Campan, FLorica - A doua carte cu probleme celebre

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E1\*TUgFI[:! 6p;{T{A {-E

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dJc*metrlo cLr 1:a{rr,r d:irneinsiuil;

'',I,rrr,trlrt,lirtl\r1)r..ri,.i-.,rrirtlrrti.tnir|ircrI[ililiil(iti.jli,.rslc.Il:.r.,rlr::r:i.i.tr.t,.r,.,r.:r,.:r,t,,r.,'r1..,,ii,,]l.ir,rr. irrlrf rf r rir trr{1iii(. iL: ohis"
trrtt'lnl ;tlltl.ll.
iJ i,' r^ir |r, .1,,1-ti1 ctt
tr Tt'i:t i lr i s i Xtl i

l,,,liirrr

, privc;ti,,ni;f rrrrs.:r irr rr!lrril;i ;i-_;,;i rr,.._rrrilrr,r ,."i.lrr
,,..,r.)f",r,
il;il;t;';i;r:'(,sit.riiir-rr,;nrr,liiiNic:iiiia"nr.rr,.rr.,ornrq-ltr^laaiinijlntrictutill;ii;rLi:lircv''irrrriilriorrrri-1la'ir-,l;fsirni,r;-iisr:i,ini1,'i,i.,i"1u.,'s,1i.r,.;",l,ag,t,rir,ii'r,-il,,t,lri'ic'i,1,r.;,itr'i;lr,"[i',onat,itrrr,,.r-tf-rri,r'i,1t,r1rl"ti:lqn{.eo_inir,rs,.itirnerruii.rr,i"ll.e,lo),,;.;
1n1

in stur.r.sir sirliirrrltt'i)('luta irr r:r::tllrltii si rttttllci itl lttc ritr
iloLrii Ittitntt;i tlt' lrr irri'llir stirtgl-r lri illtrrl'.b."tir'ti(c)lrl);(lPr(i'(rlclltriio.lrr';i
N-rttn slttts; ctl c:i"t tlllrl)li sit rrrir I

N-llnt nlLi lrlzit pirri ar:Urri cl-r s-tLr'ptrtclr fltct'{) r()til .tit irr
jrir.ul rrnui jinuruirlrrttrirltriu.,ri rprrlilLJnrriilrIc)tlrcriiltt-,luini .ijnncr-lrriill;uruirtpil,lrrtlxnrr',l
liLi.iisi rliL, r1iar1r liinrt.
inliri it, c1 rlt,s:piu-tc spl.rirrl i1 crtLi' irr.' lLllijnt i1 tl6uii pl-rr-ti
i'iirI nici tl porti{ii clc lrect'rt' rlt' la rtlla llr :r11-ir I Si lLtr-rnci,
i-Lrtn vrci -cl-L rott':rc ttt;irttrla: l)rtlrr sllar-g-inr.l llglirrtlll , llsll Crllli
ii-itnt sptis ntrri ittlrirt tt' I
i)ir'It slirti sprrrtJ tlll r,n-illll ltsit'irtltt ilL ltiilll tlitz|tlit:
,liri -lilLI)tr'1'ct'lrt'iir(r)tlutli"t(ir'si l)tilatilrtt:l
-rl;atiltit' I ilt clii-t lll llttt lrl'lsctlt
..
rllIrl, si)lltilll
5i tri icse , rrll ltr'(' ti-r-i ci irllcnsirrlr i : llrtrgirrrr', l;ii irllr si itlij l-
rittr,: l lril,' i ;r l)irll;tI
- .\ ptrtltr tiiitlcrlsittrtt', cl si lllilllllilt.1:L lri.t'irt1i, 'i' lrir" l7 BA
;tilit ttlt<'iur"ir I (.ll sit tc Iac si'r o irltt'lcgi, ;i sir lrtl tllii 111 'rl
niion, rlrtlr'r:iIlrcirr,clr,cr,ligt.'i;rIt'.)ittlslr)ci-'tlstltrCzi,iriitlttuerltl(.trrrlr:gii(u.';tritignfohtlailiictl;itr'ciluc'llltlcuiit'trtlitlcirhr,'i.r:r:ttinl"l\lcls(t,-1i"|tirl,:.1c1ct,r'llttstt'rig:llltlrili l,i(1rlr':ri.lcrrliiiotJcrls;r;s(rr.';tt(lr{tlt.ctl,i^stclra(t)l)'rlzi-.lItg\..,s\illl'-iuu-rtisrlrt)rtri(rrtIi_loirlul)i''rurllrn)liltrr;rrrli.rqrrrIt'lrrtcrr,liirrirrriri.lrr'(rlc:],.i'lqrsr\lr-lrti'rr'rrrr).'i,rztp,rrririlrrrorlrisrirri;,sl1rt,i3r.b,,i.ltirirl:;.rl,rtiiir,rqr.l;irui'r,rlrt(rr;l,iin,.s,n7r.ci,liii'1i.'lsriIrifiliir'rrrirri-rlrr,,t,rtlrI,t.rt.,i.r.,il,rlc:rlrir,r.risrt,cricl,r:r.,rlnr,rl,;uilr,trri._,r,i.r
,{,.'l,rt'.r.'i'il,)'1,'1.,,,,\,-\n'l,)i(,'cr,at,ri(rrlt'rs',t,'titti'ozrtl-ll'ririllcol'iltlt'o;iatsriirutlrlcglrsiit-lr\itl'(tt'..ir'(, l';itir titr
rr -ui:i rllr

l.l)r\lrIaii(linltllt;'ili,rhli)ltl.r;rrl,l;ritc,'2'si,ciptitjtttio'tri(r'':ll:ltltltlIitl.'tirl'tq',t'lllttllritaritlllllli{ilrp' il;lrill'tt'itt(t.''s1itt1..lt1\'(lr'ilort.rlr'ir(tilltilgl1grl'liirtr'lgllg6'll)trl.irrcIlitrti

X

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N, \t N,-,-8--l C
t
,y v l'ir. lS



dac-l in sPatiul cu ii dirncrrsiuni poziti* u'rri p.rrct t:stt:
ii
lirecizati't prin..celc r:c;orclonate cartezicnc ale-sale carc,
chipir cun) ;tii, rcprczintii rlistanfele pu'ctului l;r ct.:]c
ir.lsrclpct,l*'a-latr'tlttuicta:nrgmdcriciinicrci.iorltcorp-rurrc:iriln"onr tcsut;pt,tarc:ro..tnriu:lrlr'ictntlrorp:t.rc-ac.tnce,ldrgi.r.in?cc"u,nlasrioeuarilt,irPriocziittieu. cloui:i intrct
Punctnlui

l:'

"* Itccunosc ci. intrc,bar.clr tlr c gr-ar i ;i nu,ii lrrii r-;isltrrrrr'lc
clr acrLreasi de-'zinvolturi clc pirrir acnm. Nu ne liuienr inihipu:i
i.:rrrir arat:i sistcnrul rlc coordouatt'folrnat nidi d,-: 4 si nici rlc
n. lrlarre pcrircndicularc donii citr'<louii ilrtre ,:lc, diii forrnirl,
rlirrlurulttcar.nsueprcrlamitiLcc,,-ecsxtcisiteu:nctipl r.rlt)itr,t;rr,c,ctel,a'f.i.rfririirr-uscililsli.tr;r,roiJ1)ar.iurrlilnlrtrrrca
tlouir lcrulin'nceten.Asio(rnr,a, l)x,p.r.i.n. "trr,-Josi{Ror(m.v1ri,lIisir..,.'r)r.l,o)atlgjriri;ricrtc,sc1l,'isLpi;rctiiirni

i' .-*
spatirri cri clouir sr1l1 cLl trci climensiurti:

,i -.,. 1.,,(';"*, i1t,1_(i;-T,)r;- :.. t\ : l;F

lr:: i: {,'u aiutorul n,tiurrii dr: rli.stautii c,[intlc rlorui Dunr-'tr., r)r-rt. lii
trarlsl)use si r:cle.l al tc not iun i di n gcont.tr-i;r s1l;itirrhr i cri 3
rucnsinni, r.::r. lrrnginrr' rrnghi, , t i-
;rlir;, r'ohtnr ,.tri. in slurtirrl
..:: osclrii n <.lir.uensiun i. crt
tl':i:
-|
)::,::l: l,rrrr is Lrtgrlngt:
sslrrPr'irzii-riyciticNftnocl uraaotcnsa.irr-_toisannnrtcfru's.ialcnCa.itrnrti;,rn':i.iiiglir{r.Cr,r"'ocrr_rarnc.r.tritarca'..lsttct'ctr;(rr'.)i,,;nL,ici(tl'alit(l.r'rrlr).rr.(.oi.1lrsur,i1:lprlo11laLPrL.itrurirluinruilleori_ncdrinirn.irls,rlsr)ir;:aisrrt.igrfirtunrcr"oroiirniLn,litmrrl)reilettrlrrrnUi('srcioi:l-,il
trtrirlica tttt lrt l.it;t.,1 ilt t:ltt't't'rtiltclclr t'itt:r'a t('()l'('lIl('tlirl s1ra1 itrl
[" t."i ,lirut.n*iun i llr slratirtl crt 4 clitltrl.siuni. l):Lr clti:rrr
J-611 i-c T.aglatlgc, irl '\lccallica arrlLliti-
sinraiitriLittlc,.[,rsc1,h
rfiicfiipcliarn1rinilinri"a1errt.l.tl.,-h1rr.nc,piperrbtczsfpir".sp"liiicc.tnt,q",etr."af"rrirrur-Llfrr,ti),.,i',1rt'.,,,.ti,;cu('irr,'ci;lliiitt-ritL.r1iIr^stiri-i,r,iu\tiliritamCtarIrlr:icgi,i1glti1llcurcnpt.riirlcct,alr,i,ln:t..rrcL;'l,agi-irc,ctifrtrru-etti"ig'.cottcci,llor,:lcat'(titcitut'nulici-,cvltan,iluuprscl1te{r;ariAtp7.'i,ci.irf,8rllatot'i'l1c8(l'irln,njl;:serl-litsraitu.(icrcuotti'tillarrnloaln-'grzliurlit{gugicii.[it'llirs"ti^i.iii'ic!ccispttceittl-|l,ttoorlia,tei-t,ttarc1bnrtn:t'rr,srlatn:rttr'rc1lIt,itittici1trr'.'tct,tliii-trIr'rnorttr''-o'l'tt,llrt'o5ttrrcattttiftitciit-tr'irttttirigcrtr,il1totrtrntgrls1trrtsla6lttl'ttl;'airctlrpt.ri:ricstttlrc]l:g;ilrrltrrrir1attItc.'rtrtt1it\rnls'Oiltrr;ct1.lcltlrcllr.iitlrit'.-ir,l-'.
I)lur:kcr, ccl cltrc a intrcvizut" dtoilatl"t i:u I lrr.lev geor-nctriiTcr
,uri'rrrl.r.rvnuio'celulr.hntiuiuiiscir'saitfllurirap1r'in.clareoiu,lt",nnii,Ilu,t)saeo1ictrL,xoetnrr.prniniiPdrrauicrtomlrlckeairs',il,iicziPnrna1"cetr:r,rinnoilie.lcscl.ntilntcsalPrrstramo"tatliiijccim'csiu,,rctle,relIrnl1n)ilesoicas'rsiitc,u(lp.aet'rnn'usr:rigsrl(,p'-'ip,lDlll.tio;rotrr:artcu'.ii.:nunltclrccrlioulnlinipri.iotriollar,nirts,riuittnrrrrurrtticitirrtrri,rfzii,rrjztrir'iiricclilranlrir-.i,
1i rnatcnratic. insi. lLceasti intr.rjtic ar c(,r(,o;ttt,rrtit,cu nurlli
rn;ri sustinutli pcntru a 1i rlczvoltati. rlrcit lr(-('ra g.colnetr-ia:i'r.

107

1rl6



oIurlcni rlc ljtt'r:c, (l('.sl)1.('r:arr: ninric r]u rrri-L intlr-r-,1-lti:rtt,stt:

rp]sispr!jit)r)irtrcai_cir)it:lat(,-;srcprl'iiicrPrilrsinusiciiiiliorlitnib'r,orra.zcmr'ii':ltrinsi$1rtaiit:alrrci.cal'l.teioltopcrtttcrnrtn8blrr.rtcrire9_tttlbrirUienl5oa)ull-rt.ir.si.icnr'iban,srriaair:xiuco;itartlc-rcnc'ciolciuclaoltrlr.'asalu'tt;rini.rtrltnactstirtir'i'(.c.,truvlri,aril1l,rilri!)tracirrr\lirl.(u\sttt,n)'i.irt4,fitiittolir.urilirIlonsnislrtriatirrlc;ac)rijl,trar-:ci;oa"u,,rirtn',iiriflr-rttlai\i,rta,,rsl,.,ri,r:scilliAsrrtltzo,-,tciid,nrilrnl.n-rir,rjtirr:rtr,i;;IliirriiirIca,'t)trsCa.tlru.;lia.iarri,rr.L|lt-nllIi1tiiJtrlciiptnlnt(.ir:;,unuru-,r:ctH.lr"ttli,ritr1iru,i,.rn,nn,i,.i.riict,si."len,otrlutil;nourrmnrtpiu,rirlsir,i-olIsi.ita,,oi,.lii,.diiitrss.rsiclai,c,titiii
srl,sr,{cp]irnIlr:lir'rts.istlriiiui;cnrrllcurri'r,,.il.liimc[nn]irrclr,r-siir1ttrpirrLlilLmirrnn{i"tiuirc,tlzrnrct,isrlrlrirlittii'irr'iiPia1riia.,-tikrilc-',:zlirritlirir1nr:ra'i.tl.gilirnr-i'cirlrr1.,cn,tat.lsr.ll:r)<'i:ruuprlnntontcrvi,:n.lrrriirrill\cir\t-acr,ur',lzllill.ntsi:nljuicl',i"ia,ai-,tn"r,l,b,.i'.a.;ltrrnr*,,,,r,,r-rc..rr.a,it-,,i,
ir,jrisllt.ril'iu-;lur,rt,rllr'ri,':;-"L:rtIi,ttrrriir:It,.r:r[ir,rulrriJciL'PrlusIiricirritrriLtsirir;ttri;rils,,'l,sirr'.{citr.ilittlsirtritt'trtr1t.r'iriri..ilnr,nirlls:t}lrrr)'it[lo1u-;'rtalrv:.ihrl,]rrrrririltst'riurrrn"tlrtitc1ta.o:cnt..nscs-.n,ttr\rrrr(rrr'.r:uisnrnrcr.n."lcriitttn:'rti'rtlr,t-Lr'cicstiiraa,,l;,rrrit;c'rri,1.r,i,cai,rit,sr,i,rtP,rpirr.;c:r;a,tt,,tirt,.':iarir1n.i:l,rcisurl(ua;icjr,rr,,],,a,i).,,,c,l,,"nnt,1rr'.,cl,it.,rfci,rr.",rtrl1icto,r_..l\,r-i/,
s.(.)^2-_7r(_)1\t.n si-ti s1'rrr c.. \\-r'lls isi t.'ritlrr.t, t,f''l in ,rrrl

oclriior";i, rr1e"',rrll.ir:ir',t.--rit-t.ct'}tna)'l\'c:),i:,crri'1ra'ir,iu1'rtl1rult't'ir:{',.l',(cctn'scu1iit,cine1.iucc'p(hasn.eun,stsiitrrril-ueri.aic-iirrarujr,.uirnialP',regs'srlsaiitmiillzcrarscnaoliutucb,lLrli,csurc.lrru,z.cn'ri.niliaril-otitiisrs.;rr,,1iur^r.c;tt-iri''itir1ituril'.rrrr',rrririllli.rr.rsgr;risLioiirrrit;Crcri.failh,lplc"nl'Iub,.}r4;,r,tr:,it"-i;t,ci.,a--ui.i"t
l,t'i'lt rlt tt I t
l)r(,ir rlrtrl tir

tiirlrr,'tcri-ltitcgia]'rlrrtr.li-ula:tr'nt.liretr ulsilnllPrtrlrirsni-tl,tcitaner-ta.rt'crrsrs4irrltloltcrsta,ilttoltrcnntra,criirr.tninirrpltusl^Lornir'ata wl.;,;;

1r:rt.iu,s

--, I)r'rrtt,stt'z L\li-arl*c an.rirrtc l<xrr-tt, bi't,: s_. suit in
itt;rsirriL t irrrlrrrlrr i , iL pr-irs cclc rlclrrir rrrarrctt-, lr apir.sat Jrc

ll0 111

Zitrr y 1,r1'fi1';1 tr';i i 'l' L r'

:ll .*; l)(\l'il\:ilr' 1l! -il
'

:r',i'.ri ii lil t11i-a ti ecttt 1-l 1!l [c]' ilcci_,f;.\.
ti r l.:)Li( (i, r,!rtr':i"t
: , ii;, , ,,,ir cr 1..';1ri' - adnr itc

Il;r ri-l itrrg.. tn ltl{i?. h. iiru- il {1i. r::stl in atltl'51 lrurnar.i.
i,'i i di(r c'r, nu ne lrrltel:ul
l(rrri : ir. 1{iiliililirilll iul \\rt: i I s
5t] 1', r :,',1 ^ [;.r I t'r,r cl irner:s i r-tllri
,,,;r r l-i1iit it,lr:.iti1. tit; Il11fi11;']' :- 1 f-. ;1.r;ilt"ercil nleii'll
&3
llt-tri i;Ltlr'ir

t.!t,' u tl i:rcitatitl intrr: cxplo-
r:riilit'lt 1'lr('a sir 1t-- prezi rltc
',iltrril in calc incc1i sil-tl

[]

tr)ontL',

e,j,,tirrircltjrliit,.lrLlilllo, lt''.eaDlca. .r. altcPtn'qi ()

i. 'rr. tt tiistilulall( tt'

nil. arc rlic i tl II 11,.i

feai[?

t 1:?

112

siunilc 1-'ii.urinir.rlui no:rtt'tt, si a lol' fati tlt: ct'tr: itt itslt i'c1
cJ. bubunizii, care Pot ini;ca ticclt pc ilcL'st ccrc"
nu -co
si. aibii inrplt:sia ciii sc nri;cli. yrc r-r linit tlrctptit. llLitt ut-rttartr
aceyti buinruzi nri itu clccit hinginrc si nn ctlllo:tL- altir tli-
rnc'nsiunc rlt'cit lungirtrelt. Lllii;rr si lrtntoslei'lL lrtr itstt. r:tt-
prinsir. in uriica rlit'ttctt.:iitttc, ltt:t'c'sibilii lor': lrrrtg-itlt'it^ 5;1
achnitr-'rn ci accsl cclc sc rotcytc in julttl r:ctrtrttlui sitit -qi,
?n acclali titnp, rtrc o trtiscltrl tlc tratlslatic in spatiu. tlcs-
carc lrlrburirzii ;Lrt allltt, iit lrentii, rlt'si nn ,ui-rt
puc inragina. )l i;scan'lr lor t'it tlc-cttrsttl utisclri tLlt piititilt-

pot si ltt--t'stt'
tului lor lc-au clut liosibilitlitca sil-;i ittstt,*cast'li trtttittnr:il
clc tirirp. ;i sli st;Lbilc;tsci"I o nttistrrit a timlrttltti, tlltlli't ,.:ttttr
pi-ltu. stlrl.rilit;i ri geotnrtlic ttni-rlit.tictlsiottlr.lli txlt(rt
r"il iL
no;rstr-ii. ( a yinoi, t:i ilit {)l)srt'r-\'ilt cli tirnpul estt' ilcvL'rsibil"
-- I"irrc, ilirr ar:ersti buLrttrtizi llll all l.rosibilitlrtt'lt i,il. r'o-
munice rlir-ect, rlccit cri inclj tloi, ctr ct'1 riin tlr:t'ap1.lt si cit t'

cel clin sting-a. .tc('it:ttit :ttt lt';'ttt t'ttili tillrlt l;i,r.2r]
r:ti toc:ttrai tlitt t'llttza
-, Poatc ttoi, ct-'clt ct'rrt'Iacc sii arlrrtitcrn t-l]L ettt tlcst:opli'it. ll yi tir. irc(,r il .lrr itttplcsilL ci, r'l"rnrirrin<l pc loc, t'i irrrlrirlr"i-
ciccil.
riigzrz
.s-ui tucn'tl,sissrtt'r't'rltrltIr-rrl1t1'grt''1lr1i1rltillri'', tclccolntin irlatic I I)tr1-'iL slltc
uu \\'clls-]lubttrttz, (rollvills ,.i s$rrtisiracirc.luacn(lrlt'irci;'aro1lriirbr-riiipiri.c:tttrrpc-r'icp'i,llotitri:rinrrtioirinclrau1itl'liii.rsnlsipiliuIo3,1hlt.trrtiiLirrlto;rri,trbunuirictiriuznnirtlrlrtI'it,rtuirtr.i

{"it.

timirrrl lrseanrttir o uotti tlitttcttsittltc a spalirrlLti sliti, r'lr.
fi c:onipus nccl;L1i r()i)liIll lantastic':: ,,\lasirllr iillrllultri". ( ir
ea, orolrl sIu clispart, ilitr spatiit, t:lt si ctllll srIirlltpr.ocirs' i1'r1iin11- rkrnii , biziii,, i,*i intirtrlt' ar i1;it;lti'clr: si a clisPiil'irt. .{.ttrnci
I ilin llnrtl S()2 7()l li. ;i I.i,1 s(' \'ai(l pentrti iirtiia oarii in r.iatit lor la firtir si
nlatcr-i(), ccrcrtcaziL Ititttelr i.e rrr inurrcnzir rLnrintlrri r1t' ltccustii i'ritirrrlrlltr't' r'.rilltoltliuarii
oDttooeastrt-caiciccipta[iictlrrvlci1t;nzttcliuict.t,i'tl<clculrlt]rlrriisutrsintntir-igiiz;riiltriloirilunttttri'--1c1n1.a)i-lc.lllttttrurrtrtslrsr:titiit'i)--illIt1ri)vtlrcrliit'izcct'itliztcrtagr.iigtIl-iitcrl'c,rtpctiiloiililvltir)tt.,L'sli'tli(rir''l:lt'ts'rc;t.ii
c;rrc contrazit'c 1t'gt'lL cotrso'r'ilii iirattr.iciI Ii, lr rlislrirrrrt
rlin lunrcir lor', Itirtrt,ri tttt-l tttiri vt'clt', loctti (](rliiiitt (l('('l
I'stc gol 1i nrr-i o iirrzir liirrclcir, plitt ltcr';tstli intirt-rplrrrr', Ii"
snusllanuuiddubcpirurioinirltm)u,rcziccruil]laiznxrisl'ta,s.cxuilstu,i;tricootnltcat.rrlltmIctnc.lcttiiasitnsc\loaciarurlls-to)rnpriiittctar-ft)cltiiletial-lltll.l')ttunlItIIibr)al;t)iii'lci1aspslxir(i-rlialincitnnrltr'tl-'lnttlriril.til'itrlxriixgtrltolt-t]l1ri:t'.i-crLl('rLSltliltgirnt'ir.xrts'lior2rs'rln{lli}is.-ti li lt:r.r1it sil sr'itrtilui.as:cii ctr .13,1 llrpt cilrc nlr s-lL intinrlrlat
lriciodlltl"r dt' cirrrl csistli lumt'lr lor I Iri lcrrlitatt', li, rlupir
c('s-ir clcsplins; c[itr spatiul lui, u" t'nt'r-s dc-a lurrg-trl usci
1.'t. in srrs pinri irt tli't.1ilul rtnultti 802 7(.!1..\crilr, rr oprit
alipiorrrt'lc si opr-it irttrc lrlt i rloi
;i altli posibilitatc <1c ttrilclLt'e. Sinrlrrl lor-rL'ttt-it'tttrttt'tttl rirnuolimoLrrlrrzl "i ;i-a r-.qiittloiuls Ii,. si I).,1 . Llr s-rt
itrc lrlti clisltozitie tlt'cit ttcccll tlt'a pcrct'ptr o s;itlgilrii tlittlt'tl- dc nrarL'ri frrst rnira-
rl irr Icl
ri.lL acc'stol cioi r.ct'ini ciirtl ci rrtr s*lttr
sirrrre: lilltgirttca. Itr,sit, firr-i-t sl't irrtt'ltlllg-ir ctttlt, t'i lr:'inttit'str i.atu si. rlitr sc'nin, a apirrtrt intlc cri unlrtrtbi r|trlrrttiut'ut zr"lcicriic,lt.lirnr
carnc ;i oasc I [)upr-r ptlipcf iile <'lc rigorrrc, attitrci cin<'l irr
ci Pirniutitl, slitt lut'tlt'a lot r'x, -{c ([('Jlllls('irzit lrr rlil-i:t'tirL ,[i tost sir fir'1r'irrs, t'i a zbtrglrit-o inirpoi lrptilintl tot ir;rL

1i4 l ll



irr|:i pc drrasullriL clt.llt()lrrri 1or ;i lc irtillroiczi clollilttl, \.'(]stl ,iirllrlrr1rilr1.rj.'Ur1r.lrll'{rrrrs.rlrir.rUrr,irrr,irr,r.lirr1rrlr,ri.lr.,r.r.nt.ltt;('r;.-irl.ir..tr\ri,;rir;r,,rrr;1'u1r,.l,.j,l,;,,,lr,r'r,..rl,,,,il,.,r.qitlir.,r,i.1.;firi,i1r[r,,;t1,r.,tl,,,r,,.,:'r,.i.,ti,,rc,,[,:,1tl..r:1ro'rj;i,-.rlli,Jrr1rrrl,..jirr,l,,rr,ir,,.rr,tr.,orl,,'tr,i;,rr,rt.ri,irrr,rti
si ncviitirrtr:rt l ,\ i virilat, llc.ntrlr t'i, crL: nrai sigure li:r'i r-qrl;i,r:ti,i,,il'til,t:tc'',:r(i'r.ri:ilicr,.l'1l-rr,'r.r1lli,r'li;cri;',trrl,,rrlt;i;,(l',zlr:.ic'ti1rir(r;r,r';r't'rri.l"irir.c-l-,lrrtt:;il;irlrrriil.trrrsr\trrl.(l[.rt,rc.,r1i;rpir.ctrr.i:r.'t.lr.is1r,.Lri;l,tirlirr5szl)rsrltil.rli:t,,riqi:,l;:il..ill:ilr\,rl;r'_rjr-1rl,rs..itrri,;iinirr,rli,r,i,,r.l.t.rJ)I,rilil;r.rrrrlrirnrr>o.'rr,iilriga,r;lr;irr.lrrrrL.ti,iitt.rti,rt1.,ll,,I-i.itg.,t,i,l,r',,t,,ri,..l.1gii1rr..irr,,r.c,rr;r.,ii,r,,,r,,ftr;l,l,rt,t',r1,it,,lrli.lu,i.tia,i;;rn'::,rr,r;l,rr1ri.',l;;uiSi;T"i.rr;rii;'lr,rr;i,ri..;,r,,..,;,ratrri,,(,lr.if;..r,ti.ii",rlr\,,.,;,,r,ii,,,i,tri,i,i',1,lr.i,i;r,:iirrrr'.,l,rr;'raiirrr1,,j,,rl,;r;r_r'1il,rl,ri..lrr;;ir1t1,r,i,,,r,r:r,,:t,,irlr,1;.s,;,'r,,,r,;r;t,lcr,r{,ii,;,c.i-'r,,ja,i]r,*ls;rr_iir,,,ii,iri1,r,tsu,.,r.,rl.,.i.l,,,.'trrt_,r_:l1.l._,tjr;_:,:'
rlin tizicir: 1r'gt'lt. cr.rns,-'rvin ii nratclici, t:ici rri Iiicut si. cl is* lrlii"r,rri,'"ojrispi-rnc{-l;rlrlL':r,l,(rNr,.si"r-rtr7t,tru'':lustr,)(/,',i;rit,rlr'.llrr,tr:-r,rrtt.lrrr',l)'rru,r'a.:t'o..';'r|r.'l',1'scviit{r1n:au,.llll)rrlru(lrlaSt)i'r;,irri).isrs,tlir.,rrrrt.r'iit,n,.s'f.1c.,ji'urr1rcislt1lrl1t.rii,riiir.ri'rlsi,r,;,;irr'1iil\,.i;i\rt,ir.;r1_lt,r',l,"'ti1,t1s.',>1',;,lr,.;r;..lr,,r-;1lir:rr:i,;irg,'rstrrl;,.rl;i,rt,i,.;il,,s;,r.11rr,'r,r.l.,;r,s;r,rr,,,cri;rt,:,..:,,firr,i.l,.tli,s;a:;iir,l,l;rr,lii,,rt,pr1r,lr_gili.;.l,.,r,,'ri-blia;'ir';'li,,.ci,;;,.,,ro;.\-r,,.,rl,,,;i=,,,;,,ii:i'i
Iar.ii si :rir lLlirii'i tttlrtr.'ria. lCgc,il ltcl)cnctr:llbilitittii rnlLtt'-
lir,i solirlrr, clici, rrr'intr-iirrl pc rt;l'r , rczultir cir- lr i pirtrrrtrs rafaidrfsrssillo.taclnllc.-c.tJ'iriarmr.i'ldatacpt..calfrurle-saccltieosfcuitscni:taanrSllidrssdrtjit'rlcii-crcnilrcocrnuailircrnslctltsirr.jrtslc'i)rtiutlttr,i.J^r.cca'ocs(itil-ra:t'acitiuzL.'rtiii,tr(irsi;caii'l'tili.tincctrpu-circsiptlcr'oiriice'arlrrrisn.ltcs"irp(ilrci.en,rrriritsicrzrrr,t.(r,sit,;',asir-u,st'rrs,r:cifr,rltr.ii.sasrccrIttiitinc'irliiu'frrutircifrt,rrri{;tlri'1,i'ic)lriirirt;uirlrlrr_pnrrir.,iriiitiicr.o.rc.,rrrnrsil,-"tttrrrc.rtttrrir,1al,',,,riL,a,',i,:ra,,1r,i..i,r,.nr-r(1.r.1ta1i--i..\tiiiltuir,lii,-;.u,'i.r.iI.,.._r.),.i,'r'trru"r(t,iiiir;i)riitirt,ri.,-ir'-l.irA,ti,c"ttlrt1c,1lSli,rc,ii"e.otiiri0"_,,sc,rri.urit2Jitl,eril,rt,'cisrr1irric,ZL.rllciilt,lutrrtiOl'"i"rurirr'urflicd,icrficr'.rir,lri;i,rCi.gcnltLr..;iti.--n.r,;rl;.rc,r,t,nin,l,r,rr,',.,s;,,leprrr;c,*rifr;<i,rc'rori,,1;c;sa;ir;.,s",lii';c'rrrl;u.,tuii'-i'r,.:u,r:bt;rrs,r,',,.i-tiit'i,;i.r,:,*iaai,l1,,ir:ir.L
pq'1-f ii c'a si t:utlr 1tc't't'iii nLr al cxistlt.
ixin-- lrr ltrit'r'ir', tlltcir i:i nu rtu uotitti'tcit clc:rtls ;i jos, trtr-;i .t 1!)

nirtoitti inral;init cii- crt ant irrtlttt la ti in casir, itr ltloclul ci. l
siirrPiu si trltl.ut'rll cu putintii, llciicr'i l)c (1cll.'tr.ll)ra, 1lIilt
a trtia iiiiriln::irttrt Si ilrr', irrtclcg'cir (1, ttr-t r-tL Ji in sitt i'tr
sir li' r.'xirlit:t {tc s-a intirlrplrtt cu c1, ot'icit clt' pt't-'r:oct' lt,i- l.i.
.- .\tl l)tttr.rlt sai L' ntai liLct'nr o lttttlttt-ic lt.cttstttr ltxliirs-lirirltrii
])r'ca1-('i-lil'n nt'crijit tlcstLrl. \'czi,;i (i' t'utt co|il
r:it si t int' s i pit'rtir lrlrrru ttrlirttttsa stirigii . It itc lLcttlo itr
coll sint rlott,i" u'rittitr;i <lt: pt: accc'lrsi nrinl-r. I Iiri sit ]uiiin
rrira. rlirrtrt'c'lt', sii o irtviltirn in sltatiril 1)ostt'i1 ;i sii ti 1xr-
ncrtr iirtiroi. S-li liictrt. ])t-ivt'ste stLtlir-rllt-t'll ttttttt'ol- rrirJr-ri:
bd:i-rrrrcpa:r;lihr.ItcDnrr:rriiaf:ttirrrtt:iilricuis,lrLrsrr('n(ltttstrccliitcttllLct;ci 'ltvtairtritc'ilirnrt.vi'lz-liLti'il'rirriili
Jrl-rilnr.itsittcno! usiitr-irittitirunzpiritllrt:-c'l.'ilPtiicl: crtalalth d irraiirtr' o
I"cttontctttri aclsta clill()::i, :r .

t' si ;lirli

,:u.iti.. cici, pt lingii rlisprLrililt. si j.irSp-lltrrirtii'l;rt-l.izLlrtriltstcrtrpitc-ai,;'rrr-;t
lcili)irf i,, pr:in intagittt'iL t'i sirttertricit
n('r-('iL a tlrtitti figrlri siittrrtt'ir.lr, L'rlli L'f ('()lltrilzic(' gtolllctl'ilr I
i)r'o1i'sr-,i'ui: Nrr clr-rrgiii rttt'rt, trtt-i nici o ct.rtttt'tl:ziccrt'. JI-arir
cirrclit rrtrLll llt ct'lt' intirttplal.c ;i trtt nri l'-' irot cxl)liilll .
ilt'si , r'r'critt()sc ci"l c.rlrlicllt iil l11(:a l)oilt(' iII)ilr('it absul'(ili '
l{iirna: Oricit rler absulrlit ai: pirerL, r'i rttgiint sii nc-o irl 1ii-
tisati;i notrir. \'a li nrai rnult tlt:cit nclitti;tca.ite cirrc !i-o
picull tt'lor-ou, in strllt:t, ititttttr't'icul nt';tiirl!ei. Sotrtl irlt'rt
si cu nrittr'ar.n cl"tutlrt sit trc atriitttirn rtruite cl intrc.llr<t1rlt'-
irrclc rliscutatc 1a s,r'nrinitriilc dc nratct'naticii, tic liltizoli,',
rlar- dc'gt,rltbit ltu liltt lrtttrrt ajutrge lii nici o concitlzic. Sprr-
ncti-ttt' ctrnr vir t'xplicai i ft'nrltltttlul. l)roft'sorul: Acl Irri-

tlicncclrtncio-rasstpc:tr.tttirrtltoair-tsr itri-tc'tit:tililnrtr)ra'tl,siilrlttnl--i.()'\ttl:itr'st'1t:tt'ilttrrltottt'rclil.titptotrstc':ltllt'liir
nrtrrir. l)irr splttirrl crt ll ciiilrt'ttsitttti otitirlr'ilr l)tlt('rl \:('Ili,
lrrirr r:r,lr (lc rt tr('ia rlit'i:ctit';tici lttlttctt sitti ltliI cu sittc tttl

I rB

I

.rriilii.ail" ar'l ,\,I I i

r.rru.trt, {5r,i ,'.1 t?

rl ctresenil.t T

lrlu-l-lg&ir[aii,. 11 it rr.r l-l ;t

liiicut-o? proir:ct iir

- I)u1lii t'.c i.l. ,.rfu;,t.v',,'iLt Ii{ ,i.l]rj I l iil.tiig}lir echilil.tera.l t'stt' ttt:in - i'rrtii cii
iigula{ia care stabild-'iitq.) ( i'x 1x,,1 p.,tlucte pot fi a;czatcr tutr-L:a- 8'
lnlnirs.pI;;llJiduitsr tiallntclrllr{,r,q.srt.-Jr5,.;xrilin.l.itirl.li;lltriubrlht.leiilltticrlrcX. fii]g-.rl?p?o)s,ibailai.rSp.cta"tntcriltl
. in1i-lul cu liig. l5
Jra.tru pr-ln{rtt', ir,r ,.:il:,,iI ii-(,' 1J jin :ir;.rtilr cit tt:ei clintCnsirtni"
p;rti.rul ,i.slt lr.:l 12t
pllncitt s,,r i.,,,'ti ele sd lic, dortil.
r:i'te rJouij., lit ,r*a l5 ,,liisr,l rnt;j r,lrlrrl ,.lc altlrl. Consiclcl'in,-l
IL punciul .1,] cn nesitr;.ii ii:'r 1ii'tra.uul i)unctelr-ir A, ts, C, t:i utt-
,deva intr-o c.l irct:ii it. jr,r:CiiLxro,-qcutfr., act:sta formeazii 1'ir:ir-r]
colllun al irltor trci ir rungh iuli t"rcirilrterale care all r(rsl]rl(:-
liu; r-lrept baze cito ,l:na ,:liir"llirc iaturile triunghiuluicchi-
latcl'al. AllC. t'i'ntl.rl it jrirl,:, r:t,:tjr-r.t \,.ez,i in spa!iu, r: siniplu
l[ iir i;tc,'.,rrstJ 'i)r(l]1cr.:tii,-] itil
1i veei tttl:aedru (fig'. 231 , dal'
iiui tii ita ci ei uu ;i-X t)r:ti,l.r.u ii;leq-ilra cunl erati, el.
ci lrr.uniri
ci
;rcca-cttl estr: :i rlr..il;5t1i,ul,rirr,.irr,,r-lll,fir41,rri-ill{lt.{riial. }t;i pe un plarr i
in-ra.ginat, tililtr.i
*- fll acler,iit, r.l_e

sr:.iltlii. di: coni'igurltil :i;'r lriirrLl,:.t,,,1nr. l)ar Lln c1-1lt? Air-i ri-r,ri
,r:.l] ;Ll rirebtli sli ln irrr,l-Lllili tr-t;iir,":".u',r;t l-Jt-. f.lltnsla,tie !

