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Пифагор и его школа

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Ieienoa?noai iauaai e i?ioanneiiaeueiiai ia?aciaaiey

?inneeneie Oaaea?aoeee

O?aeueneee ainoaea?noaaiiue yeiiiie/aneee oieaa?neoao

Eaoaae?a oeeinioee

?aoa?ao ii oeeinioee

Oaia: Ieoaai? e aai oeiea.

Ieoaai?aeneay oeeinioey /enae.

Nineeaoaeue:

A?ia/aiei A.

I?aiiaeaaaoaeue:

Aeaoa?eiao?a

1999

Niaea?aeaiea:

Aaaaeaiea

Aeia?aoey Ieoaai?a e aai oeiea

Ieoaai?aeneay oai?ey /enae

Oaaeeoea 10 /enae

Caeeth/aiea

Nienie eniieueciaaiiie eeoa?aoo?u

Aaaaeaiea

Eaaeaeue ec ian ciaao oai?aio Ieoaai?a, aeieacaiioth ei 8 ouny/ eao
iacaae. Ii ia eaaeaeiio ecaanoii, /oi Ieoaai? aue iaeiei ec aaee/aeoeo
oeeinioia, o/aiea eioi?iai, e niaeaeaieth, ia nio?aieeinue aei iaoeo
aeiae. Aai aaeeeee o?oae oia? aianoa n iei e aai o/aieeaie,
?aca?inaiiuie ii anaio ie?o iinea nia?oe o/eoaey e ia noiaaoeie
ainnoaiiaeoue oeieo, /oiau nio?aieoue ioae?inoue Ianoa?a.

O/aiea Ieoaai?a ecaanoii iai eeoue a ia?aneacao ae?aaieo oeeinioia. Iie
ia iiaoo aeaoue iieiiai i?aaenoaaeaiey ia yoii /aeiaaea e aai o/aiee, ii
ec yoeo iaei/eneaiiuo, io?uai/iuo naaaeaiee iu iiaeai noaeeoue,
ianeieueei oiai aue yoio /aeiaae, e aai ciaiey iaoaiaoeee – eeoue
/anoeoea ciaiee, eioi?oth ii ia niia inoaaeoue aoaeouei iieieaieyi.

Io naiiai Ieoaai?a ia aeioei ie iaeiie no?iee – ii-aeaeeiiio, ii
aeaenoaeoaeueii ie/aai ia ienae.

Iaiin?aaenoaaiiua o/aieee Ieoaai?a ia inoaaeee eceiaeaiey aai eaeae a
ioee/ea, neaaeai, io o/aieeia Nie?aoa. Aeioaaeoea aei ian o/aiey ?aiieo
ieoaai?oeaa (eiiaoe VI – na?aaeeia V a aei i.y.) – Aeeeieiia, Aeiiana,
Iaianoi?a, Aeiiiia e ae?. – iinyo neeoeii eiaeeaeaeoaeueiue oa?aeoa?,
/oi au aeaeaoue a ieo eceiaeaiea nenoaiu naiiai O/eoaey. *oi
i?aaenoaaeyei niaie ieoaai?neia niiauanoai: iieeoe/aneee nithc,
?aeeaeiciia a?aonoai, oeeinioneoth eee iao/ioth oeieo, eeai ana yoi
aianoa acyoia?

Ia neo/aeii ieoaai?neee aii?in inoaaony iaeiei ec naiuo caiooaiiuo e oae
ii e?aeiae ia?a naiui aeeneonneiiiui a enoi?ee iaoee e oeeinioee.
Aeia?aoey ?aaio ii ieoaai?eecio ian/eouaaao nioie eiea e noaoae, i?e
yoii, iaeiaei, o?oaeii iaeoe eaeie-eeai oaeo, aeeth/ay naii enoi?e/aneia
nouanoaiaaiea Ieoaai?a, n eioi?ui niaeaneeenue au ana auneacuaathueany
ii yoiio aii?ino.

Iaeie n/eoatho Ieoaai?a iniiaaoaeai aa?iiaeneie iao/iie o?aaeeoeee, –
«oaiaiii», i?aaeaiaeeoaeai yenoaoe/aneeo eoeueoia e oaeiuo ienoa?ee,
o?aouee iieaaatho, /oi ii niaaeeiye a naaa iaa yoeo ea/anoaa. Iaeiaei e
aioo?e yoeo iai?aaeaiee oi/ee c?aiey nouanoaaiii ?aciyony. O?oaeii
iiyoiio iiiyoue, /oi i?aaenoaaeye niaie Ieoaai? e nicaeaiiay ei oeiea.

