,"h we h only to repeat a caution1 ^: neglect has l$ to mu* misund]/&+4 #ajf4 ,! di/9ction betwe5 v>i\s po9ts1 !n1 is n a result1 b a condition1 ( ! projective coord9ate sy/em4 ,! coord9ate sy/em is a :olly extrane\s1 & m]ely conv5i5t1 set ( m>ks1 :i* 9 no way t\*es ! ess5ce ( projective ,geometry4 ,:at we mu/ beg9 )1 9 ?is doma91 is ! possibility ( di/+ui%+ v>i\s po9ts f "o ano!r4 ,?is may be designat$1 ) ,v]onese1 z ! fir/ axiom ( ,geometry#aai4 ,h[ we are to def9e a po9t1 & h[ we di/+ui% x f o!r po9ts1 is = ! mom5t irrelevant2 = "h we only wi% to discov] ! nature ( projective ,geometry1 & ! k9d ( prop]ties :i* x uses & demon/rates4 ,h[1 & ) :at ju/ification1 x uses & demon/rates !m1 we w discuss lat]4 #ajg4 ,n[ x is obvi\s t a m]e collection ( po9ts1 di/+ui%$ "o f ano!r1 _c f\nd a ,geometry3 we mu/ h "s idea ( ! mann] 9 :i* ! po9ts are int]relat$1 9 ord] to h an adequate subject-matt] = discussion4 ,b s9ce all ideas ( quantity are exclud$1 ! relations ( po9ts _c be relations ( di/ance 9 ! ord9>y s5se1 nor ev51 9 ! s5se ( ord9>y ,geometry1 anh>monic ratios1 = anh>monic ratios are usually def9$ z ! ratios ( f\r di/ances1 or ( f\r s9es1 & are ?us quantitative4 ,b s9ce all quantitative comp>ison presupposes an id5tity ( quality1 we may expect to f9d1 9 projective ,geometry1 ! qualitative sub/rata ( ! metrical sup]/ructure4 ,& ?is1 we %all see1 is actually ! case4 ,we h n di/ance1 b we h ! /raight l9e2 we h n quantitative anh>monic ratio1 b we h ! prop]ty1 9 any f\r po9ts on a l9e1 ( be+ ! int]sections ) ! rays ( a giv5 p5cil4 ,& f ?is basis1 we c build up a qualitative sci5ce ( ab/ract ext]nality1 :i* is projective ,geometry4 ,h[ ?is happ5s1 ,i %all n[ proce$ to %[4 #ajh4 ,all geometrical reason+ is1 9 ! la/ resort1 circul>3 if we />t by assum+ po9ts1 !y c only be def9$ by ! l9es or planes :i* relate !m2 & if we />t by assum+ l9es or planes1 !y c only be def9$ by ! po9ts ?r\gh :i* !y pass4 ,?is is an inevitable circle1 ^: gr\nd ( necessity w appe> z we proce$4 ,x is1 !re=e1 some:at arbitr>y to />t ei!r ) po9ts or ) l9es1 z ! em95tly projective pr9ciple ( duality ma!matically illu/rates2 nev]!less we w elect1 ) mo/ geomet]s1 to />t ) po9ts#abj4 ,we suppose1 !re=e1 z our datum1 a set ( discrete po9ts1 = ! mom5t )\t reg>d to _! int]connections4 ,b s9ce connections are ess5tial to any reason+ ab\t !m z a sy/em1 we introduce1 to beg9 )1 ! axiom ( ! /raight l9e4 ,any two ( our po9ts1 we say1 lie on a l9e :i* ^? two po9ts completely def9e4 ,?is l9e1 be+ det]m9$ by ! two po9ts1 may be reg>d$ z a relation ( ! two po9ts1 or an adjective ( ! sy/em =m$ by bo? toge!r4 ,?is is ! only purely qualitative adjective--z w be prov$ lat]--( a sy/em ( two po9ts4 ,n[ projective ,geometry c only take acc\nt ( qualitative adjectives1 & c di/+ui% betwe5 diff]5t po9ts only by _! relations to o!r po9ts1 s9ce all po9ts1 p] se1 are qualitatively simil>4 ,h5ce x comes t1 = projective ,geometry1 :5 two po9ts only are giv51 !y are qualitatively indi/+ui%able f any two o!r po9ts on ! same /raight l9e1 s9ce any two su* o!r po9ts h ! same qualitative relation4 ,reciprocally1 s9ce "o /raight l9e is a figure det]m9$ by any two ( its po9ts1 & all po9ts are qualitatively simil>1 x foll[s t all /raight l9es are qualitatively simil>4 ,we may reg>d a po9t1 !re=e1 z det]m9$ by two /raight l9es :i* meet 9 x1 & ! po9t1 on ?is view1 becomes ! only qualitative relation betwe5 ! two /raight l9es4 ,h5ce1 if ! po9t only be reg>d$ z giv51 ! two /raight l9es are qualitatively indi/+ui%able f any o!r pair ?r\gh ! po9t4 #aji4 ,! ext5sion ( ^! two reciprocal pr9ciples is ! ess5ce ( all projective trans=mations1 & inde$ ( all projective ,geometry4 ,! fundam5tal op]ations1 by :i* figures are projectively trans=m$1 are call$ projection & section4 ,! v>i\s =ms ( projection & section are def9$ 9 ,cremona's ,projective ,geometry1 ,*apt] ,i41 f :i* ,i quote ! foll[+ acc\nt4 ,to project f a fix$ po9t ,s 7! c5tre ( projection7 a figure 7,,abcd 444 abcd 4447 compos$ ( po9ts & /raight l9es1 is to con/ruct ! /raight l9es or project+ rays ,,sa1 ,,sb1 ,,sc1 ,,sd1 444 & ! planes 7project+ planes7 ,sa1 ,sb1 ,sc1 ,sd1 444 ,we ?us obta9 a new figure compos$ ( /raight l9es & planes :i* all pass ?r\gh ! c5tre ,s4 ,to cut by a fix$ plane 7transv]sal plane7 a figure 7 444 abcd 4447 made up ( planes & /raight l9es1 is to con/ruct ! /raight l9es or traces 1 1 444 & ! po9ts or traces a1 b1 c4444#aba ,by ?is means we obta9 a new figure compos$ ( /raight l9es & po9ts ly+ 9 ! plane 4 ,to project f a fix$ /raight l9e s 7! axis7 a figure ,,abcd compos$ ( po9ts1 is to con/ruct ! planes s,a1 s,b1 s,c4444 ,! figure ?us obta9$ is compos$ ( planes :i* all pass ?r\gh ! axis s4 ,to cut by a fix$ /raight l9e s 7a transv]sal7 a figure 444 compos$ ( planes1 is to con/ruct ! po9ts s1 s1 s4444 ,9 ?is way a new figure is obta9$1 compos$ ( po9ts all ly+ on ! fix$ transv]sal s4 ,if a figure is compos$ ( /raight l9es a1 b1 c 444 :i* all pass ?r\gh a fix$ po9t or c5tre ,s1 x c be project$ f a /raight l9e or axis s pass+ ?r\gh ,s2 ! result is a figure compos$ ( planes sa1 sb1 sc4444 ,if a figure is compos$ ( /raight l9es a1 b1 c 444 all ly+ 9 a fix$ plane1 x may be cut by a /raight l9e 7transv]sal7 s ly+ 9 ! same plane2 ! figure :i* results is =m$ by ! po9ts sa1 sb1 sc4444 #aaj4 ,! successive application1 to any figure1 ( two reciprocal op]ations ( projection & section1 is reg>d$ z produc+ a figure protectively indi/+ui%able f ! fir/1 provid$ only t ! dim5sions ( ! orig9al figure 7 ! same z ^? ( ! result+ figure1 t1 = example1 if ! second op]ation be section by a plane1 ! orig9al figure %all h be5 a plane figure4 ,! figures obta9$ f a giv5 figure1 by projection or section alone1 are relat$ to t figure by ! pr9ciple ( duality1 ( :i* we %all h to speak lat] on4 ,i %all endeav\r to %[1 9 :at foll[s1 fir/1 9 :at s5se figures obta9$ f ea* o!r by projective trans=mation are qualitatively alike2 secondly1 :at axioms1 or adjectives ( space1 are involv$ 9 ! pr9ciple ( projective trans=mation2 & ?irdly1 t ^! adjectives mu/ belong to any =m ( ext]nality ) m ?an "o dim5sion1 & are1 !re=e1 priori prop]ties ( any possible space4 ,= ! sake ( simplicity1 ,i %all 9 g5]al conf9e myself to two dim5sions4 ,9 s do+1 ,i %all introduce no important diff]5ce ( pr9ciple1 & %all greatly simplify ! ma!matics involv$4 #aaa4 ,! two ma!matically fundam5tal ?+s 9 projective ,geometry are anh>monic ratio1 & ! quadrilat]al con/ruction4 ,ev]y?+ else foll[s ma!matically f ^! two4 ,n[ :at is meant1 9 projective ,geometry1 by anh>monic ratio8 ,illu/ration ,if we />t f anh>monic ratio z ord9>ily def9$1 we are met by ! difficulty ( its quantitative nature#abb4 ,b among ! prop]ties d$uc$ f ?is def9ition1 _m1 if n mo/1 are purely qualitative4 ,! mo/ fundam5tal ( ^! is t1 if ?r\gh any f\r po9ts 9 a /raight l9e we draw f\r /raight l9es :i* meet 9 a po9t1 & if we !n draw a new /raight l9e meet+ ^! f\r1 ! f\r new po9ts ( int]section h ! same anh>monic ratio z ! f\r po9ts we />t$ )4 ,?us1 9 ! figure1 abcd1 abcd1 abcd1 all h ! same anh>monic ratio4 ,! reciprocal relation holds = ! anh>monic ratio ( f\r /raight l9es4 ,"h we h1 pla9ly1 ! requir$ basis = a qualitative def9ition4 ,! def9ition mu/ be z foll[s3 ,two sets ( f\r po9ts ea* are def9$ z hav+ ! same anh>monic ratio1 :5 7#a7 ea* set ( f\r lies 9 "o /raight l9e1 & 7#b7 correspond+ po9ts ( diff]5t sets lie two by two on f\r /raight l9es ?r\gh a s+le po9t1 or :5 bo? sets h ?is relation to any ?ird set#abc4 ,& reciprocally3 ,two sets ( f\r /raight l9es are def9$ z hav+ ! same anh>monic ratio :5 7#a7 ea* set ( f\r passes ?r\gh a s+le po9t1 & 7#b7 correspond+ l9es ( diff]5t sets pass1 two by two1 ?r\gh f\r po9ts 9 "o /raight l9e1 or :5 bo? sets h ?is relation to any ?ird set4 ,two sets ( po9ts or ( l9es1 :i* h ! same anh>monic ratio1 are treat$ by projective ,geometry z equival5t3 ?is qualitative equival5ce replaces ! quantitative equality ( metrical ,geometry1 & is obvi\sly includ$1 by its def9ition1 9 ! above acc\nt ( projective trans=mations 9 g5]al4 #aab4 ,we h next to consid] ! quadrilat]al con/ruction#abd4 ,?is has a d\ble purpose3 fir/1 to def9e ! important special case kn[n z a h>monic range2 & secondly1 to af=d an unambigu\s & exhau/ive me?od ( assign+ diff]5t numb]s to diff]5t po9ts4 ,?is la/ me?od has1 aga91 a d\ble purpose3 fir/1 ! purpose ( giv+ a conv5i5t symbolism = describ+ & di/+ui%+ diff]5t po9ts1 & ( ?us af=d+ a means = ! introduction ( analysis2 & secondly1 ( s assign+ ^! numb]s t1 if !y _h ! ord9>y metrical significance1 z di/ances f "s po9t on ! numb]$ /raight l9e1 !y w\ld yield -#a z ! anh>monic ratio ( a h>monic range1 & t1 if f\r po9ts h ! same anh>monic ratio z f\r o!rs1 s h ! correspond+ numb]s4 ,?is la/ purpose is due to purely te*nical motives3 x avoids ! confusion ) our preconceptions :i* w\ld result f any o!r value = a h>monic range2 x all[s u1 :5 metrical int]pretations ( projective results are desir$1 to make ^! int]pretations )\t t$i\s num]ical trans=mations1 & x enables u to p]=m projective trans=mations by algebraical me?ods4 ,at ! same "t1 f ! /rictly projective po9t ( view1 z obs]v$ above1 ! numb]s introduc$ h a purely conv5tional mean+2 & until we pass to metrical ,geometry1 no reason c be %[n = assign+ ! value -#a to a h>monic range4 ,) ?is prelim9>y1 let u see 9 :at ! quadrilat]al con/ruction consi/s4 ,illu/ration #aac4 ,a h>monic range1 9 elem5t>y ,geometry1 is "o ^: anh>monic ratio is -#a1 or "o 9 :i* ! ?ree segm5ts =m$ by ! f\r po9ts are 9 h>monic progression1 or aga91 "o 9 :i* ! ratio ( ! two int]nal segm5ts is equal to ! ratio ( ! two ext]nal segm5ts4 ,if a1 b1 c1 d be ! f\r po9ts1 x is easily se5 t ^! def9itions are equival5t to "o ano!r3 !y give respectively3 7ab_/bc7_/7ad_/dc7 -#a1 7#a_/;ab7 - 7#a_/;ac7 7#a_/;ac7 - 7#a_/;ad71 & 7ab_/bc7 7ad_/cd74 ,b z !y are all quantitative1 !y _c be us$ = our pres5t purpose4 ,nor are any def9itions :i* involve bisection ( l9es or angles available4 ,we mu/ h a def9ition :i* proce$s entirely by ! help ( /raight l9es & po9ts1 )\t measurem5t ( di/ances or angles4 ,n[ f ! above def9itions ( a h>monic range1 we see t a1 b1 c1 d h ! same anh>monic ratio z c1 b1 a1 d4 ,?is gives u ! prop]ty we require = our def9ition4 ,= x %[s t1 9 a h>monic range1 we c f9d a projective trans=mation :i* w int]*ange a & c4 ,?is is a necess>y & suffici5t condition = a h>monic range1 & ! quadrilat]al con/ruction is ! g5]al me?od = giv+ effect to x4 ,illu/ration ,giv5 any ?ree po9ts ,a1 ,b1 ,d 9 "o /raight l9e1 ! quadrilat]al con/ruction f9ds ! po9t ,c h>monic to ,a ) respect to ,b1 ,d by ! foll[+ me?od3 ,take any po9t ,o outside ! /raight l9e ,,abd1 & jo9 x to ,b & ,d4 ,?r\gh ,a draw any /raight l9e cutt+ ,,od1 ,,ob 9 ,p & ,q4 ,jo9 ,,dq1 ,,bp1 & let !m int]sect 9 ,r4 ,jo9 ,,or1 & let ,,or meet ,,abd 9 ,c4 ,!n ,c is ! po9t requir$4 ,to prove ?is1 let ,,drq meet ,,oa 9 ,t1 & draw ,,ar1 meet+ ,,od 9 ,s4 ,!n a projective trans=mation ( ,a1 ,b1 ,c1 ,d f ,r on to ,,od gives ! po9ts ,s1 ,p1 ,o1 ,d1 :i*1 project$ f ,a on to ,,dq1 give ,r1 ,q1 ,t1 ,d4 ,b ^! aga91 project$ f ,o on to ,,abd1 give ,c1 ,b1 ,a1 ,d4 ,h5ce ,a1 ,b1 ,c1 ,d c be projectively trans=m$ into ,c1 ,b1 ,a1 ,d1 & !re=e =m a h>monic range4 ,f ?is po9t1 ! pro( t ! con/ruction is unique & g5]al foll[s simply#abe4 ,! introduction ( numb]s1 by ?is con/ruction1 (f]s no difficulties ( pr9ciple--except1 inde$1 ^? :i* always att5d ! application ( numb] to cont9ua--& may be /udi$ satisfactorily 9 ,kle9's ,ni*t-,euklid 7,i4 p4 #ccg ff474 ,! pr9ciple ( x is1 to assign ! numb]s #j1 #a1 to ,a1 ,b1 ,d & !re=e ! numb] #b to ,c1 9 ord] t ! diff]5ces ,,ab1 ,,ac1 ,,ad may be 9 h>monic progression4 ,by tak+ ,b1 ,c1 ,d z a new triad correspond+ to ,a1 ,b1 ,d1 we f9d a po9t h>monic to ,b ) respect to ,c1 ,d & assign to x ! numb] #c1 & s on4 ,9 ?is way1 we c obta9 any numb] ( po9ts1 & we are sure ( hav+ no numb] & no po9t twice ov]1 s t our coord9ates h ! ess5tial prop]ty ( a unique correspond5ce ) ! po9ts !y d5ote1 & vice v]sa4 #aad4 ,! po9t ( importance 9 ! above con/ruction1 h[ev]1 & ! reason :y ,i h reproduc$ x 9 detail1 is t x proce$s entirely by means ( ! g5]al pr9ciples ( trans=mation enunciat$ above4 ,f ?is /age onw>ds1 ev]y?+ is effect$ by means ( ! two fundam5tal ideas we h j discuss$1 & ev]y?+1 !re=e1 dep5ds on our g5]al pr9ciple ( projective equival5ce4 ,?is pr9ciple1 z reg>ds two dim5sions1 may be /at$ m simply ?an 9 ! passage quot$ f ,cremona4 ,x />ts1 9 two dim5sions1 f ! foll[+ def9itions3 ,to project ! po9ts ,a1 ,b1 ,c1 ,d 444 f a c5tre ,o1 is to con/ruct ! /raight l9es ,,oa1 ,,ob1 ,,oc1 ,,od4444 ,to cut a numb] ( /raight l9es a1 b1 c1 d 444 by a transv]sal s1 is to con/ruct ! po9ts sa1 sb1 sc1 sd4444#abf ,! successive application ( ^! two op]ations1 provid$ ! orig9al figure consi/$ ( po9ts on "o /raight l9e or ( /raight l9es ?r\gh "o po9t1 gives a figure projectively indi/+ui%able f ! =m] figure2 & h5ce1 by ext5sion1 if any po9ts 9 "o /raight l9e 9 ! orig9al figure lie 9 "o /raight l9e 9 ! d]iv$ figure1 & reciprocally = /raight l9es ?r\gh po9ts1 ! two op]ations h giv5 projectively simil> figures4 ,?is g5]al pr9ciple may be reg>d$ z consi/+ ( two p>ts1 accord+ to ! ord] ( ! op]ations3 if we beg9 ) projection & end ) section1 we trans=m a figure ( po9ts into ano!r figure ( po9ts2 by ! conv]se ord]1 we trans=m a figure ( l9es into ano!r figure ( l9es4 #aae4 ,be=e we c be cle> z to ! mean+ ( our pr9ciple1 we mu/ h "s notion z to our def9ition ( po9ts & /raight l9es4 ,b ?is def9ition1 9 projective ,geometry1 _c be giv5 )\t "s discussion ( ! pr9ciple ( duality1 ! ma!matical =m ( ! philosophical circle involv$ 9 geometrical def9itions4 ,conf9+ ourselves = ! mom5t to two dim5sions1 ! pr9ciple ass]ts1 r\ghly speak+1 t any !orem1 deal+ ) l9es ?r\gh a po9t & po9ts on a l9e1 rema9s true if ^! two t]ms1 :]ev] !y occur1 are int]*ang$4 ,?us3 two po9ts lie on "o /raight l9e :i* !y completely det]m9e2 & two /raight l9es meet 9 "o po9t1 :i* !y completely det]m9e4 ,! f\r po9ts ( int]section ( a transv]sal ) f\r l9es ?r\gh a po9t h an anh>monic ratio indep5d5t ( ! p>ticul> transv]sal2 & ! f\r l9es jo9+ f\r po9ts on "o /raight l9e to a fif? po9t h an anh>monic ratio indep5d5t ( t fif? po9t4 ,s also our g5]al pr9ciple ( projective trans=mation has two sides3 "o 9 :i* po9ts move along fix$ l9es1 & "o 9 :i* l9es turn ab\t fix$ po9ts4 ,?is duality sugge/s t any def9ition ( po9ts mu/ be effect$ by means ( ! /raight l9e1 & any def9ition ( ! /raight l9e mu/ be effect$ by means ( po9ts4 ,:5 we take ! ?ird dim5sion into acc\nt1 x is true1 ! duality is no long] s simple2 we h n[ to take acc\nt also ( ! plane1 b ?is only introduces a circle ( ?ree t]ms1 :i* is sc>cely pref]able to a circle ( two t]ms4 ,we n[ say3 ,?ree po9ts1 or a l9e & a po9t1 det]m9e a plane3 b conv]sely1 ?ree planes1 or a l9e & plane1 det]m9e a po9t4 ,we may reg>d ! /raight l9e z a relation betwe5 two ( its po9ts1 b we may also reg>d ! po9t z a relation betwe5 two /raight l9es ?r\gh x4 ,we may reg>d ! plane z a relation betwe5 ?ree po9ts1 or betwe5 a po9t & a l9e1 b we may also reg>d ! po9t z a relation betwe5 ?ree planes1 or betwe5 a l9e & a plane1 :i* meet 9 x4 #aaf4 ,h[ are we to get outside ?is circle8 ,! fact is t1 9 pure ,geometry1 we _c get outside x4 ,= space1 z we %all see m fully h]eaft]1 is no?+ b relations2 if1 !re=e1 we take any spatial figure1 & seek = ! t]ms betwe5 :i* x is a relation1 we are compell$1 9 ,geometry1 to seek ^! t]ms )9 space1 s9ce we h n[h]e else to seek !m1 b we are doom$1 s9ce any?+ purely spatial is a m]e relation1 to f9d our t]ms melt+ away z we grasp !m4 ,?us ! relativity ( space1 :ile x is ! ess5ce ( ! pr9ciple ( duality1 at ! same "t r5d]s impossible ! expression ( t pr9ciple1 or ( any o!r pr9ciple ( pure ,geometry1 9 a mann] :i* %all be free f contradictions4 ,nev]!less1 if we are to advance at all ) our analysis ( geometrical reason+ & ) our def9itions ( l9es & po9ts1 we mu/1 = a :ile1 ignore ?is contradiction2 we mu/ argue z ?\gh x did n exi/1 s z to free our sci5ce f any contradictions :i* are n inevitable4 #aag4 ,9 accordance ) ?is proc$ure1 !n1 let u def9e our po9ts z ! t]ms ( spatial relations1 reg>d+ :atev] is n a po9t z a relation betwe5 po9ts4 ,:at1 on ?is view1 mu/ our po9ts be tak5 to be8 ,obvi\sly1 if ext5sion is m]e relativity1 !y mu/ be tak5 to conta9 no ext5sion2 b if !y are to supply ! t]ms = spatial relations1 e4g4 = /raight l9es1 ^! relations mu/ exhibit !m z ! t]ms ( ! figures !y relate4 ,9 o!r words1 s9ce :at c really be tak51 )\t contradiction1 z ! t]m ( a spatial relation1 is unext5d$1 we mu/ take1 z ! t]m to be us$ 9 ,geometry1 :]e we _c g outside space1 ! lea/ spatial ?+ :i* ,geometry c deal )1 ! ?+ :i*1 ?\gh 9 space1 conta9s no space2 & ?is ?+ we def9e z ! po9t#abg4 ,neglect+1 !n1 ! fundam5tal contradiction 9 ?is def9ition1 ! re/ ( our def9itions foll[ )\t difficulty4 ,! /raight l9e is ! relation betwe5 two po9ts1 & ! plane is ! relation betwe5 ?ree4 ,^! def9itions w be argu$ & def5d$ at l5g? 9 section ,b ( ?is ,*apt]#abh1 :]e we c discuss at ! same "t ! alt]native metrical def9itions2 = our pres5t purpose1 x is suffici5t to obs]ve t projective ,geometry1 f ! fir/1 reg>ds ! /raight l9e z det]m9$ by two po9ts1 & ! plane z det]m9$ by ?ree1 f :i* x foll[s1 if we take po9ts z possible t]ms = spatial relations1 t ! /raight l9e & ! plane may be reg>d$ z relations betwe5 two & ?ree po9ts respectively4 ,if we agree on ^! def9itions1 we c proce$ to discuss ! fundam5tal pr9ciple ( projective ,geometry1 & to analyse ! axioms implicat$ 9 its tru?4 #aah4 ,projective ,geometry1 we h se51 does n deal ) quantity1 & !re=e recognizes no diff]5ce :]e ! diff]5ce is purely quantitative4 ,n[ quantitative comp>ison dep5ds on a recogniz$ id5tity ( quality2 ! recognition ( qualitative id5tity1 !re=e1 is logically prior to quantity1 & presuppos$ by e judgm5t ( quantity4 ,h5ce all figures1 ^: diff]5ces c be exhau/ively describ$ by quantity1 i4e4 by pure measurem5t1 mu/ h an id5tity ( quality1 & ?is mu/ be recognizable )\t appeal to quantity4 ,x foll[s t1 by def9+ ! ^w quality 9 geometrical matt]s1 we %all discov] :at sets ( figures are projectively indisc]nible4 ,if our def9ition is correct1 x ought to yield ! g5]al projective pr9ciple ) :i* we set out4 #aai4 ,we agre$ to reg>d po9ts z ! t]ms ( spatial relations1 & we agre$ t diff]5t po9ts c\ld be di/+ui%$4 ,b we po/pon$ ! discussion ( ! conditions "u :i* ?is di/9ction c\ld be effect$4 ,?is discussion w yield u ! def9ition ( quality & ! pro( ( our g5]al projective pr9ciple4 ,po9ts1 to beg9 )1 h be5 def9$ z no?+ b ! t]ms = spatial relations4 ,!y h1 !re=e1 no intr9sic prop]ties2 b are di/+ui%$ solely by means ( _! relations4 ,n[ ! relation betwe5 two po9ts1 we said1 is ! /raight l9e on :i* !y lie4 ,?is gives t id5tity ( quality = all pairs ( po9ts on ! same /raight l9e1 :i* is requir$ bo? by our projective pr9ciple & by metrical ,geometry4 7,= only :]e !re is id5tity ( quality c quantity be prop]ly appli$47 ,if only two po9ts are giv51 !y _c1 )\t ! use ( quantity1 be di/+ui%$ f any two o!r po9ts on ! same /raight l9e2 = ! qualitative relation betwe5 any two su* po9ts is ! same z = ! orig9al pair1 & only by a diff]5ce ( relation c po9ts be di/+ui%$ f "o ano!r4 ,b conv]sely1 "o /raight l9e is no?+ b ! relation betwe5 two ( its po9ts1 & all po9ts are qualitatively alike4 ,h5ce !re c be no?+ to di/+ui% "o /raight l9e f ano!r except ! po9ts ?r\gh :i* x passes1 & ^! are di/+ui%$ f o!r po9ts only by ! fact t x passes ?r\gh !m4 ,?us we get ! reciprocal trans=mation3 if we are giv5 only "o po9t1 any pair ( /raight l9es ?r\gh t po9t is qualitatively indi/+ui%able f any o!r4 ,?is aga9 is1 on ! "o h&1 ! basis ( ! second "p ( our g5]al projective pr9ciple1 & on ! o!r h& ! condition ( apply+ quantity1 9 ! measurem5t ( angles1 to ! dep>ture ( two int]sect+ /raight l9es4 #abj4 ,we c n[ see ! reason = :at may h hi!rto seem$ a some:at arbitr>y fact1 namely1 ! necessity ( f\r coll9e> po9ts = anh>monic ratio4 ,recurr+ to ! quadrilat]al con/ruction & ! consequ5t introduction ( numb]1 we see t anh>monic ratio is an intr9sic projective relation ( f\r coll9e> po9ts or concurr5t /raight l9es1 su* t giv5 ?ree t]ms & ! relation1 ! f\r? t]m c be uniquely det]m9$ by projective me?ods4 ,n[ consid] fir/ a pair ( po9ts4 ,s9ce all /raight l9es are projectively equival5t1 ! relation betwe5 "o pair ( po9ts is precisely equival5t to t betwe5 ano!r pair4 ,giv5 "o po9t only1 !re=e1 no projective relation1 to any second po9t1 c be assign$1 :i* %all 9 any way limit our *oice ( ! second po9t4 ,giv5 two po9ts1 h[ev]1 !re is su* a relation--! ?ird po9t may be giv5 coll9e> ) ! fir/ two4 ,?is limits its position to "o /raight l9e1 b s9ce two po9ts det]m9e no?