?aoa?ao ii aaiiao?ee
ia oaio:
“I?ecia”
o/auaeny 2 eo?na
Iineianeiai Yenoa?iaoa
Iineaa 1996
Iaeaaeaiea
TOC \t “Caaie1;1;Caaie2;2” 1. E?aoeee iaci? ?acaeoey aaiiao?ee
GOTOBUTTON _Toc355845835 PAGEREF _Toc355845835 3
1.1 Iauee enoi?e/aneee iaci? GOTOBUTTON _Toc355845836 PAGEREF
_Toc355845836 3
1.2. I ?acaeoee aaiiao?ee a Ae?aaiae A?aoeee aei Aaeeeaea GOTOBUTTON
_Toc355845837 PAGEREF _Toc355845837 6
2. I?ecia GOTOBUTTON _Toc355845838 PAGEREF _Toc355845838 10
2.1 Ieiuaaeue iiaa?oiinoe i?eciu GOTOBUTTON _Toc355845839 PAGEREF
_Toc355845839 11
2.2. I?ecia e ie?aieaea GOTOBUTTON _Toc355845840 PAGEREF
_Toc355845840 13
2.3. Ie?aieaea e ieiuaaeue aa iiaa?oiinoe GOTOBUTTON _Toc355845841
PAGEREF _Toc355845841 15
2.4. Ecia?aiea iauaiia GOTOBUTTON _Toc355845842 PAGEREF
_Toc355845842 16
2.5. I ie?aieaea e aa iauaia GOTOBUTTON _Toc355845843 PAGEREF
_Toc355845843 17
2.6. I i?ecia e ia?aeeaeaieiaaea GOTOBUTTON _Toc355845844 PAGEREF
_Toc355845844 19
2.7. Ia?aeeaeaieiaae GOTOBUTTON _Toc355845845 PAGEREF _Toc355845845
20
3. Neiiao?ey a i?ino?ainoaa GOTOBUTTON _Toc355845846 PAGEREF
_Toc355845846 24
I?ecia (/a?oaae) GOTOBUTTON _Toc355845847 PAGEREF _Toc355845847 25
Caaea/e GOTOBUTTON _Toc355845848 PAGEREF _Toc355845848 26
Eeoa?aoo?a GOTOBUTTON _Toc355845849 PAGEREF _Toc355845849 30
Ii?aaeaeaiea. Iiiaia?aiiee, aeaa a?aie eioi?iai – iaeiieiaiiua
iiiaioaieueieee, eaaeauea a ia?aeeaeueiuo ieineinoyo, a ethaua aeaa
?aa?a, ia eaaeauea a yoeo ieineinoyo, ia?aeeaeueiu, iacuaaaony i?eciie.
1. E?aoeee iaci? ?acaeoey aaiiao?ee
1.1 Iauee enoi?e/aneee iaci?
Ia?aua aaiiao?e/aneea iiiyoey aicieeee a aeienoi?e/aneea a?aiaia. ?aciua
oi?iu iaoa?eaeueiuo oae iaaethaeae /aeiaae a i?e?iaea: oi?iu ?anoaiee e
aeeaioiuo, ai? e ecaeeei ?ae, e?oaa e na?ia Eoiu e o. i. Iaeiaei /aeiaae
ia oieueei ianneaii iaaethaeae i?e?iaeo, ii i?aeoe/anee inaaeaae e
eniieueciaae aa aiaaonoaa. A i?ioeanna i?aeoe/aneie aeayoaeueiinoe ii
iaeaieeaae aaiiao?e/aneea naaaeaiey. Iaoa?eaeueiua iio?aaiinoe
iiaoaeaeaee ethaeae ecaioiaeyoue i?oaeey o?oaea, iaoanuaaoue eaiie e
no?ieoue aeeeeua, eaieoue aeeiyioth iinoaeo e iaoyaeaaoue oaoeao ia eoe.
Eiia/ii, aeanyoee e nioie ouny/ ?ac iaoyaeaaee ethaee naie eoee
ecaioiaeyee ?aciua i?aaeiaou n i?yiuie ?aa?aie e o. i., iiea iinoaiaiii
aeioee aei ioaea/aiiiai iiiyoey i?yiie eeiee. I?eia?ii oi aea iiaeii
neacaoue i ae?oaeo iniiaiuo aaiiao?e/aneeo iiiyoeyo. I?aeoe/aneay
aeayoaeueiinoue /aeiaaea neoaeeea iniiaie aeeeoaeueiiai i?ioeanna
au?aaioee ioaea/aiiuo iiiyoee, ioe?uoey i?inoaeoeo aaiiao?e/aneeo
caaeneiinoae e niioiioaiee.
Ia/aei aaiiao?ee auei iieiaeaii a ae?aaiinoe i?e ?aoaiee /enoi
i?aeoe/aneeo caaea/. Ni a?aiaiai, eiaaea iaeiieeinue aieueoia eiee/anoai
aaiiao?e/aneeo oaeoia, o ethaeae iiyaeeinue iio?aaiinoue iaiauaiey,
oyniaiey caaeneiinoe iaeieo yeaiaioia io ae?oaeo, onoaiiaeaiey
eiae/aneeo naycae e aeieacaoaeuenoa. Iinoaiaiii nicaeaaaeanue
aaiiao?e/aneay iaoea. I?eia?ii a VI – V aa. aei i. y. a Ae?aaiae A?aoeee
a aaiiao?ee ia/aeny iiaue yoai ?acaeoey, /oi iauyniyaony aunieei
o?iaiai, eioi?iai aeinoeaea iauanoaaiii-iieeoe/aneay e eoeueoo?iay
aeeciue a a?a/aneeo ainoaea?noaao. I?iecaaaeaiey, niaea?aeauea
nenoaiaoe/aneia eceiaeaiea aaiiao?ee, iiyaeeenue a A?aoeee aua a V aei
i.y., ii iie auee auoaniaiu “Ia/aeaie” Aaeeeaea.
