moswavlis biblioTekajemal kiknaZejemal kiknaZejemal kiknaZejemal kiknaZejemal kiknaZe

azris jaWvimoswavleTa inteleqtis

ganviTarebisaTvis

gamomcemloba `inteleqti~

Tbilisi

2001

pirveli nawili

daamtkica saqarTvelos ganaTlebis saministrom damxmare

saswavlo masalad maTematikaSi zogadsaganmanaTleblo

skolis V-XI klasebis moswavleTaTvis

2

winamdebare mecnierul-popularuli wigni-almanaxi Taviseburi damxmare wignia,

romelic maTematikis sxvadasxva sakiTxebs moicavs da ZiriTadad zogadsaganmanaT-

leblo skolis moswavleTaTvis aris Sedgenili. maTematikis garda masSi sxva

saintereso sakiTxebzecaa saubari da saxaliso maTematikis Janrs ganekuTvneba.

wignSi mravladaa warmodgenili savarjiSoebi, kiTxvebi, TamaS-gasarTobi Tavsa-

texebi da xelovnebis zogierTi nimuSi, romlebic xels Seuwyoben ymawvilebis

dainteresebas, SemoqmedebiT saqmianobas, damoukidebeli muSaobisa da garkveuli

tipis inteleqtis ganviTarebas. am mizniT wignSi Setanilia sxvadasxva testebi,

romlebic ganaviTareben farul kanonzomierebaTa aRmoCenis, analitikuri da induqciuri

azrovnebis unar-Cvevebs. nimuSisaTvis ganxilulia zogadi codnis Sesabamisi

inteleqtis koeficientis (ik) dasadgeni ramdenime testi. igi daexmareba maswav-

leblebs saklaso da klasgareSe muSaobaSi.

wignSi ganxiluli sakiTxebi misawvdomia ara marto zogadsaganmanaTleblo

skolis moswavleTaTvis, aramed am sakiTxebiT dainteresebul mkiTxvelTa farTo

wrisaTvis.

© jemal kiknaZe, 2001.

© gamomcemloba `inteleqti~, 2001.ISBN 99928-67-93-0

didi galaktionis msgavsad yovel bavSvSi aRmzrdeli

momavalis did saxes~ xedavs da minda es wigni maT mivuZRvna.

`qals patara dahyavs xeliT,

bavSvi zrunviT zrdili,

momavalis didi saxe _

Svili _ Svili _ Svili~.

galaktioni

redaqtori: arkadi qurCiSvili, Tsu-s ufrosi maswavlebeli.

recenzentebi: romanoz danelia, gamomcemloba `inteleqtis~ direqtoris

moadgile, Jurnal fizika da maTematika skolaSi~ redaqtoris moadgile,

gela kvirikaSvili, Tsu-s T. gegelias saxelobis respublikuri

fizika-maTematikis saSualo skola-internatis fizikis maswavlebeli,

guram oTinaSvili, Tbilisis 55-e saSualo skolis maTematikis

maswavlebeli.

3

Sesavali

mravalwlianma pedagogiurma moRvaweobam da Tanamedrove meToduri literaturis gacnobam damarwmuna,

rom swavlebis tradiciuli meTodebi moiTxovs gadasinjvas da sqolastikisagan ganTavisuflebas.

sqolastika aris skolis muSaobisa da cxovrebis wesis sistema, romelsac skolis gareT ar gaaCnia

araviTari faseuloba, radgan skolas ar SeuZlia moamzados bavSvebi cxovrebisaTvis.

zogjer tradiciuli skola iyenebs mxolod axsnis da naTqvamis sailustraciod cdebsac, magram

aseTi axsna iZleva mxolod zedapirul da formalur codnas, romelic ver mkvidrdeba adamianis

gonebaSi. aseTi codna im norC ylorts waagavs, xes rom Camosxepen da miwaSi CaarWoben. cxadia, igi

male iRupeba. swored aseT zedapirul, ufesvebo codnas iZleva tradiciuli skola, romelic Tavis

naklovanebebs niRbavs uSinaarso, xmamaRali, wyaliviT morakrake mravalsityvaobiT. ra Tqma unda kargi

mexsiereba SesaniSnavi Tvisebaa, magram mas xSirad SecdomaSi Sehyavs pedagogi da mSobeli. saubedurod

gamocdebzec mxolod imas amowmeben, Tu ra daimaxsovra moswavlem. dazepireba ar niSnavs codnaso _

uTqvams frang filosofos, humanist miSel de montens. igi uaryofda sqolastikosTa mTavar meTods

`CavasxaT~ adamianSi codna, rogorc ZabrSi.

saubedurod, xSirad tradiciuli swavleba emyareba mexsierebas, romlis meqanikur gamoyenebas

dauZlurebamde da gamofitvamde mihyavs axalgazrdoba. sinamdvileSi aseTi codnis usargebloba sul

ufro da ufro aSkara xdeba. axla yvelgan (iSviaTad skolaSi) bevrs laparakoben WeSmarit, Rrma

kulturisa da inteleqtis mniSvnelobaze da moazrovne Taobis aRzrdaze.

cxadia, am urTules pirobebSi, rom icxovro, saWiroa iazrovno. swored amitom gagaxseneT didi

frangi filosofosis da maTematikosis rene dekartis gamonaTqvami: `Cogito, ergo sum~.

bevr rameSi veTanxmebi da praqtikulad viyeneb meoce saukunis gamoCenili frangi pedagogis

selesten frenes (1896-1966), originaluri pedagogiuri sistemisa da `pedagogiuri invariantebis~

avtoris mosazrebebs: unda vifiqroT, rom namdvili kulturisaken da inteleqtis ganviTarebisaken

mimavali gza eqsperimentul mosinjvaze gadis. codnis SeZena miiRweva eqsperimentis gziT da ara

wesebisa da kanonebis SeswavliT (gazepirebiT), rogorc zogjer fiqroben. pirvel rigSi wesebisa da

kanonebis Seswavla imasa hgavs, uremi xarebs win davuyenoT. cocxali eqsperimentuli muSaobis

principia _ `mefolade mxolod RumelTan SeiZleba gaxde~.

samwuxarod tradiciuli skola aviTarebs mxolod abstraqtuli azrovnebis unars, rac Sorsaa

realuri cxovrebis moTxovnilebisagan. maT visac hipertrofirebuli (aranormaluri ganviTrebuli)

aqvT inteleqtis es forma, SeuZliaT SesaniSnavad imsjelon nebismier `Wkvianur~ Temaze, magram

sruliad umweoni aRmoCndebian, Tu saqme Seexeba cxovrebasa da garemosTan Seguebas.

arsebobs inteleqtis sxva tipebic da isini eqsperimentuli mosinjvis procesSi viTardeba.

magaliTad: xeliT Sromis unari, SemoqmedebiTi unari, problemaTa gonivruli, praqtikuli gagebis

unari, Semecnebisadmi mecnieruli midgomis unari, socilaur-politikuri mimarTulebis mqone inteleqti,

romlis mflobelTagan izrdebian da yalibdebian sazogadoebrivi moRvaweni.

kacobrioba yovelTvis maRal Sefasebas aZlevda inteleqtis am sxvadasxvagvar formebs; sworedac

maTi SemweobiT gvyavs xelovnebis genialuri moRvaweni, gamomgoneblebi Tu mecnierebi. aseTi xalxi

ver egueboda swavlebis tradiciul xerxebs da xSirad isini skolaSi CamorCebodnen.

Tanamedrove sazogadoebas esaWiroeba gansxvavebuli mimarTulebis kadrebi _ gamomgoneblebi da

mkvlevarebi, rac saWiroebs inteleqtis sxvadasxva formis ganviTrebas.

cnobilia, rom bavSvis buneba arafriT gansxvavdeba zrdasruli adamianis bunebisagan. gansxvaveba

mxolod gamocdilebasa da ganviTrebis doneSia. arc bavSvs da arc ufross ar uyvars, roca rames

ubrZaneben. yoveli adamiani asea mowyobili _ nebismieri brZaneba TiTqmis avtomaturad winaaRmdegobis

grZnobas iwvevs: adamiani wiTldeba, azri efanteba, da pirveli survili, rac uCndeba, es aris, win

aRudges, ar daemorCilos brZanebas. kanonzomieria Semdegi mtkiceba: nebismieri avtoritaruli brZaneba

yovelTvis Secdomaa. nebismier brZanebas, rogorc wesi, bavSvi Tu ufrosi pasuxobs uaryofiTad _

`ara~! _ mniSvneloba ara aqvs xmamaRla warmoTqvamen am sityvas, Tu xmadabla. amitom saWiroa maTTan

megobruli urTierToba.

am qveyanaze aravis ar uyvars fuWi Sroma, robotiviT moqmedeba, aravis ar exaliseba raime samuSaos

Sesruleba sxvisi Canafiqris mixedviT. yoveli SromiTi saqmianoba damoukidebeli da SemoqmedebiTi

unda iyos. erTi da igive samuSao erTisaTvis mZime monobaa, meoresaTvis Tavisufali SemoqmdebiTi

Sroma, romelic mas yovelTvis did siamovnebas aniWebs.

codnis aTvisebis yvelze efeqturi saSualebaa ara ubralo dakvirveba, axsna da demonstracia,

rogorc mimdinareobs tradiciul skolaSi, aramed Semecnebis bunebrivi, mecnieruli da universaluri

meTodi, romelic ciklurad viTardeba: faqtebis erTobliobis analizisa da problemis dasmiT

4hipoTezisken; hipoTezidan Teoriuli Sedegebisaken; Sedegebidan maTi interpretaciiT eqsperimentuli

Semowmebisaken da praqtikuli gamoyenebisaken:

maSasadame, mkvlevarma (moswavlem) axali movlena rom Seiswavlos, is am movlenis Sesaxeb agrovebs

da asistemebs cnobil faqtebs; dagrovebuli faqtebis safuZvelze Sesaswavli movlenis `meqnizmis~

Sesaxeb agebs models (hipoTezur models), romelic aZlevs ara marto dagrovebuli faqtebis axsnis,

aramed axali, manamde ucnobi movlenis winaswarmetyvelebis saSualebas; bolos, hipoTezasa da miRebul

Sedegebs eqsperimentulad amowmebs. am meTodis fuZemdeblad galileo galilei iTvleba.

winamdebare wigni, romelic gamocdilebisa da sxvadasxva literaturis gacnobis safuZvelze

ymawvilTaTvis Sevadgine, nawilobriv zeviT ganxiluli problemebis gadawyvetas isaxavs miznad. masSi

mocemuli bevri saintereso sakiTxi ZiriTadad maTematikas exeba, magram aq sxva sakiTxebzecaa saubari

da mecnierul-popularulia. masSi mravladaa warmodgenili (pasuxebis gareSe) savarjiSoebi, kiTxvebi,

TamaS-gasarTobi Tavsatexebi da xelovnebis zogierTi nimuSi, romlebic xels Seuwyoben ymawvilebis

dainteresebas, SemoqmedebiT saqmianobas, damoukidebeli azrovnebisa da garkveuli tipis inteleqtis

ganviTrebas. swored am mizniTaa wignis bolo paragrafSi mocemuli sxvadasxva testebi (zogierTi

fsiqologiuri) da maTi pasuxebi. nimuSisaTvis ganxilulia zogadi codnis inteleqtis koeficientis

(ik) dasadgeni testebi. aRsaniSnavia, rom es kiTxvebi, Tavsatexebi da testebi xels uwyoben faruli

kanonzomierebis aRmoCenas, analitikuri da induqciuri azrovnebis unar-Cvevebis ganviTrebas. wignis

saTauri misi Sinaarsidan gamomdinareobs. moswavlem es wigni ubralod ki ar unda waikiTxos, aramed

bevri ram sakuTari xeliT, damoukideblad unda Seasrulos. vfiqrobT, kiTxvebisa da Tavsatexebis

pasuxebi calke gamovceT.

dainteresebuli mkiTxveli am wignSi ganxilul sakiTxebs (zogjer isini mokled da gacnobis

mizniTaa warmodgenili) ufro Rrmad gaecnoba im literaturaSi, romelic wignis boloSia miTiTebuli.

uRrmes madlobas movaxseneb:

redaqtors, baton arkadi qurCiSvils. recenzentebs, batonebs: romanoz danelias, gela kvirikaSvils

da guram oTinaSvils. agreTve i. gogebaSvilis saxelobis pedmecnierebaTa erovnuli institutis

mecnier-TanamSromlebs, qalbatonebs: iza xuciSvils, eTer guldamaSvils, da Tsu-s T. gegelias saxelobis

respublikuri fizika-maTematikis saSualo skola-internatis maTematikis maswavlebels, baton iuri

fiCxaZes, romlebmac guldasmiT waikiTxes xelnaweri da damexmarnen dasabeWdad momzadebaSi.

gamomcemloba inteleqtis~ xelmZRvanelebs, maT yvela TanamSromels, gansakuTrebiT ilia xelaias,

romelmac udidesi Sroma gaswia am wignis Seqmnaze da kompiuterul grafikaze.

yvela moswavles, maT Soris Cems SviliSvils _ laSa tabataZes, romelTac stimuli momces am

wignze muSaobaSi, xelnaweri yuradRebiT waikiTxes, zogierTi savarjiSo damoukideblad amoxsnes _

Seamowmes da damexmarnen koreqturis kiTxvaSi.

agreTve madlobas vuxdi yvela mkiTxvels, romelTa saqmian SeniSvnebs madlierebiT miviReb.

jemal kiknaZe

Tbilisis ivane javaxiSvilis saxelobis saxelmwifo univer-

sitetis Tengiz gegelias saxelobis fizika-maTematikis respubli-

kuri saSualo skola-internatis fizikis maswavlebeli.

gTxovT survilebi da SeniSvnebi gamogzavnoT misamarTze:

� 380079, Tbilisi, guram rCeuliSvilis quCa, korp. #3, bina 21.

� 29-21-23.

modeli Sedegebi

faqtebi eqsperimenti1

2

4

3

5

dedamiwa, dedamiwaze da mis gareT, sivrceSi

mze, planetebi, varskvlavebi da sxva, realurad

arsebuli yvelaferi erTad samyaros _ bunebas

warmoadgens.

adamiani samyaros na-

wilia, albaT erT-erTi

yvelaze didi saocreba,

rac ki bunebas Seuqmnia.

bunebam mas misca sam-

yaros Secnobis saSuale-

bebi _ grZnobis orga-

noebi da tvini.

adamiani samyaroSi

yvela informacias grZnobis organoebis saSu-

alebiT iRebs, romelic Tavis tvins gadaecema

(sur. 1).mxedveloba, smena, ynosva, gemovneba, Sexeba _

ai Cveni fanjrebi~ garemomcvel samyaroSi.

tvini amuSavebs giganturi raodenobis sig-

nalebs, romlebic momdinareobs samyarodan da

adamianSi iwvevs siTbosa da sicivis, sigluvisa da

simqisis, ferisa da sikaSkaSis, xmaurisa da siCu-

Sen _ adamiani, rogorc samyaros sarke

buneba, Sen da gamokvlevebibuneba, Sen da gamokvlevebibuneba, Sen da gamokvlevebibuneba, Sen da gamokvlevebibuneba, Sen da gamokvlevebi

vidre raimes Seswavlas Seudgebode, winaswar saWiroa gaerkve, ra ginda, saxeldobr ris gagebas

da Cawvdomas apireb. zogierTi sityvebi: samyaro~, buneba~, fizika~, maTematika~, mecniere-

ba~ da sxva mravaljer gsmenia, magram yvelas ar SeuZlia axsnas, Tu ras aRniSnaven isini. saWiroa

gaerkve ara marto maT dedaazrSi, aramed nebismieri sityvis arsSic. winaaRmdeg SemTxvevaSi gad-

mocemuli azris gagebas da, miT umetes, masSi Rrmad Cawvdomas ver SeZleb.

Tavidanve zemoT naxsenebi yvela sityvis ganmartebas ar Sevecdebi, radgan vfiqrob, es igivea

vimsjeloT im qveynis Sesaxeb, sadac jer ar vyofilvarT. es TandaTanobiT sxvadasxva saswavlo

procesSi moxdeba. daxmarebas es patara wignic gagiwevT.

`Cogito, ergo sum~.Renatus Cartesius

`vazrovneb, maSasadame varsebob~.

rene dekarti

romelman Seqmna samyaro ZaliTa miT ZlieriTa,

zegardmo arsni suliTa yvna zeciT monaberiTa,

Cven, kacTa, mogvca qveyana, gvaqvs uTvalavi feriTa,

misgan ars yovli xelmwife saxiTa mis mieriTa.

SoTa

sur. 1

mis, zomebisa da formis, sxeulebamde manZilebis

SegrZnebebs.

realur samyaroSi, rom viarseboT, an rom

gvqondes saSualeba gaviumjobesoT cxovrebis

pirobebi, saWiroa rac Sei-

Zleba meti vicodeT am

samyaroze. aucilebelia

ganvasxvavoT xmeleTi

zRvisagan, saWiroa sakvebi

mcenareebis da cxoveleb-

is gamravleba, maTi mov-

la. sacxovreblebis ageba

da mtacebeli cxovele-

bisagan Tavis dacva. ratom ar SeiZleba am miznebis

ganxorcielebisaTvis daveyrdnoT Cveni grZnobe-

bis organoebs? marTlac xom ase iqcevian primi-tiuli civilizaciebi.

arsebobisaTvis materialuri pirobebis gaum-

jobesebis mcdelobisas iZulebulni varT gavi-

farTovoT Cveni codna gare samyaroze. risTvisac

saWiroa maqsimalurad davZaboT Cveni grZnobis

organoebi. magram samwuxarod maT zogjer Sec-

domaSi SevyavarT.

6 buneba, Sen da gamokvlevebi

� daakvirdi sur. 2-ze gamosaxul T-s magvarfigurebs, romlebzec gamosaxulia horizonta-

luri da vertikaluri monakveTebi. Seadare maTi

sigrZeebi erTmaneTs. Semdeg gamoiyene saxazavi

da gazome maTi sigrZeebi. a) SemTxvevaSi ver-

tikaluri monakveTi ufro grZeli mogeCveneba,

vidre horizontaluri. Seadare dakvirvebisa da

gazomvebis Sedegebi erTmaneTs. b) SemTxvevaSi

monakveTebis sigrZeebi toli mogeCveneba. gazom-

vebiT miiReb sawinaaRmdego pasuxs.

aq saqme gvaqvs mxedvelobiT (optikur) ilu-

ziasTan. amis mizezi isaa, rom Tvalis mobruneba

horizontaluri mimarTulebiT ufro advilia,

vidre vertikaluri mimarTulebiT.

� aseve daakvirdi sur. 3-ze da sur. 4-zemocemuli figurebis AB da AC monakveTebs.dakvirvebiTa da gazomviT Seadare maTi sigrZee-

bi. gaakeTe daskvna.

� ganixile sur. 5-ze gamosaxuli figura jerpirdapiri dakvirvebiT, Semdeg gverdidan ise, rom

Tvali rac SeiZleba axlos iyos furclis sib-

rtyesTan. Seadare dakvirvebebi erTmaneTs, gaa-

keTe daskvna.

� daamzade bzriala sur. 6-ze gamosaxulimuyaos an fanerisagan gamoWrili diskos gamoye-

nebiT. diskoze naCvenebia TeTri da Savi zolebi.

bzriala gaanaTe, daabrune da daakvirdi. dain-

axav ferebs!

� sur. 7-ze naCvenebia rkinigza. ori parale-luri relsi gveCveneba, rom Sors erTdebian. es

wrfivi perspeqtivis magaliTia.

� sur. 8-ze sxivebi miemarTebian Sekrebiswertilisaken (perspeqtivis centri), moculo-

biTi scenis iluziis SeqmnisaTvis. suraTze ga-

mosaxulia erTnairi zomis maRali yuTebi, mag-

ram gveCveneba sxvadasxva sididis _ Sors~ ufro

didi. am iluziis mizezi aris gamocdileba: rac

ufro Sorsaa dasakvirvebeli sagani, miT ufro

mcire zomis gveCveneba igi, amitom marjvena yuTi

ufro didi gamoiyureba, vidre sinamdvileSia.

� realisturi maneriT daxatuli suraTi or

ganzomilebiania (sibrtyezea), magram Tu isini

daxatulia wrfivi perspeqtivis Sesabamisad, ma-

Sin Cven is gveCveneba sivrceSi (samganzomilebi-

ani scena). aseTi `moculobiTi gamosaxulebis~

iluziluziluziluziluziebiiebiiebiiebiiebi

sur. 2

sur. 3

sur. 4

A B A C

a b

moric kornelis esxeri. `belvederi~*.

liTografia**. 1958.

* belvederi (it. Belvedere _ mSvenieri xedi) _ koSki, saidanac gadaiSleba garemos xedi; fanCaturi (talavari) maRlobze.** liTografia (berZ. lithos qva da graphõ vwer) _ beWdva specialuri brtyeli qvis zedapiridan, romelzedacwinaswar xataven.

7buneba, Sen da gamokvlevebi

sur. 7 sur. 8

sur. 6sur. 5

O

perspeqtivis

centri

salvador dali. `jvarcma. hiperkuburi sxeuli~.

saukeTeso magaliTebi uamravia. mas xSirad iy-

eneben konstruqtorebi, arqiteqtorebi da mxat-

vrebi.

am saxis kargi magaliTia XX saukunis erT-erTi yvelaze popularuli adamianis, udidesi

espaneli mxatvris salvador dalis suraTi:

`jvarcma~ (hiperkuburi sxeuli). 1954 w. (sur. 9).� saocari movlena: mTvare amosvlisas an Cas-

vlisas (horizontTan) gveCveneba ufro didi,

vidre maRla caSia aweuli. sinamdvileSi gazom-

vebis Tanaxmad mTvaris xiluli diametri hori-

zontTan ufro pataraa, vinaidan am dros mTva-

re damkvirveblidan ufro Sorsaa. fsiqologebi

Tvlian, rom es faqti ganpirobebulia adamianis

tvinisa da mxedvelobis TvisebebiT. damkvirvebe-

li midrekilia warmoidginos mTvare ufro Sors,

rodesac igi axlos aris dedamiwis zedapirTan

(horizontTan), vidre maSin, rodesac maRlaa caSi.

e. i. is Seucnoblad rogorRac moaTavsebs mTva-

res horizontze. radgan mTvaris kuTxuri zomebi

ucvleli rCeba, igi mogveCveneba SesamCnevad didi

horizontTan, sadac mTvare mogveCveneba ufro

daSorebuli (sur. 10).� iluziebsa da Secdomebs adgili aqvs sxva

grZnobis organoebiT SegrZnebisas: temperatu-

ris, gemos, bgerisa da moZraobis dros.

magaliTad: aiRe sami WurWeli _ cxeli,

Tbili da civi wyliT. marcxena xeli moaTavse

civ wyalSi, marjvena _ cxelSi. cota xnis

Semdeg orive xeli gadaitane Tbil wylian

WurWelSi. ra SeigrZeni?

sur. 9.