*- tr;ti filioal'tt.r .rir1.,).i',llorl (lr.:r ,!r]{.rviilr'. ;.t fcilosit tttilr.'itr,'rl.
rll.. tra.nsla.titr;i ul..ilo,.li.l L.r sr'i trhtin r:n p;itrat, it si)us
q;i, trr:buie si rrii;t'. ,rlr -si::,rlr-r I'ti-r .r\.[-], .intr-o rlilccf ie pur'1)r.n.,

dicllJn-rd. iic ,(':.1 Irtr.,'i r'ti

f', t. n') t;t ll

lrotr:ive;tc, f iindc:i clacl"r rlcscncz nrr cduinbttpnts' hiuinrtiicinaslrl.l,ai)tfirorl- H
icctirL ururi corp clin spatiuJ crr trt,i
cincl aici vlci :rir sar.i rlc //
cu dorti. rlimctrsiuni, pjac cr'1 cir dr,'rui l('rrin i)t-(.ir Ia spatiul ,,
cu illilt.(, 1)1.r-
piitr'lr rLitrtr.n:: iuni

1 r'rr t ir' I
ni_euc*t-iil:lLmilhqiplrroc.'cslcitbucibliotu:pLluutcii?ainlrsnsip-ptailoticru:tol:rnclsiu'zl,rt'srriiii:t',.itscccnl liLrn1;ilctiinrrrsc.ilririrrrrlrli1ii;rirl:sfri1,i.r,i.1dlt,t1ci-ui,r,,-
ti-1 transpLrn luintr'-o nouii prr-,ircf ic in liiatr.
vonl tl'cc'c rttiinc pc 1;t S(.rliit;rlul -\lir I)ai.- tlacii
in cantcla clc lli sir r'czi t, nurtic, lrnr sl'i tcr clLrc
1rr:dir:ctirL in slrtrtiLi lr lri-
pcr-cubuiu i si rlcohcilacrrlcli1lir ipcrtc'trut cli ulrri, anrbclt t'st,cu iatr:
in sticlii l)c altfci sl)er cir pi'nir Ja rrlltrir. ui sir rrri-1 constrrL-
ic;ti chiin' tu pc .lilr1.it'l
l\'IlLi stii ntinunc? Inccpc iLturrci, sir vct'lr'nr I
*-- Nu Iuc clcclt sii ilriit llttiorraritcnlnl gingairit'i ('r.

fltlaloalnirsciprlliccrtrrrc"ccnnlcsnintctn'circib, nc,isuuiciohl rrcilrloiPsiltlrarcrrs.ltr).{oziii'jnncc'tg:rlt-rr1-:rhr,.lricnlrir.,rc,\rc,r'{tauitrri:iJiPrrc,iai'rr1c.i1un1li1Ll1Lrl liLcp1raif.-- Itii:. A

['orpu1 cat'('s-a Iot.nttLt pi'iu;iccr-stllt rniscrrr-c itr..:iratirrl crr
pcnbaLrctrztlsrui:pL\l[-rci-cllslirulapUrrl)l-astrrjtrurr-rnrrini-usscn,pl iapccrtLs.iitinlcccl,lctiliclcritsll'rartl-acncscircalhcki.u:,i'J,snti;itncuiircapiicr,oLiiailzncrri,ltisciiliaiu'tvt.,\its'rrF.la"utIlisi'i-;,lii.1i,,ctr.rtiloi,'(-it'ri:utr-.i---,
tlatii, {iccarc clin cclc (i fci'c irlc lui lr rlcsci'is; rru norr crrb. ar:cst ctib t'itliur,rsi.'irr cor'( sl)'rrric prrrtr:rturrr i tiin slratiul

llr:-lllll

,\lirclar, hilrercubul tl. S t rrhrili, i,,. ,-;Lrc le r.czi lrici irr tectrlr.'rr,-bn'Jlr.irrrt'l{i'r)iutg;r'risr.i.nniill-lui('sti:.i.lpcr.laIlciarntur8nr.airc.gtcliesrirr-r:,rrrlbcrcruzri,rcnliIisrrilrPi:ir,1bI:.irrl,tr:c,il,'cr',rrr,,.rr,.ii,lgr,t,g:ii-c,i,,rgia"i1tri,ei,t,,"iri,*riiit.tl,tr,,c,i,rr:pi.;rc.,l!-iu,iiirf,,,^,-',;r,1;r;,,.ur,;,ii,,;..,"'c;,;,r.if,'s.;;r;;t,;,.i
pcrspcctir':l (fig'. 26) S,cc'1;s_)rti34ul.rt*.ll--bu(,r11trf;*lo',rttJiir'i.liIL.,rr)\inrricltirsri.crrlrirrri-u'ciltl1ntNtt.ilrsr:i1r:,r.n1irsrl1.tairtludniii2l-r1l-ricrcri'srrtat[r':tli[lurfrr'i,tl,rcr..r'I,l5g'lltni1ir.iacr1-'llas1cc-ryi;p{-intn-ar1ll,.atcaa,(mrc,,e)'at,lttc'lcp.i:oit\t.g'/rp.ir.bi'tn:lr:tic,u.iLr6'iptlucrftlolI]c.,ibliitr;riumtismr'ltrbo.tiril,iurirr.e.;r,r;ci'sfcilio.cslnrcircr'"pl-tlnairlf'to.aurflrr:cr:tramrant'tc;rrrJl.;ic,ir,iistzn"c,i,i.;tr;i'e;ti'ri,itr;'oltii;"Zpinh,r,,;r,r';r'nc;tani.o;r;t?'i;iune;t;
*-. B cubrrri zici? Iiu lrs int lina sil r,iir'l nrrr.rxri 6 l,.\r:cl-
eti tlcscrisc' dc cclc 6 Ietc' alr' lrri I
nn-l1rPpariltcciar rnai iiiaint,l er,r,i Ioar-t.r: nrillrt t:ii gilq'trniilr:
t'l. \.cclca cuJxll Sii. nt intourt:i'rir la t'l (f ic-. ll5i,
ciici aur inrl.rrr,sia cii uici tu tru l-lri viizrrt lrri,rr birrc I .\i
I Yl', por:ttit cic la piitratul AllCD, c'l ir t-rilnlr. ( rlrral,;::r:2i.1\,lilr'r(l1'l,)l.'iir
si cltrplasindu-sr. ln spatilr a folrnat u doulL
iar ile inl irnprcjril sc aliittu'l'r l tt'Lr nlscritc tk, c:. ic 4 la-
ttu'i a ic pir.tratLilu i .
-- Ai ch'cpta'rc" Alacllrr', culrrrl rL'llr crrrc anr iroirril ra
fc.,r'tna rsi cl bazu hipclcubului. in sliatirrl cu 4 tl irncusirrrri,

'ti 122

rl

rL.stltrX ,l tetrardre Iat,::rall, (j,..r.rr,{.r ;t-it1 (tir, bftzil. tttraechul ,jt: }a

0iwe_flnl J-'rornit. ii:r ll.rit,crt,:1r;tLqJiu, la cele * iriungl,i,,,I
,ta:lr ifutt:rale alc: tttr;rc:rli;ul,ul u:,ri'r:spunrl S tctra_eclre,
i;.rr e ,l
ii1r',t-:ntle.,l,'rttcLri(rnjlonlb(g-)th.itinruiurlritrrelrr;rlrr,iilitlrrirrEr:itx]:upilur i,r*ta:t.Ilr.lell-.{'lx;1,t]rerlitq,'::t6ramednrcuhruiii, uurrrli

rrdiii

rnr*-;-i-""sil-l:a{-uxuacmhctia.i?i$f[ioXmlimlsirccr[trpi""rr$.ir),..,rttiLl ,;rlilnt'',n,,';tiil:lirrl..rllli"rllrdrucicl.etleetr-atrecvlirrn.ftnr rril':rr,iJr,,.
'(51,i,lnel,rtr-,jtr.r-rrqiti.*t:illt-rt.artclp::clt[lrirceor,let.:laudncIleIirial5trslir.pjv.iiPl-"nl,rei\r.t'i".fl-s;actrll'sitirhrtl:t'it;riircrt,_"ilr*,-,1rr1iri..,r-:rli;.1rt,slrrr;,'i1,r'.r1-1r.i'.,,ti.r*..,ri.rr;,l,1i,slil"ij,qi.rri,r.rr-rrtrrtl:lij.slt1,i,t,"..,,,r.:,.lnr,i,.J-titlrr.e,:}u;,,tl:vli*.rtitil:l,iroidn.lrfiairitcnihridiit.pillifrcteetrrii?"',,iir*iopckr.l:t.trtill.n,:i,r
lJsjrrr,l;l,slrn-irftiirtr.-.ifUiti-st.itr;rllii;ltltiii(lrrltrti:tr,,i"r!:rlt'il{ili'tlit,iit)i..i:'l-ririr'."i':ti)ii.it'it';tii!".'iiii:lliirriLiirlii.rlrrriii:iirr:r1rl:ri1i.r:l:lri1li{Ll1,i;iilrliir'i':r;iit;-l1I_l)r1;ttL'iiirirl:r:ic'lliiii,lil'ji.rlitrlllttl"i1us(^iljll-i,iittir-tlltil'llr'tl,tll:r;-(cl;li'}li1flrtilI"lt - Acurna, rniri lasi',.-{r" bi'il.ir*,, 1l: ll r;ll"nt,r:t (l{lr.rorlliui {-a iii
ir'rl'lsidera un spatiu crl ,5 diil'lrtrnsiuni 1
5'l,pur'1r t'r1il"iltirll ('Lt i)iLl.il1 il 'i iri,.tl:"iLrtl:, l, ilt a:ll dc'i t:ll ilttiliii tir-ir ! ",,rxlli*lrrp;_rti Zii:ur:i prin s1l*tiilt, r.rr
ciii'tii.;ttl'i sii Iit'la (!,l, li'-l i-l istlruili irl fr,, r- lr'. *- Cnre rra s5. zicS

- l)a, tnt lli'ilr iillil.ilr!,ii'. ,'\r:t r:-l 1".1itli':i (li' la-lr'iilturilill riitai rnulte c.liurcnsir.llli (lc I_lal-{Li ltr-.ri tlirr in scrinciobi
:r1i,iI rlrr j)ilt]'1,1 llii11{:lrsiultrl, u-,rr.-.r.ruj, lrici ltu biinuiirrrr
ll7;. rrllrii r ii tlii{lt rlu-l plli - In aclcvir, itcunt rl.u;l-,1.-l.ri.ci,

-rilrlt s;'i iisi'z:r.l 5-ir'il i 1i)nirn-{lrll''ii n.:;1 ,:.li lt'li"lteiei,SietligilSpet.1r1\\c,rner)1itt.rsJ].]s.'i,jjil,.!iJ:lFirrii1r-iir\..rsult.c-i oli{tanitl0rtll.r_l.tpStt;Art.r":,iadrcsilCi, tp.l,n,rlnl)tin.rlt
iri ,iillir, tciliii'tli'rritl :;i-t
ll;:

ytq[,i1 cttgt.,]tiinrt'i,p,:lil(1111];i1-'11;1: ii:tllt,illli { ,1111 }ltl1{'il \'':'{iu:", i,r:i1rr1j,-ll..it*i'1rijl..ci:i\ipprsiit;ltaut'.orhlanlripu;lcurtir*rvlr'ratail:rinitiral0itnt-ti:;ir.sill.li1.{r:1ii,ut\rrlr;1',rr..ii,i..rl',lll,jrlrrr...Llj.i,ll:jr,iitlrll1t;iti.;,..;ll1.rlr.l1,irtll-ri,tiirir'11r.;r1t..1qir.r]Ii;,,l.1,n\)rielotreni:.'.,iisin:,rtiuirr:r,,;nI,i{.,s),i1i.rr,.lllrrt.rto.trlil]rillrur..,il,.
Vir-Iul tttl'acdl.ulrLi. Ac:Lttn r,ilr, ^:,' .iri::i i"ril'{ r.:ir Vir{liIi.[rt .[ir:-
'ciirrria rjirtlri: cile ''.{ tl'i,ittl;irirli I .rll tnri|itidi-ului si i:i.'!ill

-- \ii'i sli. SlrU i t-rr,i ,,.1tir, l, i r rr r.rrii ::;inl ptiyi s;i lla irrs, i
i,r'11.,.-rilt spilti'.t cu l)ii.ii il r.lii lt:,,.lrsriltlt j i
,r.,ult-il.i-irin.Vlir.sl:tzpi ia,oictinlritrrcluinrltoiir.srrtrr;i'{cr,i,srr1".r,r;i.r:,,,.1lr.;rtl,,i'rr'ririrrin,jiint.rlI.in,riiiilnlli-rii.rrr:;,rit.irtl.sL,j:r{1r.Lcrnrrsilrjr..i)lpe'nr,:tafLtfrot,crt-tlcirlli;rt;rt.nr1,s1r,i-r.li,.rirlrrl,it'11i ..
[: tr] ,ri'rr,liririt'rfll-:1tirt:ilr't]ni.,rrt:tVrpp',,:s-tla:'Jiu.tai.iiirnrnliaacni.ttst{ul!ilrv"upc-.tittri',iraa.'tS-r..o:trUtli:ili(rtl)r(llnrllr'llJsni,1(ittl-,.,r,{tr.ru,;..:.'lturrii,jl;l.,illt1,{ij'-1.,:trr'itl'.,ri,1'rlirq'rr1t,li,.jit.;"n'.r;l,r.,ir.,i;;ijiri,rt1tr.itlr].ti.l;tliri.lLlJ.ttl}-.ci.:,l;.lrili;ntitn,'triinOt;i_ostr.jciur-,ts;,tf;tifri.rcilinl.ill.i1ln'.c\iJsi:)'ltitlcatritltrltI.:s.i,li,-Ir,,

t1;l1rtrl--, r-:ar"e pOrtriSC rtll ti,l.tt.rilr,i ,)r'l il.-tt5l ,-..r15cqq-]iu rje,.tiiihll.i^

!:15

'i;i.al,,1.|',r,i,.{;.",;,l,."i.,'''l;,.i1.,,,,"1,,,-,-n.,u;r,,n,,i,.t1r;;"t,',r'..L,*t,,i,iriP.i,i,,rmiuj,;,iL:.,.iili',cnitic,,;o,',:,iili,lr,in.,i.,t..l.l,.rt,',ir,t,ont1rJP'l,",irlit,-t;at'ii,,'r,ii.,:,rt;r.,l,srir,intIri,ili,itcllut'"l,.,a'leaLsct,.rrne,'I;Ltlllaulroaur1l1sristigit;t,lcri,la-tliptl;ii,riittrl'ntir0Ltitrtlit'ri'l,tnbi.ta,irt.corJri"iou,rrJlti'u.,tia"tritnr;nrtisn.1r:tcltpcttltt.lcii:rp:licct-ilirtl:jne"Ittcasi-tstli(t(t.tt1lntnic(t-lrar*I''cot'fsia.rn'rirl:trau1iltllls'atlttl,tt''Iri;iri,acatrr]]nrld'ltelasetar'trrltllitlsl,flrlcznt_liritticrirpi'ittlsrsitrrtti1l-rcirli'1irlr-cors'"lltlsrloirlurlrririprlrilritstcitrrrrcttlcrcc'cata'rrnai;rt'liliclfttn.tiaia,alitsttior-d''l'ccn[ia6d:ecr-'ii-itclrluasli.rtte1rrinltisclitrrcol''zt:t;lt1'riiiclusliltora'-clilctictetillrfctmcattioni:_tr1itiiit't-il'aner.;latrr.litacils^,crsIc'liCrca'ti'nlclni:rpCicu'a1c:tmtr1altirtcj1n1rtitt'ilt1i;tL:cltcati-t"'t"r-'tiii'' ofcd?cil,)rliii:drntrct,llr,pos,cte:cr.sccatiiiiicriur: rrtcccal,ui.'rrfs.ioc-r_cjpcl ltri'l)stai[jr.ulr,i'\.irli:,inr,.r,iit,'t:ll,,s:,pici(,o'i1itiar'1iettt'.r:z.i,rdt',csc1r.iiac,,el',rr' a,.irn,igh'ri'lnll,Lr,t,rlri_ttrtiipiri,tct:ri i1ui.in,.n,ro;i,;r.
.J1nsa<il[rurL,,i,ta',,r,-r,t'rn;r.or,,i'r,rii,cstrrcln.il:aicnc'ci,iil1ricLc,lr,rtrL:h'i,'is:,Lli-iitc.p(grcnin(inliog)iiii'ngccuace1;llio;tlninlcL.nr,rs-iliircut.iluic,c,Linl,ili-di,ltinll-z,cuctc.ccrcirtcitsuiAanltclcbgtIli'(iiL]sucli,'niI(ltirfuiLCn'igntrOiatcn':crtno2:tli'lsnl8iai.rnac)icl'ojdccus'ntf\anprlllisr1gtarltim-{iicicircii",srtrucitetlutltipictc.urlrpuneiatslusll}tpacpsnerttcililclisicanc'ttruriirvCl-iinlictnilsrtsO-r'--rili
f'u i' .i

I

l
i,
i

I

liii

,

ii

l

iti
sc1ddfrccaatircrl-ti':ri.rc'rujiliurontiuci(t.tioiol)iil'rrsorc-miiusaii"alra.tsrtlairrir.cld.iuaibrct.rtc*siraiac.cisErofIialoatcco,pbotuflnoucodl.co's,Dssuiccnlca.lcir.l.lqiiual'rr-{rssar'ott.'ri.u.li,ie,'1tLllotanlonarrhlr,nr:,torcrit,tic.s^,ri,irrpi-iuratr,.l,iPo',.c.to,terui.ir,'t,lki,i*.r,,tc.i__l.
cnr:i'i',r:ticc, sir f ie in tioilir"s clin uou ln lunlca
ln i.'Asa s_a
intlir|lat chiar cu acle:r':iratul Rar"nstlLlrrc I rlc.si
ai. {i c.lor.it sd
i.ii.rrrinir,acolo., acceptiL siisc intoiLrcirc.:rsarcleli posibilitiLtca
r.x:rncuilor'-zr'i sI r.cri{icc lcu;ita clcscopei.ir.ii lor". (':r
tlansnritir lin sclrlll cir a rcinirat in unir.crsril s.irr i si PolLtir

lioart'l'osi(' j).carc a lost rugai sii o asczr. cxact in s-a'tllLt o
ourr' \'.r <;bsclva ci irr:ls ina iu i inccpr,: si lasc r.1l-nr c l)c
inclcPlinincl i:uginrintea, ci vecie'uimit ci" .ccai;ta i.cul i'

'aulrtucinintrr-loll.ccalirpciar,".cir";i crrr.n: irinr in t.
lL clisr;i-
,,o ntini ar.fi apirrut fulgerirtor

' .Lsta c clzrr, floa.rc'a a fost'idicuti in .fara.irir.crs.rui
s-(nr'.stiiir1r',rs,-(:.-lrk-ti<'Lrs\r--l1,ucor\\>lt.r,i'pr'b;..,Irrc"ar)st',a"rcsi,ucc'll''ci:')r.iir,li...iaiadrina'rccalcccuicerclciaaaorrr.scl,Lpenerttsir'ezinraP'aiiriltrtr'ociiprtigaitoccoi,caial,clsitsu*ccuCrntPpliracruts-larclrsc)di,ica.rilPt.isia.clicpri.sparrciticiur.laaiuinccc-plci,*ll,i,r-tiii'trrcr'p.rc.ri'iruclil.tn.irgr:.s)irrc..isiuaiicp''"lotccarlrs?ntgripruiin,r.ilc,ng,catcreuei-
.si-e v,;'ba, clcspre ucr_.la r li',,1I;r; irriL ti mfului,, sau clcsprc
lr.(jirsia. clin Lirmii in carc ccle patru clinir:nsiun j sc rcferl toitc
.llr. si;a t iu ?

Fig. 2S - Ai" crrnt pui tu lrrtrcbai..a tr-t:b,ic sii_ti spu' cei c: r,orba
ut'l.rL.liiir;irru'rrrpl-iit:';'ur'rii1lr,ts'ilincrlsil.lrcr:'cttcr:ciitaorio)rr'..nclr'acsersiulcperrl'r!erpii'c'rpr..s'.itt:rtlrrrt.aaclocictIitetuclicc,irca'air-rucredo.4er'icsrPscl'iciilpi.uilnin-cacrrteriticrsi,nulilseut-'.aotinuiamfirelrti-cnpuja,;crojlcnisirrtp:rprrsrrtacr-riceirztlimar'cuo:il'iltiIs,rtiicnat,grcttuicarl'ec.llsieirn'prsssa'ci-i.-.

):t

'il'rs,,inirt .!ungitlli l;l-t'l-l.rt;, t.i.irr1, [[rl'i t-rrLil'.':r'{'.itzir Cetr;} 1i1-,it cLiir, sI-_lu-aridlrJel{vii'itrseaert:vpcaadlreenlecuLdpeucteteivmmap,atvpoert.aineorepupunetirttaeardteealcnaucDecsaatnreeei mrsraenrcmitai[t-i!

ihLrt ii 1 i ciitt'nltqr. etlrlq1, a;a cum b-auvnemtruencumte! tlr$uumeteaslochr,iedcini ecgaaulezl
oclati trecut, e
{..rit. f iillr.k:fr tu p:'ivrstitil:rL1r,uIsl irungirnr;r aia (ru]fn.[c.
r[)i1lt,i\.rririira. jn-ri,:tptr0;rituirl:it,inr\zNi i.irncst,itritirnr.ngrcrr'r"',lt'irrll.Ilit-N; prr-Lanlnil,
prca fliirru .lln,z; cd tirnpui,
ci]rt: interviilele
mintriL:;tc: de timp folosite de un mobil ca, si parcurgii, cu o rniscare
llru'coiitit,ll cle la inarirtlt rlr citi,-;ttltiti (:ri.i:i(:ceaii cicrti:rit;l{:r-:
un iformri, acelea;i..segm-ente. de cireapti sa*u,' i n tr"-o rn i lcare
{'u (';ljr.r ttiirlrt ie niiin ir-rllril :ritilrgrlcr ,r,Jlt' 1r!n3;-['{)ng, ccrl:i/.ie1'}i, cle rotatie uniformd, aceleasi arcc de ccrc"
i:a 1i t;rirr,;, c/i accstr dl.;il;l,t tio1 rtrrl ri ,;, liL:lrgli prirt serrirltr I c{c)
r,r {t illt 1r'rr i,l . cc.elep-ce*l_tVsl[cazoirs:lci.viciee-zlcuelel.."aitdrjuteqnirlsoyrssdntuoscatl.ersinbtrtacctipenrre.nel,cpoIanucifni:ipntluetrmcn,na;iriil ln aclinc
- Iti ;tii:.1 ct pal?triil'j iii't' lriirliLiLll nit:rii rtrr ;Lr:east.I lrri""illr;[,
rt;ll st lu ur. si,guran{;i r:r;i lrirlrri-l ii;i rit l lil. rrn {ca.:i i)c clu'q:-l totril in_
ccnsr.rifri ilintilr'ai;-r c;i si.i r: r.r irriililllr lrrir c"luci arc cir,r'ilin.u-
profeso*
rului rnerr Alexandru Mvller",;rrn citit ca.rtca lui:\.S. trding-
I'lit i, luir,t ilrir', l-;ttti -r'ii;:ut t:ii si: (o[u;jr..Lt dt ul-l rur,:tl tt sri t_on: ,,Es1tace, temps et grar.itation'". ,{pirruse in tri_
francezi nurn_ai
ii IciLrttrrl-i1 cic t:n ct trt.1oi.tlr'l-j.11 I l-ltir,:"1 ;i ,rr. {r :;r:}r:inlbat r'hlc,ctil.l* tlucerc co'randat[ de de citiva ani si, bineinleles, fusese
1' j{6: si ur f i inh,i:uit cr,lrilL,i rirl u[1 rx](rtr'f r. :;-U.r ir,r{r's, tr: lrs.ip;lr;r cdtrc Ser'inarul &Iatemitir:. {iindcii
ai,l a; ir r..,lts,:rt'at scJriuibltl'r- ir , sri r-,-r; ll l e:i-r'nt o cxplLic;oi"lc lndat[

- ilr l{}r.:ul acestei tir'a.rlt,, {.' ix{'(:' irr{i Irr'it,]11er ;i iri'c,t-';tit: ca.ac-eusnt runladrcinatrsetropnuolrinni;ii filozof englez era. considt.r-at atunci
ii ironicli, as Ii pr"rfci;it oameni care au inteles cu
rr ^l;li ti, lr nrri li,r'itind, ca r-leir:rrnf ., de stiinti
tilr nrrabu..urdcluetncirlmruit;tirras:liaaii.tt;rotit,ie-nsrooeearrci"ta,iitsemnrfsefa-icla-c.aimtmtIi)ii'eicitral5uramts.ceteilliiaene. catPSit,tunccfodrhuilen.iinaiia.lrcocr-rd,c::asrlLiisini .ta'ir.pNiin-craria,mulrr;tnact.piiiesimii.i.ri.a-ncrr"luinnt;t-c;aci,lttl'l-.aitor;"mi."t
; . \'iilllr 3. sltrrl' rt I i. i:

"J,iL-oi.i st{,rALtir (L.,ri'q:-ir l',1:lr.l {t

calrr-ir['i[ ,.lt, Ir.lri!;i ,
C;r rnii i1c rLrr l i-;n.l f r',.'li.rurt
I-.Lrnii-l it sii nr'-rr {r.rit;rLri.' . Pinir acum nu s-a ierzcitaomcatizrzia-iusi,.-itrilvosiiber:scf;,rlcc..sip;rie"ip.;i,e-i"lrl;,,i
de;i
ln astd seard.,
sd urc ln c;rmera rnea si si i:obor"
S;i {:r ti opa/iizat o:ire liinri Irrtr'-ltiii illtnitia sirirIiatrX, l)c,:bicei, cind citcsc
illc'it sii fi uitat de acestc r''r:rsnri) $;ill , rlr cinri ai inrbiltr"i- re[e, a reluat prietenul o cartt: Ilc LrnrL'am lclrirat-o fit t.in*-
nat, slrilitr-rtr c1.'conh'adiclie;.[ []us rn;ti tllre mlnii pc tini:i nrcu rlisr:uf ia, ciup:i. oc s-i]. lntors,
nu mai-gaseSC a.ct'e orsi strd"lucire si nici accla-*i parlurn. Tatnl
- l\fir acuzi {-legei}ba I $i in accste l e:rsr-rri e t'orbn r-lc o I e mai dr.colorat ;i mai cvapora[. .De: data apcrei.a"sit.a".!n.is"-i.onrf-;a*
ciici, deli 1.l1iso1' timirn[, ri]n atl dr, lurniuir insearnnfi
Iungimc,, 'l fost arca.*lle la. piimclc pn.[ilri iur r"g:,rit
pentlu rnine ctristanta pal"curs:i rit lrt;:t luminoasS" in aces't
timp, adic5. inseamn[ 300 0{)0 klrrf:i trrnrulti!i cu numArun ii pvtmninirio-tiardean,.rla{dnci.ic.a,olTanlsfrrtiiedab,pltauu;dsiiteeifnaimsicnnie.dd-adtcusiisdt,sdccpudcautoana;ireitcsc;5itc.isluPismimrd.rour;l'ltcrofiiiuig'tceouegillii.t,'i",e"C.;sietitoie-intcusTft",iee.on,cg_rrl:eiiatv*oonsRipelteae";tu;lrAai;a"tm},ia.-
15LrcunLlcloi dintr-un ;!n, asa,Lla{" It[o$r::qt' tirnpul ca sil ex- ,l
Xrr*ilnlioralrcut-ragsimaecr! eC;i eu. Riinlirn,r: sl-tl rnai arzit ci sc fqllo-
:'icit{r Si lungimea ca si se €xp{'irni: tirnlrl-ri,;i atrinci slntelrl d
cirit. OLi, uit[-te te rog, lapendu[;e matale cu carc te flJepta adus-o,aculn aici, fiindcri. irimcni nu crecl ci a.r putea sE
I rispundii mai hine decit acest prolog
{ la nedurnerir:ile sau

cii r.r ai rlc la bunic. Cum nidsourd ca s['clrn]da? poate Ia irnpasul ln care am ajuns noi. Ra chiar m;li mnlt,

128 'i oo

g - A doua esrte cu pr$bleme celebre

sint sigur cir cele citeva pasajc pe carc yreau s5^le citirn aoc.le*u*e-naIg1)\icem;aasreigei,uspccrl,lsa'rtrcrdcicrsae:zofiSpari_,'ci:.t;aitscit,_u;-iea-oicdartirca:,iigrcilriud?i.n{\irg'"cliamgzliceielgrdL!i,litarui 'nsroiclgutltaiiieo?rnigicrlled este
lmpleund
tc vot ademeni si citc;ti toat[ cartea_" In Prolog lun-
estc rrldati o discutie intre ,,un l"izician experitncntal, uu
1\{atcrrratici;u1 plir' ;i un Relatir-ist, apirltor al celor mai
noi concep{ii ale timpului ;i spa!iului, in ;tiinfelc tizice" ,
Sint a;adar pttse in cliscu!ie inseli notiunile {undamentale ]appmcpncc(nf$sctr's-*-hrlcrr(cluojroiiiiut'iiTlcicvruslann.a'panvntae",uosorrrsarcrjndrttp,d'lun-mtr:i[idttlfuiier.;ii.fetnn-uipiii.aas,,atsiipcd.r.nnicl{_r-,cl.irrt'ee;r;'n{lrrdoeori:nc'lrrrrlo.rrrilcu.crionasuifrstrtacieorri.ttenltrtoipsilrt,.ar;srrnc,disarirzbcn,fortuueseoci-iru,rlbtllrlrlil:-Lrcn_citr''ti-roirlr.cCinr.:..ii.iasrirliirriccs-irrrrrr,irlrlr'1(rt-'it'(irs"rc)trldc-;rnrn.ri:ro'llrl:ilI'srtrurart'soiltll.s,piierlunllrrtptnrxrirniricirlcnruptIrrict]s,r;lcir,*oir,r-c:r,i.r(r,i',,rrir,:rbltir.sr'sr'airr,iriiirirlu:enrtp'rtr.r-r<sirc'rbrr.nci)na{ispll.tii,latrrdl1ruc'irtr:ln'n1inlii.rcisrce'm,sc1';rriiicsuips1r:cii',..atusli-s|'uriirnrtrrna'rrilitictli.tacrptrronril'ecor'aiir,-;elr-i.rirnicrlili.rr'.ts(useeerrrlsi'ajl;i;Lpriint:rsenc,'ri-o,li,,paltusi,ir,ircttn,-,r'rnc.ti.,t.is,rrtrr,ni,rororit,rrltrori.olltisriLrilinrcr,'.,,urtr."tr.,i,iiss,.tici,i,i",orciiri,,t,r,ieir,.,r'.,airni,llrn,,z:riti,ti,,ni,-,i,t,a,,,rr.,ii'r.;ll,-r;,-s,iie.i,t.,,,o.,".-ga,,-r,.,ir-,,,.oi.t,i,,,,l.ol,j,,al,acnifr;rt;,,il,rbr"iti.lt,l;;i-n,*"l*:"",-i.--:l
ale geometriei prir-ite clin pilnctul dc r-cdcrc ai fieclruia
clintlc volbitori.
un-daNmuenvtaidlc cunr al rna.i putt'n {i considelate notiunile
f
din diferite princte de veclere ! Clred c5" acestc

no!inni, pc car:c se ltazt'azva gcorttctria, sint in af ar5. r1e oricer

cliscu! ic . te in;eli, a{id cd ltelativistul incepc prin
t-ntrCebaasidvaczcidczci adel'irat sau nu ct-t suma lungimilor a
a
doud laturi dirrtr-un triunghi r:sle mai male decit lungimea
c1c a treia laturi.
cel*ei Cc mai intrebare ! Cred ci ;i Fizicianul ;i lfaternaticia-

nul i-iru ris ln nas"
* r'Nlaucirp- rteeaol"cnm'Iaatccruear.tcicleiavrinraultdi-asarduspnuun,s_dcaicni uinpopaleteaala{biri-l

miL
lru ailnite cii axionelc pe cat'c st baze.ttr,i ea siut aclevdrate
srpamcaconacuz1sitfncctc,nriCci.peEcilg.n.ebdIsc)aroii-zrcrnu,lp;tertccttrcl'llliaiicarsiFrtctirrvi-rxznipiscietotiouarilpmlnilrouaeollupratfirasiit-clpc.iiutiuatnstntctceisnc;auautr,,aaIositrdipraeicagvtu'illommilurb_llsaguaerciadcedoe;siamteotptinlp.otuerrLtoodaa--
.'t\,lglrirlrlnt,'_I'u.irissPt,aiinirr:liLrfia!tt.l\cr'e1c, atir;irr*l,,.liiittriv-,i.ttarstculra.;ir,,l'lsrrt;,suta,b,ri,l,i,t-rti,.,.,r-i,e-
irrictitilc siratiair: rczulri fliri ac.lc. ifr, ;,i-iri.1rl;,;;,,j,,i";;
niitsurii, tot, lliL li lirol-rrici.:jtilt: tirupului r,lcpinrl rlt, ;llt ct;r-
ssItvtrropccLralbtafrtiiuitcuiaeir.lrnluriarsrirlit-icaluu\flrloa:ccltnclr,iris.,rsrcndiiLtcuccrrir.acttp.irtlncetr',.rirnr-prIiL'',n,rl'iiirJrr..r;-rrrlorr:t.--i'rsrlrcii,-rr,("rlc'tnrsir,irliillsrrrl)l1ircisllnfLa.orl.in'ls'I.rlrrprircr-,arroirrfrbtirrcr,l.rr,*.r,tsrrirritriLr.n,lu-i,ctnrrrr"ticre,i*rrriisnjrilirir|rrurcrrr,r,rl:uiicri,"a.i
o rigli naterialii cincl ii dai cliferitc dirccfii". I)in obsen'a-
'tiite cc urmcazir tot el trage cotrcluzia cd Llniversul nu poate
fi expk-rrat clecit prin anumite aparate, <le aceea p"rintr-o
rigl5-gradiitl se ijunge la gerimetria naturalS, printr-un
acimag'netizai se: r,a descoperi cirnpul magnctic_etc" Reve-
ninci ia pr-oblcura mdsuritorii. Fizicianul precizeazl' cva a si nu si sinrlrJn {ra r-r ir. lrat*l r-a.riabil;i. ilcrrncnul rlirrr,rn-
lungirne trebuie s[ fie misurata cn o rigld rigidii, adic5. o iri'urri pa'c lcgat rlc o icl.'r: d. or(lirlcr. J)upii plir:erc, ,1c.,
si''e
riglir care pistrcazi a,ceea;i lungime.
*- I)csigur c:i-i aqa. Daci rigla ar f i din elastic, uu ai mai orrliirca cvclrillrcrltclol il in nrrtrrnir rstc,r c,rrlin,,,Patlu-iiinrcn-
sior:rali indisolubiiri . Fntcnr sri o
R-pue-tlcaatE.iav, visfeolnaalrint,ecirrcacce.clzeltrescttiiirncaotn,nicndiuitomicmacihoscnudir,ultcdoLua'prelda, ! irupiirtinr r rbiirar irr sPirtiu
;i tinip.. tot asa rlupii cum put.' rnrpJrti o'tlinca rrirr siratiu
cum obsen'[ Fi in lurgirne" li!ir'c si iriltinr,.. I)a. spatiui filrii. tirt,1,
un cerc vicios: tot asa rlt'r'recr.rmplct cir si o srrpr-:lfat.i liirii grosirnrr,,. "rte

r30 131

!)*

Prin aceste simple iclei incepi sri pitrrlnzi fil tcoria trcl;L- STRADANII DE PERFECTIONARE
tiviti!ii, prin care Einstein zr t'evoluf ionat nu numai ginrli-
rea;tiintifici a fizicicnilor, de;i in primul rlncl a lor, dar A PROCEDEELOR DE CALCUL
si a irstroiortr i.lor, a rnatenraticienilor, a f ilozof ilor' ;i pe rlnci
ile altor spcr,' ill i;ti. Iii indcir datoriti :r.cestrt i mare ginditor sau
s-a putut in{clege clai: ci. tlatura fctronrenelot' obscrl-abilt:
clepinclc cic condi{iile cir: De la abac la creier$l electronic
spatiu ;i tirrip iti carc sc a{li obscr'-
vatcml.
*- Arlici ntt {tr ;ivcil n ici itn seus sit ciir-rtirlrl absohrtuli
lsn'Dl^ua-vtiu\ortin-i.l;tiiriicrnri iictcriiioici'cliatltacti,tr'irctplolorulziiri;tctila'xt'ottanlrbitnsllori:elin-rttt;i il untli obii'r:t din
st'1lt:tt'ct itt ii.celasi

tirnp, arJici tlacii elc sint sinrultilll{', silll litl .
'*"* Iii, asla-i prca rlc tot i
t\-al sii ntai sltui lr5a, clripi t-i: r'ci cilj ciirit'il illi -- r\zi anr a,rrut o ntil.rc hucuric, mi_a zis

Eclirrgtcn I Teodor" Sollnar,, prietenul si acum oaspetele meu, cind i

intrat pe^ rltd-" Vin de la nepotul td.u, Nriiu, pe care 1-am vi-
zut lucrind la cnlculatorul elcclronic.
"- IIi-a spus;i mie ci Institutul lcr.r nrirrrit o nrasinj
rlc culculelectronici, mic insi nrr nri-rr ficut
invite si o admir ! t.jnstt,a sa'mij.