Aaiiao?e/aneea ciaiey i?eia?ii a iauaia nia?aiaiiiai eo?na n?aaeiae
oeieu auee eceiaeaiu aua 2200 eao iacaae a “Ia/aeao” Aaeeeaea. Eiia/ii,
eceiaeaiiay a “Ia/aeao” iaoea aaiiao?ey ia iiaea auoue nicaeaia iaeiei
o/aiui. Ecaanoii, /oi Aaeeeae a naiae ?aaioa iie?aeny ia o?oaeu
aeanyoeia i?aaeoanoaaiieeia, n?aaee eioi?uo auee Oaean e Ieoaai?,
Aeaiie?eo e Aeiiie?ao, A?oeo, Oayoao, Aaaeien e ae?. Oeaiie aieueoeo
oneeee, enoiaey ec ioaeaeueiuo aaiiao?e/aneeo naaaeaiee, iaeiieaiiuo
ouny/aeaoeyie a i?aeoe/aneie aeayoaeueiinoe ethaeae, yoe aaeeeea o/aiua
noiaee ia i?ioyaeaiee 3 – 4 noieaoee i?eaanoe aaiiao?e/aneoth iaoeo e
aunieie nooiaie niaa?oainoaa. Enoi?e/aneay caneoaa Aaeeeaea ninoieo a
oii, /oi ii, nicaeaaay naie “Ia/aea”, iauaaeeiee ?acoeueoaou naieo
i?aaeoanoaaiieeia, oii?yaei/ee e i?eaae a iaeio nenoaio iniiaiua
aaiiao?e/aneea ciaiey oiai a?aiaie. Ia i?ioyaeaiee aeaoo ouny/aeaoee
aaiiao?ey eco/aeanue a oii iauaia, ii?yaeea e noeea, eae iia auea
eceiaeaia a “Ia/aeao” Aaeeeaea. Iiiaea o/aaieee yeaiaioa?iie aaiiao?ee
ai anai ie?a i?aaenoaaeyee (a iiiaea e iiiuia i?aaenoaaeytho) niaie
eeoue ia?a?aaioeo eieae Aaeeeaea. “Ia/aea” ia i?ioyaeaiee aaeia auee
ianoieueiie eieaie aaee/aeoeo o/aiuo.
A XVII a. Aeaea?o aeaaiaea?y iaoiaeo eii?aeeiao naeaeae aiciiaeiui
eco/aiea naienoa aaiiao?e/aneeo oeao? n iiiiuueth aeaaa?u. N yoiai
a?aiaie ia/aea ?acaeaaoueny aiaeeoe/aneay aaiiao?ey. A XVII – XVIII aa.
ca?iaeaeaaony e ?ac?aaaouaaaony aeeooa?aioeeaeueiay aaiiao?ey,
eco/athuay naienoaa oeao? n iiiiuueth iaoiaeia iaoaiaoe/aneiai aiaeeca.
A XVIII- XIX aa. ?acaeoea aiaiiiai aeaea e a?oeoaeoo?u i?eaaei e
?ac?aaioea iaoiaeia oi/iiai ecia?aaeaiey i?ino?ainoaaiiuo oeao? ia
ieineii /a?oaaea, a nayce n /ai iiyaeythony ia/a?oaoaeueiay aaiiao?ey,
iao/iua iniiau eioi?ie caeiaeee o?aioeocneee iaoaiaoee A. Iiiae, e
i?iaeoeaiay aaiiao?ey, iniiau eioi?ie auee nicaeaiu a o?oaeao
o?aioeocneeo iaoaiaoeeia Ae.Aeaca?aa e A.Ianeaey (XVII a.). A aa
nicaeaiee aaaeiaeooth ?ieue nua?ae ae?oaie o?aioeocneee iaoaiaoee – AE.
A. Iiinaea (XIX a.).
Ei?aiiie ia?aeii a aaiiao?ee aia?aua i?iecaae a ia?aie iieiaeia OIO a.
aaeeeee ?onneee iaoaiaoee Ieeieae Eaaiiae/ Eiaa/aaneee, eioi?ue nicaeae
iiaoth, iaaaeeeaeiao aaiiao?eth, iacuaaaioth iuia aaiiao?eae
Eiaa/aaneiai.
Ioe?uoea Eiaa/aaneiai auei ia/aeii iiaiai ia?eiaea a ?acaeoee aaiiao?ee.
Ca iei iineaaeiaaee iiaua ioe?uoey iaiaoeeiai iaoaiaoeea A. ?eiaia e
ae?.
A ianoiyuaa a?aiy aaiiao?ey oanii ia?aieaoaaony ni iiiaeie ae?oaeie
?acaeaeaie iaoaiaoeee. Iaeiei ec enoi/ieeia ?acaeoey e ia?aciaaiey iiauo
iiiyoee a aaiiao?ee, eae e a ae?oaeo iaeanoyo iaoaiaoeee, yaeythony
nia?aiaiiua caaea/e anoanoaiciaiey, oeceee e oaoieee.
1.2. I ?acaeoee aaiiao?ee a Ae?aaiae A?aoeee aei Aaeeeaea
O/aiua e oeeiniou Ae?aaiae A?aoeee aini?eiyee e ia?a?aaioaee
aeinoeaeaiey eoeueoo?u e iaoee Ae?aaiaai Ainoiea. Oaean, Ieoaai?,
Aeaiie?eo, Aaaeien e ae?. acaeeee a Aaeiao e Aaaeeii aeey eco/aiey
iocuee, iaoaiaoeee e ano?iiiiee. Ia neo/aeii ca/aoee a?a/aneie
aaiiao?e/aneie iaoee naycaiu n eiaiai Oaeana Ieeaoneiai, iniiaaoaey
eiieeneie oeieu. Eiieeoeu, ianaeyaoea oa??eoi?eth, eioi?ay a?aie/eea n
ainoi/iuie no?aiaie, ia?auie caeinoaiaaee ciaiey Ainoiea e noaee eo
?acaeaaoue. O/aiua eiieeneie oeieu aia?aua iiaeaa?aee eiae/aneie
ia?aaioea e nenoaiaoece?iaaee iaoaiaoe/aneea naaaeaiey, iicaeinoaiaaiiua
o ae?aaiaainoi/iuo ia?iaeia, a iniaaiiinoe o aaaeeiiyi. Oaeano, aeaaa
yoie oeieu, I?iee e ae?oaea enoi?eee i?eienuaatho iaiaei aaiiao?e/aneeo
ioe?uoee. Ia ioiioaiee Ieoaai?a Naiinneiai e aaiiao?ee I?iee ieoao a
naiai eiiiaioa?ee e “Ia/aeai” Aaeeeaea neaaeothuaa: “Ii eco/ae yoo iaoeo
(o. a. aaiiao?eth), enoiaey io ia?auo aa iniiaaiee, e noa?aeny iieo/aoue
oai?aiu i?e iiiiue /enoi eiae/aneiai iuoeaiey”. I?iee i?eienuaaao
Ieoaai?o, e?iia ecaanoiie oai?aiu i eaaae?aoa aeiioaiocu, aua iino?iaiea
iyoe i?aaeeueiuo iiiaia?aiieeia:
1) oao?ayae?, eiathuee 4 a?aie, 4 aa?oeiu, 6 ?aaa? (?en. );
2) eoa – 6 a?aiae, 8 aa?oei, 12 ?aaa? (?en. );
3) ieoayae? – 8 a?aiae, 6 aa?oei, 12 ?aaa? (?en. );
4) aeiaeaeayae? – 12 a?aiae, 20 aa?oei, 30 ?aaa? (?en. );
5) eeinayae? – 20 a?aiae, 12 aa?oei, 30 ?aaa? (?en. ).