I?eaeeceoaeueaiie aeaoie ?iaeaeaiey Ieoaai?a n/eoatho 570 a. aei i.y.
Aaeaa ee naiua /anoiua yieoaou, a ie?oaeaiee eioi?uo oeao?e?oao eiy
Ieoaai?a ia no?aieoeao iiioey?iuo iao/iuo ?aaio, – yoi «eaaaiaea?iue»,
«iieoeaaaiaea?iue» e aeaaea «iieoieoe/aneee». Nicaeaaony aia/aoeaiea,
/oi aeia?aoey Ieoaai?a aeaenoaeoaeueii ninoieo ec eaaaiae.

Ia?iiiay neaaa Ieoaai?a nineoaeeea aio aeaiyeoth neoaeao: naeaeaa aai
eiy i?eoyaaoaeueiui aeey eaaaiae, oiiiaeaaoayny io aaea e aaeo, iii a
oiaea a?aiy iicaieeea aeiianoe aei ian iaiyoue i ?aaeueiuo niauoeyo oiai
a?aiaie.

Oaea io 5 a. aei i.y., a ia/aea eioi?iai oia? Ieoaai?, i iai aeioei
aieueoa naeaeaoaeuenoa /ai i ethaii ae?oaii oeeinioa. I Ieoaai?a aiai?yo
aai nia?aiaiieee Enaiioai e Aa?aeeeo, a na?aaeeia aaea – enoi?ee
Aa?iaeio e oeeinio Yiiaaeiee, aai eiy ano?a/aaony o iiyoa e oeeinioa
Eiia Oeinneiai e ienaoaey Aeaaea ec ?aaey.

Ia eneeth/eoaeueiie iiioey?iinoe Ieoaai?a naeaeaoaeuenoaoao iiiaiou n
aai ecia?aaeaieyi e n iaaeienueth «Ieoaai?», auiouaiiua a 430-420 aa.
aei i.y. a Aaaea?ao. Aeey 5 a. aei i.y. yoi neo/ae aani?aoeaiaeaioiue e
ia oieueei iioiio /oi ecia?aaeaiey oeeinioia ia iiiaoao iiyaeythony
ai?acaei iicaea e, eae i?aaeei, a eo ?iaeiuo ai?iaeao: ia?aae iaie
ia?aue ii?o?ao ia a?a/aneeo iiiaoao, ai anyeii neo/aa, ia?aue
iiaeienaiiue ii?o?ao.

Ieoaai? ia?aui ec a?a/aneeo oeeinioia oaeinoieeny niaoeeaeueii
iinayuaiiiai aio ni/eiaiey.

Ec i?aaeoanoaaiieeia Ieoaai?a /aua anaai oiiieiatho Oa?aeeaea
Ne?inneiai, aaoi?a iaeiie ec ia?auo oaiaiiee a i?ica. A aieaa iicaeiea
a?aiaia o Ieoaai?a iiyaeythony e ae?oaea o/eoaey, iai?eia?, Aiaeneiaiae?
eee Oaean.

AEeciue Ieoaai?a e aai oeiea

N?aaee i?ioeai?a/eauo o/aiee naieo o/eoaeae Ieoaai? eneae aeeaie nayce,
neioacaaaeeiiai aaeeeiai oeaeiai. Ii iinoaaee naaa oeaeue – iaeoe iooue
aaaeouee e naaoo enoeiu, oi anoue iiciaoue aeeciue a aaeeinoaa. N yoie
oeaeueth Ieoaai? iinaoee aanue ae?aaiee ie?. ii n/eoae, /oi aeieaeai
?anoe?eoue e aac oiai oaea oe?ieee e?oaici?, eco/ay ana ?aeeaee,
aeieo?eiu e eoeueou.

Ii aeee n?aaee ?aaaeiia e iiiai ociae i oaeiuo o?aaeeoeeyo Iienay,
caeiiiaeaoaey Ec?aeey. Caoai iinaoee Aaeiao, aaea aue iinayuai a
Ienoa?ee Eneaeu. A Oeieeee e Ne?ee Ieoaai? aue iinayuai a Ienoa?ee
Aaeiiena, e, noiaa ia?ana/ue aeieeio Aao?aoa, ii iaoiaeeeny aeinoaoi/ii
aeieai o oaeaeaaa, /oiau ia?aiyoue eo nae?aoioth ioae?inoue. Ieoaai?
iinaoee Aceth e Ao?eeo, a oii /enea Iaeaoen, Eiaeonoai e Aaaeeii. A
Aaaeeiia ii eco/ee ciaiey iaaia.