-.- Lasi bd.iatul ln pace cdci ei habar n-are de pasiunea
ta ptntru matematici. Chiar aclineauri, cincl l-am rugat sii
treac5 lnspre se.ard. pe la noi ca sd discutd.rn rntri pe lndelete
rlcsple masina lui tie calcul, cra co'r'ins c;i. tc-irrl plictisi pe
tine"
*,.intr-un fel e a;a,fiinclci eu am discutat cn el nurnai
probleme de literatur"5., istorie, rnitolcgie.."
-* Ai uitat sd adaugi ;i capitolul ca.re intervetea cel rnai
cles intre voi: rnorala, rnorale si ia.r morald. I A-propr:s ! Clntl

a{i incheiat uitimul capitol al acestei inrportante ilisci-
pline? Cind si-a trecut examenul cle inginer, sail a. mai
continuat ;i dupi?
Asta-i arnestec ln treburilc internc si
Cu-m a r5.mas? Vine? si refuz rispund.

--- na" iti face pld.cere?

1ta

-- X{n mai intrebi? Poli rEmine indiftrent in fafa pro- cinste abacele. Ait--i vcncau negustorii si aducean inmultirile
de ilezvoltare a calcuiului numeric? De multe gi lmpd.rfirile ce le aveau cle"fdcut iar peste citcva ,it* i"
ce'slrltii
ori mi se af undd.5lindul in hiul trecutului, cercind si prind iDt"rccau ca sd ctr1it,l('rczultatcl. !
lnceputul accstei emotionante povc"sti ce ince:pe asa: ,,A tost zcrtt.aDtt ac,aaoiapecrla'af,ii.tciixrrtdrrc,ian irrrrrrrltirilt, -i irnlillir.jlt,
odatd un om care nu auzise de la nimcni niciodati. despre sc pr.c_
numcl'c. care nici nu bdnuia ce ar putca sd. inscmtte sclcr:r- rlt, ctrritl-rlicait,. Nir rrita ,ii ..:ld."i"
arabc erau incd irt rir'.rii.{ ltc clt ccl(, 1-(rtnntrc:i nrrr}t i lr.prefc_
teala ;i care a lncr:put, a;a deodati, singur-singurel, si cpc'orurruiaanm,ucntdraisuciztnaelfi,b_oiirdd.lbceir;iiidi;enenecnut;roiiiunnrloutrtiimp,rl rir'urpr,tlcni'cr.edaucieccfenaarsiicbrtncciradriarcuczsi ialiJirst')rriuunrnnrtcrtta,r,q,nlnll.ta(s,u.stRincrp:.reueuiFda-uIcrtz.r,,;r,ri,nub.rtr.ei,.fstluinuatur;u.a,,i.iipldnoitu,aainpci:ituet"ei
socoteascl" 1... l)acd-aLi lti ce dor mi-e sd stau in fa{a acelui

frate al mcu de dcrnuit gi sir-i surprind mirarea ince{o;in-
du-i ochii ! Llum a inceput? Cu pietricele, cu rS"murele, ctt
scoiciori t u 11 intii lriimalcktt'ucise, pe carc-i pistra drt'pt
trofee sau ca podoabc? ln grdmaioara aceca de pietricele, rczolva singrrr.multirnca crruiltiilor rl jf,_,r.t rriiulc. e ;,,i.Sr"-
sau de mai stiu cu cc, r'dd in{iripindu-sc umbra abacului...
si mi intrcb: oare omul a inventat abacul ca si-l ajuter tJievl0rdseapuraoldeuacltticorstiourpnclclcdarz,calrlalcuunl, c.,r'trtt tlr.pr-rrc, -tri r,,su.c-
ia socoteli sau abacui s-a zdmislit singur, aldtriri cle oin ;i
c.ritru clc t,rr,'rrr ,.,.'.x-;
prez rnte datele rt'spectr'r'cr. g:rta prelucrate.
l-a invdlat el, pe om, sit socoteasci? - O fi, nu zic nu, dar tiecerea cle lir abac la rnasina tle
-- Incurciitc gincluri, dragd prieteue I Le-ag numi ,,poe- calcul s-a f5.cut oare aia dc ssIins-ripalupicmumaiaici,ftmin?rtaf,ini"erlri,lir.rA*
zinieti;tiualacltevaisr"t"fepleanutrtoubaiocgerlaaf;iiarnscat.iv pc-ntru care Goethe ;i-a mi-o prezin{i, pr-obabit ca
1ln\tqf-cauc,o_tcloroJ 3 sa-mi irrri jaloanele nect sa.ri, s;i nlii cdliuzeascd
Oricum ar {i {ost ln inccputuri, abacul a l5mas pinl azi
un mijloc inclemdnos de ugulare a calculului, de a-l face
mult mai lapicl . ft-rtoclatd el a inlocuit rnunca intelectualit pblain-beislFocrn?eitt1.cn'nreil.iblaiuIitaeaesegi.dnipreettari.r'ipr.ttoiCil.'arNrcsueilmng,'ahhiiccietu,;ntrirltr,alcrtnrcrlcs,1ii,g, ti',iur_oroitr";Jiinc,c_*leit,i_,
carc sc: inipune ln mod necesar ca si poti calcula bine, prin-
tr-o muncd mzri simpli", aproape lizic5. la Pitagr-r: a !
{;crteeiil-be-unleiiiccn*,i-sca;iiditlciafvinpiciaoer,ti2ecsnad+iscle2iln,cn{tauliilnrcinlduiencltrdu:iz.ii{eAi-c;totuiitasampbueiarngcctl:eetllaleabnljeauucn-usal,isssiandpregicv'upuorinie.cotr"rssfiu-tii
-- La tablir inntr"rltirii? ,\c1jcii, in {cl-l(.r.;tl . la .riil..,r,lr: lL_,

cr lcul?

regulile abacttlui ;i si ;tii sii numeri ! lntr-o jucilit: paasti5-nbc-a-sulai!'f1rleclEdeiut2eixee9lsaeccsolcootulpperb!eeacPrrfdiaotrfroitrniigrsmiln.itumisidfleladdcjarie.u.nlgaotatrrariersdbdu5cdce,udolseslcemialoisueptcarrxr:ibtirneuszlicenicpu.c,i,it'jaicfaricrttciriJncip)'1.tnfarlc)rrrc.smep(c-,zsaarriurrlroielelit.aitaru.srkstecsc:c,li,irieits,tf':riipbiucsneclrnir_tfl-ieeit-.:ii
pe-n-truIacr odpaici,dinlaldnroilei abacul s-a trans{ormat
ii din Orientul apropiat sau indepdrtat

el se folose;te pe scard" iargi di: cdtre ncgustori, care {ac cu o
repeziciune uimitoare cele ruai grele yi complicate operatii
numai cu ajutorui lui. uumerice ldsate de aceste_r'echipopoarc: tabele de pitrate,
sctaiunbgenuleurmeclueS.cfrouitlbooursuriitlleedidnfeeurn-eirtegdriepclotetr',liintianfbrcm.aleoccfcilliecrpu:urree'f'nnatcr"ciircreiittnas.rifu.iain'ucolrir*il."ori,r
i XcleV-cI a,Alcemuxlis-ct*aitcuiet\l.nuanEcdouerrevoasppacu5nuz,niprt{oreirnl cdseeenbctroierloieoulrerniXosaIaVsutr.einXdtVreecp9arilicncudhleia*rri vorbesc de tabla lui Fitagora pe care clc pi c'fsit-ca,r2,,-1,0,0nutnlor"a,-n"ii
numai c5, in locul masinilor de calcul, erau la loc dc o lnr;a!d. pe clc rost
liecare olx cu cartc

I34 135

numai clterra dintre cele mai vechi tipuri de tahele, inainte
a-{i vorbi despre cele mai importante,
d*e tabelele cle coarde,
c vorba de coarrla sublntinsi cle un
arc de celc- gi de
tabelele funcliilor trigonometrice construite de arabi. Ta-
belele de coarde, lntocmite cle greci, premerg pe acele:L ale
sinusului ;i le vom lngloba tot in masa tabelelor trigono-

rnetrice" Pentru calcularea. lor s-a cheituit multd materie
ccnu;ie si mult timp, eilorln cle mult timp. Dar cu ajutorul
lor s-a clgtigat nu nuntai incomparabil mai mult tirnp, ci
s-au realiz:.Lt metode astronornice ;i tehnice de orientare ;i
misurare a timpului carc au finpins lumea inainte pe drutnul
civilizafiei ;i al progresului.
lucruri cu atltil
--- Nu 5tiu cum po!i trece preste asernenc;r sitnt
u;urinfd dc parci a.r f i volba despre un fapt divers?' lli
f ati aceia, irindu;i, chinezi,
asa de mic ;i dc ttmil cle oamenii noapte calculiud;i iar
ara.bi, s-au
europeni, care trudit zi si
calcullncl cifrij cu cifr:i, ca si le inscrie in tabelele lur pe
stliduiau si le facd cit mai le-au ficut
cilre se allii, car:e ecxiatictteru. d$di. ;r fost cuce- Ncpcr
pentru nici nu binuiau cu
liti fiecare cifri, care nu ;tiau cum au putut fi stabilite.
Clnd le au la lndemlni"r, le consultd gi, daci sint pe lrlarc,
supsucna:tsdinetteemrmiinniicuretazriestelorncfa9iuenouriapcoudta, rdei,leicafri"atl;rucni.usi idnrtupnr:l ;i1$;rtiuuir''"irlgr-{ur-loPrlcinr'Elisdtol.,rl.unti1iir.)1;rcetj(t1l'Iruu1rsXcn_<cV9riIcsrcnIllcrI-ien-Nrlrc.ittironlcapIcriec"rl.r,rrl:rios'racr_triotcraoldpc.riirsrrrro,,rl:rrrolg*'i:pcrisrrri'.iritrtriittr',rririlllro,i(riignl,.r.ia.r\1i_i\i',rtrui,r.'riinnr.'rrprieiJrl!rn,i)raalitlinllt,i.,nu;s1cscce"or,pui1scur;tciur;all;r
strn ci.ne stie cc a,ltc date cu care se preface rrechirtl in nou !
-- Recunosc ci uu stnt a;a de sentimental ca tine. Lirmit- jyD-,,f_r"*z.i,a-ltt?,ilz.")A1ay.1rrp.neror'f--doaira.msr..pouuurbltsararInloriI.ilin,ig-i1'o,s,Irpnotror.r,.mroc,ists.,sticlc*ur1r.lijricrrt1ir,rt;,ri,r.,.-.coi.a,.l.rliro.c,grr.,narr=..ci;t;in;rt';sil;oa)r.r"tLrcjiii:cisl-,e,
rind cu mintea, rlnd pe rlnd. sirul de lnmulfiri, tmpdrfiri,
ridiceiri la puteri ;i aclundri din care se compurle, in intimi-
tatea lui, procesul de construire a une i tabele trigonometrice ,

nu mai am currr fi romantic! Dupi. cum li-ani spus. lu
afari de aaridtmuneitriciie-*er;ai uinavneersvoaioeai,sesc;Sicdle[rdeaca-u
1 operalia toate
celelaite operalii rnult

de furcd mltematicieuilor-calculatori. lncepind cu secolul al. cos (n--b) -- cos (a _i_ b) ,,,..2 sin a.sin b
XV-lea, clezvoltarea navigaliei si tot avlntul tclinic de cale
era cuprins apusul liuropei, cerea lnmulfiri, cerea tabele!
Atunci sc descoperise tiparul" clar lnci nu era dert fui folosinf a rle care sint sigur ci lti mai acluci aminte.
cat-d.U! Tnre"b,u<ielarsda. tcrcarnsstfionrnricitontulim(e,srtecr"(,Axt"riit"oir'duinriaiiri(;lc.;#c(;,;lnl)li-
curentd, totul se scria cu mina. I1i tnchipni cc lipsi cle ta-
bele ;i de calculatori iscusili era pe atunci? La lnceputul
;i apo"i,.<hrpi ce ai a{lat rr'ghi'r il,l,i iil,, ,,i cauri i' l ,b;i,:
veacuLui al XVI-lea s'a produs, ln sflr;it, un errreniment de c(,slil ,; .r)) sl cos(a b) cn. drrp;i
accca, sa lc arlrinil

136

noa-strAd;.1.U_enrueI_dd_i;ni tereu,cdeai mr lauirndeeasedaem5a.tumnactiennuatiecLieandir: pirerea Doudzeci de ani a lucrat John N.pcr la aceastf;. i'r,entie:
tgctarrcertesri.scr eir)ribruariclzeeerParczin(,.ad, plcielulaczotgrrrtc'.oso.pmlolrlnldraieconab{aisrtrc.cree'r'paeitnxdicsact,ilrfi,un,lontrsu"ii;ricAatolatuan.a"rurjirrto*-j
ii astro-
nomi din vremea aceea, Christofor Clavius, exclarna fericit cd
aplicarea acestei metode,,cste plin5. de pld.ceri;i
lns-emInnactaozulnl macuelsf tiare, pot inlelegc sa.tis{actii., !
perfect cc cs:anrc+e.iloisut,s-ar{rpmiene,te_seiuNacrp'tcatiaxoui.!vlltrrncumiandnsipiecnidprrdnnotlcudetgtrri.erigl;,lnctorrisel,r.ag,ecaedi.asce,o,iiirencipr.igaitt.eemrscigertsoaaeimtr?ottetien'mierlerSestume'piucctcdnrerrereiiicenirtce"iinnicnceuopnpnjlrorroer".uosncitl".1l[inctirr,irlreucdro{enlsu_.atcdiroclelat;a.ipiacn-rarruicsrm.or.ircli'igeellteprimair.cni."sru.e.d.t;iIritei"riceSn'";iea";rt',;.-dlo;;',ctnbtiai;rar"cttr"fz;ca"-;al
i trebnic sii Ii

ma-esVtreulzCi,la,;vi iudse, clata accasta judeci pdtima; I Nu rrita ci

?n calitatea iui dc astrononr, era Llbi5nuit
si mlnuiascd tabeiele trigonometrict,. in pJus, rm ,l,, inujf
se terminase de publicat tabelele tr.igonon"ietrice alc lui
Rheticus: ,,Opus lralatinunL" ln care crau date, cu zect zt ci-
male, cvua_lsoirgiulerafnulin, ccgiiiloocrotpriigeoanoacneresttorirctca,beellejnfu1s0e"selnpro1c0u,*.
Cred, refe,_cir_si. poti memora primetc 12 zecimale a.cstui
numlr. Iata-ler;
ratd ;i de Observatornl astronomic din Roma, uncle sealla u1,-

Clavius. A;a cd, pentru el, cra lntr-adevirr o pJicere sd. caute
ln tabele gi apoi si faci aclundri.
nrai putini incortlarc intr,lectuala O asemenea rnuric;-r ccrea ,,Pe nurnd.rul e , sar,arrtul i1 stintcazri
tl,.r.ig opcrat iir inlnrrlt it.ii. E-academic ;i formeazi
AvInesndhfisiasptrliudruliu,tclfjlrdcra1i6ua14crmc, slttiecEilcrdr.rlincmdtobatrtlrcdr.grL,crfc:ouNstrr:cpitcalur'-isli:i-::.rL,'.uUpniurl.ibiIltii.crr,:irrrr]io,t g{iuta--l Bazh pentru
Iogaritn-ri,,.
Asaclar.t e : 2, 719281828469.."
,\r.il-Z('tf(i']l"r^o.lTzriat1tv(i'elnsarp.ucttlrCs.ruaer.-ilI)!aMr.ai-tui nrcu'si irtrcc
rithmorum Canonis descr.i ptio". cle cle I ri
pre--fuiCpasauc,mdf.ii,inAacslscacrircis-u\nccplhctirurr,,r,lltiuLasiirrcor,nsriurrlfiiturrr'Jlrcrrin;.L\i'tlceiiir:lcorihgli.slaLtro-luisi_,itlaiiirrsirtciLjt.'sritic,si,i. Ic,_rt
[il(.i{itrt l}rrblicar

scr-*i-t Eram tindr, scuzi .dcstrrl rlt, l,iauzibili. _,\;aderr,

extraorclinarer r,aloare a tkrsrcol-rerir-ii -siiic.,\zi, r..trt:t:trii.ilc -in i ct'lc rlr,rri lrl.u!r.r,sii !
Iati-le:
de istoric a rnatematicilor au stabilit t:A au t:-ristat. inrrii tliir
rreacul al X\'"-lea, unclc lucriiri mateuraticr.{.;lr.(,}.rot Ii r.,1;,- I c c? c3 (-'4 0i ... .,

sideratt-. ca prenlergd.to;rrc iogari l rn i lor, dar pc \:i.cm ca rrc{rcir 0 I 2 3 4 5 ., ,"
descoperirea anuntalS dr Nc:per a liicut s.rrlratic. lltr r-.rtn-
temporan, ,!*p[ clte tin ruintr:, ;r scris ci invelrtia. lugrrrit- icnnltcn'*nlputiillAldfndcirr.'e:nNtrNr rec'':f\relrsc.octiBrvds-c.iurpd.3oNin.bg,tsrr:cc' r.veeiB(5,c3scii+ioer5in"esnecpuFcuir.ent,icar\srlci.uaiL'risalaiLst"d..dliof;na,ricr;ilrl,n.rn,a.ir"rsn;r;ie";.rt;Fa"l*c
mizoill<aitrvaacvleeonriict einalhtermrnect,o':crltu: ndetrurl5.sunrta,trecliirncasiecunliclrjisri prrceeeiuxiilzspta;puciol"ursnatpnc'e,zu,.nStlnsaet.eimtlaoonprrifn,a-til3anrcJinm,o-cfdga-saczat5iaur;Irsiliaet-imdrairsenirpijpcfoalrleuicti,dpdi_.scuu,il,tc.ec,,hnclitsrrafiiifn'iua.ltocntoreciapilmtnsetirils;a.1,,isi,a1ianrj6bnulsiiicdptcall1cdciNa.mr"c.iu"cpitz*rloeiaaecrgbaroa.ir.ulerii";tl..to,miia..rLc-ol"to--ii
ciL rlir 'sti
irrvr,ntatt:
mai lnainte, fird s:i lmprilnirrdcr nimir: cle la nirlcniI llc
altfel a rdmas un mistiJr calca pe cilre a urmat-o Nsper cir ,iJ.
ajungS" la notiuncil de logaritrnul unui numai.r. El nu r_r spllnc
si nu reiese nici din caltea clintii ;i nici clin a doua, publi-
cati 5 ani mai til:ziu clupir nroartczr autorulrri srb accla_ci
titlu, numai ciL ln loc dc ,,rlescrilltio" estc ,,t_:onstLrrctie".

138

139

niaumc ltrmillpu5i r;liireexdpeonaecueteual ;pi u:rtactruiiriiia. rLeoxgiraalgitemreiai de ::idicinri {bli:c[d"tltt(di;l-,']n;,{rl(at\rc,ur.1crercl1t"'i"'lsitlcieullpJi.lrl(lsgari,n1ltllcc.1-,ctntt:tplrr,n1i-tiltt1lt'tn(,l.c,'rt]1-.1tetIrJlei'1;llih(al,t1li.a1uel.,Frtr:tjipl!,.rtrti1n-Igttla\ni,''l.0.adnrra,l:]irle1'cr1)i,tir\,p'a,igc.lujoc,1erlti.cm1irnaccnrri.Laa'uilruutl;int1rtii"'',tarn'il6lyi'uiiiill.pe;.sr1rrrrl"i'rtolB'nr).i.ln.'re1r.ralcclrrrlr'alniL.ro.ig{ro-rci*rosir{crc;r,r,r'c-n-rt.ait;g.(r,raigr.rnrlei.,rr..ts,rs'r'rccil."llir.rTtarrn-ccrrrrnp,6;lat:r'rdlrlcrriI'eesi'itir.r;,rcr;rauiec1,;t*r,,{rr:;,urtr:irrns.o.}oie-pr1Lrra7,illipiern,HarirMte.l:,rli;r.tjcrir1..nnrrilrt.st,nisia-{t..;srcriit;\(l,iiaetartltr:atmci,rrlic.rir.rdha;pci..oo.rrccraIplui,iraliiriirtii.{rl1lrcrltgorz'u\atfrc,o.e,i:t.rnlairl;.,(t,trrie.-telr)lalri'.aeriulrtir:,tuiciri_iLr,tre.ratnl,tret,eirrb;t{tr,arar.idtt,,nJi'.snnlr,tatnaiall,tn;rrficrr.c.rlbltjf,iee;lrrt.ir,ri,,lIi.i1ccjl.ii.ct1r.,irLl.rsi.|,ic;ttl"rir,1rJo;e,rifr,:r,cl,.,;,-.r.iil,p6rr.;i;,u,i'il,,,cl.;rdtrpi,,nrtolra.i,1",2g;.;i,;,,u1t"trii.'ipirr',i,,;,'0r,3.pci,,ir,rpn,,{-.hti,-,,i'*1.,if,sg*\c,fir,,",iu.t,,uaoi,i.Kir1_,lri;,",,itf,r,i,lr".n",iil.n,^;igi;;rie,lrs,*c.,li;ritlirl"i^,;r,rtlg5i,1,,"rro'.it,;,':,.rsr;ftiii,1.i;'l,,it".r,.,,,,-*i'r'r,;il,'.lr...iir.i1.'.tniu.,"i,,l'nic.,.,:i,iat.1*iiil,,.r.,Ui1rllirir.rr',ic,r'ii"lnr,lnCilln,;ai,r,,.ririlt;,,*,,ii,a1pv*f,'rot;i,r,,r,.ite'ii,,i1,,c.',1raraJc;ri,t,iu1rr.,afod\rt.lri,.al,,r.ic,ilrn,oti.,.*,,zsrcclgrn_*n.rnn;;r.truc;erL}l,g"ria;trrr,,,.,r.,riic.',irsri1giacr.:iii.l,lroni;ictfl'rrir,nii,ilr.if.'lo,c.pira1lir;rp1)cllrr1.irs,tsnl,r;rr'.*ir,n'r;irrcfs',;riite;irii'I;1'r'bir,r!.,ntr,$irri.itfs:,;E'n1rtc,h.1iaee.I,iilrtrmiri'.riaii*tc.ia,i'iori;t,anrg,:.';aair"aiiie',g{t,ltr,emr1l5'"ouii,r".;oloCjtrdiictj,,t,,.l;sn.nfn;lr"uiil1_sg"ur,tnr;tlr."iaaoaut1nre.;jil,'ic"i:laiac';,sctl,clal_sdilt.nia;rir,0ic,eu*iit,ric,';;l1rt;*::ii-i,citi,.iLj],i
a_ltl']m"ci1:a1.llrlc1,ru.nT1hr.i,i?uPiin:r:ront-dlo:in,lstoii'g.laa{urr:ecitu,ndrvicoasr"rfbi;minnrcaalirtcioiart,prcirlreoistipecrs.cL,1r,l",*ii.,'ia.ltr*s.,"fr;ir";.r,ro;-rr"e"
se tlovedcnu
"
astfel un rnijloc universal rJe u;urare a calculelor, cu singura
conr-lilie ca s5. ai la" inclerninir o tablI c1e Jogaritmi I .*,'IsP,'c)clocgca? r(llrirllr.r(,,lln'icr':gllrr.nrirbirc;rtitr..l aIirrrrizrrloicr.ucsrrcic:.iiir_rrnnfri ,,lrircilcnr.r,rrr,teirserti

--- Cu singrtrt cr:ndif ie t:a si ;tii cc cste logaritmul unni i{1
numiir I

-- iti cer it'rtare petrtrr.i :rccast5 uitarc, rie;i ;tiu ci ai fdcrlt

obsen'afia numai elin mali'i.iozitate ! Neper nume;tr: logarit-
mul nunlSrului care sc aflil lii ploglesia geometricil numd.rui

corespunzfitor din progresia aritrnetici.. Cuvintui colril]its
ncwlinm,d,,llo,gcoxnp"r*im.di[circiu;pprioi,'1rbinariliietrs-arruoprr'i#e,tafotera{;i]o,,rarfiuthnncltaoms"enatcali.cli;,i

aesntuemaececeiiai;lii;fecrluepn[faalJtoiig, aprnitrrn;iliosr imnupnl:uerccX[olropgraorpiotrrntioiinsailnet

nllmele care au uu a.numit r.aport cn iilte lltlmere.
fac-f-ieimtrri:blnuciehispiiusi cclei cle lmplirrire ;i
seniiment citf;. sati.s-
atnnci printre matemati.-
mani{esta.t

cieni !
--- Flenry Brig6^s, cei cc ae alcnlat prima tahl[ clc logar:itn:i

zccinrali, cJupf,. ce a citit r.;;rrtcit iui Neper" e. f'ost a.tit ele c**
plelit incit nu yi-n tlor:it nirric altceva, iiecit s[-1 cunoascf;,
Siltcrn irr 16lrl 1-6"i1n5uiiln;r 1969!Ileaceea intllniretn
i!.ie Ne1',,r'. r'f;.mas conseltnatir lntr-o
airia fn ;tnul
avui loc
cronir:d. rr timilului, circi la intrcveclere a Luat parte gi un
irrictrn .rl lui Nclrlr* invital rle acesta. Eriggs era, profes*r
i1c ccornctri.: la {-amhlid8'c. llJ- a plecat la Mar:chiston, {n
Scolia, unde st'afla. casteltil lui Neper. Cind a fr:st condus
picnlerircc{aedmlrSdula'c;rlat ucvnurte'laeudn-ncu:elir;satf;1l!aisec]"io"capl.:te'ioira:t,rtrlcrrotrpr'}err5oc.b,ua,t"bcnii..lcpircoifraieplaccaisrreungilscsfpetriarivfencaul..ei
litrcpcr, rlup5. intreverlere, Briggs lncepe calcnlarea logarit- 'I
milor zccimali, c;tre, clin punct de l"eder"c practic, prezintil
avantaje insemtrlte. Kepler, rrestitul astronom, a fost gi r:n I
La fel de entuziilsntat de aparifia logaritrnilor. dar, slrac
crlm L'ra. glndul utrei intrcr.erleri cu l,Jepcr probatril c[ rrici 1
nu i ;r trer:nt prin rninte. L-lu ac-eeaFi rivn;i ca si l3riggs, fird
ls"iitn;tiiieztc-cleimacaelsi taa,;aseciai,pinnc51, 6s254c. a;rlc;ruiilreezaen si el o tabl5 de ,l
cle suh tipar, la ioga-
l-on- .l

1,10 I
lrL

ri

- Rine ai si rcfoaigctiinsshi-icmnaipi srccil.anuinuSc-et.ilselafsidrsnoiuut.mdl a,i.icifaiirer-e"-bi-uc;cl;Leroris,a,cibaiictici,csrl-r
lectuti, te
Lei
articolul r.i sir
riuit.;s.
- i)uios, spui? Iatri-Ie'"
- r\scult5. ;i ai sir tc convingili tu: ,,E,ranr ln clirsa I\i-a
pli.ntard, cincl i,'i.cl la tin anticir-o ar.itmeticd mar:c si jerpe-
liti, pc calc all luat-o po Lln pr-rr! de nimicir. IIi riui spre
crsrl , u.r'isfrrit'sc si rarl pe Ia s[ir'-it ull capitol rlt s;,r., luga-
rsritnirntrp, iilra"t.icri1le]'.'pslcin.sparclrllu.arrcarii;ai csec;itidiaerpi.oDt einblouccruririlertrcnirraltmir.ilseiL
slncacpatrlee,s{itoinrojeas.iecle'binicuiruinltsior.i!i;!ii inrpirtiri cu
nllmere cnmplexe
de palnrele .
degetcle si liniile
lor cub ice sc lnmulteari cu lei, parale si lasciri ,. . . nu .ti,, curl
atunci una. din
:rnr ajuns acasd !,..'Logaritn,iir,ii-or-iietat
fi'ur"sertile pe care maternatica lc a'e asclrnse intr:-insa si
mi-au dat irnboldul s5. mar caut si ccu' nsotsucd;iui laultneclier .c.t.rcfastpiutrniil

cd acum lmi petrec di' orele libere.

