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XIV eeH l 7. 3 j/,. '7 6 n B 9
1508,. L 2 3 4 S b 7 8 9
JlpeBHAIi .z j3 r l' l 5  ,z D l
r pel.i,MA
,
PHM I 11 11/ "I V VI VII 111/1 IX X
- 6
,e
'2
13
14
11 H 12, nocTpoeHbI He no )J.ecHTHqHOMY npHHuHny, COqeTalO-
meMY )J.ecHTKH C e)J.HHHllaMlI, H 3TH nOCTpoeHHH He CBH3aHbI
co C.TJOBOM, 0003HaqaJOlll.HM qHC.10 10. Bo <ppaHllY3cKOM H3bIKe
CJIOBa, 0603HaqalOlll.He 20 H 80, nocTpoeHbI TaK, 'ITa n03BOJ1HlOT
npe.n.nO,TIaraTb 0 nepBOHaqaJ1bHoM cymecTBoBaHHH CHCTeMbI
C OCHOBaHHeM 20. Ho KaKoe 6bI qHCJ10 HH 6bIJ10 nOJ10)KeHO B
OCHOBY n03HllHoHHOll CHCTeMbI CIlHCJ1eHHH, )J.J1H 3anHCH qHCe,TJ
nOTpe6yeTcH CTOJ1bKO 3HaKOB (umpp), CKO.rlbKO e,J,HHHll HMeeT
ee OCHOBaHHe. CHcTeMa C MaJlbIM OCHOBaHHe:\J Tpe6yeT He-
l\IHOro I.{H<PP, HO qHc.1a B HeH 3anHCbIBalOTCH OqeHb )J.JJHHHO.
JO

3anHCb >Ke qHCeJl B CHCTeMe c 60JlbllIHM OCHOBaHHeM KOpOqe,
HO I3 Hell Tpe6yeTcH 60JIbIlJe UH<pP H TpY)J.Hee 3anOMHHTb Ta6-
JIHUY YMHO)KeHHH.
DOCMOTPHM, KaK 3anHllIeTCH O)J.HO H TO }Ke qHCJIO B pa3-
JlHqHbIX CHCTeMax CqHCJIeHHH. B03bMeM qHCJIO: qeTbIpeCTa
llIeCTb)J.eCHT BoceMb.
B )J.eCHTHqHOH CHceMe CqHCJIeHHH OHO 3anHIlJeTCH TaK: 468.
HanHllIeM 3TO }Ke qHCJlO B nHTepHqHOll CHCTeMe. BbIHC-
HHM, CKOJIbKO B HaIlJeM qHCJIe nHTKOB, T. e. e)J.HHHU BToporo
pa3pH,Aa: 468'; 50:='93
-45
18
-15
o
HT8K, B HallieM qHCJle 3 npocTbIe e)J.HHHUbI H 93 e,nHHHUbI
BToporo pa3pH)J.a. Ho Ka)K)J.bIe nHTb e;I.HHHU BToporo pa3pH)J.a
COCraBJIHIOT e)J.HHHUY TpeTbero pa3pH)J.a. BbIHCHHM, CKOJlbKO
B HaIlJeM qHCJJe e)J.HHHU TpeTbero pa3pH)la:
93:5= 18
-5
43
40
Q)
OCTaToK 3 YKa3bIBaeT Ha qHCJIO e.lI.HHHU BToporo pa3pH)J.a,
a qaCTHOe 18 - HaqHCJIo e)J.HHHU TpeTbero pa3pH)J.a. Onpe)J.e-
JlHeM, CKOJIbKO B 18 e)J.HHHllax TpeTbero pa3pH,Aa e)J.HHHll
qeTBeproro pa3pH,Aa:
1815=1J'I
-15 w
@
HTaK, B nHTepHqHOll cH.cTeMe qHCJJO 468 3anHIlJeTCH TaK: 3333.
06paTHbIH nepexo)J. MO)KeT 6bITb C)J.eJIaH npH nOMOlll.H
paBeHcTBa: 3333=3.5 8 +3.5 2 +3.5+3=375+75+ 15+3=468.
BbIHCHHM, KaK 3anHIlJeTCH ':lHCJIO 468 B TPOHqHOH CHCTeMe.
BbIqHCJIeHHH paCnOJlO}KHM TaR, KaK YKa3aHO HH)Ke:
468 1 3
-468  I 3
-@ -=-156 _52 [ 3
@  _17 1 2-
CD ._5 1 
0- cD
Q)
II

TaKH1\I 06pa30M, MbI BH)J.HM, qTO B TPOHqHOH CHCreMe qHCJ10
468 3anHIlJeTCH B BH)J.e: 122100.
06paTHbIH nepexo.n )J.aCT:
122100=1.3 5 +2.3 4 +2.3 3 +1.3 2 +0.3+0=243+162+
+54+9+0+0=468.
91'0 }Ke qHCJ10 B )J.BeHa)J.uaTepHqHOii CHCTeMe 6bI.'IO 6bI TaKHM:268.
DpOBepbTe 31'0.
Ho eCJ1H 6bI BbI nOnbITaJlHCb 3anHCaTb B )J.BeHa)J.uarepHqHOH
CHCTel\le qHCJ10 131, TO )J.eBHTH HallIHX uH<pP Y)Ke He XBaTHT
H npHIlJJ10Cb 6bI BBeCTH eUJ.e .nBa HOBbIX 3HaKa - UH<PPbI MH 10
H 11. DYCTb TaKHMH 3HaKaMH 6Y)J.YT rh )J.J1H 10 H J )J.J1H 11;
TOr)J.a: 131 : 12 = 10
I- 120
11
H qllCJ10 3anHCaJ10Cb 6bI TaK: . j" .
06paTHbIH nepexo.n J1aCT:
 .12+.1 =10.12+11=131.
ECJm TpH-qeTbIpe )J.eCHTKa J1eT Ha3a}]. 3anHCb qHCeJ1 B
HHbIX CHCTeMax CqHCJ1eHHH, n01\lHMO )J.ecHTHqHOH, H1\IeJla rJ1aB-
HbIM 06pa30M TeOpeTHqeCKHH HHTepec, TO B HaCTOHlll.ee BpeMH
nO)J.06HbIe 3anHCH HMelOT 60J1bIlJOe npaKTHqeCKOe 3HaqeHHe:
./I.BOHqHaH H TpOHqHaH CHCTel\lbI CqHCJ1eHHH IlJHpOKO HCnOJlb-
SYIOTCH npH pa60Te C .:weKTpoHHbIMH CqeTHbIMH MaIlIHHaMH.
o npHqHHaX, 3aCTaBHBIlJHX cOBpeMeHHbIx BbIqHCJ1HTeJleH npe.n.-
nOqeCTb )J.eCHTHqHOll CHCTeMe CqHCJ1eHHH CHcreMbI )J.BOHqHYIO
H TPOHqHYlO, qHTaTeJ1b Y3HaeT H3 10-H rJ1aBbI :9TOH KHHrH.
Y n p a )K H e H H H. 1. JI,aHbI )J.Ba qHCJ1a: 15 H 12, HanH-
caHHbIe B CeMepHqHOH CHCTeMe. Hall)J.HTe HX npoH3Be)J.eHHe
B Toli )Ke CHCTeMe. C)J.eJlaHTe npoBepKY, 3aMeHHB H COMHO)KH-
TeJ1H H npOH3Be.l1.eHHe cooTBeTcTBYIOIII.HMH qHCJlaMV. )J.eCHTHq-
HOH CHCTeMbI.
2. JI,aHbI )J.Ba qHCJ1a: III H 101, HanHcaHHbIe B )J.BOHqHOH
CHCTeMe. Hall)J.HTe HX npoH3Be)J.eHHe B TOll }Ke CHCTeMe. CAe-
lIaHre npoBepKY, 3al\IeHHB H COMHO)KHTeJJH H npoH3Be)J.eHHe
cooTBeTcTBYIOlll.HMH qHCJIaMH )J.ecHTHqHOH CHCTeMbI.
33
Omeem. 35 B )J.ecHTHqHOH CHCTeMe. X 44
242
+ 242
3212
3. B KaKOH CHCTeMe CqHCJ1eHHH
npoH3Be)J.eHo YMHO)KeHHe:
12

PaCKpblTble H HepaCKpbITbIe TaHHbI
HaTypanbHoro pHAa
BepHeMcH K HaTypaJlbHOMY pSl.!I.Y qHCeJl H paCCMorpHM He-
KOTOpbIe ero CBOHCTBa.
HanHIlJeM 3TOT pH)J.:
1,2,3,4,5,6,7,8,9, 10, 11, 12, 13,...
HeKoTopbIe H3 qHCeJl HaTypaJlbHoro pSl)J.a, TaKHe, KaK 2, 3, 5,
7, 11, 13, ..., )J.eJlSlTCH TOJIbKO caMH Ha ce6H H Ha e)J.HHHUY,
)J.pyme )Ke: 4, 6, 8, 9, 10, 12, ..., nOMHMO 3THX )J.eJIHTeJJeH,
HMelOT H )J.pyme )J.eJlHTeJlH. fTepBbIe qHCJIa MbI Ha3bIBaeM
qHCJIaMH n p 0 C T bI M H, BTOpbIe - C 0 C TaB H bI M HI).
E)J.HHHua He CqHTaeTCH HH npOCTbIM, HH COCTaBHbIM qHCJIOM.
TaK KaK npotTbIe qHCJIa HrpalOT 60JIbIlJYlO pOJlb B apH<pMe-
THKe, TO OqeHb )J.aBHO CTpeMHJIHCb C03)J.aTb Ta6JIHUY npOCTbIX
qHCeJI. O)J.HH H3 cnoc060B COCTaBJleHHH Ta6J1HU npOCTbIX qHCeJI
6bIJI npe)J.JIO)KeH 60JIee 2000 JIeT Ha3a)J. rpeqeCKHM yqeHbIM
3paToc<peHoM (276-195 1'1'. J10 H. 3.), BbI)J.alOlll.HMCSI reorpa-
«pOM )J.peBHOCTH, 6H6JIHOTeKapeM AJleKcaH)J.pHHCKOll 6H6JIHO-
TeKH. fTpHeM nOCTpoeHHH TaKOH Ta6mlllbI Tenepb Ha3bIBalOT
p e IlJ e TOM 3 paT 0 C <p e H a.
fTOCMOTPHM, KaK HY)KHO «npoceHBaTb» qHCJla HaTypaJIb-
Horo pH)J.a qepe3 «peIlJeTO 3paTOC<peHa», qT06bI BblJ1eJlHTb H3
MHO)KeCTBa qHCeJI HaTypaJIbHOrO pH)J.a TOJIbKO npOCTbIe qHCJIa.
DYCTb HaM HY)KHO C)J.eJIaTb 3TO C qHCJIaMH:
" 2, S , 5, '" 7, 8. S, , 11. m., 13. 1'4, 
te, 17. $, 19, 2'Q, . . 23, U, 25.
Dpe)K)J.e BCero BbIqepKHeM e)J.HHHUY, TaK KaK OHa He npH-
Ha.!I.JIe)KHT HH K npOCTbIM, HH K COCTaBHbIM qHCJIaM. JI,aJlee,
BbIqepKHeM Bce qHCJIa, KpaTHbIe 2, KpoMe 2, 3aqepKHBaH Ka)K-
)J.oe BTopoe qUCJIO qepe3 O)J.HO, HaqHHaSl OT 2. C)J.eJ1aB 3TO,
BbIqepKHBaeM Bce COCTaBHbIe qHCJIa, KpaTHbIe 3, KpoMe 3.
LI.JIH 31'01'0 3aqepKHBaeM Ka)K)J.oe qHCJIO qepe3 J1Ba, HaqHHaSl
OT 3. B HeKoTOpbIX CJIyqaHX npH)J.eTCH _ BTOpOll pa3 nepeqepK-
HYTb qHCJIO, y}Ke paHee 3aqepKHYTOe, - 3TO Te qHCJIa, KO-
TopbIe KpaTHbI 2 H 3. C)J.eJIaB 3TO, nepexo)J.HM K OTceHBaHHIO
COCTaBHbIX qHCeJI, KpaTHbIx 5, 3aqepKHBaH Bce qHCJIa qepe3
1) .IJ.eJIeHHe 'mceJI Ha npOCTble H COCTaBHble OblJIO BBe.!!.eHO nH<pa.
ropOM,
13

qeTbIpe Ha nHTOe, HaqHHaSl OT 5. 3.necb TaK}Ke npHAeTcH HeKO-
TOpble GHCJla nepe'lepKHYTb BTOpOH pa3.
llJIH quceJl HaTypa.rIbHoro pH)J.a OT 1 J10 25 pa60ra no
«npoceHBClHMIO» npocTbIX qHCeJI 3aKOHqeHa, TaK KaK npH
)J.eJ1eHiiH 25 Ha 7 (Ha npocToe qHCJIO, cJle)J.YIOlll.ee 3a 5) qaCTHOe
OKa3bJBaeTCSl MeHbIlJe )J.eJ1HTeJ1H. OCTaeTCSl TOJlbKO BbIIlHcaTb
nOJ1pHJ1 Bce OCTaBIUHeCH He3aqepKHYTbIMH qHCJ1a, H MbI nOJ1Y-
qHM Bce npocTbIe qlICJla cpe)J.H 25 nepBbIx qHCeJI HaTypaJ1b-
Horo pH,na:
2, 3, 5, 7, 11, 13, 17, 19, 23.
Y n p a )K H e H H e. CocTaBbTe YKa3aHHbIM cnoco60M Ta6-
JlHUY npocTbIX qHCeJ1 Cpe,l(H qHCeJl HaTypaJIbHoro pH)J.a OT
1 )J.O 100.
fTocreneHHoe ycoBepll1eHCTBOBaHHe MeTO)J.a 9paToc<peHa
n03BOJlHJ10 B HaCTOHlll.ee BpeMH HMeTb Ta6J1HUbI npoCTblx qHCeJl
B pH)J.e Harypa.lIbHbIX qHCeJ1 OT 1 )J.O 100 000 000. 3TH Ta6.1HllbI
npe)J.OCTaBJlHIOT B HaIlJe paCnOpH)KeHHe 06IlJHpHbIH MaTepHaJ1,
Ha OCHO?e KOTOpOro MO)KHO BbICKa3aTb pH.n mnore3 0 CBOH-
CTBax npOCTbIX qHCeJJ. LLOKa3aTeJJbCTBa 3THX rHnOTe3 OKa3bI'
BalOTCH HHor)J.a O'leHb CJ1OLKHbIMH, a HeKOTopble rHnOTe3bl He
.nOKa3aHbI )J.O CHX nop.
O.nHliM H3 nepBblX BonpOCOB, KOTopble MorYT B03HHKHYTb
no nOBO)J.y npOCTblX qHCeJ1, HBJlHeTCH Bonpoc 0 TOM, KOHeqHO
JlH HX qHCJlO cpe)J.H 6eCKOHeqHOrO MHO)KeCTBa qHCeJ1 HaTypaJlb-
HOro pH)J.a Hm! qHCJ10 npOCTblX qHCeJI 6eCKOHeqHO.
LLOKa}KeM TeopeMY: Cpeou l{uce.1/. 1iamypaAbH020 pfloa cy-
Ujecmeyem 6eCK01iettHoe M1iOJICeCmeo npocmblX I{uce/l.
CnpaBe)J.JlHBOCTb .:iTOll TeopeMbI CJ1e)J.yeT H3 Toro, qTO,
KaKOBbI 6bI HH 6bIJ1H pa3J1HqHbIe npOCTble qHCJ1a: PI' P2' Pa,
..., Pn' MO)KHO nOJlyqHTb HOBOe npOCTOe qHCJlO, Cpe)J.H HHX
He cO)J.ep)Kalll.eeCH. TaKHM npOCTblM qHCJ10M 6Y.D.eT npOCToH
,n.eJ1HTeJJb CYMMbI:
Pl'P2'Pa ... Pn+ 1 ,
KoropblfI, )J.€JlSl BClD CYMMY, He MO)KeT cOBna;AaTb HH C O)J.HHM
H3 npOCTbIX qHce.1I Pl' P2' Pa, ..., P n , TaK KaK npH ,II,e.leHHII
YKa3aHHOH CYMMbI Ha KaKOe-J1H60 H3 3THX npOCTbIX qnce.l
Bcer.na no.TJyqaeM B OCTaTKe 1. HaH.nH TaKHM o6pa30M HOBoe
npocToe qHC.IIO Pn+l H COCTaBHB CYMMY
Pl',P2'PS'" Pn'Pn+l+ l ,
]4

MbI npH)J.bf K CJ1e)J.YIOIII.eMY HOBOMY npOCTOMY qHCJJY Pn+2'
KOTOpOe HB.l1HeTCH npOCTbIM )J.eJ1HTeJ1eM CYMMhI
Pl' P2'P8 ... Pn' Pn+l+ 1
H T. )J.. TaKHM 06pa30M, qHCJ10 npOCTbIX qHCeJI B HarypaJlbHOM
pH)J.e qHCeJ1 - 6ecKoHeqHO.
TeopeMa 0 6eCKOHeqHOM MHO)KeCTBe npOCTbIX qHCe.'I 6bIJ1a
)J.OKa3aHa elll.e EBKJ1H)J.OM, T. e. 6oJ1ee 2000 lIeT Ha3a.n;. HaqH-
HaH C 3TOro oT)J.aJ1eHHOrO BpeMeHH, MO)KHO Ha6J1IO)J.aTb nonbIT-
KH MHOrHX MaTeMaTIlKOB COCTaBHTb TaKYIO <POPMYv'IY C ueJ10-
qIlCJ1eHHbIM aprYMeHToM n, KOTopaH )J.aBaJla 6bI npll pa3lIHq-
HbIX UeJ1bIX nOJIO)KHTeJ1bHhIX 3HaqeHHHX n TO.'!bKO npocTbIe
qHCJ1a. 3TH nOnblTKH OKa3aJ1HCb TIII.eTHbIMH, -'TaKYIO <pop-
MYJ1Y nOJ1yqHTb He Y)J.aJ10Cb.
B TeqeHHe HeKOToporo BpeMeHH CqHTaJlH, qTO nOCTaBJ1eH-
HYIO 3a.n;aqy pell1aeT <popl\IYJ1a
n
f(n)=22 + 1,
npe)J.J10)KeHHaH <PpaHllY3CKHM MaTeMaTHKOM nbepoM <PepMa
(1601-1665). <PepMa BbICKa3aJ1 npe.n;nOJlO)KeHHe, qTO Bce
qHCJ1a, nOJ1yqeHHbIe H3 nOH <POPMYJ1bI, npH pa3JIHqHbIX lle-
JlbIX 3HaqeHHHX n 6Y)J.YT qHC.TlaMH npocTbIMII. B Cal\IOM )J.eJ1e
npH n= 1, 2, 3 H 4 MhI nOJlyqaeM:
f(1.=22+ 1 =5;
f(2)=2 2 " +1=24+1=17;
.
f(3)=2 2 +1=28+1=257;
f(4)=2 2 ' +1=216+1=65 537
- TOJIbKO npocTbIe qHCJ1a. Ho npH n=5 <popMYJ1a <PepMa,
KaK nOKa3aJ1 B 1732 r. qJ1eH fTeTep6yprcKoH AKa)J.eMHH HaYK
3MJ1ep, HeBepHa. 4HCJ10 2 2 " +1=2 82 +1=4294 967297- co-
CTaBHoe H paCKJ1a)J.bIBaeTcH Ha MHO>KHTCJ1H 641.6 700 417.
He)J.aBHo YCTaHOBJleHa npH nOMOlll.H 3J1eKTpOHHOH Mall1HHbI H
CJ10)KHOCTb qHCeJ1 f( 10) H f( 16).
11HTepecHo, '11'0 qlICJ1a <1>epMa BCTpeqalOTCH H B reoMeTpHH,
Kor)J.a pa36HpaeTCH Bonp'oc 0 B0310)KHOCTH nocTpoeHHH npH
nOl\IOllI.H lI.HPKYJ1H H .'1I1HeiiKH npaBHJ1bHbIX l\IHOroyrpvlbHHKOB l )
C npocTbIM qHCJ10M crOpOH.
1) MHoroyroJlbHHK Ha3bIBaeTCH npaBJlJlbHblM, eCJlJI Bce ero CTOpOllbi
paBHbJ II Bce ero BHYTpelIHHe yrJlbJ TaKn<e paBHbJ.
15

faycc (1777-1855) )J.OKa3aJI cJle)J.YJOlII.YJO 3aMeqaTeJlbHYlO
TeopeMY:
MOCHC1iO nocmpoumb npU nOMOt«U L{UPKY/lfl U /lU1ieUKU
mO/lbKO maKue npaeU/lb1ibte M1iOcOYcO/lb1iUKU c npocmblM I{UC/lOM
cmopo1i, Y (WmOpblx I{UC/lO cmop01i ecmb «npocmoe I{UC/lO»
n u
cJJepMa, T. e. qHCJlO BH)J.a 2 2 + 1. 113 3TOH TeOpeMbI c.'Ie)J.yeT,
qTO npaBHJlbHble MHoroyroJlbHHKH C ceMblO, C TpHHa.n.uaTblO
CTOpOHaMH nOCTpOHTb UHpKYJleM H JlHHeHKOH HeJlb3H, a C
17 H 257 cTopoHaMH MO)KHO.
PaCCl\WTpHM elII.e )J.Be <P0PMYJlbI, )J.alOlII.He MHOro npocTbIX
qHCeJl :
F(n)=n 2 -n+41
H
q>(n)=n 2 -79n+ 1601.
nepBaH Ji3 3THX <POPMYJl )J.aeT npocTble qHCJla npH n= 1, 2,
3, ..., 40, HO npH n=41 HeBepHa; BTopaH <popMYJla )J.aeT
npocTbIe qHCJla npH n=l, 2, 3, ..., 79, HO npH n=80 nOJlY-
qaeTCH COCTaBHoe qHCJlO.
113yqaH Ta6J1HUY npocTbIX qHCeJI, MO)KHO nO)J.MeTHTb, qTO
npocTbIe qHCJla He O)J.HHaKOBO rYCTO pacnpe,lI,eJleHbI B HaTypaJlb-
HOM pH)J.e qHCeJI.
ECJlH B3HTb nepBbIH )J.eCHTOK qHCeJI HaTypaJlbHoro pH)J.a,
TO cpe)J.H HHX MbI HaH)J.eM qeTbIpe npOCTbIX qHCJla; B rpynne
qHCeJI 01' 20 )J.O 30- npocTblX qHCeJI TOJlbKO )J.Ba; rpynna qH-
ceJl 01' 410 )J.O 420 )J.aeT TOJlbKO O)J.HO npocToe qHCJlO 419;
rpynna OT 850 J10 860- TPH npOCTbIX qHCJJa; B rpynne qHCeJI
01' 3990 .n.o 4000 HeT HH O)J.HOro npocToro qHCJla, a B rpynne
OT 4000 )J.O 4010 Mbl HaH)J.eM TPH npocTbIX qHCJla.
TaKHM 06pa30M, pacnpe)J.eJlemle npocTbIX qHCeJI cpe)J.H
1JHCeJI HaTypaJlbHoro pH)J.a OTJlHqaeTCH qpe3BbIqaHHO Henpa-
BHJJbHOH nOCJJe)J.OBaTeJlbHOCTbJO. Ho 3Ta HenpaBHJJbHo':Tb HC-
tle3aeT, eCJlH MbI 6Y)J.eM CYJ1HTb 0 I' Y C TOT e pacnpe,neJleHHH
n
npOCTbIX qHCeJI no BeJlHqHHe OTHOllleHHH N '
me n - qHCJlO npOCTbIX qHCeJI CpeJ1H N nepBbIX qHCe.n HBry-
paJlbHOrO pH)J.a.
11l\IeH Ta6J1HUY npOCTbIX qHCeJI, MO)KHO nO)J.CqHTaTb BeJlH-
qHHY OTHOllleHHH N npH pa3JJHqHbIX 3HaqeHHHX N. BOT Ta6-
JlHua 3Ha1JeHHH 3Toro OTHOIlJeHHH:
16

N II 10 I 10 2 I 10 8 I 10 4 I 10 5 I 10 6
n /I 4 I 25 I 168 I 1229 I 9592 I 78 498
! 0, 25 1 I I 0.09542
n I
N I 0,4 0,168 0,1229 0,078498
B % 40% 25% [16,8% 12.3% I 9.5% I 7,8%
N
3Ta Ta6JIllUa nOKa3bIBaeT, qTO n JI 0 THO C T b pacnpe-
)J.e.T/eHHH npOCTbIX qHCeJI nOCTeneHHO YMeHbIlJaeTCH, ecJlH pac-
CMaTpHBaTb B03paCTalOlll.He qHCJIOBbJe npOMe)KYTKH HarypaJJb-
HOro pH)J.a. <1>paHUY3CKOMY MaTeMaTHKY JIe)KaH.n.PY B 1798 r.
Y)J.aJIOCb )J.OKaaaTb HeOrpaHHqeHHOe y6bIBaHHe 3TOM: nJIOTHOCTH
npH B03pacTaHHH qHCJJa N. HaqaJIHCb nonblTKH C03)J.aHHH TaKoH
<POPMYJIbI, KOTOpaH n03BOJIHJIa 6bI paCCqHTaTb n no .n.aHHOMY
N. 3TH nOnbITKH He YBeHqaJIHCb ycnexoM - TOqHOH <POPMYJIbI
nOJlyqHTb He Y)J.aJIOCb. fTpHIlIJIOCb nORTH no nYTH C03.n.aHHH
npH6J1H)KeHHOH <POPMYJlbI, HO TaKOH, qT06bI oIlJH6Ka 6bIJIa )J.O-
CTaTOqHO MaJIOH npH 60JIbIlJOM 3HaqeHHH N H YMeHbIlJaJlaCb
6bI npH ..n.aJlbHeHIlJeM YBeJIHqeHHH N. KpoMe TOro, 6bIJIa no-
CTBBJIeHa 3a)J.aqa: OTKpbITb Te 3aKOHOMepHOCTH, KOTOpbIM no.n.-
qHHHeTCH caMoe OTKJIOHeHHe HCTHHHoro 3aKOHa pacnpe)J.eJIe-
HHH npOCTbIX qHCeJI 01' pacnpe)J.eJIeHHH HX, KOTopoe nOJlyqaeTCH
no TOll HJIH HHOH <popMYJIe.
3THMH )J.BYMH 3a.n.aqaMH 3aHHMaJIHCb KPynHeHIlIHe MaTe-
MaTHKH Ha npOTjI)KeHHH MHorHX JleT. Cpe.n.H 3THX MaTeMa-
THKOB MbI BCTpeqaeM HMeHa JIe)KaH)J.pa (1752-1833), 3HJlepa
(1707-1783), faycca (1777-1855), LLHpHXJIe (1805-1859),
BepTpaHa (1822-1900) H, HaKoHeu., O)J.Horo H3 BeJIHqaHIlIHX
MaTeMaTHKOB XIX B. fT. JI. 4e6bIIlJeBa (1821-1894).
4e6bIllieB B CBoeM MeMyape «06 onpe)J.eJIeHHH qHCJIa npo-
CTbIX qHCeJI, He npeBOCXO)J.Hlll.HX )J.aHHOH BeJIHqHHbI» (1849),
)J.aJl 6JIeCTHlll.ee pellIeHHe nepBoH 3a)J.aqH. 3aKoH pacnpe.n.eJIe-
HHH npocTbIX qHCeJI, OTKpbITbIH 4e6blilieBbIM, BbIpa)KaeTCH
<POPMYJlOH
n 0,43429
N = log 1o N '
17

ToqHOCTb KOTOpOH HeOrpaHllqeHHO B03paCTaeT C B03paCTa-
HHeM N. B !'nOH <PopMYJle logloN eCTb TOT nOKa3aTenb CTeneHH,
B KOTOPYIO HY)KHO B03BeCTH 10, qT06bI nOJlyqllTb qHCJIO N.
B CBH3H C pell1eHHeM 3TOll 3a)J.aqH Y.e6bIIlJeB YCTaHOBlIJI Henpa-
BHJlbHOCTb <POPMYJIbI JIe)KaH)J.pa l ), BbIpa)KalOlll.eH qHCJlO
npocTbIX qllCeJI, MeHbll1HX N, H )J.OKa3aJI nmoTe3Y, BbI)J.BHHY-
TYIO, HO He )J.OKa3aHHYIO EepTpaHOM. 3Ta mnOTe3a TaKOBa:
MeCHCOY /lJ06blM 1iamypaAb1iblM '-lUC/1OM U e20 yoeOe1iUeM
ece20a 1iaXOOumCJI xomfl 6bl 001i0 npocmoe I{UC/1O.
Y n p a }K H e H H H. 1. fTPH nOMOlll.H <POPMYJlbI Y.e6bIIlJe-
Ba COCTaBbTe Ta6JIHUY npOCTbIX qHCeJI, He npeBOCXO)J.Hlll.HX
qHCJla N, eCJIH N=IO, 10 2 , 10 8 , 10 4 , 10 5 , 10 6 .
2. fTpoBepbTe mnoTe3Y EepTpaHa )J.JlH He60Jlbll1HX qnceJI.
3aKoH pacnpe)J.eJleHHH npOCTbIX qHCeJI, )J.aHHbIll Y.e6bIIlJe-
BbIM, MO)KHO OTHeCTll K qHCJlY 3aMeqaTeJIbHbIX OTKpbITHH B 06-
JlaCTH MaTeMaTHKH. OTKpbITHe 3TOro 3aKOHa npOH3BeJIO 60Jlb-
ll10e BneqaTJleHlle Ha MaTeMaTHKOB - cOBpeMeHHHKoB Y.e6bI-
ll1eBa. He)J.apoM O)J.HH 1I3 60JIbIlJHX CnellllaJIHCTOB B 06JIaCTll
TeopHH qHCeJI, MaTeMaTHK JIaH)J.ay, B 1909 r. nHCaJl: «TIepBhIH
nocJle EBKJIll)J.a, KTO nOIlJeJI npaBllJlbHblM nYTeM )J.JIH peIlJeHHH
np06JIeMbI 0 npocTbIX qHCJlax II )J.OCTHr Ba)KHbIX pe3YJIbTa-
TOB, 6bIJI Y.e6bIIlJeB». A aHrJIHllCKllH MaTeMaTHK CHJIbBepCT
B 1881 r. nHCaJI: «JI,aJlbHeHIlJerO pa3BHTHH TeopHll qHCeJI Ha)J.O
)K)J.aTb, nOKa pO)J.HTCH HeKTO, HaCTOJIbKO )Ke npeBOCXO)J.Hlll.HH
Y.e6bIIlJeBa CBoelO npoHHu.aTeJlbHOCTblO II B)J.YMqUBOCTblO, Ha-
CKOJlbKO Y.e6bIIlJeB npeBOCXO)J.llJI 3THMll KaqeCTBaMll 06bIKHO-
BeHHhlX JIIO)J.eih>.
MHpOBoe 3HaqeHHe OTKpbITllH Y.e6bIIlJeBa B 06JIaCTH Teo-
pHH qHCeJI XOpOIlJO nO)J.qepKHYJI B O)J.HOH H3 CBOllX pa60T qneH
AKa)J.eMllH HaYK E. H. JI,eJIOHe: «Pa60TbI Y.e6bIll1eBa 0 npOCTblX
qllCJIaX 6bIJIll nepBbIMll ycneIlJHbIMll ll1araMll B 06111.eH TeopHH
npOCTbIX qHCeJI, BnepBbIe nOCJIe EBKJIH)J.a .n.aBll1He TOqHbIe
pe3YJIbTaTbI B 06111.ell TeoplfH pacnpe)J.eJleHllH J:pOCTbIX
qllCeJI.
Or 060llX MeMyapoB Y.eObIIlJeBa BeeT He06bIKHOBeHHOH
CBe}/{eCTblO, OCTpOYMHeM H CllJlOll, npHCYlll.eH MOJIO)J.OMY nep-
BOKJIaCCHOMY MaTeMaTllqeCKOMY TaJIaHTy 2 ).
3Tlll\m )J.BYMH reHHanbHbIMll pa60TaMll HaqHHaeTCH fTeTep.
6yprcKaH ll1KOJIa TeopHH qHCeJ1».
n 1
1) <I>opMYJla J1em8H)\p8 TaKOBa: N = 2,3025 10gloN-I ,0">366 .
2) n. J1. 4e6b1weBY B rOJ\ BbIXO.l\a YK83aHHoro lIIelllyapa UblJlO
28 JleT. .
18

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n. J1. 4e6blWeB..
ncl)HYTHH J1bBOBHq 4e6bIllieB (1821-18Q4), OKOHqHB
B 1841 r. MOCKOBCKHH YHHBepCHTeT, BCeI.{eJJO OT)J.aJlCH Hayq-
HbIM pa60TaM. B TeqeHHe nHTH- JleT, paBHo)J.YIlIHO OTHOCHCb
K CBoeMY nJlOXoMY MaTepHaJlbHoMY nOJlO)KeHHIO H He )J.YMaH 0
. Kapbepe, OH TBep)J.O npo)J.oJl)KaJI H36paHHbIH HM nYTb.
.lI.Be tIepTbI xapaKTepHbI )J.JlH 4e6bIweBa KaK yqeRoro:
IlIHPOKHH )J.Hana30H ero TBOpqecTBa H nOCTaHOBKa TeOpeTHqe-
CKHX np06J1eM B CBH3H C 3a)J.aqaMH npHKJla)l,HOrO xapaKTepa.
fTepBaH 'lepTa OTpa)KeHa B Tex pe3YJlbTaTaX <PYH)J.aMeHT8J1b-
HOrO SHaqeHHH, KOTopbIe 4e6bIllieB nOJlyqHJJ B TeopHH qHCeJl,
TeopHH BepOHTHOCTeH, HHTerpaJlbHOM HCqHCJleHHH, B TeopHH
MexaHH3MOB H .l(p. BTopaH 'lepTa HpKO CKa3bIBaeTCH B CTpeM-
JleHHH 4e6bIllieBa C6J1H3HTb TeopHIO C ee B03MO)J{HbIMH npaK-
THqecKHMH npHJIO}KeHHHMH. OH CqHTaJI, qTO «roJlH)K-eHHe
'TeopHHC npaKTHKOH )J.aT eaMbIe OJlaroTBopHbIe pe3YJIbTaTbI.
H He O)J.Ha TOJIbKO npaKTHK8 01' 3Toro BblHrpbIBaeT, caMa HaYKa
paSBHBaeTCH no.l( BJIHHHHeM ee: OHa OTKpbIBaeT HM HOBbie
npe.l(MeTbI )J.JIH HCCJJe)l,OBaHHH HJlH HOBbIe CropOHbI 0 npe)l,Me-
Tax, )J.aBHO H3BeCTHbIX».
BbICOKaH oueRKa 3TOH CTOpOHbI .l(eHTeJlbHOCTH 4eObIIlJeBa
,l(aHa B e,OOBax aKa)J.eMHKa 5. A. CTeKJlOBa: «feHHH 4e6bI
IDesa npe)l,CTaBJlHJI COISOH HCKJIlOqHTeJlbHbIH 06pa3eu Coe)J.H-
HeHHH npaKTHKH C TBOpqeCKOH, 01S06ll1.alOllI.eH CHJIOH OTBJle-
'leHHOrO MbICJJHTeJJ H - MaTeMaTHKa.
19

npaKTHqeCKHe 3anpOCbI npeBpaII.{aJlHCb HM B COOTBeTCTBylO-
II.{YIO MaTeMaTHqeCKYIO Te9pmO, npe)J.CTaBJ1HBIlJYJO HOBoe 01'-
KphITHe B 06J1aCTH qHCTOH HaYKH; 3Ta nOCJle)J.HHH He OCTaBa-
JlaCb B 06J1aCTH qHCTOH MbICJ1H, BOnJ10111.aJ1aCb B peaJlbHYIO )J.eH-
CTBHTeJ1bHOCTb, B pa3J1HqHoro po.n.a MallIHHbI H MexaHH3MbI,
KOTopbIe CJlY)KHJ1H KaK 6bI BeII.{eCTBeHHbIM OCYII.{eCTBJ1eHHeM
ero TBOpqeCKHX )J.OCTH)KeHHH».
fT03TOMY He Y)J.HBHTeJlbHO, '11'0 4e6bIIlJeB BMeCTe C pH)J.OM
6J1eCTHlll.HX pa60T no MaTeMaTHKe OqeHb MHoro c.n.eJlaJ1 KaK
H306peTaTeJ1b-KOHCTPYKTOp. B OCHOBe Bcex ero H306peTeHHH
Jle)KHT CTpeMJleHHe 06J1erqHTb pa60TY qeJlOBeKa, BHeCH B Hee
B03MO)KHYIO SKOHOMHIO TpY)J.a H BpeMeijH.
He Y)J.HBHTeJlbHO TaK)Ke, '11'0 MaTeMaTHqeCKHH reHHH Ta-
KOro MaCIlJTa6a, RaK 4e6bIIlJeB, He MOl' He OKa3aTb caMoro
KpynHoro BJ1HHHHH Ha .n.aJ1bHeHIlJHll XO.n. pa3BHTHH MaTeMa-
THKH Ii )J.OJI)KeH 6bIJ1 B03)J.eHCTBOBaTb, B 60J1bIlJeH HJ1H MeHb-
llIeH CTeneHH, Ha BCHKoro MaTeMaTHKa, )J.a)Ke )J.aJ1eKoro 01'
4e6bIllieBa no CBOHM HHTepecaM.
Cpe)J.H ero yqeHHKOB MbI Bl!)J.HM TaKHX KpynHbIX MaTeMa-
THKOB, KaK A. A. MapKOB, A. M. JIHnYHOB, LI.. A. fpaBe,
K. A. Docce H MHorHX )J.pymx. HeKOTopbIe H3 HHX npo)J.OJ1-
)KaJ1H pa60TY Ha)J. BonpocaMH, nOCTaBJ1eHHbIMH HX yqHTeJ1eM.
TpY)J.bI 4e6bIllieBa nOCJlY)KHJ1H OCHOBOH C03)J.aHHH PYCCKHX
MaTeMaTHqeCKHX IlJKOJI, npo.n.OJl)KalOlll.HX ycneIlJHO pa3BHBaTb
H.n.eH 4e6blIlJeBa. 3TH IlJKOJ1bI B CBOIO Oqepe.n.b (y)Ke B COEeT-
CKoe BpeMH) )J.aJ1H HaIlJeH CTpaHe MHomx BbI)J.alOlll.HXCH yqe-
HbIX-MaTeMaTHKOB; cpe)J.H HHX TaKHe KpynHeHIllHe MaTeMaTH-
KH, KaK aKa)J.eMHKH C. H. I3epHIlITeHH H 11. M. BHHorpa)J.OB.
DepBbIH OqeHb MHOro )J.aJl B 06J1aCTH TeopHH BepOHTHO-
CTeH, OCHOBbI KOTOpOH 3"3J10)KHJ1H fT. JI. 4e6bIIlJeB Hero 6J1H-
)KaHIlJHe yqeHHKH - A. A. MapKOB H A. M. JIHnYHOB, a
BTopon - B 06J1aCTH TeopHH qHCeJ1.
11HocTpaHHble MaTeMaTHKH OqeHb BbICOKO CTaBHJ1H saCJJY-
m fT. JI. 4e6bIllieBa. 4JleHbI <1>paHUY3CKOH AKa)J.ervIHH HaYK
BO rJ1aBe c 3pMHTOM np1I3HaJ1H 4e6bIIlJeBa «rop.n.OCThlO HaYKH
B POCCHH, O)J.HHM H3 nepBbIX reOMeTpOB EBponbI, O)J.HHM H3
BeJllIqaHIlJHX reOMeTpOB Bcex BpeMeH».
np06JIeMa, B03HlIKllIaH B POCCIiH H pellIHHaH
PYCCKHMH
11HTepecHO 03HaKOMHTbCH elll.e C O)J.HHM CBOHCTBOM qHCeJ1
HaTypaJlbHoro pH)J.a, H3BeCTHbIM B MaTeMaTHKe no)J. Ha3BaHHeM
«np06J1eMa f0J1b)J.6axa». OCTaHOBHTbCH Ha 3TOM CBOllCTBe HI!-
20

TepeCHO H no O)J.HOMY TOMY, '11'0 np06.'1eMa fO.'1b)J.6axa B03HHK-
.'1a B Poccmr, H qepe3 200.'1eT, B TeqeHHe KOTOpbIX pH)J. KPyn-
HellwHx HHocrpaHHbIx MaTeMaTHKOB 6e3ycnewHo TPY)J.HJICH Ha)J.
peIlJeHHeM 3TOH np06.'1eMbI, ee peIlJeHHe 6bI.'I0 npo)J.BHHYTO
Bnepe)J. B HaIlJe BpeMfI COBeTCKHM MaTeMaTHKOM, H, HaKOHeI.{,
np06.'1eMa 6bI.'Ia nOqTH peIlJeHa TaK)Ke O)J.HHM H3 BbI)J.alOlll.HXCfI
COBeTCKHX MaTeMaTHKOB.
11CTOpHfI B03HHKHOBeHHH np06J1eMbI fOJlb)J.6axa TaKOBa.
B 1742 r. q.'1eH fTeTep6yprcKOIl AKa)J.eMHH HaYK X. fO.'1b)J.6ax
B nHCbMe K 3HJlepy, q.'1eHY TOH )Ke AKa)J.eMHH HaYK, BbICKa3a.'1
O)J.HY HHTepecHYIO rHnoTe3Y:
Jl1060e 1iamypaAb1iOe I{UCJlO, 60/lbUlee nJlmu, npeOCmaB/lflem
CYMMy mpex npocmblx I{uce/I.
3H.'1ep OTBeTHJI, '11'0 OH He MO)KeT )J.OKa3aTb 3TY mnOTe-
3Y, HO '11'0 MO)KeT BbICKa3aTb HHYIO rHnOTe3Y, CBH3aHHYlO C
nepBoH:
BCJlKOe I{emf-we HamypaAbHoe I{UCAO, 60AbUlee oeyx, npeo-
Cma8/lflem CYMMY oeyx npocmblx.
4HTaTe.'1b, KOHeqHO, 3aXOqeT npoBepHTb rHnOTe3bI, BbICKa-
saHHhIe fO.'1b)J.6axoM H 3HJlepOM )J..'1fI pH)J.a qaCTHbIX c.'1yqaeB.
B TeqeHHe 200 JleT BbI)J.alOlll.HecH MaTeMaTHKH MHOrHX cTpaH
cTpeMH.'1HCb ,lJ,OKa3aTb np06.'1eMY fO.'1b)J.6axa - 3H.'1epa, HO
6e3ycneIlJHo. B TO )Ke BpeMH qHC.'10Ba5I npOBepKa 9Toro CBOH-
CTBa, ,lJ,OBe)J.eHHafl K HaqaJlY XX B. )J.O 9 000 000, nOKa3a.'1a,
'11'0 mnOTe3a fOJlb)J.6axa BepHa. 11 BOT, Kor.n:a 6bI.'I0 BbICKa3a-
HO npe)J.nO.'10)KeHHe 0 TOM, 111'0 npo6.'1eMa fOJlb)J.6axa - 3HJlepa
cpe)J.CTBaMH COBpeMeHHOH MaTeMaTHKH He MO)KeT 6MTb )J.OKa-
3aHa, O,l],HH MO.'10)J.OH COBeTCKHH MaTeMaTHK lIeB lIlHHpe.'1bMaH
(1905-1938) B 1931 r. ,lJ,06H.'1CH 3HaqHTe.'1bHbIX ycnexoB B pe-
IlJeHHH 3TOH 3a)J.aqH, H, HaKOHeu, B 1937 r. aKa)J.eMHK 11. M. BH-
Horpa)J.oB nOqTH nOJlHOCTblO pewH.'1 3TY np06.'1eMY. OH )J.O-
Ka3aJl, '11'0 cyu{Ecmeyem cpeou HamypaAbHblX I{uce/I maKoe
I{UC/lO «C», sa KomOpbtM eCJlKOe Hel{em1iOe 1iamypaAb1iOe ttUC/lO
J1BAJlemCJI cYMJ;/o11 mpex npocmblx I{Uce/I, npUl{eM 8mo I{UCAO
«C» Ol{eHb ee/lUKO.
fTPo<Peccop 11. K. AH)J.POHOB B cBoeH KHHre «ApH<pMeTI1Ka
HaTypaJlbHbIx qHce.'1» )J.aeT TaKoH HTor cOBpeMeHHo:\IY COCTOH-
HHIO np06.'1eMbI fO.'1b)J.6axa:
«BecKOHeqHbIH pH)J. HaTypaJIbHbIX qHce.'1 pacna)J.aeTcH Ha
qeTbIpe pa3.'1HqHbIe qaCTH:
1) MHO)KeCTBO, COCTOHlll.ee H3 nHTH nepBbIx HaTypa.'1bHbIX
qHCeJI, r)J.e Ka)K)J.oe qHCJIO He eCTb CYMMa Tpex npOCTbIX qHCeJl.
21

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11. M. BJlHOrpaJlOB.
2) MHO)KeCTBO HaTypaJIbHbIX qMCeJI OT 6 )J.O 9000 OM,
r)J.e onbITOM npOBepeH8 npaBMJlbJIOCTb YTBep)K.l{eHml, liTO
BCSlKoe H8TypaJlbHOe qMCJIe eCTb cyMMa Tpex npOCTbIX quceJI.
3) MHO)KeCTBO H8Typ8J1bHbIX quceJI OT 9 000 000 )J.O Becb-
M8 60JIbllIOro Jl8, H8H,neHHofo 11. M. BliHorp8)J.oBbIM,
r)J.e HU onbITOM, HU MaTeM8TUqeCKUMU cpe)J.cTB8MU He YCT8HOB-
JleHO, qTO K8)K)J.oe HaTypaJlbHoe qUCJIO B 3TOM npOMe)KYTKe
eCTb UJIM He eCTb CYMM8 Tpex npOCTbLX qUCM.
4) BeCKOHeqHbIU pSl)J. HaTyp8J1bHbIX qUCM HaqUHaH OT
IIUCJla, H8H)J.eHHoro 11. M. BUHorp8)J.oBbIM, r.l{e Ka>K,noe He-
qeTHOe H8TypaJlbHoe .eCTb cYMMa Tpex npocTbIx qUCeJI».
o TeopUH IjHCeJI
4uCJlO - O)J.HO U3 OCHOBHbIX nOHSlTl-d% M8TeMaTHKH. 3ro
nOHSITHe, S8pO)J.HBIlJUCb B fJly6oKOH .l{peBHOC'tU. nOCTeneHHO
nOJlBepraJlOCb p8CIlJHpeHlIIO U o600llleHUIO. 48CTb M8TeMaTUKM,
nOCBHlU.eHHYIO u3yqeHHIO CBOHCTB qUCeJI, Ha3bIBalOT TeopMeu
qUCeJI. KOHellHO, B TeopHH qUceJI, nOMUMO Tex CBOHCTB- I.{eJlbIX
qUCM. KOTopbIe 6WIH p8CCMOTpeHbI HaMH B iToH rJI8Be. BbI
22

BCTpeTHTe H MHOrne )J.pyme CBOHCTB8. CJleJ1yeT T8K)Ke OT-
MeTHTb, '11'0 TeOpHH qHCeJI H3yqaeT H CBOHCTBa TaK Ha3bIBae-
MhIX aJlre6paHqeCKHX qHCeJl, T. e. qHCeJl, KOTopbIe HBJlHIOTCH
KOpHHMH 8J1re6paHqeCKOrO ypaBHeHHH
Xn+alXn-l+a2Xn-2+ -.. +an=O,
r)J.e aI' a 2 , ..., an - I.{eJlbIe qHCJla.
EoraTOH cO)J.ep)KaHHeM HBJlHeTCH qaCTb TeopHH qHCeJl,
nOCBHlll.CHHafi peIlJeHHIO B I.{eJlbIX qHCJlaX Heonpe)J.eJleHHbIX,
HJlH )J.HO<paHTOBbIX, ypaBHeHHH.
I1MH ,QHO<paHTa, rpeqeCKOrO MaTeMaTHKa, )KHBllierO B
II I B. H. 3., HaTaJlKHBaeT HaC Ha MbICJlb 0 TOM, qTO y)Ke B
)J.peBHeH fpeI.{HH CTpeMHJlHCb peIlJHTb HeKOTopbIe BonpOCbI
TeopUIi qHCeJl. 11, )J.ellCTBHTeJlbHO, 3a )J.eBHTb CTOJIeTHl1 )J.O
,QHocpaHTa, B VI B. )J.O H. 3., IlJKOJla nH<paropelII.{eB H3yqaJla
.n:eJlHMOCTb I.{eJlbIX qHCeJl, BbIJIeJlH.'Ja H3 HaTypa.'1bHOrO pH)J.a
qHce.'1 qHCJla npOCTbIe H COCTaBHbIe, HHTepeCCmaJlaCb CBOH-
CTBaMH KBa)J.paTHbIX qHce.'I H HaIlJ.1Ja pellIeHHe B U.e.'1hX 'lHCJlaX
ypaBHeHHH
X 2 +y2=Z2.
fT03)J.Hee, B III B. )J.O H. 3., EBKJlH)J. )J.a.'1 CHCTeMaTHqeCKOe
nOCTpoeHHe TeopeM )J.eJlHMOCTH qHCeJI H )J.OKa3aJl TeopeMY,
'11'0 MHO)KeCTBO npOCTbIX qHCeJI 6eCKOHeqHO.
PaCI.{BeT TeopHH qHCeJl B EBpone HaqaJJCH C pa60T <ppaH-
llY3CKoro MaTeMaTHKa fTbepa <1>epMa (XVII B.), 3aTeM 3H.'1e-
pa (XVIII B.) 11, HaKOHell, faycca (XIX B.).
Eo.'1bIlJHe 3ac.'1yrH B pa3BHTHH TeopHH qHce.'1, KaK MhI y)Ke
rOBOpHJlH, npHHa)J..'1e)KaT PYCCKHM H COBeTCKHM MaTeMaTH-
KaM: fT. JI. 4e6bIIlJeBY, A. A. MapKOBY, 11. M. BHHorpa.n:oBY
H JIpyrHM.
B CBH3H C paCCMOTpeHHOf! HaMH Ha npeAbIJ1Ylll.HX CTpaHH-
uax np06.'1eMOll fOJlb)J.6axa - 3H.'1epa, 6.'1eCTHlll.e peIlJeHHOH
aKa)J.eMHKOM 11. M. BHHorpa)J.OBhIM, XOqeTCH HeMHoro paCCKa-
3aTb 06 3TOM BbI)J.alOlll.eMCH COBeTCKOM MaTeMaTHKe.
11BaH MaTBeeBHq BUHorpaJIoB pO)J.HJlCH B 1891 r. OKOHqH.'1
<pH3HKo-MaTeMaTHqeCKHH <PaKYJlbTeT fTeTporpa)J.cKoro. YHHBep-
CHTeTa B 1914 r.
CBOIO pa60TY B 06.'1aCTI1 TeopHH qHCeJI 11. M. BHHorpa)J.oB
Haqa.'1 elll.e B npe)J.peBOJlIOI.{HoHHbIe rO)J.bI. OKTHopbcKaH pe-
BO.'1IOUHH )J.a.'1a eMY B03MO)KHOCTb IlJHpOKO pa3BepHYTb CBOH
HayqHhIe H3bICKaHIIH. B 1937 r. 11. M. BHHorp3J1oB SaKaHqH-
23

BaeT CBOIO pa60TY «HOBbIM MeTO,n B aHaJI HTHqeCK0I1 TeOpmI
qHCeJl», I1MelOlll.YIO <PYH)J.aMeHTaJlbHOe 3HaqeHHe J1JJH )J.a.'1bHeH-
IlJrO pa3BHTHH TeOpHH qHCeJI. MeTO)J.bI, KOTOpbIe npe)J.JlaraeT
B 3TOM pa60Te 11. M. BHHorpa)J.oB, HMelOT TaK)Ke npHMeHeHue
B 06.'1aCTHX MaTeMaTHKH, BbIXO)J.Hlll.HX JIa.'1eKO. 3a npe)J.eJlbI
TeopHH qHce.'1.
B 1941 r. 11. M. BHHorpa)J.oB 3a 3TY pa60TY Y)J.ocTaHBaeT-
CH rOCY)J.apcTBeHHoH npeMHH I-H CTeneHH H 3BaHHH repoH
COI.{HaJlHCTHqeCKOrO TpY)J.a.
B HaCTOHlll.ee BpeMH 11. M. BHHorpa)J.oB B03r.'1aB.'1HeT Ma-
TeMaTHqeCKHH HHCTHTYT HMeHH B. A. CTeK.'10Ba AKa)J.eMHH
EaYK CCCP.
3a.n.aljH
1. lI.oKa3aTb, qTO cYMMa n nepBbIx qHce.'1 HaTypaJlbHOrO
pH)J.a paBHa n(n+l)
2
Y K a 3 a H H e. HanHcaB HaTypaJlbHbIll pH)J. qHCeJl 01'
)J.o n
1, 2, 3, 4, 5, ..., n-2, n-l, n,
nO)J.MeTHTb CBoikTBO: cYMMa qHce.'1 3Toro pH)J.a, O)J.HHaKOBO
OTCTOHIII.HX 01' KOHI.{OB ere, Bcer)J.a paBHa CYMMe KpaHHHX
qHCeJl.
2. lI.oKa3aTb npH nOMOlll.H rHOMOHoB l ), BKJllOqeHHbIX B
KBa)J.paT (qepT. 1), '11'0 CYMMa n nepBbIx HeqeTHbIX qHCeJ1 paBHa
KBaJIpaTY HX qHC.'1a.
Y K a 3 a H H e. Do.n:CqHTallTe CYMMY e)J.HHlII.{ B Ka)K)J.oM
rHOMOHe H CYMMY e)J.HHHI.{ BO Bcex KJleTKaX KBa)J.paTa ABCD.
3. JJ.oKa3aTb, '11'0 cYMMa n nepBbIX qeTHbIX qHce.'1 HaTY-
pa.'1bHOro pfl)J.a paBHa n(n + 1).
4. JI,oKa3aTb npH nOMOlll.H rHoMoHoB, BKJllOqeHHbIX B KB.n.-
paT (qepT. 2), '11'0 cYMMa KBa)J.paToB n nepBbIx qHCe.'1 HaTY-
paJlbHOrO pH)J.a paBHa n(n+l2n+l) , 1'. e.
12 + 22 + 3 2 + ... + n 2 = n (n + 1) (2n + 1)
6
1) rHOMOHOM (')YJJ:eM Ha3blBaTb tPHrypy BH.l\a
r.
24

tJ
C fJ
f.l.
c
B
4 4
J J
2 2
1 I
/l 1 4 h .0/1 I,
4epT. I. 4epT. 2.
Y K a 3 a H If e. CYMMa qHCeJl BHYTPH Ka)K)J.oro rHOMOHa
paBHa KBa)J.paTY 'lHc.1Ja, COBna)J.alOlll.erO c HOMepOM fHOMOHa.
CYMMa Bcex 1!HCeJl BHYTPU KBa)J.paTa ABCD paBHa 12+22+3 2 +
... +n2, KOTOpylO 0603HaqHM qepe3 X. fIO)J.CqIfTaeM 3Ty CYMMY.
CYMMbI 'lHCeJl BbIIlJe JIHaroHaJIH AC H HH)Ke ee paBHbI.
O)J.Ha H3 3THX CYMM paBHa:
(n-1)+2(n-2)+3(n-3)+ ... +(n-l) [n-(n-l) J=
=n+2n+3n+...+(n-l)n-[1 2 +2 2 +3 2 +...+(n-l)2 J=
. =n[I+2+3+...+(n-l) ]-[x-n 2 ]=
_ n.n(n-1) [ 2 ]
- 2 - x-n .
CYMMa qHce.J1, HaxO)J.Hlll.HXCH Ha J(HarOHaJlH AC, paBHa
n(n+l)
2
COCTaBJlHeM ypaBHeHHe
n2(-I)-2x+2n2+ n(nil) =X.
PeIlJaH 31'0 ypaBHeHue, nO.1JyqUM:
6x = n (2n 2 + 3n + 1);
n.2(n+1)(2n+1)
x = 6.2 .
1 2 -L 2 2 + 3 2 + + 2 = n (n + I) (2n + 1)1)
T. e. I ... n 6 .
1) nOJIYIJeHHOe paBeHCTBO He pa3 BCTpeTJlTCH IJJlTaTeJlIO npJl Jl3Y-
ljeHJIJI MaTeMaTIIKJI. 3TO paBeHCTBO MomeT (')blTb BblBe.lleHO JI JlJIblMH
cnoc06aMI1.
2ti

5. .IT.oKa3aTb npH nOMOI4H qepTe)Ka 3, qTO cYMMa Ky60B
n nepBbIX qHCeJl HaTypa.'1bHoro pHJIa paBHa KBaJ(paTY HX
CYMMhI, T. e.
}3 + 2 3 + 3 3 + ... + n 3 = [ n (n i...!l r .
(j
n
6. JI,oKa3aTb, '11'0 BCHKoe npOCToe qHCJlO, 60J1bIlJee 3,
BbIpa)KaeTCH <POPMYJlOH
C
T. e. npH )J.eJ1eHHH Ha 6
npocToe qHCJ10 J(aeT B oc-
TaTKe HJlH I, HJlH 5.
4 .IT. 0 K a 3 aT e.'1 b C l' B O.
;} J 6 B DYCTb P - npOCToe qHCJlO,
60JlbIlJee 3, H nYCTb npH JIe-
2 6 .'1eHHH Ha 6 .91'0 qHC.'10 ,naeT
/ I J /; B qaCTHOM q H B OCTaTKe r,
fl I 2 3 4 IJ fJ1e r<6.
/1 Tor)J.a
LI epT . 3. p=6q+r. (1)
6n::!::I,
4HCJ10 P 6Y)J.eT npOCTbIM TO.'1bKO B TOM CJ1yqae, eCJlH r= 1
HJlH r=5, TaK KaK ec.'1H 6bI r paBHH.'10Cb 2, 3 H.'1H 4, TO paBeH.
CTBO (1) COOTBeTCTBeHHO npHHHJlO 6bI BH)J.:
p=6q+2=2(3q+l);
p=6q+3=3(2q+l);
p=6q+4=2(3q+2).
npaBbIe qaCTH nOCJle)J.HHX Tpex p8BeHcTB roBOpHT 0 TOM,
'11'0 qHCJ10 P B nepBoM cJlyqae )J.eJlH.'10Cb 6bI Ha 2, BO BTOpOM -
Ha 3, B TpeTbeM - Ha 2, '11'0 npoTHBOpeqHT YC.'10BHIO, TaK KaK
p - qllCJ10 npOCToe.
06paTHoe nOJlO)KeHHe HeI!epHo, T. e. He BCHKoe H8TypaJlb-
Hoe qHC.'10, JIalOlll.ee npH ,neJleHHH Ha 6 DCTaTOK 1 HJlH 5, 6Y)J.eT
npOCTbIM. fIpOBepbTe 31'0 Ha qaCTHbIX c.'1yqaHx.
7. JI,oKa3aTb, 'ITO KB8)J.paT JlI06oro npOCToro qHCJ1a, KpoMe
2 H 3, npH )J.e.'1eHHH Ha 12 ,naeT B OCTaTKe ].
8. .IT.oKa3aTb, '11'0 pa3HocTb JIByX ue.'1bIX qHCe.'1 )J.eJ1HTCH
Ha 9, ecJlH II.H<PPbI O)J.Horo qHCJla pacnOJlO)KeHbI B o6paTHoM
nopHJ(Ke OTHOCHTeJ1bHO )J.pyroro.
2f}

9. .L{oKa3aTb, '11'0 qHCJlO 144 6Y)J.eT TOqHbIM KBa.z:r.paToM BO
BCHKOM CHCTeMe CqHCJleHHH.
3 a )J. a 'I a-wYTKa.
.5I 3a)J.YMaJl TpeX3HaqHOe qHC.'10. ECJI" 01' Hero OTHHTb 7,
TO OHO pa3)J.eJ1HTCH Ha 7, eC.'1H OTHHTb 01' Hero 8, TO OHO pa3-
)J.eJlHTCH Ha 8, eCJlH )Ke 01' Hero OTHHTb 9, TO OHO pa3)J.eJlHTCH
Ha 9. 41'0 31'0 3a qHC.'10?
JIHTEPATYPA
11. K:. A H JI P 0 HOB, ApHMeTJlKa H8TYP8JIbHblX gJlce.n, YtmeJl-
rH3, 1954.
H. K A H JI P 0 HOB, ApHMeTJlKa JlP06HbIX I./JlceJI JI OCHOB-
HblX Be.nJlI./JlH, Yl./ne.l\rJl3, 1955.
f. H. Be p M 8 H, 4J1cJIO JI HaYK8 0 HeM, fOCTeXH31J.aT, 1948.
r. H. B e p M a H, Cl./eT H I./HCJIO, fOCTeXJl3AaT, 1952.
B. K. B e JI .n 10 C T JI H, KaK nOCTeneHHO JII01J.H 1J.Ow.nJl 1J.0 HaCTOH-
ll\eH 8pJlMeTJlKH, YI./neArH3, 1941.
K. 11. ill B e U 0 B, C.naB5!HCKaH HYMep811JlH, «MaTeM8TJlK8 B
WKo.ne», 1952, .N'2 2.
51. 11. n e p e JI b M a H, )l(IfB8H MaTeMaTJlKa, fOCTeXH31J.aT, 1955.
11. 51. .lJ. e n M a H, PaccK83bI 0 MaTeMaTJlKe, .lJ.eTrJl3, 1954.
B. E. n PYA H Ii K 0 B, n. JI. 4e6bIweB - Yl./eHblH H neJl8rOr,
Yl./ne.l\rJl3, 1950.
B. B. f H e JI e H K 0, BeCe1J.bl 0 p83BJlTJIJI M8TeMaTJlKJI, Jl3J1.
AnH PCI1>CP, 1946.
«BJl6JIJlOTeK8 M8TeM8TJlI./eCKOrO KpymK8», Bbm. 1, fOCTeXJl3J18T,
1950.

r.TJABA /[
H3 HCTOPHH fEOMETPHH
«TIycn, ClO.ll8 He BXOAJlT mlKTO, He
3H81Oll!.JlH reOMeTpJlJl».
Hapm!Cb Hap BXOpOM B AKapeMHJO nnaTOHa.
I M HOrHe H3 'yqalll.HXCH, paCKpbIB CTpaHHbI reo-
.,.,. MeTpHH, )J.JlH TOro qT06bI BbIyqHTb 3a)J.aHHbIll
ypOK, He npe)J.nOJlaralOT, '11'0 KHHra 3Ta HBJlHeTCH
TOJlbKO ynpOIUeHHOll nepepa60TKOH KHHrH, C03-
)J.aHHOH 60.'Jee 2000 .'JeT Ha3a,l(, qTO B KHHre, KOTOPYIO OH )J.ep-
)KHT B PYKax, 3a<pHKcHpoBaHa 60JIbllIaH qaCTb reOMeTpHqeCKHX
3HaHHll TOro BpeMeHH, '11'0 B He! OTpa)KeH H,l(eaJl .'JorHqeCKoro
nOCTpoeHHH 3TOM HaYKH C TOqKH 3peHHH )J.peBHHX rpeKOB.
C03JIaHHe .:HOH )J.peBHeH KHHrH 06bIqHO CBH3bIBalOT C HMe-
HeM rpeqeCKOrO MaTeMaTIfKa EBKJlH)J.a, )KHBIlJerO B III B.
)J.O H. 3. B AJleKcaH)J.pHH, I.{eHTpe HayqHOH MbICJIH rpe-
KOB III H II BB. rOpO,lJ, 6bIJI pacnOJIO)KeH Ha 6epery Cpe-
)J.H3eMHoro MOpH, HeJIaJleKO 01' YCTbH HHJla. 3)J.ecb 6bIJlH 3Ha-
MeHHTaH AJleKCaHJ1pHllCKaH AKaJIeMHH-MY3efJ H orpOMHaH
)J.mI TOro BpeMeHH 6H6.'J, HOTeKa HaCqHTbIBalOlll.aH COTHH TbICflq
PYKonHceH.
MO)KeT C03)J.aTbCH npe)J.CTaBJIeHHe, '11'0 HaYKa, HMeHyeMaH
reoMeTpHeH, C03)J.aHa CH.'JaMH O)J.Horo qeJIOBeKa, qTO y EB-
28

KJlH)J.a He 6bI.'I0 npe)J.IlJeCTBeHHHKOB H '11'0 n03)J.HeHIlJHM yqe-
HbIM HeqerO 6bI.'IO npH6aBHTb K TOMY, '11'0 Y)Ke )J.aJl EBK.'1H)J..
TaKoe npe)J.cTaB.'1eHHe 6bI.'I0 6bI a6COJIlOTHO HeBepHbiM. feo-
MeTpHH, KaK H BCHKaH )J.PyraH HaYKa, C03)J.aBaJlaCb nOCTe-
neHHO, yCHJlHHMH MHOrHX .'1IO)J.eH, Ha npOTH)KeHHH MHorHX
BeKOB.
3aqaTKH reOMeTpHqeCKHX 3HaHHH TepHIOTcH B rJly6HHe
TbICHqeJleTHH. .QpeBHellIlJHe H3 naMHTHHKOB KY.'1bTYPbI-
efHneTCKHe nanHpycbI Ii BaBH.'10HCKHe r.'1HHHHbIe )J.Olll.eqKH C
K.'1HHOnHCHbIMH 3anHCHMH - rOBOpfiT 0 TOM, '11'0 Y)Ke 3a 4000-
5000 .'1eT )J.O HaIlJero BpeMeHH .'1IO)J.H 3HaJlH MHoroe H3 TOro,
qTO MbI CeHqaC H3yqaeM B IlJKOJle.
B )J.peBHeHumx naMHTHHKax BaBH.'10HCKOH apxHTeKTypbI
MbI BCTpeqaeM reOMeTpHqeCKHe <pOpMbI B BH)J.e Ky6a, napa.'1-
.'1eJIenHne)J.a, IlJeCTHrpaHHoM npH3MbI, I.{H.'1HH)J.pa H KOHyca.
I1x H3J1106JIeHHOH KOMn03HUHeH 6bI.'Ia cTyneHqaTaH nHpaMH)J.a,
B <PoPMe KOTOpOH CTpOIi.'1HCb rpoMa)J.HbIe 06cepBaTopHH.
06bIqHO TaKHe nocTpoHKIi COCTOHJlIi H3 ceMH Teppac,nocTaB-
.'1eHHbIX O)J.Ha Ha )J.PyrylO H npe)J.cTaBJlHIOlll.HX C060H KaK 6hI
cTyneHH orpoMHoH .'1eCTHHI.{bI, Be)J.Ylll.eH K He6y. l(a)K)J.aH
Teppaca 6bI.'Ia OKpaIlJeHa B OC06bIH I.{BeT, npHcBoeHHbIH ceMH
CneTHJlaM: COJlHUY, JIYHe, lOnHTepy, CaTYPHY, Mapcy, Be-
Hepe H MepKYPHIO. fIOpa)KalOT MOHYMeHTa.'1bHOCTb, npOCTO-
Ta, CTpOHHOCTb H KpacoTa )J.BOpI.{OB, nOCTpoeHHbIX B TOM )Ke
CTHJle. I(TO nOCMeJl 6bI YTBep)K)J.aTb, '11'0 3TH rpaH)J.H03HbIe
3)J.aHHfI CTpOHJlHCb 6e3 3HaHHH reoMeTpHli?
TaM, r)J.e nepe)J. HaMH B03HliKaeT CTpOHHOCTb apxHTeKTYP-
HbIX <pOPM,- TaM HeMliHyeMo npHCYTCTByeT H MaTeMaTHKa.
Ho BaBH.'10HHHe He TOJlbKO 3HaJlH 0 CYlll.eCTBOBaHHH pa3-
JlHqHbIX reOMeTpHqeCKHX <pHryp, OHH YMeJlH TaK)Ke onpe-
)J.eJlHTb HX n.'10111.a)J.H H 06'beMbI. TaK, OHli YMeJlH H3MepHTb
nJlOlll.a)J.H npHMoyrOJlbHHKa, pOM6a, TpeyrOJIbHHKa H Tpane-
I.{HH, 06'beM Ky6a, npHMoyrO.'1bHOrO 6pyca H I.{HJlHH)J.pa.
BCMaTpHBaHCb B 60raTellIlJHH reOMeTpHqeCKHH opHaMeHT Ha
KaMeHHbIX n.'1liTaX, COXpaHHBIlJHXCfI OT TOll )J.peBHeiiIlJeii 3no-
Xli, MbI y6e)KL(aeMCH B TOM, '11'0 Y HHX qpe3BbIqaHHO CH.'1bHO
6bIJlO pa3BHTO qYBCTBO CHMMeTpHH. BaBHJlOHHHe 3HaJlH, '11'0
n;yra, COCTaBJIHlOlll.aH IlJeCTYIO qaCTb Bceii OKPY)KHCTH, CTH-
rHBaeTCH XOPJIOH, paBHOH pa)J.liYCY. BaBHJlOHHHaM MbI 06H3a-
HbI TaK)Ke rpa)J.YCHbIM )J.e.'1eHHHM, H B 3TOM HX 60JlbIlJafl Hayq-
HaH 3acJlyra. 1(0r)J.a MbI rOBopHM, '11'0 yroJl cO)J.ep)KHT 20°32'18",
MbI B CYllI.HOCTH qHTaeM qHCJlO, HanHcaHHoe no IlJeCTH)J.ecHTe-
PHqHOH BaBHJlOHCKOH CHCTeMe CqHCJleHHH:
29

23 -'- 32 I 18
I 60 T 602 .
fpeKH B JlHI.{e cBoero Be.'1HKOrO aCTpOHOMa fITO.'1eMeH (liB.
H. 3.) OI.{eHHJlH no JIOCTOHHCTBY npeHMymecTBa TaKOH 3anHCH,
Y)J.ep)KaBllIeiIcH H JIO HaCTOHlll.erO BpeMeHH.
DOJIbIlJOH HHTC:pec npe)J.CTaBJlHeT HaM)J.eHHaH npH paCKon-
Kax CTaTYH O)J.Horo H3 npaBHTeJIeH lUHpnYJI.'1hI, fY)J.ea, )KHB-
llIero He n03AHee 3000 .'1eT )J.O H. 3. Ha qepTe)Ke 4 MbI
BH)J.HM nJlaHbl YKpenJ1eHHOH nocTpoHKII C Mac1lITa60M.
""t.
c8[}:
I 
r
::
4epT. 4.
HaJlH1JIIe nO)J.06HbIX nJlaHOB H Macca apxHTeKTypHbIX )J.e-
TaJleH, HecoMHeHHo, rOBopHT 0 3HaKOMCTBe BaBH.'10HHH C
npIIHI.{HnOM nO)J.06HH.
MHoroe H3 Toro, qTO MbI II3yqaeM CeHqaC B reoMeTpmr,
6bI.'10 II3BeCTHO II )J.peBHHM emnTHHaM. 3HaKOMCTBO C emneT-
CKOH MaTeMaTHKoH OCHOBaHO rJlaBHbIM 06pa30M Ha )J.ByX
nanHPycax: Ha nanHpyce AXMeca, OTHocHlll.eMCH K nepHo)J.Y
2000-1700 JleT )J.O H. 3. 11 COAep)K3111.eMY reOMeTpHtJeCKl1e
3a)J.aqH Ha 113MepeHHe nJlOlll.a)J.eH H 06'beMOB, 11 Ha MOCKOBCKOM
nanHpyce, OTHOCHlll.eMCH K 9noxe OKO.'10 1850 r. )J.O H. i. l ).
ErflIITHHe YMeJIH H3MepHTb nJlOlll.a)J.H npHMoyrOJlbHHKa, Tpe-
yro.TJbHI1Ka H TpaneI.{HH no TeM )Ke npaBHJlaM, no KaKHM MbI
H3MepHeM HX Tenepb. 113 Toro, '11'0 n.'10111.a)J.b Kpyra emnTHHe
1) TI8nJlpyc H83bIBaeTCSI MOCKOBCKJlM nOTOMY, liTO B 1893 r. (')blJI
npH06peTeH PYCC,KJlM C06J1paTeJIeM rOneHJlllWBbIM, a B 1912 r. nepe-
wen B co(')cTBeHHocTb MOCKoBcKoro MY3eSJ Jl3S11l\HblX IICKYCCTB.
30

npHHHMaJ1H paBHOH nJlOlll.aAH KBaApaTa co CTOpOHOH, paBHoH
8
"9 AHaMeTpa, CJleAyeT 60J1ee TOqHOe, qeM y BaBHJlOHHH, 3Ha-
qeHHe qHCJla 7t (7t=3,1605). OAHHM H3 caMbIX 6J1eCTHUI.HX
AOCTH)KeHHH emneTcKoH MaTeMaTHKH HBJlHeTCH HaXO)K)J.eHHe
HMH npaBHJlbHOrO cnoc06a onpeAeJ1eHHH 06'beMa ycelleHHOH
nHpaMHAbI C KBBApaTHbIM OCHOBaHHeM. fIpH nepeBoAe 3TOro
cnoc06a Ha HaIlJ COBpeMeHHbIH MaTeMaTHqeCKHH H3bIK MbI
npHxoAHM K <popMYJle
h
V =3(a 2 +ab+b 2 ) (qepT. 5).
ErHPTHHaM, TaK )Ke KaK
H BaBHJlOHHHaM, 6bIJlO npH- (j
cYlll.e qyBCTBO cHMMeTpH!,1 H
nponoPllHH. TOJlbKO HaJlH-
qHe y emnTHH BHyTpeHHe-
1'0 qYBcTBa nponoPllHH n03-
BOJ1HJlO HM B03)J.BHraTb Ta-
KHe 3.lI.aHHH, KOTopbIe no-
pa)KaIOT rapMOHHqHOCTblO
CBOHX <pOpM.
11 BCe )Ke HH y BaBH-
JlOHHH, lIH Y erHnTHH He 4 5
6 .. epT. .
bIJlO elll.e reOMeTpHH KaK
HaYKH. I1x n03HaHHH HB-
JlHJlHCb TOJ1bKO pe3YJlbTaToM HaKOnJleHHOrO onbITa Ha nOIl-
Be qHCTO npaKTHqeCKHX nOTpe6HOCTeH; HO 6YAeM nOMHHTb,
IlTO erHnnme H BaBHJlOHHHe nOArOTOBHJlH TOT CbIpOH MaTe-
pHaJ1, KOTOpbIH nOCTeneHHO 6bIJI HCnOJlb30BaH APyrHM Hapo-
)J.OM C 60J1ee n03AHeH KYJlbTYPOH. 3THM HapO)J.OM 6bIJIH rpe-
KH. 4epHOBaH pa60ra 6bIJla BbIIlOJ1HeHa Ha npOTH)KeHHH
MHomx BeKOB B BaBHJlOHe H EmnTe, H Ha .lI.0J11O rpeqeCKHX
yqeHbIX BbIIlaJlO pa3BHTb 3TOT MaTepHaJl,CHCTeMaTH3HpOBaTb
ero, .lI.aTb JlOmlleCKHe .lI.OKa3aTeJlbCTBa H yCTaHoBHTb HaYIl-
HbIe MeTOAbIreOMeTpHqeCKOrO HCCJle.lI.OBaHHH.
fIepeA HaMH ApeBHHH 3J1J1a.lI.a 3a 6-7 BeKOB )J.O H. 3. Ee
ropO.lI.a pa36pocaHbI no MHOrOqHCJlemIbIM OCTpOBaM H nOJlY-
OCTpOBal\1 Cpe.lI.H3el\1HOro MOpH. K 3TOMY BpeMeHH O.lI.HH H3 em-
neTCKIiX <papaOHOB OTKpbIBaeT CB060.lI.HbHl AocTyn B CBOIO
CTpaHY. MHO)KeCTBO 3J1J1HHOB YCTpeMJlHIOTCH TY.lI.a. EAYT Kyn-
llbI H yqeHbIe, HCKaTeJ1H npHKJlIOqCHHH H MOpenJlIiBaTeJlH.
31

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Moca (564-473). OHH CTpeMHTCH B EmneT )J.JlH TOro, qT06bI
Ha MeCTe n03HaKOMHTbCH C TeM 3anaCOM 3HaHHH, KOTOpbIH 3a
TbICHqeJ1eTHH HaKOnHJlCH B 3TOM .J1peBHeM CTpaHe. OHH BnHTbIBa-
lOT B ce6R BCe TO, .qTO 3HaJlH B 06J1aCTH MaTeMaTUKU efHnTHHe.
fTpe.n.aHHe rOBopHT, qTO fTH<parop e)J.eT elll.e JI.aJlbIlJe, H8 BO-
CTOK - B CKa30qHbIM BaBHJlOH. BCIO)J.Y H Be3)J.e OHH no KPY-
nHI.{aM c06HpalOT HHTepeCYIOlll.He UX 3H8HUH U C HUMU B03-
Bpalll.aIOTCH Ha P0.n.HHY.
KOHeqHO, <1>aJlec U nH<parop He O)],UHOKU B CBOeM CTpeM-
JleHHH nepeHeCTU H8 nOqBY 3J1J1a.J1bI MaTeMaTHqeC}{Ue 3HaHHH
J1peBHero BOCTOKa, HO UX HMeHa - Hau60J1ee HpKHe B 3TOM
OTHOIlJeHUH. TaK H npOHHKalOT 3TH 3HaHUH H3 EmnTa H
BaBI1J10Ha B rpellHlO, r)],e 3apO)K)],aeTCH reOMeTpHH KaK HaYKa.
HaqaJlo pa3BHTHH reOMeTpHH Ha nOqBe rpeI.{HH CBH3bIBalOT
C HMeHeM <1>aJleca. B 06J1aCTH reOMeTpUH eMY npHnUCbIBalOT
OTKpbITHe CBOMCTBa BnHcaHHbIX yrJlOB, onHpalOlll.UXCH Ha )],ua-
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)J.ByMH yrJlaMH, npHJle)Kalll.HMI1 K HeM; OCHOBbIBaHCb Ha 3TOM,
OH YKa3aJl B03MO)KHOCTb onpe)],eJ1HTb paCCTOS1HUe )],0 He)J.o-
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4>aropeHllbl. XapaKTepHO B 3TOM OTHOIlJeHHH YTBep)K)J.e-
Hue O)],Horo H3 )],peBHerpeqeCKHX nUCaTeJleM - npOKJla:
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qllCJlO UX, nOJlyqeHHOe UMH C BOCTOKa. Ba)KHO TO, '11'0 MbICJ1b
rpeKOB IlJJla B UHOM HanpaBJ1eHUH, qeM MbICJlb erunTHHUHa
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OHU cnpaBe)],JlI1BbI He TOJlbKO )],JlH OT)],eJ1bHbIX qaCTHbIX CJlY-
qaeB, HO U Bo06111.e cnpaBe)],.1UfBbI. fTocTaBlfB Bonpoc HMeHHO.
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nOB TpeyroJlbHuKa; nOCTpoeHHe npaBHJlbHX nHTHyroJlbHuKa
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348). OH 6blJl yqeHHKOM COKpaTa. CaM COKpaT He CqUTaJl
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 A. A. Konocoa
. 33

BXO,n:Hlll.HH B AKa,n:eMHIO, qHTaJl Ha)J.nHCb: «fTYCTb ClO,n:a He
BXO,n:HT miRTO, He 3HalOlll.Hil. reOMeTpHH». fIOA BJIHHHHeM
YLJeHHH nH<paropeHI.{eB fTJlaTOH OC03HaJl Ba)KHOCTb TOqHOrO
3HaHHH, B qaCTHOCTH MaTeMaTHKH KaI< O,n:HOH HS cTyneHeH K
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nHTb B ero IlIKOJ1Y, HO He nOJlyql1BIlJeMY MaTeMaTHqeCKOrO
06paSOBaHflH.
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npHMOH: «JIHHHH, B KOTOpOH cpe,n:HHH TOqKa nOKpbIBaeTCH
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qeCKOrO nOCTpoeHHH npHMOH Ha MecrHocTH npH nOMOlll.H
npOBeIlJHBaHHH.
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HHH H cocraB aKCHOM H3MeHHJIHCb, a Ba)KeH caMbIH <paKT no-
,n:06HOro no,n:xo,n:a K nOCTpoeHHlO reOMeTpHH.
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34

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MoyroJlbHOrO TpeyrOJ1bHHKa B pallliOHaJlbHbIX qHCJlaX; 2) no-
cTpoeHHe npH60pa .n.JlH .n.eJleHHH yrJla Ha TpU paBHwe qaCTH
(TpHCeKlll1H yrJla); 3) HCCJle.n.OBaHHe CBoikTB npH3:11'bI, nHpa-
MH)1bI, llHJlHH.n.pa H KOHyca; 4) H3yqeHHe «KOHUqeCKHX ce-
qeHHll», T. e. Tex KpHBbIX, KOTopbIe nOJlyqalOTCH B pe3YJlbTaTe
nepeCeqeHHH KOHUqeCKOH nOBepxHocTH nJlOCKOCTbIO (qepT. 6).
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ooJIa H napa60J1a. B H3Y'leHHH 3TH X KpHBbIX 60J1bllIYlO pOJIb
CbIrpaJl YlleHHK fTJlaTOHa - MeHeHXM H n03)Ke ApXHMe;:I. H
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MeTO.n.OB peIlJeHHH 3a.n.aq Ha nOCTpoeHHe. Co BpeMeH OJlaTOHa
npH peIlJeHHH 3an.aq Ha nocrpoeHHe CTaJlH pa3JIHqaTb qeTbIpe
qaCTH: aHaJ1H3, nOCTpOeHIJe, .lI.OKa3aTeJlbCTBO H HCCJlen.OBaHHe.
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MHHYTb HMH EB)J.oKca (408-355), spalla, aCTpoHoMa, Mexa-
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AJIH Toro BpeMeHH CTpOroCTblO H3J10)KHJI reOMeTpHlO. KpoMe
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.lI.pY)KeCTBe C <pHJlOcoclJHeM. Ho CBH3b C <pMoco<pHeM oropBaJI8
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KO ApxHMe).{ (287-212) TpeooBaHuHMH caMOM }KH3HH 6bIJI
nOCTaBJleH B He06xo)J.HMOCTb CBH3aTb TeOpeTHqeCKOe nocTpoe-
HHe reoMeTpHH C npaKTHKoH.
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Hb!H clJaKTHlleCKHH MaTepHllJI. Ha3peJla He06xo).{HMOCTb ero
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<paropa, KOTOpbIH HanHCaJI nepBbIH yqe6HHK reoMeTpHH. BbI-
JlH, OqeBH)J.HO, H )J.Pyme nOnbITKH C03.n.aTb nO)J.06HbIH Ylle6HHK.
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HOro B Tpy.n.ax ero npe)J.IlJeCTBeHHHKOB, )J.aJ1 .lI.OKa3aTeJlbCTBO
Tex reOMeTpHqeCKHX nOJlO)KeHHH, KOTopbIe paHee 6bIJlH OooC-
HOBaHbI He.lI.OCTaTOqHO y.n.OBJleTBopHTeJlbHO. HecoMHeHHO, IJTO
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nepenllCbIBaJlaCb, a nOTOM nepeneliaTWBaJIaCb MHO)KeCTBO paa
B nepeBO)J.ax Ha H3bIKII Bcero MHpa. TOJlbKO C 1842 r. STa
KHHra BbI)J.ep)KaJIa 60Jlee 500 H3J1aHHiL no cHaliaJIaM» H3Y-
qaJlH reoMeTpHlO KonepIIlIK, rBJHIJIeH, .D.eKapT, HblOTOH,
JIOMOHCOB, JI06aqeBCKII" H MHOrHe. .J1Pyme BeJJHKHe yqe-
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)J.OMY KynbTypHoMY lieJJoBeKY B ynpOlll.eHHOM nepecKaae
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36

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«nepe)18JJH nOCJIe.lI.YlOlll.HM Bpel\IeHaM H.lI.ea.TJbI BnOJlHe JlOrHqe-
CKOH 06pa60TKH rOOMeTpHH» (<1>. K JI ell II).
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TBopeHHe BeJlHKoro rpeqeCKOrO MaTeMaTHKa He O,'l.HHa!<OBO
COBepIlJeHHO BO Bcex CBOHX qaCTHX. fTO.lI.l\ieqeHHbIe He.lI.OCTaTKH
KaCaJlHCb rJlaBHbIM 06pa30M OCHOBHbIX nOHHTHH, onpe.lI.e.TJe-
HHll H aKCHOM ero KHlIfli. DepeC;>'IOTp 9THX OCHOB 3aBepIlJliJ'-
CH C03.lI.aHHeM HeeBKJlH.lI.OBOll reOMeTpHH BeJlIlKliM PYCCKHM
yqeHbIM H. 11. JI06aqeBCKHM (1792-1856).
B KOHue XIX H B HaqaJle XX B. nOHBv'IHeTCH pH.lI. pa60T,
B OTOpbIX aBTopbI CTpeMHTCH K 6e3YKopH3HeHHoMY <POPMYJIH-
pOBaHHlO Bcex aKCHOM H OCHOBHbIX nOHHTHH reoMeTpHH.
Cpe.lI.H 3THX paooT BbIJJ,ev'IHeTCH pa60Ta HeMellKoro MaTeMa-
THKa .lX. rH.'Ib6epTa «OcHoBaHHH remIeTpHH», BbIIlJeJJ,IlJaH
B 1899 r. B J903 r. rHJlb6epT 3a CBOlO paooTY «OCHOBBHHH
remleTpHH» nO.TJyqaeT npeMHlO HMeHH H. vI. JI06aqeBcImro.
TaKoB .lI.JlHHHbIH HCTOpHqeCKHH nYTb C03.lI.aHHH TOll HaYKlI,
KOTOpylO MbI Ha3bIBaeM «3J1eMeHTapHafi reOMeTpHH».
OJ1Ha H3 3HaMeHHTWX 3aa peBHOCTH
fTO)J. TpeMH 3HaMeHHTbnm sa.lI.aqal\m .lI.peBHOCTIf nOHHMalOT
3a)J.aqH: 0 TpHceKllHH yrJla, T. e. )J.e.'IeHHH .'IlOOOro yrJl8 Ha
TPH paBHble qaCTH, 0 KBa.lI.paType Kpyra, T. e. nOCTpoeHHH
KBa.lI.paTa, nJIOlll.a.lI.b KOToporo paBHHJIaCb 6bI nJlOlll.a.lI.H .lI.aH-
HOro Kpyra, H, HaKOHell, sa.lI.aqy 06 Y.lI.BOeHHH Ky6a, T. e.
nocTpoeHHH Ky6a, 06beM KOToporo 6b1JI 6bI BJJ,Boe oo.TJee
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60"lee 2000 JleT YCHv'IHH i\IHOfHX MaTel\IaTIiKOB 6bIJlH HanpaBJle-
HbI Ha TO, qT06bI peIlJIITb YKa3aHHbIe 3aJ\aqH C nOMOlll.blO TOJIbKO
UHpKy.IJH H .'IHHeHKH, T. e. Tex HHCTpYl\leHTOB, KOropbIMH MbI
no Tpa.lI.HUHH, H.lI.Ylll.eH 01' DJlaTOHa, nOJlb3yeMcH 06bIIIHO,
peIlJafl Sa.lI.aqH Ha nOCTpoeHHe. Ho Bce 3TH YCHJIHH 6bIJlH Ha-
npaCHbIMH - 3a.lI.aqH He 6bI.'IH perneHbJ. B X I X B. 6b1J10
.lI.OKaSaHO, '11'0 HX H He.TJbSH peIlJHTb, eC.IJH npll peIlJeHHlI orpa-
HHLJHTb ce6H npHMeHeHIfeM TOJIbKO llHPKY.'IH H .TJHHeHKH.
Ho Y)Ke B )J.peBHeH rpeuHH nOHBH.'IHCb peIlJeHHH Tpex
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KpoMe llHPKYJlH H JlHHeHKlJ, HHbIX JlliHHll, KpoMe oKpy>imocTI:I
37

H npHMoH. 06bIqHO TaKHe peIlJeHHH BeCbMa OCTPOYMHbI H OqeHb
nOJle3HbI )IJlH pa3BHTHH KOHCTPYKTHBHO-MaTeMaTHqeCKOrO
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J1a)Ke B HaIllH J1HH.
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3a)J.aqH 0 TpHceKUHH yrJla, MbI OCTaHOBHMCSJ Ha )J.BYX pelle-
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)J.O H. a.
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paBHbIe qaCTH npH nOMOlll.H nOCTpoeHHH paBHocTopoHHero Tpe-
yrOJlbHHKa (qepT. 7).
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6o.'lee 06111.eH 3a.lI.aqH - 0 .lI.eJJeHIfH Ha TpU paBHbIe qaCTH Jl1060ro
yrJla. npOCToTa peIlleHHH nepBoH 3a.lI.aqH BCeJlHJla HaAe)K.lI.Y,
qTO H BTopaH 3a.lI.aqa 6Y.lI.er peIlJeHa, eCJm H He TaK npOCTO,
KaK nepBaH, TO BCe )Ke KaK-TO peIlJeHa. 3a.lI.alJa 3Ta 6bIJla
nOCTaBJleHa elll.e B V B. )J.O H. a. H nOJlyqHJla Y .lI.peBHHX rpe-
KOB Ha3saIHfe 3a.lI.aqH 0 TpHceKI.{HH yrJla 3a ee peIlJeHHe
6paJlHCb MHome H3 nY'lIlJHX rpeqeCKHX MaTeMaTHKOB, HO, .lI.OH)J.SJ
B aHaJlH3e 3a.lI.alJH .lI.0 onpe.lI.e-
neHHOrO nYHKTa H, K33aJlOCb,
YJlOBHB nOpH.lI.OK peIlleHIIH ee.
OCTaHaBJlHBaJlHCb nepeJJ, HeB03-
MO)KHOCTblO BbmOJl HHTb no-
CTpoeHHe TOJIbKO npH nOMOlll.H
UHPKYJlH H nHHei1KH, KaK
TOr.lI.a Tpe60BaJlOCb. CTOJIKHYB-
IllHCb C TalWH TPY.lI.HOCTblO,
HeKOTopbIe Ii3 MaTeMaTHKOB
peIlJHJlH He OrpaHHqHBaTb ce-
6H npH peIlJeHHH JJ,aHHOH 3a-
UHPKYJlH H JlHHeHKH 1:1 npHIllJlI:I
4epT. 7.
.lI.aqH npHl\!eHeHIfeM TOJ1bKO
K Y)J.aqHoMY pe3YJIbTaTY.
P e IlJ e H H e H H K 0 M e)J. a. nYCTb AOB - )J.aHHbIll
yron, KOTOpbIH Tpe6yeTcH pa3.lI.eJlHTb Ha Tpl:l paBHble qaCTIi
(qepT. 8). 113 TO'lKH K, npHHa.lI.Jle)KaUleH CTOpOHe OB, Ha CTO-
POHY OA onycKaeM nepneH.lI.HKYJlHp KL Ii .lI.OnOJlHHM nOJly-
qHBWHHCH TpeyronbHHK OKL .lI.0 npHMoyrOJlbHHKa ODKL.
nYCTb JIyq ON OT.lI.eJ1HeT 01' yrJla AOB TpeTblO qaCTb - yron
AON, H nYCTb ON nepeceKaeT KL Ii DN, npO.lI.OJI)KeHHe
OK. COOTBeTCTBeHHO B TOqKaX M Ii N. EC"lH P - cepe)J.HHa
MN, TO
3ti
MP=PN=KP=OK

BbI y6e,n:HTeCb B cnpaSe,n:JlHBOCTH nOCJ1e)J.Hero paBeHCTBa, eCJlH
BcnOMHHTe, '11'0 Me,n:HaHa K P nmoTeHy3bI MN TpeyroJlbHHKa
KMN paBHa ee nOJlOBI1He, H npoC.'1(,n:HTe 3a cJle)J.YIQlll.eii 3a-
mJCblO:
CJle,n:osaTeJlbHO,
L4=2L3=2Ll=L2;
L4=L2;
KP=OK,
MN=20K.
1J
N
17
LIepT. 8.
TaKHM o6pa30M, HanpaBJIeHHe Jlyqa, oTCeKalOlll.ero OT JIaH-
HOro yrJla ero TpeTblO qaCTb, MbI Hafi,n:eM, eCJln HaM y,lJ.aCTCH
nOCTpOHTb Ha npo,n:OJl)KeHHH DK TOqKY N TaK, qT06bI
MN=20K.
I1oCTpoeHHe TOqKH N npH nOMOlll.H llHPKYJlH H JIliHeJII<H If
6bIJlO TeM nYHKTOM, nepe,n: KOTOpbIM B paCTepHHHOCTH OCTaHaB-
JlHBaJlHCb MaTeMaTHKH, peIllalOlll.He 3TY 3a,n:aqy.
HHKoMe,n: npo,n:OJ1)KHJI peIlJeHHe 3TOH 3a,n:aqH H npH nOMOlll.1J
KpHBOti, Ha3BaHHOH HM K 0 H X 0 H ,n: 0 H, ,n:OBM peIlJeHHe
)J.O KOHlla.
03HaKoMHMcH npe)K,n:e Bcero C 3TOH KpIlBOH (qepT. 9).
JI,aHa npHMaH RS Ii TOqKa 0 BHe ee. 113 3TOH TOqKIi Ha RS
onYlll.eH nepneH)J.IiKYJlHp OL, Ha npo,n:OJ1)KeHIII1 KOToporo 01'-
JlO)KeH OTpe30K LLl=a. Ha Jlyqax, SbIXO,n:Hlll.HX H3 TOqKIi 0,
01' TOqeK nepeCeqeHl1H C npHMoiI RS OTJlO)KeHbI OTpe3KH, paB-
IibJe a. reOMeTpHqeCKOe MeCTO KOHI.{OB 3THX OTpe3KOB Ii npe,n:-
39

CTaB.'1HeT KOHXOI1,Ily. fTpH nOMOllI.11 KOHXOH,Ilhl MOMHO peIlJIITb
3a,Ilaqy 0 TpHceKUHH yrJla.
.ITJlH 3TOro 113 TOqKH K, Jle)Kaw.eif Ha CTopOHe OB (qepT. 10),
onYCTHM nepneH,IlHKy.'IHp KL Ha CTOPOHY OA yrJla AOE.
CTpOHl\\ KOHXOH,IlY no TOq!<e 0, npH:l-IOH R H no OTpe:.H<y
a=2.0K.
D
R
4epT. 9.
L, 0
4epT. 10.
ToqKY N BCTpeqH npHMOH KN, napaJlJ!eJlbH0I1 OLl, C KOH-
XOHJJ,OH COe,IlHHHeM C TOqKOH 0
Tor.lJ.a
I
L.NOA =3L.AOB.
.ITei1cTBHTe.'1bHO, no CBOHCTBY KOHXOH!I.bI:
eC.'1H MN=LL 1 =2.0K; MP=PN,
TO
.10
K P=PN =0[(

n03TOMY
L-4=L5 H LI =L2' L5=2L2' L2=L3;
L5=23;' L4=2L3:
1'. e.
1
LNOA=3 LAOB .
Ha qepTe)Ke 11 BbI BH.LI.UTe npu60p, npH nOMOIIlH KOToporo
MO/KHO BhlyepTlITb KOHXOI1.L1.Y.
LJepT. 11.
11
LJepT. 12,
P e III e H II e A p X H l\I e J1 a. PeIlIeHHe ApxHMe.n.a OCHO-
BaHO Ha JleMMe:
EC/lu e OKPYOfC/iOCmU U3 nzOI{KU, Aeau{ea eHe ee, npoee-
oeHbl oee ceKYllI,ue - OaHa 'Lepe3 'leHnzp, a opycafl maK, 'lmo
eHeUlfluu ee ompeSOK pa8eH paouycy OKpYHocmu, mo y20/l
Ateoy ceKyU{W,1U U3A,tepJlemcJl mpembei1 'iaCmblO 60Abwei1 ll3
oye, saKAlOlleHHblX Aleoy elO cmopOHaJ,lll.
J1. 0 K a 3 aTe JI b C T B 0 (YepT 12):
fTYCTb MC=OC=R, vBA=mo Ii vCD=n°. TaK RaI<
TpeyroJlbHHK MCO paBII06eJJ,peHHbIH, TO
LCMD=LCOD=no;
o .n.pyroH CTOpOHhI, no Teope:\lc 06 113t\IepeHmi yrJla C BepIllIi-
11011 Bue OKPY)KHOCTU:
LCMD ===
Hi .D,Byx nOCJlC)J,HHX paBCHCTB lIl\IeeM:
mO - n°
2
m O _ n°
2
=n°
41

Ii
mO=3n°,
T. e.
1
LAMB=3"LAOB.
npH nOMOlll.H )J.OKa3aHHOH JleMMhI pemeHHc 3a)J.a'IH 0 TPH-
CCKI.{HH yrJla CBe)J.eTCH K cJle)J.YlOlll.eMY.
LLaH yroJ1 AOB, KOTOphlH Tpe6yeTcH pa3AeJ1Hb Ha TPH
paBHbIe qaCTH. OnHIlleM npOH3BOJlbHhIM pa)J.HycoM R=OA =OB
OKPY)KHOCTb C I.{eHTpOM B BepIllHHe )J.aHHoro yrJla ('IepT. 13).
ECJJH HaM YAaCTCH no-
CTpOHTb Ha OKPY)KHOCTH TO'I-
KY CTaK, qT06bI: a) MC=R,
6) TO'IKa M HaXO)J.H.'!aCb
Ha npo)J.OJl)KeHHH CTOpO-
HhI OB yrJla AOB If B) 1'0'1-
KH M, C H A npHHaAJle)Ka-
JlH O)J.HOH npS1MOH, TO Ha
OCHOBaHHH JleMMhI 3a)J.a'Ia
6hIJla 6bI pemeHa.
ITonhITKH MHomx rpeqe-
CKHX MaTeMaTHKOB nOCT-
pOHTb TOYlKY C npH nOMOlll.H TOJlbKO llHPKYJlH H JlHHeHKH He
YBeH'IaJlHCb ycneXOM. B HaCTOHlll.ee BpeMH AOKa3aHO, 'ITO 31'8
Baqa He pa3penIHMa npM nOMOlll.l1 YKa3aHHhIX npM60pOB.
f1
tIepT. 13.
'8
tIepT. 14.
ITOJlO)!{eHHe TO'IKH C MO)KeT 6hITb )J.ocTaTOqHO TOqnO onpe-
JJ,eJJeHO npH nOMOlll.H JlHHeHKH, Ha KOTOpOH HaHeCen8 J1J1HHa
pa.n.Hyca npOBe)1eHHOH OKPY>KHOCTM (qepT. 14).
LL06HBIlJHCb nYTeM HcnhITaHHH, qT06bI KpaH JlHHei'iKH, Ha
I<OTOpOM OTJlO)KeH8 J1J1IIHa pa.n.nyca, npoxOJIHJI 'lepe3 TOII!(Y A
42

H qT06hI OTpe30K MC paBHHJlCH JJ,JlHHe paAHyca, MhI H no.ny-
qHM Ha OKPY)KHOCTH TOqKY C. fTpoBe.n.H npm\1YlO AM, CTpOHM
paAHYC OE. napaJlJleJlbHbIH AM. YroJl BOE H 6YAeT TpeTbeH
qaCTblO yrJla AOB.
3aMeTbTe, qHTaTeJlb, '11'0 3a)J.aqa peIlleHa npH nOMOlll.J.1
llHPKYJlH H JlHHeHKH, Ha KOTOpOH HaHeceHa A.nHHa paAuyca.
3aJI.a'm H C04JH3MbI
CocpHCTaMH B }1peBHeH rpeI.{HH Ha3bIBaJlH <pHJlOC0<P0B-yqH-
TeJleH, 3a}1aqeH KOTOpbIX 6bIJlO HayqHTb CBOHX yqeHHKOB
«MhICJlHTb, "rOBOpHTb H }1eJlaTb». 6Y)J.yqH B 60JlbIlJHHCTBe CJlY-
qaeB rJly60Ko 06pa30BaHHbIMH JllO)J.bMH, OHH He CTO.nbKO nepe-
. }1aBaJlH yqeHHKaM 3HaHHH H3 pa3J1HqHbIX 06J1aCTeH rpeqeCKOH
HaYKH, CKOJlbKO CTpeMHJlHCb HayqHTb HX BJla}1erb HCKyCCTBOM
CJlOBeCHbIX COCTH3aHHH. 4T06hI BhIHTH no6e.n.JiTeJJeM B c.nOBec-
HOM noe)J.HHKe, COcpHCThI qaCTO no.nb30BaJlHCb TeM, '11'0 npo-
THBHHK He)J.OCTaTOqHO rJly60Ko 3HaeT npeAMeT, 0 KOTOpOM
H)J.eT pe'lb, He)J.OCTaTOqHO BHHMaTeJleH H Ha6.nlO)J.aTeJleH H no-
9TOMY He B COcrOHHHH OTJlHqHTb <paJlbllib 01' npaB)J.bI. B pe-
3YJlbTaTe CJlOBeCHOro nOe}1HHKa npOTHBHHK )J.OJl)KeH 6hlJl corJla-
CHTbCH c .n.oBoAaMH cocpHCTa H npH3HaTb ce6H n06e)K)J.eHHhIM
XOTH HCTHHa, Ka3aJlOCb, 6bIJla Ha ero cTopoHe.
ECJlH B MaTeMaTHKe nYTeM JlOmqeCKOH I.{enH paCCY)K)J.eHHH.
B O)J.HOM 3BeHe KOTOpOH HaMepeHHo AonYlll.CHa Tpy.n.HOYJlOBH-
MaH oIlJH6Ka, XOTHT y6e)J.HTb KorO-TO B HeBepHocTH BceM H3-
BeeTHoro MaTeMaTHlJ.tCKOro no.nO)KeHHH. TO nepe.ll. HaMH MaTe-
M3THqeCKHH COcpH3M.
nonp06YHTe H BbI, qHTaTeJlb, 06HapY)KHTb OIllH61<y B Tex
CO<pH3Max, KOTopbIe BaM 3)J.eCb npe)J.JlaralOTCH. 6YAbTe BHHM8-
TeJlbHbI, Ii BbI OCT8HeTecb n06e.n.HTeJJeM.
43

nepBblH COIJlH3!\1. «BCHKaH OKPY)KHOCTb HMeeT }1Ba
u.eHTpa» .
B03bMeM npOH3BOJlbHbIH yroJl ABC H B npOH3BOJlbHO 8351-
TbIX Ha CTopOHax 3Toro yrJla TOqKaX D II E nOCTpOHM nepneH-
,nHKYJlHphI K COOTBeTCTBYlOlll.IIM CTopOHaM. fTYCTb M - TOqKa
nepeCeqeHHH 3THX nepneH)J.HKYJlHpOB (qepT. 15).
IlpoBell.eM OKPY)KHOCTb
qepe3 TPH TOqKH D, M HE.
IlYCTb OHa nepecexaeT CTO-
pOHhI .AaHHoro yrJla B 1'0'1-
Kax K H L. COell.HHHM
TOqKH l( H L C TOqKOH M.
YroJ1 l(DM - npm,>!QH no
nOCTpOeHl1lO; CJle}10BaTeJlb-
HO, 6Y)J.yqH BnHcaHHbIM yr-
JlOM, OH onHpaeTCH Ha .AHa-
MeTp KM. YrOJl LEM-
TaK)Ke npHMoH H BnHcaH-
HbIH; CJleJJ,OBaTeJlbHO, OH
TaK)Ke onHpaeTcH Ha .[I,Ha-
MeTp LM. 11TaK, KM H LM -)J.HaMeTpbI o,lJ.Hoi'l OKPY)KHOCTH.
Cepe}1HHa Ka)K,lJ.OrO H3 HHX eCTb lleHTp 3TOM OKPY)KHOCTH.
OKPY)KHOCTb HMeeT lI.Ba lleHTpa.
4epT 15.
/J

E H D
8 H C
4epT. 16.
BTOPOH COIJlH3M. «KaTeT npHMoyroJlbHoro TpeyralIbHH-
Ka paBeH ero rrrnOTeHY3e».
JI,aH npm\10yroJlbHblH TpeyroJlbHHK ABC (qepT. 16).
DE - nepneH)J.HKYJlHp K AC B ero cepe)J.HHe; BM - 6Hccel{-
TpHca yrJla B; M - TOqKa nepeCeqeHHH YKa3aHHoro nepneH-
.AHKYJlHpa 11 611cceKTpHCbI. 113 TOqKl! M onymeHhI nepneH.AH-
KYJ1HphI Ml( Ha BC H ML Ha AB.
6,BKM=6,BML. (IlOqeMY?)
44

CJle)J.OB3TeJlbHO,
BK =BL,
£-:,CKM=£-:,ALM. (fTOqeMY?)
CJleAOBaTeJlbHO,
(1:
CK =AL.
CKJla)J.hIBaeM paBeHCTBa (1) H (2):
BK +KC=BL+LA,
(2)
HJlH
BC=AB,
T. e. l<aTeT npHMoyroJlbHoro TpeyrOJlbHHKa paBeH ero rHno-
TeHY3e.
fTpHBe)J.eM pH}1 reOMeTpH'leCKIiX OIllH60K, B03HHKalOIII.l1X
npH OlleHKe p33:\lepOB OTpe3KOB HJlIi <pIiryp.
1. Ha qepTe)Ke 17 nOI<a3aHO, KaK pa3J1l1qHOe oKpY)KeHlIe
BJlliHeT Ha 3pHTeJlbHYIo ou.eHKY oTpe3KoB AB H BC.
/J . /:r t
////
IJ
4epT. 17.
D
-1
1-
lJept. f8.
4epT. 19.
2. Ha 'IepTe)!{e 18 KBa)J.paT nocJle npucoe)J.IIHelUlH K HeMY
JlBYX nOJ1YKpyroB Ka)KeTCH npHMoyroJlbHHIWM.
3. 113 )J.BYX paBlIhIX npHMoyrOJlblIJIKOB, H306pa)KeHHbIX Ha
'lepTe)Ke 19, BepxHHfi: Ka)KeTCH )J.JlIIHHee, a HH)KHIIH - BbIIlle.
45

B 38KJlIOqeHHe 3TOll rJlaBbI qHTaTeJlIO npe)J.JlaraeTCH He-
CKOJIbKO }10CTaTOqHO TPY)J.HbIX 3aAaq.
3alla'la t. JI.aHbI TPH KBa)J.paTa ABCD, DCEF II FEKL
(qepr. 20). fTpoBe)J.eHbI AHarOHaJlH BD, BF H BL.
J
G D F l
lJepT. 20.
JI.oKa3aTb, 'Ira Ll+L2+L3=90 Q .
3a.na'la 2. nOCTpOHTb TpeyrOJ1bHHK ABC no TpeM TOqKaM
K, L If M, B KOTOpbIX nepeceKalOT onHcaHHYlO OKOJlO TpeyroJlb-
HHKa OKPY)KHOCTb:
a) 6HcceKTpHCbI TpeyroJlbHHKa;
6) BbICOTbI TpeyrOJ1bHHKa;
. B) BbICOTa, 6HcceKTpHca H MeAHaHa, npOBe}1eHHbIe H3 O)J.HO
BepIlJHHbI.
lIep,. 21.
3a.na'la 3. Ha 6HJlbHp)J.HOM
C nOJ1e CTOHT ABa Illapa M 11 N.
O)J.HH H3 HrpOKOB nYCKaeT Illap
M TaK, '11'0 OIl, OTpa3HBIlJHCb
nOCJ1e)J.OBaTeJJbHO OT 60pTOB
AB, BC H CD, nona}1aer B IlJa p
o N. BbIqepTl:ITb nYTb IlJapa M
(qepT. 21).
38.n.8'18 4. CTOpOHbJ Tpane-
IUiH a, b, C H d. Onpe.n.eJlHTb
COeAHHftIOLeA cepe.n.HHu OCHo-
"
.n.JlHUY OTpesK8 npftMoA,
BBHllit
48

.nHTEPATYPA
B. H. K 0 C T JI H, OCHOB8HJlH reorvreTpJlJl, Y'meJJ.rH3, 1948.
B. H. J1 e 6 e JJ. e B, O'lepKIl no JlCTOpl1J1 TO'lHbIX HaYK, BbIil. II
II [V, 1917.
B. fl. III e p e MeT e B C K JI ii, O'lepKIl no JlCTOpHIl rvraTervraTJlKJI,
Y'IneArH3, 1940.
B. B. r H e JJ. e H K 0, BeceJJ.bJ 0 pa3BJlTJllI MaTeMaTJlKJI, Jl3J1.
AflH PC<I>CP, 1946.
H. 51. J], e n M 8 H, PaCCKa3bJ 0 rvraTervraTJlKe, J],eTrll3, 1954.
J],. C. J], Y 6 HOB, OWJl6KH B reorvreTpJl'leCKIIX JJ.oK838TeJlbCTBaX,
fOCTexH3JJ.aT, 1955.
A. H. <I> e T JI COB, 0 JlOKa38TeJlbCTBe B reOMeTpJlJl, fOCTeXJl3J1aT,
[954.
"BII6J1J10TeKa rvraTeMaTJI'IeCKOrO KpY)KKa», BbIil. 2, focTexII3A8T,
1954.
B. J1 JI T 11 M a H, fJl.e OWJl6K8? .<I>Jl3MaTrH3, 1962.

1
r Jl A B A III
PA3BMniE nOH$lTM$I 0 1.JMCJIE.
IJHCJIA EJIbIE, )J.POBHbIE M I1PPAMOHAJIbHbIE
«HeT HlI'Jero YJlJllJlITeJIbHorO B TOI,
'no OTKpblTlle HeCOJl3MelHIMbiX nOTpI!C-
JIO rpeIJeCKJlX MaTeMaTIIKOIJ.
P. KYPoHm
OT u.eJIbIX tJltCeJI K JlOMaHblM
, or}1a Mhl HaqliHaeM AYMaTb 0 npoIJcxOjj{AemIH nOHHTIIH
o lJHCJle Ii )J.JlH pa3pemeHUH :/TorO Bonpoca o6pa-
lll.aeMCH K UCTOpHH MaTeM3TUKH, TO HaC npe)KAe Bcero
nOp3)KaeT Ta HCI{JllOqHTe.TJhHO )J.OJlrml H ynopH2H nCII-
XOJlOflJlleCKaH pa60Ta, IWTOPYIO npUIllJlOCL npOJJ,eJlaTb qeJlO-
BeKY, IlT06bI npIJ06peCTil 31'0 nOHHTlIe "nOJIO)KIITb HalJaJlO
HaTypaJlbHOMY PH)J.Y qUceJl. .
fTepBaH TPY,lJ.HOCTb, KOTOPYIO )J.OJJi!(CH 6bIJl npeO)J.OJleTb
Ha aTOM nYTH nepB06uITHhU'J qeJlOneK, 3aJWIOlJaJlaCb B TOM,
qT06bI 01' qaCTllbIX npeJICTaBJleHUH np!IliTlI K npe):{CTaDJleIiI!HM
06Ilero xapaKTepa; )J.J1H 31'01'0 qeJlOBeK)J.OJl)l{CII 6bIJI nO)J.MeTllTb
06111.He npH3HalUi OT,lJ.eJibHbIX npe}1MCTOB, a :iTO CJlyqHJlOCb
TOJlbKO Tor.llJl. Koma OH HaTO.nI<lIy.f!CH HE rpynnhI, no ero II-IIIC-
48

HHlO, O)J.I1Hal<OBbIX npeJJ,MeTOB. Tor)J.8, O'ieBIi)J.HO, Ii B03HHKJUi
B C03HaHHH qeJlOBeKa npeJJ,CTaBJleHHH «O,lJ.HH» H «MHoro».
.LI.aJIhHeHIllee «C03epuaHHe» C03,lJ.aJl0 nOCTeneHHO nOHHTHe
o nepBbIX 'lHCJlaX HaTyp3J1bHOrO pH)J.a; npl1 3TOM nOHHTHlO
o qHCJIe OTBJleqeHHOM Bcer)J.a npe)J.IlJeCTBOBaJlO H CHaqaJla CHUM
TeCHO CJlHBaJlOCb nOHHTHe 0 'mC.He K3KHX-HII6Y)J.b onpe)J.eJleH-
HblX npe}1MeTOB. fTepeXO,lJ. 01' nOHHTlIH 0 qlICJle Onpe)].E'vleHHbIX
npe)J.l\JeTOB K qHCJlY OTBJlelIeHHO!\lY 6bIJl TaK)Ke II OqeHb JJ,OJlfHM
H OqeHb TPY,lJ.HhIM. HarJlfl)J.HbIM noc06He;"1 )J.J1H 31'01'0 nepeXO,lJ.a
CJlY)KHJlH OKPY)KalOlll.He npe)J.MeThl. TaK, rJla3a, YIllH, PYKH
OTJI'CJlhIIOrO qeJlOBeI<a, KphlJlhH nTHI.{bI CJ1Y)KHJlH KaK 6bI Ha-
rJlH,lJ.HbI:lI noc06HeM }1J1H BbIpa60nm npe)1.cr3BJIeHHH 0 qlICJle
«,lJ.Ba». BCHKHH OT)J.eJlbHbIH npe)J.MeT: CO.TJHue Ha 6e306J13 IlHml
ne6e, JlYHa B HCHYIO HOqb, caM qeJlOBeK - BbI3bIBa,,1 npe)J.CTaB-
JleHHe «O)J.liH». fT03TOMY He Y)J.HBliTeJlbHO, lITO JllO)J.H Kor)J.a-TO
)J.3BHO BMeCTO «O)J.lflI» rOBOpHJlH «SI» HJlH «JlYHa». .LI.JlH BhIpa)Ke-
HHH 411CJl8 «,lJ.Ba» ynoTpe6JIHJlH CJlOB3: «rJla38», «!{PhIJlbH»
11 T. II.
fTocTeneHHO, nO.Tlb3YHCb CHa'18Jla naJlbllaMH PYK, a nOT0:\1
u n8J1braMH Hor, JlIO)J.H Y}1J1HIIHJ1H CqeT, H npOliIJlO OqeHb r.moro
BpeMeHIf, .npe)KAe qeM H3 Onepall!iH CqeTa, H,lJ.eU nopH)J.l\a
(nepBbII':i. BTOpm] H 1. A.) B C03H811lIH JllOJJ,eii B03HlfKJlO npe)1-
CT3BJleHTIe 0 HaTypaJlhHOM pSI)J.e 411CCJI, np0)J.OJI)KeIIHO;\f AOCT8-
T04HO )J.3J1CKO.
n ROWJIH rOAhI, cToJleTIIH, MO)/(eT 611[TI> TbICH4eJleTIIfI, II.
HaKoIIeI. B YMe qeJlOBeK8 HaqmUJeT 33pOiK,'I.3TbCH lIJJ,efl 0 )1Pii-
CTI3IIHX HaA II.eJIbIMIl qlfCJIaMH. B03HiIKIIOBeHTle ::noll H,'I.ell
,nIIKTOB8J10Cb cal\wi'I )K113HblO, pa60TOH 'IeJlOEeK3. era nOBCe)]:
HeBHbIM OnhITOM. fTepBhre waru B STOM HanpaBJleHHH 6hJJ11J
OqeHb p06KIIMH: .n:eHCTSHH nepBOHaqa.'1bHO npOH3BO)1.liJ1lICb Ha)1
He6OJJblilH!\1II qHCJlaMH. qT06LI 6uJla B03MO)KIIOCTb npaBepHTb
npaB!f.1IbHOCTb nOJlyqellHbIX pe3YJlbTaTOB. fIOCTene!HlO np1l06-
peTaJlIICb HaBbIKH, C03)J.aBamfCL npaF:THqeCfmC npallllJIa, KO-
TOpb!e C TeQelII1CM BpeMeHH pacnpocrp<UIHJIHCb IIa Bee 66J1bIlJHe
Ii 66J1bUIlie qHCJla.
CJIe.naTb npll06peTCIIIIbIe TaKII!\1 06P330;\1 31I3II1ISI TOqHhl-
MH, elll.e 60Jlblile YSCJllIllHTb 06J18CTb npIIl\leHeHIIH HX,
paCIlJllpSIH npH 3TOM IJ caMoe nOHHTHe 0 qHCJle. II CTaJlO
cO)J.ep)KaIUlel\l o.n:nOH H3 l\pCBHei'IIllllx HaYK - a p II 4) M e-
T UK U.
BMecTc C pa3nIITUeM apmpMelllKII npcTepnCn8J1a f'Jly601me
H3MeneHile H cal\lOe nOHHTue 0 'lHCJle.
I3 111 u. 1\0 H. S rpe'lCcI\ntl M3TCMClTIf/\ ECH<JlIIJI onpe.ll.eJlHR
49

'lHCJlO KaK MHO)KeCTBO, COCT8BJleHHOe H3 e,lJ.HHHl( 1 ). CJle,lJ.YH
EBKmmy, T8K)Ke Onpe,lJ.eJlHJI nOHHTHe.tlHCJla H JI. <P. MarHHll-
KHH, C03;I3BIlJHH nepBhlH Y'Ie6Hl1K apH<pMeTHKH B POCCHH
(1703). Ho erne panee XVIII B. npHMeHeHHe onpe,lJ.eJleHHH
ESKJlH}1a BCTpenIJlOCh C pH,lJ.OM TPY,lJ.HOCTeH. fTpe)K,lJ.e Bcero,
onHpaHcb H3 Hero, HY)KHO 6hIJlO npH3Harh, '-ITO 0 H 1 He HB-
JlHIOTCH 'IHCJlaMH. JI.a.nee, C 3Tl1M onpe,lJ.eJleHHeM He MHpHJlOCh
onpe,lJ.eJleHHe ,lJ.p06H K3K 'IHCJl8, He rOBopH Y)Ke 0 qHCJlaX HP-
paI.{110H8J1hHhIX. Bce 31'0, BMeCTe B3HToe, npHBeJlO K TOMY. '11'0
BO BTOpOH nOJlOBIIHe XVI I I B. 3aBOeBhIBaeT MeCTO HHoe,
06111.ee onpe,lJ.eJleHHe qHCJla, Bhl.lI-BIiHYToe elll.e HhlOTOHOM
(1642-1727): I{UC/lO ecmb OfnfLOWeliUe OOliOU ee/lUI{Ulibl K opy-
20U m020
e pooa, npuliJimou sa eouliu4Y.
3ro onpe,lJ.eJleHHe ,lJ.eJl8J10 paBHonpaBHhIMH nOJlO)KHTeJlhHhIe
I.{eJlhle, ,lJ.p06HhIe H HppaI.{HOHaJlhHhIe qHCJla.
CJlO)KeHl1e ,lJ.BYX I.{eJlhlX qHCeJl a H b MO)KHO paCCl\f3TpHBaTb
KaK HaXO)K,lJ.eHl1e B H3ryp8JlhHOM pH,lJ.e qHCJla, 38HHM8lO111.erO
b-e MeCTO nocJle a. TaK K8K H8TypaJlhHbIH pH,lJ. 6eCKOHeqeH,
TO 31'0 qHCJlO Bcer,lJ.a l\fQ)KeT 6hITh B HeM HaH,lJ.eHO. CJJe)10B3reJlh-
HO, .n.eHCTBlJe CJlO)KeHHH I.{eJJbIX qHCeJl BCer,lJ.a BhIOOJlHHMO, H
n03TOMY OnepaI.{HH CJlO)KeHHH ueJlbIX qHCeJl He MOrJla H8rOJlK-
HYTh 'IeJlOBeKa H8 MhICJlh 0 CYlll.eCTBOB8HHH qHCeJl, OTJlHqHbIX
01' qHCeJl HaTypaJlbHOrO pH.n.8.
T8K)Ke Bcer.n.a BhillOJlHHMO H .n.eHCTBHe YMHO)KeHHH ueJlbIX
qHCeJl, TaK K8K YMHO)KeHHe B .n.8HHOM cJlyq8e MO)KHO paCCMa-
rpHBarb K8K q8CTHhIH 'CJlyqaH CJlO)KeHHH, Kor.n.a CJlar8eMbIe
O,lJ.HHaKOBbI. fT03TOMY YMHO)KeHl1e I.{eJlbIX qHCeJl He MOrJlO
nOBJle'lb 3a C060H Heo6xo.n.HMOCTH BBe.n.eHHH HOBbIX quceJl,
OrJlUqHbIX or QHCeJl H8TYP3J1bHOrO pH.n.a. To )Ke C8Moe MO)KHO
CKa38Tb H 0 B03Be.n.eHHH lleJloro qHCJla B I.{eJ1YlO nOJlO)KIUeJlb-
HYlO CTeneHb.
I1TaK, npHMhle onepau.UH Ha.n. lleJlhIMH qHCJlaMH He CTaBHJlU
Bonpoca 0 paClimpemiH nOHHTHH 0 qHCJle.
06paTHhIe OnepaI.{HH Ha.n. I.{eJlbIMH qHCJlaMH - BhlqHT8HHe,
,neJleHHe, H3BJleqeHHe KOpHH H JlorapH<pJ\mpOBaHue - He Bcer-
)18 BbmOJlHHMbI, eCJlH pe3YJlbraT 3THX .n.eHCTBHH HCKaTb cpe.n.H
qHceJl H8TypaJlbHOrO pH.n.a. TaK, HeJlb3H npOH3BeCTH .n.eHCTBHH
Ha.n. lleJlbIMH QHCJlaMH 5-8; 4 : 7 HJlH HaHTH x, eCJ}H 2-
=5,
He pacnmpHH nOHHTHH 0 'IHCJle.


1) B nomlTJle MHomeCTBa EBKJIJlJI BKJIa,D.blBaJI KHoe COJl.epm8HJle,
geM TO, KOTopoe BKJIaJlblBaeM Tenepb Mbl. B HaweM nOHHMaHKJI «MHO-
meCTBO» MomeT 6b1Tb COCTOSJli(HM H3 OAHOU eJl.JlHJlllbi II MomeT 6h1Tb
«nYCToe MHomeCTBO» - HYJIb.


50




Ho ObIJIa H .n.pyraSI npHlIHHa, nOMHMO 06paTHbiX LleiicTBIiH
Ha.n. u.e.1bIMH lIHCJIaMU, 33CTaBJISIBllIaSi lIeJIOBeKa HLlTH no nYTH
paCllIHpeHHSI nOHSITHSI 0 lIHCJ1e; 9TOH BTOpOH npHlIHHOH OK a-
3aJICSI npou.ecc H3MepeHHSI BeJIHlIHH, KOTOpbIH, KaK H npou.ecc
ClIeTa, B03HHK OlleHb .n.aBHO.
B HaCTOSIllI,ee BpeMSI MbI 3HaeM, liTO, HeCMOTpH Ha oecKO-
Hell Hoe pa3Ho06pa3He BeJIHlIHH, KOTopble nplfXO,!I,HTCH H3MepSITb
lIeJIOBeKY B HaYKe Ii npaKTH4eCKOH .n.eSITeJIbHOCTH, H3MepeHHSI
9THX BeJIHlIHH MorYT 6bITb CBe.n.eHbI K H3MepeHHIO TpeX OCHOB-
HbIX BeJIHlIHH: JJ.JIHHbI, MaCCbl H BpeMeHH. BbIopaB onpe.n.eJ1eH-
HWM 00pa30M e.n.HHHu.w JJ.JISI
H3MepeHHSI Tpex YKa3aH-
/I I
HbIX' BeJIHlIHH, MOlliHO oy.n.eT
nOCTpOHTb CHCTeMY Mep JJ.JISI C I
BceB03MOlliHbIX BeJIH4HH, B
KOTOpOH Bce eJJ.!1HHu.bI HB-
JI HlOl C5J [J pOH3BO.n.HbIMH OT
e.n.HHHu. .n.JIHHbI, MaCCbI H BpeMeHH. TaKaSi CHCTeMa Mep no-
CTpoeHa H Ha3BaHa a6COJ11OTHOH CHCTeMOH e.n.HHHu..
B03MOlliHO, liTO HMeHHO npou.ecc H3MepeHHSI BeJIHlIHH H
H3TOJIKHYJI lIeJIOBeKa Ha nepBoe paCllIHpeHHe nOHSITHSI 0 lIHCJIe
If 33CTaBHJI BBeCTH B MaTeMaTHKY .n.pOOHbIe lIHCJIa. B JJ.aJIbHeH-
IlleM B u.eJI5JX HarJIH.n.HOCTH H3JIO)KeHHSI 6y.n.eM 6paTb 3a 06pa3
BCSIKHX H3MepeHHH Henocpe.n.CTBeHHO OCYllI,eCTBJIfleMOe H3Me-
peHHe JJ.JIHHbI.


I I}


I II


4epT. 22.


H
J


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.


