Campan, FLorica - A doua carte cu probleme celebre

rnisticri stabititiL tlc f inracus clintrc cc:lel 5 lolicdlc t'rgulatt' cop-ircIe{lu-clote1liracsel,fi crczut, rlal una ca asta nrr ! ,\tunci haf,
si cc-lc patnl cloillcnto priltlorclialc, la mai intiicri olicc
cale adluga, c1n1rt si-r precizirrn colil mirginit

al cincile a clencnt, chintcscnf a .cau cterul. J.iisind insiL clco- de fete lrlanr: sc rrumcltc .folicrlrtr .
-ciiAtcsrttattc;tttirur I. $tiu ;i cii felcle politchului sint r,-oligoane
parlc Iatula Iilozofici, ct'czi cii pu.titr ii dalot'i:llziL Euclitl lr,rlit.dt-tr cstc conrl,lls tlirr 1t,lrs rnrritr. si
-lclucltiectPeqlttai?r,troIc}nou?a,tt,rS.nri urcnlitareicc1aellicnastcleecuctacgoccLcoe'mincrtc:ltillneiaar.olndnatitsccphr acitltcaiutcc,l ,tsuictnuctltcrta,rlstt-lc,uiulpucirli
si

;rccste {iLlrtc nra.i t'nclcaproalrc', au lti;at polic.<lri:lc ft'gul:tt(l I t c d r tr =.. f atir planir. i\fa i ;tiu c.l un t.ch i rt Pol it'elnr l rr i cstc I in ia

rlc itrtcrsccf ic clintrc rlouir fctc, cr"r accstc clcrrrii f i:fc frirrncazi,

clc nnrnelc lui Piaton, ca un olllagiu pcutnr itttcl"cstrl cc l-lt un rntghi tlicdrtr al pcrlicclnrlui ;i cri vir Irr1 irolicclnrlui cste
arirtat gcomctrici I I)e altfcl. rlc abia tuai t.irzitt, tluliti t:t' in carc sc intilntsc rnainrrrltr'{rtc alc lrri,
lrtrnctul tlc ajrrnsi c:cl l)utin

.Platorr a iitlus atcn{ia a.slrpr{l. accstoL corpui"i rcrtrllrcabilc, tt'irj. E intii ilLtir

s-au giisit .;iiolatirutiii Ci'trsr 5i"s scric clcspre clc ori sii att'ibttit dcs- - Ntr. cliu vir:ful urr urnn,ugclrhi ipiloclicI'rah',ulb(l,igl.il:t,21'l.ill,r.s'tlt':t,i.ig..uilna
coperirea
Piiagora sau pitag,-'tricicnilor I compnsri '[',

pu-t- Nu tc contlazic. Crt aceste scricri dc mai tilzitr rL itl(:t:- nulniirclcrillcrrrt: {olrnelrzl la rinclrrl lor nnglriurilc planc
istoria matetliaticilor. De pilclir
PrcrclLts, carc lt tliiit irTb, trIc, c'fd, ...plccum siunglrinrilcdicclrc rLf irc, b'fcd .,.

plin veacul ai \r-lca al c.u., asaciitr', la o clislanfir clc vlcrr 'UI ncsutenpgchnitai.crodlricuc,hi'aur arc ccl iru.f in 3 fctc. in lignr.ir, unghiul
t OOO clc irni clc Platon, aiirmil ci Pitag'r.'ra a clerscopc'rit n t toir tc tlirclrtr"
rrscrr:ir,.rcu.f''rgttrilttr cosl//ir't:', -- r.la lltlillcil cl pr.,licclrt'lc Icgtr- unglr iur ilt' u, b c, cl,*i r si
'l'oatc: accstc unghiuri polieclrc llc cai'e lc vrzi in accastri
latc'. if.liigl ucrsltc-scrsntnrinbreitsuct;icrolenvvrr:c-x'ncnpucl ltrtlrin.rl c:i"r. niciulrrrl rlintr:c clc
lilanclc c:rr-r.-l
-- A1 r'rca siL pltrcizcz itlrlttlta. lLcclLstd. asotlier(', ilrr tlaci lr,r.utr,irzir"
am s-o gi:ciesc, te rog tni ittchcitlitai, Piiurintui cofcsprln(l[iit
cubulni, {ocnl tctt'aerlrulni, act'ttl octacdntlui, itpit icoslr-
r:clntlui ;i in f inc' chitrtt'st'tr{a, sit'r';t insi'tsi ll. ttttivct'srt1tti,

col'csirtiirilcit doclccaCtlt'rtitt i.

--- Pclft'ct. rtyrL rh i'crfcct, dtriri crtttt i1i ilchiliui1u,
rlecesarc urultr t:xlrlicatii,
iiin-dcrir\rrlt-ni ichsiinatr ctt ccir: cc ti-lrlti
irrlirul pc cle rost sI aibiL lrcntru nlitto iu{clcs clcplin.
-'- "liiirc, clar'rs1il tot as:r clc 1rt'cc:is ca rut tniuc ci"t ttoi tttt

si

ntaiput0ut1tt'iuclcirtfr.:IcsulIiIozof ic:iIaccstoLcclt'csPonllcrllt(!,
datoritii formatici noastlc actnali' I Noi ar.cur al[it cul.tttlii,
altir viziLrnc a(rillrtcrccrfritlriicoolusfiotn'nutlittcctlmca11i lti1rn1rtirt'trttri tratrsPrttrc itt
lllclltiti itatca' ilc le
civilizitt ia

acrrirr 2500 dc an i I , ci lit Itrclui'i urult tliaillt'ti-
si |cntltt lrrictcnnl llleu, ctl
zui-ce.NAu{ltirl-acimr sirprfcculittr.uIat,lirnltcro,itcitlitl
in mecl.icinir, sc pollivcltc
cluloic't[o)iraattontlr crr plit,irc lit cuuoltiufcle foartc biuc obscrr-atia.
gccrmetlici
in spal iu !

200

cl'ac"--ri1cj"-ittiiLS.tisinrari,-trtromlsoe'-iirltrr!cicciruaa.irtle;mt'rigtr*ihl*linliplialrurliictrntfaillclrozlloriurlrslirlir"tlgp'npdsholitlrialu;titig!n,tui'oicurlapt-mn'natllaeclitetnltroonu.rr'oagurrhcllreniosinnltupervecileazldxru:neeonut!eg'ds'shtttccicoIpcfneooammlileapndu)rirsluec' poligonale cit ;i unghiur.ilc policclre rc;rulate ;i cgale
,,,o,I.,-nlioior',eDii*frr,a.,fpe"cuoisdi.ttleup!r;rlm,rrr,-gi1alt'rllriiiari.isnrrtl'iirtl,cirr,liii*otipi,,t'dzo'crclnil'cictridriicttLrlic--nu1trurlur(irrt'n-icrigljr,s.suh3tterLi3ru,ac)r!tiii,'n't..dlltt'rrti,c,t)e1i.itpa'ciotrmec',s'.ctic''r*viittrl"d<'ecl,cn.rItlsar'rsct;i-rtolll,lla':-r
intre ele.
Credearn ciln aceastd
-- cunoscutd plni acum, clrrfinifie, care nu-nri era l)rea
am sir dibui rispunsul la intre-
bincr

barca..tlesple nurndml a;a dc_mic dc poiiedrc i:egulatc, clar
inselat I Definif ia carc rni-ai dat-o d'espre poii_
r ird ci m-anl
trrlelc rergr-rlate_colespuncle exact cu aceea a poligoanelor
1rlanc regula.te-fiindcir ;i-ele sint coil\/exe, au latur:i egale,
i{:?gnaclaczu(ial pro, liicnchc'aelzourlappaorl:i{eecf lreeprololri,gocnnaioc.nrnedglulsaites)e;'iinutrn-ogd[iuucrd,i
unghiui'i polieclre regulate) . Atunci, cle ce sc pot {orma o
in f in iiatc dc apollnigleo-leansed-lecgcuela, -ftieinc;ilcnluimlleaei.5tetpsotliaebclirleiser.edgeurlnaotcn?-
Platon
-
s traf ia gi i-a aritat-o. Dal, mai lnaintc, tot el l-a initiat si in
t;lina. de a ie construi. Euclid redd ace.ste dcscopeliii in

cartea a XIII-a a,,Elementclor" sale. Ill duce chiir si mai
il.parte aceste descoperiri ale luiTeetet, ardtlncl cum le pot
insclie to:rte cele cinci poliedre itr. at:cea.)i sJtti;i termini
Iur:raieu sa.ptin_denronstratia care arati. cd. nu
irlte corpur:i regulate in afar5. dc acesic cinci. irot cxista

cllro,--lauA. {lT5tnuicsrcuntuctcIcii'sc,rlnc.ettpcemoti'stuplgilebt nsirlitn,rraufcai iucdcaimisscii'lId,te.ueleu,uLtpesaedmtuna.striuic-{rpicrraeictdtaedrneruucrllooatndrs.utSr,acctocialciao!
iui Iiuclid..Dupd-cit rni-.aduc arninte, sirni cam asa: ,,Spn,n
cv l' l'{ ' .1.t in afalir cler cele cinci {iguri desprc citl.c am ciisiutat,'nu
i i,
:,r. ntiti llorr tr. conslrui nit i o lLlta, rrrir.girriti rlc l-rolig';rne
, r irilatcrlrI,' si cchirrnglriular.,., c,galt. intr:c clt,.
lrepte. Accst,.:a vot- loLttttr mttt:hiiltr ung-hiului pulicclru- nr:' ti'iCLirrncgi h.niuursi;ei poate. constlui un unghi
lreasictnlaltccirp1t.aliii;rititi'n,cl,t1'1,:.rpiri-ltctl'lz"LruttniuglhciulinJtlcrr-',lr'l.:tniaghriuarp:iclcri piane nici din alte cloui policclru clin doui
fig.ur-i
llpstt plane. l)ar cu trci
lriunghiuri se poatc constr.ui vtrful piiamidei, cu patru
,,ruchia aT cu ]rT. Or', in a.ccst caz, c cic Ia sinc lntclcs cd. triurrgh.iuri al octaednrlui, cu ciuci acela al icosaecliului.
:ulna ungliiulilor: rdllrase nu tnai rrc -l unghirtri,.clreptc,
;riinnrcrilcpiuanicttta'lclcclminarpeiaensteI suma ttltttt'il'rrng-iriurilor" rlin iurul lnsir cu 6 tliunghiLrri ecrhilaterale, reunitc lntl-unpunct, nu
s,.: va fonnn un unghi poliedru, cici unghiul triirnghiului
cchilaterai Iiind dc 60o ;ase vor fi egale cu 360o, adicii cu
, le-reg.{utluanr:ictai ttctu;-iu'srrii-m! iacliefriiuncrtsirclcindticicrlntrgslitrittinI tlr>oaclilcutlcrtt'uorefgiuunlueta, -l unghiuli dlepte, ceea ce nu se poate. Din aceea;i cauzi un
rmghi polieclru nu se va putea construi cu mai mult cleclt
a f iincl conlcx;i arilnd nnghinrile piane ;icclt'dicdre egale
,ntre ele, iar rtpoi, f ctliadnil' rcgttlat: acela carc ale atit Ie{ele ;ase dintre aceste unghiuri plane. La cub, in jurul unui virl

202 203

si;;flst,Li,ril;li,,o.t,ninr,-'*ir;ll,.t,ii.rlj,'c..r;r-i,t,.,-itl.rii,iillt:il|,lci,tIjor,trr;ir:'.i.l,iccritp.lrrt'(iltirl,tiir',ilrcttttiltriilii,qarr:rlli,ti,it')'glngrriit:rtr'rrcu,ltlr,"l;ci'1rsc.ilt{)rlli,pr.t(rllt'xit)rtiot.titl-rr'riiltlt'l{,|i't'lr'{ttgrut,.itl'-1tl:t.l.t'iIi'tirftrllpitr'.l,;rtr'tit:nr"'it.,rtriiorgtl'acd'atatulcctcr'lt.ril.ic{l.r'lpt;:ipilitlt.lrlrtal'arrlrntut,ltStri,"l,tu",'(r''rrlp':ltl1arr'll'1ooiitlltttl1r'r'it'1tlrltg'1lgs,I:lilol'rillo\:rxtui;'-li'rrl)cr;tlir:lttrttllhrtlill')l1lftltli;t-ttt'*tit'-t.1','C'ugl''t:,l(tl'1hnoa-:pillr,illl't'ulu.cirs':,'1illa.tllltlllgltltai{rcgll..l''llrl'"il.'(i'lt;t-'i,:'t)i

"cc.,.r,l'ijca)ariric-;,,oii-iIiel.'i.,nL1,.t:,:r.,,c".eti",llriritiri''i1.icii,lf-t,l',o'-i;'.,r,,.!rsrp',t.',tttri,,,n,:rtst,.lil,iiri|rt';rri-,,irltri,ntt:i,rrtilo"i-,']!t,,t):.-l,i\strllta(ort,i'rc,sicnsuicccuts",s,tit'!iiaioc,"rt,-r"rp;c(tli,lont1l)cla1iliie'ict'1ltli,1l-r\rr{1cttltro'?lln^1tcilo]tlif)v-nsilt:rui.'tc'rctlt'lr'tt;rl:'lr:tl:iitas'tll'irittntrt'tltlutirtrr,.i'.tsidjiltr-d'tr-ttrlitcul:'.1ri',t-irt:antst'ssijl1nr-ttriilaic-t'.
;i';fi;;;.t",li,,i,iiii',,'rr_r,'rirt,;',rltr"r,,it",_is,r,ntJir;itilq,rtc,t;,,,ri-,inr;1Aintlrrl,tti'rc,llcic"'t1r,llt.i,t;,e'""'l,iiol';tL":1,;,titlfr'1';ru'rt1ilr'tg.iu;li'icca1"'utt.tiitt*l1l.c'ta.r,t'rrIs"li,'til'oozl.riltu.utilill.oplntit,rcIilirtl1"nll+)r.l,'l,relcilgiiact.c;-srit(tgiijalilclrucl,ci{nttlllr,.c(,ht["titil'c;rl'Icrli.its}iirI,lcitIl'trc'iai*l}'Grie'il-rl-ccsrla')r-iIti:'lirirt.rutr'1ir''':rr'I\rlcri,srcr'rrlittrtlrri':l:'''trtloi'l'illrr:t'i'l'trii_ellsil't'lu''itrlciasl-ici(rcllc',lls'i;laasl'il;rll1cilt'1gtlult1iprl1scsaj itc;itt:ra;crs.i!sli1itcct;rstiri't'lt-iii
.Uig,. ii I
...(.urrrnu:Ii_i-\l'..}I.j9,untlcIi=-.trtrlniil.nlltl!r'}tlt.,

\', al vir[ui'ilor';i ]{, rrl nluch.iilor:' 1 ='= 6 i 2! Si aclllll'
- ,\tuttt'i .,-,',.1 trcrif icitm: 't -l
i,titlJt,r:'i,r;:icti.r.iarir.iturtdinn"i;rlrj,iiti]gii;1i,lrti,-."c;t,t.nai,rriii.t'iirnc.aru'avltllllruirrpartciuinr;occ\ut.,cBii;l'tsaisiC,lriati?rrrg-1t-lachtt'ucl\uscitijr.uu10iaalarciNilri2pic,b-SclcaiIah,clcitazt5di(dii,tfii[cils(l1giatciil!ir.,git-*tt.il'nrl.)sa;gc4Si;ocli'i)c-i5.bon.Ic)ln!,aiItlina'oils.iccttLtltti<caor:t'llcurricrld'caic.cct,ccttalc]clltcnr-zlSliaiinti:Nci'irunil'ieu''trli:t'n.tlsllt\utgiftt"ilcr,lnnrnsirc.ttglilracriri(rQnmn;,s)\,rtci"'tItttalllt-i'!i-tril
t,iil- ls

.ill iiiu"S-t., iul ToC , undtr OC:R estc cc'cullric ilcttttlscris
'aza

?c{

tritrnghiului A13Cl. Din X'I ,.= Rlr/lT, rezultd giC *- #V.1
tp t.
si clcci O1' ,,- l[l't:.. ,r u,,, r )'l tl lIt ;i l)rrlrrr t't riirlt's, -
ilc r,-iclici tiin
nat ti iulgh ir rJ cchilater"ai ,\I.',(1 . iaturi N,

centt'uJ lttI r-r lrcr'1rcn<1it:rtltr:ii r1c ltttrgirnc )l lli/l:; ;i ai Sisit
astfel celi'-l r'?r l'ul-i aler tcrtractit'trlui: 'l-, "l' l"i, ;i t' l
_.Crrratijttc|rr.icllSitrltuli,lrr.iitritttrrlir:illilric.clefr:!t
,s"ibi"iCAirlrrn.'iirrt.l;.l,e",ir)pi,ra-,-lif,,.trar.-..ss;ir,i=*:tl-itnr.raru-t.i:tl:cpliAl"ci'til6t'oii.lCfsil,t"tir"trt,ilki.cP-,,icrrtiilti(rls'c)i.iptrt:tahLt1l.il,iit1,)rlai1rlj'll.lstt1aat,tte,.'qce-iu'rrc1i't;,;,cr.ii',t'Sicl,'sril1llilo')it,ia1iuiclfet{clioitr1c.Lzt'ir2uispr,ct,'v't'tl1,lb,il^ll,-iorr,1lcr,nt.urilttr:',r-rl{;cu,li,h,;rtuiL1atttili1et.rriifel1li..ufa:rt.-rloiilrrr{'1nVrnarer:ri1i-;t-t,itiiilttiLr0t,-ilt1'l'itiIlr'1tlt,itllrp,ir':iorricrtIt'}tortrtrtarl'r-teiri;trrtitltrltnrt':ll,tic""r'ln'L1tlr.t,tlit'ilrlllltt,it'lr.Pt''rt1ei;ltiltiltll(n,t'i1lcr{ici"(tlttnlirl"'irltic}co'Ialnlrlriolitl(Pl}itlrrttllr,:)il'-l{(lll,litltill'irjlljl.ltt1lil:t'tf-lsr-iill'-eiIi:rflrItt:dmdl}nllilpllr']rrtil.iralu-,ifntltsc'itc-'lilcr-t"ili("rrtl]'l:(tt."tr:ri'ai']":lt;r-Ll(ir-Litl'l.!
Iiig. 116

:f;*;,-,r;*;iir,r,,;,,riciirrr..sf rii-inir6rucl.iri;rrr-.1fii,rri.(lri,:,(i,I r-i,r-tI'r-lir,.i1*'i-a,:r8l.lr.t\'lrir:otr-f-tltiitill. r,rr

rlrr'

^l2- nrttclrril ilig. :i71 i .\r1it.i .ltl.l, !r-lri.-t tlt' lrrittitlr'1 , trrr
" Crirr,rr,' niai titi
tri ,,it,, nluchii' ltl'c tttr ctth:
st"ti'-a.i itl^"t, rrii .,tti,irr',, tn:abtr l.1.it I(;'\.ei)i,lr1lrrirr:er1-i,*1ilirllt(ili.t' l,it't sli'rrli
f(''i'{r-
;rriitilrri' ccrralitiiti si-a cistig;tt

z"e'-llt?"1I'P,;i;i;ll,l'i'pI)i1itrrrl 1 st,Liril.itritc, prirt.sr-rlielililtr, p1 in lt:i'iilttrtrine
'tr,;r-l;t-l,,iirr.li'lri,'li1,r1t'rirl,rl,l;rtll,irt.ttcl it'lt't,titltl'i;irlrlll lltrlt l-t ltlttjrt:r'
-i. 'l' la
i' '1i;ti ltt, 't'-:r
fusi,eio;t;it_I,c.t-i;i,ft;iiN,lri.rlot'-a,-irrlintt'ilr:lurili_t-'*'tiar:l".:ttir.ipi',rrsliiriispe'n{l.xlarsccaclgn,e:'(toia'clnlsieutIri(ll[i'cl-urositrttrri,;cr'l.rr''-iiLrt't.loi(,rttl'll:'t'rtnI-ultl]i)rl.ar;rzllrilrrltiiltl;.tlrcr-J:llltr{)rcrxi(lc;iirrri(c{rrc;ri}l-llteri'irl('6u,'qtr:lrl,talrltzi']rle",tt

206

:rlitrtrinci ltttit cu Jinic ;]uuctttit ;'i altll r:tt liuic lrlini' 4F t
51ru-,n-t:\[-am1i..rolipuce(cleS, ccrcrvnt-a'zii?c;{tfiig!'-3g-B' ). t:u5, 'r'r'ri .ir-i :tnittzltt t
?n*tr-o^,lfttouznic!iic<clclsncrucratczsirtattbtrilici u! b ttllri rttillr.l ;i intr:-o ;-rozilit: -\/:'- \
rrriri stirbili I I)clincsc c.a figlittlr,,' rt:$ttlolt: racif>toce tlot.rit' I
tpi,,torr:l.)i,ciur.lr''tdi:rircrotsrtcirrs. tpirioculiit'ici:rr-'liIrtuctra:lti;l-crttt'l{cgccilllcus aiilarc',l:egt.nrrtllliactl:cci l1ailtir1etcl'lcti|tr:rlr:llc(tplgoiu}li:l,trll'pcitaiplrtrrcot:*-c I
I

i.!.r,r . ttttttiPolicdm rllrt.ltt't: IT
alitcrr [t'!.c ci1,c vir:f ui'i - /-,\/l
atit-o*irl)rr.i1rrt{uLrt1i 'lcllilltucc,fi:pfctiJaitt:'0rlrarrclrc's-:tcai1jlil:ra'r'c:c
ea,lrr1e*.,I"1)ri,ir-'l,ci8cir,Ll,rcc"iurcrl1;cicf;cit'rir'ssirli:iifl-uclrr,rlj.t,t'-Cit{c'c;ri\Lls,C'.zcc.iitiiltsiptclelctrillttinacidr'l.O'rit:illtrrcrIrnr:ictll"iprtllrrr:'t"clrctl!drttt:ttur I!

\l

, l'i't. li:l

-,- I:ttiL-1, rlesittrt lr lost 1lt'ira.siirllllrt, l!;l ('ttlll tc-rri r-r)tt-
vSiint sc,arrlautp-.iticlri-z"itrcmrsldtuercilico]c)ctaccacll'crtrllc:-aI tu{cfae.r'trltitrur-irt:gfhii.icuurti ! (tig.39)
ralc, G lirlur:i
echilati:-
nridc cu bl.rza
1tritrl2piir.nti:uact,hsii.lrpAlpaaprl:lcslrcad{ttoltrini raact rdrlisnti.dboauzi ip. irNau-
girscyti cir I'r"rrin;r ltri ar:r' itr ar,lcuiir ccvt teLiau? Parc r:i lr Ii

susPCudat cL: o l'txi-r I'i r'g'.it,t1l . gittlt si'i sr.' i'trtlrt'iotfirs fn q.-1,,
;;a.l lu r,itrtui: i !
- - Chestie cle iruagiualit.:. Oricunr, c intelcsir.nt rie rcrnrrlcat
legitrma dintrc piinlnt (cub) ;i ilcr (octaedru) stabiliti pr:in
l-r':. JJ r..cciplocitatca. cclor douir r:orpuli, luct't.i cafc il lost cuiioscirt

2{19

t4

0Ii*

l

i

Fig. Jl f'

iie ./rerlrea lui llitlgora. l)er it(r,ftnt lltnt cu Yt.r|ha si nu an1
ctahiiit cui1l sc cr:nstr.nit:;tc 1i sc tlcsfd-sriari r.ict;ri:rinrl I
m,.1-,gt;-ria-clih-rii}is)irlicidlleer'ligs'ffaisLlr;i;ir.i'rll'i.arli'rtCrtrra'e'crirczf.lircc';crpacixii;-rrr;tursrmgurio;irr{cc;airIna:iuotinirrrrri{e.'r'ig:lrsl.iiccluiiins'urtgr:r-ifhi.uitrc,rqli:.rcrc.:hialiiernla:tt,oc,lrlrr,,ar"al?ret..
,{ Iris. -tr{)r

dc,cdtr-cgf:^,:i,cacii'cittteita,\\,',aa. IJ.lcritc'teic;1. lui [rucliri rlitcli rru,t'sc -1t"o{,11lr.te('o'inl{sutlririrlcl iaAi ,.\s[it,u(-.i. ;i li 111 ii--,,,, sinqur
ninlt:t {fi-q.
e.xrstd problcnra dc a i'scrir: un t;clacci^r intl'u' cril;. 1i;ri ti.l5.st*.r J
x_I;V-P-aar;imaiXle\r--paa.,r:seeccl.onr_srilctirmr:erilccic,lu.uda. uc{iori1ti- alr: Iilc'r'rt.cl.r.,
a scrist, cic Frr',tiJi ,jrfipg-i-.taN4nirltl:i.irtllrI,).tllrco-tsr.tr;'rtitnr:,t:iz-ur:i;btiiitrti,irctat:srtrliirisicrd-aiil,l;illFaesjtLcurrcIi{itI,rdlneculmcclhaairacb-aoizcaint.aicj.otcindnrrltrrrciil;au-{i

D{caoaatrsrmi-rtb-,aacuFsoiotctmrdiaauprsrm-ttuedsupdial-ntrdf.ildneDcvl'av5reHmer;naLyarctesp,ucscltiocac;ltrrlicnlt].cuIavldi-claiHe,'a.acvXrip\lIIrs\ear9ic_xll.aaeV,,nlc,'cp_airtaaerrpeiararro,cas;arl*pir"e{.tt:fej.i*.na,:sr.islgalcAnu.iX-ir"is\u;e,c;'d_i."J.a;iiiii,lr rlusd- rlin cciltr:rii 1.i o,,.'u urnrlitia clr si nrut:hiilc laieralc
.a ii. i't tlc lungi'it,a i\L Fit,;i 'l''1",celc ci',i r,"irfuri iar
f"[r-I]r . tr)iu trirrnghirrl ('{)l' i-r:ziritii: 0l' .,.
conducea;coala din Atenapc la lnceputul veacirlni ai \rI_lea, [)( afF

frnportant este faptul c[.problemere regate rle proprietdlilei rt.lli t"l l/i. ll ,,r,, t.r-r { )'f l/ ? li. - {t ,/ -.i-"]t.
poliedrelor se riezbiteau ln c'ntinuare,"incepiild,"poate, di

210 ?ll

rlcl)ilractiur riiirluMirerlupncgr'lpitcttrcrralic(u;lXat,'a(lcT0 t' dirr ccntrul pitratului

() l)al'tcr;i clc altn il 1)1.111c-
tului C), obfin virlurilr:5;i 6 alc octaedrului j.\unc ntiii
Errcolllcmlraruumrlp.ncsiinCcmtlaeaabliocsniltuii'lr,tmiiiLcnucari:lrlctircltrhliesctcrcrocrmsacuauncrcrcclcrdhiucucfitlLcc;tciiudpaateiifpcdoiouieridlctdilinctceelsiatnimlLo. ccfiasolacicrlcotolaeusccaiadltiirnairtut.rta,lcc\l,ItlIiIspu-io"callliliclrcl-li
^ lcoslrr.ch'ri1 lLrc, 20 clc [t:te, toa,tc triurtghirrri cc'liilrr.ic-

rlr. ic .