I oll T{rllast'll cle .,stiin1;... se datore,ste in prima linie transformirii pe care
au proclus-o iri mine logaritmii lei o ctate la carc imirr:csiilc
v"i,srt,alri'r,a;t;,i;.liti*r;.'.iizl1;fi-;o-tzu';asii\a..i:-tir:is*,,,,,,rrir;,rilttr.rir'rri;..,'t,"tili;'utltit,.irt,crr',ritlsniir,ttitli'ti'lci,tigrrlbti'lai'r'rirritiblifi;nr:Jiijiir,iij iltit,lttiltltzri.InoI-ircr-..bltr,L'\nlzi,it-siit'rrriirri...]tr-Iib{iltliiI;ti)ca[triil)ii't'o'eril.r'an.L:lrltittliLttut'rt'tsl5t,':rtilirlrtiii'ltt'ir,ttrrl,'crisctl-.tn.l'tctt"atclrI-uLtc''tlitaorirdlrc'ltbrur1iiizltiriitrr1rtlirltPlciltlltlort1iiul"lll4itts1'c-.l'ttlatii-lrantllili'l'oslni)gittttLr'rgr'ti''l-i'al'-t'aloti;rcPttrtl:riraliiirttiilloL't.t'ltc''rtir-ltg't3ll';lirl(ttltirr0i'llrtrtrLiir{ril.ttil'tllto'itoliurlrltlsdalcctilr"i(tcnit',''sla,ip'irl'1}ir-rrtl:lrr'tln1I^l1131/t'tr,o:itir^oi'tii'lttils'ai-t\r"rtn1i,c,'}'rsltl,t/;;ll'l"crti]iriiiltI pglsnr.uiof-stlc*ntlruutI_nsiiumrctia'm.lsdoce,daclirt;cr-ettaeotmrirrnc-agattis,tncitcpuiutpuuIgai,tirieniGnnuioqt"r,io.zr:lcana{ergnifaaMerdnaloti;tIi'i:al-)arme,uiraiiccdlceii,rerrpaui'seict,n'lLctcLrprruaeasit,tiriltniiaprnirraacrinleiau..n-lir
citcai, mL gincieam ci. dacii nc-aL auzi un ;colar cll., lzi. carc,
a;;cc"t1l,"i;crri_.anu'a1lu-'llt.1ulti,it.rln'..t,.,c\ir,luic,il,i,urracmrlilr-dlrt1ai*oracri,gdlrtaci,:tirislri_-tinltatainnrtrlstiritiall-lc0rlrlLirrltr-lt,iiiicItrl,i)rrri.ts-cico'tticliai'lcipiisl.itlcrsrl.rlcriitris,iai-b,la.rtl,lrflc4liitclatlrua]tsri)triilirtlllraarf.ri,tr:V,lrpic'c{crti'risn't5ttici.onncsi2idtir'51rt:;ori-,1tts,r'st,t1:.ili'-rrrrtrttl,r-n'i-tll-ji nu s-a plea intc|csat dc t|ccutuI trnitl'itilor rlt'rni.'r:'rrr-ir clc llr
noi, irr riminc' cn totul rlt:zrlrit'ntrrt, auzir.rcl c,1,,: .slitt.iuri
iil-l ( ltr'{('. ,falntc, tlegelc st linii sau t1e farnlt: si ltisctii. ,

1 t'l ? --- [)a, si cle n'mer-c'conrplcxc pe c[rrr. lc-ar co'iir*c1. cu

num.rele imaginare I Irr cazul acr'sta a' trtbui sri-i cx|lictinr
ci inaintca introduccrii sistemului nct'ic 1'
-- la I ianuarie 1B66 -uceucnaistidltnilncgidmeemjniisucerdlc. ntaulac.r'aonansitcrii
erxacte ;i nici nll avelll douii princi-
patc. clcrri alcall acelca;i nurniri: stlnjen , palmd , clcget.
trinic. in plus, lntre ncestc unitaitinu exista ,rn ,aport con,
stant li nici lorbi si f ie zccimal I

I {.r

6 dIaetdg.ectlee;ci e2olilnuinigeinraeu, ncanuc1me pir icldomi:3plesxt,inajseanciu, m4 paime,
esteazi
nurnlrul care cxprimi un intcrval de timp: 3 cre, 4 minute,

6 secunde;i 2 terfe ! Si acum si revenim de unde am plecat
ccaalcsuI lpatorpnriimrnamtaaibdleipcaurteJo. gliia-aritrmn iivolirnbiiitlodretrGiguonnotemr ectarircee.a
Tot el are ;i ideea ingenioasS. de a construi rigla clc calcul"
adici o tabl6 de logaritmi transformatl intr-un instrument.
abacul ! imi fact:
o n-esUpunsdinpsltircuemreenctln;di mai ingenios decit librdriilor, acele
vdd, prin vitrinele
mici rigle albe, prevd.zute cu un cursor lat ca o cataranlS,
amestecate de-a valma cu creioane pi stilouri de parc5. ar fi
ni;te rigle obi;nuite de clesen gi iru un rnini-ialculator I
ad-miRriigtulapluriinGvuintrtienreelrea foarte departe cle accea
pe care o
librdriilor. Era pur ;i sin-rplu o
scard logaritmicd, adicd o linie dreapti pe care erau lnsem-
nate, pornind de la o origine comund., segmente proportio-
nale cu logaritmii numerelor naturale. Acestc segmente

Del,reacudiu,nploodtgacatearirtpernruiunllmnluaumiinud.nrouurl dfaiiinlnecdairuzloeirrloonrgasateurinrtamolStailnoruermipgreienrzeeelanetec2ru.,
3,.. . Fiecare interval era la rlndul lui tmpdrlit in zece pdrti
egaie. Ca sd se facd inmulfirea a doud numere se luau in corn-
Blaise Pascal
lungimile celor doud segmente
pas de pe dreaptl ;i se statri- ipnltcrlat tri;gilalaacduebvseanuit altele necesare ln topogratic:, a;a fel
lea cit este suma lor. Sub aceastS.
form5. rudimentari, rigla o veritabilir mini-tirbelir. de logaritmi,
rnulti vreme cd se mai intil-
a fost uneori;i azi folositd cu succes ;i poate vechi. O primi ca sti folosim termenul la modii acum I
ne;te in marind,
pe vasele mai

perfecf ionat:e a adus-o, cltiva ani mai tlrziu, EdmundWingate e;ixpaectesa, nntuiereex,isuntideunuuasjuintot rnemcea-i

care introducea o a doua rigld ce aluneca de-a lungul prirnei, sar-e E f oarte lndemdnoasl
in a;a fel ca cele doud segmente s5 poati f i aduse unul in prc- decit puline zecimale
lungirea celuilalt. Pasul decisiv cdtre rigla de azi 1-a fdcut ;..i..mDaari rapid, clnd e vorba de fdcut lnmulfiri, impdr-
bun cc te faci clnd ai coloane lntregi de adunat?
l?obert Bissaker care a construit rigla din trei linii aliturate
si gradate la fel,linia din rnijloc alunecind lntre ccle clou5, liri
Te lntorci la abac?
extreme care erau legate fix intre elc. In {ine, pe ia jumS.ta- la-toaNteu,mfoalgoas.ezi;ntiemlean;ionaastrdee. calcul, aceeir pt ciire o vezi pe
tea secolului trecut s-a mdrit precizia ln citirea rezultatelor,
ntruu--sitclmuoriiomisnacs;5hi.nipoludai efclu[c?a;ilDcueulpaa5ta,. receiltoeep;o;ti\irueps,rtieLm.euOiblanirinezvecanrctaczotionrcs?5-,
prin introducerea cursorului transparent, avlnd gravat

pe el o iinie cle rcper.Totodati al5.turi de scdrile logaritmice

au fost introduse sclri trigonometrice, scdri de ridicarc la

L44 145

lna-irrNtctrit, i-a luat,o inainte Blaise Fa.seatr, care s-a n5scut ])c cirre 'r pllf t. 1rr' I:arrria r'( sl( i,clt(i'Sirtirtt;ijCadctcttaOstttitltsi chitnnrlbtmar(ej
Fllaisc Pascal a. -:a
lui Leibniz. Mai mult cliiar, SC l){'itllr ol)Sr t\.J j)rilril.-U
construit mapina sa de aciunare la l0 ani pe cinci Lcibniz
a concepirt-o la 2sI5iP.raeAszccolalevlveiinr;asit6,icncim.a-iruetXoafribriira,l-ucclltaleor oorrmitaetarb.;rinnai'[-n"aimdcauei ;,:colo.
1,,ln,-sJr-rr,l,iljtl-r:r'icsi.lPrur
cornprlicatir carrc. i ,--ri. Iittcar.r, c i li'd*r coresltuurlc uuui c;rt1in

mult mai tirziu clirr car(.f lornurt rrunririrl?
,-_._lrrt.y'u.)l:sil.l;frsIti).rintnlcr.rl,"r1,,*-t:riecri-:iii"rr,rJivIitrii,rrIpt{.}iili,rlrrulItLlr.i,'rriil-uirrn,ii.i.-ctttr';lurnrurjrrii:itin1ct:r.ur,itihi-,o-i.irnr,.lira)crtui,,.ini6iiuirr,lrl0,nrpicrt;rl'ie,u..l.lit.eu,r.6;.i,,l1;ic1r;;,i7l;ii.;nclj_u
scurtir, trcbuia si se gribeasc[! De altfel, Fascatr a foit un
copil rninuner ;i a arltat predispoziqii tirnpurii nu numa.i
pentni geonletrie clar ;i pentru irrr.enti,i mecanicc. trll a n6s-
nmcouinciittrrrrti',,ftli-rslti:ltps,otata'att,easaclliatunju-raticrdigeiriidtnai\n.etalnlrtialloutri:,nsailrl*oicnpi:rrt'it,cmiinc ;rcrrlierigrr.rris,lccdra-ipcsria-t.
li.l]tuiln],.;fIilr:r,irr:rrprrt(.-rt;rr-r|iarinsrrr,irl.r,';iritit,arl,nrr.tllairrr.tilrrr.sacprtii"rrsn(n,r'.{")-it.,irr;.lu,;gr,l.1;;0;
':l,-irlr,t.rni,r-.tsafrr.pi'rlrrt'oncr:zrlcrsJ.iltupt,(lc:i\rit>'saiLts,alrl'riFtur(ir,irr;{rir-s;ls,-;rrircr-,lir,'iPlt1ldraa-ia<'t.':(ti:l.r-;a:rj:,iiiti,nfrirsilrguulilrtl.:aoculull*l}'itri.izrlrute:iiz'lcrrt.ri,jicltririaru,rtirtnii:,rti,;i'rrsrr.l1r.iirLii,ll,r,ii1l.1r]l,cti1iji,irl.l;'iirilr,u,-.,d'rriti:'T;'t*;:,ni,,-
aclune ;i szi-i ajute siL intocmeascir situatiile {inanciare ce i
calitate de inspectol- f inauciat-.
se ccrurltl ilcestuia in fsto-
rriri,a pe c:rre o incerca clup[ asemenea efortuni l-a inc'ern-
nat sii cnutc', ;i in acest caz, roaba potrivitfi .'iE, rluch e a. greu-
tatelL ccconvcae{nitriirnrciiirmcaintitradlce;ni usm5esrepuI ie rezultatul,
singuuE" tri,',lil"';'.'lllc-1p'i';X.rj.,tclt\.t.'..riit''ill"crtt1'\tlircccol.ni'llr''cz-jrrc'gi'j-'II?tr*irnia,s',toi1i.i-lat,ir-rr(ziirt.trslr:al'cui.scJIteciritt,altIIat-niiarril-l,ulu''rial.;'cutii'tlairrrrrzr'ltstlrsia:te:i':r;lpl,-c'ltfi1rr'lii*t"ult,'lci-.r-.;rtacc's-ilz:)1.srrItrl.a.lcti.'til,iui,tiiLrsP\rgttibrir,ci''cl'ai,uirrialicrui'ilrz:il'i1lattir'n]jiirirl,.ir.\-tualrr,giriirrtct.r'itli'ti.rir.ir,cc.'rraiint'riit-iiii{i\rltil;-'tl-liu.urni,,trnliitosc1\)Pl*;ilt""i'[erctc1iIa,tcniricrc-sidl.;.-Ila;a.sr:utta1scan,'ir:'t\,nillrilt.c1,_srrgulr,til1eur-irpretiiiii',i1f,irl-iilil,le,trico,.,'rrrr.1it-s']c,.,,rris'ia",a1-il,{,lll,il,tt.r:rtieS'}1lgt.r,nrl.ir],-c,r1loiirtt,c8..,irriXis'rdJr':rircsaii,l.(.,rlc1lai.cl.rre)airIjr'l',rtr,r.'uL1dettX1tbr,s.,it,ti(r.srpliiil,r-t:ie1rrltsials-,t-,rrili,zitrtt.(ii{drttlir,tr:irrlrterI,-.tirirr(,;s:.,i,n,pi:r,t;r;riia}srsi,icira'1t>at,ri.-.I:tr,1zlrL,.llrr,,itlr(rrlrtlfrr).*ti-iilnc.trii6,i.pitilrp*iicniirr,\ls.c'lrol*.l,irr'ra,tllrp-(tr,Ltle*,,-i,i-nip.illru.ulctar.tlr,.'a.g,.'.rrL.irr:u,Linri,s.icanlirntiirI..b.tJJ.iiul:,.pi,'tititra12'c2;trgini,,ratli'.'rnr.itail0ll''esnr,rr(rl,Luliitisrrrt(urr,,.zrli1iiuJilt,,rlrtn,.rl;tii,irl.inriid,ti:rl,iritrri,itii.,;.irniala;rrliii,cl,,i.,tcircrtnlir.,,.,,i.c.li,i,cnI,,sre,,i.ni;,in"t;:.,..,,,air,icli',rt\,s,.,r,u;l11,,,,cati,.rit;1l,,t.l,,rJ:li,ri.r,,h,-,.r,,rt.,r,ifi,,1i,,,s,.f:c;iguri;,1int,,,,ii,,,n,l.,,i:ri,..t-.i;',".urrl,ri.,;i,iir;,-.*,rl,i,,iiirnii,,ntc_cr,;iirif..,i,td-uijiiill;ili-.t,ei
dupi
Irnaginafia lui bogati i-a aritat ci cXe clara acea.stir" r:ul
vur ujungc o singuri roatir si, in noptilc hri rle insomnie,
n;al-eaifnju^crial9tr-:pscucaatrest5os,a5c"t-oc1ncaslctureucli[caslracoiit,titceiln. ;Aigpitnoodti,iatcoiiainitncecdprroiriitticlesailr.ceti;tiireeabn;uigiarscru"-r
sir :-.c lml-rr.rce nnul intr-altul, Fzeiecceafrti;ciiitriengcalruie n,\/ea supra-

{atir lui lateralir inrpS"rtiti in clncl un cilinclru nr.rmer-ol.atc,,
de la 0la I in asa fcl car atrrnci
er;r inyirtit
de I r lt) diu lungimea cc'r.culu i ric1e hazti , i,. ..nl*n" .liti",

Mrilina jrri l'lLscal

#ffi

i+7

tr0*

clescrisc mai multe dispozitive de calculat clintrer care unele msdrmfeeiueaaznpiu;;zdii.annltdlaoiiileeetdusepe-lescg'ucfaueaninuclldelctaduaui arlidintdtd;eme'.irmnoIucamaplasf'aelpiicrnrtuoiriasel,cr",ler'eeie.l.cxes"tsetjciuce,ctll,rru,oiabs"temd'i!a-rioencrieuc'tina;s.crant''ulLntiecirlc;cir,;late,;rAIt:tlgiee,cu2;cei)0.st,i;r,er,i;l,c;,;i"n".nAt"S;ii;[;.i;ll;";n;;-;t
au fost {olosite de citre fabricantii masinilor de calcul ce ppncccPdalfoaa;tosIaecaiect,aourl9uunmueurmaadec;olxo*ae*r_emmYscnup..rrnr-em-suc.nptcpszp-.aiinia*ua-getssctaviacasrErunraCauu-dmru,,rAiamrmriaDt.jSsmad,tnif.iulutmrueta,\nrslm:bilauaai;daapnnaemi;ncractlasn,auitimvrtaresd,a,atslucitcaf;eJataaterpooen1aiazeoJopzoloaucpvlv,tn9aoruaciasnddaccdpniccrpeoaao2n.emleeiurutoetibfcdttml,eno.0ulrcneiidifpalmRurnebetniaeonaraliaolaTicusod?tlmdeaul.nrulfft,hsetroeu-fttmiaulsa.faucemirfdutu-iaiafvirOinneraiecealacillriaclsl-deleaca5deut!eilit[rroljdcdezetpr.cnmmnditm"ise;lian.c,r;ae,-;irretdceeAen-iurieiaea!dl5enl.aigtscpir;l.ttc;tigesln,lalai:uciiipeuiroarnTreeneinaiQclzellnnicunrlaueudceercoc,oesceneaiucanaduuttt,atmlrtelaer.etnrsrmplrrolc.cmcrstNi,mpcpiuucudccevee,rscaitesa,u.tdtucenafnrilarrdtlatorieu.mrlianrdi"rccc,ip"eairafmcscnduTsCopicaudraiciptf.e;Jctiducrr.li;auepi,nra!ei.mitrnixutjSts'oi.rne,ud,djiiaicecp.cti"o,rrlicpcnscreee,;rricai"ltpia.acniileotim"eu.l5.uecrc.ualsiccn'acf;lclzer,batpadarar,late'.au.ucinulreccrz.',nset"llililicmfiececatinr?anuer,t"ccael"o.lrrarueiscitlr{aau[ml!iatrc.al,etnuinsDsaoicnop;cdanfcdsta6f;eentiaav*irtioliiul;etsrnlasncr"ir;crlro".nic;itcolds;tpgdicne;u.at;uei1.icrsiuu;tla,is'a'idba.lzudari.;i0rlrcc.fror,i;r;Nc,riseocr,nsije;ip;eic*-.l-niUxr"t;uuir","ntmpr't";.';po,si-ci,".la;a;trceu*u"in-.oucrp:,ni-rimtilsa;;i"japule"t'enitci"l*et;"dorll;p"rrouc"eui1".i*.eu;?.ealatilo"ot-culre;i..icttlti_i,sulireil"snenta*;.t,a;"_tie,,il
au inundat piata ln secolul nostru, ma;ini numite de obicei
qrilmometrc 149
rdm- aOs cilnatdpr"ceoocnuepracrieaalizn:iltaet,ecrnleastiigcuicr'nciilromr, ac;iiniale
a constmctorilor.,. nu au mai
inginerilor,

ari-tmIo)ma,edtraerloprin;ripcats' iroiuneatjtuninsdeinajaucnesas;tid"pfeazmda,tecomnasttircuieirneaii
puri. N-erm decit sii-!i pomenesc numele marelui matemati-
cian rus P.I.. Cebi;ev
care a construit primul aritmometnt
ce efectua automat inmullirea si lmpirtirea !

de-maCgeinlniklerrlcelgeictprorinnicea?utomat? A;a cum sint exercutate azi

pe-carNeuc,enleulaclthciamr aa.ssian, icclaler ciiberind atenfia calculatorului,
ecaxplcluiclactuomlinsee, a{d"lcnecaoorcllantmiL.uCllaircs:i
lnfelergi mai bine am s5-!i
lntli se scriir in mapind deinmullitul si inmullitorul, apoi
trebuia si se invlrtd de manivelS -- pe rlnd rle atitea ori clt
arata. f iecalc cifri a inmLrlf itorului; se obf ineau astfel intli
pbrmoda'.rcsfeciectpuaatltiuanlen;ui mlaiturrnc:rlcd"lnprmoculullisru:ii, total" Or, dupd ce un

atenfin lui siibe;te
qi poatc gre;i nrimirul dr-: inrrirtituri; atunci inmui-
t;ciearreleia,seldavieudunencnlcruemsziusirtluetalmtdeefadilnse"vlmCrtailt;uisnriiri se evite, asemenea grc-

existi o {erestruici' ln
cxecutat ;i astfel opera-
mctooernurseltdusei;nipoaccaoctnceatcinocuni,utrdolnulacp.oi Vrcceelzasictid.ianIrnscgcrdei,unainoteuznmitjaieatreeatlrcel'lubnui imaCaesgbf,iln;fedivc,
operatorul invlrte cle maniveld ln mod continuu fird sd. mai

numere de cite ori, pinl ce in locul inmultitorului apare

peste tot zero. l.)"n clopotcl inclic-d tcrnt inarea acestor opera!ii
ii tc,todatir alialitia rcztrltatttlrti. Acesta ;i altc modif ic'hri au

contribuit la perfecfionarea rna;inii ;i mai ales la. scurtarea
timpului de oblinere a rerzrrltatului. De exemplu, la unele
aritmometre s-a aplicat un dispozitiv cu claviaturd, ast-
{el ca numerele sI f ie introcluse prin simpla apS.sare a cla-
pelor, apoi apisindu-se pc un buton care corespundc opera-

148

-- Da, uuchiule, cSci aici progresul nn consti numai in taeze es serie de calculc" I)acd am o nia.sinl c:lectricS. de cal-
per{cctionarca accstola. ci in conccplizr cu totul noui, carrr cul obi;nuiti, lnscriu ln masind. calcriiele, fiecare pe r?nd,
;i tmi notez pe a hlrtie, sau intr-un caiet, re.zultafelc par-
cleslreLi'tc masinile clcctricc de calcul , ala-zisele arilnrotttetl,c, {iale obfinute, cu ajutorul cdrola voi crintinua si. cfcciuez
Inereu alte calcule ale cairor rezultati' lc voi no.ta. clin nou
d,e calctLlatoarclc r!ccttottiL,c ! I)intr-un anumit punct cle ve-
derc, :rs pntea pune, tibacul per acelasi plan cu aritmometrul ,
fiirir si grcfcsc, clal'a; {ace o greseal5 grosolan[ in cel[lalt pe hlrtie;i a;a, rlin a.proape in aploapr', arn si gr"rsesc soht-
!ia ploblenei. Desigur ci rna;in:r rlea tle clLlcul u fost dr:
caz. lIi-acluc amintc, rlomnlrle Solonar, de bucuria pe care un ajutor nepreluit cici ea rtri-a excr.Lrtat in citevar ntinntcr
am ar-Lrt-o ln vara aceea cind m-a luat unchiulcu clinsul la
Clmpuhing si c1-r-oastr:;i mi-ali d5.mit abacul si m-ati invd.- calcule la care poate c5. zeci de calculatori, irlosincl abace
{at sir calculez cu cl. Avcam 6 ani ;i d-voastrd mir sileati sau cltiar ma;ini de caicul mirnualc, ar"fi trebuit sir lu-
creze siptHmini intregi. ,\siidar alI f{);tt{r rnotjYcLt si {iu
sd ne lutim la tntrecercr. Atunci nu gindeam asa, dar azi stiu mu**-lfuDNmau,ict;"dti.uNn,uuo-mit,naduialcrnaniu.u.-. iplicoiaotcdafititmnri'liturilrtunnitr,itdrcaug'c.at.t_p-liiic( trc: ,nlrcr,.

binc cir vi plefii.cea{i ca pierde{i, numai ca sir caplt eu pa-
chetul tlc ciocolat;i destinat lnvingdtorului !
-
A, nu-i a;a ;i protestez cu vehemenfS"! Dovada cea pentru c5. e depSgit cie propr:iile lui inv,.n_tii,
mai cind am lnteles *
bunti a fost azi dimineald clt cle lnfrd-
{it c;ti tu cu calculatorul tdu ! Eu nu ;tiu nici cum sd pun neau
Are dreptate dornnui Solonar. Problcnrele fcialrcczoslcvall.ute-
rnina pc el, pasdmite sd-l fac sl-rni rispundi I pind acun-i 15-20 de
*- Cincl ne-a venit m:rsina asta de calcul la Institut si ani in urmi llllte au
cu succes de masinile electrice cle ca.lcnl. Silit. -.i azi dt:stul
am lost repartizat si iuo:ez cu ea, m-am gindit rjc mulie cle nulte probleme cerora ele le slnt clc ur"r r-ral ajulor. I)ar
nri la d-voastri, domnulc Solonar, si de aceea am ;i {inut tehnica moderni cere ceva nai rrurlt. Si luirn, cle excrrplu,

si l'enifi;i sir o r.cr1efi. CLr clt md obi;nuiesc mai mult cu problcma determinirii traiectorit i untri lachctc trlcgh iriate.
ea, irni aparc ca una dintre stincile acelea pe care lc lntli-
afnndam in piidure: poale,susula Lla rezultatele oblinrrte si aibii r,'akrtlri.: prar:ticii, xr-rstca.
neam noi doi cincl nc "". la ardta{i stra- trebuie cfectuate tot asa de rapid, daci nr,r chiar nilii rapiel
si cl-voiistrii. mI tincafi de mlni si-mi tlecit tinlpul in calr,: pr:oiectilul sr:;rflir in zbor irrin spatiu.
plrlia;ul Or, calculele neccs;r::c in ar-:cst scop co'nr,r nrai putiii dc t,loi
turilc suprapnsc: vezi, Nucule, irni spunea!i, ai tn fala ta
urma tnultor niii;i mii cle ani. Timpulfuge, nu-tr poliprinde,
nu-l poli r.edt:a, dar l;i lasi scrnnc. . . Stratulile acestea s-au ani intregi de lucru pentt'u o echipir, r'Ar'(,'ill fclosi ct,lr rn;ri

depus pe rinrl , n-au {ost totdeauna pe stinci aici ! Vd.d in cal- rapidcrtra;iniclectrice clc calcul , lntclcginrl :rrr nurnai ruirsini
culatorul m.eru tot o stinci, formatd prin depunereagindurilor
moaa;-amacrdnefiilccolurra;trgteim;niupmrac-iteaclrnieniirdnsro-iatmiogniiianddeauatlintaci.iiCpmrineiezilee;nti{ceaicdlauanci,ii. mi-ar f i care s[ facfcalcule numerice, ci;i pe acelca lna.i conrplicatt:
daci nu
ti sc pot care rezolvd ecuafii aigebrice, ccuafii rli{eri'trtialc: sarr crrlcrl-

dezr-iilni lntr-o clipit5, cincl c;tiii si le vezi I leazd integrale, diferenfe, cleterminatii si lrsa nrlr.i ricparii,:"
fartii-mi, unchiule, nu-!i
;tiu plac Iati clar un e,xenplu cincl tehnica nrorlt.rnii nu tlai poirte
fisatisficutd de asemenca niijloacer dc: calcril rapicl . L clcule-
sentinrentalis-
uite ci md in- Ie despre care am amintit sint insd. rlcirtlliltc cle o urasinir
mele i,amaafiirarlensatciiancnl elelead;ievviaozid"indceisrccuafisai.r Dar
ardt deosebirea de de caicul universald. clectlonici nuurai in doui ole I $i crcrn
torc ci nu-i nevoie s[ niai vorbc.sc cle ctllculrrtoitlc]r' de la i:cntru]
conccp!ie dintrc act-ste doud ma;ini de calcul, Si presu-
nulnr:'ur cii avcm dt, rezollat o problcmir care cexe u5 ss gfec- spatial cle ia Houston care au sirllravegheat rxrr';r sfaliirle

15/J 15I

Anollo-12 in zborul ci spre T-unI. Af iurmXlit aceasta din executate de om. Dar ca omul sd clea o solutie rationali
,ca-[.tlraaa-cler'i-eib-c"..aoL-eirs'Mnidiiettcsiiitolndsrisi.oenice-po"slbpansiircrertoe:ndbldgoricaremrtciaca,dumzn,sceneaces-ctilrapucacirrne'ecziicnlnzuccidtulaeItclalaurtlieacntmTcctdaceltaeulmletenacvellioaipii-'.ztaaloecirmuicurtaenamlecrtcnrbeel'aoii-tdn?slrReiaddpaurcmtaeacereinsnloidlael'ib.aalllcsninrdcleeucizlvateeucogda,lm.teepamsumot.[li
unei problerrl'e. pr.opuse trebuie sb aibi" la cliipozifiu s. o

anumiti. cantitate de informatii. Aceste informatii sint
analizate.sau prelucrate in aga fcl cd clintre ele se'aleg;i

ctsreoenvbsauidlioeerrisafiitcefi.ieniuncmocamoi pnacuseesliceidninitnif,oor3mmpadtiri;ii'tnicdaprreeinlsecincipttraoulnteiil:cei-pcrleobclaelicreuil

b^re.^steefopcclcae,ledpaoctnrirari.tceeacsietacenictlnse-piloftrmiviatei ;icliia,r s[ schim- , L. un, d,e|>ot'it de infornr,alir, pe care ll vom numirlisj..o:ritiu
ar5.ta farrorabrli'
de tnetnorare ,

Asa-i? unchiulc, habar nu aln avut pinl acum ce" ?'ace Li.a2a. ruitnn,,td,eistipcos.:iil,itt cayc sri c:tcutlc calnt,le lt', a.dici rlrz d,is11o:i-
'---- Eract,
ln ma.l"a ! Llrccleam cir spccialitatetr matale este sf,-mi {aci 3.un. clisfozitiu d.e contandd care sir indice opcratiilc ce
trebuic efe.ctuate unde trebuie ele depozilot"j A..r-
mo-ralidf I si
mai ici apdrar:ea;i tea slnt pdrlile principale" Tot asa de indiipensabile ca gi
vczi?'I'c poftt:sc sii-i de-acttm

lnainte ! m[ acestea mai slnt si altele; le vom putea urmiri mai u;or
fi fac, fie ci pof- pe schema pc carc arn s-o desenez acllm:
*- Ori c1e cltc ori r,:r tevoie, anl s-o
teqti f ie ci nu m[ po{te;ti ! $i si ;tii ci biiatul are 9i acum
dreptatc, a;a cd sput-t", Nucule, mai departe, cuill lgi asumi Fig. 29

*";irru electronici rolul de cfeier, adici ia asupra sa lntleaga
munci intelectuali a. omului ceruti' de rezolvarea unei pro* tt'/r.
bi.eme. inciusiv luu.rr,:lr unci decizii mai favorabile ln 10cul IIrlr. tlrt'd
altc.ia mai putin f aYrirabile? Prin cc proccdcu s-a stabilit ana- p/sP0/tr/ y
DtsP0//il y DtsP0/t7/y
trogia clintltr cleier ;i na.;ind? DE OE
*- fti ales foarte bniunme aciudveintoulaannaalologgieie,pcoloamteuuflier Solo- QE.
nar, circi ln aclevdr vorba' Nrnooucrne MEilofrtE
A DATETOfr COMAilDi

;;mciireecncieeualrneucilllsecumcioturooildmnaeieccnn.tteiifNvsicicotz,dirdrnlciaui rscptepurrtuterectbmetrumarllnaeccliodil.nudinsrinmtrcturdaeiruiozcapremeciedoareur;n-ilnapo-imineletoxlntgigetiiqsclci!i[ l{r f
satr rat iorlrli.
*- Cu conclitia ca sit-mi spui ce lnlelegi printr'-un rtcca- tln 6 llr: h 0/sP0ziny
DE
n'isnr' rational ! sc va numi ralional dacir DtsPzztrl y 0ts?0riil y
- i,i ttn mecanism c0ilrfr0t
getetal , scopul pentru care a fost construit" Accas- OE AMTMEilC
reu;c;te
ii atingi TTI8EfrAfrE
A

nEzaffAmtnfr

tl a.{;n11ic rciultir din ^nalogia cu operatiile rafionale

l5?

Iaii, si nctirnr cu 11r. I tl ispltz,itir,ui de introclucere a npdcPmtao-uirgrtn'nmgisnir.h,z..rcinr,a'm_:Dccrr:nlelseuadiau:itcuaelo.,,sc,pnt2ictrlu0nmouinm0gprp6.rii.l.amur3icuma,e.lrr'udIn:caelirrlsoiennvuprzrianoaacuczsllrmitniinrnaticiolavitairun,ua;l7irlrnt4emsn5icefitae,les'sctrlcnriilctnailotnomatrr'u-iiipaiunItaanscncciptplcJo;tfrirao.ou.ira'rb.g.c,niolerneos"rlr,erpunq,n,lraa"nAuitier,mc.,igs.es3'"i.SrrDrilcdr;Uare;ae{;
r]ittclcr' |robir tl.lci t'csltec1 ivc. -{cestea folrlncazii siocul de
in{crnratii p(l tiilr{r lt: alr: olttul ci: ?;i propune u pl:oblcI.ni,
si pe cirrt cl l,: noteazl in caietul sau lte hirtia sa de calcu-
le, uncle rnai transcric ;i lczultai,::ic parliale datc tle tl:rsina
eler:trici, cind o folose;tc pcntru calcule. tr)e clata ar:castit,
erl nu nrili arc sd execute nici i.rrt calcul. Ilupf, ce inscrit:
datr:1c inif iale, e] trebuie sir indicc care estt' progranlirl tfcc- ppinln0rrcoess0cspce0cr.0.r(s.i'uscudrelacn.dlecoe\rp'rlraerlzee:itrlzocator1ihs'ili'sappirp'eret.r'ro;srtapreirur1c.it'noi--rzmllc;risla.ir,,tl'iincarscclpt,aleocrop.zprrisitoeeiVmbzt,luiiecaelmici'natutm.ercji.?uatmzDltaL.cclnoteaipcoi*c..riuai i"dcpiitcolirirr_",-
tuilii calculelor. Strrrt ma.i multc tiietcicle rie intr:aducert'e in
masini a tlateicr ini{ialc pi a ,1n-,,1rcslgpruarcuia'.tll,or,:dberacrad,liclmrl:agclalrr'--

teld sau bancli perforatii dril'.'1 1111
ticii unde dattjlc sXnt rcprcrzentate sttb fr:rm5. ttrc implrlsrui
ca Ia uit magnetofrln, sau a.trtfcl.Iu;tl'tr rost sti ne pit'i'clcn-rr hd-aelf:eercntFedzaielaa?dmic.sdi,nnauiasr-a'cr ipuultt*etaelcea,pclacaurne monrent t1at, sd sc
tn amdnunte,Ilai ales fiinrlci +le put fi cititc itr olicr' crrtc
despre caJcuiat.jarc'le r:lectrotilce, ll.ciittc ai:csle in{olrnafii le iut.rir"ari se ti"

ajung 1n dispozitivul nr'. 2: $r'rnoria ca.lcltlatorului. Infr.;r- rpnfcddroercai-npnirnoettrrczoI-.lliuosalt.ap-srr$de:eeoaica.nipacfI eraClr.eunliaczntcursutlulaLeeltrlasaeecetcoe't_leo,l'ps-ofgrt.'riri'tdcreil'atiasaa,lpssarvoeetel,nzt,riiiactieaincrxne-irilrasretcist.zt;tgt.ic.ecrl.aetla:pli;ioss,rtr:epl.'if.eloios;v"\liirlhrts,ziiioivaou.b.tbsri,tcilliulonop"idrp'pairtrriAccoa,sa;g"c-sfr;eainc,er*i*,ii

malil ajunsi in ciispozitivul rie rlr-lttt.rrizarc a c;Llculatorului
r':ste relta.rtizata in diverst:le celtilc aJt: nrcniorici, ttli,r.lc' !llrlnc-
\mftc.or:olrsacnntsrrtr,eionilxrSlileii.rrtcAcrCuz-dttiuanitstlcOtclal,,ottt.uioznllianttietevesiit.rrec.lr''roitrl'rmrr.ri.mlc.crn,oGiistrzin...ilal{:rsn.-ipc<Hr.lirer;(r1:,lt,cc{cn,llraieeirrplic{rl',r'z.urinrtl;a1'.rri,1*z.ici1"i,.rc,,".trni,to,i.cd,,i.i,ltir.re..r-io-,iuiartl-t-iri-lu_,,,
rotatc asa, fci r-:;L la ncl'cir sii 1-roa.1it fi folosite. lot aic'i st:
trirnit spre i'lepozitalrr, itl. iintitniit: L:,:ltt]c, t'r:ztlltltir:Lr riltrtr:i-

carc gdsitc cle rnasinS prin r:;t1ciri.
-* Ua,r ce tnseamrt;j s;i St j.r-irnitLr ifllilrltiiilii,ritlr.r tir'1 ,oz'i-
tarc. crr si f ie fi,Lisitrr l:r ltetrtir,': Stitt ert irlltlt illt'l11()rcz
si cum inri vin inform;rliilr: ili. L'tc't *ili Sti st irlit't'ltbi oiiti'

Em in,rsr'rr:

,,D iri c.c noia t t indt';riirl'trtt ngchCrteureuiarn-neirtn,eeer'rll\z1icrltneoirru.c,alts\iae'rnmarn'.isr,1penfr,sr1pisl*1s.dsr1iic1eur,c.lb-.aneloanltdocrlrncreIi:rmcrm,rbl,esohadiihrrliulrnaiiari,nrcrltgecrnii*lirp.Iar.i.'rce!,prrnrt,.ni,Llraittrlr:riirrrrnzu::nc'adtlsretalairu]i,ircocrnrpsu.eirrraorcrrt-i'rsglaiersttrlliraoi"l,z;tmpulpirle;iLdrtrii,onrlr.la.tgu-Atitrritirria_scsatrrcr;rrt.a_s,.cr,il,-,eri,isa.,ijn1inolturrrx,aii,cr"-.r,---i,l,

-\u ri-tsl:trit in ruinl', ..."

