Figuring Space

Science and Philosophy

HlIXMhl

.1;",,, ,Ar ....

N_, J "' ........... I'm;'- .. C"".lri, ... _o;,-__ ~, C.~k~ «C_,~, wwI SrItMl ofl'oblo: 1'<>I1n. c_#'" I""", ... ofT~'" ~'I_

b ... ......... 0., ... ", .... 11 I h. r .... l''''''''.-_TtiA ... L''''~'''I'fIE_ ... _/ CtLlT-,<, C_.0.; . "«1'1' $ .. ,t ..... _of~

'.qoC ..... n ... O'pa' ".ofn'!: $"":':';'~':'; 'of';' ; .. _ __ '" J,' _"" • ...i"!··L ... · __ ft_."J~ , .. 7 ' .... pu ........ _ AI ............-_ ...... _, lnodI<)-DoAIta.C. I .......... H~_N ""$ "of5toau.

U....."..., of M"","-' M .. flo: "' .. _ .... ""r-_ n.J"'~ S/ .. Nr Sbt,J_lf_"""1 AI ... F ...... li .. V. ... , ,_"' of J>IoI'o-~ U.,,~,,"l'''' c.... '. _, M .. jo". er.... n.".,,,,,,,,,, of ~" I)."",;'" ajc ... ;/"" .... IJao;.I' :"""''''/ Ao.\nIr~ ~ oj l'IuIt>.top/h'. u",,,,,,m <J{ /'i,"""..p "'-lroo,_M...--<J{o..;-- ••. ; ... 7>,....,. H_U.,; ... ,..". ~ 'ocUeo.l.lo1 .. -/n -, ,',. l/oJomjq<J{N. 1_ P ., ... 1<14, Dr; ""of'" 1"." \I·_~ r_ .. L-..;,,'>iI.

no. .......... _ ... ISokJ •• ""'- lor. , ' ....... , ...,. ... vl ,'-' - ,' ' 01 "'II .. • _ .... "'.. '00II .. iJIo ... -..obOo vl ................ pII,u<aI _ .... """"-'<*I ............ C_,bul_ .. ,/1 _ pia« .............. ;1. ~ ... _~ _ 1<l000I vl ~~ ~ Howe, ... Lhty .. 111_ .... IIttJcf""" tho plHlooofoI>rvl ><i<nc< ...... lit IInnIy """"" '" ..... ml ...... ,,( OL1wo1 oci""f", pnC, .... ~ .hoi """ ..... I" 01", .. n.. .ill 1",1"" '"" dqooow.t ...... ro:-Jy ......... .w) ," ,,, .... iii....,. vl """"". ""'''' Of ~. 111t ~.<h~ •• ~._ "'*""'- f""" I<if"l,'" .. 0.«11 .. f""" p/H"""""," oaI hoi, • ,"' 0/ """'"

FIGURING SPACE Philosophy, Mathematics

and Physics

By

GILLES cHA'rELET

09(5· 19991

r ... "of I ........ """nll. __ '·_.

T"",>hc:d It, ROIIFR'f' SUORIl a(ld MURI EL ZAGH A

.. KL UW E R ACADEMI C P U BLI S I-IER S

OO.UlECHT _TON lO'IKl"

, .. btl.+. • by K ..... "" .. m 1 I" I'I>H ..... P.O. "". n. ))XI .... o, .... _~ U .. fIodMrlaDdo.

s..t.I .nd dDlnlto"'" .. 1'>ot1b. COllI ...... nd S<Ioo.k "-..ao by Ktu""" "",I, ... Publo .......

lOt t'lulo~ Dn .... "' .......... MA 010II1. LSA

t. 01\ GIber """~111<t, lCoIeI.nd "",,.;"" ..... by Ktu""" ... _ ......... d' ....

l' 0 Ro. lll. nOll,\1I Oord"",~~ 1 __ N<.bcrIaaoII

u. mJ<v'" ooob/J, M"'~iqw. ,..".,_ ~1"-,,,p/l1to l!l'lJ r.~ . """, ~u S<"j~ r o";,

.. n ........... ,~ 01000 "''''wtr ""Me' .. """'d' ' ro "0 pol' 01 LJo. ......... , ptOIe<I<d by tbio «>1'7'"" ..... 1« "'J' be .... o<Iooool Of I>LihLcd

.. -, "'"" '" by...,. ................. ___ "'"'''''''' It .. _J ..... _II ... '" by anT """"""_...,.... uoI ........ .t 'J'_ .... _, an'''''' PC' " i .. from

Ill< <OMnatn _oon

~FRlL~ EI>JTOR'S PREfACE

We are pleased to ha\e thIs "'ork appear In t~ Sckllu lJlld Phl/osoplr_~ series and wISh to thank Roben Shore and Munel Zagha for theIr fiflt:' tl1llUlalJon and Kenneth Knocspel for hiS IntrodllClJon. which ~Ips to situate tbe bool: for the anglo-amencan aud~nce.

Wc I"(gret that Giles ChIitelet. ",ho died dunng the final stages of pn:pardllOfl of the volume. did nOI li"c 10 see his ",orl: In pnnt.

The sigmfieant inSIghts tnlO the nature of mathrnlatics and !lClC11Ce 11 provides ~Iand as a testament to hIS widerungtng ;ntclie<:t.

•

•

TARLE O~ CONTENTS

~';SNl'Tl' ~NOC!I"-1. . O,agnunmatic "";tin, and tb< conf'lIunltJOfI of

I"'''': ;.

Ad n"" led gcments

ll'AN-TO<J:!.,SA'''''' DtCSA."Tl.'Tb< hhera.ion of the Fi'.'. and 11'1< hiu of the visible .....

, "". .mduI.' ...... ' .r ,"" .in.ol I Abslrac1ion and pOtonl;al l EIa.tic Ind """'''''e ,'inua)ity ] The "nne,plc of "nual -.IOOu.,. 4 Cauchy and Poisson' •• ", ... 1 ~ulOul> No, ..

,

" " " " " II Tht "''''''' I .... """rum and lho p ... ·.Mum: hanLOn. of

a=lorauon and dro.l."""M O .... me·. di.yam. J8

2 Spec1 ... and "",iwns: .... ' taiM<! ",l.",,'Y" ",,"'pocIj,'c P'o1:<"OO of Or<::<mc', dia,nlm. 47

] Ein.tein and de Br~ L" hlo symmetrical I.n, 57 4 I nd,,!duallOn by ;mpaCl and ind"idWlIl<>n by elect;"" 6) No'''' 7J

III "n .. f"",c of .mbi&~ity: dw...tinol ""bI"" ... I IntroducinK the ~al revolutio ... of\hr \l1511.,lIlIy poinu 13 1 It '1"",,;<>11 from KanL ",hal dIagrams for the """'U>~' 82 1 Atpnd and , ..... n~tk of.be: 1.:1, ... 1 82 ~ lnd,lfc",,,,,,, «n' .... ond knou of amb'Aulty. fukra of ,be

balance:! or Iklna &Il A Es.:lltm .... yd. di.Bram 88 R ~ p<:>W.r of imlllf ..... """ "",n!1 90 C ~ indilfemlCe cen!", a, a,,",ul.,ion 94

,'j;

'"uuorw ... ,um;

S Eumpia. '" ",WUClIO" '" anlhllUII) by. poonl_arloculluon A The splillln. In Iwo or p<>Illl>C , ... 1 ""mben II The p,obIem o(.he mea." .. men' 0( an&les C '!lue' lind 'RoIIcr', d,a,,,,m. '" P' ,.heel,

No, ..

". G.-.._', n" ... of I~ .... .. 1I<00_ry aod d ... lc:cloc I la •• odlOCllOn 2 11_ do)'OU bruI, ""Iendc<l ipIOCC' boKk 10 life! J "rlocullu and .... ,.,11: rc.nnaIlCIW , •• nd real rae ....... • C ........ aa·. q<Uodnll.cnol S TIlt ,",cnA~ C~I"""'" d .. lccr",

6 ",. iddmve I " •• '''''' of naonal ')'$Oem. 7 Crasoma ...... pmrl_

A M"'''9b,,.1IOn .pollN iddlilool:. "He in nalu .. 1 ph,lot<>pIIy

II Prod"", and orom'.'1Oft e &,..,..1 pmrlua 1_' Mvlr," ..... _J.od lho

UlI ..... ve.n.moI'c doalect", D Tbo: all"",,", "'.he ..... , d .............

" Tho: "crew"" or ,nvol"". prodllC1 [riroV>o'~""re. """,.,le 1

B Tbo: .. ak .. or the non-«>onmu •• trve Note.

v r.w.,t ..... oo .. 'ri<..-n I U,nl1h and mIJIIC'l>m 2 The a,,'onom, l>f the inlhll"""""", ocn.re: wldlh and ci«lnc:U, J Electnad hel>CH 4 The . .. al ... "h'·~,,'on 0( the I .. n.i ... ·• S TIo< ci«trll&fOll'C'rir.: "pc,iment •• "'I" ..... fOOt

" ~.",df,r'lo< pcd;iIOI)' p, ,he I" ... 0( fom: B ll am,ltOll' 5P"C"" "'Iu ... fOO' C The tl:rt"W ., bold me,.phof

6 To ..... rd<.10< kr>O' "' sccul.n'-"lO"n or the 'n .... ihlc No ...

A"....u. I NOle on ...... '.m'o""

A"....., II ""'. on II lmihon", •• I"""""ir;aI ... mplc

A".....~ III lI anul.on', opc ... or V

~ ~

" " .. '" '" '''' '''' '" '" m

'" ,,.

'" '" '" '''' '" '" , .. '" ,~

'M ". '" ,~ ,~

'" '"

DIAGRAMMATIC WM ITINO AND THE CON~ .GURATION O~- SPACe

Kcn ... lh J, Knoupcl .--,.. ... _ <1 , ... '" A d .......... COIn ... ndi •• JOIIIIu~ bnn," to -r. Ion, before" I'll ....

"p ,n'O. sip. "'Iuch" ",hy rnorlc.n p"'lCn aad _""." ~k< doqIIms w"b .boor pcrtmpLory Jl>""'U of ... '<realM>a. Thocy cape...., ~ures lIIod-"'", .. for IhoM cae-hie of •• ten ........ boy ... lhe momen .... here bel.., <I """ptcd ..... il,na. 0.."""", an: in a do:",", tho: a.a:ompbca 01 """' .. mcuopbor lIIIl Ihc)' ... a btlk lew. uapcrt .. litO. 01 III alwaY'...-hle.o .... k ~ ... lhe ",undue pIo!\lJI& 01 \/Ior:I. Ihocl b_ ..... ~ CaJlhluI: tlocy can ",oIon,llith "au !D'O lUI opef.loon ... ""'h k«pf ,hom f,om w-.. j.., wont ouL LiU lhe

mcuophor. 'he)' !tap "". In 0«Itr 10 .... ~ .. 'JW"'" .nd ndLlOC ppo: .Iooy bkrn-... ""b dotted b ..... n ordc110 t'Oplf imqct IlIaI were 1'",.,o!Uly &.ured in d .. el I ...... Rut unl,k.!be """"pOOr.he obqram II not u .... \I&IorI. \ 10)'

By bnn .. ", in.u (" IOCII .~".,.ho d;'lf1Imma'oc ",nunl IlIat """"do .... """ ...... ChiLd, ..... pkndill hooIr: fJr."'" Sp"'" I"""~ .n iDtelloctkal' spacc: for. far mo .. anocul •• e .n.hfopoi"l' 01 ..,..ntrf\r: d",,,,..ry. In fact. ,I " quite ,"""ura,e 10 ,~ ... lhe boook a, mI,kon, .he formulJ.non l>f . «i'ical pndir:e ,h.1 could he ""lied diu,.---so/ury, • 1'",<1"", 'hot would brinl mlo f.n~ the phenomenolol;ical .... Iy.'. 01 <I, ... ".",. ond d,a, .. mmatic pnd"'" ,n "';"nc:e. ChA •• Ie. r>OL only"""" q_'ons abou' fi,,,fCO ond .. Tilm& ' .d",,,I. ..... arid ht,w .... th'nk about '1*00 ... ,.h lhem. thI""'" lhem. and m Ih<m. btu off • ., tlrlt.,.... for US"n"1 thaI .ubrtequcnl d''''' .......... doa nol bc<:ome ; ...... ed ,n """ <I""l'hnc. In oont ...... '0 <I"""' ........ lha. I pprOKlo .he pracL"'" ol OCOC"'"' ffom.\or! "IOIIF'po,n' 0( natunJ Itrnpaae.. CbO.dcc u..,.. .• 100 ... " .• .... ompo. O\U" 0i<IC011J!\r: ",KI"'" In.ho 'n""'troc • .....,. JM'<"'odcd by d,ON1"'mit><: wn" .... AIIIlouP Chlt .... telll \IS thaI he hat .. -nltcn In. book u :0

plUIo.oph"",' d"""'-..... " It owothcr an o.d ... " .. tcd ... ..w diKounc nor a II ...... ph,ro,opbicol """ .... .-NOhO ...................... tcad a"",OKIo ChI..,...·. clio"'._ of ph)'llCl. nurlherna .. a .od plulotophy u a <1_"" rem'""""" or the ""''''''tVe fI .. ' ..... for ,honk,.., allou. opticol ..... ,herna .. a aDd ph,...." ill

"

• pro.""",;.., FrencII InlO .... h II ,_ 11_ II. 10:0 ~. s..:~ """,pa_ .'" IWJI u <uan .. u Ih<y unl.ially appear •• hen one recalls WI kaa Ik Me ..... .be u,hor or lhe IoOCOIId part of ,he rn..hc,,,l .. on. lullu "",,,,,,I r/ly.ks., .IIe In,,-.n''~ of I'~.i .. I .... m In omp"."" ,,,,,h In """'1"'<"00 """'1"''''0'' bo<:aUK" anUClpalfl ,"" Johber". """n".,, in .. ·"",h Chi to ..... <kploy. "" ... pec\«l ~phon .. ·"h." I """"""""ocal fidd. The comparison" uod,,' beea .... 01 mliSr.;K u. tha. ,he book ob<>ukl not he adoploo ."l<lm1l1cally 11110 one I'M'" 01 another hul .M>li<s ,,,. bro.d ludlmC'e.. In wbal roIlow1., I ."nl \0 Iook.t II", •••• ndoN ludic""", for.ho boolt. Tbe JCqllCDCe of CUI< Itmonos ,bal mike up .be 1>00II., 000 ... ·.ral or ,he ,mplicall"", mhe..,", in Chi'.""', d'lr\I>Iiloo of Ihll,'."" and m<'aphon.

'AI.TO" , UTlIS(lS

('bI.,okf. projtC1 muil be W'!I;I,ed f.om .ho "" ........ 'h'o In onl"'ol dlOCU»lOft

abou, the ""nttp1 ... 1 Or """'''". f"r" .. ,ion of .~'" pursued by ," •• !<l". d""'ph ..... Allhuuy./l i, i. iml"""blc ,n rq"'<r aU ,"" .... ""'" lit,... lhe ' .... lIm " mArk«l by. ranp: of .. ·.II·"""'I"'1Cd wurk. dr'<>Iod 10 ,h,nk'hl.boul lhe confipno\ion of .pa<t"" in I n h",<><y and ...... br'a IA'nheo .... Oombnch. S1IJI'onI\, in ""pIuo and """,,,,,I" 'n.~'fce: , .... 'P! (Tufte, Turklt. Coynt. 8oI.~rl. aohunollllOd ... ( Mildocl~ hlllC •• H ......... y~ .. n .... Ind mouphor ( RorootIr. II;oflha. l..kolT .nd John>Oll). ""Io.y 0I11C"'~ (G.l ....... Da'IOII. unu,,).' Gi ... n the .. ..,~llh .,r .uch d.s.::uosion .. the a~JICe of broad 'tudy of 110 ..... y. mllhc:rnolical 'I"K"' ..... boen COIIfiau,.."j LJ .mk,nl- Al ~I initiall Y Ihi< ""'1 tor ~.pIa,ned b) an .... rtnal oIltor ... ,.. ,n "'hocb malbrmaue. ro.Uln .... '0 ... nJ ... btillOtO of onl .. nalioI lbeooes of IICOClKX or U lhe domI", of lop;- par ~.cdInooo. NlCkt·, book .. mna,hble HKllylor lhe wa,.. II ..... uld ha ... ..-.de .. IU'OII\U~ 1110 roaloniont embodocd "'. h'~lory 01 lIIalhemllo<s "'''lien f.om lhe .·.n'.~pOInl oIlbcomm '"Ihe. Ihan rrom a CQIIlbln'lion of !bco.y. pnocIicc and appllCa.ion. In ~lI"o:et. fo. ChA'<I~1 uu. ~Iatlon""p 10 ma,htma.OCI lou Mm elplarned only /1.11' .... 1_ To I<U""\~ our undetsuncItnJ 01 '"IUUOOR. """CftllOll llId d._,u f ,n mathm"uir:s ..., .....,j

10 btooaoe ~t l/I.It lloe "I*'" 01 malhrnuIIICI lou boao 1Ded .. ~ no!

onIy.hrou~ tbtOO'Pb but alio 'Im:>u", d""&ramL I am pIoaood 10 inlfoduce !Ioe !nonsbloon .,r Cb5,~kt·. boot brca .... il ma,k,

... ""f"",nl momenl In the ...,~am,,,,,uon of lhe I''''''IICC .,r nuthc1!IluQ.· On the br ... <ksl Ie ... L It... boot roa,"e)'1I mult,pIe ~,~u .... n\~ _pillSl ,100 pnMIle~ ofdloetpb .... ry id<nllty thaI would ifola ... m~ or hlad<bo ..... 'hemal. OCI.. When If'p<oacIocd IbrouJh ... ..",'" hirl.onc:ol "'''''Y'' .... nnbto.kltd _""'plOOII> thaI "",.""",",ics <boWd tor ~ ... ,ed .. -ub lop: or lba. malhema,. oa '1 ·",nned" '0 .... u~. malbcrno,,,,,, .. pnoct"'-"ld br<:omeo dlSl"'led. ~Of Wltl. l. tlocno " f •• man: ' 0 do than honour I Ibcom'Hln'"Cn h,",,,,_ or mllhema.lics Q. ,,"e'l'""<1 ma,hematlC> f,<om the vanllJ<:.p,,"nt or ph;IOfOplt). Whal CblI ...... 1 PfopoteJ ... PfOJC'Cl r ... ,"""",nl" the .... r> ma,toemalOCS hal

Hplomd PUlOCt'" spare. lrut..od 01 pro,",,,, ,hcooc", .. CW ....... _ ,III, our ,n"""plioa or .... 'bemallCl mUll lndtodc "'" manual I~ lor "pc.senu",.pIICC mlrklof. dra";nl. .tole/Unf. ombbl,nf. ~'" Chh:kt OUr O.in intc. lIC1ion "'IIh lhe ~.u'" ,ha, .... dfllw ",,"";'Ula.. pI_ 01 ,n ... nllon and di ......... ry thaI cannol toe u('ll'ncd ...... y by the ,he<><n!u truaj app<2' 10 lock ......... n • ponocu"r mad-.anllocal proa:du .... 1loe,.., rani,"," 1>0». ..... ". conf""'" deoo,.., 10 lib up ,..,0 i. tloe IIeIh wllb lbat .. 1""Ot;.o,yd at. 1'1I0I0 lhat has b«a muulatN b~ lecho ... 1 dllpC1XlEOC1lt. ..... "' ... -.alp lOr m..p:.l """"', uaspmoled bylhe m<lpacoly of rIa"in' ral"""'~ua to 1"1 to IflPO ""h 011 tloe>e """'" or band. .il tloe>e 'm:ol""". 0111""",, Ilwlup' npen· ....n, .. lhese filu"'" .nd \I .. K ....... _II tlono: \lynulies or Pfoblcmt .... mlnJly capable of ,ho ""'flele' 01 """","''0''' (3)

I( .. po •• ",.ly the ....... " .. tlon d ".oIIlem. lbal ChI,~'" undcn.ok ... ,n the bool. RlI'" tha.n.,.,., .......... ,I ... ODIn .. or lin ....... "Tho "..,al ""' ..... 01 GlIoIooll, ~arw:b.y. M ...... U '" H~bert .... " be,.~ - but ... bio'""'" atab-lo.hod CI'lIIOnc$. ltlW'lu,", oll,oocb and fonnot do not. ThaI " whrl ...,u tie> .... c m)",II ho~ "llh the ~id 01 ........ nmpln. 10 demoru.lra"n. lhe hiotoncal dip"l)" <If problnm. till'>, dy .... lic exnnpLorol'Qllhal hal ... ,..,.. bttn exblulted by a.dt1lOWledponn or b)'J)O'ho:ai. d )n/l\Ol''t~J' ~Jl""~.w /If'OhInru_t _.,..... -'''--.,0 AH OU<r. _ ,.1MOpMn.,.,1M d., wy<.w /t4aL. [ My ..... phn"] Il) CWldcl', obsc, "'I0Il .. ""portalIl .." Ior.muacuon bu. r ... 'prcdlaty. \\-"ba, Cht,deI cmphUl21!l " tho omponance d i00i<' .... , .toe .... n ... 1 !""hnolotY Il5<>d '0 fumml_ •• 'PO"". Rulers. Olratr!tl ed ... mnqIUIaI.

pro'r.cIon .. ript anJlts, <;OOO"dJnlt~ .y,,<ml. ~C.· eacb hOI hem talen Jl) for ....n.ed thaI .... rlil 10_ lbe woy It...y ha'"e o;ontrollcd "". 'pollal Ift' cract __ "Tloooto dOl"""'" take rbnr pIa<z ,n a I .... 01 dut:tDt thaI run, from Avicntnl '0 .... tonu. and tho:o'rior< pttp&l'1' lor ... ha. HOIIlo<gcf cal .. aondo:.n • M ,M, ,'. prn;.c. of lhe ma""",,"toaoJ qo ..... IOII'nl 01 ,., ...... 11 It DO ..... gmotlOil 10 .""ak of lile ~."ona.l m«II""" of tIono: do;ojlnl"M. .. hodl abol!lob the rip:! d.., .. "n be,.....,n algeb. a. .. koch donflCd lhe del .......... l;"'n 0" .... 1l0M of , ... ri· ablco., and JC<)n><try •• '1IoM 1110=0 cn.u,.."j lhe prot",",;on d the tonlcmplau,..,: (III In o:om""'10 the nwlaplton I''''''h bro<iom< filed at " .... ~ ,n the oed""".,,_ d J.oo......-. dllJ'llbJ may be ~, .. 1Cd. "'I'"Iw: dIJ.&ram ""'''' """ OIIt 01 fub""" It .. a IXojOCl thaI Il .... 10 apply ud"",-dy 10 .. hal ~ 'kotcheo; lb. dnnond roo IU."""",y mal .. " tloe .... urol /lCWfDpIooe of lhour!tl "'pm' """',,: ( I I )

It .. not lhe para\l,gm . hift. bul !he ",,",Inu,ty 01 pmbk"" thai cor....,,,,, Chli'okt."Ttl .... 111"0 _Of}' ddf .... nt .1o)11uM underpon 'be"lmlory of _': the 6 ... LJ 'nl"""'''~n. and r:ontdu oI·ruplu .... •. of ""f1<Iitrn> and tbar """"1lOftJ: lhe MOOrtd 11 quofin bul .. "~) """,,'0 moa'''I""" Ind «- nil of. "''''' .... "Oil. a tnOIoonleso "'Off)',,, of lbe probIcmllJC cora.' (1). "11IIt .. why 11"0 dIW ...... 1l ,)oythm< undtTpt" tho 'lo .. tory of •• 1' 1M one, <:OnIpItt.ly d ,_nunu, 0 .... oI·ru"lu~.·. ·pa.ltd..-· and 'Mu ",f"' allo"" 0b>Il It... o,lIe • .,r the: problemallc J.ooI...a.. lhal an: II ... ,.. "","bIe for """"-,, .. hon and f,,11 01 ,~uu ....

for Ih"'" who can reaw.ken them: (69- 70t For CM",Ie, lhe double pro,rno"'" 01 truth in _fICC' Invol.e< not only a dcfini"on 01 trulh bul ~" n"loinl prnv=K>n of I«:hnoln&>e< whid, control I~ pro<>r" il~lf. II if p ...... sely lbe proarewon of the t«:bnolog;.. tbal bas b •• m igr>ornJ ,.ilh;n tbe otudy 01 matb ...... "QI. ·It ha. Ion~ bacn lhe calC III ",.'hemalic!. and Euclid h,m ... tf pr<mded tilt model. Ihat oIt"" Iwn IC1ie:s of drvelopm<nl' i, ;, oormal to val .... only the <>rIO thot forms lhe contenl proper. whilo lhe lrouble 01 findinB tlK otbet between the hr>es .. len 1o lho ~ader. Only, aJtOOugh .rnnJm[ellI and p=t.llJQJI of lh .. series 01 d",dopmenu a~ complele. il IS nnnellKbs ,mpOSSlbIe 10 m. ko lhe 0'· ..... ,. .;';1>1. at oach .Iap: of lhe latlo1" to ,,'I\oc, ... should fin, of.1t know ana:. and to pull1 in" ,Iato ,,' I>e~ it con ptoJP":'S fr ... ly and or 'IKlf, To do Ihi>, it is ralbe, nccesury for lho reada 10 he pUI;n the ,I;>to In ... hoch lhe di"".""" or trulh nugh! I" he in lhe moo' fa"orable ..... : (Grassman quoted by Chit.let, 1(4).

Wluil Chltolel .n.,,';o," is. _aIOa:Y 01 d,agrams lhal boJP"" 10 map the "lIY .hat m.athe"",lIat h,.. wo,ked Ib,euth w,;nnl- In eff«" "hal Cb.ilelet b"nlP hefo,_ hIS IU""';' a .tudy that "nold dm,n,u,'" bolwD<1I lhe IWtonal oven" of malhemata aud lhe le<:bnolojpC< that .coompany lhem. J l'!U •• the ,Iudy oIlangua8" Of hj,;lory may be d,,'jdrd ,nl<> •• ,iou. iynchronie aDd dis<;h,onlC ""mpon<:nU. 00 mathe"""IC. may be """-00 10 not only Io,'correct 1lU,,-. ... ' but fm 'I"'tqpos of """,n;,'e '"tullion, M.them.lico it hi'lo,ictzed al tIw sa"", "me thl II ;. ,«~d to ""n'a, n f"'ll ..... ollhol1thl. ChIt_lei 0",,,, a IKW •• "e""" n.()t of tmpes or fill""'" 'n lhe .. me ol Vi<:o bIIt fo, . new ...... y 01 ""'tbe""'t"",t topo; aDd lhe "'''Y thl Ihey pro~nkc thOUght. Sucb • propl hal val'>Ou. I""<'r~t;OM.. Certa,nly one Imond " provided by neoKanl"'n optswnolOjUl (Ca~r). ano,her by lhe _ffo,u ol phenOlll<1lOlo&r 10 deombo mathemallClllope ... IIoO.( fJunscrt.Ponly.lIl~hda.d. Il t.ger, aud Demd3t, and "n()lbe, by ,be pragnlot'" .. mlO''''' 01 Pic:mc Sucb ground. in part """'~n=l by Chitelel". bibl;"I! .... phy. all""" u. 10 n",ognile pbilooop/l<ra or 1Ct<_ "'ho .1uI",. OImila, <pm' 01 it>q"lty ...,b OJ Michel Serra. Gilks Doleule. and linall It.,tman.' E\'tn mo~ r=ntly ;t would not he mtskadinl 10 mlOJIl'7A! the bnkages bot" ..... ChI"lel and the r=nl I"'bbcation 01 dt<Cu.s';0"" bot"' ... n Chlns<ux and Conn ... • A f"nhe, .. 11mB ro.- ,be wo,k ;. pro.ic!od by V"ual""llon ... uhin ou, _vol,in; d«tlOnlC on'''''''fJlcn'' wbieh I WIll "'tum 10 h<tnw ,' AIt",,",h ,I may initially appear I m,,,,,ep to romporc: eMlckt Wllh Ucmda. I Ihlnk lhe CGml"'tloon 001 onty appropnltc buI ;nstrtlO!ive. A ma"o, ~.Iu~ of Domda', wo'k. ".ually J"", .. 'nh,n the world of A!tJlo-Amcncan ,henry. ;. Dcrrida', ongomg ""nversauon ""Ih m.lbem.l ..... whieh is ,n pan a "'''''''Iumce ol hi. O,,'n OOuca.jon ~nd besl rep .... nle<l Ihrouth hi, wor~ On Ilk com""",l.ty on H..-,t', 0.,,1 ... of G.""",,,v.' lIuI "hile DcrTid.', proj<ct tao '"\'<lIved labgu,&< and IlK: multiple """"p<1 that hl\'t boe" u>ttl 10 .ncornpa .. lanluas<, ChAI_Iet·, prop:! In.,,!>-.. II>< way, onta,lmenl hA. wD,ked ;n lhe deocnpl;"n ol pomelnc '1"'<".' IIolh.re drawn to m01Tlnot, ,n wh.,h tbe u .. 01 ta,,&\Il&1I" .... m:ltll<mal"" bono....,. ..,Jf·"*"o',,,1. IIoth .~

siii

,nlerested in anopho!o8Y or lhe l«:hllOlogI" "'0 c~.t. oot ""Iy 10 ~"",,"ber but to th'nl<. Fo, Ch1tele~ tn panlCOla,. dklgram. ,",o,t a, pIO.lltellC de ... """ tnal bccomo verucles of tnlwUon aDd Ibought. Chalelel". bout cballet>ses lIS

'0 ,",nk 01 a dllljlr""""",()jQ1:)' pred .. Jy In 5uch seltinp.

PA~T TWO, 11.~"'CTlV"'TING PROBlEMS, CAst IliSTOM1~S • • 0" ""'T1ltMATtCS

fir f,um being."""""" di,pI ")" In a muocum_hkt ""(01)' of ..... nee. CMlelet·, case hi>lon.. fu"",ion 1,k. nprnu"", of nDd<:s "'l1hin an """"n, e:ttplO .... 1I0n of obstrac1 ,pant, At'hough ,he ohopt.r< 'nl .. ..." "'l1h oao" 01,,", .. ·;truD U

hi'l!>riell con.inuum. "",di nl! Ibtm oul 01 orde, woult] ,bow IIOt only lho lheoreliQllinl nll'" a l ,,'(Or\: through"ut Ihc book bu. " ..... Id provole lbe ~.dc, lu m.ko funll<, di""","''''''' 01:><",. d, •• n.m ..

When pi-=! s.d. by ... & ,"" cblptcn deocnbe e1o'ohln, .t .... t.pa for . dd ..... ,nl t"" ut"""O" ol <pant aud the rnavam""'llC I«hoologies Ihat render ""'" e.tension pffl6ible.. l,ot lIS look bneftY'1 lhe lemoin Cbtlelet o.,pIo<es in n.-e ",,"plo ... &"ot<d to Vlfluality. aeome-tric per>ptClive. ""pli\'t numherj., tl'1< dllOkctio betw.,." rcat nd """Blood 'pace. and eltct,omlJllClIC '1"'«-- In ,ho fin, ehoplt<, Chltelcl .bow, that An.lotle·, COOCI!pt ion of m.tte. (h~k) bn.r< "lth .1 aD ,dee. ol Vlrlualny IMt mak .. 11 im",,",ible 10 fi .... ' urr wilrun a

l£OI1IC"'" Bnd. It " precisdy .uch ';nua~IY "'hien p,ond« leibni. "ith a means fo. lrunbnl of man .. U U"IIJ. 'L.;bni., ,, ""nainly am""llt..- woo tIl\'t m",1 pro61ubly pondered Ari'totle', di",..,.1 ,m,ital;"n; I>< ..... able: 10 .... i" .. nuailly a means of comblninl oc •• n<! 1'0"'" lbal .... , >u,t«l 10 a",,,-.nnllbe q\le!!lIon, Ihat We'" 1"""""""Pyinl hun,' (10) It II 110'. howe\'t •. the vlfluallty or polent;al;ly of ... 'u~ which inl.""I' CM,dcl bul lho imp<><',. hility of affirming an absolule co .... ponde""" bct,,'a:n m>.11<'t and malbtm.ttat. Inst •• d 01. on< 10 one "",,,,,pond.,.,., ben,.-""" m.themalico .nd nalu~ "boch would rend..- matMmalics OJ mo<hanic-..t. Chilt.le, <mph"""", tlte dram. lhal lak .. pl~ bot ..... n man..- and ,II< ]anguall'" uoed to <k1.mh< ,I. 'lc,bniz "' ... 'ht lhe comP!Cl_ O1alin, '" ma,btma.';':11 phy.ict in.<>l~ed lhe oon<lftH.1JOn 01' imp<><ing inf ... tructuteS which could no' hetp bot implicale molapll)"i"': (21)

In ,II< ... :ond chllpt •• , ChAt.1et tooh II ho,,' idt~, ol'honlon' functIOn. w,lh,n lhe di",,>o'" of. ma,""m.td ~nd physic<. Whal C!t!lelet ... ould hl._ h" rralk, ..., i. ,"" way''''' in •• nlion aI ge<>mnri< P""'ptd" .. """"""'" • v.lucle for "'einS .pant, What i. "" inlnguin! .boul Ch~I.let". d,OC!NJO!IS" the ,,'ay In whoch he .. n move from • d' .... ""'" of honlono In fifico"lh· cenlUry ","'pe<:" '-' draw,., to "'01""1 1<1<: .. 01111< 'pe<:u\lm aud III< honzon. .",,,ted by lhe m<»1:mcnl of a prn<lutum, [, ,,,,h ' CI,on Or n]» 'OtI UOO """",,bed by Chit_leI" crucial hec3U>e ,I creut ... on _.pcctltion aI_n .... " chonlll"1 hon", ... Foc Chi'okl " '" .he s<>lurcs OT diio_""", lhat !qUI« "",11 •• wh.,h .... ,mponanl ""I becau .. tber fiJ an undr"laud,n, bu, beca ....

," lbey bctoi,.: Ilall'lf lor an ,nlull"", '" _ lhe 110'''.001 may k .herN. n.. . ,uraord,ogry 1''''1'' fn)m lcibouz'. M~~"'pk • • "od. pro>otcd "" ", ... h toIDfDtllUlr)I by I>eIoN>:e. .1lnICI. Cldkkl u .-ell. '110', hi .... 10 pootUlil. that ,ben: II • semi! III l~ d.nt room to r=I'''' the <pc .,. and lhal .\ If "'" lUll""",, bul " dI,-nsoMd by IoIds iCptbcnllDl n ..... of ,lUgl. k""""Iod.,; and, .1"11 II _. thaL ,b .. '""ce .. or ....... brane. bc1.., u ...... Imi>OOl. hu. h :.;1 01 .... UC.ly I" loCI'" fOn:e TIno a.c\lon ..-auld OCI<KI>\ III a1U.1~ VIM ....... or ""null." ........ It~t lhooe ..... ~ ,",'!len a coni under ~nUO<l II pllICked' II it 1>01 lhe ron_olut..:! Of ._ mYiliQl "'I"'*,""'.i<tn <II 'f*'I' whICh "Illot!> l.1II.'clct bul lbe playful .... y '01 wtucb Letbn" <1 .. WI 001 or '"11m" .. ,110-p<ll,nu.hl} of 8<'<><II<'I'ic i~ Rat,,"," Ln.n off .... prOUH: 6p1~h<)n '" Ulbn,,_ ChI,.1et rcmod.b or echoes Let be",- .hrouab hi. own Oaq,puon <Ji • pace.'· 1bno: ""'" • ",,,,,,.(let" .hal the Ikl'- of """"nl from & pa,nhnl eon"'" be tl,dcd 0 ... ' IIh (be deK ..... or. thermometer bul rompcl,ht.ye 10 """"1'1''''' I"U"""" or ,be oblique; one mo ..... lul1l" _If ""1~ the depoa 01 ... """ .. 1 .... ,boUI e>'er letnn, oondf bt '",PIlOd by ... \lalJOI)' or I\o<w>n'aluy • Ione,.nd """YEIlIn, 11111 taa beconLcmplo.ed by. Amp&< nJJo". 0( lhe ........ Of by • !al ... ..."..:IJOQ only ,,'reck, II .. dopth.' 150I'Wbc,hcr II " • " ..... _ 0( I II.D,belpo Of oil domain 01 t_lcdp.. the Dbbqwly 'I\~ .... , II, lbe bonzIOn ""po1oI!II I ' ....... b'lI .... pI..,. .. _ '01LICh. by I."'.n, n,llI lbe /lpores 01 ~ ... , ... I"""ioa .. ,,,,, ... os, '" .1 t:bi'M .. ,~ ..... hI, ...... ~ lua .. belJllttn II .. ",""Ue 01 .... 'metJ.pIw>ricar and ,be already domeohCIIlCd I'1lliona/: IU) Tho hol1lDD makes ,I poooibic '0 '~Iure ,n,o ,lie ,u,bulenl ~ . -he", ~ n, an aDd phiJOIOflhy ~ .... n ........ rJOthe. "·,,hoUI ............ ' (5)) 'The horilon ...... 11 .. ". boundary lI\.IM:tf ,/la, po"";"''' Of

ooJociu .ranl,I'l$'I,u ... nor a ..... ...,. dra".-n In. do""" 1,,>I'.crou lhe .~,_ Once ,t lou boc-n de.:Kitd, one 11""')"1 amel .,....,-. hen", .. " ... y . ·,,11 OM. Thi, il ,"" ~n,pe, •• ,o ... de: 01,,,,, "')nUOII: w"""j." ltke tbe ""bh'. '~naciowo lik. I """,II, comprom,,, ... lik. louch." d"", not dres. Ibing< liP ""b "rpo'no"""" bu, irnprcp ••• "'·.,)·Ih,,,. lha • .-e ~ ",""h'cd u, .... Il'o. I ~ I ...... " m"lu ... y .1Ia. II io "'" .he potc", .. lny of ,"" 01_ .. -hd .... "'" bef_ lhe oculpt"'" lha. ''' ..... 11 N' .... bu. ,he pol.au.Ii., 0( lbe IpIIQII Mllloun,h", lhe .. """ Ind ,t.. KWptllre.

In lbe ,rurd cUpt.,.. Chltelct """ou from opUCa/llra'epn (0< ....... "" ....

o.nd a,cndt.., .'YW''''' """"" to .......... 11 .... ry b""-,,pes P""'"oIocd by , ... d,·' '''1 01 nepl" ........ ben. Tho ,..... .. mc ialW ........ ,berdore no .......... anly,'" k,nd of "",,!a. iMpcc!lOfI .. -hJdo _ ....... ,''' _."e .... '" 10 rnal .... ..... ,JCaI calcula,_ .. "b II 'U"",",," 01..,.1 ~ n it .... oo..forth ..-. ,''' pooaobili" of ........ ". by <1 __ ,110 """",,'lOn of .... """" ....... 01 "f'IICC. """.....,..1,"1 Ih ... lhe -.....:e 01 11 (III'-""''''-'N c",".I_ In 1"'0 du ... ". ......... (79) 1M d ........ 'on "'" only ,a',,,,. tho 1U<Io. ;n.o Ie .. b.o .. ·" .-lKt; by K .... , "" ,ma.,"a.,. numbe ... bu. >howl ho ... lbe U"hl, 0( ncpll.e numbco """ be hn~"" to 1110- ''Iiual "'rtaCntatlO1l of ...... romtt .... ' od.m In It.. n' .... t .. nlh a: .. ,ury. 'W""".n. of,.;ene. ha ... pro""bly ..... 1 lI'tn ~ ..... u.h lhoupl

,. 10 Ih ... urpm.n. r .... : ,be o:IcnllliaollOfl of i ....... ttary ""mben ""h the pIanr a""",n in ,''' .... ,btnta.ICI;/ b ..... y", ""Iy .'lhe bepnllln. of Itw:........""th ..... lury.· 178 79) A. 'hrOUJltoou. , ... book. C!I&,c\o:!', ,o,em, ........ '" tl>c <IeYdopntcnl 01,,,,,,,,,,,,,. or ,n pouIlln, 10 111e pe-1Ih 01 ...... moun ...... clwn in lbe tu.10f)" 01 ............ ,011 bu. rI.her In ,be .... y ,hal IInaIInlrY oumb<n rrovale ...... ",,,yoo( ................ maner, M I .'U ... ...". bolen>. Chl'dn .... y

"" lhoupl 01 ........ 'of",". ,;poaaIlop>t "lIh,~ .... 'bnnabCI JUl'I as ..... ... able 10 ...... ,II'J ~ .. ·"lnn'lIo .. udyoll,,~ralure. h om .......... n!aI'C pOUlt.

('hilclcl ""'J Ie lhou,h. 01 optn,n •• II"Of!OIOI1 of m.' he .... II" •• I" t'" fourth chapter. Chltdc. mo • .,. .0 1M .... ,hema'ieltl """iCpOS lor

. prodin. ~nd .. ",,,:bml d,""'nsia .... hl). Tho d'<C\IIllOn .haws how ideas 01 muhid;nICnl", ... 1 I", .. ,,, CItIC'8" Ih,ou.h "IC .... "I'On or ... odea of ..... BilltlCnl . In ,,/feet, 1M !XI""",pl;"n of 0'"' .. I 01 .""",~I paralllCtCfI po-o.ok .. a "......,.. by ",-hidt ."""" pa ........ ,,'" "",0 M'tNtk .", ... Ied by lnolhe. para .... "", ChiI.Ie, 11l1li ........ 00 ru, h ... na1)'" 10 O ........ nn·. T'-'Y 0( "'" b ..... KIG (ISoWI, 0 ... _ pro,'ideo an .... "l pOO"""'" 1 ..... ,k1b..:enlul')l pbysics thmqlt , ... ...,.. '" ..-h>ch be ,nlui .. 'M rdahn",J",. ...... · .. n .-u.bIe and 'n' .... bic 'pac.:, In CWtdo:t·, ..-or<b • ......, ,h ... a""",,,, ... , ... bIII undorrstaftd,,,, o.nd the "tIdtt.'aad,n, ... " ,,,.-..ible ",,_. ,,01 ) WI"" CWtelct fia,.! ....... "''''1'''''1 .boul 0""""" .. /oo.".u.;, ....... ,.hal be 'rcpr ...... , •• FdWnc pcdaJOI) of , ... Ionno of ,he ..... I'U'I 01 "1'"""' "01\1 ...... '0011 .. ,tu. I""farncr" ,be fUdcr·' .... pK'Il) foo-o." "."n-n·. -n. .. '0 ..... ' .... ' .. 11011"" d,lettante', dlStan. Wlltcatplat ....... ,1 Ulk ... pan ,n ,he .ct1OQ. " It." W.I/dl....t Wvir_ '" , ... Iotn .... ",t.nded ,n ,be pbiJO!l<lpbJ oI .... urc It lra"'ponJ US '0 Ihll p.,..,1cJCd itOM ..... '" ,nt~'llOn allll d ..... ",,,,,,, braJmc kn.oUN 'n'O' I .... ", unny. It IS !lUlbe •• rOori 110. a poo'enon; " II con.cn,ponllC<l ... ,,-,'h .-hll " ..... ,.: 11(4) In O ...... n~"·. own .. 'onl.: - ....... "11" .. "1 LAir"",,,] 'r;l<a ... fOfC'SJI '0 lhe d""""I" of ""rc ,.;e'l«. ~b<>.., .11 IQ lho dom"n or ma,btltt.ll' >c!' II ............ "'I\bo~\ II. I. " ,n'l""'"ibio 10 r.oo an, ..... ..sr. It.. ,r (0""""""" '" lhe ... hl "lIY lbe look lha! .. i= in .,...., ... any dc"'~'I''''''n' lhal leach .n lbe

""' .. "",h, bu. wmpm,n. ilUllaDI> ,'''', ha.·~ "". )'" bca1 ...... """"". It IS foo,/lo, tc.1(tJIlhal,''' pracn" .... nl """ a"'" bell"""" unly he oku", [<I.nUn 10, • .,..,..101]." 11(101) 'Tbal • why !I<XIl"r.; """,,,,'."on j> .-nloaUy a ptoI"'"IOn 011 .... ...... of d.,-dup' .. hlo. ..... oIwhid, Iadt. In ctI_,."..., , .... _ !nil. It, ,It; .,., and rom.. , ... ,""',.., prope .... h,le ,110 other """mI, ,. "",,fA II..-If aad dcl .... "'1tS lbe ""'" .: 110')

In .... Mil cUpier. Chi,"', .....,..·0 how ,''' d""".·cry 01 doct'4 ; .... ,"" 'J*'C 'n "' ..... ".>111'".""1')1 ph)' .. openf ~lra'<'IQ lor I .. "", .. d.-<:enlury pbp. ICI 81'1 lbe chapter "'.0"'''.,. ""lIh ,he po-C'o'_' dl>C\l"""'" In p<nol"l ... 11 beyond .... ,,,,,u • ., ,""I .. ouk! .,mply ' ....... . "'''" .. ''' ............ een ..sea. 01 '1*". I, .. 1'''''''''''') the 1R1c,,""""' d ... "" .... ~hlCh '>hew' .pau. 0. '0 be more po-«, ..... o.paa! II "" ... Icd by lbe upeume-nla/ !PM'" of ..... ,._[ .. ''''' di;l •• am .. ·{OJ ... mI8~1 'lW"k of" pnpcmJ'fo. uf qIK'~ '" n,.....Im<",",_ _I"", ... /Q ,/w, ,( ,Iw ... .,uJ "M "~~.wd '" _,,"lory.' l \1)' .. nph"") ,16J 1111

'" • <I.",....,.. IMI a, ~ W", lip ......... mila. 10 lbox I1altd by Bna .. ROI""'''' Chilcltt Il0l,,,,," thll !he onctI'."on poo.ldcd by ... ...... of 11 _lin .... '" d"'''F- J .... ,.. dlifermt cona:~ic>M of '0' duu"",,,,, f.udod .1><1 III< _ 1 ... ",,11 01 uruIpnary numbe .... to • dLlformt Idea of '0' oporlO'. ,n elaclr.,..o,"eLr>e 'P""" A. Wlelet 'bll .... Ibnt it. oenoe in .'hlth II II 1he ""''''''" ,t>c,f rq>Iaa::< ... idea of '0' III • point of ono"lahun. "1:10<.-1"",1 """'11>>11 ~ "'" thlll of ."",mul.uo". 1:>101 thai of ~lIr"'nl thaI loop" ,110 ("") aroor>d "flC1'<d by oq>orauon. , ',," era of ,he '''"(:\111, hn beaun and II ",iU "'""" ~I no ,..rpruc. it II". do"," aytlalhlC:l ....... '}'PO '" ,.nIulII"n ltnhd .. "b lbe <kl<oItnllJOa of oppooilioDl br Ioopo ."" bctKh."'1 So6) The modll',,,, bnk ...... _ 11 but '''''.pac< "d. (156) lC]urn:nt mIIOI apjHoplllc .......

<P'w ~"I""' __ ' (1$6) For cum",". Hamtl''''''. q ... terruom off ... not only • _ 1IIII,bcn>ltd metbod. bul '. method of ""n.kUII. and DOt. a, IcbI ror the pn:tom, F""""KWI.' me\bod of ....... , lbuu,.,,: (Muwell u qUOIN by CllJIdct. .", M .. -n 'fortco G "",,(Ii<: ".uu of~)'~..J IIw '",rooturl"'" tJ/ Pmllurl'y.' (1110) "Tho 6&".., of ",""",h or olll>ouah' by .-11"," "' .. l",n.Je. tM Ifon,\II;1&e al>(! ,<10 ... of. famIliar .a,,,,,,,, to """ ",1h .... hlCh ". an: leo!. ~q""'"to;,l may be ClIIIed Scicnllfic Me'aphor.' IMu ..... U . J q""Ied by Cbitdc .. , 1811

Jllfl ., I ncb I .. ""'" IS ""I(IWIlcd btl"""" ,be ditrclau ... IE r .. ol,ho ... onl pkM III f."Iilo.h. IC\-.ahnl vale, ........ ..,.,nd lbe: won! dillram or &.)'/,11"" ,n Glffk. The: root .-em of lIrr,,.._lHr,·~. """ ..... ply oomeIh,nl ... hIdt is marked '"'t by ~nes, I lip"'. fn,m. '" pion bul also Clmc:I a 1CaMldat)

connot~hon of m.l~ inc or crOSSIn. 0111 Th' l ...... "."1 is op<dlkally pr_nt ;0 tbe: word .1.,<)'/'.'" In oontcrnponLlY G"",k the .crb Inrun"', "",a .. 10 ... Ilt.

"""""no Q/f.) The: CQllII ... :lIOO bet""Nn d,. ,,·,o. 81Un:< and croutlll _,h,n. ou' i, more Ibn. philolopcal CUllOStty beathC il ougo5l1. word lha, ..,.. ule<! ill G .... k 10 rrla I<> .. nulII Oft I _~ noIC pad in ... h .. b ,be mart ol I •• yl .... _Id "mnl"~l CfOH ....,.. ,be raarU IMI hIId been d"""n (!U'

.--.Iy -tlr.,..,.,." '" off«l .... bod ..... pBCtJat ol 61<'0"1 and dtefipltnJ.

..,. ""I .. UltcreIIl"l 1Icn. ;. thaI lhapm I"'n""I"'I"" III • If t.,...,. of lIpn:< ,hat "",,''n from tbe .... "'bitt I<>lhecom.l"'ler .. tUIL t"1OIII a ~ gJ ... n .... "",n'. tbe Gmok ..... ntlll of dtlltam iooocata ,bl Iny ft,"1"0 tbl i , d .... 'a is lIiXOmporuN by an UpectBIIC)' ,bt " ... ill ""''''''n. hom the ",nt.,. ))O,nl of the Gruk root, dll ... m prex,," • ",m,nder 01110 ... ff'f'llllen!ly 'e<;hoolo,iQ of r-oprosenlal'on .... il1l'O .... d ""hin lbe hlSl<lf}' of malbemttt..,.. TIlt ,tat ... "f lbe d.t>.gram lI.t It .. , ;n p.o.n I ~o"St<!""noc of techool ...... 01 "'presen'.11OII .. bether III ",roIlt. rodoca. pnn,ed boob, mlln •• " .. phcd or • ..-alal thtm .. 01 hand·bdd c:aIt:uIlton or ,,",,~IOP morn'OflL. I, ,"""" no, I ~t_ bol ........ ("0 ... IUId poruIlItUd ..... 'b maDU"",,'" n:prcsmtlt_ Oft tbo ..... ol Gplmbcr" Ii rI •• ", 0>01 ...... ,,"""",p bet ....... Pot""",.-a. "otIt: .....

DlAGUNJ,U,TIC ""mMi AI<D THI!: CONtlGU .... no ... 01" IP-\CI!

a>Ior..Jubopotpi>y or- [koatna and ..... mp1l>l1ll 1~ n.... lin

q_t_ 'MI ""e aK only ..... btJIan,n,I<>"". 0.. ... _ ILottdly ".nd .. """',ed Ii'u .... but 11 ... pIaa.l .. ·"h", I nalraU~

JIeIu .... Tllty btc"".. 01 .... '''Imdod '0 booorl.. pan of I .. rua .. red III,,,,,,",'. If .i" ",lin! "" may th'nl of ma,hc"""iat I' II Jl'f'X*'fI 01 buiklia. lI)(I<b)u"," tM' enlai! the! ",,,,,,",,nlouon of 'fHI"" idil ... m.,. J" .. IS i, is ",*ibk: '0 drii ... a ,a."nomic .","",,, ... lor Fn,. oIllOry probImu. it il po.,.,bIc to idmtify d ..... mmauCil catogories. Alt"""'" I caallOt", '010 delail """". I ...... nt to a''' ..... t""1 0 btOlld .n.ttllCliDa Cln be II'IIdc bet.WtI .... a ... lllI Ind pr .. iDnal .pplQIIO .... Doodlma on I ""pi" ... or tho id""yDCntttc .)'tImII ol. Fy ....... n 1ft In • ClkJOr)' IC"J'IU'I'. from tbe d .. ""a>tt oI""Ihooir tllldn ...... E>n hrK. ...... "" .... 1 II ""' ....... hIc to..ale. nJOnIOI ~""""""'And ,ndeed it would • .,.,.... 1U' n"""", choUIOCIOOIb .~l bee _Iff d,fflCUlt til mttkc ... .., follow d ..... a>tt ,,>10 tbe c~",,"mmlal .~ ol tlIte """';tor. I .lOnt 10 .... l •• n otld,lio",,1 oboenlOuon II •• 11. -no.. dt>WlCllOll

m.de bet","" dia ... "", ,n I~.,boou and in dood~nl d one tlill hu ""dlInLicd 10 be ... i"foloN b)' • hiilOl1 of math"""""" W, """Iin_ to cmpha'~ ,lie<>< ..... Ind lheil dt~y '" I"-'tboon. Wbelhcr 0 ... loo/oJ III Tan." .... Galile<). N ... "on, or Leobltiz. c-pltt:m<lllI IIIJ.JIltlCnpl d"",n",," lnap.,n <;IoQdba,) to piMa! io the !aIm or .,..,.raphy .. ,hor llIIo .... lm ol matbtrnatl(1 n... iJ 1 mdtlkc I th'nk bea_ il 1t1l'OQ' ... 1IIt .. 1e 01 ... ,.."""", from ..... tlonnattcal

.-~ teao..., btJIn to 1hlnt thaI lhe qtoClbOftl.-e .... It./;"" lbotL. dill ......... ....,

_bo ... uniq ........ 'b'n tlte prlO<I."'" ollllllbtrnalOC&, o.ale~ moonioo Illit 1111 bern ",ported 1010 lilt Iootn"t .. or mOlsinlh. 01 phllOloOfthy I'IaIO·' u,", 01 . d,.pam 10 ,oach ,"" l'ylha"" .. n thco",n, to ~n ilnOllln, ...... 'anl. Niool .. 01 C" .. ·• "",'aplrOt ..""nmctlt .. the: nplooiofo ut di~IV"=';o mallcr dllf"le<!tltloulh tbe dtlfu· .tOII 01 pnntinl ledt""""'" pnwtdcr; .,'_ of c/fon. to r.pU.k aboUI d ...... mmatoao. In """ctol <UeJ .'e ha~ ~ ol dt ...... m""'tlClo .. ",nlrA in .".~c:moIoc and ,hc pluloaopbl c( eel""",,,o,,- lcIbn ... ror rumpic. 1><.,.,,, •• i""I""""aly Iltr-..c:tcd to IIndcn,"ndtlll'bo ;"ten<l1OO bo''''' ...... d'II" .... '" oymbol ''''<11».'' Al lilt ond of tIM: nt",",""th <m'"!)'. Ch.ofb Sande .. I'att:tr ..... t .. I ""local ....... " OIl ,be .11'''' or math ed ..... t"'" ,n the Uniled Stat .. tn .. Iti<h tbe ...... t""',htp btl","" naITlt""e Ind dIItnm MOO...., I local pmnl."

In h,~ prcf."" '" ,be fi .. , Nil"'" oI.be Pr/lt(/pla 116871. I'lCtl1"" Q-bs1o ..... tha, F''''''''Y;' 'f.:>"rodcd tn ....... hlnial practKII:" Even thou", he lit.rally "",.nll""t ,........,y ... 1,.,. on tbe nLOnu.laru. Ne"'Ion', t1,,'n d'''' .......... oho",. Ihlt br "lllmtttciy brh • ..,. tha' .ud> """,banical Iblilly dcf>md. "" """,', C'lIPK>ly to """'ptTbmd na,,,ral II ... In o,her .. <>«I. "","'llIICdLon""'! .hiltty '0 d""w. d obIpcd by one', ClJIIICIIY to t""Q,u1e Gnd" ... ph .. prcocnoo tn nalun. In .... crt. ,be NewtocuaQ _10 .. Ilk .. ,....".-cd d" .... uon. /'oC"'OII', • ... "_1 d III I. IUOfC ....... rhble p.-.be """""",,tlllt) do_ '0 .·hodl

"", he II o~ b) lhe ,,"-phie ~prnonlation of bi,.,.,n lhotIgln." 1 "'an,lo <mph.""" ,be c ... or Newlon bcca...., 'I ,UtI'ltral" lbe .. oida""" lbat philoso-ph) and lhe: h"lOry or ma,homau"" h .... Ibo .. 'n lho mcd,auol p .... """ of .. mina ,echnolo8)'. Cldtokt .. ould ha"o ... .Itt how d .. "lm$. far from boin& "mJ>lr 0 ~gos'« of a nlCChani<:a1 "'.,.,. ..... 'ork ru a tnt:>nS for malhemallCa! d,,,,,n'·.,)'. Wh;J.I Ncwlon a",rilles to mt1aph}"$lQ and ",h;J.t hru b«omc factor ,n Iho mysuficalion of "'athernati<>! ' "rather a manif .. uo"on of hIS ,"'Cr1lctJOI1

w"h ,he: ,h;J.,,"," mark .. .."l>bIJnp. . kel""",, on 'he I"'por belor ..... Whal oAC1\ II ,akon ror IIPnl<d whal "', .. rqat<l<d Q' '",rOlch I"'por' ~. "gnific>n' moan. r.,. malhemalic::olaplo ... I ... n. 1t ""'"Caling ,h., ., .. n.' ,he .. me "me Newton ... umes a ''''n><:Crid<:n' aulbonly and .... n ""tbonzanon or hI< "ork. hli o ... n manu§Crtpt ,,'o. k is filled ,,"h mh" ... of ",rateh ""por, In<kotl. from lit< ,'aolage point or Je1"Ont .. mh-«nlu,y mal io.tmatlC1l. u,bn"·' 0"'0 fully e,,,,,,,,OUJ and 1"I1n ,nte ... 1 in the ,mpo",,""" "f ,he iV~phic med,a u;cd 10 thIRk marh In ,mpo1tant diff ... """ bel"-ecn hi' ...... k and "",, .. ton

lido", oom"",nl1nl on II><: USC or metaphor in the book. r want 10 ",,'!C'"

..... "d function. Ch~I.lel 0>Cnbe! '0 d,allPm,. Although tile book w;U ."n:I,. provoke lhe ",.dcr 10 idenllf) .dd",onal d,al",n"","" opontion .. lt ..,..y be I><lpful '0 not."" six m~jo. ","111m.

Dl<lg'a" .. <""'"; .... ~ r«lrnol/lfffl. "w, _Jim. ("o'Mun ",lin /«IwX'Wu of .. ',iring, The phpicality ollhe d,.""m is .1 ... an ."""'>on of lbe ht.nr.J I..,hnol"""" used 1R tho" "on\lruo1,U~. 'I k """ I h .. c lbo",hy .. it ...... p"'l"'lI«.1 • hand b)' thollJltl and 011< miilhl I>< lempted 10 .. Y ,hal II>< P""'''' or <:On\p8.<>e5 lIVe a point <f nrw ' 0 the ~.nd. by asoociatinl an an~1< In ... h"h tbe 'nl...-val ;. · ... n· "'lilt tho ~p: llSI) 'We iel .be rull lRCUurn uf .. hat di,linlU"hco lhe ruler from lhe rompa>i<'> a nd the ansJc:. The fUl~r impl," an end·l.....,nd pl.""mcnl of iland.rd me ... uriog umt .. a manipul"linft of a piece 01 dn"""on, The ruler i •• lraml .. lhal hOI been I'ro7cn - ~ !>;or' (I SI-152 J II ..... will ,.,., belo .... the rhy,itOIllIY aswcial<d .. un di'II""Tl> is al'" associ.led ,..ilb mol.pho •. Di~gra"" <n''''~ $I'au II)( ""'11w-",,,,iru/{,,,,,,,,Io •• 'We could dcs<;, ibe thIS iI>. Ictbmqll< of •• Husion,· (10) and'A philoso(>hr of lhe: pby,ic<l-m311\c. mllical ,,"nnot 'gIlon: Ih, •• )'mbobc: p<X1i<e ~hich is mo, 10 fonnahsm, Ihl$ ",a<;Ii<e or "ondtlU'ilion and ampl,fic>uOR of lhe lOtuiu"", ( II) D,,.,.. ...... "u IlOl ,1<1,1< "", proj~, dr'o"lily QnJO ,10. 'P<i'" • ~k~ ,10.)' ,.d< 1<1 ... presem. ·F.rad.y 'urn, ph)"SICs on its head by decid101 no lop!!", 10 sec lhom lltncs of fo=) .. &nbsKli • • y 6Xurrs. bul 01 i ... U, .... "u .a protV>i<. 'P<IU .... dIOgrams, ~. d<>lrnMflIr .~I"'nm<n" alluding to · .. al· .'pon",",," thaI manlf~" I.t.nl o<lion': 1166) 'Ln><, of fUf<le I .. ncitne. · ••• r "r 'amflcial'; I""y d" not malll<mal"'" fo,en; tbey do nol rin _,n<:aJ onto ph]'>l£Ol, mine. lbey lubslilule In i<k ... bUI datI" and ,0uJll, nc\.

,,·ork. "hiclt .now. "irt~&lily to .. ed.lm II> nlhls,' ( 1661 - Du,J'u"" "'1"'''''''1 a ."""/ Jumrryfi'" .... illl_",. ["The OO.;..,n IS ne"ber

UI~G~~ \1",~'n{" w~mMl ANI) TIIB CONFIGUUTION or §r~Cti '" • boundary "...r~ ... lhal prohibtts or salkiu tran.vnsion. nor. bar ..... drawn in. dOlled line.croH the .ky. Onoo il II ... bc<:n dctid«.l. ~ oJ .... y. carries one', horizon away "'''h onc. Tb;' is the .~asporaliollJde of Ih< horizon: corrooi,. hk. lho ..;,ible, lenAcious Ir" • "'""t~ romprom,.,nl ~k. tOll<h. il does ""I dt\'55 tbinl' up .. 'nb Ippearanees. but 'mpn:gnI1tcS .vcrytrunllhal we a ... rnol1-.d 10 InuP: (54)

'To a=p 3 d,m.n"on" 10 m,-.m I dialram ... "" .. the pu .. dlSpcr.ron 01 ,..hal H~1!'l'1 call«! au. ....... I""OW" .. 1It (Ihe 'be'Da .... ts.dc: one .""theol and lhe c>ubota""" of lhe Inten,,,-. baJallCC ON .""Iher ou[; tho un!Old,oll '" III< lalter """.i,he:! lhe mobihzau<NI and "rallhteo,ng <>l tho fu~r. Ind _icc: v.na: ( 11.1) 11iQgr_ an mNWti,,& ,~id .... ~~Id --.. ... ,100, ./try ca""", only M '«:<K:~ud h." rdir<~. While they mal,al. bc:lwo,," " ' her lechnolozieo or ,,·nln'g. lhey may al", med,al~ or ",veal bnk, bel"""'" probkots. ',,"ollun~ ;, mo •• fertile. all math""",liclln, k"" ... Ihal ,1H:oc obKu~ ~nol<>P<s. these murky ~I\rction, '" OllC theory in "no'''''r. Ibcsc fu,,; ... """'''""' 1"- ''''''J>lieablc lill's; .ho nOlhi"l cI"" a. much ple .. u .. 10 lhe ..... "'he. A day "<>mes ,.. .. n lhe illusion ,.ru,hes: presonllmeol turns ;n,oent.iuly ,Luckily for ..... a'ohes, as Ih~ f"ll' dcar., 0"" po,n<. ,hey form .gam at a ""ther'IAndnl Weil as quorcnnalinll .. hR' ..... ~ p'Of1O<mg '0 <:aU d,agr;U11matic ~,po"rncn .. and ,n giVinG . >Ial u. of III" tal\l:u,go {)( 'spoak,nll ... ith li>c hand,'. {)( lhe 'phY'1Ot1 oen .. · .. II .... dr11"I'4I po ............ n~ by the ""atesl ~lcctrogeomc' .... and .. h.,h can va'p the In,(am "'hen lhe metaphor sol,d,r;.,. .. '11<0 lhe "ptrauun ronl",,<k wnh " for "''''''''anty, Thi. Iangu.go f .. d. on ge<IUTCS moch ""'''' than on al ... dy •• .tilable malh.maucal form. and P'''''-''>C$ a ponicubr .llu,i .... po,,'or ... hich C'O.n tr .. , 'C1"$C the d<dUCli, ... hai", of malltem,tical physics .nd illuminat. ~'hole >«tion. of geome'ry. "Tho IIcId has 10 inlrinsic <'porimen,.l mi .. ion Ihat proICCII il from tbe pret.n.iORS {)( the un<ler· $I~nding (in Ihi ... 1lK it i. irredUCIbl. to a lran=nd.nlaJ 1""'1') and from tho Ippotit .. of callMhll .... "Tho •• is, ;n lit< I,~ral "nS<!, nn "'pplu:al,nn' 01 lhe lines of ficl<ls. bul mtber an uIOnl1h'"g pedagOg)" by all",i"". which .1""&IIt an.y ,",,<>du.., an . 1.,,", • .,."" and place .ith ;\1 Cll"'cily to resoo"e: ( (67)

Ch.it.l.f. p',,","lahnn 01 dial1"~m. ~ inum.tely """';aled .. nh hi. dlKU"'On Iln<IlJlIt {)( ..... 'aphor. M.I_pIloN ~rc dcplu)'cd tllro\lgllout tho book In way, Ihal pU1"]l<lII<Iy ,ul aga,n'l lhe: common propt'R.ity to ~~ilc mctapbor from malhemalicaldi ............. AIII ..... Ch~I.Iot". oentCRCn .bow lum in,'Clti· galtng mathematical $p3.oc .. a hody ,..hich hold. i" hisro.y ,n the ""'Y thaI it hu been m".«.1 by iClIN 0' oiJno of muttl.uon. Such .. <JW>d< .. <en n::quire lhal add .. "", or ..... rom" of abslrilC1ion "'h«h CII;\I~I.t ....r ... 10 U PfOlitlH:oca.

Tbc d ........ ol lma~ , .... .;:0,. lhal ......... nc opaa: bas IIOt >Imply been JIn'O",1\ed '" Inlmll.ed .hroup _plio< bul .. honda .... ,llaUy _."eeI .. I h.'",. enu.y, ~or .um!*. "hradoy "' .... 100 ~JII .0 .... ekctIOFC><'''''nc ~pao!' ,Itro/I' (1611) OIlier 1I111ft ml.I\emI."", II ponra)'ed .. a ooIl«t"", of n,.w.,h., hlv. b«c><' ''' .... "'y lUlu, ... Tilt mcupho< AI ....... ha. ChI,.kl'. 1"'0 ....... IIIIIY 110 ",l"funy .ppro;c.dwd of as a Inlhrorok>iY of mlllltmlllcs. ChlItdcl'. me.a~""" hardly malk 110l\'1li ht1 ....... IIICUJIIOn of IInIOl!'''' p1oml" CUllY inl<l • domaIn of IJIIqCleu Iotoc ht!! "'" 10ppoMcd b)' ,IK: II'III1lpllol') .ha' ha • ., comaowJy been emplo)'ed wnh,n ma.hcma.ocal p .. a~. ~or UI"'· pie. .. ·hm Ampirc • ... ".mul .. ·1 concI ..... or folded lnlo a be],,'10< lile rnaanec· ilod b,", ,,,,, ... _ of IJw """ II' II bota· (ljSl "mpb.: lunudf doo""beo bOo ...... Il ... n a ,abl) 10 wh",b I ,,"ft pnn lhe ....". e!KUodynanuc 1O .... 0Id. r....,. .1It Greet word .... I ""~ .. ''''*' IIloCIInl"l nt"1!( II ptea ... ly ,ha, .. h.:h ....... pod lik •• c,,"nnd. lha. II 10 $II) .1It ",.rate of tbo Iorm .... .. ·hM:b ¥iJ lhe CIrculi .... 10 110 round.· (IH) L"".~'I d,,,,,,,",,,,, Ind .... of mc'lPI>o< J/M:I .... lhal lit Ihlnkl of m.uphar

... f<llm of Icchn<>lo&Y and 1"'. l1li1" .""n mom prcci ... ly ... kInd of ~plWlI """1f,,kllO&' .\\ •• hooid speak of u IKImoI"tJ 0/ 1M m<wp#oor 1"", pOf!m = an ulnllOflll)df ""'" and ",«Cdet fOlll\llh>alioll.: (117) Chll."'I'. 01>00"11';"" "" ..... leva"", IIOt only foO' bO'oad d,_.,1OfII of mwcphor ~ .... __ bul IhQUId 110 .. Latcd 10 nu""' ........ IlUI,hlJ I'f'O"MIcd I."", I:;unou$ CIIK h .. ,oneI m .lie ~~'OI)' olocomcco Fo •• .amrk'. ('bI1 .... •• on .. ,J", po" furtbtr .... '''un. 10 .... ,oaUO'"' made bl· C .... m I. bu woR .... lhe ~ ~~pm""'OIl 01 '.:oIai 01 C-. I. They abo pro.">CIe I ..... M 10< appr""''''''' La""". npl.;:o· lion 01 [UlIII .. O"' pm -nUIIOfI 0( I~ l'11mry of Spccial Rdat .. '"y, ,. "bo.·~ al~ ..... taphor ... .,hICIe for Ih",ll~' a .... on ...nlm _ r.,. "h~nnllllt "",y liRe .hlnits. Ralher I""" belnl limply ooIlcclrd .nd (Onlfolled .1 • .",.,;. onela· pbe. pro\ldc:l a mOlln' for Imul!in," .1111 r.'ol .. n~ id(l. ·w. m .... 1~ ... fo", dliW"or iOI!IC """bod 0( '.'·<'5li",IIO<1 .. h,ch allow, ,be mind KI •• ory >l(P ,0 Lay hold of. dca. phyllCll <IOIIC<pi"'lI. .·ilhoUl bciol cornmimd 10 ony (/loOt)' founded Of! lhe phY-' $CXIIDC from .hlCb ,""-I .......... pllOII i, ""'ro,,·cd. IjO

,,,,", 11 " 1otI,t..rdran __ ""'" lhe "' .... In pu1"IILIH of anaJ)licll ... ~kI ..... """ o;..,1t'd bq...-l the uu,h b)' a fnon~ 11)"",--.' (Mu • ..u .. "lI01ed by Chlcl ..... 1811

CONCLc.;SION

In "" ... III>ion. I w.nl 10 """'anile 11K: multIple .... Y' IhOi Cbi~"'f. boo~ rocnplcn .. n .. lhe inlcmlfied work Oft .·i> ... I~uOll llIal acxompanIN Oil' c!>l. 111 .. 1 WlpI"'" of multIple .. rten '''''~ In ",p.d 10 malllcma,,,,,- m""h d __ W""""",6.11ht peLla"""",1 unr""",,,n ... ,,,,,, of PI ................. as Malho""'uca\ ..... C"o'UI ....... lho muJ"pIo 1000po"I= fa< ",p&onq ~ round III.lie """'l ol_liDI:a ..... ' ........ , .... 1Ir ,,1oou. """'1'11- IKh""""u" frae ....

....... ,'1 would "" • ., """,,,nod lma"nod Of ,",,,,bIe. n.. redl. mel , ol mal ..... ""' ...... In npenmm.alockl or II In Importanl ""P/.:a1lO1i 01 tbooe ....... ,". l<doaolopcs. 8u. as ('bI,.1eI .ho.~ lhe ~~per;II>C1\ ... 1 ~'u.., of malllcmal"'" >hnuld ""''III)' Ix thou,ln ol .. confined !o do<lfOII'" .'1"Ironm<nll. CbJ ..... ~ bu<II; LJ """"nanl boa .... 11 demDflil .. ..,. Ito .. 1110 nptnmcnul nalu", of ..... ,ho:m~.ia can 110 ...,., •. t"md f"'Ill dlajram. "I!»"~ in lbe bi .. Of}' of ma,hematics. Finally. 110 .. '."" •. 11 LJ nO! "n&le dl.i""m. Ibl CMI.kl ",OOJd h, ...... 0« bu. 1/1. "' ...... . hey pro.ole fo •• "".1110 ....... d...",...... F""klnc ~p;!(:<: ..... Ito"· .... b SJ*Y In lum bcwmcs uoed ... poa lor llunki .... II .. he", lhal Chi ..... ·• ",e!H' ,nvol..,. IIOt limply'll ..... """ but ~ .... Il Ilk Oludy ol dIaJram. ", ... b,a ow emlYUIJ ...... "ft "*0 ..... ", Ch.I!dct .... ply d<moOlOl"' .... JIIdt lIudy ulu"",!eIy caa"", be ratncud 10 I .II"pe dlKlpb ... or dorn:!ia. 'TIom: " .n u'em' need lor philooopby 10 pmtt .. '" f~,"'n Inlo ,ho >111;11" he ... lhe fiaum lnanl conlcmpll'ioa and lbe formulx KI".h.un. op<:r;"iono 01 doo ....... ,n,u"'" . r. Inicullllt,d 8uI lne... "'pre><1I' a jU.I .... pi""" alnoady lYum~ by 'wo olher rom""l~nl' on lhe ballld.OIII ol the ob>cu"" Irt ond scicnoo. II •••• l\'III\aphnr !>qln, lhe pr"""'" of ~l"i IU ~kin Ihal .. m mt1amorphOIt It Inlt) oreralloll. Ind h ... " i. ,hal Ihis I>OOIr ...... nn. ,,"h ""<lib ,h.t I1nl., 10 in"ll. u' '0 vic ... 1 redl"""''"'' Op<:r.Ui.lly.' (I 9) It iJ Chilclc!'. oo:co"'pl'slu ....... ".,. ""Iy In lui .... nU .... book nch _llh 1fII1&I1" .boul pomnnc spao;e bill '0 ... .., ·~I_oeml npcraln"i and iD JO

doon, 10 II!I!ft ddincd • c:nliclllllO''CnIoCn1 fmno dJqf1.nl. 1<> d""pDIIID'I!ics.

l'iOTf..S

' .uc ...... ""' .. 0-.. CW ...... , ....... s,.... ", ,I, '" ....... ,.. ..... I~ ", ....... k' I .... _!11m.. ''''~ __ N ...... Lt. _ ... __ __ ....... ,._. ~I '.n~ ULI_ do. s..iI. '")1 's.. 000lI0 .... <k I. ....... Joo. <k M ...... '" ~_"Io """. ~ ... 0.1 1<1<, I~ (~.m U ......... II_CIoo.ol_ '''''1 ' . _ A ___ ".. .. 1 •• I.wy lIo .. IC I .. .t.tlr_I_E6ttaftI • . r-.r_Eo;' 01 .~_ o •• [0"" ......... , ... ' ••• 10 'm.e ....... """l'r-.l""-.,.ToRlo.l.fi" _, .. _ ..... ..., ....... .,._t~yoR _"'I ""t.'. (", ... D,.. ' .: IT ... F , .. , .. , F •• M"_ Ii,' 'Ito .... I " . -- , 1<"1 I " Mrr _ ''''~ h) Iltol'ld _, "_ ~"'" ,.,. (" I ,. H"..... ... ..... .,..

H-, '" ......... l1tolloololt. """ Jono:r f.rh_ ""~ '·J.1 ,""ICIocII."""" r-. I~L .... ~"'-'""J.H&l ...... n..~ ... .,' .. ~K I 1,.MrT_ 1J90, I_I H ..... ~ , .1Ii .... ""'-:So<-' "'-.1'_"., <1 .... """',_1_ ~"'\ .... 1< •• \ .. 1, "" .. ~, , ...... "'.-,.00 M!tIO,/j'"'f'l-, _ of ... ...-_"',. ' •• , •• .. 'II"'_I. .... <lT_ .. ~_"ttrJ..,!J!.l_lIo_ s I. ,rw. "oR Hill "'"' ,,_ ,"4 [ ' .... jk (l ., I.M<II\ .~ M .... J I • "" I I ..... u...."o • L_.<I~_,tIf)~ ...... CW "1-..,_ • __ c_ ., ... "'''' ..... [U •• L.,<I~I'r-.'..,II: ___ a I _C..-_ ""_Sc_.", I ....... ,. I .. I",,~l~'-" I ,F .St .... So .. r ..... ""' ... U.h.,I',fC .'_1 .......... ' I." •• "-,",,

'1.- .. _"..,1 ... 7 7 ....... "' ... oI. ... II: __ ...... ~aI .. ,

.... M ..... ' "-"''''_. lot_ ::' dl"''' ., .... __ .$ ".,.looo 1 I ._ ...... " 7 ' .. ' .... _111 t.o r __ c ...... lMt' ."'_I""~III 'S'~ ..... DJI, __ h~Ioo<l< .. '. ... __ , .... (_._...,n.,.",~ ........ 4 ,M "

..- <If \I " "- "U [I.' ..... RIo Gooolari, " ..... ,, ' ,, 'r":_ v-. Cd b,u.._'''''', .. " ..... '_T , .. ~ ... t ..... _

.... \' ___ .- .. " " r'~,v-o.,"''' , 5'" ~'IOw' De, """'_ ..... (n. _ MoO"" 0-_, ,...lJ., '!I9H JI9 .n ... _ ... ,u .. ' , .. .- .. l..-I, ........ 'Polo.u. ~ ..... u •• _'''''I • _-.. \"''7'2 .. .ad AIoJo C_ C_ ... _ .. \loot ,II ........ ,_.. ....

II~ ,. """" .. u .......... t"'[l ... Jl-".~ ••• _ ·1 •• y_~I"' .. .u."' __ ~I'I'.' ' > ... ,.._ ..... _,..1 __ 1o_._ ...... OJI_ .. _-"" ... _ ....... prioMr)'-.toro. __

1 ... " • ."kod .. aI ........... _ ~_..t, .. ! •• '... ''<0 _ , 'W) """'" ,_, "'_looo~ .. 0' ! .. _ 100' '' ' ..... ' ..... tLO'III" "J' .. ,_Ql ... " ..... ... h ~ .. __ ,.-. ."""",Iy, I .... _Ud .. Moo_5

, _ .Mk 1Iio_~ _'. " .. po II .... "lot. 01. ,'7 7 ,,' 1 .. .. <If ,..." .. _ .... , .,. ........ '_oor __ ... __ oa.w,_bII_."" 't, 7 AI,. rzI.I ... , ' ........ ..-01 .... _ .... ApIoo, " •• '_' • ..,.._ .... _: ' Ie SooIt_JII: , ''. ... ' .. _s..._ ........ _·.Tlo' .... _ ...... ' .," .',,'. __ "" .... _f f_ ... ...-U .... . I _do' K_ .... ' r.th' .~J",\,I'" ;m, __ ~Ir_ .... -,..., « "ZT" " '*'I_'~I" ' .. r 1t:_,"""'lJ l __ _ l'. ,a ..... " :"'l .1 6I:'llItf ": afC-: h'C, .... :..n-.".,.IoC ___ C ....... II,. ,. .. '" _ •• __ 10 ..-V_ofCkop ,....,1"H:QI).:12;."." .......... ". ..... af) ... 1 __ '. r. ' ..... ' af_ 01 "'_ (1,.:00,. ,. PI ' .. ~ofO-<..to, r .. I"". rr ... ·'-I_Oott.b.r:.. ... H_~~.,'" "' ............. _'_1_"'_ U.,~ IA( :":1, • "[-)1 .............. oil .... '. !o.o~<oI """" 50 ~ !u.o ...... 1onlJ .... "" ... :-.: ...... _

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"500 1'.-(_, n. I ,. '1 '_""'C_1o 1I< ••• _ •• "r.,'~.'.:,,-~, •. ".,_,_ 0I,,-,t. ... ,,-,m [1911 ], 7 ... '

' !i«_'-. A'+:',,,~ 01 '. -.......... ,;_. ',.,,"-" _ , ~'... • ..... Il00 ... '''' '_01,,,, .. , .... , ... _'-" .... ___ • .F.' .1' _, ... "'." !"_oj'" ~(~ l"". P ·w, IMlllrt III

For Bela-Andrcas HcnlSch, lhc frlcnd who taught me so much

ACK l'oOWLEDGE\II:NTS

My Ihan" 110 10 II<m.rd u.~notr. Gilb IXI.uu. Jtan-TOUJ.Woinl o.~nlJ and to I"" mllCh-m.SJOd HI,-, Guanan. w/w) ~" ..... hntd me to p""OS<Iplty

To lbooc ... ..., b ..... uppontd .... Ibmugh a period of UI........, d.,.,nay [no:

All,.", Fra_ .nd Eum ... 1Ia~b.ar. "hrti ... Bun\., Palnok Balldtt. {l",.uan IIeIoYllIe.. my ... 1 .... (do ... ., BourstJ'Q, J.eq_ COl"'. Hen.! COUCTJOIl. Aud~ C.,.. ~h<bcl Crn«>k. M"", Oamoo\'fI. l)omilUQ .... Oosanu. A"""-clam" ... .ud Ph,~~ I)eton! •. A",,,,,,,, Oulors, Jotl Dupo<lt. CtiUllor I:f,~n .... Stow!~'" 11:.,. Ginludoll, Ohord Gnne\, IItlCno 1Iu&a. M and ,\I .... lo<:au, ~rI"C<l" Loud<nbach. Domin"! .... Lco\Irt, t;hoabelh LWI",", Ciror,.". JOCII .... h"" and V",o"iq'" unglel. Cia". Plmet. Ala,,, f'roch,anll'. ,\I)'nam R •• aull o(i'AUonneo, Klau . Rnlo,. 8 isabelh IlIpJ. Mmc Roq ... rnrt_Vil ...... uv., JOC<! .... II nuhaud, And~ &:.Ia, ReM ScM~" Jnn-i.le'ni. StIne .. M ... 1\1 .... Guy Sorman. and ""'") fnend. al 1M Col .... in~""'lional de pIt,lnooph .. " ' /w) ha,~ u.o.,'" me 'h<>' .ymfllllhy,

Tn M .... Flann, for btlnl Ible to make me Inend 10 .. hal ,,,,,", is of pIIf$ICI UI ..... 1 ........ '- and 10 HI'oId lI ot.enber~ .fIo fIII' .... lly pidonI my .. on. 01\ ddl'ermllalIOJlOloly,

To J ..... ..a.ud< Loot .. ~. fot 10 .. loyal fnmd""r To Fra_ BalJbu, Pllnd! llaudoot. Ilan .. IIIefl"Cl!u"" Thibault Damour.

An ... Clui.ulM DacoIa. f'h,,,PI'" 1:leoooIa. Fra~ .. lal"knbkh. And..! Scali. Ind Jean-Jac" """ SlCl:OCln,arz. for lJoe.. s"mlliauni ron"~,,,,\IDn.

I also owe I .,eal deb!. rn. 110." f, .. OO,h,1' Ind ~n""U"'F""'nl. 10 a..rIco Allun'. Alain !of""'" Alain Ibd",1I, Jofl IIll1ro. l_1tO Il0l. lIa,ha", CaSsin. Ahun Chenrillc:r. CalMn ... C"", .. II<y. "lain C""-. Jun-Cloudc C"'I ..... A",y !);than. SI/-pba ... 1X1 ........... .Inn DltomR La",.nl [}up<>!. RlptIlIlI t>ouody. Dorruniq .... ~lammt. M Ind M .... Gadoll"~ l..oond Godart,l'o....w GwJbnt, MdId IlennaD., leu·Pirm K""" .... .koJt. t rall9lKl " ••• cPD., A1aUl La<1Oo .. kan-Picl1t Ldeb, ...... And~ lIch"...,..'ta, f""Q! .......... .Ico.a I'etnOl, Qu,,,lal ROlnnlt. Jean·~t K:"'" S.Lom.k ... Jean s..n."ltn, Picr'" Sch&po ... M.n: Sch"I",n~,. HoufJll SIna<eU'. MK:bcl SI"btd.~ Pa.cal To"y. U.rnard T.,,,,,,,r. 11:......, Thom. l nn-Otd .. , V"""nl. I'. ul V.rilio I nd.n Ittoo. ~ho h&ve like" pan in lbe R.nconllfl pIt;k>sop~~nn:o.

U'"

TilE! Ll8ERATION or TilE orSTURt AND THE; 81AS OF TI l E VISiBlE

To ....... hIIl .,...10,,,,,, 11"e: IhIIl Ii pe. l\al'" "hili "'VPI- II. WIwc,·e. ""-CO malbemalla. .. "II .. pcne ...... someiLoN .. _"Ir.\.adin. Gilles Chllelet'l bool.. and "tK",,' •• d .... 1101 ma~ d""",·« a fta>on I<> I,,.. lhom In li>e po..hl:UW "~y lhey hII,·. III aSiliminJ! atluude> ond mol;"n "01 il "'111 boo ........ 1)' (0 Ie~.n 10 Ihink \);a(:k.i.rdl. 'aplnil lhe ... "n·. 10 JrnUh Ih.e muillplc ""mobile ~11t in "'hkh I"" 'm.lbemalJcal Ih'Q,' .. deh"u..d. sohd ~OO oonaealcd, 1X1,,'.red Ioke a "'lul.1..d ob,..a of Ie .... 'n .. sohd Iole an Irdlll""I"'" of DDfI·

C.pll, oonFaJed bke • I),""" of ",boo<lCn' ","'Ief" In lhon.. prodllCl pad:a,.,t and II""IIN, ready 10. CO"'luml""'1L

Slop amlllm" .. rot • ""hik Ig,.",d lhe ord,....,. """' .... of lhe ",Ie$, lIy u.e card. of Ih ......... Qui <)II lhe table. 11= lhey .'" he""dQrth 'ouUod< lhe prne'. lhe)' h,o, ... _ ...... d ",10 a f"'" "alcarw;l .... yet ,,,II "$Obk.lhae ·card.'lhil "'-c ""VI! learned 10 deliK and .................... "WI> of potnl', 'o<d, ... ,)' 1eI0',

'.y1JaIU 01 numbcn'. 'sJl'lCl"'. etc. h-.c, buo by 110 ........ UOttI. OulSOdc lhe P""" wlib '" predeicmu_t ,uie:l, Ibq ....... 00 10 boo 1.1 ..... "aka and oft"ct 1I ..... oel,..,. up 10 II", ntko ' .... 1 'IIo:y 1'"""""1"" .......... Wllboul n .. , wbm,lI'''1 10 1lII)' .u,.""ly 01 VJOIcnce. If I",,", ito vi<>Ience, II. IiCCd '\ ,n IhmI .Ion«.

Al .oel il _ II) me Ihal lhis io ,1'1"", Gil"'- CMln.fl dotttaOO_1I15 of: b-reak up,IIo: rndy-made pmn: 10 ''''JHUe the rit.tna up of lhe Ih'lIi , ... If Ihat lhost pm.. ba ... been "o"",,~ w,lh bUI whICh IMy havc ended up m.ukln. In I""', .,.U.()rdtl'Cd and H'fY ta"II)'ln, o«um ulallOll The 'th,na llieir' Th ... 'J'I"'UI()n <I<>Cd$ '0 Ioc ... m,n«i. Ao "",.ad(>.ical .,.1 .ppca ........ ~ mu" rod ,I of.U ',h,nl.J,k.· 1'fO'Uf'P""IJOG$. No,,,,,,lly. 'th,n,' tkoo .... I"" "bar II lhere. aln'lldy """"ely lieu'''' ....... Iq a c'-'<land unt:ban",.. n>l"" .. r form. \ 'ou gon peon' 10 i~ and " ;. .11 .... y ....... ,"" dew.n:1I1On. '"""",,' .• i II .. ""Ilcd . ...,...nobie and oep&nllcd. La", Iho$ ... uoJ _ ... and thmk "'Iher 01 tho: I" ... OC«i. .. """ IU • .., c::aa Ir.UIIIale by 'thJn.~ ' .... h ... ad",\13 .. and try 10 Iboa1 01 llIen! .0I<thn, The lin! dnopa'eII ,1Ia1 Wlldl n made .. 01: tho: ...:tiiid thai . '11..:11 RlqUl .... all _ and ... pondl 10 'l If.., .,..... 10 1''''' lbe n:lme 'thon"udl" 10 thai wb"'" ...". .... , .. ,be 1"0 1<JFlbn, "~ <:aD .. y lhal by ,IUJ _ .. u ....... 'ood thaI " 'hodt .. ,,, ....... and ""I< ou, on""" '" motl(ln: Ihul.'h.e """ lor .. 'boo:,..,. ......... ,n i" lbe ... 1oJ ... 1IoevcT bralloa.. ,be tanh fOf wboeve, place:< thci. Ioc, 011" In . """.," ... y. ,be 1bi,.. nIdi" .. II .... ,., ,nh)btud by .,IIM ..... is mnto'med ... ,," tl. Ii manilDu i .. elf .. ito

.u · • ......,'u·, and the bin .. fin,h IIf10lf immeDCd In L~'" 10 'fleIk. II Ii lhw thaI lon, ... ,n I f."""" •• -nx ('th .. ",h1 and u .. l<'1K:e In: ,be A .... Ib"'11 Pi ........ ...., said of linn, thou 11 II the ek;"",iil a( lbotipl. '" lhe ...... war lhal the .. r .. 1he ok" .. "t of the bud, lbe ... lbal of the ..... rn ....... And tlus ""me P...-nKlel added ,,,, ,tboot the bc1nl .. '''""' II' uP"'''''' .. kx:ated. )011 couJd 110' aIId lbou"'t '

To f..c II •• "~YI! capK11y !the ~ II> UK ,he .. on! con..."..,N by IndlUOOIf rallhnl ,n llot Ihll', ,...,If ,,·be •• ma,lIrnI&lIcal .e:l" 'I) i. Immcned. Ib,. ~ 11 5ft'nI> 10 me. (iillr$ Chl.!o,Iet', proJCCI. r rom Ih .. folio .... the look. 1l!"I"' htr ....... thaI be _to ''''''''nh the '!uslo' f. in many way. common !O .1I1hlW, ofmalhoomOln ol phy.'"'! I nd or melaph)~ From Ihl> .I~ follow lhe ,CtIUl,kable p,,,,od. th.1 he pICk' Out and te .. hict\ he: dc.met h,m'lelf: Or~ and lIN: Imllw./o fonow""" (1 ,," ute""",,) of klnne): Arpnd and lhe unfold!n. of the rom", .. pLo~: Kin' and lhe 'lU\u •• hw .. ' of ntpllve ""aall;' ,...,...'; ~.",d.y Ind lhe...,.".,ry ,mma"..,"" _,""tit III:kIs; G ........ II".oo lhe jourllj'y from the pola r 10 the .. oaI II ,1>0 tef(I ol.n 1WokIJ"I of duntMiom.:

HanuhOtl ud 1hc 1",,","l'1Inee of an al¥t>n c:apoo.bk 01 "~". lhe: I'unm 01 outh," lll1fok1UIJ: Sdldba, (and ~ ,,""lalI1 ROIIUJIlic o&aJ ..... , as lhe: bn"lIa. 10 10&101 01 lhe: "nllli ralin ... lhat ....,..111« tbe IIQII""" "" ul nallU'C a,.{ the: IlIlt11eo;L

ThoM choK .. In "'" athol .. ..,., Nor an tl>o)' 1M mllll 01. pn>:UtIOOI",d lheI# on the: potrt ol GIL.., Ch.Iilc1et repro'lIJI the 00l.ll'W (or 1"1«oIo>c1 0I1i>t 'lm1Oof of OIWllhmutlQ. They lIIIi<"Led tt.,,,, .. tVfl 10 bUll.,' ...... 11 allhe ~nd IIlOtUn of hI! pano<ula .... "".itnoo as. m.lllemalQn philmophtr The} .. prosl hit u" ... "tohhl< .... y 01 ~ndlllJl hlm~r ,mrntno:d ,n Ibc: 'thll" jUrIt'. wh..::h dtmand. hi. """laU" .. I ... , c:a~ thaI In .. h..::h he In.,,;1< him .. lf p' Ulopally as.:""" "'"'" «loon (as. 'livIIIJI enol ure mdowcd .nlh Inldlo, :!'. ..... ""Ihl U)·. 10 nPUt "'th minor modllkauon Anll01~'1 .rld, .. o" .1 p,,,,,,,,i. Ii""). INU >I to say In lhe ...... ,It.cm~t iCllI . lemt,,!'

'In 'bted' JhO\ikl bc: lak"n M IT. il_ IO IfIC.ln UO tlI'Ofli<1lltnoe: in"',t....t boddy. In lhe pUce ",11(0 ... "" .. dl,...,,11 , .... n ed Inlo 111(0 hean 01 tbc: ""ibk. -rbe ",nllll demand.! 1he: Flt lln,: "" .. nl .... 11..., .. ..,....I h'n. to mah ... pot_ tor. _"I. nu. io .ho:n! ,11(0 en .. oftbe .... n .. IE in the: ullbmohbk coouoec\JOII 01 Lbo panxular bod)" las 111(0 IIOC!d of mouonL !he vin .... 1 aad ""_ ...

AIIIl.IlI (act. IbIS monad", body ... hodt.", call'ou, ........ Ii al .... ,.. .nmdtd 1»-tho mobokhoruooo 01 i" .. n ...... , ... Iovc:rytlu ... ha""""" tOr thai body IS if. conqo of ....",tly bo<l .... all ""lIIlly 110 0W1l. ahno,.. r..Ito..'ed jl .nd at..-"Y' pna:ded II ...... \1 ....... 1 11 ........ bk places of """potu"". iIttOrdln, to. r<>no oIl~a. ""I whocb .. ""(Ok!ed o"thln lI.nd by It. ThtoJ. I """'" Ihrou'" Ihe """"j 01. r.mll .. , t ........ A cro ... d oUrJoo...u one: Olhen bvd... ..110 also mo,',"" I !lee _ 0( them f,om tlx bad.: OIbr" in profile; ctll(ou race on. In tum Ihty 5H me. Or al kaot I beha'", '" ,hou.h I .. ' ••• -'«1l by Ihtn\. In my tum. I couk! put my .. l! "hen I _ lbern. in 1100 pLKO Ihat ,ht)- D<Cupltd ~nd

r,,,,,, "balMy ..... ""'- Tha, Uq ooukl ha." 1OC1I .... In lbo:: "1l} lhat I r;o.'" ,hem !II In..:nbeil .D 1100 .. .,.ulu mode ol .. mc ..... of lIlY body_ In IlIiIlIlCJtK, my bvd).(O •• lbe-<llhtr. my body .. bod! is .';';b!c frono .n c:Isn bc:rc. .. """."' ... 'I'''' of.1oo ron.. 01 po<Sd4 for mc: 01 the: bod) IMI I nod "" ..... So milCh loa

tMI In Ih .. """procal ida""" 01 ~D' .he: .. tc .. lint! lkh ""I"'" ;" m..:rib<d al the r"", ... 1 I ....... '" 1100 othc:r ..... ;nllll "I'. lho sip oIa possible p"""""'O. " 'bich roukl.1so, I .... SlailCh as;t is Ihlol of anothn-.Itidc: hi< 0'",,, The ._y_

mDII> ""' '''' .... lily oil"" Otlotr aho-.)"I inhlobit' lho mo.k oIJIR'SdI<" 01 all l"'i1lcul •• boditl; arwJ prinllfllnt! the ma,k 01 •• ';nual ubtquuy ... ,tl>ool ... hich no 'poIilionl", in s.pace' "oold be poMll:>i<

II i. in thlJ IIonzon ... hleh ;1 a l ... ays in pro<;CS$ of 00111 SpIt«d 001. Ibat tho moo.:. 01 objectification an ronlti\tl{ed "«'ord ln. til " hkh ... " I.am 10 deal "llh Ih, 'corporeal Ihinll'·. al .... )'. a5S>1"ed 10 the .. rtua~ty 01 tilt" sil", tb.ot OIl' IhooIUy bodon Inhab!!. fillUI'CI of tbe ,'irlnal ubiquilY i .. nal. to the 1"'.lJeUlu body of ,,,,,II. If ." alR'C to Ii'''' 111(0 name 01'1«<1 01 d",l«tio:' 1o .ny ""'''K'CIlon of dilfle_ ....... ;U .. y 01 Illy rdliloOll bn ... wn >m""hlits thai it """"Ii· Ma IlICh a 1ft<I. (,oo""'l"""lIy •• ny 'tho ..... bowe_ IJIIJDObok mel inert 11 .... y appear. ron ..... 01 !he uaity 011 .. "(> "'inllllttid' thaI of III Slit rdlil!'" to .notbtr ltu".: """ WI 01 iuclf rcl.au." 1(> ,IS tUe. 1loc 'von1lalt1lCf' «11<1 ..... _btr aDd.,n IfUlI!. c:acb ,ndiClltc:l lbo:: Olbtr lUId Ibc:rcrOn! IIWfltamo IUd(

only Ibrau.., llu1 ... """" .. Itd dilJ",."" .. I rOIII Ihis il 1QIJo...'!i lboal • p<>iftl .. (untluntmally dlMlUlClia ..

If I ... 'ntot'- OIraIIF .. ortlS.1I is bccltu"" I an"", my .. lfto bt carried alan. by tll(o dn;am. lbon Gilles C"bJ,ckt"1 "0.1 hu prodOKtd In ","" Indeed, I d~amcd ...... 1 ~al in tho: roll ... of my l'tadlna: I Pf"l""ltd my>ICH ;010 ml pha nlom bodie5; I .. -.oJ Am¢re', ·bonbn,.._·: I " lI. lhe Inhabt lant 01 [,,,,I<on'$ loll; I "lI~ ID '","cnf ;n M lnko ... ~kl ·' COlI<. t- ow. I ...... ct ually ne"hen other Ihan on my . hair in front of m)' table. Whal hi'" I dilit:o\C'mlI"'n. if 001 thi.: l he: oPt" tidd 011'" ",nllli uboql1ily of my Own body ... 'uhout .. Iuoh ,he 1I"<>rld il .. 1( .. ould be bill am.:"", and .'Ihout .. b..::b ronJeq"""tly 1MIlhln, thai .. e <all 'malhemalical" ... 'ook! .n, "''''' 00"", 10 ....

I'o"It.m I lInldxd nadonJ. I ..-.s ,It II dramln .. or Kant and 01 llul '6"10" 01 ... hom be .".aks. .'he ...... ). "'>-e dcmorul .. ltd the ...,..!elInlID .... ·. 01 H\IISeTI abe> .... hI> "proIopooo"" ....... And I .. Ired IIl)1ldf the q""lLOD; bow .;an I ' .......... Ib .. 1h .... ., III""'" do I pIaoc hI"'! And In...tw work!"' I do 110(_"" ~y dol.,. caJcu.b..-... Of ~ • bony IUrwyor, lithe,. __ 01 .... u .... Sln,n, ,n r""'l of !he In. u.nder III nit,,, Of fit trw. He....,.,1d boa>" .It>ottl btf .l· ... nd loqulllo dll!&m; 10 IntYd In the "nual'"I"."" 01 """ .. ,om bod .... Tbc: ,Iran .. l traces .. ouId bccOillC .,.,bIc t(> hIm. C~ """",-..1. Iinp. knou. iUplura, retraruop. ronoin~ arwJ roln ... I I, ,,'ould deliah' In tho: III*Il<k. ",,,,n .. "hin hif ""'Iionlc>. body, II" "oukl open hi. <f"" ~ would .... lhe lipt hlllt ... tho: sa<l<l and, "'lb hI> An ....... ·"ukl tnt ... I .... re O(Imo: Ii .... In ordt, to relaln """",hln, or Ihi. sant>.o<l<l I' .. ~pf'" would th,nk ,hal he

\

'.UII

IYd oIcpt and dtnnvd lIul. II""", ... I n. he .. ould I'C)OOCC a111l1c oacm tor , ..... T ~ .." .-or '* """ _i. 10< ,he dream _IIa ffOal Ze ... ·. The JOUrney. be .. ould IllInk, II """" real,,,,,,, "''''''' h&hL

Thai II ""'" Cbl,elet kado .... '" """" ... ,he 'liM' JICOmCIC. Whocb It ....... p lo ..... h""" ",,.,ICc I<> baromc 1 mI'hcma'icu. ... INT RODUCTIO N

Th"1><.><>\.. ,,·ill ll}" '<1 dr. ... lile i'iIOf\lJ 011 vai>devllk pll~ '" nh . hrtt protalO'lisu .1Ie ,....lhcrn • • iciaD. 1M ph~. Ind.1Ie phi"""'pba - "''''* dl'P"la ha,~

11IIII>Iltd ""rious dtba,ts aboul idea, rOf .,,0 =.u~ Ii " ""'. ,ha. III the 1 ... 'Ic.c:adt ,lunp ha", to kCD I doff ....... 1 ..... pc. Out ,brtt prota"""." would __ porbaflll totsolliiC a bult,."..... bd"on: tlklll, up lbar Irtdm"""l lIlnudos.

The ma,hmIa' laan m"", not be so "illin& 10 play IlIc fic-m: defender of .he au iO"""'~ 01 "ruc1Urcs, oo" ..... p'IMIui1y '&""fln, .be "Old .... from Id .. pen. "",,,ill ·, .. h,y' lhal ,,'''10k! ",akc hOi 'p<all.ollOn' .... "I>1c_

The .... ockins phy5i<;1\t' ... ould «t1 >lnly chnlln. " &hlly 10 lhe racik ",,"aln· ,..,. of. duali,m bel"'''n 'modd' and ·",.h,y'. N<>I.., lonll .",. Ib .... ' .... , no " .... 1;.," or lhe lick! "Of~" .. · •• lin. lime pIIi)i", K.OUnd "nh lhe preny "''''''ura. 100 'lOph",oolltd 10 be found ill Na""''-. """'~Itn. f.on! ..... ,hm\a. Iici:II.· braI .... ~·Of ""'"1 ",dlnal) prvI""""'" of Ib< .,:an JCl(' ;e.. _Ibemal. if;s "'-al ",,11' "1001'_ 8uI .. b)' doc. ' .... tali .... "'" 10 .. ~Jf' II "'Bllk! Ippear th.n IhlS queRoon II 1>0 10 ....... 0lIl) I "",Iter of IniCra, for Ib< _hCIa of phll ..... ""y Wipe •. lono.'",n. II",oenberi aod man)' Olhen .... ould II." cnt:aJnIy lui, .. \.IIuMod lhe 'pn.JIII.l"lnl' of OIl' 0«111111)' ,....."K><W: .. Wlui, halld i, hoIdl ". ,he ,00/1 WIui, Jellumo I~ ,nwh'Od i" 'hi> 'IUll<:lIOIIIn.' ... hlCh esub-li>ha I hi&h<r fomo of ",n" nuily bel .... n ·.tructun:· and • .... In)"~ II il .. mply I ClI~ of ·apply ... ,· ma,b<"'~I "''J 10 p!o)'~"" o. r~'hr. of aWllkcnlnll,hf ph,-,"""

in malb<rnal""" Indeed, Ibc~ " 4UII •• nough 10 C'llpc' "1< ovm ,he "'00' pal .. ", or under>'aodo"", lbe , numplo of ,he "pcra,IO<:tQloidaJy ",.'",bIt Ind ,he ·Ippba.tioa· wry quooUy fo><pU 1b< ..... 1lfC> ,hi. II """"'''--

Whodl .... _.M pIuIooophc •. ,,"" IS 11 .. .,.)'1 .... pcacd of bftnt lhe ~y III ,be OIn' ...... t.. H_~. be """""" p ... up 11)"'1 10 hIi.-e lEw bcr.e/i, 01 ,be Ia>i word Id lOY 1;3>11; be 110, ... hi.,,,, 10 "",k. up hit "'Ind. "hod! .... b)' b< 'i IiCIC\UCd of ahandono", hl", .. lr '00 rad,I)" 10 lazy mcdl,.,K>n. 00 .he 'prol>....... ,"'. and ,",~n of on ... _mbln.,o lbe 0<'<111<110,," of bemw""U\J(1. Docs he pndo him-"'lf Of! ful\d:r,mtnlall" 0 0 .""h occasiono he il " &lolly ~pf<>ac"hcd for ",,,i,,. up 10"" and .... ,bcm.lica, fOf mlucinl ,he lane. ,,,. formal pm< ond ""IIf", .. ", ;. IiI'llh ,b< ad"';ni .... ,oon ol proof< by leiun. h,,,,,,,lf bcromo Inl,,,oca,td .... ' h.b< ","a,n'''''of 'lO'U.