+ b "o /raight l9e1 ! ?ird po9t _c be fur!r limit$4 ,?us we see :y no intr9sic projective relation c be f\nd betwe5 ?ree po9ts1 :i* %all enable u1 f two1 uniquely to det]m9e ! ?ird4 ,) ?ree giv5 coll9e> po9ts1 h[ev]1 we h m giv5 ?an a m]e /raight l9e1 & ! quadrilat]al con/ruction enables u uniquely to det]m9e any numb] ( fre% coll9e> po9ts4 ,?is %[s :y anh>monic ratio mu/ be a relation betwe5 f\r po9ts1 r ?an betwe5 ?ree4 #aba4 ,we c n[ prove1 ,i ?9k1 t two figures1 :i* are projectively relat$1 are qualitatively simil>4 ,let u beg9 ) a collection ( po9ts on a /raight l9e4 ,s long z ^! are consid]$ )\t ref]5ce to o!r po9ts or figures1 !y are all qualitatively simil>4 ,!y c be di/+ui%$ by imm$iate intuition1 b :5 we endeav\r1 )\t quantity1 to di/+ui% !m conceptually1 we f9d ! task impossible1 s9ce ! only qualitative relation ( any two ( !m1 ! /raight l9e1 is ! same = any o!r two4 ,b n[ let u *oose1 at hap-haz>d1 "s po9t outside ! /raight l9e4 ,! po9ts ( our l9e n[ acquire new adjectives1 namely _! relations to ! new po9t1 i4e4 ! /raight l9es jo9+ !m to ?is new po9t4 ,b ^! /raight l9es1 reciprocally1 alone def9e our ext]nal po9t1 & all /raight l9es are qualitatively simil>4 ,if we take "s o!r ext]nal po9t1 !re=e1 & jo9 x to ! same po9ts ( our orig9al /raight l9e1 we obta9 a figure 9 :i*1 s long z quantity is exclud$1 !re is no conceptual diff]5ce f ! =m] figure4 ,imm$iate intuition c di/+ui% ! two figures1 b qualitative discrim9ation _c d s4 ,?us we obta9 a projective trans=mation ( f\r l9es into f\r o!r l9es1 z giv+ a figure qualitatively indi/+ui%able f ! orig9al figure4 ,a simil> argum5t applies to ! o!r projective trans=mations4 ,?us ! only reason1 )9 projective ,geometry1 = n reg>d+ projective figures z actually id5tical1 is ! intuitive p]ception ( diff]5ce ( position4 ,?is is fundam5tal1 & mu/ be accept$ z a datum4 ,x is presuppos$ 9 ! di/9ction ( v>i\s po9ts1 & =ms ! v life ( ,geometry4 ,x is1 9 fact1 ! ess5ce ( ! notion ( a =m ( ext]nality1 :i* notion =ms ! subject-matt] ( projective ,geometry4 #abb4 ,we may n[ sum up ! results ( our analysis ( projective ,geometry1 & /ate ! axioms on :i* its reason+ is bas$4 ,we %all !n h to prove t ^! axioms are necess>y to any =m ( ext]nality1 ) :i* we %all pass1 f m]e analysis1 to a transc5d5tal argum5t4 ,! axioms :i* h be5 assum$ 9 ! above analysis1 & :i*1 x w\ld seem1 suffice to f\nd projective ,geometry1 may be r\ghly /at$ z foll[s3 ,i4 ,we c di/+ui% diff]5t p>ts ( space1 b all p>ts are qualitatively simil>1 & are di/+ui%$ only by ! imm$iate fact t !y lie outside "o ano!r4 ,,ii4 ,space is cont9u\s & inf9itely divisible2 ! result ( inf9ite division1 ! z]o ( ext5sion1 is call$ a po9t#abi4 ,,iii4 ,any two po9ts det]m9e a unique figure1 call$ a /raight l9e1 any ?ree 9 g5]al det]m9e a unique figure1 ! plane4 ,any f\r det]m9e a correspond+ figure ( ?ree dim5sions1 & = aught t appe>s to ! contr>y1 ! same may be true ( any numb] ( po9ts4 ,b ?is process comes to an end1 soon] or lat]1 ) "s numb] ( po9ts :i* det]m9e ! :ole ( space4 ,= if ?is 7 n ! case1 no numb] ( relations ( a po9t to a collection ( giv5 po9ts c\ld ev] det]m9e its relation to fre% po9ts1 & ,geometry w\ld become impossible#acj4 ,?is /atem5t ( ! axioms is n int5d$ to h any exclusive precision3 o!r /atem5ts equally valid c\ld easily be made4 ,= all ^! axioms1 z we %all see h]eaft]1 are philosophically int]dep5d5t1 & may1 !re=e1 be enunciat$ 9 _m ways4 ,! above /atem5t1 h[ev]1 includes1 if ,i am n mi/ak51 ev]y?+ ess5tial to projective ,geometry1 & ev]y?+ requir$ to prove ! pr9ciple ( projective trans=mation4 ,be=e discuss+ ! apriority ( ^! axioms1 let u once m briefly recapitulate ! ends :i* !y are int5d$ to atta94 #abc4 ,f ! exclusively ma!matical /&po9t1 z we h se51 projective ,geometry discusses only :at figures c be obta9$ f ea* o!r by projective trans=mations1 i4e4 by ! op]ations ( projection & section4 ,^! op]ations1 9 all _! =ms1 presuppose ! po9t1 /raight l9e1 & plane#aca1 ^: necessity = projective ,geometry1 f ! purely ma!matical po9t ( view1 is ?us self-evid5t f ! />t4 ,b philosophically1 projective ,geometry has1 z we saw1 a wid] aim4 ,?is wid] aim1 :i* gives1 to ! inve/igation ( projectively equival5t figures1 its *ief importance1 consi/s 9 ! det]m9ation ( qualitative spatial simil>ity1 9 ! det]m9ation1 t is1 ( all ! figures :i*1 :5 any "o figure is giv51 c be di/+ui%$ f ! giv5 figure1 s long z quantity is exclud$1 only by ! m]e fact t !y are ext]nal to x4 #abd4 ,n[ :5 we consid] :at is involv$ 9 su* absolute qualitative equival5ce1 we f9d at once1 z its mo/ obvi\s pr]equisite1 ! p]fect homog5eity ( space4 ,= x is assum$ t a figure c be completely def9$ by its int]nal relations1 & t ! ext]nal relations1 :i* con/itute its position1 ?\gh !y suffice to di/+ui% x f o!r figures1 9 no way affect its int]nal prop]ties1 :i* are reg>d$ z qualitatively id5tical ) ^? ( figures ) q diff]5t ext]nal relations4 ,if ?is 7 n ! case1 any?+ analog\s to projective trans=mation w\ld be impossible4 ,= su* trans=mation always alt]s ! position1 i4e4 ! ext]nal relations1 ( a figure1 & c\ld n1 !re=e1 if figures 7 dep5d5t on _! relations to o!r figures or to empty space1 be /udi$ )\t ref]5ce to o!r figures1 or to ! absolute position ( ! orig9al figure4 ,we require = our pr9ciple1 9 %ort1 :at may be call$ ! mutual passivity & reciprocal indep5d5ce ( two p>ts or figures ( space4 ,?is passivity & ?is indep5d5ce involve ! homog5eity ( space1 or its equival5t1 ! relativity ( position4 ,= if ! int]nal prop]ties ( a figure are ! same1 :atev] its ext]nal relations may be1 x foll[s t all p>ts ( space are qualitatively simil>1 s9ce a *ange ( ext]nal relation is a *ange 9 ! "p ( space occupi$4 ,x foll[s1 also1 t all position is relative & extr9sic1 i4e41 t ! position ( a po9t1 or ! "p ( space occupi$ by a figure1 is n1 & has no effect ^u1 any intr9sic prop]ty ( ! po9t or figure1 b is exclusively a relation to o!r po9ts or figures 9 space1 & rema9s )\t effect except :]e su* relations are consid]$4 #abe4 ,! homog5eity ( space & ! relativity ( position1 !re=e1 are presuppos$ 9 ! qualitative spatial comp>ison ) :i* projective ,geometry deals4 ,! latt]1 z we saw1 is also ! basis ( ! pr9ciple ( duality4 ,b ^! prop]ties1 z ,i %all n[ endeav\r to prove1 belong ( necessity to any =m ( ext]nality1 & are ?us priori prop]ties ( all possible spaces4 ,to prove ?is1 h[ev]1 we mu/ fir/ def9e ! notion ( a =m ( ext]nality 9 g5]al4 ,let u obs]ve1 to beg9 )1 t ! di/9ction betwe5 ,euclidean & non-,euclidean ,geometries1 s important 9 metrical inve/igations1 disappe>s 9 projective ,geometry prop]4 ,?is sugge/s t projective ,geometry1 ?\gh orig9ally inv5t$ z ! sci5ce ( ,euclidean space1 & subsequ5tly ( non-,euclidean spaces also1 deals really ) a wid] conception1 a conception :i* includes bo?1 & neglects ! attributes 9 :i* !y diff]4 ,?is conception ,i %all speak ( z a =m ( ext]nality4 #abf4 ,9 ,grassmann's pr(\nd philosophical introduction to 8 ,ausdehnungslehre ( #ahdd1 he sugge/$ t ,geometry1 ?\gh improp]ly reg>d$ z pure1 0 really a bran* ( appli$ ma!matics1 s9ce x dealt ) a subject-matt] n creat$1 l numb]1 by ! intellect1 b giv5 to x1 & !re=e n :olly subject to its laws alone4 ,b x mu/ be possible--s he cont5d$--to con/ruct a bran* ( pure ma!matics1 a sci5ce1 t is1 9 :i* our object %\ld be :olly a creature ( ! intellect1 :i* %\ld yet deal1 z ,geometry does1 ) ext5sion--ext5sion z conceiv$1 h[ev]1 n z empirically p]ceiv$ 9 s5sation or intuition4 ,f ?is po9t ( view1 ! controv]sy betwe5 ,kantians & anti-,kantians becomes :olly irrelevant1 s9ce ! di/9ction betwe5 pure & mix$ ma!matics does n lie 9 ! di/9ction betwe5 ! subjective & ! objective1 b betwe5 ! purely intellectual on ! "o h&1 & ev]y?+ else on ! o!r4 ,n[ ,kant _h cont5d$1 ) great emphasis1 t space 0 n an intellectual con/ruction1 b a subjective intuition4 ,geometry1 !re=e1 ) ,grassmann's di/9ction1 belongs to mix$ ma!matics z mu* on ,kant's view z on t ( 8 oppon5ts4 ,& ,grassmann's di/9ction1 ,i cont5d1 is ! m important = ,epi/emology1 & ! "o to be adopt$ 9 di/+ui%+ ! priori f ! empirical4 ,= :at is m]ely intuitional c *ange1 )\t upsett+ ! laws ( ?\ght1 )\t mak+ k =mally impossible3 b :at is purely intellectual _c *ange1 unless ! laws ( ?\ght %\ld *ange1 & all our k simultane\sly collapse4 ,i %all !re=e foll[ ,grassmann's di/9ction 9 con/ruct+ an priori & purely conceptual =m ( ext]nality4 #abg4 ,! pure doctr9e ( ext5sion1 z con/ruct$ by ,grassmann1 ne$ n be discuss$--x includ$ mu* empirical mat]ial1 & 0 philosophically a failure4 ,b 8 pr9ciples1 ,i ?9k1 w enable u to prove t projective ,geometry1 ab/ractly int]pret$1 is ! sci5ce :i* he =esaw1 & deals ) a matt] :i* c be con/ruct$ by ! pure intellect alone4 ,if ?is be s1 h[ev]1 x mu/ be obs]v$ t projective ,geometry1 = ! mom5t1 is r5d]$ purely hypo!tical#acb4 ,all necess>y tru?1 z ,bradley has %[n1 is hypo!tical#acc1 & ass]ts1 prim facie1 only ! gr\nd on :i* re/s ! necess>y connection ( premisses & conclusion4 ,if we con/ruct a m]e conception ( ext]nality1 & ?us ab&on our actually giv5 space1 ! result ( our con/ruction1 until we return to some?+ actually giv51 rema9s )\t exi/5tial import--if !re be exp]i5c$ ext]nality1 x ass]ts1 !n !re mu/ be a =m ( ext]nality ) su* & su* prop]ties4 ,t !re mu/ be exp]i5c$ ext]nality1 ,kant's fir/ argum5t ab\t space proves1 ,i ?9k1 to ^? :o admit exp]i5ce ( a _w ( div]se b int]relat$ ?+s4 ,b ?is is a "q :i* belongs to ! next ,*apt]4 ,:at we h to d "h is1 n to discuss :e!r !re is a =m ( ext]nality1 b :e!r1 if !re be su* a =m1 x mu/ possess ! prop]ties embodi$ 9 ! axioms ( projective ,geometry4 ,n[ fir/ ( all1 :at d we mean by su* a =m8 #abh4 ,9 any _w 9 :i* p]ception pres5ts u ) v>i\s ?+s1 ) discrim9at$ & diff]5tiat$ cont5ts1 !re mu/ be1 9 p]ception1 at lea/ "o pr9ciple ( diff]5tiation#acd1 an elem5t1 t is1 by :i* ! ?+s pres5t$ are di/+ui%$ z v>i\s4 ,?is elem5t1 tak5 9 isolation1 & ab/ract$ f ! cont5t :i* x diff]5tiates1 we may call a =m ( ext]nality4 ,t x mu/1 :5 tak5 9 isolation1 appe> z a =m1 & n z a m]e div]sity ( mat]ial cont5t1 is1 ,i ?9k1 fairly obvi\s4 ,= a div]sity ( mat]ial cont5t _c be /udi$ ap>t f t mat]ial cont5t2 :at we wi% to /udy "h1 on ! contr>y1 is ! b>e possibility ( su* div]sity1 :i* =ms ! residuum1 z ,i %all try to prove h]eaft]#ace1 :5 we ab/ract f any s5se-p]ception all t is di/9ctive ( its p>ticul> matt]4 ,?is possibility1 !n1 ?is pr9ciple ( b>e div]sity1 is our =m ( ext]nality4 ,h[ f> x is necess>y to assume su* a =m1 z di/9ct f int]relat$ ?+s1 ,i %all consid] lat] on#acf4 ,= ! pres5t1 s9ce space1 z dealt ) by ,geometry1 is c]ta9ly a =m ( ?is k9d1 we h only to ask3 ,:at prop]ties mu/ su* a =m1 :5 /udi$ 9 ab/raction1 necess>ily possess8 #abi4 ,9 ! fir/ place1 ext]nality is an ess5tially relative conception--no?+ c be ext]nal to itself4 ,to be ext]nal to some?+ is to be ano!r ) "s relation to t ?+4 ,h5ce1 :5 we ab/ract a =m ( ext]nality f all mat]ial cont5t1 & /udy x 9 isolation1 position w appe>1 ( necessity1 z purely relative--a position c h no intr9sic quality1 = our =m consi/s ( pure ext]nality1 & ext]nality conta9s no %ad[ or trace ( an intr9sic quality4 ,?us we obta9 our fundam5tal po/ulate1 ! relativity ( position1 or1 z we may put x1 ! complete abs5ce1 on ! "p ( our =m1 ( any ve/ige ( ?+hood4 ,! same argum5t may also be /at$ z foll[s3 ,if we ab/ract ! conception ( ext]nality1 & endeav\r to deal ) x p] se1 x is evid5t t we mu/ obta9 an object alike de/itute ( elem5ts & ( totality4 ,= we h ab/ract$ f ! div]se matt] :i* fill$ our =m1 :ile any elem5t1 or any :ole1 w\ld reta9 "s ( ! qualities ( a matt]4 ,ei!r an elem5t or a :ole1 9 fact1 w\ld h to be a ?+ n ext]nal to itself1 & w\ld ?us conta9 some?+ n pure ext]nality4 ,h5ce arise inf9ite divisibility1 ) ! self-contradictory notion ( ! po9t1 9 ! se>* = elem5ts1 & unb\nd$ ext5sion1 ) ! contradiction ( an inf9ite regress or a vici\s circle1 9 ! se>* = a complet$ :ole4 ,?us aga91 our =m conta9s nei!r elem5ts nor totality1 b only endless relations--! t]ms ( ^! relations be+ exclud$ by our ab/raction f ! matt] :i* fills our =m4 #acj4 ,9 l mann] we c d$uce ! homog5eity ( our =m4 ,! div]sity ( cont5t1 :i* 0 possible only )9 ! =m ( ext]nality1 has be5 ab/ract$ f1 leav+ no?+ b ! b>e possibility ( div]sity1 ! b>e pr9ciple ( diff]5tiation1 itself uni=m & undiff]5tiat$4 ,= if div]sity presupposes su* a =m1 ! =m _c1 unless x 7 conta9$ 9 a fre% =m1 be itself div]se or diff]5tiat$4 ,or we may d$uce ! same prop]ty f ! relativity ( position4 ,= any quality 9 "o position1 by :i* x 0 m>k$ out f ano!r1 w\ld be necess>ily m or less intr9sic1 & w\ld contradict ! pure relativity4 ,h5ce all positions are qualitatively alike1 i4e4 ! =m is homog5e\s ?r\gh\t4 #aca4 ,f :at has be5 said ( homog5eity & relativity1 foll[s "o ( ! /range/ prop]ties ( a =m ( ext]nality4 ,?is prop]ty is1 t ! relation ( ext]nality betwe5 any two ?+s is inf9itely divisible1 & may be reg>d$1 consequ5tly1 z made up ( an inf9ite numb] ( ! w\ld-be elem5ts ( our =m1 or aga9 z ! sum ( two relations ( ext]nality#acg4 ,to speak ( divid+ or add+ relations may well s\nd absurd--inde$ x reveals ! impropriety ( ! ^w relation 9 ?is connexion4 ,x is difficult1 h[ev]1 to f9d an expression :i* %all be less improp]4 ,! fact seems to be1 t ext]nality is n s mu* a relation z b>e relativity1 or ! b>e possibility ( a relation4 ,on ?is subject1 ,i %all enl>ge 9 ,*apt] ,,iv4#ach ,at ?is po9t x is only important to realize1 :at ! subsequ5t argum5t w assume1 t ! relation--if we may s call x--( ext]nality betwe5 two or m ?+s mu/1 s9ce our =m is homog5e\s1 be capable ( cont9u\s alt]ation1 & mu/1 s9ce our inf9itely divisible =m is con/itut$ by su* relations1 be capable ( inf9ite division4 ,b ! result ( inf9ite division is def9$ z ! elem5t ( our =m4 7,our =m has no elem5ts1 b we h to imag9e elem5ts 9 ord] to reason ab\t x1 z w be %[n m fully 9 ,*apt] ,,iv47 ,h5ce x foll[s1 t e relation ( ext]nality may be reg>d$1 = sci5tific purposes1 z an inf9ite cong]ies ( elem5ts1 ?\gh philosophically1 ! relations alone are valid1 & ! elem5ts are a self-contradictory result ( hypo/atiz+ ! =m ( ext]nality4 ,?is way ( reg>d+ relations ( ext]nality is important 9 und]/&+ ! mean+ ( su* ideas z ?ree or f\r coll9e> po9ts4 ,z ?is po9t is difficult & important1 ,i w repeat1 9 some:at great] detail1 ! explanation ( ! mann] 9 :i* /raight l9es & planes come to be reg>d$ z cong]ies ( po9ts4 ,f ! /rictly projective /&po9t1 ?\gh all o!r figures are m]ely a collection ( any requir$ numb] ( po9ts1 l9es or planes1 giv5 by "s projective con/ruction1 /raight l9es & planes !mselves are giv5 integrally1 & are n to be consid]$ z divisible or compos$ ( p>ts4 ,to say t a po9t lies on a /raight l9e means1 = projective ,geometry prop]1 t ! /raight l9e is a relation betwe5 ?is & "s o!r po9t4 ,"h ! po9ts conc]n$1 if our /atem5t is to be fre$ f contradictions1 mu/ be reg>d$1 if ,i may use su* an expression1 z real po9ts--i4e4 z unext5d$ mat]ial c5tres#aci4 ,/raight l9es & planes are !n relations betwe5 ^! mat]ial atoms4 ,!y are relations1 h[ev]1 :i* may und]go a metrical alt]ation :ile rema9+ projectively un*ang$4 ,:5 ! projective relation betwe5 ! two po9ts ,a1 ,b is ! same z t betwe5 ! two po9ts ,a1 ,c1 :ile ! metrical relation 7di/ance7 is diff]5t1 ! ?ree po9ts ,a1 ,b1 ,c are said to be coll9e>4 ,n[ ! metrical mann] ( reg>d+ spatial figures dem&s t !y %\ld be hypo/atiz$1 & no long] reg>d$ z m]e relations4 ,= :5 we reg>d a quantity z ext5sive1 i4e4 z divisible into p>ts1 we necess>ily reg>d x z m ?an a m]e relation or adjective1 s9ce no m]e relation or adjective c be divid$4 ,= quantitative treatm5t1 !re=e1 spatial relations mu/ be hypo/atiz$#adj4 ,:5 ?is is done1 we obta91 z we saw above1 a homog5e\s & inf9itely divisible =m ( ext]nality4 ,we f9d n[ t di/ance1 = example1 may be cont9u\sly alt]$ )\t *ang+ ! /raight l9e on :i* x is measur$4 ,we ?us obta91 on ! /raight l9e 9 "q1 a cont9u\s s]ies ( po9ts1 :i*1 s9ce x is cont9u\s1 we reg>d z con/itut+ our /raight l9e4 ,x is ?us solely f ! hypo/atiz+ ( relations1 :i* metrical ,geometry requires1 t ! view ( /raight l9es & planes z compos$ ( po9ts arises1 & x is f ?is hypo/atiz+ t ! difficulties ( metrical ,geometry spr+4 #acb4 ,! next /ep1 9 def9+ a =m ( ext]nality1 is obta9$ f ! idea ( dim5sions4 ,positions1 we h se51 are def9$ solely by _! relations to o!r positions4 ,b 9 ord] t su* def9ition may be possible1 a f9ite numb] ( relations mu/ suffice1 s9ce inf9ite numb]s are philosophically inadmissible4 ,a position mu/ be def9able1 !re=e1 if k ( our =m is to be possible at all1 by "s f9ite integral numb] ( relations to o!r positions4 ,e relation ?us necess>y = def9ition we call a dim5sion4 ,h5ce we obta9 ! proposition3 ,any =m ( ext]nality mu/ h a f9ite integral numb] ( dim5sions4 #acc4 ,! above argum5t1 x may be urg$1 has ov]look$ a possibility4 ,x has us$ a transc5d5tal argum5t1 s an oppon5t may cont5d1 )\t suffici5tly prov+ t k ab\t ext]nality mu/ be possible )\t ref]5ce to ! matt]s ext]nal to ea* o!r4 ,! def9ition ( a position may be impossible1 s long z we neglect ! matt] :i* fills ! =m1 b may become possible :5 ?is matt] is tak5 into acc\nt4 ,su* an objection c1 ,i ?9k1 be successfully met1 by a ref]5ce to ! passivity & homog5eity ( our =m4 ,= any dep5d5ce ( ! def9ition ( a position on ! p>ticul> matt] fill+ t position1 w\ld involve "s k9d ( int]action betwe5 ! matt] & its position1 "s effect ( ! div]se cont5t on ! homog5e\s =m4 ,b s9ce ! =m is totally de/itute ( ?+hood1 p]fectly impassive1 & p]fectly void ( diff]5ces betwe5 its p>ts1 any su* effect is inconceivable4 ,an effect on a position w\ld h to alt] x 9 "s way1 b h[ c\ld x be alt]$8 ,x has no qualities except ^? :i* make x ! position x is1 z oppos$ to o!r positions2 x _c *ange1 !re=e1 )\t becom+ a diff]5t position4 ,b su* a *ange contradicts ! law ( id5tity4 ,h5ce x is n ! position :i* has *ang$1 b ! cont5t :i* has mov$ 9 ! =m4 ,?us x mu/ be possible1 if k ( our =m c be obta9$ at all1 to obta9 ?is k 9 logical indep5d5ce ( ! p>ticul> matt] :i* fills x4 ,! above argum5t1 !re=e1 grant$ ! possibility ( k 9 ! dep>tm5t 9 "q1 %[s ! necessity ( a f9ite integral numb] ( dim5sions4 #acd4 ,let u repeat our orig9al argum5t 9 ! light ( ?is elucidation4 ,a position is completely def9$ :51 & only :51 en\gh relations are kn[n to enable u to det]m9e its relation to any fre% kn[n position4 ,only by relations )9 ! =m ( ext]nality1 z we h j se51 & nev] by relations :i* involve a ref]5ce to ! p>ticul> matt] fill+ ! =m1 c su* a def9ition be effect$4 ,b ! possibility ( su* a def9ition foll[s f ! ,law ( ,exclud$ ,middle1 :5 ?is law is int]pret$ to mean1 z ,bosanquet makes x mean1 t ,reality 444 is a sy/em ( reciprocally det]m9ate p>ts#ada4 ,= ?is implies t1 giv5 ! relations ( a "p ,a to o!r p>ts ,b1 ,c 4441 a suffici5t weal? ( su* relations ?r[s light on ! relations ( ,b to ,c1 etc4 ,if ?is 7 n ! case1 ! p>ts ,a1 ,b1 ,c 444 c\ld n be said to =m su* a sy/em2 = 9 su* a sy/em1 to def9e ,a is to def9e1 at ! same "t1 all ! o!r memb]s1 & to give an adjective to ,a1 is to give an adjective to ,b & ,c4 ,b ! relations betwe5 positions are1 :5 we re/ore ! matt] f :i* ! positions 7 ab/ract$1 relations betwe5 ! ?+s occupy+ ^? positions1 & ^! relations1 we h se51 c be /udi$ )\t ref]5ce to ! p>ticul> nature1 9 o!r respects1 ( ! relat$ ?+s4 ,x foll[s t1 :5 we apply ! g5]al pr9ciple ( sy/ematic unity to ^! relations 9 p>ticul>1 we f9d ^! relations to be dep5d5t on ea* o!r1 s9ce !y are n dep5d5t1 = _! def9ition1 on any?+ else4 ,?is gives ! axiom ( dim5sions1 9 ! above g5]al =m1 z ! result1 on our ab/ract geometrical level1 ( ! relativity ( position & ! law ( exclud$ middle4 #ace4 ,be=e proce$+ fur!r1 x is necess>y to discuss ! important special case :]e a =m ( ext]nality has only "o dim5sion4 ,( ! two su* =ms1 giv5 9 exp]i5ce1 "o1 namely "t1 pres5ts an in/ance ( ?is special case4 ,b x may be %[n1 ,i ?9k1 t ! function1 9 con/itut+ ! possibility ( exp]i5ce1 :i* we dem& ( su* =ms1 c\ld n be accompli%$ by a "o-dim5sional =m alone4 ,= 9 a "o-dim5sional =m1 ! v>i\s cont5ts may be arrang$ 9 a s]ies1 & _c1 )\t int]p5etration1 *ange ! ord] ( cont5ts 9 ! s]ies4 ,b int]p5etration is impossible1 s9ce a =m ( ext]nality is ! m]e expression ( div]sity among ?