--' I)rtr tttiii itrc [2 rrir|rri si iti) tlt' tttli<:lLii, tlirl lit'fllt't:
rirI lrurnin<l cito 5 rluchii.
-- [a stai ! l)acir din ficciti'c vir [ 1;oltrcsc cit.c 5 rttucltii,

;i sint 12 vir:luli, ilttrnci rle cc ttu sint 60 clc muchii?
l-iorcrlt'io-lsrit"rrtfd'iscicn6ru0irt.c:,tl'-r':l5iftc;i tftctlol,cfti!iinvtlpceirntinru{cita'clact'scinr,titn'Irauilaicio2s0acccluh't{tclu1ic.
irr - []inc nr-ai ck:sctircat, tt-iLttt cc zitr'-: i LlLr ruuchiilc, tlcltcir-
sr.

nrcalgir, o scot la ctrpetc, fiinilcii cl mrtchie c'stcl cotlritni"t llt
rlonit r.ir{nri ;i <lct:i linr siii socotcsc 5 ori 6 carc fac 30 I
' l ur lu cltzttl l r,tclot'. l irtitt<1 scft l)tii ci. clc sitrl l liunpll l irrr i,
aclicir o lzttij t'str' totntrnlj llt it r ir'[Lrli, r'lirlrinc sii lnrnlrlic;ti
pt.5ctt4!
- ()otnplic:atzi Lig-ulii, rt it:i rru ;titt cttrtr ai s-o cl:scnczi I Ifi:. l9

A;lltl-ci,titcr,z.l.i.ep{ca.r.c. i1i .mrit'aImcitctlpetipintc:nletag;ionpaclcnici.nacre sc iinbuci nnclc pr' ddcarl:ltla1s;.'ria-ltclrlcrti.orusptioti:stai I Vtitl cir.stai crr r:irslrrrrrsul pc limbI"
urttczcala apci I plir"c;lc cit clc Llior sc cJescncazir. nn

- Aici vloiai ruatta si ajtrngi? Lit simbolism? llilascrtr icosacchu. lntii lac un hcxagon, ca. aclincauri la cub ! (Iig"
rlc asta, clirt uu nriL pot irnpieclicir si nu-!-i admir p_rttclea rlc:
sirm.iar ascugioli,ncpacc!,i,cipa.lliScne:(lp)iormcrimrapnaoifaclotsctlicigitaloapbcacu.slCtacriiutrp.nlgitvehncitc;utlicin{il-foaigliuedrdcercai.nru,itbsntiupnzlcrc,. s4col2rnp):.tj"tA1a',upiaocsilciulrcnrri-c;ani,Ltpu.liim1raul tncri-<xptluccrniopcrct',ninctalagin1oitnccaruilomcr,rlc-uagi;tuclvacatdcieca(iTladrbtecTccll,.cear)s""'i
Natr-ual cii tir scllmir dc laptni r.:ir totul sc vcde in perspcctivti.
nu-i clr:iar: asa c1c complicati ;i miL lntleb inglijorat cc s-itl' Apoi cl il f ii,calc vh'f , in alalii dc cclc 3 carc aJ-iar!in hc-rago-
intlmpla, tslaL'clxiiircar;gualavlccaaflacnluteizAiarhi.msire-c!ilcar*ninIt3cslict ;i dcsplc nului (adici c ;i cl), constnricsc altc doul pilauridc 1;cnta-
gonalc. Pe pitltca vizibilit am, clc toatc, 9 vfuIurj. Inti riimin
polieclrc:lc nuttle"tr'

sau irrcit , dc cele 4 policclre rcgulatc stclatc? Nu, urt tc spcria, 3 irc paltca irrvizibiiti pc cai:il arn si lc clcsencz cu linii plul(---
nu a1r'l sir alllrzt:z clc dclicata ta matclicr ccnrtlic, tnai turtlt tate. Lc vrzj? Ca sii. ltt urntr'ilclti mai ufor lc nurncrotczi
l, 2 ;i .3.
212

: t.,

-- in ;tdei'dr ! Te pomcnc;ti ci tc-ai lncrrrnr:ta sd-nri nli
tlrr:u i in ;ir-l1 polieclml reciplcc lni? !
I)c ce nu? I3inuie;ti cinc ai'c siL fic?
*- Unincl crrlc cinr:i ccrltl'e alc fctcior" triunghiularr: k:gate

<lc un virl al icosaech'ului, obiin un irentagon " Ccic 12 virfuri
sc transfolnrri. clcci in I2 lefc ia.r'celc 20 cic
fctc alc icosacdrului rlerrin 20 dliccnvtairgluonlia;ltelt,:
tksdcrattlrulu.i,
c6"ci dcsprc cl cstr'l vorb;r, iiu-i a;a? I)al cu r.lt:sfir;ularca ico-
siterdlului c0 sc audt-.?

-'Dcrsfd.irrlar'(.eI? Parcii. r:riri cuprins cle iln1czr;rlil , cr-1n.1
<Jc n-ai uitat de r:a? tiite-o aic;i, ilitr-o formir loartc <:iegaritft:
l0 triunghiuri formind un pnral:logrlrrx crr latrrrile l\{ 5i
5 It, iar celr-.laitc l() distribuitt' de o partc;i drl alta n
r:clor

douir laturi iraralclc. rnai lungi" lnd,ri clupti. rlLrchii ;i apoi 1.f
lipc;ti (fig. a3).
/
* \'-a s;i zic5, tri:iclintre poliedrelc legula.tc: au {et,,: trir.rn-
ghinlale, tuul singnr: arc fctclcr salc pitlatc;i cciilali pi)nta- i Il

goane ! I)esfir;urarea doclccacch"uJui c foarte ui()r'clo r{,'a,lizat. \

tlitc cutn o vi"rd en: {ac un pcntagr,rn rcgulat pi pt lutur:ilr lui 2ls
cot-.stLnicsc a,lte cirrci. A|oi r-:opit:e nrodclul si lipcsc ccle

doui dcsr:nc dnpi o nrtrchic. E binc? {f ig.441,

Fig. {3

nhrl-i L,in\llnaiirvicclsripsnruc,cliuttpiii,c'llni vcotioLtbaciillo?iiasrr\pnclcuuntrdctcalicglpca<a-blruaat-ici c.i itrcet'ci 1i clesenul dtdsclinloeaftri'cllr.cstrttsrrorrciiiiriitrclr"rt:Ittirics'IIrrtii)lrvcrlurliccatriartlncasiictnlntutaucivrllnuctiiriclcstiriaarit:(muatotir..iiPlrrlc)sr.nXrrtirisitt\,tcfraz.l]i-iir[-r)iatfi.ca,cctrcrc,,21aisig2,,ltrlr,itrl,r;r,tc-jrriracl,rra,cJb"ls,,iriri1'iin,,iilpi'].irl.c.irltc-rrcouiri-.iruloiiittt-"atac,i,.in',.;c,irili.rleu,tl..;-rilrh;f.c,i,cue;irur"rirt,;a'ilii,r;i"it,i;'nlc,r?ci;r',-rr;i;ioanlii'lrfi*tr"tn;i.iic,o*l,1,,"i,_,-.,,
SiL incrrc, {iirrclciL rru-i priml
1? fr:!r, atiica citc sclrlro a1'c scurtrtrl chintcscufci, iii,.'i";;i;Ll:;,1tl1Lu:,lltcr-L.l,c1'!il]i,)r,^1,ri,'r1ujt:1rtln,:ti:r:r.-ri,r.z.,<1lcrlit;t':i.rrrttir,,l1l,rrrl':.rll,"ri'c.ls'iilt-ii.lctt.si:rrr-.irat.i..r\ro,stcri'<rlrr,.ri,irlxb.i1Liii.ti1irirznrolriilJtl'inirlr,ir1..;.i1i,g6r,i.,l1;;:r-st1L",.1p.1o1i,,,dl,.r1cI,'ruLrr,iNr:i,lcb"r;lic;.;f,rolcrr
lnfirf i;arc !
1i.. 46
.di-'r oari. c?nc1 l-am clt'st'nat"
zocliacrul;ri, clintrc toatt:

lrolicch'elc rcgulatc insclise in accca-si s{clii., cl cstc acclit ca1'(l
sc apropic c(il mai titult ilc sfcrii, tlecile invirluic
pc toatc,
asa du1"'ir crlm o lacc sri ccrul. Ca si cubul, cl aLc mai nrultt":
vir{uri clecit frtc:20 <ic'vil'{uli;i nurnai 12 fr:te (fig. a5).
s;ittrc--ca-liuAPplrcl,n:epuirLoanlsirctsllctrimlllrln-'cbarloofllmllcaIrl$titnctiii{u-roaelcitrIuis(hcp('LrpuiuskI a'crsatslo;iinpoJim'iiev,pcacliis'rr,t1:.s;ir-:rcbtloaorzlccauclapccagctlirlcuolcll
clcscrolreritc clc cl, Ncrvton a putut stabili lcgta gravitaf ici
luantiver,:;risa?lcn,citi,lacliunp-airncr it;tiar,trlt cir l-au intclcsal p,rlit rllck' r'r'gu-
intclcs
clin altrziir tr, ;i rrL'ttririt
irini ;i analogiilc, stabililc ln antichitalc, ilintrt' rtcc'slc
Foliech'c si clcrncntclc trnircrsulu i...
E;tittrr ont fcricit clacli 1i so l)arc (r rttttttni ulild n-tt.i
gliL-rt ! Ai si"r si asta, cincl i-a vcni
;tii \-renlca.

- S$5iai ilnturcliirasrrItdulcril-ch;icvib,canc.ibitn?ailsDe{ra\p'cciiaa-ri.-pnati;rran'1,ir,-m'!ian'ilcldoltrc't:ciranstlit"itsIaoiLcccuotontti-'cLlilltcl{ei-taa:tit!
gis-it

dr,ck:caeih'u I !

r('{,-r-il)Cl'o:rcrcclsoodcccoatcllci?h'uCluric?zi[,ciLiconsuabeidrniuuli!es]{caci adrce-girpaobl[itinhl:-uali
4l{i.rurllnccfnucrip5aciutcrnularcvirri-licfruuorrniins;ictmcpicrarstcmtcptlfoaelficeec?cllmnArdll.irc[ciircl itpoalctrlclctn4gittrcs'tcirrsafccucullirnuu?;il
.lixact asa ! lletracclrul c singurnl polietlru asimetric
{;r t-i'i {igulii salc. Ijl r1u iu'c nici clia5l'onale r"i nici
clc cerritr-ul
rcciploc lui cstc un ali
lrirtnc rliagc;nnlc. Iar polieclr:nl acuil, aici (fig.'16), cit tctra-
er.lnr,pcr,arc anr timlr
sir 1i-lrlescncz
t.tr ai sir f icrbi
cerfcaua.
'I'c-ai tnilat auzind cr"r .,si licpkrr a t:itLcctitL 1-rroprictir-
tilt-: policdrcic,r lcgulAtc, it rcluat rliscuf ia pricterttttl tucu.

Ii'uirna lui, clr lonckLtor al :\stlonouriei moclcrnc, clatoritii
clcscopcririi cclor tlci lcgi calc-i iroaltit trutttclc, a fticut

2i6

Aya cii, dupl aceea, a trebuit si traiiasci numa.i din horos-
ceccdoda"a-*rcrpeir-riseC$ce.ac;aeaInsmasdiptomdauna.anru;a,ccl u;e!micml-$dfidjtaplidocarleuiaivaapccetldcdahlsocodearrollpausacraciiu.somcldtaerediopaeensec,toa.xdm;f:itdssi,dia.te,retN,iio.n.aicr'nli!adtcn,asrfdaoca.acccdra-ratacs'rrarte.icanalndscdrtfireinaovrltiiocncigliiteteiiea?--
ci*d a hotdrtt si se lnsoare a cloua oard., gi-a ol., ,r"uo.to ain
unsprezece candidate, Ca sd nu se inpelc, le-a faicut la toate
ahpoiortaorusivczioti.tp!LRSrlc;iizauacltaualmteus,l?-osdApceledlamacueccariarm-ecalp-raiecfaeicaauvdt uisntithisio'prnuoesncciloucpmeuteef-ri"rc*eei
a'iynrviatraTzIyy,tcacahcIcoee-eaBsat'rieanchuide,dgpoir?tibvipPsirecuermlvluaainnteidiulccinsumic'ofmtaillnei{cramiirtcioradrlu. olid,en*egTcloyislccot ,hsrifirri-ircnBlalccrn.lscaeidlLtra6veri.uresn,viutTids,rrlr
tp.ct';S.irriilo.iiatitp'izainchuscifii,iinresitcilbee_adumuaearevir!cadc.5isni.su,tmltca5dec.ludeeoaarlcltacblsoa.cegserruaEeoeaslssrlartrb'tiaenl,e.ud,uu,dcoiaanteaarovvrtilceuretoa:dalbraueipsssbgtli'oaea;ecofcrndibdtvo.eefodau[utcea!eopieilillc[roniea,irrceoe.cnsuabrNtuoamtoioftsilacessr,e'ei,titalor.reini-gl,cdrti,lc",nlraol"ti,o.notu."*l:ura''ii"c"a..iluotosuliunnpzttacas*Jittte-eiiisrtidilitifie-eieueadi
T,vch' '- llrei'r6

cl e o1:tici. I)c astronourit: s-a apllcat nlai tir:zirt i tlcrtenintl
ajutoi'ul 1ui I'ycho-l-llrl[i, cl foiose;tc obscr"r'atiil-e acesttti
destoinic astlbnom, ca sii-;i stabileascir iegilc tiupi. care
se rotesc planctelc itr jurul Soarclui.
vmdaer-filiiiiiiNcmcur;piiliisauracnotl'pralfuasirnticruoRctnuuoddrmaiootleifvlccace,sli Ttnciytiuo?ciahleCo,apr-e,Bucstrluaaucht idftan,stt'crrtatearlziclc'igrahiuilaaiircrcle'ui:s,rttscotprbo1isicntc6'orr;:-i-i, r:irts1,n_lL,.rt1l1t:f,ircicea.c..ll,lsltacl\anp'Cclrcsprolngeufaue.dio-ilrlcacci9anranraeacruueam.isr.rrler,pcei,nt.eqaesila;l1ciitsnroeirtdi-.lia-ruva,ecobiheiontdza,prornte.zarrelftneolBrtnvmtasi.raicdo9czeneoriabrlceatarbaoia.ndopiNrcdfteczedeceeliat.ceTsolc[peitldunic-cclntatrfedre'orrcucr.vsehauien.agistir'xaeeu.tuicer-xlIErcadiijmnsugeslrpttiaciaapiauoiluor,uisandbc"npr,-i,Jla,neuepiu,culiiiiir.r.caupi,rif,ptiatfnSso,tete;cida"nr-seufavltaacp;&gettrl;ouuiv.un;,ei;rn;.lteidiii:liic:_inaioe.;i
ar]} citit en odati, tindeva?
as'-a cu r[spunsul lii tt]lirlla ta iutrcbar:c: da, ii lost
Si, incep
atagat curtii lui Ruclolf al doilea si, itt nroarttra iui Brahd,
laso5tc.urfniielcrais'caeisslftialeoipi-oalcgfrioa,satetdll:taa.-tlaalsutmiroKoneddpmilre;ric."elEtrra;ai'c,auisll$ttarivr:acluotSmg,cirufciinomncultclitnl ,zp^eetl
de astrologi, pentru veniturile pe ca1'e lc aclucea' Kepler a teau prezicerile lui? bine, foarte bine chiar I Nu se potri-
r-cau ?cine gtie de ce? S-au supirat stelele? ir."o'
dus o rriiLli foarte gtea, se pare ci tr avut o nevasti' foarttl calcui? sd vadd, urmitoare, ,ru a g.re;it
ln serile cerce-
luxoasl;i o sumedenie de copii' Pe Jlng5. aceasta' a fost tdrile pi apoi va face alte caicule ! {ai mult"onrurtini.rsae
dat afar[ clc la cateclra sa;i de la curte, fiind socotit eretic. pute,a

2rg

2L

ctiatlircertcllfrioin: lclfeiotcgts'ii'o,it icclci1uucclhlratil"pcmrriallecocsaittip|cltcccqoscaasrtltrtc;cccrntl;scrrt-l:aa,aulrnliiircrt{crii.!;:ctrliulli'lor,grt'rrrn-'a,lInjiiii?cr:trr,tp1)orIl.ttiItr'ii:rlicl''clrPiicrstliicLisdrittrcl.alntgllrri:ruli-nsa--pi[itlocl'ilri.sla'tcupacttte*ilitst)leitetue-t!
lIi)aielsccaiiafrcocilr.n!*autartthrtoctparu:ui{c.zte,:arLcaaarctcrcirls:loul'tirnh;&annz1rai1lact'ci;uliu'icl-urocincgslirittr'nlcir*i-t;1ct.'arc1(aijsullrrlurlsrIocsnriii-ooirallr"lr.ot!iic-i
DIird,t,rr"r.ulr-rnae-it,tsirllrlceci.aln.til'acirccNrc,.rlilcllr(au.rlrrcrg-,cetIiiosilacneeI,s,strtalscuictltsilltlciOcinor\t-tiicpfh,nc.xcootorlpsritiijlbcnaninlsuiirdinc,cieitiPttnac1Iic:i'znetflcl,ipctiunnlicldtrrrrcar-'cc,loivrl,dlclnlaiiiolzrur,r.-ti:'-'cc;tlIrclilrilrriui.l'csrtitir;tulia\icitlzrl1licrnz;at-rc-ira.nltitiurbc,1cinla*lhrloVai:actotfloziuarcidbpcuia,ati-r:
clc tlcscoircrir-ca calcuinltii in sine, cirr:i nriltc<,la. tl iittri:zi.t istr
t)ip'ttcg.1]'"-si {ictrlcr'. Ncrvton il stabilit dcci cxpri:sia tlateina-
,ctlicac-'lcsIluirfsoti'tii'''itrr,.itrc1lI,r' iS,zgiotcalaviri.tsaciliill{i:,crcfcroacrli:"i!esiLc!ii("ullr,ls: ipprirliacttlrnccdttc'ciccili{tl.cacptoli*tlttt:iIrttl-ruti;t[c'-ltami:r;cit;at-:
in ju'nl Pi.rnlntului icr ipLir1niripnrtortrhliltcic. r-}ri-cr:ilruttirlnaercl ccl:opflisuituai i. Iitrplr.r:

l-unii alc'for{ci <1.c atr.ac! i<' -t(li(I{';rrrnldc'ro{lc\rlc(ulr-giapnl-'lu*rq,r'lts({rr)mltld'tl)'ntral.ciarcitIIAl.rrciroact)tnanloiccerrrttrT|<lst'ranac)lI.r.'arl-uiucsi,cLlc'eri;(scalttr,oiiir)i'rtitlaxno(ttstt..ccrren1rrrral*llcra'ti'iaiilrrarczlcr'trunirrraldlcrctcllrcatz'ncira.xtaicrrrcuiItaia\ic,cir,zic:1scltlltcrulorir,tttaIqrncirlll-u.prilfcicliIncrtriarcll.iP.l.irdi.iancc'nrrrnntnlarcrltti/crlpsrcrrrtrr,-<"tloP-c'irr,a.lr,cdl,l.r;,:citrrlJsitltuer.ncltlrl.cio'(.r.''asrcltutaucoui'cr:,rtn,r"lotl/rro*;ianctirirclIi,rilicscrcl-lisr,rlrrljeojoru2lrrl,tr|alnalucarirlt0ra'(c'grrclfr"bj,.ictrarnilil,r.1fia1irit'rgtttrr.)i',rsrrliisll,l,r..}crltocr,iirti^rcL,uttnrlo*r1,c)ilrii,.lrnr,-ct,ol,'it,.anr'i.riliiri"t.ciztci/'rrrtll,rj,a.rcl;irtn'l,ttcicurt(;tt,rusr.ric,rclin(.r{ra,"rfr.irtl.,ir-rii.r.icrrnllrlcr,lrrira;crc,s(rlt,lt5(l.c1L'itcniit?.'LIrl,:1cirr,l;uslirrr.rss.ibir(:uroir".ita',.rslilcci,,i,tiallrr'crc,r.rir1iitiotlI,ruiir,r,,lr-l.'slo.-.vririri-tzialr-tiri'ia:.t.:r.,nir-.r"-t,i.i.tr:-.

.:fectc
sor,iilclilia.iiflni:oi'".,sttnl.a{tril.i,u,crllla"ss,iL,liia,,bitrlr{t.crirltrirtniantcisr[i._iCsrdz'i1ittietas',|t'.xlSr,ir1Sorrtc'iart-1ttccrqls"luscr,ti'tintsIc'i\i1rict!rl,rrip1flr'rii:iltd,tncr,lt'1trf,'lc',l(cl',al'1-lrc1'of,'llocic1r(l:',o:tnrt',r.Isrr''ti;uilirctli,'l:li'zlia'nclPrn(l'eapiolccttltlllactos.rrsirl.",l[rtl]-rlltl]s{clasi:c-jituotiranpaCstgcntlaitil:s,'lc;'ltlhall(rr.'i-'---l
ni.,ui'co]niainEeiloltLt'iiiiUrcn"l1aia.tr"rt,ircjt.aurt..Dth..rrI,,)i:icrlrnit.itscrriiilsylclcrtt.aircrt1ic'ru,icon'itcnl.ur;,'rciilii{ctIrcputnag.ltalclliPcins(u)acttrtu'tiraai(1it,rr1lr'l'tlrtPti'iiIpitc'ti'.littrisjliciics;stltritoltrtretfcet:ismlloloaoirrlutrp:l'icltlrtracrerlicell.glleloiuocysrlritialorr5rt;(ler'iglppluonocillolriiietl_csttccr(ldl')alllr:o,rieiic:--t
rlirrtl astft I'irrrpr..s.irr ftrrtuo:nltti.;i tr,';ritt,l hitr:ttt'irr
iri]Leriirf,rfti, gitsit accstc pr:opr:ictitf i gcomctricc.
a p-oliCcc'.ulrcaloltrc, ccaupvcinotcp|,o1c-llt'nieltoatticveitertzriinr saccrmir oInlioirr,gperooul'rccntliincd'i
ilI9uorll'r,(t1tcc.i.r:5tntcli'rciztsla)torse':lrirp'.trcocrtirilninf,srrrcyrPrlifost,rc.rttf, ic:crimia riintrc accstc polieclrc,
din insir;i natura lor? lua <lct.it Ii vrrl,,r'i, tk,-i sinl dr:

2:0 221

po-liedArceliiocrScoran pjuogartteu?l pdstreazi aceea;i valoare ln cazu) tJcil1i'.rll(l'rllt'sr;l::rc'rL1or.riiis]i'r:c1{r.la1l'r'rt.l(r'lr,t'.{.murrr'zlrltiiilsa'r'lur,;;rttlil{(':-l;n,;'uscrir,r,Il_;iLe,lrta,s.pr.la,ttfat.rul,rc}crrirt"'.r'-c;:a/i.ilr.rtix),lcrtr!"1'arruair-liiyluir;riirrr\n.i''1,,'r,tc:ir.r'[rilt{li.tb.,,isl.lr,iti.da,uir,rriarc_lrzr(ttf[a;a,irl.runrs;ri".tf,ierlj.tSr-Arelilr;aiz[r;,..;lti.nra,uizisr,ar,rc;,t];cr.u,i,liu;,iat]"lircSrii.lii"ioi,slJra;ot.[;r.a.ireailr'i;tt".seailii
* Da.
ciic:iad-1'l(siraa,.ji',ar(r-u.t-r"lu.t:-lc]irr,d,iig.!T-isI:ialnAn,iic-elt(ratir-'.'flrcidsiirrtcecrc.li;cl.tnrccicrrriBcac'u''iitrrrrr_ihLsrtnr''c,ciczitta;lrp.riir'rniiltrrtui-rre,.aic\rsicprrrzritciicurni'rrlsrr.!ritzieitiirlar,sIrc.rirl;dtct,stbirit:l.etu,,lirse,Lr,,;srcdibti1;rs.ntloiudiu.liccidiilirctr'ccil,d.racv,icrsic,aii"rcr,.dl,erl,ru,uIJ:rncra11.r'rhiirn.c.irt,.a'r:rlrul.,;tcri,,rprlrrlii:i,lLiilocn,.a,lira_c,,ricrl,ict,ou,.lua"eurJil.6l'.a;l,',a;riiiAta-s.c1r1riricr,cIlrti,'"ic;,fasieli;sa;i.itep*rcti:;;-e,"'a.rl;ce;Ji;;ir;;,r,,;ir,,;;s;U;,titii:;crr;;,.n";l.;a*.i;l,i_i"ir.i;na.rl;sl;..1iiii",.li[
Iu-i, De;i am amintit la ficcarc policclru caracteristicile
ay i'rcit s,il intocmesc itcLi'Il;'irr]'ict(rc'ti-a1lelrs]cttglian[anr(l'ruu.tr.rse,rt;i'sl(1-liaaropprlo:i].ll.tt-lrit,l.iccnasrtctce;i,Irreordsg.titun.rsif,r;ieii-"i,pn';ct;,rriG.d;u;;iiini;r;rt;e;ia'rimuvra;c;;irrlrrurcr;lrtrh;rc.t.'cut.Fiti,tr;iisif;ctt.H;str;c;;
un tablori gencral, ca si le vld
acum in lurirh-,a iclt:ilor lui licpler: viertrptc.ir;tct;lgiearil-,mulcnsiseleiIaaIpip)saetcretlcosfceiiuc6'rudocl;enueapc'iorolcca:cmssnluccecotdslrbcblc.aiui,silrntleetrt-creilealibncdnrslcparreo_uliroa-dtcrp,iiruiieee.i-aiciscpdagiai.rrigic-alcrniioiir.rripiir!prcea'cir.i.r.eurr.rssn-i",,fi"rl:"ivlts,cli,clic.:i,resrrfi,.t;iirriior-irlaa".rn-grfi;,iiicc:,..i,sueroictil.'".,sridnn'n*iiecrriu,p'ali,rtio;',p;;rn,t;ra'iu;;,i,tii,ii;iriilrd,i"irie;i-t"-a:urau*r],
vspa.roif:dlr-risc'c6udusBlrtupe-iioJun,e'cpdK_,arr.iecncgull2aucnlrolccarPrite;eaisitpailt'ocgenloiadnirmleatcn,asutacg;oitinoppinct!urucIalianiiaritflteaiita-rr:uzcpiuinsustncia.capbntrilslmaicitc_ticcolol.n,nvricton,ncatatlsrc,uererne4"aprc.oru;ltieJvnici;ii*lt-t
ir{nr iitui rrch nefit..ipl 1isu.el.icccdr_iehastrlcirtrtaisiilersrndeaacrucpr.nlaiA.ndceicststaarn"rf6iarr,p,cie,"aiiit"ml"=ai,ii"i,nll"'r'irjcu:i;riau;,lp{lil;Srn'o;c;at;crtei lud"i,c.;.,S;ioTa"r;a;;