Xila.-sirta. ir-rrrti t-leli urr lit,rcr'.:,'irtr' -r: btirlrtir';tt' tiitrtt','li
se prodnce aga, ;i nu a ltfel ,l
trrrhnic;riirnirat cllltilI i]ilre lt frtst itr
* Cu attt mai nrn.It {iirli
si sii-i
starc '-1ep5;eilse'i. sltbcr-,n:1ir:nti,il sttrulg'il s.r1'('*
rlrl
* " l)r:sigur, dornnl-tle So]*r'r;lr,:ir'ia se irililc llrt.!1.i1(rruir. {)lli}1'1. pr:ea l,oiumint.rs?
colyaa.ns*oicc'pclTrnootgsrciremarbnoarrsiirrtif]taeirli'cPsnicrr:so.fetga'ri'ccry-c,:il-crrnullerrrilrpngitr-eii.nr"r\sr,.Ilu:timp*tcarcriltireooic.uriiraltican.ctcellrlsiirrii-,';tntlreatf:.-r;s'aliirtrrti-]tici1iLn.r,c.1',htror.d'iitPoi,c,s'.i--ii
curu a"rn spus in clispozitivelc 11c tnelnolizarc sint foat'tc rra-
riatc;i clepincl de construcrtia r:alcr:iatoruiui. f,-'t'i ce iucrcazd

cu calc'ulatorul electronic trebttie si crtnoascil codtrl rcspt:c-
tiv ca. sd poatir intocnri, cu ajutoi'iil lui, ta.bclul tl.t co'

r54 155

oh.ia"oamil*-sitr-a";itt.s-EPr',"icignxtoU"u;aiaituocoetunietel,dune'"ea1csoFic,tirirasie;ce"siaiiibotmtsinidrie"e,eicnenmcdclloeaeeit,lpleaesc'a!osulnczncl[[d:te,l,oinssdneca-ueapltcpdmeeouedrlsiecmpaeinucpistnceternueoimnpgtomoriec-asteaxeanempilrrseairleopilecraretzadlittauseecm'ltomsa;a'pttaaceie';lmbriennaitni'luipdie?ile,'t, malec-aeas;ritnefizliebpoeurulteurcitt,reiacxeteltcoudbtaiin;ndtceiaiLcttezit?,Llonvrtoricl-uultmnduetliumriapcpleidpceraclacaurlienfid, fcpoleustnc,ganaruet
ca sd..mai pr-ezinte vleun interes practic, cit ;i
sapvinei"m"dgsei.pcp6rloe".itatpi.ucuaratla'iitinppp'trairaroriianmgiittv'meraiancliclndmeiteiaoiocramicmic,Cloprisddrmruiauisnnanerttneeraad-,9ussi aipngspceeucucnsrtciiateuoeuiaSmman"psIicruliOornimaaneprd'aiatcdcartttAuetdmtgeaamocrt'cec5c'oot'.'rtanmntmuoteoemtunrn,ldziutzneirl'ii' deciziilor rapide pe care omul nu le putea clin carlza
lua'! Masina nu
poate inventl ea singurd. nimic, clar imi poatc alegi
datele la care eu nu as avea posibilitatca sd s'aijuf unrg--
niza
nici intr-o via!5 lntreagd.
Cpcieiin-rmef-eac\;rt;!ieiienuzanid,e-naduicrnaivcgoaidmt fauNiclucenicsnutucellvaeva,asedpin.uuatnesitba-aairdseiunascifietiiriiamantuvaitevonicmtuepazratoeetpllet,r-asielias,hionosgtiai.u"srtraltriaree-!
bilerasci solulii oliginale? I)ar si fim lnfele;i, nu in'mag-iniL
"hlaa"titoidi,rr"iit"iai"nc.daeraefsetalusleonpuemraeli;itleorccoedutrleobpueireafeixecic;uatalttee pdderfcialcaule- arn eu accasti. incredere, ci in om !Iat5, mi-ai adus aminte
de un strasnic articol al Acad. prof . Octav
aceleia;i comenzi arati unde se gds-esc numerele asupra yte cale Onicescu: O,mwl.
operalia ,;'i an,l,om.at,ele l-am citit, acum lln
ietoru irebuie executat5. $i locul unde trebuie sl an sau doi, in-
tr-unnl din volumele intitulate'. Prottleute aclual,e de malctna-
fie transmis rezultatul clnd operafia se termind.i acestca fzcr;. autorul ideea construirii dc
iaetx"r"e,atcu".taattr..c".a""lreu".rn,oMlrnasoihpmc;iruaolftri,di iunloengaeicodeiei existi ;i conrenzi sugereazd. f antastici fiincici masin.i
care in^ata.ltd. Ideea nu pale cleloc arc la bazd
prin carc sd se sch'etmnbtreu,
o analogie dintre cele mai plauzibile, calculatorul electronic
transmitcre a comcnzilor' liincl comparat cu un reactor atomic. Asa dupd. cunt calcula-
Atta"c.teeulios"trla"iinsetemlrnmatliemda.vpiaalnri"tai]ujro"ca"os"e-ip,rnictta"arreedsaienasotdatobru.irlde*qivtteacriirananmuterim{iscaiaarlelieizgupele-- toml reac!ioneazi clnd. omul introduce
cfaroreblmemdraim;i esatabluilei girtuc
rezultatul, ln r-eactorul atomic, ln
dcpdrsegte dimensiunile critice, daci un neutron pitrunde
din exteriol se procluce o rezrclie care se neutralizeaz:a.
p"sainr-gcuuDtrsdo.,"c,f,ncuaasiintlcgugiiinadresii-nmiliiastpivuai c;[i malina ta
ca eu sint gindc;tc insd daci mi.rimea reactorului clepi;e;te dimensiunile cri=
la chcre- tice, atunci neutronul intr.oclus ar putea provoca o reac-

mul ei ! {ie in lant, aclic5. ar putea crea ceva nou. Prin analogie deci,
srsipii-u"as"d.NJEsuuiag-fsdgc"lrdiadnedscae5h;g'eteelrnzedirptneeeu,snltceteruuicm, eumenaaicns'cheefia,ueclceueu,,i-dmpaaomuratllnetpc,nrooealguoir'aerIme)aap.;earadt tcesrue,idmbgeuicnaiel-ti unei ma;ini electronice in fazi
crearea, ce vrei sd mi intrebi, dar n-are supracriticd... $tiu;
auto-
Nucule, rost, cd.ci nici
lul ;i cu atlt mai pulin eu, nu as putca sd.-ti spun ce se
deascd, ie "cutimpsreevvailaitvt itooaltmepvraerjiuarnatreelesapuoasilbtail:c ale pro-
lnlelege prin acea fa':ii. sr,t,fracril,ici,t, care nu-i cleloc exclus
blemei, du"poi si nu se infdptuiascS. ".
Si rnai interesanti mi-a pirrut o ir doua problemd. discu-
tn cazul A se va aplica formula X, ln cazul B se va-aplica
tati in acel articol, anume daci inteligen{a umanai, crc-
fucoarrmlmcudulalratiot Vazr.bclToeeruc-alleicaptcriotetnoliect cid,uenaaivclietof]edsal ptcf,aonnliavuilnesg-aeAr spf iionpglrlurotru-'^lt2dr;lerac{L4i-Latznai
ierul omcncsc, nu s-ar pute;r depi;ipe sinc cu ajutorul auto-

tnatelcr?

_tDo 157

-- lJ inc, dar, ctt sau firr[ atttolltate, otr-rul progle seazd care ajute la lnv5larea limbilor, acelea care slnt introduse
ln unele scoli cxpelimentalc, pentru predarea anumitor
llr cf rlu .
Nu, riu-i accst progrcs norrnal, ci dc un altul,' ionbc-ieecptPuea.t,r"c.adsanuin-mgi lvuimned,ssI dcrveocrl bcimi lndecsluppre5a-abmaica;ziaacausmtaaann(.r
cle-o dcp5.lirc vorba de stabilitir, drtpai anum.ite reg'uli, date
dinaintc
dcccruelatmeeoaziA;rmtaniisdinjrlaudi nn\rfsitalerttlarrcrabnsuecilnxitc[auaac.t-lnecinucrevtmnancltt ,'calpnvutcril:t1'a:iint.h1ute9urtn'lrcltaie'iltrr.otetliaictAranec,araaatcotoljt.irrlslOilt1ni/fiicc!Doeausplecp?u:i
frimintdm mintea cu masinile electronice de calcul. \/ino-
va*t- de pldccrca asta e;ti tu, Nucule!
putea sd hibir, r-rnchiule , clc'rl
Se nu-mi gdse;ti mata o
cd s-ar face o bortd-n cer, daci. ar fi altfel!
Cmiobcelrenlelltisciastacrnciuclauiotbreicrvcot spcccrfnetc'l-fi1olnaalreliaintaeulotor:llrra'itir;:lcAr ldtiucpiiL- -- Md lntreb si eu dacd. s-ar putea si nu ai ;i tu
bernetica va. avea ca oLliect perfec!iotl:irea otrlriltti, ii it cuvint? Dc-ar f f altfel nici tu ir-ai fi tu I ultirrrrrl
altor fiinle vii, in spccial a crcicrului ca ot'girn de 3,indirc,
ult-imDuel data aceasta 1ns5. unchiule, nu vreau sd. arn cu
contr:ol si conducei:,', pc baz;r. datclor fttrnizatt' il0 tn;tsinile domnului
cuvlnt, ci Arghezi. Am sd-i amintesc

clectronict'! Solonar, f iindcd mise pare cd. se potriveste numai bine, cXteva
versuri din tr'-o poezie pe care am lnvd!at-o tot la C lmpulung,
Dr-obNlcunrc;,uncolsocmanrutilccoSitorll.odnaarr.alliri ar.rzit 1i cll il(lsl)rf ltcc;lsti odatd cind urcam amindoi pe Runc. D-r'oastri rni-o spuneaf i
foarl.c aclel'irat gi eu o rcpetarn, vd amintili?
cii 1rr:in
ttlnstruirrle urasinilot' clectrr'inictl cart 1;i ilropull sa iniite,
cel pu!in ln partr:, furrc!ir-rnilc cl'cicrujui,. pror:eslll cfcctiv
ele ionstruire a. accstor malini a fdcut ituninl in trtlr:lc pl'il- F4,,Ciirnelaltdct eozgivrimniqdi!es5rlistlndeocsuuidnggsiuin-urdn;uilossncirovdritcsloafnrieeear,.lurmnainsiann.
bleme lcclare pl1"{ atlnr-:i. in r:ceit. irtl pri\:cstc arltiyitatlll'
ccci-r-el bc$riaai tlpiir..gttlt{rta-ina. crin.iln'[ltateoiart.r:ls)ntiel}i1ti'li.1rctclslrstlatlra1st'adt itlclocrllil.ti'iit-tietlltr*'.l1in1cl.iniltt'lat'itliti,':rlci;<l'rl I\c-nclripuit mai taie ca bialul nsi spinarea".

ci, clacir poti, cit mai stai pc aici...
.- IJint'. itlll st"l aduc cartett dr: Ii,L St'rtlitrll'ul -\[att'rirlrtir-"
tlar irtr s-o t1a-u intii lui Nucu, ca Iiind dc's1lt'cir-i iitatt''.-
Crecl c:i la atrlncloi arc sii vti faci nlrtttl plilcci"e ;i iliscutia
iu, clcspuiat dc clLtifea'-rir nliracnlosulu i ; i 1rr:iVit
rlt:sprc' gt'n
pli;i sip'rpll ca. lllltitcit pt'itl lltrc st'l'tttrst:;ls si st'.prc1uct't'z'e
.si sise r.,airtrificc r:antit.lten ncccsa.riL cLr inforitratii ctr 1ri'ivi-
ie ln o lrrobiemi datii !
Ant fost totclcttuna dt pa.trtrelt lrsttr sl lrlri crlttsidcrttt
ci -o intcligcn{d se ilrati a,tit mai ascufitii cu cit ltic ct:
cu
trcbuit- sii aleagir clin nlultinrca crttloltinfclor:1le carc 1t:
poate a\rca intr-o altumitii chcstiunc.l)tr ltcceil crcd cir '\nti.-
iibernetica poate fi o problcmi a viilot'trlrti"
in sprijinul t'lsaisininviclit:ccra*liectriit'-a'ijnugtiui rpct!-'9)1Iat ss1ini ii9le-
.'vct-e, pinii.,it'ot ei vin si
fi in starc

t;)o

iNcnpuruntLe

CALCULULUI DIFEREI{TIAL

sau

Cinil doi se ceartd al lreilea cigtigfl

ici.,"e;"e"c""t"irt"i;"eai"ir:i"i"lfita--o-t-nun*tio*-o"rt-fJfitf"c"l"armiitt'ist[iidtps,um5ccsi-.aai"ia5lnr-niamtenNpetcirauintmion.-ialmti-indvda[tpiimuJemzloresalititi]iimnidln,pd,eit.bLnrteaiaeetAcrtbsltt1l(riea'inaipTtgltrlcueurdpla'oialrnuvdeotdc:iiousisesertcmtacamS'oitettlatonacuoitiitaoatnlhstmntuc6itan-anmntl;r'rtaca,aeiasn.'1t'd;itur;caiceLiritnctrlcputc"tnala{lmnsJtrlnda"-cc'r fsa*c Ne\,vton
tai;tuiD#;;Yuiilc;tf"aj"i:dt;Hl"r;*i;m:ti:cr;irvivf.aiiirntsaeriij'-!uaai.imilcetei.aincrllril;rctisrszaipd,meppiatnordtiaevdt,itnc-c,lcr-imlnfeaci-plilr;titunirelicssappcuriTitnzt'slrmcec(sllrac,ls''ts(nrlicj-..iasal]cr-
-sd. *mtreaAiile,ila,nsutcrl&egziSteicgrioi"s-doaaNtleduamparaosit'eacidbp,auerllcen5doaasttafrousa: ts,,tgiai,i,s.rti,toldicl.tfia.i nspeuornv"le"esi,rtieceaa
i;ncti"d"-i;u;-le"r"e;nTt-l"at:abrdiiu;i;ueis;ol"ar;fBc'.irmman"riuilLt;"snt;ul4oaucz;7n.eaiaer[ltteCcaeiraocm"mhnrsuoiaaulapanglin5berario"uosemsip;;nemaiati"ieaic.itrnaisroitcaFoetldmieracaiucirupstdntllitcemeild'nMnraIaicesioimvcvosarmoimei,al-frg"arooae-beiamrSilzsrrctn"eai'unlsd-ipnfiipien[anstouestrroiiriicpnuNndrnszr.rseiepclpetitd--tiurlNat'isvcaiapfhiteas;alolrid'ewmerleiltim;t!dtsPli,odreiccednntlpfu'e(,auiclaelsumenebr(rltu(lcaeeea)arult'')a*tst(ptrtct'rtpncr'reeliir-ottY-naiiieeet,a-'
ai
in legnturn cu legea gravitatiei universale?
este a$a de fantasticd., cei cloi c:rre se eertau nu erau,-cle fapt,
160 acccoli;urlaeiap':c,ri |mce_euineraiii9scneei uinac-muacaam;riteuiturlsitteiaindcuaeeriinteaatxreuics,edtceleencaaefarftcaielgiidn!tiii'getrdpeu"atuciauacmeca,..sA,rtduri^pircer.erSaurt*ted*-,i
- Mni-bedifi Tendor, ce-ar fi claci n-ai mai face-o mata
cusaapnsafucpnpaleeuaanpme,visa;inltgeeeaar,iuofm,nsuaddleia, gltcaoia,splginnoicdtrdep-rolpenusdpci.uo. avl etlunsnctierceiputcetuuplsuouisf.tpeVesnrcseemsasf_i1"mi"i*ia_*ca;ii

161

1l - A doua carte cu probleme celebre

r :,,* iti':iriirrri ri::ill :a,ll,giiiLlillnl:ll,itl'l;inr.J]ti'niltJn'ilri'rit,lTti-'.m1/;t'.:ii1t-1m.*i csrseaiprcro-cncrril:aaorDucdciient,tridearc.rl.clialirusla,'titarj,r,aicnsclcactc.iivuniciitLat'la.lilrettdlnzeberrcnollee.aeurbrdrrciecnunrr1cs-coiiiaig.tupi.ruirtrlcu'urc:i,rlantcp\aniLu'rrin'ciar'cvr1atcsp'iucr,rr-a-:elir:lrsroirtrlrr-.',rrr.tl.rocafrrie.e;rratc'rraip,crrlc,,rt'i,t,arti.irsna,eei-
>ri tirL-li
r,,, ,' iJ
,

i"', ,I,. , il,t']'',i",".',;",,,,1,'1,,,;,uii,'."."i,'',,,1-,,t",t,,,'"iti:':,,t,']t1i.l)l,lili:,,',,,1i,",',],l,''i,.',.r],i]'rilrli'rtllu;lr sti fcrlosr.sc si err rrn Provcrb: ,,'tiutt bunrj. s. cun{).rstc clt,rli-
nr incati"
,

I' rii.- l: foar-tc :rdclir;rt rlur n-us vr.t,rt sii z;ilroi.iin pr.in1rt, Iucir-
liirr i;rzir. i rr ttlut, j.,t, r:irlri rLi., a
,,'' "t l'Iai hi:r. sd rlr ulir'()p iL'rr rlt'
,,1 Miuri, a intrat crr t'lt'r- la l'rilitl,t rill.gr, trin i'unrb'irlge;
, ::t. r , .'ill, :,;,, ',i',,,1i,i ;;,;;iil'iiil:]'li;]ij;"iii rttrll\-il)(l balti clc 1-a.rt, 1,lrrt,.,r iir rr;rtqr-;-i , lir.ij1.1
.'., - \:cr-sc rllrlnci ina.nuulr,. \i.r clrill rlin llrinrul lrn
:'l'' -:ri .rll..r"i;:liri:ri-iili,i.i\l{{l_l' ,,-r....,.,,,, i-,11

',:,: ;,',,',,,,,,',,-',.-' .i1i,1',l,l ;;,'1.','ii.l ,,i,- rlar rlin t.t,l.
tii:ri'u--t1or, Ncu'toir itr(. l)oLoclll sii fit,,,lt,r,ul Lrrruia. tlrlrtr.t,r.t,i
(eluslrriaciacltiiirn:s1lIrlrrai'nrtr'lbilrntir.tri*:rivti't1.o-i*,r\siiic-;ltsl'ts;isritvat'riiLr:aitnrlririrncis'trniirsjiiila,rrruc-i'ioairfcttl'lsd.i\trc'ir..uaiin.i<cili,rlr,il,rrirtiiitajuislrnitlorf.ixrai,niiilrr)lc.rlr,,:n0tr".aLb,rl.,ni(l.iliea"()
l;; ;;,;',, ;,i',,i,'t,';;i::'i}o rtir"tl*t'i1i,'t*' "ill;.:f*: : n';

ir.,i',1 ' i, 'i' :rr 'r:l 'r it'-r ;i lt'

I,,i:it,'ti'll'i'"i"-.i'i;rlrrt"'i1-11;1r-!it iirlirolrirSlrrlcttt''croaionnnn'Psrirrro.ia'tlt:ig'(n,la1lda1..rtt:z:i{cvfgrioi-('l{'tn(.a.tr)t.(l'tlt).tccrctlrlttlll:i.inali:;itr,aliocnlilcadnalttilittli.tciticri,lrli')pll);acirtlir)(llro.rliiti(it'ialPlItzilrlll..rr:rtirrnii-';LtriisrsIt)ooccirc,,liltJi.r,lrr(rjr,itrtn:.rra}n-i

s$ ,itltttr:i: ltrnliz;r. int'initcrzilntrl.i. JJarrou" rra .lrL ittirtltt cil

:tr)i'stc fr:irnlritr'irj. t,l insur.i Iiinrl Ltnui rlintr-e irc,.,i cc infiip-
Iuiiru lrc.::istir tlrrnsforrrralt lnslrr(, rn(xl('rrli-rirr. (lur:sul jiri

rL.r gtrornclric ro'erLlcrr"nr'lrrlti'strcocnurillrrl'i(s'oirsl)irilrcld;i. l)t' pilrlri, rinri rliirtre
lrr rririrli rrtnrrci cr.a
Problultli:
.rirulccn(s'(rt''anbaiinrtircrlositnctirin,nrgrin.cciorlIni.itr'iatr'nttliiSnaculnuut0i i;liiuircliitlrri-tcli1l itrhtr'irrclirnl;i-lrri.'tasrlrirliurrctnulr;orrrd(s}.i-rrI:)nrurlJ.pbcdiLi.
t:lngcnta la un ccrc. ,\tit;i trirtric mai rnult.
-,- \'r'ci si sltLri ci la crrr:c itot rlucc cu usurintir" tangenta,
iiirrcicd clr cster pcrpcndicularti pt: ritzra la. ixinctrrl c]c crnta.t?
l)esigur, ;rscz ccher,l rtr:ucc'cft1pr'1in ri'ghiul'i rlrept pc cci.c yi
rlacii una rl intrc catetc
ccntnrl lui alunt:i ccillaltil
r-';'itctii.ste t';rngt'r'rti lrr ct'r-c.Illir-tlacil ni sc d;io curbiiua+

t-lL t),

tr1r

t"-si,nirn,ei.onTic*"-izA"t*i-t'i.;ro"tciAiND',;,,r:Ci".at";siatuin,.'a(lp5f*.nlau,i,i[iit,gilt''c,l,tnpr.i,ii,'c,i"li3eittlcr:it.,0ui'tciip).1inlaaru1<imnalnirsnl'i'u'lc.;ieanl;i?LIuE,;g-a;'crstc"elr-rttrn,iet'C'ci":iccitPtap''lculrcccT,td.'dtbt'It(ii.ltle'oicm-Nctgu.ooltta-iiriarl'ru-rircfi'pg-eicunitbaclt:e6niaItn-lgl:cittcic''trrdert?'at:jcrltatbrr[tccrri'rtoartstniPrtactfenlucll'lrgfntacSu'tc,p'(rilsaa'rtstiidtraiicti:l?rtlf^c:ri(ul!Ultiljnturnlai:cot:necst1Ptc--'i c;diaeraesp-uiliabcadtim.liA,ircdinemacsisitdi paupzirl,ioncbiolnecsmriadiueera"sfut"esmdiieettoaadlatrf.ec;la;;eo;c;ch;lei"vs;ac;ol;cpXnetridir".e asa.
pu-nct de vedere
ciin
culul.i difercnt giaelo.mEei,tridca, rcuaipurmobsledmrcavefurrnimdamlaetnatuaeldenatac.alal-

colpq (C) in punct.ui P. Ca sir putt.m dccitle car.c cste iangenta
ar lr dc alu's si sl inr dacd punctrrlrsau T'cstc celcc apa*rtine
tangentei, saLt, ceea ce estc acel.si Iucru, si stabilim .rr"_i
tsseutrabutbaa.itniiicgtieaannctacfsr1atanicucucerbrzudP., iilnel.rPdree:slcc\tolTepeFsracisureruMpaTrti.,n.E'Ot6l.i,n.q;u;maaprterutlb"*l",icnfia._t
.F;-"i<ifci'r"-,;"ir;in-rl-il'ri'ero,--t."rlt'--',,'iui.bi.,i-n;'';,ri..iil",.I'lten,";"'i.tlr\.irl'mt',ni,it";,n;n',r;,;t,:tisacrf';"i"nr"c',tt;i;'];,,"ct'i,ie':t,i."liril'tlneitc,isr,tiit'"rasrptilc,irtt.te-cna,r,ea'tlcr1ct-ouItrb*tancdnii.a"n,'cttinr.cii'larhutric"natexstlrilp.lilta.'bi'"ilcois;lmdtrlu-lct.ca-iitixe'is,errrtnic;i.rue,ct'iogo'.t|'liisr.rttlreniir,rcrmcr;lui'snl"i.;arnitiicairlcinls'iNlrcetlli*euirelnirca'tlrcrrrisid-:mtAlrtoatottrsett.criciil'rir.rtn..priei.tc'ccltilttrutarlr-llttom'itpallasitrtre,rrt_rrtn-|lltjsiaaretiana'ltrl'nri-alii,lcrsticr:rcttt''tn1t*lttiptc'tttair.-'ittctra-i.itto'lx.tlt"sc:rtttpbtDttcl;'tec'rclrltlic.:'slo'clzoildillmmtrriiilteltalailclrtaa'ltr.nielail?{tlliltpa'cctll'ctpcpnerc:r1is"ugulunlt(nettlripittellrbccntct]rltl'"tt]elrr'?or-rl!tiic--rit--rt'' nicioclatS" nimic insi, fpoct.si ternvarteeomla0tiectineinmiicdtoecalat,uanc"i*,.{.irinroclrc-l
cuti aproape dc citrc
el o conrunicase lntr-o scrisoare aclresatd lui Descart"s. in
treacdt l!ispun ci" I)escartes, supdrat de mai lnainte
prea muitd atenlie acestci metode, pe Fer-
irat, nu a dat
nou motiv de cearti! ie"a ce
a plovocat Lln

il( -ittuCnucnic!Elsrc"abinurer moda acestor neintelcgcri dintre savantii
pritcj clc distractieiscr'isorilt. lor .f-
i)ra-tc;r trrnrist't'u rapidilatc la tr,1 ict,i intt.rr.sati. "Irin"cucl
se formau rloud tabcre care rizboiau, iat
picarca piirtilor !
se pinai sc ilbfinea irn-

cstr pilralelit ctr itra x'r. sint tt:trlt.t lLla c1t clctnt'tttaL., ncy({pai;re,i:rix,oirru.fita:r.slqelng-en--tn-mdflu-stsve.i"gtciru5ipid6fsb-3-,{utLlcjEreurttl[o)li-ltaiyrr)biaii)uicd,dmsn.m.tccgis.taciltcfsneeieC-u,peBnn-utoutgmnlaogu.f.rnaau)eftd-pcrn?-.usrcaoidalnpetraeSnmtiziotenucistasble,xetcwensctprrotanucaucrsu;faciaicoipeitnandtcctrirdztleccmQeu..niiuaotnitIdcveclsenfD.aPiccveaedoarrtseulzmuclnuti.daomcadicrlrtreirIibn"ruinin)padlgafeedpeuem,derersDcre,'c:nnocasluIau-gt"aasfccualcrr"eriutucpbesetreplerpsriiot,a,uibaalaseRrrcFdn,n5.uctid,nIel'a*,g,isdesIngisnaiesrnoiFicnmhaaa1inrcQttcaedaicr,at[aiaruioiccRtpmurtrnorplNJoauii;seioafnrariaetiieun,tsrnitrtntrrderl.o.ptrcdevr.cuolr..-telu.ltoidnornaueronnfn*niattmfrclir,i"icetua:tt1.eudi"Btprl6ieucce","toa6inve.n1rlpnxfrr-em3aia,prilRiti5ine,"noinariszetriiwecniadmei")eaeia*.i,:t
-p,xiii vezil Acc,stc 'rtetlit', rr trtrtr:i ittsi"-r siL'*'tt 1lc
.i, - Iilnrdzrtc i' lLrct'ttIi
sc'rLla

Iris-. itO

165

nDt iacatirru,rra_iiiiimencauui asacttiaiplocuart.ct acorlprrironabrveliertormtcaHlcIlcaexco'ltrc'.eiccmar,lcfdu.cili;ic'rlpcoi'aiirittlua.:'nci;ti!.d'.,,'iir;a"'c.i*ic'nrc,ai,_ai
rtonrul,nu.sc pu.t.fl cL.zvolia rlccit Ia putr.l.i intr.,i.ii. \r,rlt'u

r'ril stabili. clcsigur intriitiv, adicL lirrii , r-lt'uroristi"it;.,',.i-
gur"oasfi , folrlula clezrrol tirii bailniioi'murulultricJttlrrc,.ur
1 r11fc.r-. frait io-
na'd ;j, tot pc vrcnlcit lu j, el;l .l,,rr,,rltir."i-i,,
serii dc puterli-a.k' f*nctiilor: ca sin t ;,,,,-,: tg
J'tg trairsccnrlcntL'
x ,:i itltelc.
*- L*'r s-ar slrrilr('. t'i clt'ri'lLu ir;rirrtr.{li'* iti ,'i. rlc'i-
\-('ilzil i

jris. lll t,(srrzlIlii':i-'tltiu;nnltitltr'clt-lrrriittlrb-lria'rr,i.ironltr.;(ol'trt')ritelrir,zoafrtro.,iriit.;irlr'\rr:Ai.i\rl-NiL,r'rIpI)rfrtnai-iIo{.tc1ll\ilI'ir.iuilt1ciroiS{ris;irnr-n,r;{t'i-*,i'iltrglrr'ii.c,tcrl'i'r,.aelajiri\riir-ilrIoIir)isr--\rs.n-iae.irsir1s,cprlrP1rtci.cn.1"\'i(.rrr1lirr,.{eni.sc,\ol)rrs1']rai(rl,.rrr1,cl'r.lr.si1nltlrtl,ir-cnlcr'tllrtarairurlirt1rrtll.lr.l{ir:itirc,lrursr(:lii1i'rrrlri,'-..tar6tlI,iLllt'-.6ri:rtr"'urtu;,5.u]-r.,i'.rig,(s.llur;aia'c)rrrlui,iritii(r,srl,tr,f'r1trrc6liyr,ni1;'ltlrsr.cutrel1lrI,r1t,lii1,trlt]aliifl1,titri:i"r,r;,ruttrLcil,i,.tlirru,iri]*ptr,.t.\i6opt,.l.i,illcu,rirrtt-ritrli,\rr';tus.ir',\,l;si,r,r-rLL1,rr.rir,j,.r,.rtrstur'ul,t"tir1l.ritr'"t-i*ioiic,,c.-arlirlurtln.n,uirl,i;.ilr-r-,...-_-i'

ilil1rr {'1"i1(iniiti'it :,i eibscisclc' flirctr'1o1'lr-ii O. A':cst iri-
{'tl triunghiui ltrltl i', ric
unsii.i i.::rtc li:lci:iL:i1r:il uliilt l, zrr ila:

l,.r'i= Q-l:- 9 'I'il ... {(r) ' ;

i I'i i'tl a

ittr.r.n.''itLti l' sl, aiia ulor cl i' r.,cuaiia cnr5':i clrtt' r' -.: ii::] " rttrionrnilrlpcsluij ltapo.ocaitrtsrciirrs'li<tii-nrsodlliirl:rc;iihs1a-aa(lra.rigriniarrczlruiliiilrsltg.lcIr).tir'irznrhtrniu,rtclirlNai coiurtr',Piloiirrnrleisrctnrciat,latrolcctoncrritua,-,i
Ltcri.lrr:.*axild;ir(.n,c*;ritilrrltt'Lcii,-r'r'itarir.t;tiltlt:'ltLar',tsc.'rlsettlutt(,tlt.lr*l,-i-\i''i:lottir'-r:-ir.?tiitl-tliirr:ri1rlriL:'(1tt,]nrciuttrti;:1t1'lllt'tti;t:ti;]i,lit::rtIi!'r"llt''.i:f'r.r(i1ilxc()'li)l]ilg.li'vi'ilo'di'ait-iillrtsriitiL''olailtliclaru'5rt'-'i irrtlblc'lrir'lc rnlLri ct:i :t, r''ntin'rrsr,r-:j, rrst.rrltirrrl crir.sur.ilt, lui
Ilru'r'oti' slrrr citiirrl rrlit rilrcrt'lt, rnlrlil.r. uliltr'iriirticit,rr j gr.cci
.[tr tttl,. l ritr i:l;ll:'i,'-, ,-,]'liir' u t'cttlrli' itr tl.; ;1 ; riui l)r'intr.r, ;rt.r'stclr t,r-lrri..()Ptit.lr., lui Iir,plt,r.,
.i tuo<L,rni. ,,.\rith_
-"- Stii.iirpilr*rirootuitii-ri*,sttittt;iral"csre|ii:"n;'rsieo'rNugtstieto,tijuge,m:laii,itjfzalairinSirirntc'smndapCueti:llpni.iroituaurt'troiirlr,eCa:orrttugiecilnioeu-pnlrvAiusi1,rdutr1certpene1rce1iirolr7aeiob"cia1irrlx-trei?irciigl'tti':ttesnLrt(1tena1iltaiaplttc'^c.li^Ltrs,rm.(i1aiti:iis.ii!LiL.1irri,-ipi:ir'.i1errt-l;inla1et--l
nNtc;riluuil"f;mc;c;iri;c*r;niitp',i'tniolni,atc;na;lt,ce't;ei"liaes-,i.t;Pa;c'ca;;.i;za"sQiti"-i;erus"ieu;in";ncci;og"g;nlhsiidjiuiaacerrl-rie.lr.cariniaicrfstiooetednarmsr,ttii(edp'ntcrarrliatpci',starreocretopss-piritaneeIttorcaiaitl'ulrtDCss(eeianrm'1larctclual'te'filzacuii-"!ll lr:li)rtl)it{rl(g''l-.nrtetrtitrrcnr-i,itrsl,r.ll,;}iri"i.nll.iz(rlilr,ulutrr'ittoitioofn\orcnr-rrartirn'rtl,r,lrtiirorli""plnLollrrlsztriilinrturi.oltil)'rntu,\sts\u-t;t;ru:rluiulrtrlitniscsr(st-a,lgriPalbibrtirilir)fli.tl,.,t'sctril_ljr)r('rl.,.lrn.I1rIu)liiclntc.t[rran.t-ru]rrlrna(r,tc_ir\riJrr)i(ilr.lr--ir
i)c act't,;t s('s])Llll. lrccstt'i for'tui.
\ crltorr": ,,lri,,'trr.l Irii

l'tr--.' i lt'1, rl., c. t.lrtt,ci lrr.rrtru i.lil.t-ilc ilttr.i.q.i .lr,

('\l)(lll{'rtulu for-irrula ltirr0nrrrlui tttsc'sc cun()scutil crL r.rrr}l

167

166

lfnfia;iinl;t;e^. ;N;ue-wiitnon"faiop"licnarefo,rmaudlaicicilnaclf-oexrmpounleentduel avea valoli * Iaci are ! Pentru Ner.vton orice curbi, era imaginea iclea-
lizatd. a unui plriias, adicd se forma prin mi;carei continuf,
fclul a purrctclor salc I.cna,,gTirarrtnatpuliraiisausp,rpaiicnvacdurrdgbturcriai
stropilol clc
-*!:[u.- (l * b)-n sau 1a-+- b)f : /("+$ia api. continui. a
curbelor.'
b)" spune el lnsngicam asa: ,,Eu nu consider- miirimile cr
oriclt de mici ar {i ele, ci
iffi;ii;;;" '"ia;ti*.*.;b,;c,r.i;";";-i",un;"l"f"o;ir;;";.-ff}rit;l;i.aoji"iir-arbii';ri;i;cii;e"."*"i;sli;"^fe;.'ri.f;i]ri,]r";ris.iard;i,;.;"i;ft"tt-';ri";u']e;*.ilill.*"";.'u,"t;.hi,irt-ci;;ilinaNr';"Ia;lagltp'mbi'actrle"iz""oel'inrtrr;fu"atic,aur'roi,"l-.5l-r1e'iv,nliicti.'rirt,a,';'lperl.o,iN"r;gcitincenlcin.,iclitcc.lias,nu'e'i,-ip"trlicos'Itn-.fvm[c'eoriciiitcu'-t-orntioLtproluicai'i'i-aaiaciclnanlrtticiesurtacd".ri1lrilanniertolrt,e.fi;(iNrhircttiii'iiicoutul",dsrccrurecarlinaattrpljncnrvtvt:;stuctcctiortriiict'ierdiociognoplrfiPfesesnbldipn,daricitsfllnclorllmicogeft'afoprstmsoniup'criidicrcrlrIrlcoaa-tn'ruietntfr-.irsctarlli-r-tciglcc'aisepogc;eD,e'irecrtittrdsiorstta'.cuibrcencattcdirpuncerprtinircvAtclpoit;tuci'etl.tca'ulact'irmazcii'ocltttrilne.ec,eaelrtprliu:lztgutt'r'!tctcanIt(dft'mituiiioaclclrintterarnolicclirdNcoati'silxselnaco.taiucottcicdl'trta<im'oscdrrtarileiaonplimviatopciiracr'ticrttaczplril.e.nacoelrbaae-avietlsrncmnllnt,cntor,alslelt1lntcsdtilrPaao.iltcritcuoea.aurclnd-T(ldct'sl'trl*-ac-ri-'c(-ttt'-l--r-r'l' ce ca formate din pirli matemati-
ca clescrise
printr-o mi;care continuS. Considerlnd deci cd. md.rimile carc
cresc ln timpuri egale sint mai mari sau mai mici, clupd curn
cl:esc clr o vitezimaimare sau mairnicS, am ciutat o iretodi
care sd stabileasci m5.rimile dup5. viteza cre;terilor care le
formeazl ;i nurnind Jluxitut,t: r'itezele acestor cre5tcri, ln
timp ce rnSrimile ndscute iau nurncle clc fl ttc tt lc, arl stabilit
me;:tis:et*c-eet*soatcecBAllevaat;lra.lieudfvdleauuaa,ntnxartfi,auun-ndnuiacimaiclroicsaedrtie.e"seyypnriaeua:ac.nnilaaruiil(t?fmdx,m)iftltiuccxgsliieetuenini*trfa-u:.nfiIinic)hsatdiiraicd;ficu. il;rfatclitt{uriiea,n,calaine..ruyvI.)jaituaevnbeziylaatar
cnrn obscrvi Oscar Bcclier ln,,Fundamentele ini temat ici i,,, be
care o cuno;ti, termenii aratd. influenf a liter.aturii meclicr-aic:
)lf5ticon'f0lrtt=nrndc-ivrteiafdae(ua_"pasnt)lrucies,oleEncili.osuivz.mlpocaTrrrae.rctcmecreu,b.eaammuailiaierct{trreinecs.fucikdsertexci;lutieslin,nieslcl-mecfeio5lnie.y1bignt0fnei0tecairzamdea,te'izesa?iinannapunt1nrlic6iaxnrpglp2lelin,rtfeiufcicuslniacinicirincplaciteccdilscx.ritsplc'ieafr;,eairi--fJsaroieinr_seno-ti
i;;;oa-aiA;;iltsia;;#p'.am-;;:t;d;";rlci.imntCl;lc'co;l;ui5o.icn;n,"'dp;;o{t'riu;or?aui.ri"o,lrt,bttit.stit.onc:iiiit-ilacl.ttaicc"r"iitint'cu"tthl.'dilu-t"i.gaairt.i"icice:.ttl:"ncttt,rAd'*s,cait'itciti,iii.tc"{rinclsoni'rqccllicaodisiptiritr'pctlsttcqaatuiiitatnnmilngsttscecierralnrauaicAaCet-nnod'lrmdIc{deune-cgid.ulotI',trlop'nacitadasranslvcmtoa'licr.tr-tlt'lti,cbit)ri'nac-fghillotttLatilc(rtnlullfcr'ac(u.bt\lSrttl-rlc,tr<cttCnr-abnlfinci,lu.rrI,lKunit"gc.ptc(ltr,1tln.pcalsin:-'r.li1.*bc.njiaf:-rult(til,cac'll,r^(lre.lnlt..iitnrl.re((:rrlt'' spus toate acestea ca sir vczi cum se facc matcmltica si sd
nu crezi cd. un termen, asa de banal azi, ca ccl de funilic.
ndicteszpirmc'acli'rrscauintacntof .iioirltrlcirzt*si i;flvttoxribt'tim'ni,loarc' ccotlinl lc.act{lottstt,lttituttrtnt ittncltl-: rr circrrlat cu accca;i clezinvolturd;i acrrm vrco lS0 cle ani,
nsicili_csdp. uvnacrmia.b\9ileryletodgc'lpne{enldccgnetac priuthtcnte ,prdrirni nriscule,,,