L
I


C I . ll


4epT. 23.


nYCTb .n.aHbI n.Ba OTpe3Ka AB H CD (lIepT. 22). 5y.n.eM
HaX0JJ.HTb OTHOllIeHHSI AB K CD, nOHHMaSI no.n. OTHOIlleHHeM AB
K CD lIHCJ10, KOTOpbIM H3MepSIeTcSI o.n.Ha H3 9THX BeJIHlIHH
(AB), ecJIH .n.PyraH (CD) npHHSITa 3a e.n.HHHu.y. PelliaSI 9TY
saLlallY, MbI MO>KeM CTOJIKHYTbCSI C .n.BYMSI CJIYlIaSIMH:
s) Bw6paHHaSI HaMH e.l!.HHHu.a (CD) YKJIaJJ.blBaeTcSI B OTpe3Ke
AB u.e.noe qHCJIO pa3 0e3 OCTama; Tor.n.3 OTHOllIeHlfe 'AB K CD
AB
8blpS3HTCfI u.eJIbIM 4HCJIOM n. B LlaHHOM CJIY4ae n= CD =3
(qepT. 22);


51




0) n.aHHbIH OTpe30K He H3l\lepReTCSI a6co.11OTHO TOllHO BbI-
6paHHOIf eJJ.HHHu.et'l, He OKa3bIBaeTCH ell l<paTHoH, KaK 9TO MbI
BIIJJ.HIIl Ha lIepTe>Ke 23. OTPe30K CD, KOTOpbIl1 MbI npHHHlIlaeM
3a e.n.HHHu.y, con.ep>KlITCSI B OTpe3Ke K L TPH pa3a H elll.e
OCTaeTCf! OCTaTOK EL, MeHbllIH!! CD. E,1J,HHCTBeHHo (lIepT. 23),
liTO MbI MO)KeM CKa3aTb B 9TO:\1 CJIYlIae,- 91'0 TO, liTO pe3YJIb-
Tal' H3MepeHHSI 3aKJIIOlIaeTCSI Me>K.n.y lIUCJIaMli 3 H 4. )KeJIaHHe
no.1JYllHTb oOJIee TOllHbIM pe3YJIbTaT 3aCTaBJI'HeT Hac npHMe-
HIITb npUeM, BBe.n.eHHbIl1 B npaKTHKY C JJ.peBHeHllIHX Bpel\IeH,-
pa3.n.pOOllTb H36paHHYIO t'iI,HHHu.y Ha paBHbIe lIaCTH, lITOObI
onpe.n.eJIHTb, CKObKO pa3 orrpe.n.eJIeHHaR lIaCTb .n.aHHOH e.n.H-
HHu.bI (CD) COJJ.ep)KHTCSI B H3MepSIeMOM oTpe3Ke K L. nYCTb
eJJ.HHHu.a H3l\1epeHHSI CD pa3JJ.eJleHa Ha lieTbIpe paBHbIe qaCTH,
1I 9Ta rrO.ll.'be.n.HHHu.a YKJIa.ll.bIBaeTCSI' B OCTaTKe EL TpH pa3a
oe3 OCTaTKa.
B l\1aTel\IaTIIKe no.n.'be.n.HHHu.a, rro.'IY4aeMaSI npH .n.eJIeHIIH
nepBOHallaJIbH0I1 e.n.IIHIIu.bI Ha lIeTbIpe paBHbJe qaCTlI, 0603Ha-
'IaeTCf! CHMBOJIOM
I
4'
II eCJIIi oTpe30K KL COn.epiKHT 15 nOlJ.'beJJ.IlHUIJ. (qepT. 23), TO
Mepa 3Toro OTpe3Ka BbIpa3HTCSI lIHCJIOM
15
nJ="4'
15
nOCJIeJJ.HHJ1 CHIIISO.n (4) H Ha3bIB3eTCH .lI.p06blO 1I.1H OTHO-
meHHeM. TaKHM o6pa30M, .n.po6Hoe lIHCJIO MO>KHO nanlIcaTb
npH nOMOlll.H .n.BYX u.eJIbIX lIHCeJI. O.AHO 113 3THX lIHCeJI lIa3bI-
BaeTCSI 3 HaM e HaT e JI e M II nOKa3LIBaeT, KaKYIO lIaCTb
BeJIHlIHHbI (CD), C KOTOpOH cpaBHHBaeM JJ.pyrylO BeJIHlIUHY
(KL), MbI npllHf!JIH 33 HOBYIO eJJ.HJmu.y; .n.pyroe - q H C JI lI-
T e J1 b - nOKa3LIBaeT, CKOJIbKO pa3 HOBaSI e.n.mmu.a cOJJ.ep-
)KUTCH B tI3MepSieMOl! BeJIHlIIIHe.
OCT8B8JIOCb CJJ.eJI8Tb noc.ne.n.HHt\, caMbIA rpY;J.HbiM mar-
OCBo60JJ.HTb CHMBOJI !!! OT CBR3H C npou.eCCOM H3MepeHHSl 11
n
caMHMH lI3MepSieMblMH se.'IHlIHHaMH - H paCCMaTpHB8Tb TOT
CHI\fBOJI KaK OTBJIe'IeIlHOe lIHC.nO, ypaBHeHHoe B CBOHX npaBax
C HaTypaJIbHblM 'IHCJIOM. 3TOT nOCJ1e.n.HHH Illar B HCTOpHH
JJ.P06HbIX qUCeJI ObiJI OCYlll,eCTBJIeH TOJIbKO B pe3YJIbT8Te Ha-
KOllJIeHlIR OT.n.eJIbHblX YCHJIHH B Te'IeHHe MHorHX CTOJ1eTHii,
52

TaK Ka« HaXO>K.n.eHHe OTHollIeHm'i JJ.BYX Be.'lHlIHH CDO.n.HTCSI
In
K ,!I,e.leHIIIO ,!I,BYX lIHCeJI, TO .n.pOOHOe lIlICJIO - CTaJl0 CHMBOJIOM,
n
BbIpa>KalOllI.IIM lIaCTHOe 01' JJ.eJIeHIISI ,lJ,BYX u.eJIbrX <!HCeJI m
II n, «or.n.<'I pe3YJIbTaT He MO>KeT ObITb BbIpa>KeH u.eJIbIM lIHCJIOM;
In
IIHa'-le .n.pOOb - eCTb n-SI lIaCTb lIHCJIa m.
n
BBe.n.eHHeM .n.P06HbIX lIHCeJI 6bIJ10 oCYllI.eCTBJIeHO nepBoe
paCllIupeHlle rrOHfITIISI lIHCJIa II .n.OCTllrHYTa B03:\IO>KHOCTb rOBO-
pllTb 0 .n.eJIeHim. H B Tex CJIYLJaSIX, KorJJ.a OHO HeB03MO>KHO
BO IIlHmKeCTBe u.eJIbIX lIHCeJI.
OOHSITue JJ.p06u, l\aK H nOHHTHe He.l0ro lIHC.1a, C TelleHlle1
BpeMeHII rrpeTeprreBaJ10 1I3MeHeIlIlH. EBKJIH.L\ (III B. JJ.O H. 3.)
B CBOHX «HaLJaJIax» I' HI1KOMaX (I B. H. 9,) B «BBeJJ.eHIUI B
apHcpIenfl{Y» COBceM He paCCMaTpHBalOT ;l,po6eii, TaK [«IK
HII Y TOro, lIlI Y .n.Pyroro HeT nOHSITHfI 0 .n.P06H KaK 06 OJJ.HOIII
LJHCJle. B TO }I{e BpeMSI ApxHMe.n. (287-212 rr. 11.0 H. 9.), YlIe-
HHK EBK.IlH.n.a, B CHOUX pa60Tax YI1OTpe6.'lSIeT .n.po6u. Cny(m-
JIOCb 3TO lIOTOMY, LJTO rpeKH OT.L\PJISIJIU apllcpleTlIKY KaK HaYKY
OT apHcpleTUKH KaK HCKYCCTBa ClIHCJ1eHHSI. LLpOOH C TOllIm 3pe-
HHH rpeKOB MO>KHO ynoTpe6.1SITb TOJIbKO npll pellIeHHH KaKHX-
JIH60 rrpa«THlIeCKHX 3a.Aall, HO H!\f tIel' MeCTa cpe.n.H lIUCeJ1,
[<OTopbIe 113YlIaeT apllcpMeTI1Ka.
B 60.1ee n03.n.Hee BpeMSI H BI1JIOTb .11.0 XVIII B. ClIIITa.1H.
'1TO JJ.P06b, HJIH, «al< lIaCTO TorJJ.a roBopHJ1H, JIOMaHOe lI1ICJ10,
eCTb coopaHue paBHbIX lIaCTei] eJJ.HIIHu.bI. Bo BTOpOH 1I0JIOI3IIHe
XVI II B. .n.pOOb Y>Ke pacCIIIaTpHBaeTcSI «aK lIaCTHOe 01' .n.eJIe-
HI1SI u.eJIbIX lIuceJI. 3i'IJIep, lIJIeH neTep6yprcKoH AI<a.n.eHHI
HaYK, B «YHHBepCaJIbHOI'i apHcpMeTHI<e» (1787) paccy}!{.n.aCT
TaK: «npeJJ.cTaBb ce6e TOJIbKO J1HHHIO B 7 ca>I{eH JJ.JIUIiOiO, TO
HHKaI<Oro COMHeHIISI lIe 6YJJ.eT, lIT06 HeB03MO>KHO ObIJIO pa3.n.e-
JIliTb 9TY JIHHlfIO Ha TPH paSHbIe lIaCTH H 0 BeJIHlIUHe TaI{OI1
HMeTb nOHSITHe. nO.1YlIaSI 0 lIaCTHOM lIIICJIe, B TaKHX CJ1YlIaSIX
npOHCllIeJJ.llIeM, SIC Hoe nOHSITHe, XOTSI OHO H He u.eJIoe lIHCJIO,
AOXo.n.HM 'lepe3 OHoe .n.o n03HaHlfSI ocooJIl-IBoro POJJ.Y lIUCeJI,
KOl'opbIe .n.p06S1MH, HJIH JIOMaUbIMII 'lHCJIaMU, na3bIBaIOTCSI».
11, HaI<OHeu., BO BTOpOH nOJIOBUHe XVll I B. nOSlBJISIeTCH
H TpeTbe HCTOJIKOBaHUe .n.P06H, Han60JIee OTBellalOllI,ee 06UCMY,
.n.aIlHOMY HblOTOHOM onpe.n.eJIeHlIIO nOHRTHSI tJHCJI8 KBK OTHO-
IllelIHH OJJ.HOH BeJIlfllHHbI K .n.pyroii, TOro >Ke po.n.a, npHHSIToH
3a eJJ.HllHu.y. C 9Toro BpeMeHII nOHHTue JJ.pOOHOIO '1UCJIa CTa-
HOBIiTCSI TaKIIM >Ke npocTblM " RCHblM. KaK Ii nOHRTue u.eJIoro
lIliCJI8.
53

BbIlIHCJIeHHSI Ha.n. .n.p06SIMH MbI BnepBble BCTpellaeM B O.AHOM
H3 .n.peBHerrWHX, .n.OllIe.n.llIHX .n.o HaC naMSITHHKOB KYJIbTYPbI -
emneTCIWM nanHpyce AXMeca. Ho cnoc06bi 3THX BbIlIHCJIe-
HHH y enmeTCKHX MaTeMaTHKOB 2000 JIeT .n.o H. 9. 6bIJIH OlleHb
cBoeo6pa3HbI H n.JIHHHbI. EmrlTSIHe, OlleBHJJ.HO, SICHO npen.-
CTaBJISIJIH ce6e TOJIbKO .n.OJIH en.HHHu.bI H n03TOMY Ka>KJJ.YIO 3a-
JJ.allY CBO.n.HJIH K JJ.errCTBHSIM Ha.A .n.P06SIMH C lIHCJIHTeJIeM 1.
nanlIpyc AXMeca H HallMHaeTCSI TaOJI, Hu.err BbIpa>KalOllI.eH
.n.p06n
2 2 2 2 2
5'7'9'11 ' 99
CYMMorr JJ.p06eH C lIHCJll-ITeJIeM I. HanpHMep:
2 I 1
5=3+15;
2 I I
7=4+ 28 ;
2 I 1
""9 = (; + 18 H T. .n..
Eha Ta6JIlIua COllep>KHT 49 .n.p06eH C lIHCmneJ1eM 2 1I 3HaMeHa-
TeJIeM B BH.n.e o.n.Horo H3 HellcTHbIX lIHCeJI OT 3 JJ.O 99, H 3TY
Ta6JIHu.y, nO-BH.n.HMOMY, erHrreTCImH MaTeMaTHK TOro BpeMeHH
.n.OJI>KeH 6blJI 3HaTb HaH3ycTb, TaK }!(e KaK MbI 3HaeM Ta6JIHu.y
YMHO>KeHHSI. B TO >Ke BpeMH 9Ta Ta6JIHua nOI<a3bIBaeT, liTO
erlinTSIHaM ObIJIO 1I3BecTHO COI<palll.eHHe .n.po6ei1, TaK KaK B
Ta6JIl-lu.e OTCYTCTBYIOT COKpaTHMbIe .n.P061I. Tpy.n.HO CI<a,3aTb.
CIWJIbKO BpeMeHH nOTpe60BaJIOCb, lIT06bI COCTaBHTb 9TY Ta6-
JIHu.y, HO HeCOIImeHHO, liTO OHa COCTaBHJIaCb He cpa3Y H SIBH-
JIaCb pe3YJIbTaTOM Jl.J]HHHOro pSI.n.a HCKaHHrr H npoo.
npH rrOMOlll.H :norr Ta6JIHu.bI JII06aSI .n.P06b MO>KeT 6blTb
rrpe.n.CTaBJJeHa I<aK CYMMa OCHOBHbIX .n.p06err C lIHCJIHTeJIeM 1.
HanpHMep:
5122111 I I
T=T+T+Y=T+T+w+T+ 28 =
I ( I I ) ( I I ) 1 1- I
=7+ T+T + 28 + 28 =T+2 +14=
I I I
=2+T+14°
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54

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55

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o HOBbIX lIlICJIaX, He06xo)].HMO elll,e BbI-
a
LlepT. 24. SICHIITb cnoc06 HX 3anHCH npil nD:\IOilI,lf
U,HcpP.
nYCTb HppaUHOHanbHoe lIHCJIO BbJpa>KaeT OTHOIlleHHe .n.BYX
HeCoH3:\IepHl\lbIX OTpe3KOB, B qaCTHOCTH lf3MepSIeT 11.JIHHY
.imaroHa.1I1 KBa,J,paTa, CTOpOl-Ia KOTOporo npHHSITa 3a e.n.HHIfu.y
H3IepeHHSI. By.neM HCKaTb npHOJIl-I>KeIIHYIO BeJ1IllIHHY OTHO-
lUeHHSI .n.HarOHaJ11I KBa.n.paTa K ero CTopoHe, CHallana C TOlIHO-
CTblO )].0 1, nOTOl\! C TOllHOCTbIO .n.o 0, I, .11.0 0,01 11 T. J1.. 113 tIep-
Te>Ka 24 SICHO, liTO
1 <  < 2,
a
CJIe)].OBaTeJI bHO,
d
-I
Q
(c lIe,J,ocT8TK01\I) If !!..2 (c lIa6bITKoM).
Q
60

4 T06bI HaHTH 9TO OTHOllleHHe C TOllHOCTblO 11.0 0,1, pa3..'!;e-
J1HM CTOPOHY a KB3,'J,paTa Ha 10 paBHbIx 4aCTeH H 6y,1,(,\1 )'K.TIa-
.n.bIBaTb -10 lJaCTb ee Ha .n. H aroHaJ1lI. MbI npH,rr,e:\1 K TaKoMY
pe3y.f!LTaTY:
1.4<!!... < 1,5,
a
T. e.
.E..;::::;J,4 (c He;!OCTaTKO:\I) H .E..=I,5 (c H36bITKO:\-I).
a a
JKenaSI HaHTH OTHOllleHHe '!:!... C TOllHOCTblO .n.o 0,01, MbI
a
,rr,OJl>KHbI 6bI.flH 6b1 pa3.n.e.'JHTb CTOPOHY KBa,'J,paTa Ha 100 paB-
u I.. u
HblX GaCTeli 11 YK.laJj,bIBaTb 100 lIaCTb ee H8 J]HarOH8JIH. l>lbI
npHlllJIU 6bI K C.'Ie,rr,YIOru.eMY pe3YJIbTBTY:
d
1,41 < -<1,42;
a
CJIe.n.OBaTe.'IbHO,
d d
[;;::::; 1,41 (c He.n.OCTaTKOilI) H [;;::::;1,42 (c H36bITKOM).
EcmI npe.n.CTaBHTb ce6e npou.ecc HBXO>K.n.eHHp. 60JIee TOtI-
d
Horo BHatIeHHSI OTHOllIeHlISi a npo.AO.rDKeHlIbIM .n.BJIbWe, TO
BCe >Ke MbI' He MO>Ke:VI npllHTli K !HBtIeHHIO !noro OTHOIlleHlISI
B BH.n.e tIlICJl8, BbTpa>KeHHOrO KOHellHOH .n.eCRTlilIHOH .n.POOblO,
TaR KaK TOr.n.a .l!.lIarOHaJIb II CTopOHa KB8;pBTa OKa38J1HCb Obi
COH3MepHMbIMH, a ,no npOTHBOpetIHJ10 UbI H3BeCTHOH HBM Teo-
peMe 0 HeC01I3MepHMOCTH .n.lIaroHaJ1l1 H CropOHbI KBa.n.paTa,
C.'Ie.n.OBBTeJIbHO, npe.ACTBBJISISI ce6e npou.ecc HBXO>K.l!.eHHSI 60JIee
1OlIHoro 3HBlIeIIlISI OTHOIlleHlISI ! 6ecKoHetIHbIM, MbI B P e3VJIb-
a .
TaTe npH.n.eM K lIHCJlY B BlI.n.e 6ecKoHetIHOH .n.eCSITlilIHOH .n.P06H.
EhB .n.po6b He MO>KeT 6bITb nepllo.n.HlIeCKOH, TaK KBK Tor.n.a
OHa MOrJIa 6bI 6bITb npeBpameHa B OObIKHOBeHHYIO .n.p06b II
.n.lIBrOHaJIb II CTOpOH8 KBa.n.paTB OKa3BJIHCb 6bI cOH3MepliMbIMlI,
C lIHC.f!aMli B BII.l!.e 6ecKoHellHbIx Henepllo.n.HlIeCKlIx .n.ecSI-
THlIHbIX .n.p06ei-i MbI CTa.f!KlIBaeMCSI BnepBbIe, npaB.n.a, TaKHe
lIlICJIa Mor.f!II 6bI ObITb JIerKO nocTpoeHbI HCKYCCTBeHHbIM 06pa-
80M H paHbllle, HBnpHMep 1,343343334... . Ho .l!.J1SI HBC BB>KeH
CPBKT, tITO ueCKOHellHBH Henepllo.n.lIlIecK8H .l!.p06b nOJ1YlIeHa
HBMlI B pe3YJIbTBTe lI3MepeHHSI, liTO 5TO lIl!CJlO OTpB>I{aeT CY-
llI,ecTBO Bonpoca, B03HHKlllero B npouecce H8MepeHHp. peaJ1bHO
CYllI.eCTBYIOru.ero OTpe3KB npHMO,
51

TaKHM 06pa30M, cnocoo 3anHCH Hppau.HOHanbHOrO lICJ1a
npH nOMOIll.H U,ficpp BbISICHeH, H TeM CaMbIM nOSIBnSIeTCSI B03-
IIIO>KHOCTb HHOro onpe.n.eJ1eHHSI HPpau.HOHaJ1bHOrO lIHCJIa.
H ppau,UOHa.1bHblJH If.UC.1Olrt HQ3bl8alOfn If.UCAO, 8blpaJICeHHOe
6eCKOHelf.HOtl OeCflfnUIf.HOU HenepUOOUlf.eCKOU Op06blO.
BWICHH;\1 npOHCXO>K.n.eHHe Tep:\IHHa - nppaUHoHa:IbHbliL
Ratio - npn nepeBo.n.e C JlaTHHCKoro SI3b!Ka 03Ha4aeT-
OTHOllIeHHe; paUHOHanbHbIe lIHCJ1a - 4HC.'la, KOTopbIe 1II0rYT
6bITb 3anHcaHbl B BH.n.e OTHomeHHSI .n.BYX ue.'JbIX lIHCeJ1; Hppa-
u.HOHaJlbHble 4Hcna He MorYT ObITb npe.lI.CTaBJ1ellbI B BH.lI.e Ta-
Koro OTHOllleHHSI.
I< He06xo.lI.Hl\IOCTH BBe.n.eHHSI B MaTeMaTHKY Hppau.HOHanb-
HblX lIHCeJl Mor.'Ja npHBeCTH H 3a.n.alla 0 HaX0>K.ll.eHl-IH CTOpOHbl
KBa.n.paTa no ero nnOlll,a.n.H HnH pe6pa Ky6a no ero 06'beMY.
J1erKO pemaSI nepBYIO 3a.n.allY .n.JISI lIHCJIeHHbIX 3HalleHHH nno-
llI.a.n.H KBa.n.paTa B 4 KBa.n.paTHbIe e.n.HHHUbl, 9 I<Ba.n.paTHbIx
e.n.IHHU nYTeM H3BnelleHHSI apHcpMeTHlIecKoro KBa.n.paTHoro
KOpHSI H3 9THX lIHcen, MaTeMaTHKM CTonKHynHcb C HeB03l\10>K-
HOCTblO pemHTb nOJI,06HYIO 3a;I.allY .n.nSI lIMcneHHbIX 3HalleHHH
nnOIll.a)l.M KBa.n.paTa B 2 KBa)l.paTHbIe e.n.HHHu.bI H T. .n.. DonbITKH
r- r-
onpe.n.e.'JHTb l 2, V 3 H T. .n.. BO MHO>KeCTBe paUHoHanbHblx lIU-
ceJ1 He npHBe.'IH K >KeJIaeMO:\IY pe3ynbTaTY. BcnOMHHM .n.OKa-
3areJ1bCTBO HeB03MO>KHOCTH pellleHHH 9TOH 3a.n.allH.
1J.oKa>Kel\l TeopeMY: eCAU 1 / 2 He pa8HflemCfl u,eAO.«y If.UCAY,
mo 011. He lrtOJICem pa8HflmhCfl 1I HUKaKOU KOHelf.HOU op06u.
TaK KaK )2<2<22, TO 1/2 He paBeH u.eJ10MY lIMCJlY. 1J.JISI
.n.aJ1bHeHllIero .n.OKa3aTeJ1bCTBa TeopeMbI BCnOI\lHHM, liTO KBa.n.paT
lICTHoro lIHCJ1a eCTb lIHCJIO qeTHOe, a HelleTHoro - HelleTHoe.
nOCJ1e.n.Hee BblTeKaeT H3 3anHCH:
(2n+I)27'4n2+4n+l=4n (n+l)+I.'
J],OKa>KeM Tenepb, liTO 1/2 He MO>KeT 6blTb paBeH KOHerlHOH
.n.pOOH. J],onYCTHM OopaTHoe.
DycTb
112 = .L
J Q ,
r.n.e p H q - lIHcna u.eJ1ble, q=f-) H P H q - lIHCJ1a B33H;\1H0
npOCTble.
Tor.n.a
( 1/2 ) 2 = p"
r (f 1-1 2 q2 = p2,
62

DocJ1e.n.HSISI 3am-ICb rOBOpl!T 0 TOM, 'lTO p2__ lJUCJIO lJeUJOe,
CJIe,n,OBaTeJIbHO, H p - lIHC.'JO lJeTHOe. nYCTb p=2k.
Tor,n,a
4k 2 =2q2 H 2k 2 = q2.
OTCIO.n.a q2_ lIHCJ10 lIeTHOe H q - TaK>Ke lJeTHOe. Ho eC.1H p
H q - lIHCJIa lIeTHble, TO .n.P06b !2. 6bIJIa 6bl COI<paTHMa, 'no
q
npOTlmOpellliT nOCTaBJ1eHHOMY YCJ10BHIO, II, CJIe.n.OBaTe.'1bHO,
1/'2' He paBHSIeTCSI KOHellHoii. .n.P06H.
TaKliM 06pa30M, BnO.TIHe peaJ1bHa5I 3a.n.alla 0 HaxaiK)J,eHlil1
CTOpOHbI KBa,n,paTa no ero n.'JOlll.a.n.H He Bcer.n.a HlIIeeT pemeHI1C,
ecJIli OTBeT HCKaTb cpe.n.H u.eJIbIX H .n.P06HbIX lIHCeJI. MaTellIa-
THKli 6bIJIH nOCTaBJIeHbl nepe.n. He06xo.n.HMOCTblO BbI6paTb O.n.HH
113 .n.BYX B03MO>KHbIX nYTeii.: nepBbIH  OTKa3aTbCSI COBceM OT
pellIeHI1SI 3a.'lalJl! 0 HaXO>K.n.eHHli CTOpOHbI KBa,n,paTa no ero
nJl, Olll.a,n,l! Kor.n.3 110CJIe,n,HHSI 3a.n.aHa lJHCJID:\1 B BHJJ.e HenOJ1 Horo
KBa.n.paTa; BropOH - paCllIl1pHTb nOHSITHe 0 lIHCJIe, BBe.n.SI
nOHSITHe 0 HOBbIX lIHCJ1ax. OHI1 BbI6paJIH BTOpOH nYTb 11 pacmH-
pHJ1H nOHSITHe 0 lIHCJIe, BBe.n.SI B 1\1aTellIaTHKY lIHCJ1a Hppau.HO-
HaJIbHbIe.
, 1.JT06bI n03HaTb npHpo.n.y 9THX HOBbIX lIHCeJ1, 6y.n.eM pac-
CY>K.n.aTb TaK: 1/2 MbI IIIO>KeM onpe.n.eJIliTb npl!6.'1H>KeHHO,
C He)J,OCTaTKOM l!J1H C H36bITKOM, HcnO.'1b3YSl MeTOA np06.
HanOMHIIM, KaK no .n.eJIaeTCSI:
1 <1/2<2.
EepeM pH.n. rmcen:
1,1; 1.2; 1,3; 1,4; 1,5; 1,6; 1,7; 1.8; 1,9.
Cpe.n.Hee lIHCJ10 9Toro pSI.n.a, 6y,n,YlIH B03Bep,eHO B KB.,aApaT,
AaeT
1,52=2,25;
CJIe):(OBaTeJIbHO,
1/2 < 1,5,
TeM 60J1ee 1/2l1IeHbIlle 1,6; 1,7; 1,8; 1,9.
Cpe.n.Hee 113 OCTaBWHXCH lIHCeJ1 -1,3, 6YAYlIH B03SeA,eHO
B KBaApaT, AaeT
1,3 11 .... 1,69.
63

3TO roBOpUT 0 ro:VI, [ITO
1,3 < 1/2:'
Ii 1'e:\] 60nee 1/26YJJ.eT 60.1bllle 1,2 HI, 1.
B3HB cpe.n.Hee H3 OCTaBllIllXCSI lIHce.'J 1,4 H I303Be)J,SI ero
B KBa.n.paT, no.1YlIHM
1,4 2 = 1,96.
CneJJ.OBaTeJl bHO,
1,4' < 1/2 < 1,5.
B3SIB HOBHM pSI)J, 4Hcen:
1,41; 1,42; 1,43; 1,44; 1,45; 1,46; 1,47; 1,48;
1,49 Ii npo)J,eJJaB 1'Y }Ke pa60TY, MbI npH.n.eM K BbIBO;I.Y, 'Iro
1,41 < 1/2 < 1,42.
31'01' npouecc 60JJee TOQHOro HaXO>K)J,eHH5I 1/2Mo>KeT 6blTb
npO)J,O.'l>KeH HeorpaHHlIeHHO. DpeJJ.cTaBmI ceoe 1'aKoii npou.ecc
6ecKOHe1.!Hbl:\I. T9r.n.a B pe3y.'Ib1'aTe MbI npH)J,e:\f K BbIBO.n.y,
liTO 1/2' eCTb 6eCKOHellHaSI )J,eCSI1'H'IHaSI )J,p06b. 3Ta )J,p06b He
MO)j{eT 6bITb nepHo.n.H4eCKOH, TaK KaK nOCJIC)J,HHSI .'JerKO npe.
r-
Bpalll,aeTCSI B 06bIKHOBeHHYlO .n.p06b, H 1'or.n.a l 2 paBHSIJICSI
6bl 06bI1<HOBeHHOA .n.p06H, liTO npOHIBopellHT TeopeMe: «ECJ1H
1/2' He paRH5IeTCfI U.e,TIOl\1Y lIIICJ1Y, TO OIl He MmKe1' paBHfITbCfI
Ii KOHe'lHOii )J,P06H».
TaKIIM 06pa30M, 3a.n.a1.!a 0 HaxorK.n.eHIUI CTOpOHbI KBa)J,paTa
no ero nJI0lll,a)J,Ii npHBo.n.HT K Hc06xo.n.HllfOC1'H nplIHSITb HOBbIe
lIHc.,a B BII.n.e UecKoHellHbIX .n.eC5ITlfllHbIX HenepHO,'J,H4eCKIIX
)J,p06eA, T. e. rrpHHSITb lIHCJ1a HpP'au.1I0HaJIbHble.
I< TaKollfY >Ke BbIBO.n.y IIlbI npHmmf 6bI H pellIa51 3a;I.a4Y
06 onpe.n.e.TJeHHH pe6pa Ky6a no ero 06'beMY H Boo6llI.e 1I3B.'le-
"aSI Va, r.n.e a - lIHCJ10, He npe.n.CTaBJI5IlOlI!ee no.1HOM n-li
CTeneHI! KaKorO-.'IH60 .n.pyroro 4HCJ18.
Hal\I OCTaeTCSI pelllHTb Bonpoc 0 reOllleTpHtIeCKOM 11306pa-
i!,eHlIH Hppau.lWH3J1bHOrO 1.!lfcna. Dpe>K.n.e lIeM npHc1'ynH1'b
K pemeHHIO 9Toro Bonpoca, 06paTH:\1 BHHMaHHe Ha O)J,HO CBOM-
CTBO MlIO>KeCTBa par{HOHa.'lbHbIX lII1Ce.TJ:
Me3ICoy ,11061JLJIIl oey.Mfl pal{UOHa,l"HhI.IW 'lllC,1GJIll 'I II r l
eceeoa cyuleCtnByem mpemte PQlfUOHaAbHoe 'iUCAO.
B C8MOM ).\e.TJe, nOJIYCYMMa r]r2 eCTb Bcer,J, paUIIOHaJIb-
64

Hoe lIHCJIO, JIe>KaIll,ee Me>K.n.y '1 H '2' YCTaHOBHB 9TOT q,aKT,
MbI nOBTopHbIM ero npHMeHeHHeM npHXOlJ,HM K CJIelJ,YIOIll,eMY:
MeJ/(oy pal{UOHa/lbNbLMU 'I.UC/laJ.tU '1 u '2 8cezoa 3a1UllO'leHO
6eCKOHe'lHOe J.tHOJICeCm60 pal{UOHa/lb'ibLX 'lUce/l.
ECJIH, npe.n.nO.10>KHTb, liTO BCe MHmKeCTBO pau.HOHaJ1bHbIX
LIHCeJI 6y.n.eT HaHeceHO B' BH.n.e COOTBeTCTBYlOmHX TOlIeK H3
LIHCJJOBYIO OCb, TO HaM MO>KeT nOKa3aTbCSI, liTO 9TH TOllKH no-
KpOIOT BCIO lIHCJJOBYIO OCb H Ha Heii He OCTaHeTCH MeCTa .DJIH
H306pa>KeHHSI Hppau.HOHaJ1bHbIX lIHCeJI. npOCToe reOMeTpH-
4
lJepT. 25. '
lIeCKoe nOCTpoeHHe pa3y6e.n.HT Hac B 9TOM. BHHMaTeJIbHO pa3-
6epHTe lIepTf}K 25 H B.n.YMa, i-iTecb KaK OH nocTpoeH.
r, -.r:- -.r- r- r- ru u
KOHu.bI oTpe3KoB "J;. r 0. r 5, "J; 6, "J; 7, "J; 8,... 3aH-
MYT t>:eCTa Ha lIHCJ10BOH OCH, cBo60.n.HbIe OT BCSIKHX HHbIX
TOlleK, 1i306pa>KalOllI,HX pau.HoHaJIbHble lIHCJ1a. ECJ1H 6bI KOHeu.
.n.HaroHaJJH -V 2 npH BpameHHH ee OKOJJO HallaJIa Koop.n.HHaT
H COBMemeHHH C lIHCJ10BOH OCblO OKa3aJJCH B TOllKe, H306pa-
>KalOru.eH paUHOHaJ1bHOe lIHCJ10, TO H -V 2 6bIJI 6bI pau.HoHaJIb-
HblM lIHCJIOM. CJIe.n.oBaTeJIbHo. MHO>KeCTBO pau.HoHaJIbHblX TO-
lIeK He nOKpblBaeT BceH lIHCJIOBOH OCH. O.n.HH H3 BH.n.HbIX MaTe-
MaTHKOB KypaHT rOBopHT no 9TOMY nOBO.n.y: «HaHBHOMY
C03HaHHIO, HeCOMHeHHO, MO>KeT nOKa3aTbCH CTpaHHblM H napa-
.n.OKCaJ1bHbIM, liTO Be3.n.e nJ10THOe MHO>KeCTBO paUHOHaJIbHbIX
TOlleK He nOKpbIBaeT BceH npSIMOH... HeT HHlIero y.n.HBHTeJIb-
HOro B TOM, t;T.J OTKpblTHe HeCOH3MepHMblX nOTpHCJIO rpeLIe-
CKHX MaTeMaTHKOB».
nocJle BBe,n,eHHSI B MaTeMaTHKY Hppau.lIOHaJIbHbIX LIIICeJ1
Ka>K,n,OH TOllKe Ha lIHCJ10BOH OCII COOTBeTcTByer onpe.n.eJ1eHH02
lIHCJIO, Ka>K.n.OMY lIHCJIY, pau.HOHaJlbHOMY IIJ1H IIppall.l1OHaJ1b-
HOMY, COOTBeTcTByeT e.n.HHCTBeHHaH onpe11.eJIeHHaH TOLIKa Ha
lI1ICJ10BOH OCH; Tenepb lIHCJlOBaSI OCb 3an0J1HeHa CIlJI Olllb , 11,
nepeM(Ill,3SI Cb no HeH, MbI npoxo.n.1IM HenpepblBHbIH pH.n. lIHCeJI.
1) P. K y P 8 H T " f. Po 6 (j H H C, lJTO T81we M8TeM8THKa.
3 A A KO,'OPGB
65

HppaHOHabHoe qHCnO KaK CeqeHHe
1.JlITaTeJIb, HaBepHOe. 3aMeTHJ1, liTO PYKOBO.n.SIru.HM npHH-
u.HnOM npH BBe.n.eHHH KaK pau,HOHaJ1bHbIX .n.p06eii, TaK H Hppa-
llHOHaJ1bHbIX lIHCeJI 6bIJIO >KeJJaHHe 06eCnellHTb B03MO>KHOCTb
H3MepeHHSI .n.JJHH OTpe3KOB nOCpe.n.CTBOM lIHCeJI, C 9TOH TOtJKH
3peHHSI Hppau.HOHaJJbHOe lIHCJJO Mbl paCCMaTpHBaJJH KaK lJ,JIHHY
OTpe3Ka, HeCOH3MepHMOrO C e.n.HHHu.eH. }KeJJaHHe .n.aTb Hppa-
u.HOHaJ1bHOMY lIHCJJY 60J1ee Y.n.OBJ1eTBOpHTeJIbHOe C J10rHlIeCKOH
TOllKH 3peHHSI onpe.n.eJ1eHHe, OCB060.n.HB 9TO onpe.n.eJ1eHHe OT
LIepT.26.
CBSI3H C npolleccoM H3MepeHHSI BeJIHlIHH, npHBeJ10 BO BTopoH
n0J10BHHe XIX B. K lIHOMY onpe.n.eJJeHlIlO Hppau,HollaJJbHoro
tJ:HCJJa.
EOJJbllIYIO pOJJb B .n.aHHOM CJIYlIae CbIrpaJ1a pa60Ta HeMeu.-
Koro MaT€MaTHKa .lle.n.eKHH.n.a «HenpepblBHocTb H Hppau.HO-
HaJ1bHble lIHCJJa» (1872).
He B.n.aBaSICb B H3JJHllIHlIe nO,Apo6HOCTlI, nocTapaeMcH BbI-
5ICHHTb TO.'JbKO caMYIO CYIlI.HOCTb 9TOro HOBoro, 6oJ1ee a6CTpaKT-
Horo onpe.n.eJJeHHH lIppau.HoHaJ1bHOrO lIHCJJa.
npe.n.CTaBHM ce6e MHomeCTBO pau,HoHaJJbHbIX lIHCeJ1. B 9TO
MHO>KeCTBO BOH.n.YT lIHCJJa u,eJIbIe, KOHellHbie J!.P06H H 6ecKoHell-
HbIe nepHo.n.HlIecKHe .n.P06lI. BbI.n.eJ1lIM H3 9TOro MHO>KeCTBa
KaKOe-J1H60 pau,HoHaJ1bHOe tJHCJ10 r (.n.onYCTHM, 3). 3TO lIHCJ10
,AeJ1HT BCe MHO}KeCTBO pau,HoHa.rIbHbIX lIHCeJI Ha ,ABa KJJaCca:
A H B (lIepT. 6).
K KJJaccy B OTHeCeM Bce pau.lIOHaJJbHbIe lIHCJ1a, KOTopbIe
MeHbme lIl1CJIa r, a K KJI?CCY A - pallHOHaJJbHble lIHCJ1a, 60J1b-
mHe lIlIC.'Ja r. CaMoe lIlIcno r (3) MbI MO>KeM OTHeCTH HJ1H K
KJIaccy B, HJJH K KJJaccy A. EC.TIlI tJHCJ10 (3) OTH€C€HO Har-.m
K KJJaccy A, TO OHO 6y.n.eT HaH60J1bmHM lIHCJ10M a* cpe.n.H
Bcex lIlICeJJ 9Toro KJ1aCCa; eCJJH tJHCJJO r (3) 6y.n.eT OTHeceHO
K KJJaccy B, TO OHO OKa>KeTCSI HaHMeHbmHM lIlICJIOM b* cpe.n.l1
Bcex lIlICeJI 9Toro KJJaCca: MO>KeM J1H MbI npe.n.CTaBlITb ce6e
OAHOBpeMeHHoe cyru.ecTBoBaHlIe HaH6oJJbllIerO pallHOHaJIbHOfO
,qHCJIa a* B KJ1aCCe B H HaHMeHblllero pau.HOHaJ1bHoro tJ:HCJIa b*
66

B KJIacce A? OlleBH.n.Ho, HeT. EC.rIH 6bl TaKoe nOJIO>KeHHe cyw,e-
a*+b* ..
CTBOBaJIO, TO pau,HOHaJIbHOe lIHCJIO ,3aKJIlOlIeHHOe Me>K.n.y
a* H b*. 6bIJIO 6bI 60J1bllIe, lIeM Hau60J1bWee lIHCJ10 a* KJ1aCCa
B, H, CJIe.n.OBaTeJIbHO, K 9TOMY KJIaccy He f!pHHa11.JIe>KaJIO 6bI,
H MeHbllIe, lIeM HaHMeHbmee pau.HOHaJ1bHOe lIHCJ10 b* B J(JIac-
ce A, CJIe)l.OBaTeJ1bHO, OHO He npHHa.n.J1e>KaJIO 6bI TaK>Ke H
KJ1accy B.
Ey.n.eM Ha3bIBaTb pa36HeHHe MHO>KeCTBa pau.HOHaJ1bHbIX
4HCeJI Ha .n.Ba KJIaCCa npH nOMOIll.H BbI.n.eJIeHHSI H3 9Toro MHO-
>KeCTBa lIHCJ1a r ce4eHHeM B 06J1aCTH pau.HOHaJ1bHbIX lIHCeJ1.
OCTaHOBHMCSI elll,e pa3 Ha Tex CBOHCTBax, KOTOpbIMH 06JIa.n.aeT
no.n.06Hoe celleHHe. BOT 9TH CBOHCTBa: I) JII060e pau.HOHaJIb-
Hoe lIHCJIO npHHa.n.JIe}KHT HJIH J( KJ1accy A, HJIH K KJ1accy B;
2) JII060e lIHCJ10 KJIaCCa A 60.1Jbwe JIIOOOro lIHCJ1a K.1acca B;
3) B K.rIaCCe B cyru.eCTByeT pau.HoHaJ1bHOe lIHCJ10, 60J1bWee
J11060ro H3 lIHCeJI 9TOro KJIaCCa, eCJ1H lIHCJIO r OTHeceHO K KJ1aC-
cy B; B npoTHBHOM cJ1Yllae B KJIaCCe A CYlll.eCTByeT pau.HO-
HaJ1bHOe lIHCJIO, MeHbmee J11060ro H3 lIHce.n 9Toro KJIaCca.
CelleHHe, o6JIa.n.alOru.ee BbIllIeYKa3aHHbIMH CBoHcTBaMH, Ha-
30BeM «co6CTBeHHbIM» celleHHeM.
Ho cyru.eCTBYIOT H HHble celleHHSI B 06J1aCTH paUHOHaJIbHblX
lIHCeJI. Ey.n.eM HX B OTJIHlIHe OT «c06CTBeHHbIX» celleHHH Ha3bI-
BaTb «Heco6cTBeHHbIMH» celleHHSIMH. I<aKHMH >Ke lIHCJ1aMH npo-
H3BO.n.SITCH 9TH HOBbIe celleHHSI H KaKOBbI CBOHCTBa no.n.OOHbIX
celleHHH?
nYCTb B 06JIaCTH paUHOHaJ1bHbIX lIHcen npOH3Be.n.eHO celle-
HHe lIHCJIOM q, He npHHa.n.JIe>KaIll.HM K MHO>KeCTBY pau.HOHaJIb-
HblX lIHCeJ1. HeT npenRTCTBHH 3a TaKoe lIHCJIO B3SITb J1106ylO
HenepHo.n.HlIeCKYIO .n.eCHTHlIHYIO )l.p06b; C no.n.06HbIMH lIHCJ1aMH
Mbl CTO.rIKHYJIHCb B pe3YJ1bTaTe H3BJIelleHHSI KOpHSI n-H CTeneHH
H3 lIHCJIa, He npe.n.CTaB.TlSIlOllI.ero nOJ1HOH n-H CTeneHH (TaK,
V 2- 1,41421...). MbI H B03bMeM q paBHbIM V 2 (lIepT. 27).
-2 -1 0
561
tlepT. 27.
8-
67

B 9TOM CJIYlIae MbI TaK >Ke, KaK H B «C06CTBeHHbIX» CelleHHSlX,
JIJI SI Ka>K.n.Oro paUHOHaJ1 bHOrO lIHCJ1 a MO>KeM pe1lI1ITb Bonpoc,
60JIbllle OHO HJ1H MeHbme, lIeM lIHCJIO q (1,41421 ...). J],JISI
9Toro npH.n.eTCH TObKO cpaBHHTb cooTBeTcTBYIOe .n.eCSITHlI-
Hble 3HaK1I (HanpI1Mep, 1.4142<1,41421... I1JIH 1,41422>
> 1,41421, ruIH 1,414 (2»1,41421...) 11 co06pa3Ho c 9THM
OTHeCTH Ka>K.n.oe pau.HoHaJIbHOe lIHCJ10 JIH60 K K.laccy A (lIHC-
JIa, MeHbIllHe lI1ICJIa q), J11160 K KJ1accy B (lIHCJ1a, 60J1bmHe
tIHCJIa q). TaKHM 06pa30M, «Hec06cTBeHHble» celleHHH TaK}Ke
06JIa.n.a1OT CBoikTBOM (I). OTclO,n,a CJIe.n.yeT, liTO «Hec06cTBeH-
HbIM» celleHHSIM npHCYllI,e H BTopoe CBOHCTBO «c06CTBeHHbIX»
celleHHH. Ho eCJIH B «c06CTBeHHOM» celleHHH B KJ1aCCe A 6bIJIO
HaH60JIbmee lIHCJIO a* HJ111 B KJ1acce B - HaHMeHbllIee lI1ICJIO
b*, TO B HOBOM celleHHH B J(JIacce A He MO>KeT 6bITb HaH60JIb-
mere pau.HOHaJIbHOrO lIHCJ1a. a B KJIaCCe B He MO>KeT 6bITb
HaHMeHbllIero pau.HOHa.rIbHoro lIHCJIa.
B caMOM .n.eJIe, KaKoe 6bl pau.1I0HaJ1bHOe lIHCJ10 m H3 KJIaCCa
A, CKOJIb yro.n.HO 6J1H3KOe K lIHCJ1Y 1/ 2= 1,41421... 11 MeHb-
lllee ero, Bbl HH Ha3BaJ1H (HanpHMep, m= I ,414209), MO>KHO
ceHlIac }Ke Ha3BaTb 6eClIHCJIeHHOe MHO>KeCTBO .n.PYrHX pau.1I0-
HaJ1bHbIX .n.p06eii (HanpHMep, m 1 =1,4142099, m 2 =1,41420999
H T. .11..), KOTopbIe 6y.n.YT 60J1bme m, HO MeHbme V 2=-- I ,41421... .
ECJIH 6bI Bbl Han1lCaJ1H, liTO V 2 1,4142135..., TO, B3S1B
m= 1,414212, Bbl Ha3BaJJH 6bI 6eC t IHCJIeHHOe MHO>KeCTBO .n.pyrHx
pau.HoHaJJbHbIX .n.p06eii (HanpHMep, m 3 =1,4142129, m,=
= 1 ,41421299 H T. .n..), Koropble 6y.n.YT 60J1bme m, HO MeHb-
llIe V 2,
QJe.n.oBaTeJ1bHO, eCJ1H B o6JJacTH pau.HoHaJIbHbIX lII1CeJ1
npoH3Be.n. eHo CelleH1Ie lIHCJ10M V TO B J(J1acce A HeT HaH-
60JIbIllerO pau.HOHaJIbHoro lIHCJIa.
TOllHO TaK >Ke, KaKoe 6bl paU1I0HaJIbHOe lIHCJIO n 113 KJIaCCa
B, CKOJIb yro.n.HO 6JIH3KOe K lIHCJIY V2= 1 ,41421... H 60JIbmee
ero, BbI HH Ha3BaJiH (HanpHMep, n= 1.41422). MO>KHO ceHlIac >Ke
Ha3BaTb 6eclIHcJIeHHoe MHO>KeCTBO .n.pyrHx palJ,HOHaJIbHbIX .n.po-
6eH (Hanp1lMep, n}=1,414219, n2=1,4142189), KOTopbIe 6y.u.YT
MeHbIlle n, HO 60n' bllle  1 ,41421... .
Cne.n.oBaTeJIbHO, eCJ1H B 06JIaCTH paUHOHaJ1bHbIX tIHCeJI npo.
1I3Be.n.eHO celleHHe V 2, TO B KJIacce B HeT HaHMeHblllero pa-
llHOHaJIbHOrO tIHCJIa.
68

TaKHe me paCCY>K.n.eHHSI MO>KHO nOBTopHTb OTHOCHTeJIbHO
V -
ceqemIii lIHCJ12MH 1/ 6, 5, 7t H T. .n.., T. e. JIIOObIM lIHCJIOM
q, He npHHa.n.JIe>Kaill.HM MHO}!{eCTBY paUHOHaJ1bHbIX lIHCeJI.
T2KHM 06 pa3Q:\t , .n.J1SI «Heco6cTBeHHbIx» cetJeHHH xapaKTep-
HO OTCYTcTBHe HaH6oJ1bmero pau.HoHaJ1bHoro lIHCJIa B KJIaCCe A
H HaHMeHblllero pau.HOHaJIbHOrO lIHCJIa B KJ1aCCe B.
YCT2HOBHU pa3HHUY l\Ie>K.n.y «c06CTBeHHbIM» H «Hec06cTBeH-
HbIM» celleHHSIMH, MO>KHO nepeHTH K onpe11.eJIeHHIO .lle.n.eKHH.n.2:
KaJ/Cooe Ce'leHUe 6 o6Aacmu pat{UOHaA 'HbIX 'lUCeA Mbl 6yoeM
Ha3bleamI:J pal{uoHaA"HbLM UAU uppal{uoHaA"HbLM 'lUCAOM, CMO-
mpfl no mOJrtY, 6yoem AU 3mo Ce'leHUe «co6cmeeHHbIM» UAU
«Heco6cmeeHHblM» .
Onpen.eJIeHHe .llcn.eKHHn.a 60JJee 26cTpaKTHO, lIeM onpe.n.e-
JIeHHe Hppau.HOHaJJbHOrO lIHCJ1a, H3BeCTHoe lIHTaTeJ110 H3 Kypca
a.TIre6pbI, HO, npHHSIB TaKoe onpe.n.eJIeHHe, Mbl c.n.eJIaJIH TPY.n.-
HbIH llIar - OCB06o.n.HJ1H 110HSITHe Hppau.HOHaJ1bHOrO lIHCJIa OT
CBSI3H C npou.eCCOM 1I3MepeHHSI.
3a.n.a(lH, C04>H3MbI H IllYTKH
1.JeJIOBeKY, 3aHHMalOill.eMYCSI HaYlIHOH HJIH npaKTHlIeCKOH
pa60TOH, OlleHb lIacTO npHxo.n.HTCH CTaJ1l<HBaTbCSI C HppaUHO.
H2J1bHbIMH lIHCJIaMH. He 6yn.eT npeYBeJIHlIeHHbI1 YTBep>K.n.e-
HHe, liTO B HaYKe H TeXHHKe Hppau.HoHa.TIbHble lIHCJ12 BCTpe-
qalOTCSI He pe>Ke, lIeM lIHCJJa paUHOHaJJbHble. MHorHe BbIlIHC-
JIeHHH, CBSI3aHHbIe C .n.JIHHOH OKPY>KHOCTH HJ1H nJ10ru.a.n.blO
Kpyra, npHBo.n.SIT K HppanHOHaJIbHOMY lIHCJ1Y, TaK KaK OTHO-
meHl'!e .n.JIHHbI OKPY>KHOCTH K .n.HaMeTpy eCTb lIHCJJO Hppau.HO-
HaJ1bHOe, H n09TOMY nO.n.ClIeT no cpopMYJIaM
C=27tR H 80=7t(2
6oJ1bmeH qaCTblO .naeT npH6J1H>KeHHOe 3HalleHHe HppauHOHaJ1b-
Horo lIHCJIa C onpe.n.eJI€HHOH CTeneHblO TOllHOCTH.
T26JIHUbI TpHrOHOMeTpHlIeCKHX 4>YHKunH, KOTOpblMH BbI
n0J1b3yeTecb B VI I 1 KJ1aCCe, KaK npaBHJ10, con.ep>KaT lIHCJI2,
BbIpa>KalOIll,He npH6JIH>KeHHbIe 3HalleHHSI HppaUHOHaJ1bHbIX
lIHCeJ1.
ECJ1H Bbl cTpoHTe npHMoyrOJ1bHbIH TpeyrOJIbHHK no .n.BYM
KaTeTaM, 3an.2HHbIM pau.HoHa,TIbHbItH lIHCJI2MH, TO B 60J1bllIHH-
CTBe CJ1YllaeB Bbl nOJ1YllaeTe rHnOTeHY3Y, HeCOH3MepHMYIO
C KaTeTaMH, H, CJIe)].OBareJIbHO, ee OTHomeHHe K JIIOOOMY H3
69

K2TeTOB 6y.n.eT Bblp2>K2TbCH lfppau.HOHaJIbHbIM lIHCJIOM. H2-
npHMep, eCJ1H a=2, b=3, TO c= 1/ 4+9 = 1/13.
OrpOMHOe KOJIHlIecTBO HaYlIHbIX H TeXHHlIeCKHX cpOPMYJI
CO.n.ep>KHT B ce6e .n.eHCTBHe H3BJIetJeHHSI KOpHSI HJ1H .n.p06HbIe
nOKa3aTeJIH. I1 MbI, .n.eJIaSI paCtJeTbl no 9THM cpopMYJIaM, lIacTO
3a6bIBaeM, liTO nOJIYlIeHHblH Ha:'v1H pe3y.rIbTaT SIBJ1HeTCSI TOJ1bKO
npH6JIH>KeHHbIM 3HalleHHeM Hppau.HOHa.TIbHOro lIHCJ1a.
npHBe.neM HeCKOJIbKO npHMepOB TaKHX cpOPMYJI:
2) BpeMSI t no.n.HSITHSI TeJIa, 6pOllIeHHoro BepTHKaJIbHO
BBepx Ha BbICOTY h:
t = 1/ 2; .
6) J],HaMeTp d TpaHcMHcclIoHHoro BaJ1a B c2HTHMeTpax:
y-
d = 12 JI  ,
r.ne N - MOlll,HOCTb CTaHKa B JIOrna.nHHbIX CHJ1aX, n - lIlICJ10
060pOTOB BaJJa B MHHYTyl).
B) nepHO.n. K0J1e6aHlISI MaTeMaTHlIeCKoro MaSITHHKa:
T=2"Y  ,
r) 4HCJIO i cnHu. llIKHBa:
I V - . ' V -
i = T D HJIlI t = b D,
r.n.e D - .nHa'l1eTp mKHBa B MHJ1.'IHllleTpax 2 ).
.n) CKOpOCTb v pe3aHHH CTanH B MeTpax B r-.umYTY:
30
to. 45. SO,52 ·
v=
f.n.e t - rJIy6HH2 pe3aHHSI B MlIJ1J1HMeTpax, S - no.n.aqa3)
B'MHJIJ1HMeTpax 3a O.n.HH 060POT,
-
I) Pe'lb HJI€T 0 nepeJla'le npH nOMOll\H peMHSI aparneHHSI OT 9J1eK-
TpoMoTopa K ll\KHBY. Har JIYXO Haca>KeHHoMY Ha BeJlOMbli\ BaJi.
2) 4eM 60JlbWe JlHaMeTp WKHBa, TeM CHJlbHee CTpeMJleHHe ero
K pa3pblBY. H n03TOMY Heo6xo.D.HMO 60JlbIllee KOJlH'IeCTBO CnHlJ" CBH-
SbiBaIOUHX 060.D. C OCblO.
8) Do,na1Ja - c,nBHr o6pa6aTblBaeMOH .IleTaJiH npH O.!l,HOM 060OTe.
70

3a)J.alJH
1. J],OKa, >Klue liTO lIHCJIa V2 + 113 II 2+V2 - Hppa-
IJ,HOHaJIbHbIe.
Y K a 3 a H IT e. ECJIH 6b1, HanpliMep, nepBoe H3 9THX
t{HCeJI 6blJ10 pau.HOHaJIbHbIM lIHCJ10M r, TO, HanHcaB
V3 =r - V2
H B03Be)l.SI B KBa)l.paT, MbI 3aKJ1IOQIIJ1H 6bI, QTO V2 eCTb pau.HO-
HaJ1bHOe QHCJ10.
2. 3a)l.alla HH)l.HHCKOro MaTeMaTHKa ExacKapa. nOKa3aTb,
liTO
VIO+V 24 +V 40 +V oO =V2+V3+VS.
Y K a 3 a H H e. Pa36efne 10 Ha CJ1araeMbIe, a nO)l.KOpeH-
Hble lIHCJ1a Ha MHO>KHTeJII1 TaK, lIT06bl no.n.KopeHHoe Bblpa>KeHHe
OKa3aJIOCb nOJIHbIM KBa)l.paTOM TpeXlIJIeHa, CTOHllI,erO B npaBoH
'laCTH paBeHcTBa.
Co4J:-I3M.
BCSIKoe 'lHCJIO BO BTOpOH CTeneHH eCTb e)l.IIHHu.a.
nYCTb
m 4
x=Y=T'
(1)
OTCIO)l.a
1IX" = -vY.
(2)
7)

BbIqTSI paBeHcTBO (2) li3 paBeHcTBa (1) no lIaCTSlM, nOJIyqHM'
x-VX'=y-VY
HJ1H
x-y =VX - vY-
Pa3JIO>KliM x - Y Ha MHO>KliTeJIli:
(vX' + Vy) . (VX' - Vy) = VX - yy:
J],eJIliM 06e qaCTli nOCJ1e)l.HerO paBeHcTBa Ha (VX -Vy):
VX + vY- = 1,
HJIli 2 VX = 1, TaK KaK x = 4 , TO 2 Y m; = 1,
liJ1H m 2 = 1.
B KaKOM 3BeHe paccY>K.lI.emIii )l.onYllI,eH8 omli6Ka?
3a.n.a"la-llIYTKa. 113 lIeTbIrex CnHqeK COCTaBJ1eHO qHCJIO
ceMb. KaK, H3MeHHB n0J10>KeHHe TOJ1bKO O.n.HOH cnlillKli, no-
JIYlIHTb e)l.HHHu.y?
JJI1TEPATYPA
B. K. B e .n . 10 C T H H. K8K nOCTeneHHO .n1O)1H )10m.nH)10 Ha.
CTOHlu,eH 8pn-pMeTHKH, Y'weJi.rH3. 1941.
11. 51, 11. e n M 8 H. P8CCK83bl 0 M8TeM8THKe, 11.enH3, 1954.
B. B. r H e)1 e H II 0, EeCe)1,bl 0 p83BHHIH M8TeM8HIKH, H3,1l.
ADH PC«1>CP. 1946,
72

P. 1\ Y P a H T H r. Po 6 6 Ii H C. 4TO T8Koe MaTeM8THK8, foc.
TeX1l3118T. 1947.
B. n. ill e p e MeT e B C K Ii ii, O'lepKII no liCTOplill M8TeM8TIIKII,
¥QneJl,rli3, 1940. -
A. I\. <1> 8 11 e e B II I1. C. COM Ii H C K II ii, AJlre6p8. Q. II,
¥QneJl,rli3, 1954.
B. M Ii H K 0 B C K Ii 1\, I1CTopH'IecKliii 0630p p83BliTiiSi nOHSITIiSi
IIpp8UliOH8J1"HOrO '1liCJl8, «M8TeM8THK8 B WKOJle». 1939, N 3.
r. P 0 JI, e M 8 X e p H O. Ten JI II U, 4HcJla II IlJHrypbl, <1>113.
Manll3, 1962.

--=r
r J1 A B A IV
H3 HCTOPHH PA3BHTH yqEHH 06 YPABHEHHX
«l1>opMyna nOll'lac Ka>KeTC!I 60nee
MYllpoii, '1eM BblllYMaBwflH ee 'Ieno-
BeK».
EO.llb/{aHO - qellJCKlf1! MaTCMaTIfK XIX B.
KaK peIllaJ1H ypaBHeHHSI B .n.peBHeii rpe""
Y l[eHHe 00 ypaBHeHHSIX eCTb O11.Ha H3 OCHOBHbIX
TeM BCeM aJ1re6pbl. C ypaBHeHHeM MbI CTaJIKHBaeM-
CSI npH peWeHHI1 MHOmX BOnpOCOB cpll311KH, Me-
XaHHKH, aCTpOHOMHH H XHMHH. PellIeHl1e 3a.n.aQ
Me'tO.AOl\! COCTaBJ1eHHSI ypaBHeHHM SlBJ1SIeTCSI MOrYllHM cpe..u.-
CTBOM npH pellIeHHH MHomx BOnpOCOB H3 06J1aCTH TeXHH-
KH, npm!3BO)l.CTBa, CTpOHTeJIbCTBa H HapO)l.HOrO X03SIH:-
CTBa. n03TOMY He y)l.I1BHTeJ1bHO, liTO 3allaTKH YlleHHH 00
ypaBHeHHSIX MbI )l.OJl)I<Hbl HCKaTb TaK x-<e .n.aJIeKO no BpeMeHH,
KaK H 3a'laTl<H XOTSI 6bl reOMeTpHlieCKHX 3HaHm1.
nepe)l. HaMI-! EmneT 3a 2000 JIeT 11.0 H. 9. EllI,p HeT MaTe-
MaTHKH KaK HaYKH, HeT H aJ1re6pbl, CJIe11.OBaTeJIbHO, HeT H
YlleHI1H 06 ypaBHeHHSIx. 11 Bce )j{e Jl3YlJeHI1e <;oxpaHHBllIHxCH
OT Toro BpeMeHH naMSITHHKOB .n.peBHei1 KYJ1bTYPbl EmoTa
rOBopHT 0 TOM, liTO H 4000 .rIeT Ha3a)l. pemaJ1H 3a)J.allH, KOTopbie
74

MbI Tenepb pemaeM npH nOMOllJ.H COCTaBJ1eHHSI ypaBHeHHiL
PellIaJlH erHnTSIHe 3TH 3a.n.allH HeCKOJ1bKO nO-HHOMY.
B03bMeM O.AHY H3 3a.Aall, nOMelll,eHHblX B ..n.peBHeHmeM erH-
neTCKOM nanHpyce. npHBe.n.eM TeKCT 9TOH 3a..n.aqH:
«OTHOIlleHlIe ..n.BYX lIHCeJl paBHO OTHOIlleHHIQ 2 : I ; , CYM-
Ma HX KBa.n.paToB 400. Onpe.n.eJ1HTb 9TH qHCJla». Mbl pellIHJ1H ObI
91Y 3a..n.aqy nYTeM COCTaBJleHHSI CHCTeMbI ypaBHeHI1H:
{:2:; 2  J
H npmllJ1H ObI K OTBeTY:
x=16; y=12.
4000 J1eT Ha3a.n. 9TY 3a..n.allY pemaJ1H, npHMeHSISI npaBHJlO «JIO>K-
Horo n0J10>KeHHSI». Xo.n. paccY>K..n.eHHH .n.peBHero erHIlTSIHHHa
TaKOH: nycTb nepBoe lIHCJ10 eCTb 2, Tor..n.a BTopoe 6y..n.eT 1 
1
CYMMa HX KBa..n.paToB paBHSIJ1aCb 6bI 6 4 , a KopeHb KBa.n.paT-
HbIH 113 CYMMbI KBa..n.paTOB 6blJl ObI  . Ho KopeHb KBa..n.paTHbIH
5
H3 CYMMbI KBa..n.paToB lICKOMblX lIHCeJ1 eCTb 20. TaK KaK 2'
COCTaBJ1S1eT BOCbMYIO ..n.OJlIO OT 20, TO nepBoe H3 HCKOMbIX He
.n.Ba, a lIHCJlO, B BoceMb pa3 60J1bIllee, T. e. 16, a BTopoe He
nOJ1TOpa, a B BoceMb pa3 60J1bmee, T. e. 12.
3..n.ecb HeT elll,e COCTaBJIeHHSI ypaBHeHHSI. Ho nO.A06HO TOMY,
KaK MbI, COCTaBJ1SISI ypaBHeHHe, 0603HallaeM HCKOMoe lIHCJ10
lIepe3 x, TaK H ..n.peBHHH MaTeMaTHK ..n.eJlaeT ..n.onYllI,eHHe, tlTO
onpe..n.eJ1SIeMOe QHCJ10 paBHO npOH3BOJIbHO B3HTOMY (IHCJIY
(B .AaHHOM cJlyqae ..n.BYM).
B III B. .n.o H. 9. OlleHb MHoroe ObIJlO .n.OCTHrHYTO B pa3-
BHTHH reOMeTpHH. C03..n.aflbl «HallaJ1a» EBKJIH..n.a. ApXHMe.n.,
peIllaSi reOMeTpHlIeCKHe 3allallH, n0J1b3yeTcSI MeTo..n.OM, OJlH3-,
KHM K TOMY, KOTOpblM lIepe3 2000 J1eT nOCJ1e Hero 6bIJ1 BBe-
..n.eH B MaTeMaTHKY HbIOTOHOM. Ho aJIni6pa (H, B lIaCTHOCTH,
peweHHe ypaBHeHHH) 6blJla H3BeCTHa HM nOllTH B TOM >Ke
OO'beMe, liTO H erHIlTSIHaM. 3a.n.allH, KOTopbIe Mbl ceHlIac J1erKO
peIllHJlH 6bI MeTo.n.OM COCTaBJIeHHH ypaBHeHHH, OHH peIllalOT
lIHCTO reOMeTpHlIeCKHM nYTeM.
PaCCMOTpHM O.n.HY 3a..n.allY H3 KHHrH EBKJIH.n.a «HaqaJ1a»;
«Pa3.n.eJIHTb npSIMYIO AB TOllKOM M Ha TaKHe ..n.Be qaCTH,
qT06bI nJ10lll.a.n.b npHMoyroJ1bHHKa, nOCTpoeHHoro Ha npHMOH
75

AB H H2 O..n.HOH li3 ee qaCTeH, 6bIJI2 6bI paBHa nJIOlll,2.n.H KBa.n.-
p2Ta, nOCTpoeHHoro Ha ..n.pyroH ero qaCTH».
nOCTpOliM Ha npSIMoi'! AB KBa.n.paT A BCD (lIepT. 28) H
nOJ1YOKpY>KHOCTb c pa.n.HYCOM BE(AE=DE). Ha AF CTpOHM
KB2.n.paT AF LM Ii npo..n.OJIX-<HM LM JJ.O BCTpe4H C CD B TOll-
Ke N. npHMaSI AB H 6y..n.eT pa3JJ.eJIeHa TOllKOi'! M TaK, liTO
nJ10ru.a.n.b AFLM p2BHSIeTCH nJIOUI,a.n.H MBCN.
J],OKa>KeM "TO:
AB2=AP.AK=AP.DF (Tal{ KaK AK=DP), HJIH
AB2=FL.DF.
I(
D
[
F
4epT. 28.
EY.AeM p2CCM2TpHBaTb nOCJ1e.n.Hee paBeHcTBo KaK paBeH-
CTBO nJIoru.a.AeiJ l<Ban.paTa ABCD H npSIMoyroJ1bHHKa DFLN;
BbIlITeM 113 ooeHX lIaCTeH nOCJ1e.n.Hero paBeHCTB8 IlJ10lll,a.n.b npH-
MoyrOJIbHHK2 AMND H nOJ1YllHM:
nJ10lll,a.n.b MBCN=nJIOlll,2.n.H APLM.
T2K 3a.n.2l12 peIlleH2 y EBKJIH.n.a. Mbi 6bI ee peIllHJ1H no.
HHOMY.
nYCTb AB=a H AM=x, r.n.e M - TOllKa, .n.eJISlru.aSl AB
TaK, KaK Tpe6yeTcSI B YCJIOBHI1 3a..n.allH; Tor.n.a BM = a-x.
COCTaBJIHeM yp2BHeHHe H3 YCOBHSI 32.n.allH:
x 2 =(a-x)a.
PeIllaSi ero, nOJIYlIHM:
x =  (y"5 - 1 ).
76

310 peIlleHHe JIerlle, lIeM .n.aHHoe EBKJIH.n.OM, nOToMY tJTO
MbI npHMeHHJ1H .n.JlSI peweHHH 3a.n.aQH MeTO.n. COCTaBJ1eHHSI
ypaBHeHHH.
Ho H 32.n.OJlrO .n.o BbIXO.n.2 B CBeT «H2tJaJ1» EBKJ1H.n.2 rpeKH
pellIaJ1H nOCTpoeHHeM TaKHe 3a.n.allH, KOTopbIe B HaCTOSIllI,ee
BpeMSI oe3 Tpy.n.a npHBo.n.SITCSI K pemeHHIO nOJlHOrO KBa.n.p2T-
Horo yp2BHeHHSI BH.n.a x 2 +px+q=0. feoMeTpHlIecKoe pellIe-
HHe no.n.ooHwX 3an.all aJ1eKO YCTynaeT B 06ll1,HOCTH aJlre6paH-
tJeCKlIM npH€MaM.
T2K. ypaBHeHlIHM
x 2 +px-q=0 H X2_pX+q=0,
n.
XI
8
f}
qepT. 29,
1<0TOpble MbI pelliaeM npH nOMoru.H H3BecTHoH BceM 4>OPMYJIbI,
y peBHHx rpeKoB COOTBeTCTBOBaJ1l1 .n.Be pa3J1HlIHOrO co.n.ep-
JKaHlIH aa.n.allH, TpeoYlOllI,He pa3J1HlIHbIX nOCTpoeHHiL
npllBe.n.€M oopa3eu. 2HTHlIHoro pellIeHlIH KBa.n.paTHoro
ypaBHeHlIH:
x 2 +px=m 2 ,
r.n.e m 2 B OOWlIHbIX oo03HalleHlISIx eCTb q.
PellIeHHe YKa3aHHoro ypaBHeHHSI rpeKH CBeJ1H 6bI K pe-
IlleHlIIO TaKo!"1 3a.n.allH Ha nOCTpoeHHe:
K .n.aHHOMY OTpe3KY AB=p npHJlO>KHTb TaKoif npHMo-
yrOJIbHHK, lITOObI, HMeSi H30bITKOM KBa.n.paT o.n.Hoif BbICOTbI
c npSIMoyrOJlbHHKOM. OH 6bIJI p2BHOBeJ1HK .n.aHHOMY KBan.paTY
co CTOpOHOH m.
P e 1lI e H H e. OTpe30K AB, paBHbIH p, TOllKOH C pa3.n.e-
JIHM nOnOJ1aM (lIepT 29) H Ha OTpe3Ke AC, K2K Ha cTopoHe,
CTpOHM KBa.n.paT ACDE. CTpOHM OTlleJ1bHO npHMoyrc.'IbHbIH Tpe-
yrOJIbHHK. OAHH KaTeT KOToporo MN=m, .llpyroif NP= :' Or
n

TOllKH D B CTOPOHY C OTKJIa.n.bIBaeM OTpe30K DF, paBHbIH fH-
nOTeHY3e MP nOCTpCeHHOrO TpeyrOJ1bHHKa, H lIepe3 KOHeIt
F npOBO.n.HM npSIMYIO BIG napaJIJ1eJ1bHO AB .n.o nepecelleHHR
C npo.n.OJI>KeHl1eM )l.HaroHaJIH AD B TOllKe G; ABBIAI eCTb
HCKOMbIH npSIMoyroJ1bHHK, a ero BblCOTa BBI H ecTb KopeHb
Xl .n.2HHoro ypaBHeHHSI (x1>0).
J], 0 K a 3 aTe JI b C T B O. J],OCTpOHM KBa.n.paT AAI GK
H npHMoyroJIbHHK AKLE.
DF - CTopOHa KBa)l.paTa DFGL, no nOCTpoeHHIO paBH2
DF = MP = 11 m 2 + (  r;
CJIe.D.OB2TeJIbHO, nJIoru.a)l.b 9Toro KBa)l.paTa paBH2
m 2 + (  r .
CD - CTopOHa KBa.n.paT2 ACDE, no nOCTpoeHHIO p2BHa  ;
nJ10ru.2.n.b 9Toro KBa.ApaTa p2BH2 (  r. nJ10ru.a.n.b cpHrYPbl
LGFCAE, OlleBH.n.Ho, paBH2 pa3HOCTH nJIolll,a.n.eii )l.BYX KBa.n.-
paTOB: nJIoru.a.n.b LGFCAE paBHSleTCSI nJIoru.a.n.H DFGL-
-nJ10lll,a)l.b ACDE=m 2 +( r-(  / =m 2 ; HO npSIMoyrOJIbHHK
KAEL paBeH npSIMoyroJ1bHHKY CBBIF; n03ToMY nJIOlll,2.lI.b
KBBI G paBHHeTCSI nJIOlll,a.n.H LGFCAE =m 2 .
T. e.
X 2 1 +p X I=m 2 ; (xl=BBI=KG).
C)l.eJ12HTe no.n.06Hoe nOCTpoeHHe .n.JISI qaCTHOro cJIyqaSl;
nocTpoHTe n0J10>KHTeJIbHbIH KopeHb ypaBHeHHSI
x2+6x=16.
HoaHTOBbl ypaBHeHHR
TIpOIllJIO nHTb-IlleCTb BeKOB nOCJIe EBKJ1H.n.2, BeKoa nepe-
CTpOHKH rpellecKoH MaTeMaTHKH C reoMeTpHlIecKoro MeTo.n.2
Ha MeTO.n. 2JIreOpaHlIecKHiL 3a 9TH ro.n.bI AJIeKcaHn.pH-HCKaH
IIlKOJIa nOqTH He npO)l.BHHYJIa B'nepe.n Bonpoca 06 ypaBHeHHHx.
H TOJIbKO B KOHlJ,e aJIeKC2H.ApHHCKoro nepHO)l.2, B III B. H. 9.,
78

nOHBJ1SIeTCSI MaTeMaTHK, I\moro c..n.eJ1aBllIHH B OOJ1aCHI pa3BH-
THSI aJlreOpbI. Pe% H.n.eT 0 J],HocpaHTe.
HeCOMHeHHO, liTO B pa3BHTHe YlIeHHSI 00 ypaBHeHHSlX
J],I-IOcpaHT BHec BKJ1a..n. OOJ1blliI1H, lIeM KTO-JIHOO .n.pyroH 113
YlIeHbIX .n.peBHeH fpeu.HH. Ero apl1cpMeTHK2 HMeeT OlleHb
MaJlO 06lltero C apl1cpMeTHKOH EBKJlI1..n.a H CKopee noxo.n.HT H2
2J1re6py I1H.n.I1HueB. Bo BCSIKOM CJ1Y4ae B rpellecKoH MaTeMa-
THKe Y J],110cpaHTa He ObIJ10 npe.n.llIeCTBeHHI1KOB; He ObIJ10 y
Hero H nOCJle.n.OBaTeJleH cpe..n.H HI1X, TaK KaK BCJ1e}.!. 3a ero Bpe-
MeHeM HacTynaeT 3noxa OKOHlIaTeJ1bHOrO na.n.eHI1SI aHTHlIHOH
HaYKI1. J],llOcpaHT ObIJI nOCJ1e.n.HHM H3 BeJlHKHX MaTeMaTHKOB
..n.peBHeH fpeu.11H. B KOHue IV B H. 9. fpeu.l1f1 no.n.BeprJ1aCb
onYCTOllIl1TeJIbHOMY HallIeCTBHIO pHMCKHX JIemOHepOB. fpo-
Ma..n.Hafi AJleKcaH.n.pHHCKaSI 6110J1l-lOTeKa, O..n.I1H KaTaJ10r KOTO-
pOM COCT2BJlSIJ1 120 TOMCB. ObIJ1a 6e3>KaJlCCTHO 11 6eCCMbICJ1eH-
HO YHHlITO>KeH2. fpellecKaSI HaYKa II KYJ1bTypa H2 MHOfHe
BeKa npl111IJ1H B yna.n.oK.
o 6110rpacpHH J],HocpaHTa Mbl 3HaeM OlleHb MaJlO. Lla>Ke
ro.n.bI ero >K113HH TOllHO He H3BeCTHbI. HeKOTopble CBe.n.eHI1SI 0
J],HocpaHTe COXpaHHJ1HCb B Ha..n.nHCH Ha ero Ha.n.rpOOHII, c.n.e-
Jl2HHOH rpelleCKHM YlIeHbIM MeTpo..n.opOM. 3T2 Ha.n.nHCb .n.aH2
B BH..n.e 3allallH. BOT ee TeKCT:
«J],110cpaHT npOBeJ1 llIecTYIO lIaCTb cBoeH >KH3HII B ..n.eTcTBe;
.n.BeHa.n.u.aTYIO - B IOHOCTH; nocJle ce.n.bMOH, npOBe..n.eHHOH B.
6e311eTHoM cynpY>KeCTBe, H elll,e 5 JleT Y Hero pO..n.HJlCSI CblH,
YMepllIl1H no .n.OCTH>KeHH11 nOJIOBHHbI lIHCJ1a J1eT >KH3HH OTu.2,
nOCJIe lIero J],110cpaHT npO>KHJ1 TOJ1bKO 4 ro..n.a.CKOJIbKHX J1eT
J],HocpaHT YMep?»
ECJlI1 lIHTaTeJlb nOTpy.n.HTCSI COCTaBHTb ypaBHeHHe H3 yc-
J10BHfI 9TOH 3a.n.a 1 1H H pelliHT ero, TO OH Y3HaeT, liTO 9TOT Bemi-
KHH MaTeMaTHK npO>KHJ1 .n.OJ1rylO >KH3Hb - 84 ro..n.a.
«Apl1cpMeTHKa» LlHocpaHTa COCTOSIJ1a H3 13 KHHr; .n.o Hac
.n.OIllJ1H TOJ1bKO 6 KHHL He HY>KHO I1CKaTb B paooTe Lll1ocpaHT2
TeOpeT11l1eCKOrO 113J10>KeHHSI aJlreOpbI; B HeH Mbl BCTpeTHM
TOJIbKO pH.n. 3a.n.all, CI1CTeMaTl1l1E>CKH paCnOJIO>KeHHbIX H CHao-
>KeHHbIX TeopeTHlIeCKI1MH OO'bSICHeHHSIMH. nepB2S1 KHHr2
co.n.ep>KHT rJlaBHbIM 06pa30M 3a.n.allH, pemaeMble npH nOIllOlltH
COCTaBJIeHHSI ypaBHeHHH nepBoH HJII1 BTOpoii CTeneHI1 C O.n.H11M
HeH3BeCTHbIM, OCTaJIbHbIe KHl1rM 3aKJlI0 1 lalOT B ceoe npeHMY-
llteCTBeHHO 3a.n.allH, pa3pelliaeMble npH nOMOlll,H HeOnpen.eJleH-
HblX ypaBHeHHH BTOpOH CTeneHM, r. e. CHCTeMbI COBMeCTHbIX
ypaBHeHHH, lIMCJ10 KOTOpblX MeHbllIe lIl1CJla HeH3BeCTHbIX;
TaK, HanpHMep, 32.A2l1a: no ..n.aHHoH fHnoTeHY3e c=5 npSI-
79

MoyrOJIbHOrO TpeyrOJ1bHliKa onpe.n.eJIHTb ero KaTeTbI x H Y-
npliBo.n.liT K O.n.HOMY KBa.ApaTHOMY ypaBHeHlilO X2+y2=25
C .n.BYMSI Heli3BeCTHbIMli. TaKoe ypaBHeHHe HMeeT 6eClIliCJ1eH-
HOe MHO>KeCTBO peIlleHHii, HeKOTopbIe H3 KOTOpblX MbI MO>KeM
onpe.n.eJIHTb, .n.2BaSI x 3HalleHHH H3 MHO>KeCTB2 JJ.onYCTHMbIX
3H2l1eHHii (HanpHMep, npH x=2, y= 1/ 21) . 4HCJ10 pellIeHHH
HeOnpe)l.eJIeHHOrO ypaBHeHHSI MO>KeT 6bITb orpaHHlIeHO Ka-
KHMH-HH6Y)l.b YCJ10BHSIMH. Y J],HocpaHTa HCKOMble 3HalleHHSI
HeH3BeCTHbIX .n.OJ1>KHbI 6bIJ1li BbIpa>KaTbCH paUHOH2JIbHbIMH
n0J10>KHTeJIbHbIMH lIHCJIaMH,. TaK KaK HHbIX (mceJI rpellecKaSi
MaTeMaTHK2 li He 3HaJIa. Tenepb 06billHO BBO)l.HTCSI elll,e 60-
JIee TeCHoe orpaHHlIeHHe pellIeHHii O)l.HHMH u.eJIbiMli n0J10>Kli-
TeJIbHbIMH qliCJIaMli; npH TaKOM orpaHlilleHHH ypaBHeHHe X2+
+y2=25 HMeeT TOJ1bKO O)l.HO pellIeHHe: x=3, y=4.
HOBoe B «ApHcpMeTliKe» llHocpaHT2, liTO n03BOJISIeT Cqli-
TaTb ero O.n.HliM H3 TBOpUOB rpellecKoii aJ1reOpbI, 3aKJIIOlIaeTCSI
npe>K)l.e Bcero B TOM, liTO OH OCB060>K)l.aeT peIlleHHe ypaBHeHHii
or reOMeTpliqeCKHX nOCTpoeHliii H, TaKliM 06pa30M, reOMeTpH-
tleCKliii aHaJIH3 3aMeHSIeT aJ1re6palilleCKHM. EOJ1bllIOii 3aCJIY-
rOH llHocpaHTa SIBJ1SIeTCSI TaK>Ke Hero nOnblTKa BBeCTH B ue-
JISIX ynpOlll,eHHSI 3anliceH 6YKReHHYlO CliMBOJIHKY. TaK, He-
H3BecTHoe lIHCJIO OH 0603HallaJ1 6YKBOH c: (CHrMa). 3Ta 6YKB2
o.n.Ha H3 Bcero aJ1cpaBHTa He BOlliJIa B nHCbMeHHYIO HYMepau.lilO
'rpeKOB H, CJ1e)l.OBaTeJ1bHO, He npe.n.CTaBJ1SIJIa KaKorO-Hli6Y)l.b
onpe)l.eJIeHHOrO qHCJ1a. K09cpcpHUHeHT npH HeH3BeCTHOM J],HO-
4>aHT CTaBHJ1 He Bnepe)l.li, KaK MbI, a BCJ1eJJ. 3a HeH3BeCTHbIM.
3Tli K09cpcpHu.HeHTbI TaKOBbI: ; 1, f{- 2, 1=3, o4 H T. .11..
3anHCSlM:
c:a COOTBeTcTByeT
t:J COOTBeTcTByeT
I:q COOTBeTcTByeT
X'
,
2x;
3x.
K Ba .n.p2T HeH3Becoro OH HaaJ1 O'Ia(.Lll: ()l.IOHaMHc) H
KopOlle 0603Hall2JI ou; Ky6- xuot: (KI060c) li COKpallteHHO
-- --
xu. 32nHCb xu13 0603HallaeT 2X2. CB060.AHbIii lIJIeH, KpOMe
tlliCJ10BOrO K09cpcpHu.HeHTa, CHa6>KeH 3HallKOM (.Lo - COKpa-
llI,eHHOe (.LO'laOEI: (eJJ.HHliu.a); TaK
",,8 1 =3, ""t = 1.
3H2K CJ10>KeHliSi J],HocpaHT He ynOTpe6J1SIJ1 H CJ1araeMbIe
npOCTO nHCaJI pSl.AOM. 3HaK )l.eiiCTBHH ynOTpe6JIHJICH TOJIbKO
80

.n.JIH BbItlHT2HHH Ij-, (OnpOKHHYT2H 6YKB2 Iji). BMeCTO SHaKa
paBeHCTBa .n.Be lI2CTH ypaBHeHHH COe.n.l1HSlIOTCSI CJIOBOM «p2B-
HO» (l8oc;); Hepe.n.KO BCe CJIOBO COKpaIl1,2eTCSI .AO O.n.HOH HatlaJIb-
HOM 6YKBbI l.
TaK, ypaBHeHHe
x 2 =2x-l
y .nHOcp2HT2 S2nl1IlleTCSI CJIe.n.YIOIl1,HM 06pa30M; o;c;c;rrtPp. a a;
yp2BHeHHe
(x 3 +8x)-(5x 2 +1)_=x,
x-;;c;c;,p 8Vsp.ti c;.
PeIll2SI YP2BHeHHH, .nHOcp2HT npH3H2eT TOJIbKO p2u.MOH2JIb-
HbIH H nOJIO>KHTeJIbHbIH KopeHb, TaKHM 06p230M: H3 .n.BYX KOp-
HeH ypaBHeHHH
x 2 -x-6=0
OH npHHHM2eT TOJIbKO KopeHb. paBHbIH 3. KopeHb, paBHbIH -2,
.n.JIH J],HocpaHTa He CYIl1,eCTByeT. Yp2BHeHHe >Ke
x2+7x+12=0
COBceM He HMeeT peIlleHI1S1, Tal< K2K 062 KOpHH ero - OTPH-
u.aTeJIbHbI.
PeIll2S1 3a.n.2l1H nYTeM COCTaBJIeHHSI YP2BHeHHH, .nHocpaHT
oc06oe BHHM2HHe y.n.eJISIJI BbIOOPy HeM3BecTHoro. YJI.2l1HbIH
BbI60p HeH3BeCTHoro lIaCTO .n.eJI2eT pellIeHHe OOJIee npocTbIM.
B YMeHMM BbIopaTb y.n.atIHO rJI2BHOe HeH3BeCTHoe, npoBe-
CTH 2H2JIH3 3aJI.allH, npeJI.llIecTBYlOllI,HH pellIeHHIO ypaBHeHHSI,
H CK23bIBaeTCSI HPKHH M OpMfHHaJIbHbIH T2JIaHT llMocpaHTa,
npMBe.n.eM .n.JISI npHMep2 HeCKOJIbKO 3a.Aall .nHocpaHTa H
ero pellIeHHSI.
3a)l.2Qa 1. HaHTH .n.Ba lIHCJIa, CYMMa KOTOpbIX 20, 2 npo-
H3BeJI.eHHe 96.
nYCTb 2x- p23HOCTb OOOHX lIMCeJI. TOrJI.2 60JIbmee 6y.n.eT
lO+x, 2 MeHbllIee IO-x. npOH3BeJI.eHHe HX
100-x 2 =96; x 2 =4; x=2.
I1CI<OMble lIHCJI2: 60JIbllIee J 2, MeHbllIee 8.
3anaQ2 2. CYMMa JI.BYX lIHCeJI 10, a CYMM2 HX KBMpaTOB
68. HaHTH 9TH lIHCJIa.
COCTaBI1M CHCTeMY ypaBHeHHM:
81

{ X+Y=lO;
X2+y2=68.
.nHOcpaHT npe.n.JI2raeT pemHTb ee CJIe.n.YlOllI.HM OOp230M: H3
nepBoro ypaBHeHHSI HMeeM
x+ y _ 5
2 - .
nYCTb
x-y
=n.
CJIO>KeHHe .n.BYX nOCJIe.n.HHX p2BeHCTB .n.2eT
x=5+n,
BbIlIHTaHHe
Y = 5 - n.
nOJJ.CTaBJ1SISI 3HalleHHSI x H Y BO BTopoe 113 ypaBHeHHH CHCTe-
MbI, nOJIYlIHM:
(5 + n)2 + (5 - n)2 = 68,
OTKy.n. a
n=3; x=8; y=2.
t1TOObI HMeTb HeKOTopoe npe.n.CTaBJ1eHHe 0 HeOnpe)l.eJIeHHbIX,
HJIH .n.Hocpa, HTOBbIX YP2BHeHliSlX, KOTopbIe ocooeHHO lIaCTO
BCTpellalOTCSJ B «ApHcpMeTHKe» llHocpaHTa Ii B pellIeHliH KO-
TOpbIX OH npOSJBliJI TaK MHoro OCTPOYMHSJ H TaJIaHTa, pa30e-
peM HeCKOJIbKO npOCTblX 33)1.all.
3a)l.aQa 1. HaHTH JJ.BY3HallHoe lIHCJIO, p2BHoe y.n.BoeHHOMY
npOH3Be)l.eHHIO ero u.Hcpp.
P e 1lI e H Ii e. nYCTb HCKOMoe lIHCJIO oy.n.eT lOx + y, Tor.n.2
no YCJIOBHIO 3a.n.allH HMeeM:
lOx + y = 2xy; (1)
.n.eJISI ooe tl2CTH 9Toro yp2BHeHHSI Ha 2x, nOJIYtlHM:
5 + !fx = y;
(2)
TaK KaK y .n.OJI>KHO 6bITb u.eJIbIM lIHCJIOM. MeHbllIHM 1 'J, TO H 2
.n.OJI>KHO ObITb u.eJJbIM, T. e. y )l.OJI>KHO .n.eJIliTbCSI Ha 2x. no-
JIO>KHM, liTO Y = 2xn; TorJJ.a ypaBHeHHe (2) nepenHweTcfI CJIe-
,lI,YlOllI.liM 06pa30M:
5+n=2xn
(3)
82

HJIH, .n.eJISi H2 n, HMeeM:
5
- + 1 = 2x;
n
(4)
5
ypaBHeHHe (4) nOKa3blBaeT. tlTO - .n.OJI>KHO 6bITb u.eJIbIM lIHC-
n ,
JIOM, CJIe.n.OBaTeJIbHO, n=l HJIH n=5. ECJIH n=l, TO x=3 H
y=6, eCJIH n=5, TO x= 1 H y= 10. nOCJ1e)J.Hee peIlleHHe He
y.n.OBJIeTBopSieT 3a.n.2l1e, TaK KaK Y < 10.
I1TaK, HMeeM OJJ.HO lIMCJIO 36.
31]I.aQa 2. BbI >KeJIaeTe CTraJJ.aTb .n.aTY pO>K.reHHSI CBrero
TOBapHllI,a. BbI npe)l.JIaraeTe eMY YMHO>KI1Tb .n.eHb ero pO>K.n.e-
HHH H8 12. a MecSiu. - Ha 31.
OH )l.OJI>KeH C006ll1,HTb BaM CYMMY 060HX npOH3Be)l.eHHH,
H BbI BbIlIMCJISIeTe no HeH )l.aTy ero pO>K.n.eHHH.
nYCTb Balli TOB::!pmu. pO)l.HJICSI 3 cpeBpaJIH. Tor.n.a OH .n.OJI-
>KeH c.n.eJIaTb CJ1e.n.YlOllI,Me BbIKJI2.n.KH:
3. 12 + 2.31 = 36 + 62 = 98
H C006ll1.HTb BaM pe3YJIbT2T 98.
Bbl COCT8BJISIeTe Heonpe.n.eJIeHHOe ypaBHeHHe:
12x + 31y = 98,
r)J.e x - tlMCJIO pO>K)l.eHMH, y - MecSiu. pO>K)J.eHHSI. OCTaeTCSI
TOJIbKO pellIHTb 9TO ypaBHeHHe B u.eJIbIX H nOJIO>KHTeJJbHbIX
tlHCJ1ax. BOT 9TO pellIeHMe:
6x + 31 .  = 49.
TaK KaK x - IJ,eJIoe tlHCJIO, TO  - IJ,eJJoe lIHCJIO. 0603H2t1HM
ero lIepe3 t; Tor.n.a
6x + 31 t = 49, Y = 2f;
_ 49- 31 t _ 8 _ 5t+ 1 - t
x- 6 - 6 .
I-t n I-t
6 )J.OJI>KHO 6blTb tlHCJ10M IJ,eJIblM. YCTb {i =f 1 ; OTCIO)l.a
t = 1 - 6f 1 ;
maK,
x = 3 + 31 t 1 H Y = 2 - I2f 1 .
no CMblCJ1Y 3a.n.2l111 o <x-<:31 H 0<y12. Ha OCHOBe 9Toro
COCTaWUJeM CHcreMbl HepaBeHcTB:
R:!

{ 3 + 3It 1 > 0;
3 + 31 II -< 31.
{ 2-12t l > 0;
2-12/ 1 -< 12.
PelliaSi cHcTeMY (I), nOJIYlIHM:
3 28
t 1 > - 3I H II -< 31 .
(I)
(II)
PellIeHHe CHCTeMbI (II) )l.2CT:
1 5
II < ""6 H II > - "'6 .
COnOCTaBJIHH nOJ1YtleHHbIe .AJIH II npe.n.eJIbI, liMeeM:
3 28
- 3f < II < 31'
TaK Kal{ t 1 - ue.noe lIliCJlO, TO II =0. npli t 1 =0 x=3 H y=2.
J],2T2 pO>K.n.eHHH B211Iero TOBapmu.a HaH)l.eHa: 3 cpeBp2JISI.
npoJJ.eJ12HTe TaKYIO >Ke 3a.n.allY. npe)l.nOJIO>KHB, liTO .n.aTa
pO>KJJ.eHliSI Balliero TOBaplllll,a 1 )l.eKaOpf!; B03bMHTe .D.JISI 9T0i1
32.n.allH )l.aTY B2mero coocTBeHHoro pO>KJJ.eHlif!.
3a)J.2Qa 3. HaHTH ueJIoe .n.BY3HallHoe lIHCJIO, KOTopoe B
JJ.eBSITb p23 oOJIee u.liCPPbl ero e)l.HHHu..
OmBem. 45.
3a)J.aQa 4. BbI )l.OJI>KHbI ynJI2TH1'b 3a nOKynKY B M2r23HHe
19 pyo. Y B2C eCTb TOJIbKO TpeXPYOJIeBbIe KynlOpbl, Y K2C-
clipa - TOJIbKO nSlTHpYOJIeBble. MO>KHO JIH npli HaJIlillli1i 1'a-
KHX JJ.eHer paCnJIaTliTbCH 3a nOKynKY H KaK HMeHHo?
Omeem. qHCJIO TpeXPYOJIeBbIX KynlOp: 8, ':13, 18, 23 ...;
lIHCJIO nSITHPYOJIeBbIX KynlOp: 1. 4, 7, 10
BKJIaA HHJI.HMCKHX MaTeMa1'HKOB B pa3BHTHe
yqeHHH 06 ypaBHeHHHx
TBpu.aMH aJIreOpbI nl cnpaBe.n.JIHBCTH Ha.n.o ClIHT2Tb HH-
.n.H!u.eB. Ho, CCT2H£1B.nliB3SICb Ha .n.peBHHx 3noxax HH)l.liikKOH
M2Te\1aTIiKH, MbI B3b\:e, Bpe.VlSI, Henvcpe.n.c1'BeHHo cJIeJJ.YlOllI,ee
S8 BpeMeHeM LlHocpaHT2.
B cepen.liHe VI B. H. 9. HH)l.HHu.bI Y>Ke pe1lI2JIH 3an.atlH
nYTeM COCTaBJIeHHH ypaBHeHHH, MaJIO OTJIlillalOllI.HMCSI OT
84

COBpeMeHHoro, K2K 9TO ceyeT H3 COQHHeHHSI 2CTpOHOM8 H
MaTeMaTHK2 ApH20XaTa «H2YKa 0 ClIHCJIeHHH». Y HerO >Ke
MbI BCTpellaeM 32)1.2l1H H2 COCTaBeHHe H pellIeHHe B u.eJIbIX
QHCJIaX HeOnpeJJ.eeHHbIX ypaBHeHHH BH.n.a:
ax + by = c.
I1H)l.HifcKHH MaTeMaTHK EpaM2rym2 (598-660) nHllIeT B
628 r. TpaKT2T, COCTOSllllHH H3 20 I<HHr, H3 KOTOpbIX 2re6pe
nOCBHllI,eH2 XVI I I KHHra. B HeH MbI BCTpellaeM pellIeHHe YP2B-
HeHHH nepBoH H BTOpOH CTeneHH, CHCTeM yp2BHeHHH H He-
onpeJJ.eeHHbIX yp2BHeHHH B u.ebIX lIHcax.
BOT O)l.Ha H3 32JJ.all, npeJJ.J10>KeHHbIX EpaM2rymOH:
«J],B2 aCKeTa >KHBYT Ha BepllIHHe OTBecHOH CK2.'IbI BbICOTOH h
H B paccToSlHHH OT ropo)l.a B m p23 oObllIeM, lIeM 9Ta BbICOTa.
4TOObI .n.OCTH'lb ropoJJ.a, O)l.HH JI3 2CKeTOB cnycKaeTcSI BHH3,
nOTOM H)l.eT no )1.0 pore. LlpyroH nO)l.HHMaeTcSI B B03YX H2
HeKoTopylO BbICOTY H oTTY)l.a YCTpeMSIeTcSI K rcpo)l.Y. Cnp2-
mHBaeTcSI, Ha KaKYIO BbICOTY no.n.HSICSI BTOpOH 2cKeT, ecJIH
002 COBepIllH.'IH OJJ.HH H TOT >Ke nYTb?»
mh
EpaM2ryma pelliaeT 9TY 3aJJ.allY H nOJIYlIaeT OTBeT m+2 '
nonp06YHTe H BbI pellIHTb 9TY 3a.n.allY.
B X II B. 3H2MeHHTblH HH)l.HifcKHH aCTpOHOM H M2TeMaTHK
5xaCKapa-aKapHSI (2KapHSI - MY)l.peu., YlIeHbIH) B CBOeM
TpaKTaTe «CHJJ..lI.aHTa CHpOM2HH» (<<BeHeu CHCTeMbI») H3ar2eT
2reopy B OOJ1ee SICHOH cpopMe, lIeM 9TO )l.eaJ10Cb )1.0 Hero.
ECJIH J],HocpaHT paccM2TpHBa OT.n.ebHO TPH CJ1YlIaSI KBa.n.P2T-
Horo ypaBHeHHSI:
ax 2 + bx = c; ax 2 = bx + c H ax 2 + c = bx,
TO ExaCKapa BJ1a)l.eeT Y>Ke pellIeHHeM KB2)1.paTHOro ypaBHeHHSI
B 06ll1,eM BH.Ae. nOJJ.06HO LlHocpaHTY, EX2CKapa He npHHHMaeT
oTpHu.aTeJIbHbIe 3HalleHHSI HeH3BecTHoro, rOBOpSl, liTO «JIIO)l.H
He 0JJ.06pSlIOT OTBJIelleHHbIX oTpHu.aTeJJbHbIX lIHCeJI». KOHellHO,
He HY>KHO .n.YMaTb, liTO 3anHCH pellIeHHSI ypaBHeHHH npOH3BO-
)l.HHCb B TOH >Ke cpopMe, KaK .n.eJIaeM MbI. 0603HalleHHe HeH3-
BecTHbIX H K09cpcpHUHeHTOB H 3anHCH paBeHCTB2 ObInH HHbIMH,
HO npouecc pellIeHHSI ypaBHeHHSI ObIJ1 TaKHM >Ke, KaK JI Y Hac.
YCOBHSI 3a)l.allH 5xaCKapa 06JIeKaJ1 B KP2CHBYIO, HHor.n.2
.n.a>Ke n091 HlIeCKYIO cpOPMY.
J],SI 00pa3ua npHBe)l.eM TeKCT o)l.Hoii H3 3a)l.2l1:
«B C2MbIH, pa3rap 60H HeHCTOBbIH npHTraBa CXB2THJI nytlOK
CTPeJI, QT06bI nopa3HTb KapMY. nOJIOBHHa nOIllJ1a Ha caMO-
85

32HTY, YlIeTBepeHHbIH KopeHb KB2.Ap2THbIH Cpa3HJI JIOIlla,l{eH.
lllecTb CTpeJ1 npOH3HJIH CXaJIHa, B03HIHJ.Y, TpeMSI npOpB2JIO
3HaMSI H paCllI,emiJIO JIYK Ii TOJIbKO OJI.HOH nOCJIeJI.HeH CTpeJIOH
Ka pM2 6blJI nOpa>KeH B rOJIOBY. CKOJIbKO CTpeJI ObIJIO B nylIKe
Y npliTraBbI?»
He Y3H2eTe JIH H BbI, lIHTaTeJIb, CKOJIbKO CTpeJ1 HMeJI HeH-
CTOBbIH npHTf2Ba?
ExacK2pa pe1lI2eT If CHCTeMbI ypaBHeHHH nepBoii CTeneHH
C HeCKo.'IbKHMH HeH3BeCTHblMH, OObIlIHO nOJIb3YSICb cnOCOOOM
nOJJ.CT2HOBKH, U Heonpen.eJ1eHHbIe YP2BHeHHSI B l(eJIbIX lIHC.'I2X
60JIee CJIO>KHOrO BH)].2, lIeM Y Ep2MarYIlTbI, HanpHMep BH.n.2:
ax 2 + 1 = y2.
Bce 3TO CBH.n.eTeJIbCTByeT 0 3HallHTeJ1bHOM ypoBHe 3H2-
HHH aJIreOpbI Y HH.n.HHu.eB B XII-XIII BB.
I1HTepecHo BbICK23bIB2HHe ExacKapbl 0 MaTeMaTHKe: «51
rJIy6oKo nOllliTalO M<lTeMaTliKY, nOToMY (!TO 3HaKOMbIe C HelO
BIi.n.SIT B HeH cpe.n.cTBo K nOHHMaHHIO Bcero cYllI,eCTBYIOllI,ero».
npOCMoTpHTe Ii pellIHTe HeCKOJIbKO 3a.n.a4 H3 TpakTaToB
ApHaOX8Tb!, 6paMarynTbi Ii EX2CK2pbI. HeKoTopble H3 3THX
3aJI.2l1 HanOMHSIT BaM 3a.n.2l1H H3 Balliero llIKOJIbHOro 3a.n.2l1-
HHKa.
3aJI.a'Ja 1. «11.Ba CBeTIiJIa, Haxo.n.SIlllHecSI H2 .n.aHHOM p2C-
CTOSIHHH .n.pyr OT .n.Pyra, .ilBH>KYTCSI O.n.HO K .n.pyroMY HaBcTpe-
lIY C .n.aHHbIMH CKOpOCTSIMIi v VI. Onpe.n.em!Tb TOllKY HX
BCTpellH». (A pH a 0 x a T a.)
3a.D.aQa 2 ('lepT. 30). «HaHTH BbICOTY CBe'-lIi, 3HaSI .nJIHHY
TeHeii, 6pocaeMbix fHOMOHOM B 1J.BYX pa3JIH'lHbIX nOJlO>KeHH-
HX, npH YCJIOBIiIi, liTO .n.aHO paCCTOSIHHe Me>KJI.Y nOJIO>KeHHSI-
MH fHOMOHOB». (5p2MarynTa).
OmBem. h (1 + a  b ) .
d
4epT. 30.
b6

3a.n:aQa 3. «HeKTo CK232JI cBoeMY .n.pyry: «J],2H MHe 100
pynHH, H SI 6yJJ.Y BJI.BOe 00r2l1e TeoSl» , Ha liTO nOCJIeJJ.HHH OT-
BeTHJI: «ECJIJI TbI MHe .n.311Ib TOJIbKO 10 pynHH, SI CTaHY BllIe-
CTepO 60raqe TeOS!». Cnp311IHB2eTCSI: CKOJIbI{0 6bIJlO Y Ka>K-
.n.oro?» (E X a C K a p 2.)
PellIeHHe caMoro ExaCKapbI.
nYCTb y nepBoro 6YJJ.eT (2x-IOO) pynHH, a y BToporo
(x+ 100) pynHH. HCHO, liTO nepBoe YCJIOBHe 6y.n.eT BbIIlOJI-
HeHO. I1MeH B BH.AY BTopoe YCJIOBHe, nOJIYQHM:
6(2x - 110) = x + 110.
Omeem: 40 H 1"70.
A K2K ObI BbI, lIHTaTeJIb, pellIHJIH 9TY 32JI.allY?
3a.n.2Qa 4. «H2HTH lIHCJIO, oOJIaJJ.alOllI,ee TeM CBOHCTBOM,
I.JTO OHO, 6YJI.YlIH YMHO>KeHO Ha 12, no npHOaBJIeHHH K cBoeMY
KYOY paBHSIeTcSI YlliecTepeHHoMY KBaJI.paTY caMoro ceoSI, yBe-
JIHl.JeHHOMY TpHLUl3TbIO nSITblO». (E x 2 C K 2 P a.)
PellIeHHe 3a.A2l1H npHBOJJ.HTCSI K yp2BHeHHIO TpeTbeH CTe-
neHM:
x 3 + 12x-6x2=35;
BxacK2pa pelliaeT ero CJIeJJ.YlOlltHM 06pa30M:
x:i+12x-6x2=35;
x 3 -6x 2 + 12x-8=27;
(x-2):i=27;
x-2=3;
x=5.
OT XIII B. - 9nOXH ExacKapbI - BepHeMCSI HeCKOJIbKO
Ha3aJI., R VII B., KorJJ.2 Ha MHPOBYIO Cu.eHY BbIcTynalOT 2p20bI,
nOKopHBllIHe K 9TOMY BpeMeHH BCIO nepeJI.HIOIO A3HIO, Ce-
BepHYIO AcppHKY H I1cn2HHIO. I1CKJIlOlIHTeJIbHO ObICTpO OHH
YCBOHJIH KaK 3anaJI.HYIO, T2R H BOCTOllHYIO Hay KY. HaYlIHbIe
3HaHHSI rpeKOB H HHJJ.HHueB CJIHJIHCb BMeCTe. PaOOTW Tex H
.n.pyrHx nepeBOJI.SlTCSI Ha apaOCKHH SI3WK. Bce 9TO JI.aJIO TOJI-
lIOK JJ.aJ1bHeHllIeMY pa3BHTHIO H2YK, B TOM I.JHCJIe H aJIreOpbI.
B 820 r. B Ear.Aa.n.e - CTOJIHu.e BOCTOllHoro XaJ1HcpaTa-
nOSIBJISIeTCSI nepBbIH caMOCTOSITeJIbHbIH Tpy.A no MaTeJl.JaTHKe
KaK pe3YJIbTaT H3YQeHHSI 2pHcpMeTHKH H aJIre6pbl HH.AHiiu.eB,
C O.n.HOH CTOpOHbl. H reoMeTpHH rpeKOB - C .n.pyroi'I. B 9TOM
roJI.Y O.n.HH H3 YlJeHbIX, KOTOpbIX OblJIO HeMaJIO npH JJ.Bope X2-
JIHcpa 2JIb-MaMYHa,- MyxaMMeJJ. oeH-Myca H3 XOPe3M2, Y36eK
no np0I1CXO>KJJ.eHHIO, 32K2HlIHBaeT CBOe COllHHelme 0 BbIlIHC-
JIeHMSIX C nOMOIl1,blO aJIb-.n.>KeOp BaJIb-MYK262JIa, KOTopoe OH H
87

Ha3BaJI «AJ1b-)l.>Ke6p BaJIb-MYKaOaJI2». H23B2HHe CBSl32HO C
.n.BYMSI npUe.\1aMH pellIeHHSI ypaBHeHHH: nepeHOCOM tlJIeHOB
ypaBHeHHf! H3 OJJ.HOH lIaCTU B .n.pyrylO H nOCJIe)l.YlOllI,HM npHBeJJ.e-
HUeM nO)l.OOHbIX lIJ1eHOB. 3TOT TpaKTaT, nepeBeJJ.eHHbIH B XII B.
Ha J1aTHHCKHH SI3bIK. CTa.iI TeM COllHHeHueM, OTTO.'IKHYBllIHCb OT
KOToporo nOllI.'IO JJ.2JIbHeHllIee pa3BUTHe aJIreOpbI, HO Y>Ke H2
eBponeHCKOH nOllBe. nepeBo.n.lIlIKH COllUHeHHSI oeH-MYCbI
nOCTeneHHO OTOPOCUJIU BTOPYIO lI2CTb H23B2HUSI' ero Tp2KT2Ta,
OCT2BHB TOJIbKO nepBYIO - 8JIb-JJ.>Ke6p, HJ1U, K2K MbI Tenepb
roBOpUM, aJIreop2. CaMoro 2BTOpa nepeBO)l.lIUKH nepeHMe-
HOBa.'IH B aJIb-XOpe3MH. MbI >Ke OY)l.eM nOMHHTb, liTO H2111
cooTellecTBeHHHK- Y30eK MyxaMMe)l. CbIH MycbI U3 Xope3M2,
C.n.eJI2J1 OOJIbllIOH BKJI2)1. B C03)1.aHUe aJIreOpbI.
TpaKT2T oeH-MYCbI COCTOHT U3 .n.BYX lI2CTeH. B nepBOH
aBTOp )l.aeT npaBUJI2 CJIO>KeHHSI, BbIlIHTaHHSI H YMHO>KeHHH
aJIre6paHlIecKHX Bblpa)!{eHUH, co.n.ep>K2ll1,UX HeH3BecTHoe, ero
KBa.n.p2T HJIH KBa.n.paTHbIH KopeHb. Bo BTOpOH - pellIeHHe
ypaBHeHHH, KOTOpbIX HaClIHTbIB2eTCH llIeCTb BH.n.OB:
x 2 = bx; x 2 =c; bx=c;
x 2 +bx=c; x 2 +c=bx; bx+c=x 2 .
Bce 9TU llIeCTb BHJJ.OB YP2BHeHHH MO>KHO 6bIJIO 6b1 06'be.llHHHTb
B OJJ.HH BH.A:
ax 2 +bx+c=0,
eCJIH ObI apaoaM OblJ10 H3BeCTHO OYKBeHHoe 0003H2l1eHHe K091!>-
cpHu.HeHTOB YP2BHeHHH: no)l. KOTOpbIMH MO>KHO 6bIJIO 6bI no.
HHMaTb lIHc.'Ia H nOJIO>KHTeJIbHbIe, H oTpHu.aTeJIbHbIe, H paBHbIe
HYJIIO.
npaBHJIa pellIeHHSI K2>K.n.oro H3 llleCTH BH)l.OB ypaBHeHHH
aJ1b-XOpe3MH JJ.aeT Ha lI2CTHbIX npHMep2X, onHCbIB2S1 era CJIO-
BaMH oe3 BCSIKHX cOKpallleHHH.
BOT np2BHJIO .n.JISI pemeHHSI ypaBHeHHSI BH.n.2
x 2 +c= bx.
«KB2)1.paT H lIHCJIO 21 paBHbI 10 KOpHSIM Taro >Ke KBa.u.-
paTa; cnpallIHBaeTCSI, H3 lIero 06pa3yeTcSI KB2JJ.paT, KO-
TOpbIH nOCJIe npU6aBJ1eHUSI K HeMY 21 JJ.eJIaeTCH p2BHbIM
10 KOpHSIM TOro >Ke KBaJJ.paT2?»
Bonpoc, OlleBHJJ.HO, CBOJJ.HTCSI J{ pemeHHIO ypaBHeHHH:
x2+21 = lOx.
Pew e H H e. «Pa3JJ.eJIH nOnOJI2M lIHCJIO KopHeH (noJJY-
88

tlMIllb 5). YMHO>Kb 9TO lIHCJIO caMO Ha ceoSi. OT npOH3Be.n.eHHH
OTHMMH 21, OCTaTOK oYJJ.eT 4. 113BJIeKH KopeHb; OH eCTb 2.
OTHMMH ero OT nOJIOBMHbI lIMCJIa KopHell (T. e. OT 5); OCT2-
HeTCSI 3. Eho H oYJI.eT K0J:eHb HCKOMoro KBaJI.paTa. I1JIH >Ke
TbI MO>Kellib npMJIO>KMTb 9TOT KopeHb K nOJIOBMHe lIHCJ1a KOp-
Hell, liTO .n.aeT 7. 3TO M eCTb KopeHb MCKOMoro KBa.ll.paTa, a
C2M KBaJI.p2T oYJI.eT 49» (H3 PYKonMcM 2JIb-XOpe3MH).
B pe3YJIbTaTe MHorOBeKOBbIX TPYJI.OB BbIJI.alOllI,HXCSI M2Te-
MaTliKOB MbI MO>KeM pellIHTb 3TO ypaBHeHHe ropa3JJ.0 cKopee:
x=5:!= 11 25-21- 5:1:2; xl=3; x2=7.
ECJIH KopeHb ypaBHeHHSI nOJIYlIaJICSI OTpHu.aTeJIbHbIM, TO
oeH-Myca OTopaCbIBaJ1 ero. TeM He MeHee nOHSITHe 0 lIHC.'Ie
y Hero OOJIee llIHpOKoe, lIeM Y rpeKOB, T2K K2K OHO BK.'IlOlIaeT
H lIHC.'IO HPpau.MOHaJIbHoe.
AJIre6ra npHXOJI.HT B EBpony
C X II B. B EBpone H2YllHbIe TPYJI.bI rpeKOB, HH.n.Mtiu.eB H
apa60B C 2pa(CKoro Sl3bma nepeBOJJ.SITCSI H2 JIaTMHcKMti SI3bIK.
TaKMM oopa30M, eBponeHCKMe MaTeMaTHKH nOJ1YLlMJ1H B03-
MO>KHOCTb M3YlIaTb BeJ1HKMe TBopeHllSI EBKJIHJJ.a, AnOJIJIOHMSI,
ApXHMeJI.2, nTOJIeMeSI, 2JIb-XOpe3MH H JJ.pyrHX. Ho nOTpeoo-
BaJIOCb OK0J10 Tpex CTOJIeTHti, lITOObi OT H3YLleHHSI 9THX TPYJJ.OB
nepetiTH K caMOCTOSITeJ1bHbIM HCCJIe,AOB2HHSlM.
«KHHra 00 aoaKe» HTaJIbSlHCKOro MaTeMaTHK2 J1eoHap.n.o
<1JHooHallllH, HanHcaH'HaSi B 1202 r., JI.aeT nepBoe OpHrHHaJIb-
Hoe H3JIO>Kt'HHe apHcpMeTHKH M aJIreOpbI. B TelleHHe 200 JIeT
9Ta KHHra CJIY>KHJIa OCHOBOH H3YlIeHMSI MaTeMaTHKH M nOJJ.ro-
TOBMJIa ycnexM MT2J1bSIHCKOti aJ1re6pbI B 9noxy B03pO>KJI.eHHSI.
OpHrHHaJIbHOCTb 9TOti paooTbI CK23bIBaeTCSI rJI2BHbIM oopa-
30M B MaHepe H3JIO>KeHHSI, BCerJI.a SICHOti M TOLlHOti. 3.n.eCb OH
BO MHOrHX OTHOllIeHHSlX BbIIlle CBOMX apaOCI<MX YllMTeJIeti.
B caMOM >Ke cOJJ.ep>KaHMH <DMooHallllH MeHee caMOCTOSlTeJIeH,
,lta H OH caM nMllIeT, liTO COCTaBH.'I «KHHry 00 a6aKe», «npH-
COeJI.HHHB K .MHJJ.HtiCKOMY MeTOJJ.Y Koe-lITO OT ceoSI, Koe-lITO
OT TOHKocTeti reOMeTpHlIecKoro MCKYCCTBa EBKJIHJI.a». TaK,
B OJJ.Hoti KHHre CJIHBalOTCSI BMeCTe BJIMHHHSI H BOCTOKa H ,ltpeB-
Heti rpeu.HM.
B 1494 r. nOSlBHJIOCb nepBoe nellaTHoe PYKOBOJJ.CTBO no
MaTeMaTHKe MTaJ1b5lHu.a J1YKH nallHOJIH. B p23.n.eJIe «AJIreopa»
MbI BCTpell2eM pellIeHHe ypaBHeHMH nepBoti M BTOpOH CTeneHH.
KaaAP2THbIe ypaBHeHHSI paccMaTpHBalOTcSI TpeX BMJI.OB:
89

xl!+ bx=c;
bx+c=x 2 ;
X 2 +C= bx.
np2BHJIa .n.JISI pellIeHHSI 9THX YP2BHeHHH n2l1HOJIH H3JI2-
r2eT B CTHXaX. naqHOJIH >Ke BBO.n.HT 3H2KH CJ10>KeHHSI, BbillH-
TaHHSI H paBeHCTBa B BH.n.e 3arJIaBHbIX 6YKB P, M H Ae JIa-
THHCKHX CJIOB «plus», «minus» H «aequ2Iis». nepBYIO CTeneHb
HeH3BecTHoro OH 0003H2QaeT OYKBOH R (res - Belll.b), KBa.n.-
p2T HeH3BeCTHoro - 3HaKOM B BH.n.e OYKBbI O. BHYTPH KOTO-
pOH CT2BHJ12Cb u.HcpP2 2. TaKHM o6pa30M, ypaBHeHHe
x2+2x=8
B 3anHCH nallHOJIH BbIrJ1SI.n.eJIO TaK:
@P2RAe8.
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0003HalleHHH, KOTopble T2K ynpOlll,alOT H YKOp2l1HB21OT pe-
rneHHe ypaBHeHHH H Bce 3anHCH .n.eHCTBHH Ha.n. aJIreOpaHQe-
CKHMH BbIpa>KeHHSlMH, 06bIlIHO CtlHTalOT cppaHu.Y3CKoro Ma-
TeMaTHKa <DpaHcya BHeTa (1540-1605). BBe.n.eHHeM 0YKBeHHbIx
o503HalJeHHH BHeT YKa3aJ1 nyTb )l.JISI )l.aJIbHeHllIero pa3-
BiiTHSI 2JIreOpbI. 3TH 0003HalleHHSI n03BOJISIJIH JIerlle nO)l.Me-
tl2Tb OOIllHe 3al\OHbI, YCK0J1b3alOllI,He H3 nOJISI 3peHHSI npH
32MeHe OYKB lIHCJIaMH.
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Y BHeTa B TOllHOCTH COBna.n.alOT C TeMH, KOTOpbIMH nOJIb-
3yeMcSI MbI. B npHMepax C lJHCJIeHHbIMH K09cpcpHu.HeHTaMH
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BOH Q. T2KHM 00pa30M, ypaBHeHHe
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OH He TOJIbKO OBJI2.n.eJ1 OCT2J1bHbIMH 6YKB2MH 2J1cp2BHT2, HO
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KOTOPYIO OH OlleHb J1106HJI.
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2 nOTOM H IOHOlliH .n.2J12 CBOH pe3YJ1bTaTbI - .n.B2.n.uaTH TpeX
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BepOHbI, nbSllleHUbI H, H2KoHeu., BeHeu.HH. Ero npHrJ1a1lI2lOT
lIHTaTb J1eKu.HH no reOMeTpHH, MeX2HHKe, 2J1re6pe. Ho 9TO He
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OH n0J1Yll2eT K2l!>e.n.py M2TeM2THKH B BepoHe. B 9TOM >Ke ro.n.y
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I1 BOT T2, pT2J1bSl YBepeHHbIH B nooe.n.e, BbI3bIB2eT C1>HOpH H2
nYOJ1HlIHbIH M2TeM2THlIeCKHH noeJJ,HHOK. noe.n.HHOK Ha3H2l121OT
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93

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u.y 9TOro cpoKa pellIHT HaH60JIbIllee lIHCJ10 3a.n.all H3 30, npe.n.-
JIO>KeHHbIX conepHHKOM, TOT H oy.n.eT ClIHTaTbCSI nOOe.AH-
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3aJJ.allY.
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CJIYXH, liTO C1>HOpH .n.eHCTBHTeJ1bHO 3HaeT cnocoo pellIeHHSI
ypaBHeHIIH BH.n.a x 3 +px=q.
TapTaJ1bSI lIYBCTByeT, liTO ecJIH 9TO TaK, TO C1>HOpH OOSl3a-
TeJ1bHO npe.n.J10>KHT eMY HMeHHO TaKHe ypaBHeHHSI H, CJ1e.n.OBa-
TeJ1bHO, OCTaHeTCSI n06eJJ.HTeJ1eM, TaK KaK caM TapTaJ1bSl 3HaJI
pellIeHHe no.n.OOHbIX ypaBHeHHH .n.J1S1 HeCKOJ1bKHX lIaCTHbIX
CJIYllaeB. Tor.n.a, KaK nHllIeT OH B O.n.HOM H3 CBOHX cOllHHeHHH,
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CJJ.eJ1aTb 3a 10 .n.HeH .n.o cpOKa, T. e. 12 cpeBpaJISI, 6J1aro.n.apSi
ClIaCTJ1HBOH cYJJ.boe». MbI ObI CKa3aJ1H, 6J1aro.n.apSi ero HCKJ1IO-
lIHTeJIbHOMY TaJ1aHTY. .
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HbIH CpOK C1>rlOpH nepe.n.aJ1 CBoeMY ConepmtKy 30 3a.n.all, KO-
TopbIe Bce npHBo.n.HJ1HCb K ypaBHeHHlIM BH.n.a x 3 +px=q.
I<aKoBo >Ke ObIJ10 y.n.HBJ1eHHe Bcex, Kor.n.a TapTaJ1bSI Bce
HX pellIHJI 3a 2 qaca; C1>HopH >Ke He CnpaBHJ1CSI H B 50 .n.HeM HH
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JIOB reOMeTpml H aJ1reOpbI.
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ynoTpeOJISIeMbIH HM MeTO.n. pellieH, HSI HO OH ynopHo XpaHHJI
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H paOOTaJI HCKJ1IOQHTeJ1bHO TaJ1aHTJIHBbIH YlleHbIH J],>KepOJ1aMO
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TO ero HHTepecYeT aCTp0J10rHSI, HaYKa, npe.n.CKa3bIBalOllI,aSl
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lIecKoro ypaBHeHHSI.
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TapTaJIbH, HeBepHble. nOJ1eMHKa MOrJ1a ObI 3aTSIHYTbCSI Ha-
.n.OJIro. eCJ1H 6bI TapTaJ1bSI He pellIHJl n0J10>KHTb eH KOHeu. H
He BCTpeTHTbCSI CO CBOHMH npOTHBHHKaMH JIHu.OM K J1Hu.y.
OH npHe3>KaeT B MHJIaH. Ha YJ1l1ll.aX ropo.n.a nOSIBJISIeTCSI
repOJIb.n. C repOOM TapTaJIbH. «5:1, TapTaJ1bSl,- npOB03rJIallIaJI
repOJIb.n., - BbI3bIBalO CBOHX npOTHBHHKOB I<ap.n.aHO H <pep-
papH Ha nYOJ1HlIHbIH Jl.HcnyT B u.epKOBb CBSITOH MapHH Ha
10 aBrycTa 1545 r., B 5 lIaCOB YTpa».
B Ha3HalleHHoe BpeMSI SIBHJ1CSI TOJIbKO O.n.HH <PeppapH,
«IOHOIlla C He>KHbIM r0J10COM, BeCeJ1bIM JIHu.OM, rpOMa.n.HbIMIf
cnOCOOHOCTSIMH H xapaKTepoM .n.bSIBOJIa»,- nHmeT B CBOHX
BocnOMHHaHHSIX Ta;:naJ1bSI. <PeppapH 6bIJ1 OKPY>K€H TOJInOH
.ApY3eH H po.n.CTBeHHHKOB. TapTaJ1bSI >Ke ObIJI TOJ1bKO B.n.BOeM
C 6paToM cpe.n.H Bpa>K)l.eOHO HacTpoeHHoH TOJInbI lIY>Koro
ropo)l.a. '
Ha1.JaJ1cSI .n.HcnyT. TapTaJIbSi .n.OKa3aJI HenpaBHJ1bHOCTb pe-
weHHH o.AHoii 3a.n.allH, c;J.eJIaHHOH I<ap.AaHO, H XOTeJI nepeHTH
g6

K p2360py .n.pyroro pemeHHSI, HO Tonn8 He .n.2na eMY rOBopHTb,
TpeoySi cnOBa .n.nH cI>eppapH. Hanp2CHO TapTanbSI npOCHJI
.n.aTb eMY 3aKOHlIHTb, a nOTOM .n.aTb rOBopHTb cI>epp2pH BCe,
4TO,OH 32XOlleT; .n.PY3bSI cI>epp2pH HHlIero He XOTenH cnY1lI2Tb.
nOCJIe)J.HHH, H2H.n.SI OllIHOI\Y B perueHHH TapT2JIbH, nYCTHncSi
B TaKHe .n.miHHbIe paCCY>K)J.eHHSI, liTO BpeMSI 32TSlHYJIOCb )).0
o6e.n.a H uepKoBb 6bICTpO onycTeJIa. J],HcnYT )J.OJ1>KeH ObIn ObI
npo.n.OJI>KaTbCSI H8 .n.pyroH .n.eHb, HO, cy.n.SI no HallaJIY, TapT8nbR
Mor 0>KH.n.2Tb x y)).llIero , He HCKJIIOlI8SI H CMepTH OT KHH>K2JIa
HaeMHOrO Y0I-lHUbI. TaKOH cpl1H2JI mopa He ObIJI pe.n.KOCTblO
B TO 0YPHoe BpeMSI.
HOllblO HHKKOJIO TapTanbSI Hero 6paT, 3aKyraBIllHcb B
nJI2ll1,H, nOKHHynH MI-InaH. I<ap.n.aHO H cI>eppapI-l C04nH ce6SI
no6e)J.HTenSlMH. HCTOPHSI TO>Ke OKa3anaCb HecnpaBe.n.nI-lBOH no
OTHOllIeHHIO K CKPOMHOMY T2pT8nbe - cnocoo pellIeHHSI Ky-
OHlIeCKoro ypaBHeHI-ISI )).OJ1ro OblJI H3BecTeH B MaTeM2THKe nOlI
Ha3BaHHeM «cpopMynb[ Kap.n.2HO».
T2KOB p2CCK23 0 TOM, KaK .n.B2 BeJII-IKHX HT2JIbSlHCKHX M2'
TeMaTHKa CTaJIH CMepTeJlbHbIMH Bp2raMH.
4HT2TeJIb, MO>KeT ObITb, 3aHHTepecyeTcSI, K2K >Ke T2pTaJIbSi
pellIHn ypaBHeHHe
x 3 +px=q?
nprlBe.n.eM XO)). ero paCCY>K)).eHHH.
Ey)).eM HCK2Tb KopeHb ypaBHeHHSI x B BHJJ.e
r- r-
X= II t - II u,
r.n.e t H U - HeH3BeCTHble, KOTopble H2)).O onpe.n.eJIHTb no
.n.8HHbl,1 p H q. J],anee, HO Boe OpHrHH2nbHOe npe)J.nOJIO>Ke-
HHe, liTO
(1)
_3r-
p=3 II tu.
(2)
ECJm no.n.CT2BHTb BbIpa>KeHHH .n.JISI x H p B JIeBYIO qaCTb
.n.aHHoro ypaBHeHHSI, TO nonYllHM:
(t;T - -trur + 3 ;rtil (;rr - -tru) = q;
3r- 31'1.:2 3 - 3,.-
t - 3 J/ t 2 u +3 t tu 2 - u + 3 yt 2 u - 3 Y tu 2 = q;
t - u = q.
(3)
97
4 .\ A, Ko,'oco"

PaBeHCTBa (2) H (3) .n.a.n.YT CHCTe:'.IY:
{ t - u = q;
3 rz:-
3y tu = p,
realHB KOTOpylO, lI!bI H no.'JYlIH:\1 Bblp2>KeHHSI t H U lIepea
p H q:
I ! P 3.
u=,l3)'
t - -i- (  t -= q;
HJIH
, ' 3
[2 - ql - (  ) = 0;
OTCIO.n.a
q 1/ { q ) 2 ( P ) ' 3 q 1 f' 'L , ) 2 ( P , ) 3
t = 2 + JI \ T, + 3" H U = - 2" + JI I,:! +"3 .
3 - 3--
DO.D.CT2B.'JSISI 3HalleHlUi t HUB p2BeRCTBO X = V t - Y U,
MbI H nOJIYlIHM cpOPMYJIY TapT2JIbH:
v V
q2 p'3 Q Q'2 p' Q
x= Vh-) +("3) +2- V(2")+h-j-2'
Do.n.YMaihe, liTO .n..'ISI T2pT2JIbH ObIJIO C2MbIM Tpy.n.HbiM
npH BblBo.n.e :HOH cpOPMYJII>!?
OrpmmaSi 3ac.TIyr2 Kap.n.aHO a2KnIOlIaeTCSI B TOM, liTO,
OB.TIa.n.eB peIlleHHeM ypaBHeHHSI
X S + px = q,
OR IlOIlle.l n.a.TIee H H2Ille.'J cnocoo peIll2Tb nOJIHOe KYOHlIec-
Koe ypaBHeHHe:
x 3 + ax2 + bx = c.
OKa3a,'JOCb, liTO CTOHT TO.nbKO X aa:\1eHHTb lIepe3 Z - 
T. e. nO.l0>KHTb
a
x=Z-3'
98

B nOJIHOM ypaBHeHHH YHHlITO>KaeTCS! lIneH co BTOpofi CTe-
neHblO HeH3BeCTHoro H ypaBHeHHe npHHHMaeT BHJJ. ypaBHeHHSI
TapTa.'1bH:


Z3 + pz = q.
1.JHTaTe.'Ib He JJ.o.'1>KeH .n.YMaTb, liTO npH nO
IOlltH 4>OPMY-
JIbI TapTa.'IbH - Kap.D.3HO OH OIO>KeT TaK>Ke .'1erKO peIllHTb
nlO60e ypaBHeHHe TpeTbeH CTeneHH BHJJ.a x 3 --;- px + q = 0,
K8K npH nO:\IOlll,H cpOpMy.1bI X I . 2 = -
 == 1- 1
2 - q OH
peIllaeT KBa.n.paTHoe ypaBHeHHe X2 ...L px
 q=O. 3TO cooceM
He TaR. PeIllaSi B VIII K.1acce KBa.n.paTHoe ypaBHeHHe, MbI
2
CTaBHM YC.'10BHe:Vl, lIT06bI .n.HCKpHMHHaHT
 - q ypaBHeHHSI
6bI.1 60.%Ille Hy.TIS! HnH paBeH RYJIIO. B CJlYllae .n.HCKpHl\IH-
HaHTa, MeHbIllero Hy.TJS!, MbI rOBOpHM, t.ITO KBaJJ.paTHoe ypaB-
HeHHe He HMeeT .n.eHCTBHTenbHbIX KopHeH. DbITaSiCb pellIHTb
Ky6HlIecKoe ypaBHeHHe X3 + px + q = 0, MbI npe>K.n.e Bcero
nOTpe6yeM, (IT06bI BbIpa>KeHHe (t t + (
 r CTOS!llI,ee no.n.
KBa.n.paTHbI
1 KOpHel\! B CPOp:Vly.1e TapTaJIbH - I<apJJ.aHO, 6bI.10
OOJIbllIe HY.'JSI H.1H paBHSI.'10Cb " y.'1 10. OKa3bIBaeTCSI, npH
( '2 ( P 3
t) + .3) > 0 cpop:\lyna n03BonHT Ha:V1 onpe.n.e.1HTb .n.eH-
cTBHTe.'IbHbIH KopeHb ypaBHeHHH; .n.Ba .n.pyrHx KOpHSI OYJJ.YT
MHH:Vlble. C.n.e.1ae:\1 9TO .n..TISI ypaBHeHHSI


X S + 6 X = 2.


B nOM CJIYllae
( t r + ( f r = 1 + 8 = 9;


H cpOp:\Iy.'1a TapTanbH - I<apJJ.aHo ..u.aCT:


_3; _3; _3;- _3r::
x = V 119 + 1 - V 1/9 - I = V 4 - V 2.


DpH nO\iOlll,H Tao.'1Hu. EpaJJ.Hca (CTp. 40 H 41) MbI HaH..u.eM,
liTO


;r;r :::::: 1,588 H :/2:::::: I ,260


H TOr.'la


x= I ,588-1 ,260=0,328.


4'"


99




npOBepHM HaH.n.eHHbIH KOpeHb



0,328 3 +6.0,328;::::;0,035+ 1,968=2,003.


HaH.n.H1e 9THM cnoc060M .n.eHCTBHTeJIbHbIe KOpHH ypaB-
HeHHSI


x 3 +6x+20=0 H x 3 -3x+5=0.


ECJIH ( ; r +(
 r =0, TO BCe TPH KOpHSI 6YJJ.YT .n.eHCTBH-
'feJIbHbIe H .ABa H3 HHX paBHbIe. I<opeHb, HepaBHbIH .ABYM
.ApyrHM, MO>KHO TO>Ke onpe.n.eJIHTb npH nOMOlll,H <PoPMynbI
TapTaJIbH - I<ap.n.aHO. C.n.enaHTe 9TO .n.JISI ypaBHeHHSI
x 3 -12x-16=0.


OCTaJIbHble .n.Ba KOpHSI, paBHbIe Me>K.n.y C060H, BbI onpe.n.emlTe,
ecnH JIeBYIO lIaCTb ypaBHeHHSI pa3.n.eJIHTe Ha pa3HOCTb Me>K-
.AY x H HaH.n.eHHbIM KopHeM, lIaCTHoe npHpaBHSleTe HYJIIO H
pellIHTe nOJIYlIeHHoe ypaBHeHHe.
ECJIH (
 r +(
 r <0, TO Bce TPH KOpHSI 6y.n.YT .n.eHCTE3H-
TeJIbHbIe, HO caMbIH npou.ecc HaXO>K.n.eHHSI HX .n.nSl lIliTaTeJISI
B HaCTOSllll,ee BpeMSI He .n.ocTyneH, TaK KaK CBSl3aH C TeopHeH
KOMnJIeKCHbIX lIHcen.


3a)J.allH


B 3aKJIIOlIeHHe nonp06YHTe pellIHTb .ABe 3a.n.allH npaKTH-
lIeCKoro xapaKTepa npH nOMOlll,H COCTaBneHHSI ypaBHeHHH.
3a.o.alla 1. B O.n.HOM cocy.n.e HaXO,lJ.HTCSI a (.1)p-npOu.eHTHoro
paCTBopa KHcnOTbI, B .n.pyroM b (.1)q-npOu.eHTHoro pacTBopa
TOH >Ke KHCnOTbI. 113 Ka>K.n.oro cocy.n.a OTJIH.TIH O.n.HO H TO >Ke
KonHlIecTBo nHTpOB, H B3S1Toe H3 nepBoro nepenHJIH BO BTO-
pOH, a B3S1Toe H3 BToporo - B nepBbIH. I<aKoe KonHlIeCTBO
JIHTpOB HY>KHO OTnHTb, lIT06bI B 060HX cocy.n.ax OKa3aJICSI
paCTBOp KHcnOTbI O.n.HOH H TOH >Ke KpenOCTH?
3a)J.8l1a 2. j],Ba KycKa pa3JIHlIHbIX MeTaJInOB o.n.HHaKoBoro
Beca CnJIaBHnH BMeCTe. HaHTH y.lI.eJIbHbIe Beca 9THX MeTannoB,
ec.1H O.lI.HH H3 HHX Ha 3
 60JIbllIe .lI.pyroro, a Y.AenbHbIH Bec
CM .
z
nOJIyllHBllIerocSi cnnaBa paBeH 8,75---s.
CM


100




JIHTEPATYPA


r. H. non 0 B. O'lepKH no HCTOpHIi M3TeM3THKH, 1923.
B. n. III e p e MeT e B C K H ii, 01lepKH no HCTOpHH M3TeMaTHKH,
Y'Ine!lfH3, 1940.
A. n. 10 W K e B H 'I, 0 MaTeM3THKe H3pO!lOB Cpe!lHeii A3HH
B IX-XV BeKax, fOCTexH3,naT, 1951.
H. 51. JJ. e n M a H, PaCCK33b1 0 MaTeMaTHKe, JJ.eTfH3, 1954.
51. H. n e p e JI b M 3 H, 3aHHM3TeJlbH3J1 3J1feOpa, focrexH3!l3T,
1949.
A. r. K y pow, AJlfeOp3H'IeCKHe yp3BHeHHJI npOH3BOJIbHblX CTe-
neHeii, fOCTeXH3!laT, 1951.
«6HOJlHOTeK3 M3TeM2TH'IeCKOrO KPY*K2», Bbln. I, [ocTexH3A3T.
1950,




rJIABA v


cIIYHKLlHH H fPAcIIHKH


«B nOHRTMM (jJYHKLlMOHa.7bHOfi 3a.
BJICHMOCTII, KaK B 3apOllblIJle, Y>Ke
3aJIO>KeHa BCR HlleR OB.ijalleHIISI IIBJIe-
HMRMM npMpOJ].bl H npOLlecca
1H Tex-
HMKH npn nO
IOU
M MaTe
laTMqeCKorO
annap aTa».