I'ctlacrilu I {triunghiur i) 6 9q?
(1rItr atc) t2
Clulrul l+4 (tr ierlrc ) {triLrnghiuri)
0ctaccL'u le8 ( tr icclrc) 12
[)oriecae'tlr-ri (peniagoarnci
ls6 (tetraeclrc) 30
1 <:oiliti'tlt'tr 30
lr:2r) (triedre)
12 (penttrcri lr' lz.r (triLtn6lltirrr i)

i

lit -o Acestt: n'lai.ii sltLt itrsir c(rlc 1)1a.i,sinrplt'. Iit'irlcr a stabi-

nuitirnl lrltclt', tlt' 1'lii'iii, ci stipt"lLitLa ti'tractlrttiui

cstc i-clitr ri ot:tltt'ch:tlit-iiinsclis iit iLr-:ccil;i sft'rl"t sltLti:ai vt.'lu-

rniil 1-ctrlrctlr'ului estt: f- ,flt t ht'rilet'lt'nltti, lrrlici a cubrtiui
{)

irrscrjs in at:ccali s{t:rd, sa.u r:ii stll;rata1'lt ltrtrltt'rltrthii cste

;1 cl irr slll,l'itllr 1;t icuslrt:tlr-ului car(r illt' it(llclt:i itlt-icltii',clt sii

nu-ti ttta i ttntirttcsc rlcsprt: ircrtttnr"lratClI t';rllctal'tc cc lC-a
g;[sit intlcr irili]]iiru1 r,lliuriloi' si al nti-tciriiiot' ctc. c'tc',

intre lung inrtlt nruchiilor policrdrelor insct'is'r itr acccr,li sLtrl'it

;i iar, ctc.. ertc. in uncl.e ditrtt-c accste rili;tlltrtc ci a idt:ntif i-
cat intcrvi)lL:Ic funclairtcntalc ciin nruzicii. rlar dcsl;re actrs-
tea nn ain si-!i t'otltesc ck:loc. ,\ni si m.i t"ttcl nunai la
unele analogii 1-rc c:arc le-a ptts in erviclcnlir ln lt'gitttrir cu
elcmentr:1c sisti'rrtlllrii ].'ltrr':tltt'.1yil, de pililti,lttincl ca razd

222

ntrliltlrr, si elr.lltr'tltt.lc l-rritttordilLlC, irr ttttttrllt' trttllllli rlt' spcytou_ltillceiudamrcicisrcltc'rur(llcgpourislra,t'.tbicrict;riliacsccin.ontnrcicr.c,l,usgzruriliuaertt,.cial,urcnfr,ctlcrlJccruclclrgv,rc.i1ic:irihr,,iii.a,.Ap.-tc"c.llsiciti
illrtrrr. liiindci rrttul clinttc llolicclrc trt'bttit' lilsat rlcgparlt',
,re(t[it(rz:,]claltss)llt.tctrl)gcil)lirullcail]'lrst)clli.tn't<cctlctir'Oilurisliricrnbagitlleftrauiulplrltitdaue;tliccctnitprrltaetlorlriaadccticcrc'rtattiiellJtocelrltrol[)g]U,prcelclCu(ill'tlli:rl,etgllocst'IncIcCuti_i.t"llrtS,l-li.tltclJ0'>,1."t,lt-,-I cxpllSe.

strrlctilr ircrrtegorlrlli, rla! Erir t.,rbn tlc inrpir'lilca ittrrri etltL:'itulI-*ou,nsr,r.APnraclliat,cuircrm1l''1iolcic:prirtc,ccoi;-t.sstaTriirrppbst;',aairluri,cr-'lct.orli-ix.lg*oesbipnalrstirclrin_lulouclrltsrr.coi,pocitirtic'clcsnccccut--,or{nl)iiaslrrrtirrc'.c-arrir,inciiilicutiatarcceadip-arurin?n'lc'rrnrri.i.snasaitdincrcnobfr?*iiicce'tf'ncco,trisiucndttninict'"ctio"lc.r".rcl.-.r
s(''s'-n-rcSrrttinri"lotalrr,ticr^plt;iini ctl,uipitlri"crnturtrtaf ttt'lioc.a\:ulrlu:ttptei tnciliu.
,
care ll
L]esco])t-

tr,sr]icta"-"cno"trtai*i,''iitt.aal,tg',l.'oqr;o.crtilitcartcuicloi 'rdia-liro)pl'rru-'ictiacpulg{rniop.orrpiaitcraituieesrcr(cn'iici1i,ir.iinl,actlp1iertlaseci-toraac_plvcpocscnl''aittcp^t's,rPeintcr'xi.necimtluupplsreecn-costli"liltli-r-* f.rtc-rc r-asiglunra, stii binc ra cc ln--urrr gindit -si liai, nrr n*i
rrazuri !
r'btri''tuioiv-itl,rtiztcIgi,aaltcc',rcraar'ar,st.'irturi'lcirat'iciccdffrc',o:rrlari,a*rrirgr:'h(i,llea;c.rocill1rco.Jig,riiti'e'nir"rr,trii.ni2iriilr6.Lrr;i.icr-riioil-rdcr;rci;rt;-r.noiIrt:irair;c,r;'cni;tna.;ac;<rtc;lle,l'lsr,,jlc;e,"#r,li,ierli;ii;rr;,.rl;if";ioiir,;;li.o'i"-i;'s:^,fi:rir'nic,rl;
lsclrotur-nulsuitllmiin.i cd'ta;Ji\acdclecegcivcsaicrtcrcr'tn,hec'sutoerluinC1i.1is;rl',etrarc.ug;tcrclrclliieni ttpta-rigctacsbgnlueuorilaaluic'icif;eliiinnnai'it1pc2aoupi ccacualtetalsoogccoolenapcrneaacrreir't-) 1;l{lrf'rfr-!n.ir1l1eitclt\.rirot'{.{1li.lre'Jf.tlattrcZil;io,ta,tl,r'a.,rcjr;rrc'lrllair'"culiftt'cl:gal'rotilnritifrrcrl.iullrtr,cl;r.rlrcriar.cd.ci:s(-ris<i.st(otcatlicalr.rurcritrrtrcllleonrr-rrct,imilirr.ociiilcnr,priianr(,,l.,.tti,lrii.tJ(rl),,crcrl.l,rloL,l.,r,r,sJr'arrrr.frrrriulllsil'lr[rrrrfc,r"ro.,ri'lllcsar,cIr,r,l..ci,,lr.i,rrl:(flric.lrr,rctri,1rri;l,rr.r,,r.,i:,\,iinin,i2t;irl.l("ll,,f)ri.;,,;ic,.trr-nl,l,,c,r,r,i'ii,;;,tfr..,i,..ii..iu:,ii;f:ji,r,;fi,.i"ulil,i;.';ii;r._i-,;iil
[i;ri lr.,t,tr* cI ele tlc,vddeau incotncisurabiiitatea clin-
otcrnre]ilrsIllottcicrrtrrlt'irarutt.t'ttipltttitlitirrt.i{tgiortrttinicaiiy,l'agriliiplo,crnlnartlal,'gtoatinpttutttli,u1iti',tttalodittiatctilr"icrtgrceoccsnutttrn'aotz';Sitcelclalrtttf:rt i' a*i'tit.foatz(L(pistrr'l'-'trtcrprnctcartr'.l_tcrr.rgrrtIlttUrcpz.soitse'o.franuecpcaixnelrr1rrrtl,a-eir-rn'.qrcrmiitlirfci.lalitacr1i,rteitcir'clarr56niclltcl,1Itlnrtlr,'evtr1iruc1stavds'icrc-,stic'-i,l[all/ma'i',tc.2c"sapl'arr1,iig0tscu;'.crc\,,r0.*d'(c'z'lrlPts(rprlc.dt.r'Jc'(irit.eltiuitl'.nidler:rLtJcrrrpr-zri'fo,lirnett-rciisrgir,'rk'<*iocrlirrr're1'iisaca-trnt.,!.tlilctlrrlctrt{irL'iln'ndttiiniitl's'cat.sI{cilttctmruidtc,li;r.ctasrotji1irgtrririnrc]tl|i',.{isl)rcr'ireluo.atlrc'rlrlii,nrriri.rrctlr-eies.ttarJiiri,iatc'r.lt..rsor;n,rd'lxir1ris,lr,rr*d[,r.rirfpscs1,rcrp.tic',afil1orl,tn.o-o,''i,rti;,.tl,lreiro:r',l{ira.ct,.p,cr.'(grt.-r,,\rntlrirsnrr...i"irti,r.ct.-.c.,r,lrairir'.stncti"jrliitrtl"tri,r.r;r,d<rl,rr1n,,i;r..ii,r1,S,.;uile'ig;r;'"i'ri.s,.i"ilra,,oi1,a;i,rs*run,"ai,"r';;'i,t,.rl.r.z.l,,ignd;a=''r'";al,;;'urt^",;,.,itr;nins;,*"l;ititucri..1lc.irnrtrlltlr;,;r"i'..i;i:,tt_ii,-',,;,._;.:,.:
ar^n.sr-rsi luni ii L\fk'l lnsir ciL clodecacdrul, ca obicct, n frlst 225
cu rrrrlt rnni irriintc, incl rlc pc vfr)Itlc{l ccllilor
ooirdrcnt Nil itiu dacir ti-uru r.'or'bit rrr:ciclati dtlsl-ll-ir
si r ctnrsciir-rr..
i,;r';,;ii.;i;it'iui FcrAinancl Liirclcnlaiur in car:c sc octtlrii tlc

aac.otlclce.c".hc:--i1r..ia,cal,rsAl,rN,l,ctirc'lixortioaalair'isn,t':,cuilpl-'acr'tri;1cimcabr"lrtrc.ln,iitolscic.1icrnl1stls\lc,iilcc:?ctr-ltclcc'conpars'rplcuu.t\,nl\re'cai:licliaIsrrt)rl"sru)cttcrcrip'aii.nLo,il'r,unslia,t1cscrcuharidllcmtrapciuntll.lturlilucr.\lllua'ncactinlztll{z,;Jilt,c,r*Dn,cilns-\1l1'ltuc'eic8ia.ms'-9csvpn7hoiirtlc,nntzclrt-1tnatoli'us5grr'ti"sioazlblernooAii"a)aeilutrtdtptiolcu'aorl-prutirlccgiil:i
trati ciL au fost cl.escoltcritc 28 dc cxcmplarc cle clodecacdrr:
regulate turnatc rlitr l:ronz, rlatincl clin pcrioada' Tcn ;i
ap"ar:li'ind cuitur.ii cclticc, altc ttrtrltc corpuri

r l]r,rcr:scle \icrbale alc 'rccu'"] 'i tlirr lliincitcll'

,\,clrtlctit.ici r.1c ;tiirrle

22t

15 - A cloua carte cu prohleme celebre

cImIcattohl).udtslia,e-n-t'm'tl1t'oe,ca.'siiDrlfltnt'iicftcrare:acatuga,lccl'tdrsrucedpl.tclc.y'rarcl,tp.iai-rttlru1cccciliuanltili'ra'c"ercifllttoi?rtarilr1crrtLrsrn''a.rsutnv'teralCiseiaecmtr.crtunir'c,suuip'rlirttoea,-cr'iiu-sg,id"[i,,;roiscli-tpirOrrce'c'gtrrtr.'netiilo-nircioarsrliud'tn,pill.i,nlic.rrllIjicrtpmcr,iotnittcrruit.crito'icarsoml'iactiii.iroili'itrriciastcC,iz,rals.i(icrtc,.ic.r-iL..lir,cii'r(t,rdl.s"ite,rg.]iu-r,l,nll;rntrl.,coar;,;";lts;treiier.c;tlinit;nu.;iar".t.;,f,u;Fpfu';,ior..i;ufr-';io;iiJiiiue*;s;e-;.,-t
,-- .'\tunr'i rii minr:
clcscoirclit pol iccL.clc dolrr _o irgcnclit crli pit;tgolic:iclii au

re,guJatr?'-
ipr;sj'al'u,i,rt',otstt'rt-(tr-Cin1ir:-dici,iu.alcn.ldt\a"N\nttpif.'fn.',ecllurttir'Iit/rpirrs.)kgI'(o'tii,r:iitc1n)11.',lg,idoo')c,rfiircurltl-.i.ili,r;ttni't:c'irlacrirciucittrettirri:l.-rlaai"critPc'ltg1uttlpcuric.rt.cltrl:ts,r;c'-clru1pctorsiitt,,r'lttiii{c.iialst.ritrrilcnIr,cilcci.arriorc-}rtt.ir.r'rL.rpgi:tSr'.fiisc.i'iiiiiraoc.cai(r.ils,'cilpc,rcirradtlra;-Lr'raocirrec.ctst(ts:ctr,lai,ttpi,tir1,l'.sicit''iearsiiu'.itulcct'r,thru;llral.iiucrirtleirctaun.liscislni&foput,cc:lrtrr,iriaislcs.rce",,ftu-rcrruaac:urlnirne.,iilt
ut:cr'_a;l*iri*-.-in,--1,..itl,'cIiic\,-)iispl'aorrotrtd,a.fsatl.ciiarit;ni'a;(iiuilstirrlrl(iirr,,enrlpciinto'rac.i.fts1liaiti:l'l"rttirr'rr.c.rpiziclirriilrrr-rrfrolLiL.itiplili'tc,cr.i.crri.i-'rtt\l.l;LrlliiltrgIleii-.rXrrru-irscVrriirtnrli1iircr-rc*lilriit'-ur,"rI\li'tr.,.."rc1rltsiarcuao"bllri,t".eorsfilopiire,a,".rr-epi:rni,ei!
Fig. .{? sl1-.-a,l-ir*,birliYlrLri.:frt;:iaul::i..tilirlsit-ra,c."r1iilr.!i;";*lqLrt',lr,'sfb'-i,'l't-Ll,'rl'ltg.,'i*'irrni1rr!rrrtl.1it.';ii-tf1rilaiairimlicr,itlnr,:!,as:i,tntc.t,.l.\ir:r;dtrcl:'s;rrull;-r.t.'.ii,trir,rl"arpsiiea"acsrailnrle"t
ii;:;";ili;';;i"fup.snlicurttl;lutusiirtitittarer"iLt.r,'ri'rc:ir,t.',.llia1I''oitt\cr;Ir:cr1li.rti1;t'r;'ri.'ritlc1upgaIi.l;rrit"i'tcr,2i-'b.c\r;ir^[.ri,cuir.,.ln:,rs:;iticlau.pric<r,tcleJ'.iil-icisa'ita.ttg,.r,,rt,,\ei..-;do.tIa,"im:c:tr*,as)\ci;rtl,ttih,iit'ifuicclnillr.,i1rctn.cr.rt.liil'rrra,crtccert.limcistitcLaicjt!giilctdsiia;rarcc"rnstc,ct'5,s"i.fihqo;t#a"miallra:.tcaeir"e-i
;;l;;rie;/iinlni.o1^it;;lpcruue'";,ti"d-.mt;"t-mi;tc.;ir-,;.i'cuY.;a/c;iR;Ciiarr"irn;'ns""cia.utvcc;:titc*r;plai.ilo|ro;n"rrr'circ,rtoia.grle-ci;rcctt"c.ciri'cctci'griri,la'isi"in\".trznstc'tr,tiie,i'lc.e-ttc1l'ni,Ll*rrp,'tteriit,a'uiullir'"a,ae,uri,iiii:'1m^*i"rip.tiJlllg*[oea.*u"r;ua"cilitliiji'iiclc;o'i(ii'["LJpui''l;:ctteetutt"l"lui:hs,a?sati.rrpa"lrsaiiill"ilhdiprz;$r'ctea'utsa\itli;;rtlr(dillciu'it'-editacssta'rl'cr,;cittcinri{gliiicet;eav'rouai'boex^rnllnomeaipsildr1nuccc'l'acera'i?iu'c-tr"snaatdid'cicfttecuindr,ru;ct'svttrdnmiucdt/Eor"s>'staasrarccsycbitiotlitcichat'i-;-ttlL,taaiitsycr1rlnrzclrttltittcaurnifrcvgdraai'eu'dicrern1ar.1alicreotutllcast5lio:srl-*atu-c'.t
227
de az,i,
15*
qrA

clc calculr po clrlc nriLr"ii gcontctriai antichi[irtii, cu cxccptin pr.oclhiciltairtcurial-ro-.b;li-isrrcr;rttoaartcc ca Icfe 6 spfciLrtrrj"a.tc*-;'ilaltJir triunghiuri
inscric in cri aiiiicut
iui Arhinrcclc, nu le-arr cronsiclcrat vrcclnicc clc a fi notatc
in mocl special, Helon sclia ci. ,,clc;i Alhimr:clc a dcscopcrit
cclc 13 policch'e scmiregulatc, inscrilitiiiilc irr sfcrir, Plnton ilcllrn cuno;tinlii ;i crr unul clin ccle lli poliedrc alc lui
a cunoscut clc asctrtcnca uuul clintrc accstca, anllmc policdlnl dAurlz-riinnlralclircrlcal:rt,uccpauerrnlrlaulci lrtciimcunat:sottlyt!otioncflariet.l,rstiLcl loI resc sii-tlri ltlczinti ;i
cu 14 fc'[c, clintr:c carc cxistii clonr-"r tipuri, ttnul fircut din
B triunghiuli ;i 6 piitr"atc, caclclirpliaLlmt finotluttluuitt;iclainlc acmlui, dcjit
cunoscutc rlc cci vcchi, lS,lsoiti.rt-a:-rtrrtrilt'rctt.At:<r'\g1.dtlii1trriilirLttactntrlrtuit.cic{i,yc,lrtirs,cai'trlrytiigi:,neh,.ilrnsl.rarirccntrca_rttlpcucs.qJltinlirtuittn,l,ligltc,ufpi/oit,tltat.tlc.i<rtlutrlrlccn:,ttrrl1muitau.i.iclrpti:.ll<otf't<llrc,i:llgtifiniicorp:ito.i,k:,/r.,illllgai,c.rocunrt1ltipi'i(riItoai.cllctriitciredrrl-Lgcr.ltnrierru:tltdaLorlttu-cslclitt--t:I:
I pi'rtratc si
6 triunghiuri, carc parc si f ic mai grcrr clc obtinut" I
tul-uiE;xi tlaalocrcalicnraulluI i,",.,9 tliung[iuri -:i 6 pitlatr:, alc
1x'i.rnin*
a,clicir cttbttl ;i octacclnrl?

- lil{u cnbrrl oc t acih'n l , ci tclrilrcnu l norr dc : t:tt ltoot lutrlnr , AcigcSclc(rucu)c"c1-bc1cltoarz-uarrkiolorl))cJlaltal)nuillecc.a.ur'crticNlinocrll1srrurrralttltincur.llii.irrllitns:l.a.ii-rliia6ncriIcinrtittiilr1crlr,<,t'1cgk,rilbrr1a:csru1l'ct';r1iiilor;t1ii;.rrrn1rrrt,tc.t,s*rrsrt;r'r,;,rtili:cac,ar.etnrtliliu.accn.n,lustrucPrrniuiiii'o,u]rt--tiirulrgfinils5oihiiJp-lt;oiaio1lta.r;li1si:nmg1a'iaoutadcnc"xtcmci,irtchnras.a'ri,tielt.lima"cnitgiliertIrrlrriilralt.cnatl.trcr,?i:,,-l.,t,l

ciat acestui nou c-oi:p !Pcntru cci vr:chi lrcclrstit cLa nunrit'<rtr
o1,i;nuit6, asar curr] clc. pilcli irzi estc pcnLm tinc nuttrilca clc

cub, de dctacclru si a.;a nui clcpartc I llcntnr ci. sirnbolul

se conf [ncla cu insir; i clcnutnirca po licch'tr l u i rcsl:,cc1. i r' . I n loc
sri-i zici. h,e :tscdrtt, , sau cru1.,, ii zicr:au piuuirtt. Intcrcsatrt cstc
c:i Heron clL gi tlcscrieLea rtc:csttti policch'u tlcspre calc Pla-
ton amintcra, ln tlialogul siu l{t'tto. Antttnc cl spunt:
cam ASa.: ,-.,,I)acir sc lnscric'citc un pirtlat in ficcalc lalii

pS"tratir a ulrui cub, anrrmc: trnin<l mijloacclc lai,urilol lui, ciri-'it O hc, rlc.oarc,cc' rrrr i:st i priruul itr-,It,sr_,t. rlt, rnirtt,nltlici
(fig. aS) sc ob{in 6 pirtratc;i, tlacir sc ta.ic clin villtr';ilc cu- reclu't-c lL accst rrlsurlcrit !
ultinr
bului piramirlc ar,inrl ca bazi tlitrnghirrrilc ast{i'l folttratc, po-rt-lti{r-'rlurirniracll-rl"rrjurrrrrssl.,\nr.lhiliri;triiccd.r: la polir.:ch.t:lc
i'_srisi si:icclsta, a,r car:.-i
altc lrtcriiri rh alt's:rlr', nr,, lt,,
rizrrt in I.iinclcl'L
Ior', a;l r:unr ai
s-;r lrit'r'rlut. l).in r"r lutirtilc ulnllsi-
rlc-.i-r.irr:r,a lui ll,ci:.n, rlar.nta.i irics
;ltlitia.lIictn1erc.1-raaor1.,co,p.e,btas'!ttaii'o'rftrlr1illir.llco)Ill'cioIrr.vLdtio'nt(t'u,iniltrr'.(r'rr'in'rrgrrirrririI'lirlltrct.snincprctnrri-..iiri,rn.itrn'aipsLtc1iCialPu;nrcri1iI.rPi,rrtliiisr,i,:ccrrtc":eelzlour'lc'tligi.fuc'-i-r
fii*--dcAii)i,e;p",ladt'ni!ucns'.lccrcdic'csstsiierc1sffLuiurrcrn.tritn''upafo.trifarrrtrsea.rfprie'oacucctculi,;fxr.icca,ecr.ttritp'rd,crohi''uulmiri ifjirccj ll*cruif,r
/---a dftIr.lraeurru{lbl9i,rl1i]rsrcr,1i'i<pllilloo5rcot1llraixiclci1lar.crr:rlrtrcircriiris.rrdl{ulr1eciinrcsl';ititcri'rlfrrntirllrctll'r.rrscrirrla,f,rit'nliriisir'tnrrrl<-'irrrLir.rcithriritriitnlnicicrol-licofrs,ciligiai.gtsibrtajaiuaaciuu[ru.e.rrLi^itlinIctcaauCrfteSacuhtt.Ci"nrrAlrmnait.uuuctrcomcrschttitiaii,,ii

Ii'ig,4ir

229

-<*---r*--

A

Fig. a9 F ig. 5t) Itrg" 51

sli"ue.,iotic.b"i.rtJiiaili"tLti(tilfniilgl.ir,.troatar9irtc)':lrlctsrcp5clnllilrocctiiepelicaelnre"<le5psaprrenailfemcl$c*ucclaltirtielce:el1eu' n4tclli'fvcauleandgarulhtfr'i cu-ioAt"cSmpnu, nclaec-itirai;ii giisit cc cra nrri ercn, trrnriniLlnc;rltr:a
cc fcl slnt,irin r.Izuri
clte lerte a,i'c? ci
citr: vlrfuri ;i cite muchii?
poliedm, situatc tn I a. i ilf . Corliul ohfinut are' 4 fcfe tog-ouS;fr,lcn.ulrriruiuillriim? : cle toate 14 fetc, B tlirrnglriultt.r'si 6 oc-
fat: lJ.
li oli I egal 24"cici rlinti'-rur r-ir-l si,
{cltr triurrghirrli r:t-hilateralc' iir t,.\l,rtrrll'lroiri?i.\r,'ir:rrtli"rrcril.rl'ot,lltrl'ittiirtc,tcuhrii,;iinIrtrolctar'l:;3ttf.ia(fsig(.'ll5lrlyi. .trlar,git.
htie*x"augi"ouvaitlref rtcngiutll'iit'tdt' '.-,"si c'[ d'r.rii lcxdgoa'e tlirccm nr;ri dcltarte;al trciL:a la
;i un triunghi tntn-chSiai,r!
. Fetcrle Iui {iin11 tr:irrnglriuri rinel cstt: ocltrttlru.l
e--c-hilateral" coustt'uit l.r fei, s-a li utfutl trunclria't'IJar de echiLrteralc, *{oco-
t-cal;r va li ca;:i la l.ctraedm,:rclicii sc inrp;rrtc nnrchia ln
rtt cloiiea,
adlua-i-ta"tniiair3cepiajd.stt.";rrg!ic,rcl,tg]ic"altlcei.ri;'ri-rius.'-ioi irpiruofcit'r.:'piivlaittrnesbii-trirlidmite' isrsairerttcifelinuurnceelgt luiaalaopli,tLratcldaimitcude-'
.3pi.r:ti egalc;isect ionartr;t sc l'acc, ia r: trcinic r1c: r,irf. F"etclt:

lut sint hitxagoanc: 8;i pirtratc: 6, in totai l l, r'irirrr-i e
ar,ut 6, clc "1 oli tat:2'1 . l.fuchii a iLrrut 12, ltcurn ar:c in pft.rs
f6cincl figurra sir am (fig. 50): 24, ;rsat"lar 36 cle rrruchii. Figura c cilrlr a9a (tig. 52). Llngi.
l) P,C '* LIE 9i cleci BC + 2Cf) : lK
el voi tiesena ;i icr"naedntJ lnt,nr:hia!, liindc'zi s,: supune
Irctelc lui rsi.ricnt(j0ir,cnatcalgic.bf,andee
2,,,AUF:DtiU"P CE DE c1) ilcrioras i leg i c1,-: r;t-.ctionarc. 20. lrlrfuri
M ln riumfu'dc 12;i ircxagcairc
sau nr/2 A,T

De aici se oblirrl 5 ori I2, iar muchii:30 -l- 6C . 9{, (f ig.53). Al cincilea poli-
crlru aririrnedic cstc clo,lttotdrttl lltutcltiat" I)istantar ia
: y9*V2." calc trebuie dus planul de secf iune, ln asa {cl ca rL-,cagorrrin
BC : t,+.{v.2r=: M(l/I, - 1) ;i cn 0 cc sc formcaz:i. sd fie regulat, se calcnleazf; in ruoti analcg

230 cu ;rce[a folosit ca sir oh!inem rl1 octogon legulat din irdtrat.

23!

pf ilinlinriclatlluillaapsccnltiiaurgaor"ncaulutriiusntcgthatiu, rciolotcr,sCpDuInIz;iito:r\DpFe.nAtaFgo:nul-l
iui dc lattrrir nI. !-aloarea ci sc cunolr;te. anuinc *-clacii

ll =. ; lfft-nyt5,atrr'ci l'= ,i tit,, + 2i's,r'fiintt r.a-
zA crc r(:u lLt i r: ilt:lrrn sr: r'is Irc'l tlrgonri l tr i. i. rr lrx'u itrrl ; i fitcinil cal-
Ilt/5-crrlc,k, r'ciobl irrc CI'i ..= CIi =.., *ll : ti,\B=,,Cf l=.
"4.