8 buneba, Sen da gamokvlevebi

� davuSvaT saaTma Seasrula 6 dartyma 5 wamSi.ramden wamSi Seasrulebs is 12 dartymas? intui-cia gvkarnaxobs: 10 wamSi. magram araa swori.

6 dartyma gayofilia 5 dayovnebiT, xolo 12dartyma _ TerTmetiT. maSasadame swori pasuxia:

11 wami da ara 10 wami. e.i. intuiciam gviRalata!� mocemuli gaqvT erTnair perimetrebiani

(perimetri aris gverdebis sigrZeTa jami) ori

marTkuTxedi. erTnairi farTobebi eqnebaT Tu

ara orives? albaT ifiqreb, rom erTnairi. mag-ram rogorc gamoTvlebi gviCvenebs sazogadod

farTobebis toloba ar aris aucilebeli (See-

cade damoukideblad daamtkico).

bunebrivad ibadeba kiTxva, erTnair peri-

metrebian romel marTkuTxeds aqvs udidesi far-

Tobi? irkveva, rom kvadrats! (samkuTxedebisSemTxvevaSi ki tolgverda _ wesier samkuTxeds).

� oci metri siRrmis WaSi bayayi Cavarda da

cdilobs amosvlas. yovel saaTSi 5 metrze amo-dis, magram me-5 metrze ise iRleba, rom Tavsver ikavebs, fexi ucurdeba da 4 metriT ukanvecocdeba. rogor fiqrob, ramden saaTSi amova

bayayi Widan?pirobis Tanaxmad bayayi saaTSi erTi metriT

iwevs zeviT, radgan Wis siRrme 20 metria, al-baT ifiqreb, rom bayayi Widan amova 20 saaTSi.magram intuicia aqac Segacdens. Tu kargad

dafiqrdebi mixvdebi, rom bayayi Widan amosvlas

moandomebs 16 saaTs.� intuiciiT Cven vcdebiT mravali sxva situ-

aciis drosac. magaliTad adamiani, romelic

garkveul manZilze imyofeba vaSlis xidan da

xedavs, rom erTi vaSli es-es aris unda Camo-

vardes. mas unda, rom Tofis tyvia moaxvedros

vardnil vaSls. romel wertilSi unda dau-

miznos man, rom tyvia moxvdes vaSls?albaT ifiqreb vaSlis qveviT, romeliRac wer-

intuintuintuintuintuiciaiciaiciaiciaicia

`mTvariani~ optikuri iluzia:

or Sav diskos aqvs erTnairi zomebi.

(rokisa da kaufmanis suraTis mixedviT).

sur. 10.

albaT marcxena xeliT wyali geCveneba cxeli,

xolo marjvenaTi igive wyali civi!

� oTaxSi sxvadasxva sagans erTnairi tempera-

tura aqvs. magram, Tu xeliT Seexebi rkinisa da xis

sagnebs, rkina ufro civi mogveCveneba, vidre xe. e.i.

grZnobis organoebi zogjer gvatyueben, xolo

rac erTxel mainc mogvatyuebs, sando araa. de-

kartiseuli eWvis pirveli wyaroa grZnobadi

codna.

maSasadame SegrZnebebi yovelTvis san-

do ar aris!

tilSi, radgan raRac droSi tyvia miaRwevs im

adgilamde, sadac wydeba vaSli, imave droSi vaS-

li Tavisufali vardniT, raRac manZilze daiw-

evs qveviT. magram mtkicdeba, rom gasrola unda

moxdes mowyvetis momentSi da msrolelma unda

daumiznos pirdapir vaSls da ara mis qveviT.

riT SeiZleba Sevewina-

aRmdegoT sxvadasxva sa-

xis iluziebsa da in-

tuiciiT daSvebul Sec-

domebs?Cveni yvelaze efeq-

turi pasuxia:

maTematikis gamo-

yenebiT!

rodesac aleqsandre

makedonelma (Zv. w. 356-323) ganizraxa msofliosdapyroba, man saberZneTis

oikumene* aTenidan gada-

itana egviptis erT qa-

laqSi, romelsac man Tavisi Cveuli mokrZale-

aleqsandre makedoneli.

romauli marmarilos qandake-

bis asli. msoflios dampyro-

bi mefis saxis nakvTebi albaT

gadmocemulia zustad, radgan

igi Seqmnilia samefo karis

didi moqandakis lisipes mier.

didi aleqsandre.

(pompeis mozaikis detali).

* oikumene (berZ. Oikumene _ dasaxleba) _ dedamiwisadgilebi, romlebic adamianebiTaa dasaxlebuli.

~

9buneba, Sen da gamokvlevebi

* Tavisi batonobis gansamtkiceblad igi dapyrobil teritoriebze aarsebda gamagrebul qalaqebs _ aleqsandriebs.

daarsa daaxloebiT 70 aseTi qalaqi.** hipoTeza (berZ. hypothesis varaudi) _ raime movlenis asaxsnelad wamoyenebuli mecnieruli varaudi, romlisueWveloba jer ar aris cdiT damtkicebuli.

bis~ gamo saxeli gadaarq-

va da aleqsandria* uwo-

da. swored iq, aleqsandri-

aSi, evklidem (Zv. w. 300)dawera pirveli RirsSesa-

niSnavi maTematikuri cod-

nis dokumenti _ Tavisi

klasikuri `sawyisebi~. am

naSromSi gamoyenebuli iyo

damtkiceba. garda amisa

evklides ekuTvnis naSrome-

bi musikaSi, optikaSi, meqanikaSi, romlebSic Zir-

iTadi roli gamoyofili aqvs maTematikas. maTem-

atika gamovida rogorc idealuri versia imisa,

rac Seadgenda CvenTvis cnobili realuri sam-

yaros agebulebas. samyaroSi mimdinare movle-

2347_ 298

gamokvlevebigamokvlevebigamokvlevebigamokvlevebigamokvlevebi

es aris ucnobi mxaris `dazverva~, Semdeg

`dalaSqvra~. Cveni mizania dagexmaroT im gzeb-

isa da xerxebis _ meTodebis moZebnaSi, romli-

Tac midiodnen didi maTematikosebi da bunebis

mkvlevarebi, moZraobis mimarTulebas ki Sen

TviTon airCev. gaxsovdes qarTuli andaza: xer-

xi sjobia Ronesa, Tu kaci moigonebsa~.

daiwyeb ra romelime movlenis SeswavliT, ma-

galiTis amoxsniT, ideiT an faqtiT Sen Seudge-

bi namdvil kvleviT muSaobas. movlenebis Ses-

wavlisas Caatareb dakvirvebebs da cdebs, daag-

roveb faqtebs. maT safuZvelze raRacas ivaraudeb,

e. i. Camoayalibeb egreT wodebul hipoTezas**.

dagebadeba kiTxvebi: ratom xdeba es? ra gamovaTu gavakeTeb ase? ra SeiZleba gamovides am ma-galiTidan? da ase Semdeg.magaliTis an amocanis amoxsna, movlenis Sesas-

wavlad cdis an dakvirvebis Catareba, sxva da

sxva SesaZlo variantebis ganxilva, Sedegis miReba

an imis Segneba, rom dakavebuli xar raRac Rir-

seuliT, Sen siamovnebas moganiWebs (Tu es grZnoba

ar dageufleba, maSin sjobs sxva saqmes mohkido

xeli). zogierT SemTxvevaSi Sen aRmoaCen, rom

winaswar gansazRvruli gzebidan zogierTi

miznamde ar migiyvans. aseT SemTxvevaSi unda eZe-

bo axali ideebi da mimarTulebebi. cxadia gadaw-

yvetilebas iRebs mkvlevari, e. i. Sen.

gamokvlevebisas miznis misaRwevad arsebobs

sxvadasxva xerxebi. cxadia yvelas migneba da gamoy-

eneba Znelia. sasurvelia, Tu erTi da imave prob-

lemas sxvadasxva xerxiT gadawyvitav. yvela xerxi

misaRebia, magram Sen amoarCie is, romelic ufro

mogwons. maTematikuri gamokvlevebis dros

yovelTvis asea: Sen mas SenTvis ufro moxerxe-

buli da sasiamovno mxridan Sexedav. magram

rogorc ar unda miudge, yovelTvis dagWirdeba

mondomeba, nebisyofa, warmosaxvis unari, orga-

nizebuloba da dro. magram mTavari mainc is

aris, rom gqondes amis survili da ganwyoba!

ganvixiloT umartivesi magaliTi: ra miiReba,

Tu 2347-s gamovaklebT 298-s?pasuxis miReba SeiZleba sxvadasxva xerxiT:

1. qaRaldisa da fanqris gamoyenebiT saklebsqveS miuweren maklebs:

Semdeg yvelasaTvis cnobili wesiT bolodan

TanmimdevrobiT asruleben gamoklebas:

a) 17 _ 8 = 9; b) 13_ 9 = 4; g) 2 _ 2 = 0 daiReben 2049.

2. maklebs daumateben 2-s, miiReben 300 (e.i.Seavseben mTel aseulamde) . Semdeg

2347_300=2047. magram, radgan maklebi gaz-ardes 2-iT, cxadia sxvaoba Semcirdeboda 2-iT. e.i. saZiebeli pasuxis misaRebad saWiroa

2047+2=2049.3. zogierTi moswavle (mag. laSa tabataZe, IV

kl.) cota ucnaurad asrulebs am moqmedebas,

magram sworad! ai ase:

a)

2347_ 298

nebs ZiriTadad Seiswavlis

mecniereba fizika (berZ.

Physis _ niSnavs bunebas).igi aris erT-erTi Ziri-

Tadi mecniereba bunebis

Sesaxeb. didi italieli

mecnieri galileo ga-

lilei (1564-1642) ambob-da: `maTematikuri sim-

boloebi aris is damwer-

loba, romliTac RmerTma

dawera bunebis udidesi wigni~.

gindaT SeicnoT garemomcveli samyaro,

buneba? iswavleT maTematika! da nurafers

daijerebT damtkicebis gareSe!

galileo galilei.evklide.

10 buneba, Sen da gamokvlevebi

b) 8 _ 7 = 1; g) 10 _ 1 = 9; d) 9 _ 3 = 6;e) 10 _ 6 = 4; v) 200 _ 200 = 0 da bolosiRebs pasuxs: 2049.

4. SeiZleba asec: (2000 + 300 + 40 + 7) _ (200 + 90 + 8) = 2049.

5. kalkulatoriT.cxadia nebismieri xerxi misaRebia!

magram sjobs vecadoT SevarCioT yvelaze mar-

tivi. am magaliTSi cxadia, yvelaze martivia

kalkulatoris gamoyeneba, magram winaswar

sxva xerxebis Seswavlaa saWiro!

6. sainteresoa, arsebobs kidev sxva xerxi?yovelgvari gamokvleva SemoqmedebiTi Sro-

maa. aseTi Sroma ki adamians did siamovnebas ani-

Webs... kvleviTi moRvaweobis sfero ki aris mec-

niereba, romelic gamiznulia bunebis, sazoga-

doebis da azrovnebis Sesaxeb axali codnis ga-

mosamuSaveblad.

dasmul kiTxvebs da amoxsnis process xandax-

an amoxsnis magier mivyavarT axal kiTxvebamde.

raime axlis aRmoCenisas ar damSvidde. ikiTxe

(dafiqrdi) ratom da rogor xdeba igi.

magaliTisaTvis daakvirdi da Seiswavle pi-

Tagoras (berZeni maTematikosi) cxrili _ gam-

ravlebis cxrili:

bis gadakveTa). am cxrils Sen maTematikis gakveTi-

lebidan icnob da misi gamoyeneba Zalian advi-

lia.

magram, Tu Sen SegekiTxebian: namravlebis mo-

Zebnis garda ra aris saintereso am cxrilSi?Sen SegiZlia bevri saintereso ram aRmoaCino

da dasmul kiTxvas upasuxo.

marTlac, SearCie romelime mamravli, vTqvaT

9. daakvirdi Sesabamis vertikalur svetSi anhorizontalur striqonSi daweril ricxvebs _

namravlebs (e.i. 9-ze gamravlebiT miRebul nam-ravlebs). advilad mixvdebi, rom:

a) `yoveli namravli 9-iT metia Tavis winanamravlze~,

b) Tu 9 mravldeba luw ricxvze, namravliluwia, xolo kentze _ kenti~,

g) `TiToeuli namravlis cifrTa jami to-

lia! da udris 9-s~.axla wamoiWreba sxva kiTxvebic. ratom aris

namravlis cifrTa jami yovelTvis 9-s toli?didi ricxvebis 9-ze gamravlebis drosac asea?risi tolia gamravlebis cxrilSi sxva ricxve-

bze namravlis cifrTa jami? da sxva.piTagoras cxrilSi gamoyofilia kuTxeebi~,

daStrixuli da dauStrixavi. SearCie romelime

maTgani.

Sekribe masSi Cawerili ricxvebi da miRebu-

li jami Seadare Sesabamis samravls (an mamravls)

Tavis Tavze samjer gamravlebiT miRebul nam-

ravls. magaliTad:

3+6+9+6+3=27 da 3×3×3=27,aRmoCnda toli!

ase moiqeci sxva kuTxeebis~ SemTxvevaSic. Se-

giZlia Tu ara, dainaxo raime kanonzomiereba?dagexmareba Tu ara es kanonzomiereba iwi-

naswarmetyvelo Semdegi? saidan miiReba es kanon-zomiereba?

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

3×3×3=27

5×5×5=125

cxadia es cxrili SeiZleba gagrZeldes usas-

rulod, magram yvela ricxvs xom ver Cavwer-

diT? naturalur ricxvTa simravle: 1, 2, 3, 4...usasruloa!

samravli (mamravli) ricxvebi: 1, 2, 3, ... 9, 10, ...moTavsebulia pirveli svetis ujredebSi, xolo

mamravli (samravli) ricxvebi: 1, 2, 3, ... 9, 10, ...ki _ zemodan pirveli striqonis ujredebSi.

danarCeni ricxvebi namravlebia.

magaliTad 3×7=7×3=21. amitom namravli 21am cxrilis or sxvadasxva ujraSi weria (Sesa-

bamisi horizontaluri da vertikaluri mwkrive-

piTagora (Zv.w. VI saukune).

portreti rafaelis freskidan.

11buneba, Sen da gamokvlevebi

gamokvlevaTa organgamokvlevaTa organgamokvlevaTa organgamokvlevaTa organgamokvlevaTa organizaciaizaciaizaciaizaciaizacia

gamokvlevebi, rom ufro efeqturi iyos da

rac SeiZleba meti sargebloba moitanos saWiroa

Sedegebis sistemaSi moyvana. maSin ufro advi-

lia axali kanonzomierebis SemCneva. aba, daakvirdi

9-ze gamravlebis Sesabamis svets, aq SeimCnevagarkveuli wesrigi (namravlebSi cifrebis gan-

lagebis mixedviT). es ricxvebi rom mTel fur-

celze iyos mimofantuli, usistemod (rogorc

Cvevia zogierT moswavles), maSin raRac wesrigis

danaxva da kanonzomierebis aRmoCena Zneli iqne-

boda.

bunebrivia gamokvlevebis dros dagebadeba sxva-

Tu imuSaveb gulmodgined, bejiTad, gamokv-

levebi ganaviTareben Sens warmosaxvas, inteleqts

da giCveneben rogor SeiZleba maTematikam migiy-

vanos aRmoCenamde. Seecdebi ra amoxsna amocana

an Seiswavlo movlena sxvadasxva xerxebiT, mo-

Zebni ra SenTvis sxvadasxva gzebs, Sen iswavli,

rogor unda ufro ukeT amoxsna ara marto

ra mnra mnra mnra mnra mniSvneloba aqvs gamokvlevebsiSvneloba aqvs gamokvlevebsiSvneloba aqvs gamokvlevebsiSvneloba aqvs gamokvlevebsiSvneloba aqvs gamokvlevebs

dasxva azrebi. maT Soris mniSvnelovani maSinve

unda Caiwero, imisaTvis, rom ar dakargo an ar

dagaviwydes. zogjer saWiroa suraTebis (naxaze-

bis) Caxatva, Sedegebis gaformebisaTvis cxrilebisa

da grafebis daxazva. gamoiyene SenTvis xelmi-

sawvdomi yvela xerxi. Canawerebi rom ufro mowes-

rigebuli iyos, gamoiyene ujredebiani an mili-

metrula qaRaldi. aucileblobis SemTxvevaSi

isargeble mikrokalkulatoriT. igi dagexmare-

ba swrafad miiRo zusti Sedegebi da rTuli

gamokvlevebi gaxado ufro xelmisawvdomi.

maTematikuri, aramed cxovrebis mier dasmuli

yvela amocana. ganiviTaro zusti da naTeli

azrovneba. cxadia es mxolod dasawyisia did da

rTul gzaze mogzaurebisaTvis~. yovelTvis unda

gaxsovdes kiTxvebi: ratom xdeba es? rogor moxde-boda ama Tu im pirobebSi? ra iqneboda es romase gavakeTo?

gisurvebT warmatebebs!

1 � kargad daakvirdi holandieli mxatvris m. k. esxeris suraTs `belvederi~.

aRwere da imsjele dakvirvebis Sedegebis safuZvelze.

2 � holandieli mxatvari moric kornelis

esxeri ambobs: `wisqvilis borbals var-

dnili wyali moZraobaSi moiyvans, Semdeg

or koSks Soris daxrili arxiT gaedineba

da ramdenime zigzagiT isev CanCqeris

saTavesTan miva. mxolod aucilebelia

drodadro wisqvils aorTqlebaze

dakarguli wyali davumatoT~.

Tqven rogor fiqrobT?

Seni azriT ra aris gamosaxuli am suraTze?

3 �

m. k. esxeri. `CanCqeri~. liTografia. 1961.

12 buneba, Sen da gamokvlevebi

8 � moifiqre iseTi figurebi, romlebic gamoiwveven optikur iluziebs.

cxrianis saidumloeba. dawereT nebismieri samniSna ricxvi, im pirobiT, rom pirveli dabolo cifrebi gansxvavebuli iyos. es ricxvi Sebrunebuli TanmimdevrobiT dawere da dids

gamoakeli patara. yovelTvis aRmoCndeba saintereso kanonzomiereba: sxvaobis Sua cifri iqneba

9, agreTve pirveli da bolo cifrTa jamic 9!es imas niSnavs, rom Sen sxvaobas swrafad gamoicnob, Tu gecodineba am sxvaobis pirveli an bolo

cifri.

Tu exla sxvaobas Cawer Sebrunebuli TanmimdevrobiT da am or ricxvs (sxvaobas da mis

Seqceuls) Sekribav, Sen yovelTvis miiReb 1089!cxrianis aRniSnuli `saidumloeba~ kargi gasarTobi iqneba, Tu amasTan erTad gamoiyeneb

furcelze winaswar daweril ricxvs 6801, misi SebrunebiT miiReba 1089.moifiqre gasarTobi, rom ufro saxaliso gaxdes da SesTavaze Sen amxanagebs.

cxrianis yvela `idumali~ Tviseba aixsneba im ubralo faqtiT, rom igi aris

aTobiTi sistemis bolo cifri.

romeli elifsia didi:

qveda Tu zeda SigniT?

4 �

es wrewiria?

5 �

kvadratia Tu ara?

6 �

rogor fiqrobT, aris am suraTebze kvadrati?

7 �

9 � amocanebi

1. davuSvaT, Sen da me gvaqvs erTnairi raodenobis Tanxa. ramdeni fuli unda mogce Sen me, rom Cemze 10 lariT meti gagixdes?

2. boTli Rvino Rirs 5 lari. Rvino 4 lariT Zviria boTlze. ra Rirs boTli?

3. vaWarma raRac saqoneli 7 larad iyida da 8 larad gayida. Semdeg igive saqoneli xelaxla iyida 9 larad da gayida 10 larad. ramdeni lari moigo man?

4. zoomaRaziaSi iyideboda didi da patara Tevzebi. didi orjer ufro Zviri Rirda pataraze. myidvelma iyida 5 didi da 3 patara Tevzi. Tu is amis nacvlad iyidda 3 didsa da 5 patara Tevzs, gadaixdida 2

lariT naklebs. ra Rirda TiToeuli Tevzi?

5. 10 ZaRlsa da katas SeaWames 56 orcxobila. TiToeul ZaRls Sexvda 6 orcxobila, xolo TiToeulkatas _ 5. ramdeni iyo ZaRli da ramdeni kata?

�

Tavsatexi

13

fibonaCis ricxvebi. ricxviTi mimdevrobebifibonaCis ricxvebi. ricxviTi mimdevrobebifibonaCis ricxvebi. ricxviTi mimdevrobebifibonaCis ricxvebi. ricxviTi mimdevrobebifibonaCis ricxvebi. ricxviTi mimdevrobebi

1202 wels, italiaSi gamovida naSromi maTematikaSi, saxelwodebiT wigni abakze~ (berZ. aba-kos _ saangariSo CarCo). misi avtori iyo qalaq pizaSi dabadebuli bonaCis vaJiSvili leonardo,romelsac pizelebi leonardo fibonaCis (daaxl. 1180-1240 w.), mokled fibonaCis eZaxdnen (filiusBonacci _ bonaCis vaJiSvili).am wignSi mocemulia Tavsatexi. igi dRemde iqcevs maTematikosebis yuradRebas, im SesaniSnavi

ricxviTi mimdevrobis gamo, romelic Tavsatexis pasuxis Zebnisas miiReba. saocaria, rom es mim-

devroba sruliad moulodnel situaciebSic gvxvdeba.

Seecade SeniSno kanonzomiereba, romlis Tanax-

madac xdeba kurdRlebis gamravleba!