Xu H
 U H A. fl. - COBeTCKHIl MaTeMaTHK


B MaTeMaTUKY BXO)l.HT H )l.BH*eHHe U )l.Ha
eKTHKa


71 OHRTHe 0 CPYHKUHH HB.1SIeTCSI OCHOBHblM ,lI;JISI
MHOflIX pa3,l1;e.'IOB COBpeMeHHOt'! MaTeMaTliKH. B
,9TOM nOHRTHH, «KaK B 3apO,ll;bIWe, y)/{e 3aJIOmeHa
BCR HlleH OB.la,II,eHHH HB.'IeHHS!MH npHpollbl 11
npOuecea1\1H TeXHlIKH npH nOMoru;H MaTeMaT1I4ecKoro anna-
paTa. BOT noqe!\IY MbI .n.O.'l)/{HbI co Beet'! 6eCnOlll.a,ll;HOCTblO Tpe-
60BaTb OT OnpelleJleHHS! 9TOro nOHHTliH nOJIHOH, 6e3YKopH3-
HeHHOH HCHOCTH; HlI OllHO C.'IOBO B HeM He llO.'I)/{HO BbI3bIBaTb
II TeHH cOMHeHHH; MaJIeHWaH ,lI;BYCMbIc.'IeHHOCTb 311eCb rpo3HT
C,lI;eJIaTb BCe BeJIlIqecTBeHHoe 311aHlIe, KOTopoe CTpmlT HaYKa


102




Ha 6a3e ,noro OCHOBHoro nOHHTlIH, HeCOBepweHHbIM, Tpe6ylO-
ru;lIM KanHTaJIbHOH nepeCTpOHKH»l).
C nOHHTHeM CPYHKUHU TeCHO CBH3aHO nOHHTe nepeMeHHOH
BeJIUqlIHbI. 06a 9TU rrOHHTuH 6bIJIIi BnepBbie BBelleHbI B MaTe-
MaTHKy cppaHUY3CKHM MaTeMaTlIKOI\1 PeHe lleKapToM (1596-
1650). 113BecTHo, KaKoe 60.'lbWOe 3HaqeHHe npU,lI;aSM <P. 3H-
reJIbC BHellpeHlIlO yKa3aHHbix nOHHTIiii B eCTeCTBeHHble Hay-
KH. «OOBOPOTHW\l nYHKTO!\I B MaTel\IaTIiKe 6bIJI.a lleKapToBa
nepeMeHHaH BemjqrfHa. D.1arollapH 9ToMy B l\IaTeMaTliKY BO-
lliJ1H U ,lI;BHmeHHe Ii llIia.'IeKTHKa» (<<llHaJIeKTIiKa npHpOllbI»,
1952, CTp. 206). C03llaHHbIH HeCKOJIbKO n03me JIeii6HHueM H
HblOTOHOM aHaJIH3 6eCKOHeqHO MaJIblX, B KOTOpOM nOHHTUH
CPYHKUHIi 11 nepeMeHHoli HMelOT nepBOCTeneHHoe 3HaqeHHe,
OTpa3HJI oc06eHHOCTli rrpoueCCOB, npOTeKalOllllIx BOKpyr Hac.
«AHam13 6eCKOHeqHO l\IaJlbIX  OCHOBHaH MaTervmTljqecKaH
cpoprvm COBpeMeHHoro eCTecTB03HaHHSI U TeXHIiKlI, H HeT B03-
MomHOCTH yqeCTb Te HeHCqHCJIHMble 6JIame pe3YJIbTaTbI, KO-
Topble npHHec C C060H 9TOT aflaJI1I3 B 06JIaCTb TeopIilI 11 Tex-
HHKlI», - rOBoprn Ii3BeCTHblH cp1I31IK C. 11. BaBIiJIOB.
KHlIra, B KOTOpOH BnepBble lleKapT nHweT 0 nepeMeHHOH
BeJIIiqIiHe U CPYHKUIiH, Ha3bIBaeTCH «[eOMeTpIiH». OHa BbIWJIa
B CBeT B 1637 r. CJIe,ll;OBaTeJIt>Ho, npowJIo 60JIee TpexcoT JIeT
C Tex nop, KaK nOHHTHe CPYHKllHH BOWJIO B 06JIaCTb MaTeMa-
THKlI. B TeqeHHe Tpex CTOJIeTliH B pe3yJIbTaTe nOIiCKOB HaH.
60JIee npaBlIJIbHOrO Ii nOJIHOrO oTpameflHH B 3TOM nOHHTHH
Cyru;HOCTH COBepwalOIUHxcH BOKpyr Hac rrpoueCCOB nOHHTHe
CPYHKUHIi HeO,ll;HOKpaTHO nO,ll;BepraJIOCb Ii3lY.eHeHlIHM H yTOq-
HeHHHM. 009TOMY BnOJIHe eCTeCTBeHHO B 9TOH rJIaBe, nOCBSI-
llI.eHHOH npOCTeHWHM CPYHKUIiHM H HX rpacpriKaM, n03HaKO-
MHTb Bac, qrnaTeJIb, C COBpeMeHHbIM COCTOHHlIeM 9Toro Bonpoca.
113yqeHHe MaTeMaTIiKH H cpH31IKlI B KJIaCCaX, npe,ll;.
weCTBYlOru;uX BOCbMOMY, ,lI;aJIO BaM B03MomHOCTb HaKO-
nHTb B BaweM C03HaHIiH 60raTbIH MaTepHaJI 0 3aBHCHMOCTSlX
pa3JIIiqHOrO pOlla. 3TOMY cnoc06CTBOBaJIO H H3yqeHHe apmp-
MeTliKH (Ii3MeHeHHe pe3YJIbTaTOB ,lI;eHCTBliH B CBH3Ii C H3Me-
HeHIieM KOMnOHeHTOB, Ii3MeHeHHe BeJIHqIiHbI ,lI;p06H B CBH3H C
1I3l\1eHeHIieM ee 3JIeMeHTOB, npHMaH nponopUHOHaJIbHaH Ii 06-
paTHo nponopUliOHaJIbHaSl 3aBHCliMOCTH, 3aBHCHl\10CTIi Mem,ll;y
Be.1Hqmlarvm, yqaCTBylOllI.IiMH B 3allaqax), 11 Ii3yqeHHe reo.
MeTpIiIi (H3MeHeHHe llJIIiHbI oKpymHocTH B CBH3H C 1I3MeHeHIieM
1) A. SI. X H H 'I H H, BoceMb JIeKl.lHH no MaTeMaTH'IeCKoMY aHa.
JlH3Y, r OCTeXH3Jj.aT. 1948.
103

pa,ll;Hyca, H3MepeHHe nJIOru;a,ll;eH npSlMoyrOJIbHHKa H Kpyra,
H3MepeHI1e 06'beMOB Ky6a, 6pyca H UI1J1 11 H,lI;pa , CHMMeTpHSI,
CYMMa yrJIOB Bbm)'KJIOrO IvmoroyrOJIbHI1Ka, nO,ll;06HOe npe06-
pa30BaHI1e cpHryp), Ii H3yqeHHe aJIre6pbl (qHCJIeHHOe 3HaqeHHe
aJIre6paHqeCKOrO BblpameHI1H, COCTaBJIeHHe ypaBHeHHH H3
YCJIOBI1H 3a,ll;aq, peWeHHe CI1CTeM ypaBHemIii), H 3HaKOMCTBO
C cp1I3HqeCKHMH 3aKOHaMI1, KOTOpble B Onpe,ll;eJIeHHOH cpopMe,
a qaCTO H B onpelle.'IeHHbIX cpopMYJIaX cpHKCHpOBaJIli 3aBHClI-
MOCTli Mem,ll;y BeJIH1JHHaMH, yqaCTBYlOllI.HMlI B 1I3yqaeMOM HB-
JIemiH.
MHO roe B 9TOM MaTepIiaJIe HaBO,ll;HJIO Bac Ha MbICJIb, qTO
CPYHKUl10HaJlbHaH 3aBHCI1MOCTb Me)KllY BeJIHqHHaMH HeMHHye-
MO BJIeqeT 3a C0t'50H cYlll.eCTBOBaHlie onpe,ll;eJIeHHOH CPOPMYJIbI,
aHaJIHTl1qeCKH BbIpa)KalOllI.eH 9Ty 3aBlICI1MOCTb, H nplIyqaJIO
He MbICJIHTb CPYHKUlilO BHe ee CBH3H C CPOPMYJIOH Bo06ll1.e. B TO
me BpeIvISl 3HaKOMCTBO C TeXHHqeCKI1MH Ta6JII1UaMI1, caMOCTOH-
TeJIbHOe COCTaBJIeHlIe Ta6JIlIU, cpl1KCI1PYlOllI.HX pe3YJIbTaTbi
onbITOB, rrpOBOlllIMbIX BaMH npH H3yqeHHH OnpelleJIeHHblX
HBJIeHHH B 06JIaCTI1 cpH3IiKI1, XHMHI1 11 npOH3BOllCTBa, nOCTpoe-
HHe cpHrYPbI, CHMMeTpHqHOH I1JII1 nO,ll;06HOH ,lI;aHHoj:'I, nOJIb-
30BaHHe rpacpliKaMH, CHHTbIMH C KaKOrO-JIH60 npHt'5opa (6a-
porpacpa, TepMorpacpa 11 T. n.), BeJIO Bawy MbICJIb B HHOM
HanpaBJIeHI1H, YTBep)K,lI;alOru;eM, qTO CPYHKUliOHaJIbHaH 3aBH-
CHMOCTb MO)KeT CYlll.eCTBOBaTb Ii BHe CBH3H C onpe,ll;eJIeHHOH
MaTeMaTHqeCKOH CPOPMYJIOH. Bce 9TO, BMeCTe B3HToe, nOCTe.
neHHO nO,ll;rOTaBJII1BaJIO Bawe C03HaHHe K BocrrpHHTHIO CO-
BpeMeHHoro OnpelleJIeHUSI CPYHKllHH.
Pa3t'5epeM HeCKO.'lbKO 3a,ll;aql KOTopble nOMorYT HaM HaHTU
HaH60JIee qeTKOe H HCqepnblBalOru;ee onpe,ll;eJIeHHe CPYHKllHH;
nonYTHO C 3THM Mbl 03HaKOMHMCH C HeKOTopblMH HOBbIMH no-
HHTHHMH, TepMHHaMH H onpe,ll;eJIeHHSlMH.
o n p e ,lI; e JI e H He. E yoeM H.Q3bl8amb MH.l»ICeCmeOM co-
6paH.ue 9AeMeH.m08 (npeoMemoB, 'iUCeA, mO'ieK U m. n.), KOmopoe
onpeoeJ1JlemCJl H.eKOmOpblM npaeuJ1OM, n0380AJlIOU{UM c nOAl-lOu
onpeOeAeH.H.OCmblO cyoumb 0 mOM, 8xooum Oal-tH.blU 9AeMeH.m
8 'iUCAO 9AeMeH.mOB C06paflUJl UAU He 8xooum.
OpliMepaIvm MO)KeT CJIY)KHTb MHomeCTBO Bcex HaTypa.:lb-
HblX qHCeJI, l\IHOmeCTBO Bcex paUliOHaJIbHblX qHCeJI HJIli MHO-
meCTBO Bcex TpeyrOJIbHHKOB.
3a)l.aQa 1. llaHO ypaBHeHlIe C ,lI;ByMSI He1I3BeCTHbIMH:
y=2x+ 1.
CocTaBHTb Ta6JIHuy HeKoTopblX peWeHI1H 9Toro ypaBHeHI1S1 113
6ecqHCJIeHHOrO IvmomeCTBa I-fX.
101

4HTaTeJIIO 1I3BeCTHO, qTO ,lI;aHHOe ypaBHeHHe HMeeT 6ecKO-
HeqHOe MHO)KeCTBO peweHUH 11 qTO Mbl MO)KeM HaHTli HeKO.
Topble 113 3THX peweHUH, eCJIH O,ll;HOMY 113 HeH3BeCTHbIX, XOTSI
6bl X, npH,lI;a,ll;HM npOH3BOJIbHble qHCJIOBble 3HaqeHHH H orrpe.
.n.eJIeHHbIM 06pa30M HaH.n.eM COOTBeTCTBYlOllI.ue 3HaqeHHH ,lI;JIH
y. TaK KaK B 3a,ll;aqe Ha X H Y He HaKJIallblBalOTCH HHKaKHe
OrpaHHqeHHH, TO, pewaH ypaBHeHHe, Mbl MO)KeM llJIH x Bbi-
6paTb npOlf3BOJIbHbIe qHCJIa H3 MHO)KeCTBa Bcex .n.eHCTBHTeJlb-
HbIX qHCeJI. HaHllS!, TaKliM 06pa30M, HeCKOJIbKO peweHHH
,lI;aHHOrO ypaBHeHlHI, Mbi MomeM 3aHeCTH lIX B Ta6JIHlI,Y. B
KOTOpOH 3HaqeHlIH x H y, COOTBeTCTBYlOllI.He llPyr ,lI;pyry, pac-
nOJIOmeHbl B O.n.HOM BepTHKaJIbHOM cToJI6ue.
x Ii -I I 0
Y -I I
y3"
2
3 ).... .
7 I.....
2 Jl r 3 + I
5
OOCTpoeHHaH Ta6JI1iua B HaweM npellCTaBJIeHHH MomeT
6blTb npOllJIeHa HeOrpaHlIqeHHo.
COBoKynHocTb Bcex qHCJIeHHbiX 3HaqeHlIH x npe,ll;cTaBJIHeT
c060H MHO)KeCTBO, KOTopoe 0603HaqlIM qepe3 M; cOOTBeTcTBY-
IOlll.He 3HaqeHHH y COCTaBHT ,lI;pyroe MHomeCTBO, KOTopoe 0603-
HaqlIM Qepe3 N.
OocTpoeHHaH Ta6JIHUa Ka)K,lI;OMy 3HaqeHHlO x H3 MHO)Ke-
CTBa M CTaBHT B COOTBeTCTBHe e,JHHCTBeHHoe H OnpelleJIeHHOe
3HaqeHHe y 113 MHomeCTBa N.
3a)l.alJa 2. OYCTb 1l.'IHHa CTOpOHbI KBa.lJ.paTa paBHa x, a
nJIOru;a,ll;b ero y. Tor,ll;a
y=x 2 .
CocTaBHTb Ta6.'IHUY 3HaqeHlIH !J npH HeKOTOpbJX 3HaqeHHSlX x,
1l0nYCTHMblX B ,lI;aHHoH 3allaqe.
llJISI peweHHH 3a.n.aqH .n.a,ll;UM x npOH3BOJIbHble 3HaqeHHH,
BbI6paB HX H3 MHomeCTBa nOJIOmHTeJIbHblX qHCeJI, 11 HaH,lI;eM
COOTBeTCTBeHHble 3HaqeHHSI y. OOJIyqH1 Ta6.'IlIuy
x 11 0 . 5 I 2 1 2 . 5 I JI] I.. . ..
I \6.25\ I.. . '
y 110.25\ 4 7
105

OOJIyqeHHaSI TaOJIl1ua, KaI< II B 3alIa'Ie 1, Kam,ll;oMY 113
,lI;OrrYCTHMbIX 3Ha'-leHl1i1 X CTamlT B COOTBeTCTBlle elIHHCTBeHHOe
H onpe.n.eJIeHHOe 3HaqeHHe y. 0603HaqHB nO-npe)!\HeMy qepe3
M IvIHO)KeCTBO BCeX 3HaqeHlIH X, OTMeTliM, qTO IvIHOmeCTBO M
B 3TOH 3a,ll;aqe HHOe, qeM B 3alIaqe 1. EC.TJli B 3alIaqe 1 MHO)Ke-
CTBO M BKJIlOqaJIO B ce6S! Bce .n.eHcTBHTeJIbHbIe qHCJIa, TO B
3a,ll;aqe 2 OHO BKJIlOqaeT TOJIbKO nO,'lO)KHTeJIbHbIe qlICJIa. Hli
OTpHuaTeJIbHbIe qlICJIa, HI1 HYJIb He MorYT BXOlIliTb B IvIHO)Ke-
CTBO M, TaK KaK 3TO npOTI1BOp'eqlIJIO 6bl CMblCJIY 3a,ll;aqli.
3aJl.a'la 3. OYCTb qHCJIO CTOpOH BbmYKJIOrO MHorayrOJIb-
HliKa paBHO x, a CYMMa BHYTpeHHlIX yrJIOB era y. Tor.n.a, KaK
H3BeCTHO li3 reoMeTpHli,
y= 180 0 (x--2).
CocTaBHTb Ta6JIIiUY 3HaqeHIHf y npH onpe,ll;eJIeHHbIX 3Haqe-
HHHX x.
YKa3aHHaH cpopMYJIa n03BOJIIiT 9TO C,lI;eJIaTb.
yO
\\ 3
\\ 180
\ 4
\ 360
5
1......... 1
\.........1
x
540
l(aK H paHbI1Ie, Kam,ll;o;\-IY 3HaqeHI11O X H3 MHomeCTBa M
Mbi MO)KeM nOCTaBHTb B COOTBeTCTBl1e elIliHcTBeHHoe Ii onpe-
,lI;eJIeHf{Oe 3HaqeHHe y li3 MHO)KeCTBa N. Ho MHO)KeCTBO M B
9TOH 3a,ll;aqe OTJIliqaeTCH KaK OT IvIHomeCTBa M B 3alIaqe 1,
TaK 11 OT MHomeCTBa M B 3a,ll;aqe 2. 3,l1;eCb M ecTb MHomeCTBO
UeJibIX nO.,'IO)KHTeJIbHbIX qHCeJI, HaqliHaH C 3. Opli HHbIX 3Ha-
qeHHHX 3alIaqa He liMeJIa 6bI CMbICJIa.
3aJl.aqa 4. CaMOJIeT H3 MOCKBbI HanpaB.rIHeTCH B CeBacTo-
nO.'lb co CKOpOCTblO 300 KM/LlQC. PaccTOSlflHe OT MOCKBbI ,lI;O
CeBaCTOnO,TJS! 1200 KM. B XapbKoBe (800 KM OT MOCKBbI)
CaMOJIeT HMeeT oCTaHoBKY Ha 2 qaca. CoCTaBl1Tb Ta6JIHUY pac-
CTOHtIHH CaMOJIeTa OT MOCKBbI C MOMeHTa BbI.TIeTa 11 110 MOMeHTa
nOca,ll;KM ero B CeBaCTOnOJIe lIJiH HeKOTopbIX MOMeHTOB BpeMeHli.
OYCTb X - BpeMs:! nOJIeTa B qacax Ii y - paccToHHHe caMO-
JIeTa OT MOCKBbi B KI1JIOMeTpax; TorlIa
2 2 2 2 I 5 I 'f
x 2- 3- 4- )
3 3 3
y 1\ 300 1 600 800 800 800 1900 I  200
\ I
106

CHOBa Ta6.mma .n.aeT B03MomHOCTb Kam,ll;OMY 3HaqeHHlO x
li3 MHomeCTBa M nOCTaBliTb B COOTBeTCTBlie e,ll;liHCTBeHHoe Ii
onpe.n.eJIeHHOe 3HaqeHHe y H3 MHomeCTBa N. MHomeCTBO M
BKJIlOqaeT B ce6S1 Bce ,lI;eHCTBliTeJIbHble qHCJIa Ha OTpe3Ke
[0; 6]; HHKaKlie liHble qliCJIa no CMblCJ.IY 3a,ll;aqli B 9TO MHO-
meCTBo BXOlll1Tb He MorYT. I1HTepecHO TaK)Ke OTMeTliTb, IlTO
H3MeHeHlie x He BCer,ll;a BJIeqeT 3a cot'50" li3MeHeHlie y; npli 3Ha-
qeHHSlX x Ha OTpe3Ke [ 2 ; ; 4 ; ] nepeMeHHaH y oCTaeTcSI
nocToHHHOH, paBHO" 800 KM.
3aJl.aQa 5. Oepe,ll; HaMH qaCTb Tat'5JIHlJ.bI qliCJIeHHOro COCTaBa
yqaru;HxcH lllKOJI MOCKBbI Ha 1956 ro,ll;:
N2 lllKOJlbI I lJ HCJlO
Y lI8lL\HXCfl
421 1120
422 '1040
423 957
H T. J{. If T. J{.
EC,'Ili HOMep WKO.'lbI I\1bI npU!\lel\l 3a I\1HO)K{'CTBO M, a K)-
JmqeCTBO yqalll.HXCH B OT,lI;eJIbHbIX WKOJIaX - 3a rvIHomeCT-
BO N, TO Kam,ll;OMY 9JIeMeHTy rvIHomecTBa M I\IbI onHTb MO>K{'M
nOCTaBliTb B COOTBeTCTBue ellHHcTBeHHoe. onpe,ll;eJIt'HHoe ql1CJIO
H3 I\IHO/KeCTBa N. OqeBUllHO, rvIHomecTBO M B ,lI;aHHoM CJIyqae
BI{JIlOqaeT B cet'5H ueJIble qHCJIa - HOl\1epa Bcex lUKOJI Mo-
CKBb!. Ocot'5eHHocTblO 9Tm{ 3a,ll;aqn B OT.rIU'lHe OT Bcex rrpeJJ.-
IlleCTBylOllJ.IfX 3a,ll;aq 6YlleT OTCYTcTBlfe cpoprvrYJIbl, npH nOMO-
ru;H KOTOPO" Mhl CMomeM onpe,ll;e.'IJfTb 9.rIervreHT MHomeCTBa N
no cooTBeTcTBYJoru;eMY 9.'IeMeHTY 1HOmeCTBa M.
3aJl.aQa 6. Ha qepTe)/{e 31 ,lI;aeTCH qaCTb JIeHThl, CHHTOfI C
caMonHwYllI.erO npH60pa-TepMorpacpa, pemcTpliPYK\llJ.erO TeM.
rrepaTYPY B pa3JIHqHble ,lI;HI1 anpeJIH npl1 nOMoru;H rpa<pliKa
1956 r. OnpelleJII1Te no llaHHoMY rpacpllKY Tel\mepaTYPY B
pa3JIliqHble MOl\IeHTbI BpervreHH 14 11 15 anpeJIH.
OYCTb qIlCJIeHHble 3HaqeHHH BpeMeHI1 npellCTaBJIfllOT co-
60" MHO)/{eCTBO M, cooTBeTcTBylOru;l1e qliCJIeHHble 3HaqeHHSI
TeMnepaTYPbl - MHO)/{eCTBO N. Kam,ll;ol\lY :meMeHTY MHO)/{e-
CTBa M I\IO)KHO npl1 nOMoru;li CHHToro C 6apa6aHa TepMorpacpa
rpacpl1Ka nOCTJBII fb B COOTBeTCTBlie e,ll;liHCTBeHHbIH H onpe-
.n,eJIeHHbIH 3JJeMelIT I\III00I\CCTBa N. MHO>KeCTBO M B JJ.a HHOM
107

11, onpenn 150nfJMn 160nfJenn
01y68mMmmmnOZ?68mUmmmnOI468m
to
o
, "-
,
+---4---
-10
-10
elO
4epT. 31.
CJIyqae onpe,ll;eJIHeTCSI BpeMeHeM ,lI;eHCTBHH TeprvIOrpacpa. 3,l1;eCb,
KaK H B 3a,l1;aqe 5, MbI 'He UMeel\l cpO prvrYJIbI , npH nOMoru;H KO-
TOpOH Mbl MOrJII1 6bI no ,lI;aHHoMY 9JIeMeHTY MHomeCTBa M
onpe)I.eJIHTb cooTBeTcTBYlOllI.HH eMY 3JIeMeHT MHomeCTBa N.
/I,
4epT. 32.
3anaqa 7. llaH qeTblpexyrOJIbHHK ABCD. DOCTpOHTb
HOBbIH qeTblpexyrOJIbHHK A}BlClD 1 , nO,ll;06HbH1 ,lI;aHHoMY,
eCJm ,lI;aHbI nOJIO)!{eHHe ueHTpa nO)I.06HH 0 H K03cpcpHuHeHT
5
nO,ll;06HSI k = 2" (qepT. 32).
108

OOCTPOHTb Ha CTOpOHe A.B. MHoroyrOJIbHHKa AlBlClDl
TOqKY Yl' cooTBeTcTBylOru;ylO TOqKe Xl Ha CTOpOHe AB MHOrO-
yrO.'lbHI1Ka ABCD.
llJIH peWeHHSI nOCTaBJIeHHOrO BOnpOCa HymHo TOJIbKO no.
CTpOHTb .nyq Ox l ; TOqKa nepeCeqeHIiH 9Toro JIyqa co CTOpOHOH
AlBl MHoroyrOJIbHHKa AlB 1 CID I H onpe,ll;eJIHT nOJIOmeHHe
HCKOMOH TOqKH Yl'
I3Y,ll;eM nOHI1MaTb nOlI MHomeCTBOM M B 9TOH 3a.n.aqe CO-
BOKynHoCTb TOqeK Xl KOHTypa MHoroyrOJIbHHKa ABCD, a no.n.
MHO)!{eCTBOM N - cOBoKynHocTb TOqeK Yi KOHTypa AlBlClDl'
TaKHM 06pa30M, nepe,ll; HaMH .n.Ba MHO)!{eCTBa M H N, CBH3aH-
HbIX Me)!{,lI;y C060H, HO MHO)!{eCTBa 9TH He qHCJIOBble, a TOqeq-
Hble. J(aK H BO Bcex I1pelIbIJJ.yru;HX 3a,ll;aqaX, npH nOMoru;H
nOCTpoeHHoro HaMH qepTe)!{a Ka)!{,lI;OMY 9JIeMeHTY (TOqKe)
MHO)!{ecTBa M MbI MO)!{eM nOCTaBHTb B COOTBeTCTBHe e,ll;HH-
CTBeHHblH H onpelIe.neHHbIH 3JIeMeHT (TOqKY) MHomecTBa N
H ,lI;ame COCTaBHTb Ta6JIHuy 3Toro COOTBeTCTBHSI.
Xi I x I x 2 I X:] I . .
1 I
-- , I
I 1 I
Y l I YI i Y2 Y3 . . .
I
Pa306paHHbIX npUMepOB 6Y,lI;eT ,lI;OCTaTOqHO, qT06bI nO,ll;OH-
TH K nOHHMaHHIO COBpeMeHHoro onpe.n.eJIeHI1H CPYHKUHH H
CPYHKUl10HaJIbHOH 3aBHCHMOCTH.
OpelIBapl1TeJIbHO C,lI;eJIaeM HeKOTopbIe BbIBOlIbl H3 pa306-
paHHbIX 3a,ll;aq:
1. B Kam,ll;OH 3a,ll;aqe MbI HMeeM ,lI;Be rrepeMeHHble BeJIHqHHbI,
KOTopbIe B 60JIbWHHCTBe 3a.n.aq 0603HaqaJIH OlIHY qepe3 x,
,lI;pyrylO qepe3 y; 9TH BeJIIjql1HbI HaXO,ll;HTCH B 3aBHCHMOCTH
,lI;pyr OT ,lI;pyra: nJIoru;alIb KBa,ll;paTa 3aBHCUT OT ,lI;JIHHbI CTOpOHbl
KBa,ll;paTa (3alIaqa 2), cYMMa BHYTpeHHHx yrJIOB MHoroyrOJIb-
HHKa 3aBHCHT OT qHCJIa CTOpOH era (3a,ll;aqa 3), TeMnepaTypa
B83,l1;yxa 3aBHCHT OT BpeMeHH CYTOK (3a,ll;aqa 5), nOJIOmeHHe
TOqKH Yl Ha KOHType AlBlC.D. 06YCJIOBJIeHO nOJIOmeHHeM
TOqKH Xl Ha KOHType ABCD (3a.n.aqa 7).
2. OepeMeHHoH X MbI npH,lI;aBaJIH 3HaqeHHe I1JIH nOJIO>Ke-
Hue no CBoeMY npOH3BOJIY, BbI6upaH HX cpe,ll;u B03MO)!{HbIX
3HaqeHHH .n.JISI lIaHHoi'r 3a,ll;aqH. I3Y,ll;eM Ha3blBaTb nepeMeHHYlO X
a pry M e H TOM.
l09

3. COBoKyrrHocTb Bcex 3HaqeHHH aprYMeHT8, .lI.0nYCTHMbiX
B ,lI;aHHOH 3a.naqe, npe,ll;craBJISleT co6oi'I IIfHO>KeCTBO M, KOTO-
POMY COOTBeTcTByeT onpe,ll;eJIeHHOe rvlHomecTBO N - COBO-
KynHocTb Bcex 3HaqeHHH ,lI;pyroH nepeMeHHoH y.
4. B pa3JIHqHbiX 3a.ll.aqaX Mbl CTaJIKHBaeMCH C pa3JUlqHblMH
MHomeCTBarvm M. B 3a,ll;alle 1 - 3TO MHomeCTBO Bcex ,lI;eHCTBH-
TeJIbHblX qHCeJI, B 3a,ll;a 1 le 2- MHomeCTBO nOJIO)!{UTeJIbHbIX
llHCeJI, B 3a,ll;aqe 3- MHO)!{eCTBO ueJIblX qHCeJI, HaqHHaSl C 3,
B 3a.naqe 4- MHomeCTBO ,lI;eHCTBUTeJIbHblX qHCeJI Ha OTpe3Ke
[0; 6 J, B 3a,ll;aqe 5- lIIHomeCTBO ueJIblX qHCeJI - HOMepoB
WKOJI MOCKBbl, B 3alIaqe 7- MHomeCTBO TOqeK KOHTypa
ABCD.
5. MHomecTBo M B Kam,ll;OH 3a,ll;aqe onpe,ll;eJISleTCH cO,ll;ep'
maHHeM caMofl 3a,ll;aqH, npaKUlqeCKHMH co06pameHHSlMH H Tpe.
60BaHHHMH 3lIpaBoro CMblCJIa.
6. MHomecTBY M B Kam,ll;oH 3a,ll;aqe CTaBHTCH B COOTBeT-
CTBHe MHO>KeCTBO N.
7. J(am,ll;ol\lY qHCJIeHHoMY 3HalleHHlO x I\IH0meCTBa M HJIH
Kam,ll;OMY 3JIeMeHTY ero MomHO nOCTaBHTb B COOTBeTCTBHe
onpe,ll;eJIeHHOe H elIHHCTBeHHoe 3HaqeHUe y H3 MHomeCTBa N.
8. 3HaqeHHe 3JIel\leHTa y Jl3 1\1H0)!{eCTBa N, COOTBeTcTBylO-
ru;ee Bblt'5paHHoMY 3HaqeHHIO 3,'Iel\leHT8 x 113 1\1H0)!{eCTBa M,
HaXO,ll;ltTCH no onpe,ll;eJIeHHOMY npaBH.'IY, KOTopoe B ,lI;aJIbHeH-
IDeM 6y.n,eM Ha3bIBaTb 3aKOHOM COOTBeTCTBIIS!.
9. 3aKoH COOTBeTCTBHSI 1\!0)!{eT 6bITb lIaH B BH,lI;e cpOpl\!YJIbl
(3a,l1;aqH 1, 2, 3), rpacpHKa (3a,ll;aqa 6), Ta6.rlHUbl (3alIaqa 5),
qepTema, nocTpoeHHoro no onpelIe,lleHIlOI\IY 3al\OHY (3a.n.aqa 7),
H C,'lOBecHOH cpOpMY,'lHpOBKH.
Tenepb nepeHlIervl K onpelIe.rleHHlO cp IIKHIlH.
llyctnb M u N - o8a ltfH.OJ/CeCtn8a, EJ.le.lfeHmaltfU KomopblX
Mocym 6wnb .uo6ble o6beKmbi. ECAll KaJli'ooMY EJAeMeH.my x
MH.OJ/Cecm8Q M ltfOJ/CH.Q nocma8umb 8 coonwemcmeue H.eKOmOpblU
.MeltteH.tn y lttH.OJ/Cecm8a N, mo c080pJlIll, 'uno y eCtnb CPYHKI{UJl x.
::hy MblCJIb Ot'5blqHO 3anHCbIBalOT B BH,lI;e paBeHcTBa:
y=f (x).
CHMBOJI f 03HaqaeT YCTaHoB.rleHHbIH 3aKOB COOTBeTCTBHSI,
T. e. npaBJi.'lo, n03BO,'IS!lOllI.ee JJ.JISI Kam,ll;oro llorr)'CTHMoro 3Ha-
'IeHHSI x HaXO,ll;HTb COOTBeTCTBYlow.ee 3HaqeHHe y.
J(QHeqIlO, orrpe,J,e.leHHe (YHKUHII, lIaHHoe BbTIITe, He S!B.
.'ISIeTCH e,ll;HHCTBeHHO B03MOmHb[t. BOT, Haflpmlep. HeCKO.'lbl{O
HHoe onpe,J,e.'IeHue:
110

Be.lll l lUHa Y HG3bl8aemCJl cpYHKl{UeU 8eAUI/.UHbi X. eC.lll KG."I/C-
VO.MY 3HQ'leHUlQ X, 83RfllO.\lY U3 COBOKYIlHOCmU GOIl ycmWllJIX
3HalleHUU, CIlla8UmCJl B COOmBemCm8Ue onpeOe,leHHOe 3HQI/.eHUe y.
0PHHUHIlHa.lbHO BTOpOe onpe,ll;e,'leHHe HHqeM He 0T.rUl4a{>T-
CH OT nepBoro. Ho H TO H .npyroe onpe,ll;eJIeHHe Tpe6yer OT Hac
npH Hcc.ne.D,OBaHHH Ka)K.D,OH onpe,ll;eJIeHHOH CPYHKUUH OrpaHI-!-
4HBaTbCSI TeM MHO)KeCTBOM M 3HaqeHHH BeJIHqHHbl x, KaKoe
;J.HKTyeTcH Ue.'lblO ;J,8HHOro HCCJIe,ll;OBaHH5I.
MHO)KeCTBO M. onpe,ll;eJISlel\10e CO,ll;ep>K8HHeM nOCTaB,leHHOH
3a,ll;a4H HJIH npaKTHqeCKHMH Tpe60BaHHSlMH H CO,ll;ep>Kalll.ee Te
3Ha4eHHH aprYMeHTa x, KOTopble MbI MomeM npH.D,aBaTb eMY
B VC.nOBHSlX .nannOH 3a.naqH, Ha3bIBalOT 0 6.n aCT b 10
on"pe,ll;e.neHHSI CPYHKUHH. TaK, B 3a,ll;aqe 2 00.1a-
CTblO onpe,ll;eJIeHHSI CPYHKUHH 6Y,lI;eT MHO)KeCTBO Bcex nOJIOmlI-
Te,TJbHbIX qUCeJI, B 3a.na4e 3 06JIaCTbIO orrpe,ll;eJIeHHH CPYHI{UIUl
6Y,ll;eT MHO)KeCTBO Bcex ue,TJbIX 4Hce.n, Ha4HHaSl C Tpex, H T. ,lI;.
EC.TJH 3aKOH COOTBeTC1BHS! CPYHKUHoHa.nbHoH 3aBHCHMOCTH
3a,ll;8H cpop:.IY,'10H, TO 06blqHO no,ll; 06.nacTblO onpe,ll;eJIeHHSI
CPYHKUHH pa3Yl\lelOT MHO>KecTBO Tex 3HaqeHHH aprYMeHTa, npH
KOTOpblX 3Ta cpopM)'JIa HleeT Cl\IblC,l. O,ll;HaKO qaCTO 9Ta WHpO-
KaH 06,TJaCTb 3Ha 1 leHlfH aprYMeHTa cy)!{aeTCSI npaKTHqeCKHl\1H
co06pa>KeHHSlMH.
OOHCHHM .ny MblC,TJb Ha npHl\Iepe.
3aKoH COOTBeTCTBUfi CPYHKUHH Y 3a,ll;aH q,OPMYJlOH
V X- 3
Y= -.
4-x
BbISlCHHTb 06.nacTb onpe.ne,'1eHHS! CPYHl'UlfH.
nOCTaBH:II Tpe60BaHl1e, qro6b1 3Ha4eHHS! Y 6bIJIH ,lI;eHCT-
BlfTe.nbHbli\lH; Tor.D,a 3HaqeHHSI A.'lfl x filL! AO.1)Kllbl BbI6paTb
TaK, 4T06bl ,lI;p06b
x-3
4-x
6bJJJa nOJIO>KHTeJIbHOH. T. e.
x-3 0
_ 4 ' >.
-x
COCTaB.1HeM ,lI;se CHCTellfbl HepaBeHCTB:
{ X - 3 > 0;
4 - x> 0;
{ X - 3 < 0;
4 - x < O.
111

PewaSl nepBYlO CliCTeMY, rrOJIyqHM:
X> 3 H X < 4.
BTopaSl CHCTeMa ,lI;aCT
x < 3 H x> 4;
nOCJIe,ll;HHe HepaBeHcTBa npoTHBOpeqaT ,lI;Pyr ,lI;pyry. I1TaK,
3 < x < 4.
Kpmle TOro, x MO>KeT paBHSlTbcSI 3, HO He MO>KeT paBH5ITb-
CH 4, TaK KaK TOr,ll;a 6bI 3HaMeHaTeJIb ,lI;p06H
x-3
4- x
npeBpaTH.nCH B HYJIb. OKOHqaTeJIbHO MbI l!\lee:\l:
3-< x < 4.
TaKHM 06pa30M, 06JIaCTblO orrpe,ll;e.neHHH CPYHKUHH 6Y,lI;eT
MHomeCTBO ,lI;eHCTBHTeJIbHblX qHce.n Ha npOMe>KYTKe (3; 4),
BKJIlOqaSl H qHCJIO 3.
Kor,ll;a 06JIaCTb orrpe,ll;e.neHHSI CPYHKUHH BbISlCHeHa, TO ro-
BOpSlT, qTO CPYHKUHSI 3a,ll;aHa HJIH CPYHKUHSI orrpe,ll;eJIeHa Ha
l\1HomeCTBe M.
Ope)!{JJ.e qeM qHTaTb ,lI;a.nbwe, pewHTe HecKOJIbKO 3a,ll;aq Ha
HaXO)!{JJ.eHHe 06.'laCTli orrpe,ll;eJIeHHH CPYHKUHH:
1
y=
Y x- 5
y=
I-Y l- x
2
y=
x 2 -3x-4
rpaUqeCKaH xapaKTepliCTHKa YHK""
OYCTb 06.nacTblO orrpeJJ.eJIeHHSI CPYHKUHH y=f (x) 6Y,lI;eT
HeKoTopoe MHO>KeCTBO M ,lI;eHcTBHTe.TJbHbIX qHce.n. Ka)!{JJ.OMY
:MeMeHTY noro MHomeCTBa Ha qHCJIOBOH OCH ooTBeTCTByel'
onpe,ll;eJIeHHaH TOqKa. OpHHSlB qHCJIOBYlO OCb 3a OCb a6CUHcc,
nOCTpoHM npHMoyro.nbHYlO cilcTeMY Koop,ll;HHaT. I13 MHomeCTBa
M .D,onycTHMblX 3HaqeHHH aprYMeHTa x BbJ.D,eJIHM OJJ.HO 3Haqe-
Hile x k ' a Ha OCH a6CUHcc - cooTBeTcTBYIOru;ylO eMY TOqKY.
3TOH T04Ke B CBOlO Oqepe,ll;b MomHO nOCTaBHTb B COOTBeTCTBHe
112

.D.pyrylO TOqKY KOOpJJ.HHaTHOH n.rIOCKOCTH, lIMeHHO TOqKY C
a6CUliCCOH X k II Op.D.HHaTOH Yk=f (Xk), H rrOCTpOllTb ee. TaKliM
06pa30M, Ha rr.nOCKOCTli 6Y.D.eT rrOCTpoeHa TOqKa K (qepT. 33),
Koop,ll;HHaTbI KOTOpOH KOe-qTO paccKa3bIBalOT 0 CPYHKUHH:
Op,ll;HHaTa Yk TOqKH YKa3bIBaeT qllc.neHHoe 3HaqeHHe CPYHKUHU
y'
II
 II
....... - j:
./
" !Iff ........ ,/ I
- !
a 0 xI( [j
x
4epT. 33.
npH 3HaqeHHH aprYMeHTa, paBHoM a6CUHcce 3TOH TOqKH.
VI3MeHHH x BHYTPH MHO>KeCTBa M, MbI MomeM nocTpoHTb ue-
JIbIH pSl,ll; TOqeK, KOOp.D.HHaTaMli KOTOpbIX 6Y,lI;YT 3HaqeHHSI He-
3aBHcHMoH nepeMeHHOH x II cooTBeTcTBYlOru;He 3HaqeHHH
CPYHKUHH y.
9
7
!I
'400
9
'I
6
5

3
2
1
! 6 I I I I
II 5 I j'200 I I 1/
/I I' 1'000 ! i I A
Vi ! 11 , BOO I V I
J : 600 I
I ! I 2 II : I !,DO I I' I
! Ij J: i 200 i I I
'I 1 /' I I I
; i I i
; l I 1 I I I I
I
.f 01 Z J 4
-I 0 , 2 3 "
J._
O'23G58
If en 1',
4epT. 34.
feOMeTpHqeCKOe I\feCTO TOqeK n.nocKocTH, npSll\myro.'IbHbIe
KoopJJ.HHaTbl x II Y KOTOpbIX CBSl3aHbi COOTHOWeHllel\1 y=f (x),
Ha3bIBalOT rpaqJHKOM CPYHKUHlI f (x).
3HaSl 3aKOH COOTBeTCTBUH ,lI;aHHoH CPYHKUIiH, Ibl Bce me He
CMO)KeM rrpaKTHqeCKIi nOCTpOHTb no nO.1Y4eHHbIM Koop;J,l-IHa-
113

TaM Bce roqKH rpa4mKa. 06WIHO CTpOSlT TO,T) bKO Hei\OTOphle
H3 HHX, JJ.oCTaT04Ho 6.'I H3Kue .D.pyr K .D.pyry, a 3aTe;>.1 eCTeCTBeH-
Hbll\f 06pa30M coeJJ.HHHlOT nOCTpoeHHbIe roqKU rrJIaBHOH JJIIHHeH,
3a HCKJJlOqeHHeM yqaCTKOB, HaxO.D.Hru;HXCH BHe 06.'laCTH orrpe-
,lI;eJleHHH CPYHKUHH.
fpacpHK CPYHKUHH Ha BbI6paHHOM yqaCTKe H3MeHeHHH apry-
MeHTa lI-!OmeT HMeTb BH.D. HenpepbIBHOH JJHHHH; TaKOBbl rpa-
cpHKH CPYHKUUH B 3aJJ.aqaX 1, 2 H 4 npe;I.wecTBYlOru;ero napa-
rpacpa (qepT. 34).
9 9
'40
120
lOG
130
6fJ
1,0
,(J,
01231,55
{I
{I I
0
0
'0'
u- - i
II ,
1000
500
7 {j A
4epT. 35.
I
42/ IrZZ 1r2J
It
B ,lI;pyrnx CJIyqaS!x, Kor,ll;a aprYMeHT x no CMblCJIY 3a,II,aq
He l\lOmeT npHHHMaTb Bcex 3HaqeHHH Ha OTpe3Ke [a; b I,
a TO.1bKO HeKoropble H3 HHX (3a,ll;aqH 3 H 5 npe,ll;wecTBYlOru;ero
y
I 18 II I I
I 7 I I
'6 I I I
5  I
--1 \ ; , I
I \ i I I
I 2 I i'-..., !
-5,-413;-21-/ I I I I 1-:---
-- 10 1, Z, J' 4' 5, O' 7 8,
, --
. :--....i I I
, j-Z I
-3 \ I
\
K
4epT. 36.
naparpacpa), rpacpuKO\I CPYHKUUH 6Y,ll;eT cOBoKyrrHocTb OTJJ.eJIb-
HliIX TOl.!eK (4epT. 35).
ECJIH Ha oTpe3Ke [a; b 1 H3MeHeHHH aprYMeHTa x CPYHKUHSI
cyru;ecTByeT npH Bcex 3HaqeHHSlX x, KpoMe 3HaqeHHSI x=m,
114

TO H3 rpacpHKa BbIITa.D,aeT TOqKa C a6CUHCCOH m H caM rpacpMK
KaK 6b1 pacna,ll;aeTCSl Ha OT,lI;eJIbHbIe qaCTH. TaKHM, HarrpliMep,
6Y,lI;eT rpacpHK CPYHKUHH
4
Y = x- 2'
4
TaK KaK npH x=2 3HaMeHaTeJIb ,lI;p06H x- 2 npeBparu;aeTcSI
B ° M, CJ1e.n.oBaTeJIbHO, npH 3HaqeHHH aprYMeHTa, paBHoM 2,
CPYHKUHSI He cyru;eCTByeT (qepT. 36).
B03MomHbl H rpacpHKH BI-I,lI;8, YKa3aHHoro Ha qepTe>Ke 37.
J
1;
II
liepT. 37.
3aKOH COOTBeTCTBHSI B 9TOM c.1Y4ae TaKOB:
I x npM x < 0;
y = + npH x = 0;
x+- 1 npH x>O.
B 9TO1 c,'lyqae, XOTR aprYl\leHT x H I\I0meT npHlUll\faTb ,'IIO-
ObIe 3HaqeHHSI H3 l\IHOmeCTBa .D,eikTBHTeJIbHblX 4 HCeJI , Bee me
rpacpHK CPYHKUHU He 6Y,ll;eT HenpepblBHOH JIHHHeH.
TOT cpaKT, qTO K3>K.D,OMY 3HaqeIlUlO x 113 MHO)KeCTBa 1\1
CooTBeTcTByeT e.D,HHCTBeHHoe 3HaqeHHe y=f (x), Hl\IeeT npocTYlO
reOl\leTpHqeeKYIO H.'J,TJlOCTpaLlIlIO: Ka)K.D,aH n pS!I\laH, rrapa,TJ,TJ e.rJ b-
HaS! OCII OY, nepeceKaeT rpacpHK CPYHIU{HH t (x) He 60JIee qeM
B 0,'!,H011 TOqKe.
115

TpY,lI;HO rrepeoUeHHTb 3HaqeHJie rpacpHKOB CPYHKUHH B
HaYKe H >KH3HH. OHH HrpalOT HCKJIlOqllTeJIbHYIO POJIb B MaTe-
MaTlIKe, eCTeCTBeHHbIX HaYKaX, B HaYKaX 3KOHOMlIqecKHX, B
TeXHHKe, rrpOH3BO,ll;CTBe, B CeJIbCKOM X03S1HCTBe - BClO,lI;Y H
Be3.n.e, 06JIerqaH, HJI.nlOCTPHpySI Ii YCKOPHH peWeHHe B03HH-
KalOIUHX BOnpOCOB, KaK HayqHbIX, TaK If rrpaKTHqecKUX, HHOr-
.na TpY,lI;HOpa3pellIHMblX 6e3 nOMOIUli rpacpHKOB.
«Ha rpacpUKe CPYHKUlIlI,- rrHWeT A. 51. XHHqHH,- MbI
GaCTO HenOCpe,ll;CTBeHHO BlI,lI;HM TaKlie qepTbi B ee rrOBe,ll;eHHH,
KOTOpble JIlIWb ueHOH ,lI;.nllTe.nbHbIX rrO,ll;CqeTOB MO>KHO 6b1.'l0 6bI
YCTa, HOBlITb HCCJIe,ll;YH ee aHaJIlITHqeCKOe BbIpameHlIe lIJIH CO-
CTaB.T1eHHbIe JJ.JISI Hee Ta6m-IUbl».
BOT nepe.n HaMH rpacpliK CPYHKUliH (qepT. 38)
2x
y = x 2 + 1 .
OqeHb BamHO Yl\IeTb BbISlBlITb cBoikTBa CPYHKUHlI, HMeSl
nepe.n C060R ee rpacpHK.
!J
-
-6 -5 -.;0 -3 -2 I 0/ "'f--L
'./ I 2 3  5 6
r--.
-
x
4epT. 38.
4T06bI YMeTb qHTaTb CBoHcTBa ,lI;aHHOH CPYHKUHH rro ee
rpacpHKY, HymHo HMeTb B BH,lI;Y c.ne,ll;YlOru;ee:
1. 3a,ll;aHHOe 3HaqeHHe aprYMeHTa 6epeTcSI Ha OCli x; COOT-
BeTcTBYIOru;ee 3HaqeHlIe CPYHKUliH YKa3bIBaeTcH Op,ll;lIHaTOH TOR
ToqKH rpacpllKa, a6cUHcca KOTOpOH paBHa 3a,ll;aHHOMY 3Haqe-
HHIO aprYMeHTa.
2. EC.nH rpacpHK CPYHKUHlI CJIeBa HarrpaBo nO,ll;HHMaeTCH,
TO 3TO rOBopliT 0 B03pacTaHuH CPYHKUUU C B03pacTaHueM apry-
MeHTa, eCJ1H rpacpHK CPYHKUHlI C,TJeBa HanpaBO onYCKaeTCH, TO
CPYHKUHH Ha 3TOM OTpe3Ke H3MeHeHHH aprYMeHTa y6b1BaeT.
3. EC,'lH CPYHKUHSI Ha KaKOM-JIlI60 OTpe3Ke [a; b] apry-
MeHTa, 6Y,lI;yqH HenpepbIBHOH, OT B03paCTaHliSI nepexo.n.HT K
y6bIBa, HHIO TO rpacpUK ee Ha nOM yqaCTKe ,lI;OJImeH CHaqaJIa
CJIeBa HanpaBO nO,ll;HlIMaTbCH, ,lI;OCTurHYTb TOqKH, HaHBbIcweH
116

no cpaBHeHHlO C 6J1IimaHllIHMH TOqKaMH rpacpHKa, a 3ae\f
HaqaTb onYCKaTbCH. 3HaqeHHe CPYHKUHH, orrpe,ll;eJlSleMOe Op,ll;H-
HaTOH HaHBbIcweH TOqKH, Ha3bIBalOT MaKClIMYMOM CPYHKUHU
Ha OTpe3Ke [a; b]; a6cUlicca 9TOH TOqKlI roBOpHT 0 TOM 3Ha-
qeHHH aprYMeHTa, rrpll KOTOpOM CPYHKUHH OCTHraeT MaKCU-
MYMa. .
ECJIH CPYHKUHSI Ha KaKOM-JIH60 OTpe3Ke [a; b] aprYMeHTa,
6Y,lI;yqll HenpepbIBHOH, OT y6bIBaHHSI rrepeXO,ll;HT K B03paCTaHHIO,
TO rpacpUK ee Ha 9TOM yqaCTKe ,lI;OJImeH lIMeTb TOqKY, HallHlI3-
llIYlO rro cpaBHeHlIlO C .D,PyrllMH TOqKaMH 9TOro rpacpllKa, Ha-
XO.n.HIUlIMHCH B HerrOCpe.D,CTBeHHOH !J
6.nll30CTli. 3HaqeHlIe CPYHKUUlI, on-
pe,ll;eMIeMOe Op,ll;lIHaTOH HallHH3llIeH
TOqKH rpacpllKa, Ha3blBalOT MHHHMY-
1\I000f CPYHKUHH HB OTpe3Ke la; b ];
a6CUHcca me 9TOH TOqKH YKa3bIBaeT
3HaqeHlIe aprYl\IeHTa, npH KOTOpmt
CPYHKUHH ,lI;OCTHraeT MHHHMYMB. TaK,
HH)KerroMeru;eHHbIH rpacpHK (4epT. 39)
YKa3b1BaeT, qTO COOTBeTCTBYlOru;aSl
el\lY CPYHKUHSI HMeeT MaKcHMYI\I, paB-
HbIH 5 rrpH x=O, II ,lI;Ba MHHHI\IYMa, -3
paBHbIe 1 rrpH x= 1/2 H x=....:.-.,lp).
4HTaTeJIb ,lI;OJI>KeH 3aMeTHTb, qro
MaKClIMYI\I CPYHKUliH He eCTb He06-
XO.D,HMO HaH60JIbllIee H3 Bcex B03-
MomHbIX 3HaqeHlIH ee, a TaKme, qTO
MlIHHMYM He eCTb He06xo,ll;HMO TaKoe
3HaqeHHe, KOTopoe MeHbwe Bcex ,lI;pymx 3HaqeHHH CPYHKUliH.
4. ECJIH rpacpHK CPYHKUliH Ha orrpe,ll;eJIeHHOM oTpe3Ke H3-
MeHeHHH aprYMeHTB oCTaeTcSI napa.rI.neJIbHbIM ocu x, TO Ha
9TOM OTpe3Ke CPYHKI(HH oCTaeTCH rroCTOHHHOH H He MeHSleTCSI.
5. EC.nH rpacpHK lIJIli qaCTb ero HaXO,ll;HTCSI HHme OCH x,
TO ,lI;JIH onpe,ll;eJIeHHbIX 3HaqeHHH aprYMeHTa CPYHKUHSI OTPH-
uaTeJlbHa, eCJIH BbIwe OCH X,- nOJIO>KHTeJIbHa.
6. Koop,ll;HHaTbI TOqKlI nepeCeqeHHH rpacpHKa C OCblO x
YKa3bIBalOT, rrpH KaKHX 3HaqeHHSlX aprYMeHTa 3HaqeHHe CPYHK-
UHH paBHO HYJIlO; Koop,ll;HHaTbI TOqKH nepeCeqeHHH rpacpUKa
6
5
( 
.3 \
-
2
J , 1\
-21 -, 0 , 2
-
A
lIepT. 39.
1) Ha ljepTe>Ke J\aH rpaqmK 6HKBaJ\paTHoro TpeXIjJIeHa y=
= (x 2 -2)2+ I; Y J\OCTHraeT HaHMeHbllleii BeJIHIjHHbI B TOM CJIYtJae,
eCJIH x 2 -2=O H, CJIeJ\oBaTeJIbHO, x=:!:: 112:
117

C OCblO Op,ll;HHaT rOBOpHT 0 qHC.'IeHHOM 3HaqeHHH CPYHKUHIi
rrpH Hy.neBOM 3Ha4eHHH aprYMeHTa.
7. KpYTH3Ha nO,ll;'beMa HJIH orrYCI<aHIiH rpaqJHKa Ha pa3JIHq-
HbIX yqaCTKaX n03BOJISleT conOCTaBJIHTb CKOpOCTb H3MeHeHIiSl
CPYHI{UHH Ha 9THX yqaCTKax.
8. EC.nH }{a Koop,ll;uHaTHoH ceTKe HaqepqeHbI rpacpIiKH
,lI;BYX CPYHKUHii, TO HMeeTCSI B03MOmHOCTb CpaBHHTb, KaK rrpo-
TeKaeT O,ll;HO HBJIeHIie no cpaBHeHIilO C .n.PYrHM.
BepHeMCH K rpacpHKY CPYHKUHH
2x
y = x 2 + l '
Pa36epHTe rrpH nOMoru;H rpaqmKa H3MeHeHHe CPYHKUHH Ha
oTpe3Ke [-6; +6], H3MeHeHHSI aprYMeHTa, OTBeTHB Ha BO-
npocbI:
1. OpH KaKHX 3HaqeHIiSlX aprYMeHTa CPYHKUHSI nOJIOmH-
TeJIbHa? OTpHuaTeJlbHa?
2. Ha KaKOM oTpe3Ke H3MeHeHHH aprYl\IeHTa CPYHKUHH B03-
pacTaeT?
3. HMeeT JIH ,II,aHHaH CPYHKUHSI MaKcHMYM H I\IHHHMYI\I?
EC.nH HMeeT, TO rrpH KaKHX 3HaqeHHSlX aprYMeHTa H qeMY 4HC-
.neHHO paBHbI HaH60.TJbWee H HaHl\IeHbWee 3HaqeHHSI CPYHKUHIi?
4. 4TO MomHO CKa3aTb 0 CKOpOCTlI H3MeHemIH CPYHKIlHH Ha
OTpe3Ke [-6; +6], H3MeHeHHH aprYMeHTa?
5. OpH KaKOM 3HaqeHHlI aprYMeHTa 3HaqeHHe CPYHKUHH
paBHO 0,5 (oTBeT ,lI;aTb npH6,TIHmeHHbJi't).
6. KaK pacrrOJIOmeHbi T04KH MaKCHI\IYl\Ia Ii MHHUI\IYMa rpa-
cpHKa OTHOCHTe,TJbHO Haqa.na Koop,ll;HHaT?
Ha Bce 3TI1 BonpocbI ropa3;.\0 TpY,lI;Hee 6blJIO 6bl OTBeTHTb,
ec.nH 6b1 nepe,ll; Ha:lHl He 6blJIO rpacpl1Ka CPYHKUHH.
0630p npocTeiirnux CPYHKLl.UH
B 9TOll r.naBe 1\Ib1 OCTaHOBHMCH TOJIbKO Ha CPYHKUHHX,
C KOTOpbII\IH qHTaTCJlb rr03HaKOMHJICH B VII I K,TJaCCe WKOJIbI.
BcrrOMHHM 9TH CPYHKUHH.
I. J1uHenHaSi flJYHKLl.HSI. 3aKOH COOTBeTCTBHH JIHHeHHOH
iPYHKIlHH MomeT 6bITb 3a,ll;aH cpOpMy.nOH
y=kx+b.
OCHOBHoe CBOllCTBO non CPYHKUHH Bbl nO.TJY'JlfTe, ec.nH .n.a,'J.HTe
aprYl\IeHTY x .n.Ba 3HaqeHHSI XJ H X2' HaH,II,ere COOTBeTCTBYlO-
ru;He 3HaqeHlUI CPYHI(UIHI VI II Y2:
118

Yl=kxd-b;
!/2= kx 2+ b
(1)
(2)
H BblqTeTe 1i3 paBeHcTBa (2) no lIaCTHM paBeHcTBO (1):
Y2 - Yl = k (X2- Xl)' (3)
Ha30BeM pa3HOCTb X 2 -X l rrpHparu;eHHe1 aprYMeHTa, .a pa3-
HOCTb !f2 - YI- COOTBeTCTBYIOru;HM rrpHparu;eHueM CPYHKUUU;
TOr.D,a paBeHcTBo (3) Mbl MomeM npOqHTaTb TaK:
OpHparu;eHHSI CPYHKUHH nporropuHOHaJIbHbl COOTBeTCTBYlO-
ru;IiM npHparu;eHHSlM aprYMeHTa; K03cpcpHUHeHToM rrporropUHO-
Ha.nbHOCTH 6Y,lI;eT K03cpcpHUIieHT k rrpu X B <popMy.ne, BblpamalO-
meH 3aKOH COOTBeTCTBHSI ,T]HHeHHOH CPYHKUHH.
3TO CBOHCTBO Ii HBJISleTCSI OCHOBHblM .D,.1H :\HHeHHOH CPYHK-
UHH; HHKaKa HHaSl CPYHKUHS! 9THI\\ CBOHCTBOM He 06.na.n.aeT.
BOT npHMepbI JlUHeHHOH 3aBHcHMoCTH, H3BeCTHbie Bal\! H3
Kypca CPU3HKH:
a) v,=at-!- v o .
r.n.e Vi -- CKOpOCTb paBHOMepHo-nepeMeHHoro ,lI;BHmeHHH B MO-
MeHT t, a - YCKopeHHe, t-BpeMSI H Vo-Haqa.'lbHaSl CKOpOCTb.
6) 1=l o (1+(1.t),
rAe 1 - ,lI;.nHHa I\leTa.n,TJliqeCKOrO CTep)KHS! npH r, 1 0 - Haqa.'lb-
HaSl MIiHa CTep)KHSI npH 0" H (1. - K09cpcpHUHeHT JIHHeHHOro
pacwHpeHHH.
OCHoBHoe CBOHCTBO .nHHeHHoH CPYHKUHH n03BOJIHT YCTaHo-
BHTb BU,lI; rpacpHRa nOH CPYHKUHH.
nYCTb ,lI;aHa CPYHKUH$I
y=kx+b.
llaeM aprYl\!eHTY X pSl.n. rrpm-J3BO,'lbHbIX 3HaqeHHH: Xl' X2' Xs
H T.,lI;. II HaXO,ll;HM COOTBeTCTBYIOIUHe 3HaqeHUS! CPYHKUHH:
Yl, Y2, !l3 H T. ,lI;. OOCTpOUM pH,lI; TOlleK AI' A 2 . A3 C KOOp,ll;H-
HaTaMH: (Xl' Yl), (X2' !/2), (Xs. Y3) H T. ,lI;. (qepT. 40).
00 cBoikTBY JIHHeHHOH CPYHKUHH
Y2 - Yl = k(X 2 -Xl) H Y3 - Y2 = k(x 3 -x 2 ).
OOC.'le;J.HHe paBeHcTBa J\lOmHO nepenHcaTb TaK:
Y2 - YI = k H
X 2 -X\
Ys - Y2 = k.
Xs - X 2
119

OTClO,lI;a
Y2-Y'
XjI-XI
Y3 - !/Z
=-,
X 3 - Xz
!I

!Iz
!fJ
o
1.,
)(2
4ep1'. 40.
x
1.J
HJIH (qepT. 40)
A 2 B _ A 3 C
AIB - A 2 C '
OOCJ1e,ll;HHSI nponopUHSI y'TBep)K)I,aeT, qTO
6AIA2B6A2A3C;
CJ1e)1.0BaTeJIbHO,
LA2AIB = LA 3 A 2 C
H TOqKH AI' A 2 H A3 HaXO,ll;S!TCSI Ha O.D,HOH npSlllloH.
vITaK, rpa<puKOI\! 1HHej.'IHoH CPYHKUHU 6Y.D,eT npHMaS! ,'IH-
HHH. rpacpuK nepeceKaeT OCb Y B T04Ke, Op,ll;UHaTa KOTOpoii
paBHa CB060.D,HOl\lY q.1eHy'b, TaK KaK npu x=O y=b.
Pa36epeM ,lI;Ba rrpHMepa.
1. Wap C ropu30HTa,'lbHOH nJIOCKOCTH, no KOTOpOH OH ,lI;BH-
.M -
ra.'lcS! paBHOl\lepHO co CKOpOCTblO 1 -, nepeXo.l1,UT Ha Ha-
eeK
KJIOHHYlO rrJIOCKOCTb. Ha HaKJIOHHOH rrJIOCKOCTH ,lI;.'IHHOH 4 At
AI
Illap ,lI;BHraeTCSI paBHoMepHo-ycKopeHHo C YCKopeHHel\f 0,5 - .
eeK
OpHHHB Mm.teHT BCTyn.1eHHH mapa Ha HaKJIOHHYIO rr.'IOCKOCTb
3a HYJleBoti 1\I01lleHT, BblqepTHTb rpacpHK CKOpOCTH wapa Ha
oTpe3Ke [--1; 6] BpeMeHH.
BbIS!CHH!\! rrpe)K)I,e Bcero Bpe!\ISI ,lI;BumeHHSI wapa no HaK.TJOH-
HOH nJIOCKOCTH (qepT. 41).
«1>opMYJIa
all
S=- + v!
2 u
120

$' nm
 ----
r
$////&
4epT. 41.
,II 0 5t 2
npH S = 4 At H a = 0,5  ,lI;aCT 4 = ....:...- 2 + t; oTKya
ceK.
t = V 20 - 2::::::; 2,5 (CeK).
<I>opMYJIa CKOpOCTH ,lI;BHmellHSI llIapa Ha lIaKJIOHHOH I1JlOCKOCTH
TaKOBa:
V, = 0,5 t + I.
rpacpHKoM 3TOH CPYHKUHH Ha OTpe3Ke [0; 2,5] 6yeT I1pSlMaSl,
nepeceKalOru;aSl OCh Vi B TOqKe C Op,ll;HHaTOH 1; ,lI;aB t 3Haqe-
HHe 2,5, 110JIyqHM)l.JISI Vi COOTBeTCTBYIOru;ee 3HaqeHHe 2,25.
V t (e';)
'"
J
2
1/ V
 f 2 ]' " 5 6 7 8\
-1
tfUK}
LIepT. 42.
4eprem 42 ,AaeT HCKm.fbIH rpacpHK CKOpOCTH mapa Ha OT-
pe3Ke [-1; 6] Bpel\feHH.
2. Ha qepTeme 43 ,lI;aH rpacpHK pOCTa ,lI;06blqH Hecpm B
CCCP. OpH BbIqepqHBaHHH 3Toro rpacpHKa npHHSlTO, qTO Ha
OTpe3Ke BpeMeHH Mem,ny Ka)!{,lI;blI\Ul ,ABYMSI pS!,lI;OM Ha,ll;nHcaH-
HhIMH Ha OCH BpeMeHH ro,ll;a:\IH ,lI;06bIqa HecpTH B03paCTa.'la paB-
HOl\!epHO. OpOCJIe,ll;HTe no rpacpHKY 3a DOCTOI\! ,lI;06blqH HecpTH.
121

llaCTHbIM CJ1yqael\f .'IHHeHHOH 3aBHCI-fI\fOCTH SlB.'lSleTCH npSlMaSl
npOnOpUHOHa.'lbHaSl 3aBHCHMOCTb. 3aKoH COOTBeTCTBHSI ::nOH
CPYHKLI.HH TaKOB;
y=kx.
OCHOBHOe CBOHCTBO ee:
3HQ'leHUR ifJYHKl(UU npfL\to npOl/OP4UOHa/lbHbl eOOm8em-
emeY1OIl(WI 3HaIJeHURJ.! ap2Y,\leHma; K03ifJifJU/{UeHmOM I/ponop-
lfUOHQ;lbHOcnlU 6yiJem KOacpcpU/(UeHm k I/pU X 8 q;op,tlY,le, 8bl-
paJ/CQIOIl(eii. 3QKOH eoomeememeUR.
r;J.e F - CI-fJla, m - MaCCa
H a - YCKopeHHe.
fpacp1iKOM npSlMOH npo-
nopUHOHaJ1bHoti 3aBHCHMO-
cm 6Y,ll;eT npSlMaSl, npoxo-
;{S!ru;aSl qepe3 HaqaJIO
KoopllHHaT. K03cpcpHUHeHT
paBHbIH OTHOllIeHHIO J!.., rOBopHT
x
OTHoweHHIO OCI-f a6CUHCC (qepT. 44).
qTO
140
no
200
180
160

 140
"l, 120
:t:
100

 80
t
60
40
-,
-\
4epT. 43.
nponopUHOHaJ1bHOCTH k,
o HaKJ!OHe rpacpHKa no
113 qepTe)Ka I\fbI BHllHM,
MO>KHO npHBeCTH OqeHb
MHoro npHMepOB nponopUH-
OHa.'l bHOti 3aBI-fCHMOCTH.
BOT HeKoTOpbIe H3 HlIX:
a) S=vt;
0) C=2..R;
B) F=p.P,
rlle F - CH.'Ia TpeHI-fSl H
P - Bec TeJla,
r) F=ma,
k= tg :l.
Pa30epeIlf llBa npHMepa:
1. B TeopHH pe3aHHSI MeTa.'l,'lOB OOJIbllIOe 3HaqeHHe I-fMelOT
cpopM)"JIbl:
nn
w = 30 '
122
(1)

r,ll;e w - yrJIOB8S1 CKOpOCTb llIKHBa B ceKYH,lI;Y, n - qHCJlO
060pOTOB ero B ceKYH,II,Y, H
r.Dn
v=60'
(2)
r,ll;e v - JIHHeiIHaSl CKOpOCTb B leTp8X B ceKYH,1Y TOqKJf Ha
oKpymHocTH 06pa68TbIBaeMoro npe)J.l\1eTa, D - ,lI;H8MeTp ero
B MHJIJIHl\1eTpax.
x
4epT. 44.
113 cpOpI\lY.'IbI (1) CJIe,II,yeT:
30",
n=.
(3)
KaK nOKa3bIBalOT CPOP)'IYJIbl (1) H (2), Mem,ll;Y (JJ H n, Mem,ll;Y v
H D npH orrpe,ll;e,TJeHHOM n CYllI.ecTByeT npSlMO nponopUHOHaJ1b-
HaSl 3aBHCHMOCTb.
CKOpOCTb pe3aHHSI npe,ll;CTa&SleT C060H CKOpOCTb nepeMe-
weHHSI TOqKH Ha oKpymHocTH o6pa6aTbIBaeMoro npe,ll;MeTa
OTHOCHTe.'lbHO pe3ua, T. e. CKOpOCTb pe3aHlfSl COBna.ll.aeT co CKO-
pOCTblO v. CKOpOCTb pe3aHHSI BO H36emaHHe qpe3MepHoro Ha-
rpeB8HHS! pe3ua HMeeT CBOH npe,II,e.lb1 - B 3aBHCHMOCTH OT
TBep.ll.OCTH o6pa6aTbIBaeMoro npe)J.MeT8, KaqeCTBa pe3ua Ii ,lI;PY-
rHX cpaKTopoB.
OOCTpOHTb rpacpliKH CPYHKUHH
r.Dn
V=-
60
npH n = 180, n = 360 H n = 540,
B3S!B Ha OCH x MacllITa6 ,lI;.'lSl D = I CM
50 ,\1.\1
123

H Ha OCR y MaCIllTa6 JISI [) =
1 CM
.41
2-
ceK
Ha rpacpHKe MbI nOJIyqHM pSlll JIyqeH, BbIXOllSllll.HX H3 pa-
'1aJI"! KOOp,ll;HHaT. CTpOHM rpacpHK ,lI;JISI ,lI;aHHOH CPYHKUHH npH
n= 180.
v( ;;/f )
fO 00
36 0/'!UH
8
6 00
18 0/'!UH
4
Z
o 50/00/50 2002503003501,001,50 .lJf/'!/1J
4epT. 45,
cI>opMYJIa
r.Dn
[)=-
60
rrpHMeT BH.n:
v = 3..D;
eCJII-f D=200 MM, TO v=3.3,14.200=1,88 (  ) ; COe,ll;HHSlSI
eeK
HaqaJIO KOOp,ll;l-fHaT C TOqKOH
(200; 1,88), l\1bI II nOJIyqHM COOT-
BeTCTBYIOru;HH JIyq. Pa36epHTe,
KaK nOCTpoeHbI ,lI;pyme JIY'lH
'1epTema 45.
2. Ha roHKax OllHH BeJIOCH-
ne,ll;HCT npollIe.'1 Bce paccToSlHHe
B 20 KM C nOCTOS!HHOH CKOpO-
t('IOC'
l' 20 KM u
CTblO Itac ' a llpyroH nepBYIO
nO,'lOBHHY nYTH exaJI co CKO-
. 16 KM
pOCTblO Itac ' a BTOpylO no-
K)I
JIOBHHY - co CKOpOCTblO 24 -. BbISlcHHTe npH nOMO-
Itac
llI.H rpacpHKoB, KaKoH BeJIOCHne,ll;HCT npollIM 3TO paccToHHHe
cKopeH.
P e ill e H H e (qepT. 46).
SIf(/'I!
20
5
I
0'
5
R
15
10
4epT. 46.
124

n. 06paTHO npOnOpLl.HOHaJILHaH 3aBUCHMOCTb. 3aKOH co-
OTBeTCTBHSI ,lI;JISI 06paTHO npOnOpUHOHaJIbHOH 3aBHCHMOCTH
TaKOB:
k
Y - -
- x '
r,ll;e k- BeJIlIqHHa nOCTOSlHHaSl, MOrYllI.aSl 6bITb MH OT,lI;e.'1bHbIX
'1aCTHbIX CJ1yqaeB ,TJ106bIM ,lI;eHCTBHTe.'1bHbIM qHC.TJOM. 06,TJaCTblO
Onpe,ll;eJleHHSl CPYHKUHH 6Y,lI;eT Be MHOmeCTBO lleHCTBHTeJIbHbIX
'1HCeJI, KpOMe HYJISI. OpH 3HatI€HHSlX aprYMeHTa, 6J1H3KHX
K Hymo, ,lI;p06b !. Be.'1HKa no a6COJ1IOTHOii BeJIHqHHe; Ha06o.
x
pOT, npH 60J1bIllHX no a6C0J1lOTHOH Be.'1HqHHe 3HaqeHHS!X apry-
MeHTa 3HaqeHHSI CPYHKUHH MorYT 6bITb CKOJ1b yrO,ll;HO 6JIH3-
KH K HYJIlO.
Bbl 3HaeTe MHoro npHMepOB 06paTHO nponopUHOHaJIhHOH
saBHCHMOCTH Mem,ll;Y BeJIlIqIIHaMH. BcnoMHHM HeKOTopbIe
H3 HHX.
1. llJISI ,lI;aHHOH MaCChI ra3a npH nOCTOS!HHOH TeMnepaType
llaBneHHe ra3a MeHSleTCSI 06paTHO nponopUHOHaJ1bHO 06beMY
(3aKOH 130HJISI - MapHOTTa):
p' v=const.
2. ConpOTHBJ1eHHe npoBolla npH npOXOm,ll;eHHH no HeMY
9J1eKTpHqeCKOrO TOKa 06paTHo nponopUHOHaJIbHO nJIOlll.a,ll;H no-
nepeqHOrO CeqeHHSI npoBolla:
I
R=k-.
s
rpacpHKOM 06paTHo nponopUHOHaJIhHOii 3aBHCHMOCTH, KaK
H3BeCTHO qHTaTeJIIO, 6Y,lI;eT KpHBaSl, Ha3bIBaeMaSl paBH060qHOH
mnep60JIOH. Henocpe,ll;cTBeHHoe nOCTpoeHHe rpacpHKoB 06paT-
u k
HO nponopuHOHaJIbHOH 3aBHCHMOCTH y = x npH pa3JIHqHbIX
3HaqeHHSlX k npHBe,II,€T Hac K qepTemy 47, Ha KOTOpOM CnJIOliI-
HbIMH JIHHHSlMH HaqepqeHbl KpHBbIe, cooTBeTcTBYlOllI.He 3Haqe-
HHSlM k>O, nYHKTHpHbIMH - 3HaqeHHSlM k<O. T Ka>K,lI;OH
KpHBOH npOCTaBJIeHO cooTBeTcTBYlOlUee eii 3HaqeHHe k.
Bon po C bl. a) OYCTb nOCTpoeH rpacpHK CPYHKU HH
12
y=- npH x>O; H3 npOH3BOJIbHblX TOqeK ero onYllI.eHbI nep-
x
neH,lI;HKY,TJS!pbl Ha OCH KOOp,ll;HHaT. 4TO MomHO CKa3aTb 0 nJIO-
ru;a,ll;Slx npHMoyrOJ1bHHKOB, OrpaHHqeHHblX a6cUHccOH H 013,l1;H-
125

HaTOH B3S1TOH TOqKf{ rpacpHKa H OC5tMl-I KOOplll-fHaT? HeJ1b3S1
JIH HCnOJIb30BaTb paBH060qHYlO rHnep60JIY ,lI;,T]SI nOCTpOeHHH
pS!lla paBHOBeJIHKHX npS!l\loyro.rIbHHKOB?
6) 4eM 6YllYT OTJIHqaTbCS! rpacpHKH CPYHKUHH
k
y= x+a
k
H Y=x=a
OT rpacpHKa CPYHKUHH
k
!J=x
npH O,ll;HOM H TOM )Ke 3HaqeHHH k?
y
r ! ! "II
r . 11,1
=ttttt ':
, i I 1m '
i i i 5X' 'II
!. / I
..; /,.?
'_.-J.-_' 1,,/ i't..
=-=-:::::.-.r-"".(f'
x
. .
4epT. 47.
III. KBaJl.paTHaSl $YHKLl.IUI. BbI H3yqa.'lH KB8,l1;paTHYIo
CPYHKUHlO B TaKOH noc.1el10BaTe.'IbHOCTH; CHaqaJI8 6palIlI npo-
cTeHIIIHH ee BH,lI;:
y = ax 2 ,
38TeM nepeXO,ll;H.TJH K BI-I,lI;Y:
(1)
y=ax 2 +e
(2)
126

II, HaKOHeu, pa36I1pa,'lIl 9TY CPYHKUHlO B 06ll1.eM BIllIe:
y=ax 2 +bx+e. (3)
OepBble ,lI;Ba BlIlIa S!B,'lS!lOTCSI laCTHbIMH CJIyqaSll\UI BH.n.a (3).
He OCTaHaB.'lHBaS!Cb nO,ll;p06HO Ha TOl\!, qTO BaM ,lI;O,TJmHO
6b1Tb xopowo 1I3BeCTHO 0 CBOHCTBax 9TIIX CPYHKUHH H HX rpa-
cpHKax, BCnOl\!HHl\! TO.1bKO cne).!,YlOllI.He rrOJlOmeHHS!:
a) fpacpHKOl\! KBa,ll;paTHoH CPYHKllHH S!B.'IS!eTCS! KpHBaSl,
Ha3blBaelaS! nap a 60 .'I 0 H.
6) <l>opl\la napa60,'lbI 3aBHCHT OT a6CO,'llOTHOH BeJIHqHHbl
K03cpcpllUlIeHTa a.
B) OCb napa60JlbI If.'IH COBna,ll;aeT C OCblO 0PlIHHaT (CJIY-
qaH 1 II 2). HJIH napa.'I.'le.1bHa eH (C,TJYIWll 3).
r) llJISI nOJIyqeHlHI CPOPI\IY,'lbl KOOPlIHHaT BepWHHbl napa-
60JIbI cpOpMY,TJY KBa,ll;paTHOH CPYHKUHH rrpeo6pa3YlOT TaK:
y = ax 2 + bx + e = a [X2 +  x +  ] =:II
[ b b 2 b 2 C ]
=a x 2 + -x+--;---+- =
a 4a 2 4a 2 a
= a [(x + ;a r - b 2 24ac ] = a (x + ;a r - b 2 7a 4ac -
( b ) 2 4ac _ b 2
=ax+ 2a + 4a '
TOr,ll;a KOOp,ll;HHaTbI BepwHHbI A napa60JIbi 6YlIYT:
A ( _.!..
2a'
4aC-b 2 )
4a .
,lI;) TaK KaK B BepWHHe napa60JIbI rpacpHK KBa,ll;paTHOM
CPYHKUHH ,lI;OCTHraeT CBoeM HaHBbIcweH HJIH HaHHII3weH TOq-
KH, TO KOOp,ll;HHaTbI BepWHHbI napa60JIbI ,lI;aIOT TaK)Ke MaK-
CHMYM HJIH MHHHMYM
4ac - b 2
4a
STOM CPYHKUHH, H TO 3HaqeHHe aprYMeHTa (- ;a )' npH KO-
TOpOl\! CPYHKUHSI ,lI;OCTHraeT CBOerO HaH60.'Ibwero I1J1H HaH-
MeHhlUero 3HaqeHIIfl.
127

e) KBa.npaTH8S1 YHKUHSI HMeeT M8KCHM}'M, eCJUI a <0, Ii
MHHHMYM, eCJ1H a >0.
>K) He MeWaeT TaKme YMeTb 6bICTpO. 6e3 COCTaBJIeHHSI
COOTBeTCTBYlOllI.HX TaO,TJHqeK ,lI;.'ISI x H y, CTpOHTb rpacpHK
q>YHKUHH
y = ax2
npu npoCTeihm-Ix 3HaqeHHS!X a. BOT rpacpHKH ::noH CPYHKlI.HU
I I
npH a = I, a = 2, a = 2" H a = T (qepT.48)
,
g
1/
y
\ :/1\ / \ J
\ / \ if \ II \'
\ II 1\ / "\ V
\.. / \ J \ ./ " V
o K
a-I
o
0=2
x 0
D,
0=2
x
o
,
G=;;
x
4epT. 48.
MO>KHO npHBecTH 60JIbIllOe KOJJHqeCTBO npHMepOB Ka.n-
paTHoH CPYHKUHH. YK8meM Ha HeKoTopbIe H3 HHX:
1. OJIoru;a,ll;b KBa,ll;paTa co CTOpOHOH x
y = X2.
2. OJIoru;a,ll;b Kpyra C pa,ll;Hyco1 R
S = 1tR2.
3. nYTb, npoH.n.eHHblH TeJIOM paBHoMepHo-nepeMeHHbIM
.nBH)KeHHe1
ot 2
S = vol + 2'
4. CMJIa conpOTHBJIeHHSI B03,l1;yxa ,lI;BHmeHHIO Te.1a npH
60JIbWHX CKOpOCTS!X -
F = kv 2 .
128

5. KHHeTHqeCKaSI 3HeprHSI ,lI;BHmyru;erocSI TeJIa
mv'
Ek = -r'
r)].e m - Macca TeJIa H v-ero CKOpOCTb.
6. llBH)KeHHe TeJIa, 6poweHHoro no)]. yrJIOM K ropH30HTY
C HaqaJ1bHOH CKOpOCTblO V o
y = tgcx.x - g .x 2 .
2v. cos 2 :z
7. OOBepxHocTb wapa C 5/'1
pa,ll;HycoM R
S = 41tR2.
BcnOMHHTe eIue npHMepbI
KBa,ll;paTHOH CPYHKUHH.
3a)l.a4a t. BhIqepTHTb
rpacpHKH nYTH, CKOpOCTH H
YCKopeHHSI CBOOO,ll;HO na-
)].alOllI.erO TeJIa B 3aBHCHMO-
CTH OT Bpel\leHH t, eCJIH
V o =0.
3aKoHbi ,lI;BHmeHHSI B
5TOI\I cJIyqae TaKOBbI:
ZOO
/60
120
80
40
o
6 t(efK}
!:!40
;:)
gt'l. v =gt
S=2'
Z 3 4 5 6 t /celf/
Ii g = 9,81 AI 4epT. 49.
eeK'l. .
OpHMeM llJIH ynpOlll.eHHSI nOCTpoeHHSI rpacpHKOB YCKope-
.M
Hlie CH.rIbl THmeCTH paBHbIM 10 _ 2 ' Tor,ll;a pellIeHHe 3a.n.aqH
eeK
6Y,ll;eT BblrJISllleTh TaK, KaK nOKa3aHO Ha qepTeme 49.
3a)l.a'la 2. ll,TJSI TeJla, 6pollIeHHoro BepTliKaJ1bHO BBepx C
HaqaJ1bHOH CKOpOCTblO vo=30, nOCTpoHTb rpacpHKH pac-
eeK
CTOSlHHS! S TeJIa OT 3el\lJIH, CKOpOCTH H YCKopeHHSI 3a Bce Bpel\UI
,lI;BHmeHHH B 3aBlfCHMOCTIf OT BpeMeHH t.
5 A. A. KO.'OCOB
129

ECJIH rrpHHSiTb HanpSlmeHHe CH,TJbI TSI)KeCTIf g paBHblM
.M
10  H He npHHHMaTb BO BHHl\l8HHe conpOTHBJIeHHSI B03-
eeK
P,yxa, TO 3aKOHbi ,lI;BHmeHHS! B ,lI;aHHOM CJIY'-Iae 6Y,lI;YT TaKOBbI:
H
S=30t - 5t 2
v=130 - lOti.
(1)
(2)
Onpe,ll;e.'IS!eM BpeMS! llBHmeUHSI TeJ1a BBepx H BHH3. LI.J1S1 9TOro
nOJIaraei\! B cpopl\fY,'le (2) v=O (loMeHT OCTaHOBKH TeJIa HaBep-
xy); nOJIyqHM ypaBHeHHe OTHOCHTeJIbHO t:
5 (N)
D
IJ
u/j,j,}
JO I
2 J
5
6 Ilu"}
/0
o
y(/lR ,j 1 l
- I
{Vr-i
'V, -
I
J 
S (; I fceo)
o
" ;.>
5
J 4
6 I fc:.}
10
4epT 50.
0=30 - lOt.
OTClO,lI;a
t=3 (eeK).
Bce Bpel\UI ,lI;BHmemISI TeJIa
6Y,ll;eT paBHO 6 eeK, H, CJIe-
,lI;OBaTeJIbHO, 06J1aCTblO on-
pe,ll;eJIeHHSI CPYHKUHH (1) H
(2) 6y.n.eT OTpe30K [0; 6]
a p rYl\leHTa t.
PeweHHe nOCTaBJIeHHOH
3a,ll;aqH 6YlleT Hl\IeTb BH,lI;,
KaK nOKa3aHO Ha qepTe-
me 49.
4epTe)K 50 ,lI;aeT HaM
nOJIHYlO KapTHHY H3yqae-
Moro 5IBJIeHHS!, H no Hel\lY
I\IbI MO)KeM OTBeTIfTb Ha pS!,lI;
BonpOCOB, HarrpHMep:
a) 4TO MomHO CKa3aTb
o Bcex Tpex CPYHKUHS!X npH
3HaqeHHif aprYl\IeHTa, paB-
HOI\I 3?
aprYl\IeHTa CPYHKUIHI (1) H3-
6) B6m!3H KaKHX 3HaqeHHH
l\IeHSieTCSI MeA.'leHHee?
B) KaKaS! H3 llBYX CPYHKUHH (1) H (2) Ha oTpe3Ke apry-
MeHTa [0; 3] B03paCTaeT H KaKaS'! y6bIBaeT? If np.
3a)l.a'ia 3. OOCTpOIfTb rpacpHKH CPYHKlllfH (Ha OAHOM lIep-
1eJ-Ke):
130

y= -x 2 +4x+4;
y= - x+6
(1)
(2)
H no qepTeLKY OTBeTHTb Ha BonpOCbI:
a) KaKaSl H3 CPYHKUHH HMeeT MaKCHMYM, qeMY OH paBeH
H npH KaKOM 3HaqeHHH aprYMeHTa?
6) OpH KaKHX 3HaqeHHSlX aprYl\1eHTa <PYHKUHSI (1) qi1c-
JIeIfHO 60.'lbWe CPYHKUHH (2)?
B) OpH KaKHX 3HaqeHHHX aprYMeHTa CPYHKUHH qHCJIeHHO
paBHbI HY.llO?
3a.ll.aIJa 4. lloKa3aTb, qTO H3 Bcex npSlMoyroJIbHHKB,
HMelOru;HX ,lI;aHHbIH nepHMeTp, HaH60JIbWYIO nJIOI.I..l,a,ll;b HMeeT
KBa,ll;paT.
OYCTb nepHMeTp npS!l\IoyrO,'lbHHKa ABCD paBeH 2 p.
ECJIH AB=x, TO AD=p-x (qepT. 51), H nJIoru;a,ll;b ABCD,
KOTOpylO MbI otr03HaqH:\1 qepe3 y BbIpa3HTcSI TaK:
y=x (p - x)
p-x
:D C
fI 0
H.'IH
y= - x 2 +px.
Oepe,ll; HaMH KBa,ll;paTHaH CPYHKUHSI.
OHa HMeeT MaKCHMYM, TaK KaK K09cp-
cpHUHeHT a<O. 3HaqeHHe MaKCHI\IYl\Ia
H 3HaqeHHe aprYMeHTa, npH KOTOpOM OH HacTynaeT, onpe-
,lI;elS!IOTCH Koop,ll;HHaTaMH BepWHHbI napa60JIbI, CJIymaLLl.eH
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3apmK,lI;eHHe H pa3BHTlIe KOTOpOH ,lI;HKTOBaJIHCb Tpe60BaHHSlMH
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MaTel\!aTHKOB H cpHJIOCOcpoB yqaCTBOBa.TJa B C03,l1;aHHH HOBOH
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MeTpHH, ,lI;HcpcpepeHllHa.rIbHOrO H HHTerpaJIbHOrO HCqHCJ1eHHH,
,lI;HcpcpepeHUHaJIbHOII reOl\leTpHH - H pa3pa60TaHbl HOBbIe Ma-
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3TY rJIaBY MbI 3aKOHqHM KpaTKHM onHcaHHeM mH3HH H
,lI;eSlTeJIbHOCTH O,ll;Horo H3 Ha3BaHHbIX BbIwe yqeHbIX - PeHe
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qaeM nOHSlTHSI nepeMeHHOH BeJIHqHHbI H CPYHKUHH.
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>Ke HMeHHO ero Mbl ,lI;OJ!mHbI CqHTaTb OCHOBaTeJIeM HOBOH Ma-
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ero pa3Hoo6pa3HH H n<tCTOSlHHOH H3MeHSleMOCTH nepeBeWH-
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rr03BO.rISllOru;HM OTpa3HTb npOTeKalOru;He BOKpyr Hac npoueCCbl.
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cyra nOCBSlru;aTb peweHHIO MaTeMaTlfqeCKHX BonpOCOB, HO 9TO
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OTHOCHTCS! K MaTeMaTlIKe, Bce paBHO, 6Y,lI;YT JIH OHH HCKaTb
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me 06yqaJIH H MaTeM8THKe; B qaCTHOCTH, aJlre6py OH H3yqaJI
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133

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ero npe6bIBaHHSI B WKo.TJe He3YHTOB. HaYKa Toro BpeMeHH
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YBepHJIH, qTO OHa ,lI;aCT SlCHoe H BepHoe n03HaHHe Bcero, qeM
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poro 06bIKHOBeHHO CqHTalOT 3aqHC.rIeHHe B pHAhl yqeHbIX, !\lOH
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yqeHHKOI\I 3HaMeHHTeHweH WKO.TJbl EBponbI H nO.'laralO, qTO
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,lI;PyrHe, HO, He Y,lI;OB.TJeTBopS!S!Cb 3THM, SI npOqHTa.TJ Bc.e, KaKHe
TOJ!bKO MOrJIH nonaCTb MHe B PYKH KHHrH 0 npe,ll;MeT2!X, cqH-
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rOC,TJOBHH, He Mor.TJH y.n.OBJIeTBOpHTb ero nbIT,'lHBbIH YM. na>Ke
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HaqHHaeTCSI nepHO,ll; H3yqeHHSI «BeJIHKOH KHHrH BCe.'leHHOH»
B nOHCKax BepHoro nYTH, KOTOpO!llY OH .n.OJ!meH C.'le,ll;oBaTb.
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KpOl\Ie TOll, KOTOpylO S! Mor HaHTIi H.TJH B ce6e C31\10M, H.'IH
B Be.'lIiKOH KHHre Bce.'leHHOH, S! yrroTpe6H.'l OCT3TOK cBoeH
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B apMHSlx, 06paru;aHcb C JIlO,lI;bMH pa3JIHqHbiX 06ru;ecTBeHHblx
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1) MepceHH, .L\pyr .QeKapTa, BOlle.'! B HCTopmo MaTeMaTHKH
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134

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HUH npu06pecTu HeKoTopylO onbITHOCTb SI peWUJICSI, HaKoHeu.
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TO B pe3y.TJbTaTe BCTpeqH CO CBOHM ,lI;pyroM MepceHHOM OH
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co.TJ,lI;aT B apMHH MopHua OpaHCKoro, KY,ll;a CTpeMSlTcH MHorue
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nmior eMY.
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rpynny npoxmKUX, co6paBWHXCH nepe,ll; HaKJIeeHHbIM Ha CTeHe
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qHTaTb TOro, qTO 6blJJO HanHcaHO Ha n.rIaKaTe,- neKapT
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nOma,'lYH, nepeBe,ll;eT eMY TeKCT 3a,ll;aqH, eC.TJU neKapT B03b-
MeTcSI ee pewIITb. He3HaKoMeu OKa3a.TJCH npocpeccopoM MaTe-
l\1aTHKH BeKMaHoM. KaKoBo >Ke 6bWO Y,lI;HB.TJeHHe BeKMaHa.
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CTBOM npocpeccopa BeKMaHa. 3TH 3aHSlTHH npO,ll;OJlmaJIHCb
B TeqeHHe .n.BYX .1eT.
na.rIee neKapT B nOUCKax BoeHHoro onblTa npHHHMaeT
yqaCTHe B 6HTBax HaqaBWeHCS! TpH,lI;uaTHJIeTHeH BOHHbI. CY,lI;b-
6a KJr.n.aeT ero TO B BaBapHlO, TO B BoreMHlO, TO OB cpmKaeTCS!
no,ll; OparOH. HaKoHeu, YCTaB OT BeqHOH CYTO.TJOKH BoeHHoH
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25 .TJeT. Ka3a.TJOCb 6bI, qTO YC.'lOBHSI BoeHHblX nOXO,ll;OB He
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CXOJIaCTHqecKOH yqeHOCTH H He06xo,ll;HMOCTH pecpOpMbI HaYKH,
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06'be,ll;HHHBllIHXCH BOKpyr MepceHHa, neKapT paCTeT KaK
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He3YHTOB H 60rOCJIOBOB, npOTHBHHKOB ero cpl.I.TJOCOcpHH H yrpo-
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,lI;OBaHHlO H pa3pa60TKe CBoero cpHJIOcocpcKoro yqeHHH. B
1629 r. OH KOHqaeT nepBYlO CBOIO pa60TY «OpaBH.TJa ,lI;JIH PYKO.
BO,ll;CTBa pa3YMa». B 1630-1633 rr. neKapT pa60Tae'f Ha,ll;
«TpaKTaToM 0 rvmpe», B KOTOpOM BbIpameHbl ero B3r,TJS!.n.bI
o CTpoeHHH MHpa. B 1637 r. B J1eH,lI;IiHe BbIllI,TJO aHOHHMHO ero
HOBoe npOH3Be,ll;enHe, nOCJIymHBllIee KpynHeHllIefi BeXOH B
HCTOpHH HaYKH H cpHJIOCOcpHH. 3TO COqHHeHHe cO,ll;ep>KaJIo
«nHonTpHKY», «MeTeopbl» H «feoMeTpHIO». HaKoHeu, B 1644 r.
BblXO,ll;51T B CBeT «OpHHulmbl cpHJIoc()(lmH», B KOTOpblX yqeHHe
neKapTa B ue.rIOM nOJIyqaeT CHCTeMaTHqeCKOe H3.TJO)!{eHHe.
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pa60Tbl neKapTa HaBJIeKJIH Ha Hero npecJIe,ll;OBaHHe UePKOB-
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meHHIO PYKOH naJIaqa.
neKapT 6blJI O,ll;HHM H3 TBOPUOB HOBOH HaYKH. C03,l1;aHHaH
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nYHKToM B pa3BHTHli MaTeMaTHKH, Hero BKJ!a.n. B pa3BHTHe
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MaTlJK Dbep <!>epla.
136

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reOMeTpHqeCKHe 06pa3bl B aJIre6paHqeCKHX ypaBHeHHS!X H Tel\l
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MomHO C03,l1;aTb npaKTHqeCKYlO, rrpH nOMoru;H KOTOpOH, 3HaH
3aKOHbI npHpOl1.bl TaK me OTqJIHBO, KaK MbI 3HaeM pa3Hbie
peMeCJIa HallIHX MaCTepOB, Mbl 6Y,lI;eM B COCTOS!HHH npHMeHHTb
HX TaKHM me 06pa30M KO BCS!KOMY ,lI;e.TJY, K KOTOPOMY OHH
npHrO,ll;HbI, H CTaTb KaK 6bl roCnO,ll;al\lH H B.TJa,ll;eTeJISlMH npH-
pO,ll;bI».
B KOHue 1649 r. neKapT no npHrJIallIeHHIO llIBe,ll;cKoH KO-
pOJIeBbl XpHCTHHbI nepee3)!{aeT B CTOKrOJIbM. Ho nepBaH me
ceBepHaSl 3HMa ry6HTe.TJbHO nO,ll;ejfcTBOBaJIa Ha cJ1360e 3,l1;O-
pOBbe neKapTa; OH YMep B cpeBpaJIe 1650 r.
JIl1TEPATYPA
K. - C. B a p bl 6 fI H, I1>YHKIJ.JJJJ JJ JJX rpa4>JJKJJ, «MaTeMaTJJKa
B llJKOJIe», 1952, ,N'g 6.
B. n. III e p e MeT e B C K JJ H, O'-lepKJJ no JJCTOpJJJJ MaTeMaTflKH,
Y'-lnerH3, 1940.
C. 11. HOB 0 C e JI 0 B, AJIre6pa H 3JIeMeHTapHble 4>YHKIJ.JJJJ,
Y'-InerH3. 1950,  111, 113 JJ p.
B. JI. f 0 H '-I a p 0 B, BbI'-IHc.!JHTeJIbHbIe H rpa4>JJ'-IeCKJJe ynpalli-
HeHHH C 4>YHKIJ.JJOHaJIbHbIM coeplliaHHeM, H3. AUH PCI1>CP, 1948.
A. 51. X JJ H '-I H H, 8 JIeKIJ.JJH no MaTeMaTH'-IecKoMY aHaJIH3Y, foc-
TeXH3aT, 1948.
11. n. HaT a H C 0 H, npOCTeHllJJJe 3aa'-lH Ha MaKcflMYM H MH-
HH:\,IYM, [oCTexfl3aT, 1950.
C. 11. 3 e T e JI b, 3a,I:\a'-lfi Ha MaKCHMYM H MHHHMYM, [ocTeXH3,I:\aT,
1948.

r 0'1 A B A VI
H3MEPEHHE nJIOIlJ,A)J.EfI
«Bce, 'ITO 6e3 Toro 6bJJIO TeMHo, COM-
HHTe.'1bHO H HeBepHO, MaTeMaTHKa cJ].e-
JIaJIa HCHblM, BepHbIM H O'leBH,I:\HbJM».
JlOMOHOC08
KaK H3MepSlJlU nJlOll\aJl.H B CTapH"Y
K a qepTe>Ke 52 ,lI;amSl n.TJaH llIKO.TJbHOrO yqaCTKa:
J -3,l1;aHHe llIKO.TJbI; 2 - cpH3Ky.TJbTYPHaSl n.TJoru;aJJ.-
Ka; 3 - CPPYKTOBblH ca,ll;: 4 - llIKO.TJbHbIH oropo,ll;:
5 - capaH. Onpe,ll;eJlHTb n.'Ioru;a,ll;b Bcero llIKO.'lbHOrO
yqacTi<a, n.TJoru;a.n.H OT,lI;eJIbHbIX ero qaCTeH: 1, 2, 3, 4, 5 - H
n,'loru;a.n.b CB060,ll;HOH 3eM,'IH. nJIS! onpe,ll;e,TJeHHH n.10ru;a.n.H Bce-
ro yqaCTKa ero npH,lI;eTcS! pa36HTb Ha qaCTH, n.TJoru;a,ll;H KO-
TOpbIX MbI YMeeM H3MepSlTb. Bcero y,ll;06Hee C,lI;eJIaTb TaK, KaK
9TO YKa3aHo Ha qepTe)Ke 53. Tor,ll;a BblqHCJleHHe n.TJoru;a,ll;eH
I, II, III, IV npH nOMoru;H H3BeCTHbIX HaM cpOpMy.TJ n.TJoru;a-
,lI;eiI Tpeyro.TJbHHKa H Tpanewm He npe,ll;CTaBHT 3aTpY,ll;HeHHiI.
YqTH ,lI;aHHblH Ha qepTeme 53 I\laCllITa6, Mbi CMOmel\I
OTBeTHTb Ha Bee BonpOCbI, nOCTaB.'leHHbIe B 3a}J.aqe. C}J.e.'IaH-
Te 9TO.
138

<POPMYJIbl, KOTOpblllIH I\lbl nOJIb30BaJIl-lCb, pellIaH .n.aIIHYIO 3a-
.n.aqy, BeCbMa .n.peBHero npOHCXmK,lI;eHHH; yme O.n.HO TO, qTO
9TH CPOPMYJIbl nOMelll.eHbI B «HaqaJ!ax» EBKJIH.n.a, rOBOpHT
o B03paCTe HX He MeHee qel\l 2200 JleT. TpY,lI;HO TOqHO YCTaHo-
BHTb TY rpaHb BO BpeMeHH, Kor,ll;a JIIO,lI;H COBceM He 3HaJIH
npaBH.TJ 1I3MepeHHH n,'loru;a,ll;eH npOCTeHllIHX cpHryp, KaKlIe
Mbl 3HaeM Terrepb H yme 3Ha.TJH B ,lI;peBHeH rpeUHlI BO BpeMeHa
EBK.TJH,lI;a. CO,ll;ep>KaHHe HeKOTO-
pblX H3 ,lI;peBHeHllIHX em neT- 8 {j r:
CKHX nanHpycoB H rJmHSlHblX
,lI;oru;eqeK BaBHJIOHS!H n03BOJISleT
YTBepm,ll;aTb, qTO B 'ere oT,lI;a,'leH-
Hble Bpel\leHa 1I erHnTS!He H
BaBHJIOHHHe 3HaJ!1I 3TH npaBH-
JIa. OpaB,lI;a, HapS!,lI;Y C HHMH
B ErHnTe nOJlb30BaJIHCb H HHbI-
I\IH npaBH.TJaMH, KOTopble n03BO-
,TJH,'IH 6bICTpee H3MepHTb nJIoru;a,ll;b 3el\leJIbHOrO yqaCTKa ny-
TeM TO,%KO 06xo,ll;a no rpaHHIlal\1 ero, XOTS! pe3YJIbTaT H3l\1e-
peHHSI 1I nOJlyqa.rIcS! C HeKoTopoi'i norpellIHocTblO. TaK, B
.npeBHeM EmnTe ,lI;,TJS! BblqHC.TJeHHSI n.rIoru;a,ll;H qeTblpexyrOJIb-
HHKa ABCD (qepT. 54) nOJIb30Ba,'lHCb CPOPMYJIOH:
S = a+b . b+d
2 2'
O ..  :;; .
"//:/ . ,w
- -.
() [) C" 0 l' <-""
o 10 10 30 40 JU ,.,
Nal1//TlOO O.5u' - ION
4epT. 52.
4epT. 53.
o
d
!J
c
11
LIepT. 54.
3Ta cpopMYJIa, 6Y,ll;yqH BepHOH ,lI;JIH npSlMoyrO.rIbHHKa (npo-
BepbTe 9TO), He 6Y,ll;eT BepRa Bo06ll1.e ,lI;,'lH JII06oro qeTblpex-
yroJIbHHKa. BeJIHqHHa OllIH6KH 6Y,II,eT pa3JIHqHOH ,lI;.'lH pa3-
139

JnrqHbIX TIInOB qeTbIpexyroJIbHHKOB. EmnTSlHe npHMeHSlJIU
9TY CPOPMY.'IY ,II,.TJS! BblqHC.TJeHHSI TaKHX qeTblpexyrOJIbHHKOB,
Y KOTOpbIX yr.TJbI 6JIH3KH K npS!MbIM. Ho H3MepSlSl npH nOMOru;H
9TOH cpopMy.TJbI XOTH 6bI n.rIOllIa.n.b npOH3BO,TJbHOro napaJI,TJeJIO-
rpaMMa, Mbl MOmeM nOJIyqHTb 3HaqHTeJIbHbIe nOrpellIHOCTH
(qepT. 55).
00 emneTCKOH cpopMYJIe
SABCD = AB+CD
2
c
c
4epT. 55.
AD+ BC
2
Ho AB=CD H AD=BC; CJ!e,ll;oBaTeJIbHO,
2AB 2AD
SABCD =""2 .=AB.AD.
TaKHM 06pa30M, nOJIb3YSlCb emneTcKoH CPOPMYJIOH, MbI
npHHHMaeM nJIoru;a,ll;b napaJIJleJIOrpaMMa paBHoH nJIoru;a,ll;H rrpH-
MoyrO,TJbHHKa co cTOpoHaMH, paBHbIMH c.TOpoHaM napaJIJIeJlo-
rpaMMa. 3aUITpI1XOBaHHble Ha qepTe>Ke 55 nJIOru;a,ll;H nOKa3bI.
BalOT BeJIHqHHY ,lI;onyru;eHHoH OllIH6KH B onpe,ll;eJleHHH nJIoru;a,IJ.H
fj
4epT. 56.
140
4epT. 57.

napaJIJIeJIOrpaMMa B ,lI;BYX pa3JIHqHbIX CJIyqaSlx. ECJIH yrOJI
A napaJIJIeJIOrpaMMa no Be.'IHqHHe ,lI;aJIeK OT npSlMoro, TO
OllIH6Ka MO>KeT OKa3aTbCH OqeHb 3HaqHTeJIbHOH.
DbIJIa y erHnTHH H npH6.HH>KeHHaH cpopMYJIa ,lI;.rIH onpe,ll;e-
JIeHHH rrJIoru;aJ.!,H paBH06e,ll;peHHoro TpeyrOJIbHHKa.
BOT 9Ta cpopMYJIa (qepT. 56):
s=
2
BblqepTHM H B 9TOM cJIyqae rrorpellIHocTb pe3YJIbTaT2.
CpaBHHM emneTCKYlO CPOPMYJIY C CPOPMYJIOH, H3BecTHOH HaM
H3 Kypca reOMeTpHH:
S =  H S = ab n C1. .
OTCYTCTBHe B emneTCKOH cpopMYJIe MHO>KHTeJIH sin rJ. (qepT. 57)
rOBopHT 0 TOM, qTO em nTH He nOJIyqaJIH pe3YJIbTaT H3MepeHliSl
nJIoru;a,ll;H TpeyrOJIbHHKa 60JIbllIHH, qeM HY>KHO. OllIH6Ka 6Y,ll;eT
paBHa 0, ecJIH sin rJ.=1 HJIH rJ.=90°. OllIH6Ka 6Y,ll;eT MaJIOH,
eCJII-I '.J. 6,'lH30K K 90°, T. e. TpeyrOJIbHHK OqeHb BbiTHHYT H
HMeeT MaJIbIH yroJI rrpH BepllIHHe. 11, ,lI;eHCTBHTeJIbHO, em nTH He
nOJIb30BaJIHCb nOH. npH6JIH>KeHHOH CPOPMYJIOti npH rJ., 6JIH3-
KOM K 90°, Ii He yrroTpe6JIHJIH ee B ,lI;pymx CJIyqaHX.
4HTaTeJIb, KOHeqHO, nOMHHT TaK Ha3bIBaeMYlO CPOPMYJIY
repOHa (II H I BB.) ,lI;JIH onpe,ll;eJIeHHH nJIoru;a,ll;H TpeyrOJIbHHKa
no ero cTOpoHaM. OpHBe,ll;eM _ony CPOPMYJIY:
S = Vp(p-a) (p-b) (p-e) ,
r,ll;e p - rrOJIynepHMeTp TpeyroJIbHHKa, a a, b H e - ero CTO-
pOHbI. I1HTepecHa nOnbITKa HH,lI;HHCKOro MaTeMaTHKa DpaMa-
rYITTbI (598-660) nOJIyqHTb nO,ll;06HYlO CPOPMYJIY ,lI;JIH H3Mepe-
HHSI nJIoru;a,ll;H qeTblpexyrOJIbHHKa. ECJIH MbI 0603HaqHM nJIO-
ru;a,ll;b qeTblpexyrOJIbHHKa qepes S, 'ero rrOJIynepHMeTp - qepe3
p, a CTOpOHbI - qepes a, b, e H d, TO DpaMaryrrTa npHHHMaJI,
qTO
S= 11 (p-a) (p-b) (p-e) (p-d).
<PopMYJIa DpaMarynTbI BepHa ,lI;JISI np51MoyrOJIbHHKa
(qepT. 58).
Orrpe,ll;eJIHH nJIOlll.a,ll;b npHMoyrOJIbHHKa no 9TOll cpopMYJIe,
Mbi nOJIyqHM:
S=1I(p-a)2(p-b)2 =(p-a)(p-b);
141

HO p-a=b H p-b=a, C.ne,ll;OBaTe.rIbHO,
S=ab.
<PopMYJIa I3paMarYIITbl OCTaCTCSI BepHOH H ,lI;JISI paBH06oq-
HOH TpaneUHH (qepT. 59).
8
oDe
d
4epT. 58.
.0
tj
4epT. 59.
B caMOM ,lI;e.rIe:
S = V(p-a) (p-b) (p-e) (p-d) ,
TaK KaK a=e, TO
s = V (p_a)2 (p-b) (p-d) = (p - a) V(p-b) (p-d) .
OpOCJIe.n.IiTe 3a ,lI;aJIbH eHllIHlIIIi npe06pa30BaHHS!MH:
S- ( )V a+e+d-b a+e+b-d
- p-a 2 ' 2
= p-;a V(a+e)2- (d-b)2 =
= (p - a) -V ( ate Y _ ( d / Y'
13 npSlMoyrOJlbHOrO l::,.ABE, B KOTOpOM AE= d 2 b H
a+c_a+a_
T-T-a, HI\leelll:
V ( ate Y - ( d / Y = h.
CJIe,ll;OBaT€JIbHO,
S=(p-a)h= 2 P ;2a . h = a+b+ct i - a - e = b;-d .h.
142

Mbl npHllIJIH K H3BeC1'HOH Hal\! cpopMYJIe nJIOlll.a.IlH Tparre-
UHH. TaKHM 06pa30M, npaBlIJIO H3MepeHHfI rrJIoru;a,ll;H qeTblpex-
yrOJlbHHKa, BblCKa3aHHoe BpaMarynToH, OCTaeTCH CrrpaBe,ll;-
JIHBbiM ,lI;JIfI paBH06e,ll;peHHOH TpaneUHH.
I1HTepeCHO TaK>Ke OTl\IeTHTb, qTO YKa3aHHaSl cpopMy.'la
OCTaeTCSI BepHOH H ,lI;JISI qeTblpexyroJlbHIiKa, BrrHcaHHoro B
OKPY>KHOCTb, ,lI;HarOHaJIH KOToporo B3aHMHO nepneH,lI;HKY-
JIHpHbl (qepT. 60).
Ho OHa He 6Y,ll;eT HMeTb
MeCTa ,lI;,'lS! qeTblpexyrO,'lbHHKa
Bo06ru;e. 3TO JIerKO npoBepHTb,
npHMeHHB cpOpMY,TJY BpaMaryn-
TbI K BblqHCJIeHHlO n,TJOlll.a,ll;H
napa.'l.TJeJIOrpaMl\la. 00np06YHTe
9TO C,lI;eJIaTb.
CaM BpaMarynTa 6b!.'l OCTO-
pO>KeH B rrpHMeHeHIiH cBoeH
cpOpMY,'IbI H nO,TJb30BaJICH elO
TOJIbKO ,lI;JISI onpe,ll;e.'1eHHS! nJIO-
ru;a,ll;eH paBH06e,ll;peHHOH Tparre- 4epT. 60.
UHH H BnHcaHHoro B oKpym-
HOCTb lIeTblpexyrOJIbHHKa C B3aliMHO nepneH,lI;HKYJIHpHblMH
,lI;HarOHa,TJS!!\iH.
AHaJIH3 TO,TJbKO qTO paccMoTpeHHblx npaBHJI H3MepeHHfI
nJIoru;a,ll;eH, KaK emrreTcKHx, TaK H HH,lI;HHCKHX, rOBopHT
o TOM, qTO 9TH npaBHJIa S!BHJIHCb pe3YJIbTaTOM He.n.OCTaTOqHO
060CHOBaHHoro nepeHoca npaBHJI, BepHblX B HeKOTopblX qaCT-
HblX C,'lyqaS!x, Ha 06ru;HH C,TJyqaH. I1HTepecHO, qTO no TOMY }Ke
nYTH He060CHOBaHHblX 0606ru;eHHH nOllIJIH H y Hac, B ,lI;peBHeH
PYCH, B XV-XVI BB., B nOHCKax cnoc060B 1i3MepeHHSI rrJIO-
ru;a,ll;eH. Y>Ke BO BTOpOH nOJIOBHHe XVI B. B03pocllIHe nOTpe6-
HOCTH B H3MepeHHH 3eMJIH, pa3BHTHe apTHJIJIepHHCKoro ,lI;eJIa
H CTpOHTeJIbCTBO B ropo,ll;ax npHBeJIH K He06xo)l,HMOCTH C03,l1;a-
HHSI pYKorrHceH reOMeTpHqeCKOrO cO)l,ep>KaHHSI. TaK, H3BeCTHO,
qTO uapb I1BaH IV B 1551 r. pellIH,,'l rrOCJIaTb nHCUOB «onHcaTb
H cMepHTb rocY,ll;apCTBo». Be3 oc06oro HaKa3a, B KOTOpOM 6bl-
JIH 6bI ,lI;aHbI npaBHJIa H3MepeHHS! 3eMeJIbHbIX yro,ll;IiH, 9Toro
C,lI;e,'laTb 6bIJIO HeJIb3S1, H ecTb HeKoTopble OCHOBaHHH CqHTaTb,
qTO TaKoH HaKa3 CYlll.ecTBOBaJI H qTO COqHHHJI ero «HeKTO
3HalOru;HH reoMeTpHIO C Bblqe'raI\iH rrJIOCKOcTeH», KaK rOBopHT
O,ll;HH H3 HCTOpHKOB XVIII B. K CO>KaJIeHHlO, 9TOT HaKa3 ,lI;O
Hac He ,lI;OllIeJI.
143

OepBOH H3 COXpaHHBllIlIXCSI PYKOnHCeH, B KOTOpblX H3JIa-
ralOTCH npaBHJI.a H3MepeHHSI nJIOlll.8,l1;eH, 6bIJIa «KHHra COllI-
HOro nHCbMa» (1629). B KOTOPO" HMeeTCSI rJIaBa «0 3eMHOM
BepCTaHHH, KaK 3eMJIH BepcTaTb». BOJIbllIHHCTBO reoMeTpH-
qeCKHX CBe,ll;eHHH, }WTOpble Mbl MomeM nOqepnHYTb H3 Ha3BaH-
HOH PYKorrHcH, OTHOCHTCS! KaK pa3 K HHTepeCYlOru;eMY Hac
Bonpocy, T. e. K H3MepeHHIO TI.rloru;a,ll;eH. B Te ,lI;aJIeKHe BpeMeHa,
TaK >Ke KaK H Mbl Tenepb, H3MepHeMble yqaCTKH CJIO>KHOH cpop-
MbI pa36HBaJIH Ha 60JIee npocTble - B BH,lI;e npSlMoyroJIbHIi-
KOB, KBa,ll;paTOB, TpeyrO,TJbHHKOB H TpaneUHH. ECJIH nJIoru;a,ll;b
rrpm.lOyrOJIbHHKa YMeJIH TOr,ll;a onpe,ll;eJIHTb BepHo, TO nJIO-
llI.a,ll;H TpeyrO,'lbHHKa If TpaneUl-m BLlqHC,TJHJIHCb no 3aBe,ll;O!\IO
OllIH60qHbIM npaBHJIaM. TaK, nJIoru;a,ll;b TpeyroJIbHHlm H3Me-
pS!JIaCb nOJIOBHHOH npoH3Be.n.eHHH MeHbllIeH CTOpOlibI lIa 60JIb-
llIYIO, a nJIoru;a,ll;b paBH06e,ll;peHHoH TpaneUHH npHHHl\laJIaCb
paBHoH npOH3Be,ll;eHHlO rrOJIycYMMbl oCHoBaHHH Ha ,lI;JIIiHY 60KO-
BOH CTOpOHbI.
}lpyrHe reOMeTpHqeCKHe pYKonHcH. ,lI;ollIe,ll;llIHe ,lI;O Hac,
TaK)!{e CO,ll;ep>KaT MHoro OllIH60qHOrO B cnoc06ax H3MepeHHH
nJIOlll.a,ll;eH, B qaCTHOCTH B HHX YTBepm,ll;aeTCSI, qTO cpHrypbl
C paBHbIMH rrepHMeTpaMH 3aMbIKalOT H paBHble rrJIOlUa,ll;H.
TpY,ll;HO OTBeTHTb Ha Bonpoc, ,lI;eHCTBHTeJIbHO JIH Tor,ll;a pyc-
CKHe ne 3HaJIH BepHbIX cnoc060B 1i3MepeHHS! nJIOlll.a,ll;eH HJIH Te
npaBHJIa, KOTopble nOMeru;eHbI B pYKonHCSlX, nOSlBHJIHCb B pe-
3YJIbTaTe HCKa>KeHHH nepenHCqliKaMIf rrpaBIiJIbHblX npHeMOB.
EC.rIH npHHHTb, qTO pYKonHCH OTpa>KalOT HCTHHHbIH ypo-
BeHb reOMeTpHqeCKHX 3HaHHH Ha PyClI B XVI B., TO npH,lI;eT.
CSI npH3HaTb 9TOT ypOBeHb 3HaHHH HeBbiCOKHM.
11 Bce >Ke TPY,ll;HO BepHTb 9TOMY, j<or,ll;a BcnOMHHaellIb
o TaKHX HCKJIlOqHTeJIbHbiX no KpaCOTe apXHTeKTypHblX naMSlT-
HHKax Toro BpeMeHIf, KaK c060p BaclfJIHH BJIameHHOro, no-
CTpoeHHbIH B 1553-1560 rr. npH HBaHe fp03HOM PYCCKHMH
«MaCTepal\1H KaMeHHbIX ,lI;eJI» OOCHHKOM, 5IKOBJIeBbIM H Bap-
MOH. HeBo31110>KHO npe,ll;CTaBHTb ce6e, qTO C03,l1;aHHe TaKoro
apxHTeKTYPHoro llIe,ll;eBpa MOrJIO 060HTHCb 6e3 llIHpOKHX reOMeT-
pHqeCKHX 3HaHHH. A pa3Be 9TOT c060p - e,ll;HHCTBeHHblH npe-
KpacHblH naMSlTHHK apxHTeKTypbI Toro BpeMeHH? I1x MHOro,
H Bce OHH rOBopS!T 0 60JIee BbICOKOM ypOBHe reOMeTpHqeCKHX
3HaHHH, qeM TOT, 0 KOTOpOM CBH,lI;eTe.'IbCTBYlOT ,lI;OllIe,ll;llIHe,ll;O
Hac PYKonHCH.
KOHeqHO, 6bIJIH npHqHHbl, KOTopble 3a,!I.ep>KHBa.rIH pacnpo-
cTpaHeHHe 9TH X 3HaHHH. CTOHT TOJIbKO BcnOMHHTb, qTO elll.e
B XV B. l\IaTeMatHqeCKlfe KHHrH BMeCTe C ,lI;PYrHMIf CBeTCKHMH
144

I< HHfal\n-t, BbIBe3eHHbIMH C 3apall.a, n01l.BepraJIHCb 3arrperu;emtIO.
«DoroMep30cTeH rrepe,ll; 60roM BCS!KHH, KTO JI106HT reOMeT-
pHIO»,- rOBOpHTCS! B O,ll;HOM 1-13 noyqeHHH TOrO BpeMeHH.
B 1701 r. neTp I OCHOBaJI B MocKBe «MaTeMaTHqeCKHX Ii
HaBHraTCKHX, T. e. MOpeXO,ll;HO-XIiTPOCTHbIX HaYK llIKOlIY».
B nporpaMMY 9TOfl llIKOJIbI BnepBble 6bIJIH BBe,ll;eHbl apHcpMe-
THKa, aJIre6pa, reoMeTpHH 11 TpHroHOMeTpHH. 3TH HaYKH rrpe-
rrO,ll;aBaJI BbIITHCaHHblH H3-3a rpaHHUbI npo4>ecCop-MaTeMaTHK
aHrJIHqaHHH <POpBapCOH BMeCTe C HeKOTOpblMH «nplipO,ll;HbIMH
PYCCKHMH, a He HeMqHHaMH». OO>KaJIYH, C 9Toro BpeMeHH H
MO)!{HO CqHTaTb. qTO OCHOBbl reOMeTpHH KaK HaYKH rrpOHHKJIH
I< HaM B POCCHIO. I1MeHHo B HaqaJIe XVIII B. no,ll; pe,ll;aKUHeH
<PopBapcoHa 6bIJIO nepeBe,ll;eHo Ha PYCCKHH H3bIK H H3,l1;aHO
HeCKO.lIbKO KIIHr H3 «HaqaJI» EBKJIH,lI;a.
HOBble cnoco6bl 1-13MepeHUSI nJlOl.1\a)l.eH
B npaKTHqeCKHX 3a,ll;aqaX qaCTO npHXO,ll;HTCH BblqHCJISlTb
rrJIoru;a,ll;H cpHryp, OrpaHHqeHHbIX KpHBbIMH lmHHSlMH rrpOH3-
BOJIbHOH cpOpMbl. OpHBe,ll;eM npHMepbI nO,ll;06HbIX 3a,ll;aq:
4epT. 61.
a) Onpe,ll;eJIHTb nJIOlll.aJJ.b nOnepeqHOrO CeqeHHH peKH, eCJIH
npoMepbl rJIy6HH ee ,lI;aJIu pe3Y,'l bTa Tbl, YKa3aHHb1e Ha qep-
Te>Ke 61.
6) Orrpe,ll;eJIHTb n,'lOlll.a,ll;b ocTpoBa rro nJIaHY (qepT. 62,
MacllITa6 0,5 CM -5 M).
B) Pa60TY ,lI;BHraTeJIS! qaCTO onpeJJ.eJIHIOT nyreM H3MepeHHH
nJIOlll.all.H HH,lI;HKaTopHoti ,lI;HarpaMMbl, BbIqepqeHHOH OC06bIM
npH60por.l, Coe,ll;HHeHHblM C ,lI;BHraTe,'!eM. 4epTe)!{ 63 ,lI;aeT O,ll;HY
li3 TaKHX ,lI;HarpaMM.
nJIH onpe,ll;eJIeHHH nJIoru;all.H cpHrYPbI, OrpaHHqeHROH KpHBO-
,TJHHerrHbIM KOHTypOM, CYllI.eCTByeT HeCKOJIbKO cnoc060B, ,lI;alO-
llI.HX pa3JIHqHYlO TOqHOCTb pe3YJIbTaTa H3!\lepeHHS!. Ka>KJJ.bIH H3
145

9tux cnoco60B aHaJIHTHqeCKH Bblpa)l{aeTCH Orrpe,ll;eJIeHHOH CPOp-
MYJIOH: cpOpMy,TlOH npHMoyrOJIbHHKOB, CPOPMYJIOH TpaneUHH
H CPOPMYJIOH CUMrrCOHa.
,,;
4epT. 62.
Ope)!{,lI;e qeM rrepeHTH K YKa3aHHb1M cpopMYJIaM, c,'le,ll;yeT
3aI\leTHTb, qTO B 60JIbllIHHCTBe c,'lyqaeB cpHrypy, OrpaHHqeH-
4epT. 63.
nylO 1<pIiBOJIHHeHHbII\I KOHTypOM, MO>KHO pa36HTb Ha pH,lI;
9JIeMeHTapHblx cpHryp: KpHBO,TJHHeHHYlO TpaneUHlO (qepT. 64),

4epT. 64.
4epT. 65.
4epT. 66.
KpHBOJIHHeHHbJH TpeyrO.'IbHim (qepT. 65) H KpHBOJIHHeHHbIH
cerMeHT (qepT. 66), 1i3 KOTOpbIX ,lI;Be nOCJle,ll;HHe S!BJISllOTCH
QaCTHb1l\1l1 c.rJyqaH!\IH nepBoH.
146

BOT rrpHI\1ep nO,ll;06HOrO pa36HeHHSI (qepT. 67). 009TOMY
ueJIeC006pa3Ho CHaqaJIa n03HaKOMUTbCS! C rrplieMaMH H3Mepe-
HHS! nJIoru;a,ll;eH 9JIeMeHTapHbIx cpHryp. HaqHeM C H3MepeHHSI
n.'IOlUa,ll;U KpHBOJIUHeHHOH TpaneuHli; Bce, qTO 6Y,ll;eT CKa3aHO
06 9TOH cpHrype, MO>KeT 6bITb C He3HaqHTe,TJbHbIMIi H3MeHeHHSI-
MH npHJIO>KeHO U K KpHBOJIHHeHHoMY TpeyrOJIbHHKY H K cer-
MeHTY.
4epT. 67.
{j
II
IJ
4epT. 68:
Pa3,l1;eJIliM CTOPOHY AD TpaneUHH ABCD (qepT 68) Ha n
paBHbIX qaCTeii; nYCTb ,lI;JIHHa Ka>K,lI;oH qaCTIi 6Y,ll;eT h. 00-
CTpOHM BHYTPH KpHBOJIHHeHHOH TparreUHIi n npHMoyrOJIbHH-
KOB TaK, KaK YKa3aHO Ha qepTe)!{e. Tor,ll;a nJIoru;a,ll;b TpaneUHH
ABCD MO>KHO npH6JIH>KeHHO npHHSlTb paBHoH CYMlIIe nJIoru;a-
,lI;eH n npHMoyrOJIbHHKOB. ECJIH BbiCOTbi 9THX npSlMoyrOJIbHH-
KOB 0603HaqUTb COOTBeTCTBeHHO qepe3 Y1> Y2' ..., Yn-l' Y n , TO
nJIoru;a,ll;b KpHBO,TIHHeHHOH TpaneUIiU 6YJJ.eT Bblpa>KeHa CPOp-
MYJIOH:
SABCD = h(Yl+Y2 + ... + Yn-l + Yn)' (I)
3TO U eCTb cpOpMYJIa npm,lOyrO,TJbHHKOB. OJIOlll.allH, 3allITpHXO-
BaHHbie Ha qepTe>Ke 66, YKa3bIBalOT Ha BeJIHqHHY ,lI;onYlUeHHoiI
npH TaKOM H3I\lepeHIiH norpellIHocTu. 3TY norpellIHOCTb 1I1O)!(H0
YMeHbllIHTb, YBeJIWlIfB qHCJIO n ,lI;eJIeHHH CTOpOHbl AD Tparre-
UHH ABCD.
)l\eJIaSl YllleHbllIHTb OllIH6KY nOJIyqeHHOrO pe3YJIbTaTa, l\lbI
lIIO>KeI\1 nOCTpoHTb BHYTPH KpHBOJIUHeHHOH TpaneUUH He npH-
lOyrOJlbHHKH, a TpaneUHH (qepT. 69).
CYI\IMHPYH nJIoru;a,ll;H 9THX TpaneuuH, I\IbI nOJlyqHI\1 BHOBb
rrpH6JIH>KeHHYIO BeJIUqUHY nJIoru;a,ll;H KpHBO.rIHHeHHOH TpaneUHH:
s  (YI+ Y2) h + (Y2+Y:,) h + --'- (Yn-l+YIl) h
 2 2'" I 2
147

HJIH
5 ABCD ::::::; h ( Y; + Y2 + Y3 + . . . + Yn-l + y; ). (II)
OOC.rIe,ll;HHSI cpOpMYJIa, OqeBIi,ll;HO, 6Y,lI;eT 60JIee TOqHOH, qeM
cpopMYJIa npHMoyrOJIbHHKOB, H TOqHOCTb ee MO>KeT 6bITb no-
BbIllIeHa, eC.rIH YBeJIHqHTb n.
<PopMYJIa I I H HOCHT Ha3-
BaHHe cpopMYJIbI TpaneUIiH.
Eru;e 60JIbllIYlO TOqHOCTb ,lI;aeT
CPOPMY JIa CIiMncoHa, aHrJIHHCKOro
MaTeMaTHKa (1710-1761). BbI-
BO,ll; ,:nOH CPOPMYJIbI ITOTprooBaJI
6bI ,lI;OnOJIHHTeJIbHbIX TeOpeTHqe-
CKHX CBe,ll;eHIiH, BbIXO,ll;Hru;HX 3a
paMKH nporpaMMbI BOCbMbIX
KJIaCCOB; n09ToMY MbI ero He
npHBOJJ.IiM. CaMa >Ke cpopMYJIa
TaKOBa:
SABCD""='  (Yl+4Y2+ 2 Y3+ 4 Y4+ 2 Y6+ ... ... +2Yn-2+
+4Yn-1+Yn)'
!I, Y2 'IJ !/. ... _." n. 11 M Y n
4epT. 69.
(III)
06pa'l'HTe BHHMaHHe Ha qepe,ll;OBaHHe K09cpcpliUHeHTOB BHYTPH
clm6KH B cpopMYJIe III.
BOOPY>KHBllIHCb cpopMYJIaMIi I, II Ii II I, BbI CMO>KeTe pe-
IllHTb 3a,ll;aqH nYHKToB «a», «6» H «c». Cpa!3HIiTe pe3YJIbTaTbI
pellIeHHH KaKOH-JIIi6o H3 npe,ll;JIO>KeHHbIX 3a,ll;aq, HaH,lI;H YKa-
3aHHYlO B HeH nJIoru;a,ll;b TpeMH pa3JIHqHbIMH cnoco6aMH.
MeTOJl., HJl.YIl\U" OT ApXUMeJl.a
Y qIiTaTeJISI, npOqHTaBllIero  2 ,lI;aHlOH rJIaBbl, MO>KeT
B03HHKHYTb BnOJIHe OrrpaB,lI;aHHblH Bonpoc: HeY>KeJIH cOBpe-
MeHHaH MaTeMaTHKa C ee HCKJIlOqHTeJIbHO CHJIbHblM BblqHCJIH-
TeJIbHbIM annapaTOM He H3blCKaJIa HHbIX. MeTO,ll;OB H3MepeHHH
nJIoru;a,ll;H KpHBOJIHHeHHOH TpaneuHH, ,lI;alOru;IiX He npH6JIH-
>KeHHbIH, a ToqHbIH pe3YJIbTaT? ECJIIi 9Toro C,lI;eJIaTb HeJIb3S1
,lI;JIS! KpliBOJIHHeHHOH TpaneUIiIi, OrpaHHqeHHOH rrpOH3BOJIb-
HOH KpHBOH, TO KaK 06CTOHT ,lI;eJIO C TpaneuHHMH, OrpaHHqeH-
HbIMH KpHBbIMIf, ypaBHeHHe KOTOpbIX HaM H3BeCTHO? HeY>KeJIH
H B 9THX CJIyqaSlX MbI ,lI;OJI>KHbl npH6erHYTb K cpopMYJIe npR-
MoyrOJIbHHKOB HJIH K KaKHM-JIH60 HHblM npH6JIIi}!{eHHbI:V1
148

crroc06aM H3MepeHHH rrJIoru;a,ll;eii? 4HTaTeJIb, Y KOTOpOrO B03-
HHKHeT TaKOii BOnpOC, nOJIyqHT Y,lI;OBJIeTBOpeHHe, Y3HaB, qTO
,lI;JISI KpHBOJIHHeiiHbIX TparreUHii, OrpaHHQeHHblX KpHBbIMH,
3a,ll;aHHb1MIi ypaBHeHHHMH, TaKOii cnoco6 HMeeTCH Ii qTO B He-
KOTOpbIX npoCTeiiwHX CJIyqaHX 9TOT cnoco6 BnOJIHe ,lI;OCTY-
neH qHTaTeJIIO. BBH,lI;Y llIHpOKOrO rrpHMeHeHHSI ero B MaTeMa-
TIIKe ueJIeC006pa3HO HMeHHO ceiiqac Ha npocTeiiuIHX npHMe-
pax n03HaKOMHTbCH C HHM.
l1,l1;eSl 9TOro MeTO,ll;a TaKOBa.
!J
c
'I
c
0
1 J) 1
a
4epT. 70a. 4epT. 706.
OYCTb KpHBOJIHHeiiHaSl TpaneUHSI ABCD OrpaHliqeHa He.
npepbIBHoii Ha OTpe3Ke [a, b] KPHBOii, 3aJJ.aHHoii ypaBHeHlielll:
Y=t(x) (qepT. 70a). (1)
Y paBHeHHe (1) n03BOJIHT HaM onpe,ll;eJIHTb Op,ll;HHaTY KPHBOii
,lI;JISI 3a,ll;aHHbIX 3HaqeHHii a6CUHCCbI x, eCJIIi TO.rIbKO npH 9TOM
3HaqeHHH x CPYI-1KUHH t(x) cyru;eCTByeT. Pa3,l1;eJIHM CTOPOHY
AD TpaneUHH Ha n paBHbIx qaCTeii H ,lI;JIHHY Ka>K)I,oii H3 HHX
0603HaqHM qepe3 h. Onpe,ll;eJIHB Op,ll;HHaTbl Yo' Y1' Y2' ..., Yn-l'
COOTBeTCTBYlOllI.He 3HaqeHHHM a, a+h, a+2h, ..., a+(n-l)h
a6cuHccbI x, MbI MO>KeM no MeTO,ll;Y npHMoyrOJIbHHKOB HaiiTH
npH6JIH>KeHHOe 3HaqeHHe 51 nJIoru;a,ll;H TpaneUHH ABCD:
51 :=:: Y o h+Y1 h +Y2 h +...+Yn-1 h =
=h(YO+Yl +Y2+...+Yn-1)'
4HTaTeJIb, HaBepHoe, 3aMeTHJI, qTO paHee, rrpHMeHSISI cpop-
MYJIY npHlIJOyrOJIbHHKOB, MbIop,ll;HHaTbI Yo, Y1'''' H3MepH.rIH,
a Tenepb, HMeH ypaBHeHHe (1), l\!bl HX BblqHCJ!HeM.
OOJIyqHB 3HaqeHHe 51 ,lI;JIH nJIoru;a,ll;H TpaneUIiH ABCD,
MbI Ha 9TOM He OCTaHOBHMCH. )KeJIaS! HaiiTIl 60JIee TOqHblIi
(2)
149

pe3YJIbTaT, pa3JJ.eJIHM CTOPOHY AD TpaneUHH Ha 2n paBHblX
qaCTeH. nJIHHa Ka>KJJ.OH qaCTH Tenepb 6YJJ.eT
 . OrrpeJJ.eJIHM,
KaK H B nepBOM CJIyqae, npH nOMoru;H ypaBHeHHH (1) opJJ.HHa-
h
Tbl Yo, Y1' Y2' ..., Y2n-1' COOTBeTCTBYIOru;He 3HaqeHHHM a, a+"2 '
2h
a +2' ... a6cuHccbl x, Mbl CMO>KeM BHOBb MeTOJJ.OM npHMO-
yroJIbHHKOB HatiTH npH6JIH>KeHHOe 3HaqeHHe 52 nJ!oru;aJJ.H
TpaneUHH ABCD, 60JIee 6JIH3KOe K ee HCTHHHOMY 3HaqeHHIO 5:


5 h I 2h + + (2h-l)h
2 = Yo' ""2 T Y1 . ""2 . . . Y 2 n-1 . 2


(3)


OpOJJ.OJI>KaH YJJ.BaHBaTb qHCJIO ,lI;eJIeHHH CTOpOHbI AD KPH-
BOJIHHeHHOH TpaneUHH ABCD, Mbl nOJIyqHM, TaKHM o6pa30M,
pH,lI; npH6.1IH>KeHHbIX 3HaqeHHH 51> 52' 53' ... ,lI;JIH ee nJIoru;a,ll;H,
nOCTeneHHO npH6JIH>KalOru;HxcS! K ee HCTHHHOMY 3HaqeHHIO 5,
npHqeM pa3HocTH 5-5 1 ,5-5 2 , 5-5 3 , ... 6Y,ll;YT nocreneHHO
y6bIBaTb. 4epTeLKH 70 a H 706, Ha KOTOpbIX pa3HocTb Me>K,lI;y
HCTHHHbIM 3HaqeHHeM 5 nJIoru;a,ll;H ABCD H ee npH6JIH>KeHHbIM
3HaqeHHeM Ka>K,lI;blH pa3 3aUITpHXOBaHa, nO,ll;TBep)!{,lI;aIOT :ono.
Tenepb, qHTaTeJIb, C,lI;eJIaeM BMeCTe C BaMH O,ll;HH CMeJIblH
war. BblH,lI;eM 3a npe,ll;eJIbl rex B03MO>KHOCTeH, KOTopble HervlH-
HyeMo CTaBHT HaM npaKTHKa Ha nYTH nOCTeneHHoro y,ll;BOeHHH
qHCJIa n ,lI;eJIeHHH CTOpOHbI AD TpaneUHH H MblCJIeHHO npeJJ.no-
JJO>KHM, qTO TaKoe y,ll;BOeHHe npO,ll;OJI>KaeTCH HeOrpaHHqeHHO.
Tor,ll;a, OqeBH,lI;HO, pa3HOCTb Me>K,lI;y HCTHHHbIM 3HaqeHHeM nJIO-
ru;a,ll;H H ee npH6JIH>KeHHbIM 3HaqeHHeM MO>KeT 6bITb C,lI;eJIaHa
no a6COJIIOTHOH BeJIHqHHe, CKOJIb yrO,ll;HO 6JIH3KOH K HYJIIO.
He nOMO)!{eT JIH HaM MaTeMaTHqeCKHH aHaJIH3 pe3YJIbTaTbi
9TOro l\!bICJIeHHOrO npouecca oTo6pa3HTb B MaTeMaTHqeCKHX
BbIKJIa,ll;KaX H, BMecTO CPOPMYJI, Bblpa>KalOru;HX npH6JIH>KeHHbIe
3HaqeHHH nnoru;a,ll;1i KpHBOJIHHeHHOH TpaneUHH A BCD , nOJIY-
qHTb CPOPMY.'IY, Bblpa>KalOru;ylO ee IiCTHHHOe 3HaqeHHe? Ma-
TeMaTHKa ,lI;aeT Ha 9TOT Bonpoc nOJIO>KHTeJIbHbIH OTBeT.
Ho npe>K,lI;e qeM nepeHTH K npHMeHeHHIO YKa3aHHoro MeTO,ll;a
,lI;.'I51 onpe,ll;eJIeHHH nJIOlll.a,ll;H KpHBOJIIiHeHHOH TpaneuHH, or-
paHHqeHHOH ,lI;yroH KaKOH-JIH60 H3 H3BeCTHbIX BaM KpHBbIX,
npliMeHHI\i ero K orrpe)I,eJleHHIO nJIoru;a,ll;H 06bIqHOH TpaneuHH,
cpopMy.'Ia nJIOlll.aJJ.1i KOTOpOH BaM 3HaKOMa H3 Kypca reOMeT-
pHH. COBnalJ.eHHe pe3YJI, bTata nO.TJyqeHHOrO HOBbIM I'l'leTO,ll;OM,
C paHee H3BeCTHbIM BaJ\I pe3YJIbTaTOM BHeceT 66.TJbWYIO YBe-
reHHocTb, qTO nO,ll;o6HbIH MeTO,ll;; npHMeHeHHblH K KpIiBOJIH-


150




HerlHOH TpaneUlHf, ,naCT, KaK 3TO H eCTb Ha CaMOl\I .n.eJIe, TaK>Ke
sepHblH pe3y.%TaT.
vIraK, HaM .n.aHa TpaneUHH, OrpaHHqeHHafi cBepxy qaCTbIO
npSIMofl y=kx, OCbIO x H op.n.HHaTa:\lH, COOTBeTCTBYIOllI.HMH
a6CUHccaM a H b (qepT. 71).
Tpe6yeTcSI onpe.n.eJIHTb ee
nJIOIll,a.n.b.
npeLK.n.e qeM peIIIHTb
3TOT Bonpoc, BblllHCJIHM
ITJIO
a.n.b TpeyrOJIbHHKa
AD E. JI,JISI :1Toro pa3.n.eJlHM
OTpe30K AE Ha n paBHbIX
u b b h
qaCTeH n; nYCTb n = .
nOCTpOHM BHYTPH Tpe-
yrOJIbHlIKa. npSIMoyrOJlbHIf-
KH C OCHOBaHHeM h H Bbl-
COTaMH, paBHbIMH 0P.n.HHa-
TaM Yl' Y2' . . ., Yn-l' CO-
OTBeTCTBYIOIII.HM a6CUHCCaM *, :
 ,
 ,..., (n n 1 )b
nJIOIll,a.n.b TpeyrOJIbHlIKa ADE 6y.n.eT np1l6JIHLKeHHO paSHa:
SADE;::::::h(Yl + Y2 + " . + Yn-l) =
=h [
 + 2kb + 3kb +... + (n-l)kb ] =
n n n n
hkb
=- [I + 2 + 3 +... + (n-I)].
n


1'13 nepBOH rJIaBbl MbI 3HaeM, llTO


!I


o
4epT. 71


Tor.n.a


n(n-l).
1+2+3+...+(n-I)= 2 '


n03TOMY
SADE = hkb . n(n-1) = kb 2 (n-l) = kb 2 ( 1-
 ) .
n 2 2n 2 n


ECJIH npe.n.nOJIOLKHTb, qTO n 6y.n.eT HeOrpaHHqeHHO y.n.BaH-
1
BaTbCfI, TO .n.p06b - 6y.n.eT CTpeMHTbCH K 0 H .n.JIH nJIOllI.a.n.I1
n
TpeyrOJIbHHKa Mbl nOJIyqHM TOqHYIO CPOPMYJIY:
kb 2
SADE= T"


151




COrJlaCHO 3TOMY i1J10llI.a.n.b TpeyrOJIbHHKa ABC 6y.n.eT paBHa


ka 2
SABC =
.
nJlOllI.a.n.b TpaneUHl1 BCED HaH.n.eTcSI nYTeM BblqHTaHHSI
H3 nJlOllI.a.n.H TpeyrOJIbHHKa ADE nJIOllI.a.n.H TpeyrOJIbHHKa
ABC:


SBCED = -} (b 2 - a 2 ).
3TO H eCTb H3BeCTHaSI HaM H3 reOMeTpHH <popMYJla nJlOII.{a.n.H
TpaneUHH. B caMOM .n.eJIe,
k
SBCED = 2(b-a) (b+a);


HO


b-a=CE H k(b+a)=kb+ka=DE+BC.


vhaK,


!I


o


1/
o


4epT. 72.


DE+BC
SBCED = . 2 .CE.


Tenepb MOLKHO nepeHTH K 3a-
.n.aqe onpe.n.eJIeHHSI nJIOllI.a.n.H KPH-
BOJIHHeHHOH TpaneUHH. Mbi orpa-
HHqHMCH .n.BYMSI CJIyqaHMH,
Kor.n.a KpHBOH, OrpaHHqHBalOllI.eH
TpaneUHIO, B O.n.HOM c.lJyqae 6y-
.n.eT .n.yra KBa.n.paTHOH napa60JIw
y=ax 2 H B .n.pyroM-.n.yra Ky6II-
qecKOH napa60JIbI y=kx 3 .
JI,JIH peIIIeHHSI nOCTaBJIeH-
HbIX 3a.n.aq He06xo.n.Hl\1O BcnOM-
HHTb <PO PMYJIbI cYMl\mpOBaHHH
KBa.n.paToB H Ky60B qHCeJI
KOTopbIe MbI BbIBeJIH B nepBoH rJIaBe.


naTypaJlbHoro pH.n.a,
BOT 3TH q,OPMYJIbI:
[2 _L 22 + 32 ..L + 2 _ n (n + 1) (2n + 1)
I I... n - 6


II


}3 + 2 3 + 3 3 + . . . + n 3 = [ n (n 2 + 1) r
npHCTynaeM K peIIIeHHIO nepBOH 3a.n.aqH. Onpe.n.eJIHTb nJIOIII.a.n.b
KpHBOJIlIHeHHOH TpaneUHH, OrpaHHqeHHOH cBepxy qaCTblO
napa60JIbI y=kX2 (qepT. 72).


152




CHaQa"la HaR.n.er\1 cnoc06 BbIQHCJ1eHHJI nJICIII.a.n.H qmrYPbl
ADE, orpaHHtJeHHOH napa60JIOH, OCblO a6CUHCC Ii op.n.H.HaTOH
Yn, COOTBeTCTBYIOllI.eH a6CUHcce x = b ('lepT. 73). Pa3.n.eJIHM
oTpe30K OE OCH a6CUHCC Ha n paBHblX qaCTeH .!:.... nYCTb
n
b
-=h.
n
nOCTpOllM BHYTPIi qJHrYPbl ADE npJlMoyroJIbHHKH C OCHosa-
HHeM h H BbICOTaMH, paBHblMH op.n.HHaTaM Yl' Y2' Y3" . . , Yn-l'
!J
o
Yepr. 73.
!J
D
Ii
o
b 2b 3b (n - 1) h
cooTBeTcTBYIOIII.HM aocUllccaM -, , , . . . ,
n n n n
Tor.n.a HCKOMaJl nJIOllI.a.n.b S ADE 6y.n.eT npH6JIllLKeHHO pasHa
SADEh(Yl + Y2 + Y3 +... + Yn-l};
H3 ypaBHeHHSI
Y = kx 2
HMeeM:
kb'2 k 22b 2 k.3 2 .b 2 k (n-1) 2 b 1
Yt =, Y2 = ---ri2' Y3 = n2 ,... 'Yn-l=-
Tor.n.a:
S ADE = h2 [12 + 2 2 + 3 2 + . . . + (n _ 1)2] =
kb 3
= --a [12 + 2 2 + 3 2 + . . . + (n - 1)2].
11
.53

BOCnOJIb30BaBIIIHCb 1iJ0PMYJIOH CYMMbl KB8.n.paTOB HaTypaJIb-
HblX qHCeJI OT 1 .n.o n - 1, nOJIyqeHHbIH pe3YJIbTaT npe.n.cTa-
BHM TaK:
S = k  (n - 1) n (2n - 1) _ kb 3 (n - 1) (2n - 1)
ADE n 3 6 - fi2 6
= kt (1 - + ) (2 - +}
1haK, nJIOllI.a.n.b ADE npH6JIHLKeHHO HaH.n.eHa. OIIIH6Ka,
KOTOpylO Il'lbI B .n.aHHOM CJIYtJae .n.onycKaeM, Ha tJepTeLKe 73
3aIIITpHxoBaHa. 4T06bl YMeHbIIIHTb 3TY OIIIH6KY, y.n.BOHM
tJHCJIO n.
Tor.n.a
kb 3 ( 1 ) ( 1 )
SADE::::::' 1- 2n 2- 2n '
H oIIIH6Ka, KaK nOKa3bIBaeT qepTeLK 73, 6y.n.eT 3HallHTeJIbHO
MeHbilie. JI,aJIbHeHIIIee y.n.BoeHHe tJHCJIa n, YBeJIHtJHBaSI npH-
6JIHLKeHHOe 3HaqeHHe nJIOIII.a.n.H ADE, .n.eJIaeT OIIIH6KY Bee
u u 6 I 0
MeHbilieH H MeHbIIIeH; npH 3TO:'I .n.po b Ii: cTpeMHTcH K ,
a BbIpaLKeHHH B cK06Kax (I - + ) H (2 - + ) CCOTBeT-
CTBeHHO K I H K 2. nOJIaraH, qTO tJHCJIO n y.n.mmBaeTcSI
HeorpaHHtJeHHo, Mbl H npH.n.el\1 K TOtJHOH 1iJ0pMYJIe nJIOllI.a.n.H
IiJHrypbI ADE:
kb 3
SADE =.
COrJIaCHO 9TOMY nJ10llI.a.n.b IiJHrYPbl ABC 6y.n.eT paBHa:
ka 3
SABC='
nJIOIll,a.n.b KpHBo.IJHHeHHOH TpaneUHH BCED 6y.n.eT HaH.n.e-
Ha nYTeM BbItJHTaHHSI H3 nJIOIll,8.n.H ADE n.10Ill,a.n.H ABC:
kb 3 ka 3 k
SBCED = - = T(b 3 -a 3 ).
PH

npl-lMep. BblqHCJIHTb riJIOllI.atlb KpHBOJIHHeHHOH TpaneHli,
OrpaHHqeHHOH cBepxy napa60JIOH
X 2
Y=T'
OCbIO x H Op.n.HHaTaMH, COOT-
BeTCTBYIOllI.HMH a6cHccaM 2
H 4 (qepT. 74).
If
B
8
7
6
5
'"
3
2
01234jlf
4epT. 74.
7
6 II
J I
"
3
2 I
1
-4 -3'-2 . -I 0 V
/' -I 1 2 J 41 5
7 -2
I -J
-4
-5
I -6
I -7
-8
4epT, 75.
BbIBO.n. CPOPMYJIbI .n.JISI nJIOIIla.n.H KpHBOJIHHeHHOH TpaneUHH.
OrpaHHqeHHOH cBepxy napa60JIOH
y=h,
qHTaTeJIb MOLKeT c.n.eJIaTb caMOCTOSITeJIbHO. 4T06hI BcnOMHHTb
<PoPMY Ky6HqeCKOH napa60JIbI YKa3aHHoro Erma, nocTpoHM ee
.n.JISI qaCTHOro CJIyqaSI, Kor.n.a OHa 3a.n.aHa ypaBHeHHeM
X3
Y=S'
JI,JISI 9Toro COCTaBJIHeM Ta6JIHUY 3HaqeHHH x H COOTBeTCT-
BYIOllI.HX 3HaqeHHH Y:
x I: 0 I 1 I 2 I 3 I 4 I -1 I -2
Ii
,I
;1 1 1 I 3 1 1-+
i!
y 0 8 1 I 8 8 -1
nOCTpoeHHaSI no 3THM TOllKaM KpHBaSI, OqeBH.n.HO, 6y.n.eT
CHMMeTpHqHa OTHOCHTeJIbHO HaqaJIa KOOp.n.HHaT (qepT. 75).
155

4T06bI BbIBeCTH 4>OpMyJJy .n.JISI nJIOllI.a.n.H KpHBOJIHHellHOH
rpaneurm, OrpaHHqeHHOH cBepxy Ky6HqeCKOll napaOOJIOH,
B03bMHTe ypaBHeHHe 3TOll napa60JIbI B 06III.eM BH.n.e y=kx 3
Ii npOBe.n.HTe pacCY:H<.n.eHHSI, aHaJIOrrrqHble TeM, KOTopbIe BbI
npOBo.n.HJIH, onpe.n.eJISISI nJIOllI.a.n.b TpaneUHH, OrpaHHqeHHOll
cBepxy KBa.n.paTHOH napa60.'10H. i1cnoJIb3YHTe npH peIIIeHHH
9Toro Bonpoca <POPMYJIY:
1 3 + 2 3 + 3 3 + . . . + (n - 1)3 = [ (Ii -; 1) n r-
OTBeT .n.OJI:H<eH 6bITb TaKHM:
k
S = T(b 4 - a 4 }.
3a)J.alJa. Onpe.n.eJIHTb nJJOllI.a.n.b, OrpaHHqeHHYIO napa6oJI0fr
x 2
Y=T
(1)
H npSIMOH
x
y = "2 + 2.
(2)
Y K a 3 a H H e. a) Onpe.n.eJISIeM TOqKH nepeceqeHHSI napa60JIbI
H npSIMOll, peiliaR ypaBHeHHSI (1) H (2) COBMecTHO; 6} CTpOHM
qepTeLK 76; B) onpe.n.e.IJHeM nJIOllI.a.n.b TpaneUHH ABCD; r} on-
pe.n.eJIHeM nJIOllI.a.n.b AOB;
g ,Q} onpe.n.eJIHeM nJIOIII.a.n.b
OCD; e) Haxo.n.HM HCKOMYIO
8 nJlOIII.a.n.b nYTeM BblqHTaHHSI
7 H3 nJIOllI.a.n.H TpaneUHH
6 ABCD CYMMbI nJIOllI.a.n.eH
AOB H OCD.
Om8em. 9 K8. ea.
Pa306paHHbIll HaMH B
3TOM naparpmpe HOBblH cno-
c06 H3MepeHlISI nJIOllI.a.n.eH
, MOLKeT 6bITb npHMeHeH BO
MHomx CJIyqaSIx. 3.n.ecb,
OqeBH.n.HO, 60JIbiliOe 3Haqe-
HHe HMeeT HaJIHqHe ypaB-
HeHHH TOll KPHBoi'I, KOTopaSI OrpaHHqHBaeT KPHBOJIH-
HeHHYIO TpaneUIfIO cBepxy. MbI He MomeM 3.!1.eCb KacaTbCSI 06-
156

lll.eH 3a.n.aqH .n.JlSI CJJyqaSI npoH3BoJlbHOl! KpHBOH. PeIIIeHHe
3TOH 3a.n.aqH COCTaBJISIeT npe.n.MeT 60JIbiliOrO pa3.n.eJla BbICilieH
MaTeMaTHKH, H3BeCTHOrO no.n. Ha3BaHHeM HHTerpa.'1bHOrO HC-
qHCJIeHHSI .
3a)J.aql-l 1-1 COtl>1-I3M
1.JHTaTeJIIO npe.n.JlaraeTCSI peIIIHTb HeCKOJIbKO 3aJJ:aq Ha
BbIqHCJIeHrIe nJIOllI.a.n.eH pa3JIHqHbIX <pHryp.
3a)J.alla 1. JI,oKa3aTb, qTO nJIOIII.a.n.b <PHrYPbI, OrpaHHtleHHOH
npHMb!MH, COe.n.HHSIIOllI.HMH Ka)K.n.ylO BepIIIHHY KBa.n.paTa C
Cepe.n.HHOH npOTHBOnOJIOLKHOH CTOPOHb!, paBHa nSITOH qaCTH
nJlOma.n.H BCerO KBa.n.paTa (qepT. 77).
4epT. 77.
78.
3a)J.alla 2. 4epe3 KaLK.n.ylO BepIIIHHY qeTbIpexyroJIbHHKa
npoBe.n.eHbI npHMbIe, napaJIJIeJIbHbIe .n.HarOHaJIH, He npoxo.n.H-
lII.eH qepe3 3TY BepIIIHHY. nOKa3aTb. qTO nOJIyqeHHbIH TaKHM
06pa30M napaJIJIeJIOrpaMM HMeeT nJIOlll.a.n.b B.n.Boe 60JI bIIIY 10 ,
qeM .n.aHHbIH qeTbIpexyrOJIbHHK (qepT. 78).
3a)J.alla 3. BHYTPH TpeyroJIbHHKa HaflTH TaKYIO TOqKY,
KOTopaH, 6y.n.YQH coe.n.HHeHa C TpeMSI BepIIIHHaMH, pa3.n.eJlHJIa
6bI TpeyrOJIbHHK Ha TPH lIaCTH, nJIOllI.a.n.H KOTOpbIX nponopHo-
HaJIbHbl TpeM .n.aHHbIM qHCJIaM p, q H r.
P e III e H H e.
A H a JI II 3. nYCTb TOqKa 0 - HCKOMaSI, Tor.n.a:
SOBC : SOCA : SOAB =p : q : r.
157

Y TpeyrO.lJbHHKOB OCA II OAB (qepT. 79) eCTb 06111.ee
OCHOBaHHe OA, H nOToMY HX nJIOIII.a.n.H OTHOCHTCSI KaK COOT-
BeTcTBYIOIlI,I1e BbICOTbl CQ II BP. Ho I13 nOA06HH TpeyrOJIbHH-
KOB CQK H BPK HMeeM:
11
CQ : BP=KC : KB.
TaKHM 06pa30M:
KC: KB=q: r;
LA : LC=r : q;
MB : MA=p : q.
Cne.n.OBaTeJIbHO, .n.JIH no-
cTpoeHHH TO'lKH 0 .n.OCTa-
TOqHO pa3.n.e.IJHTb .n.Be CTO-
B pOHhI TpeyrOJIbHHKa B .n.aH-
HbIX OTHOIIIeHHSIX H coe-
.n.HHHTb TOqKH .n.eJIeHHH C
npOTHBOJIeLKaIll,HMH BepIIIH-
4epT. 79. HaMH. Opo.n.eJIaiiTe nocTpO-
eHHe .n.JISI KaKOrO-J1Jf60
tIaCTHoro cJlyqaSI. f.n.e 6y.n.eT Hax0.n.HTbCSI TOqKa 0, ecJIH nJIO-
llI.a.n.b TpeyrOJlbHHKa Tpe6yeTcH pa3.n.eJlHTb Ha TPH paBHbIe
qaCTII?

 a 
4epT. 80.
3a)J.aQa 4. TpaneUHSI CBOHMII .n.IiarOHaJISIMH pa36HBaeTcH
'Ha qeTbIpe qaCTH. Onpe.n.eJIHTb nJIOllI.a.n.b TpaneUHH, eCJIH nJlo-
ma.n.H TpeyrOJIbHHKOB, npHMbIKalOllI.HX K OCHOBaHHHM, paBHbI
SI H S2 (qepT. 80).
OmBem. 0/ Sl + 1 S2 )2.
3a)J.aQa 5. TpeyroJIbl:IHK' ABC - paBHocTOpOHHHI1 (qepT.
AC
81): AM=BK=CL=3' KaKYIO qaCTb nJIOllI.a.n.H TpeyrOJIb-
nHica ABC COCTaBJISIeT nJIOllI.a.n.b TpeyrOJIbHHKa DEF?
158

P e III e H H e. nYCTb AC = a:
. V a2 V a2 2a.a
1) AL= a 2 + --2aCN = a 2 + - -- =0
996
=  'V7;
2) /::" ADM ro 6 ALC;
=  :  :V7 = ; 7 ;
3 ) S _ 8 ABC
ALC- 3
AM
K03cpqmU H eHT no.n.06HH K = AL =
S ADM
8 ABC _ 8 ABC .
3(Y7}2 -,
8 8 68 2S
4 } S = ABC _ ABC = ABC = ABC .
MDLC 3 21 21 7'
28 ABC 1
5) S DEF = S ABC - 3 . -----r-- = 7 S ABC.
PeIIIHTe 3TY 3a.n.aqy .n.JISI CJIyqaSI, Kor.n.a CTOpOH8 Tpe-
yrOJIbHHKa pa3.n.eJIeHa Ha 4, 5,.. ., n paBHblX qaCTeH H c.n.e-
JIaHO TaKoe LKe nOCTpoeHHe, KaK H .n.JIH n = 3 (qepT. 82).
8 8
()
LJepT. 81.
LJepT. 82.
CO(jJM3M. «nJIOIII.a.n.b KBaD.paTa, CTopoHa KOTOpOro paB-
Ha 8 .IJHHeHHbIM e.n.HHHuaM, 6y.n.eT HMeTb He 64, a 65 KBa.n.paT-
HbiX e.n.HHHU». Pa3pelKeM KBa.n.paT ABCD Ha qeTbIpe qaCTU,
15!;)

KaK YKa3aHO Ha qepTeLKe 83. PaBeHcTBo TpaneIJ.HM I H II
cOBepIIIeHHo otJeBH.n.HO, KaK H paBeHcTBo TpeyrOJIbHHKOB
III H IV. H3 qaCTeH I H III COCTaBHM npSIMoyrOJIbHbIH Tpe-
yrOJIbHHK, TaKLKe COCTaBHM npSIMoyrOJIbHbIH TpeyrOJIbHHK H3
qaCTeH I I H IV. 1'13 nOJIy-
qeHHbIX .n.BYX paBHblX npSI-
MoyrOJIbHbIX TpeyrOJIbHHKOB
(qepT. 84) CKJIa.n.bIBaeM npSI-
MoyrOJIbHHK co CTopoHarxm
13 e.n.HHHIJ. H 5 e.n.HHH
(qepT. 85).
nJIOllI.a.n.b 3TOro npSIMo-
yrOJIbHHKa, OqeBH.n.HO, 6y-
.n.eT paBHa 65 KBa.n.paTHbIM
e.n.HHHIJ.aM, CJIe.n.OBaTeJIbHO,
H nJIOIII.a.n.b KBa.n.paTa ABCD
6y.n.eT TaKLKe paBHa 65
KBa.n.paTHbIM e.n.HHHaM.
OTKy.n.a B3SIJIaCb JIHIIIHSISI KBa.n.paTHaSI e.n.HHHa?
'i-....
r--..... :--......
....... ..........
......... ..........
.......... ........
o
/
u IV T
5
.......... ......... I
....... f-..... I
J Jm
3 I
II
:J
5
iJ
4epT. 83.
I
l
If
16
c
D
D
4epT. 84.
f1
N
4ePT. 85,

PeKOMeH.n.yeM llHTaTeJIIO BbIpe3aTb H3 KJIeTqaTOA (5YM8m.
KB3.n.p3T (B3HTb CTOPOHY KJIeTKH B I C.M) , p33pe33Tb ero, K3K
YK333HO H3 qepTeLKe 83, H H3 nOJIyqeHHbIX q3CTeH CJIO}{{HTb
npHMoyrOJIbHHK.
JIVITEPATYPA
D.. 10. n a HOB, BbI'IHCJIeHHe n.'1ow.a,l(ei1, TexH3113T, 1946.
B, f. n a H K paT 0 B a, Cnoc06bl npH6JIHH<eHIiOrO onpelleJIeHHSI
nnow.a,l(ei1. «MaTeMaTHKa B WKOJIe», 1956, N 3,
51, 11. n e p e JI b M a H, 3aHHMaTeJIbHaSi reOMeTpHSI, focTex-
H311aT, 1955.
5, B. f Hell e H K 0, 01JepKH no HCTOpHH MaTeMaTHKH B POCCHH,
TexH311aT, 1946,
H, A, f JI a r 0 JI e B, 3neMeHTapHaSi reOMeTpHSI, 'I. 1, Y'Ine./l-
I1H3, 1954.
f5 A, A. 1I:000eCOB

d.P
ilL
n. ?
n -J- g I'L:::== C ·
a 2:
6 2 :;$ (C
r JI A B A VlJ
TEOPEMA nH4>AfOPA. PABHOBEJUiKHE 4>HfYPbl
«nmparop BnepBble pa3pa60TaJ!
npHIIUHnbl reoMeTpHH H HccJ!elloBan
TeopeMbI HeBelUecTBeHHbIM, pa3YMHbiM
nYTeM».
npOK.II - rpeqeCKHI! IjmnocoljJ
H MaTeMaTHK V B. H. a.
EOJlbUIOfi MaTeMaTJfK H HeY)J.aI.JHbIH nOJlJfTHK
C HMeHeM nH<paropa CBSI3aHO MHoro pa3JIJfqHbIX
paCCKa30B H JIereH.n.. B HeKOTopblX H3 HHX npH-
nHCbIBalOTCSI 3aCJIym, SIBHO eMY He npHHa.n.JIe-
M8I11.He. TaK, B o.n.Hoii H3 CJIaBSIHCKHX PYKonHceii
XVI I B. YTBepM.n.aeTcSI, qTO TImparop nOJIOMHJI HaqaJIO apH<p-
MeTHKe. 4TO KacaeTCSI <PaKTOB H3 MH3HH TIH<paropa, TO 3.n.eCb
OT.n.eJIHTb npaB.n.y OT BbIMblCJIa OqeHb Tpy.n.HO, TeM 60JIee qTO
ero yqeHHKJf npHnHCbIBaJIH eMY MHoroe, qT06bI B03BblCHTb
CBoero yqHTeJIH B rJIa3aX Hapo.n.a.
PaCCKa3bIBaIOT, qTO B MOJIO.n.OCTH nH<parop I\moro nYTeIIIe-
CTBOBaJI, n06bIBaJI B EmITTe H npOHHK qepe3 MaJIYIO A3HIO
KapaBaHHblMH nYTSIMH B BaBHJIOH. 4TO 6y.n.TO 6bI BCIO.n.y H Be3-
.n.e OH no KpynHU8M C06HpaJI 3uaHHSI .n.peBHeHIIIHX Hapo.n. OB
162

no MaTeMaTHKe, aCTpOHOMHH TeXHHKe H llTO, BepHYBIIIHCb
Ha CBOIO pO.n.HHY - OCTpOB CaMOC, OH TaK nOpa3HJI npH06pe-
TeHHblMH 3HaHHSIMH CBOHX COOTeqeCTBeHHHKOB, qTO ero CllH-
TaJIH nOJIy60rOM.
LI.aJIbHeHiliHe CBe.n.eHHSI 0 LKH3HH nmparopa CTaHOBSITCSI
60JIee .n.OCTOBepHbIMH. BepHYBIIIHCb Ha OCTpOB CaMOC, nH!JJa-
rop c06HpaeT BOKpyr ce6SI IOHOilieH H3 6,;laropo.n.HbIX ceMeH
H Be.n.eT C HHMH TaHHbIe 6ece.n.bI. HHKTO He 3HaeT, lleMY Oli
HX YllHT. nOJIHKpaT, npaBHTeJIb oCTpOBa, 60SICb, qTO no.n.
npHKpbITHeM 3THX TaHHblX 6ece.n. 3peeT 3arOBOp npOTHB Hero,
BeJIHT CBOHM JlIO.n.SIM CJIe.n.HTb 3a HHMH. TIH!JJarop, B03MYIII.eH-
HblH 3THM, HaBcer.n.a nOKH.n.aeT pO.n.HOH OCTpOB U nOCeJISIeTCSI
B O.n.HOM H3 rpeqeCKHX ropo.n.oB IOLKHOH HTaJIHH - KpOTOHe.
3.n.eCb npOHcxo.n.HT 60pb6a MeLK.n.y 3HaTbIO H Hapo.n.OM 3a
BJIaCTb Ha.n. ropo.n.OM. Y 3HaTH eCTb BOLK.n.b - aTJIeT MHJIOH,
HO HeT qeJIOBeKa, KOTOpbIH, Mor 6bI !JJHJIOC0<PcKH 060CHOBaTb
He06xo.n.HMOCTb nepe.n.aqH B.naCTH Ha.n. ropo.n.O:\1 B PYKH 60raTbIX.
YI BOT nOSIBJISIeTCSI nH!JJarop. CnaBa 0 HeM .n.OCTHrJIa H ropo.n.a
KpOTOHa. nu!JJarop ytJHT: «TIOCMoTpHTe BOKpyr ce6SI. Be3.n.e B
I\lHpe nopSI.n.OK, Bce nO..'I.qllHeHO rapl\lOHHH, Mepe. LI.a>Ke 3BYKll, H
Te nOn;tJHHeHhI tJHCJIaM... Be3.n.e B npHpo.n.e rocno.n.CTByeT
CTpOHHbIH nOp5l.n.OK, YCTaHOB.leHHbIH 60ra1\lH. LI.a)!{e He6ecHble
CBeTHJIa H 3 Be3.n.bl nOiJ.tjHH5IIOTCSI eMY. KaK LKe MOLKeT He nOk
tJHHHTbCSI el\1Y tJeJIOBeK? fope TOI\IY ro, po.n.y r.n.e uapcTByeT
xaoc, r.n.e BCe peiliaeT To.lJna, r.n.e HeT nOtJTeHHSI ,QpeBHel\lY
CTpOIO» .
3HaTb KpoToHa tJYBcTByeT, tJTO ytJeHHe nH!JJaropa MO>KeT
06'be.n.HHHTb BOKpyr ce6SI Tex, KTO 3a cTapbIe nOp5l.n.KH B ro-
po.n.e, KTO He XOtJeT, tJT06bI BJIaCTb 6bIJIa B pYKax Hapo.n.a.
Bce 60JIbilie H 60JIbilie yqeHHKOB y nH!JJaropa. OH11 06'be.n.H-
H5IIOTCSI' B COlO3, COlO3 nOCBSIIII.eHHbIX, Ky.n.a He MorYT npOHHK-
HYTb npOCTble JIIO.n.H.
B COI03e uapHT .n.HCUHnJIHHa, nOCJIYIIIaHHe, CJIOBO YllHT€-
JIH - Bce. «COI03 .n.pYLK6bl» CTaHOBHTCSI nOJIHTHtJeCKHM COIO-
30M e.n.HHOMblIIIJIeHHllKOB, 3aHHMaIOIII.HXCSI He TO.lbKO HaYKOH,
HO H MetJTaIOIII.UX nOXHTUTb MaCTb y HapO,J,a.
H OHll .n.06UBaIOTCSI 3Toro. BJIaCTb Ha.n. ropo.n.OM B HX py-
Kax. Ho nH!JJaropeHII.bl He ycnoKaHBaIOTCH Ha 3.TOM. o HI!
crpe:\JSITCSI YCTaHOBHTb TaKOM,LKe «nop-SI.n.OK», TaKYIO LKe «rap-
l\IOHUIO» H B .n.pyrux ropo.n.ax. 3TO HM TaK>Ke Y,J,aeTCSI.
Ho H.n.eT Bpe:\ISI, H B caMO:\I KpOTOHe 3peeT He.n.OBOJIbCTBO
npaBSIIII.eH 3HaTblO H COI030M nH!JJaropei'IueB. nOSIBJISIIOTCSI He-
.n.OBOJIbHbIe H cpeAH 'lJIeHOB COI03a. MHorne Tpe6YIOT H3fHa-
6-
163

lIHSI rm<paropeHu.eB. TIH<parop nOKH.n.aeT ropo.n.. B 3TY )!{e
HCtlb pa3rHeBaHHaH TOJIna Hapo.n.a - pbI6aKH, peMeCJIeHHHKH,
ropo.n.CKaH 6e.n.HOTa - OKpYLKaeT .n.OM MHJIoHa, r.n.e C06paJIHCb
nmparOpeUIJ.bI, H YHHqTOLKaeT HX.
A caM TImparop 6e>KHT .n.aJIbille - B MeTanoHT; HO II TaM
ero npeCJIe.n.yeT rHeB Hapo.n.a, H .n.eBSIHOCTOJIeTHHH yqHTeJIb
nom6aeT B O.n.HOU H3 HOqHbIX CXBaTOK.
TaKoBa nOJIHTHqeCKaSI Cy.n.b6a TImparopa H OCHOBaHHoro
HM COI03a. HaM a6COJIIOTHO qYLK.n.bI nOJIHTHqeCKHe B033peHHSI
Omparopa-apHCTOKpaTa, HO K HCKJIIOqHTeJIbHbIM 3aCJIyraM
TImparopa-yqeHOrO MbI .n.OJILKHbI OTHeCTHCb C yBaLKeHHeM.
TImparop Hero IIIKOJIa nOJIOLKHJIH HaqaJIO TeopHH qHCeJI;
OHH 3aJIO)!{HJIH OCHOBbI rpeqeCKOU aJIre6pbI, H3yqaSI nponop-
Il.HH H nporpeCCHH. B reOMeTpHH nmparopy npHHa.n.JIeLKaT,
nOMIIMO TeopeMbI, HOCSIIII.eU ero HMSI, yqeHHe 0 npaBHJIbHbIX
MHoroyrOJIbHHKaX, TeopeMbI 0 CYMMe yrJIOB TpeyrOJIbHHKa H
MHoroyrOJIbHIIKa, H, HaKoHeu., nmparOpeUIJ.aM MbI 06SI3aHbI
OTKpbITHeM HeC0I13MepHMbIX OTpe3KOB.
LI.aJIeKO onepe.n.HJIH BpeMSI H aCTpOHOMHqeCKHe B3rJISI.n.bI
TImparopa.
COI03 nl:I<paropeHu.eB paCnaJICSI. 4JIeHbl ero paCCeSIJIHCb
no BceM ropo.n.aM rpeIJ.HH H, 06yqaSI MaTeMaTHKe .n.pymx,
nOCTeneHHO paCKpbIJIH Bce HayqHbIe 3HaHHSI, H3BeCTHbIe HM.
To, qTO 6bIJIO paHee, npH LKH3HH TIH<paropa, TaHHOH .n.JISI .n.py-
rHX, CTaJIO .n.OCTOHHHeM BceX.
)J.OKa3blBaeM H o6061.l\aeM TeopeMY nuclJaropa
1) npe.n. am1e rOBopHT, qTO Kor.n.a
TIH<parop npHIIIeJI K TeopeMe,
H3BeCTHOH B reOMeTpIIH no.n. ero
HMeHeM, TO OH npHHec B LKepTBY
60raM CTO 6bIKOB, OTqerO 3Ta
TeopeMa H, Ha3bIBaJIaCb B cpe.n.-
HHe BeKa «reKaTOM6a», qTO 03-
HaqaeT npH nepeBo.n.e C .n.peB-
HerpeqeCKOrO «CTO 6hIKOB».
Ho He HYLKHO .n.YMaTb, qTO
L TeopeMa nH<paropa He 6bIJIa
4epT. 86. H3BeCTHa paHbilie. Eme B .n.PeB-
HeM EmITTe, 3a 2000-3000 JIeT.n.o H. 3" erJ1ITTHHe 3HaJIH,
qTO TpeyrOJIbHHK co cTopoHaMH 3, 4 H 5 e.n.I1HHIJ. eCTb npSI-
MoyrOJIbHhIH, H IIIHpOKO nOJIb30BaJIHCb TaKHM TpeyrOJIb-
lJ
1 ,r;I
I
c.