.[:.lc.stc forur.t cli. 12,t,,,,"*,,1'lr?'l]r-{r\'('ri1. c1i, ,.t:!';1r;.
tagoanc initialc, I;L carc. st :r.tllrrrgi '1[) rlc' triirui:;liirn i t,chi-
lalcllr_lj., a.srrrllrr li2 rlr: fr.tr', G(l rlc vill'rrr.i si f)t| rlc rrrrrr:lrii.
(f is. 55).
.l:L-Pol)lilrrcsltr'cr'r-.lccrici g1r,rolllitccc1cr<'cilnrrjusilrarrtteirclitrgrrrcliLritrrc' t('llllcl(cii;(r(.rli-rt,rrirlinLrl,l-l,(rl
il i;1. 52 1lig' 53 ltttr*slnt'l'-lrttrttetltvIlr)ticic'icrrhs-'r{xi1uetup'-r'll-rl'lrtiiclc,rci'lt.rai[lr,cubAc{lirctl|iil,oi.'1tc,iuiri7.ri'.rturu'ittarlrPl,c,o,ihlarltclitie[ii'tcl'tltfcilhlrI.ro,[)'tc':eicrrll.Lu,oai'r.t.rl','tr,'nlcclr:lricii1ic,rsrirlr.'i,r-cj'r,(l-crLprr..r.,acJssc.icrrcchcn,icsrim<auc.llci-blrzrbar.ilniutrt)ititftci(ir,i,rs:titrrllriiLl.scj;r1nrccrilarlnnc"cirl

li i,' tttltiagolltl I l'( iltili'l '\F rr(striLr ll'"5I t) ' lfttchia II s-a impiLrtit
t';t li(1, ( l:;il]('
i'ii'trtiicturilc.\nti'iCllitt clctcrntinc Alj I2.4li =
- :tt. A tlotta , irrlrgit' din catu sir sr: ;i CI! se
i c c'scri cr! t t,rtc d rt t,

-- ILris:otlodccactlru, spr-ri? [iL si-i \'('(1(]ll] r:t:-rrri lrratl numi-
r,'rrl icosirt,rlrrr, rlar uu,i icosacc'lrul
inlii cir Ir''r'ir.ic il in

Iis-. 54

ti:unchiilt, ciLci atutrci nlt aL titlti avea trtci un sen-c nulnlt:ea i;. ;i;itrd.,rt,cu-nor.lalsu,;rt.pr'.rdlaca-lcslR'!t(scleer'ttcitip'4:rIcra,i,i,ripisnc{fc,crri,'ilrrorr'ccrf\sm,irhpu,oe.iovr,irc'dr.,l,arii,,ci.g,,t1ii.-o,lLo,,t,i1sr.aictrfi,nniir,bic,,e,rc;c,p'h;,';ca-qitp'rerdarr-i"c,rirad;,occl;;itl.at_,",;"l;riiirio.,.'r.,itl;;r',;r,ir;;,;i;c';";a,i;'b;,;i;i;,:c;,p,;,;;;',o;i,i;;iu;all,h.i;,ret.urc,d,siturh;r.u,-*n.,ep'li;it(lpr;i,,ekas;r,;rrf,(zl,a.ririirafiiipa;iA'tl,,rr.t.rlia.iilrrD;cir.1i"iis,naic"U,il:rlri'.rr"c;ihuimm;iidilJeiueliia_i_r;ii
clt: dodecacdru. Ace a-.t5 nurtt irc cct'c f ele pcntagonale alituri
i:;1,:tllli,',i,:;,,,l1lil.n ',',,ii',ii'',i,..i'..,lr,"lr,u,i ;,,,ic,,rrt,t., r1,51116
ecJlee*ca-ltlJtseuildeoecatar'pitul1Lnicgtiahpiecuifdatcrreceea.apat[t.mCnacsltiitsriti:,rbpilrce:z;tei ustiantitrectloedHae,lrto-;nti.
ade-vdA.r,;afdiinadr secfiunile sr: {ac la jumirtatca rntichiilor:? fn :;l,ljl i* ::*;q1si\rI;;{ili:rli)zift;s-;cul;rrt':ll1lt.(Fal'!,o.(t|';rr);:tir,irrclr,l'cr';r{tt')'s,r-.,ztil,,ocnl.;irc;ri.-t,t(.si1lrir-lt:lliro.r.;,,;r:,Irr,.ildi1.f,,rr,(,r;,,'r..r,i;n,lrir,ill.rl,-iir.i,1lrr_':,i)ii,,r(itlir;iil,rurl;rtti,c'i,lr,i;ar];,ti;,itri",,;di,l,1,l'r,l,,ul1'1i;l,',,,clil.,liif,ilt;,jr,iJir1I,tr;rt,Ali,.,ii.;.ri;(l't,,.i,ir.nrJIttr.lrtriut".i;crI,..x,tiAca.iurc"ntlt
vorba cle feft: trinnghiirlarc, echiiatt:r:ale, sec-
iiiJit{ilr{r.IS;'rrlr,'ttrrrrrr,r,terpr}r:s,c.rturr'r:art'ir,iia*rprlf't-rlctl,iIc.nsof'nrPt,c3elihericctS$a'cslaIl:2ri'ltsetia"2tttiiL{rdl:'4lrcniftectbcrrjc'riel's-!i-a..Bitgctatsp'lf'iriir:polu-clirLtstireecrrrr,(laos.florlrrnln1coirirlsrs!gcpa'ilt2agrs,(irilaciriiutntil)tcrirn'herelectnltderern.rf:*i:c,.illlnru'.br;.r)r{rddrnf.ilivuilrrarrcscucf,l:oucititr,ri*u.neiir!soruh.srr;.uflilit,til'lrrr'pi.i(.rri:ctiiiJt'rta'a'r-rl.i,daliirla,1r..isrrlza)'li*cliniflrBrtrauritr,u,i,ci.r'rlrrruItrjiCurriir:i"rrliut;lrrtrtrcl.lc.lrra.c.rc.tut.ri'l"n:.;^:lr.rrl.irini(e.prt"b.riu;n{n'.-iL.;i:lr,r"LrnLrt,ut"--;ati1rc.'-rri'ir.-',';:l_iai\i\;t..iut,1i;r'{rarf.tirI,Lr"Jia,,y"ii*i,lci"i,t\zf;,jrt,.i,:eir,iotil1.,;;"i"lrliir,lic;."i,i;,to{-,.iii'r,i'(;i'll;ar,if,t,ro,ri"jp,;;,,1tl\u,re,r,.r";,-;,it,r,,'r*(,r,1l:,,*a,,o-.f,,,.,*,"'f,icds,1,r,.n;criri.itur,'e.o;,,1iLt,."rieurr*,Srt-irfc.z,rr,nJ'!arltcn"itri,tar.rutr,1.l-l{lte,l,-iprecrcii,,i,crlcrt1lirlirJ,tuU;nlrnr..c:ol'ric*c.i,u..,lrgsti,.lzlp.s,t;ln6lr_np_artirrii'lnLlr!_1"cc.;l.rt2oaiiu;fii.atrelhi1rtrcr,So'rninci;iel]'retririieilcc;nyrri'r'.lir2.,r_urorcgulutrpieid,.tcnl;inrl"adtca*leoio'ltlh'.ua:s"br-hi"dair.ur;uiIerao-]-i.e;irtir.ir'-i,;-ij
!linunaicleelanscicstitmc:lpc,oru'dirufcurlailtirsltcctr{ea{nesftorirulrn.g{ hiniulpaernr:targcogau.lnirete: ;1,
12, 235

cllcrvlrftrri rtlr.r icosnt'tL'trl. LIL ilctlslti! sr:ll(latlgd i:ele20cle ltoi
triunghiuri t'chiirLtcrak:, tlcr:i itr trttaliilZ cle fete;i l2'5 .,,,. 60
mucliii. \:irluri pe junriitate, rdicii iJ0, cici in {iccale vlrl
sc lntllnc-"r: citc douii inuchii ({ig.56)" A;a ci, pe -supra{a}a
accstui policclru, ltltct'neazii l2 pentagoalte cu cclc 20 clt:
triunghiuli. Ii singura figur',i fornr;ttii tlin lrctttagoanc si
triu*ngDhaincirri
I o proelciciciplrietuclalisttb',utiantrp';;riinrrriinatacu1n-rlo, liag\o7acnlrtl':liolrr-

flcettr 7

de pe fefelc cclor

lndtoarea r"eparti{ic clc poligoane:

lely*t:dv'u l. lt'rtuclt,'it:tl: 4 tt'iringh iuli , { ltc-ragoanc ;
cubu,l, tyttrtch,ia{, ll tr:iunghiuli, 6 octogcani:
au,booc!.at'tlttt,l' ll triunglliuri, 6 pzitrate; ;

octaedrttl t'rrtacltiut {i l.riiti:rttt', B hcxtLgoan'.r;
icosaedl'tt/ lrlutthitrt l2 pcnta.gtiant', 20 hc,ragoattt:;
rlodecaedrttl lnt,nchittt 20 tr:iunghririri, l2 clccrt.goanc, :l
icosodorlemcdrul 20 trirrrigTr ir.rli, I2 1rr:;rt'agoane.

l'i,t 56

su aila l2 pltlatc ilig. qt]1. Asl ci (,1 rrr.c 26 rlt, ft,fc,4tt clc
*i'fur-i
5i 27.dc nnrchii. l\li-a rnai .rrnas rrrr poli,,cliu, ac*ia
ocbsfteirriccoasPoldicoirdrerclaIcldryrurcl ct(r[ucrni crlhcist{P';ci bi,uc;ti, crecl, ce'."i.i
cdnilrri.
carc'drr,lrr.,, icnsn,lnil.ca-

16gi'1coLcrs"irr-fca-rcl.l[aIc3e{la'0ac. icPcvusalocdl;li-lIi2iltitloit';rfniitaur'rb2clcocc0ircr-lrorccllscccol:arcigctulooitoctat,lucctirl.rrrrccliacf.l.'r.ccuaci'llui],lrlo)sunilcliiL_rr,clt'cceuc.ealncrlrolIlni0nUlaiOrilcr3cltc,lcit-[,rofccarrt'lciptcrc.,ill,cua.*r*rri.iuai6nnrsr2e--:
lni{rtccrtal{c-iccsi tdrm:tuU-.tan'nAuilclrclsc1ci1mnmcirnapualchitcchl:ritirrictrirsil-tiiic_latrt.u'c.inisurigffag,riir,sur)rao'rlis-,uicO.(iLjn,lrtr,irhaltcerr".c\a.riit"ntrgt.r,n,odic;a*iilitnrr1icti-rnrcs.Lu;rniiimt'arcl2rgliiror0tiaraaoicrtcrilc.,coiicp.rvl.cri.ercfcuiaonrcllcitl
Iiic-.5S (f i9.60).
1;;!. 57
pcsl{pioe.uc-t-l-cii'i-clicud$cUlatulteunnircl'rur,arcisiccnuccllccalcngv'niirrui'urus-:rrcuirintsitucamtrrcncnunaucalrit'riri'n'crcgdnilri(i-.iccIcrrrlulrc;uci\).lsalrcgcltrlohclatcluilt..x*ii'ispI()iilpt_lrlci.t'c'crrlaaai'sc',rctsidcrtrrcdrriitLctcrrio,':witissatccris|ri.fincktun,i:rrirslauiicrrtgl.grfarcuisriclilcircalrldl_nstocil'crrl,i!!
c;.rul;rli,l;Li;rOi;tlc'i,'r..l;l'i;,t,ir;l.;"tlLr,,'t.;,i,f'i(i,:llr',ltn=t.t*,-',c'tl"c.,.rirt..ri1ln..lrilg,r1.si.',l(t|ti;irtrrrtiI.triiit,trci'il!cltrri1ltal.rti'r;.itrllillrIlt;s,lo)Irrili.;'r.i1titci,rt'tgl3'trll'c't,a,lsi(1rlirlrlr),rr'.trttLll'tt'l1iciirfilltil'tis1ct,r.rl;,i(lrliprirl{i{cttrrl)'rrtoitlirtltrairtrri.ln;tc0:tlttrtltc'irlt.-rtsir::''tt:rtilr'-tr't.-tri.iiIt1tlri{tl'il,it:lllrttajcnllc)titrcltl'a'ri'tlrpccl'}:lit(c'sr}sll'(r-i:ittsntlrltilttp)pit'rttrA''0:sttrtrelltltlrt'itr'ctarri{t{:rlit,rrlao']'rJ,ltt'anr-'tilt3llallt'-tr{lgrit);tcrIll}rti';tttltl'1rt)rll1rull'(r6ic[,i2cl.clic'-it.-tc-'l-t1ii1cio:sll'ct-Llt-r:t: sii-1i spun cnnr sc chcarlii si caric lc sint caractcristir, il,l, Iics-

ii"i,t,.i-'''-r.iiAu-,ucJt.,rt,,n,,1rq1'sr,:'iiinrrrtrlnr;:lrLu,i|,Li6Ol0rsi.ril'icrrll'tilclrllc{iliiip'.tiaiit;ili?at1cc2i0i;iarliciilc{t-a'cutlrccl hptcri:'irn':(triftligigrot'aa5trBrtcl)c'' t'rl .111 1..:,.. rtrr
c'lCit;lr""t;ui(jtf:,;,_ui;r;lilU;;-l-;ctrCiliItll;{"ttiliu-ttii;r,;l'.c'"II.i,,"'!1,isrtci.rri,iil::ls',irsi,itun'.'rfl,i,g'gilrl,uctr$-.c:',n'cr,us"i'lgtj't,,1rii'i'rhvi,,'.lir|{,tui.lct:',.'glililcal-,zt6irrtra',ll:,9,;c,o-,rs(!-,at(fi.i"ctt',o't1il:orncig,trlcvorfi{Lnii,lnrli(i!rttl,i.tost;,'""asspi.rntrrftstisgtnte,c-'oclLoilftotolrfi)rconi'olciunnDlcrtla{t'c''trc"rrra.c'JtIlr'1ail'lral6rrrt;cturlilirct:1'lfclt,nc'2rlicctarrrvillttlallostlrtlrol,itt-rlfrt.e*-m'atrt-'lil''o--'ii

-: rU

Iiig.61 spArtia'"orlra.:lr,ix(cicaaacduu.l'rdgtc'prcrl,oo.6bvr.tnr-'isic;istit'trtasdtn,ic-ri-.llteei,liaog'rtdu,l'/rliittrsaiconttrrtr"iuci.dgraX.iitIl,Jric.rii.uctmlizru-ni*iaarooKtccsic*'clr'.plaar't.lticc"uiccirnoli.ceADntaarci-cbiradcirrsliortntrfcrcerazrl-l,iclcaeiflrnmest;irlanireionl;b;;ati;ui;s_"iit
af7I.(l5!el]t*1Jt'ilrsSll,erIo,oleS1rI'.cnplltr'ti,ns'ltin-'l,rtiiirir.,ral_ad*trihtt:,cir1'cicacnc"urr.i:c{csic!crrpLl:aaburtecirrac.slie;i,afprirc:ift'orJpsrrtiris,,rni.li;,;ii,;f;c;n;-;'rb,ia;liz-le;rdcr"
trris. 62 pucpacV:.lenocr,al,nul_l*']ncit(lgt-l'ei.ep,cod,lDtl.1loa'i.c,tsicurntr.f,n"ic]tietilir..l,'.celi1.br;l.ir.tc',i''idz,'irssl.i.(r'.^li.:rietpiuuhrpu.llr.rrela(*te.i'.italtis{ee'a.r*sluLanrrc;c1r.nccinienioclc*r's1ii*crrtl.r.rieplt.ela:adcegrrcrcl"tihrcs,bpl,r'rloc,naiIrc'mrnr'zir,[c"d,iilc:-iecriucTc1ugclllur\ii,nfpilc,i.tt.,.fiirlrt,C-lraa:rt,oriinr,rtrta;,a.:errrcsicrrsArl*Cteecc:,."erO1rr,iualrinrIritm;.in"I..c;)j',uin;u'Iit;;il'rriiul;;;,r:lirr'nii"ir.;;rcl.i,;i:C*;*i",pcrllOilc;ren;i""ufiinrrrl;.;Sidrit;;irirl,;;die;u;oi;C;;s;lo;ur;tf:ee,i.
sii, crr nn rrnghi r-'qaI cu l:,9;9t, n rii'rl *uur;ir*llat'r,ilor
tul, trcbuic st1-l a{li singur diu citr|i. }i votba <k: tubul Lc..it,
(lig" 61) Ctrc ;lrtr 38 dr:1e!e, clinire cart-'32 sint tlirrngliiui i
i$ritrii+ptidtctigaittcc;:iars,c:: obtine din cftetbl'cs,auB0ocftiaini'(ltll'rrt,li;rirnclgcldriortdlti'r;-i.
arc 92 cle
l2 pcntagoarrc (fig. (i2).
*-- T)rt"r'ir hrli si nrttch ii rrrr liu?
Ba au, tipicarrrle ! arc 24 rie lirfuri;i
Cr.rbni te;it"
6{} dc muchii' ia' rloclecar:clt'rti te"si1 lt'c 6tl d. r,fi fnri ;i 150
mdea*mticAucicohuiid.tao,rrintcc;dtltact:calcirortir[aati:t,i:llnptrrtrttlbakn-trct't t:o:rlirlt't;i siste-
* Creil ci t'lacl tt-at' itttzi I{ep1er i-at' ct'clte ittiilla tle sp;rrrlitgtco. nt.r'lhlurit$;ihtiruplri.rti,uch'eils.icLtrv:riarflru.l',iriernaasitlfienrtrcea r.efere ratcrare

ttiuchit: a hrrzt'i. latur:i fiind o

brrcurie ! 0p1rriLs{',unritr|eiui, lpdt(er',si{te'lrnntrel-zrs. ,(,ari}iitcgiti'{.q.6g-qrlrriiisa;_itn1tlri1r{.)ha.irrrhIriiccmrPliecrdts'irecd*1topinaittar,,aiitndrrs.rti:girtiiacot lisattinr-ctO!d..

* I)ar t1c cc tocniiri licplcr? I)c i:e rtr :\thirtlr-'c.lc srtu

Pappus?
* .lP.recnot1ruirlcrdir,liril,s,'itluciln.p[tfitcl"u,m*,t.,P, lplip.uts;iaasltcucilriiitlot r1iir a sclis tit's'
tlinaiut' a
pr"
idpHulrnaie,clmllaalocaenrlsinicrtrcceelin,rlL:\lo{ldu5::Anrpnuiuluirf,{litepecpu?l;erbsmrilriipcnauateustr.rllitm:csinliaeirt5irPttclai'uopicllirrulocumapslttc';riieoi1prIeitaa!tl'nupnttro:rorila'lr.1rtte^8rlad6ot4tttt,l-tosI:rir,*-l'
gisesc, allumc in volumul cioi, toate calcule lc clc' c;rrc noitlit
ne-am incuructat sii ne atingern, pentlu a se rlctermina mirli-
mca latur:ilor, iuclitrarcaplantlor de scc!ittncIi alte le.Cclcc-
-o)***
tirliic clt: nrai tilziu, alc iui Enlel', Poinsot,Cauch;',Poincard, Fig. 63

z)o

/l \ n!rJovnualoamtiln'n{'aiuaceisie,nif,ticliunhlaucdrcscrtftiuiulriurccs-lfi'lcalolc*tacriejrildutfccsct.iigiirct,rlesrclcu,.trl[,'i.pocitacKrrsn,ullietrp:rliupttlo.lrru.trcc,,lcg'crbii)tseaicrlt-rclceFi.r5rrr:ia'nvzprodiilcrelluii*rsvnel irw,te"j;oaljoairglirritiuiti;.r;i,*il]*r"-n,i]i,i*.;-

,, \ actual.

II

.lui-N,(c'rr.''tto:tnl ,srlia.rr''lir'i! lir'1 rlt'l' lru 1('lu'ollr(rJ,:itrrr.i:rl ,il(,iSirrr.,;tir-]li.'ritrVifiitilutt,j,t,,,i ,-;r

rrr'Ifcrirr{-rr,

clrt siL rlovt.,l,.;rst.i :rrilr{,r};:r :;i:lt,lnrrlrri Dl;urr,l;rr. llr.}llzrr url,r_
irrir ti!ilo-t't','1,-':tqi"r pqrligrIr.1,. lt,irll;r1,1 , :
ani tlc zilc crr asriur,:nca clrlr:uJr: i' irr1r.,, l,i S.i f,i,i,. l,.,

lrj,r. ${ slrtirini:.s:r!--cifho;rlLli\ncrl.ec;ttInziailil.p,cb.ao,cr;ll!ivcciliul"surrtiurttPr-n!siicnu[csicclo.adloilriavscelcc]ruilil'ccmrcaualslr-ncc.csitiiicsfilchiiri\irazc\iniil.iir;iclntrceiarlclocii'rn;lrnieironllaolnu.trc"c:;tiliu*rr..tptridn"leliLud'lfiacnnolot.r,ir"risLrieticci.itt:"-,,]sa-,,-.
Ilnlrrsvis.trc-lrLniuit1'1srlr;-rilhrtL'l,op.ric:rr,rirrztrirrtlilii;isi.i'qsicsict-iirDcrlo-inlirL,.,iprrl.ri.rrr" ,"rucmeje"isioiidno,,--
sf..ri couc't'rltr.icit sft''t'r'i r:irctlnls!-'rir'(' politlrlllllrti l:c'g{rli}t' siilarntectjplra,:o;trerllircclr{d)e1jorgct,:rs.lirlcl)a,o:t;clriluo(LIt)'Pfrt:Itcrcrixr,rc'{trr,tr'r3lr'.rririlrlir:trt'{Jr,lrr,r.stzlilrcrl}rilii'lc-igi:r,lrrllllrua,trrrri,,g. ti.iti'r:,,s.i,r"plr,oi,Luliie,jj,celitIreocltrr;',r,,
Cqil'r'SllllllziltC-,r' ltt i.
'.1lc'rs-t1rr-roIc/g:aPssioi.rzl-iicinri"{i-t'istr/rpilzi'lrrsli.tlii:o't/,,trrr/c..-r,,r-tfiirrririL.iurrlnt,il.t,s,r;lt.'iirth'rtr,:,ir|,.rraJp' lto:}iltN'rlircr'i!ut:usctruilvt,.i,l.ltf
.rt,;.l-ii:llit,i:*ttr;:-;ilt:ir.it,F',;ti'l,ir'li.lrilr.li'-;r,''r'.rirl'ic,-c"l",,:l[tct,'i,r'r;i.:r'i;"1aipt.,ilri"rtt,r.''i'le:t,rl.,tirt:'"'rt'''tiitiitr''''':]i:'l''ri'''r"r't'llr"''rl'''-'r';1-il";ttttL:;r'lbl"Iiri';'"tti'1'1''il'-1'ii1\''it1r:'i'1r1l"i;>1r;tiirrt'ol'irttlti.}'t't'li'il:trlt'rtr:'tllt''l'illtcl''lt:l:l:ilttisl)ltt1f"i.t:rt1ttr'1.l"itlcitsl{';lg'iprlrli},ro'utt;tli[]lit]slrardr'r"'r(rttl'rcic';trittt'tur',tlr'.';rt]'rit't' zpbooii--ici'ijsclcccCNldripi.ccits1rlitci:i.)rrrrrc,.tcr\icncirliicu'rrr.'l"'lrlt.rlsrl'g'i'taLJitnI,cL'drl;rr,tir'r'sritrcc1lI'zrrIi.S;.ir;lI;tLil,rlulrLtlz)r,'r(.,i:)rlpii.sl'ttrIiirrii'uirp$rir[ic_ri-arrntris]r'rlr'-*'riri urr1li;lllilrrt.i,,rr.tlIrir.erls1tciliirrertcu,rtrr-riatuirliruiriirtcu.lr,1rolrrrtilrrlrlcsr_il,:i
nrr.aJlcirrtg\l;irrlcrlrrn\ilili.rhrtinrtai-iic1rt{,ali;'rp:Lct'(.c1r.rit,n}'tlll;lil,9.,ictrFlL:i[ii;ll1la'tilmr,clrrithrt;ic5L'ipitircrrr;oi'.a:(?lttnrl:l',foilulilIitttrcrct)itrll,rlrtle1rcirc,to(ttltirc1(''dp'tt'l,,ii1r)tr\r'tr;icu,,i:t'itr,,rc[i-sl'',"ltivli..,lrI(ttI,,,tiicc,iol'rll,,reIlrn.,itg'ir'tlii-'t',nr,'-ir.""'r,,ii*Ji'l1"si;"',tl.ril";;'ti".rirli't,'::t,fliitii'9"';;;;t.i',r'-':i"1't',u'tti'i'1tt,,'i'""ct'ii;''f"i0,;i*i-";tut,'tcii';;'i'"lt'"t"'''i'';"rsiil'C;"i.lfitt'til'ittt:pi'aifi"a''titp'till"rr'i"tt"alr'1trtl'\tiiatilctsiri*""t]-ilutcltrula")iirl"u'i"ute't'i-i"'trt'ri:*sp''tl't'i.illtit1rtilfu'ttc1rh'itr'ilatr"tli]bire''{n'rlu;llp,lll)l'lt.ri(iihlr's-t:cttr'}t'(rr'ssiIlsrtar'd'.(]fe;?nti''Hltrtiritt)itrilrrt(1l['lr\rcrliiiri'(tstun'Itiri)rrr'slrt"rrr'tblllirllritar''ilr1rrui'''r:clcil"Ioo9-citttgttnlttitt4t,l'citll';ifttcfr2rtriIvt'rc"ttl,tilrtilJt't;criN''lltmti1iis"crtss1n'otIiu11ti'rati1l1stMpillir'cillc1r,ll}r$luP'c]tlrrair"n'rt1irtt'1tlflt1'rcrvil"r'orttc'r51rtl'e'iri-itt--rsi'sl''l-'t'-'i
r:a asta ?
210 atpooia-n*st-ciiibeifzinlalriuucsatnji,,pruanri>inrirt.i.tcit;li:tninati-culu.er,1rsrcncrni:iilr-m'iictlLuc:nisllclrl-arcrr._idjurrrarr:ir.siriiipiisrric-lrlc,;rli)lpior)(ooiIlriharlticetiPaccci.iJlrlsrNltcli]jcs'luit?cr;'sr"t.rrti,'{euo]jnlatau"tsetcrl.u',drre:rcudlceitlic.[[r,.;aritsr,cifli

1,\ctualitirs sci. g1 iurl.. l):rris, lgl2, rrr. t)12, -t.. J..=-.2_

2|L

|;b;-i:f;;ltin;tfiiiil;cu;;iJ,;;lirnil;";i;'in;r;trl;;l;it;tiie"*,s;';i;iici;i,,it,ibio;,tc:,raia*.;iiirt"'l,iest;i"l.;[li.uu'cipoli;ei*rtu;iopiii'pta-bitilo.cgil',e"aei;tium"-;"f"lc;c"a.eijarl,;ulpftct;et.rnu=r"c,cri;,brii-nc'uic,ct{raa,iutiellgiecs',r;.a5aa,ir'.c*rilllaepeaaadletsipcfdIhcpaareacieui;eiiapisitcetc!ueamrllil]ece-ei'uncnrl'iciiomtillirutortrcaartrmirlt.tnuip:rins-lrc-ianrseoconidhrSalnrcraicpnrinirutftnieniiizerdrai-ii!i svgcrllr-tuoiuoer**nnrlslgrruaeril.iuNtlJluoegfb:uiIaa\,,f,BitipIloc'rCaueaetudtDii.uuepdasEralS"ictilquclF{cdoc5n,ecGjrta,sa,ss,ccr.ecdrnlitt,,rdneri1in?oim1'cubl'a3r.unt,'z.i.inliptrunrofoludlh3lpoirgoc,ecopia4nrt-liac,atnapl'tsgigiicoctnoi"lginomlgaia'ob(crpftcalit,u;cinn;ni,.cref"l6isagc5asroaru)tir"ie'sicaitlJia..ops.ntrric,clirt6r.-olurnlgriCiqr;ioi,it"snign"*fcurur;pi,;p5,cirtiro.*l'irii,Ji*il-,il
;n'ri;'iit;u-;i;ri.i-r1;tr"coq*fct-C,tt,.c;Iiia'n.ecminlte'irtgi,;aatrlnl[uti:rrrit't,uli,nl.itstnatvactooirtcleueislstcicr;ioliaieiucepsrpciac1-roar'el1-rt'.\cntpi'il']l,a.oel'csC.lti4tieoccc.dxniilntret:ensltdep?t!rrti''n'-,o'arcpataclotiralserdigtpersouLcacstccaonahrtre'iellacara;nstr\tte''ipricl":aaR'p:iseueoa-t-t :I,.rl,rlzlt.,;lfltjtrC,"tdi.tCfS, r,,aIrd'i].r'lrrrr.irlilriir.laI l,rr,rr.ilnr'oliur
Lir.ci'rrigiriinr
Crr lrCenA dirr r ir.l'rri i , re.

