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BpeMH nplIBJIeRaJIa aRCIIOMa 0 napaJIJIeJIbHbIX, KOTopaH IIpeACTaBJIH-
JIaCb MeHee OqeBlIJJ;Holl II 60JIee CJIOmHOH, qeM APyrlIe aRCHOM:bI II
1I0CTYJIaTbI. Ee CTpeMIIJIIICb JJ;OKasaTb yme B RJIaCClIqeCROH ApeB-
HOCTH. BamHeHIIIlIe 1I0IIbITRII AORa3aTeJIbCTBa pa3JJ;eJIHIOTCH Ha JJ;Be
rpYIIII:bI. B OJJ;HIIX CJIyqaHX aBTOp:bI He3aMeTHO JJ;JIH ce6H IICnOJIb30Ba-
JIll AORa3:bIBaeMYIO aRCIIOMY IIJIII JIOrlIqeCRlI paBHOCHJIbHOe ell IIpeJJ;JIO-
meHHe, TaR qTO IIX paccymAeHlIe cOJJ;epmaJIO JIOrHqeCRlIii Rpyr 1I
6bIJIO HeCOCTOHTeJIbHbIM. B APyrllx CJIyqaHX V nOCTYJIaT OTRpbITO sa-
MeHHJICH KaRIIM-HH6YAb npeJJ;JIOmeHlIeM, Ra3aBlliHMCH 60JIee OqeBII,n;-
HbIM IIJIII IIpOCTbIM. B 060lIX CJlyqaHX, eCTeCTBeHHO, HCIIOJlb30BaJllICL
OCTaJIbHbIe aKCIIOMbI H 1I0CTYJIaThI «HaqaJl». B HOBoe BpeMH OCO-
6eHHO BamH:bIMII HBHJlIICb Tpy,n;bI no TeoplIlI napaJlJIeJIbH:bIX JJ:m. BaJI-
JIHCa (JIeRqIIH 1651-1663 rr., ony6JIIIROBaHHbIe B 1693 r., CM. [68]),
m. CaRReplI (1733, CM. [62]) II M. r. JfaM6epTa (1766, ony6JI.
B 1786, CM. [55]), a TaKme A. 1\1. JIem aHJJ;p a, yqe6HHR reOMeTpHH
ROToporo (HeoJJ;HoRpaTHo lIepeHs,n;aBaBIIIHllcH B 1794-1823 rr. [56])
IIOJIyqHJI lliHpOKoe pacnpOCTpaHeHHe.
Yme B cepeJJ;IIHe XVIII B. JIHTepaTypa 110 TeopHH napaJIJIeJIbH:bIX
6bIJIa CTOJIb 06IIIHpHOii, qTO r. C. RJIlOreJIb, yqeHHR A. r. RecTHepa,
B 1763 r. ony6JIHROBaJI JJ;HccepTal HIO, cOJJ;epmaIf(YIO HCTOpHRO-RplI-
TlIqeCKHH 0630p OROJIO 30 H3BeCTH:bIX eMY 6e3ycneIIIHbIx nOII:bITOR ,n;o-
K83aTeJIbCTBa aRCIIOMbI 0 lIapaJIJIeJIbH:bIX [54]. MMeHHo 3TH TpYAM
OKa3aJIHCb, XOTH H B paSJIlIqHOH CTelleHII, OTllpaBHbIMH IIYHKTaMH .D;JIH
TBOp OB HeeBRJIHAOBOH reOMeTpHII. rJIaBHbIM CJIeJJ;CTBlIeM HX HBIIJIOCb
B03HIIEHOBeHIIe YBepeHHOCTll B TOl\I, 'lTO ,n;ORa3aTeJJbCTBO V 1I0CTYJIaTa
TOJIbKO C 1I0MOIf(blO JJ;pyrlIx aKCHOM «HaqaJI» HeB03MomHO.........
R TaROMY B:bIBOAY nplImeJI yme I\JIlOreJIb, XOTH eIf(e AOJIroe BpeMH
1I0CJIe Hero OT,n;eJIbHbIe KpYIIHbIe MaTeMaTHRH TIf(eTHO IIpeJJ:npIIHIIMa-
JIll pa3JIHqHbIe 1I01l:bITRII TaRoro po,n;a. EOJIee Toro, CaRRepH H oc06eH-
HO JIaM6epT IIpHIIIJIH R PHAY IIpe,n;JIOmeHHH HeeBRJIHnOBOH reOMeTpHII,
He CYMeB, o,n;HaKO, IIpaBHJIbHO o eHHTb HX 3HaqeHlIe. B CBOIO O"tJepenL
cy ecTBeHH:bIM OTnpaBHhIM nYHRTOM pa60TbI BaJIJIHCa no TeopHH lIa-
paJIJIeJIbH:bIX HBIIJIaCb 1I01l:bITRa }J;oKa3aTeJIbCTBa V 1I0cTYJIaTa, npe,n-
JIOmeHH3H B COqHHeHlIH, H3AaHHOl\1 B PHl\le B 1594 r. rro,n; IIMeHeM
HacHp 8A-,[(HHa aT-TycH, BbIAalOIT(erOCH MaTeMaTHRa H aCTpOHOMa
XIII B. Yme 3TO 06CTOHTeJIbCTBO IIORa3bIBaJIO BamHOCTb BRJIa.n;a,
BHeceHHoro B TeopHIO rrapaJIJIeJIbH:bIX 110 RpaiiHeii Mepe OAHHM 1I3
yqeHbIX MHpa apa6cKoH RYJIbTypbI. OAHaRo BIIJIOTb )];0 HeJJ;aBHero
7
BpeMeHH HMH HaCHp a - HH8 aT-TYCH o.CTaB8J1o.CL eJUIHCTBeHHIdM
apa6CRHM HMeHeM B IICTo.PHH AaHHo.U' npo.6J1eMLI, eCJlH He CqHTaTb
Ro.MMeHTaTo.pa «HaqaJI) 8H-HaupH3U, mHBmero. B X B. H H3JIo.mllBmero.
B CBDeM Ro.\lMe TapHH HCCJIe o.BaHHH CBo.HX npe,nmeCTBeHHIIRo.B -
CHMIIJIHRIIH H HeKo.erO AraHHca. To.JlbRo. aH-HaupH3H II HaCHp aA-
HH aT-TYCH yno.MHHaJlIICb B pa6o.rrax no. IICTo.PHD Teo.PHH napaJlJIeJlb-
HbIX l1.o. 1935 r., HanpHMep B o.qeHb Xo.Po.meM D63DpHOM TpYAe P. ED-
Ho.JIM [5].
B 1935 r. aMepHRaHcKHH HCTo.PIIR MaTeMaTHKH p:. 10. CMHT o.ny6-
JIHRo.BaJl CTaTblO [63], B KDTo.PDH BnepBble RpaTKo. YRa3aJI Ha IlCCJIe-
AOBaHHH DO. TeDpHH napaJIJIeJIbHhIX OMapa Xa:UHMa; CMHT DCHo.BhIBaJI-
CH npu aTDM Ha CBe)];eHIIHX, co.,n;ep.maII:(UXCH B D,n;Ho.H TDrAa eII:(e He
o.ny6JIHRDBaHHD:U PYKonHCH HaCHp aA-,[(HHa aT-TYCII, C KDTDPo.U nD-
3HaRDMHJICH BD BpeMH nOe3)];RH B MpaH. BCRo.pe, B 1936 r., upaHcRH:U
yqeHbIi TarH 3paHH ony6J1HKDBaJI apa6cKHH TeKCT «RDMMeHTapHeB
K Tpy,n;Ho.CTHM BD BBe)];eHHHX KHHrH EBKJIH)];a) [51] XauHMa, B nepBDu
KHHre Ko.TDPLIX H3JIOmeHa o.pnrlIHaJIbHaH Teo.PHH napaJIJIeJlbHbIX
9To.ro. yqeHDrD, 6DJIee npDCJIaBHBmerDCH KaK nD3T, HD He MeBee 3a-
MeqaTeJIbHDrD KaR MaTeltlaTHK. K CDmaJIeHHIO, D,D;HaKD, 3paHH He
o. eHIIJI B ADJImHo.:u Mepe H3JJ;aHHbIH 11M TPYJJ; XaHRMa BDD6 e II erD
8HaqeHHe B HCTo.PHH yqeHHH 0. napaJIJIeJIbHbIX B qaCTHo.CTH. Eo.JIee
nDAPD6HhI:U aHaJIH3 3TDH TeDpHH 6MJI npo.H3BeAeH aBTDpaMH 3TDH
KHHrH B 1953 r. npH nepBDu ny6JIHKaqHH PYCCKDrD nepeBDAa. TpaKTa-
Ta Xa:UHMa [31], nDCJIe qero. Heno.JIHbIH aHrJIII:UCRU.U erD nepeBD,n;
B3naJI A. P. AMHp-MD:33 B 1959 r. [52], a A. M. Ca6pa [53] H3naJI
apa6cRHH TeKCT TpaKTaTa no. )];BYM PYKDnHCHM.
ITo.cJle To.ro. 6LIJI o6HapymeH II H3yqeH qeJlhIu PH}]; TpaKTaTDB no.
TeDpHH napaJIJleJIbHbIX )];PyrHx yqeHbIX apa6CKDrD BDCTDKa, II Te-
nepb MDmHD CKa3aTb, qTo. pa6DTbI B 3TD:U D6JIaCTH CDCTaBHJIH D}];HY H3
3HaqHTeJIbHbIX rJIaB MaTeMaTHKH B CTpaHax apa6CKo.H KYJlbTypbI.
3TI! pa6o.TbI npDBD)];HJIHCb Ha D6mHpHo.:u TeppHTDpHH DT RaHpa )];0.
CaMapKaH,n;a C IX)];D X IV B., II Mem,n;y HHMB CYII:(eCTBDBaJIa npHMaH
npeeMCTBeHHDCTb. I1pHHqHnLI, RDTDphIMH PYRDBO}];HJlHCb aBTDpbI, He-
pe1J;Ko. 6bIJIH CYII:(eCTBeHHo. pa3JIHqHbIMH; npH 3TDM 6bIJI npeAJID;I\eH
PH}]; Ko.HCTpYKIJ;H:U, npHeMDB, TeDpeM, BHDBb DTKpLlTbIX B EBpDne B
XVIII B. MLI npHBeAeM cpa3Y me cnHCDK MaTeMaTHKDB, pa3pa6aTbI-
BaBmHX TeDpHIO napaJIJIeJIbHhIX, CJIeAYH, HaCKDJIbKo. Do.3BDJlHeT Aa-
THpDBaHHe, XpDHDJIDrHqeCKDMY nDpH)];KY. BDT 3TDT cnHCDR: aJI-
AiRayxapH (IX B.), Ca6HT H6H RDppa (IX B.), aH-Ha:upH3H (X B.),
M6H aJl-XaHCaM (X-XI BB.), 116H CIIHa (nepBaH nDJIOBHHa XI n.),
aJI-EHpYHII (IIep aH IIOJIOBHHa XI B.), OMap Xaii1lM (BTopaR IIOJID-
BHHa XI B.), ac-CaJIap (HaqaJlD XIII B.), aJI-A6xapH (XIII B.),
HaCHp aL(AHH aT-TycH (XIII B.), aBTDp ynDMHHYTDrD CDqHHeHHH, H3-
1taHHo.rD no.)]; HMeHeM HaclIp a1J;-,L(IIHa aT-TYCH B 1594 r. (XIII B.) 1,
aJl-MarpH611 (BTDpaR nDJlOBHHa XIII B.), aJl-CaMapKaH,n;H
(XIII-Xlv'" BB.), am-illHpa3H (XIII-XIV BB.).
1 Bonpoc 06 aBTopCTBe 8Toro Tpyp;a paCC laTpHBaeTCH p;aJIee Ha c. 2.
8
3AeCb Ha3BaHhI TOJIbKO aBTOpLl C01JHHeHIlii:, 06HapYiKeHH:bIX K
HaCTOH eMY BpeMeHH; KpOMe TOrO, H3BeCTHO, 1JTO TeOpHeH napaJIJIeJIL-
HhIX 3aHBMaJIHCb H APyrBe Y1JeHhIe apa6cKoro MBpa, HanpllMep aJI-
RHHAH (IX B.), aJI-Xa3BH II am-IIIaHHII (X B.), aJI-XaHa.pB (XIII B.),
KOTophIe ynoMBHalOTCH B B3BeCTH:bIX HaM C01JBHeHBHX, HO COOTBeT-
CTBYJO Be TpYAbI KOTOpbIX I!Q1\a :ae o6HapymeHhI.
R naJIbHenmeM M:bI 6yneM nOCTOllHHO pa9.filttiafb nOnhITI\H )J;oKa...
3aTeJIbCTBa aKCIIOMhI napaJIJIeJIbHOCTH, COAepiRa He JIOrlIlIeCKBe
OlliB6KB, H JIOrH1JeCKH 6e3ynpe1JHhIe BhIBOAhI, B KOTOphIX 3Ta aKCBOMa
paCCMaTpllBaeTCH KaK TeopeMa, AOKa3hIBaeMaH Ha OCHOBaHBH APyrBx
npeAJIOiKeHBii:, npBHHTbIX 3a 01JeBHJJ;HhIe IICTBHhI, He rrOAJIema Be
060CHOBaHBIO. M3JIaraH pa3JIH1JHhIe )];OK 3aTeJIbCTBa apa6CKBX Y1Je-
HhIX, MhI 6y)];eM no B03MOiRHOCTII IITBpOBaTb COOTBeTCTBYIO Be
TeKCThI B PYCCKO I nepeBO)];e, conpOBOiK)];aH BX nOHCHeHBHMH HCTO-
pB1JeCKOrO II MaTeMaTB1JeCKOrO xapaKTepa. B 3aKJII01JeHBe MhI pac-
CMOTpBM Bonpoc 0 BJIBHHBB aToi apa6cKoii: JIIITepaTyphI Ha TPY)];bI
no TeopHII napaJIJIeJILHhIX MaTeMaTBKOB EBponhI XIII-XVIII BB.-
BBTeJIO, mBBmero B llOJIbme (XIII B.), JIeBH 6eH repmOHa, npOiKH-
BaBmero B TIpoBaHce, Ha lOre <DpaH BH (XIV B.), ero COBpeMeHHIIKa
AJIL.pOHCO, mHBillero B McnaHBB, HeMn;a Xp. RJIaBHH - illJIIOCCeJIH
(XVI B.), BTaJIbHHD;eB TI. A. RaTaJIbAB (XVI-XVII BB.), JJ:m. A. Bo-
peJIJIB II B. jI(OPAaHo II aHrJIB1JaHIIHa Am. BaJIJIllca (XVII B.).
reOMeTpH1JeCKBe 3HaHIIH, HaKan-
JIHBaBIIIBeCH C rJIy60Koi )];peB-
HOCTB, nOBCIO)];Y B03HHKaJIH B
.popMe npeAJIOmeHB:U, npBHaA-
JIeiRa BX JIB60 K TaK Ha3hIBae-
Moi a6COJIIOTHOi reoMeTpHH B
CMhICJIe BOHH, T. e. He3aBHCIIMO:D:
OT eBKJIH)];OBa nocTYJIaTa 0 na-
paJIJIeJILH:bIX, JIH60 K eBKJIII)];O-
Boi reoMeTpBB. OCHoBHaH Teo-
peMa MeTpH1JeCKOi eBKJIBAoBOi reoMeTpBH, HocH aH HMH TIB.paropa,
6:bIJIa H3BeCTHa eCJIH He. B 06 eH «popMe, TO )];JIH 1JaCTHhIX CJIY'IIaeB,
HanpBMep TpeyrOJIbHBKa CO CTopOHaMIl, OTHOCH BMHCH, KaK 1JHCJIa
3 :4 : 5, B ApeBHHX ErBnTe, BaBIIJIOHe, MH)];BH H RHTae; aHTII1JHhIe
aBTophI npBnBChIBaJIB )];OKa3aTeJIhCTBO aTOH TeOpeM:bI, KaK H Teo-
peMhI 0 paBeHCTBe CYMM:bI yrJIOB TpeyrOJIbHHKa, HMeHHO TIB.paropy,
mllBmeMY B VI B. )];0 H. a. BecLMa JJ;peBHHM 6hIJIO H Y1JeHBe 0 nOAo6l1n,
06pa3YIOID;ee BamHYIO rJIaBY eBKJIH)];OBOi reOMeTpBB: B03MomHO, 'liTO
C HBM 6hIJIH 3HaKOMhI MaTeMaTHKH BaBBJIOHa, H HeCOMHeHHO, 1JTO
OHO 6hIJIO B Toi llJIH BHOH Mepe pa3BBTo B mKOJIe TIB«paropa. TaKme
HecoMHeHHo, qTO TeopeMbI 0 C.BOHCTBax nO)];06HbIX <I>Hryp IIMeJIHCL
B «Ha1JaJIax» rBnnOKpaTa XBoccKoro (cepeABHa V B. )];0 H. a.),
KOToporo ITPOKJI, mHBillUH B A<pIIHax B cepeABHe V B., 01JeBBAHO onll-
paHCb Ha AaBHlO1O Tpa)];BD;BIO, Ha3hIBaeT aBTopOM nepBoro CIICTeMa-
TB1JeCKOrO Kypca reOMeTpHH, - OT nOCJIe)];HerO )];0 Hac )];omeJI JIBlIIh
He60JIbmOH OTphIBOK 0 KBa)];paType JIYH01JeK B B3JIOmeHBH CHMnJIn-
KRH 113 RBJIBKBB (Ha1JaJIO VI B.).
2.
T eopuR
napaJlJleJl/b1-lbtX
00 E(3'K,Jluoa
9
CBCTeMaTHqeCKaH pa3pa60TKa reOMeTpRR, Ha,n; KOTOpOH pa60Tan
enblR pH,n; rpeqeCKRX yqeHbIX V-IV BB. ,n;o H. 3., BKnlOqaJJa H pa3BR-
TBe TeopHH napaJIJIeJIbHbIX, B BbI,n;eJIeHRe Tex OCHOBHbIX rrOJIOiKeHR:U,
KOTopbIe ,n;OJIa\HbI 6bIJIB CJIymHTb OTrrpaBHbIMH npH nOCTpoeHHH ,n;e,n;YK-
TBBHOH HayqHOH TeopRII, 06 ylO CTPYKTYPY KOTOpOH BCCJIe OBaJI
II OnHCaJI AplICTOTeJIb (384-322 ,n;o H. 3.). MbI 3HaeM, 6JI ro,n;apH
IIpoKJIY H HeKoTopbIM APyrlIM n03,n;HeHllI1IM aBTOpa1\I, BMeHa HeCKOJIh-
KIIX TaKHX yqeHbIX H rro pH,n;y npHMblx IIJIB KOCBeHHbIX YKa3aHR:U
MOiKeM C 60.ITbllIeH RJIII MeHbmeii BepoHTHoCTblO On;eHRTb BKJIa,n; He-
KOTOpbIX B3 HBX, KaK, HanpBMep" TeaTeTa A<pHHCKoro, EB,n;OKca
RHII,n;CKoro (nepBaH nOJIOBIIHa IV B. ,n;o H. 3.) B em;e HeMHorHX ,n;py-
rHX. K COmaJIeHHIO, MbI BOllce He OCBe,n;OMJIeHbI 0 nOCJIe,n;OBaTeJIbHOM
paSBBTBII TeopHH rrapaJIJIeJIbHbIX II 0 ee COCTOHHRII ,n;o EBKJIH,n;a MO-
iI eM Cy,n;HTb TOJIhKO no OT,n;eJIbHbIM TeKCTaM ApI1CTOTeJIH, KOTopbIe
BoepBble BeCbMa nonp06Ho II rJIy60Ko H3yqlIJJ M. TOT [66]. 8TII
TeKCTbI pa36pocaHbI no pa3JIlIqHbI 1 COqIIHeHRHM, nOCBHm;eHHbIM pa3-
JIllqHbIM BonpocaM <pIIJIOC0<pHII, JIOrRKH R eCTeCTBeHHblX HaYK, II He
COCTaBJIHIOT e,n;RHoro n;eJIoro, Hepe,n;KO HBJIHHCb TOJIbKO reOMeTpB-
qeCKBMB npRMepaMR, HJIJIIOCTPIIPYIOII\IIMU Te IIJIR HHble nOJIO ReHIIH
ApBCTOTenH. TeM He MeHee B CBoeH cOBoKynHocTH BbICKa3bIBaHIIH
AplICTOTenH n03BonHIOT OTqaCTR BOCC03,n;aTb COCTOHHHe TeoplIU na-
paJIneJIbHbIX B" anoxy, Henocpe,n;CTBeHHO npe,n;llIeCTBOBaBllIYIO HanB-
caHBIO «HaqaJI» EBKJIH,n;a. MbI no,n;qepKHeM CJIOBO «0 TqaCTH»:
CJIe yeT HMeTb B BB,n;y, Bo-nepBbIX, qTO TeKCTbI, 0 KOTOpbIX R,n;eT peqh,
OTPbIBOqHbI; qTO, ,n;aJIee, <ppaahI IIHor,n;a ,n;OllIJIB HerrOJIHOCTblO H IIepe-
BO,n; HX ,n;onYCKaeT pa3JIBqHhIe BCTOJIKOBaHRH 1; HaKOHen;, qTO ApIICTO-
TeJIh, KOTOpblH He 6hIJI reOMeTpOM, HO JIRllIh MaTeMaTRqeCKII 06pa30-
BaHHbIM MblCJIBTeJIeM, BOBce He BMeJI n;eJIbIO n03HaKOMRTh qRTaTeJIH
C COBpeMeHHbIM eMY nOJIO ReHlIeM TeoplIH napaJIJIeJIhHhIX.
TIOCKOJIbKY caMa np06JIeMa ,n;oeBKJIB,n;oBOH TeopBB napaJIJIeJIbHblX
He HBJIHeTCH npe,n;MeTOM HallieH KHBrB, MbI He 6y,n;eM IJ;HTlIpOBaTb MHO-
rOQRCJIeHHble MeCTa, aHaJIB3 KOTOpbIX npOB3BeJI M. TOT, a aaTeM II
npyrHe IICTOpHKII MaTeMaTBKH, H OrpaHBqlIMCH TeM, qTO HaM He06-
XO,n;BMO. MOiKHO nOJIHOCTblO COrJIaCBTbCH C 06 IIM BhIBO,n;OM H. TOTa,
qTO,n;O .pOPMYJIHPOBKB nHToro nOCTYJIaTa 0 napaJIJIeJIbHhIX B «HaqaJIax»
EBKJIH,n;a 6hIJIH npe,n;npBHHThI nonhITKR ,n;OKaa8Tb aTOT nOCTYJIa T IIJIH
HeKOTopoe paBHOCBJIbHOe eMY npe,n;JIOiKeHBe cpe,n;CTBaMII a6COJIIOTHOH
reOMeTpRH B CMhICJIe BOHII. IIpB aTOM B031\IOiKHO, qTO nepBble nonhITKB
BMeJIB xapaKTep npHMhIX ,n;OKa3aTeJIbCTB, a nOCJIe Toro, KaK y6e,n;H-
JIRCb B BX Hey,n;aqe, 6hIJIB npe,n;npBHHThI nonbITKII HenpHMoro ,n;OKa-
3aTeJIbCTB.a nYTeM npRBe,n;eHBH K HeJIenOCTH Tex HJIU IfHbIX ,n;orrYlI\e-
HlIii, rrpOTIlB0110JIOmHbIX IIHTOMY IIOCTYJIaTY HJIII paBHOCB JIbHbIM elVIY
1 TOJIbKO B KaqeCTBe IIpHMepa YKameM Ha pa3JIHqHOe HCTOJIKOBaHlle O,IJ;HOrO MeCTa
B COqHHeHHH A pHCTOTeJIH «0 He6e» M. TOToM, ROTOphlll: BHP;HT B HeM HeKOTopoe
IIp"e,IJ;JIOmeHHe HeeBKJIH,IJ;OBOH reOMeTpHU, H T. XHTOM, m. JI. CTOKCOM II
iK. TpHKO, IIpe,IJ;JIaraIO HMH COBeprneHHO ,IJ;pyroH nepeBO,IJ; H 1l0HHMaHlle 3Toro
me TeKCTa [66, c. 317 -323].
10
npe,nJI()meHIIHM. B xone aTHX IICCJIenOBaHHH 6bIJI o6cym,neH Bonpoc
o CYMMe yrJIOB nJIOCKOrO TpeyrOJIbHHKa II paCCMOTpeHbI HeKOTopble
CJIe,n;CTBBH, BbITeKalOm;:oe H3 ,nonym;eHRH, qTO 3Ta CYMMa 60JIbme
HJIB me MeHbllIe ,n;BYX npHMbIX. HaCKOJIbKO ,n;aJIeKO npo,n;BIIHYJIBCb
p;peBHBe B aTOM HanpaBJIeHBB, Cy,n;BTb C YBepeHHOCTblO Tpy,n;HO; BO
BCHKOM CJIyqae, HeT OCHOBaHBH npe,n;nOJIaraTb, qTO ,n;peBHBe 6bIJIH
6JIB3KB K C03,n;aHBIO TOH HJIH HHOH HeeBKJIH,n;OBOH reOMeTpHqeCKOB
CHCTeMbI. C ,n;pyroH CTOpOHLI, MomHO YBepeHHO CKa3aTb, qTO em;e BO
BpeMeHa ApHCTOTeJIH, a MomeT 6bITb II ,n;o Hero, npB pa3pa6oTRe aK-
CHOMaTHKH eBKJIB,n;OBOH reOMeTpBB BeJIHCb nOBCKB IIall60JIee no,n;xo-
)J;HmeH aKCHOMLI napaJIJIeJIbHbIX.
B03MomHO, qTO O,nBH H3 BapBaHTOB TaKOH aRCIIOMbI npllBeJJ;eH
6bIJI caMHM ApBCTOTeJIeM; BO BCHKOM CJIyqae, 3TO YTBepm,n;aeT OMap
XaHHM. B yme ynoMHHaBIIIBXCH «ROMMeHTapHHX K Tpy,nHOCTHM BO
BBe,neHHHX KHBrB EBI\JIBJJ;a» XaiHl\1 nBCaJI, qTO «npHqHHa OIDH6KB
II03]J;HeHIIIIIX yqeHbIX B ]J;OKa3a TeJIbCTBe aToH npe,nnOCbIJIKB 1 COCTOHT B
TOM, qTO OHH He yqBTbIBaJIH npBH lInOB, 3aHMCTBOBaHHbIX y <pHJIOCO-
<Pa», T. e. ApHCTOTeJIH [32, c. 119], 11 3aTeM npHBo,n;BT nHTb TaKHX
npBHIJ;BnOB:
1. «BeJIHqHHLI MomHO ,neJIBTb ,nO 6eCKOHeqHOCTH, T. e. OHH He
COCTOHT 113 He,n;eJIBMbIX)}.
2. «TIpHMYIO JIHHBIO MomHO npo,nOJImaTb ,no 6ecKoHH HOCTH».
3. «BCHKHe ,n;Be nepeceKalO BeCH npHMbIe JIIIHHH paCKpbIBalOTCH
H pacxo,n;HTCH no Mepe y,n;aJIeHBH OT BepIIIBHbI yrJIa nepeCeqeHHH)}.
4. « Be Cxo,nH BeCH npHMLIe JIHHHH nepeceKalOTCH, H HeBOSMomHO,
qTo6bI ]J;Be Cxo,n;H BeCH npHMble JIHHBII pacxo,nBJIBCb B HanpaBJIeHBH
cxom]J;eHIIH)} .
5. «Hs ,nBYX HepaBHbIX OrpaHHqeHHbIX BeJIHqBH MeHbllIYIO MomHO
B3HTb c TaKOH KpaTHOCTblO, qTO OHa npeB30H,neT 60JIbmylO)} [32,
c. 119-120].