164

HHKOM .n.JIH nOCTpoeHHSI npSIMbIX yrJIOB Ha nOBepXHOCTH
3eMJII<! H npH nOCTpOHKe 3.n.aHHH. Mbl He 3HaeM,' KaK .n.OKa3bI-
BaJI 3TY TeopeMY nmparop. Ho YLKe 3a 500 JIeT,,QO Hero B
KHTae 3HaJIII .n.OKa3aTeJIbCTBO 3TOH TeopeMbl. TaKoe >Ke .n.OKa-
.3aTeJIbCTBO MbI BCTpeqaeM H Y HH.n.HHCKoro MaTeMaTHKa 5xaCKa-
pbl (l114). OpHBe.n.eM 3TO .n.OKa3aTeJIbCTBO.
ABDE - KBa.n.paT, nOCTpoeHHbIll Ha mnOTeHY3e npSIMO-
yrOJIbHOrO TpeyroJIbHHKa ABC ('lepT. 86).
DI(i..BC, ELi..DI( H AMi..EL;
£:::,ABC= £:::,BI(D=£:::,DEL=£:::,AME;
I(L=LM =MC=CI( =a-b.
H3 qepTeLKa HMeeM:
2_ 4ab +( b) 2
C -2 a- ,
HJIH
c 2 =2ab+b 2 -2ab+a 2 ;
c 2 =a 2 +b 2 .
CYllI.ecTByeT MHoro .n.pymx .n.OKa3aTeJIbCTB TeopeMbI OH-
<paropa. C HeKOTopbIMH H3 HHX, a TaKLKe H C HeKoTopblMH 0606-
lII.eHHHMH TeopeMbI, MbI ceH'lac H n03HaKOMHMCSI.
n e p B 0 e .n. 0 K a 3 aTe JI b C T B o CYHOCTb nep-
Boro .n.OKa3aTeJIbCTBa COBepIIIeHHO HCHa H3 .n.BYX npHJIarae-
MbIX qepTeLKelI (qepT. 87, 88).
1
4epT. 87.
m iff
4epT. 88.
BTopoe .n.oKa3aTeJIbCTBO. nOCTpoeHHe qepTeLKa 89:
I) Ha rHIIOTeHY3e AB TpeyrOJIbHHKa ABC nOCTpoeH KBa.n.-
paT ABDE; 
2} EF i..BC H DMi..BC;
3) DI(i..EF H A Li..EF.
165

!
11
b
c
4epT. 89.
F
j}
Il 1\ \ \ \\ \ \ \ 
" 1' 11 11 \1\ \
I ' IIII I III I
1 1 1\1111111\
,11 1 1\\\11\\1\
111\111111\\1\
1\\ III III I I 11\
I II I IIII \I III
II I \I \ I I \I \ I I II \
II 111\ I \ I 1\ 11\1
1111111111111111 \
11 '11,1\1 111 \111\\
\ II I
L
4epT. 90.
[J
t1
c

I1. 0 K a 3 aTe JI b C T B O.
ALE= LDKE= LBDM = LABC.
ECJIH 0603HaqHTb qepe3 S nJIOIII.a.n.b <lmrYPbI, TO
SACMDE-(SAEL+SDKE} =a 2 + b 2 ,
r.n.e a H b - KaTeTbI TpeyrOJIbHHKa ABC.
SAClI1DE- (SBDM + SABC) = e 2 ,
r.n.e e - mnOTeHY3a Tpe-
yrOJIbHHKa ABC.
TaK KaK JIeBble qaCTH .n.Byx
nOCJIe.n.HHX paBeHCTB paBHbI,
TO paBHbl H npaBble qaCTH.
TI03TOMY
e 2 =a 2 +b 2 .
T peT b e .n. 0 K a 3 a-
T e JI b C T B O. nOCTpoeHHe
qepTeLKa:
I} Ha CTopOHax npSIMO-
yrOJIbHOrO TpeyrOJIbHHKa
ABC nOCTpoeHbI KBa.n.paTbI
(qepT. gO);
2} LKLM = LCEF =
=LABC.
I1. 0 K a 3 aTe JI b C T BO.
[-
4epT.
ADGB=DEFG;
ACLM =CBKL;
ADGB=ACLM;
SDEFGBA-(SABC +SCEF )=a 2 +b 2 ;
S ACBKLM-(S KLM+SCAB) =e 2 .
113 .n.BYX nOCJIe.n.HHX paBeHCTB CJIe.n.yeT, qTO
e 2 =a 2 +b 2 .
4 e T B e p Toe .n. 0 K a 3 aTe JI b C T B O. 3TO .n.oKa-
3aTeJIbCTBO SIBJISIeTCSI 06pa3l1.0M O.n.HOro H3 MHOrHX .n.oJ\83a-
TeJIbCTB TeopeMbI nmparopa, OCHOBaHHbIX Ha pa3pe3aHlHI
167

KBa.n.paTOB, nOCTpOeHHblX Ha KaTeTaX, H YKJIa.n.bIBaHHH nOJIY-
lleHHbIX qaCTeH Ha KBa.n.paTe, nOCTpoeHHOM Ha nmOTeHY3e
(qepT. 9l).
BbISICHHTe, KaK pa3pe3aH KBa.n.paT ACDE.
0606eHHe TeopeMbi nHclJaropa - TeopeMa EBKJlH)J.a
TeopeMa nmparopa cnpaBe.n.JIHBa .n.JISI nJIOa.n.eH KBa.n.paTOB,
nOCTpoeHHbIX Ha CTopOHax npSIMoyrOJIbHOrO TpeyrOJIbHHKa.
Ho KBa.n.paTbI npe.n.CTaBJIHIOT C060H no.n.06HbIe qmrypbJ. He
6y.n.eT JIH cnpaBe.n.JlIma 3Ta TeopeMa .n.JISI nJIoa.n.eii npoH3-
BOJIbHbIX no.n.06HbIX <pHryp, nOCTpoeHHbIX onpe.n.eJIeHHbIM 06-
pa30M Ha nmoTeHY3e H KaTeTax TpeyrOJIbHHKa? EBKJIH.n.,
>KHBIIIHH Ha TpHcTa JIeT n03.n.Hee nH<paropa, .n.OKa3aJI cnpaBe.n.-
JIHBOCTb no.n.06Hoii TeOpeMbI. OH YTBepLK.n.aJI:
eCJIH Ha KaTeTax H mnoTeHY3e npSIMoyrOJIbHOrO TpeyrOJIb-
HHKa nOCTpOHTb KaKHe-JIH60 no.n.06HbIe <PH rypbI, B KOTOpbIX
KaTeTbI H rHnOTeHY3a SIBJISIIOTCH Cxo.n.CTBeHHbIMH CTopOHaMH,
TO Bcer.n.a HMeeT MeCTO paBeHCTBO:
Sa+Sb=Sc'
r,Qe Sa' Sb H Sc - fIJIOIII.a.nH 3THX <pHryp (qepT. 92).
.lI. 0 K a 3 aTe JI b C T B 0:
Sa =._!C.
Sc e 2 ' Sc e 2 '
SaC2 = a 2 S c ; Sb.c"=b2Sc.
CnO>KHB .n.Ba nOCJle.n.HHX
paBeHCTBa no qaCTSIM, no-
JIyqHM:
'-IepT. 92.
168
c2(Sa+Sb} = (a 2 + b 2 )Sc;
8
II
4epT. 93.

TaK KaK c 2 =a 2 +b 2 , TO
Sa+ Sb=Sc'
3TY TeOpeMY MO)KHO Ha3BaTb 0606111.eHHeM TeopeMbl nmpa-
ropa, rpeqeCKHM nHCaTeJIb npOKJI reBopHT, qTO no.n.o6HOH
Q:>opMe TeopeMbI TIHQ:>aropa -OT.n.aBaJIOCb npe.n.nOqTeHHe nepe.n.
.n.pymMH KaK BepHO BblpaLKalOIlI,eH caMYro CYTb .n.eJIa, «XOTSI
SI npeKJIOHSIIOCb H nepe.n. TeJ\1H, KOTopbIe BnepBbIe .n.onblTaJIHCb
.n.o HCTHHbI ::nOH np06JIeMbl, HO ern.e BbIIIIe SI CTaBJIIO aBTopa
«HaqaJI» EBKJIH.n.a He TOJIbKO nOTOMY, qTO OH cHa6.n.HJI Teope-
MY caMbIM y6e.n.HTeJIbHbIM .n.OKa3aTeJIbCTBOM, HO TaKLKe H no-
TOMY, qTO ern.e 60JIee 06rn.yIO np06JIeMY, co.n.epLKalll.yIOCSI B
6-ij KHHre, OH YTBep.n.HJI Ha HeonpOBepLKHMbIX OCHOBax HaYKH».
HHTepeCHO HHoe .n.OKa3aTeJIbCTBO TeopeMbI EBKJIH.n.a, He
OnHpalOlII.eeCSI Ha TeopeMY nHQ:>aropa, a BKJIIOqaIOrn.ee ee
B ce6H. PaCCMOTpHM 3TO .n.OKa3aTeJIbCTBO.
B npSIMoyrOJIbHOM TpeyrOJIbHHKe ABC onYCTHM BbICOTY
CD Ha mnoTeHY3Y AB (qepT. 93). 06pa30BaJIlICb TPH npH-
I\lOyrOJIbHbIX TpeyroJIbHHKa: LABC, nocTpoeHHblH Ha mno-
TeHY3e AB, LACD, I10CTpOeHHbIH Ha KaTeTe AC, LBCD,
nocTpoeHHbIH Ha KaTeTe BC. Bce OHH no.n.06bI. Hx nJIorn.a.n.H
OTHOCSITCSI KaK KBa.n.paTbI Cxo.n.CTBeHHbIX CTOpOH HJIH KaK nJIO-
lII.a.n.H KBa.n.paToB, nocTpoeHHbIx Ha mnoTeHY3e fI KaTeTax.
HMeeM:
SABC : SACD : SBCD =c 2 : b 2 : a 2 ,
HJlH
SABC =kc 2 , SACD =kb 2 H SBCD =ka 2 .
Ho
S ABC =S ACD +S BCD.
OrcIO.n.a
c 2 =a 2 +b 2 (TeopeMa nHQ:>aropa).
ECJIH LKe Ha rHnoTeHY3e II KaTeTax nOCTpoHTb KaKHe-JIH60
I1HbIe no.n.06HbIe MHoroyrOJIbHHKH, TO Bce paBHO HX nJIOIII.a.n.H
6y.n.YT OTHOCHTbCSI, KaK nJIOIII.a.n.H TpeyrOJIbHHKOB ABC, ACD
H BCD, H MbI npHxo.n.HM K TeopeMe EBKJIH.n.a.
PaccMoTpHM ern.e O.n.HO 0606rn.eHHe TeopeMbI nH<baropa.
LI.oKa2KeM TeopeMY:
Ha cTopoHax AB H AC npOH3BOJIbHOrO TpeyrOJIbHHKa ABC
KaK Ha cooTBeTcTBYIOrn.HX OCHOBaHHSIX nocTpoeHbl .n.Ba Ka-
KHX-JlH60 napaJIJIeJIOrpaMMa ABEF Ii ACOff (qepT. 94),
pacnOJI02KeHHbIX BHe TpeyrOJIbHHKa, a B OCTaJIbHOM COBep-
IIIeHHO npOH3BOJIbHbIe; CTOpOHbI EF H Off npo.n.OJI2KeHbI .n.o
B3aHMHoro nepeCeqeHHSI B TOqKe M.
.169

110Ka3aTb, qTO napaJIJIeJIOrpaMM, nOCTpOeHHblH Ha CTO-
pOHe BC H HMelOllI,lIH .n.pyrylO CTOPOHY, paBH)'1O II napa.'JJIe.lb-
HYIO AM, paBHOBeJIHK cYMMe nepBbIx .n.BYx.
ff 11 0 K a 3 aTe JI b C T B O.
no YCJIOBHIO AM=BK;
npOBOJ(Hi\1 OC II MM1H BP II
I/AM;
Tor.n.a
SCLM,N =S AMOC=S.4HGC;
l
f1
4epT. 94.
npSIMoyrOJIbHOrO TpeyrOJIbHHKa
PY>KHOCTII (qepT. 96).
4epT. 95.
no TeopeMe nmlJaropa
Q2+b 2 =C 2 ;
170
SBKJU,N=SA1PBA=S BEF.4;
CJIe.n.OBaTeJIbHO,
SBKLC =SAHGC+S,']BEF,
LI.oKa3aHHaSI TeopeMa
BKJIlOqaeT B ce6SI TeopeMY
nlHparopa KaK qaCTHbIH
CJlyqaH. LI.oKaLKHTe 9TO npH
nOMOlll,H qepTeLKa 95.
JIYHO'IKH fHnnoKpaTa.
DOCTpOHM Ha cTopoHax
KaK Ha .n.HaMeTpax nOJIyoK-
LI.oKa)!{eM, qTO cYMMa nJIO-
lII.a.n.eH .n.BYx 3aIIITpHXOBaHHblX
JIYHOqeK paBHa nJIOlll,a.n.H npSIMO-
yro.'JbHoro TpeyrOJIbHHKa.
o
/J
4epT. 96.

ft
YMHO>KHM ooe qaCTH nOCJIe.n.Hero .paBeHcTBa Ha 2":
1ta 2 + 1tb2 _ 1tC 2
 -.
EC.1H OT nJIOIII.a.ll.H nOJIYKpyra, nOCTpoeHHoro Ha mnOTe-
HY3e, OTHHMeM n.10Ill.a.n.H cerMeHTOB I H I I, TO OCTaHeTCSI nJIO-
lII.a.n.b Tpeyro.TIbHHKa ABC. EC.1H OT CYMMbI nJIOIII.a.n.eH .n.BYX
no.1YKpyroB, nocTpoeHHblx Ha KaTeTax, OTHHMeM nJIOIII.a.n.H
Tex >Ke .n.BYX cerMeHTOB, TO nOJIyqHM CYMMY n.10r.qa.n.eH .n.BYx
JIYHOqeK. npHHSIB BO BHHMaHHe paBeHCTBO (l), MbI MO>KeM
CKa3aTb, qTO CYMMa n.10r.qa.n.eH .n.BYX .1YHOqeK paBHa nJIOIII.a.n.H
Tpeyro.1bHHKa.
3TY 3a.n.aqy BnepBbIe nOCTaBH.1 H peIIIH.1 rpeqeCKHH MaTe-
MaTHK fHnnOKpaT, .n.OKa3aB TeM caMblM, qTO n.10Ill.a.n.b KpllBO-
JIHHeHHOH <PHrYPbI MO>KeT 6blTb paBHa n.10Ill.a.n.H npHMo.1HHeH-
HOH <PHrYPbI.
o >KH3HH fHnnOKpaTa H3 XHoca (V - IV B. .n.o H. 3.)
H3BeCTHO OqeHb HeMHoroe. no npo<peccHH OH 6blJ1 TOprOBIJ.eM.
HeKoTopoe BpeMH yqH.1CSI y nH<paropeHIJ.eB. Bc.1e.n.CTBHe Ka-
KorO-TO HeCqaCTbSI OH nOTeps-w Bce CBoe COCTOSIHHe. KaK
.' CKa3aJI 0 HeM ApHCTOTe.1b: «XOpOiliO H3BeCTHO, qTO .1IO.n.H,
r.1ynbIe B O.n.HOM OTIIOIIIeHHH, OTHIO.n.b He r.1ynbI B .n.pyr_HX.
B 3TOM HeT HHqerO CTpaHHoro: TaK, fHnnOKpaT, XOTSI H HC-
KYCHbIH B reOMeTpHlI, Ob!J1, nO-BH.n.HMOMY, B .n.pymx OTHOIIIe-
HHSIX CJIa6bIM H 6eCTOJ1l\OBbIM Qe.10neKOM; H OH, KaK rOBopSIT,
no CBoeH npOCTOTe nOTepSIJI 60JIbIIIYIO CYMMY .n.eHer 6.1aro.n.apSI
oOMaHY C60pr.qHKOB nOIII.1HH».
fHnnOKpaTY npHHa.n..1e)!<HT nepBbIH HanHcaHHbIH yqe6HHK
no reoMeTpHH, H qepe3 .n.Ba CTO.1eTHSI EBKJIH.n. MHoroe HcnO.1b-
3yeT H3 3TOro yqe6HHKa npIl COCTaB.1eHIIH <;Boero 3HaMeHHToro
TPy.n.a «HaqaJIa».
fnnnOKpaTY npllHa.n.JI.e>KaT TeopeMbI 06 yr.1ax B OKPY>K-
HOCTH, 06 OTHOIIIeHHH n.10r.q3;J,eH .n.nyx KpyroB.
B oc06eHHocTH MHoro c.n.e.1a.1 fnnnoKpaT B 06.1aCTH Tpex
3HaMeHIITbIX 3a.n.aq ,QpeBHocTH: 3a.n.aqH 00 y.n.BoeHlIH Ky6a, 3a-
.n.34H 0 TpHceKIJ.HH yrJIa H 3a.n.aqH 0 KBa.n.paType Kpyra. Co
BTOpoii H3 3THX 3a.n.aq qHTaTe.1b 03HaKOMHJICSI. C nepBoH II
TpeTbeH 03HaKOMHTCH n03>Ke.
(I)
OT nH4>aropeHcKHx IJHCeJI K 3ara)J.Ke «I>epMa
nml)arOpeHcKHe "mcJla. Cyr.qecTBoB3Hl1e nps-nloyro.1bHoro
Tpeyro.'!bIHlKa co CTopoHaMH 3, 4 II 5 CT3mlJ10 Bonpoc 60.1ee
o6Il1.ero xapaKTepa: KaKHe eIII;e cyr.qecTBYloT np5Jl\Ioyro.1blIbie
171

TpeyrOJIbHHKH, CTOpOHbI KOTOpblX Bblpa>KaIOTCSI u.eJIbIMH qHC-
JIaMH? Ka>K.n.yIO iaKYIO TPOHKY l1.eJIbIX qHCeJI Ha3bIBaIOT nmpa-
rOpOBOH TpOHKOH. MO>KHO BbIBeCTH <POPMYJIbI .lI.JISI nOJIyqeHHSI
BCeX TaKHX qHCeJI. 3TO MbI CeHqaC H C.n.eJIaeM.
nYCTb l1.eJIble'tmCJIa a, b H e 06pa3YIOT [IH<paropOBY TPOHKY,
T. e. CBSI3aHbI COOTHOlileHueM
a 2 +b 2 =e 2 .
Pa3.n.eJIHM 06e qaCTU oHoro paBeHCTBa Ha e 2 :,
(r +(r = 1.
nOJIO>KHM ,QJISI KpaTlWCTH:
a
-=X
c
H
b
c=y;
Tor.n.a
X2+y2= 1,
UJIU
y2=(I+x)(I--x}.
HanHilieM nOCJIeAHee paBeHCTBO B BH.n.e HenpepbIBHoM npo-
nOpl1.HH:
y _ I-x
l+x --y
npe.n.CTaBHM KaK OTHOIIIeHHe .n.BYx l1.eJIbIX quce.!} n H
CTaBHM CUCTeMY ypaBHeHHl1 C .n.BYMSI HeH3BeCTHbIMH:
OOIll,ee 3HatleHHe .n.nyx oTHOIIIeHHH B 31011 npOnOpl1.HH
fll
co-
I Y _fll.
l+x -n'
I-x In
---y - n'
KOTopaSI MOLKeT 6bITb npHBe.n.eHa K BH.n.y:
{ mX-flY =-m;
nx -I- my = n.
172

YpaBHSIB K03rprpHIJ.HeHTbl npH HeH3BeCTHO:vl y H 3aTeM
C.10LKHB no tlaCTSIM ypaBHeHHSI CHCTeMbI, Mbl nO.1yqHM:
OrqOJl,a
(m2 + n 2 ) x = n 2 - m 2 .
n 2 - m 2
X = m 2 + n 2 .
nOJl,06HbIM 06pa30M
Ha, u.n.eM qTO
2mn
y = m 2 +n 2 '
nO.n.CTaBJIHH  H
c
b
B\1eCTO x H y, 6y.n.eM HMeTb:
c
a
c
n 2 '- m 2 b
m 2 + n 2 ' c
2mn
m 2 + n 2 ,
H.1H
c c
a = ( n2 -m2 ) b 2mn
m 2 + n 2 ' = . m 2 + n 2 .
0603HaqHB 06Il1,HH .lI.J15I a H b paUHOHa.1bHbn'1 MHOLKHTeJIb
2  2 qepe3 k, T. e. no.10JKHB, qTO C = (m 2 + n2) k, MbI
m n
npH.n.eM K TpeM COOTHOIIIeHHSIM:
! a = (n 2 -m 2 ).k;
b = (2mn).k;
C = (m 2 + n2).k.
TaKHM oopa30M, eCJIH qHCJIa a, b H C 06pa3YIOT nHrparopOBY
TPOfIKY, TO OHH COOTBeTCTBeHHO nponopUHOHa.1bHbl qHCJIaM
BH.n.a:
(I)
n 2 _m 2 , 2mn H m 2 +n 2 .
JIerKo npOBepHTb H 06paTHoe no.nOJKeHHe, qTO BCSIKHe
TPH qHCJIa a, b H C, onpe.n.e.1eHHbIe paBeHCTBaMH (1), 06pa3YIOT
nHrparopOBY TPOMKY. LLml :OHoro CTOHT TO.1bKO COCTaBHTb CYM-
MY a 2 +b 2 H cpaBHHTb ee C e 2 . B caMOM .n.e.1e:
a 2 + b 2 = k 2 (n 4 _ 2m 2 n 2 + m 4 + 4m 2 n 2 ) = k 2 (n 2 + m2}2,
C .n.pyroH CTOpOHhI,
e 2 = k 2 (m 2 + n 2 }2;
CJIe.n.OBaTeJIbIIO,
a 2 + b 2 = e 2 .
173

COCTaBHM npH nOMOllI.H iPOpMy.1bl (I) HeCKOJIbKO nHiParO-
pOBbIX TpOeK. nYCTb m= I; n=2 H k= I. Tor.n.a:
a = (4-1).1=3; b=2.1.2=4; e=(1 +4}.1 =5.
nOJIyqHJIH H3BeCTHYIO HaM TPOHKY emneTCKoro TpeyrOJIb-
HHRa.
nYCTb m=3; n=5; k=l.
Tor,Aa:
a=(25-9}.1 = 16; b=2.5.3.1 =30
e = (25 + 9).1 = 34;
34 2 = 16 2 + 30 2 ;
1156 = 256 + 900;
1156 = 1156.
B CBSI3H C paccMoTpeHHeM nniParopeHcKHx rmce.1 oOJIee
HJIH MeHee eCTeCTBeHHO B03HHKaeT Bonpoc 0 B03MOJKHOCTH
c.1e,QYIOIlI,erO 0606111.eHHH 3a.n.aqH: cYIlI,eCTBYIOT JlH l1.eJIble no-
JIOJKHTe.1bHble qHCJIa a, b H e, KO'1'Opble y.n.oBJIeTBOpSIJIH 6bI
paBeHcTBY
a 3 +b 3 =e 3
HJ1I1 paBeHcTBY
a 4 + b 4 = e 4 ,
HJIH, BOOOllI.e rOBopSI, paBeHcTBY
an + b n = en,
rJl,e n - l{eJIOe qnCJIO, oOJlbIIIee 2? OTBeT ObIJI Jl,aH iPpaHl1.Y3-
CKI!M IIWTeMaTHKOM DbepmI $epM3 (l601-1665).
cPcp.la HMeJI npIfBblqKY, qlITaSI COqUHeHHSI MaTeMaTHKOB,
Jl,e.1aTb Ha nOJISIX KHHr CBOH npHlVIeqaHHSI, BbICKa3bIBaTb pa3-
.1H4H.>Ie Teope:\lbI B CBSI3H C npOqflTaHHbIM. LIaCTO, BbICKa3aB
TY HJIH HHYIO TeopeMY, $epMa He npHBo.n.nJI He06xo.n.HMoro
.n.OKa3aTeJIbCTBa 310H TeopelVlbI.
TaK CJIytlHJIOCb II C .n.OKa3aTeJIbCTBOM HeB03MOJKHOCTH
peIIIHTb B LLeJIbIX tIHC.'laX ypal3HeHHe:
an+bn=e n
npH n>2. Ha nO:IflX O,'J,HOro H3 COqlIHeHIIH rpeqeCKOrO MaTe-
MaTIIKa J].HOiPaHTa ObIJIa o(5HapY>KeHa 3anHCb cPep:\1a: «CYMMa
o.n.HHaKOBbIX CTf;ijIl,ei'r .n.BYX qHCeJI He MOJKeT ObITb TOH me
CTeneHbIO KaKoro-JIH60 TpcTbero ql!CJIa. I1CKJIIOqeHHe COCTaB-
.1SIeT JIHilib BTopaSI CTeneHb, ,Qm! KOTopaH 3TO B03MmKHO.
174

Y MeHSI eCTb 3TOMY nOHCTHHe y.n.HBHTe.1bHOe .n.OKa3aTeJIbCTBO,
HO nO!ISI C.1HIIIKOM V3KH. 4T06bl BleCTHTb ero».
ECJIH .n.pynie TeopeMbI, 3anHcaHHble cPepMa Ha nOJIHX KHHr
H He .n.OKa3aHHble HM, BnOC.lJelJ.CTBHH 6bWH ,lI;OKa3aHbl .n.PYrHMH
MaTe:'WHma:\1H, TO Teope:\Ja 0 paBeHCTBe
an+bn=cz
B 06Il1,eM cpopMe HHKor.n.a H HHKeM BnOCJIe.n.CTBHH He 6bIJIa HH
Jl,OKa3aHa, HH onpOBeprHYTa, HeCMOTpSI Ha TO qTO IJ.eJIbll'I pH.n.
KpynHeHIIIHX MaTeMaTHKOB nb1Ta.1CH 3TO c.n.eJIaTb. TaK 3Ta Teo-
peMa, H3BeCTHaH B l\IaTeMaTl1Ke no.n. HMeHeM TeopeMbI cPepMa,
H OCTaJIaCb 3ara.n.KOH.
o LKH3HH cPepMa, npOTeKilieH OqeHb cnOKOHHO, MbI 3HaeM
Ma.10. nO.n.rOTaBJIHBaSICb K Tpy.n.OBOII .n.eSITe.1bHOCTH, OH H3Y-
4a.1 IOpH.n.ntteCIme HaYKlI, 3aTeM CTa:I a.n.BOKaTOM H, HaKOHeLl
COBeTIIUKOM npOBUHIJ.HaJIbHOrO nap.1aMeHTa. 4C::Cbl .n.ocyra
cPepMa OT.n.aBaJI r!IaBHbIM 06pa30M 3aHSITHSIM MaTeMaTHKOH.
B 3TOH HaYKe OH .n.OCTur HCK.1IOqHTe.1bHoro ycnexa. OH He
Hane4aTa.1 npH LKH3HH HH OlJ.HOH CTPOqKH CBOlIX Hcc.1e.n.OBa-
HUrl. OHH COXpaHlIJIHCb B cpop:\le PYKonHceH, OT.n.e.1bHbIX 3anH-
cel'l Ha K.10qKaX 6YMarH H.1H Ha nO.1SIX CTpaHHIJ. Tex KHHr,
KOTopbIe OH H3yqaJI. Ho HeCMOTpSI Ha 3TO. ero OTKpbITHSI B
06JIacTH MaTeMaTHKH YLKe npH LKH3HH aBTopa CTaHOBSITCH
H3BeCTHbIMH H OKa3bIBalOT B.'IHHHHe Ha HayqHyIO MbIC.1b.
HeCKOJlbKO BonpOCOB "3 TeMbl «PaBHOBeJlHKHe cp"rypbI»
cPHrYPbI Ha3blBaIOTCH p a B HOB e JI H K H M H, eC.1H OHH
IIMeIOT paBHble n.10Ill,a.n.H.
4epT. 97.
4epT. 98.
4epT. 99.
Ha OCHOBe H3yqCHHbiX Ba:\111 reOMeTpHqeCKUX COOT-
HOIIIeHHH .n.OKaJKlne paBHOBe.1HKOCTb 3aIIITpHXOBaHHbIX CPH-
ryp Ha KaLK.n.OM 113 .n.aHHbIX qepTeJKeH (97-103).
4eTblpe nOCJIe.n.HHX qepTeJKa n03BOJISIT nO-UHOMY C T04KH
SpeHI1H paBHbIx nJIor.qa.ll.eH C<PoP:'IYJIHpOBaTb H3BeCTHbie 83:\1
175

4epT, 100,
4epT. 102.
l.JepT. 101.
l.JepT. 103.
TeopeMbI 0 nponOpl1.I;IOHaJIbHbIX oTpe3Kax B Kpyre. C.n.eJIaH-
Te 3TO.
I1HTepecHo, qTO MMeHHO B 3TOH <popMe Bce 3TM TeopeMbI
H 6bLnH Haif,QeHbi .D.peBHIIMM rpeKaMM.
3a)J.aQH H rOOBOOMKH
11: B e r 0 JI 0 B 0 JI 0 M K M. 1. $lirypa ABCDEF CO-
CTOMT li3 Tpex CnJIOIIIHbIX KBa.n.paToB (qepT. 104). Tpe6yeTc5I
pa3pe3aTb 3Ty <pMrypy Ha .n.Be qaCTli TaK. qT06bI M3 06pa30BaB-
IIIliXCSI qaCTeH MOJKHO 6b!J10 COCTaBliTb KBa,ll;paTHYIO paM KY.
OTBepCTMe BHYTPH paMKH .n.OJ1JKIIO TOJKe HMeTb KBa.n.paTHYIO
<POPMY, paBHyIO K8)K.D.OMY li3 Tpex KBa,QpaTOB, COCTaBJISIIOllI.YIO
.n.aHHYIO <plirypy.
2. B03bMliTe 5 paBHblx KB3.D.paTOB. KaJK):lbIA H3 HHX pag-
peJKbTe Ha TpaneLLliIO li npSIMoyroJIbHbIH TpeyrOJIbHliK TaK,
qTOObI ero KaTeTbI 6b!J1li O.n.liH paBeH CTopOHe KBa.n.paTa, a
.n.pyroH nOJIOBliHe ee (qepT. 105). 113 nOJIyqMBIIIliXCH 10 <pHryp
COCT2BbTe O.n.HH KBa.n.paT. (3TH rOJIOBOJIOMKli B3S1TbI: o.n.Ha H3
KHHrH' B. A. Kop.n.eMCKoro «M2TeM2THqeCK2SI CMeK2JIKa»,
176

.n.pyraH H3 KHHrH 5: A. KOP.n.eMcKoro H H. B. PycaJJeBa «Y.n.H-
BHTeJ1bHblH I<Ba.n.paT». B 3THX KHHrax llHTaTeJJb HaJ"I,a,eT MIfOro
no.n.06HblX 3a.n.aq II 3a.n.aq 60JIee Tpy.n.HbIX.}
113 BCeX nJ10CImX MHoroyrOJIbIlIIKOB, OqeBH.n.HO, CaMblM
npOCTblM H no CBOHM CBollcTBaM II no cnoc06y H3MepeHHSI fWO-
r.qaH 6y.n.eT KBa.n.paT. Mbl 3HaeM, qTO JII060H nJIOCKHM MHoro-
c
"
I
I
f I
I
I
I
I
I
/
II
4epT. 4epT. 105.
yrOJIbHHK MOJKHO npeBpaTHTb B paBHoBeJ1HKHH TpeyrOJIbHHK,
a nOCJIe.n.HHH - B paBHOBeJIHKUH KBa.n.paT. D03TOMY BnOJ1He
3aKoHoMepHo nOSIBHBIIIeecSI eme y .n.peBHHx rpeKoB CTpeMJIeHUe
HaHTH cnoc06bI npeBpar.qeHHSI JlI060H IPurYPbI, .n.a>Ke orpaHH-
qeHHOH KpHBOJlIIHeilHbIM KOHTYPOM, B paBHoBeJ1HKHH eH KBa.n.-
paT. 113 TaKHX IPHryp HaH6oJ1ee qaCTO B npaKTHKe BCTpeqaeTCH
KPYr. Ho nonbITKH rpeKOB 3aMeHHTb .n.aHHblll Kpyr paBHOBe-
JlHKHM eMY KBa.n.paTOM He yBeHqaJIHCb ycneXOM 1 ). 3Ta 3a,Qa4a
no.n. Ha3BaHHeM «3a.n.aqa 0 B
KBa.n.paType Kpyra» CTaJ1a
O.n.HOH H3 3HaMeHHTblX 3a-
.n.aq .n.peBHOCTH. I1cTopHH ee
OXBaTbIBaeT He MeHee qeTbI-
pex TbICSIqeJ1eTHH. EIO O.n.H-
HaKOBO HHTepeCOBaJIHCb KaK
JII06HTeJ1H, TaK H 3HaMeHH- /I
TbIe MaTeMaTJIKH. 3.n.eCb MbJ
3TY 3a.n.aqy nOCTaBHJJII B
CBSI3H C BonpOCOM 0 paBHO-
BeJIHKOCTH IPHryp. DO.n.P06HO 0 KBa.n.paType Kpyra qHTa-
TeJIb npOqTfT B rJlaBe XII.
4HTaTeJIb nOJIyqm 60JIbiliOe y.n.oBJIeTBOpeHHe, eCJIH peIIIHT
CJle.n.YIOr.qHe 3a.n.aqH Ha nOCTpoeHHe:
F
l'
lJ
m
4epT. 106.
1) 3J1,eCb rOBOpJlTCSI 0 HeYJI,a'lHbIX nonblTKax pa3pewHTb 3aJl,a4Y
o KBa,/!parype Kpyra npll nOMOIl.lH UlIPKYJlSJ II JJHHeHKH.
177

I. ltaHHbIH TpeyrOJIbHHK pa3.n.eJIHTb Ha .lI.Be paBHOBeJ1HKHe
qaCTH npHMOH, nepneH.n.HKYJ1SIpHOH K OCHOBaHHIO (qepT. 106).
Y K a 3 a H H e. nYCTb EF .n.eJ1HT TpeyroJ1bHHK ABC Ha
.n.Be paBHble no nJ10Ill,a.n.H qaCTH. CocTaBbTe CHCTeMY ypaBHe-
HHH C HeH3BeCTHbIMH x H y, HCXO.n.SI H3 Toro, qTO nJ10III.a.n.b Tpe-
yro,,1bHHKa ECF paBHa no,,10BHHe nJ10III.a.n.H TpeyroJlbHHKa
ABC H qTO TpeyroJ1bHHK EFC no.n.06eH TpeyrOJlbHHKy BCD.
Om8em. x=v m.
2. ltaHHblH TpeyrOJIbHHK pa3J1,eJ1HTb Ha TPH paBHOBeJIH-
l<He qaCTH npSIMbIMH, napaJlJ1eJIbHbIMH OCHOBaHHIO.
3. npoBe.n.HTe caMOCTOSITeJIbHO eme O)JHO .n.OKa3aTeJ1bCTBO
TeopeMbI Dmparopa.
o
E
I! L
l.JepT. 107.
ltaH npSIMoyrOJIbHblH TpeyroJ1bHHK ABC. Ha ero KaTeTax
AC, BC II rnnoTeHY3e AB nocTpoeHbI KBa.n.paTbI (qepT. 107).
KBa.n.paT, nocTpoeHHblH Ha KaTeTe BC, pa3pe3aH TaK, qTO
FN II AB H NO .lAB. KBa.n.paT, nocTpoeHHbIH Ha KaTeTe AC,
p23pe3aH Ha .n.Be qaCTH npSIMoH AM, HBJIHIOIlI,eHCH np0.n.OJIJKe-
HHeM EA. nOJIyqeHHbIe nocJle pa3pe3aHHH nHTb qaCTeH YJIO-
LKHTe Ha KBa.n.paTe ABDE TaK, qT06bI OHH nOJ1HOCTbIO 3anOJI-
HIWH :>TOT KBa.n.paT..
178

JIHTEPATYPA
B. JI H T U M a H, TeopeMa nmparopa, OHTH, 1962.
B. H. JI e 6 e .ll e B, KTO aBTop nepBblx TeopeM reoMeTplili, M.,
1917.
B. H. JI e 6 e Jl, e B, 3HaMeHliTbie reoMeTpH4ecKHe 3aJl,a'lli Jl,peB-
HOCTH, M., 1917.
B. n, III e p e MeT e B C K Ii ii, 04epKH no HCTOplill MaTeMaTHKli,
Y4nellrH3, M., 1940.
p, K y p a H T Ii r. Po 6 6 H H C,' 4To TaKoe MaTeMaTHKa, foc-
TeXH3AaT, 1947.
5. A, K 0 p Jl, e M C K II 11 li H. B. P y C a JJ e B, YJl,IIBHTeJJbHblii
KBa,n,paT, focTexlI3Jl,aT, M., 1952,
5. A. K 0 p Jl, e M C K II ii, MaTeMaTH4ecKaH CMeKaJIKa, focTex-
H311aT, 1963.
J\\, 11 JJ b li H H E. C era JI, KaK 4eJJOBeK CTaJJ BeJJHKaHOM,
JJ.eTrH3, 1946,
r. P 0 A e M a x e pliO. Ten JJ II 11., 4HcJJa H <pllrypbI, <1>113-
MaTrll3, 1962.

L'
,
1::: 1 2
1+3=2 2
1+3+ 5=3 2
,\
r Jl A B A VIII
METO MATEMATH4ECKOA HHYKQHH
«...DOHHMaHHe H YMeHHe npaBH.nbHo
npllMeHHTb npHHUHn MaTeMaTHqeCKOll
HHJI,YKUHll HB.nHeTCH XOPOWHM KpllTe-
pHeM .norll'leCKOW 3pe.oCTn, KOTopaR
cOBepweHHo He06xoJl,HMa MaTeMaTllKY».
A. /l. KO/lAlOlOp08
06 JlH)J.y;cLlIIII, CMe.lIOM c1JH3HKe II He MeHee CMeJiOM, HO 60Jlee
OCTOpO>KHOM MaTeManfKe
91 pn U3YtJelUIH SIBJ1eHHH B JII060ii 06.naCTH 3Ha-
HHSI - 6y.n.b TO MaTeMaTHKa HJIH HCTOpHH, cpll3HKa
IWI1 I\fe.n.ULLHHa, aCTpOHOMHSI IfJ1H 3KOHmmKa-
BCIO.n.y H Bcer.n.a OCHOBHbIM 3TaijO:\f SIS.ITHeTCfI YCTa-
HOBJleHHe onpe.n.e.neHHbIX 3aKOHOMepHOCTen, CBH3bIB81OIII.HX
OT.n.eJIbHble 3JIel\leHTbl If3yqaeMOrO SIBJIeHHSI. llano y.n.aeTCH
OTKpbITbIe 3aKOHOMepHOCTIf oT06pa3HTb B onpe.n.eJIeHHbIX <PoP-
My.lJaX, H tJeJIOBeK, H3yqHBIIIIIH SI3blI<, CPOPMYJI, MOJKeT cpa3Y
YBH.AeTb pe3YJIbTaTbI Tpy.n.a, HHOma .n.OJIroro II ynopHoro,
KOTOPblil nOTpeOOBa.'JCSI .n.JJ5J OTKi:>bITHH 3THX 3al<OHOMepHOCTeii.
Ho, .AmI Toro tJT06bI BOCn!lTaTb B ce6e IfCCJIe,'(OBaTeJIH,
ne.n.OCTaTOlIHO nOHHMaTb TOJIbKO KOHe1JHble pe3y.nbTaTbI HC-
180

CJle.n.OBaHlH!. He06xo.n.HIIIO npOHHKHYTb B caMbll1 npoecc
HCCJIe.n.OBaTeJIbCIWII pa60Tbl Ii YCBOHTb TOT MeTO.n., KOTOpblH
6b1JI HCnOJIb30BaH, qT06bl npHHTH K y.n.aqHOMY pe3YJIbTaTY.
«He TO Ba>KHO 3HaTb, qTO 3eMJISI KpyrJIaSI, a TO Ba>KHO 3HaTb,
KaK JIIO.n.H .n.OIIIJIH .n.o 3Toro», - nHCaJI JI. H. TOJICT0I1. DOHC-
IUI BepHoro MeTo.n.a - 3TO HaqaJIbHblH 3Tan H3yqeHHSI JII060ro
SIBJIeHHSI. 4aCTO Bbl60p MeTo.n.a peiliaeT ycnex Bcero HCCJIe-
.n.OBaHHSI. 3HaKOMCTBO C pa3JIHqHbIMH MeTo.n.aMH HCCJIe.n.OBaHHSI
H OBJIa.n.eHlfe HeKOTopbIMH H3 HHX eCTb o.n.Ha H3 B3fKHeHIIIHX
3a)1aq, KOTopble .n.OJIfKeH nOCTaBHTb nepe.n. C060H Ka>K.n.bIH,
3aHHMaSICb B lIlKOJIe.
Pa3JIHqHbie HaYKH nOJIb3YIOTCSI pa3HblMH MeTo.n.aMU pa-
60Tbl, H .n.arKe B 06JIaCTH O.n.HOH HaYKH" HanprIl\lep MaTeMaTHKH,
CYIlI,eCTByeT MHoro pa3HbIX MeTo.n.OB 11CCJIe.n.OBaHHSI. 0 HeKO-
TOpblX H3 HHX MbI CeHqaC H norOBopHM C BaMH, qHTaTeJIb.
DpeLK.n.e Bcero BcnOMHHM O.n.HH H3 MeTo.n.OB HCCJIe.n.OBaHHSI,
KOTOpblM qaCTO nOJIb3YIOTCSI iPH3HK11. BbI LKeJIaeTe YCTaHOBHTb
3aKOH 0 COnpOTrIBJIeHHH npOBo.n.a npH npOXOLK.n.eH11H qepe3
Hero 3.1eKTpUqeCKOrO TOKa. 4T06bl OTKpblTb Te CBSI3lf, KOTopble
cYIll,eCTBYIOT B .n.aHHoM CJIyqae lIIe>K.n.y oT.n.eJIbHbIMH 3.'1ell-leH-
TaMH H3yqaeMOrO SIBJIf;HHSI: CIIJIOH TOKa, .n.JIHHOH npoBo.n.a H
nJIOIll,a.n.bIO ero nOnepeqHOrO CeqeHHSI, Bbl 6ephe HCTOqHHK
TOKa, al\mepMeTp H npOBOJIOKH pa3.'1HQHofl .n.J1HHbI H pa3JIllq-
Horo nOnepeqHOrO celleHHSI, c.n.eJIaHHbIe H3 pa3HbIX MaTe-
pHaJIOB. 3aTeM npHCOe)1HHSIeTe I{ HCTOqHHKy TOKa XOTR 6hI
lIIe.n.HYIO npOBO.'10KY, BKJIIOIIHB B u.enb aJl.mepMeTp, H 3aMblKae-
Te TOK (qepT. 108). CTpeJIKa a!\WepMeTpa OTKJIOHIITCSI H nOKa-
>KeT BaM, KaK BeJ1HK TOK, H.n.YIll,HH no u.enH. BbI y.n.J1HHSIeTe
npOBOJIOKY B.n.Boe, BqeTBepO II T. .n.. H CJIe.n.rlTe 3a nOKa3a-
IUlSIMH aMnepMeTpa; 3aMeqaeTe, qTO CHJIa TOKa YMeHbIIIH-
JIaCb COOTBeTCTBeHHO B 2, 4 pa3a H T. .n.. 11, CJIe.n.OBaTeJIbHO,
COnpOTrIBJIeHrle npoBo.n.HrlKa yBeJIHqHJIOCb BO CTOJIbKO LKe
pa3, BO CKO.lbKO pa3 YBeJlHqHJIaCb ero .n.JIHHa. Bbl .n.eJIaeTe
BbIBO,'J,: conpotnU6/le1-tUe npo6oi:J1-tllKa nponopI.{UOHa/lb1-tO ezo
o.lUHe.
LI.aJIee BbI npHcoe.n.HHSIeTe K UCTOQHIIKY TOKa Me.n.HYIO }j{e
npOBOJIOKY onpe,ll;eJIeHHOH .n.JIIIHbI H c nOnepeqHbIM CeqeHHeM
I K6. .MM 11 OTMeqaeTe CIIJIY TOKa, nOK33bIBaeMYIO aJ\mep-
MeTpOM. 3aTelli npHKJIIOqaeTe K HCTOqHIIKY TOKa MC.n.HYIO
npOBD.TIOKY TaKofl LKe .n.JIHHbI, HO nJIOIII.a.n.b CeqeHllSI KOTOpOH
B 2 pa3a MeHee, 3aTeM B 4 pa33 MeHee, qeM paHee, 11 T. .n..
3al\IeqaeTe, qTO TOI< TaKLKe YMeHblIlHTcSI B 2 pa3a, 4 pa3a H,
CJIe.n.oBaTeJIbHO, COnpOTrIBJIelllIe YBeJIllqHBaeTCfl COOTBCTCT13eil-
IBJ

HO B 2 pa3a, 4 pa3a II T. .n.. Bbl .n.eJIaeTe BbIBOJ\: conpomu8AeHue
npo8ooHuKa o6pamHo nponOptUOHa,1bftO n/lou{aou ezo none-
pe'lHOZO Ce'leHUR.
4epT. 108.
HaKoHeu:, Bbl >KeJIaeTe BbISICHHTb, 3aBHCHT JIH conpoTHB-
JIeHHe npoBo.n.a OT MaTepHaJIa, H3 KOToporo OH c.n.eJIaH. lJ,JISI
3Toro BhI 6epeTe npoBo.n.a o.n.HHaKoBoH .n.JIHHW H o.n.HHaKoBoH
nJIOIll,a.n.H nOnepeqHOrO Ce'leHHSI, HO c.n.eJIaHHble H3 pa3Horo
MaTepHaJIa: H3 Me.n.H, >KeJIe3a, HHKeJIHHa, H no Oqepe)I,H BKJIIOqae-
Te 3TH npOBo.n.a. OrMeTHB no aMnepMeTPY KaLK.n.bIH pa3 crlJlY
TOKa, BbI .n.eJIaeTe BbIBO.n., qTO conpOTHBJIeHHe ;Ke.1Je3a 60JIbIIIe,
qeM COnpOTliBJIeHHe Me.n.H, a conpOTHBJIeHHe HHKeJIHHa 60JIbIIIe,
qeM cOnpOTHBJIeHHe LKeJIe3a. CJIe.n.OBaTeJIbHO, conpOTHBJIeHHe
npOBo.n.HHKa 3aBHCHT OT TOro MaTepHaJIa, H3 KOToporo c.n.eJIaH
npOBo.n.HHK. BBe.n.SI nOHSITne 06 y.n.eJIbHOM conpOTHBJIeHIUI,
Bbl yme MOLKeTe C<pOpMy.'lHpOBaTb 3aKOH: conpOmU6/leHUe npo-
800HuKa npRMO nponopu,uoHa/lbHo' ezo O/lUHe U o6pamHo npo-
nopu,UOHa,1bHO n,10U{aOU nOnepe'lHOZO ce/.{eHUR.
OCTaeTCfI no.n.MeqeHHYIO 3aKoHoMepHocTb Bblpa3HTb MaTe-
MamqecKoH CPOPMYJIOH. 3Ta <popMy.lJa TaKOBa:
1
R=ks'
r.n.e l - .n.JIHHa npOBo.n.a B MeTpax, S - nJJOIII.a.n.b nonepeq-
Horo CeqeHHSI B KBa.n.paTHbIX I\UlJIJIHMeTpax H R - COnpOTlIB-
JIeHHe B OMax. K03cp<pnl1.neHT nponopU:HollaJIbHOCTH k Bblpa-
LKaeT y.n.eJIbHOe cOnpOTHBJIeHne, pa3Hoe AJISI pa3JIlIqHblX l\Ja-
TepHaJIOB.
182

BOT O.ll.HH H3 MeTo.n.oB HCCJIe.ll.OBaHIH1, KOTOpbIM qaCTO
n0J1b3YIOTCSI B iPH3HKe l ). B qeM LKe ero CYllHOCTb?
DpO.ll.eJIaB .n.OCTaTOqHOe KOJIHqeCTBO OnblTOB, HCCJ1e.n.OBa-
TeJ1b no.n.MeqaeT onpe.n.eJIeHHbIe 3aKOHOMepHOCTH, CBSI3bIBaIO-
IlI,He OT.n.e.lbHbIe 3JleMeHTbl H3yqaeMOrO 51B"leHHH; yCTaJWBJ1rI-
BaeT, qTO 3TH 3aKOHOMepHOCTH OCTaIOTCSI CnpaBe.ll.JIHBblMH llJISI
Bcex Tex CJIYLIaeB, KOTopble 6blJ1H HM pa306paHbl, H, HaKOHeIJ:,
Haxo.n.HT B03MOLKHbIM pacnpOCTpaHrITb n0.ll.MeqeHHble 3aKO-
HOMepHOCTH H Ha Te CJIyqaH, KOTOpbIe He 6blJIH npOBepeHbl
onblTOM. CTeneHb YBepeHHOCTH, C KOTOpOH OH MOLKeT c.n.eJ1aTb
TaKOH nepexo.n. OT qaCTHblX YTBepLK.n.eHHH K 06Il1,HM, If, CJIe-
.n.OB3TeJlbHO, CTeneHb YBepeHHOCTH B cnpaBe.n.JIHBOCTH BhICKa-
3aHHorO 3aKOHa 3aBI1CHT OT qHCJIa npOJl.eJIaHHblX onbITOB.
4aCTO no.n.06Horo pO.ll.a paCCYLK.n.eHHSI 6bIBaIOT BnQ.,lHe y6e-
.n.I1TblbHbIMII .
yITaK, HCnOJIb3YSI nO,ll;06Hbli'I MeTO.n., MbI no.n.MeqaeM onpe-
.n.eJIeHHYIO CBH3b Melli.n.y 3.1eMeHTaMH H3Yl.Jae:\Ioro SIBJIeHHSI,
cnpaBe.n.JIrIBYIO .n.JISI MHOnIX IIaCTHbIX CJIYl.JaeB; 3aTeM pacnpo-
CTpaHSIeM ee Ha Bce CJIyqaH Bo06Il1,e, YCTaHaB"lHBaSJ TeM caMbiM
06Il1,HH 3aKOH, paCI\pbIBaIOIlI,Hi'I cymHOCTb .n.aHHoro SIBJIeHHH.
TaKoH MeTO.n. pacCY>K.n.eHHH, Kor.n.a OT qaCTHblX YTBep-
LK.n.eHHH nepexo.n.SIT K 06111.H:\I, B JIOrHKe Ha3bIBaIOT H H.n. Y K -
IJ: H e iL MeTO.n., HCnOJIb30BaHHbIH HaMH npu peIIIeHHH Bonpoca
o conpOTHBJIeHHJ} npOBo.n.a, 6bIJIO 6bl IJ:eJIeC006pa3HO Ha3BaTb
MeTo.n.OM «3MnHpHqeCKOH HH.lI.YKLLHH», 3MnHpHqeCKOH nOTOMY,
LITO B03MOLKHOCTb H3IIIHX OKOHqaTeJIbHblX 3aKJIIOqeHHII BblTe-
KaJIa H3 Ha6JIIO.n.eHHH Ha.n. pe3YJIbTaTa!\UI HeKOTopblX onblTOB.
4aIll,e 3TOT MeTO.n. Ha3bIBaIOT MeTo.n.OM «HenOJIHOH rIH.n.YKIJ:HH»;
TepMHH «HenOJIHOH» CTaHeT nOHHTHblM H3 .n.aJIbHeHiliero.
MeTO.n. HenOJ1HOH HH.n.YKIJ:HH IIIHpOKO npHMeHSIeTCSI He
TOJ1bKO B 06JIaCTH iPH3HKH H eCTeCTB03HaHHSI Bo06me, HO
H B 06JIaCTH MaTeMaTHKH. B qaCTHOCTH, MHOrHe CBOHCTBa
qHCeJl 6bIJUi CHaLlaJIa OTKpbITbl nYTeM Ha6JIIO.n.eHHH H HenOJ1-
HOII HH.n.YKl1.HH, OTKpblTbI 3a.n.OJIro .n.o TOro, KaK HX HCTHH-
HOCTb 6blJIa crpore .n.OKa3aHa. 11 Bee LKe B MaTeMaTHKe HenOJ1-
HaSI HH.n.YKIJ:HSI 06blqHO He SIBJISIeTCSI 3aKJIIOIHITeJ1bHOH iPa30H
HCCJIe.n.OBaHHSI. EC.1H iPH3HK B pe3y.lbTaTe pSI.n.a Ha6JIIO-
.n.eHIiH H HenOJI HOH HH.n.YKIJ:HH Haxo.n.HT B03MO)KHbIM YTBep)K-
,QaTb, qTO 3aKQHOMepHOCTH, BblSIB.leHHble HM TaKHM nYTeM,
1) He HY>KHO JlYJllaTb, 'ITO YKa3aHHblH MeTO.lJ. HBJlHeTCH eAHHCTBeH-
HO B03J110H<HbIM B 1jJ1I3HKe; cymecTBYIOT 11 APyrlle MeTO.lJ.bI, n03BOJIHIO-
mHe YCTaHOBnellHble 3aKOHLI Z!,OKa3aTb nOfll'leCKH, Ha oa3e oonee 00'
U{HX lICXO;(HblX nonOlKeHlIH. .
183

'>".e.. 
-
", ...: ".


..1
'11
. f
f
",.< ..'
.-
,
i.
(.
n£ep cI>epMa,
MorYT 6e3 .n.aJIbHef%IIIHx HCCJle.n.OBaHHfr O!;>ITb npHHSITbI 3a 06.
llI.HH 3aKOH, KOTOPOMY no.n.qHHSIeTCSI SIBJleHHe, TO MaTeMatHK
He pHCKHeT TaK nOCTYILHTb. OH tJO.n.BeprHeT CBOH OTKpblTHH
.n.onOJlHHTMbHbIM, 60JIee TOqHbIM HCCJle,QOBaHHSIM, nOCJIe KO-
TOpblX 3aKJIIOtIeHHSI, .n.OObITble nyreM HH.n.YKl1.HH, HJIH CTaHO-
BHTCSI saKOHOM, HJIH OTBepraIOTCSI, KaK OIIIH60qHbIe l ). 3.n.eCb
nOJIe3HO npHBeCTH CJIOBa JIeoHap.n.a 3frJIepa: -«... MbI .n.OJILKHbI
npOSIBJlSITb 6oJIbIIIYlO OCTOpO>KHOCTb, IlT06bI He npHHHTb 3a
HCTHHHbI& TaKHe cBOfrCTBa t:!HCM, KOTOpble MbI OTKpbIJIH nyreM
Ha6JIJO.n.eHHSI H KOTOpbIe nO,QKpellJIHJOTCSI O.n.HOH JIHilib HH.n.YK-
HeA. B .n.eHCTBHTeJlbHOCTH Mbi .n.OJImHbi IJOJIb30BaTbCH TaKHM
OTKpbITHeM KaK B03MO>KHOCTbIO OoJIee TOtIHO- HCCJle.n.OB8Tb
9TH OTKpbITble CBOHCTBa H .n.OKa3aTb HX HJIH on pOBeprHYTb;
B OOOHX CJIYQ8SIX MbI MomeM HayqHTbcSI KOe-tleMY nOJIe3HOMY».
lIHTaTeJIb nOIJYBCTByeT BCIO npaBory CJIOB 3iiJIepa, eCJIH
BMeCTe C HaMH pss(5epe-r HecKOJIbKO npHMepOB, Kor.n.a HenOJI.
H8ft HIUYKU.HH Be-r K OiliHOotmbIM S8KJIIOIJeHHSIM.
I) .n. B 4>HSBK, n0J1h3YHCh TonhKO HHJ1YKlI.HeA, HHorJ1a oIIIH(\ancH.
BcnoMHHTe SaltOH .npeBBHX: CnplipOJ1a ooliTCH nYCTOTbI». 8TOT saKOH
B Te'JeHHe Aonroro BpeMeHH C'JHTanCH Benpe.nO>KHhlY, H TonhKO II
1640 r. TOppH'Ie.n.nH Y.ianoCio onpoBeprHyn er.
184

6Y)J.eM ytIHTbCR Ha pey)J.atIe
<PpaHIJ.Y3CIWn MaT€MaTIIK nbep <pepMa (1601-1665), C
HMeHeM Koroporo BbI Y)Ke BCTpeqaJIliCb B nepBoH r.naBe,
paccMaTpliBaJI tJIlc.na BH.n.a
n
f(n) = 2 2 + 1.
npH n = 0, I, 2, 3, 4 tJlIC.lJa f (n) - npocTble tlHCJIa. B caMOM
.n.e.lJe
f(0)=2 20 + 1 =3, f(1)=2 21 + 1=5,/(2) =2 22 + I = 17,
f(3) = 2 23 + I = 257, f(4) = 2 24 + I = 65537.
<pepl\m npe.n.noJ101KEJI, tJTO Bce tJHC.lJa TaKoro BH)J;a - npoc-
Tble, 1I 6bI.IJ HaCTOJIbKO YBepeH B cnpaBe.n.JIHBOCTII CBoero
npe.n.nO.IJOLKeHHSI, GTO npe.n.JIOLKH.1J HeCKO.IJhKHM aHrJIHHCKIiM
MaTeMaTHKaM .n.OKa3aTb 3TO npe.n.nOJJO)KeHHe. Ho tJepe3 He-
KOTopoe BpeMSI 3iiJIep nOKa3aJI, qTO YLKe CJIe.n.YIOllI.ee tIHCJJO
f (5) = 2 25 + I
SIBJlSIeTCSI COCTaBHbIM. B caMOM JJ,eJIe
2 25 + 1=4294967297=641.6700417.
PaCCMOTpHI eme npHMep.
B nOHCKax iPOPMYJIbi C IJ.eJIOqHCJIeHHhIM aprYMeHToM,
.n.aIOIII.eiI TO.'1bKO npOCTble llHc.'1a, 3}'IJIep no.n.Bepr IIcnblTaHHIO
TpextJ.neH
 (n) = n 2 + n + 41.
hOT TpeXqJ1eH .n.aBaJl npOCTbIe tJIIc.na npE Bcex 3HatJeHESIX
n OT I .n.o 39:
ff (I) = 43, cp (2) = 47, cp (3) = 53, .. . .
Ka3aJIOCb 6bl, tITO, IICnOJlb3Yf! Mero.n. Heno.IJHOH HH.n.YKIJ.IIH,
MOLKHO YTBep>K,lJ.aTb, qTO YKa3aHHaSI iPopMy.na npH .nI060l'vl
u.e.nOl\I n paCT npocroe qHC.lJO. B iPlI3HKe tlaCTO .n.e.'l3IOT MeHb-
IIIee KOJIH'leCTBO OITDlTOB H Bce LKe OTBa)KHBaIOTCSI .n.eJIaTb
nepexo.n. OT YTBepLK.n.eHIISI cnpaBe.n.JIIIBOCTH tlaCTHhIX CJIYtIaeB
185

..........
"t on
.
: -'1j
\
JIeoHap.n 9Anep.
-
K YTBep>K.n.eHlIlO. cnpaBe.n.JIIIBOCTH BCeX CJIYtJaeB Bo06III.e.
B .n.aHHOM cJlYtJae nO.n.06HOe YTBepLK.n.eHHe 6blJIO 6bl OIII1I60tI-
HbIM; nplI n=40 <l><>pMy.na cp(n}=n 2 +n+41 Y>Ke .n.aeT cacTaB-
HOe qHCJlO:
cp (40)=402+40+41 = 1681 =41 2 .
HOBblH nplIMep MO>KeT 6hITb 60JIee y6e.n.HTeJIbHbIM, tIeM
.lI.Ba npe.n.IIIecrnYIOIII.HX.
!laHo tJHCJIO BH.n.a
F(n}=99In 2 + I,
r.n.e n - HaTypaJIbHOe tJHCJ10. nO.n.cTaB.IJSISI BMeCTO n tIHCJla
HaTypaJIhHOrO pSI.n.a, '1IatJHHaH C I, Mbl fie nOJIyqlIM tIHCJIa, 51B-
JlHIOIII.erOC5I nOJIHblM KBa.n.paToM KaKOrO-JIH60 .n.Pyroro qHCJIa,
He nOJlytJHM ero H B TOM CJlyqae, ecJlH 33XOTeJIH 6bI nOCBSI-
THTb 3TOH pa60Te BCIO caoIO /K1I3Hh. 11 Bce >Ke MaTeMaTHKa He
.n.onYCKaeT B03MO>KHOCTH YTBepJK.D.aTb, q.ro Bce qHCJla YKa3aH-
HOro BH.n.a He SIB.IJSIIOTCSI KBa,3,paT8MH. TaKaSI CTpOroCTb 6y.n.er
npaB.n.88HOH, TaK KaK npH
11=12055735790331359447442538767
tJnC.lJo
F(n)=99In 2 +l
6y.n.eT nOJIHblM KBa.n.paToM.
186

11, HaKOHeIJ., paCCMOTpHM 3a.n.aqy YLKe Ji3 o6.naCTH reo-
MeTpHH.
Ha CKOJIbKO qaCTeH .n.eJlSIT npOCTpaHCTBO n nJIOCKOCTeii,
npoxo.n.SIIlI,HX 'lepe3 o.n.HY rolIKY TaK, qTO HHKaKHe TP" H3
HHX He npoxo.n.SIT qepe3 o.n.HY npSIMYIO?
O.n.Ha nJIOCKOCTb .n.eJIHT npOCTpaHCTBO Ha .n.Be 'laCTII. .lI.Be
nJIOCKOCTH, npOXO.IIJIIII.Heqepe3 O.lI.HY TOqKY, .l!.eJISIT npOCTpaH-
CTBO Ha qeTblpe 4aCTH. TpH nJ10CKOCTH, npoxo.n.HIlI,lIe qepe3
O.n.HY TOLIKY, HO He npOXO.l!.5!III.He 4epe3 OAHY np5!MYIO, .n.e.l5!T
npocTpaHcTBo Ha Bocelb 4acTeH (qepT. 109). OqeBH.n.Ho, MOlli-
Ll epT . 109.
HO BbICKa3aTb YTBeplKAeHl1e. qTO .l!.JISI n= I, n=2 " n=3
qHCJIO qaCTei%, Ha KOTopble n.IJOCKOCTH .n.eJ1SIT npocTpaHcTBo,
paBHo 2 n . EC.IJH Bbl, nO.'lb3YSICb !\leTo.n.OM HenOJ1HOH HH.n.YKIJ.HH,
C.l!.e.7aeTe 3aK.'1IOqeHMe, qTO II n.'10CKOCTeH pa306bIOT npocTpaH-
CTBO Ha 2 n qaCTet'r, TO BbI 6y.a.eTe He npaBhl. B ;J,a.7bHei:iIIIeM :\1bI
BepHeMCSI K peIIIeHHIO 3TOH 3a;:I,aqH H YBH.l!.HM, qTO qIlC.IJO qaCTell
npOCTpaHCTBa BLlpaLKaeTC5! CPOP:-'IY.'10i-I:
cjJ(n) = n(n - 1) + 2,
r.a.e n - J1Io60e ue.noe, nOJIOLKHTe.1bHoe qHCJIO.
nOqeMY LKe 3aKJ1IOGeHH51, c.n.eJIaHHbie B qeTblpex nOCJIe.n.-
HHX npHl\IepaX, OKa3aJ1I1Cb HenpaBH.rlbHbIMH? HCK.'JIOqHTe.'1bHO
nOTO!IIY, qTO 06III.ee YTBep)I{.lI.eHl1e OTHOCHTeJIbHO J1I060ro n
6blJl0 BbICKa3aHO TO.'IbKO Ha OCHOBaHrIH Toro, qTO 3TO YTBep-
>K.n.eHHe OKa3aJ10Cb cnpaBe;:I,.1I1BbII\'1 .lI.JI5! HeKOTOphlX 3Ha4eHHH n.
HTaK, 3ii,'1ep 6bl.7 npaB - npOCTaSI HH,J,YKlJ.HSl IO)KeT npH-
BeCTH K OIIIH6Ke.
187

B MaTeMaTHKe 0 «jJaKTaX CHaIJaJla )J.ora)J.bIBaIOTCSI,
a 3aTeM UX )J.OKa3bIBaIOT
B tJeM LKe 3aKJIIOtJaeTCH CYIII.HOCTb Tex «60JIee TOtJHbIX»
IIcc.lJe.n.OBaHHH, 0 KOTOpblX rOBOprlT 3i1JIep H KOTopble n03BO-
JISIT .n.OKa3aTh HJIH onpOBeprHYTb MaTe!\IaTHtJeCKHe 3aKJIIOtJe-
HHSI, OTKpblTble C nOMOIII.bIO HenOJIHOll HH.n.YKIJ.HH? OTBeT Mbl
HatJHeM C pa360pa O.n.HOH 3a.n.atJH.
rIepe.n. Ha:\1H nOC.'le.n.OBaTe.IJhHOCTb HeqeTHblX tIHCeJI HaTY-
paJlbHOrO pSIJI.a:
1,3,5,7,9, 11, 13.
(I)
4eMY paBHa cy!',nIa n nepBbIx tJJIenOB 3TOll nOC.'le.n.OBaTeJIb-
HOCTH?
CocTaBHM CYMMbI o.n.Horo, .n.BYX, TpeX H T. .n.. nepBblx tJJIe-
HOB .n.aHHOH nOCJIe.n.OBaTe.r!bHOCTH.
81= I;
S2= I t3=4;
S 3= 1-\-3+5=9;
S4= 1+3+5+7=16;
Ss-= 1-\-3+5+7+9=25.
ECJIH Bbl BHHMaTeJIbHO npocJJe.n.rlTe 3a CBSI3bIO Me}KJI.Y HaLK-
.n.OH lI3 nOJIyqeHHblX CYMM H 3HatJKOM nplI S, YKa3blBaIOIII.HM
na tIIICJIO CKJIa.n.blBaeMblX tJJIeHOB nOCJIe)J.OBaTeJIbHOCTII (I),
TO 6e3 oc060rb TpY)J,a YCTaHoBHTe, qTO Ka}!(.n.aSI lI3 CYMM paBlla
KBa)J,paTY qHCJIa CJIaraeMblX. B caMOM .n.e.ne
SI= 1 = J2;
S2= 4 = 22;
Sa= 9 = 3 2 ;
S4= 16 = 4 2
H T. .n;.
Ha OCHOBe 3THX Ha6J1IO)J.eHJlH Mb! MO>KeM BbICKa3aTb npe)J.-
nOJ10LKeHIle, qTO
Sn= I +3+5+ ... + (2n-I)=n 2 ,
(2)
T. e. tlTO CYMM8 n nepBblX netJeTHb!X tmCeJI HaTypa.lJbHoro PSl-
.n.a paBHa KBa.n.paTY HX tJHcna.
BepRO JIH 9TO npe.nnO.10LKeHHe. T. e. Bepno JIH OHO npH
JII060M IJ.eJIOM nOJIO>KHTe.'lbHOM n? cI>OPMYJI8 (2), KOHetIHO,
BepHa B qaCTHbIX CJ1yqaSIX np" n= I, 2, 3, 4, 5, H3 KOTOpblX
188

OHa B03HHKJIa. BepHa JIH OHa npH n = 6? Cne,ll,YIOIII.He Bbl-
KJIa.n.KH
1 +3+5+ 7 +9+ 11 =36=6 2
nOKa3blBaIOT, LITO cpopMYJ1a (2) cnpaBe.n.mlBa H npH 12=6.
MbI MOrJIH 6bl nepeHTH K npOBepKe <POPMYJIbl (2) npH n=7,
n=8 H T. .n.. H nOJIyqHJIH 6bl HOBbIe no.n.TBepLK.n.eHHSI ee cnpa-
Be.n.JIHBO';TH, KOTopbIe YBe.IJHqHJjH 6bl YBepeHHOCTb B HCTHH-
HOCTH 3TOH <pOpMy.lJbl Bo06III.e.
HaKoHeIJ.. Ha OCHOBe HenOJIHOH HH.n.YKIJ.HH MOLKHO YTBep-
LK.n.aTb, qTO HaMH OTKpblT 3aKOH: CYMMa n nepBblX HeqeTHblX
tIHCeJI HaTypaJIbHoro pSI.n.a paBHa KBa.n.paTY HX qHCJIa. Ho
6y.n.CM OCropOLKHbl H BcnOMHHM CJIOBa 3tiJ1epa: «.. .MhI BH.n.eJIH
CJIY'laH, Kor.n.a npOCTaH IIH.n.yKIJ.HSI BeJIa K OIIIH6Ke». BcnoM-
HHM TaKLKe qeTblpe 3a.n.aqH, C KOTOpblMH MbI n03HaKOMHJlHCb B
npe.n.IIIeCTBYIOIII.eM naparpa<pe. npHHSIB HallIe npe.n.nOJlOLKe-
HHe 3a 3aKOH, He yno.n.06HMCSI JIH Mbl TeM 300J10raM, KOTopble
.n.o OTKpblTHSI ABCTpamm YTBepLK.n.aJIH, qTO Rce JIe6e.n.H Ha
3eMJIe 6e.lJhIe?
JIyqIIIe noii.n.eM no HHOMY nYTH B nOHCKax 06III.ero .n.OKa-
3aTeJIbCTBa BbICKa3aHHoro HaMH YTBep>K.n.eHHSI.
Dpe.n.nOJI, OLKHM qro <popMYJIa (2) BepHa .n.J15¥ n= k, r.n.e
k - KaKOe-JIH60 HaTypaJIbHOe q1fCJ10, T. e., tITO
1+3+5+... +(2k - I) = k 2 . (3)
,lI.oKaLKeM ee CnpaBe)I.JIHBOCTb H .n..IJSI tIHCJIa, Henocpe.n.cTBeHHo
CJIe.n.YIOIII.erO 3a k, .n.JISI qHCJIa n = k + 1, T. e., qTO
1 + 3 + 5 + ... + (2k - I) + (2k + I) = (k + 1)2. (4)
LJro6bI y6e.n.HTbc5I Ii 3TOM, nepenHilieM JIeBYIO qaCTb paBeH-
CTBa (4) CJIe.n.YIOIII.HM 06pa30M:
(1 + 3 -)- 5 + ... + (2k - I) J + (2k + I),
H 3aMeHHM Ha OCHOBe paBeHcTBa (3) BCIO CYMMY BHYTPH KBa.n.-
paTHOH CKo6Kn qepe3 k 2 , nOJIyqHM
k 2 + (2k + I) = k 2 + 2k + 1 = (k + 1)2,
JleBaSI qaCTb paBeHcTBa (4), .n.eikTBHTe.'lbHO, OKa3aJlaCb paBHorr
(k + 1)2.
Mbl npHiliJIH K BeCbMa BaLKHOMY BbIBO.n.y: eCJIH HallIe
npe.n.nOJIO>KeHHe BepHO .n.JISI HeKoroporo HaTypa.lJbHoro qHC.1Ja
k, TO OHO Henpel\leHHO OCTaeTCSI BepHbJ:\I .n.JISI C.1e.n.YIOIlI,erO
IJ.e.10ro tlnCJJa k+ 1.
189

Ho Mbl YBepeHbl, tITO npe$OJIOLKeHHe BepHO .n.JISI n= I,
2, 3, 4, 5 H 6. Ey.n.yqH BepHblM .n.JISI 6, OHO H8 OCHOBe nOJlY'leH-
HOro BblBo.n.a .n.OJlLKHO 6blTb BepHblM H .n.JI5I CJIe.n.YIOIII.erO l1.eJIoro
qHCJIa 7; 6y.n.yqu BepHblMMSI 7, OHO .n.OJDKHO 6bITb BepHblM H
.n.JIH 8 H T. .n.. Dpe,ll;nOJIOLKeHHe BepHo .n.JISI Bcex u.eJIblX qHCeJI;
HaM y.n.aJIOCb .n.OKa3aTb HCTHHHOCTb ero B nOJIHOH 06III.HOCTH.
Dpo.n.eJIaHHble HaMH paCCYLK)J.eHHSI MorYT 6blTb HeCKOJlbKO
YKopOqeHbI.
ECJIH HaM y.n.aJIOCb npoBecTH .n.OKa3aTeJIbCTBO npaBHJIb-
HOCTH iPOPMYJIbl
Sn=n 2
npH nepexo.n.e OT K8KOrO-JIH60 npOH3BOJIbHOrO HaTypaJIbHOrO
tIHCJla k K CJIe.n.YIOIII.eMY 3a HHM HaTypaJIbHOMY qHCJIY k+ 1.
TO HymHo JIll npOBepSITb HallIe npe.n.nOJ10LKeHHe .n.JISI n = 6,
5, 4, 3, 2? JI,ocTaTOqHO 6bITb YBepeHHbIM B TOM, qTO OHO cnpa-
Be.n.JIHBO npH n = 1, H Tor.n.a Mbl CKalKeM: eCJIH npe.n.nOJIOLKe-
HHe BepHo .n..rISI n = I, TO OHO, Ha OCHOBe .n.OKa3aHHoro, 6y.n.eT
BepHO H MSI n = 2; eCJIH OHO BepHO npH n = 2, TO OHO Bep-
HO H .n.JI SI n = 3 H T. .n.., T. e. 3TO n p e.n.n 0.'1 OLKeHlle Bep HO
Bo06III.e H HaTlle npe.n.noJloLKeHHe CTaHOBUTCSI 3aKOHOM: cYIrvla
n nepBbIX le'leTHbIX qllce.'1 HaTypa.lJbHoro pSI,'I,a paBHa n 2 .
PeiliaSI TY 3a.n.aqy. Mbl n03HaKOMHJlHCb C OqeHb Ba>KHbIM
IIleTo.n.OM .n.OKa3aTe,rlbCTB8. OH n03BOJIHJI HaM y.n.ocroBepUTbCSI
B HCTUHHOCTH npe.n.nOJIOLKeHHH, B03HUKiliero y Hac B pe3YJIb-
TaTe Ha6JIIO.n.eHHH H no.n.Kpen.IJeHHOrO HH,lI;YKl1.HeH. T aKoH
MeTO.n. MomHO 6blJIO 6bI Ha3BaTb «nepexo.n.mI OT k K k + 1»,
HO 06biqHO ero Ha3bIB8IOT I\leTo.n.0ll1 «lIlaTel\laTHqeCKOH HJ1H
nOJIHOti HH.n.yKl1.HH».
'B OCHOBe 3Toro MeTO)J,a JIeLKHT nprlHl1.Hn MaTelllaTJ1IIeCKOii
HH.n.YKl1.Hl1, KOTOpbIH MO}KeT 6b1Tb ciPO(my.ll1pOBaH TaK:
ECJ1r1 HeKoTopaH 3aKoHoMepHocTb (B HailieH 38.n.aqe: 1+
+3+5+...+(2n-I)=n 2 ) cnpaBe,J,.'lUBa .n..'1SI LIHC.'1a 0:, r,ll;e
0: - HaHl\leHbIIIee H3 HaTypaJIbHblX quce.'!, npH KOTopml
3aKoHoMepHocTb HlIleeT CMbIC.'1 (B HaIIIeii 3a;I,8 L Ie CI.= I), H
eCJIH cBepx roro OHa OCTaeTCSI cnpaBe;I.JIHBOH )J..'1SI 4HC.la k+ I.
BCSIKHH pa3, KK OHa cnpaBe.n.mlBa .n.JISI qHCJIa k (B HailieH
3a,J,aqe paBeHCTBO: I +3+5+...+(2n-I)=n 2 OCTaJIOCb cnpa-
Be.n.J1HBbIM npH nepexo.n.e OT npOH3Bo.lJbHOrO HaTypa.TJbHoro
qHCJIa k K qHCJIY k+I), TO 3Ta 3aKOHOMepHOCTb CnpaBe;I,J1rlBa
MSI BCSIKor o HaTypaJIbHoro 'lHC.'la n, 60.lJbIIIero 0: 1 ).
1) flpllHIJ.llfi laTelanj1jeCKOil IIH!l.YKUllH Mbi npHMe1 6e3 ;\OKa3a.
Te.ijbCTBa KaK aJ(1I0IY. Ha ;noil TO'JKe 3peHHH CTOIIT pH;J. BHJ:\HbIX
MaTeMaTIIKOB (KypaHT. K.eilH. flyaHKape 11 p'p.).
190

l..h06bl BaM CTa.1 HCeH CMbICJI nOC.'Ie.n.HHX CJIOB: «n 60JIb-
IIIe CJ.», 3aMeTHM, qTO qHCJIO CJ. He 06SI3aTeJIbHO .n.OJlLKHO 6bITb
paBHblM I. HanpHMep, npH .n.OKa3aTeJIbCTBe Me-ro.n.OM MaTe-
MaTHQecKoH HH.n.YKl1.HH TeopeMbl 0 cYMMe BHYTpeHHHx yrJIOB
MHoroyrOJIbHIIKa
Sn= I 80 0 (n-2) ,
(5)
r.n.e n - IlHCJIO CTOpOH MHoroyrOJIbHIIKa. HaHMeHbIIIHM 3Ha-
qeHHeM 1?, npl1 KOTOpOM <}JopMYJIa I1MeeT CMbICJI, 6y.n.eT ql1C-
JIO 3, CJIe.n.OBaTe.'lbHO, CJ. B 3TOM cJIyqae 6y.n.eT paBHSITbCH
He 1, a 3. ECJII1 ctopMy.'la (5) cnpaBeMHBa npH n = 3 (a OHa
cnpaBeM, I1Ba) TO, .n.OKa3aB cnpaBeMHBOCTb <PoPMYJIhI (5)
npl1 nepexo.n.e OT np0l13BOJIbHOrO HaTYPq,TIbHOrO k K qHCJIY
k+ I (=1T0 MOLKHO c.n.eJIaTb!), MbI .n.OKaLKeM cnpaBe.n.JIIIBOCTb
<PoPMYJIbl (5) Bo06III.e MH JII060ro HaTypaJIbHOrO n, 66JIb-
IIIero 3. 4HTaTeJIb, KOHeqHO, nOHI1MaeT, qTO npH n= I H n=2
<jJopMYJIa (5) TepReT CMbICJI.
Me)}{.n.y Mero.n.OM Heno.IJHOH HH.n.YKII,HI1 H MeTo.n.OM MaTe-
MaTl1qeCKOH IIH.n.YKl1.I1H CYIII.ecTByeT rJIy60KaR CBR3b. MeTo.n.
nOJIHOH HH.n.YKl1.I1M HBJIHeTCSI KaK 6bI «MaTeMaTHqeCKHM .n.o-
nOJIHeHHeM K HH.n.YKl1.HH». 3THM B031\10LKHO 06'bHCHI1Tb, nOqeMY
nOJIHYlO HH.lI.-YKl1.HIO Ha3hIBaIOT eIII.e MaTeMaTHqeCKOii IIH.n.YK-
l1.HeH. TaKIIM 06pa30M, MeTO.n. MaTeMaTHqeCKOH II H.n.yKl1.IIH
He HBJIReTCSI KaKHM-TO HOBblM MeTo.n.OM no OTHOIIIeHHIO K HenOJI-
HOH IIH.n.YKl1.HH, a CKopee HeKOTopbIM ee paCIIIl1peHHeM. B CBR-
31I C 3TMM lIHTepeCHO npHBeCTM He6oJIbIIIYIO l1.HTaTY H3 KHIIrH
fl. DOHa «MareMaTHKa Ii npaono.n.06HhIe paccY)}{.L(eHHSI»:
«MaTeMaTIIqeCKaSI HH.n.YKl1.IISI Qacro B03HHKaeT KaK 3aKJIIOqII-
TeJIbHbIH IIIar HJlH nOCJIe.n.HHH <}Ja3a HH.n.YKTHBHOro MCCJIe.n.OBa-
HIISI, H B 3TOH nocJIe.n.Heii <}Ja3e llaCTO HCnOJIb3YIOTCSI HaBO.n.SI-
III.He co06paLKeHHR, B03HHKIIIHe B npe.n.bl.n.YIII.HX <jJa3ax».
npHo6peTaeM onWT B HH)J.YKTHBHOM MaTeMaTI1IJeCKOM
HCCJIe)I.OBaHHH
Pa360p 3'a.n.aqH 0 CYMMHpoBaHHH n nepBhIx HeqeTHbIX
qHCeJI HaTypa.'lbHOrO pSI.n.a, peiliemIOH B npe.n.bI.n.YIII.eM naparpa-
<}Je, n03BOJISIeT HaM YCTaHOBHTb OT.n.eJIbHbIe <}Ja3hI .n.OKa3aTeJIb-
CTBa MeTo.n.OM MaTeMaTHqeCKOH IIH.n.YKl1.HH.
06bIqHO npHMeHeHHIO Mero.n.a MaTeMaTHqeCKOH I1H.n.YKl1.IIH
npe.n.IIIecTByeT pSI.n. onbiTOB, B pe3YJIbTaTe Ha6JIIO.n.eHHR KO-
TOpbIX no.n.MeqaeTCSI HeKoropaSI 3aKOH01\lep, HOCTb CBR3blBaIO-
III.aR OT,QeJIbHbJe .9JIeMeHTb! .n.aHHoro HBJ1eHHSI. "YaCTO Ha OCHOBe
191

9Toro 6b1BaeT B03MO>KHO BblCKa3aTb npaB.n.ono.n.06,HOe YTBepLK-
.n.eHIIe 0 CYIII.HOCTU caMoro HBJIeHHH. 3TO YTBep>K.n.eHue, eIII.e
He .n.OKa3aHHoe BO BceH CBoeH 06III.HOCTII, Ha3bIBaIOT r H nOT e -
30 H. TaKoH mnOTe3e HHor.n.a MO>KHO npII.L(aTb MaTeMaTUqe-
CKYIO <POPMY B BII.n.e <PoPMy.'lbI C IJ.eJIOqHc.'leHHblM aprYMeHToM
n. OOJIyqeHHaH <PopMy.'la 6y.n.eT cnpaBe.n.JIHBa .n..'lSI Tex qaCT-
HbIX 3HaqeHUH n, KOTopble 6bIJIU HCnOJIb30BaHbI npII YCTaHoB-
JIeHIIII mnOTe3b1. OYTeM HenOCpe.L(CTBeHHoH npoBepKII MO>KHO
pacIIIHpHTb 06JIaCTb 31WqeHIIH n, npH kOTOphIX <popMYJIa
oCTaeTcSI cnpaBeMIIBoiL Ho Bce 3TO eIII.e He .n.aeT HaM npaBa
CqUTaTb BbICKa3aHHYIO mnOTe3Y cnpaBeMIIBOH .n..'lSI JII060ro
HaTypaJIbHoro n. BOT Tor.n.a-ro H IJ.eJIeC006pa3HO nonp060BaTb
npuMeHIITb MeTo.n. MaTeMaTHqeCKOH, HJIH nOJIHOH, IIH.n.YKIJ.HH.
3TOT MeTO.n. qaCTO n03BOJIHeT .n.oKa3aTb UCTIIHHOCTb rIInOTe3bJ
Bo06III.e II.'lH oTBeprHYTb ee. B nepBoM cJIyqae mnOTe3a CTa-
HOBUTCH 3aKOHOM.
Cal\lOe .n.oKa3aTeJI'bCTBO MeTo.n.oM MaTeMaTUqecKoH UH.n.YK-
nUll COCTOHT H3 CJIe.n.YIOIII.IIX qaCTeH, UJIII <pa3:
I. OpOBepHIOT cnpaBe.n.JIIIBOCTb BbIcKa3aHHoH rHnOTe3b1
MSI HaIIMeHbiliero 3HaqeHUH n=a., npH KOTOpOM 3Ta runOTe3a
HMeeT CMbICJI.
2. C.n.eJIaB npe.n.no.nO>KeHIIe, qTO nmOTe3a BepHa .n.JIH He-
KOToporo 3HaqeHHSI k, cTpeMHTcSI .n.OK83aTb CnpaBe)(JIUBOCTb
ee MH k+1.
3. ECJIII TaKoe .n.OKa3aTeJIbCTBO y.n.a.'lOCb .n.OBeCTII .n.o KOHIJ.a,
TO, Ha OCHOBe npIIHIJ.Una MaTeMaTUqeCKOH II H.n.YKIJ.HH , MO>KHO
YTBep>K.n.aTb, qTO BblCKa3aHHaSI rHnOTe3a cnpaBeMuBa .n.JISI
JII060ro HaTypaJIbHOrO qIICJIa n, 6oJIbilierO a., II, CJIe.n.OBa-
TeJIbHO, 3aKOHOMepHOCTb, no.n.MeqeHHaSI B Haqa.'le IICCJIe.n.OBa-
HHH, .n.eHCTBIITeJIbHO paCKphIBaeT CYIII.HOCTb H3yqaeMOrO
SIBJI eHII SI.
OpUMeHIIM MeTO.n. nOJIHOH HH.n.YKIJ.HH K peIIIeHHIO HeCKOJIb-
KIIX 3a.n.aq.
3a)J.alJa 1. .llaHa nOCJIe.n.OBaTeJIbHOCTb:
3'
1 1
15' 35'
1
4n 2 - 1
, ... .
HaihH CYMMY n nepBhIx qJIeHOB ee.
4T06bJ YCTaHoBHTb mnOTe3Y, KOTOpyIO nOTOM 6y.n.eM .n.o-
Ka3bIBaTb Mero.n.oM nOJIHOH IIH.n.YI{IJ.UII, COCTaBHM fleCKOJIbKO
qaCTHblX CYl\IlI1 , nOJIaraSI n paBHblM I, 2, 3, 4. OOJIyqUM
192

1 _ 1 .
3-3'
1 1 6 2
3+15=15="5;
++= 35+7+3 ==.
3 15 35 105 105 7'
 +  +  +  = 105 + 21 + 9 + 5 140 4
3 15 35 63 315 = 315 = 9" .
B03HHKaeT OqeBII.n.HaSI .n.Ora.n.Ka:
1 1 1 1
""3 + 15 + 35 +"'+, 4n 2 -1
n
2n+l .
nOJIyqeHHYIO TaKIIM 06pa30M nmoTe3Y 6y.n.eM ,lJOKa3bIBaTb
MeTo.n.oM MaTeMaTHqeCKOii HH.n.YKIJ.IIH.
I. rHnoTe3a BepHa npH n = I.
2. D.e.'laeM npe.n.nOJlOLKeHHe, '1TO rlinOTe3a BepHa npII n=k,
r.n.e k - KaKoe-.'lH60 npoII3Bo.'lbHOe qHCJIO, T. e. qTO
1 1 1 1
:3 + 15 + 35 + ... + 4 k 2 - 1
k
2k+ 1 '
H, HCXO.n.H H3 3Toro. 6Y.n.eM .n.oKa3blBaTb ee cnpaBe.n.JIHBOCTb
1 1
npII nepexo.n.e OT k K k + I. OpH6aBIIM K cYMMe 3" + 15 +
+  + ... + 4 k2  1 c.'le.n.YIOIII.HH (k+ I )-ii qJIeH nocJIe.n.o-
1
BaTeJIbHOCTH, T. e. 4 (k + 1)2 _ 1 . nOJIY'1HM
1 1 1 1 1
3+15+ 35 +...+ 4k 2 -1 + 4(k+l)2-1 =:2
=[  + 115 +  +...+ 4k21 ]+ 4(k+\)2-1
k
CYMMY BHYTPH KBa)J.paTHblx cK060K 3aMeHHM qepe3 2k + 1 .
BY.n.eM HMeTb:
k 1
2k + 1 + 4 (k -+ 1)2 - 1
k
2k+ 1
1
+ (2(k+l)+I] (2(k+l)-I] =
193
7 A, A. KOJlOCOR

k 1
- 2k + 1 + (2k+3) (2k + 1)
2k 2 + 3k + 1
(2k + 3) (2k+ 1) ==
_ (k + I) (2k + 1) _ k + 1 _ k + 1
(2k+ 3) (2k + 1) - 2k + 3 - 2 (k + 1) + 1
fHnOTe3a OKa33JIaCb CnpaBe.n.JlliBOH H npH n = k + I,
eCJIH OHa cnpaBe.n.JIHBa npH n = k. CJIe.n.OBaTeJIbHO. Ha OC-
HOBe npHHIJ.Hna MaTeMaTHqeCKOH HH.n.YKl1.IIH Mbl MomeM YT-
BepLK.n.aTb, qTO BblCKa3aHHaSI HaMH BHaqaJIe rHnOTe3a BepHa
npII JII060M HaTyp3JIbHOM n.
3a)J.alJa 2. llaHa nOCJIe.n.OBaTeJIbHOCTb B BH.n.e apml)MeTII-
tJeCKOH nporpeCCHII
-:- a 1 . a2' as. ..., an'
pa3HOCTb KOTOpOH paBHa d. HaHTH BbIpaLKeHHe n-ro q.neHB
nOCJIe)lOBaTe.'lbHOCTH tJepe3 a 1 H d.
Y CTaHaB.'llIBae:\J rIInOTe3Y.
llJIH 3Toro HanIIIIIe:\J pSI.n. paBeHCTB Ha OCHOBe onpe)J,eJIe-
HIISI apmln1eTIIqeCKOii nporpeCCIIH:
a 2 = a 1 + d;
as = a 2 + d = a 1 + d + d = a 1 + 2 d;
a 4 = as + d = a 1 + 2 d -I- d = a 1 -I- 3d.
HanIIcaHHble paBeHCTBa n03BOJISIIOT BbICKa3aTb fIInoTe3Y: JIIO-
60H tJJIeH apII4JMeTHqecKoH nporpeccrm paBeH CYMMe nepBoro
tlJIeHa ee II npoH3Be.n.eHIISI pa3HocTII nporpeccrm Ha tJHCJIO
qJleHOB, npe.D.IIIecTBYIOIlI,HX onpe.n.eJIHeMoMY. 3Ta fHnOTe3a
MomeT 6bITb BbIpa)KeHa tVOPMYJIOH:
a n =a 1 +(n-l }d.
1. fHnOTe3a CnpaBe,ll;JIHBa npH n=l, n==2, n=3, n-4.
2. JI,eJIaeM npe.n.no.noLKeHHe, tJTO rIInOTe3a CnpaBe.DJIHBa
npH n=k, r.n.e k - KaKOe-JIH60 npOH3BOJIbHOe HaTypaJlbHOe
qHCJIO, T. e.
a k -=- a 1 +(k-l)d,
H, HCXO.n.SI H3 STOro, .n.OKa:>KeM CnpaBelIJIIIBOCTb ee npHn=k+J.
1I.JIH nOJIYtJeHHSI (k+ l)-ro qJJeHa nporpeccHH. no onpe-
}J.eJIeHHIO nporpeccIIH, MbI .n.OJImHbl K npe)I,IIIecTBYIOlIJ.eMY tjJIeHY
npH6aBJlHTb pa3HocTb d:
194

ak+l=a k +d=a 1 +(k-l }d+d= a 1 +- kd-d+d:=z
=al+kd=a1+ [(k+l)-IJd.
TaKHM 06pa30M, (k+ I )-H 'lJIeH apHiPMemtlecKoH nporpec-
CHH paBeH CYMMe nepBOrO '1JIeHa II npOII3Be.n.eHIISI pa3HOCTH d
Ha tJIICJIO tJJlelIOB, I1peIIIeCTBYIOIII.HX on pe.n.eJI51eMOMY , eCJIH
k-H qJleH ee nOCTpoeH no TOMY LKe 3aKOHY. Cne.n.OBaTeJIbHO, no
npHHIJ.IIny MaTeMaTHtJeCKOH HH.n.YKI.{HII, BblCKa3aHHaH BHa-
qaJle rIInOTe3a cnpaBe.D.JIHBa npH JII060M IJ.eJIOM, nOJlOmH-
TeJIbHOM n.
C nOMOIII.bIO MeTo.n.a MaTeM3THtJeCKoi! HH.n.YKIJ.HII MbI Mor.'lH
6bI .n.OKa33Tb H cnpaBe.n.JIIIBOCTb COOTBeTCTBYIOIlI,eH iPOPMYJIbJ H
.n.JISI reOMeTpHqeCKOH nporpeCCHIi 7;-b 1 , b 2 , b 3 , ..., b n ...
b n = b 1 qn- 1 .
3a)J.aQa 3. llaHa nOCJIe.n.OBaTeJIbHOCTb B Bu.n.e reOMerpH-
qeCKOH nporpeCCHII
-:7b 1 , b 2 . b 3 , ..., b w ...
3HaMeHaTeJIb KOTOpOH paBeH q( q=l= 1). JI,oKa3aTb Mero.n.OM
MaTeMaTHtJecKoH II H.n.YKIJ.II II , qTO cYMMa 1l nepBbIx tlJIeHOB
noii nOCJIeAOBaTeJIbHOCTH Sfl no.'lYtJaeTcH no iPopMYJIe:
s _ bi (1 - qn)
n - l-q .
( I)
PeIIIaSJ STY 3a.n.aqy, HaM HeT Heo6xo.n.HMOCTH YCTaHaBJIH-
BaTb KaKYIO-.'lII60 mnOTe3Y: rHnOTe3a yme npe.n.JlOmeHa H
BblpamaeTCSI iPOPMYJIOH (1). OCTaeTcSI npoBecTH TOJIbKO l-H
H 2-H nYHKTbl .n.OKa33TeJIbCTBa.
I. rHnoTe3a, OtleBH.n.HO, cnpaBe)J.JlHBa npH n = J H n = 2.
B caMOM AeJIe
8 - hdl - q) - b .
1 - 1 _ q - 1.
hI (I - ql)
8 1 .... 1 _ q == bd 1 + q>.- b i + hI q - b 1 + bl'
2. D.onycTHM, tlTO THnOTe3a cnpaBe.n.JIHBa npH n s:: k,
rAe k - K3KOe-JIH60 npOH3BOJIbHoe HaTypa.lbHoe qHCJIO, T. e.
s.,.... blI-. l
 !......'
,.
IDa

Ii, HCXO.n.51 H3 SToro .n.onYIII.eHHSI, .n.OKameM cnpaBe.n.JIHBOCTb
ee npH n = k + 1. noc.'Je.n.YIOulHe BbIKJIa.n.KH nOKa3bIBaIOT,
tlTO 9TO .n.eHcTBHTeJIbHO TaK:
Sk+l = hi + b 2 + . . . + b k + hk+l = Sk + b k + 1 =
_ hi (1 - qk) + b k _ bl - blqk + blqk - blqk+ J
-, l-q lq - l-q
bdl - qk+l)
l-q
CJIe.n.OB3TeJIbHO, Ha OCHOBe nprmIJ.Hna MaTeMaTllqeCKOH lIH-
)J.YKlI,HH MO}!{e!\i YTBepLK.n.aTb, llTO <pOp:\1YJIa (I) cnpaBe.n.JIlIBa
npll JII060M IJ.eJIOM nOJIO}!{IlTe.'lbHOM n.
3a)J.alJa 4. BblqllCJIHTb CYMMY n nepBbIx 'lJIeHOB nOCJIe-
.n.OBaTe..1bHOCTll
{ n (n  I) }.
(2)
JI,JISI nocTpoeHHH nmOTe3bI COCTaBJISIeM CYMMbl:
1 1
Sl = T2 = 2;
1 1 4 2
S2 = T2 + 2.3 = {; == 3;
1 1,1213
Ss = t:2 + 2:3 -1- 3T = 3 + '12 = T'
OqeBH.llHO, MC>KHO BblCKa3aTb rllnoTe3Y, (lTO
Sn = n  1 .
1. rHnoTe3a cnpaBe.n.JIIIBa npH n = I. 3Toro }J.OCTaTO'IHO,
IJT06bI nepeHTlI K nYHKTY 2 .n.OKa3aTeJIbCTBa.
2. npe.n.nOJIOLKHM, qTO nmOTe3a cnpaBe.n..'lHBa npH n = k,
r)J.e k - KaKOe-JIH60 npOH3BOJIbHOe HaTypaJIbHOe qHCJIO, T. e.
'ITO
k
Sk= k+l '
196

HCXO.n.SI H3 3Toro" .n.OKaLKeM cnpaBe.n.JIHBOCTb ee npH n=
= k + 1. LLJISI 3Toro K cYMMe k nepBbIx qJIeHOB nOCJIe.n.o-
BaTeJIbHOCTH (2) npH6aBliM eIII.e O.n.HH HOBblM (k + I)-if tlneH,
1
paBHbIH (k + 1) (k + 2) . nOJIyqHM:
1 1 1 1
Sk+l = t:2 +  + . . . + k (k + 1) + (.k + 1) (k + 2) =
1 k 1
= Sk + (k + l)(k + 2) = k + 1 + (k + l)(k + 2)
k 2 + 2k -I- 1
(k + 1) (k + 2)
(k -I- 1)2 __ k + 1
(k+l)(k+2) - k+2
k+l
(k + 1) + 1 .
OOJ1yqeHHbiH pe3YJIbTaT rOBopIIT 0 TO'>1, qTO cYMMa (k + I)
nepBbIx qJIeHOB COCTaB.'leHa no TOMY )Ke 3aKoHY, qTO H CYM-
Ma k nepBbix qJIeHOB, T. e. 3Ta cYMMa BbIpa)1{aeTCSI .n.P06bIO,
qIICJIHTeJIb KOTopaH cOBna.n.aeT C HOMepO;v! nOCJIe.n.Hero qJIeHa,
a 3HaMeHaTeJIb Ha 1 60JIbIlle qIICJIHTeJIH. CJIe.n.oBaTeJIbHO, no
npHHIJ.IIny MaTeMaTIIqeCKOH I1H.n.YKIJ.IIH, BbICKa3aHHaSI HaMH
BH3qaJIe rIInOTe3a 6y.n.eT cnpaBe.n..'lIIBa IIPH JIl060M u.eJIOM
n
nOJIOLKIITeJIbHOM n, T. e. Sn = n-fT'
3a.n.alJa 5. LLoKa3aTb, qTO cYMMa KBa.n.paToB qHCeJI HaTY-
n (n -I- 1) (2 n + 1)
paJIbHOrO pSI.n.a OT I .n.o n paBHa 6 ' T. e.
12+22+3 2 +...+n2= n(n+l)6(2n+l)
rIInOTe3a yme .n.aHa.
I. npOBepSIeM cnpaBe.n.JIIIBOCTb rIInOTe3bI .n.JIH n = I H
n =2.
npH n = I, SI = 1 (1 + 1 (2+ 1) = I;
npH n = 2, S2 = 2(2+ 1)?2+ 1) = 2..5 = 5;
npII n = I Ii n = 2 rIInOTe3a OKa3aJIaCb BepHoH.
2. o peJl,nOJIOmIIM , qTO rIInOTe3a cnpaBe.n.JIIIBa npH n=k,
r.n.e k - KaKOe-JIH60 npOH3BOJIbHOe HaTypaJIbHDe qIICJIO, T. e.
npe,l!nOJIOmHM, lITO
197

SII"""I+22+32+...+k2= k(!t+l(2k+l) .... (3)
HCXO.n.SI H3 STOro, .n.OK8>KeM ee cnpaBe.n.,nHBOCTb npH n=k+
+ 1. JI,JISI 3Toro npH6aBli\1 K CYMMe 1 2 + 2 2 + . . . + k 2 HO-
Boe CJl2raeme (k + 1)2. nO.'lyqHM:
S k(k+l)(2k+l) +(k+I) 2 (k+l)[k(2k+!J+6(k+l))
",+1- 6 ... 6 =
(k+ I) (2k l + k + 6k+ 6) (k + 1) (2k 2 + 7k 1- 6)
.... 6 == 6
PaCKJIaJl,bIBaeM TpeXqJleH 2k 2 + 7k + 6 Ha JIHHeHHbie MHO}KH-
TeJIH. KOpHH !:1TOfO TpeXtlJIeHa T2KOBbI: k 1 = - 2; ks =
3
= - ""2' CJIe.n.oBaTeJIbHO,
2k 2 + 7k + 6 = 2 (k + 2) (k -I-  ) = (k + 2)(2k + 3).
Tor.n.a
Sk+I =
(k 1- 1) (k 1- 2) (2k 1- 3)
6
nO.lJyqeHHOe paBeHCTBO eCTb pe3YJlbTaT 3aMeHbJ k B paBeH-
CTBe (3) qepe3 k + I; TaKm.1 o6pa30M, .n.aHHaSI Har,1 nmOTe33
ocraeTcSI cnpaBe.n.JJHBOH npH n = k + I, eCJIH OHa cnpaBe.n.-
JIHBa npM n = k. npHHunn MaTeMaTnqeCKOH HH.n.YKIJ.HH no-
3BOJIHeT c.n.eJIaTb 3aKJIIOqeHHe, TO 3Ta rHnoTe3a BepHa .n.JISI
BCSIKoro H2TypaJIbHOrO qncna.
HTaK,
11 + 22 + 32 + .., + n2 = n (n + 1 (2n + 1) .
3a)J.alJa 6. JI,oKa3aTeJlbCTBO MeTo.n.OM MaTeMaTHqeCKOH HH-
.n.YKIJ.nn MomeT 6bITb C nOJIb30H npJIMeHeHO H B reoMeTpHH.
JI,oKameM, qTO n nJIOCKOCTeM, npoxo.n.SIllI.HX qepe3 o.n.HY
TOqKY TaK, qTO HHKaKJIe TPH JI3 Hnx He npOXO)J.HT Qepe3
O.Il.HY npSIMYIO, .n.eJISIT npOCTpaHCTBO Ha
qacTeA.
fHnoTe3a .n.aHa 1).
4> (n) = n (n.- 1) + 2
(4)
1) B CBR3H C 9TO" 3alla'Jel! llHTaTenb Harl,neT 01leHb HHTepeCHbiA
MaTepHaJ! B KHHre .lI.. nO"R cMaTeMaTHKa H npaaJj.onoAoOHble pac:-
CYH<,neHHR», CTP. 65-73.
198

1. npOBepSIeM CnpaBe,n:.nHBOCTb rIInOTe3bI .D.,nsl n""" I, n-2,
n-3, CpaBHMBaSI Ka>K.D.bIH pa3 pe3YJIbTaT, nOJIyqeHHblit K3
$OpMYJIbI (4), C COOTBerCTBYlOIII.IIM tIepTe>KOM (qepT. 110, 111
H 112).
<P(l)= 1(1-1 )+2=2;
<P(2)=2(2-1)+2=4;
<P(3)=3(3-1)+2=8.

4epT. 110.
4epT. 111.
4epT. 112.
2. npe.D.nOJIO>KHM, tlTO rIInOTe3a CnpaBe.D.JIHBa .D.JISI n=k.
r.D.e k - KaKoe-.nH60 npOH3BOJIbHOe HaTypaJJbHOe llHCJIO, T. e.
qTO k llJIOCKOCTeH .D.eJHIT npocTpaHcTBo Ha k(k-1)+2 qaCTII,
11, IICXO.D.SI II3 3TOro, .D.OKa>KeM cnpaBeMHBocTb ee np" n=k+1.
npHMeM 6e3 .D.OKa3aTeJIbCTBa TaKoe YTBepLK.D.eHHe: n pa3-
JIHtlHbIX npHMbIx, npoBe.n.eHHblx Ha nJIOCKOCTH tlepe3 O.D.HY
TOqKY, .D.eJISIT n.IJOCKOCTb Ha 2 n tlaCTeH (qepT. 113 1 ).
/ >< " M Y
I i 
IV
4epT. 113.
nYCTb P eCTb (k+ I )-SI n.nOCKOCTb. HMeH OOllI.Y1O TOqKY
C nepBbIMH k nJIOClmCTSIMH, nJIOCKOCTb P nepeCeqeT KaLK.D.Yro
H3 HHX no HeKoTopoH npHMoH, npoxo.D.SIIlI,eii qepe3 STY TOqKY.
Ha nJIOCKOCTH P no.nyqaTCH CJIe.D.bI TaKHX nepeCeqeHHH B BM-
.D.e k pa3J1HqHbIX npSIMbIX, npOXO.D.SIllI.IIX tlepe3 O.D.HY TOqKY.
3TH k npSIMbIx pa3.D.eJIHT nJIOCKOCTb P Ha 2k qacrefr; Ram-
.D.aSI qaCTb npe.D.cTaBJIHeT c060ii nJIOCKHH yroJl c sepIIIHHoi!:
B TOqKe ne peCeqeHHSI nJIOCKocTefr.
1) 910 YTBepH<AeHHe nerKO llOKa3b1BaeTCI'I TeN x<e NeTO,UON.
199

nepBbIe k nJIOCKOCTeH, nepeCeKaSICb, .n.mOT B npOCTpaHCTBe
MHOrOrpaHHbie yrJIbI. HeKOTOpbIe H3 3THX yrJIOB .n.eJISITCSI
nJIOCKOCTbIO P Ha 2 qaCTH (qepT. 114). TaR, Ha qepreLKe 1141)
nJIOCKOCTb P, BH.n.IIMbIe qaCTII KOTOpOH 3aIIITpIIxoBaHbI, .n.e-
JIHT Ha 2 qaCTH TpexrpaHHbIe yrJIbI I, 2, 4, 6, 7 H 8, HO He
3aTparIIBaeT yr.'lOB 3 II 5. 06III.aH rpaHb .n.BYx TaKIIX qaCTeH
eCTb o.n.HH II3 2k nJIOCKHX yrJIOB, Ha KOTopbIe pa36HTa nJIOC-


4epT. 114.


KOCTb P B pe3YJIbTaTe nepeCeqeHIIH ee C nepBbIMII k nJIOCKO-
CTHMH. Cne.n.OBaTe.nbHO
 qIICJIO MHororpaHHbIx yrJIOB, pa36II-
BaeMbIX Ha .n.Be qaCTII 'nJIOCKOCTbIO P, paBHO 2k. 003TOMY,
ecJIH k nJIOCKOCTeH pa36HBaIOT npoCTpaHcTBo Ha [k(k-I)+2J
qaCTeH, TO (k+ I) llJIOCKOCTb pa306beT npOCTpaHCTBO Ha
[k(k-I)+2 J+2k= k2_k+2+2k= k2+k+2=(k+ I )k+2.
nOCJIe.n.HIIH pe3YJIbTaT MbI nOJIyqHM, eCJIH B BbIpaLKeHHII
k(k-I)+2 3aMeHIIlVI k qepe3 k+1. aTO roBOpIIT 0 TOM, qTO
rIInOTe3a, BhICKa3aHHaH BHaqaJIe, OCTaeTCH cnpaBe.n.JIIIBOH
npH nepexo.n.e OT k K k+ I. Cne.n.OBaTeJIbHO, no npHHIJ.IIny Ma-
TeMaTHqeCKOH HH.n.YKIJ.IIH, OHa BepHa Boo6III.e.
3a)J.aQa 7. Pa36epeM eIll,e o.n.HY 3a.n.aqy YLKe II3 06JIaCTH
TpHroHoMeTprm, r.n.e MeTO)); MaTeMaTHqeCKOH IIH.D.YKIJ.IIH TaKLKe
MOLKeT 6bITb IICnO.'lb30BaH C ycnexoM.


I) HYMepaUHSI yrnoB H!l.ih npOTHB 'JaCOBOll CTpenKH.


200




npoH3Be)J.eHHe
COS a . cos 2:l . COS 4a . ... . CDS 2 n a
BbIpa3HTb lIepe3 sin a H sin 2 n +1 a.


YCTaHaB31MBaeM rHnoTe3Y:
2 sin a.cos a sin 2a
COSa= 2sina = 2sina l


sin 2a
cosa.cos2 a = --:- .
2sm'l


2 sin 2a.cos 2«
2sin 2:z


sin 4'1
2.2sin :z


sin 22«
2 2 sin a


ConOCTaBHHSI JJ,Ba n031YQeHHblX pe3YJIbTaTa, MbI MOLKe'\1 Bbl-
JJ,BHHYTb runoTe3Y:


sin 2n+1a
cos a . cos 2a . cos 4a. ... . cos 2 n a = 2 n +1' .
sma


1. rMnoTe3a BepHa JJ,JISI n = 0 H n = 1.
2. npeJJ,n0310>KHM, ':ITO BbIJJ,BHHYTaSI HaMM runoTe3a cnpa-
Be)J.JIMBa npH n = k, T. e.


. k sin 2k+1;x
COS a. cos 2a. cos 4a. '" . cos 2 n = 2 1i +1' ,
sma


H, IiCXOJJ,SI H3 3Toro, ,lI,OKa>KeM ee CnpaBe,ll,JIHBOCTb npH -n =
=k+1.
,n31H 3Toro K npoH3Be,ll,eHHIO cos a. cos 2a. cos 4a. .... cos 2 n a
)J.06aBHM ew.e O,ll,HH COMHO>KHTe31b cos 2k+1a; Tor)J.a
COS a. cos 2a. cos 4a. .... cos 2 k a. cos 2 k +1 a =
s in 2k+1a 2sin 2 k + 1 «.cos 2k+1a 2sin 2 k +1 a.cos 2 k +1 a
2 1i +1 sin a 2 sin 2R1-' a 2.2 k +1 sin a
sin 2 k +2 a
= 2 k +2sin a .


CnpaBeJJ,JIHBOCTb rHnOTe3b! npH n = k + I ,lI,OKa3aHa. nOJIb-
3YHCh MeTO,lI,OM MaTeMaTHlIeCKOH HH,lI,YKllHH, MbI MO>KeM YT-
BepLKJJ,8TL, ':ITO
sin 2n+1a
Cosa.cos2a.cos4a. ... . cos2na = 2 n +1. .
sma


201




Bar nHJI, Ha3aJl.


nonpo6yeM c,n,e.naTb HeKoTopbIe BbIBO)J.bI 83 npOQHT8H-
HOro.
1. EC31H KaKBSI-JIHOO - 3aKoHoMepHocTb nO,ll,TBepLKJJ,aeTcH
Ha KOHeqHOM 'lHCJIe CJIYGaE'B, TO, KaK ObI :no 'IHCJIO HH ObIJIO
Be31HKO, HliKOHM 06pa30M He31b3H CtJ:HTaTb, tJ:TO nepeJJ, HaMH
)J.OKa3aTeJJbCTBO B MaTeMaTHGeCKOM CMbIC31e. OpH TaKHX 06.
CTOSITe31bCTBaX MBTeMaTHK MO>KeT paccMaTpHBaTb nOJJ,Me'leH-
HYIO 3aKoHoMepHocTb TOJIbKO KaK Bn031He pa3YMHYIO rHnoTe3Y,
KOTopaSI B pe3YJIbTaTe nOC31e,ll,YIOlll,HX HCnbITaHHH HJ1H oma-
,lI,eT, HJ1M nOJJ,TBepJJ,HTCH. YCTaI-IOBHB B pe3YJIbTaTe HcnblTaHHH
onpe,ll,e.TJeHHYIO saKoHoMepHocTb, MO>KHO no JJ, 03 P e BaT b,
'ITO OHa cnpaBe,ll,31HBa BO BceH 06lll,HOCTH, M, qT06bI 3TO noJJ,o.
3peHHe nepeIll310 B Y B e p e H HOC T b, C31e.nyeT CJJ,e.TJaTb no-
nbITKY npHMeHHTb ,lI,OKa3aTeJIbCTBO MeTOJJ,OM MareMaTHqeCKOH
HHJI.YKUHH. EC31H nOnbITKa Y,lI,aeTCSI, TO TeM CaMbIM cnpaBek
JIHBOCTb 3aKOHa YCTaHaB.TJHBaeTCSI; B npOTHBHOM C31Y'Iae BO-
npoc OCTaeTCH OTKpbITbIM.
2. CpaBHHBaH Me>KJJ,Y CODOH pa306paHHble B STOM naparpa-
<lJe 3aJJ,aqH, MbI YBHJJ,HM, qTO B HeKoTopbIX H3 HHX (3a,natJ:H
1, 2, 4, 7) HaM caMHM npHXOJJ,M310Cb CTpOHTb nepBOHatJ:aJIbHYIO
runoTe3Y, B ,lI,PyrHx (3a,ll,a'lH 3, 5, 6) :)Ta rHnOTe3a JJ,aBaJIaCb
HaM B rOTOBOM BH,lI,e. 4HTaTe.TJb, HaBepHoe, nO)J.MeTHJ1, 'ITO
nOCTpoeHHe rHnOTe3bI He BCer,ll,a npocTO, H eCJIH OH C)J.eJJaeT
nonblTKY caMOCTOSITeJJbHO npHihH K nepBOHatJ:aJJbHbIM runo-
Te3aM B 3a,ll,aqaX 3. 5 H 6, TO, HecoMHeHHO, CTOJIKHeTCSI C onpeJJ,e-
JIeHHbIMH TpYJJ,HOCTSIMH.
MeTO)J. MaTeMaTHGecKoH HH)J.YKI1.HH He )J.aeT HMKaKHX YKa-
saHHH, KaK nOCTpOHTb rHnOTe3Y. «Bonpoc 0 TOM, KaK B03-
HHKaeT rHnOTe3a, npHHa)J,JIe>KHT K TOH 06JIaCTM:, B KOTOpOH HeT
HHKaKHX 06IllHX npaBH31, 3,l1,ecb ,lI,e31aeT CBoe )J.e310 3KcnepH-
MeHT, aHBJIOrHH, KOHCTPYKTHBHaSI HHJJ,YKI1.HH» (K Y P a H T).
K YCT8HOBJIeHHIO O,ll,HOH H TOM >Ke rHnOTe3bI pa3HbIe HC-
C31e)J.oBaTe.TJH MorYT npHHTH pa3HbIMH nYTSIMH. 3,l1,eCb HeT e,nHH-
CTBeHHO B03MO>KHOrO nYTH. 4acTO H36paHHbIH nYTb ,lI,aJIeKO
He npOCT H Tpe6yeT OCTPOYMHOH BbIJJ,YMKH. 4T06bI y6e,nHTbCH
B STOM, p3CKpOeM KHHry LI.. nOHSI «MaTeM3TIiKa H npaB)J.ono-
,lI,06HbIe paCCY}K)J.eHHH» Ha CTpaHHl1.e 134 H npOCJIe,ll,HM, K8K
3BTOp CTpOHT rHnOTe3Y 0 cYMMe KBaJJ,paToB tJ:HCeJJ HaTyp8JIb-
Horo pH)J.a OT I )J.O n.
npHBe.neM xo.« ero paccY>KJleHHu.