Itig. 65

-gceic.euliaaac-l-i,ra'uist,rrr-nea"vF=uEp.octnro1nruNiil1i.-ntp,'rro1cltci.oooruo,rop1ilit"it.?ccg)t,"eu.'esiOcl"in,a"5rn,eo..fl-iorronticaslt-craltatcrppriilnnnoetatitr,nurrioeneticrttbu1larcuaeeslsiotbttpanl";nlteiiacitcngtc1ictcsuuaidpursioiinntnailrgiiueig1eeLdl8don'r8t5emde2{Ch,le9ieci'1ttlpilertl.niienccntddipdpt-rmlmoaealinllmgrlc{sgoo-'ai.nanpA,{cpi1ltlaea1linrt-t,
5notef,.qil:"l*iru,o".a,.t.i',*,oir'pflrirlbtnt:otai,oeuc,,-l,nnis'targ"iatalci^Jtolriafr'fIrci;tl,.nfiti,i*.Ncci1..;,s'irll-ra;sricclalr"-ilft;Jlntrdctc-cittettlgeifct'oplttaa,tr;trllon-i'Liatoat,.tai'ltgcticditiii.a--rsnptsiyeolciluocel-seanlaxlttbapiiucvz"ternacuudesor,ctltgpcaicncsaitctttldtll'litl.cnnhiaacno!tautciucrcgicixciennliseiunasgcttncreatrtsiiian"clrccatredgitirlir'imeic!amrtrcia,d-.ucippuer-at"enlalorctricibetcrtebfaoardaln,uodabdrlcrtciidgaulemcS'crcslseI.e.usEnpUna'labla-rocasnucrieslb.iunun.sccaal1acanuis;i-oc:.-rtgtit,Bic:atqi'1tl

pnc-'n--tadgr;a;ruasd! nu ! Odir,rioarl sernnul pitagoricienilor; azi
sterna ![rii noastre !
irnjp-ondo6brten;dte ;i pltratul, orice poligon
la o paric triunghiul
uo*or,*-n*i*[rlat adna;tere la rin*I sau Ia mai multe poligoane

stelate regulate.

242

lrr a.culasi tirurr. t,,r'in uttilca t'ilfnriLir AIiCDIIIiG sall a tNi uurcirs;thcirlrrrcicstrtilrt'ruilncrl,toliliicridttn:rrlletgtu-'clalt:tIstfctJriu'ct?.rr: .ctrrcd'rlui I
tilluriloi eLrcclcfg'sc obtin altc clouir hcptagoanc' collccll-
i.i"u .u ptirnul,"-1i ac.ri." tot- r'cgulatc. Dai sc mai polte:
arlrtinc rrir t,oligc,p stulat lcgrrlat'tliltr-uu lloligol rcgttlitt - -\*,, ,lirr acc-lasi nrotiv pi:trtru ,i(:i lr..rago'ul
alllllll(': sc clttcit <lilgt'nalcle ltt i. ccca ct' i" nl(;rri cill.(r n*
toatc uroihi|ilc posibilc Virfu- rrrr poJigolt str.lrrt !
<:r,iilcs ;i littfci, nncasciL sii )l
irr
.tnsc,amdir si sc ',r-il-.tivui-,i.'*ctltrIe.)iccczuir1rt*rrct.lllrlicin.rtorahngfaca,"x'ic';iyinridgcoi"'cp;stilocc',gssro'tcnrL'ru?rsr'itt(,:aa.iiz''lcIriirl,ii,rrrrlh-uiaic'rcixoalaargdj'oli.t''ttr.lirubtr.is'chIinicrxccrr,Liritigioiro'tpsneiourti"li,ti
rilc lui. 1n primul caz sc zicc cir pc"'ligc,.nul co-n\icx.gellcrft-
rtnr;,ar"trc: s,tcit'r'i.nrrtili.tcf IlocilkcJpoclllitg--ocnirlrcl lsa/cu tlc poligoanclc stclatc for-
iitt'clcslr '
.Flsact la fc'l sc irttimplii ;i cu 1.loliech'clc r-:ouVcx(r r{igr'r- ltitrair.goi.ul-ttirc,-nr;'i3rsiirrirlnrrnrriuritsirrIicartls,Irol)tuecilr:til-c..:[;{.ltrctii.llc]rl'itliurlt:nnirctg'tirlir,clrilui(fLt.r(t--t1-'irc'ltise1i/r:r,rr:iprrrlr.lrrurrrrllit-.r:riSrrstiotcrdll,oi;,r' n'z,trrrior.lirrrsrriznti(irqfrli,a,.q,lti'.ir"rlirr{lrii6n:)ir\..rlr:l..(lersat;l;ij-.i
latc, I)iri r,ic st- pciL coristrui policclrc stelatc-r'cguliLtc, licr
unin<l , irr toatr: rnoclnr.ilc pr,silrile , i. irfril'ilc policdrtriui cot-r-
vcx ll-rspccti\" aclicii tiiinclrr-l Prin fcgclc.tlittgonalc- carc:
rczrritir --, ii,:pr"clilnE;inilmuchiilc, sau, ttlai binc zis, fefclc:
rrolir:clrului. Dcsi nrr lc-a stabilit tcoria. I{cplel a cunoscttt - $i din lrrr,.:as1ir t:rluz_, t.l sr,s()rt(,t.si,t l)()ligo, stclat?
uqlrr.,ltttr-Xt rdt titnloclsotrrrirlcdscirr'1rtrrrtI' illrto"licsdltrttt'1r,c' rsltt'scclat'tJcrcrr(i'tSr"llllfltlt'lc.;tLirl,r'lcblairlrttr"'
-.t.r'-o Fiiric:inir-'lcs, iiirr<lc;i {ig'ni'l uccastir rri s. ()irti'r: pr-in-
r:orrtrrr:irln i .sirr
clc cirtrc lloiusot irr l80t) si clc lccca dc multc ori vci .intilni trii,sirturri colrtiirui lr , ilsii crulr r'rcsenr:zi
1-rolitdrelc rcgnlatc
stclatc' sillr numclc tlc policclrclc I(c,- un_ti:iuui;iri, sarr pcirtagriinra. ,\r,'r,.lusi lur:r-ri sl irrtiurpli -.i cri
L,i,,sttr
lrrilirtlntl (l I'slrr'ral'c ltli,aj i"oi-bit, cunoSi.rrt snlt

'Erlcr- Poinsr-rt. cttrn vorbciti intclcg cii sittt nunrai patt-u

Dupir fclul
polierclre regulatc, stclatc. l)c cc rtuurai patru-,' l"ii-ai ariitilt,
cloar,, ci, uil poligon poatc rla naltct"r: lit tnlti multc poli-
isrooaliellccl,st,tc:rlaetgc,.laPtcr,in1rai nsaclogic, <l:iin'i lt.ndt.csbratimsiiil ciL sint cinci

rniL altcp1 la

nr;ri multe Poliedrc stclat'ac.rI.r: t:it utt pot ii mai mult
-- Nu, Cauchv n clovt-'clit pliurul
a.s,ltieiercnlaittt"priautetnrt.iguahepicoollrtiue;lc,ilrrpc:uilrtrcrrrgi]tuu1lail tocncutsotlgcrctlalrntgcuc.lurDcrurcatfaonrl.timfceci lar,ztuait lnpacicltilgiu;cpolincr

'$rrili-edinrt:asdtecvlaiutc', I ura-r ginricsc urlri ltinc, v'ircl car nici prin

cllrcii nici prin a diagonal-elor--nn
l* cub. un poligo-' stclat. l)ar
'ft)jro'ct lfuonignirae,al*fetleetlroarcdojp.uusscau;i
dc ce nu s-a' fotma nnul1 Prcl*ngincl
ica?tocc3tarelinclrfuefe'ule,otnplneslre''ge arc inc0njcrarir pe a patra se fOrmeazt"t

t'tr I; i;. tiG

nulnele clc steaua octogonali, lnsf, nu-i un poliedru stelatu
ci o lnl[n!uire cle doui tctraedre !
- Niciodatzi n-am binuit ci polieelrele pot lnchidc' in
ek*: atitea tainc !
Stai tainclor iucl uu ti-am descoperil-o.
cir taina
Flstleazl-{i uimilea, ca sir o ai intactii pentru clipa aceea I
Dupd. cum vezi, rru ti-a mai r5"rnas, ca materie primi pentlu

poliedrclc str:latc dccit rloclecaedlul ;i rercilrt'ocul lui:
icosaedrul I
-- Atunci si
cdoinirtsri:dcecr edloccl1e2ca{cefcelrnolb. f.Dinaclinptriei lucnli5nlefsiei:cmartet-
chiilc f iecirlcia
{alii cittr ur pc;rtagon stelat, cr.r collurile ln arfala c-'lr:clccacr-

c1rirlui. insir cite 5 clintrc acestc colluri sc ultesc formind Iiig. tiS

un unglii poiiedru pentagonal rcgulat.
Foartc a.dcvirat ! Acesta este r+t''icttl d,tttlt:cnt:tlnt stt:ktt
fele stclate . T)upX crrn sir-.grir- ai .s;-lus, arc 1,1 {r,tc cliago;
- ;iai(l ig.67) , sc t'r1 1).ul11i.rclg dotiecaedt'ul tle spccia n Lrtict ctt,

Afnuanrlucirn;sictcr3slai0rts,c:clccl.i1rrp'curcsccluhiuinig.tllisrtr-cIa-ierr:c,{rli:ritr1c, trl5*itrriirntt'rf:,,ircr''icri'crtcie.rrc,tliLru.ilii'l.rsl.*iLt,'\tas.tl,icIaZtrktuIrclltetrr-..
caedrnlni Prcisnpiri..ccctir,lent(ifiing..{aiti)c.clirr:ra5ccfts'ttt:cst{i2ti5r1lL, tit,,b!P.int,ceamrirsrrfta.r}at
lprisirfcfei
rchloaclleccasetcehla'utcs.teNluart.rdf.routlcfui:tcI2lorr.'il-rLlu*r'iinril'atrocttrl2.irsgi;ilriilr,,*rilut,c1irri,i,iltiti,r-
30,.clar:, fafI ctc prinrui tlocleclrt tlru sLt. la.t, nr,c,st^ al)ill.r] (r;l
reciprocril lui, cziciacestlr
fete ucn Iagona.ic coulr... xr', alr: unqhiuri lrolicrll.alt. srciatt: ;i
pr i ilu l rtr:c lrtr,l,: puri agon a
ia r. ie

stelate si virfrrrile policrL"alc oonlrrxt.. li j su nuritc.ctt: .,rtttt,tle
tjri.le{rrrilaodtcrt-r.at,e.,t:,l,ru(l(u}tnj'cd)ee
claclacaciit'rt. san tlodt:caciittt rlt: sjtr:t:irt.
l)acti se lxclungcs,c rnuch
(fig. 6Sl.

speciit a 3-a cu fct* conr,cxe (iig.70) sc obf inc rrr no. crrirc.-
eaedru, rtttreIc rfodtcttt:,1 itt .s/i/)z/ sar.r ,lcdecntdtttr, cIe speeia
tt 7-a, care ate vlrfuti, avind citt:3 pentagoanc stt:-
20 dc fefelor lui fiincl tit 12 iar
later ln fiecare numirrul
virf,
muchii 30 (fig'. 71). in fine, ultimul policclru s{.clart, care
se nutne;tc tns.yclc 'ieosaetlnt stttt ,icosat,d,yul, tle slt:cia a Z-6
Fig. 67

2't-1.

lris. t 0

friilI"riultnititi"lixrf.,.r'7s"c,.'.it2o,i.t,rtnI,lasalsiiirtitrr,'tlr.('tlcsi;t'tcrir,t:ltlt'2tio(rr:bll&ircltr{r:rii}crrr'lltcIti2rrr:ralrLtIdlzitit.1.iiru:n,tli(ot:l'l1tlltiir;rtncinl'.roiiiiirrl'ls'tttstat'irglrcc'rcflditlnntiin'ttcttlrrrainllulrlpritttitrttLc''i;,cr'lt'uiciutttl'tltti:gyt<ltrit'ritttrtt-tltiltf'lclutrtutrl'rttntiilloltogtoll'httrcll'ftigla:t;tp'iti2'ril.{lr.:(icnt]l,tc.a--I ir' i I

ct'lol ll fr.lt' .'llt',' ittt:otl.iolrli tlrl?t o1rtlr;"1 ltt:cstt i:t '
.,,t-,i-itcSiricc,Lltr,rasta, ! irrst'lttlttli'i ci lti i'clrr:iat
lxrsta t{rittt', atit c'tryt'\c cit si')tsl\t:(c:slatctca'

't'gtr1atc,

Itis-.7U

Fis. 72

t!nirrinrl,.fa')cntcsula"o"lirr..,.itaal-_.tt''lsl{,ru1ii.ir-in,rlr!.'i;irt';itrtut'cm,li,(a't-I*2ci<ii'st,)a'nr.;tlftui.ctrii'i),,enr"i.sfl't1r-su;*tol)rvt!iit"l,rlgii:"sla{r"sre'r,t.ult,'i,ruti-ilu'-i';,ltrcetcl-,at"llttr'rill,aciln'tliI"tl,r1:lccgtctf"atigttcs,'.Curriir(htr.tl.l'c'lii12(im,l1rt,lrr.(.1tro,'l'til"muttt,riirl.,L,,lt"il';a"i,l?llqlar,.';ia'it'1ltbailtr"uitrirr^i:i"tialct.t:"i",'itoi;'''iu;t;"l.:i""lp'tt;Ltlict'c;i;lutctt1;e;lt'ili"eaitti.t;Acqtnticcr.i''o''lpar"ieot''iitat,llmect{it)iLtrt'f^;tcttr'rceop-illalaici!l,cst,d-art:tt,a'licci'5cucriciidlctesl.',cdll-ttr'e.ltil.htntctlt-sicrtl'mlr'Ir-t''tstl'tttrllrc'lrtcto,la'lil:g'L"i"acaii(asrtst'cfctan'itr,i(csinsriet,ncl'cnttaciucccc'tva-d[rioolsr.rliaa)rruttt-enctlnr'rriatritl1'icslrttttr'ic1rrlisc(r-t1rutnar'ittPsrr((il'rltsrsriia''luio(cig-pespact1),cirihi'psloit.oorlull1hecduvitlH:ticn'il1rrtnccalia.l'lrmrrar?dac\ertlsllvrt^)ctrr-o.t(ite[c1iltttlc'rt''r'Li*l''ll;'a':''.r]n)itt(li-tr--' -ilVlr).)!)rrloricurTnrt'sbtec,atarillr.igocococt.lculdrap)pl,uoiitilil)luulictenscrcclrtotttcar;?culigra,rocli,n.cirnulr{enJ1tues',anlitesuildianiiti",ge'itJu;r.l,"r;.ic"cI.rn;Ii;;i"t;ilr'ai"l;ric"rr;ctiii,irs,s"tsfacdpraIrcruzil.n,ei,rtcpa.eucdcmdctlreaarscraa'tdccraer,afeaiirmsilItua_ii
I.)i.-rrduaC;ti{ariitaiarctt.-rletirrancsiit;larr}sri,,}to:i,i:.,'irn,)rcr.,:iir,ini_,ui.i,ri,iiir,_;'i;x,:i;:s'';tf,l:i,'lrc,.ie,irlt{,(,.l;ir,c',ia,.r,c,:L1tme,ra..i"h1irlrra;:a"1d,xite,sj..

ctrgLlaen"r-9zpaotcl,icdce''esirrt.gurlrJnuirutcrc,tri_rctlto-r,:i ,t., i,,n't.l ,,,,,, rriai dcglabd

4,69,1?-si z0!

tI iti*tc i',. tLr , riaci1 it, pu u toatr: ltolirrrlrelc pc r.\tcl:
T'ttr. ('ub. ()r.r .Dorler lnts. .l)od. ,,j,st
J;it,atr[li,ttlc'tst'll,',',llr-,it'ilrtp1lulrr,tlrgt,'i;l.'ll'\:rtpP;t'llnc,iualt;tird;t't.;tttr,,,sttc''1it'ittiln'lill,r;iril'llr'i(pr;ri'itr.rtlt't:t'[it1tr-itt(llrs.t'"'t('tairi;'l"ririiit'l{liiiiltti"ett''i;s'iing"t'i'-'itii;t;'uii:''ilit-i1i't''''-"tot'^;1ll'tt;a'tt;t"'iiltlc't'tt''ttatl";t'6t5Si'1:\o'"tl't:1)?r'tsi.l'l-p'oSli'l1'tts'rtt'^.l't1)irrivzttl'ril'?'crii'l..t:riri'1lo'tr't\tllr'trcrltlsIlirt5ilgitrtlnt":t'rcrl'ntrltrictal1rliarrrt'1tti'iloilrlc(lr'tit'ac"(ilIrr'r'ilrliec1tlss1rrl'strt1lir'iSi1'nttr:'-'liiolrtulli'lt(r'i[Ii1rtll'rrrc't.lr;tt'llil-t'i--' l.)od. 3tott. .l)ttd .,i I t:os.. st.
I;tlc: "t 6 N l? ')rl tt)
12 12 2(J
l'ir{uri: tt 8 6 2{) 12 t2 lrj 20 12

rLll"tirt1.,:1lr"tf.l,r\,.rc"lirirrt,c:lt'itc-ttlllni,sii,trl,c.tititii"i,,,t,i.'i,igti'rsifli<tn't:rr'"ra'rc,rn,aai!',i;r'l,i"lr'rptitrt[st".piltncrcitliti.rr''t I''zi iltt 'rt'lt' loltjl!lriIt)'rir'irI l;i,itr:\trrrrir'cr.rr''rc(r(-i-.ltlatar-(Bri'i'Tcs'mdaaItto0Ici'rc.iniisir{iil$rrjiI'rorIrfitrriur.iLtitlirltn*rtrart,urj'irb;il,irtrzcapctdricrsrcr;cta,rol'isrtirgri;l.,ttr,.rt.ri\,;ir,r,,rt,,';),urILI,arrr,rcPrr,ilutr.s(nroirilts,itc,..,lrrsrrsiur.rt.;lr,;icr1r,r,r;rl;z,ir-;rr-,lut;.i,;iu.r,r,l..o,th,ttr;,;trrr,.rii,;.r,prt;.r;/.i.li;i'i.Ei.!i;i\rir,.i0;'l.r,I;'r;.l,]diul;;irosi-i:.rtt',fn,,a,cr,c.pi.lrr;tilo,ei"irrl;r,r,rur-r'i,ru1,t!ti,lrr.'ii,o;,inu,rri;clrr)cciro1''pra;arcs.rlcr'roiil),usc;'ittr1rf,cA1sirct-l,.taci,eIt$glc:tr|.mcrt;liv)lrtt:cuarule:rs-.lri
'l''rc' tlt lc, t'()11\'t'.{() ji,ll:;c.iI'rlrr)o).rlsors(:asmr*ic_otc.)i'c1ppctcbtilthr.-giiumanlisl*iai;lsleit;tn.lirr-inro.nuiiirst,rcrrnnreaxi-imciaccsslis,ushpea-raotc:lurcmaac:Dte_rr'icarr..ellcincsLl,rianAr,cd:,isf,,"c'r.fiantari"iearatc.-rcunroLriur^r,rcc",.rniodpdi,dl.rcJ,tc.la''ntp.mOrg,itm?e..iii;iJigibc;,ru.6"riuen"r;'l,ii;*t:t-;""l;ori';l,;trtr-;s',r;,fii;,rL,iil;rii:,i;_,;';'oc2raidli.i,arnr.r_r;r"atr";ecg,"'izlarlc.ccn,orlnroscf?rduorc.cc.cmissousebtpid;trlilrd.ouira_aiarr{cr.i.ictli4c:Lf,tf,alie,tuzdNcntoeniurluueuiiIpel_aoirc.ii-,ri

Pr;licdr"t'

s5{

l,,,lli,ii).uol,rr"i-)s.-a.l,i'i,l*,)rlir(n'idls'c,r1,\ly,i,a,grii.cctrlirtnrrIl;ttric,it,iiarslcr.iutlt'cllrcu)itsc*'lrcvciLts[t,lc,t'Nt.cr1'lrai.i-trip\'r:iuii_cnn3ital-iaosrsrIoscl'ftaiir$r'rifcCscCoahsalcttlltravnrbtl'ocll(1'u']lt1o)r1l"glalit(tIe'ceirJaS"ue.'tcnsIleliioccrilililu'itc,cl'l]lcrierel:tzcclinrce, r.t'o 45 dr au isirl'' ulti 'ratcmaticir,rri ir*;tri ca.]. r3c.r.tr.'cl

ii A. Car.lt.r.. '
flt.cttfIrt;rrritcctcrieitrir'io.'t,rot-a-c'ricli'-du)Imrr:clrr,l-grcri.cc,lrrrir.Jr.iltan,cll\irilrr'carciisir'rrgillcr'a,trcccatlt'irl1)strtcririir,latrsnc6rirpicc''rrarnr;rcc-tracr1l:ic'ctrtrrlrrru,ltrirfai<.r'.,raccrtllrc'tsci:lrrlrr(.irLcc.-(r-rrrt'tirrir'ni.lcar,.ur"iill,Pirtnstl<rt'str)tli1(caclrt-iro'i/rii,riss1ts;o'rIi"',iniccftr,r.,,irrcrrnrrti.?crlrrscc;cili(i?r'cr[c'naitirr,oit,rr'tt,o,.trrt*piairiir.rrrt'r,es.r7g-'crr0''2pcittrrgi^rnn.itrlcc't"0liir<.ocn.gs'ic';ac'nr,r!sf.ri.",t:{",n,g.P{rtr,,rs)l:,itcaciraii-1.rt'.u,c,t.nlxrn,iitcil,liftii..(tr*ru'ltiiic"c.rior!i.i,o",;rIg;i-a,"rc''r,"i;.is,c,;iljriir.iiji(i;t;tigri.ci;_,"",i';".o'a",rar-.i,i",iIriiarril;.i;gliiil'ur,iitot_i";;j;ir.rlri;r;1a,clrnli'prctj";,r.lrr,r.ir,;',t.rrl,i,i,"i.rrirrfi:.;clt.c"u"-ni;ri"rtvirciir;Slrtii,a,clto;G''si.,itic,ti,.irtrl"tcrc'gct".irt-";i":l,,t'tsri'ri:r,,nt.r..'iniil(ii',"utrt:i,rtr.,,'r;r",[t{ii!;",sor"g"r(,:i..i'rtt.r,,'c:.\;,rsriii.,4gc,,;.",rir"r,gtcri];r,':Lri,...ri;ib'ii:c,rlrr;r.r;r',rsr.:L;rt,'titc,ii.r'l""ti(rcrisl,ci.l.,cri_cirirc:.:ri
i",i,utri.i:.*"oin*-bi,trl-i"*l',i",,tritur,.firi:;ilrl*crci.i-,r,naltraIillotiitlsr.a\tl,rltncor.eli\i.g,l:nratia--trioiLrn,',t,[c,a,trcrlcol"ilccrs.iilrltiitltcatelcsitallt!trtiactat,btl.iLatliia.tsgourcrpacirrnlacgi'ceJ,ciabccfcsalrItl:cc'5e.i.
,,j',fcflir,.o"*ic",r-eiamf.Ooa"lsui.ifcsitt..tttri'i.ti'ictCcfrti,ieopai;pr,utbrrrcc'cifrsoillcrlirrc;:)trcrd,:lltltia.c''sructc,ais:tnlc{,rternecatruo,bit.t',vleoIgtic'lasaiilxi*lrlLrrcplcml,ar_llcr.ital:crzcarii.ilirrlircf1nl.lrr'1iraotstarulliil1pit*c'ru{ordac(ncli.r'lttaiiec'.i'cn-illcrxcclrli1.itsu"llclrllt.ctaucirin.scItcpalit:iotccl''tncc':ecctsi-tllncci-cva,l.nrcCl'tl'Pilrll:gctr'elai1rsnl1lfe*rt1prtcctt;;r-:iliii
,,g,,iiI.)ioi"nnui t'-r.iiu:,'gip1f ii'.rii strit c'galc <1.cii art laturilc rcslrcctiv
rr.1ri prrrlt clccit trci i5tlr:-''ti-r#5u1"m1I11)c.11,ru.c;cit,clic.?cagafsi)ias,ccitidiurfaocc'rnn:cnrc.ci,azrsu(irr.ppfrooiii,llniipcc\c;icillll.;.c,.i,,iiriiielJilg,e-"u'g;ili;a.ualtrclrratuIct,Iaiorarif_airlcanici'ta;ac;rslc;cticc!.i;tN,! s;*r;i
lii ,rilg,rat.r cli
lrtttiri I
tr]()l-L'1]1i-r ar- Iio ga'ilCf lr.lizarc i-rtllltt'lr e ar.ttl
-- ,,\parlar lrCcast;-r it t(..)rt,l)lcitlcsilr,l i'g-ilIitat('il tritrrrgltitt-

lroliec|lclor"Colt\'('x(.,
li 1ot"?
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, Trr. t';ti tlil for.rna ct.l rlintti j.

l'.st r r.i srr icli itr itilln0,./

:,';,,'[;;";iila abs,rcl I Accstei oncbcicacstgiriicCaaurztcdir'yc- l'euli-ide'*mtoen*-
;;i;;;t";$i;il.;ii,lt;r.irlrioinntsi,ritfpi,cpirola;itlec,tliu a fost r.luatii pr:ste