TpeM B3 IJ;BTHpOBaHHbIX npBH BnOB COOTBeTCTBYIOT B ,n;ome,n;IlIIIX
,n;o Hac COqHHeHBHX ApBCTOTeJIH Tom,n;eCTBeHHbIe no cO]J;epmaHHIO,
XOTH H HeCKOJIbKO HHaqe C<POPMYJIHpOBaHHble npe)J;JIOmeHIIH. MMeHHO,
npHHIJ;BnbI 1 B 2 BblCKasaHbI no cymeCTBY B CJIe,nYIO BX YTBepm]J;eHHHX
apBCTOTeJIeBOH «<DB3HKB»: « JIBHa B BpeMH H Boo6me Bce HenpepbIB-
Hoe HaSLIBaeTCH 6eCKOHH'IHbIM B ]J;BOHKOM CMbICJIe: :UJIH B OTHomeHBH
,n;eJIeHBH, HJID B OTHomeHBD KOHD;OB» [2, T. 3, c. 183]. «HeBosMomHo,
1ITo6bI qTO-JIB6o HenpepbIBHoe COCTOHJIO 113 He,n;eJIHMbIX [1IacTei],
HaIIpBMep JIBHBB B3 TOqeK, eCJIB JIBHBH HeIIpepbIBHa, a TOqKa He,n;eJIH-
Ma» [2, T. 3, c. 179]. TIHTbli npBHIJ;Bn, BOCXO]J;HmBH, nO-BB]J;BMOMY,
K EB,n;OKCY RHB,n;CKOMY, HeCKOJIbKO nO-HHOMY C.pOPMYJIHpOBaHHbIH
B «HaqaJIax» EBKJIB,na, a saTeM ApxBMe,n;oM (287?-212 ,no H. a.),
TaKme, 6-e3 CO IHeHBH, 6bIJI B3BeCTeH ApBCTOTeJIIO, KOTOpLIH B TOH
me «<DB3BKe)} nBCaJI: «BCHKYIO KOHeqHYIO BeJIHqBHY [Bcer,n;a] MomHO
IICqepnaTb JII060H oIIpe,n;eJIeHHOi BeJIllqBHOH)} [2, T. 3, c. 118l. Tpe-
TBH IIpRHIJ;Hn C«POPMYJIBpOBaH ApBCTOTeJIeM B ero TpaKTaTe «0 He-
! «(STOM npe)J;lloChIJIKoii» XaihIM Ha3hIBaeT V nOCTYJIaT EBRJIII)J;a.
11
6e»: «qTO TeJlO, nBHmYJ:I(eeCH no Kpyry, no He06xOAHl\10CTH AOJImHO
6hITb KOHe1JHbIM BO BceM CBoeM 06'beMe - 3TO HCHO H3 CJIeAYIOJ:I(ero...
ECJlH 'TeJIO, nBHmy eecH no Kpyry, 6eCKOHe1JHO, TO JIHHHH, npOBe-
eHHbIe H3 n;eHTpa, [ T. e. paAHYChI], 6YAYT TaKme 6eCKOHe1JHbI.
A eCJlH OHH 6eCKOBe1JHhI, TO H npOMen-\YTOK MeiHAY HHMH 6eCKOHe1JeH»
r2, T. 3, c. 275]; 8TO YTBepmAeHHe ApHCTOTeJlH IJ;HTHpyeT TIpOKJI
(OK. 410-485) B CBoeM AOKa3aTeJIbCTBe V nOCTYJIaTa, 0 KOTOpOM MhI
6YJ(eM rOBopHTb HHiKe. MhI He 3Hae I, BameJl JIll XaBHM 1JeTBepThIH
npHHD;Hn B KaKOM-TO He AomeAllIe -I .a;o Hac TpYAe ApHCTOTeJlH IIJlH
me B3HJI ero H3 BTOphIX p'YK. MbI em;e BepHel\IC.H K 8TOMY rrpeJ(JlOiKe-
BHIO, COCTO.Hm;eMY H3 ABYX YTBepiRneHHii, KaiKAoe H3 KOTOpbIX paBHO-
CHJIbHO V nOCTYJlaTY EBKJlHJ(a. B naHHbIH MOMeHT CYJ:I(eCTBeHHO 3a-
MeTHTb, 1JTO «1JeTBepThIH npHHIJ;Hn» OCTaeTC.H AJIH Hac nOKa eAHHCTBeH-
BOH <POPMYJlHpOBKOH aKCHOMbI napaJlJleJlbHbIX, KOTOpylO MbI MomeM,
eCJID J(OBepHTb HCT01JHHKaM Xaii.HMa, OTHeCT:n K AOeBKJIHAOBOH MaTe-
MaTHRe. BCJIen 3aTeM EBKJIHA C.pOPMYJlHpOBaJI aKCHOMY napaJlJleJlb-
HblX B .popMe YiKe D;HTHpOBaHHoro V nOCTYJIaTa, If YlKe AOBOJlbHO
CKOpO 8Ta <POPMYJlHpOBKa, CYJ:I(eCTBeHHO YCTynalOm;aH. J(PyrHM nOCTY-
JIaTaM H aKCHOMaM «Ha1JaJl» B OTHomeHHH CBoeH HarJIHJ(HOCTH :n npoc-
TOTbI, nopoJJ;IIJla nonbITKH nOCTpOHTb nJlaHHMeTpHIO Ha J(PyroB OCHOBe
II KaR-JIH60 AOKa3aTb nOCTYJlaT, nepeBeAH ero B KaTeropHIO TeopeM.
HaCROJIbKO H3Bec.THO, nepBbIM CO -
1JHHeHHeM, nOCBH eHHbIM 8TOMY
Bonpocy, nOHBHBmHMCH Bcero
1Jepe3 HeCKOJlbKO AeCHTKOB neT
nocJle «Ha1JaJl» EBKJlHAa, .6:bIJI
OAHH He J(omeJ(mHH AO Hac TpaK-
TaT ApXHMena «0 napaJIJleJlbHbIX
JIHHHHX», ynoMHHYTbIB B cnHCKe
C01JHHeHHH ApXIIMeAa, npHBeAeH-
BOM B «RHHre 6H6J1HOr-pa«pHH
HaYK» (RIITa6 aJI-<PHxpHCT aJl-
'YJIYM) A6Y-Jl-<PapaAiKa MyxaM-
MaJJ;a U6H aH-HaAHMa (YM. 993), rJ(e Ha3BaHbI TaKiKe coxpaHHBm:neCH
TOJILRO B apa6cKHx nepeBOnax TPYJ(bI ApxHMeJ(a «JIeMMbI», «0 ceMIl-
yrOJIbHHKe, BnlIcaHHOM B Kpyr», «RHlIra 0 KacalOm;:nXCH KpyraX»,
«KHHra 0 TpeyrOJIbHHKaX», «RHlIra npeJ(nOJlOmeHHH», «RHHra 0 CBOH-
CTBax npHMoyrOJIbHLIX TpeyroJlbHHKOB» H «KHHra 0 BOAHHbIX 1Jaco-
BbIX npH6opax» [50, c. 266]. TOT .paKT, 1JTO meCTb H3 ceMII coxpaHHB-
mRXCH apa6CKHx nepeBoAoB TPYJ(OB ApxHMena npllHaJ(JlemaT Ca6HTY
1I6H Koppe (836-901), n03BOJlHeT npennOJlaraTb, 1JTO TpaKTaT ApxlI-
Mena, Ha3b1BaeMbIH M6H aH-HaJJ;HMoM «KHlIroH 0 napaJIJleJIbHbIX JlH-
'HHfIX» (RHTa6 aJl- Y"fY1' aJl-MYTaBB.3HHiia), TaK}He 6bIJI H3BeCTeH M6H
Koppe II, MOiReT 6bITb, 6bIJI HM nepeBeneH Ba apa6cKHH fI3bIK. IIo-
3TOMY He HCKJII01JeHO, 1JTO OnHH H3 nBYX coxpaHHBmHXCH TpaKTaTOB
H6H ROppbI 0 napaJlJleJIbHblX JlIIHHH , KOTopbIe MbI paccMoTpHM
HHme, 6bIJI HanlIcaH non BJlHHHlIeM TpaKTa Ta A pXHMena. R COmaJleHIIIO,
HII1JerO onpeneJIeHHOrO 0 TeopHH napaJIJleJIbHbIX ApXHl\lena CKa3aTb
B HaCTOHm;ee BpeMH HeJlb3H.
3.
AumU"t1-l/bte
001Ca3ameJlbCmf3a
a1CCUoMbt
napaJlJleJlb1l0CmU
12
HeKOTophIe CBe,n;eHIIH 0 AaJIbBe:nme:n IICTOplIH BOIIpoca coo6 aeT
TIPOKJI, TBep,n;o y6emAeBHbIH B TOM, 1JTO V IIOCTYJIaT ,n;OJImeH 6hITh
lICKJIlOqeH lIa 1JIICJIa nOCTYJIaTOB II npeACTaBJIHeT C060H TeopeMY,
no,n;JIema ylO JJ;OKaaaTeJIhCTBY. TaK, npem,n;e Bcero, no ero CBII,n;eTeJIh-
CTBY, .pIIJIOCO«p, aCTpOHOM II MaTeMaTIIK IIOCII,n;OBHH (135-44? ,n;o
H. a.), ypomeHe ClIplIlI, pa60TaBilllIH B PIIMe, IIpe,n;JIOmHJI OTJIlIqHOe
OT eBKJIIIAoBa oIIpe,n;eJIeHlIe napaJIJIeJIhHbIX KaK aKBII,n;lIcTaHT: «TIa-
paJIJIeJlhHbIMII HaaLIBalOTCH TaKlIe npHMble, KOTopLle Haxo,n;HTcH B
O,n;HOH nJIOCKOCTII, Be C6JIIImalOTCH II Be YAaJIHIOTCH, TaK 1JTO Bce
IIepneHJJ:IIKYJIHpbI, npOBe,n;eHHble 113 TOqeK OAHOH lI3 HIlX K APyroH,
paBHhI Mem,n;y C060H» [58, c. 287]. 8TO onpe,n;eJIeHlIe napaJIJleJIhHhIX
JIIIHIIH, OCHOBaHHoe Ha npe,n;nOJIOmeHIIII 0 TOM, 'ITO reOMeTpllqeCKOe
MeCTO T01JeK nJIOCKOCTlI, paCIIOJIOmeHHbIX no O}];HY CTOPOHY OT npHMOH
Ha paBHOM paCCTOHHlI1I OT Bee, HBJIHeTCH npHMOH JIIIHlIeH, .paKTlIqeC-
KII COJ{epil IIT'.YTBepmJJ:eHlIe, paBHOCIIJIhHOe aKCIIOMe 0 napaJIJIeJIhHbIX;
Ha rlInep60JIlI'IeCKOi nJIOCKOCTII, B KOTOpOH 8Ta aKCIIOMa He BblnOJI-
HHeTCH, TaKoe reOMeTplIqeCKOe MeCTO HBJlHeTCH KplIBOH JlIIHlleH.
AKCHOM:Y 0 napaJlJleJIhHbIX MomBO JlerKO BbIBeCTlI KaK CJle,n;CTBHe II3
3Toro onpe,n;eJIeHIIH napaJlJleJIbHhIX JlIIHIIH.
TIpOKJI co06 aeT II 0 }];pyroM ,n;OKaaaTeJlhCTBe V nOCTYJlaTa,
IIpHHa,n;Jlema eM aHaMeHIITOMY aCTpOHOMY RJlaB,n;1I1O II TOJleMelO
(oK.90 - OK. 165), HanlIcaBilleMY, no CJIOBaM IIp KJla, eJIYIO KHHry
06 3TOl\{ nonpoce. ROMMeHTIIPYH npe}];JlOmeHlIe 29 KHlIrlI I «HaqaJl»
EBKJlII a, B KOTOpOH BnepBLIe npHMeHHeTCH V nOCTYJlaT, H 3aMeqaH,
'ITO 8TO rrpe,n;JlOmeHlIe HBJIHeTCH 06paTHLIM K ,n;BYM npeAbIJJ;Y IIM
npe,n;JlOmeHlIHM [58, c. 307 H 310], TIpOKJI npHBo,n;IIT IIOIIbITKY TIToJle-
MeH ,n;OKa3aTb 8TO IIpe)J;JlOmeHlIe, He onlIpaHCh Ha V nOCTYJlaT.
COrJlaCHO IIpoKJlY llTOJleMei paccym,n;aJl CJle,n;YIO IIM 06paaOM: «II pH-
l\IaH, nepeceKalO aH napaJlJIeJlII, He06xo,n;IIMO ,n;OJliRHa 06paaOBhIBaTL
BHYTpeHHlIe O,n;HOCTopOHHHe yrJlbI, IIJllI paBHble ,IJ;BYM npHMbIl\{, lIJIII
l\IeHhilllIe, IIJlII 60JIbilllIe lIX. IIycTb AB II CD napaJlJleJIhHhI II C HHMlI
BCTpeqaeTCH GF (pIIC. 1). TorAa H YTBepm,n;alO, 'ITO OHII o6paaYIOT
BHYTpeHHlIe O,IJ;HOCTopOHHlIe yrJIbI, He 60JlbmHe ,IJ;BYX npHMblx. ECJIH
yrJlLI AFGII CGF 60Jlbille,n;BYX IIpHMbIX, TO JJ;pyrlIe ,n;Ba yrJla MeHbme
,n;BYX npHMbIX. Ho, TaK KaK AF II CG Be 60JIee napaJIJIeJIbHhI, 1JeM
FB II GD, 8TlI ,n;Ba yrJIa TaRme 60JIhme ABYX rrpHMbIX» [58, c. 421].
KaK BIIAHO, TITOJleMeH, npe nOJIaraH, qTO !{Be AaHHble napaJIJIeJIhHble
npHMbIe o6paaYlOT C TpeTbeH npHMoii ,n;Ba yrJIa, B cYMMe 60JIhmlIe
}];BYX npHMhIx yrJlOB, HeHBBO npe,n;nOJIaraeT, qTO 3TO ,n;OJlmHO HMeTh
l\feCTO nJIH JII06bIX ,n;BYX IIapaJIJIeJlhHbIX npHMbIx, B TOM 1JIICJIe II ,n;JIH
Tex me }];BYX rrpHMhIX ITO ,n;pyrylO CTOPOHY OT ceKy eH. aJIee TIPOKJI
nplIBo,n;IIT CJle,n;YIOID;He CJlOBa IIToJIeMeH: «IIo3TOMY eCJIII npHMaH,
BCTpeqalOm;aHCH C .4F H CG, 06pa3yeT BHYTpeHHlIe yrJlbI, 60JIhIIIIIe
,n;BYX npHMbIX, TO 3TO 6y,n;eT IIMeTb MeCTO II ,n;JIH npHMoii, BCTpeqalO-
eiiCH C FB H GD, HO OHII TaKme MeHbme ,n;BYX npHMbIX, TaR KaK
yr,JIbI AFG, CGF, BFG H DGF COCTaBJIHIOT RMeCTe 1JeThIpe npH-
l\fbIX. Ho 3TO HeBoaMomHO. TaRiKe MhI nORameM, "liTO npHMaH,
nepeceRalOm;aH IIapaJlJleJIII, o6paayeT BHYTpeHHHe O,n;HOCTopoHHlIe
yrJIbI, He MeHhme ,n;BYX npHMhIX. Ho eCJIII 8TII yrJIbI He 60Jlhme
18
A F B K
/ c
C & Jl
Puc.l Puc. 2
A
A F
K
8
-
En Hn 0
-
JJ
r; l C
C
Puc.3 Puc.4
E
"-. F A
A B H
"'- G K
C .D C F 1l
Puc.S Puc. 6
II He MeHbme )];BYX npfI1\fbIX, TO OCTaeTCH, 1JTO ceRy afI o6paayeT
BHYTpeHHHe O)];HOCTOpoHHHe yrJ1LI, paBH:ble )];BYMfI npfIM I f» [58,
c. 421-]. « oHaaaB» 6e3 nOMO H V nOCTYJIaTa YTBepm)];eHne npe)];JIO-
meHHfI 29, II TOJIel\feii B:bIBO)];HT 113 Hero V nOCTYJIaT CJIe)];YIO IIM 06-
paaOM: «nOCJIe Toro, KaK 3TO )];OHa3aHO, Tpe6yeMoe )];OKaaa TeJIbCTBO
npOBO)];HTCfI aBTOMaTH1JeCKII. MT8R, fI YTBepiHnalO, 1JTO eCJIU IIpH faH
Ce1JeT nBe npHM:bIe TaH, 1JTO 06paaYIOTCH BBYTpeHHHe OnHOCTOpOH-
BHe yr JIbI MeHhIIIe AB yx npHM:bJX, TO npfIl\fbIe npH npOnO.rrmeHIIII
BCTpeTHTCfI C Toii CTOpOH:bI, C KOTOpoii yrJIbI l\feHbme )];BYX npH1\fbIX.
TIpennOJIOmHl\f, 1JTO OHn Be BCTpe1JaIOTCfI. Ho eCJIH OHH He BCTpe"tJa-
IOTCH C TOll CTOpOHbI, C KOTOpOB yrJIbI MeHbme nBYX npHMbIX, TO Tel\1
60JIee OAn He BCTpe1JaIOTCH C ,n;pyroii CTOpOHbI, C KOTOpoii yr T}bI
60JLbme ,n;BYX npH1\IbIX. TIo3TOl\fY npHM:bIe Be BCTpe"tJaIOTCfI HH C Toil,
14
HII C ,n;pyroii CTOpOHLI, II, CJIe,n;OBaTeJIbHO, OHII napaJIJIeJIbHbI. Ho
6bIJIO AOKaaaHO, qTO eCJIII npHMaH nepeCeKaeT napaJIJIeJIH, TO BHYTpeH-
Bile O,n;HOCTopOHHHe yrJILI paBHbI BYM npHMLIM. TaKIIM 06pa30M, 3TII
yrJIhI [OAHOBpeMeHHO] paBHbI ,n;BYM npHMbIM II MeHbille ,n;BYX npHMbIX,
a 3TO HeBoaMomHO» [58, c. 421-422], aJIee TITOJIeMeii Cne lfaJIbHO
OCTaHaBJIIIBaeTCH Ha TOM, nOqeMY npHMLIe BCTpeqalOTCH C Toii CTO-
pOHbI, C KOTOpoii BHYTpeHHlIe OJJ;HOCTopOHHHe yrJIbI MeHbille ,n;BYX
npHMbIX: OH npe,n;nOJIaraeT, qTO OHII nepeceKalOTCH C JJ:pyroii CTOpO-
HLI (pIIC. 2), II nOJIyqaeT TpeyrOJIbHHR, CYMMa yrJIOB KOToporo 60JIb-
me ABYX npHMLIX. CaM 3TOT <paKT TITOJIeMeH He 6ecnOKOHT, HO OR
aaMeqaeT, qTO eCJIII CYMMa yrJIOB AFG II CGF MeHbIIIe ,n;BYX npH-
MbIX, TO, TaK KaK CMeiRHble yrJIbI AFG II GFB B CYMMe paBHbI
ABYM npHMbIM, yrOJI CGF MeHbllIe yrJIa GFB, II, CJIeAOBaTeJIbHO,
....
8HeillHIIH yrOJI TpeyrOJIbHIIKa MeHbllIe HaKpeCTJIemam;ero BHYTpeH-
Hero, qTO HeB03MOiRHO. TI03TOMY, TaK KaK ,n;OKaaaHO, qTO ,n;Be rrpH-
Mble BCTpeqaIOTCH, OHH BCTpeqalOTCH co CTOpOHbI yrJIOB AFG II CGF,
MeHbilllIX JJ:BYX rrpHMbIX.
YKaaaB Ha OillH6KY IlToJIeMeH, TIpOKJI nplIBo,n;IIT Tal\me paCCYiH,n;e-
HIIH lie Ha3BaHHbIX HM rpeqeCl\lIX MaTeMaTIIKOB, COrJIaCHO KOTOphIM
,n;Be npHMble, nepeCeqeHHbIe TpeTbeii, He MorYT BCTpeTHTbCH II B TOM
CJIyqae, Kor,IJ;a 3TII npHMble o6paaYlOT C ceKym;eii ,n;Ba yrJIa, MeHbIIIlIe
]J;BYX npHMbIX. IIpORJI rrepeAaeT If X CJIOBa CJIe,n;YIOII(IIM 06paaOM: « Be
npHMLIe AB II CD, II AC, KOTopaH nepeceKaeT IIX II o6paayeT C HIIMII
BHYTpeHHlIe yrJIhI, MeHbilllIe ,n;BYX npHMbIX (pIIC. 3), II OHII ,n;YMaIOT,
qTO MorYT ,n;OKaaaTb, qTO AB II CD He BCTpeqaIOTCH. AC ,n;eJIIITCH
nOnOJIaM B TOqKe E, II Ha npHMo:ij AB OTKJIa;l{hIBaeTCH OTpeaOK AF,
paBHhIii AE, a Ha CD OTpeaOK CG, paBHhIii EC. TorAa HCHO, qTO
AF II CG He nepeceKalOTCH B TOqKaX F II G, TaK KaK eCJIII 6bI 3TO 6hIJIO
TaR, TO ,n;Be CTOpOHbI TpeyrOJIbHIIKa 6bIJIII 6bI paBHbI CTopOHe A C, qTO
HeBoaMomHO. aJIee npOBO,IJ;IITCH coeAIIHIITeJIbHaH JIIIHHH FG, ,n;eJIIIT-
CH nOnOJIaM B TOql\e H, II OTKJIa,n;bIBaIOTCH paBHble OTpe31\1I F K ,
paBHblii F H, II GL, paBHhIii HG. 11 8TII OTpeaKII Ha TOl'tI me OCHOBaHHH
He MorYT nepeCeqbCH. aJIee, npo,n;OJIiRaH AO 6eCKOHeqHOCTH coe,n;lI-
HHTb HeCOBrra,IJ;alO He TOqKII, ,n;eJIIITb nOnOJIaM Coe,IJ;HHIITeJIbHble JIII-
HIIII II OTKJIa,n;bIBaTb OTpe3KII, paBHble IIX nOJIOBIIHaM, OHII YTBepm,n;a-
lOT, qTO ,n;OK8aaJIII, qTO rrpHMble AB II CD HHKor,n;a He BCTpeTHTCH.
TaK rOBopHT OHH» [58, c. 423-4241. TIpOKJI onpOBepraeT 3TO YTBepm-
;n;eHHe, coe,n;IIHHH TOqKII A If G Ha pllC. 4 JIIIHlIeii II nOKa3hIBaH, qTO
yrJIhI CAG II ACG, MeHbllIlIe yrJIOB CAF II ACG, nOJJ:aBHO MeHbille
,n;BYX npHMhIX, a.npHMble AG II CG nepeceKalOTCH B TOqKe G. 3aMeTIIM,
qTO, npOBo,n;H npHMble AB II CD TaKIIM 06paaoM, qTO yrOJI ACG
6JIIIaOK 1\ npHMOMY, a yrOJI CAB ;O;OCTaTOqHO MaJI, JIerKO nOJIyqlITb
TaKoii cJIyqaii, Kor,n;a npHMhIe AB II CD nepeceKalOTCH yme Ha rrep-
BOM mary. O,n;HaI\O paccymAeHlle IITHpyeMbIx TIpOKJIOM reOMeTpOB,
rrO-BII,IJ;IIMOMY, rrpeJJ;nOJIaraeT paBeHCTBO yrJIOB CAB II A CD, Kal\
3TO lIa06pameHo Ha plIC. 3. B aTOM CJIyqae TaKme JIerKO nplIBeCTII
nplIMep BCTpeqalOm;HXCH npHMhIX AB II CD, o,n;HaKO rrpo ecc, 0 KOTO-
pOM rOBoplIT TIPOKJI, MomeT ;n;eiiCTBHTeJIbHO 6bITb npO;n;OJImeH ,n;o
6eCKOHeqHOCTH: npH 3TOM OTKJIa,n;hIBaeMhIe OTpeaKH CTaHOBHTCH Bce
16
MeHhme II MeHhme II BX ,n;JIHHbI HBJIHIOTCH lIJIeHaMII cxo,n;H erOCH pHAa,
CYMMa KOToporo paBHa paCCTOHHlllO OT TOlIeK A II C ,n;o TOlIKII nepe-
CelIeHIIH. 3TO paccym,n;eHBe BeChMa 6JIH3KO K B3BeCTHOMY paccym;n;e-
HBIO 0 TOM, lITO 6hICTpOHorBH AXIIJIJIec He MomeT ,n;orHaTh Me;a;JIIITe Tlh-
HOH lIepenaXII: IIpo ecc, lIrYPIlPYIO IIH B paccym;a;eHBB 3eHOHa,
TaKm.e He ,n;aeT B03MomHOCTII )J;OCTBlIh TOlIKII, B KOTOpOH AXIIJIJIec
AorOHIIT lIepenaxy.
B CBH3II C npllBe;a;eHHhIM 11M paccymAeHlleM IIpOKJI 3aMelIaeT, lITO
«XOT.H MHorlle 113BeCTHhIe npHMbIe, BhIxo,n;H Be nOJ]; yrJIaMB, IeHhmJI-
:MII BYX npHMhIX, nepeceKalOTCH, OCTaeTCH IICCJIeJ];OBaTh, 1Il\leeT JIll
3TO MeCTO BO Bcex CJIYlIaHX. II03TOMY MomHO 6hIJIO 6LI YTBepm,n;aTh,
lITO eCJIB YMeHhmeHHe ,n;BYX npHMhIX HeOrpaHHlIeHHO, TO cy ecTByeT
TaKaH CTeneHh MaJIOCTII, Kor;a;a npHMhIe He BCTpelIaIOTCH, a npH J];PY-
rOH, e e MeHhmeH; CTeneHII OHH BCTpelIaIOTCH» [58, c. 424-425].
TaKBM o6pa30M, 3;a;eCh J];onYCHaeTCH, qTO J];Be npHMhIe l\lorYT He
nepeCelIhCH II B CJIyqae, Kor;a;a CYMMa BHYTpeHHIIX O,n;HOCTOpOHHIIX
yrJIOB npH IIX nepeCelIeHBB C TpeThen npHMOH MeHhme J];BYX npH fbIX,
lITO UMeer MeCTO TOJIhKO B rlln p6oJIBlIeCKOH reOMeTpHH.
O,n;HaKO caM TIPOKJI, npBBe;a;H 3TO paccym;a;eHlle, COBepmeHHO He-
COBMeCTBMoe C V nOCTYJIaTOM, B03Bpa aeTCH K reOMeTpHII, B KOTO-
POH BhInOJIHHeTCH V nocTYJIaT, II npHBo,n;BT co6cTBeHHoe ,n;OKa3aTeJIh-
CTBO 8Toro nocTYJIaTa. TIpOKJI OCHOBhIBaeTCH Ha ynoMHHYTOl\f Ha III
YTBepmJ];eHBB ApBCTOTeJIH 0 TOM, lITO eCJIB 113 TOlIKII BhIXO,n;HT J];Be
npHMhIe, TO npH HX HeOrpaHBlIeHHOl't1 npo,n;OJImeHIIB paCCTOHHlle Meffi-
,n;y HIIMB CTaHOBIITCH 60JIhme JII060H KOHelIHOH BeJIBlIBHhI. «II PHHH-
MaH 3TO,- npo,n;OJIiKaeT IIPOKJI,- H YTBepm,n;alO, q1:0 eCJIII npHl\faH
nepeceKaeT O;a;HY 113 napaJIJIeJIhHbIX IIpHMhIX, TO OHa nepeceKaeT H
,n;PyrylO. TIYCTb napaJIJIeJIII - AB II CD, a EFG IIepeceKaeT AB
(pHC. 5). Tor;a;a BF H FG - J];Be npHl\fble, BbIXOJ];H He 1:13 TOlIKU F
H MorY He 6hITh npO;a;OJImeHHhIMB ,n;o 6eCKOHelIHOCTII. rr03TO IY pac-
CTOHHHe Memny HIIMH 60JIbme BCHKOH p;pyroii BeJIBlIIIHhI, n03To IY OHO
60JIhme paCCTOHHHH Mem;a;y IIapaJIJIeJIHMII. .II03TOMY OHH OTCTOHT
;a;pyr OT ;a;pyra ,n;aJIbme, "tJ:eM paCCTOHHHe Mem,n;y napaJIJIeJIH}IB, TaK
lITO FG nepeceKaeT BCD» [58, c. 425]. aJIee IIpoKJI ,n;OKa3hIBaeT
V nOCTYJIaT: «IIycTb ,n;Be npHMhIe - AB II CD, II EF IIepeceKaeT HX
TaKHM o6pa30M, lITO yrJILI BEF II DF E, o6pa3yeMLIe elO, :MeHbllIe
J];BYX npHMbIX (pIlC. 6). Tor,n;a H YTBepm;a;alO,lITO 8TII npHl\lbIe BCTpe-
lIalOTCH C TOH CTOpOHhI, C KOTOpOH yrJILI MeHhme ,n;BYX npHl\lbIX.