202




«CoBceM HeTpY)J.HO HaifTH CYMMY nepBbIX l..leJIblX lJHCeJI.
«1>OPMYJIY
1 + 2 + 3 + . . . + n _ n (n 2 + I) ,
KOropyIO MO>KHO OTKpblTb H ,lI,OKa3aTb MHOrHMH cnOC06aMH.
MbI 6Y)J.eM CQHTaTb H3BeCTHOH. TpY)J.Hee HathH cpOPMYJIY )J.JISJ
CYMMbI n nepBbIX KBa)J.paroB
1+4+9+l6+...+n 2 .
HeTPYJJ,HO BbIIJHCJIHTb :ny CYMMY JJ,JIH MaJIbIX 3HaIJeHHH
n, HO YJIOBHTb npaBHJ10 HeJIerKO. BnoJIHe eCTeCTBeHHO,
O)J.HaKO, nonbITaTbCSJ 06HapY>KHTb KaKorO-TO p0,ll,a napaJIJIe-
JIH3M MeLKJJ,y 3THMH ,lI,BYMH CYMMaMH H paCCMOTpeTb HX COB"
MeCTHO:
n I 1 I 2 I 3 I 4 I 5 I 6 I. ..
I I I I ! I I.. .
1+2+3+...+n I 3 6 10 15 ! 21
11I+211+3 11 +..-+nll I I I 5 I 14 I 30 I 55 I. 91 I.. .
I
KaK CBH3aHbI ,lI,Be nOC31eJJ,HHe CTpOKH? HaM MOLKeT npHi:iTH
B rOJIOBY HJJ,eH HCCJIeJJ,OBaTb HX OTHOIlleHHe.
n I 1 I 2 I 3 I 4 I 5 I 6 I. ..1...
111 + 2 '1 +...+ nil 1 5 7 3 11 £3 ..F
1+2+... +n "'3 '3 "3 3'
3JJ,eCb npaBHJIO OqeBH,lI,HO, H, eCJIH OTHOIlleHHSI H3 BTOpOH
CTpOKH sanHcaTb CJIe)J.YIOIII.HM 06pa30M:
3 5 7 9 II 13
3'3'3'3'3'3'
203

ero nOqTH HeB03MOLKHO He 3aMeTHTb. E)J.Ba JIH MbI CMO>KeM
YJJ,ep>KaTbCH H He CcpOPMY31HpOBaTb npeJJ,JIO>KeHHe, qTO
1 2 + 2 2 + 3 2 +... + nil _ 2n + 1
1+2+3+...+n ---g-'
nOJIb3YHCb 3HaqeHHeM 3HaMeHaTe31H B JIeBoi\ qaCTH, KOTO-
poe MbI c'unaeM 1I3BeCTHblM, MO)KeM BbICKa3aTb Hawe npe)J.-
nOJIO>KeHHe B cpopMe
1 2 + 2 2 + 3 2 + ." + n 2 = n (n + I (2n + I) ».
BOT TOT nYTb, KOTOpbIM WeJI 11. nOHH K YCTaHOBJIeHHIO
rHnOTe3bI. 4HTaTe31b, HaBepHoe, qYBCTByeT, qTO 3TOT
nYTb HenpOCTOH. Ho H )J.PyrHe H3BeCTHbIe nYTH He MeHee
CJIO>KHbl.
3aJl.atjH
1. LI.oKa3aTb MeTO,ll,OM MaTeMaTlIqeCKOH HHJJ,YKltHH, qTO
cYMMa Ky60B n nepBblx qHCe31 HaTypaJIbHOrO pH)J.a paBHHeT-
CJI KBaJJ,paTY HX CYMMbI, T. e.
18+28+38+...+n 3 = [ n(n;l> r.
2. HaHTH CYMMY qHce.Tl
5 11 = 1+ 2 + 2 2 + 2 3 + 2 4 + ... + 2 n - 1 .
Y K a 3 a H H e. LI.JIH nocTpoeHHSI rHnOTe3bl paccMoTpHTe
Ta6JIHuy
n
2
3 I 4 I 5 I...
7 115 131 I...
8n
3
3. HaHTH CYMMY npoH3Be)J.eHHtl
Sn= 1.2+2.3+3.4+4.5.+... +n(n+ 1).
204

Y K a 3 a H H e. 11JIH nOCTpoeHHH rMnOTe3bI paCCMOTpHTe
Ta6JlHl1Y
n
2
3
4
Sn
2
8
20
40
nOnp06YHTe qHCJIa HHLKHeH CTpOKH Ta6JIHl1bI npe)J.CTaBHTb
B BH)J.e GaCTHoro OT AeJIeHHH npOH3Be,ll,eHHH Tpex nOC31e)J.o-
B8TeJIbHbIX GHCe31 HaTypaJIbHOrO p5l,ll,a Ha QHCJIO 3.
4. HaHTH CYMMY npOH3Be,ll,eHHH
Sn = \.2.3 + 2.3.4 + 3.4.5 T '.. + n(n + l)(n + 2).
Y K a 3 a H He. LI.31H nOCTpoeHHH rHnOTe3bI nOAyMaiiTe HaA
Ta6JIHlteH
n
2
3
Sn
6
30
90
4HcJIa HM>KHeH CTpOKM Ta631Mltbi npeI\CTaBbTe B BHAe
llaCTHOro OT Ae31eHMH lJeTblpex nOCJIelI.OB3TeJIbHbIX tJMCeJI Ha-
Typa31bHOrO pHAa Ha GHCJIO 4.
5. 110Ka3aTb MeTOAOM MaTeMaTMtJecKOH i!H)J.YKltHH, [ITO
1 1 1 n
W + 3.5 +... + (2n-l)(2n+J) = 2n+l .
6. LI.oKa3aTb, liTO CYMMa Ky60B Tpex IIOCJle;tOB8Te.'IbHbIX
HaTypaJIbHblX qHCe31 ACJIHTCfl Ha 9.
7. .l1aHa nOCJIe)J.OBaTe31bHOCTb IlHCeJI c1>H6ona Il11Hl)
0, 1, I, 2, 3, 5, 8, 13, 21, ...,
r)J.e U o = 0; U I = I H un = U n _ l + U n _ 2 . LI.oKa3aTL, '-no
1) l!Ju60fW'l'lU - lIT3J1bHHCKHii MaTeM3THK tla'laJIa XIII n. f1oJIPo1.
tlee 0 ct>H6oHa'l'l1i 11 ero 'IlICJIax 'lliTaTe,1b Y3HaeT H3 JJ.eB!lTof: r naBbl.
205

U n = V { c+ ;5 r- c-;5 f}.
P e III e H Ii e. l{aHHaSI cf>opMY31a CIIpaBeJJ,JIHBa JJ,JISI n... 0,
n = 1 H n = 2 (npoBepbTe!). OpelI.n0310>KHM, lITO OHa cnpa-
Be,ll,JIHBa JJ,JIH n = k, rlI.e k - JII060e HaTypa31bHOe qHCJIO, Ii
)J.OKa>KeM ee cnpaBe)J.JIHBOCTb npIi fl = k + I.
I {( J+V5 ) k ( I-V5 ) /1
U k + 1 = Uk + U k - 1 - VS 2 - 2 -+-
+ c+5 r-l- c-r5 r-l}= ;5 [ C+;5 r-'x
X VS :- 3 _ C - r5 r-' . 3- s ] ,
HO
c+;5 r= 1+2S+5
3+ V5
2
H C - r 5 r = 3 - r s .
CJIeJJ,OBaTe.nbHO,
U =  {( I+V5' ) k+1_/I_V5 ) k+1 } .
k+l -VS 2 l 2
opMY31a OKa3a31aCb cnpaBe)J.31HBOH )J.JIH n = k + 1 npH yc-
JIOBHH ee CnpaBelI.JIHBOCTH jl,31H n = k. D031b3Y5lCb npHHIlIi-
nOM MaTeMaTlilJeCKOH HliJJ,YKUIiH, YTBep}KJJ,aeM, qTO
u =  { ( ' I + V5 ) '1_ ( 1- VT ) 'I }
11 Jrs 2 2
npH .T1I060M ueJIOM, nOJIO>KHTeJIbHOM n.
8. l1oKa3aTb, qTO n npHMblx, pacn0310>KeHHbIX B nJIOC-
KOCTH, 1I3 KOTOpbIX HHKaKHe )J.Be He napaJIJIe31bHbI H HHKa-
206

KHe TpH He npoXO,ll,HT lJepe3 O.QHY TOtIKY, pa3,l1,eJIHIOT nJIOc.
KOCTb Ha
N-l + n(nt l )
qacTefi.
Y K 8 3 a H H e. ECJIH K k HMelOlliHMCH npHMblM npHcoeJJ,H-
HHTb HOBYIO (k + 1)-10 np5!MYIO, TO OHa npH YC310BHHX, BbI-
CKa3aHHbIX B TeopeMe, nepeCeqeT Bce k ,lI,aHHbIX npHMbIX H
npotl,ll,eT no (k + I) KYCK8M nJIOCKOCTli.
9. Ha CKOJIbKO Tpeyro31bHHKOB npOH3B031bHbIH n-yrOJIb-
HHK MO>KeT 6bITb p836HT CBOllMH HenepeceK8lOmllMllC5! ,AHa-
roHaJIHMH?
10. j{OKa3aTb, qTO
sin x + sin 2x + sin 3x + ... + sin nx =-
(Om8em. n - 2.)
. n+ 1
smx
- . sin.
x 2
sinT
Y K a 3 a H H e. TIpll JJ,OKa3aTeJIbCTBe HCnOJIb3yiiTe <pop-
MYJI TpHrOHOMeTpHH
sin (k + I) x = 2 sin k 1 1 x. cos k t 1 x
H
. k+2 . k+l 2 2k+3 . x
sm 2""x-sm x = cos  x. sm T'
II. J{oKa3aTb, QTO
2n+ I
I. sinx
"2 + cosx + cos2x +...+ cosnx= x
2sinT
Y K a 3 a H He. npH npOBepKe rHnOTe3bI npH n=1 npe,ll,-
. 3 . ( + X ) u rt-.
CTaBbTe sm2"x KaK sm x 2" H HCnOJIb3YHTe 't'0PMYJIY )J.J]H
207

sin (a+). B ,lI,aJIbHeHWeM npoH3BeJJ,eHHe 2 sini.cos(k+l)x
3aMeHHTe pa3HocTblO CHHYCOB cooTBeTcTBYIOIII.HX ymoB.
12. 113BecTHo, qTO
(a+b)2=a 2 +2ab+b 2 ,
(a+ b)3=a 3 +3a 2 b+3ab 2 + b 3 .
l{oKa>KHTe, qTO npM: J11060M n l.{eJIOM Ii n0310>KHTe.TJbHOM
cnpaBeMHBo paBeHcTBo
n (n- I) n - 2 b 2 +
(a + b)n = an + na n - 1 b + 1.2 a
+ n (n-I) (n-2) n- 3 b 3 + +
1.2.3 a . . .
+ n (n- I) (n-2).... (n-k) an-lib" + . . . + b n
1.2.3...k
(cpopMYJIa 6HHOMa HbIOTOHa).
HeMHoro HCTOpHH
I1H,lI,YKTHBHbIH l\IeTO)J. HCCJleJJ,OBaHHH CBOHMH KOpmIMH yxo-
)J.HT B r31YOOKYIO ,lI,peBHOCTb. Be3 IiH,lI,YKI.{HH 6bI310 6bI HeB03-
MO>KHO TBOpqeCTBO HH B l\1aTeMaTIiKe, HH B cpH3HKe, HH B 31IO-
60H HHOH 06J1aCTli HaYKH. B !\1aTeMaTHKe HHJI.YKTHBHbIH MeTOJJ,
qaCTO nOMoraeT HaMeTIiTb TeopeMY, KOTopaH 3aTeM )J.OKa3bl-
BaeTeH BO BceH CBoeH 06lliHOCTH KaKIi:ll-.TJH60 IiHbIM MeTOJJ,OM.
Ho )J.JIH YCTaHOBJ1eHHH HCTHHbI HH,lI,YKTHBHbIH MeTOJJ, He-
)J.oCTaTOqeH. B CBH3Ii C 3THM IiHTepeCHO 3aMeqaHHe B. 11. JIe-
HHHa B era «cI>liJIOCocpCKIiX TeTpa)J.5!x»: «CaMaH npOCTa51 HCTHHa,
caMbIM npOCTbIM IiH)J.YKTHBHbIM MeTO,ll,OM n031, yqeI-IHaSI BcerJJ,a
Hen031Ha, H60 OnblT BCer,ll,a He 3aKOHqeH» (roCn031HTH3JJ,aT,
1947, CTp. 154).
BOT nOqeMY B MaTeMaTHKe npH06peTaeT 6031bWOe 3Haqe-
HHe no.TJHaH, liJIH MaTeMaTIiqeCKaSI, MH,lI,YKI.{HH KaK 3aBepwaIO-
lliaH cTyneHb HH,lI,YKTHBHoro HCCJle,ll,OBaHHH, H BOT nOl/eMY B03-
HHKaeT >Ke31aHIie YCTaHOBHTb, KTO H3 MaTeMaTHKOB H31H KaKaH
H3 MaTeMaTHqeCKIiX IllK031 BnepBbIe HcnO.'1b30Ba.'1a MeTO;I. l\Ia-
TeMaTHqeCKOH IiH)J.YKUHH.
YKa3aHHH HeKOTopbIX aBTopOB, qTO ,lI,OKa3aTe31bCTBa MeTO-
)J.OM n031HOH HH;I.YKUHH !\1O)KHO BCTpeTHTb Y>Ke Y EBKJI, HJJ,a
e,ll,Ba JIH COOTBeTCTBYIOT HCTHHe; TaM MbI CKopee BCTpel/aeMCH
C 3aKJIIOqeHHSIMH no a HaJIOfIiH.
208

.t.., . .\i:
t if. ,; . "' #
,;"" '"'.,
.i.. ,f. j,#' ;, '>
"
<:::-A_
' , _no ....
 '\ -" ,
.. . ... . _..
; --. .
.
.. . "k;:r"-:::, . . ,

-. , )
..
'-' <\
,, i
.. i
!
!
L
i
:1

..JI
EJIe3 naCKI!JIb.
I1MelOTCH HeKOTOpbIe OCHOBaHHH npHnMCaTb H306peTeHHe
MeTOJJ,a MareMaTHqeCKOH MHJI.YKI.{HH 51K06y BepHYJIJIH (1654-
1705), H HBJIHeTCH COBepweHHO 6eccnopHbIM, qTO 3THM MeTO)J.OM
WHpOKO nOJIb30BaJICSI cppaHI.{Y3CKHH MaTeMaTHK, cpH3HK H
cpHJIOCO<p BJIe3 naCKaJIb (1623-1662). Ero 06bIqHO H CqHTalOT
H306peTaTeJJeM YKa3aHHoro MeTO)J.a.
HeCOMHeHHO TaK>Ke, lITO B HCTOpHH OTKpbITHSI MeTO)J.a
nOJIHOH HH)J.YKI.{HH onpe.rr.e31eHHYIO p031b CbIrpaJI H .lI.eKapT
(1596-1650), HCn031b30BaBIllHH npH H3yqeHHH 06rn.HX CBOHCTB
aJIre6paHlIeCKHX KpMBbIX CBOe npaBH310 3HYMepaI.{HH. CYlll,-
HOCTb 3Toro IIpaBHJ1a COCTOHT B TOM, qTO «)J,JlSI H3YlIeHMH He-
0603pHMoro MHO>KeCTBa CJIyqaeB HX Heo6xo)J.HMO paCnOJIO>KHTb
«KaK 6bl no CTyneHHM», H3yqHTb CBOHCTBa om,eKTa, HaXO,ll,H-
rn.erOCH Ha nepBOH cTyneHH, H cnoco6 nepexo)J.a OT O)J.HOH CTY-
neHH K CJIe,ll,YlOrn.eih> (c. 51. 51 HOB C K a SI).
He06xoJJ,HMO oTMeTMTb, qTO npaBHJIO 3HYMepal.{MH .lI.eKapTa
HMeeT HeKOTOpbIe 06rn.He lIepTbl C MeTO.D;OM MaTeMaTHlIeCKOH
MHP.YKIlHH.
209

JI I1 T E PAT Y P A
l1. C. COM II H C K Ii Ii, MeTOI1 M8TeM8rli'leCKOA HHAYKUHH,
fI1TTJI, 1960.
JI. l1. f 0 JI 0 D H H a II 11. M. 51 r nOM, HH.nYKUHJt B reOMer-
pHH, fI1TT.n, 1956.
n. 10. f e p M a II 0 B Ii 'I, BOnpOCb! H sa.na'lH Ha co06paJKeHHe,
Y'Ine.nrIl3, 1957, CTp. 41.
P. K Y P II H T Ii r. P 0 6 6 Ii H C, LITO TaKoe MareMarHKa,
rHTTJI, 1941.
n. c. M 0 .II. e H 0 11, C(0)J1H1K soa.na'l no MaTeMaTHKe, 113.11.. «Co-
oeTCKaSl HaYKa», 1954.
C. H. 3 e T e JI b, MeTo)t MaTeMarHqeCKOii .HHl\YKUIiH H reOMer-
pH'IeCKHe nOCTpoelliUl.
«113 onblTa npeno.naBaHH!l MaTeMiTHKH B VIIl-X K,1accax:t. YIJ-
ne)trH3, 1955.
H. 51. n. en M a H, MuoJl MIIT8MaTHlleCKOH HH.IlYKlI,HH, Y'me.nrHII.
1957.

rJlABA IX
npOfPECClUI 1-1 HEKOTOPblE ,nPYfI-lE
nOCnE,nOBATEnbHOCTH
«npe.nMeT MaTeMaTHKH Ha-
CTOnbKO Cepbe3eH, 4TO none3HO
He ynYCKaTb cnY4aH .nenaTb ero
HeMHoro 3aHIiMaTenbHblM».
nacKaAb
OT HeKOTOpblX aaJl.aq Ha nporpeCCIlH BeeT rny6oKoH
ApeBHOCTblO
B Cer,ll,8 HHTepeCHO 3HaTb, K8K ,lI,aBHO, B KaKoij
CTpaHe II B pe3YJIbTaTe KaK1-IX nOTpe6HOCTeH B03HIlK
TOT H31H HHOM pa3)J.eJJ MaTeMaTHKH. EC31H qlITaTeJJb
nOCTaBliT Bonpoc 0 TOM, r,ll,e H KorJ{a B03HHK310 yqe-
IDle 0 nporpeCCHHX H nOC31e)J.OBaTeJJbHOCTHX, TO eMY npH.n.eT-
CH nepeBepHYTb Ha3a,ll, MHorue CTpaHHUbI HCTOpHH H paCKpbITb
HX Ha Oll.HOH H3 ,lI,peBHeHWHX 3nox EmnTa HJJH ACCHpHH.
JIeT CeMb,ll,eCHT Ha3aD. aHr31HqaHHH PHH,lI" npOH3BO,ll,H pac-
KOnKH B ErnnTe. 06HapY>KH31 nanHpyc, KOTOpbIH 6bI31 COCTaB-
31eH 3a 2000 JIeT ,lI,0 HaweH 3pbI, HO H 3TOT nanHpyc 6bI31 cnHcaH
C .Apyroro, erue 6o31ee JI.peBHero, OTHOCHlll,erOCH K TpeTbeMY
211

TblCHl1eJIeTHIO.AO HaweH 9pbJ. YqeHbIe, KOTOpbIM J{ocTyneH H3bIK
erHneTCKHX HepOr31HB, paCWH<tJpOBaJJH TeKCT nanHpyca H
np01IJJH HeCK031bKO 3a,ll,al1. CO,ll,ep>KaHHe HeKOTopbIX H3 9THX
sa,ll,al1 n03B031HeT OTHeCTH HX K 3a,ll,aqaM Ha nporpeCCHH.
npHBe,ll,eM O,ll,HY H3 TaKHX 3a.n.aq.
«nYCTb Te6e CKa3aHO: pa3.n;eJJHTb ,lI,eCHTb Mep HqMeHH
Me)K)J.y 10 l1eJJOBeKaMH TaK, qTo6bI pa3HOCTb Me)K.n;y Ka)KJ{bIM
l1eJJOBeKOM Hero COCe,ll,OM COCTaB.1H31a 8" (T. e. i) MepbI Hq-
MeHH».
.,<:  . ,; .,,-
--< < 
1 .' 1.d.
""'" CtJ
'::: 1. . <4'''' .t
 "'....it 1 tAl... . .1
." """'4' .#.?' tt ...
='2.Z...Vl ..__ ____. .)n..t.1.
tuU'
J..,
ErHneTCKHH MaTeMaTHqeCKHH nanHpyc.
npOl1TeM TeKCT 9TOH 3a.n;aqH B pMe, 6031ee npHBbIl1HOH
HaM: «10 Mep HqMeHH pa3,l1,eJJHTb Me)K,lI,y ,lI,eCHTblO 31HLlaMH
TaK, IJT06bI .AOJ1H 3THX 31HLl COCTaB31H31H apHcpMeTHqeCKYIO npo-
u 1
rpeccHlO, pa3HOCTb KOTOpOIl paBHa 8" MepbI».
B nanHpyce ,lI,aeTCH He TOJIbKO TeKCT 3a,ll,aqH, HO H npe,ll,-
JIaraeTCH npaBHJlO )J.JIH BbIIJHCJ1eHHH ,lI,0J11l nepBoro H3 ,lI,eCHTIl
JIHLl. 3ro npaBHJlO, BbIpa>KeHHOe H3bIKOM COBpeMeHHbIX p-
M)'JI, IlMeeT BH,lI,:
S d
a 1 = n - T (n - 1) . .. .
(I)
GHTaTeJJb, OIJeBH,lI,HO, nO,ll,MeTHJl, l1TO nOCJIe,ll,HHH pMYJIa
MO)KeT 6bITb nOJ1yqeHa H3 H3BeCTHOH eMY <tJOPMY31bI CYMMbI
1IJJeHOB apH<tJMeTHqeCKOH nporpeCCHH
S = 2al+d(n-l)
2 n,
212

eC31H ee pa3peWHTb OTHOCHTe31bHO a 1 . B caMOM ,lI,e31e
d
S = a 1 n + T(n-l).n
H
S d
a 1 = n - 2 (n -I).
KaK T031bKO ,lI,031H nepBoro H3 ,lI,eCHTH 6y.n;eT HaH.n;eHa C nOMO-
m.blO PMY31bI (I), ,lI,O.1JH OCTaJIbHbIX ,lI,eBHTH O 2 , 0 3 , ..., a lO
31erKO HaHTH:
. 1
O 2 = a 1 + d, a 3 = a2 + d 1 , . . . , a 10 = 0 9 + d, r,ll,e d = - g'
BbI3bIBaeT Y,lI,HB31, elUle KaKHM nYTeM ,lI,peBHHe erHneTCKHe
MaTeMaTHKH, He HMe51 pa3pa60TaHHOH TeopHH nporpeCCHH,
npHlIl31H K npaBH31Y, BbIpa>KeHHOMY cpOPMY310H (l). B 3TOM
Bonpoce HeT e,ll,HHOrO MHeHHH HCTOpHKOB MaTeMaTHKH. Olle-
BH,lI,HO, npH nepBOHalla31bHOM peweHHH 3TOH 3a,ll,aqH 6031blIlOe
3HalleHHe HMeJJ «MeTO,ll, np06» (TOIIKa 3peHHH B. B. Ba6bI-
HHHa H M. 51. Bbiro,ll,CKOro).
B nanHpyce PHH,lI,a HaXO,ll,HTC5I H e,ll,HHCTBeHHaH 3a,ll,aqa Ha
reOMeTpHqeCKYIO nporpeCCHlO, COCTaB31eHHaH H3 CTeneHeH
IIHCJIa ceMb: 7,7 2 , 7 3 , 74, 7 5 .... PH,lI,OM C 9THMH qHCJIaMH
nOCTaB31eHbI HepOr31HcpHlleCKHe 3HaKH, cooTBeTcTBYIOm.He C310-
BaM: ,lI,OM, KOIllKa, MbIIllb, HqMeHb, Mepa. CMbIC31 paCWHcppOB-
KH 3THX HepOrJIHcpOB TaKOB: HMeeTCH 7 ,lI,OMOB, B Ka>K,lI,OM
,lI,OMe no 7 KoweK; Ka>K,lI,aH KOWKa C'be,ll,aeT 7 MbIllIeH; Ka}l{,lI,aH
MbIWb C'be,ll,aeT no 7 K0J10CbeB HqMeHH; Ka>K,lI,bIH K0310C, eC31H
noceHTb ero 3epHa, ,lI,aeT 7 Mep 51I1MeH5I. BaHTH CYMMY 06lll,ero
qHC31a ,lI,OMOB, KoweK, MblWeH, K0310CbeB H Mep.
B nanHpyce ,lI,aeTCH ,lI,Ba peweHllH 3TOH 3a,ll,aIlH; nepBoe-
nYT€M HenOcpe,ll,CTBeHHoro YMHO>KeHHH H noc31e,ll,YlOlliero C310-
>KeHH5I lJJleHOB nOC31e,ll,OBaTe31bHOcTH: 7+72+73+74+76; BTO-
poe - nYTeM YMHO>KeHHH IIHC31a 2801 Ha 7. 4HC310 2801 no-
31yqaeTCH B pe3Y31 bTaTe CYMMHpoBaHHH 1+7+7 2 +73+7 4 .
06m.eH PMY31bI CYMMbI q31eHOB reOMeTpHqeCKOH nporpec-
CHll ,lI,a}l{e ,lI,31H C31yqaH b i = q erHnTHHe, nO-BH,lI,HMOMY, He
3Ha31H.
ha ,lI,peBHeHlllaH 3a,ll,aqa Ha reOMeTpHqeCKYIO nporpeCCHIO
He pa3 BCTpellaeTC5I Y pa3HbIX HapO,ll,OB C HeCK031bKO H3MeHeH-
HbIM TeKCTOM. BCTpeqaeTCH OHa H Cpe,ll,H ,lI,peBHepYCCKHX Ha-
PO,ll,HbIX 3a,ll,aq. «11,l1,eT CeMb cTapyx; Y Ka>K,lI,OH CTapYXH no
ceMH nOCOXOB; Ha BCHKOM nocoxe no ceMIl CYIIKOB; Ha BCHKOM
213

cYQKe no ceMH KOWeJJeH; BO BCHKOM KOWe31e no ceMb BOpO-
6beB; BO BCHKOM Bop06be no ceMb nynKOB. CKOJIbKO Bcero
npe.lI.M eTOB ?»
HCCJIe,ll,OBaHHe BaBH310HCKlIX KJIIiHOnHCHbIX TeKCTOB 3nOXH
XaMMypanli (XVIII B. ,lI,0 H. 3.) rOBOpliT 0 TOM, qTO H B ,lI,peB-
HeM BaBH310He peweHHe HeKoTopbIX BonpocoB X03HHCTBeHHoro
H HaYlJHOrO XapaKTepa npHBO,lI,H310 K aplicpMeTlilJeCKOH II reo-
IIleTpHqeCKOH nporpeccliHM. BOT O,ll,Ha 1i3 TaKHX 3a,ll,alJ,
2
10 6paTbeB; 1"3 MIiHbI cepe6pa; 6paT Ha,ll, 6paToM nO,ll,HII-
IaeTCH; Ha CK031bKO nO,ll,HliMaeTCH, H He 3HalO.lI.OJlH BOCbMoro-
6 WeKe31eH. BpaT Ha,ll, 6paToM Ha CKOJIbKO nO,ll,HHMaeTCH?
(I  MHHbl COCTaB31HeT 100 weKeJJeH.)
Pa360p peIlleHHH, .n;aHHoro ,lI,31H 3TOH 3a,ll,alJH ,lI,peBHHM Ba-
BH310HSlHIiHOM, YKa3bIBaeT Ha TO, lJTO eMY 6bIJ10 1i3BeCTHO
CBOHCTBO lJJleHOB apHcpMeTHqeCKOH nporpeCClili, KOTopoe 06bIlJ-
HO YCK031b3aeT OT Hawero BHIiMaHHH H KOTopoe JIerKO n031Y-
qaeTcSI H3 PMY31bI
S - al+a n
n - -r- n ,
a HMeHHO
S
a +a =2--!!-'
1 n n'
s u  u
Ha3BaB Ii cpe.n;HeH ,lI,O.TJ€ll H 3HaH, lJTO CYMMa KpaHHHX lJJleHOB
nporpeCCHH MO>KeT 6bITb 3aMeHeHa CYMMOH ,lI,BYX 31106bIX
lJJIeHOB, paBHooTCTOHWHX OT KOHUOB nporpecCIiH, MbI MO}l{eM
::ITO CBOHCTBO npOqeCTb TaK: CYMMa ,lI,BYX qJIeHOB apm!JMeTH-
lJeCKOH nporpeCCHH, o.n;HHaKOBO OTCTOHIl1.HX OT ee KOHIlOB. paBRa
y.n;BoeHHOH cpe.n;Heii ,lI,031e.
KaK >Ke C nOMOlliblO YKa3aHHoro cBoMcma .n;peBHeBaBHJ10H-
CKHA MaTeMaTHK peUlHJI 3a,naqy? PaccMoTpHM XO,ll, ero paccY}I{-
lI.eHHH. Cpe,ll,HHSI )J.031S1 Ka>K)J.oro H3 10 6paTbeB COCTaBJ1S1eT
10 weKe31eiL CYMMa ,lI,031eH TpeTbero H BOCbMOro 6paTbeB
COCTaBJIHeT 20 weKeJJeH, a TaK KaK ,lI,OJlSl BOCbMoro paBHa
6 llIeKeJIHM, TO ,lI,O.TJSI TpeTbero COCTaB31HeT 16 IlleKeJJeH. Pa3-
HOCTb Me>K,lI,Y ,lI,031HMH 3-ro H 8-ro 6paTbeB paBHa 14-6=8
(weKeJJSlM). TaK KaK TpeTHi\ 6paT «B03BbIIllaeTCH» Hall BOCb-
MbIM Ha 5 «CTyneHeH», TO O.Ll,Ha CT}'neHb (pa3HOCTb nporpec-
8 3
CHH) paSHa 5" = Is'
214

A KaK ObI Bbl peWHJl H 3TY 3a).!.a lJY,
B 3noxy XaMMypanH BaBHJ10HCKHe MaTeMaTHKH HMeJJH
npe)J.CTaBJ1eHHe II 0 reOMeTpHqeCKOH npcrpeCCHH. Ha11,l1,eH8
r31HHHHaH .n;Olll,eqKa C K31HHOnHCHbIM TeKCTOM. paCWHcppOB3H-
HbIM O,ll,HHM aHf31HqaHHHOM-aCCWpH0310rOM. 3TOT TeKCT pac-
CKa3bIBaeT 0 TOM, KaRaH qaCTb JIYHHoro ,lI,HCKa OCBeeHa
C031HueM B Ka>K,lI,bIH H3 15 ,lI,HeH OT HOB031YHHH ,lI,0 nO.1H031Y-
HHH. YBeJJHqeHHe OCBelll,eHHOH qaCTH 31YHHoro ,lI,HCKa B TelJe-
HHe nepBbIx nHTH ,lI,HeH nO,ll,qtIHeHO 3aKOHY reOMeTpHqeCKOH
nporpeccHIi C 3HaMeHaTe31eM 2, a B nOC31e,ll,YlOlliHe 10 ,lI,HeH -
3aKoHY 3pHcpMem'leCKOH nporpeccm! C pa3HocTblO, paBHolt 16.
lllupoKHH HHTepec, npOHBJ1HeMbIH B ,lI,peBHeM BaBUJIOHe
K aCTpoHoMHH,' ,lI,e31aeT BnOJ1He nOHHTHblM B03HHKHOBeHHe 3roH
3a,ll,aqH.
Co,ll,ep}l{3HHe pSl,ll,a HCTOpUlJeCKHX 3a.n;aq Ha nporpeCCHH
OTpa>KaeT TO Y,lI,HB31eHHe, KOTopoe pO>K,lI,a310Cb Y 311O)J.eH nepe).!.
6bICTpblM pOCTOM BeJJHqliHbI q31eHOB reOMeTpliqeCKOH nporpec-
CliH C 3HaMeHaTe31eM, 6631bWHM I, KOr,ll,a OHH conOCTaB31H31H
,lI,Ba HBJ1eHHH, H3 KOTOpblX O,ll,HO npoTeKaeT no 3aKoHY apH-
Mem'leCKOH nporpeccHH. a ,lI,Pyroe - no 3aKoHY reoMeTpH-
lJeCKOH nporpeccHH. npHBe,ll,eM O,ll,HY 1i3 TaKHX HcroplilJeCKHX
3a).!.a'l HH)l.IiHCKOro npOHCXO>K)l.eHHSl, B03MO>KHO Y}I{e 3HaKoMYIO
'lHTaTeJJlO.
no npe,ll,aHHIO, HH,lI,HHCKHH npliHU CHpaM, BocxHmeHHbIfi
OCTpoYMHeM H pa3Ho06pa3HeM B03MO>KHbIX nOJ!O>KeHHH wax-
MaTHolt HrpbI, npH3Ba31 K ce6e ee H306peTaTeJJH, yqeHOrO
Cera. H CKa3a31 eMY: «51 >KeJJalO ,lI,OCTOHHO B03Harpa,ll,HTb Te6H,
CeTa, 3a npeKpacHYIO Hrpy, KOTOpylO TbI npH,lI,YMa31. 51 ,lI,0-
CTaTOqHO ooraT, qTo6bI HCn031HHTb 311060e TBoe >Ke31aHHe».
«nOBeJJHTeJJb,- OTBeTH31 CeTa,- npHKa>KH MHe BbI,lI,aTb 3a
nepBYIO K31eTKY waxMaTHoH ,lI,OCKH O)J.HO nWeHMqHOe 3epHo,
3a BTOpylO - 2 3epHa, 3a TpeTblO - 4 3epHa H TaR 3a Ka}l{.n;ylO
HOBYIO K31eTKY B,lI,BOe 6031bWe 3epeH, qeM 3a npe,ll,wecTBYIO-
mylO». «TbI nO.TJyqHWb CBOH 3epHa. Ho 3HaH, 'ITO TBOH npocb6a
He,ll,ocToHH3 MoeH llie,ll,pOCTH. CTynaH. Cnyru MOH BbIHecYT
Te6e TBOH MeWOK C nweHHueH».
Ha ,lI,pyroH ,lI,eHb npH,lI,BOpHbIe MaTeMaTHKH HBH31HCb K
npHHuy CHpaMY.
bI ,lI,06pOCOBecTHo HCqHC31HJ1H,- rOBOpHJ1H OHH eMY,-
Bce KO.TJHqeCTBO 3epeH, Koropoe >KeJJaeT n031yqHTb CeTa.
4HCJJO 3TO TaK Be31HKO, qTO 3epeH He XBaTHT HH B TBOHX }l{HT-
HHU3X, HH B }l{HTHHuax UeJJoro uapCTBa. He Haii.n.eTcH TaKoro
qliCJJa sepeH Ii H3 BeeM npocTpaHcTBe 3eMJIH».
216

C H3YM31eHHeM BHHMa31 npHHU CJIOBaM yqeHbIX. «HanHWHTe
}l{e MHe 3TO lJY)J.OBHIlI.HOe qHCJIO»,- CKa3an OH.- «BoT 3TO
lJliCJIO -18 446 744 073 709 551 615». 4HTaTeJJb MO>KeT npo-
BepHTb, TaK 31H 3TO B .n;eHcTBlITeJIbHOCTH, eC31H Hcno.1Jb3yeT
cpOPMY31Y
S = b. (qn_l)
q-l '
nOJIaraH B HeH b 1 =1, q=2 Ii n=64.
B ,lI,peBHeH rpeuHH BO BpeMeHa EBKJIH,lI,a H ApXIiMe,ll,a
(Ill B. .n.o H.3.) cBoikTBa nporpeCCHH paCCMaTpHBalOTCH He
TOJIbKO ,lI,31H peweHHH npaKTHqeCKHX 3a,ll,aq, HO H B CBSJ3H C
TeOpeTHqeCKHMH HCC31e.n.OBaHHHMH.
TaK, B KHHre EBK31H)J.a «HalJa31a» BbIBO.n;HTCH <PopMY31a
CYMMbI Tpex q31eHOB reOl\1eTpHqeCKOH nporpeCCHH. I1HTepecHO,
lJTQ K reOMeTpHlJeCKOH nporpeCCHH EBK31H,lI, nO,ll,xo.n;HT, HCnOJIb-
3YH cBoikTBa nponopUHH. 4 HTaTeJJb, HaBepHoe, nOMHHT,
u a b u b
qTO B HenpepbIBHOH nponopUHH l)=c cpe.n;HHH lJ31eH
HB31HeTCH Cpe,ll,HHM reOMeTpHlJeCKHM ,lI,31H KpaHHHx qJIenOB a
H c. nYCTb ,lI,aH pH,lI, paBHbIx oTHoweHHH:
b
c
d
a-=b=c
CocTaBHM npoH3BO,ll,HbIe npOnOpLl,IlH, BbIlJHTaH H3 Kaoro
OTHoweHHH no I:
b-a c-b d-c
----a- - "f)" c
I1cn031b3YH CBOHCTEO lJ31eHOB pH,lI,a paBHbIx oTHoIlleHHH, Ha-
nHweM:
(b-o) + (c- b)+ (d- c)
a+b+c
. b-a
=-
a
c-b
= "f)" =
d-c
c
d-a
a+b+c
b-a
d-a
a+b+c=a- b .
-a
--
a
MbI nOJIyqHJJH CYMMY nepBbIx Tpex lJ.TIeHOB reOl\1eTpllqeCKOfi
nporpeCcm-I -;-:-a, b, c, d; qT06bI IIPHHTlI K H3BeCTHOH BaM
tPopMY31e CYl\ll\1bl q31eHOB reOMeTplilJeCKOH nporpeCCHH, CTOHT
TOJIbKO pacnpocTpaHHTb e Ha n C31araeMbIX
i-a
a+b+c+...+k=a- b '
-a
216

r,ll,e 1 - 'lJIeH nporpeccHH, CJIe,ll,YJOIlI.HH 3a k,
npHMeHHTb ee K nporpeccHH
.;-;.-1, q, q2, q3, ..., qn-l,
H MbI n031YQHM:
1 + q + q2 + q3 + ... + qn-l = qn_ l l .
q-
ApxHMe)J. TaK}I{e HCn031b3yeT nOHHTHe reOMeTpHqeCKOH npo-
rpeccHH H cyw\lIIpyeT qJIeHbI ee. B pa60Te «0 KBa,ll,paType
napa6031bI», B KOTOpOH pa36HpaeTcfI Bonpoc 0 HaXO>K,lI,eHHH
n310lll,a)J.H napa60m-rqeCKOrO cerMeHTa nYTeM BnHCbIBaHHH B
Hero MHoroyrOJIblIOH npflMOJIHHeHHOH cpHrypbI H nOC31e)J.OBa-
Te31bHOrO Y;I.BOeHHH qHC31a ee CTOpOH, ApXHMe,ll, CTa.1KHBaeTCH
C He06xo,ll,liMOCTbJO HaXO,ll,HTb CYMMY q31eHOB 6eCKOHeqHO y6bI-
BaJOllI.eH nporpeCCliH:
1,, (+t(+y,....
npHeM, KOTOpbIM npH 3TOM n031b3yeTcH ApXHl\1e,ll,. He HBJIHeTCH
06lll,HM H npHrO)J.eH TOJIbKO )J.JIH HeKOTopbIX KOHKpeTHbIX 3a-
,lI,aq. 06lliee peweHHe Bonpoca H H3BeCTHaH cpop1l1y'31a CYMMH-
pOBaHHH II31eHOB 6eCKOHeqHO y6blBalOllI.ei1 reOMeTpHqeCKOH
nporpeCCHH 6bIJ1H nOJ1yqeHbI MHoro n03)J.Hee.
Y Hac B POCCH1-r 3a,ll,alJH Ha nporpeCCHH BnepBbIe BCTpe-
lJalOTCH B O)J.HOM H3 )J.peBHeHWHX naMHTHHKOB PYCCKoro npaBa -
B «PYCCKOi1 rrpaB,lI,e», COCTaBJ1eHHOM npH .S!pOc.aaBe My.lI.-
pOM B XI B. TaM eCTb cTaTbH, nOCBHlll,eHHaH BbIqHC31eHHIO
npHn310,ll,a OT ,lI,Ba)J.uaTH )J.BYX oBeu 3a 12 31eT npH YC310BHH,
qTO Ka>K,lI,aH OBua e}l{erO)J.HO npHHocHT O)J.HY OBUY H O)J.Horo
6apaHa. OTBeT, OqeBH,lI,HO, 6Y,lI,eT TaKHM: 22.2 13 . TaM >Ke
MO}l{HO npOqeCTb JIJ060nbITHbIH paCqeT, COCTaB31eHHbIH Ce31b-
CKHM X03HHHOl\l 0 npHn31o)J.e 3a 12 H.TJH 9 31eT OT Bcero CKOTa
H nqe31 ero ce31a, npH6bI31H OT BbIceBaeMoro xJIe6a H nHTH CTO-
rOB ceHa, a TaK>Ke paCqeT n31aTbI 3a l2-JIeTHJOIO Ce31bCKYJO
pa60TY }l{eHlliHHe C ,lI,OqepbJO.
BCTpeqaJOTCH nporpeccHH H B PYCCKHX MaTeMaTHqeCKHX
PYKonHcHx XV-XVII BB. BOT O,ll,Ha H3 3a)J.a'l, CBfl3aHHbIX
C nporpeccHHMH. «BbI310 40 ropo)J.oB, a BO BC5IKOM ropo,ll,e no
40 Y31Hu, a BO BCHKOH Y31Hue no 40 )J.01l10B, a BO BCHKmi )J.OMe no
40 CTOJInOB, a BO BCHKOM CTO.TJne no 40 KOJIeu, a Y BCHKoro
K031bua no 40 KOHeH, a Y BCHKoro KOHH no 40 qeJJOBeK, a
Y BCHKoro qeJJOBeKa no 40 nJIeTeii, HHO MHoro 31H nop03Hb
6yeT?»
17

3HaIJHTe31bHOe KO.TJHqeCTBO SB)J.aq HB nporpeCCHH HMeeTCH
B SaMelJaTe31bHOM nBMHTHHKe HalJaJIa XVIII B.-«ApH<fJMeTHKe»
JI. cD. MarHHUKoro. B TeqeHHe n031YBeKa :na KHHra 6bI31a OC.
HOBHbIM MaTeMaTHlJeCKHM YIJe6HHKOM B POCCHH. MHXaH31 Ba.
CH31beBHIJ JIOMOHOCOB OlJeHb BbICOKO oueHHBa31 KHHry
JI. cD. MarHIiUKoro H Ha3BaJI ee «BpaTaMH yq€HOCTH».
nporpeCCIil! B «ApHcpMeTIiKe» MarHIiUKoro BK31JOlJeHbI B
nHTYJO qaCTb nepBOH KHlirH «0 nporpeCCHHX H pa.ZJ;HKcax
KBa)J.paTHbIX Ii Ky6HlJeCKHX». npHBeJJ.eM .ZJ;Be 3a)J.aqH H3 9TOro
pa3,l1,Ma, noc,rre,ll,HHH 1i3 HHX BeCbMa 31J060nbITHaH no cO)J.ep-
}l{a HIIJO.
o e p B a H 3 a .ZJ; B IJ a. «Kyneu HMe31 ]4 qapOK cepe6pH-
HbIX, npHIJ€M Beca lJapOK pacTYT no apHcpMeTHlJeCKOH npo-
rpeccHH C pa3HocTbJO 4. OOC31e,ll,HHH lJapKa BeCHT 59 310TOB 1 ).
Onpe,ll,e31HTb, CKOJ1bKO BecflT Bee lJapKIi».
B TOp a 51 3 a ,lI, a q a. «HeKHH IIe310BeK npO,ll,a31 KOHSl
3a 156 py631eH; pacKaHcH >Ke, Kyneu Haqa OT,lI,aBaTH npO,ll,aBUY
06paTHo, r.rrar031H: HKO HeCTb MHe 31eno B3HTH cHueBoro KO-
HH He)J.ocTOHHoro TaKOBble BbICOKHe ueHbI. 0p0,ll,aBeu >Ke npe)J.-
JIO>KH HHY KynJIJO r31aro31H: alll,e Te MHHTCH BeJJHKa ueHa ceMY
KOHJO, y60 KynH T031bKO rB03,l1,He, IiX >Ke ceH KOHb HMaTb B nOk
I<OBaX CBOHX Hor, KOHH >Ke B03bMH 3a TOJO }l{e Kyn31eJO B ,lI,ap
ce6e. A 3a rB03)J.H B Ka>K,lI,OH nO,ll,KOBe no IlleCTH H 3a O,ll,HH
rB03.ZJ;b ,lI,a}l{,lI,b MHe n031YIllKy2), 3a ,lI,pyroH >Ke ,lI,Be n031YIllKH,
a 3a TpeTHH KoneHKY, H TaKO Bce rB03,l1,H KynH. Kyneu >Ke,
BH,lI,H CTOJ1b Ma31YJO ueHY H KOHH XOTH ce6e B ,lI,ap B3HTH, 06elll,aJI
TaKY ueHY eMY n31aTHTH, lJaH He 6031bWe ,lI,eCHTH py631eH 3a
rB03,l1,He ,lI,aTH. 11 Be)J.aTeJJbHO eCTb: K031HKO Kyneu npoTopro-
BaJICH?»
PewHTe 9TY 3a)J.aqy H npOBepbTe OTBeT, ,lI,aHHbIH JI. cD. Mar-
HHUKHM,- 4 178703 3 / 4 Kon.
A nO'leMY apH4>MeTH'IeCKaSl H reoMeTpHllecKasr?
3HaeTe 31H BbI, nOqeMY O)J.HH nporpeccHH HB3bIBalOTCR
apHcpMeTHlJeCKHMH, a )J.pyrne - reOMeTpHqeCKHMH?
B03bMeM aplicpMeTHqecKYIO nporpeccHJO:
-:- 01' 0 3 , a 3 , a 4 , ... , ak-l' ak' ak+l' ... , an'
1) JIom - PYCCK3fI .npeBHSU! Mepa, paBHaH 12,8 "
2) llOIlYWKa - p,peBHHH PYCCK3H MOHeTa, p3BH3H 1. KoneAKH.
4
218

113BecTHO, IJro
Q,,-l == al + d (k - 2); a k = a 1 + d (k - I) H a k + 1 == Ql + d,,;
HaH,lI,eM nOJIycYMMY a k -l H aMI:
Gk_l + ak+l GI + d(k - 2) + al + dk 2 G( + 2 dk - 2d
2 = 2 = 2 =-
_ 2 [al +: (k - I)J :z al +d (k-I).
CpaBHHBast n031YIJeHHblfi pe3YJIbTaT c a k = Ql + d (k - I),
MbI BH,lI,HM, liTO
ak-l + ak+l
a k = 2 '
T. e. Ka>K,lI,bIH II31eH apHcpMeTHqeCKOH nporpeCCI1H, KpOMe
nepBoro" H nOCJle)J.Hero, eCTb Cpe,ll,Hee apmpMeTHlJeCKOe )l.31st
BYX coce)J.HHX C HHM lJ31eHOB.
B03bMeM Tenepb reOMeTpHlJeCKYIO nporpeCCI1IO:
77- b 1 , b 2 , b 3 , ... , bk-l> b k , bk+l> ... , b n ;
H3BeCTHO. qTO
b k _ 1 = b 1q k-2; b k = b 1q k-l H b kH = b 1 q".
CoCTaBHM np0l13Be,ll,eHHe b k - 1 Ii b k + 1 :
b k - 1 ,bk+l = b 1q k-2.b 1q k = b 1 2 q 2k-2.
H3B31eKaH KBa,ll,paTHbIH KopeHb H3 n031yqeHHOrO npdH3Be,ll,e-
HHH, HMeeM:
V b k - 1 . b kH = V b.q2(k-l) =b 1q k- 1 .
CpaBHHBaH nOJIYlJeHHbIH pe3Y31bTaT C b k = b 1q k-1, Mbl YBH-
HM, t.lTO
b k = Vb k - 1 . b k + 1 ,
T. e. '11'0 Ka>K)J.bIH lJJIeH reOMeTpHqeCKOH nporpeCCHH, KpO-
Me nepBoro H nOC31eHero, eCTb Cpe,ll,Hee reOMeTpHqeCKOe
BYX COCe,ll,HHX C HI1M q31eHOB.
Tenepb MO>KHO CKa3aTb, OTKY,lI,a nOHBHJIHCb Ha3BaHIiH-
apmpMeTHt.leCKaH H reOMeTpH'leCKaH nporpeCCHH.
219

o CYMMe 6eCKOHe'lHorO MH01KecTBa y6b1BaIOllI,llX CJlaraeMblX
BOJ1bIllOe 3Ha1Jemre B MaTeMaTIlKe H ee npHJImKeHHHX HMeeT
6eCROHe1JHO y6bIBaIom.aH reOMeTpll1leCKaH nporpeCCHH. B lllKO-
31e BbI Y>Ke n03HaKOMII31IlCb C onpe)J.e31eHHeM CYMMbI Bcex ee
'131eHOB. BcnoMHHM 3TO onpe,ll,e31eHlre.
OYCTb ,lI,aHa 6eCKOHeqHO y6bIBaIom.aH reOMeTpHqeCKaH npo-
rpeCCHH:
ai' G 2 , a 3 , ." , an' '" .
Ha30BeM CYMMbI
SI =a l ,
S2=a l +02'
S3= a l +a 2 +a 3 ,
Sn=01+ a 2+ a 3+ ... +a n
'JaCTHqHbIl\Ur CYMMaMli.
CYMJvWU ecex 'iZeft08 6eCKOfte'iftO y6bl8alOUfeu e.eoJvrempu'ie-
CKOU npocpeccuu ftQ3bLBalOm mom npeoe.I, K KOmopw,zy cmpe-
1rtumCR. nOC/leoOeame/lbftOCmb ee 'iacmU'iftblX CY1rtM SI' S2' S3'
... , Sn npu n, cmpeMflUfeJvrcfl K 6eCKOftC'iftOcmu.
n03HaKOMHBIllHCb C onpe,ll,e31eHHeM CYMMbI Bcex qJIeHOB
6eCKOHeqHO y6bIBalOllieH reOMeTpH'leCKOM nporpeCCHH, MbI
HMe31H B03MO>KHOCTb BbIBeCTH li cpOPMYJIY Jl,31H HaXO>K)J.eHliH
CYMMbI 3TOM nporpeCCHH.
cI>opMY31a HMeeT OqeHb npOCTOH BlihU
S=
l-q'
r,ll,e a - nepBbIH q31eH 6eCKOHeqHO y6bIBalOlll,eH reOMeTpH1Je-
CROM nporpeCCHH, a q - ee 3HaMeHaTe31b, no a6C031IOTHOH Be-
31H1JIIHe MeHbIllHM e,ll,HHIIUbI. Ecmr, HanpHMep, ,!laHa nOCJIe,!lo-
BaTeJJ bHOCTb
1, 2' 4' 8'...'
(1)
TO CYMMa S BCex ee 1JJleHOB 31erKO HaMJI,eTCH no BbIweYKa3aHHOH
PMYJIe:
1
S = --r = 2.
1--
2
220

DOJIytum TaKOM OTBeT B03MO>KHO BnepBbIe, t3bI CTOJIKHe-
TeCb C TeM cpaKTOl\1, qTO CYMMa 6eCKOHellHoro lJMCJIa CJIara.
eMbIX OKa3aJIaCb paBHoi'I KOHeqHOMY qMCJIY.
HanMllleM )J.pyrylO nOCJIe)J.OBaTeJIbHOCTb:
I 1 1 1 1
1, ""2 ' 3' T' 5' ... , 11 '
(2)
Eha nOCJJe)J.OBaTeJIbHOCTb, KaK M oecKoHeqHO y6bIBalOlll,a
reoMeTpMqeCKaH nporpeCCMH, eCTb nOC31e,ll,OBaTeJIbHOCTb y6bl-
BalOru.aH; HO eC31M )J.31 nporpeccHM (1) MbI MMeeM cpOPMYJIY
.n:JI nO,ll,CqeTa CYMMbI Bcex ee q31eHOB, TO ,lI,31H nOC31e,ll,OBa-
TeJIbHOCTH (2) MbI nO,ll,06HoH CPOpMY31bI He MMeeM. 11 Bee me
nonpooyeM nO,ll,OiiTU K Bonpocy 0 cYMMe
1 I I 1 I
1 + ""2 + 3 + 4" + 5" +. . . + n +... (3)
BCex llJIeHOB nOC31e,ll,OBaTeJIbHOCTM (2). MbI He MomeM cpaK:
THqeCKM C310>KUTb 6eCKOHeqHOe «qHC310» CJIaraeMbIX 9TOH
CYMMbI, HO Mbi MO>KeM nOHTIi no nYTli cpaBHeHM CYMMbI (3)
C ,lI,Pyroii CYMMOii.
Ham!llleM CYMMY (3) B BIi,ll,e
1 +  + [  +  ] + [ -.l +  + -.l + J.... ] +
234 5678
+ [ + + /0 + ... + 116 ] +...
Ii nO,ll,nMllleM no,ll, HeH ,lI,pyrylO CYMMY
1 +  + [  +  ] + [  + ..!.. +  +  ] +
244 8888
+ [ 116 + /6 +... + 116] + '" .
n031yqMM ,lI,Be CTpOKM:
1 + -.l + [  + -.l ] + [ -.l +  +  +  ] +
234 5 6 7 8
[ 1 I I ]
+ "9 + 10+'" + 16 +...
1 + J..... + [ J..... +  ] ...L [  + J..... + J..... +  ] +
244 1 8888
[ I 1 I ]
+ 16 + 16 +... + 16 +... .
H
221

Ka>K,lI,bIH tlJIeH BepxHei'I CTpOKH 60JIbille HJlH paBeH COOT-
BeTCTBYIOlll,eMY 'lJIeHY HIi>KHeii CTpOKH; HO CYMMa qJIeHOB,
crpynnHpoBaHHbIX B Ka>KJ!.OH Ii3 CKo60K HII>KHeH CTpOKH,
I
paBHa "'2; n03TOMY CYMMY n nepBbIX qJleHOB MO>KHO C)J.eJIaTb
KaK yrO,ll,HO 6031blllOii, eC31H T031bKO B3HTb ,lI,OCTaTOlJHOe lJHCJIO
'lJIeHOB; CJle,ll,OBaTeJIbHO, II CYMMa n nepBbIX q.neHOB Bepx-
Hero p,lI,a HeOrpaHHlJeHHO B03paCTaeT npll HeOrpaHHtIeHHOM
B03paCTaHHH n.
TaKHM 06pa30M, MbI y6e,ll,H31HCb B TOM, tITO CYMMa Bcex tIJIe-
HOB nOCJle,ll,OBaTe31bHOCTH (2) He MO>KeT 6bITb BbIpa>KeHa HIlKa-
KIiM qHCJIOM.
11TaK, ,lI,JIH noc..le,ll,OBaTeJIbHOCTH
I I
1, 2'""3 4'...
MbI He MO>KeM onpe.n.e31IITb CYMMY Bcex ee il31eHOB, B TO BpeMH
KaK ,lI,JIH 6eCKOHetIHOH nOCJIel1.0BaTeJIbHOCTIl
I 1 1
1, 2' 4' 8''''
Ta CYMMa paBHa KOHelJHOMY lJIiCJIY
1 I, 1 1
1+"2 + T -I- 8" +... = --r = 2.
1--
2
B MaTeMaTIlKe, eCJlH ,lI,aHa 6eCKOHeQHa5I nOCJIe.n.oBaTeJ1b-
HOCTb
U l , lll' lls' -.. , un' ... ,
TO CYMMY Bcex ee 'lJIeHOB, HanHcaHHbIX C COXpaHeHHeM ox
nop)J.Ka, T. e.
U l +ll 2 +ll s + ... +U n + .., ,
Ha3bIBalOT pH)J.(jMl). 0603HaQOM tlepe3 Sn CyMMY II nepBbIX
'lJIeHOB pJ!.a
S.-lll+U.+U,+ ... +U..
1) B 'I8CTHOCTH, C:YMM'
1 .. 1 I I
1+2" + T-r--t +,-+... + 11+'"
na3b1SaeTC« r8p\lOiilol1Jf,:LJnl 011.110101
222

....
,;',-!'r. _-,'
": .J.- . -.
".",
7:.
.:;.: :-:...,
'"
. '.
'v. ....
;:!
-  ;'
":" 4
''' , , ""
, ).:
\
.. ;
". '"
:  >1 .'}.
". 'OJ;
.-.::;
'('
f. .

"
...
J
;,--
l .I
, ,
..,
. 
f
<":
, ,
.:-
L.
".\r. ';!" A ,.::..:.: -
". y ..:
,.,J
. ........ . Y.(
,n'AJlI!.M6ep.
ECJlH 8n CTpeMHTCSI K onpe,ll,e31eHHOMY npe,ll,eJJY 8, Kor)J.8 n
HeorpaHHlIeHHo B03pacTaeT, TO pH)J. H83bIBaeTCSI C x 0 )J. H-
lll, H M C SI H lJHCJIO 8 - era CYMMO.if. ECJIH }l{e 8n npH n-+ 00
He HMeeT npe)J.eJJa, TO PH)J. Ha3bIBaeTCSI pac x 0 ,lI, H lll, H M-
e SI; B STOM CJIyqae OH He IiMeeT CYMMbI. Cne,ll,OBaTe31bHO, 1i3
BYX paCCMOTpeHHblX pHOB
1 1 1
1+"2 + T+T+'"
H
1 1 1 1 1 1
1 + T + "3 + 4" + "5 + '6 + T + ...
nepBbIA pSI,lI, 6Y,lI,eT CXO)J.Hlll,HMCSI, I!I BTOpoA - pacxo)J.Sllll,HMCSI.
OtIeBH.D:HO, BCSlKaSi 6ecKoHe t IHo y6bIBalOlll,aH reoMeTpHqecKaSi
nporpeccHSI npH CYMMHPOB8HHH ee tlJIeHOB ,lI,8eT CXOASllll,HHCSI
PM. B caMOM lleJIe
m

Sn = b + bq + bq2 + . . . + bqn-l = b 1 ;- blq" ,
-q
HJJH
b bqn
S =---
n I-q I-q'
TSK KaK .nJUI 6eCKOHeqHO y6blBalOlll,eii nporpeCCHH Iql<l, TO
lim  = 0
n"" 1- q
H
1 . S 1 . [ b bqn ] b
1m n= 1m --- =-.
n-"" 1 - q 1 - q 1- q
B p8CnOpH>KeHHIi MaTeMaTHKOB HMelOTCH HeCK031bKO npIi-
3HaKOB CXO,ll,HMOCTH pH,lI,OB, T. e. npH3HaKOB, n03BOJIHIOlliHX
CK83aTb, CXO,ll,HTCH ,lI,8HHbIH pH,lI, H31H paCXO,ll,HTCH. [J03HaKO-
MHMC C O,ll,HHM H3 TaKIiX npH3H8KOB, OTKpbITblM cpP8HUY3-
CKHM MaTeMaTHKOM 'u'A31aM6epOM (1717-1783). MbI ,lI,aeM
ero 6e3 ,lI,OKa3aTeJIbCTBa.
,UaH pH,lI,
UI + U 2 + U 3 +...+ Un + Un+1 + ...
ECJIH
lim ( U n + 1 ) < 1,
n_oo Un
TO pH)J. CXO)J.HTCH; eC31H
Iim ( U+l ) > I,
n- 00 n
TO pH,lI, paCXO,ll,HTCH. ECJIIi
Iim ( U n +l ) = 1,
n- co Un
TO npH3HaK ,U' AJIaM6epa He peIll8eT Bonpoca.
I1cnOJIb3yeM npIi3H8K ,U 'A31aM6epa ,lI,JIH BblHCHeHHH CXO-
)J.HMOCTIi HeKOTopblX pH,lI,OB.
a) J1.aH pH)J.:
1 1 I I 1
1 +"2 + ¥ + 23 +... + 2 n =t + 2 n '
224

LLJI 3TOrO pH,lI,a
I . ( Uf1+1 ) I ' ( I I ) I . ( 2n-l ) I
1m U = 1m 2" ; 2n-1 = 1m V =""2'
n-+(X) n n_oo '1-+=
lim ( U+l ) < I; C31e)J.OBaTe31bHO, no npIi3HaKY Ll 'AJIaM6epa,
n- 00 n
)J.aHHblH p,lI, CXO,ll,HTCH. Mbl Y>Ke BIi)J.eJIIi, qTO CYMla Bcex
lJJIeHOB pH)J.a paBHHeTcH 2.
6) HCCJIe,ll,yeM CXO)J.IiMOCTb pH)J.a:
I I I I
1 + T + TI + \.2.3 + 1.2.3.4 +... +
+ 1.2..\n-l) + 1./... n +...
HaH,lI,eM ,lI,JIH 3TOro lilT! ( :t.! ) :
tl-+ :x> Un
: ( U:I) =,,'I [ I .2.... n : \.2.3,1. (n-I) ] =
I " [ 1.2.3... (n- I) ] I . ( I ) . 0
= tnl = lnl - = .
noo 1.2.3... n noo n
Y1TaK,
lim ( U II I.I ) < I.
n -+ 00 Un
CJIe,ll,OBaTeJIbHO, no npH3HaKY 11 'A.TJaM6epa, p5lJt CXO,ll,liT-
CH. B O,ll,HOH H3 nOC31e)J.YlOlliHX r31aB MbI Y3HaeM, lJTO pH,lI" CXO-
.nIiMOCTb KOToporo Mbl T031bKO qTO ,lI,OKa3a31Ii, IiMeeT HCKJIiO-
lI1iTeJJbHoe 3HaqeHIie B MaTecIaTIiKe. CYMMa 'weHOB ere paB-
Ha HenepOBY 'IHCJIY e, npHHHTOMY 3a OCHOBaHIie HaTY-
paJIbHbIX JIOrapHcp10B. [JOKa MbI He peIllH31H Bonpoca 0 CYMMe
3Toro pH,lI,a, HO IMbI BbIHCHH.TJIi, lJTO TaKaH CYMMa CYllieCT-
ByeT.
npe,ll,.TJarae:\1 qHTaTeJ1!O caMOCTOHTeJ1bHO BbIHCHIiTb Bon-
pOC 0 CXO,ll,IiMOCTH Ii.TJIi paCXO,ll,IiMOCTIi pH,lI,OB C nOMOlll,blC
npH3HaKOB LL' A.TJaM6epa:
8 A. A. KOJIOCOB
225

1 1
I) 1.2.3 + 1.2.3.4.5 + 1 .2.3.4.5.6.7 + ..,
1 I I
2) 2 + 4" + 6 + ...
1.2 1.2.3 1.2.3.4
3) 10+ I(j2 + 103 + .., .
HHor,ll,8 Bonpoc 0 CXO,nHMOCTH H31Ii paCXO,ll,HMOCTH pSI,lI,B
JIerKO peIll2eTcH nYTeM cpaBHeHIiH 'lJleHOB ero C COOTBeTCT-
BYlOlliH,\-!H qJIeHaMIi pH)J.OB:
I I I I I
"2 + 22 + 23 + 24 + ... + 2'1 + ."
H
. I I I I
1+ 2 + 3 +"4+5+'" ,
H3 KOTOpbIX nepBbIH CXO)J.IiTCH, a BTOpOH paCXo,ll,HTCSI. TalC,
HanpIiMep, pH,lJ.
I 1 1 1
I + ¥ + 33 + 44 + ... + nn +...
CXO)J.IiTCSI. TaK KaK K3>K.lJ.bIH lJJ1eR .9TOrO pH,lI,a, KpOMe nep'
Boro, MeHbllle COOTBeTCTBYfOlliero lJ.TJeHa pH,lI,a
I I I I I
2" + 22 + -2 3 + 24 +... + 2 n ' + ...
PH,lI, >Ke
1 1 I 1
1+ - +- +-+... +- +...
Y2 Y3 Y4 yil
paCXO)J.HTCH, TaK KaK ero lJ31eHbI 6031bWe cooTBeTcuYfOIll.HX
lJ31eHOB pH,lI,a
I I I I
1 + 2 + 3 + T +... + n +... .
"226

reOMeTpHH npHXOJl.HT Ha nOMOU(b are6pe
BepH€MCH ee pa3 K 6eCKOHeQHO y6bJBalOeH reOMeTpH-
'IeCKOH nporpeccHli. BbI y}({e 3HaeTe cpOpMy.TJY CYM\lbI qJ]e-
HOB 3TOH nporpeCCHIi. OCTaeTCH 03HaKOMHTbCH C reO\leTpH-
QeCKHM BbIBO)J.OM 3TOH cpOply.TJbI. 11JIH BbIBO,ll,a £pOPMYJI
paCCMOTpHM qepTe>K 115 1 ) .
N
c
f1
4epT. 115.
.u.OKa>KeM cpOPMYJIY CHaQa31a )J..TJH nporpeccHH BH.na:
1, q, q2, q3, ... ,
(1)
r,ll,e q < 1. Orpe3Kli OC, AB, AIB1' A2B2' A3B3' ... rpacpH-
lJeCKli li306pa>KalOT BeJIHqliHY lJ.TJeHOB nporpeccHH (I). 3TO
YTBep>K,lI,eHIie c.TJe,ll,yeT li3 nO,ll,06liH Tpeyro.'lbHIiKOB: ACB,
AIBB1' A2BIB2' .. _, eC.1Ii npIiHHTb OC = 1 H A8 = q H
AD = 1.
B caMOM ,lI,eJIe
A 1 Bl AlB. A 1 B l _ q . A B _ q 2-
AB = AC ' q - T ' 1 1 - ,
A 2 B 2 A2 B l. A2Bz _ q2 . A B 3
- - 2 2=q
AIBI - A1B 'T q ,
li T. )J..
1) DOCTpoeHHe llepTeJKa aallHHaeM C M3J10rO KBaJ!paT3, OTKJla.nbl-
BaeM OTpe30K AB = q H CTpOIlM BeplllllHY M 6oJIblllOro KBa.npaTa.
8*
227

CYMMa lJJIeHOB ,lI,aHHOH nporpeCCHIi rpacpUqeCKH Ii306pa-
3HTCH CTOpOHOH ML KBa,ll,paTa OLMN. HaH.D.eM ML.
6 CBD ro 60CM.
n03TOMY
ED CD AD . ED _ AD .
ct- = uM = AiL' OC - ML'
l-1TaK,
ML= CC.AD ==
ED l-q I - q .
I + q + q2 + qa+... = _ I I .
-q
ECJIIi nporpeccHH HMeeT BH,lI,
b. bq. bq2, bq3, r,ll,c q < I. TO
b + bq + bq2 + b(l + .., = b (I + q + q2 + qS + ...) =
b
l- q'
PewaeM 3aJl.a'lH
BbI XOpOIllO C,lI,e.TJ8eTe, eC31H peIllHTe HecKO.TJbKO sa)J.aq Ha
nporpeCCHH H IiHble nOC31e)J.OB8Te.TJbHOCTH.
1. ,UaHa qUC310BaH noc.TJe,ll,OBaTe31bHOCTb:
 01, O 2 , aa, ..., an' a n + 1 , a n + 2 ,
a2n' 02n+l' 02n+2' ... , 0an' aan+l ...
(I)
H ,lI,pyraH noC.TJe)J.OBaTeJIbHOCTb:
Sl> S2' Ss,
(2)
r.D.e
Sl = a 1 + 02 + ... + an'
S2 = a n + 1 + a n + 2 + ... + 02n'
S3 = 02n+l + 02n+2 + ... + a3n
H T. )J..
'uOKa3aTb. lJTO nOC31e,ll,OBaTe31bHOCTb
Sl' S2' Sa, .,.
eCTb TaK>Ke ,apIHp',leTIiqeCKaH nporpeCCIiH, paSHOCTb KOTOpOH
B n 2 pa3 6031bWe paSHOCTIi ,lI,aHHOI1 nporpeCCIiU. .
228

Y K a s a H He. HanHWHTe nOCJIe,ll,OBaTe31bHOCTb (2) B BH)J.e
(al + a,.)n (a n +l + a 2n )rz
2 2
H npoBepbTe, BbInO.TJH5IeTCH .TJH ,lI,.'1H Hee CBOHCTBO qJleHOB
apHcpMeTH'IecKoii nporpeccHu, pa306paHHoe B 9 2.
2. B reOMeTpIiqeCKOH nporpeCCIiIi:
-;-:. b 1 , b 2 , b 3 , ...
b m + n = A, b m - n = B. HaihH b m H b n .
3. HaHTH CYMMY n lJUce.l BH,lI,a:
1,11,111, 1111,
IO n -1
Y K a 3 a H Ii e. H!3nIiIllHTc Ka}K.lJ,Oe 'IHCJlO B BH)J.e -g ,
r,ll,e n - nOp5l,ll,KOBblH HO:\!ep 'IIiC31a.
4. ,UoKa>KHTe, qTO eC.TJIi a 2 , b 2 , c 2 - 'lJleHbI apHcpMeTIi-
lJeCKOH nporpeccIiH, TO Ii lJHC31a
I I 1
b+c ' c+a ' a+b
TaK>Ke 06pasYJOT apHcpMemqeCKYIO nporpeccIiJO.
5. ,UaHa nporpeccHH:
-;.:.-b 1 . b 2 , b 3 , ... . b n
I I I I
Sn = A H b; + b;" + b; + ... --I- -,;;; = B.
HaHTH P = b 1 .b 2 "b 3 . ... .b n .
Y K a s a H Ii e. Ope>K.lJ,e lJe1 pewaTb 3a,ll,aqy, BbIBe)J.IiTe
cpOp1YJIY
n
T
Pn = (b 1 .b n ) .
6. ,UaHa nOCJ1e)J.OBaTeJlbHOCTb:
I I I I
n' 2.3 ' 3.4 ' ." , n(n + I)
, ... .
229

HaMTH npe,ll,en CYl\1MW n lJ31eHOB 3TOH nOC31e,ll,OBaTe31bHOCTH
npH n -+ 00 .
P e III e H H e.
k=n I I I I
S" =  k(k+ I) =1.2 + 2.3 +... + n(n+ I) '
k=1
nYCTb
I A B Ak+A+Bk (A+B)k+A
k(k+ I) = Ii + k+1 = k(k+ I) = k(k+ I)
HTaK,
I
k(k + I)
(A + B) k + A
k(k+ I)
Onpe,ll,e.TJHe:\1 A H B H3 YC.TJOBHH; qTO ecnH ,lI,p06H c paBHb'.
MH 3HaMeHaTe31HMH paBHbI, TO paBHbI TO>K,lI,eCTBeHHO H HX
qHC31HTe31H, T. e.
(A + B) k + A = I.
3m TO>K)J.eCTBO B03MO>KHO TO.TJbKO B TOM C31yqae, eC31H
{ A + B = 0;
A=l.
PeillaH n031yqeHHYIO CHCTeMY, HaH,lI,eM, liTO A = 1 Ii
B=-1.
()Je,ll,OBaTe31bHO,
1 I I
k (k + I) = Ii - k + I .
Tor.D.a
k=n ( I I ) k=n I k=n 1
S,,= T- k+1 =. T- k+1 =
k=1 k=1 k=I
( I I I ) ( I I I )
= 1+2" + 3" +... + n - ""2 + 3" +... + n+ I =
230

I I ! I 1
= 1 +2+ 3+'" + n-T-T-'''-
!! I
----=1--
n n+1 n+1
II
!imS n =!im ( 1- n + ! I ) = l.
n-+ 00 1l,- 0C .
7. HaHTH TaKH:\1 }Ke npl1eMOM npe,ll,e31 npH n oo CYM"
MbI p5I,lI,a:
I I I I
}.2.3 + 2.3.4 + 3.4.5 + ... + n(n + 1)(n+2)
3aJl.aa 0 KpOJIHKax H Hcna <l>H6oHaH
B caMOM HaqaJIe X II I B. B lhamm nOHBH310Cb MaTe MaTH-
4eCKoe COql1HeHl1e «Liber abacci» - «KHl1ra 06 a6aKe». 3Ta
06WI1pHaH KHl1ra, COCTOHlliaH 113 15 pa3,l1,e310B, HBJlHeTCH nep-
BblM B EBpone Opl1rHHaJIbHLIM 113310>KeHl1eM apHcpMeTHKI1 11
a31re6pbJ. Bce, 4TO 6bIJIO 113BeCTHO eBponeiiuaM K Haqa31Y
XII I B. B 0631aCTH 3TI1X HaYK, HaW310 OTpa>KeHl1e B «Liber
abacci», npl1qeM B cpopMe HCHOH, TO'IHOH 11 caMoCTOHTeJlbHoii.
«KHl1ra 06 a6aKe» 3HaKOMliT eBponeHcKoro 'IHTaTe.1H C ynoT-
pe631eHl1eM HH,lI,I1HCKIiX ul1cpp 11 ,lI,eCHTl1qHOH CIiCTeMOH cqI1C31e-
HI1H. B HeH MbI BCTpeqaeM pa360p Bcex apl1cpMemqeCKIiX ,lI,eH-
CTBl1ii C UeJJbIMH 11 )J.P06HbIMH ql1C31aMIi 11 peweHHe pH)J.a Tl1nO-
BblX 3a,ll,aq. Cpe,ll,H ee pa3,l1,e310B eCTb pa3)J.eJl 0 HaXO>K,lI,eHI1H
KBa,ll,paTHblx 11 Ky6HqeCKliX KopHeii 11 ,lI,eiiCTBHH Ha,ll, HI1MH.
y qeHl1e 06 I1ppaUI10HaJI bHblX Be31 I1ql1HaX H3310>KeHO 631 113KO
K H3JIO>KeHI11O 9TOro BOllpoca B KHl1re EBK31I1,l1,a «Haqa31a».
11a31ee pa3611paeTcH peweHHe pa331114HbIX BH,lI,OB ypaBHeHHii
11 3aAaq C nOMolliblO COCTaB31eHI1H ypaBHeHI1H B MaHepe, 631H3-
Kon MyxaM;\le,ll,Y 113 Xope3Ma. B 06llieM «Liber abacci» KaK 6bI
nO,ll,Bo,ll,I1T qepTY no,ll, BCel\1 TeM, qTO eBponeiiUbI Y3Ha3111 113
apl1cpMemKI1 H a31re6pbl OT rpeKOB, I1H,lI,I1HueB 11 apa60B 11
C qero nOTOM OHI1 Ha4aJIH CBOH c06CTBeHHbIH HaY4HbIii nYTb
B pa3BI1TI111 3TI1X pa3,l1,e310B MaTeMaTI1KI1.
ABTOpOM 3Toro 3aMeqaTe.TJbHOrO TpY,lI,a, nO,ll,rOTOBIiBwero
ycnexli HTa31bHHCKOH a31re6pbI B 5DOXY B03pO>K,neHH1J, 6bI31
ypO>KeHeu ropOlJ.a DH3bI JIeoHap,no CblH BOHaqql1, HJ1H, KaK
ero npliBbIK3111 H13bIBaTb, JIeoHap,;."o <l>1160Ha I JlIH. 06WI1pHaH
231

:\1OpCKaH TOproB31H, KOTOpylO BeJI ropo,ll, DH3a B XII H XIII BB.,
3aCTaBliTJa OTua JIeoHap)J.o nOKHHYTb CBOH pO,ll,HOH ropO,ll, H
nepeexaTb B A31>KHp. 3,l1,ecb JIeoHap,ll,o n031Y4IiJ1 MaTeMaTH-
'IeCKOe 06pa30BaHHe H n03HaKOMII31CH C apHcpMeTHKoM H a.TJre6-
pOH apa60B. B )J.aJIbHeMWeM no po,ll,Y CBoeH pa60TbI (JIeoHap,ll,o
6b131 TOproBbIM ynOJIHOM04eHHbIM ropo)J.a DH3bI) eMY npHw-
310Cb 06"be3,l1,HTb ErHneT, CHpHIO, fpeUHIO, CHUIiJ1HIO H <l>paH-
UIiIO. COBepwaH 3TH nYTeWeCTBHH, OH CTpeMH31C51 paCWHpHTb
CBoe MaTeMaTHqeCKOe 06pa30BaHHe. B03BpaTHBWHCb OKOJIO
1200 r. B DH3Y, JIeoHap1l.o c03)J.aeT CBOH Tpy,lI, «Liber abacci»,
KOTOpblH BbIXO,ll,HT B CBeT B 1202 r. B 1228 r. KHHra <l>H60Haq4H
nepeH3,l1,aCTCH B nepepa6oTaHHoM BH,lI,e. B npe,ll,HC310BHH K 3TOH
KHHre JIeoHap,ll,O pacKpbIBaeT TY 631aropo)J.HYIO ue31b, KOTOpylO
OH nOCTaBH.TJ IIepe,ll, C060H, C03,l1,aBaH CBOM TPY,lI,. «51 peWIiJ1CH,-
nHweT OH,- npHcoe,ll,HHHB K HH,lI,HHCKOMY MeTO,ll,Y KOe-qTO
OT ce651, KOe-qTO OT TOHKOCTeH reOMeTpHqeCKoro HCKYCCTBa
EBKJI, H,lI,a COCTaBHTb TP Y)J. , KOTOpbIH 51 Tenepb 06Hapo,ll,YIO
B 15 OT,lI,e31aX, ,lI,a6bI pO,ll, JIaTHHHH He OCTaBaJICH 6031ee HeCBe-
,lI,YlliHM BO Bcex 3THX Belliax».
TeOpeTHqeCKHH MaTepna31 B «Liber abacci» IIOHCHHeTCH
Ha 6031bWOM qHCJIe 3a)J.aq. OCTaHOBHMCH Ha O,ll,HOH JIl060nblTHOH
3a,ll,a4e - 3a,ll,aqe 0 KpOJ1HKax. «HeKTO nOMeCTIiJ1 napy KpO-
, -, l'Ietl1a (M<na,
+
. .
fJucno r<.pol1uHt*f
I-Q -
+
..
- 2
.. 6
.. 5
5-D
h a
Ci-tJ
..tJ
4epT. 116.
.TlHKOB B HeKoeM MecTe, oropO>KeHHOM co Bcex CTOpOH cTeHoH,
'ITo6bI Y3HaTb, CK031bKO nap Kp031HKOB pO,ll,HTCH npH 3TOM
B Te4eHHe ro)J.a, eC31H npHpo,ll,a KpOJIHKOB TaKoBa, 4TO qepe3
1ecHU napa Kp031HKOB npOH3Bo,ll,HT Ha CBeT ,lI,PyrylO napy,
a pO>K,lI,alOT Kp031HKH co BTOporo l\1eC5Iua CBoero pO}K,ll,eHH51.
CKOJlbKO nap Kp031HKOB B O,ll,HH ro,ll, OT O,ll,HOH napbl p0>K)J.aeTcH?»
232

OOCTapaeMCH nO,ll,MeTHTb 3aKOH, no KOTOPOMY npOHCXO,ll,HT
pa3MHO>KeHHe Kp031HKOB. BBe,ll,eM 0603Ha4eHHH: pO>K,lI,alOlll,YIO
napy Kp031HKOB 6Y,lI,eM 0603HaqaTb 3HaqKOM , HepO>K,lI,alO-
lliYIO napy . Ka>K,lI,aH p0>K,lI,alOlll,a51 napa lJepe3 MeCHu
)J.aeT caMoe ce6H lid H HOBYIO HepO>K,lI,alOlll,YIO napy .
4epe3 MeCHU HepO>K,lI,alOlliaH napa  nepeXO,ll,HT B pa3p51,ll,
pO>K)J.alOlll,HX  ' HO He ,lI,aeT HOBOM napb!. Tenepb BHHM3TeJ1b-
HO paCCMOTpHM qepTe>K 116. BblnHweM nOC31e,ll,OBaTeJ1bHOCTb
H3 lJHC31a KpOOHKOB B Ha4a31e Ka)K,lI,oro 113 llleCTH MeCHueB:
1,2,3,5.8, 13.
nerKO 3aMeTHTb, 4TO 3-M q31eH nOC31e,ll,OBaTe31bHOCTH n031Y-
qaeTCH H3 l-ro H 2-ro 431eHOB nYTeM CJlO>KeHHH; 4-ii - CJIO-
>KeHHeM 2-ro H 3-ro, 5-M - C310>KeHHeM 3-ro H 4-ro, T. e.
Ka>K,lI,bIH HOBblH q31eH, KpOl\1e l-ro H 2-ro, nOJIyqaeTCH nYTeM
C310>KeHHH )J.BYX npe)J.wecTBYlOlliHX eMY q31eHOB. 0603HaqHB
06lliHM q31eH nOC31e,ll,OBaTe31bHOCTH 4epe3 un' 3anHilleM no,ll,-
MC4eHHoe HaMH CBOHCTBO TaK:
U n =U n - 1 +U n -2'
I1pO,ll,031>KHB 3TOT pH,lI, qHCeJJ ,lI,a31ee H OCTaHOBHBWHCb Ha 13-M
lJ31 eHe
1,2,3,5,8,13,21,34,55,89,144,233,377..., (1)
Mbl H n031yqHM OTBeT 3a,ll,aqH, nOCTaB31eHHOH <l>H60HaqqH.
EC31H nepe,ll, nepBblM q31eHOM nOC31e,ll,OBaTe.TJbHOCTH (I) HanH-
weM ellie O,ll,HH 431eH, paBHbIH I,
TO 3THM Cal\1bIM MbI He H3Me-
HHM 3aKOHa COCTaB31eHHH 3TOH
nOC31e,ll,OBaTeJI bHOCTH.
I1OC31e,ll,OBaTe31bHOCTb
1, 1. 2, 3, 5, 8, 13, 21,
34, 55, 89

E::J
B 4eCTb aBTopa 3a)J.a4H 0 KpO-
31HKax Ha3bIBalOT nOC31e,ll,OBa-
Te31bHOCTblO <l>H60HaqqH, a
'I31eHbl ee - qHCJIaMH <l>H60-
Ha4lJH. 3TOT pH,lI, BnOCJ1e,ll,CT-
BHH OKa3aJICH nooe3HblM B
233

HaYKe. OH H3BeCTeH He T031bKO MaTeMaTHKaM, HO H eCTeCT-
BOHCnbITaTeJlHM. TaK, HanpHMep. eCJIH .a-epeBO pa3BeTB31HeT-
CH Ka>K,lI,bII'i ro)J. H Ha BTOpOM ro,ll,Y HMeeT )J.Be BeTBH, TO Ha
TpeTbeM ro,ll,Y qHC310 BeTBeH 06bIqHO )J.OXO,ll,HT )J.O 3, Ha qeT-
BepToM -,lI,0 5, Ha nHTOi\1 -,lI,0 8, H TaK Ka>K)J.bIH pa3 B
nOC31e,ll,OBaTe31bHOCTH qHCeJJ pH,lI,a <PH60HaqIIH.
PaccMoTpHM ellie HeKoTopbIe 1i3 CBOHCTB qHCeJJ <PH60HaqlH.
BbI6epeM JII06bIe TPH qIiC31a H3 nOC31e)J.OBaTe31bHOCTH <PH60-
HaqqH H cnpocHM ce6H: MmKHO 31H nOCTpoHTb Tpeyr031bHHK,
1{31HHa CTOpOH KOTOpOro Bblpa>KaeTCH BbI6paHHbIMH 4HC31aMH?
11aHbl )J.Ba ypaBHeHHH:
x 2 _xy_y2=1 H x 2 _xy_y2= -I.
B03bMeM .TJlO6bIe ,lI,Ba COCe,ll,HHe 4HC31a H3 pH)J.a <PH60HaqlJH.
BbI6paHHaH HaMH napa qHCe31 06H3aTe31bHO Y,lI,OBJIeTBOpHeT
Ii31H nepBoMY H3 HaIIHcaHHbIX ypaBHeHHH, HJIH BTOPOMY
(x - 6031bWee, a y - MeHbwee H3 3TH X qHCeJJ).
MO>KHO 6bl310 6bI YKa3aTb elll,e ue31bIH pH)J. 311060nbITHbIx
CBOHCTB qHCeJl <PH60Ha'IqH, HO MbI nOH,lI,eM no HHOMY nYTH.
DYCTb ,lI,aHa nOC31e,ll,oBaTeJ1bHOCTb
U 1 . u 2 . us' ... , un' ... ,
(2)
B KOropOH, KaK H B nOC31e,ll,OBaTeJJbHOCTH <PH6oHaqqH, Ka}K,ll,bIH
431eH paBeH CYMMe ,lI,BYX npe,ll,weCTBYIOlll,HX 431eHOB, T. e.
npH n> 2
U n =u n - 1 +u n - 2 .
113151 TOro qro6bI HanHcaTb KaKOH-31H60 qaCTHbIH C31Y4aH no-
,lI,06HOH nOCJle)J.OBaTeJ1bHOCTH, 04eBH,lI,HO, He06xo,ll,HMO H ,lI,0-
CTaT04HO Ha3BaTb ,lI,Ba ee nepBbIX q31eHa; HanpHMep:
1,3,4, 7, 11, 18, ... .
.2, 7. 9. 16, 25, 41, ... .
-1, -3, -4, -7, -11, -18,
PH,lI, <PH60Ha44H MbI n031yqHM, eCJlH U 1 H U 2 npHpaBHHeM
e,ZI.HHHue.
I, I, 2, 3, 5, 8, 13, ... .
BbIBe,ll,eM npe>K,lI,e Bcero <PoPMY31Y CYMMbI n nepBbIX 'IHCe31
nOC31e,ll,OBaTeJ1bHOCTH (2).
234

Do OCHOBHOMY CBOHCTBY 'lJIeHOB nOC31e,ll,OBaTeJJbHOCTH (2)
HMeeM:
U 1 =U S -U 2 .
U 2 =U4- U S'
u s =U 6 -U 4 ,
U n _ t =U n + 1 -U n ,
Un = U n + 2 -U n + 1 .
CnO>KHB BCe 3TH paBeHCTBa IIOq31eHHO, MbI n031yqHM:
Ut+ U 2+ 11 S+ ... +U n =U n + 2 -11 2 .
1131H pH,lI,a <l>H60HaqqH
1, I, 2, 3, 5, 8, 13, ...
CYMMa n nepBblX lI31eHOB paBHa u n + 2 -l, TaK KaK U 2 B 3TOM
p5I,lI,e paBHHeTCH I.
I1TaK,
Sn=U n +2 -U 2
.lI.J1H nOC31e,ll,OBaTeJJbHOCTH (2) H
Sn=U n + 2 -I
)J.31H pH,lI,a <l>H60HaqlIH.
Tenepb MbI nepeXO,ll,HM K Bonpocy 3anMCH 311060ro q31eHB
p5I,lI,a <l>H60Haq'IH. DOKa Y Hac HMeeTCH B03MO>KHOCTb C,lI,eJI8Tb
3Ta TonbKO peKyppeHTHbIM cnoco60M, T. e. nYTeM B03BpaIll.e-
HHH K )J.BYM q31eHaM, npe.llweCTBYlOrn.HM onpe)J.eJJHeMOMY, H
CYMMHpOBaHHIO HX. Be3 3HaHHH (U n - 2 )-ro- H (u n _ 1 )-ro lJ31eHOB
MbI He CMO>KeM HanHcaTb Un-H tJJIeH. ll31HTe31bHOCTb 3TOro cnO-
c06a 01IeBH,lI,H8. 003TaMY )Ke31aTe31bHO HMeTb cpOPMY31Y 311060ro
'I31eHa p5I,lI,a <l>H60HalIlIH KaK cpYHKUHIO ero HOMepa. B r31aBC
BOCbMOM 6blJ1a )J.aHa 3Ta <PopMY31a
Un = ;5 [( I +2]15 r - ( 1 -2 115 r]
B BH)J.e rHnOTe3bI H 6bIJIO npe,ll,JIO>KeHO npoBepHTb cnpaBeJI.-
31HBOCTb ee MeTO,ll,OI\1 MaTeMaTHqeCKOH HH,lI,YKUHH. B03MO>KHO,
lITO TOr,ll,a nepe,ll, H8MH B03HHK Bonpoc 0 TOM, K8KHM nYTeM
Y)J.a310Cb npHHm K CTa31b CJIO>KHOH mnOTe3e. BOT 3TaT nYTb.
npe>K,lI,e Bcero paccMoTpHM TPH cBoHCTBa nOC31e.n.oBaTeJJb-
HOCTH
U t . u 2 , Us' ..., Un' ...,
235

B KOTOpOH
U n =U n -I+U n -2'
1. ECJIH YMHO>KHTb Bce qJleHbI nOC31e)J.OBaTeJJbHOCTH (2)
Ha O,ll,HO H TO >Ke ,lI,eikTBl!TeJ1bHOe 'IHCJIO k, TO nOOY4HM no-
C31e)J.OBaTe31bHOCTb TOro >Ke BH)J.a.
11JIH )J.OKa3aTeJJbCTBa 3TOro cBoikTBa )J.OCTaTOqHO npOBe-
PHTb, lJTO
n p 0 B e P K a.
HO
kUn=kun-2+kun-I'
kU n -2+ kU n - 1 = k (ti n - 2 +U n - I );
U n -2+ U n-I=U n ;
C31e,ll,OBaTe31bHO,
ku n =ku n - 1 +ku n _ 2 .
2. EC31H ,lI,aHbl ,lI,Be nOC31e,ll,OBaTe31bHOCTH:
VI' v 2 ' ['3' ..., tin' ... ,
(2)
(3)
U I ' U 2 , Us' ..., Un'
H
MH KOTOpblX Un = U n - 1 + U n - 2 H V n = V n - l + V n - 2 , TO, C310-
>KHB nOC31e,ll,OBaTeJlbHO O,ll,HHaKOBbIe no HOMepy q31eHbI 3THX
IIOC31e,ll,OBaTeJJbHOCTeH, Mbl n031Y'IHM HOBYlO nOC31e,ll,OBaTeJIb-
HOCTb
Yv Y2' Ys' ..., Yn' ...
(4)
TOro >Ke BH,lI,a, 4TO H nepBble ,lI,Be, T. e. qTO MH nOC31e,ll,OBaTeJJb-
HOCTH (4) 6Y,lI,eT CIIpaBe,ll,31HBO paBeHCTBO
Yn=Yn-l+Yn-2'
OpOBepHM CHaqa31a :::ITO CBOHCTBO ,lI,315I 4acTHoro C31YQaH.
11aHbl ,lI,Be nOCJIe,ll,OBaTeJlbHOCTH:
H
I, 3, 4, 7, 11, ...
2, 4, 6, 10, 16, ... .
CK31a)J.bIBa5I nOCJle)J.OBaTeJJbHO q31eHbI, O,ll,HHaKOBbIe no HOMepy,
nOJlY4HM nOC31e,ll,OBaTeJJbHOCTb
3, 7, 10, 17, 27, ....
nOC31e.ll.HH5I nOC31e.ll.OBaTeJJbHOCTb COCTaBJIeHa no TOMY )Ke
saKoHY, qro H nepBbIe )J.Be.
236

110Ka>KeM 3TO CBOHCTBO B 06rn.eM BH,lI,e.
11aHbI ,lI,Be nOCJIe,ll,OBaTe31bHOCTH:
UI' l/2' US' Un
H
VI' ['2' VS, ..., Vn'
)J.JISI KOTOpbIX
Un=Un-l+Un-2 H Vn=Vn-l+Vn-2'
DYCTb n031YQeHa HOBaH nOCJIe,ll,OBaTeJJbHOCTb
Yl' !h, Us' ..., Y n
nyreM C310>KeHHSI 'lJ1eHOB C O,lumaKOBbIMH HOMepaMH nepBblX
,lI,BYX nOC31e,ll,OBaTeJ1bHOCTeH; TOf,ll,a
Yn = Un + V n = (Un_1 + U n _2) + (V n _ 1 + V n _ 2 ) =
= (U n _l + ['n-I) + (U n -2 + V n _ 2 ) = Yn-I + Y n -2'
y!TaK,
U n =Yn-l+Yn-2,
T. e. nOC31e,ll,OBaTeJJbHOCTb
YI' Y2' Us' .... Y n , ...
COCTaB31eHa rIO TOMY >Ke 3aKOHY, qTO H nOC31e,ll,OBaTeJIbHOCTH
{Un} II I Vn} . CBOHCTBO ,lI,OKa3aHO.
3. ECJIH )J.aHbI TPH nOC31e,ll,OBaTe31bHOCTH
U I , U 2 , US' '" , Un' '" ,
VI' V 2 , V s , ... , V n . ... ,
ZI' Z2' ZS' ... , Zn' ... ,
)J.JJSI KOTOpbIX U n = Un-I + U n -2' V n =V n - I +V n -2, Un H V n -
HenpOnOpUHOHa31bHbI H Zn=Zn-l+Z,,-2' TO BCer)J.a MO>KHO
HaHTH TaKHe ABa qHC31a C 1 H C2' YMHO>KHB Ha KOTopble COOT-
BeTCTBeHHO q31eHbl nepBoM H BTOpOH nOCJle)J.OBaTe31bHOCTeH H
3aTeM tJIO>KHB nOC31e,ll,OBaTeJIbHO q31eHbl C O,ll,HHaKOBbIMH HO-
lepaMIi HOBbIX nOC31e,ll,OBaTe31bHOCTeH, MbI npH,lI,eM K nOCJIe]I.o-
BaTeJlbHoCTH
Zl' Z2' Z3' ... Zn ... ,
T. e. ,lI,OKameM cyrn.€CTBOBaHHe TaKHX lJHCeJJ C 1 II C 2 , lJTO pa-
BeHCTBO
C 1 U n +C 2 V n =Zn
6y,ZteT CnpaBe]I.31HBO npH JIIo60M HaTypaJIbHOM n.
"237

.l1. 0 K a 3 aTe JI b C T B o. nOCJIe YMHO>KeHHH Bcex tJ31e-
HOB nepBblX )J.BYX nOC31e)J.OBaTe31bHOCTeH COOTBeTCTBeHHO Ha 
H C2 Mbl no CBOHCTBY 1 n031YtJHM nOC31e,ll,OBaTe.TJbHOCTH Toro >Ke
BH.lI.a. CnO>KHB B n031yqeHHbIX HOBbIX .lI.BYX nOCJIe)J.OBaTeJJbHO-
CT5IX tJ31eHbI C 'O,ll,HHaKOBbIMH HOMepaMH, Mbl no CBOHCTBY 2
onHTb npH,lI,eM K nOCJle,ll,OBaTe31bHOCTHM Toro >Ke THna, 'lTO
H ,lI,aHHbIe HaM BHaqaJ1e nOC31e,ll,OBaTe31bHOCTH. EC31H TaK,
TO MbI MO>KeM nOCTaBHTb Tpe60BaHHe, tJT06bI
c 1 u. +C2V2=Zl H C.U2+C2V2=Z2'
3T0 npHBe,ll,eT Hac K CHCTeMe
{ UICl+VIC2=Zl,
U1+V2C2=Z2
C HeH3BeCTHbIMH C 1 Ii C 2 , KOTopbIe H Ha, H)J.eM pemaSl STY CH-
CTeMY.
nOJlYtJHM:
V2Z1 - Vt Z 2
c. =
UIV 2 - U 2 V l
UIZ2 - U 2 Z 1
H C 2 = ,
UIV 2 - U 2 V l
r1!.e U.V 2 -U Z V 1 +O, 'ITO COOTBeTCTBYeT Tpe60BaHmo
.!!J.. =1= .!!:! ,
VI V 2
T. e. nepBbIe ,lI,Ba tJ31eHa nepBbIX )J.BYX nOC31e)J.OBaTe31bHOCTeA
He ,lI,OJI>KHbl 6bITb nponopUHOHa31bHbI.
I1TaK, npH YCJ10Bl-iH U 1 V2-U2Vl =1=0 lJHC31a c. H C 2 BCer,ll,a
MorYT 6bITb HaH)J.eHbI. CBOHCTBO 3 )J.OKa3aHO.
nOHCHI-iM ero Ha qaCTHOM CJIyqae.
ILaHbI TPH nOCJIe,ll,OBaTeJIbHOCTH:
1,3,4, 7, 11. ...,
2, 4, 6, 10, 16, ...,
7. 10, 17, 27, 44, ....
KaK H3 ,lI,BYX nepBblx nOC31e,ll,oBaTe31bHOCTeA n031YlJHTb TpeTblO?
YMHO>KaeM q31eHbI l-H nOC31e,ll,OBaTeJIbHOCTI-i Ha C 1 ' a q31eHbI
2-jj - Ha C 2 H nOTpe6yeM, 'IT06bI c..I+cz.2=7 H c 1 .3+
+c 2 .4= 10. npl-i,ll,eM K CHCTeMe
{ cl+2c2=7,
3c.+ 4c 2=1O.
,238

PeW1leM ee:
c 1 =7-2c 2 ; 3(7-2c2)+4c2=1O; 21-6c 2 + 4c 2=1O;
J II 11
-2C2=-L: c2=2; TOf,ll,a c 1 +2.- 2 -=7 II c 1 =-4.
YMHO>KaeM 'lJIeHbl I-ii nOC31e,ll,OBaTeJIbHOCTH Ha (-4), a 'l31eHbI
11
2-it Hli 2' nOJIyqHM:
-4, -12, -16, -28, -44. ...
11, 22, 33. 55, 88, ... .
CJIO>KH8 nO'l31eHHO no.'1Y4eHHbIe nOC31e,ll,OBaTeJIbHOCTH, Mitt H
npH)l.OM K TpeTbe nOCJIe,ll,OBaTeJIbHOCTH
7, 10, 17, 27, 44.
Tenepb Mbl MO>KeM nepeHTH K BbIBO,ll,Y cpOPMY31bI un ,lI,JIH
pH,lI,a <l>H60Haq'IH.
Ope,ll,nOJIO>KHM, lJTO lJJleHbI nOCJIe)J.OBaTe31bHOCTH
UI' U 2 ' Us' ... , Un' ... ,
re Un'" U n - 1 + Un-2 npe,ll,CTaBJIHIOT C060H reOMeTpHqeCKYIO
npOrpeCCHIO
1, q, q2, ... , qn- J
(Ha OCROBe cBoikTB3 I Mbl MO>KeM paccMaTpHBaTb u1=1).
tora
qn-2+qn-l=qn
nJIH, ,lI,eJIH Ra qn-2, n031YlJHM
1 + q = q2.
PemeHHH 3Toro ypaBHeHHH TaKOBbI:
I+YS
ql = 2
I-Y5
H q2 = 2
TaKHM o6pa30M, B3HB nepBbIH 431eH paBHbIM I H 3HaMeHa-
I+YS .
TeJlb nporpeccHH O,ll,HH pa3 ql = 2 H BTOpOH pas q,,=
_ 1 - ys
- 2
, MbI npH,neM K l{BYM nOCJle)J.oBaTe.'IbHOCTHM:
I + YS (1 + YS)2
1, 2 4
(5)
l-Y5
1, 2
(J - YS)2
4
t ... .
(6)
239

11m! nOCne,ll,OBaTe31bHOCTeH (5) H (6) BbIn031HeHO YCJIOBHe
un = U n _ 1 + u n - 2 ; C31e)J.OBaTe31bHO, OHH IIpHHa,ll,JIe>KaT K TOMy
>Ke THny. qTO H p5l,ll, <PH60HaqqH. HaH,lI,eM ,lI,Ba IIHCJIa Ct H
C2 (CBOHCTEO 3) TaKHX, lJT06bI B pe3YJIbTaTe YMHO>KeHHH
'l31eHOB nOC31e)J.OBaTe31bHOCTH (5) Ha C1 H q31eHOB nOC31e,ll,OBa-
TeJIbHOCTH (6) Ha C 2 H nOC31e,ll,YIOlll,ero nO'l.'IeHHOrO CJlO>Ke-
HHH nOJIY'leHHbIX ,lI,BYX I:OBbIX nOC31e)J.OBa'reJIbHOCTeH MbI npH-
WJIH 6bI K pH,lI,Y <PH60H8lJqH:
1, 1, 2, 3, 5, 8, 13. 21, ... .
LL31H 3Toro ,lI,OCT8TOlJHO, Ha OCROBe CBoHcTBa 3, 'lTo6
j C 1 ' 1 . + : 1 = 1 . _
I + l' 5 I - V5
c,-+c2-=I.
PaCCMaTpHBaH JJ.B3 nOC.leJJ.HHX paBeHcTB1 KaK CHCTe\IY ypaB-
HeHHl1 C HeH3BeCTHbIMH C} H C2 H pernaH ee, n031Y'lHM:
1+V5
C1 = --- H C 2 =
2 }/5
I-V5
2 VS
OTKY)J.a
( I + VS ) n-I ( 1_ VS ) "-1
Un = c} 2 + C2 2 =
= I+V5 ( I+V5 ) n-I_ I-VS ( I-TrS ) n_,=
2 V5 - 2 2 VS 2
HTaK.
= V [( +;s r- c-5 r).
Un = V [ C + s r - c - r s rJ .
(7)
Pe3Y31bTaT HeCKOJlbKO HeO>KH,lI,aHHbIH. PH,lI, <PH6oH3 'l'lli , 'l31e-
HaMH KOToporo 51B31£IOTCH -Ue31bIe 'IHC31a, IiMeeT T8KYIO C310>K-
HYIO cpOPMYJIY 06w.ero lJ31eHa. <popIY31a (7) Ha3b1BaeTCH cpop-
.MY310H BIiH3 B 'leCTb MaTeMaTIiKa, ,lI,OKa3aBwero ee.
240

3aAa'lH
1. BbIBe,ll,HTe cpOPMYJIY, nO,ll,06HYIO cpopMYJIe EHH3, NIH
nocneAOBaTe31bHOCTH
I. 3, 4,7. II, 18,29, ...
2. 110Ka>KHTe, qTO cYMMa KBa,ll,paTOB nepBbIx n qliceJ1
cDli60Ha lJqH paBHa Un. U n + l .
Y K a 3 a H H e. Uk' U k +1 - Uk_I'U k = Uk (U k t-l - Uk-I) = U2k"
3. 1I.0K8>KI1Te MeTO,ll,OM MaTe:\laTHqeCKOH liH,lI,YKUHIi, qTO
U n + m = Un_I' U m + Un' llm+l'
Y K a 3 a H li e. DpH)J.eTCH npe)J.n0J10}I{HTb, qTO YKa3aHHoe
paBeHCTBO cnpaBe)J.J1liBO npH In = k Ii npH m = k + I, Ii ,lI,0-
K33aTb ero cnpaBelI.JIHBOCTb npH m = k + 2.
4. PCillHB 3alI.aqy 3, npe,ll,nOJIO>KliTe, 'ITO m = n, H )J.O-
 a!KHTe, qTO
2 2
Un+1 - Un_I = U2n'
T. e. pa3HOCTb KBa,ll,paTOB )J.BYX qHCe.TJ cDH60HaqlJH, H0:\1epa
KOTOpbIX OTJIliqalOTCH Ha 2, eCTb CHOBa q11C310 cDH60Ha 1 lqII.
JIHTEPATYPA
H. H."S 0 po 6 b e B, YHCJIa <PH60Ha4lJH, fI1TT,11, 1951.
10. A. K H K e u, l-iCTopUlJeCKlle CBe,QeHIIH K pa3.1i€JIY «Dporpec-
CHII» , «MaTeM3TIIKa B WKo.e», 1956, N2 I.
5. A. K 0 P .Ii eM C K H II, MaTeMaTlIlJeCKaH CMeKaJIKa, fHTT JI,
1956.
A. l-i. Map K Y weB H 'I, S03BpaTHbIe nOCJIe.nOBaTeJIbHOCTH,
rl-iTTJI, 1951.
SI, H. IJ e p e . b M a H, 3aHmlaTeJIbHaH aJIre6pa, fHTTJ]. 1950.

, L" rmr "" "'''' 
- r.--- l - '). : II
 flI L s . ' · · , 1
tf} (}) 1£ "7l tJP t1l\
x =" IJ..}'&D JJ C' /fr«JJ  ,
- ""0
llU3 ,,3 d"1J
J(:.:: IV 118,;:;cr
rJlABA x
KAK EblJUI C03.!J.AHbl nEPBblE T AEnlUJ.bl
nor API1<1>MOB
«KIIHra 0 .D.OCTIOKeHHSlX HaYKH...
.D.OJJL!ma .naBaTb He TOJJbKO KOHelJHbte
pe3YJJbTaTbt 'I"'JJOBelJeCKOii MbtCJJH, HO
H BIJO.nHTb lJIIT3TeJlR B caMblH npOl\fCC
HCCJle.nOB3TfJJbCKOH pa60Tbt. nOK33b!-
BaR nOCTeneHHoe npeo.nOJleHHe Tpy.II.-
HOCTeH u nOHCKH BepHoro Men)jJ.a».
rQP6Kul1
C ."leJI.YH COB€TY A. M. ropbKOro. BC'IKHH, H3YQalO-
UJ.HH MaTeMaTHKYOJI>KeH He TOJIbKO B06paTb B ce6H
rOTOBbIe nOJIO>KeHH5I 3TOH 3aM€qaTeJIbHOH HaYKH, HO
H B03MO>KHO r.ny6>Ke n03HaTb Te nYTli, no KOTO-
pbIM WJIa 'IeJIOB€qeCKa51 MbICJIb. C03,l1,aBaH 3TH n0310>KeHHH.
HCTOpH5I C03lI.aHl1H nepBbIX Ta(')31HU JICrapHcpMOB BeCbMa
nOY'IHTeJIbHa B 3TOM OTHoweHHH. Mbl lJaCTO n031b3yeMcH ,lI,JI5I
BbIqHC31eHHH C.TJO>KHbIX BbIpa>KeHHH Ta6JllUlaMH JIOrapHcpMOB
HJIH CqeTHOH 31HHei'IKOH.
003TOMY Bn031He eCTeCTBeHHO, qTO nepe)J. HaMH B03HHKHeT
Bonpoc 0 TO:\I, KaK 6b131H COCTaB31eHbI nepBbIe Ta631 H UbI JIora-
pHcpMOB. KTO 6bIJIH Te yqeHbIe, KOTopble 110,\apHJIH lJe310Be-
242

qeCTBY 3TOT HCKJIlOqHTeJJbHbIM HHCTpYMeHT ,lI,JIH BbIqHCJIeHHH.
KaKHMH nYTHMH lllJIH 3TH yqeHbIe. COCTaBJIHH Ta6JIHUbI, C Ka-
KHMH TPY.IlHOCTHMH OHH BCTpeTHJIHCb.
3a'l3TKH JlOrapHct>MH'IeCKHX Ta6J1HI.\ H HX p01KJl.eHHe
HaM HSBecTHo. qTO H,lI,eH JlorapHcpMHqeCKHX Ta6JIHU KpoeT-
CSl B COnOCTaB31eHHH JJ.BYX nOCJIe,ll,oBaTeJIbHOCTeH - reOMeT-
pH'IeCKoH n apHcp:VleTH'I€CKOH nporpeccHH. onpe,ll,JleHHbIM 06-
pa30M nOCTpoeHHbIx. l1,l1,eH TaKoro conOCTaB31€HHH B03HHK31a
OqeHb ,lI,aBHO. Elll,e ApxHMe,ll, (287-212 ,lI,0 H. 3.) B CBoeM
COqHHeHHH «OcaMMI{T» paCCMaTpHBaeT )J.Be nporpeccHH:
-;':-1, 10. 10 2 . 10 3 , 10 4 . 10 5 , 10 6 , 10 7
-;-0, 1. 2, 3. 4, 5. 6. 7. ... .
(1)
(2)
H
nepBaH COCTaBJIeHa H3 nOC31e,ll,OBaTe.IJbHblX cTeneHeH qHCJIa
10. a BTOpa51 - H3 HOMepoB q31eHOB nepBoH nporpeccHH. B pe-
3Y31bTaTe COnOCTaBJIeHH5I YKa3aHHbIx nOCJIe,ll,oBaTeJIbHOCTeH
ApxHMe,ll, npHxo,ll,HT K HHTepecHoMY BbIBO,ll,Y: eCJIH 6Y,lI,eT ,lI,aH
pH,lI, qHCe31 B HenpepbIBHoM nponopUHH, HaqHHa51 C I (nocJIe-
,nOBaTeJIbHOCTb (1), H eCJIH ,lI,Ba ero lJJIeHa nepeMHO)j{HTb. TO
npOH3BelI.eHne 6Y,lI,eT 'IJIeHOM Toro >Ke p5l,ll,a, HaCTOJIbKO y,na-
JJeHHblM OT 60JIbWerO l\1HO>KIITeJIH, HaCKOJIbKO MeHbWHM Y,lI,a-
.1eH OT e,ll,HHHUbI. HJIH O,ll,HHM q.TJeHOM MeHbwe npOTHB Toro,
HaCKOJIbKO Y,lI,aJIeHbI 06a l\IHO>KHTe3151 BMeCTe.
n05lCHHM MbICJIb ApXH:VIe,ll,a. OYCTb He06xolI.lil\10 nepeMHO-
IKHTb ,lI,Ba qHCJIa 10 2 H 10 4 . BbI6paHHbIe Cpe,ll,H qJIeHOB nOCJIe-
,nOBaTeJIbHOCTH (I). ApXH:\1e,ll, npe,ll,JIaraeT ,lI,3151 nOJIyqeHH5I
np01l3Be.lleHH5I B351Tb q31eH TOH >Ke nOCe;I.OBaTeJIbHOCTH, Y,lI,a-
31eHHbIH OT 6 JIbwerO MHO>KHTeJIH 10 4 Ha CTOJIbKO, Ha CKOJIbKO
MeHbWHM 10 2 y,naJIeH OT e,ll,HHHUbi. MHO>KIiTeJIb 10 2 Y,lI,a31eH
OT e,ll,HHHUbI Ha 2 qJIeHa; CJIe,ll,OBaTe31bHO. 10 2 .10 4 = 10 2 + 4 =
= 10 6 . 11m! ,lI,pyrHM cnoc060M: pe3YJIbTaT YMHO>KeHH5I 10 2 X
X 10 4 6Y.lleT 'fJIeHOM TOro >Ke p5l,ll,a, HOMep KOToporo MbI nOJIY-
'lHM nYTeM BbIlJHTaHllH 1 n3 CYMMbI HOMepoB COMHO>KHTeJJeH.
TaR KaK 10 2 no HOMepy 3-M lJJIeH, a 10 4 -5-M q.TJeil. TO HOMep
'I31eHa, CT05llliHH B 3TOM >Ke p5l.lle H lI.aJOlliHH pe3YJIbTaT YMHO-
>KeHH5I. 6Y,lI,eT TaKHM: (3+5)-1 =7. Ho 7-M q.TJeH nOCJIe,ll,OBa-
Te31bHOCTH (1) paBeH 10 6 . l1TaK, 10 2 .10 4 =10 6 .
MbI Bll.llHM, qTO ApXHMe.ll BnOJIHe B.TJa.lle.TJ Ii nOHHTHfM 0 CTe-
neHH, H TOH CBH3blO, KaKaH CYlli€CTByeT Me>K.IlY ,lI,eHCTBH5IMH
co CTeneHHMH H ,lI,eMCTBHHMH C nOKa3aTe31HMH. OTCIO,na. Ka-
243

3aJIOCb 6b1. eCTeCTBeHHO nepeihH K HCnOJIb30BaHHIO nO,ll,MeqeH-
HOH CB5I3H M€m,ll,Y reOl\leTpH'IeCI<OlI H apHcp:\leTH1JeCKOH npo-
rpeCCHHMH .IJ,3151 ynpOlll,eHI1H Bbl'Il1CJIeHHH - 3a:\leHbl y;\IHO>Ke-
HIi51 H .IJ,eJIeHHH COOTBeTCTBeHHO C.TJO>KeHHeM If BblqHTaHH€M. Ho
:::/TOT nepexo)J. KaK IIOC31e)].HHH war K COCTaB.TJeHHIO Ta6JIliU
He 6blJI C;I.eJIaH HH ApXH;\le,ll,OM. HH nOC31e,'I,YlOlliHl\1H lIIaTel\laTH-
l<aMH Ha npOTmKeHHH 6031ee qeM 1700 .'leT.
B 1544 r. BblXO)J.HT B CBeT KHHra HeMeUKoro MaTel\IaTHKa
MHXaHJIa UlTHcpe3151 (1486-1567) no.IJ, Ha3BaHHeM «06llia51
apHcp:\t€TliKa» I). B Hell lIlnlcpeJIb, conOCTaBJIHH )].Ba p 51lI.a
lJHCeJI B Bli,ll,e reOMeTpHqeCKOH Ii a plicp:\'eTH'IeCKOH nporpeCCHH,
npe)].CTaBJI5IlOlliHX C060H nepBaH - B03paCTalOlllH€ CTeneHH
O,ll,Horo H TorO >Ke OCHOBaHH5I, a BTopaH - COOTBeTCTBYlOlllHe
nOKa3aTeJIH CTeneHeH, paCKpbIJI 3aMeqaTeJIbHble CBOHCTBa 3TliX
qHCJIOBbIX pH.IJ,OB. HanJlWeM. qaCTHblH CJIyqaH TaKHX nporpec-
CHH, npHH5IB B nepBOH nporpeCCHll 3a OCHOBaHHe 'IHC310 2:
-:-;-{. 4"' . 1,2,4.8, 16: 32. 64, 128, 256,512, 1024;
--;- -3, -2. -1, 0, 1, 2, 3.4.5,6.7,8,9,10.
npe)].nOJIO>KHM. 'ITO 3TH nporpeCCHI1 npO)].OJI>KeHbI B TY H ,lI,PY-
rylO CTOPOHY; Tor,ll,a ,lI,ei-icTBHH yMHO>KeHH5I, ,lI,e31eHH5I. B03-
BbIweHHH B CTeneHb H H3B31e'leHH5I KOpH5I Ha,ll, qIiC31aMH-
qneHaMH reOMeTpHqeCKOH nporpeCCI1H MorYT 6blTb 3aM€HCHbI
COOTBeTCTBeHHO ,!J,eHCTBHHMI1 CJIO>KeHIi5l. BbI'IHTaHH5I. yMHO>K€-
HIiH H )].eJIeHlI5I Ha)]. COOTBeTCTBVlOlliHMH qHCJIaMH - q31€HaMH
apucpMeTH'IeCKOH nporpeCCHH. HanpH:llep:
-..!... .128 = 2- 3 .2 7 = 2- 3 7 = 2,4 = 16
8
HJIH
10
V 1024 = V 2 10 = 2 5 = 2 2 = 4.
npaB)J.a, 3TH p5llI.bl CO,ll,ep>K3T lJHCJIa  0'IeHb 6031blllHMH
HHTepBa.laMH. HO MbI Mor.'1H 6bl YMeHbWHTb HX, BCTamiB Me-
)K.IJ,y )J.ByM5I coce,!J,HHMH qHC31al\tH nepBoro pH,!J,a HeCKOJIbKO
Cpe,ll,HHX reOMeTpH'IeCKHX, a Me>K)].Y COOTBeTCTBYlOlllHMH 'IHC-
JIaMH BToporo p5l,!J,a - TaKoe >Ke KO.1HlJ€CTBO Cpe,ll,HHX apmpMe-
1) HaHMeHonaHHe c06w.aH apHljJMeTHKa» 6bl.10 B TO BpeMH CHHO'
HHMOM cnoaa anreOpa.
244

THqeCKHX. no CYTH ,lI,e31a B napaJIJIe31bHOM HanHcaHHH ,lI,BYX
BbIllleYKa3aHHbIX nporpeCCHH MbI nO,ll,MeqaeM Y>Ke H;I.pO Ta6-
:mu 310rapHcpMOB; CTOUT TO.nbKO pa3MeCTHTb qJIeHbI nporpec-
CHH TaK. KaK 3TO C,lI,e.TJaHO B Ta631Hue, nOMellieHHOH HH>Ke,
H nepe,ll, HaMH - Ta6.TJHua 310rapHcp:\lOB qlICeJI npH OCHOBa-
HHH 2.
OCHOBaHHe 2
I I 'I I !I I
N log2 N I N log2 N II N log2 N
,.
I I :1 I II I
8 -3 I' 4 2 ;i 128 7
.,
,.
i I :1 I i  I
I ) -2 8 3 i, 256 8
I i r
I 4 "
.1
I I I !I I I' I
-I , 16 4 I' 512 9
2 1 "
" ! I:
I ,: I I
I 0 " 32 I 5 II 1024 10
: 
I , I oj j
2 ) !; 64 I 6 II 2048 II
r
! ,I
IlITHcpeJIb He HCnOJIb30BaJI CBoero OTKpblTlIH ,lI,.'!51 ynpoIlle-
HU5I BblqUC31HTeJIbHbIX onepauliH li npe,J,nOqe.l npOHTH l\1HMO
c 3aKpbITbIMH r.TJa3al\IH. B KHl1re «06lll,a51 apHcp:VIeTHKa» CBOH
3aMeqaHH5I no nOBO,ll,y conOCTaB.'!eHII5I nO.IJ,06HbIX nporpeCCHH
OH 3aKJIIOqaeT TaKHMH C310Ba!llH: «MO>KHO 6bl310 6bI HanucaTb
u,eJIYIO KHHry 06 3TliX 3aMeqaTe31bHbIX CBoHcTBax qHCJIOBblX
p5l,nOB, OHaKO 3THM OrpaHHqycb H npoH,ny MI1MO c 3aKp-
TbIMH rJIa3al\1H».
tIo Tpe60BaHH5I npaKTHlJeCKOH >KH3HH H 6blCTpO pa3BHBa-
IOlll,HXC5I HaYK. TaKHX. KaK aCTpOHOMH5I H MexaHHKa, HeMH-
HyeMo TOJIKaJIH yqeHblX H BbIQHCJIHTeJIeH Ha nOI1CKli nYTeii
,n31H ynpOlueHH5I npouecca CJIO>KHbIX BbI'II1CJIeHHH, OTHHl\1alO-
w.ero TaK MHoro BpeMeHI1 H CH.'!. HeCKOJIbKHMH .IJ,eC5ITHJIeTHfI-
!\IH n03Hee n05lBJIeHH5I KHl1rH IllTHcpeJI5I B HH,lI,ep31aH.IJ,3X
HH>KeHep CmwH CTeBI1H (1548-1620), nepBblH B EBpone
n031b30BaBIlIHHCH ,!J,eC5ITH'IHbIMH ,!J,P0651MH, H3aeT Ta6JIHu'bl
CJIO>KHbIX npOlteHTOB, T. e. Ta6.'1lilty 3HaqeHHH '-IHCC:I (1 +r)"
245

npH pa331HqHbIX 3HaqeHHHX r : r=0,02; 0.03; 0,04, . ..1). B03-
MO>KHO, qTO 3HaKOl\lCTBO C 3THMH Ta631HUaMH HaTOJIKHYJIO
,npyroro 3aMeqaTeJJbHoro BbIqHCJIHTeJI5I -l106CTa BlOprn H8
MblC31b COCTaBHTb CBOH Ta6JIHUbI, KOTopbIe 6bI.TJH 6bI npHrOJI,Hbl
,nJI5I 06JIerqeHH5I BceB03MO>KHbIX BblqHC31eHHH, a He TOJIbKO TOp-
roBo-cpHHaHCOBbIX. KaK Ta6JIHUbI CTeBHHa.
l106CTa BlOprn (1552-1632) pO,ll,HJIC5I B lIlBeHuapHl! H 6bIJI
HCKYCHbIM qaCOBlliHKOM H MacTepoM aCTpOHOMHqeCKHX llHCTPY-
MeHTOB. OH He nOJIyqH31 HayqHOrO 06pa30BaHH5I H 6bIJI TaJIaH-
TOM caMo6bITHbIM. OqeHb' HaCTOHl.jHBbIM H ynopHbIM B TpY,ll,e.
COBMeCTHa51 pa60Ta C 3HaMeHHTbIM aCTpOHOMOM l1oraHHoM
Ken31epOM. YCTaHOBHBJlIHM 3aKOHbl ,lI,BH>KeHH5I nJIaHeT BOKpyr
C031Hua. npHBe31a ero K Heo6xOJI,HMOCTll 3aHHMaTbCH 60JIbillOH
BbIqHCJIHTeJIbHOH npaKTHKOH. Te TPY,ll,HOCTH. KOTopble OH npH
3TOM BCTpeTHJI, 3aCTaBHJIH ero HCKaTb npHeMOB 6031ee CKOpbIX
BbIqHCJIeHUH H BOnJIOTHTb B >KH3Hb H,lI,eH, BbICKa3aHHble MH-
XaHJIOM lIlTHcpeJIeM.
O,ll,HOBpeMeHHo C BlOprH H cOBepllleHHo He3aBHCHMO OT
Hero nO,ll,06HYIO 3a,!J,aqy nOCTaBHJI nepe.IJ, C060H JI.>KOH He-
nep (1550-1617) B lIlOTJIaH,lI,HH. Henep TaK>Ke He 6blJI MaTe-
MaTHKOM no npocpeCCHH. Ho eMY Y.IJ,aJIOCb nOJIyqHTb xopOlllee
06pa30BaHlle. H OH 3aHHMaJIC5I aCTpOHOMHeH H MaTeMaTHKOH
KaK JII06HTeJJb. TeM He MeHee, nOMHMO COCTaB31eHHH Ta6JIHU
1) ECJlH Bbl >KeJlaeTe COCTaBHTb npe.nCTaBJleHHe 0 CJlO>KHbIX npo-
ueHTax, TO c.neJlaihe npe.nnOJlO>KeHHe, IJTO BaMH nOJlO>KeH B coepKaccy
Ha xpaHeHHe I pyo. Ha n JleT. CoepKacca HaCIJIlTblBaeT B KOHue Ka>K.noro
ro.na npHObIJlb. COCTaBJlHIOIllYIO p% OT Toil CYMMbI, KOTopaH XpaHIIJlaCb
B Kacce B TeIJeHHe .nallHoro ro.na. K KOHUY I-ro ro.na !Jaw PYOJlb npe-
BpaTHTcSI B ( 1 + &) Pyo. K KOHUY 2-ro r01J.a coepKacca oy.neT paCCIJH-
TbIBaTb npHOblJlb co Bceil Toil CYMMbl, KOTopaSl 6b!Jla B Kacce B TeIJeHlle
2.ro ro.na, T. e. C'CYMMbI_(I+ IO )PYO. TaK KaK Ka>K.nbIHPYOJlbIJepea
ro.n npeBpalII.8eTCSI B (I + IO ) PYO., TO K KOHUY 2-ro ro.na B coepKacce,
BMeCTO BJlO>KeHHOro PYOJlSl, oy.n eT Haxo.nHTbCSI CYMMa B (1 + IO r Pyo.
PaccY>K.naH TaK >Ke, MbI npH.neM K BbIBO.ny, IJTO B KOHue n-ro ro.na
( p ) n p
BJIO>KeHHblH Py6JJb npeBpaTIlTcH B I + lW Pyo. DOJlaraSi 100 = r
Mbl nOJIYIJHM Bblpa>KeHlle (I+r)n. 1.JHTaTeJlb, H8BepHoe. aaMeTHJI, IJTO
nepe.n HHM reOMeTpHIJeCKaSl nporpeccHSI:
I, (l+r). (l+r)2, .... (I+r)n, r.neq=l+r.
246


"
.........-.,
s
<r
ll>KOH Henep.
310rapmt>MOB. OH BHeC B MaTeMaTHKY pH,lI, ueHHbIX H)J.eu.
no nOBO,ll,y pa60TbI HaJl COCTaB31eHHeM Ta631HU Henep nHca.n:
«51 BCer,ll,a CTapaJICH. HaCKOJIbKO n03BoJUI.nH MOH CHJIb!
H cnoc06HocTH. OT,lI,e.naTbCH OT TPYJI.HOCTH H CKYKH BblqHCJle-
HliU. ,nOK}"IHOCTb -KOTOPbIX 06bIKHOBeHHO omyrnBae-r MHornx
OT H3Y1J:eHHH MaTeMaTHKH». TpY,lI,HOCTH He HcnyraJIH caMoro
Henepa; B Te1J:eHHe JI.BYX ,lI,ecHTHJIeTHM OH pa60Tan Ha)J. COCTaB-
31eHlleM CBOHX Ta6mfU.. nepeJI. HHM CTOH31a 631arOpOJI.HaH lte31b -
06Jler1J:HTb pa60TY 6Y,lI,YlUHX BbI1J:HC31HTeJJeu.
l106cTa BlOprH H l1>!<H Henep - Te 3aMeqaTeJIbHbIe 311O,lI,H.
KOTOpblM MbI o6H3aHbI COCTaB.neHHeM nepBbIx Ta6JIHU 310ra-
pHcp:\iOB. MbI JI.on>KHbI YJI.HBJIHTbCH TOMY rHraHTcKOMY TPYJI.Y.
KOTOpblH OHH BJIO>KHJIH B CBOH nepBble Ta6JIHUbi. Be,ll,b 340 31eT
Ha3aJI. He 6b1.n0 HHbIX BblqHC31IiTe31bHbIX cpe,ll,cTB, KpOMe Tex.
KOTopbIe npe,n.naraeT 331eMeHTapHaH MaTeM3TliKa. B qaCTHOCTH,
eure He 6bJ.na nOCTpoeHa CTp9raH TeOpHH pH,lI,OB, KOTOp3H no-
MOrJIa 6b1 YCKOPHTb COCTaB31eHHe Ta631HU. H nepBblM COCTaBH-
TeJIHM HX npHw.TJOCb nonb30BaTbCH HCKJIlOqHTe31bHO annapaTOM
9JIeMeHTapHOH MaTeMaTHKH.
247

OT OJl,HOH CHCTeMbi JlOrapHlj>MOB K Jl.pyroH
113151 COCTaB.TJeHH5I nepBbIX Ta631HU JIOrapHcpMOB OqeHb
Ba>KHbIl\I 51B.n5leTC5I TO 06CTOHTe31bCTBO, qTO, HMe51 Ta631HUbI
.10rapHcp:\1OB qHce.l no OCHOBaHHIO a, MO>KHO npH HX nOMOlliH
n031yqHTb cpaBHHTeJIbHO npOCTO Ta631HUbi JIOrapHcpMOB 'IHCeJI
no JII060MY .IJ,pyrOMY OCHOBaHHIO b. BbIBe,!J,eM cpOPMYJIY ne-
pexo,!J,a OT O,!J,HOH CHCTeMbJ JIOrapHcpMOB K .IJ,pyrOH.
OYCTb
logaN = a H logbN =;
(1)
Tor)J.a
a"" =N H b =N.
(2)
npOJIOrapHcpMHpyeM nOCJIe,ll,Hee paBeHCTBO no OCHOBaHHIO 0:
 logo b = Icg a N;
3aMeH5I5I  qepe3 logbN. nOJIYIIHM:
10g b N .laga b = logaN.
OTCIO,!J,a
logaN
IcgbN = _ I b . (3)
og"
MbI nOJIYIIH31H cpOPMYJIY nepexo,!J,a OT CHCTeMbI 310rapHcpMOB
npH OCHOBaHHH a K CHCTeMe 310rapucpMoB npH OCHOBaHHH b.
3Ta cpopJ\IY31a rOBopuT:
Cj mo6bl ftQ(lmu JlOcapuljJM 'iUCJla N no ft080MY OCft08aftUIO
b, ftYJlCftO pa:3i'JeJlUmb Jl02apuljJ,w 3mocO 'illCJla no cmapoMY
OCft08aftUIO a fta 1l02apuljJ,w ft08020 OCft08aftUfl b no cmopoMY
OCft080ftUlO a.
npHMep. HaHTH log210:
lcg 2 10 = 11lo0 o = 0,;010 = 3,3223.
nOHCKH YJl.alIHoro OCHOBaHHH
EC.TJH nepexo,ll, OT O.IJ,HOH CHCTe:\1bI JIOrapHcp\10B K )J.pyroH
Bn031He B03MC>KeH Ii CBOJI,HTCH K YMHO>KeHHIO 31crapucp:\loB
1
'IHCe31 npH cTap01l1 OCHOBaHHH Ha MHO>KHTeJIb _ I b ' Ha3bI-
oga
Bae:\lbnI 1\1 0 )J. Y JI e 1\1 n e p e x 0 ,lI, a, TO npH COCTaB31eHHH
nepBOH Ta6JIHUbl JIorapmp:\10B BbI60p OCHOBaHliH ,lI,HKTyeTcH
248

HCK31lOqHTe31bHO HaH60JIbilleH npOCTOTOH COCTaB31eHHH ,nOH
Ta631HUbl. Bbl5lCHHM. KaKHe C006pa>KeHHH npHBe,!J,YT HaC K
HaH6031ee paUHOHa31bHOMY Bbl60py OCHOBaHH5I.
Ta631HUbI 310rapHcpMOB ,lI,031>KHbl npe>K,lI,e Bcero y,ll,OB31eTBO-
pHTb ,lI,BYM Tpe60BaHH5IM: Bo-nepBblX, MbI ,lI,031>KHbl 31erKO
H 6b1CTpO HaHTH B HHX TOqHOe H31H XOT5I 6bl npH631H>KCHHOe
3HaqeHHe HeKOToporo nO.TJO>KHTe31bHOrO QHC31a. 310rapHcpl\l KO-
TOpOro Tpe6yeTC5I on pe,ll,e31HTb , H 3,!J,eCb >Ke pH,lI,OM npOqeCTb
H caMbIH 310rapHcpM ,noro qHCJIa. H, BO-BTOpbIX. OTblCKHBa51
lJHC310 no ero 310rapmpMY. MbI ,!J,031>KHbI 31erKO HaHTl! TOqHOe
H31H XOTH 6bl npH631H>KeHHOe 3HaqeHHe ,lI,aHHoro HaM 310ra-
pHcpMa Ii 3,!J,eCb >Ke p5l,!J,OM npOqeCTb H qHC310. COOTBeTCTBy10-
lUee ,lI,aHHOMY JIOrapHcpMY TOqHO H31H XOT5I 6bI npH631H>KeHHO.
l.lT06bJ 6hI310 BbIl1031HeHO H nepBoe H BTopoe YC310BHH. Ta6-
31HUbI ,lI,031>KHbI 3aK31IOQaTb B ce6e, C O,ll,HOH CTOpOHbl, qHC.TJa,
H,lI,YlUHe B B03paCTalOlliel\l nOp5l,ll,Ke t Ma31bIMH HHTepBa.'IaJ\1H
Me}K,ll,Y HHMH (0.01, H31H 0,001. H31H 0.0001). H, C ,lI,pyroii CTO-
pOHbl, 310rapHcpMbI qHCe31, pacnOJIO>KeHHbIe TO>Ke B B03paCTa-
IOM nOp5l,ll,Ke H TO>Ke C Ma31bIMH HHTepBa31aMH. OKa3bIBaeTC5I.
MbI Y,!J,OB31eTBOpHM 06m!M 3THM YC310BHHM, eC.1H «Y,lI,aqHO»
BbJ6epeM TO qHCJIO. KOTopoe npHMeM 3a OCHOBaHHe CHCTeMbl
.'IOrapHcpMOB.
B 10 P r H H Hen e p He3aBHCHMO ,lI,Pyr OT ,lI,pyra npH-
W31H K He06xo,ll,HMOCTH npHH5ITb 3a OCHOBaHHe nepBoH Ta6-
.'IHUbl JIOrapHcpMoB qHC310 onpe,ll,e31eHHOrO BII,!J,a. KaKoe >Ke
:::ITO qHC310?
fHraHTCKHH TPYJI. ElOprH H Henepa
Ope}K,ll,e Bcero YC310BHMC5I 0 TOM. KaK MbI 6Y,!J,e1l1 COCTaB-
31HTb Ta6.TJ H UbI 310rapHcp:\1OB. Y,lI,06HO nOCTynHTb TaK:
6Y,!J,eM B03BO;I.HTb BbI6paHHoe Hal\1H OCHOBaHHe B pa331HqHble
CTeneHH, nOKa3aTeJIH KOTOpblX 6Y,!J,YT nOCTeneHHO B03paCTaTb.
OT.rmQa5lCb ,lI,pyr OT ,lI,pyra Ha Ma31YIO Be31HqHHY (HanpHMep,
Ha 0.0001). KaKoe >Ke OCHOBaHHe a Bblro,ll,Hee Bcero B351Tb
.n31H COCTaB31eHH5I nepBblX Ta6J1lIU? Ka3a310Cb 6bI, 'ITO npH
HailleH ,!J,ec 5ITHq Hoi-i CHCTeMe CqHC31eHH5I ue31eC006pa3Hee
3a OCHOBaHHe a B351Tb qHC310 10. 3TOT BbI60p ,lI,eHCTBHTe31bHO
.J.aeT 6031bWHe npeHMYlUecTBa (BcnOl\1HHTe cBoHCTBa xapaKTe-
pHCTHKH H MaHTHCCbI ,lI,eC5ITHqHbIX 310rapHcpMOB), HO a6C031IOT-
HO HeBbIro,ll,eH lI.3151 COCTaB31eHH5I nepBbIx Ta6.1HU. Op06Y51
COCTaB3151Tb TaKYIO Ta631HUY. I\IbI cpa3Y BCTpeTH31HCb 6bI C He-
npeo,ll,OJIHMbIMH TPY.IJ,HOCTHMH. CBH3aHHbIMH C Heo6xo,ll,HMO-
249

-::TblO H3BJIeKaTb KOpeHb 10000-H CTeneHH H3 qHCeJI 10. 100.
leoo H T. ,lI,.


OCHO!1aHHe a = 10
IgN I N
0,0000 10 0 = I
0,0001 100.0001 =looo
' 10
0,0002 IO O . COO2 = 1000
 " 100
H T. .n.


nonpo6yeM H3MeHHTb OCHOBaHHe TaK, tIT06b1 ,lI,eHCTBHe
H3BJIe4EHHH KOpHH He Bbl3b1BaJIO 3aTPY,lI,HeHHH. TaKHM OCHO-
BaHHeM MO>KeT CJlY>KUTb IIHCJlO:
a= 10 10000 .


O:HOBaIlHe a = 1010000
logaN I N
0,0000 (10 10000)0 = I
0,0001 (1O IOOOO }o.OOOI = 10
0,0002 (10 IOOOO )0.C002 = 100
0,0003 (1010000)0,0003 = 1000
H T. .n.


nOCJle,ll,H5IH Ta6J1HIla TaK>Ke He MO>KeT Y,lI,OBJleTBOpHTb Hac.
B Heif JlOrapHcpMbl H,!J,YT, B03paCTaH Ha MaJlbIe BeJlHqHHbI
(Ha 0.0001). HO 'UfCJla B03paCTalOT CJlHlllKOM 60JlblllHMH CKatI-
KaMH.
npo6yeM YMeHblllHTb OCHOBaHHe H B3HTb ero paBHbll\l
XOT5I 6bl


a=210000.
Tor,ll,a MbI nOJlyqHM TaKYIO Ta6J1HUY:


O:HOBaHHe a = 210000
logaN I N
0,0000 (210000)0.0000 = I
0,0001 (210000)0.0001 = 2
0,0002 (210000)0.0002 = 4
0,0003 (210000)0.0003 = 8
H T. /1,.


250




YMeHbUIIiB OCHOBaHlIe, Mbl .n;0611JIIiCb YMeHbllieHIIH IIHTep-
BaJIOB Mem.lI.Y IlI1CJIaMII cToJI6ua N, HO BCe me 3TII I1HTep-
BaJIbl eIlle CJIl1lliKOM BeJII1KII 11 6bICTpO B03paCTalOT C B03paCTa-
HlieM JIOrapII<pMOB. ConOCTaBJIeHIIe .n;BYX nOCJIe.n;HI1X Ta6JIliU
HCHO YKa3bIBaeT Ha TO, IlTO IIHTepBa.TlbI Mem.n;y IlHCJI8MH CTOJI6-
ua N 6y.n;YT TeM MeHbllie, lJeM 6JIlime K e.n;HHHue 6y.n;eT B351TO
OCHOBaHlie JIOrapH<pMOB. f103TOMY nonpo6ye\l B3HTb 3a OCHO.
BaHlie JIOrapII<pMOBIlIICJIO, OIleHb 6JIII3KOe K e.n;I1HHL(e:
a= I ,000110000.
f10JIYIJIIM HOBYIO T86JIIIL(Y:


OClWBaHlie C1 = I,O:Jopeooo
log"N N


0,000)
0,0001
0,0002
0,0003
0,0004


1,000000
I,O'JOI00
I ,00.) 200
1,000300
1,000400


0,0009
0,0810
0,0011


1,000900
1,001000
1,001001


0,0050


1,005012


0,0060


1,005018


II T. JI..