Aoaeo/e iinayuaiiui ai ana ae?aaiea Ienoa?ee, oeeinio aica?aoeeny a
A?aoeeth, aaea iniiaae oeieo, a a. E?ioiia. Oeaeueth Ieoaai?a auei ia
oieueei ia?aaeaoue naia o/aiea a?oiia eca?aiiuo o/aieeia, ii e
i?eiaieoue yoe o/aiey e ainieoaieth thiioanoaa e aeecie ainoaea?noaa. A
eiioea naiae aeecie Ieoaai? iiiuoaeny i?iaanoe ?aoi?io ainoaea?noaaiiiai
ono?ienoaa, eaeaae eioi?iai ninoiye a ii?yaeea e aa?iiiee e aue /oaeae
ieeaa?oee. I?eieiay aei?eeneoth eiinoeoooeeth eae oaeiaoth, ii
no?aoieeny aianoe a iaa iiaia ono?ienoai: naa?oiieeoe/aneee aeanoe
(«niaaoa ouny/e») nicaeaoue aeanoue iaoee n niaauaoaeueiui eee ?aoathuei
aieinii ai anao aii?inao («niaao o?aonio») . «Niaao o?aonio» iiiieiyeny
eneeth/eoaeueii ec /enea iinayuaiiuo. Oaeei ia?acii, Ieoaai? no?aieeny
iinoaaeoue ai aeaaa ainoaea?noaa oiiuo i?aaeoaeae, «iie?athueony ia
aunoaa ciaiea». Oi, /oi oaeaeinue naeaeaoue Ieoaai?o, oioy au ia
ei?ioeia a?aiy, ia oaeaaaeinue ieeiio ec iinayuaiiuo. Ii yoa ?aoi?ia
i?iecieaeao 30 eao nionoy, a ii i?eauoee a E?ioii oeeinio iuoaaony
ioe?uoue oeieo, /oiau ia?aaeaoue naie ciaiey neaaeothuaio iieieaieth.

Aaeeeay A?aoeey

I i?eauoee Ieoaai?a a E?ioii nio?aieeinue eioa?aniia naeaeaoaeuenoai
Aeeeaa?oa. Eiaaea Ieoaai? i?eaue a Eoaeeth e iiyaeeny a E?oiia. Ii
?aniieiaeee e naaa aanue ai?iae eae /aeiaae iiiai no?ainoaiaaaoee.
Iaiaueiiaaiiue e ii naiae i?e?iaea aiaaoi iaea?aiiue naiae noaeueaie. Ii
iaeaaeae aaee/aaaie aiaoiinoueth e aieueoie e?anioie, aeaai?iaenoaii
?a/e, i?aaa e anaai inoaeueiiai. Nia/aea, i?ieciany aeieaoth e
i?ae?anioth ?a/ue, ii i/a?iaae noa?aeoei, nia?aaoeony a niaaoa, caoai ii
eo i?inueaa aeae ianoaaeaiey thiioai, iinea yoiai aeaoyi, nia?aiiui
aianoa ec oeie e, iaeiiaoe, aeaiueiai, eiaaea e eo nicaaee, /oiau aai
iineooaoue.

Ii aue /aeiaaeii n neeueiuie nioeeaeueiuie e iieeoe/aneeie
i?aaeoaaaeaeaieyie e aeoaieei /oanoaii nianoaaiiie cia/eiinoe. Ii aue
eca?aiiui eeaea?ii, i?i?ieii, ii ia aac oeo?inoe e oi?ioaai ciaiey
i?aeoe/aneeo aeaoaeae e n?aaenoa, eioi?ua oieueei e iiaoo iauynieoue aai
iineaaeothuee oaiiiaiaeueiue oniao.

Iaiaeue oniao nua?aei oi, /oi Ieoaai? i?eaue a E?ioii a oyaeaeue aeey
ai?iaea ia?eiae e eaeei-oi ia?acii noiae eniieueciaaoue yoo neooaoeeth a
naieo oeaeyo. Neaaa Ieoaai?a eae ainieoaoaey ianoieueei aaeeea, /oi ana
thiioe oioaee noaoue aai o/aieeaie, a eo iooeu i?aaeii/eoaee, /oiau iie
i?iaiaeeee a?aiy n iei, iaaeaee caieiaeenue nianoaaiiuie aeaeaie. Ieaoii
a naiai aaeeinoaaiiii oiiieiaiee i Ieoaai?a iacuaaao aai
«i?aaeaiaeeoaeai thiioanoaa», nicaeaaoei iniaue ieoaai?neee ia?ac
aeecie. Iieeoe/aneia aeeyiea noi?iiieeia Ieoaai?a aic?anoaei iinoaiaiii,
ii ia?a eo aicioaeaiey e aeeth/aiea a ainoaea?noaaiioth aeayoaeueiinoue.

Iao ieeaeeo naaaeaiee i oii, /oi ieoaai?aeneia niiauanoai, niaa?oea a
E?ioiia ainoaea?noaaiiue ia?aai?io, caiyei ianoi caeiiii i?aaeoaeuenoaa
a niaaoa noa?aeoei. Ieoaai? ia caieiae ieeaeie caieiaaiie aeieaeiinoe.
Yoi ia cia/eo, /oi Ieoaai? noiye a noi?iia io iieeoeee.