I. t.hould he ack"""'Itd.,ed ,"", lhe p/ll1osopbo'-' ,Ill< 1$ not at ,he .. "''''''''"

,

,

, ISTlOOUCTlO'I

;1 i. d,i'lkull !O d" ....... ~F «1,,= ,he 'lmtaphoncal' ,Old ,'''' 'op<nni<>!IlIr. H~ ""II thererore f«I the pt""u"" of the eombil'l«l ""- of In .nd ~ ond ..,u be lhe ob)«'l of,,,., cppcml I_pea""",", 10 """ """ .. It <)\Iff .<> lWUJllDI ... O~ 11.,1.>111& abtohed I\uIddf of II>< m;ponIoibility of.be pluml .... ..,i .... 11OfI of.Io<~. or . ... lIdlos much tnOI't' oil." II,. """" ol'd mud! .. one. becQmc lhe humble major d",,", of !lCienli,.!., ",I>o.re ... "mt<l10 be in f""'''Sfion ol.h< only ... Uy 'teriou,' I,Ulh .. and .hu. 10 confi ... h,m5elf 'modtstlf L<I. !'Ok d~l poIicin&- Sad dcsllny !'or philooopby; lhe d.fClpI, ... thai f<>nnnly .. ' 0' The head of 1M lIbIc find. Itself 1'Cd1M.'ed 10 , ... role of. (\ndtr,",la 'al .... up . 'lIh 'wnl'lauion' .rod ,IJ tbrillInJ ",able", .. d dlrea .... the tI"",lIloon of romnl(ln~ ideal'

Cirnkn:llll ",n .no _k (0'1"''''''' for lit. n,,.,.y .,hen raa:d "lIh 'calcul., lions' II) .h"",n, t..r dnOilon throop "" .... ,......". "",II ".'iua,d JUJlIfia, lions ,_1"'1 .tdtnoque .. 1t1l bu .... """'. ThiIs ,.,...,tifico..cluontablc I"",,*," da"".lo int.od_ 111< 'educa.od' pubbc 10 QtfUU! IpcaacuLo. rewJa. .. lid .re dremod 100 .ho".ct and whi>h KomI 10 col~de head_ Willi com""", ""!Ie. Mr Pn..th"",""" adorel Oil' Cillde.db and hc:. '.thicar ... "" .. 1I7Ihonl. The lal~' infonn Fnlly .1Iho~1 II";". 011" ...... and bnllJ .,;0<""" in.o ."" ... Im 011"" 'n"tf)'dly", n.. """" oftl'l< ball...., iIII !a>I,_,k<! .... Ih.he oea.nly 01 """"moll wnJe' 0.... JOOd man dr:daIa IlurudI' $W pliid and n ... ·on •• reuod' by llieK 0"'.1 r.obInno., r-.llCUlarly b) It.:. lilll< bltn<rll ... lhal call de,.,· mIn,,,,,, in'o q...sllon a •• "" Nonh Pol< by ~"n5 lhe'l1' ,"'rn" in I"" lfOl'lC'. An •• hi> 'lSIl 10 11M: .. bilK'! of curiosi."", M. I"Tu<lholnm< kaln .I,onul.t<d and IUIOUml. he has bc>rn.hI< 10 conllrm ~ h,mlo<if' .'hal 1M: ha.IUIJ'<ICI«I ror" Ioq I....., after all It.:. 0 ..... f'nlblmu ~ 1>01 to ru """",'«I from bis.,..n IUlIe I"'obicms; alr. welt 11>0, tfId. ordI "Dd .wrYOIW.,.1f lkqI easy. III ro<H. I~" ,. jIIJIlikr 1/wII

To ooMOh.b.e lhele .1UI~lhel" and ,mhopoKdic functions, C;"Je",U .. t;O,n off •• her .... ica in" mo", iub(1e "",nner, b) Ih""!\j lUI P""". <klIpn. j" ......., ........ ""rub ...... IIIi&ln >'U)' .'dl be lamed '" lbe ~a' prdcft '" ph,Jo.. topII.:.J .:alop ia. The publ,., IS Ihlll mlnod to ",,""'PIlI< ,n • pm< of l-Spy '0_ .bo', heK' Who', bid<" Look' H='IIIM: ".,.. ftl"'" of Subs.a_. Com< Ihll .i.y The T.ulIOC1ldmlal lIflII'plinl lhe nlne-huded h).ln. or 11'1< E:mpiriQI to lhe ""und will M familia. II, ... ryone And 0 .... "" ... ·1 lhe DIaI«uI:. 11'11 "'I cbippnl • Narut1lt)'. llId IUodtd lOll. "'.Iki<"Q al1l1lYO a>nd~ ,,'ilb a ,Iu. II JIIII I,ke lhat' 0... '"lionabMll', an JOl "P '" ,1:$

5u:tIday beot. OOnn', ca", mucb ror lUrprue>; for ,1>CI!·.nd·_h ... noo:tu~ • ucb •• nd· .... h " ""Io,o.y' E"<ry\hinl .. ill hi ... itt place, frum .he IILllJanl! 10 11M: """«term,n;,' bUllcrlly

Bill do ..... tully wanl ClItdt.d" to lit al ... )'.", ~ al".yo "" ,mpa"cnl 10 pnn" her ·,,"odo:. wly' IJ1d her lIIullid"""pl1lwy JOOII "dl' Do ..., moU)' ... ". her .0 <>If., "' .. 1>00 a ,upPc:" .. nl "'IOUI and pia)' lhe part '" healrr ... """'" by lat,n. up """JUlln •• nd '"""n"Iio"" Sboold .he ded .... l. htudr 10 CCImIIIunicl.i<>n and dil~lic, .nd ploilp he, oIlcjJa""" 10 """1>001 Cin<lc,..,lla

, p<obol>ly II", .... ' ylhm •• o lose: bY'JrOeUI.lo lhe "!b of bnnl"'l ......... 10 hut ptugin. ppo and IlOllcn'lIJ paradJgmo, .1It find. Ioersdf .hrook 10 lhe lUI", 01 ' .. f~ ror ICaD& ,he .. orld· and """' liked 10 ....... up .llh ........... 10 do:oioncd 10 ",.00..., !he anilIery of the dlff ...... 1 'xhool' '" inl.,. PfC"'I"",·: .. Ib ... In)' neal 10 ""'''I OUI tha~ .. IIm. queslion .... 1ottn Killed by Upori ...... 1 or anlhml"lic, ou ... ",anI II 0>0 Iontn ncedcd7

The ... remain .. h" ..... r. a """ru"",, deoi", 10 I.ke up .pin in the fI<sh ""UI "bal Ii perao,"ed 1I ... boIe lhal hu """" """,laltd by technical d.,persemm~ I pu .... ......will lor .'1"'"1 """"r. e''"f'''''&1td by tilt 'M1IPIl'C"Y of dw ·.t nliouliMll 10'" 10 ,"po .. uh aU "'"'"' """'II d lland. aU lhew 'm:.pn'. aU lbow lhou,,11 upenln<nlo, .t.:. fi,um .nd d,.""mt. ali llocw dYtWliei 01 problcmi .... ruln"y ""pablo of 11M: ·m.nd.' of ... ct'''''lion, ThO! ",.c"""nOn it ~n!orma"vc' in lhe prope. "' .... and """DOl be ml.......t 10 lhe ron"oyin. of • pre ........... vcd forln rrom one IfIlI>m>l1er ". IIIOIher ""'""'eJ, It lYOldo lilt dd,h of vulpnolUOII and .",.,ntfio )ournallun'

It II doar . 'by o;IaH.,1 "Iional"'" dono'l muocb ~ke 10 ,gb ...... naI 1M '''')'110'1' of lheoe prot",:" and lhee ... ,.. of _'''I' lhe lall •• II. applied 011 pr«i .. ly the Jenoil"o, bul blind, poonl' of lhe Ultdcntand,na. II lilt lun ... bonuNII . 'here indlOlllC 1)""_ boJon '0 "n/old, in 011 tItoote pi ....... here onetIlalioa ""'1 be bad lor r .... and wilt ... , he ""' .. lIOI .)'_ymow .,tb !be .m!i.able. The pal ........ of Golois, randoy. Mnnll Of lI......,bo:r& I''' .... I'bo:n.lC - i>ul ""btll""'" ntab!1$1>od ""I...,. .... 01""'111 ..... objoLc1J and form. do 0>0" Tbal il ,,'hy I ... ,11 do'Ol~ m)"''' he .... .,.,tb lhe lid of ""me "um ...... 10 demo"",,,,,.,. lbe hlllo" ""1 d"ndy of problem" thoo. d)'llll'lic oem.,w.- that hu ....... bcm .. hausted by 1d ...... 1ed"""'" 0. b)'pOIbau '" )all,l"D""", ~~m aod I"'<>I>I<mJ marl< "" ~h and Wlk1lOW1IlO, 7

., • •

'en and pbilo<op/Iot> ,utde lhe oyo: aDd hand. A .hruoltn. pJuJotOphy Cl>nnol br renlen' "',h """unual I1Il1oc"'llIom on

lilt '01&',.' '" JOon'oflC obp:tl. II l!aj 10 powl,,", i...-Jf It 1M OIIIpotI' '" the ot.:u~ IooIL:IIII upon lhe irrauoul_ .. ·diabol .... · and ...... llnllo articulo • .iDa. .... 1 .. !her !II lbe -FTM by whICh,,", <iunen ____ ",10 brsq.

A >coena: !ba, mnIIDcd oh • ."od " 'IIh refe ........ IUId p<c<hctooa .. oWe! ~ qulte illClpablc of ,. l in. up HOIde...,r'. flmow .ballen ... whICh II .'hy ,,~ ""'nl'o Jl"o due "'oi",1 bo", II, 11>0 func!ion ",,,,,,,,,,,10:< """, ..... ,w"'" referred 10 by G.loi.. F .. d<b). t:uut .... and .... n)'otben. Any pbilooopby lbal deb ..... Ildy Dqkctcd !be CIOCIlempia"'" pok "ould IlOOII ha,~ 10 110 .. 10 " 'hal II OptfllllOll and 'fcuihl<'. ".,."" 10 ........ )' llIl<m w,th ''''''In'eJ PIlnr/lf1loa.Dd d .. tQ .re 10 han.J •

S<:om. conlompol1ll)' dobole< oet:nl 10 ... 11M: relot,ooship Io." ..... u JoCI!>noe W pltiiooophy oaly In Ie""" '" ,..,,'ito.)' '" of I"'rrcuw wId .• houldennJ, f"'",,,n, thaI ror I"" plot '''0 and a IWJ' ""Ucnnoa ,n lbc \\eo ........ p/lywco.. .... lhonnIUCl and ph) ..... baw at .... )'O twill)' wily) JOno haDd in 1Iaad_ J.;. a obobbolelb "'1M hlJlOf)' of ideao 10 lta'~ I ..... each ...... I'r;., """oIullOII prod ...... .. new p;ontd,lI'" ... ilh lhe.."".. 0 ' )" .. "pI""l ."'''',,_ of """aplly ...... Tbo

,

• .<*&, .:.rOil§!> rI III< ..pl«nlh Ctrn~ry lIalcd ok.d) [be lutonomr ... <1 poo'i"'Uy of rmtlrhp""', inIO"",nl>Oll In lhe" work. II ...... bout • romm<)D problema"" 1""1 m:IlMma,ica, physoa aM mclaph)'PCIIl opccul~ li"nj wen: "",",,1I1ed. ExfI Il'Pf'08<bcd the qll .. no,.. in '\1 .... 'n r..tuon. .. ,,""", ","III, Ib,...1eIOOd by OI~ "'_ Soeft, .. 1l and philotopbtn olll!< ",,,,, .... nb tClnury ....... ndcd web. COIIUI'I"""Y 01 qllC:Ol_ DOt '" ""'~ lack oll<eh""l"" .. "II • · ..... ,.pII)'1K"1 >lop pp', b<ll lit .......... at prc,,,nlml lbe ..".nllloo olll>< rt'IO<Iet ollppn:hmd,n .. I.Un1 and "",hnl-

8.11 i, may be lhat modtm pltikJoophy .. ,he only d,sscnun, I'oia> on • w..losue bn .. · ..... m.ubcmalQ .ud phpi<:;! .. 'II",h 'llould olh< ...... bc '00fI

lIr\1C!i.,,· ...... I -r .. 1. A k_~ "''''' •• umm:uy oae. ol.M h$ory d .. ".["'" <iemonolnll'l

IMI lilt reI.,""' .... p bet ....... mll i>mwoUCll . '.'It,"1 maocr' ,boll also 'I""'" «tbe ..:.tn«<) Ind physia 1M If"bby, tffiaml """" ..... hardly !xcn !d)'IIi<:. A'Ulo,k *"I"cd b,m""f • 1~1 \If balM. by kecP"'8 lhe .".my iii"." under lbe . u ,,", 1Pt1'lop: 01 theolOl)' But it IS a Ionl lime pow ... ,"" ","y"'" .1Id _,bema,,,,, libonolcd lhemoeha b<&i""'''I. r .. "", """,I<>pcal n ... lry !I111 ....... ""'" ........ ' " II for III 10 Ih",. 01' ..... ",Lou"",1II1' .n _ OIher lh.." ,..bo<dll .. IIOD, .ppI""'hOIlIM w,ib ... t .......

01' """nc. 1i1lh Of .. "hool ·metapltl1oia·. pnano\tf dcalo «In "" ~,ruck; '111 .,,~ ~"" lhe ~~I.1f yt>u life .... lhe funda .... OI.L· ,... l'e3uh. "p'I!cmol"l'"' more: 01 Ie$S e~pK'I«I It> confer olllcilli !I1a'.a prohfcnllO: ·ron'·o""o""I;,m·. I»rt....n> '" malhe!nal...-.l Jrmbollsm. panl$:l1tS '" '~al,,,,,' . 11>0)' ""'" 1»"" pIttu< 'fonnttlx·. hun, 00.11":" mct.Ipb)'SlClJ pn<'CIpie 'allhe '0'" 01 ,I &11'. and tnI""" mathmlatia '0 a· ........... • lhal II ,,,c"lf.rm' 10 ,IS ..... " ronl<'nl and pitY""' 10 • 00fIJ1""""'" 01 'mII<leb' fu","bll', p .... JoCl ......... It .,..".~. a rholoooplt) Iha, "'<'}uk! lhink pIt)'JIro'lNItilrola, ,,,, mu,l breok Ihrou.b lhe ·al>umpl...,l11· ,nocfl bcd In thIS ronlrxt and undo"ta"d. for «.mpl~ , hat "" "g • m.ol i>nnll...-.l nJl.l....,. to lhe fulJU"'1JOoIH of m.o.l1.r IS n<>110 ron6 ... a ~"I ..... Ihln a ....... lneaI cdl bul UlJI.~ 10 Iea.n 10 ... k_ all the pot<1It.aliltn '" patltdr:s. '" tho .pIt)~' Lcibrli1lP,a dural "'1" ,On to tb .. """proatl rormlllallClOl; 'My mo:taplty5XI II ","my _"' ...... Ilal.., 10~. " . """lei boromt ",,', ~nd II II ~I) lhe ....... phystatJ """""P' 01 'mOood' ,hal makes il poulble 10 undenl.t>d d'jr ..... lial catculuJ.

H_ caD malbmu,ia. .. an '.un te.<,,""'·. a\'Old bfro .. ,,", 'flpped bol "'''''' .he Idcntocal at>d tho at..olu,cly Othet" II ..... do we .... ke • "'OI'Id from the ........m '" pit)....,.. IlllllhemattCI and meuphyItdT II.".' arc: ... 10 OOil(>tn" Ih .. ,rtpIc wi '00 ."bou, adop"", Arnloll.'. 'lcoloPal ooIu'lOn1 11.....,..1 i. rapoRIlbit ror unckflin'", .he pllilo!lc'>pltiaol 1I.kcs , .. . he .labofl"<>n 01 mode,n phys;oo.ma'l\nn.O.l",,- Analy"", oI.he _ri>b lII<Iw .h.~ ,f . be worlJ .. 'bou,ht 01 •• ~ malhemalical "~i"'''w," (u 5<"""'" ha. in,j"od 101""" G~Itk:<!I. t/loloft m ..... bt 0 mode of co,~_ bel ....... ·""~11 pmc.riad· 0('1, .. hdl "'~oI .. lhe .. orld ofbou_" _ and fiJU" oM.be 0('1' oIm.o.,maJ 1iII1 .... '(.tun. ... calculable ~lIIr..,_ .... , ..... lhe "~jN nJeoled vllliml%

, oIa .,.<>IOCol of ", ... 0 ... 11 bd .. ·..,n lbe """'OIU""T of fonns and .he """"m ~n"y

"''''''~ An)'1k,nl lha. <111"011 1I~lr Ii ~al ,,"hin .be 'p«ir", OCD"bk qUIIIII'" ibould ha, .. "I ...... ,hematlQ) Illdu .. ',hln lhe protCi ... of the opbcre: '" .he form .. and.llInItt, ... ind""", """bt""'IIl"ioo """'Id btuiolX poosiblt in e>nIct '0 0DI\J''lOC'I u J.riJ alI.be I"'''' 5 'OIl lbe IoIde oIlhe fi.ll.Jnp!

NI'''fllly, ' bit ,"oIUlionary """P .... ma lbe ru.tocalOOll 01 AmloUc', I. ,d a .. II 01>0 _rna ,he t<ln<epI 01 'lbi,,0('1 'PO""" 0<>1 lhe ... , .. lhe Greek. capablt of lboorb"'l 11110 III homotc,,",ly I"" •· .. rIeI of ronnl and lbat 01 bodoa Ind, finally, .he ronsI..ruon oIa ddferml.al cakool .... able 10 Ipptebend motion by"""' ..... 'hemaloc.l ............ 10 en"'. cIdi ..... by. ...00. of radl~ nw. coonplet~ J1aei", 01 plty..om.malbemallCf .henf ...... .Ioa/;lI

10 lhe ~'fIM:1l<)n oIlmpOtlol infnlsIrUCIu .... hlCb """"unly ,n .......... lht: ",hole oI ....... rhj'$lCl.

Al fi .. t """ .bis pltyWro-mlliIrollliC' dOCI aPfICI.' .0 be an UIOOIlII>f;I; ,,"n, .,. ....... form 10 lbe I)'"'''' 01 "",,,i .. 1cnoo bcI . ·ct\I .... , he .... l...-.1 00il(>tP" and pit,....,.. coouiAI- To "tKIn1l1"" .he ,"",allonary ooup lhal mrtaIl7.1>11 "'tomll"" ;, to cinco,., 1I:l0l pt'OlJDU' J '" lwo horuooo InIIic up '" mllli dctcrmm311Of\J wh>c:h Q~ tho ""1ntI1 """ of Hpj ... , dclenn.nallOM and .. hICh .1111 n,mam a."iLobie fu. u.am;nl'iorl. Coro;:ei>-.d ;u In .ppmn ..... h'J'o .. a l'romclh ... n impul§O and b"' . f a combma lorial "",mPIIlal;"" or OOn, .. In·'he· .. · ........ 01' U ... 'abJIrxt p","', m.o.\bc:Iruo,,,,, _ , orily rulm. lid tn pIt)'JOCS. II II m u..... lenn •• bac we obouId undtnlind GaJiko', """""'5y ,n danrq to bnq ,be boundary _ '" """' ........ 11)' ""0 _'*" ,,,,h lbe .. "rid 01 """"111 bod_

In', .. , .. some H amplcs. Quanliflauion confi.......J .he rommuruly of the parliclll.o, Mr''''''' or Ina·

I)',,,, mocha....,. aDd '" the lheory 01 .b ...... , ,~(JtI the oroc hand and 01 lbote 01 dIcmiJIry and a.omic phr-s on tho <>Ibot. The d_( aIti",,,,,, '" clcctrodo:mllilry '"'~) ""IU,mI. field 01 opcr1I""" WI .:as ....uJ inrd .... ible 10 " 1>0011'.' ....... ronlmu"," I Bob.'. orbots, when: tho eIoct_ docs DOt

fld .. ", aJlI'ldy lmpliady ,,<knowledged lhi.~ AI ,he bepnrun. "',I:IoI, .. · ... IIIC"h ""nIU'), fp""' ....... 1 .poco coMmule a lheme '" whlth lhe """/Iomt.o,,,,,,n', ,"cnlioo f""u",", m""n ,"",c on tlk ploy of.he 'I"'mlorr " .. If than <>n Ilk ... bs'tllC lhat ."" lauer .t. IUPJIOI<'II 10 tAIl"'- A F'l'III:lnc:a1 to.., ...... I H;I"",,', .......... ' _"plnoa Ih .. play and tmdo 10 .. ",," 1.11 '~I_ U1n ..... plaU"" pole and to ror .. lhe ..... 01 a f"""'I(JtOtII)pol r""",,' 01 fltoo;o>lNU, ..,.."rum. eoc). h II pnocoody II011I.I"""",y ,ha ......... "Ied '0 lbe """fllors .ilal aliOWI lilrol '0 be rr<>.IIJ>...,.;I IS '~bIa' '0 qllanlum mecllanlCl. I' a.adcxi<::ol ","uit: i. i •• he tn(>lIon il"U oI.he .mphf),!", absll1OC-hoo oI ..... hemalicslha ...... m •• hei.I ......... IKHI .. p/lywcaJ bei .... ' ,,... ...".,. .""" ....... ' _"'-'h ~ 'M HtflrTlr ........ J" ~ppbcQli<M. nu ... It> ..u.blioh a __ 1""",".teaI pIl)'llC'l" to "" "",po,,," ct '_'1mII to wha •• xlrnl ouclt

,

• In clfon of ..-docalluto ..... my U\ """hemlltQ 1\(' "'ily in.Dlve. 1~ bonwIl of the Vlnualil><a <:J p/lyiIG

Lel', take.1K>lber example.: lho _lHbt CllMd by ,he W"""""'" oI'M!>OIlIouclidun """"", ... in the nU";I«IIlh =IUI')' II .... 11 tn""'''' W. can llail'lus tveIll a. 1"" """,plclc libenllOa of ........ '1')'. 't'n'Od' "' luI from the pbysoeal ~ru.-cne: i .............. 10 tM m'1aph)~1 .... ,." cI r ........ u IG IeIIX ,he ....... ,,"" 0( .. 1\0. ..... "' stake In II .. O1lIleDOO "'....:II ."""'nes. In !art. EIItbdcaa ,cxmy' ......... ptlOtl 01_11 .... ,atwm ..... rrom !he co,,,nanl of dI...,.J ma.bnnabCIl ph)'SG: aJ!bIe [,,"' .. hn Of*" • '(HI" ..,..,. lrod the plw;di.y ....... ..,t_lal .... oljleOOletno; worlds uapbod 11 .. ,\ the.ery elron '" Ia'_'''''' .h"'" t..d """", ....... lIlm> shO,,1d .... pen<lno,. tho bonroa of pbysQ- c.,..,.,,,,,, by R_ aad GI_ as bema Tn .. '"<e<i ..,11 •• lbe .. , oltbe inhabl,"'I 01 MOtfaas. tho eOM I pi' of model d • ", ..... 1')' ICOMCCbOll. """"'''I<') can be fuBy ,..1uaI .. p<)W1I, ...... 1Id .. "" ..... II is from .hd poult of new Lhal .... lhouki ._ the d'l'oru 01 b"p'. TlIcory. Wid.L II II out. III !be IIone, .. IlOl to III"'" the Ihico ... y of tho ror-ulll [Iu" .. 'ill 'finaII)' aNide.* !he <qUl'-.knoe '" natural ~ b," Ihc 'phyoic:al' lIi."talllL", of IE UOIt'ry, in other .. <Hth the """,,."1 up of the dl"on of ...... ,. .. """'" of "orkb in .... ", •• 1 10 pun: mobil,ty, lUI dl'on den.eeI rro... 1he ,~h( ... of the CllqOnQ oflhc 'purely ~ncal' and the '""rei)" physical'

11 If ''''''' 110 ........ 1""1 IMn If: .I ..... ys IOfMlIti", _Ddalout ,boul ""uhlljl lbe physoc:ai ~."I.""" al III .. or t""l 00"""1" I""t ..... ~iowJy 1"'1"'" .. 'purdy m.tbnnaticlll· M.Ihomf,,,,,. "", .... lunctionl .. a limplo: ,001 for the physicist and .. indilro •• nl I" 011 ".tlll .. 'qu«n allhe uac:t .... ~. ,n "blCh ;, is _i ..... 1<'1 up 0<11 ofm'II ..... '.n.d envy. 8«UiC ill 1.;""'ltict ,,"-nnat be .wl>e«l 10 • otode of do" .n.d rulH 10' f •• alin, fo. mal .n""",,"lion .. mock,n mot ....... " •• becomet more . n.d mo •• uplicilly ,,"-pablo of inhibllin, lhe honMn of ito ,'i' I[([\\mes .nd lhem,," of ",acti""lin, n .. II, We ahoukln', fo,l'" lhal l lulm .. n ,ho",ed lha' the moo, t«hl\koJ lheorem. CIllO Invoh.., .:aICJorito Ibal In' Ilc:cmod 10 be ,trictly phllosoplucal oDd Ih¥\ Ilachdatd k""" t""l 'the .. "", ... 100II of """",,IOn prod_ Ii",,,, .,ord>, ,,,,,,,,bt ...... 1 .. ntcnooa, Irunk,"I .. nt • ...,..~ In t~" 50""". lhe phy.;ro.lIUllhema • ..,.J iovol.". lhou&ln OIbe ........ IlI.In "ouk!tlle 'PUn' .nd .. mplc· 1I'.nolanon 01. pbYSlCOl ..,,,.,,,,,,t· ,RIo I mIIthenwttlCOl ·1.00.11& .. ·• Parucies do "'" pi 'nl tlW:lII .. hu as Ihlop. .. het"P'I .... ~"o.1d 111.11 'n' .. mply """ilablo' ( HrxIoaP't'. ~i><"""""""ir~ ,nd mockm pity_lead .... dOM' to.he F""",,Ineal <>nIJ.D of tboo&bt If leibn ... ",n InICULo'" ",.....Joey and dJII'«tnua!. il it boca .... the I.ou« .. "'" 0 .. mplc tool OJ ._ 0 poi""""" lJ>d If poI ... ,,"L ",0 doJm lbe rnoaal II ..... tll.lt If: ~ ......"ded 10 it ,n ph)'$OCIIII'" ,,'os i. It boca .... II hat .I ..... y •• n.,1ICd I CIOIWn ""Ioloa 'I dJ"",y. e-..", If it ..... until qun. .-nt\Y, ~ ... InCl. ont........dluy d,_nl In .akubu-.

n.. ',n"01 foooodcn of ph)_ ....... Ied .n lhe order of lhe okuIobie tht: a:" . nil ol botb phywgol Ind malbe .... uocal ~ by F '-no ol .... """",,1, .... """""' remain uanploml _ PloN< .... uc corel WI In' l1<li dim;tly

, in.lII,..,.. tn tbo ITJ<'f\a1 of 1HUml" ...... 111Od "'!U"UbWI made """"bLc b)' ,bl< .....,."..\J<:t. ThUs. t.o vet)' d,lrtmll .hythmo undt.pift lbe ·h;'lory olKku': lhe finl ito intt'mUUflll ,lOll oom,m ol ·rul"u ... ·, oJ. poradllPM and lbe .. mubl\JOllJ; tbe HoCOnd it quiet ... hul;' .. ,II.",.., 10 11!a<U .......... and ooMisb all rumihltl;.m. nIOIlOftlcol _"""'In, oJ. lhe probInn.o"< 00 .....

Thoo ~bil'ly al reacu .. un. and •• kxw","1 tbe prohI<mauc as ...... imploos lbot loday', physlro.rnl.tbemlUtl Iftlly h ... '_physocs' ulKlot II. ohn ond tbcrd'ono it 01 concern 10 thot plti .......... He m .... l""'" 1""1 tbne ........ 1Il" .... le INW (tI""" II .. IlOl lbe trut,," bul 1M PInt Ie • thaI on rcact1, .. ted) .nd i ....... ony IlraeauIII all 'ruI', lbe INlb ol_1uch "",,",uk ,txIf '!:odnnd' tht: cakul.ollOftl. What beCOJIIU ~allnJly IInlildy. by lbe £a.'OIU 01 God, is tht: JlOOAOhWty al but)'tlll ca_uont .... th our .... ,.....,.. ....... ud of 'Ioa';o, ......... ,thlhem once ud 1Or.u.

M ... ph1""l does lui .. , Cltal)'t.: d'I«t. "hod! "'" btao d • ibed 'ft • 'In)'

bt.ttutlfultc>.1 by 1100 .... ~bemI\lCill1I And.t ""cil.

No1hln, .. moK k.liie. .u malbmlah.Iol'" ~...",.. lhin the><' obocun: .... Jos!eo. lhelw: mur~y I dl ... tlOM al Ol'" I"""'> ,n anothu. ,""'" run,.., <=IIn:»et. thac i .... pliCllbie !tire .Iso nothllljl'"'tt .. much plcupr. to lbe ..,...,,,,h« A day .... ,.: •• hen the iUuliot! VIlA"""'" ~""""'t turns ,nt" .:ena .. I}' ludll) ro. """,.rcl>c ... as lhe fop ,,"at II onc !"rint, lbey form ".,un It ..... lhor·

Andri Well cu .. to the quw:k hen:: he <ORJu", lhelw: "mu,ky .e&c1t00l·. lhe><' iun". "' ........ 1M ~ ...... y olO.lol> lho. 1..I ... nll" loocheo ... with his fi".,. tluou.b ........ n th •• be d_ MIt "" ...... 1" pi<~: Ho .. o oonnoisscu. oIlMoe _laphY.1<I.1 'fop' whoto d11llJ'l'11on II 0"" poinl horak!, lhetr morm' '''I at . nolhe. It "ould be hen ... 10 I>lk ben: of I horizon Ihat lillS lben:b)/ n:,ulin,.1IC'W 'plce of , .. tura which hI! not u yot been. eLucidaled aDd cuI <Jul ... muct un:.

A m.lhematocl.n of thOl ",hhre" .... ' ... Ihl! &<ic""" oup\><>,ts iudl on Iwo ....... Iellet:

lho oniaal crul~h of 1M J".,oJ tut .• .-h.:h oa:ounlJ lOr lbe Cltf)lng oul or I'" opt .. uonl aDd .. hic~ I"l .. ntea the lra""",",oo nt k_lodJ<; I more .ubtlc CNt<h. n'$<', ... d rOf ,n'tlat ..... 110 'n' able to I.",,,, .... I>0I0 n." .. "'~ of ,,11-'0111 inlfrlaced .. "h tloc h",ral I.~I .Dd OOIthnLlllly ""'" nO''''",IL

nu. II tht: p<Mnt .·heR lhe rooocuYlI"", altbc I"$t""'" IlI.It mllluriy know" ed .. is "'led 0111 ... d . 'here lbe p;a-F from P<mIonlllOtl to ""nau"y is ....." ...

In ,to onI'",ry ..... ~ ond """""""1<1.""'" ... ,...., FlRt ...... 10 _ mamly "" the 6 ... ""'tolt.. II....,.. il " qu". pouible .0 "'"iap;at", ~io:t. """' lip data and ,Ioo»e aU 10 aort)' out operat_ WI II In .. y 10 .... ke lbo: _ of lbe InL6aal dl$J~""""" al prod ...... l1li prOOIiCU>1ty:.-e lOr .... lII< ..... "", '"

\

, IKTlooucnoS

of tl>< ronq"",t of ~'Y ""IUI~ by tho openllOCl In order '" eMU", ,be ~.Imonl~ of ,t. rtlllH>nsh,p to ill p"n",,,,,..L Tho",;, • ."rt of ~n",,, .. ,d· '''I 1""00'''''''''' .bout the d;\nlY of the opt, ",;"'n; i , II' ... ,.. oIn ... 10 ""r.oeaI doe FlIIIK WI ""'0 oul • "'rIll oIlmculouoo • li>ifoo or I ru""'ion; II ..,..., that thai; ..-CUR ,,,,,.... .. bile lIo .... nlDJ!y atOIl$IlII .DOtbo:. InocsallllOll lihe fulcrum 01 ,I' \e..a HIII~ Tb< IrlL6ce lhal =0'" 11 ..,...., al_YI lou • bbnd opoc ·If. ' .... ~ "''- I""n~ i, rrio",. on anotho. '

F .. , oft"" irral>OlWll uDdoobl«ily f«th Of! OIl' lact of knowlodgr. boo, above ,II On ,boose pOi .. 1< of omilSlOCI. on I "mOlll impal~1IC< ,b.\ 1...,... u' f,om pnWt""'''I 10 the huon '" the ope""''' by """["""1 learn,,,. ""Ill."" I.lcnt for ... podly COIU<IIIIiq I .. """,",,"11OIYl """enl" IIu1 10 karn .. 10 prepl.rt OIOeII:IfIO am "tuu O~ 1<1 ...... _, aIrc:MI) t""", ... and 10 put ' r r ·eIf,,,.,, • ""h • " ... w~ ,1M: ron .. o"""" be<ween tluop Ie'" berlles ID the OOIIfIIOCIlOmc: """ltd .nd F\1iJIJIcd up "it"""l ""nJ pol>ted by p'rml~

Tbill .. ,hoe MIU"I bean ot ortf'llll'''Y. , .... is . 'hert lbe relanon ,,~ ""p!>tilly U I ·Cllm:' Q(o;" _ 0 can be ,,,aid .. he", dttennUIalioo nlodopt uto<'lf.Dd then I,bmlles. ,n U ... nl~II, peonL<. .uerrW ...... I00I p;t"'meI""'· ",1;0,.,.., Of • .". .... oon· Q( ,he ' yre 1'" PI,,), .. be", .be oep;!,a.1OCI bot.""" ,he '''0 >ide< Q( . he eq~aUon - lhom~ bolW1:C1l d'l ~m .nd m;ull i •• !n'.dy ron.ummoted. li e'" .he plUrcI ot (UIU~ .1Id .. ha , lhey carry off" uh ,hom can he , ... pod. JeJ.uYIn,lh •• ~ dots DOl Ihu,k Ioh philooopll). but 'MI " io CtKk1 .. o<>d w,liIthoupl . ',.""", .uo."'1 ".<lft" he ""'uccd 1<>. m'ullblo prqu.;ba:. H..,., ,110 I ... ' be, 'Ioon. the am'o, Q( amiftl- CIJI he .1.W .... h<d; Iwn! ,ha, ,If.n. cap •• ny Q( SeIC""" ,0 dn .. ". po"''' from ,Ii 'U)' 1"'''11111), can "" ,lI u",,,,.'e>cI; he .. ,he d,ffe",_ beI"-,,cn .he ~nowkdF Q( botnp Ind whl' ma ..... It ..... erber.le beoornelllal'lln •.

Th .. ;. ",by"""""", I,t ...... Ihcmalics . pplo«l ' 0 , he cakula,ion Q( , he 1,,- beI_ ploprMiliom and.o ,he 8dm'ni ...... '''''' orproot canDOl e""m '0 n, .. "" ,hc ..... ,," 01 ,III, . '1Nch .... rks.D ~ .".h,n tbc dt,'CIop ..... , of "",......."'.lhcma' ... 1 kno .. led ... T~ aoo indord um ·obJects· .. booc I'ormIl defin"IOn .. im.-~ .nd .no.. .• problcm-r .... <>1"'", .. 0 ... b.n .. booo """",Oll!LO ... by ........ 1IO'"100n •• hak", up a "h6Ie m .. tocm"iClIol 1;(1811, .... " . An obvOOUI e.ampie i< Ibal or lhe ·redi,...,.e,)" 01 imagll'''ry numl>e<t by ,be in' rod",,'ion of ,he plane (_ dlapt..- Ill ). It onIdy from ,he poin' ot ___ of ""'1bcm.. .... 1 Jotoc. ,he d ...... '·cry is l11'e. ,h .. fresh mcounl<'r or ,he F""""nc ,,'uh ,he .. ,m..oc ,and lherrio .. of ,he .... bIo . 'lIb .he C2IculabloJ..u dJcta,e ,he work. oIlIam,lton and R""""nn (.nd many ot ....... ~., lbe ~ tulIC .. ""'1", an impact on lho very nOioon Q( lbe 'pphca,ion or ""'li>nnIucs: pIoy.;c.. P"I""lry and a 'I<'b", "ill ,,"101M''''''' 1(> ..... h up • real ekctrOJCOll>t1"" .~ wb.,b .. '~l """I",,,,, 1(> ''''PO''' tbc Il>00, COfllcmJOOfl;fJI rueardl.

The",".n U ..... I ....... lor pIoik:>ioophy 1<1 po ........ ,. f.o1'" ,Ol<>.he ...... 'u ........ ,Joe fi&u!1$ 111ICI.,'coB' .... pIr.._ &ad the bmul.oc ... ooal''''''.DflCnL' ....

• of doturm,nal"'" a", .00ocuLuod. Bu. I ..... rtprcocd' ... "lIepc pIa<% already houo,«j by ._ OIher cornbl.'&nl> on Ihc boottJe(t(lDI utlJoe ob><;u",: an and .... noe. H.,e. meI.phot be"ru lhe proc:eJI 0( lhedd,nl il> >k,n lloal "ill ...... morphuoc i. mIn ".., .. 1I1otI. and bence i, iI ,ha. Ih .. IMJOIi ''''lOffDS ""lh die""" 1111, ., .. "" '0 ,a." .... 10 ....... ""'.-W1'fd """'''''')

C'bdai ~ LIoc JI"Iu"" thaI 1'I1Im1ly llllnaaed 10 , ....... QU' an idenJ"y or ..... ,..,... 01'0 ...... 'IOCI of """""~ 0 Ihou ........ '0 .of'Inl,1oc l""h.ll.Q.) Dalu"" oil"" <>I"' ..... ioo and ,I, 'coId' u.""""'Y by • ..,., m wnhos>on. BUI I""h elle"",, .1 ... )'1 come 100 Ia",! They ,,',II .lw.)'I be ckj)rOClllod II ·m~." phooc:ar and denounced lor u'wpin, lbe olM'''' ...... ·• p...s11JlOUS ploooo. SuptnUlIo"" ba ... aotIiliI. I<> fea, rrom . h .. poor IRIJI'J openou""y "'hodo IoaYft ,hc opcnooo.". ulmo",y 'n ..... CIdo! potc5 DO dWlm. '0 the at'f1)o-

pna: of 'M fomo:lJODal. bu, ..u QUI '0 d....tlow .... y ,Iooulill I ..... ...... . "","""pu,aL wi,_ ''''' puerile di<play eW>lod by .oodo ""''''pU u ,he bi, ban, ( .. 'h.,h I'I<V'" "<>PII,ulioC)" "".o. I .. h,d. ncul",li ........ ylhin.). 0111,. It(lpioe (whleb no k>n~, rriJh l .... ""'bY pooople). lhe froclal l f .. ", ... lon, ."",~ all '0 trunds of ... mpler U_I'1"'ndIl1l!~ mancc (of whid! ., "' oml ho" • ., ""',, God to.. • hoorl'Of~ ,,"-

In tbc .... _ of h .. MIt,.,. ua-,.,.... If ~ • ...-..-. dod .... «j '0 pb""" 9pcnmcnl .. Jan C .... dlh demand. Ih., "" &hooald .... 'iCOWr ~be <'m'''' ;n'UlUon .. ·h ..... e<:>n'lo ,u,eo lbe prQ(ound un"y bul.blS II .... lfUI"'bIo 01\

action or • Il>rory:' Th .. "" ...... pt or ,..".,., ........ 10 ul n .... al ;n nu , apprOA(h 10 lbe amphfy,n,"""lfaCUOft 01 ""'Ihematocs. ... blCh dUlb ra'ianalu:" "" .... raplo"'''''" - an.·I)'IIo<> ....... - mrllIpiMwJ .nd lbotr conf.",..;l Fa""""1001 ud ~ ... U1 •• bone III Iorm.1,,., ...... 111, would Ioke.o huclle , hu\ a 1rlI"U'II' 01 reMuros: 00:Idd hal ttc"'OIII.",,«j lha. td!dl"" ...... , ........ " ones ,ha •• ", I""". b,n 1101 prnnblc .'" ,mmediately proo,JlO<lOd by any .,nll~ .ha ...... 11 ambo,;ou.,·

Til .. PS'u", doc:! nol allow ,toclf,o be 'Q1ptuml" in (ronl or .. - ,' ""'" DOl throw 0<1. bod .... be, ......... and thUIP nor 'behind' III no al,onlbm cort ..... tI .......... II ....... 1eI bo 11<1'''-.0 .poak of. ~ wlo .. h .. , ....... , ... If up .... n ,n .n ompu'" 01 • ... reM"'" tNtl .... ~ • iiI ..... "'" 1:.", .Dd ._\e",;n '" o.he< feIIu .....

It ~ .1 .. , l1<tt> .. ry 10 empha",:,. Ihat

lhe F'IUR ~ nOl ... bwonllll: n pIllS Implotudc by ..... 'mlJlltll ,todf. It. .hcoClS'!'f ito «Iooal ' 0 iU pctICI"'''''''' .ad lbal .... hy .... rd..- '0 .be fell" .. ·• " ...... flOC)" lhe"., ,' ''"0 Q(.he otnkc .. proof 01 the """bo .. """ of'It &t,ll The FlI"'" , .... ull"taoa" r..m~y or FI'u",," .. '''''''''' lhe ...... only nl"I>Ciaoa ·;nll"""' I(0"'"., protocol lor da:"IIIp<>1.tll& lhe IlClIOfO ,nlll mdk$ll.ly ,epralable rae ... The F"",e pow'e",. hi"",iI;:oJ e.emp'an....., if 0 ... can .peak of on acc-/ulw.. 01 koo"'lcd.., o ' -a the """1'1< ot ..w, 'lYe FfIC .......... 0II< ....... 1eI sl""'k of JI"Iu", ,,,,u,U"'loq. dyna ..... oIPlobItuoo;

,

"

"",.., .. "kit ........ - ..... '" ....... k .... OIl>« .... Unoi: 01 ~ ,bit '0 ,,"'" up.u the .. n ........ ', pr"..,..""" nrt .. alllies, WII""", 1kt..",,11 ,nlO abbrnJ.o'OOIl.

II ... m 00"'" .. "" Itlfpl'U< then .haL .i. 1"'Y pamell'" In.nHM 10 lhe d}'na •• a 01 Pi""," <>f CUIU", oul, 10 dilll"m. lhal cap."," tbem m!d-nWn. ,0 ,IIooPI up.fllntn\<. w ..... 11 hunl 0111 ,he hUlge"l pOIlU "' .. h"'h pron.apies b<com< WIllable. .... "';:11 _k 0111 1M con>lrllC'tcd ,),,,,,,,,,lfl:(Jn, dc"""" and ",,'uJ tho lao; upoacd d,,")mmelf)' INoI 0I'1efI1I .. mo", rail"",' pro",",.

A d..aanm can """,Ij, ..... ,ure, hoI", II 10 ru1.\ona; bdon: Ii eun. up In1(l .......... 'h.:b .. "'h ID(l(ImI JCOfDdtn and cosmoIot<n like d ..... ms .nl~ ,bot, I"'(o.plOI)' "" •• , 01 <VOOOiJOo. Tb<-y cap ........ ,ura nud-llIatll. lor ,,,,*, .. ~bIe 0( IntnllOll, .hey ..... he "..,.".." ..... """" boo", il aJlDlp!Cd ",,;10"1 lManm ..... In .. dql'fC .be IICCOmph"'" '" poeIOC: n •• uphor. aUI lhey ..... lillie leu ,mpcmllml ,\ iI.I .... )'> pouiblo 10 ~k 501= in .1>0 mu"d ..... ploitlflB of th." thick 1.1\eI . and moro r.uhfut the)' nn prolol\j! Ibc",~I.cs in,o In 01"'I1I.hOd which keqllllilml from 1:"':01"''', "'",n oul. Lib tbe 1IlO,lphoi. !hey lap oUt ,n orOt. I" <rcl' •• 1' ..... aDd reduce PI'" the)' bk 110m .,lb doIlld hne> '" 0100 10 ttll"lf unaan !hal .... ~ 1'fC."",wy Hlun:d 1II 11ut~ h-. lion IlJltik. I~ moI.pIIo< lbe dLloW- .. _ ",ha ... ,ed; If il ...unoboh .... a plu~ iJt order 10.lOCl 1io.'1I11J1 opontioo, II d/,)Q M> by l1t1cl1"'l • ~,,~ Ihilll""" cull ""I ...... be, The doItcd k ... ...ron .... ,htf \,1Ibe pornl and II> dllCf<'k ~tJon, DOl' 10 lbe 1I ..... nd ,11 conUn_ lrace. builO It.' P""U~ ol the "nuahly 1_ duopt., II lbal .'orNO It.' alfUdy ..... 'labk ,m. on onk. \0 <T<:at •• ",,,,. fo< ...... dl1nomion: lho d,av"m', modo ol oxlll.na: ",u.:h ,'''', "" FIIO''';' I>(Mllpnoed ,n ito bet"1- W .... uld deocnbe Inil III I Itthniq ... ol oJ1U$ioni,

1'.. k_ ho .. Gilbal S .. nondon malll.l"'l '0 ,lluoninll. p,.... .. ol """"" ... nee aDd .ctaplatJon ollho iIQI"o"c ,n'o • l<dm"l"" .bleb .... l< II pO> .. "",

10 "!'Uk ol I lI.'clrrucalliJ>c: ol d 1111 • Tberc .~ lho~ f."""", ol dlql1lrnl oll~"""'Y JIfOICI'IO and .... bll_ aUlllla ... JIISI .. ,hero .... b_ ol.. .", lhal a~ "''''''' .... nll) IU''''''''''''''' .nd l'OiICI.IC, and.J ...... lho lI.'clrnoarJ obp:t

" docs 110( foIIooo ~kd ... 10 the d......., doa 1>01 "",ply ;0 ... '111. or lramble an altady ...... bIe _'<:DL The ~.n: Ibt .. fo •• ~ .,Ih "'1"'''' -. aDd _allhe"""h.,. capo.'* ol.""'O" ... I .... and conVC)'lQJ; 'aU llus talkin, Wt\h lho iIaJIdJ' ,.'n"", 'I would ..... 1No,. bt btlt.,. \0 calIlhu talkinl Irr tho banlb) or ... h~ ph,....".. •• '" "" ""oud .nd .. hte:h lhoy ,,"lunUr ~ for the 11Ii".I ..... A philosophy ol'h. phyoico-malllCmalka] ""nnol ilnore Ihil I)'mbo(;e p.IICUOI ... hidl i, poor 'n ,,,,,,,"111111, In" proCl;~ ol con<k!UolI;on"'~ lImpMaolio:m or lbe InIU";""'.

W. will -. ;11 clIaPlO' I In po.nitular. how OrnrM', diqrams doIy lbe Iowfllllc:oi - .1 cowtc uI"rcd by Armolk • ..,"" alilhouahl """""rD1lII mollon. ru .... ad,'In ..... ""Ihe 1111""""'"' ....... by A.""", .... 1Id Giin "" R.,.,.. btt'a ....... ",,\Ill a..t eIi ...... _·] " ..... "IOCI, lhaI: .u,r- ..... 0KI:t .......,.. bk. lhe am:uialooa ollbrcc po. ..... ,. '\npl)'dt) and .......... the dottnllllll""" .,lb p/aIIici'y. Lik. the ... tUR, they.ao uoid lho: dilemma 0( • "''''''[10[' whICh only pinl ia oom""'hea ....... bat il '-" In ClI~n ....... They also obow llNol onlaloP::;rll1lClI~m'K"1l Oi nol ... mple o;IlCTlIIl obtnvablc.. hut lhe ~vinl relation.hip lhat ;nlmn;n&\H In LIlte""ly (Iho 'oolololicar polo of "", .. ",e· menll and Iwo oxl ........ q~.nlilito,

It .. nol only. q .... lioD or hlllonc:aUy .. ~no"Icd'I'" •• ublld,ary 'tool' for rep< ... "tLllI <q",,1I0,,1. hul oC a .... "ouauo,lhe ron,idon.blc 1Utlt ....... 0 00'0-

.........1 di...,ty panotul.ar 10 lho "un.! .'iucb ....... klncmlltc:l and lllliyUCIIJ II )I,ury pao6ible loa, before tbe lliocoway 01 .wr."",1111 cakuIus. These dlilfl/Dll lib lbe" pIIooe iJt • b"" 01 d • "I thai rwu from A,lCmDI 10 LcOt.uz.nd lbe .. ,.".. prq>Ire far .'hat If ..... U .. aollIlDOde,a ""[10['-, ""'Jen oll'" mal ....... llOIIl "_10 ..... ' ol N.,ure. It .. 110 """ ...... IIOD '0 ,peat 0(

tho ""Ioneal .aonanee ol tbac d..."....,... ..-hodl .balM the npd dlmi"" bel ....... oJ.,.bnl. ,wlDeh d.rilkd tho dol.l1l11lIItion operollon ......... nahks, alld &eOflIOII)', wh"", ~Iure, ..... urcd lhe p<OI<:Cli011 of the rontcmpLoth't

The dtaaram an.,. ..,.. ""I of [ .. Ilion: ;1 .. a ,.....;.c. Inat .,nu 1o apply oxdu .. ,'dJ '0 .'hal it,k.,c!ta; Ibi< demand for autonomy male ... " lhe nal"r:!! aewmplioe .... tlIOulhl e.>.pm .... nti.

The llIOupl nponmmt does 110( dam> 10 prulrc:l 0' ...... Y • law, _ 10 ~k lhe ao...Jj'y ollhiDao had.-: il _kJ I1lhor.o ... oid I~ '0 do ._y ... lIb wrialtly lrul/u.ad .VIIlabie ;"''''tiom. If'l II 'OOr or~." II "'" bl ... " ... i, is ;m~bIIe 10 CIrf)' 0<11 . 'j.h·....r "'U"'"'1!n1f.. bu. b«:o_ ,I ct.""" 10 "ueolion or "lIC'U'''' ~ ol ickalQluO<l, I .. IU ".r " 1$ W'ty ".....In!:;, illho tal by .h",n lhe p/I),;cis,--J)hikKophOl' I.k .... pan h,m ... 1f 1o 1 .... hi/; beannp, to know tbe pt. pIoily ,/IM ... nl 'n .ny ." ... \lon ,,'he ... d,..,.rument i. by no ...... "" oclf .. "donl, It il' lila, .. , for him of .'11"1'''1" .ub,'."""", of ,Iw; babi" .. fIJci.led "'Ib .. ""bJe o(;elreo 1"0]111' Ih .... _inl for luntllnd at I' • .upon,") htm .. 1f Ihrouah tbouahl into pnallCl. he)'ood "1111$11101). IbdlCrcd from 10--. '0 lot h' ..... " ft ... , ~t.' ..... ma,ban.il'" aod .. ,-

T'baI II how (jahJeo ' ...... _ .. body !o .... d ......... honJoMaJ ........ '" lho:

\

" INTJ.ootJCTlO'<

.hienee ohU .-tad<' and n::rnafu .b>."he mol ....... die body _ill _lui ... urufonnlJ and "enally on 'lID ....... plane, pro.idod ,bo, ~ is .minndy pro.-Ionpd" In lilt same way [ina«io io.,"II ... hi ...... If'" a loft 10'-11 ... ,,,,, ddIi 01. r<apol1lr-forlll.' ........... li~ .... U ... and 10 _ ,ho r"", .... , .. ' m 01 ,II< ten_ III,.. ..... , joiM IhmL II .... ",UeI' 01 ""1l"l.1It IIMN!..omplew iI,1e 01 doonmlallOll III onIer 10 orimt aDd .....n .. pbYSOCO-lIIIOlbem,nicol POOji:! ,ho, ,,',n .ppt.or. r~~u •• as tI •• IDOI' ............ We hI."C a1'-y_td LhaI\eorn."I .'11 above all. prqoanotiotl for IearNo, WhaL or>o IlfUdy tDnl" II COIIkI ... said ,h.. ,11Ie tboup, expo ....... n. aloo 1imI., 111"1111;1 OlIO in su<h .. 'lUre ,N., II 1>«_ ob''tOIIllha. lhi< pnllClpk 01 thai klrnl,ly ( •• en . 'M ... OQIIlradiCtN by ' ... 1.1' opmenoe) i, ,he moo;. ·11'It ~rll·. Thit ' ''Ilunm.s.' ICCW '" defy all ro,ninFf'C)' ~nd 10 bolon,l" U"" "r\kt of lilt i"""IIOble! a.me .. ~"""Iltal ·"'."'ably' objOOlI Ion lhrou,h . ~um Illh" lime '~ ... i,hout h." .. n, upeneDOOd lL

One could even .. y ,har the rad..:allllou&.hl upco .... ", II In •• pcnmml .. hen:: !'Ia, .. n:: and 'M Ulldel'!ilood,,,, ... n~h ph ,. O.hlco ""mcL,nICS 1"'11 h._If U\ 1'1.,11 ... •• pIai;oo. .. bd In ,10 .. mphcoly. could 110. ha •• cboocn 10 '"""", ., a., 1"""""11 ...... OJ!ft'd an '''''''" ..... numlln' or Yn}' Ia...., bod .... to

rrod""" • -.alt for .. luf:h lho n>O<IcQ~ IIIO>'ftIIeIIt or • 'II. body tW'1llD.l • rou .... t\.l _~ OO/l1re wo.aId 1<IIIio:c-'," EonsI .. n ..-u.n tho hahll or saY'''' that .1 .... " "ry '0 PII' {,,,.KIf ;n God·, ........ to Dndcnt.nd "'.tun, Thtn: II noth.n .... '1l""a. lherri_ ,n thex1dctK>naIlO ..... 1om: E.~n Lak .. "' ...... ror. phoI ........ (!NOl",," h.nuc/f ... thor horuon of YdQcIt_ (fft chapI<T III ,n thae "nn...,,,1OnJ .'1Ien: AIdl.oKdti. '0 hOI b'l1h,~b, una" .... ,hal hOI bod)

II nothinl boul 'lJDU,d of"lIte' Th .... to ~nde"tand lhor phot<1ft. II iJ ',",",I'ry to b<com< a pilat"" .nd. 10 ~nde,qand "",,"",. II II n<' IN.,. to lurn o_lr ,n,o a .i,_k,nl The lh()\t.11l e,pe,unmtl.k"" 10 '" ronclu"on is a Jt"p_ """'" "'pet'rnt'nl In ",hoch It """" ..... dea, Ihat • d' .... '" il for ,twlf ,II ""'(1 upenmenl. The ftStu, ... that" ""!'lures and f'I.,.,culany those lhat it .rouws a .. /10 11)I11C" d"«Ied lo"'ardJ tlunp. bout take the., place ,n • I",., of d,aanms. '''th,n a ,oehn"",1 de-el"l"""ol. thus Ortsmo', recuntla.re PllIllo) .. 'td up by ,hor 1'01'If'OCI'''' Pfo,ea"'" or .... 'ra'ned relamuy i_ chapter II). ,hor "",pool ck uPOl1 ,td ,n I CIrculi ( .... chapler V). Innpo tra"lkrm! IMmMl_ ;'uo ltd ....

111m: are ceftaul dt.\CIC$, ddibetatdy prodOCl ..... or .",bo",IIY, .. htdl JI'~ ..,. to tJoou&llt "penmenll fhat ,,",,","'" c ....... to the ..... n or mall ... and op<ran"lY; n .;:aU thtm fIIJ .... ,~ J"i<v.> ..... lhe Ia,. three chaPlm Uli. 1 v. VI ~ \lIk ... up ","h thel. study. An allu,",,,, 01 .. 1 ....... com~ ud unfolds. bout is not 1M oame lOS ""her.n .bbre ..... "'" or.n ",planatl(Nl.. tronl lM end or tho Cla\lloenth century. pb~ and ... ltnlll" .... 11 compelC .. ,Ib OI'K

.""'/ttr on 1'",_ england and G<nn>ny '0""'" the m)"llery 01 lilt blnh 01 poIanhc. and d,lIOO"'" M 'P"''''' rttjahVlty .. hoM f«und"y KMnt bd _oed ,n t",o en .. :.,,1 Irtlclcl fWhll I, II t" Orttn! Ortaclr In Tho.o",,'" Ind ,ho ·AU."'rl 1(1 I"trod""" ,n'" l'h,l"""rh) tho C<>~n! of N'1!lIuve Quanmyl

t""'OOt!CTIO~

Th." 1fII11.I1 ..... tl\,ty. too IICJIe<:t<d "', the hlltOl'}' 01 pbilooloph) tcsp"aally ill it. rot>Cern .,,,b IlK """'" 01 Lbo *"""'" .n • penpoclJ'~ oIt .... po"'11I1 Iud bw.IOf1CIt)). play •• crua.al rok Ulthe .............. 01 dt",."p'll," "'''''''

In ".WI' ".",.,., lOS III IlK "'""""ft')' 01 _ coan«tl(lftl IKt"''ftlII ........ ar.d ,""",Iry or ID IlK ..... ' .......... 0\ of d«tromapttttc lheond. II wa ....,.,.. ary.IIoo>"·'er. to DOte WI IlK oaax:o: of IlK IUIIeICCIIlh CCII\Uf)·. f""'" ArpI>d 10 G"'MmaD Ind M .... ·dI. could u ........ otaud the 4tpti"" lOS I po5ill<MW m.titm In order In dn'", on III ruloobtabk otrnl>o'''' qau>1l nt<rldcd 'pl.halot) .nd its tIoad dtlUllICCI. AIlu ... ·c devoc:a; .re ""pablo of MI'P<<tpN.U4~ all 1M Ialc",1 ",obtl" ... ,nvoh"ed In IJI ".itnl.litm IIId of thK .... 'l'n''''''''',,''',ul fol'l;U rn~turally' idrntlfi+od " 'ilh an ope .. " ... ty that mO ... from 0 .... boundary to anOlhe, Of f,,,,,, one poinl 10 3001"''''1 in I.,ou. of 'la tltud,nal' (tra_'UWI f"""", Ih.o. are lin~ed to a dilfu.iOll of d.mrl1ifOf\, Thae de.'m can he tooughl olIOS ·dial«!.1 balanaoo': utntablc balain> that II\' I)I1ly Upocl .. -beD ""rryin, otr. Il10 .. aml*'pKL They bnq 10 I'P.I an ,ndilfuUlc" ceolrc: (the t')'" 01 1M <)'(Io .... ~ • po>'Ol pollIl thaI comnno.od< • lICt"ork of '),mJl'Klna .. hoie break ... bdo to • Ii ..... degree of d~nI_l.

SIII:h '" <>W '''''''Iwu"uoa 01 dtaIGI.uI _MIKe: doakctoe ;. DOl tbc .) .... t .... oe ..... ' .. ~I."OII or IW<>~"" and opi>£' J tenM. but tM dixoVU)'

oIlhe IrtOCULo,IO" thal.,.,.... .... t ,he d .... " '10'1 aIont .. bidt Lbo) willlU .... I'ke 'wb' Thi. apia;,.. <>w ,ntCfDlt_ chaplU VI,n diqra .... uooaaled

.. "h lhe --.""" 01 d,-.-."" """ horiaII ....... poCIl. Wlbcrc: .. ~ horl ...... Clc. They ........ that orimlaUon cannot he ruluccd h, • ounpie '<:boice' and ,- "" eIf.." .. "y that ,q'l)h-a ..... 'Klry and ph)_ ... m""" .. mwo· ph)'1OCl. They IU""", d""'l'Ii,.. of ftStu"", Iud I'''' .. ..., 10 '''''''a\lt expcri. ...... " thlt .'" • • I"",lal«l loo,lKt'".., ""Y JormaJlfJn ..... 'Po

Wc ,,',II )Iooy 'n f'I.''\Jcula, 111.lOIlo",n. e1am~:

a .......... anQ·. t"'o,>, or the .,tcn.,on t_ cb~ptc. IV) and the delin,,;oo 01 hO~«>"'mutM!i ... product>. "be", It.. II_lion of o,ieolati"" p/.1}'I a cnICUIl role': Hlmilton', .",,,,,,1 ....... '" roo" <_ dup!e, v~ _hid! O'''''WID< lhor d ... lny of , .... "".u" ... and rotatIOn>. lhe ...... Ioop 1)"<1111 01 Am~ Farad.o) _",:I \!u"""l ... hicb ...., .. - (hal lhe ...,..,. 01 ",hat " .. II ..-ell to .;:aU cloct'Cj 3O'Klly .. linked 101M ,",",bol,ly 01 """,un",IM IIIllltllllty lhat I""", •• beD __ t .... to pmdllCC the ....... tfO(a''''''''llatero.i ro ....... j from till poIto. lb .. am ...... I)' ....,. Cftdl. by ott ''',ru and u""""""'np. kA Ioooiuuac up the 'nanunof rn.llICIer 0I,1K ~1"'''''''' dn"..,.,. .. hidt acctlfI1pony the IIiOdIanallIC undent.ndi ....

The ""'" ....... I""'P tlK 0"," .. "", ., ito 'UJ bon'" .. hidt i. lhe "",bo_ of any tlloory of "'n.ulari' .... Thn tbcoty p<t>poHf 10 m ... ter tbe poin" It .. hoeh lhe ..anonkal cho"", of. """'lMt.,, .. h"," .11" ..... d"ta ...... 0 he L .... '·dlcllind , * j«I'o IK .. nnen no !on,., ....... ,,',,"ou, ")',nl. n.... poi"" ,he.et"", _'t!I!C tIM: on,<ntkon cI • con!inuum .. hkh If .bIc '0

" •• Iwu .. ,lIlhe JOIulIOOU .... .., alFbrU: IN fIIncclOllal cq .......... no.. 'man;rokII'.~ dotnshed by 1110" .... 1eI" ~k •• 1D, .... lu ... Nature 1M! fftalnJ MId """h,...,. • ..-,nc:al Fl.w.ly .. Iud! ;! II fino"l 10 IpproJIfIII!1t. no ... . _ ' lypt 01 ,nllllhOIl • bono ",hdI """" DOl 'rdy ",,'.110 .boIlXI lip .... of Iloe..-bk worid, bu' ..... II .... penmtllu ... 1II11b owf .... (Go .... •• ·.clryl or ~ II dtfomull, ....... ( II. ....... M·. mod ..... ). Anll"'~ de.""", .... "'" Ium!ed I" ,he r.dl.h .do""" ,n the po" 7 of 'fUnct-.'",. lhal II 10 .. y 0( proI ...... un"nlO Iotqllll_ of ,emil ... llIch lhould be wrapped in plm"""-=' """,rd,,,. 10 • rn urlbltSlled ruk, 001 of 1larnaoI", 1Il00,,,. e1""""-~,..,....uical or p/lytie:o! !lUll fu.1IJ!h 11M: """loon u. Cu" .. or u. pr<jl>d>o< in""""'u of III cauu.hly.

W,lh lbc:te d.,_.a.roce ha"n ,I< ~lOn pi«u or •• !den<:e that IU.", belte. Ihln 11>0 Q""01l11."" 'n~rul" IN Lhe """, UOLic mulllphcih .. or ~I 11>0<>'1. 1NI1 it mall,,, lho Infirul<'. nil. it ·, .. bo •• ",,,I is ~ .pI by 1hc _n 0(

lie"", Ind Implt F51U"", ",hod! .'" ...... 'produced', bul ... IhH all 0\1\ lbe form .. oct lip .nd W~le Ill< uea, """_ lb"",,,,lves "" tho hC><UOl>S, unmobil,IC lhal wtPfh, 'by ddirut>on', is lIIP1'" I 10 be IIIoOboIc. ""'" the ..... '.' ."", >CfllnlIH .nd ~ou II .. 1'"'0 "SIdeI' of In "'11>11-. ,mp<* I rhythm ollUlfurillll ')\IlafIIII'ttr' II OIl 1"" most t.ack1r0tlj of OOIIIpACI "",tIeL

III u. onIIlW')' fIolttIl/)l\Jnlo ~ _ 10 IImII uld( 10 lile pstuns thai • ...",nlft lile PI Ifl"'I1011 oft_ltd ... and leave IlIIdllnut'ed lhe pal ......... ' oflbotc 1h.l1 WIll Upl ud mldtiplJ iL l'bo« art lito the _thai ave II from iadcflR'lc _Ulllu!alioa and iI",u/icallon, f.om the childishnou of au ... bWd poiIUYlloa., from tho comfort of tho 1","StIi dIM 'openuona!' and. flBaJly, from .M IM1plaLion of .Um,in. IIICIf 10 be buckled up III a pammar. TIley IUIISl"'le 1M 1I11"lICy of an au.benul: way of <O~YI'" informauon .. 1Iid> .. ·ook! 1101 be romnuuod ooidy to oommulIlCItlioa. bill III'OIlld .Im 01 a tIOhII ... larnp of .Uuaion and of 1M lea.n'"11 of learn;"," Thel.n.r. of «KIlle.

",auld /w: lar tmIO,-ed I""" lhe Deu."""l barbltnsn! .. h'I:II nil. ",," it,,",! in hunll"l <In .. ·11 lhe rOClpie'" of lhe 1hought and in o;onlulIl,Iea,"inl .",11 • pilla .. ", of ,nf"""",UOIW booIy !khell'lIJ pe,h.tpt UW OlIO'" datir he k ...... lhal IhOllpL "&\I II1II oJ .. ",.. .ncapouLa,od .. ·"h,1I ,lie brain. ,hal il i:Ould be '"'"ef)· ... be"'. 0II....x in lhe mom;". dew.

l\OTB

. A '"'J ,_ ::' :_;:' :;,:' ::'" ;.:'~':'::. __ ........ """ ....... , _.~. " , ..... ~ ... _.-,. I_ ... _". ' ....... " "A_C_~_~ __ ' '[n_~1.oc_"".o"",_, · _."Io", "". '0' . , "'

C' ,..".,.,,1 '10. "' ... ·0.10 ".,.,..... ...... _" . '4 , .• .:&.n., .... 1I.p.* "' ...... _ ...... ...... ~_...,. _" "",'00 .......... ",,"" ~ '11<" .... ~ ..... ' ... ,' ... ,.- <it ... ____ ". oJ, ... _""r->"' .... ,Iooo<y<of_,_' n..,- .• " .. _.., _."""poJ ....... ", _ ............ dIIkwI' ................ __ ., I ............. , ............ , ..... "....... "._ ........... _ .....

INTIOOUCTION " * "',"' ... ....,' ' ... -....' _ .. ,', ~hoo'.''''''_IOI. __ __ _ ____ lwiJ""_ .. ' ....... _ 10..,_ .. _"''''''' .. 10.' ,o' ..... __ I ............. _ .. -...r_ .. , .,=, ,. ___ ...... _1_.Ikooo.-, ___ , .• __ ', • , ",_-,., hn . ..... __ __ ........... _.' 4.-_I' ' .. ""'. . " ..... -._ .. , • .. --., ,._,1" ... ~_ __ ....... "..; .... '""" ' _ _In. ''II''' . 1'- ,,""' __ ........ ,..... ... ' _,' b=""" .. __ .......... _. __ Il00''''_ ........ _ ..

-n.. .. ~ ..... _ •• a , _~ .. * ____ ot,, __ • ... ,-. _ ..... __ • ""'~ .... bo"' •• ~·'-, . .. _ .... _..,. .......... _ ... 0 £' . __ ... _ ..... " .. I0_"'~I.o&t_ ..... f', . __ ,... ... .... "" ...... """ rood ..... ~ ___ oJ .. • , """ ... "",,_' ._ "" '-4<1 ....... _ Jfl_, __ ,...,..,.. .......... bod-. ,'.,1 ....... _,1 .. ,Ioot_ .. "" _.-.1 ..... _.,-.. .. ~O!2Iy_IlI<ot __ r" "',,"YeO ...... .. _ ,I:ar "'I,ou .... , -'1 .......... "'''piI<>o1 of.W, ""fI .. _ .. _ ... , ' 0"

............... )_10'_ ... _.- .... £ ' .. ' __ ""', ...... , ........... __ " " .... _ ... 1Ina& ....... _ ... _ .. "", .... ,,10 ... -. ....... _Joo .... __ 01 -, -r-. .. "'" _ .. _ oj "'" ..- ub <MIo:oo .... , _ ... " ....

. .. , ... - ,

- • .. m ....... ,

" ,

I

, TH E ENCHANTM f.NT OF' T HE VI RTUAL

I ~~sn"CTION MiO POT~IH1At

Did Arislotle ha'e ony ""'picion or 11K pol<DC)' or the IK>IIndary be had lb • ....." up ~" .... n phyJicaJ aDd malbem~licaJ lI:ilura1 Whale ... 1M ClI ... be re,,,I. in nppoolnl lhem point b~ polO!: ph)~ical Ulures are dedic;md In mobility ami .ubject 10 fi .... 1 couses, but enjoy a sepa.ote ni"rn<e; mOlhemalkal nOlUres, immobil< and 11"""""''1. ore 5<l"'fllbk only by Ihougln, and 0'" ...-;1 hal 10 I.nd ,hem an •• ;'1.,..,."

Many mo<km cpillcmolopr:. .Iil! ""rry the mark. of , ... "phi d'fcrtro by "'rulolle 10 f"und hI< cl",.if..,.lIon of It.. ... icnen, and .-em.in fa""nalcd by tbe relat,,,,,,h,p of fttlproc!ll predatIOn thut th<:;c I",,, nal" .... """" to r"'let ,,,,,,,,,,10 one anod ... , For these natu .... ~thl "" ... onlo",*,,,,,1 1'",«<IOlloe. Mi<tolk, •• It "" happens. ....".,1\. ambiguo,," on In,. q".",nll. he refe ... to phYI;';' ao a ..rond philosophy. beh",d metaphysics. makin, 1M! ..... lIl1On of mathematico;' he .".n ,1.,eo thai if there wne no ..,boton= other than Iltooe Ibal .,i'l in N..t" .. , pnr ..... would be 11111 ' monR ,..;."'"'~! nUl matllem .. !;':'l n.alu~ aR nOt prey 10 mobilny and. '" an intdhilbk ..... "cr, mllllcfnalil;:s can ~ dtd..ocd frem a 6«1 prilK"Jplc by ~ dlaln j}( t><O<nIlK$. llIe tall< of ohoosing i$ Ihcrrl"RO far from .. mplc.

It i, ""I ouy 10 .... eigh in l .,nglc bal.""" a '~l" ~I."""" .... ""'b i. indopen<lc:ol of <I" bUI mnhtlc and OOfTUpllbIt, and lUI ;oldltgtbl< mallet. wbiclt .... u only by pr ..... y Ih.""gh IIIe .... '1 of I~ jIW ..... , ... lI ow do we cbooic bel .. ...,n pno«lenoc in lhe orde. of Being (·.,-hich physical nll"teI can loy claim (0) and l<>&ical p~"",,", i .... hlCh malhematical .... t,,"'" can lay d.im InlT' We know ho'" A,;jlmle ~JKiIcs t~ Iwo riul" b)' ."bordonatins lhem 10 m<1aph),>ia lhe first phtlooophy "·h .... oop:tive i, lhe the"')' of immobile and ",,,I bema. .mmutable ..,botana:.

8u, "'mtotlc d<>e:!t "", ront ... , himself ... "h ,he tUleJage of mc,ap/l),>1Cl; hIS theo.y or abstract"'" al$O \.Oku cat<! '0 JIUll. a more """"",te pHOa~ IIe,,,'oo:" ,he t ... o nat"re ...... bos.o OI'flO'II"," he bad SO ."'",&1y "ntkr~n«l Tbc Ari,totelian theory of abolraClJon.ad.!Jllon, .... hICh ",tat .. bo,,·., .. ""I only to ,be ",Iationship \>c''''""n mathemaliesandphyoics. .... bIi.bcs a ,~ ..... ';b1~ oct"," ,b.t m.k", II ~ble 10 odd Or ."bl~ dc'cnniution' at lcismc. Stril>l""1 a ph)~;""l being of II, mane. and J!. moInlily, I con olso prod""" by

"

" ,bot ..... OCl I malhcm.o,Oco] bt>n. 10 . 'h",h the .OCII(' .. ' ..... , ."\Illond an ~~"le1lOl aDd.hen limply mn.qrate til .. ban. inl" the order 01 physical ... t~ret, by mlOilal ,be dctcnnination> '" wh"," r !\ad dcpn>u1 ,I. Ihos IS the ...uo" oI'proIbftlt° or 1I<ld1~ 01 detemu ... ll<)Ilt.· Tho lhtoty oI.bo''''''' ...... proIhc:wJ alto makes;' m.ddy poMihic to bmok ,bot t .... 'th _bIe ma" .... La Ii;MFt .boII' mobib'y. \0 opt tor U". or !hit IIIOdoo of <'OCI' .... platoon. 10_ .. ';,_. d,lfe1fllllaU"I ., tho pm pcrlI.ar- of the 'Io"'lln,' oil" ~."'et>C'r • I"""" of alUlbie malte. or • d,oembodoed pone\ncolloOllll.

8~1 an one Qlrac1 • pan f!'(>tll •• '1>010 .. 1Ibool 1ea"'''1 _n' Cao 0 ... .....,.,. dc\erm'lIIllOlU back on "' .. 'ill'

We hIo .. , JUOI tol>Chcd on the weak point or ,/II: theor)' of ab>lracl""" a <:<".,"m _"".""" "'nh rcprd to ... ""1 makn 11 poslIble to Inacll or Ikl""b deu:rm.""li""., For lor allsl'-":';' ail •• ay. to mUIlIale. Arulol\e',amllblc nalure (~,'lJl ilI.IWII)'1 pan of a "MIe: .b<lracI_ removes. po«e from the fInh 01 ,ho ~blc.' 11>11 ,hi. opcraliof\. olWJIY' <n>d. ""nnoc be mluord to all ;nolfrn.,"" .ubl.Klloa of delffMIna' .... 1ll, whi<h <>lOr an ..... ualon u and .'IIom ..... cbootol. Aay""," . '1>0 ha. ptaCtliCd ma,hemal""" Oi lhc phyw:al ","""'" I bn .. c:aano< hdp b.1I dclCCt ben. ok:I&h1 01 Mad.; .. " ..... Y 10 ' ..... 0' .... h • ph)'JOCal beI~ 'punfy' " "' I ..... II mlo • malhanllloc:tl belnl .11<1 .... 'remllk,' II rblmt>!.b or eanch ........ , ""0 • plt)· ..... 1 bct.nl' Ot>e _'M 1M. I_ reaJ ~.~ 01 bw>p: .,.."nol be pahcd (IG b)' ...... ,. .. 110m 01._ philo-..,phtr, lor ...... hIe .... ut< .. UWICIaled ... ,h <'II'ou" 01 pIKed or .....,md tnd,,"' ..... r.. •• roo- m.llhemt.teal bel. lhelr dllJ'l'Y as ........ 1 and : "') belli&> _ ... lbo..., all . .... y 01 OOIIIpc1Iiall"lior lbe fJIrm>e povm~ ..rtbeU" de!et1IUJ\IIIOns. \to'"",," •• bs".,.ioo ru,1>od '0 III lunl" p • .,....,11 III .. itlt • I"'radollCl.I .. tWill,",: immobile ... t><t.nee:. 11M: ooJCCI 0( nwupltJ'1CS. " 'ould tbm be oIlbe pOO''''' ronl .... ,l

It il ,a.y.o dillCO"er .. bal ph.callli! ""'in On tIM: thc:o.,. ..rabstractlon: II. ron",mplior the FOtu", tb.t mad< tbe dc'.rnltNl,ion ('DSl',b\e. It ........ tM' .he .bcory of .bJtnt.<llOOl·prol ...... ' r...,.,.M,eII tM oppotlltion ol ,he , ...... lba, II ..... 6Uppootd 10 bolld boo<k I"I<Ib<. n .. "uti be ..... "". ", .. ".1>1<, Ml.hImIlucal btlnp appur ~ all ....... ""' ........... phY'IaII ",",np. alld lbe "'11" mnam pruonetJ .... Iho order 01 finaJ .,..""'" a mo.i", body is "",<red tottOfd"'l 10 lho Ionn .kI •• , ...... oId lit"" and am> .. hnl 10 the plan: " 'he ... ,, ovaht 10 """'111,

Tbcoc oho ... :""" ..... t leul ba", the ..... 11 or dnoWOlI a" ... _ to , ... 0 tnIOIl poont>.

_ II ...... "",pIa 01 cun,ol oul I1I<Ilbnno.teal b<1 .... by thoulh" ",hoch llio" .. ,hem 10 be .... 1"' ..... f.om tIM: mobili'~ oItbe .. ..n.r uO¢r ....... ill ,Ioow .be c .... ,,,. fora 01 ,It .. pnnciplc.' .. or~ In tbe '''''''Ih' •• pen ........ of modem phJ'!CIlch. "' .... II .1><1 Ill)'"

_ lbe .hk''''''''ru,,';on ollhe physico! as ... n"blo nUll .... monl m!1'u'" of fnokincu' ud . ubmi,>ioD to tOe oroe. of floo) cou_

" Sc.'eBI q .... ,1OI\J Ibm "'* 1hnnJo;1_ COllI one _,,'C 01 pity .. ",,.,,,.,,,,,",, m.o1JClJ be,op thai are 1>01 """ .. Ioollly ntUavW 10 tbe Ipprtill/t .... the 1InIrld. ... thou. '''''''''11 ...... 111'0 ,MI. .... lip"" RICIl>QId 10 uiJlmce b, proIi'To am_ .. tllne ""I""WM!1'1i. """' __ E 0 pltysoaHlllthaNtllCf hal bon'owtd • ..,.,.. ..... from AriMOIlc', w..o... ddi:lwuoo:""," lultiJ .... ". 01 .. hi ... potntloaD" it mot..,..," MotJOrt, aa:onb ... to"~ IS ,he pt.... of ... ",.I,'I""" oIa poIc.",,: fonst ""hoc!r. _ be ..... ua.d to lite .. III"" _taIpc yamt ... 01 , fonst In traMi, 10\10""'" a Ingber form. For to mI_ /Il0lI00I 10 .n .... n immtd .. le1, 10 qUOfldl il: 1M """'i", hal 10 be Mft:1ed In rcspecc 10 IJO" .... .. nee aU mn .. on ;" .bo"" all • """" 01 mOl", ... II I. Pfn'U'<ly lhi. pl.Utn<t' 111 tho fulfil"",,,, ol,he loCI ,Ilat ........ Ih •• thou"" .,..nnot rid h.df ol motlM by ~ ... mi1."n. II to. Iran';lion be'",u:n ... t .. diel'led by appetrt>Ctl,

RUI neither il motioo ,he p ... i, .. ""amnl ol • lonn, Motion" a ... y ol knll"", 1CI.1td pO""., 'o",tbe. and. if poIcnual" 1\0. rtd.-l 'o tbe m:<:i.inl oI-.knlt,. '''''''p' of lito .... tut.obl. bcconIl/t """"boIII: lho .... 1" ... or ia: .. hOI 01 ice ,1>1, . .,..... melt. bul ol ia: ,/tal is • ..... Dy· In the proeu ..... mdbna; ...tcr .. 01 toll .... ·poI.nuar 111 ..... bul .bo.·. ID il ""II:llllCl 1 .... lf I~ PotcnloaJ IS _Itat.. In _ion. ano ... . ho l ..... "", 1000bOf 01 an 'aJrudy' and • 'not yd',,, 11_ """" resrn ... 10 ,J,.. ..... " It "hal mt~rcs thaI .... docs "'" ...hagll motIOn aad.; 111 11''''1 """'" '""'" '0 1111 If"'fptol 01 .be motioIl. ;. ~,1tDd ""tok tbe b"I""" I coiled in lite bod""" That .. " 'hy pcl fe .... moUoo muil be undentood ... an in<kllmtdJ '''E:~ltded ",I<II.I".tlO", ddaplllllli no pO,,'," and /!lqwm'I"" d"pla' '"0::"'41\0 °loeal """lOa'): pI" .... , and ..,"C1C,,,, 'he "",hlluy of the 8..., mo.ot does "'" d ..... """Iion '0 ;IIdf by ea'l1I, "I' 'pICe

Poteau.r 1M pl.rtK:II.: •• ","' .. nee: alla.ched tQ each 1110'1'1"1 body _ io .. acdy lhe lh1n, .Ut "'ltd., ,Ite clu'ches olin .bstra<:lioo ,h., ,.U<>d mobib,y from. o ..... nted mobil;ly 10. MIOISo Th •• houahl ol'''' pol.nr,.1 doeo 001 IIph"" mob,ltly of!" f, om lbe molor 10 lhe moved. it tiocJ nol pOur n from I full ~pt .. 10 in,o .~ ""'pt) one:. TIM: mol", and t .... mo.ed .... not 1"'0 ,nel1 be'n., "1'1""" ...... ...."h<,. '''''~'''''tlinl a quallty: ..... mo\'Cd II "'" lhe onl1 CHW:.o ~b.ft .. , lito: motor F I ...... ,be form. but can orIly IC' in tbe ptT",,", oltbe mom:!. 'The ~ II ""'""'..." to mobihl)'; lbe .. tI ... boIe preparalion of the "",,,nl body 10 lite IlIpmo' form aad.; It Iny .. Ie, ..... j .... lilY form <XMIId IlIIprtIIl jOtlI aay ~'tcr; tbe US cannot learn. but lbe pupi~ """ an .......... ' ...... .,..ll. ThaI '" mou", .. r, _h. II .lltakc In ka'IlUII' 10 """,tc. bnd or tcrulOft l/tal .... "",.. to Ib< ~1 call of tbc pupiL

To Jearn or!ead\. 10 ocoord or cede mobiU.y p ..... Uy 10 I body. II alw'aJ' '0 in'....,1 a ...... It.on>oamei'y a poIelltiol alld In noES! lhe Clpod"""" ptOtt..: o. of lbe ~ra...rcrmoe 01 infonno"",,'

1 Fl"STlC "NO DECt~tYl YIRTt'''UTY

In wnlin, .ltal phP'':' ''''II,Iw:"""la wti beEnI pl'l)«l o~t ., tbe kI..,j 01 an •• ~""n.., 0( mobl!l!y tu>d 'hat .he ........ red • k,1Id 01 ",ofoun<! affinity bct....,.,n

III< pupl .. ha 10:0 .... &Dd lhe ..-,n, bod} lhal l'fOII'CW..ay r=I_ ~ tI .docouc> or 10ftft, A"",.de ~n;:bod ""I lho ,.Ih of aDOthc, I"",,,y of -t .. ,flICI_. <>< .norhe, manno, or cUllin, ""t by lhaugllt. "'hidI doa nol IIm'l ;loeIl ", makmg a 'prOI""",,' Ihat can be adJ u<lod al .... ill l.eibnit i, «'Ialnly lmoo>. Ih_ ... 00 h .. " """I profillbly ponde"'" ... riIIOlk·, d...,=1 in.n~IKln: he ..... al>l< ,,,...,.n mlllllluy a n .... M of oomhin,n. loCI and po .... lluil .. "I ...,lod 10 ...... "nn. 1M '1_''''''' lhat '''e", preoooupy'''I Iu.m:

By ,,'ha, '""radr' '''' ..... lbtma1Jl;al bmIp. 'llb&tn(1' and """"ulable. ~"kod 10 11>0: ""'_.-mIlia lion of physoal belIIIJ from 'I"alu",' "hlOOfIC>Ie lIlultipbCl,y and map"""'" How i. uno 10 carlUn: tl>< ph)'$ioo-mllI><lIUIl .... 1 II"'IU'" Can oM Imp <.ltV<" and qu~nllty ..,muLla ..... "",)"

",n..ably "" ''''''' and dca<IoI on l modt tI elullaty; " ~I"'m.. "'''!MI' and pooptk ..... 1"'>11< unll .. V>n..ab'y Inbnt Je'lura: n JOIkltJOttemuna'1Oft, II doa DOl. ..... tch n. II " not I"" WI1< ., lhe 12~ of the poMIbIe$. bu. allowl for II. docwo!I and unfokl' n~ I.eibnl1 k ....... hat .1Im: It an ,n"LlallOfl in 11K: "nual lha. escape> .1>< phi~' ,,'hu il In a hurry; 10 lhe latter .• bled of "n .. fined ma,ble and lhe $la ll>C Ihal 'romn .,.,1' of ,I ",11 ocem .. nnlff!' 'e one anoIMr. ,f he d.,., nol kno .... the medll.1I0"' of Ih~ Theophilu! oIlM .\'~ B>D,-.:" IS hard 10 ,mag"'" lhatlM .. a ... fins In 1M marble un,il

""" dlKO'''''' th<m. In • "1I}. A."""""", Iud allUdy toroupl "nuallly ;nlO lhe wOl1d of bocbeI

to) ,h'I"'I'n.m"g bee"""", lhe dwlLa ...... that II KIIWIlIOd ia ka&th, nIh Ind depth (.,....,.,If'" d, ...... """"") and IIIL ,ndc1tm1' .... '. d"nens>Oll ,,,,,,,,,fl,.., ,,,,,,,,IU) I""t pn:d""""",, ,IK: unrclin<d .... lIer 10 ..-;." ,h .. Or ,hOI deI,,",,\n~· I" .... TIK: q""nliltu 1'1111"'#' Ii lhe,dore lin.cd 10 .IK: Iu..<c<:plibihly of 11K: mailer 10 acqu" .... 00 • ..,,, and h' b«<>nI< graspable in Ihm: dun."'''' ..... nd I ... ha' .. , ... " ,,~chara<:t .. a,. ro. pe>rtal form ,1«_ "",pond",).'. TIl< .,....,ulltu ,itt"""", •• """ I ""I""ud!:. bill aa u,.......,1 b) .. hdt th< ....... ph)'llcian poopol. coena,n componmb 0I1ti<.,..,. bod) 'OVoaI\J$ the ..... ,kr be mocribes .. hal _ would loda) all I .ok,o ...... Inhedron ,n ad. pot'" of 1M r ......... 1 0( man", ... hooe mollOfl he .. ,;IlL< '" itudy. A a>onl",,,uol> <Illb" l,nd 'htll ,11o .. 'S h,m to ..... ke Mire." Jal.""" <IlIM 'Il0l,,,,· lhall'" "" many ... nu' .... rudlmen'ary a",lic..) hod,.,., .. b",h ha,' •• ,na: beto,"" idtot,1\cd ";Ih lbe objeCl <Il cI~~sical m«1I.o"b, The """".I/IU ,'1" ... /;" " "'" an ·.""ractl",,· it "'1I>OVf/$ no ""'crmu",,,on. It "'*" 11K: <lIended body., thaI "'h",b .In,,,,IIt plhe" and a»n1t"",,, the detenn, ... IIOO. 01 ,he .,....,../,,,, ~h The Ian .. CUI nry. boi." .. lm~bIc '0 >lnp .... u .. of an indeI ..... ,1Ia1< d, ...... • -. of ..... <Il,a .,-U"'nnn"""'" .. h",b nco,"'" """""""'" not fMk

Tb .. wa~ 01 o" .. "nl • field <Il ""'nhplK>ft .. cha..actcrul", 0( lhe "n..al. ..,,,,,<>IlIn,'ho: di,ttlIly of ,he 1DCIden .. of • lulK.ra .. by ;n,roducinl I .ubjClC1

"!I" • ran.., ... flicienll~ eo'''''';''' and , .. lliciently d .. ,jc 10 unfold Ih ....

" ", •• d.;'II!S lit _ ,.. ""hoa, aboul"n, "d ttl 'rqt>tmnl llinn .. mulll..........ty and SIC' SI>'eIy,

II"",. 'n m,nd lha l ,I IS 1100 ""' ... /"" """"I .. lhal alIo .. 'S ACSid,u. 10 Jlv., con,·i""'"I..,lu,i"" 10 Ih. '"",,login.1 Jl'I'obkm of ,,,,,,,.,b6l1",,I"oo. II I""'· denlJ .... n only be , ....... ned _mbcddod III .... bJ«:o, . hen . he ''''''pendenl '."'<nQe of .M OGIour 01 ,he bread .,.d the "'j"", .. hen I"""', .. .w..r" ....... be"",,., 1M body and blood 01 Chnfl. '-'''d' .... oa.,""ILMlbIe. But. If 11K: OCQdcn'" lonted ,,'IIh only II>< .... """"y 10 IIdbcrc 10 a "'''''In~ I. Fudoanol CUI IlL 'M ot>,ec1 tI • .. ,KI<Ial tlKoIotY To pup IhOI ·,,"pecoly'. 0""....",1d ha .. 10 _"'"'. "", .. a: lhal .ph .. 1M qWl~I"y ifllM ..... " .... rln...J1I and .,-j,,,, ~/Men,",..,Ii> al ,he ..".., "tnII a •• (\Ieula"", them, ,~ I""h ..... y t~t t!le finl iJ ~bIe 10 unfolJ lhe ~trid<nlJ of lbe i«on<!. II " Ihu. lhat. '0 take...,."."t of Ih. ptrmallCl>tll: of ""ruuo tt<ritkOlJ (.uch "' the ~ of eondctuallOQ and nn:floCl_ or flu",,) .... ...,wuo .. """'lnoe1l I ,nptyd! 1M. II.. VJ ... 'tom: .\1 IS IIIL -,is.,. _,_ ( ... ·UI ..... today call lbe ...... 1. d ,he .s.:.u.ly and I tloe vol ........

TIlL dnuoly d pr0\1daI a h" .. bel ......... IIIL ....... 1/ and lhe volume I . lhe for""" "",,,IoJ'l"l ,ho ....... 1""'. oIlM bit.. Indeed. eacb poe« of lhe In",,,,,"

1"'" - ' ..... '"",,,Ii<: on'oJosocal \III .... A.,.;oJ.", ""Ieo:

It ohouk! IlL Undc"'.-\Ihllt in ,he mailer of Ih<: b ... d ... <1 lbe """" .. ..... 11 as in III ... nhly man .. (bert "0 Iwo q .... o"t"'" and 1"'0 k""b 01 dnncnsioou; &"nn""," and ind<1..,ml .... '. duDemiooo. I'or man.r .. 10 and "" midi and 0C<1lpoc:t """" and _if •• 00ume. If .. an IlL siooIIn thaI 01 " "'" I"" ...... '1I11J1"'Y b)' .. Iud! mall<, .... and 00 m""" and by "Iu!;h " has $lOCh and .0Id! a . "IIOme. and. 0>11 d.o OIhe, Ioand. of_ eta ""t< lha, ,he qWloUly ,n ,,,lIoc: of .. 'hlCb mallor .. 10 Ind "" mllCb pn:<lO<lcs lhe qWloltly ,n ,',nl>C of .. hlCh I, occupies 'lICh Ind jllCh •• 'olume and that in lhe fi"l kInd of qU'''lily .•• in I subjeCl. tbe o«ond k,nd of qU:lOhl)' 01 and.on:<!. ,hen 11 .. .. ,y"

II< 1M mallOli (""!ol bee .. "", '''0 q""nbllLS, 1M ...... ,' .... k .. 11 ~bIc '0 """'pcrt$OIf fol 1M ... """"". or voIu",", uod 10 """'" lho 001 ....... · .1 I"otal<""" of the ....... iuch ~lCO\_ can ",maon ......... "1 .. hik unfoldi"l ddJ ..... I de,,_ ofrona:n'ratiotL The d!>lUW;bOtl b .......... 'dtmcmlOllal' quail--Illy Ind ','In,,,,1' q ... nhly lead. Iherd'On: 10 d", ,n,'_nuon of a de>'''''' 'hit. mokln, u"" of lIN: _Jail ICily p.ned hy lhe Inlrooi""bon 0( an ,nICn, .... q ... nmy (I"" den ... )· ,n OU, • .ample). proYKko ;1",,11 ...... b lhe mUM 10 a .. "",lalo lhem b)' envelop"l " .. h O .... M ... nltiun. of lhe OIMr.

" .. ould "'" be cugml1cd .0 00kI lb •• ,be dISCO ...... of "'viDm"" and """",II< Ib",,",' oo..n 'M r.,..nda.OOM ur. phyoieo-matbtmatlCll poaM'. ,"" undemanoh". of .. hoch bu CIIadJ,1OnC dod no' """'PC ~bllLc 'M de __ for uflfoldm. .peelra mll.1 be on ,ntfMJlIJ PIt" of • n1«b""", 111;0,. bocouoc: i, doc< n",

" cb...-prd poImhal,'oes. doa IKM .Il010 illelf I" be ... boN" .. , ... 10 S' -.metry. !lei", _'IC, ma .. II .... .,.,.. IfI"'IIIO lbiap; O ..... MOI _ply • .alIow II Up ;., • kll ... naloc. .. Of +rI"" dod. Tbu .. probably . 'tllll Leiba .. ""'"'~ 111m for. "">1111 redlll;Od the Galilru plOlI(! I" • ","","pUOD <II ''''p rIO .1Id , ............... of ''''1'',1 11 ... pC;«' e.-r'IlIIIIIy hid ,he .... bou'"' 01""", ...... ., ... the ...... Id ofbod ... u .......... 'nc.JtI,., .... bul ...... aDd t,be aU rite. of .,..,.,.111 ,n • son ... IOMU" .. 1M u",,8"'O(' of Lbo: c.....,ptJ of acom<lry and i11I' eI,he _ of .... lnial ~mnllJld connecl>OftS of ca....Jity. This last requlremea! ce.,.,oly does 1101 Imply qU'1e ,ho conlrary ,be m",,,,,uon of ,n ""leotiah". II II lIO ... thc:1eu ill .. ,hal Oeocart .. pomlliO U tarT)""1 'UlpocI"melapllysical' caon"'.,io .... boCIu~ lhey a re al .... Y'IUSpoct<d of "'" &i"nl tbem""",,, up bound band and roo. 101M d lmy of the UI,nSlon.

1"0 DncarIn. who clalll\li 10 pup ,be ph}'sieaJ hcinl"n·lhe,,,olld undor lhe dclrnni"lliolll of Ion.,t.. brud.b lJld deptb ""'y. L.eilm~ rtlopondilbl1 l ,he ba .. JU% doa DOl nil' &lone and ,"", ',he poin" .. ~.h noth,nl', lie doa nol ...... ," lhal ,he q!WIllly 01 IDOIKNI II ,he 0010 deI.muRaliofl ,hal .. not cnmpleLely JUIII'W'IC. wbK:h KI""'; ,1Idf .. 1IN IIDpK" ""'"'" In facl. the ""1*'1 II d .. pn~ 'll'< 01 ph)'lOCal <_, in C.rl ..... " ""'""""'" whdl .IIIII!t tbe body all .. ~. po ... ol aniooIllIId __ w.>oWllnl by .... UD~_ ol K1 ... lwollOn. I.cibn", saw ibo.t the COOIIpltlc ' ..... n' ol ...... """""· "'"' phP"'" illvoI...,;! the com"'",""" ol ""pool'" o.nflUtn.:l~," ... 1ld! rouId DOl help 11<1, imp/>al,e """.phpics. In pamellla,. he IS ro ... ,nttd ibo.. ,baoo OOlion> ol ~Dh.b'y .nil ol "umW mobW.y. fII, •• bly lnl~Lcd. lillie II potiibk to o(>[lCCn~ ,ho profouDd >&Icnbty he.wun ml,he"",t"" IlIId ph)'IICI ",Ihwt raorI ,0 the o"loloP:oJ fllbordilll'OO" prOPOlfd by "'n$lode.

To "vc cLlmeny.o 1" .... to .... ,. '1> irr«l~bk d,ff ..... _ f,om ,,"0,",,'';'; me." ." eon"" In o nlolOJPCll di",i,y on;1 ~nd ,<> .. -.a'e" ,,·nh lhe ICI,ve po,onuahly of "hoch "'mlotle bad ~I.o.dy h~d In inklin" h IS clajtocny thaI II"" spdn •• 0 ml"I" h"'h 10. Oescan .. is .. simple ra.1<1I' 01" incnil) IDd ,hal IMkes il pou,ble r". the l)r>deBlln<i1Jl1 to PIP the I"'0lfb'lve """'1'1"'" 01" .. rca 01 ""Ioaty_

I, lhe", .ny <bill" ol.hll pi ... ".;., 01"",. ',u!lym,' tbe ..... """,,,-,Ilia"' II 11 ..., •• be OOfTUpt;b!e """1 ollhe <d,~~ QUlle ,he "ppoo"'. lor u;bni.t .... ""

Allhouah I .... eOlmnttd ibo.. c ... ·l')'lhI ... IS ox- mo<;h.omr;:ally," ""'1"".&1 Na." ... I De"I'tflbelcw 100 be/iM., ,hal tb. vet) pona.,... 01 ~ ,hal i. 10 ""Y .be 6rsl .... ., 01 """""" blo~c • _'" .... bI ...... _ .. " lhaa pure m.thom! •• """ con furnish .. Then: is in .... na _h,,,, Otbo:. lhan

,he p"",ly ."""""'0, 'M'IS, tJwo ,"" ~II.n>ioIIllIIiI i., altal.ion. and ,ts bo,.. all"'IKl1l .. OM rnJIUS lhat _ fIIptnOr eM ""',.ph,.. •• 11 110 .......

,hit'" 01 ,ubellllCe, actIO" and roltt, III .... bo a<klcd. and thc:oc IIOltDIH

...... n Ih;o'."l')'thlrt.IUI wlr ....... ""10" ",ull in 'u.n ..,1.1D<l.''er)'llull8 tb, ... co. m".' ."ffe. some """",on ... I "'InA,n pcnua.led 'lull an bodia

',11£ E..'<CIIASTMEI"T Of" THij ,"lnJAL

.... ... lanUy eumdcd, IlIId IMI ,hen: " 110 atenlOOCt ""boo. body '1"nethcksa. ... m .... _ conf ..... lbe: not ...... ol pIaoc. .paa: or pure ~J'<1I$ioD .. ,ib tbe: _ oI.ulIo .......... hoth, '" ..Jd,1IOa 10 u,., ULnIIIoOA. also ..... tam< tbe """",,1lOII, ibo., .. 10 ""1. ,ho """"", and Ill< """"",,-"

II lllIOI.--ah '0 durn .. Ansl()l.le', ddjl1luoa lIS 'ob!<:ure and mctaph)'>ocaI': " II '011(' 'III')' .0 ..,k""",lodgo:: Iha, it wu obi< 10 _ motloa lIS • ,,-., ol kl1l'M .... and po..-u 'OlI<1ber:

Iliad ptlltlnlod deopl, in'o tb< land 01" ,be ScllolHhet, "'h(n m.tbema,D and modtw I .. ,b<)rl"'- "'" .. "btln .. from ,1 ,,'luJc, I .. ~.till youn .. '1"hr" bo .. u,;rlll ""'~I 01 nplo.ininl NI'''''' """,hanically ~h:umod me. and " 'jlh 1000 rrllSOn 1 &'p"",d tbc mcllt.od ollh.,... ... bo u~ only ton". or fac-ul!iH of" hkh .,,,,hiDI I. "nde~lood. I/U\ I.'.f, ano. il)·'''I'o e~plo.~ tbe: principlco or rnc:dIl""'" "lClr '" o.der 10 accoun, for the Ia..-. 01 Nalure " 'h",h "'e learn ("'''' ""J><ncnoe. I J><.t ..... od 'ht ,be IOJo e<>mO:nlion 01 faltoldcd rnaoo: ".,.. "ot """"tit bII, ibo., II wu •• " AI). i11I111d"", ... '0 u~ Lbo .0"""1'1 01 (0""" "luch "full)' ,n,.llIpblc. allboop ,I Cau. ."h," .be J~ of .... tapb):oic::s. ,.

[T JIM: _pi 01 """"" OC' pO....... . Ior ... boor Uplanl'IOa I ha.-e 10Cl lip a dl>l ...... .,ic_ ol d),..,.,oa, bnnp .be ,crou ... , 1Iah' '0 bar IlpOII our Uodcl'Ollnd, ... ol,"'" I".. DlNlCXpI 01 ... boI ....... Act;, .. 10"", diII' ..... from ,1M: men: ~ ..... Cam,h" 10 tb< Scbool .. for ,ho """" .. p""'C'r OC'

faculty or lhe ScholMllCS ts notluna 11<11 I doM ",,",bili'y "'"""""" "'hoch .-I. "" ... cmal nellal"", eM • "lmuJVI, .. 'I "en:. In 1M: tnnWc1rod inlO ""'10ft .... Cl .. ·• 101'<%. ' " conH';U~ rollLlulf a """.,n """,0" lit ",ukC/ ... ;" and i. Ihu, mod ..... y bolWttn .Il< I_h)' of """m) and the """"," ,,,,,I(.nd invol .... a ",,,,,,'n It II thus o::o.ncd iato IIcloon by ,t.elr and need. no h.lp bul ooly lhe ... mo,·a! 01 an imprd 'n><n.,"

[TheJ ... hole nalure of body d_ 110, roN;'\ ..,lely i" ""en"" ... Ih.l " '" My ,n .I1e. ~gu ... "nd mObOo., bul th~t 1t.:r'1: 1n".1 .'",,,,, .. ily bo ....,....,,,od ,n " somelitma ... boch ,,~latod 10 ...... 1$ I.,d ,,'hoch II mmmonly called ~ulloun" .. 1 form. ,-

II .. ,herdo,.. ...... ,,.,,. ol nd"",11y .. Itndtna lhe CNt, .... pejoct Ii ... """" al ..... oft' lhe .. oriel oll"'" boondary lOt lId ol pu, t 1I00000'ry, IbM" fiau .... 11101 1IIm",_tn ........ lhecaYAlcO" ... .:UOM ol,bec"'orid ofbo<hrs_ In labaiz. , _ 'JPO ol ..... lhanllocaJ bnn& ,he doft'C'rCllIUI~ """'- """" tnpped ""'.-..em ..... lIIy .nd ah>olute 01,,"., ,,0;" IIf cmC'rJtIICC 'f fOIIIplridy ..... te:m. pot1IlIt<)lIt "·,,h tho ..... 01" ,he bn IOrte ,I ........ Iud! ......... ,be c_", locale ... """." bo .... 'f"" pnCM 'n .... y 1Ibpt111O\1J K1\ialll.ObOII. The , .. ' .... 1 Df lhe dt ... ents 01 ~'-e Iocu .... kes" poHJbIc '0 roll"",,1 lhe ..... 1 oIlbe body I, ,he .. "'" mile u ... ,'CIopn, i.l fu,ure 'h" II ..... 1 perm'lf {he pe<:Uloariy ph)·tko-mathmu.tic:o t notion of 'Il< pa"""""p: ol l r.~"' .... Qua"u'y Ind .. "n.ion ... ,. no po ...... tn aet aDd .'" ,1Ie,..r" .. no' '"'011"01 '0 lbe "",,,,i,u,io" of plty...,.1 '",,".

" The I"0IftIII'" .cq .... mofl ..... , ... ro.a 00a _ raj""" Lbo: 'u4uo .... ,""'. of •• mallt< 10 •• boo", ..... of rmpcc .. by ImpKI .. n... he ron.: ;, _ ·pumetnc~ " doeo IlOl ...... ' 1IIIy appal from I: Gd')' Ibroup pardy ma.he.-malocal <led_I; "IS pecul .. "ly pb~lhmlabCIL.o tho "'''''' lha.,,,,, (:IIUW roolDfCI1OCI lILa. I'nk< tb< d,If ... nt ,n.'anU (mea.uraJ by ,be clerne". or live fora-) ;, 10 be ldenl,f"'" ... "h II>< I)'nlbcw clfecu,d by .he dllT ... ollal 1>0, ... _ ,he lpace aa •• hod 10 ,he in"an, I 100 .be .pa<>< .u""Md 10 ,be inotanl ' " Dr II II 1, ,he,-d"., .. IWO ,pry 10 •• habilitate ,hi. force. 'II'''",h .he Canes'lIIs ... n~ 10 F' "d 01100 qUICkly. 10 IOXn iu ..... ocal ;rmlll<1b,hty 10 ,.omr<ry. by lltocuiahol ,1 .. "II .. lboory 01 'imaalny. and not 10 rat CO.UUI ,mb '...."dll •• n .... 'fad .... N ..... 'on ..... do. Labnu, ha""1 .. ad 1'0""011" book. ..... fid<d 10 HuytclU Ius trpUl""N'" roo- , .... ocaal ........ .... brdded III 'ho"p'"1 do _ u ........ talSd M lXNMplIOII ol .... my ..... nrxtlOL 11 would WO/II\ lha~ ac:wnim, to h"n, ., IS IKIlhiDB b.n ..,.... ,"""'lH'f'CIl and IDClpbnlbIe property' " To 'h .. Huy~ rcplio:dc 'Rep,d,nl .'ha. Mr SC"A'IOII ... ft U .hoe e;>\Iie or ,be udal ebb. I 1m "'" I., all ... ,i.1Ied "'jlh i •. ...,.. .uh any or hll 0'11<1' 'hMnn. ",h"'" "" ha"" G" hi. prillC1p1e or amacuon. "'h"'h seeml absu,d 10 "",,'"

AI a~ "n<:Of'PO .. a1 .nd ,,,,,.pliable pl'openy·. tbe ro"", ij Ihu. tro«n ,n I llIu .. 01 pu .. onbcrancc ... ".:h cIa,m. Snally 10 JI'UP motion """",,nlly. a' Ibc "' .... 11"", U rncrvtJ\11or ,urI,. pin 01' 'M" 'illle ,nlenonl, <O_Q'

If.ted II ,II 1'01.111 oI.ppI.:"bon; lbe Ioroo .. a1"",,. ,mplanltd '" .... Il ... I .... ' ;. II\) .... ~ oadowcd,.,\h Iorca."

TIt.. Urw.I<..wtdtolalwayo a1 ....... IIdto be loe' I b"b .. burll oh,-..Iy' f.om motiod. tm",,"'''' to pro< d. II __ to .. hl.t is moot U'pl by propelh.,. In . rro ........ boK m""on is to 'njCCI """" octlvily in'o .... tlon PrelC1l1ed H ~nc,,' IIhu.R tht-rtby hI.' .. ..,me roroc. bul . ro..., ..... 1,...... 'nlO .""nonly b~ tile .rro ..... ,..IIboUI ... ""d fo, wh.1 Uibnl1 ",,1I0:Il ,Ioc · .... nne' or ope"'lion'." "'hieb cunously bnnll' " c[ojc, to lhe ;nc<:><po .... l prop,,,toeJ 01 KlIol ... IIQWI! The foroc JCem. ""' ........ 1' Ind <'Tn 'truer' tllan ,he Ih,n, iulClt The UndcD,.ndonl ·belic' ... · '0 lhe 10f'C.'C. bII~ .. Iocn il boItI>o$ up .""h a ~ 'n order 10 """" done ."b ca...ely. I\ .... Wd ... tI •• oo eaoily ""th tbe plI'II'ef' oIIClIOI ollhe mati .... "'[tbowp "IIoCl ,,,10 • polO' of ,be lh.tIJ. ,be am>'+"

Uhl. ... llllocII' '" Iaptlll .... wdc it erAlZ ol u'~.y. In ",,",III I" poinl ol IppIoCIIlOOll, II ... p;to all II. in~lISIly.

,

nit ("iCt"'NTIoIL'l'T OF nlE ~Il fOAl

The Ioroo arrow..".,.w hk. to PQlludf o,rr", a .... bly. bul th .. llbel.y> 100 rnuch and no!"""'alI: " _..,. >1"1'& rq>dl"'1 rtJdf .