3Ta Ta6JIIIL(a yme y.n;OBJIeTBOp5leT TeM Tpe60BaHII5IM, KOTO.
pbIe MbI nOCTaBIIJIli npli BbI60pe OCHOBaHII5I: II lJIICJIa 11 JIora-
PI1<p'lbI IIX IIlI.YT, B03paCTa51 Ha MaJIbIe IIHTepBaJIbI.
COCTaBJIeHIIe TaKoH Ta6JIliUbI 06.TJerIlaeTc5I TeM, 1110 JIO-
rapH<p:\lbI IlI1CeJI YBeJIHIlliBalOTcH no 3aKoHY apl1<pMeTIlIleCKOH
nporpeCCliH C pa3'HOCTblO d=O,OOOl, a IlHCJIa N 113I\1eH5IIOTC5I
no 3aKoHY reOMeTpHIleCKOH nporpeccliH co 3HaMeHaTeJIeM
q= 1,0001. f103TOMY nOJIYIleHIIe HOBoro IlI1CJIa B CTOJI6L(e N
CBO.n;I1TC5I K YMHomeHIIIO npe.n;bI.n;YIllero lJl1CJIa Ha q, T. e. Ha
1,0001, IlTO B CBOIO oIlepe.n;b CBO.n;IITC5I K YMHomeHIIIO npe.n;bI-
.lI.YIllero lJIICJIa Ha 1 H npII6aBJIeHIIIO K 1I0JIYlJeHHoMY Pe3YJIb-
251




TaTY 0,0001 IlaCTII 3Toro npe.n;bI.n;yru.ero tJHCJIa. T 8K, eCJIH
log N=0,0051 , TO:
N= 1,005012. 1,0001 = I ,005012. (1 +0,0001)=
= 1,005012+ 1,005012.0,0001 =
=1,005012+0,000101 = 1 ,005113.
HMeHHO npil TaKOM OCHOBaHliH 6bIJIa COCTaBJIeHa E 10 P r II
H BbmymeHa B 1620 r. B CBeT Ta6JIIIL(a JIOrapII<pMoB, Ha3BaH-
Ha51 11M «ApII<pMeTilIleCKa51 II reOMeTpIIIleCKa51 Ta6JIIIL(a npo-
rpeCCIIH)}. TaK KaK nOJIb30BaHlie .n;eC5ITIILJHbIMli .n;P0651MII TOJIb-
KO HaLJIIHaJIO BXO.n;HTb B npaKTIIKY eBponeikKIIX MaTeMaTliKOB,
TO ElOprH, 11 T06bI II36e}!{aTb .n;p06eH, YBeJIIIIlliJI BCe IlIICJIa JIe-
BOrO cToJI6ua B 100 000 pa3, a IlIICJIa npaBOiO cToJI6L(a B
100 000 000 pa3. 3TIIr.l Ta6JIHua ElOprH OTJIHIlaeTC5I OT HallieH
Ta6.fl II UbI.
Ha.n; COCTaBJIeHIIeM CBOHX Ta6JIIIL( l106cTa ElOpm Tpy.n;IIJI-
C5I 8 JIeT, C 1603 .n;o 1611 r. f1oIleMy}!{e eMY nOTpe60BaJIOCb
TaK !\fHOrO BpeMeHII? B.lI.YMaiiTecb B CJIe.lI.YlOlllee: .lI.JI5I Taro
LJT06bI OT 0 B cToJI6ue log N .n;OHTil TO.TJbKO.lI.O 1 npil HHTep-
BaJIe B ,0,0001, Hy}!{HO BblIlIICJIHTb 10 000 IlIICeJI cTo.TJ6ua N,
H, BbI1JHCJ1IIB IIX, MbI .n;OH.n;eM .n;o 11 IICJI a , npH6JIII}!{eHHO paB-
HOro 2,7181. f1paB.n;a, IleM .n;aJIee, TeM IlIICJIa cTOJI6ua N 6y.n;YT
pacTIl 6b1CTpee, H BCe }!{e ElOprii B 06111eH CJIO}!{HOCTH nplilli-
JIOCb npoII3BecTH CBbIllie 230 000 000 nOCJIe.n;OBaTeJIbHbIX YMlIO-
}!{eHHH Ha 1,0001.
BbIBO.n; II3 Bcero CKa3aHHoro TaKOB.
11JI5I Toro LJT06bI COCTaBHTb nepBbIe Ta6JIliUbI JIOrapmpMoB,
npIIro.n;Hble .n;JI5I RpaKTilKII, He06xo.n;IIMO 3a OCHOBaHHe JIora
pII<pMOB 6paTb tJIICJIO BII.n;a:
a=(I+-.;r
r.n;e n - L(eJIoe LJHCJIO, 60JIblliOe no BeJIII1JIIHe (n= 1000,
10 000...). Ta6JIIIUbl 6y.n;YT TeM TOLJHee, IleM 60JIbllie n.
Ta6JIIIL(bI ElOprii He nOJIYLJHJIli llilipOKoro pacnpOCTpaHe-
HII5I, TaK KaK paHee 1620 r., a IIMeHHO B 1614 r., B AHrJIIIII
6bIJIli ony6.'lHKOBaHbI 3HaIlIITeJIbHO 60JIee cOBepllieHHbIe Ta6-
JIliUbI. I1x aBTopoM 6bIJJ 1111mH Hen e p.
Henep np" COCTaBJIeHIIH CBOIIX Ta6JIHU nOJIO}!{IIJI n paB-
HbIM 10 000 000, T. e. npIIH5IJI 3a OCHOBaHlie JIorapmp:VIOB
11 IICJI 0
a=(1 ,0000001)'0000 000.
252

CBOII Ta6JIIIL(bI OH Ha3BaJI «OnHCaHlie y.n;HBIITeJIbHOll Ta6JIHUbl
JIOrapII<pMOB». Ha.n; 3THMII Ta6JIHua!\1Il Henep Tpy.lI.IIJIC5I OKOJIO
20 JIeT. Ho II 3TO eme He BeCb BKJIa.n; Henepa B MaTeMaTHKY.
He MeHee ueHHbI II Te H.n;eH TeopeTIIIlecKoro nop, 51.lI.Ka KOTopbIe
3apo.n;IIJIliCb B C03HaHllH Henepa npli COCTaB.'IeHIIll Ta6JIliU.
Ta6JIliUbI .lI.eC5ITll4HbIX .TJOrapII<pMoB n05lBll,TlllCb B 1620 r.
OHll 6bIJIli COCTaBJIeHbI .n;pyroM Henepa, npo<peccopOM KeM6-
pII.n;>KCKOrO YHHBepCllTeTa feHpli 13 p II r COM (1556-1630).
3TII Ta6JI II UbI 14-3HaIlHble II co.n;ep>KaJIli JIOrapmpMbI IlHCeJI OT
1 .n;o 20000 II OT 90000 .lI.0 100000.
Pa60Ta, HaIlaTa51 I3pIIrcoM, 6bIJIa npo.n;OJI>KeHa II 3aBep-
weHa rOJIJIaH.lI.CKliM MaTCMaTlIKOM AH.lI.pIIaHOM B JI a K K 0 M.
KOTOpblH B 1628 r. ll3.n;aJI 1O-3HaIlHbIe Ta6JIHL(bI JIOrapII<pMoB
4IICeJI OT 1 .n;o 100 000.
Ta6JIliUbI BJIaKKa 6bIJIll nOJIO>KeHbI B OCHOBY Ta6JIliU JIO-
rapII<p\lOB, BnepBbIe n05lBIIBlliHXC5I B POCCIIII. :3TO 6b1JIO B
1703 r. OHII HOCliJlII .n;JIIIHHOe Ha3BaHIIe: «Ta6JIIIL(bI JIorapmp-
MOB II CIIHYCOB, TaHreHCOB II ceKaHCOB TmaHHeM II 3a OCBII-
.n;eTeJIbCTBOBaHlieM MaTClIIaTII4eCKliX H HaBHraUKliX lliKOJI Y4II-
TeJIeH AH.n;pe51 <l>apxBap.n;COHa, CTe<PaHa fBIIHa II JIeoHTII5I
MarHIIUKoro».
TaK Ha npOT5I>KeHllli 20 JIeT B pa3Hbix cTpaHax, pa3HbIMH
JllO.lI.bMII 6blJI C03.n;aH HOBbIH BbIIlHCJIIITeJIbHbIH annapaT B BII.n;e
T36JIIIL( JIOrapII<pMoB. MHOrOJIeTHHH TPY.n;, nOTpa4eHHbIH Ha
COCTaBJIeHIIe 3TIIX Ta6JIIIL(, He nponaJI .n;apOM. MaTeMaTliKII
BbICOKO OueHIIJIli 3TO II306peTeHIIe. KenJIep lliHpOKO nOJIb30-
BaJIC5I JIOrapII<pMIIIleCKliMli Ta6JIllL(aMII B CBOIIX aCTpOHOMII-
IleCKliX BbI4llCJIeHII5IX, a <ppaHL(Y3CKliH MaTeMaTIIK II aCTpOHOM
JIanJIac rOBOpIIJI, 4TO OTKpbITlie JIOrapII<pMOB Y.lI..TJIIHIIJIO IleJIO-
Be4eCKYlO }I(II3Hb, eCJIli 3TY >KII3Hb pacueHllBaTb He KOJIIIIle-
CTBOM npO>KHTbIX JIeT, a KOJIHIleCTBOM c.n;eJIaHHOH 3a 3TY .>KII3Hb
pa60TbI.
4lICJIO e
lI.a.TJbHeHWa51 IICTOpII5I Ta6JIliU JIOrapH<pMOB TaKOBa.
T p y.n;aMli H blOT 0 H a (1642-1727) II JI e i1 6 H H L( a
6bIJIli 3aJIO>KeHbI OHOBbI MaTel\laTHIleCKoro aHaJIH3a. :3 H JI e p
(1707-1783); A 6 e JI b (1802-1829), K 0 lli II (1789-
1857), .D.'AJIaIll6ep (1717-1783) paClliIIpIIJIli II yrJIy6IIJIli 3TH
OCHOBbI 11, B IlaCTHOCTII, C03.n;aJIli Y"leHIIe 0 p5l.n;ax, II TO, Ha
1110 blOpm II Henep TpaTliJIll MHorHe ro.n;bI BpeMeHH, CTaJIO
B03MO>KHbIM c.n;eJIaTb ropa3.n;0 6bICTpee.
253

npe.>K.n;e Bcero OKa3aJIOCb, IlTO IIIICJlO BII.n;a
(1 + +r
npH HeOrpaHIIqeHHOM YBeJIIIIleHHII n CTpeMHTC5I K OlIpe.n;eJIeH-
HOMY npe)1eJIY. MbI CMO.>KeM 3TO YTBCp}!{.n;aTb, eCJIH, KaK Toro
Tpe6YeT TeopeMa BeHepUlTpacca, .n;OKa}!{eM, qTO Bee 1JJIeHbI
nOCJIe,lJ,OBaTeJIbHOCTH
{(I+-;f}
MeHbllie HeKOToporo onpe.neJIeHHOrO IlIICJIa M, T. e. .lI.JI5I
JI60ro HOMepa n IIeeT MeCTO HepaBeHCTBO
(I ++r<M.
npliCTYlIaeM K .lI.OKa3aTe.TJhCTBY 3Toro HepaBeHCTBa.
f1oCJIe.n 0 BaTeJIb HOCTb
{ ( 1 + -1z f}
B03paCTallI.a51 1), KaK 3TO BH.n;HO II3 HH.>KenOMemeHHOH Tau-
JIIIL(bI:
II 1(1 + -} r
1 2,00
2 2,25
3 2.31
4 2,44
5 2,41
B lIepBOH rmme ObWO IIpe.n;J10}!{eHO .n;oKa3aTb MeTo.nOM
MaTelaTHIlecKoH IIH.n;YKUHII <POPMYJIY 6HHoMa HbTOHa, no-
1) lLemHl BblBO 0 TOM,' QTO YKa3annaH nOCJleOBaTeJlbHOCTb-
B03paCTaiOmaH, MbI HCnOJlb3yeM HenOJ1HYIO HH,AYKI.{HIO. 3.n;eCb 3TO c.n;e-
JlaHQ .n;JlSi ynpomeHHSi.
254

3B('JI1IOIII.YlO B03BO.n;IITb B L(eJIYIO CTeneHb .n;BYIIJIeH (a + b).
3Ta <popMYJIa T8KOB8:
(a + b)n = an + -T .an-1b+ II (lIl I) a n - 2 b 2 +... +
+ II (n - I) (n - 2 ).. (n - k + I) n-k b k + + b 
1.2.3 .. k a....
BOCnOJIb3ye1c5I eIO .lI.JI5I p33JIO.>KeH1I5I 6lIHoMa ( 1 + + r .
MbI nO.TJYIlH:ll:
(1 + + r = 1 + -T . + + II (Ill  I) : 2 +
+ + n(n-l)...(n-k+l) .  + + 
. .. I . 2.3 .. k 1I 1l ' . . nn
IIJIIi
( I +  ) n = 1 + 1-1--  ( 1 _  ) +
n I 1.2 n
1 / 1 )( 2 )
+ 1.2. 3 11 - n 1 - n .+..,. +
+ 1
1.2.3...k
(1 - -* ) ( 1 - + ) . . . (1 - k n I ) +... + :n .
CEoikTBa me nOCJIe.n;HeH <POPMYJIbI CJIe.n;YIOUlIIe:
1. K2>KJJ:'151 II3 pa3HocTeH. CT05lUl351 BHYTPII cK060K, eCTb
IlIICJIO no.nomIITeJIhHOe; 3aMeH5I5I 3TH p33HOCTII Ilepes HYJIb,
l\:bI npaBYIO IlaCTb nOCJIe.n;Hero paBeHCTB3 YMeHbIllH:ll II no-
nOMY M02Kel HanlicaTb:
( 1 + + r > 2.
2. K8.>K.lI.a51 II3 Tex >Ke p83HocTeH MeHbllie 1; 38MeH5I5I
BC5IKyIO H3 HIIX e.n;IIHHueH. 8 K8m.n;oe H3 IlIIce.TJ 2, 3, 4 ...,
IIaX0.lI.51UlIIXCH B 3H8\1eHaTeJI5IX nepe.lI. cK06KaMII, - .n;BoHKa-
MII, I\1bI npaBYIO IJaCTb paBeHcT3 YBeJIH, IlIIB8eM II Bee me
OHa OK2>KeTC5I MeHl'.llie HeKOToporo qIIC.TJa M. LITO .>Ke 3TO
"a qIIC:IO? C.lI.eJIaB BbIweYK338HHbIe aaMeHhI. 1I::WYIlIIM:
( I ) n I 1 I I I
1 + n < 1 + 1+"'2 + 22- T 23- +... + 2n .
255

ECJIH TelIepb npe.lI.nOJlO>KIITb, liTO IlIICJIO n Ha1JIIHaeT 6e3rpa-
HIIIIHO B03paCTaTb, TO, B35J.B II3 npaBOH GaCTII nOCJIe).l.HerO
HepaBeHCTBa CYMMY BCeX CJIarae:\lbIX, KpOM nepBOrO, MbI
npH.n;eM K CYMMe 1JJIeHOB
I 1 1 1
1+ 2+22 +23'+ .., + ¥ +...
6eCKOHe1JHO y6bIBalOIII.eH reOMeTpHIIeCKOH npOrpeCCHH C 3Ha-
I
'!eHaTeJIeM q = "'2 H nepBbI:\l IlJ1eHOM, paBHbIM 1; :na cYMMa,
I
KaK H3BeCTHO, paBHa ---r = 2.
1-- 2
YIIHTblBa51 3TO, MbI npII.lI.eM K HepaBeHCTBY
(1++r<3
.lI..TJ5I .TJlC60ro L(eJIoro nOJIO>KIITeJIbHOrO n.
l-!TaK,
2 < (1 + -; r < 3
npli JIlO60M L(MOM nOJIO}!{IITeJIbHOM n. Tor):!a Ha OCHOBe Teo-
peMbI BeHeplliTpacca MO>KeM YTBep>K.lI.aTb, IlTO nOCJIe.n;OBa-
TeJJbHOCTb
{(I + +r}.
npH n-+ 00 HMeeT npe.n;e.'1. f1pe.n;eJIOM 6Y.lI.eT CJIY>KIITb lippa-
L(IIOHaJIbHOe lIHCJIO, npH6JIII>KeHHO paBHoe
2,71828182845... .
2ho IIppaL(HOHaJIbHOe IlIICJIO, BCTpeIlalOIL(eeCH B MaTeMaTliKe
TaK >Ke IlaCTO, KaK II lIIlCJIO 'It, B IleCTb 3HJIepa o603HaIlaeTc5I
6YKBOM e H Ha3bIBaeTC5I B IleCTb Henepa <<HenepOBbIM 'lIICJIOM».
I1TaK,
. ( 1 ) n
hm 1 + - = e.
n-...oo n
256

B HaCT05lIL(ee BpeMH nOJIb3YIOTC5I .n;BYM5I CliCTeMaMH JIOrapH<p-
MOB: HaT Y p a JI b H 0 H, OCHOBaHlieM KOTOpOH CJIY>KlIT
tllICJIO e, II ,IJ, e c 51 T II II H 0 H C OCHOBaHHeM 10.
HaTypaJIbHbIH JIOrapII<pM IlIICJJa N 3anIICbIBaeTCH TaK:
In N, )J.eC5ITHIlHbIH >Ke JIOrapII<pM - Ig N.
Oc06aH pOJIb, KOTOpylO IIrpaeT .n;eC5ITIIIIHa51 CliCTeMa JIO-
rapH<pMoB B npaKTIIIIeCKliX paC'leTaX, 06'b5lCH5IeTC5I TeM, IlTO
IlIICJIO 10 eCTb OCHOBaHHe .n;eC5ITHIlHOH CHCTeMbI CIlIICJIeHH5I.
Ho IIIITaTeJIb 3HaeT, 1JTO B03MO>KHbI H .lI.Pyme CHCTeMbI CIlIIC-
JIeHII5I II He TOJIbKO B03MO>K, HbI HO 1JTO OHII CYIII.eCTBOBaJIli.
.D.o cero BpeMeHII B .n;eJIeHIIII 1Jaca Ha 60 MIIHYT, a MIIHYTbI Ha
60 ceKYH.n; II B .lI.eJIeHlIli rpa.lI.yca Ha 60 yrJIOBbIX MIIHYT H MIIHYTbI
Ha 60 yrJIOBbIX ceKYH.n; cOXpaHHJIliCb CJIe.n;bI llieCTII.n;eC5ITepII'l-
HOH CHCTeMbI CIIIICJIeHII5I, npIIH5ITOH B .n;peBHeM BaBIIJIOHe.
TaK >Ke B 513bIKax HeKOTopbIx Hapo.n;oB cOXpaHHJIHCb HaMeKH
Ha cYIL(eCTBOBaHlie .n;Ba.n;L(aTepIIIIHoH CliCTeMbI CIlIICJIeHII5I.
llllIpOKO IICnOJIh3yeTc5I B HaCT05lIII.ee BpeM5I .n;BOIIIlHa51 CliCTeMa
C1JIICJIeHIl51 npli YCTpoiicTBe C'leTHhIX MalliliH. ECJIII 6bI ceH-
1Jac rocnO)J.CTBOBaJIa IIHa5l, He .n;eC5ITII'lHa51 CliCTeMa CIlIICJIe-
HH5I, TO )J.eC5ITII1JHhIe JIOrapH<pMbI He IlrpaJIIi 6h1 HHKaKOH pOJIII,
Me>K.n;y TeM KaK HaTypaJIbHbIe JIOrapH<pMbI, OCHOBaHlie KOTOpbIX
He 3aBliCliT OT CIiCTeMbI CIlIICJJeHII5I, nO-npe>KHeMY HCnOJIb30-
BaJIliCh 6h1 B pa3JIIlIlHhIX TeOpeTII'leCKliX Bonpocax MaTeMaTII-
KII, <pII3 IIK II , MexaHliKH II .lI.Pymx HaYK.
f103TOMY B npaKTIlIleCKliX paC'leTaX nOJIb3YIOTC5I npe-
HMymecTBeHHO .n;eC5ITHIlHbIMII JIOrapH<pMaMII, B TeOpeTH'leCKliX
>Ke oonpocax npe.lI.nOIlTeHIIe OT.n;aIOT JIOrapIl!pMaM HaTypaJIb-
HbIM. f1epexo.n; OT O.lI.HOH CIiCTeMbI JIOrapII<pMOB K .n;pyroH
OCymeCTBJI51eTC5I C nOMOIII.hlO II3BeCTHOH HaM <P0PMYJIhI:
logaN In N
10gb N = loga b IIJIIi 19 N = In 10 .
BbIcmaSI MaTeMaTIiKa .n;aeT <POPMYJIbI, no KOTOpbIM HIL(YT-
CH HaTypaJIbHhIe JIOrapII<pMhI IIHCeJI. BOT o.n;Ha II3 TaKliX
<PoPMYJI:
In(N+I)=lnN+2[ 2N 1 +1 + 3(2+1)3 +
+ 5(2N1)6 + '" ].
9 A. A. I(onocop,
257

HaH.lI.€M 1I0 3TOH cpopMYJIe In 2, .lLJI5I :JToro 3aMeHlIM B HeH
N qepe3 1:
, [ 1 1 1 1 I ]
In 2::::::; 0 + 2 ""3 + ""3 . W + "'5 . 35 ::::::; 0,693.
HaH.lI.€M In 5:
[ I 1 1 I 1 ]
In 5::::::; In 4 + 2 '9 + 3" . V + 5 . 95 ::::::; 1,3862 +
+ 2.0,1126::::::; 1,6094.
HaH.lI.eM, HaKOHeu, In 10:
In 10 = In (2.5) = In 2 + In 5::::::;2,3026.
BcnOMHHa51 <POPMYJIY nepexo.lI.a OT O.n;HOH CliCTeMbI JIoraplI<p-
MOB K .n;pyroH, MbI nOJIyqIIM:
InN InN
19 N = In 10 ::::::; 2,3026 ::::::; 0,4343 In N.
MHO>KHnJIb 0,4343 (1JHCJIO I1ppaL(IIOHaJIbHoe) oy.n;eT MO.lI.Y-
JIeM nepexo.n;a OT HaTypa.IJbHoH CIICTeMbI JIOrapII<pMoB K ).I.e-
C5ITHQHOH H 0603HaIlaeTC5I OYKBOH A1.
A1 = 0,4343 (c T01JHOTblO .n;o 0,0001).
C'IHTaIOT MaWHHhI
TaOJIlIL(bl JIOrapHcpMoBIi H306peTeHHa51 H€CKOJIbKO n03>Ke
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258

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.:JeH, ObIJI HenpOCTbIM.
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K 8 JI b C03.n;a.n nepBYIO CIleTHYIO MaWliHY. 2ha MaWHHa ObIJIa
OIJeHb npOCT8, COCT05lJIa B OCHOBHOM liB By6IlaTbIX KOJIeC
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TaKOH npHMHTIIBHOHMalliHHbl IleJJOBeK C1JHTaJI H TOqHee H
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B 1617 r. H3BecTHbIH HeMeL(KHH MaTeM8TlIK JI e H 0 HIlI{,
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npeJIJIO>KeHO aHrJIHi1ci<IIM MaTeMaTHKOM b e 6 0 H .lI. .>K e M B
1842 r. EMY YJl8JIOCb p33pa6OTaTb npoeKT MalllHHbI, KOTopaS1
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259

B 1847 r. PYCCKIiH IIHmeHep O.n; H e p c03.n;aeT apII<pMo-
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neIlaTHbIM YCTPOHCTBOM.
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B KOHL(e XIX B. A. H. K P bI JI 0 B, MaTeMaTlIK 1-1 Kopa6-
JIecTpoIITeJIb, pa3pa6OTaJI Teopmo MallilIH, peWaIOIII.I-IX CJJom-
HbIe 3a.n;aIlI-I BbICllieH MaTeMaTIIKII.
Ho Bce 3TII MalliIIHbI He MOrJIli H.lI.TII B cpaBHeHIIe C COBpe-
MeHHbIMI-I 9JIeKTpOHHHMll L(II<PPOBbIMli MalliliHaMI-I.
f1epBa51 TaKa51 MalliliHa - 3HI1AK BCTYlIliJIa B CTpOH B
1946 r. aHa 6bIJIa nOCTpOi::Ha B f1eHCliJIbBaHCKOM YRIIBepCH-
TeTe B AMepliKe 1-1 MOrJIa COBeplliaTb B O.lI.HY ceKYH.n;y .n:o 5000
apII<pMeTIIIleCKI-IX .n;eiicTBliH. 4epe3 Tpli rO.lI.a 6blJIa C03.n;aHa
lIepBa51 3JIeKTpOHHa51 Mallil-lHa B CCCP no.n; PYKOBO.n;CTBOM
aKa.n;eMliKa C. A. JI e 6 e.n; e B a, 1'1 eIII.e Ilepe3 4 ro.n;a, B 1953 r.,
no.n; ero me PYKOBO,lJ,CTBOM 6bIJIa CKOHCTpYlipoBaHa YHIIBep-
CaJIbHa51 II YHHKaJIbHa51 no pa3MepaM MalliHHa E3CM (6bICTpO-
.n;eHcTBYIOIII.a51 9JIeKTpOHHa51 CIleTHa51 MalliI-lHa).
BOT IJTO nlillieT 06 9TOH MalliliHe ee KOHCTPYKTOP a Ka.n;e-
MIlK C. A. JIe6e.n:eB 1 . «3a O.n;HY ceKYH.n;y MalliliHa COBeplliaeT
B cpe.n;HeM 7-8 TbIC5IIJ apII<pMeTlitIeCKIiX .n;eHCTBIiH. aHa Ok
ApllcpMOMeTp.
1) C. A. .n e 6 e )J, e B, 91.€KTpOHHble BbI'IHCJlHTeJlbHble MBlIIIIHbI,
H3)J,. AH CCCP, 1956.
260

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261

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npOBe.lI.eHIIII HaYtJHX IICCJIe.lI.OBaHliM B 06JIaCTII 51.lI.epHOH <pII-
3IIKII, XIIMIIH, 6IIOJIOrIIII, npII IICCJIe.lI.OBaHIIII nOJIeTa CHapH.n;OB
H paKeT C 60JIbWIIMII CKOpOCT5IMII, npH COCT8BJIeHIIII npOfHO-
3a nOrO.lI.bI. Heorp8HIItJeHHa51 TOtJHOCTb. paCtJeTOB n03BOJI5IeT
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rapH<pMIItJeCKliX Ta6JIIlll, Ha KOTopoe Henep nOTpaTIIJI 20 JIeT,
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3aTb.
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OCHOBaHH851 Ha CtJeTe .lI.eCHTKaMII, Tpe6yeT ,DJI5I 38nIICll tJIIceJl
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CT8TOtJHO HMeTb TOJIbKO flBe IlH<PPbI: 1 II O. C nOMOlllbIO IIX MbI
MO)j{eM HanHcaTb JII060e tJIICJIO.
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262

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H L()I BToporo p83pH.n;a COCT8BJI5IIOT O.lI.HY e.lI.IIHHILY TpeTbero
p.13p5l.lI.8; CKOJIbKO me e.n;HHIIU TpeTbero p83PH.lI.a 6Y.lI.eT B
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6 2
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e.n;IIHIlllbI TpeTbero p83PH.lI.8 COCT8BJI5IIOT O.lI.HY e.lI.HHIIL(Y
QeTBepTOrO p83p5l.lI.8. CKOJIbKO me T8KIIX e.lI.HHIIL( 6Y.lI.eT B
H8weM 'lHCJIe? BbIlIOJIHHM HOBoe .lI.eJIeHIIe:
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f1oCJIe.lI.H5I5I 3anliCb roBOpIIT 0 TOM. qTO B H8IIIeM I./IICJIe Ok
H8 e.lI.IIHIIL(8 TpeTbero p83p5l.lI.8 II O.lI.Ha e.lI.IIHIIua l./eTBepTOrO
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C'lHCJIeHH5I 3anHWeTC5I T8K:
1101.
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263

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Ta6JIIIIJ.Y.
3anHCb '<MCJIa
B eCHTHHOH CHCTeMe
3anHCb HCJIa
B BOHHOH CHCTeMe
o
1
2
3
4
5
6
7
8
9
10
11
12
H T. JI,.
o
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10
11
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101
110
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1000
1001
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1011
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Ha.lI. 'lMCJIaMM B .lI.eCHTII'lHOH CHCTeMe. Hanpmvlep,
5+6= 11 12 -7 = 5 4.3 = 12 12 : 4 = 3
+101 1100 X IOO 1100 100
110 111 II 100 IT
1011 101 + 100 100
100
HOO
C 1I0MOLltblO L(II<PP 1 II 0 MorYT 6bITb 3alIHcaHbI JII06bIe
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HanpIIMep,
3 3 11 5 5 101
T = 22 = 100 = 0,11;"8 = 23 = JUDO = 0,101 II T. II.
KaKlie >Ke npeIIMymecTB8 .n;aeT .n;BOH'lHa51 CHCTeMa IIpli
pa60Te Ha 3JIeKTpOHHblX MallilIHax? BbI60p .n;BOHIlHOH cliCTe-
264

MW CqHCJIeHII5I 06YCJIOBJIeH B .n;aHHOM CJIyqae 3HaqIITeJIbHM
ynpOLlteHHeM «apII<pMemqeCKOrO YCTpOHCTBa» MaWIIHbI. 3TO
YCTPOHCTBO 51BJI5IeTC5I O.lI.HOH H3 MHOfIIX IlacTefI MaWIIHbI.
U.II<PPbI 1 II 0 Ka.>K.lI.OrO pa3pH.lI.8 y.n;06HO npe.lI.CTaBJI5ITb B
BII.n;e HaJIIIqIIH IIJIII OTCYTCTBH5I 3JIeKTpIIIlecKoro CHrHaJIa,
nepe.lI.aBaeMOro 1I0 L(eIIH. E.n.IIHHue MO:>KHO nOCT8BIITb B COOT-
BeTCTBIIe HaJIHIllie IIMnYJIbCa 3JIeKTpIIqeCKOrO Hanp5l:>KeHII5I
a HYJIIO - OTCYTcTBlie TaKoro IIMIIYJIbca. TaKHM 06pa30M.
JII060e qIICJIO, .n;aHHoe B .n;BOIIqHOH CHCTeMe, MO:>KHO npe.n;-
CTaBIITb onpe.n.eJIeHHOH nOCJIe.lI.OBaTeJIbHOCTbID IIMnYJIbCoB.
T8K, H8 qepTe:>Ke 117 npe.n;CTaBJIeHO B BII.n;e IIMnYJIbCOB qIIC-
JIO 1001 (9 B .n;ec5ImqHOH cIIcTeMe),
L(elIb ,lJ,JI5I 4-ro pa3p5l.n;a A
L(enb .lI. JI 5I 3-ro pa3p5l.lI.a "
uenb .n;JI5I 2-ro p83pH.n;a "
A
L(enb ,lJ,JI5I l-ro pa3p5l.n;a,
tIepT. 117.
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ero OTCYTCTBliIO. Ha qepTe:>Ke 118 MbI BII.n;IIM qHCJIO 57. MbI
MO:>KeM npoBepHTb, TaK JIII 3TO H8 caMOM .n;eJIe.
f1pII TaKOH CIICTeMe 3anliCH qIICeJI Ka)j{.n;OMY pa3p5l,lJ,Y,
OqeBII.lI.HO, .n;OJI)j{Ha COOTBeTCTBOBaTb CB051 L(enb, no KOTOpOH
lIepe.n;aIOTC5I IIMnYJIbCbI. E3CM pa60TaeT Ha .n;BOIIqHOH CIICTeMe,
H qHCJIa Ha HeH npe.lI.CT8BJI5IIDTC5I C yqeTOM nOp5l.n;KOB OT O.lI.HOH
MIIJIJIIIap.lI.HOH .n;o O.lI.HOrO MIIJIJIHap.n;a.
CqeTHa51 3JIeKTpOHHa51 Mawlma HMeeT cJIe,lJ,YIOmIIe OCHOB-
HbIe YCTpoHCTBa:
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apH<pMeTHqeCKHe .lI.eHCTBH5I.
2. 3anoMliHaIDmee YCTPOHCTBO, cooTBeTcTByIDmee JIIICTY
6YMarlI, Ha KOTOpOM BbIqIICJ1HTeJJb 06bIqHO 3aHOCIIT npoMe-
)j{YTOqHbIe pe3YJIbTaTbI BbIqIICJIeHHi't.
3. YCTPOi'ICTBO BBO.lI.a, qepe3 KOTopoe .n;8eTcH 3a.n.aHHe B
BII)J.e IIporp;:lMMbI BbIqIICJIeHHi't. CaMa IIporpaMMa, KOHeqHO,
265

COCTaBJIHeTC5I M8TeM8TIIKOM-BbI'lHCJIIITeJIeM H 3anIICbIBaeTC5I
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4. YCTpOHCTBO BbIBO.lI.8 pe3YJIbT8TOB peWeHII5I 38.n;a'lII.
5. "TPOt'ICTBO MeX8HII3fvI8, H8np8BJIHIOmero pa60TY
M8wr:'bJ T 11\. IlT06bI BbInOJIH5IJICH 38.n;aHHblt'1 npOrpaMMOH no-
pH,lJ,Oh ..ep8UIIH.
Ho M8WIIHa BpO.lI.e H8WeH E3CM 38HIIM8eT MHOrO MeCTa
(.lI.0 200 KG. M) II Tpe6yeT Tm8TeJIbHOrO yXO,lJ,8. D03TOMY B
nOCJIe.lI.Hee BpeM5I n05lBIIJIOCb
60JIbWOe 1\0JIII'leCTBO C8MbIX P83-
H J06p83HbIX cne1lII8JIII3IIpOBaH-
HbIX M8WIIH, npe.lI.Ha3H8qeHHblX
.lI.JI5I peWeHIIH O.lI.HOH IIJIII He-
CKOJIbl\IIX TIInOB 38.lI.8'l. OHII
IIMeIOT OTHOCIITeJIbHO He60JJbWIIe
pa3Mepbl, H8MHOrO .lI.eWeBJIe H
KOMnaKTHee YHIIBepC8JIbHbIX Ma-
WIIH.
MbI 3,lJ,eCb KOpOTKO OCTaHOBH-
JIUCb H8 IICTOpII4eCI\OM nYTII
C03,lJ,8HH5I CqeTHbIX M8WIIH II
nO'lTII He 38TPOHYJIII npIIHL(IInOB
IIX YCTpOt'ICTB8. ECJIH IlIIT8TeJIb
3aIIHTepecyeTc5I 3TIIM Bonpo-
COM. TO OH Hat'l,lJ,eT ,lJ,OcTynHbIH
.lI.JI5I Hero M8TepII8JI B KHIIr8X
M. r. PeHH6epr8 «D.YM8IOIllIIe M8WIIHbI» (D.eTfII.3, 1957) H
H. A. ApX8HreJIbCKOrO II E. 11. 38HIleBa «ABToMaTH'leCKHe
L(II<ppoBbIe M8WHHbI» (<pII3MaTfII3, 1 958) .
A
A
A
-
-
!\
4epT. 118
3aJI.a'lH
1. D.oK83aTb. LITO
Igxy.lg y x = 1.
l' K 8 38 H II e, VTCnOJIb3YHTe <POPMYJIY lIepeXO.lI.8 OT O,lJ,-
HOH CliCTeMbI JIOrapIIcpMOB K .n;pyroH.
2. 3Ha5l. qTO Igln64 = a. BbILlHCJIHTb 19lO Y 25 .
Pew e H H e. IglnlO = 1; Iglo2 + Ig 10 5 = 1; Ig5 =
= 1 - Ig2.
a
6 Ig 2 = a; Ig2 = 6 ;
266

a 5-a
CJIe,lJ,OBaTeJIbHO, Ig 10 5= 1- (; = -g-;
1 V - 2 I 5 2 6-a 6-a
g 25=- g =-.-=-
3 3 6 9'
3. 11.aHo: a 2 + b 2 = 7 ab.
.lJ. 0 K a 3 8 T b:
a+ b 1
Ig  = 2' (lga + 19b).
4. .lJ.oKa3aTb, tlTO eCJIli a H b - .lI.JIIIHbI K8TeTOB. 8 c-
.lI.JJJUla mnOTeHY3bI npHMoyroJIbHOrO Tpeyro.'lbHHKa, TO
logb+ca + loga_ba = 2 Jogb+ea .loge_ba.
l' K a 3 a H II e. npIIBe.lI.HTe JIOrapmpMbI K OCHOBaHli1O
(b + c).
5. .lJ.oKa3aTb, tlTO eCJIII
logk x , logm x , !ognX
06pa3YIOT apII<pMeTIIIlecKYIO nporpeCCIIIO, TO
n 2 = (kn)log k m.
Y K a 3 a H H e. no YCJIOBHIO 3a.lI.atJlI
. logm x I IOmx
'7 I06m k ' ogm X ' logm n
II .lI.aJTE'e IICnOJIb3YHTe CBOHCTBO tlJ1eHOB apmlJMeTIItlecKoA
IIporpeCCIIII.
Collm3M
.lJ.ea 60JIbWe qeTbIpeX. OtleBII.n;HO. qTO
1 I
T > 16'
linn
(+r>(-r
267

JIoraplI<pMHpY5l, IIMeeM:
21g ( ) > 4 19 ( + ) .
lIeJ1H Ha )g( -}), 1I0JJyqaeM
f.lI.e ollmoKa?
2>4.
JIHIEPAIYPA
A 6 e J! b C 0 H, PmKAeHlle JIOrapll<pMoB, focTexll311aT, 1948.
A. H. Map K Y III e B II 'I, PHllbl, focrexll3AaT, 1961.
A. 51. f IJ P III B a JI b A, HCTOpllH OTKpblTllfl J!orapll<pMOB, 1J3lA.
XapbKOBCKoro YH-Ta, 1952.
51. B. Y C n e H C K II ii, OlJepK HCTOpllH J!oraplJ<pMOB, neTpo-
rr) aA. 1923.
. B. n. ill e p e MeT e B C K II ii, OlJepKll no HCropHIJ M8TeMaTHKll,
YIJnellrH3, 1940.
11. B. A p H 0 J) b A, JIorapH<pMbl B Kypce 9.ijeMeHTapHoii aJIre6pbl,
H3)J.. AnH PC<1>CP, 1949.
r. f. U. e ii T e H, I1cTopHH M8TeMaTHKIJ B XVI-XVll EB., fTll1,
1961.
51. 11. n e p e JI b M a H, 3aHliMaTeJlbHaH aJlre6pa, f111.1 JI,
1949.
M. r. P e ii H 6 e p r, LIYMalOlltlJe MalllIJHbl, LIerrH3, 1957.
H. A. A p x a H r e JI b C K II ii IJ D. 11. 3 a ii u e B, ABToMaTIJ-
lJeCKlle Ull<ppoBble M8Il1HHbl, <1>1J3M8Trll3, 1958.
H. D 0 6 P H H C K II ii, B. n eKe JI H C, DbICTpee MbICJlH,
1J3J],. «MOJlOll8H rBapJl,IJH», 1959.

o
rJlABAXI
HOMOfPAMMbl 113 BblPABHEHHblX T01.J.EK
C nAPAJ1J1EJ1bHbIMI1 UJKAJ1AMI1
«PaHD nJlJl n03JjHO BCHKaH npaBJlJIb-
HaH MareMaTHlJeCKaH HAeH HaXOAHJIa
npHMeHeHHe B TOM IIJ1H HHOM JJ.eJIe».
A. H. /{pblAOB
1.J.TO JKe TaKOe HOMorpa1JnSl?
O IJ:eHb IJ8CTO I1HeHepy, H8YIJ:HOMY pa60THIIKY,
3KOHMIICTY' TeXHIIKY, M8CTepy npH p8JIII'lHbI
paC'leT8X npIIXO.lI.IITCH nOJIb30B8TbCH O.lI.HOli II TOII
me <popMYJIoil, onpe.n.eJI5I5I II3 Bee 3HaIleHII5I Ka-
KOH-JIIIOO nepeMeHHot'j no .n;8HHbIM 3H8'leHII5IM .n.PYfIIX nepe-
MeHHbIX H nap8MeTpOB. ECJIII :JTa <POPMYJI8 60JIee HJIH Me-
Hee CJIO.>KH8H, TO BbI'lHCJIeHIIH no Heil OTHI1M8IOT MHoro Bpe-
MeHH H O'leHb YTOMJI5IIOT pa60THHK8. P83.lI.eJl M8TeM8THKH,
H83bIB8eMblH HOMOrp8<pHei1,,IJ,aeT_B03MO>K-HOCTb 06JI,er'lIITb TPY.lI.
BbI '1 IICJI IITeJI 51 , npe.n.JI8r8H eMY IIC1I0.nb30B8Tb .lI.JI5I BbI'lIICJIeH!lM
oc06ble IJ:epTeH-HO\>1OrpaMMbI, C nOMOlll:bIO KOTOpblX MOHO
ObICTpO II C .n;OCTaTQt{HOH .n;JI5I ,lIpaKTllKH TO'lHOCT,bIO c.n;eJIaTb
Heo6xo,AHMblH DQAC'leT.
269

--:
m:'. .
....:: <
H.
}..
>.,.<-'
\ ::.:
S......
", .'\ '.;'"
*',
,
f.:..
....'.
r
:. ...:s-.
.,
,,'Ii
'-",,7
..
.;1f: .
it,./
racnap MOH>K.
CJJOBO «HOMOrpa<pIIH» B lIepeBOAe Ha PYCCKIiH H3bIK 3Ha-
'lHT «rpa<pH'leCKOe H306pa}KeHIIe 38KOH8», a onpe.lI.eJ1eHHe
HOMorpa<pHH MO}KHO J{aTb TaKoe: HOMorpa<pH5I eCTb YGeHlIe
o MeTO,lJ,ax rp8<pIIGeCKoro H306p8.>KeHH5I <PYHKL(IIOHaJIbHOH 3a-
BIiCHMOCTH Me.>K.lI.Y HeCKOJIbKHMH _nepeMeHHbIMII BeJIII'lHHaMIi.
I1cnOJIb30B8HHe rpa<pH'lecKIiX IIplieMoB1J.JI5I perneHlI5I Bon-
pacOB npIIKJI8,1J,HOH MaTeMaTIiKH 1I0 BpeMeHII BOCXO.lI.IIT K JJ.e-
KapTY, nOCKOJIbKY MeTO,lJ, KOOP.lI.HHaT 1I03BOJI5IeT CTpOHTb rpa-
<pIIK <PYHKL(HII O.n;HOH He3aBliCHMOH nepeMeHHOH. TaKaH KplI:
Ba51 c06CTBeHHO H eCTb O,lJ,Ha 1I3 lIepBbIX HOMorpaMM.
f103}Ke rpa4>H'lCKHe npHeMbI.lI.JIfi peweHIIH npaKTlItJeCKHX
BonpocoB IIpHMeHH.II <ppaHL(Y3CKHf% M8TeMaTlIK II IIH}KeHep
racnap M 0 H}K (1746-1818),' 3H8MeHHTbIA TBopeIl Ha'lep-
TaTeJJbHOH: reoMeTpHH H O.n;HH H3 .n.eHTeJIeA 4>paHL(Y3cKOH
peBOJIIOL(HH. MOH}K OK83aJJ pewalOmee BJlIISlHlIe H8 nOCT8-
HOBKY TeXHH'lecKoro o6pa30BaHHH IIPH opraHH3aIlHlI nOJIH-
TexHlItJecKoA WKOJIbl - npo06pa38 TeXHIIt.lecKHX Y'Ui.lllIm
XIX e.
Tpy.l1bI MOIDKa no.nO}KIIJIflHaqaJIO pa60TaM B 06JIaCTIi HO-
Morpa4>HH, XOTH B TO BpeMH HOMOr-pa<pIIH KaK cneuHaJIbHoro
p83.lI.eJJa MaTeMaTIIKn llie He CYmecTBOBa.fIO. TaK, MOH}K BCBoat
270

C04IIHeHIIII «rpa<pIIIleCKa51 aJIre6pa», BbIlIYI.I.J.eHHOM B CBeT B
1795 r., BnepBbIe .n;ab HecKOJIbKO cnoc060B rpa<pII4ecKoro
II306pa>KeHII5I ypaBHeHIIH C TpeM5I HeII3BeCTHbIMII.
B Te4eHIIe CJIe.n;YIOI.I.J.HX TpeX-4eTbIpeX .n;ecHTHJIeTIIH rpa-
cpII4ecKliH MeTO.n; Tpy.n;aMH raycca, Me6IIyca Ii .n;pyfHx 6bIJI
YKpenJIeH, 0606111eH H ynpcmeH HaCTOJIbKO, 4TO, Kor.n;a B co-
pOKOBbIX ro.n;ax npowJIoro CTOJIeTH5I B 3noxy CTpOIITeJIbCTBa
nepBbIx >KeJIe3HblX .n;opor BO <I>paHUIiIi HBHJIaCb nOTpe6HocTb
B C03.n;aHHII HalinpocTei-iwHx cnoc06oB pac4eTa 3eMJI5IHbIX pa-
60T, TaKIie cnoc06bl 6blJII-J C03.n;aHbI. f1epBbIe 3a.n;a4II no.n;06Horo
po.n;a 6blJIH nOCTaBJIeHbI II peweHbI cppaHuY3cKIiMl1 HH>KeHepaMII
JI a JI a H 0 M Ii 11 a B e H 0 M, KOTopbIe II nOJIO>KHJIIi Ha-
4aJIO HOMorpacpIIIi. OC060H 3aCJIyroH 3TIIX .n;BYX I-JH>KeHepOB
HY>KHO C4IiTaTb BbI.n;BII>KeHIie IIMH II.n;eII aHaMop<p03bI rpacpII-
KOB, T. e. npe06pa30BaHHH KpliBOJIIIHeHHbIx rpa<pHKoB B
np5lMOJIIIHeHHble Ii IIHor.n;a KpyroBble.
MeTo.n;bI JIaJIaHa H 11aBeHa 6bIJIli pa3BIITbI H ycoBepweH-
CTBOBaHbI M 0 C COli JI a JI JI e M a H.n; 0 M, H, HaKOHeL(,
o K a H b .n;aJI 06mylO TeopHIO II KJIaCcHcpIIKaUHIO HOMorpaMM.
B 1890 r. Ha Me>K.lIYHapollHoM MaTeMaTl14eCKOM KOHrpecce
B f1apII>Ke OKaHb C006111I-JJI Kpyr 3a.n;a4 3Toro HOBoro pa3.n;eJIa
MaTeMaTliKII H npe.n;JIO>KIIJI caMO Ha3BaHlie «HOMorpa<pIIH».
B HaCT05l111ee BpeM5I HOMorpa<pHH WlipOKO HCnOJIb3yeTc5I
BO MHOfHX npaKTH4eCKHX HaYKax (TeopHli pe3aHII5I MeTaJIJIOB,
3JIeKTpOTeXHIIKe. npIIKJIa.n;HOH MexaHHKe, 3KOHOMliKe II .n;p.).
3HaKOMCTBO C HOMorpaMMaMH, yMeHIIe IIX CTpOHTb H IICnOJIb-
30BaTb .lI.JI5I nO.n;C4eTOB Heo6xo.n;HMO Ka>K.n;OMY npaKTH4eCKOMY
pa60THIIKY.
CIJHTalOll\He IJepTe)l{H
t.h06bI nOIlYBCTBOBaTb Te npeIIMYI.I.J.eCTBa, KOTopbIe .n;aeT
HOMorpaMMa IIO cpaBHeHHIO C .n;PYfHMII nplieMaMH nO.n;CIleTa
no .n;aHHOH <popMyJIe, paCCMOTpliM HeCKOJIbKO HOMorpaMM.
OepBa51 II3 HIIX nOCTpoeHa .n;JI5I <POPMYJIbl
1 1 1
z=-X+-y,
lIaCTO YlIOTpe6JI5IeMOH B OnTIIKe.BoT 3Ta HbMorpaMMa (4epT. 119),
f1YCTb He06xo.n;I1MO onpe.n;eJIHTb 3Ha4eHHe z npII x=6 Ii
y=4. AJIre6pal14eCKII MbI nocTynHJIII 6bl TalC
x+y . z= = 6 1 ' 0 4 = 2,4.
z xy' x + y
271


ci


,..;
'" I'..J c.
OJ
::r

Qro6bI nOJIYlIHTb 3TO qIICJIeHHOe 3HaqeHHe Z C nOMOmblO HO-
MOrpaMMbI, .n;OCTaTOqHO npHJImKHTb JIHHei'IKY K OTMeTKaM
6 H 4 Ha OC5IX X H Y H B TOlIKe nepeCeqeHH5I JIHHeHKIi co llIKa-
JIOH Z npOqeCTb TOT .>Ke pe3YJIbTaT 2,4. BbIro.n;a nOJIb30BaHH5I
HOMorpaMMoH CTaHeT eme 60JIee 51CHOH, eCJIH x Ii Y npliHIIMalOT
.n;p06HbIe 3HaQeHH5I, HanpHMep x=3,5 H y=2,75. f1poBepbTe
3TO, HaH.n;51 3HaqeHHe z npH .n;aHHbIX 3HalIeHH5IX x H y .n;BYM5I
cnoc06aMIi: B.bIQHCJIeHHeM no <popMYJIe H c nOMOll\blO HOMO-
rpaMMbI.
KaK .>Ke nOCTpoeHa 3Ta HOMorpaMMa H KaKOBO TeopeTH-
qeCKOe 060CHOBaHHe 3Toro nocTpOeHH5I?
n 0 C T P 0 e H H e.
I. CTpOHM OCb Z H Ha HeH IlIKaJIY B MaCllITa6e 10 .M.M 3a e.n;H-
u 10 MM )
HliL(Y (MaCIlITa6 B .n;aJIbHelillieM 6y.n;eM nl-!CaTb TaK: m z = -1- .
2. f1pOBO.n;HM OCH X H Y TaK, qT06bI OHH BbIXO.n;HJIH liS
HYJIeBOH TOqKII OCH Z H 06pa30BbIBaJIH C OCblO Z yrJIbI B 60 0 .
Ha 3TI-!X OC5IX CTpOHM IlIKaJIbI B TOM .>Ke MaCllITa6e, QTO H IlIKa.
JIa Z,
10 MM
m x = my = m z = ---r- .
TeopeTHlIeCKoe 060CHOBaHI-!e nOCTpOeHH5I.
Ha OC5IX X II Y 6epeM npOl-!3BOJIbHbIe TOIlKI-! A I-! B C OT-
MeTKaMH Xl II !/!. Coe.n;IIH5IeM 3TH TOqKI-! np5lMOH AB. nOJIY-
ql-!M TpeyrOJlbHI-!K ABC. ,lI.oKameM, lITO TOqKa nepeCeqeHH5I
np5lMOH AB C OCblO Z HMeeT TaKYIO OTMeTKY Zl, KOTOpa51 CB5I-
3aHa C OTMeTKaMH Xl I-! YI <POPMYJIOH
...!..=...!..+1
ZJ Xl YI
nJIOma.n;b TpeyrOJIbHliKa ABC CKJIa.n;bIBaeTC5I H3 nJIOma-
,lJ,eH ,lJ,BY.x TpeyrOJIbHI-!KOB:
nJI. 4ABC=nJI. .6ADC+nJI. l;.BCD.
AC.BC sin 120 0 = AC.CD sin 60° + BC.CD sin 60°;
2 2 2
TaK KaK
sin 120° = sin 60°,
TO
AC.BC=AC.CD + BC.CD.
HO,lJ,JIHHa OTpe3Ka AC 1I3Mep5leTC5I npOH3Be.n;eHl-!eM BbIopaH-
Horo HaMIl ,lJ,JI5I OCI-! X MaclliTa6a m x Ha OTMeTKY Xl' npOCTaB-
273

JIeHHYIO B KOHL(e A OTpe3Ka CA; TOliHO TaK JKe BC= myYl
1-1 CD=mzz\.
CJle.n;oBaTeJIbHO,
m...xI' myy\ = m...xl' mZz I + myYl' mzzl'
TaK KaK B .n;aHHOM CJIYlIae
m x = my = m z ,
TO
rj
10
9
8
7
6
5
II
.3
X\Y\ = X1Z 1 + y1Zl"
V," V.
BOO
600 20b
7100
100
80
200 60
40
100
80
60 20 .---
40 ..--...-.........--
..- 30 - 10
.--- B
20 6
II
10
B
6 2
"
2
2,
4epT. 120.
!I
10
9
8
7
5

--.-
"
.3
2
,
=+.
Zl Xl Yl
.ileJIJI 06e qaCTII nOCJIe.n;Hero paBeHCTBa Ha npOH3Fe,lJ,eHlIe
XlY1Zl' nOJIYIlHM
CJIe.lI.YlOmlie TPH HOMorpaMMbI MbI .n;aeM nOKa 6e3 Teope-
T1-IqeCKOrO 060CHOBaHH5I HX nOCTpOeHII5I. 3TlI HOBble HOMorpaM-
274

MbI BMeCTe C nepBoH (liepT. 119) 1I0KamYT qllTaTemO, Ha-
CKOJIbKO WlipOKO nOJIe IICnOJIb30BaHIl51 HOMOrpaMM Boo6me
II HaCKOJIbKO OHH pa3Ho06pa3HbI no <popMe.
O.n;Ha II3 HHX (liepT. 120) nOCTpoeHa .lI.JI5I nO.lI.ClIeTa 06'beM':JB
LlIIJIHH.lI.pa H KOHyca H, CJIe.n;OBaTeJIbHO, cooTBeTcTByeT <pop-
MYJIaM
r..d 2 h 7:d 2 h
[!q=; tJ1(='
r.n;e d - ,lJ,HaMeTp L(HJIHH.n;pa llJIli KOHyca H h - BbICOTa 3THX
TeJI.
to
o
7
J
2
9
8
6
r

_7Jl---
air
! :
7
5
4
3.
2
8
9
o
10
a
4epT. 121.
lI.pyra51 HOMorpaMMa (qepT. 121) .n;aeT B03MO>KHOCTb pe.
waTb ypaBHeHHe
ax
r= lOO-x '
C KOTOpblM IIpHXO,lJ,lITC5I HMeTb ,lJ,eJIO npH pa60Te C MOCTHKOM
Y HTCTOHa.
Pewa51 3TO ypaBHeHHe OTHOCHTeJJbHO x, IIOJIyqHM:
100r
X = a+r .
275

11, HaKOHeL(, TpeTb51 HOMOrpaMM8 {qepT. 122) pewaer KBa,lJ,par-
Hoe ypaBHeHlIe
x'+px+q """ 0
B TOM CJJyqae, eCJIIi XOT5I ObI O.n;liH H3 KopHeH 1I0JIO.>KIITe.neH.
nOp5l.n;OK HaXO}!{.n;eHII5I HeH3BeCTHOH HJIH HeII3BecTHbIX Ha
Ka}!{,IJ,OH H3 3THX TpeX HOMorpaMM TaKOH .>Ke, KaK H Ha lIepBOH
HOMorpaMMe (qepT. 120). K OTMeTKaM Ha .n;BYX lliKaJIaX npH-
p q
6 I/i
5 
4 l;
3 .3
2 2
.;'
","'1
'"
0 .;' 0
- 1 0.5 - 1
/'
.2 /'o - 2
3.0 - 3
4.0
- I;
.5.0
,5 6.0 - 5
-6 - 6
-7 8.0 - 7
-8 9.0 -8
-9 1QO -9
'10 fI,O -to
qepT. 122.
KJIa.lI.blBaeM JIHHefiKY Ii B TO'lKe nepeceqemul ee C TpeTbeH
UlKaJIOH qllTaeM OTBeT.
TaK Ha HOMorpaMMe OO'beMOB L(HJJHHJI.pa II KOHyca H8H,IJ,e-
HbI 3HaqeHH5I 06'beMOB 3THX TeJI npH 3HaQeHli5lX d=3 Ii h=5.
(JIIIHeHKa, KOTOpa51 II306pa.>KaCTC51 Ha qepTe.>Ke nYHKTIIpHoit
JIIIHlleH. IIpHJIOmeHa K OTMeTKe 4 Ha IllKaJIe d Ii K OTMeTKe 5
Ha IllKaJIe h.) Ha IllKaJIaX vi< II V x B TOqKaX nepeCeIleHH5I IIX
276

c .nHHeHKOH qHTaeM OTBeT: vl{=35,3 H v«= 11,8. C nOMOI.I.J.blO
9TOH .>Ke HOMorpaMMbI H TeM .>Ke npHeMOM MO.>KHO onpe.n;eJIHTb
d no .n;aHHbIM 3HaqeHH5IM v II h HJlH h no .n;aHHbIM v H d.
Ha HOMorpaMMe .lI.JI5I ypaBHeHlI5I
ax
r = lOO-x
HaH.lI.eHO 3HaqeHHe X IIpli r=5 II a=3. 3HaqeHHe X, paBHoe
62,5, MbI npOqTeM B TOqKe nepeCeqeHII5I JIHHeHKll co WKaJIOH
X, eCJIli npHJIOw.M JIHHeHKY K OTMeTKaM 5 H 3 COOTBeTCTBeHHO
Ha llmaJIaX r H a.
HaKOHeL(, Ha HOMorpaMMe ypaBHeHH5I X2+ px+ q=O HaH-
.lI.eHbI KOpHII Xl :::::3,4 H X 2 :::::0,6 ypaBHeHll51 x 2 -4x+2=0.
3.n;eCb JIIIHeHKa npHJIOmeHa K OTMeTKe 4 Ha lliKaJIe p H K OT-
MeTKe 2 Ha WKaJIe q; 3HaqeHl-lH me KopHeH IIpOqTeHLI HaMH Ha
KpHBOJIHHeHHOH lllKaJIe X, B TOqKaX lIepeceqeHH5I ee C JIHHeH-
KOH; ECJIH ypaBHeHIIe IIMeeT TOJIbKO O.n;IIH nOJIOmHTeJIbHbIH
l\OpeHb, TO JIHHeHKa nepeceIleT lllKaJIY X TOJIbKO B O.n;HOH
TOqKe, r.n;e MbI II npOqTeM 3HaqeHl-le 3Tor,O KOpH5I. BTOpOH,
OTpHL(aTeJIbHbIH, KopeHb MbI MomeM HaHTH nYTeM BbI1JHTaHH5I
HaH.n;eHHoro nOJIO.>KHTeJIbHOrO KOpH5I H3 (-p) , TaK KaK H3-
BecTHO, qTO Xl+X2= -po
ECJIH MbI npo.n;eJIaeM HeCKOJIbKO BbIIlIICJIeHIIH C £JOMOI.I.J.blO
HaqepqeHHbIX HOMorpaMM H 3aTeM TO .>Ke caMoe C.lI.E'JIaeM C
IlOMOI.I.J.blO COOTBeTCTBYIOmHX <P0PMYJI, TO, HeCOMHeHHo, OTMe-
TIIM 60JIblllOii BbIllrpbIlil BO BpeMeHH, KOTOpbIH .n;alOT HOMO-
rpaMMbI. B03MO.>KHO, n05lBIITC5I .>KeJIaHlie He TOJIbI\O HayqIITbC5I
nOJIb30BaTbC5I rOTOBbIMli HOMorpaMMaMH (qTO OqeHb npOCTO),
HO H caMHM CTpOHTb HX. 3.n;eCb MbI .n;a.n;IIM npaBIIJIO nOCTpoe-
HH5I HOMorpaMM TOJIbKO o.n;Horo THna, BeCbMa qaCTO BCTpeqaIO-
meroC5I B npaKTHKe. Pe% lI.n;eT 0 HOMorpaMMax H3 BbIpaBHeH-
HbIX TOqeK C napaJIJIeJIbHbIMII IlIKaJIaMH. f1pIIMepoM HOMO-
rpaMMbI TaKoro THna 51BJI5IeTC5I HOMorpaMMa, H306pa.>KeHHaH
Ha qepTeme 120. Ha3BaHHeM «H3 BbIpaBHeHHbIx TOqeK» no.lI.-
qepKHBaeTC5I, qTO peWeHH5I .n;alOT TOqKH, KOTopbIe JIemaT Ha
OAHOH np5lMoH.
nepexo.n;51 K HOMorpaMMaM C napaJIJIeJIbHbIMH lllKaJIaMli.
MbI ,lJ,OJI.>KHbI IIpe.n;BapHTeJIbHO n03HaKOMHTbC5I C IIOH5ITHeM
<PYHKL(HOHaJIbHOH lllKaJIbI, nOH5ITHeM OqeHb BamHblM He TOJIbKO
.n;JI5I HOMorpa<pHll, HO H .lI.JI5I nOHHMaHH5I OCHOB YCTpOHCTBa JIO-
rapH<pMHqeCKOH JIIIHeHKH.
27,7

«I>YHKI..(HOHaJlbHaSi UJKaJIa
nYCTb .n;aHa <PYHKUIIJI nepeMeHHOH x f(x} II COCTaBJIeHa
Ta6JIIIua 4I-JCJIeHHbIX 3Ha4eHIIH 3TOH <PYHKI..(IIII }J;JJ5I .n;aHHbIX 4IIC-
JIeHHbIX 3Ha4eHIIH aprYMeHTa
x I XI I X 2 I Xa I
f(x) I, (Xl) I' (x 2 ) I, (xa) I
B03bMeM OCb II Ha4aJIO OTC4eTa Ha HeH  T04KY A C OTMeTKOH
.M.M
o (4epT. 123). BbI6epeM MaClilTaO m T H B 3TOM MaCUlTaOe OT
HaIlaJIa OTC4eTa A OTJIOJKHM Ha OCII OTpe30K, rpa<pH4ecKli II30-
6pa.>Ka1OIlLIIH 4IICJIeHHOe 3Ha4eHHe f (x,). 1I.JIliHY 3Toro OTpe3Ka
MbI HaH.n;eM nYTeM YMHOJKeHH5I. MaClllTa6a m Ha 3Ha4eHHe
<PYHKUHH npli 3Ha4eHIIH x. paBHOM x" 1. e. .n;JIliHa I, =mt (Xl)'
B KOHue nOCTpoeHHoro OTpe3Ka nOCTaBHM OTMeTKY «Xl»'
/JO E I I '" - j' ;CJ-
-- m/{"'2) --'
mt( "'3) -
4epT. 123.
COOTBeTCTBYIOmYIO 3Ha4eHIIlO aprYMeHTa. TaR JKe CTpOIiM
OTpe3KH .n;JIliHOH 1 2 = mt (X 2 ), Is=mf (x: i ) II T. .lI.. H BCHKHH
pa3 B KOHl..(e nOJIY4eHHoro oTpe3Ka npOCTaBJI5IeM OTMeTKY,
cOBna.n;alOlUYIO C 3Ha4eHIIeM aprYMeHTa.
nOJIY4eHHaH TaKHM 06pa30M illKaJIa H Ha3bIBaeTCH <P Y H K-
I..( II 0 H a JI b H 0 11 ill K a JI 0 H; ee OCHOBHoe CBOHCTBO
TaKOBO:
Jbo6oil ompe30K q;YH.1W,UOHallbHou WKallbl, 63Rmblu om H.a-
'lalla omc'Lema, i!pacpU'LeCKU u306pa:Jl(aem 'lUClleH.HOe 3H.a'LeH.Ue
l/JYH.KU,UU npu 'lUClleH.H.OM 3H.alf£H.UU api!YMeH.ma. npOCma8lleH-
H.OM 6 KOH.u,e 3mOi!O ompe3Ka.
.ilJIliHa I k OTpe3Ka <PYHKUHOHaJIbHOH UIKaJIbI. B KOHl..(e KO-
TOporo npOCTaBJIeHa OTMeTKa X k . paBHa mt (x k ):
lk=mf (x,J.
278

JIJIlIH8 me OTpe3K8 d <PYHKL(llOHaJIbHOif UIKaJIbI, 3aKJIIOIleH-
Horo MeJK.n;y .n;BYMH TOlJI<aMli C OTMeTKaMII XI II X k , OlJeBI1.l.1.HO.
paBH5IeTC5I m It (xk)-f (Xi) I:
d=m [f (Xk)-f (Xl) J.
Ynpa:JIClteItUJl.
I. nOCTpOHTb <PYHKL(IIOHaJIbHYIO lliKaJIY .l{JI5I <PYHI\L(HI-I
f (X}=X2.
.lJ.,n51 nocTpOeHH5I lli.Ka.'1bI Bbl6epeM MaclliTa6 m; nycTb
5 MM
m = -r- ; Tor.lI.a <PYHKUHOHaJIbHa51 illKaJIa .lI..n51 tPYHKUliH
f(x) = x 2 6Y.lleT HMeTb BII.ll (qepT. 124),
2. f10CTpOIITb <PYHI<L(IIOHaJIbHYIO lliKaJIY .n;JI5I <PY HI<UI-Jll
24
f(x) = -.
x
0 ' , 2 .3 "
, 1 I I
4epT. 124.
5
I
COCTaBIiM Ta6JIHL(Y:
x 11 1 2 1 3 1 4 1 5 1 6 I 7 1 8 I 9 1 10 I. . 'I' ..1 100
I(X)124112181614.81413.41312,712.41...I...lo.24
5 AiM
LI..nJI nOCTpOeHII5I BbI6epeM MaClllTa6 m = ---r-. nOJIyqIIM
IllKa.ny (qepT. 125).
, 6 oS 4
!
2
,
4epT. 125.
3. nOCTpOIITb <!>YHKL(HOHaJIbHYIO IllKaJIY ,lJ,JI5I <PYHKL(HII
f lX) = IglOx.
279

.LI.JI5I IIOCTpOeHlI5I IlIKaJIbI HCnOJIb3yeM TaOJIHL(Y:
I x I Igx I x I Igx I x I Igx
1,0 ° 2,0 0,30 5,5 0,74
I ,1 0,04 2,2 0,34 6,0 0,78
1,2 0,08 2,4 0,38 6,5 0,813
1,3 0,11 2,6 0,42 7,0 0,845
1,4 0,15 2,8 0,45 7,5 0,875
1,5 0.18 3,0 0,48 7,0 0,983
1,6 0,20 3,5 0,54 8,5 0,929
1,7 0,23 4,0 0,60 9,0 0,954
1,8 0,26 4,5 0.65 9,5 0,977
1,9 0,29 5,0 0,70 10,0 1,00
E 100 MM
CJIII BbI6paTb MacllITao m - 1 ,TO Tpe6yeMaSi
llIKaJIa 6Y.lI.eT HMeTb TaKOr! BH.n;, KaK Ha lIepTe)J(e 126. By.n;eM
nOMHIITb, IlTO JIJ060H OTpe30K JIOrapII<pMIIlIecKoH lliKaJIbI, B351-
?
.
I> II f> 1 e 9 II]
JO O
:/0
lJepT. 126.
TbIiI OT OTMeTKH «1», rpa<pIItJecKIi B olIpe.lI.eJIeHHOM MacllITa6e
1I306pa.>KaeT JIOrapII<pM Toro lIHCJIa, KOTopoe npOCTaBJIeHO Ha
lllKaJIe B K OHL(e ,noro OTpe3Ka I).
1) B YKa3aHHOM MaCWTaoe peKOMeHD.yeTclI 'IHTaTeJlIO caMOCTOHTem,.
HO nOCTpOMTb IlIKaJlY. Ha '1epTe>Ke 126 MaCIlJTa6 YMeHbllleH. M3MeHeHHe
MaclliTaoa HMeeTClI H Ha D.pyrHx '1epTe>KaX 3TOH rJlaBbi. JIorapmpMIt-
280
lJepT. 127.

B ,lJ,aJIbHeHllleM ,lJ,JI5I nOCTpOeHIl51 HOMorpaMM HaM 6Y,IJ,YT
He06xo,lJ,lIMbI JIOrapll<pMIIIleCKlie lllKaJIbI B pa3JIH'lHbIX Mac-
wTa6ax ('lepT. 127 a, 6).
OCHOBHble OPMYJIbI AJISI nocTpoeHHSI HOMorpaMM
C napaJIJIeJIbHblMH OCSlMH H paBHoMep HblMH IllKaJIaMH
B OCHOBY lIocTpOeHlI5I HOMorpaMM C lIapaJIJIeJIbHbIMIi lllKa-
JIaMIi 1I0JIO.>KeHbI .n;Be <POPMYJIbI, 113 KOTOpbIX o.n;Ha rOBopliT
o B3alIMHOM paCIIOJIO.>KeHHll lIapaJIJIeJIbHbIX oceH C HaHedH-
HbIMIi Ha HIiX lliKaJIaMll, a ,lJ,Pyra51 - 0 BbI60pe MaclllTa6a
Ha O,lJ,HOM 1I3 HIiX.
BbIBe,lJ,eM 3TH <POPMYJIbI.
D.aHbI TPIi lIapaJIJIeJIbHbIe OCIi X, Y II Z ('lepT. 128). Ha
HHX HaHeceHbI paBHoMepHbIe lliKaJIbI (<PYHKL(IlOHaJIbHbIe lllKaJIbl
.lI.JI5I <PYHKL(1l1l f (x) =x). MaClllTa6bI 3TIIX lllKaJI COOTBeTCTBel{
HO: m x ' my II m z .
f1YCTb OCb Z .n;eJIIiT paCCT05lHlIe Me.>K,IJ,Y OC5IMII X II Y
TaK, tITO OTHOllleHlle
AC
CB =k,
nplIqeM IIp5lMa51 AB nepneH.n;IlKyJI5IpHa R HanpaBJIeHHIO oceH.
Coe)J.IIHIIM TO'lKY D OCIi X C roIlKOH E OCIi Y. f1YCTb B TO'lKe
D IIpOCTaBJIeHa OTMeTKa Xl lllKaJIbI X II B TO'lRe E - OT-
MeTRa Yl lllRaJIbI Y. f1p5lMa51 DE lIepeCe'leT OCb Z B HeKOTO-
pOH TOqKe F. f1YCTb B T01JKe F CTOIIT OTMeTKa Zl lliKaJIbI Z.
BbI51CHIIM, KaKOH <POPMYJIOH CB5I3aHbI 'lIlCJJa Xl> Yl H Zl'
ECJIIi lliKaJIa Ha OCIi X 1I0CTpoeHa B MaClllTa6e m x II TO'lKa
A 51BJI5IeTC5I Ha'laJIOM OTC'leTa Ha 3TOM lllKaJIe, TO ,lJ,JIIiHa OT-
pe3Ka AD=m.xXl' ECJJll MaClllTa6 lllRaJIbI Y=m y , TO ,lJ,JIIiHa
OTpe3Ka BE=myYlo TOIlHO TaK .>Ke CF=mzzlo f1pOBe,lJ,eM
ljeCKHe lliKaJIbI nOJIO>KeHbI H B OCHOBY nOCTpoeHHH JIorapmjJMHljeCKOH
CljeTHoi! JIHHei!KH H HCnOJIb3YIOTCH TaK>Ke npH H3rOrOBJIeHHH JIorapH<p-
'MHl.JeCKHX cerOK (ljepT. 127), Heo6xoAHMbIX AJIH BbInpHMJIeHlI H rpa<pH-
KOB <PYHKUHH BHAa: y=x n H y=a x .
281

JIorapH<pMHlJeCKHe IlIK3JIbI B M3CllITaOax
30 M.II 200 .11M
OT -y-.n;0 1
10
9
8
7
6
5
4
3
2
o
m=
30 40 50 60
70
4epT. 127a.

80 90 100 110 120 1.30 140 150
4epT. 1276.
10
9
8
7
6
5
4-
3
2
283

npHMYfO MN lIapaJI.,'IeJIbHO AB. Tor,lJ,a TpeyroJJbHIIK DFM
lIo,lJ,06eH TpeyrOJIbHHKY EFN H, CJIe,QOBaTeJIbHO,
DM ME AC
EN = NE = CB = k,
IIJIH
mzz.-mxXt
myYt- mZZt
=k.
113 3TOrO paBeHCTBa Bblpa3HM 2 1 Qepe3 Xl H YI'
mzZ l - mxx 1 = kmYY l - kmz21' m z (1 + k) 2 1 = m xXI + kmYYI
II
m x 1 , my k
2 1 =-' -'Xl+-' -Ylo
m z 1 + k m z I + k
(1)
14 c
4epT. 128.
v TaK KaK BeJIH'lHHbl mx' my,
m z H k - ,lJ,aHHble, TO Mbl
MO.>KeM 1I0JIO}!{HTb
m x I (2)
m z . 1 + k = a,
my k (3)
-.--b
m;: I +k - .
B Tor.n;a paBeHCTBO (I) IIpH-
MeT BH.n;:
2 1 =ax I + bYI'
:c z
"I 1J
f10JIYQHmr <POPMYJIY, KOTOpa51 CB5I3bIBaeT OTMeTKII Xl' YI
H 2 1 , paCIIOJIO.>KeHHbIe Ha OC5IX X, Y H Z H Ha ceKYmeH IIp5l-
MOH DE.
B03bMeM IlaCTHbIH cnyqai'I {'IepT. 129). f1YCTb Ha 'IepTe.>Ke
5 .MAl AC 5
m x = my = m z = ---r- II CB = k = T'
Tor.n;a C 1I0MOIIJ.bfO .popMyn (2) H (3) MbI Hai'r.n;eM, 'ITO a= ;2
5
H b=12'
II caM IlepTe.>K IIpeBpaTHTC5I B HOMorpaMMY ,lJ,JI5I <POPMYJIbl
7 5
z = 12 X + 12 Y,
284

eCJIH HYJIeBbIe OTMeTKH, Ha OC5IX X, Y II Z paCnOJIO.>Ket-Ibl H3
O,lJ,HOH np5lMOH. 3Ta HOMorpaMMa n03BOJIHT no .n;aHHbIM 3Ha-
qeHH5IM x H Y HaHTH COOTBeTCTBYIOlllee 3HaIleHHe z. D.JI5I 3TOro
npHlliJIOCb 6bI K OTMeTKaM x H Y Ha OC5IX X H Y npH.l0.>KHTb
JIHHeHKY H B TOqKe nepeCeqeHH5I mlHeHKH co UIKaJIOH Z IIpO-
qeCTb oTMeTKy Z 3TOH lliKaJIbl.
K
10
9
6
7
6
5
<-
3
2
0
z
0---
y
10
9
3
6
5
"
J
2
0
4epT. 129.
HRlIpliMep, eCJIIi BbI 1I0JIO.>KHTe x=8 II Y= 1, TO ,lJ,JI5I Z HO-
MorpaMMa .n;aCT 3HaIleHHe, HeMHoro npeBbllliaIOlllee 5 (oJTO IIpe-
BbIrneHlIe MO.>KHO 6bIJIO 6bI OL(eHHTb ,lJ,OCTaTOIlHO TOqHO, ecJIH 6bI
paCCT05lHHe Me.>K.AY OTMeTKaMH 5 H 6 Ha lliKaJIe Z 6bIJIO pa3-
J].eJIeHO Ha 10 paBHbIx qaCTeH). f1o,lJ,cIl(h no <popMYJIe
7 5
z = 12 x + 12 Y
,AaC't B ,lJ,aHHOM CJIYIlae T01JHblH pe3YJIbTaT, HO 6y.n;eT COlIp5l-
.>KeH C 6CJIbrneH 3aTpaToH BpeMeHII:
7.8 5.] 1)6 5 6] ]
z = "12 + "T2 = 12 + 12 = 12 = 5 12 .
PerneHHYIO HaMIl 3a)J,aQY MO.>KHO C<POPMYJIHp0BaTb TaK:
no .n;aHHOH HOMorpaMMe C TpeM5I lIapaJIJIeJIbHbIMH OC5IMli X,
Y II Z II HaHeCeHHblMIi Ha HHX paBHOMepHbIMli lliKaJIaI\fH
nOJIYIlIITb <POPMYJIY, BbIpa.>KRIOIIJ.YIO 3aBlICliMOCTb Me.>K,IJ,Y lIe-
peMeHHbIMIi x, Y II Z.
285

OKa3aJIOCb, 'lTO 3Ta 3aBHCHMOCTb npHHa,lJ,JIemliT K THny
z=ax+by,
npHti napaMeTpbI a H b onpe,lJ,eJI5IIOTCSl H3 <PoPMyn:
a = :: . 1  k . (2)
my k
b = -. - (3)
m z 1 + k .
Ho .n;JIH lIocTpOeHH5I HOMorpaMMbI MbI ,nOJ])KHbI peIllHTb
06paTuylO 3a.lI.a'lY, 3aKJlIOIlaIOmYIOC5I B cJIe.n;YlOmeM.
.QaHa .popMYJIa
z=ax+by.
f10CTpOHTb Hor.lOrpaMMY .n;JI5I paC'leTOB no 3TOH <popMYJIe, T. e.
OlIpe.lI.eJIHTb m x ' my, m z I-J k.
.QJI5I HaXmK.lI.eHH5I YKa3aHHbIx BeJIH'lHH B HaweM paclIo-
p5lmeHHI-J HMelOTC5I TOJIbKO .n;Ba ypaBHeHHSl (2) I-J (3). n03TOMY
113 'leTblpex HeH3BeCTHbIX BeJIHlJHH: m x ' m)" m z H k, .n;Be Be-
JIH'lHHbl BbI6epeM caMH, a OCTaBwHec51 .n;Be HaH.n;eM I13 ypaBHe-
HIIM (2) H (3). 06bIllHO npOH3BOJIbHO BbI6HpalOT MaClliTa6bI
Ha OC5IX X H Y, 1. e. m x II my, I-JCX0.lI.51 TOJIbKO H3 y.n;06CTB lJep-
Tema. Tor.lI.a ypaBHeHH5I (2) H (3) .lI.a.lI.YT JISl k H m z BeJIHIlHHbI
k=
mxb
ma'
y
(p»
mxmy
m z = mxb+mya . (II)
ECJIH B 'laCTHOM CJIy'lae mx=m y ' TO <POPMYJIbl (I) H (II)
npMYT 60JIee npOCTOH BII.n;:
k = : ' (I)
m z = n'; b . (II)
TaKOBa TeopeTHIleCKa51 OCHOBa IIOCTpOeHH5I HOMorpaMM
C TpeM5I lIapaJIJIeJIbHbIMH OC5IMH H paBHOMepHbIMH WKaJIaMIi
Ha HHX.
1) D.JI!! onpeJJ.eJleHH!! k YJJ.o6Hee Bcero ypaBHeHH!! (3) H (2) pa3JJ.e.
JlHTb no '1aCT!!M.
286

Y npa3ICHeHUJI..
I. f10CTpOIITb HOMOrpaMMY <POPMYJIbI CKOpOCTII paBHO-
MepHO-YCKOpeHHOrO ,lJ,BI-J)j{eHII5I
V t = va+at,
.M
ecJIH YCKOpeHHe a=3 .
ceK
20 M.M
BbI6epeM MaCIIITa6bI Ha OC5IX Va II t paBHbIMH ----r-' T. e.
20 .MM
rn'll. =mt =-1-. Tor,lJ,a 1I0 <PoPMYJIaM (I) H (II) nOJIYIJ:IIM:
k 3 20 5 MM
= 1 H m z = 1 + 3 = -r
n 0 C T P 0 e H lie. a} f1pOBO.n;HM .n;Be napaJIJIeJIbHble npSl-
\IbIe, KOTopbIe IIpHMeM 3a OCI-J va H t. H HaHOCHM Ha HHX lllKaJIbI
20 .MM
B MaclllTa6e m'llo = mt = -r (lJ:epT. 130).
6) f1pOBO,lJ,HM OCb VI napaJIJIeJIbHO OC5IM va II t TaK, IlT06bI
OHa pa3.n;eJJHJIa paCCT05lHHe Me.>K.lI.Y OC5IMH Va II t B OTHOllieHHH
3:1.
B} f1pe)j{.n;e lJ:eM CTpOHTb Ha OCH V t lllKaJIY B MaclllTa6e
5 MM 6 u
T ' Heo XO.n;HMO «YJIOBIITb» O.n;HY H3 OTMeTOK ::nOH lllKaJIbI.
flJI5I 3Toro BCnOMHI-JM. IJ:TO OTMeTKU Ha OC5IX Va, t H Vt. y.n;OB-
JIeTBOp5lIOIllHe ypaBHeHl-JfO v t = v a +3t, .n;OJI)j{HbI Haxo.n;HTbC5I
Ha O.n;HOH np5lMoH. va II t .n;aeM npOII3BOJIbHbIe 3HalJeHH5I H3
IJ:HCJIa Tex. KOTopble HMeIOTC5I Ha OC5IX Va H t. II no <popMYJIe
V t = vo+3t HaH.n;eM cooTBeTcTBYIOlllee 3HalJ:HHe Vt. f1YCTb,
HanpHMep. v a =2 11 t= I, Tor.n;a v t =2+3=5. OTMeTKI-J 2,
1 H 5 Ha OC5IX V o ' t I-J V t .n;OJI)j{HbI 6bITb Ha O.n;HOII np5lMoi'r.
f1pHJIO.>KHB JIHHeHKY K OTMeTKaM 2 H I Ha OCHX Va
I-J t. MbI .n;OJJ)j{HbI B TOIJ:Ke nepeCelJeHIIH ee C OCbfO Vt nOCTaBIITb
Ha OCII Vt OTMeTKY 5 lllKaJIbI II OT Hee BBepx Ii BHH3 CTpOHTb
.MM
llIKaJIY B MaclllTa6e 5 To
r} f10CJIe IIOCTpOeHI-J5I HOMorpaMMbI He06xo.n;HMO npOBe-
pIITb ee. JJ.JIH ::noro 3HaIJ:eHI-Je Vb nOJIYIleHHoe Jj,JI5I napbI npo-
H3BOJIbHbIX 3HalJeHI-JH Va H t IIO HOMorpaMMe. CpaBHI-JBaeM
C 3HaIleHHeM Vt' 1I0JIYIJ:eHHbIM .n;JI5I TeX .>Ke 3HaIJ:eHHH Va H t
287

v...!L H
a eeK V t ceK t eel(
 20 5
18
It. 16 4-
14-
3 12 3
1
2
2
o
o
llepT. 130.
no <popMYJIe v t = v o +3t, cOBlIa.n;eHlIe pe3YJIbTaTOB rOBopHT
o IIpaBHJIbHOCTIi lIocTpOeHH5I HOMorpaMMbI.
no HOMorpaMMe (qepT. 130) 3alIOJIHHTe Ta6JIIIL(Y
tlo I 4 I I 2,5 I I 3,5
t I 2 I 17 1 4 ,751 4 I 1,25
Clt I I 4 I I 12 I
2. nOCTpOHTb HOMorpaMMY CJIO.>KeHH5I
z=x+y.
ECJIH B .popMYJIe z=ax+by nOJIO.>KHTb a= b= 1, TO MbI
IIOJIY1JHM HOMorpaMM)' CJJO>KeHII5I (qepT. 131). B 3TOM CJJY'lae
288

k = ...!L = 1 Ii III = m,,< = In x
a z 1+1 2'
T. e. lllKaJI3 Z 6y.n;eT Haxo.n;HTbC5I nocepe.n;lfHe Me}!(.n;y IUKa.
JI3MH X 11 Y, II J\:laCllITa6 ee 6v.n;eT B.n;Boe MeHbllIe MaCllITa60B
Ha OCHX X I-J Y. ..
)/ Z v
6 r' -6
 10 5
t, 8 "
oS 3
2 "
, 2
0 .----_. 0--- 0
4epT, 131.
n p H Me'! a H \I e. B Jl,aJIbHeHllIeM npll "YJIaR.1lllJamll1» OTMeTKH
IUKa,1bl Z Heo6xoAlBIO Rbl6HpaTb TalUle 3 lJa1JelJlH! nepeMeHHblX x H y,
npll KOTOpblX <tJopMYJJa z=ax+by J\ana.la 6bJ AJJ51 Z l.leJIoe 3Ha1JeHlle.
HOMorpaMr-.loii CJIOJKeHH5I MOJKHO nOJIb30BaTbC5I I-J npH BbI-
'-/HTaHIIII 'lIlCeJI. LIJI5I ::noro YMeHbWaeMOe (HanpI-JMep, 6)
6epeM Ha illKaJIe Z, a Bbl t lIlTaer-.lOe (HanpHMep, 4) - Ha illKa-
JIe X, TOr.lI.a pa3HocTb npO'lTeTCH Ha llIKaJIe Y B TO'lKe nepece-
lJeHII5I ee C JII-JHcilimH, npl-JCTaBJIeHHOH K OTl\1eTKaM 6 II 4 Ha
IlIKaJIaX Z 11 X.
Ho MO.>KHO nOCTpoHTb H ClIeL(llaJIbHyIO HOMorpaMMY .lI.JIH
<POPMYJIbl
z=x - y.
Hanl-JWeM 1I0CJIe.n;H10l0 <POPMYJIY B BH.n;e
z=x+(-y),
II Tor.n;a HOMorpaMMa
TOMY .>Ke npIIHlI.I1ny,
TOJIbKO HanpaBeHIIe
HanpaBJIeHlI5IM illKa.a
BbIIlIlTaHII5I MOJKeT 6bITb nOCTpoeHa no
lJTO II HOMorpa:VIMa CJIO}!{eH1l5l, CTOIIT
IlII{aJIbI Y c.n;e.1aTb npOTl'IBOIlOJIO.>KHbIM
X H Z (lJepT. 132).
[) .&.. A. KOJ!OCOD
289

HOMorpaMMbI c napaJIJIeJIbHblMU OCRMU U JIOrapmIJMH<IeCKUMH
llIKaJIaMH
BbI51CHHM, KaKYIO 3aBI-JCIIMOCTb Me>K.AY nepeMeHHbIMH x,
Y H Z 6YJI.eT BbIpa.>KaTb HOMOrpaMMa, H306paJKeHHa51 Ha 4ep-
Te>Ke 132, eCJIH paBHoMepHble lliKaJIbI 3TOH HOMorpaMMbI sa-
x z
5 {J
4
.3
2
. 2
.3 [,
- 4
0 -5 !J
4epT. 132.
MeHHTb JIorapH<pI\1HIleCKHMH (4epT. 133).
3aMeHI-JB paBHOMepHbJe III KaJIbI JIOrapl-J<pMH4ecKHMH, MbI
,lJ,OJI)!{HbI Hy.1eBble OTMeTKII Ha OC5IX X, Y H Z B T04Kax A,
I
v
()
/J
o
[
o
8
YepT. 133.
B H C 3aMeHHTb OTMeTKaMH «1», TaK KaK HYJIIO paBH5IeTC5I
191. f1YCTb I\IacwTa6bl JIOrapl-l<pMIIIleCKHx WKaJI Ha OC5IX X,
Y H Z 6Y.lI.YT COOTBeTCTBeHHO paBHbI m x ' m H mil II OTHO-
AC
llIeHlIe CB = k. Tor.lI.a
290

AD = m x 19XI;
BE = my 19YI; CF = m..IgzI'
113 no.n;06II5I TpeyrOJIbHIIKOB DF M II EF N IIMeeM:
DM MF AC
EN = NF = CB = k,
IIJIII
m z Ig Zl - m x Ig Xl = k
my IgYI-mzlgz l .
113 nOCJIe.n;Hero paBeHCTBa onpe,'1.eJJ5Ie;\1 19 Zl:
mzlgzl-rnxlgxl = kmylgYl-km z Igz l
II
tn r 1 , 11lv k
Ig Zl = m z . 1 + k Ig Xl -I- m z . 1 + k 19 YI'
0603Halla51 nO-npe.>KHeMY
m_<
m z
I
. I+k =a II
m .
m z
k
1 + k = b,
nO.1yqIIM:
19 Zl = a Ig Xl + b 19 YI'
Pe3YJIbTaT IIOTeHllIIpOBaHII5I nOCJIe.n;Hero paBeHCTBa ,lJ,aCT:
Zl = x a l . yb l .
f10JIYlIIiM <POPMYJIY, CB5I3bIBaIOmYIO OTMeTKII X.. YI II Zl'
HaXO.n;51mIIeC5I Ha O.n;HOH np5lMOH DE HOMorpaMMbI C JIora-
pII<pMIIlIeCKHMII lliKaJIaMII. clJ0PMYJIbI .n;JI5I onpe.n;eJIeHII5I k II
rn z OCTaIOTC5I npe.>KHIIMII:
k = mxb
myu
(I)
m = mxmv
z mxb+mya
(II)
Ii npli m x = my
b
k=-
a
(1/)
m x
m z = ---.- b . . .-
a,t-
(11/)
10"
291

l'lTaK, HOMOrpaMMa C napa.TJ.1eJIbHbIMH JIOrapII<pMIIqeCKIIMH
IDKaJIaMH n03BOJI5IeT .n;eJIaTb paCqeT no <popMYJIe
z = x£l!l.
CTpOHM HOMorpaMMbI

1
Igg 
EC-
7
f,['
50
liO
E:
-7
z=xy.
':3
l' HOMorpaMMa YMHO>KeHHJI
7
f-
-6
ECJIll B <popMYJIe
"
z = Xayb
5
j
 nOJIO.>KIITb a = b = I, TO
npe.n;llieCTBYlOma51 1I0MO-
rp3l\1Ma .lI.OJJi-KHa .npe06pa30-

- BaTbC5I B HOMorpaMMY YM-
HO.>KeHH5I
"
z=xy.
BbI6paB AJI5I WKa.1 X H Y
paBHbIe MacuiTa6bI m., =
100 MM
= my = 1 ,MbI no-
4epr. 134.
. k b
JIyqIIM .n;JI5I =- 3HaqeHIIe,
a
paBHoe I, T. e. k = I, II
- mJr _ 50 ..I!.t EC. " II k = 1, Z
,lJ.JI5I m z - 2" -- -j" TO OCb pacno-
JImKeHa Ha paBHOM paCCT05lHllll OT oceH X II Y.
}lJIH «Y JiaBJIIIB3HII51»
X = Y = 1. Tor.n;a z = I;
OC5IX X. Y H Z ,lI.O:DKHbI
(qepT. 134).
100 ..11M
JIorapII<pMHIleCI\He lIJKaJIbI 13 MaClliTa6ax I
6epeM C IlepTe/Kefi 127 a II 127 6.
OTMeTKII Ha lliKaJle Z nOJIO.>KIIM
CJIe)10BaTeJIbHO, OTMeTKII «1» Ha
Haxo.n;IITbC5I Ha O.lI.HOH np5lMOH
50 MAl
H-
I
292

x
2. HOMorpaMMa AeJIeHIHI Z = y'
f1 0 P 51 .lI. 0 I{ no C T po e H II 51. 1. JIorapIl<pl\H1pyeM ,lJ,aH-
HYIO q;OPMYJIY:
19z = 19x-lgy = 19x + (-lgy).
2. BbI6IIpae1 MaclliTa6bl
100 MAl
In" = Iny = ------r- .
.n;JI5I lliKaJI X II Y; nYCTb
3. Onpe.n;eJI5IeM k
tll z ; TaK KaK lal = 1
jbj = 1, TO k = : = 1,
J(
10-
M 9
II f3
7-
6
Z
10
9
8
7
£3
5
4
v
T. e. OCb Z HaXO}1IITC5I no 5
OTHOllJeHHIO K OC5IM X II Y
TlOCepe.n;I-JHe: "
:3
2
2
11l r 50 JUt
m z = '"2 --r-' 3
4. CTpOliM illKaJIbl X II
Y, YIlIITbIBa5l, IlTO Hanp8E-
J1eHIIe lliKaJlbI Y .n;OJ)KIiO 2
6bITb npOTIiBOnOJIO.>KH bIM
J-JanpaB leHII5IM lliKaJI X II Z.
5. f1pOBO.n;IIM OCb
Z (k = 1).
1
0.9
0.8
, 0,7
0,6
0,5
0,1.
. 0,3
3
-4
[)
0,2
6
-7
8
9
10
6. «1' JIaBJmBae\l» Ka-
KYIO-JIII60 OTMeTKY Ha illK8-
JIe Z. f1YCTb x = Y = 3;
Tor.n;a z = 1. f1pllcTaBIIB
JIIIHeHKY K OTl\leTKa\t «3» II «3» Ha illI<a.TJ8X X H
nepeCeIleHII5I ee C OCbIO Z CTaBIIM OTMeTKY «1».
0.1
4epr, 135.
Y, B TOI1Ke
50 MM
7. CTpOliM illKaJIY Z B MacUlTa6e m z = - .
8, f1poBep5IeM HOMorpaMMY (qepT. 135)'.
2W

3. HOi\lOrpaMMa )J,JID onpeAeJIel!HD n,Oll.\aJUI TpeyroJlbHHKa
S = bh
2 .
11 0 P 51.n; 0 K no C T P 0 e H H 51. l. JlorapII<p\1Mpye:\f <pop-
MYJIY:
Igs = 19b + Igh-lg2.
0
1: 3 s
71 30
20
6
5
10
9
L. 8
7
6
J 5
q
3
2
n
2. BbI6IIpaeM MacuJTa6b1
10 .A.151 llil}a.r:l b H It. f1YCTb
9
100 MM
tn b =fIlfz =-.
B
7
:}
3. Onpe.n;eJI5IeM k H ms:
k= = 1 H
a
6
4
nib 50 .111M
tns=-y ----r-
1
4. CTpOinI IllKaJIbI b
H h.
5. f1pOBO.AH\I OCb S
2 (k=l).
6. «Y JI8BJlliBaeM» OTMeT-
KY «1» Ha IllKaJIes. f1pH b= 1
l llh=2S=1.
7. CTpOIII IllKaJIY S B
1 MaCIllTa6e:
I'
L[epT. 136.
50 MM
ms='
8. f1poBep5IeM HOMorpaMMY (IlepT. 136).
4, HOMorpaMMa AnD onpeAeJleHHH nYTH, npOHneHHoro TenoM
npH paBHOMepHO-YCKopeHHOM )J.BH>KeHIIH 6e3 Ha'laJlbHOH CKOpOCTH
at 2
S=2'
B .n;aHHOH <PoPIl1YJIe a - YCKopeHHe B
JI.BII.>KeHII5I B ceK, S -. paCCT05lHlie B .M.
294
j,(
ceK2 '
t - BpeM5I

f1 0 P 51 .n; 0 K n 0 C T P 0 e H Ii 51.
1. JlorapH<pMIIpyeM <POPMYJIY .n;JI5I BbISICHeHl151 HanpaBJIe-
HHH mKaJI:
Ig s = 19 a + 2 19 t - 19 2.
2. BbI611paeM MacwTa6bI .n;JI5I llmaJI a Ii t. f1YCTb
ma = mt =
150.!IIM
1
3. Onpe.n;eJIHeM k II mO'. TaK KaK nOKa3aTe.rJ11 CTcneHeH y
2
a 11 t - I JI 2, TO k=T; c.'1e.n;OBaTeJIbHO, OCb S )J.eJIliT paCCT05l-
HIie Me>K.n;y OC5Ii\1II a H t B OTHomeHIIH 2 : 1. C'-HITa51 OT OCM a:
ma 50 .lJIt ( rh II' ).
mO'= 1+2 =  'FopMYJIa
4. CTPOHl\J llmaJIbl a H t (llepT. 137).
5. f1pOBO.lI.HM OCb S (k=2 : 1).
6. «YJIaBJIHBae!\I» I{aKYIO-JIH60 oTMeTKY lilKaJIbI s. f1YCTb
at 2
a=2 H t=2, Tor.n;a no cpopMYJIe s=2 s=4.
7. OT nOJIYIlcHHOH OTMeTKII «4» lliEaJIbI S CTpOHM lilKa.TJY
50/lUl
B M8ClilTa6e -1-'
8. f1pOI3epHeI HOMOrpaMl\iY. HanpM\Iep, npH a=4 M t=
rh at 2 ..
=4 'popl\1YJIa S = 2 )J.JI5I S )J.aeT
_ 4.4 2 -  -3')
. s - 2 - 2 -.
ECJIII K OTl\1eTKaM «4» II «4» Ha lilKaJIaX a 11 t npI1JIOLKHTb
mlHeHKY, TO B TOIlKe nepeCeIleHH5I ee co lilKaJIOH S MbI npa-
IlTeM TOT )Ke pe3yJIbTaT - IlIlCJJO 32.
5. HonlOrpaMMa }1;.D cJIUPMYJIb)
v=
30
t o ,45. sO,52
B :IToii <lJopMYJie t - rJIy6lIfJa pe3aHII5I CTa.'1II B .MM, s-
295

01'1 :'m tCf!If,
W 2
/0 500 10
9 400 9
300
B B
200
1
6 100 6
90
BO
5 70 -5
60
50
lJO
l, 30 t..
20
3 3
-10
9
8
7
8
5
1 u 2
J
2
I -
I
0.9
0.8
0,7
0,6
0.5
lJep1'. 13i.

nO.lI.aIla B
MM .M
060POT II V-CKOpOCTb pe3aHH5I B MU/-l (IlepT. 13£)
0,5 <.t <. 10;
0,2 <. s<. 5.
t.lepT. 138.
f1 0 P 51}]; 0 K no C T P 0 e H H H.
1. JIorapII<p\mpyeill (POPMYJIY
Igv = - 0,45Igt- 0,521gs -I- Ig30,
2. BbI611paeM M8clliTa6 }];JI5I WKaJI t II S. fIYCTb mt = fns =
100 Ml>!
1
3. Onpe.n.eJI.HeM k II mv'
k = 0,52 = . filt _ 100 _ 103,1 MM
0,45 45' mv= 0,45+0,52 - 0,97 - ----r-
4. CTpOIiM lliKaJIbI t II S (IlepT. 139).
5. f1pOBollIIM OCb V (k = 52 : 45).
Ocb HaXO}];IITC5I Ha paCCT05lHllii 52 AtM OT OCIi t II 45 AtM
OT OCII S.
6. «Y JI8BJIIiBaeM» OTi\!eTKY UlKaJIbI v. f1YCTb t = S = 1;
TOr}];a
V=
30 _ 30.
1°,15.1°,52
103 ,1 MM
7. CTpOHM lliKaJIY v B M8cwTa6e 1
8. f1poBep5le.\I HmmrpaMMY.
f1pll t = 2 II S = 4
tI=
30
2°,40.4°,52 .
29;'

f10JIb3Y5lCb Ta6JIIIL(a:VIlI .10rapII<p,!OB. HaH.lI.eM
v = 10,5.
HO\lOrpaMMa .n;aeT TaKOH .>Ke pe3Y.1bTaT.
CMM V H
0.5
100
0.5 90
80
0.6 70
0.7 60
O.B
0,9 50
MN
oS ooop
o,a
0,.3
0,4
2
ZD
B
 I?
'0
< 9
: B 3
7
t- O 4
 5
6
4epT. 139.
"6. H0:l10rpaMMa )J,JIH pac.Ii!To3 AHaMerpoB BaJIOB 113 .ijliTOro >KeJIe-
3a no c1lOpMY.ije
-:r-
d = 12 V  .
298

B 3TOH <pOpMYJIe d - .n;IIaMeTp Ba.'1a B CAt, N - qIICJIO
JIOllia.n;IIHbIX CHJI, nepe.n;aBae\!bIX BaJIO'>l, n - qIICJIO 060pOTOB
BaJIa B ceK. f10CTpOIITb H01IlOrpaMMY npli YCJIOB.lIli
1 -< N -< 200 II 40 -< n <- 1000.
f10p5l.n;OK nocTpOeHII5I (IlepT. 140).
de" n
NHP HuH
40
200
50
60
roo
90 70
BO
70 80
60 90
O 100
40 9
30
8
20
200
6
fO
9 5
8 300
'I
6
5 " 400
(;
500
3
3 600
2 700
800
900
1000
2
4epT. 140.
1. JIorapII<pMlIpyeM <P O P\lYJIY
1 1
19 d = T 19 N - 4"" 19 n + 19 12.
299

2. BbI(5npaeM MacuITa6 }1,JIH lliKaJI N II n. nYCTb
60 !HM 100 .MM
m N = -r II m n = 1


m b
3. OlIpe}1,eJHIe
l k II ma. IIO <Pop\lYJIe k = -----"-- IIMeeM k =
myu


1
mN.T
1
m n - T


15 3. A-.
= 25 = 5' IIO ,!:,opMYJIe tll z =


m,r-my
no-
mxb+myu


JI YIlIIM


mN-mn
m d = 1
(tnN+mn)- T
4. CTpOIIM lliKaJIbI N II n.
5. npOBO.lJJIM OCb d ( k =
 ) .
5 /


60.100-4
160


150 .I1M
--r--


6. «"YJIaBJIlIBaeM» oTMeTKY lliKaJIbI d. nYCTb N =n =100,
Tor}J;a d = 12_
7. CTp0I1M lliKaJIY d OT nOJIyqeHHOH OTMeTKII «12» B Mac-
6 150 ,11M
UITa e I' .
8_ npOBep5le\1 HO:\lOrpaMMY. }1,aB351 N I-J n KaKIIe-JIII60
3HaIJeHII5I II HaXO}J;51 cooTBeTcTBYIomee 3HaIJeHHe d (MO.>KeT
6bITb, c nOMOmbIO Ta6JIIIL( JIOrapII<p.\10B).


7. HOMorpaMMbI C HeMbIMH IlIKaJIaMH.


HOMorpaMMbI C HeMbIMII lliKaJIaMII CTp05lTC5I }1,.151 <POP\lYJI.
B KOTopbIe BXO.IJ.IIT 60JIee TpeX nepeMeHHbIx BeJIIIIlIIH.
DOCTpOIIM HO.\lOrpaMMY }J;.151 <pOP:VIYJIbI
1 !.!

N = 36 .v.tI5 S 4
r}1,e N - MOIL(HOCTb CTaHKa B H p. v - CKOpOCTb pe3aHII5I B


AI AIM

 UH . S - nO}1,3IJa B _ 6 II t. - rJIy6IIHa pe3aHII5I B .M,U
,n 0 OpOT


lO,5"", t -< 5; 0.2 -< s -< 2: 0,5 -< t -< 5).


300




n 0 P H,A 0 K n 0 C T P 0 e H H H H o !vi 0 r p a :-'1 M bl ('1epT. 14\).
1. npeoopa3yeM cpOPMYJIY
1


N = 36 v. t 15 . S 4
TaK, QTo6bI B Ka>K,AOH 'Ia,CTH 6wIO no )J.Be nepe'vIeHHbIX
36N
 2-
- =t J5 . 8 4
v .


NHP


t,.m z
b
'-: .9
;
0,6
3 0.5
0."


v

5



 o

p
2


I)


7


6


0,8


0. 1


0.6


0.5


20


O.l,


0.8
0,7


0.3


0.6


O.2 + V
t oS


N


:J


0.2


0.5


4epi 141.


2. BBO)J.HM HOBoe nepervreHHoe z, n031araH:
14 3
Z = t 15 . s 4;
36N
z = ----v- .


(l)
(2)


301




3. CTpOHM HO1OrpaMMY 1p0pMY,f1bI. (I):
a} JIOrapmpMHpyeM:
14 3
19z = T5lgt -I- 4 lgs ;
6) BbI6IIpaeM MacIlITa6 Ha IllKaJIaX t H S, nycTb
100 111M
mt = ms = I
B) HaXO)J.HM k H m z :
k _ b _ 3 . 14 _ 45.
- a - "4 . 15 - 56'
111 = 100 _ 100.60  60 M1I1 .
z + -l6l--r-'
15 4
r} CTpOHM HO!\lOrpaMMY, He HaHOCH )J.eJIeHHH Ha OCH z.
4. CTpOIIM HmmrpaMMY cpOpMy.!Jb1 (2), COBMeCTHB IllKaJIY Z
co IllKaJlOH Z nepBoH HmlOrpaMMbI:
a} JIOrapIIcp:\mpyeM:
Igz = IgN -lgv + Ig36;
6) onpe)J.e31HeM MacIllTa6 Ha lUKaJIaX N H v, )J.eJIaH HX
O)J.HHaKOBbI!\m:
mN m/V
60= 1+1 =
120 111M
1
11lN = Ill'll =
B) onpe)J.e.'1HeM nO.TJO>KeHHe IllKaJI N II V OTHOCHTeJIbHO
Y>Ke HMel{)lll.eikSl IllKaJlbl Z:
1
k=-= I
I
Ii CTpOIIM IllKaJIY v, Y4HTbIBaH npoTIIBOnOJIO}!{HOe HanpaBJIe-
H11e ee no orHOIlleHHI{) K IllKaJJe. N;
r) CTpOHM IllKaJIY N, npe}J{,n:e «YJIOBHB» Ha HeH KaKYI{)-
JlH60 oTMeTKY. Hanpl1Mep, eC.TJ11 t = 1, S = I, v = 36, TO
N = 1. Coe)J.HHHB OTMeTK11 «I» 11 «I» Ha IllKaJIaX S 11 t, nOJIY-
Li11M OTMeTKY «1» Ha IllKa.TJe Z. Coe)J.HHHB OTMeTK11 «36» H «l»
. Ha IllKa31aX v H z npH!\:oti, !\:bI B T04Ke nepece4eHHH :iTCH
IlpSlMOH C OCblO N )J.031}!{HbI npocTaBHTb oTMeTKY «I», OT KO-
120 111M
TOpOH 11 CTpOHM IllKa31y N B MacIllTa6e 1
302

)J.) BbllJeplJJ1Bael K31lOlJ )J.J151 nOJlb30BaHHH HO\.JOrpaMMOH.
nYCTb t = 2, S = 0,8; v = 9. Onpe)\e31IITb N. npHKJIa)J.blBaeM
JIIIHeHKY K OTMeTKaM «2» II «0,8» Ha IllKa31aX t H S.
B TOLiKe nepeceLieHHH JIHHeHKH C HeM 011 IllKaJIOH Z Ha
noc31e)J.Hei:i )J.e31ae\1 oTMeTKY KapaH)J.allloM. DpHKJIa)J.bIBaeM
.r1HHeHKY K OnJeTKe 9 H3 IllKa.1Je v H KapaH)\aIllHOH OTMeTKe
Ha He\10H IllKaJIe H B TOlJKe nepeCelJeHH5I .1JIIHeHKH co IllKa-
310H N lJHTae11 Ha IllKaJIe N IICKOMbIl1 OTBeT )J.315I N (N =
= 0,4 HP).
CaMOCTOHTeJIbHOe nocTpoeHHe HOMorpaMM
3a)J.aLiU
nOCTpOHTb Hm-wrpaMMbl JI.JIH cpopNlYJI:
1.
4 r. 2 R
a=72'
f)J.e R - pa)J.HYC B CM, T - nepHO)J. o6parueHHH no oKpym.
CAt
HOCTH B ceK, a - ueHTpOCTpeMIITeJIbHOe YCKopeHHe B ceK 2 .
2.
100
n=sr'
r)J.e s - no,n:a lJ3 B tAf ,t - BpeMH o6pa60TKII 100 MM
o OpOT
)J.eTaJIII H n - lJHCJIO 060pOTOB.
3.
1tR2H
v=,
r)J.e v - 06'beN! KOHyca, H - BbICOTa ero H R - pa)J.IIYC oc-
HOBaHIIH.
4.
6
K = M bh 2 '
r)J.e M - H3rII6alOw.HH I\lmIeHT B KF. CAt, b - WIIpHHa npHMO-
yro.1JbHOrO CelJeHIIH CTep}!{HH, h -- ero BblCOTa B CAt, K - Ha-
Kl
npH}!{eHHe Ha H3rH6 CJ.!2 .
5.
R = -./ V
r 7tH'
r)J.e R - pa)J.HYc UHJIIIHJJ,pa, V - 06'be;\1 UH31HH,n:pa, H - Bbl-
COTa era.
303

Koop,n:HHaTHaH CeTKa TBOpHT «y,n:eCa»
B TeXHliKe BeCbMa 'IaCTO npHXO)J.HTCH BCTpeLiaTbCH C
<l>YHKI.IJieii BH)J.a
y = CXn,
r)J.e C H n - napaMeTpbl. MbI MOrJIH 6bI nOCTpOHTb rpa<l>HK
nO)J.o6Hoti <l>YHKUHH B BH,n:e I{PHBOH Ha paBHOMepHOH KOOP)J.H-
HarHoti ceTKe- H nOJIY4H.I]H 6bI napa6031Y Tora UJIH lIHOra
BH)J.a B 3aBHCHMOCTH OT 3Ha4eHUH C U n. BbI4ep4UBaHue Ta-
Kora rpa<l>lma npH CKOJIbKO-HH6Y)J.b CJIO>KHbIX 3Ha4elUlHX c
H n (HanpHMep, P=88 P,842) )J.OB031bHO TPY)J.HO, )J.a H no)J.-
C4eT no HeMY He )J.aeT )J.OCTaT04HOH TOLiHOCTH.
B 1843 r. J1 a 31 JI a H npe)J.310}!{H31 H)J.elO npe06pa30BaHHH
KpHBOJIHHeHHOro rpa<l>uKa B npHMOJIHlIei1Hblfl, KOTopaH )J.ael
6031bIllOH BblHrpbIlll U BO BpeMeHH Ii B TOLiHOCTH OTc4eToB.
OCTaHoBHMcH Ha TeopeTu4ecKoM 060CHOB3HUIi nO)J.06Horo
npeo6pa30BaHHH rpa<l>HI<OB.
LLaHa <l>opMYJIa
y=cx n ,
310rapHcpMHPyeM ee no OCHOBaHHIO 10:
Ig y=nlg x+ 19 c.
(1)
(2)
ECJIH B paBeHcTBe (2) 3a aprYMeHT npUHHTb He x, a 19 x, a
-3a <PYHKUHIO He y, a 19 y, TO OTHOCI1Te31bHO 9THX nepeMeHHbIX
ypaBHE'lme (2) 6Y)J.eT ypaBHeHHeM I-H CTeneHH. nOCTpOHM na
Koop,n:HHaTHbIx OCHX BMeno paBHoMepHbIx IllKaJI IllK3JIbI JIora-
pH<pMH4ecKHe Ig x H Ig y B O,n:HOM H TOM }!{e MaCIllTa6e H BbI-
lJepTHM coorBeTCTBYIOw.YI{) Koop,n:HHaTHylO (eTKY (lJepT. 142)
(laKHe Koop,n:HHaTHble ceTI<H HMelOTCH B npo)J.a}!{e). Tor)J.a
rpa<pHKoM ypaBHeHHH (2) Ha JIOrapH<pMH4ecKot'I Koop,n:HHaT"
HOt' I ceTI<e )J.OJI}!{Ha 6bITb npHMaH JIHHHH, )J.JIH IwropOH n 6y-
,n:eT yr 310BbIM K03QJ<pHUHeHTOM, a 19 C - OTpe3KOM, OTceKae-
MbIM elO Ha OCH 19 y, C4HlaH OT HalJaJIa Koop)J.HHaT (I, 1).
npouecc npe06pa30BaHHH KpHBOJIHHeHHoro rpa<pHKa B
npHMOJIHHeHHblH J1aJIJIaH Ha3BaJl a HaM 0 p <p 0 3 0 ii.
nOCTpOIiM npHM031HHeHHbIe rpaqmKH <PYHKIl,UH
y=x 2 H y=VX-
(3 " 4)
Ha JIOrapH<pMHqeCKOH ceTKC.
304

n 0 p f! JJ.. 0 K n 0 C T P 0 e H II 5:1 ('IepT. 142).
I) J1orapH<pMl1pyeM <POPMY31bl y=x 2 H y=-VX.
1
Ig y=2 Ig X 11 19 Y= T 19 x. (3' H 4')
2) YrJIOBbIM K03cpcpmUleHroM k npflMbIX 6Y)J.YT )J.JIH nep-
1
iJoro rpaqJlma k=2, )J.JIH BToporo k=2-
OTCYTCTBHe cB060,n:Horo 'l31eHa B cpopr,rYJIax (3' H 4') roo
BOpHT 0 TOM, liTO npHMbIe npoi:'J,1J,yT 'lepe3 Ha'laJIO Koop,n:HHar.
rpa<pHKH nOCTpoeHbI Ha 4epTe)!,e 142.
2
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4epT. 142.
DpH conOCTaBJIeHlIH I{PHBOJIHHeIIHOro rpacpHKa C npSlMO-
31HHeMHbI!\I )J.31H O,n:HOH H TOll }!{e cpYHKUHU (LiepT. 143)
k
y=-X
UP" k=4; 8; 12; 20; 40 II x> I !\IbI BH,n:HM, <iTO nocrpoeHHe
I3hmpH!ltJleHHbIX rpacpHKOB npOlll.e, tHM 1l0CTpOeHHe npSlMO-
I1HUeHlIbiX.
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nOKa3aTeJJbHOH YHKJ1H
y=a X ,
LiaCTO BCTpelJ.alOIl.{eikH B TeXHlIKC. C)J.eJJaeM 3TO J],JTH lJ.aCTHOro
CJIYLiaH, KorJ],a a=2, T. e. )J.JlH YHKJ1H
y=2 x .
DpoJIorapHMHpyeM nOC31e)J.HlOIO OPMYJIY:
Ig y = x 19 2. (5)
ECJlH B paBeHCTBe (5) x ClJ.J1TaTb aprYMeHTOM, a 3a YHI{IUIIO
npJ1HHTb 19 y, TO OTHocmeJlbHO 3TllX nepeMeHHbIX ypaBHe-
HHe (5) 6y,n:eT ypaBHeHJ1eM nepBOIr CTeneHH. Ha OCHX KOOp,n:H-
HaT crpOUM: Ha OCH x paBHOMepHYIO tilKaJlY J1 Ha OCJ1 y JIora-
plIMHlJ.ecKYIO 19 y H coorBeTCTBYIOW.YI{) no.nY:lOrapl1l\IHLie-
CKYIO ceTKY. Ha :HOIr ceTKe rpaqJJ1K CPYHKLlHH 6Y)J.eT npHMOH
JIfIHHeH (lJ.epT. 144). PH)J.OM BbILiepLicH KpJ1BOJIJ1HeHHblt'I rpaJ1K
TOU }!{e YHKLlHH (Maclllra6 Ha OCH x B3HT B)J.Boe MeHbIlle, 4eM
Ha ocn y).
To, liTO BbI Y3HaJlH 0 HOMorpaMMax 113 :)TOH rJIaBbI, HB31HeTCH
He60JIbillOH LiaCTblO ueJIoro pa3)J.e31a MaTeMarHKJ1, Ha3blBae-
Moro HOMOrpaJ1eu. norvlHMO HOMorpaMM C napa.n31eJIbHbIMH
IllKaJIaMJ1. cyruecTByeT MHoro HHbIX BHAOB HOMorpaMM, C KO-
TOpbIMJ1 Bbl MO}!{eTe n03HaKOM11TbCH no YKa3aHHbIM KHHraM.
JII1TEPATYPA
A. A. r J\ a r 0 J\ e B, HOMorpa<p1U1 ,!lJlH IIIKOJlbH1IKa, Y'WeI\fl13,
1958.
B. M. B P a 11. 11 C, CpeI\CTBa 11 cnOCOOb\ 3.eMeHTapHblx Bbl'l1lCJ\e'
111111, Y'InerIl3, 1954.
Ill. B. n e Ii TKO B C K 11 H, C'I1ITalOll1.1Ie '1epTeiKll, [l1TTl1,
1953

rJIABA Xll
o KBA,lJ.PATYPE KPyrA It 4ltCJIE r.
«MHorlIe 113 IOIIX COBpeMeHHIIKOn
HJJII nOC.e)1OIJ<JTe.1ei\, 03HaKm1HBIUHCb
C 3TlIM MeTO;!,OM 1 ), 6YAYT B COCTOHHHH
HaXO,'J,HTb HOBbie TeOpeMbJ, AO KOTOpblX
Heme He AOJJ.YM3JJCH».
ApXIIMeci
1-13 HCTOpHH IHICJJa 7t
a,n:alJY 06 onpe.!I.eJIeHHH OI'HomeHHH )J.JIHHbI OKPY}!{-
HOCI'H K ee )J.HaMeTPY HCI'opHLieCI\H rPY)J.HO OT)J.e-
JIHTb OT 3a)\a'UI «I{Ba)\paTypbI KPyra», I'. e. no-
CTpoeHHSl KBa,n:para, paBHOBeJIHKOrO )\aHHOMY KPY-
ry. B03MmKHO, 'ITO nOCJ1e,IJ,HHH no BpeMeHH B03HHK.'1a paHee
nepBoH, Tal{ KaK rOBopHTb 0 C03HaI'e31bHbIX nOllCKax orHO-
UleHHH lIJlHHbI OKPY}!{HOCI'H K )J.Ha:\IeTpy XOTH 6bI Y )\peBHHX
erHI1I'HH H BaBIWOHHH e,IJ,Ba JIH B03MmKHo, B TO BpeMfi KaK
Bonpoc 0 HaXO}!<)J.eHHH n3101U.aAI1 I{pyra no ero )J.HaMeTpy HBHO
1l0CTaB31eH B HeKOI'opblX 3a,n:aLiax, cO)J.ep}!{aw.HXCSl B )J.peB-
HeHUlHX erHneTCKHX nanHPycax.
3
1). Mew;! He;J.e.HMblX,- A. 1(.
809

npHBe,n:eM TeKCT 3a,n:allII H3 nanHpyca AXMeca (oKono
2000 ,n:o H.3.).
«1I.aHo KpyrJIOe n031e )J.HaMeTpoM 9 Mep. KaKoBo cO)J.ep}!{a-
HHe ero nOBepxHocTH?»
3)\eCb }!{e pH)J.OM )\aeTCSl H peIlleHHe :=nOH 3a)J.a4H. «Bb14TlI
1
H3 )J.HaMeTpa "9 ero, T. e. e)J.H, HHLl.Y OCTaTOK 8 B03bMH BoceMb
pa3. nOJIYLiI1Il1b 64. 310 H 6Y)J.eT nOBepXHOCTb )J.aHHoro n031Sl».
TaKHM 06pa30M, n31ow.a,n:b Kpyra C ,n:HaMeTpOM B 9 Mep
OKa3aJlaCb paBHoH 64 KBa)J.paTHbIM MepaM. B03MO}!{HO, 4TO
peIlleHHe YKa3aHHoH 3a,n:aLiH no BpeMeHI1 HBJIHeTCH O,n:HOH
H3 nepBblX nOnbITOI{ pewHTb 3a)J.a4Y 0 KBa,n:paType I{pyra.
0603Ha I IHB )J.HaMerp Kpyra Liepe3 d, MbI, CJ1e,n:YH 3a ,n:pes-
HeemneTCKHMH MaTeMaTHKa!l1H, nOJIy4HM ,n:31Sl DJIOW.a,n:H Kpyra
cpOPMY31Y
So = (  d ) 2 = d2.
9 81
3TO )J.aeT HaM B03MO}!{HOCTb HaHnI emnerCKoe npH631H}!{eHHe
LiHC31a , ynoTpe631HeMOrO OKOJIO 4000 JIeT Ha3a,n:. ECJIH
nd 2 64 d 2
4 81'
TO
256
 = 8T = 3,1605.
TaKHM 06pa30M, efHnTHHe, He CTaHOBHCb Ha nYTb C03Ha-
reJIbHbIX nOHCKOB OTHOllleHHH ,n:JIHHbI OKPY}!{HOCTH K )J.Ha-
Mrpy, nO)J.OlllJIII K TaKOMY npH6JIH}!{eHHIO LiHCJIa , KOTopoe
pa3HHTCH OT Halilero 06bILiHOro npH6JIH}!{eHHH (3,14) T031bKO
BO BTOpOM ,n:eCHTWIHOM 3HaKe.
HeT COMHeHHH, liTO Bonpoc 0 KBa,n:paType Kpyra ObIJI
pelileH )J.peBHeenmeTCKHMH MaTeMarl!KaMlI 4HCTO OnblTHbIM
nYTeM. MO}!{HO, HanpHMep, npe)J.DOJIO}!{HTb, KaK 3TO .n.eJIalOT
HeKOTopbIe HCTOpHKH MaleMaTHKH, 'Iro Kpyr H KBa)J.paT CO
CTOpOHOH, paBHofl )J.uaMeTpy, efHllTHHe nOKpbIBaJlH cn.1JOIll-
HbIM CJ10eM ceMHH (3epeH) B O,n:HH pH)J.. npOCTOH nO)\CLieT LiHCJIa
ceMHH Ha Ka}!{)J.oH H3 3THX cpHryp cpa3Y y6e}!{,n:aJI npOH3BO)J.H-
W.HX OnbIT, liTO Ha I1JIorua)J.H KBa)J.paTa ceMHH 60JI , bille LieM Ha
n310rua)J.H I{pyra. nOcreneHHo YMeHblllaH CTOPOHY KBa,J,paTa
H Bce BpeMSl nOBTopHfl TOT >Ke OnblT C ceMeHaMH, OHH npHIllJIH
K T8KOMY BbIBO)J.Y. 'ITO 'mCJIO 3epeH Ha nJIorn..a.ll.1l KBa)J.paTa
310

6Y)J.eT COBna)J.aTb C LiHC310M 3epeH Ha nJIOIlI.a)J.H Kpyra TOJIbKO
u 8
B TOM CJIYLiae, eC31H crOpOHY KBa)J.paTa B35:1Tb paBHOH "9 )J.JIH-
HbJ )J.HaMeTpa Kpyra (LiepT. 145) H, CJIe)l.OBaTeJIbHO, B 3TOM CJIY-
'lae n.'101L1.aJ,iI 3THX qmryp 6Y)J.YT paBHbl.
3al1.a'lH 0 J<BaJI.paType Kpyra H BbmpHMJIeHHH OKPY>KHOCTiI,
eCTeCTBeHHO, )J.031}!{HbI 6blJIH 3aHHTepecoBaTb )J.peBHerpe4e-
CIUlX MaTeM3THKOB, MaTeMaTHKOB TOH cTpaHbI, r)\e, c06-
CTBCHHO, H 6b1J1H 3aJIO}!{eHbJ OCHOBbI reoMeTpHH. D..eHcTBIITeJIb-
.d
9
d
. 0
-. w...
'1:J
<oJ",
< ?'
z.
So
4epT. 145.
HO, MlIorHe 113 rpeqeCKHX MaTeMaTI1KOB: A H a K C a I' 0 P
(V B. )J.O H.3.), A H T H <p 0 H (V B. )J.O H.3.), Ll e II H 0 -
C T paT (IV B. )J.O H. 3.), r H n n 0 I{ paT (V B. ,n:o H. 3.),
B pH 3 0 H (V B. )\0 H. 3.) H ,n:pYrHe, CTpeMH31HCb peWHTb
3TH 3a)J.a4H. HeKoTopble H3 HHX y)J.aLiHo cnpaBH31HCb C HHMH,
HCn031b30BaB npH nOCTpOeHH5:lX, KpOMe L1.HPKYJIH H JIHHeIIKH,
11 )J.pYrHe npH60pbI II B:>lecro OKPY}!{HOCTH HHbIe KpHBbIe. Ho
HX peweHHH 6bl31H oTBeprHYTbI WK0J10H DJIaTOHa KaK peIlle-
HHH «MexaHHQeCKHe».
D31aTOH Hero WKO.'1a n03B031S!31H )J.JIS! BblllOJ1HeHI1H reo.
MeTpHLieCKHX nocTpoeHHII 6paTb B PYKH TOJIbKO )J.Ba HHCTPy-
MeHTa: L1.HPKY31b H J1llHeHKY, H OTBepraJIH JII060iI )J.pyroH
.HHCTpYMeHT. TaKoe Tpe60BaHHe npHBeJIO K nOHBJIeHHIO B
reOMeTpHH TaK Ha3bIBaeMbIX «HeB03MO}!{HbIX 3a)J.aLi», T. e.
3a)\atI, KOTopbIe He MorYT 6bITb peweHbI C nOMow.blO TOJIbKO
YKa3aHHbIX npl160pOB. DpHMepaMH TaKHX 3a)J.a'l HBJlHIOTCH
3a)J.aQll 0 TpHCeKL1.I1H yrJIa - )J.e31eHHH yr31a Ha TPH paBHbIe
4aCTH, KBa,n:paType Kpyra H )J.p.
OrpaHHtIeHlI5I, HaKJla.lI.blBaeMhle IlIKOJIOH D31aTOHa Ha BbI-
Oop MexaHH3MOB npH BblnOJIHeHHH reoMeTH4eCKHX nOCTpoe-
311

HHA, HaM Ka>KYTCSI CeH'laC He060CHOB3HHbIMH. MbI B HaCTOH-
w.ee BpeMH nplIBeTcTByeM peweHMC 33JJ,3'lH JIl06b[\!H cpe)J.-
CTBaMH, .'1HWb 6bI 9TO peweHlie 6blJIO np3BlJ.TIbHbIM. Ho eCTh
II HHaH CTopOHa B 3TOM Bonpoce. ECJIH 6bI BO BpeMeHa 11313'
TOHa 6bI310 )1.onYULeHO BBe,n:eHHC npOH3BOJIbHOrO LiUC31a fIHCTPY-
MeHTOB )J.31H pemeHHH 33;J.aLi H3 nOCTpoeHHe, TO lie 6bI.'lH 6b1
npe)J.npHHHTbI MHOflle HCC31e,n:OBaHHH B 06313CTH «HeB03l\IOfKHbIX
3a)J.aLi»; 3TH HCC31e,n:OBaHHH 4aCTO npHBO)J.H.'1H K He06xoJl,JIMOCHI
C03.lI.aHHH HOBbIX MeTO)J.OB B MaTeMaTHKe H YKa3bIBa.'1H -pa331HtI-
HbIe nyTH ee )1.a31bHeHWero pa3BHTHH. «.MexaHuLieClme» pewe-
HHH 3a)J.a'l 0 l<Ba)J.paType I{pyra npe)J.CTaB31HIOT 6031bIllOH
Hlnepec.
MbI OCT3HOBUMCH T031bKO Ha O)J.HOM TaKOM peweHHH, IW'
Topoe )J.a.'1 lLeHHOCTpaT. a }!{H3HH H pa60Tax lLeHHOCTpaTa
MbI 3HaeM 04eHb Ma310. 113BecTHo,
c liTO OH 6bI.'l Y4eHHKOM D.'1aToHa H
c. 41'0, npUMeHHB oc06yIO KPHBYIO-
C l K Ban. paT p H C y, CYI\Ie31
nOCTpOHTb KBa.lI.paT, paBHOBeJm-
KHH )J.aHHOl\1Y Kpyry.
Dpe}!{l\e Bcero n03HaKOMHMCSI
C KpHBOH, KOTOp3H Bow.'1a B M3-
TCM3THKY no)J. Ha3BaHHeM 1<B3)J.-
paTpHCbI.
DYCTb MbI HMeeM KBa,n:paT
.ABCD (LiepT. 146). Dpe.lI.CTaBHM
ce6e, liTO CTopOHa AB paBHOMepHO Bpaw.aeTCH OK0310 T04KH A,
KaK ueHTpa, H KOHeu B ee OnI1CbIB3eT )J.yry ED. O.lI.lIOBpCieH-
HO C HaLia310M Bpaw.eHHH p3)J.Hyca AB Ha4HH3eT nepeJ\IeULaTbC5I
BHH3 CTopoHa BC KBa)J.paTa T3K}Ke paBHoMepHo H n3pa31J1eJIb-
HO caMOH ce6e. Kor)J.3 Bp31llalOULHHcH p3}J,IIYC AB COBMeCTHT-
CH co CTOpOHOH AD, BC TaK}!{e COBleCTHTCH C HeH. o pe.lI.CT a-
BHM eULe, liTO npii 9THX )J.BYX O.lI.HOBpeMeHHbIX ,n:BHJKemiHX
Bce BpeMH OTMeLialOrCH TOLiKH nepeCe'leHH5:I )J.BH}!{yw.erocSi
BHH3 OTpe3Ka BC C Bparn.aIOULHMCS! pa)\HYCOM AB. reOMeTpn-
tIeCKOe !\teCTO YKa3aHHbIX TOLieK nepeeeLieHHH H eeTb KBa)J.pa-
TpHca BE. Ha pHcYHKe nocrpocHbI CHatIa.na OT.lI.e31bHbIe TOLiKH
MI. M2"" 3TOfI KpI1BOH, )J.31S! 'lero npHMOH yrOJi BAD
pa3,n:e.'1eH Ha 8 paBHbIX LiaCTeH H Ha CTOJIbKO }!{e p3BHbIX
qaCTeH pa3.lI.eJIeHa CTOpOH3 KB3.lI.paTa AB. 4epe3 KOHbI OTpe3-
KOB, n031Y4eHHbIx H3 cTopoHe AB. npOne.lI.eHbl npSlMbIe B,C}>
B 2 C 2 . ..-, nara.'1JleJIhHbIe BC. II OTMe4eHbI T04KH MI' M 2 ....
nepeCe'leHHS! HX paJ(HycalViI1, n.eJIHw.HMH yro31 BAD Ha BoceMb
8
B.
B.
8 2
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'-IepT. 146.
D
312

paBHbIX 'lacTeil. COe)1.HHHB 3TH TOt.JKI1 IIJIaBHOii Kpl1BOA, Mh:
II nOJ1YLiHM KBa,'J,paTpncy BE. Ee OCHOBHoe CBOHCTBO, BbiTeKalO-
uJ,ee H3 nOCTpocmm, OLieBHlI.HO, TaI<OBO:
.c: BAMt BBt .L. D,4Mt AS,
- = --- HJIH = -
.L. BAM k BBk .L. DAM k ABk.
nOCMoTpm.1 Tenepb, KaK C n01\10w.blO KBa)J.paTpHcbl MomBO
BbmpHMHTb )J.yry OI{PY}!{HOCTH. LlJIH 3Toro )J.OKa}!{eM TeopeMY:
EC31H A - eHTp )J.yrH BFD C pa.n.HYCOl\1 AB, BME-
KBa.n.paTplica, TO HMeeT MecTO COOTHOilleHHe (LiepT. 147).
l AD
AD = AE '
rJI.e 1- )J.JlHHa )J.ym BFD.
.lloKa}!{CM 3TY TeopeMY I\'IeTOJI.O:\1 OT npOTHBHoro, cJle.n.ySl
B .n.3HHOM c.TIYLiae 3a rpelieCKHM MaTeMaTHKOM Dannol\l, }!{HB-
WH:\1 B A.'!cKcaH)J.pHH B II I B. {3 C
II. 3. Dann B CBOe1\1 COLinHeHl1H
cd\1aTe:\laTHLieCKlie KO.'1JIeKUHH»
co6paJl Bce 3al\le4aTCJlbHb:e OT-
KpblTHH B 06J1aCTII reOMeTpHH,
c,n:e.TIaHHble )1.0 HerO, H, B LiaCT-
HCCTH, paccKa3a.TI 06 HCTOpJ1i1
Tpex 3HaMCHHTb!X 3a.n.ali )1.peBHO-
CTn: TpnCeKUJIH yr31a, Y)J.BoeHHSl
Ky6a H KBa)J.paTYPbl Kpyra.
PaCC1\IOTpH1\I 3TO )J.OKa3aTeJJb-
CTBO (1IepT. 147).
Ha IlepTe}!{e 147 nFD - )J.yra OKPY2KHOCTH c ueHTpoM
B BepwHHe A KB<1)J.paTa ABCD, BE - KBa,n:paTpHca. DOKa-
l
>KeM, tITO OTlJOWCHHe AD He MomeT 6b1Tb HH OOJlbllIC H HH MeHh-
AD
me OTHOIlleHHH 
.llon, YCTHM 'ITO
l.
t!,
R
II
Q
4epT. 147.
J)
I AD
AD ""'" AK '
r)1.e AK > AE'). 113 BepIllHHbI A, KaK H3 UCHTpa. pa)J.HycCI\
AK npoBe.n.eM )1.yry KL, KOTOpa51 nepeccLieT KBa;J.paTpHcy BE
n TO'IKe M. Tor)J.a
I
, K1\, L
(1)
AD
- ItK .
(2)
t) TC'IKa l( -- npOH3l1o)J'Mlall II!I OTpe3KIt! 15:,).
II!

CpaBHHBaH paBeHCTBa (l) H (2), nO.1YLiUM:
vKML = AD= AB.
(3)
4epe3 TOqKY M nepeCeqeHlUI ,n:yrH KL C KBa)\paTpnCoH npo-
Be)J.eM pa)J.HYc AF )1.YflI BFD. 113 cBoikTBa KD3)J.paTpnCbJ
C31e,n:yeT, qTO
I AB
uDF = AMI .
113 qepTe}!{a TaK}!{e C31e.n.yeT, qTO
I uKML
uDF - uKM
(4)
(5)
3al\leHHB, Ha OCHOB3HHH paBeHcTBa (3), v KML 4epe3
AB, paBeHcTBo (5) nepenHIlleI TaK:
I
u[)F
AB
uKM '
(6)
11, HaKOHe, cpaBHHBaH paBeHcTB3 (4) H (6), npH,n:eM K
paBeHcTBY
AMl=vKM.
ECJ1H H3 TOLiKH M na CTOPOHY AD onYCTHTb nepneH1I.H
}{YJISlp MQ, TO AMI = MQ H, CJJe)J.OBaTe31bHO,
MQ=vKM.
nOC31e)J.Hee paBeHCTBO HeB03:\W}!{HO, n03TOMY Haille npe)J.no
I AD
JIO}!{eHne, liTO """""A1J = ------xK' r,n:e AK > AE, HeBepHO.
I
I1TaK, !bl )J.OKa3a31H, qTO OTHOllIeHHe AD He 1!0}!{eT 6blTb
AD
MeHbllIe OTHOIlleHHSl AE '
tITO OHO He MO}!{eT 6bITb H 6031bIlle
Tenepb )J.OK a}!{eM ,
AD
OTHOllIeHHH AE .
.IJ:onYCTHM 06paTHoe, tITO
I AD
AD -- AQ ,
(1 ')
f)J.e AQ < AE. 113 BepIllHHbl A. KaK 113 LleHTpa, npOBO)J.HM
)1.yry QR, KOTOpa5:1 nepeceLieT pa,n:HYC AF B TOqKe S. Tor)J.a
I AD (2 ' )
\.jQSR - AQ .
314

CpaBHHBaH paBeHcTBa (1') H (2'), HMeeM:
vQSR=AD=AB.
Do CBOHCTBY KBa)J.paTpIiCbI:
I AB
DF = MQ .
(3')
(4')
C )1.pyroH CTOpOHbI,
I \.J QSR . ( 5' )
DF - vQs '
3aMeHHH, Ha OCHOBaHIiH (3'), v QSR Liepe3 AB. n031ytJIiM:
I AB (6 ' )
DF - vQs .
B
D
IJ E D
4epT. 148.
CpaBHIiBaH (6') Ii (4'), HMeeM:
MQ = vQS.
DocJIe)1.Hee paBeHcTBo HeB03MO}!{HO; CJIe)J.OBaTeJlbHO, He-
I
B03MO}!{HO H paaeHcTBo (1') Ii OTHOllIeHHe AD He MO}!{e1' 6b!Tb
. AD
6031bllIe OTHOllIeHHH AE '
TeopeMa )J.oKa3aHa. .
TaKH!l'1 06pa30M, 3a)J.atJa 0 BbmpHM31emIH )J.yrIi BD OKPY}!{-
HOC}H CBe,,1aCb .I{ Haxo}!{)J.cHHIO T04KH E KBa)J.paTpHCbI. HaH)J.H
ee, l\Ibl 1\1O}Kel\l BbmpHMHTb .n.yry BD, nOCTpoHB TpeTblO npo-
nopHOHa31bHYlo K AD H AE. Ha tJepTe}Ke 148 TaI{Oe BbmpHM-
31eHne )J.YrH I C)J.eJIaHO.
Cpe,n:H rpe1.leCKIiX MaTeMaTIiKOB, 3aHHMaBIllHXCH 3a)J.aLieJ:i
° KBa)J.paType Kpyra, Mbl )J.OJl}!{HbJ Bbl)J.e31HTb )J.BYX: fHnne-
KpaTa H AHTHcpoHa. DepBoro nOToMY. 1.110, )J.0I{a3aB TeopeMy
31.'5

o JIYHOtJ.KaX, OH K3R 6b1 BCeJIlIJI B YMbl M8TeM8THKOB 66.'JbIllYIO
Ha)\e}!{)J.Y Ha B03MO}!{HOCTb pellleHHH 3a,n:a4H 0 KBa)\paType
t<pyra (B CaMOM ;I.c.rJ.e, eC31H fHnnOKpaTY y.n.a.rJOCb «KpyroBble
.rJYHOIIKH» npeBpaTllTb B KBa.n.paT, TO n04eMY He31b3H 3TOrO
C,n:eJlaTb ,n:J1H Kpyra ?). BTopoll}j{e CBOHM pellleHHeM 3a.n.a4H nOJI.-
roTOBH.'1 n04BY K B03HHKHOBeHllIO H.n.eH ApXH:\le,n:a. YI fHnnO-
KP3T H AHTHcpOH «BCpH31H B TO, 4TO OHH )\eikTBHTeJlbHO Halll.'1H
KBa,n:paTYPY Kpyra, H 06a 0IllH6aJIHCb»,- nHWeT EB.D,eM I )
(IV B. )J.O H. 3.) B cBoei'I «YICTOpI1lI MaTeMaTHKH».
DpHBe1J.eM TeKCT TeOpeMbI fHnnOKpaTa 0 31YHOLiKaX: cy.HMa
n.'70/v,aod J neyx /lYHO'leK, nOC11lpOeHHblX Ha C11l0pOl-lax KeaOpa11la,
paSHa nOJl06UHe nAOUl,aOU K6aOpa11la (LiepT. 149).
Q
-
lit B,
4epT. 149.
4epT: 150.
4epT. 151.
3ry TeopeMY, ,TJerKO 0606u13eMYlO .n.JIH 31YHoLieK, nocTpoeH-
HbIX Ha KaTeTax npm'vIOyr031bHOrO TpeyrOJlbHHKa (LiCpT. 150)2,
rHnnOl{paT pacnpOCTpaHHJl H Ha 31YHOtJKH, nocTpoeHHble Ha
cTopoHax npaBlIJlbHOrO BnHcaHHoro llleCTHyr031bHHKa (4epT.
151), 4TO HeBepHo.
He 3mleTHB 3TOll 0IllH6KH, flmnoKpa-f .n.e31aeT nonbITxy
HaHTH KB3)\paTYPY Kpyra. PaccMoTpHM XO,n: ero paccY}!{)1,eHHH
(tJepT. 151).
DycTb AB - cTopoHa npaBH31bHOro BnHcaHHoro WeCTlI-
yro.'lbHHKa, AIBI =.4B. TIJlOw.a.n.b n031YKpyra C )\HaMeTpoM
AD paBHa CYM!lle n.10w.a.n.eH 4eTblpex nOJIYKpyroB C paBHblMH
)J.HaMeTpalv!I1 AB, BC, CD II AlB13). S TpanewIH ABCD=
=S no.'1YKpyra C ,!JJ!aMeTpoM AB 6e3 35 cerMeHTa A1HB=
=4S lv -3S cefI\!eHTa AMB=Slv+3 (SIV- S cerMeHTa
AMB)=Slv+3S 31YH04KH.
1) EeoeM - rpe'leCKHll MaTeMaTIIK, Y'IeHfIK ApnCTOTeJlH.
2) CM. CTp. 170.
AD2 AB2
3) nJlQma.I\b nOJlYKpyra C .I\HaleTpoM AD paBHa T = T .
OJlOlna.I\b 'IeTb!peX no.1YKpyroB 'c paSHbIMH )J;HaMeTpa1I1 AB, BC, CD
( AB ) 2 r.AB2
H AIBI paBHa 47. - : 2 = --;;- .
2/_
316

OTcl{))J.a Snr = S TpanelHr ABCD - 3S JIYHOLiKU.
.D.onYCKa5l, 4TO 31YH04}<H I, II H III HMelOT KBa,n:paTYPY,
Mbl )\031}!{Hbl )J.onYCTI1Tb, tITO 11 n031YKpyr IV TaK}!{e HMeeT ee.
EB)J.eM, pa36Hpa51 perneHl1e 3a)J.a4H 0 KBa)J.paType Kpyra
.n.aHHoe fHnnOl{paTOM, 3aMeLiaeT:
«TpaKToBaHHe 3a)J.a411, )J.ei1cTBI1Te.'1bHO, reHl1a.'1bHO, HO
Ornl160LiHOe 3aK31IOLieHl1e npOI1CXO)1.I1T BC31e.n,CTBl1e )J.onYllleHllSl
laKoro n0310}!{eHI151, KOTopoe He )J.OKa3aHO Bo061lle; He )J.JI51
BC5!KOH 31YHOLiKI1 CYUlecTByeT KBa,n:paTypa; OHa cyuleCTByel
)\.'151 31YH04KH, KOTopaH onHpaeTCH Ha CTOPOHY KBa)J.paTa,
Bnl1CaHHoro B Kpyr; JIYHOtIKa }!{e B )J.aHHOM Bonpoce Onl1paeT-
C5I Ha CTOpOHy npaBI131bHOrO BnHcaHHoro rnecTHyr031bH11 I<a».
Do nOBO,n:y perneHI1H 3a,n:aLiI1 0 }<Ba,n:paType Kpyra AHTH-
<pOHOM EB,n:eM nHWeT: «HaLiepTaB Kpyr, AHTH<pOH BnHca.n B
Hero TaKofr npaBI1JIbHblH MHoroyr031bHI1K, KOTOpblH MbJ YMeeM
Bnl1CaTb. DycTb 3TO 6Y)J.eT KBaJI.-
paT (LiepT. 152). DOToM OH pa3-
,n:eJIH.n KaJK.lI.YIO CTOPOHY KBallpa-
Ta nOn031aM H 4epe3 TOLiKH )J.e.ne-
HHH npOBeJI npHMbIe, nepneH)\H-
}{Y315!pHbIe I< CTopoHaM, ,n:o nepe-
CeLieHI1H C OKPY}KHOCTblO. 04eBM)J.-
HO, 3TH npHMbIe )J.e31HT cerMeHTbl
Kpyra Ha ,n:Be paBHble 4aCTH.
.D.a31ee OH COt')J.HH1131 n031YLieH-
Hble TOlJKH C KOHuaMI1 CTO-
pOH KBa,n:paTa, TaK 4TO n031Y-
4H.nHCb LieTblpe Tpeyr031bHI1Ka H BCH 06pa30BaBIlla5lC5:I <pl1rypa
CTa31a npaBH31bHblM BocbMHyr031bHHKOM». TeM }!{e cnoc060M
.lI.eJIH CTOPOHY I1paBH.nbHoro BnHcaHHoro BOCbMl1yr031bHI1Ka,
nOTOM weCTHalluaTH}T031bHHKa H T. )1.., OH n031YLiaeT MHoro-
yr031bHUK C t.IHC310M CTOpOH B)J.Boe 6031bWerO 4I1C31a. «DoCTY-
naH Tal<, nOKa He 6)')J.eT IJCLiepnaH BeCb Kpyr, OH 3al<31IO'laeT,
'lTO, HaKOHe, 6y,n:eT BnHcaH lV1HOroyr031bHHK, nepHMETp KOTO-
poro MmKHO paCCl\JaTpUBaTb KaK .lI.J!HHY OKPY}!{HOCTI1».
TaKUM 06pa30IlI, AHTHcpoH n031ara.n, '-ITO nYTel\! nOC31e)J.o-
BaTeJIbHOrO )')J.BoeHH5I 4l1c.na CTOpOH BnHcaHHoro MHoroyrOJIb-
Hl1l<a MO}!{HO B KOHe KOHUOB )J.OHTH )J.O OKp)'}!{HCCTH Kpyra
H TOLiHO onpe.lI.e31HTb ee )J.31UHY H3111 nJIOrna,n:b orpaHUt.JUBaeMoro
em Kpyra. KpUTHlJeCKI1 OTHOCHCb 'K H,n:ee AHTH<poHa. O,n:I1H
U3 rpe4:eCKHX nI1CaTeJIeHI) nHIlleT: «DOCT05lHHbIM .n.e31eHHeM
1) CUMnAUljUa. KOMMeHra"fOp ES.!I.eNa.
4epT. 152.
817

rrpOCTpaHCTBa Me}!{.lI.y'XOP)J.OH H .lI.yroH OHO HllKor)J.a He 6Y)J.eT
HCtIepnaHO, l\IbJ HHKor)1.a He )J.OCTHrHeM OKpY}!{HOCTH }{pyra,
)\a}!{e eC31H 6bI )J.eJIeHHe npo,n:031}!{a310Cb ,n:o 6eCKOHe4HOCTH».
COBpe;..!eHHHK AHmcpOHa BpH30H, T3K>Ke nbITaBIllHHCH
Haihn KBa.lI.paTYPY Kpyra, npe.lI.31ara.'1 HHOH MeTO)J.. C)J.eJJaB
npaBH31bHOe 3a K.TllOtleH He, 4TO nJIOw.a,n:b Kpyra So 6031bllle
nJIow.a;J,H S n npaBHJJbHOrO BnHcaHHoro 1\1H0royr031LHHKa
H MeHbllle nJJow.a)J.H Sn' O,n:HOHMeHHOrO onHCaHHOro, T. e.
Sn<So<S/,
OH, p.OnycKaSl 0IllH6KY, YTBep}!{,n:aJJ, liTO
So == Sn  Sn'
e
i%f////
4epT. 153.
4epT. 154.
XOTH nonbITKH AHTHcpOHa H BpH30Ha H3ihH KBa)\paTYPY
Kpyra H He BCTpeTHJJH 631arOnpHSlTHoro OTHOllleHHH Y 60JIb-
WHHCTBa rpe4eCKHX MaTe1\l3THKOB, Bce }!{e 3aC31ym HX He-
ocnopmVIbl. AHTHlpOH nbITa31CH HCqepnaTb Kpyr, BnHCbIBaSl B
Hero npaBHJJbHble MHoroyr031bHHKH, BpH30H }!{e no,n:MeTHJI
He06xo)J.IIMOCTb npHHHMaTb BO BHHMaHHe npH perneHHH )\aHHOH
3a,n:aLiH KaK BnHcaHHble, TaK H onHcaHHble l\IHOroyrOJJbHHKH H
TaKHM 06pa30M HaTaJIKHBaJJ Ha MbIC31b BBO)J.HTb BblCIllHH H HH3-
IllHH npe,n:e31bl )J.J1HHbl OKPY}!{HOCTH H31H nJJOw.a,n:H Kpyra.
Bnoc.'1e,n:CTBHH C H306peTeHHeM MeTo,n:a HCLiepnblBCIHHH ,n:Ba
npe)J.e31a Bcer)J.a paCCMaTpHBa31HCb BMeCTe, KaK 3TO l\Ibl BH)J.HM
Y EBKJJH)J.a H ApXH:\le,n:a.
C04HHeHHe. B KOTopmi ApxHMe,n: (287-212 ,n:o H. 3.) CTa-
BHT 11 pelllaeT Bonpoc 06 OTHOllleHHH )J.31HHbI OKPY>KHOCTH K
CBoeMY )J.Hal\leTpy, HOCl1T Ha3BaHHe «ybl\lepeHHe Kpyra». Pe-
3Y31bTaTbI, n031YLieHHble ApxHMe,n:oM B pellleHHH nOCTaB31eHHOH
3a)J.a4H, TaKOBbl:
1. D.'1ow.aJI.b Kpyra paBHa n310Illa;:IJI npSll\1oyrOJIbHOrO
Tpeyro.'lbHHKa, O)\JfH H3 KaTeTOB KOToporo paBeH )J..'1HHe OKPY}K-
Hocm, a ,n:pyroi'j - pa)J.HYCY 3TOro Kpyra (qepT. 153).
318

2. DJIOllla)1.b Kpyra OTHOCHTCH K n31ow.a)J.H l<Ba)J.paTa, no-
cTpoeHHoro Ha ero ,n:HaMeTpe, n04TH KaK 11 : 14 ('1epT. 154).
3. LL31HHa OKPY}!{HOCTH MeHbwe Tpex ,n:HaMeTpoB, C310}!{eH-
I 6 -
HblX C "7 3Toro )1.HaMeTpa, H 031ee Tpex )J.HaMerpO.B, CJlO}!{eHHbIX
10
C 7i Toro }!{e J].HaMeTpa, T. e.
J  d<C<3 d
PHC. 155.
Mero)J., C nOMolllblO KOToporo Apxmv!e,n: npHWe31 K YKa3aH-
HblM pe3y31bTaTal\I, 6bl31 TaKHM. OH nOCJIe)J.oBaTe31bHO onpe)1.e-
.TIHJI nepHMeTpbl npaBHJIbHblX BnHcaHHbIX 6-yrOJIbHHKa, 12-
yrOJIbHHKa, 24-yrOJlbHHKa, 48-yro31bHHKa H 96-yr031bHHKa.
3aTeM TO }!{e cal\loe OH npO)\e.'1blBa31 )\.1HI Onl1CaHHblX l\IHOrO-
yr031bHHKOB. Doc31e :)10ro nepH:\ieTpbI npaBl1J1bHblX O,n:HO-
mleHHblx BnHcaHHblX H onHcaHHblX I\lHoroyrOJIbHHKOB cpaB-
HEBaJl Me}!{llY C060H. 3THM OH HaxO)J.HJI HH}!{HHli H BepXHHH
npe.n.e.TIbi )J.31H OTHoweHHH )J.JIHHbl OKPYLI<HOCTH K )\U8l\!eTpy,
nOCTeneHHO c6.'1mKa5J 3TH npe,n:e31bl.
HaM l\W}KeT nOKa3aTbCH, '1TO 1\IeTo)J. ApXHl\iella He HOB,
Be,n:b AHTmpOH npl1 peIlleHHH TOH }!(e 3a)J.a411 n031b30Ba31C5:I
nO)J.06HbIM l\1eTO)J.OM. Ho 3TO He TaK. B TO BpeMH KaK AHTH-
CPOH HaI1BHO nOJIara31, liTO nYTeM nOCJ!e,n:OBaTe31bHOrO y,n:Boe-
HHSl 4HC313 CTOpOH BnIlCaHHOro B Kpyr l\1Horoyr031bHHKa MO}!{-
HO B KOHue KOHUOB )J.oiiTH ,n:o OKPY}!<HCCTH Kpyra, T04HO onpe-
D.e.'1HB ee )J.31HHY, ApXHMe,n: C TOHKHlVI l\laTeMaTH4eCKHM '1YTbeM
)\aeT )J.JIH OTHoweHHH )\.'1HHbI OKPY}!{HOCTH K llHaMeTpy npH-
6.'1I1}!{eHHOe 3Ha'leHHe, HCHO C03HaBaH, '1TO «HCLiepnaTb)} Kpyr
HeB03MO}!{HO, 4TO n.'low.a,J,b I<pyra Bcer,n:a 6y,n:eT OTJIl1'1aTbClI
OT n31ow.a,n:H BnJ1caHHoro H.1H OIIHcaHHoro MHoroyro31bHHKa Ha
HeKOTopYI{) KOHe4uylO Be.TIl1'1HHY, KaK 6bl Be31I1KO HH 6bJJIO
'1HC310 CTOpOH TaKoro MHoroyroJlbHHKa.
OT31H4aJlC5J l\leTo)J. ApXHMe)\a H OT MeTo)1.a EpH30Ha, KOTO.
pblH npHHHl\Ia31 .n.JIHHy OKPY}!{HOCTH paBHOM cpe)J.HeMy apH<p-
MeTH'IecKoMy nepHl\leTpOB ,n:ByX O)J.HOHMeHHbIX MHoroyro31b-
HHKOB, BnHcaHHoro H onHcaHHoro. Y ApXHMe)J.a conOCT.aB31e-
HHe nepHMeTpOB npaBH31bHbJX Ol1.HOHMeHHbIX BnIICaHHbIX H
onHcaHHblX l\iHOroyroJIbHHKOB )J.e31aeTCSl T031bKO )J.JIH onpeJ].e-
.TIeHHH TO'lHOCTH IIO.'1yqeHHbIX pe3y.'lbTaToB.
819

TaKHM 06pa30M. ApXH:vle)J. nepBblH H3 MaTeMaTHKOB no-
CTaBH.'1 3a)\a'lY 06 113MepemfII )J.31I1Hbl OKPY>KHOCTH II nJIorn.a)J.II
Kpyra Ha BnOHe HaY4HY nOLiBY.
CTOHT OTi\len1Tb H BTOpYI{) 3aC!lyry ApXHMe.n.a. B np0ll.ecce
cBoeii pa60Tbi ApXHMeJI.Y I1pHW.'10Cb npOII3BOj1,HTb BeCbMa
C310}!{HbIe )J.JlH TOro Bpe:\>leHH BbILiHC.'1eHHH. «11 MbI, npHBbIK-
lime K a.'1re6paHLiecKH1 0603Ha4eHHHM, Ta631HuaM KopHeii H
JIOrapllcp:\!OB, He rOBopH Y>Ke 0 1\13TeMaTULieCKlIX Bbi4UC.'1HTe.'1b-
HbIX MaWHHax, He B COCTOHHHII B03;J.aTb )\O.'})KnYI{) )J.aHb BOC-
XHllteHHSl 3TOii CTOpOHe pa60TbI ApXHMe,n:a»l).'
C03.l1.aHHbJH ApXIIMe,n:OM l\leTO.ll. BbI4HC31emUl )1,31J1Hbl OKPY)j{-
HOCTH nocpe)J.CTBOl\! BnHcaHHblX H onHcaHHblX l\IHOroyr031b-
HHKOB )J.a31 B03:V10}!{HOCTb noc.'1e,n:YI{)!lJ.lIl\1 MaTeMaTIII<3l\! 6031ee
T04HO onpe.ll.eJlHTb OTHOllleHHe ,n:JlHHbI OKPY>KHOCTlf K )J.Hal\!eT-
PY, T. e. LiHCJlO 'it".
O.ll.HH;\l H3 nepBblX MaTeMaTHKOB, KOTOpbIH .lI..'1H qHCJIa 'It
rJO,7b30Ba.'1CH 60.'1ee TOLiHhIM 3Ha4eHHeM, qeM ApXHMe,n:OBO
3HaqeHHe. 6bJ31 rpe4eCKl1f'I r.:taTel\laTHK H aCTpOHOl\I n T 0 31 e-
M e H (II B. H. 3.). Y nTOJIel\ieSl
377 3 17
1C= '20 = .20
Pue. 156.
I1H,n:HHCKHe MaTeMaTHKlf VI H VII BB. TaK>Ke CTpeMHJII1Cb
HaHnl 60.'1ee TO'IHoe 3Ha4eHHe 4HC.'13 'It. TaK, A p H a 6 x a -
T a (VI B. H. 3.), nOJIb3 YSlCb cpOPMY.'10H
Q2/1 = V 2R2_R V 4R2-a '
,n:aeT )J..'1Sl 'it" 3Ha'-leHHe, paBHoe 3.1416,
1C = 3,1416
PIle. 157.
I1pHlleM Jl31H ,noro el\lY I1pHIllJlOCb onpe.ll.eJlHTb nepHMeTpbJ
MHoroyrOJIbHliKOB BI1HCaHHOro H onHcaHHoro C 384 cTopoHaMH.
1) C. 51. J1 Y P b e, ApXIIMel\.
320