SevTanxmdeT aseT aRniSvnebze: kurdRlebis

wyvili, romelic badebs axal wyvils, aRvniSnoT

aseTi niSniT , xolo wyvili, romelic ar

sadRac, yvela mxridan SemoRobil adgilSi viRacam erTi wyvili axaldabadebuli kurdReli

gauSva, rom gaego ramdeni wyvili kurdReli daugrovdeboda erTi wlis ganmavlobaSi. kur-

dRlebis buneba iseTia, rom erTi Tvis Semdeg wyvili kurdReli yovel TveSi aCens axal wyvils,

xolo axaldabadebuli wyvili ki _ ori Tvis Semdeg Tavisi dabadebidan. ramdeni wyvili

kurdReli dagrovdeboda Tormeti Tvis ganmavlobaSi erTi wyvili axaldabadebulisagan?

amoxsna

Tveebi

I

ianvari1

1

2

3

5

8

II

Tebervali

III

marti

IV

aprili

V

maisi

VI

ivnisi

VII

ivlisi

VIII

agvisto

IXseqtemberi

wyvilebisraodenobas q e m a

badebs _ . TiToeuli dambadebeli wyvili

erTi Tvis Semdeg aCens Tavis Tavs da axal

aradambadebel wyvils . erTi Tvis Semdeg

aradambadebeli wyvili gadadis dambade-

Tavsatexi

14 fibonaCis ricxvebi. ricxviTi mimdevrobebi

belTa TanrigSi, magram ar iZleva axal

wyvils.

exla yuradRebiT ganixile qvemoT moyvanili

sqema (ukeTesi iqneba Tu, Senc daxazav da imsjeleb

zemoT moyvanili aRniSvnebis Sesabamisad).

aba, scade gaagrZelo sqema. Tu sworad im-

sjeleb VII TveSi (ivlisSi) wyvilebis raodeno-ba aRmoCndeba 13, VIII TveSi (agvistoSi) _ 21,IX TveSi (seqtemberSi) _ 34.axla amovweroT miRebuli wyvilebis raode-

noba erT mwkrivSi: 1, 1, 2, 3, 5,8, 13, 21, 34, ...es aris fibonaCis mimdevrobis ramdenime sawyisi

wevri. am mimdevrobis gagrZeleba SeiZleba usas-

rulod. aseT ricxviT mimdevrobebs usasrulo

mimdevrobebs uwodeben (cxadia arsebobs sasru-

li mimdevrobebic).

Seecade fibonaCis mimdevroba gaagrZelo

damoukideblad. amisaTvis saWiroa rogorme miagno

is kanonzomiereba, romlis mixedviTac fibonaCim

miiRo igi. aba, daiwye mesame ricxvidan. rogor

aris is miRebuli wina ori ricxvisagan? asevedaakvirdi am mimdevrobis nebismier wevrs da mis

win mdgom or wevrs (ricxvs). kanonzomiereba

miageni? kargia! e. i. fibonaCis mimdevrobis

yoveli wevri, dawyebuli mesamedan, udris

wina ori wevris jams.

maSin Sen miiRebdi aseT gagrZelebas:

... 55, 89, 144, 233, 377, 610, ...axla Tavsatexis kiTxvaze pasuxis gacema ad-

vilia. cxadia, Tormeti Tvis ganmavlobaSi

wyvilebis raodenoba iqneba 144.maTematikidan SenTvis cnobilia, rom ricxve-

bi SeiZleba aRiniSnos asoebiT. mag. a, b, c da a.S. ricxviTi mimdevrobis wevrebi aRvniSnoT ase:

pirveli wevri a1-iT, meore _ a

2-iT, mesame _ a

3-

iT da a. S. xolo nebismieri wevri an-iT _ me- n-e

wevri, sadac n naturaluri ricxvia. maSinmiviRebT ricxviT midevrobis zogad Canawers:

a1, a

2, a

3, ... a

n, ... sadac (n=1, 2, 3, ...).

fibonaCis mimdevrobis SemTxvevaSi: a1=a

2=1,

a3=a

1+a

2=1+1=2,

a4=a

2+a

3=1+2=3,

. . . . . . . . . . . . . . a

n=a

n-2+a

n-1 , n=3,4,...

an ase: Uk+2

=Uk+1

+Uk , k=1,2,..., U

1=U

2=1.

rogor moiqcevi Tu SegekiTxebian: ras udris

fibonaCis mimdevrobis magaliTad me-100wevri? cxadia, mimdevrobas ganxiluli wesiTgaagrZelebdi, magram me-100 wevris gansazRvramdekalkulatoris gamoyenebiTac ki didi Sroma da

dro dagWirdeba. maSin rogor unda moviqceT?gamosavali, ra Tqma unda, arsebobs, magram amisaT-

vis saWiroa mimdevrobis zogadi wevris gan-

sazRvris, e. w. bines formulis:

⎪⎩

⎪⎨

⎧

⎪⎭

⎪⎬⎫

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +=nn

n 251

251

5

1a

codna. magram misi miReba (an misi WeSmarite-

bis dasabuTeba maTematikuri induqciis meTo-

diT) maTematikis ufro met codnas moiTxovs.

amis Sesaxeb qvemoT vimsjelebT.

fibonaCis mimdevrobam SemdegSi gamoyeneba hpo-

va mecnierebaSi. igi cnobilia ara marto maTe-

matikosebisaTvis, aramed bunebis mkvlevarebisTvisac.

magaliTad, Tu xe itoteba yovel wels da

mesame wels aqvs ori Sto, maSin meoTxe wels

Cveulebrivad totebis raodenoba izrdeba 3-mde, mexuTe wels _ 5-mde, meeqvse wels _ 8-mde,da a. S. fibonaCis mimdevrobis Sesabamisad (sur. 1).

SeiZleba fibonaCis mimdevrobis pirveli ori

wevri ar iyos erTianebi, aramed nebismieri ricx-

vebi. magaliTad: 4, 7, 11, 18, 29, 47, 76, 123, …aseT SemTxvevaSi mas uwodeben fibonaCis gan-

zogadobul mimdevrobas.

1 � gaamravle fibonaCis mimdevrobis TiToeuli wevri nebismier erTi da igive

ricxvze. Seamowme xom ar miiReba isev fibonaCis mimdevroba? daamtkice zogadadaRmoCenili Tviseba.

sur. 1

1

1

2

3

5

8

I

II

III

IV

V

VI

weli

15fibonaCis ricxvebi. ricxviTi mimdevrobebi

1 � dawere nebismieri fibonaCis ganzogadebuli mimdevrobis pirveli 10 wevri, Sen aTive wevrisjams swrafad miiReb, Tu bolodan meoTxe wevrs (Tavidan meSvides) 11-ze gaamravleb. (ra-tom?) Seecade daamtkico.

es WeSmariteba Sen SegiZlia gamoiyeno gasarTobad: SesTavaze vinmes zemoT ganxiluli wesiT

Seadginos ganzogadebuli fibonaCos nebismieri mimdevroba da ipovos pirveli 10 wevrisjami ise, rom Sen ar icode. sTxove, giTxras bolodan meoTxe wevri. maSin 11-ze gamravlebiTmomentalurad etyvi saZiebel jams! magaliTad:

4+7+11+18+29+47+76+123+199+322=836. 76×11=836.2 � a) daxaze nebismieri marTkuTxa ABC samkuTxedi (sur. 2) da gaatare kaTetebis paraleluri

Suaxazebi. miiReb AA1C1B1B texils, romlis sigrZe kaTetebis sigrZeTa jamis tolia (Seamowme!).analogiuri ageba Seasrule AA1C1 da BB1C1 samkuTxedebSi. miiReb AA2C2B2C1A3C3B3Btexils, romlis sigrZe kvlav kaTetebis sigrZeTa jamis tolia. gaagrZele aseTi ageba

miRebuli oTxi patara samkuTxedisaTvis da a. S. usasrulod (warmodgenaSi). cxadia, Tanda-

TanobiT yoveli axali texili mcirediT gansxvavdeba hipotenuzisagan da zRvarSi SeuTav-

sdeba mas. am texilebis sigrZe yovelTvis mudmivia da kaTetebis sigrZeTa jamis tolia.

maSasadame, AC+CB=AB! e. i. kaTetebis sigrZeTa jami hipotenuzis sigrZis tolia!

b) daxaze tolgverda samkuTxedi ABC (sur. 3) da dauSvi, rom misi gverdis sigrZe erTierTeulis tolia: AB=AC=CB=1 erT. Seasrule wina magaliTSi ganxiluli agebebi da Sengeometriulad daamtkiceb, rom AC+CB=AB! e. i. 2=1! ra aris winaaRmdegobebis mizezi?

2 � miRebuli mimdevroba dawere pirveli mimdevrobis qveS. Semdeg pirveli da meore mimdev-

robis wevrebi wevrobriv Sekribe (e. i. pirvels miumate pirveli, meores _ meore, mesames

_ mesame da a. S.). Seamowme miRebuli mimdevrobac, xom ar aris fibonaCis mimdevroba?daamtkice zogadad es Tviseba.

3 � es davalebebi Seamowme ramdenime ricxvisa da mimdevrobis SemTxvevaSi. ra SeiZleba vTqvaT

fibonaCis mimdevrobis Tvisebebis Sesaxeb?4 � daamtkice, rom: a)a)a)a)a) fibonaCis ganzogadebuli mimdevrobis pirveli n wevris jami

ganisazRvreba formuliT: Sn = an+2 _ a2; b)b)b)b)b) miiRe analogiuri formula fibonaCis mimdevrobisaTvis.

5 � daamtkice, rom fibonaCis mimdevrobis pirveli n wevris kvadratebis jami udris an. an+1.

miTiTeba: 2

1111 )( kkkkkkkk aaaaaaaa =−=⋅−⋅ −+−+ .

6 � daamtkice, rom mocemuli P = 4a perimetris mqone yvela marTkuTxedebidan udidesifarTobi aqvs kvadrats.

Tavsatexi

sur. 2

BCsur. 3

�

A

A1

A

A2

B1BB3

B2

C2

C1

C3A3

C

16

oqros kveTa. fidias ricxvebioqros kveTa. fidias ricxvebioqros kveTa. fidias ricxvebioqros kveTa. fidias ricxvebioqros kveTa. fidias ricxvebi

fidia (Pheidias) iyo Zveli welTaRricx-

vis V s. berZeni moqandake, maRali klasis war-

momadgeneli, Zveli berZnuli xelovnebis erT-

erTi udidesi ostati. daiba-

da aTenSi daaxloebiT Zv. w.

490 mxatvris ojaxSi. jer

swavlobda moqandake gegiasTan,

xolo Semdeg saxelganTqmul

skulptorTan ageladTan, rom-

lis saxelosnoSic Cawvda

brinjaos Camosxmis xelov-

nebas. misi qmnilebebidan gan-

sakuTrebiT popularulia aT-

ena promaqosis (aTena _ me-

brZolis) brinjaos Svid metriani kolosa-

luri figura, romelic sparselebze gamar-

jvebis aRsaniSnavad Zv. w. 460 aRmarTes aTe-nis akropolisze sparselebis

mier ganadgurebuli taZrebis

nangrevebs Soris (akropolisi

_ berZ. akropolis _ antikuriberZnuli qalaqis gamagrebuli

nawili, ganlagebulia ZiriTadad

amaRlebul adgilze). e. w. zeda

qalaqi, romelic warmoadgenda

cixesimagres. akropolisi _

qarTulad Sidacixe, zedacixe,

zedaqalaqi _ hqondaT Zvel qar-

Zveli Taobis maTematikosebs kalkulatori ar hqoniaT. amitom maT didi Sroma uxdebodaT

angariSTan dakavSirebiT. Sen ki SegiZlia gamoiyeno kalkulatori da ipovo fibonaCis mimdevrobis

ori mezobeli ricxvis Sefardeba:

1:1=1; 2:1=2; 3:2=1,5; 5:3≈1,666 ...; 8:5=1,6; 13:8=1,625; 21:13≈1,615384 ...Tu ase gaagrZeleb, Sen zRvarSi miuaxlovdebi usasrulo aTwilads 1,618034 ..., romelsac ΦΦΦΦΦ

asoTi aRniSnaven: ΦΦΦΦΦ≈1,618034 ... ≈ 1,618. mas fidias ricxvs~ an oqros ricxvs~ uwodeben.

albaT bunebrivad dagebadeba kiTxva: fidia vin aris?

cxenis Tavi parTenonis

aRmosavleTis frontonidan

parTeEnoni

17oqros kveTa. fidias ricxvebi

Tul qalaqebsac: mcxeTas, Tbiliss, quTaiss,

arqeopoliss, anakofiss, vans da a. S. berZe-

ni istorikosi dion kasios kokeiani (155an 164 - 229) Tavis `romis istoriaSi~

akropolisad ixsenebs mcxeTis Sidacixes

(zedacixes), igi mogviTxrobs, rom Zv. w. 65romaelTa sardali pompeusi iberiaSi Sei-

Wra da miaRwia akropolisamde. mcxeTis am

nawils armazcixe ewodeba).

ZvelberZnuli xuroTmoZRvrebis isto-

riaSi gansakuTrebuli adgili uWiravs aTe-

nis akropolisze nagebobaTa kompleqss. maT

SeqmnaSi wamyvani roli ekuTvniT gamoCenil

saxelmwifo moRvawes, brwyinvale orator-

s, Zveli elitis mxedarTmTavars periklesa

da udides moqandakes fidias, romelic Teq-

vsmeti weli emsaxura akropolisis mSeneb-

lobas. aTenis akropolisis mTavari nageboba

aris parTenoni (berZ. Parthenõn<Parthenos_ qalwuli), qalRmerTi aTena _ qalwulis

TeTri marmariloTi nagebi taZari, romlis

zomebia: 30,89m ××××× 69,54m; simaRle 10,43 m;svetebis raodenoba 8 ×××××17. igi warmoadgensZveli berZnuli xelovnebis, msoflioSi erT-

erT yvelaze cnobil nagebobas (Zv. w. 447-438, xuroTmoZRvrebi iqtine da kalikrate,

skulpturuli morTulobis xelmZRvaneli

fidia). Zvel berZnebs swamdaT, rom taZris

Senoba warmoadgens RvTaebis saxls, es ki moiTxovda Caketil Senobas, centrSi ZiriTadi

sakulto qandakebiT. amitom mis centrSi fidiam dadga Tormeti metri simaRlis qrizoele-

fantiniT (xeze gadakruli spilos Zvlis da oqros furclebi) Seqmnili aTena parTenonis

qandakeba. Zveli mwerlebis gadmocemiT masze daixarja didi raode-

nobiT spilos eSvebi da TiTqmis 1160 kg oqro (q. S. IV s-Siqandakeba waiRes konstantinopolSi, imperatoris sasaxleSi, sadac

xanZris gamo ganadgurda).

mdinaris RmerTi ilisi parTenonis dasavleTis frontonidan*

* fronton-i (frang. fronton) _ (arqit.). Senobis fasadis zemoTa, Cveulebriv samkuTxa nawili, romelic Semo-sazRvrulia ormxriv daqanebuli saxuraviTa da karniziT.

2. j. kiknaZe

`aTena promaqosi~(rekonstruqcia)

fidia. `olimpieli zevsi~. romauli asli,

inaxeba saxelmwifo ermitaJSi.

18 oqros kveTa. fidias ricxvebi

parTenonis fasadis frizi* Semkobili iyo

metopebiT**, romlebzec gamosaxuli iyo ber-

Znuli miTologiis Sesabamisi gamosaxulebebi:

RmerTebis brZola titanebTan, gmirebis _ ama-

zonkebTan***, lapiTebis _ kentavrebTan, berZ-

nebis mier troas dangreva da sxva uamravi

qandakebebi. isini gamoxataven sinaTlis, sikeTis

da civilizaciis brZolas simxecisa da Camor-

Cenilobisadmi. es miTebi alegoriulad gad-

mocemdnen berZnebis brZolas da gamarjvebas

sparselebze. drom ar daindo parTenonis kazmu-

loba. jer kidev Sua saukuneebis gariJraJze

taZari gaxda qristianuli eklesia da am dros

ganadgurda parTenonis bevri qandakeba. 1460wels Turqebma daipyres saberZneTi da par-

Tenoni gadaakeTes meCeTad. 1687 wels, Tur-

qeTTan omis dros venecieli meomrebis gasro-

lilma Wurvma aafeTqa parTenonSi Turqebis

denTis sawyobi. afeTqebis gamo daingra par-

Tenoni da ganadgurda xelovnebis bevri nimuSi.

1801 wels britaneTis elCma TurqeTSi

lord jeims brius elginma sulTanisagan mi-

iRoO nebarTva, antikuri qandakebebisa da aTenis

taZrebidan reliefebis inglisSi gadatanis Se-

saxeb. amotexes TiTqmis yvela figura, rac

iyo darCenili parTenonis frontonebze da

fasdaudebeli marmarilos xelovnebis nimuSebi

gemebiT inglisSi gadaitanes. didi xnis dis-

kusiis Semdeg isini 1816 wels parlamentma

SeiZina britaneTis muzeumisaTvis. amJamad iq

inaxeba 15 metopi, 17 Zlier dazianebuli

qandakeba da 75,3 m sigrZis frizi.gansakuTrebiT didi aRiareba moutana fidias

`RmerTebisa da adamianebis mamis~ zevsis qanda-

kebam, romelic idga olimpiaSi, zevsis taZarSi

(igi erT-erTia `msoflios Svidi saocre-

bidan~). welamde SiSveli figura damzadebu-

li iyo spilos ZvliT dafaruli xisagan, tan-

* friz-i (frang. frise) _ 1. (arqit.). antablementis Sua nawili (arqitravasa da karnizs Soris); kozmidi. 2. arSia,bordiuri kedlisa, xaliCisa, parketisa.

** metop-i (berZ. metõpon are Tvalebs Soris) _ (arqit.). Sualedebi triglifebs Soris frizze, amovsebuli filiT,Cveulebrivad Semkuli reliefuri gamosaxulebebiT. antablement-i (frang. entablement) _ (arqit.). Senobis zedahorizontaluri nawili, romelic kolonebs eyrdnoba da Sedgeba karnizis, frizisa da arqitravisagan. arqitrav-i

(frang. architrave) _ (arqit.). mTavari koWi, antablementis qveda nawili. triglif-i (berZ. triglyphos sami WriliT) _(arqit.). marTkuTxovani qvebi, romlebsac sami vertikaluri Rari aqvs amokveTili.

*** amazon-i (berZ. amazõn amorZali _ miTiuri meomari qali) _ (moZv.). 1. cxenosani qali. 2. qalis sacxenosno kaba.

sami RmerTi parTenonis aRmosavleTis frontonidan

19oqros kveTa. fidias ricxvebi

sacmeli ki _ oqros

furclebisagan (qri-

zoelefantiniT Sesru-

lebuli). cameti met-

ri simaRlis taxtze

mjdomare zevss mar-

jvena xelSi eWira ga-

marjvebis simbolos

qalRmerT nikes figura,

xolo marcxenaSi Zala-

uflebis simbolo _

skiptra Zvirfasi li-

Tonisagan naxelovnebi,

romelzec ijda arwi-

vi. q. S. V s-Si olim-

pieli zevsis qandakeba

gaanadgures qristianu-

li religiis fana-

tikosma mimdevrebma,

rogorc warmarTobis

kerpi (sanimuSod warmodgenilia fidias ramde-

nime naqandakebi).

saocaria, rom umadurma Tanamemamuleebma ar

daindes udidesi moqandake da misi sicocxle

tragikulad dasrulda. perikles politikurma

mowinaaRmdegeebma, misi uaxloesi TanamoRvawe-

ebisa da maT Soris fidias winaaRmdeg wamoi-

wyes rigi sasamarTlo procesi. aTena parTe-

nonis qandakebaze muSa-

obis dros fidias oq-

ros moparva daabra-

les da daapatimres.

igi Zv. w. 431 cixeSigardaicvala!

parTenonis fasadi

kargad jdeba im mar-

TkuTxedSi, romlis

gverdebic miRebulia

e gr eT wode buli

`oqros kveTis~ mixed-

viT. am marTkuTxedis

sigrZe mis siganeze

metia miaxloebiT

1,618-jer _ fidias

ricxvjer. aseTi wesiT

agebul marTkuTxedebs

`oqros marTkuTxede-

bi~ ewodeba. cxrilSi mocemulia zogierTi

maTganis zomebi metrebSi.

modiT ufro dawvrilebiT ganvixiloT oqros

kveTa~. xuTqimiani varskvlavi Tavisi srulyo-

filebiT gamudmebiT iqcevda adamianTa yura-

dRebas. piTagoras (Zv. w. VI s.) skolis moswav-leebma _ piTagorelebma swored es varskvlavi

TavianTi kavSiris simbolod airCies. maTTvis is

kentavris brZola lapiTTan

parTenonis samxreTi metopidan

Wabuki mxedrebi parTenonis dasavleTis frizidan

20 oqros kveTa. fidias ricxvebi

janmrTelobis amuletad iTvleboda (sagani,

romelsac atareben tanze). xuTqimiani varskv-

lavi amJamad mravali

qveynis gerbebze da

droSebzea gamosaxu-

li. ratom aris igi

mimzidveli? daak-

virdi sur. 1-s, Zneli dasajerebelia ma-

gram dam tkicebulia

monak ve Tebis Sefarde-

baTa to loba:

AD:AC=AC:CD=AB:BC=AD:AE=AE:EC=... (1)

isargeble varskvlavis simetriulobiT da gaagrZele monakveTebis SefardebaTa tolobis

mwkrivi.

albaT gainteresebs, ras udris ricxobrivad

es Sefardeba? amis miReba SeiZleba, magram saWiroa kvadratuli gantolebis amoxsnis

codna (Tu Sen es jer ar ici, SegiZlia gamo-

tovo). AD aRniSne a-Ti, AC ki _ b-Ti, e. i. AD≡a; AC≡b da CD = a-b (naxazidan). maSin (1) toloba ase Caiwereba:

a : b = b : (a - b). (2)es toloba warmoadgens proporcias. gamoi-

yene proporciis ZiriTadi Tviseba _ kidura

wevrebis namravli udris Siga wevrebis nam-

ravls da Sen miiReb:

a2 _ ab _ b2 = 0. (3)

am tolobis orive mxare gayavi b2-ze da

aRniSne Φ-Ti. e. i. Φ≡ , maSin miiReb

kvadratul gantolebas:

Φ2 _ Φ _ 1 = 0. (∗)

am gantolebis fesvebia:

cxadia, meore fesvi uaryofiTia. is Cven ar

gvainteresebs. amitom miviRebT:

(4)

xSirad ganixilaven patara monakveTis (b) Sefardebas did monakveTTan (a):

da aRniSnaven ϕ asoTi.