;i independente. Dor,;rclir, l'
cxernplul considelat dc el ca sd stabil-easci derivata functi-
eixn"Itl spunea: ,,fie cantitatea care curge uniform x, sii. st:
r{iiseasci f luxiunea cantitflti i x'i" " Meritul lui Ncr,vton ,
luiLeibniz, nu std in ld.rnurirea acestor notiuni, nici ca 5i al
micar
n aceleia de continuitate, pe care o folosea a;a de des, ci cd
"Nfeeltewgto[1cnc". "a, teclacrulrngelarteintc.rsittt:c, i.lfclttrtclruel-ininsfeinaitmezni[maatlr?tl'g{t' Nu in- au introclns calcu,lul, ur mti.yim,ila inJinit
dc lrecere Ia li.mild, cincl aceste mdr:imi de rnici ;i t'rocesul

infinit cle mici, cu

168 169

care s-au f[cut operaliile algcbrice pini la un rnomeut clat' fi"$i.'ffi ffitrri*lf,lr

d-nsu#ni*;:;-anerutn:;J;uri;v-cFt.;u;-i_;m}i-ea-v"iaem;*ldn;e;c;iIa"cCc.iicniecrcS.li?;oistai;tns;ar,ltcin;au;c;ttitadm";recr's;lailedcr,iic,b;Jd1ze,sngJaobe,i6i;aul;ae.tp"irlll6""ans'aodlt,xiLg..c'i5i,trrpri.i'ti'1lr,c,.,cr;,llc;reratloo,odtii'r"ui.mrnIl,rixedi,iint,titul,a?zgrit,ttcs'tl";".-*e''pll,i;"i"cctu";-':ir";',it''ue""".ooi'u,tJ1i.'it5j'ptit;f,lpf"tii'i"uii;,t't'iiud-i.toiiri"i,et;iieri1l.t,lrpii"si,-Nttr'til,aitcNli"n't,;iiais'e:tit't"rtcietit.''twlt'trttt1.crttrrlcrtini.eti'uil'cn1trrtlteoltr'Iolp't'ilrnipi.talic\rr'tnau.-t|trmjrclci'r'mrl'ti"l'iitattttrrt":tnraaerletatlti.'t.iiocta<rllrti:iaafci'Ltlraatt1ctllcicm'oa3cl1edlnrcIcdlttl'uileoa'tctrtuteearctrdstaittltlcsorr'i1nurItiicetnlrrr;r""ai1lllra'"ionlta.I6ir'ilibaaIrc1tcinnts'itt'eiircrnla'7fllrr./og-tnrt'l"nos"i'ottnsnrnrt''rltr;clarrlor'alii'iianlrrr'tirirlrtrlrtqi,aia;iut'i'ra;iciz;'rtrrlmnt'z'a(trililru'rt'm'prlaglita.titcitntElti1npsi'tieraer;icoict-t1l-''ii'l';jl-ir .',t,. .g ,
Cat"cba;"r*"irla;t],u--*f_ji.;iit';inc";--?;iH"oiuil.P""jota'tJ"lcr'sra..u,t'ia.bPt-tiltl'a;'trIt,t'llr,{ptll;l.t(a..i;i"ll,rtc(rltP'iliirs-ac'",[liolrAll,il'rll'icrtu,',t,'r1,ts,il]rr,rat'\;tt'l'rt,r',t'';lrir.'ul;lt't'lcl;"ltltrtltill""rutt';t-iltt.'t'tt"l"itelttl''t(l,-rl{ll"tlt.il1r'.rt)rtr.lrrer'i't"1in''it-t'"s'rlilcc''l"i-"itt.c1tttt1.trttrl'rlt1'jttL'ltlrrl''l"t;tiit1ic"-t'1tt'it'1lrIr.:t.l1i'rl|pt.iilrtii".tstlt'rl.;r,t'l'll.ritp'tlr"t'1ttntrutuPl..altbit'"tii.titi;'irt.tcitl:trt{t'.liiit':lltl)*'tscldii.t.rri.i"-.'r'l-l.t;llt:ir,.",.tlt;nilictt.rl'lli.'1riir.;*i;'!':'1i-illlrl:ttrl'rl]r,r.t:.]]tl".'.itl']lir,,r,'[iitrr,ti''rtrar"'.-..t'.t-'
"cPc4.;--u;;i2eu-r;;Nr6_ntlrsir;-e-"uuetiaii;'uarrccinirlrlaclutrn^nr"?"etrsirt"cOsidurpruprumtsue{st1l,tinic,tcctla'ai'uar,rctl,irmlniruaotrcBr''isl'cu"iailuG'ei1rPdb'';rotroiaZositiu"tt,it"rNtt'rcb,1l'tea'{.trct-rir-to1atvilfJcrtti{lg'toolrtirtnttria'1:a''a'ts-lpitautii|ta'nnt.ilrl:i''urtrucu'sirefclctitinittiilrmot<t:ln1t1l'ertor,6ubi-lii6errnalrr-n9irtnc-rgri:|isrt1cat.urcr6-iactebl.6clr.ei9i.uLtt..i'rul-lit'mJlllnuuIral)ei\l.vl,ulrra"Per'p.izc-'tgr*al-.el-''- ;.1::l .
il #:rii:i.iii..
170
lt *tri;,#m lS##

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r
li:{]::lt ti-

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;: .:.'**;t'ri:' ,:' , ,,':

'l ': l - in.i r lirr ,rl,r.r.:, l: ji li le. h{J1!-xl}is$i ::il+:i.
lui -\el'ton,:rr,:'.li'::r:i:l'

rlciciraiulo'r.r*l..liiirl('rrr:)re-m-npi'torrititnrti..tcriiic.iIrauc,lrlns.raiosstti'rta:spc.riliir:erccsirtnieiziisccn1ri:rtiiauirrirlmirua7-iral'naidcic4.aLete.i:s,cit'ndt,iNrzrciilaniol6ceumt.aeraci?ien6mni{nt.ael.it1irtnu,ii.zc.,qdi\1,c-,6i,.pniri7i_r,0tclaifi,naoeian.rlrp,.,s1o.ip,r.a",-.ur.uipcs;i;'lrrr..ie.,"nl;uiL:,l,;orr;pii"ttt\;:.l;i;,ceir;:;u:t;r;eJ,rlr;sn.iiiunl;u.*r-:i
otad{s],ac[:ornneogliciiSrunllernitirt.rr,,.tJu,-lanilcalpiar:rniiolfl:(ioetiti,_lciir-c-tcNleos\d.)aaelrrd,a[rcu{dr.('.e)uupu.,rrfata.'exurtuimtncocsplo.slaetconuncictair,ser.usc;l,ia,filppca,oapcbclra_irrarereniducernecdsdlaeaattaliaocoillc!dtLezlilmarugciu[slit,is,euit,lmcrcren.snallInaceereiisiipirslnmnifacirriczadriiourheooio,ct:,b.naimznc;amplrpieooeelr;,u[ll.frmiru,o)eirn,ccrtens:rd,eeixr.1uca;rat?,rrlrr,t-runrt.(trrra,s..arlms.a"nIroidioor](ra.lc.ildn.trcin;"atiru;rncrile,tr;l"r,earnhil"uvr-;roum.o.i-ai,s"ce*,r",;ilabaiiiniii.mcs'ez"irtt'gl;la;tic*;aactAu;ss;ur.igicr;clrai".io':o;iicni;pi..i"ipiez.rai".d.s'e,ai"ot;;cr;t"srrfu.ei;a;t.ilra-l_iiii,-

17L

;a*o."",.nic'"*"-or;.b-6iir"llt*.*s;,t"i;t"ir""suisi-aaivpttu6lpcsrSlrisit.foreoianicilluleiulealu.it"*tleJItsi"lua-ie'ap.tsRuucigtoernet"pgriii.,bal,-ucrliectoaurstasdrtler[inlo-nnf;lnecvioLe!xpdmoi.ueenIurbn.dleclriauJ1aco6c'vch7can2drrrce.alfsr.'lfc2ctotdmoasr.ptsprr.etrsrerlu.licuerrcinugldedi.lt,l' Iacclfrasnerdnei.i,t_1ail6rLs8u4dp*i*os1rp6ta8'zt7i.cfihCeiuacrtli.poiSrproaircreiteeatacltuleiniasc_Rhaeeagltnuag'lidaejia"i .t.,tl;"nosnu.fL;;i.s*Hi;,ai";l_l:crl",
cclouuc'-V-r*.tiarrAPrutfetlcoutss:enftu_mPcrironaoeic_rblrepLtcr,i,,c'rmve<c'aulscnt-ueNgi rican'aertvrvirnnteoiittaspaeeg.iritnvcci,eeirannasfniuitillpt,Fl;ce,Ioia*-qnlij1lie,1-ii.oip.v.c3;"a1,mrriiiI-tpaiaricti"efr,,rll,re',l"r"trfr'mrii,_lm,tuiir,ltlids?l",i
atunci pe"ce baz[ s-a {icut accasti numire?
P'(sicL)'sllu'lllooielutlorti,trr'.ia^(nuctetl:i,l'-luDdrracuxl,.cn,_roeliu,ca.rljlturmirr:i,rcccg1lu(rului.u"itlitt;'t*.len.Ilir^.K',ls.d1:ari\ttcicer'sirt1;nt'raaSrlt'r^lruvRlklto)trtLl-t.eroc.rrrcarlr.rrnbgri.lirc.lrecc_rS;fctrriAirat.tloriitu',ur,'a.tlatHcziil'iartrid(rato.i,r,trrcitcloc.ir,ctrsccrak.rig;tr.Iaeitliaiafrr,(roto,{l'nlb.,r9i(,ruc..lP.'iratrlfl(tzni1ri,1m'triiNuf,gcrt<.btrlbdr,,,t.ra..,,deiritg.r"rt,virrii".cignri,5ttin'rlt,,rrcgiiar,n.iirirra,"feiissrtlt.rrr,iru,t,i'i,:5i,pm't.ii.tri1a,t,,rori.rr'.e1orfrtriri-(*rrm1cp.l',Ii;isr.l,6";ors"rbr;*l6t<r.r';iiitrr.:;,6,;riip,ti;-it*i_;o..r.uc;ir;r"rn;,s,:'teitir;c'ta''1,;fiii;.a.r;,r,liru;rrircn.nr;r".;"i,.rsri;aii;,s,r;.";i;ii-til
;i #;#i,t;;-aii;iTo;eeHtott;.iinm;-r"atrr;;;l-".t."r!;'laxe'ti.i;li;lu;PttptiuiFiit";"telcnir;fi"t.c,oa"briiatae"lnLiraur;nzasuz*eiuruta_tcritnle-al"ruteeutdaaieirorittlarilsie"nrecesl."iGloetaasep"nscLcT"tlrcasoitoen*,le;._opdrxindis;atlarnupmep;idif-olep"tuetulici.iar;sunripill:ianai;ci,o"brl;ptrunteadtiiloenilaliSzaneactrericetioltan!.tdn,oaxilctt'fecrrecceeiii.limiadtnrsc.taSesDltarp-eilLiioatlufelieic'oan.olimshc,ocrsidndiingooetiiilpdaurrnutndtr5tedrtruse'ieatdsigeictccircr.ipca;mmlhuiibainurtAeuieiuratcimrtas',ciPuc[gfletaaV-enlNm'tair;cateceiil1cr-oecssi"ctiwasttamrptiotNdeoti"atr'mscetlobaoetcrois.htpueunrric'dutsosu'xi.cirt,a5iorsniadeisilanteLe;ae-o-oi"il
ltN(s.sjr.:1a'.:cau.c:'rrllrtlr'.litcc.s,s;lir)Tou'r:ta.trlo,l.j-enu-rtlnr'llnn.ie,ar.pcrs\ttri,'-lsy.uc\ler'cc;asr-ltriir'e.grl'ucie,srr,acl'bmrr!titsr,..rSu.ltarillna'liarszrisirn,.rbiriP{d--froecomiierrnitiaor,rrovm^rubi;cc;nlifirire.torr-i,flrtl-lrrcitfili'acair'liicvia"ecgioroiitcrcrr*n'r.nuit.Ptirrr,i,ntur.rrirccie.ir"atiebrrlcateliraicipti.in'iliticcLtit:ilnar,ari-inuv..carcnpcztirlori,urluer*:niplrLtscrriPu'p,citrtrlritroiouiurrtoccr"nrrl,.prtlic,cbct,ol,riur,lrHrioi;ztli.citri'o,li;.irl,s*Ln"e;cci,i;ainlli;lli;,rlan,_ii-
.iHi^s"";lr{;oii;toi;r;"i;a;i*;oJln;-"ilik*;.o.ti.el";i"Xt;i,'tip-c't"Dc'ril*c;iue"lti;aii,zcmit:.'tttrrir'lmannf,;,i,icitviplatsniliscicd=d,eclc'"slirecatiacpiti[.urv5'ocrpo;r_,"seieimrrtoine1nsoutduui4rtldeuscvpit!avoniiaeepninnnlbc1idmrai,aur6ctasuci7aueeec6inrsetle,(as-cpa'aaaroivui,duclulb's.ircualleelp{iidculsu-odaeapavrnestegutiandccrcstoctuiinomodaplistrninerit'nacioriusill.rrr.aas,iite;lrc'u.1i'ocrits(cl}esaol-dTsd[rtbetali,ennnlunIllereeluu--t-' r'li-.ll'rrurr,is't,'lcri:ti,rra-r:al-tri3nte'tNs'altf,tlp'e;tr'lujtpclc.ioe.,i-s,r^dliIr<ct.l-ccctze;5sacr;ccidrriapet.iatraeraulaeatnrriiscttdatffflc.ocpritialacsccmcvctctuf-rculcaoaruinhbsocstde.tcaniciossseHtltn)-tiiieia'narlvaucczclrehnllziaeIciruacrytsacwealc.tcerariP.re$mc"*trrrsamioSi,ptnie,ictpcroieiai.rocfps,Jzi1o"reodor.llprn!tn,ri!dss6nou:dbtiic.nirl.lpiicor?a.i1oDliui1c.tc.1"laer.crizis.r.l.ae,c,,;,irrtlric1aea"olna,.iirrc,l'itre,rcf.,,r8nt:oLgt""iLrrubiiaiirltlricriosl,-it

ca si le Poatd aP6ta". 1.73

;n;i;S;;;Hii;;m-t*.;lp;"i;i;.;ctf"i$i;;l-t,i.*id;;r;.o"li;ii-nt;iT-aiutuc"tdi,ot;liiueapfn.-iriinpcuoifd.iohutlniepcasld"u1"a"ei1tuili-u-tocaipial.u"nreidvscia,lsaManospictnllet?e-ervilcon,inna5n;csuni.oiu.nimcsra^9siitrunr-ipc6fl-ti'.8e;_eEci7irodnce,imtnanaenrvtsurueovicnnc_iIad's-mebanmHceaipta[zaarasel-inlpctedpip'rcayueipei'tbctpciEicninlcrunttIiapnrr*uedtal-

172

]"clii;Hp1ivfitltalriat;lno[6eecuct;iacSiallC"m,arbiIt]arln8;r;.tlruli.r;;tiuigt5:i1(.r;.mNft".1,i'libaai.cili;ae;i.ibi1]cfirc.)iir{rdirr"i1-vo=r'ii'iaiioi'1ltdtinrotYt.on;rs,cila$,i,ili'tcet,itri,.iriutiegttlil(iii..1ri)l,li-:c'c,llsrge'lr6,:'zr,ivain,'-rcalo;-a,treoir,t.tp't5i,.sctl.ial!v,iniiJl:ctliNiWl.:r[ctllrir{i|:lrr'rdo'o,er(Itic"srli'rcltlazll.lS"ii:,)t)l&lclri.ut{n<ft(stucoritrlac)ilu'innuru.tl1luieut'lnaotls"ro:fi'{pcuaatneta;r.ucsirdrriulra.!ilmltciteiuliian(e'prrt-lo'crmnr{csldrtgiicah(Pnrala-trtullai'ial-tuinsarltiaillrt.tittric\rriirit.r1clcnt1lLricrpuiul((u\t''ct-ils,\i'slinulz-tti'iic:ltrltNi'tiul{i'rrstpt,]):cs:fc:i)caial'lttlcctl'lr'iialiut'lirtrc't]ia.btcrt:t'flutunotri"tlul'1il,ltltiiisa:sl\ttltlarut'llaiulitsi'.i,ue\ljrs-ii!t',strteelrtlc.uri'ttpt{tacrlstszcutcnuitc;lo'talo(rtla'lttrit[LtsuIstl"r'l',l:'ttIitil;!a1c'Ltta','tu(Li,rrf'tt'rtriLt,r\clptluu''ltpu.l-t'lltlrcrsrisiitc:ucn]oat---iiitlerlaiot-,sllt-cl"''l-t tcc:ptcnCcliopi'mcccr.i-li:li1Sipniorszas_irc*srlfc;,dle*sJ:ia'iro'd,roertrouidri,l.nuiicloiii),ibUii't),."'ilrlr{'aaL.ol,.rritr-rs-sm,uecLc,i'i'rt,r{i.r','nrf'l-;c,i_.;n.i.l,Jic:.:,:iri,jr-re1tiic,ir;.'e1ilm}ta',:':i[ri'trrrriils.:.cr.:rcgn:-l}u:r,ati,i5iati:orcrrl:,1,u1rttcii,e,t\ij,diln.a"i'ni.ii.i'el,isr'dil\i.rutrr'irdlljcprjllLllc,m.areruiLietaurr,.narcil'mlce)cretirlr"'ttl.:cilIttilarrcli*.tieil..acir(dii;,aijc:nanmeiCr,o{iei5:piil'tl1urri:oj*r,r'lc:cilriiiitt..n,,utle-gm*rlrnrciictr..aral1;cti(salIs.;,tririnpaiioIrttiocemx,tl1ii'rdi1rira.pai.,t._aii"riirIcac-'ntp/obir,t.;.r\c-i.llarci,;l,*i]lific.riethi,rasetUlriri3ta-iililtatrdle:r'inIir,o-iriii,lrsire,li.r.rJ-.nnedir-rn..rishtl.;iminni1a]iev,1t,t1dctics,prldita,r,djnalio_i..liur;is,ttn,itrclciciceci.ctt-trrZrlt,tu.rltjru,;ctccnri\itie,alirpftero{-:iIprUftiirll?r;seicrJiic.,il,s._lct.:lie.irrlelrlrafbslrcc::rnzi:irvltrurri.i,ivtzdln.,rrip1iS;.Ciriirlurtrrn'i.!aoiriCl!ilarrteraii,rl,.r:d'rl,.-rtcnLis,m;nrrcmrrtr"alnJUrrnirnar-lllp*frate)Lcio-,r.p:ruieau!aam.ipi;-rtrrrnlaets:iiutr,',l-ir{rsLrnieorriiLac-i.ui.:itrlreannia.ijprtilrn.rlcirir.'rrcp,i-i.cceci)eii,-PllrclvCroe.l:,jr.el6drrl'lt;iirpe.oSd'ttlec1Lectnrip,niltlii,i1'lih.l-eno.cN,tr)rcrl:ilairiciueup-,,ric,sni",irt,,ri._idilrxni:rrc;raitrq.jn;.,till;i.i1.',urri,;ic?,icci,pc"''r,rcp,':;ttir:,.nrlatuari.d,.U.jrcbr.,cilctiirriiJri,1,]i,r'._.eii,,,a'.sicnnets-r.l,i{rurt,,i,rntcaAr..-ifmilr1.ri,,ria;,',iarcjt,it,l.ra,tt,ri.nrzt.iii1narblr,mu,e1rcnS.irrc-ricqn.i<r,lasiroiiirsripN,,,ip,ur,iin,rcI.t.n,ae,i,c'p,:111.macram,.rania,,s1:.iene1.triii,l.a,nr.in,lir1s:i,ii,.n.l"rd.c,'r'r1,,:.nCuuvra,;.,.,eai,orsarrdzreliitl.nlaic,el;eo,,.r.i;actirls,irndtrsiii,tu.re'ciriaeair'rj.._al,fo*.tlui,iriirc1,rsu.lcd.i-.r,'l_p,,mSp;.;,:iniuu,rii.iurprCt',l,.;{tli,iia.n'm.i:tjiri-ril,i;ri..pjiincici;*r,niO;e,ialt,i,u,,e,rii.r,iill,4.r,,ir1rco,;r;ir,,.r)it.n,"ltrn;miro,.i-.:jr.Uo1ira*:.ii,^alO,c{,t."lr1lr,t;rici.rji,.rtrbL,i;n,rr'suli,,,li,D;.tci-_t,,li;,i:e,'sLo-'cu.,,rli,,ti..,lrrri:il,gien,'ric,l.rfl.it,i,,la-'r,iri,rti',in,.rerrc,"],l=;r:-i1,.imra,,ti*i:.i':,t_i1ii.r,''t''r.;r.;r.r,,,i'.ti....r;,.r,c-,i,,.e,,,..iirs,,r:,i'1'e,o-lr,,.,,m,li:,',.,,,;,ni..,;i1,:ari,Air;Li1,,,ir;,,i;r.pa;;i,it)l,imL,,tlii,-ai,,.i:{1tii;i.,pirsar,,",-i,i,ls;,rt",;1;rlin"-cutric"ai,'ic;Ci_;:,ii;e1;,"A''cir_Jii.ie-Alnl,':rn1r;.n__.;,1er,tU.r".-xi1*--s.,"i,i;,,,-:;1l;,ili,:ti_-.;e,j-it-rj,
,.",; ,,i'.'i,,1ri,, rrll rlr u ir.',rr.tir'1i,'rltii.rlj 1r, rrlr'rr lll_':, -.,.!
,ipit*-ftiiii,;.irtto-ri1'l'i:ta.i,e"dr,toocri"r,-titii-.'ricirn-l"u,rir1(.,"enr'',,-ptir'ri,urclta';;-},i,E'ilil,t,ic;,i]tni."''';.c;,.fni,rx.ni.Lctii,"ltt,vr,"iar:toiti,i,lr.t,iscrti"itct,rialrl.''t.,riuatipNc.,i',nst'i"i*alt,et'l,cE,t,'tDiiiitpi.",iLi,,r,l',s,u',r'nirr"r.l,ct1ltlltt.tilrrl,'.,r'tirit-rt.ii,''ttltsrrstrat,t.,ste'c'rt.,rss'ti'bt1(:lrllrttiell'ra):bt1cr.liflrl;r;auattllutrriillir,rltii-n,lrtr-til'<;.tiiag'll.liii'rturi,r",":ii1,rit'a;arriir1tiritiarri(sl'ittilr,.l1\i,.,'il'0''iirl.UrJxrrii'ria:'tl;rIiiu'ciil-"t1"ttlict,tcltrt,rat'1lr,s'lccl'titl.tr1'Clrrti2;atrttn'r-'i;;{i''a'taltnOirirJ.tlr'rzi'ti';'arlrttrsru-i)lr''itIrsii.tc:rtiilllttlrllilritr'itaclli:'Iirt'tttoa'r'Ll:allr\iS)lLtl(r-nlturg;rttttuircr;'oi'i1rtiru'iciiilr.-tlrt.ltiii.ii,-"lllr'tll.rlLi-cirllr1lairt'laLpaltiti:'li).tiii;t'ltlIur:tciillc(crs'rtrcilrl'n.oa:'l,rttir-ar(il)et,.rriLirtcisr:s]ltf)-l:t'aitrlltl,lf.tn\ttltitl1itagli:Ii1i-li.ztr.r'ltt1tl.tir1ll'tirtt''iu.t\i1.tl1,tist':iicfi:lla;1'l1rlr'1tLl.lt.atl;'gtt''t'3l1"zrr'pltil:'g-'rr,;iilr1i1tttS;1ltocc1i:'lt'iti'it1rrji-'rllrI)ltl;:i'tllrtirllrtllrsrrrr'lrr.rssrt'1"i'rtt'.r'-i(l-utl!u.i'rttr)ltr'vl'iio:apattilrltt'i(iaA1:lili'rrStl'ii-ll--1'i--l-atil,'rt'i'ir---'tl ITit
i;.t;,; ; jrt,,n"t.tot, it pus bazt'it' tt't.'t it.'i per:triiiratiili)r- [r!'i,,-

,:Jusc lr-r nrelsttl plancte lcir clt: eil'ltct ia c-rr-'rt: itrttii ll:rtpt'lr
1'ui tdt"cttl'ritla'tust:itt-:'rJts'dttipnurslpnf(6)ci)lif!rifi(t'cli
1or, nu llun-lai r-lc souL,,, <ri
,n csplicat fuuonlt:uul
l,rr;

174

i' sN;iailafnXnniiGa4tDf;o[,dmNe;tvt;lucairt;ellrrtmhttli;c;iluaw:eotaieee*;eocdI,anueduit,;il;o:):aa"G"craentzx;l;irste;oewt:;:irlfcuroio,aiuaca;fuodtl;ci;Gcr;i-ctspntrc;itittnc;inaif;ieeatc"imcor;Ielaec"'tu;lvm.,;rJsi"iaie.rsrpiafi;nim-cotJieinxlefnta.ii;i.c1lr.te"eimdut,rlmnduJeis'ul"t"ifieiiIsnn'mnevt*lIeiuoee"iir;eWen'ttoam)ln,Iduesfiiplbop."inrd,saor'ee"artr*iorHpecui-ealemaoit;l"iisiars?trcaaul*oefll.tal,,bcltud"aetuialha",,iprcs'iatf,",;uac;uiro-tadtl*5cetacaturthf.ltcgonraqzrpittliXtie"eatl'l.,lree,.l^ad,r"egcae''d."-lm",r",,ie-l.lctdunt,nai'f'cal"tcee""'"rt"iiiiLit,iinil;cdutt;-nd"kf'i'u;f1,;plnt"tt-ar'a7#;;i,tin,teri"aio;;i;o-1rrT;";<"e'lnr'6tfl'"*nF,fi-io'a1.;,r'":i;'-in"';rlirlX"ot't'"il"t'in""s;'"nreilt;;iiir'irit,^i."t",'iiiaa"a;pt'oint;o"i[r'oiie;JU;;titt'y'ti"i'";l"ilttCici'rtiittai;liidri'iflut;t"ld''itfu"";,sm;i;ed"t"eiinp;bi";'i*,nc1?tcre'i"tt-rClnllltd,i;leiu-'r"ia"ieoa,ti;t-.c'ie;a''"1"pelira*-ibnrrtoaucniclattODrirof'ti;ni''zppli'e!l'ptntstliii'elfxiscctleroir",nterv-asiillaitutiucisoeioa."aciu[ia"mtsaoitcte["itt'.lacitttnrttnr"-i^eninettal1asllr-lasltiuc"lt"aece"lntipimtirplaat-ad(m.-ritc6cifaio],iptdxleeldpls"tell".t-lloie3iu{nbsripflrpisiirc'pta,mautc-rirc,6ieaic1neiitnlttc'ierltcri.irn"aadai'ti'r"elo.'ar.ee'"ibifoc'tnebailnlziacuNmooennnpiailvii-zerp)cuaebnrfaclacclrriaaicp'r;latsurcl'ei"ar'csramcatardjraeiscllitirurP"lsetcad1tlaortcttoazrrlttciarorasnioIlicinaea'eeueoili6mnusatutd(l.atlocrlcirer'crob'(ccdiizrmnrrnp6l(nitrrt'rlr"ultaseaelaeotcitr'ulu'tcn'sttlitr,ai6r'ctu^scimalc'lsirr'ocrrveslrrlla'ncaitnguoai|eil'tcd-aaleetleatdaitictttnirlal"r'no':c'ra'a'olsctiaajLa'i''ireipiBveuIpt'illccrc1tcT'i'autcrt','lcIi?a'torucm[(1l1alile*anta"eft'las1'flo'mriedludl,ccslG'rin9uu1c^a'rcuti.{trctp'usagtscrurnjeoluuasc.nOlar\t:oeiacretnssulr:i\ndpclenr?r^inltuumnt-unuac*tutrmiaiJaruoritcruo.aeiatliadcO'o'rbvti^claead:lro''tdcetlnut;iutiru'iinocc:celrrcalllnurleui'Suoi"e'ecoi-t'si:.ua-]rtflglr-nlocirdeela;lr-a',ie--1cl"i'"-'-ec''l aluli-amcDeasetetsumiigacuatri,clciLluoelrlcbaunpitlzioctaeutselte"mDcooedneasrlnitd,feeclra,artteocatapdaeurnatintupenrecdmio,emirlgedpntriouer--
ocupa siplanul de a construi o ma;ini de calculat. De;i el
nu minuia matematicile cu aceea;i dexteritate ca lttrer.vton,
iar preocupirile sale erau foartc variate ;i din domenii
cu totul deosebite, el avea un cult pentru aceastd" gtiin!5".
in L672 Lcibniz, care intrase ln ser"viciul Electorului de
l\,Iayencer,a fost trimis cu anumite chestiuni diplomatice la
Paiis. Aici l-a cunoscut pe Huygens, s-a lrnprietenit cu el
;diotriema peld.eInunaannuiaucremstdatol-ratrlenrvbudileats:mi paie.tccemlaatiLcoilen,draas.a cunl o
la Acolo,
el
face cuno;tinta matematicienilor cnglezi, asistti ;edin-
!cle Societi!ii regale si la una clintre cle plezinti lnsu;i

ma;ina sa cle calculat. tsaptui impresioncazd lndeajuns ca
si {ie ales membru al Societdf ii regale. Relntors la Faris,
acluce cu sine cartea de logaritmi publicati de Mercator
din care lnvatl cum se poatc face cvadratura hiperbolei
cu ajutorul clczvoltdrii ln serii de puteri a unei {uncfii.
Problema aceasta ll prcocupa pc Lcibniz de mai lnainte.
El a stabilit o scrie numericd" alternatrti, care exprima

l./aloarua 1r,ti A prin inversclc numerelor lmparc. La Londra

-l'
ca;i 1a Paris,I-eibniz a
tangeutelor ia o curbl. avut oc.azi:- sd discute ;i problema
Ea {usr:se fornrr-ilatd" rle Fcruiat ;i
rPadevai.liruni(sca'rlrtlrinccacalIer6t5i5Pt"l9.ua:s:,c,,Ta,Slr,acltrnaisttor-droicsacplredr:rhtsiciiriiirciAsi lrr1nrrioke-s:c?ls)trfeeetrtotoupnluuvrl'ilidiicceascccdcttrcIea"
ai-luIi{iPasescpala?re ci Amos Derttonville erstc un pseuclonim
an-umfe)a: .LIo)euisfacplet cstc o anag.rrLnriL a altui pseuclonim al lui,
Montaltc. .tn ac,-Lsti lucrare a iui Pascal,
I-eibniz a gdsit tn 1673 triungh,iul in,Jirtitaimal, cdluia
ii spunea ,,triangulum charactcristicurn". Cia ;i odinioard
Newton, Leibniz lntrevizu aici miezul calculului di{eren-
rteiaspl u-nzcdutovairnetuinl ilfi apar!ine. trl a introclus notatiile co-
ialci acestui termen: tl.v, dy, etc. ca si

