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,*apt] ,page ,i4 ,my/icism & ,logic #a ,,ii4 ,! ,place ( ,sci5ce 9 a ,lib]al ,education #cc ,,iii4 ,a ,free ,man's ,wor%ip #df ,,iv4 ,! ,/udy ( ,ma!matics #eh ,v4 ,ma!matics & ! ,metaphysicians #gd ,,vi4 ,on ,sci5tific ,me?od 9 ,philosophy #ig ,,vii4 ,! ,ultimate ,con/itu5ts ( ,matt] #abe ,,viii4 ,! ,relation ( ,s5se-data to ,physics #ade ,,ix4 ,on ! ,notion ( ,cause #ahj ,x4 ,k by ,acqua9tance & ,k by ,description #bji ,index #bcc ,,our ,,k ,,( ,,! ,,ext]nal ,,_w ,,z ,a ,,field ,,= ,,sci5tific ,,me?od ,,9 ,,philosophy1 ,by ,b]tr& ,russell ,,cont5ts ,,lecture ,,page ,i4 ,curr5t ,t5d5cies #c ,,ii4 ,logic z ! ,ess5ce ( ,philosophy #cc ,,iii4 ,on our ,k ( ! ,ext]nal ,_w #fc ,,iv4 ,! ,_w ( ,physics & ! ,_w ( ,s5se #aja ,v4 ,! ,!ory ( ,cont9uity #abi ,,vi4 ,! ,problem ( ,inf9ity consid]$ ,hi/orically #aee ,,vii4 ,! ,positive ,!ory ( ,inf9ity #ahe ,,viii4 ,on ! ,notion ( ,cause1 ) ,applications to ! ,free-w ,problem #baa ,index #bdc ,,index ,absolute1 #f1 #ci4 ,ab/raction1 pr9ciple (1 #db1 #abd ff4 ,a*illes1 ,z5o's argum5t (1 #agc4 ,acqua9tance1 #be1 #add4 ,activity1 #bbd ff4 ,allman1 #afa n4 ,analysis1 #ahe1 #bjd1 #baa1 #bda4 legitimacy (1 #aej4 ,anaxim&]1 #c4 ,ant9omies1 ,kant's1 #aee ff4 ,aqu9as1 #aj4 ,ari/otle1 #dj1 #afj n41 #afa ff41 #bdj4 ,arr[1 ,z5o's argum5t (1 #agc4 ,ass]tion1 #eb4 ,atomism1 logical1 #d4 ,atomi/s1 #afj4 ,belief1 #eh4 primitive & d]ivative1 #fi ff4 ,b]gson1 #d1 #aa1 #ac1 #bj ff41 #acg1 #ach1 #aej1 #aeh1 #afe1 #agd1 #agh1 #bbi ff4 ,b]keley1 #fc1 #fd1 #ajb4 ,bolzano1 #afe4 ,boole1 #dj4 ,bradley1 #f1 #ci1 #afe4 ,broad1 #agb n4 ,bro*>d1 #afi n4 ,burnet1 #ai n41 #afj n41 #afa n41 #agj n41 #aga ff4 ,cald]on1 #ie4 ,cantor1 vi1 vii1 #aee1 #afe1 #aij1 #aid1 #aii4 ,categories1 #ch4 ,causal laws1 #aji1 #bab ff4 evid5ce =1 #baf ff4 9 psy*ology1 #bai4 ,causation1 #cd ff41 #gi1 #bab ff4 law (1 #bba4 n a priori1 #bbc1 #bcb4 ,cause1 #bbj1 #bbc4 ,c]ta9ty1 degrees (1 #fg1 #fh1 #bab4 ,*ange1 dem&s analysis1 #aea4 ,c9ematograph1 #adh1 #agd4 ,classes1 #bjb4 non-exi/5ce (1 #bje ff4 ,classical tradition1 #c ff41 #eh4 ,complexity1 #ade1 #aeg ff4 ,compulsion1 #bbi1 #bcc ff4 ,congru5ce1 #aie4 ,consecutiv5ess1 #acd4 ,cons]vation1 #aje4 ,con/itu5ts ( facts1 #ea1 #ade4 ,con/ruction v4 inf]5ce1 iv4 ,contempor>ies1 initial1 #aai1 #abj n4 ,cont9uity1 #fd1 #abi ff41 #ada ff41 #aee ff4 ( *ange1 #ajf1 #ajh1 #acj ff4 ,correlation ( m5tal & physical1 #bcc4 ,c\nt+1 #afd1 #aha1 #ahg ff41 #bjc4 ,c\turat1 #dj n4 ,dante1 #aj4 ,d>w91 #d1 #aa1 #bc1 #cj4 ,data1 #fe ff41 #baa4 h>d & s(t1 #gj ff4 ,dates1 #aag4 ,def9ition1 #bjd4 ,desc>tes1 #e1 #gc1 #bch4 ,descriptions1 #bja1 #bad4 ,desire1 #bbg1 #bce4 ,det]m9ism1 #bcc4 ,d\bt1 #bcg4 ,dreams1 #he1 #ic4 ,duration1 #adf1 #adi4 ,e>li] & lat]1 #aaf4 ,effect1 #bbj4 ,eleatics1 #ai4 ,empiricism1 #cg1 #bbb4 ,enclosure1 #aad ff41 #abj4 ,enum]ation1 #bjb4 ,euclid1 #afj1 #afd4 ,evell91 #afi4 ,evolutionism1 #d1 #aa ff4 ,ext5sion1 #adf1 #adi4 ,ext]nal _w1 k (1 #fc ff4 ,fact1 #ea4 atomic1 #eb4 ,f9alism1 #ac4 ,=m1 logical1 #db ff41 #ahe1 #bjh4 ,fractions1 #acb1 #agi4 ,free w1 #bac1 #bbg ff4 ,frege1 #e1 #dj1 #aii ff4 ,galileo1 #d1 #ei1 #aib1 #aid1 #bci1 #bdj4 ,gaye1 #afi n41 #age1 #agg4 ,geometry1 #e4 ,giles1 #bjf n4 ,great] & less1 #aie4 ,h>v>d1 #d4 ,hegel1 #c1 #cg ff41 #df1 #aff4 ,"h1 #gc1 #ib4 ,h]$it>y prop]ties1 #aie4 ,hippasos1 #afc1 #bcg4 ,hui ,tzu1 #bjf4 ,hume1 #bag1 #bba4 ,hypo!ses 9 philosophy1 #bci4 ,illusions1 #he4 ,incomm5surables1 #afb ff41 #bcg4 ,indep5d5ce1 #gc1 #gd4 causal & logical1 #gd1 #ge4 ,indisc]nibility1 #ada1 #adh4 ,indivisibles1 #afj4 ,induction1 #cd1 #bbb4 ma!matical1 #aie ff4 ,inductiv5ess1 #aij1 #aie ff4 ,inf]5ce1 #dd1 #ed4 ,inf9ite1 vi1 #fd1 #acc1 #adi4 hi/orically consid]$1 #aee ff4 true1 #agi1 #ahj4 positive !ory (1 #ahe ff4 ,inf9itesimals1 #ace4 ,in/ants1 #aaf ff41 #abi1 #aea1 #baf4 def9$1 #aah4 ,in/9ct v4 ,reason1 #bj ff4 ,intellect1 #bb ff4 ,intellig5ce1 h[ display$ by fri5ds1 #ic4 inadequacy ( display1 #if4 ,int]pretation1 #add4 ,james1 #d1 #aj1 #ac4 ,j\rda91 #afe n4 ,j[ett1 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#ajg1 #adf4 ,"o & _m1 #afg1 #agj4 ,ord]1 #aca4 ,p>m5ides1 #fc1 #afe ff41 #agh4 ,pa/ & future1 #bbd1 #bcd ff4 ,peano1 #dj4 ,p]spectives1 #hh ff41 #aaa4 ,philoponus1 #aga n4 ,philosophy & e?ics1 #bf ff4 & ma!matics1 #ahe ff4 prov9ce (1 #ag1 #bf1 #ahe1 #bcf4 sci5tific1 #aa1 #af1 #ah1 #bi1 #bcf ff4 ,physics1 #aja ff41 #adg1 #bci1 #bdb4 descriptive1 #bbd4 v]ifiability (1 #ha1 #aaj4 ,place1 #hf1 #ij4 at & f1 #ib4 ,plato1 #d1 #ai1 #bg1 #df1 #fc1 #afe n41 #aff1 #afg4 ,po9c>1 #abc1 #ada4 ,po9ts1 #aac ff41 #abi1 #aeh4 def9ition (1 vi1 #aae4 ,pragmatism1 #aa4 ,prantl1 #agd4 ,pr$ictability1 #bbi ff4 ,premisses1 #baa4 ,probability1 #cf4 ,propositions1 #eb4 atomic1 #eb4 g5]al1 #ee4 molecul>1 #ed4 ,py?agoras1 #ai1 #afj ff41 #bcg4 ,race-c\rse1 ,z5o's argum5t (1 #aga ff4 ,realism1 new1 #f4 ,reflexiv5ess1 #aij ff4 ,relations1 #de4 asymmetrical1 #dg4 ,bradley's reasons aga9/1 #f4 ext]nal1 #aej4 intransitive1 #dh4 multiple1 #ej4 "o-"o1 #bjc4 reality (1 #di4 symmetrical1 #dg1 #abd4 transitive1 #dh1 #abd4 ,relativity1 #ajc1 #bdb4 ,repetitions1 #bcj ff4 ,re/1 #acf4 ,ritt] & ,prell]1 #afa n4 ,rob]tson1 ,d4 ,s41 #afj n4 ,r\sseau1 #bj4 ,royce1 #ej4 ,santayana1 #df4 ,scepticism1 #ff1 #fg4 ,see+ d\ble1 #hf4 ,self1 #gc4 ,s5sation1 #be1 #ge1 #abc4 & /imulus1 #aci4 ,s5se-data1 #ef1 #fc1 #fg1 #ge1 #aaj1 #ada1 #adc1 #bac4 & physics1 v1 #fd1 #ha1 #ig1 #aja ff41 #adj4 inf9itely num]\s8 #adi1 #aei4 ,s5se-p]ception1 #ec4 ,s]ies1 #di4 compact1 #acb1 #adb1 #agh4 cont9u\s1 #aca1 #acb4 ,sigw>t1 #ahg4 ,simplicius1 #agj n4 ,simultaneity1 #aaf4 ,space1 #gc1 #hh1 #ajc1 #aab ff41 #acj4 absolute & relative1 #adf1 #aei4 ant9omies (1 #aee ff4 p]ception (1 #fh4 ( p]spectives1 #hh ff4 private1 #hi1 #ij4 ( t\* & sight1 #gh1 #aac4 ,sp5c]1 #d1 #ab1 #bcf4 ,sp9oza1 #df1 #aff4 ,/adium1 ,z5o's argum5t (1 #acd n41 #age ff4 ,subject-pr$icate1 #de4 ,syn!sis1 #aeg1 #ahe4 ,tann]y1 ,paul1 #afi n4 ,teleology1 #bbc4 ,te/imony1 #fg1 #gb1 #hb1 #hg1 #if1 #bab4 ,?ales1 #c4 ,?+-9-itself1 #ge1 #hd4 ,?+s1 #hi ff41 #ajd ff41 #bac4 ,"t1 #ajc1 #aaf ff41 #acj1 #aee ff41 #aff1 #bae4 absolute or relative1 #adf4 local1 #ajc4 private1 #aba4 ,uni=mities1 #bag4 ,unity1 organic1 #i4 ,univ]sal & p>ticul>1 #ci n4 ,volition1 #bbc ff4 ,:itehead1 vi1 #bjg4 ,wittg5/e91 vii1 #bjh n4 ,worlds1 actual & ideal1 #aaa4 possible1 #ahf4 private1 #hh4 ,zell]1 #agc4 ,z5o1 #abi1 #acd1 #acf1 #afe ff4 #a ,deliv]$ z ,l[ell ,lectures 9 ,bo/on1 9 ,m>* & ,april #aiad4 #b ,london & ,new ,york1 #aiab 7,home ,univ]sity ,libr>y74 #c ,! fir/ volume 0 publi%$ at ,cambridge 9 #aiaj1 ! second 9 #aiab1 & ! ?ird 9 #aiac4 #d ,appe>ance & ,reality1 pp4 #cbcc4 #e ,creative ,evolution1 ,engli% translation1 p4 #da4 #f ,cf4 ,burnet1 ,e>ly ,greek ,philosophy1 pp4 #he ff4 #g ,introduction to ,metaphysics1 p4 #a4 #h ,logic1 book iii41 *apt] iii41 #b4 #i ,book iii41 *apt] xxi41 #c4 #aj ,or r a propositional function4 #aa ,! subject ( causality & induction w be discuss$ aga9 9 ,lecture ,,viii4 #ab ,see ! translation by ,h4 ,s4 ,macran1 ,hegel's ,doctr9e ( ,=mal ,logic1 ,ox=d1 #aiab4 ,hegel's argum5t 9 ?is portion ( 8 ,logic dep5ds ?r\gh\t ^u confus+ ! is ( pr$ication1 z 9 ,socrates is mortal1 ) ! is ( id5tity1 z 9 ,socrates is ! philosoph] :o drank ! hemlock4 ,ow+ to ?is confusion1 he ?9ks t ,socrates & mortal mu/ be id5tical4 ,see+ t !y are diff]5t1 he does n inf]1 z o!rs w\ld1 t !re is a mi/ake some:]e1 b t !y exhibit id5tity 9 diff]5ce4 ,aga91 ,socrates is p>ticul>1 mortal is univ]sal4 ,!re=e1 he says1 s9ce ,socrates is mortal1 x foll[s t ! p>ticul> is ! univ]saltak+ ! is to be ?r\gh\t expressive ( id5tity4 ,b to say ! p>ticul> is ! univ]sal is self-contradictory4 ,aga9 ,hegel does n suspect a mi/ake b proce$s to syn!sise p>ticul> & univ]sal 9 ! individual1 or concrete univ]sal4 ,?is is an example ( h[1 = want ( c>e at ! />t1 va/ & impos+ sy/ems ( philosophy are built ^u /upid & trivial confusions1 :i*1 b = ! almo/ incr$ible fact t !y are un9t5tional1 "o w\ld be tempt$ to *>act]ise z puns4 #ac ,cf4 ,c\turat1 ,la ,logique de ,leibniz1 pp4 #cfa1 #chf4 #ad ,x 0 (t5 recognis$ t !re 0 "s diff]5ce betwe5 !m1 b x 0 n recognis$ t ! diff]5ce is fundam5tal1 & ( v great importance4 #ae ,encyclopdia ( ! ,philosophical ,sci5ces1 vol4 i4 p4 #ig4 #af ,?is p]haps requires modification 9 ord] to include su* facts z beliefs & wi%es1 s9ce su* facts app>5tly conta9 propositions z compon5ts4 ,su* facts1 ?