Llt.fIIIed""""" .. no IUl'"., 10 I"" rahNn of Iorcc: ;'1<1*,"00\ II I dlUllna "I ""'Y ol crallq' lymboollC rdatiomlup bet ..... n an ",.n ~plltal ,nl." .. 1

• raod~ ol ,,,,,,,urn bet.·_ ltunl' and.n ;nl"",,,, f,cdr p ..... by the f"""", 01 11InKlion (>r ,",,"I .. nn. MUI In th ....... "'dle. lbe ru"", .no .. ~ " .. If 10 he ...... lIo .. ·td 1.11' In lbe 5p1l1Lo!. I..,bn~ rccogR'lCd Ihll il .. ' ....... nlie moll"" llIal would 8uod.fy o.paa: Ind do,u!>sl.ll.ntllic (0fCC 10 Ihro ... bnd", bet",'<C1\

'1*" and act"' .. il .. ....,.,..,ry 10 pmmo~ the MIca oil p:ncralilCd cI.IlsbC,ly lhal OJ Ippbeoblc 10 thIn", tIJu .... lor""" d .. UI.DCCI and culOllIO 01 ><1M.....,. ..,. lbe L;ndcnuodlnl-~ .... the tde&o lhal kd IA>bn~ 10 <be _lOll ollhe .,.._ ol """"""""

"tbe .,...,_ iI DOl -.botrac1cd' r ...... _iN<: ~11 It .. "'" In .h!.oIulc opIIa!

llIal Clll:rnl.Hy del ..... ,,"'" .... lmaI 1'011 ..... ",hic:b, pcril.lro ,,,,,,pod '0 ootid· (>I[\)' let lhem .. t.n he Il'l""hc:ndod .. ""'i'''''''" ....... I ... Mo road •• .., '"",1.11.. ph)'lot(:ttl poinlo'. Ihunde,bolt> r,,,.., the o."oily.nd not .1.10 c/lhlOlP_ They . ... ",,,hOllI paru "nd do nol 'nl ...... "'Oleri.lly lc,bn" It~"" In ...,.U lbal .""", ... JIIP n'" JI" ""I lid .... bJI~I''''''·. TlK:y.'" bo.iic:ally d, .. y"''''''Inc..1 .nd a", nut ,m"",n.od ," lhe .,.,11 ...... The .,.. ....... """I~'IU !I(I """,iblo: ",,11.1; ,I do>o nol eo<rE (.0nI lbe Ib."oct"'" ollila, .hiett doa I" ... : pi"" bul r.ont U lI,..itJrl m...tJOn hc1""«n ..."...,.~ 'hinp.

The _IU',.,D 01 the 'pICC ol.,n""I" ... 0111, '" r.c. I'f""'ol'pooo I 10 .... ol roord, .... ,_ 01 tilt "'1 ..... 1 "1""'1.11."""'" 01 mo ....... Th .. II_llhe JI<1p1e 01 ha,mony) ..... UfD -=II It1oOfIaOl lhe. 1'""\$ q_uly of CI1"OODC. and '0

co .......... 1Ie .,... ....... 10 CUfIC'CI"" the f .... unfold"'l oIlbe 'nl ....... 1 ""-' po ..... 01 act,nl- Th ... the """",n"lion of ,to. I~ quanllty ol h, .. for<>: " '11"",1<:" hy ,hi, "" ...... plc Ind no! b)· ....... ind""I,o" 0' ..... lIlied ".bili,y 01 .ul>otance>. Th.", arc ne,lhe. 1""'" 00' · •• ternal ""n.t""""' fo. 1110 .• p/JIium. TIK: <000,110111 of ~ "",bkon .re an inl.,.al pari or I .. ",l<lIilibtlily .nd "" ""'.' ",I 11'Icm..t ... up ... impod,mcnlO u:oooo lUI '"'" ~1I.mpU 10 see lhem ...... kiPI pouiblc lhe r ..... uofokj.". olllle Y\f\""li' trI u .... ~ by;t.

lcibnil tonlinuaUy dr_~ tI ........ , 0( Ihin~I"1 Pd tu wOOnWIIlc 01 'M'" "';1" ...... t"'l the ",hole iOlo pam. .bocb _~ ... bd>e-o-.:;" II .. ,..11""'" ol"utmutr IoR:a. Nil Pt~ tak. pIaa:.n lhe .,.._ cvat if lbey ... III.IJI" •• ' ,n tbe .. l<m-. hucna.oo. pan. tbe I.d: 01 .....,.illlllc. ClI ....... llmpacl; lhao: "" ant} the m~lilaled n:s.d ..... pro,ccted "'III II", "anl1 r:A lbe .. I."""'" 01' aboolulrty In'cntlOl fo""", thai.,.,. only be und<: .. tooPI'I;'- by ·OOI>Cro'. ph~'. oU>c'IO',SC b,o",·" •• mcllph)'Jin.

~ or Ihi, ~~ phy,ic:s. 10 dcfino .... nlio dOnla"" r:A lbe: '1".1"", " IIN:cl$al~y ,0 !>nil, !nlo be",. an .. ' ....... 1 'a>n.lrll1lll·. "blCh io lbe \'ft1'I\! <Jl a l<>Iabl, .'1Ioso ,n,emal cohcooon "'" bcc:o lInK ... and . 'toic:b Nil 10 be .... orcd dapil. the ""niallty of lbe lfMIfll-&'lOn. Th .... & Ioroo Ilia, it • .. I.'lIar u> • 1" ....

• , ...... ,,,,,,'I'd to. con," "d an ."" • ....,. ~ 10 In &I'I"Of'OIIie. """" ample .,......, ,he", " • kll,d ol .... 'erialwotJOQ cL ,he """ ........ 1 01. 'otolil, ,nU>

m[ o..cllANTIoIE!'IT Of" llU; '"tUlIAl

pant. lc,bnll'. concrete ph)"SQ mvluIK""m 1M cheory 01 Iblc~ cho O;UCIMIU .nd ... huxu ...... t df..:cs IJ"C no 10..,.. _ ct."' .... 01 \hov."'~ !he u ......... 'J 01. force II __ ledt j;\ll mn& • ........ ·,ed ,,"h cllt ",guJau"" 01 ... · .....

J TH£ paINCt!'!.!! OJ" VlaTUAt VELOCITIES

W •• '" ,oon, 10 tft how cho ptI"""pi< 01 '-tnual >'eIOOhCS npoundcd by Upn. rcspolldt 10 che ... "hcs oIlcibnoz:

olld""" n, the fora: ",Ih palle""" by ~nd,"" ,n the ~Iu"clt)· of a pol<nU.t I 'mHldle bel,. ... n pow~r and .,;t,oo·; dcmyollf)',n, tIM: r""", by a,5<"I1,nll ,he Ir ... unf<>ld,n, of lponcallC'lJa alld by ICUpI,n, Ihal .'hICb Inleract ...... "Ito o. pros!a in 1M nl.n,ion a. "",,"ra'nU aJ>d rond,tlo ... com,nl Imm ",na,n pI. llllOn. oIlho .,..,i_ tM ptllIClpi< 01 vtr111&1 ,""""'1 ... os ... !:>1, ..... a I .... .....:IWII<l11u.1 no IoD~ """~ dooIn bcfoR the pan,,,,,,,, or ,be ''''''''''"10", the 'dtl;t.all· 01 cbc .,.C ... Of 1M d'l'et§tly oIroro...

F", Founn. a tnt"' ... ·ptUlo$oplaK1r .......... nu had '1"'1)1111 hem bont.: 'HII """,,,pal ""'k. ,he AIKII,"finII M«1wmIa.. could .. CUlly he caIkd !he PlUlooophocal \1"""""",,, rot it r«I~ oJl ,he 10.., 01 cq .. ,~bnu", aDd ",olio" 1O'"'''Ilc ,,"""'pic'" l.th lcibn ... LatraoF ~U 10 cum, ... Ioroes ... bert thrr a", not Imodt,td

In ."" .. l by force Of p"""ef " un<krslood .'halov~, ct.UM Ilttplu" Of tenlh Co ,mPl'rt mollo" 10 Iho body 10 .-hoch II i .... """'" 10 be Iwlotd, and ;c II al", by the q" .nllly or mollo" impa.led, o •••• dy 10 he imparted. Ihac lhe lo'<:e 0, po"'O' m\lOt ~,pr ... " .. If. In lhe Ila,o of.qu,lIbn....., lhe f~ hal no ~IU41 eIf",,~ It prod...,., only a IImpic lonlkncy 10 molJon: b"l " m,," .1 ..... 1' be om ... uml by lhe eIf""l 111.11 It "'oold prod""", if il .. -e", nol """pal."

A tyot .... 01 bodoes bcln& liven. I can by lhoupc ~ndow ic WIth ,mapur)' rn.£Inef\I. 01 mobilur .' .. I....r d •• "hymenc .. "ThaI: d .. pI.."tw:~U att by no _Of c:omparlblt 10 'smalr'md' IIIDtIOIU 1ft clle e"onTl'll;" il ......... ~ I,berale 1M """ ... fi"ottIln\Jnf:nOOn ,n the spaloa! Cn lranofi ..... thrm ,0 mobilt: uoll. of IIIe .,..,;-.

St.led F""'aUy. the """",pit aIIowI bod;" 10 mowe .houl on "'" al """ a"",he •• 10 dI>'~ aDd ncn not 10 1I:$pe.'t ,hi< Ilttplial Of uphcn hlt_ of lhe .,..1..." cl"l1 lbey IJ"C ... pt'I 110 ".".."c~, .. n.cs.. dlfpla"" .. ilu caa OlaF aU cbc powell 01 acllD, aDd obi,. che """haruc CO rootSldc. tYCrylhiul I .... ' a mal ..... I.)'Ilem " cal'llbic of: 10 make 0 ... 1'1111 ,liJc on lnolhe •. 10 lid """ ma .. apan (,om anotheo- Ind ~"'" '0 uplodo< the ty.lem by Ihou",1 by b,ukio. ,hlJ 0, Ihol I,Won, Ltmll<d Co ,ho II~I"" ,110 pn"",,,,,, of .;,,"'" powc:n ton he lilted I, rollow" 'Far 11m" " of rom:. ' u hold I malerial

TKI! D'OIANTMDo,. Of" nlE '·IIWAI.

, ' ~.

"

,)"I.m ,n oquihbn" .... ;c io lit· ,,')' aDd .,,1Iiacn1 that any ,06",col) small ." , ,,,,1 d",pll,,:cmcnl impnocd on cbe ')'Ic.m mako lhe .um of the \'i1l ... 1 lI'otkJ o r ,lie (",en lake lbe ~.h .. ze,o: The: lIalC1!lC1ll IherefO" ",11'os to tho ."m 0( the .. ..,..b of the forces:

I ' ) (tile 6q, I .. the 'l'irtual' displacemettu).

It it to be ~ 1,,"ltbe: ".t<men' ,d.lI" 10 ,he _~ of the ron... aDd if theref"", • I>OOOtIlhal m,.lopo thrm by "!CPI"', lbe poin .. of appllCllloa co the boock£lllUnd. Whll io doruomllJUal .. ,be "Oft Ot lhe paa-.... IIIaI ... ....... ' thrm 10 .. nfold '"lboupt; it .. III >Itowtn. 00' wI.ft CIIpflbIc 01 ....,gallllt IIIe ..... h deI'cIopod by che mobibex, ""b ... Inch ~ endow the bod ... lhal .., CIIn dod"" tbc tquilibna aDd IIIe ...... .....,._ We Iu. .... leu Ied 6.,,) aad an ,nlett""" quan .. ty (f,) ,ato play. W. can apeak 0(. 'rlpty<~:

", .. how a't"IInlCfllCn1 II .b:Iolul~I)· idenlical co Ihal of "",,d,u,' 'l""n"' ..... ,,,trIM 1.11, d. VI_

Il • • e I&IIln, !he "n/oIdillt 0( ,Joe ph)"IIOO--malbenlallOll '''IUllbly i. ~hic>"Cd by diQI oI!he ,n\e1\Jtl y T, .. hl f, a .. IlOl ,It< Aril) Ion:a ,n lhe ord,"'ry klIIO aDd if IIIe 6.,. arc 1101 _.-<il, dim ........ 10 rIC!. 10 kno., all che elf ....... 01 tbe fom:o 'bal loCI 011 cbc .)">1 ..... ,I u ,., p,),.Dd ..".-.... ;,,11\ co know cbe up" •• on 01 tbar .. nual ... ork .Itd, co k_ I .... ClpraRon." .. '" '')' aDd 0IIIIi0u01 10 t"""", ,he lUes F • .. blC"b CIIn he atlled...,....tUed Ion:es. It .. in be """"" thaI Chi< nIIl ..... DI. F ..... 1u:ed fora, f dcl'" .. do 011 che ... 1 .. ", altM , .. riablc 9 10 .. -hicb " ..-la!"" for IIIe" prod..:! f6q mu>I alwaJl "'pr."'" • "'Ofk ~w"

In <:e".in ClIO<$, ,be applieallon DIllie p<illClplc 01 Vlttuat ... ",k< Irads '0 cbc ,nl.odUCIIOn of a pot.nlioJ r"","11011 flq,. ~" ... , ~.) cf the pflfUIe\en. makin, il potl,l)k '0 'M~I.", !he ' ol.hly of lho acl""" of <:e'IIIin 10 ..... c.l.mal m lhe _1'lcm.

c;e .... ",11l:ed fon:ft lnd polnnial allow. mudl more « .. It"" II'UP of Ii., ~uilib"um 'han ,ha, "hlct> confi .... 'IKlf 10 lbe purely ".'oe con.JdcnI,non of a pa,"llelOll'lm Q/'fO<Cft." Wllb lbe p.i"",pIc or "." .. 1 powcn. lbe f~ndamtotal ",lIuo" of I,," dYllimoe is ellablishcd ... ,I .", for ,110 scI11<. by re""hon, an ~Ulbbnum ...... IMI 1110 "nual .. od. d" ; of ,be '.,,,,nl,,", f<>m$' f, ""' a>mpeMaled r .... by tllo work> of lbe l'orce. of 'neI'IJ& J. proo;wd,n. fi'Olll _lenl_" T1w>c 10" an boo np.u.d ,a 1110 Iurm EJ,""" IJ>d. ,n ,loot baht Q/' 1110 fonJlOlIIJ. ,I " IUIl"nl 10 II.Jme IMm ",,"~II!n /arrt,.f 1 ...... 'Ia. TIlt funda"""".1 fo.mull of 1110 d)'namic ""I"'''' Ihto Ih.1 lbe lum of Iht IWO '1U1n!!!MS MI, ,[ I',olt, and JII', _ [J,&I, $bouk! ~ ... 11Cr<' I,,, IU ,ho 'U'1uaJ d"f'lK ""'MI impoootd on ,be OJ'ltem ..... ia Ulbrr It ....... lbal one Iw.. a''''J _I:

'" l'l>ese are ,he famo". /~pU"l~ ~",,'I"wJ. ... hien .'.'e ,h., II Iny m<>monl 1,,".1ItlI of ,II< f"...,.. .el.n. (NO lbe 'Y"ffll ~nd lbe ",m 01,,,,, fort:a of ,nenw. wwld be o:o.pable of k ... pon. ,II< OJ""" in ~ui~b<iwn ,n ,he 'Of)' .1&10 '" .'boeh ., .ppca.t'lll Ih .. """""ot: 11 II <aiy ,0 dod_ fl'Olll them !he Ihcoum of I .... f<>m:!l: 'The . 'orl <1<:> ...... " a panxubr alDOWll 01,-. by lbe rW Io<CtS lbal tollCl' a .) .. ,cm It ~Ull '0 the ,,,,,reaS<! uOOc."""" ., ,be AInt ,,,"" by lbe .y.,om', I .. e fo . .... ·

We ... now in a p"',uon '0 app'oa.,o lbe whole Io'<:e of ,be P"IlCIple 01 VlnuaJ ."k'c" .... by """,ion.", IumstH .. ",j.." uIf ,II lbe L>ebnwan parad,se of lhe >p<II'-' 11 .. , of ,he ..... unal dllplacnw:nl of ,ho: powm of """ ..... lho: """''''IUI: o:o.n In.-enl new de, ...... of equil'bnum and "'" lot h,nud/ be iubjU-p,ed by ,ho: ,nor" , I,,,,,,, of 'real Joroa', .'luch .... ~uo.naIIO II", ')'I'1:1Il.

In ,n.. """" of lbe d)'nlmlC, Ibcot fon:a oould be compemaled b)' foren of IMn, ... ,~ M .... do'""" of ~~i!ib.i"m n,,,,1 lhem"", be ,,,Hinenl!) amplo '0 Illow lhe oqu,.,.,,1e«o: and acha"", 01 mob.hlit> boIort"", '0 U ".l<1l'i Df'Ik" .... radocaUr hrte'ol"''''W''' lhole oI...,.net')' and luMtI""" ~. "'" __ IOIal '0 .... boooI. in lbe .ale of lcobn .... ,ho: pnoctplc .... t ..

"po6IIbIe 10 V&$P lbe Iorceo .... ,emoJ '0 a .)'SIcm 01 bnda .. lbe ICOn 01. m~,~~""" of II, • • pur,_,

Mar. proc, .. ly. ""b lbe ~Id ollne """",pie, ,lit meo:hanic .,11 be Ible I"

"

rortoCc1 ... of lbe Iota::o e..,em"1 10 11)'>1_ r •• ,"" Iooas or bai!iOll "" .... 111 10 l.yJlem r .... b"'h i' ........ ample ,hln L, H~ will .1", be able '0 br,n, ,''''''' bact ~I ,,·m by ""n;ol oul };' IIp,n in I ."".bIe mlnM'. whoeh " ,bon uf1<kDlood ... ,he 'U"'PI<: oil. and 0,,,",, » .. ,.m •. Constdcr ,bo.dore ,'0'0

")'.wms 01 bnda $, &Dd S,. U! &11, _ [ri&'o' (...."...u .... y dll; ' [1':&,:) be !be mttJ&! .. OI l of ,he Joroa ,ba. act "" S, ''''''pccll.'dy 5, ~

TIlt....", of,,,,,,,,._ .'OIl<. al,ul1Illy

'" If. DOW, 1 COII$IlI"'. II 1)'11"'" of b<xiIH ""h ,~ of S, and S, by j""¥p<IJln. ,hem ,n oucb I .... y '1lII ....... bodMS oIlbe 6f'1l l)'SIan lind , ........ " ... In

"",1lICI _lib ...... bOIl ... oI'be wrond 'J'''''''' e.ch mtual <I,spl.,,,,,,"' of lhe ..... llanl .)'1, .... ,"'in imP""'< on ,be "..m'""", 9: and 'If vana''''''' ~ and

oi<6. Ind on ,he """,nl """"" I "''''\

MY .. 6W, + ,lII~ .. r. f I oII!,' .. r. f;~f 15)

lIuI """ .. ill not 11"")'IobWn I ~in ... 1 d",pl" I "",., of ,ho: .... u1'.nll)'Jl .... b) OlKIIbin,"1 J"" Iny nn.w d,.pla<zmml 01 oJ'llefl1 I .1Ih J"" Iny Yln.w dlSpbcelllWI of ~"em 2: , ... , .... P'*'bie .-hw each of lho: JJ1'Cft'lS ! &Dd 2 ..... ,ed alooc; bu, ,I\tt. combtnlo,,,,,, .... y ............ ImpoMible .. hw .,.,_ I and 2 arc ju.1I"pooed. bo;a.UK II """k! .... " Iho .Jfoa of brln"nl.' ,he ... _ Hme. 10 a .mij.lc poim of.pact. ... ,nc bodIeS oIly,lem I Ind ....... bodies 01 I)\,em 2.

Tho ju."P"""OI' 011)".1<:",. I .Dd llberefo", Impooa. ...... """""",n. on !be di:lplw .... a .. 01 each; "'" 1",,_ and I""'" , .. _ do "'" lea"" ,he >'aI_lloal OIl< c;o.n .lInbu'e b)' I .,"' .. , d""",", ... al 10 &,! and 101, enurely M.b.,,,,'Y, They de~ 1IlI' lheM .al .... ""nr."" (HIot o ... nnl «] ... 1""", <>lI\olJ ~quu'lo<u oIJ llal ...... "hoeh u ••• ,"" lorm;

r.C,'&l~ .. r.(':M-O (l)

To the COdIh'lOftII of canccllallOG of lho: """ I ~ ~ 011< oIIOotld lbe ........ add lhe romItIlDlt 1' ...... by I'~ lho: dlmlDll"'" 0I1ht cI,tplw, menU &,) and &,J bel-.: ... n lhe 0\1"''''''''' (51 and 16, bnnp " .. 0 bela. _ q,wIIl'''''' '~. ,he ~Icd L.u.t'~"""" ""'Iffp/ ..... lbe number ol .. b",h II "'lUll '0 l loal 01 !he «]"'Iion. of 1""!IIIn Mnd which o:o.n be mlorpmro U ~ '" IM,W<!, Tbeoe .. 0 ,ho: """"" lloal compol S, Ind S. 10 """",,,lute a ...,.. l)'item Ind "bicb "''''' "", .... all)' be alOP", ,ed U .0' ..... '10 lho: .YIIftII 1:,

1 0:0.0 ....., "'" Ille .)'1, .... I: 001 ..... n In'o S, and S, and ,h ..... ,<> 1ft lban ,,"u,..-..oUI"1\oIes 10 .b>Cb I ooould apply ,be pr,lS/;IpI&ool .. n ... 1 ,doa_ In ,be fo"" .,...,n on ""I:< 27. pro.'oJed r e.p1;"'lIy add "" ,ho: "nWlI ... "". of ,he fOTClCl <>fllal ..... - .. hICh ore 1.'''''''''/10 L bu, ,,'h"'" "'" ,""W ronco,,,,," Ii forta "",,.,."'" ,0 S, IDd S,.

We an .1111<1111'" 'I\i< double ,a"rprdlliool ... ·JlIt ....... tnmpb..

Two "-I'll " and U ..., artJ<:Ub!l ...... .-.. ID " .. ~.

"

Tho:x ban c-.o" be ilmlPlIOil a. bciOI /j>t1.J a1 0 and at M. n.c .. lOR tw" eqU;llions 0( IlluM/n, "hkb 1P,'e 1'101" Lagn.Ojpafl muluplocn. Thete qlWl"h". """ be ;a~rprelN as the ~ thaI compel the ban 10 rei.,.." lMl. joinin," Tbnc boB all liso be """pDOd ... btl", 1"",', bul each beuII ",~1 10 the aclK>l>l 01 Iu.- A, ano:! ~, (A IlDIl<rJon lbeto actioaa a, 0 lI>d II ,\/, B a1 M ollly),.

Tbc mwuplla IllClbod (IOn abo be applied 10 dy.....u.: tquohboa. Tat" !he ,,;uompk of. [,oUt) IOOlI do." &II mcIinocd pIaao f_ Ii&. 'J,.

,. • • • • • • , 1~"'" • • • • • • 0 ------------------

,,""'

•

We I:IIn IIdopIlbc dw -I ~ ...... <>adI by _".dena, 1M trolley as lIIbJC><l \0

'''''' M<d; pI.'>!y and the raosta""" of .ho VOIUld. tbe rawlant of willch a<1.$

an 1be can. .1uctI1S .. • ""h sub,.., \J) • bDd ", artdic:a&Ily .... uccd gn..uy h. .... GaljJ.oo'o dr.'", Intk 10 !Judy the -.I"" ...... dooooo' of fllh"l bcKhctl.

One oould.lto ..... """ Iht 1 ..... • trolley as IUbJect 10 _ ron:. only (,..,ural F""l)~ hut ~ 'ofol'- ,/or .. (the coDdluoo "<>u1d bere be ..-nn ... ,-I, ... ,. h .. ""lei be _y to """"",,,,-"'Ie thai _ lhl'l ibid wrod_ ..... n wnb the "'IlIInon ..... n b) the Ii .. ! app.o»cl\.

Ln ... I\01It tlk" a fiMl ..... mpk from ' ... ""nF-'" n", pru,""'" an <!<pn. lil.u.trat,OII 0( 1M "'1 in "'h,h "'''''' """""I'll rtum m«:hulCI ('n thIS .... prao> .. rel .r. comparoblo 10 .mual culOuU. Consld., a ftu,d In .quihbnum

('"' n" 6).

" ,

Stpar.Il" by lhou"" • pan A from ,ho mt 01 tho Iwd. l 'ho iatttr is n ..... di'tldtd ;0'0 .wo pans A and II oepantted by ........ lumoe $. If" undel'lOO'I I"'""'. dliplaceaIl!O'''hoc:!! ,,.,...,.,'" tIII_. till! '.0 pIlttt A and II do "'" QllJIIk; Ihn d' .... aw:n' could •• .,,,1000 IlOl raull r.om just uy disp" 'aKat of pan ... ",k ... 10 itoIa'ion. in COIIIu"",,,,,, .. "h JlISl .ny dlJplz ,,,.,0' 01 port B. "'ken .n isoII.tioD; lor each oItho t..., patU ,he COftlact <;:OO$IilUI ... Iiai _ _

koeplhe form.1Id poIilion 01 pan .. but .... p(Y OUt lho ftWd by t""""'l. 110 WI pan ... bcoomeo • eo";ly. Tho ftwd from INn B • .. mh IIII! <>b!.u.<i< 01 coolact ,,"h ,ho ftuod 01 INn .. ",,,,,,ftd. IJ no kM~. nalurally;o equihbnum lrid ha •• 'enUe""y to now into tbe i"'eno. of lie eo,,'Y.

The muillpli., method now m.k.,. il pmsobl< 10 silo .. ,hat i, "ill be .... ,on:d 10 "'Iuihbrium if. Ion:e thaI it norm.1 10 lhe IUtl_ S it al'J'licd '0 each .Iemonl ~S ""he O"tf...,. S. 1k"O''''''''I 10 the 10IC'io. ollbe: "jpOrI A .nd 01' II\qrUlutk "",S. the fOCl« • • .,,"""'"'''' the hyd.ost.lItiC PfC"'un:.' Ihll poInl.

Wbm lhe IWO hod pam A and II arc;n conlKl, ,III! hyd.DWltiC ...... " .. cion IlOl raJly act on .. ,1II!r one: bul If. by '''''''&hl. one 01 Ibtm IS s"pp" I 110 lha, the OIIII!r II lruted .. if It alont ".\dIed. lite bydrooAlic poo '''' .. mRII lor .pploed 10 Ihll ..... 10 ",place the obotado lhal lite ",he< 0Pl'" ·.1 10 no _""'_ F.om 'Iuo. Ard.n,"",."· IlIo:omll an be rudily dtu ........ tal If • body IodJe> ludf nactly III till! COVl', A. tI will be SUb:locllo the act"", of lite resultan, of ,he ro.o.s ol PO' 'un:. oImouoIy coqual .1Id 0J0I1CI ,d '0 ,he .... &hl of III volUme" of .... ,er ""'" ,he auid ,,-u ,n.hally in equiltbnum.

TbIJ pmo of ampliflcauon 01' the l)'tl ...... rid cu''''''' ., ..... l In Ilc op"lIcoIIO" 01' ,Ic princlpl< of ";flWIl .. do"uCl;" ..... 'o •• oe,,, ol,hal oIol!o,noo· ,ion.rid prot""'" t_ I. I). bul here,lhe Cl.n""".nd adJunct""" an: n"ulll!r h •• mle» nor arbitrary. H.",..., <all find 11K n .. onl of d""p"tln, ,he ilh"i<m nr the trl't'd ... ,ble .... nonly ol ce<1.1" 10"",," "", .. un. I .y" .... E by .h,nIt""

" .... ,brm as folCU of huon 01, bu .. " .ys, .... 1:' II:. 1>010' ~ ... PO" on:. ~I ~ out.,cct 10 COlI' .......... 11 II •• othtr ports).

111" II tho va' menl of LoobD ... ·'IKd,t.100Q1: 10 ha,~ ohooon M. 01 ... ,),. boule naod ........ 1 the ""'10.1 .... ' abo. baule .... 1I"d ap,,,,, !he .....:haplClOl fOfUll of lMltracuOII Rnd 'h,' ;1 could only be .... "" by .n''eIlun, . he cun<ep< of ..... rl~ .... wh",h dellroy> "r ul"'nllle 'ml"I' 01 CUIOut .. In Now .... , In

lhauthl. n .... fora: all""'" i ... 1( ,<> boo " ' iPfl<'! of il< m>'11U)' by ,ho.in. ".,If clelio.ly I •• <;(Indlaon. pori . nd pIIrcel ollhc ,n1dbllblh,y of • ')'>I~m; il i> born ""'b lhe ceslu", "'h'h .. pooraltt alld lInn. aDd lh<r~ boro...., In InlClllo\lOll bet ..... " Id HI"",,," and I n U\l.....,.. No Ion .... , •• he fora: In .now drlVlO, to • ma!cmaI pOInt ,lid 1;("1110"","", pusIu", 11 b*:k, ,, limbo<!· ... 1M elrOtt ol.houart ...... ,. r..,. of ... ,,,. ,1XIf lbe mr""" of H''''''''' II ... or lha. II"-n. ud doos .....Irm porion.,. •• III tbe (011< .... 1110( : 01 OI:>IIIo'"U''''1 • fYJlf!I1. The ro.c.. """"mI (1'0<11 .1Iead..,.," hUI$ limp/}' 10 be e.p!.o"lrd by • pamll ,0 both.."...,. 0( ,he . 'onl poml "' ........ ",llIeh ;.,. ","",Iy the J'H.I""hl"Mm oil" utlpol>cJll monad f-.l ... ;,11 p'OC,,",I'''''Y·

This hal ,.porco_ .... 11 boy"nd lho """,hanism and .. ,abh.,," a"",,· !'I'lauon!hill btl"~ ""n.iblt .1Id ,nlclh"bIc ..... ,l<>". Inlrlh,lbie ",lUC' .. no \o11J11' obla,DOd b} "J"bl",CI,utI 01 """c.mi"."o",·. bu, by. !!til" Qpatlly 10 ,,:up lbe ..... ,,\5 in lile mafbl!!' or lhe ..... 0 ..... of "",,"lIian· Tbe Ltobrul.1&n Ihcor} oIlbe oqw""""'" al.I>II"""'_ and "'"'" ...... ko ... ~hi .... k,n 10 ...... >/tW _,.,. In ........ Ir)". ~no: hemp an:"'" . ·hlt rtm.lrllll wbtn all illdnld.,.,_ is ........ trl. .... 1ead tbey "'11>1 he ~ .. pa" al~ amplt pb)""","maIIomu.UCIJ bn ....... Iud! ron.:.. ... 10 !'I'<"OI1<oda lbe rdallOnJII,p beI .. ~ Iop::al ""pllCtllOll ..... !'I'al,mp(,calooa.

• l ' ACCHY ANO PO ISSO" 'S VI RTUA L CUTOUH

1'0" hlV& JItU _n 100 ... in ... luy pRpIlm ,be voulld lor 11Id,,·III ... ,1OII . nd hnk. Ion:e to. , • ....,.. al pa",.Iw.llon. 10 • doned·I"", CUtoul • hod! .e_1s I'" ...... ,y alall oobdily by dnooI"", ,''' II''''' p;uU. Doned bnet bna,'''' ron.. lOIn. beQI_ ~COIIII"UIC lbemfd_ q:".,S1I'" li:l!um. .bdl red""" " 10. pun ""piAnuolXlO Df upu .....

Ru" .. aber: l.e!.bru>: dn::omed 01. m«hlnQ 111>., .. ould WI POln" .bIue·. ,..".. ambol""'" la lbar a""" Cauday .1Id POIUOII p" I. dod. ... 'e shall .... • a Juch ••• y III>., I'" JL""klr. would ItOIorIIirt be bappy 10 IndlCtl. pointS .... tloo'" "'pm 10 ,''',, poi"nl .. 1 ""KilO'" They.' lUI pto'>lled an .n • ....,' 10 lhe ptobkm: ho1\' 10 male. ",,,,,hie I"'inl OUI 01. pomL.

".tll",lI) . 'can ·111"'· myodr a point by limply i1allol' ·let -"0 he .""b-a"d· w<h I raJ nllmber oak ... 10 be on ,he .. ",,&h' Ii ... ·. or: ·let ... be IlItb·llId· w<h I poIOI on Lbe pbM : 'fluI; 6esIPliod ,,"""'b ·11' .... • "'" the "",nt .... eo. • 1 .... ,b 110 ~oon; I limply .... t. il a""ilable 10 "'tv!(. I dlllm 10 .. II>.UOI u by lluo .. """ ~ J ·roonl·, I ,,_ bu, I mobWlc _inri,: ,ho

n o.ltaC'l>O<I allh .. poI"l w",,1d be I maner allOtlJ ind, ....... """ 10 the poIDIi

"""'" " Tbe "1""1"'" II oompltldy d.ifCftllI it 10. e.urnple. I _!oIde. lhe fnet"'"

Ilxl"--"~ . , II .~ lendllo"",,,,,, .... I\.~) lend! lo"",rds ;nfinny (heroaflc, ... iI .. hal m.,herno.· IIUal\f wI • »Ok). N .... lhe \K$plion ·i, »tObItmal"'", it iI no Ion .... qunUOll al """pi, ,ubllu"u", ,be I~ Inler -"0 lor lhe In.,. '" u " ·",,Id ""'y pmdllOl! In .n.""III,. 10. ~ COn unly be VUPtd lIS lhe irwoa:ao.iNo..,.1 "h mobile poIOI 1; ..... be_forth. polo tluol ..... ,,'.,. 'ntlnu' ..... Ibl> in6";., cal! be ...... C"nOCI. Tho> .. Caudl,-. &mil If;\Q: 10 po.mol'l!H" I'll" '" lhe ...... pk>. pbDc 10 F1 ro,,,'" the polo$.

Talen IDp"""'-', ...... ". '011·. ,I..,&htl,,,.,, the "",nl·poIe .. limpiy ,\bal .'luch mu" not he bumpod ,nlu·; InfinilY .. · ... n. Ilia, ~h,n. lui. 10 bo ,nasal to .",-•••• 1 ... ·,0/ ... '.

• )( l(

• '.

• x ' .

HilI. j., lhe rompiol plane, I Uln F1 round lIoc oboIacIc (WI: fit.. n TIu$ iimplc obsnvalton IuodJ '0 Cillthy'l mid ... aoJcW ..... lhe ....... , .. 1

-'.~\I<_ <II .-h",h caa ...... he .1i1N. In • doawo D ... marded by. aI"" r 1_ 6~ B J and 111 I tun""",, be dc>dopablc 1010 • complete -a In lhe 'iaJuty al caott poonl d Ih .. """"'In D. A pom' Po of D bel". ", ...... he ..h .. oflhe {.n",,'oIl" I I ,h .. poonl .. l "".'n .. lOOn .,1'" .aJ .... of the filna.on 01\ lhe aI ..... r an: kno,,"n. Tbe cboooo of Ion r cd,..,.. Lh'''''Sh .he .. "bolo don ••• " Ihi. jllsuficJ the IIH of lhe te"" 1wiomo><pbf.

Th'~ '''''''rem .no ... ~n ,1«""".,,,, f.,""ulal;",,: POltilOn·"hoorem .... h",h .. moreo ....... ltd fo. olhe. d,met\JlOIU: tbe rko:1",.""", polen"ol,n a do"""n ole"""," OIIly on 't> , .. J .... 011 lile edt< allh .. doma,n.

w •• -,11 """ .. "'". lhe """""I part allho cIrmon"'.'K,n .hkh '"" ..... ho-oI Eo. C.oa:hy aDd Powoo .... oIot"""-l lbe: """'"" """""'P'_ of ,he: pOtDl .. I .. mplc 1"'"""'" .,.",bmhcd Ihrou&h de>t1""'1OII.

,

How do we bruIa 10 hk )11>1 an,.. "",01 ,Il00..,,, ,n .be inlcno< olllw: donwn t1I

If I 1m 0DfI~'" mom, 10 ..... ',Ral. Ill .. poiol. I pili'" In ''''''II'''''' finFr II .M pion£. Tho cOlllOuro or .hot plane "'" ",,,,",Jly Imina' ..... 10 {hi. apprOKI\. .'luck <:OI>kllx ~rncd ."h )1111 lor 0'''"'' ""'01. I can ma~c ,h .. )u.t .ny' Ibophu by tltann, I linle space .rooM p.

A. In LA,.on .. ', ... mpJe. Ih" """,,,IJ In alliin, 001 I part by ,,,,,,,,Ill Ind PfOpo1hni ~I' ,he"" ,,, venll,.. I p ..... ge btt....,." In Inler- Ind c>lcnor d~ on by ,h. 1<"""'«. The d" ... "<lon. of the cav",.re or Iotlle impel' taI>CC': (0 complcu. ,he demon.,nouen. II will In .ny calC be nt(4 "'Y 10 make III rad,U. I.nd 10.,.,,11 1,,,,,_ W"", is Impo""n. he ..... for p. 10 mllke an "peruo, 10 ,he pbno: and bc<:omc 1M l1sibic ocal oft"" ;II<,.'on. Th.u lManl ,hal lbe «I. oIlhc ......u camy II .. 10 cnc'- It comJ'l<1cJy. The ct.""", of .hi. ~u" uy' poIn' IS no km .... _n e"..,...1 preocnp""" .'hl<:h ~1II'la .pchlnok""b " POIIII .... uIfed'n ,he pto.,.. arnon, o.1Ia$. \I. lib Po ... rt .... Dy ""'talltd It!he .... " 01. mebe.11I< pta ..... rnad< jlr ... II"~ the carrl1&! ..... ""'" .,0 ..... "" po,n' .. ooOOUnJ:lJI nU""Uon .. -hlCh II ,n nn ..... Y ~ ""'''''' ol mdllf.", .... 'n ,lit dnawn D.

I'o~ ..... COII,,",,;,y ol ... ,""," dtlcllml '1f<'II' nu, I><~ II'" • pocn'llIN:k ..... aDd ha,'~ i, '~IC OIl, by UIJ'KI"" ~ omalI due aplblc ol ... ,nil", a ,,' "ric _, ..... ol"'lIo:h i, rrual" .1"'0"' be oa>d ,Iu, 'hi. ·cotta'i ... Iraa,plan!' cmk .. " • .I"a '0 Arulo,!.·, ..,Dslblc .... 'CC.

We .... n nO_ 11'J'f"CI&1. how far....., ha, .. rome'

...... "al out f,,,,,, ,he pol'" ""IIOOi....a .... 1""'" IN<I1Olro; 6,u .... mado ""',,'l<Iunl by ,he .rbitra. y 'mpatl ol,h. """.na,l"l fln ....

THII E.'fClfAl<TMENT Of' Tin "llTUA~ " - Mal, U • polo on tilt .,ra")n]" ... ,, _, aM dtreO:l alPI"" by prod_I ttIdeImni .... ,N>nI;

- .. I polo In Ibt plane. " .00 .... penotiar "1OIIf' il II ""'" ./coat _..J """'A one <IoD HClroCC and which CILII be II ..... dms"y

In e'l'OIIndi",,1>< IhMI..., of re5Id ..... n o~ Ih,,, lbe curves bnrdori", lbe d"",.," D r..d In .. usty a c:nt1J1I CICNId,hna. Spoaf\cI"y lhi. roodllioo i£ lbe til""" ha. '0 embna: Ibt pn;nl p. ""'" ""!,. In I _ complica,ed CtiC

for ea.mpk. lhal obo .. " in 6." ... 9 .... hue,o .pmf, tbe n"",be. ofloopa loCtually """'!'Ioled by r around Po.

,

' .... In ,Iu .... ) II .. culO<ll "'. OIIIaII..,n:Ic ""P'*l' ..... pIa.tI ol .. ..,..k 011 lite

'LU'ltl<!r By conOKkriq I poult u. ,.,. .... W,.", _ I doIl<'d4u>< IIok. '" ,n .pn.I: tbe F""'d'" bas aIteody 'HOjI<.1ed iw:Mdf inlo _bet i5eId 01 ,n'lUl""" lha, 01 ,be numbers of nnboau.nd dtfoltmlllOll. '" _,0IIn.

A ma,lItml1l<U1n familia' .... Lh Lhc lheory of re5Id ........ Ion .... hoidIlO!be 6", ... 01 ,h .. or lha, cun. ... 001 rol"'" 10 lhe p.lUIbWly 01 <OnualtOllll) dri'ornu", ....,h·.nd.....,h • cu ..... uno .""h .. IId, • ...,h l/IOIhe •. Suo:b • ""' .... COOn 'hrow 1I.1oopo nul ..... ry faT from 1'.110« fll- 9 J. 00' i1 ;. no Ina dd'<>nnIbic by., .... inlo ....... ,1 <j~1t Co. ~hkh K do .... mouo,,, •. uo;dinl,l>< po,n, P. whern.s ,,,. 'lilllt' <~ ..... C, by 1001""1 a I".. "".1Wled fltkllban r . """DO, be dolQ. mod inlo Co ."<lid,,,,}',.

II!E ENCIIANT)tENT Of TIn; \lHUAl

1 .. term 'mid",,' i. fta....c<l !>o"",,'cr btcall~ " lead< '0 "".unders'and,ng. (Of II I$I>~ DO mI'l"" • qlX'S';"n 01....,.,.., .. 'ha, """'" rcll"Uns •• 'ailabk '0 my action. The I2Icul ... 01 ,..,.;'10>« lead< U'I '0 ,he ant,podes 01 ,he trivial ""';\1,i!$ of markinl :rnd ",·.I ... UIII 'he """I1"',ioll 0111'""" or ,he passiMi,y of .hl$ gr 'hII' mo'ion. A spati.1 in.on·al upstt< no kno"'Ic:d,,, nd ..,Iict" 110 d,mc"';on. i, dofi_ 0111)' a l'mit '0 my moUOn hetween the I>odtel. II rrltrs 0111)' 10 a control .... ,he poos.ibk plO£e5; ,he ... ,due makes 1",1e of the ellonsion and. by appropria'inll' ·h.,·inl turned'. cap'u", a HICk of,he wri". a modali,), of II><Itl<'" ca""ble .... "i,rinl ,ho vicini,..,. of. point

In ,mill a poin, ,hickllCS$. 1 ."" ... hy ""u~ i, to re,'«berate .hrough .he ,"'1101. pLa.nc: 1hi," ,he minodo 01 1><>1""'0'1'hy. Tbe d ....... :ued. 'purely ~mc!' ncO peinl opp<»<:<l " ... If """ ... ly II' .he pLa.no .. a wholc: in de.«llnR. more .ub.1e dua~ly bo.wecn lhe whole and ,ho p'1f1. ,110 pe,n' . .. a rlrl"al ~" .. iry. ,n,rod""",.he lrome'.' in,o. new lan<l<cnpo. aim"" "Pm>! hi, 'IIIi\l: tho bole ,nVlIQ • number 01 tum> and th;, numbo, of til ...... nlCulal .. a po;nl and (.md;" 01 loops. C~uch)' and Po; .... ". or cou ...... did 110' know aoout a1lebnu< topeloll)'. bu'.

by decidlnl '0 ...., a poin, .. a prOVO<:.l1lon to utrICuon. 11>01' fo<x«J a "'"",pline of lhe .)'0 and hand 'hal .... capable of releg:llt.., the .""".i,.., pla ... 1o ,he wings and !If ...... 1 ... 11,,"11< COnn,V"""" he''''"«11 ci"",;" ,h ... hn .. "" •• iJclll sim;lari.y, In lho" wo'k. all the: ncb".,. !If ~1."iciIY and cu'ou" of the ,'inll&l i,.o be found:.he ,..,n'u,,"' of a FlI"re ,bat libe,al .. a new pI . "ic uni, and ,hat. 1"0vok'"11 rellC""'''ions 'hrou~b a wbole field . • 1Iude< ' " "'hrr 11""" ..... which ... how."cr ol .. ",y< detachable ... ·'Ihin a .i!llk horimn.

'The foU .... inl ~h.I".' ."."'pIS to olIow Jpccifically how vin o.li.y and ho,;>on af. l>nked. by <;Qn"d<:nn8 ,be pr<:SefI"'ion oI.be kinema'ic diagno"" "r ,Ite .",!>nol or r.ri. and it< .pphca"on .0.M "1I~ particle duality.

NOTES

'A,'_'" ""'.""" VII (ll. II. 1'131. 111 __ 'IL 0. Ill< __ of II>< 1'dI, ......... ""' ...... .. ,...oaII_.-....... ,,-M_I .. , I ·' ... ~I00 ,.,......_ .... How'". 'loa~.;.-r ..... ''''Ill''' 'rdupl'" V!. ' ,",·,04,,,",.'110-' ,( 1I'''IIJ' ___ IoP>.! .. ,,,J, 10Ft,.... ...... , " ..... r A_ br """""'""'" , ... , .. ,"n_ duo, ." ,loo, .. Ill< "'''''1 d ,ho , .... o . , ... """''''''r. hod _..I , • ...u """ ..... __ ........ p/I,... .... ".bIt. Tho"ncr _ ,Ou I ..... __ ... ', • 01 p/lJ'RC' .... ","""""to. " ~ _) ", ... duo, Il1o """",",_ of "" '''''' ""F'itM>" _. n-..l _ .... ""l' ~1 ,10>' ....... ,~ -...,,'" I", ~ , ~bo""""",,,; 1 .,u,oololl_ .. ,....100 ... '_ A. rot._, .. ,,><.,_11.,._ ....t .. ",,,_.f'!> '.7_501-, So. ,hooj, ~_ m . • "" ...-.po ,( "''' ........ b) '""",,,,' _ ,,"0""1""'" . ,LIo .... ... ' ................ kd '<10,_ ... ",_', <&0 .. YO>tod i ..... """""" .. <=<..,. .. _ ,,!>Io, ~ ,1Iit ....... ,"_, Ion .. . .. " .... "..,.,.."."oIIy br ~ .... " -" .. , 8.00.. .... fi. ofOcl;IIo"" ""..,. , • .,.... _' . .... "'1 • ........ ' ...... , """""" ..... 1:11 K loPloa ........... , ...... _1<0 • .... ndu·. C ........ c .. 1Jo< .... ,I.,,,, ' .... _7.""U LlI-

TI'~ E/'fCIIANTMENT or 1m: V'ITUAl " , Tho _J .. ll<te\ ... 11.1 ....... ""'" , .. , ~''''' ""r-o<a ..... , .. , ",Mt\'eI ""' ......

• AhIt ....... r~_ It" '. ;-0, ..

' ''-'''_·t: ... " ... ,IotH_L ... , ',"".L I. " .. "00 ,II< -.... _ rot . J ...... , C_, </.11_ .. n..It-oI ... 14oMto r',..' lIuvaro1 11 ........ u., ..... , ....... '~I~.II .......... A. M ..... Do". r~ ... _ ..... 1_ _-,.-.I~

" 00 Aqi<li .......... '" Odn .... --.' ,...,($< '"'"- It< P ~ U s""_ .. _!"""" ''''- 191h ...... V~pp.VI JU. " r"l '_ .. o( .... T ..... . ... ' .. poo,e_ ..... , ,.',''''''"' ..... 10 .. J ........ e_, .. , of , ........ c ... . .I ... , ...... "'J-" " , 1 "U_W Lab.<. ·ld ... " ''''Q;ooc ... """""'" "" _ ... , ... Il0<l, C_m """ b" .. __

(' I"-'''I~ "0."" lei ...... ·A Nooo-S"""" "'''''''''., .... '''''''''''c.:-... .,,_ .. SOlm, • , .. _.,11< u_ ""_ , ... So.-...J 1Iood)' ('~). • r"l '·h",..,.. ...... u....~ u .... la,") U ......... l<. \a..c...o' u ........ , ,( Ckqo r._ I~l. Vol11. fill. 7 ...... ' "U,"" ',a"" ... 'Qa ,ho C",,,,,,_ "/1 ........ ,..,, on<! .... e ' pi "'Subo<.o_'_ io ~ v .... ,~ .~ " G."" ~_ ..... " .. , ........ " ..... r,O,'--""LUnoI, .... ,., ' .... _b.

I~)l~" ,I. III tho ................ o. 1ldt=o.1>fIh __ .. "' ..... _\r.n'-PUF. 'm~

,

... " .. ,..,.,ty .... __ ,. ld ... "- I. t ' ... 00nt01U 1I • ..- 0ct0IIe0 'O'!O '" 11"1''''''' _"'0 1,.;"" ... ,. ,,_bot 1_

,

.. II« Ht .... ' mtoq .. III ""'" $ano<. of ~~_ 1" _,_ of ~, SA It. lod __ '*'0" • .1.11. "'Aod ~ .... ;" IlIiI _, "'" 1dI_'"" -. .... "'" ""_ III _ "- ""~ ""I'Ou, • ..... pI<. _'110\1 , .. , ... , __ , .......... "'." .. III _ '" _iooAiott ........... q ... li, ... ""_ ...... .. ,. _,( <.WIll .... , ... _.,... .. _'boA;" 1_......, _ .. , • " ._ ""' ...... ot. II,.... 'bot • -.: .... ,II< .... ,.jj"" ........ , ,."..[' ... _ ............ _ wo + .......... ,01) ""'_,·(Uiloau. _ ... .,.",1,...,." m "·~Io.Mtoriq .. <l<\l.lc_"' I .. ~.. '40 ........... IIllqao ... r • . I'

""...... ,0 Ie II;";' 'lIlY.poI M, Ie __ .q_ia P o.b<m.U:. .... _"' .. T ', •• (I!ltUl' Po,,,, v .... '''l~ p ... ., J I. ~" ... . """"'-_,,_, 11"11 r.,;,; 114"""'" .. '96.! ~ v .... ,. " , ,,~ ... ""'.". ..... ~ .. 'K' _ r"" ....... r .. , ... .., , 111""'1_ .... "" on ..... "00 .... """',,', ..,.,... 01 ___ duo", .. tH III til< .............. k><> .. <1<.- .. --..... ~ " t--.."W;. 'I' _fw ..... lT. I'I" ll_lIl,

" THE SCItHN. Tlt l! SPECTRUM ANI) THF PENDULUM Horuoal or Accelerall"" ud I>«d~,.n"n

w. have 10 poJI ulal. ,"", thtre it • ""rem on ,hi. du~ 100m 10 ,ctt>"" l~ 'p«ia, Ind lhll 11 II 11(l' omlom> but i. d,,,,,,,ficd hy loki. ' "1'I'l"""0"1 11 ..... oIinnl1. knowltdF; and. ... ·M' " m"~ thOi ,hi, ICrectI <II mnnbraD<, bo,." undH '."!.IOII. hi •• boo 01 d»I.;U1 or teU", ro"", Tbio 1<1,0n ....... Id _" .. In oent'n Y1bra,,., ... or OK"ilb,""'" lIke ,hose .. "' .... wben • rord ul'ldtr ItmN'MI II plucttd "' ((i. W leIbn~ N,.. "'-fI "" ,Iw "_ t· ... "'.....,...,. 11.12. I )

1 O.ESME 'S DIAG .... IIoI S

w. "lW 11.2) bow .Il00 ,,,oen""" 01 tho Inpl)'l'b ~II'" _'""" 111.01, V) p'" • 'hid 01 !'aIbmcc' .0 .... b<tatICa.' M,,,, r I '. Indeed three ..... ""'IldQ., bul ..... as;, .. _ arucull~ by tho ",Iatoon M _ .. , Their corr.pIc" .. mamy musl _. bt lo • ..,I1m. even If il .. IlOl .... plaJnod In 1M pnle"a, of Cl.1cuIa,..,.., the ,D<itt(1l!l,NO'. q .. aomy !of the ·...,vtlopna .... plm.d.· "c"'-n because ;, Ii copob/e 01 ""....,Iop". ,be •• ",mon. of 1M PUllrntt •• Y by _I ... DI.be oI ... ,,,,lI y I"ovw..:t by.be iote".lIy d

w • • ro ,oin, to "'"w how lucb a de,·ice Joe< '''If'll« "'il h any .~r .. """ 1 00 "", .. ldon, of dlll'l<"';"" by 6 .. 1 of all an..I)'$l"i holll' SOm< k'DmIlIticianpbilooopher. 0111>0 School of Pari .. Ore.me ift ""ttiollla., "l1'e able to clarify rhroUlh 11>011 dl .......... " .. hat .. a1 .. on In 1'" .... lion J. .. 17 .nd lind a ..... an,... or.HlCk kor lhe d.mc..h problem of!be molloo of mollon.' In 1CCOr<I.nce wnb I'" .ochol.,l", rrad,hoQ, It i. M""U'Y III """"'" 10 d .. hnplM; bool ....... lho 01 ...... .., poIftl of...... I_ 101 ... -..1 ICllOalIy ltavdlecl. 1 .. d ..... I"'" .. lime

and !be 'nl ..... "" potnl of ,'_ lhol man", 10 ill quod_ or""",_ {iII '\al--.,.

no. dlaata.., ... ...", by Orr.m. ....... Dt-~t_ 4"<'111", ..... .a:U:UI '" CRalUlJ a pau,,,, urut Clpoblo: of k ... ,u", t ... ,..., potn" 111,'_ 1"F1 .... Thoy appear ... ,ft ft,u •• I'

The otIno II wry .. mple: • mobiIc subJect ma .... te.' I" , .. .., .~IOfISKl1I 10 1'" qualll) of .... hlCh il ,," .. ak" {M .. ",)oclly reprnenred oenqJly. by 11';1 • ..,110"" • ",,"ain ;ja"""" dunllJl I'" uml of "me. w. _ Ito .... learly ,be d •• v~m

c

c

" 1 , c , I

" • , • , '-_1

ull<krl._ the dmlncl_ be .... ..,n Cl,.n';".,ub]ea (h<>ri~"nlal) -...1 inlen"; .. , lubjoc!l",ruc:alj,

TIle 1mJ1.-h " ak:ulaled ... I_ ami """J)riIinl bool ... ..," I"" 10M or tbe: deS' ..... a1!d I"" 11M of ,'''' •• tended ... bjo;I. 11!d lbe dlll''''"1 "",II",," appear I~ 1I kf~ of lhe .WIdard "",an'" (_ ft" II,

ar.. Ih'lIJI ~ i> an _ .. ,."" .... of .... """ .. 10 II .. modern .... Ihema'i-cou. .. "" ... ou\d otmoasIy ha"" d",.'<> I pap/! 1<,. "lUll the 11 .... Ton tho: .... 111 and !be IntaIh J. Oft !be,..... .. and ,,-!to .... O\lId "<",,,.nit loa"" .. oodal ....." .... Ihllloqtb., an arcol_ 6 .. 21-,

r..,.: The IIM"L.n I"lpII,mm .. batdy orrl, 10 ~ all lu •• nK>d on lbe Ib"a"""

1<:lIOIIIy 1"' .... 11ed by t ... mO''-'''1 body by redllC1q .1 10 an """pIIl. l..m a qllalllll) of ,;",. II .uc:h a vdoc.IY ~ "''''''''II' web I qlWlUly..t ~ and the rell1llon L .. 17 II "'I .. tied "lib ..... lln, Ibis bol oflbc "' ..... '~ ....Ih thaI "I..t lhe ,... ...... Ih"'"lroplIY'nllhe hoNOCIlably of lhe ", .. , ... Dd lho: '..,n ... llly or lbe:,.....,..

The I",. tin only ITpreoeot a "/BUI, of fonu. .. berea. lho: area hal oll()Ulh an' ilIlIlKk 10 O"vt-lop tho un." ( belT. nor p&tamew alfCl&dy """.umod by ,he: 1pn:~J onll oul of V lIy lTfK1""ntlll, ,be kntJIh .. I n .,,*,,, 0. ...... >howed Ihll ... had .~ in lrupin. Inlon,;I'" anJ "Iomioru by mean, or.

.. <In'" ;"IUII""" ",,,hout dep.,hnl from • lradm",n 11\;01 carolull) di,hn_ 1',,., t them.

II oIIoukl ~ ...... mh«ed llu" Intcmoloa Ire ...... um:! by ..... aDd .... oaIr SUICCf'I,bIt of · ........ or loss' n. .... 51 Tbomu -'qalna .... , ... "",& Iq'Inl Lo .... ....,..... of ull_t_

In form. lhal 'nlCD.a~ and ~ ........ lho: FOWlh and dlJll.lnUIIOII h .. .., • II1II" .... or prodll<"UOn .. hieh <loa ..... ma'" 10 ,he "'rm i .... ~ ia , ... If, "". 10 ,he t.a lhol tho IU~ P'"'''''f*''' In lhem d"fl\dy. No.' the pnl'cct_ 01. ,.,...., ..... y be """",,""red In IWII .... )'1' fiDi. in rosp<el of the ru.m tld: -=Dd. in fl'Spt'<1 of the 1"".lCIl"'lion of I'" f()l'm b)' II •• ub1"'l. In J<> fa<'" ,,-. """."dor tho: 1"''''''1;0''. of a (",m in ""'fJ"".1 of ,h. l.mn hodf, ,be f<mn i ... ,d 10'" hlllc." ",,01; In. oumple. val " . lillie heollh 01' ... ioonce. 8u. In.., fa. II .... ron>llkf ,he pc.fectlon 01. form In , .. ""'" of II>< panicipotOO1I 1t",, 001 by tIM: lul>j<a. iI .... id 10 be m .... Of leN: b HAmple. """" Or Jo:. .. h,.e or "".lIhy·

In.'''' I2iC of ............ 10 m",,"",' ,'doaty .. 10 """'!;lire bow. lUbjcct .... "ieipo, .. In 1h .. ""'1I0Il. and Sl Tl>omu l<kIo:

A _lOll" ........ 01"'" by nnytbl1', ...... It .... kd 10 I~ be •• llllhe rom. of ,"" 111M iI WI .. be il In tbe form of ,_ po'" ;1 lab,.: 11101 bowevn boQo_ oil"" ually oil"" ~ II .. oIwap 1110< ....... 'I"'- _, oflllOlion, ",,,,,,.,t " ... moIoon ~ I'ows ,n ,nlaU"Y _d,nl.o.he ... bjca'. p"nlCll""""'- .hal II 10 .. " ,n '" far IS lhe ....... moloon can he producal ""'"' or loft It&iI.ly or promptly'

Inlet>Slua III" .. , n<l ,uch ,b,n. IS In ind,fl'errn' Ju.uposihon. ,hey do "". odd .hrnuel>os 1.,heI" ,hey . "anF one 1110,1>0" lbey i"lnaso or Ies .. n one Inoll>o,. II M"" Iha, ;, ;nfe.;", 10 on01ho. IS "", ,,-.:lodOl<l ,n .n. 10" •• a •• pan ml.h. be ;n I "hole;

Thenfo.e't-.e.e ,,,,,,rea ... [oflnl ..... lly] do 110. ,~uh from ,be oddlllOll of """ form 10 ."",he., tN, from lhe m<m or leu perfect ponlCll"""'" of the IIIbjecI,n I lIn&le form.'

TIwI ftf)" .. net ,nl.'I""""""'" oflhe "uailly prohlbll' ,IS dl~ 1ft ...".,rol ... bjeclo; • '1l1.1ll1y .. IS II ,,-ete tmpruoned 1ft lhe 1IIbJ«! and IW) 1dd"1011 can be IUde <.lapt by mea .. of lhe bnn&ln. IOI<"Iher of levfflIl .ubjecu. ~·s dl4R"'l JCI round the ddlitully by mo"", ... nahlt '0' ..... ' ...

&Ion, • I" .. rrof_rtfl ill rw lI0jm ' One <;:all speak hen of • rqubtod mod " lallOll of iD'..wI .... aDd " IS ... nli< pIa<b< will' ,hal lrupolhe .0\0<:1., ... ,-enicony .·btdla~ Ihc<rl"<>n:........, <If .... Iu,h and lbe durallOn> Ih'a~ .. ,~ ·~Icd horuo,,, oIl), ,he perpendlClllanly ul!de,hn,nl ~ ,he Of'IIOOilioo bet"-een ,nlon"'e .nd ........ '"C. Or~ II.os II>< follo ... ,nllo .. y .""P' hi, dLlpllmt , .... Ii .. I ~

I t ...... ouch In in,,,,,.lOo if 10 he ;ma&lned b) Ionet: Ind ........ oon.'.nicn,ly

-

" aad Inl ly. by Iioa appDN '0 t1 .. lIIbp:t. whodo an .1'CC1 ........ ,...... .... ·I·"J I<> iI. Tho co "11dmI1lOII of ""'"" I ..... II of _""" aad IooIod< "", .. raIIJ 10 .he k_IaIF of any III'''''''''' .. ·ha._".. . 1Iena:"'I"" iqlnlolOnl

............ 1011 by "'I"" Jux:.. a double .. ,_ by • double II-. aad to

..... bntWIy ,n. proportJ<>D&I m.annn 1S..,....,.,....d·

o...m.., d' ........ make the nl<CWOll of lhe qualolM!f posullle: lhe ... bj«t .. not ,n ol.onF' of beiollnpped bel ....... I PUI'O ,-enicol"y rpu~' intemill·' and • pu .. honlOOlIllly thaI of an alrudy raJu:cd OIl ........... lib 1\(\ pw.o;boh,y of Iny runho, incrcase.. 8y "'pre>enltnl In. \enllh IS In .~a. Oresme .bow, Ihll ' hOI lenllh envclopo lbe mobole .ubjod and a,·oicI, ,he d"perlion of the '1ulla...,

Tho '1uanlily or lOY Ii ..... ' "uah,y a, .11 i. In be im~.,ned by ..... ra.:.. .. 1\0« lonJi'ude or btie is. I,,.. prOlraclod in IoOme wa~ in ,'''' .ub"", ._. ADd I u"",,",and b~ ·IIIIQI" qual",. lhe ".wilY ofJOnlO h ... 'n the ... bj<CI in/"n"",r "lib lha, qualay. 1'0. ,hal ,lit ,,""'DIllY of ,,.,., • "uatity can be , .... "ned by I ... rfacc of Ihtl kJnd II otmoooo. bf(:J,.,. ~ ""'" be k>oond • "'~ ... Ia ... b/e IOlhal qIl.lOIllY,' III~ "'1"' ....... 1 III In\atb 0< nl~ and or"'*' altllude illIJIUlar 10 the u,"e_"

Tho !en"h II "'" ohwned "Diy by .... '"n • .,aOldanl m p ",reo cool I" eod -,11.0, .. oukl be a ..... 1* ao::a.mula""" tNl motal""" ,tJd(.nd .... 1 .. ;1 <>b ....... 111.01 ' d~ ~D""." hcitf"F"<"'''"'o ,he "me parlmel., MOVlDJ . line lbo • .., I moboIe I .. b"", 10.......,P 0 .... a ... rf .... ,n,',ta ._her .)'pe of operalion ,II.on ,11.0, or ,be limp\< , ..... paso'''''' of bits of.po« Ih., ha.., oI.....,y been NI ""I; il would he bette. 10 .peak lot", of. ~",,',,/Qoo of .mpa ....... ' ... n. .h" o. ,11.0, ,un_ and ruMn, • dtrkv /'" ... """;,,, d" . te>_ In rombo", •• .enicolily Ind hori>onla~'Y. Ihew: deylcn.", nol COIIlen' )U" 10 ·Ii ... ""';l~' 10 .. '.nd .... lpace. tN, .nable us 10 wa,ch llot P'''Ife:JS'''' acquIJ,,,on of de""",, of ,-elocily and .~.o immodoal.J) 10 <>bu,n ... n,"""nl "",uh~ oo,>cc""n& ,lit ' unlfonnly cIefo.nnod motiono· W1l"",,1 kno""OI any d,If .... n"oI "'1cuI .... IS ~IU~ 3 1Ioows.

, ,

"_I The k' ....... "" ... n phil!J<Opher ... n Ibu ...... 1 oul loon. or """''''' """"ned

,h,ou", oon"n...,... y.naho ... of , "" ... _nI .. n,'. Iha, ' .. by ,he """anp: ....:ICI&,nt "",'b uniform moIiQn (_ 6 .. )), Such !kj1J ..... ,tll"'" I",,.,de '""""'" 10 ,lot motIon ,n motion. whodt It ho~ •••• \ricll~ l""oIIibj' .... in 11';"01"'·,

.. phySICO." Wr already lrIow ,ha, the ...... \JOn of ""'""" I, .1Ido " • detICIt. .- ror • lIWI~pII""" '" ...... a ...... a .... .., 1ft all 1M ad .. ~ ... ., plnal b)' theot dlqrams. whtdl an: .. pablo of pc ..... l1IIl """"'" .. I realllat<:d ....rold,,,, of ft]oclly, U. pa.tic and ull4lndcd ..... , tb •• wtudo • !Ubp1Ippt<>-"...ta .pI«

II oboold be '." .. mbt~ thaI, for • 5C~bC. tho: W 'lit'"'''''''' .. I .. m' ,bal .... le. ,t ",,*,bIe 10 kIlO" ... """ml' 0.- I ...,. ... of 'nltrlSlly. II muOl . b, ... , aU be ul>Okntood If a .. ..:\ of kn,,"'Ioedge. .. In IrtlC'Ullu"" of I do1ft'< of 'o, .... libil,ty bcre I IlIOn: '" \nil ""II ""nq_' or ....... by lbouah' , and a .h),.bm of j ..... 'aPQSoluoft of pall> nlrackd from I .. hole. Fo. Oreune. I. for u .. 10 ~",,"ro. veJ(:Q'Y is of COIlr$< '" k~o ... t .... dIstance lra .... tlcd ,,,. urut of 111M. bout 11 II "bove III 10 JIidF ,be mOre ," IllS! pea' 1""'<'<'11"" of 'Oe ,""p,n, of ,pace ud it. uplanahon by. subject.

The dol'" I of "ckxit) (ah$olu'e mobllny) KI1.U IJ'I"O! ,n the b"n~'n, 01 ID <)e: u wch SPlICe ;, ... ~ u U1fin,~ly ""mp"".....:!. IbKll u.0I1 '"erloaol. It taa .110 ho Ii_ u (OQl.pl<idy ut<1>dN: th .. lS.t tho!."..,,u,.w ol...toclly. aDd tberc' is • whol$ roIIllouum of do:JP1OC:$ lhat unfolds helwem , b .. deJru (""and absolule mobtltly 14101.

"

' ...... I\cl'"", n is quanti!icd ..:cord."@ ,<> tho number 01 fil...t mark~f'I ,h,o., ,be

"""'i", body pu.es In ,he uni, of 'ime. ,he ,n,u,l1nn of ,be' 'III<Ire or '"" .".tf\' au""hoa" .. 1f 10 M cenaln d.~ of """,,I .r .. ~ of Ip.tIce. .1Id .h ....... " In

l.oct ... urnes 110 po;..,. d~, of lho lubje<:~ AI • kll't'lTUO'I(:LOn ., ~l<,''''' Col ..... on< of Ommr·, ~ .. ~ 1'0 rmy df~ of VelOCl'Y ..... ' .. ~.,M d"'M~ ,ha, ... -U be deocriMd IMUrN/lla 11\0 ... ""'01 .'Ih .. do ." "'"'. TIIII II _1m1 .. ,,11 ,lit ><k;t tb:i, U,. I:Omp;tf15Orl. 1M eornpoilll lOll 01.

beu .... 1M C." •. hIlOIl 01 ........ do .... 4 unpl ... a A a/I'IM/I " lrup ollheoe dcj1'u. aDd IIui. ... ·"lImI. _"~. 0... ol'M _tors olll .... 10001 01 Pari! IwI no heII'.'_ '" IfS<f'I'", thaI I ,tho ...... poCIChlal1(Nl ollhe lalle. "",,. all (Ooupa'''''''''

A"' ..... rorttl of 'n' .... "y IItdudoo. 5e"e'1ll dqm:J of 1he""," ~"Id .""h .. ,he po","",,,, anod lhe foIl""'DI de""", " ,be lono"',n, dcar«. lhe po ...... mrUI ol d,\III'iu .. hlll, il frnm ,he I'f""'dinl .......... " ... "he. mu~ perfecl IlOl' ira perfen th,o.n lhe lallt •. 11. OIl lbe ronln'Y. on< ron,,<lr~ ,II" <lqrao .. iMhHlm, '" i~1f ''''' inferior d.vee. ., taken II

" the ...... lime .. tlw' """"" devee. II II mo", perfect thaa ,"" Rake, deJrce 00 Hirnd ill iioIahoa. "

Then: ......... an .... C<"b.'ily oladdJlioon; lOr it IMre II .M oJ dlSpoBlOll ol

I..., ,n'eMI'''' ""'1ft '" "'mcr w"Jo'<"ta. tlIn, .PIII.,.hOIl ill .. """'" subFt ral • It .... potn,' lOr 0" ..... ) aIIot<~ lOr IIIn. oIJ~''e coabbnft. TIl .... !he add,"on ol lepodncooci becomes p""'bie .. be", "'" la.ln- fum""",,, !he dawc .... mpk ,hal,lhl1licd lbe rqecI>OII olaU ...... m' ... ol '"'(nil"'"

When on< .dd. 0 ... ICc*! ..... 10 anoiller I.pod ...... IheM lwo faded he.u "m.in In "pa"" ... bjC'<'>. .. be/o~ 'h'" lhe heal .. DC>! ,he~by au" mtn,t<!; bul " .. 'ould be 'lICI'Oaocd il lbe addItion of I'" 1 .... 0 lep>dn, 55F1 .. ·.r. mlde ,n I"" same .ubJCd."

MOf\'O.'t •. i, I!I no' diftic\,ll.o ... lb. , ,h .. il Ihe ronduion for aU Inluition of lite JIOlion oIlCCeln1o'l(Nl; lhe """'pa....,n or de""" of ... Ioaly _uired by • wnale .ubjec' can onlJ be made 'on lhe apGI' ., ,"" ....... i"*t:.anl Tbr m«lianoc.-IC""'U-I" can en<knr ,h .. mobile .ubFt "·,,h • man: Or Iot$ l .. -ift COIOlnnplluoo: '" .. :( ir"n"l- ,b .. IUbFt would .... .".rlr.cn 10 pas! ,~ .nlly npodly, tlIt dl!ltatW;>: 0CI...ur , ... ,,1Ied bern. ronIpil:tely ....... Ie "'" pou>L

n .... IWO ')"" or~. an: ,..lo .. 1Iy '_' oci w,lh lbe odditioot--jIn,apotitiotl!l ol nit' W ..... u and ,be ~.1>OftI III dqr«o of >doa'y. I" Ioio ........... tary. O .. sm. tal ...... to dI!I"opHh bet ....... 1"0 tcnn< ro."o _u .. ·' MI", ,,"' .... nd ......... " ...... Im.-ar"," ·"".,lloci " '''h IJI eUa ..... lion Ib"""" II""IIOIL Th2s .. I'" ""rdy " ... nllt.lI ... _"""'.'" of ... '" ..... , ... pond'". '0 • lpatial or '..."pon.l ... tnISKNL, .. hicl> 1M .. '" lherd'on: .. helbn ,'" motion i1 more 01 Itss Iq 01 IDO", 0. IeIo .ItorL M ..... OCUk n' ,n , "" oen .. 01 ~.uoul'r." aims 10 ..... p ,'" cban ... of .. I«"y. ,''' ,ump from one deSra: 10 a""''''r. and ,,,,",,,Ion: ,be PI' 'hat OCfHI"'" lhe ·deform"' .. • Ind lite un;, ofmollon.

Mnr<:Oc'''. 0"",", ..."pbasi ... Ib.1 ,""" 1 .... 0 ,nteM,,·. Ind ~" ..... ,ye or Of1I0i0akal .nod "uanli'.l1"" - meal"reUkO'. h ... " 1"0 ro .... J1OIIIIlnl filura· ' H>M. II. fiB' d ,ibn. 'borilOlllar JI>IIm'l'

110'11' I"" ""'Iu"",,,,,, of • " uali,y ;. '0 be ima"nrd. SlIIXeSIo>on ;0 lhe -.<qu""_ "h qualilycan ,ah pI;toe in ,"0 _lIya. _d,n,lon........", and acoc><d"'l'o '01.......... . And to ul .... A'·. -.<q""'loon ofl ....... q ... b,y is 10 be ..... ,,_ by I'" mot;"" of a potnl lI0'II, ... 0' ..... W,I 1UbjC'Ct 6"",

10 lha. lhe pan [011'" ijnrr] '''''nKd hu ,be " ... 1111 aDd , ... pari nol )'" ' ..... nod hal not I'" """Ii,)'. Euruplrr: it po.nt c ........ ....,.-ed over I .... AB • ..-hal"'Of pan ...... 'raverw;! by lha, PO"" .0\lId ..... h,,,, IlDd

.. balC'Of ..... no! yt1" .... ....! " 'ould 110, rtf be .·Iut.'· The ute"",cJI>II""'>' i. therefore If Ed ... \ _n: by .... DOntrall bo,w,",n

• pen ,bal h ... Irudy "'''1«",,=1 by the "ualIIY IIId anot ..... ,till "rIP" pan. On tbe OIlier hanod. inl ..... tfica'ion aon he ,mal''''''' .. Ih. mo'lOn of a potnl rilinl contlnUO~llf . t..we ,h. ""nc'",,' ~j«I . h>r ... mplt.: Ie, A 8 be ,he I, ...

of tho .. t .... ,,~ ou","",- then ,he ..... UIJIIIOd of dot" ........ t .. porods to the uc.'nSOlld of the POInt ( .... fia:> I and S)

01""' .. •• ~m" &reIlat ..... ,' II ,bn. """"'"' ... ~Mlt'- '''''(1

_b"" .. nl ........ 'rvI!lIII tbe.. Of'PMI,_lJId Ullin, Ibr pnlK>Pk!bat ~m ,ha~ ,n ,he COf1'O'Ul order. on,~ ... .....,. and "~,...,,,~ Slas ODCIfI'!i'1I,e ""Ihool bcoom"" ..... &«1. Of COlI .... Ih ... uC><'l"""ition n« SOUlI",br dtIor <kmonotrmlion of an .. 1 ..... lOn """,mun IU lhe IWO m(lt", ... : here lhe .,nllil Ilreadlt",,,,,11ed .... . bIes I,," qUOnt"." .... CUlnpori><.>n of lhe ,'e IO"uioJ or ''''0 mobile """ .. , M, Ind MJ • We con .. y ,hal V, > )~ if durin, I unll of ' i ..... lhe IIICIbik body .\1 , ""ven L, and ,he m<>bile bod)' \I, "",·e,. L, 1O'lln L, > L.,

'Ac h ... jllJl ...... lhat 0....... .... 1 .. , ..... '''''1'''''~I''''clea. by ...... , ..,ILII, .. hat 1Iu ....... 'ta,.lled aDd ... hat "yet 10 he , .. .-died. n... 1)1'" of ovid.""" wtl\IId Nuily I ... MlI"T}'OI" .... bo .. "'""y....my '0 e>-al1II1C bow eDCUmbcrtd .... (It,","" .. >pkeI an: and .. ho k ...... OfIly J"'''"poWlK>OI (""al or 'reaI) The ..... bohoa dlhe d""",mo 111 of....,.m k,nd Ih"",lhcr bl "".....senn" ,n dcfuo""" nI """"" Ifllu;t;"'" the Iea.!Ilh ••• n area. 0....... ...... n. 10 mike ,I • lear Ihal "<:On\l""'hen<h I,," patll ..... , ... II II 001 only lhal "hoch IS a!t<ady d''''l'' lod in I,," ",I.nsmn. bu, abo" •• 111""1 ,. h,.h mo k"" it pOj;\,bl. 10 II". oul lhe 'OII'oloa;:ic:oJ' ~" .. " .. "I by de.rteo and the ·qlllnlll." •• · m .. lure· mnll by ltond.,d IU.'" by ~bfx"" ,n' ""lie 1<'>1"n: lhe .pn:~d,,,, OUI nI lbe dlJCl'<tc unlU by "''' ..... ' .. 'ion .nd lhe ""Iecu<m of .......... ollhe roc"n",,'" 1p«Itum of ,he ,e\oaU ... ''' TD ronNdtr tho Ien'th ........... iltD mike dtIo, ,ha. tile cooperatlDll of the I"''' mealu • ......,u invohu ,"'" on" .. ,,"'" of a conunuum capable of ", ... :allllJ Ii COdt..",,,,,......,... lhat .. 'b.do appean ... already d,,!ded and ,Iu" .. ·It;oh -.u Iud( •• an andn"lded ... lIly

lei '" ~II sum a a>nltnuum ,he _pi ...... " ... u.bflI1. <k><$. l""btI",,1huoo1 'hI ouc:b U _pI"", ."""'~ ... 10 undtn,.nd 1 .. 0 d!\' ....... ~ .... n u .. " .... hoello .. · .... I.m •• """'" ,n P<1".;p!e ..... ~ •• n,lal.bIe· ,,," Dn. lII'O I .... 01 ..... To """ IfDI Ih" 'hel .. "",nc"y', .... ""nmlOCto di.I •• n .. thl anieul ... ,,.D "'rico.. """ of "'~,ch (I,," '~ppliq" .. :lli" ",,,,"n,ed as d;',:'~le and t .... ot ..... (the '1l'f'IKluanq II. fam,ly of dtlfDOl. TIl .. ram,ly ~ apptoe.t ~a comp<Aed reasotl'" It_h pornt in Ibr dllCt<1e lenos. thaI " 10 Ny by foI~,n. ,be: o!'dr. of I"'" ~1DlI oIlhe "" . ... and .. "'PO '<lted by conlin"",," hna ,n !itu,." .. hod! .. a copy of U,bmz·. dto ........

., •

"p."

n... d>qn.m clearly .... lI lhe "PJIO'I'IDII t..t ........ lbe d,~rete lJId ordered pouentol"'" of tbe IppIJq""l llhe ·r-t ..... q aDd the uDd,\'Wkd "I'OW'I of lbe IppI .... u.n. (tho ...... 1'). .. btch CflSUre lhe I""I""Id"",Ia. d"lnbltbOlll of lbe 'nl ..... ' ... (01,,,,,, ..... ioa) Tbet'o "alIo. noltcable reIMlana", of ,be roles. I. ""ere::" "' lhe ~1I •• "thal .tonda'd' .... 'he orde. of lbe 1 ....... 011'" "" ...... ,be unlOlJ'nl oft"'" <kjp'CC'L "The p" .. IIcI"", of the . ,""" .. IVood, ,be dlSl"'l"OK1!I 011 ....... dq:~ by 10m"'", ,0 lllnll< ~"Iur. the d .. ",bulI<ln oI .. hal LeiblUl eal!. III< ,.."", ... " and which p..-nl. ,"" ~~op/~", 80. ",ialN • ..,~. like a ra~ oIll".-

Tho.., i. nD <UJIl'ntllOn in ~p<.k,1I,I of an ordered coml"'''''''''IIDn oIlhe 1"0"'''' 01 [.0",,,,,, ,be COllI"", P'" pt,Id>Clli.my 0( tbe .ppliq .... 1 do..x.urqa uy ...... IIlIIi"''''''' 0( lbe: J«OI\IJ by lbe 6,., and lbe,ero.. '~y d .. 'Ib ... ';oa of tile dotJla'l lIu" .. ould lead 1(1 Ibn. Impol"- ,."'" ~"I"' dfctt ucn •• .......... n tho ... me d.""",,,,,,, by b .. 'lIn,' ........ lbe d,JCm< aDd u.. COlI

ltft"""'" ,I tornocb lbe p.I:C Nt)" I I:' rut croatll,l of lbe Ih- lbol """k! red_ " 10 a Ju".po<tllOll nI patI .. Leibruil _pl_ ••• d.apam of the 'pban. acoc>rd'lIJ 10"""~ _;on'.,.. a IlnaLf<!.rea. makoc ,I olear lha, 1M ........... " .. ,y "'U,I 1M' "'<>" and ,""~ (or rtO", mululII lrom. n<U''''J .... tlO .. of ,"" d" ... "<o",, •• il a"'X"nl"n~ tht" tmc:.gcnor. nUl dillf&"' . In 'hr I",di, lion 01 0 ......... ,1>0..-. I"', 'Ihe "", I"", of on< d,menSOll" ,n .nollo.ot i. I"" ral m;orubUl"Od of nlm,.1 "'~lIlp1kl1"on·· by <wlr$Onl ,"'.., ..... «" .. ,"" lnodl' tionaUy opf'1 J 10 """ .~"""

In ·nl ....... • 0/1"«\1\'1", ex",od d!$C1"<ldy in ..... al .... ~1 (the ddfcr=' potnll of ,ht <Laa:ram~ In ~n' ..... '~· ctrctt""y. a/l"e<;'!Ul, a uIUII"" .... b,..,. b) ddf ..... ptOSO ......

, -.,. . .:; ,

., . __ . _. ....... " .... -., ...... .. "ILl" ... . ,

'--' \........ IJ " ___ '''' __ '

ntt ,cum mE 5J'1)C, .liM AND Tlt~ PE1"OOLUM

The """'01 fllmplum ,""eII pnorny ,0 II .. en;do ....... , In III lIratqy fI """'red ullko .... QI fI,llte .... ' ..... -bote ..... ,btma,oc::al up .. r .... coWd be COd_ " .. ,d inlO IIIe formula (I~

... - f/lll, '" wben: P II. po,.."., .... I an illl ..... ,y and ...... ha, "c c:allod II lhe 001 .. , 10

.0odoP'1II amph .... Je. The d"'plm fI lho -tJ- mal .. n dear Illal lbe iOI .. ,.1 !.i,ll f ror .. hlC~ leIbn ... "pled only an.r lon, 1Ie>lIIIKtn." - hi. 10 b< und<",loo.IlU In ."'<>Wi..,. Ihll i.,o oay ..... I;,wt!opm, OOJ1I<mpGnl"""'" wilh • Rb.llioll. TIlt IrIplyt"h I .... I. p) i. I hin",: i, "I"'IIS I dimcll.ron "' lbe .. me hme., dOlIn, II.

The .ip f .. nO! In 'Ibbrn,lllon' Ihlt de"l""t<> It.> II'!"".imllcd ",ilIUm". up oltb. _pl ... by 'hllie realnaJcs', ralbe. It" bcuu .. n u,,(oId. Ihll il an be rcloled 10 weh '""nllllnal""". One: win tbwdooc not be: .u,priKd 10 ~"I"" .\lCh d..,.a"" and ronnWx .11 It.> "p<IlOl>OII of. phy>ICIlhat <ka "'" ~ne itodl' to dra.,n, up a....,pIe ....... d fllbe r.a fI di~IY

H-. '''''Il' can be doIIned II:

It-IF';' I " ~

01"_ d .. pamrnahQJly, U III 1iJ= II;

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'-' 11Ic Jan..- II ~I,ned IU lhe ·coml" I 'mu",,' of lhe ""'"..,. alon. I polb

llId oll rorce Ipplled II ...... "",ntll

11Ic <nd fI ,Jut ehapler and lbe COfIImetLllry "" OrMSlnann'. A~dnoo",*~. IJI'~.n <hapln IV) will allow ... 10 dnn~I"''')''fI more oorty ,ho ",,","IU .. on, oll"" dqram of lho ""'PI"", Ind ,ho m,ure of ;nlqn.1 tIIve"'Pt ... nl Ihal il mad< pouiblo.

n.. p.-.....d.R. ""'mpleo mlJlll .u~) lead u. 10 Ibiok 'hit ,he .""",Iation (A. 1,,1 "'" he .'(>I"Qtcd by • JI.'.~"I of lbe k",d 'p H", ... " A "boca" .. of" 1'_ 0 .... w""ld ."""",,. f,om Ik l*.an>Ctu, I" "hI.in A, ,~'" .. ·o~ld be a kind

ruE K1tEEN. TlI~ S...:cnUM A"D nlr ~D'DtJlU"

ol 'Mlurar 1",""II."y (rom p 10 A 1<,Hd""lcd lor , ,-,

"

Ba, till. 1IkCha""", dtc:hi o-mooQ lluIl lho <:\1'''''1 ol,he Inptyd> II ,..K""I, the ......!111"" ol cluJ 1 •• 0, .. ,..,,). "Iuch, aflo:nr>o.d!,' nil ",,-... d·UL 11Ic porankl<1' ;, by ..., mems lhe pooa, ol dtpo.f1ptc_ lhe 'ob>-iouo ""u.a: from .. hlCh A lion u in I '''''<"\loa; lhe """"'1 "" • .,.,pIe. bom .... ·cd (JOdI thermOs dynamlCl. ~ Ih.at lho ""'''10 ....... ' d .... not , ... , unly 10110 .. ·· from lbe pora .... '.r.

In the relauon dS _ ~ (".ben: 6Q i, I "'OIIlIon of'l"".';'Y of hoaL T the t.ml"".'".' Ind ~S lbe ,·ari.ation of ""Ir01lY~ we ·btlm· frum 6Q. "'hlCb ........ d"ded by T. WID '",,,' tbe variolion at In. pa,ame,e. S H,,~, lbe CIl,"]opm<nl by lbe quanuty of .... , ........ nLthtr IU prl:Cil<k tbe Int"""· ...... 1' of.he ""ramct<r S. w!>Me unil io oot .\ aU re,·calod by .. n"bl< dlChIs and. ,n llul po""",II. CII"'- the I"'rpendJCuJa.nly ,1a:wIc.1 by lbe Inplycto (Q. T. '\"1"""" to -d 'lid· from Q 10 S

In ill Ilrat'lD of breakill& .. ·ilh m ilnlCll dlChls. mode,o pllyucs hal IK"""Im! lhal lbe undwidod chanct..- fllbe _~ tnpl)'Ch fA I. p) and lhe ,..OUiUllOII ollbe ,"''' ......... ' .. 111 (JlCflRltlcd h, A) pLo) .... auaal mIe. To pUp opKlra of '·arialioG, 'I 6nl poom""s Ilorif ,n an £ p= fI potho. fields and ... en YIU'nxl" Ind SIn"", 10 d,..." II lbe mIlIt deI.m"nabie IoId fI !hit ~ Ai w .toaU ..., al It.> <nd or lbe Wpl<1". lhe ~ pnmi,. an Indl,·I\IUlt>Oll ~amcd by pmmrnion fI tM oon.ul,,,, In ·optlCl!" ;,.C .... idualion - and not by lot.h7)"ion of pRVIOOwy <ut-oul pam.

! srECUA AtolD HO.IZO~S, HS·UAINlD Hl~IIVln AS r ,-U PI' CTI\· t, r IlOJ EC'lION O~ ORt S" II'! D! AGIlA '-I S

w •• ,,, fure".,,,",,,,, '.r><!n." .. eml...., only In It. !.t"gl< sub"...1 and. to be . W<>:I,,-.. , he dei"'"' "'" only be d .. II"l!uISIlod by rnnIllnm, II n .. er. ~J\5b,;""",, In , .... < 'pe<1fUIIL OM could 1>01 al'f"OpNllc Ibtm by """"'·'1lJI lhom lil~ lho pdak ol. 110 .. ·." u " IW" ary '" lind a deo·..., lMI untold. Ibtm u roatempo-ra_ .kl1 .. m ..

Thll II ,*<1lCUiarty m...lIn' to the do",,", ol 'rical) .. ,,,,IoaIJ ... l"" on ,tlo.-n lou 1,,11. 10 ,dl .... btu • .-- '" f .... ol nJoc"otS ... epa"" lbe InILlIl10Il mlld! belllr rot a leap. llIII IS "",y II " "'" ...... up ..... ply ' 0 UlI.c I" ,a .... ''''} ol c:>ramo.., IKI&IIPa .. bocll WI I.Spond!; \0 lbe vano ... ...,...... ol """""y 1Ie!,,·em lbem I""'" <"11>$ .... kind of COf11>C'<"UOII IMI prcoodon 11 ... dol"""; ... I"", oltlu.o ... ", thai CDC. 110< ,.d<JCJUQ lbo.uld _ be I<ldtd up Iooe rukn pbcrd <nd '0 end. bul Pould he paoped III I ....... y 1n.1 .. ~ope j.JJb]ect. wcoWd untold 'loom In I"""&h' .n or ..... 10 I ... po<:!" IpIICC mure or Ie. promptl,. We oIw>uId "",.<fure '''"'''~ of ,be A"""'"",, of lho. '·e"""l~ and, In 1m, .petl""'" tho m.~lmll tit""", (111< de!!ree ! 0( •• loC1II ... ) (oee fi .. 41 .lcLLfI) 1'1><1' •• cry ptllI .. ular rol<:"

.. m£ 5O:E£N, TH~ $1'lCTI.1:!oI AJ"D n.t; PF .... DULUl.I

,. Jubord .... 1a '0 lu" aU d~ OIbn ~ .'lucb e:wt <lilly IhrouJh IMlr prO.lUDlly 10 ,1, " Ibndon: run.bO", .. lhe omlno! pot(lO eI .he ·OOlolo&lcal ......... "'m.'1It'; Ih .. siluauon. wbICh " prOfnIll.n. In ,he h"'",rchy ellbe de",,"" ~Mf mod.1 eI petf<CIj",,~ forl>l<b .11 IUlIm"auon '0 .. UlcMi .. un" iOl' k"llih or t,me. lor uamp!el whoic choo<c .. am.ln'Y' ' he dqrcc I 0( vdocil1 .. ",," a ... hid! aU d"""",,,,,, bee.'em " ..... and .pact is ~Iod ,0 be illl Jl'O'Ioibie.

I, IIlh .. deJree .... , 0ptIIJ and do8e:i lbe _"... llhe ~~'} and ,he dill"'''''''' bee ........ II ..... and Ip.ooe II poosoble only fo. bodies Cl.Jlllbie 01 """ ~1MIfI-nro maucs).

To ",1Kk. ,h. OOI.hl;on ellhe velocilici diaaramm.'ically. il i, nOI cfKlllJII iual 10 UM>CIa'.' n:t:IMnak ,h., .... mply 'bI""" Ihln ,he OlbaJ 10 the ~, .. de" •• ( .... fi .. 'h~ n,bcr il iJ , .. ' '."y '0 m.h II In o"""'at or penpta" .. "",,...1I0Il (_ 6 .. 910).

, ,I

•

• • • f,...... Po,; 'U .. """","",,,o.-'.-...-We hi ... IftCI ...... oadI of On:sme's rttUonab IIW\aJCd 10 ..... ulm1>Oa

10 I dcpI>r 01' "doaty. by ooatI<Ienn.1lIot Imllk at 0,.. -1 011' '" .be mo'lll& body. ,be .",.,Ieqt/I ... 1IImCd 100000ber I~ eI >'deIa'r .Dd made dntr d~ IX><lnectlO<l "",b tbt mobile !<Ot,ea i_ 6" S and .~ If .... _ "-UI lhe unfold,n, or'he Iptarum of ,he vrioc!, ... and lhe ~Ji.1O<I of ,he MIIfOIlJ'O J-p .... dl w •• "'e m .. ' make .,,,.wl_ ClPI'bIe 01 K (""''''P/~'Iort ''''1 , .. leo in all .be dillram. in 0"'" 10 ~nd makeo lhe ."bold''''"on 01 ,he ,mperfa:, dcarces '0 ,he dq.n: I JIll''''' and "",a.;~

n...vee I ,bill oppears .. ~~bIe' 'rom u IDbD- dqn:e. IN! .bo ... a/l .. lhe dowra: II .. , MItftpIU"""'y "",,"010 all .he .. pIon.lioDs ClmId "", lOa .. _ .Ai ..... lIy .. ""'1,"' ..... , lik •• "" .... from .ny 01 ' ...... who WOIIId It)' '0 6. If opf>OSI'. h,m hh u ·ob,.a·. bf Ii"i", ,Udf", · •• ympiOliC lor all imp" ...... .sell .•• he deS"'" I finnly ffilbl .. bcJ ,.""'f II ,be ........... 'bcre i! II .... "'"""'Iy "",uilttl 10 1'O",;"n nnesoolf n,h. olf If one "-,,hos.o embtaco!.U

.. .".a III ... nile look, The ......... I oflhe >doci'IU mUS' ilwtriore funruon .... fIatr:ooI,n o....m. .• dia&nmo:: Ihlf II poobably ,he ~I",I ;,UUII"'" of ",lou",y,

W. hi •• alre:ad) bftt1 able '0 judI" cvery.h,n. ,h., l1'al pined by ,i>t<e dl&J1llm. 'hit comblllttl "" .. loponen, a nd perpcndiculan'1 and ho .. ,be prewnoeptiorI of ...... mply COM1rllCtttl by ju"opooooon 01' dead .x ........... found Itself dotmyod 1'0 \ntaIh .. "hov\ >doc.i.)' To ,1m doli"" 01. tbt kmmIa,oaan pitt .............. 'K penp«I''''' ",c;eCI>Oll of O,,,,~'I reCIalIpe. ... boa III reply. no <f*'O "",bou. drpth. .. ,.bou, appeal '0 the ~1, ""thou, lhe """ ... '0 ...... Iop Ih,n,,'

Now. lhe elf"". of """"pta"'. ~und, for the: ."""Irum of .... IOCIIJU ., drptb doco 'or Ih. ~I""" of remMol 'fum 0 pamunll ",h .... ', he .~ 'hat OlIO im, ...... .....,./Iow. on aU OKb.he _ra« "p",,,,,,,,od, tho< ftniwde ,1Sdf of tbt ""iolin) m.~ ", •• he ",fin"1 of .J'ItIOIO and i., ron"nlDt} pcmpo,bIe:<>

, o--~---•• =-----"""--~:

'"""'/' As honlOn of ,.10."'", il "'nlojn:l IIop:k>.sIy ';u~m]"lIOlIC' for ma, ..... 1 bod ..... ,'.n whe", <ndrfinllcly ...:ekfa,ttI. It iJ .",".-.10", 'rully' tn" I ui bIe. bu, ,he 11<..- of ,-eIoci') JeenI no Ic:M "",nxd oIr by • mot_ of cmerahted Cf\I\lI!mml TIli .... he """,II ofobl: ,_ ·'tM",/4fI of pcnfXC'

,;ve itbt 'fuu""x' "" .. Iuch PlooI'IoI:Y"'''''<j. Tbu CORIpKltfica'lOn domc5'K::It'ct lhe _bk: 10 dradc on a bonzon ,. '0 make OfICieJf m .... f of ,he un". oIlenll.b aOjOcialod ,,'j,h lhe dtlf.",o. Ii .... of d",.~ the"" 10 docIdc Oft. de.,..., I ol ,'cioo;,irI il'o dom"' .... d.",,,,,.ly III ,'''' 1I< •• loprnc:n .. of 'nl ..... lly pormitlcd by 0retme·1 diaJl'l."" a nd '0 dillnbu, •• I><m In I "' ..... fl&nF 1Mf- g~J )tIf<ntlJ ...... M~ ..-p/ft..J1

8", lila ronlcmplo._ 001lId _ boo mu.1la1lOd by t<d"""'l" '0. look thai d""'Prded hotuoo,ab,)' and atnleCl1<d ,lid( .lIh marl, .... wp ... k_ bo .. "", • ., tOp o,' ..... he 1leJ..,.,. 01 • ... Ie. "The obIiqu"y tM' ....... he dftcl of dcplh ,n • "",nlln, II ach .... ~ by a .... uLo'ed knll"nIIOSC:'hoer of .... "cab.y .nd honwnllh,y" 1' ..... y'tunB ' 0$11 on , ... monne, In ",hid> • <l<i'I'" of

m~ scar£. .... Tltf.SPt:<:al:M ANUTlIE Pt:SOUlUM

rmIO,'al and an al'l'"=' d,m,"u,""" 01 1M "anda.d ...... u~ ollen.lb an: .",cubltd: pcr>p«11'-' C.n Ih~m,,", be "ndc.~ood U I oompcnoa1inJ de,'ice bel"ecn I .'n .... l leap f.om one dIM"""" I,DO 10 anolhe. Ind • virt ... 1 ron· tnnW)ll of the ~,td ... ntbrd" .u.eI. Obliquity ,"""""" oa I poet bel"ecn tho: "ct11C11 and lbo """-1.11: i, obi".. ,he eye '0 •• ran ... JI"UlpoII' 110M ,II • tcaIo: and ,hemore 10 ..... a Mit'" 10 an iteration.

8uI 10 •• II,hll. lhere is no q~hO" of _'i lktn,lIotLlonla~ly. and lru,,, odla! ,he JIH' m •• t~ .. of ptnp'~I ... e bad lVupcd 110 compIe1dy III ott." "".',nl IIonzont:il moW. like Itla ., t""'. dil.,.....1 . , the 'bo:' oflhnr plUlWlP. n.... " ..... mntndtr tho! tho: de .. ,,, of ftIIIOvai from I plU""'" cannot be ali<k!lhe p.:tt~t>a of the obhQue: 0 ... mutl Mtulllle onetclf w;.h 1M ~ oIl'1'tIlovoi .itbout ...... lentnl ....... U be lrapped by ..... ""'Jay '" IIorilorttoltty atone. and e"<f}1.ht", lha. """ be t:Of\lnnpialed by I simple raisml of the look or by • lateral "'"1""''''''' only" m:t;, lbe depth.

y" know lhat tl>e horit.oa ;olhe \oca""" <>f 'M subord,natl"" po,nt! of .11 wnilaritltl, the "'"lralu;, oc. I»lanrill, p .... he,we<ft ,'."teal and lIo, ilAlIII.I whlCb conta,n, them aiL l.atlem •• po .... fuluth .. IIO. ,n dan...,. of ..... ainl .11 tM de ....... ,nto I oinaJc ....... t-;o. beat .... tho< tle ..... I if aOl I Xn"_ ... "" ..... "" 0I1iw:ae dear= bul lho< ~. lhat W.1I"l"i>beo litem by unfOld,". .hem; ,he oomjWCI'fica ...... oI,he sumLrtnhn d .. I,,,,,,w... bctWftfl lhem mIlCh bener ttw. I""'. """",,nlaltOn one by one.

NOI .... ·.n thaI tho< · .... Iric·. dcpct1dc:nt on • neutral ot..crv .. dod'"ttd 1<1 mea,,,nnll'" .ntcn'&ll ho<t ..... 11 Ih.op. the d"la ...... from poinl 10 poo"~ ""n OIIly be the a.\>Itef 01. """,I»I lhal bu alrc:ady booo .on by lbe ~""l To lru. drital or I"" metric. II If 7 ,uy 10 01'1""" 11M: 'ictory oItIM: .proJDCI'''C.

lhat 0I1ho< 1Ktna>n IIIaI I .... bcor t"""aJol 10 o,"Cf1um III ,ho< ftOIl. dcar .... or """",.,,1. W,lh I'" horiLun. lho< ,.ftn.le II 1. ,1 flnd. a "" .. pl,nl plaa: .... th In. ft"il" h. ahJ<:_ till", Ib~ malln IIOm".,nel1y pro"itlod by depth. Thi. lIo"IO ..... e"y " by .......... '" the ........ u ........ "'" operat""" R,,, .. ,,,hct' 1"'1 I"" !at .... """"'pI! 10 1W\flrlJll:l' d'tn .he fact lho< doJfW1l, or. collr:ct"", or objocct. by prod"", .... ilatld.ard ..-oU", ... boor: " """I0Il u ...... " lru. rolie<:IWto cAlcnl.-ely, H it exactly ' hIS 'YPC or Optrll"'" Ihal II demanded hy tIM: finit ude IhO! "'" .... n d,vorQld f.om Inftnity, H ~ui"l , homop: .... ly IlLIot don .""'y ... Ih Jclw:f ,lb .... relid "' dcn$Ory ., Iftal 01 • "I or dtlf.ltiKU and "l1h lime makfl all thmlS oqIIal. "lbr: 6 ... 1e.1IOla .... (r"", tho< ,"6Mc. rmtIIIftI bllnd . nol dtnI:""'-1 OIl .he ,UIIOUry pannI. 01 ,he ...... 1 ... one. Firutll<k I<tl!htlcs il"",lion: 'I """ .. Iy d'hl> to • mantplliabie u"" Ind ... oul<.l ,hu. b<coo .. ,nlo.<;""ltd ""Ih C01>CrcI~~ ... hile: fu'lI"lun.,balll.u ...... · ders II>ri( bound ""00 and Iuot to lho< demand. 01 tho< 'lII:RaI~ "boch II''''' 11 no ....pIe: II <ndlady "" .. to lho< nesl -+- I II . 'iII 1ft .-. 11'0, tb_, ,~ poplan. b," 1l1lI ....-.r ma ...... ,.... of pOplar> (_ Ii&. (hI," ~or. to,,' ",,,Id.,od f.am lho< IIOriLun ,hal ""folded II .. Jlm.laflly. ;mmodi·

'!ely break> up inll> 0 .... IWo. ~ popl .... ond .. "" (..., fiJ. l~bJ' the pOpLt"

"

! • ! l • I i --i •

" + +

'" "'" 0' 0tICC 100 dI~nv and ,"" OIImuQJ tnd. eaPlCd In """" I calculallOOL, .,... """" linda: onaeif "",,"oJ for I '1~ d,II'< ... ,"', forne .. hounded beyond lbe fin'l'" 111m: bes the ( ..... lnlIIOO of the """'Halle. a falx ,nfinne.

An Ittn.11OfI depn'-..d 01 """- min""" lIP mUln, "'" or lhe ., ... "I"l'm.,!! ollluDjp. .. Ind. as fi@ure 12b oIlo ..... by U'IH'lCllUnllh. obI",ue. " ... copt quo:kly ,nlo the Invl~hl) of the j"'lO~l;on and lhe Iran'ponallOfl "r "nolL

We oould, cl cou .... 80 10 t"""k 'lIn "'t' thai tbe .. gmcnlO .n: nol ·",.lly· 0' \be "" ..... dIStance. btI ~ "'/,/"",, a MrL"(M. aD de,i« a1lude:o II> il TI>< ",,,,,ioa io !<kIm"", lor the I" .. ofp«>pk ,n ft,,,", 13: 11 ••• )'< po.moned.l 0 oboefva only on "pp"mn oqlLlhly of....., bet"..,a II , •.. • 11., '"'"rllI) II .. ell .... '" lO 'n,,,'C the e,'C • bid. to dwoI .... lhe ~I~ of ,Ins oqAlny ..-hoth _. obu"nod f.om. random p,,"m"D and "Iad! ob'iou>Iy ""'" no! i1lS1I10l. the IfUP of Iny ""' dll" .......... n....: nampb 1I1""ra,. th< mllade of the "....p:ct, ... prn,..:tlon, capoble ol unfokhnl ,. .... I&n' ........ ,lbout .""till dIJpIkt"""1 and II"'" '0 In) wbmiHlOll 10 In ""'111<11<: of ",nJioClobon. n.. jOin 'D, of lhor finn. and the mftnht provided b)' the "" .. am ,,,,,'e<. TrIO'" a pil'Ofi"4l,han a lron,fer or deduc'um. Th" jQiruAS "P d_ nOI mea.ute tbo d,s'.ntt Ibl' rem.i", bel ...... n Ih'n.>. bul in,'iles ,he dc:,tftlI uf '.m""a~ "]"'al" 'UuesIN •• only '''''''''/I) dJjhn<l. 10 d;".nod

'[,h ..... blk <k>ice, .. hich .... ,hcJ d .. eas) "ict(H'>« of 'M tqualll~ or dill'o .. u<", ,ha\.", ul ... lor IfInlod. ...,.,.,helo:oo dnnand .. '0 Ullffi .... In ,t.. ... ~do .. "'u' (}/ lhuop.. obarp cull,.,,_ An) IUIUd"Y III ~U1' ,he I\onmIo llpo lbe: Infill'lCC ",t" OlD ",,"ji/tJ,~ I ""MI a ....... 11m tJtddim •• _Il00 _II",,""'" I"" mllllllde, .Iudo. "bm i, IS limply placod t- ", thc utfirul., IIdOid ,"" d..."pbllO of tho obi>q"" anod brnk. up. .ulI n""'DBI"" hop" (}/ """'pI"nn. ,It r""nod.""" lilt Ulfinil. .·hlCh .1 ... ,. •• , .. 1, I link flll1Mr ..... y ThIS 011 .. h), III ,110 lndc:6n"t, <""'l'h,". 'finllh!:s' by belDS ,"" .. me ,hlnK: ,,,*.,,, lhe 'd","","',lnd Illesc 'di<'.'''''.' _limn '~n'u",., rll a. the lo"",oun<l. • bot l,k •• 'andonnl....,. .. f,om lhe inflllu. lha, ...... , •• " ...",Iy <I«Id«I 0". .... y '" ~not""'.

II ... hndon: "'" '''')', ,n onkr 10...r- Illy ............... 10 d .. ,nddinnc ~nd to "l'P"'I""'lC • ceoo1lO1nc "'6 ...... 10 ~1Ik ,w ....,: ..... II) .. , .... "nl _If .. ,,~ ,t.. dn.cpli". (}/ d .... '" " ...... , .ha. " pi.,." c O and • hoch ........ ..

lhe Inflnil. ,h,,,. ,n """Y flnll. Ihln(l. Whe1her i, il a qll<S, oon Q/. landilCOp" u' Q/ a du"",;n 01 knowicUar.. the obliqui" Ii .. " ""I by the horizon imposa I f .... le bu. ""placable "".m, ... hlCh, by ,,'"nl n,hl ,be lIJur .. of finll"'" (II,""""",,, lram,'" .. _ chaiN .. ~ e<rabh,.,. a pm._ hlllJC' "'".-em lbe ....,kle(}/,he '1neIlphoncal' .ro:I,be Ilrndy ~IN .. ,,,,,,;01. R........ta how 1AIbmr. masl ...... lhell,.,., ... by lbe oq<Wo,cd npoo .. ", ""be ~ l'IIo .... ,.po'>onal leap prod_ 1<'*, .... """cd '" unldlllll ..,...,&:a,..,.,. I,be 'daz1j,.., u ..... (}/ mellpOOn' "'" IudI J""'I<>o laaIn 1rolkcd ,; lbey "Jl""'d ""I .-elI bo)'lHtd flmlllir """""",, bu. only 10 bu.n .... , Ind 'l'illi ba.;l ' .. podIr ,aID the ....... ",cd Ino .. l mean,n" 'f ,hey ha,-.. not bten able 10 I"o .. d~ ,1",,,,,,,1,,,, .. 'nb I ""'17"" 'hit open!. I1ek!, I contnlu"'" (}/ dc: ........ bel ... ..," lhe m ... ",,, of a 1ran.pa~nl ",ad,n, .nd ",'ha, klokJ .. if i, will IIC\-..r be wllh,,, ",aob beau"" n ma,nta,n •• 11 t~ 'rea.,,",,' Th" lick!, wk"'h .o'~'n"be <l,,'nbullO" 01,,,,, do(TCe:'l Q/ di.cffnnwn', likes 0'· ... f.om ,he "..,.phc><, .. b",b, .. SOOD U II no Ion ..... uaspl ,_ odea5 ,n ...... ouc:cumt. 10 lhe 1fI' ''a lJoaal pull .... onIl!tal}' 1I ......... nd auo.', .rsetr 10 he d1\'1Ilcd ,a,a 1_0 do"blful uP,r;,..., ,.-no. horiwa 'MmOre ~ tho dll§lC' (}/,be: ""pnt, ... lAl .... lIIElIpbor by IJRPOUlDI I protocol WI fold. it .... , Of do HI " t.d; 0" like a faa 0.-

...... of dop«" Th .. protocol p ..... I ......., r-I.etI' aPl"""",h ,,, ,be unarlJl:u. 1I",d and, althoush II " ne_ rcdllCtd '0 " fonu&l pl)itc), no ..,.. c:brly ~""'. r",k1 of "'ptnmen', U lhe ",amples III <""plCIl I\'.nd V lbo .. -

The horilOll mak .. it l""',hIe '0 ''CtIm", inlo the I",huknt "".,., ,,·be ... .... nee. all Ind phil.,..,phy hru.h "P,n" ono lOOt"", " .. h",,' """1'''(1. Th .. ""","" IUI, "rally m .... h rOll",t ""'" and.he ohhq llny Ind <:(I""""" of,licld (}/ in!"""'''' ~.t """" ... uoo ~p.1N1 ."" den$I'y otlhe 6~ .. raI .0Id ,It Ialmll pi 111m &tid .p ...... lhe ddCf<1. """ ... , • i of ,he doxu .... -e. no. boruo<o 01 • lanclk:ope Of (}/. doawn of t"",,1odp: Ioa<b 10 .... n..,....'N ...... tomplatioa of tho IIcI.I , .... t .c. k .h.n ... "'" l001I000., .... t tho .-.- 01 d .............. ' •

THE salEL'. nit 51'£CTIUN "''''D TIIf PE."Dt;LUI

1.100 ... ha. JOII'I up me standard m nuns oIlenllh or me ulUlS 01' ","'bbOn. The bonlOII C>fIC1'" .M Iidd 01' all tlw. ... " .... lIy ..... tbill ",ad. ~Pfoble 01' boon. dtduord. IiIIcd, but that ~ all Lhei< Flum that II .. olltdlt'lcu MaW"" Or.:: ""n_ uuxt.he boru:"" r,,,,,, ,11<0 ·POOj«L;.c· ..;.IIov' mulilat· In, the l\tId·fl.ll·' capaaty of Unfoldl"" The hO"'lJD is not ·_ .. ,IIk·. but adheres no inI '0 !he arasp that II "','U'""""1o """ con'ruls:;t is as unremiu,", aI BltlCbrard'l key.