+s1 f :i* x foll[s t ?+s _c occupy ! same position 9 a =m1 unless !re is ano!r =m by :i* to diff]5tiate !m4 ,= )\t ext]nality1 !re is no div]sity#adb4 ,?us two bodies may occupy ! same space1 b only at diff]5t times3 two ?+s may exi/ simultane\sly1 b only at diff]5t places4 ,a =m ( "o dim5sion1 !re=e1 c\ld n1 by itself1 all[ t *ange ( ! relations ( ext]nality1 by :i* alone a v>i$ _w ( int]relat$ ?+s c be br\ght into consci\sness4 ,9 a "o-dim5sional space1 = example1 only a s+le object1 :i* mu/ appe> z a po9t1 or two objects at mo/1 "o 9 front & "o beh9d1 c\ld ev] be p]ceiv$4 ,?us two or m dim5sions seem an ess5tial condition ( any?+ wor? call+ an exp]i5ce ( int]relat$ ?+s4 #acf4 ,x may be object$1 to ?is argum5t1 t its validity dep5ds ^u ! assumption t ! *ange ( a relation ( ext]nality mu/ be cont9u\s4 ,bo? to make & to meet ?is objection1 9 a mann] :i* %all n imply "t1 seems almo/ impossible4 ,= we _c speak ( *ange1 :e!r cont9u\s or discrete1 )\t imag9+ "t4 ,let u1 !re=e1 all[ "t to be kn[n1 & discuss :e!r ! temporal *ange1 9 any o!r =m ( ext]nality1 is necess>ily cont9u\s#adc4 ,we mu/ reply1 ,i ?9k1 t cont9uity is necess>y4 ,! *ange ( relation1 9 our non-temporal =m1 may be safely describ$ z motion1 & ! law ( ,causality--s9ce we h already assum$ "t--may be appli$ to ?is motion4 ,x !n foll[s t discrete motion w\ld involve a f9ite effect f an inf9itesimal cause1 = a cause act+ only = a mom5t ( "t w\ld be inf9itesimal4 ,x involves1 also1 a validity 9 ! po9t ( "t1 :]eas :at is valid 9 any =m ( ext]nality is n1 z we h already se51 ! inf9itesimal & self-contradictory elem5t result+ f inf9ite division1 b ! f9ite relation :i* ma!matics analyzes into vani%+ elem5ts4 ,h5ce *ange mu/ be cont9u\s1 & ! possibility ( s]ial arrangem5t holds good4 ,9 a "o-dim5sional =m o!r ?an "t1 ! same argum5t mu/ hold4 ,= some?+ analog\s to ,causality w\ld be necess>y to exp]i5ce1 & ! relativity ( ! =m w\ld /ill necess>ily hold4 ,h5ce1 s9ce only ^! two prop]ties ( "t h be5 assum$1 ! above cont5tion w\ld rema9 valid ( any second =m ^: relations 7 correlat$ ) ^? ( ! fir/1 z ! analogue ( ,causality w\ld require !m to be4 #acg4 ,! next /ep 9 ! argum5t1 :i* assumes two or m dim5sions1 is conc]n$ ) ! g5]al analogues ( /raight l9es & planes1 i4e4 ) figures--:i* may be reg>d$ ei!r z relations betwe5 positions or z s]ies ( positions--uniquely det]m9$ by two or by ?ree positions4 ,if ?is /ep c be successfully tak51 our d$uction ( ! above projective axioms w be complete1 & descriptive ,geometry w be e/abli%$ z ! ab/ract priori doctr9e ( =ms ( ext]nality4 ,to prove ?is cont5tion1 consid] ( :at nature ! relations c be by :i* positions are def9$4 ,we h se5 already t our =m is purely relational & inf9itely divisible1 & t positions 7po9ts7 are ! self-contradictory outcome ( ! se>* = some?+ o!r ?an relations4 ,:at we really mean1 !re=e1 by ! relations def9+ a position1 is1 :5 we undo our previ\s ab/raction1 ! relations ( ext]nality by :i* "s ?+ is relat$ to o!r ?+s4 ,b h[1 :5 we rema9 9 ! ab/ract =m1 mu/ su* relations appe>8 #ach4 ,we h to prove t two positions mu/ h a relation indep5d5t ( any ref]5ce to o!r positions4 ,to prove ?is1 let u recur to :at 0 said1 9 connection ) dim5sions1 z to ! passivity & homog5eity ( our =m4 ,s9ce positions are def9$ only by relations1 !re mu/ be relations1 )9 ! =m1 betwe5 positions4 ,b if !re are su* relations1 !re mu/ be a relation :i* is intr9sic to two positions4 ,= to suppose ! contr>y1 is to attribute an int]action or causal connection1 ( "s k9d1 betwe5 ^? two positions & o!r positions--a supposition :i* ! p]fect homog5eity ( our =m r5d]s absurd1 s9ce all positions are qualitatively simil>1 & _c be *ang$ )\t los+ _! id5tity4 ,we may put ?is argum5t ?us3 s9ce positions are only def9$ by _! relations1 su* def9ition c\ld nev] beg91 unless x began ) a relation betwe5 only two positions4 ,= suppose ?ree positions ,a1 ,b1 ,c 7 necess>y1 & gave rise to ! relation abc betwe5 ! ?ree4 ,!n !re w\ld rema9 no means ( def9+ ! diff]5t pairs ,,bc1 ,,ca1 ,,ab1 s9ce ! only relation def9+ !m w\ld be "o common to all ?ree pairs4 ,no?+ w\ld be ga9$1 9 ?is case1 by ref]5ce to fre% po9ts1 = x foll[s1 f ! homog5eity & passivity ( ! =m1 t ^! fre% po9ts c\ld n affect ! int]nal relations ( our triad1 :i* relations1 if !y c give def9it5ess at all1 mu/ give x )\t ! aid ( ext]nal ref]5ce4 ,two positions mu/1 !re=e1 if def9ition is to be possible1 h "s relation :i* !y by !mselves suffice to def9e4 ,precisely ! same argum5t applies to ?ree positions1 or to f\r2 ! argum5t loses its scope only :5 we h exhau/$ ! dim5sions ( ! =m consid]$4 ,?us1 9 ?ree dim5sions1 five positions h no fre% relation1 n d$ucible f ^? already kn[n1 = by ! def9ition ( dim5sions1 all ! relations involv$ c be d$uc$ f ^? ( ! f\r? po9t to ! fir/ ?ree1 toge!r ) ^? ( ! fif? to ! fir/ ?ree4 ,we may give ! argum5t a m concrete1 & p]haps a m conv9c+ %ape1 by consid]+ ! matt] arrang$ 9 our =m4 ,if two ?+s are mutually ext]nal1 !y mu/ s9ce !y belong to ! same _w1 h "s relation ( ext]nality2 !re is1 !re=e1 a relation ( ext]nality betwe5 two ?+s4 ,b s9ce our =m is homog5e\s1 ! same relation ( ext]nality may subsi/ 9 o!r p>ts ( ! =m1 i4e4 :ile ! two ?+s consid]$ alt] _! relations ( ext]nality to o!r ?+s4 ,! relation ( ext]nality betwe5 two ?+s is1 !re=e1 indep5d5t ( o!r ?+s4 ,h5ce1 :5 we return to ! ab/ract language ( ! =m1 two positions h a relation det]m9$ by ^? two positions alone1 & indep5d5t ( o!r positions4 ,precisely ! same argum5t applies to ! relations ( ?ree positions1 & 9 ea* case ! relation mu/ appe> 9 ! =m z n a m]e inf]5ce f ! positions x relates4 ,= relations1 z we h se51 actually con/itute a =m ( ext]nality1 & are n m]e inf]5ces f t]ms1 :i* are n[h]e to be f\nd 9 ! =m#add4 ,to sum up3 ,s9ce position is relative1 two positions mu/ h "s relation to ea* o!r2 & s9ce our =m ( ext]nality is homog5e\s1 ?is relation c be kept un*ang$ :ile ! two positions *ange _! relations to o!r positions4 ,h5ce _! relation is intr9sic1 & indep5d5t ( o!r positions4 ,s9ce ! =m is a m]e complex ( relations1 ! relation 9 "q mu/1 if ! =m is s5su\s or intuitive1 be itself s5su\s or intuitive1 & n a m]e inf]5ce4 ,9 ?is case1 a unique relation mu/ be a unique figure--9 spatial t]ms1 ! /raight l9e jo9+ ! two po9ts4 #aci4 ,) ?is1 our d$uction ( projective ,geometry f ! priori conceptual prop]ties ( a =m ( ext]nality is complet$4 ,t su* a =m1 :5 reg>d$ z an indep5d5t ?+1 is self-contradictory1 has be5 abundantly evid5t ?r\gh\t ! discussion4 ,b ! sci5ce ( ! =m has be5 f\nd$ on ! opposite way ( reg>d+ x3 we h held x ?r\gh\t to be a m]e complex ( relations1 & h d$uc$ its prop]ties exclusively f ?is view ( x4 ,! _m difficulties1 9 apply+ su* an priori d$uction to intuitive space1 & 9 expla9+1 z logical necessities1 prop]ties :i* appe> z s5su\s or intuitional data1 mu/ be po/pon$ to ,*apt] ,,iv4 ,= ! pres5t1 ,i wi% to po9t out t projective ,geometry is :olly priori2 t x deals ) an object ^: prop]ties are logically d$uc$ f its def9ition1 n empirically discov]$ f data2 t its def9ition1 aga91 is f\nd$ on ! possibility ( exp]i5c+ div]sity 9 relation1 or multiplicity 9 unity2 & t our :ole sci5ce1 !re=e1 is logically impli$ 91 & d$ucible f1 ! possibility ( su* exp]i5ce4 #adj4 ,9 metrical ,geometry1 on ! contr>y1 we %all f9d a v diff]5t result4 ,al?\gh ! geometrical conditions :i* r5d] spatial measurem5t possible1 w be f\nd id5tical1 except = slight diff]5ces 9 ! =m ( /atem5t1 ) ! priori axioms discuss$ above1 yet ! actual measurem5t--:i* deals ) actually giv5 space1 n ! m]e intellectual con/ruction we h be5 j discuss+--gives results :i* c only be kn[n empirically & approximately1 & c be d$uc$ by no necessity ( ?\ght4 ,! ,euclidean & non-,euclidean spaces give ! v>i\s results :i* are priori possible2 ! axioms peculi> to ,euclid--:i* are prop]ly n axioms1 b empirical results ( measurem5t--det]m9e1 )9 ! errors ( obs]vation1 :i* ( ^! priori possibilities is realiz$ 9 our actual space4 ,?us measurem5t deals ?r\gh\t ) an empirically giv5 matt]1 n ) a creature ( ! intellect1 & its priori elem5ts are only ! conditions presuppos$ 9 ! possibility ( measurem5t4 ,:at ^! conditions are1 we %all see 9 ! second section ( ?is *apt]4 ,section ,b4 ,,! ,,axioms ,,( ,,metrical ,,geometry4 #ada4 ,we h n[ review$ ! axioms ( projective ,geometry1 & h se5 t !y are priori d$uctions f ! fact t we c exp]i5ce ext]nality1 i4e4 a coexi/5t multiplicity ( diff]5t b int]relat$ ?+s4 ,b projective ,geometry1 9 spite ( its claims1 is n ! :ole sci5ce ( space1 z is suffici5tly prov$ by ! fact t x _c discrim9ate betwe5 ,euclidean & non-,euclidean spaces#ade4 ,= ?is purpose1 spatial measurem5t is requir$3 metrical ,geometry1 ) its quantitative te/s1 c alone effect ! discrim9ation4 ,= all application ( ,geometry to physics1 also1 measurem5t is requir$2 ! law ( gravitation1 = example1 requires ! det]m9ation ( actual di/ances4 ,= _m purposes1 9 %ort1 projective ,geometry is :olly insuffici5t3 ?us x is unable to di/+ui% betwe5 diff]5t k9ds ( conics1 ?\gh _! di/9ction is ( fundam5tal importance 9 _m dep>tm5ts ( k4 ,metrical ,geometry is1 !n1 a necess>y "p ( ! sci5ce ( space1 & a "p n includ$ 9 descriptive ,geometry4 ,its priori elem5t1 nev]!less1 s f> z ?is is spatial & n ari?metical1 is ! same z ! po/ulate ( projective ,geometry1 namely1 ! homog5eity ( space1 or its equival5t1 ! relativity ( position4 ,we c see1 9 fact1 t ! priori elem5t 9 bo? is likely to be ! same4 ,= ! priori 9 metrical ,geometry w be :atev] is presuppos$ 9 ! possibility ( spatial measurem5t1 i4e4 ( quantitative spatial comp>ison4 ,b su* comp>ison presupposes simply a kn[n id5tity ( quality1 ! det]m9ation ( :i* is precisely ! problem ( projective ,geometry4 ,h5ce ! conditions = ! possibility ( measurem5t1 9 s f> z !y are n ari?metical1 w be precisely ! same z ^? = projective ,geometry4 #adb4 ,metrical ,geometry1 !re=e1 ?\gh di/9ct f projective ,geometry1 is n indep5d5t ( x1 b presupposes x1 & arises f its comb9ation ) ! extrane\s idea ( quantity4 ,nev]!less ! ma!matical =m ( ! axioms1 9 metrical ,geometry1 is slightly diff]5t f _! =m 9 projective ,geometry4 ,! homog5eity ( space is replac$ by its equival5t1 ! axiom ( ,free ,mobility4 ,! axiom ( ! /raight l9e is replac$ by ! axiom ( di/ance3 ,two po9ts det]m9e a unique quantity1 di/ance1 :i* is unalt]$ 9 any motion ( ! two po9ts z a s+le figure4 ,?is axiom1 inde$1 w be f\nd to involve ! axiom ( ! /raight l9e--su* a quantity c\ld n exi/ unless ! two po9ts det]m9$ a unique curve--b its ma!matical =m is *ang$4 ,ano!r important *ange is ! collapse ( ! pr9ciple ( duality3 quantity c be appli$ to ! /raight l9e1 because x is divisible into simil> p>ts1 b _c be appli$ to ! indivisible po9t4 ,we ?us obta9 a reason1 :i* 0 want+ 9 descriptive ,geometry1 = pref]r+ po9ts1 z spatial elem5ts1 to /raight l9es or planes#adf4 ,f9ally1 an entirely new idea is introduc$ ) quantity1 namely1 ! idea ( ,motion4 ,n t we /udy motion1 or t any ( our results h ref]5ce to motion1 b t !y _c1 ?\gh 9 projective ,geometry !y c\ld1 be obta9$ )\t at lea/ an ideal motion ( our figures ?r\gh space4 ,let u n[ exam9e 9 detail ! pr]equisites ( spatial measurem5t4 ,we %all f9d ?ree axioms1 )\t :i* su* measurem5t w\ld be impossible1 b ) :i* x is adequate to decide1 empirically & approximately1 ! ,euclidean or non-,euclidean nature ( our actual space4 ,we %all f9d1 fur!r1 t ^! ?ree axioms c be d$uc$ f ! conception ( a =m ( ext]nality1 & owe no?+ to ! evid5ce ( intuition4 ,!y are1 !re=e1 l _! equival5ts ! axioms ( projective ,geometry1 priori1 & d$ucible f ! conditions ( spatial exp]i5ce4 ,?is exp]i5ce1 accord+ly1 c nev] disprove !m1 s9ce its v exi/5ce presupposes !m4 ,i4 ,! ,axiom ( ,free ,mobility4 #adc4 ,metrical ,geometry1 to beg9 )1 may be def9$ z ! sci5ce :i* deals ) ! comp>ison & relations ( spatial magnitudes4 ,! conception ( magnitude1 !re=e1 is necess>y f ! />t4 ,"s ( ,euclid's axioms1 accord+ly1 h be5 class$ z ari?metical1 & h be5 suppos$ to h no?+ p>ticul> to d ) space4 ,su* are ! axioms t equals add$ to or subtract$ f equals give equals1 & t ?+s :i* are equal to ! same ?+ are equal to "o ano!r4 ,^! axioms1 x is said1 are purely ari?metical1 & d n1 l ! o!rs1 ascribe an adjective to space4 ,z reg>ds _! use 9 ari?metic1 ?is is ( c\rse true4 ,b if an ari?metical axiom is to be appli$ to spatial magnitudes1 x mu/ h "s spatial import#adg1 & ?us ev5 ?is class is n1 9 ,geometry1 m]ely ari?metical4 ,=tunately1 ! geometrical elem5t is ! same 9 all ! axioms ( ?is class--we c see at once1 9 fact1 t x c am\nt to no m ?an a def9ition ( spatial magnitude#adh4 ,aga91 s9ce ! space ) :i* ,geometry deals is inf9itely divisible1 a def9ition ( spatial magnitude r$uces itself to a def9ition ( spatial equality1 =1 z soon z we h ?is la/1 we c comp>e two spatial magnitudes by divid+ ea* into a numb] ( equal units1 & c\nt+ ! numb] ( su* units 9 ea*#adi4 ,! ratio ( ! numb] ( units is1 ( c\rse1 ! ratio ( ! two magnitudes4 #add4 ,we require1 !n1 at ! v outset1 "s crit]ion ( spatial equality3 )\t su* a crit]ion metrical ,geometry w\ld become :olly impossible4 ,x might appe>1 at fir/ sight1 z ?\gh ?is ne$ n be an axiom1 b might be a m]e def9ition4 ,9 "p ?is is true1 b n :olly4 ,! "p :i* is m]ely a def9ition is giv5 9 ,euclid's eigh? axiom3 ,magnitudes :i* exactly co9cide are equal4 ,b ?is gives a suffici5t crit]ion only :5 ! magnitudes to be comp>$ already occupy ! same position4 ,:51 z w normally be ! case1 ! two spatial magnitudes are ext]nal to "o ano!r--z1 inde$1 mu/ be ! case1 if !y are di/9ct1 & n :ole & "p--! two magnitudes c only be made to co9cide by a motion ( "o or bo? ( !m4 ,9 ord]1 !re=e1 t our def9ition ( spatial magnitude may give unambigu\s results1 co9cid5ce :5 sup]pos$1 if x c ev] occur1 mu/ occur always1 :atev] pa? be pursu$ 9 br++ x ab\t4 ,h5ce1 if m]e motion c\ld alt] %apes1 our crit]ion ( equality w\ld break d[n4 ,x foll[s t ! application ( ! conception ( magnitude to figures 9 space involves ! foll[+ axiom#aej3 ,spatial magnitudes c be mov$ f place to place )\t di/ortion2 or1 z x may be put1 ,%apes d n 9 any way dep5d ^u absolute position 9 space4 ,! above axiom is ! axiom ( ,free ,mobility#aea4 ,i propose to prove 7#a7 t ! d5ial ( ?is axiom w\ld involve logical & philosophical absurdities1 s t x mu/ be class$ z :olly priori2 7#b7 t metrical ,geometry1 if x refus$ ?is axiom1 w\ld be unable1 )\t a logical absurdity1 to e/abli% ! notion ( spatial magnitude at all4 ,! conclusion w be1 t ! axiom _c be prov$ or disprov$ by exp]i5ce1 b is an priori condition ( metrical ,geometry4 ,z ,i %all ?us be ma9ta9+ a position :i* has be5 mu* controv]t$1 especially by ,helmholtz & ,erdmann1 ,i %all h to ent] into ! argum5ts at "s l5g?4 #ade4 ,a4 ,philosophical ,argum5t4 ,! d5ial ( ! axiom involves absolute position1 & an action ( m]e space1 p] se1 on ?+s4 ,= ! axiom does n ass]t t real bodies1 z a matt] ( empirical fact1 nev] *ange _! %ape 9 any way dur+ _! passage f place to place3 on ! contr>y1 we "k t su* *anges d occur1 sometimes 9 a v noticeable degree1 & always to "s ext5t4 ,b su* *anges are attribut$1 n to ! *ange ( place z su*1 b to physical causes3 *anges ( temp]ature1 pressure1 etc4 ,:at our axiom has to deal ) is n actual mat]ial bodies1 b geometrical figures#aeb1 & x ass]ts t a figure :i* is possible 9 any "o position 9 space is possible 9 e o!r4 ,its mean+ w become cle>] by ref]5ce to a case :]e x does n hold1 say ! space =m$ by ! surface ( an egg4 ,"h1 a triangle drawn ne> ! equator _c be mov$ )\t di/ortion to ! po9t1 z x w\ld no long] fit ! great] curvature ( ! new position3 a triangle drawn ne> ! po9t _c be fitt$ on to ! flatt] end1 & s on4 ,?us ! me?od ( sup]position1 su* z ,euclid employs 9 ,book ,i4 ,prop4 ,,iv41 becomes impossible2 figures _c be freely mov$ ab\t1 inde$1 giv5 any figure1 we c det]m9e a c]ta9 s]ies ( possible positions = x on ! egg1 outside :i* x becomes impossible4 ,:at ,i ass]t is1 !n1 t !re is a philosophic absurdity 9 suppos+ space 9 g5]al to be ( ?is nature4 ,on ! egg we h m>k$ po9ts1 su* z ! two ends2 ! space =m$ by its surface is n homog5e\s1 & if ?+s are mov$ ab\t 9 x1 x mu/ ( itself ex]cise a di/ort+ effect ^u !m1 q indep5d5tly ( physical causes2 if x did n ex]cise su* an effect1 ! ?+s c\ld n be mov$4 ,?us su* a space w\ld n be homog5e\s1 b w\ld h m>k$ po9ts1 by ref]5ce to :i* bodies w\ld h absolute position1 q indep5d5tly ( any o!r bodies4 ,space w\ld no long] be passive1 b w\ld ex]cise a def9ite effect ^u ?+s1 & we %\ld h to accommodate ourselves to ! notion ( m>k$ po9ts 9 empty space2 ^! po9ts be+ m>k$1 n by ! bodies :i* occupi$ !m1 b by _! effects on any bodies :i* might f "t to "t occupy !m4 ,?is want ( homog5eity & passivity is1 h[ev]1 absurd2 space mu/1 s9ce x is a =m ( ext]nality1 all[ only ( relative1 n ( absolute1 position1 & mu/ be completely homog5e\s ?r\gh\t4 ,to suppose x o!rwise1 is to give x a ?+hood :i* no =m ( ext]nality c possibly possess4 ,we mu/1 !n1 on purely philosophical gr\nds1 admit t a geometrical figure :i* is possible any:]e is possible ev]y:]e1 :i* is ! axiom ( ,free ,mobility4 #adf4 ,b4 ,geometrical ,argum5t4 ,let u see next :at sort ( ,geometry we c\ld con/ruct )\t ?is axiom4 ,! ultimate /&>d ( comp>ison ( spatial magnitudes mu/1 z we saw 9 introduc+ ! axiom1 be equality :5 sup]pos$2 b ne$ we1 f ?is equality1 inf] equality :5 sep>at$8 ,x has be5 urg$ by ,erdmann t1 = ! m imm$iate purposes ( ,geometry1 ?is w\ld be unnecess>y#aec4 ,we might con/ruct a new ,geometry1 he ?9ks1 9 :i* sizes v>i$ ) motion on any def9ite law4 ,su* a view1 z ,i %all %[ bel[1 involves a logical error z to ! nature ( magnitude4 ,b be=e po9t+ ?is out1 let u discuss ! geometrical consequ5ces ( assum+ its tru?4 ,suppose ! l5g? ( an inf9itesimal arc 9 "s /&>d position 7 ds2 !n 9 any o!r position p its l5g? w\ld be ds4f7p71 :]e ! =m ( ! function f7p7 mu/ be suppos$ kn[n4 ,b h[ are we to det]m9e ! position p8 ,= ?is purpose1 we require p's coord9ates1 i4e41 "s measurem5t ( di/ance f ! orig94 ,b ! di/ance f ! orig9 c\ld only be measur$ if we assum$ our law f7p7 to measure x by4 ,= suppose ! orig9 to be ,o1 & ,op to be a /raight l9e ^: l5g? is requir$4 ,if we h a measur+ rod ) :i* we travel along ! l9e & measure successive inf9itesimal arcs1 ! measur+ rod w *ange its size z we move1 s t an arc :i* appe>s by ! measure to be ds w really be f7s74ds1 :]e s is ! previ\sly trav]s$ di/ance4 ,if1 on ! o!r h&1 we move our l9e ,op sl[ly ?r\gh ! orig91 & measure ea* piece z x passes ?r\gh1 our measure1 x is true1 w n alt]1 b n[ we h no means ( discov]+ ! law by :i* any elem5t has *ang$ its l5g? 9 com+ to ! orig94 ,h5ce1 until we assume our function f7p71 we h no means ( det]m9+ p1 = we h j se5 t di/ances f ! orig9 c only be e/imat$ by means ( ! law f7p74 ,x foll[s t exp]i5ce c nei!r prove nor disprove ! con/ancy ( %apes ?r\gh\t motion1 s9ce1 if %apes 7 n con/ant1 we %\ld h to assume a law ( _! v>iation be=e measurem5t became possible1 & !re=e measurem5t c\ld n itself reveal t v>iation to u#aed4 ,nev]!less1 su* an arbitr>ily assum$ law does1 at fir/ sight1 give a ma!matically possible ,geometry4 ,! fundam5tal proposition1 t two magnitudes :i* c be sup]pos$ 9 any "o position c be sup]pos$ 9 any o!r1 /ill holds4 ,= two inf9itesimal arcs1 ^: l5g?s 9 ! /&>d position are ds#a & ds#b1 w\ld1 9 any o!r position p1 h l5g?s f7p74ds#a & f7p74ds#b1 s t _! ratio w\ld be unalt]$4 ,f ?is con/ancy ( ratio1 z we "k ?r\gh ,riemann & ,helmholtz1 ! above proposition foll[s4 ,h5ce all t ,geometry requires1 x w\ld seem1 z a basis = measurem5t1 is an axiom t ! alt]ation ( %apes dur+ motion foll[s a def9ite kn[n law1 su* z t assum$ above4 #adg4 ,!re is1 h[ev]1 9 su* a view1 z ,i rem>k$ above1 a logical error z to ! nature ( magnitude4 ,?is error has be5 already po9t$ out 9 deal+ ) ,erdmann#aee1 & ne$ only be briefly repeat$ "h4 ,a judgm5t ( magnitude is ess5tially a judgm5t ( comp>ison3 9 unmeasur$ quantity1 comp>ison z to ! m]e m or less1 b 9 measur$ magnitude1 comp>ison z to ! precise h[ _m times4 ,to speak ( diff]5ces ( magnitude1 !re=e1 9 a case :]e comp>ison _c reveal !m1 is logically absurd4 ,n[ 9 ! case contemplat$ above1 two magnitudes1 :i* appe> equal 9 "o position1 appe> equal also :5 comp>$ 9 ano!r position4 ,!re is no s5se1 !re=e1 9 suppos+ ! two magnitudes unequal :5 sep>at$1 nor 9 suppos+1 consequ5tly1 t !y h *ang$ _! magnitudes 9 motion4 ,?is s5selessness ( our hypo!sis is ! logical gr\nd ( ! ma!matical indet]m9at5ess z to ! law ( v>iation4 ,s9ce1 !n1 !re is no means ( comp>+ two spatial figures1 z reg>ds magnitude1 except sup]position1 ! only logically possible axiom1 if spatial magnitude is to be self-consi/5t1 is ! axiom ( ,free ,mobility 9 ! =m fir/ giv5 above4 #adh4 ,al?