17V

t2

no-sc Im-aairdcs'lcpudresnrouiitcai-crcriiits,t\.riroi-ptor,nras{iie.I|irI^n:rratsnGurl:ru..grronlv:itocrrl,-
1676, Leiltniz a stat c'l irt nou lu Lonclra clruir siiptiimini.
-{tunci tr1 a cunoscLlt plin.fohn Cr-rlijns, consiljerul
nrateuratic tl Socicta! ii rcgalc si pasionut rlr.r nratcnta-
licile atunci la rnorla, llll nt:runscr-i,' al lui Ncu'ton ln
Jcglturi cu c.lrzvoltart'rr f Linctiilrir il st'r'ii rll pu1.eri. tlar
p.{trl\:)aLurrld'rnuuNtlu'r'intelrrohiiutrrtul.,t"ctiriurtigitcoit:rrr"llllra"crI'rei-s'tLripl,slriuerbtort,ttnnrn,or.uitrzldllJiu'1lIniiii'rrrrIrii's'irzrcrrii-oirntrrlirr-ultil,tttlL,ltlrl,turnl'i:lo..i-irtijsis\rnpicrin,ircr;;oili:tiror'.h:'t,r1Ju'l:rlaclrrinnrsiutncl-'rtqt'l;'rl(i1iii,rrrlittrril'iritrartl-c';urrilria-l.srilpirorclroaril:ilrlt:iir'rattrtr.irorlrii'lblrLil.ciujti;rriiii:..
rleocarldirtir nlr i lt,;rlati t.lrcit plirr rc'zultatr'. l\Ictctda
sci,-\etr{tli-cl-(car'lt(rr'):lstn'utcriclnlu(t'tcia.rtiqrt.ft,:;{iiri,i)tj,irznnllulflornus,ttlcttoa'ncictirll.i'arittloutidiJht1t'r.,\ssis'\icurr'lrr-b.iir,r1's-Ilrrio.tgrls'litnlrrir:rtnri'I.Lioarligot1L't,r:raslrcLtn-,icl;roniilrrrn-i,rtll[liri)trte:tl'rl'.cr'aC.asciit,ll(rutllrr(.i'r':aojic-staoeaaplrrlctci-ioiirtsp-lutlritrr^ru:i:oiaj.--.
n ical l-lulgens lui \\'alli-. r:ii lr rlesrrrpt:rit iitr,lul lui Saturn I
-- l\llri tii rniltt,r 1r't'buil sii lit' ,:;Lrn rr.rrrjti5r;r' clr. ciitd
t i-run .-rltris-o !
-- I)ii' ct lriti tlt'ubti lirt ntiltt'lr ,llc it si pirzr'usc;i. sil.
.f lr r;rt,.s (ilcgol1. nu-i zb()u.rc pr-,-o lurpnrlcllrrcrr,.lcrs(:ll,'ir-isrcilnistollarllt'l:rc cr,rr.llj1li.? ('nnr
sinrt
it Lailtniz. d -*-a

notezL. :rcele miirin:ri diferentiale sau ilriinit dc' nrie i. lr, *- ('rt'rl tit [oitltr'ilcinllrt,liirrrica irr -\lglia, ilr.i Nt'rv-
aceea;i cdrtulit' a girsit ;i calca cdtre problemil invcrsi
tangentei, aceea a cnl'cululwi integral , ce se referS. la cva* tori nr.r iiulrlicrL nintic. r(,1)Lltitl itr Iui ,lt, sur"arlt crll licutii
rlr.r ccl putin -l ani, dt,cirrcl lusc'st'ill(^s 1)ti.l]tltru Ii Societitii
dratura curbci considerate.
*- Drtr cursul lui lialrorl' nu i-a cdzut in mind? r-cgair'. Sr' l)()iltt (ronstirtil rluliii st'r'isr.iilrc:t (i(' ris:1rtlns, rlin
* Ba dir. Aceia;i un altul la {ei de interesa.nt, ;rl iut
1677, in car-r'I-c'ibniz ii a.i:ita 1ri:or.lc.tlrrrl si:Ltr clc ;r clucr.tan-
sJg-aiamcuicirnstoo(r;l:stcingttolairn"yc,LuaoOundlfdoreasn.tbIt)uceringoe, isndcieccorcsstteaubrduiitall tpiemerdpireoetlLutaeunibtaniiiz;St ocficonistc'-li q('lttil Ia curbii, notatiili, l'olo-sitt' tlt. cl 1i totoclatti ii dt'zr"i-
luia pe fatir, liu"i nici urr {cl dc ascunziyuri, rrtltocla itrr.r:rsd'

t5!ii regale, cll care a rimas in coresponcltntS. Intr-o scri- ,rcclt'iir tlc g-isirc a tilngcut('i, nuiltitil clc, 0l lituuc'i ca,ltttluf,
:ttutal,orirts, ial urai tilziu ittlcgrul. Tinrp tlc 7 lui, clt' :rii:i
soare clin 1675 trimisd de ia Paris, Leibniz ii comunica
faptui cd a clescoperit o metodd anaiiticii, gencraii, bazatd" inainlr-., I-cibniz nu va puLilica nirrric rl in ;rcestc cl:scopcriri
rL si, rlupi crirn ai virzut, iirca lui rleschisl , r'esnic ag.itatei
pre seriile inf inite, prin care poate face ct aciratura curtrelor, ii rntuzilLsta, tlt'lir-ric, sI li ucc:t'littLt cu g^rr"u lccasti tiiccr-e.
- $i Oldenburg ce a fdcut?
17t)

1?$

lgr

nci;;ioi;aais-;ticr"iial;;nA,ni;i.r"ie[a:cric;ii"l,'a'i1"rnrciidc*l";icn.i'-ailm"l;p"ot"rir,;vot1;eai.l';ep,z;i*irc1.irir'rtmivrsu'stii.iaueJrLttnr],eaiuiset.ui.icl;tbiluofiundicn.diilrLzteurddaeapgtsizbnceciBcr-nnhuc!rei.itzilmfiaaSnivlrb;oseeaiatg.'lstiiansciCittn^iki'noglin'dnispeauls"nlctlciXtce.eiclnsaelTettctriP'elt-a.ata-dtmIvsl1ucctili-aLarl nFei rmeaqliizastuilccpelsnei.alncu;mtiinafdr "l.nsemna'n imbold pe'tru noi
paat.iac.ur-o-cuz.eleng.SI?ucn^Nitd$^uudnciimJrtiLncrlaoptaireiasurdtlnnnrrrieaef?aadEe_ccll;.fvtiemoel)pnfeatiaaatoeti,ecnNaIrmaaemitwniBdcttiaoeielntanci i,c.lur.i.cannsrd'tticica'tit,ioilt.etodlczcrurIlat;u1ailic"rlcr;;si;;st-tet";c,;ar;r;;s.;-i
"iiiai"siia;l"di;a"io;i"c;ti;f'."u;";vteo.riit;ifA.;irrisiu;n"t;l""aitoto;r,rc;id";ti*tt";t.sa.',te"itn"-rct",utpr"l"oei'rlf"c;-trtcet,Du"nl"it"*ieiv";olsa'ta*i;telListtcci...itrI'nt1laeetl"ul;otaticcis;iisi.c"ri,tulbe,:pineeir:cP,v<n,(artpa,at,'.ili'atfinnrr:zu.asicie'."c{-ulli-#eiiicialr.pt;IgGoftfit;laiitlliao"illztli'croi','c"rglo,".cretr,iziicllftr.lpictoittn'afneloiaceocc''fslbra'ca'ecnria.ictt.trAIlticnrift''.ln",det;tss,viiiin,ecclamfdeAhfsacio(irpa5c')acuitaacir-mrn-ctlirrstzaomcn-isunimtuau,nttrtissrnp'i'uiludt.i'rau,flrilierclmiton.t-qtrtlrssenauiaiaicppdatcptlistcitoqtecrtclatloanitnaeertfmtinclttta<arziteoftallsllptl-tai;a'aut"tte.dtcdcltu:riraclecPo:iele,dc;u:scPfi.cptrlbaicctrinauitrltiteimn'rcf^lbu"alnre'cirtttamrltri-lctn:!isllietolriielcaaLeecIlca--rn,us-',i cal.teu-n.nTrc,.iea;pigloplantnpddee-a;ttzoii,fliaccufeavlemreboriliia1in20Bnceulemrnmaoiuatlrtleei?imgCietuin.icg;nramiitiaii,t;e*iam;dra.dt:ie*
t,Dnir;itairci";lieiiinaei.,roto\ve'tor*'..;trclauamr;"ut.;r"r'uo.lt,]ttltiilD,;i"i{tnt'"";srtt'iiaoi'uacfnlrCra1eum"etl'ea,t"ol1,unia'.rliqeinglal-,"lsiuicntt7,tatu.lirhe'tooauccie.rornrclrra:ttu[irrcilcAcciti:ne.l,"canleLtls,u;ieiicArstctastisinIloaie'tcratiuistntdna1u'iiaaLfriatihiu,jrit:lliur,}:n.npcli,tatr',ft,sc,lr,u1lazoitl1tum.icm6ilsibctrn,)it8:eu.iunlid'''4aiin"icnNllrrt.,itextJlaailt:PsseDi,icntlcfmNescweotidtc--Iotrsatataoiaci'lo,tsir,rulcrlprrotnscoEu"ctigiuno"cilcrny.la'"abtinlpIiurrottni1"ilr'dndaiuiic'i6oacsilaitc8tL{bnjlac,Iuirodu4,ncnacciln'rmerpiiainblr'ibupscatnuatialntcrli!e'rnbAialimsi,mr'-slaz,lsuaticn1iueucfimltciee.6dn-rttt{.lr'orl8lei-sftnci1rmo6displlrfirularc'1noilctuieralnloiaoisnrm{amiprlrpaitccscrvicsaoclaceeatairpCbrlrl'asi-allaeriu$eacaulc$sclcais1raLla:a-it''ue,o-l cdp?adrlv"na_ae5erl"ter.:rr1aXIrv'-,unueo:VsuacIlncaIntrIaeiuu-InBsl[me-lrp-(tei'xu]mdarcpreinitinnoevamrntcnovi,metnpreerurealsfcaetulattaistenicfe,adseadipmlcencdsceatctteaeigtraooeeir.tll,erriuricsiLcitanemibiolaa.iluftllraaonu-drismtgsr,iiens{l_tc,ilIarnu!otA;vuiuiansitsncumccnatuearldneir(ie"crs,sfnvae-driafdai.li".di'l{Zlralca)tuns.loScsieltBsieliccsmieaen;auiu.i.rtr;ptnlaetnrl.;"oteaur"c;r"ita;;ri,trr,iaiicc,dirn*,r;u;lr";s;;liv"frEld";_e;ti";;'talan*c.<-rcr-a*i_;'iueb";z;sta-;ir-a
tTIrllDonn5eedla.snl,lrnolirardp1cizyet-lnn.cnsglela.raBEg_-cqIteteeei^snrlippdlnsd,urloaairnce,lu-tair;"clrcs,licin"dl-ur"iabn.icmap-Aotrosjlucoode,emcc,aaAspoumsttrirtteueeiat,cnozpulicir'tn-nitiaptteeaaifaJudscscecstunesi,uitaplueitnusn.iprttLif;.l)e)aialrii.tntlnD,tr1ic:ruiuidaitr,.m.,rlcMiaXpeocdcm*tld.aer,'"ee.srnT"n"tufot,amait'"ii;lrmi;n;ei",;usdi;J,ruc:.
"oiopf"t"t" ccrdcllcvloi_mneaoilrait;tepntinletcdtioguueee.iBl.t1^li.a3aASlsmz0itclB[lneiaaucd]ecicrsueaesdrusicncu.airmanaeeDolvcclznluaeaeu_eiigaltsr.n.idiilctoiioEnBe,tum,mieralumaafarBeepir-gepmonberfrieiutwarrgooicinllaicdriio;tirohepiaicliurieaentaautlesmloridilril.utumiua;aalniicai;irezareLcsuap'|oeddipit.beinie;e"bdvi,sru"snetrilccgidctriindaztire,ooei-d:r,srea.rmfaemtmr,stoianaeuiaa,tsnoei_ct"piaeiaeled*o"lsmsenaa*titpprdd"aldlaiL.uiutn,.riasc;lctiueti;iiuv;.dAa.l;oronceCe;ei;bcuinnou;li"et"-.--
rB0
I81

.:i : -- .-:;:..1:i

' rliill:lrl I ,. I . i: : : rr : i: .r

'i ::i irli,, lr,,,l :,:1

I')aniei IJ,i'rl'rlllli

f!, .

intiiLrl c,oniitcnaaigleltt:toiiJrmrilaic'1tc1riulaillroatainisadd,clirit:il?cetdl,cc1zc,oaAhlcr'nuailrnucdl oFri'eaanrritlarguti[isnioiutlrlslnti''rarpccertolsa-l trotilr l lt:r-rtotr ii i
oaitut
*.\io i.lc artrhjtios cllr?
[,ri,:mc -ui I Afll cri, liirrd Iir PiLli-., si-a a.luir-
al probabiiiti,tii .ate nu-9i af lafouapitiendsuast'tuinnucitedeczulegLaeriebani.zt'l -- Nn;rsa, ci si rnai
au'inccptr{ o corcsponilenli lui at'asi"r si trtr;t nlti ltrtt si-r ;tir'dc tl
qat l)1'ol)r'iu.l fiu rle tlt'rlin-.trl, la li
pt'ntru <:ii, a concurat,;tlitru'i utt plr:ntiu
i*ntfntIemmlucrit,.irn*truor;ai.-or'riBrpri.t^biLl;"etntrtt,er"i.mnai,i"iorit,fiiizinl,iaioiiirfttt.ot,otercui:loca"t"auhrretmuiiciCleLn.all'iaitdiDrrlia1ciea-lic6crliruarn"t8{LeLcrf-r7psrihor1tt.slstit'ri6eulglieiliot;og9viil;ocreei,0mctilitr;asit-{ubeslaeeasdniitrlaiol(Br;coalpiild,niLt;lntaetce!cpr'"iu'g!acaslonFiiciil.antoilut-eAeRtr;tinlirlisclcerdtliatln(adcrur;tLcnitn'itloc'izlatEFeoeraadli'alcLugiilc.nicarn.nlerealiuea.siuid'rlbla..tfepaIcuo-nIrnscoorspiczlifatlutoeo,caertasprlruurrtodeemtrB,crucrscesiueep"rtltcnitratnllil1aeo'nzr,rsptbslorn,c'ruad'.d.luecaolcu1ealLac--ir' ,\cadcnrir:i {rancezr- pr:ntr-Lr rnatcnlttici :i l-a t:lstig^at llttn-
'f.rn Ioan I Belnouli i rtiiitctnaticit c'rlt o ])ltt itltii , ilsrl r'Llnl
;ru altii patina. jocultii srLr;r vinLrlLri. in scrisolilt'tlirttre
*ci cloi {rati sint ancstcclttc, alituri rlt' ricscopcriri tttatt.r-
rlatice tle o rarri srbtilitate, crlc nni grosolilne injurilturiL
;i acuzatii dc furturi dc idci I
p-a.rD'teucpldincrldtesplnintcrllicr.tg'a, llalculul infinitt'zitnal isi clalorr:;te
o clintrc accpti
ltri tocnrai inlrrijbirii

cloi ft'ati pdtimapi I

182 roo

d[aiLc-c-orbel'lnIutJia-n]lcu;limirionauitlelriigbsurpiblinlaidcrtlirlocosast,eorcpiecir:;cailecadcreltoziclvodoleoltai lrfneraaAlic.Ctlaa\'lIcaEuirltuundlutit-ii pecmcL*t0nS)'rle,uarapcilsbrtr-,..i'r>es-n-c:;esaa'iiirzcpria(c.nftnlo5zocie"ra.oa),crp-Lr:vlccec'itHr,t".'te..rucidpnls.nprsaor-eciirintncapsacollItascac-ic,.ltbccceirgc.lucrsaapitsnrucarctncnbpelraiorlripltirnpa;anirrcaocrlgccaicdi'rprrcicsmcrltacaupttiiaoiiilinlcacirorlt:uaclcioilioirspu.naurtrcrrohcc.lbeetofl*oc,dtrmniu,pI-,irn,at,iaiic.ren-:i,
iornur cnr(. au clurul sd tdlmiceascS. multe probleme ln noul
limbaj ;i s[ arate clt de simpld apare astfel so_lufia lo-{'
D in 1694 la Acta Ernditorum colaboreazi 5i Jean
carc pinir atunci c1Icluse lec!ii despre mctoda. I Bernoull i'
la I'alis. curs cle calcui tcc;r1crdSrctt1[rrculouc;olclir]riprrt_ln;cl-r*"-raiernrrulrrttsz-r-iarireui-cititrlc,-|eitalrPderTlSDIrcIlil'xlitsroatracrrpoxrciptlclianrraz.ciuurc!*truulect,riunlIctgrmc.iirisrisltltlai.oe.trear'1'{IacNai.i'ai,lrrrr(a'tcti'iair;r'rriec.rfr'riurlivsclrrnil'eutlotrwruaIriPlc{curr'r.c'ricHiiiurlintr.enitlc,a,na.triofcta,Orroo-rrln.rdccon,ur'cm.tnsr.L,urneiapuraaiou-',,tl{aaLput.'icirn,l-oinir'tedire(tticla;r''rcsrrt(nu,ll-eiter,.,iarllbinminniprtl,ir'rancPau.trBriaoiel'rl.;rcrauirusruzr'iren.cdnntnfittti(lrii.caic.krtlccJco'uai[.i:hll',a,ans)pacim'l*rpriororcic,.".raltdr'ttlfieet,acaoimrii.iil,rc1."riit.gtrirt|ril;t'cll,eiiirccrriiuil'cciluliretirr;ccr'cs-lcr'tialrriteitaagrairldst.r:lttrielrI.arrne.ru-1,[tus1pr..irit<l,llp.ctidcrolruiir,s0lrlrcili:.n,rcifi*a'rrr,:n'il.nsrr'nati1cr,tfnre'pt'icrc"l[1.Iuin;i.-ctcr,nbnio.ttcuc,e6ur'r.lccbIcr,t,rxarirl{i9lir?a.s*lcratnd-nicsi7tirt,.mlf,cirrr1ipllrus*liit'apipi.sndl""ai,.icm,,'tg.a;.,lrt;ltirtai,cir,.o-r,ii"a,,e.jilnlp!g,vi;ttiih'l"ini.,."nreainr1pIrc]r,Arcl"e.1rnnsrfJoai"ibi.oltfd"?ii-rz,-;r,t;.n;niBi"Jnii.tu;iiiis.iDno-tr*rei;lirl;ai"ii"r;tcu"it,i"_,.,ii"-".;i-r,_.:i-i:.-ir\r
Iil er compus chiar un lui Lcibniz
hlanusclisul cr,insiclclat pierdut a fost regdsit
clifcrenfial.

in 1920 ;i
ipul'aanuirdPiliicsacsratiscsteiilnpaaicrfillm3amauas;etlielcmmiun;arLsrt1icc9thi2iapi3nz.i.urPilAtrGsinad.teFrssec.pdfcraeleceeavls'iclie{_66llnuipneiin.te1aJ6ae,9la,6p-ncrao-lIball.npedsdrenuruelgati,
intitulat: ,,:\naliza infinililor mici pentru intelegerea li-
niilor curbe", plin car"e sc {ace cuuc6cutl opela lui Leibniz
cn rrotaf iile folosite cle cl. Cartera e scrisl intr-un stil sim-
plu;i clzrr, sprc deosebire de articolele-de oilinioarir alc lui

lcrlnl'cnulrnlltri:!c.l.!c1alClr1rjprillucrNira,Ji,crncc'r,r,lg.taIroai.'nlpdoag{cni.ueua2ruoe0drin:6-aasi,rttdiIatat,.t.cilcLoLcr.ccrucrnstitnrruasiapi,irtrl"is[r,.c,aa"t.cc',rd,crrer,m,ttar,nccnedu""ssrutL.rndfca.ii'Jpbb.tinn.-;ie,i.irec"ru'"iinui;io;t.r.si,,-,
Vld: ,,llornentul unci
tl-c*- ,,genita', cstr: egill cu monlen_
rl ivt'r'sclor'..."
t

P:LginX dc titlu riin crr.i.ea -, ANiuinatestlcasiicricndpcoritacsntte, o tcorcml <lcr calcul clifcrerr-
ci Scolin cart: nrncazi dupir
,[ui Ioau Bernoulli tial?

rtceastfl tborenr;i la. pg" 20ti.

185

* Vrci si o cite;ti? aceasta de aici nu-i- cca originald, ;nsijlilllam'sur*t"riti.itol'ri.rl;;pr,_'.viiil*ir'ii,[,rrrs-r*;.i-c"iirrl:rrL'n1iLLl.,1rtl'rr:rr,ir':{"1riiIiriijir,'nc1lji,'lri.rlrrli_rirri,;'D.r.ml,cirriir.ttrtjtiipu(;p:to.aciLSrrrrlil1nrt'ii1lnrtlrsPnrti,i'tL1rlcsrta'tLtadaoiu.'dllrlr1,-'rc<r't.,t!ti.rrr11;rir'nit.am'ir'ris,:ciisiil,ii,,irt.,1ir.lir1.rt,-ir,'rii,i.c,,sn-"rti,i'iliriirr,tIhl_i:l.'c;r,l;ltsp:-ln:lrrir:ctStc-r:ir'irur.i.,iult)l.iia,.;ei.',asi;iii1i1trlrN:,jtlcsoilu.rreit-j,i1.rti'itrln.e\1,i,i,jr'ialc;arr.,l,)iu,r;r,\'l,u:l,,c',n.ilrri,,,'rr<)r1,'t-trriri,lItrrilr'aic-r1cr.r.r;n''sJlf,t.i"rt,ir,L..tm,1ti('rl]r1,brnrr,,i.i;rii.i,nit:rt"silr;isi,i,pr,.rL.i,,ui.rl,isr,..,i,lcsllat:rr:re,/"ic'rr_1r.,iili.it..:trq,,.',ilteiirri.pL2,rrr.ltti,.r,{,ar',li-,-i,t1,,,,t,ti,a,r,ir,r,t,ri.i;,1c,ii,,ul.Jrc,,,r1,.1tii1;t,r,e,rilt,ii,i..,.lp.r:,-'ir,,:.uiriri.*rij-ii,lnr;,ir,r:l,,rilr1,,li,cr,:,,rlii,\,.r,.i,.fi,i.cI,itr{ir,ri,si,.ii'rr,",iii:',r:r.r,rrr,.ti.:-l"in,ra,ri'i-.*;n,,lr:,'s,i,iri.,,;r;rl,,r,;rl..i.,,arir,ii,i,t,,i,,,,,jr,t,,'iL,,,i,t1ir'rii{i;.i,.,.li,,r,;.,.;,i,.t,-b1]-,iar'1,,),,:r;,l1.'1ii,.i,-;*,;,'lr,s'';._i[i.i,___,t-'i,Li,,:l"t,i
* l1,*liili.1rLr1uoi''rigir7irtriil'iii'it1ariiiuii'rLt;tll,lirir,tczit.i.:.drii\uil.rnlssiIliii,r-)irrl.liradr,d1)ir'i(.rLts,ii.tmco'.jt;atIirznorls,u'ttcrillifam1ia,t-:irr:l,.-:..stcrciiir:,l'ci'lrllrlui5lr,l,s:il,,'r:'1loiIc1r.ci.-rlao:.i:\l,rlpil,.-nr1rii.iroirr'Sirt-titl''r-Ll,tcraciic,iaclcrrll*1..rili:rrra.icl.iilt,,a:ri,tc,,,rl,tr,rr-s,l,iisr;i1IIr't:..iun.r*c,ar,.'ii,'i,i:;tr.r',mii.iirarrrcJ,nsnilit.i;{,l*rnnisi.ir:.'iir,)i.ini,il{,,):;,i,,.,,i,jI;,,,*ii";rt-rt';,rrl;.r.na.l,iir,,"r;i.;i,h;rt,.,=,,,1itn.,,;iifiil,tl.oiir;;rrr,ir".,i..>...;..;lo.ur,tia!,urn.ii,.,;J;c,i,,.,lii,r,r.ri;i-"n.t.'il.;'1,irT.:r',,,r,rj,,i,r-ir-.;i-"
nnc5. nu, fiinclcd
aoiruti in prina editie a Princ ipiilor, ci o alta, ce nu se- gd"- ,'i(f'rntrr.i'ia\e.pa'e\a(c'-tn-.ncrttrt;oLfurarlr,,idliAr.rAlorlrr.iai'.-lsiaiiea-cilpc,asdipar:lds.atgi'eztli-cnccrrd,rtrst',i.-tor'lrlauoiu.anl'zrsalaj:l:ia'ttensnmrc,ttLa,rcsre1arbcial,ir"ca)+,ii-liprtnncvi:::rdsvliiot-icttirrcLaaLc;.sar.rsooicrtatust:ira'acrauiiciilsrc.rchl;t_t^ritil.n9rudoii,cuiGin,uiul.,arcidi-ria'l\niilcet,ili,t,v,.icsa.,,s,.etpinicrc.|uiteiiti.ieie.p'iiii'xirrcar'clrcdcii.crjrnprtiealiiccanrioci,nilasr.rra.ir,tJOtl,itruiiiieroiliCil;a,/".'naiitct-c'ciaizi-*coiec"ir^Nii:iroib._iirlh,,p"se"Ii,i""rlt.lsi;ap].rifptlc,".rti'riaJti;;rnic;l;;..i[;i1ra;".r;d;l;o-..;ri;a,s";,;*i
t=i6iiritc* t"tueid\i)liiactoarIIlIl,a'driiann1.7V26e.ziScdensecd-ruiecetnrar dd*ecchi ltao*Atcl,lnola-
'pI.t-Xi;e-;I-"r.-:iaxt,-6c,.l5en\,t,iiot"'ten,nlcicl.tusli,ip'llioi-ragcbceISuiotcct;ltoisrclcaitac:rulIe,tI,l-ScsG-picl.r.rre\rit\scr;-.orr.rlgelIti-uoc2esl{i}eiia'brSbcc'lieirzaes,tacrt.irnl,cSua.t-icntlo.otxelciti'uleclirsuctttce'cl1itt0trrir.cna'in:in"1ci.-t,' 187
ra,-o;.iiinriiira,"clrJgtaiuu-raita-cr.anrt.sinealeee.i,mmcoliceiraac:nacxrucuctit:inl,oit,}rcc.nrtplueiotltrnicldrrcoi-cn,s;'irctic,;iluiunrn.rt!rrnic-nitnl:lcliiirn'ii.tdltnoualtemltcriil-ctoulrarnalna,{tr]piarnrta:lrtoi-isp!roacoiI.zr'fmn'til'riusiaetltcistcraoatc:tdrt-t(ti,aaFra,{snciisotilgancnitdanriall'ct.tcfetrllcta-cirtto;in-dti
ti.:lit".;lo;fi.nin;o;l;at,,,fisi;ti,toi;n.,yr;nci,reis,t)o,-cral,ci,rc:Jsle't-'ba,,citr.ebblaa";tpt^"i 'ccgilicts't'ilnrs.smi crntArii--i;anr rlisi'rr-rs cil

c''runicat

{ortltilat"et

,-,rrr;"t"tot'1i ir sinrbolnr-i1,rr". llriz;r ;Lrlltclrtl" sc cltlit inclc in

lctrril!ri)ritlnl-rr-lai:1a:a6iPil9ntl',i3cit,ac.aeta.htidnclte,crllir,tlca'sltitnii'illt"nnt'i"ldiIrl7lcrttic6ialr-llr,rl;"'rirtbritrn.,Nt,i.zitc'.,trilit'tr-'ll't1tt',oit:in'|tu,'ilsl-llat':icOnriiatpcin.\[cittmr$'l-to''toa-stcitlrriir.spoacta'rslllll'ti'(tr-'''
a.diuger 1i d6uir a]lu-\C, una tlcspr'(' .n11lllt'l'111'(.lI Lrtlrll('l()r (l(:)
,;rtliriul al trcilca ;i ct'alaltti tlcsP|t' cvaclrattua crtt-bcltlr,
accsta ii explici'i cic cr.r ll uplit-ut problcnra cva<'lrlttrtrii in
lLri \Valiis, tlitptiirliNtct'\itnt.ratcl tl'llc-'lat'cns.ltItal-ltiljLolipla-rr'ticlrlr(ilizr.:r
ri )pert.le scriSoirrt' a
accaStir
n;,ittr,nrr-t\i-\tactnric.nannrar,lir:iriilpsasocll.ran,r.anttisAlria-mnonssaoimlisfnncrtciitrccvuvunirrrtti,ie:mlt1trnitlric'lcrcs'csoiscclcugrii-rulncc-l'tclti')s(-it'1tiiItt1a:lctitct:rllicper:ti-cunnoclrerntlt-pt1tartaliltcltcn[cctlsizculcllc'iitt,t1torrllP,1Parciti'crt;trlol'ttlrittc"rll'iirrctlvSrclpatcp'icttl''lcitilircarpratst'rctir-itltai--'i'

xlletic('"
"

186

rttLen"nit"a,,1;.iaic7l;-;m0;i4sac.Cptluiirenctnei"t{earreeancilnuiutdliemneceascpbtirveueuelrncnctbuicsapiriincbrc,ieasectFteteopeaanatzbtrifrdrolcaiacuplidtrrurmiesoslipiaDoidtrze.iuebtemoiutuldtlaclin.ienttfi?lcirrlse,ineeDo-c"bnbaurLiairl-l-pehaoiiispr'lnotnroeocucnlviurmtgootitalniecpreeazrlitaa.itl,' * ltre*,ton a ticnt chi"tic, ca si cum rrici n_ar fi ;tiut cer'a

rlcsprc accasti acrrzat ic I
--
Si dacd, cu a,clei-iir.at, nu;tia? Er.a un ont asa clt_, ab-
setrt de lumea clin a{ala lui, cri n-ar mirarel Xu mi_ai
slrus fi cle
ci <laciL avt'ir l)rictt,ni la masi .si mer.gca in nir.niti

jcel, ma'i sutYf? ci ar trebui sd prccizezi cd" e vorba de lp<ahpthlvriltsnareept.iaao.aapcnc'''tictoriiircrcn?vocie,'Einnlru'dciIrlraiunsn{uofiitunesprsgrtccer''.lrirsae'iasntsc{eeor,il'llciaoui-t,rrLsmaaeigiiri-rnbl:cliinrn..ca,ccrietzilrcutr*etrlscenvtliariiastcn;eti'criilnculsacciiic.--c;.Nitras.ies-slrcltnpi*actrii,cnrrnlrnptncc',ia.e'':,'ziiicfc*rrrfrnt.ilii"ii.bri.e,srlnizuuilru_i-i,

Alu-i, numai fixate dinainte ;i cd mi;carea
un drum cu extremitltiie
asta are ;i un cintcc'
r" f"." fdri. frecare. Cuiba
pr.--e-u.ACnituuctnurcefiiuca.alnntiititlm.B-Iae. irnlnoauilnlitea,ualdaincsi aint p1ro6b9l6e,mLacibbrnaihzisitmo-' ,F atio !
ii"ci-pcsiiua,Li"o-luruc]atione"l".ulIt,fs"utuaaSdeli.*zr,ceunud"a"unl,f.caii,rt{iivacprerPtiatctcnte,.ucpd,rc{o,nehttca"trsiNc.nittcfirntci!aebrddincrAaiuu.sa,se,lt'si1i,ptagrslebcalLusmLinfidtnl(nla-ml,,'as,rjztao,u!o,aifoncofradtspcilgierveiuieramdrumsxozaitcr'tbiaiat-tnrgs?laioteer.diu'capcr;miei-arviuua;emTasianninlltdcddccaiaecpoltctrloolnai.uucifzrnsalotanv'lt,drasiBc-caLezatet-l:I6;siuplpamnoiii,rttueu,olnuafCuc.tsbieltoactnIi)elt-turtrtiiadr'elnttmniurt'rifactonliinea.rrltsii,rl1'sit.itdnc;t'tef.o.elIpradcaisrso(lturucr'[lriti'utesrrliccpArrrfri,rrrarbctireilrrtIrliaiuatna-lt--li- -- Arf-itLtca sir fie;i a;a. Clll pu!in acunt la lnccput" Ori-
rvruld;ztl"uft""s.*ot""lufinttiiarti,ucIilui,ie,nCibuInnBoizse.crnIlitcoruuulbiliidnauurpicadcuagn'htoues'cacuri ctc"iI'i"rr,IcJicunltiaactcullralu"it'iot-i;-,r cum,I-eibnt, nlrera motir-e si cf iincdarsncirii,"t,epsetesteocra,izsiiatr.irR;izi fgiuir_-
i,.icobEia.ar"ua.rlo_ulieri"t*zc..uspra,tiuip"iv-tdmtuuil,"ui1drstn"it"lee.oi-i"t"urtr.iuaoart,ta.lciiafcrcelBdceilafincldrneaseasseurplhae7sgpciiCintscrhn;ia)tofice0oiLpal,pascceacelrrr.ocir!irbaionirAtiniiprlarinntiiiizstrr,ldiii..a-{pii-aa.bilseictr,.peefiiofiiuninaillnbiodncpnls-uiSsllcoriamti-ci'nngg;onliii<iumtlsuiiuilaNtmiira"uie1caNnnIras-ivecereaaeteuirorr,bai''tntstcridontecrlierinrraoz:gesO,rlcuic-sstorccluicnterldc}'l"o;csert-'pis;ueirr\rttnc.cerlntttcl.iairltrtnr--rtr'
Principiile rlin 1687. se;te cu_ cale sd se rizbunc-,
Annplln'aaocrrttcletaaecatgarisIaelirutauulibducrtsr.lii.iieletdolrc-asreudi5d.ire.cb'mcaInfailcalictozpl-uozo;aoilgr;i:frrcilriCl,uil,iroudnrinaifdricnrN.cicirce.tcccxa*enzt'ptrriioi.emdcnmor,ua,r.tilrlgcenOlaisensud'apir:frrkttortroiilrsct,ai*rtiri,t.ttioclrl.a.acniencrscfN-teci1eprr0criiscvln4renteto,tipnim?feut.n,i
* $i Newton? p;[r)ricrrsiccidgau'ltra'lc'traasncils.nsagccl:ctriczohiniaisttr,r_ccpliirnt.irv1ni7at1i a0cc, cluia.nspt'iaLlilntfa.'elArllcsljva1.ancltir,c'luio,i
insrrltii si

philos<,_
l'{ervton,
.J'ohn Keil, ;i el'rcrnbru al Societitii regalc, publiclncl un
idanrirt'eiicm'ctialcttcioile.dsp.perferlunftoxruri[utpnlcriliomcr(.'.r]at.rtaa?rltciid,,esficrcrr-ciiitl..e:ib',c,rTiliot.raiaastleti.il,azcic,clecst*Serc'icrlaefcsfuaoarsgci
Nc,wton, lucru clespre cilrc sc poate convin5;e cu u;urinfl
sale, p'blicate cle wallis. AcL-
o'isicinc ar citi scrisorile

lasi calcul a fost clat prut"rlicitiifii mai tirzit clc Leibniz ln
r\cta Eluclitorurn, ln ca'e el a rnoclif icat nunrai clenumirca,
for:ma si moclul cle a lnsemna".