A?aie aeiaeaeayae?a yaeythony i?aaeeueiuie iyoeoaieueieeaie. Aeeaaiiaee
aea i?aaeeueiiai iyoeoaieueieea ia?acotho oae iacuaaaiue caacae/aoue
iyoeoaieueiee (?en. ) – oeao?o, eioi?ay neoaeeea yiaeaiie,
iiiciaaaoaeueiui ciaeii aeey o/aieeia Ieoaai?a. Ecaanoii, /oi
ieoaai?aeneee nithc aue iaeiia?aiaiii oeeinioneie oeieie, iieeoe/aneie
ia?oeae e ?aeeaeiciui a?aonoaii. Niaeanii eaaaiaea, iaeei ieoaai?aaoe
caaieae ia /oaeaeia e ia iia ia?aae nia?oueth ?anieaoeoueny n
ooaaeeaaaoei ca iei oicyeiii aeiia. Iineaaeiee ia?eniaae ia noaia naiaai
aeiia caacae/aoue iyoeoaieueiee. Oaeaeaa /a?ac ianeieueei eao yoio ciae,
ae?oaie no?ainoaothuee ieoaai?aaoe inaaaeiieeny i neo/eaoainy o oicyeia
e uaae?i aai aiciaa?aaeee.
Aeinoiaa?iuo naaaeaiee i aeecie e iao/iie aeayoaeueiinoe Ieoaai?a ia
nio?aieeinue. Aio i?eienuaaaony nicaeaiea o/aiey i iiaeiaee oeao?. Ii,
aa?iyoii, aue n?aaee ia?auo o/aiuo, ?anniao?eaaaoeo aaiiao?eth ia eae
i?aeoe/aneoth e i?eeeaaeioth aeenoeeieeio, a eae aano?aeoioth
eiae/aneoth iaoeo.
A oeiea Ieoaai?a auei ioe?uoi nouanoaiaaiea ianiecia?eiuo aaee/ei, o. a.
oaeeo, ioiioaiea iaaeaeo eioi?uie iaaiciiaeii au?aceoue ieeaeei oeaeui
eee ae?iaiui /eneii. I?eia?ii iiaeao neoaeeoue ioiioaiea aeeeiu
aeeaaiiaee eaaae?aoa e aeeeia aai noi?iiu, ?aaiia Oe2. *enei yoi ia
yaeyaony ?aoeeiiaeueiui (o. a. oeaeui eee ioiioaieai aeaoo oeaeuo /enae)
e iacuaaaony e??aoeeiiaeueiui, o.a. ia?aoeeiiaeueiui (io eaoeineiai
ratio – ioiioaiea).
Ieoaai?aeoeu ia ciaee ae?oaeo /enae, e?iia ?aoeeiiaeueiuo. Iino?iea
aeeaaiiaeue eaaae?aoa, noi?iia eioi?iai ?aaia 1, iie eiinoaoe?iaaee, /oi
iia ia iiaeao auoue au?aaeaia ieeaeei /eneii, oae eae aeey ieo ia auei
ae?oaeo /enae, e?iia oeaeuo e ae?iaiuo. Yoio oaeo i?eaae a aieueoia
niouaiea ieoaai?aeoeaa, oae eae a iniiaa eo oeeinioee eaaeaei iiiyoea i
/enea eae iniiaa anao aauae e yaeaiee i?e?iaeu. Ii aio yoa aaeeeay
iniiaa – /enei – ia a ninoiyiee au?aceoue aeeeiu i?inoiai io?acea a
i?inoie oeao?a – aeeaaiiaee eaaae?aoa. Aio ii/aio ioe?uoea ianiecia?eiuo
aaee/ei yaeeinue aieueoei oaea?ii ii o/aieth Ieoaai?a e ieoaai?aeoeu
aeieai aai aea?aeaee a no?iaie oaeia. Niaeanii i?aaeaieth, o/aiee
Ieoaai?a, ?ane?uaoee ioaee/ii yoo oaeio, aue iaeacai aiaaie e iiaea ai
a?aiy ei?aaeae?ooaiey. Ioe?uoea ianiecia?eiuo aaee/ei auei aaaeiui
iiai?ioiui ioieoii a ?acaeoee aioe/iie iaoaiaoeee. Ociaa, /oi
nouanoaotho ioiioaiey aaee/ei, ia au?aaeaaiua ieeaeeie ?aoeeiiaeueiuie
/eneaie, ae?aaiaa?a/aneea o/aiua noaee i?aaenoaaeyoue aaee/eiu ia
a?eoiaoe/anee, a aaiiao?e/anee, ia /eneaie, a io?aceaie. Oaeei ia?acii,
aicieeea aaiiao?e/aneay aeaaa?a, a iioii e oai?ey ioiioaiee Aaaeiena.