\gh ?is axiom is priori1 its application to ! measurem5t ( actual bodies1 z we f\nd 9 discuss+ ,helmholtz's views1 always involves an empirical elem5t#aef4 ,our axiom1 !n1 only supplies ! priori condition = c>ry+ out an op]ation :i*1 9 ! concrete1 is empirical--j z ari?metic supplies ! priori condition = a c5sus4 ,z ?is topic has be5 discuss$ at l5g? 9 ,*apt] ,,ii41 ,i %all say no m ab\t x "h4 #adi4 ,!re rema91 h[ev]1 a few objections & difficulties to be discuss$4 ,fir/1 h[ d we obta9 equality 9 solids1 & 9 ,kant's cases ( "r & left h&s1 or ( "r & left-h&$ screws1 :]e actual sup]position is impossible8 ,secondly1 h[ c we take congru5ce z ! only possible basis ( spatial measurem5t1 :5 we h be=e u ! case ( "t1 :]e no su* ?+ z congru5ce is conceivable8 ,?irdly1 x might be urg$ t we c imm$iately e/imate spatial equality by ! eye1 ) m or less accuracy1 & ?us h a measure indep5d5t ( congru5ce4 ,f\r?ly1 h[ is metrical ,geometry possible on non-congru5t surfaces1 if congru5ce be ! basis ( spatial measurem5t8 ,i w discuss ^! objections successively4 #aej4 7#a7 ,h[ d we measure ! equality ( solids8 ,^! c\ld only be br\ght into actual congru5ce if we _h a f\r? dim5sion to op]ate 9#aeg1 & f :at ,i h said be=e ( ! absolute necessity ( ?is te/1 x might seem z ?\gh we %\ld be left "h 9 utt] ignorance4 ,euclid is sil5t on ! subject1 & 9 all works on ,geometry x is assum$ z self-evid5t t two cubes ( equal side are equal4 ,?is assumption sugge/s t we are n s badly (f z we %\ld h be5 )\t congru5ce1 z a te/ ( equality 9 "o or two dim5sions2 = n[ we c at lea/ be sure t two cubes h all _! sides & all _! faces equal4 ,two su* cubes diff]1 !n1 9 no s5sible spatial quality save position1 = volume1 9 ?is case at any rate1 is n a s5sible quality4 ,!y are1 !re=e1 z f> z su* qualities are conc]n$1 indisc]nible4 ,if _! places 7 int]*ang$1 we might "k ! *ange by _! col\r1 or by "s o!r non-geometrical prop]ty2 b s f> z any prop]ty ( :i* ,geometry c take cognisance is conc]n$1 ev]y?+ w\ld seem z be=e4 ,to suppose a diff]5ce ( volume1 !n1 w\ld be to ascribe an effect to m]e position1 :i* we saw to be inadmissible :ile discuss+ ,free ,mobility4 ,except z reg>ds position1 !y are geometrically indisc]nible1 & we may call to our aid ! ,id5tity ( ,indisc]nibles to e/abli% _! agreem5t 9 ! "o rema9+ geometrical prop]ty ( volume4 ,?is may seem r a /range pr9ciple to use 9 ,ma!matics1 & = ,geometry _! equality is1 p]haps1 be/ reg>d$ z a def9ition2 b if we dem& a philosophical gr\nd = ?is def9ition1 x is1 ,i believe1 only to be f\nd 9 ! ,id5tity ( ,indisc]nibles4 ,we c1 )\t error1 make our def9ition ( ?ree-dim5sional equality re/ on two-dim5sional congru5ce4 ,= s9ce direct comp>ison z to volume is impossible1 we are at lib]ty to def9e two volumes z equal1 :5 all _! v>i\s l9es1 surfaces1 angles & solid angles are congru5t1 s9ce !re rema9s1 9 su* a case1 no measurable diff]5ce betwe5 ! figures compos+ ! two volumes4 ,( c\rse1 z soon z we h e/abli%$ ?is "o case ( equality ( volumes1 ! re/ ( ! !ory foll[s2 z appe>s f ! ord9>y me?od ( integrat+ volumes1 by divid+ !m into small cubes4 ,?us congru5ce helps to e/abli% ?ree-dim5sional equality1 ?\gh x _c directly prove su* equality2 & ! same philosophical pr9ciple1 ( ! homog5eity ( space1 by :i* congru5ce 0 prov$1 comes to our rescue "h4 ,b h[ ab\t "r-h&$ & left-h&$ screws8 ,"h we c no long] apply ! ,id5tity ( ,indisc]nibles1 = ! two are v well disc]nible4 ,b z ) solids1 s "h1 ,free ,mobility c help u mu*4 ,x c enable u1 by ord9>y measurem5t1 to %[ t ! int]nal relations ( bo? screws are ! same1 & t ! diff]5ce lies only 9 _! relation to o!r ?+s 9 space4 ,kn[+ ^! int]nal relations1 we c calculate1 by ! ,geometry :i* ,free ,mobility has r5d]$ possible1 all ! geometrical prop]ties ( ^! screws--radius1 pit*1 etc4--& c %[ !m to be sev]ally equal 9 bo?4 ,b ?is is all we require4 ,m$iate comp>ison is possible1 ?\gh imm$iate comp>ison is n4 ,bo? c1 = in/ance1 be comp>$ ) ! cyl9d] on :i* bo? w\ld fit1 & ?us _! equality c be prov$4 ,a precisely simil> pro( holds1 ( c\rse1 = ! o!r cases1 "r & left h&s1 sph]ical triangles1 etc4 ,on ! :ole1 ^! cases confirm my argum5t2 = !y %[1 z ,kant int5d$ !m to %[#aeh1 ! ess5tial relativity ( space4 #aea4 7#b7 ,z reg>ds "t1 no congru5ce is "h conceivable1 = to effect congru5ce requires always--z we saw 9 ! case ( solids--"o m dim5sion ?an belongs to ! magnitudes comp>$4 ,no day c be br\ght into temporal co9cid5ce ) any o!r day1 to %[ t ! two exactly cov] ea* o!r2 we are !re=e r$uc$ to ! arbitr>y assumption t "s motion or set ( motions1 giv5 u 9 exp]i5ce1 is uni=m4 ,=tunately1 we h a l>ge set ( motions :i* all r\ghly agree2 ! sw+ ( ! p5dulum1 ! rotation & revolution ( ! e>? & ! planets1 etc4 ,^! d n exactly agree1 b !y lead u to ! laws ( motion1 by :i* we are able1 on our arbitr>y hypo!sis1 to e/imate _! small dep>tures f uni=mity2 j z ! assumption ( ,free ,mobility enabl$ u to measure ! dep>tures ( actual bodies f rigidity4 ,b "h1 z !re1 ano!r possibility is ma!matically op5 to u1 & c only be exclud$ by its philosophic absurdity2 we might h assum$ t ! above set ( approximately agree+ motions all _h velocities :i* v>i$ approximately z "s arbitr>ily assum$ function ( ! "t1 f7t7 say1 measur$ f "s arbitr>y orig94 ,su* an assumption w\ld /ill keep !m z ne>ly syn*ron\s z be=e1 & w\ld give an equally possible1 ?\gh m complex1 sy/em ( ,me*anics2 in/ead ( ! fir/ law ( motion1 we %\ld h ! foll[+3 ,a p>ticle p]sev]es 9 its /ate ( re/1 or ( rectil9e> motion ) velocity v>y+ z f7t71 except 9 s f> z x is compell$ to alt] t /ate by ! action ( ext]nal =ces4 ,su* a hypo!sis is ma!matically possible1 b1 l ! simil> "o = space1 x is exclud$ logically by ! comp>ative nature ( ! judgm5t ( quantity1 & philosophically by ! fact t x involves absolute "t1 z a det]m9+ ag5t 9 *ange1 :]eas "t c nev]1 philosophically1 be any?+ b a passive =m1 ab/ract$ f *ange4 ,i h introduc$ ?is p>allel f "t1 n z directly be>+ on ! argum5t1 b z a simpl] case :i* may s]ve to illu/rate my reason+ 9 ! m complex case ( space4 ,= s9ce "t1 9 ma!matics1 is "o-dim5sional1 ! ma!matical difficulties are simpl] ?an 9 ,geometry2 & al?\gh no?+ accurately corresponds to congru5ce1 !re is a v simil> mixture ( ma!matical & philosophical necessity1 giv+1 f9ally1 a ?or\ghly def9ite axiom z ! basis ( "t-measurem5t1 correspond+ to congru5ce z ! basis ( space-measurem5t#aei4 #aeb4 7#c7 ,! case ( "t-measurem5t sugge/s ! ?ird ( ! above objections to ! absolute necessity ( ! axiom ( ,free ,mobility4 ,psy*o-physics has %[n t we h an approximate p[]1 by means ( :at may be call$ ! s5se ( duration1 ( imm$iately e/imat+ equal %ort times4 ,?is e/abli%es a r\gh measure indep5d5t ( any assum$ uni=m motion1 & 9 space also1 x may be said1 we h a simil> p[] ( imm$iate comp>ison4 ,we c see1 by imm$iate inspection1 t ! sub-divisions on a foot rule are n grossly inaccurate2 & s1 x may be said1 we bo? h a measure indep5d5t ( congru5ce1 & also c\ld discov]1 by exp]i5ce1 any gross dep>ture f ,free ,mobility4 ,aga9/ ?is view1 h[ev]1 !re is at ! outset a v fundam5tal psy*ological objection4 ,x has be5 urg$ t all our comp>ison ( spatial magnitudes proce$s by ideal sup]position4 ,?us ,james says 7,psy*ology1 ,vol4 ,,ii4 p4 #aeb73 ,ev5 :]e we only feel "o sub-division to be vaguely l>g] or less1 ! m9d mu/ pass rapidly betwe5 x & ! o!r sub-division1 & receive ! imm$iate s5sible %ock ( ! m1 & s f> z ! sub-divisions ( a s5se-space are to be measur$ exactly aga9/ ea* o!r1 objective =ms occupy+ "o sub-division mu/ be directly or indirectly sup]pos$ ^u ! o!r#afj4 ,ev5 if we waive ?is fundam5tal objection1 h[ev]1 o!rs rema94 ,to beg9 )1 su* judgm5ts ( equality are only v r\gh approximations1 & _c be appli$ to l9es ( m ?an a c]ta9 l5g?1 if only = ! reason t su* l9es _c well be se5 toge!r4 ,?us ?is me?od c only give u any security 9 our own imm$iate neighb\rhood1 & c\ld 9 no wise w>rant su* op]ations z w\ld be requir$ = ! con/ruction ( maps c41 mu* less ! measurem5t ( a/ronomical di/ances4 ,!y might j enable u to say t "s l9es 7 long] ?an o!rs1 b !y w\ld leave ,geometry 9 a position no bett] ?an t ( ! ,h$onical ,calculus1 9 :i* we dep5d on a purely subjective measure4 ,s inaccurate1 9 fact1 is su* a me?od ackn[l$g$ to be1 t ! foot-rule is z mu* a ne$ ( daily life z ( sci5ce4 ,besides1 no "o w\ld tru/ su* imm$iate judgm5ts1 b = ! fact t ! /rict] te/ ( congru5ce to "s ext5t confirms !m2 if we c\ld n apply ?is te/1 we %\ld h no gr\nd = tru/+ !m ev5 z mu* z we d4 ,?us we %\ld h1 "h1 no real escape f our absolute dep5d5ce ^u ! axiom ( ,free ,mobility4 #aec4 7#d7 ,"o la/ elucidatory rem>k is necess>y be=e our pro( ( ?is axiom c be consid]$ complete4 ,we spoke above ( ! ,geometry on an egg1 :]e ,free ,mobility does n hold4 ,:at1 ,i may be ask$1 is !re ab\t a ?or\ghly non-congru5t ,geometry1 m impossible ?an ?is ,geometry on ! egg8 ,! answ] is obvi\s4 ,! ,geometry ( non-congru5t surfaces is only possible by ! use ( inf9itesimals1 & 9 ! inf9itesimal all surfaces become plane4 ,! fundam5tal =mula1 t = ! l5g? ( an inf9itesimal arc1 is only obta9$ on ! assumption t su* an arc may be treat$ z a /raight l9e1 & t ,euclidean ,plane ,geometry may be appli$ 9 ! imm$iate neighb\rhood ( any po9t4 ,if we _h n our ,euclidean measure1 :i* c\ld be mov$ )\t di/ortion1 we %\ld h no me?od ( comp>+ small arcs 9 diff]5t places1 & ! ,geometry ( non-congru5t surfaces w\ld break d[n4 ,?us ! axiom ( ,free ,mobility1 z reg>ds ?ree-dim5sional space1 is necess>ily impli$ & presuppos$ 9 ! ,geometry ( non-congru5t surfaces2 ! possibility ( ! latt]1 !re=e1 is a dep5d5t & d]ivative possibility1 & c =m no argum5t aga9/ ! priori necessity ( congru5ce z ! te/ ( equality4 #aed4 ,x is to be obs]v$ t ! axiom ( ,free ,mobility1 z ,i h enunciat$ x1 includes also ! axiom to :i* ,helmholtz gives ! "n ( ,monodromy4 ,?is ass]ts t a body does n alt] its dim5sions 9 consequ5ce ( a complete revolution ?r\gh f\r "r angles1 b occupies at ! end ! same position z at ! beg9n+4 ,! suppos$ ma!matical necessity ( mak+ a sep>ate axiom ( ?is prop]ty ( space has be5 disprov$ by ,sophus ,lie 7v4 ,*ap4 ,i4 #de72 philosophically1 x is pla9ly a p>ticul> case ( ,free ,mobility#afa1 & inde$ a p>ticul>ly obvi\s case1 = a translation really does make "s *ange 9 a body1 namely1 a *ange 9 position1 b a rotation ?r\gh f\r "r angles may be suppos$ to h be5 p]=m$ any numb] ( times )\t appe>+ 9 ! result1 & ! absurdity ( ascrib+ to space ! p[] ( mak+ bodies gr[ 9 ! process is palpable2 ev]y?+ t 0 said above on congru5ce 9 g5]al applies ) ev5 great] evid5ce to ?is special case4 #aee4 ,! axiom ( ,free ,mobility involves1 if x is to be true1 ! homog5eity ( space1 or ! complete relativity ( position4 ,= if any %ape1 :i* is possible 9 "o "p ( space1 be always possible 9 ano!r1 x foll[s t all p>ts ( space are qualitatively simil>1 & _c1 !re=e1 be di/+ui%$ by any intr9sic prop]ty4 ,h5ce positions 9 space1 if our axiom be true1 mu/ be :olly def9$ by ext]nal relations1 i4e4 ,position is n an intr9sic1 b a purely relative1 prop]ty ( ?+s 9 space4 ,if !re c\ld be su* a ?+ z absolute position1 9 %ort1 metrical ,geometry w\ld be impossible4 ,?is relativity ( position is ! fundam5tal po/ulate ( all ,geometry1 to :i* ea* ( ! necess>y metrical axioms leads1 & f :i*1 conv]sely1 ea* ( ^! axioms c be d$uc$4 #aef4 ,?is conv]se d$uction1 z reg>ds ,free ,mobility1 is n v difficult1 & foll[s f ! argum5t ( ,section ,a#afb1 :i* ,i w briefly recapitulate4 ,9 ! fir/ place1 ext]nality is an ess5tially relative conception--no?+ c be ext]nal to itself4 ,to be ext]nal to some?+ is to be an o!r ) "s relation to t ?+4 ,h5ce1 :5 we ab/ract a =m ( ext]nality f all mat]ial cont5t1 & /udy x 9 isolation1 position w appe> ( necessity z purely relative--x c h no intr9sic quality1 = our =m consi/s ( pure ext]nality1 & ext]nality conta9s no %ad[ or trace ( an intr9sic quality4 ,h5ce we d]ive our fundam5tal po/ulate1 ! relativity ( position4 ,f ?is foll[s ! homog5eity ( our =m1 = any quality 9 "o position1 :i* m>k$ out t position f ano!r1 w\ld be necess>ily m or less intr9sic1 & w\ld contradict ! pure relativity4 ,f9ally ,free ,mobility foll[s f homog5eity1 = our =m w\ld n be homog5e\s unless x all[$1 9 e "p1 %apes or sy/ems ( relations1 :i* x all[$ 9 any o!r "p4 ,free ,mobility1 !re=e1 is a necess>y prop]ty ( e possible =m ( ext]nality4 #aeg4 ,9 summ+ up ! argum5t we h j conclud$1 we may exhibit x1 9 consequ5ce ( ! two prec$+ p>agraphs1 9 ! =m ( a complet$ circle4 ,/>t+ f ! conditions ( spatial measurem5t1 we f\nd t ! comp>ison1 requir$ = measurem5t1 c\ld only be effect$ by sup]position4 ,b we f\nd1 fur!r1 t ! result ( su* comp>ison w only be unambigu\s1 if spatial magnitudes & %apes are unalt]$ by motion 9 space1 if1 9 o!r words1 %apes d n dep5d ^u absolute position 9 space4 ,b ?is axiom c only be true if space is homog5e\s & position m]ely relative4 ,conv]sely1 if position is assum$ to be m]ely relative1 a *ange ( magnitude 9 motion--involv+ z x does1 ! ass]tion ( absolute position--is impossible1 & our te/ ( spatial equality is !re=e adequate4 ,b position 9 any =m ( ext]nality mu/ be purely relative1 s9ce ext]nality _c be an intr9sic prop]ty ( any?+4 ,our axiom1 !re=e1 is priori 9 a d\ble s5se4 ,x is presuppos$ 9 all spatial measurem5t1 & x is a necess>y prop]ty ( any =m ( ext]nality4 ,a simil> d\ble apriority1 we %all see1 appe>s 9 our o!r necess>y axioms4 ,,ii4 ,! ,axiom ( ,dim5sions#afc4 #aeh4 ,we h se51 9 discuss+ ! axiom ( ,free ,mobility1 t all position is relative1 t is1 a position exi/s only by virtue ( relations#afd4 ,x foll[s t1 if positions c be def9$ at all1 !y mu/ be uniquely & exhau/ively def9$ by "s f9ite numb] ( su* relations4 ,if ,geometry is to be possible1 x mu/ happ5 t1 aft] en\gh relations h be5 giv5 to det]m9e a po9t uniquely1 its relations to any fre% kn[n po9t are d$ucible f ! relations already giv54 ,h5ce we obta91 z an priori condition ( ,geometry1 logically indisp5sable to its exi/5ce1 ! axiom t ,space mu/ h a f9ite integral numb] ( ,dim5sions4 ,= e relation requir$ 9 ! def9ition ( a po9t con/itutes a dim5sion1 & a fraction ( a relation is mean+less4 ,! numb] ( relations requir$ mu/ be f9ite1 s9ce an inf9ite numb] ( dim5sions w\ld be practically impossible to det]m9e4 ,if we rememb] our axiom ( ,free ,mobility1 & rememb] also t space is a cont9uum1 we may /ate our axiom1 = metrical ,geometry1 9 ! =m giv5 by ,helmholtz 7v4 ,*ap4 ,i4 #be73 ,9 a space ( n dim5sions1 ! position ( e po9t is uniquely det]m9$ by ! measurem5t ( n cont9u\s indep5d5t v>iables 7coord9ates74#afe #aei4 ,s mu*1 !n1 is priori necess>y to metrical ,geometry4 ,! re/riction ( ! dim5sions to ?ree seems1 on ! contr>y1 to be :olly ! "w ( exp]i5ce#aff4 ,?is re/riction _c be logically necess>y1 = z soon z we h =mulat$ any analytical sy/em1 x appe>s :olly arbitr>y4 ,:y1 we are driv5 to ask1 _c we add a f\r? coord9ate to our x1 y1 z1 or give a geometrical mean+ to x#d8 ,9 ?is m special =m1 we are tempt$ to reg>d ! axiom ( dim5sions1 l ! numb] ( inhabitants ( a t[n1 z a purely /ati/ical fact1 ) no great] necessity ?an su* facts h4 ,geometry af=ds intr9sic evid5ce ( ! tru? ( my division ( ! axiom ( dim5sions into an priori & empirical portion4 ,= :ile ! ext5sion ( ! numb] ( dim5sions to f\r1 or to n1 alt]s no?+ 9 plane & solid ,geometry1 b only adds a new bran* :i* int]f]es 9 no way ) ! old1 "s def9ite numb] ( dim5sions is assum$ 9 all ,geometries1 nor is x possible to conceive ( a ,geometry :i* %\ld be free f ?is assumption#afg4 #afj4 ,let u1 s9ce ! po9t seems ( "s int]e/1 repeat our pro( ( ! apriority ( ?is axiom f a slightly diff]5t po9t ( view4 ,we w beg91 ?is "t1 f ! mo/ ab/ract conception ( space1 su* z we f9d 9 ,riemann's diss]tation1 or 9 ,erdmann's ext5ts4 ,we h "h1 an ord]$ manifold1 inf9itely divisible & all[+ ( ,free ,mobility#afh4 ,free ,mobility involves1 z we saw1 ! p[] ( pass+ cont9u\sly f any "o po9t to any o!r1 by any c\rse :i* may seem pleasant to u2 x involves1 also1 t1 9 su* a c\rse1 no *anges occur except *anges ( m]e position1 i4e41 positions d n diff] f "o ano!r 9 any qualitative way4 7,?is abs5ce ( qualitative diff]5ce is ! di/+ui%+ m>k ( space z oppos$ to o!r manifolds1 su* z ! col\r- & tone-sy/ems3 9 ^!1 e elem5t has a def9ite qualitative s5sational value1 :]eas 9 space1 ! s5sational value ( a position dep5ds :olly on its spatial relation to our own body1 & is ?us n intr9sic1 b relative47 ,f ! abs5ce ( qualitative diff]5ces among positions1 x foll[s logically t positions exi/ only by virtue ( o!r positions2 "o position diff]s f ano!r j because !y are two1 n because ( any?+ intr9sic 9 ei!r4 ,position is ?us def9$ simply & solely by relation to o!r positions4 ,any position1 !re=e1 is completely def9$ :51 & only :51 en\gh su* relations h be5 giv5 to enable u to det]m9e its relation to any new position1 ?is new position be+ def9$ by ! same numb] ( relations4 ,n[1 9 ord] t su* def9ition may be at all possible1 a f9ite numb] ( relations mu/ suffice4 ,b e su* relation con/itutes a dim5sion4 ,!re=e1 if ,geometry is to be possible1 x is priori necess>y t space %\ld h a f9ite integral numb] ( dim5sions4 #afa4 ,! limitation ( ! dim5sions to ?ree is1 z we h se51 empirical2 nev]!less1 x is n liable to ! inaccuracy & unc]ta9ty :i* usually belong to empirical k4 ,= ! alt]natives :i* logic leaves to s5se are discrete--if ! dim5sions are n ?ree1 !y mu/ be two or f\r or "s o!r numb]--s t small errors are out ( ! "q#afi4 ,h5ce ! f9al c]ta9ty ( ! axiom ( ?ree dim5sions1 ?