(5)

(Φ da ϕ aris berZnuli fi~ asos beWdviTi

da xelnaweris formebi. igi miRebulia fidias

pativsacemad).

kalkulatoris gamoyenebiT Sen advilad

Seamowmeb, rom Φ ≈ 1,618034... da ϕ ≈ 0,618034... miiReba saintereso da saocari Sedegi: es

ricxvebi usasrulo aTwiladebia (algebruli

ricxvebia), gansxvavdebian mxolod mTeli

nawiliT, danarCeni ricxvebi ki erTmaneTs

emTxveva. es faqti gamomdinareobs TviTon

kvadratuli gantolebidan. marTlac gar-

davqmnaT igi (∗) Φ-ze gayofiT da ase dava-lagoT: Φ − 1 = 1/Φ.am tolobis marjvena mxare ki tolia ϕ-si.

maSasadame Φ − ϕ = 1.(*) gantolebidan Φ2 = 1 + Φ, aqedan 1Φ = + Φ

. am tolobis fesvis SigniT Φ SevcvaloT misi

toli 1+ Φ -iT, maSin miviRebT: 1 1Φ = + + Φ

gantolebas. Tu wina proceduras kidev da

kidev gavimeorebT usasrulobamde, miviRebT

fidias ricxvisaTvis lamaz formulas:

(**)

Φ ricxvisaTvis SeiZleba miviRoT kidev erTi lamazi formula:

(***)

es formula miiReba Tu gardaqmnil gan-

tolebaSi:

Φ = 1 + 1/Φ, mniSvnelSi Φ-s SevcvliT misi toli gamosaxulebiT:

da a.S. usasrulobamde.

rogor gavyoT monakveTi Φ SefardebiT? far-

siganesigane sigrZesigrZe

123...

1,6182 × 1,618 = 3,2363 × 1,618 = 4,854

.

.

.

sur. 1

Φ = ++

Φ

11 11

.Φ = ++

+

11 111 ...

±=

1 51 5Φ

2

+=

1 51 5Φ

2

1ba

=Φ

1 5 12

ϕ −= =

Φ

ù1 1 1 1Φ = + + + +

21oqros kveTa. fidias ricxvebi

gliTa da saxazaviT aseTi gayofis wesi mocemulia

evklides sawyisebSi~, romelic dawerilia Cven

welTaRricxvamde 300 wlis win. es wesi aseunda gamoiyeno: aiRe

sibrtyeze raime ABmonakveTi (sur. 2). gaa-tare B wertilSi AB-s marTobi BC monakve-Ti ise, rom BC = AB/2. Semdeg SeaerTe Ada C wertilebi monak-

veTiT. C wertili miiRe centrad da SemoxazeCB radiusiT wrewiris rkali AC-s gada-kveTamde N wertilSi. Semdeg A wertili mi-iRe centrad da Semoxaze AN radiusiT NMrkali. M wertili AB monakveTs gayofs ΦΦΦΦΦ-is toli SefardebiT:

Φ Φ Φ Φ Φ ===== AM : MB = AB : AM.mravali mkvlevaris azriT, proporcia, romlis

Sefardebebi tolia ΦΦΦΦΦ ricxvis, yvelaze ufrosasiamovnoa adamianis TvalisaTvis.

didi italieli mxatvari leonardo da vin-

Ci Tvlida, rom adamianis sxeulis idealuri

proporciebi dakavSirebuli iyo ΦΦΦΦΦ ricxvTan.am Sefardebas man `RvTaebrivi proporcia~*

uwoda. termini `oqros kveTa~ xmarebaSi XIXs-Si Semovida. radgan ϕ ϕ ϕ ϕ ϕ ===== 1/Φ1/Φ1/Φ1/Φ1/Φ ≈ 0,618.0,618.0,618.0,618.0,618.amitom

ϕϕϕϕϕ = MB : AM = AM : AB da AM ≈ 0,618 . AB,maSin cxadia MB ≈ 0,372 . AB.amrigad, `oqros kveTis~ nawilebi miaxloe-

biT mTelis 0,618-sa da 0,372-is tolia.aRorZinebis xanaSi `oqros kveTa~ Zalian

popularuli iyo. mas xSirad iyenebdnen mx-

atvrebi, moqandakeebi da arqiteqtorebi. oqros

marTkuTxedis~ formas aZlevdnen suraTebs,

wignebs, magidebs, safosto baraTebs da sxva.

SemdegSi wignebsa da sxva qaRaldis nawarms

misces marTkuTxedis

forma, romlis gver-

debis Sefardeba udris

2 . es imasTanaa dakav-Sirebuli, rom misi

Suaze gakecvisas miiReba

ori marTkuTxedi gver-

debis imave fardobiT

(sur. 3).am azriT oqros marTkuTxedsac~ aqvs sain-

tereso Tviseba. Tu misgan movWriT kvadrats,

maSin darCeba isev `oqros marTkuTxedi~. es

procesi SeiZleba

usasrulod gavagr-

ZeloT! Tu gava-

tarebT pirveli da

meore marTkuTxe-

dis diagonalebs,

maSin maTi gadakve-

Tis wertili ekuTv-

nis yvela SesaZlo

`oqros m arT -

k u T x e d e b s ~

(Seamowme damoukideblad).

arsebobs `oqros samkuTxedic~ (sur. 1) su-raTze GEF an AEF. cxadia es aris tolferdasamkuTxedi, romlis ferdisa da fuZis sigrZee-

bis Sefardeba ΦΦΦΦΦ-is tolia. agreTve arsebobs`oqros kuboidi~. igi warmoadgens marTkuTxa

paralelepipeds, romlis wiboebis sigrZeebia:

ϕ, 1, Φ ϕ, 1, Φ ϕ, 1, Φ ϕ, 1, Φ ϕ, 1, Φ (sur. 4).Zveli drois mrvali maTematikosi cdilob-

da, moeZebna kavSiri πππππ da ΦΦΦΦΦ ricxvebs Soris,magram maTi cda marcxiT damTavrda. da ici

ratom? es ricxvebi sxvadasxva bunebisaa: ΦΦΦΦΦ _algebruli ricxvia, xolo πππππ _ transcenden-tuli**. magram miaxloebiTi Tanafardoba maT

Soris moZebnes:

πππππ ≈ 1,21,21,21,21,2 ΦΦΦΦΦ22222.....

� `oqros marTkuTxedis ~ ageba

fibonaCis ricxvebis gamoyenebiT.

fibonaCis ricxvebis gamoyenebiT Sen Segi-

Zlia furcelze aago `oqros marTkuTxedi~.

pirvelad SearCie erTi patara kvadrati da

Semoxaze igi, daumate meore aseTive kvadrati.

aa

a√2

aage grZel gverdze kidev erTi kvadrati,

Semdeg analogiurad kidev da kidev (sur. 5).

sur. 2

sur. 3

sur. 4

sur. 5

* 1509 wels leonardo da vinCis ilustraciebiT italiaSi gamoica luka paColis traqtati saxelwodebiT DeDivina Proportione~ (`RvTaebrivi proporcia~).** transcendentuli (laT. transcendentis) rac zRvars scildeba), (maTem). iracionaluri, rasac algebrulad ver

gamoiangariSeben, rac algebrulad ar gamoisaxeba.

2a

a a

22 oqros kveTa. fidias ricxvebi

rac ufro didxans gaagrZeleb am process, miT

ufro miuaxlovdebi oqros marTkuTxeds~. aba,

daakvirdi ra SuaSia aq fibonaCis ricxvebi?� .`oqros marTkuTxedis~ ageba

fargliTa da saxazaviT. misi

daSla kvadrtebad.

1. aage kvadrati da gayavi or tol marT-kuTxedad (1).

2. gaatare AB diagonali (2).3. A wertili gamoiyene centrad da AB

radiusiT Semoxaze wrewiris rkali (3).

4. gaagrZele fuZe rkalis gadakveTamde.5. aRmarTe misadmi marTi kuTxiT monakveTi

kvadratis zeda gverdis gagrZelebis gada-

kveTamde (3). miiReb CDEF `oqros mar-TkuTxeds~ (4).

6. oqros marTkuTxedi daSale kvadratebad

ise rogorc suraTzea (5). daakvirdi kvad-ratebis wveroebi razea ganlagebuli?oqros marTkuTxedis aseTi daSla war-

moadgens mxolod erTianebisagan Sedgeni-

li jaWvuri wiladis (***) geometriul

saxes _ interpretacias*.1 2

3 4

1 � ganixile wesieri aTkuTxedi, romlis gverdis sigrZea 1. misi centriSeaerTe aTkuTxedis or mezobel kuTxis wveroebTan da miiReb `oqros

samkuTxeds~ _ daamtkice.

2 � SenTvis cnobilia, rom wrewiris sigrZe tolia 2π2π2π2π2πR-is. gansazRvre wesier aTkuTxedzeSemoxazuli wrewiris radiusi da am wrewiris sigrZe.

3 � gamoiyene perimetris cneba da dauSvi, rom wesieri aTkuTxedis perimetri da Semoxazuli

wrewiris sigrZe daaxloebiT tolia, maSin Sen miiReb miaxloebiT kavSirs πππππ da ΦΦΦΦΦricxvebs Soris. rogoria es kavSiri? da aqedan gamomdinare ras udris πππππ?

4 � ras udris `oqros kuboidze~ Semoxazuli sferos zedapirisa da oqros kuboidis

sruli zedapiris farTobis Sefardeba?

5 � ras udris oqros kuboidis~ diagonalis sigrZe? misi sruli zedapiris farTobi?

6 � risi toli iqneba oqros kuboidze~ Semoxazuli sferos radiusi?

7 � cnobilia, rom sferos zedapiris farTobi tolia 4π4π4π4π4πR2-is, (sadac πππππ ≈ 3,14...3,14...3,14...3,14...3,14... _ wrewiris

sigrZis Sefardeba mis radiusTan araperioduli aTwiladia _ transcendentuli ricx-

via). ras udris oqros kuboidze~ Semoxazuli sferos zedapiris farTobi?

8 � daamtkice: a) ...2222 −+−+=Φ . b) .

...21

2

12

12

+−

+−=Φ

milionerma erT-erT bankSi gaxsna angariSi da Seitana gansazRvruli x Tanxa. meored ki _ yTanxa, ise rom orive Setanili Tanxa warmoadgenda dolarebis mTel ricxvs. meanabris mier yoveli

momdevno Senatani Tanxa warmoadgenda wina ori Setanili Tanxis jams. rigiT meoce Setanili Tanxa

Seadgenda zustad milion dolars. Tavdapirvelad ramdeni x da y dolari Seitana meanabrem bankSi?

9 � amocana

5

* interpretacia (laT. interpretatio) _ risame azris axsna, ganmarteba.

�

23

egviptis paramidebi da oqros kveTaegviptis paramidebi da oqros kveTaegviptis paramidebi da oqros kveTaegviptis paramidebi da oqros kveTaegviptis paramidebi da oqros kveTa

sarkofagidan* dgeba mumia.** ra siCumea. haeri lurji abreSumia . . .

da saukuneT rigs Tvlis mumia: mziani dRea, Tu samumia.***

galaktioni

daaxloebiT saukunenaxevris win qristes

Sobamde saxelganTqmulma berZenma maTematikosma

filon bizantielma dawera mcire traqtati

`samyaros Svidi saocreba~. am patara rito-

rikuli stiliT daweril TxzulebaSi man mok-

led agviwera Tavis mier SerCeuli samyaros

Svidi namdvili saocreba _ antikuri saoc-

rebebi:

didi piramida _ xufus (xeofsis) gizaSi,

aleqsandriis Suqura,

babilonis dakidebuli baRebi,

rodosis kolosi,****

mauzolusis akldama halikarnasSi,

artemides taZari efosSi,

zevsis qandakeba olimpoSi.

Suqurisa da dakidebuli baRebis gamoklebiT,

yvela es saocreba religiuri Zeglebi, anu

RvTismosaobis qmnilebebia (magaliTad, rodo-

sis kolosi mzis RmerTis, heliosis qandakebaa).

amaTgan, Tu ar gaviTvaliwinebT mcire nangrevebs

mauzolusis akldamisas (aqedanaa sityva mav-

zoleumi~) da artemides taZrisas efosSi,

SemorCenilia mxolod erTi _ xeofsis pira-

mida. es erTaderTi SemorCenili saocreba, amave

dros saocrebaTa Soris uZvelesicaa. igi sxveb-

ze ori aTasi wliT adre iyo aRmarTuli,

amasTan, sayovelTao aRiarebiT, yvelaze metad

gansacvifrebeli, Tumca ara mxolod Tavis siZ-

veliT. klasikur avtorebs, herodotedan moyo-

lebiT, vinc moinaxula piramidebis zegani 4

sur. 1. piramidebis kompleqsi gizaSi.

marjvniv, Sors xeofsis piramidaa, SuaSi _ xefrenis, marcxniv _ mikeranis.

* sarkofagi (berZn. sarkophagos) _ Zvel qveynebSi, kerZod, egvipteSi: qvis kubo; saflavis qva kubos formisa; pataraakldama.

** mumia (arab. mumija) _ gaxrwnisagan dabalzamebiT (an bunebrivad) daculi adamianis gvami (an cxovelis leSi);gvamebis mumiebad qceva gansakuTrebiT gavrcelebuli iyo Zvel egvipteSi.

*** samumi (samumma) _ (arab. samum) _ mSrali, cxeli qari arabeTisa da CrdiloeT afrikis udabnoebSi, romelsacgadaaqvs silisa da mtvris Rrublebi.

**** kolosi (berZ. kolossos) _ uzarmazari qandakeba.

~

~

24 egviptis piramidebi da oqros kveTa

saukunis ganmavlobaSi qristemde, ar gaaCndaT

garkveuli codna siZvelis Sesaxeb, Tumca

icodnen, rom igi Zalze Zveli iyo. maTze STabeW-

dilebas axdenda misi uzarmazari zoma _ qvis

mTa, kacis xeliT nagebi da saocari xelovneba

konstruqciisa.

xufus (xeofsis) piramida, xafras (xefrenis)

da menkauras (mikeranis) piramidebTan erTad

qmnis Zveli egviptis nekropolis* kompleqss,

romelic Zveli samefos epoqaSi (Zv. w. XXVIII-XXII ss. III-IV dinastiebi) gizaSi iqna agebuli.

(sur. 1) am ansamblis Seqmnis dros, adgilis

SerCevidan dawyebuli yovelgvari wvrilmani

didi sizustiT iqna gaTvaliswinebuli. igi

nilosis dasavleT sanapiroze CrdiloeTis

ganedis 300 paralelzea ganlagebuli, sarwyavi

miwebisa da libiis mkvdari udabnos qvian grun-

tian sazRvarze. Zveli egviptelebis rwmeniT,

Camavali mzis mxare sikvdilis sasufeveli

iyo. snofrus (IV dinastiis, daaxl. Zv. w.

2613-2498 pirveli faraoni) memkvidreebma Tavi-

anTi sasufevlis adgilad dRevandeli giza

airCies, radganac es adgili akmayofilebda

geodeziur, astronomiul Tu sakulto xasiaTis

yvela moTxovnas.

micvalebulis maradiuli saxlis~ _ pira-

midis konstruqcia religiuri warmodgenebis

Sesabamisi unda yofiliyo. samive piramida sam-

yaros mxareebis mimarT sakmaod zustad aris

orientirebuli _ maTi samxreT-aRmosavleTis

kuTxeebi praqtikulad erT wrfeze mdebareobs.

xeofsisa da xefrenis piramidebis cenrtebi

Zevs erT wrfeze (xeofsis piramidis fuZis

diagonalis Tanxvdenil wrfeze), romelic

sur. 2. piramidebis ansamblis agebis

sqema gizaSi.

sur. 3. xeofsis piramidis samxreTidan CrdiloeTisaken Wrili.

3. mdinare nilosis maqsimaluri done;

4. mdinare nilosis minimaluri done

0 10 20 30 40 50 m

* nekropolisi (berZ. nekropolis) _ Zveli aRmosavleTis qveynebsa da antikur samyaroSi: sasaflao, samarxi.

a varskvl

avisaken

A O F D O′ B

12

3

5

a d

5

25egviptis piramidebi da oqros kveTa

momcro piramidis (mikeranis) Crdilo-dasavleT

kuTxis wveroze gadis (sur. 2). Sesasvleli

moTavsebulia CrdiloeTis mxridan, saidanac

Casvla SeiZleboda miwisqveSa senakSi ise, rom

masSi SeRweuliyo gveleSapis Tanavarskvla-

vedis alfa varskvlavis sinaTlis sxivi (sur. 3),

piramidebis mSeneblobis epoqaSi es varskvlavi

iTvleboda samyaros polusad.

konstruqciis gaangariSebis dros saWiro

iyo saaRmSeneblo wesebis mTeli kompleqsis

gaTvaliswineba: piramidis fuZis kvadratis

farTobis gamoTvla, gruntis SesaZleblobebis

gaangariSeba, rom gaeZlo uzarmazari qvis mTis

wnevisaTvis. amis Sesabamisad iqna dadgenili

fuZesTan gverdebis daxris kuTxe: xeofsis _

51052′; xefrenis _ 52020′; mikeranis _ 510.

piramidis wiboebi wverosTan gadaikevTebian

63026′ kuTxiT, romelic miaxloebiT Seesabameba

fxvieri sxeulebis _ mocemul SemTxvevaSi

silis (RorRis) mTis mier Seqmnil kuTxes.

varaudoben, rom xeofsis piramida aago

xuroTmoZRvarma xemiunma (faraonis naTesavma)

daaxloebiT Zv. w. 2 600. mis agebaze daixarjadaaxloebiT 2 590 000 m 3 granitis qva.

(2 300 000 qvis blokebi da filebi; saSualod

TiToeuli qvis blokis moculoba 1m 3-istolia, masiT 2,5 tona; zogierTi blokis masa

50 tonaa, sul saWiro iyo daaxloebiT

6 000 000 tona masis qva _ e. i. 600 000 vagoni.granits ezidebodnen axlandeli asuanis

midamoebidan, romelic daaxloebiT 1000 km

manZiliTaa daSorebuli piramididan), misi

Tavdapirveli simaRle 146,59 m (amJamad 137 m)

iyo, gverdis sigrZe _ 232,05 m (amJamad 230 m).

piramida agebulia duRabis gareSe. blokebis

gadasatanad mxolod Tokebsa da berketebs

iyenebdnen, radgan sxva saSualebebi maSin ar

arsebobda. mopirkeTebulia gasaocari sizustiT

erTmaneTze morgebuli kirqvis samwaxnagovani

filebiT. drom Secvala piramidebis garegnuli

saxe, mxolod xefrenis piramidis sul zeda

nawilzea SemorCenili mzis mcxunvale mxares

sxivebiTYgaxunebuli mopirkeTebis niSnebi.

gizas piramidebs klasikurs uwodeben. egvip-

tis Zveli samefos arqiteqturaSi klasikuri

piramidis forma erTbaSad ar Camoyalibebula.

mas win uswrebda grZelvadiani kvlevebi. pirami-

debis fuZemdeblad iTvleba III dinastiis fa-raonis joseris safexurebiani, erT-erTi

pirveli piramida (daaxl. Zv. w. 2700, simaRle

60 m, mis aSenebas ukavSireben xuroTmoZRvar

imhotempis saxels). IV dinastiis pirveli

faraonis snofris akldamebi medumSi da

daSurSi warmoadgenen gardamaval etaps gizis

kompleqsis klasikuri piramidebis Seqmnis

gzaze. piramidebis formebis Camoyalibebis

procesSi xuroTmoZRvrebma upiratesoba fuZeSi

kvadrats mianiWes, vidre marTkuTxeds (joseris

safexurebiani piramida agebulia marTkuT-

xedovan fuZeze). qvis blokebis damuSavebisa

da dawyobis (duRabis gareSe) teqnikis damu-

Savebam mSeneblebs saSualeba misca nageboba

ufro swori geometriuli formis aegoT. maT

konstruqciebSi miRweulia srulyofis zRvari

_ mwyobri proporciebi mTeli moculobis

masas awonasworeben. nagebobas uryev sidiades

aZleven. egviptelebma erT-erTi yvelaze mar-

tivi da mkacri geometriuli figuris far-

glebSi ulamazes maTematikur amoxsnas miagnes,

romlis safuZvelzec gizas mTeli ansambli

(piramidebi, taZrebi da faraon xefrenis didi

sfinqsi) oqros kveTis proporciebzea age-

buli (sur. 2). konstruqciis arcerTi elementi,

qvis blokebis zomebidan dawyebuli, Siga saTav-

soebisa da faraonis dasakrZalavi sakaniT dam-

Tavrebuli zogadi TanaSezomvidan ar gamo-

dis. sur. 2-ze naCvenebiaBmonakveTis dayofa

oqros kveTis Sesabamisad:

AB/AO′′′′′ = AO′′′′′/ O′′′′′B = ΦΦΦΦΦ = 1,618... aseveAB/BF = BF/ AF = ΦΦΦΦΦ = 1,618.. sadac

AD = DB = 1 erTeuls.

amave suraTze naCvenebia: kvadratis diagonali

2 , samkuTxedis hipotenuzebi 3 da 5 .

gizaSi piramidebis ansamblis sqemaze am propor-

ciebis gamoyenebaa naCvenebi.

Zveli egvipturi Zeglebis Seswavlisa da

analizis safuZvelze aRmoCnda, rom yvela isini

geometriuladaa agebuli. Tanamedrove arqiteq-

torebma, mxatvrebma gazomes ra mTeli rigi

Zeglebi, daadgines, rom praqtikulad nebismieri

Zveli egvipturi kompoziciebi mkacr maTe-

matikur sqemaSi Tavsdeba. amasTan geometriuli

figurebi, romlebSic Caixazeba kompoziciis

TiToeuli elementi, ar gamodis umartivesi _

samkuTxedis, kvadratis, wrewirisa da sxva

nakvTebis sazRvrebidan. Zveli egviptis xelov-

nebis harmoniuli erTianoba haromiis algeb-

ridan~ gamomdinareobs, sadac konstruqciuli

da mxatvrul-esTetikuri azrovneba urRvev

erTianobas warmoadgenen.

radgan Cvenamde ar mouRwevia sistemur

Teoriul debulebebs, savaraudoa, rom isini

26 egviptis piramidebi da oqros kveTa

TavianT SemoqmedebaSi albaT praqtikuli Sede-

gebis ganuxreli logikiT midiodnen. esaa gza

dakvirvebebisa da Sedegebis. Catarebulma ana-

lizma aCvena, rom Zveli egviptis saxviTi

xelovnebis Zeglebi, gamonaklisis gareSe, pro-

porciebis struqturis Sesabamisadaa agebuli.

igi warmoadgens rva sididisagan Sedgenil

skalas, romelic miRebulia mecxredan _

sawyisidan, romelic mTeli kompoziciis maqsi-

mums Seesabameba. piramidebisaTvis aseT sidided

misi fuZis kvadratis gverdis sigrZea miRebuli,

qandakebisaTvis _ simaRle, reliefis kompo-

ziciisaTvis an moxatvisaTvis _ mTeli velis

maqsimaluri zoma, romelSic scena horizon-

talurad Tu vertikaluradaa moxatuli.

piramidebis konstruqcias safuZvlad udevs

oqros kveTis proporciebi (sur.4). warmovid-

ginoT, rom misi fuZis Wrilis kvadratis gver-

debi dayofilia oqros kveTis Sesabamisad.

dayofis wertilebidan gatarebuli diago-

nalebi gadakveTisas warmoqmnian or kvadrats,

romelTa gverdebic agreTve oqros kveTis

Sesabamisad daiyofa. am kvadratebis mopirda-

pire wveroebisa da gadakveTis wertilebis

SeerTebiT miiReba rva monakveTi. es rva sidide

RDH* (Rapports Divine Harmonie _ RvTaebriviharmoniis gadmocema) proporciul skalas Sead-

gens, sawyisi me-9 monakveTi (M) sidide moce-muli kompoziciis maqsimalur zomas Seesabameba.