TaK KaH yrJIhI BEF II DFE l\1 Hhme ,n;BYX npHMbIX, TO nYCTh yrOJI
HEB. - IIX ,n;onOJIHeHUe ,n;o JJ;BYX npHMhIX. TIpo;a;OJIiKBM Tenepb HE
,n;o K. Tor,n;a EF nepeceKaeT KH B CD TaKBM o6pa30M, lITO BHYTpeH-
HHe yrJIbI HEF II DFE paBHhI ,n;BYM npHMhIl\f, TaR lITO npH}lhIe HK
H CD napaJIJIeJIbHbI. Ho AB nepeceKaeT KH, n03TOMY B CII:IY npu-
Be,n;eHHoro ,n;OHa3aTeJIbCTBa OHa nepeceKaeT II CD. TaKBl'tI 06pa30l\I,
AB II CD nepeceKalOTCH C TOH CTOpOHbI, C KOTOpOH yrJIhI MeHhme ;a;BYX
npHMhIX. TIocTaBJIeHHaH u;eJIh, TaKBl\I o6paaoM, ,n;OCTHrHYTa» [58,
c. 425-426].
3,n;ech TIpORJI HeHBHO IIOJIb3yeTcH npe;a;IIOJIOffieHHel\l, lITO pac.CTOH-
HHe l\IelliAY ;n;BYMH HenepeceKaIO UMBCR npHl\IbIl\fH Ha nJIOCROCTB, KO-
16
TopLIe OH B COOTBeTCTBHH C onpe,n;eJIeHlIeM EBKJI:n,n;a Ha3:bIBaeT napaJl-
JleJlbHLIMH, KOHe1JHO; 3TO YTBepm,n;eHlIe paBHOCHJlbHO V nOCTYJIaTY
EBKJIHAa, H B rHnep60JlH1JeCKOii reOMeTplIH, B KOTOpOD 3TOT nOCTYJIaT
He BbInOJlHfIeTCH, cym;eCTBYIOT HeilepeceKalOm;:neCH npfIMLIe - TaK Ha-
3bIBaeMbIe paCXOAHm;HeCfI, paCCTOHHHH Mem,n;y KOTOpLIMH MomeT 6bITb
c,n;enaHO 60JlbllIe JII060ii KOHe1JHOH BeJlII1JHHbI.
M3 He ,n;OllIe,n;llIHX ,n;o Hac aHTH1JHLIX TeopHii napaJlJIeJlbHbIX OTMeTHM
TeoplIlI lIo,n;opa (I B. ,n;o H. a.) II AHTIIHllaTa (?), ynoMIIHaHlIe CII}fn-
JlHKIIH 0 KOTOpbIX COXpaHHJlOCb B IIepe,n;a1Je aH-HaiiplI31I (CM. paa,n;. 4).
B03MomHo, 1JTO Hl\feHHO 8THX MaTeM:aTIIKOB IIMeJI B BHlJ;y TIPOKJI.
B apa6cKoM 1I3J10meHIIH coxpa-
HIIJlHCb TaKiRe ,n;Be IIonLITKH ,n;o-
Ka3aTeJlbCTBa V nOCTYJlaTa, npH-
Ha,n;JIemam;lIe B:n3aHTHBCKHM Y1Je-
HLIM V -VI BB. AraH:ncy :n CH In-
JIIIKIIIO. M. Rypu;e II 11. 31. reii-
6epr, Ha,n;aBilllIe Jla THHCKHii II
apa6cKIIii TeKCTbI KOMMeHTapHeB
aH-HaiipH31I K EBKJlJI,n;y [37, 47],
OTom,n;eCTBJlfIJlH AraHHca C IaTe-
l\IaTHKOM 1 B. ,n;o H. a. reMIIHOl\I,
rrpOTIIB 1Jero B03pamaJi fl. TaH-
HepII [65]. r. E. TIeTpocfIH
BLICKa3aJI IIpeJJ;IIOJlOmeHlIe, 1JTO
AraHlIc - rpe1JeCKaH TpaHCKpHnU;IIH 1:Il\leHII apMHHCKoro IIpOCBeTHTe-
JIfI II 3HaTOKa rpe1JecKoii KYJIbTYPLI AraHa, Y1JHTeJIH 1I3BeCTHbIX ap-
IfIHCK:nX Y1JeHbIX BTOpOH nOJIOBIIHLI V B. raaapa <PapneU;1I II Bara-
Ha MaMHROHHHa [19, c. 153-154]. oKa3aTeJIbCTBO AraBHca coxpa-
HHJIOCb B II3JIOiHeHHII CHMIIJIHKHH, Ha3LIBaBmero ero CBOHM ,n;pyroM;
aal\-IeTHM, 1JTO CHMllJIHKIIH 6bIJI Y1JeHHKOM al\faCKHH, 6bIBmero B
CBOIO 01JepeAb Y1JeIrnKOM TIpOKJIa. M3JIOmeHlle CIIMIIJIIIKHfI npIIBe,n;e-
HO B KOMMeHTapHHX R «Ha1JaJIaM» EBKJIH,n;a IIpaHCKOro MaTeMaTHKa
X B. A6Y-JI-A66aca aJI-<Pa,n;JIa :n6H XaTIIMa aH-HaiipH3H. CooTBeT-
CTBYIOID;Rii paa,n;eJI KOl\IMeHTapHeB aH-HaiipH3H OaarJIaBJIeH «TIpe,n;-
IIOChIJiKH H npe,n;JlOmeHHH CHl\-IIIJIIIK:nH H AraHHca, He06xo,n;HMhIe ,n;JIH
,n;Ba,n;u;a Tb ,n;eBHToro npe,n;JIOmeH:nH nepBoii HHlIrll». MbI npHBe,n;eJ\.1 era
TeKCT nOJIHOCTblO, TaK HaK B HeM :nMeIOTCH paccym,n;eHRfI H IIpe,n;JIO-
meHHH, Heo,n;HOKpaTHO BCTpe1JalOm;lIeCfI B n03,n;HeHllIeH, B TOM 1JnC;rIe
apa6CKOH, JIHTepaType:
«B ,n;oKa3aTeJIbCTBe rrpe,n;JIOiI eHlIfI 29 KHHrll 11 IICnOJIb3yeTcfI llpeJ];-
nOCLIJIKa: BCHKlIe ,n;Be npHl\IbIe, rrpOBe,n;eHHbIe no,n; yrJIal\llI, MeHbIIIHl\III
,D;BYX npfIMLIX, BCTpe1JaIOTCH. 3TO He o6menp:nHfIToe YTBepiI ,D;eHue.
CHMnJInKHH CKaaaJI 06 aTOM: aTOT nOCTYJIaT He 01JeBII,n;eH, HO He06-
XO,D;Ifl\1 ,n;JIH ,n:oKa3aTeJIbCTB C nOMo blO JIHHHii. AHTIIHHaT II ,I:(IIO,n;Op
,D;oRa3aJIIf 0 Hel\f l\IHOrO pa3JIII1JHLIX npe,n;JIOmeHlIii. n TOJIeMeii TaKme
,D;ORaabIBaJI ero, B ero ,D;oKa3aTe TIbCTBe IICnOJIb30BaJI:nCb IIpe,D;JIOmeHRH
4.
BU3a1-tmUl C1Cue
001Ca3ameJlbCm6a,
COXpa1-tU8mUeCR
8 apa6c1COJW
U3JlOJj(,e1-tUU
!- l\'1w 3aMeHHe 1 3,n;eCb JI ,n;aJIee CJIOBeCHYIO HY lepa 1l1O npe)J;JIOmeHlIll 11 RH:nf
«(HaqaJI» II4>pal\lII.
17
13, 15 II 16 KHlIr:n I «HaqaJI». O)];HaKO 3TO HeB03MOfHHO, TaK KaK
EBKJIII)]; nOJIb30BaJICH 3TIIM nOCTYJIaTOM TOJIbKO B npe;O;JIOrKeHlI1I 29
3TOll KHlIr:n, YTBepm)];eHBe KOTOpOH onBpaeTCH Ha YTBepm)];eHlIe 0
TOM, qTO, eCJIII )];Be npHMble npOBO;O;HTCH nO)]; ;o;BYMH npHMbIMB yrJIaMII,
OHII napaJIJIeJIbHbI, a eCJIII OH:n npOBO;O;HTCH no)]; yrJIaMB, MeHbllI:nMB
;o;BYX npHMbIX, OHB BCTpeqaIOTCH» [19, c. 154]. 0 nOKa3aTeJIbCTBaX
Ano;o;opa II AHTBHHaTa MbI He 3HaeM HlIqerO, KpOMe ynoMHHaeMoro B
1 IITlIpyeMoM TeKCTe. «qTO KacaeTCH Hamero ;o;pyra AraHHca,- npo-
;O;OJImaeT aH-Ha:HpII3:n npHBO;O;lITb CJIOBa CIIMrrJI:n BH,- TO OH He
BIIJ{HT, 3aqeM CTaBIITb 3TO YTBepm)];eHlIe B KaqeCTBe nOCTYJIaTa, TaK
KaK OHO Hym)];aeTCH B )];OKa3aTeJIbCTBe. OH nOJIb30BaJICH ;o;pyrlIl\III
npe;O;JIOmeHIIHMII BMeCTO npe;O;JIOmeHIIH, IIMelOm;IIXCH B «HaqaJIax»,
)];JIH Toro, lIT06bI )];OKaaaTb npe;O;JIOmeHlIe 29 6e3 3Toro YTBepm)];eHIIH.
3aTel\I, no ero MHeHIIIO, OH ;O;OKa3aJI 3TOT nOCTYJIaT reOMeTp:nqeCKIIM
CIIoc06oM. BOT ero CJIOBa B ero [c06cTBeHHbIx] BblpameH:nHx.
AraHlIc CKa3aJI: COrJIaCHO HameMY 06em;aHIIIO 1I3JIOmlIM ;o;oKa3a-
TeJIbCTBO 3Toro IIocTYJIaTa, COCTOHm;ero B TOM, qTO, eCJIII )];Be JIIIHIIII
npOBe)];eHbI no;o; yrJIaMB, MeHbllIIIMII )];BYX npHMbIX, OHH BCTpeqaIOTCH.
8TO )];OKa3bIBaeTcH, HaCKOJIbKO B03MOfI\HO, C rroMoID;blO reoMeTplI-
qeCKOrO ;o;oKa3aTeJIbCTBa. HeKOTopble noplIIJ;alOT reOl\feTpOB 3a 3TO II
rOBopHT: eCJIII BLI Tpe6yeTe, "tJT06bI MbI COrJIaCIIJIIICb C TeM, qTO 3TO
He ;O;OKa3aHO, nO"tJeMY fRe BbI nOJIb3yeTecb 3TBM npll )];OKa3aTeJIbCTBe
)];pyroro? IIo3TOMY H npoBe;o;y 3TO [)];OKa3aTeJIbCTBo], II, XOTH 3TO
YTBepm;o;eH:ne II BeCbMa BamHO, OHO He Hym aeTcH B TOM, qT06bI pe"tJb
o HeM 6LIJIa ;o;JIIIHHOi.
.H YTBepm;o;alO, "tJTO MbI onpe;o;eJIHeM napaJIJIeJIbHbIe JIIIHIIII KaK Ta-
Rile, KOTopble HaXO)];HTCH Ha O;o;HOH nJlOCKOCTB, :n eCJIII npO)];OJImaTb
IIX 6eCKOHe"tJHO B 06e CTOpOHbI, paCCTOHHlIe Mem)];y HIIMII 6y)];eT O HO
II TO me. PaCCTOHHlIe 3TO - KpaT"tJaHmaH JIIIHHH, COe)];HHHIOID;aH HX,
TO me rOBopHTCH II 0 ;o;pyr:nx paCCTOHHIIHX.
06aBHM aTH npe;O;JIOmeHIIH K KHlIre I «Ha"tJaJl» nOCJIe rrpe;O;JIOfl-\eHIIH
26, TaK "tJTO 8TO npe;O;JIOmeHlIe CTaHeT npe)];JIOmeHlIeM 27» [19,
c. 154-155]. aJIee aH-HaiplI3H npIIBO,IJ;IIT "tJeTblpe npe;O;JIOmeHIIH
AraHlIca. IIepBoe Ba HBX rJIaCBT: «EcJIII ;O;Be JIIIHIIII napaJIJIeJIbHbI,
paCCTOHHlIe Mem;o;y H:nMII nepneH;O;lIKYJIHpHO Kam;O;OH 1I3 HIIX». 3a-
MeTIIM, "tJTO AraHIIC, KaK II Bce ;o;peBH:ne :n cpe;O;HeBeKOBbIe l\IaTeMaTII-
KII, nOHIIMaJl no;o; CJIOBOM «paCCTOHH:ne» OTpe30K, a He qMCJIO, KaK
3TO nOHHMaeTCH B HaCTOHm;ee BpeMH.
IIpo)];OJIlliIlM H3JIOmeHIIe aH-HaipII3B: «IIpIIMep 3Toro - ;o;aHbI ;O;Be
napaJIJIeJIbHLIe JIIIH:nB AB, CD, :n nYCTb paccToHHlIe l\Iem;o;y HIIMH -
EG (pBC. 7). .R YTBepm)];alO, qTO [JIRHBH] EG neprreHJ{IIKYJIHpHa Kafl-\-
;n;OH 113 ;o;BYX JIHHIIH AB, CD.
,lJ,oKa3aTeJIbCTBo. ECJIII EG He rrepneH.n;IIKYJIHpHa JIIIHHHM AB,
CD, yrJIbI nplI To"tJKe E - He npHMble. IIYCTh OCTpbI:H 1I3 HIIX -
yrOJI AEG. OnycTIIM 1I3 TO"tJK:n G neprreH)];HKYJIHp Ha JIIIHIIIO AB,
3TO GH. B CIIJIY .n;OKa3aTeJIbCTBa 19 [npe.n;JIOllieHIIH] I [KHHrlI]
GE )];JlHHHee, "tJeM GH. Ho GE npe;o;rrOJIaraJIaCb KpaT"tJaHrneH npHMOll:
JIIIHlIei Mem.n;y JIIIHIIHMH AB, CD, n03TOMY 3TO Heneno. TaI{IIM 00-
pa30l\1, JIIIHHH EG nepneH;O;lIKYJIHpHa Kam;o;oH 1I3 JIIIHIIH AB, CD.
18
A .H E 8 A E B
\j
C & .D C & .D
Puc.7 Puc.8 H
8TO II eCTb TO, "tJTO 1\1:bI XOTeJIII ]];OKaaa Tb» [19, c. 155]. 3 a 1\1 eTII}f , qTO
B npe]];JIOmeHIIII 19 KHlIrH I «Ha"tJaJI» EBKJIII]];a ]];OKa3aHO, "tJTO B Tp "
yrOJIbHIIKe 60JIbllIIIH yrOJI CTHrlIBaeTCH 60JIbmeH CTOpOHOH.
TI pe]];JIOmeHUe 2 AraHlIca. ECJIH npHMaH JIIIHIIH IIa]];aeT Ha ABe
npHJ\tlbIe JIIIHIIU II nepneH]];HKY JIHpHa Kan-\]];OH 1I3 HIIX, TO JIIIHHII na
paJIJIeJIbHbI, a nepneH]];IIKYJIHp HBJIHeTCn paCCTOHHlIel\1 Me»-\]];y HIIMII.
TI plIJ\tlep aToro - JIIIHIIII AB, CD, Ha HIIX naJ];aeT JIIIHIIH EG,
COCTaBJIHIOm;aH C Kam]];OH 1I3 HIIX npHMble yrJIbI (plIC. 8). .H YTBepm]];alO,
"tJTO JIIIHIIII AB, CD napaJIJIeJIbHbI, a JIIIHIIH EG - paccToHHlIe l\le RLty
HII1\11I .
oKa3aTeJIbCTBo. ECJIII aTII .n;Be JIIIHIIII He napaJIJIeJIbHbI, "tJepea
TO"tJKY G npOXO]];IIT JIIIHIIH, napaJIJIeJIbHaH JIIIHIIII AB. TIYCTb, eCJIH
BOa1\IO»\HO, aTO JIIIHIIH GH. B aTOM CJIy"tJae JIIIHIIH EG HBJIHeTCH pac-
CTOHHlIe1\1: l\Iem.n;y JIIIHIIHl\III AB II GH, TaK KaK OHa - KpaT"tJaiiulaH
1I3 Bcex JIIIHlIii, npOBe.n;eHHbIX 1I3 TO"tJKII G K JIIIHIIH AB. TIoaT01\IY
yrOJI HGE - npHMoii B CIIJIY ]];OKa3aTeJIbCTBa npe]];bI.n;ym;ero npe]];
JIOmeHIIH. IIo rrpe]];noJIaraJIOCb, "tJTO yrOJI DGE- npHMoii, nOaT01\IY
aTO HeJIeno. TaKII1\I 06pa30l\f, JIIIHHII AB , CD napaJIJIeJIbHbI, a JIUHIIH
GE paCCTOHHlIe IeiI,J];Y. HIIMII. 8TO II eCTb TO, "tJTO MbI XOTeJIII ]];O
KaaaTb» [19, c. 155-156]. aJIee CJIe.n;yeT «npe]];JIOmeHlIe 3 AraHlIca..
TI pHMaH JIIIHIfH, npOBe]];eHHaH K ]];BYM napaJIJIeJIbHbIM npHJ\tIbIM JIIf-
HIfHM, o6pa3yeT paBHbIe HaKpeCTJIe»-\am;lfe yrJIbI, BHeII1HlfH yrOJI,.
paBHbIH COOTBeTCTBeHHOMY eMY BHYTpeHHeMY, If BHYTpeHHlfe OJ];HOCTO
pOHHlIe yrJIbI, BMeCTe paBHbIe J];BY I npHMbIM».
BbIme IbI BII]];eJIII, "tJTO aTO - npe]];JIOmeHlfe 29 KHlIrlf I «Ha"tJaJI»
( C I. c. 6).
«TIplf1\Iep aToro,- npoJ];OJI»iaeT aH-Haiipualf,- .n;Be napaJIJIeJIbHble
JIIIHIfIf AB, CD, K HIf1\1 npOBe]];eHa npHMaH EG (plfC. 9). .H YTBepm.n;aIO,
"tJTO yrJIbI, o6paaYIOII lIecH nplf aT01\I, TaKOBbI, Kal\ MbI YKa3aJIIf.
oKaaaTeJIhCTBo. TIpoBe.n;eM 1I3 KaiH]];OM 1I3 To"tJeK E, G paCCTOHHIfH
1\Iem.n;y JIIfHnH IIf-AB, CD"aTOJIIIHIfIfEF, GK. qeTbIpe yrJIa, o6paay-
lOmneCH npII aTO I, npH1\IbIe. TIoaToMY JIIIHIfH EF napaJIJIeJIbHa JIII-
HUll KG B C"JIY .n;OKa3aTeJIbCTBa npeJ];bI.n;ym;ero npe.n;JIOmeHIfH, a .,1JII-
HUH EK napaJIOI1JeJIbHa JIIIHIIII FG. JIIfHIIIf EK If FG - paCCTOHHIfH
Ien'.n;y JIIfHIIH III EF II KG. llOaT01\IY OHII paBHbI. TaKIfM o6pa30 1,
JIUHIIH FG paBHa TlIIHIIII EK, a JIIIHIIH EF - JIIIHIIII GK, II aTII JIIIHIIII
o6paaYlOT paBHbIe yrJIbI. [JoaT01\IY J];Ba TpeyrOJIbHIIKa [FGE II KEG]
paBHbl II OCTaJIbHbIe .n;Ba yrJIa paBHbI OCTaJlbHbIM ]];BYM yrJlal\1. TaKII1\1:
o6paaO!\f, yr.oJl FGE paBeH yrJIY GEK, a aTO HaKpeCTJIemam;lIe yrJIbI,
19
Ho yrOJI FGE paBeH yrJIY HGD, TaK KaK OHII - IIPH nepeCe"tJeHIiII,
B CHJIY ]];OKaaaTeJIbCTBa [npe]];JIOmeHHH] 15 (KHlIrH] I. TI09TOMY yrOJI
GEK paBeH yrJIY HGD, BHemHHH - COOTBeTCTBeHHOMY eMY BHYTpeH-
He:M:Y. TaK KaK MhIAOKaaaJIH, 'liTO HaKpeCTJIema He yrJIbI paBHbI, TO,
eCJIII MbI npH6aBHM K HHM 06 HH yrOJI DGE, 110 JIYlIHTCH , 'liTO ]];Ba
yrJIa FGE, EGD, paBHbIe ]];BYM IIPHMbIM, paBHbI ABYM yrJIaM KEG,
D(JE. TaKIIM o6paaoM, BHYTpeHHHe OAHOCTopOHHHe yrJIbI paBHbI
.n;BY f npHMbIM. 8TO H eCTb TO, "tJTO MbI XOTeJIH]];OKaaaTb» [19, c. 156].
OTMeTHM, "tJTO B [npeJIJIOmeHHH] 15 [KHHrH] I «Ha"tJaJI» AOKaaaHO
paBeHCTBO BepTHKaJIbHhIX yrJIOB. .D:aJIee CJIe]];yeT «npe]];JIOmeHHe 4
AraHHca. ECJIII npHMaH JIHHHH npOBe eHa K )J;BYM npHMbIM JIIIHHHM
II HaKpeCTJIema He yrJIbI, 06paayeMbIe Mem]];y Heii H ]];BYMH JIIIHHHl\IH,
paBHbI, HJIH eCJIH BHemHHH yrOJI paBeH COOTBeTCTBeHHOMY eMY BHYT-
peHHe fY, HJIH BHYTpeHHHe O]];HOCTOpOHHHe yrJIbI paBHbI JJ;BYM npHMbIM,
TO ]];Be npHMbIe napaJIJIeJIbHbI».. 8TO npeAJIOmeHHH 27 H 28 KHHrlI I
«Ha"tJaJI», TaKm.e ynoMHHaBllIHeCH HaMH BblIIIe KaK npe]];JIOmeHIIH,
06paTftLIe npe1(JIOmeHHIO 29 KHHrH I.
«TIpIIMep 9Toro,- npOAOJImaeT aH-HaiipHalI,- JIHHHH AB, CD,
Ha HIIX na]];aeT JIHHHH EG H 06paayeT C HHMII yrJIbI, KaK MbI YKa3aJIII
(pHC. 10). H YTBepmJJ;alO, "tJTO JIIIHHII A B , CD na paJIJIeJIbHbI.
.D:oKa3aTeJIbCTBo. ECJIH JIHHHB EG - nepneH]];HKYJIHp, 01IeBH1(HO,
"tJTO JIHHHH AB, CD napaJIJIeJIbHhI, 9TO ]];OKaaaHO BO BTOpOM lIa 9THX
AOnOJIHHTeJILBbIX npeJJ;JIOmeHHii. ECJIH me EG - He nepneHJJ;HKYJIHp,
onYCTHMHa TOlIKH E Ha JIHHHIO CD nepneHAIIKYJIHp EK. Tor]];a eCJIH
yrOJI E - npHMoii yrOJI, CHOBa 01IeBII]];HO, lITO JIIIHIIII AB, CD napaJI-
JIeJIbHbI, 9TO ,lJ;OKaaaHO BO BTOpOM Ha aTIIX ]];OnOJIHHTeJIbHbIX npe]];JIO-
HeHHii. ECJIH me yrOJI E - He npH fOH yrOJI, BOCCTaHOBIIM B TO"tJKe
E nepneH]];HKYJIHp K JIHHHH EK, KaK AOKaaaHO B ]];OKaaaTeJIbCTBe
{npe]];JIOmeHHH] 11 [KHHrH] I. TIYCTb 9TOT nepneH]];HKYJIHp. EL. Tor]];a
JIHHIIH EL, CD napaJIJIeJIbHbI, KaK AOKaaaHO B TpeTbeM Ha 9THX ]];0-
nOJIHIITeJIbHhIX npe]];JIOmeHHH. TI09TOMY KamAbIH lIa yrJIOB GEB II
GEL paBeH yrJIY CGE, a 9TO HeJIeno. TaKHM 06paaoM, JIHHIIH AB,
CD napaJIJIeJIbHhI. 8TO H eCTb TO, "tJTO MbI XOTeJIH ]];OKaaaTb.
TaKOBbI npe]];JIOmeHHH AraHHca. BbICKaaaB HX, OH nepeXOAHT K
npe]];JIOmeHHIO 31: MhI XOTHM npOBeCTH Ha AaHHoH TOlIKH npHMYIO,
napaJIJIeJIbHYIO ]];aHHOH npH foH; K npeJJ;JIOmeHHIO 32: y nJIOCKHX
.pHryp C napaJIJIeJIbHbIMH CTopOHaMH npOTHBOnOJIOmHbIe CTOpOHbI
paBHbI; K npe]];JIOmeHHIO 33: JIHHHH, napaJIJIeJIbHbIe O]];HOH JIHHMH,
napaJIJIeJIbHbI Mem]];y c060ii; K npeAJIOmeHHIO 34: npHMble JIHH II II ,
COe]];HHHIO He paBHbIe napaJIJIeJIbHbIe JIHHHH, paBHbI H napaJIJIeJIbHbI,
H K npe]];JIo» eHIIIO 35: eCJIH npHMaH JIHHHH na]];aeT Ha ]];Be IIpHMble
TlIIHHH H BHYTpeHHHe OAHOCTopOHHHe yrJIbI MeHbIIIe ]];BYX npHMbIX,
TO 9TH ARe npHMbIe, eCJIH npO]];OJImHTb HX B CTOPOHY yrJIOB, MeHbillHX
]];BYX npHMbIX, BCTpelIaIOTCH» [19, c. 156-157].
llpe]];JIOmeHHH 32-35 AraHHca cy ecTBeHHO OTJIHlIaIOTCH OT HOCH-
HX Te me HOMepa npe]];JIOmeHHH I KHIIrH «HalIaJI», npIIQeM npe,IJ;JIO-
meHHe 35 eCTb, OqeBH]];HO, V nOCTYJIaT (C f. 06 aTOM ]];aJIee).