\gh 9 "p due to exp]i5ce1 is ( q a diff]5t ord] f t ( 7say7 ! law ( ,gravitation4 ,9 ! latt]1 a small inaccuracy might exi/ & rema9 undetect$2 9 ! =m]1 an error w\ld h to be s consid]able z to be utt]ly impossible to ov]look4 ,x foll[s t ! c]ta9ty ( our :ole axiom1 t ! numb] ( dim5sions is ?ree1 is almo/ z great z t ( ! priori elem5t1 s9ce ?is elem5t leaves to s5se a def9ite disjunction ( discrete possibilities4 ,,iii4 ,! ,axiom ( ,di/ance4 #afb4 ,we h already se51 9 discuss+ projective ,geometry1 t two po9ts mu/ det]m9e a unique curve1 ! /raight l9e4 ,9 metrical ,geometry1 ! correspond+ axiom is1 t two po9ts mu/ det]m9e a unique spatial quantity1 di/ance4 ,i propose to prove1 9 :at foll[s1 7#a7 t if di/ance1 z a quantity completely det]m9$ by two po9ts1 did n exi/1 spatial magnitude w\ld n be measurable2 7#b7 t di/ance c only be det]m9$ by two po9ts1 if !re is an actual curve 9 space det]m9$ by ^? two po9ts2 7#c7 t ! exi/5ce ( su* a curve c be d$uc$ f ! conception ( a =m ( ext]nality1 & 7#d7 t ! application ( quantity to su* a curve necess>ily leads to a c]ta9 magnitude1 namely di/ance1 uniquely det]m9$ by any two po9ts :i* det]m9e ! curve4 ,! conclusion w be1 if ^! propositions c be successfully ma9ta9$1 t ! axiom ( di/ance is priori 9 ! same d\ble s5se z ! axiom ( ,free ,mobility1 i4e4 x is presuppos$ 9 ! possibility ( measurem5t1 & x is necess>ily true ( any possible =m ( ext]nality4 #afc4 7#a7 ,! possibility ( spatial measurem5t all[s u to inf] ! exi/5ce ( a magnitude uniquely det]m9$ by any two po9ts4 ,! pro( ( ?is dep5ds on ! axiom ( ,free ,mobility1 or its equival5t1 ! homog5eity ( space4 ,we h se5 t ^! are involv$ 9 ! possibility ( spatial measurem5t2 we may employ !m1 !re=e1 9 any argum5t z to ! conditions ( ?is possibility4 ,n[ to beg9 )1 two po9ts mu/1 if ,geometry is to be possible1 h "s relation to ea* o!r1 = we h se5 t su* relations alone con/itute position or localization4 ,b if two po9ts h a relation to ea* o!r1 ?is mu/ be an intr9sic relation4 ,= x foll[s1 f ! axiom ( ,free ,mobility1 t two po9ts1 =m+ a figure congru5t ) ! giv5 pair1 c be con/ruct$ 9 any "p ( space4 ,if ?is 7 n possible1 we h se5 t metrical ,geometry c\ld n exi/4 ,b bo? ! figures may be reg>d$ z compos$ ( two po9ts & _! relation2 if ! two figures are congru5t1 !re=e1 x foll[s t ! relation is quantitatively ! same = bo? figures1 s9ce congru5ce is ! te/ ( spatial equality4 ,h5ce ! two po9ts h a quantitative relation1 :i* is su* t !y c trav]se all space 9 a comb9$ motion )\t 9 any way alt]+ t relation4 ,b 9 su* a g5]al motion1 any ext]nal relation ( ! two po9ts1 any relation involv+ o!r po9ts or figures 9 space1 mu/ be alt]$#agj4 ,h5ce ! relation betwe5 ! two po9ts1 be+ unalt]$1 mu/ be an intr9sic relation1 a relation involv+ no o!r po9t or figure 9 space2 & ?is intr9sic relation we call di/ance#aga4 #afd4 ,x might be object$1 to ! above argum5t1 t x involves a petitio pr9cipii4 ,= x has be5 assum$ t ! two po9ts & _! relation =m a figure1 to :i* o!r figures c be congru5t4 ,n[ if two po9ts h no intr9sic relation1 x w\ld seem t !y _c =m su* a figure4 ,! argum5t1 !re=e1 app>5tly assumes :at x _h to prove4 ,:y1 x may be ask$1 %\ld n ?ree po9ts be requir$1 be=e we obta9 any relation1 :i* ,free ,mobility all[s u to con/ruct afre% 9 o!r p>ts ( space8 ,! answ] to ?is1 z to ! correspond+ "q 9 ! fir/ section ( ?is *apt]1 lies1 ,i ?9k1 9 ! passivity ( space1 or ! mutual indep5d5ce ( its p>ts4 ,= x foll[s1 f ?is indep5d5ce1 t any figure1 or any assemblage ( po9ts1 may be discuss$ )\t ref]5ce to o!r figures or po9ts4 ,?is pr9ciple is ! basis ( inf9ite divisibility1 ( ! use ( quantity 9 ,geometry1 & ( all possibility ( isolat+ p>ticul> figures = discussion4 ,x foll[s t two po9ts _c be dep5d5t1 z to _! relation1 on any o!r po9ts or figures1 = if !y 7 s dep5d5t1 we %\ld h to suppose "s action ( su* po9ts or figures on ! two po9ts consid]$1 :i* w\ld contradict ! mutual indep5d5ce ( diff]5t positions4 ,to illu/rate by an example3 ! relation ( two giv5 po9ts does n dep5d on ! o!r po9ts ( ! /raight l9e on :i* ! giv5 po9ts lie4 ,= only ?r\gh _! relation1 i4e4 ?r\gh ! /raight l9e :i* !y det]m9e1 c ! o!r po9ts ( ! /raight l9e be kn[n to h any peculi> connection ) ! giv5 pair4 #afe4 ,b :y1 x may be ask$1 %\ld !re be only "o su* relation betwe5 two po9ts8 ,:y n sev]al8 ,! answ] to ?is lies 9 ! fact t po9ts are :olly con/itut$ by relations1 & h no intr9sic nature ( _! own#agb4 ,a po9t is def9$ by its relations to o!r po9ts1 & :5 once ! relations necess>y = def9ition h be5 giv51 no fre% relations to ! po9ts us$ 9 def9ition are possible1 s9ce ! po9t def9$ has no qualities f :i* su* relations c\ld fl[4 ,n[ "o relation to any "o o!r po9t is z good = def9ition z m w\ld be1 s9ce h[ev] _m we _h1 !y w\ld all rema9 unalt]$ 9 a comb9$ motion ( bo? po9ts4 ,h5ce !re c only be "o relation det]m9$ by any two po9ts4 #aff4 7#b7 ,we h ?us e/abli%$ our fir/ proposition--two po9ts h "o & only "o relation uniquely det]m9$ by ^? two po9ts4 ,?is relation we call _! di/ance ap>t4 ,x rema9s to consid] ! conditions ( ! measurem5t ( di/ance1 i4e41 h[ f> a unique value = di/ance involves a curve uniquely det]m9$ by ! two po9ts4 ,9 ! fir/ place1 "s curve jo9+ ! two po9ts is involv$ 9 ! above notion ( a comb9$ motion ( ! two po9ts1 or ( two o!r po9ts =m+ a figure congru5t ) ! fir/ two4 ,= )\t "s su* curve1 ! two po9t-pairs _c be kn[n z congru5t1 nor c we h any te/ by :i* to discov] :5 a po9t-pair is mov+ z a s+le figure#agc4 ,di/ance mu/ be measur$1 !re=e1 by "s l9e :i* jo9s ! two po9ts4 ,b ne$ ?is be a l9e :i* ! two po9ts completely det]m9e8 #afg4 ,we are accu/om$ to ! def9ition ( ! /raight l9e z ! %orte/ di/ance betwe5 two po9ts1 :i* implies t di/ance might equally well be measur$ by curv$ l9es4 ,?is implication ,i believe to be false1 = ! foll[+ reasons4 ,:5 we speak ( ! l5g? ( a curve1 we c give a mean+ to our words only by suppos+ ! curve divid$ into inf9itesimal rectil9e> arcs1 ^: sum gives ! l5g? ( an equival5t /raight l9e2 ?us unless we presuppose ! /raight l9e1 we h no means ( comp>+ ! l5g?s ( diff]5t curves1 & c !re=e nev] discov] ! applicability ( our def9ition4 ,x might be ?\ght1 p]haps1 t "s o!r l9e1 say a circle1 might be us$ z ! basis ( measurem5t4 ,b 9 ord] to e/imate 9 ?is way ! l5g? ( any curve o!r ?an a circle1 we %\ld h to divide ! curve into inf9itesimal circul> arcs4 ,n[ two successive po9ts d n det]m9e a circle1 s t an arc ( two po9ts w\ld h an indet]m9ate l5g?4 ,x is true t1 if we exclude inf9itesimal radii = ! measur+ circles1 ! l5g?s ( ! inf9itesimal arcs w\ld be det]m9ate1 ev5 if ! circles v>i$1 b t is only because all ! small circul> arcs ?r\gh two consecutive po9ts co9cide ) ! /raight l9e ?r\gh ^? two po9ts4 ,?us1 ev5 ) ! help ( ! arbitr>y re/riction to a f9ite radius1 all t happ5s is t we are br\ght back to ! /raight l9e4 ,if1 to m5d matt]s1 we take ?ree consecutive po9ts ( our curve1 & reckon di/ance by ! arc ( ! circle ( curvature1 ! notion ( di/ance loses its fundam5tal prop]ty ( be+ a relation betwe5 two po9ts4 ,= two consecutive po9ts ( ! arc c\ld n !n be said to h any correspond+ di/ance ap>t--?ree po9ts w\ld be necess>y be=e ! notion ( di/ance became applicable4 ,?us ! circle is n a possible basis = measurem5t1 & simil> objections apply1 ( c\rse1 ) increas$ =ce1 to any o!r curve4 ,all ?is argum5t is design$ to %[1 9 detail1 ! logical impossibility ( measur+ di/ance by any curve n completely def9$ by ! two po9ts ^: di/ance ap>t is requir$4 ,if 9 ! above we _h tak5 di/ance z measur$ by circles ( giv5 radius1 we %\ld h introduc$ into its def9ition a relation to o!r po9ts besides ! two ^: di/ance 0 to be measur$1 :i* we saw to be a logical fallacy4 ,moreov]1 h[ are we to "k t all ! circles h equal radii1 until we h an indep5d5t measure ( di/ance8 #afh4 ,a /raight l9e1 !n1 is n ! %orte/ di/ance1 b is simply ! di/ance betwe5 two po9ts--s f>1 ?is conclusion has /ood firm4 ,b suppose we _h two or m curves ?r\gh two po9ts1 & t all ^! curves 7 congru5t int] se4 ,we %\ld !n say1 9 accordance ) ! def9ition ( spatial equality1 t ! l5g?s ( all ^! curves 7 equal4 ,n[ x might happ5 t1 al?\gh no "o ( ! curves 0 uniquely det]m9$ by ! two end-po9ts1 yet ! common l5g? ( all ! curves 0 s det]m9$4 ,9 ?is case1 :at w\ld h9d] u f call+ ?is common l5g? ! di/ance ap>t1 al?\gh no unique figure 9 space correspond$ to x8 ,?is is ! case contemplat$ by sph]ical ,geometry1 :]e1 z on a sph]e1 antipodes c be jo9$ by an inf9ite numb] ( geodesics1 all ( :i* are ( equal l5g?4 ,! difficulty suppos$ is1 !re=e1 n a purely imag9>y "o1 b "o :i* mod]n ,geometry =ces u to face4 ,i %all consequ5tly discuss x at "s l5g?4 #afi4 ,to beg9 )1 ,i mu/ po9t out t my axiom is n q equival5t to ,euclid's4 ,euclid's axiom /ates t two /raight l9es _c enclose a space1 i4e41 _c h m ?an "o common po9t4 ,n[ if e two po9ts1 )\t exception1 det]m9e a unique /raight l9e1 x foll[s1 ( c\rse1 t two diff]5t /raight l9es c h only "o po9t 9 common--s f>1 ! two axioms are equival5t4 ,b x may happ51 z 9 sph]ical space1 t two po9ts 9 g5]al det]m9e a unique /raight l9e1 b fail to d s :5 !y h to ea* o!r ! special relation ( be+ antipodes4 ,9 su* a sy/em e pair ( /raight l9es 9 ! same plane meet 9 two po9ts1 :i* are ea* o!r's antipodes2 b two po9ts1 9 g5]al1 /ill det]m9e a unique /raight l9e4 ,we are /ill able1 !re=e1 to obta9 di/ances f unique /raight l9es1 except 9 limit+ cases2 & 9 su* cases1 we c take any po9t int]m$iate betwe5 ! two antipodes1 jo9 x by ! same /raight l9e to bo? antipodes1 & measure its di/ance f ^? antipodes 9 ! usual way4 ,! sum ( ^! di/ances !n gives a unique value = ! di/ance betwe5 ! antipodes4 ,?us ev5 9 sph]ical space1 we are greatly assi/$ by ! axiom ( ! /raight l9e2 all l9e> measurem5t is effect$ by x1 & exceptional cases c be treat$1 ?r\gh its help1 by ! usual me?ods = limits4 ,sph]ical space1 !re=e1 is n s adv]se z x at fir/ appe>$ to be to ! priori necessity ( ! axiom4 ,nev]!less we h1 s f>1 n attack$ ! k]nel ( ! objection :i* sph]ical space sugge/$4 ,to ?is attack x is n[ our duty to proce$4 #agj4 ,x w be rememb]$ t1 9 our priori pro( t two po9ts mu/ h "o def9ite relation1 we held x impossible = ^? two po9ts to h1 to ! re/ ( space1 any relation :i* w\ld be unalt]$ by motion4 ,n[ 9 sph]ical space1 9 ! p>ticul> case :]e ! two po9ts are antipodes1 !y h a relation1 unalt]$ by motion1 to ! re/ ( space--! relation1 namely1 t _! di/ance is half ! circumf]5ce ( ! univ]se4 ,9 our =m] discussion1 we assum$ t any relation to outside space mu/ be a relation ( position--& a relation ( position mu/ be alt]$ by motion4 ,b ) a f9ite space1 9 :i* we h absolute magnitude1 ano!r relation becomes possible1 namely1 a relation ( magnitude4 ,antipodal po9ts1 accord+ly1 l co9cid5t po9ts1 no long] det]m9e a unique /raight l9e4 ,& x is in/ructive to obs]ve t !re is1 9 consequ5ce1 an ambiguity 9 ! expression = di/ance1 l ! ord9>y ambiguity 9 angul> measurem5t4 ,if #a_/k#b be ! space con/ant1 & d be "o value = ! di/ance betwe5 two po9ts1 #bkn d1 :]e n is any integ]1 is an equally good value4 ,di/ance is1 9 %ort1 a p]iodic function l angle4 ,?us su* a /ate ( ?+s r confirms ?an de/roys my cont5tion1 t di/ance dep5ds on a curve uniquely det]m9$ by two po9ts4 ,= z soon z we drop ?is unique det]m9ation1 we see ambiguities creep+ into our expression = di/ance4 ,di/ance /ill has a set ( discrete values1 correspond+ to ! fact t1 giv5 "o po9t1 ! /raight l9e is uniquely det]m9$ = all o!r po9ts b "o1 ! antipodal po9t4 ,x is tempt+ to g on1 & say3 ,if ?r\gh e pair ( po9ts !re 7 an inf9ite numb] ( ! curves us$ 9 measur+ di/ance1 di/ance w\ld be able1 = ! same pair ( po9ts1 to take1 n only a discrete s]ies1 b an inf9ite cont9u\s s]ies ( values4 #aga4 ,?is1 h[ev]1 is m]e speculation4 ,i come n[ to ! pice de rsi/ance ( my argum5t4 ,! ambiguity 9 sph]ical space arose1 z we saw1 f a relation ( magnitude to ! re/ ( space--su* a relation be+ unalt]$ by a motion ( ! two po9ts1 & !re=e fall+ outside our introductory reason+4 ,b :at is ?is relation ( magnitude8 ,simply a relation ( ! di/ance betwe5 ! two po9ts to a di/ance giv5 9 ! nature ( ! space 9 "q4 ,x foll[s t su* a relation presupposes a measure ( di/ance1 & ne$ n1 !re=e1 be contemplat$ 9 any argum5t :i* deals ) ! priori requisites = ! possibility ( def9ite di/ances#agd4 #agb4 ,i h n[ %[n1 ,i hope conclusively1 t sph]ical space af=ds no objection to ! apriority ( my axiom4 ,any two po9ts h "o relation1 _! di/ance1 :i* is indep5d5t ( ! re/ ( space1 & ?is relation requires1 z its measure1 a curve uniquely det]m9$ by ^? two po9ts4 ,i might h tak5 ! bull by ! horns1 & said3 ,two po9ts c h no relation b :at is giv5 by l9es :i* jo9 !m1 & !re=e1 if !y h a relation indep5d5t ( ! re/ ( space1 !re mu/ be "o l9e jo9+ !m :i* !y completely det]m9e4 ,?us ,james says#age3 ,j z1 9 ! field ( quantity1 ! relation betwe5 two numb]s is ano!r numb]1 s 9 ! field ( space ! relations are facts ( ! same ord] ) ! facts !y relate4444 ,:5 we speak ( ! relation ( direction ( two po9ts t[>ds ea* o!r1 we mean simply ! s5sation ( ! l9e t jo9s ! two po9ts toge!r4 ,! l9e is ! relation4444 ,! relation ( position betwe5 ! top & bottom po9ts ( a v]tical l9e is t l9e1 & no?+ else4 ,if ,i _h be5 will+ to use ?is doctr9e at ! beg9n+1 ,i might h avoid$ all discussion4 ,a unique relation betwe5 two po9ts mu/ 9 ?is case1 involve a unique l9e betwe5 !m4 ,b x seem$ bett] to avoid a doctr9e n univ]sally accept$1 ! m s z ,i 0 approa*+ ! "q f ! logical1 n ! psy*ological1 side4 ,aft] dispos+ ( ! objections1 h[ev]1 x is int]e/+ to f9d ?is confirmation ( ! above !ory f s diff]5t a /&po9t4 ,inde$1 ,i believe ,james's doctr9e c\ld be prov$ to be a logical necessity1 z well z a psy*ological fact4 ,= :at sort ( ?+ c a spatial relation betwe5 two di/9ct po9ts be8 ,x mu/ be some?+ spatial1 & x mu/1 s9ce po9ts are :olly con/itut$ by _! relations1 be some?+ at lea/ z real & tangible z ! po9ts x relates4 ,!re seems no?+ :i* c satisfy ^! requirem5ts1 except a l9e jo9+ !m4 ,h5ce1 once m1 a unique relation mu/ involve a unique l9e4 ,t is1 l9e> magnitude is logically impossible1 unless space all[s ( curves uniquely det]m9$ by any two ( _! po9ts4 #agc4 7#c7 ,b f>!r1 ! exi/5ce ( curves uniquely det]m9$ by two po9ts c be d$uc$ f ! nature ( any =m ( ext]nality#agf4 ,= we saw1 9 discuss+ ,free ,mobility1 t ?is axiom1 toge!r ) homog5eity & ! relativity ( position1 c be s d$uc$1 & we saw 9 ! beg9n+ ( our discussion on di/ance1 t ! exi/5ce ( a unique relation betwe5 two po9ts c\ld be d$uc$ f ! homog5eity ( space4 ,s9ce position is relative1 we may say1 any two po9ts mu/ h "s relation to ea* o!r3 s9ce our =m ( ext]nality is homog5e\s1 ?is relation c be kept un*ang$ :ile ! two po9ts move 9 ! =m1 i4e41 *ange _! relations to o!r po9ts2 h5ce _! relation to ea* o!r is an intr9sic relation1 indep5d5t ( _! relations to o!r po9ts4 ,b s9ce our =m is m]ely a complex ( relations1 a relation ( ext]nality mu/ appe> 9 ! =m1 ) ! same evid5ce z any?+ else 9 ! =m2 ?us if ! =m be intuitive or s5sational1 ! relation mu/ be imm$iately pres5t$1 & n a m]e inf]5ce4 ,h5ce ! intr9sic relation betwe5 two po9ts mu/ be a unique figure 9 our =m1 i4e4 9 spatial t]ms1 ! /raight l9e jo9+ ! two po9ts4 #agd4 7#d7 ,f9ally1 we h to prove t ! exi/5ce ( su* a curve necess>ily leads1 :5 quantity is appli$ to ! relation betwe5 two po9ts1 to a unique magnitude1 :i* ^? two po9ts completely det]m9e4 ,) ?is1 we %all be br\ght back to di/ance1 f :i* we />t$1 & %all complete ! circle ( our argum5t4 ,we saw1 9 section ,a #aai1 t ! figure =m$ by two po9ts is projectively indi/+ui%able f t =m$ by any two o!r po9ts 9 ! same /raight l9e2 ! figure1 9 bo? cases1 is1 f ! projective /&po9t1 simply ! /raight l9e on :i* ! two po9ts lie4 ,! diff]5ce ( relation1 9 ! two cases1 is n qualitative1 s9ce projective ,geometry _c deal ) x2 nev]!less1 !re is "s diff]5ce ( relation4 ,= in/ance1 if "o po9t be kept fix$1 :ile ! o!r moves1 !re is obvi\sly "s *ange ( relation4 ,?is *ange1 s9ce all p>ts ( ! /raight l9e are qualitatively alike1 mu/ be a *ange ( quantity4 ,if two po9ts1 !re=e1 det]m9e a unique figure1 !re mu/ exi/1 = ! di/9ction betwe5 ! v>i\s o!r po9ts ( ?is figure1 a unique quantitative relation betwe5 ! two det]m9+ po9ts1 & !re=e1 s9ce ^! po9ts are arbitr>y1 betwe5 only two po9ts4 ,?is relation is di/ance1 ) :i* our argum5t began1 & to :i* x at lea/ returns4 #age4 ,to sum up3 ,if po9ts are def9$ simply by relations to o!r po9ts1 i4e41 if all position is relative1 e po9t mu/ h to e o!r po9t "o1 & only "o1 relation indep5d5t ( ! re/ ( space4 ,?is relation is ! di/ance betwe5 ! two po9ts4 ,n[ a relation betwe5 two po9ts c only be def9$ by a l9e jo9+ !m--nay fur!r1 x may be cont5d$ t a relation c only be a l9e jo9+ !m4 ,h5ce a unique relation involves a unique l9e1 i4e41 a l9e det]m9$ by any two ( its po9ts4 ,only 9 a space :i* admits ( su* a l9e is l9e> magnitude a logically possible conception4 ,b :5 once we h e/abli%$ ! possibility1 9 g5]al1 ( draw+ su* l9es1 & !re=e ( measur+ l9e> magnitudes1 we may f9d t a c]ta9 magnitude has a peculi> relation to ! con/itution ( space4 ,! /raight l9e may turn out to be ( f9ite l5g?1 & 9 ?is case its l5g? w give a c]ta9 peculi> magnitude1 ! space-con/ant4 ,two antipodal po9ts1 t is1 po9ts :i* bisect ! entire /raight l9e1 w !n h a relation ( magnitude :i*1 ?\gh unalt]$ by motion1 is r5d]$ peculi> by a c]ta9 con/ant relation to ! re/ ( space4 ,?is peculi>ity presupposes a measure ( l9e> magnitude 9 g5]al1 & _c1 !re=e1 upset ! apriority ( ! axiom ( ! /raight l9e4 ,b x de/roys1 = po9ts hav+ ! peculi> antipodal relation to ea* o!r1 ! argum5t :i* prov$ t ! relation betwe5 two po9ts c\ld n1 s9ce x 0 un*ang$ by motion1 h ref]5ce to ! re/ ( space4 ,?us x is intelligible t1 = su* special po9ts1 ! axiom breaks d[n1 & an inf9ite numb] ( /raight l9es are possible betwe5 !m2 b unless we _h />t$ ) assum+ ! g5]al validity ( ! axiom1 we c\ld nev] h rea*$ a position 9 :i* antipodal po9ts c\ld h be5 kn[n to be peculi>1 or1 inde$1 a position :i* w\ld h enabl$ u to give any quantitative def9ition :atev] ( p>ticul> po9ts4 ,di/ance & ! /raight l9e1 z relations uniquely det]m9$ by two po9ts1 are ?us priori necess>y to metrical ,geometry4 ,b fur!r1 !y are prop]ties :i* mu/ belong to any =m ( ext]nality4 ,s9ce _! necessity = ,geometry 0 d$uc$ f homog5eity & ! relativity ( position1 & s9ce ^! are necess>y prop]ties ( any =m ( ext]nality1 ! same argum5t proves bo? conclusions4 ,we ?us obta91 z 9 ! case ( ,free ,mobility1 a d\ble apriority3 ,! axiom ( ,di/ance1 & its implication1 ! axiom ( ! ,/raight ,l9e1 are1 on ! "o h&1 presuppos$ 9 ! possibility ( spatial magnitude1 & _c1 !re=e1 be contradict$ by any exp]i5ce result+ f ! measurem5t ( space2 :ile !y are consequ5ces1 on ! o!r h&1 ( ! necess>y prop]ties ( any =m ( ext]nality :i* is to r5d] possible exp]i5ce ( an ext]nal _w4 #agf4 ,9 connection ) ! /raight l9e1 x w be conv5i5t to discuss ! conditions ( a metrical coord9ate sy/em4 ,! projective coord9ate sy/em1 z we h se51 aims only at a conv5i5t nom5clature = diff]5t po9ts1 & c be set up )\t introduc+ ! notion ( spatial quantity4 ,b a metrical coord9ate sy/em does mu* m ?an ?is4 ,x def9es e po9t quantitatively1 by its quantitative spatial relations to a c]ta9 coord9ate figure4 ,only :5 ! sy/em ( coord9ates is ?us metrical1 i4e41 :5 e coord9ate repres5ts "s spatial magnitude1 :i* is itself a relation ( ! po9t def9$ to "s o!r po9t or figure--c op]ations ) coord9ates lead to a metrical result4 ,:51 z 9 projective ,geometry1 ! coord9ates are n spatial magnitudes1 no am\nt ( trans=mation c give a metrical result4 ,i wi% to prove1 "h1 t a metrical coord9ate sy/em necess>ily involves ! /raight l9e1 & _c1 )\t a logical fallacy1 be set up on any o!r basis4 ,! projective sy/em ( coord9ates1 z we saw1 is entirely bas$ on ! /raight l9e2 b ! metrical sy/em is m important1 s9ce its quantities embody actual in=mation z to spatial magnitudes1 :i*1 9 projective ,geometry1 is n ! case4 ,9 ! fir/ place1 a po9t's metrical coord9ates con/itute a complete quantitative def9ition ( x2 n[ a po9t c only be def9$1 z we h se51 by its relations to o!r po9ts1 & ^! relations c only be def9$ by means ( ! /raight l9e4 ,consequ5tly1 any metrical sy/em ( coord9ates mu/ involve ! /raight l9e1 z ! basis ( its def9itions ( po9ts4 ,?is priori argum5t1 h[ev]1 ?