Aeey anao – e aunoeo, e iecoeo – o Ieoaai?a auei ioae?ia ec?a/aiea:
neaaeoao ecaaaaoue anaie n?aaenoaaie, ionaeay iaiai e ia/ii, e anai, /ai
oieueei iiaeii, io oaea – aieaciue, io aeooe – iaaaaeanoai, io aeaeoaeea
– eceeoiaai, io ai?iaea – niooo, io aeiia – ?acaei?u, e io anaai aianoa
– iaoia?aiiinoue.

Niiauanoai, nicaeaiiia Ieoaai?ii, ieacaeinue aanueia aeecianiiniaiui.
Ianiio?y ia aai aaanoai a Iaoaiiio, iii noiaei i/aiue auno?i ii?aaeoueny
io oaea?a, iaianaiiiai aunooieaieai Eeeiia.

Aeeyiea ieoaai?aeneeo aaoa?ee ia iieeoeeo inouanoaeyeinue ia a oi?ia
i?yiiai i?aaeaiey, a iooai o/anoey ioaeaeueiue ieoi?aeoeaa a
aeayoaeueiinoe i?aaeoaeuenoaa eaaeaeiai ec ai?iaeia.

A oa/aiee anaai yoiai a?aiaie ieoaai?aeoeu inoaaaeenue aeoeaiuie
noi?iiieeaie a?enoie?aoe/aneiai ia?aca i?aaeaiey. Iinea ia?aoiaea aeanoe
a ?oee aeaiie?aoee, ia?aue, e naiue iiuiue oaea? aue iaianai ii
ieoaai?aeneei aaoa?eyi. Ai iiiaeo ai?iaeao Aaeeeie A?aoeee auee
niaeaeaiu aeiia, aaea niae?aeenue ieoaai?aeoeu, /anoue ec ieo auea
oaeoa, ae?oaea aaaeaee a eiioeiaioaeueioth A?aoeeth, aaea aicieeee
oeaio?u ieoaai?aecia. Ii oaia?ue ieoaai?aeoeu naiinoiyoaeueiie ?iee a
iieeoeea ia ea?aee.

Nia?aa a?oiio o/aieeia, Ieoaai? iinayoee eo a aeoaieoth ioae?inoue, ei
ioe?uooth, a oaeaea a iniiau ieeoeueoiie iaoaiaoeee, iocuee e
ano?iiiiee. Yaeoa?ae Oth?a ienae: «Yoa iaeaiueeay iaueia eca?aiiuo eae
au inaauaea niaie ?aneeioaoeeny aieco iiiaiethaeiue ai?iae. Aa naaoeay
yniinoue i?eaeaeea aeaai?iaeiua einoeieou thiinoe, ii ia eaaei auei
i?iieeiooue a aa aioo?aiithth aeeciue, e ana ciaee, eae o?oaeai aeinooi
a n?aaeo iaiiiai/eneaiiuo eca?aiiuo».

Iieiaeua ethaee, aeaeaaoea iinooieoue a iaueio, aeieaeiu auee i?ieoe
iaeioi?ua eniuoaiey.

Nia/aea iiae/ie iiiaaeae a aeiianoe/aneee cae, aaea ii aianoa n ae?oaeie
o/aieeaie oi?aaeiyeny a ?acee/iuo ea?ao. N ia?aiai aea acaeyaea ii
caia/ae, /oi yoio cae ia iioiaeee ia ana inoaeueiua a ai?iaea: caeanue
ia auei ie a?iieeo e?eeia, ie ouaneaaiiai aunoaaeaiey naiae neeu eee
ioneoeia aoeaoa. N?aaee iieiaeuo ethaeae oea?noaiaaee aaaeeeainoue e
aeia?iaeaeaoaeueiinoue. Ieoaai? cai?auae a naiae oeiea aaeeiiai?noai,
aiai?y, /oi ?yaeii n ?acaeoeai eiaeinoe yoi aaiaeeo a aeiianoe/aneea
oi?aaeiaiey yeaiaio ai?aeinoe e iceiaeaiey.

Caoai ieoaai?aeoeu i?eaeaoaee iiae/ea naiaiaeii auneacaoueny, ia
noaniyynue inia?eaaoue eo iiaiey. A ainoi?aa, /oi aai oae ethaacii
neooatho iiae/ie ia/eiae ?acaeaaieuenoaiaaoue. A yoi a?aiy iiyaeyeny
Ieoaai?, /oiau i?ineaaeeoue ca aai aeanoaie. Ae?aaiee oeeinio i?eaeaaae
iniaia cia/aiea niaoo e iioiaeea iieiaeuo ethaeae. «Niao, – aiai?ee ii,
– naiia ianiiiaiiia oeacaiea ia oa?aeoa? /aeiaaea». Ii n/eoae, /oi niao
ia niiaeao ne?uoue oa?aeoa? ceiai.