2. I?ecia
?anniio?ei i?iecaieueiue iiiaioaieueiee, iai?eia?, iyoeoaieueiee AANDA
(ni. /a?oaae ia no?. 25), eioi?ue eaaeeo a ieineinoe a. ?anniio?ei
oaia?ue ia?aeeaeueiue ia?aiin, ii?aaeaeyaiue iaeioi?ui iaioeaaui
aaeoi?ii V, ia eaaeauei a ieineinoe. Ia?acii ieineinoe a aoaeao
ia?aeeaeueiay ae ieineinoue b. Ia?acii iiiaioaieueieea O aoaeao
iiiaioaieueiee O1=A1B1C1D1E1, eaaeauee a ieineinoe b. Iai?aaeaiiua
io?acee AA1, BB1 aoaeoo ia?aeeaeueiu, oae eae eaaeaeue ec ieo
ecia?aaeaao iaeei e oio aea aaeoi? V. Iiiaia?aiiee ABCDEA1B1C1D1E1
iacuaatho i?eciie.
Ii?aaeaeaiea 1. Iiiaia?aiiee, aeaa a?aie eioi?iai – iaeiieiaiiua
iiiaioaieueieee, eaaeauea a ia?aeeaeueiuo ieineinoyo, a ethaua aeaa
?aa?a, ia eaaeauea a yoeo ieineinoyo, ia?aeeaeueiu, iacuaaaony i?eciie.
Iiiaioaieueieee O e O1, eaaeauea a ia?aeeaeueiuo ieineinoyo, iacuaatho
iniiaaieyie i?eciu, a inoaeueiua a?aie – aieiauie a?aiyie.
Iiaa?oiinoue i?eciu, oaeei ia?acii, ninoieo ec aeaoo ?aaiuo
iiiaioaieueieeia (iniiaaiee) e ia?aeeaeia?aiiia (aieiauo a?aiae).
?acee/atho i?eciu o?aoaieueiua, /aou?aooaieueiua, iyoeoaieueiua e o.ae.
a caaeneiinoe io /enea aa?oei iniiaaiey.
Anee aieiaia ?aa?i i?eciu ia?iaiaeeeoey?ii ieineinoe aa iniiaaiey, oi
oaeoth i?ecio iacuaatho i?yiie; anee aieiaia ?aa?i i?eciu
ia?iaiaeeeoey?ii ieineinoe aa iniiaaiey, oi oaeoth i?ecio iacuaatho
iaeeiiiie. O i?yiie i?eciu aieiaua a?aie – i?yiioaieueieee.
Ia?iaiaeeeoey? e ieineinoyi iniiaaiee, eiioeu eioi?iai i?eiaaeeaaeao
yoei ieineinoyi, iacuaatho aunioie i?eciu. Ia ?en. io?acie A1O –
aunioa ecia?aaeaiiie i?eciu.
Ii?aaeaeaiea 2. I?yiay i?ecia, iniiaaieai eioi?ie neoaeeo i?aaeeueiue
iiiaioaieueiee, iacuaaaony i?aaeeueiie i?eciie.
Aieiaia ?aa?i i?yiie i?eciu, a oii /enea e i?aaeeueiie, anoue aa aunioa.
Ia ?enoiea ecia?aaeaia i?aaeeueiay oanoeoaieueiay i?ecia e aa
?acaa?noea; aunioa yoie i?eciu ?aaia aa aieiaiio ?aa?o. Io?acie, eiioeu
eioi?iai – aeaa aa?oeiu, ia i?eiaaeeaaeauea iaeiie a?aie i?eciu,
iacuaatho aa aeeaaiiaeueth. Io?acie B1D (ni. ?en. ) – aeeaaiiaeue
i?eciu. Na/aiea i?eciu n ieineinoueth, i?ioiaeyuae /a?ac aeaa aieiauo
?aa?a, ia eaaeaueo a iaeiie a?aie, iacuaatho aeeaaiiaeueiui na/aieai
i?eciu.
2.1 Ieiuaaeue iiaa?oiinoe i?eciu
Iiaa?oiinoue iiiaia?aiieea ninoieo ec eiia/iiai /enea iiiaioaieueieeia
(a?aiae). Ieiuaaeue iiaa?oiinoe iiiaia?aiieea anoue noiia ieiuaaeae anao
aai a?aiae. Ieiuaaeue iiaa?oiinoe i?eci (Si?) ?aaia noiia ieiuaaeae aa
aieiauo a?aiae (ieiuaaee aieiaie iiaa?oiinoe Saie) e ieiuaaeae aeaoo
iniiaaiee (2Sini) – ?aaiuo iiiaioaieueieeia: Si?=Saie+2Sini.
Oai?aia. Ieiuaaeue aieiaie iiaa?oiinoe i?eciu ?aaia i?iecaaaeaieth
ia?eiao?a aa ia?iaiaeeeoey?iiai na/aiey e aeeeiu aieiaiai ?aa?a.
Aeaii: AN1 – i?iecaieueiay n-oaieueiay i?ecia (ia ?enoiea a ea/anoaa
i?eia?a ecia?aaeaia /aou?aooaieueiay i?ecia), a^AA1, A2B2C2D2 –
ia?iaiaeeeoey?iia na/aiea (na/aiea i?eciu ieineinoueth,
ia?iaiaeeeoey?iie aieiaiio ?aa?o), l – aeeeia aieiaiai ?aa?a.
Aeieacaoue: Saie = ?*l, aaea ? – ia?eiao? ia?iaiaeeeoey?iiai na/aiey.
Aeieacaoaeuenoai. Saie= SAA1B1B + SBB1C1C + SCC1D1D +…
1444442444443
n neaaaaiuo
Eaaeaeay aieiaay a?aiue i?eciu – ia?aeeaeia?aii, iniiaaiea eioi?iai –
aieiaia ?aa?i i?eciu, a aunioa – noi?iia ia?iaiaeeeoey?iiai na/aiey.
Iiyoiio
Saie=lA2B2+lB2C2+lC2D2+…=(A2B2+B2C2+C2D2+…)l=P*l.
Saie = ?*l.
Oai?aia aeieacaia.
Neaaenoaea. Ieiuaaeue aieiaie iiaa?oiinoe i?yiie i?eciu ?aaia
i?iecaaaeaieth ia?eiao?a aa iniiaaiey e auniou.
Aeaenoaeoaeueii, o i?yiie i?eciu iniiaaiea iiaeii ?anniao?eaaoue eae
ia?iaiaeeeoey?iia na/aiea, a aieiaia ?aa?i anoue aunioa.