«TIplIl\Iep 9Toro,- npO]];OJImaeT aH-HaiipH3H H3JIOmeHHe JJ;OKa3a-
TeJIbCTBa AraHHca,- ]];Be npHMbIe JIIIHHH AB, CD, Ha KOTopbIe na,IJ;aeT
20
'"
. .l.,.;
A
E
K
B
A E 8 lJ
C F .D
Puc. 9 H
T
Puc.l0 l I P
/If 0
A
H
Puc.11
C K G .0
npHMaJl JIHHHH EG, rrplI"tJeM BHYTpeHHHe O]];HOCTOpOHHlIe yrJIbI co
CTOpOHbI BD MeHbIIIe ]];BYX npHMbIX (pIIC. 11). .H YTBepmJIalO, "tJTO
JIHHIIII AB, CD BCTpe-qaIOTCH C :3TOH CTOpOHbI.
OKa3aTeJIbCTBO. IIpoBe]];eM "tJepea TO-qKY G JIHHIIIO, rrapaJIJIeJIbHYIO
JIIIHIIII AB, KaK nOKa3aHO EBKJIII]];OM B 1J;OKa3aTeJIbCTBe [npe)];JIome.
HHH] 31 [KHlIr.n] I. IIycTb 9TO JIHHIIH GH. TIpoBe]];eM paCCTOHHlIe Meffi-
JI.Y HHMH B COOTBeTCTBIIH C )];OR3aaTeJILCTBOM [npe;a:JIOmeHIIH] 11 [KHII-
rH] I, :3TO JIHHIIH GE. IIpe]];noJIomHM Ha JIHHHH CD npOlIaBOJIbHYIO
TO-qKY, Ky]];a 6:bI ORa HII nonaJIa, nYCTb :3TO TO"tJKa F. OnYCTHM lIa
TO-qRII F neprreB]];IIKYJIHp Ha JIIIHIIIO GE, KaK ]];OKaaaHO B )];QKa3aTeJIb-
CTBe [rrpe)];JIOmeHHH] 11 [KHlIrH] I, nYCTb :3TO JIHHHH Fl. P a3]];eJIHM
nOnOJIaM JIHHIIIO GE, RaK ]];OKaaaHO B )];OKaaaTeJIbCTBe [rrpe]];JIOmeHHH]
10 [KHlIrH] I. Pa3]];eJIIIM nOrrOJIaM TaKme ee nOJIOBHHY II 6y]];eM nOCTY-
rraTb TaR IIOCTOHHHO )];0 Tex nop, nORa [TOqRa] neJIeHHH He nOrra]];eT
HHme TOqKII I. TI YCTb [To"tJKa] )];eJIeHIIH nOna]];eT B TOlIKY M. O"tJeBH,n;-
HO, QTO TOlIKa M nOna]];eT Ha TY qaCTb JIHBIIH EG, 0 KOTOpOH MbI ro-
BOpIIM. IIpHMeM, HanplI lep, -qTO qaCTb, KOTopaH na]];aeT HHme TO-qKH
1,- lIeTBepTb GE. TIpoBe]IeM "tJepe3 TOqKY M JIIIHIIIO, rrapaJIJIeJIbHYIO
JIHHHHM: GH, AB, KaR ]];OKa3aHO B ;a:OKaaaTeJIbCTBe [npe]];JIOmeHIIH]
31 [KHHrH] I, :3TO JIIIHHH MN. TIpO]];OJIiI\HM JIHHIIIO GD ]];0 6ecKoHe"tJ-
HOCTII II nOCTpOIIM: [JIIIHHIO] GQ, CO)];ep)Ra yIO GN CTOJIbKO me paa,
CKOJIbKO paa EG CO]];epmHT BeJIII-qHHY GM, T. e. -qeTbIpe paaa. .H YT-
Bepm]];alO, "tJTO JIHHIIH AB, CD BCTpe"tJaIOTcH B To"tJRe Q. .D:oKaaaTeJIb-
CTBO. OTJIOmH f Ha JIIIHIIH GQ JIHHIIIO, paBHYIO GN, KaK L\ORa3aHO B
.n;OKaaaTeJIbCTBe [npe)];JIOmeHHH] 3 [KHHrH] I. IIycTb :3TO JIHHIIH N X.
II pOBe]];eM 1I3 TO-qKII X JIIIHHIO, na paJIJIeJIbHYIO GE, :3TO JIHHIIH X S ,
II rrpO]];OJIiI\IIM: JIHHIIIO M N ]];0 TO-qKII O. Tor)];a B TpeyrOJIbHHKaX
GMN, XON ]];Be 1I3 HX CTOpOH paBHbI, :3TO GN II NX, a yrOJI GNAf
paBeH yrJIY ONX, KaK ]];OKaaaHO B ]];OKa3aTeJIbCTBe [npe)];JIOmeHHH]
21
15 [KHnrlI] I, II B CMJIY ;n;OKa3aTeJIbCTBa TpeTbero tIpe,tJ)Io)HeHHH M3
YCTaHOBJIeHHMX AraHlIcoM B aTIIX npe;n;nOCWJIKaX yrOJI AlGlV paBeH
yrJIY NXO, TaR KaK aTO HaKpeCTJIemall{He yrJIhI. B CMJIY ;n;OKaaaTeJIb-
CTBa [npe;n;JIOmeHMH] 26 [KHlIrM] I OCTaJIbHbIe CTOpOHbI [nepBoro Tpe-
yrOJIbHMKa] paBHI>I OCTaJIbHbIM CTopOHaM [BToporo TpeyrOJIbHIIKaJ.,
Kafl\;n;aH CTOpOH3 - cooTBeTcTBeHHoii, M oCTaBlIIMHcH yrOJI paBeH
OCTaBlIIeMYCH yrJIY. TIoaToMY CTopOHa GM paBHa CTopOHe XO. Ho
CTopOHa 08 paBHa CTopOHe GAl KaK npOTMBOnOJIOmHaH CTOpOH3
IIJIOCKOH <PMrYPbI C napaJIJIeJIbHblMII CTOpOHaMfJ;. lloaToMY JIIIHMH
X 8 - y;n;BoeHHaH JIMHIIH GAl. ECJIII npOBeCTII 1I3 TO"tJKH Q JIIIHIIIO,
napaJIJIeJIbHYIO JIMHMHM EG M X 8, II npOBeCTH "tJepe3 TO"tJKY X JIIIHMIO
X T n HanpaBJIeHMM, napaJIJIeJIbHOM AB, OHa BCTpeTMTCH C JIHHlIeH"
npoBe;n;eHHoii M3 TO"tJKII Q napaJIJIeJIbHO JIMHMII EG. TeM caMbIM ]);0-
Ra3aHO, "tJTO OHa OTceRaeT OT Hee JIMHMIO, paBHYIO JIMHHH GT TIpoBe-
;n;e1\.1 ee, aTO JIHHMH PQ. TIoaTOMY JIMHIIH PQ paBHa JIHHIIH TG, TaK HaK
XQ paBHa XG, yrOJI TXG paBeH yrJIY QXP, yrOJI PQX paBeH yrJIY
TGX KaK HaKpeCTJIej-l am;Mif, TaK "tJTO PQ paBHa GT B CIIJIY ;n;OKa3a-
TeJIbCTBa [npe;n;JIOfI-\eHMH] 26 [KHlIrlI] I. Ho GT paBHa T E, n03TOMY
PQ paBHa T E II JIIIHMH A EB BCTpe"tJaeTCH C JIMHlIeH PQ B TO"tJRe Q
B COJIY ;a:OKaaaHHoro AraHMcoM B YCTaHOBJIeHHOM 11M npe;a:JIOfl-\eHMII,"
B KOTOpOM OH YTBepfl-\J{aeT, "tJTO JIMHMII, COe;n;IIHHIOIIJ;Me KOH:U;bI paB-
HbIX II napaJIJIeJIbHbIX JIMHMH, napaJIJIeJIbHbI II paBHbI. TeM caMbIM
;n;OKa3aHO, "tJTO eCJIM npHMaH JIIIHMH na;n;aeT Ha ;n;Be npHMbIe JIIIHMII II
BHYTpeHHMe O;n;HOCTOpOHHlIe yrJIhI MeHbme ;n;BYX npHMbIX yrJIOB, TO
aTII ;n;Be JIMHMM, eCJIII npO;n;OJImMTb IIX B CTOPOHY, B KOTOpoii yr JIbI
leHblIIe ;a:BYX npHJ\tlbIX, BCTpe"tJaIOTCH. 8TO II eCTb TO, "tJTO MbI XOTeJIH
;n;OKa3aTb.
Bce, qTO OH YCTaHOBMJI B aTOM npe;n;JIOmeHHM II B npe;n;lIIeCTBYIOm;IIX
eMY npe;n;nOCbIJIKaX, rrpMHHTO KaH Heo6xo;n;IIMbIH nOCTYJIaT B nepBOH
KHHre, HO npeLJ;CTaBJIHeT C060:H npe;n;JIOmeHIIH, ;n;06aBJIeHHWe AraHM-
COM: K npe;a:JIOn-\eHMH?vI EBKJIM;n;a, M HII B qeM 1I3 C;n;eJIaHHOrO IIM HeT
HMqerO 3aCJIymMBalOm;ero nopMn;aHMH.
CMJ\tlnJIMKMM CKaaaJI: TaKOBbI CJIOBa AraHlIca, II, nomaJIyii, EBKJIM;n;
nOCTaBMJI aTO YTBepi-l\;n;eHMe B Ka"tJeCTBe nocTYJIaTa J{JIH Toro, "tJT06bI
CJ{eJIaTb nYTb 6JIMiRe, qeJ\tI nYTb, oCHoBaHHbIH Ha TOM, "tJTO napaJIJIeJIb-
HbIe JIMHMII - aTO TaKlIe JIIIHMM Ha o;n;Hoii nJIOCKOCTII, "liTO, eCJIU
npO;n;OJImaTb MX B o6e CTOpOHbI, paCCTOHHIIe 1\lem;n;y HJI IH Bcer;a:a
6y;n;cT paBHO. ECJIII 06paTMTb aTY pe"tJb, TO 6y;n;eT Ha060pOT, T. e.
eCJIM JIlIHIIII Ha OJ{HOH nJIOCKOCTM TaKoBbI, "tJTO paCCTOHHIIH MeiR;a:y
HII1-11I Be paBHLI, OHII He napaJIJIeJIbHbI, a TaK KaR OHII He napaJI.rreJIbHbI,
OHH BCTpe"tJaIOTCH. EBKJIM;n; nOJIbaOBaJICH aTIIl\1 YTBepm;n;eHMe 1 B npH-
1\101\1 npe;a:JIOmeHMM M3 YTBepmJJ:eHIIH, npMHHTHe KOTOpbIX He06xo.n.Ml\10,
a 1I1\leHHO, "tJTO aTH JIMHMM TaKOBbI, "tJTO npH IIX npO;n;OJImeHMII [B CTO-
POHY] yrJIOB, MeHblIIIIX ;n;BYX npHMbIX, OHII He coxpaHHIOT O;n;Horo
paCCTOHHIIH II, CJIe;a:OBaTeJIbHO, BCTpeqaIOTCH. O"tJeBII;n;HO, "tJTO HX BCTpe-
"tJa npOlIcxo;a:MT B CTopOHe HaRJIOHa, C ;n;pyroH me CTOpOHhI OHM pac-
XO;a:HTCH, II paCCTOHHIIe :Mem;a:y HM1\III YBeJIMtJIIBaeTCH. TIoaTO!\IY ro-
BOpIITCH, tJTO eCJIII ;a:Be JIIIHMM He napaJIJIeJIbHW, OHM He06xo;a:MMO
BCTpeqaIOTCH, eCJIII TOJIbKO aTO He ;a:yrM, TaR KaK ;n;OKaaaHO, "tJTO KO-
22
IIH eCKlIe ce eHIIH He napaJIJIeJlbIlbI II He nCTpeqalOTCH. EBKJIHJ(
ynoMHHYJI 3TY npe nOCbIJIKY H npHM:eHHJI 3TH npeJJ;JlOmeHJlH, a TaKme
(J(OKa3aJI] YTBepm;n;eHHe, 06paTHoe YKa3aHHOMY npeJJ;JIOiI-\eHHIO: eCJIH
Ha ;a:Be npHMMe JlHHHH na;n;aeT npHMaH JlHHIIH II BHYTpeHHHe yrJIW
paBH@ ABYM npHMMM, OHH napaJIJleJlbHW. ECJlH 3TO npe;n;JlOmeHHe
;a:OKa3bIBaeTCH, TO YTBepm;n;eHlIe TaKme Hym;n;aeTcH B ;n;OKa3aTeJlbCTBe.
MbI H3JIOmHJIH Bce, "tJTO MOiHHO CKa3aTb 0 napaJIJIeJlbHbIX JlHHHHX»
[19, C. 157-1591-
KaK fbI BR;n;IIM, AraHHc HCXOAHT H3 onpe eJleHHH napaJlJleJlbHbIX
IIpH ,fbIX KaK 3KBHJJ;HCTaHTHbIX, npH"tJe I nOJJ; paccToHHHe f Memny na-
paJlJleJlbHbIMH npHMbIMH OH IIOHHMaeT KpaT"tJaHllIHH npHMOJlHHeMHbIH
OTpe30K, coe;n;HHHIO HH 3TH npHMbIe. Q)aKTUqeCKH TaRoe OIIpeJJ;eJle-
HHe napaJlJleJlbHOCTH, KOTopoe MbI yme BCTpeqaJlH Y rrOceii OHHH H
KOTopoe B XVI B. 6bIJIO nplIHHTO Xp. KJIaBHe f (Cl\f. c. 115), npe;n;-
IIOJIaraeT, "tJTO reOMeTplI"tJeCKoe MeCTO To"tJeK IIJlOCKOCTII, paBHO OTCTOH-
m;HX OT JIemameii Ha Heii npHMoM H HaXOnH HXCH no onHY CTOPOHY OT
Hee, eCTb IIpHMaH, a reOMeTpH"tJeCKoe MeCTO Bcex To"tJeK nJIOCKOCTH,
paBHooTcTOH HX OT JIemam;eii Ha Heii IIpHMOH, TaK Ha3blBaeMoii
«3KBHJ(HCTaHTbI rrpHMOM», IIpe):{CTaBJIHeT co6oii napy napaJIJIeJIbHbIX
IIpHMbIX. 8TO YTBepm;n;eHlIe paBHOCHJIbHO «ycHJIeHHOH aKCHOMe napaJI-
JIeJIbHOCTH» (aKcHoMe IV* rHJIb6epTa), B HeeBKJIIIJJ;OBblX reOMeTpHHX
9KBIInHCTaHTbI npHMoH HBJIHIOTCH He npHMbIMH, a KpHBbIMH BToporo
nopH;n;Ka (B 3JIJIHIITII"tJeCKOH nJIOCKOCTH - OKPYiI-\HOCTHMII, B rlIIIep-
60JJHQeCKOH IIJIOCKOCTR - cneU;H.pHqeCKIIMH KpHBbIMH, nepeXOJJ;Hm;n-
MH B ce6H nplI CKOJIbiReHHII npHMOH - «6a3b1 3KBHnHCTaHTbJ)-
B;n;OJIb ce6H). 8TO 06CTOHTeJIbCTBO HBJIHeTCH npHqHHOii HeCOCTOHTeJIb-
HOCTH Bcex nOCJle;n;YIOITJ;HX ;n;eAYKlI;HH AraHHca, Ha"tJIIHaH C ero npen-
JIOmeHHH 1. MbI OCTaBJIHeM B CTopOHe Bonpoc 0 TOM, QTO caMO nOHH-
THe «paCCTOHHHe» Memny ;n;BYMH npHMbIMH JIIIHHHMH nOnJIemHT
TO"tJHOMY MaTeMaTHqeCKOMY onpe;n;eJleHHIO, KOToporo He 6bIJlO H He
MOrJIO 6bITb HH Y AraHlIca, HH Y ;n;PyrHx MaTe faTIIKOB, BnJIOTb AO OT-
KpblTHH HeeBRJIII;n;OBOH reOMeTpHH. ORa3b1BaeMOe AraHIICOM cy ecT-
BOBaHHe "tJeTblpexy"rOJIbHHKa C qeTbIpbMH npHMbI fH yrJIaMII - u;eHT-
paJIbHbIH nYHKT Bcex n03;n;HeHllIHX AOKa3aTeJIbCTB V nOCTYJIaTa, IIHOr-
,I\a aaMeHHeMbIH paBHOCIIJIbHblM e fY ;n;ORa3aTeJIbCTBOM cy ecTBoBaHHH
napaJIJIeJIOrpaMMa. JII06oIIbITHO, "tJTO B rrpe;n;JIOmeHHH 4 AraHHca
BbIBe;n;eHbI npe]IJIOmeHHH 27 H 28 KHHrH I «HaqaJI», ROTopbIe caM EB-
RJIII,IJ; ;n;OKa3b1BaJI, He onHpaHCb Ha aKCHOMY 0 napaJIJIeJIbHblX, Mem;n;y
TeM KaR 3AeCb QHH nOJIY"tJeHbI Ha OCHOBe npe;n;bInY IIX npe;n;JIOmeHHH,
onHpalOmHXCH Ha YTBepm;n;eHHe, l'aBHOCHJIbHOe 9TOH aKCIIOMe. 8TH
:qeTblpe npeL\JIOmeHHH AraHIIC, nO-BH;n;Hl\fOMY, C"tJHTaeT 3aMeHHIOm;II fH
npe;n;JIOmeHIIH 27 -30 KHHrlI I «Ha"tJaJI», TaR KaK ;n;aJIee pe"tJb lI;n;eT
o ero npe;n;JIOmeHIIHX 31-35, H3 KOTOpbIX nepBoe COBIIa;n;aeT C IIpe;n;-
J.lOmeHHeM 31 RHHrH I «HaQ8J1», BTopoe - C nepnoii qaCTblO npeJJ:-
. .omeHHH 34, TpeTbe - C npeAJIOmeHHeM 30, QeTBepToe - C npen-
n:omeHIIeM 33, a nHToe npe,IJ;JlOmeHHe eCT.L V nOCTYJlaT. AraHlIc AO-
Ra3blBaeT 3TOT nOCTYJIaT )];-TIH CJIyqafI, Ror,a;a ceKYII aH nepneH,a;HKYJIHp-
Ha OAHOM 113 ABYX AaHHbIX npHMbIX (B 3TOM CJIY1:Jae nOCTYJlaT rJlaClIT:
.«nepneHAIIKYJlHp H HaKJIOHH8H nepeceKaIOTCR»). AraHHC pac-
23
A
Puc.12
B
Puc.l.3
8
Cl\IaTpMBaeT npH)'Ible AB M CD II CeRY YIO EG, nepneH,n;MRYJIHPHYIO
AB II HaRJIOHHYIO R CD. BbI6paB Ha npHMoH CD npOH3BOJIbHYIO TOqRY
F, OH onYCRaeT M3 Hee nepneH,n;HRYJIHp FI Ha ceRym;yro EG. ,I(aJIee,
ceRymaH EG HeCROJIbRO pa3 ,n;eJIHTCH nOnOJIal\I ,n;o Tex nop, nORa
o,n;Ha H3 TOqeR ,n;eJIeHMH M He ORameTCH M:em,n;y TOqRaMH I II G (H3
'tJepTeme AraHHca TOqRa Ai/ IIOJIyqaeTCH ,n;eJIeHMe f oTpe3Ra EG Ha
'tJeTblpe qaCTM). B TOqRe lVI BOCCTaBJIHeTCH nepneH,n;HRYJIHp M N ,n;o
nepeCeqeHIIH C npHl\IOH CD M CTpOHTCH oTpe30R GQ npHMoH CD,
6QJIbllIMii OTpe3Ra GN BO CTOJIbRO me pa3, BO CROJIbRO OTpe30R EG
60JIbme oTpe3Ra Ai/G. aJIbHeiirnee nocTpoeHHe AraHIIca paBHOCHJIb-
HO nocTpoeHil1O npH IoyrOJIbHHKa co CTopoHon EG H ,n;HarOHaJIbIO
GQ. TIocTpoeHlIe AraHlIca nplll\leHHMO II B CJIyqae, Ror,n;a ceKym;3H
He nepneH;:I;HKYJIHpHa HH K O,n;HOH H3 ]];aHHbIX npHMbIX - B aTOM CJIY-
'tJae npHl\IoyrOJIbHMR CJIe]];yeT 3aMeHMTb napaJIJIeL lOrpaMMoM. TOT aRT,
liTO OTpe30R EG MomHO ]];eJIHTb nOIIOJIaM ,n;o Tex nop, IIORa He DOJIY-
QMTCJI OTpe30R MG, MeHbllIMH OTpe3Ra IG, HBJIHeTCH CJIe,n;CTBHel\l
8RCM01\IbI EB]];ORCa - ApXMMe]];a, ROTopaH y EBRJIH;:I;a nOHBJIHeTCJl
JIMllib B V RHHre «HaqaJI» B RaqeCTBe OIIpeJJ;eJIeHHH 4.
AH-HaiipH3M He npHBO;:I;HT ]];ORa3aTeJIbCTBa aKCMOMbI 0 napaJIJIeJIh-
HbIX, IIpIIHa]];JIemaru;ero Cal\IOMY CMMIIJIMRHIO, aaMeqaH TOJI bKO, qTO
COrJIaCHO CMl\fnJIHRMIO, «eCJIH JIHHHH Ha nJIOCROCTH T3KOBbI, YITO pac-
CTOHHHH :Mei£,n;y HMMH HepaBHbI, OHM He napaJIJIeJIbHbI» [19, c. 159].
8TO ]];Onym.eHHe ;:I;OBOJIbHO 6JIH3RO R qeTBepTOMY IIPHHII;HIIY, npMIIM-
CbIBae IOMY ApM.cTOTeJIIO XaiiHl\IO 1 (CM. c. 11). HpaTRoe II3JIOmeHIIe-
TpaRTOBRM TeopMH napaJIJIeJIbHbIX CMMllJIHRHeM npHBe]];eHO B nBCbMe
AJI81t1 a]];- MH3 Haiicapa aJI-XaHa.pM (1178-12'58) K HacMp a,n;- HHY
aT- TYCH (1201-1274), RonMH ROToporo B 60JIbillHHCTBe c.TIyqaeB
npHcoe,n;MHHJIaCb -K CnnCRa}f TpaKTaT3 aT- TYCH 0 napaJIJIeJIbHbIX JIH-
HIIHX, 0 ROTOP01{ lbI 6y]];el\1 rOBopHTb HHme. AJI-XaHa.p1I nMC3JI:
«113 Toro, qTO CJIeAyeT npe]];JIOiHMTb BarneMY BbICOROl\IY BHMl\IaHHIO,-
TO" qTO BCTpeTII:IOCb IHe B npe;:I;JIOmeHHH, ynoMHHYTOM CHl\lllJIBRHe 1
B ero RO I leHTapMHX R BBe,n;eHMHl\I RHHrn «HaqaJI» Cpe]];H npe,n;nOCbIJIOR
R MaBeCTHO IY YTBepa\]];eHMIO, a Hl\IeHHO: eCJIH npHl\IaH .JIHHHH na]];aeT
Ha ]];Be npH}[ble JIHHMII H o6paayeT ;:I;Ba BHYTpeHHHe OJJ;HOCTopoHHBe
yrJIa, IeHbllIlIe JJ:BYX npH lbIX yrJIOB, TO 3TH JJ;Be JIHHHH, eC:I1I npo-
,n;OJImHTb I1X B aTY CTOPOHY, BCTpeTHTCH. OH rOBOpHJI, "liTO ;D;JIH Kam-
,n;oro yr JIa M IeeTCH 6eCROHe"llHO MHoro XOP)J;, O]];Ha 113 KOTOpbIX 60JIb-
2 1
me .n;pyroH, Ram));aH H3 KOTOphIX OTceKaeT Ha ]];BYX JIHHHHX, orpaHH-
qlIBalO IIX yrOJI, paBHhIe [JIHHHH). OH IIOJIbsyeTcH 3THM CJIe,nYIOII(Hl\1
06pa30l\1: eCJIH JIHHHH AB na]];aeT Ha ));Be JIHHHH BD H AC H yrOJi
CAB - npHl\rOH, a yrOJI ABD - OCTphIH (pIIC. 12), TO ,nBe JIHHHH
AG Y II BD BCTpeTHTCH co CTOpOHhI C H D. JIH 3Toro OH CTpOIlT npH
TOQKe B JIHHHH AB yrOJI ABG, paBHhIH yrJIY ABD. Tor]];a HMeeTCH
6eCKOHeqHO MHoro XOP));, OJ];Ha H3 KOTOphIX 60JIbme ));pyroH, CTHrHBalO-
mnx 3TOT yrOJI, H o.n;Ha JI3 8THX Xopn nonanaeT BHe TOqKH A, KaK
xop.n;a GED. II03TOMY ]];Ba yrJIa A H E - npHl\lhIe, II eCJIH npO]];OJI-
;I\HTb JIHHHIO A C, OHa He BCTpeTHT JIHHHIO DEG. II03TOl\IY OHa BCTpe-
THT JIHHIIIO BD» [60, c. 8-10 II 19-20J.
TaKIIl\1 06pa30l\1, COrJIaCHO CHl\fnJIHKHIO H3 Toro, qTO B npHMOJIU-
HeHHOM yrJIe MomHO npOBeCTH 6ecKoHe"tJHo MHoro «XOP]];», OTceKalO-
HX Ha ero CTopOHax paBHble OTpe3KH, CJIe]];yeT, qTO ]];JIH BCHKOH
BHYTpeHHeii TO"tJKH yrJla cy eCTBYIOT npOBe]];eHHhIe TaKHl\{ 06pa30M
«XOP]];hI», IIOna]];alOmHe «BHe» 3TOH TO"tJKH. Tenepb MhI 3Hael\{, "tJTO B
rHnep60JIII"tJeCKOH reOMeTpHH TaKme MomHO npOBeCTH 6eCKOHeqHO
IHoro «XOP]];» npHMOJIHHe:UHoro yrJIa, OTceKalO HX Ha ero CTopOHax
paBHhle OTpe3KH, HO cyw;eCTBYIOT TaKHe BHYTpeHHHe T01JKH yrJla, 1\0-
TophIe HaxoAHTCH «BHe» 3THX «XOP)J;»,- TaKHMH TOqKal\{H HBJIHIOTCH
BHYTpeHHIIe TO"tJKH yrJIa, JIema He no ]];pyrylO CTOPOHY npHMO:U, lla-
paJIJIeJIbHOH B Cl\lhICJIe JI06a"tJeBCKoro 06eHM CTopOHaM yrJIa. II03TOMY
npe.n;nOJlOmeHlIe CHl\fnJIHKHH paBHOCIIJIbHO V IIOCTYJIaTY EBKJIH]];a.
A. M. Ca6-pa o6HapymHJI B BO]];JIeHHCKOH 6H6JIHOTeKe OKC<pop,lJ;CKoro
YHHBepCH.TeTa pYRonHCb KOMMeHTaplleB K- EBKJIH]];y COTpy,nHHKa
MaparHHCI\OH 06cepBaTopHH aT-TYCH MYXbH a]];- HHa aJl-MarpII611
(YM. OK. 1290), co.n;epma ylO ));OKa3aTeJIbCTBO aKCHOMbI 0 napaJI-
,,'TJeJIbHbIX, 6JIH3KOe no H));ee K npHBe,lJ;eHHOMY paCCYlli]];eHHIO CIIMIIJIH"
KHH. OHO COCTOHT 113 qeTblpeX npe]];JlOmeHHH. B nepBOM IJ;OKa3hIBaeT-
CH YTBepm]];eHHe CHMnJlHKHH 0 cy ecTBoBaHHH 6eCKOHeQHO MHorHX
«XOP]];» yrJIa (3TH XOP));bI HBJIHIOTCH XOp]];aMH ]];yr, onHChlBaeMhIX Ha]];-
JIema Hl\rH pa.n;HycaMH H3 BepmHHhI yrJIa, pHC. 13). BTopoe rrpe));JlO-
meHHe COBna]];aeT c npe,lJ;JIOmeHHeM CHMnJIHKHH, npHBe));eHHhIM aJI-
XaHa.pH. HaKoHeu;, TpeTbH H lIeTBepTaH TeopeMbI npenCTaBJIHIOT co-
60H ,lJ;OKa3aTeJIbCTBa nOCTYJIaTa ,lJ;JIH CJIyqaeB, KOr]];a ceKymaH na,lJ;aeT
Ha ,lJ;Be ]];aHHhIe npHMble no)); ]];BYMH OCTpbll\fH yrJIaMH II no]]; BYMH
yrJIaMH, O]];HH 1I3 KOTOpblX OCTphIH, a ));pyroH TynoH. A. M. Ca6pa
CqHTaeT IJ;Ba nepBblX H3 3THX rrpe]];JIOmeHH:U HeCOMHeHHO npHHa,n;JIe-
iRa IIMH CHl\fllJIHKHIO, nOCJle)J;HHe ]];Ba npe]];JlOmeHIIH, B03MOiKHO,
npHHa]];JIemaT caMOMY MYXbH a]];- HHY aJl-MarpH6H HJIII p;pyrOMY
aBTOpy XIII B. K 3TO IY Bonpocy MhI BepHel\fCH HHfHe.