Joan· T"">SI.in' Iksanti uyl '1"110 "a!ltly tha' u i, 'impiicit and ro-posited', .. The I>onwn " IICI.IIa .. boundary math • • ha, prohibilS or lOIicilS '''' .... peMM>II. lID ... bama" d",,, ... n .. IIotlod I" .. ..,....be sky. o..::c;. hal bom decoded. one al ... ,.. cam.. one'.1Iorw>oo ...... y .... tb ...... nw is the uUJ>nW'· "" _ 01 me bonzoD: cor"";,,, ~h .be ,.We., trnvio ... w.o a ...,.11, cornproo m""" I'ke touch. It cIoa _ d .... tlunp up .. ,til appearallo<ft. but ... pteJII&t .. ~ that ""...., raok'ld til I""'P-

Any do,a,ied ,n'fI"CIKMI of. Land ..... "" or 01. dom..a.tn 01 k"""'iedlC ,,",up-poteII. ", ..... y. and.be bonnH> ..... hat mdo"" ","UUM "'nh 101M "yle, ' fth .. w,vcy Wffe .edllCC<l merdy tu """""'bin, .. 'borizOInal' ",ad,n. and, ·vcnlCl.l' ",ad,n" i . ... ould lead only tu. filed IInu/kation.. The I ...... ,y ind","" an .... 01 dnmwtiOll oIlhe mobilih •• 'hit .....,,, .. t. k"""'N:dF: lbe st .. 1I. bcc:omo • ~eW .... ...,.. up!on.l;"" imph", a I>OI'IQ oI.it!toal ,,!Up 01 the ""''''mrnll. Tlu. _Ill"""" e>&n ,,,,,If Up~) lor ro.nal)'lll and al ... ,..evadeJ 1.lI)'OM

... bo wollld JeK ,I 'DD tho -' or r.r 1I\k' of the honzoII WI roIItrob ,L I"

... tbdta ....... al ... ,.. ~ .. fuI aDd is pP,Ilor by. 'mrtapburlCl.l' d"w,_ 01 tile IieId or by ,II rWuc:IIOfl 'o ..... Ifkaa, OIfII,,·ork. "'bo<d .... ted 10. <:<Xk.

TIlt ,mpiOCl' 1""1'''''' or lho 1t.on.zoa .. !ike ,he.,. 01. FI'~rt' ... _ h b. ~""'t lhat hal bom able .0 ,,'abl,lI> • d.:lICl.'c paC! het"ff1\ 1M .. ' .... 1 prQfum of the ,mage 'end,q to (UK 1M dq .... " and .he fOlft! .i",,". 10 e.pIode the .... ,n'o ';JIIl.focanl ~nIllL This ""'0:\ " IIDI IIOCh,I'la; It II po,hap" """ oIlhe hy, to .ho PI1'pilr&lIOft. .I ..... Y' allied 10" pr~ of ~arn,n,!_ [.2~ b\l1 ,he horuon..,.,. io alWllyl implieu 10 JI~ some lalil" '0 ..... ",<lCOI 01 aIlllliotlJ U .. , m ....... ~. """FII in.o Ibbrn'tal.!<>na.. "J"1It'r1, .. ,ndc>ed aft

u..w.b,l"y oI.he I>oriron; 1I " ...- tlta. or.be """puUlbk; " ponpoonu and Drp", .... Ilraf.lFllll 01 """'P _ of dtd ..... ~ dwu" ~ '~IU""''' "'C dtaII _ Il> .,,, ... ·p ... IIlplct' ..-bodL, If they .... _ lei 'aIIOn'lf'. do """ dooapbllt kw ftckh 01 .Ipen ......... Tht> ."'; .. 1Iil"1 ... 0I1Xl1tnc. 10 ... """,,,,,,rod ... ,h lhe Ioo<i1.on or oeIoa ........ hod!. boaule " " IlOl wnl~ by any Pfonicl,o'" fi~Il. dqru, can unfokl ... ntoally . lllhe doll'ftl of d'..:em ...... t or "".. .nd 01 'JIa<C. .u ,ho h'"1ft ... k",!t. ,a lhe $lime mo""n. di"et>d the dell""" lad join up ,ho >tondard unit. of Icn~th or lime. The ,,,,"bOllly ,n,onled by I'" I>omon .. in<l<ed ,hot 01 111<0 fan ... lticb ulTc .... an dtUtic: rm.,. 10 ,he int ..... lies: 00t can Ih ....... _ .hoI I '. bhenl. Ibcm or put .hem ,n ......... e. TIm if .. tw alIooo .......... phon til a>mc 10 b5e _in Inti dlill._ .0 ....... ,he, .... _ ., depth. TM Ian " "'" , .. ,lable lile • t1u"ll' I fioI that"""" about " onI) """",pia iI, III ....... ,KMI "of qiD'C _ ..... kInd. " tip..,. .0

" >quia", ... ",po, ... "" by ,n..:uLa""" ROSol ... _ ,1110' _lOll anti I ,"""UJOlt

'nln ... ex''''''''''' n.. bont<KI orchalralCl U II ......... J<&b>"""", 01 .... a"" ... 'hen'.,..,b

pin ;. ."'SO"'ld ... "b I n rlalllllly WI i"vile! " to o,erfto.., lIoclf and I .. ... .,.,..,.. , UJUly Iltal II more lha, of a cI.", of ",,,,,bIa [0 •• ha, 01 the ,.,,,100 .. ) 'han .ha. of the pilmel.

W • ..., 10 ... hat ... en. ,be SO\'.'.'JIIl and d..",,,, """""" .. ·.Ik> IwId ,n haltd ..."h the "nual and piI.llk"1o 0( rome: or ,he Characl.''''1CS of. Iypo: 0I.tr<'C'l. 'vIly ,hal c:on utlt. propublona. di]a,;atialll: in ilion, .~ 'hou&It' expcn .........

Thu" "'by ElMt .... doococVd 10 POi""'" Iumodf nabl ...... ) .. the Iiorv;orI

or ,he "doalln by po:rchtq 011 a pboton. JO t .... """ ''''''''111 • Itliola$l", lnod,,,,,,,, "'boch uoened ............ 1 <1<", ... 01 pub,oOoI. ~ of tim IfIn,D'

erod'ill ,I 1 ...... nk.1III: inli","'1y obo<"ure.he zero dqno: \_ lip 14 and 19~ ThIS ir po.ha", the ccnlt;>!,n,wlIon al ""Lric:led "''''''''y and lhe or<nuon.. lUtttton. Iltat ~b4t hold. fOf ,he okM"," 01 ",ma .. 1 11M> hold. for ,he ok ......

of ~'OCI.y,

•• -

-.-

------------ ..

U .... I I"""'~"'''' ,-... _t ... ·." .. "

•

.. , Thut. lhe vdocHia can be ""nl""'rioled ."Ildly on I ,,,",,um lile the

d,ff ... n' d .. t.""" It .... 0( • p,n""" .. lIul .. ." k_ thaI ,l1li: """'p:>el JfI5I' .. 1ICI1IKICd by .... pn>pICI"" ..... ",....iOII .. oruy ..:h .. ....J by .bOlule poII._i .. , II the hoo'w>n al,he ,'doa .... : "",b,,. '0 propel lumtdf lhe", "nip. <>If. iho ph,"L .. too """,kl""" In hOI ""nd to ~. "'fY hop ... n Ii",,,, ... 10"':"1, """"'M a I ......... of the d""~' 01 ............... and ........ "'Iy po ...... ,.

,. -",'c/lln, up' ..."It tho lMxuoo "-lihout ""'" *,'" rum,.lf "' the horizon. too. (be h<>r;,,,," piays a cruaaI role; " pro>'m. ... nd conl«>l" houJ/u UpalmentL A ,0000,ln ">peri",e,!! doa not ~ Qui 10 prNoa. fIocl Of 10 IramI'orm lit .. or ,ha. <>bj<a. 1>11, " '00 10 "IF lhe ph)'llCi., "",,, .. If I"'ppiJ"I .. -,[h II .. menial h.obllJ. and lbo,~ .U 10 Ia} bare tho ..... Iftl' ,""I he IIXI .. -ben he cia' .... u> be merely "comment,,,, Oft' or ~rb .. h"n.' lbe """",Ii.., po .... that io lUI.,.. , 10 he ..... ned In lbe Cllculatio ....

We h", .. I"', bnslnD bli:M to ' ........ lumtdf beln, dill""", aIom, in I dlarKM "' tho opoed al h&hl Of totally itoIalC!d 10 • bft; Lo ha"" >bown ,"", lben: it 110 ,ft'I(I((OOC ol IO"Ullon .. ","lIN, 001 01' Jut ....... dtait"" """Iribu

t"- to lhe DOIIImunllJ of ph~ A"""" tho dIClobo po'ddkd by"" - ,I ft ' hID .... the .dknd:_ '" ooIidlly was undoubtedly ODe altho most , .............. ....t tho ......, 01 lhe duuc I'OPICII) of ,"" bonzoa...lnro.,. wu

n wry 10 diloolw il and 10 mould ports.1Id ........ u fl'Olll 1M same cia,. "The .'doci., honwa dew/», lhe pm...,..,.. oIlny "."""«1 m ' ''"n! ... bordi, .... cd 10 It; llus .. I,.(n.,,~'. ra_ «)nlnellon. Thus. JUSt .. dtpIh betraY" the roll"""" betWfttl the "PI*mn ~ of In MlLel aDd 1M dmanoe of oh!orrYilion, 10 lho el<JMmCe of I .. loa,,! !\or ....... nil dqru I oil"" ........ o( . """", .. pIodn the .,lIlIran_ Ir><! ..... 'u .. lncg· 01 ' M IWldanl meuu"", 01 """Ib and

V'" '~U'I.poo.Ihonl .......... led ","h lII<on: u <IV w" • ......,'tcs

III< number I - .. nh • "' ... pawlIJ in fronl of him I I III< vdoeoty • if be '.lready ~ 001Ii"I lhal 01 II idcnral 10 I ",Itr of "'''JIb I placxd m<lflonlcso Oppoo.ilC him' lorI'nu'. """" K'IIO" mUll be Undctll...oo IS I lhouJJlI c.pcri. mctll lhal .1101"$ ,he "'Itr.jl" ...... rtl ...... 'u'" 10 like On lbe whok '''Ierval of .... 1 .... bel ........ 0 and I 'I' It 1\0 Ion",r ~ "nil bul III< "1"'''''ly of • ,,,,,,,Itum,

W. I", ",.dy 10 lackle lbe Ih,n! IIIif' or Ihi' ProtreU by d,oJT1\"" .. "h,,:b

Ie.d, t.> lbe "' ....... "on oIlhe d""hly belWtef1 lhe: JT1\'P of I nll",al body as I ""flIcle ~nd iU ,rup a, I ....... In lbe HUI ".,., O~"",·, d"'5rammonl of molle" allowed it to be CUI uul I. lhe pj •• lic unit.,f th= ""'u"'" and _de I AII" .. r opprooch p<>OJiblo. Then. '" III< IeOOnd !'hue. lhe ptf"lptai ... proj<ctJu". or tb .... dilaram.1od 'U the ''''''''''on of ,he ')l<X"ioc cWh""y Ihal llIoOoT<l I",", tM<rtlOfl 'n I .ptatum opened OUI f,em " horiwn. Each k'nemalic dL01fI1D ;, artlculaled w,lh mc<:bal\X1 lhank. 10 Gali .... ,. c.pcnmcn~ "-h,,,h ~'im of Il1lin'n .. 10' ~ de&= of vdoeoly, I ,,1 ... lKNI .. -here moohan"", II ,nd"l!nlullhabio from pure kUlC1l1luao Ind dexnbcd by lhe pnnaJ>lc of .... n"" 'In lhe lboo..,. oIlIIY eol,,","1 korte.. a body eotIU" .... 10 mo"" II...p1 ahead III unU<N'1II Sf'CCd '

ld' , cab Ihi. flctlilOU. hi".,. bel ........ ","",ban"" Ind .omclly lhe ",In of -.u. The ruler 01 ISO mo. lbe bonzOft IItcrefon! unfold. all tho rvJen 01 ...... Ibal .... bod' lbe dq.- ofd"unruon bel.....,., II ..... nd.paoo. from lhal of I"" ·If'IIUDdbne', &.ed by. -.... ror.-J IIInIfon _ uro. 10 thal ... ben: lbey ~ '''''~.,..,'' '101 ... and l/us io -r omportanl r", " -UI foIJow$. ,hat I"" dal ..... 01 ....... It:.lncled from mohiltlJ • Ihnefono "'I"'Yalcal 10 lbe 1D1,,'\ioa 01 I rod_' "'-\I0Il bel_ II .... lAd ..,.". and ddi".. an

" i"fi"il. dcp'a: 01 clot1onctlon bel"""" lbe tan .. , .. hcJUJ ' lCr<. _111"'>0·

.Ied "'"h I :wo <ktrte 01 d""nctlOfl. w. kl'(\lOl Iltalllte ttonzon.. '" la'oUll'D, clclf'11Cl 01 d.,hcll" funnlOIlII hke I PO"" FuI MIa.,.,.,. of ,,,lull,,., dtdtes: II mak .. aU ... Ioctua "..,01. The UlIWIl(M1 "QO'''~'ed "-lIh.n ",definll" aa>elcn!. lina.Uowa 01111 a Pllttlll ncUlralllJlbOll oIlbe dK:itt oil"" .... 11"""""" ",len, 01 'i","., .. bIc IcoI&lb': " confincs 11 .. 11 to JOINn, them up ulernaUy by itavutl

lbern lile pul "'~ a lld "'" .. lorentz'. conlraruoa .,..,.."" 10 '"""'I"'"' I .. mew

We are -IOPII 10 """. ,n lite Ih,n! SlaF. how cIc II .... ", lat .. Ettul .. a II hll......r_ IU be potIltOned..p1 .... , al lbe .doa,y IIonzon 11 10 ba"" lhe pho,Oft (a lCro __ I wi, ... r¥Kol 11,,1 ..... 1 lboul .."..·",1 ........ ~ "lid oJ rat . 'ere onJ)" • cbcW' 11 .. <k BrOIl .. ·, ..,." .. 10 ha", .., thaI I"" lope 01 PIInck', "lid F.1MI .. n', prnmpllODS led 10 mala",,,,.. ,udf pootb1l&bc by u,.JenIaIld'1IJ il "." Ioomoa. by unfolth"lIUI .. Iuch _ 1M moll ',,'-ea' I,,,,,, luI" Iudo l00I0:, .. lbouth I, iI i~""_

l 1! I NHEIS "NO 011 "OG~I1, TWO 5YM "'El ~ IC"~ I'M'5

w. t_- Ihal Oresme', d .. pams 1I1rf .. -J • .,loc:!ly by """Ir;ull", I """'" lhat had .Iready bcCfI V\f.Iled by I 1110,.,,,, body . '1lh IJIOlhc,- .hol had )'ella be .. pIoo-cd. Thone dill,.."" llIcrrlol'C prI"Ie", I 'pIIlllli cb"'*- lhal or .n 186ni'1 of m.uk ... 11k.., 10 be already practd II III equal d,,~ from ono "'01"'" and ro,cd in rtloll"" to an oboon' .. F", lbe loller, lbe ....,..,alro dia""m hy M,nk"",',k, 1"-. illo'c cIlOloOn a" o.i"" 0111"'" 6",mr " 'llh a dolled h ... ) fI'OO<"b'pace I" .... ". Ir ,'nped . -,'h .,.nical,_

No'. lluol if Ihi' ,)"1.", il 111)" Ini",a~ by " ,'docil)" '";n reI.IIOft 10 lhe

, L,--. , -

litE 5O.U;N. TIlE IPECRIJ'" A":J) Tltl! rESl>tJLU)I

..tIk ..... !lot b_ of the dlaanm bt<>;)o, .. obi", .... 1Id I ,. INtrNl ouiocs r ...... IlJ< axot,nul;Ofll.

f, ...... 16 INk., i' possible '0 follow doody ,he """" "''' 81>1 ........... of drJT<a ot ,'docIIY .. hio:b fOn'eOpOnds 10 lhe IOkhn, t.d of lbe anp. oa ,he d' ..... ·1 4. A 1iou",. ,·doo:o.l, (0 S ~ < I) doco Iloorer"", 0l0ITafI0I0'I1O a dtcru ot <hsIODCbOn btl'""'" l1li"< and.~ 1-0_ 1""1 lb. dqn>e no fDllumal .. be1I 1100 ...... "., body "ot:to:rO vrioal,. lhe up. () ban,lbe1I a ,;pI .",10 In Ilns SOIIllIlOn. 'tpJR-,"",' .. onl, a .... ~ 'pKC •• WIII~ly wbja:1 10 lhe d.,bt ot ;ulllpCIIed .1Okri lor .I .... UI' ...

" , , , , , , , , , , ,

, , ----T' , , ,

,

.. , , , , , , , , , , ,

, , , , , , , , , , , , , ' \ : , , , , , , , , , , ----r-------r------,----, , " , , , , , ,

'-" no.. 'I~n.,b of Minko .... ki·. d!.l.ram is '" tM .... ) it captu ... o'.'OCIty as

an anal<. Ihl mlk,nl .... ,IM the piro/I~, or v<\ocitieol a!Ound the .. horizon. In mok;nll 'pu.~ ~pIICO' rommonploce u a pc-rpc-nohcul~ r. v<loI:lly 1>0 10"",' aptJ<ll" u .. hoI" m'M,".in ordet '0 .vood lhe ol.,hi! or the filed ma.k. rs: IlJ< intunoon of an ',nfino'.· v.IOCIly (n .. ouW be bell •• 10 ... y ·;ndefin'te", del<rtf lbe "'",,kneu or u ,n'unton "'hoch. lOll "uk 10 cut ou' the depoe I. •• h ...... 't .. 1f in do, ........ , 1I11h space by rema,n'lII ",isonet ot I Jfoulldlin< lhal,' a.oo.okllIOI ..... arod ,n """111 ,be ml.l en 610 ~ .. 1"lUl" IOId r .. ' ... ·

TlI.I, to .. by II " brlln 10 " .. , ou .. from 1M dOl.,.,.1 t\. open III< I. () ",,,,,,,,,,,,,1, and ,hu~ R'1:al the P"'" .. " ..... ad, ... ou, btIOOft11'M Jooctiom of pure."""" \I_~lfl atooIlht ........... 01 ""re II"", (I _fl .. / n.. d ......... or IlJ< "",.,nl hod, II lUI. Ib ... ,,,,,,,an IIh I ,-..." ~ .~mmctncaJ .... th ,II< IoI, ... t"", ""be complel.'j doad ..... 1< (de, .... I ~ 111 10'10"1 ~Ioclly open OUI 10k. In .n ...... " '"'' " ·' ... ·0 <odes' aM dllCOll . ... any .. mplluon 10 .... 'I U "",.dy d;u""n.1O lboIi"". opaual "".,,'Il.

To nu.l:. 1M .)"' ..... ry of ""'" aDd 'I"""" al tM k''''mllUC 100'01 "M~ !I.e ~ IIlm .. lr .1111 """" patcnt, de B' <>&I" <k<:o&o 10 1"" ki ..... tr • pure 'ponodic p/J<no"",non·. tut "'0 .. y. to .""t~h lhe: roles or ""'" ~nd"""",,;n tbe prcudm. di.",.",. Spooo-tune IS no'" m,ped ,,'ilk /oorl."",~I ..

" , , , , I

11p« "

Ti __

' po ,,0101 ~ .. , .-

Spice-Ii"'" is reducrd 10 pu re Ii""" 10 •• )'dlool un, ...... 'hat I .. ...., ..... ·"a t .. '.Dd petrl!u,cs ptoa ..... ,iboul .p.o.l' al dill" ... IOon.

l. ... I"'''''''*' ""w lbal thos 'poriodic p/Ien<>rnmooI' ""' .... II lbe vrioal)'. 1m ~la,"'" 10 ,he oborn'er the ·be>'Cl1rd od .. • &, .. up aDd I ImP IpptaB..

• , , , ----1-----, , ,

,.) , , , _._----, ,

I'Ip«J'

--\ - --

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ThIS 11""'- IpllOO " 1M pa .. ......" Ind ume " .. ·""1 .... loa." callrd lhe mvdopuIJ .... plnlOde 1_ lUI. Wi,1I tbe ..... k.n of 6,ure 15. "" had foI<h4" Sf*'I' \ImI in tho 'don:ctJ<>n 01 ~." We b .. ~ J":" IoIdcd I~ la,,,,, ',n lhe 'cJ"teClioa oIu"",· 10 obw.a an iAtnwl of ~ ~(.~ but tltH Ito&th .. no! "n:II': il is_ otor.,.wI by actually ............ I .... lmaI JIIodlord _,,,,re aIon. I bod)' Thno Ito&th to .W'ri •• od woslt wbat pt.)! .... , call the pIwoe .... ,,<OIy oIa _ ..... TlUIHk":OI, "Ihal 01 ... 'l1l&i'l1li)' obIc . ........... , .... ,II web a_y WI lhe .y ........ , .... ,.. apptan 10 hom .11 tbe....,. .11 ..... De llroclie obo1o. .. ,""I .11 .. ~ .elocn), ( .. -hd ... " dnoole M 1')" eq.w 10 I,. Ind .he,eb • .."."" ,ha .. I. An ""'1Ca1 'uodoI· 00«)- 1 I' can be nwode 10 oo,,"spoad 10 'I. SOl"'" II. (I ~'l > I . n(c)" «Jmprued be,,,,'«11 0 aad I; .h .. ,n<\el .. Iict,,,,,,,,, ..... " no! I,nked 101M p.o._", """'-. '",ai' materoal cm',ron"",nl,

l'be d' ........ in fi,ure 19 unfold. ,he: .pecI.um of pILI .. """"" ...... 1I.,h i, .Iso ,ut or the d."""" or d .. t"K:!',," of '01'" 10001pace.

We ~ .. " ,hal lhe rna» II rCS' II 're .. d",J ' '<Of an l~ftnl" ,'all>< of lho

,. , •• ' ... d .....

_.- -• v_{ __ .. ,

'.-" .. <10 .. 111 I' nil ;1 the ... ~ I. ro.I>odcnlll ~I a lion."" ,lie: c1'l'la l unl~r$t .. ..,.".lcd by the obxrnr .. ',rh lbe .... rio.lll: pht""""' ..... "· ,,'hid! it.1 tnt ,n rdIUO" to bun",,( II read,,,, 01,,,,, ~1IIj; dJ;irram ohow. 11101 the ~IIJ 0( the 'n' u'l .... "'i Iin1td .. nh .11" p/xll(lm(nOll" rnI .. not., .U 1Idf .. ~"wk'" lor an oboef'"C1' ...... >'C, alwa)'l "" the!DO ...... 1Io. to readlll ... nJlI, from .he ,,000.'11 ""~ I~ _ • . _ I) .'htch ..... 'Qj)O....u for hIm .,lb I • ... ,ural' 1JICb>\l1l-lU,wlolory btt"l><1Illl11C aDd....., will ha~ \(I '~"""'I/ aIII'l_fifIJ'tIy II r - h . J ) '" p"'IIIOII h,m .. 1f In tbe 'lUI' ($Q< fil- 41. (In lennI of 1.IIaJe. tbe pe'pendoculan'J at d,aaram 16 11.1 lilt boruon 01 Iht ipccln.m 0( .... nal;"" oln

De 11.0&1" Il'" In exampie lhll illu\Lra,es lhe >'InuaJ 'pallo, d",inclion rbl! anld "lien 11>0 'p<nod", ph<nomcoo,,' i. 5eI in mOlion on mllion 10 lbe .......... "

COfII>dcr •• y>t .... of.pnap OIaIJaU"1 In phaoe In rdluoa to ",Iud! ..., ~'" ""moblle. Til .. Jir .. 1JOI> II ddincd br.be d"'IJI"'" (fia, 171.11d on h"."' ..... laJ -"'npet. The 'yltern appear> '''''<'II III ........ ~.1<> ",and _ 'PEL" "Ie ... uh " ,he ima~ 01. myrDd odniliell 5f11',ap vilnn ....... ""'loOn. to!' .... ~ - .. ,UII • la ... dod aDd I"" fY'JI .... off ..... wp "" .,.m..I IIi/ktmrr f,.,... "1wdI, WlIlt 0( Itrqth roukl 1M: ""I 0011,

II",. ","" ~ IS.., In """IOn If ,he .... "., •. b •• " .. 0( fiJllf* 18 elMo up tho '""np no Iollter aPf'C'" to WI ",.11 .-.bnUng ,n phase. We can d",u"gW<h

the 11_ uj' Hlcftticat phue and oI«idc to _ ....... mo.io", ok • • <hinS thcnu<lv .. out. n... ...... ". moll",," . '" MI' 'rnJ~ .... hive ,,, ... il .... '" pnnt ,hem vim,ally .n • IpKe and cit 11,1>11 .. incited ,poalia ,hili they .", ...." '0 be ""'(uoed ..... ~ U "ppearaAa!. The ,"moh 0( <iulvam III. .. bleb I .. hc"""""'-h obIiq .... co~ to ,100 dtff"""" """' .... 0I.IM: .Y ...... m'O the __ ",'e ...... ..., caa. .hrou'" • ..- ol i," .............. lHOaI'e ,I >n,h an ;n'ertli ...

rut !iJCIEUI, TI+~ Ira. leN ANU TIlt. I'£"OOLLM " IN!" ~ " ..... .u The .plem 01 ......... III pIw£ ...... imaM)br", IIIlpnoo<><d ~ ,.- i&, .. ~pe rhYlhrll. "low." ms ,,, be awIc irid '01 t.,-" .. 'I'" ",bote """'''" 0I~" ..... met", " • 'naturar tu """"'I, ... O'"",IIC" dLopam< _uml the "urn,," oIma,kon pu-' per S«Ond.

De llroa1~·' &mil ~m COdjl<U in lui ha ... II, "'OOc:t$'ood ,ha ••• lhe aKO_ !!ally kl~a"c 1c: ... 1 ..... '" he placN hi"...,lf hi: ,,'QUId no! OJI.bI .. h , hi: -wa,'. »lIrti<;k dUlhty by mllk"",", "mplc a"alogies. IndllDOd by con<l<\c:rinl W1l1ift IMI '''' 'rul ""ll", In "",,,,,,,', .n loon'ml""' manner, and ,ha. thl: _ p/ly~ "''''$Il1w II, "11'"'''''' to.be .ntwt"", 1'"1'1'> .hem,.,....., .... pbDt ... "" it "'" a tol\l1b'< or causal docbC. ,,,,,,,,",ed .nth ,he actIW 'ra ... " 01 10m! Of crrcrg ( .. .,..n ,II ..... 'n. """"" ... _ dc<:1r""".",cllc .. .,. ... 01"

OIhtrs •. ~ .. /'<oar .. -as " I qllo$JQa ,." ham ol rdytn, OIl I '<:C'l1.Ilb .IIIJoc)' ~111111& ,'''

pnlltlpIott 01 M ... pn-t.... 'I panldo deploys lbe mmtmum 01 act ........ """""Ill i" \I'JJtCIOI)" ar>d (II ~erma' 'j."" crosses .... ' .... 1 ......t," .0 • mommal h~' Jr>da:<i. it'l '''$) to 1ft ,hat, .fl.&h' .... re comp<)K ~I)' .. " .. Id "..k •• IM: "y, (I .. ' ...... "hXb COII' rotiM:1j tbe "' .... or m""",,,,,, (we II .. 20) .

., ..... ,

.,

""' . '-'

TIlt crude anaJou ""h rnJ......t .. (w ...... r>d ...... ", wrn- 10 tIM: """) ~ '0 IJI """"""" Ir>d , ............ hn .... ou' an 0jijJ0iII1"'" Ix'_ ,he "" ......... ""La. "" .... 01 '1e!I' IDd !be "lIdulo,ory potnt ol_ f ... pc,1(IIOt "nder.tood II .... rk confronta.ron "'IIh ' .. Jit1' dmdeo .R f •• o", 0( ,he undulatory poont of ,,~. champroo«! by I-r_l

To "'''0''1';10 ,h . . ... 0 "",nl' of ,_. II to ll«"U'"ry '" oon<rdct firr"ioNJ ..,,' '''' ,·.Ioc;,..,. and mdicet. .mplYlng no ca .... llly. To bn:ak tbe ",,,,,,. 01....,.,,, moIlOn, ~n>Ss\D1 rnl medIa. de 1I rQj!lIc .<", .. ~ O.lJ!eo'. bokl .. rote; to conoe!W the com""", <me.I"""" of ...,....try ...... modoalllCl, Gililto h.od rome up ,,'uh the r.._ ,110",,,, ... pc ........ , of ,he ',oIlcy d • ",1d1~I"d ~I ~ 1IIopo. 10 'llrf t\cbora. tbe ... 1nBIJ" """" diol OIl' .,..tuall) G"li~ ''"' I. in .... 11 .... """"' ..... """,lid ,lor "'~17 01 the world """ !be

auooJ,lios .uaooa,NI ... nh ,L We k"" ... tu, (h, • ..,....rio deatNllhe ... , fa< lbe "."""'''1 of ,he pnllClpk ut'nnllI by IA'KI" I ... Dc BtOIl .. r<COInired ,UI of. in lbe ,..roum, ,n lbe aboence of.1I ro..-.. "".., quan'lty of motJOII

prod~."nIi<r "I m ... •• I ~nl of ""'''". mcch.on;'" ,,, lbe utmoo' p<IO'"

H' J<IUI n up ."h JCOi'''II')'. '" lho .. "'" way • tipously MOiKlCfI"""""" """"". ,..1<11", I,,,", aU etC'fJIII,. I< IMII dl<ptnaI ... panicle II INK cIdIcc1cd in ,ho Ib.,,,,, of"""" and a.eru iIMIt' U IU' .·hd aa paop lbe """'.. "., ... "", of «pt""" otnd i<Om<1l')'. Nl lurall,. lbe pan......,.. of Ilns .... "" (fnq........,.. w:a"" len"", etc ..... d"wtl, calcul.oblo f,om Iho!o: 0( a ntk, ut ........ • ,thou ... ", ... Itt ." .ny «Iemal modo ..... 1 .. .........tJ1\& 1M pwhIL", ...-Ii

In pmuqlnl u .... U pa ........... ~ ",,",INllbe UDI1 Jorm for III _"" unuonn ~ili...., IIIOtIOII (Ga.Ioleo· ......... "" nlltr'). ... b.d. IIlIdt .1 r-o-1<k to ~ fora: • Ilw CI_ of ...... 1 .. ot loNer .x.. ... _ ,n rda,- '0 .his _>all rom.. By III,IU", __ as pa ........... tho phaw """" UIIIOrt< ;"11' as Ilw un'l lonn fa< "bn<UCIaL

Thio .. Anstotlr·, ..,1'ftI1" (>II albleo! O.Mol .....,u,,"'" Mel dJ>Li<arpd 1In,lo"" amlla, mol",,, front iIJ ",,,,,lied plWI'lOCI as Il.Ind .. d ..... ~ ",,,.o~-ro.,,,, .unh 10 ,be I""nod~ p/Ioraotnena .......... ltd WI(h ....... 'nl body by I"" pr~u..,. romtuJalNl by de Ihoa! ... It.. cin:k 1i<>dJ ....... OCIloIoIicaJ d, ... "I,_ A. a 81u,.. ut tho Cf<'hc, n ,mph .. "" .. a1 lrajectory: il it. """ mipl .. y. lbe uo/, fon« <f ribr.a!1oou. By <ld'o'm,nl a.hle,,'. ruin> of may, ..... CI.II

obtam 0.11 lbe mo,iont 0( ,1.0...,.1 """un"," llid <>quaUy. by moduLoIl", jll """II.,,,,,, .... an In pnllClpie rapolld 10 .11 lbe problc!M JI'*Il by ... , .. IIlC'C h.o nlCl.

W. h."" JUII ...... 'hal Ihe LIIOCIaI1Of1 ut a" OfCIllllor IlId • rulet of mass ln no .."y dep<ndi On • liJural .... I"IY .btill1lCttd lrom lhe <>P<',,=occ of 'n:a/' mtdla. ,",0 lIIOI1: could i, ~ cakul~u:d f,O<lI lhe ptinctplrJ of anllyl"",] mccbao. ICS (!'Ianck', int,oduaion of """11.1" .. fur 1M pboton is alrudy I ",.oI"'OOII"Y .~roko). w. bl". II..,ady .. n~ Ikll IkillW(lcialion IJ pllyed I I ,he It,d of d<al'?".nd lbe antP of ,hoe unlukh"p ut del""" dUll lhey lU,horize..

U I .... back 10 lbe perspra". p,ojloo.1iun d;'v.uns of lhe ,,,I,,al ,,,," of lhe """":' body ~ and the phuo ... Ioal ... " _ l.. ( .... lip I. and l~~

]0 JU'"pO.",& IheK ''''0 dll ... nu, ,I bKoo.lt. p.>t<'fl' luI ''''' ~1Ifokl'-"i of ,he .,...,.rum of rulent of <nUS Cia be ... perJl" .d lymmctnc:aUy ... llb lbe uololdtq 0( lbe Jptarulll of OKIlbllon (pn1ta dI""""lic .... "").

Th..to IUpnpllOI"'"' "»OOIhIL .. h,1t ~'''' lhe blCfa",hy of dqRa of ~ IlId of V. It .. _~ 1o _\ha( lho: .. Lo",'''11t COfIII"*lion 0( lhe oelool;'" of tbe ""'''I'', body" olio lUI of 1M pIt.I .. 'dodl .... In panocuLo •• lhe boruoa of "" .... ual ,doa~ 1Up<'pe lei Iudi' ""lhII' of ...... ¥doall .. (muo.1 .... 1. nu. <I ...... ".."" ' ..... '--1"'_ ...... II I d :,Ie' field than lui oI'lIf11ClWU' or fannionaJ malJOM. I. "..t .. Sa,Jraal 1M Jll'O~II1IIIYo(bi'lund.a1')' ""pentIItIIll wtuch, by ..... 01 tt .. ",Id_ beyorwl lhe QUA1'1'" and 1Iu,.......,... lhal a/f.a JllllIIdaK mobdo.}. mat .. II J>O$PbIe 10 propox. fa< cocb -.- 01 d ............... 1

_dkal.,. of Il10',,,, ' I ••

;1 , ."""_"'_<'0)"" _______________ •• '1. 1 -----------

c , ...... ,....,,. (00 .... _

r.:.- ....... "'_1

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.. bo ' ..... --, .'1.1 ... ...... .... .. 0 .. -_.-

Feo ... • ... _i.h (oo.h( liAr.:.

_.... "'"""'l

d· ............. .- I ' . or .... .......-. ... _ .. _or""·1 0 ... _ ....

••• '.'1 .... '

of lhe .p.>I .... lt1nPQflll. II""" be'ween 1""","11')' and mecbaD"", and I lruce bel..--een awmetl')' and opllcl. Thw. II iI lbe 1Ioru0fl ... bich. 10 ]IIo,'id,n. a h,"1!" bel"-"'" .. hal mUJI be coiled dl .... mm.l~ "'p<nmcnu." 000,,01. them b)' "nfold,n,. common el>lllci'y: Ihll ut,be """'pa"",1t.

W ..... how [in.IOI" .ma>htd ,he ",,,,,,,",soble ehara<:te, 01' lbe <Ic&r« I 01 ,.Iocilie ... By ,,,,,,,,,,''In, lho ma .. "' ..,., U a ho<imn by mea", of on 'inti"itt' 'elocily. de S'aall. n:M"'I lhe ""bola.l .. l!'IIdu,on ..-hieh posiled • m,,,mal dell"'" t- dell""" I ) Ind ,hon . haded u/f 10 lhe infinu.ly ob$cu,..[lho '0'" <kl!rt'O). LIke [,n>tei". de Sru&!,,= undomood lho ... !tole ''''IIM;; I""", of lbe l\(lruon and ". CIOpacily 10 unruld Ind 1o oomp'''' "' ,.-.11. ~ IlndOClJl< 01 pb.~. " honcd'otlh ,,,,nod un I .. hold; tbe ,'doctly of lipi IS no ron..,. II>< "ymp,ott 1M' e,"des 0.11 mOlcria) • • up arid lhe m... al ""I " M longer ""i,d, pall'llt<k. made "' .illblc. ,be .... ,n frunt of me. by .... 11-e>1de," In'\llhon. F.lR" .. n. ,n perclunl on. phOlon. and de 11'0&1'" ,n .. 1id'"1 ,I>< ...... llll'$l 'infirulely r ... ·• donn&lY oIl'ortd 1"0 lhoop' .. penmenll .. t<osc rcp<'mus",1lS cannot be o'''''<SIU!taled

• INDIVIDUAtION In 1\lI'A('1 ""0 l"OI ... mUATlOJ< ., ELECTIO'

Gto_'1')' .ad opt.,. hOld ...... /wid In Iwwl fa< lloDa 'Ulle ~ Gal;1eo _pnod lbe boo ..... l}· ""'1"'''_' 01 "'" U>d" .. d !'k ... llIId c'·",,"'" tk d ......... '" lbe ntIcn of .... t_ Ii .. I') .. hlCh .. l>CIIlala III<Ch.oaocs llIId i" 'I')' br makla, ,..pllal lbe b .... wbm: 1!Iq- . , .. llIId r""" .. boob u..y

bQurQ'''' n.. mrdIannl "",n l of ._ pn\'ilqn 1M .... paa of .ndmd..al

"",lib; "'" the ulrriorny 01 ~ lcibruz', .,...,_ ".' Ie! ,~ " ... al a o .. ta.a .... IIa.' !O mrdIal\OQll oonlmlplallO" .. bu! !.I!o: 1;10"., ' ..... "111 """"""'" of .he .,. .. _1 cbcbt ofl/le bed martm. .nd. '0 ~~Ir the p'"'''' 01 &<;t,nl 01 bod_ m~ .. o;:o.n •• nI IIw:l( "'!I~ a«elerlhal ,ndh-.d .... I. II .... b1k/llo1uly I ... ""kI ~k •• 0 be able !O di S60h. c:ompk •• ly (allea't on t"""lh. J!he lCna<:iou. Intunion or the JUIII"""tion 0( $La ndard .... asu'e; 0( len~th by "'In, p;oil lhem 'infinnely qu><kly"" '0 I<"ach Ih" p.:trad,,~ "-he,,, t""" .nII .,...,. .1<" • nd>ll!n.wslLal>it and ,,-hlOh ....... !O hold lile' """"-'I of ",.dlln!Clo1 ind"'OIIua' 11011. Yr ., kno .. INt it ..... "'" .. y '0 "lOU lor Ihr dawn 0(.1Ie' ' ...... uHh conl\lf)' for ~ ' 0 "'~ Iool,q .IIe Illfinrlo: in the lila:. ~un'nl oul ,,,

horu:on.nd mastrn"l! ,he dc._ of .".....",tJ ~"pUuioto ... hICh unfold. ,1. For opi"'" lu n>-.J. all tl"u IIId boa! ..... ,Un.! .. lor marly .m "'""IU ..... For , lie'", p .m I .. >coup duUal, of ,(lluillOftl dJtpo.wd by b&hl; opt .... don 110' "begin" from all'lOady JI'-'" " .... , 10 ,bm PN ... I nod!" In endowin, lhem ""' h """" 11 .. ,Muy .. ,n maklnl b)'mpiOhaollJ I ,,",isba.1 o;:o.n"nuum " hoch II 11 .... yl fl' ",,,,,,,,",, from the peri"" pjtn,ludt: of a '"""rum.

Th1$ '" no doubt .. 'hy. II lhe hori~n of perl«t,on of formll. KholullC op. w;:.an-ploik>oophen placed pure. Ind.' .. mi .... ,. lillo' (lot.rJ.. pnDCIple of III , he d •• unccionl.. 8", " ..... nd",tcd h&b' t/-..t !hot ...... kentd ,he ... " •• pol<1lUlbl ... of man .... . nd aU bod_ 'nd nd",~ ,lie """" bwnblc. 1'1110010: of 1M lum,1>OI1l. ~ "b AI-- ",nd~ G"""",,,C$l~ and 8' ,"~ oplia iI'ho~111l of .. ph)'OlCO-tna'hcmaliooal; lhe)' lilt", I ".,naple of conapoTMkI!Cll bc! ...... ,he lipm of "'0111011')' I"ral"" 1.-. ups. tim,la"", IDd !he ~ nl..a~,near propap'>OI! ("I\cct~ rrlract~ d,lI"ractod _),- Pu", hahl doe. IIOl ff~r 1M ,nftnlle 1-111Ipie, uDdw1dcd.. ;, new, dluoh'''' .n,o lhe ,ndriin,lc 11>11 ""hem! reoolutd) 10 ' " hooV~n - Ihal .. -h oo:h (l'",",,'c<I. had """,,,,;«<1 lI.he hel,n .. nln, of (.' , mo;

Lia/ll, I. r.oc. .. . pre.tdo 00' f' "", "Jdf ,n III dU""'1011J.. in . .... iI. "'J Ihal. fl'O!t'l. lu~ i><>1n~. luml ........ phe"" of , 0J ~ Is IIbl~n •• -'" ,.,m,.,ed, "" .... '1 .. NoeLed by ........ hin. opoq .... 1>0_. rorpornl"r it .... , his ... "', 'II) contOq......,. 1M 'X""''''''' of """'''' In 'h .... dU"",*3n. L'pl is 11<11. '~ I form ,''', ...... 111 from """""""I"r. I! Ii I!id' "",,,,,rcahly .. 10 ""Ib. ~ah' .... of 011 ""'pote.llhinp. IIIaI . 'hoch .... I """" ,,·onh). nobIe.nd ~_I""I "' ..... «:" '" m~ I"'o.n Olher bod_ lMHnilablc 101M topIrile f3rm< thai ... . he ,nltllogrrotn. Lillo, " lhe"""", 1M fin.1 '''''''''''01 f"""

~o< If l i&hl. by ''''' 'nfiru\~ plurol\aollOR 01 ;, .. If. I""" .<,,_ l<> m."., [in ' ''''h I ""'J .. I" C()nSl,lUlc II] in d,,,,,,,,,ion 01 , .. .., cubu .. 'ken. b} doulllin, Ih .. ' Rfinll' pluntkahon.. ,I I"'" ~x"n""" ' 0 ...... " .. in du .... """'" of lour ,""brlL ..

" ........ 1*- I 'plunlicall"" " .. hoo:b does no! nba .. J! " .. If", colli"", ltoe deiinsurulO .-J I III .... 'OIo1n1, 1'1,,, pluntkaooa fltal ", ,I poMIl>Ie ' 0 P'" "",II .. ."eN""'- 10 con/ron' IClUlI ,nliMe; and ... -eu 10 ha", an illl"oa 011 lI .. m:

It io ...,..,bIe lb • • In .nf,,"'e CQlle<:lion of numbr,,,, may be lied 1""0",,<1Ion. ally 10 an ,nfinlle """n""""" """",d ing 10 all nu"",, >c&1 [",l;Iollon.b lps] and ~I oo aero, d,nllo all non· nume' .... 1 ["'lal1Omh, .... 1- And lhere .,. ;nfinol"" 111:11 . '" 1"~le' lb • • Olher infinite; and l.fin,l", Ihal.", ...... 11e. Ihan ",he ...

Tl>e rolle<:loon of all n~mbe .... odd U " .. II 1S ..... n. .. Infin,1C:; Ind II " ,herrlo", rna'" INn 1M roIkaioa of all ,,'<11 n~mhe ..... Iuoh " none!beN .0MIie. For " .,cred. i, b) lhe colIeaiort 01 IU odd numbe ...

FUrlbem>or ... lbe «>Ilea"", of oon'inuaUy doubled numben l>eJ!arun. .. ub lhe ~o" II InftMe: Ind ,n lhe ..- ""phe wllea_ of l lllbe IW>-.. _pond!n~ !O , ioHe douhlo> i. infinile ,And 11 .. tlear INI lhe .. me i. lrue: 101' all k,ndl of numcric:>-I ",oponion. lin« lhe In~n". an be hnhd p,oponlo ... lly 10 IIIe ,nr.,.;,. """"ding 10 any "ne 01 I ..... p'oport,OIlJ.. ..

l"h" pion" ..... d .. pm>ed by lhe I ..... and ,"""lfeI,ed by lhe d,W ..... "", of 'M ~~ ......... 0I1on , ... 11'0 be ro.-rllpl<d by .he In<kfinn .. 1<>01 'N' .... by"

II ... II"" ... Itd .. Bou"" ' ''1 Ind c:I1"""""II-" leu .... lCflIlli1Ct <h'ne:".toe._ """k I ........ n Ihr "lOy lho, a ".mp .... pnn" ""'1<. w ... ~ ~ ""I ..... -dru.. Ibm (Hbm. aod. mud! bc!1 ... INn lito: ..... "nal "",a' . u.k .. up Ihr """""" lIIdI' ol ml.u.,.bWty by d"""Iuoohon. Ihrooop""""" "",r'" and ....... mud> m<I""han .h,ouaIo con',,,,"

("hl'lI'Il mcchoalCl.nd lI'-"""lCI'r mo&hl .... U ...... udfe 1M lor" ''''0 I""'VlIqn: 10 be al",ad1 po'''I,oned at lhe I\orilOll of (0"'" and 10 ,nd,.OIIual. pa"tnlly Ik!"'" by d<BIeC by ' .... nM f"'" ""n 'o lhe pia) 01 OIU<l",," Dnd dellet:llQ""

•

("o.....ser 'M "",al. P, and P, ,n n,1l'" 21.

"

I.n",,,,,, •

Tl>e · ....... Iu .. C .... -.- oIlhc:oo "",nu CIlD"'" M COplu"'" by ... mple

u'«nal .... i ..... ,ion. bill must be pa.pcd by ,bo ""1 nIOIlOft of t ,," 10<>1: Th .. p<I'bt mak .. "" oon.ru, like an ",,,l.,ed point In I plane bul 1"'1~IIlJr ~porc:s for ,If, dc,«,;"", and Ih" " .. hy .uch · .. ""'r~~bI.· pooou ah ... y> k..,p in ,heir dc6rution ....... 'h'lII tJI the mol"", IIIaI bnnp them dMCl' and of ,be look lha •• by delay,nl """" Of Ie.., .U able to poy ancn,lOfI to II •• ", and •• ai .... p<>\erlllabllel.

Thn d"unKu,,"" ,,," pocn .. 1', and P, f,om ordl ... ry poon •• Ill. p, .. hodl can be ,exiled in ' ... ·0 compltld) ,n.kper>deD. m.nne .... One Cln rollow ,he cU"'. wilb "",,', ~ni<' Ir><! <Io.'lnll. r .. ,he: point ,h., ",orb ,hoe (Ont"''' bc1 ..... " .. II.a, ha> been co...,n:d and .. 11<0, """"'iii U, be t,,, .. raI; <>lit nn Ill ... lei • Slrai,lll ~ ... lhde ponllcl.o l,oeIr ..-bil:b is "'1", ... 1 ... 1'0 aboorblnl ,bo look IJI II COIli,an, dllU1"", "Dull bt.OdXS ..... ply 'l<M:h,nlfta"F"ual) 10.be C'tIn"e \>inh aa ordi .... f) poonl. the hand thaI d", ... rompleldy ''''''_ ,he ioolllla' c" .... The .11 .... 'lOn II di(fCfC1l' for II IUUORaI}' poon. Ill. 1', u, ~ 1""01 ur inlleuon lIke P,; the deft""';on of ,he hne «hoes Ih" pous'"l of ,be: look .• be mmions of the b.:ind and .) .. be""", •• at~nllled ,1>< ....... ""

SI~lJOI1ary point. or pomu of ,n""'K'" .~ ,beret" .. ,,,,*, ... h",b poobk ..... -10« the .... ,~raJ· lunctoonal COI'rnpondomce be,,,cctI ,_ pGlQI ".uIn'1II lion, Its I~ oDd the 1lI11#f1\ WI rn"",,,po;. ,,"""'. aU the """,IS Df 1110 CUrYO,

,he ""luLa. p .. UIlO .'" IIotrcfore ....... nI II ,,,,- " 'hen ...,vdopn>mt by .he Ioo~ " "'" eq .. (\\.lml to a axnra., and 1'0""", .. I 10 mJl«\ ,ho $p<IQ~IlY of "puaIID.hvwJLUlion. Th. Ian., illbl. ,<> be palion. Ind Is a-..are of [lie vaMy of ct .. m'III 10 ioo\a,o lh,. or lUI colour frocn • """bow or 10 ClLrKt ,blS or ,"". ""I"bo. pouIl f.om I spectr\lm; II "bet.«,., lao.. ,lot _ M.. ' .. N.", IN ritnN by dnco>W111& • Ik"., .. Qpjlbk d <>peIlUI, 0101 ill "nltlbues. and ,Iu, II why helll m'rod..,.. lhe pnllClpiet 01 >UlOa .... y do:'erntlJ1lILIOII from .1Ie ""1><1.

In <>rdor I., d,,;unJUI>h Ihit or tlt.lt mdl>Ld .... L ,,- principle< d" not uUow tl>tmltlV<'$lo he 1,,!<Ied by Simp\< ronu"su or by nude quanuu>'C illC'ljOl;lllloe1o. rroc: ' ,dinl I.om pal1'\101101 or IIIOIIII*LaIIOfli oIl'n::ody aaxNIIplHb<d ind ...... · uab, .. 1udI1N __ "mo. ..... ll0. b·! ...... .- "' •• ;hod! ..... hopn II .. ;0

be able ' " pther illclll lIItO. ~1.bIo totabty, Tho pnnapb "'''''IOIW} doIrnrulUlKNI do not hc:allllc 10 rocddle in !be ''ef)' 1"nc:cM or ,ndil'ldUlI>Oll, ., tho 1'1.,0 1<>I1., ... ,n. "alancnll. o;:omlO, re:\pecl"~ll from F~rnllli .nd ~ .)nrnlll1. ill" ... ·

'Amon, all ,'''' OIl' ",_<d,~, ffOOl ,'''' poonl A IlnIl rompcJloIl IU ~ Ibl'Ol.l", D. ,''''''' .. ODe lhal accompliibos the ,o.,,1IC)' on I m.nlmum ''''''' .nd 11m .. ,'''' polh actiWl) f.,IIo."Cd by IIgIII • 'No po.h i. p"nlqed ~ priori. ,My consiliul. I """lOnu"", .. htn: lhoy .'" ,,"ually equ,,·.lonl. A photon pr(O(tt<linl from A and absorbed Inlo 8 can tok< any ' l'lOjonory 0( Ih,~ """IInoum, bu. lhe .... obabibly 0( mU11No1 Ioca."", i. tho 0<..,..11' .1Ia. join. A .0 D."'"

no.. ...... OIa' ....... 1>.. in panoaoiar ~CJ"""",· ... "lIJo:h ,n" ........ tbo melhod

'::---),

" / \ I , __ , ~', , -, ' I,~ • ___ ~_,

, , , ' " , W---- - -- " ... -{.. _______ =~ .. B -- ., , -- 0.---' , ' , / , , , , , , ~'

.,

oI.he ·on~I ... 1 01 tho pol ... ·• olio .. lha. lhe lIa.""",')' Pili"" ...... n: .1"")1 .obp •• o 'formal ,nl<".I. or-mph,ude' or ·ampl ..... In'"IlOlI.· I"~n by formulae 01 1M ,),,,,,.

.... f -":-:.., '" N",,,,,, ,mm<dla.ely .M. ,he Inl<",,1 "P' r doe!! nol al ... ". rokr '"~ •

",OI'Oo,ly ddiDCd malbcmlllni heinB. but 101M <n""lopm<n' 01111 ,10< "",h. 01 • Mnllnoom D subj<c1<d to ""nain cons ... ;n" (for 'nslance •• 11 oI,ht po'lu powinJ by A and 8 '" lhe <IJIO oI,hc ",«<dUll >lalommul Thor .-t>I'n ..... ;do· should IlOl he ur>d<nloocl u a ""kulan,," ~n,q..e Of u 11 .. ball""" oh<et ol,Io< j, .... I'<"4I11011 of Ill''''' palM 01 D: tboy _pi"" and finnh lhe linea,. 0I0resmc·. d>apam. by I'nk inl i. 1o. ptIr1C>pi< ol orouc:al illll,,,,,I1 ... ,_ ,,'hid! 6<>1 01 all en"". '" d<iormi"" .110 ..... "'11 ...

In (1). lhe .orm I..m i. nol an <:H'di .... ry ful>Clioa bu •• Inplych (L I. y) obt.o;ncd by mo"nl Ofesme', ,n.~"';,y 1 Ikml' path (itt 61. 2~ ). ,

I.(", .. S , ... ,

r,....1. '1a,ul'lIlIy. l.4i'f '",1<1 " '''''n y is dd'onn<d. .00 11 .. [Mu.dy .h", '<BUI';""

, ...... ..,. .... o..no.,. ,be .<MdJ....,. pou,. '0 .,.<1, ....... ,'* 'd from.he ..... ida, palM: fOf an oro, ... .,. '''''''-MfI.ar y, 1ft, ... ,ual d ... ouban<c ., of, ,nd\JCll:l ....

/.4 ,'. I 'an&uon 011"" .............. In tJus ca50 0'"' ca~ lbe...r_ o;ompo •• lbe InptydlIL .. I. yllneilM rria~lIlp btl ........ )' a...! t hl wllh. 'f"r><1"",' •• ·llh • ",,,,pi< rea",),,, •• ", IlIf !of'll! ftOUl ... Inll.,nllllcf dId,n;lWlllo • nttIVCI' Ind ..... d....t "lib no ' P"'pof1tIlOO' ,'" 1M ~ IlIf '''"'''''''''00II of lbe Il10''''1 bo.ly', ..., ~b.lptel' I. P, 19 •. Tbo ",lal"""""p btl"",,, 1 IJ>d 1.(;'1 .. oodI. ,b.I. lhe ",,,..1,,,,, 0( """ of the I ...... an be?" ·a ffOfl'l lhe nl1lll"", '" lbe ott.... The", n I IrlnSpamIrJ at lhe m..1""""',p beI"'mI ,he 'ca .. ",· lhe bnn of i'

and lhe dfC<'l L.M "luc:h n .!Khlfe""l1 I<> ,"" <Ontl",,,10 '" ,I>< (\omlI.ln D. TIl< l .... ntll .. 'C and .eClprocal ...... act •• oIlhe rclal"" .. hlp lhal Ilnl, ), 10 1.(;'1 II lui of 1Ia"'. \lele....,ln,,,,,'. Tbe ' .. nlllll( IOO",dual. Y .... ,"""" •. ,,Yln"al d,Slu,b.1MCCl do 11<>1 Ind""" a unllion 011.. oIlbe .... "'. onIe<. n..,.. ·"auon,IIy'lndmdll.I •• .., lhere"' ... ,,,- ,bo, d,,,,,,,,,.8" .ny '.".'~'''k fu",,· IIonai I'UP of ,'''' I)'po! '1"" one " .... 1M o,he,' u( the Inplych 1I .. I. y). Moreo ...... Ihto, UlSl.""" I...! ,btl. """'rnlIJlallon . '" analyli<lolly boo...! up . ·uh ,"" oomlnllnlJ of lbe do"""" 0."

w ', murn _ 10 lhe ~"I","I' 411 I ... """,,Ilk 10 WI.' IJutI " n pm:t>dy lbe ·".IIOfII.y' I...!,. .. ,haal, IJutI prtm<k I "pubnl ronlnbul"'" to (J ). We Ibm:fon: now .... ,. 10' ...... '" lion Iype '" Intqnol. Iu ""'"" «MaUl otnJUla. modes '" "'1"""'11On of deJ'ft'I'''' <nU.in)ounxl" 01 0.-). i""",,,, lbe ordllal'> moda rf ............. r Inptyc:ho.l Tbc ..,I«tiuol at lhone ~ it; _ lUI •• b", • .,. ." ..... 1 ........ Ioon; I"" ""qnol (.II io """'ltuC.'l<>Il,O IIl'c, .... 11'" 'n.ilurnne!o'· 10 lhe <'InefF""" '" lhe oinluW )ou""'r> of~· .• luc:b ........ 10 d.C1,h l""iII",I ... 0111 .'11" lhe oclr~..." of lbe foIdl '" I..<ibwz·. .... "'brane

w. have lirady "",,n able 10 obJImc ,be ..... 1 .Uu.[\.., PO""" '" lbe "",,,peel"'c p' ........ 1itm "f 0 ........ ·• dialfllmo;.he Inlev.1 of the poilu (.11 If I .,,11 ntIl'" """,,"blah'" .".1 ..... '. "hich "" .. I.ly link> lbe roMIIIO'" lhat d.fi"" ,he domaon I) I'po'" by 1'<""" A And 8' in the "".., of ~.)nman·. ".1.""",0 and lhe .Io;t""1 b)' fold • .., ".nain ,ndi,id""l. in optic.( fl.

-I'htl.c fuld ' ...... utur.1 and "nificial ••• Uw.iont '" deplh can be. N~I"'" oma,nly m.k ....... " ..... l"""alIile •• pro,·;'ion 01 iICIItl. jpeelr;l '" VI"III;oo; and InIOf",I. of deJm:s 11011 ""'.~ ..... ny JU~ r", IN, jou""'Y u"eI' 1kuI" bul I ..... 1'f'I'ciou. h,,,1'" ~n: miud up," 11 io...."...ary I ...! wul> wbal <h1li.:.tll"'! I" "",,"',\IeI "" •. ",.. 01 ..,..f1Ilioa ;and hall<m .luch ""~''n ,1I<kpmdett1 parame1,...,. and aloo ... u", .. '" lbe muI<'r)' of mil,.., tamWn '" ,nell""'ual, and lhe """"cr 10 pIOIIalI .0.1 • ,n lbe """",, ... 0( '.nlIIOn .. bdo "1'0111' lhem. It " Ih"" 1""1. b) a ..... ~i~ iU.....,... .." .'" .<ftIPlnl 10 contC'l.c a ",a"fioJ 'mOOer of lbe Mlural _,,;-. W1kIy d .. ,od<d "'10 .Irata of Inpt)'Chs . ·h",h • IIh ,ei..,''''''''''''' """ wOliId finall) •• 1""."

Bu, .be uample '" ,he """""'1)' poin. 0I81 .. n: 22 """ .. , I"" na"'C10 01 tbe ""fWI')' hind (If Iool 1'.. ha~ also al=ody ."'ph.",.:d Iluli the .. ",ubI! Inpty""" Iha. I.., dlli"n ... "h«[ by.be _Ibod of 1M ,nlel .... 1 of II>< .... 'ho an: 110"", 11111 ""nool he reduced '0 • runellonal la .. tolOl)' and thai lhe ... fo,.., prohlbl1 ony " •• ur"", "",dllli or the ""'l'iu .... an)' r<d .. ';(lon of ill unfold'''I IO

.. • poramClOC 101Io.-up·. to. 'nIIlurn' .iou""'l' ,Jut, "0II1d ,.note tbe rondo""". 1111. doiIne IhI:: dom:oin D .

Tbc _,s- 1,,1.,.at (.II " adJlnted ,n _h • ""y thai 1M """""""'" utOI:IoOlod .. "b 0 bcooobC erall'" '" •• m,nl nlallllalllnd""""" 0.' .... ceruno I'I'I\WItltabie 1!Kh.ld ...... ~ dId .. od ..... arc eIoco.:d bcc:a_ lbey .roIicu -./ _, ,,,, _"". Ii obi'p''''l' .. .,.,. of OIIr }OIl""')' ,tuuup 8co"&' Hrn:. lhe ' Hllula, Ct..,,,,,, III n,''''': the duaL,... bCI ...... n .. Jutl i, ok1cnnln«!:lJld . ·10.:01 tICI.nn",.,., lel"~ lhe condlllODl '" unlokl, .... and the aclual ..... uo .... an: abQI"hod 10 Ilc prol'l, of . coupled fCSOnI""" """"""IN ru. "or)' • ..,n by.1e "," Ihemalocal I"""'>' 011'" drum: lhe mode:< 01 .,bf'llllO" IIIe " pt"''1''''''') Ill". inlOfmnil"" .boUI'''' f""" of lhe drum Ilh. rondllion~ "hl<h define 01_"

H ... "uld <ytn bc leml"lnl 10 ... y Ihat ",,1I.,n ",ode, .PI'''"' ",,,,,bi< 01 ~ .. ailn,lhel! 0 .. n mOOll;on of u;',.""" by 1001"01, .roond lhe,rud.· ... !nte«st in lhe Int .. nol. of ""'PiwK. oonlemponory ph) ..... no Ionacr I,mlll"ll ," ambition 10 ~",dJllhn. qc<:lrII u( ... ,ia.iont and Jlnollfy.", • body of ~n<>wIoJJ!I'. bul IIlmln.,o I"'P" dootly '" pouible tbe ,..,u," ,hal limal< Ir'pl).'" lha,

unfald.nd """'''''''''' thcoc will come .. no lU'p''''. The CT\IaIl .ole .... ,~ 1»' "",·olop, .. nl lhould be ......... wcd. it fUDa"""

bl •• ' hof11.Olt' lhal COIIIf ..... lidd of .n.ulIM>M. and dnet 10 11_ aI .... l" before ,Ie 11..,.IIUIl, oJA' "" IDllhema"all drfinn"", ollhe Iype of fll ............ Tbc ~ ,olqral c:aptu," III upmmmi '" ,ndl,'odlllliuol "hod! """<Iops .n nrdc1 '0 ",,".m"ne" . .".. 'h" ~Iu'" fliP"''''''''''' ,n onlolop:al dll"'ly ,be lummi", 01 aIrcad) acrompl .. hod ,nd .. ldlllb. It I. not. 'I""""'" ol .... 1lnllO 10111111< lhe d, ........ t,· '" the n:al (If 0I '~""inJ:' Ioo .. oduak from lbe honmn ... mpI_. but of allllni oul lbe """linuu", (0I1f1}C<:1<>nn. bul aloo ""J'a'Cl. oI .... lria and of mulilplocilin 01 all k,ru,I,] SO ,1111 ,be ~on'''"'n'' ... hu In the mOl'l perfcel .'Y 'n lhe form of c..,all'-' ""Uahhl'l Ind 10 Ihal 01'" thu> nil"," tho 'nat u,aI' .cl"",ion uf ind,v,dua), lhal a", 1"" ... ""nAlr .. ""bIe of ..c~lng tl><" o.'n COiId,litm 01 """e"",",

This I, "hy de ll.osJ"'·' rnetl,ll"un on ,he In~IoJY '" ten"""~ and Maupc,W;I'. ",,""Ipie:o Juts .uc:h fa,·""",hln. ~'''''.,n. .,.n bt)'ond ," ttlk .. in fIIt)~ !n l .. bord''''''"1 opt"",1 ,,\di,-Id""lioo and mtcl>anlcal,nd,· "ldual"'" 10' ~''''no" b<>r.lOIL, be"" led (",Ih I'Iaocl and r."" .. o) In O\'crWmln. lhe n\'alry of tbe 1_ poIet whodt, "" n •. .,(y 1 .. '0 """"Ilnt$. load horn fiall""1 " .... lie dlrc<1"'" oIlbe lwo ...... 1 '" lbe ";""uflc an<P of tic "",,,,. ,n·IJoc..-orid:

• ·ho .. """tar poIr. ,,-hdt r...,...,., .. I...! _hal! pr~ an .1Iu'; • ..,. , ... l.1eal .peel .... ( pole. lhe pri.'ilrto<d IIdd '" ~.cmoc: "" "" .. I lay

l 'bl .. "h)' I"" d,ff .... " rb)·'hm. underpon lhe 'b,"o,y '" 1dc3o': tic onc. I;Omp(e.dy dJ!iCQnunu., .... oI'ruptu",,', ·""'.-adl ..... · and lhei, ",rulnli.on ... nd liloo olhe. III Ilc pr" blema.", I,'ond.. Ihat I.., al" .. }" .. ~ilalHc lor 1'CICt .... ,j,on and full of I ...... u'" I", Ihose .. hu COn .u .... kcn 110.",_

" 1II f. SCII]:", lllH SPOCUlJM ,,"SD THti l't''IOI,;L.UIoI

sons

''''' .................................. .....,...'1'''''''.$' ......... ' .. "'1"'''' .-,. .......... ' of...,' 5 T,,':-,.-. " .. ,..... of w .. ~-.- .. __ ,-I'4I'JdI ......... : TIp! 5 ,_"Jb., a_"-llI~oI ....... )fdo·! J ' , .,. ..... , .... I' r ""WI .. ...

_ ...... _,_ t ......... _ .... 'tnpW ........ ' Li" ., ...... _ ..... • .. , ..... _~,_ ...... C'_ .... .,_(A, .. n '''''...... uoI ' pU_ ......... pI<r .... .-"""_ ...... c '''''''H'-J oJ~ ' ,."." .. "" ,',IC : ! Coi.aoo,,,,- ,,.,, • 'W._M a. ,~ ""_....... . ...... _,.,.,IU.h=oio/of " · "'-'",~).O1- ")0),10< . ...... , ........... _01 ........... _"' : ' .

• bot ,Il00, 0.-1 ... lJl)..l.1lll o. boA<9 of ........ lli, Or< jp1 b",~ '" _ .... _Ie> d """,",,' _.oc! .. i , __ m" {' .. _ ..... -. ft. ,401 " .. """'*' .... . . """",,'" ,_,,. .. _ (I/o _or _ _ , ..... )

00 _ q I,. ... _I' ~ Ctooh _~ .. , ..... ,..."., I~ Il~) _ .... '\''''_.'''''-__ $(' ,' 14 .... ~I._L.,)I,_IItt" "w_ "". ; "'_11_ 19)0) '!io~'''''_. n. cN .. I-I~: -.ll_1 .. ~ .-·""' ... f·., .. _,t. ....... , ... .., ... _~- ____ .. ' .,._ p ...... ',",,$0'_,,('" ', ............... "-.. J7I~ ... " .10" 2 '1000_...,. ........ 77 ' 01_ .. _,,,,, ' 7 .of ....... h>d, ... """ ... .,., .... _ ... _'" .............. __ 'l'JIO_ ... "2_~_j

'c:,...... J4 OO!hol.p'.

• ,,_ """"''' v, :w , ''''~. 1:1. a ',,' ' .... ,.p<t .. _ .. (.w.,. Ie ... I . .. _toi ..... ', .... _..,., _ .... So , I , I_~'- ,1:00 Ior ... _ ...... ....w .. ""' .... _cl ___ ,' • 01 ....... I ...... k._... ''1 lor '''' ""'- ..... )'1'0 I, ... 0:1 _I>r .... ""' .... ""y",::, , . ... ........... ,. .......... ~ .,<10 ........ .. .... , .<Pui:I _'eu 'h< ciuo.., ..... " .... " . ;." ......... ""

'n.: _oil,,,," 0110_) .,ft .... M....,., ( ...... OU'''''' o..,! 0.- ~_ ton,. "opp«l .... __ 100 ... _ , no. , ... ~ ... _ .. it .. :t ......... ".:Iad , ...... ~,;,., ...... "'_ _.... ' ... _ .. _ .. , ..... 1,-'·...,,_s..,_', ' 'f • .....,~0I_'1 .... _ ............ ", _,~,_ -'"'TO I ........ ' •• __ ... I : 'o .... So_"" ,10 ... """...,_ .......... '~-O:'..,. M o..,..,n.. .... ~ Y ' ''' .... .II_~'l .. I "Qo::ooo:t - , ... Qoo cI ........ 000 ..,. , ~ 1(l:,zI,o _ I~ ~. "_ , l41~

"0 - ... "'0,''', ..... ,lU. -* .... ..,. ... '1 .. I .. ~n.S.-..<{.wh' ...... _ ....... ,.ll1 "c ' ....... 1" ",_ .......... _,,' I 1 __ • "0: .... 1 ,1 ~_. I , _ '0 " u... __ I, , .... ~ "H~""'I'I~,,, . ...... _ I. ~, _._."., ............ .

- .. ' " ,_ .. ,I 15 • ., .,., 2·· .. ' ... _"' ...... " h.""" .-. _ • ..,.., io . , 'b. ___ .... __ .... ! I '), .. io i." ... ::, .... ,.too .. -o..'f'" " 10._ ..... ' __ ....... , .... p .... , ....... "'-. ___ ....

I." ............ ..,. _,_~; " ""''''' _ •• ." ..,. ""''''''''' .. -....,,ud _h. :l:o _01_ ... ,too J " • • _ ....... 'l1It,t; .... "' ..... -"u,_ -... F L' "'dOC;' •• _~<>" ... _ ........ _""io_"' ....

" .. ..... 'Ii I , .. "' ....... "'._ ........ I _ ... '", __ I •

....- .......... .,.,.... ."..... JoN , ... _ ... • .. _Ii' . 'D) .''', " ..... ___ .... '0,

........ _ ... ' I'"

'.tU ...... IU, .. m I" .0 ." • .-_ ....... 100 •• 2 '0 .. ( Of "" _* .... _h ........ 0

.... I'L .' ' ..... wiII ......... '

.. ;,.,-. ,_ N. "l1,,.}I 01........... .• 0/ 1,0 , Ii

",c.,on.H_,«" ' ..... . : lG .. 000 ..... '_1' " _or I. '. ·-i,L

.. 1";'.'-''''''''' 're I ,./ ... __ .... _

.. "" .... bI" io"", no:. I l). E,_t', :,Iooo,y"'--. ....... OAd I ... ..".', __ q..,,' II ........ s."'" Ilo. ~"", I'IIJ>. .... , I"" D It 196Q, S '" iI ••• .,., 11_ '" k,,_ 1 I"J~ c. Do: ""'"" 1OdJ1_ V __ "" • ...,.. I"',

(10 ,110 .-, "' ___ iI. ~ ....... II" ,II.~ 1'1,.. 1911(zt, .... B.S 0" "".",

0Ad N (" ... '" ,... II .. V'''_ "" .............. '" ~_ ~""_> i l'h '._" u .. _ .. , "'-1"", "0. _ ........... ; _ M a.-. ..... _ !;o, •• ri'.'""" La .. w ............. , , .. ', ~ it,;o. ..... t.,." 1'1 tl "[ P-,,~-.,... .. _"""",; Ie, !'"".M_'..,lli • "500(1(; ' ',122'''_.' •• t. !' ''' I"", ...... ·,·. 1 __ "' ............ __ I<OIIi' ... 1ooo ·ho)".", 0 ,I"-n.. 0 ........ .... -_ ............... _ ..... "' .. 0 ' ..... , , •• 1"1' I .... "', 0 2 .. _..n.; ........ "'.... ,...... '.U"_'" ,_. :1:0 I .. ........ . ,. ~ ..... , •. ____ A'· _., ," ........ . ._ -ou)' • _ • _

• • t • I .. ' ...... -.09lO~

.. 1_ .......... ' 0 •• ....,.."' ......... _ .... 110 ..... l __ ~s.-•• Io

ior.-l.'lBo.allod.. "'0 • Z,C, .. 0 I""''''''''''''. 09>lUl, .. .. ',TO"' .... tnw" ... _, · "0 ., ..... _, .... \!))

..~ M, .... I_ ... s.-.."!W~I .. _·"'· ·. '.IIM~"""""'h"" " TOo .,_ 01 • .... ..,., __ • _,_ .. fIOO'OLI ... 0 ",,-"<'do"', 50< GoWbiou. 0..-'. '", ..... Spo<O:n.o. c, h, I~ 1'<11 . .. "!lfl . "Oa , ... _.JIIo, ... . . ,ope. /I"'''''' .... _ .. ,_ .. _ !i~'!'o I I P....:: .. _ 09'11. if 'I .• oci ow." ,'1 '" "IP."" n ... -.v.u. ... itl' •• I' .. 0. , ... I0I:l, _ (I 0., ,M 111'1/, J..-" " ... ,. I I'.", .. , . .... '9iC, A, ~ /,.0 1ifooht.~ ..... H, ... U,,'/Oo1 .. ' ... ' .. Do: ...... a.. .. , .... _10_ .. , '' 0Il0l,,2'

,,, 11

"500: ... " :,-, .. I" .. _ ................ d' ..... __ ...... .. rl I', ' 11..-'(' •• ,.,_,), .I'lt...,.Vl.ooIVll ... l .••• _1oo " __ iN, .. _"' • .-'''''''' ........ _ ... _ ... _ .. _ -"A''''_'''''' .. 0 , ......... _"' • .,.. '.-....Moro .... "' .... ...- .... 1000 .. -.,10., ......... , 0. ... ,0' If""' -..-..J""_ ...... _O.DtiNIw,I21 .10" ( """" O,,,IUII. I'iI~ •• 0 hXO...,lUl, .. 0. ...... " ... ,H·e. .. . ..... ' __ ._of".,.. .. ~ il"" ''',Odood.''I~~n:l ./,.on' ,er I._a .... <1_ ..... ,,2' 1, ":Ilt __ • __ ... _',", L 110., Il001 I I 0' " ... M ..... """ ............ ' '' _.lti2~!l &1 --".,0 b" , ... , .... "'''''''' ..... _ .......... I<N ..... II! ." __ .. io_ ..... d ")'0

... ...... "",,,, ...... lit, .... '" ~..,. __ ... , no".

,

. , "" .. 2J .......

'" TIlE FORCE Ot AM81GUIn 1)IA LfCTICAL BALANCES

I lI'HROI)UCING H I ~ GR~"'T Rf~OL\' TlON 01 THE INSTABILln rOI,ns

l'o',tf."ldl"..,pIolr bas k.od • bad pl'dJ. It S«m! Ihal hisla, ion. or phik>oop/ly h."" 'l' oN 10 hold onlo the poIm",,' ..... tellhetM: 1"" nllM ""U'1'eO 0( Flchte. Schellia. ~I>IJ IkJrl. bul ""V~ livm up on tho: dca or ... .,., Oc ...... ide"~IWn·1

I'lul",Ioopl,,<'I or n;"u~ In o...many. hov;e'lTl". the hqI"ni"l 0( , ... ninc' .. nlh CCfltUI)' was • pcnod ollenKlhy adYnltl bc1"f1"ftf1 .atn'I>1J and pltiloocphc ... SchcUia. had. "puoion lor the """",,,,' and II.,... IOIw "" '''''''1'1' .. 1* mom,. 10 .p«uIl,IOII '0 pwlli,,'y_ e.".In....,.b In the U.,lCd SI ..... C"..,......"y and Enpnd 01..,.. 'h" "poe! oIllIe pl"",""""y of nature.nd the imp<1<Ul""'" ,"", .houkI hc kWrdcd 10 mcupltpoa • lbe · .... In' of ode .. '; ,,,",,, .nalyxt. hoa'e>'n, .. ,.".lly rmwn ., tbe Ievd oIlOcio1oa7 C"onmon itbJ ..... ""","Joe.. .... allemplod 10 mh.nk ,ho O""",laIlOft bc1 ....... the euct .... ' ...... and plulooophical qxcuL1nofl. retu.inl to lunn I,d 10 the problnnauc or 1M rhilooophy 01 k"",,-iedll' and .. , ... ,,", lhe hanron 01,,," 'hlnh,*, wdl beyond tbe ob;rt<:l', rond"",n. or poss,bilm ...

Con.idered IIIe gokIon .,. of lhe pUt pl"l"""phoaol .~.tm1'" that 01' tho: ...... 'lFfK'" 01' tho: ""lion 01' structu .. , the be"nn,nl"l' tho: n,ncl .. ntb cemury ,,"I ... I"',h.ps th. m0lllC111 "'ben tho: "'nl .. ch.n ..... IOIU,. tho: infinny polnl o( tho: d .. Jical ... aivn " .. , to the ... uuKI ""htr ... """ poin" .... hicb lrunk <1<1 ....... 01' ~u'hbrium llId r,om .. 'hlCh poIantlCf ront"'oowJy oJW1n ..

W. know tm.. mIlCh malbcmaltaaM olld pi .. """""" .. 01' lhe cho" ..... l "" load bonked on lhe I>ornotn>ttty lhat uruokb. I>onron. "!lIlyl""'l philo.ophen In pIInicula, k,... hot< 10 rnal~ u .. 01' ,t. they hod ......... :ltd ,n IImLIIJ lbe ~ndoeImm ... t. fonns· d tho 1)1'IC 011 and 0(, >:", which au.h tho: cnMk finile ot.WUUitU (by rormulae WI amount to tho I)'''''

_lid rc,nJ Ihmloci_ I'ormdablo modI,_ (0, prod"",,,. ' (\oaf' (,,, .. ,,,,,,,,11 "umben.' .J he .. "memc ,nfin,le ..... . Iou prnptrl) WIitd by 1Jotl.art ..... per_ >f1<'C\'>'e, .. hOlle ... n .. h,ng poon1. pUI lhe """,,,un ""h'" ..... h .. j. _r-e. lIuI

TI1~ FOlCEOr A"~IGlllTY IlIAUCTICAl .... UNCU

If, fo, lbe analy,ical plul""'pbe, 0' ~." tho '1'"''11''.';''' ,nfinne aDd II>< ,';IniJb; ... poin" .,.'" "" mo", lhan i""rr.n." .. singulam .... if 001 ael .... Uy domc!llica'ed. lbe n>e"""mg oi ngular illhal "h,eb «",ai, "rkn «<IOUy ~ lhe hIIi,-.," of ,he 'on ,he nn< ""00, on lhe olher nooo' (>II " 'hleh lhe: undo"I .. ",!.n, of M, I'rud""m~ li~"i '0 fl>Od Tb. 13".r btl"" '0 note ... "h I ""rtajn dre.d Ihlt Junction.1 """""'pondonces are "",'.r ""If..,¥idcnl .• hal lhey had In bt remo~ f. om 1IIu"io", "'hen: the rule:! of ''''''''hOn Ihal .... notJ 10 ISS<:rt ,bcm .. h," .... ,u'"Uy' and lherd""" 10 kgibtn.:ll. <:4non~1 pro.ocol. of d,,,,,,rn, menl <:011.01""- MIJIlcti<; and .lectrical polanlJd and lhe amb"""". ch""'" Ih., guj.d", IIIe indoul i(m of lhe roo" of an <quau(m Tt~a l lh" teandat lhe '""nonital' ...... <l«KIm 'hehind lhe back' of .b. ~I.' o. ph ,looophe., wi,h • I"",ure made ",ilhou, hi, k/KJ .. ina.

Cono.i<k. ,lie diall",m oI.he magnet: lhe: d,rr.",,,,,,, be1ween the No .. h pole nd .1>< Soulh pol<: i. cleor al ,I>< poin" N and Shu. becomco .1"" .... ' .he poinl 0(_ fill)_

N

• f.nctiooo,

o ,-,

, •

'Tho IKknlS of ''''' Nonh b .. ncb can be <"pelposo: J on 'hose of.he Sou,h Mall<:h and " func.",,,,,' orro", Immod,.trly e""",.,. 10 oomponsa'. fo. Ih" 10"'" d'lferenoe, by ,nducing . roTreopon<ien« bt'",ocn a ,,"CUll or I"" t'oonh brallth .nd a "'n:UJI of I .... Soulh b",,,,,h. A. ,lie pom' O. 'hi. '""nnn,caJ' ""' .... pond.""" w.nist...:.. 0 >\>Om, '0 be lhe po,nl ",bt .. ,ho ''''0 fU""IIonal """ "poDdolJ£eS .. para ... a Dd jOtn.

o i •• ra .. jfic~'iooI po/nl ,,!>tre: ehe 1>.-0 pa'."""""11<'" lin~ed 10 ,he NOrlh hrInch ~nd t!>t South brunch oIl' .. r II>tm .. Jvcs, II><",by makln, .",-.",1 [u ..... I;onal .rap" P<""DIe. This nn:ess;ly of opunS far lhl< Or Ihal del.,,,mna,,on, .,'h.oh ., .... yo """"n'panics ,lie el\COun~"nl of an ,"d,ff.ren", ",ntre, lrouble$ I""""".'" ,I 1>«_ Dhviou, ,h. , lhex POlOts promu" " m",h !>tad .. r (and

" much moo: difficult .0 DOnlrol! ) mull,pi""'tio" of rormpondcnoes IbaD lhe ',nde1cm"nal. forms'.

ll>e ",nll-.," an ;nfullly poin~ 00 do..b, made il posoibl<: 10 control lhe q..anlieat"-c dllfcrcnoes ,bOl w:pan". finisbtd bcings. bul did.., by.ubj"g..lln! Ihe pamculan,ies ~X"T,.,J'y. n.c ioodl'e",,,,,,, """Ire poilllon. u'.' lhe !>ta rt of ind,vidwllion .he in'labi)jl~ de....,. "bic:b do:lC<'U i, mal ... ,t ".,...bIe 10 envelop a ~bOIC f."",a of vul ualilieo ...rlboul mi,inl! lhem up. Symmelry i. redlSCO",red as a prud""", of diff..--ctlCC:li.

Tho n,<Ie"", of .""b an indilfcr= ",nlre: in .ltancal and mal!ll<llC di., lVami faociula pby$i<i,,. . nd .pa:uiatiY<: philosophe .... A. the No .. b and Sou.h poles of. magnet >t<:m 10 pror=d f,,,,,, • ""ulral po;n,. il is I.mpl,nl! .0 Ih,nk 1M. l!>t ,it",n differc:noes cannoc in,,;..,. ,he pr"'%".e' of U>d;';duallon. The" uv."beano! he .. ro~rn<1ly prod""'" no dimension' .... form. only I"" who" .. klDd of <>Jog""'''''' ,ketc""" iUcl[ OU', and .his ""ngroe""" """m. ,ndeed as>Oci.lcd ",ilh lho Inn",ion. o[ lhe hand Ih~I draws a cu ........ "h lhe .mb"ui'y of lho .b.,;a: of ,be ' 001. of an "'lualion. ... ith ,h ... "",nge pain" ,,·horc tho ch.""" hoi,.,..,. 0"" anolbor """ ;n .horl, wi.h Iho .. pi...,.. ..-h ... tho undo:nl.Ddinl fall« .. A, II>< ,ndirr""""", ",nlra 1M hiBheoI u""""a;nly i • .chic ... :d, ",hlCh ,herdo", rtqu,reo ,ho moot ,rre",,,,ib!e deCl .. on,

TIK:>c « .. Ireoopo'" neil"", I"" .denl;,1 nor lho philosop/ler.nd the ,ribul • lhoy lay cbim 10 is c<,moUlcnl .,,r,h ...-hal '""y ,"",rcomo and d"mma'e: oPl""""i poln. An mwff"",_ ",n,re ""n ' '''''Y' be dcfjned a, I"" princi"",lltinge of. r""'".. ..... 'w", do:.iotc (0< ""';"W pool""ly. ~ ambilully pr<.>duced b)' 'hose de.ica ,h.o ... , us inlo pc:rpk.ily .• Dd i, i. p ........ y Ihi, ambigu'ly ,bo 'iriumpin 0 ,,,. an O~"OD by p.udaiminl i' •• tho tea, of an ·.round· ..-hich 01",.)" aCC<)mpanlCO ''''' bi"h "fa pa", Fo' i, i. """."Id a cr""" Ihal .u.ch a,rangemenl< 1M' 0(> i .. i ... nlty oolicil ou, im.,,-.,,'io ... pi.o,_ The) manage '0 I!'''P eV<f)'· 1hm8 .n.·dopc:<! ~Dd ""ntamin'l~d by .ho ..... ,gelltt of. T,,'o, This T .. ·o ......... ;, !KII an ,,> ..... ,'" "",d;nal numbe, . bu. a """pie, and .berefore obi.,.,. the .... ching of an o,der in no way pr ... nl ;n .paoo: in 0110 .. , orientation.'

Thanh I".""",, a,range"",n'" ,ho pb),sicl .. , "';Ih hi. axis.'loop duali.y. lhe SW""' ...... " h hlJ on.ntalion. .he pbilo>opher, ... ith his dialectic. can. a, last. Clplu", "" ha' lanllU3)1O bad rtCOgruzed "'hm it del"""W Iba, th.al ",Inch Ii'" ,wI! .. mpl)' '.ide by side'. like ,ho '~ . .' mu,e have ari .. n I"gelb ... IJld Iba, il .. ,ho umhtpj,y of. propooition IbJl in';,,, lhe ",plo,""on of a conlexl. Whal '~ QUI l<>gc'h." i. nolo"" IbinS arul another lhin~.!>t 11"<1' ~,iJ~ tho ~n-ncr; Ih., "QUid be no n,ore Iban .he pLon.ude of the 'gde b)' ,ide' on In already arnon,ru baclv"unli l1>e cboioe of an on","a,ion indted silo,,', Iba. a TIVO ,,,,,,,,,,,bi), knil> i",elf .. i.h an en"eJopmenl. ConOlde, ,be dIagram 1>01""-111"" fiI2 ), "'h"'h mle....". a ,,'a)' of en, .. loptna "'ilh. "'-ar of,,,,n';';ol It " lhe JOurllC}' of the n,o .. nl body, ."kulated in ''''' . n,'.lopmrnl. Ih .. mok .. 'he ~ .. a.ion of I .... I.,eral poss,bIe by "".m,lIin, ,ho deoiJll"lJOn of . '!dl sid .. aDd. 'tilh1 sid" .

The OIlJ'O>IIeo a .. ,n<ietd '.'I"'.sed, [0. ,ho plure ,hlOl broujlht 11>tm iMro

+ • ,

ui$''''''' """ be<m nil ..... ~ bu. Ih" tuUn'l <>II' .. ""ill r",r by II>< ,nnu.,o" or anolber dimc:nsion, ... hloh "".m'li "" 10 """ lhal ,he "'0" All eMn be '"'~ "",lid inlU 8" 1" ,_., WlI)'I (lite fi,. JJ 1M'''', the ",clt'Oll1in. of In Of'llO$"IO" is .1 ..... )1 paid ror lOuh ,ho _.e""" of. " n!ulanly

• • •

,...., III_ lui"" !let" bo. ,be implicit aDd cop<><cd ho",,,,, m.nqod I con.roIlnI

III.....,n CO the _hllOO '" ik~ 10 '""'-Ylen.h<>n. Tho 1",I>I,UII1 and olio,",. amonl!<'m<"1! .'hoch "",,'ok< it 'OmJelll tb< un<kn,"ndl", ~nllllt os rudy", buro,; Ihi. ;, the 'pound 01' Bah' lhal m"sl h< laken to ""II<UII. r"rlher '",,, lbe unanl< .. III«1. Th., ..... hy Kines .rude 'On Inc c,,~'" or Nc",';'. Q".OI'II .... · m.'~f/'d ." "ptIC" by ... '''''~ni lh. crue lly of Ih. <I",,,,,,, Ih~, Schelhn. ,"ould Ia,c, all '.'1"'''''"''' l1>oy d,sd;un lbe Ir .... 1 <killen .. of iopcal cOfllnldkllQf1S I"'hoch llllmcl .. ,.I) ;',..,11114,. ""Y .'~rll,,"I I" u,dlc It.; """""''''''' "'lib babnoen .. hero • poinl CX>nIIII;ondi lhe 1)''''11<>11 lhal ruin lhem

W .... I(,uch,o, 11m: on ,he 1.1Kkr IoJIOI <>lIb< ""'''''''''''r of ~Iure; lha, cI ."" 'Olu"lOn of lhe dtJuo at anocublio!l, t"'*' • ~ lI<,n. o",.hanp. .·bon " ... n"t •• ,.." d,/fmnca and ·", ... onar c"""," 01 ~"'m<1."'" 'aN .... AmbtMuny a((OOllpaNOS .hex m'lall~ for II aIoat m.o," a dJ .... mlC _u'''t on 1M ...... '.,io)a. T .. o amWy pDr<I ... 1 ,,"'Qm,.. &U .... 11h ,he ttrUin.y ,,,.ho .. '~bloohrd d ... h ..... Pyrrhic: ",""y' 0 ... ~"" not O""onlC an In •• ,.,.,,,,./I1 b) pIo),nl on ,he """1 ... neI)' of Iwo '., mll, bioI by a,o" .. ", tho -,., I16""y Ih., Imh lhem. b)" oIidi,.. ........ 11 m", lhe """'0<1 b) .. hid! ",,",li"ily hoIk" .. 'OIl' a .hlCk .... ' for ..... If. The simple dOlUm of. 1~""'JoNJ/

~'" p. f(·.1 (O J

n ... umet 11"11 a ..... '1' tepa ... """ bt\"'em I 1"*'''''' and ... ""1 hao "'""" 'I potSil* h"'" """" a(hln~ I"" qUICkly." 0\" ntlbbW«l ~ of tho: Iype of ( 1\ " 01 .... 1' hom of a I.... \l..U '" idYll/ICe of lhe funcborl, tho: "pOi"'" caplun:. I POom" of opmI'''''' "111\ou, .... " ........ _ To ""1"* II DOt

10 •• pIam. bul to ';:u'llrn~. noJ., conllnuum 10 the .. " .... Ia. by " .... Iopo"1 all lhe .,nual do:1o.rm'JlllIOIII proacd,n. f."", • prvbk,U I1h. IS thr ~ "l'Ib 'multlform fu""'.o .... or .. nh GaIoto· bod .... !UdI "'" IUIIdcd to au equ • • oon~.·

In the """ I,Ott x'. 1 funct""" ... IIher I, a 0<C0tId "'" Ii I + I. bul II luI 1110'" "ma1'"1Ydc to ,"I<nIC1 "·IIh 11 .. lf . Whr"" ·"r uponcnll appea •. In Imh'luny 11"11,.. '",." ... ~ lh" " lhe prlOO 'e> pay for 'J'II,IJ prod"'J aNI ",oJ""k/'" IC>jt1h., .rId for ''''"' fe>rm n~ory. 11", " lOmory is ",,. pboccd In lhe f,...m; 11 """ly lwakened hy lile In'U,I,,,II of . Ile ' nlUh ... ,,_ lhe ImIHlu,'y 'rIIe I"f)n\<IOc ,nlu",.". of F""IO"Y ""n"""n>d .u Il"""".h .. alwlys appeal. 10 "",,"m'y. to ,be "",",un", ''''''' .. of I ..,..u.ol If1Inqu,lI) .·nlPJIld around Ih,np. ror 1Iclp, TI". ,nlultlon of lilt ,lIlu",OII 0I.hN:h .1ot )'OlIn. H.,.s drcamtd blKlcflak .. thr Ihsmaotl'''1 of faltdy ob.~ Ippea ... nca and ",.,1<'1' II> ", trotS lhe Ihrcshe>ld> of .... un>alIJllbl,lUt' to """""'I"" "read) "'IlhI,,1wd k"""lcd ... In the oample 01 lhe _ ..... ".,,11 of allJl<s. ... §hall

_ Ihil " IS JlftCHdy lhe «IS!"""" of ''''' 'Y"'lIkIncal r-'''' lha, ....... '. thaI lhe m"lIIK>n bu I" ""II ,He!( up ,"'0 .he ~ ullhe r-.Ih 10 .....,..Ie a ""lIIbn ""h. l1li1 In&!<

If I do IlOl ,,"" 'WU 11<>1,,1'0'" 10 ,ho ~"'I "'" .' ~ 1 I...,...M. ]" · ...... loo'~ ...-J ...... ,he ...... o,lude .. ""'- mohtlanl. ,I roI .. ....,. ,n on ' .... If aDd 11M: prod ..... ""Iy" •• hon,ltd ,II ,he ptodOC1 In ,he .. me "lIy. Gr ......... nn·. prOOOC1 ""II lbo,,· Ih~I jf" d,d not fundamentally fCS' <I" IIle ImboIU" Y ul. clto .... 1m orlkr ". hoc ...... ,he tb",~ "", of lhe rum" in'""'''n', "'.:t""11e lu 1..-" po""l, tl ... "uld """ lhe ""plu", of.be . "e",jon, 011<", "r .c ..... lin. lhe .......... Irnl '" " 'he,,, •• ' •• y.hi"l " loin,'o he Ik<:Idcd hll. c,,",plcr IV), Throu]lh lhe amhlg.II') 01 lhe tII"I""""""nl, ... ·ha, "'..,. """,.O<! in lbe u""y ulan"", . apply 1 ",It R

unfold, ,n ,he "",,'inuily ul 11 .... IUre. Tho •• 'Ili .... 1* "'""" .. Iy. ingcn"""'''' r-Ir-..... ~"t1ICh to en.""'p 1 '''"''tnom 01 001 .. """" fI,II".·,,,. the r""""Lo. m' - R The 0.<>0',," had • "" l'R.imrn. of Ib" I"orm"la. they b .. ,,- .ha. 11>0 .. ...,.,. I"' ..... ,lit PO"'" of , .. squa"," (}pp.>fIllUII. IeInna a.,..n. unJlOSlilbtbl): lhac COn be the .ympt",," of. Ikcree 01 HI,u,l"", '" be crosoed. but .he ...... dune"_ ...... nol ...... ·bca_· of \be opptiSOI"'" Oi' lhe mil""""-btbly_ "J"heo;,: bLt' ....my- ,n"" .... 10 ,bPK tho «nl,nly of lhe ·UIlIUII'~· d .. bel, bul lhey do not "lead· me te> lhe 001"'I0Il: the Im.."l.ol"", " a leap. whld! ,. no"hcf dod""'i.· •• xh .... "on ..... '1l;Ktrao;i' ,nduct,.". from com""", lealu,""

f'" ,he ,nd,lI'trell« """ .... lhe •• p<>n<tII ...... )'. J"f"I'lICnt. II",,", .. a d<p« of 1It1~. In it. prod ..... and p<oduct'Y1ly are ;nd,s"na~"""hIe and.hr forms .... 111 1<1 ha,'. Inumrhcd 0 .... Ii>< ombolu,lJ. Th. "ponen'_'oo, offo,., i., "",m<l')· .,'" It. Ikr'b, It, """'1,"",1 IS JI"" ul,he ·l~""'"'''TI.''' · ,of .. hlclo Gr .... m" .. n

" THE FOICl! OF A_'IBIGUITY' DJAI.F.cnC.U BAI..\NCl'.S

'peak .. ,,'hll:h &<1. " 'ell 1><[0'" tbe Ip""'<lin, 001 of III tbe momed" of tbe new'rulh.

All .. , ,be ...... nl 01 a oepan"ion become& con..,,, .. bIe. Too often ,nle'Pre,od ... "'pI.n, and quickly eDllu!kd In lbe fraca. of quantltath'. dapariU¢J, tni, "'I"-""Ion of form! i. nO .. ' p"yod oul ,n the iofin" .. ;mal "<1' b.o.oh",rds ", ••• Ied by tho .'pone"1: lbe hither tbe .ymlllt'lry, til< more im:vucabl< lbe doci,oon. T"-.", nnt lbe Ie ... , par.ulo ... 01' ,,,,,",, .. ."nge JI'It'.ldn;es. whcK form. _m 10 onllioal. wilbout injUry [ra", Ibcmsc!.'os- In uoocrstandlng lhe nepli, .. . . a "'lati"" of .. ,ualion, Kant opened tbe: doo .. 10 "" .... paradl$C3 ,hi'''''''"''''' and """aph)'i>ciii.n. "'oold 1>0 Ion,., ..... b U, ignore. We "'ill sec "" ... «pol><nl. ""li<."Il lbe intuItion in the ""'" ..,,,1. manner .00 ho", they .~ In "'''' lumiol lb. o;J~ss;ClIl IW'I ",.Ied Ior',....,n gcom.try and .l geb~ by Iyinlthe imall"" of lbe first {'filum') 10 tbe Incralily of tbe I<'«Ind (tlM: .. """ of formulae ud of eolc:ut..',on. 01 ""'I"ilude).

A. an allu"", .rrangtmcn~ lhe <>ponml and ,11< ballnce roncenlrale all lhe .. ~,.., power in . poinl or an a~is, and it is in . ""h poinu or .... thaI 111<

~Iurc Ihal P''e$ "orIb to the poIaril, ... n ..... lI ul,h. III.,... 111< CIIloul ~I · .... Iural' ujl<,..."Js, ,he mo", It has made us .. .,IIL I, YI'U b) '~pj>In' dec""r Ihat lhe n<"" hom"ge~ly WH fouoo: "",h i. lbe d,alect"", . ... 'ere Iew>,,-

Oncntalion ha. 10 di. ""y do<p. for il i. only ., lhe kainning or lhe nin<leenlh ""nluf)' ,n.llile lu. ok or It>< 100crai erupt .. ".., """,live. a. in .... ed and "'I • .:ned. " """'&JIiud IS a fort!< al ,110 ""me lime f"""ng Ilacl><a from ""'''''''''''ily and JIt<>II>CIry from '1"'Iioi clieht\; " al I ... !rnUho:o the boundary Ihal p""'iou!ly "'I"'ra'ed 'pI.""] inluition and deduct". u""""tandi.., OlIId "netly defillOll ,lie IIomain, or .IFbra (lbe $cicnce of masniloo..i and ,C<lmelry \Ibe licit""" of fiIU"",. L';!miz ""It> "", aWdr. of 1m. divo..., and had "'~ lbe p<>uibilily or ""kul.,in, on lbe ·,ria"onl of ';Iuation!' (1I11a1}',u JIIU'~ It>

• IeUeT 10 Cbri .. ian Hu~acl'\ll boa ...... itness:

I hou diKQ, .. rr;l .... MOJn tkmcnll of I IrW en.",<teri.tic wh",h ... ""rely diffen:n, rrom IIFbm and ... h,<h "'lI ........ 1=1 .. hlnlages ,n "'pr""'"'ml '0 lbe mind. elactl~ .nd ,n a way fOJlhful lu i" n.lu",. ercn ... "h"ul filUm, everylhinl .. hkb dc:pend. 011 ;m.pnllion. Algobm II Ihe char.e •• r_ iJllc ror u",,"le.mined numbers o. malln,t ud .. only. bul il don "Ole.prOSS litualion. ona!" and mOhOn d" .... ly, Heoct"" oil." dHlkult 10 a""l~ lhe proptnin of a fillU'" by ""kul.lioo..

Or . ",in'

I bel .. ,·e 'hat by 111.1. method OtIC could "T~I """,""nICS .Imosl li~e gcomc"y. and OtIC could e,·.n lesl lhe qualn ... of malorial .. becau .. Ih" ordi"""I)' <!opend. On ""<lain 68"'" 'n the .... n.,ble I"'fI<. Finally. I h •• e no bope Ihat " .. ""n w<1 .... y fa. In phy .. ", unlil .... e h .. " round "'n>< melhod (lr abridgemenl 10 lilhl." I" burden or 'mag,"a"on, '

t liltom", of .... nce 1Lt, .. probably nnl "ven enou", tbo",hl I(J I"," .urpnl_ inll rae" Ibe J<k"ulicot"'fI of ,""'lPn.:Iry numben .... ,lb lbe p"''''' "1'1"'''1'> in lhe

Tll~ H)RO' or AMBI(]Vln OIAlf.CTlCAL BAl.ANCES

lIl<Ilbenlaucal lile ... lu .. only al It>< bei'nll,nl of lhe "",«ecnln """IUry. The 5)"nlbcsio of gcomtlfie £CSIU"" effecled in Ihc 1 ... (K\lIl1<n.bonal conlinuum at>d ""kula"on. implying in'p<><Wble '1uanlil ... · lbe .. ro .. I(>O~ rIeIdy Ihree cenlu,"" 10 complelo! Tho malbe:maticians or lbe: cWtlec,lIb centu,y had 1Io .. · •• c' . l",ady I""klcd $O"'" p<obkrn. Ihal we"" r.om • lochrucal poinl of .~. ulrcmclyelabo""e: but by comiJcnng I •• ubold;;"y. o. quile .. ",ply as stlf· evident. these ' .. IU .... III onenlallOn. or indcoLalion _n fect

classical malbtmotios rondemnod al~.bnI and gcom.lfy to main1.ining I curious ",lo'ion,h,p of reciprocal predauo", algcbnl ;, e~=ted by a,'oidlng l11e ",tfall' of fillu,,", .nd gcomc'')'. ,0 rc1UfIl. &" ... nlets lbe · .. ar. lan"bIe cha.acter or alsebra", calrnl."on .... 'h",,,, Wllhout It. "",uld be pu .. ·aMl...,.. lio",,·. Gcomcl'}· makos Ihem ·leap 0,,1 .1 )'Ou·. In. '",al" oh;rcl. of our.~ and In.ir ",Ia,ions of d"tance embody a MI/ ·~d4 .. ,u Ihal com""l. ~on,iellon and IbUll mru.m lbe agenl. oro """ilCmal;"'1 lruln .lIn 10 a oimplc fOJlh: Ih .. fallh 0100 noun.hed ""'If on lbe <"",I' e~1SI<ncc or km<mallCl ~nd dynamICS, whi<:h 'ndeed .h",,·ed Ihal algeb •• ", MrnlS ""'0 "01 'pmo 'blOOJ of tbe mind· and lhat IMy C<lnlroltru the la .... or n.ttu.e.

Wi!h Af!lnd-W .. ",r. cOlItpkx plane, lhen Ham,llon·, ~""leT",ono. Or ...... mann·, .1jJ1Obfa und tbe .. wk. of M6b.ius. a ",volullon i. broughl about: one II lut '""",. It>< muhiplico,io" 0( ;maginaric'l"hieh lben bc.:omc ·allegcdly tma,'"8nos,). an&lcs and ,....:lOri, The: mulliplocabon or ma~nitoo.. "'Iuiml only d,lalalioM; hencefonh. a ,C<l""'''''' 1 .. I'lllfo,,,,,,,,,,,, ,n 110'0 dl"",n"onJII """G';"" a=mp;t."in lbe my"e"oul .)'tIlbol Vl.

The geomelric ;oIH"ion U Ih.",fo'" no lonlt. onl)· the kind of ocular in.peeI;"" ... hi<:h ~sa:"ained lbe ~grcemenl of malllcmatical aICII.lalion< .... ,Ih """"'ure"",n" 01' ",.1 objc<:ts: il i. bencefoMh mo .. lbe """ibilll) III .... ''''08 by 4i~'"", Ih< .'.-llicn of lhe "'''1<1''.'' 01 11", .po ..... ron"", .. I;nl,b ... tbe nne.gence pi" a P~)'si<o-8'Qm/!'ric con';""um In Iwo dut"""ion ... a> rar =no"ro rrom Euclid·, plan< os il .. rrom lbe allotf1lcl ,·oclo",1 .~ !I. ,,!I. of Ou' ronnah .....

8u, lhi< moruliuliun of '1'= also "'.>'<5 phy.icII and .• bove all the ... hole or the .1M.0phil""",h) of lbe besinnJn~ of lhe erotu,y lC' lOller; in undc"lan<l· i"ll Ihal a magnoti<: bar conccnlral .. no boreal 0' au"raJ nui<! I",Ond ," j'IO\n. in collCC,vin& of ,I .. I I000p .pl,m"I'pact into 11'1'0 I""", and •• an ui. ""l"'bIe "f l"odllCing a third d1m<n'KI". Ampere and h ,"day 'n'·~nl I JI"""ICn, thaI;' ."Ili<:icntl) "",,·<:",,110 im"", and I,,·e definile fo'm 1o Ihe whole ele<:tromagnclk '",on from whi<:h Mu\\"nand Einst"n drew .1ICi, tns","'I;on,

2 A OUI:SYIO" I"ROM KA"T W UA·r I>IAG ~ AM5 ,·OJ. TtlI': N Etl ... Tl Y 1';1

To IOt,oduce II>< concept of ""gat"". nlOgnllu<lc o. quan"'Y. Kanl ob>oe,,·es.:

T",o Ihlnp Me o!,po,cd lU oach o,he. ,r one Ih,nl ca ....... out lho' ""bl<h

" TIl! rOtlC1'.OF ""lI(iurn DlAI.£CIIC"L IA .... NCH

to poull'-C In thor I)''''''. 0pp0$I.00II..,... he: lop;:oIor .ut Hnhcno Il1rnl"", hal betn ""nl~ On k>&o<al ~"OJ\_ Th" ~on""u In affimHnI and dony'", IOmelllloa ni,,,, "'me "'bj«!. Thl, Iop:aJ oo,,,><Ctioo i> ... ,thour wmeql>elK:C r .. II;/ ...,..:inuor ",,, ... _<>bow], IS 1M pnOC1p1e of tO~lradJh,. .. 1UItca.·

Tho .. ",pi< lapcal COfItrod.""on .. "'" 'real~ it ... bcn. no .nlunoon.l"oo",," no .ron<cq.....". •• nd lead< thou&h! m,o u ,mmcd ... ,. 'mpa .... The ·roal' ol'J'O"">I)n .. hod! .,. .uch tbal ho" 1'",,11"<1'" of •• ubp.~ a ... oppol !ld. but ... ·lIhou' ronu"dkt;un' don imply des,.ue •• "", buI du. <k;'ru<Ii(>rl i, Ih,nhble ('111e CUDK<j..rnc:< i. ><>me th,n,11nd doe. IlOl bful.ny d"""""'1< all phllowph_ icaI clfort.