Ieoaai?aeneea niiauanoaa auee ?aca?inaiu ii aeanyoeo ai?iaeia THaeiie
Eoaeee, a caoai nianoaaiii A?aoeee, e niaianoiua caiyoey, ?aaii eae e
iauaa ?oeiaiaenoai auee a yoie neooaoeee iaaiciiaeiu. Iaeiaa?iyoii,
/oiau aeaaea a E?ioiia a?aiaie Ieoaai?a caiyoey iineee ?aaoey?iuo
oa?aeoa? e eanaeenue anao /eaiia iauanoaa.

Ieoaai?aeoeu, eae e Ieaoii, i?eaeaaaee aieueoia cia/aiea ainieoaieth e
ia?aciaaieth thiioanoaa e ?ac?aaioaee iaoe?ioth nenoaio iaaeaaiae/aneeo
iaoiaeia. Ii oioy a ieoaai?aeneii iauanoaa e i?aeoeeiaaeinue iao/aiea,
iii auei nicaeaii ia aeey yoiai. Ia niaianoiua caiyoey ?aaee
aeinoeaeaiey ioae?inoe auee aai aeaaiie oeaeueth – aaaeue iiaeaaeythuaa
aieueoeinoai ieoaai?aeoeaa ia eiaee ioiioaiey e oeeinioee e iaoea.

Ae?aaiee oeeinio no?aieeny ?acaeoue a naieo o/aieeao, i?aaeaea anaai,
eiooeoeeth. Aaaeue ioae?inoue anoue iiieiaiea enoi/ieea, eee i?e/eiu
anao aauae, e iiaeao auoue aeinoeaiooa oieueei iiaeiyoeai eioaeeaeoa aei
oie oi/ee, aaea ii eiooeoeaii iniciaao iaaeaeeiua yaeaiey, iai?aaeaiiua
/a?ac aeaeeiua, noaiiaynue, oaeei ia?acii, niiniaiui e iauaieth nei?aa n
aeooaie aauae, iaaeaee n eo oi?iaie. Yoa niiniaiinoue e anoue eiooeoeey.

?acaeaay a naieo o/aieeao niiniaiinoue eiooeoeaii iiciaaaoue ie?,
Ieoaai? enoiaeee ec anoanoaaiiuo /oanoa /aeiaaea e iniiaiuo iaycaiiinoae
i?e ii anooieaiee a aeeciue e iieacuaae niioiioaiea iineaaeieo n
ie?iauie caeiiaie. Caia/aoeyy a na?aeoeao iieiaeuo ethaeae ethaiaue e
?iaeeoaeyi, Ieoaai? ?anoe?ye yoi /oanoai ioiaeaeanoaeaieai iooea n
eaeaae Aiaa, a iaoue – n eaeaae I?e?iaeu.

Ii ia aeaiiii yoaia Ieoaai? n/eoae, /oi eaeay Aaeeiiai Aiaa, Aa?oiaiie
enoeiu, aoaeao iaiiiyoia o/aieeai. Iiyoiio ei aeaaaeinue eeoue
i?aaeaeaeaiea aa, ia?aianaiiia ia iocueo e /enea.

Ieoaai?aeneay oai?ey /enae

Ieoaai? ia caienae naiaai o/aiey. Iii ecaanoii eeoue a ia?aneacao
A?enoioaey e Ieaoiia. A?enoioaeue ienae: «Ieoaai? i?eciae iaoaiaoe/aneea
ia/aea ca ia/aea anaai nouaai».

Oeeinioneay enoeia ia?aiineony ei ia iocueo e /enea. *enei iiieiaaony
eae oa?iei, i?eeiaeeiue ei anai oeeo?ai e eo eiiaeiaoeeyi. Ieoaai?
ii?aaeaeyy /enei eae yia?aeth e n/eoae, /oi /a?ac iaoeo i /eneao
?ane?uaaaony oaeia Anaeaiiie, eai /enei caeeth/aao a naaa oaeio aauae.

Eiaiii iaoea /enea iiaeao iaeaaeaoue eeth/ii aeecie e nooe auoey.