252 't:'Slenra

NOTUNE/{ DE ORUP irrppcsliivt-in;aolir,iccniirirt;lteneiD'rrr,ciii;.rcctctluldlijf:,cial;_ca,i;lccoiac;tidcr:,:c,ro:;sr,aa'r,af'ti1'rsi.rs".r,"rcir:s,t:c,ri.ir,ul;tljriJ."T;uurrrc;:rd:ni,rn';nro...tiso,rnrjuIba,"iirrtcro.li*tir,cli,,c,i.irf-r,r,llr,a,.rir:.,.t;iutra,,ltr."irc;'ic'ti:r,;iiciu:-,ct-:;i*e,i;'",oi,i;.;.",,)r"l;r"iu..i;'_,r.orlii'_r;lin-",';ir;;i';iii":l;r";;-1l'l,il;,;;.'i;rr;l,;'1;t'i",inr;i";,:'ri*cni,i.i.e':iic;,,.i.r,.nrrr,$.iflLr,,ujlt,'..t;,c,c9ial,iro,s!t1trc'rr,troi,,c,ir,ia1iriitijt,:,ro,n,i,,rsJia,sita,r'nlri'i'i.,ti,i,"cai"lt"lircrmo,sigro,i;c'rir[tti.gc:i;ilicirtni".rir:ilcso'aout,umtii"ll,ir;t"r_,:rr;fr

6au ii ;i.lrzpcemiiinsIt,9:;nnspnusit*-ullc;oPtaria.ipro-:trcn:*r"ilraa_i,f,.,tilisuiitdulrigr,rrt'/idlcp.'n,nsntrnel,eAair.IC,tlttr'.c?gpcirzcr"cc,tDct;rndnnlt,ct.cetiijl".too'd'?Itiainnaiil'aiiito.rSrn;cnamciclc,t.tlr'p.iuirr;,srctlctlrc.cc';,n''cie,nealcil.1.utariclcia.trrtlinspi'iupr.ialrlrg*rccf',acen.cdeuPor,rssuebiiicaiortsruar1.isriitaiis.ttsedirrbLcnc_i.t,n,idttaen.,t"iiiuisccitn-tca;ertlmi.s.ra.lcrcccr.i,ltl[.at,ltlitrolnc,,ic"uunrric.ir1,ln'i,]irincr,r-rt,ziley,i.1lmcrsu;r.einr.1"saitrr;i.**sgn?iat.ciilniii.;*,1^i,r,_':raj.rsaa[aa-i,irfrpC,*lrita,,c':l"f1i,t,ilriir.ixr,,p'eae,i1lt-s\ao,iioiitrf:af.a,'iolilrtc.r.f.il.,,bir,_liirvitrii,ilsi.c,ci-ct.cr,Tl..rie.,,e,l,;cri"icltruar,_s,rii;p.nblm::Sulic.e".,p;..if"."ii,.r,i,;t;t,tci:.bt;eiii_i";:;i.lics;rii,ri;ui,ifi,:r;i,i.rrrt'_fi,ri;;g:.;1.ircir;),;*rr?,idc,airi._rtirLi;ll,,rii'juiipi.,l,tr.id"ii..i,"lrs.ir"""rilaic;i,l,,r,,s're,Iies1tf.ci,;rl,i-,.,-rt';,,l*cor"'snaci1ioi"d.k;lJ,.;'e:rzailr,rUl","frsrnpjr,.'iiail.i.rnrtf'r;crrru,_,t,rn,inril,o"si.rll,tn_ecir.,a"vr;irthcruJitls_t;rnm'ir".ceiiccri_.i;.r.iu,aiilculrtriuol,a;rilarrel!eb,rd;s,nit.lt;,rinmtt;rlc;a'rifct,i:rrLiar;itteigutlaiur'cerrisci;a:rti.,alpitlaicraiel,fiZJuartsia:,.;nuircnr.i"rortpjiv:.fr,,"crn,ttua'rintcst:,Ll,usu,'e,,cit;dr.c"fe::r;,.;tlnoraascl..e;s.,,ril.ircis=is,risoori:inat.s,HtrrA;crr'1ta,,vc_riuou#tair,t;r:t:r_,1rri:aeai1ri.os*rtlsel:n:cc.1i,s.i-iua-I?,.cn-;-idEe,i1,.r

Unile dai gi unde uap6

nbaa*so-,.fSoci.nticS'ltetuluor?ncaaIr-l'l:-,t'ncsl1Guir'p:rt,crit1iccc,lucirs,ctlitltlc,?-i-\i'tcIrn'Lliln--laucsllclmpnultl'-aliccattbmecanlctltclplatlriiii'menrti'aertc,snmlu' nel aamsstt'luun-'
dlc*n-tiNei-,uvi rgeohira:iittit! cuno;tinll de rnai-tilziu?
Ctt nepotul- tin, i{ur;rr I in
* X"-nii t'itrc sz-t- ct',:ci ciL bucur'irr cart'-ti strirhicrste

c"i'c-b*i iiujnn1,agrncc11t'cN1.ritcri:antrrzri_r liinitnllcnhirrpcaui..,ln Cc incrirr-:itiurir nt-a
l)lln(' lr.{)t( ;ll)il.l)Ul !
xL'l-lr-i.ll't'l clnd cl-uaiio'itin.zl.'t:\ulu-c1rlr'rr1rt.i la r'api.a
,ir".ir,g.,,,ri ! Dc r:ozi tlccil
clectroriici, Tr:-a irnbroltoclit
sc-in-ae-vt-usc:lni',rqtlxuai ri;tolili;rt,Uiinisgittai;uiitr-rrrtale.eu!rta,' cb*tu'.ac,r'nnr|b:lu"avaa,etlrnct:-'c,i.g''eacchlb-oicaiat'ri''.irnfbtiiluouttlznlcof ndlfieimI nclanup-diupllirlo'io'{cbtodsai'irt
clupii lt orc de nruncii ;i cind llr-a r-iizut', cit'spre cr:r crezi cd
rni*-a vcribit?
I)espr"e rnalinai clc calcul !
cc bit:r:-rrri pare e:;'i' viL lrrtll-
* I)a' cle nnde I ]\[i-a spus:
tttl 1'c!i scoatc ditrtr:-o grea
nesc, cloinnule Solonar'" Dv.
$ncurcitur[ ! car<,'? l-am inircl-rirt cu " Dttmnear.oitstrS. singur
i* ipirt"l i spunc cle nncle v ine' cuvint ui m ntcttt rt lirri. C lncl mi-a

254

j?sil;;rri;1liri"Tnnr_ii"-.a;i;la";ll,iAlbsintinfi.tr;""t'i,;rl;S'"ilr";.rc'ri.il;.--cc)"aulA!iil{i;n.Srr"JiNar,l-,'o-aSt;;i;',uuir,ii,\rric:itrrat,ci_ti:ut,5tc'itciisltiurr-itlnrit,t,itc,a-;z,.iuctliitalrecr'tllcs'nciclfsctiirrattii'ec1riatzoci:'.Cil:cncnJiCaICllt1-Szs.amrtdlirtii",itcLaii,-r's'rrartc:lrscirtIitr;iuettirirrtra.toll:acs5lrtrrallcr"arl'uc{s-nirztItnatttattur*ri1tttrtatrr:itlrit.t.rcilairal1irttl'tcillit,lpprpnlrrlltliloll'tiiel,ttc-lil')tcAct'JartlsrrllrillotlcL('':li'i; nlntfarmumnrdppoartdeeuoueenuncerizlielsnlecubnlmru.sag:inr.c,.dpupdl.iltm,ptm"oro-iedD-ie,ecetc\Drculd{[ileieluccafiizugmesu.iursirilepdcvzapdupApileod.ircrf.ciipvierai2mtceczipreduaccns,i.luiiaeaedecnrcgttaArnosirstrecdmideioauud;lpcittmoeoeemrercasuu,rtie,tclsceuidoairlcsatil.eiccacnvuaeiptlcr.uuI^aoitzcoiprrtcoemgmtuoveibrrmiooein.tm,ui,zmainilnirpvcueiaitoeue'eittchiurcuitvunriipriiegsrnal-mrciigbn.e"iu-tto"r_iira_eniaeerD-n,c.czo;tli'"l*acFalaaeialer.neft.rleoiteiflSs,moezhe'za""u"piiretiir.cif,ocma"e"ra"."esIxicri-ri.f"r.iaIce""i."ini"a,sciss;to*b."drdrau"tldaa"inirc,r"xelei".rics;.diii;"omiciliI.na;u".t'i.s'rr;ru.u;pnpa;rrei;";intt"c;dtlrire","t:oa.rt,poT.Ll;sdIrdnsfi;n.;".ercie;iu;;ia";l;ii"t;i"-;;npaet[;lr"itiil;;c;;acni5p,;.^tti;i;iudd';ullu;;]ii*;r.-.n;.;;;-si,ef.i;"tfadri.n.rulu;;rfji;;;;;uim];inl,lia..ll:'Jm;',:rdTi;ct"ii;r;d.?aJerln::;e:le;p-;-r::,
s' -i *unBcilrcr.L,"pLiri.tiorr,ciisIcnai'ivv..ciachlai., Iii'clci ae .st.c Incrur.i rrr-lur i'tc-
tg,T?u,,t,,;oo,"'r;;,,rii.;,i.ar",,ijit"i.:;]i'"..-i,,ci,i-lr,la;.ra,,i;rl1ri;icciiiiJr'il;J'o,tc'),,*fri,t|r'ltii.lr:ti;n"trt"ir[cti,;gr"oia;.5t'i,incrlpitrt1,.l'u'ii,cn.rtl,Y-o"'crt.iiat"liltcilialii,lic,Ll'c,l'.tt.iairititr.tctt't,tet'cli{l,-cetrcLi/c.,,nlrlIiscatr,tlItl".rltcrt,\rivtDc'.:(ulati'c<'iiit'Cr:ira:lrii;ritttlrii,rt'r"tfir,estilllrc*ci)l:t,ccz:lc,iar\?\c.lt-rcintr.llhctst:trc'l,rc,rolttclltt'uota:tudiltI-,iirl,tn,tttt1-]urcll)ttialri1t-llatsiit:rli,ltl.r.'' 1erpa>nvrvLiedn-cuntrzttidu,miviAan.ifli-zd_elsoe.iair.rt.irndtnIeitarerc.a-r'gant,ftdanAisotaurdsn.euirainsirgiitetinot'ttturttnruc?cclii,tct.rgiiau<aunri1ni,aurctm"c;tid,creen;s";tvtui;iece;Aifdadt+,ie;ts"figioc;cpinisiii*ipnt"itpirmo;iirn;ria;nifttieJ;d;n;i;e*i;rsl"ia;cr'#toi"irpc;.;i"ci;lar'_i"nr1ei.,a,A:
i'i;av.,iiu',,t,j[Ei:es',.tuicrifli,t,i;ls)ltr"rtoi,ilclkr'i.i;'oi""'ti"rtiro,,1n"t';,,r,irtsi;nr'a,i,,';r1i-.llti"rpi"ts.crr'l..s"l,ti,,c,;;j'r,filli".'cltnl'riri'e;,[ii;at,,itJlc"lr,ir"ii',cttro,.rilrr,,.ila'ril-.lit,,';n:ttukut;ar,.:,;icit,irrir,rr;tiltrtoit:ntfirttt,ttr:"tdtt,,lrdr.t'.t,,,vt,.cPnu,rr,ii.csrir.r--"tlin""i.rrl:r,ac:alitst,,ic,"iariltciartir,iotr;'.,r-ctci;ra'loto;at.rlsirttc,tghr.a".:brtiilcrcnirlltcniioittie'"rlaacir'iinnl,tti-,,s,lirlili,rdie,ror.}u,utr1.tc,rutiri'l,ciirrinu'lruatr,i''lnt'r,i,,1:.rttLrtlicl,,itl;[vcr.l,i'lirti",lL,ll'l:l;'r.,iailialvitriJrilnt,r'inlri;nlntt.letllttiilroilt1tticlttl'tursgctttutrhsrum:t.,l,talriritto5"lil;innslllil'tr;rloirl.ii;ltltr'tpi';csst1lliul(lnb'air'cltii'tta.illt1lcttipitrlra'clticl1it(lrcLv(i1ieIifccpiccll'ulildotrtci|'it,aulrilrtrnctlrei'Cacc:rirniotL)5tr',tlil;.gfi("tt[tcastllL1osii''',t'racnatr-ct"tis:irsrlcatr,ii:l;.tidilbrcltl,."(mtctttrttsationatcut.itrl''trticli'rlr'c'tsr0trnt\tc"'ta'tliits;rltt1oclu1c.t\otltirjtpxtt'ir'',tllr-;ctSctonrri{'rtsbr'icror'lt'tlaintccilclioil:Ilarsr*tcitcinrltresallncci;tutltl1tlJriltl'-piscett1rtrlrcla:1r/tlirl:.ir!itainosnt.c'tcslr'ual1tidactlu'tlr'ii'ltrl1snc'litiriatiti.1titnactsr(sttntl]-stiLo(1ud'](ptdcolAtiN'pc[tttIltl1lroecrtlloc"lruoeitri|liccv:mcai!il.csloe(Cutrilhiji:(a-.rnrlcr'lmll'lirilcaP-ecpPlr1o-ttil:uat,cr'r1etngt.ir1:rilala(l'tl1start1'iir1lnit-artia*-is.''I'l"-'-'- strpeagtiia),l.ea,diacdi iacidiasccreotnugluini u;iiugiirei !ometli;;,;;t,";;i_;'i;;i"iit;"r;-"?ri;i_;;'
Te intreb
2J{i da-ci. nu asscuiudcmuitrAeas, caa$sra1fuuiirisaf,seuun"Ocua-i""p'tirrio"nUstcuoul;,v"ai,n;m:tr;rsF;priui;ns;diicc;ud,
cumva
mi s-a intimpiat
sa., ruatetnatici? Cu alte cuvinte, existi un o;g";i,;;'";;;;,
cv"ixig]l.usirt,aos,ucinanrcecho-egngigaglotit,snecegaratreat cddteeifremizraaittiepmrminuotltrde_uudrniisdocbeipieleincxtelpg,iiiamov"i_n;"d;;";f;ie;_;

srnuipa2lcArSgiorfiecrmhrdiciutianazsice;carcoa,itae,tdalcaezrdnml"uueairnnipciutietilacf,ioegnlrooeresapiv:taugtinreluomlmooebceciltierduicaetq,eauasaartclrietoimrniivoneimutvimciieta.i.m,asi'ttr'olu'oi_

257

l7 - A doua carte cu probleme celebre

care mctocla sinati:l)rrro.Pllclitpl trlien ccrcetiLrc c;iiii de exprirnltrc,.ilsir rir'trrr--r.udaullilrs'npliti.l,amvdsietnciccstincniniu.rrr:DsunciliLnp-Etnucvrt'iuaesnraiiscettehxuceGmm'raralill.onrA,i,sxnccuian'nrrrptJc1lu,gla3ir'0lorr,,arcicgii',ui,pntlc,,ai,aos,1rthriririic,iii,l'
ltrsate toatc
clifcrite clc' comunicafie?
-- lli-ai pus o iutrcbarc la care nu-!i pot da un r['spuus
;li)_aln'cir-ci anum;ctaiu, dacl existti ! Dar, mai inainte de,a-!i
prccis trcbuie
sDuuc lin si-!i atr:ag aten{ia c5. nu
rcstcsdtseiuimcuntsiritJo' ilictruarelcl;iaaccnautastitreomitnualtoeircxiccileeoplc]alioocmntuac-l- nt<iit1rlit.il'rorat'airccpilaclrol'iingrrr'c:;o:rbrtlilsri,^iccitrrrctrarrrirsro.rlicrcarciiuirtrrtucqtrlnti'-'iisiriltrorrrrirrptcltur.r,claili.crrr-tb'ct,'rrni.Irct',iccsrzt,llc.iisicripriansrr(ccclt,..r"unrr:tl;otiu.ricl"rtincr'orti,rr,aborplarrn-".a,iiallitr1,rr,.c.--
s[ corrfir nz i t'x tla olcl iria

aIc cu unilatca ci. Azi

cit un nratcmatician sit
niu rrl mtrtclnnticilclr sau si abordczcl, cu Accuasiugurin!5,
elouir sau rnai multc rlomenii diferite. I)e celc n-riri multc cir n-a {ost rcmarcatl'r. --\z.i ;r ([1,\'r]nit eel urui putcrnii: itiilr
eiri, cl sc tntrltuttteltc sir Iit: lrt cttrtrtlt;i sir-;i dt'a contribufia l)r) carc sc
slrrij ini\ tc'oria clcrspr"c rrnitutc'a rrial.t,nratir:ikrrl.
ncluonr.tnaeiriirnt.trL-u-uptr<,ralrlttcitgacl tcrslittcratr-cgtlactl'nclmcl ,itiint tpuairtti(v' ,a Ittt-tttc[i)l;-(Lctuir)i()rst(,ir';sr lti.-;rtni irtrnoll.llrrrrir.;lIillr,lltr,ts1icrr.:,. lt.urttt ".,t.rthtu.il,,r
rturti attuttrit

;i n-ag putea
s5-ti sptrrr cle cc, crii existi-t tt sittgurlt ;tiiutd nt'at(lnalicii carc tpalin.st-riu_'irnrPlttactititl.stc(roll(c'insnplatlrtrol'c(,n,brt1riLi-'s{tr,rizcrtrtua}rltvn,iilulricc'{(c.ir;icrura'rc(n,r).trurrin-girra--tnu,iirlrb,,i,te[ri.rcaPrlt'aLirr.rc,itr-irinoruarr.n;tru-ntlrrninri.c_;itil.dllu;cnirn,n,<[ral ainrrlr,i-l
isi cl..zi'ltu ic, pt' zi cc trccc, tot rnai precis f iin{a. iVIi-o arati
cta; riefaiupt-ui l rcnit i1it2to'ttltalctoics5cb, i un capitol cle matcmatici
buim l s inr pilcld, logicir. Am
clc cle
c1e altul
oablciescltoagicciai ir;'cicnstucafisziinctrl;tsiiarurttm' ccrdpiccrinitan,cn{itianclelc,ir;i.mapi-oaii,plougtiecaa rii-lrur-,nliccrPirlnci isri-rcap'irr''ti,r.',rrci,i'ltr"lrrr1tr,ai<ul,'r-ll.i)rrc'.iz'u.oillsqr.r'lat'riccrrL(1'r:clrs,lcl.rr1.lal1!:rti\ti[t,i.i-i,ar,lnrc.ir'g,'icn,.Lrat,l1ur,'l,i,^,5,tic-l,r.r"ci'n;'i.
este clcstul tlt' allrtlaPc tlc tlrirtcuratici, dar logica nu-i ma-
tenraticiL ; i rric i'i rr r'&s, de;i cxistl un cap itcil clc logiceY mate-
maticri.Iic aittt'l, inri amintcsc clc lrn llrca {rumos articol n5;gsc[i.:krrlLsct-'-4i'uartn.'s,ptu,pi'tuA1trsnrrr)iorsliarrlrcitor,rrrirr.*rLy-t.ritilHa.rrSrPcliil.lr-l'lrr(iirl(tc'trs'i'cl'x'cr;.{tiii1:lirr-L',saIct;lL,r..s.epis,iintzlr.rtrsorrc.cl;iplrttrtrtritna'.risi.lrsrztt-lr;_rarrrL:,lscLtltlr,ti'bcLaic'Llsotirli,l..tnLr;lcrLlgcrnirrsnb[rlrti.turLi.ircsilpn'cliicitgiipa,rrtuc,r,ztszln'rtctctrr<[(iilocLiikjl1colririrricrtror1tj':,.tulrcirps.rac-rilrinrtrto.,riiiglncrn'r'lr{.sii:,ipicu1,,n.ur1,.,ocr,,r'"r,,crl'r,ti:git,r,ci,itI,r,t,rcrl,,l-'rc2,ogir.,.critrr,ii,,lilii,:,t,c;"-.It,-i
al lui N. Tlourbalii itccutttilistltui lp:l-o',bAlrehnitrcirc. t$uiracIrlalactcocmntauteicaizloirc"I[
in care sc tlisctrti
:rcolo nnclc ttn obscn'ator stt/tcrJic'ial corlsiclcrir ci sint cloud
sall miri multe teorii clistinctc, intrc ctlrctlll inatcmatician
7gtea[n,oiyanl taai;tailuL.tlt{s' fd)rrsntea. b?irltelausmciinpi uirnle!ii"lttcclconltettrg?icrtausrcl u,nusen i.nobsf6err--
mcle cxtcrioarc, complct cliferite, ale acclor teorii.
-- Da, iatil cir au ttccut 330 dc ani dc cincl l)cscartcs gt(.Tttlr-'(:-tlr;'LrLrigidrartt'rtlrLrirclr'arrligr'rrirrrr)l-sa.ln\rtr-tL;itir'l'cl,rg(ilcrrcr-.all:sttrrct:clrclr,uucictl:l"s2ncrtr,ic:tsril,iiiraiifcesncia'acertrusoi"acattlrotticsrn:rt,sr:aticb,rctllaoci&,rzlrrl'tctiat,nal'tozizrtrir.ir:ifc?lArciltic:lccociuilrscozlsisturttiiii,rliciiln,lieilricct.crrair,n,,altiieztl,ietc"irrt..i
a pus ln iumini iclcilc colllune ale algebrei;i alc gcometlici
si'a creat in tclul accsta un nou instrnmc'llt rle ccrcctarr:
nisrlr[ua--slt-.c,amDDttataaittciitcdtflrii'a;n:ttic{l'clJccoicittlstutApltlumliclaisrilrpioicnprtctal{alii]illatitc'ictumlcid!vaciieir,,urlflaiicssitetonri:auiclt',iatir!.pfci
trebuit :ljutir.toarc, {orinatir clin riiclicitrilcr ccuatici clnte, lL fost
'{pollo- rrumitri clt'citrc Lagrange ecrtati.e ro::ol,vutitri. in cirzul unc!

t F. lc l.,iorrrlltis: l,cs gr':ttttls ('()llrill'lli (lc lil llcrts[e ttratlri- t,r:nafii clc graclul 5, Lagrange girscltc ci gra<lul rrcuatiei
nr;rtiqrrc. l':rris, 19J8. c'cltz'coiltvaa1ntcccu.sattcici6c,lautcc.licii nu cstc nlii rnit.l ci nuri ,rrur"

ti{r

pro-ceDdereupii tns-air-i!riinstepleusnmimarei pare prea. complicat nt:-:! #,";;.."t ridicini, dupi cite imi amintesc, slnt date de for-
luclu. N-ai vrea sd ne oprrm
rsr ra:*i ar[!i'mai clar, pe un exemplu, despre ce-i
"voi.r"ib-a?Ba da. Cel mai simPlu estr: si pornim de la ecuatia -t- N;/-t-Jeq, '*lf'+- n1-- V _q tv/ii-lnE'

2

de gradul al doilea rac-l.icai-tllii. accugb"si{ci.6sne2p, oLaatgcrascnrgfee sta bi 1e;te cd. fiecare dintre
sub forma
f{"J;x2* px + q : 0unclexr,, : I t
ii I ,{*, c.rx, -}-- r,.r2x"}
Dupi cum ;etii,i,xrin;tirex,coeexifs.itcdicrnelriilalll.ii?lci :q ai acerstei ecuaf
ricilcinile
;i iffi ;;c!5vuaienni^lCIceilfnliluae,cldl.ncccfluncoqsufejclsielaaasttla9tmcter{izEc.couiri_sndrmzlaaeessaao'dr-,tibecdislI'pamtc.irulrlieiavnitlie.c,0irraa.iicdiilsnee.adrst,.rrcctclefuerirm'.piSIr"elei"ac.ieaaricei.tJueiut-bi,si"Xiicli"iteeuirs,dapd;oliioentfrtiaud.cd',fifsaticicL*tieainig,is,rianacngadguiescr,,ii

xr*xz:-P

XrXz:Q

Tnlocuind sub raclical, coeficienlii p ;i q prin rlcllcinile

ecuatiei, a"vem:

t/r"F: v -l].-"r'+, lV/ i'-' -;.-u-' - t:{xr*<oxe+torxa)g
4
: b;! nu ia decit dou'd aaro,i distincte, cind se fac toate permutd-
si asa mai d.e-
rile.*ceAldoric3d riddcini x,, x" si x.. r, :;o,, cu x,
se inlocrri6;te'x, .,i
parte?
Asaclar, raclicalul s-arcelxaplireimcaatreprpinretrz-ti.nrtirjc. lua.nticelelinpira'prirriienttirfei
"."""fl.Jiu""1a.-.i;Nc1Ia5i cinintii,i ci elte inclifercnt carc dintre rdclicini iioi((n31att,,-'e2e1'c,i,3oaI)2r)raepu)",ol;Cdtiqi.ifrnr(a:uo3dcsp,esee2ebi,,pidteiel:';i r.iomr6*grurclu'te-iprerdiutl"erd,ie,, 1a3at,c"neSups, m1tZe1)eaa;raefsZe.s,.aiipu.,oiid"teJsS-c""hie";ilm*;e?.mb*:aer"n"i;ltine;
i! ;;";fi;;:f";;;f!cer,i^uana-ai..,-rr,Crgterecc?c,:uszl.i/ntdaol1oenpenl&ivrarfsipcaeiiutut:i;rlr,silofe1eucuiL.calug"taain.esg,n,puunt,rtyrnm1cattcdtlenenezsrg{m,ordca,eeetcllc"vr,ocentas,yorszerrertroaae7ei'pnaltatvzi1uio1num.o,dctu.2letc;fe"fd.iaiut5;.g"etracae..rruraeaeati,nLi.dei1tc"dp:uAiu,,.al"cca2;li"ainuu"ei.tl3mb-t.gr'nrd;ei._cr,4z"ooaeaor:didll-_aaesr1uaical6cntldg.ai{pr,rdpd'reer.sa&r.idro-iomeu'bE"icnluai"c"fitbu"midrid"ilr.iieirirc"lclrsatiiii
uri;;*ein";;l.a'";r;l".o;_o;o";cac1infl;u"de"A1grpi"tei"s"opdi"nis.t"tid"adc"ed,eav'rlealupamaoai'cgaerflliilixilaieocb1erpacaersdcrllirr:ietuluomecel(lg?.",af-rtroioat,ai,r;anric;ai'esicr-lp;uate.aepl*irairruiin"mc)nn,n"'ilptatnae*ibllridaiist;tltIIad;ocreSCcldiutccta[euaputazc"aac;eodlililnacnlti')ecustspaa.ieiarccorcdplleueiacidciaertaogaai'diacrst'aciataecriefudcvgl-a,amtuear'Ja;ial'rpotese;rdarciaetzeltuI.rruoesbeleacilcavu'li.udo.\ani.pceaaicrl:trec'1ctetisicaiadea-.'ii
261
cubici

x3*px+q:o

250

exa,ct lh)()litsi luLrrtl -\c illtillll)lit;i cu ccuttfia tlcr graclul lll {)r'irr lLt't,lrslir oitbcs,trr.lrcttztrit.,u_l.tiLLtilfrl'c.1r,.,lri.n*{*(c.lilr_lt!.t;in; s1r"r.rtr.ir4.iznu"lrt.ctJo'ridc;i
p' at-r':ulDcrttr.. La.grangc aratl ci, in accst caz, funcfia rezol-
varrti este u": x, x, -{- xB x4 cale nu ia dccit lrei ualor'i g-r'up,ri lor f i.