\gh ,i believe x to be q s\nd1 is n likely to c>ry conviction to any "o p]suad$ ( ! opposite4 ,let u1 !re=e1 exam9e metrical coord9ate sy/ems 9 detail1 & %[1 9 ea* case1 _! dep5d5ce on ! /raight l9e4 ,we h already se5 t ! notion ( di/ance is impossible )\t ! /raight l9e4 ,we _c1 !re=e1 def9e our coord9ates 9 any ( ! ord9>y ways1 z ! di/ances f ?ree planes1 l9es1 po9ts1 sph]es1 or :at n4 ,pol> coord9ates are impossible1 s9ce1--waiv+ ! /raightness ( ! radius vector--! l5g? ( ! radius vector becomes unmean+4 ,triangul> coord9ates involve n only angles1 :i* mu/ 9 ! limit be rectil9e>1 b /raight l9es1 or at any rate "s well-def9$ curves4 ,n[ curves c only be metrically def9$ 9 two ways3 ,ei!r by relation to ! /raight l9e1 z1 e4g41 by ! curvature at any po9t1 or by purely analytical equations1 :i* presuppose an intelligible sy/em ( metrical coord9ates4 ,:at me?ods rema9 = assign+ ^! arbitr>y values to diff]5t po9ts8 ,nay1 h[ are we to get any e/imate ( ! diff]5ce--to avoid ! m special notion ( di/ance--betwe5 two po9ts8 ,! v notion ( a po9t has become illusory4 ,:5 we h a coord9ate sy/em1 we may def9e a po9t by its ?ree coord9ates2 9 ! abs5ce ( su* a sy/em1 we may def9e ! notion ( po9t 9 g5]al z ! int]section ( ?ree surfaces or ( two curves4 ,"h we take surfaces & curves z notions :i* intuition makes pla91 b if we wi% !m to give u a precise num]ical def9ition ( p>ticul> po9ts1 we mu/ specify ! k9d ( surface or curve to be us$4 ,n[ ?is1 z we h se51 is only possible :5 we presuppose ei!r ! /raight l9e1 or a coord9ate sy/em4 ,x foll[s t e coord9ate sy/em presupposes ! /raight l9e1 & is logically impossible )\t x4 #agg4 ,! above ?ree axioms1 we h se51 are priori necess>y to metrical ,geometry4 ,no o!rs c be necess>y1 s9ce metrical sy/ems1 logically z unassailable z ,euclid's1 & deal+ ) spaces equally homog5e\s & equally relational1 h be5 con/ruct$ by ! metageomet]s1 )\t ! help ( any o!r axioms4 ,! rema9+ axioms ( ,euclidean ,geometry--! axiom ( p>allels1 ! axiom t ! numb] ( dim5sions is ?ree1 & ,euclid's =m ( ! axiom ( ! /raight l9e 7two /raight l9es _c enclose a space7--are n ess5tial to ! possibility ( metrical ,geometry1 i4e41 are n d$ucible f ! fact t a sci5ce ( spatial magnitudes is possible4 ,!y are r to be reg>d$ z empirical laws1 obta9$1 l ! empirical laws ( o!r sci5ces1 by actual inve/igation ( ! giv5 subject-matt]--9 ?is in/ance1 exp]i5c$ space4 #agh4 ,9 summ+ up ! di/9ctive argum5t ( ?is ,section1 we may give x a m g5]al =m1 & discuss ! conditions ( measurem5t 9 any cont9u\s manifold1 i4e41 ! qualities necess>y to ! manifold1 9 ord] t quantities 9 x may be det]m9able1 n only z to ! m or less1 b z to ! precise h[ mu*4 ,measurem5t1 we may say1 is ! application ( numb] to cont9ua1 or1 if we pref] x1 ! trans=mation ( m]e quantity into numb] ( units4 ,us+ quantity to d5ote ! vague m or less1 & magnitude to d5ote ! precise numb] ( units1 ! problem ( measurem5t may be def9$ z ! trans=mation ( quantity into magnitude4 ,n[ a numb]1 to beg9 )1 is a :ole consi/+ ( small] units1 all ( ^! units be+ qualitatively alike4 ,9 ord]1 !re=e1 t a cont9u\s quantity may be expressible z a numb]1 x mu/1 on ! "o h&1 be itself a :ole1 & mu/1 on ! o!r h&1 be divisible into qualitatively simil> p>ts4 ,9 ! aspect ( a :ole1 ! quantity is int5sive2 9 ! aspect ( an aggregate ( p>ts1 x is ext5sive4 ,a purely int5sive quantity1 !re=e1 is n num]able--a purely ext5sive quantity1 if any su* c\ld be imag9$1 w\ld n be a s+le quantity at all1 s9ce x w\ld h to consi/ ( :olly unsyn!siz$ p>ticul>s4 ,a measurable quantity1 !re=e1 is a :ole divisible into simil> p>ts4 ,b a cont9u\s quantity1 if divisible at all1 mu/ be inf9itely divisible4 ,= o!rwise ! po9ts at :i* x c\ld be divid$ w\ld =m natural b>ri]s1 & s de/roy its cont9uity4 ,b fur!r1 x is n suffici5t t !re %\ld be a possibility ( division into mutually ext]nal p>ts2 :ile ! p>ts1 to be p]ceptible z p>ts1 mu/ be mutually ext]nal1 !y mu/ also1 to be kn[able z equal p>ts1 be capable ( ov]com+ _! mutual ext]nality4 ,= ?is1 z we h se51 we require sup]position1 :i* involves ,free ,mobility & homog5eity--! abs5ce ( ,free ,mobility 9 "t1 :]e all o!r requisites ( measurem5t are fulfill$1 r5d]s direct measurem5t ( "t impossible4 ,h5ce inf9ite divisibility1 free mobility1 & homog5eity are necess>y = ! possibility ( measurem5t 9 any cont9u\s manifold1 & ^!1 z we h se51 are equival5t to our ?ree axioms4 ,^! axioms are necess>y1 !re=e1 n only = spatial measurem5t1 b = all measurem5t4 ,! only manifold giv5 9 exp]i5ce1 9 :i* ^! conditions are satisfi$1 is space4 ,all o!r exact measurem5t--z c\ld be prov$1 ,i believe1 = e sep>ate case--is effect$1 z we saw 9 ! case ( "t1 by r$uction to a spatial correlative4 ,?is expla9s ! p>am\nt importance1 to exact sci5ce1 ( ! me*anical view ( nature1 :i* r$uces all ph5om5a to motions 9 "t & space4 ,= numb] is1 ( all conceptions1 ! easie/ to op]ate )1 & sci5ce seeks ev]y:]e = an opportunity to apply x1 b f9ds ?is opportunity only by means ( spatial equival5ts to ph5om5a#agg4 #agi4 ,we h n[ se5 9 :at ! priori elem5t ( ,geometry consi/s4 ,?is priori elem5t may be def9$ z ! axioms common to ,euclidean & non-,euclidean spaces1 z ! axioms d$ucible f ! conception ( a =m ( ext]nality1 or--9 metrical ,geometry--z ! axioms requir$ = ! possibility ( measurem5t4 ,x rema9s to discuss1 9 a f9al *apt]1 "s que/ions ( a m g5]al philosophic nature1 9 :i* we %all h to des]t ! firm gr\nd ( ma!matics & ent] on speculations :i* ,i put =w>d v t5tatively1 & ) little fai? 9 _! ultimate validity4 ,! *ief que/ions = ?is f9al *apt] w be two3 7#a7 ,h[ is su* priori & purely logical necessity possible1 z appli$ to an actually giv5 subject-matt] l space8 7#b7 ,h[ c we remove ! contradictions :i* h haunt$ u 9 ?is *apt]1 aris+ out ( ! relativity1 inf9ite divisibility1 & unb\nd$ ext5sion ( space8 ,^! two que/ions are =c$ ^u u by ! pres5t *apt]1 b z !y op5 "s ( ! fundam5tal problems ( philosophy1 x w\ld be ra% to expect a conclusive or :olly satisfactory answ]4 ,a few h9ts & sugge/ions may be hop$ =1 b a complete solution c\ld only be obta9$ f a complete philosophy1 ( :i* ! prospects are f> too sl5d] to enc\rage a confid5t frame ( m9d4 ,,footnotes3 #aaf ,see infra1 ,axiom ( ,di/ance1 9 ,sec4 ,b4 ( ?is ,*apt]4 #aag ,?us on a cyl9d]1 two geodesics1 e4g4 a g5]ator & a helix1 may h any numb] ( int]sections--a v important diff]5ce f ! plane4 #aah ,cf4 ,cremona1 ,projective ,geometry 7,cl>5don ,press1 #bnd ed4 #ahic7 p4 #ej3 ,mo/ ( ! propositions 9 ,euclid's ,elem5ts are metrical1 & x is n easy to f9d among !m an example ( a purely descriptive !orem4 #aai ,op4 cit4 p4 #bbf4 #abj ,"s gr\nd = ?is *oice w appe> :5 we come to metrical ,geometry4 #aba ,! /raight l9e a d5otes ! /raight l9e common to ! planes & a1 ! po9t a d5otes ! po9t common to ! plane & ! /raight l9e a1 & simil>ly = ! re/ ( ! notation4 #abb ,cremona 7op4 cit4 ,*ap4 ,,ix4 p4 #ej7 def9es anh>monic ratio z a metrical prop]ty :i* is unalt]$ by projection4 ,?is1 h[ev]1 de/roys ! logical indep5d5ce ( projective ,geometry1 :i* c only be ma9ta9$ by a purely descriptive def9ition4 #abc ,!re is no correspond+ prop]ty ( ?ree po9ts on a l9e1 because !y c be projectively trans=m$ into any o!r ?ree po9ts on ! same l9e4 ,see #abj4 #abd ,due to v4 ,/audt's ,geometrie d] ,lage4 #abe ,see ,cremona1 op4 cit4 ,*apt] ,,viii4 #abf ,! correspond+ def9itions1 = ! two-dim5sional manifold ( l9es ?r\gh a po9t1 foll[ by ! pr9ciple ( duality4 #abg ,x is important to obs]ve t ?is def9ition ( ! ,po9t introduces metrical ideas4 ,)\t metrical ideas1 we saw1 no?+ appe>s to give ! ,po9t prec$5ce ( ! /raight l9e1 or inde$ to di/+ui% x conceptually f ! /raight l9e4 ,a ref]5ce to quantity is !re=e inevitable 9 def9+ ! ,po9t1 if ! def9ition is to be geometrical4 ,a non-metrical def9ition w\ld h to be also non-geometrical4 ,see ,*ap4 ,,iv4 #aif-aii4 #abh #afc-age4 #abi ,on ?is axiom1 h[ev]1 comp>e #aca4 #acj ,= ! pro( ( ?is proposition1 see ,*ap4 ,,iii4 ,sec4 ,b1 ,axiom ( ,dim5sions4 #aca ,! /raight l9e & plane1 9 all discussions ( g5]al ,geometry1 are n necess>ily ,euclidean4 ,!y are simply figures det]m9$1 9 g5]al1 by two & by ?ree po9ts respectively2 :e!r !y con=m to ! axiom ( p>allels & to ,euclid's =m ( ! axiom ( ! /raight l9e1 is n to be consid]$ 9 ! g5]al def9ition4 #acb ,t projective ,geometry mu/ h exi/5tial import1 ,i %all attempt to prove 9 ,*apt] ,,iv4 #acc ,logic1 ,book ,i4 ,*apt] ,,ii4 #acd ,cf4 ,bradley's ,logic1 p4 #fc4 ,x w be se5 t ! s5se 9 :i* ,i h spok5 ( space z a pr9ciple ( diff]5tiation is n ! s5se ( a pr9ciple ( individuation :i* ,bradley objects to4 #ace ,*ap4 ,,iv4 #ahf-aia4 #acf ,*ap4 ,,iv4 #bja ff4 #acg ,x is important to obs]ve1 h[ev]1 t ?is way ( reg>d+ spatial relations is metrical2 f ! projective /&po9t1 ! relation betwe5 two po9ts is ! :ole unb\nd$ /raight l9e on :i* !y lie1 & ne$ n be reg>d$ z divisible into p>ts or z built up ( po9ts4 #ach #bjg1 #bjh4 ,cf4 ,hegel1 ,naturphilosophie1 #bed4 #aci ,see ,*ap4 ,,iv4 #aif-aii4 #adj ,see a =?com+ article on ,! relations ( numb] & quantity by ! pres5t writ] 9 ,m9d1 ,july1 #ahig4 #ada ,logic1 ,vol4 ,,ii4 ,*ap4 ,,vii4 p4 #baa4 #adb ,real1 z oppos$ to logical1 div]sity is ?r\gh\t int5d$4 ,div]se aspects may coexi/ 9 a ?+ at "o "t & place1 b two div]se real ?+s _c s coexi/4 #adc ,on ! insuffici5cy ( "t alone1 see ,*apt] ,,iv4 #aia4 #add ,geometrically1 ! axiom ( ! plane is1 n t ?ree po9ts det]m9e a figure at all1 :i* foll[s f ! axiom ( ! /raight l9e1 b t ! /raight l9e jo9+ two casual po9ts ( ! plane lies :olly 9 ! plane4 ,?is axiom requires a projective me?od ( con/ruct+ ! plane1 i4e4 ( f9d+ all ! triads ( po9ts :i* det]m9e ! same projective figure z ! giv5 triad4 ,! requir$ con/ruction w be obta9$ if we c f9d any projective figure det]m9$ by ?ree po9ts1 & any projective me?od ( rea*+ o!r po9ts :i* det]m9e ! same figure4 ,illu/ration ,let ,o1 ,p1 ,q be ! ?ree po9ts ^: projective relation is requir$4 ,!n we h giv5 u ! ?ree /raight l9es ,,pq1 ,,qo1 ,,op4 ,metrically1 ! relation betwe5 ^! po9ts is made up ( ! area1 & ! magnitude ( ! sides & angles1 ( ! triangle ,,opq1 j z ! relation betwe5 two po9ts is di/ance4 ,b projectively1 ! figure is un*ang$ :5 ,p & ,q travel along ,,op & ,,oq1 or :5 ,,op & ,,oq turn ab\t ,o 9 su* a way z /ill to meet ,,pq4 ,?is is a result ( ! g5]al pr9ciple ( projective equival5ce enunciat$ above 7 #ajh1 #aji74 ,h5ce ! projective relation betwe5 ,o1 ,p1 ,q is ! same z t betwe5 ,o1 p1 q or ,o1 ,p1 ,q2 t is1 p1 q & ,p1 ,q lie 9 ! plane ,,opq4 ,9 ?is way1 any numb] ( po9ts on ! plane may be obta9$1 & by repeat+ ! con/ruction ) fre% triads1 e po9t ( ! plane c be rea*$4 ,we h to prove t1 :5 ! plane is s con/ruct$1 ! /raight l9e jo9+ any two po9ts ( ! plane lies :olly 9 ! plane4 ,x is evid5t1 f ! mann] ( con/ruction1 t any po9t ( ,,pq1 ,,op1 ,,oq1 ,,op or ,,oq lies 9 ! plane4 ,if we c prove t any po9t ( pq lies 9 ! plane1 we %all h prov$ all t is requir$1 s9ce pq may be trans=m$1 by successive repetitions ( ! same con/ruction1 into any /raight l9e jo9+ two po9ts ( ! plane4 ,b we h se5 t ! same plane is det]m9$ by ,o1 p1 q & by ,o1 ,p1 ,q4 ,! /raight l9es ,,pq1 pq h1 !re=e1 ! same relation to ! plane4 ,b ,,pq lies :olly 9 ! plane2 !re=e pq also lies :olly 9 ! plane4 ,h5ce our axiom is prov$4 #ade ,a detail$ pro( has be5 giv5 above1 ,*ap4 ,i4 #crd p]iod4 ,x is to be obs]v$ t any ref]5ce to inf9itely di/ant elem5ts involves metrical ideas4 #adf ,cf4 ,section ,a1 #aae-aag4 #adg ,contra/ ,erdmann1 op4 cit4 p4 #ach4 #adh ,cf4 ,erdmann1 op4 cit4 p4 #afd4 #adi ,/rictly speak+1 ?is me?od is only applicable :]e ! two magnitudes are comm5surable4 ,b if we take inf9ite divisibility rigidly1 ! units c !oretically be tak5 s small z to obta9 any requir$ degree ( approximation4 ,! difficulty is ! univ]sal "o ( apply+ to cont9ua ! ess5tially discrete conception ( numb]4 #aej ,cf4 ,erdmann1 op4 cit4 p4 #ej4 #aea ,also call$ ! axiom ( congru5ce4 ,i h tak5 congru5ce to be ! def9ition ( spatial equality by sup]position1 & %all !re=e g5]ally speak ( ! axiom z ,free ,mobility4 #aeb ,= ! s5se 9 :i* ^! figures are to be reg>d$ z mat]ial1 see criticism ( ,helmholtz1 ,*apt] ,,ii4 #fi ff4 #aec ,op4 cit4 p4 #fj4 #aed ,! view ( ,helmholtz & ,erdmann1 t me*anical exp]i5ce suffices "h1 ?\gh geometrical exp]i5ce fails u1 has be5 discuss$ above1 ,*apt] ,,ii4 #gc1 #hb4 #aee ,*apt] ,,ii4 #ha4 #aef ,*apt] ,,ii4 #gb4 #aeg ,contra/ ,delbuf1 ,l'anci5ne et les n\velles gomtries1 ,,ii4 ,rev4 ,phil4 #ahid1 ,vol4 xxxvii4 p4 #ced4 #aeh ,prolegom5a1 #ac4 ,see ,vaih+]'s ,comm5t>1 ,,ii4 pp4 #eah-ecb esp4 pp4 #eba-b4 ,! above 0 ,kant's :ole purpose 9 #agfh1 b only "p ( 8 purpose 9 ! ,prolegom5a1 :]e ! intuitive nature ( space 0 also to be prov$4 #aei ,on ! subject ( "t measurem5t1 cf4 ,bosanquet's ,logic1 ,vol4 i4 pp4 #agh-ahc4 ,s9ce "t1 9 ! above acc\nt1 is measur$ by motion1 its measurem5t presupposes t ( spatial magnitudes4 #afj ,cf4 ,/umpf4 ,ursprung d] ,raumvor/ellung1 p4 #fh4 #afa ,z is ,helmholtz's o!r axiom1 t ! possibility ( sup]position is indep5d5t ( ! c\rse pursu$ 9 br++ x ab\t4 #afb ,cf4 #abi1 #acj4 #afc ,?is d$uction is practically ! same z t 9 ,sec4 ,a1 b ,i h /at$ x "h ) m special ref]5ce to space & to metrical ,geometry4 #afd ,! "q3 ,relations to :at8 is a "q involv+ _m difficulties4 ,x w be t\*$ on lat] 9 ?is *apt]1 & answ]$1 z f> z possible1 9 ! f\r? *apt]4 ,= ! pres5t1 9 spite ( ! gl>+ circle involv$1 ,i %all take ! relations z relations to o!r positions4 #afe ,wiss4 ,abh4 ,vol4 ,,ii4 p4 #fad4 #aff ,cp4 ,grassmann1 ,ausdehnungslehre von #ahdd1 #bnd ed4 p4 ,,xxiii4 #afg ,delbuf1 x is true1 speaks ( ,geometries ) m_/n dim5sions1 b gives no ref]5ce 7,rev4 ,phil4 ,t4 xxxvi4 p4 #dej74 #afh ,9 criticiz+ ,erdmann1 x w be rememb]$1 we saw t ,free ,mobility is a necess>y prop]ty ( 8 ext5ts1 ?\gh he does n reg>d x z su*4 #afi ,cf4 ,riemann1 ,hypo!s5 wel*e d] ,geometrie zu ,grunde lieg51 ,gesammelte ,w]ke1 p4 #bff2 also ,erdmann1 op4 cit4 p4 #aed4 #agj ,?is is subject1 9 sph]ical space1 to ! modification po9t$ out bel[1 9 deal+ ) ! exception to ! axiom ( ! /raight l9e4 ,see #afh-aga4 #aga ,9 speak+ ( di/ance at once z a quantity & z an intr9sic relation1 ,i am anxi\s to gu>d aga9/ an app>5t inconsi/5cy4 ,i h spok5 ( ! judgm5t ( quantity1 ?r\gh\t1 z "o ( comp>ison2 h[1 !n1 c a quantity be intr9sic8 ,! reply is t1 al?\gh measurem5t & ! judgm5t ( quantity express ! result ( comp>ison1 yet ! t]ms comp>$ mu/ exi/ be=e ! comp>ison2 9 ?is case1 ! t]ms comp>$ 9 measur+ di/ances1 i4e4 9 comp>+ !m int] se1 are intr9sic relations betwe5 po9ts4 ,?us1 al?\gh ! measurem5t ( di/ance involves a ref]5ce to o!r di/ances1 & its expression z a magnitude requires su* a ref]5ce1 yet its exi/5ce does n dep5d on any ext]nal ref]5ce1 b exclusively on ! two po9ts ^: di/ance x is4 #agb ,see ! end ( ! argum5t on ,free ,mobility1 #aee ff4 #agc ,9 ,fris*auf's ,absolute ,geometrie na* ,johann ,bolyai1 ,anhang1 !re is a s]ies ( def9itions1 />t+ f ! sph]e1 z ! locus ( congru5t po9t-pairs :5 "o po9t ( ! pair is fix$1 & h5ce obta9+ ! circle & ! /raight l9e4 ,f ! above x foll[s1 t ! sph]e s def9$ already involves a curve betwe5 ! po9ts ( ! po9t-pair1 by :i* v>i\s po9t-pairs c be kn[n z congru5t2 & x w appe>1 z we proce$1 t ?is curve mu/ be a /raight l9e4 ,fris*auf's def9ition by means ( ! sph]e involves1 !re=e1 a vici\s circle1 s9ce ! sph]e presupposes ! /raight l9e1 z ! te/ ( congru5t po9t-pairs4 #agd ,nor 9 any argum5t :i*1 l ^? ( projective ,geometry1 avoids ! notion ( magnitude or di/ance altoge!r4 ,x foll[s t ! propositions ( projective ,geometry apply1 )\t res]ve1 to sph]ical space1 s9ce ! exception to ! axiom ( ! /raight l9e arises only on metrical gr\nd4 #age ,psy*ology1 ,vol4 ,,ii4 pp4 #adi-aej4 #agf ,?is /ep 9 ! argum5t has be5 put v briefly1 s9ce x is a m]e repetition ( ! correspond+ argum5t 9 ,section ,a1 & is ins]t$ "h only = ! sake ( logical complet5ess4 ,see #acg ff4 #agg ,cf4 ,hannequ91 ,essai critique sur l'hypo?se des atomes1 ,p>is1 #ahie1 passim4 ,,*apt] ,,iv4 ,,philosophical ,,consequ5ces4 #ahj4 ,9 ! pres5t *apt]1 we h to discuss two que/ions :i*1 ?\gh sc>cely geometrical1 are ( fundam5tal importance to ! !ory ( ,geometry prop\nd$ above4 ,! fir/ ( ^! que/ions is ?is3 ,:at relation c a purely logical & d$uctive pro(1 l t f ! nature ( a =m ( ext]nality1 be> to an exp]i5c$ subject-matt] su* z space8 ,y h m]ely fram$ a g5]al conception1 ,i may be told1 conta9+ space z a p>ticul> species1 & y h !n %[n1 :at %\ld h be5 obvi\s f ! beg9n+1 t ?is g5]al conception conta9$ "s ( ! attributes ( space4 ,b :at gr\nd does ?is give = reg>d+ ^! attributes z priori8 ,! conception ,mammal has "s ( ! attributes ( a horse2 b are ^! attributes !re=e priori adjectives ( ! horse8 ,! answ] to ?is obvi\s objection is s difficult1 & involves s mu* g5]al philosophy1 t ,i h kept x = a f9al *apt]1 9 ord] n to int]rupt ! argum5t on specially geometrical topics4 #aha4 ,i h already indicat$1 9 g5]al t]ms1 ! gr\nd = reg>d+ z priori ! prop]ties ( any =m ( ext]nality4 ,?is gr\nd is transc5d5tal1 i4e4 x is to be f\nd 9 ! conditions requir$ = ! possibility ( exp]i5ce4 ,! =m ( ext]nality1 l ,riemann's manifolds1 is a g5]al class-conception1 includ+ "t z well z ,euclidean & non-,euclidean spaces4 ,x is n motiv$1 h[ev]1 l ! manifolds1 by a quantitative resemblance to space1 b by ! fact t x fulfils1 if x has m ?an "o dim5sion1 all ^? functions :i*1 9 our actual _w1 are fulfill$ by space4 ,b a =m ( ext]nality1 9 ord] to accompli% ?is1 mu/ be1 n a m]e conception1 b an actually exp]i5c$ intuition4 ,h5ce ! conception ( su* a =m is ! g5]al conception1 conta9+ "u x e logically possible intuition :i* c fulfil ! function actually fulfill$ by space4 ,& ?is function is1 to r5d] possible exp]i5ce ( div]se b int]relat$ ?+s4 ,"s =m 9 s5se-p]ception1 !n1 ^: conception is includ$ "u our =m ( ext]nality1 is priori necess>y to exp]i5ce ( div]sity 9 relation1 & )\t exp]i5ce ( ?is1 we %\ld1 z mod]n logic %[s1 h no exp]i5ce at all4 ,?is /ill leaves unt\*$ ! relation ( ! priori to ! subjective3 ! =m ( ext]nality is necess>y to exp]i5ce1 b is n1 on t acc\nt1 to be decl>$ purely subjective4 ,( c\rse1 necessity = exp]i5ce c only arise f ! nature ( ! m9d :i* exp]i5ces2 b x does n foll[ t ! necess>y conditions c\ld be fulfill$1 unless ! objective _w _h c]ta9 prop]ties4 ,! gr\nd ( necessity1 we may safely say1 arises f ! m9d2 b x by no means foll[s t ! tru? ( :at is necess>y dep5ds only on ! con/itution ( ! m9d4 ,:]e ?is is n ! case1 our conclusion1 :5 a piece ( k has be5 decl>$ priori1 c only be3 ,ow+ to ! con/itution ( ! m9d1 exp]i5ce w be impossible unless ! _w accepts c]ta9 adjectives4 ,su*1 9 outl9e1 w be ! argum5t ( ! fir/ half ( ?is *apt]1 & su* w be ! ju/ification = reg>d+ z priori ^? axioms ( ,geometry1 :i* 7 d$uc$ above f ! conception ( a =m ( ext]nality4 ,= ^! axioms1 & ^! only1 are necess>ily true ( any _w 9 :i* exp]i5ce is possible4 #ahbagh4 ,! view sugge/$ has1 obvi\sly1 mu* 9 common ) t ( ! ,transc5d5tal ,ae/hetic4 ,inde$ ! :ole ( x1 ,i believe1 c be obta9$ by a c]ta9 limitation & int]pretation ( ,kant's classic argum5ts4 ,b z x diff]s1 9 _m important po9ts1 f ! conclusions aim$ at by ,kant1 & z ! agreem5t may easily seem great] ?an x is1 ,i w beg9 by a brief comp>ison1 & endeav\r1 by ref]5ce to au?oritative criticisms1 to e/abli% ! legitimacy ( my div]g5ce f him4 #ahc4 ,9 ! fir/ place1 ! psy*ological elem5t is mu* l>g] 9 ,kant's !sis ?an 9 m9e4 ,i %all cont5d1 x is true1 t a =m ( ext]nality1 if x is to d its "w1 mu/ n be a m]e conception or a m]e inf]5ce1 b mu/ be a giv5 elem5t 9 s5se-p]ception--n1 ( c\rse1 orig9ally giv5 9 isolation1 b discov]able1 ?r\gh analysis1 by att5tion to ! object ( s5se-p]ception#agi4 ,b ,kant cont5d$1 n only t ?is elem5t is giv51 b also t x is subjective4 ,space1 = him1 is1 on ! "o h&1 n conceptual1 b on ! o!r h&1 n s5sational4 ,x =ms1 = him1 no "p ( ! data ( s5se1 b is add$ by a subjective intuition1 :i* he reg>ds z n only logically1 b psy*ologically1 prior to objects 9 space#ahj4 ,?is "p ( ,kant's argum5t is :olly irrelevant = u4 ,:e!r a =m ( ext]nality be giv5 9 s5se1 or 9 a pure intuition1 is = u unimportant1 s9ce we neglect ! "q z to ! connection ( ! priori & ! subjective2 :ile ! temporal priority ( space to objects 9 x has be5 g5]ally recogniz$ z irrelevant to ,epi/emology1 & has (t5 be5 reg>d$ z =m+ no "p ( ,kant's !sis#aha4 ,if we call intuitional :atev] is giv5 9 s5se-p]ception1 !n we may cont5d t a =m ( ext]nality mu/ be intuitional2 b :e!r x is a pure intuition1 9 ,kant's s5se1 or n1 is irrelevant to u1 z is its priority to ! objects 9 x4 ,t ! non-s5sational nature ( space is no ess5tial "p ( ,kant's logical tea*+1 appe>s f an exam9ation ( 8 argum5t4 ,he has made1 9 ! introduction1 ! purely logical di/9ction ( matt] & =m1 b has giv5 to ?is di/9ction1 9 ! v mom5t ( sugge/+ x1 a psy*ological implication4 ,?is he does by ! ass]tion t ! =m1 9 :i* ! matt] ( s5sations is ord]$1 _c itself be s5sational4 ,f ?is assumption x foll[s1 ( c\rse1 t space _c be s5sational4 ,b ! assumption is totally unsupport$ by argum5t1 be+ set =?1 app>5tly1 z a self-evid5t axiom2 x has be5 sev]ely criticiz$ by ,/umpf#ahb & o!rs#ahc1 & has be5 describ$ by ,vaih+] z a fatal petitio pr9cipii#ahd2 x is irrelevant to ! logical argum5t1 :5 ?is argum5t is sep>at$1 z we h sep>at$ x1 f all connection ) psy*ological subjectivity2 & f9ally1 x leaves u a prey to psy*ological !ories ( space1 :i* h seem$1 ( late1 b little fav\rable to ! pure ,kantian doctr9e4 #ahd4 ,we h a "r1 !re=e1 9 an epi/emological inquiry1 to neglect ,kant's psy*ological tea*+--9 s f>1 at any rate1 z x di/+ui%es spatial intuition f s5sation--& att5d r to ! logical aspect alone4 ,t "p ( 8 psy*ological tea*+1 :i* ma9ta9s t space is n a m]e conception1 is1 ) c]ta9 limitations1 suffici5tly evid5t z appli$ to actual space2 b = u1 x mu/ be trans=m$ into a mu* m difficult !sis1 namely1 t no =m ( ext]nality1 :i* r5d]s exp]i5ce ( div]sity 9 relation possible1 c be m]ely conceptual4 ,?is "q1 to :i* we mu/ return lat]1 is no long] psy*ological1 b belongs :olly to ,epi/emology4 #ahe4 ,:at1 !n1 rema9s ! k]nel1 = our purposes1 ( ,kant's fir/ argum5t = ! apriority ( space8 ,8 argum5t1 9 ! =m 9 :i* he gave x1 is conc]n$ ) ! ecc5tric projection ( s5sations4 ,9 ord] t ,i may ref] s5sations1 he says1 to some?+ outside myself1 ,i mu/ already h ! subjective space-=m 9 ! m9d4 ,9 ?is %ape1 z ,vaih+] po9ts out 7,comm5t>1 ,,ii4 pp4 #fi1 #afe71 ! argum5t re/s on a petitio pr9cipii1 = only if s5sations are necess>ily non-spatial does _! projection dem& a subjective space-=m4 ,b1 fur!r1 is ! logical apriority ( space conc]n$ ) ! ext]nality ( ?+s to ourselves8 ,space seems to p]=m two functions3 on ! "o h&1 x reveals ?+s1 by ! ecc5tric projection ( s5sations1 z ext]nal to ! self1 :ile1 on ! o!r h&1 x reveals simultane\sly pres5t$ ?