Catarebulma agebam Cven mogvca mxolod

proporciuli skalis gamosaxvis geometriuli

forma, sadac sidideebi ganlagebulia zrdis

mixedviT _ R, I, E N, O, S, C, A.** maTi ricxviTimniSvnelobebis dadgena imave agebidan SeiZleba,

Tu gamoTvlebis gamartivebis mizniT sawyisi

kvadratis gverdis sigrZes 2-is tolad mivi-

RebT. es ricxviTi mniSvnelobebi saxelde-

buli ar aris, amitom isini iZlevian saSualebas

nebismieri sigrZis sazomi erTeuli gamoviyenoT,

es iqneba Zveli: idayvi, terfi, muSti, romliTac

Zveli egviptelebi sargeblobdnen, Tu Tanamed-

rove metruli sistema.

aRsaniSnavia, rom ar arsebobs araviTari

dokumenturi monacemebi, iyenebdnen Tu ara Zveli

egviptelebi proporciuli sidideebis sistemas,

magram maTi monawileoba SeimCneva Zvelegviptur

yvela ZeglSi dawyebuli piramididan da damTav-

rebuli mcire formis nebismieri nawarmoebiT.

maTi aRmoCena SeiZleba gazomvebiT, es ki

moaxerxes arqiteqtorebma: am mxriv pirvel

sur. 4. geometriuli agebebi, romlebic

gansazRvraven rva proporciul sidideebs

RDH sistemaSi.

sur. 5. moxele anxirptis gamosaxuleba.

Zveli samefo.

*abreviatura (ital. abbreviatura < laT. brevis mokle) _ sityvebis Semokleba.

** es asoiTi aRniSvnebi aRebulia frangi arqiteqtoris furne de koras naSromebidan, romelic Tavis mxriv man

aiRo petrus talemarianusisagan, romelic iyenebda formulas, sadac miniSnebuli iyo saxeli ARSENICOM.

27egviptis piramidebi da oqros kveTa

rogSi aRsaniSnavia frangi arqitetoris a. furne

de koras naSromi `egvipturi proporciebi da

RvTaebrivi harmoniis Tanafardobebi~. avtorma

SeZlo yvela Zvelegvipturi xelovnebis Zegleb-

Si gamonaklisis gareSe rva proporciuli

Tanafardobis monawileobis aRmoCena.

mjdomare mwerali

Zveli egvipte. V dinastia daaxl. Zv. w. 2500.

SeRebili kirqva. simaRle 53 sm.

luvri, parizi sur. 6. kanonikuri tipis mwerlis qandakebis

ageba oqros kveTis proporciebSi.

sur. 7. sfinqsi gizaSi.

A O D

2

3

5

C B

28 oqros kveTa. fidias ricxvebi

radgan agebis safuZvels warmo-

adgens oqros kveTis proporcia,

romelic iZleva oqros ricxvs _

fidias ricxvs ΦΦΦΦΦ = 1,618..., igi

warmoadgens usasrulo aTwilads

_ iracionalur ricxvs, amitom

RDH skalis proporciuli sidi-

deebic iracionaluri ricxvebiT

gamoisaxeba, romlebic agreTve usas-

rulo aTwiladebia. Zveli egvip-

telebi icnobdnen iracionalur

ricxvebs da maTTvis miTologiur

warmodgenebSi ganuzRvrelobisa da

usasrulobis cnebebi arsebobda.

proporciuli Tanafardobebi

Zvelegviptur plastikaSi xelov-

nurad ar iyo Setanili. es TviT

bunebam gansazRvra: adamianis figura

mis anatomiur agebulebaSi Seicavs

proporciebs, romelic gansazRvru-

lia oqros kveTiT da gamoisaxeba

oqros ricxvis (ΦΦΦΦΦ = 1,618...)xarisxebiT (sur. 5) da (sur. 6).

Cveulebrivad sfinqsis Tavi

warmoadgens ama Tu im faraonis

konkretul gamosaxulebas, rome-

lic Serwymulia lomis tanTan

(sinkretuli saxe). gizis sfinqsSi

gamoxatulia faraon xefrenis saxe

(sur. 7). mxolod zustad damu-

Savebuli proporciuli Tanafar-

dobebis sistemis saSualebiT iyo

SesaZlebeli monoliTuri kldi-

dan veberTela monumentis, 57 m

sigrZis gizis sfinqsis gamokveTa.

proporciebis sitema safuZvlad

daedo arqiteqturis, qandakebis, mxatvrobis organul sinTezs da kompoziciis Semadgeneli

nawilebis urTerTSerwymas. es ki imaze mianiSnebs, rom Zvelegvi pturi kanonis sitema

Camoyalibebulia. zogadad kanoni aris proporciuli Tanafardobis skala, romlis nawi-

lebic gamonaklisis gareSe monawileoben yvela Zvelegviptur xelovnebis ZeglebSi,

ganTavsebulni arian yvela tipis gamosaxulebebis mkacrad dadgenil sistemaSi da

kompoziciis yvela nawilebs Soris Tanazomierebas uzrunvelyofen.

sur. 8-ze naCvenebia RDH proporciuli skalis sistemaSi faraon exnatonis, dedofal

nefertitisa da princesas gamosaxulebebiani stelis* kompoziciis agebis sqema. aq stelis

sigane miRebulia ZiriTad sidided (FB monakveTi), romelic oqros kveTis Sesabamisad aris

dayofili: FO = A da OB = A + S monakveTebad da im wrewirebis radiusebs warmoadgenen,romlebSic exnatonisa da nefertitis figurebi gamoisaxeba.

sur. 8. faraon exnatonis, dedofal nefertitisa da

princesas gamosaxulebiani stela. kompoziciis agebis

sqema RDH proporciuli skalis sistemaSi.

* stela (berZ. stele ) _ qvis fila, romelzedac warwera an reliefuri gamosaxulebaa.~ ~

C

F BO D O′

FB = MOB = A + S

FO = A S

Φ==OFOB

OBFB

29

ricxvebis idumali samyaroricxvebis idumali samyaroricxvebis idumali samyaroricxvebis idumali samyaroricxvebis idumali samyaro

`RmerTma Seqmna naturaluri ricxvebi, yvela danarCeni adamianis mogonilia~.

leopold kronekeri

naturaluri ricxvebi dawere Tanmimdevro-

biT, miiReb naturalur ricxvTa mimdevrobas:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...am mimdevrobis wevrebidan zogierTi:

2, 3, 5, 7, 11, 13, ...iyofa mxolod Tavis Tavze da erTze, maSas-

adame aqvs mxolod ori gamyofi. sxvebi:

4, 6, 8, 10, 12, ...iyofa ara marto Tavis Tavze da erTze,

aramed sxva ricxvebzec. e. i. aqvs sxva gamyofebic.

ricxvs, romelsac aqvs mxolod ori

gamyofi, martivi ewodeba. danarCen

ricxvebs ki _ Sedgenili.

ricxvi 1 ar iTvleba arc martivad da arcSedgenilad, radgan mas aqvs mxolod erTi

gamyofi _ Tavisi Tavi.

ricxvebis aseTi dayofa SemoiRo berZenma

maTematikosma piTagoram (Zv. w. VI s.).

martmartmartmartmartivi rivi rivi rivi rivi ricxvebiicxvebiicxvebiicxvebiicxvebi

ricxvTa mwkrivSi (mimdevrobaSi) kanonzom-

ierebis aRmoCena sasiamovno movlenaa maTema-

tikosebisaTvis. es kanonzomierebani SeiZleba

gamoyenebuli iqnas garkveuli varaudebis _

hipoTezebis agebisaTvis, damtkicebebisa da Teorie-

bis SemowmebisaTvis.

maTematikosebi martivi ricxvebis gansazRvri-

saTvis iyeneben sxvadasxva xerxs, magram dRemde

erTaderTi WeSmaritad efeqturi xerxi mogvca

berZenma maTematikosma da astronomma era-

tosTenma (Zv. w. 276-194). es xerxi 2000 welzemeti xnisaa. gairkva, rom martivi ricxvebi ar

emorCilebian garkveul kanonzomierebas _

swored esaa maTi Tviseba.

eratosTenes meTodiT

SeiZleba SevadginoT cxri-

li da ganvsazRvroT, mar-

tivia Tu ara romelime

ricxvi. mas `eratosTenes

saceri~ ewodeba. vnaxoT

rogor SeiZleba am sacer-

iT `gavcxriloT~ Sedgeni-

li ricxvebi naturaluri

ricxvebis mwkrividan: pirve-

lad gadavxazoT ricxvi 1.is arc martivia, arc Sedge-

nili. Semdeg gadavxazoT 2-is jeradi (luwi ricxvebi),

garda 2-isa, amisaTvis gada-vxazoT 2-is Semdeg yovelimeore ricxvi. Semdeg gada-

vxazoT 3-is jeradi Sedgenili ricxvebi dawye-buli 3-dan _ yoveli mesame (zogierT SemTx-vevaSi mogvixdeba ricxvebis orjer gadaxazva,

esenia 2-isa da 3-is jeradi). amis Semdeg gada-vxazavT 5-is jerad Sedgenil r icxvebs: dawye-buli 5-dan yovel mexuTes. (aqac zogjer me-ored gadaixazeba zogierTi ricxvi) da natu-

ralur ricxvTa mwkrivSi 1-dan 25-mde darCebaSemdegi martivi ricxvebi:

2, 3, 5, 7, 11, 13, 17, 19, 23.amis Semdeg igives gavimeorebT 7-ze, Semdeg

11-ze da a. S.`eratosTenes saceris~ erT-erTi varianti

warmodgenilia cxrilSi: aq

1 dan _ 50-mde martiviricxvebi rgolSia Casmuli.

Sen SegiZlia gaagrZelo

es cxrili da 100-mde ipo-vo danarCeni martivi ricx-

vebi. cxrilSi 5-is jeradiricxvebi dalagebulia di-

agonalze marjvnidan marcx-

niv, xolo 7-is jeradi ricx-vebi _ diagonalze marcx-

nidan marjvnisaken.

yvelaze didi martivi

ricxvi romelia?TandaTanobiT eratosTenes

meTodi srulyofili iqna da

dReisaTvis arsebobs martivi

ricxvebis cxrili 1 dan 100

30 ricxvebis idumali samyaro

000 000-mde. maTze dayrdnobiT gamoiTqva sxva-dasxva hipoTeza (varaudi). zogierTis damtki-

ceba aRmoCnda rTuli. zogic dResac daumt-

kicebelia.

erT-erTi pirveli kiTxva martivi ricxvebis

Sesaxeb aseTia: sasrulia Tu ara maTi ricxvi

usasrulo naturalur ricxvTa mimdevrobaSi?jer kidev evklidem daamtkica maTi usasrulo-

ba. e. i. yvelaze didi martivi ricxvi ar

arsebobs. davuSvaT piriqiT, vTqvaT arsebobs

yvelaze didi martivi ricxvi da aRvniSnoT P-Ti. gadavamravloT yvela martivi ricxvi erT-

maneTze da namravls davumatoT erTi. miviRebT:

2×3×5×7×...×P+1.cxadia, es ricxvi ar gaiyofa arcerT mar-

tiv ricxvze. maSin an TviTon aris martivi

ricxvi, an aqvs P-ze didi martivi gamyofierTic da meorec ewinaaRmdegeba imas, rom Pudidesi martivi ricxvia. es winaaRmdegoba

gviCvenebs, rom Cveni daSveba mcdaria.

xarisxebi

23 aris ricxvi, igi ase ikiTxeba: 2 xarisxad3~ da niSnavs 2 Tavis Tavze gamravlebulisamjer. e. i. 23 = 2×2×2 = 8.analogiurad `2 xarisxad 4~ iqneba:

24 = 2×2×2×2 = 16.ipove 210 _?maSin 222

= 24 = 2×2×2×2 = 16.aseve 223

= 28 = 2×2×2×2×2×2×2×2 = 356.

uZvelesi droidan moyolebuli maTematiko-

sebi cdilobdnen iseTi formulis Sedgenas,

romelic mogvcemda yvela martiv ricxvs, ma-

gram aseTi formlis Sedgena ar moxerxda.

frangma maTematikosma pier fermam (1601-1665) gamoTqva azri, rom formula

f(n) = 22n + 1, (1)sadac n naturaluri ricxvebia (1, 2, 3, 4, 5,

...) mogvcemda martiv ricxvebs. marTlac, rocan = 1, 2, 3, 4, fermas formula iZleva martivricxvebs. magram roca n = 5 _ ara (1732 w. esdaamtkica leonard eilerma 1707-1783).

f(1) = 221 + 1 = 22 + 1 = 4 + 1 =5.f(2) = 222 + 1 = 24 + 1 = 16 + 1 =17.f(3) = 223 + 1 = 28 + 1 = 256 + 1 = 257.f(4) = 224 + 1 = 216 + 1 = 65536 + 1 =65537.

5, 17, 257 da 65537 martivi ricxvebia.

f(5) = 225 + 1 = 232 + 1 = 4294967297 == 641×6700417, ar aris martivi (e. i. ferma

Seacdina arasruli induqciis meTodma. am sa-

kiTxze dawvrilebiT qvemoT vimsjelebT).

eleqtronuli saangariSo manqanebiT damt-

kicda, rom f(10) da f(16)-ic Sedgenili ricx-vebia.

� ganixile formula:

F(5) = n2 - n + 41. (2)roca n = 1, 2, 3, 4, ... 40, igi iZleva martiv

ricxvebs (ramdenime Sen TviTon Seamowme). ma-

gram roca n = 41-s _ ara. Seamowme damoukide-blad.

aseve formula:

ϕϕϕϕϕ (n) = n2 - 79n + 1601 (3)roca n = 1, 2, 3, 4, ...79, iZleva martiv ricx-

vebs, xolo roca n = 80 _ ara. SegiZlia Seamow-mo.

dReisaTvis cnobili yvelaze didi martivi

ricxvia: 286243 _ 1.am ricxvSi 25962 cifria!Tanamedrove kompiuteriTac ki didi mar-

tivi ricxvebis gamoTvla Znelia da moiTxovs

did dros.

286243_1 es martivi ricxvi rom vipovoT 2-

iani Tavis Tavze 86 243-jer unda gavamravloTda Sedegs 1 gamovakloT. es Zalian didi ricx-via!

marTlac warmoidgine, rom wertilma pirv-

el dRes gaiara 2 mm da yovel momdevno dResmis mier gavlili manZili orkecdeba. ra man-

Zils gaivlis is:

a) 5 dReSi?25=2×2×2×2×2=32 mm

b) erT TveSi? (230 mm)

g) or TveSi? (260 mm)

d) erT weliwadSi? (2365 mm)

2 mm

1 dReSi

2 dReSi

3 dReSi4 dReSi

5 dReSi

(viciT mzidan dedamiwamde manZili daax-

loebiT 150 000 000 km-ia).dReisaTvis cnobili yvelaze didi martivi

ricxvis sidide rom vipovoT, es igivea vupasu-

xoT kiTxvas: ra manZili eqneba gavlili ganx-

ilul wertils 86 243 dReRamis Semdeg?sainteresoa, rom 86 243 dReRame ufro

metia vidre 236 weliwadi da ganxiluliwertili gava CvenTvis cnobili samyaros

sazRvrebs gareT gacilebiT adre, vidre saZie-

bel manZils gaivlides!

cota ram istoriidan: 1742 wels peter-

31ricxvebis idumali samyaro

burgis mecnierebaTa akademiis wevrma x. gold-

baxma imave akademiis wevrs eilers miswera

werili, sadac Camoayaliba erTi saintereso

hipoTeza:

nebismieri 5-ze meti naturaluri

ricxvi warmoadgens sami martivi ricx-

vis jams.

eilerma upasuxa, rom mas ar SeuZlia am

hipoTezis damtkiceba, magram SeuZlia sxva

hipoTezis Camoyalibeba, romelic dakavSirebu-

lia pirvelTan:

yoveli 2-ze meti luwi naturaluri

ricxvi warmoadgens ori martivi ricx-

vis jams.

200 wlis ganmavlobaSi mravali qveynis cno-bili maTematikosebi cdilobdnen goldbax-eil-

eris hipoTezebis damtkicebas, magram uSede-

god. amave dros XX saukunis dasawyisSi amTvisebis Semowmebam 9 000 000-mde aCvena, romgoldbaxis hipoTeza sworia. 1931 wels axal-

gazrda rusma maTematikosma lev Snirelmanma

mniSvnelovan warmatebas miaRwia am problemis

gadawyvetaSi da bolos 1937 w. akademikosma i.m. vinogradovma TiTqmis bolomdis amoxsna es

problema, magram sruli damtkiceba jer aravis

mouxdenia!

Semdegi problema, romelic goldbaxis prob-

lemaze gacilebiT sainteresoa, Tavis gadawyve-

tas jerjerobiT mcirediTac ver miuaxlovda.

SemCneuli iyo, rom martivi ricxvebi araiS-

viaTad gvxvdebian P da P+2 wyvilebis saxiT(aris erTi gamonaklisi 2 da 3): (3, 5); (5, 7);(11, 13); da a.S. aseT martiv ricxvebs tyupebihqvia. ai, sakmaod didi tyupi ricxvebis wyvili:

(1000000009649, 1000000009651).gamoTqmuli iyo da sarwmunod iTvleba hipoTe-

za: tyupi ricxvebis simravle usasru-

loa.

magram dRemde ar aris miRweuli araviTari

warmateba am hipoTezis damtkicebaSi.

1 � daakvirdi piTagoras cxrilSi gamoyofil kuTxeebs. daamtkice, rom am kuTxeebSi

moTavsebuli ricxvebis jami qmnian kubebis mimdevrobebs: 13, 23, 33, 43, 53, ...2 � ras udris 25-e kuTxeSi moTavsebuli ricxvebis jami?

1 � cxadia 1+2+3+4+5+6+7+8×9=100. qvemoT moyvanil magaliTebSi tolobis marcxena mxares Camoweril ricxvebs Soris Casvi

saWiro maTematikuri moqmedebis niSnebi. ise rom tolobebi WeSmariti gaxdes:

1 2 3 4 5 6 7 8 9 = 100, 1 2 3 4 5 6 7 8 9 = 100, 1 2 3 4 5 6 7 8 9 = 100.

2 � 100; 97; 81; 68; 58; 43; 32; 21; ? romeli ricxvi modis Semdeg?

3 � aRmoaCine romeli wesis mixedviTaa mimdevrobis wevrebi SerCeuli:

a) 11, 31, 41, 61, 71, 101, 131, ...b) 13, 23, 43, 53, ...

4 � Seamowme goldbaxisa da eileris hipoTezebi ramdenime ricxvisaTvis.

5 � aba, scade da gaarkvie ramdeni tyupi ricxvia pirvel aseulSi?6 � ipove yvela martivi ricxvi, romelic erTdroulad aris ori martivi ricxvis jami da

sxvaoba.

7 � ipove yvela martivi ricxvi, romelic miiReba gamosaxulebiT:

21

(n - 1)(n + 2), sadac n naturaluri ricxvia.

8 � daamtkice, rom namravli 1.2.3.4.5. ... .19 (mokled iwereba 19! da ikiTxeba `19-isfaqtoriali~) iyofa 820 125-ze.

Tavsatexi

�

32 ricxvebis idumali samyaro

nikomax geraski (Zv. w. I s.) piTagoreli,cnobili berZeni filosofosi da maTematikosi

werda: `srulyofili ricxvebi lamazia, magram

lamazi nivTebi iSviaTia da araa bevri, uSno ki

gvxvdeba mravlad. uamravia TiTqmis yvela ricxvi,

maSin roca srulyofili ricxvebi cotaa~.

Zvelma berZnebma icodnen mxolod ori

srulyofili ricxvi: 6 da 28. ra aqvT maTsaerTo da riT gamoirCevian sxva ricxvebisagan?aba, ipove maTi gamyofebi da Semdeg Sekribe

isini. miiReb igive ricxvebs: 6=1+2+3, 28=1+2+4+7+14.

maSasadame, TiToeuli maTgani udris Tavisi

sakuTari gamyofebis jams. aseTi Tvisebis mqone

ricxvebs srulyofils uwodeben.

evklidemde (Zv. w. III s.) mxolod es orisrulyofili ricxvi iyo cnobili. evklidem

moifiqra formula:

2P_1(2P_1),sadac 2P_1 martivi ricxvia. am formulis

gamoyenebiT man kidev ori srulyofili ricxvi

aRmoaCina:

22_1(22_1) = 6;23_1(23_1) = 28;25_1(25_1)=24(25_1) = 16×31= 496;27_1(27_1)=26(27_1) = 64 ×127 = 8128.mexuTe srulyifili ricxvia: 33 550 336,meeqvse _ 8 589 869 056;meSvide _ 137 438 691 328;merve _2 305 843 008 139 952 128.mecxre srulyofili ricxvi gamoTvlili iyo

mxolod 1883 wels rusi soflis mRvdlis i.v. perkuSinis mier. am ricxvSi aris 37 cifri.

2 305 843 009 213 693 951×260, sadac

(misi Sroma gmirobis tol-

fasi iyo).

cxadia aseTi ricxvebis xeliT angariSi Zalze

Znelia (TiTqmis SeuZlebelia).

meoce saukunis dasawyisSi gamoiyenes meqanikuri

saTvleli manqanebi da 1911 wels aRmoaCinesmeaTe srulyofili ricxvi:

618 970 019 643 690 137 449 562 111× 288,

sadac

am ricxvSi aris 54 cifri.1914 wels aRmoaCines meTerTmete (65

cifriani) srulyofili ricxvi:

162 259 276 829 213 363 391 578 010 288 127×2106

da meTormete (77 cifriani)2126(2127_1)

srulyofili ricxvi.