2.2. I?ecia e ie?aieaea
Iiaeiaii oiio, eae o?aoaieueiee a iiieiaiee Aaeeeaea ia yaeythony
ionoui, o. a. i?aaenoaaeyao niaie /anoue ieineinoe, ia?aie/aiioth o?aiy
iaeiieo?aioiuie (o. a. ia ia?anaeathueieny a iaeiie oi/ea) io?aceaie,
oae e iiiaia?aiiee o iaai ia ionoie, ia iieue, a /ai-oi caiieiaiiue
(ii-iaoaio – /anoueth i?ino?ainoaa). A aioe/iie iaoaiaoeea, iaeiaei,
iiiyoey ioaea/aiiiai i?ino?ainoaa aua ia auei. Aaeeeae ii?aaeaeyao
i?ecio eae oaeanioth oeao?o, caeeth/aiioth iaaeaeo aeaoiy ?aaiuie e
ia?aeeaeueiuie ieineinoyie (iniiaaieyie) e n aieiauie a?aiyie –
ia?aeeaeia?aiiaie. Aeey oiai /oiau yoi ii?aaeaeaiea auei aiieia
ei??aeoiui, neaaeiaaei au, iaeiaei, aeieacaoue, /oi ieineinoe,
i?ioiaeyuea /a?ac ia?u iaia?aeeaeueiuo noi?ii iniiaaiee, ia?anaeathony
ii ia?aeeaeueiui i?yiui. Aaeeeae oiio?aaeyao oa?iei “ieineinoue” eae a
oe?ieii niunea (?anniao?eaay aa iaia?aie/aiii i?iaeieaeaiiie ai ana
iai?aaeaiey), oae e a niunea eiia/iie, ia?aie/aiiie aa /anoe, a
/anoiinoe a?aie, aiaeiae/ii i?eiaiaieth ei oa?ieia “i?yiay” (a oe?ieii
niunea – aaneiia/iay i?yiay e a oceii – io?acie). A XVIII a. Oaeei? aeae
oaeia ii?aaeaeaiea i?eciu: yoi iiiaia?aiiee, o eioi?iai ana a?aie, e?iia
aeaoo, ia?aeeaeueiu iaeiie i?yiie.
Ie?aieaeo Aaeeeae ii?aaeaeyao eae oaeanioth oeao?o, ia?aie/aiioth
ieineinoyie, eioi?ua io iaeiie ieineinoe (iniiaaiey) noiaeyony a iaeiie
oi/ea (aa?oeia). Yai ii?aaeaeaiea iiaeaa?aaeinue e?eoeea oaea a
ae?aaiinoe, iai?eia?, Aa?iiii, i?aaeeiaeeaoei neaaeothuaa ii?aaeaeaiea
ie?aieaeu: yoi oeao?a, ia?aie/aiiay o?aoaieueieeaie, noiaeyueieny a
iaeiie oi/ea, e iniiaaieai eioi?ie neoaeeo iiiaioaieueiee.
Aaaeiaeoei iaaeinoaoeii yoiai ii?aaeaeaiey yaeyaony eniieueciaaiea
iaii?aaeaeaiiiai iiiyoey iniiaaiey. Oaeei? ii?aaeaeee ie?aieaeo eae
iiiaia?aiiee, o eioi?iai ana a?aie, e?iia iaeiie, noiaeyony a iaeiie
oi/ea. Eaaeaiae? a “Yeaiaioao aaiiao?ee” oae ii?aaeaeyao ie?aieaeo:
“Oaeaniay oeao?a, ia?aciaaiiay o?aoaieueieeaie, noiaeyueieny a iaeiie
oi/ea e caeai/eaathuayny ia ?acee/iuo noi?iiao ieineiai iniiaaiey”.
Iinea yoie oi?ioee?iaee ?acuyniyaony iiiyoea iniiaaiey. Ii?aaeaeaiea
Eaaeaiae?a yaeyaony yaii ecauoi/iui, o.a. niaea?aeeo i?eciaee, eioi?ua
iiaeii auaanoe ec ae?oaeo. A aio aua iaeii ii?aaeaeaiea, eioi?ia
oeao?e?iaaei a o/aaieeao OIO a.: ie?aieaea – oaeaniue oaie, ia?ana/aiiue
ieineinoueth.
Aua a ae?aaiinoe nouanoaiaaee aeaa iooe ii?aaeaeaiey aaiiao?e/aneeo
iiiyoee. Ia?aue aae io oeao? aunoaai ii?yaeea e oeao?ai iecoaai. Oaeie
oi/ee c?aiey i?eaea?aeeaaeny, a /anoiinoe, Aaeeeae, ii?aaeaeythuee
iiaa?oiinoue eae a?aieoeo oaea, eeieth – eae a?aieoeo iiaa?oiinoe,
eiioeu aea eeiee – eae oi/ee. Aoi?ie iooue aaaeao, iaiai?io, io oeao?
iecoaai ecia?aiey e oeao?ai aunoaai: aeaeaeaieai oi/ee ia?acoaony eeiey,
aiaeiae/ii ec eeiee ninoaaeyaony iiaa?oiinoue e o. ae. Iaeiei ec ia?auo,
eioi?ue niaaeeiee iaa yoe oi/ee c?aiey, aue Aa?ii Aeaenaiae?eeneee,
ienaaoee, /oi oaei ia?aie/eaaaony iiaa?oiinoueth e aianoa n yoei iiaeao
auoue ?anniio?aii eae ia?aciaaiiia aeaeaeaieai iiaa?oiinoe. A
iiyaeaoeony iicaea ia i?ioyaeaiee aaeia o/aaieeao aaiiao?ee i?eieiaeanue
ca iniiao oi iaeia, oi ae?oaay, a eiiaaea e iaa aianoa oi/ee c?aiey.
2.3. Ie?aieaea e ieiuaaeue aa iiaa?oiinoe
Ii?aaeaeaiea. Iiiaia?aiiee, iaeia ec a?aiae eioi?iai – iiiaioaieueiee, a
inoaeueiua a?aie – o?aoaieueieee n iauae aa?oeiie, iacuaaaony
ie?aieaeie.