Kiol ,hen II"" • few oJ.ampks .. hocb Iohow ,hi. ,'''' ............ UOfl ll .. lOYl • ",cJopo ;udflll "'_51 in tboo",!) ."h In <quil,bnum .h:ollfbPl tho ~n"'" of lhe <k>lrucholl of. Illln, or of. mO'lOn;

no. dn.-,", fom: nI" body ,<ndin, ,,,, .. rd,. o<n.,n pmn! .,Id I \,""la, elfon of llu. body 10 100>'1: ,n .be 0f'P0I'I!e d ,m:tion .... 001 """,""1<1.,.,. and ~ poM,bIe .. "rub".1eI in • OInak body 11 ,lie ...... lime. n.. OOIUI:<I_ or \h .. II Lbo: ....... IIIcb is oomet.II"" I "pnotM1II • .ot><I.r] 8 ", .... ha~ ,0 do ... lIh " Fdu,nc "PJIO'I,ioo; __ Iendm<y "'PI"" ,~

~:ol clf«'l of ,he C>lher, ,he ''11'0 lendmaes are re.1 p. MICO' .. 0( .. "nile body ,,1>(1 refer (0 ir ~r ,"" U"'" IH,.., n.. oorueq,,,,,,~ of II ;, eq .... lly l>iotltml< bu, in .. d,II'"""" ..",'" r.om con,ndM:UOn [.;/011""'.11,,_ ,..,..,............,.kl·

1'"",", io ,"" UlIlllpie olllx: _ ..... nLlllf' baIa ..... \heel ' en crea'e" LIb!< "',lb ,'''' "*""'~ 'nlO,y,bod on ,he "1111 In<! ,he loCI"" 0" 1100 IdI , " ill ,b", lui", ,,,,,, .. dol ,n rt"."",n''''8'' __ ,.....""i<I<I, '" unde •• und'nl ~ uro capi'a! &. lhe "'~""""" 0( I .. " aOl"lIieo llIal a.e p<>S!,,,,e '" Ihe""",I,'", I C,~t.li '" he "",O\'t'rM or debt. '" be ""'" b>.ck), 'r~ , h", • • ,.mple. ,ha, ,,, lhe ship Iha, ..... rrom P",'upl '" Blud. It probably ,Itt _ ,llum,na",,&, I "''' ma.l ... " dlSta""", " .. died by ." CD! ",nd ",ib. "P • t' .nd ,b""" , .... ,'e/ .... br. " 'at 'ft'IlId. _. ~o iCe ,ha, ,he _IN ·+·.nd· ' ''''''.'0 mart lho I'>J'I>O"U'OCI ol .... " ma"''' ...... "hlell. ,... lOj/tllltr, "'I'JIR'" 0""' ."",hol I'O";l""""'lIy,

K ... t' .... mplo.! a • ., nO, pICked ., .. Dd(>m; ,~y.U "'udo 10 a de""", \, • .,.1' 0' '1<kaI"J ""QeI""I" anceD,n, 0\1, .1Id a boJ:lDCIJII, I mUM ollUJ.,.., , ... 0 prcd",",l<$ ,ha, .re poaI'"'' by 'te"' •• h .... 'hu, IlI.weII ..... D ... ' ,hoot. i" ,Itt ... -. '~ax'lM'q""""'" "'WI : ""'" .1IOtbot "".,procally :" ,hoc _ ............ , Th .... ,n ,bcoc ,h ...... uam"",", :0 aned OIl' .. """ '0 '"",Lo a vK\lum·. 11 " !IOI 10 "",lubil all ,n,UlI""" ,n ,he .... y ,ha, a \oaiCa1 ,..,.,,,adlCliOll. ,mp"IICRI ,0 ·h.,,, doLlO' ,*"k lul l . wOIIJd Ii .... " ... ult ub<u.dlfY, '" ,nllo .. ,I up ""h "" pot.OI"'hlyof rodem""o,,,

In lho a.. ol ouJ oppo>IlJOIl. ,IMo "'" • -' ,lMhcalef I =""0 a,"""y '" ... '0II11d ,ha, ",lucb &IKft. uri .. pum) P"'" and d-.dtdt. to roll .... l ...

" ~n.ol~nl< ,0 """'~ ,ho ,"" .... oil pun: .'PO"-' nil ap"eny. com"'",n. mobili,y alld OnenlalLon. " " '15 a ,um,nJ moILO,,:n rbe m:i>d "blCh li""nllel i, rrom 'a~ .nl :"", KOO\In, " 'ba, u somply poIoOl«i lhore.:. foroc embedded ifI ""'''"", ~ rom 01 """"'f. I cbs:~~ ,rani"", 1M !.II'u.1IOn Ih ... """"pes rrom :~ kind 0( ,u",UPIi", >lupod:ly olllx: 11< .. ,10<1 ol ' '''' sum, r.om ,hoc Iy .. nny of,~ c.l<1U1'· .... ·bid! UIIP<*l 'l5<'1f OIl .... " 'Ilh ,he """"J 00._ ..... 01 lid. ord,~

" an' a(kb:

It. IUjI"lIud< " """" .... III «!a"on '0 a""U~. "",,,,"ude ,n ... ra, as " can only ho)OU>td 'ott':Mr ""h j, by an opposIILOn. lha, is '0 oay. m ... ~. at ,he <me mn ... _,ollude «1""] 10 115<1f dlJ3rpo,ll' III ,ho other. "

It. "'pm., ....... '_ ,~ fIJei III' " ·,,b ,he ..... : ... _ ola so"",la • pm", 1000tllln .• 'Iuch .. by rIO moan. Jdl.."ldOll' hu, can bo brou"" a_' by M ... "ohle .1I'0n. p<IS"lt)ne,j "n ,he bndF 01 ,he 'hip. J can .,mpl) allow nly .... ' m hoc led. I tan oca:pt '" <IoU: .... '" J'If'OII'CW- 8u , I can "'WI)" caoa:! ,h" I'fOII""M virluaLly by 'h<lIIah~ by ;"'apmnl' ron,ruy "'lDd, v.c ..... 'hal IS al .... ~ .. ,he all ........ '0. rno''Uli body "hid! could move:n ''''' OJIPO<II. d"~"",, ...... no! dOIlY'''' cu.,,,,,,,,, ol,h" ...... ;nl bod). b.n ,nlloal<l • :_ah, loop a""bIe ol ronl",II"'I'ho ,p._IIn. olan •• 1ension .. hero, .hoI''''· ,og. diS'"nce ,hal Iw: ., ....... y beeo ' .... ' elitd l. Lefo io ,~ .. d on: "0, 'nolh,n,. <lOCI' " ,""lid •• ubjcICI·1 ap"'';'y '" in,'nI' 3 roun'< .... ',,;&ll' do,'",", , ... , ""ill produce IbiS joIn'lII ' ...... ""'. aDd tho Ib ... I"O(:Odinl .>ample •• ho .. lhal """",LI,,,,,, Ind mobilill ""'1. CO"""" mit in lho """,1",,1_ ol 'M don:t. RC*! OJIPO<I'''''' lhordi:'Io'C 'n .... "'" 1I....p JIItJj«. 'n • at"' .... bot.""" '''0 alii"""",,,, prcdOCil:Q. llnI .,"' ....... DOl .... ," lhe '''~ ,.,act; lhiS ..

pcrlIap6 .. hal diSlInpWla I'QIlIJIIIO<illOo rf'OlT\ \oaiCa1 """'rWlCIion. .. hldo a.ually put; ,110 propoSilion A a Dd '" ",bu".) on lhe .. me rOllllnl an.,! is

\'C1')' hilI< _mod 10 k"",,,' "hid! 0( ,hoc ,,,.,, I'rcdlCl'" " ''''Iy l /l\muti"": ' Ral' OJ)pO"<lion. ,Itt '<0""""<' In<>, «IlIIhbnum: Ibcoc ... no' ~ br

lho .. mple J''''''''''''''on oll"" poaIU'"a. bu, br ,ho" orinrl~ .... l<'1li ... _ It. II""'" ~ tad OIIIy « 0I'C'm>mI< b) ."",he .... hlCll ucrt> ,lOdf ,n • MdltItJ dif'<Ct, ...... rounlc".·oi&h, Li "'" I 'dead" wriIh' hul ..,..,.'.111 ........ ~j"'''hcd .... 1, To ",_, _ Ir)' ftu •• " " MCCI""r)' '" ''''''"' a douhit-rn,r)' t...h""", • ......,. for ,he lallcr 'mplios a du. di",ncllOfl be,,,"'",,,,,, '''0 ..u.s ",,,,,""M br • , ..... icall:,.., IIc'crsinl ,Itt oou ... or a ,hlp '",all)" 0. by ,hou&lu i!; by no Inalns rcd"",bIc '0 • oimrk dOlI .. , ol,,,,, .. We"'" ollhlll ""1" t1b~ dOlI ... ... auld I<IOblJ ... "" ac\iooa.. .,,,boo II. ""lUK or :" ,ho mmd, l"hon:" """""''(1' '" ,ho oqah .... I k,,1d ol ""p<f'I.'''''' or pn>cIIIOfI or aUl'ort:ft. ol all pooiObI< rou.oco. ,bnc •• txactly """ ,h., ;. capobl< oIbolancin,lIx: one Ih., hal ....... ,.-

S""h a ""I.""", is by no ..... n' Ii'·oo. In.,! 'liLS. """",d,n, '" Kanl. is """" role> .. n, '0 lho fora> ,ha, all'cct Ihoc ..... L

One rod .............. Iy lhal '0 mH ... 'bouV>t 1lbod ... "b btll<l_ d .... """".

" aDd .... " .. 0)'.1 demaDds a F""'roe, and ofIcn _"Ie .• hIe, ao;Uyuy H~ .00 " .. "u to I'QLO~' soletnn to"" ,,'~J <lnly Yluh ,real elf 01'1 .trace tbe im.ge ,h.1 ... cued InJ,hl.r,"

The wb)ea mUll lhomor< t<IM I. Mp',," i''''1/ I<> .""rcome ,I>< <>PP<>O"''''' .Dd 'M ""1<111111011 .hat ;1 mus, 1"'00""" Cln only be II. O"nl- To ,ame ,I)

,a,,,,, ... 1 ro..:e... II mlUl "" up • oen'~ 1Il1"'lIy "' the ''UY heln nIt.. Ih,nk,n.""b,ec1.

The", an be M d,ff .... """ "", .. "Ctn ,he """,dcnb <>/ .pon' .... l ".lure and 1M ronooqUCn<:e:l of the 10,,,,,, 0C1'''1 in ,be COfpo< •• 1 .. odd. lh;jl i . 10 ".y ,hal .bey can ... "'" be .... lfOlcd "'C<'pI by ,he .-nu;oe oppoood d",',nl rot"" '" aDOIbe, bod). and tlull In inlmW x<. " ~" I lhou&hl of I"" sout ('aJlDOI .,.... 10 . , '" om""," • pull!otly M'bn, fora: or It.: _ Ihlnkoo. '1IbJ«t.. Onl) II ........ thaI I." ..... , ..... , ... ., k"wI. or bcinp dlffor 'M m.n ... · .... Ie can only he moo,fiod by "."" ... 1 <2UIiQ, 100 ,I.,,, of ,,,,, mu>d can equally be 110 by an i.,mIIl/ ""uw. the ".,.,.,.,ily or lh. real opposition nor.etbclno a1"",)'1 mnai,," (be .. me In the ''''''' danwns."

II ... ndca! 11m drOOl or aoobotl1l'_ ,hot o<>"".MCI lhe on.oIop;aJ d'PU1 III lbo 1Xp",~.nd J(u, <;;alb the ..... _onhnl 10 ''''''''b ........ "" ""'Inll\ldel; .'<)<lld bo: lao . ba~ "')lh,n,' .tr.unl, l"cp' '"c nurnw.. .~ ,,(>I IfnlllFd 0fI ..

lodde, ~biceh ;. Jlrnply 'belo ... • Ilia' 0( p<>,i l"" nunt bllrs. .£ern.. 1h" ",Ial"'. "",hIRI. i. prod-.i by a .1Iow,!r., f" ";-",. by • compt",...ilm dc,,", Cl.p>bIt 0( <11.-.:1<>1"1111 T.'o .','h , • .,""",,1 mu .... l"<pu'~ rna&n>'~ ... pal lhe·. "'00 domoufto(b<>O\OF1lll,"'''''' .h",b IlSUPlnof .oenl<k d,ff.' .. .... '" q""hl) or ~ ... nhlr " Ii US}' '0 d"h"luoflanllhal I "tid -A .hoold b< "'''''II)' "'Iual in 'absool ul< "01""'.,, 1 .... lli'Ic: I<ce", i, pla«<l neh.m'ely on lhe Jl">Ihon~1 rd~"1)11 (oppos,tion). lIuI if I ..-ani '0 d " .. ",ui.h be" ... ~n A and A. I mU>.1 "~IP' a mlrro. or .. bollnoe .,..j <kodc 10 pt .. ,,,,," ooe ado 'Ill .. "al'''1 ..... unI) allo ..... """ '0 he unoknlood as "1NIkI .... as ,"" p<OOUCI of ,be neull'lllllallOfl 01 +A bJ A. ""I 1Ill<> ..... los " """";b .. '0 open u'o 00' Inlo " .... Iw<w,.". aDd " .. J .... Ihn aUU>ion 10 ope"'", Ihal ptrmm lhe CO",!""'I 011"" dice""" asoocu.lcd "lIh il<ral",n.

1 AlnASO AND THl AITAC"" O ' TUI LoI.TElol.l

\\< ~_' how cnlhu .... lically tciomlOllll'ft1ed per>"":1'" and the """"Ian .... -,_ 0( tbe lJI>I>m<'flC ,,,6,,", tbal " 1110,,«1. 11"'I)'../)<1l. tbe lal ... 1 and .. aromcuice ... pt"'ly capable of fo.mally m~.· I<nol pt" ..... nd nl<lapbors Ihilluod prc.iowJy .... mod ,"<:,,-...1 for "",n!"" .nd poeU. n.c ....... rct. CODd .... "" b) oI.rpnd. I\ c:ood. QauB and otlIoo-fJ ••

o _.

., OJ

Ia.I made i. poo.siblt 10 II". aJI ·'OUllU .... """Ienf to 1110: """111'"1} .WIlbe. In .. """""'. publi""'" In 1806 " .. ""'h ""TJ .. nnot""'" at 1110: lime. oI.rpnd p<opooa I" Iklclop 'a meaN of .. pr' 'or,lIn"ntal""'>" qUltn" ..... in ....,.,...tflC DCln.lfucllooi·" He h<:",nl "'IIh til< aCOOunl of I <oo»trucu"n ." "" ... "v< ma", .. udei. from whicb .~ 1"'" a r.:... "," ",,11:

lei .. be any a."'1r:1I1lprt ... i • ....w..r/J ~""n lllJ, It 10 th .. 'I""nll1y ...... .... equal!o II h<: IMtd, .... ma1 n(ll'OO5 ,lie ,.,."Ihn, ... rn 1It:!oo If .... rq><:lI '

1M op<o"&l>On. tht .... ul! .... 11 be 3<l. and to ..... ..... e tbu. obilln , ... _ a. 2a..\If .......... udt l<,rn ...r ,,'hteh II donved from t"" p".,ed,,,, b~ ,II< ",me oporation. capable cf IndtnOl'. "I"',,,l<>n."

W. _ themore Ibl'. "any 0'''''''''1 quanUlf h<lna adopted. the ...... of j>OIOl"~ n"",l><n OIrnpl, b«:o .. .,...tcrr.11fIod ""b ~o 't.<ra""" lint.". a, .... dabonr.""" '" 00 aweJI'. tha, Ii oot:, 6niohcd Tbi. IQR'PlC &fOWt onddI.. nll<.ly and the oena .. rcd-.i '" ,I>< paTlLmel'IJC "'. jU'lliPOSO''''" 01 "DILL To uruol<l II. 1\ 'It enouaJo' 10 let on.:". IwId oJide 10 tbe naIll, 1.1 J, .1<".

NiluntUy, 1" '0 Ihorny quc>""n."", ld\ lu "ftC .. de: thai of Ih. choice 0111>< Imil IDd lhal 01 ,be: t ... HWCI! lhe .. """ "1'f'C&'" m,,"" up in the l""""",si.lIy <JI ti ...... tdnI,1I\nl .,th ,he tlck,n, <JI. d<><L II.""" all tht inoaiioDl 01 ih< ..... "" ___ .''''''h "J>OOI'" (,.., ... J L 111m lthrn J. etC, apprat com_ fortod by .bl> lazy poIIl""J .... 'hll. mna,n'''IIOIIU) co!"i .. ' 0 lhe docht 01 on \nfinlle' coIkction Iblol ... uttld ha .... 10 be: •• ha .. >!"", .. "houl II. b .... in. beetI da.ly defined. h ><CtflJ the .. fo", Iha~ 10 obta>n ....... '" .. , nuno"" .... il i. =0",," 10 DClIUIOO Ih" ......... ,.", INM:k ...... nt., lhal I" ..... j~. lo>.~, ""lind add",

lach lftm '" tbi> __ oe .... mal be .. prded ....... >-...1 ff(llll ,be procodlq by an """"1100 .. '"dl it I ..... ,,'"'" uti'" fuo-mer. )et. heo"'anII ,_

filE .UIlUOt" A~lIKil:m' OIALf£ll("At .AUI'O:S

eeneo III<"", " Ih .. d"r.n:ncc: lhe ~'" ml) be Inde8nnd¥ .'lended. hul lhe ><ron<! 0I1Ino!_ An., lhe lerm '" "' •• hould oblKIII O. bul beyond Ih .. poInl lhe q ... nmy ~ mlUl be 01 sud> a nalu", .. 10 permn OIl. operlllln, 00 nro ..... .., did Of! II .. od .... · UfmI ...... M. 2u. • . Bu •• Ius .. _ &1",") ..

""",hie " Afpnd Ibrm .... shows lillho 11"'"' oIll>r 'Ureple. ... h,ch .xh.au,,, ilself

'IImullaooowJy and 'U<>XUI...,ly ["", .... " ~ ..... $UT~]', Tho .. ph .... OInnOl 110 red"""" to the II!a# 01. booly oIl'rm1 in lbe fllen" .. ,," H .... ".hi, obsllftaq 011110 '+1' .-h~h alnys ",''', ID lb: ... me p:u, 10 be "",...,.".! It IS IIO-r"C lhal Arpad bnol'l'" !fie _'"- • hoob ..... k. !l pou,hIe to hll.o""" .he ltc'''''''''' lIIddioile 10 lbough. In ,1>0 ""Y Ihol lhoy do .nh 'rnal' nt.O»c<;

Sut 1!\Slood oI ... rica of ... .,~h1S, Ie1 U' ron>i<k< lhem u tIC""' In I pan A of 0 boIJa""" con.alnltl,f .'~t. In 1110 <>Ihrr PO" .110; ~nd f.,. .he purpooo 01 ~t"""'lion. lei III IUi_ lhe di ... ...,. P>*d o.'er by I,," Inn> oflbo boll..., io proporIlOnat 10 lhe "'aah" added Of .111><1", .. ' .... -hodl ir>d«d ""OIIkt be lhe c.!IO If I opnn ••• ,., odj ... ,.,;! 10 Ih. axl~ I( It.. .dda >on of llIe weigh' n 10 the pan ... m.,..,. 11M> oll,.,milr oI.he arm ... • dlS,ance .' •• he addi.ion of 11>0 .. ..,Ihu z... 3 ........ __ .... iII ""_ th" ....... e",.,mnr 1<> .,..,..., 0,," It.. donao<:n ~'. 3M'. ""-. ___ . .. 1uo;Ir. .... 1 be uk .... II -nu,,~ oIlhc .. ..,ttl III Iii< pan 4 Iht< . 'CO"" iI D/ro "'boll lho ""M .,., bol.nccoI lIy pluin, lbe ... .,~h1> ... ~.l>t. __ -. in tile pllI .... on lI\.Iy obtaIn lho ","ull> ~'. 211', J.'. _'-. or, by .. awn, .. i,h 3"' ud .. "I><1 ..... 'n' .h ... ..,ghl .. lhe ,",,,I,. 211'. ~'. 0. 8u' ,t>t<.e " '<t.hU """y bt reached 001 mlly by lakin, """"11 ou, rA lb: pan ~. bu. &I"" by add"" them ,o.hc paQ II i'ww ,"" adda"", of .. "'du. 10 the pan II coo bt coobo\lOd ""ldin".Iy: Ind In to 1Iout, "" UlaU oblmn """III e~l"" 'd by - ~'. ~', - ]~' . .- '. Ind Ihe>e Innu. ,~11ed ncp'"C. " 'ill npres£ '1Ul1nll. .ie:< a, ~"Ial drd the p<»tH'C 0 .... W •. lherofo ........ 'h., .. hoo 'wo lerm .. """"""",11, flI"'~ t.. •• 0J'P0'l" ","" .. 100'. _ 310, Ihcl dnIJn;ole dttrnml poto._ 01 tho t.l;t""" ar-ms. tuch ,""t ,he ...... """"y io<hcallll( .ho .. ..,at" iI '" bolh CI~ equally <btunl from ,1M: poonl 0.'·

w • ..., ,be ,nl.,,'OoUOo or lhe bl.nll "<1)' ,,"'".,Iy I. &110 .. . IIIe '"'"'Iwlrb,u!t"n .>f an "era'rnn J"'" a. It rompcnsall$ fo' an aa:umul .. ion of poin ... hu,,',,"' oonWdc-r:lbl< II ""'1 ho [con thus. by a<!<JClII!t"l <It,,«, 01 .. -.:ialt' ,,"h _h t<Npk of ...... hl'IP" P,).,. muDd .he rmpa»e 10 .. h",h any tleductiott leNs aDd pr-o\<>rI. the JUXl.p<lOJ ...... or wbtnoctMlll Indefinlld)

A <><p'i, .. numbe, ma~ .. ,t ltan 10.1Ie Icf\. 4 I""'tl''' number mok .. alun '0 the riSh" MobIle im"", and n"pt;". number .~".tU por(Nluoc a .. hok pta"" .. hicb btar$ I .sI,.tt1 ,,,,,,linl'ion 10 OM $t<Ir:: ..... he OIhtr A oumbt. ("""me.,. "'II"") thUlIpp:aB hie. ><l rA po<§OhIe rndrnal>on. Imc. .. "tW IIO-r"C by lho I .......... 1ft !fie lfII1"l ..tJU>kd 10 lhe UII): [I), Al _ -t A.IA. 0)

, ... -.nd In "",",cula. ( ....... , _ 0. It o .. . hould we rona:". thl , YICIo'Y 0'· ... r"""I1,,"~ Whot can the Irl~'

Tltl' t'OIl~OF A .... l()l; lTl" DIALECTICAL ..... '- of ""to: 'lr,bu .. ry' '" """. (Ih't &I"",.. oIiIJhdy lIw»cful ."~ borne by ['O>Ili,e Dumbrn.),;, • ...,n. I ..... r bl:r ··"':>nh ... fulc.um. lIS l!wl ~"",1Id .'/ok. lho ''''0 .'Im p"'<>l ",'hile .mdoptn, a "'bole pl."" ..-h",h '$ III'" (. "m the 'p"lIal Intumoo ... here rt " ... """'ly ..... It ;, Arp(Nl'J pat .......... "" lalmtlll.1I cnabk:$ b,m 10 VUP D/ro .. ,lle lr..",.."" bocan. as lba. " 'hdl otU UP' n •• "on ",hilt uri .......... '01 ,rrunobtlc. AD llIe ambtJullY and JtO-' .. ofen."doprl .. m oItllO T.o _ "..w ," =n A baia_ Ihc.of"", ""'~'" rl pomblc In .ltntrJhI and rn NI'''''' '0 <lUl.n"f) ,toe nqal,,'e by a,. .. lnll l1li. /", :ml I .. , ...... by m~I"r .. ta,.",ltly .proul. The bot.occ . .,. m<>I'C 1'fe\.~loOly ;1$ fulf;rum, ,herd.,..., rUl'el'lion> I> • muhiphor rA ";nUl.ll1Ie •• U III aJlu~~ de....,.,

As lOll III""", de, ....... a Infaill"" tU11Oba1J ttoe ,alulllOfl ,a In ,,'trn1C ........ It>d <"f'Wocs chcilk. ",ro II M4 U\ nalu .... t>Of" ...... I rukrum, t>Of" Ibt numblr · O. The baJ~..., d""" n<I' 'Ioad' to the '!dca' rA lhe .. t!"1,,·0 Zero;.

.. o"hor 'n<><'",' "'" ·n<><""'tic·. 11 IbIo.hft,," aeu" .. ltr'. in .... "I""""'" "''''' 6naUy d,,,,,,,, .. n I bh .... ul pb<oc .'1IO-r"C mobdrhc. I,., pRCiid1 pdutqnl III ""tu .. and lhov.ght A.pDd ""'" .-..y durly thaI all.bI: """CPU of .ymmct ... lIl"'" """,""".ted In ""ro. the d",,,,,,,, in 1.'0 of,hc IU. rA.he Inm 11M ·.-..t !f"'.aIt' 'i"",) a n ",nt>m",,". 1 " 'hole pia .... It i. o .. y 10 '"' In.. ""'0 ,osol.os 1M cqu'''on I~. <) _ (X.d). ",' ~",h;, ..-rII'"" x_a _ d <, 7"" 0 i •• 'moddle', II II .hr..-r"", I .... plinl 10 "".n:b for I ' middle' 10, mUh,pha'r./)"'!hOl .. 'hr ....,.....ric .... an Ihat. ""lllln, A,pnd'". DOUlio".. .. ddinod by G ' ,: ,: It. Ind, '" modem ""' ........ b)

• • - ~ • •

.. l ilE rOl:Cf.: Of' A"'IKlCITY OI,O.ucnCO\L IlAUloIC'IS

If " &lid b .n: ,_ numbers o.ucb ,ha, .. > 0 and b > 0. _""'" • '",utupi),,,, middle' iJ oqU"lIienl 10 dclcrmiolnllM ..,ame1ricalprol'<'f1k> .... 1 "",an ~( ... em ,,0'0 " .... "'ilies of d,II'.,o"l RaIII. Ilia! 1$ to Jay ,be quantl'Y J: ,..11",11 Alalia 11",,'01'0111011 +1:+..-::+..-: 1.

Arpnd COflUnueJ;

H m: "" _lUI difficulty .... 'hta "'C"1ihod 10 CUIII;" ... lbr ~ . ntlIilKta;:al prOfi 'i ", b<J'<>IId l«O, for % Clnno! be made "'I1LlI '" 'n) q ... nu,y .... , her Pl»111"" or "" ... hl"\:; bu~,.. before. the " ... omy ... h ~h .... ' ..... .,nary ... lion applied 10 ~nain ma,nnudes, bcalmc ",a1 who" 10 the KIea oIltro.olulf number ... ..Jdod ,10>, ot di ... :1I0ll. ""'Y i. DOl bc poe4iblo 10 !reO. Ib .. q ... nlll, • ..tudI II repnIed IINIJlIW)'. boca ... "" canJIQI ....,.

" • ~ '" lhc: tc:aIo of .... 11 ... .ud ""1'111'" '1 ... "111 .... "'lh tho _ .'0:' 7 On n:1kcuoo Ihil bu 5 "ted poosible. 1"' ........... " can 6e>ue • k ... d 01 " ... nlllY 10 which .. " .... y apply Iho ><lea III d" ...... iOll, "" WI /11."'''' ~h(l<rn Iw/) OfIp<>Slle dim:tionl, <l"" for pOSlll'. ,lid one for .... 1I"1.i\'C vaJus, l~ oJu,U 01>1 • tbird _. liwJl r~ "....,;, ... d,,«.bt MwU1 .,DNI Do ,/0,

_ '"'-"'" I" It ,. , ... ,.,,., JQn III/lot .."..,W "

n.. fi ....... 1><10 .. , .... "Id ....... by ArpDd. """ .. how ,ho nnu.b,.,IOft willch T' m"d '" bc -",""ed ID 0 Ilmchcs ,Id.""" tho: _bole . . .. OF.. p<'f!It'n,hcular .0 . he arm, oI .he ""'"nce: perched .. n oE, I can wn,empl" . ,he poII l1'''' nd ,he ""P"' •• "'" ,he A_ mdi lf.n"""" (...: HJ. 6): , ,

, ----~. ., --- , .,

D , ,~.

A r .... II ..... funhe. Of\. Atpnd ."'.,.....«:$ a~aln 'hI! ~he d,rea;"" oI(1A ... ,n "'pro." ,he d,,,,,,li .. n OE. wl'lll ,ho laltt< i$ In rqa.,J 10 (I /, ,he I>O,;on of ' '''''an 1""""01' un,,, IIl>d been bo.n. I,nked '0 ,he idea 01 d",'':IH)n;Ol Ii ..... " The _n unot II J"*ObI< in "",,,,il...,..nd.n """hon. T,.... uru .. iporad.),,. caOy cal'"e! 'm.pn ..... t"", """.....w) only pa.optd IJJ«oJ/J (by ,110 IYmboI

" v::I ,. ' 11 .. hlrn>h..., pe. mlned. "" doobl. II..,b.,.w: "",,",I0Il> ,I'll, ....... r",,,,,,lIy heyon<! ,op,<*h. b\J, ,,,volved no ,mullion. No. '. II .o:m. ,ha, lhe: 11'""".0( ,hue un,U IS ",_led 10 .. ,", .I'II,.he UIllU an: loca.na,(d. "OA lllhe: O<1,",1I 01 poIIlI"·. "n,1. (1/ It . he: "",,,live "",I. (If. u.J OV.", ,he: mean 01Ul.:

y, ,, ..., "'''h .. hal care Arpnd proser ... bae un.U;! I is .. I'll, III powled fi~. _I .. ,be: <nttnler>l.ll.nd _.j an:.be: ,", .... 1 ""'U. TIIe: ... .o ~n'rr .n ...... '0 a con".lla'lOR 01 .... ' .. "'" ... Iud! /orm a con'''''''''''' dwn bel ..... " lhe: pI.,n dog.nalion 1',," PU'" po<>,K>Il 01 + I ) and ,,," ..... ,....,..,.. 01 ,,," "',,,ral un,,,,, ",Iu<h ,numphctl ".·.f . he <>rf>l>"ilon of + I and I Zero. .. a:n,rr '" roIauon. ,n,,'" II. to ..... rr In a k,nd of ...,..,.,,,,,.,,,,, ... hkh lea,'" ... in II<> doobl .. 10 Il!c ..... bly of dim:u" ... ll,na;

Rul lhe:-!< I"... .rr ~ ... "hllC:O ~1I"e •• """ II .he """, ... "III~ \1><) .,. dc",'Od f.om i, 10)' lhe ... oei.l,ion 01,10:: ...... 01 dim:,iun "''''' ,II,,, 01 ""'I",t ud .. and .'" In .h,. re>pccI lik. tilt ........ 'i'e I", .... !lOcb ho, II<>

,,,,,,,,n.,y ",",fica"on. Tho term. fflJl .nd ~i_, do "'" 11>< •• r" •• ..:co.d .. "b ,he obo"" "."""""'"" It .. DOCdku '0 rcm;o,k ,I'll, the ".~ _." ·Nt and~. 1<lme1,""'" ........ 1th. Irr.ruU .... appropriate. The ...,. of ,hror ,cnn. In ,he e.act "'''' ..... In anr O1hcr '."'tI' than w, of",.. In. .. oJ potl'llr< 'orpn"nl-"

I.nd ·1 .", ,nd<:ed '"",r f"r the) can he 1I'''''' a,ed ... i<lc>. and lha, K far ",on: "'I ...... n' ,n u. ' h.on tile ...... u""""'" 0('. "nll,1t Ito ...... ,. c:crtain or Alpnor. con, ..... pon.""" d,<CtInN ,n ,hi> m" .... ) lhe in' .. n' ..... 0(. p>mcular ."""""",'. H. h,modI" '",,,kly odm,u ,ha, n m'&II' be ....., .. notlunlmofC ,han ,he .. mplo .... 0('. pan ....... , ....... 'i(>n; '~OI m)-.df. r ., .... ' 'hll [ do)fl

_. Id Ihll """,,10 ... only •• ,,,' of 1"""""'" ml~""",pendded 10 .... l)'ue formo "'000c d,,,,,,,,,,., ,,, .. """" .. mplo.r.<! .. p<d 'lio ......... 11m lbe ,n,erooi of 'his ·1"0 .... "'. mOJquc· dod 00' .scope J. ll . (lc'1l0""". "' ho .. joIoed to. Soo<' 'IIFbI"", "'1II1~ ., Ia" dJoencumbe~ 0( Ihctc un,n,rllil'hIo and m)"'."",,, Ionru.. of , hctc ".,...... ... wlUclt lpo>oJ " and make of '. a ..,11 ..r cato.J .. "c K I' " .."" !peak:"

Tb .. III .. hy Arpn,f. , ... ,111< marta! . n ~It C ............... rr ., Lu. difCIK'Umbtml or the rormal m)1u6ct""" ..r V, I.

From the $i"",ly """,.1 Ind opcl'l'''. p<~m nr v"' .... the nOla"on =-l<+V I "·OltN potfcoctly .. ·.Il b.II;' .., .... "ed ooIy 0 .... 11'"''''''' ,he t"lu..,,,won. 01 \' _ I "\ I by I . Th, .... ""Iy ll ............ ,h ... ROU'IOII ooIx:ilf. f'tT~InI'" propul_ ol 1110 ."'", .. , .... bmok.nl ",th all ,I>< pia". ,1Idc:i 01 the...... ·i •• aad oKffl'" aU ,I>< .mbo • .,I1 .... and no .... ol.1>< ... ' .... l

- I and + I. bel"1 C(>fItrary ....... ,",." •• " I>ftIF ".,.., ...... I nd J.D. GCf'JOfl""")"I vcry ... "d) th.t111c p.opO. llo .... 1 mean ol "",,,,,,\Ide btl>"«11 +~ Ind d" and "",uk! only be <r. fo, .... hen 0 .... is ,peak,,,, oo"l~ 01 malrut""'. """ "'lII' om" IIwo ... ". V .. · II_~. Ru, " 'hen 0 .... 'Ml .. t aV - ' .. ,I>< mea ... 0dC """"'""" ,bn'cby lha, <NW hili ,.kon ",to ~,_ .1>< IIn'CrOC .,.,..., ...... of ... " and . II: ,he: mean "'~ .. tlottdore ,hen _. ,I>< ''''''

01 1M ronllMlnaloon: " • thrrri<>rc. M I_a. • ..... n 01 posillOll .,.d ., 01 ma"IIIIIdc: the ,nl" I""'II_ 01,... J)'lIIboI v'=111 Ihcrdote red...-ed 10 bWlUIIJ 001 a ~mpl b"., of .. htob " eon be ..... ,'''', " II pooolCd III rdo,'l011 10 +a, .. -(I. posiled 10 rdo, ...... 10 11-·" II II the '<'3llll<lmlt_ (01] ,he "" ...... powuo .. · ,'''', '1"1 I. '" b.ttk1n,'he b,,1Id InlllSlll'''Y 01 ,"" ~ ""'Y' <IIkula1lOlU, .'hIi;h eo .. finally ba;:OOlk romplu..

I, II Impooublt 10 .... UC"'''' tho: ~ oI5U('h an t'\'OII; dc:gd'1lJ 10 ICC J and -I .. I """klnc m ... .11 ""Ill up r",m Ilk-llben'" Ind OOIII. olkd .mbolulI~ and resolu,",,) '''''pulJ ze.., as a pwhIt_J< ~t..,.. ... the (>::n'''' of a 1k-,K1' .. hlCh does "'" wnfi", IISOI! 10 ..... ""'n' nq.II •• nun'bc: .... hu, rel<."1i"'01 IlIIclr In an a"IOCk of the b_.tf1ll COJlllhic of JPh'"ng a ",holt p/;oM, It I> lhe •• r"", .,;.h ~ ~'I#i n "'pl"re IhOi .. or,...""", _lid I'hysklAIU " '111 ..... lIlh' •• " hun. ou •• hOloC nle", 0. It<.< hKklen k.y poinl •• hat con!p'''' ." ,uide our V~,p 011 .ru.h,

~ 1'<; 01 t , ~. f. "IC ~ C t_ NT. F.S ~ '10 K 'IO'rs 01 A Pol 110t; 11 'Y. " , ' C.A or TfI~ BALANCES o t 1","0

A &Iw_)~:.Jiuat_1J

T1WI CO""""') '" mk"'ed all~t In .... ,,,Jo, paon .. had al.., , .. III'1okd phLk). ooplKn r", _ II ..... Ii .. YC2l$ ~ "'pnd'$ . -0I1t. Uchcnmayn ""hlilbed .... fiBI Ru .... ..-/or, I ....... h "'" " pri«i DHw._ '" "'QI"'''~- P .... ..,,, .......... Wc mil>! 110(. ol coun.e, Jo.w,I .. Ihcoc R.,.."..-Ior, aooonlln, .0 """' <mpOnry oticnll1ic cn •• na. bul "'Iher "od< ..... nd how '''''YI~od In Inoculllll"l lhe m ... ph)....:aJ ""k", 01 1'Qio.~ .. poncn ... 1Ih Ih_ ol the cornpk,dy IkW ·ckc! ,pphIIOSQphy'. In JIIInicular. , ... appel"''''''' In lhe R ..... urrlH> .• ,II a «.laln dIa".m 1«1111 ,.uClal W u. in onlc. 10 Jf8-.p lhe p,..ciloO mot .... l KI ,,'hlCh mol_ph". II,. (II'C' .n~ c. y.lalh«$ • " 'hole ol"'''"i''~ fitld,

rhi. dlaV~nl laYI bore I'" p'oc<ss ol d""",u,,hln. and """nl", OUI I .. " .·,Ulloc< ,. 000 I bet."",n. rrom an cquiJlbr,"m paon. rO_ 'n.e.. ~nfllin ~rc )nfinll.'. bu. C9JI(ic1""""",. fOf'm!, of lhe I) po.. ~ J.

~ I " ( , I " --

" I' I' " I· ... I "- I I I I I I I -•

I" ,~,

1loc '",,,nr , • • ~pio;Ics uti" oPP' §' d ocquc....,. 01 cqu,hbna. ,"'""" cquilibna ""' ..... , the Jl"'If<'UI'. dca>ml""'UO" of.he enh'y ' •. I:.<htnmayer .,...1:. of a 'proo<><' In ,he p."""" of """"mlnS " quot .. nl' , . n .11 c"I IIY of dq;.ec:

It<O, .nd from " and ,. are <!n,m! • InD of '"",1' fia"'" (snoc-bic: .. !/lmp) arod qualified by fi1liu.-.I .... j,.11,,, ,.~"_ Tl>c lalla III"'" he: ~ndo, .. ood .. the dqra:s of dJo.bnc:uon arod ~"'IJOD oflhe';"- b)' . -Iuc:h I 011'0'" iudl'lo raJ nDlCDtt. 0- iliuM IhcreJon: 'lkl~ oIlhe upotlCtl' ... "

,- and I arc loWly houDd 10 "'" upao"'" 0 and ..... 1JICd .'l~ .-another IIY moboh(,"1 t<J>II .. "n. appu>ll""'" 01 ,ncrc ...... p" .... r.~ f ............ <Ii 1Ic:I1I&' of ...."..,. .. ". IoC1Wt"'ly~ en ..... pn. cqu,hbnl 01 ""'rt""", .ubtlety, I con .......... O1II",..-hem ,b<: poIanl ... ,· and' a ...... ' ... mloOl' ..... 'l~ ,ncn:a •• in •• fId<1tcnd<- ('h<o ""j"idwtl I' lK,n, Inerea .. nsJ)· domlna .. , , ... "Uds 11K Icf\ and' 1.."....~sJy <Iorn1l'W:lI I",,-..nl. the nlShll Th"1" opec!",m ol "fII""Id q""/Ilit. con "nfold rrom lbe "",n' ,. (poin' 01 "boolu,o tndminlui,hahoilly) .iJllt "p ' " lbe • • ymptoll< poin .. I '" .nd I " . .. IKre' ~ ud' .. .'" !pully IkJlllr .. ~. w. m ... ' rrmcmlK. I"'" a ku>d "'.ffin;ly p" .... m .... wecn lbe opp""", PO""" '" and I '. wh",h i, <lem<>mlr~.nJ by'''' b •• d .. "'hkh jnl ... ,hem ,.,.., ..... in r,.,.", 1. The ",'allOlU '" 6, ... & •• , can he COIKeI''''' 01 •• tIkII.u.il1(l • po"'" ol racluhon 01 Lhe dt'\'ICCJ ol equilibri um .·hich prod""" lhe dlif<rcnl r.,.,.,.. a nd Ihi, """"" 01 mooIu.inn II ,00 Lbe """"" ol Mkfty" In rebl"'" 10 I ".

Tho d ...... m pc , In amar.u>a; 'Uap:Sli"" capICIl) and p".hapo "'pla" .. .hy EM:hcnmaya'o .",h .... rl:ed In q)lXb. S<:hclhnt 101 .. ad< ..... kdil<d hi> debt .. hen be dlbo .. 1Cd Ino con<><pI 01 p" •• .-.... W. ~""'" ,""1 0Ik 01 ,"" majOt ambo,,,,,,, allbe plulosopb)' ol""u'" ..... LM JIIIlk1It upla"'I"'" 01 lhe "booIu .. It .. no .......,.. • q .... ,"'" ollolll", OMtdr he wbJup,ed by 1'" "booI" .. ( .. 'htIhtI 1ft relcp"n. it 10 tb<: iMlfabie. 01 "",11"1 it oo.cn """"" Iy~ bu' ol d""IIJ .... hlnJ "<>pO. probkrn.1'''' hlnp$ • hem nal"", and lbe "ndtr· "andinl cr_ Olk anolher. " -he,,, .Iar 6", lum, IlIoOIr ,nlo ·,"ible Undt,,'aOOIRB ."" tbe 1i<COf1,J ;nvilible N.lu ... ·!' " 'bere on .... rolatioR ... ,....,.n I"" ind,,'idu."on ol Reinl 10 be ktlOJ .. ·n and Ih., oI,IK l"""'lnl ,,,t;r..1 bc<:omt$ ""parent. Escbcnmay .. •• d, • ..-am .. 'o.h Ilk< a po .... rful 00<1'

<len,.. and ....... 1'" cl ;nIU;LIOftS. whICh W<X'f'N1 iJl knll,Ln, 'OJ<.b .. all,,,,, rorttJ 01 ambia""Y. 01."" unstable poinl Ind 01 ,he ;.,11.,11 •• ha. ,,-•• "cmpled Lo d . .... 0111 ,., ,he prcttd"" ~s""JIIn-

8y 00<111011"" LIK .... """;11 tofIOC1ILnled ;n ,', .. 'hcR ,"'0 D>fiaite. ~nu"" confronl ""'" _her)." pi"'" .n. COGltnUlly In lhe opn>I"J 0111 LlI IiJurca 01 LIK _lablt.1uc:h are ....-~. COIIIplctdy RpIIRk lor ,bo " .... t ,-.I""" 01 lbeupoMftt_ tach cqwbbn",," ... mn .... "' ..... 'hit oll_~ 10 "'" nes, plu.1oO ol """'pttISIIllOII.. lid dcsl1l>Cd 10 IK brol .. by a )'tI .... JjItr Ik-oo:<' la finn ·_" ...... nl by ",..,'1_· t:.c ....... yn .• <hqram II. ""'1 hloohomt exampl< of lbe .11" ... ., ok>"", ThiulraJOF """'pr lOCI ol , .... "\iioltnll.·o 'uIdc •• "",. nat.' q ... n"''''''1 .t lbe pom, , . mqIl, he lhoop' u'()m)h'''1- BUI II" .IK equ.hbnum .. = haJ>l1)' ju., to 1kU1I1l"",,, flnllc rorors 1dt1.nnlna1t in ....... elJlIOfIOt), I~ would be mtfd)' lbe Immed,.le "",,,It ,II an odd" ... n 1/' .. I ' +, I. and would ... O<tiy ... m~ ',,,,," .. '. t.ha""ed .,lh no po ... boh.y 01 rt0<-11 .. 11O", ro ... h"'h , ... rOmlul. ,""uld be I' + I , •• 0.

IitR' · o,...... ... r Inlo .... 0 do-.="t bfal>Cbca, bur . 'b.;:h ue """,,""Cd do .. " In!O ,he ,n6n1t ... / . and I "'l'H"" IlIcmsolva ''''''''''Milr d"unclly and lhe Iffinlty 1h>.1 hnks them ... n only I>o<:omc ku and "'- !Oi'!lI/. o..hOllId h.1.rdly Jprak of a fill,nl up of InlUltion in rotIn«Iion . '111. fi&1In 1, bul n.lbe. of. mobtIc wul C.u,.,upocd unn'. as SdwilUlI .... 10 .. y~" Once more: it ~ be ~rnd WI lbe -""" ... ofthc butd IJoei no! be: I ""MOt dn. .. 1M d"'lRm oflho opn1lD1 out 1_ 6a. 11 "'Ihoul piI .... DJ 10 d", .. lhe brackel or "'Iboul dtwm""''''1 ,nlO , .... ~ lho oJ[",.'h",.lI of lhe " ...... '1. To cap4UA! lhe ,UIPI; • .., po'O<" of lbe .. d,",ram>. I mUit mobIl"" the h~nd and lhe eye II lho .. me u ......

The tnl">11"" .... l.,. ;' Kif , .......... o&ly poneln.lln, ,"',III ,he ert8In. of lhele 1"'''". of eqlllhbr",,,, (_100 ",11 .. ,1 ""mal pooo!!. 1In../r,.....tIt). ro. lho) i",,,.u • lunu"" lbe:w .... lbe poon" of ...... mal am .... "'ly . 'bore • _ pK1 .. ll'tled bet .. · ... " tbe "ndon'.""'n, aad the umllllon. thor • .e the poon" of ... Iuctt SclH:1hnl .. wid $Iy .hll they atp4ure lbe 'Inlunion In ,be ""UUNln', ... hicll·" """her diff.,ellL'" n<lf lndi{f.....,..,. bul Ibat in .. hich b../,b bo'comc: Klellll!lcd.· ...

/I 1lor,. ... ,"''''''Ifn~,."ws

n- potn •• of t.xt .. -ard ... Jld.ro...lI,d mohon Cl)"Olalh"" ""'''':'';nlly IU ..... "".""h,n .. ou' of lho unll and ,,11uw UI to think of. pro.hlC1 .. ity that d<>ell not ,ol"ely ",hIU51 iutlf In II> pr.x!uet and therdu .. Ie,t< • <1Ia""" of ", •• :c".IIon. n.c.c poln" 61U1Uy una.tJt lhe d~ of 1110 mtthanwn.. lhe In.n .. ,,,lI,,,,NI .u",upte ... , .... Ind lhe "r>da.ut""'"" under the ....... yok". Ii.,..., • .., ~ pefllapo.1 lho bran or lbe pbilooopb,", of utu .. thaI follooo-.d h<benm.».,.·,......t ., ... blmll potnll nal"", <:an osa.pe lbe ......:t...rusm lhal .. bou." 11 ,n lhe .. mrle In."""",_ of ImpaC! or in til< monolody of!M ."",,.i,·,, PO'UI"',,; in lhe ....... ,mpu~.11I< undt' .... ndm' It no lon,et r<du~ to od"""ISI .... n. daI""uon. '" tlllUft""""" of InfOOr\lldoa. The!t ""tnll .Uow ... 10 hope 10 <on/;:<"e 01. Nit .. ,," '" Uadontand,1l, thll d,lf ..... t .. lt< d,,,,,,, objrtti and • DO loot ... tn.pp<d br lhe """~ S _ 0 or Q _ S. Ad",,,,",,,,,, ItOCh lha, a had: ... rd..ond-lon ... rd motloOll ""'Y he """,bk .nd .he", the KIt ... t) «the pauqe of ,he prnduc:tmly IIlIo the produa ) P"'" In ampillude III it "",nlf""" IUtlf.

DO<1ro<:hemini f~ . ... ""lh.'"''''I'''',I)y disoo~ ~, the begl"n.n, ,,1111< """,t ... nlb <>:nlury, W,th these. nalure ~ Rlb;roct ('tIl.,. Itkin", oo,.ct'~ ..-bustlOn .tr....,. IIIdf b)' rmeo>l"l'I' OW" 1XPnd1l1O .... hal tnCUYlteJ lbe plo •• "", .. _ "" lhe 'flO!. W" ha,'c here. WHta1lIod dlt<qullibriwn. hut tQkoIy • 1 the pcnplM:., I .. In, ... Ut i .. "'11k. !>nJy. "',,''' .rth .,lb DO "",,1b!J,ly of rt'IoC".,lltoll, • h<",u h~ atn p;th"nlly Wn.trllC1. fur ;.wlr ""),If"",,,,,,, ctn',..,. .. h",b allow n In ""he. n .. lf In leap funhe •.

W" ...... b"... the bola""" <leo/roy. lbe d",M of Ilenllon by &lludtn. lo.~ equilibrium I .... ' I atn break and ~,,".I I c"""'*-. prrot ....... """ ..... Ie""" f.om .h>o nOl bt: reduood \(, an I hhf'o,." .. ion of conl~"" 001 oItoould ",Ibt:, be """"'11''''' of ... 'Immel""n, o" .. "to. "hic:h ... "« mil.,. Imt,..Ully .""""mb 10 F""f1I~ly. lhe "'I."'''''' o. lhe "" .... a"oca';.,n or thlt or Ib.t <:aM:. Theoc opon.lon make;t poMible Iu prqII" 1pact5 fur lito: ptt.cn"l up of daluc"", chaml . 'here liIq' ClOD be: MI Of ~"",tod 10 be w.-m:d .t .. rna paUl .doclly. n.. .,-mmelnul"'" ""-'tnt> OIl. """".SIlInd" .... 'hIdt .. at .... ,.., .... ptcd b)'.be faaillf ortbe In.n .. " ...... "'" 10""", i • • o w .... ,uct a ..... baIa. ..... of !k"" .... be .. other form .... ,11 be deci<Iod. 10 .~tnk ou •• 110" indl.idua!ion, "

Tbe follo.';"1 aample "",l., i! POSSIble tn u!Klenland 00 ..... }mmcl'y . LIooo·, III 10 sllak~ up l ...... tiVlly TIlt Ihtotem ocrood .... to .. It>Oh ion ........ II """"" .. ,.,.-.d "h," under ....... perf"cdy drl'" >hocks can be ...... III lhe <hqram-.... Ioe equl>'l.ien. ol fi." .. L

r_, W" """" f,om • "'ry ill."",lched sitU. IIO": Ihll of 1.'0 O<d'nu)' ball, tu,bin,

Ic..,u d. OlIO anolhe,., possIbly d iff .... "1 ,'d.""IQ;

,

r_' " " a.e .... n. '0 _ I\ow' ... ne< of d ... """", ."""red by a dtmoMI"""'"

by tt uYlltnl. "_",," ,.d. In ",)rIquen". lh .. hel"' OSC""uy. produel.' • hnmo,.neo", ... lIly .b",h can only be lpill Inl0 .ymmclncal pa""

Tbe wU..,videtl<c dIll< Ia .. of shock. " lbe ..... .".., madt all tl>< ""'"' to by lhe ,,""""'try ~."'bo,cd ,,, .... n •• n trie=.Jal .,..ttm. 1I1th t:ertllD "",I>..

(II I CIa .... u ,\,, I , ... M , I , _ \' L lbll II eq"" ...... , '0 jot"'''1 lhe .'" ... boOt t<>F\hor by lbouahl and .l><Jmtnl. COIIImon unJ/ ol,mpu .... I a.n. lor .... mple. "'f' ......... , ,he q ... nl1ly ol tmpuLst M , I, U. "",.a"," R, and '\'.", 11$ a".,tlttr .""'an"" II ,

1·1tt total impulw: " l ilt alll"bnlJc .um of lhe ..... ol ll , and II , •• -bich • ." .. potId$ In II •.

I ~ I Uhlm", GUiIeo·. pt111C1pk; I 1"01'01 m)w:II' by ,-.'" ""n • ..ruu",",

" poIn. IIIO'-;nl a •• 1It .'doa,y r: I obIalo of cou..., ,"" chlV.m of an t E obi" boll'

•

• " • •

• " •

1npI}<.'II---- ,_'V

B~ bro"l'nl ,nlo pI.y ,~ "n><>,h<'Qi or perf""" da,'''''}', ly""''''I<) II ... "red, I-i,"", n comlclUlet all J'<II'<iblo p;lmllOno, and I ~'" "l\lI,wnolly' compello!d 10 "'" J2InJrjJm""'J" IJa 0. Db.

• • '_'J

~ I ..... of ....... .tI«h ,herd'<>", _ aPPQr1 '1Illlu,aj' I .... ....rIOI,o <It,.i] .be ""'K'" of. ,/1""lh. nf"'n .... nl ;n " '11,,,11 • 11<>'01)' Qymmtlnal .nd hel~fflJC_ "tUI" "" ....... n:baloncnJ II ath;',',," • ~",alL .... "omoac""l!). .. he.., lhe PIU'''''pk ol.utliaenL "'."'" """PI "'" ," ,_ 1'h" if perhopo ,he Inlf<" II .. " pre .... ..., Iny oJlu"'-< "mon."", .. " to bal"e IItt. I'nnoaplc 1M: deadt;I .n In ,II pun.),: 10 obItt< lb< m<>bIluatlOtl or. put& lbal don _ deadc

0IloO ...-ay or d .. ",he. by 'nlulily .drlBl on fact, M''''''Iualny (If formal ,)11011_

W. DOW undemand ./I) ,he ")rnmctrWD, ope!"llor m,an:l.fC' 10 >pIn the dtllV ...... of ,alenhO"']U) S _ 0 .001 Q_ S I'he . ubjoc1l"'" ,,,,,,,, .. II. 'M 00,..:' or lbe oqnt JOCI "0 ~ • • be oub)tn) and 10 dwoh~ ~n..,.1 1I ..... ".,ua. olon, .... t/I aLi ,100 cbchb .ha. ,hoy any "-1111 \btrn, 11". 1 ...... "'.- " .... d .... riIy ,,,If..,,,,,,, .. 11 and rollo ..... """,",'Cd Klc:;u and """11ft. bu. " h<co ......... ry I'fOb)"", . .,'" wh<n .... annl'h< pI.o'ea~ ut.n ""uil;tmum. 0 .... " 'Quid almOi' he I~mpled 10 uhl.", lhe ,Mml .. J ""'ai,,," S .. 0 to .mp~I;. fully 'he ~m"'iu'ly of ,line ",;m ... r»Ol\II of lho! "'I u,,,oa, 0( ,t.e... 'OOUPII "P<'~" .. -he~!i "PU" ;, ... If In 1M "'""" "'" 0, S 'f"wll llidf"," O.Dd ...--.... h 0 "haUD', S" 0 "J,I,>lur""" S

lb: mil"" r1pcnmnL' "Of)' dl"ca;."dy demo.Ii''''," "'Tf)'lb'n. ,1LlI. IS ., >ut. In .he >rin"'" '" '''I'L' 0( ,be,.,.".. ... I can look a. my' ....... In a nurtor II .... I Of I_I'), .. h..:h i. 001) In "objC<'t" M""h '""~ d,,,mbon, Ii.be m.....em "he ... ) "'" /OL,,><'/f in.he m'rtor, ,,-M •• "'Jump" ou,.' "",",ha, ,. IS m) .... II .. ' I'k) Illook,nl.' "'" II ..." or 1 ,,")" -n >t lUll brl .. ...,n Iht>C ''''0 I. can"'" be bndiW and )'<1 i. i ••• Ihll 'II>I.n' lha, \boy ore hnled b) 1M h'VaI ro,m '" uruty" I propd my .... ""... ,be mlln"..&. I 10_1 ,ho .... "'" """'"' .. h;h .laln ., "'" in • 'rnU)" ,roubltnl IBm ...... 0.. .. ,U DOl M .... ,p ... ..., b) ,ho dnLmo"c rharKIrr 0/ thl> propulsion 01.1>< I, ... 11"," " .I ... )~ • .....:i.,ed .. ,th I",.t d;..,.,,"<~ Arclumrd<l put> ~i....nt ,n ,ho pt..,. or ,he I\oolln. boUy .. 'hen h< cl'lC$ '[ur.b.!"" [ inst.,,, 'ak .. 1U_lf r",. pholOft or t~ /wtt.woV i.n • lift. Tbetc poin .. ma.~ ,ho . ....... ' to a .upenor horno-...... ,y and.", aI .. 'II)' aa:om",nled by '}mmc1nl1nloponl'''''' 11 ....... CSiaI)

'0 cooJure up. ~ .. bor-o>.be prtlICIpic '" JUlt\aenI ~_ rnwItI~ ....... . llse1f. Thco< OP""""-' or .11u~"" •• raD",,,,,,,U 1"" ..... "..Ieq ... boo .. ,,, ,"" ,n,ua", ... '" hieh ;~ a singl. pJ co.n ""'" hold cI 'mm ...... dod",,!! ... ~h." •• 10 ,ub", i, ,he,,, 'I> ,ho Ia .... at ~ co~p'"

Tbt>C po'''b kftp ... 'ure as ,"'rU IIt.e and.lll~ nd'nl from ""'''I ,rapptd by ,be URlt) ........ ;1 .... </'''' ,Iv {«" ma.upulabl< btlt blood .... (I"" ".lnIiaIl mtaJllOf by "'" compot;"1 ~ft"Y ",bocl> .wa.I....., .'''fl,h,nl up ,.IOd,-...a:. A. t ..... poonu. • can ",,"1I)" lorl'l\ ,,,,, "'u""'na ....... lha. bestn front • "" """1 .0><1 • can fur.,1 lhe bciLtllel 0( cd.......:.",. They.", tfuly pmn., of m .. "nal IIObo'.'>O" .. I.ho '"'UII'O''' btl, • , ,,,a, firmness II ,....,.. • .,Y ,I> ",,. .. ,on .nd ma,nlJi,n "n.".,1f ,he, . aDd '0 ."}Of.he ....... oon/""" lrun thoy .. 1T~r_ F .... ,hose .'" abo ]><>In" at ""' ......... dbcomlOn. .. ""'" lhe taro. an: ,cdal~ .""'" .100 j2 'dul .. "'"Iia' ...... buill up by old k"",,-led .. btromo: Jumbled and unn'-ri.

........... ,100 captu", 0( ..... of Ibeoe ..... "u II ......,."...~ ~ '.". htu ... "my been adapted 10. ,"'",* """''''III ." ~_IedF .rrea • • nd ...... mt ", .. h " .. ual,un, impal .... tly . .... ,JlinB III< F"UI'O \~t .. ~ ... '0 hbonl< ,be " .. , ~,mt oI.nicul:o· '''''' by ltah"l a nn. pacl bet'"'"'' opcr.lhon .nd .n'~."o ... 'Tho .ym,,,,,,n..>lb, <>p<1I.'on Mtmmort • p"naplc ,,,. >o,ffiaen, ro:uoDn ~nd <I<m&nd ,1M <ftotoo of. discqtnhbntllll, ''''''' ..-bod! • new "'"" will ..... W,lhoIt, , ...... potJIU. ~kdt<" ..-ouId be """", .. ,oIay" pb)'SOCl would ~'C <pe<tnt '" ...... ,oon. .. ..-

I I

.. tlpcriona", .oy maltm<11l I(lonen! IUd( IlId '0 UI~. lip new intclleccwoll'lahotJ. AI there rue JIOUlU of toomcioalll\ou,ln ",'hf/n, S IlIt", up t.pln "'lib O • •. ""," .... ",IoPKdil CUI hi: rucI'I'I,N.1>d made In ",'-erbrf'lllC.

C T/wo i..J,ffrrtllU ~_" ... Q,lIttol",1ooo

Thll bn ....... 10. dow. ~UJlUnalioft of the q...,'lOn oCthe ...... 1no/QriM.

AnoculllJOft doa not daJru 10 m:ollCllc 'wo oonuuu A .nd _ A, il 1"1$ ..... nd , .... , ",,",""'talLOI'I. It .n, ,,,,, • new mod<' 01 n",<lny b)' unlold"" • no",. .... drvcu. or .. hoch I II>d I are the .'Unnc bruocbes Il~ He no Ion .. , ol'P'*d but ir,up b) 180 Ikpml . ..... , .. " ..... ift."JOUSIy krled Ill. '1QOUS confro,II.U"" is !lOW "'''(,;.0:1 01 at 1M ''''0 peds of. dJptyd!_ Tho! '"l<\Ib,IOII prdIpra 1l1li form.nd moI:oilim II ••• """" /«."bn ...... , .. iar..",. 'fonnar. II ... "~ ,I reo ... nd 1IOI'l .... 'Sthen'n • .".,... of the ,.,...", of lrll.u1al_·~ U It pmml. ""'''Iiooo .... 'hou' detachmm~ ."Iumml wilbou! tn.t, .... It clll .. 11M! connoct or Iam$ w;thou, .uboni,na,ioG; poiarualiom ,.,... suhRnulM '" OI'l'O'lIIODl. SlIoch.1Id lmpuhn Jed ..... only ,1M: lhoushtJ w.:.n;,lCd b)' ........ Iily .nd compktcd ,!Id,,-oduab; 10 .nocull, .... I .... l" 10 aDo .. on-Hif I """ an~Q" 10 I/' __ 'rr. III.,..w,,,,,, is """" tlwtik IWlhal 01, ... Iideo. An anlC'lllouon d_ nol bnk 10F1b<. '''-0 COlli...". or 'WO IqIOnn. tqmenl, which prcuu,cd il; II pups Ih1< .tty a' .. ' ....... III ,lac" from In ind,ff_ poin ••

In 'Ii lPO; .• hI< Imcula.1OII a...., ,"" rrudU<'l and prodU<'li,,'y_ It I.hl'llj'S

pamDp"I" in ,"" bliC'ra.ioII of I d'me'n"",," In ok<1Cll,n. ,n .be air • whok $JlOC1rum of vin ... I" .... " ,iotr,,:ms, ... os ,nat i ... . hrays I .. ,ure lha. mwupln kno ... lodge. II <klcs Mt dllll/iify, " dll1m ..... 1I<> <karen Ind ..... ~rn. rhythms, Tbe omcuiah'", ,kCle",," ""'If OIl. in .1It I ...... m;na of bC\;ollllna; wi,hou. ,~ cons.ant)' tklrnon .. ., ,RIo .ubsl~_ (.mlt, ..... ch~pI'" IV). Tho .nirulabon PO"EZ"'S • mobil, pari,,,,,,, "hioh i'.~ '0 .. nl8;n O\l'ud< c!<rni"", .hat are "';.Itout <"fill 1,Id pr<l'''- Hili. _lid ,00 lOon. II ;. indood ,he "",cula""" Ih., mak., il Jl'I'$li~.o A'",'e """""If 1It)"ond III 0PJKlO>'"'''' and th<rdore 10 o,n" rome all """"",,.io.I. l:or .... I 1IUI1l •• "."hr, ,." "''''1 ,lit old d"~""",. '5ut;«:1 ~. tOo-m""""'rnl, .'el "'" """'l1In, """""If he .ubmerF<l in .he confu_ sion ,.,,"""'" »nmo.d .... """p" II ,u".hIe articuJlllOft no clout.. aIloWIII ,...,,, •• llIt<vah"" olin .be Iotooo: .mpruoned br """'rall, 00' " , .. h'lI)~ """'CImponlOd by .... birth,." ......... lan.y Thu \I prorudy .ba, dI$llIll;U'V>ct ., I'rom a '"W",' · .. id.". lwit-t .~ ot ,lie dubooloJ "","". If.hr . .. "",Ia,,,,,, ~"""',"' .. "",.. mode 0I.11e ~"" .. I ... )~ br 'd'roch.an, In .mbo&w.y.

I rXAMPI HOI '.ODl"CnO,," 01 A .... rGUln IV A POI' I"-A.TlC\;LAT,O,"

II TIlt .,t;""" ........ <If,..,,,, f"ftII_·~IJ ""'" "''''''pIe cioarly Ill ... ,,,,, .. ,be ..... bo ... lly III "<.",ed . ·,,11 _,n>! ...... _ "'I _k .... iudr n.. ,"""U" mol d"",ben bel". .. , ...... 1 <"1.11 oI>taIlI ,be

TIt~ FO_CeOr AMBKlllrO" O'ALlCT'Ul BA .... ,...CU " ...,." .... by m",," ,ymll><"y .nJ ,hon _J ,he -«ro ."''''''"''" The lau ••• hm tI<1adIes i.oeII' dcanly Iok~ an ntrtiCel>Ce and .. luI I, """".nK1ed pos;oe:sse. only .he medJo,: ... co""""" III .he a""",,'e. Bu. 1 COB aIoo 00IKlt0 .. ,he """" .. "wnbcrl •• 011< ol 'M o.bcct. ",occcdllll; from :I foklo,,! 01. ~'nucb' 1'1Ie; ,be po.n. O'hm .PJlCln. .,. I"¥o, f'OIlI~.nd .herd'"", .. "" 4Jrlinr/.at1ooo.

•

• r, .... '.

Here apln. ,lit .... blp.,.. ,lit .n.III"OII ol."S«Ond ~"" ...... _ claim 10 "toII~nn" • caJculo,,,,,, ,lie "",pie ocmsodenuoo olllle ,,""u oIlbe cl<finu. opn;o""'" OfIlhe ",,",uve nllmben .,Il lIeI"n ..... bIe II>< II, ".plIl them ,n 'wo· bu." Iol •• ., .. ~Ol,tJ""" F- :1\ ~ ""' -ay 0(....".. n.... :I'" I'd cd ._ ""'J' 0( Jprad,ol or 0( rokI,nl biId: ,be 'SIdes'. '" other .. onts ol uposj"ll.he ",al numben.

It ;, clea. lha. 0 .. not .. simple po"'. No ohlnl<" In f'OIn. 01 ...... , a' ~ r,,,,,. u. innn". d""_lby "",pee.,,., proJCC'-~ II" ... ~ '" lilt """"" '"y', 'pl""nl- 0 Ii .be r.m,flC:O.ioll poIn ... h ..... ' .. 0 fun<"UonaJ PIpS b • V ~.1 • V~) ~ ~ nolled. II _ d.me...,,,,, appo-ars.bC"rd"o", " ';Ih II,. I,at......" oI.be ,...bIt .. ,

(11' • ..-

Th. lo" ....... ,. by nu lIl<~nll" ... n In .be >trol'",' I",. oo"."uc,ed axioml"· all). The puP'l"C n:~1 numbon mul' be &raiped .. Jp~1 jnl" tilenudl'CS; .hey OIXJUlK lhereb)' a Ih",k_

11 no. proIH, .. 'f'It. _"""'..- qf~. 1"1>0 p.obktn 0( .1It rneasllmnen, 01 .... """ fu,n.!1ia ,n uample ollbe ".,.. .... 0( ,Mst «n ..... I belln'C J all ~.q,ly lrasp ,be PI' 0111" .. po.I1\.1<$ .he ".....,. ol .... -.cu'. lI\iIe:" Kem. '0 he .... ".fed by ' hr "mpk «>n<:ept ol dUlWl«_

If I "" .. "'Ib '0 hi .. I dear idea 01 ,he hand mo'-ernen' I"', croaled " "hod! UIIOIInu '0 "''''I <N Jr I conUII_ ponl"",'ra~ ,be c ... '....,. 01 ''''' ""J' oI'u"""l lfldlCl.et '" fact tho, ,lie adl'" bet_ ''''' Jlr.u!h' ""'" .. "'" ... ilab/<. '" .he _ .... ) at.be dllta""" ........... ,.'0 "",nil "". hot.

An anlloo ....... ;" .... ,._ pUpl'; II onil d .. pb)"I '1I<1f....... ';,'dy atr .... choon- w b<en ......Jc. An illfiruty pIJInl '".01 ..... "" diIem .... f .... Ih< .. __ tcr." aa eallly 110 redlOCed ,0. ftn,lC d ...... a' b) .. JUIU~ U:lnslomui,lOft; a 'p<IIIIUOD' " .. _ ""'" clearly' .. hera. ,be • ..., I'Ot)' qlUCkly oho .... ".

• •

• ,

.' ....

o

..

• 1'_16

0,

The num~" •• nd - ~ ~nnOl be u...J Mi arc n~mc,,,,,,1 magni,udn or d","noc:cs. Thcooo arc ~,po""n'" IMY U'f In""IYIII~ 1/wI, o~'~ "",a,w,..-", O ne could "') l ilol1he "''''''''"' ... bAI by IOn"," , he bdn.·'n·tllc-.... ."Id 1al .. I.' 0""" Ima,un:",.",'; I "'Uj! 0"". Ih. " .... to,," IMl l!'I'" n boMb. by Slumblinl on ,II< ,lKklcrm'R'uon .ehd>, "hen. thre<hokl If bel", ,",<»>cd ,n",I,"" , I>< onal0menl 0( .... ~n(\f .... ' ''' 01 inlU,hon. Tn u~ K anl's I."":' lhe .ub,.." 'm"iI 1.01"" "pofI UstJr' II, f"II,,", ""h hI' C)~ tile upowrc of,,,,, anele 10

chooot bof"""n _ and l'; ,lui , •• [ ... 01 COUll<; """" ""''''''1< Ihan ,be .n' .... , opt- ()( rhe IlCUlc anpe I'Mullt''''.') prtknblo boa" .. o:<MlVUI,

w. 0« here "lui, II is ,ha, IqIIInu .. ,.'0 C1Iln"'~ d ,II< numb<r ,hal .. hlcb ""h","" " ... 11 ' ..... 6"". al'fJl'UAlmallOll and ...... mm.n, <II • ""'1a and ,Iu, ... hIch de.OCb II ......... 1110..",. u",",rMnt.nd ... ord' .... .,._.i>J ,ntUlllOfL

C s.. ... '8#/Il·roIk, . I"W .u(tnor.lwl.

"'. ""'" "n:ad) bocn .bIt 10 ftC oa: IIU.1I1 bow ,lie "'-II ... p"" .... 01 M ·opc .... ,,) roukl be ~I"""'.cd ..... cl .. "n"' ....... 1.'0 SIal<"

" {., • fla .. of qua"m&u,. 1I<>mOp'fU .. "on """ ... ...1 111 ju,'a""",,,,lnplydu of ,mpuJ .. (P. I . .\f lexlon" • .,,), 10 ... heJlhmlIOF1 .... . 0 a oiolle dilpam ('10,"1 Impul .. oil'" ')'II .... ·~

(II) .110 ')'I, ...... ,mmobdoml by chanlll'" .d'ama: poin" no dll'CCl.K1a is pri.';1qod aay ""'"' " 'lIodi Iq.umim .he ,a'"""o,_ of lbe prinaplt of oafficl<al __

We wiD ...... ' III'" "" uamplo .. here lhiese '''1) "_..., rondtm;cd ul1o • ........ i".1nch I- '.''' .......... 1UId llllmobilwlnoll a", conlnnporancoul.

Clift · •• '''0 cu ...... C, and C. (1ft fia. In ,he fim 01 .. 11"'" (.be ......... ') ,oil!. .. illlOul toIid.", no , ... ........d (1'" 'toaK):

<,

\ <.

r_" w I, be.be POln' 01 CQn'''':1 01,110 I"" ~u .. "., , ... In".n' ',ronlMlerod

Ii I 'mobile poi,,!', .11" poinl II oo. h .. I pI)Inl 01 C, no. I I"',nl of C, . leI I, and I, be ,1M: 1"""" '""~rlably linked '0 C, Ind C, '' ' III ... hld! il coilKidts . , .he ' ''I,aOl I. A f .... ;"""nl~ !ale., I, I\l&~ mo."", ,,,. po,nl /' 0" C, Ind 0" C ... berea. I, and I, become di.11oluilhablc, , ... " ~poct" .• I"""ioo, be'''1 .""h Ih., .110 IenJlII L, 0/ ,he arc "', "'Iu. llihal 01/'/, (one could say ,h., lhe (lO,n, 01 con,lCI I dc •• lop' in 111 mohon 1"0 ."1 .......... L, and L, whi<h ...." '0 '1"'''' fo.lh I'rom 11), ~o, lhe imlan" r.'I!'ylh,ol Oppooe! lhe '"" .. ' C. 10 ,be 'rollor' C, ",hid! confillCal'" .be mobihl) , .. ,he lImpic a.. of a hoop C .. hid! roll. alon, a IIr1lshl ~ ... 01 olIO"" (toe 6 .. 1 8~

To rounler 'M cbc:hC of ,he mob .... bed, .... could IlInnp' 10 I'C\'eIK .be ..,... by 10m", 011 .... 1_ 10 1ft ,110 hooJp bod .ad .... doped by ,be 1IIObi .. "~Ilhl line 01. Bu, ,hal WOIIId onoo .... " be In ~ I, ...... 1< 4'" 10 QIIII

of (he """"'" 10 ' ,,,hd,fy' " by pri ...... ", ,he «.lllnl"" of & 'buc' &1 lhe "'paHe of • 'roller'

An1 d;,'YOlJllCII)' br,wcea .n COIrnll"ny '1i.0<I' ""n'~ and • cu"'. lhal ""''''''''''''''" mobil,'Y comrrom_ • do... ",1","", JBlP of I, u.IIIIaIII." C U PI,otal 1""'" WI hodi muOI d ... nbu •• ",loco ..... cqu ... bI)· \\. mlUl IlOl pn, ...... ..,1Ier the hoop 1If!be m.,,'11 bll< for I,,. ......... 1IId0<l lillO !be ...... ,.,. of C DOt '"10 ..... , of 01. " " • pouI' of 0011'""" .bldo ....... I.IIe> lhe ,'*'"" in>p'"r~ by C and 4 by VUPUIIlhcm •• """ ..... ~ _k _ .........

.. , - .

,

muslll>onf,,~ be un<i<I'$Iood AI ore""'lnl'", I ..... I>orr>oFD<1,) .. I I .. nccnA.r) 10 ~b«al' ,he 0)1' , .... , IoNia'" bel ....... lwo pouibiluono 10 d'lIJ 10. point fJf 1I 0.- 10 n .... u .. indf ID 1O/kJ.1IIJ • poiOI on d .. 'roilier'; _ ""UI raol"lCly dlOCOlOrJ .. ny .olld,!icI.,IOfl oll, b)' lind,", lhe 01 ... """, lhllln"",_

1.1" C and 1I .)_"'-a/lj'. II i. he", 'hI' ,"" condillon of .olhn. "'lIhoot slOlin. "blOh .... ra Dul

oquaIly lhe e".nl""" L, aDd L., WI.KS ilL II IS .~'Y 10 1ft lhal ,. ""'~ .. il p(M.iJbIc. Ib,,,,,,,, •• unple tn.noIonn.n" ... 101m"" "' 'M .UuallOfl dcocnbed by fi~u", 19 d ,wo C<I"'" circles Co and C, ' ... run, .. opposn. direct""' .. "' obr.a- "-"'Plio. """"'.).

¥ \

"- ./

" '. \

'_I' In ouch. walfll .... DO "mUo:r' _1oI .... te '" cbu", lllbe Upm5c of. "bol<e',

then: n 1>0 •• ode_ oil ''''nw. of mobdny (,om C, ", .. """, C, or f.om C, IQ .. 'a,4. C, lhlt would II""" 1M n l.hluhmon' ol . pnvikged fUrJCI,,;m.1 mrrnpolldenct hetlO'''''" lhe .~. Jln~f/tllo " ... or the cirdC'i and lhe O\kcr, n ......... tbe caw when .. .., d.lHnfd 10 dl<tJIlIWOh. 'b.ue' frocn I ·roI .... ·' the _ """Id be Wldelllood .'"p" 'dlD, OIl' It<. dctr- 01 ."doc:;I), 011 ,he otbtr and thord'_ .. ""1"'"'''' .lIeU' .. tho acI,n .,all ill. , ........ ". 'A 01 ........ __ W< a", prett'n ....... '., II .. Iomh 01''''0 mobolc J'OI"'" and lherrlon: 01 lwo or Orame'. "",n 'nlMyell .... hid! ",,"nl for,h ~[jnualJy r.om I" .. hid!

lid"", (he ,)'mm«O\ZU\I 0,,""'''011. I O«It>«l eon<.Ie'mnttl 10 be, ... Lorn bet ........ ,01'0 prea. ... ia. opaca. ~w il .rpcan .. (he poinl ,hal ,,'''illa (heoc <r-u. .... "", ..... .,. "'!"'iY(>(:a1 ........ nl bet_ , ... 0 p!*ibk Iahclf Ind

" ru .... """. be ......... "'"''''''" .mho ..... 'y, ..... I pot"( 01' _'IICt. I has ., w, round. plll« wonby or ,L.he ~,"p/~ .. (ICC 1I.1).

Ii .nd /, we", merely mobtle point! inlh",," , loc'e"'ll"!IoCOII' InlMj'clK 01 Oramc: /, .. ,he 11"4'_ ,",wok _ .. hid! p<o"i:la ,he p""',!qod .OaIO&Y from .. hi<:b .boy ....... ..., I, .. '0 I. "'ha, I, " .0 '" ...... rpn -r"~·"".'-• _ ... I. ......... , • ..-t 'C'nL '" ... ,_ • • ... ,.dod_ ..-..... ....,-. n. ...... k .... ' "od ~,,~ • .,.,.. .... '"Ii -. " .. q_"o,,",'" ", ....... ,_ 01 "",pIoo _, --.10<>4 no _.' ... pIo • \lor. ~iIII 0.-', ~ T100 ........... "' ... '"

ii' .... lLIni af U,. t , .. ,' ..... "-' _, ,1:0 "l""'bOO •• .1 ..... ' <If

""" ..... 1:0 ........ Ii TI:o ....... 01 "'- on. " ... ~ 0 .. r .. n.".oo:' '""" Of'" 1'_1 '::: ' ...... DH". ,,,.,",,.~.-s,.."''''-'''I' • .0. ...... ·boO_ ..... · 'w,af","_h" of __ '" ,', ).0.l000' tInooor). _ ., 1 a' T """,,"" __ ,,~', CIS..ll, C_ ... .... I.«<w<>. ",

,s,. .. \'m.a,'''''. 100 ., .~ ..... "'I.n_.~ ... _tu', ~ iii. ~ .. /of, V'~p.)6.0

, u. ... _ !.-. .. H.,_ .. .".!Iot '." . .11. I~ "" lD-l' ·500 1( ... "_., '"", ...... " .. Q w .... "' ;'I'UJtJL."-_ <MIo ......... IOo'o«tClw ' ,,_I,,"~ " ·Iw',.,. ... , , n..:.. .. ,.., " """ _ r 111

" """. p. " ~ J. ",pM. , __ ,. """"'""" n ... "'-"'''' ". .. ,.._". ... A,S, II • ...,. II<o<w V",I [) ... ~"" ... ,UI • .• """ -,. jJ

"boI" I. oolbonl., .. , ... ,1< "t\<JoooI" _ ....... ~""_._ ..... _"' __ .. ,I' ' ..... "'.010_ I ek' n " JI. ", ...... ' " , OM-I:/n., Ill. "n.: .. _ ....... _101 s.. ..... J_LAta-l. L.- _ _ " ..... ,.1=- .... _h', .. _ .......... , " h, .. 11106'( ....... I ...... 'h'M. H'DG:>. ',=, .. ""_Soo iIIoL.pp.,O:OJ "-" I:. ... "",*hl .,.. Ii""':' 'el l __ .......... .." I I .. , •• " .. ,,0< I -'b ," .... , ofrll, l'lolTi""S " ...... '" "~.~i!ic17. ~ .. 11400", 'S'l ....... _00) ,. -. _. "' .... ,,,.,.., Jnr. ~,,,,,,,,,,,,,,II"'.,,,,.

~ .. , ...... r" rll'"Il:. .. ,_""'_"' .... ~.. • ........ ot""" .... lI~. '::::::~ __ ............ of ........ >!' ....... _ ... "' ... , 'L" •• ",-,"",,_1:0 ... .,., ... '''"'._,_. I TIooo" ..... __ oI"""'.-.... .

100 TIlE r ORCI! Of AM81UUm O'''I,EmCAL BALM<CEll

"n..~" "_,,,},.f M""",'''_'_oo5'' "". U!Hn."'-';"",,.,., ...... (J.n;-.,.j. '9"1." 111 r "W) 50; ... ' .... 'W"', "" ... de .. " _ ",," . , ... ",.. ( tw.:. "" ....... "'<!. ,10. ...... 0( "'" ....... ..., ' ... ' "~ .. r..... .. _'ill __ Ill, ' ... ..... of ..... _ . ... "," _ .... r ...... iI_ boo. , ...... .- .101 ,1\0, • ~ .......... cakd ""'" ...., ,",,\«60. II 010<0 boIlI. :-; .. _,."" .. .... 1Ik M ......... M,ooI 00, ...... !<O .. """ " ......... !o ""_. ookowo, of ,ho 101.,.) ......... 01 "".f< __ .. , ... , ,10. ........ 10 ,I>< .. t: "' of , .. ,_ ... , _ .. ., ....... Iound,.

.. TlIoo " .. oll_ '0 .... ~ <loc. of ,100 " I'< of \\_', .................... _ ......" ..... , ..... , ........... ,-...... ---, .. _". "f WJ s . • P' • JIno ..... o.,". __ tIw _l'>Wf,oIooo(1 ..... "'IIO.I~Id. M_ U. V.,.., ("" .... r SuI< U"' ...... , oft;.. V ... "'- 'fU l"I. .. -....'a..t.. p. ,a -' 'oJ ... )0« ... . .......tab!< ... 1yoio of ''''''' 10',"""", .. _ .. boo. .... «IIIf1l>Od ~_ <lio<t." ........ doot> ......... " _..oJ m. .. _ "ho:h io '.""""", MIooI". of''''''''P' .... betq....., 10'"';''''' " 0. "'I .... tono. ... It _, ~dIt ........... ."p'_ Nlwl .... ,%" I~n " _ ... _ pp. "1---" .. , ... " "" "'Iuolilor1_ pob.I 01 "'" tonto oI'd,,,,,,,,,,, __ 0, ,ho ''''''', "~""""';"" "" _.11_ .. ScIo.!I"", 1lIt ....... o/tIw _01, s... Yod, 1%1).

" GR ASS;>'I ANN"S CA PTU RE 01' l ' II E EXTl!r-.SION G"om.try and D,.I.."ic

, INTRODUCTION

We <>Ifer • ccmmen,ary OIl ,he fi", .,h';<m (1844) u( GrJlnmaM', T1wtJry of lite £xlt>ul<>ft.' lIta' ;n mind 'ha' ;11 ;n"oo<l(:' ''''' ... on link admir:l.lion f,om ,he ma,hcm.,iria ... u( GTtiSJrlann'. ,ime: i, oon'I'as,«l Ihotply " 'i'h ,he ,,"10'11', hegemony <>f ,he ax,oma'ic "rle al><! h;' puhl;,;"', . d.'iocd G'anmann 10 Ie .... ;, On' of ,he sorond <:d"ion, In hi< ",.<ly u( ,he dc>'.lopmen, u( ml lhomau"" in ,he nillClttnlh ""nlury, F, Kk:in ","",ni"", hun I . I plne'.r of p, .... bul d .... 110' mil hi' " 'ON, 'n 'peliXIn, "home or h .... udenl ... a ...ct of' crJlnko! G ...... mann'. ml,hema,ical d"'''''''erieo were only really unde",<><><1 ,owards 1880 1100 .pmod by Peono and lIu"'I;_ FonJ~ and i, 'oot un,il 1983 fof "erni" oo'''ompo'''ry m .. homo'''''''''' (R" ... and h .... uden,,) '0 do IU'''''' '0 lhe "mpl lO'd' prooUCt l.ymm. ,n",,1 ... ·lIh ,h. ' . ... ,oar prod ..... ) and '" lind ito possible WieI in ...,mbma,one ..

I, .. ck .tT rna, Grasllmann had • phU<KOphicJl background and adopted ,he ,dea u( ,he Gennan ideali". ,b., saonoe can by DO m ... n. he reduced 10 ,he adopllon of ope,a"O""Uy f.,ule ruI .. ~nd '0 ,he proo ..... i<m of limply dedua:<J tru,1>£. A ma,hema'lCal ,''x, should MI ... pu' ,he reade, in I po."tion '0 ha ... an '0.'."'''''' rOM,,1<hll, Tbil o,.,rvlew;' no'" . ,,, .. ral pen"",,""" ,h. , ,kim. ""e, lhe del.il" nube, i, feeds on ,he .pe<:ifici,y u( ,he <ingubtr '0 ",..,h ,he pia"" wben: ,he 'nI";"''' and ,be dis<;J";"" arc b<>rn .ynclm>n'miily, Tbi, ,)'pe of denuond idenlifies Gr .. wno"n wi,b ,he philooophict <>f .... 'ure ScheI~II' .ad Hegd ... j«.:'<:d ,he fini,h<:d "»UlOtion u( <>bp' ... <1 ... bject and ".II,ed 10 C<H'Ia:, ... In ->tDbfrc""" k no .... lruJt< u( ,he "",n,_,"_,he-world "Iu<:h ""-•.•• would nO' he o<>I.hin~ 11 is G lUUlIonn'. "",,",ula • ..,h .. \letllen, II> h .. ., un<l<1".. <><> crOMlOd by IUs undel!'anding.

Spaoo ,hul " W"al!" a .... 'ble undenlandHlI and lhe unde" ... ndi~, .. an ,nv;.ible IJIIOOO.' Tbc po,"1 ;. ,<> darify ,be l,nk ~"""'n IhI; fI<""<$I of ioo",dUl_ lion oflhe k"""inl .ubject and lhe pl"OCCU of 'rli<:uI~IIOf\ of lho form. u( ,he "",,,,n& of <pKC. II will come u no ,u~. 'hen, '" ..,. • d'.Ioc!", u( the 101etal Ind ,he ,,,,n<moe unfold ;n Gra ...... nll'. 4Xi<'~,tJ mwllip/frtll/ofo (lhe

'"

'" I.mo"," ..,ticommutatl~e prod"",~ Foll,,",n& on from am..,.., O""''''ann ""'" the full po"~r of th. dIB""m ~ ~. helin, ...... plitude (..., eh.pta " I Ind sh""" Ih'" 1/0(0 po.I,,"I of".... J ''''''Mien <~II. ~""'hn. Ho can Ih", fn:c h'm~Jf from ,hoc: OOns'ramt.< of the Il"I".1 (,,'be .. dime"o.i,,!l.S an: paral)'OCd ..w lake account only of ,he inlc"al·UJid ..... be,,,,,,,," things). Th" " .... hy he manqes to V~,P a re"'"'" ,"", ~c""rales whl' he can, the "."c",,..,, lonruo_ lion. of ordcr ... witllou, ,he brlp of any l<Omtloc · .... Ii.m·.

Thi. d,al<tliClOI 'gen ... I;",,' [E. :n.gung) undoubtedly comtit"'" Gnu." mann', m(><l Inlerntiol <XInlribullon 10 1Ift'tI'\<,,,,, phiJooop!ly. H. ohow. thai oncnl11ioo .. on " .... banllin, do",,,,, ..-hieh don "'" oink in,<> lbe ilKkfinitc .1Id "'h",b an reactlvol. i ... 1f In"", times by I>rc:akinl IJld I'l'gra'l"na ,t.. allu",' • • mtnll~nl Iha. " pooll«l: the capt .... of Ih. ""0""00 openl up by Am of all confcrrin, I mobility <IfI the point in onlcr 10 gcncnnc. dln,.n,;""; " cOhlm""" wil h tbe in,'cn"on 0( lbe. uternal ",ooU<:!. 1II!u~h perm;1O an .!lOOndinl! JOU""')' Ihroulh lbe. dimcn;;o"lI; il 6n .. """ by ",halalll:lnR IhlJ "'"' ""ilh a symmtlric~l proces$ ",1a1l.S 10 ~n 'impilOd' prodllCl.

2 HOW DO YOU BR I'IG f:XT~N[)tO SPA(:2 BAC K TO I.IFF'

Morku -PonlY IIInler: 'AD DOlology thaI anJid! mcnlioo!Jlg nature .hut< 1I .. lf """Y ,n thoe i"""'po"",1 and, fu, Ihi. very ""$On. olf ... a ramasocil image 0(

mankmd: or tbe. mmd, of hi'lory:' tl e could h.", add",...,. the.,.me .... minA 1o aU ph,losophy Iud perhlp< 10 all Kie"",,!) ... hich claim, 1o kno ... only Ih ..

eJlle~<kd s~c:e. 11 ..... y. ai'''' op"""'le .... Ih .. =Cpladc conl' ;ning. wi1I~.nilly. ponlClll";me. ,ndmd ... led by Ik$c;oru. aDd N"",,<>n·. mons'", ... "n"".poin!.

Whal '" IdI aI.!,.., in 'his 110 ... "'hi<:h " f,,",ver """'og OUI 0( il.self . ... ho, """.,".0( Ih .. pe .... , ."''''''' In ."hausoon and ,mk~n;leneu? The ·part",ulari. I.e.'. suhJupted then lbond<>r>ed by the .bool",. from ""lueh tbe.y Ire f"", , « <klachod. Clon only enler I re,uctOU, oonl .. 1 for pla<"n. "Spaced' .• ".nded spa..: II only I final prool>Ct. a '",o'pareol .ggJq;Ile 0( abo<tl"ldy ... I ...... a~ rIlIo"""ly CII"ent, deiorly linked par". No en"dopmcnl ",. Irticulal10n is IICCOrded 10 ~hc-o< emply ,heU .. They ""lIide "'ilh one anolher 'mpat~lly: utcnded space .. UId«d '1"" oboolole bein, outoHle 'Isclf "'h",h " also pu,dy I nd .,mply urun'orrupled .• perpclua!ly other IInng "'hoch If .den'"",I,o ilsclf'.' For lheoe bloodkM boinp, i"""",wton' wilh IlKnr """""'pt pcrmane~1 ex<"""",nca. . 1 .... ys orphoru oI!he F"IU'" .. hich posilJ lhern. kno ... only lho «uelty 0( lhe law. 0( """,,<1 . nd flIrily. All _ilio"" f<)u' into peIlIlO"<' art<! il II .'lcbolhng"1 IK~I "",nl 10 ha,,, .... n thai lbe homogen""", o"e. iorily, IUPJIIISCdl) ensunng 11>0 peaceful coexiste""" of ihlOp, ou«umh< '0 dapet>,on .rt<! e .. cerbol .. ""n" .. tittle.. &I~ 'pa<e. OJ lhe .... fled fortll oI . hiltJ:! llhn .. ,"",,"Ic. metel) .. " ... 10 IU'laio a treachefOWl 11"0' belll"tt1l dead ""Iu= A .....,.

morco,·.T w"hout " .. ~e: all pl_ are i>a1kolly IlIe """<' DI.ndcd 'P''';'' ,. ino.pable 0( k""pinl a pmn" .. , lhe poss>bk< ,bat " hand.

.... 1 l-hamc:le<-.ly .'" only poujl>m,w~ lat," QJ p<>>.Ubilitl ... , ""illloul hOpe of

'" being mJeemed hy impendiollKtualizat'o" . for tltese.", ""Iy Kluahled poo .. -",~tic-o - I~kcn a. poosibilitiel. Dt.ndcd 'pa"" ""eo itself "n ...... "·edly becau,," ,I i. 0\'.'. finile

110"''''.., •. Plato ...... lhat Ih. rcceplack: is a parl or lbe in,dhlJll>Ic. but ,n a very obscure wa)". Sp&e< is thaI "'ilhin ",·hi.;:h '""""'thin. io !:rin. hitched and Ibis i. 001 necessa.ily chaotIC. If he ,,-i.he, 10 try with lhe poot '5",,<% was splend id!' II>< philosopher mull remain •• 11", .... lpo>l> 01 tl>< ~re. ocupt full ..... lbe ,·enicalil)· of Beina. tbe. .pringiol forth 01 dimen,iooo: be m",,1 let hlmsclf be haunted by .pace: II>< ""p'"'" of,h<: .. Iension ....,,,Id nol he .. Iistied ... itb·. kno""IedF ... hich. like l!un 01 ord' nary gcomctl)'. i. t;Ompietely .t.macl and deprived of life'" Thi. o. dinal)' SCOmt,t) conlcnl. "",If "jlh memy "",",urinllbe. IIi".""", bet~n .. «nded 6 ... _ .. 1>1: ....... ",hoI" ""'- so,)" is 1o apprehend lhe p<>rilil"")' of 'M """, D." of i .... 1/. 0( th .. ptrmlnent act of ""UI<>I>rOOllClion. "llh""1 idolinn~ il and ,,",loom f01"F111nllhlllhe l rue~

ti, .. i. lh'I, ocmrdin& 10 Hegel "'hi<h """""brr. IN"" pot/Ild Thi. posil"', i. no longer lhe hrad'I',,", iMine""" of a petllion of e.i".n<%. and . rue 1""1IICtry must gra<1' 'he ;"" I nl "'here .""<% finally qu,v ... " ,lh the VI"ualil;" Ibal ,nhlb,l ,I and ,""lOS u, 10 e.pe.icnct d ime .... "" ill the in,.cnb"" or an "'ICtIlahOn. It lead. u. by ,h. hand .... 10 ."""k 10 ,d ...... the mOlioo Iha! al 0""" scpar.l .. and 1mb and to be .bk 10 ",pIUn: in 3 ,;ml* rra~nt lhe d"..,aion .nd continuity 01 a 1I"'u",. 1b" gt"Orntlty Ihu, demon>!"I" i\> abtlny 10 ",. tI" ... ~ a proollCl,,,,ly ,hal i, ""'e' "lIn~ui'hod in 'Ii proou<t.nd re,·oal. lhat lhe ... me: mind is II ... urk in lhe produclio<u 0( nalu'" and ,he (frat/o ... of li berty:

Tbis is whal m.kei Grawn""o', lheory 0( lhe "".n,ion m" .. "'able. It "'~"IS • gt"nuine pedagogy of lhe form. of tbe. IVRSP'"& 0( I",,<e. and for the apprenli<% geomele. ,,110 'can Ihink fredy, Ihe tU$Jrnllat101l of knowledF then b<i:om<1 I true rcprodoo,on pro"ded 1><;. In,u.,ed ,n,o the ,ubject ;0 a "'scientific" [~;>", ..... luifllj;"'] man1lC1"" L,ke phtJo.aphy. geomelry oI!"efS a '>ci<:nlific' th.ntCler if il lead. the "",dc, 10 tht necesstly of admiltin. each Ilngular trulh (ded""uon) ""d .1>01"1: .11 if ,I pl. oes !um in a .iluati<m " .""", he an hal"l: an ·o, .. ,,·t""'·; wha' G raum.nn calls I 'puning in "rd.,...· This i, ",hal di'lI"guisn.,. Gm.,ma nn·, (and <%,,",n 0( hi< conlemporaries) ph'losophi • cal ",in>« from oU .... .. ht<h " mo", <%atred on verification and predi<otion. Graumann demands an .'pllClt articulation btl,,· ... operalion o.nd ,n,uilion. This den,and is hy no means 10 be confused ... ·ilh a subsidiary ',upplcmenl 0(

ooul". bul It is cloa.ly lrumpeled I. lhal ... hich aUo .. ·• ,.;i<m:c 10 lay da"" to being ,)·"o"'"'tio. and 10 «1",,111)' "'ilh philOSOph)"

Malhemal"",1 .... , hud procud. f."", II>< Slmpie" 10 lhe moot compOO;ilo 00"""1>10 .nd lhereby obu,ns, by lhe liaIson of lhe P;IfI"'UW, ne .... nd RlO'" gen.nd OO .... pl .. Wb'~i in plulOliOPhy lho "obol o,·~,,·ie .... [di~ U.hnokl" Wba Ju.

Gwo:~] predumina ... . ... in ma,horno'''' " i. Iht p"' ...... 'pn from p;t.r1icu • lar to I"'rticuJar ,h.1 predomi""t ... and •• ob compleled "" •• Iopeneo'

". OKASSJoLlNS'S CAI'IUllo ()t T'III EnfJ<SION

rorm. in the .. 1IoIe ""'Y' bnl< '" It..: rou.:...,"1 po.,. si"on. Thll dUfeoeoce ,n ... bod .. pmoenl in the ~ 10< In phil""~by .he PIII'Y d the Wb ..... he I'00I. puUCubnly" dod"""'- by ""fUfUI. If! ""uhemau<:s plll~rHY " ., 1110 tool. ... bo:._ .he ...... is ., tIw rad.. 1\ is tbe ''''lCIpIIlfti .... _

0II ...... 1Im:.n: dcmontI'ra"" ..... -hn. II .. "'" """"bIo 10 t...,...., .he b<JuIn,n •• 'belher lhen: ...... noll p«al.", Iheormo..O .. I\i(b liter Iud. do_'nlllo". by",bicll .""denly and ullUpoctcdly lilt dmIonstrabk '1U1" .. fuuoll, Im.cd .~ _h oIep I\&";n, r ..... ""n&ln '''110 ~n follo.,.ed blindly""" by cha ...... Such. detnonsIra'Km .. "'''''''PI .nl' ..... ' .. nslacr '011 from the poin! 01' .""" of nlOur. b,n II .. 1\0\ &<:IOnn!k (,.;,. ", ..... .Iug1lklt); ,he ICCUnd ""IIU"' ....... I io mlni1lJ' ,he o""""~ I.

T"l. ·ove",,"'" it 1>0. the dilenaolo', distant wnltf1lpialion. 111al .. part In the aclioo. II I. an h,ulk<' .... 1 ;~IMil;""''' in ,"" ~_ m.ended in lhe plulosophy or n;o\ure II 1ralllPOOU us to IlIal P"'iIep z ........ ""'" ,nlUII"", aDd dlJCU ..... _ "1 bcoome l 0011«1 ;nlO • b>'1IJI unny. II i, "enher Q ".n...; IlOl' G ,....triort. il iI w'n ..... pon._ ..,th ... "", ;, pups. II uk .. oa(:h bnn. II ,II "''-'' Ind. .. "bo'll ....... "poIIIII.1 ;nlo ,k",,,,, .. or pbaa. 01 ,n a _lei' iIOd 01 ~aht)'.

Th,. a:icnulic pi"""," \2Q by 110 means be reduad 10 ,be' adopuon 01. pao:k .. 01...... 110 ..... ,1eI' bow ol'"'l'iouII), fer"'" Thank,to it.. the fader nllI$t be toIpooble: of m:ocruzml- '0 the IIIOIk of ~~,..,..tiooI 01 tbe ron.... .......... boonoIop>w to ,1M: ..... ..,. 01 "" 0 .... prop .... TItt .. d,.. ..... II( I'" u ........... JtlB _ ....... ..." oN!'''' l_t..IKt ",'" - ~jm. ;, fuJI 11ft/( /Ito .'" IltJi.w...... ."'" ",,,, l.Jto..,lrq., ".", ... bj#'r', Ono doa not I.'OIlftrllCl ,pan: by In mbIiJI, fo" ... lile the p n ol. M""",,,,, "." (0"" ... o~1d lhen •• k •• he artoculatwm for I1Utcd~ Morco.' •• , i, i< nO' • m.ttor ol co ... " "",,". ,pace. bul of leninj Ollfttlr be bewi'ched by a rhythm: .h., .. hieb 1no .. Ir><! and "~'"'' homo,.,lell;" ",rgal wlIh .enolon .. Much more .""" HamH,,,"·, 'Order in I'r~'," Gn ... mlnn', toIplun: .... the .~'C1IAon i, IUl ind ... d""I'on i" PfOI'C»ICI<I, ,"", oI.he n:lder, .. h" <:an Dl> ..... n .. 'J,,,,,,,ly the lui",,"',", .... ,h', he bcC'o""", ...:noltomallO ,ke.ehn" nul linlUlor """ion. of lpace in ~ '01""",," .. 8m RMNnry.o be 1'"111:,",«.1 b, the .h)thm of ... m' .... It ;,; '" ,m. .. n,. ,III., .he ,,"plu~ 01 the ... , ......... p'asrmivdy "",1 ... he ,,,,nd ,I< ""-..

p, -n"","" [A_",l.ppnn folTllJD '0 ,he domai" 011"'''' .. klKX:.

above.H '" ,he don .. "" of _,hema,,,,," H" .. c ..... ..,,""". il. 01 .. ""~ Able: ,,, 6nd IDJ new idea I ... - 'f cona:o..od ,n the ..p, .... , _ ,Ito< loot .hlt ""'" '" .,... ... la, devdopmen, ."", Iead~ '0 the new "u.b. but rom",,,,n, , ... ,""" ,""', hive "'" y<1 be ....... ell'<' d, ., ,,10< til .. ~I_ , ... , ,Ito< p<eIC"'nn .... ' can .. , ,Ito< bcl'nn,". only bc """"" .. [""Mid ' MrtfwItJ 1

Tlul. i •• h1 ocitnttlk """","t"ion i. """Hllly I proao-;"o 0( , ..... ... ies of de.t!opmen,,', 0 ... ol whICh load. ,n """""'I1I0no. [.um <HOr rru,h

'" '"~ ,Ito ttU1 and ........ ,he CO<IInI pt"""" ."hile tbe oIher ... """ r'" "o<dJ Irsot'.nd cImnnuta IheJor-.

It ""./oq ....... lito< ""'" ,n awbmwrUCI. lod E...tod IwmdI" pnMdcd the modd. tba, of the '''''' oona 01 de,~lopo .. n .. " ............ to val ... only .be ...... ba, ......... ho """ten. proprr, " 'Ilile the l,."..bIe: of &00 .... .he OIher bc1....,." lhe b"", itt kII.o tho, Ie,. Onl" IJthoouab :otraOJI'IIII:I' and 1M ,,11'I0Il of IILitt ...",. of ..... "l'Iopm<"" ... """,pIetr, " " tIO".,itt.

"'" 11I11'D"'ibi< '0 make ,ho <;WCn'1CW .",bIe:., ... d, ..... of ,he /a.1eI' 10 whOClo-c' slro1l1d 6N! 01.11 kno ... lot",,,",,,, and 10 pul " in. ""0"~ ,. taO proJJaII I~ly and 0( i".'I. TO) do ' h", il ;1 .. ,he. '"' .. ,10' 1M tUdt. '0 bc put in lhe "3'C in ... h"", .be d",,",,, ..... of I1u.h oualn to bc in .he nIOJt lavourable c .. e."

l~ke ;"'ollct:,oW ,n,ullioo. 'M KXOtific: me.hod dcmllldod by Grawmnn i"IrOO...,.. I kllO .. 'Itd .. tha. don DOl leave.be ,ub.l"Ct;obj«t duali,m In'act. bu •• on ,lot conl .. ry, ","'ura '0 crea'c .he ob;!ct ,.self. 10 IM<f1'M fUM .. """'· tal ....... ,,'y of prod"'" and prodllCl"'lly. lhe ~ precise ."'" mode oIanirula· 1t0ll ,ba. wwn' .... ...,. ..... ....,." , ....... lhe Ito<nn ,bll ....... ",y itt concri.'Cd ....... ..~IIIOI be .. "pt ... d ,bat ,he not..- of d,_ lrod ..... 'lIall00 play suc:h I nvctIl ... III lhe ""plu" 0( the . ............ Tloq ."'-." 5IU"TOp1itiwsly threat .... ,be ........ Io'y of the ...... , "C, faced .. nb hOI 01:; L no.,. """"", bc _",,,,..od III the IIRIC WI)' tha, ..... can .... ke ...".,,, 01, .b,"&- Thq""UCS' tbc n,,''''''''' of an. u~i.'" h!,,,,,Icd., 0( the btt .... 'n.tJt.e.. .... icI. ... 1ndt .. 11(1', ~_, "",h"'110

Oimtnsion itt "';,Itn I poOpLl t1 110. In I"n"".o of lhe fi"t<_ 00,..:1. II . ff"eru It ~ lhe object and its IIIOIk 01 pNduction, r <:an, '" cou"",,,- kn""" .he n __ u,c",.,," 01 tIL .. o. li>atlh,n8 hy ",,","n8 I ""rut;n ",'U .. (.1"",)'1 .... Ia"'" """j, m<Winl olo"l ao (olready ehoocnj a.bi,ra.y ,,~ndard·unll. Thi. co""",,;"n . un,l ;. gl'p1W (Mia lbe o;>bJCC'. bu. i, J_ ro/)j fftrol"" _ ;n ill proa:n of kno..-Icd., 0.- ptOOuruon, A _poio" 01 ,he dimen<i<>n .... ould DOt

Ito< A'JtDod by 00 .';';"1. q .... ""'.,,..., pOI~1 01,_ tire 6,,,,........, it DOl ooly "11U",bn', i ... . doJne.. run •• ha, had '0 be: d,mb«:i. I. is part 01. IhoqILt "'-en ..... bcte>pKe " ma.ured by lhe .......... , ... further unfold, ... "'a pbk. _ r ........ a ~lrc( at allo .. "" tbc poIan/.uon oil ""'1JD"1Ide. lbrooab dimm· ....... the inler>at "'hooIL ...... merc:I)' the rQldtaal "I*'" """ .. , ..... bct .... 0 two Ib.n ... bee"",oa dt.IiJi<d .. ,lilt .=>ions ...... '.. I. tlldI' ... "" """" .. cond.r""" tor .he borth of. Wuc1ure. Tbc d~ """",",,,".' ' 'o.n WX>."""'" de., ... of 1Ico""" tak .. Ilk -...,,"""". of. "'.,.. • or bc<:o<n, ....

Nor docs ,be onmUt""" <pan: 'ho d<;od ""m>wn; by .11 .... 1 ....... ·_ by lOde· "'nh _ ~op' and _ ·hOIl""" , i, ..,..".Is .hat ,he JU~""""""'" ""'" ....... made: ,n. ",,"a,n o.der In demon.'"",,",_ pn:fcm>t'O, II • • "" .... I penna""nl 1"'_' oIJpllIt;n. a, ,I>< 00 ... of wlul. hit> bcr:n """"",*, ,I CIIn react,,,,tc ,~f ,,'i,hou\ bcin. "~a"nro; >, ClIft lhem" .. be an rn'opal poor! of _ d,.k<:local I"",ollon.