SemdegSi gamoigones eleqtronuli saTvleli

manqanebi da 1952 wlis 30 ianvars amerikelmamaTematikosma robinsonma, kaliforniis univer-

sitetSi gamoiyena eleqtronuli saangariSo

manqana 2P_1 ricxvis simartivis Sesamowmeblad.man Tavdapirvelad gadawyvita 2257_1 ricxvissimartivis Semowmeba da miiwvia lemeri, romelmac

20 wlis win TiTqmis erTi weli moandoma amisangariSs. lemerma miiRo didi siamovneba, rodesac

dainaxa, rom manqanam miiRo igive Sedegi 18wamSi. imisaTvis, rom mieRoT Semdegi srulyofili

ricxvi, manqanam daiwyo axali martivi ricxvebis

Zebna. man or saaTSi 42 ricxvi Seamowma,

romelTa Soris yvelaze mcires hqonda 80-istoli cifrTa ricxvi! yvela es ricxvi aRmoCnda

Sedgenili. axali (mecamete) srulyofili ricxvi

manqanam aRmoaCina 1952 wlis 30 ianvars saRamos:2520(2521_1), (P = 521).

am ricxvSi 314 cifria.imave dRis SuaRamisas manqanam aRmoaCina

meToTxmete srulyofili ricxvi. Seamowma kidev

13 evklidiseuli ricxvi, da man miagno martivricxvs 2607_1, romelsac aqvs 183 cifri (aTobiTsistemaSi). es srulyofili ricxvia:

2606(2607_1), (P = 607).meToTxmete srulyofil ricxvs aqvs 366

cifri.

1957 wlis seqtemberSi Svedi maTematikosisg. rizelis mier napovni iqna meTvrmete

srulyofili ricxvi. eleqtronul-saangariSo

manqanis gamoyenebiT man xuT naxevar saaTSi

daadgina 23217_1 ricxvis martivoba da miiRomeTvramete srulyofili ricxvi:

23216(23217_1), (P = 3217).masSi daaxloebiT 2000 cifria.Semdegi srulyofili ricxvebis Zebna moiTxovs

ufro meti da meti moculobis angariSs.

saangariSo teqnika TandaTanobiT srulyofili

xdeba da 1962 w. moZebnili iqna ori axalisrulyofili ricxvi, 1965 wels kidev sami. amricxvebs evklidis formulaSi Seesabameba

P = 4253, 4423, 9689, 9941 da 11213.srulyofil ricxvSi 211212(211213_1) aris 3376

sssssrulyofili rrulyofili rrulyofili rrulyofili rrulyofili ricxvebi.icxvebi.icxvebi.icxvebi.icxvebi.

angarangarangarangarangariSi da saangariSi da saangariSi da saangariSi da saangariSi da saangariSoebiiSoebiiSoebiiSoebiiSoebi

33ricxvebis idumali samyaro

cifri!

udidesi maTematikosi edmund landau aR-

niSnavs : `... dRemdis ori problemaa

gadauWreli: usasruloa Tu ara luwi

srulyofil ricxvTa simravle? _ ar

vici. _ arsebobs Tu ara kenti

srulyofil ricxvTa usasrulo

simravle? _ me isic ki ar vici

arsebobs Tu ara erTi mainc aseTi~.

cxadia iseTi damxmaris saSualebiT, rogoricaa

saangariSo manqanebi adamianma SeZlo aseTi

veberTela srulyofili ricxvebis dadgena.

dReisaTvis cnobilia 30-ze meti srulyofiliricxvi romelTa Soris udidess Seesabameba:

P = 216 091.srulyofili ricxvebis Zebnis istoria,

TvalnaTliv gviCvenebs, Tu rogor zrdis saan-

gariSo manqanebi adamianis SesaZleblobebs.

maTematikosebi yovelTvis ocnebobdnen iseT

daxmarebaze, romelic gaamartivebda maT Sromas

xangrZlivi da damRleli gamoTvlebisagan.

erT-erTi gamomTvleli mowyobiloba iyo

abaki. es saangariSo CarCo dResac gamoiyeneba

zogierT qveyanaSi.

logariTmuli cxrilebi XVII saukunis

dasawyisSi daamuSava jon neperma da aadvilebda

gamoTvlebs TiTqmis 4 saukunis ganmavlobaSi.msoflioSi pirveli saTvlel-saangariSo

mowyobiloba (manqana) aago 19 wlis blezpaskalma 1642 wels (sur. 1).dRes TiToeul Tqvengans aqvs saSualeba

gamoiyenos miniaturuli saangariSo manqana-

mikrokalkulatori (sur. 2), romelic saSualebasiZleva gamoTvlebi Catardes swrafad. igi

namdvilad iqca maTematikosebis (da ara marto

maTTvis) megobrad da mxsnelad.

� mikrokalkulatorze, Tu daakvirdebi ric-

xvebis klaviSebis gan-

lagebas Zalian advilad

aRmoaCen saintereso

ricxviT kanonzomiere-

bebs: romeliRaca ric-

xvebis sxvaoba yovel-

Tvis 111-is an 333-istolia. aba, moZebne,

romelia es ricxvebi?� agreTve rome-

liRaca orniSna ric-

xvebis sxvaoba yovel-

Tvis 27, 198 an 594-istolia. zogierTi jami

yovelTvis iyofa 11-ze.maTi moZebna ar gagi-

Wirdeba, Tu gulmod-

gined daakvirdebi da

sxvadasxva variants Sea-

mowmeb.

� klaviSebze kidev sxva saintereso kanon-

zomierebebis aRmoCenac SeiZleba! ivarjiSe da

aRmoaCine! magaliTad:

3+7=10 1+5+9=151+9=10 3+5+7=152+8=10 2+5+8=154+6=10 4+5+6=15

1 � dawere kent ricxvTa sasruli mimdevroba 1-dan 999-mde.a) ras udris am ricxvebis jami?b) Seadgine analogiuri amocana luwi ricxvebisa da naturaluri ricxvebis

SemTxvevaSi.

2 � moZebne da gamoiyene yvelaze martivi angariSis wesi Semdegi magaliTebisaTvis:

a) 99_97+95_93+91_89+. . .+7_5+3_1.b)

1 1 1 1 1 1 1 1 1 .1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10

+ + + + + + + +⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

3 � a) swrafad ipove 1 .2 .3 .4 .5 .6+1 ricxvis 5-ze gayofisas miRebuli ganayofi danaSTi.

b) ipove zepirad 1000000–(1000000–{1000000–[1000000-(1000000-999999]}).4 � frangma maTematikosma sofi Jermenma (1776-1831), roca moswavle iyo, daamtkica, rom

K4 + 4, sadac K erTze meti nebismieri mTeli ricxvia, aris Sedgenili. Sen daamtkiceb?

sur. 1

sur. 2

3. j. kiknaZe

198:9=22.

594:9=66.

34 ricxvebis idumali samyaro

cnobili Sveicarieli maTematikosebis,

bernulebis ojaxis warmomadgenelma iohan IIIbernulma 1773 wels berlinis akademiis SromebSigamoaqveyna cxrili, sadac mocemuli iyo nraodenobis erTianebiT dawerili ricxvebis

martivi gamyofebi:

miuxedavad imisa, rom man ver SeZlo am saxis

zogierTi ricxvebis martivi gamyofebis moZebna

(n=11, 17, 29), xolo sami ricxvis (n=20, 25, 27)gamyofebi ar iyo dayvanili martiv gamyofebamde,

miuxedavad mis mier daSvebuli Secdomebisa (n=22,24, 26) Cven dRes SegviZlia pativi vceT misgigantur Sromas am didi ricxvebis martivi

gamyofebis moZebnisaTvis. cxrilSi bernulim

varskvlaviT aRniSna misTvis saeWvo SemTxvevebi.

am cxrilis gamoqveynebidan TiTqmis 200 wlisSemdeg, 1964 wels niu-iorkSi a. beilermagamoaqveyna wigni `saxaliso ricxvTa Teoria~

sadac am ricxvebs miuZRvna mTeli Tavi

saxelwodebiT `111 ... 1111~, da Semoitana

maTTvis termini `repiunitebi~ (`repunit~,Semokleba inglisuri sityvebisa repeated unit _gameorebuli erTiani). es termini qarTul

cnobarebSi jerjerobiT aRniSnuli ar aris.

aTobiT sistemaSi repiunitebs aRniSnaven R-iT.mag. R1= 1, R2= 11 da a.S.maTematikosebi muSaoben am cxrilis gafar-

Toebaze da 1975 wlisaTvis cxrilSi n-ma miaRwia3000 (s. eits), magram masSi sakmao raodenobisuzustobebia (amJamad am uzustobebis nawili

likvidirebulia da yvela n = 162 ricxvisaTvismoZebnilia martivi gamyofebi).

`111 ... 1111~ _ repiunitebi

gansakuTrebul interess iwvevs martivi

repiunitebi. ukve damtkicebulia, rom R1, R19

(1918 w.), R23 (1929 w.), R317 (1978 w.) da R1037

(1985 w.) martivi ricxvebia. e.i. repiunitebisojaxidan jer-jerobiT cnobilia mxolod xuTi

martivi ricxvi.

aRmoCenilia repiunitebis mravali saintereso

Tviseba, albaT bevri jer kidev ucnobia.

namravli Ri×××××Rj (-i, -Ji indeqsebia _ natu-

raluri ricxvebia), rodesac ji ≥≥9 , war-

moadgens palindromul ricxvs (12... j ... 21saxisas). ricxvi palindromi ewodeba naturalur

ricxvs, romlis Canaweri emTxveva misi sarkuli

anareklis Canawers. magaliTebi:

11111×××××111 = 1233321,1112 = 12321.

Tu 9>≥ ji , maSin Ri×××××Rj namravli ar arispalindromi.

35ricxvebis idumali samyaro

1 � romeli ricxvebiT SeiZleba Sec-

valo asoebi, rom cxra Sesak-

rebTa jami gaxdes repiuniti?

3 � romeli ori repiunitis namravlia ricxvi 123455554321?

4 � avtomobilebis gayidvis Semdeg firmis Semosavali gaxda 1111111 dolari. ramdeni avtomo-bili gauyidia firmas, Tu TiToeulis fasi erTnairi iyo?

5 � cxadia: 121 = 11×××××11, 11211 = 111×××××101, 1112111 = 1111×××××1001.e.i. palindromebi 121, 11211, 1112111 ar arian martivi ricxvebi. daamtkice, rom

������

erTeulierTeuli nn

11...121...11 repiuniti nebismieri n-isaTvis ar aris martivi ricxvi.

6 � SegiZlia oTxi 2-ianiT da Svidi 5-ianiT miiRo repiunitebi?

7 � Tu 12 345 679 gaamravleb 9-ze, miiReb repiunits (Seamowme). romel ricxvze unda gaamrav-lo mocemuli ricxvi, rom miiRo a) mxolod xuTianebiT da b) mxolod cxrianebiT dawe-

rili ricxvebi.

8 � dawere formula, romliTac miiReb repiunitebs.

` `

` `

` `

` `

` `

` `

` `

` `

+

2 � romeli ricxvebiT Secvlidi asoebs?

R R R R R R RR R R R R R R

×××××

REPUNITINUPER

Ð Å Ï Ü Þ Í È Ò

36 ricxvebis idumali samyaro

`zepiri angariSi~ _ cnobili rusi mxatvris bogdanov-belinskis farTod cnobili

namuSevrebidan erT-erTia, romelic moskovis tretiakovis galereaSi inaxeba. mxatvarma daxata

Tavisi skolis erTi gakveTili da maswavleblis rolSi warmoadgina cnobili pedagogi s. a. raCinski.

kargad daakvirdi suraTs. masze Sen aRmoaCen Zveli saxalxo skolis moswavleebis cxovrebas

maTematikis gakveTilze. dainaxav maT sxvadasxva saxes, xasiaTs, maZiebel gonebas, sazrianobas da

niWierebas. mxatvari gadmoscems maT gatacebas skoliT da amasTan erTad maswavleblis unars

daainteresos bavSvebi ... magaliTi, romlis amoxsniTac isini gatacebuli arian dafaze weria. ai

isic:

Sen mas zepirad amoxsni?

102 + 112 + 122 + 132 + 142

365

n. p. bogdanov-belinski. `zepiri angariSi~

s. a. raCinskis saxalxo skolaSi. 1895.

�

37

maTematikuri induqciis principi*maTematikuri induqciis principi*maTematikuri induqciis principi*maTematikuri induqciis principi*maTematikuri induqciis principi*

nebismier mecnierebaSi movlenebis Seswavlisas,

es maTematika iqneba Tu istoria, fizika Tu

medicina, astronomia Tu ekonomika yvelgan da

yovelTvis mTavaria kanonzomierebis dadgena,

romelic erTmaneTTan akavSirebs Sesaswavli

movlenis TiToeul elements. xSirad SesaZlebeli

xdeba aRmoCenili kanonzomiereba gamoisaxos

formuliT. adamianma, Tu esmis formulebis ena~,

SesaZloa uceb dainaxos kanonzomierebebis

aRmosaCenad, zogjer Zalian grZeli da

Sromatevadi saqmianobis Sedegi.

magram, Sen TavSi rom mkvlevari aRzardo,

araa sakmarisi gaigo mxolod kvlevis saboloo

Sedegebi. aucilebelia Cawvde TviT kvleviTi

muSaobis process da aiTviso is meTodi,**

romelic gamoyenebuli iyo saboloo Sedegebis

misaRebad. mecnierebi muSaobaSi sxvadasxva

meTodebs iyeneben. TiToeul mecnierebaSic ki

maTi ricxvi uamravia. Cven ganvixilavT mxolod

maTematikuri induqciis meTods da mis gamoyenebas.

induqcias*** uwodeben msjelobis meTods,

romelsac kerZo magaliTebidan raime zogad

daskvnamde mivyavarT.

magaliTi 1. kenti 1, 3, 5, 7, ..., (2n-1)ricxvebis SekrebiT n cvladis sxvadasxva

mniSvnelobisaTvis (n = 1, 2, 3, ...) miviRebT:

_ hipoTeza SeiZleba ase CamovayaliboT: yvela

naturaluri n-saTvis marTebulia toloba:

1 + 3 + 5 + ... + (2n - 1) = n2

amrigad, xuTma ganxilulma magaliTma migviy-

vana hipoTezamde, romelic mtkicdeba, rom

marTebulia.

magaliTi 2. maTematikosebi eZebdnen ra

mxolod martivi ricxvebis misaReb formulas,

gamoTqvamdnen sxavadasva hipoTezas. am mizniT

leonard eilerma Seamowma formula:

412 ++=ϕ xx)x(

Tu am formulaSi (samwevrSi) CavsvamT

naturalur 1, 2, 3, 4, 5 ricxvebs miviRebT:

.71)5(;61)4(

;53)3(;47)2(;43)1(

=====

ϕϕϕϕϕ

mocemuli samwevris yvela miRebuli mniS-

vneloba martivi ricxvia. Tu x-is nacvlad

CavsvamT 0, -1, -2, -3, -4 ricxvebs, miviRebT:

mocemuli samwevris mniSvnelobebi x cvladis

miTiTebuli mniSvnelobebisaTvisac agreTve

martivi ricxvebia.

ganxiluli msjelobis safuZvelze iqmneba

hipoTeza, rom samwevris ϕ(x) mniSvneloba x-isn ebi smi eri mTeli mniSvn elobis aTvi s

(0, ±1, ±2, ±3, ...) martivi ricxvia. magram

gamoTqmuli hipoTeza mcdaria, radgan roca

x = 41, maSin

.

e.i. miiReba Sedgenili da ara martivi ricxvi.

� damoukideblad Seamowme hipoTeza, roca

x = 40.

amrigad, msjelobis erTsa da imave meTods

zog SemTxvevaSi swor daskvnamde mivyavarT

(pirveli magaliTi), xolo zogSi _ mcdar

daskvnamde (meore magaliTi). amis mizezi isaa,

_ `jami , romelic erTi

Sesakrebisagan Sedgeba~, aseTi

gamoTqma maTematikaSi

miRebulia.

2

2

2

2

2

52597531

4167531

39531

2431

111

==++++==+++

==++==+

==

yvela moyvanil magaliTSi advili SesamCnevia:

pirveli kenti naturaluri ricxvebis jami

SesakrebTa ricxvis kvadratis tolia. bunebrivad

dagvebadeba kiTxva: xom ara aqvs adgili am Tvisebas

SesakrebTa nebismieri ricxvisaTvis? Cveni varaudi

* principi (laT. principium dasawyisi, dasabami) _ romelime Teoriis, moZRvrebis, mecnierebis da misT. ZiriTadiamosavali debuleba; saxelmZRvanelo idea.

** meTodi (berZ. methodos) _ 1. bunebisa da sazogadoebrivi cxovrebis movlenaTa kvlevis, Secnobis xerxi. mag.,eqsperimentuli meTodi. 2. xerxi, wesi, wesebis sistema raime moRvaweobaSi. miznis miRwevis saSualeba.

*** induqcia (laT. inductio) _ azrovnebis meTodi _ kerZo faqtebidan, calkeuli debulebebidan zogadi daskvnebisgamotana (sapirisp. deduqcia).

38 maTematikuri induqciis principi

rom am msjelobebSi daskvna ramdenime magaliTis

ganxilvis Sedegad keTdeba, romlebic yvela

SesaZlo SemTxvevas ar moicavs, amitom msjelobais

am meTods arasrul induqcias uwodeben.

rogorc vxedavT, arasruli induqciis meTods

ar mivyevarT sruliad saimedo daskvnamde (swored

amitom Secda ferma, gv. 30), magram is imiTaasasargeblo, rom saSualebas gvaZlevs Camova-

yaliboT hipoTeza, romelic Semdgom SeiZleba

davamtkicoT an uarvyoT. magram Tu daskvna yvela

SemTxvevis ganxilvis safuZvelze keTdeba, maSin

msjelobis aseT meTods srul induqcias uwo-

deben. es meTodi maSin gamoiyeneba, roca

SemTxvevaTa ricxvi sasrulia (da ar aris

`metismetad didi~).

magaliTad: davamtkicoT, rom yoveli natura-

luri ricxvi n, romelic 2 ≤ n ≤ 15 utolobas

akmayofilebs, an martivi ricxvia, an gamoisaxeba

araumetes sami martivi ricxvis namravlis saxiT.

damtkicebisaTvis ganvixiloT TiToeuli

naturaluri ricxvi 2-dan 15-mde. 2, 3, 5, 7, 11,

13 ricxvebi martivia. 4, 6, 9, 10, 14, 15 ricxvebi

SeiZleba warmovidginoT ori martivi ricxvis

namravlis saxiT, xolo danarCeni 8 da 12 ricxvebi

_ sami martivi ricxvis namravliT.

davubrundeT tolobas:

1 + 3 + 5 + ... + (2n - 1) = n2.

es toloba SeiZleba ganvixiloT, rogorc

cvladis Semcveli winadadeba da moxerxebu-

lobisaTvis nebismieri naturaluri n-isaTvis igiaRvniSnoT A(n)-iT. maSin A(1), A(2), A(3), A(4),

A(5)-is WeSmariteba niSnavs mocemuli formulis

marTebulobas, roca n=1, n=2, n=3, n=4, n=5

(rogorc dasawyisSi vaCveneT).

radganac A(5) winadadeba WeSmaritia:

2597531 =++++ , amitom

e.i. WeSmaritia A(6) winadadebac.

amrigad damtkicebulia, rom A(5)-is

WeSmaritebidan gamomdinareobs A(6)-is

WeSmariteba. Tu nacvlad sityvisa `gamomdina-

reobs~ gamoviyenebT niSans ⇒, maSin winadadeba

`A(5)-dan gamomdinareobs A(6)~ mokled ase

Caiwereba: A(5) ⇒ A(6).

davamtkicoT, rom A(k) ⇒ A(k +1). es

niSnavs, rom 2k)1k2(...531 =−++++ to-

lobidan gamomdinareobs

toloba.

marTlac

axla ukve cxadia, rom A(6) ⇒ A(7). zustad

aseve A(7) ⇒ A(8); A(8) ⇒ A(9) da a.S.

aseTnairad SeiZleba mivaRwioT nebismier n

naturalur ricxvs da misTvis davamtkicoT

A(n). winadadebis WeSmariteba. sxvanairad rom

vTqvaT, A(n) winadadeba WeSmaritia nebismieri

naturaluri n-isaTvis, Tu igi WeSmaritia n=1-

isaTvis da nebismieri k-saTvis A(k)-dan

gamomdinareobs A(k+1). magram miuxedavad

naTqvamis sicxadisa da damajeroblobisa, aq saqme

gvaqvs axal maTematikur principTan, romelsac

maTematikuri induqciis principi ewodeba:

Tu A(n) winadadeba, romelSic n natu-

raluri ricxvia, WeSmaritia n=1-isaTvis,

da iqidan rom igi WeSmaritia n=k-saTvis

(sadac k nebismieri naturaluri ricxvia)

gamomdinareobs misi WeSmariteba momdveno

n=k+1 -isaTvisac. maSin A(n) winadadeba

WeSmaritia iqneba nebismieri naturaluri

n ricxvisaTvis.

maTematikuri induqciis principi naturalur

ricxvTa ariTmetikis erT-erTi im aqsiomaTagania*,

romelsac farTod iyeneben maTematikaSi. am

principebze dafuZvnebulia mtkicebaTa meTodi,

romelsac maTematikuri induqciis meTodi

ewodeba (sruli induqcia).

am meTodiT damtkiceba ori nawilisagan

Sedgeba: pirvel nawilSi amtkiceben (amowmeben)

A(1) winadadebis WeSmaritebas; meore nawilSi

varaudoben, rom A(n) WeSmaritia n=k-sTvis da

amtkiceben mis WeSmaritebas n = k + 1-isaTvis ,

e.i. asabuTeben Semdegs: A(k) ⇒ A(k+1).

Tu damtkicebis orive nawili Catarebulia,

maSin maTematikuri induqciis principis

safuZvelze A(n) winadadeba WeSmariti iqneba

nebismieri naturaluri n ricxvisaTvis.

axla SegviZlia CavTvaloT, rom zemoT

ganxiluli (1) toloba marTebulia nebismieri

naturaluri n ricxvisaTvis, radgan igi

m arTebulia n = 1- s aTvi s da am a sTan

A(k) ⇒ A(k+1).

* aqsioma (berZ. axiõma) ama Tu im mecnierebaSi amosavlad miRebuli debuleba, romlis WeSmariteba daumtkicebladaris miRebuli, magram aucilebelia sxva debulebaTa dasamtkiceblad.