25
«HaqaJIa» EBRJIH]];a 6bIJIII Bnep-
BbIe nepeBe]];eHbI Ha apa6CKIIH
H3bIK aJI-Xa]];mma.n;)ReM 1I6H IOcy-
<pOM 1I6H MaTapOM. COrJIaCHO
61I6JIlIorpa<py X B. M6H aH-Ha-
.n;HMY aJI Xa]];mma]];m nepeBeJI
«HaqaJIa» ]];BaiR]];bI: nOJIHOCTblO
nplI XapYHe ap-PallIlI]];e, rrpa-
BIIBmeM C 786 no 809 r ., II B
COKpa eHHOM BapHaHTe npll aJI-
Ma:MYHe, 6bIBllel\I XaJIII<p0M B 813-833 rr. M Y}Re BCKope HaqaJIOCb
IIHTeHCIIBHoe KOMMeHTlIpOBaHlIe H )J;aJIbHeHllIee pa3BIITIIe «HaqaJI»,
II B qaCTHOCTII KplITllqeCKIIH aHaJJII3 rrpe.n;JIOmeHHbIX B aTOM Tpy]];e
OCHOBaHIIH MaTeMaTIIKH - aKCIIOM II orrpe]];eJIeHIIH reOMeTpHII (I KHlIra
«HaqaJI»), TeoplIlI OTHOllIeHHH II nporropn;HH (V KHHra) II TeoplIlI
KBa]];paTllqHbIX IIppan;IIOHaJIbHOCTeH (X KHlIra). Bo Bcex aTIIX Ha-
npaBJIeHIIHX yqeHbIe apa6CKHX CTpaH ]];OCTHrJIII 3HaqlITeJIbHbIX ycne-
XOB: MbI 3]];eCb OrpaHlIqlIMCH TeoplIeH napaJIJIeJIbHbIX.
TIepBaH 1I3BeCTHaH HaM nonbITKa ]];OKa3aTb aKCIIOMY 0 napaJIJIeJIb-
HbIX B apa6CKOH MaTeMaTlIqeCKOH JIIITepaType 6bIJIa c,n:eJIaHa aJI-' A6-
6aoo1\1: 1I6H Ca'H)J;OM aJI-,r(mayxapH, pa60TaBllIIM B Ear)J;a)J;e B nepBOH
nOJIOBIIHe IX B. II, no HeKOTopbIM YKa3aHHHM, HBJIHBIlIHMCH ypomeH-
n;e1\-I <1>apa6a (OTpapa) B HbIHelliHeM IOmHoM HaaaxcTaHe. aTbI pom-
,n:eHIIH II CMepTII aJI- mayxapH HelI-3BeCTHbI; MbI 3HaeM JIHllb, qTO B
829-833 rr. OH nplIHIIMaJI yqaCTlIe B aCTpOHOMHqeCKHX Ha6JIIO)J;eHIIHx,
npOBO]];IIBIlIIIXCH B Bar.n;a.n;e II aMacKe. OH KOMMeHTHpOBaJI KaK
nepBYIO, TaK II nHTYIO KHHrlI «HaqaJI». I1CCJIe]];OBaHlIe ero 6bIJIO
03arJIaBJIeHO «Y COBepmeHCTBOBaHlIe KHHrH "HaqaJI"» (McJIa aJI-KII-
Ta6 aJI-YC;YJI). 8TO COqHHeHlIe nOKa He o6HapymeHo n;eJIIIKOM, HO 0
co,n:epma eiiCH B HeM TeoplIlI napaJIJIeJIbHbIX MbI MomeM Cy,n:IITb no
1I3JIOmeHIIIO B ynoMIIHaBmeMCH HaMII TpaKTaTe HaclIp a;n;-,r(IIHa
aT-TYCII 0 napaJIJIeJIbHbIX JIIIHIIHX. «qTO RacaeTCH aJI-,r(mayxapH, ]];a
nOMIIJIyeT ero AJIJIax,- nlIcaJI aT-TYCII,- TO eMY nplIHa)J;JIemlIT
« Y COBepllIeHCTBOBaHlIe KHlIrlI "HaqaJIa"», r)J;e OH BHec IICnpaBJIeHIIH BO
Bce npe]];nOCbIJIKII II npe]];JIOmeHIIH KHlIrH. B ero CJIOBaX no 3TOMY
Bonpocy HMeJICH npIIHn;lIn: eCJIII ]];JIH BCHKIIX )J;BYX pa3JIIIqHbIX JIIIHIIH
OT 60JIee ]];JIIIHHOH OTHIIMaTb nOJIOBIIHY [KOpOTKOH], 3aTeM OTHIiMaTb
nOJIOBIIHY nOJIOBIIHbI MHoro pa3, TO qlICJIO pa3 MomeT 6bITb TaKHM, qTO
OT nOJIOBIIHbI 60JIee nJIIIHHOH OCTaHeTCH [JIIIHIIH] KOpOqe 60JIee KOpOTKOH
JIHHIIII. TIepBoe 113 ero meCTII npe.n;JIOmeHHH OH HaSbIBaeT 28-M [npe.n;-
JIOmeHlleM I KHHrH «HaqaJI»]. B nepBOM H3 npe]];JIOmeHIIH OH 113JIO-
ffiHJI TO, qTO aBTOp «HaqaJI» H3JIOmHJl B [npe]];JIOmeHHII] 27, II ,n:06a-
BHJI K aTOl\IY [eru;e O.n;HO YTBepm,n;eHHe], n CJIe]];Hee 1I3 aTllX npe.n;JIO-
meHlIll [OH Ha3bIBaeT] 33-M. pyroe .n;o6aBJIeHHOe 11M npe,n;JIOmeHHe -
npe.n;JIOmeHHe 23 «HaqaJI», r.n;e YKa3bIBaeTCH, qTO eCJIII 113
BCHKOH TOqKII npOBeCTH B pa3HbIe CTOpOHbI TPH npHMbIe JIHHHH, TO
TPII nOJIyqeHHbIX yrJIa Bl\IeCTe paBHbI qeTbIpeM npHMbIM. IIIecTb yno-
YHHYTMX npe,n;JIOmeHIIH ,n;o6aBJIHIDTCH K «HaqaJIaM» nOCJIe npe]];JIO-
rneHIIH 27» [28, c. 501].
5.
T eOpZtR
napaJlJleJlb1-lblX
aJl- A*ayxapu
26
«IIpHHIJ;lIn», C<POPMYJIIIpOBaHHbIH aJI- mayxapH, COBna aeT C npeJ];-
JlO)ReHIIeM: 1 KHlIrlI X «Ha"IIaJI» EBKJIH.n;a, }];OKa3bIBaeMbIM C nOMO blO
onpe.n;eJIeHHH 4 KHHrH V, T. e. aKCHOMbI EB.n;oKca-ApxHM:eJJ:a.
Ao6aBJIeHHOe aJI-AmayxapH npe}];JIOmeHHe 23 RHIIrII I «Ha"IIaJI», nO-BH-
H fOMY, BBO}];IIJIOCb HM nepe.n; npe.n;JIOmeHHeM 23 EBKJIII a, T. e. aa}];aqeii
o nOCTpoeHHM Ha AaHHO:fI npHl\fOH npH aHHoii TOQKe yrJIa, paBHoro
.n;aHHOMY. B CBHaH C 9TIIM npeJ(JIOmeHHH 23-26 EBKJIII a
y aJI-,L(i-KayxapH O"Ra3b1BaIOTCH IIpe.n;JIOmeHIIH fH 24-27. llIecTb HO-
BbIX npe.n;JIOmeH _H, KOTopbIe aJI-AmayxapH BCTaBJIHJI nOCJIe rrpe.n;JIO-
meHHH 26 «HaqaJI», COrJIaCHO aT- TYCH TaKOBbI:
«IIpe.n;JIOJI eHIIe 28 "HaqaJI" B ero aR3e frrJIHpe, co epllia eM meCTb
yrro fHHYTbIX npe JIOmeHIIH. ECJIH IIpH faH JIHHHH na.n;aeT Ha }];Be
npHMbIe JIHHIIH, HanpHMep JlHHHH HF na aeT Ha JIIIHIIH AB 11 CD
(pHC. 14), II IIPH aTOM o6paaYlOTCH paBHbIe yrJlbI AHF II HFD, TO
JIHHHH AB H CD napaJlJIeJIblIbI, a eCJlH OHH napaJIJIeJlbHbI., TO pac-
CTOHHHe Ha)R OH TOqKH JIHHHII AB OT COOTBeTCTBeHHOH TO"IIRII JIIIHIIH
CD O HO H TO me, T. e. paCCTOHHHe OT nepBoii TOqKH JlHHIIII AB .n;o
nepBOH TO"IIKII JIHHHH CD paBHO paCCTOHHIIIO [OT BTOpOH TOqKII
JlIIHIIH AB }];O] BTOpOH TO"IIKH JIHHIIH CD, H TaROBbI me paccToHHHH OT
TpeTbeH TOqKH }];O TpeTbeH II OT qeTBepToH TO"IIRH .n;o "lIeTBepToH.
YRa3aHHbIe yrJIbI - HaKpeCTJIemaII:{Me yrJlbI JIHHHH AB II CD.
ECJIH npO OJImlITb IIX B o6e CTOpOHbI, OHH He nepeceKYTCH, TaK
KaK eCJIH OHH nepeceKYTCH B TO"IIKe K, TO yroJl A H F, BHemHHH yrOJI
TpeyrOJIbHHKa H F K, paBeH BHYTpeHHeMY yrJIY H F K, a 9TO HeJIenO,
KaK .n;OKa3aHO B npe.n;JIOmeHHH 16. TOqHO TaK me ORa3bIBaeTCH, "lITO
OHH He rrepeceKalOTCH II C .n;pyroH CTOpOHbI. IIoaToMY JIHHHH AB II
CD na paJIJIeJIbHbI.
.H YTBepm.n;alO, "lITO paCCTOHHlIe OT Ra}l\ oii TOqRH JlHHHH AB o
cooTBeTCTBYIOII:{eii TO"IIKH JIHHIIH CD - O HO H TO me paCCTOHHHe.
AOKaaaTeJIbCTBO. YrJIbI AHF H FHB [B feCTe] paBHbI }];BYM npH-
MbI f, KaK }];OKa3aHO B IIpe JIOmeHIIU 13 KHHrll I. YrJIbI CFH H HFD
TaKme [BMecTe] paBHbI .n;BYM npHMbI I. YroJI AH F no IIpe IIOJIOn,e-
HIIIO paBeH [yrJIY H]FD. IIoaTo fY yrOJI FHB paBeH yrJIY CFH.
OTJIOmlI f paBHble [JIIIHHH] FQ H HO H npOBe.n;e f JIIIHHII QH II OF.
Tor.n;a JlHHHII OH H H F paBHbI JIHHHHM QF H F H II yrOJI F HO IIO rrpe -
x
F
IJ
A
F
E
K
B
Puc.15 C
A
o
H
L
B
c
Puc.14
27
nOJIOmeHlI1O paBeH yrJIY QFH. II03TOMY OCHOBaHHe FO paBHO OCHOB3
HIIIO HQ II BCHKIIH yrOJI paBeH COOTBeTCTBYIO eMY, KaK )];OKa3aHO B
npe]];JIOmeHIIII 4 [I KHlIrlI). II03TOMY yrOJI H FO paBeH yrJIY F HQ. OT-
JIOmlIM [paBHble JIIIHIIII) H L H F X II npOBe.n;eM JIHHIIII XO H LQ. Tor]];a
JIIIHIfII LH II HQ paBHbI JIIIHIIHl\1 XF II FO, a MbI ]];OKa3aJIH,lITO yrOJI
LHQ paBeH yrJIY XFO. IIo3ToMY OCHOBaHHe LQ paBHO OCHOBaHlI1O XO.
TaK KaK OH OTKJIa.lJ;IuBaJIaCb paBHoH FQ, a HL - paBHoH XF, TO
OL paBHa XQ. II03TOMY paCCTOHHHe OT TOlIRH L .n;o COOTBeTCTBYIOm;eii
TO"tJKH Q TaKoe me, KaK OT TO"tJKII 0 ]];0 COOTBeTCTBYIO eii eH TO"tJKI1 X.
110 3TOMY 06pa3U;y ]];OKa3bIBaeTCH, QTO paCCTOHHlIe OT Kam.n;O:B TO"tJRII
)];0 COOTBeTcTBYIO eii eii TO"tJKH TaRoe iI\e, KaK OT )];pyroii TO"tJKH ,n;o
COOTBeTCTBYIO eii eft TO"tJKH. 8TO H eCTL TO, lITO MbI XOTeJIII ]];OKa33TL)}
[28, c. 501-503J.
TIepBan nOJIOBIIHa rrpe,n;JIOmeHHH 28 aJI- mayxaplI COBna]];aeT C
npe]];JIOmeHlIe 1 27 KHIJTH I «Ha"tJaJI». Bo BTOpOH nOJIOBHHe npe,n;JIO-
meHHH 28 aJI- mayxapH XOlIeT .n;OK33aTb, lITO 1I3 paBeHCTBa HaKpeCT-
JI ma IIX yrJIOB, o6pa3YIO lIxCH npH nepece"tJeHIIII ]];BYX npHMbIX
TpeTbeH, BbITeKaeT, "tJTO YKa3aHHbIe ]];Be npHMble - paBHOOTCTOH He,.
OJJ;HaRO <}>aKTH"tJecKH OH YCTaHaBJIIIBaeT JIHIIIL, lITO YKa3aHHbIe .n;Be
npHMbIe CIIMMeTplI"tJHbI OTHOCHTeJIbHO cepe,n;HHbI OTpe3Ka TpeTLei npH-
MOB, 3aKJIIO"tJeHHoro Mem,n;y HIIMH. IIpo]];oJImaeM H3JIOmeHHe TeopHII
aJI-AmayxapH B TpaKTaTe aT-TYCH:
«TI pe]];JIOmeHlIe 29. ECJIII B Kam]];OM TpeyrOJIbHHKe )J;Be 113 ero CTO-
pOH pa3J].eJIeHhI nOnOJIaM H [cepe,n;HHbI] coe):{IIHeHhI JIHHHeii, TO CTO-
pOHa TpeyrOJILHHKa paBHa ]];BYM 3THM JIHHHHM.
IIpHl\Iep. B TpeyrOJILHHKe ABC Kam]];aH 1I3 CTOpOH AB 1:1 AC pa3-
]];eJIeHa nOnOJIaM B TO"tJKax Ell G II npOBe,n;eHa JIIIHIIH EG. .R YTBepmJ];alO,
"tJTO .BC paBHa ]];BYM EG (plIC. 15).
oKa3aTeJILCTBo. TIOCTpOIIM B To"tJKe C IIPII JIHHHII AC yrOJI, P3B-
Hblii yrJIY A, KaK ]];OKa3aHO B npe]];JIOmeHHH 23,. 3TO yrOJI ACF
Tor.n;a JIIIHHII AB II CF napaJIJIeJILHbI, KaK )J;OKa3aHO B npe)];JIOmeHIIII
28. TIpoAOJImlIM JIIIHIIIO EG B ee HanpaBJIeHIIH ]];0 TO"tJKH D. Tor.n;a
yrJIbI AGE II DGC - BepTIIKaJILHble npH nepeCeqeHHII JIIIHIIH A C
II ED, n03TOl\IY OHII paBHbI, RaK ,n;OKa3aHO B npe)];JIOmeHHH 15 [RHlIrlI
I], yrOJI A C F, KaK li3BeCTHO, paBeH yr JIY A, II nOJIy"tJeHHaH nplI ,n;e-
JIeHIIH IIOnOJIaM CTopOHa AG paBHa cTopoHe GC. TI03TOMY TpeyrOJIb-
HIIKH AGE II GCD paBHbI Ii EG paBHa GD, CD paBHa EA, a yrOJI
AEG paBeH yrJIY GDC, KaK .n;OKa3aHO B npe)J;JIOmeHIIH 26 [KHllrlI 1 L
TaK KaR GE OTJIOmeHa Ha ED, ED paBHa ]];BYM EG. YroJI AEG
6bIJI paBeH yrJIY GDC; TaR KaK 3TII yrJIbI HaKpeCTJIeiRa He, paCCTOH-
HIIH OT TO"tJeK JIIIHHII EB ,n;o [COOTBeTCTBeHHbIX) TO"tJeK JIIIHIIH CD
paBHbI, KaK MbI ]];OKa3aJIII B IIpe.n;bI.n;YIIJ.HM npe,n;JIOmeHIIII. n03TO IY
ED paBHa BC. Ho ED 6bIJIa paBHa ]];BYM EG, n03TOl\IY H BC paBHa
]];BYM EG. 8TO II eCTL TO, "tJTO MbI XOTeJIII .n;OKa3aTL» [26, c. 503-504J.
8TO npe.n;JIOa,eHlIe 0 cpe.n;Heii JIHHIIH TpeyrOJILHIIKa paBHOCHJIbHO
«ycIIJIeHHOH aKCDOMe 0 napaJIJIeJIbHbIX».
CJIe)J;YIO ee npe,n;JIOmeHlIe aJI- mayxaplI 6JIH3KO K npIIBe]];eHHOMY
BblIIIe npe.n;JIOmeHIIIO CIIMIlJIIIKIIH: «II pe)J;JIOil eHHe 30 & B Kail\.n;OM
yrJIe MOil\HO npOBeCTII 6eCKOHeQHO MHoro OCHOBaHHH.
28
C A
K
AF
IT
B
Puc.16
K
HE H
l
.D B
E
Puc.17
TIpHMep. ,1].aH npOHaBOJIbHLIH yrOJI ABC (pnc. 16). .H YTBepmnalO,
llTO Y yrJIa ABC HMeeTCH 6eCKOHellHO MHoro OCHOBaHEH.
oKaaaTeJIbCTBo. TIpo,n;OJImEM JIEHEIO AB B ee HanpaBJIeHHE aa
B ,n;o TOqKH E. Tor,n;a yrJIM ABC E CBE [BMecTe] paBHbl ,n;BYM npHMMM
yrJIaM, KaK ,n;OKaaaHO B npe,n;JIOmeHHH 13. II09TOMY yrOJI ABC MeHbme
,n;BYX npHMbIX Ha yrOJI CBE. OnEmeM E3 n;eHTpaB Ha paCCTOHHEE BD
nOJIYKpyrGDK. TOllKE G E D Ha ,n;yre, n09TOMY JIEHEH, coe,n;EHHIOID;aH
GD,- OCHOBaHHe yrJIa ABC. OnHmeM TaKme ES n;eHTpa B Ha pac-
CTOHHEH BX nOJIYKpyr FXH. TOllKH F E X Ha ,n;yre, n09TOMY JIEHIIH
coeJJ:HHHIOID;aH FX,- OCHOBaHEe yrJIa ABC. IIo 9TOMY 06paaI(Y
npOBo,n;HTCH 6eCKOHellHO MHoro OCHOBaHEH yrJIa ABC. BTO H eCTb TO,
llTO MbI XOTeJIE ,n;OKaaaTb» [28, c. 504].
aJIee c.rre]];YIOT ]];Be aa,n;allE Ha nOCTpoeHEe:
«TI pe]];JIOmeHlIe 31. Paa,n;eJIEM npOEaBOJIbHbIH yrOJI Ha ,n;Be llaCTB
JIHHHeH H npOBe,n;eM npOHaBOJIbHOe OCHOBaHHe, TaK llTO 06paayeTcH
TpeyrOJIbHEK» .
«TIpe,n;JIOmeHHe 32. Paa,n;eJIEM npOEaBOJIbHbIH yrOJI JIHHHeH H OT-
IeTEM Ha 9TOii JIIIHEH npOHaBOJIbHYIO TOllKY, Tor,n;a MomHO npOBeCTE
llepea 9TY TOllKY B 06e CTOpOHbI JlHHIIIO, HBJIHIOID;YIOCH OCHOBaHEeM
]];aHHOrO yrJIa» [28, c. 505].
8TH ]];Ba nOCTpoeHHH paBHOCHJIbHbI npe,n;JI01HeHEHl\f 1 II 2 CIIMIIJIH-
KHH. TIpe,n;JIOmeHlIe 30 aJI- mayxapE OTJIEllaeTCH OT nepBoro npe,n;-
JIOm HEH CHMIIJIHKHH OTCYTCTBEel'iI Tpe60BaHEH, llT06M «XOp,n;bI» HJIB
OCHOBaHEH» yrJIa OTCeKaJIII Ha ero CTopOHax paBHMe OTpeaKH, 0]];-
HaRO TaKHe «OCHOBaHIIH» aJI- mayxapE PPIIBO]];HT B CBoe I npe]];JIO-
:HieHHH 31. IIpe]];JIOfHeHHe 32 aJI- mayxapH, MHOrEl\f KasaBllIeeCH
COBepllIeHHO 01leBEJJ:HbIl\f, Ha caMOM ]];eJIe paBHOCHJIbHO V nOCTYJIaTY
(OHO, KaK E npe]];JIOmeHHe CEl\fnJIEKEH, He BbInOJIHHeTCH B rHnep60-
JIlIlleCKOH reoMeTpHH) BnOCJIe]];CTBEE HeoJJ:HoKpaTHo BbIcTynaJIO B
KalleCTBe «nocTYJIlIpyeMoro OCHOBaHIIH» npH ,n;oKaSaTeJIbCTBaX 9Toro
nocTYJIaTa; TaK, JIemaH]];p [56] OnEpaJICH Ha Hero C 3-ro no 8-e H3]];a-
HlIe CBOIfX «HallaolTJ reOMeTpEE» B 1800 E CJIeJJ:YIOID;lIe rO]];bI (1-e Ha]];a-
Hue 3TOH KHl:lrH BbIllioITJO B 1794 r.). O]];HaRO y aJI- ma yxapE «nOCTYJIE-
29
pOBaHlIe OCHOBaHHH» npOH3BOnIITCH e e B npenJIOmeHIIH 28, B KOTO-
pOM, KaK MbI BH.n;eJIH, nOKa3bIBaeTCH COBceM He TO, llTO YTBepm.n;aeTCH;
BTopaH nOJIOBHHa rrpenJIOmeHHH 28 aJI-,IJ;mayxapH, BbIIIOJIHHIO aHCH
TOJIbKO B eBKJIHnOBOB reOMeTpHH, paBHOCIIJIbHa YCHJIeHHOH aKCIIOM-e
o napaJIJIeJIbHbIX, H, CllHTaH npenJIOmeHHe 28 nOK838HHbIM, 8JI-
}1{8YX8PII «BbIBO.n;IIT» H3 Hero npyrHe YTBepmneHHH, p8BHOCHJIbHbIe
9TOH 8KCIIOM:e, B 1l8CTHOCTII B npenJIOmeHHH 29 - TeopeMY 0 TOM, llTO
.cpenHHH JIHHHH TpeyrOJIbHHK8 p8BH8 nOJIOBHHe ero OCHOB8HIIH,
.a B npenJIOmeHHH 32 - TeopeMY 0 B03MomHOCTII npOBeneHHH llepe3
BHYTpeHHlO1O TO"tJ:KY yrJI8 npHMoH, nepeceKalO ei 06e ero CTOpOHbI.
. 8JIee 8JI-,IJ;mayx8plI BbIBOnHT H3 CBoero npenJIOmeHIIH 32 nOCTYJI8T.
«11 peJJ;JIO}KeHHe 33. ECJIII npOnOJImlITb nBe JIHHIIII OT JIHHIIH B CTO-
POHY yrJIOB, MeHblIIHX nBYX npHMbIX, TO OHI1 nepeceKYTCH C 9TOH
CTOpOHbI.
I1pn:Mep. JIHHIIII AB II CD npOBe;n;eHbI OT JIIIHHII BD B CTOPOHY
yrJIOB ABD H CDB, MeHbmlIX nBYX npHMbIX (pHC. 17). .R YTBepmJJ:81O,
qTO, eCJIII npO;n;OJImHTb HX B HX HanpaBJIeHHH, OHII nepeceKYTCH.
[,[(oKa38TeJIbCTBoJ. TIponoJImHl\f JIHHHIO BD B ee H8np8BJIeHIIII no
TOqeK E II H II OTJIOmHM BF, P8BHYIO BD, K8K ;n;OK838HO B npenJIO-
il\eHIIII 3. "YrJIbI ABD II CDB no npeJJ;fiOJIOmeHIIIO [BMecTe] MeHbllIe
]l;BYX npHMbIX, 8 yrJIbI ABD H ABE [BM:eCTe] paBHbI nBYl\{ npHMbIM.
1109TOMY yrOJI ABE 60JIbllIe yrJI8 CDB. 110CTpoHM B B Ha JIIIHIIII
BA yrOJI ABK, p8BHbIH yrJIY CDB. Tor;n;a yrJIbI ABD H ABK Bl\feCTe
J\IeHbme nBYx rrpHMbIX, 9TO yrOJI KBD. TIpoBeneM II3 TO"tJ:KII F JIIIHIIIO
KL - OCHOB8HIIe yrJI8 KBL, KaK nOKa38HO B rrpenJIO}HeHIIII 32.
Tor;n;8 yrOJI KFB, BHemHHH yrOJI TpeyrOJIbHHK8FBL, 60JIbIUe BHYT-
peHHero yr JIa F B L, K8K JJ;OK838HO B npenJIOjl eHIIII 16. 1109Tol\fY
nOCTpOHM B F H8 JIIIHHH BF yrOJI BFO, p8BHbI:H yrJIY FBL. YroJI
EBA, KaK H3BeCTHO, p8BeH yrJIY CDB. 1109ToMY yrJIbI BFO II OBF
p8BHbI yrJI8M ABD II CDB, KamnbIH COOTBeTCTBeHHOl\fY, 8 BF OTJIO-
n eH8 p8BHOH BD. 8JIOmHl\f BD H8 BF, P8BHYlO eH, Torna yrOJI
CDB HaJIOmHTCH Ha yrOJI OBF, paBHbIH el\fY, 8 yrOJI ABD H8JIO}KIITCH
Ha yrOJI BFO, p8BHbIH el\fY. 1109TOl\fY npHl\fbIe BA H DC, eCJIII npo-
nOJIn IITb IIX B HX H8np8BJIeHIIIl, H8JIOmaTCH H8 JIHHIIH BO H j?O,
nepeceK81O HecH B TOllKe O. 8TO H eCTb TO, "tJ:TO l\fbI XOTeJIII JJ;OK838Tb»
[28, c. 507 -508].