Ia ?enoiea ecia?aaeaiu iyoeoaieueiay ie?aieaea SABCDE e aa
?acaa?oea. O?aoaieueieee, eiathuea iauoth aa?oeio, iacuaatho aieiauie
a?aiyie ie?aieaeu; iauoth aa?oeio aieiauo a?aiae – aa?oeiie ie?aieaeu;
iiiaioaieueiee, eioi?iio ia i?eiaaeeaaeeo yoa aa?oeia,- iniiaaieai
ie?aieaeu; ?aa?a ie?aieaeu, noiaeyueany a aa aa?oeia,- aieiauie ?aa?aie
ie?a-ieaeu. Aunioa ie?aieaeu – yoi io?acie ia?iaiaeeeoey?a,
i?iaaaeaiiiai /a?ac aa aa?oeio e ieineinoe iniiaaiey, n eiioeaie a
aa?oeia e ia ieineinoe iniiaaiey ie?aieaeu. Ia ?enoiea io?acie SO –
aunioa ie?aieaeu.
Ii?aaeaeaiea. Ie?aieaea, iniiaaiea eioi?ie – i?aaeeueiue iiiaioaieueiee
e aa?oeia i?iaeoe?oaony a aai oeaio?, iacuaaaony i?aaeeueiie.
Ia ?enoiea ecia?aaeaia i?aaeeueiay oanoeoaieueiay ie?aieaea.
2.4. Ecia?aiea iauaiia
Iauaiu ca?iiauo aiaa?ia e ae?oaeo nii?oaeaiee a aeaea eoaia, i?eci e
oeeeeiae?ia aaeioyia e aaaeeiiyia, eeoaeoeu e eiaeeeoeu au/eneyee iooai
oiiiaeaiey ieiuaaee iniiaaiey ia aunioo. Iaeiaei ae?aaiaio Ainoieo auee
ecaanoiu a iniiaiii oieueei ioaeaeueiua i?aaeea, iaeaeaiiua iiuoiui
iooai, eioi?uie iieueciaaeenue aeey iaoiaeaeaiey iauaiia aeey ieiuaaeae
oeao?. A aieaa iicaeiaa a?aiy, eiaaea aaiiao?ey noi?ie?iaaeanue eae
iaoea, aue iaeaeai iauee iiaeoiae e au/eneaieth iauaiia iiiaia?aiieeia.
N?aaee caia/aoaeueiuo a?a/aneeo o/aiuo V – IV aa. aei i.y., eioi?ua
?ac?aaaouaaee oai?eth iauaiia, auee Aeaiie?eo ec Aaaea?u e Aaaeien
Eieaeneee.
Aaeeeae ia i?eiaiyao oa?ieia “iauai”. Aeey iaai oa?iei “eoa”, iai?eia?,
icia/aao e iauai eoaa. A OI eieaa “Ia/ae” eceiaeaiu n?aaee ae?oaeo e
oai?aiu neaaeothuaai niaea?aeaiey.
1. Ia?aeeaeaieiaaeu n iaeeiaeiauie aunioaie e ?aaiiaaeeeeie iniiaaieyie
?aaiiaaeeee.
2. Ioiioaiea iauaiia aeaoo ia?aeeaeaieiaaeia n ?aaiuie aunioaie ?aaii
ioiioaieth ieiuaaeae eo iniiaaiee.
3. A ?aaiiaaeeeeo ia?aeeaeaieiaaeao ieiuaaee iniiaaiee ia?aoii
i?iii?oeeiiaeueiu aunioai.
Oai?aiu Aaeeeaea ioiinyony oieueei e n?aaiaieth iauaiia, oae eae
iaiin?aaenoaaiiia au/eneaiea iauaiia oae Aaeeeae, aa?iyoii, n/eoae
aeaeii i?aeoe/aneeo ?oeiaiaenoa ii aaiiao?ee. A i?iecaaaeaieyo
i?eeeaaeiiai oa?aeoa?a Aa?iia Aeaenaiae?eeneiai eiathony i?aaeea aeey
au/eneaiee iauaia eoaa, i?eciu, ia?aeeaeaieiaaea e ae?oaeo
i?ino?ainoaaiiuo oeao?.
2.5. I ie?aieaea e aa iauaia
Oa?iei “ie?aieaea” caeinoaiaai ec a?a/aneiai “ie?aien” eee “ie?aieaein”.
A?aee a naith i/a?aaeue iicaeinoaiaaee yoi neiai, eae iieaaatho, ec
aaeiaoneiai ycuea. A iaie?ona Aoiana ano?a/aaony neiai “ie?aion” a
niunea ?aa?a i?aaeeueiie ie?aieaeu. Ae?oaea n/eoatho, /oi oa?iei aa?ao
naia ia/aei io oi?i oeaaoeaa a Ae?aaiae A?aoeee “ie?in” – ?iaeue). A
nayce n oai, /oi oi?ia ieaiaie eiiaaea iaiiieiaao ia?ac ie?aieaeu,
iaeioi?ua n?aaeiaaaeiaua o/aiua n/eoaee, /oi oa?iei i?ienoiaeeo
a?a/aneiai neiaa “ie?” – iaiiue. Aio ii/aio a iaeioi?uo o/aaieeao
aaiiao?ee XVI a. ie?aieaea iacaaia “iaiaoi?iiia oaei”.
A Ae?aaiai Aaeioa a?iaieoeu oa?aiiia eiaee oi?io ie?aieae. A III
Ouny/aeaoee aei i.y. aaeioyia nii?oaeaee nooiai/aoua ie?aieaeu,
neiaeaiiua ec eaiaiiuo aeieia; iicaea aaeiaoneea ie?aieaeu i?eia?aee
aaiiao?e/anee i?aaeeueioth oi?io, iai?eia? ie?aieaea Oaiina, aunioa
eioi?ie aeinoeaaao ii/oe 147 i, e ae?. Aioo?e ie?aieae iaoiaeeeenue
iia?aaaeueiua neeaiu e ei?eaei?u.
Niaeanii A?oeiaaeo, aua a V aei i.y. Aeaiie?eo ec Aaaea?u onoaiiaee, /oi
iauai ie?aieaeu ?aaai iaeiie o?aoe iauaia i?eciu n oai aea iniiaaieai e
oie aea aunioie. Iieiia aeieacaoaeuenoai yoie oai?aiu aeae Aaaeien
Eieaeneee a IV aei i.y.
A “Ia/aeao” Aaeeeaea aeieacuaaaony, /oi a ?aaiiaaeeeeo ie?aieaeao
ieiuaaee iniiaaiee ia?aoii i?iii?oeeiiaeueiu niioaaonoaothuei aunioai.