39maTematikuri induqciis principi

ganvixiloT magaliTebi.

magaliTi 1. mocemulia mimdevroba:

...,14

1...,

35

1,

15

1,

3

1

2 −n

ipoveT misi pirveli n wevris jami.amoxsna: jer CamovayaliboT hipoTeza da

Semdeg igi davamtkicoT sruli induqciis

meTodiT. gamovTvaloT saZiebeli jamis ramdenime

pirveli mniSvneloba:

S1, S

2, S

3, S

4, ...

;5

2

15

6

15

1S

15

1

3

1S

12==+=+=

;7

3

35

1

5

2

35

1S

35

1

15

1

3

1S

23=+=+=++=

;9

4

63

28

63

1

7

3

63

1S

63

1

35

1

15

1

3

1S

34

==

+=+=+++=

Tu davakvirdebiT am jamebs, SevamCnevT, rom

mricxvelSi dgas saZiebeli jamis nomeri, xolo

mniSvnelSi _ gaorkecebuli nomers damatebuli 1.amrigad, iqmneba hipoTeza, rom

1214

1...

35

1

15

1

3

1

2 +=

+++++

n

n

n

(2)

(2) tolobis marTebulebis dasamtkicebladvisargebloT maTematikuri induqciis meTodiT.

es toloba naturaluri n-isaTvis aRvniSnoTA(n)-iT.

1. A(1) WeSmaritia, radgan

1

2. davamtkicoT, rom A(k) ⇒ A(k+1).maSasadame davuSvaT hipoteza WeSmaritia rome-

liRac nebismieri k-saTvis da vamtkicebT misWeSmaritebas (k+1)-saTvis.

.1k2

k

1k4

1...

35

1

15

1

3

1S

2k +=

−++++=

[ ][ ]

1)1k(2

1k

3k2

1k

)1k2)(3k2(

)1k)(1k2(

)1k2)(3k2(

1kk2k2

)1k2)(3k2(

1)3k2(k

)1k2)(3k2(

1

1k2

k

1)1k(21)1k(2

1

1k2

k

1)1k(4

1

1k2

k

1)1k(4

1SS

2

22k1k

+++=

++=

=++

++=+++++=

=++

++=++

+

++

=−+++

++

=

=−+

++

=−+

+=+

maTematikuri induqciis meTodiT mtkicebis

orive nawili Catarebulia. maSasadame, (2)toloba WeSmaritia n-is nebismieri naturalurimniSvnelobisaTvis.

1 � mocemulia mimdevroba ariTmetikuli progresiis saxiT:a1, a

2, a

3, ..., a

n, sadac

a2 = a

1+d;

a3 = a

2+d = a

1+d+d=a

1+2d;

a4 = a

3+d=a

1+2d+d = a

1 +3d.

d aris ariTmetikuli progresiis sxvaoba. Camoayalibe hipoTeza, progresiisnebismieri wevrisaTvis da misi WeSmariteba daamtkice maTematikuri induqciis

meTodiT.

2 � mocemulia mimdevroba geometriuli progresiis saxiT:

b1, b

2, b

3, ..., b

n, . . . sadac

b2 = qb

1; (q≠1)

b3 = qb

2= q2b

1;

b4 = qb

3= q3b

1.

q geometriuli progresiis mniSvnelia. Camoayalibe hipoTeza progresiis nebismieri wevrisaTvisda misi WeSmariteba daamtkice maTematikuri induqciis meTodiT.

3 � daamtkice: geometriuli progresiis pirveli n wevris jami q

qbn −

−=1

)1(S

n1

4 � mocemulia mimdevroba:

⎭⎬⎫

⎩⎨⎧

+ )1(

1

nn

.

40 maTematikuri induqciis principi

es niSnavs, rom mimdevrobas aqvs saxe: .

)1(

1,...,

43

1,

32

1,21

1

+⋅⋅⋅ nn

gamoTvaleT am midevrobis pirveli n wevris jami.

5 � daamtkice, rom 1-dan n-mde naturalur ricxvTa kvadratebis mimdevrobis jami udris:

6

)12)(1( ++ nnn e. i. .

nnn

n...

6

)12)(1(321 2222 ++=++++ hipoTeza mocemulia.

6 � daamtkice, rom .2

)1(...321

2

3333

⎥⎦

⎤⎢⎣

⎡ +=++++ nn

n

7 � gaixsene bines formula da daamtkice misi WeSmariteba.

miTiTeba: .;2

53

4

5521

2

51

2

53

4

5521

2

5122

−=+−=⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=++=⎟⎟⎠

⎞⎜⎜⎝

⎛ +

8 � maTematikuri induqciis meTodiT daamtkice, rom fibonaCis mimdevrobis pirveli n wevris

kvadratebis jami udris .

1nn +⋅ aa

m. k. esxeri. `sami sfero~. 1945.

mormormormormoric kornelis esxeric kornelis esxeric kornelis esxeric kornelis esxeric kornelis esxeriiiii

holandieli mxatvari, grafikosi, mravali graviurisa da liTografiis avtori m. k.

esxeri daibada 1898 wlis 17 ivniss leeuvardenSi (niderlandebSi). yrmoba gaatara arnhe-

imSi da iqve swavlobda sko-

laSi. adreul asakSi cxadi

gaxda misi miswrafeba xat-

visadmi da swavla gaagrZela

arqiteqturis fakultetze.

magram holandieli mxatvris

de moskitas rCeviT, arqiteq-

turas Tavi daaneba da mu-

Saoba daiwyo grafikaSi. swav-

lis damTavrebis Semdeg, esxeri cameti weli cxovrobda

italiaSi. igi gansakuTrebiT moxibluli iyo romiT. bevrs

mogzaurobda fexiT samxreT italiaSi da espaneTSi.

1935 wels, italiaSi faSizmis damkvidrebis Semdeg

gadasaxlda SveicariaSi. 1941 wels dabrunda samSobloSi _

baanSi, xolo 1970 wlidan sikvdilamde cxovrobda larenSi.

gardaicvala 1972 wlis 27 marts.

msoflio aRiarebas esxerma 1951 wels miaRwia, rodesac

misi ramdenime namuSevari gamoqveynda cnobil JurnalebSi:

`The Studio~, Time~ da Life~.1954 wels amsterdamSi Catarda maTematikosTa saerTa-

Soriso kongresi da masTan dakavSirebiT moawyves esxeris

didi gamofena. mecnierebma mis namuSevrebSi SeniSnes bevri

saerTo maTematikis zogad ideebTan, maSinve aRiares da Seiyvares

`TavianTi~ mxatvari. am droidan moyolebuli esxeris naxatebi

xSirad ibeWdeba fizika-maTematikur gamocemebSi.

esxeri ar malavda, rom cudad icoda maTematika, magram

ambobda: me ufro xSirad maTematikosebTan vgrZnob did siax-

loves, vidre kolega mxatvrebTan~. leonardo da vinCisa da

albrext diureris msgavsad esxeri didad afasebda wesier

mravalwaxnagebs, xuT platoniseul sxeuls: tetraedri, heqsaedri,

oqtaedri, dodekaedri da ikosaedri. igi gaocebuli iyo maTi

srulyofilebiT da ambobda, rom isini ganasaxiereben harmoniisa

da wesrigisadmi adamianis swrafvis somboloebs. es sxeulebi

ar arian adamianis gonebis nayofi, radgan isini arsebobdnen

bunebaSi kacobriobis Seqmnamde gacilebiT ufro adre.

esxeri Tavis naSromebSi Taviseburad asaxavs samyaros

usasrulobas, sivrcisa da drois erTianobas, ar arsebul

obieqtebs, iluziebs da sxv.

sainteresoa, rom m. k. esxeris naxatebis didi koleqcia

Seagrova aSS-is yofili prezidentis, Teodor ruzveltis

SviliSvilma kornelius ruzveltma.

avtoportreti. 1943.

41

arqimedes aRmoCenaarqimedes aRmoCenaarqimedes aRmoCenaarqimedes aRmoCenaarqimedes aRmoCena

Zv. w. 323, daaxloebiT 33 wlis asakSi

gardaicvala aleqsandre makedoneli _ didi

aleqsandre. misi monarqia daiSala. aTenSi

Zalaufleba xelSi Caigdo antimakedonelma

reaqciam da misi maswavlebeli, udidesi moaz-

rovne, filosofosi, platonis moswavle aris-

totele (Zv. w. 384-322) iZulebuli gaxda

TavSesafaris saZebnelad gaqceuliyo. igi evbe-

aze q. qaldkideaSi gadasaxlda, sadac male 63wlis asakSi gardaicvala. aTenma Tavisi poli-

tikuri mniSvneloba da pirveloba rogorc

inteleqtualurma centrma TandaTan dakarga.

am dros mecnieruli interesebis centrma egvip-

teSi, didi aleqsandres mier daarsebul qalaq

aleqsandriaSi gadainacvla.

ptolomei I soterma (`mxsneli~) lages

Zem, aleqsandre makedonelis mxedarTmTavarma

da megobarma egvipte sammarTvelod miiRo didi

aleqsandres gardacvalebis Semdeg diadoqosebs

Soris atexili brZolis Sedegad, Zv. w. 305Tavi mefed gamoacxada (Zv. w. 305-283) daptolomeebis _ lagidebis* dinastias safuZ-

veli Cauyara. man Tavis sasaxleSi miiwvia

aristoteles moswavle demetria falerski

da daavala Seeqmna aristoteles skolis _

`likeis~ (liceumis) msgavsi skola. ase Seiqmna

aleqsandriis museioni, romlis biblioTekaSi

Tavi mouyares aristoteles Sromebs.

ptolomei II filadelfosi (`dis mosiyva-

rule~) (Zv. w. 285-246) xelovnebasa da

mecnierebas mfarvelobda da misi mmarTvelobis

dros q. aleqsandria, kerZod museioni elinis-

turi kulturis centri gaxda. aq saxelmwifos

xarjze mecnierebi erTad cxovrobdnen. maT

gankargulebaSi ori didi biblioTeka iyo, sadac

Zv. w. 48-Si 700 000 tomi inaxeboda. male

biblioTelkam wignebis gamocema daiwyo, rasac

xeli Seuwyo egviptis papirusis arsebobam

(saweri masalis damuSavebaSi egviptes buneb-

rivi monopolia hqonda). es pirobebi gansakuT-

rebiT uwyobdnen xels mecnierebis ganviTarebas.

msoflios yvela kuTxidan aleqsandriaSi Tavs

iyridnen mecnierebi da mTeli antikuri**

periodis ganmavlobaSi aq mecnieruli skolebi

gaifurCqna. amitom elinistur periodSi, bunebis

movlenebis kvlevaSi garkveuli warmatebebi

aleqsandriis museionerebTanaa dakavSirebuli.

es warmatebebi garkveul wilad arqimedes

saxelTanacaa dakavSirebuli.

Zv, w, 287 sirakuzSi (sicilia), cnobili

astronomis, fideas ojaxSi daibada Zveli

berZeni mecnieri arqimede. didi drois ganmav-

lobaSi igi aleqsandriaSi swavlobda da mTeli

Tavisi sicocxlis manZilze museionebis

swavlulebTan mecnieruli kavSiri ar gauwyve-

tia. am udidesma moazrovnem: inJinerma, astro-

nomma, fizikosma, maTematikosma Tavisi SromebiT

xeli Seuwyo kacobriobis ganviTarebas da

waruSleli kvali datova istoriaSi. am geni-

osis naazrevi yovelTvis aRfrTovanebas iwvevs.

man aRmoaCina sxeulTa curvis kanoni, berketis

wonasworobis piroba da simZimis centris

`momeciT sayrdeni wertili

da dedamiwas avwev~.

arqimede

* ptolomeebi, lagidebi (berZ. Ptolemaioi, Lagidai), samefo dinastia elenistur (epoqa, rodesac ayvavebuli iyoSereuli berZnul-aRmosavluri kultura _ aleqsandre makedonelis monarqiis daSlidan romis mier saberZneTisa da

aRmosavleTis dapyrobamde _ Zv. w. I saukune) egvipteSi.

** antikuri (laT. antiguus _ Zveli) _ Zveli berZnuli an romauli (kultura, xelovneba, sazogadoebrivi wyoba da sxva).

42 arqimedes aRmoCena

mTavari Tvisebebi. magram, cotam Tu icis, rom

man faqtiurad aRmoaCina meTodi, romelic

safuZvlad daedo maTematikis erT-erTi Tana-

medrove dargis _ integraluri aRricxvis

ganviTarebas! misi gamoyenebiT arqimedem gansaz-

Rvra paraboluri segmentis farTobi (parabo-

lis kvadratura), birTvis moculoba da sxv.

arqimedes saxelTan dakavSirebulia sxvadas-

xva legenda. magaliTad legenda mefe hieronis

oqros gvirgvinis Sesaxeb~, abazanidan amosvlis

Semdeg cnobili gamoTqma heureka~ _ evrika~

(vipove).

Tqven ukve iciT, rom zogjer intuicia

gvRalatobs, magram xSir SemTxvevaSi igi udidesi

aRmoCenis wyaroa! swored fizikuri intuicia

udevs safuZvlad yvela drois erT-erT udides

maTematikur aRmoCenas _ arqimedes mier integ-

raluri aRricxvis aRmoCenas. igi ambobda:

`ramdenime maTematikuri amocana meqanikis

saSualebebiT gamovikvlie~*.

arqimedes gadmocemiT demokritem gansazRvra

konusis moculoba: igi tolia im cilindris

moculobis erTi mesamedis, romlis simaRle da

fuZis farTobi igivea rac konusis. magram

araferia cnobili misi meTodis Sesaxeb. evdoksi

iyo pirveli vinc daamtkica demokrtes mosazreba.

Zveli berZnebi garkveuli azriT koordi-

natTa meTods~ icnobdnen. isini sibrtyeze

wertilTa geometriuli adgilis SeswavlisaTvis,

moZravi wertilidan winaswar fiqsirebul aTvlis

or RerZamde manZils ganixilaven.

sur. 1-ze naCvenebia e.w. dekartes marTkuTxa

koordinatTa XOY sistema sibrtyeze. wertilTa

geometriuli adgili, romlebic akmayofileben

pirobas: 222

R)()( =−+− byax (piTagoras

Teorema), warmoadgens R radiusian wrewirs

O(a,b) centriT.axla gavyveT arqimedes `azris jaWvs~ da

ganvixiloT rogori meTodiT gansazRvra birTvis

moculoba: sibrtyeze XOY marTkuTxa koordi-

natTa sistemaSi R radiusiani wrewiri exeba

koordinatTa O saTaves da misi O1 centri OX

RerZze mdebareobs. iqvea marTkuTxa samkuTxedi,

marTi kuTxiT O wertilSi, agreTve 2R da 4R

gverdebiani marTkuTxedi (sur. 2). arqimedem

warmoidgina wrewiris, samkuTxedis da marTkuT-

xedis brunva OX RerZis garSemo. Sesabamisad

miiRo birTvi, konusi da cilindri. cxadia

konusis simaRle da fuZis wris radiusi igivea

rac cilindris da 2R udris (sur.2). Tanamedrove

aRniSvnebiT am wris gantolebas aqvs saxe:

x.yx

yx

R2

R)R(22

222

=+⇒

⇒=+−(1)

warmovidginoT nebismier A wertilze OX

RerZis marTobulad gatarebuli MN sibrtye.

igi gadakveTs ra birTvis, konuss da cilindrs

kveTaSi miiReba wreebi (sur. 2, marjvniv), romelTa

farTobebi Sesabamisad tolia:

22, xy ππ (radgan ) da .)R2(

2π(1) gantolebis orive mxare gavamravloT

πR2 -ze. davinaxavT, rom gardaqmnil gantolebaSi

gaCndeba ganxiluli farTobebi:

(*)

cxadia, radgan MN sibrtyis gatareba SeiZleba

nebismier wertilze, amitom x da y cvladisidideebia. maSasadame birTvisa da konusis kveTis

farTobebic cvladia, xolo cilindris _

mudmivi.

arqimedem gamoavlina genialuri azrovneba

da intuiciis araCveulebrivi unari: man (*)

gantolebaSi SeniSna Tavis mier aRmoCenili

berketis wonasworobis piroba (momentebis** wesi).

man dauSva, rom Wrilebi Zalian mcire h sisqiserTi da igive masalisagan damzadebuli diskoebia,

e.i. sxvadasxva simZime aqvT. Tu Tanamedrove

ganmartebebs gamoviyenebT, masalis simkvrives ρ-Ti aRvniSnavT, xolo simZimis Zalis aCqarebas g-Ti maSin, (*) gantoleba ase gardaiqmneba:

* Cp. The Method of Archimedes, edited dy Thomas L. Heath (arqimedes meTodi, tomas l. xitis gamocemuli),Cambridge, 1912, P. 13. am patara wigns uwodeben `Method~ (`meTodi~). arqimedes es naSromi xeibergis miermxolod 1906 wels iqna aRmoCenili (ix. Ãåéáåðã È., Íîâîå ñî÷èíåíèå Àðõèìåäà, Îäåññà, 1909).** berketi wonasworobaSia, Tu brunvis RerZis mimarT masze modebuli Zalebis momentebis algebruli jami nulis

tolia. Zalis momenti Zalis modulisa da misi mxaris namravlis tolia.

sur. 1

~

43arqimedes aRmoCena

gx.hghyghx ρπ=ρπ+ρπ 222 )R2()(R2 (2)

sadac ghy ρπ 2 sibrtyiT birTvis kveTis

Sesabamisi diskos simZimis Zalaa, ghx ρπ 2 _

konusis, xolo ghρπ 2

)R2( _ cilindris. amitom

R22 ghy ρπ , d a

Sesabamisi Zalebis momentebia O wertilis mimarT.

maSasadame (*) gantoleba gamoxatavs im faqts,

rom ori Zalis momenti gawonasworebulia

mesamiT. amitom arqimedem OX RerZi warmoidgina

rogorc uwono berketi da kveTaSi miRebuli

diskoebi ase gaanawila: cilindris ganivi WriliT

miRebuli disko Tavis adgilze datova, e.i. O

wertilidan x manZilze (B wertilSi), xolobirTvisa da konusis Wrilebis diskoebi berketis

O wertilidan marcxniv 2R manZilze erTi da

igive D wertilSi warmoidgina dakidebuli(sur. 2).

cxadia, rodesac x icvleba 0-dan 2R-mde,

miiReba cilindris yvela ganivkveTi, romlebic

cilindrs mTlianad avseben. cilindris yovel

ganivkveTs Seesabameba ori ganivkveTi, romlebic

berketis D wertilSia dakidebuli da isini

birTvsa da konuss avseben. amitom berketis D

wertilSi dakidebuli birTvi da konusi O

wertilis mimarT imyifebian wonasworobaSi

cilindrTan, ise rogorc maTi Sesabamisi kveTebi.

birTvis moculoba V-Ti aRvniSnoT da konusis

moculobasTan dakavSirebiT demokrates aRmoCena

gamoviyenoT. gadavideT diskoebis momentebidan

Sesabamisi sxeulebis momentebze, miviRebT:

,R2)R2(RV3

R2)R2(R2

2

2

π=⎟⎠

⎞⎜⎝

⎛ +π (**)

saidanac birTvis moculoba

.

3

R4V

3π= (3)

ganxilul msjelobaSi gadamwyvet nabijs (*)

gantolebidan (**) gantolebaze gadasvla

warmoadgens, e.i. ganivkveTebis mier mTeli

sxeulebis gavseba. maTematikuri simkacris

udidesi berZnuli tradiciebis warmomadgenelma

arqimedem Zalian kargad icoda, rom es mixvedra

iyo da ara damtkiceba. amis Sesaxeb igi aRniSnavda:

`faqti, romelTanac Cven mivediT, sinamdvileSi

gadmocemuli msjelobiT damtkicebuli araa.

magram am msjelobam mogvca Taviseburi miTiTeba,

rom Sedegi sworia~. war-

modgenili idea ganxiluli

amocanis moTxovnebis saz-

Rvrebs scildeba da ganu-

zomlad didi gasaqani aqvs.

ganivWrilebidan mTlian

sxeulebze gadasvla Tanamed-

rove enaze aris usasrulo

mcire nawilebidan mTlian

sidideze _ diferencialidan

integralze gadasvla. udi-

desi moazrovne arqimede ase

afasebda Tavis meTods: `me

darwmunebuli var, rom es

meTodi arc Tu mcire sar-

geblobas moitans maTema-

tikaSi; saxeldobr me vxedav,

rom vinme axlandeli an

momavali mkvlevarebi, gaec-

nobian ra am meTods, misi daxmarebiT daaskvnian

sxva Teoremebs, romlebic me jer TavSi ar

momsvlia~.

mogagonebT, rom arqimede Zv. w. III saukuneSi

cxovrobda, integraluri aRricxva ki q.S. XVII

saukuneSi Seiqmna, e.i. TiTqmis oci saukunis Semdeg.

es fati cxadyofs arqimedes geniis sidiades. igi

samarTlianad iTvleba statikisa da hidros-

tatikis fuZemdeblad. eWvgareSea, rom man mraval

gamogonebebTan erTad aago zusti meqanikis

`margaliti~ _ planetariumi, romelic Cvenamde

mouRwevel arqimedes erT-erT naSromSi iyo

aRwerili.

romaelma sardalma marcelma sirakuzis aRebis

Semdeg planetariumi, rogorc samxedro nadavli

romSi gadaitana. SemdegSi misgan markus talius

ciceroni (Zv. w. 106-43), Zveli romaeli poli-

tikuri moRvawe, oratori da mwerali aRfrTo-

vanebuli iyo. cnobilia gadmocema romaelebis

sur. 2

44 arqimedes aRmoCena

Setevebisagan qalaq sirakuzis Tavdacvis Sesaxeb.

sirakuzelebi sami weli igeriebdnen marcelis

xelmZRvanelobiT romaelebis Tavdasxmebs. am

periodSi arqimede igonebda da agebda sxvadasxva

samxedro manqanebs, romlebic momxvdurT SiSis

zars scemdnen. magram sirakuzi mainc daeca.

legendis Tanaxmad romeliRac uxeSma romaelma

mebrZolma, marcelis brZanebis miuxedavad moxuci

arqimede mokla im momentSi, rodesac igi silaze

geometriul figurebs xatavda. Tu es epizodi

mogonilia, is mainc sakmaod damaxasiaTebelia.

arqimede Tavisi Sromebidan yvelaze mniSvne-

lovan miRwevad Tvlida Semdegs: birTvis mocu-

loba erTnaxevarjer naklebia masze Semoxazuli

cilindris moculobaze

(sur. 3), e.i.

cilbV

3

2V = (4)

arqimedes anderZis

Tanaxmad misi saflavis

qvaze cilindrSi Caxa-

zuli birTvi iyo amok-

veTili. gardacvalebidan

TiTqmis saukune-naxevris

Semdeg swored am niSnis

mixedviT eZebda ciceroni geniosis saflavs,

romelic amJamad kvlav dakargulia.