I?iieeay a naienoaa /enae, iauyniyy eo ?acee/iua ni/aoaiey, Ieoaai?
iuoaeny nicaeaoue iaoeo anao iaoe. Ana /enea ii ?acaeaeee ia aeaa aeaea:
/aoiua e ia/aoiua, e n oaeeaeoaeueiie /ooeinoueth auyaee naienoaa /enae
eaaeaeie a?oiiu. *aoiua /enea iaeaaeatho neaaeothueie naienoaaie: ethaia
/enei iiaeao auoue ?acaeaeaii ia aeaa ?aaiua /anoe, iaa ec eioi?uo eeai
/aoiu, eeai ia/aoiu. Iai?eia?, 14 aeaeeony ia aeaa ?aaiua /anoe 7 + 7,
aaea iaa /anoe ia/aoiua; 16 = 8 + 8, aaea iaa /anoe /aoiua. Ieoaai?aeoeu
?anniao?eaaee /aoiia /enei, i?ioioeiii eioi?iai auea aeoaaea,
iaii?aaeaeaiiui e aeaineei. «*aoiua /enea, aeiioneaaoea ?acaeaiaiea,
eacaeenue aieaa ?acoiiuie, ieeoeaoai?yee iaeioi?ia iieiaeeoaeueiia
yaeaiea», – ienae A?enoioaeue. Oae /enei iieo/aei oa?aeoa?, oa?yei
aa/iia, aano?aeoiia ia/aei.

*aoiua /enea Ieoaai? aeaeeee ia 3 eeanna: /aoii-/aoiua, /aoii-ia/aoiua,
ia/aoii-ia/aoiua.

Ia?aue eeann ninoaaeytho /enea, eioi?ua i?aaenoaaeytho niaie oaeaiaiea
/enae, ia/eiay n aaeeieoeu. Oaeei ia?acii, yoi 1,2,4,8,16,32,64,128,512
e 1024. Niaa?oainoai yoeo /enae Ieoaai? aeaeae a oii, /oi iie iiaoo
aeaeeoueny iiiieai e aua ?ac, e oae aeaeaa aei iieo/aiey aaeeieoeu.

*aoii-/aoiua /enea iaeaaeatho iaeioi?uie oieeaeueiuie naienoaaie. Noiia
ethaiai /enea oa?ieiia1, e?iia iineaaeiaai, anaaaea ?aaia iineaaeiaio ca
au/aoii aaeeieoeu. E i?eia?o, noiia /aou?ao oa?ieiia (1+2+4+8) ?aaia
iyoiio oa?ieio – 16 ieion iaeei, oi anoue 15.

?yae /aoii-/aoiuo /enae eiaao e oaeia naienoai: ia?aue /eai,
oiiiaeaiiue ia iineaaeiee, aeaao iineaaeiee iiea a ?yaeo n ia/aoiui
/eneii oa?ieiia ia inoaiaony iaeii /enei, eioi?ia aoaeo/e oiiiaeaii naii
ia naay aeano iineaaeiaa /enei a ?yaeo.

*aoii-ia/aoiua /enea – yoi /enea, eioi?ua aoaeo/e ?acaeaeaiu iiiieai ia
aeaeyony. Iie ia?acothony neaaeothuei ia?acii: aa?aony ia/aoiia /enei,
oiiiaeaaony ia 2, e oae aanue ?yae ia/aoiuo /enea. A yoii i?ioeanna
1,3,5,7,9,11 aeatho /aoii-ia/aoiua /enea 2,6,10,14,18,22. Oaeei ia?acii,
eaaeaeia oaeia /enei aeaeeony ia aeaa iaeei ?ac e aieueoa aeaeeoueny ia
iiaeao. Ae?oaay iniaaiiinoue yoiai eeanna /enae ninoieo a oii, /oi anee
aeaeeoaeue – ia/aoiia /enei, /anoiia – anaaaea aoaeao /aoiui, e
iaiai?io. Iai?eia?, anee 22 ?acaeaeeoue ia 2, /aoiue aeaeeoaeue, /anoiia
11 aoaeao ia/aoii.

Aeaiiue eeann /enea i?eia/aoaeai aua e oai, /oi ethaia /enei a ?yaeo
yaeyaony iieiaeiie noiiu oa?ieiia ii iaa noi?iiu aai a ?yaeo: 18 anoue
1/2 noiiu 14 e 22 (/enae noiyueo io aeaiiiai /enea ii iaa noi?iiu).

ia/aoii-ia/aoiua /enea yaeyaony eiii?iienniuie iaaeaeo /aoii-/aoiuie e
/aoii-ia/aoiuie /eneaie. A ioee/ea io /aoii-/aoiuo iie ia iiaoo
iineaaeiaaoaeueiui aeaeaieai i?eaanoe e 1, e a ioee/ea io /aoii-ia/aoiuo
iie iicaieytho aieaa /ai iaeiie?aoiia aeaeaiea iiiieai. Ia/aoii-ia/aoiua
/enea iieo/athony neaaeothuei ia?acii: oiiiaeay /aoii-/aoiia /enei
(aieueoa 2) ia ia/aoiia /enei. Ae?oaea ia/aoii-ia/aoiua /enea
ia?acothony oiiiaeaieai ?yaea ia/aoiuo /enae ia 4 e aeaeaa ia aanue ?yae
/aoii-/aoiuo /enae.