Ia?aia iaiin?aaenoaaiiia au/eneaiea iauaia ie?aieaeu, aeioaaeoaa aei
ian, ano?a/aaony o Aa?iia Aeaenaiae?eeneiai.
Eioa?anii ioiaoeoue, /oi a ae?aaieo aeieoiaioao ano?a/athony i?aaeea
aeey ii?aaeaeaiey iauaia ona/aiiie ie?aieaeu, i iao i?aaee au/eneaiey
iauaia iieiie ie?aieaeu. A “Iineianeii iaie?ona” eiaaony caaea/a,
icaaeaaeaiiay “Aeaenoaey n ona/aiiie ie?aieaeie”, a eioi?ie eceaaaaony
aa?iia au/eneaiea iauaia iaeiie ona/aiiie ie?aieaeu. A aaaeeiineeo
eeeiiieniuo oaaee/eao oaeaea ia ano?a/aaony au/eneaiea iauaia ie?aieaeu,
ii caoi a ieo anoue iiiai i?eia?ia au/eneaiey iauaia ona/aiiie
ie?aieaeu.
2.6. I i?ecia e ia?aeeaeaieiaaea
A iaiyoieeao aaaeeiineie e ae?aaiaaaeiaoneie a?oeoaeoo?u ano?a/athony
oaeea aaiiao?e/aneea oeao?u, eae eoa, ia?aeeaeaieiaae, i?ecia.
Aaaeiaeoae caaea/ae aaeiaoneie e aaaeeiineie aaiiao?ee auei ii?aaeaeaiea
iauaia ?acee/iuo i?ino?ainoaaiiuo oeao?. Yoa caaea/a ioaa/aea
iaiaoiaeeiinoe no?ieoue aeiia, aeai?oeu, o?aiu e ae?oaea nii?oaeaiey.
*anoue aaiiao?ee, a eioi?ie eco/athony naienoaa eoaa, i?eciu,
ia?aeeaeaieiaaea e ae?oaeo aaiiao?e/aneeo oae e i?ino?ainoaaiiuo oeao?,
ecaeaaia iacuaaaony noa?aiiao?eae; Neiai yoi a?a/aneiai i?ienoiaeaeaiey
(“noa?ain” – i?ino?ainoaaiiue, “iao?ai” – ecia?yth) e ano?a/aaony aua o
ciaiaieoiai ae?aaiaa?a/aneiai oeeinioa A?enoioaey. Noa?aiiao?ey aicieeea
iicaea, /ai ieaieiao?ey. Aaeeeae aeaao neaaeothuaa ii?aaeaeaiea i?eciu:
“I?ecia anoue oaeaniay (o.a. i?ino?ainoaaiiay) oeao?a, caeeth/aiiay
iaaeaeo ieineinoyie, ec eioi?uo aeaa i?ioeaiiieiaeiua ?aaiu e
ia?aeeaeueiu, inoaeueiua aea – ia?aeeaeia?aiiu”. Ooo, eae e ai iiiaeo
ae?oaeo ianoao, Aaeeeae oiio?aaeyao oa?iei “ieineinoue” ia a niunea
aaca?aie/ii i?iaeieaeaiiie ieineinoe, a a niunea ia?aie/aiiie aa /anoe,
a?aie, iiaeiaii oiio eae “i?yiay” icia/aao o iaai e io?acie i?yiie.
Oa?iei “i?ecia” a?a/aneiai i?ienoiaeaeaiey e aoeaaeueii icia/aao
“ioieeaiiia” (oaei).
Oa?iei “ia?aeeaeaieiaaeaeueiia oaei” ano?a/aaony aia?aua o Aaeeeaea e
icia/aao aeineiaii “ia?aeeaea-ieineinoiia oaei”. A?a/aneia neiai “eoain”
oiio?aaeyaony Aaeeeaeii a oii aea niunea, /oi e iaoa neiai “eoa”
2.7. Ia?aeeaeaieiaae
Ii?aaeaeaiea. I?ecia, iniiaaiea eioi?ie – ia?aeeaeia?aii, iacuaaaony
ia?aeeaeaieiaaeii.
A niioaaonoaee n ii?aaeaeaieai ia?aeeaeaieiaae – yoi /aou?aooaieueiay
i?ecia, ana a?aie eioi?ie – ia?aeeaeia?aiiu (?en. ). Ia?aeeaeaieiaaeu,
eae e i?eciu, iiaoo auoue i?yiuie e iaeeiiiuie. Ia ?enoiea ecia?aaeai
iaeeiiiue ia?aeeaeaieiaae, a ia ?enoiea – i?yiie ia?aeeaeaieiaae.
I?yiie ia?aeeaeaieiaae, iniiaaieai eioi?iai neoaeeo i?yiioaieueiee,
iacuaatho i?yiioaieueiui ia?aeeaeaieiaaeii. O i?yiioaieueiiai
ia?aeeaeaieiaaea ana a?aie – i?yiioaieueieee. Iiaeaeyie i?yiioaieueiiai
ia?aeeaeaieiaaea neoaeao eeanniay eiiiaoa, ee?ie/, nie/a/iay ei?iaea.
Aeeeiu o?ao ?aaa? i?yiioaieueiiai ia?aeeaeaieiaaea, eiathueo iauee
eiiaoe, iacuaatho aai ecia?aieyie. Iai?eia?, eiathony nie/a/iua ei?iaee
n ecia?aieyie 15, 35, 50 ii. Eoa – i?yiioaieueiue ia?aeeaeaieiaae n
?aaiuie ecia?aieyie. Ana oanoue a?aiae eoaa – ?aaiua eaaae?aou.
?anniio?ei iaeioi?ua naienoaa ia?aeeaeaieiaaea.
Oai?aia. Ia?aeeaeaieiaae neiiao?e/ai ioiineoaeueii na?aaeeiu aai
aeeaaiiaee.
Aeaii: AN1 (?en. ) – i?iecaieueiue ia?aeeaeaieiaae, A1D – aai
aeeaaiiaeue, oi/ea I – na?aaeeia yoie aeeaaiiaee.
Aeieacaoue: Z0(AC1) = AC1.