1 � Seamowme arqimedes mtkicebis WeSmariteba, rom birTvis moculoba erTnax-

evarjer naklebia masze Semoxazuli cilindris moculobaze: Vb=2Vcil /3.

2 � sur. 4-ze gamosaxulia birTvi da

cilindri fuZeebTan ori toli konusuri ZabriT.

cilindris simaRle da fuZis diametri birTvis

diametris tolia. vTqvaT es sxeulebi damzadebulia

erTnairi nivTierebisagan. romeli ufro mZime iqne-

ba?

sur. 3

sur. 4

3. amocanebi

arqimedemde cnobili iyo kavalieris principi~: Tu sivrceSi mocemulia

ori sxeuli da mocemuli sibrtyis paraleluri nebismieri sibrtye

am sxeulebis gadakveTisas qmnian or figuras, romelTa farTobebi

tolia: S1=S

2 (am dros TiToeuli sxeulis kveTis farTobi sazoga-

dod cvladia), maSin am sxeulebis moculobebic tolia: V1=V

2.

am principis gamoyenebiT daasabuTe arqimedes formulis: Vb=2Vcil /3WeSmariteba.

miTiTeba: R radiusian wrewirze SemoxazeT kvadrati (sur. 5) gaatare diagonalebi. warmoidgine

am nakvTebis brunva vertikaluri AB RerZis garSemo. miiReb R radiusian birTvs, masze

Semoxazul cilindrs, cilindrSi ormag~ wriul konuss wveroebiT birTvis centrSi.

sur. 5

sur. 6

45 arqimedes aRmoCena

daamtkice

sur. 7

sur. 8

6 `meTodis~ me-9 winadadeba: sferuli segmentis simZimis cen-

tri mdebareobs mis RerZze da mas yofs ise, rom wverosTan

mimdebare nawili ise efardeba danarCen nawils, rogorc

segmentis simaRlisa da damatebiTi segmentis simaRlis gaoTxkecebeli jami Seefar deba

segmentis simaRlisa da damatebiTi segmentis gaorkecebl jams:

4(2R ) .2(2R

+ −=

+ −x h h

h - x h h) x sferuli segmentis simZimis centris abscisa, _ sferuli

segmentis simaRle, R _ segmentis radiusi.

7 samyaroSi Cvengan yvelaze ufro daSorebul obieqtebamde, kvazarebamde, manZili 5-13 miliardi sinaTlis weliwadia. sayrdeni wertili rom hqonoda, SeZlebda arqimede

dedamiwis `awevas~ berketiT? dedamiwis masaa 6.1024kg, radiusi _ 6400 km, s. w. aris manZili, romelsac sinaTle gaivlis erT weliwadSi. sinaTlis siCqarea 300 000 km/wm, g≈10 m/wm2.

ujredovani gumbaTi da fulerenebi

cnobilma amerikelma arqi teq-

torma da inJinerma riCard bakmin ster

fulerma daamuSava msubuqi da magari

`geode ziuri TaRebi~, kun struqciebi

foladis swori Rero ebi sagan. am

meTodiT man moskovSi, sokol nikebSi

1959 wels aago saga mofeno pavilioni.

igi aris `}jqon-67~ pavi-

lionis ujredovani gumbaTis av tori.

mas burTis forma aqvs.

80-iani wlebis Sua periodSi

mecnierebma aRmoaCines naxSir badis

giganturi molekulebi.

gumbaTis avtorma gamoiyena igive

princi pebi, romlis safuZ velzec

`aigeba~ es giganturi mo le kulebi.

yvelaze cnobil molekulas C60 aRmoCenis avto rebma Searqves bakmin-

sterfu le reni (futbolino), xolo

nax Sirbadis giganturi mole kulebis

mTel klass _ fule renebi.

dReisaTvis bevri mecnieri am

molekulebis intensiur kvlevas ewe va

da yovelwliurad aTasobiT sta tia

ibeWdeba maT Sesaxeb.

4 meTodis~ me-7 winadadeba: birTvuli segmentis moculoba ise Seef-

ardeba imave fuZis farTobisa da simaRlis mqone konusis moculobas,

rogorc birTvis radusisa da damatebiTi segmentis simaRlis jami

Seefardeba damatebiTi segmentis simaRles:

5 `meTodis~ me-6 winadadeba: naxevarbirTvis simZimis centri (O) mdebareobs mis RerZze da am RerZs yofs ise, rom naxevarbirTvis

wverosTan mimdebare nawili (a) danarCen nawils (b) Seefardeba ise, rogorc 5:3 (sur. 8).

bs

k

V R (2R ) .V 2R

+ −=

−h

h

46

magiuri kvadratebimagiuri kvadratebimagiuri kvadratebimagiuri kvadratebimagiuri kvadratebi

� ujredebiani rveulis furcelze daxaze

kvadrati (3×××××3 = 9). gamoyofili kvadratis

ujredebSi TanmimdevrobiT Cawere ricxvebi 1-dan 9-is CaTvliT (sur. 1). kuTxeebSi ganlagebulricxvebs adgilebi Seucvale ise, rogorc sur. 2-zea naCvenebi. axla am kvadrats forma ise

Seucvale, rom ricxvebi 2, 5, 8 da 4, 5, 6 diagona-lebze ganlagdes (sur. 3) miRebuli kvadratimoabrune saaTis isris moZraobis (an sawinaaR-

mdego) mimarTulebiT 450-iani kuTxiT da Sen

miiReb e.w. mesame rigis magiur kvadrats (sur. 4).amboben, rom is pirvelad gamoCnda CineTSi daax-

loebiT 2800 wlis win Cvens welTaRricxvamdesaxelwodebiT lox-Su~. is dRemde gamoiyeneba

rogorc Tilisma da gamoxatulia amuletze ise,

rogorc sur. 5-zea naCvenebi.sityva rigi~ am SemTxvevaSi aRniSnavs kvad-

ratis erT gverdze ujredebis raodenobas. bu-

nebrivad dagebadeba kiTxva: riT aris saintereso

magiuri kvadrati? aba, daakvirdi mTavar diago-nalebze, TiToeul horizontalur da vertika-

lur mwkrivebze ganlagebul ricxvebs, maTi jami

erTi da igivea da udris 15. mas mesame rigismagiuri kvadratis mudmiva ewodeba (mudmiva aqvs

yvela magiur kvadrats). meore rigis magiuri

kvadrati ar arsebobs, xolo mesame rigis _

erTaderTia (Tu ar CavTvliT im 7 magiurkvadrats, romlebic miiRebian ganxilulisagan

mobrunebiT da arekvliT).

� axla scade meoTxe rigis (4×××××4 = 16 _gverdSi oTxi ujra) magiuri kvadratis Sedgena.

Tu zemoT ganxilul wess gamoiyeneb, ufro

advilad miiReb erT-erTi saxis meoTxe rigis

magiur kvadrats (damtkicebulia, rom maTi

ricxvia 880!). magram, Tu mainc gagiWirdeba,daakvirdi didi germaneli mxatvris albrext

diureris mier 1514 wels Seqmnili graviuris*ilustracias (sur. 6).mis marjvena zeda kuTxeSi warmodgenilia erT-

erTi meoTxe rigis simetriuli magiuri kvadrati.

mas diureris magiur kvadrats uwodeben

(aRsaniSnavia, rom diureri maTematikiTac iyo

gatacebuli). Tu Sen mas kargad da guldasmiT

gaecnobi aRmoaCen, rom ara marto nebismier hori-

sur. 1

sur. 4 sur. 5

sur. 6. albrext diureri. `melanqolia~.

graviura spilenZze. 1514.

sur. 2 sur. 3

* graviura (frang. gravure) _ 1. raime magari masalis (liTonis, xis, qvis, minis da sxv.) gluv zedapirze amokveTilian amoWrili naxati. 2. aseTi naxatis anabeWdi qaRaldze.

47magiuri kvadratebi

zontalur, ver-

tikalur da mTavar

diagonalur mwkri-

vebSi ganlagebuli

cifrebis jami

udris 34-s (mud-miva), aramed kvad-

ratis SigniT kidev

ramdenime patara

kvadratSi da ara

marto maTSi!

sainteresoa, rom zemoT ganxiluli wesiT

miRebul meoTxe rigis simetriul magiur kvad-

rats diurerma meore da mesame vertikalur

mwkrivebs adgilebi Seucvala (sur. 7) magram amiTsimetriuli magiuri kvadrati ar Secvlila

(mudmiva _ 34 igive darCa). daakvirdi qvedahorizontaluri mwkrivis or SuaTana ricxvs

da Sen advilad mixvdebi am Secvlis mizans.

sur. 8-ze naCvenebia XVIII saukuneSi didimaTematikosis leonard eileris mier Sedgenili

me-8 rigis naxevradmagiuri kvadrati,

yoveli horizon-

taluri an verti-

kaluri mwkrivis

ricxvebi jamSi

260-s iZleva (dia-gonalebis gaswvriv

ara, amitomaa naxev-

rad magiuri), xo-

lo mwkrivis na-

xevari _ 130-s.� warmoidgine,

rom es kvadrati saWadrako dafaa da mxedari

iwyebs moZraobas ujredidan, romelSic weria

ricxvi 1. igi agrZelebs moZraobas. ra xdeba amdros? Tu am kiTxvas upasuxe da miageni raimekanonzomierebas, gamoiyene igi da aage me-8 rigissxva naxevrad magiuri kvadrati. ramdenia aseTi?aravin icis (Zalian bevria). dReisaTvis isic ki ar

aris cnobili ramdeni me-5 rigis magiuri kvadratiarsebobs, Tumca zogierTi SefasebiT is 13milions aWarbebs!

magiuri kvadratebi arseboben yvela rigis,

4-ze meti, im luwi rigebis garda, romlebic ariyofian 4-ze. magaliTad, meeqvse rigis magiurikvadrati ar arsebobs. aseve ar arseboben me-10,me-14 da a.S. rigis magiuri kvadratebi (isiniSeiZleba naxevrad magiuri iyos).

magiuri kvadratis rigi aRvniSnoT n-iT (maSinkvadratis ujredebis ricxvi toli iqneba:

n×××××n = n2. am ujredebSi ricxvebi unda Caiweros

1-dan n2-is CaTvliT TanmimdevrobiT), xolo

mudmiva _ C-Ti. zemoT ganxiluli magaliTebidanCans, rom roca n = 3, maSin C = 15. xolo, rocan = 4, maSin C = 34. agreTve cnobilia roca n = 5,maSin C = 65.

mexuTe rigis magiuri kvadratis ageba

imoqmede Semdegi ganawesiT

(algoriTmiT):

1. gamoyavi kvadrati

5×××××5 = 25;2. nebismier ujraSi Cawere

ricxvi 1;3. am ujridan gaakeTe

saWadrako mxedris verti-

kaluri svla (erTi ujra

marjvniv da ori ujra zeviT)

da dasvi ricxvi 2;4. gaimeore mesame punqtis

miTiTeba da dasvi ricxvi 3.da ase Semdeg dasvi ricxvebi

zrdis rigis mixedviT, sanam

ar Seavseb mTel kvadrats,

5. Tu mxedris svlas

gamoyavxar kvadratis zeda

napiris gareT, dabrundi

qveviT da Seavse mxedris daw-

yebuli svla. dasvi momdevno

ricxvi (4);6. Tu ki svlas gamoyavxar

marjvena napiris gareT,

dabrundi marcxniv da kvlav

Seavse mxedris svla. dasvi

momdevno ricxvi (5);7. Tuki ujra dakavebulia,

daubrundi wina daweril

ricxvs da pirdapir mis qveviT

dawere momdevno ricxvi (6);es moqmedebebi unda Seas-

rulo kvadratis srul

Sevsebamde. sur. 9-ze naCvenebiaam algoriTmis Sesabamisi

svlebi.

sur. 7

sur. 8

sur. 9

1

2

3

4

5

6 7

48 magiuri kvadratebi

1 � rogorc aRvniSneT, meoTxe rigis (n = 4) magiuri kvadratebis ricxvia 880,xolo mexuTe rigis (n = 5) _ camet milionze meti. sainteresoa Sen ramdenispovnas SeZleb?

2 � daadgine, rogori Tanafardoba iqneba n-sa da C-s Soris (e.i. maT Soris kavSiri warmoad-gine formuliT). Tu Sen amas SeZleb, maSin advilad mixvdebi, ratom aris me-8 rigisnaxevrad magiuri kvadratis mudmiva 260. ufro metic, Sen SeZleb nebismieri magiurikvadratis mudmivas gansazRvras Sedgenis garaSe.

1 � mxedriT svla. saWadrako mxedarma Semoiara

mTeli dafa da dabrunda sawyis ujraSi. aRadgine

mxedris mTeli marSruti, Tu cnobilia mxolod misi

16 svla, romlebic mxedris svlis rigis Sesabamisaddafaze naCvenebia ricxvebiT (sur. 10).

2 � Seadgine: a)a)a)a)a) mesame rigis (n = 3) magiuri kvadratimxolod martivi ricxvebisagan, ise rom centrSi moTavsdes

ricxvi 73.

b)b)b)b)b) mesame rigis magiuri kvadrati, mxolod martivi

ricxvebisagan, minimaluri mudmivaTi _ C = 111.

3 � ricxvebiani seqciebi Casvi kvadratSi (sur. 11) ise, rommiiRo magiuri kvadrati n = 8, C = 260 (yvela striqonSi,svetSi da diagonalSi ricxvebis jami udrides 260-s).

4 � mravlobiTi magiuri kvadrati. Seavse bade

(sur. 12) darCenili ricxvebiT 1-dan 81-mde, rom miiRo:

a) a) a) a) a) Sinagani magiuri kvadrati (SuaSi) 3 ××××× 3, C = 123;

b) b) b) b) b) Sinagani magiuri kvadrati (SuaSi) 5 ××××× 5, C = 205;

g) g) g) g) g) Sinagani magiuri kvadrati (SuaSi) 7 ××××× 7, C = 287;

d) d) d) d) d) mTeli magiuri kvadrati 9 ××××× 9, C = 369.

5 � Casvi darCenili ricxvebi 1-dan 25-mde (sur. 13 a,b),rom miiRo magiuri kvadratebi C = 65.

Tavsatexi

sur. 10

sur. 11

sur. 12

sur. 13

a b

�

49

dedamiwis radiusis gazomvadedamiwis radiusis gazomvadedamiwis radiusis gazomvadedamiwis radiusis gazomvadedamiwis radiusis gazomva

* astronomia _ (berZ. astron _ varskvlavi da nomos _ kanoni) _ mecniereba ciuri sxeulebis moZraobisa daTvisebebis Sesaxeb.

** zeniti _ umaRlesi wertili cis sferoze. Tu mnaTobi (varskvlavi) imyofeba zenitSi, maSin is anaTebs pirdapir

damkvirveblis Tavze, vertikaluri mimarTulebiT.

dedamiwis or wertilSi, romlebic mdebareobdnen

egviptis CrdiloeTiT da samxreTiT (qalaqebi:

aleqsandria da siena _ axlandeli asuani), erTi

da igive dRes _ SuadRisas gaezoma mzis

zenituri** manZili gradusebSi (sur. 2). esmanZili dakvirvebis punqtebis geografiuli

ganedebis sxvaobis tolia. Semdeg saWiro iyo

dakvirvebis punqtebs Soris manZilis gansazRvra.

am manZilis gradusebis ricxvze gayofiT

ganisazRvreba dedamiwis wrewiris sigrZis nawili,

romelic modis erT gradusze. Semdeg advilad

ganisazRvreba dedamiwis wrewiris sigrZe da misi

radiusi.

mnaTobis zenituri manZili ewodeba kuTxes

vertikalur mimarTulebasa da mnaTobidan

damkvirveblis Tvalisaken mimarTuli sxivs Soris

(sur. 3).mzis zenituri manZilis gansazRvrisaTvis

eratosTenem gamoigona specialuri xelsawyo _

skaTisi.

skaTisi warmoadgens naxevrsferel Tass,

romlis fuZeze xistad mimagrebulia liTonis

Rero naxevarsferos radiusis gaswvriv (sur. 4).

skaTisis Siga zedapirze datanilia danayofebi

gradusebSi. skaTiss ayeneben Sveulis (Zafze da-

astronomiuli* dakvirvebebis safuZvelze jer

kidev Zv. w. IV saukuneSi Zvelma berZenmamecnierebma daamtkices, rom dedamiwas aqvs

birTvis forma (sur. 1). am daskvnis safuZvelzeeratosTenma gadawyvita gaezoma dedamiwis sferos

wrewiris sigrZe da radiusi.

eratosTene (Eratosthenes daaxl. Zv. w. 276-194) Zv. berZeni mecnieri. daibada kireneSi,ganaTleba miiRo aleqsandriasa da aTenSi. kali-

maqes sikvdilis Semdeg ganagebda aleqsandriis

biblioTekas. dainteresebuli iyo filologiiT

(es sityva pirvelad man ixmara `gramatikis~

nacvlad), maTematikiT, astronomiiT, qrono-

logiiT, geografiiT, filosofiiT, musikiT. era-

tosTenem pirvelma gazoma dedamiwis wrewi-

ris (meridiani) sigrZe (aqedan dedamiwis radi-

usis sigrZe); Camoayaliba maTematikuri geogra-

fiis safuZvlebi; SeimuSava martiv ricxvTa

moZebnis meTodi (e-s saceri); daadgina saber-

ZneTis istoriis ZiriTadi TariRebi (termini

`geografia~ mis mier aris SemoRebuli). mas

ekuTvnis sxvadasxva naSromebi (sul 24 wignad);misi nawerebidan nawyvetebia SemorCenili.

eratosTene cxovrobda egvipteSi, q. aleqsan-

driaSi. misi idea SemdegSi mdgomareobda:

sur. 1 sur. 2

zeniti

zeniti

aleqsandriasiena

(asuani)

800 km

dedamiwis

zedapiri

mzis sxivebi

~

zenituri manZili

ϕ = 7,20

ganedebis sxvaoba

ϕ = 7,20

A BR O

4. j. kiknaZe

50 dedamiwis radiusis gazomva

ki debuli tvir-

Ti) gaswvriv ise,

rom Rero mi- mar-

Tuli iyos sfe-

r o s r a d i u s i s

gas wvriv, zustad

verti kalu rad, e.

i. mi marTuli iyos

zeni tisaken. mzis

sxivebze ganaTebisas

liTonis Rero ska-

Tisis fuZeze iZl-

eva Crdils. Reros

fu Zidan Crdilis

bo lomde gazo mili

rka lis sigrZe gra-

du sebSi mzis zeni-

turi man Zilis to-

lia.

eratosTene eg -

viptis Crdi loeTiT

cxov robda. vaWre-

bisagan da aqle-

mebis gamci leb-

lebisagan man ico-

da, rom egvi p tis

samxreTiT qalaq

sienaSi (axlandeli

asu anSi) mze, 22 iv niss SuadRisas

ana Tebda Rrma Webis

fu Zeebs. maSasdame

mze imyofeboda ze-

nitSi. imave dRis SuadRisas aleqsandriaSi

era tos Tenes gazomvis mixedviT mze zeniti-

dan daSorebuli iyo daaxloebiT 7,20-iT. man

agreTve icoda, rom am qalaqebs So ris manZili

iyo 5 000 berZnuli stadia*. aRniSna dedamiwis wrewiris sigrZe x-iT da Seadgina proporcia im mosazrebis safuZvelze, rom dedamiwis

wrewiris sigrZe imdenjer metia mocemul

qalaqebs Soris manzilze, ramdenjerac 3600

metia 7,20-ze:

0

0360 .

5000 7,2=

x

aqedan gamodis, rom dedamiwis sferos

wrewiris sigrZe tolia 250 000 stadias.berZnuli stadia zustad araa cnobili Tu

ra sigrZis tolia, magram Tanamedrove gazom-

vebiT aleqsandriasa da asuans Soris manZili

800 kilometria. maSin0

0360 .

800 7,2=

x

aqedan x = 40 000 km.amrigad, dedamiwis sferos wrewiris sigrZe

(meridianis sigrZe) daaxloebiT tolia 40 000km-s.XVIII saukunis bolos safrangeTis mecnie-

rebaTa akademiam moamzada ori eqspedicia

eratosTenes gazomvebis Sesamowmeblad. erTi

eqspedicia muSaobda fineTSi da SveciaSi, Crdi-

loeTis polaruli wris maxloblad, xolo

meore _ peruSi ekvatoris ganedze. gamoirkva,

rom eratosTenes gazomvebi mTlianad eTanxme-

bodnen eqspediciebis miRebul Sdegebs.

amave dros gamoirkva, rom dedamiwis pola-

ruli radiusi 21 kilometriT pataraa ekva-torul radiusze. amrigad dedamiwa aris odnav

Sebrtyelebuli sfero _ brunvis elif soidi.

Teoriulad aseTi Sedegi miRebuli hqonda yve-

la drois erT-erT udides inglisel fizikossa

da maTematikoss isaak niutons (1642-1727).

1 jer kidev arqimedem icoda, rom wrewiris sigrZe (l) SeiZleba vipoviT, Tu mis diametrs (AB) gavamravlebT 22/7-ze (sur.2). am monacemis safuZvelze ras udris dedamiwis sferos R radiusis sigrZe?

2 daamzade skaTisi (sur. 4) an sur. 3-ze asaxuli xelsawyo. misi saSualebiT gazome mT-varis zenituri manZili saRamos da Ramis sxvadasxva dros, magaliTad 20 sT-ze; 22 sT; 6 sT; 8 sT; gazomvebi Caatare savse mTvaris dros.

3 Zvelad sigrZis erTeulad iRebdnen manZils, romelsac mozrdili adamiani gaivlida

mzis badros horizontze mTlianad amosvlis drois ganmavlobaSi. mas `stadias~ uwo-

debdnen. SeiZleboda aseTi sigrZis erTeuli zusti yofiliyo?

* stadia _ (berZ. Stadion) ZvelberZnuli sigrZis sazomi erTeuli, igi icvleba adgilmdebareobis mixedviT 150-dan 190 metramde. aqedanaa sityva `stadioni~.

sur. 3

zenit

uri

manZil

i

mnaTo

bi

vertikaluri

mimarTuleba

Sveuli

Z

Z

Rero

Reros Crdili

zeniti 7,20

sur. 4

Z

Z 7,2020

100