*aoiua /enea ?acaeaeythony ia o?e ae?oaeo eeanna: naa?oniaa?oaiiua,
ianiaa?oaiiua e niaa?oaiiua.

Naa?oniaa?oaiiua /enea – yoi oaeea /enea, noiia ae?iaiuo /anoae, eioi?uo
aieueoa eo naieo. Iai?eia?, 24 eiaao noiiie naieo ae?iaiuo /anoae
12+6+4+8+3+2+1 /enei 33, /oi i?aauoaao 24, enoiaeiia /enei.

Ianiaa?oaiiuie Ieoaai? iacuaae /enea, noiia ae?iaiuo /anoae, eioi?uo
iaiueoa aai naiiai. Iai?eia?, /enei 14 noiia aai ae?iaiuo /anoae
7+2+1=10, /oi iaiueoa 14.

Niaa?oaiiia /enei – yoi oaeia /enei, noiia ae?iaiuo /anoae eioi?iai
?aaia naiiio /eneo. Oaeea /enea /?acau/aeii ?aaeee. Anoue oieueei iaeii
/enei iaaeaeo 1 e 10, a eiaiii 6; iaeii iaaeaeo 10 e 100 – /enei 28,
iaeii iaaeaeo 100 e 1000 – 496, iaeii iaaeaeo 1000 e 10000 – 8128.
Niaa?oaiiua /enea iaoiaeyo neaaeothuei ia?acii: ia?aia /enei ?yaea
/aoii-/aoiuo /enae neeaaeuaaaony ni aoi?ui /eneii ?yaea, e anee
iieo/aaony i?inoia /enei, iii oiiiaeaaony ia iineaaeiaa /enei ?yaea
/aoii-/aoiuo /enae, o/anoaiaaaoeo a ia?aciaaiee noiiu. Anee neiaeaiea
/aoii-/aoiuo /enae ia i?eaiaeeo e ianinoaaiiio /eneo. Iai?eia?, ia?aua
aeaa /enea /aoii-/aoiiai ?yaea (1,2) a noiia 3, eioi?ia oiiiaeaaony ia 2
e iieo/aai 6, ia?aia niaa?oaiiia /enei. Niaa?oaiiua /enea, aoaeo/e
oiiiaeaiiuie ia 2, aeatho naa?oniaa?oaiiua /enea, a aoaeo/e
?acaeaeaiiuie iiiieai – ianiaa?oaiiua.

Ieoaai?aeoeu ?acaeaaee naith oeeinioeth ec iaoee i /eneao. Niaa?oaiiua
/enea, n/eoaee iie anoue i?ae?aniua ia?acu aeia?iaeaoaeae. Iie
i?aaenoaaeytho niaie na?aaeeio iaaeaeo eceeoanoaii e iaaeinoaoeii. Iie
i/aiue ?aaeee e ii?iaeaeathony niaa?oaiiui ii?yaeeii. A
i?ioeaiiieiaeiinoue yoiio naa?eciaeeueiua e ianiaa?oaiiua /enea, eioi?uo
neieue oaiaeii iiiai, ia ?aniieiaeaiu a ii?yaeea e ia ii?iaeaeathony n
iaeioi?ie ii?aaeaeaiiie oeaeueth. E iiyoiio iie eiatho aieueoia
noiaenoai n ii?ieaie, eioi?ua iiiai/eneaiiu, iaoii?yaei/aiu e
iaii?aaeaeaiu.

Ia/aoiua /enea ia iiaoo auoue ?acaeaeaiu ?aaiui ia?acii, oi anoue
ii?iaio. Ieoaai? iauyniye ianiiniaiinoue oaeeo /enae aeaeeony iiiieai
neaaeothuei ia?acii: iineieueeo 1 anaaaea inoaaony ia aeaeeiie, ia/aoiia
/enei oaeei aea ia?acii ia iiaeao auoue aeaeeiui. Anee ia/aoiia /enei
iiiuoaoueny ?acaeaeeoue ii?iaio, oi iieo/aaony aeaa /aoiuo /enea, a
iineaaeiaa ec ieo aaeeieoea, eioi?ay yaeyaony iaaeaeeiie. Iai?eia?, 9
anoue 4+4+1.

Ia/aoiua /enea eiatho e oaeia naienoai – anee eaeia-eeai ia/aoiia /enei
?acaeaeeoue ia aeaa /anoe, iaeia anaaaea aoaeao /aoiie, a ae?oaay –
anaaaea ia/aoiie.