iIr""e(---iubznaDriezea?cdlEaetapd.laoagacieraar;sei taee,slcterhrcecsmittinlsrne*crpao.laeteSlncolccacicclevt\irrcii-llrciringrtcrdag. vacdlee, ciici
I-orr<lla !
farcc
ciiLr

l8p

188

;in1lfrr7i.r-ri(,p)m3"ta,ail{-,ltia"ncDiifpucgfr.iltri)latnitut,v^'url.cic,tc,s,i,iitt,;ic'cfrcio.ilciooiitialrrc.StrNli'tes'etciiiirc"il\rr.s,cccticeo.,turt,ntcltz.;ti.'ur1tPrilsrCliecrpzXrliri-'s,bicdpiclJasenrtjrluaclcl.'..ccaalctietciucaiict1rnsrui,t.'lnca1Csol7rlaesia1c.f2kit-t'.ierctfi'raitsiall'rtt.pi"tuia'loccfralcu'o.tlunscuint'-ill. ;;"'iii;;i;.i-i:la.l'jaillt-lilr':n'.'r'7:.o{(l.l,tir2,ai'lrur..irac.il'-''miLi.pn..rrb:\;ors.cil{r;ru1L1nt1s.rui(ii,."l,i,:'il.b,tis.rci..,,il.ille.i,-}rtn.cili:r.iiclirinibaltaii,ri)li,r,r,tltc,n,.1X'fiiriai|1ca.rttl{;iJIl/,iriit1iSuyi.riltlz1j..l.a'lr.utcdrnnE(uyi;,:1C,r.llr_e..ttlrJii!s1I.cu{rrilrtu,sr..l:lctL:frrrS.,clii;oiiu'r;.it.'iirltjai:aj.iliioouins'do,'{irnir'iINcr;;rcn"srrrsr,rdrBulr.lufliu_Fni'rr..tlotctlra,;,ji;llIrnl.;ztaaririrlrrl'ncrcttJrn.c,{rr.ll:,r,,ruarj,tio;c1I,sJrcterirpSir'b.i,,u.L-ruaelrio,ae.rL:trib..{r,tirrr,lrrlii..ao.ltsuiriiliircuirilna'cr,r,.r,riil[a,rrril_,istiiAt,,*r.i,1-i,,rli.i:ii.,,,{,rllnrn1.nirit.,.ir,c.1rrrirri.,rrn_,n1rrs,,,(r,nl.lii1._ir,r,irtiir.,,r,i;1it,t":.L.i,,'i,li.,,i..1r.*1J,;ilt.i:l!.a,,ii1,i,ii,r,1rr,r..;i,,i1i'o.,ir,li-1rinr1'f\rrl,'.1"ii,:1pp.r,ii.rr-rli1n,','.rrrr,]rii.j1l,,irrrr_,nrlr,t.1i,,<lsi.nii,i,n:il,ir-n{.ir,.lftlz,"ri,rrt.;trl,n,:,..,r,1il,i,r,t.'1i.,,.i..t,1t:r1ifl.r;:.-,llf,,6;,;r,ri,i_,,L,.,"t..,isIr;,,.l.;lr.i,.;..,r"t.t'rj.ii,,,;r,r,lt"irl,i,;it.ir,.1.Ili1;rifrit.l'",'rrr,i'fr',,r.1tk,,l..,1l,.la,,.l1l,,;l,,fr1i,"t"iiit,,',',t,,;r,.-,,,rrrl;;,;l,l,L,itr;_,l.,.ra"j'.,.,ii;nr;1;r;--,r,i.ir1,.i;j,,,i,-;1,il!,f.ir',-;.;ni,,',,,ti''_,iiin_,.,t:r.j;iit.r,;r.1l,.-ll,,-i;;,,,,ir.r
"s:b.i.ci.iiprc,',rr,itd.rt.iitnr."ricil.ilclr,('r.lsa(t'' t-'ottstatd cL cristir o itie'titatc intt't' ant-
ir.i,lrt,,t,r,, ,,uj,, ,r,,' clt'scollcrrirc'a 1ui Nt'ti totl cst'rl alltt'rirlltr-il"' rr,''i:i:r:l:ri'il,,(atu.l?.ifriiat..oj,:ara!g;(,r.,trt.,.;ea.:iJ\:.s'[11iil.crrS."rsis.onra1d;r.1,di,.in,-rpsa,nnu1ld]ta3,ri;ms.c,uririumrr.siiqttlia.za.p'-.uNsci"a*piegiaInu\prttnirs.igiifutrrsoricetl-,iitiannnrciialit,tdtr",s;er.irirc*rn..rrre,,s.,.urnlili,.,rrlri,r.,irjcr-.in.nrli"Jnir.lt.liiisi*rBisr'ti,ai.i.iiti,"i.rmrr"li'ir.;.i,i-,;,"r,,",n;,r.,f,'ln,-1irfa;rirr,;t,m,,r.i:'r;,,ii.i,r_icXil,lir",,_u,i,i;tli,it"l;:,i,a'n;r"rt,-,t,
nr,,r. ,lilcrit, l.i i inrlirtrtantir. at"ctlt lt turi<lriltti clt' t'rpli- phLoA,Lhi"[r\=.iurp.cuoiliSci.lC"ncu$ne:e*.,aiin,d,ntuCt.cc-Tir,t.uMilu"eaao.u.o:.vi,r,rbn.c,tavliuecluNriceli.iiend.\lurnd.en\dricg-.,O-re:attlv"aeocoolarltTce;in1oinetiiie;nlesGeau-tsan"ocvor-ua:bailrrnAfLcdr^ocutraocce1trelsrivetue1gitudbcas-ial,sndfcaunmsilHeucnntliiaa;zaeiscapistrileeittracnmdpditsirJeenu.ritldaartie_uicrairrn.xio:rlrc"itntpavnief,rpsr_aso.,diattd,iiae,lclsr_.btoatccrjee"jLu,.e,rrerr,'ii.aJfosaanlauopoidr-rueuizreaidieunlulrtstsmrricnir.inccrrm"dieurodlal.ipLsrrp.uaierarin"Bai_,,r"p,rlrcil"nIiu{cufef:Xcili.tl.inei""nrie;s"rac;;cx*c*pA;oteitrrs,siraoi,le'iit;.s;ndarm;;ci,pctl;;r*.uas;ii.;ia-e.;ii:l
lrlteia tlintrc cr'lr t'fi,:iicit.'r-tii ;i avarttlijulrli nll"iit sltt.t..ll
rlr.,tiit rtol:r1ii, ni.l r.slf' tlt loc ritttlsJ, t;rt Kt'tl 191

tiu cstt,gri.sit r,irtoVlLL, ,,iiiLrrlcit Nt'u'totl trst.t'lrrirllttl itlr-t'n-
tiit'r irl ,iniLliz,,i". .\rr'l a lo,rt prrblii'a1 ;i *n ,,('olrt-
i'-rnit'r t'\tta.'i' dill rplrir.rritati'a
sallnti, ntai trtt scatl'ti
rtrsrciul'n t,pistolicrr:l'' tr]td(. cSl iefe{ll'aituji
scrisoriloi' schinllrirtl
intre
cnglczi, c:t.t pri| ire la lurrrlal'('a .\nalizt'i. $i din act'astil c'arte
sc ili:spt.inclir intpre_*ia t:l:r I-r,ibniz tr inr'rut.tt1tta1 ct'L'lt c(' ('l a
cnpt'iIrmsiiriucIric-iprniltr,it-ii,tctut,P.ic'laictililittc,rcii,'crizc.r,rla0tc,l.uitncut:1l-crr]n1LctaiIlllo1arrNaiirl,il.iLr{ti\lal',,.i$c1r.ttilltl:r1lrlttbit'll]Cnn1rliC.t.pttrn"itllii1t:ti:(l(:,ti,it(ssrltlrtit,irliltrrtf,-itrjiit'i']tS:,iiciitiuiilillitcrtc'S.'ttnti(t''"iilrrt,tttr)crt.(i1sa:t'tt11,,'l'i-ititi'fir'llt,isri1liA\j'ial,tt..'-l'dulilrr-t''-ctllj'i'ricl,;t'trltltlit-ltnuiliti.lt,rir'iilitllrrtitlli'lili'l.itllist'rul:'riligttiirlu{iriltl,itt"1l"
ttlls1rtr:lr-r,Lu6-l,l'.rrri1iciA,c,rt,tt1i'ii,tr.Let,,r.i,iipl,1rllltittpiianlr;i.,lolrlrlt,tXli,.fi.,era1tit(l.'Itl,lsrt,\7,'srN{,c,,{}-e;,c1}rulri,.5rc.trhi,nllr:,iadiIii,'tvtt;lltaittucrrat[tl'tllittrilltvciit'cl.icizittrtto,lI'l,ri;s.)ri,1f"ilu.t;'tt',tii,'rtJttr:ril';rrrrir,i,Ls,rl'ttitiit,icIra1,,t'[.;'l,r).tl,',l;oa1ililtirtilililltil'tr,lrrltlat.l\"t:t',',i,1,rttcr,l;*litrtSr:lli,.i..ritlvio\l\si"itlti1llrlcrllilltr;r'iiNt"ttrllr:'l,eii1irrl'it;-r','inrr*,r'',r;1tlli1rrritl\iliotGrllrrri"i.lir'^lrlritrilttlrl,lrfitt.ir"i,r'('.r'ic,'irl;'rirt'iritltc''ii!'ii-tlritr1)lii,tllI4,.r,lt,t.'f'rrttiil,rilsittriclltilgl-i.i,.'r.ir'.sl"lr'rl:rtarlcr,',i(lt''ii'Ilr;l'i.l.rnlitrri,,it'iirrttil|'\r:l:rlit.r.'rt!..'ltoltllir:Tiilltlri,tltt-.llc:ltlrrir;_'licrltiIlrI---,-'
{uz;i.se,, insu;i tra.fcltizul d'e)r'ln'fsiiiii.tiilil, cl]l'r, il rlln(l:^('ut cal-
cului infinitczintal rtl lui J-t:ibrliz, :r-t'i:t ' lt : ,,tltt-:i ]lort te
inchipiri c,{ trttloltrl Plincilriilor rnlilllncit, iri.'lr r'i iLlltl'tlle

1t0

aacincauvertetoleanrisutaillaut-snioc.J"Lst.iieR"n:iagb.fuDnhrisu,zoncnmui:oLIaasertreoiebr.iinmaCiZpfohona.trf'ltxilacmiuntnupliilrrlounarmu,bsuasoetpaldultnrtducm,thdeecttiioeocdtiaaaaictsi'aind'uoclniia-, cint',' rLcclclt aie lui Popc:

lea inventatoinu are drepturi", iar Newton, dupi moartea ,,l{a!gra ;i legilc natur.ii sc ascunrlclru irr uoap{t,
*Sir {ie Ncs:toni) a spus l)nmnezeu, ;i totul s-a huninat J',

tui telbniz, admite si se ichimbe conf inutul aScPorliincici.ptqii.lPotri,- L;r:lac$rrcgtieauscbcuii'ii.tsaccnLc-uoilrtrflccillaerc:mIi.uoait,sicllmnsgiataiidclucmaeirrdgietcicaolsittilrul-tilii|ylcai1n-cln-soucumcdclallaela, a;
cina. Sd ne lntoarcem acum la pagina 208
ca sd o citcsti in noua sa furmS: pir
tterilo-otru,,plinentocr-saotrreuscodrilss.ooJaio.retCeadomilnlinitd,rOeddneesticcceriimincbdurieorne1mt6oc7d2taocdidiutraic compa- lr)uariinntcind;ini sii r:ecit lir nr-.ytirc acclcati \.-crsuri, pe carc lc
tangcn- tincr:ctc irriilirnca liri
Slusius, I Llrcd cir pe oamelr i tlc
f in

iul
Ncu,ton nu-i putem juclt-.ca cu unitirtile noastrt-. clc mirsur"ri.
i20---cCleaasn[i!Dscc-aajbuinugiinla1.8a,1c6c, acsutniiocsocnuct]urrlzmir:aatcrrintarcticciuatninecnl-i
pe atunci lnsi nepublicat5, am addugat: ttAceasta este un
taz particular, sau mai binc zis un cc)rolar al unci metode
geneiale,.care se extinde f5"r:[ nici o g-iraeuctuartebedeoacrltrclcauril,, glcz :\ugustus dc }lorgau pnblica, in Ilhilr;solihic;rl Nlaga-
nu numar asupra ducerii tangentelor zine, un alticol ln carc sc analizetzir cc:rrta, r'lintr:c l(ei[si
I-ei.bn_iz in IcgS"tru:ir cu calcuhrl fluxiunilor .,.i se r'..,trro";i.
fie geometrice,-fie mccanice, fie privind intr-un mocl oare- cir judecata cle oclinioarir saalleo,stalltidturtrin.iittrltca. rNe,crivatrcLnc. iPbncniztluil
cigoaerrne,"urlluiinnmigiaiimdcreiolopmtrepl lcisceaanutetracdllteoerpcrdouebrblegemr,qecui!a1;stiu-eplraaa-rleceuzcorbulvrraubrieleolaor,raaeltrtoicir-'
fost rcpus lrr rhepturilr:
matcmaticicnii cnglezi, accst alticol ;r adns o lltar.c destin-
<lere. Ccalta. aceasta pro\:ocilse izr;lalca uratcrnaticienilol-
nici nu smZ miniirfgreinloer;)tenu(cmaami elataodccalcluti'cHuuactldiiecnala'(s'i"slpinrat m.axi- ctr:ianvrgeilc{crz-l,cilocrslletercrumcilna1ojortolait$tiii,attcollnaciiirbiu]lniai.itzeI)imacl;iair,tlitcni.ircealn,i iccioocrnrcrroclcttliiri1li;tleiccmcolacnitiisinnteotunrieti-,
mclorsi
libtrc
c1e cantititi iratioirale. Aceastd metodd am inLrr:tcsut-o
cu cealaltii'in care institui erplicalca ecuatiiior retlttcin-
la serii in{ini1s.ri Pini nlti 5s1i-st)area. Si accste cu- rr matcrnaticilor', dezvoltatc trctrrtat, inccltincl dupir publi-,
du-le Lralca cclor doni-"r t'oluntr: dtt Istorir'a. ntatcirraticilor alc
vinte din urmd se referl la tratatni ]le care l-atl scris ln rnai alc.s dupii apar.itia, ftr
rNS1a6cel7eol1vleitaaosstnesu'a'ipcrnuaapucr-alimmncseiissvlntsiondrleeslumseciafracupclerr5eid..caIececddmeeaanemstiitu"ic'lmiAati.onctdrecsitfcteieceriaamtrlctrirneeatioaa'tdrncd1ecg0ceitauiittenit-i! lui tr. lfontucla, iu 175ti, si
lJu86i 8lJ, .alipoiinmceomi tp:ra:rg'instic: rlc istor:ic il ntatutrrrticilol clator-itd
lu stabilit c5. aceastir
rrici uu tctnei. lriuunlirti:rtriinirrr:r,rrlii.jlIrSiri.bcliorrgi'r'rarivircracdSetofraiap,t

p-dl1u&9rui6ivpt8.iiVn,a1mcomlloutdaas-iiirtrntaemet:aWoolreumtsliirtlmnnLietinuniebsl nat;eiitzdr?rA,inNcbtriebetwi-enatyogd!nd-,ii?nncCugleturlnemnndail,numrucaitntraadditeaveccrmimrsrroum-rtrniileulue,-i - Bine, cla.r' atunci t:trnr stl-rn.r cu ;rrovt,rbul: ,,PiuiL rru
foc, ,fum nu icsc"?
iaci
-* Ijocul a lost Jircut,;i inci cn ni;tc r.r'cuscrui uclc, tle-a.
lnai mare clragul, dar tocmai ilsta-1i spllneal)1, c5. istolia
nalalltaetnfalrtiicinloicri a stabilit, ln ultima vlL'r)r(., ci cl a fost
L,,LdeveceomnfpinasledegrNanedwtvoonil,elneetsluersanctiel'uuxnisvoenrst ouverts"l. ll con-
un rost. I)oui. argumcntc puicrnicc
<lamni, si am si lncep cu al <Joilca f iinclci accsta a provocat
I ,A,Corircnpliac.asut1vllluuil Newton, misurind. uliversul tliscutia noastrd.
si cerul s-a deschis."
-- I"-ai gisit ln cartca lui llcll, spnnt'rii. nu-i rya?

192 193

13 * A doua carte cu plolllene celebr:e

* I)a, in articolul siiu clcsirt:c Fcrmat. nu-maNiulapuctleommesnliruil acum Ia acnasti :r;-,,rble.nI, rnf,, refe r
* Desprc Ferniat? Nu clesprc Nervton sau Leibn iz? geollctriei. in acest domeniu, puter3a
ntrisea--rmcNNal;ttuuige., [cmA"??ii;tuilg-iadtaitostrpu1ul.sllrCocilc'ocardbloucvll:eird,l,aiCr lainrridcgeciprlsouiitt,,scidnacael canclutaits;,ttiiaialcnattrrctccii-,t spune ci prinrii pasi lnspr.e metorla infinitez,imald uh
fost ficu!i c1c Arhimeclc. Si chiar <lacii maternaticicr:ii
sdldduuccbpumitio-anriimcsti-liarrtzuupilurolnit*aclrsnltililiilnamaliniiiitn.testSltre,ic-glnpi-pmiioleacitltececleuccilorc_cnurlsipfnitidraineRtadreleinrt,e1aairnt;etten,lcrueslecimi;.rinnir-uiudllenc,,
de*teU-aitct-uol.buinrabtioagtllta?lia pc carc o sclic L.T. Llr:i'c tlespt'e
Iielr,ton,ln 1934, se afii o sclisoarc t1t'a 1ui Neu'ton' nectr-
noscutii pini atunci, iu carc el alatir cl idcca rn e{odai calcttltt' aulrutc la lnccputlrl vclrcrului ;Ll XVJI-le)a, cil aceea i[
lui \''alerio: ,,Despi'e cr,-ntrul r1c glcntatc :Ll corpur:ilor"' sarr
Lui fl,ttxiurtilor i-a .fust srtgeralri da rttelodn de tlttcete a tan,gtnlc- din rnonuucntala incr-ar-t' a lui (ir.t,aoi.ius a St. \/incer:tio:
lrty'a Iwi Fcnrtnlliicr:mai a fost nnul cliutre nrettetnaticicnii
cei rnai geniali, darcai nu chiar: cel mai, alirturi cle Ncrvton, ,,L)pus gcometlicum" tlin I628, car.e introducc si ter-menuI
dirr vqacirI aI X\rII-lea. A;aclitl iatir Fte:onntrrarrtrr'icFuelrmsiatlttai asa clc rnult folosit, rlupl ot:gg11 ; ,,urrtor1a exhaustiv5".
fost tri corespondcnli;i cu Pasca[ sursa '. -- '\cliciL
,,rnctodir cle c1;tiz:irc", circi cthau,rirc lnseanrntL
1i i-a if;ui.zrr .
nretorja sa cle ducere a iangcntclor. Accsta o facc cunoscutit a
fn cartea despre care !i-am amintit, pubiicatiL Ia Paris ln
1659, care i-Cservit ;i lui Leibniz drcpt izvot" de inspiratic' ---- Da, tiinclci ur.nriitd ic.lcea lui Al-himeclc, cl stabilcste
ariiJe figurilor: .slu I'olurlc1e incliisrr rle' suprafetc, cd.uttircX
"It'ej ntrAusoc.alultlu, lculcldeipfti'irnt'rlrdl mialu!rit ci lui Fertllat ii lcvirrtr gio- s?i 1er epuizezt-. prirr pal'alclrlgrant,rlc, respcgliv par:alcliiri-
pcderlc, cer scl p.r1,,uu insclic ln acc,'stc figriri. I.uCrdrjle ciire
rirr rle'a fi ionsict'crat clreirl pf,rit-ttcit: Analizei infinitlzi- ulrrrcaz:i pcralti. antl)retltc noi'" ..itndrr:cs ll'acqrtel in Elamente

male? d,(: scom.a,lrit si Jolu; Wullis liinrn^it'iitr,,i/ttptcrli,rincaulit,itrfcincitai/l,oorgeinoltnrLor--
vrr-bul.ll-c*Firepustpcuancalcir;it.n: a,,Nniimciiclrsntua-,i splijirlirrclu'tnir. pc pr()- duc noti.unca. {lc trectrc liL
t-tott srib soale".
trir:i,- cclSialt pc caic analiticir. Asta in ceea ce piir,e;te
--- FAodaicrdte;iacFl'eevrmiraatt,I i:r r lnciul lui. . crk:ulul intcgr:a1. Ca,lculul ciifrlcnf ia1. tot dc ia Aihinre'cle
inceputtrrilc .

calculuii,ri inf initezi- ;i Apoilonilrs pot'nrlftc" Din 162i) Iiei-nrat inventcazii rnetoda

ural coboarl ptn5. ln secolul al trt: ilea itraintea clci noastlc' crstalttrlsdtccctsitm.irIaaintxiirtm,iitisl-ui:rmimIilnoalim.nttirjins'oti1itc,,1lltr,nstlluli6cil];l6rii,i.n:srtiint|acrnp.tglate'rtorrltocnii ul;LinnotiaIfiaucciletIe)ccvsu:r--
ulnf, la Alhimcdc. El cstt: acela care a formuiat si a lca-
tizat ptimclc cvaclraturi, aclicit ariile r-rnor suprafele P.1a3e
rnlrgihite de curbe, e] a calculat primelc volume mirlginitc lari.rjiii:nNiliucnliitteIoic;i:rpinl;eitar-ll,-,eGidbuanlrdizri-,niu,upC;uc:lrr1.ti.'naf--l
ic;l-e"-_sg-*;uAip;iir;acflicricfeputca"ut"el.b,betim'psiito:tro'cietlu,cci"atamtaainrctdelctctItitttccr1ellrllcaluslccI ocuosntsrttrruitietsatter- irlilirc care au rictezit culc'tr.
riu-1c dcscupcriri:a calcirltrlrr

rritr,tr: nici altclc ca r-r,-r:1r:it
lit.'r'i,'lorriccili, (ircgnrv
lrlalrrilv, clesprr LtLioubrir"c, rnIIauivgr-to:irrb::i,t..f'ailcs sau
pq-enta la spirala lui.
carc: li-alr
-- Ei rrezi ! Or,Iicrmat, ca aproapc to{i matematicicnii *--,\Dtuelut :zpgeontntnort iLilrtic, riL' Ii
de atunci, a fost un admirator al matematicilor grece;ti, l),-';ttrtt llacii nu ar.c slt11s ShiLitt-.Speal'c.
pc care insi nu le-a imitat, ci le-a depi;it. r:iruic?
nici un r.ost sri se
- ln teolia uumcrclor ! pLruii. chestirurca 1;r'iolititii, se l)rllte accea a rrrt:ritekrr !
I''cl'nlat 5i l)escartcs, Ne\Vtot't 5i I-eibrr iz riir.itiu {Iu|itorii

194 195

I3r

mctodci rno<lcrtre crr cArc ;tiin!a rttatematicl iyi continua\ PROIJLEIIIA ARMOi\IEt
cercct[rilc lnccpincl a1 XVII-lea fnaintc, in F'ORMELOR GEOR{ETTIICE
cliu veacul
cEXuuVrluoIIplIa,inrXf:icnIcXiitcclczsniirtaninlaicrleipnautitiinuls;icvmcaanpcaoutiltpptiecXntXotrtungnvlociabncusult.rruiClcmircXcniYtcIacIllr-r, sau
inr-estiga!ic cn totnl clcoscbit clc accift al antichitirlii, dersi,
atnintit, r:l isi ar".' r'irdiicinilc acolo. Ace;ti Dcspre poliedrele regulate
dupii cum !i-am genitr lttt stiut sit
mari savanfi;i oarntrrti clc lliLsrascti r-a- ;i semiregula{e
lor.i noi ln no!irinilc ver: ri, eri ati introclus formrilclc ntatcr-
rnatice;i au folosit calculttl Iit{rli}r ca s;i e.rpt'in'rc prin t:I ,
in tnocl siml,rolic si lirlii citvinti', Ic'qialc natulii, itt'oplie-
tir!ilc lol matc'maticr', sau alt' terttciutcnclor', tLlc I'igtirilol
vi5leaipi:aiealcnbdi1cceon,rl'i1rttru{nuli'ninktcllo't'i.ro1lLinnras nlciraatiluncrutirtlcnIlrIslitii,,ustia'irrr-ruciati"itttlitcrlrotci:tnlltirisunirrnuir'ii,tnictitrimtliercitiarnimfdit-i"
nit mici, itn cclcctat prol-ilciut ntit;'iniiior irrf iuit tlrari, it
dezvcill'ft:iifnnc{iilol iuintu.siticr,liiir,;ipat'Linr ltciilrctlirsair..tcirl,cstlpi{lcinlccniiif;iialsir[ * I)rcpt siL-{ i s;ir.rn vrcl}r0a a.sta rrti-r slclic,
Solonai., fr.Aata Jupe
calculezc itr loli-nir i-unr zis prictenuini mcu Tcollor Llincl s-u rnai porr"riit
sau intcguLlir, nultc tttitrittri Iizict' sau geclurctlicc clt t'i-
(:o nc-il.nr clat binetclc <lc diminc_.atii.
tezii, acceleratit'. t;Lugcnti"r siltt cui'ltur-i"L lt trttci cttlbc, tlctr- gps{{iiliroleduaaciifercamni,ru.i'-uilci8'i.itpaoiSnpciuii.ttatlneir,riepid:?rleigraC,ebsu'rurmtlcr-.iuo*ci,rtlt,rr"r.-alta"orrt'bd..i'-nzc,uiI'c*lig"puIitrl'*nliercir"hi.tillca"al"r?.gilrlicoIaz.t"ie;!.e.lc;;cilr;d_rri"rirr,
tml cDlcagit'dcciitcactetrr..-tarai sr'i|rr,'it\'ol;ltuiiirtrtul|luiiitial lascmcoalitrrltlreip,a-\t,t\r?'..r\clllll
--
nu-i tot ala?
'-* Acuttt' int:c1rc c:(\'ir lr(rLl , tttr r ttpitt,l tttlti llbstl'lrt:t, tlt'
structnrl lnatcmaticir, clcritrc ctrl'c \'ot r"ot'bi ctntlr'r, rlni [accn] lrcum l
nr:llsdr.ul,tlanelc'litlr-lupmR-.{i;p.aa.iuCaooiIrnp)fcrccIrc:cottl:tiriiselanl-achcicdor,tsacrcas;cati_cArlpscscancseptiis,crs-clurrfprititmlanrpructstlo,c.iili.niimitrrslldlcc.tlci_.;.nchau.;isuig.iil-?ifii.urs;ioniiil,rilRr'lctloca.ipuoi'isrclcslg,ac<la-rltiria::tLiii{p,ricp.tiri..isc.icpa,rl-atuoriuac.n'cs!.tat{itricle,rri.-v-aaaccnrari,'a.tusiinriftlcnttur'ilar-aeiroicrs'lced<!toicra,arlrrcsrnr-ec.ntiralmclsiouln)it,n."cuot;.rcfa.ofocut,tact:ri;cstsmiel-.pet;z-iu;-Ri;oi-lccl;e;c-:arlj:*Ceezli;i-crs;acu-rtitcrr'tc-ioirril
tilziu, alti pr-ictc'n i crt uoi doi. cc1oi.rnm-zc-iocnalat'rcbapltitabc;reiutlcga!uscbc.l\iuipin!nr,_.daoeracnphrrrcoiiacncrii.aitmrle-uauraiu-imaicvieiu,fr,lirstlaccnlioil)cn!civnimn.lsaasicfptlautcaii:rinrnlcutlelmuccil_rt.aurr,i

tr7

aia-i'[ Greu sit-fi clezlcg larrrda, fiindcir Pz!bi.rmci1tcsac-lemvercam, eln, $VcJ.a:llmg-laleebsaut-f"sel.lcea,s.lendl"aoi rss,tpe uuIenrg;ecf-ui.nn;u;om;srceiulie.td,p,c.itfapi;go;lo;ieridclrrieiatilnreg'-ocrreeligniucvolae,,taeric"inuMcl tcaaili

cle accst cub clc sar-e', cu fctclc olcacl
nu vlcl triirric cat'cr si-ti f i atras atenf ia I
Tx9s2^.-.*6-ODsBc-anaauarc,ltdaolu-!csll,nlr;triolcieatiltrnnlraitiell,s1iitriilmri'aneetiicprt-'elcltl'foceaairtdr:Ict0.Nrvoloua."-f-i5l,i1i,psp.visuizttrnrriitcctpnzniciirmlnintsitcal?icliilnorcntliec'ui'tcnilntitn-
."i", .i f iinrlciL-mi t'irrt. clc urit etrrrrci iirrcl glntltrril,' Iuttt c.i e:nf(dttrya3retsr,'':auat{l,'r,etli-zit,tilInu.n..c.r.M9aoirlluluror,iciocrn.'lz.,(cdrit'noir''rtrcJl/er{(c.Ilimos5'xiIrltrdduP;uar!dr./i.rdziiaesdngrcsss-iPueurrptclrdcrI-u,uoaittt''acsic.piat;c,oite_s;lilaniir,riirrdnci.rpact*ccri-rDclili,hafstir,iaiiipnpiinlei.picauursilracrnlitacidicr,n1tc",cii.m(,,rti".izl:ri.tricctct"ia|,,h!,ruiatclti'oii'apipJ-tcr;rirnar1iiliiptl.irtoiuir.'Pprc<rpin,'tl:oac,liaoiprr,cpir,etrocoig,rlit-iisrriuin.chtr,lt*eacci'lor,,rir,n'r.cu,*i*rrt'":.it,,ilir,ii.,:t,;i"iti.itnl.,;'1ric;,ii;s;si.;-u"el'c.;..n.n;,tia'cdb;i,."lpe;i"ia,;i;;icrrt'"n;,;o,*-r_;iiai_t,l
esD'i irt:,-.rttlt-'tcniic"ttr.t.ttas. na-Dsldanri,tt1ou.'rnIiacerlltLl'nic1aluemttiriiarnctlnieitiezlelut,tin;'ittitlp'arti;o'iaarlpiir,iroPrctc'ai(i,]iictnricmraaeidvndaitritt! fllulei,ll-,'.at1,u1lcliul'itiililurPrcrtrtlirpci rurvn'a' lcox'bocuac'slpctn]ri!ieariteg.rcnoordmricec-narliliu.ri,Uet,uirri,t'.iunzni-ii"nt,rt,ec"c:ht.ric.crecixirircire.c_,
"dIt".ignl-aftcoNtcu*l,":lclllr-u.aiisrt:nlriln'grsrllnilLcmcliictsstcealeptllauolccsolcati'Cr c,alcetngi cilrrarlbtilattr'inbl.atrotnteaiualc;catl'lusc,al-sotstinci gapalrcjii.-"
r-r-:.r*rr5-toiaiptlaLi!'I\rirrlircniia'iir lline si o iau clc-ir clreptul' yrt|1utnt.tr:r'itts-l',nri1tsl1o.{:'lnllt)rlcr'lelrir{lcl'doenttnsistr'i;lleit-iinc,srd.p,'rpu,itrnrir";rsctme,i..j,eaIsctceesutl'.nCuprn:orrplI eicnrrclili"cua_ar.,-il,sir."iii,i,Miii,Ti:-airiiitimiirj.iirr:_,,
.u:r,st-t,a.'nnrbi-:lrrteIartilgi'c'acot rcpcn'rltilt1r tpi'ilnatolnnitictLcc?:
<;uti.." o cincimc... ,i.t r"ergultLte, clar nu-$
rlc-*acIoI))laccl,cceuur?JLiOrrgati.lnr':ecirlneeuaa-mifoPlltaorastcoietnIlecal5ecctplianoclci.cadlerea sctis pentru prima

o"'a-*l[ ctresiuc cle ln accl vcstit dialog ,,Timaeus"? cle ri5iI:csalI_rttil:Ilinlirrrrcliniirs,.'rt;lnel,u'rapl*ri;\.'llpldls.r^dlulnulfnt(t'j.InsLep..rlia.cesa'dd'iz.nt,rlttugxterl_e:rt(t-ntrlcii...rr'iIc'ngpAllcprtpilt:tDor1tlrctFr:ih'ccnttlr(al.i,luco.l:'lre]rlernr:rri'tsbaildrtrjrnsr1rPi.clcc,.adarei.rrriacuLiIstelomeraa's)oc,ilura,ctpnlpseed'roIsceSitJattlamrrioli'nr.aocncippor.cndta!nriur.'brur(tpocrdli)rlnlrs,iaguneL-t-rts,tastrdti,r'.i-,,prrtelrnf.iai,-elresra.aiiltd"bea]urC1,elrucscdr(.tnlrtcc.a.uitrfrrcsJuilci,agiisrlrlist-o,clraicirreit.cr,.izrc,s..iuc.Loi;i,;otiTiit1hasuirrraf.tiresaitsirllraifeteeelt;r:trtxrirr,ur,,trLoi,retarsrriiniamr_ltirit,iudsiollr,r'jlnra.aarrutilirbrpJnactcurcirrrclciulo.ie::it'lsr.ccnm.1sitr,l_ici,nc_tlrrs1'arat,iaui,flrilsstn,,.a,t"ia,aliricir(fcri,pil_Pr.cz"dsc,uthfi,rliiio,iice,u,"izp;lpr,raidi,*t.,pf;ul'si,oarot',",u,t:ic,p,*t;iel,i_stro;n.tiat-ai.-rra.lir"r,ic"-rri..,i;:,rc'i.tiiril_

n".i*,Ir, aia-i. Piaton a tostllLrtcrnic irnirrt'siotat
faotul .;'.'.i"toi lui prcfcrat, Ict'tet, n'iatcmzrtician plitr-de
;;"Dbao"ail;;lari"icIi"iIturri.;"*ti_rc(tl;u,irr;l;,;errli;rc;"riliailnhDa",ollcsott,.riurisci,fcc-r.epaauci'1e",rr!'.irgaeii",^ar'oonir,Sccrsrl:auri,oidciigtr'l'tst-ismtcrpsair!at.r,uiittie'tntuaftectrstec'ld.lpatllcs[iau.l'aiu'gliuiDpmldftirauptuoercclctlallisLnciruagt"epe1("ticinotdicluxlc";aeaslircurotrtnritnc;iterdrppti1Ir'sci,ro.,ne..atcia(r,s't)'gr'er;raclricgcitria','e'pecuexsi.stlccl'id,'irarsrtlcriue'.rii,.atttcitr:saslcpa,l-rr,luiiesib'.dstri'ctpldiiriillc.itxalLtcciir-dlsfirsetisiscir,''rin-rItcxurter"irrllcrl.nCuut"lp'itt.rtb'r'ii{'t)n..i<ilsl'trl,l(TI.rtsipl.s.itzl.tciiLrl'liaml1.g,,lLiil.t'{tr)rttctirl(t'iilulrtr)uaI'trtrc'tlt:atrls({resctn.i.ntdo'ri'c.irttt'tisusittill'rcuitiulctJn'ld,'at-yi-]ul

existe decit cinci polieclre regulate? sl accentuez
-PlAatsontan-iuaaltriegntsiccii ln ceea traistl!Eu {in *citua
ci in slgcliul lor ideile obiectele matematice intre rucrurire senzoriarc si
o contribufie original[
purel -* cl a acc.,i,at ;i ;i-a i,.,r,i;if .;;;rl-;;i;'i,;

r98 t99