( i;r ltr i s.

tlistitt'cte, atunci cincl se fac toate perrnutiirilc posibilc dintrc (Sr.Ctcss(s;'.llr'r-tar't(;irci'('.-rr-t-s:cr''-alrr((rr.niirl''cir.ltsri(lr.i'blAjr(N,lsur1-sr.*.t)irr.tcur[-,il1r'-lr;ri,s(,g,ragirc".'ir'rrsti-lfii(Pit,'atriris,ll'niirisr(:'isvsrcLrrr!'iio;tr.irtr;irirrl..ricli-iirl)tnlrr{1.r'.irir,rtirrul.t;7tr.t)r.n'.,l''c8.x(,rrsilr)l.i(iL.h9ltia.lsi(i.)ra.ardr'iiel(r:.r(ls;rc'trri',i'rJt;r..,c.ittI.irttrrr4Jto;rtar'rrrtrniir:l.rnrrrr---rt"tttrrcras,1.t;L-lzlru(lrur1r.1ii,.<rsrs.ttr;i1iPi'rttl;lriir;,-circri.rttg,rr.r.ri.u'.rilscilci:rr;lrttlrnrr'u.ri.ravtgiLtr,rru_,['ciirrlillfI,t^t'rrtrs,)rrralsrrttrri,{l.',:'u,u.:riiss,r;ilrr'L'rsiil',ilrpilrti:iligti,rrirr-.rstr,cilmisjl.rrilltn.i.itrir1cc'r''n.,,ru,,itr'rgt1irrt'igc.rt,lt;c1iJinrs"r.trit1,c,tr"f.a,,,sc1bl:ruinii''ai?t,.gtlccrf,rr',f,l,.rr;-lrio'itt1.lt.-riiiri,i.ini,'prt-,gir,lcrA.,ibriia-treitcir(,"i:ii;*.,c.,",ri'"sn'il.ii,r.n,i,.,.ltc1-;.i,c,.i'tisi,;.r.tv,,i,.-..,p,ait1'e;o.JiJrr".icct-e";li,;,l..ur,,',lc,mc;ir-u';';stii.J.nc;"iuli"oipc"lrs"iirri.n"i;iiuri"eir'_".r-i',-il
cele 4 ridicini" De aici el clccluct: ci valorilc lui ,,u" r'crif icl.
o ecuafie cle gradul al treilea, care are dr:ept cocficicnfi
iuncf ii rafionale dreticcliroccin{iiJcoicr:n.tici ocncsutait!uieiiedianttree.;r,g,A-ac-eslitl-opzroortcics
de pcrmutare a
a rczolviirii ccualiilor clc gr:adul 3 pi 4", scric l-agr"angc,
si r'si propunc in contiltuare si stuclicze care sint conclifiile
e rratici lcz.oli
in i"i* grarlrrl e attte estc ntai nric dccit
al ecuatiei tlate, tlcci si trxistc' posibilitatea rezolvir:ii
ecuatir,:i' prin radicali. irt ircest scop.pentr.u.o (rcualic clc
gradul n.cl for:meazi o funcf ie t, clerfinitiL prin rclatia:

t =,,- Xr *- cr)xg -r- togx" 'f- ,.. {,)t'--1x,,,

unclc \1 i,id-yi2c,ixn3il.e..Xu,' nsiitniitlirii,diarccliinciilreaceccuuaattiieti id: azte"'-;i o; trrla rrn a fost
d.irrtre
I =- (). .,si ctrp- rlr-'
Aceasta estc functia rczolvanti.. Gradul eutnliei rc':oltante vtt rrurrr.i un suhti'l , rir.iii,-ri ,ri"rro .rc.l.'t p;irinte
f i inclicat de numirul valorilor pe care le poatc' avca fullc-
tia t. lnsi. t'l) g('tt('l'll , iti accltstii lttnctin t sr'potsclrirrtha irr- fanrilit'. in occlL rrPor:ii rlc ,ounrctc .,i,r""g.L;;,1;;:
taa;t'SrrCiia;ct'iicir-iaott''t1llsru.eal':ceri,,rfr)r1'.,stln.a..nll'rtalcr\ti)ii'crsc..(.tru.l'i1rr;ulr'ut..irrtd',l,r.tnlpu,'eplllsssrir'clras-:n)'-n-ilctl'sn(r;r.litrrerlrr.r-rrclPfrl.sloir;tsl'clur.ilr,iiIaiv'b:r'rs,.{ilrfnlrI.i'rio.ti(aiiz;s'rrlsrr(i;rIrr:rlgs-Lr'i)':r)lri,t;rrcc'(u.nicrl.c.isr,rsLau'cii1iruzIortlel;aurl.(trrt.rzrr..ae=nirp-titdsrci'ro,i'lPls.onrtcic(rftcuuit'.itilrrcfAing,irnio-Inilist.irl,-ier,t'lir'',c,,<ptroii,iscth.aur.rr,(rlr;r.ri.iracinnr.ii,issriuctr,ti.pui,iarirrrlclcL:;l,)cri'r-(i\,oi(ict..tll,r,r,riiir,l;elrl1'i"rstL,g-t:ti(4lllitrrr."t'..t0iruc;,zAri,rgrjrlir0t;ir:1c,.s;icuirst.i.,rrir'(il.t]ic,rt,,t,rri.,rl-'tt-.rprn.irri;rtiljsrrrr{-lc,cirgrl,tc,t,c.u,ur,.*aiirs],rb,rc,nar.rc.rp,,at*t,,gcfu.tir,.',ecclet"nll,.r,ale.t,irnrt,nst.;,itul,ra.ip\:'itirtsr,ir'i.r.t,cim,r;sait.:iritiis;Ltcr,i,.".rcipr,tdisruii"orsim.,;jiutil.ir,s,grir;ia.stii;r.cca-sinLda;r;;ticslcit;tin"brt;leei,,er,p1vi;crrin,l'sn;goib.rgi,aii,;rt;cc,nct,.ni;att,p*."tltri,,,;tic;aa..ot"t,lrr;r.-nairt'l:eicctud,i;i_uu-a]'r^;i-.ldil-
ire olaltd taolatitcvlai.lcolaicrein.ilIcicure' r,tiXa2r..c.zxo,,lt;,ia, nlatirf iccalc pclnttttalt,
t va lua o \ra avea. in gcltc-

ral,gradul n!, a;adal cu nult.ntai nrare clecit acr:la al ccua!iei
propuse" Problema se rcclucc la a gisi cazurile Palticularc,
ietiinccudia,ltcipaZroicnhletargn-ultaenclpiarolrta3iftesicafiuui 4cs.oa,u,bAopsiuriotp,craielsdaaecuccuastlnunui mslu-iatc,rgiunratmidruni-lpcl-agecrteuuala-,
clac5.'nu imposibil, si jurlec aprt_or_l" scric Lagrange. Cici,
pcntru ca sd sc poatii Coboli gradul ccuaf ici tczolt'attte,.at-
'siri"llrudmi cianiIui'nnculniarialcnzl,oi lltaatar ntit'tsnirit/c''tisplercr;rclruactciraio;a'i l-ez'ttllloca-lricci,n.ailcolirc,it
,, ,1Dvc,acI.rc1+"rlcntedtc,-t,slitpicosut?guffup,olsncsctecrscaicarvertiacodtplutsrccLieocabargla:arizreaclucancnzzgoiuf.cnel,dttl'acduetzlceacoelcclcvaiinalnlcartuucr-alcmaaaccrliceocuirloatotcfmliti'pilbdl,ooinualriicpr;larcilillvtoacie.rrurpt'aampttalcirzlisen!ae"
rii-:*^p]lii.rcj]c,'];]r,1c:lrc.i.lit,orrloaatirt,'cP,rci cirrrl ,.r.itr irr birt rr inrprilur,,i .u
202 la obserr.atia riclicat6
de ;tiintli,

16:j

\rrqu-t itt (-.Llrr lrv (csrgtvatt:llc|,rea(toela',ui'^t:aniram!tllC'-r,to,ilfticjitiglsrarzr1-(lritilla'-rrgrrrc1r("r,pb.a.rafd1,lraeo,utLiic'ur:rzinrrr"rdcaaitlsll'Tulina:cO.ftrJtcitrrcsiercl8cnr.ls,ronait_-Rton.lrrtet5errcisgi.eiionmiilirpribpernrrret,rgiiie-rrrlnlt,,rdtrcrtrcrueis.otTctn)1ctpu;cam),.taucgon.illcrrruhrttisi.,i.s1alentiiorttruarrituslrbo'lorud-i.iro,a.s;{iiifiiau?liauin,falprui;,ingiid;,"ei'''rtr;,cc,c'vt:rrc,tv,i.,r',iiaLlcc.ccjr-cttrr,a-r.,,.,1neiir.rimut..cc.c.ci,J,rp;iprs,ir1nIer,rir,"it.icd,,-nc"i,ip,.r.r,i,asom-pc,'ne,ce;ir,,,lunost,.t,;1np.t,lrimzo,iirrieuc;ou,.,d.,,;sririna;.rn,.r:;b,arn,t"l.ir*uIu.sn;aeu,t;til,tit'ud;.?eii,Ei1,irmrD.Lrrrl;rd;it.iraaii-;;i-e--rij-r

cttorcrrfli;aaeucltrcrttiLti"u5.,i,'aio,rtppdSiitr"idlaaan.e;.lccgie"fcmfa.eni"nltianusdvuatplceaclcuemtlcgcufcsdialt:iLrcav,tvt,tiiui,ecl1eit-1ciinaaevleignavicumrcisudicncnaalabeuctmlftiellrretaimaasltrticipiiaeiu"-teiti.-itinrtcmitacilodllcllcruarlcr-tsii'ecstatnrotn.tl\lnaitticti-aIisgi:lcriutiroiilo.-cr,rltilr:firso'iat\r:tltrliiitdntrlii-l]r(ptrtlisc<uciipnlllul;:ir.uLiietsrnr'ir.a-a:rl1naaat1frttduca1ldi7ect:evacipoarlltaiinirttgnu;eeausinit-,--1,l c-()n,ttnCu"a_.pscpaussLaai grange cind a citit act:.lsti luc_.rrrre carc o
s[ scric'corect in iiinba lui". I-a iiceu obtiile [oa't'L- prcmiile a1s.*'t,nL_u.a,pr1,nt-tr'an'ii,t''s-'.euu[o:l.,zanNt:cnit.crualounlcI:ui.l]iead:1ar,-,lou,r-1g'crirn2.llii,erars6rs"r,tnrifljlc-iripiillayrrlcmuut,riPnctlc,aulrinrelt.ritnoeib.vicsirnl.crciipcecaraiuttla:ci'irrnnr,ctoeln.trcer\'|aann.c;airlIisrsnimiaDirclia:ui'oranclrul(ciislinf{riiti.rtrirlrinruarrrc,rcr,tr.:triqrltnc:r_ft.rri;,,l,ja,t,,tt,,spirrr'iiiic.iiar,,i]iir,ltrrr:]r;,rrii,irr.rs,rii-,i1.t'i,,sr,,r,i,-_,c,-c.,,"t{lrr:a_',.i,tc",";ne,i;,;#,,r;;;a"i.;;i;r;f.s";;,r.tI]ti
plaosiSbcioleal;'ia1Pa 1o6liatnehi sneripcricgiidtet,;,toer-os.Itt<eIeaaltrcalmoielcnaulpcc1c adm itere
ir!;rllra{ :r l-a iisat si rl()sJ)(.ilsc:r.lit,lliziril,.t,,i,t.u,.t.',r,!i,ri
listd' I-a
;;"lU ;"-;;, cuioit, ia ltj ini, cu lucrdri maternatice :]fll:.:nGtrlrrlrv,ail:l.cu.rii.is[a.ana;]t.m:i*Cl1-st]lurola"la:oac)li,ndsl,rlue1rc(P.ei,>ld.c\re-(eslrtrirh'euttr$rteiittcry.-nln;unrtBl.,a,G.iiier1tqqnitmtl2ns'rfug:aitsci1fer9.^all.ltceqc*1elrrirn.nrallcrbcin+isru,sLltcl.idrrt:saI.".pcuiitdiac.:rtnpcrlilai;crtrrrprr,Ge.ieln?,irio,liiresDlr-rieaiacs,csaIfrnJacillrrdeiilorm.ii.r'rrjsnr>ss-ririCa.ps,li,,tirsuriilisita,ei;;tia..et.eiiaPu.,iun1rtriErrrlc;rupcitc,aliri,r,rrirhn;air,,c:u,ai1l.r.'y't,,ztGsatHa,'-<iilrriir.r.gntrr;epar,;,jir,'.cbsicf:.rl.rni?eo,riae'rcti,rrrrrcrir,pr:n-z,rs5r-ci.itorr.iinriG1et.trra,iii,,nrs.t,ar,adl1tr,2eis,airrtlrr,51coiitii,icir,t;;,'n.1nnrirrrsiis-.irrjn.erjrJi,,u.rr'ai;i,i1iril,,1pir.t.Ci1i,nrr.r,.ririru.u,,Sr,;,,;r.'jf.i;.m,.r,,rii,trir;,:d1-",i",,.il.;,scr..r,;;ri.,',,,ll;hm'Jo,;'.,,c;,{iiiiet.t;uai";,l.lai;';;lJf"rt.;i;[iiraJ't;i_;ilI)"trr;:ar"ri.e_i",si""
oaiSnrigii."iirIm.aclalcc-f,$iicvtotaaeflsaiterceltesr:piinmingisisneasri-diioncudiaveipoiil:irn;rieiiclalsCecvaiiuntctll'hliynsLge'sirnvtetccr"ll4llal-m]2:fl
iu 26t

c1c

or-

264

[F: I a.i Acatlc'rici, rccunoscut astfcl printrc ce i ruaimarinrate.;
r(nillla,rtiicsiernliil ai tirn;rului. I)al n-ain tcrnrinat
[il ltl')9: irr irrlic sc sirrucielc tatil cu rlurerile lui.
siu, din
h;:il urlor l)r'rs.culii;i per ling.i clurerca acestei piercleri d"e*rr,rre.*.
l[:]t,ilii ,lirrrrrlr;tlt:irt-ttpr"'lccilccrnri,til-lni.rer:;rnsrucitr:rallr'rir;lcrLirrrr,t'r'lrjcgc'iiilll)l''trlc'ielsl'ararl.tiuucrna,inrtccalnialrL(t:lnls(c'erl)rcarsrrcncirtno:aisgrIu:aur.ill-jrhip\ccccarrclnd-arntac'rmraaicsnicnir,atpIrrsaoraclbrarAii,bc,.iiialne*,
l3rrr rii;r sii-l c{-rrc('tazc. r.lroarc inrLintc rlc lr-l fi rcstitrrii !l'i-ani
lrqsiLii:lr-irrrs,ru:.nl:mosrrrilr:ort{irtaurlrriilririnictcrtlrrirr;llitcl'tctir:cci6nr.{;s.rnr!lil(rr1'lulltcrilrrrcip:laiuialr"'.r.sgr,2l.iic,ln1asvii!c<azlatzticirccrlurrluatnlrri;irrilil)ctror,liorr.rli.ttncsIai:rlrriti(,\'tl'.itiolrirlrnrtc:s;hrGirll.tra,rrcln:oa.ri'rtsii.aTcSi.nriili
ii.l
s-ii-r:..g\c.isuinrltac i,rr:irjrrnrr:.clit-:ri.nvtoirr.l;rrat.nsuiur-ui1l_'tirc:t,innr ll]trl2.ogc:c(sialuuci iCtya,uoc.h,iy*j
':irl*.::{'j,ir:,1,i , jlr inr-';,i d.strri rlt'hi'ii.iri, clr sri s. lust,irrrluiosat r.lc iincmttra

\ Ilitinrr pirtilrii su:isi r:lc
ilta,'intc r.lr: ducl
r.li,'::i.. (l;r.lojs

:i,]i:3'
:!:

" l.lr.,r'r.!, t..,i,'i;

i,lut,.;c,i_,r;,a;1,tr1'-CPf.e"uc'a"Srlit.i;"r]i,;ui,:;'iic-.iiit,,i,j,G!.,t.it.-aiNi,;lioi,ric"oilisaiiii,l'r'*ilsiil1stil'ca,st'iatslintLn(ut(-''lrritcrlr)ficalipclr[ct(tulr'.ti(;i.'t'tticrtusl'aruintrti'iiisrlrlii1rliii'irtllr.t"cir'rui'troft.tlbrttiitrl."'tlt''rrtl_ri'rtrt;;li'tiLii;rlti(lttIiaIriis.'rtg(at'tllr''4irtlr{i.'rr:,)1tr:itl(rr.i[ir.r.lliril.crriirrl-tt'l

Sirrir'pinlcCs'ri.oatjtacrzec,ilt.1'-(rr','rcuacrlrt\''i-';ratrlallitt-tolin[-ttlgtr'-ril<rnltg''t]'l(l']L",lrrtt1i '-1s1'tl{t)1trt-1rr5rJf.rprlrl'rtrt'-

,,ll;;'iln";iio;i,.t""-tir1-"r npl''";:;"*;";"i.",if'"'.-,ui;rr,r.c:-s.ix'rui.ii-isiilr.ti(a"'cfn-tflrltt-,n'o.l^,ai1ci,,aitig"-r,frro:,nq.sia^tr:tlictaoa"foccrPeftrteaotsastlliietrccsCg[h':iuerrutircrilacch'sa5r,,1ci'ttsutranprrli*ltlrtrt{t's.".l_t{l:.rilrrfuur;iltiriullrotlttritrtt'c'(rrfttt'i'utaionrt.loi'hcfo.atrt"ilri.u:,

266

l:: : i: lcaaC{aaiarsiranuuludercndrptlccac,arhcdamtcy.ilo,suuisciucen-hornAeaoipuiibrriaceacrsapnzur*uidoinoemr-tatalcuiArtniece.fcCria.uurcadall$maue2sis,cm7odehaa,idre5sd..;aij,i-e,.la,c,,alcSAfncuahi*i;nl.n,t'tiitz=Sic*nn*fltt,'tiiiri,bic*c,s,n,t:erdri,iantuuossntrcpns-"rou'ici.cn,aeum;,An'iilai;ldbu;i;r";.iccJet;li.-r'nl";u;id#.t;"ce";;r.;s;a;a.ip.ir";:re;*e
gt :'.: : cccoIE:.a:ixc'!rradinaetce.u,r,lr,-cm,lrni,l1lmvana1eadinn"nc,uteiuia.lar'.szr.iccn"nditnriiot.acia,rsm.rirauicolrl-,a',vvllgriacAlr*si_C-b,ttlcrae.cc,,'clacpGhcri5rsa.i,pdr"llon*_cuiii.sAa.I,iirb..ri,iije;mtilplicr.\pi_n-i.rt*"aira.np-ic:r;tu.tu;r.tesuIer*t;"lnoe'',ra""omulr,f;ri'zuei'..i"r;iqniirttiio.i"c..ririro-riiiti_-,-!

};:," l, dpridn:cueTs-ntnlef-riearnlnElfsjtac,.uiatilLalnvnadlreeut:r'c,vsilfuda'ireprirl_caiiin[.dtcarrec,p-criesfe-sasscoifrpoica5stroUueicnraddlroiceicn.iiudtairrei-cnnrnn.ie"t;af(,rr,,llt,acciote"ucivrr..ia""ol""a)d]is,be;,cr;'ct,:ii;;liii;n,n;;t;iri';t.u.oi.;eitgl-ui;";";i.ri;i;,iil";;s";;_;;a_r

.l!ir" ' ,,E i doar au steic cu noroc"

ltaL :t:i:::,:t'l pjsdl1uno.ii^rcln-ta-ipeitrs.l1raeeiccDIcgcia)caeeraernrqslreittna?iacqdaltletneeinfpexlcsciilr,ciilrul?nSr.iartiilIearcifprdarcniirccuiig",g(er1'\anaoai,ner,unecpcncaeihrauapi5rrtle,adtri,ercegrfc-leoJscr:co;silltm,i-ahtn"l-r-:itsjCa;daat$c.-ra.nu.frstr'ott".iou',ii't.tsaraaedrttjeIaau>fsflseiicOfirr_reesgdiuRf;eearrlllm-;uist;;oei;i;?las;iia;;crE;;e.itrcl;lsea'ri-il
Nl:t:a!r:::.:a:
i;;i; ;H;lCDceec:a:axaaxirncUnri-:eso.eu1ntlEliaeiniNAcvllnv:rsec^luateb"hri',f:seefi!uaniillratGrtaric1m,c,1htpea_eicJ1estcaeaauItaeurrcn.azerddanili-sliriaiurnetnnlieoaaii.plc,Tseia,atrircauoctcf-liaopcrarva"is[coln-ittlcrauzal.aicn.i"nrltEoimucBuIinldn"naatrss;pcat;isine'Jdrrcm:iaseir;c,;i-*ursCiai;ao.iFtn,i;ia;-cuslb.;rrirn,e;rruiah"a,ao.f,r.tpca._rn;"ecl.;^evoee;aaceal#,rutaaae;pct"'atmer;qo1f;dul"fpar;uaSenjiut"c:iip"nArrce'eriedibege.rueeiesc"lnatlle.ee.i
ii,ii.liiiti,
265
iii:: tr:jr:: ii:
tii t:::i:i!!i

Abel

"lsuliti.loGi"aair,loieios"r.srraliiu'tsd4c0dtad,ameii.irnnctiiiL,ricopobenlr;isgtoieannlaitiledso!eEcsilaueiepr.aerSaiodturinntacutiepianurflriuot'itanrdcal'aar

io.o,",fl',lccit cur.srrrih' cflri trebuiau preglt.i1e, lcdintelc

d.zoeirlnl"a,icAacnma.audeijimuedieoercclai,mcmadareui ptardelebsvuric'radluspmdeual\srlirsoetrme,,etlaruecabrcauerierelaesnPiu-ei.resf^xioisnlutaalaeu,t
m-nlJil"*"a"r^i-*ui"lfa:t--ii-oBisX"*lif.i*i"tnait;"iArr""i?"id"a.t."ii"cmr;"d".p.i.ue,c.ii'aiirr"isiuiiirtsa,s;si"irbncn;ni"urruaie]i{llcslt""caloa.ts,.dcuna;aoivfifiiretoienentCcdluluiuuear'aemdtnt"fyacpodfrulcartiesiaculatrtuyadetinreie.pteeecclniu-!aignui{rtliemOlsec'ineincuird,urimiauliacdamlo,toaealu,furarairii_dlcipmeofiuiscdetaAuppib;lclsii[uberalc(clsemlv'lri;ueoolltl.mo,i!rcicieaSmrnucardpittlaearilalftiouavsletefuirimace.l,piuc.lr,aouOr,ailaItdiiaiA3lcrstfu.tdiboicaennsehtnnceityl-,li,il

264

.!isc irprtcil sir invetc iirtl:biL itrulianl. l'iurp clu tloi ani line :Li13?r,oitlr,l1._dItop!,r"Ssictrju"",.:!u-Ug,na,_cua-aczh,lvoayastpit;t_iaiaprs-ao:ait-l,i.ltifipcciocrie,smzccinaiuttrreeccrloa,czlcauuevctpru,autcrtcace.maiBmbfiainfpcciora,,rmits_aaiaiu*istuti,enliaar,vtilratd_?_ee
,,,,..otil" inlirrtlra italiarlir si ar mai {i stat ac:olo claci fos-
tLrl lrrirege, carc sc af la la i?raga, n-ar f i.avut niistlu'pica f:f:eIG4qIllrr,c-nco0c'rlanucra-a,-al(tdlt'_tr'ccitscirs-ieleuil'fl.alraro-cv.pam:";nrc:i1o-n'litr.lnba"rr,,mm_u?)rrCadta'i.ratea4etatlr(riir3srr,'scor'crt!iicnlullair)eaiiazrrtsricccitilJuiarrac;c'it,isritiercio,tttnol,io,.cainn,lpc;i'urtnciiitorltt)lritt,rnatLs,i.t'rprachtalrr,npcriiina"ciiiguci:t.ti,bari.r!diriCl'gJiritsDt.i,:,"Prli,rJ.iii.ta-i1,Pta'ts,ccf.riniti.i;iara.ii.,rnti.n',ii-iiae;.''ao:uri;r:'r;iJicicii;fltlrn;l,e,ilr.iiB;"rno,;iac'irrr7,r.oi:l"c;OiJer:ae,,;'1ir;ia,"ans.;i,,,;r,'.dr.-r,ris"i,;;rrri;a,r,..ct,ro,c;;,;;;;r;li;iac;ar;i;.'-ul;s;t';;;L,.s;fi;at;;p;-ir;c.,;:n;r.>;;-;i-rl
iclee si lirclccli'{czc t:clttcarca Iiuluisir', i1 virstii dc 13 a1i,
ilccstui suptls creclincios ! o_,c"-I-.,cPtE,Drc..mai\nrrc;nlcrgtti(,.r.or,Stfcri!'atisgrtrc-r[cri,dnlst,,ai,gi,tccifs"apgftatr.ticfr>;r,ur-gr;,i'r;!rt;it*c;/iitii?{loinrarorrttt.a,lf,t.iirrApira)ilcrs.arrrrL.ut..'rifrtarclc.licint'isiit:ji
io:r?,ccLr:'a:(':atlas]r.,'n]o,t,uj,il.tItultf.nbr,d',9(l,ucu-a,l tcccillrtir'nn-lrcars-cntrc_i,t,.,acsootinar rropcrrc:.craslinruiel )tclicurlrnornariilrltof L1ultlcn:m,.e" n"ut i,drli_u
* Asta-i colac pcstc ptrpd.zii I
* Avea omul phc;rtc ilc-isllit;it, a.slt c'i-l trcbttilt sri ;i le
I,liirlsriarcipc'ccill5icarcrssna;trilsu-iacrniIri-ci,rahiInnrclrti,)i\li-ri1rlirriastrtsjr6>riarroit.tal.aliiitrtlo-ii1htrccrrion,]li,alaco;g,issiritni:iri,-tl.surcccloiianlniucctr1llitcu.'r;gillltnia;iiiillir:\iii:lcl.Lriir:tAlp<rcltllrrc:r:irc1tcflatiihtitrcaclnacori.ialnlir[rit\ccri.nlaailcug-liiiaulttut,,i

chermat 1n Iir.antii ar:i.tlntl ciL locrrl lui crist.iL, fiirrrlcir e I rru-ii
clirclusc clcmisilt, si nici ntlet'il obligirt,irt clrlitatc dc acade-
rnician, sii dcpunti jrtr:irnllttt noulrti nrgc !
Ioc-*ullSJa-llriLc, irltriro.t,ro\-tcrisstltit<'aricrrcc.ciaiicLnrirstlrlr'trcil-rfnlirtritnLn?guiri
s;ii,utrr.or 'citiiLlttiini , dlrr".ntr-;i are

ctri l9 lLrli cit

ilal)olrrmrtiianei nctrilc-iii5ait(,l)c0;i-lctal-ciasntc(lo)ctstnt'rarontircriiii,li,'ilfll|itiizrrrict,,n1ii,,'Pctt"lt1cr"',-ys,ilitnr(r.iIcirni{il,ir,lairts'lrtirntotrttnlorttt[n,('rilr.r.tl:ati"ei
Prirrtre lucrirr:ilc lui Cauch-r' rlin lLr:c:tsti pcr-io:Ltlit, sint citcrril
l- I-egea cstc asrt-i.ttI iJil;lon.,r .i,',r i'tc.r,.i' 3 s;ru
cilre tratertzii din tlou tt'oria sr"rllstitLrtiilt-'r, slLtt, cun) se
.Isrtisrrt<rr'ncrsttttt,ttiitz;it,r,tciorlrt.ii:lr,"glttt'tirll[l,l,tr'i!llo;trt'tIgirtr'i;ti,.,;rsl'irrttttlt.rti)zrircr{r'll(,('tiit-litrl-(t}tl'lit' rnai rnultc elcnrcntc clin grup, no.,rii,

Iil clgrtrgnstr(tirz:r;i tr.)orrrnli Iriridltirrgntu.li"r, crttrll,tlLtii de (a.4b) Ac:a,\(b,\c),
cGicr-L-zvloCi1Nsrl)taufdiruCrai.lr'tm,u"a,cnh1cI51ii1.C'cgsll1cac'.snu:itc'tcrlh'o!o.trl-'rrr!cs'liLr-t-liicitit.nutlpirn.tortlrsr{cacnicttsrlcrttttitlLsn()crtils}rcilrtlitst'r(tt''t'lIil;lrutlc)iL,(rita(t'll'tttrrLilsu?(l)' ofutairniitnclcellee.dsaiepifxalaeii.rsmactnleditancutzrnua.iacetllrlni-cn'.titltinki.curoutptnletotirrm:etaautetf:imca,uirc, 'folcci"tacei,rtiucrra-ais,t,c'i,l:.'n,l;rs,acnirnn.'luufm"Iicdist".te-oip;$e;r;at;uttii_e
(irtlpis si tlct ilr puttrt:rirrngc,'l sittgtrr llr ltt't':tslii tt'rrt't:lllii'
r\rr-i li's, rrtt luii'ttut:lr' rtici s1,il'ittrl tlt' t5'i3i nttti'c.'iziitclrltct'i,lc-'.tI'llu
t'stc ltc'ellt clirc il intlmlucc ltl-analizit -- it.dc:eAit.:it.

crcd ciL cl ..itat.il intituir"a c' Gal.is'cliar clc at.unci, g.j 4.. Fir'Carc ('l(tnteJlt adrtrifu uD t,ltttLt,tt I inur,t.s;r-r, nrtm it
utclrctarl*r.oicrariIlcc)lnsacei,icltnifitoncaphc.urir'ptjtciiu!nipPterc:crlu.narpbtrafiuobasiccltclc,lcnaiAsnabtica;cirl'ieptrncisruriirbrsaudiictctzrIisli:iririnanunftiuittaikevr,srctlnoLcnvrlaicttlfni iicl,eer' astfttl ca
sitnelric,

afia-l:il-lAa:c.