\gh n /rictly atomic1 mu/ be suppos$ includ$ if ! /atem5t 9 ! text is to be true4 #ag ,! assumptions made conc]n+ "t-relations 9 ! above are z foll[s3 #ah ,! above p>adox is ess5tially ! same z ,z5o's argum5t ( ! /adium :i* w be consid]$ 9 our next lecture4 #ai ,see next lecture4 #bj ,moni/1 ,july #aiab1 pp4 #ccgcda4 #ba ,le cont9u ma?matique1 ,revue de ,mtaphysique et de ,morale1 vol4 i4 p4 #bi4 #bb ,9 :at conc]ns ! e>ly ,greek philosoph]s1 my k is l>gely d]iv$ f ,burnet's valuable "w1 ,e>ly ,greek ,philosophy 7#bnd ed41 ,london1 #aijh74 ,i h also be5 greatly assi/$ by ,mr ,d4 ,s4 ,rob]tson ( ,tr9ity ,college1 :o has suppli$ ! defici5cies ( my 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p8888881 cont9u\s1 & t8 p8s881 discrete quantity1 see ,iambl41 9 ,nicoma*i ,g]as5i ,ari?meticam introductionem1 ed4 ,t5nulius1 p4 #adh47 ,cf4 p4 #dh4 #be ,ref]r$ to by ,burnet1 op4 cit41 p4 #abj4 #bf iv41 #f4 #bac;b1 #bb2 ,h4 ,ritt] & ,l4 ,prell]1 ,hi/oria ,philosophi ,grc1 #h? ed41 ,go?a1 #ahih1 p4 #ge 7?is "w w be ref]r$ to 9 future z ,r4 ,p474 #bg ,! ,py?agorean pro( is r\ghly z foll[s4 ,if possible1 let ! ratio ( ! diagonal to ! side ( a squ>e be m_/n1 :]e m & n are :ole numb]s hav+ no common factor4 ,!n we mu/ h m#b #bn#b4 ,n[ ! squ>e ( an odd numb] is odd1 b m#b1 be+ equal to #bn#b1 is ev54 ,h5ce m mu/ be ev54 ,b ! squ>e ( an ev5 numb] divides by #d1 !re=e n#b1 :i* is half ( m#b1 mu/ be ev54 ,!re=e n mu/ be ev54 ,b1 s9ce m is ev51 & m & n h no common factor1 n mu/ be odd4 ,?us n mu/ be bo? odd & ev51 :i* is impossible2 & !re=e ! diagonal & ! side _c h a rational ratio4 #bh ,9 reg>d to ,z5o & ! ,py?agoreans1 ,i h d]iv$ mu* valuable in=mation & criticism f ,mr ,p4 ,e4 ,b4 ,j\rda94 #bi ,s ,plato makes ,z5o say 9 ! ,p>m5ides1 apropos ( 8 philosophy z a :ole2 & all int]nal & ext]nal evid5ce supports ?is view4 #cj ,) ,p>m5ides1 ,hegel says1 philosophis+ prop] began4 ,w]ke 7edition ( #ahdj71 vol4 xiii4 p4 #bgd4 #ca ,p>m5ides1 #abh ,a,d4 #cb ,?is int]pretation is combat$ by ,milhaud1 ,les philosophes-gomtres de la ,grce1 p4 #adj n41 b 8 reasons d n seem to me conv9c+4 ,all ! int]pretations 9 :at foll[s are op5 to "q1 b all h ! support ( reputable au?orities4 #cc ,physics1 vi4 #i4 #bcif 7,r4,p4 #acfaci74 #cd ,cf4 ,ga/on ,milhaud1 ,les philosophes-gomtres de la ,grce1 p4 #adj n42 ,paul ,tann]y1 ,p\r l'hi/oire de la sci5ce hellne1 p4 #bdi2 ,burnet1 op4 cit41 p4 #cfb4 #ce ,cf4 ,r4 ,k4 ,gaye1 ,on ,ari/otle1 ,physics1 ,z ix4 ,j\rnal ( ,philology1 vol4 xxxi41 esp4 p4 #aaa4 ,also ,moritz ,cantor1 ,vorlesung5 b] ,ges*i*te d] ,ma!matik1 #a/ ed41 vol4 i41 #ahhj1 p4 #afh1 :o1 h[ev]1 subsequ5tly adopt$ ,paul ,tann]y's op9ion1 ,vorlesung51 #crd ed4 7vol4 i4 p4 #bjj74 #cf ,le m\vem5t et les p>tisans des indivisibles1 ,revue de ,mtaphysique et de ,morale1 vol4 i4 pp4 #chbcie4 #cg ,le m\vem5t et les argum5ts de ,znon d'le1 ,revue de ,mtaphysique et de ,morale1 vol4 i4 pp4 #ajgabe4 #ch ,cf4 ,m4 ,bro*>d1 ,les prt5dus sophismes de ,znon d'le1 ,revue de ,mtaphysique et de ,morale1 vol4 i4 pp4 #bjibae4 #ci ,simplicius1 ,phys41 #adj1 #bh ,d 7,r4,p4 #acc72 ,burnet1 op4 cit41 pp4 #cfdcfe4 #dj ,op4 cit41 p4 #cfg4 #da ,ari/otle's words are3 ,! fir/ is ! "o on ! non-exi/5ce ( motion on ! gr\nd t :at is mov$ mu/ always atta9 ! middle po9t soon] ?an ! end-po9t1 on :i* we gave our op9ion 9 ! e>li] "p ( our disc\rse4 ,phys41 vi4 #i4 #ici;,b 7,r4,p4 #acf74 ,ari/otle seems to ref] to ,phys41 vi4 #b4 #bbc;,,ab ,r4,p4 #acf;,a3 ,all space is cont9u\s1 = "t & space are divid$ into ! same & equal divisions4 ,:]e=e also ,z5o's argum5t is fallaci\s1 t x is impossible to g ?r\gh an inf9ite collection or to t\* an inf9ite collection "o by "o 9 a f9ite "t4 ,= !re are two s5ses 9 :i* ! t]m inf9ite is appli$ bo? to l5g? & to "t1 & 9 fact to all cont9u\s ?+s1 ei!r 9 reg>d to divisibility1 or 9 reg>d to ! ends4 ,n[ x is n possible to t\* ?+s inf9ite 9 reg>d to numb] 9 a f9ite "t1 b x is possible to t\* ?+s inf9ite 9 reg>d to divisibility3 = "t itself also is inf9ite 9 ?is s5se4 ,s t 9 fact we g ?r\gh an inf9ite1 space 9 an inf9ite "t & n 9 a f9ite "t1 & we t\* inf9ite ?+s ) inf9ite ?+s1 n ) f9ite ?+s4 ,philoponus1 a six?-c5tury comm5tator 7,r4,p4 #acf;,a1 ,exc4 ,p>is ,philop4 9 ,ari/4 ,phys41 #hjc1 #b4 ,vit471 gives ! foll[+ illu/ration3 ,= if a ?+ 7 mov$ ! space ( a cubit 9 "o h\r1 s9ce 9 e space !re are an inf9ite numb] ( po9ts1 ! ?+ mov$ mu/ ne$s t\* all ! po9ts ( ! space3 x w !n g ?r\gh an inf9ite collection 9 a f9ite "t1 :i* is impossible4 #db ,cf4 ,mr ,c4 ,d4 ,broad1 ,note on ,a*illes & ! ,tortoise1 ,m9d1 ,n4,s41 vol4 xxii4 pp4 #cahi4 #dc ,op4 cit4 #dd ,ari/otle's words are3 ,! second is ! s-call$ ,a*illes4 ,x consi/s 9 ?is1 t ! sl[] w nev] be ov]tak5 9 its c\rse by ! quicke/1 = ! pursu] mu/ always come fir/ to ! po9t f :i* ! pursu$ has j dep>t$1 s t ! sl[] mu/ necess>ily be always /ill m or less 9 advance4 ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #acg74 #de ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #ach74 #df ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #aci74 #dg ,loc4 cit4 #dh ,loc4 cit41 p4 #aje4 #di ,phil4 ,w]ke1 ,g]h>dt's edition1 vol4 i4 p4 #cch4 #ej ,ma!matical ,disc\rses conc]n+ two new sci5ces relat+ to me*anics & local motion1 9 f\r dialogues4 ,by ,galileo ,galilei1 ,*ief ,philosoph] & ,ma!matician to ! ,gr& ,duke ( ,tuscany4 ,done into ,engli% f ! ,italian1 by ,?o4 ,we/on1 late ,ma/]1 & n[ publi%$ by ,john ,we/on1 pres5t ,ma/]1 ( ! ,academy at ,gre5wi*4 ,see pp4 #df ff4 #ea ,9 8 ,grundlag5 e9] allgeme95 ,manni*faltigkeitslehre & 9 articles 9 ,acta ,ma!matica1 vol4 ii4 #eb ,! def9ition ( numb] conta9$ 9 ?is book1 & elaborat$ 9 ! ,grundgesetze d] ,ari?metik 7vol4 i41 #ahic2 vol4 ii41 #aijc71 0 r$iscov]$ by me 9 ignorance ( ,frege's "w4 ,i wi% to /ate z emphatically z possible:at seems /ill (t5 ignor$t 8 discov]y ant$at$ m9e by eighte5 ye>s4 #ec ,giles1 ,! ,civilisation ( ,*9a 7,home ,univ]sity ,libr>y71 p4 #adg4 #ed ,cf4 ,pr9cipia ,ma!matica1 #bj1 & ,introduction1 *apt] iii4 #ee ,9 ! above rem>ks ,i am mak+ use ( unpubli%$ "w by my fri5d ,ludwig ,wittg5/e94 #ef ,?us we are n us+ ?+ "h 9 ! s5se ( a class ( correlat$ aspects1 z we did 9 ,lecture ,,iii4 ,ea* aspect w c\nt sep>ately 9 /at+ causal laws4 #eg ,! above rem>ks1 = purposes ( illu/ration1 adopt "o ( sev]al possible op9ions on ea* ( sev]al disput$ po9ts4 ,,an ,,essay ,,on ,,! ,,f\ndations ,,( ,,geometry ,by ,b]tr& ,a4 ,w4 ,russell ,fell[ ,( ,tr9ity ,college1 ,cambridge #ahig ,,cont5ts ,,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 viii #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 ix #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 x #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 xi #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$1 #aji #ih4 ,"o ( :i*1 ! abs5ce ( simil>ity1 has be5 made ! basis ( attack by ,delb\f1 #aaj #ii4 ,b does n =m a valid gr\nd ( objection #aaa #ajj4 ,rec5t ,fr5* speculation on ! f\ndations ( ,geometry has sugge/$ few new views #aab #aja4 ,all homog5e\s spaces are priori possible1 & ! decision betwe5 !m is empirical #aad xii ,,*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 xiii #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 xiv #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 xv #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 xvi #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 ,,:y ,,m5 ,,fight ,a ,,me?od ,,( ,,aboli%+ ,,! ,,int]national ,,duel ,by ,b]tr& ,russell ,,cont5ts ,,*apt] ,,page ,i ,! ,pr9ciple ( ,gr[? #c ,,ii ,! ,/ate #db ,,iii ,w> z an ,in/itution #gi ,,iv ,prop]ty #aag ,v ,education #aec ,,vi ,m>riage & ! ,population ,"q #ahb ,,vii ,religion & ! ,*ur*es #bae ,,viii ,:at ,we ,c ,d #bde