+s z mutually ext]nal4 ,^! two functions1 ?\gh (t5 treat$ z coord9ate & almo/ equival5t#ahe1 seem to me widely diff]5t4 ,be=e we discuss ! apriority ( space1 we mu/ c>efully di/+ui%1 ,i ?9k1 betwe5 ^! two functions1 & decide :i* ( !m we are to argue ab\t4 ,n[ ext]nality to ! ,self1 x w\ld seem1 mu/ necess>ily raise ! :ole "q ( ! nature & limits ( ! ,ego1 & :at is m1 x _c be d]iv$ f spatial pres5tation1 unless we give ! ,self a def9ite position 9 space4 ,b ?+s acquire a position 9 space only :5 !y c appe> 9 s5se-p]ception2 we are =c$1 !re=e1 if we adopt ?is view ( ! function ( space1 to reg>d ! ,self z a ph5om5on pres5t$ to s5se-p]ception4 ,b ?is r$uces ext]nality to ! ,self to ext]nality to ! body4 ,! body1 h[ev]1 is a pres5t$ object l any o!r1 & ext]nality ( objects to x is1 !re=e1 a special case ( ! mutual ext]nality ( pres5t$ ?+s4 ,h5ce we _c reg>d space z giv+1 prim>ily at any rate1 ext]nality to ! ,self1 b only ! mutual ext]nality ( ! ?+s pres5t$ to s5se-p]ception#ahf4 #ahf4 ,?is1 !n1 is ! k9d ( ext]nality we are to expect f space1 & our "q mu/ be3 ,w\ld ! exi/5ce ( div]se b int]relat$ ?+s be unkn[able1 if !re 7 n1 9 s5se-p]ception1 "s =m ( ext]nality8 ,?is is ! crucial "q1 on :i* turns ! apriority ( our =m1 & h5ce ( ! necess>y axioms ( ,geometry4 #ahg4 ,! conv]se argum5t to m9e1 ! argum5t f ! spatio-temporal elem5t 9 p]ception to a _w ( int]relat$ b div]se ?+s1 is develop$ at l5g? 9 ,bradley's ,logic4 ,x is put briefly 9 ! foll[+ s5t5ce 7p4 #dd1 note73 ,if space & "t are cont9u\s1 & if all appe>ance mu/ occupy "s "t or space--& x is n h>d to support bo? ^! !ses--we c at once proce$ to ! conclusion1 no m]e p>ticul> exi/s4 ,e ph5om5on w exi/ 9 m times or spaces ?an "o2 & aga9/ t div]sity w be itself an univ]sal#ahg4 ,! importance ( ?is fact appe>s1 :5 we consid] t1 if any m]e p>ticul> exi/$1 all judgm5t & inf]5ce z to t p>ticul> w\ld be impossible1 s9ce all judgm5t & inf]5ce necess>ily op]ate by means ( univ]sal4 ,b all reality is con/ruct$ f ! ,?is ( imm$iate pres5tation1 f :i* judgm5t & inf]5ce necess>ily spr+4 ,ow+1 h[ev]1 to ! cont9uity & relativity ( space & "t1 no ,?is c be reg>d$ ei!r z simple or z self-subsi/5t4 ,e ,?is1 on ! "o h&1 c be analyz$ into ,?ises1 & on ! o!r h&1 is f\nd to be necess>ily relat$ to o!r ?+s1 outside ! limits ( ! giv5 object ( s5se-p]ception4 ,?is function ( space & "t is presuppos$ 9 ! foll[+ /atem5t f ,bosanquet's ,logic 7,vol4 ,i4 pp4 #gg-gh73 ,reality is giv5 = me 9 pres5t s5su\s p]ception1 & 9 ! imm$iate feel+ ( my own s5ti5t exi/5ce t goes ) x4 ,! real _w1 z a def9ite organiz$ sy/em1 is = me an ext5sion ( ?is pres5t s5sation & self feel+ by means ( judgm5t1 & x is ! ess5ce ( judgm5t to effect & su/a9 su* an ext5sion4444 ,! subject 9 e judgm5t ( ,p]ception is "s giv5 spot or po9t 9 s5su\s contact ) ! p]cipi5t self4 ,b1 z all reality is cont9u\s1 ! subject is n m]ely ?is giv5 spot or po9t4 #ahh4 ,?is doctr9e ( ,bradley & ,bosanquet is ! conv]se ( ! epi/emological doctr9e ,i h to advocate4 ,ow+ to ! cont9uity & relativity ( space & "t1 !y say1 we are able to con/ruct a sy/ematic _w1 by judgm5t & inf]5ce1 out ( t fragm5t>y & yet necess>ily complex exi/5ce :i* is giv5 9 s5se-p]ception4 ,my cont5tion is1 conv]sely1 t s9ce all k is necess>ily d]iv$ by an ext5sion ( ! ,?is ( s5se-p]ception1 & s9ce su* ext5sion is only possible if ! ,?is has t fragm5t>y & yet complex "* conf]r$ by a =m ( ext]nality1 !re=e "s =m ( ext]nality1 giv5 ) ! ,?is1 is ess5tial to all k1 & is ?us logically priori4 ,bradley's argum5t1 if s\nd1 already proves ?is cont5tion2 = :ile1 on ! "o h&1 he uses no prop]ties ( space & "t b ^? :i* belong to e =m ( ext]nality1 he proves1 on ! o!r h&1 t judgm5t & inf]5ce require ! ,?is to be nei!r s+le nor self-subsi/5t4 ,b ,i w endeav\r1 s9ce ! po9t is ( fundam5tal importance1 to reproduce ! pro(1 9 a =m m suit$ ?an ,bradley's to ! epi/emological "q4 #ahi4 ,! ess5ce ( my cont5tion is t1 if exp]i5ce is to be possible1 e s5sational ,?is mu/1 :5 att5d$ to1 be f\nd1 on ! "o h&1 resolvable into ,?ises1 & on ! o!r h& dep5d5t1 = "s ( its adjectives1 on ext]nal ref]5ce4 ,! second ( ^! !ses foll[s f ! fir/1 = if we take "o ( ! ,?ises conta9$ 9 ! fir/ ,?is1 we get a new ,?is necess>ily relat$ to ! o!r ,?ises :i* make up ! orig9al ,?is4 ,i may1 !re=e1 conf9e myself to ! fir/ proposition1 :i* affirms t ! object ( p]ception mu/ conta9 a div]sity1 n only ( conceptual cont5t1 b ( exi/5ce1 & t ?is c only be kn[n if s5se-p]ception conta9s1 z an elem5t1 "s =m ( ext]nality4 ,my premiss1 9 ?is argum5t1 is t all k involves a recognition ( div]sity 9 relation1 or1 if we pref] x1 ( id5tity 9 diff]5ce4 ,?is premiss ,i accept f ,logic1 z result+ f ! analysis ( judgm5t & inf]5ce4 ,to prove su* a premiss1 w\ld require a treatise on ,logic2 ,i mu/ ref] ! read]1 !re=e1 to ! works ( ,bradley & ,bosanquet on ! subject4 ,x foll[s at once1 f my premiss1 t k w\ld be impossible1 unless ! object ( att5tion c\ld be complex1 i4e4 n a m]e p>ticul>4 ,n[ c\ld ! m5tal object--i4e41 9 ?is connection1 ! object ( a cognition--be complex1 if ! object ( imm$iate p]ception 7 always simple8 #aij4 ,we might be incl9$1 at fir/ sight1 to answ] ?is "q affirmatively4 ,b sev]al difficulties1 ,i ?9k1 w\ld prev5t su* an answ]4 ,9 ! fir/ place1 k mu/ />t f p]ception4 ,h5ce1 ei!r we c\ld h no k except ( our pres5t p]ception1 or else we mu/ be able to contra/ & comp>e x ) "s o!r p]ception4 ,n[ 9 ! fir/ case1 s9ce ! pres5t p]ception1 by hypo!sis1 is a m]e p>ticul>1 k ( x is impossible1 accord+ to our premiss4 ,b 9 ! second case1 ! o!r p]ception1 ) :i* we comp>e our fir/1 mu/ h occurr$ at "s o!r "t1 & ) "t1 we h at once a =m ( ext]nality4 ,b :at is m1 our pres5t p]ception is no long] a m]e p>ticul>4 ,= ! p[] ( comp>+ x ) ano!r p]ception involves a po9t ( id5tity betwe5 ! two1 & ?us r5d]s bo? complex4 ,moreov]1 "t mu/ be cont9u\s1 & ! pres5t1 z ,bradley po9ts out1 is no m]e po9t ( "t#ahh4 ,?us our pres5t p]ception conta9s ! complexity involv$ 9 duration ?r\gh\t ! speci\s pres5t3 its m]e p>ticul>ity & its simplicity are lo/4 ,its self-subsi/5ce is also lo/1 = beyond ! speci\s pres5t1 lie ! pa/ & ! future1 to :i* our pres5t p]ception ?us unavoidably ref]s u4 ,"t at lea/1 !re=e1 is ess5tial to t id5tity 9 diff]5ce1 :i* all k po/ulates4 #aia4 ,b we h d]iv$1 f all ?is1 no gr\nd = affirm+ a multiplicity ( real ?+s1 or a =m ( ext]nality ( m ?an "o dim5sion1 :i*1 we saw1 0 necess>y = ! tru? ( two out ( our ?ree axioms4 ,?is br+s u to ! "q3 ,h we en\gh1 ) "t alone z a =m ( ext]nality1 = ! possibility ( k8 ,?is "q we mu/1 ,i ?9k1 answ] 9 ! negative4 ,) "t alone1 we h se51 our pres5t$ object mu/ be complex1 b its complexity mu/1 if ,i may use su* a phrase1 be m]ely adjectival4 ,)\t a second =m ( ext]nality1 only "o ?+ c be giv5 at "o mom5t#ahi1 & ?is "o ?+1 !re=e1 mu/ con/itute ! :ole ( our _w4 ,! object ( pa/ p]ception mu/--s9ce our "o ?+ has no?+ ext]nal to x1 by :i* x c\ld be creat$ or de/roy$--be reg>d$ z ! same ?+ 9 a diff]5t /ate4 ,! complexity1 !re=e1 w lie only 9 ! *ang+ /ates ( our "o ?+--x w be adjectival1 n sub/antival4 ,moreov] we h ! foll[+ dilemma3 ,ei!r ! "o ?+ mu/ be ourselves1 or else self-consci\sness c\ld nev] arise4 ,b ! *ief difficulty ( su* a _w w\ld lie 9 ! *anges ( ! ?+4 ,:at c\ld cause ^! *anges1 s9ce we %\ld "k ( no?+ ext]nal to our ?+8 ,x w\ld be l a ,leibnitzian monad1 )\t any ,god outside x to pre>range its *anges4 ,causality1 9 su* a _w1 c\ld n be appli$1 & *ange w\ld be :olly inexplicable4 ,h5ce we require also ! possibility ( a div]sity ( simultane\sly exi/+ ?+s1 n m]ely ( successive adjectives2 & ?is1 we h se51 _c be giv5 by "t alone1 b only by a =m ( ext]nality = simultane\s p>ts ( "o pres5tation4 ,we c\ld nev]1 9 o!r words1 inf] ! exi/5ce ( div]se b int]relat$ ?+s1 unless ! object ( s5se-p]ception c\ld h sub/antival complexity1 & = su* complexity we require a =m ( ext]nality o!r ?an "t4 ,su* a =m1 moreov]1 z 0 %[n 9 ,*apt] ,,iii41 ,section ,a 7 #ace71 c only fulfil its functions if x has m ?an "o dim5sion4 ,9 our actual _w1 ?is =m is giv5 by space2 9 any _w1 kn[able to be+s ) our laws ( ?\ght1 "s su* =m1 z we h n[ se51 mu/ be giv5 9 s5se-p]ception4 ,?is argum5t may be briefly summ$ up1 by assum+ ! doctr9e ( ,bradley1 t all k is obta9$ by inf]5ce f ! ,?is ( s5se-p]ception4 ,=1 if ?is be s1 ! ,?is--9 ord] t inf]5ce1 :i* dep5ds on id5tity 9 diff]5ce1 may be possible at all--mu/ itself be complex1 & mu/1 on analysis1 reveal adjectives hav+ a ref]5ce beyond itself4 ,b ?is1 z 0 %[n above1 c only happ5 by means ( a =m ( ext]nality4 ,?is e/abli%es ! priori axioms ( ,geometry1 z necess>ily hav+ exi/5tial import & validity 9 any intelligible _w4 #aib4 ,! above argum5t1 ,i hope1 has expla9$ :y ,i hold x possible to d$uce1 f a m]e conception l t ( a =m ( ext]nality1 ! logical apriority ( c]ta9 axioms z to exp]i5c$ space4 ,! ,kantian argum5t--:i* 0 correct1 if our reason+ has be5 s\nd1 9 ass]t+ t real div]sity1 9 our actual _w1 c\ld only be kn[n by ! help ( space--0 only mi/ak51 s f> z its purely logical scope ext5ds1 9 ov]look+ ! possibility ( o!r =ms ( ext]nality1 :i* c\ld1 if !y exi/$1 p]=m ! same task ) equal effici5cy4 ,9 s f> z space diff]s1 !re=e1 f ^! o!r conceptions ( possible intuitional =ms1 x is a m]e exp]i5c$ fact1 :ile 9 s f> z its prop]ties are ^? :i* all su* =ms mu/ h1 x is priori necess>y to ! possibility ( exp]i5ce4 ,i _c hope1 h[ev]1 t no difficulty w rema91 = ! read]1 9 su* a d$uction1 f ab/ract conceptions1 ( ! prop]ties ( an actual datum 9 s5se-p]ception4 ,let u consid]1 = example1 su* a prop]ty z imp5etrability4 ,to suppose two ?+s simultane\sly 9 ! same position 9 a =m ( ext]nality1 is a logical contradiction2 b c we say z mu* ( actual space & "t8 ,is n ! impossibility1 "h1 a matt] ( exp]i5ce r ?an ( logic8 ,n if ! above argum5t has be5 s\nd1 ,i reply4 ,= 9 t case1 we inf] real div]sity1 i4e4 ! exi/5ce ( diff]5t ?+s1 only f diff]5ce ( position 9 space or "t4 ,x foll[s1 t to suppose two ?+s 9 ! same po9t ( space & "t1 is /ill a logical contradiction3 n because we h con/ruct$ ! data ( s5se out ( logic1 b because logic is dep5d5t1 z reg>ds its application1 on ! nature ( ^! data4 ,?is in/ance illu/rates1 :at ,i am anxi\s to make pla91 t my argum5t has n attempt$ to con/ruct ! liv+ weal? ( s5se-p]ception out ( bloodless categories1 b only to po9t out t1 unless s5se-p]ception conta9$ a c]ta9 elem5t1 ^! categories w\ld be p[]less to grapple ) x4 #aic4 ,h[ we are to acc\nt = ! =tunate realization ( ^! requirem5ts--:e!r by a pre-e/abli%$ h>mony1 by ,d>w9ian adaptation to our environm5t1 by ! subjectivity ( ! necess>y elem5t 9 s5se-p]ception1 or by a fundam5tal id5tity & unity betwe5 ourselves & ! re/ ( reality--is a fur!r "q1 belong+ r to metaphysics ?an to our pres5t l9e ( argum5t4 ,! priori1 we h said ?r\gh\t1 is t :i* is necess>y = ! possibility ( exp]i5ce1 & 9 ?is we h a purely logical crit]ion1 giv+ results :i* only ,logic & ,epi/emology c prove or disprove4 ,:at is subjective 9 exp]i5ce1 on ! contr>y1 is prim>ily a "q = psy*ology1 & %\ld be decid$ on psy*ological gr\nds alone4 ,:5 ^! two que/ions h be5 sep>ately answ]$1 b n till !n1 we may frame !ories z to ! connection ( ! priori & ! subjective2 to all[ su* !ories to influ5ce our decision1 on ei!r ( ! two previ\s que/ions1 is liable1 surely1 to confuse ! issue1 & prev5t a cle> discrim9ation betwe5 fundam5tally diff]5t po9ts ( view4 #aid4 ,i come n[ to ! second "q ) :i* ?is *apt] has to deal1 ! "q1 namely3 ,:at are we to d ) ! contradictions :i* obtrud$ !mselves 9 ,*apt] ,,iii41 :5ev] we came to a po9t :i* seem$ fundam5tal8 ,i %all treat ?is "q briefly1 z ,i h little to add to answ]s ) :i* we are all famili>4 ,i h only to prove1 fir/1 t ! contradictions are inevitable1 & !re=e =m no objection to my argum5t2 secondly1 t ! fir/ /ep 9 remov+ !m is to re/ore ! notion ( matt]1 z t :i*1 9 ! data ( s5se-p]ception1 is localiz$ & int]relat$ 9 space4 #aie4 ,! contradictions 9 space are an anci5t !me--z anci5t1 9 fact1 z ,z5o's refutation ( motion4 ,!y are1 r\ghly1 ( two k9ds1 ?\gh ! two k9ds _c be %>ply divid$4 ,!re are ! contradictions inh]5t 9 ! notion ( ! cont9uum1 & ! contradictions :i* spr+ f ! fact t space1 :ile x mu/1 to be kn[able1 be pure relativity1 mu/ also1 x w\ld seem1 s9ce x is imm$iately exp]i5c$1 be some?+ m ?an m]e relations4 ,! fir/ class ( contradictions has be5 enc\nt]$ m frequ5tly 9 ?is essay1 & is also1 ,i ?9k1 ! m def9ite1 & ! m important = our pres5t purpose4 ,i d\bt1 h[ev]1 :e!r ! two classes are really di/9ct2 = any cont9uum1 ,i believe1 9 :i* ! elem5ts are n data1 b intellectual con/ructions result+ f analysis1 c be %[n to h ! same relational & yet n :olly relational "* z belongs to space4 ,! ?ree foll[+ contradictions1 :i* ,i %all discuss successively1 seem to me ! mo/ prom95t 9 a !ory ( ,geometry4 7#a7 ,?\gh ! p>ts ( space are intuitively di/+ui%$1 no conception is adequate to diff]5tiate !m4 ,h5ce arises a va9 se>* = elem5ts1 by :i* ! diff]5tiation c\ld be accompli%$1 & = a :ole1 ( :i* ! p>ts ( space are to be compon5ts4 ,?us we get ! po9t1 or z]o ext5sion1 z ! spatial elem5t1 & an inf9ite regress or a vici\s circle 9 ! se>* = a :ole4 7#b7 ,all positions be+ relative1 positions c only be def9$ by _! relations1 i4e4 by ! /raight l9es or planes ?r\gh !m2 b /raight l9es & planes1 be+ all qualitatively simil>1 c only be def9$ by ! positions !y relate4 ,h5ce1 aga91 we get a vici\s circle4 7#c7 ,spatial figures mu/ be reg>d$ z relations4 ,b a relation is necess>ily indivisible1 :ile spatial figures are necess>ily divisible ad inf9itum4 #aif4 7#a7 ,po9ts4 ,! ant9omy ( ! po9t--:i* arises :]ev] a cont9uum is giv51 & elem5ts h to be s\ght 9 x--is fundam5tal to ,geometry4 ,x has be5 giv51 p]haps un9t5tionally1 by ,v]onese z ! fir/ axiom1 9 ! =m3 ,!re are diff]5t po9ts4 ,all po9ts are id5tical 7op4 cit4 p4 #bbf74 ,we saw1 9 discuss+ projective ,geometry1 t /raight l9es & planes mu/ be reg>d$1 on ! "o h& z relations betwe5 po9ts1 & on ! o!r h& z made up ( po9ts#aij4 ,we saw aga91 9 deal+ ) measurem5t1 h[ space mu/ be reg>d$ z inf9itely divisible1 & yet z m]e relativity4 ,b :at is divisible & consi/s ( p>ts1 z space does1 mu/ lead at la/1 by cont9u$ analysis1 to a simple & unanalyzable "p1 z ! unit ( diff]5tiation4 ,= :atev] c be divid$1 & has p>ts1 possesses "s ?+hood1 & mu/1 !re=e1 conta9 two ultimate units1 ! :ole namely1 & ! smalle/ elem5t possess+ ?+hood4 ,b 9 space ?is is notori\sly n ! case4 ,aft] hypo/atiz+ space1 z ,geometry is compell$ to d1 ! m9d imp]atively dem&s elem5ts1 & insi/s on hav+ !m1 :e!r possible or n4 ,( ?is dem&1 all ! geometrical applications ( ! inf9itesimal calculus are evid5ce#aia4 ,b :at sort ( elem5ts d we ?us obta98 ,analysis1 be+ unable to f9d any e>li] halt+-place1 f9ds its elem5ts 9 po9ts1 t is1 9 z]o quanta ( space4 ,su* a conception is a palpable contradiction1 only r5d]$ tol]able by its necessity & famili>ity4 ,a po9t mu/ be spatial1 o!rwise x w\ld n fulfil ! function ( a spatial elem5t2 b aga9 x mu/ conta9 no space1 = any f9ite ext5sion is capable ( fur!r analysis4 ,po9ts c nev] be giv5 9 intuition1 :i* has no conc]n ) ! inf9itesimal3 !y are a purely conceptual con/ruction1 aris+ out ( ! ne$ ( t]ms betwe5 :i* spatial relations c hold4 ,if space be m ?an relativity1 spatial relations mu/ involve spatial relata2 b no relata appe>1 until we h analyz$ our spatial data d[n to no?+4 ,! contradictory notion ( ! po9t1 z a ?+ 9 space )\t spatial magnitude1 is ! only outcome ( our se>* = spatial relata4 ,?is r$uctio ad absurdum surely suffices1 by itself1 to prove ! ess5tial relativity ( space4 #aig4 ,?us ,geometry is =c$1 s9ce x wi%es to reg>d space z indep5d5t1 to hypo/atize its ab/ractions1 & !re=e to inv5t a self-contradictory notion z ! spatial elem5t4 ,a simil> absurdity appe>s1 ev5 m obvi\sly1 9 ! notion ( a :ole ( space4 ,! ant9omy may1 !re=e1 be /at$ ?us3 ,space1 z we h se5 ?r\gh\t1 mu/1 if k ( x is to be possible1 be m]e relativity2 b x mu/ also1 if indep5d5t k ( x1 su* z ,geometry seeks1 is to be possible1 be some?+ m ?an m]e relativity1 s9ce x is divisible & has p>ts4 ,b we saw1 9 ,*ap4 ,,iii41 ,section ,a 7 #acc7 t k ( a =m ( ext]nality mu/ be logically indep5d5t ( ! p>ticul> matt] fill+ ! =m4 ,h[ !n are we to extricate ourselves f ?is dilemma8 ,! only way1 ,i ?9k1 is1 n to make ,geometry dep5d5t on ,physics1 :i* we h se5 to be errone\s#aib1 b to give e geometrical proposition a c]ta9 ref]5ce to matt] 9 g5]al4 ,& at ?is po9t an important di/9ction mu/ be made4 ,we h hi!rto spok5 ( space z relational1 & ( spatial figures z relations4 ,b space1 x w\ld seem1 is r relativity ?an relations--itself n a relation1 x gives ! b>e possibility ( relations betwe5 div]se ?+s#aic4 ,z appli$ to a spatial figure1 :i* c only arise by a diff]5tiation ( space1 & h5ce by ! introduction ( "s diff]5tiat+ matt]1 ! ^w relation is1 p]haps1 less mislead+ ?an any o!r2 z appli$ to empty undiff]5tiat$ space1 x seems by no means an accurate description4 ,b a b>e possibility _c exi/1 or be giv5 9 s5se-p]ception6 ,:at becomes1 !n1 ( ! argum5ts ( ! fir/ "p ( ?is *apt]8 ,i reply1 x is n empty space1 b spatial figures1 :i* s5se-p]ception reveals1 & spatial figures1 z we h j se51 involve a diff]5tiation ( space1 & !re=e a ref]5ce to ! matt] :i* is 9 space4 ,x is spatial figures1 also1 & n empty space1 ) :i* ,geometry has to deal4 ,! ant9omy discuss$ above arises !n--s x w\ld seem--f ! attempt to deal ) empty space1 r ?an ) spatial figures & ! matt] to :i* !y necess>ily ref]4 #aih4 ,let u see :e!r1 by ?is *ange1 we c ov]come ! ant9omy ( ! po9t4 ,spatial figures1 we %all n[ say1 are relations betwe5 ! matt] :i* diff]5tiates empty space4 ,_! divisibility1 :i* seem$ to contradict _! relational "*1 may be expla9$ 9 two ways3 fir/1 z hold+ ( ! figures consid]$ z p>ts ( empty space1 :i* is itself n a relation2 second1 z d5ot+ ! possibility ( cont9u\s *ange 9 ! relation express$ by ! spatial figure4 ,^! two ways are1 at bottom1 ! same2 = empty space is a possibility ( relations1 & ! figure1 :5 view$ 9 connection ) empty space1 ?us becomes a possible relation1 ) :i* o!r possible relations may be contra/$ or comp>$4 ,b ! second way ( reg>d+ divisibility is ! bett] way1 s9ce x introduces a ref]5ce to ! matt] :i* diff]5tiates empty space1 )\t :i*1 spatial figures1 & !re=e ,geometry1 c\ld n exi/4 ,x is empty space1 !n--s we mu/ conclude--:i* gives rise to ! ant9omy 9 "q2 = empty space is a b>e possibility ( relations1 undiff]5tiat$ & homog5e\s1 & ?us :olly de/itute ( p>ts or ( ?+hood4 ,to speak ( p>ts ( a possibility is nons5se2 ! p>ts & diff]5tiations arise only ?r\gh a ref]5ce to ! matt] :i* is diff]5tiat$ 9 space4 #aii4 ,b :at nature mu/ we ascribe to ?is matt]1 :i* is to be involv$ 9 all geometrical propositions8 ,9 criticiz+ ,helmholtz 7,*ap4 ,,ii4 #gc71 x may be rememb]$1 we decid$ t ,geometry ref]s to a peculi> & ab/ract k9d ( matt]1 :i* is n reg>d$ z possess+ any causal qualities1 z ex]t+ or z subject to ! action ( =ces4 ,& ?is is ! matt]1 ,i ?9k1 :i* we require = ! ne$s ( ! mom5t4 ,n t we affirm1 ( c\rse1 t actual matt] c be de/itute ( ! prop]ties ) :i* ,physics is cognizant1 b t we ab/ract f ^! prop]ties1 z be+ irrelevant to ,geometry4 ,all t we require1 = our imm$iate purpose1 is a subject ( t div]sity :i* space r5d]s possible1 or t]ms = ^? relations by :i* empty space1 if space is to be /udi$ at all1 mu/ be diff]5tiat$4 ,b h[ mu/ a matt]1 :i* is to fulfil ?is function1 be reg>d$8 ,empty space1 we h said1 is a possibility ( div]sity 9 relation1 b spatial figures1 ) :i* ,geometry necess>ily deals1 are ! actual relations r5d]$ possible by empty space4 ,our matt]1 !re=e1 mu/ supply ! t]ms = ^! relations4 ,x mu/ be diff]5tiat$1 s9ce su* diff]5tiation1 z we h se51 is ! special "w ( space4 ,we mu/ f9d1 !re=e1 9 our matt]1 t unit ( diff]5tiation1 or atom#aid1 :i* 9 space we c\ld n f9d4 ,?is atom mu/ be simple1 i4e4 x mu/ conta9 no real div]sity2 x mu/ be a ,?is n resolvable into ,?ises4 ,be+ simple1 x c conta9 no relations )9 itself1 & consequ5tly1 s9ce spatial figures are m]e relations1 x _c appe> z a spatial figure2 = e spatial figure involves "s div]sity ( matt]4 ,b our atom mu/ h spatial relations ) o!r atoms1 s9ce to supply t]ms = ^! relations is its only function4 ,x is also capable ( hav+ ^! relations1 s9ce x is diff]5tiat$ f o!r atoms4 ,h5ce we obta9 an unext5d$ t]m = spatial relations1 precisely ( ! k9d we require4 ,s long z we s\ght ?is t]m )\t ref]5ce to any?