". D...w,i< .. "'= 10 be uDdenlood ... ~be IIItelbJCflCO lb. wpanlln and. br

,"'}' ... ~ of this ""P'Illltioft., 'olrod ...... an order dllO ,hUIII.1Id ""'pole> •

form 0<1 lbem~" d'"MIC d_ not abal,'" doff, ... -, ,,;UW'IIInal<'S and 00."' .. IL G ....... "'""" mull be I,nked ,",'lIh the ClpKlly 10 d<mOft$' lllte.he <WUOly of .....,.th'~' . In order 10 ,hoo. up , .... """~n&l of .""ex. I mUSt "'~P " .... IntI' ..... '" 10 ~mplifica1Jon I<n<·~j"'wl. Space no lOdge' 'xh~USll ilself po .. ron& 1>11111>< ~" .. ""i",-oTher, it "'pin, conl10' or ;, .. U by , .. Iooly and ot>Cnlalio1l, and ,ho: boner do DO' ""rush 1ft • ru ..... pon<iort Sp<a<>: ,,"II IhlOl br &nlpe<! ,lfal.ln oK U UIS'ence.. 10 . be .. _ of'"""",," be)'Ond _If. aU10!N0rul<$,..

lion • ...! """,,'",,, lo0nc0etrUl.h .. n,~lJOft·." 'n.be same .,ay thaI Schdl"'l ..... in our UDder$I.ond,nl n ,n,',libI< ... ""~ in "'1l"I'1 ntf<I of ..,. ... l otl''' .. G."'sm.Jon ,," , dJ ,n protol"". ow- ,"veibie ~ ud In olfact''', f'''''' 11 mcreum&!r ,harp noetIC po-opul<Ions IlulL allow VlSlbIr 'I'M'O to ral'P"'pnllC mdf. DWcaocalfCncratlO<l .. """"'" ",,1Ufi<d ,,,,Ib .he po$I(oOIl> ,h., II h~. ju., """Iuered Ind will 11", .. y. oII"er .be key. rot "'''''''1''101 ,he ,~;. aa:ompanico:

(~) The " ' Y ""';1;"" ot the fi.ed poonl uJl...!" 10 ,he m<)/),ie PO'n!. i, ,J ,he ",nerlllon 01. d,men"",n.

! hi TIM: lII,u,lIon oflh .. """" ... prodkl<;!"'" of rum,,,,,,,,, 1,,'iIa' we ,,111 ..,.. aU 'pOSIll~ dilata'IOns") m ... ' be 011" .... by II< .)"mmetncal j"",., fnoep.· II .. , dibl.1l1ons"), bu, lbr ....... of,heoo <hIata,,,,,,, by ~ soop. U"111""" lovoh-n a rotatioa ia • plan£. W~ can ..,., on !hi< thr wry pnnapk of

........ 'io ... Tha, .,bodo alU """ Ionn IM:re. tha, .,b.dIloo", .bou' the Iddou ... • y ....... ......... ,ed .,·"h 'M ......... ...., 01, domrn ........ <Uri up • plU", lha, cocapcs rorrnal dooure. It .. pn:cisdy 1M ""'''"h 01 ,he """,,"ioo 10 ". , .d in mak,n,...,oo usc oI,h" in,ulfoao""y ar>d 10 aUude: ,u,"'pliliously 10 a MllOn Ihal f",maio:ll"ion hod ""slcclod Ian.n. ' . "on) u~ 'hot ho"-~'· ... makCl i! ~"bIe 10 defi"" the ·~ .. ~mal pro--~...:t' w. ,.-m lee 1>0 ... llu. ronn fKil".'~. the Jeap from "ne dimen""", ,0 ."",her. 110< all ........ ba, I~ ai,-etI .-., II> 1M "'""- .. how d'!lll/ll$ion$ IN <~ .ed inddlrutely.

lei h .. ~ Ilia, Gr.lWIIaDn", dlaloclocaJ Fru ... a>fIICO in 'hIS inluilion """II(!<! "iDdcfjni,ely" can boo o/I"lIOt by aI'IIn, 0<11 , "duotbdonl' ro.m tht · ..... on .. ",,' or 'implled' prOOIlCl(tf...v-__ '" I'roohoI.t)"

) ,UTKL LA"!"!. ANI> OEI'E KAfE I O . MAL SCI ENCl'S A,'I> Kt:A L SCI r .. cps

Gral-lmlnn', ,nUOO""'ion ..... 'poM:S '0 ·(ledU(>l·(abl.i' .... ) , he a)_pi 01 ""'" .... ,Iorma,1Ci. , hmlhal 01 A....t.A ... qUtIw.

II ;, _ • q .... ,IOfI. Il0>0 .. ,.,. 01 (ledllClo,swemento rrom, body ol .. _ bul 01 pII'''''' ,a'o opn:I;"'''' • doIo;.pI" .. 01 "'-ramall lha, ....... , .. do,,_ ol'''ocula,,,,,, UI'JIK1I'.nd '" 1M m,nd.nd .ilal can be .ppI .... '0 di.olottics:

"" lha, 01 ,10< dJscn!Ie and dte COIIUlIIIOIa and .ba. ol ,be IO~""'" and 'M u,"""~ e ......... "a dlSlla.,'!J)wc (UI ,be rna ...... 'pp<owd $loa ......... 011.) beI ........ he 'fta!" _,,,,,,,.nd ,Ite "forma/" ""no .. AIII' ... ltItI<md_1I Gnce

If; lhe n:$IlI' 01. C<lnfrotl'""'" """h 'M beutJ ,"p'CkAled .nd produced I» Il>oulht. bu', f...- tlte lirst. the bela. ",p.....,nted .... 11 by 1I .. 1f . ·Ite",u, fe. the toron4, il is p<>t"t(! by ,hou,he

Tbuugbt , ... 0lS only on rel:ilion 10' being ,ilal ratelo,( ["t<"""""'lrill} .nd lhal" "'1'tamled by it. For , be ruI_lhi, be,,,,,, inckpr, doni .nd e .... 11 I» "KIi" beyond '''''''PI, But lor ,he IorttW oe>enteS. i, " JIOiI,od 1»-lbou"" .. hodo ,\kif I10W faces. io 'U " .,,, .. ' .... 0. a _'" act 01 '''''''chI. If. _. lite 'Mb ruclI II" II "'"' "" lhe c:oaconbna: ol lbou&b' .",h !xi .... b lite IOrmaI KK" " 11 ........ 0 pAf11CUbr OIl • ,">o...,,,boa 01 ,be """"";IoCI cl tl>otrpl WIth lite betn, K' doa-n by ,be 61'11 act. lba. II 10 .. y. on (II<: a)lIC:OI"d ...... otlhe '.0 acII 01 tbow,hL Th .... in ,he formal ooenea.. dcnIonst .... ,ion doca "'" lea ... lhe ~ 01 Ibo"",,"nd rem.,nl PI" ~)Y in lhe oombiDl.lion oI.,nou, a<:U oIl""wgllt To do Ihi" ,he rorm.) lit.,,,,",,, mini "01 .. , 001 ' ,om .. iolI1$ a. lhe ",.1 ..;.""" do; bu, , he defin,'''''''''''' .. bat I""" lhe" bo!jU."

TIte probktnallC clutflClrr otlhe conlron ... '''''' induaod by,he '"",r .. e,",," .. obvJ<>ta!,. t -or ,be 10nnar Ie"'"",",- 'hr "'" is ""niter 'Iil~ ,bi. II ..... ml .. i"" ,houchl rnwb Itself" 10M 10 .....,ua .... "II ';" '''"''''h ..... whodo ....... ~ up ... lronl cl 11 .......an",. h'tlLL'" "' ..... if lite (Wool .. produocd tor ,...,ndard _ cl IhoqIII act" ,be _"",.," ebarvlt1. ,n .hr ....... ,ol<n h i b)' (jf"l.f$ml ..... demandi '0 a«om ... nJUIII ,,>lenllOn. • ')cllln, ,n on;In-' ollhl: pomc1. ..... ·htc:b Onenli h,m In h .......... rch. Tha, " why, ,a;ordml 10 ern<malIn. ,he formal K .. nces, almOSI before the) Iut." had II"", to be """,ed. .~plod. in a d ,alect., dello(od '(0 1100: .. arch I". "nur" ItId a ma,i>tm.o'ic:t commrned '0 'hr II OOy oIlhe parttC\lla. {I" ,'''' .ndmd ... 1 in ill ",Ie os part~

Tb< lonnal ".,;rn ..... UmlllC onlter tht .. , .. oal Iu" oI'l>ou&hl or ,he po.rtio;ub., PO$"rd by ,boIIch'; lhe 61'11 is d,;tlect>c (1op:1 tht oo:ond pit'"

.... ,bema"'" "-,, comoqlteftoOC, ""re .... 'hemalia is lite -..co cl plrucular bet ...

at; ha"na baxHo", ,b.....p IIrou&hL We call lite pAf1icu1a. betn&- taken •• th .. "'~, ro"" cl ,hou"" [~1 or «.ute "",ply. form. Thus. 1"'''' ","Ihm"""" " .he theory of Ionns [F_rJ...vtj."

I'rom Ih,,, 0"" ''''aM bol",,,. (bal G, ... mlnn ",,,,,,n •• priWlncr otlhe d ... lIs .... ,·.h,,,,,,ntly d.noun«d in Ihe phU...ophy of R.'U'O. n.. .rulh .. ,b,,, he .tKaOds prtaJely in reformon, .e-omc:'f)' by di_v"';." ' YI'" 01 ont11,od li.oioon ,.hich diomil-lcs 11;"""" ••• In " ,"Our ot AouJ_",wlw .

TIoc ""''''''<1 ~ "'0 cIu .... ,muollaJteOtlwr. It~. all mao!temabCtan .. he Iw 10 IIdr.It: lhe blInd proti .... '''''' of.bo po.rtJCubor, bu. be mu,," aho \KkJe ,t.. "P~ oI,btl 'nW' 'I*'"-:r 0)", .. fJfO' .... 01 ,,,,,,.111 dodo&:

",. From lhe cor>eqlts <:I,ablisl>ed aoo."" it is "b,';"lIS th" o,dJD'ry ge<lnl<lry l R"~..,jrlll'~]. ~k. =""n,cs, <ekn 10 • ~al be,o" "', ",dinar)' ge<lnlClry. .his ;, 'pace; .nd it is dear th., lbe """""I" or lpaoo can in no ""'y be """,,,,,ooj by thou"'" it " ah .... )'> faced by • " ..... ~" .. Anyone who ,,'Q<lkl "",inll'" the "ppooilt .hould immediately lUbjoocl ,hem .. l"", 10 ,ho 1.0>\ of dedllC:ina the 'h"'" .I'm ..... "". of.pace from the pure 1."" 0( ,huuJlb'; • taR whose result imm<:diatdy 0110 .... nlelf 10 be ImpossIble. "

II to ,100 grtJ'O$' proof ofGnuomann', talent that he was ablt to m:ogn;", this specificity of .... "metry. Jr the Lall~r ISJIIte:I 10 • place in the lhwry of form. (0<1 "" equal rOOll"1 ",ilb anlbmc'.:l. it mus, for .... a new 'ypc: of in,uition. We w,ll..., thaI the ,,-hole innodllCtion and i, •• 'yle of p'r~D1a1l0n imply an "n"....,.oknled elf,," '" g .... p in ftJ@ht th. ·und."'l>Ind,ng of tOe uDdenllUldiog'. In lhe 11'11dlllon of German idoaJisrn. whIch !wi .. , ;\51011 thoe lasl: ohlle ,..,,1mJ upo';rl"" of ,ill AI>w/uI •• Gra .. mann mobilile< roiled fo,ce:! in 11M: ".'Y cart 0( .pace. For , he \>ru'al face-to-r""" ... ,,11 the: 'patlaL ,.'hK:II d"' .... tile prulosophe'·IW!Mtc. inlO .. Jlenl. amfrnntalion witb II< 6gu,..,., Gr.""","" . Ub$','ut .. the d"""",io" 01 JC, .. ,aj degrnoo oljoln" .. nd io'.S" .... lhew: diSlllK'tions ;n,o the continuity or a p.<>ccs;s, of On 1.ullSl"" i..rolu,;"" in " hieh .'''f)' ,urn "n\'dol'" a JUpcrioo- dcgr<e or bI<IUOmm!! or ,be fonru.

111< in' .. o'ion olIn arliarlallon I'''''' the ge<>m<1ri<: on,," '" ,hough, ... i,h I.,a' , •• ubt!<o'l"

The mos, tnUmalC ,I< 01 the forca only \Jl~ .. rOmt IlIlll! th< ""II' or a P'ogn:<$1V" Ofl"ning ou, or ,fie btu ... , ClIolI degree or "'paMi"" or ,be fo","" ""inK ma.ked by ,he: 'I>JI"llr.""" or. new beln,,"

~ GIASSMANN·SQl]AORl LAT~.AL

It i. ea.y '0 p~;c, thot the d .. k>;:IlC '" ,be conlinuOIIJ and d,.."..l. af ~1lCnl' lion and pfOc<:O$ 01 disliDChOn 01 ph • ..,; (,.hieh p""upposes no ,,,,,,,,,,",vi'1) will play. cruci.1 role io llus 'deduction' 01 ,be AwJ~h"" ...... , .. h ... Conlin""", and discret .... nd opposil. one 'no, be. ~k. OOrllraries., and " i. """"""y to

lake up lh" h .. "ili,y fully '0 tUumina'. lhe P'''''''''" lha'l\<nera,ca and d""n"u. i>.bcs ,hem:

Everyth'n"ha, h •• corne: mto beinS by ,hough' [)ftI~J Ju.rh J", D,nk,. !!'n\'OO'J ..... j co.n hav. come into bei •• in 'wo ""yOl: cilber by. Olm,* &<""",I;ve act. o. by. double act of 1""11ion ilJId liai."n. Wbal baJ "'<)f%I<

inlO belnl! by ,he firs, melhod" lbe continuo,," form Cd;' ~~'iK" P"""'l o. magm'.de ,n the narrow .. rue; ... hat h •• come ,nlo bein, by 'he..-oond me,hod i. ,be dilCft'. fo.m 0. fomt_I"""", [r"'knapf~ng-1f"""'l"

The lri.ia] I"'mta""f>CC of ,be d ... ""y 01 ,he beoom'DI 'Wi, ,1><more "",dily tu,""""' .. U1 the """muouo io'm and i •• ""'·OUts, ""'~nltude In ,he narrow JCn!iE. ThIs ~"I'lf ellUmo ou, thlt 1''''''''''' of """,,",,"1 "·,,h tbe "'''nOlO''J

'" <If co"'tan, d ....... 'y. To th ... "og-fu<: unroU,nJ, we .hould oppose the ruptu_ lhe d,<oon".cuon. and all 'hat tan "",n,f .. , ,he ,,~Ious quickness of .be d= and <If ,be act. ,ha, POSIt and link." GnI"Imann tben mobih ... 0 dIal"""" ,ba, ru!lSOlvos thIS r'''.-IIuous 01'1'""'''''" of d,,,,,,,,t. and conlinllOlll ond "'gta<1'" " :u ,he mOments of. procn'i""'Y:

The conlta<' bet\\«n II!( 0;"""", and tbe Conunu(><u" 1~~ . all Irue

con""'''1 ftuld [;ft. ..... ..;,,} ... ""'" tile o;.e""c ""n oJ"" lie >ft" .. Conllnuouo and. O<ju.lly, ,he Coo"ntJOuJ M. 0;""",.. The n.s.:r.,. is seen &1 con"nll<>u< if "ha, i< hnkctl .. ",.sped a • .....,.th,o. tbat hao bcoorroe. tbe :00, "I ],alS<l<t bem~ unders,ood Q, a n>Orne:n, or becom,olO- And tbe Con'lnuous if; .... prdcd .. d""relC if ""gula, n""""n" of tbe ikoom'"1 .'" un<k",ood a~ pure 4<1, 01 "",""" and 'f ,.ha, .. hnked io til ...... )' .. S<rn a • ..,mo,hin, "lItn for ,be 1 .. ...,0."

W •• hould under!,,,,, Ittr. bow ruff .. reo, the conlinuny .uti,,, from th .. of j1ii.u~ #,,,,.,,,J'M. The >1"'11 i. bolt th< SI",pIc acl:oO\\'ledgmco, 01 an Ii .... y .. ,1><.e ,.h",h ee"",J,e,.]y ex""u<" It .. lfin lIe,ng anolb." .hi, mediocre con1l0wly. ,ha, of .... nded .pace. oIf ... it ... lf ,.."boUl resi".nce 10 indd,.n" di~bility It would "",umly lie i",,"pobl. 0( o'-ercom,ng con.rast .. The I." ... ~'" 001)defeated by the .. "" .... ny of thejW,,~ ~d ..... • -h"'" ""plures all ,he n<hco 01 the ik<:oll1log. Only.he abslract .pI;"lnl oIlh. Hu. of lbe bocomml,n'oo. before and an afl ... le"timalcd the 0pposllion con,inu<>u,-di,;crc,o.. Th., ,.-IOCI! i. """ted ilJId ]inkctl .be dl5CfCte """ be un<k"tood ". ""nlinuous fa. become) a nd that ,..hich 'flo"', con,in uaUy' ""n be "-,p",,,ted oul by lhe unOcr· .. ~nd"" b)' reo.orin!! the di,juo<,ion bel""""n ,"" Ii"""," .od II .. """tion. In lh .. Ia" case, ,.'ha' 0'''''''''0" calls lbe 'pngul., monlCnt. 0( III< be",,,,,,,,,, .. ]>l<raJyscd ,nto pure a<l. ollt~_n being .,,,,,,,led 00 " ..... " '",m ..

Orassmann "" .. denoUI>OCS II!( alo>l"'<cund lenn "" co"lempo'aneous; ,he laUe. don "'" p"""i>l t"" li~;,oll, l.<ibn .. had already OCCn ,ba, tbe dilf .. o"uli clem<nt J.< or • varLllbH: ~ '-"mool be ju~lap<>sed "",b .. . .. 0 .oo""""nce 'muld be. hII' .r1icul;o",. an •• lcruion ,hal" al.-...d~ ..... ilahle and ." Inlen'"l ,ha, '" 0" the point or .preadlOjj out" (_ II .. ] ~

• ,-,

Tile fluidi,y of tbe proce« 01 becommg ""II m3k< " """,ble 1o O".n:ome the Opposilion or d;¥.rs")- and equ,lily. 3. It h .. just do"" for lhe "~,, •• nd ,1>< di."'""

Lach ~rticular bewmes .. Ih,ough ,he CO""'pI of the d,venc by ,.hioh " II coo.dm .. ed ""In .DOlI ... part.,:I1!.., .• nd ,brough tbe con""pI or ,be

'" <qual by ... luch 1\ .. ..,bordlAlLai .... ,n mho, POt1la1lan 10 I COII1fl1U11 typo'· WIuI, bu I>ocomt by IIIe lIqual "'.., CIOn wlall"bnllc rotm. and .. ·hal hu. bccon .. by lhe d,v.rw form. The ooaHall ~l .... ecn the "'I"'" and 1M d'H'" .. 100 ft.ud. lbe "'I"'" i. bnUd 10 tho d"tt"K: b¢Iouse ,,,'0 "'I"")lnmJ .... ~I«I ,n ................. RJ>d. WIthout thII wpanuon. tho equal.,ouJd be bIll -. Ihndofc oq..allo lI0II11,,,..'''

L'lc ./10< _,rut c:onhnl>Ouo'diKn:'e. the..,ua1 Mnd the dIverse an' ,ho ","ull of. polanrallon, i. i. lhlli ,ha. al,.bra;': fann, 'booI;ome ,hr(>uKb ,he "'1",,1' .nd CQmbinl'Of)' fo, m. 'ba;om.: throup ,IK: di,'"SI" t.ln be diSlinluitbed.. II It ...... u •• offiOO"'llhe .n .... laUOfl llull makes 1\ pms,bIo 10 po .. conlin_sly f.om the «IuaJ 10 the 1iI~ II COfOCoe/e oq"']"1 r>nn Ita, .. the SlllOOIII outface 01 Pat1IIeIUdet' One L1111oOt; I ptrf«t oqualllY .. ould hc irmric>1lbly , ... IJoo<,"Cd up 'n.he IIlenUly II .. A, alld lhe doakcllCll boolanca of .""pler til ,ho .. "Cd ," Ihal .,qual""" 11 .... )'1 prO<ft<l r",,,, • prooeu of equallnltion .nd d'Jlppta.-.""" of '"",,uahlies. from an o~m.n"nl deIIi«.

G .......... n" deli".. equality Il$ "that ,,1udI QJI he mu,oatJly ",boIllut«l In ad! judl'oerl!- c.,w.lIlY 1s.1tnqon ... "".., ' ...... "",iI",",""" tbe ImD A, CIIn n!place ,,," lct1ll II, . '11""'" darnqe. at oao mndard ..."",. aD be ... botJ1~1ed b lnol"". on lhe bo.m '" a t.1a ...... BUI Ibtl ~u,I,bnum mUit 1I('\l!t mab ui _ Nlhl '" lhe prolocol of compm ... 1I0D Thai made It pos.ible: The teeOlni· lion of In ~ .... hl~ II ... a)'l,m!,l"" _kto(~ o_eo-. di .. ymmcll')'.

This il ... ·hal ,,&bo'Io'D ;0 the I'oml~1a A, _ A, . • 'ltich d"u",wsha I nJhl. hIInd ..... and. Id'\·hand ....... Tho. "'4,,"lny can be omdtnIODd .. :4, " A,'. )4, founds A,' Of It\en .~, is <1m....:! from A;. n.. IonIIJ of lho <quai aDd lho iii,'",," eI, ..... hkl! lho op<'tI,n. of. rom.,.. .. ben: A, ~.lo A, ,,~NSliPl!ly. L,ko ptrfcd ~~ahty, rWlCal di'n1oIlY" only an . "" .... 1'011; ".04, .. IJ- a, dLH~"'nl from A,. th,. """"plod .. parJllon muol hue mobiilUd an IICt"ily liuol 'rticulaled lhem ,n """"- !nUl1ft' tiuo, ... luth tned ltonn Of! • bllana: 0( Bun"

To amplote hos 'diodua"",' 01 lhe lheory of lho ~~I""""" G~ inlrn«"lf lhe 1"'0 alellhlll bt hi. J",l chIcoP#<l We ..... ,hat oxb of IhHe net fUll""',,,,", hle a ........ "on-I .. '011 rom"""" an ·eml>nH:o:·. n.... perl'''''!>''' ~Lar J""K'IItahon. hk~ u~ d'.ionll. of a oqure, Cllplur., the bu"tlnl of ,he malhe""'t"",1 rOml Inlo lour diiUlICI br:r.nd>es (_ fi" 1),

F""", lhe ,nl......aioa of IbQe tw" _IIUI" lhe Ilrn of ... hieb mat., 10 ,"'" manner of ._1""1 [An"" &, ..... ] and the .......... 10 lhe ck",.m. of .... '.11011 [~"'~ hr & ......... J. DCIfI'Ie the IUur I)'pet of fOmll and the bn""heo of lhe Thco<y of fomu ..,.re,,,.,,.d,o, 10 lbem ThaI .. '" say. lhe D''''''''t. form .. paflll .. 1""lf ft .. t '0 Numbo •• nd CombLoat""n ll bll ,,·h ldl .. 1I,,~ed IQlCtbt.,. The Number 1> lbe ducrele aJ~ form. thaI .. 10"Y it i. lhe plbeo-ulJ of lbal whieb .... "' .... ... ..t; the combu\a""", .. the ~ combu>ltory IOrm. WI .. lO ... y lhe

'" ••• ,tN.

---,.of .... ~' .. 7 '. ---

plhen", '" lhat .. hJCh ........... d"·."" n...I<Xn<:<1 of lhe d""""le.", Ill ... the Ihcofy of o\Utll'cn.rod tho lMory of """'bJ .... ,ioM.

I. lhe Am< ",."y.lhe Nnlln ....... Iorm or _,uo;k tcpIflUS ,lItIfilllo LII",-",,",'" <:On""""'" fonn Of ,nl"" .......... ,...s. and ,nlo a combLOII' lOry _lIn_ ronn Of ~.I.n .. v. mllnll*_ 1"" ml",,",', m .... llu<k .. lbul ... hal has boo""", by the 1''''''1'''" 011'" equal; lbe •• 1"."", ""'gni. I""" Or ~~tenS&On .. ",iuol h.u be<"",~ by the ....... &1100 of lhe d,,=-. Tho fo"",. coruhlUta. ... a ...... bIo onal"J!tde. lhe (""rod.hon of ,he lbr«y of fUJl(1J""<' of dlffeRnnd and ,ntqral calcul", lhe Iol1er .,.,."h· lule!; tbt In\lndahOO of I"," ~"""'Jon<"-'IsStllrf.·

n.. d,!o<ret. $ide of the fo.ml producN '""" by one' by p<>SIhOO and ha-. "umbo. and oombin&IIOII. ;1 !nul b.>lancnl b) Ih~ ·oonllnuo ... • lid .. lhal of.he (Of"" lhat hi'. ba:omc: prOlm.&' .... Iy. tbe 'dJlOOh-N numbo"· ... (fu""uorul and ,1M! 'd,..mved cumblll.'iom· ...

, THEI"HI'>SI\(F.XnNStllOlALECTIC

We mulL no ... 11"-' "1"'<'.01 Inenllon to the ,nl""".., ..... n,, •• d,.lectic .'!rich Gr ........ nn UJI'd '0 ~QOd elf.." ,n hIS """"In":I;"" of lIN: Dlrmall'f'Olluct. W. ba,. alrelld) met t_ chap"" II) lhe 111 .. 11110 .... 1 "PI"""ion bet ...... o inleasi •• and .'l ..... "~ and hI.-" oJIow" how lhe "",,1,01 of.uch u "f""",Lion by dJaJI1'IDf ,lhow of<>ra.me and leIbnll) had ~ ~'.....,alCd .. nh lbe pbyY<o-matbom.11Cal .... 'P of lhe flM<F""" of lbe dur .. n ........ R.""""...,. ,lutl lbe ,men,,,. de ...... ptm,r tbe plural ,nto a .. n&lo delerm,nallon; ,lie) cla,m 10 ~"""",>l 00 muhlpl;c,lr. hut _""''''. olOre or Jcoi.o I"'''' Io.t!I 01 comple.ion , .... QIdt. of perfcd" .... 0.. lhe eon'''''1, ,"" ~".".; •• '1,,"nlll\amoo .0 ...... "II"'JI"'" ,nlo ,I>< _n", by coIlat" .. un,b. It naJ ... , .. lhe ",,,,,,~ by do< ..... ,P"I"IKIn' Ihspe ........ II awm"al 10 ,'-

'" Gra.moM • _ ...... -an WI ,Ius ".",_ bel ......... die plhennt aad dlipenoon pcnc'ra'" .... low rot .... 01 ..... tbtmlloa:

A" ,nlen.". magnitude: i$ ... 10 speak • d;"",I"N number. ,"'",,",u !he .uen .. ,,, 1IIqIUtu<k iI the w-lYCd combination. The diopa<al 01 rIe· .... nU ....... 110.1 10 .he uICIIOI"" 1NI1"l!ude. iii II the fi"n. 01 thll ..... l"lIudc u. ""fIll"". betnl,'n'l~w"'k1; ,he .'kUII"" clc:mcnl (da. "u"pllli. £1 .... ~rJ prcocnu ilXlf hc~ u oomclll'lIJ lllal "h:onge5 [all ........ """"Ma']. .hli II 10 ... y u ~hinJ lUI PO' . .hrouab' ........ y of ,'a'''' and lhe lei olllleR d,ff."OI ,'alel collllllu'" •• "",Iy .1>< oomlJn of.he .,. •• ""...., ""'80ill.de. !lo .... ' ·.r. for 'M 'nt ....... ma&JIl1ude. il n III FIICr.IIKNI'Iuo, prl>Yl<kl. conlin"""" oenes of llaleO lUI mna,n eq..al ~Ilh.mleh ... and .'hoec " ... nllly "fA«' 'y.be InlrGA~. _",nude"

The c.umpl< of .he run:.: and .1Ie """",,I. llYn! by G~1\lI. dearly .u ....... lCIlhc roalflm beI_<'nI,hi< d ......... oo......"ed . ',," ,he Inl ........ " al>ll ""'"" con"C':led .... 'h the .. ,.nsi,-e;

TIIc ~ "'n,pI< 01 ell ...... "" ..... ""tude is thaI 01, .... ".nl [Slr .... b] . 'hofc ....... :1111 ..,., eM'IIlD.lly dr;pen.cd. and , ... "" .. ilL", .. ouch ,I"" ./w)' """"IlUI<: ,he 111" U an •• ,.....".. ~Of an .. ample of ,nte",,,. maaruuxl ..... can W< •• "","I.OO"",~ .. lith a .. .,U,n force. Meau" her. tlw .k" .. "" do 1>01 .. .,ar:u • • btl! only "I" ,t lhalbd.a on tlw ...... ·111 (S"." .'Wj. tlley tlwrdc". IOtm • <:aWn li.I ... 01 the ~tlL ..

1·he Iwo doaV.m. below IIlCC HI- l) OPl><'* the inL<1IJj,·c aIId Ihe .".""'I!! Ibe: I"orcc applocd at ".,.. poInl .... il"""t aay dtopO"l1ioa. n 'OftOnJn a pu .... Yef1M:a1ny" and cootrut • ...,111 Ihe n .... SO •• ~ .... -""'" is """"lialI) dispmod.

-_. -. '-'

Rut. a$ JO<>n a l lhe qU"IlOCI ..,( I<f\<noliO" .. pooed, these ,,,Iulm. d,<i16. d....,l.~ r on:. can aloo bc 1B>P<d .. beiDa IB the I"ocess of ddf .... "a " .. It" and the chspct'"loftl PO"'if oltbe: """,,"ot can bc pthe .... IostIber ..... " on a .... "h,

AU rul ....,.,ltu<k can be .... n in t .. o .... )'$. as ;nlcn.oo..., 0< .. lCDSO'"e, lllat " lbe: IlDc .. aIoo ............ ".n .. ..., .... pit .. 1f the .... ) '" ... ""'" ill

'" .lal_1> al • .m,.. ....... d;~ and only tbe ..,.anlll1 oIck,,",uII .. !lOIN and. ~l ........ ,he poinl ...oo..~ .. 11k a force an he co_nat of •• an ""!II'" magrutudt by tma8'n,nl lhe Ion:. in tM f""" of a line. ...

We k ....... lha, tlw .......... 1JOII olthc ,,,law',, and "'nuo",a" be O'UWI"" by In"'''Ii''l. wmprn""n""lpnsilJOOl de> ... ",hlOh dJ'''"lulShes, com........,. and u .. fuldl ih< de...,.,.. I"""PI th<1r ..... mbly and d",pc,slon """Iempo,a-",""""y Thi. mUn I! """"bk to d;onl"y ....... ol Gnwna",,'1 ddinOlIOa$. For tumpk. lloa! 01 ,he number 'The number tt lbe .. thennl ", .. ,ber ol.1oa1 ",·hich " poo.ited .. <qual.·

'" The 'In ... r (onnulll (I) diSli"lIut>hei and hnb I"..'''''' II>< ·<mll ... !-in' ... · .. ,·c· aDd ·taro",al..,.,." .. ,·.· SId .. oIlbe numbc, ·' M';I/I .• ' _I.d u kbop bel..-.at lbe tWO """" 01 1M fonnw.. I I ) and silo .... 1Ioa, ~ llh< onb ... polo) ~ II>< jlUi.lposiuoa oIlbe un,U. W. ap,n litlll ~·llnpt~ . ....... ""ualilY arlocuIOlrs ~ number as • "ep (n • del'ee. ... laid) and u thai which rounif 4as a canli .... L as.n 11..: .. '-1)_ One <;an <'""" ..... ill ihiI. mean. nI ~lnl .M1l_ oIlS .. -,liI,,*. a "' ... , of "".-dDpo", all onItnl.lioa ..... an ~'IIO" In..., and """ and one. Ole),

Fil" .... mU .. lbe .n ...... lluon 01 lbe db ""nallOll atlll ;n'mAfi,tabCMI oh>i<>tn. Thtt Iype 01 dlal .... m ""'~ .. In .l1uoi<>lllo lhe unci*""'" of 0 d,men. loin", 0 plane: i, needed 10 combo ... lho .erlkal .nd hQro(t)lltot To &'"UP a duno:noi<nl it II) In"",' • d~", ... ~ I", """" d,,~n 01 .. ·bal Hq:d ",11c.! ""''''' ow ..... .on~ (Ihe 'bonl au" .... 0"" I _ber")" Iud lbe uubna""" oIlbe ,nt.n~"e bal ..... D"" anothe, ani . the unfoldln, of lbe lotter nourube. lbe """")"alion and """~lllerun. 01 lhe fonner, and ............ _

A coa,,"_ 'mooQ ol f"""ull ( I I .... " . S.ola· formula. ",·kidt both_ onlinal ..... n"_ and ,,"rd,11&! malaot!ldo:

1.- r'"lJ. '"

'" G~~ .. 1>01 ........ ~ that thIS • ..,,'''''11 beI~ 1M pthena, aad

ddpenoon pend""" aU ro..r rot"" or ....ux......lla:

An "'l~o,,". magnnudc is "" !O opeak • d, •• oh'N number. "',,,,",u the .~~"''''C ..... JRitlNk .. 1he m.ol~ -.binatiocL. TM di"","", 01 .... IJI(IIIJ II COImIl&l 10 tbe u\CnAve "",,,,,Iude. .. II the h",. or lhll

..... ""11kk ... ""1"'1111< betnj"R.tl» .. ",k1; the F ...... lIV<: ."''''''"1 (da. tr:".,. .... £I ...... ~IJ "",",,aU ilJ~Jr 1loeK .. oomclllm, I"", <,,"ngel [at. trW ....... ........ .wn1. ,"". II 10 oar u _iii", lhal pem Ihrou&h' ......... ) of lIata. aDd II>< oct ollheK d,II"."OI .1.1 .. con,Ulul .. eUCIly tM domam olthe u1eo""" malnlll.de. Ho" ... ,'.r. for 1M mIen" .. mall/Illude. " II III 1'."".I>0Il1""1 pr .......... coni",,,,, ... ornes of Rat .. lha. mil;"" (q\l&l amorIJ Ihcm· .. ha and .. hole qunllty iI P" ..Iy the mlcQA'. "..""h .... !'

Tlw: nampk of 1he IUn:c aIId the ....... at. I!I~ by GrulmlllIIl. ~rly m .... "'lCIlbe COntrall beI"'t'CII1M d .......... oo_ed .'Ih Lbo ialtftSl., •• 1>II ""'"" connected .... th the C1C.....,'-e,

The lint •• ample 01 ut ..... ..., maamt..x .. WI of ....... "'"1 [Sl ...... b] .''"''* ... "", .. 11 ..... ftWDually dllpcn.al. and .1 II p .... /y .. -" 11 .. " lMy """,,, .. me: ,"" I",.. Of U <"):.' ..... 00 ~M an .. ample of ,n1.n_,,'. m.""">do .... can take. p"' nL md"",ed "';th a l:C11.O,n force. beau .. he.e Ihe elL",."u do 1101 separ:uc. bul <If\Jy lEI" 'I lht""eh .. In 1M lf01"Ih (SroIJ ..... I. Iboy Ibtrdore IOnn a cetWD Sla.., oItho """,IlL"

The IWO d"'Mnom. beto'" 1_ r. .. l) opp<* Iho imonli,·" and Ihe o".n";,'!'! Ihe force applied al ODC "",nt. "U"""I aay doopenioa. nlenonft> a .... n: vertlcahly" and a>nlralt. WlIIt Ihe nleMI'oe "'Immt. .. Iud! " ~I .. g) di.pnHd.

-. IXXXXXXXXXXXXXI

'-' 8ul. n JOO" •• lhe qu.sllOn oIl" .... "uinn IS paocd. Ih ... ,nIULliv. clL<Ms

dl •• "h.,. ,",ora! caD ~l"" be ~pcd .. beiDI.n tho I'f..:ess 01 dlff .... nl " .. If and Ihe d .. pel'loftl pnultl 01 the "'Imm" caa be PIhemi I ...... brr "PIn In • Ien.,h:

AU .ul maaDil ..... can be 1Oc .. in 1",'0 .... 4)". as in ....... " or "'''''''''"eo Iha' .. Ihe I ..... abo .... as ... "' .......... m"p'i'a!e It lbe WIt) on which III

'" ck"_,, ale <iaspnJed • dioroprde<l alOd only the Q1WIIIIY of ok"",,,,, is DOlt<! a"". Lil~ the poIn. md<noot<l ""nh a fon:c can he ""-."'" 0( as an UIenst •• m'g<uIOIdc by tmall'nlDl lhe fora: in Ihe f""" of. line."

We know lhal Ihe ,,,,,,,,,,Il01l of the onlmst'" ud ........ "" can be ,"," "Ane by In, .. nll", a DDmJlfC'lio • ...,Jpo5I~on d.""e ""ICh dl!,Ii"lunllcs. rom",",,", and unfuldJ Ill< de.n:n. &raSPS lh<tr uocmbly and doJpe.lion <:onl=I'O"" ......... y Tbil mak.,,, """,blo to d~nty some olG"' ...... M·' definotions- For namplc. Ih'" of the nwober 1be numbrr II lbe .. !hmnl I<>S<Iber of .hal ",luc:h i. poo.itt<! as «jual:

'" The 'In"ar 1Onn,,1a (I) dIloU"&""he> lUId hnks I<>s<tber .... ·onlIMl·on'on· .. ,,,' KDd ·tlI"hnal~ .. 'n';,·.· ltd .. of lbe number ·1· ... lip '.' _""~ u ~bov bet......,. .be t ..... """" oftht kmnula I I ) IU>d IIInon thaI. Ithc: orwoW polo) ~ "'" Jouupoo.itlOQ of tbe "n,U. We a",," fiDd Qrnme', tnpt~ """'"'IualilY IrlKUlal" • number as ... ep I" a de ....... " • Z"III) and .. that ... luc:h rown" I"" ca.di",,~ III In A",,,,w). One.."n cfttl.eo ill m.. I mean. tI~ ... ..u.s_ oIlS _ -,Iii., a "''''y of e, .. """"", an OfdtaallOCl and an IIddll><ln 1o"",, and ",... aDd one. etc).

Filii"'"' malra lho: ar1lclILluon of lho: distem.nallOCl .Dd intmtdiocauon oll\'iotn, Thll IYpe of d,alrlm makes an al1ulion lO.he a>I(1a<""" of I d,men. "'''''' • pI~nc i, n«dod IOJ comb"" lhe ,erlkal and """'<)111.01 To .... p • duntnoion .. '0 ,nvt'lll • d",palll •. !Itn: .he ""'" d,,~oa tI.bal Hq:d coiled _ ... " i ...."... .. i. i"he ' betnl oullotlt 0 ... al!OlhetT" Ind .be uubera""'" of the ,nl.n~ .. c .... 1 • ..,. on< ... 01,,". OUI . lhe unfold.n. tllbe lalle, nounslle< .be mob,,,,alion ~nd 1Ir1i&h1eninl tllhe fonner, Ind .-..:e """"'_

A ""'.,"_ >er1OOa 01 formllia ( I ) nut .. Stol..,.· rotmula. "'hoch bola_ ordinal mqrutoo. and "'''''.naI """lIInude:

' .. i"lJ< '"

'" II would be no ..ug,mo1Jon to ""> that, .. "bout ,h .. Ionnol&, modem

"",..,. -'d tulllpse. II mil>! oJ"o be Wlkcd ...,,1> • o....::e 01 aonUnuous "P'*uoa '" dqlftO a 'IJue.rnlItt' devw;e If« dtloplC'r 1111 .. bod! 0""",11 ... tWO .'~n_. and ovrroomel ,he OJIl'O"tlOll 01,,,.1<,..., and qm<1I' empha. <WId by a_a"n.llIe Ian"" <>bMc"'N 4"'" n,lId)' ,hal,,,,, inle"",. ml",itude 'd<>C'S 001 rome undone' ,n i[!>ell. 11 .... ~ly .p,,,,,dl; "", In the mode of in~~~M. Tile ,n'e",h .. i. nu Ionil<'t priwncr of ,., indln",,"'; I'fI>m<>lod '0 ~ ' .. "....Jrr for .".""inr""'. il 1211.,4.1 'come ~nd"nc' a. a Im,.h. A ..... u1 •. • ho: Ie<II,h of 1M oegmenl can IKI 10 .... , be tcd""'«l ,,, • dead d","nce; II .sI=n! ,100 rompictlOD of. JOUtl'tl tho ,,.;«tory II ,1\01 ""ocll .... l.,. .... of II'""'" The nt.n .. ..., qllanlll). La l e" ~'. ron ....... IIKI! .. II ..... "itb ''Ullin, linady II'"'' pl. ", .',," tho: ;~t odopIIO" of d"" .... ',I' In', .. horns ""'" II "'plures the: ......... , '" III 'prUIIP'" k>r1h.

Gruimann", defiruTlon of the CIOIII\II .... 'iorI, ",. pthmnt ' <>FIber or !lull

... ""''' •• """led [poW] u d........,', iJ lito billed "Ul> • comPC""'II<Jn~_ dlu,UlIlN by lhe fo",,1I1I I] ):

" I The "mbol I: does Ira,!, ,n the liMY 01 aP act whll tM II,ht.hand . ide

........ IS lhe Ourqa.I. '" d,ff .... n' .1<"1<11,.., Tho", apH" ,he ... n '.' ""'kes 11 "....bIc 10 dlSpby a plu",]"y ," mllhl11i1k. [q~"y IPI'"B I. lhe form of lho ·doIo.· and 'undoio,·. Tho 1""",111>(111 . _. *,"""",1<) .... Ill"" ohum""'I00-dcooml"""llOIIlI1 . ,lucb ,he., 'J;'.I>II ,be d.: .... 'ol$ 1",.· .. ".1 .n: boll ,he .... , ...... mommlL

6 Tltl! ... OOITIVf (;HH . "nON 0 1 VECTOIlI"" .SY$ lT\U

w. kno,,' Ih>l lbe comhma'",n "'''u~ r, lhe hei.dmOl'l 0011111'1 of a q ...... , LA, "' ,]. bul >hould dmou_ 11 lllhe men: ~.aI)'loed unl1 of. moIlOIl, lbe Slump'" d,,,,,,,,,,,,,,, Ib.u has b«n lhond.>ned by • mobole poonl unfokiJn,. Ien,,1t

It ,~ 'n f.." lhe ror .... pI of lhe mobile pooal ... bid> i. II lhe hean of lit .. pr<lCClo '" d,...,!ulioa of the .NIX conll'l>l bet .. ..,o the "allC POInl Ind lhe pi"""" .I .... d} formed and I"'"""nl"'" " .... '"." ,0 lhe COIt1pl1Cl loon of lhe

"-, The COOl'""UOU! p.oau or """'"nina. "' ..... IN''''O "' l1l(I'" ..... 'pjlCa" as. ""nlinuo", fO""aIlO" t.wrl,,, £."""'nJ . W. ornv .. al ,he CO"""PI of COnll!lU()tlS change. Wf 011 '"' ... h",1t un<k'loa clu. chuge lhe ... lItfll"·' .Ie .... o, 1U>d ... h:lIe ...... cacc ,he F'I><nli", eletoenllakCl OII,n 111 wn,e. " .. on "''''''''' of ,he COIIlin ....... f<>mL COrucqueollty. ,he ." ......... form, .. the ' <>tabl} of allihe demenll ,alo .. hd lhe F ....... " .... .10:"''''11 mUM'> dd ,Itdf III obaal"'l ...,.,Im_y.-

'" The .tallO pooal d <MIly &II .1o:,waL It IS <MIl)' """ '" 1M clJf ....... ' >tarD

... ,~ by ,he mobik pooa!. whal G ... ·mann called lhe te' iCr'''''''' "'''_L Th .. te'''''''''''' ~Iem<nl IUnto lhe ... pnettl ",,0 , 1 ... p:l0f,.

w. ~Ium 10 Ott.mc·1 and Lctbtuz'llriplycbo:

In ,,"hoAf}' """"""y [RututlldIrr). ,be po,n l ~Ium II clemen'- lhe .h:lnJll' of place or moUon pr¢;<n" 1I1I<1f I. OO.,Il"""", ~han .. , n.d lhe d .. f .... m poiIuon. of lhe p",nl In .pilOt ... preocn' ill d.Jf0Tl!1l1 IlIll.OL·'

It .. n ......... f}' 10 Ih'nl oIlhe lo<1"""" ... COIIhOllUUi flmlly '''' mo<Ic:fo affi"" ....... uon: A. _ A, -t , ... . ) ( .. ,. h O:S;' S I J. wlbch.. u Gr." .......... "",pt..-we&. n:IaID the ule1I_ IOnn 10 lhe ~ _btlllllOn and • .,umales lhe .......... tiO lhe lboo<J of cornbtni.tion. The dtffc ...... , .lementl on be UlllkntOCod .1 lhe .... mo ,n!le'" d.Jf ••• nt "at .. of lhe .. rne Fn .... li.'. tle"kO!. .mI Ih ... bslfK"l d'fT .... """ be,,,..,,, lhe lilIes i. " 'hll "".respondl 10 lhe d ,lf.r~""" of lilt pi ....... W. eall llIt change 01 lilt I~ ll'" Ihe m.nll1l0n rUtb..'.....,l of lhe ... 1It. ali ..... k"k'" from """ >l ot.< 10 lnotlw:r. and Ih ... bau,,,.! r iJ/n"db JIO ch:I..,. of lhe FIIt ... ,,,,. tiu, .. nl Ih us Wl'Up<>nW 10 1M chan ... of place Of 10 lhe lII0II0Il '" lhe PO"" ,II (!I'd"",.,. 1"0<rIC1.,. ...

Thoo "..,."k "",dt """,,10M ic.di' ,I lhe hum of lilt IIn~ dcl'InnI '" tip'" 2-.. 'bose."", 'p<,,,,ndlCUlar' doaJoctOC'lli 'mplies: lhal oIlbe 0. ....... aJId Eq~ and dill. of the Oucmc .nd Conl,n""" .. II 'h'" makes 11 """"bk II. "''', ....... lbe pooal,'luw: """"",,100,

To obtain .he m .... .Iuluc1e of lilt n rerulOI1, ] b<Jln "'lIh , he Fne",'ion of lilt lin •. Il .~ . • 8I'1It. lli .... ])D,nllaku; dlff ....... 1 poiJl;on' ,n • "",,' mllOU> Wlq\ICn<>:: [In Urll, .. F~l. and lhe 1011~'Y of lhe po,nlf ,nlo .. -bieli the ..,...,."1",, pOint If~nsr"rml il ... 1f in 1M chan .. con.ll1ulU lilt line. He""", ,he pooa .. of. II"" "",",,nl themoch ... H <SXn11ll1) d,fI'c~n' a..t.no lit ... do:!.i ..... ~ ... _ II by dllf.",nl !tu ...... BUI JIIiI U lhe eq10ll IS &/w';&}.

",herta, ill I"" di ....... (all"""'" in • mllord, ... I. Iono~ 1M di/l'cmll potOlll"'" pCknl themwll-n Ion-"*' from 0 ... plue 10 .nolher .Ire;od,' pn:e:t''''1lJ pla~_ Th .. fotmlunn douo nlll.""" lhe ..,.I U~ of lhe cu10ul and. 'n I"" lradilioa of lb. philoool'hia of nature. il ~)CrtO lhe di,'o"", of lhe 1"00"" and prOOuel"lIY," 'I' ll .. """""i"" muOl ........ bocomc 1<»1 in lhe ;""""h""""" of ,he o<\d"kMif wluclt .x.baU<l ,,,,,",,,,,,' .. Jumm,nl lhe opI1l1aJ dn'eo"I, 11 pc lhe ,""",",,00 of lhe lriptycb U ... '. n lbal .. ·t cuml" ,

earl ... l_ WplCf Ir~ Sud!. '"plP;!> pupla WIll of mol_ ,nd ..... """,..,. • tuIo:d prodllCbOrl of dlYef'III} IS the ",,,,",CfIfICC "'. tlnal< ...... 1o: lubJccI;

,,. <III dot ...... pOoIlI1 II ,"-IIDO'a. II >I no longer. ~n ... h><:/> IS.' ,ho tlelm. hut the "t{<lid,,,, "I dqrtts c{ a 'IW''''''' of ~/&ifln:

n", D..' ...... mUll de,mp _0«1,11, 10 .I:ow lor , he Fnm>tcd 10 ~ Sud. f or ,ho .. mplo rOI'lIl. ,"" bou .... mit" he .... Am< lor all ' ..... 41111 011'" "' ......... III ""'''''''III- The SImple "'LtIHoOn form IS \11 .... he rona thaI II dmvcd rrom • chait,. 'n ,he F""ntll .... <k",.m a1.i,,,, ~"" lO.he ...... II,,; .. ,e aU ,he S£I or.n .he dc~'1 .h,u tan he JC<lmIlod b)' the san .. Ill",. Iy.'tm or do"",,""

T1>< •• ho.","_ oi,"" ~ud poin!. ol ,he mart ..... "' ..... y 10 tbe lUI""""'}

of. r..,....,.,. of ' M '"'...... • Inpl)"h and.bit" why G AMma .... dw' ..... . .....,. Clrriully becWft'll JiIyI.r t+ gy (A....kno~ • .. hlCh II ju" .be '''' .... !loa rrom """ ddo'rm'nN 1,.1< ", ."",00. _nd f~od" '2f'.'" r~ • ..,.. (G",M"""'...t, .. hoch dir=t. 'M pu~ ofl ",nnl r,O I point 1' ... JI' "'hicll can "" .. dote hI .( •• one .. ·.n's. Thi. fundamentsl chaoII'< ctIrrespond. ,n "'hot .... "-!>\Ild today ..... U an infinile5lmJIl '",n.JIlIon I. '>ftd or d,.place"..n!) 'The unfold,", (0/ ,b" >D«I pemu •• 8M . cha..,a. and Pfoouca ..... pIc ."miiua rom.. h_

"" $1_ . --- " [ ]

•

The _ament. A and 8 a • ., 1M d,fI"nm. 11.I0Il (II ... ho 6nal .... u and 8 ,'''' Innw .,a,.) obl.,1ICd by ,'''' 'conllnOOlll """,,ui' Q/ ,he IIamt funda""," •• J ~~on",· 1 (.r.,;,~ f",,,., ... t<UI1g J ... ..,I"," G"' .... drIJ.,"~. By permuu". lhe .. of A ~nd 8. we pbl~;n lbe OJIPO>-I'~ funda .... "'.1 """n,. _ V S)" "po,.,."n.· 11M: IIK>Ck$ of tIIan,. ," and _r and by p",n. lhem up. we obuo,n lhe rompJctc ~ne:

Fmally. _ aU lbe lOiahly of 11M: .kmenl> lho. arc FIXI""&L<"d hf ,he I"'''UII Q/ I Jlap funda .... n"'l chan,. and by", OI'jlO<n< • ,>, I. m (01

domain) 0( fin, OIcp [rm S,· ..... (odtr Gdlk, i .. JIlT S,. fr]. l be ",.n .. n,. ,,,"I """m.,o lbe """"" .),.,tm 0( fi .. , ".p Irc ,berefore ~Iod hf lilt pu"',"1 of IInller ,he oun< dIan,.. or of 1M Of!PO'I'c fun.lamtnw cun,.. ..

The I,,,. 11>", .. '_ ,N ",. ltd all. tolalll, InICllIII<d and cIoKd 0CI1I!df. ...... ,..,.." or do.I\."n· ",bid> ill«: II ,n o ... room,n. lbe "1'J'<I'II11OII of ' ''''0 ")"IIlmct ..... 1 dill •• uon .. wb",], would o,lIer."", ·pu r.",,· 0"" lno,hcr fOf IlItlllfCh_

0..... "PIn, .. " wmo ~d< '0 """''''' .... ''on <Ie"1en f,n lhe lrad,,,on of II( In' and ArpDd). 111 CUll .... "'" ,ho: all'lbn.o: od<!i,.on, ,be .~), .. ..". ftlvtiopf. 'h< lnllTor .JIDI1>OI .... 1Id 01'11"" ...... I pooIu,., bomot"",..,.)

'" ., ~. ==---.--==="-

o r .....

w. ,bu. oOtain, lor oach fund."""".1 d,anlC". I o".-d,,,,,,mIOflal auniion f"rm ( A"sJ,h~"".I'''_):

The """""pi of oimplt UIet\J>Ol1 iI dc,nmlncd hf "'" .. Q/ ,100 """" III ... tOt . 11 ,he """""nil of ,100 clw!Je. CO<1~lIOntJy. tbc .. mplc u''''-__ "",,.he rouo.r.nl C111ahly [& .... /w,f ......... j-If f.om In clLUkD' .. Q/ lhe ........ Ilmpie alnlSlon rcouit. ano,her 11 ........ , II Ihr"",," U .." of chan .. [Ak, d", .04,l1li, ... ..,]. 110m from ~ ,"ulli • Ihlrd ~k""D' ~ from 'his "mpk .,.,~nsio~ ' /lI"OIIP I~""""" ofehan,..

In ",01, ... '1 """""" y [1t"",0101."",]. ,ho: 110 .. Q/ «I""h11 or dlUCI"'" COI]\;I11111 ... y partlCULa. cun~ In ord,,,,,," ,..,....,17. " 1\ ,htroforc ,100 ~JIII"'I' ,hal COl ,upon& (0 ,he oimplt 61",_. the onfiMl """"", I .... ~ \0 the toIallyol .......

T .. 'o ·fund.",.",.J ch." ... · bema • .,,,mod u 1' ..... tnci • .&l" ...... l..,;\o';oJ oddJlionj can "" okfi ncd in te.m. 1>1. po.ral~I''Iram,

'-' TIm addl'lOn, aocon!lDa '" certa.III b)l""L" ,," .. ld1ict ,100 dOI~ ...... of

arid""""" llId. by """'I"IInl", lhe OJ>IlO"'\lOQ of lbe IIH"b of challF 1111< ')'m .... 'ntSj ""h odd"10,,, we nbum /""", ... ,." ...... '" du>I.' of f"n.b .... " •• 1 du,"",. being p..-..,od, ,nne ~ ..... I unlq"" )">OIOr 'Y""'" Ih., ml>!" ",,,.tl1 ~n'.,lopo i, (:~_1uI """<lJJtIfd~ S,..u .. ~ OM can ,,""<al' .111 ..... Q/ IIOf' I. 2.3. n •• he pla ... 01 of"<'". and .,'Cd J).., ..... Q/ arbitr.,,)' IIOf"

U ...... pp1id ._ d'lfen:nlla .... or duo ...... , ..... ,h< ,atall'Y...r the clc:",.~U

'" thot can be I"lII"rlIlal form • ~~j.I<m of tcOInd ... ep. "I"I>C' 10.,.., 0( cillo"",. by ,"'luclIlbc cktnen .. oIlhlS 'y.lem can be produo:ed from O1lt'llI01hor, • re dcptndcnt .,., .1>0: 6 .. 1 1 .. ,(> I ... .,; iJ. third Indtptndelu I . ... II (bm ..tdod. • 1)"110m of third OI.p " oll"",.d. aDd "'" Oft.

OrdJ ... ry JKI'D<"'Y [R"""""Iw) ..... y "In .. ,.", as ....... mplt. H~ aU tile tk" .. nIJ 01. pIaoo an ...... "'Ied from • .. ~&k ok""011 on (, .. , dilution ... 1M "".,,"U.., de"","l ~"" at ,",'ill." the ' ''''' d'<ta1On> ..... 1ft .... lhoe 01110 •• ,,-ilb tho totahly of the pointlldtmcn~J "'" II\'Ilcrated forminl Ii plant. n.. pl3nc i. thul the oy.tern ",,,,,,,,nd step: an inn""" number of d,,...,!,,,,," 'hilt dtl"'nd on tbe fir .. tWO"r<! roaLlined In" If. thIrd ",dq",ndont "><<<110" II "cIdod, llln! the . '1IoIe 01. ..,6rule 6fII('C "'

.. , .. ,aled tu. Ib,r<k1C'p .YOI .... ) by = -u .. 01 t'- thrtc dll'~1OIII; a...t, ~ OM can"", 10 beyODd lbm: ,!Ilk"" .... "". "',",,1oIif 11a.". 01 chan .... ,,'bel'Ql Ln port Aw ..... ""~"'$It,,,... the num\x1 of d>~horu can lUO'" ,n6rut,ly,'"

CnoS<m.Onn don .mph~ tNt IhlS IndefinIte tJlr-1IJion of dimm'..,1II II only pouibl< it' ..... <:aD <I",,,,' I, .. ;0 on an unhmn..d .. , ollDdcjX.idenl lilfttlioM. lbts '-eclooa!.r.Jc:i><'odmn- of_ fu,,,lamt,,,"1 cha..".. 1:". • li. IS ma...r..ted by the _.b<!onp;inl 01 OIM' 01 ,hem 10 1ho ]i .... ' ",,'em unfoMeoJ by ,II< o'l><fr.

llU4ume 6n1 of all ',,"0 d,lrcten, fund~m<ot.ol ~h"'an and ,f J P<()'0I<:1 II .. ill [b.-IIr/Jlt f,,"fClluIl6l ) .... kmeo' of tht ~!11 fundanlotn'al chon.., lor ,UI "PI"""i,o) .nd ,ben proIrxI al .. iU lilt .100 ... " !hili o;hananl. IoIIoWUII ,he AGHili .... 0 .... of chan.,. ] ri then boo able 10 ....... ,. ao """",)' 01_ okuknl>. and ] ealIlJoe loulily of d ... lkalllh ....... WO, .,ed a Jj'tl ....

01 to<:<>I>d Jlop. If. n_. I li ke. ,h'N fuDdammlal dian", and if I P'OIroCl a , " 'ill any o"men, oIlhc 11',om oIlCC"Qnd "op rollo .. ·,o,'Iu. ,hurl c"'" ..... I .... iu oppos"o~ ,lI<n ,lie ,o'a~ly of lJoe .kawooI .. " ...... ,.IN "'ill form • 1)'11 .... of ,hlN Slep; and .1 Ib" meao. 01 "'''''.'ion. in ........ pt. Ii., 110 ""'" [S<lraMd. I " 'lD th ... he able 10 doev<: .1" ..... of .ny SI~p .. 'Iu.. __ "

One "'0 th .... defi ... '),>1."" ofllC1' I, l. .... k. ~~. A 11".10 oI.tcp k (Iotloy .... "'!>Uk] say 01 d" ... n,.oo k);1 dofincd al J'OO"C"1n, k tndcll"n<kn1 mean, 01 ch~o .. ',.. . t, . ... ,h., ."cry .ltrMn. I 01 tht ')ll<1"11 .;:an b<: "-nllen ... "'10 oI.ka",n" ', •. ,',. "'poe"' .. ' y relaled .0 .I><.)'t~ of"<])1 .. _a,ed by '1" .••. v. . h.o ... lha, ,he plnL-1""''' ollll'U ....... a .)'1, ...... an ·"""n"",·. a hull' lhal ",bo<d'III'" 0100 h ...... '0 " .. If III onk, 10 pul aa cnd ,0 lhe probfcn.._ 01 ,he 1O.--m6 by anOCULoll"1 ,hem.

Th\&S, II>< II<ldtlion baLo,..,., ,II< ,nd<:flftllc luxlaposlIKm;

11>0 ~ne ·COI»"''''''. l)'t' .... · by lI"ol,n. >inak mok. a11 1he fCIIIICIt" p<o<Iuccd from a "'. thon.,., '0 k_ a '1'1= 0( ... p k. il i> 'n n'Y 10 ..... <Hk>df l mdcpcndcn'

'" nIOdeII of .. nerol"'" .0 .n''CIop and unfold all ,I>< IeJIMOII ol,~ ,,"om. l1Ic:oe .. ~n" arc not. U in lbe ~ of ,be dmlCn<ion I. !limply lul><",l;· ... ,a! amonll,l><m .. l"" in • 'pe<'nlIO. but Irc anl<ubla! by ,I>< II<ldn""' .

Tbmt. .hcrd".,.. ~I1OIJ • bot"".,wo"y ,Ila, .. rommoa ... the .,.. ...... ol difl"CI"<n' t.cp" 'M,"1bd1 I ..... ' I' rrom MJ"r(~ ,..IWNI_. II,. ~y ,lui 1>tJmoF"""Y ,''''. mal""" """,hie.,, d,,,...,,. ~~ and ", .p .. l ol""cp or O)'''nt1$. d," ioKU"hlllJ JqrN$ rf IMtpooJrftN"" f'''''' Ih." mode or .. ....,.,;<m. F....,h 'yllem olll." k CIIn he co~,,'Cd ollU' plat .. u of equ,lIb. ium ror ,he .'1"",11,," or forms:

lOt .,."h ')'I'm! 1: of 1U-P'" ,here .. on OfKtIIm ....... 1,') if I , and I, bcIonJ .0 "",h a oyslem. I ',. - v, and )', + I~ hd<woA '0 11 equaJl); ,he "or k ca.n he defined ,nd,ff ...... ,I) .. lho mirumal numOO or.lemcnu IlCCI!'UlIry '0 ,II< ll'"De'"'iOn or I: Ot' .. , he m."mal "umhe. ut ,ndoll"n<\ml funda~nl.1 ..... "" ionl of tbe Iy".m l:. Tho ".p k ",n • ..,n be ~ 10 app'" 1_ Ii .. 81 IItM ....... 'nl "",nt ol.n _nJiIII: clla,n 01 ,r\d<pt'l' <1m •• y ....... and • "1C~llfI _h""n oI .. , .... u •• oyotemt..1Id ,herri_ n ,II< "",n, of anlC\l~'1OJ\ tw:t .. ...". .wo .uhorWna,a! and ............ al ·.u ... ,,..,.,u ... ·.

Th .. m"'''111 I"-',nl .. ,bemo, •• n InJ.g .. , ,,,·, U"'ff .. hero lbe ""0 free: '1"."'" p<lICftd,"I from l.: and lhe ,,>1"'" whodl ........ ,. r a'" in equilibnwn. To llIl. up OroUDWIn'l u"'~ tht .~p l" 'M a"",her .h.o, .. """"""" '0 ,he .):01 ..... ,ha. "milll dowly Ct1,"Clop' r",,"""" wf __ I,1Ie .ystelll I:n,;" equ.bbnwn .bcrrion' ·u'uraUy" ...... IIIIU .. ba ,he S1ep l IlJlOllllJoe do .. or bbel1y o. co¥dopmcn~ b\!'. aJ "",ply dill''''' """'d. lho L-Il.r rcmaia lOrmally boll,a! from one .,..,.1>0, and ti1CIIIbkd ' ''FIber", ... nak ,nd<:6nny ,\ ci u§! •• oliO,....n" heiR, olf.,ed. the,.. .",1 •• id"~ ')1''''''' .. 'hl<h ........ do".ly ctlveIo",,, """ POUIII"" a dct.nn,ned ".p. "'hodl .. """" other ,ltan .hio d",~,'1 dot" .. 01 ,ndt"pt~ lIuI lhi. 10' ... " ..... '" COIllinn itd '0 Iobelli.., lhe dotJfC'ClIUId lemll'.,. ...... '0' kInd ol 1hermotne1ry'. ,napahi<' ol """"d,"I ,hem out or I'CpIC .... a. lhem ,nddinllci) "'i,bO\I, 'bo",' oil 'roIler' 1_ 1II ••. CI.

In oru.:. '0 m.k~ ,1>0 coallllOn of d ....... clea,. i, would he ~ry 10 deli .... IOn optnllioo 1hal all.:" .. lhe r.onou,. "~P' ", he <I' ,"; i, .. ould be '4' "If), '0 ru' OIl' a more pmctrallnl .,b,-,.I~I""" ,bro .. in. ,he addib\'c oysu"" ou, 01 boh ...... ~,om ,he J>OID' 01 ,'I£"l10' oIlJoe '"" .. ' OfKtII'iorIl<Ila, .. ltich KlJ 0II1hc 'erm, lhin are .'->Iu,dy ." ..... 1'0 i.~ lhac .)., ..... an: .. 1t""gtlhi~_ I I .. "'" 'fonnally" , .... =.1)' '0 II",,,, pmIOnl;n one', ""nd.1I< K:uIl "'phd haL-nee , ... c""p1~' 111 1 '" COIn""''"'e roo lbe J1oIl""" d>lal' a'ia .... II i •• uflicicn' 10 cone.:;.., 01 ",ro ... 'M f)'mhol rrodUO«! hy ...... ,,,,1 ..... • '''''' ... + I· . ~ ) _ 0 .."d n'" .. ,he <m,te of. ""WlI PlVOL IIUI .. '" kIlO .. thai lor <:;"' ....... 1\11 !he ·d«Iuct,,'.· ..... "'" "",nnl"" ,he ......... !ic" [wisICII>cl!.onlKhl c"'"_ (.)'<I .... ,1U1JOI1). n... 'plCI<IOtuallOtl COIIld "'" he

'"'~'''---.-..... 1)_ .,

.......... 0( ..... 1'0 ...... _.n .. if"''''' r_ I

•

I coo.liBOd 10 11I<.00y .n.u" ... Ihal. proIocoI oIlWao:n be1 .. · ... n .Irudy ",uobItsI>ed fo...,." WI> ... .--.""'.1 m ..... dart 10 " .... 11Ofi 1111: F>lura lhal "'" out lbose I'<or!m .nd .kelch ..... . rtiaIbtJoo>&: in ohort. II'" • premoaot"'" dlbe ........ 1.,.' wo<k Cp. 10.). AddiLioa IUIdoioboedJy .... t'" " """,bit: to dc:/Ino <lev- oI li\dtpondence, bul by MCn&tna "'.0 U .a oJllIIrOve pole, II ""Dquon iKi .nlwuoa QlplbIt: 01 • .....,;'" • dlmnHlOiii 'rom anotbol. II • .... ,o'a' only by noui1!:hllllltJiC'lf on fioDdooonn'llai ch.aa .. drloveud free oI.ha •• 'Jimull •• _oIyand," ... ,11" _., rllotoo " (d" •• OS in},pthdenl 01 r,~ ,1m., (d"., ""'" .I~.dy.n 1111: 1)'lI<m Ir •• • ,I~ Clc. II Qnlll'it live Wllhoullho help dlho formal adJuro::llon. oIlhe CIa Ind ''''"""",,n poinls.

GrlWnlM l:IIy. "".11: ri",II, lhal .ddition " 'foun6ed" on lhe , .. ul 0C><>C:q>I of ~h'l ... h",h .. "'".,~ .. ed lop;tbt.'. II " 11 ... )'1 reI.led 10 lho .. hi . ," lion dtotaillies I h~1 Ir. ""PJ)OIt<\! 10 o/I"e. no .ui.la ..... 10 be.", booken inlo p ...", If, lhe follo"',nl .lam!>" sho....:

lor lluil, ,-, III look al IWO maptllildc:o lfomu lhal ~"II (rom. ungle IIlelln. "l.at"""", "nd IQI .... all '''''In'l,,,,,," ",,,,,,,.Ied ,n lbe _ dJft<:I""'~ Ihu. ODe a.Q ~'-arly Itow lhey ClII be JU"al"*<I 10 ro.rm a .. 'bole; lhal ... one Ib ... u oIlbeir 'a ...... hve _'enl .......... thaI to 01 lbe P*lU lhol lbe 1.0 .... ""'iides mIoIcI; and Ih •• whole Of .hen ,n 'urn ~.'Cd,", bel", ...... "'10:<1 ,n lbe ...... <b11l(t""", I""'" 'wo .... 1IU-I ..... It II ea.y .- 10"""", 1""1 'hio "" ...... II .n odd""", FiB! or a1~ I Clin Plbe. I.,.., ...... nd ptnII"k .. I 1QJI1. be<:a_ lbe pllt> lho, om<

_,'ed oIl"F'Iber n'm,un tbc <aDX. and thnr ''''( "'" can """nF _h.n ....

Col>lOdrr liJu",9. I can 11., oIlbe 1"'0 CI",1es I"" lbey.re '>Imply ..... by lOde'; .he ,1to"",1 KI 1"'1 ),no 1""'" 1"Flher • .-.... ,,'" rom",",ely dlXOn ..... --...l from 1M'" lbil poll" IMm, I can d .... one. lhen In. OIl1fr, "I,Iopend<-nliy: lbe In<kulit>n il 10laUy •• bol",.,. Ind i. ,.mply there 1o """,II Ih..>, 'hoy . n: IwO 10 bo "hi ... ",' 10.''''''' and Ibal lhoy .... , and /:lode "n In ind.If.","!

<. _ . 'fuundlllOn' U."... lbe iUltapositJon .,nora all pml"fl"""'Oll. 1, can be oonood· .IUI .1 lhe ..... y diqnm 01 add",on . ·.Ib ,11 10001Iy ~ ... bIo 'lnlbaio.. lis COfIImullloo-.,y ......... 111"1 a IiniWd topantliOa and lbe...s.aol ~rr_n\ 01 lhe hand lhal d,", il. n.r... .. lbo:,cIo ... 1IOthlni 10 ... hoped ro.r from tbe add .......... hoclo .. nbou>led, if """ .... Sf"'U 10, by ddi",UOOL To ", ... 'U 'UNOI""" ",.Ih IItt .·i" ..... ....-o IIIUlrI 110 i<MIatt ........... '!oettlral ol<" .. al' ,hal ...... ply "'*'''''' bul ... poin ..... ,..Iu., •• an UIIlabili., "",01. caplblc d ""ball"", ... lbe I'ormatxon> .""' ..... 1Cd in tbo tame d,...aiota· ud '''" ... '''''''' u tbe omtre oI.be cine thai allo .. 'Cd 1M "I .. 'p .... cl ..... ,,"" " ..... ben ~* chap1<" Ill).

Con"""" lip'" 10. wlUo;:b ...., .... how lhe point ... ro open. 001 1'''0 I",,, bra..chto.. inlo "", .J ...... ,ri<-aI "."n,!r "m .....

.--.:~=.,-~--• .. 'I

o F_'~

Zoto " lbe poonl .. be", lbe liB! "" ...... oI •• ' .... lIon II \On[ 'pAn Cthal IbfOll'" "mobile """,It')., Ind il io lbio .... ruch, 0_ apon. iolOl'" 10 ",d .... ",,,, • dooi!qlUhbnum in otdn- '0 pup in a .... 110 '""'"iota Iw() pol'au ...... alotol • II ... "', I, ... In 0f"P0Ilri'" dil1l(tiouo. To 1IIIJ1U' IhOl """""liOft, ..., Un 10 ""' .... 10 _ 1 ___ ... _ . by -obih .. ". • C<laloo. wbach "'_po !be 1oIdt", 0I11If 0 ... on lho 011><,.. ud lhu-dore 10 IC1 and'U' be d .. U d 11110. _l1li dune"""", .plI1I"'1 ,td 1hI! """'"Ie pot"1 "''''eI moll><,. d' ..... a."'" 1_ ~haptu- III ~

The .dd", .......... liorJ malt ... II ...,...bIt: 10 d"l,nKu"h okJn>n 01 iowk-ptn_ den:cc ...,1110>.>1 bo'on. c:ap;lbio 01 JOon'"1 lbem. " i. only. ·1 .. ,>on 0111", $ltp', 1~ jump> bet ..... n dim.n...,., dcmao>d ...... r •• ioln and a hallon 01 OCCOIId .I~p: muillploe;>IlOfl,"

GUSS.,A>1N'5 c ... ·It:1[ 01' TllE EXT'E.'6IOoj

1 GUSSt.lA/oI"'S ~.OOUcTS

..... ", ... 1 ""'" nuke a decaded ellmr ... u,,,, ofll>c DDfI$I...,,,,,, d ........ 11'JI;:ll prodllCU: II>c 'POOS> ',e' ptOOlIC1.nd lhe ""~,,,' prodllCl.

Tho ."",IIea"", of Ibex pr,xlo •• u •• Iend • ...-ell I>c)'and lhe b'ilOf) d nult..m~"c$, They ~Ih enl a "", ... hIe uamplc in ulllknl.r.d,n. llIe ...... lypt of 1fOn>e1 ..... 1 rnlullion den,.,.j from lhe phil.,...,prue. of n.:tlU,o of lhe bo"nn,ng of Ibc ""RIUt)'_ We ,,-;11 Illy ~n"'ut., nnpllam on 11>< follo,,,,n8 ""IRIII'

Gru.mann·, pnxillC1l a~ FOU, ... mulllplrcauon. "'hlCh break d ... ,ly ,,'uh lho ad<iJI,Y\' form. .'Iuch .. "' .... )' • ..-. orlesJ ~nl~ 10 lhe 1~1"'" cia .. nalo ""'L Tbty do not jU'Urpo>< nugruludoc:!l, .... 1 IIIIb U p<mlblc la ,pnnl frum one: .lep la aOl(Mbtr

They "'. ed 'Q ........c.O>C{lnla OOIIIIQOO ... clwn frtllO the po"u 10_"" ""'-" hoked 10 0.. drlre~QI .)-.tant. 1'bQe prodO>C{, aruculale de.."..,. d ....s.pt'ndcnoo!; ,bey them"", anoculale lhe 1)'1' em •• 'uh ono _ber, 1'IIr:ot .yil""".~ ""u!lobnum plaleau. fOt ,be lr""a. ~b"" lions. Oul an be un~lanad Il$ I<",n &i lbe nollon d ....... 1'''" b)' molHlo poi,,' " ~1 .... 1ed: I>}' deformi"l in IhoUllhl • .olurn. nn'ed om III optlOOllC; I .I~ady posilron "'y,\tlf III a ')11m! d IUpt.ior Il<r It b<ocoma '""" ....... lhal ""'" dllDClHOon .lIudn '0 anothe. (;"' ...... on·. prodllC1 II non_mIH"U'~ alFbnr .. 6 ... lIy bbcra'ed from I_ 11"'ly. and II -I-It)' r ...... ,,. oIN-;''oa Wllb d ..... ....",

Thll pmdllC1 fulfth 'lIIIM em .... d lalma. .. ho .... nled aD "'lObnr ,1Ia1 would ho able 10 cakulo,~ lho mallon, d .pal'" I""n""", dlftll;'dy. "'HI>oOI teOOII"", lu C.n .... " COO"lt"'IOl. It ... , noauat)' 10 in" •• u ..... operalions lhal made 11 """,,hie 10 anlC'Ulalc lhe da..,;i<: .ymOOI. of ""mel , ) H>o>ol. ~ .... olcl. _nd lhe~fo~ 10 forre a .... Ul"" F"fIIOlnro.alFbraic I.n...... • ·cloa ....... ImOlt,,' thai "'mold psp ."Itnded 'P'"'' dynamICally .....

II I/~I""'i<g,,,,,, IWIII .... IIJJrrI",,: g il""~ j~ ""Iural pIoJiMI,"', We know IIIaI GI'II~n .....a.led addlilOll only wnh II .. In,'1I1 lnlU'I ...... ol 0.. jU"lap"""1Oi11 ol pan .. 1Id'o ... GI'II .. m'nn ....... ptu ............ ol nalu~ had "'~ady mrlLlCd thai add,'''''' ""pIMn ",ulllpircalloa '" 11Ie....,.. 1II1IY lhal ,be mecban ..... •• IfUISI.~ ... , .... nd colen .. ,', d",_ 0pJI0I0I: , be rul dlnam ..... ollho ""'UI'III fafOOl. ~or 8.adrr. to. coamp!e:

TIle "'orld .. DOl a coikclion ol """o. a ron"",,,e. al. ol.IOml.. !luI ao orpnlc m .... go,,,med by. ,in"" pnr>r:op/e. TIN"", "'" .. ~,Uj'. I/lal i. 10 JjlY, lhe kI •• of lat"lny .. lhe f .... "dlui" .. for lhe Idea of ""II. If, " .. ,.ad of laklD' lhe 1,,1I.h11 or unuy II my "~"ma "",nl, I Ilk. lhe "'di.'W .... ' 101henr.UC Qllallh11 """'h;1 collduiotrcdL I . 'rIl/W"1II rn ...... 10111" /:Io.ct la d ... loulny, Thu ...... ",U ... ,'111 adIie", 0.. 'ynlheJq d ad ,nd"od ... 1 0$ a .,.111 ...... 01 .. " .Ian from • mli/upliClly .... n .... w ,

"'''I), The <ya.btm mid' 110 ,,"00""" aU al oooe. bke.II coplo6oc> .. Tb ...

'" ,bet'o grn be no o;ODSll1UIJOft Id ... ,ndmd"'" '."J1IIb$i. d. mwuploa,y lubord,na1ed 10. uony .... ., ,hal may be ",oduad In. I· ......... ''' ..... m><t;

n m ... 1 appea. In a .. nslo..,' 100 10 opeak. by .. ~.oIUlIOO or upkMi<MI.l, In olber 'mns. I ...... ,I no """"","rucaI CODIlrucLioo of lho bod) •• obJCOl OIXUp)1nl''''''''' ___ ; and a pnllC,pIc O!ho, lhan 11M: IUIIpIc .....,hao~ of n","~ .. ".,.,.jed, nO! only fnr ,lie con,lIlullon d o .... nlC bod'es. bul Ihro for all bod .... 'n Fnwol."

!bad., rorrtru\. lho u lornall"'ps of 11M: mn:ban .. m. "mplc a.do!,!iot» and SUblrao:'l'OiU of pan .. ,,'nh lhe ,nlerna! .... ps. .. hlCh a~ mucb """" penttralinl .nd dr.lu,b Ibc uldonlan,]'", 11)' r-0ClIedJ1I& '''' muhlplre.uoa. ra""'l by """m and utnrctJon 01' rooto:

0. ohoold nol be ""I I'" .. d oIlho phy'!><1$1. ""h hl$ mocbanrcal ",plana\10 .... IuJ dead .ntlunc1lC.nd hI> ook!y ",..,Planrcal WIll d juua""",n .. -.ddro, and .ubU'M'\,n .. an",,1 ........ p' to C<)IUpt'IO ",lh nalu~ or lhe " .\urab" ",ho romlrut1. d)'N!rnrcally .nlb h" d)'urn", Inthmeuc: hIS mul1l1'hcnuon and htl nl_Iion.

Abo. 8a.Jde, odds in .IooIOCHC:

II ... mrnako of our ""'I hematic'" maouab 10 bopn .'lIh adduioo aod .ulMrao:'loon. whICh .~ cIead and de>oid d roaapI, and IlOl .. ~tlt ~, ..... .,.,.,.,iooII; raiOlPI by P<>"~ or alnaioo d root ..

Addlloon. Ibadc< 11.,,,," add ....... , ... 1"';11& r..,.,..., IJod lbt;. 0.,..,1)' d prodllC11Of! .. "b dead m.lI .... l. and lhot •• -eiPI ... horcu fo,,- ... h",h unlle or oomhl ... dn no, OIIly add on 10 one another I;~o f ",01' dead maner. bul muillply and ,no:ro""" lho .. 1"',,'0'" Ind, on tho ",/lo, 1Ia!\ll, ... hen lhoy ",,,,,rale. 001 only dC<lOCl Ih""""I"o. from 0 ... a""lber, bul .~ ,rduad In lhe man ... r 01' lile ~'1r.C1lon of ,001$.'

Ten }~~rs before lho publICa';"" nf lho ""..s,~""""".Irr ... Rand.r "'""Jln ul' lho roll moanm, oI'oddlIJon .,.,n, M"" nu,hrntah",an' .... nn&ly ron.od .. , be saod, mulupbcauon al • 'ropealed addHion' .nd d .. """, ••• ', opca1ed ,ulMractlOll', MwupbcalMl" II. ·.....,"''''''1 pn>OIraUOll d lacton:. il prodllCa an ~nIC'lOI1ZItioII·. an'tnlenJtllealion· C"",' .. odl. d,VUOO!I r-od ...... aD 'n""'" """'Uoa' An ",'en ·.....,r-ocal penttrauon <Jl fan<tn' ... '1111 Gra ...... nn .. ..,ill& 10 domaad d IuJ r-OOUCI for II 10 be: • JOrlu, ... mulll~ICI""n, .n opt'nnion produeu,'. 01' pluralny, AddlllOn ""n' C1lI, ,I,\tll .. lib I>c,nl al* 10 'Ihln k 1"F'ber' I mul111lKle d p~"i"nl OnIH," by jU~'.por.!n. lhem, in 'M .... y Ih., 0"" m'lIh. pi~ coun.m On a \able, II ,"",11-.. _lIonn. only 'formally". a"d lhe "",,,,' Iho.l il ptUd_ ... e "' .... ,.. rn dlIIF' of Icclonn, n",r ""0 d ........ ' ... lion: .ddu"", proJ"",," ",,",y Ibc dosra<icd fonn 01' lho mulo,pIc. lhal whICh" <lmved from lho dowmp<>Oi. oon 01' u already .... bJllhed "Ob\.

U"hh addJlIOfl, wblCh "' .... ppolOC> no '.mproa! """""ra,ron d facton·lho ~I_ d Ibc racton. 10000ly Ind,Ir ..... , '0 ,hot, ~ .",hrneuc:al multlphca""'" d ................ "'pi"_ a»d a .,lpivr ,. _ 1',1', II not

O,",ft! a, by J .... u ..... ' ... 1', and 1',. bu' hy tiOWIU ... 0... N<k of ,he «1""'100 "" un,L """"'r- ,ba, .V" ,t.. prodl'Cl 01 P, by 1',- Numoncal muillph,," uon "or "Ou .... '"ommu,"m'.·. bu~ OS we ,,',11 _. 'lnlbcis lIwII),' pn~'~K'" one V<k or ,he «lUll' '''''' .. hldt is.he opl''''O<. II , .. OS Qrusmann AMI •• '11.,,,,,,, of....:and >UP' ( __ • I ~ I, IS """p"uble .. i,b ,he potAble JeWlDp .. "'O,," (ib$lnbu'n'lly la ...w.on to oddl1lollt It 6...Jly 1II>I~eJ i, _bIe 1o..,.,.,.,.., • mulupl.",y lilal, ra. from unde.m,n,n, ,i>< un;,y. JOei (.,......rd Inlo d;'~.-.ily lhe he" •• 10 \bo .. n. uolokl;na.

8 ,.""' .... , allll _",allIM'

QruomlJlfl', ra'lIet_ '1» .nalag Wllb !be .nlh ... eh"oJ pr"",,,>1;'" brod alrudy ddinal" .. "" ... Inal product:

The m:lanpe II~:M IS ,I>< 'nK' ,.o",.1<ical prodUCI if mult,plK:IIlion .. ,akon ,n III pur~~ "">I-'", .... .-J Itn"' ... Mulllr>i""'I1"" i, OIIly" conl1<1JCo lion 01 h,sM' d..-. In ......... .,.. 1110 poinl " ,he prod"",;, ... lemool. and 1M lme 'OI .... S r.oon i, hy" COflSlructKKl The reaan&lc ....... ... If." Il'QI .M ""'-&h' h"" ill ,he """''''''1 lhal we 'mol lhe poult.

The 111 ... ,100 .. odenlic:al ,n In,h~ 10 llu. ax. 1M uml .. lhe o""nal p .. ",1I1<:1n', .k" ... nl. The IInll IS limply oeea II, liven If. no ... ,he numbor IS la ken as lbe 1\< .. bII", fo, .Ileul,"on Iby .a~lnl II IS ,he ""w un,,) .. mult,pl;""""n don .ppoar OS , tic production of II numbo. of • "'I""," ordtr, ,, num"". "holt UD;' is already" Dumber A m:taAJIe" 1110: 1"""n..,rictJ pood .... of ill baIe . nd Milhl. and ,Ius prod .... boh.roes hk.!be IIrnhmnic:al prod ....

10 "0111,-'.,.. ,he poinl i. ,110 ,10",,,". ,he f)nlheoo .. is 'M lIIOIioto 01" "",nl ,0 OIOme d\l,«,mn, a"" ,he mul' p.od.-J by Ihi, .ynlhnl .. lhe ,nl)C<10ry 01,,,,, pcnnl. ",be Ii".,. If. no .... , lie 11M Iho. Iili,.illt'd rrom ,bi, fiN. 'j"o'hnIJ piar- lbe role oIlM "",nl .. nd If ._It'd in 'M lame ..... y I,ha ... I" >OJ"'' MO\-ft! ,n Ino1l1cr drrttho"~ "If " ... .&"" ,ha'" 1>0'<1 _ .. """,

., i. ifl<lo:,,:d • q"""100 01. FIIu, ............ ,rul product 01, • ., Ii"",", racl"" 1f lbe ,urfaa: lal~ lbe rolr: of the po,nt .... e ohl.," • ..,.'" «>""";'00 •• tbe product oIth"'" rac'''''' W. <:In 10 no f~"ber in pon"'ry. fOf our "1"''''' boo, only three d,me" .. Q .... No hrru,~"on 01"'" kind oppeo" III llrilh""'tic "

QIID'""," G ............. !be ra, ..... bad thlll .......... 10 "'pe !be ..... p 01 !be ........ ,rieaI addI""" 011110: P"raIJdosnlll 01 ror.:.:s: olot !'KI l n,le. hke ,he ocher liaurn oIlhe pl."" """.paa. '" not" ,h,n, thaI fIooo,. ,n."""" and ',uppom' the "'t"'t'M. bul m .. , be ""","",.-..I •• ~ mobil< unIl, Such u,"". ore ""n'''luI~ U In It.. d"""m 01 ,he number I .... Ii&. 4~ ",hI"', ""(,,d. In ..... "' ... " a, 0JItC as cardl...J aad otduW. Ho'nUIM G,......I" ... ,I>< -. JOn flllt"". by _.., t.hal n " I"",",""ly 'bis '~procal penotrlll_ 01 r.....,..,.· 'hal radoally

"f'PO>C> mul\,plo;:;r.lion t" tWdUIOD and .ha. thIS 'm,«knu"n.· can 110 obtalllt'd by p,od""'''1 on """,,.ed mobile unll. ' '<:0=11' .

1''''' _mo oIlbo: rro""JIe "Uli ha .. """ hcitlu M ,he AII1<' roo""&: ,ts cak"la,1On io 01 """roc commutali,-e: S _ llll _ "8 ud. by radUl,l 8/1 oc /Ill. I can 1d:_'k<IF ... ltiI;h loCI. CIII ... 1tiI;1I. 8~ docod.nllO nh,bn rl>r Ofdct 011"" I)n'heoe>.. Q.."."....~ f,-lnBIIeIfr""", !be ~t or. UlK'no:aJ mol I» underoland'n. ,ha, tit>! 'Teali .... · .. lhe mOfl abstract oI.Il " foraeu !be p!Umllhal pe.m\ll..! lIIe <:QII<lrUClioto of ,IK filum.

Q'OS!JlUlnn. lik~ A.pnd, JI"IIlped how p«lIIl/1"be .. """.Uon 01 lbe nep. li ..... nd lhe bll"ra1 ..... for ,-11")"'

It .. ,he "" ............ IICIII 0( ... pu.a;n ..-.."trl' thai sho..-$ me ,be _y, 1 ..... ,n the !labi, 01 :Ift'iaI,be 11'''3II"a All and 8A u OPP'*4 macnnlldoo$. From thn "boonllUO!\, one CIUI ded~ 1!Ia' if A. 8.C ,,'" poonu "" a """P' Ii ...... lhen. in .n ca,.., A8 + lle _ Ae. 1!Ia' """"',Dins ,,,.. ,f A8 ."" /lC arc po;nled 10 lhe ... "", d,rroion or,n oppot.ru duttllnn. (in 'N: ca .... he", C if br:l_n A and ll! In Ibis caw. AB and 8C Ibwld 1>01 bo ..... U "10"" Ioni\h .. bUl.heir d,rf("OO'" ( .. -hM:b .... opposed ;n !be .. .. .....,. "' .... abo br: tlk ... ialO _A, 8y ....wtlU", CIII !be ......... pi 01 ............... 1 prod .... , 1oI1000;nl;n my ra,ber', 1oot".pI, I came to the ooncluwon ,ha, nol ""Iy roct.Inl!la. bulah.o "" .. 1Ie1opa1lll CO\Ild he .... o •• ,he prod""l< or , .. " .djacen' oiJe:f. pm>wed ,h .. ,h~ ,kin ~'" DO' .eon "mply II Ir:nllM. bu, II.,.rnIOO "",,,,ilkHlu."

/U ,he 'hobo. or,."";.,, A II "nd llA II <>Plot LId .... "'"odes· mal:es " pouible to "Tlte AC" _ A8 + Be .'1thou, '.k ....... '0 .poo,1III iDlullion. .. .., .1lI....., '0 ..., ,h., ,he ""'''''' 01 "" .,....n'od "" .. lIr:I<>pom mal:a" poNI~ '0 demon"'",Ie the d .. ,ribuliVlty rd.,,,,,, loe< (Qi1J\pll (4J) G ......... nn rod<1!o:

So. whon I muluply ,Iw! ,um of, .. " ,'ee,ol'l by. ,hird oj,,,,,,cd in ,be ... me pl.l\nr: • • he prod"", obI.,r><t! coi!lcidel la"" m",' aI ..... ~~ rooncide) .. ,'" lha, obt.illt'd by ",ul!'P!)"101 oq>a .. ,dy _h or !be • .,,, ... :IOtl by the Ihud.rid ,,,",, add,", (bn ... ""ndtul 01 ,he ....... !be."'" produ<U.

'" ,01111 + C) .. All + AC ", Formula (4) _ crucW. .. Iud! it ... ·hy GntH""'"" "1Inll 10 lIlII$ullle "

Fouktnally. bdoo'< offennl" .,..,..,...IIefiMioto of ,bo prodl'Cl;

~c obi .. " • """"""qllpI'iallonru.'KK1 (rom lbe .. "ILtDl if n .....te tbe .,b.oIc It"""nl. 'hili i. '0 say _" poon' 01 II. d .... · • ...,...~, or • d,ff.",n' ""tu, •. In iU<h a .... y ,'''', "n ,h. poin .. ro ... "",,' • ling\< .. """nl. The iUrf= !Ilk .. lhe fo.m of I ",,"Ikloparn [S".., .... ~ l

We .. ill "'l,b .. '''0 IUr1lIca bolonl'"' 10 1110: sam. plane II,. dta.,n.n an "'!u,V\tknl maJl ..... J, dutuL. rl>r d".,. ... m.-ul ol!be d,m:t_ 01 the ~ .. ",""'1 ill ,lac d,~_ do6nod by ,he ............ " .. IW' ' .. y LJI

,,. boI~ """" 10 lu,n III lh< ....... dll'CC'IOft Ik>r cumpl£. l(n>llr<h the IdI. '" botb _~ rhey .. iU be AIII.n he UlI<'<ju:ol UI tho "1'1'05". cur..

W. thllJ ,mmcdla,dl obwn tho 1oIlowm, ..... who<:ll it ....... I*= .. II II ".It.lll/ ...... fII-. "'"8=-111 _> ~" /,..,.. 1M, ! ,

0/ ... X_ .. ,""" "., ."'. 'JWfon ~ ( .. f a~ , • .--........ II{ IWU"'" u.J .... "! b..,.,.J ""~ ~"",otN., .toowJri.., ".~ ...... d. 1_''''_ t{11wM .. xo .... u.

Or If. i" IN pIil<ft. " ... 1*'''' U ~ M~WIO , .... , .. " """,1111 billa. /II Jld ~ -ay IU '" fo,..! juri! ... 1M '''Xi"",,,, of _ /DtJ '" IN! .,.., 0(11" Oflt.r •. ,"'" ,'" '''',,' tur/{JI;~'" p""",,J I, """')~ Iitt _ . ......... , .... 10. /Nl,h (.II'~"~' "" broJ:..,; '''~tll. ";,h ,'" 10k rl>l\J(,/M ,h~111w: 1l<f~...J luw '" JIM ... js "",1"'~ltted.

Thi, I hN'~ln foJllow. immNiatdy r, ,,,,,, the knnw" .I\rorcm p<lralkl ... ,.. ..... Slu,,'''' / __ ./w ....... IHJM ullll ,i,;OIK I" IIw ........ """,/1" Nu', II" "'- _I"", "

' __ 'I

f" ilure 11 ,I1I1>1l1u" how, by • <:n<n"""""I01Y phel'HlmCnon an.'<lJUuj 10 Ihal .. h",h mlk., " ", ... ibk 10 dnrll)n,"",c tho, A Ii + Be _ AC'. .... " ;n 11K: ca .. .. lit", 8 II OIlU..Je .he "'8ftIC"" AC'. Ih. ,urf,.." .,ta d&-nbcd in I"" pl ..... by a broI<rn ,,,,.,, "'1,,"1 to thl' ddc;ribed by the 4traliln line 111111 hal "'" lime onniollnd ~n.ol potn!> II n, ·r ... ru .. nbu,,,o') f"rmulll ~ l.hul mmo .... \'lOW and .bo "tnir.lde. '" , .... 1&tbnoio -.idnion ",.be r in'" onen.m I~_ II n • ..r.kd.o .be oocond .. kp mobik unoB. TIl< mulliploc."on '" ..... """" " ""'-I • m .. hopiocalJOl> in .be IiCIOIt oft.... • .al lheory '" form. 1M ..... 6ci a Cha ....... 'o-d, ....... ...or.a] .bMtmll_ bdooot

A<COI\ILII"o lho """"'P" dc>eIopcd In ,"" ,co •• allhoory '" form<, f,,,,,, tho.1ooorems '" Ih" po"'."'pb and ,"" ",.....ton. one n tou.,...., lha, ,he! 101.5011 "' .... ,..., JCJIIICIIll IJ and It. "' .. hodt ''''' mull II the ,lin...., tloa, " ""mned ""Mn the fi", "PIIL"II1 is dlsplaord f,om lhe .. cond. " I mllllipl .... ".e 1 .. ,,lI0II; btcou.e. as iI immcdaltly "pP""'''' . ,hi' rola""n 10 oddm<>11 .hl' mak .. I • • m"hip/ical"'" ",'aM for lhe 10''''0".10

If lhe .. nonl ... ' ....... an n .. ch.,...., 10 de>ill .... t Ih .. Iype '" 1",,_ ~ml

o::;~ .. " .. "S"S CAI'W"" OJ· TIl~ UTl"'!iION

01 the m<>.-N RpI'IC"D' II "V~D fim. OIl' b .... e:

. nlb + ')_ lJ n l> + an.

'" '" lI. e _hQ .. Imp"'llInl lbe I1I1,ud"",,,,,, 01 o.,attccl ... n_ 10: .. _ lbe pam:

'" A,pnd·' bola""" '<10k ibc of'IXWlt ...... 10 ....",mulalion ..... "eno, ...... the)' 0p<'tI. d,_"'" koq> ,bo ..... "".'" from bootom,nl o.erwboclmcd h) the dlillm '" the II ....... y nlcn<Icd .urfaca. "The oompon..,., oilhe- vu."'" lheIa .. '" d"lflb",,,,"y .. bICb .. <led"""" from n lhe-""'''''' nllCtly ''I_ad...,. lIoe «Ia" ..... bel"'f/<"P III<~ ""'Ii'IILKIcf; whICh ... d In',,lflld G nwmann. AH .. 8{_AC.

II i. Imporlanl 10 onllylot lhe powet '" mobih'''''''n ,,"led ,,"h oflOlllm m'lJ\ilud"- It IS .. '.11 kilO"" 1 ... 1 ... lmt .hree poon" A. Ii. C .r. oJIl!o>td. lhe ,.1.110n bel"wn d"lanca AB + IlC_ AC " 11<>1 Iinys Inll:. n.o I1I.On .... ' In whICh lhe . It ..... "",nil aro dlllnbuitd II 'mpottd on me, 1111111' no relalooruhlp bel"'_ , ... dc/ft","1 pmi1l<>nJ. Theft .. I\il l .... iSC>lI bel .. ' ..... he! Ilal ......... C .~. B. r Ind 0""' of ,he """'(ICI 01 lho IhR>l' "",nlS IlIQnl me' <10 I,,,", or rdlollOnlbJp II ell .... ", ... "bcd Of "' ..... m by ,he SCIICl"l\Cl"lt AU I ....... 'bore 11 SdIlply cluu pounl dHcnbuled on • JlnuPI I,,,,, o.od I tortftnt m,odI'lo uha\lOl.l"l ,he .. coI .... II"" and 10 8ddt ... up Ucc dllCf-1ICiIII: lbey an: .1 ......... 1ed by. tblrd po'" "'hQ .. mdJlfemll 10 lhe .,..,.u ..... 1 reIa,1OfIIIup. for ... hom I'Olhlt\lll .1 pLay bet ...... n ~. B and C ,Ius .. tlie POOOI of ...... of.1ie "''''rror ... 100 carduJl, JU1UpottI alrndy r.(Ua1 boll d •• lmt>OII.

Qllile ",her ... he ..,.,.. ,bal M >lo<lrr. ,he "",au A. B. C u pIaca v""~ by • mobok "",nl \f. Iht:y lh<1> awear ~l. lhe llOl'P'''I potnl.l of lho _ mocion. B)' scann. A. ,1It-rr B. llit-rr C. 0. ,n dtflnlnllhe nola" ..... '" lho <1 ........ 1 ABC. I at<:om,,:InY lbe mobole "",nl ;n ChooJbI ."d "Br 00",,""" on ;nlll.lie", lhal of. painl M lbal .• _. from A 10 D. lben I,,,,,, B 10 C. A. B. C att •• plknly It"~cd IOl!<lh .. aDd lbe jC.ltmmt"llilC<:eu"il) il Ultctly Ihal of Iii< .i.,lo; Ali d_ aOI refer la a dala""" measured by .. c.ltm.llltm,! 1"'"). bill \<> I journey rna"" front A 10 B Thi, juUf1lC)l CULlid be lha, 0( 1M P'(NII< . ..... .,ttl mc'anturplo~ lillo I mobil< "",al ."" )«1 capablt '" ko:q>tnl ibc 10l.llo.l)dlbt JIlII' IICY btl"", lui mmd. TIl< rolll;"" AB+ Be _ AC .... '" .... thi.o!. """" lhan I ... 1»<1 •• 0 ... 1) of I look ."" .... """lbiJUy. flnaU, otr<"ll'<J htcnlly. '" JI"UI'Inl mobtlc poml.l,

" • " C

I_ J~

" • A mohol< potnl ........... ru>-en;,'y. A. B. C.n: 001 .. mpic .. II" d"lriboolcd

on eli< 11"'1..,1 1" .. , ,be .. d"~AtlY is on lnl .... ' pon of ... nJlt (riOC • of bo:wrn, .... Ai .he ntlrod"""on ""m.ndcd. the di,uncuon of lboot pot"" II 1 d"I;I>Clillfl Nr".",. ~ ·dl...,..,m """,bono'io,,·

'" II it ~>dea, lhal .\ is q""~ easy 10 <aDCd 1M JOm1'CY A8 '" lhou",. by

", •• '111", lhe ""'., ..... AS Ind SA and 10 .... lhal the Inn .. , 01 84 "I'JIOI"I ,"" nlmi! ol.4 Iowan!. B. So" ''''Iy ill wne 11410 ,millie pr<aody '1*1.al """, .. Iencc. This ;. tho IfIIWtioa III the ~ory: .......... 0 .. U\ line!

... bord"""ioa In. clod. l'hIl .... by the _Uon .18 00 Mh. ,'\aU) M ...... U • joo""'Y" the ""', ... and Ilot btcnl_ pcrfoctly """Fd ,n il.

C ... ,:dlIn, lho: lDlwtioD oil paralJdosram II CXlnJldcrably """'. dmcorc: II to 0(>1 wry clear "". 10 mdow II ""Ib nlObthty. Gran"",,,n bo<""". M !!btu.' kb 01,,,,, oriented I"'rallrlopam.. The OOIohonl ABC and lie B "*' tapee· ""ely lo,he poIilionll rdauOMh,p!I ond OII'CWI< dr .. ,'n ,n ~.U"" D_ 100 Ub;

• • ____ 0

Tbae I .. the ....... ~Med by Gnwn,nn ID biI ddiNlIOII "'Ibe prodUCt. Thn "",.1"", ... u ....... aoe1ic propuIIOoa .1>0, 01 much lublin lhall ,bat 01 .hI: !ramI ... 1Im: ' I ... "'~fp>iol from A w..-..rdJ B' In tho: _offi.u", 13, 'I h. d ..... on AS lkel "' A and bad "' B) and I ..., .Ie 10 my""'''. A "'ngk notl'''''' :"8(,,";" ao,.bIc "h,ali", . ,1 ,hac 1K"11On11l)'in, do.· ... "'"-"5) ..... of l!!IOCIllIn,. circuli '" the pl."" wilb thom,

The or~led ""I" ..... AileD is dofined In ,"" Ame .... y:.n obKr'I'tT '1)'101 01\ 18 ,reel., A ).ooeo C on h;' nsh' Ind () lC .... rdl ,he lOP'

I.

We II.",,, ..,pe4lbr on" QI,he tramI~"'Y QI ',a"""OI1a. The Lan., rd>ed 100 ...... "" lhe Klftt,t.&::allOll Qllbr b,<nJ and !be IplUal. l"hey lobcnod only d .. """le",plallOLl 01 ...... 100 ........ , ..... -ard. B. II ..... 11 II IM"'bok body IU' approp< .. l.,. a.p.occ By UllJiulIJ on 1M oro... QlIIoo 101l.n, .100 ....... uon ",11("

Of ... BCD ""plureo my"""' .... 01 con •• mplallOfl by <poctfytOI a modo 01 'rl~1f,uop <A 1M ......"bun. An """"Iod ,.,Jun .. Of. Mne •• U ,,",""'d "",pli. IwI~ doe. 'Mn unclion ao .,.ubh,hod .... m.on. In 'o"'f1',o"'d'Pl' QI ......... t,u, ••• her a "'''"\l,1 or. v,nual ."volopmtn" , ,,. Jla.ed o,d.,.nd lhe am'"

". ...... IIIWPo',ablr;. I .... B.C-DIII ....... pk auretAI. QI kite1'$, ......... :.!BCD· WI,ICIIOM a ~~ QI,,*,,, thaI II ",,,,,II ~ concm. ,b.aJI lbe .....,...,1MIe.

1'" onooled ft!\'dope abo ..... Ita ,. poGbic 10 .... 1In.w ......... 1o CIII oul ·SOIIn·. 10 lbe oorulntCtJOll <A.be .oecond-slOJl n'm ....... lhu II .ut md ... • 11 JIOf'Ihk.o o:""""'"from .. hal""" tbe mollOIl , hal amII""" ... B II made 10 ... WfI 0' ...... pa .... 1Ido&ram- In t'" cur <A ftpI", I~. lhe rnotIOCal mWI1 and an « .. dlod buc ...... can «I1UIdn lhe qmoOl"'C the -he'I"" , ... luI;h dlrecl!'Rw moloon. u ,ha. ,,·II .. h a" .. ull,.,. and .n ... ""'" lbe o".Plod N.w. Gr:IIssm.no'. ,.nctv.lion and mult'l'lkallon uk. up o..,m." m<>bIlr uOll aplO by 111,l(!C1Ld, il on,h ,II< dynamic: of the o".nl>.llon.

• r_"

C £.<rtrMi proJWI (Iu ...... MuluphhllOfl J 0,"" 'M I~.NUJ,.., .xr ... >I .... I.~!«'~

G1"lI<lmlnn III.n I've.. clearc:, defin't,on o(lhe (u, m <>f lhe: tro .... 'ncoll"odl>Cl by .1"",,," • • he: dl<!inc:l,on hot"'"",n lha' .. h,<11 II "",~cd andlll!ll "'blCh m""<:S:

Fu"~, dc\~lOf'lllC'nl no .. dom.>nds lho ,. ........ uon or nc""ypeS or utrn<>011_ The mann<l" of Ihis 1I' ........ liot! is an ,mmod,.tc mult. by allllOV'. or .he .... n ... r In wbkll .he lim· .... " n1<l\SIOn """ I""'~!od fR)1ll !ho: ........ n •• It) ...,.. ",t>jeainl ,n tho: wnf ."Y aU lhe: ek:menll or I ""pDtIIl '" anotlter ., .. ,.110<1; 10 he mo~ ",*,fIo:, the ..... p1..,ly <A the 1IIIII'I"uck .hal ;,. .., .... 10 he ,., .. ,.1«1 dtmanI;b eq .... bly oa lhe: mann<l" ",,,,lin,,,,,

alilho: <1<_0'" lUI is 10 AY alllbr .......... 11 <A."", ....... '" II dc:>criho: '''IIP ItpDtlll b. no., ",,_""II' "is po .. "tcd hen: u lUI .. Iu<:h 1<1Xial<S. ,he "' ........... Ot II .. 1br _'0"'''1 01 I,," 1<""'.'1OIl. .1111,," Inult Qllhe ., .. , •• ""' ... If" aflll 1> .... ,",UM of d"feren! ., .... ,,""_ a PO" or the: IC'COnd-tlCp .p, ..... <Iet.rn""",, b~ II and II. whICh muot lben be unclen100d u • -..ncl ..... p .".mmn [ ... ".u/UI~ ........ 111" .. !ill!/<" 1 .. _

tor the momoO'. IU, is 'O"'Y unl~"~ "", .. lk .. rm"",(1 more pa"~. 10,1) lhe: n.>tw ... of lhi. ha,,,,,". '0' dc . ...... '. lh ..... ulll"lIp<oo"'" of II and /0 by II (\ h. and [0' lbe m.mICnl by ,,(\ h. "he ... " .,1<1 h arc: lhe: "~~I"

'" ..., ...... n ,he e.u ........ 'Iuch II """',"led of eadI dc" .. ou of" I".""'" ,"" ....... iil" ..