HecMoTpH H8 JIOrlIqeCKlIe 0IlIH6KH, nOK838TeJIbCTBO 8JI-,[(m8yxaplI
npe.n;CT8BJIHeT 3H8"tJ:HTeJIbHbIH HHTepec. B Hel\f BnepBbIe .n;OKa3bIB8JI8Cb
B03l\fOrRHOCTb npOBeneHIIH qepe3 JII06ylO BHYTpeHHlO1O TOqKY yrJIa
npHMoH, nepeceKalO e:H 06e ero CTOpOHbI, H OTClOna BbIBOnHJICH V
nOCTYJI8T.
· 38CJIYJ-l HB8eT T8Kllie BHHl\f8HHH TO, llTO B KOH e IJ;OK83aTeJIbeTBa
npeIIJIOn eHHH 32, rne rOBopHTCH, llTO OTpe3KH OX II XN nplI npo-
J];OJI}I eHIIII nepeceKYT JIIIHHII AB HBC [28, c. 507], npe;n;nOJIar8eTCH,
qTO npHl\f8H, He npOXO;n;H aH llepe3 BepllIHHbI TpeyrOJIbHIIK8 II nepe-
ceKalO aH OnHY H3 ero CTOpOH, nepeceK8eT e e onHY CTOPOHY Tpe-
yrOJIbHIIKa. 8TO YTBepmJJ;eHHe HeOJJ:HOKpaTHO HCnOJIb30B8JIOCb nplI
J];OK838TeJIbCTB8X V nOCTYJI8T8; M. l1am BbI;n;eJIIIJI ero KaK oc06yIO
aKCHOl\IY, H83bIB8el\fYIO B HaCTOH ee BpeMH 8KCII0l\fOH 11811I8.
30
ABa TpaKTaTa, CneUHaJIbHO no-
CBHm;eHHbIe nonbITKaM: nOKa3a-
TeJIbCTB aKCHOMbI 0 na paJIJIeJIb-
HbIX, npHHanJIemaT yno:MIIHaB-
IIleMycH Ca6HTY H6H Koppe.
1I6H Koppa, ypomeHeu; XappaHa
B MeconoTaMHH, rne rJIy60KIIe
KOpHII nYCTHJIa rpe"tJeCKaH KYJIb-
Typa, pa60TaJl B Barnane. 116H
Roppa nepeBeJI MHorHe rpeqec-
KlIe II CHplIHcKHe COqHHeHIIH Ha
a pa6cKIIH H3bIK, B qaCTHOCTII
pHn He coxpaHHBillHXCH B oplIrHHaJIe TpaKTaTOB ApXHMena H V -VII
KHHrH «KoHlIqeCKHX ce"tJeHHH» AnOJIJIOHHH, H3BeCTHbIe TOJIbKO no
ero nepeBOJJ:aM. M6H Koppe np:nHaJJ:JIen aT MHorHe u;eHHbIe TPYJJ:bI
no MaTeMaTIIKe, B KOTOpbIX OB paaBHJI c«pepll"llecKYro TplIrOHOMeTplIlO,
o6m;ee Y"lleHHe 0 neiiCTBIITeJIbHOM [rrOJIOmIITeJIbHoM] "II:nCJIe, TeoplIlO
"tJHCeJI H MeTOnbI KBanpaTYP II Ky6aTYP; OH nHCaJI TaKme no BonpocaM
aCTpOHOMHII H MenHIJ;IIHbI. He MeHee BaiHHbIMII B CBoeH 06JIaCTH 6bIJIH
HCCJIenOBaHHH M6H ROppbI no TeopHH napaJIJIeJIbHbIX JIHHIIH. OTMeTHM
em;e, "liTO M6H Roppa 6bIJI penaKTopOM HOB oro nepeBOna «Ha"tJaJI»
EBKJIHna, BbInOJIHeHHoro ero COBpeMeHHHKOM H npyroM McxaKOM
H6H XYHaiiHOM aJI-'M6anH (06 apa6cKHx nepeBOnax «Ha"llaJI» CM.
[57, c. 438-443]). OnHH H3 TpaKTaTOB M6H ROppbI 0 napaJIJIeJIbHbIX
JIHHIIHX Ha3bIBaeTCH «RHHra 0 nOKaaaTeJIhCTBe H3BeCTHoro nOCTYJIa-
Ta EBKJIlIna» (Mal aJIa .pi 6ypXB.H aJI-MY B.napa aJI-Maillxypa MHH
Y JIHnHc), HJIH «RHHra 0 TOM, "liTO eCJIII npHMaH JIIIHHH nanaeT Ha
nBe npHMbIe JIHHIIII, TaR "liTO O.D;HOCTopOHHHe yrJIbl MeHbIIle nBYX npH-
MbIX, TO 9TH JIHHHH, eCJIH npOnOJImHTb IIX B 9TY CTOPOHY, BCTpeTHT-
CH» (RHTa6 «pH aHHaxy II B. BaKa' a a ' MycTaKHM 'aJIa aT,!,aHH
MycTaRMMaiiH «pa caiiiiapa aa-3B.BHiiaTaMH 3aJIaTaiiH «pH nllilIxa B8xHJJ:a
a aJIJI MHH 1:'B.IIMaTaiiH «pa IIHHa aJI- a,!,,!,aiiH H a y pHnmB. «PH THJIKa
aJI-nmHXa HJITaKaiia). Ha3BaHIIe BToporo TpaKTaTa HBJIHeTCH cOKpa-
m;eHHeM BToporo Ha3BaHHH nepBoro TpaKTaTa: «RHHra 0 TOM:, "liTO nBe
JIHHHH, npOBeneHHbIe non yrJIaMH, MeHbillHMH nBYX npHl\fbIX, BCTpe-
THTCH» (Mal:'B.JIa HHHa aJI- a'f'!,aiiH y pHnmB. HJIa a3-3B.BHHaTanH al JI
aJI-I:\B.HMaTaiiH HJITaKaiia). MbI He 3HaeM, KaKOH 113 nBYX TpaKTaTOB
M611 ROppbI 6bIJI HanlIcaH paHbIIle. TIepBbIii 113 Ha3BaHHbIX HaMil TpaK-
TaTOB COXpaHHJICH B naplImcKoii pYKonIICH, nepenHcaHHoii B X B.
1I3BeCTHbIl\f :MaTel\faTlIKOM ac-C:unmH3H, BTOpOH - B KalIpCKoii PYKOIlH-
CH, nepenlIcaHHOH B X,TIII B. HeCl\fOTpH Ha 60JIbillYIO pa3HIIlJ.Y
B naTe nepenlICKH, 06e PYKonIICII OTJIH"IIalOTCH OnHIIMII II Te IH iRe
oc06eHHOCTH IH :MaTel\faTH"tJeCKHX CO"tJHHeHHii M6H ROppbI, B "tJaCT-
HOCTII 0603HaqeHHHl\fll Ha nOJIHX HOMepOB npenJIOmeHHii C nO}IOm;blO
6YKBeHHoii HYl\fepaIJ;HII. B nepBOM TpaKTaTe IIHTb npeJJ:JIOffieHIIH, BO
BTOpOl\f HX Ce}Ib. 06a OHH 6bIJIII ony6JIHKOBaHbI B aHrJIlIiicKO I IIepe-
BOne A. M. Ca6poii [59], a B PYCCKOl\f nepeBOne - aBTOpal\fII [22, 23J.
TpaKToBKa Bonpoca B 9TIIX nBYX Tpynax cym;ecTBeHHO pa3JIIPIHa:
B nepBOM 116H Koppa CTOHT Ha "IIHCTO reo:MeTplI"IIeCKOH no"tJBe, B npy-
6.
A6e meopuu
napaJlJleJlb1iblX
C a6uma
u61i KOPPbl
81
rOM:, nO-BH]J;HMOMY, BIIepBbIe B HCTOpll1l naHHOrO BOnpOCa HCnOJIb-
syeT KIIHeMaTllqeCKHe co06pameHHH. M paCCMOTplI 1 CHaqaJIa IIe.p-
BMU TpaKTaT, paccym]J;eHHH B KOTOpOM HMelOT 60JIee TpaAIII(IIOHHbIH
XapaKTep. IIepeIIIICqHK aTOro TpaRTaTa nHIIIeT: «OH [T. e. M6H Kop-
pa.- Aem.] CRa3aJI: «ECJIH npHMaH JIHHIIH rraAaeT Ha nBe npHMbIe
JIHHIIII H HaKpeCTJIemam:He yrJIbI paBHbI, TO 9TH ]J;Be JIIIHHH He npH6JIH-
malOTCH H He Y]J;aJIHIOTCH HH O]J;HY H3 ]J;BYX CTOpOH.
IIpHMep. JIHHIIH AB H CD, Ha KOTopbIe naAaeT JIIIHHH EG TaKHl\{
06pa30M, qTO yrJIbI AEG 11 EGD paBHbI (pn:c. 18). .fI YTBepm alO,
"'ITO JIHHIIH AB H CD He IIpH6JIHmalOTCH II He Y]J;aJIHIOTCH HH B CTO-
POHY AC, HH B CTOPOHY BD.
OKa3aTeJIbCTBO aToro. ECJIH MbI HaJIOmlIM EA Ha GD TaK, "tIT06bI
"TOqKa E COBIIa]J;aJIa C TOqKOH G, JIHHHH EG - C caMOH c060:u, yrOJI
AEG - C yrJIOM EGD, TO JIHHHH CG COBIIa]J;aeT C EB, a yrOJI CGE-
c yrJIOM GEB. JIIIHHH CG Bcer]J;a COBnaAaeT C JIIIHHeH EB, eCJIH JIII-
HIIK GD COBIIaAaeT C JIHHlIe:u AE. ECJlH 6bI 6bIJIO He TaK, TO yrOJI
6bIJI 6bI 60JIbme paBHoro eMY, a 9TO HeJIerro. TeM caMbIM ]J;oKaaaHo,
qTO eCJIII JIHHHH EBI 11 CD IIpH6JIIImalOTCH B CTOPOHY BD rrpH JIX
npOAOJImeHHH, TO JIIIHHII AE II CG TaKme npH6JI1IiRalOTCH B CTOPOHY
A C, TaK KaR OHH COBlla]J;aIOT C npH6JIHmalO HMHcH npH HaJIOmeHIIII.
Ho COBepIIIeHHO OqeBHAHO, "tITO eCJIH npHMaH JIn:HHH rra]J;aeT Ha ]J;Be
IIpHMMe JIIIHHII 11 eCJIH nBe JlHHHH nplI6JIHfHalOTCH B OAllY H3 ]J;BYX
CTOpOH, TO OHH YAaJIHIOTCH B APyrylO CTOPOHY, H HX rrpH6JIlIn\eHHe
CO CTOpOHbI IIX IIpll6JIHmeHHH II YAaJIeHHe B CTOPOHY HX YAaJIeHHH
YBeJIlIqHBaeTCH. B qaCTHOCTH, eCJIH l\fbI nplIMeM, qTO JIIIHHH .EB II
GD rrpll6JIHmalOTCH B CTOPOHY BD, TO JIHHHH AE H CG COBna]J;aJIH
C JIIIHHHMH EB 11 GD B CTOPOHY BD, II eCJIH 6bI EB II GD rrplI6JIHma-
JIHCb, TO AE H CG YAaJIHJIHCb 6b1 H He COBrraAaJIII 6bI C HHl\flI,
a eCJIH 6hI OHH COBIIaAaJIH C HIIMH, TO OHB He MOrJIH 6bI YAaJIHTbCH B
CTOPOHY A C. 1l0aToMY, rrOCKOJIbKY JIHHIIH AE H CG rrpH6JIHmaJl1lCb
6bI B CTOPOHY AC TaK me, RaK JIIIHHH EB H GD rrplI6JIIIiRalOTCH B
CTOPOHY BD, OCTaeTCH rrpe]J;1I0JIOiRHTb, qTO 3TH ABe JIHHIIH He rrpH6JIII-
-maIOTCH H He YAaJIHIOTCH B CTOPOHY A C, TaK KaK, eCJIH 6bI OHII rrpH-
6JIHiRaJIHCb B aTY CTOPOHY, 6bIJIa 6bI JIOmHO:U npaBHJIbHaH npe]J;rrOCbIJI-
Ra, rrOTOMY "'ITO [B aTOM cJlyqae] HMeJIIICb 6hI nBe JIIIHHII, rrp1I6JIH-
.iRalO HeCH C 06eHx CTOpOH, XOTH H coxpaHHIO lIe paccToHHHe Mem]J;Y
c060:u, [TaR RaK HHaqe] EB H GD He COBnanalOT [C CG H EA], HO
paHbIIIe OHH COBrra]J;aJIII. IIoaToMY npeAnOJIOmeHHe, qTO eCJIII HaKpeCT-
JIemam:He yrJIhI AEG H EGD paBHbI, TO EB II GD nplI6JIHmalOTCH B
.cTOPOHY BD,- HeJIenO, H TOqHO TaR me onpOBepraeTCH, qTO OHH
YAaJIHIOTCH. 1l0aTOMY OHH He npH6JI1ImalOTCH II He ynaJIHIOTCH,
11 TO me nOKa3bIBaeTCH H 0 JIIIHHHX AE H CG. 8TO 11 eCTb TO, qTO l\fbI
XOTeJIII AOKa3aTb» [22, c. 593-594].
llepBaH qaCTb YTBepiK eHHH aToro rrpe JIOmeHHH nplIHa JIemHT
K a6COJIIOTHOH reOMeTpHH, HO CJIOBa «HX rrp1l6JIHiKeHIIe co CTOpOHhI
HX npH6JIHmeHIIH... YBeJIHqIIBaeTCH» 9KBHBaJIeHTHhI V nOCTYJIaTY,
"TaR RaK B rllllep60JIHqeCKOH reOMeTpHH ABe paCXO]J;Hm:HeCH IIpHl\lbIe
npH6JI1ImalOTCH npyr R Pyry 110 Mepe IIX rrpll6JIHmeHHH K IIX 06m: eM Y
nepneH]J;IIKYJIHPY, a nOCJIe AOCTHmeHIIH MHHHMYMa paCCTOHHHH Ha
92
H C
A
I
A
c
A
c
B
11
6
.D F
B
lJ
Puc.18
P uc. 19
Puc.20
00II1e 1 nepneH]];IIKYJJHpe OHM HaqUHalOT paCXO,DJITbCH Apyr OT APyra.
8TJIM YT.RepfI,JJ;eHlIel\1 M6H Koppa, BnpOqel\f, He nOlIb3yeTcH, 110 HHme,
rOBopH «TOqHO TaK n,e orrpOBepraeTCH, qTO OHM YAaJIHIOTCH», OH HeHBHO
rrOJIb3yeTcH YTBep}I\AeHHeM 0 T01\{, qTO eCJIH npHMaH rr8JJ;aeT Ha ABe
npH:\lhle II 3TH rrpHMbIe YAaJIHIOTCH APyr OT JJ;pyra B OJJ;HY H3 JJ;BYX
CTOpOH, TO B JJ;pyrylO CTOPOHY OHH npM6JIH}I{aIOTCH, qTO TaKme 3KBH-
naJIeHTI-IO ,r rrOCTYJIaTY. TaKIIM 06pa301\I, n rrpeJJ;JIOa,eHIIH 1 3Toro
TpaKTaTa M6H hOppbI, <P0PMYJIlIpye:MOl\{ TaK fHe, KaK rrpeJJ;JIOmeHlIe
1 aJI-J jl\ayxapH, TOlI,e npOII3nOJJ;IlTCH «JIOCTYJIUpOBaHlle OCHOBaHHH»,
HO He CTO TJb rpy60. B npeJ(JIOmeHHM 2 J(OKa3bIBaeTCH 06paTHoe YT-
Bep}I AeHlle, conrraAaIo ee C nepnoiI qaCTblO rrpeJ(JIOmeHIIH 29 KHlIrlI I
«HaqaJI», n KOTOp01\1 EnKJIHJ( nnepnbIe IICIIOJIb3yeT aKCHOMY 0 napaJI-
JIeJIbHbIX: «ECJIH npHMaH JIIIHHH rraJ(aeT Ha J(ne rrpHMbIe JIHHIIII, He
npH6JIllJJ,alO HeCH H He y,n;aJIHIOIIJ;HeCH HH B KaKYIO CTOPOHY, TO
HaR p eCT JIejl\ a Me [yr JIl)I] p BHbI.
TIpnI\lep 3Toro. JIHHMII AB II CD He npll6JIIImalOTCH II He y,n;aJIHIOT-
CH HII B KaKYIO CTOPOHY, II Ha DIIX rraJ(aeT EG (pIlC. 19). H YTnepiRAalO,
'lJ:TO HaKpeCTJIeiRaIIi;lIe yrJIbI AEG II EGD paBHbI.
)loKa3aTeJIbCTBO 3Toro. ECJIII OHH He panHbI, TO rrYCTb yrOJI AEG
MeHblue, II nYCTb EGli' paneH yrJIY AEG. llpone,n;eM [JIIIHIIIO] FGH.
TorAa JIIIHllH FGH II AB He npll6JIIlmalOTCH H He YJ(aJIHIOTCH, TaR
KaK HX HaKpeCTJIemaru;He [yrJIbI] paBHbI B CIlJIY Toro, lITO MbI ,n;OKaaaJIH
paHbIIle. Ho JIHHIlH AB H CD He npll6JIHmalOTCH H He YJ(aJIHIOTCH,
a [JIllHllM] CD II F H rrepeceKalOTcH B TOlIKe G, II Kam,n;aH lIa HHX He
npnO:IUa"aeTCH K [JIIIHMIl] AB II He YAaJIHeTCH OT Hee. Ho GF 6JIHme
K EB, qeI\1 CD, TaR KaK HaXOJ(IlTCH MemAY HIIl\IH, a 3TO npOTHBOpelIlIT
[CKa3aHHOl\IY). CJIe.nonaTeJIbHO, AEG II EGD paBHbI. 8TO II eCTb TO,
lJTO 1\ILI XOTeJIH AOKa33Tb» [22, c. 594-595J.
QeHTpa&lIbHOe MeCTO B J(OKa3aTeJIbCTBe 1I6H KOppbI aaHIIMaeT ero
npe]];!.IOn,eHHe 3, B KOTOpOM 1I3 ,n;BYX IIpe,n;bIAY IlX rrpe,n;JIOmeHHH OR
nbIBO,IJ;MT cymecTnonaHHe rrapaJIJIeJIOrpaMMa, lITO, KaK MbI BHJ(eJIII,
3KBHBa!.leHTHO «YCHJIeHHOA aKCHOMe 0 napaJIJIeJIbHbIX»: «ECJIH CO-
2 B. A. P03eII eJlb)1" A. II. IOwHeBH1J 33
e]J;IIHIITb KOHl.\hI ]J;BYX npHMbIX JIIIHIIH, paBHbIX II He npIIOJIlIiI\alOIU;IIXCH
H He y.naJIHIO IIXCH, npH lbIMIf JIIIHIIH nI, TO OHII TaKfI\e paBHI)I JI He
nplI6JIIImalOTCH H He ynaJIHIOTCH.
IIplIl\fep 3Toro. ABe npHMhIe JIIIHIIII AB II CD paBHbI II He npM-
6JIIImalOTCH H He ynaJIHIOTCH, II HX KOHU;bI coe]J;IIHeHbI JIHHHHAIM A C
II BD (pIIC. 20). .R YTBepmnalO, qTO [JIIIHHM] AC II BD paBHbI n He
npH6JIHmaIOTCH H He Y]J;aJIHIOTCH.
AOKa3aTeJIbCTBO 3Toro. HaKpeCTJIefl\a lIe .yrJIhI ADC 11 DAB paB-
lIbI. II03To1\-IY, TaK KaK JIHHIIll AB II AD cooTBeTc BeHHo paBHbI JIM-
HIIH I Cl) II AI), a yrJIbI ADC H DAB paBHbI, TO [TpeyrOJIhHMKII]
ADC II DAB paBHbI. n03TO IY JIHHHH AC paBHa JIHHMH BIJ, a yrOJI
ADB paBeH yrJIY CAD. TaK Kal\ 3TH nBa [yrJIa] HaKpeCTJIen aIU;He,
TO TaK iHe, KaK He npH6JIHllialOTCH H lIe ynaJIHIOTCH JIIIHIIH AB II CD,
He npU6JIIIfl\alOTCH H He ynaJIHIOTCH II paBHbIe JIIIHMII A C II BD.
VTO II eCTb TO, "tJ:TO MhrXOTeJIH nOKa3aTb» [22, c. 5U5J.
B npenJIomeHIIH 4 H6H Roppa, KaK II aJI-,r.{mayxapII, paCCMaTpH-
BaeT CpenHlO1O JIHHIIIO TpeyrOJIbHHKa II nOKa3bIBaeT, "tJTO OHa paBlla
nOJIOBHHe OCHOBaHIIH II napaJIJIeJIbHa eMY:
«ECJIII pa3]J;eJIIITb ,IJ;Be CTOpOHbI Karl\]J;oro TpeyrOJIbHIIKa nOnOJIa [ H
coe]J;HHIITb TOlIKII neJIeHHH npH IOH JIHHHeR, TO 3Ta [JIHHHH] - nOJIO-
BHHa OCT8BmeHCH CTOpOHbI H He npH6JIHniaeTCH K HeR If He ynaJIHeTCH
OT Hee.
IIplIMep 3Toro. B TpeyrOJIbHIIKe ABC [cTopoHa] AB pa3,n;eJIeHa
nOnOJIaM B D, a A C pa3]J;eJIeHa nOIIOJIaM B E H npoBeJJ:ena npHAlaH
[JIIIHIIH] DE (pHC. 21). .R YTBepmnaro, "£ITO OHa - nOJIOBMHa BC If He
nplI6JIIImaeTCH K HeH II He ynaJIHeTCH OT Hee.
AOKa3aTeJIbCTBO 3Toro. IIpo]J;oJImHM DE AO G TaK, lITOObI EG
6bIJIa paBHa DE, II coenHHH I CG. Torna TpeyrOJIbHIIKM ADE II
CEG paBHbI II JIIIHHH AD H CG paBHbI. II03ToMY JIIIHIIH DB H CG
paBHbI. TaK KaK yrJIbIADE H EGC paBHbI, a 3TH ]J;Ba yrJIa - HaKpeCT-
JIema He, TO JIHHHH AB II CG He npH6JIHrHaroTcH II He y]J;aJIHIOTCH.
II03TOMY JIHHHH B.D H CG TaKrl\e He npH6JIHfI,aIOTCH H He YJJ:aIJTJHIOTCH,
a TaK KaK 3TH ,n;Be JIIIHHH - npHl\fble, TO coe]J;HHHIOIU;I1e MX KOHI bI
JIIIHHll BC H DG paBHbI H He np1I6JIlIn alOTCH H He ynaJIHIOTCH. Ho
DG - ynBoeHHaH DE. II03ToMY II BC - y.nBoeHHaH DE If He npn-
6JIHiI\aeTCH K HeH II He ynaJIHeTCH OT Hee. 3TO H eCTb TO, IJTO MhI XO-
TeJIM ]J;OKa3aTb.
8THM me l\IeTOAOAf nOKa3blBaeTCH, "tJ:TO eCJIII JbI pa3neJIlI 1 KaiH,n;ylO
1I3 [JIIIHHU] AB II A C Ha qaCTII npOH3BOJIbHOH ]J;:IIIHbI o,n;I1HaKOBbI1-1
06pa30M II coe]J;IIHIIl\I TOlIKII. neJIeHIIH JIIIHMII AB H COOTBeTCTBYIOID;lIe
Hl\1 TO"tJ:KII JIHHHH AC npHMoH JIUHlIeH, TO OHa-TaKaH a\e lIaCTb OT
BC, KaK "tJ:aCTb JIHHHH [AB] IeiI\]J;y A H TOqKOH [AeJIeHIIH] Ha AB OT
AB» (22, c. 595-596J.
IIocJIenHee YTBepm]J;eHlIe H6H ROppbI, no cy eCTBY, paBHOClIlJTJbHO
YTBepm]J;eHHIO B03MOlliHOCTII nOCTpoeHIIH TpeyrOJIbHMKa, I1o,rJ;06Horo
n&HHOM:Y TpeyrOJIbHIIKY, Ha JII060M 3anaHHOAI OTpe3Ke. 3TO YTBep-
mneHMe TaKme 3KBHBaJIeHTHO YCIIJIeHHOH aKCIIO le napaJI.;rIe TJbHOCTH,
TaR KaK HH B rlInep60JIH"tJ:eCKOH, HII B 3JIJIIInTHQeCKOH reOMeTplIn
TaKoe nOCTpoeHHe HeB031\-IOiHHO. B pa3JIH"£IHbIX <pop lax 3TO YTBepil-\.ne-
34
A
G
A C
c
B
Puc.21
B E
T
A
.D H F
K l
Puc.23
Ii C
Puc.22
HUC TaKiHe I1CIIOJIb30BaJIOCb B ,n;aJIbHeHllIHx ,n;OKa3aTeJIbCTBaX V nOCTY-
JIaTa, a rI OH BaJIJIHC B cBoeH TeopHH napaJIJIeJIbHbIX npHHHJI ero
33 IIOCTYJIaT, aaMeHHIOIIJ;HM V nOCTYJIaT.
11, naKoneu., B npeJJ;JIOmeHHH 5 M6H KOppbI H3 ,n;OKa3aHHbIX paHee
BbIBO,n;HTCH C IICnOJIb30BaHHeM aKCHOMbI EBJJ;oKca - ApxIIM:e,n;a nOCTY-
JIaT 0 napaJIJIeJIbHWX:
«ECOITJII npHMaH JIIIHHH lla,n;aeT Ha ,n;Be npHMbIe JIHHIIH II BHYTpeHHlIe
O,n;HOC.TopOHHI1e yrJIbI l\leHbllIe ,n;BYX npHMbIX, TO 3TH Be JIHHIIII, eCJIU
npo,n;OJIj]\JlTb HX B 3TY CTOPOHY, BCTpeTHTCH.
npl1 Iep 3Toro. JIUHHII AB II CD, H Ha HIIX na,n;aeT JIIIHBH EG,
npl1"lIeM yrJIbI BEG H DGE MeHbllIe ,lI;BYX npHl\fbIX (pHC. 22). .fI YT-
Bepn ,n;aIO. "liTO JIIIHHII AB II CD, eCJIH npo,n;OJImHTb IIX B CTOPOBY
BJJ, HCTpeTHTcH.
,[I,oKa3a TeJIbCTBO 3Toro. n pOBeJJ;eM OT TO"llKH G JIIIHBIO GH, He
np1l6JIIIa alOru;YIOCH H He y,n;aJIHIO YIOCH OT JIIIHHII AB; OTMeTIIM Ha
Cl) np0l13BOJIbHYlO TO"llKY F H npOBe]];eM OT Hee K GH JIIIHBIO F K,
He npH6JIna-\alO YIOCH H He y]];aJIHIO YIOCH OT EG. Tor]];a MomBO
C"IIIITaTb, "liTO [j/K] 60JIbme EG, eCJIII me HeT, TO OTJIOmlIM F L, paBHYIO
Gl", H KH, paBHYIO GK, H COe]];HHIIM LH. OKa3aHO, lITO" LH-
Y]];BOeHHaH //K H TaKi-He He npH6JIIImaeTCH II He y]];aJIHeTCH OT fEG].