MeHee T04HOe, 4eM y ApHa6xaTa, 3HatleHHe tlrrCJIa 'it, HO
JUiTepeCHoe no cpop!\le )J.aJI )J.pyroH HH)J.HHCKm1 MaTel\laTHK-
B paM a r y n T a (598-660). Y Hero
1C=VW
PI1C. 158.
MO}!{HO npe)J.no.TJO}!{IITb, 4TO BpaMarynTa no.'1Y4HJI TaKoe 3Ha-
IleHHe, paccMaTpHBaH noc.'1e)J.oBaTe.1JbHOCTb npH6.'1JI>KeHHbIX
3Ha4eHHH tlHc.'1a it B BII)J.e nepH!\IeTpOB 12-yroJIbHHKa, 24-
yrOJIbHHKa, 48-yrOJlbHHKa li 96-yr031bHliKa, BrIliCaHHblX B
OI{PYKHOCTb C .u.HaJ\!eTpOJII. paBHblM 10. 3Ta nOC31e)J.OBaTeJIb-
nOCTb TaKOBa:
1/965, V 981 , 1 / 986 , 1 / 989 ,....
ECJIH npHHHTb, IlTO 4liC.TJa, CToHw.lie no.n: KOpHeJII, nOCTeneHHO
YBeJJliqHBaHCb, npH6JIH}!{aIOTCH K 1000, TO npH D= 10 MbZ
nOJIY4Hr1f:
'it =..f. = 111000 = 10 '/10 = VT6.
D 10 10
He OCTa.TIHCb B CTopOHe OT nOHCKOB 60.TJee T04Horo 3HaIle-
HHH .. H cpe)J.Hea3H8TCKHe YLieHbIe. TaI{, call1apKaH)J.cKHH - YIle-
HbIH ,[L}!{el\lIllH)J. li6H-Macy)J. a31b- K a III li (nepBaSl nOJIOBliHa
XV B.) BbILiHCJIHJI 17 )J.eCSlTlitlHbIX 3HaKOB.
ECJIH nepeHTII Ha eBponeikKYIO nOImy, TO CTOHT ynoMSI-
HYTb rOJI31aH)J.CKoro MaTeMaTllKa H HH}!{eHepa AH)J.pHaHa M e-
 H H (XVII B.). KOTOpbIH, BCe eJlJ,e nOJIb3YHCb JIIeTO)J.OM
ApXHMe,n:a, HaIlle.'1 TaKoe 3Ha I IeHHe 4liCJIa ", KOTopoe no T04-
HOCTH )J.aJIeKO npeBOCXO)J.HT Bce paHee H3BeCTHHe 3HaeHHSl.
y .leli5:1
1r = J 55 = 3 141592 9 I)
l1J '
PlIC. 159.
I) To i'Ke 3H8'1eHl1e .I\.'IH n 6bWO .l\8HO rOp83i1.0 paHee KIITa;!cKHM
M8TeManlK"O\! LI.3Y 4YH-'1lKJI. lKHBWH! BO Bropofi nOJIOBHHC V B.
II A. A. KO.l0COD
321

HeCKOJIbKO paHee I}JpaHY3CImH MaTei\IaTliK <ppaHcya B H-
e T (1540-1603), nOJIO}!{liBIllliH HaLia.'10 yr.TJy6.1eHHoH pa60-
Te Ha)J. lIfaTeMaTliLieCKlillIH COIJ:HHeHliSlMH ApxHl\Ie)J.a, TaIOKe
He npOilleJI Mlil\1O HaXO}!{)\eHHSl npli6JIH}!{eHHOrO 3HaLieHliSl
IJ:HCJIa 7':. EMY Y)J.aJIOCb He TOJIbKO HaHTli LiliCJIO r. C 60JIbillOH
TOIJ:HOCTblO, HO H npe)J.CTaBliTb ero B BH)J.e 6ecKoHeIJ:HorO npo-
li3Be)J.eIUUI.
1C- 2
- W .Vi (I+VJ). V (I +Vf (I+VD...
c
4epT. 161.
PHC. 160.
KaK }!{e nOJIYIJ:liTb nocJIe)J.-
HIOIO I}J0PllIYJIY?'
,[LJISl 3Toro npoc.'1e)J.liM 3a C31e-
)J.YlOllI.liMH 3anliCSllI1li, HCnOJIb3YSl
IJ:epTe}!{ 161, Ha KOTOpOM AO-
pa)J.HYc R; AB - cTopoHa BnH-
caHHoro npaBH.'1bHOrO n-yrOJIb-
HliKa; AB=a n ; OD - anoI}JeMa
n-yroJIbHHKa; OD=K n ; AC=a 2 n-
S S2/l S Sn
'- to, Aoe ="2n H '- to,AOD = "2n '
r)J.e S:,. - nJIOlll.a)J.b TpeyrOJIbHliKa H Sn li S2n - nJIow.a)J.H
MHoroyro.'1bHliKOB C YKa3aHHbIM LilicJIOM CTOpOH.
,[LaJI ee,
Ho
SAOD : SAoe = Sn : S2n"
SAOD : SAoe = OD: OC,
TaK KaK TpeyrOJIbHliKH AOD 11 AOC liMelOT 06w.ylO BbICOTY
AD. TaKliM 06pa30i\1,
Sn : S2n =OD : OC,
HO OD=K n H OC=R; CJIe,n:OBaTeJIbHO,
Sn Kn (1)
S2n -R .
no aHaJIOnnl c paBeHCTBOM (I) HanHillellI pH)J. HOBbIX paBeHCTB:
_  .  _  .  _ KM ._ 
S2n - R ' S4n - R . S8,' - R ,..., S21 - R .
822

rIYCTb n = 4; TOr)\a nOCJIe)J.Hlie paBeHcTBa npli:\!YT BlIJt
S4 _ K 4 . Ss _ Ks S16 _ KI6  _ Ki
S; - R' S16 - f[' S32 - R' . . ., S2i - R .
Oepe;\IHO}!{ll\I 3TH paseHCTBa no LiaCTHM:
S4 _ K4.Ks.K16...K;
S2i - R. R. R . . . R
TaK KaK S4 = 2R2 Ii npli. ,n:OCTaTOLiHO 60JIblllOM i, S2i 
 ..R2, TO
2R2  K4 Ks KI6 Ki
r-R2 R'R'R""R'
OocJIe cOKparn.eHliH liMeeM:
2  K4 Ks KI6 K;
-;:R'R'R"'R.
(2)
Rlf2 R v
MbI 3HaeM, liTO K4 = ' K8 ="2 2 + V2
HaffJ\H no <pop:\1Y.'le Y)\BOeHliH. liTO 016= R V;=-v 2 + V2 .
l\lbI no <Pop\ly.'1e
Kn = V R 2 - af
OIO}!{eM HaHTIi Ii K 16 :
Rlf / ..[-
K 16 = 2 r 2 + V 2 + J' 2 .
Onpe)J.eJIHB a32' l\lbl HaH,n:e\1 Ii K 32:
K 32 =  V 2 + -V 2 + V' 2 + V2
11 T. D..
OO)J.CT3BJI5UI 3HA4eHliH K4. K8' K 16 . K 32 ... s paBeHcTBo
(2), nOJIytIHM:
2 112 V 2+Jf2
1; 2' 2
1/ l [ -=-
r 2+.. 2+ r 2
2
V2+V 2+V2=i=Y2
2
11.
323

C)J.e.'1aB npe06pa30BaHHSl, nO)J.0 6Hble c.'1 e)J.y ro w. lIJ1 1:
1/2 _.-.1 2 _ -.IT . V2+ 1/2 =.. r 4 2 + f 4 2 =
-r--'" 4 -"'2' 2 V
=.. r..!..+..!.. V 1
V 2 2 2
MbI H npH)J.eM K tPOpMYJIe BHeTa )J.JISl LiHC.'1a ...:
H T. )J."
1t=
1)
V  V {(I+V  ). V ; (l+V {(1+ v I ))...
2
nOCJIe)J.yrow.He reOMeTpbl, pernaSl 3a)J.aLiY 0 KBa)J.paType
Kpyra, CTapa.'1HCb BCe TOLiHee H TO'lHee onpe)J.e.'lHTb LiHC.'10 ....
TaK, COBpe\IeHHHK BneTa ro.'1.'1aH)J.CKHH JllaTe:\JaTHK BaH-
r y M e H )J.aeT ... C 16 3HaKaMH. J1Y)J.OJIbtP BaH-U. e H .'1 e H
H3 KeJIbHa (1539-1610) npHXO)\HT K ew.e 60JIbilleMY npH6JIH-
}!{eHHIO, )J.aBaSl )\JISl ... 35 3HaKOB. HaKoHe, r 10 H r e H C,
H)J.H B HaXO}!{)J.eHHH LiHCJIa ... no nYTH ApxHMe)J.a, )J.OKa3aJI pSl)J.
TeOpel, KOTopbIe n03BOJIHJIH eMY C JlIeHbIlleiI 3aTpaTOH TpY)J.a
no.'1YLiHTb )J.ocTaTo'lHoe KO.'1HLieCTBO 3HaKOB )J.JISl .... rlOHreHc
B TO }!(e BpeM5I 6bIJI noc.'1e)J.HH:\I H3 IIIaTeMaTHKOB, KOTopbIe
nOJIb30BaJIHCb B Bonpoce YTOLiHeHHSl LiHCJIa ... IIIeTO)J.Oi\I Ap-
XHl\fe)J.a. nOc.'1e Hero BMeCTO cnoc06a BnHcaHHbIX H omIcaHHbIX
MHoroyro.'1bHJ!KOB OCHOBHOll 3a)J.aLieH CTa.'10 pa3blcKaHHe aHa-
.'1HTHLieCKHX Bblpa}!{eHlIll )J..'1Sl OTHOIlleHHH )J..'1HHbl OKPY}!{HOCTH
K )J.HaMeTpy, BC.'1e)\CTBHe Liero cTapbIe apxHlIIe,n:oBbI IIIeTO,l]:bI
6bllIH 3a6paKOBaHbl.
Pa3BHTHe MaTeMaTHLieCKoro aHa.'lH3a, B LiaCTHOCTH TeOpHH
pSl)J.OB, )\a.'1o B pYlm :.IaTe:\laTHKOB 60.'1ee cOBepIlleHHbIe IIleTO)J.bI
)J.JISl HaXO}!{)\eHHSl npH6J1H>KeHHOro 3HaLieHH5l 'it. B KaLieCTBe
npHMepa lIIO}!{HO YKa3aTb TaKoe pa3JIO}!{eHHe LiHCJIa r. B pH)J.:
I) OpH BbIBo.[\e norl cpOPMYJlbI. tJTOGbI He 3aTpYJlHSlTb BaIlJerO BHM-
MaHHSI, MbI oe3 lIOKa3aTe."JbCTBa npHHSI."JII, tJTO
- !i V 1 ,/ -.; I ;-
K 2 n- 2 2+ 2+ V 2,...+1 2 .
aTO JlOKa3aTe.!JbCTBO IO}KHO npoBeCTM MeTOl!O1 M3TeMaTHtIeCKOH 11H-
lIYKl\MH.
324

: = 4 [+ - +.  + +. 5 - +. 7 +...]-
[ I I ]
- ---+
239 3.239 3 ....
B o6eHX KBa)J.paTHbIX cKo6KaX IIlbI UMeeM 6eCKOHeQHilJe
3HaKOnepeMeHHble pH)J.bI, 3aKOH nOCTpoeHHH KOTOpblX !\ImKHO
c1erKO nO,n:MeTHTb. EC.'1H B nepBOH CKOOKe OrpaHHQJITbCH Lie-
TblpbMH HanHCaHHbli\iH Li.'1eHaMJI pH)\a, a BO BTOpOH TOJIbKO
nepBbIM Li.'1eHO:\I H c)J.eJIaTb BbILiHCJINIHSl C T04HOCTblO ,n:o
0,000001, TO ,n:JJSl QUCJJa  Mbl nOJIY l ll1M nSlTb 3HaLiaw.HX IlII<pp.
BbJIlHCJIeHHSl MO}!{HO pacnO.'1araTb no CJJe)J.YIOw.eH cxeJ\le:
I
5 = 0,200000
1
5.5 5 = 0,000064
I
=0,002667
1
7.5' = 0,000002
+ 0,200064
- 0,002 669
0,197395
X 4
1 0,789580
- 239 = -0,004 184
0,785396
X 4
7t;::::::3,1416
Doo6Hble l\leTobt n03BO.'1U.'1JI ,n:OCTJIfH)'Tb B onpee.1JeHHH
LiUC.1a  TOLiHOCTH, YOB.'leTBOpHlOllI.eii He TO.'1bKO ca:-1bI:-I
CTpOfliM Tpe60BaHUSlM, HO U )J.aJJeKO npeBOCXO)J.HW.UX UX.
TaK, Hanpmlep, aHr.rJU<JaHHH III a p n HailleJI 600 )J.eCHTHLiHbIX
SHaKOB 'lHc.'1a. Balli, KOHeLiHO, fIOHHTHO, liTO TaKoe BbILiJIC-
.'Jemie UMeeT TOJIbKO cnOpTII'3'lbIH HHTepec, KaK peKOp)J..
nOTOMY 4TO B npu.'1O}KeHHHX HeT HMKaKOH lIe06xou!IlOCTH
3HaTb  C TaKOlJ TOLiHOCTblO. lI.JIH npaKTHLieCKMX nOTpe6HOCTeH
oCTaTOLiHO 3HaTb, qTO
17:=3,14159
PHC, 162,
325

Ho H H3 3THX )\eCSlTHLiHbIX 3HaKOB HY}!{HO OpaTb 'taKoe KOJII1-
qeCTBO HX, KOTopoe onpe)J.e.'JHeTCH TOLiHOCTblO npOH3Be)\eH-
HbIX H3MepeHHH.
npHMeHeHHe COBpel\IeHHbIX BbJtIHCJIHTe.'lbHbIX III3111HH )\aeT
B03MO}!{HOCTb BbILiHC.'I5JTb npU6JJll}!{eHHbIe 3HaLielU!5! 'it C or-
pO!\lHoi:'J TOLiHOCTblO. B HaCTOHlll.ee BpeIH U3BeCTHO 60.'lee
1000 ,n:eCHTHLiHHX 3HaKOB HC31a 'it.
B KOHIe X VII I .B. He1\IeIKHH MaTe:\laTHK J1 a M 6 e p T
H cppaHllY3cKUH MaTe:\laTl1K J1 e )K a H ;J. P YCTaHOBU31H, liTO
'it - qHCJIO H P P a  II 0 H a .'1 b Hoe, a B 1882 r. HeMeKuH
lvIaTeIlIaTHK J1 H H )J. e III a H )J.OKa3a.'1, qTO OHO T P a H C  e H-
)J. e H THO e, T. e. He MO}!{eT CJJY}!{HTb KopHeM HHKaKoro a.1l-
re6paJIIleCKOro ypaBHeHHH C ueJJblMII K03cpcpHUHeHTaMH. Teo-
pel\Ia J1H1I,n:eMaHa K.'Ta)J.eT npe,n:e.'1 non blTKa 1\! peIlleHHSI 3a,n:aLiH
o KBa)\paType Kpyra C nmlOlll.bIO HpKy.'IH II .'IHHelIKII.
OCTaHOBU!\ICSl HeMHoro Ha 9TOM Bonpoce.
nYCTb pa)J.IIYc )J.aHHoro Kpyra paBeH r. )Ke.'laH nOCTpoHTb
KBa,lJ.paT, paBHOBeJJllKUH )J.aHHOi\IY Kpyry, 0603HaLiml CTOPOHY
KBa)\paTa Liep3 x. Tor)\a x )\OJI}!{eH Y)J.oB.'1eTBOpSITb ypaBHeHHIO
x 2 ='itr 2 ,
OTCIO,.1.a
r-
x=r} 'it.
TaKH1\I 06pa30l\J, 3a)\aLia 0 KBa)J.paType Kpyra CBO)J.HTCSl
K nocTpoeHFIO C nm.JDlll.blO HpKy.'1H H JJHHeiiKH, B pe3YJIb-
TaTe KOToporo OTpe30K r 6bIJI 6bI YMHO}!{eH Ha )J.aHHoe LiHCJIO
/ 3a,n:a l Ia He Km!eTCSl c.JIo}!{Hoi:'J, nOCK031bKY HaM He pa3
Y)J.aBaJJOCb rpacpHLieCKH y1\IHO}!{aTb OTpe30K Ha LiUCJIO. no-
,lJ.06HaSl 3a,n:aLia peillaeTCH COBce.I npOCTO, ec.'Iu MHO}!{HTe.'Ib
nepe)J. r 6Y)J.eT. tlHC.'10M paHOHa.'lbHbIM. HanpuMep, .TlerKO
. 3
nOCTpOHTb OTpe3KH 3r, 24r, I, (3)r, eC.'II1 OTpe30K r )J.aH. Ho H
)J..'1Sl HeKOTopbIX HppaHOHaJIbHbIX lIIHO}!{HTe.'1eH nOCTpoeHHe'
BbIllO.JIHHeTCSl. HanpHl\Iep, Rl'2 IIIbI CMO}!{eM JIerKO nOCTpoHTb,
BcnO"IHHB, tITO R1 '2 eCTb ,n:HarOHaJIb KBa)J.paT a co CT opmIOH R.
MbI C:\1O}!{eM nOCTpOHTb OTpe3KH Rl/2- l""2 H R l / 2- 1 . ec-
JIH 6Y)J.eT )J.aH OTpe30K R, TaK KaK Rl / 2-"}.'-211 RV 2-1/3
ecTb COOTBeTCTBeHHO CTOpOHbI BocbMHyro.'1bHHKa Ii )J.BeHa)J.uaTH-
Rlf3
yrOJJbHHKa, BmIcaHHblX B Kpyr C pa)J.HYCQ!lI R. TaK KaK -z
326

RV 2+ 1/2
H 2 eCTb anocpel\Ibl npaBHJIbHbIX BnHcaHHbIX IlleCTH-
yrOJIbHHKa H BOCbl\1Hyro.'1bHHKa, TO H 3TH npOH3Be)J.eHHSl MorYT
6bJTb nOCTpoeHbI, nOCKOJIbKY nOCTpoeHHe IllecTHyrOJIbHHKa H
BocbMHyrOJIbHHKa, a CJIe)\OBaTeJIbHO, H HX anocpeM He npe)J.-
CTaB.'1HeT TpYJJ:a. ,[LJIH nOCTpoeHIIH Bcex 3THX f/pOH3Be,n:eHHH He
nOTpe6YIOTCH HIIKaKHe HHbIe IIHCTpYMeHTbI, KpOl\1e LlHPKYJISl H
:nIHeHKH.
O)\HaKO rpacpULJeCKOe YMHO}!{eHHe OTpe3Ka Ha LiIICJIO BbI-
IIommMO IHpKYJIel\1 H .'1IIHeHI(oH )J.a.'1eKO He Bcer,n:a. OHO l\1O}!{eT
6bITb ocyw.eCTBJIeHO npn c.'1e)J.YlOlI1.eM YCJIOBHW 'IHCJIO, Ha
KOTopoe MHO}!{HTCSl r, )J.O.'1}!{HO 6bITb KOpHel\! a.'1re6paHLieCKoro
ypaBHeHHH C eJIbJIvIH K03cpcpHLlHeHTaMH, pa3pelllHlIIOro B KBa)\-
paTHbIX pa)\IIKa.'1ax. 4HC.'10"lt, 6Y)\YlIH Tpa, HCu.eH)\eHTHbIM He
Y)J.OB.'1eTBOpSleT nOCTaB.'1eHHOl\IY YCJIOBHIO; He Y,lJ.OBlIeTBopHeT
eMY II LiHC.'10 V;: CJIe)J.OBaTeJIbHO, 3a)J.aLia 0 KBa)J.paType Kpyra
He MO}!{eT 6bITb peilleHa C nOMow.blO U.lIpKy.'1H H .J1JmeHKH, ,n:JISl
peweHHSl ee He06xo)J.HMO paCIllIIpeHHe CpC)J.CTB nOCTpOeHII5!.
nepBoe TaKoe peilleHlIe, BbInoJIHeHHOe .llCHHocTpaTOM C no-
MOW.blO OC060H KpHBOH «KBa)\paTpHCbI», y}!{e 6b1JIO pa306paHO
HaMH B 3TOH rJIaBe.
06 ApxHMe1l.e
ApXIIMe)J. - O,n:IIH H3 BeJIHLiaHIllHx MaTeMaTIIKOB H l\!exa-
HIIKOB - 6bI.'1 pO)J.Ol\1 JI3 ClIpaKY3, ropo,n:a, paCIIOJIO}!{eHHOrO
Ha BOCTOLiHOl\1 n06epe}!{be oCTpoBa CHH.'1JIJI. Ho YLiII.TJCH OB
B npoc.'1aB.'1eHHOii A.'1eKCaH)\pHHCKOH aKa)J.el\IlIH. noc.'1e He-
CKOJ'IbKHX JIeT )!{H3HII B AJIeKcaH)J.pHII ApxHMe)J. B03Bpaw.aeTcSl
B CBOH pO)J.HOH ropo)J. H 3)J.eCb c03,lJ.aeT Bce CBOH TPY,'J,bl no
lIIaTel\laTIIKe H MeXaHHKe.
OcTaHOBIIl\1CH npe}!{)J.e Bcero Ha Tml, lJTO c,'J,e.'1a.'1 ApXHMe)J.
B 06.'1aCTH l\IaTel\1aTHKH. ECTeCTBeHHO, liTO l\IbI l\IO}!{eM TO.'1bKO
nepe4HCJIHTb Te BonpOCbI, KOTopble ero HHTepeCOBaJIH B 3TOH
06.TJaCTH.
Mbl Y>Ke BH)J.e31H, liTO ApXHlI1en. rJIy601<O HHTepeCOl3aJICH
3a,J.aLieii 0 KBa.n.paType Kpyra H HaIlle.'1 HH}!{HHii H BepXHHH
IIpC)\e.'1bI )J.31H lIIIC.1a "It, ynoTpe6HB ,n:.1H 3Toro TaKoH MeTo,n:,
KOTOpblM nOTO:\I nO.'1b30Ba.'1HCb )\PYfHe MaTeMaTIlKH H8 npo-
TH}!{eHHH MHOrHX BeKOB. vbYLi3H KOHHLieCKHe ceLieHHH, OH
nOCTaBHJI H peWHJI 3a.rr.allY 0 KBa.n.paType napa60JIbI. HaH,n:eHO
ero c04JIHeHHC 0 nOCTpOeHHJJ npaBHJIbHOrO cel\lHyrOJIbHHKa.
'627

Dpe)J.CTaBUB ce6e Tena, no.'1YLieHHble Bpaw.eHUe?\I KOHH-
tIeCKHX ce4eHUH BOKpyr onpe.rr.e.'leHHbIX ocen, Apxu!e;I. C03-
)\aeT TeopHIO 0 CBonCTBax, nOBepXHOCTHX II OO-beMax 3TH X
TeJI. OH TaK}!{e peillaeT pH)J. CJIO}!{HbIX 3a)J.aLi Ha KOM6MHaliIo
TeJI.
HeKOTopbIe TeapeMbI nJIaHUMeTpHU TaK}!{e BnepBbIe 6bIJIH
)J.OKa3aHbI . ApXHMe)J.OI\I. TaK, Teopellla 0 n"'Jow.a,n:H TpeyrOJIb-
HHKa no TpeM ero CTopOHaM
S::.,=Vp(p-a)(p-b)(p-c) ,
npunHCblBael\laSl fepoHY, BnepBbIe 6bI.'1a npe.rr..'JO}!{eHa Apxn-
I\Ie.n.oll!l).
MbI )\OJI}!{Hbl TaK}!{e BblCOKO oeHHTb Te lIIeTO)J.bI, KOTOpbIl\lH
nO.TJb30Ba.'JCSl ApXH!lIe)J. npu peilleHuH KaK lIIaTe!\taTuLieCKHX,
TaK Ii MexaHU4eCKUX BonpocoB. B LiUC.'1e ux MO}!{HO YKa3aTb
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}!{eH» , - KaK MeLi CO.'1)J.aTa nOpa3l1JI ero».
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w.a)J.H- MeHbillero Kpyra (1.JepT. 163).
4epT. 163.
331

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a
ueHTpa ero pa)J.HycoM '3 onHcaHa OKPY}!{HOCTb. Onpe)J.eJJHTb
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(LiepT. 164).
qepT. 164.
4epT. 165.
3. IJ,eHTpbI LieTbIpex KpyroB pacnOJIO>KeHbl B BepIllHHax
KBa)J.paTa co cTOpOHOfr a. Pa)l,J1ycbI Bcex KpyroB paBHbl a.
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KpyroB (LiepT. 165).
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4epT. 166.
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Cmpl13M
LJHC.'IO 'it paBHO 2.
332

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110.'10BHHe, KaK Ha )l,HaMeTpe, BHOBb nOCTpOHl'vI nO.'1yoKPY}!{HOCTH,
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4epT. 167.
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KaK 6b1 HH 6b1.'1a Y3Ka n0.'1 oca, 06pa30BaHHaH npHMbIMH KL
H MN, napaJIJIeJIbHblMH AB, HaH,n:ercH B HallleH nOCJIe,n:OBaTe.T!b-
HOCTR TaKoe MeCTO, Ha4HHaH C KOTOpOro Bce BOJIH006pa3HbIe
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333

J1MTEPATYPA
11>. P y ,n Ii 0, 0 KBa,npaType Kpyra. rTTM, 1934.
B. n. ill e p e MeT e B C K H ii, O'lepKH no IICTOpHH MaTelaTIIKH,
Y'me,nrll3. 1940.
C. 51. J1 Y P be, ApXIIMe,n, 1I3J(. AH CCCP, 1945.
C. H. ill pen J( e P. Tpll 3aJ\a'lH ,npeBlIen reOMeTpl!l1. M3 OnblTa
npoBe;J,eHIIH BHeKJlaCCHOH pa60Tbi no MaTeMaTHKe B CpeJl.Hen WKOJle,
Y'lne,nrll3, 1955.
B. 11. JI e 6 e ,n e B, O'JepKH no HCTOpHH rO'JHbiX HaYK. BbIn. 4,
3HaMeHHTbIe 3ap.a'lH Jl.peBHoCTH, M., 1917.
B. JI H T U M a H, CTapoe H HOBoe 0 Kpyre, I!JH3laTrH3. 1960.

r Jl A B A XIII
o rEOMETPHH 06AqEBCKOrO
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B03bMeM Ha npHl\foii CD TOtIKY M II coe)J.IfHIIM ee C TOLi-
KOH A. TaK KaK, no YC310BIIIO, CYMMa yr.'10B O)J.HOro TpeyrOJIb-
HHKa paBHa 2d, TO, no BTOpOH TeOpel\fe J1eLKaH)J.pa, CYMM8
yrJIOB TpeyrOJIbHIIKa ABM TaKLKe paBHa 2d II ?+=d (LiepT.
169), HO ',+;3=d, n03ToMY
ex="[ .
Dpe)J.cTaBHM ce6e, tITO TOLiKa M nepeMeUJ.aeTCH B.TJeBO no npH-
MOI1 CD H 3aHIIl\WeT pH)J. nO.'1Q)KeHIIii Ml' M 2 II T. )J.. DpI
3T01\l nepel\felIeHlIlI y.roJI ex, a 3Ha4IIT II paBHbIH eMY yro.'! ',.
6Y)J.eT Yl\feHblllaTbcH H MO>KeT 6bITb C)J.e.'1aH MeHbille JII060ro
3a)J.aHHoro yrJIa, a CJIe)J.OBaTeJIbHO, II l\IeHbllle E. KaK TO.'1bl<D
yrOJI I C)J.eJIaeTCH MeHbllle 2, npHMaH KL nOna)\eT BHyTPb Tpe-
yrOJ1bHm<a ABM" II nepece<JeT CTOPOHY BM" 3Toro TpeyroJIb-
HHKa. TeopeMa )J.OI{a3aHa.
Mbl BCTynaeM B "OBbl" reOMeTpH'IeCKH" MHp!
«B reoeTpliH Sl Hallle.'1 HeKOTopbIe HeCOBepweHCTBa, KO-
TopbIe H C<JIITalO npH4lIHOH TOro, qTO 3Ta HaYI<a )J.O HaCTOHUJ.ero
BpeMeHII He BbIlllJIa HII Ha o)J.IIH lllar 3a npe)\eJIbI TOro COCTOH-
HHH, B' KaKOM OHa K HaM nepelllJIa OT EBK.'1II)J.a».
TaK HaLiIIHaeT CBOIO KHHry «feOMeTpHLieCKile HCCJ1eJJ,OBa-
HIlS! no TeOpHJl napa.1J1e.'lbHbIX npHrllbIX» H. H. .T106aqeBCKUH.
KaKOBbl )Ke Te «HeCOBepweHCTBa», 0 KOTOpbIX rOBopliT
J106aqeBCKHii? 310 npe}!{,1J.e Bcero l<acaeTCH nHToro nocTYJIaTa
EBK.'1H)J.a 0 napa.'1JIeJIbHbIX npHMblx. NbI y}!{e roBOpliJIH 0
TOM, liTO 9Ta aKCIIOMa 6biJIa e)J.IIHCTBeIlHOH H3 Bcex al{CHOM
EBKllH)J.a, BbI3BaBwaH nOTpe6HOCTb )J.OKa3aTeJIbCTBa II, CJIe-
)J.OBaTe.'lbHO, He oC'Ia)J.a.'1a )J..1JH TOHKoro MaTeManfLiecKoro !lfbIlll-
JINIHH Toii HenOCpeJI.CTBeHHoH OqeBH)J.HOCTblO, KaKoH 06JIa,1J.a.1IH
Bce OCTa.'lbHbIe aJ{CHOl\Ibl.
J100atIeBCKIIii, C03)J.aBaH CBOIO reoMeTpHIO, npHHSIJI Bce
aKCIIOl\lbl EBKJIH)J.a, KpoMe O)J.HOH - aKCHOMbl a napa31.1IeJIb-
HbIX. OTcIo)J.a C31e)J.yeT, qTO Bce TeopeMbI reO;\IeTpHII EBI{JIHJJ,a,
KOTopbIe He 6a3HpYl{)TCH Ha nHTOl\l nOCTYJIaTe, 6e3 H3MeHeHHSI
344

nepeXOJHT B reOMeTpHIO J106aqeBCKOrO, HO cO)J.ep}!{aHHe BCeX
OCT<1JIbHbIX TeOpeM )J.OJI}!{HO 6bITb HHbIM.
DpocJIe.D.HM paccy}!{)\eHl1H J106aqeBCKOrO, npli nO!\!oUJ.I1
IWTOPbIX OH nO.D.BO)\HT LiJITaTeJlH K CBOeMY nOCTYJIaTy 0 napaJ1-
JIenbHb1X 1 ) (4epT. 170). .
«113 T04KH A onYCTHi\! Ha JIHHHI{) BC nepneH)J.HKYJISlp AD,
K KOTOpoMy B CBOIO 04epe)J.b BOCCTaBHM nepneH)J.MKYJIHp
EEl' B npSIMOM yrJIY EAD npSlMbIe, BbIXO,Lf,HUJ.JIe H3 TO'lI<JI A,
JIII60 Bce BCTpeqalOT JIJ1HHI{) BC (HanpHMep, FF I ) , JIH60 }!{e
HeKOTopbIe, nO)J.06HbIe nepneHJI.HI{YJIHPY EEl' He BCTpe4al{)T
1'.,
11
H,
G,
E,
c
8
4epT. 170.
JIHHMIf BC. He 3HaH, eCTb JIH nepneH,n:HKYJIflp EEl e)J.HH-
CTBeHHaH .'1HHMH, I<OTopafl He BCTpeLiaeTCfI C BC, 6Y)J.eill CtIH-
TaTb B03MmKHbIM, -liTO CYllleCTBYIOT H )\PYrHe JIHHIIH (HanpH-
Mep, GG I ), KOTopbIe He BCTpe'lal{)T BC, Cl<OJlbKO 6bI Mbl HX
HH npo.D.OJI}!{a31M. DpH nepexo)J.e OT nepeceKalOUJ.Hx .TIl1HHH
PP l , K HenepeceKalOUJ.HM GG I MbI )J.OJ1}!{Hbl BCTpeTHTb JIJIHHI{)
H HI - rpaHH'lHYIO JIHHHIO, no O)J.Hy CTOPOHY KOTOpOH HH
O)J.Ha JIHHHfI He BCTpeqaeT BC, Me}!{)\y TeM KaK no )J.pyrylO
CTOPOHY Ka>K.D.aSl JIHHHSI PF 1 nepeceKaeT BC».
TaKHe }!{e paccY}K.D.eHHSl MbI I\IO}!{eM npoBecTH H OTHOCH-
TeJIbHO npfiMbIx, npoxo)J.HUJ.HX 4epe3 TOqKY A M HaXO)J.HUJ.HXCH
BHYTPH yrJIa DAEl' M Te!\l caMbIM BBeCTH HOBYIO rpaHHLiHYI{)
.'1HHHIO KKI'
fpaHII'lHbIe .'IHHHH HHI II KKI J106aqeBCKHfI H Ha30IBaeT
napa.'1JIeJIbHbIMH npHMoH BC B TO'lKe A. TaKHM 06pa30J\!,
tIepe3 Ka}!{)J.YIO TOLiKY A n.'10CKOCTH npOXO)J.SIT )\Be npSlMbIe,
I) D:aJJbHeHllIlu1 OTpblBOK B3S!T 113 KHHfH JIo6aIJeBCKOfO «reOJeT'
pl!lJ€CKHC HCCJJe./.\OBaHHH no TeopHlf napaJJ.'1eJJbHblX .'1HHHf!".
341)

napaJIJIeJIbHbIe )J.aHHOH: npHM35:1 H HI' nap8J1JleJIbHaSi BC B
HanpaBJIeHIIH CB, II np5:lM8H KKl' napaJI.'1e.'IbHaH TOH JKe
npHMoH B npOTlIBOnOJIO}KHOM HanpaBJIeHHH BC. 06e 3TH
npSIMbIe paCnOJIO}!{eHbJ CHMMeTpHtIHO OTHOCHTeJIbHO nepneH-
)J.HKYJIHpa AD, onymeHHoro Ha BC.
I1Tal<, Mbr nO)J.OillJIH K aKCIlOMe 0 napaJIJIeJfbHbIX Y n06a-
'leBCI<oro. 3Ta aKCHOMa LiHTaeTCH TaK: «flf:)cmb (3 Oaf/HOU nAo-
Cfcocmu oaHa npRMaR u /le:J/CaU{G!i (3He amoii npRMOU mOl/1m;
m020a 'iepC3 Bmy mO'lKf:) JrtOJICHO np08ccmu oee pa3/lUCLHble na-
pa/7/le:l bHblC n p5l.MblC» 1 ) .
Bee npflMbIe, npoxo)J.flW.He qepe3 TOLij{Y A, n06aQeBcKHtr
pa3)\eJIHeT Ha TPH rpynnw:
a) nepeCeKal{)!l1.He npHMYIO BC;
6) )J.Be napaJIJIeJlbHbIe npflMoii BC;
B) He nepece,KalOw.He npHMYIO BC.
«DpH3H8KoM napaJIJIeJIIl3Ma CJIY}!{IlT TO, LiT9 JIHHIl5:1 CTa-
HOBHTCH nepeCeKal{)w.eH npH MaJIeHilleM OT/{JIOHeHHH B TY
CTOPOHY, r)J.e JIe}!{IIT napaJIJIeJIb, TaKHM 06pa30M, eCJIH AH
napaJIJIe.'lbHa CB, TO Ka}!{)J.aH JII1HHH AF, CKOJ1b 6bJ MaJI HI1
6bIJI yrOJI H AF, nepeceKaeT CB})2).
YrOJI HAD Me}!{)J.Y napaJIJIeJIbl{) H A H nepneH)J.HKYJISlpOM
AD ('lepT. 170) J106aLieBcKHH Ha3B8JI Y r JI 0 M nap a JI-
JI e JI b HOC T H H 0603HaLiHJI Liepe3 D(p), rJI.e p - )J.JIHHa
nepneHJI.HJ<YJISIpa, onyw.eHHoro H3 TOtIKH A Ha npm,rylO BC,
}!(eJ1aSI nOKa3aTb TaKHM 0603Ha'leHHeM, liTO yro.'1 napaJlJleJIb-
HOCTH eCTb cpYHJ<H5:I p, T. e. qTO BeJIHtIIlHa 3Toro yrJIa onpe-.
)\eJIeHHWM 06pa30M 3aBIICIIT OT paCCTOHHHH TO'lKII A )J.O npH-
MOH BC.
ECJIH cpYHKHSl n(p) He H3MeHHeTCH npH H3MeHeHHH p
H Bcer)J.a paBHa d, TO yroJI HAE paBeH 0 H He cyw.eCTByer
HHblX npHMbIX, He nepeCeKal{)li\HX BC, KpOMe npHMoH EEl;
TorJI.a cnpaBe)J.JIHBa aKCIlO:'Ia EBKJII1)J.a H Bce TeopeMbI, OCHO-
BaHHbie Ha Hell, T. e. nepe)J. HaMIl npHBbI'lHbIH HaM MHp EB-
KJI l1JI.a .
ECJII1 n(p) <d, TO HY}KHO )J.onYCTHTb )J.Be napaJIJIeJIH:
O)J.HY no O)J.HY CTOPOHY nepneH)J.HKYJIHpa AD, )J.pyrylO no
)\pyrylO ero CTOPOHY, II MbI nona)\aeM B HOBbIH reOMeTpHlJe-
CKHH MHP, MHp n06aLieBcKoro.
I) <1JopM}'.nlIpoBKa D. C. A.neKcaHAPoBa (<<4TO TaKoe HeeBKJllI1\OBa
reOMeTplIH», 113.1\. ADH PC<1JCP, M., 1950).
2) J] 0 (5 a IJ e B C K H ii, feOMeTpllIJeCKHe HCCJle!J,OBaHHfl no leopHU
napaJ1J1eJlbHblX JIll HHi'i , M.-J1., 1\31\. AH CCCP, 1945.
346

OT!{a3aBIllHCb OT nHToro nOCTYJIaTa EBKJIU,n:a Ii 3aMeHIIB
ero CBOUM, J106aQeBCKUH C03)\aJI HOBYIO reOMeTpHIO, KOTOpyro
OH Ha3BaJI «Bo06pa}!WCMOH reOMeTpHeH». DOCTeneHHO pa3BH-
BaR CBOIO reOMeTpHI{), J106aLieBCKHH )J.OBeJl ee .LIP Tex }!{e npe-
,n:eJIOB, ;'],0 KOTOpbIX 6bIJIa ;'],OBe)J,eHa EBKJIH)\OM ero reOMeTpHH,
H, qTO Bcero B3>KHee, npu 3TOM pa3BHTUH J106a'-!eBcKHII He
CT031KHYJICR HH C O)J.HUM npOTHBOpeLiHel\l, KOTopoe YKa3bIBa-
310 6bl, liTO HOBblH nocTyJIaT HeBepeH. J106aqeBCKUH H)J.e1
)J.aJIbwe: OH CBR3bIBaeT CBOIO reOMeTpHIO C HeKOTOpbJMU pa3-
)J,eJIaMU BblCllleH MaTeMaTHKJI H 3)J.eCb TaK}Ke He BCTpeqaeT
npOTHBopeLiUH. feOMeTpHH EBK.rm)l,a H reOMeTpHR J106atIeB-
CKOro B paBIWH Mepe COBepIlleHHbl. Ho eCJlII EBK.'1H)J., C03)J.a-
BaR CBOII «HatIaJIa», rJIaBHbIM 06pa30M nPHBen B CHCTeMY
°Bce, 'lTO 6bI.'10 (03)J,aHO B reOMeTpHH MHorHMH yqeHbIMH )1,0
Hero Ha npOH!}!{eHI1H ,'J,BYX-Tpex TbICSlLieJIeTHH, 10 J106aLieB-
CKI1H C03)J.a.1 CBOIO reOMeTpHIO B LiaCIH, CB060)J.HOH OT nOCTY-
JIaTa EBI{JII!)J.a, O)J.UIl B TeLieHHe HeCKOJlbKHX JIeT - TPYJI. no-
HCTHHe rnraHTCKHH. Pe3YJIbTaTbi 3Toro TpY)J.a HaCTO.1bKO
onepeJI.I!.'1H pa3BHTHe MaTeM8THLieCKOlI Mblc.nH Toro BpeMeHH, Ha-
CTOJ1bKO 6blJIH HemKH;'],aHHbIMH, tITO nOtITH HHKTO 113 MaTeMaTH-
KOB - COBpeMeHHHKOB .TI06aqeBCKOro - He 6bIJI rOTOB K
BOCnpHfITUIO l1;'],eH «B606pa}!{ael\IOJI reOMeTpHJI».
DOKa:KeM HeKoTopbIe H3 3THX pe3YJIbTaTOB B BH)\e TeopeM
neeBKJIH,c/>BOH reoMeTpHH J106a L leBcKoro H 6Y)J.eM TBep)J.o
nOMHHTb, LITO eC.'IH MbI npUH5:l31H nOCTYJIaT J106aQeBCKOro, TO
;'],OJI}!{Hbl npHHHTb H Bce BbITeKalOw.He 113 Hero CJle,n:CTBI1H
KaKHMH 6bJ CTpaHHbIMH OHH Ha nepBbui B3rJI5:I)J, HH Ka3a.I1HCb.
3)J.ecb YMecTHo npHBeCTH CJIO,Ba J106aLieBcKoro: «Ha nepBblH
B3rJIH)J. MO}!{eT nOKa3aTbCH, liTO Teope;vIbl reOMeTpIlH, Ha3BaH-
HofI MHOH Bo06pa}!{aei\mH reOMeTpI1eH, Boo6w.e npOTJIBOpeLIH-
Bbl, O;'],HaI<O 3aMeLiY, liTO XOTH OHH II npOHmOpeqaT npe;'],CTaB-
JIeHHI{) I' npl1BbI4Ke, HO, O,'J,HaKO, COBepIlleHHO 6e3ynpeQHbi CO
CTOpOHb! JIOmqeCKOH».
HOBbli1, HenpHBbl'IHbIi1 AJISI Hac Mlip
TeopeMa 1. Cy.1IAW Y2-,20B mpeY2o-,7bHUKQ ,1leHbUle 2d u
MeHR.eIllCfI om mpcY2o.1bHuKa K IllpCY20,1bHUKY.
.JJ. 0 K a 3 aTe .'1 b C T B O. CYl\Il\Ia yr.110B TpeyrOJIbHlH<a
He MQ)KeT paBH5:lTbCH 2d, TaI{ KaK OTCIO,n:a BbITeKa.'1 6b1 nocTy-
JIBT EBKJUI,n:a 0 napa.'l.'leJIbHbIX, HenpHeM.'IeMbIH B reOMeTpHiI
J106alJeBCKoro. Do nepBoH Teope:'le J1e}lwH;'],pa STa CYivE\Ia He
847

MO}!{eT 6bITb H 60JIbWe )J.BYX npHMbiX. CJ1e,AOBaTeJIbHO, CYMMa
yrJ10B Tpeyro.T1bHHKa MeHbWe 2d.
CYMMa yrJIOB TpeyrOJIbHHKa
 He MO}!{eT OCTaBaTbCH O)J.JIHa-
KOBOH JJ,JIH Bcex TpeyrOJ1bHHKOB,
TaK KaK 3TO YCJIOBHe npHBeJIO
6bI K He06xo.D.mIOCTH npHHSITHH
nHToro nOCTYJIaTa EBI{JIH)J.a.
lloKa}!{eM 3TO (qepT. 171).
1l0nycTHM, liTO CYMMa yrJIOB
Ii C TpeyrOJIbHHKa ocraeTCH nOCTOHH-
171. HOH ;'1,.'151 " Bcex TpeyrOJIbHHKOB;
0603HaQHM ee qepe3 1: 1 ). Tor,'J,a
a +  +'( = E;
a+V+o'=E;
r+W'+o" = E;
0' + 8" = 2d;
a+:1'+2/ 1'+W'+o"=2E;
(r.r:+;:S' +2"+ 1') + W +0") = 2E;
E + 2d = 2E;
E = 2d.
TaJUlM 06pa30M, npH YCJ10BI1H nOCTOHHCTBa CYMMbl yrJIOB
BCex TpeyrOJIbHHKOB CYMMa yrJIOB TpeyrOJ1bHHKa ABC OKa-
3bIBaCTCH paBHoH 2d, a OTCIO)J.a BbITeKaeT nocTy.'1aT EBKJIH)J.a,
Hen pHel\J.!I eMbI11 B reOMeTpUH J106aQeBCKoro.
Teope.Ma 2. Cf:)MMa Y2.;10e 6Cf/K020 6b/f1f:)K/1020 'lemblpex-
Y20.;lbHUKa A1eHbUle 4d, U n0311l0Jr/f:) He Cf:)Ujecm6f:)em npflMOf:)20/lb
HUK06.
lloKa3aTeJ1bcTBo 3TOH TeopeMbI npe)J.OCTaBJ1SleM LiHTaTeJII{).
Teope.Ma 3. B reOMeTpHH J106aQ€BcKoro He C!}Ujecm6YlOm
noiJo6Hble mpe!}20.;1bHUKU.
II 0 K a 3 aTe JI beT B O. o pe)J.nOJIO}!{IIM, QTO cyw.eCT-
BYI{)T ,ABa HepaBHbIX Tpeyro.lbHHKa C nonapHO paBHblMII yr-
JIail1II: LA=LAI' L.B=LB}, LC=LC I H AB>AlBl
(LiepT. 172). Ha cTopoHe AB OT BepIllHHbJ B OT.l0JKH!VI oTpe30K
BD, paBHbIH AIEl' LJ.epe3 TOQKY D npOB€;'1,eM npHMYIO DF
TaK, IlT06bJ LBDF=LBlAlC l . DpfllltaSl DF He l\W}KeT nepe-
ce% CTOPOHY AC, TaK KaK npH paBeHCTBe COOTBeTCTBe-HHbIX
yrJIOB BDF II BAC 6Y.ll.YT paBHbI II HaKpeCT JIe}Kaw.He yrJIbI,
a Hal\UI )J.OKa3aHa Teope:\1a: eCJ1H HaKpeCT .'1e}!ww.He yrJIbl paB-
Hbl, TO npflMbI€ He nepeC€KalOTCH ( 2). C.'1e.D.OBaTe.'!bHO, npH-
I)  _ ClInJa; 6pma rpe'leClwro aJl(paBIITa.
348

MaS! DF )J.OJI}!{Ha nepece% CTOPOHY BC B KaKOH-TO TOqKe £1).
TaK KaK 6ADE= .6A I B I C I H, CJIe)J;OBaTeJIbHO, LBED=
=LBIC1Al> TO B LieTblpexyrOJ1bHHKe ADEC CYMMa yrJIOB
paBHa 4d. B caMOM )J.eJIe,
ex+o=2d H "[+r:.=2d.
Pa3)J.eJIHH 310T qeTblpexyroJ1bHHK )\HarOHaJIbl{) CD Ha )J.Ba
TpeyrOJIbHHKa, nOJIytIHM, qTO B Ka}!{)J;OM H3 HHX CYMMa yrJIOB
paBHa 2d, TaK KaK, npe)J.nOJIaraH B O)J.HOM Tpeyro.'1bHHKe
CYMMY yrJIOB MeHbilleH 2d, MbI npHIllJIH 6bl K BblB0JI.Y, qTO B
B
C f
II
e
4epT. 172.
)J.pyroM 3Ta cyMMa OKa3a.'1aCb 6bI 60.'1bwe 2d, a TaKoH BbIBO)J.
npoTHBOpetIHT nepBoH TeopeMe J1e}!{aH,n:pa. ECJIH JKe cYMMa
yrJ10B TpeyrOJIbHHKa paBHa 2d, TO OTCIO)\a CJIe)\yeT nOCTYJIaT
EBKJIH,n:a, HenpHeM.'IeMbIt[ B re01IerpHH J106aLieBCKoro.
¥lTaK, He MorYT cyw.eCTBOBaTb ABa HepaBHbIX Tpeyro.TIb-
HHKa C nonapHO paBHbIMH yrJIaMH; CJIe,'J,OBaTeJIbHO, He cyw.e-
CTBYI{)T H no,n:06HbIe TpeyrO.'1bHHKH.
TeopeMa 4. EC.lU mpu fj2.;za OOH020 mpef:)2o,;1bHWW coom-
een1CmeeHHO pa8Hbl mprAl fj2:7aM opyeo2o, mo maKue mpeY201lJ;-
HUKll pa8Hbl.
JI. 0 K a 3 aTe JI b C T B O. ECJIH 6bI TaKHe Tpeyro.lbHHKH
He 6b1J1H paBHbl/l!H, TO OHH 6bJ.'lH 6bl nO)J.06Hbi. Ho nO)J.06HbIX
TpeyrO.lbHIIKOB B reOMeTpHlJ J106aLieBCI\Oro HeT. TaKHM 06-
pa301\1, K TpCM H3BeCTHbIl\I npH3HaKal\l paBencTBa Tpeyro.'1b-
HHKOB B reOl\IeTpliU J106aLieBcl<oro npu6aB.1HeTc5:I 4eTBepTbill
npH3HaK.
I) B remle1pHH KaK eBKmUlOBoii, TaK Ii HeeBKml,!\OBoil npllHHT no-
cTy.aT nawa: eC.1M npHlaH, He npOXO;HJU{afl 'lepe3 BeplllHHY Tpeyro.'Ib-
H.HKa, nepeceKaeT O.I\HY 113 CTOpOH 'fpeyronbHlma, TO OHa HenpeleHHO
nepece'lih O;l.H)' 113 .I\BYX .ilpyrllx CTOpOH TpeyrOJlbHIIKa TalOKe Me>I\J{Y
BepwilHaMH.
349

TeopeMa 5. IX/IR /l106oeo 3adaHho2o f:)2,1a ex JrLO:J/CHO Haumu
maKOU 011lpe30K p, 1{11l0 fl(p)=Cl.
B )\OKa3aTeJIbCTBe 3TOfI TeopeMbI 6Y)1.eM CJIe)1.0BaTb n06a-
LieBCKOMY.
nYCTb yrOJI BAC paBeH '1.. Ha AB BbI6epeM npOH3BOJIb-
HYIO TOLiKY BI' H3 KOTOpOH onYCTHM nepneH;'J,lIKy.'1Hp BIA}
Ha AC (tIepT. 173). OTJIOJKHlVI OT T04KH Al oTpe30K AIA2'


fJ


F


II


c


4epT. 173.


paBHbI11 oTpe3KY AAl' H BOCCTaBHM H3 AI!. nepneH)1.HKYJIHp
A 2 B 2 K AC. TaK 6Y)1.eM npO)1.0.'1}KaTb )1.0 Tex :lUp, nOKa npn;'J,eM
K nepneH,n:HI{y.'15:1py CF, KOTOpbIH He BCTpeTlIT AB. nOKa}!{eM,
QTO TaRoH nepneH,n:IIKYJ15:1p ,n:OJ1}!{eH cYUJ.ecTBoBaTb..
DycTb B TpeyrOJ1bHHKe ABIAI c)'MMa yrJIOB paBHa 2d-z;
TOr)1.a B TpeyrOJIbHHKe ABIA2 3Ta CYMI\W 6Y)1.eT paBHa 2d-2<.
(L0.ABIAI = L0.BIAIA2; CJ1e,n:OBaTe.'IbHO, C)'MMa yrJ10B B )1.B)'X
3TL1X TpeyrOJ1bHHKaX paBHa 4d-2s; OT6pacblBa5:1 yrJ1bI npH
T04Ke Alo HMeeI\l4d-2<.-2d=2d-2r:.). B Tpeyro.'1bHHKe AB 2 A 2
CYMMa )'r.10B 6y,n:eT MeHbille 2d-2e (nycTb 3Ta cYMMa paBHa
2d--:-x, a cYMMa yrJIOB TpeyrOJIbHHKa B I B 2 .4 2 paBHa 2d-'fj,
f)J.e x>O Ii 1J >0; Tor)1.a 2d-x=(2d-2r:.)+(2d-'fj)-2d=2d-
-2e+2d-"fj-2d=2d-2r:.-TJ. .Mbl npHIllJIH K paBeHCTBY:
2d-x=2d-2..-'1j; OTCIO,'J,a x=2s+1J, T. e. x>2s). Dpo;'J,OJ1-
}!{HM TaKoe nOCTpoeHHe ,n:o Tex nop, nOKa CYMMa yrJIOB KaKO-
rO-TO n-ro Tpeyro.'1bHHKa He CTaneT OTpH
aTeJ1bHOii II 3THM
06HapY>KHTC5:I HeB031IO>KHOCTb 06pa30BaHlISl 310ro TpeyrOJIb-
HHKa. DycTb TaKIIM nepneHJJ,HKymIpoM, He BCTpetIal{)UJ.HM CTO-
POHY AB, OKa3a.'1CH nepneH;'l,HKymIp CF. OH MO)KeT 6bITb
nepBblM cpe)J;H Tex nepneH)1.J1Ky.'1HpOB, KOTopbIe He BCTpeqalOT
350




CTOPOHY AB. ECJIlI me OH He 6Y.LI.eT nepBbJM, TO, BO BCHKOM
c.'lY'lae, npH nepeXO.ll.e 01' nepneH;:I.HKYJIHPOB, nepeceKalOrn.HX
AB, K nepneH.lI.HKYJIHpal\I, He nepeceKalOmHM e€, T8Ko1"! nep-
BbIH nepneH.lI.HKYJIHp DG .lI.OJI}!(eH cymeCTBOBaTb. 1-13 TOlCKH
o npOBe.ll.€M JIHHHIO DL, 06pa3YIOrn.ylO OCTpbIH yroJI GDL.
H3 K8KOi'r-.'1u60 TO'lKU H JIHHHH DL onYCTHM H8 AC nepneH-
).l,HI(YJIHP H K, npO.ll.OJI}KeHHe J<OToporo .lI.0JI}KHO nepece%
AB B KaKOH-TO TO'lKe N (nepBbIM He nepeceKalOrn.HM AB nep-
neH.lI.HKYJIHpOM, no YCJIOBHlO, 6Y.ll.eT DG). OH 06pa3yeT Tpe-
yrOJlbHHK AN K, BHYTPb I<OToporo BXO.ll.HT npHl\1aH DH U n03-
TOMY OHa BCTpeTHT fHnoTeHY3Y AN r.ll.e-JIH60 B TOlJKe M.
YrOJI GDH MO}KeT 6bITb B3HT Cl<OJIbJW yrO.ll.HO M8JIbIM, H BCe
)!(e JII06aH npHMaH, BbIXO.ll.Hrn.aH H3 D H B3HTaH BHYTpl1 yrJIa
GDH, nepeCe'leT CTOPOHY AB, U DG OKa}KeTCH rpaHH'lHOH
.TJHHHeH, KOTOpylO MbI Ha3Ba.'1H napaJIJIeJIbHOH K AB C yrJIOM
napaJI.'leJIbHOCTU 0.= D(AD). TaKUM 06pa30M, B HarueM CJIY'lae
p=AD. TeopeMa .lI.OKa3aHa.
3TOH TeopeMe MO}j<HO .lI.aTb JIpyrylO JII060nbITHYIO
OPMY-
JIUpOBKY: KUK08 6bl oempbtil YZO/I HU 6WI,. Beezoa eyU{eemByem
mUKOU nepneHOltKy.1Rp K OOHOII emopOHe amozo YZAU, KomopblU
He nepeeeKuem opyzylO cmopOHY.
C JI e l1. C T B H e: JII060ii OCTpbJH yrOJI HBJIHeTCH yrJIOM
napaJIJIeJIbHOCTH .lI.JIH HeKOToporo OTpe3I<a.
CnpaBe)J,JIHBa 11 06paTHaH TeOpeI\1a: KUJlCOOJ,ty nO.WJlCllme/lb-
HOJ,ty 'illCAY p (npu YCTaHOB.TJeHHOH e)J,I!HHue MepbJ, B KOTOpOH
P BbJpa}KaeT onpe.ll.eJIeHHbIH OTpe30K) eoomeemcmeyem oem-
pblU YZO.1 a., eB.R3QHHbIU e p coomHOUIeHUeM. D (p)=o..
TaKHM 06pa30M YCTaHaBJIUBaeTCH <;BH3b Me}!(.lI.Y OTpe3KOM
11 yr.'10M. Kam).l,bIH OTpe30K B reOMeTpHH J106a'leBCKOro 0.ll.-
H03HalJHO onpe.ll.eJIHeT yrOJI. TaKOH 3aBI1CUMOCTIi B reOMeTpHU
EBK.1H.lI.a HeT.
Teope.Ma 6, CPYHKI{IlR ll(p) RBiIRemeR J,tOHOmOfmo y6bl-
eafOUI,etl, npUHUJ>IaIOll{eU Bee 3HU'ieHltR om d ao O.
3TY I<OpOTKYIO
OPMYJIHpOBKY Ha.ll.O nOHl-1MaTb TaK: eCJIU
p nOCTeneHHO yBeJJU'lHBaTb OT 0 .lI.O 00, TO yrO.1 napaJlJleJIb-
HOCTH n (p) nOCTeneHHO y6bIBaeT, Bce 60JIee npU6JIH}!{aHCb
!{ HYJIIO npu p
 00 (lJepT. 174).
LI. 0 K a 3 aTe JI b C T B 0 ('lepT. 175). nYCTb All napaJI-
.'1e.'1bHa BC, AD - nepneH.lI.HKYJIHp Ha BC 11 AD=p. Tor).l,a
a=n(p). YMeHbruuM aprYMeHT p, B3HB OTpe30K AE <AD.
npOBe.ll.eM EEl nepneH.lI.UKYJIHpHO AD; EEl nepeCe'leT AH.
B caIIIOM )J,eJIe, npOBe.ll.H DK napaJIJleJIbHO EEl' MbI MO}KeM
YTBep}K.lI.aTb, '-1.1'0 DK nepecelJeT AH H nOJIY'lHTCH TpeyrOJIb-
351




HHK AKD. EEl' 6Y.lI.Y'lH napaJIJIeJIbHOH ..DK, He MomeT e
nepece'lb; c.lIe)J,oBaTeJIbHO, EEl .lI.OJImHa nepece'lb CTOPOHY
AK TpeyrOJIbHHKa ADK (nOCTYJIaT narua). 1.JT06bI nOJI)''lHTb
npHMYIO AM, napaJIJIeJIbHYIO EEl' J\lbI )J,o.lImHbI YBeJIH'lIlTb
yroJI a.. TaKHM 06pa30M, I1(AE»n(AD).
T 2
N Z
P2
A
=tv
D
"':":' /-1
A
H,
---L-
Dr
Dl
4epT. 174.
TeapeMa 7. l.JeM aa.we napa.vze,1bHbLe .l/.UHUU npOaO/lalOm-
Cfl B CI1l0pOHY napa.vze.ZbHocmu, me.1t 60/1ee OHU apy2 K apyc.y
npu6.ZUJ/CalOmcfl.
tJ
{
H
c
JJ
B
4epT. 175.
LI.oKa3aTe.lbCTBO (4epT. 176). nYCTb.AH na-
paJIJleJIbHa DB. 1-13 TOqI(H E, JlemamelI Ha AH OT A B CTOPOHY
napaJIJIeJIbHOCTH, onYCKaeM nepneHJlHKYJIHp EF Ha BD. TaR
li
D F
4epT. 176.
KaK L.HEF+L.AEF=2d, a .DAE+L.AEF<2d (CYMMa
yrJIOB QeTblpexyro.'1bHHKa 1\IeHbrue 4d), TO L.H EF> ..:---H AD,
T. e. n(EF»I1(AD). OTclO.lI.a, Ha OCHOBaHHH TOro, 'lTO IpYHK-
UHH n (p) - y6bIBalOaH, HMee:'l: EF <AD.
II
8
352

PaCCMOTpeHHbIe HaMH TeOpeMbI COCTaBJISIIOT TOJIbKO He-
60JIbruylO 'laCTb TeX nOJlOmeHHH, KOTOpbIe pa3BH.'1 H. 11. JIo-
6a'leBCKliH KaK B 06JIaCTH reO!\leTpHH, TaK H B .lI.pymx pa3.l1.e-
JIaX MaTeMaTHKH: TpHrOHOMeTpHH, aHaJIHTHqeCKOH II .lI.He-
peHUHaJIbHOH reOMeTpHH H np.
LI.JIH Heno.n:rOTOBJIeHHOrO qHTaTeJISI 3TH npe.ll.JIO>KeHHSI Ka-
}KyTCSI a6COJIIOTHO HenpHel\lJIeMbIMH H a6CYP.ll.HbIMH. Ho OHH
nOKamYTCH eMY 60JIee npHeMJIeMbIMH H MeHee a6cy, P.ll.HbJMH
ecJIH OH npHMeT, KaK 3TO H eCTb Ha caMOM .lI.eJIe, qTO «nJIOC-
KOCTb» H «npOCTpaHCTBO» JI06aqeBCKOro HHble, 'leM nJIOCKOCTb
H npOCTpaHCTBO EBKJIH.lI.a H, CJIe.ll.OBaTeJIbHO, CBOHCTBa reo-
l\IeTpH'leCKHX Hryp, pacnOv'IOmeHHbIX Ha «nJIOCKOCTH» JIo-
6aQeBCKOro, .lI.OJImHbI OTJIH'laTbCH OT CBOHCTB TaKHX me H-
ryp, pacnOJIOmeHHbIX Ha nJIOCKOCTH EBKJIH.lI.a. nOHCHHM Ha-
wy MbICJIb.
nOMHMO nJIOCKOH reOMeTpHH EBKJIH.lI.a, MorYT cyrn.eCTBO-
BaTb H .lI.pyrne reOMeTpHH, B KOTOpbIX paCCMaTpHBalOTCSI
CBOHCTBa Hryp, paCnOJlOmeHHblX He Ha nJIOCKOCTH, a Ha HHbIX
3aKOHOMepHO nOCTpoeHHbIX nOBepXHOCTHX, XOTH 6bI Ha no-
BepXHOCTH wapa. MomHo JIH rOBopHTb 0 reOMeTpHH "'I cpe?
MomHo, H TaKaH reOMeTpHSI C03.l1.ana. J3Y.ll.YT JIH CBOHCTBa
Hryp, pacnOJIOmeHHbIX Ha cepe, HHbIMH, qeM CBOHCTBa H-
ryp, paCnOJlOmeHHbIX Ha nJIOCKOCTH?
1.JT06bI OTBeTHTb Ha nOCJIe.ll.HHH Bonpoc, n03HaKOMHMCH
C HeKOTOpbII\-m H3 3THX CBOHCTB, YCJIOBHBWHCb npe)J,BapHTeJIb-
HO B c.1e.ll.YIOrn.el'l'I: Ha KaKOH 6bI nOBepXHOCTH Mbl HH CTpOHJIH
HrYPbI, COOTBeTCTBYIOHe npHMOH, . yrJIY, TpeyrOJIbHHKY,
IIIHoroyrO.:lbHHKY Ha nJIOCKOCTH, Mbi 6Y.ll.eM, B CJIY'lae HeB03-
MOmHOCTI1 nOCTpoeHHH Ha .lI.aHHOH nOBepXHOCTH npHMOH, coe-
.lI.HHHIOrn.eII .lI.Be TO'lKH, 3aMeHHTb ee JIHHHeH, KOTopaH HBJIH-
eTCH KpaT'laHruHM paCCTOHHHeM Mem.ll.Y 3THMH TO'lKaMH. Tmme
JII1HHH HOCHT Ha3BaHHe reO.ll.e3H'leCKHX JIHHHH. feO.ll.e3H-
qeCKHMH JIHHHHMH Ha nOBepXHOCTH ruapa 6Y.ll.YT .n:ym 60JIb-
WHX KpyroB.
Ha qepTemax 177 H 178 MbI HMeeM .lI.Ba TpeyrOJIbHHKa.
Ha nOBepxHocTH ruapa - CepH'leCKHe TpeyrOJlbHHKH; CTO-
pOHaMH HX HBJIHIOTCSI .lI.yrn 60JIbWHX KpyroB - reO.ll.e3Hqe-
CKHe JIHHHH. Ha 4epTeme 177 MABN - 31<BaTOp, PAP} H
PBP I - MepH.lI.HaHbI. Tor.ll.a CTopOHa AP TpeyrOJIbHHI<a P AB
nepneH.lI.HI<y.:1HpHa K CTOpmJe AB II PE nepneH.lI.HK)'.:lHpHa K
AB; CJIe.ll.OBaTeJIbHO, .lI.Ba nepneH.lI.HKYJIHpa K O.ll.HOH H TOll me
npHMOH MN (reO)J,e3H'leCKOH JIHHHH) Ha cpe nepeceKalOTcH
12 A. A. KOJlOCOB
353

B .lI.BYX T04KaX PH Pl' ECJIH LPAB=LPBA=90°, TO CY.\1-
Ma BCeX yrJ10B TpeyrOJIbHHKa ABP 60JIbrue 2d.
CpaBHHBaH CYMMy yrJIOB TpeyrOJlbHHKa APB C CYMMOt'i
yrJIOB TpeyrOJIbHHKa PBN, Mbl CKameM, 'lTO CYMl\1a yrJIOB
TpeyrOJJbHHKa PBN 60JIbWe CYMMbl yr.aOB TpeyrOJIbHI1I<a
M
4epT. 177.
'/
4epT. 178.
PAB, T. e. liTO 3Ta CYMMa MeHHeTCfI OT TpeyrOJlbHHKa K Tpey-
ro.'1bHHKY.
nO'leMY HaM He KamYTCH HeJIenbIMI1 3TH BbIBO.ll.bl? MO}!(eT
6bITb,. nOTOMY. liTO MbI 'laCTO BH)1,HM wap, .lI.ame mHBeM Ha
ero nOBepXHOCTH? 5Y.ll.eT JIH CnpaBe.ll.JIHBa reO:\leTpHH EBKJIH.lI.a
Ha nOBepXHOCTH 3eMHoro wapa, eCJIH Ha HeH CTpOHTb 60JIb-
WHe TpeyrOJIbHHKH HJIH MHoroyrOJIbHHKH, CTOpOHa!\lII KOTO-
pbIX 6y,n:YT )J,ym 60JIbruHx KpyroB (reO.ll.e3H'leClme JIHHI-IH)?
Ha nOCJIe.ll.HHH Bonpoc MbI OTBeTHM OTpml.aTeJIbHo. TaK)!{e H
aKTbI reoMeTpHH 3106a'leBcKoro He 6Y.ll.YT He.,lenbIMH, eCJIH
HX He CBH3bIBaTb C nJIOCKOCTblO H npOCTpaHCTBOM EBKJIH.lI.a.
OTKa3 OT nHToro nOCTYJIaTa EBKJIH.lI.a H 3aMeHa ero nOCTYJIa-
TOM 3106alJeBCKOro HeMHHyeMo npHBO)J,HT Hac K OTKa3Y OT
nJIOCKOCTH H npOCTpaHCTBa EBKJIH,n:a H 3aMeHe HX «nJIOCKO-
CTbtO» H «npOCTpaHCTBO:\I» 3106a4eBCKoro. npUXO.ll.HTCH Y.ll.HB-
MlTbCfI T0:\1Y, 'lTO 3106a'leBcKHH, He Hi\leH nepe,n: rJIa3aMH OCO-
60H «nJIOCKOCTH» H oc060ro «npOCTpaHcTBa», Mor BbIHCHHTb
CBoHCTBa qmryp. Haxo,n:Hrn.HXCH Ha 3TOM «nJIOCKOCTH» H B 3TOM
«npoCTpaHCTBe», nOKa3aTb, 'lTO reoMeTpuH EBKJTH.lI.a eCTb 'laCT-
Hblii CJIY'laH HOBOH, 60JIee o6m.eI'-i reOMeTpHl1. nO'leMY me 310-
6a'leBCKHH OCTaJICH O.ll.HHOKHM B cBoeH 60pb6e 3a C03.l1.aHUe
HOBoH reOMeTpHH? OO'leMY nO'lTIl HHKTO 113 MaTeMaTI1KOB-
354

COBpeMeHHHKOB JI06aQe8CKOrO-He no.n.ll.ep}KaJI ero B 3TOH
60pb6e? MaTel\1aTH4eCKaH CTopOHa pa60T JI06a'leBCKOro 6e3-
ynpe'lHa, 11 He OHa BbI3BaJIa COMHeHHH 11 .lI.a}Ke HacMeruKH yqe-
HbIX. COMHeHIfH 11 HaCMellIKI1 BbI3BaJIO )J,pyroe. JI06a4eBCKHH
He 6blJI YBepeH B TOM, 4TO MbI >KHBeM B npOCTpaHCTBe EBKJIH.n:a,
r.ll.e nHTblH nOCTYJlaT .n:eHCTBHTe.TJbHO HBJIHeTCJI nOCTYJIaTOM,
BOT TaKoH nOCTaHOBKH Bonpoca 11 He XOTeJII1 npHHHTb MaTe-
MaTI1KM - COBpei\1eHHHK11 JI06a'lenCKoro. CTaHOBHTcH BnOJJHe
nOHHTHbIMI1 nOnI>ITKI1 JI06a'leBCKOro onbITHblM nYTeM npo-
BepHTb CBOH npe)J,nOJIO}KeHHH, nonbITKH, He .n:aBllIHe onpe.ll.e-
JIeHHbIX pe3YJIbTaTOB. CTaHOBI1TCH TaK}Ke BnOJIHe nOHHTHbIM
HCKJllO'lHT€JIbHOe 3Ha'leHHe lIJJH np11HHTHH reOl\leTpHH JI06a-
4eBCKoro cpaKTa nOCTpoeHHH B npOCTpaHCTBe EBKJIH.lI.a pe-
aJIbHOH nOBepXHOCT11, Ha KOTOpOH BbIBO.ll.bl ero reOMeTpHH 6bl-
JIH 6w cnpaBe.n:JIHBW.
HTaJIbHHCKOMY MaTeMaTHKY OeJIbTpaM11 lIepe3 10--12 JJ€T
nOCJIe cMepTH JIo6_a'leBcKoro y.n:aJIOCb nOCTpOI1Tb TaKYIO no-
BepxHocTb 11 TeM caMbIM )J,OKa3aTb BceM, 4TO nOCTpoeHHH
JI06a'leBCKOro .lI.eikTBI1TeJIbHO npe,n,cTaBJIHIOT peaJIbHYIO reo-
MeTpH'lecKy'1O cHcTeMY, a He TOJIbKO Bo06pa}KaeMYIO.
nonblTaeMCH BblHCHI1Tb, 4TO }Ke 3TO 3a nOBepXHOCTh.
Ha 4epTe)t;e 179 lLenHaH JIHHHH ABC, Ta KpI1BaH, OPMY KO-
/J C
M N
4epT. 179. 4epT. 180.
Topoi-i npUHmfaeT lLenb, 3aKpen.leHHaH B .lI.ByX ee TOqKaX
A U C nolI. .lI.eHCTBUeM C11JIbI TH>KeCTI1. DpelI.CTaBHM ce6e HHTh
BC, YKpenJIIlmylO B TO'lKe C H nJJOTHO npH.,1eralOrn.ylO K
uenHoH J1I1HHH. ECJIH B3HTh 3TY H11Tb 3a KOHelL B H B HaTH-
HYTOM nOJ10}KeHI111, Kor.na OHa BCe BpeMH 6y,n,eT KacaThCH
I*
353

IvnHOH JIHHHH, CBepTbIBaTb C nOCJIe.ZI.HeH, TO KOHeu Bonn.,
weT B BepnIKaJIbHOH nJIOCKOCTH KPHBYIO BD, KOTOPYIO Ha-
3bIBalOT T p a K T P H CO fl. BpallI.aH nOCJIe.ll.HlO1O OKOJIO TOH
npHMOH MN, K KOTOpOH aCHMnTOTH'leCKH npH6JIH)!{aIOTCfl
BeTBH BD H BE TpaKTpHcbI, MbJ nOJIY'lHJIH nOBepxHocTb;
3TY nOBepxHocTb Ha3BaJIH nee B .lI. 0 C  e p 0 H ('lepT. 179,
180). Ha nceB)1,OCepe BbIDOJJHHIOTCH eCJIH H 'He Bee, TO MHO-
rHe nOJIO}KeHHH reOMeTpHH JI06a'leBCKOro, npH YCJIOBHI1 3a-
MeHbI npHMbIX JIHimfI reO.ZI.e3H'leCKHMH JIIUmHMH nceB.lI.OCe-
pbI; TaK, Ha HeH cnpaBe.ll.JIHB nOCTYJIaT JI06aQeBCKOro (qepT.
181), TeopeMa 0 cYMMe yrJIOB TpeyrOJIbHHKa H MHome )J,Pyme
ee nOJlO}KeHHH.
G..,
G
1\, B C
nQpannenbHQ C
{j HonpatJne HuU
H
c
B
4epT. 181.
<PaKT peaJIH3aIJ.HH reoMeTpHH JI06a'leBCKOro Ha nceB)J,o-
cepe HMeJI «He TOJIbKO nepBOCTeneHHoe MaTeMaTH'leCKOe, HO
11 HJIOCKOe 3Ha'leHHe: reOMeTpI-IH JI06a'leBCKOro He ecTb
KaKoe-.'IH60 YM03pHTeJIbHOe, HppeaJIbHOe nOCTpoeHHe, ee.3a-
KOHbI OCYllI.eCTBJIflIOTCfI Ha nOBepXHOCTHX, JIe}KallI.HX B HarueM
peaJIbHOM TpeXMepHOM npOCTpaHCTBe» (n. c. A JI eKe a H .ll.-
po B, lITO TaKoe HeeBKJIlI.lI.OBa reOMeTp11H, 113.l1.. AnH PC<PCP,
1950) .
3a Y.ZI.a'lHOH nOnblTKOH 5eJIbTpaMH nOCJIe.ll.OBaJI11 .ZI.PYrHe
HCTOJIKOBaHHH (KJIeHHa, nyaHKape) no.r!O)!{eHlIII JI06a'leB-
CKoro, KOTophle 3Ha'lHTeJIbHO cO.ll.eikTBOBaJIH npHBJIe'leHHIO
HHTepeca MaTeMaTHKOB I{ HeeBKJIH)!.OBbIM reoMeTpHHM. B pa3-
pa601Ke HOBbIX reOMeTpH'leCKHX H.lI.eH npHHHl\I8IOT Y'laCTHe
KpynHeHw11e IItaTeMaTHKH. «PaMKH Y'leH11H 0 npoCTpaHcTBe
pa3.l1.BHralOTcH npH 3TOM runpe H wnpe, B03HHKaIOT 06UIHP-
Hble HCCJIe.ZI.OBaHHH 0 pa3JIH'lHbIX MbICJIHMbIX reoMeTpHHx.
HaKoHeu, H.lI.eH HeeBK.TJH.ZI.0BbIX reoMeTpHH npoHHKalOT B :\Je-
xaHHKY H H3HKY H CTaHOBHTCH O.ll.I-IOH 113 npe.ZI.I10CbIJIOK COB-
356

peMeHHbIX <p113H'leCIUlX TeopHti, npUB0.ll.Hrn.HX K BbIBO,lI.Y. qTO
npOCTpaHCTBO <pU3114eCI<oro MHpa OKa3bIBaeTCH cBoe06pa3HbiM
HeeBKJIH.lI.OBblM npOCTpaHCTBOM. J106a'leBCKHH CTOHJI ellte B
Ha'l21JIe Toro nYTH, Ha KOTOpbIH BCTaJIU B Hawe BpeMSf MaTe-
l\1aTH4eCKHe 11 H3H'leCKHe BaYKH, HO OH nepBbIH co06rn.HJI
HanpaBJIeHHe HbIHeruHeMY HX. pa3BI-iTHIO» (A. n. 10 w K e-
B H q),
«KonepHHK reOMeTpHK»
HHKOJIaH ¥IBaHOBH'l J1 0 6 aile B C K H H, O,lI.HH H3 reHH-
aJIbHeHWUX MaTeMaTHKOB CBoeH 3noxu, 6bIJI B TO }Ke BpeMH
qeJIOBeKOM HCKJIID'lHTeJIbHoro )J,yrueBHoro 6JIaropO)J,CTBa, nopH-
.lI.O'lHOCT.!L HCnOJIHeHHbIH naTpHOTH'leCKOrO )J,OJIra H JIlD6BU K
pO.ll.HHe. «1.JeJIOBeK 06H3aH nepe,lI. C8MHM c060ii H nepe,lI. ,lI.py-
rUMH JllO.lI.bMH CTpeMHTbCH K rapMOHH'leCKOMY COBepweHCTBO-
BaHHIO Bcex CBOHX cnoc06HOCTeH - B 3TOM COCTOHT ero 'leCTb,
.lI.OCTOHHCTBO H CJIaBa, 3Toro Tpe6yeT ero naTpHOTH'leCKHH
.lI.OJIr», - rOBOpHJI J106a'leBCKHli, H He TOJIbKO rOBOpHJI, HO
H ,lI.OKa3aJI BceH CBoeH }KH3HblD, pa60TOH U 06rn.eCTBeHHOH
.lI.eHTeJIbHOCTblD, 'lTO OH 6bIJI TaKHM rapMOHH'leCKH COBepweH-
HblM 'leJIOBeKOM.
JIo6a'leBCKHH 6bIJI He TOJIbKO BeJIHKHl\I MaTeMaTI1KOM, HO
H YMeJIbIM, Tpe60BaTeJIbHbIM BocnUTaTeJIeM HOBoro nOKOJleHHH,
C KaKOH 6.'?cnorn.a.ll.HOH CUJIOH OH KJIeHMH.l HeBe}KeCTBO H
p)KaBlIHHY BpeMeHU, KOTopaH C r0.ll.aMU MO}KeT nOHBHTbCH B
,lJ.ywe 'leJIOBKa. «)KHTb - 3Ha'lHT... 'lYBCTBOBaTb Henpe-
pbIBHO'- HOBoe, KOTopoe HanOl\UmaJIO 6bI HaM, 'lTO MbI }KHBeil1,
HU'lTO TaK He CTeCIlHeT nOTOKa }KH3HU, KaK HeBe>KeCTBO:
MepTBOH npHMOH .lI.OpOrOlI npOBO}KaeT OHO >KH3Hb OT KOJIbI-
6eJIH K MOrHJIe», - V'lHJI JI06a'leBCKHH. ECJIU, no CJIOBaM
YWHHCKoro, Y'lUTeJIb' .lI.Ol}KeH 6bITb 06pa3UOM naCTOHrn.ero
'le.l0BeKa, TO J106a'leBCKHlI 6bIJI HilI.
CbIH MeJIKOrO tIHIIOBHHKa, HHKOJIav. BBaHOBH'l JI06a'leB-
CKlIlI P0.ll.H.I1CH B HH>KHeM HOBrOpO.ll.e B 1792 r. 1.Jepe3 5 JIeT
Yl\mpaeT ero OTeu. OCHpoTeBwaH C€i\!bH, COCTOHrn.aH H3 Ma-
TepH H Tpex cblHoBeH, nepee3>KaeT B K.a3aHb, B ropo.ll., r.ll.e
nOTOM npOTeKJIa nOJIHaH TBop4eCKHX YCTpeMJIeHI111 }K1f3Hb
J106a'leBCKoro. MaTb H!-II(OJIaH I1BaHoBH'la, }KeHmHHa He.
KJIlDtIHTeJIbHOH 3HeprHH, yCTp<!HBaeT CBOHX ,lI.eTeH B K.a3aB-
CKYID rHlIHIa3HlD Ha Ka3eHHbIH C'leT.
nOCTymm B rHlIIHa3HlD B 1802 r.. JI06a'leBCKHH y}!(e 'le-
pe3 nHTb JIeT, nHTHa.ll.U8TH.TJeTHHM IDHoweii, KOH'laeT ee H
357

CTaHOBHTCH CTV,lI.eHTOM Ka3aHCKOrO YHHBepCHTeTa. B YHHBep-
nneTe J106a'leBcKHII 6bICTpO Bbl.ll.BUraeTCH KaK JIY'lWUII no
MaTeMaTHKe CTY.lI.eHT. B 1811 r. OH npoH3BO,lI.HTCH B MafHCTpbI,
a B 1816 r. J106a'leBCKI1H Y}Ke npoeccop MaTeMaTHKH.
Ha'lHHaH C 1818 r., nOMHMO pa60Tbl B ymlBepcHTeTe,
J106aqeBclmi-i: Be.'l.eT 60.1bWYIO npOCBeTHTeJIbHYIO pa60TY. B
Te'leHHe HeCKOJlbK11X ,lI.ecHTHJIeT11i-i: OH PYKOBO,lI.HT BceH Y'le6-
HOH pa60T0I1 061lIHpHOrO Ka3aHC(<oro oKPyra. HHKOJI<:lH
¥IBaHOB114 .lI.eTaJIbHO 3HaKOMI1TCH C nOCTaHOBKOH npenO,lI.aBa-
HHH B WKOJIaX 11 .lI.06HBaeTcH KopeHHoro YJIY4weHHH npe-
nO.ll.aBaHI1H Bo06me H MaTeMaTHI<I1 B oc06eHHOCTH.
¥ICKJ1roQHTeJIbHO l\moro C,lI.eJIaJI J106a4eBCI\HH 11 .lI.JIH Ka-
3aHCIwro YHHBepCI1TeTa.
B 1819 r. Ha Hero B03JIO)KeHO 6bIJIO np11Be.ll.eJll1e B nopH-
,lI.0I\ ymmepcHTeTCKOH 61-16JIHOTeKI1. J106aQeBCIU1H nOHI1MaJI.
Kalwe orpO!\1Hoe 3Ha4eHHe .lI.JIH HaY'lHOH pa60TbI HMeeT 3Ta
6H6JIHOTel\a. 11 npOBe.1 1l0PY4CHllYlO el\JY pa60TY 6.1eCTHw.e H
c 60JIbWOH JI1060BblO.
B 1820 r. J106a4eBCK11H Ha3Ha'laeTCH .lI.eKaHOM <p113HI<O-
MaTeMaTH4eCKoro <}JaKYJIbTeTa. I-1 3,lI.eCb OH nOKa3bIBaeT ce6H
KaK HCKJIIO'lHTeJIbHbIH OpraHI13aTOp YQe6HOrO npouecca.
3a KaKYIO 6bI pa60TY HH 6pa.1cH J106a4eBcKHH, OH BCer.ll.a
BHOCHJI B Hee MHoro JII06BH, orpOl\IHYIO 3HepflflO H, He >KaJIeH
CBOHX CHJI, rJIy60Ko npOHHKaJI BO Bce ,lI.eTaJII1 ee. TaK, OH. pa-
60TaJI B CTpOHTeJIbHOM KOMI1TeTe YHHBepCHTeTa, CHa'laJIa KaK
ero YJIeH, a 3aTeM 11 npe.ll.Ce.ll.aTeJIb. J106a'leBCKHH aKTH'leCKH
CTaHOBHTCH CTpOHTCJleM Bcex 3,lI.aHHH Ka3aHCKoro YHHBep-
CHTeTa. OH He }!(aJIeeT BpeMeHI1, 4T06bl CneIJ.I1aJIbHO .lI.JIH 3Toro
H3Y'lHTb apXHTeKTYPY, 'lT06bI He ynYCTHTb H3 BH.lI.Y HH O,lI.HOH
.lI.eTaJIH nopY'leHHOrO eMY Ba}KHOrO .lI.eJIa.
B 1827 r. J106a 4eBCKoro 1136H pmOT peKTopoM Ka3aHcKoro
YHHBepcHTeTa. Ha 3TOM noCTY OH np06blJI n04TH 20 JIeT.
HecMoTpH Ha OQeHb 60JIbruOl1 H MHororpaHHbIH TPY,lI. Ha
noCTY peKTopa H CJIO>KHYIO HaY4HYIO pa60TY B 06JIaCTH Ma-
TeMaTHKH, J106a4eBcKHH OT.lI.aBaJI MHOro CHJI BocnHTaHHIO
cTY.lI.eH'leCKOH MOJIO.ll.e>KH. 4T06bl nOHHTb TOT nYTb. Ha KOTO-
pbllI 3BaJI J106a4eBCKHH CTY.lI.eH'leCTBO, .lI.OCTaT04HO BcnOMHHTb
CO.ll.ep}Kalme ero 3HaMeHHTOH pe'lH «0 Ba>KHeHWHX npe.ll.MeTaX
BocnHTaHHH», npOl-l3HeCeHHOH HM B 1828 r. nepe.ll. nOCTynalO-
rn.!-IMH B YHHBepcHTeT CTY.lI.eHTaMH. 3Ta pe'lb npoHHKHYTa
rJIy60KHM YBa}KfHl1eM K 'leJIOBe'leCKOH JIH'lHOCTH, K npocBe-
rn.el-!HIO H HaYK 11 rHeBOM Ha «HecnpaBe.ll.JIHBYIO cJIY'laI1HOCTb»
npHBHJIerHH O,lI.HHX H «HI13KOH .lI.OJIH» .lI.pyn-lx.
358

«Pe'lb J106a'leBcKoro HHTepeCHa H B TOM oTHorueHHH, 4TO
B HeH HBHO 3aMeTHbI BOCXO.ll.Hrn.He K nepe.ll.OBOH HJIOCOcpHH
XVIII B. MaTepHaJIHCTH'JeCKHe H,lI.eH J106a'leBCKOro, C O.ll.HOH
CTOpOHbI, a C ,lI.pyroi-i - ero He06bIKHoBeHHaH cTpacTHocTb B
OTHOllIeHHH K HaYKe H B03BbIllIeHllOe npe.ll.CTaBJIeHHe 06 06-
HJ,eCTBeHHOM )J,OJIre H 'lYBcTBe 'leCTl1 nO,'J.JIHHHOrO Y'leHoro.
3TH .lI.Ba 06CTOHTeJIbCTBa BeCbMa cyrn.ecTBeHHbI ,lI.JIH nOHHMa-
HHH HaY'lHOrO TBop4ecTBa J106atleBCKoro. Ero MaTepHaJIHCTlt-
4ecKHH no.ll.XO,lI. K OCHOBHblM nOHHTHHM HaYKH HBHJICH O,lI.HOH
113 npe.ll.nOCbIJIOK OTKpblTHH HeeBKJIH,lI.OBOH reOMeTpHH, a ero
npe.ll.aHHOCTb HCTHHe, B I1CKaHHH KOTOpOH OH BH,lI.eJI CBOH cno-
c06 CJIY}KeHHH HapO.ll.Y H lJeJIOBe4eCTBY, nOMOrJIa eMY He
OTcTynHTb B 60pb6e C HaCl\1eUIKaMH H.TJH paBHo,lI.YllIHeM,
KOTopbIe BCTpeTHJIH era HOBaTOpCKH€ H,lI.eH cpe.ll.H cOBpe-
MeHHHKOB»l) .
3HaKoMHcb L .LI.eHTeJIbHOCTbIO J106a'leBCKOro Ha noCTY peK-
Topa, HeJIb3H He cor.TJaCHTbCH C n. C AJIeKcaH,lI.pOBbIM, KO-
TOpbIH rOBopHT, 'lTO, eCJIH 6bI J106a'leBCKHH .lI.a}Ke He OCTaBHJI
«HH O.ll.HOH CTpO'lKH caMOCTOHTeJIbHbIX HaY'lHbIX HCCJIe.ll.OBalIl1H,
MbI .lI.0JI}KHbI 6bIJIH 6bl BcnOMHHTb 0 HeM, KaK 0 3Ha'lHTeJIbHeH-
llIeM HarueM YHHBepCHTeTCKOM .lI.eHTeJIe, KaK 0 'leJIOBeKe, KO-
TOphlH BbICOKOMY 3BaHHIO peKTopa YHHBepCl-ITeTa ,'J.aJI TaKYIO
nOJIHOTY cO,lI.ep)J(8HHH, KOTOpOH eMY He npH,lI.aBaJI, nO-BH.lI.H-
MOMY, HHKTO .lI.PyroH H3 JIHI..l, HOCHBruHX .3TO 3BaHHe, - ,lI.O,
BO BpeMH HJIH nOCJIe J106a'leBclwro»2)
napaJIJleJIbHO C .lI.eHTeJIbHOCTblO Jl06a'leBcKoro Ha noCTY
peKTopa paCT€T H HaY'lHOe TBop4eCTBO J106a4eBCKoro.
B 1845 r. J106a'leBCKHH 6bIJI B llIeCToii pa3 H36paH peKTopoM
YHHBepcHTeTa Ha O'lepe.ll.HOe 'leTbIpex.neTHe. O,lI.HaKO B aBrYCTe
cJIe,lI.YIOrn.ero rO,lI.a .lI.OJI}KHO 6bIJIO HCnOJIHHTbCH 30-JIeTHe ero
npocpeccopcKOH pa60TbI, 3a KOTOpbIM, no YCTaBY, .lI.OJI}KeH
6blJI nOCJIe,lI.OBaTb BbIXO,lI. B oTcTaBKY. COBeT YHHBepCHTeTa
B036Y.lI.HJI XO.ll.aTaHCTBO nepe,lI. l\II1HHCTpOM npOCBern.eHHH 06
OCTaBJIeHHH J106a'leBcI<oro ern.e Ha nHTb JIeT C TeM, 'lT06bI OH
OCTaJICH H Ha nocTy peKTopa.
«COBeT n04l1TaeT, - r.:laCllJI npOTOKOJI, - 3a oc06eHHYlO
'leCTb ,lI..1H YHHBepCHTeTa HMeTb B 4HCJIe npeccopOB CTOJ1b
OTJIHqHOrO Y'leHOrO H OnbITHOrO ne.ll.arora». Mmll1CTepCTBO He
1) A. n. 10 ill K e B II 'I, MaTnlaTI1l\a II ee npenO.D.aBaH"e B Poc-
CII" XVII-XIX BIJ.
2) C60pHHK «HHKQ_Tan l1BanoBH'I .r106a'leIJCKHf!>>, 113,.'.\. AH CCCP,
1943.
359

COlJJIO, O.ll.HaKO, HY}KHbIM HCnOJIHHTb npocb6y COBeTa. BMecTO
3TOrO JI06a4eBCKHH 6bIJI Ha3Ha'leH Ha BTOpOCTeneHHbIH noCT
nOMOmHHKa nOne'lHTe,'1H Ka3aHCKoro oKpyra. HenOKJIa.ZI.H-
CTbIH II npHMOH, CJIHWKOM npOCBemeHI-IbIH, JI06a'leBCKIIH. BH-
.ZI.IIMO, He 6blJI yrO.ll.eH Ha'laJIbCTBY HII Ha noCTY peKTopa, HII
Ha nOCTY nOne'lIITeJIH.
. JI06a'leBCKHH O'leHb rJIy60Ko 11 TH}KeJIO nepe}KHBa.'1 CBoe
OTCTpaHeHHe OT JII06IIMOH pa60TbI. Ero 3.ZI.opOBbe 6bICTpO
YXY.ZI.waJIOCb, OH Ha'laJI TepHTb 3peHIIe. B nOCJIe.ZI.HIIH ro.ll.
}KH3HII, Y>Ke COBceM OCJIenWIIH, OH .lI.IIKTYCT .lI.BYM Y'leHHKaM
CBOIO nOCJIe.ll.HIOIO pa60TY «naHreOMeTpHH». YMep JI06a'leB-
CKHH 12 BpaJIH 1856 r.
JI06a'leBCKI1H OTKpbIJI HOBYIO cTpamlUY B reoMeTplHL
O.ZI.HH 113 aHr JIHHCKI-fX MaTeMaTIIKOB Ha3BaJI JI06a'leBCKOro
«KonepHHKOM reOMeTpHH». H .lI.eHCTBHTeJIbHO, TBOp'leCTBO
JI06a'leBCKOro OKa3aJIO He MeHbwee peBOJIIOUHOHHoe BJIHHHHe
Ha pa3BHT11e reOMeTpl1H, lJeM B CBoe BpeMH CHCTeMa MHpa Ko-
nepHHI(a Ha pa3BHTHe aCTpOHOMHH. LLa}Ke TaKOH 60JIbWOH
MaTeMaTHK, KaK faycc, nOJIb30BaBruHHcH rpOI\ta.ZI.HbIM aBTO-
pHTeTOM Cpe.ll.H COBpeMeHHHKOB, KOr.ll.a Y Hero nOHBHJIHCb
H.ZI.el!, 6JIII3KHe H.lI.eHM JI06a'leBCKOro, n060HJICH HX 06Ha-
pO.ll.OBaTb. OH nHCaJI O.ll.HOMY H3 CBOHX .ZI.pY3eH: «B03MO}K-
HO .ZI.a}Ke, 'tTO H He peWYCb Ha 9TO BCIO CBOIO }KH3Hb, H60 H
60IOCb Kp11Ka 6eOTHHueB, KOTOpbIH nO.ll.HHMeTCH, Kor.ll.a 51
BbICKa}KY CBOH B033peHHH ueJIIIKOM». TaK faycc H nOCTynHJI
B .ZI.eHCTBHTe,'1bHOCTH. HaCKOJIbKO ueHHJI faycc HOBbIe H.ZI.eH
JI06a'leBCIWro, BH)J,RO Y}Ke 113 Toro, 'lTO OH Cnel..lHaJIbHO H3Y-
'lHJI PYCCKHH H3blK, 'lT06bI B nO.ll.JIHHHHKe 'lHTaTb CO'lHHeHHH
JI06a'leBCKoro. faycc }Ke npOBeJI 1I06a'leBcKoro B 'lJIeHbI-KOp-
pecnOH.ZI.eHTbI «feTTHllreHCKoro Y'leHOrO 06rn.eCTBa», KOTopoe
TOr.ZI.a HOCHJIO xapaKTep aKa.ZI.eMHH, H caM H3BeCTHJ,! ero 06 nOM.
JI06a'leBCKHH, B npOTHBOnOJIO}KHOCTb fayccy, He n060HJI-
CH «KpHKa 6eOTHHueB». OcTaBaHcb B OHlOweHHH CBOHX H.ZI.eii co-
BepweHHo O.ZI.HHOKHM, OH ynopHo 3arn.Hrn.aeT HX, ny6JIHKYH
CB0I1 pa60TbI. H HY}KHO nO'lyBcTBoBaTb BCIO Tpare.ZI.HIO II B TO
}Ke BpeMH BCIO CHJIY BOJIH BeJIHKOrO MaTeMaTHKa, KOr.ZI.a OH,
OKpY}KeHHbli'i nO,1]HbI:\1 HenOHHMaHHeM, a nO.ll.'lac H HaCMeWKa-
MH, y)!(e COBceM OCJIenWHH, OTCTaHBaJI CBOH B3rJIH.ZI.bI.
n06e.ll.a nplIw.'1a n03}Ke, HO, K CO}!(aJIeHHIO, Y}Ke noc.'1€
CMepTH JI06a'leBCKoro. Ily6JJHKaUHH nepellHCIm faycca, B KO-
TOpOll H.lI.eHM JIo6a'l€BCKOrO .ZI.aHa caMaH BbICOKaH OI1.eHI<a,
I1 pa60TbI OeJIbTpaMH npIIBJIeKJII1 BHIIMaHHe K ero TpY)J,aM.
KHHrH JI06a'1eBCKOro HalJHHaIOT 113Y'laTb, nepeBO.ll.11Tb Ha
360

,Apyme H3bIl\H. B 1867 r. BbIXO)J,HT B CBeT HTaJIbHHCImu ne-
peBOJI. «naHreoMeTpllw>, H B 3TO )j{e apeMH HalJHHaeTCH pa3-
BHTHe HOBOrl reOMeTpIUI. B ee C03JI.aHHH npHHHIVIaIOT Y'jaCTHe
JIY4illHe 113 Y'leHbIX ,MaTelltaTHKOB: PHMaH, KJIeHH, nyaHl<a-
pe, rH.'Ib6epT H JI.p. Ho MbI )J,OJImHbI Bcer)J,a nOMHHTb, lITO nep-
BbIM Y'leHbIM, He n060HBillHMCH 60pOTbCH 3a HOBbIe I1)J,eH B
reOMeTpHH, nepBbIM MaTeMaTlIKOM, 3aJIOmHBruHM YH)J,aMeHT
AJIH pa3BHTHH HeeBKJIU.lI.OBbIX reO:\1eTpHH, 6bIJI HalIl COOTe-
'leCTBeHHIiK HHKOJIaH I1BaHoBH4 JIo6alIeBCKHit
J1HTEPATYPA
B. B. K Y T Y 3 0 B, feOMeTpuSI J106a<renclwro U 9JIeMeHTbl OCHO-
BaHUH remleTpllH, YLm e,n,r 113 , 1950.
H. 11. J1 0 6 a 'I e. Be K H ii, reOMeTpH'IeClme IICCJJeAOBaHHH no
TeopHU napaJJJIeJILHblX JIUHuii. KOMMeHTapUH, BCTynHTeJIbHaSl' CTaTLSI
II npUMeLjaHIISI B. <D. KaraHa, U3!1,. AH CCCP, 1945.
r1. A. ill U P 0 I{ 0 B, KpaTimn o'lepl{ oeHOD reoMeTpllu J100a'leB-
CKoro, rHTTJ1, 1955.
n. C. A JI eKe a HAP 0 B, 4TO Talwe HeeBKJIu,nOBa reOMeTpllH,
113.1\. Ar1H PC<DCP, 1950.
A. r1. to 1ll K e B H 'I, MaTeMaTIlKa II ee npeno,n,aB'aHue _B POCCUH.
H. 11. J1ooaQeBclmii. «MaTeMaTIIKa D lllKo.e», 1948, N2 2.
B. <D. K a r a H, J106a'lenCKlli'I, M.-J1., 113,1 AH CCCP, 1948.
11. 3 a 00 T II II, J100a'lencKtJIJ, TaTlIHllroIl3,!laT, Ka3aHb, 1954.
361





.;;,-
-:::::::-::;..
r J/ A B A XIV
TPl1fOHOMETPHS1
«C6.'1H>KeIHle TeOplill H npaKTlilm
A8eT calble 6.ijar01BOpHble l'e3YJJbTa-
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IIrpbIBaeT; C8Ma HaYlla pa3BIIBaeTCH
no;! amlHIIIIEM ee».
n. J/. '1e6wUJeB
CnYTHHK aCTpOHOMHH
B 03HHKIIOBeHl1e TpHrOHD:\feTpHIl CB5J38110 C pa3BH-
THet O.lI.HofI 113 J1.peBHeflUlHX HaYK - 3CTpOHOI\II1H.
.C CaMblX OT.lI.aJI€HHbIX Bpe:\I€H fIOTpe6HOCTH TOp-
rOB.nH, C O:I,HOH CTOpOHbI, Ii .110603H3TeJIhHOCTb-
C )J,pyroii, TO.1KaJIH .mO:I,eH H3 )J,JIHTl\nbHble nYTeweCTBHH no
J\apaBaHHbIM .lI.OpOrml, 'lepe3 He0603pH:\fble nycTbIHHbIe H CTen-
Hble npOCTpaHCTBa A3HH H ApHIm, H.IIH MOpCKHMH nYTHMH,
B)J,Olb H3BHJIHCTOH 6eperOBOH .nHHIIJ1 Cpe)J,H3el\1Horo MOpH,
BOKpyr ApaBHHCKoro fIOJ1YOCTpOBa II }1.a.1ee K 6eperal\l I1H;UIH
H KIiT3H. B 3TIIX nYTeWeCTBHHX Bcer)J,a 6bIJIH He06xo,J,mlbI
l<aKlle-TO BeXH, "lT06w He OTKJIOHHTbCH OT BepHoro nYTII. Ta-
"I1:\1H BeXal\1H C ,[laBHHX nop C.'1Y>KH.'1H CO.1Hue :I,He:\1 II 3Be3,J.bI
;i:2

HO%IO. OHH YKa3bIBaJIH npaBI1JIbHbIH nyib KaK KapaBaHaM
B I1X MHOrO.ll.HeBHbIX nepeXO.lI.ax no 6e3JI1O)J,Hbr\, npOCTpaHCTBaM
3eMJIH, TaK H Mopexo.n.aM B I1X )J,aJIeKHX CTpaHCTBOBaHI1HX
B.lI.OJJb 'lymHx 6eperOB.
n03TOMY eCTecTBeHHo HCKaTb 3apom)J,eHl1e nepBbIX aCT pO-
HOMI14eCKI1X Ha6JIIO)J,eHHH H CBH3aHHbIX C HI1MH npl1MHTHBHbIX
Tpl1roHoMeTpl1'leCKI1X CBe,'J,eHHI y .lI.peBHeHlImx HapO.lI.OB A3HH 11
ApHKH, Be.lI.YU.ll1x TOprOBJIIO .lI.pyr C .lI.pyrorl'l TaKHMI1 Hapo-
.lI.a!\II1 6bIJIH: KHTaIUbI. HH.lI.IiftUbI, BaBHJIOHHHe H erHnTHHe.
KOHe'lHO, B Te .lI.aJIeKHe BpeMeHa He MOrJIO 6bITb H pe411 0
Tpl1rOHOMeTpl111 KaK caMOCTOHTeJIbHOI BeTBH MaTeManIKI1;
OHa HOCIiJIa Tor)J,a COBepweHHO HHOH xapaI<Tep, qeM TOT,
KOTOpbIH npHcylU eft B HaCTOHlUee BpeMH, llame Ha3BaHI1H.
«TpHroHoMeTpHH» He cyrn.eCTBOBaJIO. 3TO CKopee 6blJIH HeKO-
TopbIe BcnOMOraTeJIbHbIe npl1eMbI, n03BOJIHIOrn.He IJpOH3BO-
,'J,HTb BbI4I1CJIeHHH, B03HImalOrn.He B pe3YJIbTaTe np0I13Be.lI.eH-
HblX aCTpOHOMI14eCKHX Ha6JIIO)J,eHI1H. TaK, B O.ll.HOM H3 .lI.peB-
HeUIlHX MaTeMaTI1'lecl\HX CO'lI1HeHHH KMTaH, OTHOCfIlUeMCH
K TpeTbeMY TbICH'leJIeTHIO .lI.0 HarueH 3pbI, Cpe.lI.H .lI.pyrHX pa3-
D,eJIOB !\taTeMaTHKH eCTb pa3.lI.eJI, BKJIIO'lalOrn.HH HeKOTO-
pbIe BbI'lHCJIeHl1fI TpHroHoMeTpH'leCKOrO xapaKTepa. B Ba-
BHJIOHe, r.ll.e aCTpOHOMHH .lI.OCTHrJIa 3Ha'lHTeJJbHOrO pa3BHTI1H,
r)J,e 6bIJIa COCTaBJleHa O;J,Ha 113 nepBbIX KapT 3Be3.lI.HOrO He6a,
r.lI.e YMeJIIi npe.lI.CI<a3bIBaTb COJIHe4HbIe Ii JIYHHbIe 3aTMeHlIfI,
TaKme He MOrJIH 060ITHCb 6e3 npOCTeHIIH1X Tpl1rOHO!\leTpli-
'leCKHX CBe.lI.€HI1H H YMe.l!11 nOJIb30BaTbCH HMI1 npH pa3JJH'lHhIX
aCTpOHOMH'leCKHX paC4eTax. llpeBHHH ErHneT Tome naKOmIJI
HeKOTopbIe CBe.ll.eHHH no aCTpOHOl\IHH; TaM 6bIJ1 C03.lI.aH KaJIen-
.lI.apb, KOTOpblM IIH1pOKO nO.'Ib30BaJIHCb npl1 npOBe)J,eHI1H pa-
60T no ce.'IbCI\OMY X03fIHCTBY.
vITaK, MomHO c'IHTaTb, 'lTO TpHrOHoMeTpIiH 3apO.ll.llJIaCb
B CTpaHax .lI.penHero BocToKa H, 6Y.ll.Y'l11 TecHO cBH3aHHoH C
aCTpOHmmeH, c.lI.eJIaJIa 3.lI.ecb nepBbIe warl1 B cBoeM pa3-
BHTHI1.
B 60JIee n03.l1.Hee Bpe!\lH, KOr.ll.a Ha IiCTOpl1'lecKYIO apeHY
BbIcTynHJIa fpeLl,l1fI, KOr.ll.a pa3BHTI1e MaTeMaTI1KH .lI.OCTl1rJIO
BblCOKoro YPOBHH 11 y}!(e 6bIJII1 C03.l1.aHbI «Ha4a.'Ia» EBKJIH)J,a,
HaKonHBwHHcH MaTepHaJI aCTpOHOMI1'leCKIiX Ha6JIIO)J,eHHii no-
Tpe60BaJI 60JIee CTporoH MaTeMaTH'leCKOH 06pa60TKI1 MeTO.ll.a!\lH,
OTJIH'lHbIMH OT 411CTO reOMeTpl1'leCKIlx. 3TH HOBbIe MeTO)J,bl
B KOHLI,e KOHLI,OB 11 npHBeJIH K C03J.I.aHHIO HOBoro pa3.l1.e.'Ia Ma-
T2!\laTHKII, KOTOpbIH MbI Tenepb H33bIBaeM T p H r 0 HuM e T-
p 11 e H.
363

Ho nYTb C03.l1.aHI1H 3TOH HOBOH BeTBH MaTe;\laTHKI1 6bIJI
OtleHb )J,OJIfliM. y!3 rpe'leCKHX MaTe:\1aTHKOB, CTOHrn.HX B Ha-
tlaJIe :)TOrO nYTI1, MbI JI.0JI}!<HbI Ha3BaTb f 11 n nap x a (II B.
.lI.0 H. 3.) 11 n l' 0 .!J eM e H (II B. H. 3.), C03)J,aBruHx nepBble
Ta6JII1UbI XOPJI.. B Te'leHHe nOCJIeJI.YlOml1x 10 CTOJIeTHH nOCTe-
neHHO HaKanJIHBaeTCH HOBbIH TpHrOHOMeTpl1'leCKI1H MaTepl1aJI,
rJIaBHbIM 06pa30M B CTpaHax BOCTOKa 11 bJIHmHerO BOCTOKa,
H YCOBeprueHCTBYIOTCH paHee COCTaBJIeHHble Ta6JII1UbI. B KOH-
ue 3Toro nepHO.ll.a TpHrOHO:\IeTpHH OT)J,eJIHeTCH 01' aCTpOHO-
MHH 11 CTaHOBHTCH caMOCTOHTe.1JbHOH HaYKoiL 3TO OT.lI.e.!JeHHe
06bI'lHO CBH3bIBalOT C HMeHeM a3ep6aHJI.manCKOro MaTeMaTI1-
Ka Hac 11 pOI JI..lI. 11 HaT y C 11 (1201-1274). 11, HaKoHeU,
nOTpe6oBaJIOCb eme 500 .!JeT, 'lT06bI Tpl1rOHOMeTpHH no opMe
H cO.ll.epJKaHl11O CTaJIa 6JII13KOH cOBpeMeHHoH Tpl1rOHOMeTpHl1,
KaK C03J1.aBaJ1HCb nepBble Ta6J1Hl..\bl TpHroHoMeTpHlJeCKUX
Be.rJHIJHH
TpHroHoMeTpHH eme B JI.peBHeH fpeUHH, nepeXO)J,H OT
CBoero nepBOHa'laJlbHOrO 3Tana pa3BHTI1H K HOBOMY 3Tany,
AOJJmHa 6bIJIa npefK)J,e Bcero paCKpbITb 3aBI1CmlOCTlI MemJI.Y
pa3HOpO.ll.HbIMI1 3JIeMeHTaMH TpeyrOJIbHI1Ka l1)J,aTb B03MomHOCTb
pellleHHH ero, T. e. onpe.ll.e.'IeHHH Bcex 3JIeMeHTOB ero no TpeM
)J,aHHbIM, 113 KOTOpbIX no KpaHHei-i Mepe OJI.HH )J,OJI}!(eH 6bITb
JIHHeHHbIM. 31'0 np0l130ruJIO He cpa3Y.
nepBoH_ 3aJI.a'leH, KOTOPYIO nOCTa BlI.'I 11
nepeA C060H rpe'leCKHe MaTel\1aTHKI1, 6bI.1a
3aJI.alla 0 perueHI1H npHMoyro.1JbHOrO Tpe-
yrOJIbHI1Ka. I1CTOpl1'leCKI1 CJIY'lH.!JOCb TaK,
'lTO c03)J,aHHe npHMOJIHHeHHoi'I TpHroHoMeT-
pHI1 Ha'laJIOCb C COCTaBJIeHI1H Ta6JIH XOP.LI.
OCTpbIX yrJIOB. Bcer)J,a B03MomHO 3MnHpl1-
C ' 'leCKOe, rpy60e onpe.ll.eJIeHl1e cHHyca OCTpO-
ro yrJIa nYTe1 nocTpoeHHH npHMoyroJlb-
Horo TpeyrOJIbHI1Ka C .lI.aHHbIl\l yrJIOM A
('lepT. 182). ECJII1 nOCJIe 113MepeHI1H o)J,HoH
11 TOH me MepoH KaTeTa BC, JIe}!(amero
npOTHB yrJIa A 11 mnoTeHy3bI AB, B3HTb oTHorueHHe BC
K AB, TO MbI 11 HaHJI.eM npH6.nHmeHHoe 3Ha'leHHe CI1-
Hyca yrJIa A. Ho )J,peBHue MaTeMaTHKH He nOIII.'IH no 3TOMY
nyn!. 113BecTHble 1I3 reOMeT'pHI1 EBKJIHJI.a COOTHomeHHH Me}K-
.lI.Y CTOpOHaMI1 npaBIIJIbHblX BnHcaHHbIX MHoroyrOJIbHHKOB 11
pa.n.HYCOM :.lTOll oKpymHOCTH TOJIKaJIH I1X Ha I1HOH nYTb onpe-
B
4epT. 182.
364

.lI.eJIeHI1H CI1Hyca )J,aHHOrO yrJIa - nyTb B bI '1 If C JI H T e JI b-
H bI H. 3TOT nYTb 6bIJI 6.'lee TpY,AOeMKHM, HO n03BO.'lHJI no.
JIYlll1Tb pe3YJIbTaT C onpe)J,eJIeHHOH 3a)J,aHHOH TO'lHOCTblO,
BcnOMHHM Te COOTHOIlleHHH Mem.lI.Y .lI.JIHHOH cmpOHbI npa-
BI1JIbHOrO MHoroyrOJIbHI1Ka H pa.ll.l1YCOM Onl1CaHHOH OKpy}j{-
HOCTH, KOTopble MbI H3Y'laJIH B Kypce reOMeTpHH.
Xop.ll.a, CTHrHBalOlUa5f .lI.yry B 120 0 , 5fB.'lHeTCH CTOpOHOH
npaBHJIbHOrO Tpeyro.'lbHHKa. I-'bBecTHO, lITO a 3 = R V3; Tor-
.lI.a H3 TpeyrOJIbHJlKa OBC (qepT. 183, a) HMeeM:
si n 60° = RV 3 , R = va
2' 2 0
TaK KaK a 6 = R, TO (lIepT, 183,6)
. 3 00 R R 1
SIn =T! =2 0
a)
l.)
II
6J
4epT. 183.
8
dJ
I13BeCTHO. lITO a4 = R V2; TOr.ll.a H3 TpeyrO.'IbHHKa OBC
(qepT, 183, B)
. 45 0 R 1/2 R 112
sm =: =.
TaK KaK aij = i V 10+ 2 V5. TO (qepT, 183, r\
265

).
;.,
. .
'\
-,."
..
....
,.....
',.
..
4
KJlaBJlHn nTOJleMeA.
-sin 36° =  1f 10+ 2 V5 : R= 11 1021(5
. Jls + I
HaKoHeLl. 3HaH, 4TO Q10 = R 2 ,lbI CiO}!(eM H3 tIep-
Te>Ka 183,.lI. nOJIY'lli fb:
in 18° =  (V5 + I): R = V 5 4 + I .
TaKI-IM 06pa30M, 3HaH 1-13 reOMeTpliH .lI.JII-IHbI Xop.lJ. ..lJ.JISI .lJ.yr
B 36°, 60°, 72°, 90° 1-1 120°, MO>KHO Bbl'lHCJII-ITb sin 18°, sin 30°,
sin 36°, sin 45° 11 sin 60°. OCTaeTcH TOJIbKO H3blcKaTbcnoc06bl
onpe.n.eJIeHHH .lI.JIHH xOP.ll., cOOTBeTCTlIYIOrn.HX 60JIee MeJIKHM
yrJI3M. 4T06bI CTaJIO B03MomHbIM COCTaBJIeHlie Ta6.'1liUbI CI-I-
HYCOB.
TaKYIO pa60TY BnepBbIe npo.ll.eJIaJI rpeqeCKI-IH aCTpoHoM
r 1-1 n nap x BO I I B. .lJ.O H. 3. Ta6JI H UbI Xop.lJ., BbI41-1CJIeHHbIe
\
rlinnapXOM, .lJ.0 Hac He .lJ.OruJIH, 1-1 Mbl MomeM COCTaBI-ITb ce6e
npejl,CTaBJIeHl-le 0 HHX no H3BeCTHOH KHI-Ire nTOJIeMeSi «AJIbMa-
reCT». ACTpOHOM H MaTeMaTl1K KJIaB,AHH n TO JI eM e H )KI-IJI
366

3Ha'lHTeJIbHO n03me rlinnapXa - BO II B. HarueH 3iJbl.
OH HBJISIeTCH C03JI.aTI'JIeM TeopHH )J,BHmeHHH He6eCHblX TE'JI,
KOTopaH B Te'leHHe MHOrliX BeKOB, BnJIOTb )J,O BpeMeHH Kcnep-
H}JI(a, . C'lliTaJIaCb HaH60JIee npaBliJIbHOH. ECTeCTBeHHO, 4TO
npH C03)J.aHHli 3TOH TeOpliH li HeO.ll.HOKpaTHblX ee liCnpaBJIe-
HHH nTOJleIelO npliXO.ll.li.TIOCb npOli3BOJI.liTb l\1Horo aCTpOHO:\IH-
qeCKliX Ha6.rllO)J,eHHH, conpOBom)J.aeMblX BbI'lliCJIeHliHMH, B
KOTOpbIX TpHrOHOMeTpli'leCKHH BbI'lHCJIHTeJIbHbIH MeTO.ll. lir-
paJI nepBOCTeneHHylO pO.lb. «AJIbMareCT» nTo.neMeH CO.ll.ep-
mHT Ta6JIliUbI fHnnapxa B HeCKOJIbKO yJIY'lrueHHOl\1 BH)J,e li
paCKpblBaeT HaM TOT nYTb, no KOropOMY ruJIH H fHnnapx li
nTOJIel\leH npH COCTaBJIeHlili CBOliX Ta6JIliU xOP.ll..
Ta6.rmUbl XOPJI. nro.rleMeH )J,alOT oTHorueHlie .lI.JIliH xOP.ll.,
CTSIrliBalOrn.Hx onpe.ll.e.rleHHble .lI.Yrli oKpymHocTif. K paJI.HYcy
3TOH oKpymHocTli. CJJe.ll.OBaTeJJbHo, 'lT06bI OT Ta6. r mu nTOJle-
MeH nepeHTli K HaruHM Ta6JIHuaM CHHYCOB, nplilllJIOCb 6bI 'lHC-
.na, nOMelUeHHble B Ta6JIHuaX nTOJleMeH, YMeHbruHTb B.lI.BOe,
H Tor)J,a Mbl nrJIY4HJIH 6b1 CHHYCbl )J,yr, BJI.BCe l\IeHbrulix Tex,
KOTopble CTHrliBalOT XOp}.l.bl ('lepT. 184).
4epT. 184.
4epT. 185.
COCTaBJIeHHe Ta6JIliU xOP)J, nTo.neMeH OCHOBblBa.TI Ha Teo-
peM,e, li3eCTHoH B reO:\IeTpHli no)J, ero Hl\leHeM. n03HaKoMHMcH
C 3TOH TeOpei\IOH Ii ee )J,oKa3aTeJlbCTBOM ('lepT. 185).
EC.1U OKOAO ttemblpexyeo.1bHuKa ABCD AIOJlCHO onucamb
OKpYJlCHOCmb, mo npou3eeoeHue ouaeoHalleii paeHo cyMAte npo-
u3eeoeHuii npomueonoAoJICHblX. cfnopoH, m. e.
BD.AC = AB.CD + BC.AD.
367

nOCTpOMM yrOJI ABE, paBHbIH yr.TJY CBD, Tor.n.a
6.ABE N 6.BCD,
CJIe)J,OBaTeJIbHO,
AB AE
- = -- H AB.CD=BD,AE,
BD CD
.llaJIee, b::..E BC N b::..A BD; n03ToMY
BC EC
- - -- H BC.AD = EC.BD
BD - AD .
CJIOlKHB paBeHcTBa (1) H (2) no QaCTH:\1, nOJ1Y'lMM:
AB-CD + BC-AD = BD(AE + EG),
AB.CD + BC.AD = BD.AC.
(1)
(2)
HJIH
3Ta TeopeMa n03BO.'1li.'"Ia nTOJIeMeIO no xop.ll.a!\l )J,BYX .lI.yr
onpe.ll.eJIHTb xOP.lI.Y pa3HocTH HJIM CYMMbI 3TliX .lI.yr, a TaKme
no xop.lI.e .lI.yrH H8HTli xop.ll.y nOJIOBHHbI .lI.yrli.
npoc.'1e.lI.llll.l, KaK C nmiorn.blO TeopeMbI nToneMeH onpe)J,e-
JIHeTCH xop.ll.a .lI.yrM BC, npe.ll.CT8s.mllOm8H pa3HocTb .lI.yr AB
HAC, eCJIH XOP.ll.bI, CTHrHB810lUHe .lI.yrH AB HAC, .lI.aHbI
(4epT. 186).
B
D
.4
IJ
4epT. 186.
4epT. 187.
.llaHO: AB = a; AC = b H AD = 2R. Onpe.n.eJIMTb BC = x.
no TeopeMe nTOJIeMeSi It'\Iee:ll:
a.DC =x,2R + b.BD.
OTCIO.n.a
x=
Q.DC-b.BD
2R
Onpe.ll.eJJSleM DC M BD M3 npH!\wyro.'lbHbIX TpeyrOJIbHMKOB
ADC H ABD:
DC = V 4R2 - b 2 li BD = V 4R2 ' - a 2 ,
368

C.ne,ll.OBaTeJIbHO,
a V4R2- b 2 - b V 4R2",- a 2
x= 2R
(3)
HeCKOJIbKO CJIOmHee, HO TeM me nplieMOM onpeJ1e.nSleTCSI
H xop)J,a CY:\I?\IbI .lI.BYX .lI.yr, eCJIli XOP.ll.bI nOCJIe.ll.HHX ,lI.aHbI
(qepT. 187).
nYCTb AC = b; BC = cHAD = 2R. OnpeJ1eJIHTb AB = y,
no Teope:\le nTOJIeMeH HMeeM
y.DC = b.BD -+ c.2R.
1-'13 TpeyrOJIbHHKOB ADC H ABD onpe)J, eJlHeM DC H BD:
DC = V 4R2 - b 2 H BD = V 4R2 _ y2 .
lI.JIH onpe.ll.e.'leH HH y MbI n0.'1Y'l ae:\1 ypaBH eHHe:
y V4R2-b 2 = b V4R2 - y2 -+ 2cR,
PeruaeM ero:
b V 4R2 - y2 = y 1/4R2 - h 2 - 2cR;
4b 2 R2-b 2 y2=4R 2y2-b 2 y 2-4c.RY V 4R2-b 2 -+ 4c 2 R2;
Ry2 - C V 4 R2-b 2 . y -+ R (c 2 - b 2 ) = 0;
c 11 4R2- b 2 +b 1/4R2 - c 2 ( 4 )
y= 2R
HaKoHeu, HaH.lI.eM xOPJ1y no-
JIOBHHHOH .lI.ym no xop.n.e ue-
JIOH ('lepT. 188).
lI.aHo: AB = a;v AC =
= v BC; AD = 2R. Onpe.n.e.'lHTb
AC = BC = z. no TeopeMe nTO-
JIe:\leSl
lJ
lJ
o
4epT. 188.
a.DC = z.BD -+ z.2R.
113 TpeyrOJlbHHKOB A CD H A BD lillleeM:
DC = V 4R2_Z2 H BD = V 4R2-a 2
H .n.8.'Jee
a,V4R2- Z 2 =z.(V4R2-a 2 +2R).
369

Perna eM ypaBHeHue OTHocItTeJIbHO z:
4a 2 R2 - a 2 z 2 = Z2 (4R2 - 0 2 + 4R 1/4R2- a 2 + 4R2);
Z2 (8R2 + 4R 1/ ,ffi-c(2 ) = 4a 2 R2;
4Rz2 (2R + 1/4R 2 -a 2 ) = 4a 2 R2;
z = a VI? (5)
B, 1/2R + V 4R2 - a 2
TaKH:\\ c6pa30\I, Op'\ly.'lb1
(3), (4) H (5) .lI.alOT B03:\lom-
HOCTb no Xop.ll.aM .lI.BYX .lI.yr HaH-
TH xOPAY CYMMLI H pa3HocTH
3THX Ayr H no xop.ll.e AaHHoH
Ayru onpe.ll.eJIHTb xop.ll.y nOJ1O-
BHHHOH .lI.yrul). nTD.'I€:\\eII,. 3HaH
XOP.ll.bI Ayr B 72" H 60°, Ha-
meJI XOP;I.Y Ayrn B 12°.
nOBTopHoe .lI.eJlemle :nOH .n.yrn
1 0 3 0
nOnOJlaM n03BOJIHJIO eMY HHnl XOPAbI .lI.yr B 6 0 , 3 0 , 12" ' 4" '
Ho DToJJeMefi nOCTaBHJI- ce6e ue.1b - COCTaBHTb Ta6.rmu.bI
1 0 -
XOP.ll. .lI.yr, B03pacTalOrn.ux Ha "2 ' a AJIH :#Toro eMY HY}!(HO.6blJIO
3HaTb XOPAY .lI.yrn B 1 0 . OH peruH.n u 3TY 3a.ll.a'lY CJIe.ll.YIOrn.HM
npHe:\IO'\1. npemAe Bcero OH .lI.OKa3a.1, 'lm 'OTHorueHue Me}!(Ay
.n..1I1HOH XOP.iI.bI u .lI..1HHOH ee .!J,yru C YBeJIH4eHHeM AyrH YMeHb-
ruaeTcH, T. e. ('lepT. 189)
>> - A 2 B 2
vAB '-'AIBI ,-,A2B2
III
4epT. 189.
Tor;I.a en paBe,ll,.rmBbl cJIe.ll.YI01I.J.ue HepaBeHCTBa:
3" 3"
xop.:\a"4" xop;I.a 1° xop;i.a 2""
3" > > 3"
.llyra"4" .nyra 1° .nyra 2""
\I.1H
xup!la 1° <
3"
xop;i.a 4
.nyra 1 °
3°
,Z\yra T
4 xop,'la l"
3" H 30
xop.:\a 2""
> .nyra }O
3"
l!yra 2""
2
=3.
I) IlpH H3MepeHHH .!I.1IlHbl XOP.:\bl 3a e,Z\IIHIIUY npHHHT pa!l.HYc
01< PY>KHOCTII.
370

TaKHM
oGpa30M,
2 3'
3"" XOp.ll.bI 2"" <
4 "30
XOPJJ.bI 1" <3"" XOPJJ.bI""4'
3° 3°
3HaH .lI.JIHHY XOP.lI.bI B 2" H 4" ,nTO.1eMeH c!IIor BbItIHCJIHTb
2 3° 4 3?
3" XOp..1bI 2" H"3 XOp,1J,bl 4" .
IlOJIYlJeHHbIe nOC/le BblLIHCJIeHHH pe3YJIbTaTbI nOKa3aJIH.
4TO nOCe.lI.HHe .lI.Be BeH'lHH BeCbMa 6JIH3KM JJ.pyr K .lI.pyry.
BbI'lHCJIeHHble B ruecTH.lI.eCHTbIX JJ.OJIHX pa.ll.Hyca, OHH nOJIY-
4alOTCH paBHbIMH. CJIe.ll.OBaTe.TIbHO, .lI.JIHHa XOP.lI.bI B 1°, 6Y.lI.Y'lH
3aKJIIOlJeHa leJJ(.lI.Y 3THMH Be.1JH4HHaMH, paEHa HM C JJ.OCTaTOtI-
HOH ,lI.JIfl npaKTHlJeCKHX Ha.lI.o6Hoc-Tei-i TOro BpeMeHH TO'lHOCTblO,
3 3 0 1 0 n u
HaH XOp;I.bl .lI.yr B "2 H , TOJIelelI onpe.lI.eJIHJI XOP.lI.y
1°
JJ.ym B 2" H TaKHM 06pa30M nOJIytIHJI B03MomHOCTb no op-
MYJIe (4) BbItIHC.:1HTb Be.1JM'lHHY xOPJJ. .lI.JIH JJ.yr B 10; 1,5°; 20;
2,5° H T. .lI.., .lI.0 180 0 BKJIIO'lHTe.1JbHO.
TaK 6bIJIH COCTaBJIeHbI nepBbIe TpHroHOMeTpH'leCIme Ta6-
.1HUbl. CKOJIbKO TPY.lI.a 6bIJIO B.1JOmeHO B COCTaB.1eHHe 3TMX
Ta6JIMI.l, MbI npe.lI.CTaBHM ce6e, eCJIH eme pa3 B3rJIHHe!\1 Ha op-
MYJIbI (3), (4) H (5) H Y'lTeM, 'lTO BO BpeMeHa fHnnapxa H nTO-
.'1e!\leH He cyrn.eCTBOBaJIO HHKaKHX ynporn.eHMH .lI.JIH 3anHCH npH
npOH3BO.lI.CTBe .lI.eHCTBHH Ha.lI. 'lMCJIal\IH.
HHJl.HlfI.\bl, HapOJl.bl CpeJl.Hen A3HH H 3aKaBKa3bH
BHOCHT CBon BKnaJl. B pa3BHTHe TpHrOHOMeTpHH
Ce)WIOrn.HH BamHbIH ruar B pa3BHTHM TpHrOHOMeTpHH 6bIJI
C}J,eJIaH HH.lI.HHI.laMH B nepHOJJ. C V no X II B. XOTH HM M He Y.lI.<i-
JIOCb OT.lI.eJIHTb TpHrOHOMeTpHIO OT aCTpOHOMHH H OHa no-npem-
HeMY HBJIHJIaCb TOJIbKO nO.lI.C06HbIM BbI'lHCJIMTe.1JbHbIM anna-
paTOM .lI.JIH Hee, HO OHM Bee me 3HatIHTeJIbHO 060raTHJIH co-
.lI.epmaHiIe TpHrOHOMeTpHH.
npem}J,e Bcero HHJJ.HHI.lbI C03.lI.a.1JH TpHrOHOMeTpH'leCKMe
Ta6JIHUbI 60JIee TOtIHbIe, 'leM Ta6JIHllbI nTOJIeMeH. npH CO-
CTaBJIeHHM 3THX Ta6.1HU OHH .lI.eJIHJIH .lI.HaMeTp oKpymHocTH
He Ha 120 paBHbIx qaCTeH, KaK 3TO JJ.eJIaJI nTOJIeMeH, a Hd
6876 'laCTeH. OTKY}J,a nOHBHJIOCb 3TO 'lHCJIO? Bbl OTBeTHTe
Ha 3TOT Bonpoc, eCJIH BcnOMHHTe, tITO HI-i}J,HHI.lbI npmIMMaJIM
1t, paBHbIM 3,1416. Tor}J,a eCJIH JJ.JIHHY oKpymHOCTH. pa3}J,eJIHTb
371

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sin 2 0; + cos 2 0; = I, cos (90 0 - 0;) = sin 0;,
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374

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SIn X = X - 1.2.3 +  2.3.4.5 - 1.2.3.4.5.6.7 + ..,
M
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x 6
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pa;J;HaH8X.
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375

. 1 1 1 . I
Sin ::::::: - 1.2.3 T 1.2.3.4.5
1
1.2.3.4.5.6.7
= 1 - 0, 16667 + 0,00833 - 0,00019 = 0,84147.
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376

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He Y.lI.HBHTeJIbHO, 'lTO KOr.lI.a KOpOJIb <DpHJJ.pHX II, npe06-
pa30BbIBaH B 5ep.l1HHe AKa.lI.eMHIO, npHrJIaCIi.l1 B Hee 3HJIepa,
nOCJIE';I.HHH npHHHJI 3TO npHr.'larueHHe H B 1741 r. nepeexaJI B
BepJIHH, npmKHB B nepBblH pa3 B POCCHH 16 JIeT.
5YJJ.Y'lH B 5epJJHHe, 3H.I1ep np0.ll.o.'l}!{a.'l COCTOHTb 1l.'leHOM
netep6yprcKoH aKa.lI.eMHH He TOJIbKO Hor.mHa.TIbHO, HO caMblr-1
.lI.eHTeJJbHbIM 06pa30M, emerO.ll.HO .lI.OCTaBJJHH .n..I1H ee «3anHCOK»
J\1Horo MeMyapoB H 60,TIbruHX TPY.lI.OB, cpeJJ.H HHX «MopCKaH
HaYKa», «.ilHqxpepeHUHaJlbHOe HC4HCJIeHHe» H .lI.p.
nporuJIO 25 ,neT. 3a 3TO BpeMH 3H,TIep HanHCaJI COTHH CTa-
TeH KaK no llHCTOH MaTeMaTHKe. TaK if rJIaBHbIM 06pa30M npH-
KJIa.ll.HOH, H TaKHe KpynHble pa60TbI, KaK «BBe.ll.eHHe B aHaJIH3»,
«BapHaUHOHHoe HC4HCJIeHHe» H «TeopHIO JIYHbI».
B 1766 r. EKaTepHHa II npHr,'laruaeT 3H,TIepa BepHYTbcH
«Ha .'l106bIX YCJIOBHHX» B neTep6ypr. <DpH.lI.pHX, YCTynaH Ha.
CTOHHHHM EKaTepHHbI, omYCKaeT ero, H 3HJIep, BMeCTe co
CBOHMH CbIHOBbHMH, KOTOpbIM TaKme 6bwa npe.n.JIOmeHa pa-
60Ta, B03Bparn.aeTcH B Poccmo. .
3H,flep B TO BpeMH no cnpaBe.lI.,TIHBOCTIi C'lHTaJICH nepBblM
MaTeMaTHKOM B MHpe.
DOTepHB 3peHHe Ha O.lI.HH r.'la3 B 1736 r. OT nepeYTOMJIe-
HHH BbI4HCJIeHHHMH, B 1766 r. OH cOBeprueHHo .'lHruHJ1CH 3pe-
HHH, HO HaY'lHaH pa60Ta ero He npHOCTaHOBHJlaCb H np0.lI.o.1-
maJJaCb .lI.O car-IOH ero CMeptH.
C 1766 no 1783 r. HI npo;I.HKTOBaHbI COTHH cTaTeM H .lI.eCHTb
rpOMa.LI.HbiX TOMOB no caMblM pa3Ho06pa3HbIM BonpocaM llHC-
TOH H npHKJIa.ll.HOH MaTE'MaTHKH.
18 ceHTH6pH 1783 r. OH B nepepbIBe Me}!{.lI.Y 3aHHTHHMI1
WYTHJI co CBOHM nHTHJIeTHHM BHYKOM, nOl/yBcTBoBa.'l ce6H
cpa3Y .lI.ypHO H, no CJIOBai\l KOH.!I,opce, «npeKpaTHJJ BbI4HC.1HTb
Ii m HTb»,
KaK BH.lI.HO, He BHeWHHMH npOHBJIeHHHMH 60raTa mH3Hb
3HJIepa, HO 3aTO ero TBOplleCTBO H3YMHTeJJbHO H 6ecnpHMepHO.
nepy 3HJIepa npHHaJJ..'lemHT 865 pa6OT. B HHX B O.ll.HHaKOBoH
Mepe OCBern.eHbI KaK npaKTHl/eCKHe BonpOCbI, TaK H BonpOCbl
qHCTOH MaTeMaTHKH. 5YJI.Y'lJI B Bonpocax npaKTH'leCKHX npH-
JIOmeHHH BeCbMa pa3HOCTOpOHHHM, 3HJIep B CH.ny .:Horo .lI.OJI-
meH 6bIJI 6bITb pa3HocTopOHHHM H B 06JJaCTH MaTeMaTHKH.
i78

11 .lI.eHCTBHTe"'IbHO, 9HJIep ycneulHo paOOT-aJJ H B OOJIaCTM .lI.H-
epeHUHa.'IbHoro H mnerpa.l1bHOrO HClJHC"'IeHHII, H B 06JIaCTH
.lI.H<pcpepeHlJ,Ha.'JbHbIX yp.aBHeHHH H aHaJJHTII'leCKOH reOMeTpHH;
OH l\fHOrOe C.lI.eJlaJI B a"'Ire6pe. .lI.HepeHl.lHaJJbHOH reOMeTpHH
. H CepHlJeCKOH TpHroHoMeTpHH. 1-1;\7 6bIJIH C03.l1.aHbI 11 HOBbIe
pa3.l1.e"'IbI MaTeMaTHKH.
faycc, O.ll.HH H3 KpynHeHulHx MaTeMaTHKOB XIX B., YKa3bI-
Ba"'I, lJTO H3Y'leHHe pa60T 3i'wepa BCer.lI.a 6Y.lI.eT JIYlJruHM cpe,lI.-
CTBOM )J,"'IH n03HaHHH MaTeMaTHKH.
06 3HJJpe )Ke roBOpHT H paHUY3CI<HH MaTeMaTHK H aCT-
pOHOM J1anJIac: «OH Haru 061.1.(Hii YlJHTe"'Ib». 11 ;I.eHCTBHTeJIbHO,
Ha TpY;I.ax 3HJIepa BocnHTblBaJIOCb ue.'Joe nOl<O.rJeHHe MaTeMa-
THKOB. OH H CerO.lI.HH MHorOMY Y'lHT Hac.
TeopeMbI H 3aJl.a1.(H, cBSl3aHHbl C HMeHeM 3HJ1epa
TeopeMa 3HJIepa 0 CBOHCTBax MHororpaHHHKOB. B JII060M
BbmYKJIOM MHororpaHHHKe HMeeT MeCTO COOTHorneHMe
B+r -P=2,
(1)
r}J,e B - lJHCJIO BepruHH MHororpaHHHKa, r - 4MCJIO ero rpa-
Heii H P - 'lHCJIO pe6ep.
LLOKameM 3TY TeopeMY MeTO.lI.OM MaTeMaTH'leCKOH HH.lI.YK-
umr.
I. Il pHHHMaeM YCJIOBHe TeopeMbI 3a rHnOTe3Y H npoBe-
pHeM cnpaBe.ll."'IHBOCTb ee }J,JJH lJeTblpeXrpaHHHKa H nHTurpaH-
HHKa ('lepT. 191). fHnOTe3a OKa3bIBaeTCH cnpaBe)J,"'IHBOH,
QJ
6)
4epT. 191.
2. ..QonycTHM, qTO cooTHorueHHe (I) cnpaBe.lI..'IHBO .lI..I1H MHO-
rorpaHHHKa C n rpaHHl\m; }J,OKmKe11 ero cnpaBe.ll."'lHBOCTb
;J..'lH MHororp3HHHKa C (n+ 1) rpaHHMH. ..Q.'lH TOro lJTo6bI no-
3',9

.rryqHTb H3 MHOrorpaHlIHKa C n rpaHHMH MHOrOrpaHHHK C
(n+ I) rpaHH!\IH, cpemeM qaCTb nepBoro MHororpnHHlIl<a
nJIOCKOCTblO OKOJIO KaKoll-JIH60 H3 BepruHH TaK, qTo6bI 3Ta
nJIOCI<OCTb nepeCeKJIa Bce pe6pa, BbIXO;I.Hrn.He H3 3TOH BepruH-
Hbl (qepT, 192). Tor,ll.a K 'lHCJIY r npH6aBHTcH O.ll.Ha HOBaH
rpaHb ABCDEF. nYCTb B Bep-
N ruHHe M CXO.ll.HTCH k pe6ep; B
HOBOH rpaHH ABCDEF l\IbI nOJIY-
qllM k HOBbIX BepruHH H nOTepHeM
O.ll.HY H3 BepruHH (BepruHHY M).
TaKHM 06pa30M, 4HCJIO Bcex Bep-
ruIfH B HOBOM MHororpaHHHKe
6Y,ll.eT
4epT. 192.
B + k-l.
4HCJJO pe6ep YBeJIHqHTCH Ha k H
HX 6Y,lI.eT
P+k.
CocTaBMM pa3HocTb Me)K.lI.Y CYMMOH 'lMCJJa BepruHH H rpaHeH
HOBora MHororpaHHHKa C (n+ I) rpaHHMH H 'IHC"'IOM ero pe6ep:
(B+k-l)+(r + 1)-(P+k)=B+k-1 +r+ I-P-k=
=B+r -P=2.
TaI<H1\1 06pa30M, COOTHorneHHe Mem}J,y qHCJIOM BepruHH, rpa-
HeH If pe6ep, CnpaBe,lI.JJHBOe ,lI..'1H MHororpaHHHKa C n rpaHHMH,
oCTaeTcH CnpaBe,ll.JJHBbIIII H .lI.JIH MHororpaHHHKa C (n+ I) rpa-
HHlIIH .
3. Ha OCHOBaHHH npHHUHna MaTeMaTHqeCKOH HH,lI.YKUHH
YTBepm}J,aei\!, 'ITO cooTHorueHHe
B+rp=2
CnpaBe,lI.JlIIBO ;I.JIH JIIo6oro BbmYKJIOrO MHororpaHHHKa,
TeopeMa 3l'L'Iepa n03BO.J1HeT onpe)J,e.J1HTb qHC.rIO rpaHeH
npaBli.TlbHOrO BbmYKJIOrO l\IHOrOrpaHHlIl<a B 3aBHCHMOCTH OT
Op:>'lbI ero rpaHeH.
113 Kypca rem.leTpllI1 Bal] H3BeCTHO, 'lTO cyrn.ecTByeT TOJIb-
KO nHTb BH,lI.OB npaBH"'IbHbIX BbmYKJIblX MHororpaHHIfKOB.
fpqHH!\m IIX MorYT 6bITb HJIH npaBHJIbHble TpeyrOJIbHHKH,
H.11I KBa.LI.paTbI, IIJIH npaBH.J1bHbIe nHTlIyrO"'IbHHKH.
a) nYCTb B BepruHHe l\!HororpaHHJIl<a CXO)l.5JTCH Tpeyro.'Ib-
I! !1 Kii , 06pa3YH k-rpaHHbIH yrOJI, k MO}!{eT 6bITb paBHO 3,
. 4 HJIH 51). HaH,lI.eM 'IIIC,,10 rpaHeH MHorOrpaHHIfI<a.
1) aTO CJJe.n.yeT 113 TeopeMbl: (;CYMM3 n.QOCKIIX yr.10B lHororp3HHoro
yr.13 ;,leHbUle 4d».
380

ltonYCTlIM, 'lTO rpaHeH l\tHOrOrpaHHIIKa 6y.n.er n, T. e,
T  
r = n. or)J,a qlICJIO BepruuH B = Ii: II qlICJIO pe6ep p= 2" '
nO,ACTaBJIHH B OPMYJIY 3HJIepa 3HaqeHlIH r, BlIP, nOJIyqlIM:
n + n = 3; + 2,
PeruaH ypaBHcHlIe OTHOCIITeJIbHO n, HMeeM
4k
n = 6- k '
ECJIH k = 3, T. e. Y BepWHHbI MHororpaHHHKB CXO,AHTCH
TPH TpeyrOJIbHIIKa, TO
4.3
n = 6-3 = 4,
T. e. MHororpaHHHK IIMeeT 4 rpaHH,
ECJIH k=4, TO
4.4 16
n = 6-4 ="""2 = 8,
"
ecJIH k = 5, TO
4.5
n = 6-5 =20,
I1TBK, npaBH.'!bHbIH MHororpaHHIIK C rpaHHMII B BlIJ],e Tpe-
yrOJlbHIIKOB MO}!(eT 6bITb HJIII lIeTblpexrpaHHHKO:\-1 (TeTpa3pO!\I),
HJIII BOCbMlIrpaHHIIKOM (OKTa3,ApOM), H,Im ,ABa,AuaTlIrpaHHII-
KOM (HKOCa3,ApOM) (qepT. 193, 194, 195).
· 6) nYCTb B Beprul-IHe MHororpaHHIIKa CXO.ll.HTCH KBa.ll.pa-
TbI; qHCJIO HX, no YKa3aHHoH BbIrue TeopeMe, Mo.>KeT 6bITb