Gl'al4IIIann'. mobok unll IIIonri<m: rombiDes tbr onnnihon IlIId the oolmw'" ulmSi'" Ihalecl:.: alWlll-' culo:r I...., p. III .1Id rollowt~~ "'~ CIlII oIf .... d.a,nmmac.c 101"JW1llioa of .It" prodl>Cl. lei uS hnl up lhe d .. an .... of the d~ KVflml and of the applied forw ,_ """, ... 8" 3 ~ ,bu. ""''''III 1/110 ."e"""" (onNllion r, n" prod~ ,~ rltJJ ",lin If""" I J I>y I,) I. on ftlU'" II>.

n [)XOOOCX] ::

" , , ~_'I

I ........ ,. ncb potnl <Ii f. (_""'"" P"'''u • .., """"'1111,. d"Pt. wd I th"""" .he pu •• do<pbcftnml d,n:cud aa:ordinl to I,. ~ An, in'cn$"" , ....... J~ .. led .. llIal .. hoclt """"""'. 1M l"1W!nllon and .he ~ Inm .. II .. bo .. ell"" ..... nlioa.

n.. ntcntioa fO.IRIlIon ',n" pemlU'" tbo .~ of I,.nd I,: /, btwmes ,nom,;,.. Ind" u, ..... ve..

, ,

" •

fp,."

10 10 )I II )1 II

• • • • • •

y,. - .hlt ,h .. cI .. tlDCO"", til !he order of .u, ........ ~ " ,,.,.,, I, Of " • ..,,,

I" ~y <eprOd ... ,be ,.'0 dcct<oon$ _hit fa< " ... L'III lhe rca_. >nU<aIly Of "","""",all): .be ,." mod« "''librauon' '" lhe ~"""I ra:t ....

.... '" Ihn'ekIn: eu.oIly .... pcd by tbt IIOIal>Onf 1, ('1 /, Of I,rll, Tbeoe ""'"'K>RI ... Qlon:c the d .. """,,,,,, betwcm tbr ..... ' that ........ "" lib<. mulHplltf) and ,"" Pluen, thaI""""""" the .',,"'._ 11M mulnpl..,.nd), by demaod,nl liNn .be ordort orro."u.u"" oIlbe prOOIlCU be .. rtfuUy speafied; ~ n il ;1 obI.,,,",, by dlJpW:tmcn1 ol.be potJIlO oI.be 6'1' Iqrnm' (,be 'buc'~ by .he Kt"", or ,be lieC(Ioo (Ibe 'oogh,~ no .. r'odllCl IS nol .... iloble.,. ,run, I .. ~n ."m""" lhe .... I. 'IHI<'" 11 ".0' a R",u,d~"" r.,ure. nlCt ",ok, of lbe .)'mOOIJ load. ,'''' 1\0.00 ond p~nbcs an 101:,,,,,1 C"""n>etlon 0( ,Ilt ·ficu •• ·,

'" A ·,...,. ... """ .. 1 word' • n. (, In ","CIy . n a) .. coonod. n""~", a d)'i\amXS '" ,he """ ....... 1M poo~u of JeIiII<'III 6 ( .... ~ • .-dy a). ... t.o.. "lWbLi JS. is aJrady .... blWsecL .......... _ by ,be unFo!d"1iI '" ,be "p>nIl"n1ftl5!1y • 'reo~l.-dy .,_ 1M ., ..... 010 pan"' ... '" In • teIa,iomIup WI iD'onectJ

,hom. ... hcteq jtt ... poJInon """, ..... 11Ie1l"',lh ....-rdy 'llu.w.., l...,..torr· ...... eku.,ul lb., I'mWJ! $lrll ngen '0 0 ... _nolbn- '. + 6' ;1 noI ... aIly a .... 01'<1. Hore, ,be I\o.nd if impbcalod on ,be pmdlM;"!; " .... O'd .. ' .. ,tS. '1>CaMI'~IY and • loIdc.by.oodo; it bnng< about the ...... d",,,,1UOOa by ""'bod~'nl an oo<nl.1o""'" ,hank, to lhe F"fIIOlncal prodllCl. lhe pia ... "l .. pol_ina of a rnoblll1y lhal ..,."".,.. ","""N fM ,·""o~

Oraun.unn adllO"ledges his prrplnl1Y before Ih" lIranl" mul~p"""~"n:

Al R", r "'a. "'I')' d~nod by IhlJ llran.., .-...;uh lbe olhor la .... or ordm_1')' mulllpticallC>n .. ~'" all reopeaod by lh, ....... mult,,,,,,,,,,on 'in ""nocut.r Ih. ",ta, ,,,o<ln,, 01 mutuptlf:Ol\lon 10 _<l,1l1;0") , BUI ~ay """,,U' 111"",01 the ordc. of tho: facton mlatltd I cit ..... of "'lin _."

Itt tirna:s IhlS auaat "WJnltioa 01 tbc: crooo."I f'"", loll 10 nlltl (and "*'" '~IM' .",h .he dult.,. 0( Up:

l 'he )urfaoo oIlhe ""'rlIlIolo ........ Ihus r~ 'Mtd 'Pm .. tbc: utnml prod..., of ''II'O .. porn ... and tho: ,0/"_ oflbe ponlltloppcd .. "'''"'''~ in lbe .. me .-ay .. ,It.. u,....w prodllCl of'h .... "",,,,,11. """"'" ... 0 fi ... ...-'. lhen make ~p lbe sides 0I11t.. ""ntk\osnm.nd lho Ib..., III,,", lhe nd ... of a pl.nlklcp;pcd, To he mo •• >fICCI1k. ..... ,oot lit.. ........ 0' .. h_ mPllon formed lbe "",,,lilt,,,, ..... u , be ~rst fa<;1"".nd lho """"011 .. h",h mr.,.,N lho mat,Oft U lho """00. ,,"h lho "'lull 01 .. "", ". call '''0 "'1 ... 1 por .. llt!oanm< if, .... n from lJoe ft,-" facto •. lhe """00 II on lbe ... me .><l •• -and lhe ...... li or ,,·h ....... call ,,,'0 I •• '.' .. porotkJos.am. if lbey .... on ,h. Oppoo.tlC "Jo,

from lJut Ol'U>mann Jonv .. lbe .ult:

'OJ 'for'. ho M)'J, 'if b. -.. from ... II on ,be Id\, lhen .... ~n from It. m",' be on lhe nJh~ and n" ...,ra.'

'Tlt .. II lhe "","",ulas IOnn 01 a ",""at Jaw 01 SI&JIS .1ucIt G .. '_"na I_UP: , to dtd_ I"_.neall)"

11 .. dtat from tho: COO<epI 0( lhe '"'Pm" lit.a, If'the .... and lbe bori&bl 0(. pl.rlIlltlopam keq> lho gmt: dllMlOD$, ,ho .ufflce preoc.vn '" oi ..... non tf 110 wdn IIICfCllo< .,.. dcctea.., I( mo"",...".. ,be .~trm"') ",.he oolhl di.pIaca ;, .. If on.o • 10"" porotlcl 10 lho bqc, ,,'!uk lhe Jalltr ... mo" .. the .. me..hm 1M surf_ ... yo """b.IIJII<l; n the ... fo", fn"'Cn'n 115..,n. w. bo,ln f.om IIoeoc 1"'0 b)'p</lhoon 10 I"" .Ie F<'!Ik"ical JU.lltk1tllon oIllIe getlCrall.", of ",IUI,

, All pI.",U.Io,n" ... wh<w: hollk" ~n from 1he ba ..... main on .tv:

'" ....,. ...x mIlA kftJI \hr1r SlJ!n. II> a1tot<tha IUlofoUs ",h" ... or ..., dlS!>o","" h ... 1M 6nl. ...... ><1 ;&mIlhanl rid,. ... .., .. ';U Ix able 10 Q",bh.b for p.>ralklepopeds [S"",.] lhe la .. - tho volwmeo of ,01'0 pol1llltlc'ppcd. '" .... .., or chan,. ,""', ....... ~ 10 ",hetha OM IIIU1011wra 10 d~ ......e ... Of 10 • d"J~mu sodc II) pa .. from !he <1,,"'1100 of d •• lCCOI>d 00., 10 ,""1 of the third "",.. it .... obotr>"CI' iii .lIIqIooo<J 1)11110t1 . he 111101 "dac lIeo, ., the ,,",,0 and be:Id., tho "'mn"}I.·'

The rule (6 "::10 be pro."" by OIlier mean,. h,n"" .. ,nllhc ' • ....,.,'n. CO/Ilotr""· 11'1<1' .• '. ha,. 10 '~I' .. ;th ~nG I .. ",lIdD,rom of I."" ...... .. nee Q II al once ",rull i. mo' 'W . 1>11 whal me...,.. II II III ... n.y'" 10 ..., Iha, the .. Iauon ~ n" _ 0, )0",,,<1 10 Ih. propcny of dim;hul;' il) \ $), .1 .. , ,,,eo; u n ~ '"' b n a.

T""'" proportlN Imply thl tbe .. gmtn' " i, wt" .. ~r .. "h m bolonS' 10 tbe ."In! Fneralcd by III< ,."""nl" i( and <>nl) rl'. ,,1"1,_0 1' ...... ,o,nal prod"", I&lIC1lons unit the ac' Wll KIIU;';' '')'' 01",,,,,,,>11 , .... , ""'leo ', he factON """. lpolfl" 10$ OraamaoD Itinudl ... Y"

I PfC"iowJl"" nib! II". prodllCl _ In ",,~rnaI prodl>tl, alll>llu'l b, ,hit; ... me 10 ,10: tact ,,,,", ,h .. lnld ot .... od ..... only lou. non·1lIl nJ ... if.be facl.on ...0 .... port .. 011 ri.be prooun .ben conOl,'ula I flltlbe. u~ Tblt produc1 'i, (NI ,be o.he< hand. ~uallO ttfO ,r lbe ooon btIocIIlo .be .. 1M ul~_ (",m."

no.. l"'" CIII ..., .. be euC'Odcd 10 lbe ro/Uuuetioo or ,be ponliidepoped or "'p .. ,be IaU., bcI~. unden.ood II ,hr prI>II'OI>'" prod"'" 01 ~ POO"'" 'fb .. II G"bm.n~·. trvctal Ideo.: 10 la,,,, rourod .hr liai.on ,hat mike< n po",1>Ic '0 leap from 0 .... dunenoion '0 an<>,hc:t ."

The 'el,omal' or .pr(>j .... j,"' ... prod",,' .,.~n be dcfono.l r", uy 'y".m 01 >lOP " ~';'h "h,~h an one,ned vol."", h .. bc:<,n .... "',.,cd. 11". o,oe",ed ,·olumo." called tbe 1"'/11<1",,1 ...",,,,,. by O"",",."n. m~'O$ a po,,""e ,>t .. p"ve numbet COl'f1t$pond 10 ,he dalnm 01 " .. """,m ~, ..... Uo u h n In ' hIS o'der aDd dtototed (~,., .~.l Thi, pnIlClpol "",.,ure i .... 0 .. mLp' oay .• ooq ... _ of ~ leiters !Mdt mobiJe; i, 61es " .. or ,he , .. 0 ... )' oIe.,·.IoI""1 ..,,1 ... lIy • ponollcloppod bqj"oio • .."h " ""cmm'" 'Out prod"'" romb,,'" mobole ')'"L~ anaJotou> '0 ,ha' 01 ~JII'" 17. iii O ........... nn qu.;kJ)

mlphM'

\\" no ... .,end 'h" ddin""", '0 I"Y """,ber or Wfml .nd lor ,he 1II<I<IIrn' .. '" ,o,end 11)'" ( • ......, ... /Lo.t" .. IlK any number" oI""",,"Ii).be """""0" .ha, .. (onn('(j .. ben each deDl.n, 01 • • ''''"'0 ,be ICJlIICn, h. each or ,he ....... nli.., obtaur.ed .he" FIIt"'''"I.he ",,,,,,,,,, c. etc."

The (O''''rut,i<Nt oI'hD product. "'hoch (in .. m.n" 'hen ~ba; ,n d<:to.iJ. would ... bcy"rod ,he rtll"'" 01' DU' ..... I)'ld. "' . ""II cooft ... ".,,,,,1...,. ,,, r<a.lho. , ....... nHAI loa,.,",

I.e' t. be • ')'f,.m or .. ep " ~ .. umed '0 be ,,liii0'''0\1 "".h • l"i .... ,poJ "",Uu",

m (oneo,cd """me 0' """,,Icd ""'pb ...... ,lknotcd r." ...... 1 II illbeo possible '0 defi ... 1 prod"'" 1"J>roIt1::sto",,' or ." .. 1 ......... '1 'lOCb lhal.

" ., ...... an: "II''''''''' ., n ... n •• II IlI\ e""";"" form o,-"cp I:; (be P'OI"OW"" product .... ka .1 powble '0 <TOM (,"'" ,",Ie"""" form> or"<"P!'o Ilcp ~-t I; .be 1"01'"""'" product $II'O$lKo: Q,n .. nQ, ... ,n", _ _ ••. n., .• ,n ... n... In pro.n"",la. ri ..... oIlbe ....... .; .. ,. 11<10,,1' 10 Iht ')'flem .... ,,,ctI tor Iht 0Ibcn. ", n .. n " •• 0; .be onen'ed .'oI~m< 01 (~, ..... ".J is • I JIOII;"''' Of ... pl,.",f number ~ ... , '"",n. n ~.; ,1M: pr",r~"'. product .... isfK.s Ihe • ... ,ural' propc""" ..r a......".~.'I.y ~ nd d i"nb\tUl1ly fo." mulupliooou"",

Qn(bnr) _(a nb)nr Qn(b +-<1_ .. nb-t .nr

(b +<)n. _ .n~+ .n" IIu< prod"", os not ...-mu'.ln~ .. n •• (- 1 'f b n ... If " (,.,.p •• IJ~.Jy b) " • fonD 01 ~<"p ~ (""pttto.d, 11-

Tb" mulhpbc:a,,,,,, I~ makes ,I Il"'Wbie '0 propel <NO .. .If from one "<"pIO aftO'Ibtr lin< l"!" fios Iht dc$I ..... I"'" 01 ....... 11 or prol"'''''·. product ThIlS. i~. 1)'5'''''' 01 "'p 1;

II " • "' ....... iI •• "n h D • "" .. IIcl"""m; II n ~ n" ... 1"'",lklcpopcd.

II iI,nlerHllnl'0 ••• mlll¢ In delo ,l .he "mo.'on ... " f""" (III< "01' '0 .""'oc, ".,m'lI"" tor m.'llphca1lon. W. '0',11 be able: 10 """"" . "" .. ,he n nted _n"elOpe m.l a " pou,l>le 10 dltect ,he cha'" rA Iyn' ..... Ind ,he mnbolu.ab" ... ",I.,ed .0 lbern. no.. movemen. from Ilop ~ I ,0 ".1' ......... kJ "'" be utldo;r-;.~ If lbe formal IIdJunctlOlL of. ~«'''' .0 a It~ ')S''''''' Ibt .dJu ..... '''''' limply mak" ,t possible '0 ',h,nk I .... 'ber· ...... , .. ,,' .. 1l1li I .yst .... rA .. - J "«I ..... L ,-1"'" .m ..... ' I, .be,_ ,be ,,'"'u ..... ,he ............ amph-.1Ode. { ... , ....... 1 .. not (:(m'm' "mply .,,11 pulhnl r.., '''I''ber ..,,11 ~ bu, lin, ,~ 'bcm. '0 ...... "'npooo' ,utI( fIfi~. u: r.'I' .. ,.'.1-{"', n . ,n .... ,] n ....... , n ... n .... ....,,·sa.»u ..... be prodUCf .', n .. ,n .... ' .. b.;h ,rod""", on orien.cd ,-0[""", [ ... " ...... ,] for ,be 'Y'lem Eo" (01 d,nwn_.. I ~ 1f an orienled •• "",Iope I.,. ' ..• 01 or ordet .. IS 31 ....... " dcfi".. ,.0 '.u!n' ,n ,be oyo""" r., · .. po .. ,ed· b)' E. ,(I<"< fir;. 18 lor w • l ~

The lode from ... ·h.ch .he .yo'''''' E._, is '11«',,' in ,he pooH ;'", d",,<",,<>n i, "i'fIOIed '0 111:0, f, om .·h.ch .he 'YJI"'" ;,...,n in .......... 1 ... d",.c"on. Th .. '..,., ... r,om ,,",,><1<';, no .. !l" '~n tor ,be """,,,«I .. , .. lope or o<d", n, l 'be", e.,,",.,o cia ..... or "' ...... n'."

(jv.ss"IA~~"'S CAPTIJlE OF nlE EXTU<SJOS

... "af;'" /' I»' III',. 11',. 11',1

I ~ ....

w, Illl pi .... .,.;..., .. ,... w, ., ,W,. W,' V , ""' ........... I,

I , II' ~ ,

loo.e" hieh Ire ,o;n, 10 mo,. t._, 10 G(ln'lnK'l • ·"",illv.· panollclcpopod: Ih_ "'luch are gOIng to mo .... t. , 10 conmuet a ·IIe,.",'c' panoUckp,pcd,

We sec that tlo<: oncnlal em'Clop"",nl prn<-rot< lhe "cl ual conmllClioo of lhe eXlen,ion form, of 'mler n 'beginmng from' I""t or orde. n - 1.

The nclation [ .. ', ....... ·.1 01 the orienlal "'lome ... ....Jh. po .... ,u/£>II: it permil' the compreukm 01. whole .\ttn.orio, W,th the onent"'" volume under' stood a, pfoollCl. 110<: g<:t>mclcr propel. an ,n'",ibl<, body " .. hieh ha. complet. freedom 10 conl.mplale from lhe outside the dreu, jJ conlalned in lhe hon.ontal pl."", defined by Ihi. body. Th< formula or the PfoollCl ~·,n",n ... _ ["',. ,"" ~.l permil, • dcrornp""'l,on of th .. sudden posse."on .1Id .meulat" il "'ilh lhe .]rr:.ody t<labi,1htd .... 1 ..... "" fornlL

In ortk. 10 define numerical mullip!lo:alion. n was suffie~nt to comidor • numher ., a uni. alld 10 Ilk< ;. ... a new base for calculation, With a",,,,mlnn', ptodLlCl., .... ,,";1' 10 ."./rnla' unll and .... complc1.;t on • pan;colu ,ide in order lu ..... 11 lhe: already " .... n orient"'" .n,'Clopc, 11 iI theRforo by P"l,okUlIl an an,b;llMlu, .tal. cf alfl'''' .. ·hieh IS then di'l'lpatal by on uncnl"lKln Ihat a dimension cao be conq""rod, O,knu4 ""Io""s ""I",, ... 10 Nt ..... 0 ml""",. I. Qtt#:', T .. e,tlen""n is lroused by ,pllltmg. IIId lila' 1J whl'. as .. '. ,h.oU no,,' .... a dImension JlOSIled ... mobil. allud., 10 anot~.

D T/o" aliWllon 10 ,/0" ",xl dl"",nliolt

Dy ],nk;nll rnuillpli<:a,oon and on<:nta'ion. Gnw.mafl" <Irik", a d"u~Jc blo .. ·:

oric:~t<d 'olumo:s mako it possible 10 ~nk e .. t.ru.I"" magniludes Idislan«'>. areu el"l .. hioh by lbern .. "'"' arc .,...,n".lIy posil,"" ,n ,ue" . .... y lhal ;n 'M liai""" 'the """'"'Ill<:I>Cei ,uJI!'I'''' oach "'her ...:ip",,, .. lIy in ,he ""me ,ubject· (i« Kanl. Arrt .. fl ,,, 1~lroJ..ct "'Iuti!:. Q"" •• III ... "'. finl """tion. p. 86. .1Id our com"",nlary, 111.1~ <.I<rnal muJUpileali<>n mak .. ;1 pDo$J\>k! to jump f.om "lie 'Iep 10 anolher

Th. is th • ...roOT I de""", for I rtH.'UloUD, d.grees cf In<lc:pcrodc:n<.>:. Tho. lbe comPCMlluon cf ,pat"l ,nluition of .rbnra,)· dogr« i. a ... Jly .\lo"'al. ,"'hie".

qune obvoo"oIy. _",h"""leal multiphca,ion couid not do. The radIC. 1 ,,,'Cr' .hrow Mou,h, _bout hy Gr ... mann', prod"'" ,n,'"I,'O$ Ih"'" key poin,,:

I_I mjoctinl rnobilny 'nlo ~ ,t.p Iha, alrcady has •• t." .. On lunn il ioto aD .Uu"on 10 lbe no" ".1':

(b) Ihi, 1\10"010 is caplU""" by • f<lt'" d<cidc<l In a .II"""ro" of ,""x;"",f "",hip;.> P,..".;ecdinl fr"rn a balance cf Bcmg and e"<mlpllfi<, the reacu"ation of the .,otion of gcncr~""": tbe .... mal producI ,nauiura ... the era of ,be non"""",mulaln'C and .. , .... finaUy, .Igcbr~ a pc"'pc<:tl"< by f""''''8 il from lbe ,rrr>' .... iboluy oI'formul=

How d"... a dunen'''''' aIlud. to lbe cooqut<, of a suppl=entary dimc"';on~ We kno ... ItQw Grommann. in comboru", tbe dJ.l31a,ions and refl«uon. of the mooes of change, 010",,,,. a 'f'."m of slep k 'bcainninll from' fundamenlal modes of ,,"""lIion, D"I dJ.l.atal'On alld opposllioo ne,·.r aUow ,be poss.ib;lily 01 juml"og from 0"" dnn.nsion to anoth." ] .hall.hu, n.'.f be able to 'dal""" formall)" lhe plane be""nl", from a 'Iraight lio~

i\ h"" co." he ,rav.lled in one dIrection or in ,he otlo<:. It ~ ,'dudty 1'. bill, if] want to I'"'P the t .. ,o moUon, in • ';n!!,l. ,ntui'ion. I mu", ·.talld bad', I ""nDol cootent myself wilb .x<h.onl'ng ,II and M by r.Ikc!LO!I, I mool cun,truel a un" Ihat gr.-I" lhe Iwo mobil .. or rM"""" rl .... 'Cr.,..inK' from .If 10 M ' by reOe<:tion leaves me defenoole .. ""hen f~ ",.b Inm,il: ,hlo e,,,,""" cannol . Uow me .0 pkIU •• myself in Ille ;1<1 of CQIllomplall"1I. M and M ' at "nee. I mu", therefor. drform .he mobile poin •. I f onlo lhe moMe polnl M'. and not ,)un from.lf 10 M' Thi, d<form~tion ........ ·,'lfily ioy,le< anolbe. J,men>oon. I can. for enmple. fold .II back on M' (oa: filJ,. ! 9). .' .

--=·=-_O----=::::::c·~ ••

f __ 10

We are in a position of .xposilion and Ctt.o .ppcar~ os K 1",'01. Here "'C "po:rielK'C .pin tbe m.tic of ll1iculatj"". ,"'h,eh mal." it 010,'10", ,hal .toclt, . nd .... uch Kn oppooilion ;. ma'tered in I ""ide. conl;nuum: he ....... hen I link lbe t"o mobile poin'> OIl a "noi&ilt Ii .... I am already positioned ,n a pl'n<:

In ~n idenlical tnIln ...... if 1 COIt,ider 1,"'0 drooi" 01' Oppusilr: J,rtttion ,n " pia"" 1_ fig.1O)nd if r ~uue.o JITl"p lhem in" ';ngle ;nl"ition, I ,mI1l«li, .tel}' "",Ilion m~]( in "r"""'" 1'hi> """"".,.;'" inmhn.n "ep" a <)'"em of 'te", beonll .. t.hh.hed. lhere •• i.l 1" '0 lyp<O of orienled .nlume"" linkal w,lh thIS .tep r 110<:", a ... ".'nal 1"001"'10 01 k .. """",I'J .nd. by deform,,,, lhe otIC

1n'0 Ihe other by .hou&hl '0 ~'",p them in a .",gIe 1n'umon "'hleh i. oot an arba",.), ju''',posi''"" bu. an I ,,",u!allon. I J>OS.hOn m)' .. lf on .he "" .. t dimen,ion

H .. ""'_0'0"0)' 10 emp""'''': thaI:

,~

/ , r--.. c::::::' ) c:::'::)

' -__ .'0

II\e d,lfertncc bet ... ...., a p6S.tJ"" ~olufN and a I\<llallV~ vol"",," mullS ont~ from • deci,;on rel.Il,,, 1<) the "",Ilion; the ambilu;ty is maxunal between 1'0'0 oppo.>:IC<l ... Iu""," and .. by 11<)

mean. established in the o1len!ion by «UM ~u.n';". intquaJ,ll<$ 1M eh<l"", i, nol made betw ... ., Ih .. "rid th.l "nlume, but bel""""n 1"",,11 •• 0/.1401'''''''_' 01 volo"",,, or citcl11lE,

These deformations imply the 'OVmhO" of" .upen", humogcni1.OIiooIL l1>t lllrory <11M .. tension "', ou l 10 ma,ltt Ih. l>1r1h of the cunhnuouoly d"l:ts<.. Thll d""rslty must nOl be taten B. Ih3\ of bloc; di>pcrsal In the extenSIOn. bm '·'''''r/rul .... ~ .,'" .. "" . c(\fI."lent defonnaiu", mu,1 proouoc il. Thi, arnbiS'" 11y therefore demand, the I1lO:'lI molot. 01 DOCIle propuJ,k" ... nd lhe: one thlt "most orthoconalto tran"livil"",.M <,,01. m",[ highly the Jl"h.re lhal cuI< oul and unfold. lh. form. The d,alcc1ic of the molnle poont ,uccce<led ,n o'"C .. co.",na II>c I>pJ>O<l1Ion bet'., .. " the .tatoc point and the di,'cnity of places. DIalectICal renefl"on demons, ... , .. ill PW<O'''' 0( ~actova,ion b)' on'en"ng I IUpeno. m<>b<1c unO!. In ,hoc .. me ..... y ,ha, Mgand ·, balance rompeMa,ed fo, i'=t,,~ '"Iultion. "'jlh ns nq: .. "~ I.m Ind alluded to ,hoc ..... Ight lin. of the ,ma"nan.... lhe ""rall.logram o. ,he te,rahed.on ."h, lhem .. !>"" into two ponible deformal1oM.

T"o ",mila must be ,ketd>«l In dont<! lines and a~ nor ouIdcd on 10 1I1e figu~ like e.C ... ~nces Of prolonpllOn1. They aroo .. ""'emion Ih.oogh ,ptilling. Two tnAngl .. g"flt a! ~ual ""Ii"" only a 5upcrposil1nn e."",uled In ,I>< pia".., .hey 10"," no supplcmonlary d,mctl.ion. The Ian", becomes neccuary If t rlccidt '0 11<1' ',"'0 .n.ogles al ~ual a. figures.. bu. opposed a. nTtUII,]U

..... y of ." .... Io"'nx Ihe do"",int By <kforminl these ci.eoiu inlo on. a""II><. "ino.U). I ]lO'i"On myself in anol!I<T "'mensi"" whi<h .11""'1 me "' "",I<m· pia •• ,hem ""Iovdy and to op' for <me of .h<m ""Ibnut being tcndernl pw<o'<r~ by .he i,l'Ilducible fllC1 of lhe nglrl hand and III< left hand.

lindo,,-C\! "'i'h ,he" orien,ed .-01"",,",- tht- ')~I.mt of dllTe ... nl lIep< an (herrin", unfold tn.m",I,,,,, inlo I double .. """ (see 6 .. 21 I

·I'he dllTor",,' bracLCI"~ hnked '0 I . ingle 'y"cn, of ".p~, .pHI ,n,o I""

- -_-i~'''-_-== __ ;'>'''_-='I' __ ,O E_, E_. E_. I I v'--C, '--+, '---~';', -

'" cln<a of <>ric",ed voIu"", .. Th" d"'~ recalls that of F--'Cbc!,ma)"el (<<:<: 111.4). Each ... " delinc:s In ~uihb"um pial .. " .,Id ""'tk. a ""use .. I>e~ only ,I>< flrol d.""" of J!C1IC .. ~"on is a, .... o rk : ,ha' whICh unfokl> ~ ,)'.um ""lmnm, from a clU~IO' of .. gmcn .. by oombrnm~ add",,,,,, and opI""IlIon .. Then """"" ,ho second degr« of ",1I(r.lhon: ,1M! ~u,lob"Ym of degree k la Iy.tcm of .. "" l)",," be brohn ,r one d",,., l de\o'ioct impl)'ing & mot< Implt ""n"nuum: """ mu>! be po:utioncd ," ,ho followtng d'rne"'lion '0 <kf"nn 'w" Ol'l"'.ro "olumos inlo ODe .nolha. We '"' Ibc"'forr In. dynamic p ........ pI. Ih •• undtrf;'" ,ho ""p,ut< of tbt e",cnsion: ,n. #,...,ra1l0n Iha, d ..... ,ndeed .ceomp:,")' a bal .. ".,. of BeIng.

Ext."""l multip/,alion ;, 'horefo .... In. fonn Ihal CUll oull h .. ·...,rnrd deg .... ' of gcncnuiott p~ly: il ",ma,n. ,ndiifcm" 10 ,he form. that bave alreody been lIO.julrul by add"ion and thc prndl>C."l' of dependenl or opposed fllC1oT> an: equal .0 ,.ro.

The progrcssi,'. prod""l ccf1;unl)" m.k .. II pon,bt •• o ."..,.,' fr"'" k 10 k + J. bu, Ih.i. 'cros,;ing muSI abo ... atl no' be undcn;tood I. ,hal of. e"nhn...,u, e~p:insiOJl 0' a dil.lalion. The dlme""on does "'" tun 0111 pc;Icefully from a poinl_""""'" ] Ih. ·0.-."n110 an .... "'"on form of "'1' ~ (30 oncn!ed ,'olume of o,dor n~ '0 po:ulion on.self in a !y'l<m 0( ".1' k + 1. ",!h an Oricn lC\! b..", .I .... dy c>!.blishcd, an .mb'~u")" ha. to be cleared up bet",'""n two familra of dcfonnatioru. TIN: InlbrgUJt)' llSSOCIaled wIlh !he Of","I.I"", 1",,-n r., "''''" jrMJn,jbl<' of J«i!fim .. and pannlmr lhat ,he leaps of inlu,"on. ",hlCh lard from <leI' 10 "'p, do nol dq.ncral. ;nlo pCOS ..... i ..... If...,,'!dcn. r.)~nJ. pLoin ·Iugmcnlatiun· TIN: d"''''tU'on mllSt be .~,,,,J and IS acqUIred by ".p. bUI """h "a", is a 1",1 and lhe decision ,h., It ,nvof"", """not ''"''~ th. <>n<nled "o.clope tMI apl"," Jpace bcforr an)· fi lII", ,akes ptoo:.

We BCt on idea of lire ', .. m",,]" po".'" 0( Gra ... ".nn·, o""n,ed mult,pl'''''"01l, TIle add, I,ve 'honmnla!' flmn can onl)" ""ptu~ .he inlll,I,O" of ,he ·'hinkin~ ,,,,,,tlle(; I"" muhipll""",-e oric,"ed fonn gnrlPS ,he .,rtual m,ilalooo 0( a dimcru.ron. Ih~ ~mbi~ul!~ induced b)" !he mobr~lrr:s Ihat In.vel "" ..... lhooc Ih~, b",·c .hady been po:si.ed, It .",".anlee< tM' ,he din>rn>ronJl . r~ 1><)1 reduced 10. "'mpk ..... ucncc of SlCps. In a"'lract ordmal <en", of dev= 01 'B<lc:pendonoc. I"belhn~ lhe ')"tem, Irk. ",ark.< on • loormomct<:r ,,-tt" I><)

inlcmoJ COn1l<:<:I1on beI" .... n lhem. We knQw lhal ~ach d,mension can ,n """",pl. be di'lm~ui.hc>d by ~ ptlnciple

of .. k:ctio". bUI Ih.y .'" ,n." delachable Itk. ""lahl. oam~ Such de""", jnvne I rOdifi<a"on de,i<:c cap;obic of ,,,,cadml!hem ou' .. hi",. addi" , 'hem up (>tt ch.pter 111 ). GTass"",nn·, prnduc' ploY' Ib" 1010 by untotdinl ,n. deS""'" of inrlcpcodettct. u if. .hown b)' ,he formul. ',f " .. ~ nh, u and b bon. of ... p k and k'. ,""n '" ;. 01 "ep IF. + k· Ind ~mOlI cI_ly <n"dops* the "";n,ol 'o .. ""'r olIn. fo",.. a and b·

TN: o""nrC\! mulllplicallO" a"lCUIA' .. ,he de ....... "lIb Ont anor"", and f'gure ll.uggest< I/,a' lhe .n,dOpe plaY' In. rolr ofa hori",. 01'flOO". 'n the pom,..."urtt origin. In fac,. we lno .. · ,hal ,he horizon canno, be ~ached by

'"

.~.

............. <tI i,.,.... .... _ by ..... npli< .. ioo

,

,

............. <1; 'F t ' ..

)

)

------'. ------'" --------- t ...

"na.,,. IIddIlIOlWo, .""', " ""''ef raulu from I '11unlnAllOFlhcr' I""'lllll~y P''''' I" to rltl'la ,"" .......,j, .... 0( .JUdI • pthe""JI. II ~ 1Ioa..,,'.,.. I"""",i"" • ".',Iian..,..., bo<u(>lltal band< thal.rc bww .. __ "'Il~ cxh otho'r. 11m:. ,. II ,"" ... " .. bIe hand lho: oncn'M CII,"rIopc ,1Ia\ ..... he WI"'" pi .. ", I''''' 'rn:..",' f,am 1M onen.ed bale 0( ardor k \O'M enl'dopt' 01 onln ~ + II aDd mikes poblbie ,he <:<)""1 .... , or linur 5j'il<'fII. orarbnra,) d"".1I>H>o.

~ l'~ ,,/(.uS$l"~ or '"'"olrinl prod..." [.," .. ".ndl~ I'roduklj

W,th lhe p<opeM". prodllCl. II>< F<lmrtcr make, h,m .. 1f mu .... olaU poosLble dl1""nsiQ.n~ be Uplll"" .pace (11)< prod"'" II n 8 ,1"U1"I .nd ' ... on d~r ... , .. Iops' ,be .~I.noi.oo Ikn>od r,,,,,, ,be n:tIII"'" 01,,,,, UICfU./On 10.,," II aod B~ 0 .... """'00·. p:Mlnlll ..... 1_ him 10 oento .111 •• hOl ... ,ve pnonllZlnl O( ,he •• po"'""" ~ "" .be ........... 01 "ockpeoKlt,1Clt, had to "" bola""'" <MIl II) • ,)mmc1ncal f"l'<"'" • muluplica ....... ,hi, .. ov.ld ........ the spaual COfIIfKtJ<>n .nd "f'Iy 10 .be q.-tioa.: .. ha, form .wld en,'eIop onl)' \bJ.. .. hod! .. com""", 10 ...... f" ..... II ..... II'

O.aumaan .hem"", dofinD. ''''' .. ~ 1I""h'n , .... , n.'dop ,110 ID'('Il<Cuon 1ft ,he ... , ,,,*, 1M prOfra&l'" prod\JCI tn""\optd ,he limon, W~ lui,,, just -.! ,lui, lbe one.ned en,"dope t: .(WI:~ hl~ I horuon ".",n, f,om I "",ftl-<>nll" (jn ...... nn brillt.ntly ~ llIe "'. 'I) of b.LIal>Clnl Ih .... ' .... lion. 0 mu", DO .. be Jem ., • horu:oo .,.n'nl fr"'" f , .nd Ib., Ih, .... ,," I J)'mmt'f)' Ihat "" Ion .. , ''''tlches 0 ..... ,he ftlll," of'J>II« bII, ,n,oIva llIe '....-y dy .... m'" <If IhllUglU,

(jn ..... lnn pos ....... very IUn: d,al«ttcal ..,"or any notion, IR)' 'nlu,'ion.

'" IRr moIlod of tbc: ",uxl "''''' be m,,,,em! .1Id b.l1a1lCCd by •• y"'motncal de\oia. In lbe ,.-..hlOOlt of ,he pltilooophy of na!l'''' alld the bala"", of B"lnlo i, ;"Ihe .oil""a·, du,y 10 roatrol ,he pro(tfcnuoa of fomI, 1»' dwtoaw""'" ,be motlliKH ............. ,Iit ","""lIOII and dmIoIUIn""I;I ,n ,he form. Afiot the d .. loctic: of ,lit mohle P""" aod w, of,he defonna' ...... of opj" d .... an>a .. 'h.>Ch made ,he n<¢ Lb.-.....Jt ,be dunetwonI poMiblc, ........ ..... uoe<U ,be "'pCSSlW product ,,,*, allo .... ,be d , • 01' from I .) .. , .... of ''''P ... fl' .. ,be pol"'. The ....... ron<:em (0, ,)mmet'Y II beh,lId ,he incl,nl'ion 10 InlCUlale lbe dc" .... of dcpmtknce ,n lbe "'-.1 Ih., 1110 ........, ol,ndepc-n<kuce ....." Irticula,ed..

O ..... lmlnn apl_,ns:

Tile co_pi iUKlCUl«I .... ilh.n e.,.,nal prndtICl ..... till" each pieoc of. (..:tor, <Icponden' 00 lbe Olllct f.otor, oouId he tpoml .. '"hou, chanl'''1 lho 11I1tIC of lilt product r",hl<h unpb<d 1"*1 tho product of , .... dt:""o<Icnl mal""uda .. ....-0).. Real .... ..,nude!. lhal " ' " .y ,b_ ,,,*, caa be ..,praml«l "' produc.u COItl ... n,nl 01111 ",,,.Ie lacton. ....... I ..... 11111 10 he ',nclcpc .1enl of orx lnother', If cadi f .. ,,,, 01 lilt .. " ....... oomplnely OQutde ,he .j ....... "",."",ned by the ..,....,,,,n. f.ottnn Sifta: 1m. <1<1 ......... "" .. ,11.1, maIr. ... pood ..... nlll 1ft .. lemal prod .... ill IID1I' not pan of llit """"'pi of lhe prod ...... web. il m .... I ..... lit poooible ", k~p,1It ...... :01 tun<q>I of.he prodlOl:l Ind j'el .... ndoa ,h. d<10:nnin0 • lion Of rq>Iaoc ,1 .. -uh """'!itr. 10 tilOd , ... ""'" Ilelotm,nalioa. ... " muM I""",", .""onIL"1 10 ;1. llit product 01 ! .. "o dq:ICot.Ien' .... "'i' ...... mull lifO ~ .bIt '0 ba ... I """ ....... , ... 1"". nlftUOlC ,lic "'If ... o, dqretci 01 Ilepm<lcnte. ,.

"1'1It ~".rna! p,od",,' 'aa~' Ih. de"""I1 .. "" of lIN: flteln,>; ,..., "I""''''c p,odLlC'l, con ...... ly, 5<eko '0 meuUf< " .1".,.,. 11 il "'" I qu..,,,;"'" he.., of I',',n •• dela.iled cons1ruct;on of Ihu product, btLl of lUJlQIL"1 I ... lIi<:ICn,ly """"~ _ of il thaI ""' ..... "" ... ,I bolla ....... OUI IliC ~"Lly 01 tb< ~.1 ...... 1 product In llit ;D'U;""" 1_ 6&. H)

... II I mapmude ...... "" k. B l map" ..... 01 .. "" t', Ind. if A _ en A • B _ en B', il II ""Iu .. l '0 lIunk , .... , llit ilql ... of C t",~ich II' ,he ...... 1 factor com ....... to A aDd Bll1ItUu"" the dc-tlft of dcpc.tdc:II,e, UcqK lor tIwt ...... C .;U io I"" parucuJar .... lit tb< 'cpnlL'" prod .... of A ODd B ,n cadi I)'JICIII e-a4n.'«1 .. ,," liS oncn.«I cnwlop<:. 0... con I" ... II" I ""'"' 1<,,,,,01

dcfj",,,,,,, 01 ... ,fUll>" prod"", ...... ,«1 10 1"0"'~ f ....... """CPO k.nd t " (J\lCh lha, .. + t ' '" ". A _nd 8 1M prod"'" ... n .1' ... _ form t + .t' - " IOIhtt:h cn"1ojAo ,lit ')~''''''' """"",,0 '0 A . "" 8 .nd dmolcd A· B"

Let'.Il'c ....... <umpld.

(II ,n. plane: endo .. "" ",i,h an orlCt',«I ",""ui, (ASC) ( .... ti .. 240)

AII'AC _ [A BC1A,

'''' t:: 11on ... ·

,-0 '-0 , - 0

,.,

........ -OftI!IIi.-" .... .....

Ck ... E

,

A,

" " '"

, , , , ,

--"

"

'"

"

F.-, --. .. - --I)

-- , •

•

" ~~--

"

'" lhor ;0101'«100II or II", , .. " St.,,, .. a,, 18.11d AC .. Indeed d~ rain! A. 10 ... h!dlchc. 'vr [ABC];$ .,..,hotd MK .

lb. In on ordInary '1*'0 <'tIdo1o-..J .,lb an oncnltd ,'oI"me (I« fia. 2~bl.

.he 'al .. ...,._ or.~ ddu><d by a ''''"PC P, P,P,.nd b) • .,..., /"", 0\ drfinod b). ""[K~I A, A,;'. ~nl Q ""'.illed .... ,

Q ... A I A.' 1', p,p. -lA, P, P, p,J"', .. A, H,P, 1', P,]

Ie, It ~, l<npa:l""')' ~.) If .be: pb. .... dcLtrm.'1Ied hy ,he 'nanale ,oInc (respecI,,"dy PQR~ .... "ill b..vc ~w.lly I~ ~' 2"-".

18C'I'QII .. rA /'QRJIIC + l ll PQRJC A ... [CPQR] AIl

l~ lnlcr>«:!lon of 11M: pia""" ""''')''' •• 1>0: In~n,," AIK and POR ,. ,he 'In,,hl lIne ,\ arryon. Ih. combo"",ion 01' "lI"'<nll 0( Ihc nab,·"'",d .. de 0(

lhe tquam .... lhl> I ... , <.ample ""'los II powble In ",",SI' ,"" "'nu",1 KIca or the rorIl"UC'I",".

•

\ \~

I. _. of<loo_~ "" Moe"',,,.,,,, ' .. n_ »()If -"'"

<

- ,

TOI" ut ......... ronn •• ", chosen; he ... I"" ,no""" ABC • ...J PQR. ODe 01 .... ',.0 .. prnenN and lilt 01 ...... MoUn ,n • ., r .. ,n _h ...... ) ~ 10 obta," an """,,,0<1 oolume (here.. l<'IrabodJ'Oll~.1Id Ihnrion:.· t I ("""_ .,,'< IN ...... '''''1 ,"", .. muh,p/tod Wlell ebe n:tnaIR"'I i--= 01 oncncc.l mapt, ..... Th<Ko b)' pr=r>'i,. lbe enaarJe l'QR and bfuk,.., ABC ial.., dun: po 1+

(,be ~nu ,018. Be. CA~ Ira".., I poont cadt Ctme: on ehe "<Ie: ( ... pcac>-d) C. A and B~ I nbtaln ehe .......... [AI'QRJ. [ BPQll j. (CI'QRjln be muluplocd hy ,be O<jII,o;nll Be. CA and ,018

Wo muSl aol I.,..., ellae AiJC" not. " ",pi< pi<a: of C'lC'fIUQO bue a ri'r~ll lone ""'" nn., A. It..O B. ,h<. C) and ,hl,'""",, a,. 'h"", ''''Y' oI .uh'rac.1inl • pOm, r",,,, It.. <:irwll A flC

Ir AIl( .... f. unly A .Imp'" J>Hl«', by """.,.,In,. pOin ......... ouJd be.1Ik '0

(]lA$SloIAS"15 CAI"ruU: Of' mE 1'..'"'"""SlO'l

" , • •

tcionllfy 1M n!IIIdll&l poca: WI'''''''' .m!>tAu,,) HeR, I oOIJJn I CIrculi by mutilaliDllll(llhor CIrculi 8y taking .... y A. I obI.III''''O P""iblc oqment .. TIl<" .mbigu,ty '. dl>pollod if I demand thaI ,1>< choMo C1rcull pontln ,he «<O"imul ..... of ,he initial circuil by rompooinl i, io ,he doMicioI m.n .... (00 tM lefi) "'ith tl>< C1rcull lbe ... point) "hich hal boen rcmOl'a!

£,rampl .. : Ll:, the: ",rcull be Df.f' U.n"llupp<eSlCll D. "'C mull choose F.f'. for D romp<l\l t>da! ... i,h f.F (011 tbe 1dI) PI-n DF.F If .. o "<I'tII IUpp~ f.. .... musl COftnca up lbe circu,' I'D. Ii""" f.FD _ - r.DF· D£.F iNIe of Jl",.).

•

'-~ T~le tM <:IrtUlI A,II.A,II. II • ., ........ ., A,A,. A.A. io ·1efI·. II .....

II, .I ,II,A. ;1 unmed,ately fCCOru;Ututa! by jWl,apot,hon on lbe Icft ... i,,,,,,,, , .... Med ,0 s'.p 01'" uy leuc .. '-1iJ. 26b)

r.- .... n" II 1M> ~ ,hi: ,...., if I CU, oul lhe circuli A , .... , ~ 10 """,n.tUule

A,II.A ...... from "' ,A, aad A. A" I m ... t ",,,d!..- .... IM A •. The uritnttd ........... ...... Ihcn:f" .. br A,A,. - A.II.

\\. Qn 1'Cf>f)' lhat. A, ... . A.A, '" - .4, A.A. II. '" A, ... . A.II •. w. no ... un.d ... ,~nd ben •• tbe .uk ,"", h .... ,he C01'f'«' "",,"la! ... id....: "

.. ~ry IU un(k .... '.nd ,lie Ill""", pi"""" U ","ultin. f,om ,be , .. n.,,",,,on

'" of I .. lat_ 01 po<>lK>D, from the mutilauOII oil . ·boIe 1M onert,ed resod"" 1ptC16e$Il ............. 1...,;". of JCSturellNI1 .·ould .... h" poooibic to ,tuNbt ... 'u" tlus ,",'hoIL ~ ""'"' " iI. 'I1JeI'_ 01 >hoWIn. lhe Imcuiallon oI.be ...... 'IOII ...,0,1 ......... D<ia,ioG. U}" "",ply .nllll, F. .. (11, •• A,I. I merely ,ode:> ,he compot"'.u oC all .prepu ",,,Itou, .... rd ' 0 the ""'''ton I ha~ 1Io."'"C!" Iud 10 co ......... '" sua:esoi,-.iy ... ,. , .... II •• , ... ~ 11,. etc. .. fa. u II,. bill ..... 1'''''"''''''1, .~ , ..... U, arbt .... ry .Dd In"", ... ed I" I'I"P no mot ..... of po,., ......

1'he Jq>Il .... _ beI .. ""n ..-Im<\<d .grr ... ,~.,.d C,rcull " ,lind"", all ,he cle.", ...... o."""n! rntdlJel <mpha$l$t .... C<;Intru, Ihat "ppoteo; ",rcull,.nd ."cnded ....... '<>; if one ";ITui, ~ ul ... cta! from lno,I><,. ,he ,cold ... "'ITuII mllll CIItr) lhe so' (J( ,hi. muu"'I,,,n. "''''''''al in an llII"'1!a,. ja. c".nded ,ololoty) a pll1 A oDd h. c<>mplemont I" hOi ",,,,,,inl .'hen A Ilu boen ,,,,,,sedl .'" put Oft . he .. me Ioot,,,& aDd ,n " .datioo of iDd,ff"'Q' Jux ... ~"'on • Graunutnn', produeu capt"'" mobjlotoes: Illne I ... < ... (uJII .' ...... pI~ and , ..... COanetallOn, ,nl"II ... tnC.

Th< onc1lla! ...... 1JeI 1ppc2r thtn:(orc .. ..,~',ul <_11"-"'1 O,,-co • • Y'i= 01 IUp" and .,. ........ tcd .<>I"me. .be onen.ed compl ",o,,j' of • Cl/'CILI' II ",U he dellnal U"""l""oaU,. 1M onertta! 'olu"",, (.he IIWlDCf of o""""""",laptn pia} ... CfUCII.l !'Ok ...... 'u ,_"m, preoe ...... ....w.. 10

,",ha"nd,m:d) cond"""", ",,"alii .... 'u ..... bke. ""nJ"a (_ cha ...... II ~" tIootnt) .. ,n ad, ........ ny dl"on Ittem ... "" to ...... """ lhe mobilil~t""" oC ,he .. """"'" '0 • modes! iUpob)-.. tep. Thll 'm<I<b'f (J( the ".p-b)-o;,op II .... , .. y ,he camoulla ... 01. ph,Io$ophy CIt'lLa, .... I{J tnt .... """" •• cI,nl'''g '0 '1$ Iin",..Je Ind incap;oblc (J( C<;I ......... ng " Ioonron.

The orienta! oompkmc", 1'<11',,'''' ''''''Ir~ a tJlo qUI]")' ,he "I"",,'ion • .... "e"" the ""'1IOn1 n Ind . and ,nd"""", • c(>fmp<><t<kn<r lIu .. .,."",1)' to..lam:<,j ,be diaJrllm. ,n figure n. W. ha,'c ,n otf ..... ( ... ('t II)' "' ..... II' and (A' 8)' _II ' ('t B' fu, ''''0 .... nolon fo,m. " am! II." .. 1I>eh InmJo.m. the Ill<:Ont of the (kg"," (J( ,rtde.,..,de ..... in,n. deKetl, 01 the dc:~ 01 dcpcndenoe. Ind vn \'"",1_ ~1-17~

, , • • I·

• /' ,

n ./ ,

. -•

• , o 0

","""

,~

B)' "1'IJI1l"I .... hai$ono n .nd • like ,.'0 ~)lIlIDt1ncal .lOp 0( ..... p motion 01 LI!ouP<. tJx ..,.._ • """'p u .be k.,... ..... of ............. . .. -

• nE~ ST"'~ H Of Ttl E >;O"·{"O~' MIJTA lIVI!

We know LM' G"""""ann had «<o,,,,ud ,h., ""naln rIOt. lions "'" be Itnil< tho~"'t e'ptn ..... nu: '" All can be ·~n·~,.he "f'IIOOt. or II~. '10'0 opposed ,., ....... 11 InC'ftd.hoe readin, ol. n b or bn .. Onwrnann tbctd'",c $C' ~ry 1M from Lho ~"llOdc of ~p;oh.1 unui,,,,,, and I1Jd>n I,om .he bbnd Ira/tlI''''''Y of formulae '" ,,,. _ ,_ Tht .. ...., ...... sI.oIrt IS 'hid "' once 'analyuc' and "J,.."hftJc', the dad _ .ha, .be "",nlS, II .. "", Ii ...... clc bad bo'om<. ..... '"I IH"':I by u ..... CJl .-" ... tvII,rmrbuo ... "'hic:b ..... poutll' '" tbe way mOllon ol,l'Uptn' ,100: ""n-sign, ~ IS ISO lew.." In) pm1bili1t or haYlnl m:ou ..... lil . Descartes and hI<

, _ .... 10 • ""'''''''' ma. ktr .hal 1'«,,1<1 .. the coonhn..ID nc<'eJ~ry [(I

~re>lc an . Ircbra;" I ... - .he <01"'11<>00 Qtl', )'1 - 0 or .he IW'tIC'nc.1 places in 'U pastliOll 01 <:OmmeD"'U, aMI ...,anmo, or.he '000'''-'( or the ,,,,,lIal ,n,wno,,! f,om "h"'h ,. rnn~ln. ooa,!Ied, Th .. I." n.ll"",U)" "".,n, ~.ry "" ,....,., of the o;:on<tlllctJon ol.be "'''''" lha. I, 'It,. -nt,', ,he ",:MImi of ;'" ",""ul ..... hod! ~ m.'1OI\J k,WWtI m;\K1u,ude!land """"" of mapli. , ....... up _hUll dUU1Jy of ,hot bneo dno ... " .nd JOu,~ rrre<:t<d •• , .... " m ,hou"'~ h)' ,lot • 'H,d • • GnoMmaIU" ~ o""nu .... oil ,hi,' ,lot "'ORb' flbnca,cd I,ke pnxIlK'Il of ~,"·pouotf .niaolatcd h) ,he ",ult,ploa· ,ionI n Ind ' may be prnd""'<l ,n ,he _y ,hythm of ,he "a'cmm' of. problem. WhI, """ """Id ..... 0 Gfl ... tuonp'. oI}_if: /oI",~p/tl .how. i, .. lf pon;c"larly IpI '0 .,lIdy ,I>< at ...... IUo,;,alcd .... 'h ronmllCti<ln ... ill '0 be 'toneal' 1c/f",,1Cd b)' ,n,el"l«l,un, of """ih, loll<'< "nd h.''<lnl of pmn'. by

"'"""ntJ', AI"" I ,heo<mi of ,he Iype. 'If tbe IrI<lot of. ,nang!<: ,urn a.oulllllhn:e 6.cd

potnt .... II. c "'bilo ,..-0 "ctlocao ........ """" ''''(I liled ura,,,,, 1,,11$, ,he ,hon! , 'M .. IkKTobel a conoal _1"",':' an be iOUl"", ......... >iy dmIoto>, ... ,ftl by 1e11lI., _ "word' 10 ........

I, .. the __ ""utall'''' ,ha, l"',m,lO ,Ius ""....:It by bU'ttnl 1"1'"\"""" ... ,,_""" and, posillorW """,MIn, The _...,...".""ta',ft does "",.will '0 .rup l'I'b of '1'1<'" by labell,,,, ,hen, .. lIh Io"e .. Of 63"""""" tl • plane. Ihi. dew.na, .. . i!l",JIo' It .... 01<1. but .. rouo;co, ,be Ipll'oal by makin, "I' a1acbrlll: m,n,.',,1'b or _"on. Of to" .... ,he im pofl ''''''' of ,he onlc, of.he , •• ml In • f",n,ul. ha, 11mB been .~"od; C'''. "'culd hI". been ,.hn no' 'u ""nfuM ' I.) _ . in{Ioa.<1 "'lIh ~,.) _ losl"" .1, In ,hi, .. am",", .he d,.""", ,iua IImrly i'HUIU from ~1in. I .ulo of alculal,,,,,-'he d,.r • ...,,,,,,, IM:1'iiIttII f and I ;. ""'k""""1rdp! and .ubmolled 10 h)' .IM: 1tIa1 ........ lJa.ln, n". dllre •• """ _u ;1>dJ' ,~ ..... bly ."h .he ''' 0 .,. .• of tht rnu"","UOII, ..... lII'u""", _,..n .. ,1M: dit,i ...... "'" of ,be. , .... II.).", b) tupCClil" ,he orde. oflIM:

'" ... nllnJ. lhe .. onc'", It "'mjily itulTW'fd of,be dill'." .... t of ,be ..... 1 .. 1""' Il$

tht tckpapJo"t dmphers • m, ....... .,.,;1 for "ord. In Ihlt u~mplc. .IM: . OW}' .. 110 ..... before the ""","",,"'y'" ,he ord,nary I<O""'C' ( 1M: of ,Ie R""""' ........ ) k,. h,m .. lf 1M: ."","p,ed by ,be .... ' ni.odes of tht paral, ttd roc.~ of ,h,np.

!\alu,:ally, 0 ... can '"ke pklo." .. In rodb,.ly romptlCllln. ,he ... mpk 01' ,he rune"nn' f and ,. Such """'p".i,lon. >ccumulate d,ff ... nl ,'at" .. hy .... ""r. . .. Iy mlrnl<kin,lhe i"""<Mbie <hara<."ltt of mlp, 1 .... ""'.,IIy, J"" •• ,"" OI'dlnll)' ,....".0, ... ""Iy kilO'" how I<> "",. ,be fal;t or ,be drsptnoOll. l 'IM: non· tuftlmulloU ........ pend. Ih .. tanporJo~,y.Dd ''''''''" ,btl d~ by,,",fI.., on ,be order of ,be 10"." '0 prud~ 1M 01' ,hal "'motl Yln...ul). By hnru., ,be llft-ll,lo <11 ..... of ,1M: Itt'en '0 • """,,,,PilOt ....... <;11'('\111, (21!. dcfottll ...... ano>lher. ,he "OI'C'ff p,," l ,..,.;c fon:ro .. be; 1M: "..,;;apn \.be iJa,.,"'). of 1M lraNtll'" and 'PIIl:al or all iii .. ,,,. aDd motJOI'I!i r....,h. 'h'", '"'''' '" .""h I dlr«I,onl.hlCh mWIt '''_pI .0 ,mJ""Ol' ' I ... , ........ Q "~'c 111m hkc. bet of ,...",..

GilIllmann .hoW<'d ", ho .. lhe liml'lc: oppos,tio" of A8C.nd 8AC cuuld ... ""r~ ... iIks: h" ~plil and mob,lt .(!t"M kn .. " hv ... ,~ n'~~. ,IK: mo., of ,he . mbi.u,,), of 1M non:('(lmmut.l''''O; 10 JI'tiP" la" .. hal i.n:nltd",bly ...,.pcd ..J~"oon. . ' .... , a nnot I,,,,dy be "houal" ' .... ,her· '" ·,IK: ...... fIcr 11M: ",,,,,,'. "ha' pon i!K1f ImOdllClbly .... pi ....... a node: in >JIII'.'C"1tIIC

no.. .. "ha, .. at .... tc ,n clc:ctropluto.ophr .... h",h.e tKl;lo in tht do>.pt ... lha, foI/owI: '0 opb, or ..... ctop ........ by .nocula"n. ,he AA>:<_ ••• 1Id ,lot Ia'eral 10 .... thaI tht ... tWa,,,,,, an be ,IK: , ... ,bk IrIC< of In 'n'· ... bk rot.h"n and dlfllOlve ,be N ..... 'lon ... n ctlCh&.

,

, '-,_ .... "" , II

.... ' " ,.-'k

I ,, - _ "r-, ', .... k_ ..... _ ...

"h.of.... ..

iii; a [i 1'1 i =H i [i in II ~ji~ [~i! W:~"I fhl ~~ H· h~ ,< •

I I ' f , j

• • •

I 1 • • f • I • • -

•

• ~

I t ?~ .. • 5 fuil <:" U"'~ t I-I.! ~ I.J .. ~f ., i"'~ - q:>~I~q ~I l:~ 't

¥ ~ t! - ~ I r ~ r ~ trW ~ ~ i f:jl r;~ t' !I'W ' , If1ia!I,1 j '~i .. ,. j

,. u ~ f 1/ J ~i ... l~rl=<"l f r - t-

iliol I' !"J'E I J'lrl~';lii i i f t 11 ,. • J .1 I - -I i " II ~ ,r J ., r • rr i ~!I" F; I .. :... ~, l .=t

•

•

il,'ttj'fiill 'r'!iril'ii('iiili~;iiil I; 'I .1 I .• 11 " •. i.1;

i!' ',~, 11-,,1 ' .' 'tt ]J'" ='l~ r , lilH' [frl,' rt'~ 'H~ 0 " . .;~;.;: t! hl. .. .. iF (I . " 1 ~- ~ 1,.- i -:ii ~-t. . j I. 3 ~!I i, llii .E ••• ; HI. 9

-I j ',I~! ,~.! " . •

-,.. f ~f !' 1 -' • :P i .1,,,,_ R

••• 1 • i .' : f' '.' ; l ,I ~ ~' I ' I oC

t' JIi, g I f .1 1.1I ~ ~ -"

If;fH!'. j- ~ - j'! I ,.i'H>!~~ ; -! J~ Fi'!H 1 ~Ir , . i J !i!jll~th~ 1/ • -(. ,

i i'-l ~ r=1-1 · .. lIt ~~i is: ~

1 "i,hl~. tIJ-,' t ~ l • UI H; .!~·l ' ; !, i.! lallr::: ~ q'" [S" e .- !I = tit! jP' I ~ - . .... I'Plt- ~-

'>ii"- -tl Ii l • j ", Ii ". -=->:-,- , • t - 1 " ~

'1 0 '1-'- , t I I' f ' {-I -I -, , ;.1 11: :1- 1'1 It ! l [ 'I': I. H I ..... .. ErH h~ rt. I. It ... h i"f

1!" r· ~!! , f ;1"1":1 8

11 .., • ::!j;. ilil~>: iii ? ,.,

'H"'}' 1>01 t ;;:;., ~ ~." . i~! ~ ~:r. ,I'. r ~ ~ ; 5. ~ ~

hI ~ " III I' !1,t r ~ ~fl t § [tl ~ g ,Il I' .' -,i r r i l ! i

!it t.

, F.LECTIIOOLOM£TIIJC SPACE

0.. ......nJ oa:II,..,."., Ind In ""m.:uta"n Iu'...,..l on lhe dedllCloon 01 dylllml<' .p ..... J, ' SChd~n' ....... , •• «1m"""'", 'i lor ~lIal <I, ........ ""'" and _Ill: and rico:trw;:.oJ phc:nodlIIlIA. H. dno .. '. up ,he chan;

~hl"."'m [Ie('lnal),

Len.,b totm.:hln.) Wodlh ("",oad,all

Scbdlifl. obocnu tn., ~n"h and !NOpe;.." appear In lIN: 'ollliuoa as. PlI1Ilpcd <hn:ct\Qn ta bor~ whereas ... i .. llb and clectncny ... 'mploy I poIanu_ lion .. ~ lhe ..... ""iooI ~ .ha"y 6rut.hed (,nIUllion all_.-Ieo or 011"-0 oep;anobk '11'" a1ma ... ;o n.'u",~ To O,'toco .... thOi oppoo.!hon ~WttII the ''''0 po/lrin.OOIlf and 10 ~"" !hem In' ...... "UI"' IOn. Schelhn.iI lin....., .0,,"11, in I Lhtrd d"ne~ .. hoch mak",', powble, .hloop pm<Inolloa.'O lo"",to,. lbe "'mchin,' .'IIh lhe '."", •• 1.0", and hera"l1 .he 1m",,) or the cloclr{)JeOmC1ro<: ilM"< of /U1k'1«nlh~nlul)' ph).",,-

The f"'''''''''1 analr'" comple,. Ih_ Wl\duclcd in IIIe prm:dm. ~h.pI." Once """e. • dlalocl"",1 bIol.nee ,,',II ..... k. " poM.bk to .... '1' u .Iec' .... d)umjc, round.tt>e-Two _1>\1 10 """",,,11011<1 bow r ... d:ay Ind Muwdl dl,mi .. <li.""m. mol'O or In, .. plially •• Iau,d 10 InrllCtion ~pul!;on and ,"" ","",_ monr ofl" Ioo(llll>\l;nli. in fa"our of nlher inlu;lions Ind Olher ~1""I,'f' tJ"'i«. (like lhe "'1~nDhl and the UII loop .y"oml .. hkh • .oJ i)(hellin.·. D.J~l.1iwo dtmand .. '""PP' ipAn: comp/ftl'I)' and .. ""fy ,he "':lury of , he Iale •• L' Tho ;n,htre",,,", ~ntr. ~ 110 ionlit'" po!.lIecl IU tIM: F"'melnuJ moddl<: of I"'" molenal prum,·. bul <ml ... I .. 10 tile heat! u( the ;nter!",," and embrac.,o of CI ""'; II.

I l["G" 1I .0.'<0 MAGNETISM

Tho !eallh PfC'Cnl> 11 .. lf to lilt ,nlUIIIC)d II II .. 'btgo!sl" of tilt dUIlftlJion .. ,hat aloft. ,,'h"'" my look can I .. "r!. qUIt" happy '0 ha .... found,'hoId' n.. ... "lth oIfn"lltoclf lIfttUL>lln,ty IU lilt ~ramcI"' .............. of n>CaSllremntt totally .. ' ....... 1 to ,he <>hJ<a' II "Ilot domm""," 1CO>nl1"l to . 'b .. b tlu. obJect . ppeaB _ bulky and ,he"""'" il II 1M one lhal any t.ndcllC')' 1",,1U'Ck

'"

'" d><placcmcol .ill _; " II molllih I., Ihlnk u( ,he PCrpk-~1I1Ihal &q ...... or '""boo' ohJ<cll £1C1l1t. 1~ Ioolth haJ 110 """,'Cr of .o .... loytn""1 ..... "tmtp" I., mlb< .... the obp:1

by 1a)"lIIJ IIntll tod 10 ~nd I. Ih .. .....e. ,I I'tcO hand In hand .,th addition and IS ftry o/'IaIl.IfOf;II,lcd "'IIh lbe IOIn'\'11 of u .... fC<IUI~ for the completlOlt of. JOIIrney, 11'oQI"Li,lbe Im"h to ott.n d • ... reeI front,,,<aI 0""" O. from .breh aa Inccn'al has lIretdtcd " .. If all' \_ AI- tl.

~

o , ,-,

The len.,h haJ "CT) ~HIt: 10 cell ... ,boul lbe JeSlIl'" thai unfokb It: il d ..... ...eo.net IIccs lbe pOInl 0 from ,,'htch " '''!Ie'! and xbolbn& t:OUId .. ...u II ...... , ..... "IS I pn>"i.Iqo>;I lip .. of finilt>dc. for ,be dlJtq~ibb",u" b<, .. 'Cft!

1M 'form' (Ibe .accn'J.I 01'1 tond lhe ',,,,,1" 10} from .11"'" OP _ I.,·CQCIIC 0IIt' .. pwJoed to u. luntt' II .... I .. lhe ",in be, ..... " 1/>< 0""" aDd ,he Inl.mtl 10 br rri>.tb~ Tho dtc~ of 0 OlllCk In 1ht pi""" hk a .... il. on "htch a ktllih (lin br lI~n .. mull ht >maShed. once . 'd. ,[ ;. I'C"""'If) 10 ,nven, aQolbcr ..... >CtIlallon [can Ihcm".~ cItadc to..., 1,,0 ",,1"""'lics opmid. ,nl oul 'Illite A"'" I;"",·.nd lbu. CfClII. lhe d .......... of lhe op<1I'''I out "';Ih .It Indlff<"""" o;>;nlreo lhe Ionllh '11 ,n pr_ 01 brln. born'. il .. 110 Ionrrolite SImp!< ""lpoIInn. ", I lran"u",y ,n t\>.lun: ~nd in lhe mind. To b< mealum.! ~"",n:I<ly. Ihl> "flCnll', nul laumtS Ib.- mobtl!.t.illion of an . """,Ia_ ILOn .. llh I ... ., ., .... ""n,pII»tS or I"n« .... Hertce 1 hilI. lhereby .. n ,,"'" pru""Uecl ~ h~nd by Ih""lbl.nd om mllhl he I""'pleel hI My ,Iuollh. ptnOI":' Of ""nlp;li~ 1"< • (llJim 'f I'h-~' 10 lile IuInd. It)' ........ Im. In "n&le In .. h .. h lhe inlena! b 'Jct:n' ,,',,[, lhe .... p. "l'lIe .nll<: ",I,:r. 10 . <IcJ..,. u( ~mbntC1nlt and allo" . lhe I't."d 10 rtliU"'" III nan!!!. It in"let. • ...:end di<'l'l<"n"",a. lnolbcr .'orld .'he •• lhe ~I<'f ""'Y "i"""Uy proptl h,m .. lf lhe: hand inhalltll l he 0",10 and lhe I n&le IIIC!!'" lhe 11;000 10 llpen 0\11 inlO ,.'0 ...... j_ 61-n

w. act lhe full ,...,..u'" 0( ... hal d'Sh"aui,bcr lbe rule, from tbe comp;l"-'

I ~2

and tbc ~nJlc. TI>< oM. ,mpl ... ~n end-'''''''''' plaa:n~"t ohlllndard ""'~unn. un'U. a man,pul,1.t"", of a """'" of d,m:lIoo_ Tho rulo, it • tnonsftt"""t hal bono frozen I bar, 11 can bo OIIIy the ~pltca of the d_ that .t claimJ to _~ II IIIfI>ta ' ,mply to pr ..... 1 oneodf ,he Ion3th of ,he ~ In tltl>u3bt_ e<ju,ppn! ""Ih . refCIC'kX oampk ~.,,"mcd '0 "" pc,*",-I) rip.! _rod "",rupulablo) and • mealu",me"t it "bullied by ju.'ta""",uon, .'Iu<:b multiPD_

Tho IeaJth III Wfl1<1It 10 "" cmbod~ ,n dutn ... I>.nd .. ,,,,,nU .. hioh are .,.,.to.>ed .. 'nh t .. o dl>lIne. n'retrut .... IN, ., il .... .., arc ,,'cldod 10 lhe: $pltiOd-OUI rloone,,' rmbod~ by ,he bo., \I. •• .., thu. in I ""ncr !""i"_ I" ioiIIY pn!(l§<ly .. hy 11>< ",,*., ,he bq>nn,n. of th .. chapter pi...., ..... th "nd INpteWm '''F1her In hI> DdttlIit1ft. Schc-tho. <:1.11 " 'nle • ..,tilou1 h,r,tiblll"":

Un,lb Cln .bsoIutely ooly « ... In ""I"", .n the form of ""' ...... ' ..... "'he ... m~""" " aoncntlly .. 'ha , ." .... n. Icn«th in tho """"""" 0" of "",ltc' •

lll.c the inlemll. the ......... it I Slmplo pica: of d_. bo ... lIh t ..... pob cmb .. Med ,",t:. ",,"h poIo.nd. Soulh polo n... I ...... ''''''''c .hoot~ lho """.".11"",.1 CbantoC1C1 of I"'" pob,trltlOfl. .. hlC'h i< "mply JIUIncd to maHer like on o".nlllion label.

Th .. poIanuh<'" ... nll",)y .ubj«! to 1..-" dlll.nl m>.rk .... ha, 'rt:lielflb!e' ,._ the "I""" /'O<K1.h pole ud ,he 1""'- Sou'h pok 'n.. "" ... 11'''''' it _ .herd'ore decided by ad lul_. PI"""'" bu ... somply ,mponed from IC'OIIlIphical.Ol ..... Ind •• i, •. .,'" d<f>QlIlo:tl In lhe bortal.nd lu,",,,1 cha'l<1. Nalura lly, Ihtl loan",," fail' ... IIOOfI os the h'l< " I.~ea to one of ,ho poln: I ""' ..... "",1 .. i, ~bk '" PlCJ<,v ••• eloe of poIanut"'" bul not 10 plDd..,. it. Tb.:>o: -clw-Ja- In _ ,lQIablc; of I btea~ • ""'ploteci Mr, I mcreI) obI.,n l .. oo,her 1M", ......... '" pubrizll1lOll ,,","11 d,,"_ by '''0, rho baro ""n Iherefo ... he 'pht .. one r""""" ~_ fi&, .),

"_4 T .... poin, 0_ "'hctt ,hew ~ba r"," .. 0\I1d .uppott<Uy ncul. ali« 0 ... ,,,,ocher_

• Jl'I' • !lmplo ~."'tntoC1) ",,,,,,,,,na.1 middle:_ "'h",h can ho "'pr06lkX1l al .,IL b)- ..... ub""I.I"nial d",_ 'tiIdcfihllely', 0 ..... jr ...... .., _1m., ... hioh II .. hy " """ be -M'W In lhe ntcn"on II ..-ill, "lulo rema,ntnl I pr, .. , ..... of , .... ma"""IC "'""." 'Thi ... 'o'";~ d,,',soon of Q ;n. ;' " nO d",.,.,n""", 0 ia ,he rukrum of "(>lb'''1 and ""tbtt """'" a .. ..,y .- l'<"I""pl" .arlin,,&- NO! """" IOOIablo; ,Joe beMal .nd .""no! cIu,l<' ,nil..,. 100 """",II DO< "'y

'" lupc.i<>< mobllo un!! I""t n"lh, ho ... mw 0 .ppc~' u ,ho pOIn ... ho~ lho north ..... th poIa.-,,,,,, OjIC'" out.

2 Til! AI'TO'lOMl 00 TUI' 11'1)11 HMO· M -t. CENTH WlOlIl AND ELECTUCITY

A_ted "'Ilh lho .. od,h is.ho ,nluition 01 n"o aul""""""" sodc<o. .. h:II .... obtI,nod by dnt .. ,ftl' .... """,. .. or oci>5o'-.""" .,..1. n.eoe arms Ortn and p,,'()t .,ound In .""",1;o1l0n .nd l .... refU'o an ind,lf"o",nee """" , h.' propopI .. mobility. unhko lbat of. N •. whICh it tunk ;nlo II>< ""'" .. ' Wnh .1<Iln, Jpotial pob,ualioa oocbin'too 0"''"''''''1 In lhe oamo •• y_ ClcnrlCll1, "'Ih ,IS dut ..... thai .... ...-panollio by .. perimnI'_ prodliC'tt dill""'" '''''I are ''"1' d,lf"ercnl r,om ,II< .... of ..... 1"""..".

W"n .Ic<:tnclly .rol li. ,,,,1.1>10 ch.r .... il i. II l.~, poso,bIt to embody.he ,,",ptJ\'C. The:te .""'F< r"neil .... like ... I..,bntic- mUla, capabl< of .. oulr.tl..,,,, OM a_her and a""''' .U of J"""'i'llnI In ~I«,""", yJ"''''lt ~ :fPO. Tho cq .... t"", 0 _ Q' ~ Q- f .. here Q' .nd Q .... _ ",rols)...,,'" tM' ,he .Ioctncal ",ro ope ... ou, into an cq",hl'flum ~ be,,,. Iblo 10 indua:: • "" .... nl. ~ belo..-, fi&- 11 and can by lID me.oK be redoccd tl> an aboe""" of maU .. \1, •• 1tI>u1d ""'p/t.:lw:e lMI ttKh lUI "'1 .... 1"'" .. ""Id "'" perm,1 an) I00I .. 0011> Ulvoh'tfli '1e1fi"" ... n ml""- .'hioh acc.unWa,,, I .. "'''''''Y .bllIC: """,Ill"}

n.c ..... morr 11m elcancal ope.lin, of lhe ttru .. VI'pohle b) • ICChtueal do."", ,1\;0, 1$ ""pablo of prol""l'na it. ~ bonc'l' ,",.nled b) "oit. II baoed 00 t .. o ob.cTn.,lono:

i, II P'*'bIc '0 di"odc ""'0 '010 '.0 Oppt>lollC poIanh .. 1Iy plunl'lIJ: ",,0 al.-.ady o"dlm:! moct.ol pia, .. (bnI ... nd .(U>Ci inlo "'i!U; b)' ..-nanl'nl ,uch •• y.!ono in ocr"", il ,. pDI'>,bIc III ID<TcaJ< the elf"::'1 t>ec 61- 5).'

1lot diap-am in lip ... 5 is ....... L .. nee" ...... k~ .... ,1mooII ali i"" ........ of

/, +

• --- -

.... 1II~u"y .nd of ,,,,, reo:hlUCOlJ IfUoII of 1M •• oond·,"'" Two, Any .. ""tkan, OJICII"" of. It<O indlll'Cl ,lJ own """WI~,y,.n 'around' J"" at ,be !ftqutbbn, .1I0n "'the ,,,,,,,,lOCI 0. accumula,,,,,, 01.......,. ,n>o""". ",hoic pIaM, ,he "",tal euIC'I"'''''' of 101'0 0 PiM I poIan~ 01 '''0 '<>PIIOIi.a·. an be tumJlCll' .. ,td lor hy • ."....." .ha, e.>.CIOOIik them by ..,. ... ,oolld lhom. I:;l«:IncaJ "OS"""Y • _ WI oIlOtUfr1u1auon, bu, ,ha, of."....." ,hal loopI,he .round OJ,,,wd by se"u,"on. The era 0Ilh< cin:w .. hal be,un .nd n ""~! come as IICI .... prue 'f 1111. dena, Cf)'Iwllld • IICOI' 'H'" of inluillOCl Ilnktd "" h ,he dom' .... uon 01 Ol'(lOl'lllOfU by loops and ""ndo.

VollaO<: "'talllI oel'\O\l.t~ dam • ., .he prUllp: of lbe meehanl .... .. ·hoch il dc. ,,-ed frQfl1 II/Ilple transfer and <>len';,·. division. 10 'M Mnefil 011., ... 1 "an!tll""~ invol",n,. nlul" .... 'men.;o ... 11 CII USIIli,), hbenutd f.om , .. imaae dIM mob,1e .'ho<:h ',,>d from A 108 (_ 6c. 6f,

The melI.phor of,be 1""'1I\.,.nd ncpu", "fluid. »d IlimllMed 10 roIIn"'. 'Ik ' .. If ...... "",' 01 the ~od aOid marupulabl< uno" of m«.hIllICt; II had 10 ... bo:cn l no .... ,ha, an ',nftnlldy ... btlc:' Auid eould 'I"'n, in Ilia'" be,ween,wo polIntL 8u, lhe ~It' ... aDd ""pI;'" poIIriu., •• llbough '1!wd', mnalned emboddtd ,n m>1,f't like fI'I'CK>U' jrwd. Ih., only dtiped ,0 .how lhemstl .... by Up,n,n, In • IAyd<n 1'" ~ poiarilies Q' .nd Q ""re .bohd wd ,n •

I\u.h oIlill1l .. , .. " We undel1,.nd Iha' Ihe "it",,) or lho 'fluid,' 0 .... 'M m«hon"m ..... ,Ihu.

• h..olf·"K'o.,., TM laue. ha. m""h more '0 !'ea. f.om Volta', boll •• y, " 'hoch don ... ,iatCi Ihe around ,n.t .lway> ~ ... Io", 1M binh of a T"o. l'l1c dioequ,· libri~m prodllCod hY ' M <xU and itt ro""" polll,iallon. il 1>01 .ully ..... tabb.htd 'f one IIat beea ",<(Out ... in ron>HtlClinl I loop ,ha" by ind'rectly "'>MCC\'''I uP,M oPl*"ion l'fodllCod by ,ho cdL"""" ._her d,mension. the loop eft.'et In '~""',.....,J ..... ' for the NlIny. ~OI' the loop.. ,be .... ral"'" II ....... nnt.hob'ec! ,n 'M "'pI~~ 01' neul",linlion. A ........... nd has bo:cn IJUped ~~'ocaly.n ""aro.nd lhoopl .nd .. " do iI" :J red...., ..... the ~1I'C .mboll'ny 01,,,,, 1I"hlf~""""" a."", di""", • unllY 10 rach ,It< on, piln .. u 01 lieu1&. "'" , I>< ,""iKe"," "",,'re it no J.oo..,. poont, I. Iou, r""" h>l(llIlo. ""'" aIM! PfO,'ol ...... oIu,ion ,n lbe """".1 ,n'wll<>n.

fJMrod)""m .... ,he 'hear, or rna ..... 'n """-. is hom or the cleruw;aJ e:.pIowon oI.IIe ""no. ",'ho<:h ..... no med L.m.:.] "'Iu""1<,,'· • """" 1nU' hi. 110 ,mpol.c" and onlylho aa:umulahOl:> of m ..... <::on ,rw; ....... ,he elf"",. Two

chltl" of "'<0 'um p.lM!uce pIIyslCltl <if""'L "'hon ,hoy """" in (lpt>OO'" direc"unl, ,hey ""h .... ~n .k<;tncal .... ,.,.,n' ( .... fi.I, 71,

•

-~-,.

o o

- . .......... .....,. .. -_ ..... " . .. .". ...,

1 I! LECTR IC~L U [ LlCH

'"

'r ile "'.c,neal •• plooion of lhe zero Ind , .. loop'''. I .. ~,n,,() "'IT)' pII~<ia .... 'ttWlrdl ""hat .houk! be called.n t!«I,.,,-I'J. U SOOn I. ,bey vcnfy tha, ,ho .1eer ..... 1 loop ,,,11<1"11" dfec!i . , oolld ","'1', JU" •• A ... IId', balaoce arrioed '''') ,h,n, 01'110 • pia ..... ,oolld , ... 11IU. lI .ncdonh " .. lbe clemen .. 01 ."" ml. cto.d ",mull, ,ntim,e I""" or ,n,.,n.",np 01 "'..,. .. hIch hold ,ho .,tenll<>n. ThaI .. why Amph'C """"" ... lIy ",,,,,,tu'a • 'conducror folded utlo • belli' lor lhe .......... ,...;1 bar"ho dllpun oflhe ..... nood .. born:

an .ucmbly 10 "'hodt I h ..... "om ,be name d«trod)'nam", ...aeoo.d. from tbe O .... k word sdlh«W~ .. -110M IMamn, '.po 2"1'( "' ..... ely .... , ...hidllllobaptd lik." channel. rha, II ' 0 »y the IUfface oI,ho form 01'1 ... hod! III ,ho "'!'CII'''.re 1<, be found,'

The ~1ICIId ~ft~"n.b ~ loop arowlld ~ rloullMl; " 1.,ICIIlJ,,<!o .1o<1 ..... Uy " kna,h and u .nJ!< or. U we .. ',II ... ,n .. ha, rolk>wl. • 'randallon .nd I fO,""on, Thi, Inocula,,,,,, i, a<>,n& ,0 make" pnulb ie 10 ",001 ... lbe k.y qllClllon oIelectrophiloool'hy. lho unity 01' eleclnealilld ma, ... ' o<: fOfUl ( .... Ac. S),

'-' Schell, ... hid oboened ,p hi! DnlOItIl .... IhI' malJM'l_.1Id ,Iec'''''''y om'<

"wo,oJ in I k,,1d of d ..... 1i<m par;did 10 ,1Ia1 oIltn"h alld ..... :hlL A ba, al'f"Cl'" U an und"idcd 'otlb,} '0 ,he SCllSlbIo: ,n,union.. It lho ma"""", eh ..... lha, I"nduce no seporable ror=. .. ho", ... ho ,.,..,.", d<ri,-ed from

PDlil'''' <It nep"'''' ct ........ '" KJ"'ra!ul Ire n<l'''' oo,,'n ., Klont","1 for lho inlu",on. The ""'(lpII)~ocal problem .,.,., ,1><",1'0'-': 'HttW can ''''0 rOITe> be dyn~m""'lIt '""""',. and .. ! do"'n II id<nl","1 rll. lbe in'UOIion ., the

.. n .. ",r .. '··

1'1$ ' ,2')' 10 artocuL:ue ,lie kupb (,I>< fit, ..... pob,wllloa) and ,''' .. ""II (1'" tonHpon pow.nhOOI Tbd can ~ 1<11 .. ,'001 LlODC <k/i".. 0 ~Md ,nl",al In ordc,· '0 prod .... '" hOp" The ""'" onok" " p<t><iblo '0 caplur~ • "I"" d''''''""00l:

The pomcula, rntd .. \Jn. I,o~ by .. h",h , ... '''0 {QIQ$ go be "" <10000 l' , ..... "'" lIme af -.od<:o'lClll Ind Y<" at rombtr>cd for , ... ,n,u,Uoa i> "'" ,be ..... or lilt ... rf.a, b.1I 1110< >~ i.wlf. ' .... 1 .. 'o"} I"" u, ..... '" _,.IIIude LII Ib ..... d, ..... " ....... [ ........ Jr.; D.......w-.. -.vh"'"" (in"..) •

The dec'nal ..... , and 'M IoQk-""Kl do "'" lUll onlh " pIJfO,ble '0 Snluh ,lie: .poll.1 c)"' .... , or II ..... ",",,"""nlra,«l ",,"un(! pole< or poin.>; ,hey dcon"", "ta'. 'h~1 • CU'T"'" onu.' .pproprl.1'e • ne,.. ,po".1 d"""" .. on. An .. 1«1'00II' 1<>Op. I"'W. plane. in'oI ... a"" 0"",,," lhe 'JIK" compltll:ly.

O'.rucd"$ famous ",pe'''''''''''. demonSl ... ,"" ,bo rioancaJ If"tip 01 an Iround, lhe-T .. o }'fi ""'"' du<jwc',n •• han ,he Mia""" or .he Ie..,. 1_ IiJ. 9). W. undeNud ,lie" ....... of(F.",ed."" lui COII' .... ponne< faced ,,·i.h , ... ,mol" """ ,II'. r"""" Ill' ..... ,} Amp;.. recalh tbere ..... In ;n" .. 1 _neonplLo '""'" lht 10"0' .ja ,he cheba of ,he 'o"e •. "

",. ,11""n"" • ..:icnull (<J:'M«l j "'ho flllo' ""W" magr>et", po ... moye<J by lhe OC'IOO or. rond""UOK ,,"re 'nl0 d,,,,,,,, .. oo. I"'.pendicu)a, '0 Ih_ of ,luI .. ,re concludoed 1.0Il! "'h.;" oIeclncaJ """Her ,umed .,"""" II .nd puobed ,heoe po&.. ... ,he d,rccuon d,U "",.oon, eucoly ... o..-nC1 had

,''' ""'''cr of h" ">tim '"rn ,n the d""",,,,,, ci ,he pia ...... !) In1)IU110 .... "

Tho. 'hld' W3' \he""'"", 'UI'POlo<l '0 <IeI\oct ,he ....... 1e \\lIh II. ,ItIpi1(1 liP' A"'fit<! ,d~_ IhlS t.me ""n'p""m .. bel\\' .. n """h.n'cs, ",hich alway' IJ;,il.

El.ECTI.OGt:O\.l (TIle S'ACE

elmo" La. ,,,,...so.- 01 ",'pu1oo. and ,he new ekctrod)'......., loren;; be wos • da:::I.!.I", Plllbir: 10 sra", IhI: .. ,,,,,,Illy ci,he """,', ~nd .1Ieo. QPllC"y to .ubJUKote. Ihi,d \I'mt1I""n by Jl«IposIn. an u,al int<rpre,a'ion ofmagnetic pol.;"',""",, . ..

W< >boukl ftrsl rC""''''her tK,w • ",teUII draw" ,n" pLane ,ndlKU oidC1,n Ib .. "De II« alto ~IuoPlCr IV L All ...... "'.', .. 11_ bem. pcrm;Incn,l} placed tAmp;..', ·bonham ..... '), Ih< p<»<ln-c ..... It ,_Cln"cl) *P' .... I" ,hi: ..... vi lho pomll .. ....., ,he 0:> ..... " .ppan 10 Ium ......... t"".,1lal anll-dod: ...... ".spenn'<:l)' dod"',It! 1_ 6 .. I O~

-. • /--'..:--=--

0"" ""n. hlc\\'i!C, uniquely define an Irrow perpendocul .. ,,, lhe em:w' II 'ooM'CtltlODll n,,,,nan .IOCb w, A"',~",.. 'boohomme', Ibot ,I\r""p from

had ' 0 foo' by ,ho ~"''''''''. <cc> the •• row 10 ,n on hoi nail' '0""- "'" III'" on h .. Idl , ... 61- I I ), A """"",,"o.ul nonnaJ :Wo IIIIIke ,I ....... bIo: '0 chooo< bt1 .. ..,.., a "front" r..,. and. '''''''k' f_ 0<. whocto _mou.nll'O ,ho....,. th.ne. btl .. · ... ., a '11<p, ..... ' f.a ,Sou,h f_. "'" .... ·bocto ,be I".,.... JOCIlnd. 'po!;i" .... · f."" '''Mh r."". from .. hocb It r",I"lCS ''''''' fil- Ilo.l. "h"l Ie"'" '0 lhe eq u' ..... nl ",I. <If 'he "", .. creW: an u. dmary .:<I'k"" ...... ' ''med 'n lhe d""",ioo vi ,he amlll. plun.,. into 'M Soulh faa: '0 rome OIl' Apl" .hro ...... lhe "onh f""l_ 61- 12h).

, ..

ElOCT .. (lG(o .... n.1(' S~ACE

y,~ hl'~ 'bcreJore j ... , ~nUd • mobilny oonfincd "'UNa • pia ... to I """" ... hOIl .iInOd perpendicularly '0 .hos plane 1_ cb.pler IV). II is easy to ..., W\, _.'CfleIy, ,be dalllll1 or • body or •• " ...... btln tot • ' .. "" ot ........ ~.ioII) ..... k .. i, """''* ....... lJdy '0 bnL • l'OUI,ion .11b. D\IIrt mad.> on an LUI ClI' aa on","~ Jt .... ah. I' .... We 6nd .... n be~ lor .be '''aJoos. • "'II&'IOG lb.. lui, alrady ........... Iyxd ,n dwlpl~1 IV W"b", ..... rulo$. ,. is the mobillll.,. 01 .n MUtt body (Of ,hooe 01 aa ela".n'ary .e>bot Ilk' • co.k ....... <'Ipa'* 01 <'Ipll.lnn& and rrodllCllll ,bcml .ha, a", propclkd: Amplrc" bonhommc mlU' 'urn 101_" from "" nlhl '0 ... IcI\ in order '0 follo .... 11< ';""lIil ""lib his ..... ,he cork,""", Irwo¥'_. ~ Jlirl: of 1M M~ln Info 0 ,...,...Irullutl 0' . ",ha' amollR'" 10 lhe .. me Ihon .. I prOl ...... '. spr,adio, au. onlO. heillll.

lin. "1," we lilt . /It l«!IIUlIlon or I d,n>eraion. which """""pa"," lhe dlt",1h", of an Imbi.";.y by an oncn'alloo marko. 11 ; •••• y '0 ..... t>a, tn

lbe Ibt;t,,,", oIlbil 1a>1 1M an..-uL:ollon or ,he "''''''"l and Uti o.'auld be «IW'oo;:al, , l;lrtU1I o;aa to.. S«n II »osIli'~' f.om """ Old< and a. 'ncplJ\~' f.om ,ho: Ollie. 1_ ftI. JJ).

/ ( •

)

1;. ..... FlphU

Bu •• '~ h.;t" •• I, .. dy S«n b"" clu.pler lin .IIa, ,a •• ann,lhe «I1I.vocal • _ d"""""",, Cln be bo'n. •• n if I b1 '<i"iII dcaston is .... IC't,ed .nd • more: IIttbt'lOIIIlnoarJallOG It ........ ,ru.

Am.,tto Ullllcn.,andl , ... , II .. "' . "ry to bnnt ,boul ,be btnh oI,ho: <itlr,,,,,,,,,, bet .. wa Nonb and SooLb pole:> by ." .. ~ .. arid .ha •• l,nl C.tllI bml ...... 'he ~1Id"odtd ooI,darlly oI,besc poIn .rId .110 "'U"1ICUDfI be,,,,'fftI them ... Iudt ... 1lI ..... by •• otally '"-'Iernll oon''''''10'', To d._I." tJr.e.e 'borcaJ' and 'llWI"'" polan' ..... he dc<-!des II> .... ,hem lI.he Nonh ,rid Sooth facet of .ic<:'naol crrcw .. Th,. u;,,! P"'P or ma""II,m aou;cr. ,he 1"'0 poI<$ '0 '''''''1'1 oon ..... ""nrt>COUw)'.o>d .Jcc'ro""""' f)' tak .. 0"", "pion", ,he !><inc< o' form ~ ... ,od by lhe dl .. ,m='nc. or ro,.,ion ..

A. H>nl" ,110 ""'"tIC. itt "Iri ..

'" 1Iot .... pled.nd ....... "'" k,nd! 01 ro""", In: '''nW,ed '0 "",I<cuics of 'IlI'ral and bore.1 ftlD<k, " .. ,mpol"bIe '0 .odlK'C' Ibtm '0 • "'Ilk pn..aple: bul II......, II "'J WIy of .an. the COlUlnotl"'" oI,he m ....... III a4optftI, " .. 5<'aI by ,be boqao,. Clk:ut.,,_, I"" ,_ ",0 'IP"' 01 .~"OIiS.nd ,he VII .... oI.he "",_ 1It.I. """'" from ,hem ..., .ma><thotdy dedlll:ed from ",y formula ,o>d .IIII\, '0 6nd ,,- VII...,.. " .. <11011'" 10 subs",utc: fo, ,110 ,,,,o-,,,.,I<culc ."""",bIy, """ uitnl ft ..... ""*""' ... ,he OIbcr 01 boreal nun. ooIenood . 't- '"-'.tem" ....... h",h .n: ,be ... 0 .. erro ...... po'nll '0 which ,be f<MUS.n '1""'1"'" bo:Io"" In: pba:d ., '''Iclly the oame "",nu "'he .... be mol«u'" 01,110 .. ,'0 nuid. a .. ,uppooed

" ...

~. ,

F ......

!'rom now oa, 'wo 'YOtemS of l'ff)' illIlll loOk"""" .. 111 Itt oa .llIOIher, 1CCOfIi'1II 10 my IOrmIlla. like 1 .. '0 .... IJIMU compo d 0111 many mqn<\'" ekmoQu as one .. 'auld ""1'1'1'* ,here '0 be ............... '" , ..... '.0 .yo ....... one: of ,hac: loam< 'YJltnu "'~1 al.o .." OIl ,he danen, 01 .Icc.nc ~u""n', as. ""'II""I doH. and l;lOOUCqueody all ,be oalcula,ooos. all the O.P!;ollllllO"" ,hat .... bued .. m~h on • CQn .. den'IOn of lhe allr-.c1 ••• lo>d ... pul .... forces of lhew mo!feu'" 1ft 'n ...... ",opo<t"'" '0 ,he "'1I1~ of dllla""", OS on In .... of .... 'oluhO ... t)' fa«:e. be' ..... n """ 01' ..... moic<:uJcs and .n .lemenl of .ic<:mc.al ...."nlnl. CI)OOMnona o.'hlCh I h .... jUll n:a.lled .he 10,,· ,IIa, os IttCp.ed by ,he ph).io;:ul< "'ho do ",>! adop' my lheory, ... neccuanly.he Ame, .. 'IIo,lIor hk. lilt one .. pWn< ,he phenomena ptrlducru by.he malt\tll on ,hac: ,wo _ by dtttncal c,u,..,,11 Of "bether one plok" ,he h)1lOlhc:!o., oI lhe ,11'0 ftu.d .. "

1'. ............ far.be old oon.""'IOn .1.11"".. from ,ha, 01 Ampln:', ~ IhI: 6 .... "butd OIl ,be.,......cnce and ,be ."""",.,ion 01 DtfIa,n ."lilies T.he ~onh ..... , ... Soo,h poIc~.II<...,.,.,.j oa ,be "",,"nui'y 01,110 ptopubooa r:I .......... ,"'" ! I o;aa ctooo.c a """''''1 ODfI''''''IOn .n ialerprcl "S, bu, I "",~ '0 .. ",k '0 III The uill 0DfI''''''''''' and ,to p OF 01 crido_ (/l.mphe'. boD-hommc, corlo, ... w, etc, ~bcnole lhe full 11 • ...,,1 .n,uIlIon. he, .. ;:d .... " frno 01 ,he _ .. of ..... IW" .... aDd elect • .,."",,,, '''''' ..... n" .... hJ.be lIill polanD'''''' moboh""" d<l1", .. n. aJl u"",,,, from mle'''''IO~.'. dill'''''''. The la" ...... Utred only ,he JlllJO''''' con ..... pIa.1OII of. lioN",,"}, ~ho .. al kepi .pan from ,I>< pll,. or ,he fol<'n. H""" ,he ·.hi,.,. patlf il much rnO'" .~" "" he lu,no h'm><1[ ''''0 a ~.k", ... w, oompoocd of a heod w"h M nah' Mild. k'tI onn, ho propd>

,., ELfcn.oo:; EO~ nilC SPACt!

.n onmlalJOll and .... Dds MlR11 ro. the ....... rqbiluy 01 1M .ample '" onnII3l1(>1L

Tbe 1a'",",iJI1 U!"'RDX1II d-.l"", the old ,,,Im,,,IC': lhe Nonb and South poles Ire bu, .he >an; oiebe old ;m ... ry: the ..... e .. ' ."" r\«tnc:al loop Iobtn .. ~ II .. loren .,,01: ""(11M han {LIM: pi<M of len,ch). AJ Sohelhn, ""d tmstd. tl>t J"I'f""1 of ,kc1neal ... <1 mq .... t>c ~ In. lin .... ,nlulllo" dot> nOI cons"" In ,dcmlf)'iog bonal ," ""Oil'lll ..... ,,,""' .• ';th ~ilj •• Or .... "'uve <I<I1U."" but In undersLandmllhl1 n ... y,runl ~"",h-cs .'''''00. !leW .",'oIve....... of II*", arouDd a waJ of mok,n. <iM.>O«< of or .. nllb"" rcvt,brral4. We rodlfcont ,"" role of lbc ind,lf'c......" ,",orn:: Lo.ncreax ,ho tolnll ilIId I,oonl bt'"ffI\ .... ,,, .... nd though, by I'fUl"'pClal amt,..uny In ",dc. 10 ... 0 ........ d«utoft l/Ia. ,nm"..,. • hI.'F\' fIcid. II ...... II .. rictory ollhc '-Ina! ""C, lhe ioowillduw orm> oul • Vall field ;nle> .. ""'h the """",,"1 _lid ·ph".I<'" mt~p/lon '"' "hod! dorhQ,: lmCl"", .. e;U 1«<110 10'Flh"" .boo."" can """'-

\\I ... ',II """. Ulm' .... the .,. l ., of II><: "~,al In ..... ICT donaol and """" how dc<:1"'F'm><lry .. ""n of rhe ""'''1) oIlkhbrrate I mbil ully m:..k usc of by ""and to .w..., .. "'" 1;0,,,,,,1 uM. f,.,.., ...... "'call"" urul. lilt lJIouId ~mern"'" !hoI I! ..... HamillOil who wu lhe fin! 10 cktnorulralt tho: ronll«llOn 1101 ........ poWr '<CIon and ".,1Ia! (., .... 01 ""J'II'It' or "CIlia! Id"pIao:uknlt rrom OlIO!

pboc: 10 lnol""'l JfIlps." In lhe ,ulrodl>Clion 10 the UN-rJ"" Qom ... -. he n;pia.n. (to ;Dltod,," .. · ... 1 .. " .. ou .... lOday alllffift<..,..,1

In III" .... ). lhe .ymbol 8 " IIu come .. ,th ... 1(1 den",. lhe .1"';P1 I"", born ... 10 B. lhe "",nl'" btl"l (al 1;"'1 OODSld<rl:d U I ~_n Ih,,,g. <It • dJIUm in some awm<1~al '"""'''1''1>011. and Ih~ "",nl R he,"I III) cunl"'''.'''"""", .... """,hI Ih'"*, Of . q"""oimm; ",btl. jj ... i':'1 fir<! lUI'P"""'l 1(1 hc ......... nl.11On 01 ,he onlt .... 1 ... lal lOfl ,n 'JX'CI' 01 II>< ""'1111 ""' .. , 8 II, IIw ,"v.., "",nl ... , [orJ 01 l ilt p:omclrical [)I ~ fE R ['-("E- oIlh_ 1 ... ·(1 ",,'Ul.l. Ibl ;. 10 ""y. lboo dtfl".......,. ur Ih." lwo I'OSITIOI'>S'D <JIKC ,.

Ibllllll"" I ..... r;i''''' III tumplc. borro .. -e<t from .... ""Dom). ,,""'" ~ '. lilt b.,h and B lilt Sun "''''ed f""" Ibt [a.,1t.:

n. .... ,n 110< ."I"OII<Jm,.:;,1 .>lmr!c ur tanh and RIa. lboo to.,. 8 ... hal heen ""nl 1(1 t"end f""" ,lit 1'1- of ub><:rvlIiK>n ~ III>< ".rl h!, 1(1 ,he pI_ of ,lit <>bso:rved body jj liho .u,,~ .00 \(I ... r"" 10 oonlocel. ,,1 lea" on lhoujllt" the lan.r "";11<1" ,,"Ih II>< fon"e,

All''' you may ha..., ...... ,h:tl ""tlil me III< primary I!~<""., ncat 0J>Cf1I' liOn dcllOlod b) tht milk ..... 00 callod In lhe n ..... odd" ...... Of lI"IOf' fuU) .• ymbolicoJ Add,,_ ""~ .... In • ~ COf""'fJOn'lm1 Ofd, ... t ')'n,""" '" lhe """""'" ur • mal ........ 'iooaI pom' ,n 'P"'" ID>I_ 01

'" I;.'Ompon"l .1ICb 1 r-t11Oll. B, ",.h 11101"'" """",_ ....... """" rqud OItrtd....,. It den" .. lilt OlIO! fIO'IlIOII f,om 1110 01""'. n •• poonl B Iud ~ Ida", 1 ,.. ... ,_ ~.n-oI_ " .. """'. ,... ..... _ """' .. ,-. Jt """ I,udy lhe .~bjIccI 0( .... ly .... 11 II """" ' M , .. uh 0(. ')'01""'" 11 .... • m',,\; '0 be .,med .1; 11 i!i ''''''i III< end 0(. /I,pn. (I' 0(. JOufM)'. It "11'. ,h'nM 10 he in'CSI ' g.o.lod (analj'llCOJly) by ou, ,,\Id)'l'" or ""lITIlPm, ;Ii 1"'>'''''''; il i, no'" • Ih, n, which h .. been I"\ldIlCNl by ou, OflCT:I!mB l.ynlbdocaUy) un • ...,Iho< p<)I'" .... .. nh the . ... ur. cet ... n ""Irumrn~ twndY. lbe '''&lJIII ..... 8 4 ..... rdod..- q. V<CIO<. o. ca.ryt"1 pilh. •• i< ,,~prni<d b) IhI: ""'p!uj ..... o. oIlhe ..... 0( m:tKlft. t. lilroup the .-... ,at and idc::ft11Cl1 rom."l"

18 -"'1+- ... _ 8

Thai othe. poonl .... ;oSlea<l 0( be.nl _ a ,... ... , __ .,_ DOOICS ' O

hi: coft$fdered and ,pok." 01 II • ,...""' ........ Jr.oU_ "bIle . be potOI II . .. 'h .... had ~ .... natyntOO. l1li C<lrro: 10 bt ""lied • 'OI:Ium. IMXOIdi,.. 10 lhe ",,,,,,,,1 fo,mul~:

1' ... , ....... r.~~nd _ 1" ... 1 ....

"' ..... PlUJ If (id abo~ mcallOll<'d) lhe 51 ... 0( \'«11<1", or 1110 do.:I ...... n<lI<

0( nnbnal ')'lI.ItocI.L ~rum "'"" .... '" lhe ISlfOflOflllCOJ ..... mplc. ... posl 01 obInn ...... lhe "al1h. A, coma '0 be lbouahl 01 .. lhe comD"""" mea' ur. 1raJIO<00I1. 8 ~ . ,,'bdc Ib ....... arunl ... the eaI1h .. co""",,.d 10 'enmmtl' II I'" ",fl. .nd """'.· • • ody. ,..., ""0. 8. II Ih<>u&hl 0( II «CUP)'. a .. " .. '"", '0 ~.",. "h .... "no' ""'" prcp<l).d \(I hc 11ud>cd br ot.:nllliou. botl Ii .. Ihcr _,,'od III 0fI< ... 'Io .. b Iuos bet .. rexhod. or ,,,hod 01. by 1 joumo). I"",,!IOII. or ,,,,n.port 0(..- nIOVe.>bIe poinl 0' body from the eltnh. alon. lhe scoce"tr", """" 0( lho ,"n."

Th" 10111 qootallOfl from lI amilion de.rly demon'I"'I" lhe d""" tint betW<'CII ",,1o, '-..;'on .nd ' ..... '1 .. A pola' ''''''or at .. a)"l ron_II. ..... Iouoly abotlobco. pp bel .. «n , .(1 poIanl ... . hal ha"" l irady ....", p>" .. ""'" ",I; ,t 11 .. ,,)"1 rrien. to ,hi: .boI,IIOII 01 • 'ronfron"'''''''~ IrlMPOI' 01 cftarp b'I"I 001 oIliquaJ. lha. "n~ to fill • d>1f"er=cc ... pole1ltial. 1 PP bet....,., cw-"",.lod dcp'ea. In IhI: loUIe _y II fon:t (ore 1.1~ ",.. le"" "!df .. "bou. odont ..... nl ••• "'''""' ~Iu" 0( the , .. """ ..... IS • ·""ssa,e· ••• a f<>l'<ll:d ,m""",""", ("ilil0 be """"",p ... hod or alreadJ acoom"hJbod) ... IhI: ".. • ""III 1 dOlled 01 ~al 1",JfC1ot}'. ·1· ... potu, >"O<1Of zealolbly accounli fOi. br m.)",,,," banal. 'h:tl .. ht<h .. 00",..,.'«1 from a ... rum"'., 10. """'m" . • hol "'IIt<h "rommun~~lod belw..,n mark .... and. Ilk. all coonm unt<lllion. i, """. U<CIf unty 10 UCIIlnhze "9poosil""'" '(I inl ..... lal. ;1 .. 1f bet"""" lhe <ltpces 01 """", ... 10 or 10 jo'n up 1",'(1 'ltvtb'

l'oItt •• ..,. ..... nd 1 ........ 1_ red""".hc .... " ur..- '0' d.."taa:mnot. ." •• IIDD.n. oul alofts. d,recuun. WIth 00 """",,n ." • WId'h; 'bey .~ conlftl' "'''h ...... Jy lnmDU. n"l 10 ,he ,'WICtcr .... UI>ClIO<U _III"' mobil"l

'" ll~CT.OGLOMI".IC S~ACI!

whK:h C<>DIe from ajrudy C'Ol>IIuend d"l>t"SIOn~ W;lb the 'KIlL. Fnu;DC IJW"I<'11C humour fiftally buf1l~ In WIth l ilt budd, ... 01 -..other d"""","",,: btI,'It .... "h I"" hoi .... Ih,dtIQI .'ub ,he _~1C1nI TM uial _ector doeo. DOl daim ."'''''''' li"'''lpoort, doeIlIOI I'd .. . o Inl '0"",,",'''' .''", n .. ",,,WIIIII,. ... um.l"', .... II UIII! only 101 .... thCIlIaUOII 011 I'Olahon ..... , ..... ,;.,., .. -b;:b eo"'pc'" 1M Jt'OO' .. , ..... 0 oncnL huruoelf .. ,[h,,, lilt "bo:>It ........ II owen, .. pooo",.~ II .. _.eononocal thl.rxtet ohOO •• iJ<"ulluoa .. b:h mull. r"lID ,hot d .. ""U"" 01 ... .... bI ... ny IJ>d IIQI (rom llopQLI dod_ or the ,,, ..... ,,,'" 10 em~ a IrI)llCtllf)'_ 1110 .. l1li • i~_1 "' .... ,. u".'mdJ.,.: 11 ,,", ods ia roam,,"1 I deMOI) tor.he .u ....