271

270

-lr-r-t-'e-le-Tg6rie.ipb."ruiinevreseleii-mmteuni,ledeIastiemcaui multe lSmuriri' Mai intii ce grupul sc nume;te com,r,ttaliu. I)ar operatia, pe care am
totul indiferenti natura lor: nfnaousllc*eronlmsrue.dItn);-ntoaeics.ltedeosgmrcec,mndv.Inneourcblaoacimnedmspetouciznailt!mziier,u,ii1la.tanirrlcgploaeuctbepurarilcoefsp.i,eriimu.aa-drnzniuiswclduil.li,iipcgiaaientusncnateuucrliaccsel lavesuae-
'oT;nstiiotda,;tltniit-'uscI;iiet.oiiiniccrtoilteeeitn,fineeiccoimrlirrefIicieciedcenl,eaat,erinnlaptooootrrautitcirfalIdeaui;'a,,rSodi,;ldr€,id'tre'ncferabortercaiInnoeiceiiruenitlcpne,itica-atrdeitl'cctloittevntnraaitouptIrrtroiietaIccItcbeicdrtnszel,;eaditted,l.;lincxpo-infnlosawuettertn'tanlseAlleasrisiet.pu.1cg'unspar0onruroauce-'
mentul neutlu estr: 1, iar inversul luia este chiar ] ?

'pamieuso.u--liuc-"litiaeiBImcraeaiprrai'"ldrao"ttsat.:reriArddeccoce.euulivnmanauamem{leocuemealliirieenmten,leteoevl.,peJDeggiierp,iap-aepfrrisiaienltdeaa5lerol,gppetirndueeetriceiaceaao!azimauf,pilptonoazruaidi'nlmitueen?ecartaerrrceleoai.r',.indsdaisanuit - A;a-i. 1.a: a". I : a ;l.a1'1-:-.a= L

tot-iAunore, de;i nu-i volrba'de adunarea numerelor' s-ar t iv-itaI)teaar? in cazul inmultirii "rt" rl.ra"piinitir ;i comuta-

nu-. Numai clacd elementele grupului sint nunrere, altfej
De pildn, daci elemcntele ar fi vectorii, existd. o inrnul-

j!eir;eteaavccecatostrii.ionreccoarmeuntaut-iivcitaotme,utaamtivsl.dI-)tai rdafiuinudnci te necd._
exen.rplu
lelo;$p"ec"uicala1sieieJmalrnen'olcuaairzegucelancreelnraailsleacms1eolcnoiaperreeeraac!ueielaesdmeuenfonaltroeesiaoe'r;^tIensccafhcaeicasert rcbealsarpcreetcr:tairr..au.ell"demFi iumeriir.xinSa-i.rciea, acilammri5tceluimmbeci,mr cnicdo;it.oeerziaarccsuada-ciouapp3elriaactiasmuddcecreim'dsuivi_i
;;i;i asrmoatppl!ie,lil,eirrcbia.mtg!ilniAuiilcieisx2i-marxos:-poi._'2plbec1.;rir:xPaat-atriii3ilapnetl.bpiscuOraiicnmvcrtee,alsmuti'r-ot,i'i.petlaetabcrrrraestsiczeittvia.sli.tae2ap.otx.r:"b-in"St2inxphge-r"25i2m(n.xxa"A._lAco3orpuip)ml.."i'cr,suuinili_*dl"-i

dupn,i

ii[.,-ii
;ia-se-;il"at---a""cJalu"tAiSmmu)uminai-..,;c.tclto"iat6mUbeicisltlseeoeicnlnmlrarvuimtssaeumitnuldntecaliutnei,idivsln$doneci-uheraasaaiucmcttieeuorbslsuneratt,cige*tci,eaoancaepczuiseendt:mrlecaeaal +iztacriccauoira(lom-inrn:eacopet)tcrioana:zs:tzielocuim0ep(el-'eln-eecnrsaise)4tfceinfahccaiomncdonutub-udna9uliatu{?n:i-ii
icc;;r;;oei;1,-nu;,"ctlr;;lnu"iAt1ice;uce;iattaaialztm-s1upttnalisnrapslvecuteslesnrrtsgescitaoul"aorl"urcbrdiocil,oitgi[;"ciauloterittieoaaccbeori?lcu"ueleirm,eeeafebiipnicintnmtaedeeccnpolitearbemrsttorieecearunrbveltauaucrtialeelur.iio,'srdpi-dnoaeinpptc'eerificlaimacc!acilaanat pnluoac--ztruuifliSDaedil"eaecis.n.hpAcirla-eesorrtrgcarandr-cuei.lm;ufaainslczailNditr-m.elup:aarr;citn;eciiudalacesarctmi..urNmmnJ-',Siautdisilntaci-pmioieunpet,ascirmudacednusrcnpglatuuriu_mnlpelc.?giaeimnd,sreiiun. rcaro-rcmi.;eoJplaioios_,i
ce.le 4Tuco;ntdililt.iiA! r trebui sd veriiic Oac; iin? indepiinire
I Bine, lncepe ! Pr-ima?
I - Aceasta este indepliniti, prin definirea adunirii cdci
fiin-d
I ;"tisbra;eiu.;tl"t"inevlltsJi'rIttteac"ctecbocplocnt:mntms'suiucdcb-ntihiarotd7ietmb1ilvelcbi"dge,id'laa"eiiitntauotdctorgA-riurcttua.niptbac'enilI+icScilnibnmi.r;?ciegAnnes+ttccnae,rcl,i5eerfaCiis-lln,cil-ndelz7gcd-ieca2asoplilcTidecigneractnr7accctotadlmaaNzeburpS'lil1oleoa-1zTdtlatiluula:;o--irr date doud. numere naturale oarecari suma lor: aiS:S
este uccndeclneiulaamdltdue.nrcanioraeuatdunreauslt-.es,cahcsdioaccrinia;uitivaadmd:oe2ulaefmc(eo4nn+td6niteiie:u(te2rsut,e+dien4od)ae+repc6lei-.

niti

Insd

zero nu face parte dintre numerele naturale ;i nici "I"*e",t

272 antnot

sinretlic, circi nu il.tlr uitli truurcielr: ncgative! inri parc

Jortrtc rdtr ! cc? ;\krge:!i tu alt[ mulfime care' in conclifiilc ladicali akr ecuiltiilor' ;ilgcl.rr"i<:c dr. onliu supcrior? $i c,ii
-'" Da'dc suc;cdsnl lui J-agrange sc datorc;tc faptulrr i cir r,l a L-.gat plo-
stabilite, sii formezc un gruP ! blcma rezoivirii ccuat i {lc a
tirn-eaAcniummae-ir:mloar iinut;rocrg:.i,Lia'ercliicciitactllaltrcrigaIn)cr sL considcr mul- lircliicinilor ci, stabilincl j lor: ;rlgerbricrr lJC.rtrnr tliii lt,r
cir pr;t cxista functii lur iorrrlt. rle
zero si ttumertl: llccstt: rirclircinj c;rre sij lirniinir invariantt. la p'-'l.nlrrtirrilc

ior rregativc, aln tot cd-rni tt'ttbnie ! fi considcrart innrulti- tlintlc trcidoilticaingic!'uc{)raattliciltl1r .sarrlf;is"ttcitrutttiuiltotrl'),a.i-asrtl{aiailodiespiaLratcr,ir.tsittat,-
r(.a-
Iiar: tlirc.i ilr lot:r.rl acluttir"ii, ai lrilincl
cit legc clc conrltozitic a multirnii nutnerclor intrcgi,
1l('r'ilstil ntrrltitttc ritttittt'a lrrtlt? llirrcrnlitincrinIiilcc:ccail,e ecuritit', c:rlt' r'stt' gnrpLrl rlc licrrnutiiri rlintr.e
--- Iar nui incttrt'i i l)c rtltcl: r'rt,i sit ;tiu, ala pc dinlt[:rr:i? clupir cill(' s(' r'r'cultoSC plolrrictrililc culir<.:1r'ris-
clouii conclitii sint inclqllinite... chiar ticc alc,
l)r"inrclc ;i a tr:ciit pr:Lrroicnmelelaai ircclcgicrc-auolLi.tiri,(rrsarrtllsrtibiij)itlni(n'1rl[l(lLi)rr(",'lisnucicrtudiiar<cn.'lri"irLunsrrra:iil1tdroilr);rrl't(rtt.lll.ts:ari.nurtr'ilr.rrt'eu-
c-xistd I... ciar Ilu alll elctllclrtul invcrs numirului illtrcg. rt.'zolva
rlici pot sclic cillcrrlc,
Nu 5'\=-x" 5 -'. l. I)cntt'u ;rccasta mi-trr tlt'llui ;ltlorlcciLcntscttc(['ctarlau,nrbislcf<;itprrortrrs,iil.bli)i(k':',(lLi'inrllr'l(lp,ccnlrr(,r,lrr'rl.)rrt\(ilr)t.lr)rluitiltrlr'rl.rrrin.laosriurnr:lracLr'prirribb'r"tieirrurcuutrttlirirl.'nclrrtr:irir-rri
sli ahaug acestor nulllct'c si ritttrlcn'le friictionart: i
dc--inmAJlutuultln.tiricerilc.trral gcnctlottttrecrluczloiar
-* itr accst caz isc spullc ! Iirtir cctiirii noasire.
oal'c -* \jrci sri spui cir t. lt'ttrr,rrtt, lc grrrlrrrlrri sirrt for.rn;rtc rl iil'
rationalc t:stc un gt:u1; tr;rttc I 1'('rl'iI'rItnI tIuttIltirt:'ri1'llrj clc tlcir'lcrtrt'rrtt, si t'ri lr'gt,rr rlr
si tt'uf ttttt'll'iflicat'r'u. I)ar (': I I I)lL. L)c accca tt,sfiittricsc s;'1-rrriscrii air.i contlrrizif ie

acin.e's\--'a-(f'ars'csCllici)itxtrucfntirrIe?rIar?Io?rOrri"Amczl;.iirrn'lliaoieti:psJulrNtpuutttsnglgtrrr:ircrlt;;ic?llrisi!tlrl-tl:lurolrg*asnirl-tgl.itni csl1it'ru! i'\tcratrr:l,c:-ii -- cr, lr' 61rr:r-
sniriulctiinr-tlil,i,zc,ii.cpieacilostrtiriLrb:inilirtlurrtri:iriinirurriniutltrrr,lrcLt'isusisrrtc6lc,nr-ts,inatrn:lut,nitrer
rirul irrerr lor naturlLlr'r si irt rirrrlrrl tlt' jos sii irrrl ici traus{or-

mc-'il it aai afarti pi: zct'o, ca irrclczir"abii,li sPuit'ri rtlrrlti- illii.l'('ir rrspcctirr;i. I)t,n1r-rr lrlcscrn ttLlt' sti nunrr:sti f ir'iiir-e
n)ilt'('.
l]lltncl'clor rationalt', clilr caro s-a t'xclus z('l'(), c-(tc 1)('rrnutilrc cru citc o litlrir
-- Sir iuccl'c. [\:ittuL l)('l'nlutrrf('o l;rs nt'schinrblrt:l . Stii?
ul] grup rnultiplicatir'. I';rrcir irrt't.li sii irrtrt'ziilt'.sc t't'r-lr I I.\Ir.'r\'rirnstciiltl'rcei:rrnutlLrc r.a Ii
-i Vi,zil lii-im spus ctl cii-rtri scltpd gnrpul lirintrt: tlt- cIr,nrt,rrttrl irctttLu si o loi rrritlr crr
gcte?
si nu-!i mai scitPt' clt-loc A-- 1 :{:: i:),, : (;: :i;j, . :(l::J'
lrli-rrtL|r:'r si tc farrrilitrizczi cu cl, ;i lIll grttp dc trottsJ'orttttit'i,
(1)
rlogctc, nc \'()l1l |lrz.boi cu
iau ,/c su,biLi/'utii, dupii. nttmirca datir de Cauchy. Cuvintul I u=-(l :;l' "-(:i;J ":(:i;)
transformarc sau suitstitufie, aplicat aici, lnseamni cf,,
datd f iind o multimc f initl iau infinit5.de elemente, fiec:irui * -llrebuic si" 'stirbilc;ti dacir a(:r(rst(' 6 eicnrcntc fr'rr,mcazX
elenrent clin muilirne ii corcspund^e gnanumit.element binc
rleterminat din a6eea;i multimc. I1i aduci aminte cil aserne- un grlt[i. Sc ribil,nui:cstt' lr st. nrrrni rrcclstii ciperatier,proclus.
nea problenle s-au intiodus plin incercirrile de rczolvare prin
AXIL cir fo.lose;tc sr:nrnu1 r'cs1-rcctiv.
27t

275

cttairanlteecr-cesbpeca,i.lilandna\pc'dgobcoai,bipicsab,:upIijpmupb,nrriieenntmrsleclt-aBclecce)coctir.ii:onnta'ancdlctcci5lc.il'riSiSfnipin,ececr:priBimpeinarrduroaritndna9inruridrepsta,c!urc.ciel\ra-reJarolale!iD;didctomoftcuurpoc5cird.cti.nputpcea^einnrrpnf;m)eairrrutpirmt;eti.ifricnu,irc,c-ii .t;;;;;;;i;rrlale"ou,_zr*ribun-:ltie$rtnlaallvimstzeeaohirlnesrlucb:ednin,a,retmgdar-riuulorco,lpitcet{snn.ib,etatsrrrcreid.om:c,mmcitmurxAietaianc"ctcte,iteviJccinIIui"maMiurni-;satti'eisii,iIli,ae;sariirdn"c'_t"b,erm'tevfnciaiestrcti.iirieniiiin;ta;cg;c;il;iidsr;uit#i8g"caa:itine?
vl.aa-,ab9fido,.il,eotisc!iacmuvinindit.rdrztnperfo!dCues?;Ai icpiiJamrf"ictu"t,.i,n';niiir;u;;it.i:;.;'i;';;"
acum aln: fddfacaacieccinea-etsDountpreon"Bir,uaopBap,dr!.,our;:SiidssminuuniFsuulge?alduovsriiarn'eadtitrnturfcedri-noeeoinccecclivot,drieatutuecnadbtefaroieleeitrlr,en!ip"mcaeriNatteii"lunisvnvtd.er.ooaie?il,reiis?citsi"rianetuSaipei,bbt..uiotlii"oelciiA,.itlleoar"c",piod.rripue"ppB;.it";-";e.;rr;.;tDi;.sr;ro;;"di;e""r:;si;i"";crJt.Cri_ir?ti

B D :ir : ;) (; ::):(; ::):'

- lncezrrci pi produsul D.B.

DB--(;::) (t :;):(: i:J '
Rezulti cd lradt'tsttL t't'tt-i cowrtlttli;", clcci gntpui' 'laci"
va fi gruP, nu-i abcliarr ! Asta din canza necomutativiialii prohusului. As nrai
N-mc5ue.--cnitfpeCIuarl"icmunnamaeitiniuuIsts'riPouucIl;rb-iis"eegrlfi?lu"lotoicueacnlcs;riauta;lrsdtoepeccicarumxf;iosiuetrt:rn.iidru-Iei.1tsIr9ipesrr5tecciiiateil)rsc:screliottr'tt-lrtctittrulitt'(fta:ari'i1cti1ivir'eitI.al\iE.t{eiIce1?3-,i c-d. acest grup se n'me;te de ordinul 6, dupd ,r;#;i;t"?aa;us;
de operaiii distincte car-e sint p."r"ri" in
dar, te rog acest .ur. aro_
sd. formezi tabla inmultirii I

A lB lcl"ln lF
e- c:laNrcuSmAa'iCc:uCu'Aiti:mC:ir conclitie nll 1n('1'ge llrioc i :jtiu cl lt ll;;li;;;;l;lll l;i.Fl,,\ [;
sd fie gmp, cI cloar clc aceea mi i-ai^ IiiLuuspiIudlocftlilntqi-'
trebuie
darcum vine asia? trlcrnent simctric sau i'vcrs? pcrmuta*
tiear trebuicaprodusul a douii ptlrnrutirri sd tlen
iea ic1,.'tici , oiicunr le-a; filce cle 1a clre itpta sirrr cle la sti'ga. l" t_

inseamni ci trebuie sI iau pc rir-rc1 ;i sir iuci:rc t,atr-: DttrrnLt- (r,) ull* i"r
st-iirm*iAlee.$,'triTpciteiil-.n?oiin^.f*aci-efcaluarlredcrnetiaezt?utultaUapstriutotelfd?c,ulasmcuql s-DiB.tte.sr-ai Ciutstiun:tl'sl'iFpi r-o:p(r'i:iCi l,o:r
Fl .'l _c--il
;I;l;l_lnjeic3-ll;Bl l
D tr:(;:T(: :;) :A:EI)
DI
* AAmmsatiardbmitiatsaFs-.tfDelairnevesrismeplelue, lcedmcei n;iteelol reAste, B, C, D
pi propriul zl I
E.

276

',,,,-'=urL,rlr,,ercl lccap6crrcnsutet;icrle,l mai lric orilin pc car"e-l poate irvr:a it.ur'rslr.i'A"r,ri.,n..criU'Llcrc,{'iri:cri rllirtfniirri'l,rsprirltgrrr:rul,cn|mrsri(lc:rlnctraunsiigtarturlpr,rrl's{(tr,.r.'.ai tlrrrl.i,nlrrt.,lierrulies;i'L*l
tinind scanrl eu douij clcttrctllc
ii

t6C,iir"':ue."ai'ii*o,,j,-J.i;iirli].Ndi,,tiii,*i'ii.n,i'na,limt;tpji,,cu.tc,,alltair...ihrtil,,rair,mbnruint.latic",cislo"isidcri'-(s,.Jl.TiereI"ic'),l'ri,ri"ntn.g*ftia6iriirnil*e'uttJciTpf*asieuriccnrli*elciicn!'mlido,ts'ecarxurni<tosfttlicltiadt,esiiislrpnr;et.eliu:rpt'lep'o.Yl3rttcta.-lmf'tanbt2cccttrlln-:cc1tg-tirs(triclallogtlfurtir|,tluivalenpesr?our.ltillsltll:ct;i, ,\ -.:l\iI

_._-*,{

':c.--i\it'"'il-tt)i-Do"l"Jiti""rsorsi*itieri*Fi.an--lr.f)e.ounIa)rnnsr1-trcc=,ndrcatEzi.i.i.:oitprn"crt.biitiiu:1i"ii'ei-t;itic"stl,e3i m':i;ziciconil-r1t:ne?zt,r-itr1retli:tIlia"ruhtra-ar-tb:citleicrIa)ir'icncIlicidDtterirt-cro-inltlt-:rc\tl-c-'iq'i s{'N-,u\fliin;.rrsc cil :rsr,nr('lt(.a sgir-trurlrrtreri'Jlrrrrttltri.rru:pirirlcr,lt.rilicrirr gr.rrpLrl tlat

srrlrgl'111;1;1;1, ii i stabilit

trjr{,r{:'nra turrtllrrrrt'rrtali-r lr gnrpuril,,r:, enrrntatij cl<,(iat.is-;i
c{rmortstrlrtit rlc ( lrLur-:h \'. llntrrirc t:ir :
,,(li:ice grup rrl r'irrLii ,rtl i' . rli'izibil pri'tr-un nu'riu-
;rr itrr P, t'onlirrr. t','l l,rrlirr rrir srrlr{r.rrlr rlr. ,,litlirrrr l n.'
6^. -aiy.\tsiuitt(lictrl'i,'^Iiiirrtrt,l'tt:rlirl-tr''lrrilrrr.rr.lbtu.oircrssiciiltariliiiri'lisrlrtr,l.lrr6gir"u,.pa. -,lll;..;or;rci;l;iui ul
:r3t.isl,r:-r2,,tr,5ltclci,irfiIrrliottt'i:ririiitkclll,itsllr,lrrllc';-cltrr,r.sr*,.t1lrir'',ru'.nrii.lruutlJrrl)l(rr,r.riili.r;r(rr'lstisriririrrrrrr.r;tr.r,irrrlrr.l:rirrt,r,p1rizr,;;i<rslrat.irltli.iirr'iir,rul"r-ii.rt't,C,'ir,.,Ir,i.ii,;;rp;!;,..."1i;1i;r);;ilt;,--,
.stractizarcr nrfiruriirk',qrrrli.liiirrrrt.lr,,tripr'rcurnaitrtistr.r,at,

Iu dottil r:ltlttttlntr], tr li ,\ SI ;ttunc i cinrl ;r I'rst s:'i v.rilic.;irn t:alit:Lti'lr ,i,' gru1, a rrnci mul-
tiriri, irrn c:orrsi<lt,r-lrt llrl g-nrl) corrrrrt,t, sprrciiicirlri clirr cc [cI
l,\l] Ii.\ . l]
.\ :- I I , I:.\ aacrr:-klsel'tlto'r.ig''lsrclrcnutrgnuptc,curlxnililitrrtrr(rcal'it:slcraicnili.u'ncIpn'rsslclcr(tour'sznt:s(irrt,tr(,ar'i(tcl)a;rlirnriillirilic)tLlatl.'-luoct1rr.lrrr'rrrrr,ranonrrtprtrurarrlzrcticrrirur,trni'rr,Bt.stri:arilct.r,stciic.iick'aArtr:rclrs.ra-r'tt(..crfeltt,)rr.,,larinitsirLa'tsrrlnriic.li,Liacrl
si _ C,\ F
l,\l;
C. ar:t'astir tablir de irrntultire odati st;ibilita, sr.])uittLr {,1)r.1.ii
[,16 lutrnr4i cu;rjutorul ei, i;jrij a sr- nrairccllrs(. la L,u-nr.,1,,l.j(ll.

9i; A ,Or, c.rperierrJa- nc irrati. ci cxistd fc,aii,r nrultr: gru1,,i,.l

rcucl.neo*nn'g,fcrrtc{iuelbpctisc.'fni.i{nicavilt'o*i,'rdcrsdca;.nnrctltacelellcanc;cilleosnmrfuieriui;nri'tttiercrcci.nlae'clecltleigptnu'o.rrtie,pfnii*itrrclili;ul,dcs.aicics,l.irir,nr"*tic,,,io,n;rrict.1tsinrlpu;ioticn,.ti_.t,

279

i n][;,ociio:,l;_r""arI.or"iitl'm",i,;ner'.c.;''r,;;,xlaiel"rlee.iittg"c",mTa"-'our;rt"ptr,mii.t';h.*;p,icip,.ritn1ai1e,iorirculm;qzc1;tn,i"rt,l,suii1oeeceili.st"".rri,a,psnt|eftrreult.izlt5ntnnlul('.r:'tttinrpidtdtglrrt'.leli,rlei1tu'bf(lll\'ilaillie"nLHo)lrtllilrt|gltcr'natdrnisuai(rrtcrr'atb-rnl-r|ci,iluaiallxerll'lIr-tillrl.liau|'e.rilt;;''rAcrti.iil.til'cli'rtLi.Icatrtla)r.tLclbirrLtlt'lo)ctrtoci(i'cclricliuit(:zismrrc'illtzitt-e(ct'1''l'1.ii-{r\ttp'g)it''ireLr'r'rt}it-lorft'tu;"rircetsusiLr!li'u)tr-pllsu:rinas'lcurtr'erluilrlpdc.lr.ilgrio:rillnun'ct-itr'
u tttl.rel.rt' i'r.lilt'.
-_ I)ar t'otaiit;'LtciL lttllllell€.lof l:cal(] Si 1-roziiir'il itii-nrcllzii

un gr-r,rp?
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bulanli I li' tr:-rrji nt1 ilre
itl)ral.t''srt!r'\ptruintccai
.ci,ia 1te
c";r_rt,,*hi .\ijo[(c]ii c:rr.*1 cui.i';ii clin tc'r"i._ ]r:,git' illn iltir:

irr loc tlt' inlrmltirt' se f ace aciunare, lrclix l(lcin
in iiillliit'tilc sc {ircc sciicli t-r:,
ln loc clcr iilmrril-ilt" Stai cir-s n;iuc ck: tot: ,, I)r,si
riti'ir:ai-t: l:r putt'rc sc {ace tc pr)t
t,r. clc lru- []r'iccp(]". i.or.lrc-sti inteles,
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5 ;i cle

functii
ce se numesc eliprtice. latd cleci, p" il., o parte,,.cccsitatea
de a studia proprictd!ile grupuhii abstract care se degaji
cicrlniinnacgaiciluteacs,:tego3efergorrr'ri-teriptrudirceidc.rooantcarrifeiicitleled;cii,cinopeislo:crrleed.a,r.ul.trliurp.i!!ai"r-ited, eo ,pt"r.jidl

graclul al

280 28L

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mls*carOe.lr."t,, c[ alirturi dc notiunea de grup apare mereu;1 trunde in sal5, ci trebuie c* cele citeva
poze expuse in vitrind !
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284 ci" nu-i rlr,cit o singurl matentatiiA j
e_-a.re i.ie,sc clai

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