Ieoaai?aeoeu ?anniao?eaaee ia/aoiia /enei, i?ioioeiii eioi?iai auea
iiiaaea, ii?aaeaeaiiui e ioaeneei, oioy ii iiaiaeo 1 (aaeeieoeu) n?aaee
ieo nouanoaiaaee ii?aaeaeaiiua ?aciiaeaney. Iaeioi?ua n/eoaee aai
iieiaeeoaeueiui, iioiio /oi, anee aai aeiaaaeoue e ia/aoiiio /enei, iii
noaiao /aoiui e, oaeei ia?acii, ?anniao?eaaaony eae aiae?iaaiiia /enei,
niaiauathuea eae ioaeneea, oae e aeaineea ao?eaoou, cia/eo iii e /aoii e
ia/aoii.

Iau/aai o ieoaai?oeaa auei i?eiioaiea aunoei aiaai ia/aoiiai /enea
i?aaeiaoia, a oi a?aiy eae aiaeiyi e iiaecaiiui aeooai i?eiineoue /aoiia
/enei.

Ia/aoiua /enea aeaeyony ia 3 iaueo eeanna: ianinoaaiua, ninoaaiua e
ianinoaaiua – ninoaaiua.

Ianinoaaiua /enea – yoi oaeea /enea, eioi?ua ia eiatho ae?oaeo
aeaeeoaeae, e?iia naay naiiai e aaeeieoeu. Yoi /enea 3,5,7,11,13,17 e
o.ae.

Ninoaaiua /enea – yoi /enea, aeaeeiua ia oieueei naie ia naay, ii e ia
iaeioi?ua ae?oaea /enea. Oaeeie /eneaie yaeythony oa ec ia/aoiuo /enae,
eioi?ua ia aoiaeyo a a?oiio ianinoaaiuo. Yoi /enea 9,15,21,25,27,33,39 e
o.ae.

Ianinoaaiua-ninoaaiua /enea – yoa /enea, ia eiathuea iauaai aeaeeoaey,
oioy eaaeaeia ec ieo aeaeeii. Anee acyoue aeaa /enea e iaia?oaeeoue, /oi
iie ia eiatho iauaai aeaeeoaey, oaeea /enea iiaeii iacaaoue
ianinoaaiuie-ninoaaiuie /eneaie. Iai?eia?, /enea 9 e 25. 9 aeaeeii ia 3,
a 25 ia 5, ii ie iaeii ec ieo ia aeaeeii ia aeaeeoaeue ae?oaiai, iie ia
eiatho iauaai aeaeeoaey. Ianinoaaiuie-ninoaaiuie iie iacuaathony iioiio,
/oi eaaeaeia ec ieo eiaao eiaeeaeaeoaeueiue aeaeeoaeue, a iineieueeo yoe
/enea ia eiatho iauaai aeaeeoaey, iie iacuaathony ianinoaaiuie. Oaeei
ia?acii, ianinoaaiua-ninoaaiua /enea iaia?oaeeaathony oieueei iiia?ii
ae?oa n ae?oaii.

Aeey ii?aaeaeaiey ninoaaiuo io ianinoaaiuo ia/aoiuo /enae aue i?eaeoiai
Y?aoinoaiii1 iaoaiaoe/aneee i?eai.

Nooue yoiai i?eaia ninoieo a neaaeothuai: ana ia/aoiua /enea
oii?yaei/eaathony ii aaee/eia, eae iieacaii ia aoi?ie aieco oaaeeoea,
iacaaiiie «ia/aoiua /enea». Ec oaaeeoeu aeaeii, /oi eaaeaeia o?aouea
/enei, ia/eiay n 3, aeaeeony ia 3, eaaeaeia iyoia – ia 5, naaeueiia – ia
7 e o.ae. aei aaneiia/iinoe. Yoio i?ioeann ionaeaaao i?inoua /enea, oi
anoue oa, eioi?ua ia eiatho ae?oaeo aeaeeoaeae, e?iia naay e aaeeieoeu.

Caeanue 5 oiiiaeaaony nia?aa ia 3, caoai ia 5, caoai ia 7 e o.ae.

?yae ia/aoiuo /enae i?inayiiuo /a?ac 5

15

Caeanue 3 oiiiaeaaony ia 3, caoai ia 5, caoai ia 7 e o.ae.

?yae ia/aoiuo /enae, i?inayiiuo /a?ac 3

9

15

21

Ia/aoiua /enea 1 3 5 7 9 11 13 15 17 19 21 23 25

I?inoua /enea

5 7

11 13

17 19

23

1 Oa?iei a aeaiiii neo/aa icia/aao /enei, neaaaaiia

1 Y?aoinoai aeee ieiei 230 a. aei i.y.

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