ECJIH F K IeHbllIe EG, TO Y]];BOHM ee H 6y]];eM nOBTopHTb Y]];BOeHBe
]];0 Tex nop, nOKa He nOJIyqlI 1 KpaTHYIO, 60JIbmyIO EG. TIYCTb 3TO
LH. Tor]];a LH ]];JIIIHHee EG H He npH6JIIImaeTCH K HeM II He y]];aJIHeTCH
OT Hee. OToITJOa\II [ Ha LH [JIHHIIIO], paBHYIO EG, 3TO HM. Tor]];a JIB-
HI:III GE II !l M paBHbI H He rrplI6JIIImalOTCH II He y]];aJIHIOTCH. I103TOMY
[JIHHIIH], COe]];UHHIOII(He HX KOHU;bI, paBHbI B He nplI6JIHmalOTCH H
85
21=
He YJJ;aJIHIOTCH, KaK 6bIJIO paHbme. {-fo GH coeJJ;HHHeT G H 11, n03TOl\fY,
eCJIH npoJJ;OJI)I\HTb EB B ee HanpaBJIeHUH B CTOPOHY B, OHa npOll,n;eT
Qepe3 M. ECJIH He TaK H [JII1HI1H], COeAJlHHJOI aH E II lit!, He EB, eCJIII
ee npOJJ;OJI)HHTb TO [JIIUIIIH], coeJJ;IIHHIOw;aH E H lVI, He npH6oJ"'JHiJ aeTCH
II He YJJ;aJIHeTCH 01' GH. lIo II EB He npn6JIMi-I-iaeTCH If He y,n;a:JfleTCH
OT GH, a [JIIIHHH], coeJJ;HHHlOmaH E II .L 1, lIaXOJJ;IITClI 1\leiRJJ:Y EB II
GH, lITO HeJIeno. CJleJJ;OBa TeJIbHO, eC.J1:U npOJJ;OJIa,HTb EB, OHa n poiiJJ;eT
qepe3 ... 1. n03TOl\IY OHa He06xoJJ;IIMO BCTpeTUT nepeJJ; TO IROiI . I He-
I(JTOpylO TOQRY JIIIHHU C/J. 1103TO}.IY, eCJIII npoJJ;OJDt\HTb AB II C.l)
B cTopony Bl), OHM BCTpeTHTcH. 31'0 n eCTh TO, liTO J\tIbI XOTC.!11iJ AOl\a-
3aTb» [22, c. 5BG-597J.
BTOPOH TpaRTaT 1I6H l\OPPbI HaqHHaeTCH C 60JIblIIOrO llllC;J;CHIIH,
n KOTOpOM OH BblcTynaeT KaK peUIHTeJIbHbIH CTOpOHHIIK npHl\leHenlfH
JJ;BHiHeHHH K reO IeTpHII, qTO OCY RJJ;aJIOCb ApHcTOTeol1el\l, rOBoplIB-
IllHM, lITO « IaTe laTlIqeCKne HaYKII JIHIIIeHbI IIBlIiHelIHH, 3a J1Cl\JIIOqe-
Hnel\1 Tex, KOTOpLI1\IM 3aHI1l\laeTCH yqeHUe 0 He6eCHI:»TX CneTUJIaX»
[2, T. 1, c. 85], H qero CTpeMHJICH no B03MOmHOCTI1 1136eiI,aTb B
CBOIIX «I-IaqaJIax» EBKJIHJJ;. COrJlaCHO MOH Ko ppe ca loe 1I3MepeUlle
BeJIHlIHH HeB03 IOn\HO 6e3 nepeHeceHHH II lIaJIO"I eHHH.
B Cal\IOM HaqaJIe TpaKTaTa rOBopnTCH: «110CKOJIbKY B reO leTpHqeC-
ROH HaYKe B 60JIbIlIHHCTBe C TJyqaeB Hl\lelOT JJ;eJIO C BeJIII IIfIla nf, nx
paBeHCTllOM, lIX pa3JIHlIeHlfe l, IIX H3MepeHHe:M H TOMY IIOA06HLIM:, TO
nepBbIl\1 npllHIJ;MII01\1 B YTBep)HJJ;eHHHX 0 CYIT(HOCTH AOKa3bIBael\'IOrO
B aTOM: CMbICJIe HBJIHeTCH TO, IIPH nOMO II -qero noHIfMalOTCH nce
H3l\fepeHHH II l\lepbI, T. e. HaJIOiReHHe BCHKO:U paBHOH BeJIH II1Hhr Ha
paBHYIO eii» [23, c. 3H3]. II neCKOJIbKO JJ;aJIee: «BOJIbIlIIIHCTBO JJ;OKa3£l-
.... G
TeJIbCTB 3aJJ;aq H npeJJ;JIOmeHHH reOl'rleTpHlIeCKOlI HaYI\H HYfI\JJ;aeTCH
B YKa3aHHOM nepBOM npJlH Hne, T. e. npHM:eHeHHII yno IHHYToro Ha-
MH JJ;eiiCTBHH - npJlBeJJ;eHHH B JJ;BumeHHe OJJ;HOH H3 nBYX Be e:u, cpaB-
HHBaeMbIX JJ;pyr C JJ;pyroM, T. e. nOJJ;HHTHH ee C ee MeCTa, nepeHoca ee
II T. JJ;. 6e3 H3MeHeHHH ee q.OpMbI npH JJ;BHil eHHH H HaJIOil eHnH Ha npy-
rYIO B COOTBeTCTBHH C ee q.OpMOH. EBKJIHJJ; Hyn,JJ;aJICH B aTO f npn JJ;O-
Ka3aTeJIbCTBe npeII.TIOil\eHHH 4 KHUfH I ero COqHHeHMfI «lIa IaJIa» H
npH JJ;OKa3aTeJIbCTBe npej1}IOmeIIHH 8 TOll il e KHHrlI. 8TIf Ju a fIpeJI;-
JIOmeHHH - ca lhIe nepBble [npeJJ;JIOiHeHI1Hl «I-Ia-qaJI», Ha KOTophle oIln-
paeTcH ero HaYKa npH JJ;oKa3aTeJIbCTne OCTaJIbHbIX [npe)IoJT}Oil,eUI1H]>r
[23, c. 363-364]. M6H Koppa 3JJ;eCb IIl\:leCT R BH,n;y TeopeMhI EBIiJIIJ,n;a
o. paBeHCTBe ABYX TpeyrOJIbHHKOB o ABY f CTopOHaM II yr. IY :Mea ;J;Y
HHMH [-10, T. 1, c. 18] H no TpeM CTOpOHa!\1 [10, T. 1, c. 22]: JJ;ORa3an
paBeHCTBO 3TIIX TpeyrOJIbHIIKOB nocpeJJ;CTBOM HaJIOiJ\eHUH O,J;Horo
TpeyrOJIbHHKa Ha JJ;pyro:u, EBKJIHJJ; B JJ;aJIbHeJIJIIHX npeJJ;JlOiIieHI1HX YiIie
06XOJJ;HTCH 6ea HaJIOllieHIIH, HO CCblJIaeTCH Ha 3TH npeJJ;JlOllieHJlH.
aJIee BBOJJ;IITCH nOHHTHe «OJJ;Horo npOCToro JJ;BHmeHIIH» re01\leTpn-
lIeCKOH q.urYPbI, T.. e. ee nepeHoca (TpaHCJ[HI IIH), npJl ROTOPO!\I Bce
ee TOlIKII rrepeHOCHTCH Ha OJJ;HO H TO iRe paCCTOHHlle n OIJ:HO I HanpaB-
JIeHHH 6e3 H3AleHeHHH paCCTOHHHH Me)I\IJ:Y ee TOlIK8l\IU, .u <1)0 pl\IYJIIf-
PyeTcH CJIeJJ;YIOIT(aH «rrpeJJ;nOCbIJIKa»: «fl HalIHHalO AOKa3a Te.TJbCTBO
C Toro, lITO rrpeJJ;IlOCLIJIalO eMY CJIeJJ;Ylom;ee: BCHKoe TeJIO, } OTOpOe lbI
npeJJ;CTaBJIHe1\t1 JJ;HlliI(YII(HMCH n;eJII1KOl\1 B OJJ;HY CTOPOHY n pOCTblM
96
BHa\eHlIeM B O,n;HOM HanpaBJIeHHH, TaKOBO, "liTO BCHKaH ero TO"llKa
,n;BHmeTCH B 3TOM HanpaBJIeHH.n, n03TOMY JIHHHH, OnHCbIBae laH 3TOH
TO KoiI, HBJIHeTCH np.Hl\IOM JIHHHeiI B HeM» [23, c. 365].
B 3ToiI «npeIJ;nOChIJIKe» II CKpbIBaeTCH YTBepm,n;eHHe.. 3KBJIBa.rreHT-
Hoe V nOCTYJIaTY: ,n;nlliIieHlIe, nplI KOTOpOM: Bce TO"llKH ,n;BHmYTCH
no npHM:bIl\1 JII1HHHl\I, B03MomHO TOJIbKO Ha eBKJIH OBOH nJIOCKOCTH
n Hen03MOlliHO HII Ha rHnep60JIH"tJeCKon, HH Ha 3:IJIHIITJIlIeCKOH n:.IOC-
KOCTJI, 11 n03T01\IY «npe,n;nOCbIJIl\a» 116H l\OppbI paBHOCHJIbHa «YCHJJeH-
nOH al\cnO?\le 0 napaJIJIe.rrbHbIX».
flocJIe 9Toro 1'16n I\oppa ,n;OKa3bIBaeT npe,n;JIOn eHHe 1: «EcJIU }le-
m,n;y BC.HRHMJI .n;BYM.H npn1tlbIMH JIIIHI1.Hl\IH, lIaxo]J;HITJ;JIMHC.H B O]]:HOH
nJIOCROCTH, MO)I\IIO npoBecTH ]J;Be BCTpeqalOITJ;He IIX npHl\lbTe JJJIHllJI,
paBHbIe II oopa3YIOI lIC C O,n;HOH H3 nepBbIx .n;BYX JIHHHii paBHbIe O.n;HO-
CTopoBHHe yrJIbI, TO ,n;na nepneH.n;HRYJIHpa, onYIqeHHbIe Ha 3TY JJU-
HUIO II3 ]J;BYX TOqeR ,n;Pyron JIHHHH, paBHbl.
TIYCTb ,n;Be .np.Hl\lble JIHHHH AB II CD Haxo,n;HTCH B O.n;HOH llJIOCI\OCTll
H MeiH]J;Y HllMH npOBe,n;eHbI .n;Be paBHbIe npHM:bIe JIHHHH A C II E}i'.,
II nYCTb yrJIbI A CD II E[i'D paBHbI. Tor,n;a H YTBepil,;a:aIo, 'tJTO 06a
nepneH,n;nRYJI.Hpa, onYII\eHllble Ha JIHHHIO CD JI3 .n;nyx TOqeR JII1Hnn Al;3,
paBHbI (puc. 23).
)];oRa3aTeJIbCTBO :JToro. Bo06pa3HM ce6e TeJIO, orpaHIl"tJeHlIoe JIH-
IIHeR A C H OTpe3ROl\1 CG JIllHHH CD, H nYCTb 3TO TCJIO ABHmeTCH eJIH-
RO l co CTOpOHbI C B CTOPOHY D O.n;HHM npOCTbIl\f npHMOJIHHeiiHbIM J];BH-
n,eHHel\1 B HanpaB.TIeHHH JIlfHHH CD BMeCTe C Ha"lleplIeHH:bIl'tIH Ha He I
u306paiI\eIlHHl\IH JIHHIIR A C II CG, coxpaHHIOITJ;lIMH CBOIO £pOpl\IY.
flplI aTO J TaK3H JIHHJ1H, KaR H306pameHHe JIHHHII AC, HalIep"tJeHa Ha
TeJIe He B HanpaBJleHlllI ero ]J;Bl1il\eHHH, a TaKaH JIllHHH, KaK H306pa-
meHHe JIHHI11I CG, HalI plIeHa Ha TeJIe B HaIIpaBJIeHHII ero ,n;BHa,eHHH,
n03TOl\IY H306parl\eHlIe JIHHHII .CG BO BpeMH Bcero ,n;BHmeHHH TeJIa
B,n;OJIb JIHHHH CD 6y,n;eT pacnOJIOmeHO Ha Hen.
ECJIH MbI npe.n;cTaBuM ce6e, -qTO TO"tJRa C 1I306pail\eHHH JIHHUH AC,
HaqeplIeBHOR Ha TeJIe, nepexo,n;I1T npH ,n;BDmeHHH TeJIa B TO"tJRY F,
TO H306pail\eHHe JIJIHlIlI CG, HaQeplIeHHOe Ha HeM, nepen]J;eT B JII1HHIO
[i'H, TaR RaR OHa npOBe,n;eHa n HanpaBJIeHHH ,nBlfil\eHHH. Ho yrOJI
EFH paBeH yrJIY ACG, n03TOMY H306pameHI1e JIHHllH AC. HalIep-
QeHHOe Ha TeJIe, eCJIU ec TO IKa C nepeii]J;eT B TOqKY F, nOnaL(aeT Ha
JIHHJfIO FE, TaR l\aR JIIIHHH A C paBH8 JIHHIIH f"E. II09ToMY olin COB-
na]J;YT, H ee TOqRa A nona]J;eT Ha TO"tJKY E JIHHHH j"E. TI09T01vIY T01.JRa A
TeJIa nepeBo.n;HTCH B TO"llRY E npHl\lOJIIIHenHbIM ,n;BliiHeHHeM, T. e. onH-
C:bIBaeT npHM:YIO JIIIHI1IO, nOTOMY lITO TaROB nYTb l\a)HLtOH TOql\U TeJIa.
TI09TOMY TOqKa.A OnI111IeT JII1HI11O AE"B, TaR KaK TO"tlKH A H E He onH-
IllYT .n;pyroil npH loiI JIJIHI1H. OT:MeTHM Ha AEB np0l13BOJIbHYlO T01.JKY T
II onycTH1\11I3 Hee nepneHJ].I1KYJIHp Ha CD. 3TO 6y.n;eT TL. YroJI ACD
JIIl60 6Y]J;eT npHMbIM, JIJI60 3Toro He 6YAeT. TIYCTb 011 npHMon. Tor]J;a,
eCJIH TOlIRa A nepeii]J;eT B TO-qKY T, H306pameHue JIIIIIHH A C, HalIep-
eHHoe Ha T(i)JIe, COB1\leCTIiTCH C lIepneH.n;HKYJIHpO 1 T L. TI09TOl\IY
OHII paBHbI. ECJIII 6bI OHO Be COBMeCTHJIOCb C HHl\f, TO OHO COBIIa.1I0 6hI
C HeROTopoii JIIIHIIeii T K II yroJI T KL 6bIJI 6bI npHl\IblM, TaR l\aK OR
paBeH yr&lIY ACD, IIOCROJIbKY IJa06pameBHH [JIIIHHii] AC 11 CG H3
87
TeJIe nOCJIe nepeHoca HMelOT TY j!\e <pOP:MY, a H306pa}I eHlle [JIHHlllI]
CG BcerJJ;a Heo6xoJJ;UMO 6YJ];eT HaXOJ];HTbCH Ha JIHHHH CD. Ho yrOJI
T LK TaKme npHMo:u, TaK KaK T L 6hIJIa IIpOBeJ];eHa nepneHJ];HKYJIHpHO
K CD. IIo3To:MY B TpeyrOJIbHHKe K T L OKa3aJIOCb J];Ba npHMbIX yrJIa,
a 3TO HeB03 IOjHHO, TaK KaK BCHKHe J];Ba yrJIa B TpeyrOJIbHllKe MeHblue
J];BYX npHl\;fbIX yrJlOB, eCJIU MbI CJIOmUM IIX. IIo3TOMY JIUHHH A C COB-
MeCTHTCH C nepneHJ];HKYJIHpOM T L H paBHa eMY, II TOlIHO TaK )Ke Ka}H-
,n;OI\IY nepneHJ];HKYJIHPY, onym;eHHoMY H3 TO"tJ:KH JIHHHH AB Ha JIII-
BHIO CD.
ECJIU me yrOJI A CD He npHMOH, II eCJIU onYCTIITb 113 TOlIKU A
nepneHJ];UKYJIHp na JIIIHHIO CD, 3TO 6YJ];eT AG, TO II306pa)l\eHHe JIH-
HUH A C, Ha"tJ:eplIeHHOe Ha TeJIe, B CJIY"tJ:ae, KorJ];a ee TOlIKa A nepeiiJ];eT
B TOqKY T, COBMeCTIITCH C JIHHHeH T L, a II306pameHue JIHHIIH CG
COBnaJ];aeT C JIHHHe:u LK. IIo:3ToMY TpeyrOJIbHHKII CAG II LT K paB-
HbI, TaR KaK paBHbI IIX CTOpOHbI A C H T L, IIOCKOJIbKY OJ];Ha lI3 BIlX
COB leCTHTCH C J];pyroH, a yrJIbI ACG H AGC OJ];Horo JI3 H X paBHbI yr-
JIaM T LK II T KL ,n;pyroro, Kail\J];bI:U COOTBeTCTBeHHOMY. IIo3Tol\IY
CTOpOHI.)I II yrJIbI 3THX J];BYX TpeyrOJIbHIIKOB paBHbI KamJ];bIH COOT-
neTCTBeHHOMY H nepneHJ];HKYJIHp T K paBeH nepneHJ];IIKYJIHPY AG.
TO"tJ:HO TaK jl e J];JIH KamJ];oro nepneHJJ:HKYJIHpa, onym;eHHoro 1I3 TOlIeK
JIHHIIII AB H JIHHHIO CD. I103TOMY Bce ynoMHHYTbIe nepneHJJ;HKYJIH-
phI paBHbI. 8TO II eCTb TO, "tJ:TO MbI XOTeJIII J];OKa3aTb» [23, c. 365-367].
,[(aJlee M6H Roppa J];OKa3bIBaeT HeCKOJIbKO npe,n;JIOmeHHH 0 'tJeTbI-
pexyrOJIbHHKax: nepBbIe J];na - 0 paBHo6o'tJHOH TpaneIJ;HH:
«[IIpeJ];JIomeHHe 2]. Bo BCHKOM lIeTbIpexyrOJIbHHKe, B KOTOpOI\1 ;UBa
yrJI8 npH OJ];HOH 11:3 ero CTOpOH paBHbI H J];Be CTOpOHbI, npHMbIKalOm;He
K 3TOM CTopOHe, paBHbI, OCTaJIbHbIe J];Ba yrJIa TaKme paBHbI.
IIycTb lIeTbIpexyrOJIbHHK - ABCD, ero yrJIbI BAD H CD A paB-
HbI, a CTOpOHbI AB II CD TaKme paBHbI (pIIC. 24). .fI YTBepiHnaIO, qTO
yrJIbI ABC H DCB paBHbI» [23, c. 3681-
«[I1peJ];JIOmeHHe 3]. Bo BCHKOM 'tJeTbIpexyrOJIbHHKe, B KOTOpOM J];Ba
yrJIa npH O)J;HOH H3 ero CTOpOH paBHbI 11 J];Ba J];pyrlIx yrJIa paBHbI, J];Be
CTOpOHI>I, IIpIIMbIl\alOm;He K nepBo:u CTopOHe, paBHbI.
TIYCTb lIeTbIpexyrOJIbHHK - ABCD, ero yrJIbI BAD II CDA paB-
HbI, 11 yrJIhI ABC II DCB TaK)Re paBHbI. H YTBepil,J];alO, qTO ero CTOpO-
HbI AB H CD TaK)l\e paBHbI» [23, c. 368].
,[(aJIee M6H Roppa npHBOJ];HT OCHOBHoe B 3TOM J];OKa3aTeJIbCTBe V IIO-
CTYJIaTa npeJJ;JIOmeHHe 4 0 cym;ecTBoBaHHII npHMoyrOJIbHHKa, KOTO-
poe, KaK II cym;ecTBoBaHHe napaJIJIeJIOrpaI\IMa B nepB01\{ TpaKTaTe M6H
KOPPbl, .paBHOCHJIbHO YCHJIeHHo:u aKCHOMe 0 napaJIJIeJIbHbIX II HeB03-
MomHO HD B 3JIJlHIITHQeCKo:u, HH B rHnep6oJIuqeCKOii reOMeTpHU.
3aMeTHl\I lITO C 4>OPMYJIHpOBKOH npeJJ;JIOrKeHIlH 4 JJ;aHHoro TpaK-
TaTa COBna.n;aIOT CPOPMYJIHpOBKa npeJ];JIOmeHUH 1 JJ;OKa3aTeJIbCTBa
I16H aJI-XaHCaMa H npeJ];JIOmeHHH 3 TeoplIH napaJIJIeJIbHbIX Xa:UHMa;
npH 3TOM: M6H aJI-Xa:ucaM TaKme OTInpaJICH Ha KUHeMaTHlIeCKUe co-
oopameHHH, XaHlIM me, XOTH 11 He IIpUMeHHJI npll aTOM JJ;nlImeHHH
H ,n;ame OTBepraJI ero B npIIH Hne, Ha JJ;eJIe cym;ecTBeHHO HCnOJIb30-
BaJJ HaJIOiKeHHe 4>nryp.
IIpIIBeAeM npeJ];JIOmeHHH 4 H 5:
a8
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A
D A D F 1/ 8
Pue.24
Rue. 25
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Puc.26
Pue.27
«TIpe}];JI011\eHIIe 4. ECJIH H3 }];BYX TOlIeK O}];HOH 113 }];BYX npHMbIX JIH-
HHH H3 nJIOCKOCTI1. onym;eHbI }];Ba p3BHbIX nepneH}J;HKYJIHpa Ha }J;PY-
rylO, TO OHII nepneH}J;HKYJIHpHbI JiI nepBOH JIHHHH, II BCe nepneH}J;JiIRY-
JIHpM, onymeHHble 113 K3mnOH H3 3TJiIX nBYX JIJiIHHH H3 }];PyrYIO H3-
JIJ060H TOlIKH, nepneH}J;HKYJIHpHbI II I1CXO}];HOH JIHHHH H p3BHbI ;J;BYM
nepBbIM nepneH}J;JfKYJIHp3M.
n YCTb AB II CD - }];Be npHl\II")Ie JIHHIIH H3 ITJIOCKOCTJiI 11 1I3 TO"tleK E
H F JIUHHH AB onymeHbI IIepneHJJ:HKYJIHpbI H3 CD, 3TO EG II FH,
H nYCTb OHH paBHI)I. .H YTBepfl-\}J;alO, lITO OHn nepneH}];IIKYJIHpIlI)I T3K-
1He 1\ AB H "tITO I\HJI-\}J;aH JIIIHHH, npOBe}];eHH3H H3 TOlIl\H o}J;Hoii 113 JIJiI-
HHH AB 11 CD nepneH}J;IIKYJIHpHO 1\ }];pyroii 113 :JTHX JIHHIIH, nepneH-
}J;JiIKYJIHpHa H K JIIIHHH, OT KOTOpoii olla npOBe}];eH3, II paBHa 1\3/K}J;oii
1I3 JIIIHIIH EG II F II» [23, c. 369].
«TIpe}J;JIO/HeHHe 5. ECJIH 1I3 TOlIeK O}J;HOH 1I3 }J;BYX npH:MbIX JIUHl1ii,
npOBe}];eHHbTX 113 I\OHU;OB npHl\-Ioii JIHHI1:H B o,lJ;Hoii n.,TJOCI\OCTII H 06pa-
3YIOru;HX C Heii npHl\fble yrJIbT. onyru;eHbI nepneH}];IIKYJIHpbI H3 npyrylO
1I3 HIIX, TO OHH rrepneH}J;IIKY&TJ:HpHM nepBOH 11:3 3THX JIIIHnH II paBHbI
JIJiIHHII, II3 KOH OB KOTOpoii npOBe}J;eHbI 3TJiI }];Be JIHHUII.
n YCTb H3 1\0HD;on npHl\IOH JIHHHII AB npOBe}J;eHbT B O}];Hoii n&TJOCKOCTU
no)]; npH lbIl\III yr JI al\IH JIIIHIIH A C H BD H H3 BHYTpeHHeH TOlIl\H E
O}];Hoii 11:3 HIIX [T. e. A C] onym;eH nepneH}J;HKYJIHp E}' Ha }];PyrylO, T. e.
BD (pHC. 25). .H YTBep1H}];alO, "liTO OH HBJIHeTCH neprreH}];UHY&"1HPO:M
T31\me 1\ A C II paBeH TJIIHUI1: AB» [23, c. 370-371].
39
aJIee AOKa3bIBaeTCn npeAJIOmeHlle 6: «Ec..TIII npH:Man JIIIHlfH naJ(3-
eT Ha ABe npHMble JIIIHlllI, HaxoAH llecn B O}];HOH llJIOCI\OCTU, JI lIl\leeT-
CH nepneH}];IlKYJIHp K HIIM 06e1IM, nepeCeKalOIqI1H Ka1H}];YIO M3 3TIIX
npn:MLIX JIIIHIIH, TO 3Ta JIIIHIIH 06pa3yeT paBHble Hal\peCTJle}I\a He
yr JIbl II llHellIHHH yrOJI, paBHbIH COOTBeTCTBYIO e IY el\;IY BHYTpeHHeMY.
IIycTb AB II CD - nBe npnMbIe JIIIHHH B OAHOH n.rrOCKOCTII II Ha HIIX
na}];aeT JIHHHH EF, nepneH}];HKYJIHpnaH Hl\1 06elIl\I, H nYCTb ux nepp-
ceKaeT JIHHlfH GHT K (pHC. 26). .H YTBepn-\AaIO, 'ITO Hal\peCTJIema Ue
yrJIbI AHT H D T H paBHbI II TO BHeIIIHIIH yrOJI BHG paBeH COOTBeT-
CTnYlOmel\IY eMY BHYTpeHHel\IY yrJIY D T H.
,I(0I\a3aTeJIbCTBO 3Toro. Pa3AeJIHl\I JIHHHIO HT nonOJIal\1 B T01Il\e L,
onYCTlll\f H3 TO"tJ1\1I L'rrepneHAHI\YJIHp Ha AB, 3TO LM, II npO}];OJIfI\IlM
ero B ero HanpaBJleHUH AO N. Tor}];a OH BCTpeTllT JIIIHIIIO T F, Tal\ Kal\
eCJIH 6bI OH He BCTpeTHJI ee, OH BCTpeTHJI 6bI JIIIHHIO EF, a 3TO HeB03-
MOi-HHO, TaK Ral\ Tor.n;a BCTpeTHJIHCb 6bI }];Be .rrHHIIH, npOBeJ:{ellHble OT
ME no,lJ; npHMbIMH yrJIaMH. TIYCTb OHH nepeceKYTCH B N. TorAa Jl;f N
nepneH,lJ;III\YJIHpHa TaKme II JIJiIHHJiI CD. TI03TOMY yrJIbI LM H II Jl;f LH
TpeyrOJIbHHKa AfHL paBHbI yrJIaM LNT H TLN TpeyrOJIbHHKa NTL,
Ka1HAblii yrOJI CBoe IY COOTBeTCTBeHHOMY, a CTopOHa LH nepBoro Tpe-
yrOJIbHlIl\a paBHa CTopOHe L T BToporo TpeyrOJIbHHKa. TI03TOMY oc-
TaJIbHbIe CTOpOHbI II yrJIbI paBHbI CBOHM: COOTBeTCTBeHHbIM II yrOJI
M H L paneH yr"TIY LT N, a 9TO HaKpeCTJIeiI\aIqJiIe yrJIbI. Ho yrOJI
JlfHL paBeH BepTHl\aJIbHOMY C HHM yrJIY GHB, n03TOMY yrOJI GHB,
T. e. BHemHHii yrOJI, paBeH yrJIY DTH, T. e. BllYTpeHHeMY COOTBeT-
CTBYIO eMY eMY. 3TO H eCTb TO, "tJTO MbI XOTeJIH AOKa3aTb» [23, c. 371-
373].