. .-//.' .:.'''::\;-,
i
:: ::. !
: ! ::::.. \ .:
, . - : =::.::  II
, - - .
4epT. 193.
4epr. 194.
4epT. 195.
381

TOJIbJ<O TpH. HaH,'J,eM 'lHCJIO rpaHeH ,noro MHororpaHHlII<a,
LI.onYCTHM, 'lTO HX 6Y)J,eT n, T. e.
F=n,
Tor.ll.a
B-
- 3
H
4n
P=T=2n.
nO TeopeMe 3HJIepa:
4n
n + 3 := 2n + 2.
oTKy.ll.a
n-=6.
HTaK, npaBHJIbHWH MHororpaHHHK C rpaHHMU B BHe KBa.ll.-
paTa 6yeT HMeTb weCTb rpaHei1; ero Ha3bIBalOT r eKe a 3 )J,-
po M HJIU K Y 6 0 M ('lepT. 196).
4epT. 196.
4epT. 197.
B) nYCTb B Bepwinre MHorOrpaHHlII<a CXO,'J,HTCH nHTI-I-
yrOJIbHI-IKH, 4HC,,10 UK, no YKa3aHHOH Teope:\le, MO}f\eT 6bITb
TOJJbKO TpH. HaH.lI.eM 'lHCJIO rpaHeH 3Toro MHorOrpaHHHJ{a,
llonycTHM, 4TO I-IX 6Y,l.(eT n, T. e.
F=n;
TorAa
B=
3
H
' I
P v.
=
352

no TeOpel\,!e 3HJIepa:
5n 5n
n + """3 = 2 + 2.
OTClO.lI.a
n=12.
11TaK, npaBHJIbHbIH MHororpaHHHK C rpaHHMH B Bl1.l1.e npaBHJIb-
HblX nHTHyrOJIbHHKOB 6Y.lI.eT )]. B e H a )]. u a T II r p a H H H-
K 0 M (,'J,0)].eKa3.l1.p) ('lepT. 197).
npHMaH 3HJIepa
):(oKa3aTb, 'lTO B JI!060M
TpeyrOJIbHHKe TO'lKa nepe-
Ce'leHHH Me.ll.HaH, opTOueHTp
H u:eHTp onHcaHHoro I<pyra
J1emaT Ha O.ll.HOH npHMoH.
3Ty npHMYIO Ha3bIBaJOT
npHMoH 3flJIepa ('lepT. 198),
4epT. 199.
4epT. 198.
OKPY)f{HOCTb 3HJIepa
):(OKa3aTb, 'lTO B JIlO60M
TpeyroJ1bHHKe OCHOBaHHe
Tpex Me.n.HaH, Tpex BbICOT H
TPH T04KH, .lI.eJ1Hrn.He nono-
JIm,! OTpe3KI1 BbICOT OT OpTO-
u:eHTpa .lI.O BepruHHbl, JIemaT
Ha O.ll.HOH OKPY}KHOCTIi,
3Ta OKPY}KHOCTb HOCHT Ha3BaHHe OKPY}KHOCTH '.lI. e B H T Ii
T04 e K HJIH OKPY}KHOCTH :3 H .1 e p a (pHC. 199).
Ta6JIHua MaTeMaTHLJeCKHX CHMBOJIOO,
BBeJl.eHHbIX 3HJIepOM
K MHorHM JaTeManI'leCKm\'1 CHMBOJIaM Mbl HaCTOJIbKO
npHBbIKJIH, 4TO He CTaBHM ce6e Bonpoca 0 TOM, KOr.ll.a OHH 6bI-
JIH BBe)].eHbI B MaTeJ\-IaTHKY H KTO HBJIHeTCH aBTopOJ\I HX. TaK
383

KaK HeMaJIaH 3aCJJyra B BbIpaOOTKe COBpeMeHHOH MaTeMaTH-
qec!wii CHMBOJImm npHHa.ll.JIe}!(HT 3i1JIepy, TO HeJIb3H He
BcnOMHHTb 3.n.eCb 0 TOM, 'lTO C.ll.e-laHO HM B 3TOM 06JIaCTH,
npllBO.ll.IlM 3TH CBe.n.eHHH B HHmenOMemeHHOH TaOJIHue.
CI!MBOJI
foJ!.
3H8'1eHl!e CI!MBOJIa
l
sino:
coso:
tg 0:
v
fj,x
f (x)
a, b, C
A. B. C
7t
OTHOlUeHl!e JIJII!Hbl OKPY>KHOCTI!
K JIII8MeTpy
OCHOB8HI!e H8TypaJIbHbiX Jlora-
pHIjJMOB
KopeHb KB8J!.p8THbIii 113-1 ,
CI!HYC lJ,yrlf'
KOCI!HYC JIyrl! ,..,
T8HreHc J!.yrll
CYMM8
P83HOCTb X 2 -Xl I!JlI! npHp8111e-
Hl!e nepeleHHofl BeJIII'I I!Hbl
<DYHKUIIII aprYMeHT8 ,
CTOPOHN peyrOJIbHI!K8
Yr'lbI Tpeyro.bIHIK8, npOTI!BO-
JIe>Kallll!e COOTBeTCTBeHHO CO-
pOH8M a, b He. . . . .
1736
1736
1777
1748
1748
1753
1755
1755
1734
B. n. III e p e MeT e B C K I! ii, OqepKn no I!CTOpl!1! M8TeM8TI!KH,
YqneJJ.rIl3, 1940.
B. B. B 0 . r 8 p C K I! ii, OCHOBHble 3T8nbl pa3BI!TI!S! Tpl!rOHO-
MeTpl!l!, «M8TeM8TI!K8 B lllKOJle», 1956, N9 I.
r. r. U. e if T e H, 11CTOpl!lI MaTeMaTI!KI!.
B. B. n p Y ,If H I! K 0 B, Pyccl<l!e nC,lJ.8rOfl{-MaTeM8THKH XVIII--
XIX nel<OB, Y'IneJ!.rI!3, 1956.
A. H. K P bl JI 0 B, J1eoH8PJl 3flJIep. 113J!.. AH CCCP, 1933.
11. r. M e JI b H I! K 0 B, JJeoH8PJl 3fiJIep I! 3JIeMeHT8puall MaTe-
M8HI<8, «M8TeM8TI!K8 B lllI<OJIe», 1957, N 4.
11. 51. .n: e n M 8 H, P8CCK83bl 0 MaTeMaTI!Ke, J1., 1954.
.n:. O. III K JI II P C K I! ii, H. H. 4 e H U 0 B I! 11. M. 51 r JI 0 M.
1136paHHb!e 38,/(8'11! I! TeopeMbl 3JIeMeHT8pHoii MaTeM8TlIKI!, '1aCTb 2
(feOMtTpl!S!), [11T1 JJ, 1952.
384
e
JII1TEPATYPA

\
'--11= lIfi'n(J(-c;/
.....p- - .--- "/
rJlABA xv
KOJIE6ATEJIbHbiE )J.BH)I(EHIHI H TPHrOHOMETPHSI
«<D 11311Ka H M8TeM8TIIKa BCerll,a no-
MOraJlII Oll,H8 )l.pyroH, II pa3BIITHe IIX
qaCTO He pa3l1,eJlHMO».
C. H. BaBUAOB
MaTeMaTHKa npOHHKaeT B 06aCTb CBeTa, 3BYKa
H aeKTpoMaruHTHbix HBeHHii
 HTaTeJJb xoporuo 3HaeT, KaKoe 60JIbruoe 3Ha'leHlIe
HMeeT TpHroHoMeTpHH B perueHlI1I MHorlIX npaKTII-
'leCKIIX 3a.ll.a'l. Cpe.ll.lI HIIX 3a.ll.a41I Ha Haxom.ll.eHHe
nOBepXHOCTeH H 06"beMOB B reoMeTplIH, Ha CJIome-
'me H pa3JIOmeHHe CIIJI B MexaHIIKe, onpe.ll.eJIeHHe He.n.oCTynHbIx
paCCTOHHIIH B Tonorpa1I1I H 60J1bruHx paCCTOHHHH Ha nOBepx-
HOCTII 3eMJIlI MeTo.ll.OM TpHaHrynHU1I1I H, HaKoHeu, pH.lI. 3a.ll.a'l
83 06JIaCTlI aCTpOHOMHII. B aTOH rJIaBe HaM XOqeTCH OCTaHO-
BHTbCH Ha IIHOM npaKTII'leCKOM npHJIOmeH1I1I TplIrOHOMeTplIH
B 06JIaCTH BeCbMa BamHOH H, MomeT 6bITb, MeHee 3aTparHBae-
MOH B WKOJIbHOM Kypce MaTeMaTIIKII,- MbI rOBopHM 0 CBH3H
TpHrOHOMeTplIH C BonpOCOM KOJIe6aTeJJbHbIX BID:KeHHH B
ct>H3HKe,
1/" 13 A. A. I(OJIOCOB
385

Kor.ll.a Kal<aH-JIH60 TOqKa BH}KeTCH no npHMOH JIHHHH
nonepeMeHHO TO B 0.ll.HY, TO B .lI.pyrylO CTopOHy, TO rOBopHT,
'lTO TOI.J.Ka cOBepruaeT KOJIe6aHHH. KOJIe6aHHH TO'lKH MorYT
ObITb BeCbMa pa3Ho06pa3HbI, HO MbI CHa'laJIa paCCMOTpHM CJIe-
.lI.YlOmHH 'laCTHbIH CJIY4aH.
nyCTb no OKPY}!(HOCTII c ueHTpOM 0 B Ha'laJIe KOOp.ll.IIHaT
II pa.ll.HYCOM, paBHbIM A, BH}!(eTCH paBHOMepHO TOI.J.Ka M,
onHCblBaH nOJJHblH Kpyr B HeKOTopoe BpeMH T (ceK.) ('lepT. 200).
nOJIOmeHIIe TO'lKH M Ha oKpymHocTII MbI MO}!(eM onpe.ll.e-
JIHTb yrJIOM a, KOTOpbIH 06pa3yeT pa.ll.HYC OM C BbIopaHHbIM
HaMH Ha'laJIbHbIM .lI.lIaMeTpOM, COBna.ll.alOmIIM C OCblO X. YrOJI
a 6Y.lI.eM C4HTaTb BbIpa}!(eHHbIM B pa.ll.HaHax.
YrJIOBaH CKOpOCTb W TO'lKH M, KOTopaH onIICbIBaeT yry
21t BO BpeMH T, 6yeT, OqeBHHO,
v
2'1t
w=y,
M
BeJIH'lHHa W TaKme paBHa
yrJIY, KOTOpbIH palIYc OM
onHCbIBaeT B O.ll.Hy ceKYH.lI.Y;
n03ToMY, eCJIH pa.ll.HYC OM
OnHCaJI yrOJ1 a 3a t ceK.,
OT
)(
2'1t
IX = -r-,t = wt.
JIIIHeHHaH CKOpOCTb V
llepT. 200. TO'lKH M no OKPY}!(HOCTH
on pe.ll.eJIHTCH .lI.eJIeHHeM
JIHHbI oKpymHocTH 2rrA Ha BpeMH nOJIHOrO 060pOTa T:
2'1tA 2'1t
v=r=A'r=Aw
JJ:JIH Toro 'lTOObI T04Ka M paBHoMepHo .lI.BIIraJIaCb no
OKPY}!(HOCTH, He06xo.ll.HMO, 4T06bI Ha Hee eHCTBOBaJIa ueHTpo-
CTpeMHTeJIbHaH CHJIa, co06rn.alOrn.aH eH ueHTpocTpeMHTeJIbHOe
YCKopeHHe a. 31'0 YCKopeHHe, Kal< H3BecTHo, paBHO
v2. v2. ( 21t ) 2
a = Ii: = A = A T = A.u 2 ,
BY.lI.eM CJIe.ll.IITb O)J,HOBpeMeHHO He TOJIbKO 3a .lI.BH}!(eHIIeM
TOllKH M no OKPY)j{HOCTH, HO H 3a nepeMeru.eHl1eM ee npo-
386

eKuHl1 My Ha OCb Y. npH lI.BHmeHHH TOIlKH M no oKpymHocTH
ee npOeKI.{HH My 6YlI.eT cOBepruaTb KOJIe6aHHH no OCH 0p.ll.HHaT,
3TH KOJIe6aHHH HaM H npelI.CTOHT H3YQHTb.
OepeMern.eHHe y npoeKUHH My Onpe.ll.eJIHeTCH H3 60MMy:
y=A sin cx=A sin wt. 3TO paBeHcTBo H 6Y.lI.eT ypaBHeHHeM
KOJIe6aHHH.
OpH paBHoMepHoM nepeMern.eHHH TO'lKH M no oKpymHocTH
CKOpOCTb npOeKI.{HH T04KH M He 6YlI.eT OCTaBaTbCH nOCTOHH-
Hot!; 4T06bI onpe,ll.eJJHTb BeJIH'lHHY 3TOH CKOpOCTH B JII060H
MOMeHT t BpeMeHH, HaH.lI.eM npoeKUHIO CKOpOCTH TOqKH M
Ha OCb Y (IlepT. 200). 1'13 TpeyrOJIbHHKa MKL HMeeM
npyv = v y = v cosa = Aw coswt.
1.JIITaTeJIb, 3HaKOMbIH C nOHHTlieM npOH3BO.ll.HOH, Mor 6bI no-
JIY'lHTb TOT me pe3YJI, bTaT HaH.lI.H nepBYIO npOll3BO.ll.HylO no .t
OT YHKUHH y=A sin wt
dy A
v y = (jJ= w.coswt.
YCKopeHHe KOJJe6JIIOmeHCH TOllKIi M MomHO TaKme HaHTH
nYTeM npoeKTHpoBaHHH YCKopeHHH a TO'lKH M Ha OCb 0P.ll.H-
Har. 113 TpeyrOJIbHHKa MN P nOJIY4HM:
npya = a y = -a sinz =- A OJ2 sin OJt.
3HaK MHHYC He06xolI.HMO nOCTaBIITb nOToMy, 'lTO HanpaBJIe-
Hlie npoeKUHH B )J,aHHOM cJIyqae npoTIiBOnOJJOmHO HanpaB-
JIeHlIlO OCH Y. TOT me pe3YJIbTaT MbI MOrJIIi 6bl nOJIYllIlTb,
dyZ
H3H.lI.H BTOPYIO npOH3Bo.ll.HYlO J{z- OT Yf!KUHH y=A sinOJt.
PaccMoTpeHHoe HaMH KOJIe6aTeJIbHOe .lI.BHmeHlIe T04KH My
no OCH 0PlI.HHaT Ha3bIBaeTCH n p 0 C T bI M rap M 0 H H-
'1 e C K II M K 0 JI e 6 a H II e M II npe)J,cTaBJIHeT C060H OCHOB-
HOH Tlin KOJIe6aHliH BCHKoro pO.ll.a. npe.ll.JIaraeM BaM caMO-
CTOHTeJIbHO y6e.ll.HTbCH B TOM, liTO eCJIIi 6bI MbI npoeKTHpOBaJIH
TO'lKY M, .lI.BH}!(yrn.YIOCH paBHOMepHO no OKPY}!(HOCTH, He
Ha BepTIIKaJIbHYlO OCb, a Ha ropH30HTaJIbHYlO, TO nOJIy'lHJIIi
6bI npOCTbIe rapMOHlI4eCKlie KOJJe6aHliH ee npoeKuHH Mx B
TaKIiX BbIpameHlfHX: paCCTOHHHe KOJIe6.'!1Orn.eHCH TO'lKH OT
Ha'laJIa KOOplI.IIHaT
x =A cos OJt,
(1 ')
CKOpOCTb ee
V x = - A w sin w t
(2')
14 A. A. KOJIOCOB
387

H yCKopeHHe
ax = - Aw 2 coswt,
(3')
I13Y'leHHe KOJIe6aHHH BCHKOro po)].a Ba}!(HO y}!(e no O)J,HOMY
TOMY, 'lTO C KOJIe6aTeJIbHbIMH ,ABH}!(eHHHMH HJIH BOJIHaMH
MbI CTaJIKHBaeMCH BeCbMa 'laCTO B oKpy}!(alOrn.eM Hac MHpe
If C 60JIbruHM ycnexoM HCnOJIb3yeM HX.
npOCTOH OnbIT y6em)J,aeT Hac B TOM, 4TO BCHKHH 3BYl.J.a-
rn.HH npe)].MeT (nJIaCTHHKa, CTpYHa H T. n.) HaXO)].HTCH B KOJIe-
6aTeJIbHOM )].BH}!(eHHH. I1CTO'lHHK 3BYKa TO ynJIOTHHeT npH-
JIeralOmHH K HeMY CJIOH B03)].yxa, TO pa3pemaeT ero. YnJIOT-
HeHHbIH CJIOH B03.l1.yxa .lI.aBHT Ha coce)].HHH CJIOH B03.l1.yxa,
a TOT B CBOIO O'lepe)J,b .lI.aBHT Ha eme ,AaJIbrue pacnOJIOmeHHbIH
CJIOH, 1.Jepe.ll.YIOrn.HeCH .lI.pyr 3a .lI.pyroM ynJIOTHeHHH H pa3pe-
meHHH B03)J,yxa paCXO.ll.HTCH OT HCTO'lHHKa 3BYKa BO Bce CTO-
pOHbI. mo H eCTb TO, qTO MbI Ha3bIBaeM 3BYKOBbIMH BOJJHaMH
B B03.l1.yxe. 3BYKOBbIe BOJIHbI HBJIHIOTCH npO.ll.OJIbHbIMH BOJIHaMH,
T. e. OT.lI.eJIbHble 'laCTHUbI B03.l1.yxa KOJIe6JIIOTCH B.lI.OJJb HanpaB-
JIeHHH )].BJ-I}!(eHHH BOJJHbI.
HeCKOJIbKO HHbIMH 6Y)J,YT KOJIe6aHl1H 'laCTHU BO.ll.bI npH
pacnpOCTpaHeHlIH BOJJH no nOBepXHOCTH ee. BO)J,a He nepe-
,lJ,BHraeTCH C BOJJHOH, HO qaCTH'lKIf BO)].bI KOJIe6JIIOTCH, TO no)J,-
HHMaHCb BBepx. TO onYCKaHCb BHH3 - nonepeK TOro HanpaB-
JIeHIfH, Ky.ll.a 6emIfT BOJIHa. 3.l1.eCb MbI HMeeM .lI.eJIO C nonepe'l-
HbIMH BOJIHar.m.
DbICTponepeMeHHbIH TOK B npOBO,AaX nepe)].alOrn.eH aHTeHHbI
06pa3yeT nepeMeHHble nOJIH - MarHIfTHOe H 3JIeKTpH'leCKoe.
OHH CymeCTBYIOT O.ll.HOBpeMeHHO If 06pa3YlOT TaK Ha3bIBaer.me
3JIeKTpOMarHIfTHoe nOJJe. 3TO nOJIe pa3Mern.aeTCH He TOJIbKO
y nepe.ll.alOrn.eH aHTeHHbI, a pacnpOCTpaHHeTCH B npOCTpaHCTBe
BO Bce CTOpOHbI C rpOMa.ll.HOH CKOpOCTblO B 300 000.!5!!- .
ceK
TaKoe nYTeruecTBYIOrn.ee nOJIe Ha3bIBaeTCH 3 JI e K T P 0 M a r-
H H THO H B 0 JI H 0 H.
3JIeKTpOMarHHTHbIe BOJIHbI, HJIIf 3JIeKTpOMarHIfTHbIe KOJIe-
6aHHH, 3anOJIHHIOT BeCb MHP, BCIO BCeJIeHHYIO. C 3JIeKTpO-
MarHIfTHbIMH BOJIHaMIf HaM npHXO.ll.HTCH HMeTb ,AeJIO no BClO.lI.y .
CBeT - 3TO Te me 3JIeKTpOMarHHTHbIe KOJIe6aHIfH. 3",eKTpo-
MarHHTHbIMH BOJIHaMH HBJIHIOTCH If peHTreHOBCKHe JIYllIf H
HeBl1.l1.HMbIe rJJa30M HHpaKpaCHbIe, TenJIOBbIe JIY4H.
Pa3Ho06pa3HbIe BIf.ll.bI 3JIeKTpOMarHHTHbIX KOJIe6aHlIH HMelOT
O.ll.HHal<OBYIO npHpo)].y H O.ll.HHaKOBYIO CKO, pOCTb HO OHM OTJIH-
LJalOTCH )J,pyr OT )].pyra .lI.JII1HOH BOJIHbI If CBOIfMH CBOHCTBaMH.
388

CJIe)J,CTBHeM 3TOrO 0Y.ll.YT H pa3J1HlIHb:e 06JIaCTH HX npaKTWIC-
CKoro npHMeHeHHH.
ECJIH K nepellHCJIeHHbIM BH)J,aM KOJle6aHHH npH6aBHTb
KOJIe6aHHH OT.lI.eJIbHbIX 'laCTe3 pa60TaIorn.IIX MaruHH, IJ.eJIbIX
60JIbruHX coopymeHHH: MOCTOB, BbICOTHblX 3)J,aHHlI II op.,
TO CTaHeT HCHOH He06xo.ll.IIMOCTb H3Y4eHHH KOJJe6aTe.lbHbIX
.lI.BH}!(eHHH. B 3TOH He60JIbruoH rJIaBe MbI OCTaHOBHMCH lid
rpaHqeCKOM HCCJIe.ll.OBaHIIH rJIaBHbIM 06pa30M npOCTbIX r3p-
MOHlI'leCKIIX KOJIe6aHHH.
<1>OPMYJIbI
y = A sin w(,
v y = Awcoswt,
a y = - A w 2 sin wt,
x = A coswt,
vx=-Awsinwt,
ax = - A w 2 cos wt
(I)
(2)
(3)
( I')
(2')
(3')
B )J,ocTaTO'lHOH Mepe paCKpbIBalOT xapaKTepHble 'lepTbI rapMO-
HII'leCKoro KOJIe6aHHH TOqeK M v H M x' <1>opMYJIa (1) YKa3bI-
BaeT Ha paCCTOHl-lHe KOJIe6JIIOlUeHCH TOqKH OT Ha'laJIa KOOP)J,H-
HaT B JII060H MOMeHT BpeMeHH t. <1>opMYJIa (2) rOBopHT 0 TOM,
qTO CKOpOCTb KOJIe6JIlOlUeHCH TOqKH M}', MeHRHCb CO BpeMe-
HeM, ,AOCTHraeT HaII60JIbrueH BeJIH'lHHbl B Te MOMeHTbI, Kor)J,a
wt lIJIH, 'lTO TO me, yrOJJ a. paBHHeTCH k1t, r.ll.e k = 0, 1,2,3....
B 3TH MOMeHTbI TOllKa My npoxoAMT qepe3 Ha'laJIO KOOP.ll.H-
7t
HaT. CKOpOCTb TO'lKH My 6YAe paBHa 0, KOr.ll.a wt=k 2
7t
HJIH a.=k'j. B 3TH MOMeHTbI KOJJe6JIIOlUaHCH TOllKa HaXO)J,HTCH
Ha HaH60JIbrueM paCCTOHHIIH A OT HaqaJIa KOOPAIIHaT. HaKo-
Heu, iPopMYJIa (3) YKa3bIBaeT Ha TO, liTO YCKopeHIIe KOJIe6JIIO-
lUeHCH T04KlI nponopUHOHaJIbHO ee OTKJIOHeHIIIO.
HaH60JIbruee OTKJIOHeHlIe TOqKH My OT ueHTpa, T. e. Be.
JIH'lHHa pa)J,Hyca A, Ha3bIBaeTCH a M n JI H T Y A 0 H K 0 JI e-
6 a H H H.
BpeMH T, B KOTopoe T04Ka M onlICbIBaeT nOJlHbIH Kpyr,
a ee npoeKUIIH My npoxoAHT AHaMeTp ABa pa3a, Ha3bIBaeTCH
n e p H 0 .lI. 0 M K 0 JI e 6 a H H H. npOCJIe.ll.l-ITe TaK}!(e 3£1
1I3MeHeHHeM x, V x II ax'
YrOJl a.=t=wt Ha3bIBaeTCH  a 3 0 if K 0 JI e 6 a H H H.
ECJIli MbI HMeeM .lI.Ba rapMOHIIQeCKHX KOJIe6a, HJ-IH TO OHH
MorYT OTJIII'laTbCH Apyr OT .lI.pyra aMnJIHTYAOH, nepHO)J,OM H
iPa30iL
14*
;389

B 3BYKe H CBeTe aMnJUITY.lI.a 06YCJIOBJIHBaeT C060H CllJIY
3BYI<a HJIll CBeTa, TOr.ll.a KaK nepHOll 06YCJIOBJIIIBaeT BbICOTY
3BYKa II UBeT CBeTa.
nOJIO}!(HM, I.J.TO HaM )J,aHbI KOJIe6aHIIH )].ByX TO'leK YI H Y2
(tJepT. 201) C O.ll.HHaKOBbIM nepHO,AoM, HO T04Ka Y2 Ha'laJIa
CBOII KOJIe6aHHH Ha HeKOTopoe BpeMfI t l n03}\{e TOqKII YI.
3a 3TO BpeMH TO'lKa YI yme ycne.'la npOHTH a3Y
v
o
"
4epT. 201.
II Harull KOJIe6aHIIH
2r.
(X = ---r.tl'
TaK KaK nepHo.ll.bI T o6eHX
TOlleR O)].HHaKOBbI. TO pa)J,IIYCbI
OM I II OM 2 6Y.ll.YT )J,BHraTbCH C
O.ll.HHaKOBOH CKOpOCTblO II yrOJI
K cp Me}!(.lI.Y HIIMII OCTaHeTCH He-
H3MeHHbIM. BCJIe.ll.CTBHe 3Toro
Bee <pa3bl TOI.J.KH YI 6Y.lI.YT 60JIb-
rue <pa3 TOI.J.KH Y2 Ha BeJIH'lHHY cP
MomHO H306pa3IITb TaK:
YI = A sin wt;
Y2 = Asin(OIt - cp).
YrOJI cp Ha3bIBaeTCH p a 3 HOC T b 10  a 3 .lI.BYX .lI.aHHbIX
KOJIe6aHIIH.
TaKIIM 06pa30M, 3aKOH npOCToro rapMOflHI.J.eCKOrO KOJIe6a-
HHH, HanHcaHHbIH B caMOM 06rn.eM BH.lI.e, BbIpa}!(aeTCH op-
MYJIOM
Y = A sin (OIL + cp).
fpmlmKH KOJle6aTeJlbHblX Jl.BHlKeHH"
CBOHCTBa KOJJe6aTeJIbHOrO .lI.BIImeHHH TOtJKII, paCCMOTpeH-
HbIe HaMH B npe.ll.ruecTBYIOrn.eM naparpae, MO}!(HO 60JIee
HarJIH.lI.HO H306pa3HTb C nOMorn.blO rpaHlWB COOTBeTCTBYIO-
lUHX <PYHKUHiL lI.eJIaH 3TO, MbI ern.e pa3 y6e.ll.HMCH B TOM,
KaKOH IICKJIIO'lHTeJIbHbIH annapaT B Bl1.l1.e rpaHKOB HMeeTCH
.Y Hac B PYKax .lI.JIH HCCJIe)J,OBaHI1H CBoikTB H xapaKTepa nOBe-
.lI.eHHH pa3JIHlIHbiX YHKUl1li 11, B qaCTHOCTH, CBoikTB rapMO-
HH4eCKI1X KOJIe6aHHH.
rpaHqeCKl1e HCCJIe.ll.OBaHHH YHKUHH
Y = Asin (OIL + cp)
390

MbI Ha'lHeM C npocTenruero CJIY'laH, KOr.ll.a aMnJIHTY.lI.a I<OJIe-
6aHHH A = I, <p=O H w= I. Tor.ll.a ypaBHeHHe .lI.BHmeHHH TO'l-
KH 6Y.ll.eT TaKOBO:
y = sint.
0603Ha'laH B .lI.a.nbHeHrueM aprYMeHT t 'lepe3 x, MbI npHxO.ll.HM
K tPopMy.ne npoCTeHruero rapMOHH'leCKoro KOJIe6aHHH
y = sinx.
nOCTpOl1B rpaHK 3TOro .lI.BHmeHHH, MbI nOJIY4HM H3BeCT-
HYIO CHHYCOH.lI.Y ('lepT. 202). npH nocTpoeHHH CHHYCOH.lI.bI,
o
)(
-,
4epT. 202.
KaK H Bcex .lI.aJIbHeHruHx rpaHKOB, Heo6xo.ll.HMO MacruTa6bl
Ha OCHX Koop.ll.HHaT 6paTb O.ll.HHaKOBbIe. Ha HarueM pHCYHI<e
3a e.ll.HHHUY npHHHT l' CM. nOCTpoeHHe npaKTH4ecKH Y.lI.06HO
Ha'laTb C pa3BepTbIBaHHH oKPymHocTH c pa.ll.HYCOM, paBHblM
1, no OCH x, Ha4HHaH OT Ha'laJIa KOOp.ll.HHaT, H nOJIY'lHTb Ha
OCH a6cUHcc OTpe30K, npH6JIHmeHHO paBHblH .lI.JIHHe oKpym-
H<YCTH (2.3,14. 1 =6,28). 3aTeM pa3.l1.eJIHTb 3TOT OTpe30K CHa-
'laJIa Ha 4 paBHble 4aCTH, a nOTOM Kam.ll.YIO H3 nOJIY4eHHbIx
'laCTeH ern.e Ha 3 paBHbIe 'laCTH. TaKHM 06pa30M, pa3BepHYTaH
oKPymHocTb 6Y.lI.eT pa3.l1.eJIeHa Ha 12 paBHbIX 'laCTeH; .lI.JIHHa
o 6 27t 7t ( ) 1.1 0
gam.ll.oH 4aCTH y.ll.eT paBHa 12=6 CM. aJIbHeHruee no-
CTpoeHHe rpaHKa HaM H3BecTHO, H MbI He OCTaHaBJIHBaeMCH
Ha HeM.
nOCTpoeHHblH Ha 4epTeme rpa(}mK KOJIe6aTeJIbHOrO .lI.BH-
meRlU! y=sin x n03BOJJHeT Y3HaTb, Ha KaKOI\l paCCTOHHHII OT
Ha'laJIbHOH T04KH (Ha4aJIa KOOp.ll.HHaT) HaX0.ll.HTCH KOJIe6JIIO-
rn.aHH T04Ka My B .n1060M MOMeHT BpeMeHH. 1.1JIH 3Toro .lI.OCTa-
T04HO Ha OCH x HaHTH cooTBeTcTBYIOrn.HH MOMeHT t BpeMeHH
H onpe.ll.e.nUTb 0p.ll.HHaTY rpaHKa, cooTBeTCTBYJOrn.yIO a6c-
UHcce x= t.
391

CKOpOCTb KOJJe6JIIOlUeHCH T04KH B JII060H MOMeHT Bpe-
MeHH onpe.ll.eJIHeTCH no opMYJIe:
V y = cost HJIH V y = cosx.
Ha 'lepTeme 202 nOCTpoeH H rpaHK CKOpOCTH (nYHKTHpHaH
KpHBaH) B BH.lI.e rpaHKa cos x. nOCJIe.ll.HHH rpaHK 04eHb
HarJIH)J.HO OTpamaeT TOT BbIBO.ll., KOTOpbIH MbI C.lI.eJIaJIH paHbrue,
"ITO HaH60JIbrueH CKOpOCTH KOJJe6JIIOlUaHCH T04Ka .lI.OCTHraeT
B MOMeHT npOxom.ll.eHHH 'lepe3 Ha'laJIO KOOp.ll.HHaT, HaHMeHb-
rueH me CKOpOCTH - B MOMeHT HaH60JJbruero paCCTOHHHH OT
Hero, npH4eM B 3TOT MOMeHT CKOpOCTb KOJIe6JIIOrn.eHCH TO'lKH
BCHKHi'I pa3 MeHHeT 3HaK.

o
J(
-1
4epT. 203.
lIaJIbHeHruee HCCJJe.ll.OBaHHe pa3JIH'lHOrO BH.lI.a KOJIe6aTeJIb-
HbIX .lI.BHmeHHH npOBe.ll.HTe CaI\1OCTOHTeJIbHO, peruHB pH.lI. 3a)J,a'l.
npem.ll.e 4eM npHCTynHTb K HCCJIe]:(oBaHHIO, BaM npH.lI.CTcH
nOCTpOI-ITb onpe.ll.eJIeHHbIe rpaHKH, cBepHTb HX C COOTBeT-
CTBYIOrn.HMH rpaHKaMH B TeKCTe H TOJJbKO nQToM Ha'laTb
Hcc.le.ll.OBaHHe .lI.aHHOrO BH.lI.a KOJJe6aHHH, Bce BpeMH YBH3bI-
BaH CBoHcTBa }mJIe6aTeJIbHOrO .lI.BHmeHHH C ero rpaHKoM.
3aJl.qa 1. nOCTpOHTb Ha O.ll.HOM 'lepTe}!(e rpaHKH KOJIe-
6aTeJIbHbIX .lI.BH}!(eHHH:
y = sin x; y = sin 2x; y = sin .
nOCTpoeHHe BbmOJJHeHO Ha 'lepTeme 203. PeruHB 3TY
rpaH'lecKYIO 3a.ll.a'lY, He06xo.ll.HMO conOCTaBHTb rpaHKH .lI.BYX
nOCJIe;I.HHX KOJIe6aHm'f C rpaHKoM npoCTeHruero KOJJe6aHHH
y=sin x, 'lT06bI BblHCHHTb BJIHHHHe Ha rpaHK H3MeHeHHH
27t 27t
napaMeTpa w=1' (.lI.JIH nepBoro KOJJe6aHHH w=1' = 1,
27t 27t I
.lI.JIH Bmporo w=1'=2, .lI.JIH TpeTbero w=1' = 2)' npH
392

BIIHMaTeJIbHOM H3Y'leHHH rpaHKOB H COOTBeTCTBYlOlUHX HM
ypaBHeHHH MbI, HeCm,meHHO, npH.QeM K BbIBO.ll.Y:
21t '" '" .
npH B03paCTaHlIH W='j' rpa'J!HK 't'YHKUHH y=sm x C}!(H-
MaerCH no ropH30HTaJIbHOMY HanpaBJJeHHIO, OTpamaR YMeHb.
weHue nepHO.ll.a KOJIe6aHHH T.
3aJl.aIJa 2, nOCTpOHTb Ha O.ll.HOM qepTeme rpaHKH Kone-
6aTeJIbHbIX .QBHmeHHH:
y = sin x; y = sin T.X.
nOCTpoeHHe BbmOJIHeHO Ha qepTe}!(e 204.
v
o
x
-I
4epT. 204.
3aJl.a l ia 3, nOCTpOHTb rpauKH KOJIe6aTeJIbHbIX .lI.BH1KeHHH:
y = sin x; y = 1,5 sin x; y = 2 sin x.
nOCTpoeHue BbmOJIHeHO Ha qepTeme 205. ConOCTaBJIeHHe
rpaHKoB C COOTBeTCTBYlOlUHMH opMYJIaMH n03BOJIHT CAeJIaTb
CJJe.ll.YlOlUHH BbIBO)J,:
v
2
x
4epT. 205.
393

npH B03paCTaHHH aMnJIHTY.lI.bI A KOJIe6aHHH rpaHK y=
=sii1 x paCTHrHBaeTCH no BepTHKaJIbHOMY HanpaBJIeHHIO.
I13MeHeHHe napaMeTpa A He BJIHHeT Ha nepHO.ll. KOJIe6aHHH,
I1MeH rpaHK YHKUHH y=sin x, MbI MO>KeM 'lHCTO reo-
MeTpH'leCKH nOJIYIlHTb rpaHK YHKUHH y=A sin x nYTeM
YBeJIH4eHHH Kam.ll.OH 0p.ll.HHaTbI rpaqmKa y= sin x B A pa3.
v
2
x
o
-I
-2
4epT. 206.
3aJ{alJa 4. nOCTpOHTb Ha OJ{HOM qepTeme rpa<pHK KO.'Ie-
6aTeJIbHOrO ,ABH}!(emlH
y = sin 2x
H rpaHK CKOpOCTH 3Toro )J,BHmeHHH
v y = 2cos 2x
nOCTpoeHHe C.lI.eJIaHO Ha pHcyHKe 206.
3aJl.aIJa 5, nOCTpOHTb Ha O.ll.HOM 4epTe}!(e rpaHKH KO-
JIe6aTeJIbHbIX .lI.BH}!(eHHH:
,
)(
-I
4epT. 207.
394

y=sinx, y=sin(x-l) H y=Sin(x- ; ).
nOCTpoeHHe BbInOJIHeHO Ha 'lepTe*e 207. 1.JepTem HarJIH)J,-
HO OTpamaeT, 'lTO BO BTOpOM cJIY'lae KOJJe6aHHH T04KH 3ana3-
7t
.lI.bIBalOT B a3e Ha 1, a B TpeTbeM - Ha "2 no cpaBHeHHIO
C KOJIe6aHHHMH B nepBoM cJIyqae.
3aJl.aIJa 6. nOCTpOHTb Ha O.ll.HOM 'lepTeme rpaHKH KO-
JIe6aTeJIbHbIX .lI.BHmeHHH:
y = sin x H y = 1,5 sin (x + I).
nOCTpoeHHe C.lI.eJIaHO Ha 'lepTe*e 208.
LI.o CHX nop MbI paccMaTpHBaJIH TOJIbKO KOJIe6aHHH C
nOCTOHHHOH aMnJIHTY.ll.OH. B npHpO.ll.e, O)J,HaKO, O'leHb 'laCTO
BCTpe'laIOTCH KOJIe6aHHH, aMnJIHTY.lI.bI KOTOpbIX MeHHIOTCH co
BpeMeHeM, YMeHbruaHcb. nepe.ll. HaMH 3aTyxalOrn.He KOJIe6a-
HHH. 06rn.eH npH'lHHOH 3aTyxaHHH KOJIe6aHHH HBJIHeTCH no-
'l"epH 3HeprHIf. nOTepH 3HeprHH MomeT npOHcxo.ll.HTb, Bo-nep-
BbIX, OT TpeHHH, H, BO-BTOpbIX, TO'lKa MomeT nepe.ll.aBaTb KOJIe-
6aHHH .lI.PYrHM TO'lKaM, panpoCTpaHHH KOJIe6aHHH B oKpyma-
IOlUeM npoCTpaHcTBe B BH.lI.e BOJIH. YpaBHeHHe 3aTyxaIOlUHx
KOJJe6aHHH TaKOBO:
A '" 21t t
Y = e-'" cos ----.,- .
npH nOCTpoeHHH rpaHKa 3aTyxalOlUlIx KOJIe6aHHH
MbI BHeceM B 3TO ypaBHeHlIe HeKoTopbIe yn, porn.eHHH 3aMe-
HHB HenepoBO 'lnCJIO e 'lHCJIOM 2, II nOCTpoHM rpaHK .lI.JIH
YHKUHH.
3aJl.a l ia 7, y=O,5.2- x cos'itX (lJepT. 209).
1.JT06bI nOMO'lb BaM npH nocTpoemm rpaHKa 3aTyxaIO-
lUHX KOJIe6aHHl1, MbI .lI.aeM Ta6JIHUY 3Ha'leHHH aprYMeHTa x
H cooTBeTcTBYIOlUHX 3Ha'leHHH YHKUHH y.
x 1-41-3.51-3\-2.51-21-1.51-11-0.51 G 1 1 1 )51 2 )51 3 13051414:5
yl81 0 1-41 0 /21 0 I-II 0 1°. 5 1-+1 0 I 0 I-I 0 II 0
KpaTKO OCTaHOBHMCH Ha CJJomemm rapMOHH'lecKHX KO.1e-
6aHHH.
395


I
j
4epT. 208.
v
8
- 1
1"7
I
6
5
4-
3
I 2
-I' -
 I /\. i -
-4  ' 2 \VO .-I
... X
J.-  2 3 '- f -
I -, 
I-
, I -2 -
-
I - -,3
--I '-4 -
l- I -1
-5.t
--.J
4epT. 209.

3aJl.aQa 8, nOCTpOHTb rpaHK KOJIe6aHHH
y = sin x + cosx,
nOJlYIlHBruerocH B pe3YJIbTaTe CJIO}!(eHHH .lI.BYX npocTbIX rap-
MOHHlIeCKHX KOJIe6aHHH y=sin x H y=cos x.
npH nocTpoeHHH rpaHKa YIlTCM, IlTO 0p.ll.HHaTa HCKOMoro
rpaHKa npH .lI.aHHOM 3HaIleHHH x MomeT 6bITb nOJlY4eHa
nYTcM CJIOmeHHH 0P.lI.HHaT y=sin x H y=cos x npH TOM me
3HalleHHH aprYMeHTa.
nOCTpoeHHe BhInOJIHeHO Ha IlepTe}!(e 210.
v
2
o
-I
-2
4epT. 210
3aJl.aIJa 9, nOCTpOHTb rpaHK KOJIe6aHHH
y = sin x + sin 2x,
nOJJYIlHBruerocH B pe3YJIbTaTe CJIOmeHHH .lI.BYX Ko.ne6aHHH:
y = sin x H y = sin2x.
nOCTpoeHHe BbInOJJHeHO Ha IlepTeme 211.
B hI B O.ll.. CYMMa .lI.BYX npOCTblX rapMoHHIlecKHx KOJIe6a-
HHH C O.ll.HHM H TeM me nepHO.ll.OM eCTb npOCToe rapMoHHIlecKoe
KOJJe6aHHe C TeM }!(e nepHO.ll.OM. CYMMa .lI.BYX npocTblX KOJIe6a-
HHH c pa3HblMH nepHO.ll.aMH ecTb CJIOmHOe rapMoHHIlecKoe
KOJJroaHHe, .lI.JIH KOTOporo xapaKTepHO TO, qTO aMnJIHTY.lI.a
caMa HBJIHeTCH nepHO.ll.H'leCKH H3MeHHIOmeHCH YHKUHeH.
nepBblH BbIBO.ll. MomHO nOJIYIlHTb aHaJIHTHIleCKH, CTOHT
TOJIbKO CYMMY sin x+cos x 3aMeHHTb npoH3Be;r.eHHeM sin x+
397

+ sin (90 0 - x) = 2sin 45°. cas (x - 450) = V2 cos (x-45°),
IlOJJYlIeHHbIH pe3YJIbTaT roBOpl1T 0 TOM, Ilm B pe3YJIbTHPYIO-
meM KOJJe6aHl-IH nepl10.ll. OCTaJICH TaKHM *e, KaK H B CJIarae-
MbIX HOJJe6aHHHX, HO H3MeHHJIHCb a3a Ii aMDJIHTY.lI.a, KOTO-
paH OKa3aJIaCb paBHOH V2 (CBepbTe C IlepT. 21 ().
v
2
o
)(
-<1
4epT. 211.
fpalf>HKH pellialOT ypaBHeHHH
PaCCMOTpHM B noif me rJIaBe, BHe CBH311 C KO.TJe6aTeJJb-
HblMH .lI.BHmeHI1HMH, Bonpoc 0 rpaHlIeCKOM perueHHH HeKOTO-
pblX ypaBHeHHH.
1. PeruHTb rpaH'leCKH ypaBHeHHe:
. x
SIn X = -"2'
v
o x
I : : 
.,
4epT. 212.
398

Peru e H H e. npl1paBHHBaeM JIeBYIO 11 npaBYIO qaCTH
ypaBHeHIiH K y, nOJIyqHM )J,Ba HOBbIX ypaBHeHHH:
. x
Y=SInX Ii Y=T.
CTpOI1M rpa
HKH Ka*)J,oro 113 HHX Ha O)J,HOM Ii TOM *e lIep-
Te}!{e (lIepT. 212). A6cUHCCbI TOqeK nepeCeqeHIiH nOJJyqeHHbIX
y


Q


I(


-1


4epT. 213.


v


5


4


-


6


3
2 \
- -- -- - - 1\f- -
5 -4 -3 -? -, , I/r<::..!-.....
-t----..V 0 , 2 3

 --
 -
 - \-- -
-2


}(1


4 5


6


- ---


-3


-4


-5
4epT. 214.


rpa
HKOB Ii 6Y.LI.YT KOpHHMIi .lI.3HHOrO yp3BHeHIiH. 3TH KOpHH
TaKOBbI: Xl=O; x 2 =1,9 Ii x:J=-1,9.


399




2. CKOJIbKO peruemIii HMeeT ypaBHeHHe
Igx = sin x?
Ha nOCTaBJIeHHblH Bonpoc HaH.lI.eTe OTBeT Ha qepTe}!(e 213.
3. PeruHTb rpaqmqecKH ypaBHeHHe
x.arc tgx = 1.
P e III e H H e. nepenHrueM ,AaHHOe ypaBHeHHe CJJe.ll.YIOlUHM
o6pa30M:


I
arctgx=-.
x


npHpaBHHeM Ka}!(.lI.YIO 4aCTb nOCJIe,AHero ypaBHeHHH K y.
nOJJYlIHM ,ABa HOBblX ypaBHeHHH:
1
y = arc tg x H Y = - .
x


CTpOHM H3 O,AHOM 4epTe}!(e rpaqmKH 3THX ypaBHeHHH
(lIepT. 214). A6cuHCCbI T04eK nepece4eHHH nOJJY4eHHbIX rpa-

HKOB H 6Y,AYT perueHHHMH .lI.3HHOro ypaBHeHHH.


JI 11 T E" PAT Y P A


B. JI. r 0 H q a p 0 B. Bbl'lHCJlHTeJlbHbIe H rp8!jJII'IecKlie ynp8m-
HeRIIS! C !jJYHKLlHOH8J1bRblM COjlepm8HHeM, 113)1.. AnH PC<DCP, 1948.




"


-'


. \\
.--'" It : /-'
:' i
j'/ '!et-,/ //

 /

,
\ .
 K./ _
> ,



- /

1 -V/-(lJinoc)
 7 ;J


\.


'I


HA


X='Z(1-


1


.


r JJ A B A XVI


nPAKHtLJECKI1E npI1MEHEHI1S1 MATEMATI1KI1


«TeopliH Oe3 npaKTIIKH MepTBa H.qll
6eCITJJO.nHa, npaKTHKa 6e3 TeOpHII He-
B03MOlKila IIJiH nary6Ha».
A. H. HpbVI08


A Ka
eMHK A. H. KOJIMOropOB. 06palUaHCb K Y'Ia-
lUHMCS! cpe
HeH llIKOJlbl, fIHllIeT: «3H3'1eHl1e MaTe-
MaTH'IeCKMX MeTO.l(OB B TaKI1X HaYK8X, KaK Mexa-
HHKa, q>H3HKa MJIH aCTpOHOMMSI, XOpOIllO M3BeCTHO.
T aK>Ke BceM H3BeCTHO H TO, '1TO MaTeMaTMKa He06xo.l(MMa B
npaKTH'IeCKOll pa60Te MH>KeHepOB H TeXHMKOB. 3J1eMeHTap-
HbJe 3H8HHH no reOMeTpHH MJIH YMeHHe nOJlb30B8TbCSI 6YKBeH-
HbIMM q>opMYJIaMM He06xo.l(HMbI nO'lTH Ka
OMY MacTepy MJlH
KBaJIHq>Hli.HpOBaHHOMY pa60'leMY...
Erne B 60Jlblllell CTeneHH caMOCTOSlTeJIbHOCTb H cnoc06-
HOCTb nO-HoBoMY nO.l(OHTI1 K MaTeMaTM'IeCKOH 4>oPMYJII1pOBKe
3a
a'lM He06xo.l(HMa TOMY, KTO 6epeTcH npMMeHSlTb MaTeMaTMKY
K pellleHl110 TeXHM'IeCKHX np06JIeM. 3TO no CYlUeCTBY OTHOCMT-
CSI K pa60Te Ka>K.l(OrO HH>KeHepa, no.1Jb3YlOlUerocSi MaTeMaTI1-
KOll. Ho TaK KaK Tpe6YlOlUMecSi npH 3TOM MaTeMaTH'IeCKne
3HaHMS! M cnoc06HoCTM MMeIOTCH He Y Bcex, TO 60.1bIllMHCTBO
HalliHX H8Y'IHO-MCCJIe.l(OBaTeJIbCKHX TeXHM'leCKMX MHCTI1TYTOB
H .l(a>Ke HeKOTopble K.PynHbIe 3aBO
bl llIHpOKO CTaJIH Ha nYTb
401




npMBJIe'leHHSI Cneli.MaJIMCTOB-MaTeMaTMKOB J(JlSI pa60TM BMeCTe
C MH>KeHepaMM Ha.l( TeXHM'IeCKHMH np06JIeMaMH».
3TI1 CJIOBa COBeTCKOrO Y'IeHOrO, HeMaJIO C.l(eJI3BllIerO H B
06.'13CTM npaKTH'IeCKOrO npHMeHeHMSI M3TeMaTMKM (A. H. KOJl-
MoropOBY npHH,8J,I.fe)KaT. HCCJIeOBaHMH no TeopMH CTpeJIb6bl
M no CTaTM'IeCKHM MeTO.l(aM KOHTpOJISI MaCCOBOH npO.l(YKli.I1H),
nO.l('1epKI1BalOT rJIy60KYIO CBSl3b Me)K.l(Y MaTeMaTHKOH I1.l(PYrMMI1
HaYKaMM H TeXHMKOH.
Dpe>K.l(e '1eM nepeHm K OCHOBHOMY CO.l(ep>KaHHIO 3TOH, no-
CJIeHeM rJIaBbI - HeKOTopblM npaKTM'IeCKI1M BonpocaM, pa3-
pelllaeMblM Ha 6a3e TeOpeTM'IeCKOrO MaTepI1aJla, OCTaHOBHMCSI
HeMHoro Ha Tex MaTeMaTI1KaX, KOTopMe BceH CBoeH pa60ToM
YTBep>K.l(aJIH He06xo.l(MMOCTb rJIy60KoM CBSl3H Me>K.l(Y TeopHeH
MaTeMaTI1KH M npaKTHKoH. 13 60JlblllOrO '1HCJIa TaKMX MaTe-
MaTI1KOB MbI BbI.l(eJIMM HaI160JIee SlpKMX B 3TOM OTHOllleHI1H H
60JIee 3HaKOMbIX BaM: ApxHMea, HbIOToHa, .n. 3ii:JIepa,
D. .n. 4e6MweBa 11 A. H. KpbIJIOBa.
HaM CeH'IaC Ka>KeTCSI HeBepoSlTHbIM, '1TO 6bJJIO BpeMSI, KOr.l(a
Y'IeHble H IlmJIoco<pbJ C BeJlHKHM npeHe6pe>KeHHeM OTHOCI1JJHCb
K nonblTKaM CBSl3aTb TeOpeTH'IeCKHe 3HaHI1S1 no 1\laTeMaTI1Ke
C ee npaKTM'IeCKHMI1 npMJIO>KeHMSlMH. Ho TaKoe BpeMSI 6bIJlO.
06 3TOM CBl1.l(eTeJIbCTBYIOT BbICKa3bIBaHMSI H ApHCTOTeJISI
(384-322) H DJIaTOHa (432-348), BeJIMKHX MbICJlMTeJIeH M
oCHOBaTeJIeH .l(peBHerpe'leCKIIX <t>MJIOCcxpCKI1X llIKOJl.
API1CTOTeJIb, BnJIOTHYIO nO.l(Ollle.l(WMH K MaTepMaJIMCTH'Ie-
CKOl\1Y nOHHMaHMIO MaTeMaTI1KM KaK HaYKH (<<MaTeMaTI1K nOk
BepraeT paccMoTpeHMIO 06'beKTbl, YCTpaHHBllIM Bce '1YBCTBeH-
HbIe CBOHCTBa I1X H coxpaHSISI TOJlbKO KOJIM'IeCTBY onpe.l(eJIeH-
HOCTb 11 HenpepbIBHocTb, I1JJI1 pa36HpaeT Te nOJIO">KeHHSI, B
KOTOpbIX OHM CTOHT .l(Pyr K .l(Pyry»l), YTBep>KJ(aJi B TO >Ke
BpeMSI, '1TO «MeXaHI1Ka - He HaYKa, a peMeCJIeHHbIH HaBbIK,
.l(OCTOMHbIH pa6a H 113JIHllIIIHH .l(JISI <t>I1JIOCO<pHH»).
DJIaTOH no nOBO.l(Y pa60T MaTeMaTI1KOB ApXI1Ta 11 MeHeXMa,
nblTaBllIMXCSI pelllMTb KJIaCCl1'1eCKYIO 3a.l(a'lY 06 Y.l(BOeHHM Ky6a
nYTeM npMMeHeHI1H I1HCTpYMeHTOB H MexaHH3MOB, rOBOpl1JJ:
«OHM ry6S1T M pa3pYllIalOT 6JIaro reoMeTpI1M, TaK KaK npM
3TOM aHa YXO.l(HT aT 6eCTeJIeCHbIX H YMonOCTHraeMblX BellJ,eH
K '1YBCTBeHHbIM, H nOJlb3YIOTCSI TeJIaMH, HY>K.l(alOlUHMHCSI B
npHMeHeH1111 OPYJ(UM nOllIJIOrO peMeCJla».
DJIYTapx, rpe'leCKMH HCTOpI1K II B. H. 3., nO.l(BO.l(SI MTor
nO.l(06HbIM B3rJISIaM, nmlleT: «DocJIe, DJlaTOHa MeXaHI1Ka,
1) A p H C TOT e JI b, MeTaq,H3I1Ka.
402

il3rHaHHaSi 113 reOMeTPH11, OT.lleJIHJIaCb OT Hee 11 .l(OJIfOe BpeMH
HaXO.l(JlJIaCb B npeHe6pe>KeHI1H Y reOpeTJI'IeCKOI'i HaYKH, CTaB
JjHl11b OHOH H3 BCnOMOraTeJJbHblX npaKTH'IeCKHX OTpaCJIeH
BoeHHoro HCKYCCTBa».
T eM C 60Jl blllHM YBa>KeHHeM Mbl .l(OJl>KHbI OTHeCTHCb K re-
HHaJlbHOMY MaTeMaTHKY H MexaHHKY .l(peBHO::TH, BeJIHKOMY
ApXHMe.l(Y (287-212 .l(O H. 3.), KOTOpblH He n060HJICSI nOHTH
npOTHB B3rJISI.l(OB, npHHSlTbIX B H.l(eaJIHCTH'leCKOH MaTeMaTH-
qeCKOH HaYKe. CBOI1 3aMe'laTeJJbHble OTKpblTHH B 06J1aCTH
MaTeMaTHKH, H3 KOTOpbIX HeKOTopble nO.l(rOTOBJISlJIH nO'lBY
.lI.JISi MeTO.l(OB, BBe.l(eHHbIX B MaTeMaTHKY TOJlbKO '1epe3 2000 JleT
nOCJIe Hero, OH He CTeCHSlJlCH npliMeHSlTb K pel11eHHlO MHOro-
'1HCJIeHHblX BonpOCOB aCTpOHOMHI1, MexaHHKH, q>H3HKH, Tex-
HHKH H BoeHHoro .l(eJIa. PaCCKa3bIBalOT 0 C.l(eJIaHHOM ApXHMe-
)1,OM He6eCHOM rJI06yce, Ha KOTOpOM MO>KHO 6blJJO Ha6JJ1oaTb
He TOJlbKO .l(BH>KeHl1e CBeTHJI, HO H 3aTMeHHSI COJIHli.a H JIYHbl,
06 OCTPOYMHOH Mal11HHe .l(JISI nOJlI1BKH emneTCKHX nOJleH,
o CKOHCTpYHpoBaHHblx HM CJlO>KHeHl11HX BoeHHblX Mallll1Hax.
Kor.l(a HaCTaJIa TSI>KeJIaSi nopa JISI era pOHoro ropo.l(a
CHpaKY3bl, KOTOpblH pHMJlSlHe oca.llHJIH C BYX CTOpOH, ApxI1-
Me.l( npH nOMOlUH nocTpoeHHblx HM MalllHH B Te'leHHe .l(oJiroro
BpeMeHH OTpa>KaJI Bce HanaJ(eHHSI pHMCKHX COJJ.l(aT, BHOCSI
B HX pSlll,bI naHHKY H HaHOCSI HM rpoMallHble nOTepH. MapUeJIJI,
nOJIKOBO.l(eli. pHMJJHH, ll1YTHJI Ha.l( CBOI1MH TeXHHKaMH H Mexa-
HHKaMH H rOBOpHJJ: «Y>K He nepeCTaTb JIH HaM J(paTbCSI C Ma-
TeMaTI1KOM? OH, CI1.l(SI cnOKOHHO 3a CTeHOH, TOnHT Hallll1 KO-
pa6J111 H, 6pocaH B Hac pa30M CTOJlbKO CTpeJI, OCTaBJIHeT no-
3a.l(H MHq>H'IeCKHX CTOpYKHX BeJIHKaHOB». «lleHcTBI1TeJlbHO,
Bee OCTaJlbHble CHpaKY3HHe CJIY>KHJlI1 CBoero pO.l(a TeJIOM apXH-
Me.l(OBblX MalllHH,- nI1ll1eT 06 3TOH oca.l(e DJlYTar:x,- O.l(HH
OH 6blJl YllIOIO, KOTopaSi Bcex .l(BHraJIa, BCe HanpaB-
JISlJIa» .
He MeHee SlpKOH B 3TOM OTHOllleHHR q>HrYPoH 6bIJI H 3Ha-
MeHHTblH aHrJlHHCKHH Y'leHbIH 11caaK HblOTOH (1642-1727).
<<.MaTeMaTIiKa B era PYKax,- nHllIeT H3BecTHblH PY.CCKHH q>H-
3HK C. 11. BaBI1J10B,- 6blJla MeTOOM H MaTepHaJlOM MSI pe-
llIeHHSI OCHOBHblX 3all,a'l aCTpOHOMRI1 H q>R3RKR, H TeM He Me-
Hee B 06JIaCTI1 MaTeMaTHKH OH CJ(eJlaJJ HCKJllO'IHTeJlbHO MHoro.
HOBbie 3anpOCbl q>113HKI1 Tpe60BaJIH H HOBOH MaTeMaTHKH, HO-
BblX MeTO.l(OB. AHaJlR3 6eCKOHe'lHO MaJlblX 6blJl COBepllleHHO
He06xoRM J(JlSI 3aa'l HOBOH MeXaHHKI1». 11 HblOTOH C03.l(aeT
9TOT aHaJlH3. B Hallie BpeMSI Tpy.l( .l(Hq>cPepeHUHpYeTCH. <I>H-
3HKH, HH>KeHepbl, KOHCTpYKTOPbI CTaB51T 3a.l(a'lH, MaTeMaTHKH
11. 13.
403

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" .
.,.
,
I1caaK HbIOTOH.
AalOT MeTOJJ.bI H peIlleHHSI, «HblOTOH OJJ.HOBpeMeHHO JJ.enaJI H TO
If JJ.pyroe».
C HMeHeM TOro BeJIHqaHUlero M8TeMaTHKa Bbl He pa3
BCTpeTHTeCb, H3yqaSl MaTeMaTHKY, «)JH3HKY, MexaHHKY H aCTpo-
HOMHIO. CH 6b1J1 .l(eikTBHTelIbHO OHHM H3 Tex Y'IeHbIX, KTO
Taep.l(O YCTaHOBHJI CBSl3b Ne MaTeMaTHKOH H .u.pyrHMH Ha-
YKaMH, Mey MaTeMaTHKOH H TeXHHKoiL
CpeJJ.H PYCCKHX Y'IeHbIX MbI TaK>Ke MO>KeM YKa3aTb
Ha UeJIblii pSlJJ. Y'IeHbIX, KOTOpble He MblCJIHJlH MaTeMaTHKH BHe
CBSl3H C npaKTHKoii. OHHM H3 T3KHX yqeHbIX 6b1J1 lJJIeH neTep-
6yprcKoH AKa.l(eMHH HaYK JJHap 3HJlep (1707-1783).
HeI13MepHMbI ero 3aCJIyrH B pa3BHTHH MaTeMaTHKH. noslB-
JieHlle ero OCHOBHblX MaTeMaTHqecKIlX pa60T: BBe.l(eHHe B aHa-
JiH3, .l(H4xJ>epeHli.HaJIbHOe H HHTerpaJIbHOe HC'IIICJleHHe npoH3-
BenH Ha cOBpeMeHHblx eMY MaTeMaTHKOB He TOJIbKO rJly60Koe,
HO, MO>KHO CKa3aTb, OIlleJIOMJISllOlUee BneqaTJleHHe; He)l.apoM
JI.'A.naMtep l) B O.l(HOM 113 CBOHX nHceM JIarpaH>Kyl) Ha3bIBae-r
I) n . A '" . M (j e p H n a l' p a H 111: - HssecrHue 4>P.HscKHe
lIareMarHIUI.
M)4

911. IJe pa «3TOT J(bSlBOJJ», KaK 6bI >KeJlaSi BblCKa3aTb 3THM, 'ITa
c,D:e.1aHHoe 3H.1epmI npeBblIllaeT CHJIbI '1eJIOBe'leCKl1e. u
Ho Hac nOpa>KaeT TaK>Ke H HCKJlIO'IHTeJlbHO 1lI11pOKHH Kpyr
ero I1HTepeCOB H BHe MaTeMaTHKI1. CTOI1T TOJlbKO nepe'l:1c.'1HTb
Ha3BaHHSI HeKOTopbIX era pa60T, '1T06bI y6e.l(HTbCH B 3TO:\1.
Cpe.l(H HI1X MbI YBH.l(HM: «MexaHHKY», «0 .l(BH>KeHHH Tsep.l(bIX
Tem>, «MOPCKYIO HaYKY», «.II.l1onTpHKY», «APTHJIJI ep 1110» , «Teo-
pliIO .l(BI1>KeHHSI nJIaHeT H KOMeT», «TeopHIO MY3bIKH» H MHoro
,D:pyrHx TPY.l(OB.
Y.l(I1BHTeJIbHO, '1TO, Y>Ke 6Y.l(Y'I11 CJlenbiM (3HJIep nOTepSlJl
3peHI1e 3a 17 JleT .l(Q CMepTI1), OH He npeKpaTI1J1 CBoeH pa60TbI
Ii npO.l(l1KTOBaJi CBoeMY CbIHY H CBOI1M Y'leHHKaM COTHI1 CTaTeH
J-I 10 rpOl\1a.l(HbIX TOMOB OTeJlbHbIX CO'll1HeHHH no caMbIM pa3-
Hoo6pa3HblM BonpocaM '1I1CTOH H npl1KJla.l(HOH MaTeMaTHKH.
XIX B. .l(aeT HaM TaK>Ke He O.l(HH npHl\Iep nO.l(06HOro CTpeM-
.'leHI1S1 1'vlaTeMaTI1KOB CBSl3aTb pa3BHTHe CBOeH HaYKH C pa3-
BIITI1eM TeXHl1KH. Cpe.l(H HHX oc06eHHO BbI.l(e.TJSleTCSI BeJlH'IaH-
WIIiI PYCCKI1H MaTel\laTHK DaqmYTI1H JIbBOBH'I 4e6bIllleB (1821-
1894). «C6J1I1>KeHHe TeOp1111 C npaKTHKoH .l(aeT caMble 6J1aroTBop-
Hble pe3YJlbTaTbl, H He O.l(Ha TOJIbKO npaKTHKa aT 3TOro BbmrpbI-
BaeT: CaMI1 HaYKI1 pa3BHBaIOTCSI no.l( BJII1S1HHeM ee»,- nHca.TJ
Ye6bIllIeB. 11 no He 6blJlH TOJlbKO CJlOBa. BceH cBOeH HaY'IHOH
H npaKTH'IeCKOH .l(eSlTeJlbHOCTbIO OH nO.l(TBep.l(HJI He06xo.l(I1MOCTb
H B03MO>KHOCTb TaKoH CBSl311. KaK H 3H.!Jep, 4e6blllleB nopa-
)!{aeT Hac HCKJIIO'II1Te.TJbHO OOlllHpHbIM KpyroM CBOHX HaY'IHbIX
H TeXHH'IeCKI1X HHTepecoB. C03.l(aB TeOpl11O BepOSITHOCTeH KaK
HaYKY. OH npHMeHHJI ee BbIBO.l(bI K pellleHHIO MHomx npaKTH-
'1eCKHX BonpOCOB: 3.l(eCb BonpOCbI 113 06.TJaCTH apTH.'1JIepHI1,
H3 06JIaCTH YCTaHOBJleHHSI Q>I13H'IecKHX nOCTOSlHHbIX H .l(p.
B 06J1aCTH npaKTH'IeCKI1X npHJlO>KeHHH MaTeMaTHKY 4e6buneBY
npl1HaMe>KaT pa60TbI: «0 3y6'1aTbIX KOJleCax», «0 nOCTpoe-
HHH reorpaQ>H'IeCKI1X KapT», «0 U.eHTp06e>KHOM ypaBHHTe,lJe»
I1 .l(p. 40 pa3J1H'IHbIX MO.l(eJIeH llIapHI1pHbIX MeXaHI13MOB CKOH-
CTpYHpOBaJi H nOCTpOI1JI 4e6bIllIeB H HanHCaJi a HI1X 15 Me-
MyapoB. Cpe.l(H HHX: apl1Q>MoMeTp, rpe6HoH MeXaHI13M, CTono-
XO.l(SllUaSi MalllHHa, COpTHpOBaJlKa 3epHa. «CBOHMH 3aMe'laTeJlb-
HbIMI1 pellleHI1S1l\fH pSlJ(a KOHKpeTHbIX 3a.l(a'l a MexaHH3Max
4e6b1IlleB 3Ha'lI1TeJlbHO onepe.l(l1J1 Bcex CBOHX cOBperlleHHI1KOB;
60.'1ee Taro, OH nOCTaBH.'1 nepe.l( HaYKOH a MeXaHH3l\1aX TaKHe
np06J1eMbl H 3a.l(a'lH, K KOTOpbIlI-I3Ta HaYKa CTaJia nO.l(XOHTb
BII.'10THYIO T OJIbKO B caMble nOCJIeHHe eCSlTI1JIeTHSI»l).
1) 11. 11. ApT a 60 JI e B C K II H H 1-1.. H. neB II T C K H fl.
MexaHH3:VIbl Ye6bIweBa. H3.1\. AH CCCP.
15 A. A. KOJlO"O
405

.,. .-
.. ...
... .
..
..
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:
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A. H. KPbIJlOB.
HaKOHe. HeJIb3S1 3.l(t'Cb He OCTaHOBHTbCH Ha MaTeMaTHKe H
CTVOI1TeJIe Kopa6JIeH, fepoe Cou.HaJIHCTH'IeCKOrO TpY.l(a, aKa.l(e-
MHKe AJIeKcee HHKOJIaeBI1'1e K p blJI 0 B e (1863-1945). EJ..U.e
6Y.l(Y'IH Y'IeHHKOM MopcKoro Y'II1JJHlUa, OI npOSlBI1JJ 60JIbUlOH
HHTepec K MaTeMaTHKe. MHoro pa60TaSi caMOCTOHTeJJbHO, OH
TOrJ(a Y>Ke nOHSlJI, '1TO HMeHHO MaTeMaTHKa SlBJlSleTCSI OCHOBOH
HaY'IHbIX MeTO.l(OB Kopa6.necTpoeHHSI. D03.l(Hee, B MOpCKOH
aKa.l(eMHH, KpblJIOB c03.l(aeT HCKJIIO'lHTeJIbHO OpHrHHaJIbHYIO
H no COep>KaHHIO Ii no rJIy6HHe MaTeMaTH'IeCKHX paC'IeTOB
«TeopHIO HenOTOnJiSIeMOCTH». BY.l(Y'IH OCTaBJIeH npH AKa.l(eMHH,
OH BeeT co CTY.l(eHTaMH 3aHSITHSI no BbIcllIeH MaTeMaTHKe H
_ oJ(HoBpeMeHHo '1HTaeT KYpC TeopHH Kopa6JISI. CTaJIKHBaSICb
B CBoeH npaKTH'IeCKOH eSlTeJlbHOCTH C He06xo.l(HMOCTbIO npo-
H3BOJ(liTb 6QJIbUlHe -BbI'IHCJleHI1S1, A. H. KPbI:TJOB nHmeT KHHry
«0 npH6JIIi>KeHHbIX Bbl'lHCJIeHHSlX». «4HCJIeHHble BbI'lHCJJeHHSI
. HaM nOHa.l(06S1TCSI Ka>K.L\bIH eHb, n03TOMY MeTObl HX npOH3-
BOCTBa °H J(OJI>KHbl 6bITb .l(aBaeMbl B nepBYIO rOJIOBY».
Ta.n.aHT A. H. 'KpbIJIOBa oc06eHHO llIHpOKO paCKpblJlCSI'
nOCJIe peBOJIIOli.HH, H Bce C.l(eJIaHHOe HM ellJ,e'pa3 nO.l(TBep>K.l(aeT.
KaK0I1 BblCOTbl MO>KeT .l(OCTl1rHYTb npaKTII'IeCKaSi pa60Ta.
ecJIH OHa nOJIb3yeTcH MeToaMH MaTeMaTHi:<I1.
406

«PaHO I1J1H n03.l(HO BCSlKaH npaBHJIbHaSl l\IaTeMaTl1'1eCKaSf
J-j,[\eSl HaXO.l(HT npHl\leHeHHe B TOM 11J111 HHOM .l(eJIe», - roBO-
pJ1JI OH.
4eM 6JJI1>Ke K HalllHl\f J(HSlM, TeM 'lame MO>KHO BH.l(eTb MaTe-
MaTJIKOB, paOOTalOlUHX Cneli.HaJlHCTa:\IH B CMeiKHbIX 06J1aCTSIX
HaYKH. BypHoe pa3BHTI1e 4>113I1KI1. XI1MI1I1, aBHali.l1l1, paKeTO-
CTpOeHI1H, CeJlbCKOrO X03S1HCTBa, MalllHHOCTpOeHI1S1 11 Bo06we
TeXHI1KI1 B Hallll1 .l(HI1 6blJlO 6bl HeB03MO>KHbIM, eCJII1 6bl orpOM-
HaH ap:\IHSI 1\IaTeMaTI1KOB He npI1ll1J1a Ha nOMOmb Y'IeHbIM,
HH>KeHepaM H pa60'lI1M. 4aCTO MOiKHO Ha6JIIO.l(aTb, KaK 3anpOCbI
HaYKH 11 TeXHI1I<H BbI3bIBa.1J11 K iKI13HIl HOBbIe pa3.l(e.TJbI MaTe-
l\13THKI1 II c6bIB.aJlI1Cb CJIOBa D. JI. 4e6blllleBa: «COJlI1>Kelme
TeOp1111 C npaKTI1KOH .l(aeT caMble 6J1arOTBOpHble pe3YJlbTaTbI,
1I He O.l(Ha TOJlbKO npaKTI1Ka 01' 3TOro BbIHrpbIBaeT: caMH
HaYKH pa3BI1BaIOTCSI no,iJ: Bm1S1HHel\! ee».
Tenepb nepeXO.l(l1M K paCCMOTpeHl11O HeKOTOpblX npaKTH-
'1eCKI1X 3a.l(a'l.
3ana'lH no aJIre6pe
a) C 0 C TaB JI e H 11 e Y p a B H e H H H
I. 11MeeTcSI JIOM CTa. ' HJ .l(BYX COpTOB C CO.l(ep>KaHlieM HIlKe-
Jlf! B 5% 11 40%. CKO.!1bKO HY>KHO B3S1Tb K8iK.l(Oro 113 3TI1X Cop-
TOB, '1T06bl 'nOJlY'II1Tb 140 m CTaJlI1 C cO.l(ep>KaHHeM HI1KeJJSI
B 30o?
Om6em. 40 m H 100 m.
2. I1l\1eeTCH 600 Z llIeCTHa.l(li.aTHnpOli.eHTHOro paCTBopa
HO.l(a B Cnl1pTe. HY>KHO nOJlY'II1Tb .l(eCSlTl1npOli.eHTHbIH pacTBop
llo.l(a. CKOJIbKO rpaMMOB Cnl1pTa Ha.l(O npl16aBHTb ,1J,JlSI 3TorO
K Y>Ke I1MelOlUeMYCH paCTBoPY?
Om6em. 360 z.
3. CnJIaB 113 Me.l(H H li.HHKa BeCOM B 24 KZ npH norPY>KHHI1
.. 2 8 0
B BO.l(Y nOTepSlJl B CBoeM Bece "9 KZ. npe.l(em-!Tb KOJII1'1eCTBO
Me.l(l1 11 li.HHKa B 3TOl\1 CnJlaBe, eCJII1 113BeCTHO, '1TO Me,iJ:b TepSieT
B BO.l(e II -}% CBoero Beca, a li.I1HK 14  % CBoero Beca.
Omeem. 17 KZ Mel1,
7 KZ UI1HKa.
4. B O.l(HOM cocY.l(e HaXO.l(HTCSI mil p-npOli.eHTHOro pacTBo-
pa KI1C.1JOTb!, B .l(PyroM n A q-npOli.eHTHoro paCTBopa TOH >Ke
KI1CJIOTbI. ('13. K8iK.l(orO cocY.l(a OTJl11J1I1 O.l(HO II TO me KOJIIl-
15*
407

tIeCTBO JlHTpOB, tI B3S1TOe H3 nepBOrO nepeJ1HJlH BO BTOpOH, a
B3S1Toe H3 BToporo - B nepBbIH. KaKoe KOJlH'IeCTBO JII1TpOB
HY>KHO OTJlHTb, '1T06bI B 060HX cocY.l(ax OKa3a.rICSI paCTBOp
O.l(HOJ1 H TO" >Ke KpenOCTH?
P e III e H H e. ECJlH 0603Ha'lI1Tb KOJlH'IeCTBO JIHTpOB KHC-
JlOTbl, OTJIHTbIX H3 K3)!{OrO cocY.l(a, '1epe3 X. TO KOJlI1'1eCTBO
'lHCTOH KHCJlOTbI B nepBOM cocY.l(e nOCJIe nepeJIHBaHHH 6Y.iJ:eT:
nzp px qx
100 - 100 + 100 '
a BO BTOpOM
nq qx + px
100 - 100 100 .
KOHll.eHTpali.HH nOJlY'IeHHbIX nOCJIe 3Toro paCTBopOB: B
nepHOM cocY.l(e:
nzp - px + qx
100nz
BO BTOpOM:
nq - qx + px
lOOn
no YCJlOBI1IO 3a.l(a'lH nOJlY'IHM ypaBHeHHe:
mp- px+ qx Tlq- qx + px
100m lOOn
PelllaSi era, npH.l(eM K oTBeTY
mTl
x = nz+n .
5. t.h06bl HaHTH Maccy Jl1060ro He60Jlbllloro TeJla, MbI
06bI'IHO KJl3.l(eM era Ha '1alllKY BeCOB H ypaBHOBeIllI1Baellf era
Bee mpeH. ECJIH npHTH>KeHHe 3eMJlH O.l(HH3KOBO J(JlSI 060HX,
Bec, a T3K>Ke H MaCCbl paBHbl.
A KaK npl1 nOMOmH BeCOB onpe.l(eJlI1Tb Maccy 3eMJlH?
OnHllIeM cnoc06 )KOJlJlI1, npI1Bo}J,SllUI1H K COCTaBJleHHIO ypaB-
HeHI1S1.
BecbI, cnoc06HbIe BbIHeCTH 3Ha'lHTeJIbHYlO TSI>KeCTb, HlIfelOT
.l(Be napbl '1allleK, H3 KOTOpbIX HH>KH11e '1alllKI1 npI1BellleHbI K
BepXHHM Ha .l(JlHHHbIX npOBO"10Kax (npH nepBOM onpe.l(eJIeHHH
MaCCbl 3eMJlH B 1881 r. J(JlHHa npOBOJlOK paBHSlJlaCb 21 M H
HH>KHI1e '1alllKH 6b1JlI1 pacnOJlO>KeHbI HeCKOJlbKHMH 3ta>Ka!\UI
HI1)Ke BepxHHx) ('1epT. 215).
ECJlH Ba mapa ml H m 2 o.l(HHaKoBoif MaCCbI nOMeCTHTb H3
}J,Be BepXHHe HJlH Be Hl1)!{HUe '1alllKH, OHH 6Y.l(YT T04HO ypaB
408

HOBeIllI1BaTbCSI, TaK KaK CI1J1a npHTSI>KeHI1H 3eMJlI1 Ha 06e
laCCbI O.l(HHaKOBa. ECJIH nOMeCTI1Tb tn 2 Ha BepXHlO1O 'lalllKY.
a fl/l Ha HI1>KHIOIO, TO m1 OKa>KeTCSI TSI>Ke.'1ee, T8I{ Kal{ 6Y.l(eT
6.IJIHKe K li.eHTPy 3eMJlH 11 6Y.l(eT CI1JJbHee np11TSlrHBaTbCSI; HO
aeCbI onSlTb MO>KHO ypaBHOBeCI1Tb npl1 nOMOlUH MaJleHbKOro
npOTI1BOBeCa C. ECJlI1 nocJle 3Toro nOMecTHTb 60JIbllloii llIap M
113 CBHHli.a no HI1>KHeH '1alllKOH, paBHOBeC11e onSlTb 6yeT
HBPYllIeHO BCJIeCTBHe B3al1MHOrO npHTSlH<eHHSI l\Iacc M H m],
II Ha,110 nOMeCTHTb 06aBO'lHYIO He60JIbllIYlO Maccy n Ha Bepx-
HIOIO '1alllKY, '1T06bI ypaBHoBeCIfTb BeCbI. TaK KaK BepXHSlH
LJBllIKa HaCTOJlbKO Y.l(aJIeHa aT M,
'1TO B3aHMHoe npl1TSI>KeHl1e Me>K-
.ill' HI1MH He3Ha'lHTeJIbHO, npHTSI-
;'h:eHl1e M Ha m1 paBHO npl1TSI-
meHHIO 3eMJIH Ha n. Ho no 3a-
KOHY TSirOTeHI1S1 nepBaSi H3' 3THX
CHJI I1MeeT BeJl11'1I1HY
O Mm l
d 2
rJ!e d - paccToSlHHe Me)!<,llY li.eHT-
pa1\Il1 llIapOB M H nIl, Tora KaK
BTOpaSi CH.la paBHa
OXn
R.2 ,
r,lle X - HeH3BeCTHaH MaCCa 3eM-
JlI1, a R - paCCTOSIlme Mem.l(Y
n H lleHTpoM 3e1\1.'1H H SlB.'1HeTCSI
3eMHbIM pa.l(l1ycoM. MbI HMeeM
TaKHM 06pa30M:
G Mm1 _ G Xn
d 2 - R.2'
OTKY.l(a:
n .
0' Q.
YepT. 215,
X _ Mm1R.2
- nd 2 .
B onbITe. npOH3BeeHH0l\1 B 188] r., nIl=m2=5 ICZ, M=
=5775,2 KZ; n=O,589 AtZ; d=56,86 CAt H R=6366 K.M. EC.1H
'1HTaTe.lb npO,'J,eJIaeT He06xo.l(I1MbIe BbI'II1CJleHI1H .l(JISI HaXO>K-
J(eHHSI X, TO OH nOJlY4HT TaKOH OTBeT .l(JISI MaCCbl 3eMJJH:
X=6,15.10 27 Z.
409

MO>KeT 6bITb, BaC 3aI1HTepecyeT, KaKOH iKe pa.l(HYC lIMe;,
CBHHli.OBbIH llIap Becml B 5775 Kc? Bbl HaH.l(eTe ero no (POP-
MYJIe 06"beMa llIapa:
V -   R 3
mapa - 3 h ,
H y3HaB B Ta6JIMli.e y.l(e.1bHblX BeCOB, '1TO Y.l(e.TJbHbIH Bec CBHH-
lla paBeH 11 ,34.
6. Dapoxo.l( neT H3 KneBa B .II.HenponeTpOBCK B Te'leHHe
JJ.BYX CYTOK, 06paTHo - B Te'leIlI1e Tpex CYTOK. Onpe.l(eJ1HTb.
CKOJlbKO BpeMeHI1 6Y.l(eT I1J!TM napoxoA H3 KMeBa B .II.HenponeT-
pOBCK.
Om6em. 12 CyTOK.
7. DapaIllIOTHCT nOKMHYJI CaMOJleT Ha BbICOTe 4000 .lit H
'lepe3 4 MHH. npH3eMJIHJlCH. CpeJ(HSISI CKOpOCTb CHMiKeHI1S1 Ha
napaIllIOTe paBHa 6 . Onpe.l(eJII1Tb Bpel\IH CHI1>KeHI1H 6e3 na-
ceK
pallllOTa H BpeMSI CHl1iKeHHf! C napallllOTOM. (Dpl1MeMg = 1O2
ceK
M He 6Y.l(eM Y'IMTbIBaTb conpOTHBJleHl1e B03.l(yxa Ha Y'IaCTKe
CHn>KeHI1H 6e3 napaIllIOTa.)
PeIlleIIHe 3a.l(a'll1 npHBe.l(eT K COCTaB.1JeIIHIO CHCTeMbl
{ 5X2+6y=4000,
x+y=4,
B KOTOpOH x - BpeMH CHI1>KeHI1S1 6e3 napallllOTa B ceKyH.l(aX
M y - BpeMH CHI1>KeHHSI C napalllIOTD:\1.
8. Ha' onope M JIe>KHT 6aJlKa .l(JlMHOH 1 C.IIt. 4acTb 6aJIKM,
HaXOSlllJ,eHcSI BnpaBO aT onopbI (4epT. 216), paBHoMepHo
A 8
P M
4epT. 216.
3arpY>KeHa TaK, '1TO Ha Ka>K.l(blH noroHHblH CaHTI1MeTp ee
elkTByeT HarpY3Ka q Kr. Ha KOHeli. A 6aJIKH .l(eHCTByeT CHJla
P Kr. Onpe.l(eJlHTb nOJIO>KeHl1e onopbI M, npH KOTOpOM 6aJlKa
6Y.l(eT B paBHOBeCI1I1.
4iO

P e 1lI e H 11 e. 0603Ha'lHB paCCTOHHHe aT KOHli.a A 6aJIKH
.l(O onopbI M '1epe3 x 11 HcnO.TJb30BaB YCJIOBl1e paBHoBecliSi Bpa-
malOllI.l1X YCHJIHH, npH,lI,eM K ypaBHeHHIO
P . _ q (l_-X)2
X - 2 .
9. Ha CKO.'IbKO HeOuxo.l(H1\IO YKOpOTI1Tb l\faSlTHHK '1aCOB,
eCJIH .l(JIHHa MaHTHHKa paBHa 40 CAt 11 '1aCbI OTCTaBa.'UI Ha
12 "IHH. B CYTKH?
YKa>KeM nopH.l(oK pellleH11S1 3TOH 3a.l(a l lH.
I. Onpe.l(e.'!SleM nepl10.l( To Ko.,'1e6aHI1H OTCTaIOlUl1X llaCOB
no cpopMYJle:
T=2.. Jl l -
g ,
re 1- .l(.nHHa MaSlTHHKa B 01 H g - YCKopeHl1e CWIbI TH-
>KeeTH, KOTopoe npHMeM paBHbI:\1 980 01 2 ,
ceK
2. Onpe.l(eJiSIeM OTCTaBaHHe '1aCOB 3a I MHH. (no.'IY'lI1M
0,5 ceK.).
3. Onpe.l(eJISleM lIl1C.1JO n KOJIe6aHHH OTCTalOllI.HX '1aCOB B
1 MHH.
(n=  ).
4. Onpe.l(eJISleM OTCT8BaHHe a. qaCOB 3a O.!l.HO Ko.neoaHHe
(a = O5 ) .
5. Onpe.l(eJIHeM nepI10.l( T KO.le6aHHH npH npaBH.%HOM
xO.l(e '1aCOB (T=To-o.).
6. Onpe.!l.e.'IHeM .l(.nlmy x, Ha KOTOpylO He06xo.l(HMO YKO-
pOTHTb MaSlTHHK no cpopl\fy.'Ie:
T = 21t V lo g x .
f,ll,e 10 = 40 CAt.
Omeem. 0,8 CftI.
411

6) rpa<pHKH
to. Ha llepTe}!{e 217 .l(aHbI rpa<pHKH pa3BI1THSI '1epHOM Me-
TaJJ.1IYPrHH OT 1913 r. no 1965 r. (nOCJIe.l(HHH ro.l( CeMHJIeTKH).
150
li10
130
120
110
10
 9
c
E: 8
 7
C)
::, 6

 5
II
3
2
/Ken.
e-- pyrJo
a
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0
0 Cmollb
0
-
0
0 I/Y2YH
0
0
0
0
1913 1932 1941J 1950 1959 196; zod
YepT. 217.
Bon pac bl.
1. KaKoBbI .l\OJllKHbI 6bITb l\06bIqa lKe.ne3Holl PYl\bI, Bbm.naBKa qy.
FYHa II CTaJiIi B 1963 r.. IJT06bI 6bIJI BbInOJIHeH n.naH CeMHJIeTKII?
2. CocTaBbTe q,OPMYJlY, BbIpalKalOll\YIO 3aBHCHMOCTb MelK.ny KO.1F-
qeCTBOM BbmycKaeMoll CTa.nH II BpeMeHeM Ha npOT!\lKeHHII' 1959-
1965 rr.
t t. HOMorpaMMa JI.JIH 4>OPMYJIbi npoLleHToB OT 'IHCJla A
Ap
a = 100
"" .4p ..
opMYJJa a = 100 npH onpe.l(eJIeHHOM LJI1CJleHHOM 3Ha'le-
HI1I1 p BbIpa>KaeT npHMo nponOpII.110HaJIbHYIO 3aBHcHMOCTb
412

Me>K.l(Y aHA. JIY'IH Ha .l(aHHOfi HOMorpaMMe ('1epT. 218),
BbIXO.l(HIlI.He 113 H'JIeBOH TO'lKI1, Ii SlBJISIIOTCSI rpaqJHK£l1\1I1 3TOH
3aBHCHMOCTI1 npl1 p£l3J1I1'1HbIX 3Ha'leHHSlX p. flOCJIe Taro KaK
20
18 18
15 16
14 14
12
10
8
2
4
D W ro  w  w  w ro
YepT. 218.
BbI pa36epeTecb, KaK nocTpoeHa HOMorpaMMa, Bbmo.,'1HI1Te npl1
nOMOllJ,H ee ynpaJi<:HeHHSI:
1) onpe)J.e.rmTb 12% aT 70;
2) onpe)J.eJIHTb '1HC.'10, eC.'1H 14 % ero COCTaBJlSlIOT 8;
3) HaflTH npoli.eHTHoe OTHOWeUI1e 8 K 50.
12. rpactIHK 3aKOHa 60UJlH - MapHoTTa (p. v-nocToHHHoe).
Ha '1epTe)Ke 219 KpHBbIe .l(alOT CBSl3b Me>K.l(Y .l(aB.'1eHHeM p
H 06'bel\lOM (I O.l(HOro KH.'10rpaMMa ra3a.
JI.JlSI BO.l(Opo.l(a (H2) BbI'lep'leHbl .l(Be Kpl1Bble: O.l(Ha - .l(.'1S1
TeMnepaTYPbI 0°, a BTopaH - .l(.'1S1 77°. JI..'1H KHCJIOpO.l(a (02)-
O.l(Ha KpHBaH .l(JIH 27°. 3TH KpHBbIe B iPH3HKe HOCSIT Ha3BaHHe
1130TepM (KpHBbIe paBHoil TeMnepaTYPbI), .l(.'1S1 M8TeMaTI1Ka
3TO mnep6o.,'1bI, rpaiPl1'1eCKH BbIpa>KalOllJ,l1e oopaTHO nponop-
li.HOHaJIbHylO 3aBI1CHMOCTb Me)K.l(Y p 11 V.
413

no rpa4n1KY .l(JIH BOOpO.l(a npH 0 0 COCTaBbTe <POPMYJlY,
Bblpa>KaIOllJ,YIO 38KOH COOTBeTCTBI1S1 Me>K.l(Y 06"b€Mo:\1 ra3a H
.l(aBJIeHHeM Ha Hero.
B) 3 a .l( a 'I 11 H a H a H 6 a,,'} b 1lI H e (M a K C II -
M a JI b H bI e) H H a U M e H b ill H e (M H H H M a JI b H bI e)
3Ha'leHHSI YHKI1H
(j Z7°77°
fZ I.
iI
10
\'
{j
',\
\" ":-2
" I
o H? (........t..-..J...
2
'\ I ""-.,;;,,-1..-1.
I I .=.-!t-O°
o (  b {} 10 It 14 HC/CI'1?P
6
"
YepT. 219.
I-h Kypca VI I I KJlaCCa 'lHTaTe.lb nOMHHT, 'ITa rpaI1KOM
CPYHKI1H
y=ax 2 +bx+c
y 0"0
CJlY)KHT napa60JIa, OCb KOTOpOH
napaJIJleJIbHa OCH Op.l(l1HaT ('1epT.
220).
KOOPAHHaTbI BepillHHbI A na-
x
y
4epT. 220.
4epT. 221.
pa60JIbI, KaK H3BecTHo, onpe,1I.eJIHIOTCSI no opMYJJaM:
b 4vc- b 2
X A = - 2a H Y A = 4u
414

Ec.'111 a>O, TO KBaJ(paTHaSi <PYHKli.HSI I1MeeT MI1HHMyM,
com a<O- MaKCI1MYM (Ha 'lepTe)!{e 220 JIeBaH napaOOJIa),
3Ha ' leHHe aprYMeHTa x, npH KOTOpOM KBapaTHaSi <PYHKllI1H
jJ,ocTHraeT MaKCI1MaJlbHOra HJIH MI1HI1MaJJbHOro 3Ha'leHlIH, COB-
naaeT C a6Cli.HCCOH XII BeplllHHbI napa60JIbI, a BeJIH'IHHa ca.
Mora MaKCHMYMa I1Jm MHHHMYMa onpe.l(eJIHeTCSI Op.l(l1HaTOH
y A 3TOH Bepllll1HbJ.
3a)J.a ' j8 13. OKHO I1Meer <PopMY npHMoyro.'1bHI1Ka, 3aBep-
meHHoro nOJlYKPyroM ('1epT. 221). D:aH nepl1MeTp Q)I1rYPbJ.
KaKoBbI .l(OJl}j{HbI 6bITb pa3MepbI ee, '1TOObl alma nponycKano
naH60.TJblllee KOJlWlecTBO cBeTa?
P e 1lI e H H e. DYCTb pa.l(l1Yc nOJIYKPyra paBeH X, a nepl1-
l\!eTp -2p. Toma, 0003HalJ11B BbiCOTY npSlMoyrOJJbHOH '1aCTI1
OKIIa '1epe3 y, no.nY'I11l\1:
2p=2x+2y+1':x
11
1tX
y=p-x-<j' (1)
n.'10ma.l(b S OKHa BbIpa3HTCH TaK:
1tX 2 ( 1t X ) 1tX2
S = 2x lJ + -'- = 2x p - X - - == 2px - 2X2 - -'-
2 2 2
H.'1H
S = - (2 + ; ) X2 + 2px. (2)
PaBeHCTBO (2) rOBopHT a TOI\!, <ITa S eCTb KBa.l(paTHaH CPYHK-
li.11H OT X 11 I1MeeT MaKCI1MYM, TaK KaK a<O. Onpe.l(eJlSlH no
rpopMY.'1e
b
X A =-20
TO 3HalJeHne X, npH KOTOpOM S .l(OCTHraeT MaKCHMYMa, MbI
nO.'1YlJHM:
2p
X A = 4+7t .
Ho Tor.l(a
_ _  + 1t2p 2p
Y - P 4+1t 2 (4+..) 4+1t '
T. e. y=x. TaKHM 06pa30M, .'1S1 Taro '1T06bl OKHO YKa3aHHOM
rpOPMbI nponycI<a.'10 HaI100.'1bWee Ko.'1H'IeCTBO CBeTa npH .l(aH-
HaM nepl1MeTpe, HeOOXO.l(l1MO paBeHCTBO BbICOTbI npSlMO-
yrOJJbHOH lJaCTH OKHa pa.l(HYCY nOJJYKpyra.
415

PaCC'IHTaihe, KaKHMI1 .l(OJI>KHbI 6bITb pa.l(Hyc no.nYKpyra If
BbICOTa npSlMoyrOJIbHOH '1aCTI1 OKHa, eCJIH era nepHMeTp
paseH 6 M.
Om6em. 0,84 M.
3a.naIJa 14. B HarneM paCnOpH>KeHHH I1MeeTCH 113ropO.l(b
.l(JlHHOH B 200 M. Tpe6yeTcSI OrOpO.l(HTb elO Y'IaCTOK 3eMJJI1
B Bl1.l(e npSlMoyrOJIbHI1Ka HaH60JIbllleH nJIOIl{a.l(l1. KaKYIO
QJOPMY .l(OJI>KeH HMeTb 3TOT Y'lacToK H '1eMY oY.l(eT paBHa
ero fIJIOllJ,a.l(b?
Om6em. 50x50 M 2 .
3a.naQa 15. B Ksa.l(paT co CTOpOHOH a BnI1CaTb KBa.l(paT,
I1MelOIl{HH HaHMeHbrnylO nJIOllJ,a;I.b (llepT. 222 ).
1
1
o
YepT. 222.
Y K a 3 a H H e. BbIpa3HTe nJIoIl{a..'J,b S BnHcaHHoro Ba.l(-
paTa '1epe3 a 11 x.
3a.naQa 16. TOK CJUlbl 1 pa3seTsc1S1eTcSI no .l(BYM nposo.l(aM,
COnpOTI1BJleHHe KOTOpbIX r1 11 r 2 . no KaKoMY 3aKOl-!Y .l(O.'1>KHO
6bIJIO ObI n pOHCXO.l(l1Tb pa3BeTB.'1eHl1e, '1TOObI BbI.l(eJIHlOlUaSiCSI
B 060l1x npOBO.l(ax Ten.'10Ta 6bl.'1a HaI1MeHblllei1?
P e ill e H H e. nYCTb CI1JIa TOKa B nepSOM npOBO.l(e paBHa
x; TOma CH.'1a TaKa BO BTopmI npOBO.l(e / - x. BbI.l(eJlSlIOllJ,aSlCSI
Ten.'10Ta y, no 3aKOHY JI.}j{OY.'1S1 H JIeHli.a, paBHa:
y=rlx 2 +r 2 (I-x)2= (r 1 +r 2 )x 2 -2r 2 1 x+r 2/2.
11TaK;
y= h +r 2)x 2 -2r 21 x+r2/2.
nepe.l( HaMI1 KBa.llpaTHaSi <PYHKli.I1S1, HMeIOmaSi MHHHMYM,
TaK KaK a=r1+r2>0. Onpe.l(eJISleM TO 3Ha'leHHe x, npH KO-
TOpOM Y .l(OCTHraeT !\1I1HI1Ma,llbHOrO 3Ha'leHHSI no «popMYJle:
b
X A =-'2Q°
416

B .l(aHHOM CJIY'lae
x - -2'2/ _ rJ2
A - - 2('1+'2) '2 +'1
Tor.l(a CI1JIa TaKa BO BTOpOM npOBO.l(e OYAeT paBHa:
l-x=I- = '1/
'1+'2 '/+'2.
OTCIO.l(a:
x '2
I-x =-,.-,
T. e. CHJla TaKa AOJI>KHa ObITb pacnpe.l(eJleHa no npOBo,l1,8M 06-
paTHo nponOpli.HOHaJIbHO HX COnpOTI1BJIeHHHM.
17. 3aJl,aQa 0 Jl,Bu*eHHH TeJla, 6pollleHHoro noJl, yr.noM
K ropH30HTY. DYCTb CKOpOCTb TeJIa B Ha'laJle nOJlera paBHa
V(! 11 ee HanpaBJleHI1e COCTaBJISleT C ropH30HTaJIbHOH nJIOC-
KOCTblO yrOJI (J. ('1epT. 223). JJ.on, YCTI1l\1 'ITO Ha TeJIO He .n.eH-
2 !
YepT. 223.
CTBYIOT HHKaKHe BHeJl1HHe CI1J1bl, KpoMe CI1JIbI THiKecTI1. Pa3-
JIO>Klm Ha'IaJIbHYIO CKOpOCTb V o TeJIa Ha .l(Be COCTaBJIHlOlUHe:
ropH30HTaJIbHYlO V x IJ BepTHKaJIbHYIO V Y ' J;13 '1epTe>Ka HIIIeeM:
v x = vocos (J. H V y = vosin 'J..
Do HanpaBJIeHHIO nepBoM CJIaralOllJ,eH HlPlTO He MemaeT
TeJlY .l(BHraTbCH, 11 OHO no 9TOMY HanpaBJIeHHIO, .l(Bl1raSiCb
paBHOMepHO. 3a t ceKYHlt npOHAcT nYTb x,
x=(v(;cos a)t. (1)
B BepTIIK8JIbHOM HanpaBJIeHI1H CKOpOCTb V y He MO>KeT aCTa-
BaTbCH nocToSlHHOIJ, TaK KaK Ha TeJlO .l(eHCTByeT CHJIa TH>KeCTH.
417

lIBH>KeHI1e B 3TOM HanpaB.neH,llll 6Y.l(eT paBHOMepHO-3aMe.l(JleH-
HbIM. COCTaBJIHIOllJ,aSl V y 6Y.l(eT H3MeHHTbCSI no 3aKOHY:
v)'= vosin'J.-gt,
H nYTb, npOHeHHbIfi TeJIOM B BepTI1Ka.nbHOM HanpaB.neHI1I1,
6YAeT TaKOB:
. gt 2
Y = vot Sin a. - 2 .
Onpe.l(eJiHeM 113 ypasHeHI1H (1) t:
{2)
t=
x
Uo COS a
n' nO.l(CTaS.nSleM nOJIY'IeHHOe 3Ha'leHI1e t B ypasHeHHe (2):
x sin:>: gx 2
y= cOSa - 2v2cos2a'
o
sin a
TaK l<aK - = tga., TO
cosa
y = X tg ex - g X 2
2VCOS2 a .
(3)
MbI no.nY'lI1.n11 <POPMYJIY, BbIpa}j{alOllJ,YIO 3aBI1CI1MOCTb
Me<J1.Y BblCOTOH y, Ha KOTOpOH HaXoAHTCSI 6pOllleHHOe TeJIO
HaA 3et>1JIeH, [1 rOpl130HT3JIbHbIM paCCTOHHHeM era x OT Ha-
4aJlbHOH TOtIKH no.neTa (4epT. 224). 3Ta <PoPMYJIa npiIHa.l(JIe-
i!\I1T I": TImy:
y=bx-ax 2 .
C.'Ie.l(oBaTeJlbHO, nepe HaMH KBaJ(paTHaSi «PYHKli.HH H rpa<pH-
KOM ec 6Y.l(eT napa60JIa, OCb KOTOpOH napa.'I.ne.'IbHa OCH 0PAI1-
K
YepT. 224.
418

HaT H BeTBI1 KOTOpOH 06palIeHbI BHI13 aT ee BepWI1HbI. Koop-
.l(lIHaTbI BepWI1HbI napa60.nbI MbI Haii;I.eM no 4!0pMYJIaM:
b
X A = - 2a
4uc-b 2
H Y A = 4a
B aHHOM CJIY'lae
tga.2v6coS2 a
X --
A - 2g--
tga VCOS2 a
vsin2a
2g 1).
g
,n,a.1bHOCTb nOJleTa apTI1JIJIepI1llClWrO CHa, pSlJ(a Ha'laJIbHaSi
CIWpOCTb KOTOpOrO V o , O'leBI1.l(HO, 6y.aeT TaKOBa:
r?sin2a
D=2xJj = 0
g
IIocJIe.l(HHH 4!opMYJIa nOKa3bIBaeT, 'lTO' aJIbHOCTb nOJIeTa
33SI1CI1T OT yrJIa. a. (yro.n B03BbIUleH!IH), MHOlliHTeJIb sin 2a.
;jOCTHraeT HaHOOJIbWerO 3Ha'leHI1S1 npI1 0:=45°, KOr.l(a sin 20:
CTaHeT paBHbIM 1.
TaKIiM o6pa30M, !\laKCJi'lM3JIbHaH J1aJlbHOCTb nOJleTa CHapT!-
.u.a npl1 .a.amlOll CKOpOCTH V o BbIpa>KaeTCH cpopMYJIoii:
v 2
Dmax = ; .
360000
ECJII1 v o =600 MiceK, TO Dmax=1O = 36000 = 36 KM.
npI1 lJ.BI1llieHI1I1 cnapHlJ.a - B B03.l(yxe cyw.eCTBeHnylO pOJIb
HrpaeT conpOTHBJIeHHe B03yxa. 113-3a COnpOTHBJIeHHSI 803-
.l(yxa CKOpOCTb CHa pSlJ(a BO
Bpe!\1SI nO.neTa YObJ8aeT 11
TpaeKTopHSI He HMHeTCSI
60JIbIlle napaOOJIOH, ee HI1C-
XO.l(HllJ,aSl BeTBb l1.l(eT KPy-
lie, '1eM BOCXO.l(SlllJ,aSl (oaJI-
JII1CTH'IeCKaH KpI1BaSl) ('Ie pT.
224); .l(aJIbHOCTb nOJIeTa 11
BbiCOTa nO.l("beMa CTaHOBHT- 4epT. 225.
CH MeHbwe.
t 8. T SI>KeJIaSi li.enb Jle>KHT Ha no.ny. Ee 6epYT 3a O.l(HH
KOHeli. 11 nO.l(HI1MalOT BCIO .l(O BepTHKaJIbHOrO nOJIO>KeHI1S1.
sina 2sina.cosa
1) tga cos 2 a = COS a .cos 2 a = sin a .coSa = 2
sin2a
=
419

Onpe.l(eJlI1Tb 3aTpa'lHBaeMYJO npH 3TOM pa6oTY, eCJIH Bec
Bcell li.enII paseH Q I1 ee .l(JIHHa I ('1epT. 225).
P e III e H H e. ECJlI1 '1I1CJlO 3BeHbeB li.enI1 0603Ha'II1M '1epe3
n, TO Bee OHora 3BeHa 6Y.l(eT paBeH SL II era lI.JIHHa .!- .
n n
DepeMellJ,aSl nepBoe 3BeHO H3 rOpl130HTaJIbHOrO nOJIO>KeHIISI
B BepTI1KaJIbHOe, MbI li.eHTp TH}j{ecTI1 3BeHa nOHHMaeM Ha
l
BbICOTY 2n ' 11, CJle.l(OBaTeJIbHO, paoOTa A, cOBepIllaeMaSl npII
STOM, paBHa:
'.2
n
YepT. 226.
Q I QI
A = - . _ 2 = 2 2 ('1epT. 226).
n n n
DpH nepeMemeHHH BTopora 3BeHa H3
ropH30HTa.rIbHoro nO.TJO}j{eHI1S1 B BepTHKaJIb-
Hoe MbI BHOBb .l(O.TJ>KHbl 3aTpaTHTb pa60TY
2 H cBepx 9Toro nO.l(HSlTb nepsoe 3BeHO Ha BbI-
COTY -. CJIe.l(OBaTeJlbHO, BCSI pa60Ta npH nO.l(HSlTHH BTO-
n
poro 3BeHa COCTaBHTCSI H3 .l(BYX CJ1araeMblX:
A =!{.+-=+= 3Ql
2 n n 2n 2 n 2 2n 2 2n 2 .
DpH nepeMellJ,eHHI1 TpeTbero 3BeHa H3 rOpI130HTaJIbHOrO
nOJlO}j{eHHSI B BepTHKaJlbHOe MbI B TO }j{e BpeMH nOHHMaeM
" l
nepBbIe .l(Ba 3BeHa Ha SbICOTY -. CJle.l(OBaTeJlbHO, COBep-
n
maeMaSl npH 9TOM pa60Ta A3 6Y.l(eT TaKOBOH:
A = 2Q .  +  = 2Ql +  = 5Ql
3 n n 2n 2 n 2 2n 2 2n 2 .
DO.l(06HbIe paCCY>KJ(eHHSI MbI MO}j{eM npO.l(OJI>KHTb .l(O
nero, nOCJlell.HerO 3SeHa li.enI1. Tor.l(a Av A2' A 3 ,..., An
COCTaBSlT apHiPMeTI1'1eCKYIO nporpeCCHIO:
Ql
2n 2 '
3Ql
2n 2 ,
5Ql
2n 2 , . .. ,
(2n-l)Ql
2n 2
H BCSI pa60Ta A npe.l(CTaSHTCH CYMMOH '1J1eHOB 9TOH npo.
rpeCCHH:
Ql
A = 2n 2 [1 + 3 + 5 + '" + (2n - I)) =
420
_ Ql (I+2n-l) n
- 2n 2 2
Q12n 2 Q!
2n 2 2 - 2

3a.u.allH no reOMeTpHH
19. OKOJlO MopCKoro oepera eCTb onaCHaSi ,lJ..JlSI KOpaOJleH
30Ha nO,lJ..Bo,lJ..HbIX CKaJl. r,lJ..e HY>KHO YCTaHoBHTb Ha oepery Ba
MaSIKa H KaK HCnOJIb30BaTb HX, '1TOO@ ooe30naCHTb nJiaBaHHe
Kopa6J1eH OKOJIO 3TOH 30HbI?
4epTe}K 227 nOMO>KeT BaM pewHTb 3TY 3a.l(a'lY. M 1 H M 2 -
MaSiKH. KI. K 2 H Ks - pa3J1H'IHbIe nOJlQ}KeHHSI KOpaOJlSI
OTHOCHTeJlbHO onaCHOH 30HbI.
06paTHTe BHHMaHHe Ha yroJi
M 1K 1 M 2' I13BeCTHbIH KanHTaHY
Kopa6J1S1.
20. A, B H C - TPH pa,lJ..HO-
MaSIKa, KOTopbIe nOCbIJIalOT CI1r-
HaJibI onpe.l(eJleHHOH J(JlH Ka>K-
.l(Oro MaSIKa JlI1HbI BOJlHbJ. Do-
JlQ}KeHHe MaHKOB HaHeceHO Ha
Ka PTY, HMelOllJ,YlOcSi Y KanHTaHa
Kopa6J1S1. KanHTaH C Kopa6J1S1, 11 2
HaXO.l(SlllJ,erOCH B nYHKTe D ('1epT. YepT. 227.
228), nOJlO>KeHHe KOToporo Ha.l(O.
onpe.l(eJlHTb Ha KapTe, HaXOHT npl1 nOMOllJ,H paMO'lHOH aR-
TeHHbl HanpaBJleHI1S1 DA, DB H DC H TeM caMbIM Y3HaeT yr-
JlbI ? H . 3TH ,lJ..aHHbIe .l(a.l(YT KanHTaHY B03MO>KHOCTb HaHeCTH
Ha KapTy nOJIO>KeHHe Kopa6J1S1 B .l(aHHbIH MOMeHT.
YepT. 228.
421

ItJIH peWeHIIH 3aJI.a'lil Ha OTpe3Ke AB CTpOI1M cerMeHt,
Bl\lew.alOllJ,I1H yrOJI 7., II Ha BC - Ceri\1eHT, BMew.alOw.I1iI yrO.1
. TorJI.a TO'lKa nepeCe'leHIIH yr 3TIIX cerMeHTOB (To4Ka D)
I1 onpeeJIlIT nOJlO>KeHlIe KOpaOJlSl.
21. Y.HTaTeJlb 3HaKOM H3 Kypca reoMeTpHIl C nOnepe'lHbIM
MacIllTa6oM, n03BOJlSlIOW.I1M C OBO.rJbHO 60JlblllOH CTeneHblO
T04HOCTH OnpeeJIHTb JlI1HY aHHoro oTpe3Ka H OTKJlaJI.bIBaTb
OTpe30K aHHoH .l(JIlIHbJ. DOJle3HO 03HaKOMHTbCH C J(pyrllM
npH6opoM - HOHHYCOM, KOTOPbIH TaKme CJlY>KHT J(JlH onpe-
J(eJleHHS! ,lJ,JlIIHbI oTpe3Ka. BOT era YCTpoi1cTBO ('1epT. 229):
I .o'5
,B(
/
YepT. 229.
AB - MaCllITa6HaH JlHHeMKa, KL - CTep}KeHb, .l(JlI1HY KO-
Toporo H3MepHIOT. CD - HOHI1YC. D.JlHHa HOHHyca paBHa 9
J(eJleHHHM MacllITa6HoH JlHHeHKH; 3Ta JI.JlHHa pa3JI.eJleHa H8
IO paBHbIx '1aCTeH." TaKHM 06pa30M, Ka>K.l(Oe .l(eJIeHHe HOHHyca
paBHo 0,9 J(eJleHHSI MacllITa6HoH JlHHeHKH. HOHHYC no)].-
BI1>KeH. .
)I(eJiaSi H3MepHTb J(JlHHY CTep}KHH KL, cOBMeuJ.aeM HYJle-
Boe eJIeHl1e llIKaJlbI AB C KOHli.OM A CTep}I(HH, a HYJleBoe .l(e-
JleHlle HOHHyca - C KOHli.OM L ('1epT. 229). JLJlHHa KL paBHa 5
nOJlHbIM .l(eJIeHI1HM llIKaJlbI AB C HeKOTopbIM H36bITKOM, KO-
TOpbIH H HY>KHO onpe.l(eJlHTb npl1 nOMOw.H HOHHyca. JLJJ5!
3Toro OTMe'laeM, KaKOe.l(e.rleHI1e HOHI1yca COBna.l(aeT C .l(eJleHlleM
lllKaJIbI AB. nYCTb 3TO 6Y.l(eT TpeTbe .l(eJIeHI1e HOHHyca. YT-
Bep}!(aeM, '1TO 1136bITOK paBeH 0,3 J(eJleHHH llIKaJlbI AB.
B caMOM .l(eJIe, eCJlH TpeTbe .l(eJleHl1e COBnaJlO C .l(eJIeHHel\I
lllKaJIbI AB, TO BTopoe J(eJleHHe HOHl1yca OTCTOHT aT 6.rII1>KaH-
mero J(eJleHHH llIKaJlbI AB Ha 0,1, nepBoe - Ha 0,2. HyJle-
Boe - Ha 0,3. TaKHM 06pa30M, .l(JlI1Ha KL=5,3. HOHHYC (HJlI1
BepHbep) ynoTpe6J1S1eTcSI TaK}Ke II Ha TO'lHbIX yrJloMepHbIx
HHcTpYMeHTax.
22. MeH3YJlbHaH C'beMKa. MeH3YJla COCTOIiT H3 .l(ByX '1a-
cTeH: KBa.l(paTHOH .l(OCKH pa3MepoM 40x40 CAt, Ha3bIBaeMOH
nJlaHllIeTOM (nJiaHIlleT YCTaHaBJlI1BaeTCSI Ha llITaTHBe), H aJlH-
.na.l(bI C J(BYMSI .l(HOmpaMH ('1epT. 230).
JLJlSI C'beMKH nJiaHa Ha nJlaHllIeT npHKpenJiSleTCSI JlHCT
6YMam Ii B li.eHTpe n.1aHIlleTa YKpen,,1S1eTcH HrJla; OCHOBaHHe
422

HrJIbl HBJlS!eTCS! li.eHTpOM no,noDHSI; nOJlOmeHlle eHTpa no.l(O-
6J1S1 OTMe'laeTCS! TaK}Ke 11 Ha 3e1Jle npH n0l\10llJ,H OTBeca, any-
meHllOro I13 rOJlOBKH IllTaTHBa, Ha KOTOpOM YKpenJleH nJiaHIlleT.

:..- I _ _ --
.  ..   - ;
-- - -- - - -
-_.0- _ _  ,
/I/lu8alJa c
iJuo n mpal'1LJ
fi/lOHWem c a/luoaooii
YepT, 230.
DpHKJlaJ(bIBaH aJlHJ(ay pe6pOM K HrJle, HanpaBJlSlIOT ee
BH3HpoBaHHeM '1epe3 .l(HOnrpbI Ha Bepllll1HbI cHHMa"eMoro Y'lacT-
Ka, B KOTOpblX YCTaHOBJleHbI Bexu. Do KpalO aJlH,naJI.bI npOBO.l(HT
OT eHTpa noo6HS! Jly'lH H Ha HHX B BN6paHHoM MacIllTa6e
OTKJla,nbIBaIOT paCCTOHHI1S!, 113MepSieMNe pYJleTKOH, OT li.eHTpa
c
B
{J

YepT. 231.
nO.l(06UH .l(O Kam.l(oll BepIllHHbI. TaK, Ha JlY'lax, nOJlY'leHHbIX
Ha mlCTe, OY.l(YT nOCJle.l(OBaTeJlbBO HaMe4eHbI Bce BeplllIlHbl
CHH!\IaeMoro Y'lacTKa. COe.l(HHl1B 3TI1 TO'lI\I1 oTpe3Kal\lH npH!\lblX,
nOJlY'Iaf{\T nJiaH Y'IacTl<a B 3a.l(aHHOM l\IacwTa6e ('1epT. 231).
423

nOYMallTe 0 TeOpeTH'IeCKOM 060CHOBaHHH perneHHS! 3Toi1
sa,na'lH. Bo BpeMS! JJeTHHX KaHHKYJJ CHHMHTe fIJIaH KaKoro-
JJH60 Y'IaCTKa npu nOMOUJ.11 MeH3YJJbI.
23. CHHTb nJiaH Y'IaCTKa ABCDE MO>KHO TaK}Ke H nYTe1
o6xoJI.a C MeH3YJJOll no KOHTYPY .l(aHHoro Y'IaCTKa. 4epTe}K 232
YKa3bIBaeT, KaK H B KaKOM nopS!.l(Ke peIllaeTcS! 3Ta 3a.l(atJa.
YepT. 232.
Ha KaKO:\f 3Tane o6xo.l(a 1I-1O>KHO npOBepHTb npaBHJJbHOCTb
nocTpoeHHS! nJJaHa Y'IaCTKa?
nepeHOCS! MeH3YJlY no KOHTYPY Y'IaCTKa, Heo6xoHMO
.l(06HBaTbCH COBna.l(eHHS!
BeplllHH A, B, C,... Y'IaCT-
Ka C cooTBeTcTBYIOlUHMH
BeplllHHaMH Ha nJiaHe.
24. Ha HaKJJOHHOlJ nJJOC-
KOCTH .l(.rJHHOli I = 3 oM H Bbl-
C COTOll h = 1,8 oM JJe>KHT rpY3
p = 60 K2. Onpe.l(eJlHTb
CHJlY, He06xo.l(HMYlO .':IS!
Y.l(ep}KaHHH 3Toro rpY3a, H
CH.':IY .l(aB.1JeHHSJ. TpeHHe He
Y'IHTbIBaTb ('1epT. 233).
P e III e H H e. Pa3.TJO>KUM BeKTop CHJlbI P no .l(BYM HanpaB-
JleHH5IM, H3 KOTOpbIX O.l(HO napaJJJle.1bHo HaI<JIOHHOll O.TJOCKO-
h
YepT. 233.
424

CTH, a .lI.pyroe nepneHHKYJlSlPHO K Hell. nO.IJY'IHM .l(Ba BeK-
Tapa: F - BeKTOp CHJIbl, KOTopaH CKaTbIBaeT TeJIO BHI13 (CH-
JIa, npOTHBOno.IJO>KHaS! H paBHaS! ell, He06xo.l(HMa, '1T06bI
Y.l(ep>KaTb TeJlO Ha HaKJIOHHOll nJIOCKOCTH), H BeKTOp Q-
BeKTOp CHJlbl .l(aB.'IemISI TeJIa Ha nJIOCKOCTb. Onpe.l(eJiSleM
BeKTOpbl F H Q:
6, A 1 B 1 C 1 00 /:" ABC;
B I C 1 _ AICI _ A1BI .
BC-i1C-AB'
F Q _ P .
h V 12-h 2 - I '
OTCIOJ{a:
F = hP . Q = P lflCh2
1 ' 1
25. llaHbl J{Be Cli.'IbI: F 1 = 8 Kc H F 2 = 6 Kc. Onpe.l(eJlliTb
paBHO.l(ellCTBYlOlIlYIO 3THX CHJI npH yr.lJax Me>KJ{Y HI1MH B 30°,
60°, 90 0 , 120", 150°.
PelllHM 3a.l(a'lY .l(.lS1 yr.lJa B 30° Me>K.l(Y J{aHHbIMH CHJIaMH
('1epT. 234).
p 8
I
o F. C 0
1
YepT. 234.
113 /:" OBC: R = V F + F + 2F l' CD ;
113 6,BCD: CD= VBC2-BD2 = V F - (y = 2 va.
R = V F + F + FIF2' V3 .
nO.l(CraBJlS!S! '1HC.IJOBble .l(aHHble, nO.1YllHM:
R = J;r 64 + 36 + 481/3 = ]1100 + 48 . 1,73 -
= V 183,04 = 13,5 (Kr).
425

3aKOH'IHTe 3a.na'lY. B3HB J(JlS! yr.'1a Me>Kj:(Y COCTaBJlSlIO-
llJ,I1r;m CH.TlaMH HHbIe .'1UCJlOBbie .l(aHHbIe.
26. OnpeJleJleHHe .l(aJlbHOCTH ropH30HTa.
4I1TaTeJlb, HaBepHoe, nOMHHT TeopeMY: eCJIH H3 TO'lKH,
B3S!TOH BHe OI<PY)!{HOCTlI, npoBecTH K 3TOH OKPY>KHOCTH Kaca-
TeJIbHYIO I1 ceKYllJ,YIO, TO OTpe30K KaCaTeJlbHOH eCTb cpe.nHee
reOMeTpH'IeCKOe J(JlSI BceM ceKYlIleH H ec BHeUlHeH '1aCTH.
3Ta TeopeMa .l(aCT B03MO>KHOC1'b onpe.l(eJlHTb ,naJlbHOCTb
ropU30HTa H3 TO'lKI1, Haxo.nSllIleHCH Ha aHHOH BbiCOTe Ha
nOBepXHOCTblO 3eMJlI1 (llepT. 235).
0- li.eHTp 3eI'I'I.'1I1; M - TO'lKa, HaXO.l(HlIlaSiCH Ha pac-
CTOSlHHH h Ha nOBepXHOCTblO 3eMJlH; MA - OTpe30K Kaca-
1'eJIbHOH, npOBeeHHOM 113 TO'lKH M K nOBepXHOCTIi 3eMJlH:
J(JlHHa d 3Toro OTpe3Ka 11 eCl b .l(aJlbHOCTb ropH30HTa TO'lKI1 M.
Do BbillIenpHBe.l(eHHOM TeopeMe:
d = V(2R -/- h).h = V2Rh + h 2 .
CnaraeMoe h 2 no cpaBHeHI110 C 2Rh O'leHb MaJlO. n03TOMY
npu onpe.l(eJIeHI1H d CJIaraeMbIM h 2 MO>KHO npeHe6pe% H .l(e-
JlaTb paC'le1' no npH6J1H}KeHHOH <popMy.lJe:
d:::::;V 2Rh . (1)
r.l(e R - pa.l(HYc 3eMJlH H h - BbICOTa TO'lKI1 M Ha.l( nOBepx-
HOC1'blO 3eMJlH. Dpl1HSlB R npHOJlmKeHHO paBHblM 6400 KM,
npI1 nOMOIIlI1 <POPMYJlbl BbInOJl-
HIiTe ynp8>KHeHI1S1:
a) KaK .l(aJleKO BI1HO. eCJlH
CTOHllIb Ha 6epery MOpH Ha Bbl-
COTe 5 A-t Ha.l( ero YPoBHeM?
6) K8I<OBa aJlbHOCTb ropH-
30H1'a, eCJlH >KliBelllb Ha 6epery
MOpH Ha BbICOTe 100 .At?
B) KaI<OBa J(aJlbHOCTb ropH-
30HTa C BeplfJI1Hbl 3J1b6Pyca
(BbICOTa 3J1b6pyca5 K.At)? BH-
.l(eH JlI1 C seplllHHbl 3J1bopyca
Apl\18Bl1p? BaTYllm? TOHJlI1CH?
(BocnoJlb3YliTecb KapTOH.)
r) KaKOBa .l(aJlbHOCTb rOpI130H-
Ta KOCIOHaBTa, Kor.l(a Kopa6J1b
«BOCTOK-5» HaXOj:(HJlCSJ B Hal160Jlee y.naJleHHOH TOqI<e CBoeM
OpOH1'bI (h = 235 K.At).'
t1
YepT. 235.
426

27. 4T06bl H3Mepl1'rb HaMeTp 60JlbllIOrO llIKHBa, YCTaHo-
BI1J1H llITaHreHli.I1pKYJlb T8I<, KaK nOKa3aHO Ha '1epTe}Ke 236.
,aJlHHa HO>KeK llITaHreHli.HpKYJlH 8=25 CAt, paeeTOSIHHe Me.iK.l(Y
KOHli.al'l'lI1 HO>KeK 1=200 MM.
a) Onpe.neJlI1Tb J(JlI1HY .l(lIaMeTpa D.
6) BbIBeCTI1 <POPMYJlY, BbIpa>KalOYIO 3aBHCHMOCTb D
OT 8 H I. (P bI 6 K I1 H, 3a.l(a'lHHK.)
Omeenz. a) 425.
YepT. 236.
28. H"JI.HKaTOpHaSl JI.l-tarpaMMa. 4acTo .n.'1H I13MepeHHH pa-
6m:bI .l(BlIraTeJlSI K HeMY npI1COeHHS!IOT OC06bIH npH60p iJiSI
H3MepeHHSI J(aBJleHHSI napa B li.HJlHHJ(pe 3Toro .l(BUraTe.1H.
DPH60p HOCHT Ha3BaHHe HH.l(HKaTopa. ,aaBJIeHHe napa nepe-
1'l'lellJ,aeT nopllleHeK HH)J.HKaTopa. JLBI1>KeHHe nopIlleHbKa npH
nOMOH CHCTeMbI pbI'IarOB nepe.l(aeTC}I 3aKpenJleHHoMY KapaH-
alllY. .l(BH>KyeMycH no OeH BpallJ,alOerOcH li.I1J1HHJ(pa, Ha
KOTOpOM npHKpenJleH JlHCT 6YMam. DPH BpallJ,eHI1H L(I1J1HHJ(pa
KapaH.l(alll 6Y.l(eT '1epTI1Tb HeKoTopylO KPHBYIO JlI1HHIO, KOTO-
paS! nOKa3bIBaeT )J.aBJleHHe napa B .'11060M nOJlO>KeHHI1 noplllHS!
L(HJlI1HJ(pa. DOJlY'IeHHaSI KpHBaSI Ha3bIBaeTCH HH.l(HKaTOpHoH
.l(HarpaMMoH. nJlOllJ,a.l(b <PHrYPbI, BbllleplleHHoll HH.l(HKaTopoM,
.l(aeT Bem1'1HHY pa60Tbi .l(BHraTe.'1H 1 ). Ha LJepTe>Ke .l(aHa liHJ(H-
1) Pa60Ta CIIJibi 1I3MepHeTCH npOll3Be.neHlIeM CIIJibI Ha .l\JlHHY nYTH.
ECJIIi CIIJia nOCTOHHHa no BeJIHqHHe, TO pa60Ta rpaq,IIIJeCKH 1I306pa-
3HTCH nJlOUa.l\blO npHlOyrOJIbHHKa. EC.1H CII.la C TeIJeHlieM BpeMeHH
MeHHeTCH no Be.'1I1IJHHe, TO rpaq,IIIJeCKII pa60Ta 1I306pa3liTCH nJlOll\a.l\blO
KpHBOJlHHeHHOH TpaneUHII.
427

f{gTOpHaSi ,nHarpaMMa '1eTbIpeXTaKTHoro JI.BHraTeJlH 1I.H3e.IJ\1
('lepT. 237).
Onpe.l(eJlHTe nJlOllJ,a.l(b ,noH JI.HarpaMMbI HJlI1 npff nOMOUm
1}J0PMYJlbI TpaneUl1ll. HJlH no cpopMYJle CHMncoHa (rJlaBa VI).
.. 30
c::,
 Z5

 ZO
:t
E: 15
c::,
"'"
",to
::i
 5

c& 0
l.}epT. 237.
3aAaqH no TpHroHoMeTpHH
29. I1cC.J1eAOBaHHe ABH)I{eHHSI n0J13YH8 B KpHBolIIHnHo-
lllaTYHHoM MexaHH3Me. KpI1BOlllHnHo-waTYHHblll MeXaHI13M
CJlY>Kl1T JlSI npe06pa30BaHffH paBHoMepHoro BpaI..lJ.aTeJlbHoro
,nBH>KeHHSI KOPlUa KpHBolllHna B HepaBHoMepHoe npSlMOJlHHell-
Hoe ABH>Kemie nOJl3YHa, H 06paTHO ('1epT. 238).
180 0
270.
YepT. 238.
B Ha'laJlbHbIH MOMeHT, KorJI.a KpHBolllHn 3aHl1MaeT nOJlO>Ke-
HHe OAt, TO'lKa B llIaTYHa HaXOHTCSI BBl' ECJlH B aH-
Hbill MOMeHT KpHEOlllHn HaXO.l(HTCSI B nOJIO>KeHHH OA, 06pa-
9YSI yroJi a. C JlHHHeH MepTBblX TO'leK, COOTBeTcTBeHHO 'leMY
IllaTYH 3aHl1MaeT nO.IJO>KeHHe AB, 06pa3YSI C TOll >Ke npSlMOH
428

yrOJI ?, TO. CJle.l(OBBTeJIbHO, na.'Ieli. B nO:'I3YHa 38 BpeMH
rlOBOpOTa KpHBOlllI1na H8 yro.'I 'J. nepeMeCTI1J1CSI Ha Be.rII1'1I1HY
X = BIB. BbIpa3HM nepeMellJ,eHI1e x B 3aBHCHMOCTH OT .l(aHHbIX
BeJIIi'111 H.
OnycTHM nepneHJ(I1KYJlSlP AK Ha OBI; TOr.l(a:
OB= OK + KB.
113 TpeyrOJIbHI1KOB AOK Ii ABK liMeeM:
OK=OAcoso:=rcos:x H KB=ABcos=lcos;
CJle,lJ.OBaTeJlbHO,
OB = rCOsCt + I cos
H
X = r + 1- r cos 0: -I cos  = r (1 - cos 0:) + I (I-cos ). (1)
BbIpa3HM cos  B 3aBliCHMOCTH OT yrJla Ct 1I3 TpeyroJlbHHKOB
AOK H ABK; HaH.l(eM:
AK = rsina H AK = lsin.
OTCIO,lJ.a:
r sin a = I sin  H sin  = -7- sin a
H
cos  = V1-sin 2 p = -V 1- (+ sin at .
nepexoASI K BbIpa>KeHHIO (1) J(JlSI X, nOJlY'lI1M:
X = r (1 - cos a) + I [ 1 - -V 1- ( + sin at] .
BOT TOT 3aKOH, no KOTOPOMY H3MeHSleTCSI nepeMern.eHHe
nOJl3YHa B 3aBHCHMOCTH OT BeJlH'IHHbI yrJla. Onpe,lJ.eJlHTe X,
eCJlH 0: = 0°, 0: = 90°.
30. P8C'ieT .l(JlHHbl peMeHHoii nepe}J.a'iH, coe}J.HHSllorn.eii }J.Ba
UIKI1Ba: Be}J.yrn.Hii H Be}J.oMblii ('1epT. 239). DYCTb pacCToHHHe
Me>K.ny ueHTpal\!I1 lllKliBOB paBHo d H pa.l(I1ycbI HX - R I1 r.
JLmmy peMeHHoH nepe.l(a'lH pa306beM Ha '1aCTH AB, BC,
CD = AB, DE, EF, EA = DF. Onpe;I.e.TJI1M J(JlHHY K3>Kl1.0H
OT)I,eJlbHOH '1aCTI1,
429

113 TpeyroJl bHHKa 01KO H MeeM:
01 K = AB = V d 2 - (R - r)2; L AOE = L B0 1 G = L K0 1 0;
0603Ha'lI1M L AOE '1epe3 ex H HaH.l(eM ero BeJll1'iHHY.
113 6, 01KO
. R-r
stn(J.=.
F
YepT. 239.
3HaS! sin (J., MbI CMO>KeM no Ta6.1Hli.aM onpeJJ;eJlHTb 11 yroJi ex
v AE = v DF = 2;7. = 'i7. ;
AEFD R ' ..R7. 2 R + ..R7.
v =1t' Tl80 =1t' ----go-;
v BG = v CH = 2;o: = ;o7. ;
vBC=o.:r- ...r7. 2=o.:r-
180 90 .
llJlI1Ha Bce ro peMHH =
-2 V d2 ( R )2 + R + r..R'J. '-R ..r'J._
-. - -r 1t' ---00- 1 " ----00--
=2 V d 2 -(R-r)2 +1t'(R+r) +  (R-.r).
31. BHHToBaSi JUlHHSI. Dpe.l(CTBBHM ce6e, '1TO Ha 60KOBYlO
nOBepxHocTb li.HJlHHJI.pa C ,i:(HaMeTpoM d HaMaTbiBaeTCH npHMo-
yroJlbHbIH TpeyroJlbHI1K ABC ('1epT. 240) C OCHOBaHlieM AC =,,(1
4;()

TaR, tlTO OCHOBaHHe To COBnajl,aeT C OKpY>KHOCTbtO OCHOB3-
UHH li.I1J1l1Impa. TaK Ral( AC = rrd, TO TO'll(a C Ilocne Taro,
Kal( BeCb TpeyroJlbHHK OY.l(eT HaBepHYT Ha OOKOBYIO nOBepx-
HOCTb li.I1.rmH.l(pa, COBna.l(aeT C TO'lKOH AI' TOqKa B 3allMeT
nOJlO>KeHI1e B 1 Ha oopa3YIOllJ,ell A 1 B 1 li.HJIII H.l(pa, a mnoTeHY-
/l
(3
YepT. 240.
3a AB 3aHMer HeKOTopoe nOJlO>KeHHe H8 OOKOBOH nOBepXHO-
CTH li.I1J1HHJ(pa H npHMeT OPMY BHHrOBOH I1HHI1.
MbI nOJlY'lHJlll O.l(HH BI1TOK BI1HTOBOH JlHHI1H. L!.JlllHa Ka-
TeTa BC (h) Ha3bIBaerCH IllarOM BHHTOBOll JlHHHH. YroJi
BAC (7.) Ha3bIBaeTCH yr.TJOM nO.l('beMa BHHTOBOHJlHHHI1. Hall-
.l(eM 3aBI1CHMOCTb Me>K.l(y h, d H a..
113 TpeyroJlbHHKa ABC HMeeM:
h = rrd tga;
nOJlY'IeHHaSi opMYJla n03BOJISIeT
TaK>Ke onpe.l(eJlHTb yroJi nO.l('be-
Ma no .l(aHHbIM h H d
p
YepT. 241.
h
tg? = - d .
.-
32. Onpe.l(eJlHTb HaI100.TJblllHH yro,TJ H8K,TJOHHOH nJlOCKOCTH,
npn KOTOpOM Te.TJO. HaXO.l(HUJ,eeCH Ha HaK.TJOHHOH nJl0CKOCTH,
6Y.l(eT HaXO.l(l1TbCH B paBHOIJeLHH. Bec TeJla P H K03Hli.HeHT
rpeHHSI paBeH k ('1epT. 241).
OmBem. tg cp = k.
Y K a 3 a H H e. CI1J1a TpeHHH H3MepSleTcSI np0I13Be.l(eHHeM
CH.'IbI .l(aB.'1eHHH Te.la Ha HaKJloHHYIO nJlOKOCTh Ha K03qm-
li.HeHT TpeHHH k.
431

33. OnpeJleJleHHe KoacJ>qmu,HeHTa TpeHHH. T eJIO Beca P
nOJlO>KeHO Ha HaKJlOHHYIO nJlOCKOCTb C yrJlOM HaK.TJOHa a.. Ie.IID
no.l( .l(eHCTBI1eM CBoero eOOCTBeHHoro sera npOlllJlO YCKopeH-
HO nYTb S B t ceKYH.l(. Onpe.l(eJlHTb K03q>qmli.HeHT TpeHHSI k
(tIepT. 242).
P e 1lI e H H e. CHJla J(aBJleHHS! TeJla Ha HaKJlOHHYIO nJloc-
KeCTb = kP cos a..
CH.lla, KOTopaSl TS!HeT TeJlO BHH3, =F = Psina.-kPcosa. =
=P(sincx.-kcosa.). (1)
ECJIH TeJIO JI.BH>KeTCSI no HaKJlOHHOH n.IIOCKOCTH, TO YCKO-
peHHe
28
a=i2'
F F
C .l(Pyroii CTOpOHbI, YCKopeHHe a = m = P = gF; cJleJl.O-
g
BaTeJlbHO,
gF _ 28
p-f2'
(2)
¥I3 paBeHcTB (1) I1 (2) CJIeJ.\.yeT,
'1TO
g(sincx.-kcoso:) =  .
OTclOJ.\.a:
k _ _ gsina. 28 28
= g tg-
cosa. gt 2 CDS"'. gt 2 cOS(,t
34. Ha 'lepTe>Ke 243 .l(aeTcS! BHHTOB8S! HapeSl<a noplllllSi
saTBopa apTHJlJlepHHeKoro 0pY.l(HSI. BbI'II1C.IIHTb yroJi  J(JlSI
SaTOGKll pe3a, cJlY>KallJ,ero J(JlS! HapesaHHSI 3Toro BHHTa.
Omeem. 78°48'.
p
YepT. 242.
J .
-d
5
1.{epT. 243.
432

35. Onpe.l(eJlHTb BbICOTY 3.l(aHHS!, K KOTOPOMY HeJlb3SJ
Henocpe.l(cTBeHHo nO.l(oHTI1 ('1epT. 244).
P e III e H He. 1) I13MepSiIOTcSJ: 6a3Hc AB H yrJlbI iJ. H ;
2) H3 /:" AMN: MN = ANtgiJ.;
3) 113 6, BMN: BN = MN ctg a.;
4) AN = AB + MN ctg2;
5) MN = (AB + MN ctg2) tga.;
MNctga. =AB+MNctg.
AB
MN = ctg - ctg .
OTClO.l(a
N
/1, --:-..J_
YepT. 244.
36. Orrpe.l(emITb paccToHHHe Me>K.L\Y .l(B}'MSJ rrYHKTaMH M
H IV Ha nOBepXHOCTH 3eMJlI1 MeTO.l(OM TpI1aHrYHu.I1H.
P erne H H e. I. B nYHKTax M H N YCTaH8BJlLlBalOTCfI
BbIrnKH - reO.l(e3H'IeCKHe 3H8KH ('1epT. 245). Me>K.l(Y nYHK-
TaMil M H N HaMelJ"aIOTCS!. pH.l( nYHKTOB A, B, C, D TaK,
'1TOObI C BbIllIeK, YCTaHOBJleHHbIX B 3THX nYHKTax, ObIJIH
BH.l(HbI: H3 A - BbIllIKH M H B, I13 B - BbIIlIKI1 M, A, C H
D, H3 C - BbIllIKH A, C, N H T. .l(.
2. I13MepSleTcSJ oa3HC AM.
3. 113MepHIOTcS! yrJlbI B TpeyrOJIbHHKaX ABM, ABC H T. /1...
4. BbI'IHCJlHIOTCH MI1Hbl CTOpOH 3THX TpeyroJlbHliKOB no
TeopeMe CH.HYCOB B TaKoll nOCJIe.l(OBaTeJlbHOCTH: CHa'laJia B
TpeyroJlbHHKe ABM, 3aTeM B TpeyrOJJbHHKe ABC H T. .l(.
5. 113 TpeyrOJJbHilK8 BDN onpeeJJS!IOT BN no TeopeMe
KOCHH}'COB (HJJH no TeopeMe T8HreHcoB).
6._113 TpeyrOJlbHHKa BMN onpee,;lHIOT MN no TeopeMe
KOCI1HyCOB.
433

U'
I -: -. k... l'
\'I':Q...
'

-.J.\'
t '
\.;-
i
t
I;" N
reOOe3U'IeCKUil
JHOK


YepT. 245.


1'1


,f;\i



c


D


37. OnpeAeJleHHe paccTOHHHS! 3eMJlS! - JIYH3 ('1epr. 246).
)Jo np0I13Bo.n.CTBa BbIlJ"HCJleHl1ll HeOO
O.l(I1MO npo,u,eJlaTb
CJIe.l(YIOllJ,YIO nO.l(rOTOBHTeJlbHYIO paoOTY:
1. BbIopaTb .l(Be oocepBaTopnH A H B. pacnOJIO>KeHHbIe
Ha O.l(HOM MepI1,!I.I1aHe.
2. Oripe.l(e.'1HTb reorpaqm'leCKHe nmpoTbI Cfl H 92 9THX
o6cepBaTopHH.


IOn


YepT. 246.


434




3. Ha Ka>K.l(Oll oocepBaTOplll1 B O.l(HH H TOT >Ke MOMeHT
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