A ''''....,an. ...... _ 11(1, pmHrolef 1I011Hna: ,he Fit"'" .hal d,,,,,,,> n .. II" ••• ,,,hll,d,. ..... no..'OId up III II .. ImJt/l, .. ","~ ... be oW Ilr ..... , .. _her d..........,n by al'l'<Opnahlll • mck prxlllCd ,II • petpc1ldJcular pIa~ Tbc .... "I'rondha I faa: WI,bou, ... ·pballl n • • , ."" tam< II ...... ca_", oidos '" email"' Ibm! " _tu", moo' ud po,pltIUlI ,bou, .1It iCfeWlOP .1Id tho: u~ 1lIo ""Ia. 'Mor IlIIts ....... 01 "",011'131. "'bcreu tho! Dial 10.....,.'. ,Il00 spo".' I., euract <kif8£' 01 ",,""UI'oon. 0. ...... <OI'i<tcttw, tbe Illal unKr .... 1M 1oo"11Id",,,1 ' 0 tilT)' n 0/1" >nl0 'M b<i1COllbJ. h IMrrl~ b<H. wcceWut tlkuJal«llub,""""" o('M ,...,.,,;, .. by bn"l 11>10.0_ ,..., "'U "om '~"'~h<d ,n d01l«l1; ..... ;n an .uII '«"10. and 10..", ro.. 01"1< 01 lhem by. J'1"0<"n$ 01 oneOllltOll. JUl' II "'I".ltr. bol.~ mIIld el1rxt 11\0 Il leraI uni" IDd ~I>oow """ 0( l hem 10 luKlt I '<7101>0<1.

b,,,,«,,,, 1101"" 0( pen'lra,ion by IAk,". ,he ."1 ..... roof 01. iranI",,'" of I r .... """ w:r: .. '" I ...... "' .<> .... llul ~araday, Hamill"" I nd Muwc:lt could IIl lh .... h .... '<CoJI!I.<\l Ihornt .. l~ in Ih" p .... "'m ..... Atllhm: .'peri. mented ., nh ,,-,~ I,.,. coak:sc,". ~'''''' (tee 11 .11 "11" 00 long<' ""ly unJOid lbem lion, • 1"',!KIOry by maltnllhe for= ... ork. ""I Ih., "';n lbem ror I ro"quesl of lhe IIonl"y •• "11<1,, .. ,,", co,k~'1 (<II lbe lrolk-)'I Ind ru)e" 01 mall of mochaniCI I .... 6,. 1.11,

, TII~ ~Lrn~o{,l,o" fTll{" EX ~I 1 ''' '- ''T A$SQUAlr ROOI

Itrncdnrth lbe .. laL '" n>On: pr<CtICly lbe all .. loop ,nt.rbta:. t.tk .. lhe pIa<:c of lbe In<Wf ... _ ll<n ... and IUlW"S .IK: .. ·1toIo ....... roac ..... IOC ed,iioc ,uOl U lbe ""nl .. 011"'''., _I....-..d N"'1onlan ""'",""",y W. kIlO .. how lhe overfl'nll'n, """Iret; eIIn n"'''Ur" .nU ... ~UMf1" ofkno .. 1cdJc by wbord,nal-'a, rclIdl_ hele ....... I'" 10 ')·m .... 'llWIt ck>."""" " II ,II ... ,1\;1, ""'lOG. d.....,,-..,", .he a>mPftlIOaUon 01 <'mlnlup! ro.- by _Inpetll ~ 1110-Ito.bes tho lronl.,.. 11\;11 ltolated lbe .<>rld ."" hlllt'l .be "'*'>1181 " 'or\d ) from lbe . 'otId ' bcIoo.·(lhe wbJUIW}' .'Oridl A. ro.. lhot _ ph~ " drmaodod ,he d, ... ", .. 1 oIlhe ar"",""' "U1M oI.nl<'no<'l>O<1 "' I d .... _ and la]wrcd

tho "' ....... ""'" olin ft(ui"'",,,,,, Ix,....,.." 1"",",,11OlIII and roLl' ........ Wi.b lbe u .. ·1oQp .n'n\aoe .... 0" .,one 1(1 mllICO' ........ ,he 1"11" ,b." cbarxt ........ ... ~ <kn« '" 'M d.qnun 01 1M botlulco (_ ~baplCl III ~ I ""II

'"

V'?". ::;;;." •

VA8 "

capaCll,10 conde .... Ind .. "",ad 1M .,IIU,'101I. .ho COItlbtonl ~U'" """'pm_III oIlhot ........ ,.,I0Il and ,be Iiq ...... O<M Ind. JinaUy. In opCIlI",.~ PUi!'*"'Y "'luch QIOalnuia 'IS pIaa! ,n ,he mool Ul"" ......... I'Oe;Irch.

t'Il'1OdIt}. 1I1",.hOll and "bl .... UIP Ih,,,, 0."" .... y Ille.\perunoen,ed .. "It ,be ,n'CI""".nd d ' JCo"emllo II • ''''1')' "",rtlC\lla, ro""""'" Ind ""IODOIDY. from wlt",1t , he lnaly1ft IIIaI folio ... iIa •• laton ,It .... lundamc:nlll realu",",

t I) w. I .. 001 deal,". lit .. "lilt I ,.al jotn! ... in u.. """"l\;Iu"",j 'boise...lId ... ' "heOU (_ II t.5.C). "hoch ."K."UI~I<'t ,wo •• 1"";",,' 'h,ough 00""'" d'~I. conI"",. The ui, l ad lhe loop inl.,llI(, ht. lhe hnk. 01 a ella,,!. bul knoUtn& 'Ol",her ,.Ia"on. of I""",on. w .. " no mll."all,"kL Tit. F'lrin, an<! ,II immedial. ll<nl'" b" rJ. ,ntO linc> <I fo ....... ,,"''' do nol simulal .... al oUlfio ... ;"!l. bUI ,b.ch ,he lClion; .hll em. nil. ,i,lullly f'''''' · .. al' circulI< 1_ 6 .. 16).

(2) n.. It ..... of Ioroo • . ~ •• I ·fidtl·. I .)' ....... of ... v.,IK: ... ,on .ad fOIOnaJ><C

bo,"wn lhe .~ lad lhe aeome,.r or rh)1lC1ll; onc m'&h' .... a~ 011 J>ftf'I'"-lion 01."""", f.,.. OApe""..nllllOll, "",IoIO"S 10 ,11.01 oIlhe """'ed a,,"~c:nctl '0 IIMlbih,y t_ 1,1) n.. m«1..,,,cll V1IIOn rna"- Il0l111111 .. '''Ix .. ...,'' .... happy JII" '0 dealn. tmpuloc. .. he ..... dccl l"Oj«>nl<ll')', by...c' ,t'.tlly """""Ill; ,he ordeal oIambiptl). ",0 p'n ror .lOdf I ..,.., ..... "I*'" 10< lbe ullfoldimg 01 VUluab'," ~ I 1I<1I!. 1I temlOf}' IhIt! IUlho""" lad .. _ "'" 10 unpn:udmted e'penmmlS: .. hit, bappml if one lu, .. llul "n-uil • ~1lk, if oar mn« thao """ and IhI' .",. up!

TlUI ..... lnoJ ICInlory II _ III ,nddl ..... '" ... ", ... 1 ...,.,.', II ........ "011 II

.n",,",ed from • (:Cf\1l" ...... pto<ll) of ,he pobru:a.loon •• ."".,·ted .... Ib tho InnIr-to 01 dlupIlcu ...... u). To lhe 1"' .... ".l'J and lho: real,., 01 theK =u np 7 ntcd by,,,,,,' .... "!Un ~ lbe noal d\1ntct<T 01 11M: ..... p<1'" IWd ""llho) Lnd.-.

- r • ~ ' I ,

I •

\ '" • , • / •

/·, ..... I~

()~ 11>0: ",,,,,Laall, cI thot "' ...... , .. _lid .ho: Iinrs tI titId 1_ Ii" 16) dcasooI· \l"'."" ""'''"Ur ,ho dooute by loop". cllho: 1t2l and dlol' Y1n..al, cI,he: poLo. aN,l ,hot.noL 'fbo: "1l~' ro,..,. an CIItlA,ly _"Bub. fink bo' ............. .,.-doer of , be Clu ... l,ha and Ihll oil'" fi ........ bc'.'-ffIIlhe 'physical' and .... 'p<>m<inc', and . "ar801 .... the u!! loop l)'Stnn In o"lndy rmlarhble CO~,on.

A f'", .. wy: IIw ptMrxOflJ ofl'" h_ off«r,,· 11us _JUncllOfI of .Ia ... aty and I'CCIpr<ICIIJ Inn......., does no! escape Fa"""'y It lui It> be obI.,1l<'d Ihroup u pon .......... """ the r,,~naunl world P<"""""" by a; .. ,ed", d, ,,,,,,,,01)i rould nol .... ,.mnl IIke.he .. mplc ph )' ..... l "traMlalOon' Ill. m.lhtmat .... llhro<y ,10.0, il .'widy .\"I;"'blo. FantdlY ,"k .. an ,n'port<! ",,",''''''' the d",palm ofio,.rla<:ed 1111 and loop ~. ~iar lutonomy and an an ....... pOWrr lba. III .... "" ~p«I<d and IMII ImOl"" lih In OK"''' In .. 1ucIo I """heII\IIhclll 'fonn" is INlioed 10 p/I)'JOaIl .p/1a.ou., .... IKe "... In

Faraday'. " 'pcnm<nl> are MI' eMIa"ed 'a "P<NKlKI1II", but ,nslelld l Imo 10 ost.bl,w. I ....... "'tfMnpnl pOint In phy .... by donm", • mual oIlIl'SllIm whICh i. &1 .... )'1 .,..,<full) aocoml'lnied by d"'l ... m~ Tbtir aim i, \0 I'rOO_ 0 1"'01<1<01 fur app<'*'" 1"";1 ... ffklently br>Id and suffia<oUy "mc"ialed 10 .. \thuD the """"IW re.nbcn.,ioI\s 011110 ... Iera-. To Upcnmc:nl is ,1IOeTeIOft • II .. en 10 COMpO'" ",ib <10..',,,,,,,1 ud ""'JIkbt """" .n onkr 10 rn.._1I" IUMI nalu", frum lhe)'Ole 0I111e In.n .. l1~ and ,be d,.1.I.nce Faraday "'a.II. 10 ila.., ,be """1I0,w and the IIICral in order 10 oalural" hln''lClf "lib tdf .... ,idcfI( (..:10, whirh il .'hy, Ir"", 1821 -22 10 )'<'ar after lEnled ', •• ",,"men,., he publ"hco .. ""ral an.::1t!. ... ,tII ","cahn. ",I!:>: 'On Some ~ Uoa' ODIII"",lIClI MOllO"" and Oft lhe Theory 01 Mal!'d""" tQwmm, ~ c/ ScM'_, 11. 1821, PI' 70( 96~ ·FIoct.omapIC'IlC Ilocal_ " ....... 1'1' t ibid.. 11. 1822, Po l&ti~

,., •

'"

, '"

,os

'Note OIl ~ EIoct""""",""u","J MOl""'" (Ibod.. 1'P. ~16-2n lI ore. rocr· 1II1IC1lRI an ....,. ........ 1!'d1C MO.or is ..... ,be .. me •• .,..,,,f)'tn. a """",pit or til"""",",. I ...... ry; nt ....... 111< .... 1 .. ", dcmonotratc the a"lonomy of .1I< uis loop .Yilcm by ,J,o",nM Iti ClI.p.><lI)' 10 CIIpl"" and abo". ~n 10 "'I"'o<h..:e ""rt.o.o p!S,,,rn al ... ill 11'Hicod. Faraday..,.. Vl.'ry clearly'''"' lhe 001 ; ... ........... 110. from lbe lolctba ..... beyODd"'J 1'ormw.IH>It.n ''""'''' of·drd..,... lion" tho molor can ull1lCl """Uon from lbe U" loop. . 'IIItou. _,0 for belD, ·dcdua:<l" and . -blle ....... ,ntn. 011',* any lubordlnale "".,"'" 10. p<C

..... bll>bed tk.::1n .... I\cf,,,,, faradl)'. lb. rol.OIlonal and pi .... tal ,,-c .. either capll~es oIlbe pby..,.

w t's OOdy. or .. . lible: by contcmp!al;n. lhe mol"", of d i ..... 1 ...... .... 'tln Faraday. ,bank. '0 an 'Opt'fOl"""" m<Uphot" Ihli ...... not ml,m) bclon. Cltbcr to nalu", or 10 1bou.11I, ,he r<,''''1OftOll '_pc< l!at b ....... • 0( l!at pbyooosl

". aod I<f"""""" ItidI' ~l it pIeates. dimIoIulnol,nl the full autonomy oflhc .lccItomI ..... '" .. <>rid.

faraday .Ito ..,.nll!d I" mark itill III(m ckaflr the ,ulf ,hln .. parlIn! ek<:""""'I".""" r""" the medll"lCS of ,nlnacliom al a d,,,,,_ .. hod! ... ,,,,Iltre .... ' 10 ,be rcla'Oont of rmpmaJ pc'jX,idlCUlanly and omlalion. II .,"u IlIcn:fo ........ n.r of oomo U'Fner 10 m~k. lhe ,rlanON of """i,,,," ",ihle aDd ruin"";"" ..

II had lon, __ .. known ho .. to iIIus,ra,e the aCliom of I ""' .... ' on iron fiJmp by I"ICO oliofa' II ........ rfffwlfl" flr"""r IUIm ph)'1ol<3 on II, lI .. d by dead"" "" Ionp II> .... lhmI .. JUbsidiary ~IU"'" bul II ,nalmle1ll. 10 pn:nukc "'*'"- .. d<qn.llll., as .w.rnl·bllt uP' r (" •• u &1100'''1 10 oral' upen...... , ... , .... "'b' 101<111 actiotn. una 0110..,. .... ,,",Ibn 'ral' nor ·.,.,oDciaJ~ the)' do _ malhemallze ron:a; the) do _ PO" Ftn.lncaL I>nIO pbyso<:al. nu,," tMy >ullom",. an idcal, but dame and 'ouJh. netOlOl'l ... Iuch .lktw> ,';nll&lIl,IO reclaIm ill rip'"

U""" of 10"", pa,n, • qUllc ddl'.mn IIndllCajX from lIN: .. ..,.., ,mplomod in """,haniall man ... by an impallC1l\ """mlandln, IS. " II' of .~uhc",oce (_ 1.2). Mn ..... 11 uw them animalln. SPIce hk •• jungl.:

[N 10lhi", II de."" than J'OUf deocnp""IU "fall """''''''" of CoM- klq)lnl up. OIl •• of ..... rllY In a11.ha. fUITOlind. lhom, ... hid! pa'e by ill ,~_ or dlmln"""" mcaura the "ork done br any <h." ... in .he 11"10'11>. You >em! '0 .... lhe b .... of r""", c.o",,"1 ..,..,... obtbdes aad dri"", plump ., collllll<l<m. alllllOlffYllq 1000000Ns crrtoJn di=:lion. III ft)'tUIh.. .ad corry.", ..... 'h 1hmI ~'Of)-..'heft 1M ......, aa-1I1 of ,tUa<!,ft po • ..,.., Ipn'ad .'i<\cr or deMrr II 1M hnes w..sc.. or <:OIl''''''

The hllCS of Foro< from the SIIn .p""" 0'" from h.m, ,1\11 .hen lbe) ~mc "" •• a pia"", """'~ 0'" from ,I, 50 ,ltat ~ftl}' pia"", d, ..... !! • number dtptnd'"l on it. ""' .. f.om ,ltdr <0""", al\ll.ol)l.hhlt", a .y,'<m of;1O ".'n 50 -"0 become 5O ....... bmll,k. a rome', ,n,,.... offo",.. _", vi"ble.·

r.rl<by ...... 'be lirs, .0 .... deolro""""",nc 1I"'<"'MoO:;' ~ no' . _UlUIl that " m"«n 'h ....... and I" ... offortt .", 11<>1 ""nl",,' ""","Iy .. lib lImuJ ... nl atlracl""" or ~bioru. The "It"" ,ha, ,bey _ •• ;. I«m'''' ..... 'h pROO""'" allll .. nctlOa. and,.u",... ,he ory<'o _ .... oht>ddtn f, ..... IIleral,.....om. n.e ""'" ot~ 'hal ........ '" from a _-' or a loop do DOl 1"""'" ,Lt"'_I.'a

.. , ... CI'IC3'Un: of ",,"'''lry, bu, "'00 as ,bill .1beIt dtmon:Il .. 1e> ,be 10000. omy aad cobcsioa of a n:aprocal ''''.rtoa ....

ConlOdtr II .. Ii.Jtln: of. ctrclo w, /loa .. In .ho: .. ,1Illl C.neoun 1pKC; i, .. limply marked by a pIII", ... n"" .nd rema,n. ,otally ,,>drlfCfCflI to itt OUm>llnd· ,"p! .... fit. 1 9~

c ,

Ru, an elct:1naJ an:w •• 'ea,,, ,bo\d , ... 1( • ... ,."(Ift of "Ol1ed..),~ rirdn "'hl<h ""La".,. 00" the tnodlly of lhe IOImpio ,ranSl' of (ha'l"$l.he 11,:",", .11 bold In dill"",,, 16) and a,OII1e 0I1ocr ~'io ... on lhe nciftlly. II "thuo .hal lhe I ... ·• of tndu<UO" bocome 'na'"",I', .ny ;n'."'tnllO" ;n.o a tt' of loop! ,.,11 1<"" .. """ onl5 ,hal lim 10 MUlrall« i., Tho: ],,11:5 of Iofce I.he 'dotted·lino Ion:eo') f""" , t,nd of _ rornp),g,y be ...... " 1M p/!)'$IC1l1 and .be ."" .. ,,~ cal; thor ,nntlll ..... , "",n ... ,urably"

C-.Ior 6,u",,2Il. Unnl'n, I mapI01 ,,,,, .. ,d. ''''' "'onh faa: lit Lho VIet"'''y of. an:u" C

provok" !lit "PJl<'llI""" of an ""'uctd rom:IIt and "",teo" ""';bk ,,, lilsun",6It I • ... onh r..;..'.nd a 'S.""h r.".-, ""L.l1IIo: 10 <'OUn'" lbe ",,,"anen' of ,he mapet. ..". I""", of furor: an: ""'thor 'rear .. ·,Ift nor ilhoslra1l0M of \rI".'kd", already dtpoo.I.ed ,n on <1IC)dopudio: lhe) It,d (he ",plGra""" of • field .. ·luch os ""'Ihe, 611C11y ,n N.,uft nor c~""ly in UndtnUlnd,~

d , ( , ,- , "

-I ~,<

~ '~--7 (, '".

,~/~ ,

" , "_lIl FlIIlIday". I,,,"," of force,,~ in ~"p"'pnall". ,.'hal "' •• '" propo'''"J'o

""II J1"K'""""",1(: ~xpt'~J and in p"nJ ' ... , ... '0 lhlt I:onJUIIF of '"P"a~. ,n. "llh lho ho"",', of tho 'f!h)'iICI1 ...... , ... Ioot< dnVUII """'tr ....... osed by lhe ..... 1 .. ' tIta",''''nct .... 1III whICh an .... " lhe jaMU' .. hen ,ho mrta· plio, aoIodm.., "beD 'he .."....,I{>D _ltDd< .... tIt. " for OOftm.,.,y. Thu Laa ...... feeds 011 F" ........ ",1Id!..-. ' .... a on .toady ., ...... bIo ..... ' ........ tocal

, .. ~ and po "'.,. panocular .1,,,,,, .. ""' .... . WI bd Cln lrt.verx the deductlv"d""n. of malhoma.ocal pII)'Iia and ,lIum"""o ",,,,,," _ of ......... '.,.. The fItId b ..... ;nlnm:K: elpm .... nllli m ... lIln ,lull "",'<CII " !'rom rho Ift''''';"'nj of lhoe un<le"!llIIhnJl (In IhlS len .. ;, " ,rralllCibi. 10 a l .. n~nl.1 ""Ip) and from the Ippemes 0( ca.ual'h'" T1>ore it. in tho hI .. ,.) .. n!le, no ... ppbo.:It ..... • of tbe I" ... 01 1iekI .. bUI .. ,he. In •• tl>lll.ihmJl ptda'''IY by all.....,... ",Iud! .. ",ipr ... y ""rod...,. In dalric o.piIClO and play 'nth II. ClPf"."Y 10 1'CIOftILe..

II(I'tIICVCf. Faradafl .,...,lm!.pontlC' wen: ,u.".oou. ..... bcsc """,'n>O\H'

II,," of Jon:ell and pnk"'''' thc 'limp!<- and ckat' odu "hell"" ., • d","nce, "'hocb """' ...... r d""" the InMtlOl1 ,,,,,, .. ,d.,he bouOII\. I" .... rds dead .1m ....... lIut the "" ... ckc\roma"",'1< world wIth ,I. bIlC .. hll"" jostleo lhe old habm. ,"" peaceable va"'~lion>l ebe,-", N"", ,. "'n nry 10 cmbrKl! I" ... of """" ... d HI l>< or IInUe >I<';D. of .... "" or """",". The 'simple and de,,' Idea ofKt_ III dlSU ..... involved OftI)'. d","n. and plftift obot:, ,.., It II ...... the ""lid and annt IU' .n ...... 11 1010 the .... '" ........ 1006. and lha. ooem ""1'1"<$ of the IUD loop ')'$tml; puado. <II 111<'1;10 IfIID'!tOM: they an rhe _ .......... ,le· but they.re lloo.lM: ...... 111.0 ..... 8CtlOpIcd ""Ih the &mI1at ~""IIOfl>.D

MlUwcli '''''''gruud ' . ' y •• rly on the l'C'oluuon\ brought .00.' by I""",, macllll..,. for prod"""1 numben by secoon . ... '11",11 pr"",,11ed lhe .xpmmml ..... .. n"blilly lo lhe hU M of It... field. II. C<IfHIdcred lhal " ..... f,rsl oI.n ,.... or)' ' 0 ,nllll'e 00Cidf ,n the "' .... 11 01 thi;r ...... ph,....,.1 ,n' . I1"", before "I""0l"oa,,n," IOIIlbcma'ic:al l ...... led .. Ind do:l ftOl hde II,. pa' ..tminllioa..

hono lhe Rr;llJ:bl ,,,It of F.lldo:l lo ' t... b ..... ulli.>= of Farad;oy 11111 Iw hem lhe c~'.r of,t... o:Ieao by .. h"'" >CIC""" Iw; been advanced, and by lhe fre.: u"" of d ynamical .. ""eU u ........ 'ncal idea .... 'e may hope for • fU, lhe, ad .... ...., .. The aeomctrr 0( ...... ',"" i. aD Hample 0( a ma.t...ma., IeII .., .. """ nt:abhJh,ed ...-uhou •• t... ,0:1 of • ';op: ""Icu"'"on Now h,aday's Ii .... of Ioru occupy .he .. me powllon in d.afO.lfUIJIC1'" KIlO""" lha. pcnall of I ..... do In .he .. omotr), or pow.m l'bc)' i'um,"", a mechod 01 build, ... up lUI e:ucI meouol ....... of I'" ......... e an: n:&OOOlngaoo.rL The .... y;o "'hocb hraday made .... oIb ........ ul bnn ul lOra ,n "....,nl"wlln ..... pbtlKllflCna 0( maloel....,lccInc milut'l;"" .110"'$ him 10 ba," been ,n tab.) I malhc:malociloo or a "cf)' h'aJi " ,dc,

one from .. h"", lhe m.lhemaliciansol.t... (Ul u", mo)'dm'c valu~bIe and 10:,,;10 ..... 1\0<1 .. ..

In O<dcr to !deple dl ... """ 10't... _ l pou.nd driiolll""ly and.o euoam> h,-EEIt n<:!wndy "'Ilh Ikpfto ofl ...... _ COllI", 'tl .... F ...... y ...... .\1 ....

coall<lcn"1 , ..... 01 b"", 01 """ WI .,., arPl><d to a urul 01 outfxc lhal " Il .... '''' ... to .... 6eId ..... ft,. 21),

We ohould le"",,,,he • • hal ,he .... ' ... ,'" I,,,,,, 01 ro"'" d ill"u"" It... .. lICIlOR .. h,1e lust.ioin," <on'l~nl nux al>lltherclo .... ... ,I ."""d,nllo okJ,.,., of

EL£ClJ:OGEOloIn III' ~~ACE

_ ... r..w , ..

""",....,..;on 0' abolemcnL TIIil r~t!damenla] IMOjlUly ilIum,,,,',, lhe doooao "'Iuhcs: .t... numbo. 01 UlI.,.....,lioos oflne.. II ...... IIh Int .... "~ .... u,!ac:r> i. ron.lane Th<rer~ .... 10 .. u~nLi[y lhe ,"1.nOlly of lhe F>tld 'n 1p.ilCl!, ,I .. ~ufticionl '0 roll"1 lhe I,,,,,, ,olera:pted pe' uni, or 'pa<:C.j" <I U IM.peal "'. l1li»'.0, OOdy""" be ..... 'un:d by lbo Rumhe. 01 marlcn ,I J'IUICS pc. "nil of IIIDt S,*" be""" OM ,,",111 .. y, .... fir") 0I1.hc luta of Iotcc. .... o of • .hey all

onl)' he """" by.t... 'mu>d'. O)",',.ad f ... , ma k .. lhe "'Ie,a] .ll;W;l 01 .... .... pdJC II". almost ~lpo.ble.J""" 0ranIe bad boen ,bit '0 "..rold Ikpe .. 01 "dotily by ono.·.a •• mobiJo: .ubjea and t<cprncm,;n. 1f\'~1'" a, lain ,cl"";IY by a mon: Of lct.I. .. ode ,",.nale. RUI " .. """'""'I'} 10 .mr/tason In cuenlia.] d,ff.n:n"", hraday'$ lII"'al ronmOOI;"" , "'hich Willi 10 mako " !lenS" i •• 10 Oel'= of ' ntM", ... romp, .. s.on_

Wc _ wlllll >q>;tnI'" hraday [,"'" lhe ma,bnn3IK:a1 pII)...,. 0I,t... Frettdt. lrad""",. Fur .... mrk. ma'hclnallall.hcory of doublet.o (POI_'ll" don •• 1'1 W • .,aka _nl' .,( lhe elf""" or an ........... 1 loop.. bul ,I ........ n. "'bjea '0 .... old IIltbm oflbougb. of a",*",""'-rq>tIhion. The doubkl. of fIII&J>l"lic cba,p .......... and 111J\",1! ." .. lhe .. me fOfmulac. bul ,he) ..... " '''Iac •• he po .... , 011"",, 'nuKlI' "'hlCh ..... uJlPOiC<l '" «IOlhe the ImJ&j""tion. of which 5<helllo. ,.;l,d 11Ilt. il " 'al_'ay~ 11<1 .. 0 .... of ."bolfll'. 11>11 hal IOIIIC dlllkuhy In IhUll ,n. 01" .. mrie ~nn .. "houl Jomrlh.". ,n ... h1Ch lhey!ll.hcn:. "'luch iI no doubt 001)' c:omlonoble. bul al .. ",.. """up" , he ...".,w..u, .. pc""""""",,'"

The "1Iwd' ""m .. ba~) oIf"u .......... n.y ofl'" .., ... ,"' .. 10 ~ lbo ~ "",a, of ,he ......... 1C ...,j drancal poIInzal""''' lhe ... btl"'l" bumn '" drl • ....,.. up alNtodJ:roand on ... ·h .. b ....... "'i<Jfo .. dqraded bu, ..... ""', ... f ....... of .lh .. l0 .... role p;iOl. Con.ral'} 10 , he ""1<0_ , hal "",nal Arpl>ll '''''-.." .. (be ",,,1,,,1 un '" lhe ,mOllnane.. ,he ·nu ... • """I. bu' ... "he, moholu:c< nor

n. EUiCTIOGJ;O .... ':n I(" 5~ACI! ~nlCllll\a la)ltl1ns: " "","be< tho- """III"'IIOQ. bul by "" """,n, rttP"""" lor [he .evoIu""" 01,,,,, lI,cral,No' Oyntun><d oI«1roph'k>s<>phy.

u ..... or f"rce ... n ""P'o ."""" wnhoul ealhnl on ft~od. ro' help: ihc <:onoop' or n ... wholly envelop. the ui. loop arlicullll;OI1 and. IU SOOn n the Ill""" be"nl '" throb. the '.AI ... ,' 1hal '1ilIed 11 up' .,,"Ilh. W. kno .. ,ha,. """'"n>. uona]1o ...... lwn· '" IlOl .""up 10 rid ,hc op<'l1IhOlI 01 .he$< ~~ poonb; i .... pnar,.. alllllben bee .. "",, 'a>mpleJ;' oa/J .. '11m pooiiC,m"" I.nd _I .. • if )mIlne meaII wnqurn..,. '''l>0l00 pIone. TIm planc c:an_ be redlloCtCl '0 lhe ~wo-d,_1 U' .... IOII· 01 our InlbootS; II OS ... orbnl ."'" ,"", "".,~ Idcu on .,.Jo ..... ho "'ri6« 10 pcoSIltOn h,m$df ,,,,",,

The .....,hanies 0/";111 ...... 1100.' a d .. tal>CC hu",..,o "'pe OUI ,he ;mmm .. y...:uum Iofl bt1 .. ""," lhe I'mnlo, and II IS known IIow m~h thIS hto!ul heIne. 1"'''' """""'1)' .nd Qlu .. h,y irritated """'" of N ...... ' on·. C<)n' '''''p'' ... tIelI ( ....

1.21. 10 (I1Um, the Mlnaol repulsiom and lU>XIIOIl' do not 'phYJlC'"IIy' .boII ... lhe·...,...nncar '"Inn\; they .... ,~ 1M pOoI4.Ibdllr 0'" play ""'qu,~t.. ,,"'" betwcetl [he """lIonal rebll"",", ,he rep"", )O)J 1~(I<Cb.dy .he amoe-1l000tl o~ ,bo .,,,, ... nb if they aK prodlOCfd b)' 11'1'0 lnlnl..ene "" ........ t. of OIIPC"',e dilution (rcspca,,'dy of the "'me d"U' JOnI15<>< 6,. 221_

• • . . - •

' ...... It

El«tricalloops co", f.om lhe point of • .,111 of .. bat the) "'«~Jl ,f.om thr °d"'lJrllmmat., point 01 ...... 1. hr <XIftlI<I<.-.d II ..,.,~ "'",.: Ihry a .. lOOt co."""t merely . 'l1b lIC'UuIJ"'''1 the """"'0111.;0,,,,,, oIltl1aCl;'- 01 "p"b.oUlb biI, "" • .,.ttxt from It 1"0 I.ual _I".... '''.01 ...... ' a ""JIC'DWO" of Ibt tntUit"".nd ,MlltUllOI "MI ,'"sh' bt ""lied. "",""onUtII '0 !be pnII: of CIt ..... lil_ In tho: ",me ..... ) •• oqow-e toOt <loa lOOt aim to fill up lhe >91'<'" th~t ' .. m.,n' bel"~n ,1110 ~~,,"nl''''m' bu, fll1hr ' ....,k. to unlle .... Olher d,men .. ""o;;' " tb,.. llIat I b m'lton InlOUttd 10 • . "" .. :1 III"", din.:I:I;oO, from one neplt"" "n,l.

B lI_h_ .prx~ II> >qIIIJ"""" In Ihr autiol matbcmattal pllpoa. 11 ,. the mot""" 01 t ..... <lahon [Ihr tul<:,. 01· ...... • of lhe pnnaplc 0I1Mfl .. ' t""U""",l>.t. medlan,cund pu .. .."......,). ",.,. ""'Uon, ""'~ ""rwo,ktcd '.I<:m.ouu'l Ind nat ural'. for lhey docoml""'" read,ly like pi: " of ule,,,',,n (.po.t .. 1 and lempo",ll. taken ou, 'from lhe

on ouui<\e' .11"""1 tqIrd for 1111: powllon and. Wlbk. lhe 1JIp: and toIl1lOf\, 1* m, •. RI, no, .n,·o/no.lbe Jf1Iq> of. prope$lol'" .prud,". oul !lOt ,III: ordeal of ambljully, T b tu d'Al<-mben .. '",co io Ihc ,ot,OO\l(.'I,,,,, 1o his Traltl d. DJ""",","'"

AJlIIIII ........ ...... 'Iy '" tbe tnoll"" of' bod,. io lilal ;t In,'d,. t;nUiR Ipla .nd l/utl II !au. I <%flaiR lime 10 do ..,. It • lberd_ &-om lb. idea alone lhal aU lbe pnfla.pIot of media""" "'UOl Ix den.'..! n

Mlny of hradafl """",116.: ronltmpon ........... bopnn.n .. like h"", to ftnd Ih" """ahly fOl lra ...... uOM 0 .... rol.,ions e,"""';,'c. 101M not. "Sur" dual;~ dan. 1c:J oat""'" tnI,bfm>ltqllCI' of b" AI'<"I'" ltUIarl~ ...... I"ori,rw tI k d".,.lop/lf'''''''' M$ "';'Mh~ ... ,lnMJtrf., CIln1c:J IOIlM lha! ~Ien ...ould lu.n 1hc1. a".nlt"" more 10 .0mOf)' m<'ltIOOJ .. "", Ibroe rep,u.:nled 1M mosl oqlcocIed polo: of. d ..... ,~ lhal dom .... ted Ihr '),>1"'" 01 tM . 'or\d and """,h.an>a:

We .,U draw ou, H" ...... eumplc 01 du.lny f ...... lbe 'l"""" 01 !be .. odd and lbe .... '. 01 mechanics. AU bocl>c$ arc end"",'ed .,tb I...., moliom. the one be'R, ".nlialt"n. tbe ot"", rolalton aooul In ua. T k .. dout>lc mOlina ;.. r""nd IP'" ,n lbe .kmcnllt)' moUoo of a ",ltd body. lut .. 10 .. y in aoy and III .nftnitcly ..... 11 moUont of ,h .. body

ThII _~.It."'" of ,wo IIIOIKIfLI " "" ",,~.hblc raa loday ... beII roothalll,oealllw:ones .,.pII.o 11 and .wld uox;()'" " -Ilhe k"",,1cdsc WI ..... ha~ d>ol _ mult f.om ,"" obKn .. uom 01 .. '..,..,........ Rut " .-oIa1Of}' mouon " ,. lhe cyG oIlhe obtrrvcr a property of.be "demal bod_ as markedly .. tlot motion 0IIra",!aI10n, and • ...., ,nhcrml '" < .... '}'I)"nllh" " JUb,ra to lbe act'Ont or , he ror= oIlhe lln,,· •• oc. tbm grumolo" hi •• Il(l' Irra,e<! Ihese t.'o kindt or motion ... "h lhe ... me m1poru.o.h,y. They b., .. ~ tbe mOltoo of lrantl."o" lbe natllral and dC" .. IWY ~"'" of lhe body

Bu, c.n OlIO "'" ouppooo<. _, tbal tile ,.-0 lMtpo. .. bIc mot;"'" 01 .... ""d>a of 1110 U",.., ..... mu~ I"'" n ... 10 mIIlhcmaloal thcorieo. Ul "'hi<h lhose 1I0I0 n>OIIOf\' would play the idcnltal .. me role' And then. the pnncirlelhal <:auld u"nclhc>c 10110 11 .. ",_ lhal rould .nable '" to puo f,am lhe oM 'O.he Ol he •. like lhe Il>rorcm on .. htd! .... Iut ... b.1osed I"" pmclrical dual", of lhe ulo",;on ., .... t. and th., .. hich "". i<'n'ed 10 l.nk bel.Wl\ thrm .ho: t • ., model of """,hani",,, domption or t"" bodic:l. ,h. prifla.pk. _ .. y. could Ihrow IRII liPl on 1111: pnncipics or nalgral phikw>phy.

C,,, OM: fo~ C'\"ftI when: the ... '*"1 ......... or 'IIC~ I pnllOpi<" of dWllllJ • ""Id IQd" HI"". hnked ''''0 by two all ,be phmomma 01 matun: .IId. tbr "",I"""",,tallo .. , ... h .. b ""'ern than. .. ould Ih', pnDL"Iplc 110' 10 t.ot .'"C1l to ,'''' C.ll>CO of Ih ... p.'>c nomeoa? And tan one lhe" Ay Ihlt lhe"" .-ould be "" otM< I.w ..." ..... ""ndioIIO lhe ..... or .ra."" ""

'" ",be. "' .. Ihli WO\Ild ..... y , he gmt role;u N ... ·\on· .... clliko;1 '"'rYe 10 .. pili" ""I"mo, phtnomc:l\I~.

II .... Ihnri"", • mati .. "" WlmC: ulJ"'ICY ."", one "" tbe poles "" lhe 'Y''''''' oIlbe ... O<ld """"1<1 be <:<>n>odcred ,,"h '"".Ie. impa.ual!ly' and lhil Iht an",ula""" link,,,, 'n WoiIkl be ......... II .. on ,emu 01 \bit """"lY .,.. .)'mmolfu"""", bel ......... '''''' .... 11011.1 Mnd ,otallOft' luI ..... -../I. 10 .-.. Ibm,I",,," ;""",,11011 0( qUIl<:mlO>M. II. as I lam~lon ...... "tI,O>«!. .~ ill I .. 'OrdoT In ~', II '"'" not moo'" .. mply\(> com~ lho 01"'" "",,,to 10. "rc",iCI! poLD' libe ·onp.'), "'" <"UI II> product lhom by , ... a$.Ia, ...... ... lbe p<)mC,IQI ronnuLor 8 .... + (II .-I I and 1M Lnalll§tJC Iomoalo .... ,_+ l'W1 .... _ "001<10."'" _""' ... p- 161 ~ 11 _.uo II .ty IO •• ~ iI ••• ""adaboulO of II by PO>'OI'''I Oft 00_ elf I. the m,1'OII1>CbOft In "" lid ........ lIamih"" Kk",,1ia "nb .. "'n be ddi_ .. A<JUd', ..... 'hod 1_ dIaptrr JII). bul .",. : if ,n ,awhon, ,100 .. !10k 01' ,'poa UI the bal ...... .., 00 __ be ~ .he ./tow pcrpmdONbr un," I.J •• 01 • lnbtdl'Oll. .. JOO<b<,rQI .......... Thr !nbod"", II. j. AI .hut "I'I'"rI ., ."'1 d""", • o;ni$..:tOS'<UIl 01 ..,.... (Deocan .. ' mple C<><Jf>.l""'1"~ bul 0100 and abo ... In I place of ""hlf..-_ ena: bett.« .. ,be ....."" •• 111<11"*''''0 unl! ....

Ltk. A~nd. Hanul,.,,, 110)10 ....... Imoo 01\ COftIoIdmn. propon ....... bly .. OIl,mat PfM)IlO 'M doftn,uon rA In) mulllplk~I""'- II. I,an, by" ... ". h.Jmsd1 .. ",,!anh'" boIoreno d'r«-11Of1 raliof.. 'b"""os', rA.he ~ind, 'The d.r«-11011 r .. .0 .... 1>11 1M d,m;1Il)"'" 10 w', .. luch, u,,". his no,".,on. ,. "'"lIen 'ro, ,, , .. ,

'f N]ot "SSUm,,,, lhe k ....... led .. rA ~n) II"" ""1'""""0, .hoc:I, muluplkauol>, I ...... 111 to drl.,m,,'" "'/0.01 OUlh. 1<> be NII.l<lcml as lb. FOt:k TH rROPOkTIONAI.. ". to 1M . hm: "",,_nlula, d'r«-1H>n. J.I, t':'" ... - J is Ih ... 1be fou"h prOpOrhoul ...,11'C'd 10 I. J, ~. ie. i I: k:., ~nd, .oo~ all, , bc>:c Ih .... unll, apPC"' I, ,he ..:>lu,lo". 10 • ]lfobkm or I:>al.~ 10 find lbe unn Df di"""io .. .. ...,hlhall,,'o' ... hll'illo_1 Lc:11: . :: ,<: I

A mulliplica1"", ~ ~an lhen be <kfi"'ltl '"'''h Ihr hdp '" the I"'=<lina bo,al'"" T hut. j ~ J .. k i. "fIOlller ,,"y or "''''n8 (I ,1 :;:j I, We .1.., obll,n lhe (oIlo"',o~ ro. mular.

ixJ _ t .J~'

J~k.J. J~l

.'- lromplcx numbc.,

J' - -I

hd_JTix, 1' .. '" Thnc rormul .. darl) oppJ6C lhe JpIIlM ulUI. I. J. k '0 ,be • ... ''''.".lIal'

unll, I Jnd I. and Hlm~'0<1 upl" .. I) I'ok> ,be IOllcr ""'h lhe ·u .. '" 1'f'1II1l'<IoI'" and unod,n><'fUIO<IllI u,,(u,Il ... ' '" ''''''' and OJ..... Ihc:rt 10 lbe -'0I'1lIl "0'" 1..). l. .. h"h IIH>IIld lot ...", 10 lIC'JI'"'' .. IU. "IlIOn: """5."

Tboo mdbod 0( Ibm,I,.",·, IS IUd") ,ntqn'cd ,nlo ,be II.nlq)' be",1ded by Alpnd lot ... In .... '''.'bc InMlU>"': ,he ..... " no/" mol"", ,ho • • toe>' r"""

-1 10 + I .. ,ho, 0( ,be !low 0( u .... f.ocro lbe lid"", to."", ahc-r and ..... be • "PO" to oJ by Iolnlll ""'t>" (_ Ii&- 231-

'"

, .,

o -- .

" __ ,1

SI*'" .. IlOl $OOIIpiy joInod 10 11 ..... I,l ... ilOC(Ind !lap'" 1110 e>l<: ......... bu, IIPPUl1 II. an II/bclilincd ,,,pod of oqua", """" Ibmihon·, 1o.1'Ll"'" btln. abo«l .. up!"", ,nlu .. ...,,,Iable ..... nclncal lpaJUJ " ·hdllibera",, Ihmc:n""' ... thus mlleO'Cd rrum ,1>0 "...,~tnlbIc order of tbe .'<nld. Thio .... pc"""'" of ,_ Ira",,, ,,,,, unlc:. lu ..... « ' 0. "'hlCh "'lI' JU.I .... ",pic ·_n', 'n10 an inSUIlI 1i<iu«1JItd be, ...... " .. bcf<>a and an ali ....... toppon, plio"" " 'hen: lran .... llOn. and rOUUOn .... ~f1Iculalcd lhe >'«10 .. lj.l ar<: "'" I""",. ilmply 'basoo \'«10"· bul aI"" the .... '" ~,,,ual '01.110111 .'Iuch ' ..... "h lhem round (_ fI,.1."

,

11 IS no ionF' 1hot m)'ltcnOUl painl whcno 1be cuonbnal'" .. _"bnI by Dcoall", 1I"ffC iWIIllowed up lind ..... fIt'd IOFIhcr, bu ..... , . 'ben: ~ J. t """QI up hu ...... 1 Of polo. --,,,,,, and .. ben! Ih_ 'rod ..... ',. llK .m""bled a, lhe QJnC ...... It .... ~,oa ,btreror. p ... a ~ind of ..... __ __ r '"

'" (koca"",,' (nhoilmn aod lll'" cank" on it • d)"""",,,, .rId .. completIon in kllC'llIn, ",i,h ,be univ~u .. J dUAhl1 ""raided by Cha*'- Hamillon', mpit , ... labllity pIlOnl m .. ",tcs .."u, .. t .... ~ uatll thi, Lome, Md "","m.d <:apc;'-" or ,"" body

n.e,., It .. lI>ulllplJca~"" and ._tOf)' ..wolos~ or ,he IotnIvJa or lbe u,"kwI ar, IIIt'rl ... + rtlw..J '"' ..... ,_). "'bN;;h i, Ill< ,~Jlow "", l""rned WI b}' • ,..,._) and .. hoch ,;",," rise 10 lbe f".mut..:

__ " <nII"""' __

H:un~lD11 IP_ the .sample 01 .be t ... """, 01 .. u.~. Thio eurnplr ill ...... '''' dUlly lbe mUhJpllCII""" or "0<1011' l!>d Ibn. cIoul>l< role:: "OC\Ol'I <l"""",,, •• ,n ... 1 .im lod ' OUI",n U,,", ,Iuol iu',tch ,b.m (')II til .. poonl. tee Il¥ndi. 11)_

Hamil,,,,,',",OCUb.lCd tnple! (I.)..) has U obvK>ul.~prnmml.ll _'" it ICh,,",~ .. "", I-U perhaPl' A .... , n'" ..em d",am; 10 (WOpd • body by lh<>u.ahL 1/, n""," the u""~rw. H.",illon co.td'uIly dl~"a,'''lhrd~ .. pre>dll(>!(! by lho mllltoplication o:>f ''''0 "",..1101 dlf..ctio", ,11<1 "'hose "I""'. It .I .... y. """"" .... In<,! ,"ttIun. lh ... nUl'" .""Iial urulilltbe qw>J",,,,,.j __ ,~ """""1<'<1 I"h P'''' 'M 01 oneII"'h"".nd . 'booc "I"",.. " oJ ... ys ...... ';.., ( .... AI'JII'I'd" r ~ QuaI01lllOll$IIIetd'on: ..... to;, poiIIbIt 10 duu""' ..... nd.o .... ide .... lII'O ... n ... Iymhol , .. (1. ,,) ... 0 ~ upen ....... ts; tbe 1CIIlo. npenm"'" 01 dillllllOllllr>d tbe ,"ee."""] .... pen"'''.'' ","'ted.o perpendlCula. loniom <_ fi .. 2S and Append •• ]).

• I • .'M_

I (_plMiUDI

I!oy. Iabt)

1I.{.~(w)

"\ •

•

, ';"" lJ '''''po i ' ;., _'. ";i ._., .

. ...... I "'" •• ""',.-. ......... ~",.- ....

Hlmi1ion .... pc:cic:d 'M ,IIILl1!!_ ..... 'I\canoe 01 hll dulC<wny and amIOCCf;llcd lbe Wt I,,,.,"If ydLN 01 hi. 1,1. to it" Mnwdlrcwpll.t.<ld immcdLo,ely

'" rh., 4"~, ...... i.onl ,n' rod"", m,o the lI"omctry and pbyOcs of lbe fUI"'" a .. volpI"'" .. ''''pot"lanl'' Ihl! of I ......... n .. • DOOI"dn.alOS:

l'Io .. Qt&aknl"",S is. mllhnulJClll mclbod. but II is • """hod 01 ,lunklnJ. and not. al last lor Ihr prnnil .,'it'_"on. a mtlliod oI"'lna lhoopl ... II (;1111. upOn lJS ....... ny "~p 10 fonn • ml'nral UIlII" 01 1M aeom<tnc>.1 feature< "'1lr...n.1'lo,'. Out mind. enpp;d .... 'h I!OOfIl<'1OQI idru,><

BUI ro. .... ny purr ..... 01 ph~ ......,..,0 .. Ii d,u,,,,,,1SIw:d I ...... calcula· 'iOn. " is dc:io.rable 10 .,"OOd e.plocltly ,ntroduc,n8 thr Cart"';,n coord.· nl 'e!. .,,01 '" ft. Ihr mil>ll II "'"'" on I point of I,,""" ,"""ad 01 ill Ih ... roordln,'''' Ind nn ,be maan"ude ... d dir«LlDf> 01. 'Of'IlO illl,.><1 01 ilt Ibm """'p<mm," Thll mode of _ ...... plaunl "wK'ncaI and ph)-.i<:al qllltL", .... i, tnOi'O prmuU'~ 01>11 ""'"' nalural lhan ,be ot""'. _lIhoop ,hr odeao oo"'>I!C,cd .. '"b " dod not """i ... Il&ctr lull de.'d~1 ,iU I I.m,lton mlde lhr """.,u'",ep.n d""liol "l,h IP'i«: hy II>< onvonILuo of IIu Cakulu! of Qt&a1""l.Ion' I '"'" ro~"notd. I>o ...... ~r, ,"", lbe iftUotl""""", of Ihr JdcaJ, , WlII hr 01 pc!I' '* L<I ... g& l )&r ,hLdy <ll all put. <ll OUr .... b;,<ct. aDd t:SfI" pally 1ft d«I.otl)1!1mM'S, .. 'hr ... ..., lui,,, 10

deal ""Llh _ number 01 phyO&Cl.l qlWlbl_ lbe rdaIlO ... of .. 'hleh '0 each OIbo. can hr .~p.t<ocd la r mo", .. mply by • I ... "1"'=10"" of lI amllton· .. ,""" by.I>< ord,nary "'I ... 1ion ....

And:

II ""'" =r-UULI dts"''':lIon ..... dnn I» liamil"", ,"'11m hr dt.-.dcd ,he! quannt"" ",Ib "ho:h be had '0 do In,o ScaLo. q .... nIL'ICO .nd VKiO ... ..

The ;n...,nh(H1 of .be> ""kulpi 01 Q ... ,cmloo:>. II .... p 1<1 .. ,.,,<1< .IK klLOwlod .. 01 qlWl,; ..... rdated '0 "'""" .. Iuch can only hr """'1'111,,01 "" lIS lIIIportati<Z, .. ,\h 11K in,~1)(H& 01 'nplc co-onI ...... "" by DacanQ.-

Mu,"",n ...... '" .Iui, q ....... """'. prom;" lIomolFruza,1IWL .. penmenll ,hat.", much roche. ~I>II mucb mo", .ui,cd.o .,""'Irical'"""" Ihan Dcscanos' dlll'L< .... """ Indeed. lI~mill""·I'nplc: oorl""" ... on:hatratcsl.pllCC .... peri"""''' 'hoo, ,upe'!><* I~h .. euctJy OO! ,Ioooc oond!lClcd by Faraday'. boet of r- rota.ioDo ololdoiod nu' .. dolled lines by Ihr ""' ..... ..: field and lbe 1a1 ..... 1 _"txt", ... <ll C,UKIl" maIm 'n~"n ... maJ 10"","" .nd spIIl,., "'I .... '" rool • . To rombi"" ~'I"""y and it am,lton " .0 """'pLe •••• randiDo< P''''' aram"", and to undc",.nd .1>< lleld al ,hn Jl'R'C'O .... 'poin,' whr ... """""'1 and phy';"; bola_ one __ be>r ou, D.C1 • .....,......nc ~ II ",.l~ by lbe ......",11,,_ <ll .be .... ,,~""''' """I)(H& d...Jny wllh ,hal of the ....,......J 6dd aod ........ t..: field, The tnnWtlOlll ... "".,..... ,hrm ..... .,. e.urctJy on tho mol", ... of ,he ch . .... molti"l du .. n ,he oIope or pOIen,;"I, and mUll",,", ,ho. mah ,be .. pia"". ,,"i,h, pDllnnlilln pivot.

-~t~f fL~~~:l~ ~r~s~_ ~ e-itjilS-" 1Q,~28" 8~S~> i ~l!! II <1>_ ! r '<"'-8- ""il Hl~il.""P!: f= i 2."1 0 §~3 i 1"'- I -- .-~ , .. , 13 " it;;:,; 1"~ a Pii' 'iI'it"hL, 5 fJ1 = i;lf:- ~~. -, i '-! ~ ..• -Al'il!..~i .!!.A;. :l o--i l.s- K "05§ ~:iK"tR-"5;'

~~;ii; IQ,~_II .. >" ' ft~ tIl~~ ~i i!il ('f' ~a~~-i,:,,"ea.~1ir _" ~ ijta;';l !i!~ .. jf~r~ ~ -_of 'f~"· >·!tl' d'~~"!"- 8 !.s~ ~ ~SI!!!D ~_ 1~~'~[ 7,,[-01·8_~o! '" l'~l 'j-<,".l-,!;;'it 'Vih;ritH ~ . - 'i "; i [';t e ~ ;I. ~ ~ ; !;' •

"~H,, r: hi H, I " ;r!itt i!lpl~" V'- -a ~ ~':I- a&.~ .. f !l. ~ :;: f r '''-=! "'!~I .. :!".~ .. E-=i. i'l ~!iif ,,~ ! g.- :;. .. ~

~cl. i"n d~" ! ,;-;; Ira. ~ 1~ goa 1 =~ ri._=

ij~ p t!iimH~ a~lI" a:~ e:9~=' ij .. :r-~ ~i-r~~[3:~[ - Q. "I: ~t ~~ !"

I~' il L-n i '~q~ 01<1'.·,., lOilt :' ,->2, ~ ~"" ,-,~ J: -<t1 ~i,a-~.i! ~-r" o~ =.2 -!1l-

.., Q:--if -15 c::. -" "'Ii ill> if

';11;r <litH ~'I" i~~ '"'- laL, 11"" ::r~ "~:'i

f~[~( l'ti,~"u: a:is[ "! - -,,--i!. ~;; !2, .... ~~

OJ" ,-',.," . fiO- 1'-~ ~r-_r~a~- 'i~ !~e~ i ~ ~

i li~:;~J~I~i~[i ;~I~~J~~ ii~!i!i ri;i~i~ir f![ ! iil3~Ii"~ii;l:" g,;;'!§~8.e ~~~a.:"i_ i~l&Si=:= (Iii" ~ ri E9 ,i ,~:gi:at §31!;~:€~' i;~! !a..lla.r~1 ~~~ r i, .. ii,;; !~!;g8~[ !!~o~.i~!~:~ ! :a -1~ .. =~r~E :i~ i · ' .. h 'iP~"" ":go,!' " '!Jf <~'!H ," .,-r ii~i; ia, "'~~ii ~i~~;'!:- ~!ii l~!i E~~lait!! ~f~ ~ :I !fiiii I'J!~~!~i r~i~l!ii t~ ~1~.r f~~:~f~t; l;l ~ \I .. iC''lI~ ... 11 or'! i! 2.!l: ="'3&~.,..n;t1t. ;.E~ JI·'!!I-~ t ~., W ~iZ";;- g fi",~~;~ ~~':::'::r;;:5' ".-;! i!1l: rlll~ ;~l~ !~!1 ~

, .. " ~ ~ .. ~ "ire..- ... '"""' .... __ "

-~=-~ "8:'a!3:~':i" ';;1'8!8 _ <:. a,if ~~~ ii-i,1 II~~:' !~8. i'i if- "lIe- Ill:: ; ---- "_. -. -. I"j 1 ~ i,.< .,! . i ~fip ~~.r!~~~~ ~f~e-~~ir ~~ :l"i .~i,!~ i~!~ ~!i ~ i, =::~. !I !g.i!.-a;,~" l!lll=i!:l'iia. -a n.1 8. r r~_ if II

I 1m H~!~li! [!iji.a H flU !!'~!lH !~ 8. !ii- K·;:;I· ~ii~ li~i~~~ =! qr i~~! ~1~[lal!' :~ I . =1 "'; 8~;~ ~!!.~i~~i, i, 11l-3:ir ~,,~·.s[~ ~.

~~i~ l~lf;,~~ ~,~r[~~El i ~~i~ Ui~f~~~ If· ~

,~

In .. med.wn. and """par[e] ,"""" .,Ib ,he "'-n'Cd phe_.MO ... of 1tIllJ;lIIII:I ..... and d<:ctnaty _ I (WOI"* ...,..- 10 .... m",. ""'JI'tIIC ......

0IMftII from I """,haa",*, pOlin' of ..... and 10 del .......... '11.1, , .... ""'" In. Of moloonJ of, I lTItd.um Ire alpobit of produciJla .be meduonlCal p/><nOlllnuo ob<crwd If. by .he .. ...., hypolbesis.. ... CI.l! Wfln«1 tbe phcnomc:no 0( rna"",,,,, l!!rKutm ,.j,h ok<lrom"l""li<: phenomena Ir>d .. ilh ,h_ of ind~ ...... rn"' •. "'. ,,,,,II h .. ., found ... heOly "hieh. 'fno. '""'" can only be prond 10 be" "" .. ", .... by upcrilllC1l" .. h",h "'ill,Ially ... 11111 ..... ' kno .. ltdll" of lh'" PO" of ph)"ll<1"'

M ..... U IU","," thlnk,,,, of./Iae moI_ ,,,",,," , ........... III .he mcWu .... .. -a.cl\a",.;aJ tII ... ,r:ouons 10 atilOI the """1'''"00II''' II",,· .. !be ICfn' ..

In aII",,>'e i-Iratq<m ,oio, 10 ..-.',110 IfMIIIWIllOD~ lIow do .be old mectuon· IeII '1M", Pr<JI'I1I' thnn ... "," 10 rctaJ .. ,. ",,,h.n 1be ni. loop ')'>""" d,oeo ... • ~ by F ... dar and Arnptro! ~fs .'sumo .... 'h \ta ... ·.U .hal .he un",'.". It filled ""h I ma"",lic •• hor made up of m"lecula. >"".,.., .""h .hal .be d,,"cllon of 1he ui. of l~ VO,,1Ca I! ,an.onlla! ,<>.he I)""" "forte ond 1M,r .111"""'peed pr<>porhon~l lo . he intons"y of 1"" field ( .... fig. 27).

' """,17

W"h hit oy.lmI or JUr "hocl ... M .. "",n .I .. ady manaJ~!o droco •• r many ol!he 1i"'1 and p/Ie<I<>m<1lII olo!.!ic ml"",""" No!e tI"" !hal lbe mllerial nil" .. ollhcst &<:a .. omUm mu<h 100 It..on lheir elis!1C coni""'. In<! .he <I" ... '" rnodt:lo or «>opera!1On or .nlm"ly Ib,,, lbey impl)'.

An lfI1por1an. II • ..,. lhal or lbe <I) ...... ;.;, ........ " •• ,

'\1.,... I .. I ...... """"" "" .n rot •• IOII" .od 'Wby are lbey amap lOtOrd,n. 10 1M kno ..... Ia"" oll, .... of 10m: .boul Impell IlIId o;ut

.......... Wc ha.." '" fact. now _ ' 0 mqwn: Int" tbe ph)1.Ini OOnnUJOD 0( ,hooe .onlCe> "',lh .l«tl'1< (II ...... ' .. "h.,", ..... Ire ,hI! In doub' 1$ to the n.ture of ckctl'1<ity,"

It i, he .. thaI ..... loa.'. I" re"",mber 111:11 lhe", ·,..ts· In: by no "",aM ,u,>p<)JCd to be '-r. but of th. "'me I)PC 1$ 1"""" "n"lyxd in I!J.~.C A ..... t .. hoc\, ,<11< .... "" Iu 0 ..... ,lh"'",I""hc .""1oiY 11>< in."n'''f illO the f\fkI.h>1 tIw IIJIp"" 'rcod illo • """110"', It .. """"dled .. llh 1110: 11 ..... 1 tnpt)U U. I. ,11_ III. Iormula I I I~

It " _ IE" &'1 10 ... U" .be dtloo:.1t po"""'" of EM Irt""'''''''"' ol tI ••

n, .on ...... Ruucmber tI"'11be ....... or chap\el' III .. ' ,be mobib.y .j " ..

r..u, IlIId that from 1M poonl 01 OOIIUICl .here .-.. .. 1"0 1.,n 'npt)'C,," .. tuclt I .. iacI""hlc ol """I c"" ,n. 1 .... ""1I0Il. We kno .... ha' 1t ... hIt symmetry bet ........ the 'baM:' .nd tbe ', oller' .1\.01 II ...... til,,.. 10m: '0 ,hc ,JLOpm and pre'cntJ il fMm 00". <k .... dcd ,nto. 1""", ,,,,,01 KIIo"o".

We ... tbe d,lIicully: 1 ,h",. I",,,.lition ;mrod""", I d,")'m"",.'1 {,om.he ou .. ode tlLa, annot mult (rom ,be combil1.luon ol '1WJ o~ ro'an"", ._ ft" ~).

or cou""'- Ihi. u ucmpJ,I\td'," ""tu .. • by frltloon ..... hicb iI .'by Mu. .. ",11 .inta lhe 101..,..',,,,,

I """" iOIIod ..... t dofticltJly ,n _,..",. 01 tbc nlSlmCC ol ,on""" III II oncd ............ by lOde. .... '01 • ."" ,II I~ _ dl=uoa lboul pa~ u.a. no. COII'J&UlMII ~ 01=, -"" •. " 'VI'!" "' .... be "", .. n,1lI "PP'" .. Ie direct ....... aod It " d,f6cuh to uodcnund bow lbe motion 01 OM r-n 01 ,be medium ClII """"" .'IIIt..od e ....... prod..ce..n 0PP"l'I~ """"'" 01 • pon in contaCt ... nb ,t.

Tho only COfII."Cpllon . ·hoclo Ita. at til Il><lcd me in roO>«>Ylng of ,b .. kmd '" mOIm is that or the .oruoeo 00"1 """,,",,0<1 by .l:l)'C'r of ""nodes, .... oI.·;n. eacb 0fI ,I. "wn u.u ,n ,be oP"""'te du..:t'OfI '" ,1\.01 of lbe ro,'""" .., tbal tbe conulUOIi .... rfao::o ol ,be p<uuo:la .nd of ,lot • .,...,.. hi"" 'M ....... _lOa..

h •• ooochano_ I'·boa 1_ wlxdo are ,",<ftdcd .0 .... 'Ok, ,II <bo: same d" .. ,ct_ •• ·bul if pIxed bet ....... lhem >0 .. '0 be LII pt .. lib both. IlIId 'bn..-bod is aolled ... 'idle .. bod: Tho: It.n>ot"""'.rou. <bo: .on""", I"hoch I 10.00", 1<, IUgeit "that .. layer ol parlocics. actlnl a • ..u. "'hcob. .. ,nletpo«:d bo'w«tI eaeh \o'Of1t1 and 1110: .... ~ to lha, _II. "lt1t:l has. t.ndency 10 mak. tbe ""'Ihbounnl V,,"1ea ""01..., I" thc..,me d,,,,,,,><,,, with "5<1(."

MU.'e/llh"" Dbta,no 1.)'Stem 0( bulky wheel. (. 100 'yor,_l joined '''IC'ber by """ll b.lIIs lihe 'ponic .... ). It II hcre lhal Mu .... n "'oota.n impor"'''t thrnhold, be """",,lies lho: mOhon of thew: p"" ....... llh an ~w'nr ~_

II appears .... , ........ boot. """"""" 10 OUt bypolho:ooo. "" """"nc curmo' '" ' ...... Bled by ,he tnn'S , .. ...., ollM _bIe panda ,",,".," <I

,., be! ..... ~ tbe ..... "'boun Q. _on",",- We IIIII~ ~,'e thai ,""'" pamelts • '" vr:ry ...... U comr-mi .'nh the """ <>f •• or,ex, and I~I lbe ""'" of all tbe p(lnJ<'ic> lOF\be. II IILIjIj~..: .. bk wmpam! ...-lIh lbal ollhc ....n""" _ . n.. pHtidoa muu be co"e .... ed \0 roll W1lboul .w.", be",.. .. " lbe _onlC>et' "boch they IqMnue. ... d MIl 10 lO""b eado 0I1In. so 1M!. •• IOUI as they ....... in "'Ih." tbe..arne wmpkte molecule. there is no I"". of .... rar by re!oI<lan<e."

A ..... pol..,. tbrlhn 1)11. ' In .. 'e11 apln do><:nbes !be IW ..... bee......, lhe "La .. _lkth" llId \IM: '''''Ufl:

The po<1!d1:s 'om""1 , he 110) •• Ire In rollon, rcwtI""" .'lIb bolh tbe von..,.. .,II1!"'''. bu l do not ru b .1"''''' .""h o,her They I •• perfectly free ,0 roll be • .....,,, lbe ,onlOOl and so 10 .h;o .... tht1. pi..,.., ll>ec panicles. In .,... th«:><}. pLoy ,he pon ordc<:1 . ici.)· Their IIIOUOn ainu""· lion ronilllutn u den';';' ........... "

nu. is 10111& 10 make n pooo.obk to deK"be ""'"' "",:; tc:I, 110 .. MILl .. ·.1I fori'" • K"lC'nutk pr.> ... of ..... 1<>1)' .nd the ,n,.oduc\'<>fl ol "milamy. Hlltt, It MOhlonl. aDd OIMr m<>d<rn ,,,,,,,,,,, ,,,,",,. of IIIn. ,,"'" tI .. ", §lI.o .... Iha, me'.pha< dors MIl ",..,""" • J)rOUJOI"'1 _~ bu. 0(11 b) <R<IIilw ..... ,1000-",.. .. ~ IUtbon 10 ..... l'\on <pOl ... of tbe '''''nru,,,,,,' of ulhomy by ...... apbor and olllle \11'-.-' 01 "'-a. ... 01 U I(IKJOnf.. .. lie«, lho: PrJ, tho ball boan np ll>d I~ 1""0(>11, rUJ'>eI'On ~s rnctaphor<. tn the ,;tn .. lhol ~ tl>eory Iha. i, already old 10 ram,t ... , '''Iuition< in_II Ind ,llnm,,"'I<,> a ""w doma,n

.1cct ........ pe1'§IlI by pirun •• rn:<h YOIIlhfW,..... 1"""'- II to ' loU I"" ariICU"l"", poonl oIa htt.bo""" oIlk1n. the Ild loop 'l"''"'" - IMI Of~/>e$-lralel I"" "'nnual"'o' by d • .,..."", 01 .... "- 1lM: Loll.,. come (rom • pure t,ncmallC:£ , .. , I" ... , ....., "" .. camulll M .... tli .."..ral<s 0111 rnctionf and d .. m" .••• as ,"bo;KI'~'y tho: q ... ",on ..r lhe nature 01' lbe 'm •• hum' nu. b,,"p UI m ... h doter I<> lho: dl3sramnutl", ...,.1"" .... already prKlI1Cd by <l< Broalir I" dnnunslrllC lhe dably oIlM IoorIlOll 01 ... 1.na1 )'OInt vdnctla UId 1M horl.!Oll 01' pIw;e .dual ... I .... II.J~ nu. punlicalion as...,h ........ u&1 10 mn<enlral •• ,1 IUenl",,, "" lhe arua,tll_ and OIl n. c&pKlly to find 1"'0 m<>I"'ru; to OM mobile point, JU'I IS the mel'rho, copt um; two idea. in 0 ... .. nh\Htt eVC, m" ,n, them up.

Mn.'elI e»mpeh .he "m<u"t"," I" play a double role:

'I iI; 6 ... oI.U.....,....w 10 bo 'meehanteal', ,,, ki>"ng[f lho ,n'rodllCl>Ofl nr lho bill boa'tnl .nd ,I\etd'"", I" "",k. ,he .y ... ", 01 'Ort.:c<o '1"""""",', " .. lhen "u.!O<l to the lc:"d ol a model of «IoI><:I.1ioo. Wnacmtnl ", hlOh Ma ..... 11 .mpitame. I~t .. may nol •• ;'1 ,n nalure.

TIt .... aobod) >boWd bo lal m in b)" Ibll mech:o.noeal ... "0,, W. d~_ lhe force oIlhe """.pltor, . 'btd> no """0 NNWlO 'htH OJ)/'!''''''' .... i'h ,cal ....... tluln .. cllwoti<S • danF"'" mao saKi I" be · .... Id· I, a .. mple an,m.1. It • ..." mellp/>l), , •• T",J"" hooo: 'hoi lal .. on lhe IU'" 01 r<><m<r mec .... nocal

'" b3btll In 0«11:, 10 ()fXIlP)' tile no,,' eillC1ncal lc'tllory. nUl n .... """ ".y 10 ~re for lhe .n.lC'UlIon by • d>&JI'Im ..... liI/ ~ in pilfllcula, he ... the ealatla'- bMed 011 ,""" ...... of quan~l"", ,1\.1, IICfI'll"lIe lhe KtualizatlOft ,n~o f\Vo&IlIl..a... Tb .... _n .'No, to really""" .......... " the 1'C'balanci", ,n I""",h, 01 lire .J1I= 01 to, ,,-lied. by lhe ~y!llem 01' pnaIl ball." alto cll'lble ..r feSlonn) III< •• Ili loop '}~1"'" ,f.",. ol , .. cumpontn .. to mutolated. 110e analolY 1$ '.rtf..,vKlco,· .. l>en lhe idea 01' lbe """pi< ,...."u it.1f -.nd lhe .... hNl:)' ol'bt """""'" bewm<S ""bea",bk, Cllli", fOf til< """,pk1lOfl oIlbe ~ ..... , A _.'01 _ '1Illalk the .naIo&Y ... · .... h to worded Ibus; 'The .... "",1>< KIlOI' .. ,a the rn<)U()II 01 tbe lar .......... k .. 1ul1.lc<:trnJ ",,",,01

iI 10 lhe d"'J'IIaomt""b 01 the lmall boollt.' T he dlSSymn""ry in"udllCCd by lhe larze .''"'''1> " Ihuo """,pollS.lled for

and .ny d,,,u,_ Afr ... 111111he 6n1 'lItem hM repeMI"""'" for lho ~nd, cooversely .... mI,,.,., ... , 1ft the Ollie, .. Ibe ball""" Ih .. i.oo'olIH __ """"-ru-I", .. hdt .... ." .tl ,l\ ~ la the UlJ loop ,)olf<n, .·bocIt mabl<S " '0 !>I_rib<. com",l.nl mual of l"""alt1 .'penmenlO .nd 01 dilJl'l'llS .pinln. o,'e, . .. hicb aUlhonlCllll>e mel.phor 'n ph),,,,,,, II a calcul.,"" cllellO";"'al . .. or "ron Haploonal ""'undlly. Son", clhe, . ,'j)/'!oca; I t... helo<ollo • IcchoololY for """'''I up .. m,lanty, .. hid! I ..... to promot •• '"",, M ... ....n lum>rll callod /IoU _r."""" .... Ied 'a 1U"l;", the ~ 01 ~bt ..,.,.,n:ho!. bet.-..... a ph)SlCl already complrtcly "",1""""'1un! and pb)'stCll hypOlltQ.t~

The ""ula of Ihi, <lmplillc'il 'on m.y 'nk. lhe '<mn ..r a pur. ty mal""""'". ~I formul. or 01. pbJ1ica1 hYJI'l,beoi .. In lhe fiB' _ -..c OR"MY """' _",Itl oIlhe pbe~ 10 b< .. pb,ned, and 'iII>uIb'" INY I""" O.,lhe co"'.q ......... 0I1''<m ..... " .• un ""' . .,. ""'a ........... P.lttIded .' ..... ~ 01 lho ......... ""'" of ,he 'Ut,«1 If. an ,be ""bet IoIIDd. ". adopt • rh~1 hYJI'lIhe:!1S, ".""" lire phenomena only Ih,oUlh . med,um .• nd are b .. blc to tlollt bllndne .. 10 racl. and .... h.,... ,n ."umplion ",bleh a ""MIll .~planat"'" """""ra ..... ...... mIlS' lherrf<>R dltro.c.- oorne tlldbnd 01 ... ...,.. upuon .·Iuch allows lhe mind II r.ny "0\'1 II) ta)" bold oll clear ph,....,1 ......... "'on, ", .. h"", hoinl romm,"ed .O uy 'heory rOtt1tded on lhe pb)'ll> cal ocirnce from .. hdllhal ooottl'''(>n i$ botm"'«I. '" lJutI " is nttll>e, dl1lwn ASIde: r",m ,~ Stl bjeet in pu .... " 0IInl1llicai l ubllc:l>ct. nor c.m«l boyond lhe ,ruth h}. (avount. hypo' hcsil.

hi oRlo, to obuoin ph)'IiCIJ Idol! "'lboul ad..,..,OI • ph}',oeaI theory ... mU$l mike OIIl"1Cl,,,, ramil!.lf ... "h lhe ~~'\lerK>C ol pb)-..c:ol .nakl ...... By. pltyPCal '''''ioaY I mean Iha, pan!.ll .. m,latlly be1w ... n the Ia"$ of " ... ..,.,ott and llIooc or a nolher .. bkh m3~" tach ollhem iIIy>!,.,.I. lhe olhe,"

I· ........... yean ta ... , M .. ,,'dl "mo~ upl",,' .bon, ,10,. """'n"l""' of tilu .. ",· ,,,,,, .. 10m !at duM.u" .... 1"'0 lyra oI_l~pIM>r

The figure oIlpcotCh or ollh"" altl b) "'he . " tron.fer- lhe bonluap and ldt •• or. familia, 1CXoa: 10 Of\\! " 'lI h which ... are ... at:qua,nled mil

'" £UCllloo.OO" [f11C S'M:1!

he ""lied Soml!fic Mota""'" ThIlJ lhe .. <>rd. VcJo<ily. Momc:nlum. Force. &C, h."" K<ju!red ccn"n

preci$o ...... ninp in EIomCtlWY llynannCJ They In: olio omployN in lhe ny ... mu 01 • COl1llCCloci S)'JltrII II! ... """ .. hio;l •• thna", pnf«lly ........ 10 lbe do; ....... tarr., _ 1I ....... 1lDd !1IOfe .... "'1

11>0 cUrKlmlUo:: 01 • IrWy IOmIIfl< $}11m1 of ~ II ,b." _b lorm II! ", mcupholKal .... maillf aD ,lit fonnal ldalH>m 101M 01"",, lams of lhe: ')'I1m! .. h",h " hod ,n j" onl'nal "'" .•.

Tb( ... In: ""nlm .lectrical ph.nomena . .,all\, .. 'hkh an: con_,oci '<>FIher by "'IlII!OOI of lhe _ form .. lhooc ... hkh connect dyumKaI (ll"'Mbtllll. To Ipply '0 1'- iho ~ old,.,.."""" ... "h prOJlt1' d""nc-• ~ IlDd "..,.-1 '-"'1"0111 IS ... cumplc of. mtUlpIIot 01, bolder kllld; bu. il II' .... 1I .... >emctaphor Ifll con~OJ'" lrue..teaof'ho .... riI:aI ldal""'" 10 I"",," .. ho 10 .... boea aln:ldy ,,,,,noel In dy""mia.-"

SIl><l m<!_phon.re I~rcfo", m ..... 'oh"OII" lhaa 'boId' ","",phon. S""" """'~pbon dopond on formal '''''~ II ;. tho ........ Io.>miub 'h" "",1 .. ;1 I'O'I'bIc 10 ,prak 01 ..... "'l...:d dlspboge .... alO and of p-acrahzed forca. and II if.he _ "'1'f1<A IA.I"l lhalil" .. o.k ... ttI<""""_ ,ball>"'" ...... 11 L _ f r J. Of , ....,.1 II" '" f F JI {_ II,L l, Tbe "'ncI' """'pbot IIIcrJo.no I.ka few n.k" ,I .u~ ' ... 0 prcc .... l!n' collectiolll 0( formulae and ",hila • proof Ihal !$ al"",dy available in K ,ha,,,,m. The 'boLd' mcIK~I!O' !orr.J 1M """'Ofy.nd ".ps Over d<iJrtt' t>C p.rool. ito oror1d II lhal t>C ...... ures. aUu"onl Ind d"'l ....... II InnipOflSlbO<llllI uptn ..... a ... aD_from iho old IbcorJ' 11110 lhe new: lhe !aile. &allil •• 'hoLe ~ of bab,,$, ."" f.,."",... , _ ..... !U

lIuI Ihll Inn'p<>nllion .. nol arbiln"., lhe", i •• dioaph", III I<Xnbfic mtUI· p/>o., II 1110 i ...... e ICI"""N by an npc.ICna: oIlpilhnl o"or and complru.on of an';n d"'H"'m~ lhe melaphorical · .. " .. alion' II 10gmmaiN by lhe: fCCOII\" IItIn of. dualny 1hll d ... p!a~N ;, .. 11"",,,, 0' br; d"ndc,"ncly in lhe doIlN b .... of II .. di;qn.nu 01 lhe old 1bcofy

Ik>Id I!It!aphor musl ,Iit<dore be ",laiN 10 '"" <If""" 0( homoa<'lblIlRMI.nd otnppo", """n oI.nlC'lllalioo in order 10 be scimllfio:. If •• }'nun,.'II}''' ready .0 ",vo.henlt, " romeo ru"""", II an 'n' ......... madellly I. _ ·part .. r Jim· 'limy, bul .I .... y. ery.I.IIl1:cd IlOUnd kry fIOInl" lhe fulcrum 01 a IIo.I_noe. an lu.llial}' ... hal aU' lluil oommlild a .'hole nc, ... ",k of f.milia, om;oF' or f.mnula£. ~ pOlnU CtlOu,.. lhe: ma'nle .... ...,. JJl.hc ",Lal"''''lup 01 inlerpm<'" 'nllon .. Iu<b _ru.bef """,phor. iboy PI ....... , ". dlllocallOft inln 1_0 mUlu, ,Uy indoffCKnI """'Itt lhal connnU&Uy (~'" 10 IUbo,,,,,,,,,

Elher II an """"""' ..... ""'~I lhal lin""""" ..... pahal d""""""", Ind llit elccuom.olnecJC fork! II lhe lame "me. M .. ""II bod bot-n ri"" in dmI~n(1I11,1 '..,,,,, .peciol m«han,,1n. hke liull of a screw', bill " ..... the fault of n, ... ,..."h· cm'lU')' ph)'Ji<:J lhal " OI'U ooIodlfled '"'0 I .ulls",,,,,,", Th .. """''', ballooca mqnetJC ,."'" 0( IoI<tt ..... cl«tncal ~n.i .. 01 r""", IlDd I'.o""",n. d&saoI .. n, ,,,", elm . ....,..1Ii _ In " ,"" ....... ..a..mm.. 01, _ kl"""",,,,,L

"J Sacna: lui. ~no"-n lew .""h "IJI'CSI'''' '""I"p/>on;.he: .lKtocal "' ..... cnrob

'tself In a lone of d...,.nl of inc.oa.inaly "rulenl diallamJ. lhe: prnJrtM of ""hieh ... Iuo,< fol lc"'N ,h.ou'" a fe ... "a,.,,:

lha. 0I1he: pot1It. $Implr Jea. 01, dc:\oJIIIlioa oull;cct 10 lhe..,oo ...... 01'. rdcrmcc Inlitdro. 10 ~'" lIS dctcrmill&llODI; tha. of lhe: po,nt.a,'''y •• 'hlCh can "",kt"n .. trKIion IInllblc: lha. 01 lhe pOInl.lwnzon ... h"'h rompm6tO.nd unfcid. dqrffS of <krIll.; Ihal of ,he "",nil 0( conlacl and p,vo'-PO'"u. apabLe of I~m""' n>.nl lhe di"imila, and of IfOUstnll d,,,,,nsion.

£ ... ylh,n, lberrion: -.no 10 be pb)'N OUI '" lhe lII'erIaoo. ""' .... 1'> I . .. _ 1'1}"0 f'IKIi ,'''' n:aprucal nnbnloo C~p<rulltnl .. hidl hid lriumplitd 0'''' I,,", cbchCs JJl.n ..... :I_ ., • du!a"'" 'hll further and 10 And ,ru.LlbWly polOIl liull I .. }'t1 mo", dlOOOncrninl Ihu I .... ""b~' II IS ;o:nlloly "lie lhal lhe lpinl caUse!! • den.ily 10 """ up by .,,,n, knM.h ba<:k 10 lhe .. Kllh and by Inieul:tllnl "n elcvflllon.nd. 'ICP"",lio!l. II .......... k .... ,he n:Id lulled IU oJccp in ,he .ld.Il.; by IMn"IItlnnl'''"' """"'"' of 111 anp.: .... an follow iho ponclnl"'" JJllhe co!bc"... . 'nh OIl' ~ or oto.c" .. WI iho _!Ch, Iaad on • Inn'J"l.ml plane. lurns ,n One d,,,,,,ion or lhe 0Iht •. accord,n, 10 .'hel,,",' i. i ..... n from 'bo>~ or bela .. ,

flu', In IlL lbe>c ailu:t';Otll, •.• ,,'in .t,1I """.in _peal Ion t>C lhe bIMh of I~ Ib,rd dm .... n,iCll, "hlt;h "mallb 'spI.IUlI·, Tbc 00110 t>C lhe llMlI lea'~ '" .!!II _ pcrpIo<d; lhew a •• ...,.u,nc ' ..... " ....... 1 evcnts lhal II""". VCT}' IuIrd.o • .."hn •• of iho body'. ",Oblhly. 1a1Cral"y.or verlitali.) .... 'by litho krM SO

do.q ..... l!I ..

, TO\\ANUS 'I It!; KNOT AS SFCULA" 'ZATIO~ or lHf. 1"\·ISlftl , ~

Tbc knot doo'<'fmI"'" 10 ~"""""'OL There .. , lind of .111 .. ,.." ~'"y .• r;.",., _'"""" to aoy ",le!1Iall"'-..UllIllOfL There .. "",her ""'lib. nor d. .... ,y. nor " 'KlIIl be"" all d,me ... """ '''' ....... r. bul Ihl> ~uah'y it lhe: dlrn:1 or"",lIc of the CanllS,an dun"''''''nl Ihal ","pled lhe DOO.d,n.,1IS inlo I pOIn •• II IS Impo<<ib!c 10 Looahz<. 10 I."!,eum""ihe Of 10 """,,"" •• '"",1' '''''''Of')' ruled 0 .... by lhe ,nle.iaa: The", II no "",0Nk or ,n...x- lhe kllOlllO$ " .. If .nd II II ....... iill ....... " hnh. Then: .. no al'>uLe ... Iud!. >I " .. "'" pocn:cd. would lIa""d ,II .ICC1'tI; iho ~I!OI "'*" lhe l".pkl1nl ""' ...... 01 inl"":""" "'''h " .... f " I,nkj " .. If .'!lh IIsd1', A knD! .. no, I ''''JCC1D<Y .. '. do nol m~" ... 11 by pu",nl OII""h", ~n lhe: rlaa:' 011 InOb!k "",nl Ihat would d"",,,he lhe: eu,ve. 'rho knot pu.Ion lhe !Cn"on beI.' .. n ,he molkk:. lIN: 1a!C .. ! a nd ,be tIcni!ly .0 I,,", .n..",.., lhal ... hy ,I i. ,mP"l"hic'o "">I> j. by an .. nd,n .... only """ t>l! .. compo"'''''' On ,he oIher unet. "" ... ' ....... .op IlDd bouOOII. nallt &lid Id\ _ to .." .. I 60tce conln! '0 ad"", "'"

EU.CUOOEQ",,,,,,-IC SPU:e

Inlun""" e,'en if " ~n'ail. I'QllKllall"1 11>0 1", ... .,lIoM' and POO' ''''''" 0"." Ihrown by lhe ;n"'Iace."

'Tho Inl.rboI muOI be """",piN lor .. ha, II '"' a ,blft·Ju:adcd h.dno ,ha. has [0 be wned .."b ... "tie bIo,,' The Inl...taco doe< DOl """'P) .)Uct·;o our .~ " "'IIlI/ldl; ~ ,hal ~ .. ,he b«,_ "'I""U'Y of.M d, .... <IJI<JIU. The Ia" .. ka,," _'" In m'<rlace po""'''' ,"ugh ambogui.) " hoch pruhlbi ... 11 Ra"on;nl .."hou. IOlc","",IIon (~fia. 29).

' ... " Coil, and ~ not. ror.,. • dlJClplll .. of ,"" in,ullloo Iha, ma~ .. " poISIbie 10

eICI~ .he cumbe"""", .. 11, ........ here .oy ,IV'" volume <If ",lid bocomeI mIred. The 0001 lead, .0 ~ rarr,,1 upe""""" of the ,m",.."ion o(.be Ipu,a) and lhe ClpIooiofr of the ."'n" ' '' .... II ... 1>0 .... b.>ps. wly of F'"". 'ou nd ,he m>Oion of _pilon .. Iuch IK aIwa)ll under Lhrea.. from ODfIO'ftI""", ,he Inlorlaa: "",kco" poISIbit r~"'IJ"Y 10 C'bfI,hI, lu.bul"''' ........... ' .. here ,he mrtapbor 1r>eml0M n, ..... , b)' findl", I ... , ida. in one.

Thi ...... al",.dy • ma"e, 0( f"""'nllion (0. ru,.., .. nlh .... II."'Y . 10.:10'(>< rwmt'OTJ, "h<l ,!"ile ri&btly • ..."..ted lha. dll1lnms on no .... "'" not .,mple Ilh ...... 1iono. So.... hid e'o ........ wxlvul '!"". complocaled modek .0 deocribe Ihcoe 'vonwxs' k",,"ed In the ........ boll .n lheoo model. ~ dlJClnlcd . W.

_ """" a belt" """"' .. 'and .... of the proIound ",a ..... behInd ,hlO: ,he """'"' of,he kIlO. JOe5 ~I beyond the space ,n " 'hoch " is .uppooed 10 Mlhc: ;1 I<'"" M) fa, u.o SRllsQ lhe dUi1ea1 ",'a'IOMI,,!, bel"',,"O Ie" •• and II .....

Mt:clwImcai modeb of ....... I".......,.." In ..... 1ed .Iroex 'ral" YOf'1Ca. .. h",h ~ 1 ........ bIe of d 111"1 "lib the "".IIe propuki, .. captured by ,he h". 1hn< IIIOIIeb hid _ hrnl"" .. "b the npdll ... <Ii "",,,.llnl,,n,,,,, and mecMn· ical ''''""111",",,,

lIy WIly of .""" ... n .. u" ... mode'" 8«K1'1e' 1")' anJ ..... d'!C\Ivery of rhe UflJU" p«:led "''''10m bet",...,o klMll Ihcory and ".rn.a aIm '0 apt u", the "'hoIe all .... '" POW" oflhe knot. 1<> ""'lIion lhemoriYft ., ."" _""1 bean of .ho 1II1<ni:ClM1n. An ,nl.....a:uon OJ _. poon., boll ... pIodM u •• _lneaI_1; _Junllw. 1uIppenc:d .. 'hleh '''"""". ,,,",,,,n,.,.,, 1_ fia. XI,.

I! .. pe.hlIl'I'hu klild (>f •• penmenl ,hal irISh mono. "..., _klIll u ""'y

'" •

" ' .... d'

" . H~OU)G(ONfT.1C SPACE

•• 1110 it:\'CDlb ",.uury .. hm lbty ..... 'ural ... '" lbet, 1ob)'notM. II .... loom ....... ·n thaI die .n' .... IN leuen olld IIJumlnll>Ons 01,,,,, .... ou"'"p'. _", "'" added (1ft In tbc IC>.I a •• ·da:onol ...... • IN" I '14Ipple .. w:ul'. bul ,,~'" ,ulmdcd 10 MlClt ,lit 1>"'''''''''' of II .. rcadtr, "'00 " 01'..,1)'1 ,II • hurty 10 ....,.,.., lhe ,moon."'"' """,".D«! ,n I le~I." w~ 1_ Ilia! ,"""';.. l ll\d at .,.."...'ncal m<>&!IY .bout tile inlcrlo"" "' hieh ,n .. m (011 U c/for1 In 1<a.P 1110 .. he( lb:lt " uofoklo. Or DO mean, lhe", 10 'fill m', ,he COlis '" lhe iIlumonalions are Int~n<kd 10 mnmd U$ "'"'1110 ,poecullm'" "'nom be mlua:d 10 the 11l~ ... J and .. 01(11 ~hau .. "" by ohI'IIOIIll11,,' ) In. Unitt. Contemporary ....... 11:" io IlOl f. r fftI!(I\~ I,om Ihis. II,. mlnco'''"n. d"""lIsioa as luI ,,'1uc!I rtVOlu ........ " aU •· .. bIol 1"''''''' add oaly.uows ,1I.lf h> be 1J'Up<tl 1»' ..... Il0l11",,,,, ~ loll ... a.., II<>! 'or_illS', hul mba enJOY the pn ........ of OOfI1n>lhn. lilt all....ol>J and lbe ~m'mCl" of Iormt.

"O TES

'Sool;''n .. Au"oo ... _ .... ~)_ .... /'to,..., ........ ,. ... ",.... ... 1'1\, .. t Irl<,../wi{!" .". ... ,..·.l ' 10 ........ 1.<111',," UlOOlO ' ........ 1 ... , I'll "j '161 . . " ', 100 .. : !, .. ~ _,' , 1100 ~_IOII 01 ,too 10 ..... '" .".... , ... "",I of I ........... "' ...... 0. ,_~ __ .. "- 0. ..... , .. ,., ____ ... 1 __ ... llU' I~4II''''

-. ... '._."-.. ... - 1_ .... ' ........ ,..,. ..... .,,'" II ..... /·-.01,...... ~ U .. '_yof< ____ It." ' .. ' tl ........... '" _ ............. ~1OOIr """" _ .. , ............... j • ..,.,....... ' .... _. R.., '''111'1"._' 's:' .,.".".-. ..... ,,,

'0. ... _'01 ... _, ..... '. .of.... "_,._" ._Lot, in..., ~''''~'''''''''''_/l' .. ~"." . '. '9<1, ··,· ............ = ... • .. - ..... _ .... "' .. tt""~ .. __ of .... " ' .. '~ ... ~..,n.ww, ,'I ,,,, -.lnlli •• IIn-,oW, 1.,II!$)lI'ahLH ......... I~, -•

• S<Iod~ ... ~lIa""'_"",IJl • IbN" ....

'" 110.lI0 o.r- (J,nIOiI 11m· """ • ~ ,.,""" 0...1,_ L"~ 10' * .. , .................. 0...1_ ... ;s,." of , .. 0., .. ,.,_ .~ ... _... ' ......... _ """' ... """ .. ,_ ...... """ .. I " ......... _.of I '_,' ... _ ._ ..

__ .... "n· .......... ,..,. ......... " .... "D '." .... ' .. ',' .... ., ~- - ..... " "' ... ~- .... ,., .... ""Li'A''',, H I I", C , .. ,111, ,,-,. .............. ," ;"of_ ....... ___ .. Cia .'-_ • .....-, ___ ............. ' ......... _of ............... __ .. """"Ip III

F_ 111~ <;1; ............. _ "''''''''''''' __ ."' ... ,." .. "'~" ,. ,.lO 0. ,. W. '" (b~ .... IlItan- thud .... ,'" f>r>._~ 0( '",,, ... , ... ,<._"", ......., u .... .,.. 191\" fo" ~. -""" .. , ......... I1 .C <E ...... , s,~ ...... ,...... .(''''''' .... , .. A 1 "000 >UId Sea. "Ill!, J.oIa "I .................. " ..... d 'W) rl!>.... .. .. "" .. fJIoon <of. C ..... , of 1.10.. """ ... tho "' ...... " _'. ,,0000I. 0( no' , ..... Ii.lO. I .. "".1) "",0. ....... n ... ''' ' .. L ......... "' ...... ",joooNMllzJi<: __ ' " A .... A",,.., :rw.... loi ._, ~ L " __ ,~ i'

n.. __ .. _ .'~_" ,," ___ , ,.~.~. ___ '. __

'"

Lt· I .... ',~

" Iw." j.<

Iw.,p to "So< ... __ .. l-.. So< • . r Gta_I.IIO.,SO- H ' .. .. ' ,.~-- t , ~. -~ _ i._Vi< I .... ' ... -' ,.."..... I~ I.C. i\lu ..... '0000", IL",.---. •. " ...... _ r:t:'\1 'l ~_ "I.e M .. ~cIZ. A l_ ..... tI .............. ''-_ 1_ W .... I' w..-d. C1.o ........ "'_ '''l),0-01",9 -'_ "IC .......... -OO .... M." ..... 10·.' _"'""_Qouo: .... '.~. IL" .. I ' :-1 .... _,.II"-l,p1<'l

. ''''''' ~ :u

.. IC ", •• wI, n.s.-tI< ....... ~ !Ell. "I~ "" ~Jz Sl.

,u "(:S. _ t.a.. _" _. '- _ 19~ ."., pIo, '_....,.. ton_. """ 01 ""' ..... _"'~IoodI_ ...... __ ·, _,' ~ It' M ....... n... 50-_40- I\o,on"~ "" ' jl IOJ .. II:-!. , oll o,~

~ , ..... ,.,~

"tW .• .." ·lboLr .... .. M __ ._ -4'''' j' • Ie ..... , ... """" ...-. ,..:~ I.~ . """' ...... ,.., •• .,. • ..,. ,",,_< ' : ' "1,19161 "I'< (;- ' '", '..., ... ".,. ..... ... A_~ ... ~, .. s_u _ ', 7 Odo«Itr_...,.., "'_I, .. w ....... n. s.-..- ,....... .. u" t, 1W ..... I~p.2.,., 0, ._. , _~~.~~ _~ . " " _ . = ... _01 .... ~._." , __ ... . ' .... - . .~.:,,;.,_o("""_:;..: ... -:=. . .:;:"'...:;;;" .. . - .. : ;.......... e' .... ., "y ""LI\I "" • k,,", 01_"", ..... itIo '"",,' wm"", Ok, _p'"'M''' 'OPO' or .. ""' ..... , UINI 01 ........ ,. '1' lcolllo loy. 100 """ . .i <&Idol ... .. t .. ,II<..,. , .. _,_........................ uf_OOO ......... _ ••. s..,. 'A_ of 1m' • .., -.I I 7 ' • " 1_' _ 01 ... 10 ; ". "" I.t ... . ~ I.- _ .• _ II.' ." .... " __ I"1.J.~ _ ..... ' '"''''''_., ..... ' ,,,,-...t D' , ...1 __ '-'1 ... ,'10-1111

AP PI!NDlX I

NO H ON QLATERSIO '< S

" qllOlern_ Ii an "rtlo:..,.. quadruple! q .. ,.",. ;<,. x,. ~.)_ An ..JJ,,/(M ';,n. ao.l, be defined;

,>+ x, .'" x,) +, ... xi • .ri. ,,;)

-1>.0+.:0, Xl + .. ;. <, +4", +xi) ' 00 U an a mulllP/It:",;.. b)I • .."Iar la rcal num bo'rl tI>!"KOpon</,n. 10 dian..,. of ..... 1 rdllo .. ,,,,,,,l

i.t ... ~,. ~,. x.l_ ! ...... ~, ..... ,. ~.I

"Tbeoo opel1">OM "",b " lIO".bIt '0 .. n,e 4""nniollt on ,,,," form:

'(' (.~. I ,x, I ..• ,J .. " ' )

I (1 .0.0.0)

; ,40.1.0.0)

J'lo.o. l.OI

t - 4o.o.o. l)

'"

'The qUl,,,.,,io<Jm Df , be rom. 4 .. , 0. 0. 01_ ... I Ire .....t 10 be 're.I' fi<'mponol' oo:wrd,,,, '0 lIamil' .... j.nd can be posi';'e Of lItP"vc ...... ,II ~l Df ,,,," ~I Ju.,.h, 1,,110 aw>aII,cd '''''~ I

The qUl,.,n""", Df ,be Iotm (0. x,. x" x ,I In' loa'" III be "pure' 101' ·'pahar ,,,",, Ire , he 0'"'" Ihl' rel" .... n'.pace acrotd ,nl ,0 Hloruiton), In """,,,,,llr, lhe .. )" .... ntl 1, ). 1< are ",td 10 be ',,,,,,ia( unill.

",., I"...""nl remork. k~d In ' 0 dcoromJlO!oC' q ... ,,,,n,,,,, by ,he formula:

'" ,",here ..... he .JaW' pan Ind .,. ,he 'W<:looar port. We CIIn <k~"" • m")"pllOll,on " or'lu .. ' .. m'o .... ,",'hodt would be d,,'nbo"vt.

,n ",11"<'" ," ,be pr=d,nl .dd",on and """'f',,,,l>Ie ... ,," diln "" IOIII'

.. II 'Hi' +(1. q) ~.. 4 ~ 11. '('I

'"

"" APPENDIX I

The "'rmull 01 " olKe""" lbal '

bet ... compl;..Itd, .. .., .'ill I>OI ,,~ Illlcn:. bul ",'. Ihoukl

III 1M mutllphClt"'~ " 1>0( .. , .. , ally commullb>'e; ~I 1KlfI·ttn) quate,_ (dlll'ttmi r,,,,,, (0. 0. 0. 01. ""nut In in...,,,,, for X;

(el U coollOideo .,lb tbe muh.pbClIlOft 01 raj nwnbcn lor the d"'..,1115 or lbe ......J .... ,&h' I" .. ;

rd, 1 )< q .. .. " r for .11 '111.11,,",_ (e) I" j .. -}" / • .to J" A .. - i "J- to k )< /. - / >< II -t. / ' -1'-""

_ 1: ....,." .... , a1I) rI .. .. Co. .. , •• , •• ,) io 'pu .... or "pollal' q' .. .. " ., .. ( - r.·,· + .. ,' + ",'1. 0. 0. 0) T1Ic !qua .. olq'NI'lal '1 .... Lemi: 0111 III 1WffIt1w. Moo ...... , tbe proptny "'· ... pti,., ,,<We' rltw«.~s .... 11.1 q .... 1Cf1l101U;

(f) of .. llId " ~ spfIhll q ... lntlIOftJ,

., " f ' .. " x f .fand only 01 , and f' .re '"""did' (f ..... · wl1h I ~ ~ "" ,' .. -Of x., 01.l1li 0IlIy if., and " Ir.·~

(,I the • .... ".1 uruu' I.J. " _.lIIule • 'ruCCl"IIIub. tnhtdroa .nd , ..... p place throup ..,,"'_ l.oaalCd .'Ih muh'r! .... l1OIIl. p .. mcull. c:aoe 0( the 'I'''' ptadlcular 10000000' ot 8",,* 2l. p. I U).

"011" 0" U'" "'lLIO"'S ASnO,,"OMICAL EXAMPLE

IIlmillon illUi'rlta the mlilupl.(:moa bible of Lbo unItS I.J.k thlll: k1 tMre he tbe uwo] 1'00, am]" •• 1 JIOIau. t"'" 6" A 1_

•

•

,

" , ,

,

W,lh lhen, 11 .",,110" .... '.' •• ,d ''''~ d"«lo<Jnl of .im (I",m I"" ..,utb (o.,a,d, ,he nOrlh. lei lhi. ho •• and 1'<>11\ the "" •• lo1l'll,ds.h< "I~ \cl Ib" b< yl ~hk:h be roon"Itm<n11 ,,'jlh • d""".ion ~ (f,om the I"'uOO .ow:o.ros ,II< ttnllh): '" P. ~ ' hu. i~f)ln, Ihra: d"'llCIion. lUI' Inc ,eleocope (.<eo Ii" 8).

Th<'>e d,,«rions of ~,m non be pr\dc:n'ood "" ",Ioled.o .=on(ln lhe .. nse of lhe q..crmlOn 011 p. 161)

ltomillon lhen c~ 10 ICPCLJ nl lhe 101'0, .. 1,,0," i. j . k 000 to ....,.,al • • horn ""h arr<l'nal "."nll." filU'" C .tro" ...

Adop"", >unable ImJlhl, lhe d,,~ion. x. ~ 7 "" Irnked 10 i .j. k by lhe .. 1a11OM:2 " I.} _ ;-. __ '

lIy ... nable ,01&,,,,,,,, ",.he •• ''"'''~, ••. ,be dlrca"",. of rum arn be """n@Cd ro. one .""'Mr ",.". ",laIlOnll .. onenl«i by .he ...... i.J- k and ... lMoogh • • oo.n .. folio''''

(I ) A Ii .... ",UlIlOo Ill}) 1"1111 lhe ... • ...... around J l- Ii .. [) 1Ind E). '"1m I.' do<I:".se d.rca_ from II'

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• , ,

o j

w

, ,

• , , ,

j

ThIS p'·OS 11M: lC<>n>rt". .. 1 lIJ\I!1f>lion of 'M muluphClI1i<>n b~ j of ptrpolldJCuiu VttI" ....

Tho: t.bcopo: ... hid! .4> "",olin) to ... "" 11M: nonh. now potnt. 10000rth tho ...... th Z. IlI>d .... t ...... "

_· 'I(Jlhl - RUK-;j-J.'-I)- I~j .. jb) The ... by carr,;.., O<I[ I "nO! R(ij 1_ Ii&- Lb) around I (II .. lII0I_

ho,,,, .IfMcd ia I c\ocl,;~ dtra:IlOG if II illftII ,,!>In tbo ""'Ih~ Lbo 1<1 ,~ .. "",OU I_."rds lhe ... l (alrml'ly of.'1. .... c !>Io .. , [lot"*'""

}-7- RIo'H,)- I,,1I or '''<1- I x i _ }

Thll iI; tIM: posiuon of Hlmlll" .... fll""' f..." lei II II p<>oiIlIk t" 1M" point 11M: 10""""1'< lo .... ' d. lhe ..,,,.11. h) brinS'",

.bout • '",.\1"" Rlt ) of l,tS k I"'" flJ- Fl.

" -~~

"'¢ba"~ R(Ir)_ :LU:,"ab<>bo.Iu.L I " j _+Jorlo' J~'

TIl .. >«qu,,,,,,,, or '" .... povot. RIb. then RIi~ then RltJ h ... hrrdOn: mOll .... II ... '010""""" in. d,,,,,,'ion I from 11M: !>Onh I" .. .,.nI.the lOull1l ,11.1 .. of'P'l'>'le to ;1 •• 0.1;"1 d".C:1I,," ol &1m I r,um III< ""ulh "" .. ,d. the northl. II ,JI",,,.t .. the: rrlan"", iij _ I

'" Ihmihon lal .. aln: 10 d"",,,~llh ,hoe J"",.iofu <f.u.. dc:!.I"""t;d by 'M

G"",k kll." t .. (I. if."" the nl ..... u. do..IlfUllro by lho Ullin Io:Ucn Ii, j . t ). n.. Ian •• ,nd...,. 'OIallOlll. ThUJ lhoe Ute 0( lhe IntIll __ ror • dircaiotl

01 OJ'" and _lOr. ~1I .. 1 Unll """""""" ..... " '" ... lIm, my>d(lu YOU_) of the d .. """,,,,,, be, .. '«1\ the r<>m.ln and lbe i •• liI: alpll<obols" '0 ...... k •• tleaol ~pon.nl). tho diouDCIIOll bet.w.en the '''' o(.~ ...... • ,.",...J and ... , .... """ l!un&: ...... _ and I .....-; I sub;lccl of open.lioa and In ""muO<' (u.-r ........... QIIIII....uo.... II. PI>- 61 62).

"Tho ., ...... n""lalc the _-. bet ........ lhom. ",boc:bJ",,,1IeJ IIx "*~ Ip. 112) '0 aD ',nicul&lI'd tnhld,OII' II ",'ill be ""',,,.ed tbat IIlmill",,'11,;1I<dron. ....-,lUI."ICd "'llb cardinal POO"'" .. ,"" 0""",11" of !be 'd,,,,,,,' lJihcdroa 01 m<><Io. b .,<Antlen I .... fI.,. Qa lad Olt).

• )

• ,

H .... hoo·.m_ Indl_ fut . f

'-'X-C.I>

There ore In (aa , .. ,., "" a)'. ()/ "mbod)-,n, In lpaa ,"" oHltr prc..,tibed by 111< '\ICUS,j". onurv;ta,Iu" or I, Ihe" J. IbM • . They ~h indua: d,ff. r.nt con' .. nliono m 0f'd0T 10 , ... (orm 10 (he mulripioClllion 111*

• , •

) ,

AI' I' LNI)IX II I

II"MllTOt.U ornATOl v

H .... ;It""·. "I'muot V ~ defilWd br , , , V_ I - +J-- +.t -

l-. ?,- i:

I. combona ,110 don> .. ,,,,,,, and mlLlupbaol1ODS bt l.)." 0., ''''' eo "11';'-01 funcIJ<NU I(.y. y.:~ V .ow:tt .. Jo()o..:s:

,1 '1 ,{ "fI- I-~J-+ '-,', ,', ,':

On tho _.onaJ func.ooru f(~. y. :1" xr + 11--- Zi ... h«e X. Y" 70 "'" tho vanahle rooodlllltH (depend,". on ... 1 Ind :) ot v: V ....... ron"",-,.,

VI?!- d,.V:'ot!'l ... -he", VI? i. " 'lUll'.' "''''' 1_ dorornllO'llion (l l ot "PI"'nd .. I ~ Il!Id di," iT,be numnkal fUnclion;

.'X ,'Y i70 -1--'+-~x ,'J ~:

[" II ,he Kala. pori ot ,he ,"nab/<: QlllICtnl0n viT~ I nd ..-he, . ro, iT ii ,he >'..:tonal fUn<:'lOn:

(~~ "7.))+(", t. ~1

- . "') " Tho eo,"' l'irnI '-d,~ iT" ;, called "oon.erxo:nc." by M.I .. 'e1~ ,be: U rri "l1I

"tOt-I" ,I allied twl l by M .... dl . " If iT It. IXlflCelved ... fidel '" rorces II('l1n, on ""rI><1a. ,be dlvnJefICC ot I

II a pooD' P ....... u ..... he mlnron ,n ..-1I>'irinuy_ot I!'..

Tbe 'OIa,>OnIl ot 1""(1'01 I ~ abo aalled the "curl'. 1- "'ibn ''''' local rotalory _ion. '" plnicle$;" 1110 ''irinllY "',. MI." .... ",pial ... ,ba, l1he ....... 1Ot curl] ~ .. ,be di""",_ Il!Id , ....

Itle ot ,"" «)1.011'1<> ot ''''' .ubJcct IfIII'II •• moved by ,be ".,.or' IIId he ,,_ ,he IoJIowiIlJ ~h .. , .. ,,,,,,, t_ ftp" .nd II I

'"

'" .. """,mix !II

_ _ " X,·_---

/ ------ -, , , , , , X , ,

/' , - , ~~---'

t ..... _~._ .... __ ",i',,08J'" -",

NOl. th., If i1" II an uilJ oCCl", ('of ,ho< .~ of rOlalw .... ·~ 1'0.-' i. a polar VU'lOf foltho SpKICS 01 ulIntJatioml • ..a ,.;.,.. __ n."Juotifie Mu_D In .... _lOll: '1'110 Haorult""""n ort11l10.-. I""hod 10 In) ."eeI", full<"tlOll. con' en! II from I",Mla""" to ""'""on. or fro", rolltiont" m,,\~all."n.lIoOCt).d. '"I ,,, 11>0 \"",, or vca", 10 .. hi<h ;1 IS appl>ed' rOn lhe: Matbemah.,.l (1 ..... 1" .. _ 011'11)"",,1 QII,a"u".", P. 26j)

"rVf';f!)I~ III '"

b~.flt£ _ [f! .,,,, + r lit. II>< 1raJC<lon<!< .... ",,,,,,In , ..... "",,,11eI to 0:. rot V .. ....

.. ial .-ea", ... "'* InplOl'1C$"'" a",1os "",bnaq 11\0 .f.>, .. " .. m",nrd

."tV.21t 2:'; - -If ji if .;. rot I _ zF.} be m:"I" .. ~ Ln,.aonn of mI I .r .. ~b .. .a:d by tho: ,nojocl<>r)'-arcles of \

Figuring Space

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