+ m ?an space1 ! self-contradictory notion ( ! po9t 0 ! only outcome ( our se>*2 b n[ t we all[ a ref]5ce to ! matt] diff]5tiat$ by space1 we f9d at once ! t]m :i* 0 ne$$1 namely1 a non-spatial simple elem5t1 ) spatial relations to o!r elem5ts4 ,to ,geometry su* a t]m w appe>1 ow+ to its spatial relations1 z a po9t2 b ! contradiction ( ! po9t1 z we n[ see1 is a result only ( ! undue ab/raction ) :i* ,geometry deals4 #bjj4 7#b7 ,! circle 9 ! def9ition ( /raight l9es & planes4 ,?is difficulty ne$ n long deta9 u1 s9ce we h already1 ) ! mat]ial atom1 brok5 ?r\gh ! relativity :i* caus$ our circle4 ,/raight l9es1 9 ! purely geometrical proc$ure1 are def9$ only by po9ts1 & po9ts only by /raight l9es4 ,b po9ts1 n[1 are replac$ by mat]ial atoms3 ! duality ( po9ts & l9es1 !re=e1 has disappe>$1 & ! /raight l9e may be def9$ z ! spatial relation betwe5 two unext5d$ atoms4 ,^! atoms h spatial adjectives1 d]iv$ f _! relations to o!r atoms2 b !y h no intr9sic spatial adjectives1 su* z c\ld belong to !m if !y _h ext5sion or figure4 ,?us /raight l9es & planes are ! true spatial units1 & po9ts result only f ! attempt to f9d1 )9 space1 ^? t]ms = spatial relations :i* exi/ only 9 a m ?an spatial matt]4 ,/raight l9es1 planes & volumes are ! spatial relations betwe5 two1 ?ree or f\r unext5d$ atoms1 & po9ts are a m]ely conv5i5t geometrical fiction1 by :i* possible atoms are replac$4 ,=1 s9ce space1 z we saw1 is a possibility1 ,geometry deals n ) actually realiz$ spatial relations1 b ) ! :ole s*eme ( possible relations4 #bja4 7#c7 ,space is at once relational & m ?an relational4 ,we h already t\*$ on ! "q h[ f> space is o!r ?an relations1 b z ?is "q is q fundam5tal1 z relation is an ambigu\s & dang]\s ^w1 z ,i h made con/ant use ( ! relativity ( space )\t attempt+ to def9e a relation1 x w be necess>y to discuss ?is ant9omy at l5g?4 #bjb4 ,n[ = ?is discussion x is ess5tial to di/+ui% cle>ly betwe5 empty space & spatial figures4 ,empty space1 z a =m ( ext]nality1 is n actual relations1 b ! possibility ( relations3 if we ascribe exi/5tial import to x1 z ! gr\nd1 9 reality1 ( all div]sity 9 relation1 we at once h space z some?+ n itself relations1 ?\gh giv+ ! possibility ( all relations4 ,9 ?is s5se1 space is to be di/+ui%$ f spatial ord]4 ,spatial ord]1 x may be said1 presupposes space1 z t 9 :i* ?is ord] is possible4 ,?us ,/umpf says#aie3 ,!re is no ord] or relation )\t a positive absolute cont5t1 und]ly+ x1 & mak+ x possible to ord] any?+ 9 ?is mann]4 ,:y & h[ %\ld we o!rwise di/+ui% "o ord] f ano!r8444 ,to di/+ui% diff]5t ord]s f "o ano!r1 we mu/ ev]y:]e recognize a p>ticul> absolute cont5t1 9 relation to :i* ! ord] takes place4 ,& s space1 too1 is n a m]e ord]1 b j t by :i* ! spatial ord]1 side-by-sid5ess 7,neb5e9&]7 di/+ui%es itself f ! re/4 ,may we n1 !n1 resolve ! ant9omy v simply1 by a ref]5ce to ?is ambiguity ( space8 ,bradley cont5ds 7,appe>ance & ,reality1 pp4 #cf-g7 t1 on ! "o h&1 space has p>ts1 & is !re=e n m]e relations1 :ile on ! o!r h&1 :5 we try to say :at ^! p>ts are1 we f9d !m aft] all to be m]e relations4 ,b _c ! space :i* has p>ts be reg>d$ z empty space1 ,/umpf's absolute und]ly+ cont5t1 :i* is n m]e relations1 :ile ! p>ts1 9 s f> z !y turn out to be m]e relations1 are ^? relations :i* con/itute spatial ord]1 n empty space8 ,if ?is c be ma9ta9$1 ! ant9omy no long] exi/s4 ,b su* an explanation1 ?\gh ,i believe x to be a fir/ /ep t[>ds a solution1 w1 ,i fe>1 itself dem& almo/ z mu* explanation z ! orig9al difficulty4 ,= ! connection ( empty space ) spatial ord] is itself a "q full ( difficulty1 to be answ]$ only aft] mu* lab\r4 #bjc4 ,let u consid] :at ?is empty space is4 7,i speak ( empty space )\t necess>ily imply+ ! abs5ce ( matt]1 b only to d5ote a space :i* is n a m]e ord] ( mat]ial ?+s47 ,/umpf reg>ds x z giv5 9 s5se2 ,kant1 9 ! la/ two argum5ts ( 8 metaphysical d$uction1 argues t x is an intuition1 n a concept1 & mu/ be kn[n be=e spatial ord] becomes possible4 ,i wi% to ma9ta91 on ! contr>y1 t x is :olly conceptual2 t space is giv5 only z spatial ord]2 t spatial relations1 be+ giv51 appe> z m ?an m]e relations1 & s become hypo/atiz$2 t :5 hypo/atiz$1 ! :ole collection ( !m is reg>d$ z conta9$ 9 empty space2 b t ?is empty space itself1 if x means m ?an ! logical possibility ( space-relations1 is an unnecess>y & self-contradictory assumption4 ,let u beg9 by consid]+ ,kant's argum5ts on ?is po9t4 ,leibnitz _h affirm$ t space 0 only relations1 :ile ,newton _h ma9ta9$ ! objective reality ( absolute space4 ,kant adopt$ a middle c\rse3 he ass]t$ absolute space1 b reg>d$ x z purely subjective4 ,! ass]tion ( absolute space is ! object ( 8 second argum5t2 = if space 7 m]e relations betwe5 ?+s1 x w\ld necess>ily disappe> ) ! disappe>ance ( ! ?+s 9 x2 b ?is ! second argum5t d5ies#aif4 ,n[ spatial ord] obvi\sly does disappe> ) matt]1 b absolute or empty space may be suppos$ to rema94 ,x is ?is1 !n1 :i* ,kant is argu+ ab\t1 & x is ?is :i* he affirms to be a pure intuition1 necess>ily presuppos$ by spatial ord]#aig4 #bjd4 ,b c we agree 9 reg>d+ empty space1 ! inf9ite giv5 :ole1 z really giv58 ,mu/ we n1 9 spite ( ,kant's argum5t1 reg>d x z :olly conceptual8 ,x is n requir$1 9 ! fir/ place1 by ! argum5t ( ! fir/ half ( ?is *apt]1 :i* requir$ only t e ,?is ( s5se-p]ception %\ld be resolvable into ,?ises1 & ?us involv$ only an ord] among ,?ises1 n any?+ giv5 orig9ally )\t ref]5ce to !m at all4 ,9 ! second place1 ,kant's two argum5ts#aih design$ to prove t empty space is n conceptual1 are inadequate to _! purpose4 ,! argum5t t ! p>ts ( space are n conta9$ "u x1 b 9 x1 proves c]ta9ly t space is n a g5]al conception1 ( :i* spatial figures are ! in/ances2 b x by no means foll[s t empty space is n a conception4 ,empty space is undiff]5tiat$ & homog5e\s2 p>ts ( space1 or spatial figures1 arise only by ref]5ce to "s diff]5tiat+ matt]1 & ?us belong r to spatial ord] ?an to empty space4 ,if empty space be ! pre-condition ( spatial ord]1 we _c expect x to be connect$ ) spatial relations z g5us ) species4 ,b empty space may nev]!less be a univ]sal conception2 x may be relat$ to spatial ord] z ! /ate to ! citiz5s4 ,^! are n in/ances ( ! /ate1 b are conta9$ 9 x2 !y also1 9 a s5se1 presuppose x1 = a man c only become a citiz5 by be+ relat$ to o!r citiz5s 9 a /ate#aii4 ,! uniqu5ess ( space1 aga91 seems h>dly a valid argum5t = its intuitional nature2 to reg>d x z an argum5t implies1 inde$1 t all conceptions are ab/ract$ f a s]ies ( in/ances--a view :i* has be5 criticiz$ 9 ,*apt] ,,ii4 7 #gg71 & ne$ n be fur!r discuss$ "h#bjj4 ,!re is no gr\nd1 !re=e1 9 ,kant's two argum5ts = ! intuitional nature ( empty space1 :i* c be ma9ta9$ aga9/ criticism4 #bje4 ,ano!r gr\nd = condemn+ empty space is to be f\nd 9 ! ma!matical ant9omies4 ,= x is no solution1 z ,lotze po9ts out 7,metaphysik1 ,bk4 ,,ii4 ,*ap4 ,i41 #ajf71 to reg>d empty space z purely subjective3 contradictions 9 a necess>y subjective intuition =m z great a difficulty z 9 any?+ else4 ,b ^! ant9omies arise only 9 connection ) empty space1 n ) spatial ord] z an aggregate ( relations4 ,= only :5 space is reg>d$ z possess$ ( "s ?+hood1 c a :ole or a true elem5t be dem&$4 ,?is we h se5 already 9 connection ) ! ,po9t4 ,:5 space is reg>d$1 s f> z x is valid1 z only spatial ord]1 unb\nd$ ext5sion & inf9ite divisibility bo? disappe>4 ,:at is divid$ is n spatial relations1 b matt]2 & if matt]1 z we h se5 t ,geometry requires1 consi/s ( unext5d$ atoms ) spatial relations1 !re is no reason to reg>d matt] ei!r z inf9itely divisible1 or z consi/+ ( atoms ( f9ite ext5sion4 #bjf4 ,b :5ce arises1 on ?is view1 ! p>adox t we _c b reg>d space z hav+ m or less ?+hood1 & z divisible ad inf9itum8 ,?is mu/ be expla9$1 ,i ?9k1 z a psy*ological illusion1 unavoidably aris+ f ! fact t spatial relations are imm$iately pres5t$4 ,!y ?us h a peculi> psy*ical quality1 z imm$iate exp]i5ces1 by :i* quality !y c be di/+ui%$ f "t-relations or any o!r ord] 9 :i* ?+s may be arrang$4 ,to ,/umpf1 ^: problem is psy*ological1 su* a psy*ical quality w\ld con/itute an absolute und]ly+ cont5t1 & w\ld fully ju/ify 8 !sis2 to u1 h[ev]1 ^: problem is epi/emological1 x w\ld n d s1 b w\ld leave ! mean+ ( ! spatial elem5t 9 s5se-p]ception free f any implication ( an absolute or empty space#bja4 ,may we n1 !n1 ab&on empty space1 & say3 ,spatial ord] consi/s ( felt relations1 & qu felt has1 = ,psy*ology1 an exi/5ce n :olly resolvable into relations1 & unavoidably seem+ to be m ?an m]e relations4 ,b :5 we exam9e ! in=mation1 z to space1 :i* we d]ive f s5se-p]ception1 we f9d ourselves plung$ 9 contradictions1 z soon z we all[ ?is in=mation to consi/ ( m ?an relations4 ,?is leaves spatial ord] alone 9 ! field1 & r$uces empty space to a m]e "n = ! logical possibility ( spatial relations4 #bjg4 ,! app>5t divisibility ( ! relations :i* con/itute spatial ord]1 !n1 may be expla9$ 9 two ways1 ?\gh ^! are at bottom equival5t4 ,we may take ! relation z consid]$ 9 connection ) empty space1 9 :i* case x becomes m ?an a relation2 b be+ falsely hypo/atiz$1 x appe>s z a complex ?+1 necess>ily compos$ ( elem5ts1 :i* elem5ts1 h[ev]1 n[h]e em]ge until we analyze ! pseudo-?+ d[n to no?+1 & arrive at ! po9t4 ,9 ?is s5se1 ! divisibility ( spatial relations is an unavoidable illusion4 ,or aga91 we may take ! relation 9 connection ) ! mat]ial atoms x relates4 ,9 ?is case1 o!r atoms may be imag9$1 diff]5tly localiz$ by diff]5t spatial relations4 ,if !y are localiz$ on ! /raight l9e jo9+ two ( ! orig9al atoms1 ?is /raight l9e appe>s z divid$ by !m4 ,b ! orig9al relation is n really divid$3 all t has happ5$ is1 t two or m equival5t relations h replac$ x1 z two comp\nd$ relations ( fa!r & son may replace ! equival5t relation ( gr&fa!r & gr&son4 ,^! two ways ( view+ ! app>5t divisibility are equival5t3 = empty space1 9 s f> z x is n illusion1 is a "n = ! aggregate ( possible space-relations4 ,to reg>d a figure 9 empty space z divid$1 !re=e1 means1 if x means any?+1 to reg>d two or m o!r possible relations z sub/itut$ = x1 :i* gives ! second way ( view+ ! "q4 ,! same ref]5ce to matt]1 !n1 by :i* ! ant9omy ( ! ,po9t 0 solv$1 solves also ! ant9omy z to ! relational nature ( space4 ,space1 if x is to be fre$ f contradictions1 mu/ be reg>d$ exclusively z spatial ord]1 z relations betwe5 unext5d$ mat]ial atoms4 ,empty space1 :i* arises1 by an inevitable illusion1 out ( ! spatial elem5t 9 s5se-p]ception1 may be reg>d$1 if we wi% to reta9 x1 z ! b>e pr9ciple ( relativity1 ! b>e logical possibility ( relations betwe5 div]se ?+s4 ,9 ?is s5se1 empty space is :olly conceptual2 spatial ord] alone is imm$iately exp]i5c$4 #bjh4 ,b 9 :at s5se does spatial ord] consi/ ( relations8 ,we h hi!rto spok5 ( ext]nality z a relation1 & 9 a s5se su* a mann] ( speak+ is ju/ifi$4 ,ext]nality1 :5 pr$icat$ ( any?+1 is an adjective ( t ?+1 & implies a ref]5ce to "s o!r ?+4 ,to ?is ext5t1 !n1 ext]nality is analog\s to o!r relations2 & only to ?is ext5t1 9 our previ\s argum5ts1 has x be5 reg>d$ z a relation4 ,b :5 we take acc\nt ( fur!r qualities ( relations1 ext]nality beg9s to appe>1 n s mu* z a relation1 b r z a necess>y aspect or elem5t 9 e relation4 ,& ?is is borne out by ! necessity1 = ! exi/5ce ( relations1 ( "s giv5 =m ( ext]nality4 ,e relation1 we may say1 involves a div]sity betwe5 ! relat$ t]ms1 b also "s unity4 ,m]e div]sity does n give a gr\nd = t int]action1 & t int]dep5d5ce1 :i* a relation requires4 ,m]e unity leaves ! t]ms id5tical1 & ?us de/roys ! ref]5ce ( "o to ano!r requir$ = a relation4 ,m]e ext]nality1 tak5 9 ab/raction1 gives only ! elem5t ( div]sity requir$ = a relation1 & is ?us m ab/ract ?an any actual relation4 ,b m]e div]sity does n give t indivisible :ole ( :i* any actual relation mu/ consi/1 & is ?us1 :5 reg>d$ ab/ractly1 n subject to ! re/rictions ( ord9>y relations4 ,b ) m]e div]sity1 we seem to h return$ to empty space1 & ab&on$ spatial ord]4 ,m]e div]sity1 surely1 is ei!r complete or non-exi/5t2 degrees ( div]sity1 or a quantitative measure ( x1 are nons5se4 ,we _c1 !re=e1 r$uce spatial ord] to m]e div]sity4 ,two ?+s1 if !y occupy diff]5t positions 9 space1 are necess>ily div]se1 b are z necess>ily some?+ m2 o!rwise spatial ord] becomes unmean+4 ,empty space1 !n1 9 ! above s5se ( ! possibility ( spatial relations1 conta9s only "o aspect ( a relation1 namely ! aspect ( div]sity2 b spatial ord]1 by its ref]5ce to matt]1 becomes m concrete1 & conta9s also ! elem5t ( unity1 aris+ out ( ! connection ( ! diff]5t mat]ial atoms4 ,spatial ord]1 !n1 consi/s ( relations 9 ! ord9>y s5se2 its m]ely spatial elem5t1 h[ev]--if "o may make su* a di/9ction--! elem5t1 t is1 :i* c be ab/ract$ f matt] & reg>d$ z con/itut+ empty space1 is only "o aspect ( a relation1 b an aspect :i*1 9 ! concrete1 mu/ be insep>ably b\nd up ) ! o!r aspect4 ,"h1 once m1 we see ! gr\nd ( ! contradictions 9 empty space1 & ! reason :y spatial ord] is free f ^! contradictions4 ,conclusion4 #bji4 ,we h n[ complet$ our review ( ! f\ndations ( ,geometry4 ,x w be well1 be=e we take leave ( ! subject1 briefly to review & recapitulate ! results we h won4 ,9 ! fir/ *apt]1 we wat*$ ! developm5t ( a bran* ( ,ma!matics design$1 at fir/1 only to e/abli% ! logical indep5d5ce ( ,euclid's axiom ( p>allels1 & ! possibility ( a self-consi/5t ,geometry :i* disp5s$ ) x4 ,we f\nd ! fur!r developm5t ( ! subject entangl$1 = a :ile1 9 philosophical controv]sy2 hav+ %[n "o axiom to be sup]flu\s1 ! geomet]s ( ! second p]iod hop$ to prove ! same conclusion ( all ! o!rs1 b fail$ to con/ruct any sy/em free f ?ree fundam5tal axioms4 ,be+ conc]n$ ) analytical & metrical ,geometry1 !y t5d$ to reg>d ,algebra z priori1 b held t ^? prop]ties ( spatial magnitudes1 :i* 7 n d$ucible f ! laws ( ,algebra1 mu/ be empirical4 ,9 all ?is1 !y aim$ z mu* at discr$it+ ,kant z at advanc+ ,ma!matics4 ,b ) ! ?ird p]iod1 ! int]e/ 9 ,philosophy dim9i%es1 ! opposition to ,euclid becomes less m>k$1 & mo/ important ( all1 measurem5t is no long] reg>d$ z fundam5tal1 & space is dealt ) by descriptive r ?an quantitative me?ods4 ,b nev]!less1 ?ree axioms1 sub/antially ! same z ^? reta9$ 9 ! second p]iod1 are /ill reta9$ by all geomet]s4 ,9 ! second *apt]1 we endeav\r$1 by a criticism ( "s geometrical philosophies1 to prep>e ! gr\nd = a con/ructive !ory ( ,geometry4 ,we saw t ,kant1 9 apply+ ! argum5t ( ! ,transc5d5tal ,ae/hetic to space1 _h gone too f>1 s9ce its logical scope ext5d$ only to "s =m ( ext]nality 9 g5]al4 ,we saw t ,riemann1 ,helmholtz & ,erdmann1 misl$ by ! quantitative bias1 ov]look$ ! qualitative sub/ratum requir$ by all judgm5ts ( quantity1 & ?us mi/ook ! direction 9 :i* ! necess>y axioms ( ,geometry are to be f\nd4 ,we reject$1 also1 ,helmholtz's view t ,geometry dep5ds on ,physics1 because we f\nd t ,physics mu/ assume a k ( ,geometry be=e x c become possible4 ,b we admitt$1 9 ,geometry1 a ref]5ce to matt]--n1 h[ev]1 to matt] z empirically kn[n 9 ,physics1 b to a m ab/ract matt]1 ^: sole function is to appe> 9 space1 & supply ! t]ms = spatial relations4 ,we admitt$1 h[ev]1 besides ?is1 t all actual measurem5t mu/ be effect$ by means ( actual matt]1 & is only empirically possible1 ?r\gh ! empirical k ( approximately rigid bodies4 ,9 criticiz+ ,lotze1 we saw t ! mo/ important s5se1 9 :i* non-,euclidean spaces are possible1 is a philosophical s5se1 namely1 t !y are n condemn$ by any priori argum5t z to ! necessity ( space = exp]i5ce1 & t consequ5tly1 if !y are n affirm$1 ?is mu/ be on empirical gr\nds alone4 ,lotze's /rictures on ! ma!matical proc$ure ( ,metageometry we f\nd to be :olly due to ignorance ( ! subject4 ,proce$+1 9 ! ?ird *apt]1 to a con/ructive !ory ( ,geometry1 we saw t projective ,geometry1 :i* has no ref]5ce to quantity1 is necess>ily true ( any =m ( ext]nality4 ,its ?ree axioms--homog5eity1 dim5sions1 & ! /raight l9e--7 all d$uc$ f ! conception ( a =m ( ext]nality1 &1 s9ce "s su* =m is necess>y to exp]i5ce1 7 all decl>$ priori4 ,9 metrical ,geometry1 on ! contr>y1 we f\nd an empirical elem5t1 aris+ out ( ! alt]natives ( ,euclidean & non-,euclidean space4 ,?ree priori axioms1 common to ^! spaces1 & necess>y conditions ( ! possibility ( measurem5t1 /ill rema9$2 ^! 7 ! axiom ( ,free ,mobility1 ! axiom t space has a f9ite integral numb] ( dim5sions1 & ! axiom ( di/ance4 ,except = ! new idea ( motion1 ^! 7 f\nd equival5t to ! projective triad1 & ?us necess>ily true ( any =m ( ext]nality4 ,b ! rema9+ axioms ( ,euclid--! axiom ( ?ree dim5sions1 ! axiom t two /raight l9es c nev] enclose a space1 & ! axiom ( p>allels--7 reg>d$ z empirical laws1 d]iv$ f ! inve/igation & measurem5t ( our actual space1 & true only1 z f> z ! la/ two are conc]n$1 )9 ! limits set by errors ( obs]vation4 ,9 ! pres5t *apt]1 we complet$ our pro( ( ! apriority ( ! projective & equival5t metrical axioms1 by %[+ ! necessity1 = exp]i5ce1 ( "s =m ( ext]nality1 giv5 by s5sation or intuition1 & n m]ely inf]r$ f o!r data4 ,)\t ?is1 we said1 a k ( div]se b int]relat$ ?+s1 ! corn]-/one ( all exp]i5ce1 w\ld be impossible4 ,f9ally1 we discuss$ ! contradictions aris+ out ( ! relativity & cont9uity ( space1 & endeav\r$ to ov]come !m by a ref]5ce to matt]4 ,?is matt]1 we f\nd1 mu/ consi/ ( unext5d$ atoms1 localiz$ by _! spatial relations1 & appe>+1 9 ,geometry1 z po9ts4 ,b ! non-spatial adjectives ( matt]1 we cont5d$1 are irrelevant to ,geometry1 & its causal prop]ties may be left out ( acc\nt4 ,to deal ) ! new contradictions1 involv$ 9 su* a notion ( matt]1 w\ld dem& a fre% treatise1 lead+ u1 ?r\gh ,k9ematics1 into ! doma9s ( ,dynamics & ,physics4 ,b to discuss ! special difficulties ( space is all t is possible 9 an essay on ! ,f\ndations ( ,geometry4 ,,footnotes3 #agh ,comp>e1 ) ! foll[+ p>agraphs1 ! admirable discussion 9 ,mr ,hobh\se's ,!ory ( ,k 7,me?u5 #ahif71 ,"p ,i4 ,*apt] ,,ii4 #agi ,i speak ( s5se-p]ception in/ead ( s5sation1 s z n to prejudge ! issue z to ! s5sational nature ( space4 #ahj ,see ,vaih+]'s ,comm5t>1 ,,ii4 pp4 #hf-g1 #afh-aga4 #aha ,see ,caird1 ,critical ,philosophy ( ,kant1 ,vol4 ,i4 p4 #bhg4 #ahb ,ursprung d] ,raumvor/ellung1 pp4 #ab-cj4 #ahc ,see ! ref]5ces 9 ,vaih+]'s ,comm5t>1 ,,ii4 p4 #gf ff4 #ahd ,comm5t>1 ,,ii4 p4 #ga ff4 #ahe ,e4g4 by ,caird1 op4 cit4 ,vol4 ,i4 p4 #bhf4 #ahf ,i h no wi% to d5y1 h[ev]1 t space is ess5tial 9 ! subsequ5t di/9ction ( ,self & n-,self4 #ahg ,see also ,book ,i4 ,*ap ,,ii4 passim2 especially p4 #ea ff4 & pp4 #gj-a4 #ahh ,logic1 p4 #ea ff4 #ahi ,= ! ,?is1 on su* a hypo!sis1 has a purely temporal complexity1 & is n resolvable into coexi/+ ,?ises4 #aij ,*apt] ,,iii4 ,section ,a1 7 #aca74 #aia ,cf4 ,hannequ91 ,essai critique sur l'hypo?se des atomes1 ,p>is #ahie1 ,*ap4 ,i4 ,section ,,iii42 especially p4 #dc4 #aib ,see ,*apt] ,,ii4 #fi ff4 #aic ,see ?ird ant9omy bel[1 #bja ff4 #aid ,?is atom1 ( c\rse1 mu/ n be conf\nd$ ) ! atom ( ,*emi/ry4 #aie ,ursprung d] ,raumvor/ellung1 p4 #ae4 #aif ,see ,vaih+]'s ,comm5t>1 ,,ii4 pp4 #ahi-aij4 #aig ,see ibid4 p4 #bbd ff4 = ,kant's inconsi/5cies on ?is po9t4 #aih ,! f\r? & fif? 9 ! fir/ edition1 ! ?ird & f\r? 9 ! second4 #aii ,cf4 ,vaih+]'s ,comm5t>1 ,,ii4 p4 #bah4 #bjj ,cf4 ,vaih+]'s ,comm5t>1 ,,ii4 p4 #bjg4 #bja ,cf4 ,james1 ,psy*ology1 ,vol4 ,,ii41 p4 #adh ff4 ,cambridge3 ,,pr9t$ ,,by ,j4 ,,& ,c4 ,f4 ,,clay1 ,,at ,,! ,,univ]sity ,,press4 ,,transcrib]',s ,,note ,italic text is d5ot$ by und]scores4 ,bold text is d5ot$ by equal signs4 ,a sup]script is d5ot$ by x4 ,= example1 y#d indicates y to ! p[] #d4 ,a subscript is d5ot$ by x4 ,= example1 aik indicates ! ik-? elem5t ( array a4 ,algebraic expressions 9 ! orig9al text usually _h ! v>iables 9 italic4 ,! italic und]score m>kup has be5 remov$ = readability1 s = example3 dxi dxi dxi has be5 *ang$ to3 dxi dxi dxi ,obvi\s typographical errors & punctuation errors h be5 correct$ aft] c>eful comp>ison ) o!r occurr5ces )9 ! text & consultation ( ext]nal s\rces4 ,except = ^? *anges not$ bel[1 all misspell+s 9 ! text1 & inconsi/5t or ar*aic usage1 h be5 reta9$4 ,= example1 co-exi/1 coexi/2 e-day1 ev]yday2 connexion2 ass]torial2 apodeictic2 premisses4 #hb1 's ,erdmannn confid5tly' replac$ by 's ,erdmann confid5tly'4 #aej ,footnote #aeg1 ',delboeuf' replac$ by ',delbuf' = consi/5cy4 #aeb1 '"o subdivison mu/' replac$ by '"o sub-division mu/'4 #aei ,footnote #afg1 ',delboeuf' replac$ by ',delbuf' = consi/5cy4 #bjd1 '& homog5\s2' replac$ by '& homog5e\s2'4