Aeieacaoaeuenoai. ?anniio?ei oeaio?aeueioth neiiao?eth Z0 n oeaio?ii a
oi/ea I. Oeaio?aeueiay neiiao?ey – ia?aiauaiea (nio?aiyao ?annoiyiey),
ioia?aaeathuaa eaaeaeue eo/ ia i?ioeaiiieiaeiue aio eo/. Iiyoiio
B1 = Z0(D), B1C1 = Z0(DA), DA = B1C1, C1 = Z0(A).
Aiaeiae/ii iiaeii iieacaoue, /oi oi/ee D1 e A, A1 e N oaeaea
oeaio?aeueii-neiiao?e/iu. Oaeei ia?acii, neiiao?ey ioia?aaeaao
iiaa?oiinoue ia?aeeaeaieiaaea ia naay. Aioo?aiiinoue ia?aeeaeaieiaaea
oaeaea ioia?aaeaao ia naay (ia?aeeaeaieiaae iiaeii ?anniao?eaaoue eae
ia?ana/aiea iieoi?ino?ainoa, ia?aciaaiiuo ieineinoyie aai a?aiae, a
ia?aiauaiea ioia?aaeaao ia?ana/aiea oeao? ia ia?ana/aiea eo ia?acia).
Oaeei ia?acii, oeaio?aeueiay neiiao?ey Z0 ioia?aaeaao ia?aeeaeaieiaae ia
naay: Z0(AC1) = AC1. Oai?aia aeieacaia.
Ec oai?aiu iaiin?aaenoaaiii neaaeotho aaaeiua naienoaa ia?aeeaeaieiaaea:
1. Ethaie io?acie n eiioeaie, i?eiaaeeaaeaueie iiaa?oiinoe
ia?aeeaeaieiaaea e i?ioiaeyuee /a?ac na?aaeeio aai aeeaaiiaee, aeaeeony
ath iiiieai; a /anoiinoe, ana aeeaaiiaee ia?aeeaeaieiaaea ia?anaeathony
a iaeiie oi/ea e aeaeyony ath iiiieai.
Oae, ia ?enoiea A1O=OC, B1O=OD, D1O=OB, AO=OC1, a oaeaea MO=ON, aaea
M`A1B1C1D1, N`ABCD, O`MN.
2. I?ioeaieaaeauea a?aie ia?aeeaeaieiaaea ia?aeeaeueiu e ?aaiu.
Oae, ia ?enoiea AA1D1D=BB1C1C, (AA1D1)I(BB1C1).
?anniio?aiiuie naienoaaie iaeaaeaao i?iecaieueiue ia?aeeaeaieiaae.
Aeieaaeai iaeii naienoai i?yiioaieueiiai ia?aeeaeaieiaaea.
Oai?aia. Eaaae?ao aeeeiu aeeaaiiaee i?yiioaieueiiai ia?aeeaeaieiaaea
?aaai noiia eaaae?aoa o?ao aai ecia?aiee.
Aeaii: AN1 – i?yiioaieueiue ia?aeeaeaieiaae, /AB/= a, /AD/=b, /AA1/=c –
aai ecia?aiey, /AC1/=d – aeeeia aai aeeaaiiaee.
Aeieacaoue: d2=a2+b2+c2.
Aeieacaoaeuenoai. Aaaaeai nenoaio eii?aeeiao oae, eae iieacaii ia
?enoiea , i?eiya ca aa ia/aei aa?oeio A, ca i?iecaieueiue aacen
o?ieeo aaeoi?ia V, b, c. Oiaaea aaeoi? AC eiaao eii?aeeiaou (a;b;c), e,
neaaeiaaoaeueii,
?
/AC / 2= d2=a2+b2+c2.
Oai?aia aeieacaia.
3. Neiiao?ey a i?ino?ainoaa
Oai?aia, a eioi?ie ooaa?aeaeaaony, /oi ana aeeaaiiaee ia?aeeaeaieiaaea
ia?anaeathony a iaeiie oi/ea I, a eioi?ie iie aeaeyony iiiieai (?en ),
iaiiieiaao aiaeiae/iia i?aaeeiaeaiea ec ieaieiao?ee: aeeaaiiaee
ia?aeeaeia?aiia ia?anaeathony a oi/ea I, yaeythuaeny eo na?aaeeiie (?en.
). Oi/ea I – yoi oeaio? neiiao?ee ia?aeeaeia?aiia. Aiaeiae/ii
iacuaatho e oi/eo I oeaio?ii neiiao?ee ia?aeeaeaieiaaea, oae eae aa?oeiu
A e N1, A e D1, N e A1, D e A1 neiiao?e/iu ioiineoaeueii oi/ee I.
Aia?aua iiiyoea oeaio?a neiiao?ee ano?a/aaony a OVI a. a iaeiie ec
oai?ai Eeaaeona, aeanyuae: anee ia?aeeaeaieiaae ?annaeaaony
ieineinoueth, i?ioiaeyuae /a?ac oeaio?, oi ii ?acaeaaaony iiiieai e,
iaiai?io, anee ia?aeeaeaieiaae ?annaeaaony iiiieai, oi ieineinoue
i?ioiaeeo /a?ac oeaio?. Eaaeaiae?, eioi?ue aia?aua aaae a yeaiaioa?ioth
aaiiao?eth yeaiaiou o/aiey i neiiao?ee, aiai?eo oieueei i neiiao?ee
ioiineoaeueii ieineinoe e aeaao neaaeothuaa ii?aaeaeaiea: aeaa oi/ee A e
B neiiao?e/iu ioiineoaeueii ieineinoe a, anee iineaaeiyy
ia?iaiaeeeoey?ia e AA a na?aaeeia yoiai io?acea. Eaaeaiae? iieacuaaao,
/oi o i?yiiai ia?aeeaeaieiaaea eiathony 3 ieineinoe neiiao?ee,
ia?iaiaeeeoey?iua e ?aa?ai, a o eoaa 9 ieineinoae neiiao?ee, ec eioi?uo
3 ia?iaiaeeeoey?iu e ?aa?ai, a ae?oaea 6 i?ioiaeyo /a?ac aeeaaiiaee
a?aiae.
I?ecia
Caaea/e
Eeoa?aoo?a
1. Aeaeca? A.Ae. Aaiiao?ey. O/aaiia iiniaea aeey noa?oeo eeannia. I.,
I?inaauaiea, 1994.
2. Iiai?aeia A.A. Aaiiao?ey. O/aaiia iiniaea aeey 7-11 eeannia. I.,
I?inaauaiea, 1992.
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