If, HaI\OHe , M6H I\oppa, UCnOJIb3YH HeHBHO aKCHOMY EB}];OKCa -
ApXHl\leAa, AOKa3bIBaeT V nOCTYJIaT:
«ECJIH H3 KOHqOB npHMoH JIIIHHH npOne}];eHbI ABe npHl\fbIe JIHHHH
n O}];HOH nJIOCI\OCTH no}]; yrJIaMH, MeHbIlIHl\IJiI ABYX npHMblX, TO OHH
BCTpeTHTCH C 9TOH CTOpOHbI.
llYCTb H3 I\OHqOB npHMOH JIJiIHHH AB npOBeneHbI npHMbIe' .,TlHHHJiI
AG H BD B onHOH nJIOCI\OCTJiI, II nYCTb yrJIbI BAG II ABD BMeCTe
MeHbme ABYX npHMbIX (pIIC. 27). .H YTB.epmAaIO, qTO JIIIHJiIII AC II
BD, eCJIH npOAOJImlITb IIX B CTOPOHY BD, nepeceKYTCH.
.D:oI{a3aTeJIbCTBO 9Toro. To, tITO OAHH H3 yrJIOB BAG H ABD l\;leHbIlIe
npHMoro, He06xoAHMO. TIYCTb 3TOT yrOJI - ABD. OnycTHM 1I3 TOqKII A
nepneHAHI\YJlHp Ha BD, 3TO 6YAeT AE. OTMeTIIM Ha AC npOH3BOJIb--
HYIO TOqKY F H onYCTHM H3 Hee nepneHAHI\YJIHp Ha AE, 3TO 6YAeT nep-
neHAHI\YJIHp FG. TorAa ABe KOHeqHbIe JIHHHH AG II AE TaI\OBbI, 1ITO
JIHHHH AE AJIJiIHHee, 1IeM JIHHIIH AG. Ho MeHbmylO H3 HHX, T. e. AG,
l\fOmHO 6paTb KpaTHoH AO Tex nop, nOKa KpaTHaH He CTaneT AJIHH-
Hee, -qeM AE. TIYCTb KpaTHaH, KOTopaH AJIHHHee, 1IeM AE,- lllIHIIH
AH. OTJIOfKHM Ha GK [JIHHIIII], paBHbIe .r.IHHIIH AG, 3TO GT, T K II
KH. OTJIOmHM TaKme Ha JIHHHH AG [JIHHHH1, paBHbIe JIHHIIII AF,
CTOJIbKO »,e pa3, Kal\OBO 1IHCJIO [JIJiIHHH] GT, T K, KH; 3TO F L, LM,
MN. ECJIH AG l\leHbme Toro, TO AOCTaTo"tJHO AJIH 3Toro, A06aBHM
K ee AJIUHe HeAoCTalO ee. .H YTBepmnalO, 1ITO ., HHHH AN nepeCeqeT
JIHHHIO BD. .D:0I\a3aTeJIbCTBO 3Toro: BOCCTaBHM B T01IKe T nepneH-
40
AlIRYJIHp RAE', 3TO TS II onycTH}[ HB HerO nepneH}J;IIRYJIHp 113 TOq-
KH }\ 3TO f'S. COeAlfHUl\[ TO RY L C TO ROii S JIHHlleiI LS. Tor]J;a 113
ROH OB npHM:oiI JIHHIIII G T npOBe}J;eHbI ]J;Be npH:MbIe JIHHHH Gl? II T S,
oopa3YIOID;He C HeM npHl\Ible yrJIbI, a H3 TOQRH I? O]J;HOH H3 3THX JIlfHH:U
onYI.IJ;eH nepneH]J;IIRYJIHp F S Ha ]J;pyrylO. TI03TO IY JIIIHHH [/S nepneH-
AIIKYJIHpHa II H f'G H paBHa JIHHllH GT Ho JIHHIIH GT 6bIJIa paBHa
JIHHHH AG, n03TOl\[y JIHHIIH FS paBHa JIHHHH AG. l.ITo me KacaeTCH
JIMHlIll }'L, TO HCHO, "IlTO OHa nona]J;aeT BHe Toro, lITO HaxoJ];HTCH :Mem-
)J.y JIHHHHMH F S H G T. .HCHO TaRme, "£ITO yrOJI GF S npHMOH H "IlTO yrOJI
AFG- MeHbllie npHl\IOrO, TaR RaK yrOJI AGll npHMO:U, a n O]J;HOM Tpe-
yrOJIbHHKe He MO HeT 6bITb ABYX npHMbIX yrJIOB. JIHHHH }'G T81\JHC
na}J;aeT Ha JIHHHH AT H F S, npH eM OHa nepneH]J;IIKYJIHpH3 H1.1: 06eHM,
H Ha HHX na}J;aeT TaRn e npHl\IaH JIIIHHH A C. ll03TOMY BHelliHIIH yrOJi
Ll"S paBeH BHYTpeHHeMY yrJIY FAG, COOTBeTCTBYIOm;eMY eMY. TaR
KaR 3TH ABa yrJIa TpeyrOJIbHHROB AFG HIlLS paBHbI, IIX CTOpOHbI
AG H F S TaRme paBHbI, 3 CTopOHa AF O]J;Horo H3 HHX paBHa CTopOHe
[f'L] }J;pyroro, TO H OCHOBaHl1e paBHO OCHOBaHHIO II OCTaJIbHbIe yrJIbI
paBHbI OCTaJIbHbIM yrJIaM, K8rH]J;bIH yrOJI CBoeMY COOTBeTCTBYIOm;eMY.
TI03TOM:y yrOJI f"SL paBen yrJIY AGF. Ho yrOJI AGF npHMOH, n03TOMY
H yrOJI F S L npHMoll. 1-10 yroJl F S T TaRme 6LIJI np.HMbIM. n03TOMY
JIHIIHH TS n SL _0 B O]J;HOl\1: HanpaBJIeHHH, T. e. 3TH ]J;Be JIHHIIH npe]J;-
CTaBJIHIOT co60iI O}J;HY npHl\IYIO JIHHIIIO, coe)J.IIHHIOru;ylO TOqRH T H L,
T. e. JIlIHHIO T SL; OHa nepneH]J;HRYJIHpHa K AH. TOlIHO TaR He l\:IbI
]J;ORamel\f, lITO npHMaH JIHHHH KAf, coeJJ:HHHIOm;aH TO"£lKH K II M, nep-
neHAHKYJIHpH3 K AH H lITO [npHMaH] JIHHHH H N nepIIe )J.HKYJIHpHa
R AH. TI03TOMY yrOJI AHN npHl\1:0M. Ho yrOJI BEH TaKme npHl\lo:u,
TaK RaK OH paBeH yrJIY AED. ll03TOMY npHMa.H JIHHHfI AEH na]J;aeT
Ha }J;Be npHMbIe JIHHIIH H N H BD, npHlIeM HX naKpeCTJIell\arIl.He yrJIbI
paBHbI. TI03TO}IY 01I1I napaJIJIeJIbHhI, T. e. He BCTpe"£laIOTCH, ]1,8rI\C €CJIII
np0]J;OJIrRHTb IIX ]J;O 6eCKOHelIHOCTH. Ho [JIHHH.H] HN BCTpe aJIa
JIHHHIO A C B TOqKe N, a JIHHHH A C pacnOJIOmeHa no APyryIO CTOpO-
HY 01' JIIIHHII BD. TI03TOl\IY JIHHH.H HN BCTpeQ3eT JIIIHHIO ALCN,
T. e. nepeceKaeTCH C JIIIHlleH F LCN. 3TO H eCTb TO, "IlTO lhI XOTeJJH ;0:0-
Ra3aTb» [23, c. 373-374].
l\Io RHo npe]J;nOJIaraTb, 111'0 06pam;eHHe II6H I{OPPbI K KHHel\'IaTHKe
6bIJIO CBH3aHO C ero l\fHOro06pa3HbIl\IH 33HHTIIHl\1:H MexaHHRoi1, B TOM
1IIICJIe leX3HHRO:U He6eCHbIX nBHlKeHII:U; nO-BH]J;IIMOMY, HeMaJIYIO pOJIb
CbIrpaJIO 3]J;eCb II TO 06CTOHTeJIbCTBO, 111'0 M6H Roppa 6bIJI RpynHeii
llilIl\1 3H3TOROl\I ApXHl\le]J;a H nepeBO]J;lIHROM MHorHX ero COlIHHeHl1ii..
,[(OKa3aTeJIbCTBO 116H ROppbI B ero BTOpO 1 TpaRTaTe OR33aJIO 60JIb-.
llIoe BJIHHHHe Ha nOCJIe}J;Ylom;ee pa3BUTHe TeoplIH napaJIJIeJIbHbIX. MbI
Y He YR33bIB3.rrH, 111'0 lI}J;eH npHMeUeHH.H «O]J;Horo npOCToro }J;BllmeHHH»
nplIMeHHJIaCb B TeOpJIH n3p3JIJIeJIbHbIX M6H aJI-XaHCal\fo?t1. 11 XOT.H
XaiiHA-1 peUIIITeJIbHO B03pailiaJI npOTIIB npllMeHeHIIH B reOl\leTpHH
ABII ReHHH, a M6H Hoppy H33bIB3JI «TOJIbRO rrepeBO]J;"IIIIROM "H a aJI",
BHeCIllII 1 B HIIX JIIfIllb HeCROJIbRO Hc.npaBJIeHIIM» [32, c. 119], eCTb Bce
OCHOBalIHH nOJIaraTb, "liTO OCHOBHoe cO}J;epmaHlIe TpaKTaTOB M6H
KOpphI, B03l\-IOiI\HO, lIbeA.f-HH6y,n:b H3JIOmeHIIH, 6bIJIO XaUHMY XO-
pOlliO H3BeCTHO.
41
TpaKTaT 0 napa.J'1 TIeJlbHbIX JIIIHIIHX ObTJJ HanUCaH n J];pyrH: 3Hal\le-
HIITbI 1 yqeHhI -[ IX B. fl(Ky60M 116H McxaKoM HJI-l\HHAIf (y 1. 874)
TaK a,e, l\aK M6H Koppa, pa60TaBllllIM B BarAaJ(e. AJI- [\IIH,I:Uf ObIJI
1I3BeCTCH rJIaBHbIl\1 06pa30M RaK .pIlJIOCO.p (ero Ha3hIHaJIII (I)ai1.na cy <p
aJl-(apa6 -- «.pHJlOCO.p apa60B»)}, HO OH 6bIJI O,[{HUl\1 U3 KPynHeiinn x
yqeHbIX-aHl(IIK TIOl1eAIICToB II aBTOpOl\1 l\IHOrIlX TpaKTaToB 110 pa3JIlIq-
HbI 1 06J1aCTH 1 MaTel\/[aTIIKII, a TaKn e 110 aCTpOHO [1I11 II .pu3IIKe. Tpal\-
TaT a.rr-J\IfH.n;1I He AOIIIC.rr .n;o Hac, HO OH IJ;IITlIpyeTcH B TpaKTaTe aJI-
BHPYHU 0 napaJIo1"1eJIbHbIX JIIIHUflX. n03TOM:Y :MhI paCC:\IOTPIl [ Te cne.n;e-
HUH, KOTopble COXpaHIIJIIICb 0 TpaKTaTe aJl-I-\HH,L(U, B pa3!1.eJ:e. IJOc.-
BRrl eHilOM pa360py TpaKTaTa aJl-BlIpYHM (CM. c. n5 '.
AH-HaiIpM3U, KOl\Il\:leHTapIlH KO-
Toporo K EBI\JIHAY co.n;ep»<aJIH H3-
JIOmeHlle nonbITKII .n;ORa3a TeJIbCTBa
aKCHOMbI 0 napaJlJleJlbHbIX, npll-
Ha.n;JIema eH AraHIICY, 6bIJI aBTO-
pOM H caMOCTOHTeJIbHOrO TpaKTaTa,
nOCBHmeHHoro TaKOMY .n;OKa3aTeJlb-
CTBY,- «TpaKTaTa 0 nOKa3aTeJIbCTBe
H3BeCTHoro nOCTYJIaTa EBI\JlH.n;a»)
(PHCaJIa .pfi: 6aiIaH aJl-l\ly anapa aJl-
Mamxypa JlH YI JlHAHC). <DaK-
CHMHJle TerepaHCKOH PYKOlllICH 3Toro TpaKTaTa ObIJIO He.n;aBHO ony6-
JIHKOBaHO Ryp6aHII B KHllre «MaTeMaTHKH MpaHa OT aJI-XOpe3l\IH .n;o
1I6H CHHbI» [14]. TeKcT TerepaHcKon: PYKOITHCH COBna.n;aeT C TeKCTOM
PYKOITHCH ITapHmcKoH HaIJ;:noHaJIbHOH 6JiI6J1HOTeKII, KOTOpbIH B He-
CKOJIbKHX MeCTax HCllOpqeH, HaqaJIa 06eHX PYKonllceH COBna.n;aroT
TaKme C HaqaJIOM PYKOllHCH 6bIBmeH TIpyccl\oi-i rocy.n;apcTBeHHoH 611-
6JIHOTeKH B 3arra.n;HoM BepJlHHe, YTpaqeHHOH no BpeMH BTOpOH MH-
pOBOH BOHHbI (HaqaJIO PYKonHCH nocnpOll3BeAeHO B KaTaJIOre AJlb-
Bap.n;Ta [35, c. 314]). B napHmcRon: II 6epJlHHCI\OH PYKonHCHX aBTOp
HMeHyeTcH aJI-<1>aJ(JI H6H XaTHM aT- Ta6pH3H, qTO TOJIbKO O.n;HOH TOq-
Kon Ha.n; «H» H O.n;HOH TOqKOH no.n; «H» OTJllIqaeTCH OT HMeHH aH- I-IaH-
pH3H; B TerepaHcKoH PYKOllHCH no.n; «H») II Ha)]; «H» HeT HH OJ:{HOH TO -
RH, T. e. HMH aBTopa B 3TOH pYKonllCH M01HHO npOqeCTb u «aH-HaH-
p:U3H») II «aT-Ta6pU:3ih> (Ryp6aHH qlITaeT «aH-HaiIpH3H»)). BeCbl\[a
B03MO}RHO, qTO B PYKOllHCH, C KOTOpOH crrllcaHbI rrapHlliCKaH H 6ep-
JIIIHCKaH PYKOllIICH, JiIMH aBTopa 6bIJlO HanHcaHO TaK me, RaK B Tere-
paHCKOH, H TOqKH 6bIJIII paCCTaBJIeHbI nepellllCQIlKOM, 3HaBmllM ro-
po.n; Ta6pH3 II He 3HaBmHM ropo.n;a Haiipu3a (TaKoii ropo.n; IIMeeTCH
B6JIII3H illHpa3a).
TpaRTaT aH- HaiipH3H, TaK me l\aK BTOpOH TpaKTaT M6H ROPPbI,
HaqHHaeTCH C .pJiIJIOco.pcKoro 060CHOBaHJIH cy ecTBoBaHHH panHo-
OTCTOH HX npHMbIx, o.n;HaKO co06pameHHH aH-HaiIpJI3H, OCHOBaH-
HLIe Ha TOM, qTO «paBeHCTBO no CBoeH rrpHpOne npe.n;mecTByeT
pa3J1HqHIO», ropa3,ZJ;0 MeHee y6e,ZJ;HTeJIbHbI, lIeM I\HHeMaTH1JeCKlle
coo6pameHIIH 116H ROppbI. I1pHAH K BbIBO.n;y, qTO «He06xo.n;HMO, "IlT06hI
cy ecTBOBaJIH JlIIHHII, Mem.n;y KOTOpbIMU coxpaHHIOTCH paCCTOHHHH»),
aH-Han:pH31I Cq:nTaeT, qTO «K rrepBJIQHbIM YTBepfI AeHHH 1 OTHOCHTCH TO,
7
T eopuR
napaJlJle.llb1-lblX
a1-l- H aupu314
42
qTO BCflKlle npHMbIe lUIHJIII Ha nJIOCI\OCTII, MerHJJ;Y KOTOpbIl\lH COXpaHH-
IOTCH paCCTOHHIIH, He BCTpeqalOTCH npH npo.n;OJIfI eHIIJJ,I(O 6eCI\OHeQ-
HOCTH, l\aK AB II CD B 3TIIX eTbIpex npe,I(JIOmeHHHX. TIpH 3T01\f pac-
CTOflHI1e Me,H,n:y HIIl\111 - nepneH,I(IIKYJIHp 1\ HUM 06e1lM; H3 l\amJJ;OH
TO IRI1 OAHOn: [JIIIHMII] IOjI\HO npOBeCTH 1\ .n;PyroH 6eCI\OHeQHO !\-IHoro
JIIIHUii, HO He06xOAI1 [0, QT06bJ paCCTOHIIHe Men-\AY JJI1HIIHl\IU 6bI.TI0 6bI
ca IOH 1\0pOT1\0n: 113 HHX, I\al\ EG (pIIC. 28). 8TO paCCTOHHlle Mem)J;Y
HIIMII HBJIHeTCH Cal\10H KOpOTKOH 113 Bcex JIHHHH, BbIXOAHIIJ;HX H3 O,I(HOH
TO"tJI\M [OJJ;HOH .TJMHIIII 1\ .n;pyroii], Tal\ I\al\ paCCTOHHHe Ol\pymeHo [IIMIl).
ll03TOM:Y 3TII paCCTOHHIIH paBHbI» [14, c. 86].
,I:(aJIee 8H-Haiip1l311 nplfBo.n;HT ceMb rrpe.n;JlOi-ReHHH:
«rrpe,n:JIOrI\eHHe 1. .H YTBepm.n;alO, "'ITO 3TH paCCTOHHHH nepneH.n;Il-
KYJIHpHbl 1\ 06ellM JIIIHIIHM. ECJIH 3TO He Tal\, TO nYCTb EH nepneH-
nHKYJIHpHa K AR, Tal\ "'ITO [yrOJI] II rrpHMo:u (pHC. 28; B 3TOM II ABYX
CJIeJJ:YIOII{HX rrpeJJ;JIO}l\eHIIHX Ha 3TOl\1 "'IepTe}l\e aH- H a:upU3JiI, Ha HOTO-
p'Ol\f COBl\fem;eHbI '1epTe,I\H 1\ HeCI\OJIbl\HM npeJJ;JIOrHeHIIHM, IIl\IeeTCH
B BIIJJ;Y TO l\a H, HaxoAHm;aHCH Ha npHMo:u AB). Ho [yroJI] G 60JIbIlle
Hero, Tal\ I\al\ E'li ,I(JIHHHee, II G II H BMeCTe 60JIbIIIe' JJ;BYX npHMbIX
[yrJIOBJ, a 3TO HeJIenO, Tal\ Kal\ l\amJJ;bIe .n;Ba yrJIa TpeyrOJIbHHRa M:elIb-
HIe HIIX. ECJIH [yroJI] II npHl\IOH, TO G MeHbIlle H II EH KOp01Je EG,
HO no rrpe,n:nOJIOiI\eHIIIO EH 6bIJIa .n;JIIIHHee, a 3TO HeJIeno. fI03TOl\fY
[EG] nepnen,I(HRYJIHpHa K 06eHM [JIIIHIIHl\f]. 8TO II eCTb meJIael\10e.
TIpeAJIOrHeHHe 2. JlI060H 1I3 nepneHJJ;HKYJIHpOB, HaXOJJ;HIIJ;IIXCH l\fe-
iHAY ,n:BYMH JJIIHMHl\fll, 1\8K EG,- Cal\faH KOrOTKa.H H3 JIHHIIH lHaxo-
AHIIJ;IIXCH l\fen AY HIIMH]. ECJIH 3TO He Tal\, TO nYCTb caMoe 1\0pOTKOe
paCC1'OHHne - Elf (pIIC. 28). ECJIH 3TO - caM8H KOpOTl\aH JIHHIIH, OHa
nepneH,I(III\Y JIHpHa 1\ ooeUM JIIIHIIH!\-I, II yr .JlbI G II H B TpeyrOJIbHHl\e
EG1I npHMbIe, a 3TO HeClTJeno. Il03TOMY EG - paCCTOHHlle l\Ie,"AY
HJI III. 81'0 II eCTb rl"eJIaeMoe.
fJ pe,I(JIOrl\eHHe 3. l arl\JJ;bIH nepneHJJ;HI\YJIHp K OAHOn: lIa .n;BYX JIIIHII:U,
TaKoii, Kal\ EG II AB, HBJIHeTCH nepnen.n;HI\YJIHpOM K ,I(pyroii. ECJIII
3TO He Tal\, TO nYCTb EH - nepneH.n;HKYJIHp K CD (Cl\tl. pllC. 28),
TorAa 3TO - paCCTOHHHe Mem.n;y HIIMJiI B CIIJIY rrpe,I(JIOi-ReHIIH 2,
a E'G - TOjJ e paCC1'OHHHe Mem.n;y HHMH. il03TOMY [yroJI] H rrpHMo:u,
KaK H G, a 3TO HeJIeno. TI03TOMY EG nepneH.n;IIHYJIHpHa K HHl\f 06eHM.
3TO II eCTb rHeJIael\loe.
npeJlJOmeHUe 4. EC !JU I OCCTaB.JleH nepneH,lJ;IfHYJIHp H ABYM: npH-
l\lbIM JIIIHHHM, TO OHH coxpaHHIOT paCCTOHHHe Me»-\JJ;y c060:u. ECJIII HeT,
TO nYCTb coxpaHHeTCH paCCTOHHHe Mem,I(y EH H AB (CM. pnc. 28,
r.n;e IIMeeTCH B BHAY TOQK8 H, He JIemam;aH Ha npHl\fOH AB), a EG -
nepneHAHI\YJlHP K AB, HO He K CD. Tor.n;a 3TO nepneHAHKYJIHp R EH
BBIIAY npe,I(JIOfI eHHH 3. Ho TorAa "'IaCTb GEH - npHMoH yrOJI - paB-
Ha n;eJIOMY GED, a 3TO HeJIeno. TI03TOl\IY CD He coxpaHHeT paCCTOHHlle
C AB. 8TO H eCTb Tpe6yeMoe.
IlpeJJ;JIomeHlle 5. ECJIII onYCTIITb 1I3 OJJ:HOH 1t3 JJ:BYX npHMbIX JIIIHH:U,
TaKoH, HaH CD, Ha ApyrylO, TaI\YIO, l\aK AB, l\fem.n;y KOTOpbIl\tIH co-
xpaHHeTcH paCCTOHHlle, .n;Ba nepneHJJ;HKYJIHpa, TaKlIe, RaK EH H GF
(pIlC. 29), TO [paCCTOHHlIe] l\le n.n;y HX OCHOBaHIIHMH Ha BTOpO:U [JlH-
HIIH] H H F paBHO [paCCTOHHIIIO Merl\AY IIX OCHOBaHHHl\flI] H3 nepBOH
.48
.D E l'
/I 6 $ F A
B H C A Y
Puc.28 T
]} G E C II K E
N z N 0 l I? X
Puc.30
B F H A
Puc.29
[JIHHHH] E H G. 3TO nOToMY, lITO ABa nepneH}J;HRYJIHpa coxpaHHIOT pac-
CTOHHHe MeiHAY c060ii B CHJIY npe,I(JIOmeHHH 4. 3TO H eCTb Tpe6yeM:oe.
IIpe}J;JIOiJ\eBHe 6. ECJIH AB II CD coxpaHHIOT paCCTOHHlIe MemJJ;Y
C060H II Ha BHX naAaeT GH (pHC. 29), TO ABa OAHOCTOpOHBHX yrJIa
[BMeCTe] paBHbI ABYM npHMbIM [yrJIaM]. 3TO nOTOl\IY, lITO ABa rrepneH-
ABI\YJlHpaGF H EH paBBbI, a paCCTOHBHe MemAY HM1\fH HF T8KOBO, I\al\
EG, [CTopOHa] me HG - 06m;aH. TI09TOM:Y yrOJI E'GH paBeH [yrJIY
G]HF, HO [yrOJI] EGH BMeCTe c [yrJIOM] HGD p8BHbI ABYl\1 npHMbIM
[yrJIaM]. TI09TOMY [yrOJI] DGH BMeCTe C [yrJIOM:] BHG paBHbI AnYl\1
npHMI;d:M [yrJIaM]. 3TO H eCTb TO, lITO MbI XOTeJIJI.
ITpe,I(JIOmeHHe 7. ECJIH rrpHMaH JIHHHH A C na}J;aeT Ha ,I(Be rrpHMIJIe
JIfIHHH AB H CD (pHC. 30) II.CYl\IMa ,I(BYX BHYTpeHI HX O.n;HOCTOpOHHHX
[yrJIOB] CAB II ACD MeHbrne ABYX npHMbIX [yrJIOB], TO OHH BCTpeTHT-
CH C Toii CTOpOHbI, C 1\0TOpOH ABa yrJIa MeHbme }J;BYX npH1\fbIX r yrJIOB].
BbIJIO AOKa3aHO, lITO AB II CD He coxpaHHIOT paCCTOHHHfI MeiI\.n;y co-
60ft. TI03TOMY onYCTIIM neprreHAJiIKYJIHpbI AE Ha CD n AG Ha AE.
Tor,n:a AG coxpaBHeT paccToHHHe c CD. B03hMeM Ha AB BeJIJiIqHHY
AH, onYCTHM neprreH,n:HKYJIHp HF Ha AG H [npOAOJI}I HM] ero ;n;o [TOlI-
KIl] K Ha CD. Tor,n:a HF IIJIJiI 1I3MepHeT BeJIHlIHHY F K, IIJIH [6y.n;YlIlI
D3HTa KpaTHoii] npeBbIrnaeT ee Ha JI36bITOK LK. TIYCTb HF H3MepHeT
FL, nYCTb OHa - YTpoeHHaH Ii [TOlIKH] AeJIeHHH - H H M. BOCCTa-
BUM neprreH,n:HKYJIHp LQN K FL n rrpO,IJ;OJImHM AE AO R, a AH.n;o Z.
Tor;n;a ZN H AG coxpaHHIOT paCCTOHHJIe l\fen,.n;y C060H. TaR iRe Ral\
XN II CD. ECJIH 6bI AB BCTpeTHJIaCb C XN, TO AO 3Toro OBa He06xo-
J];HMO BCTpeTIIJIaCb 6hI C CD II CD rrepeCeRJIaCb 6bI C rrpHMoH [JIUHIIeft]
AB [B3HTOii] BMeCTe C ee npo,n:OJIllieHHeM BZ B ee HanpaBJIeHIIH. [OT-
JI01KHM] paBBLIe AH, TaRHe, KaK HO II OZ, H coeJJ;MHIIM XQZ. 8TO npH-
MaH JIHHHH, BCTpelIalO aH AB B Z, TaR KaR paCCTOHHlIe MeiRAY AG
II XN rrpoxo,n:RT lIepe3 0, 9TO SQ. TI03TOl\IY AR, FL II SQ - paCCTOH-
44
.
HUR [P;I,.n;y AG II XN, 3TH paCCTOHHHH COXpaHHIOTCH, II lJIIIHlIlIJ He
BCTpetIHfOTCH. llpOBeJ];eM Qepe3 H paCCTOHHl1e l\leil\JJ;Y LTIUHIIHMHJ AR
]I. L')Q, 3TO fJflY, a Tal\fl-\e qepe3 0, 3TO OAfT n03TOMY Afl paBHa
JIO. a [yr..-IbI] Yu P rrpHMble. 11pll :JTOl\1 oTceRYTcH Ana [TpeyrOJIbHII-
f\31 iI/II) n 0 H}'" rrplIqel\1 OY 6YJ];eT Tal\On ,He, I\al\ A I)., T. e. B CIIJIY
JJ;OHa3HHI-IOrO B npe,n:o!TfOmeHIIH 5, I\al\ S}T Ho SIT - TaKaH i-I<e, l\aK
}/11. },,,,O - KaK HAl. TI03TOMY SO [paBHa] Ilk!. ll03TOMY OCTaTOK
OQ - Tal oH iI\e., l\aK M L, M H - Ral\ OY a yroJI ZOQ - KaK
ryroJII flOY TI03T01\;IY [JIHHIIH] QL - TaKaH II e, I\al\ Y H, npHl\fOH
yro..l OQZ - KaK [yrOJI] HYO, a ["yrOJI] OQL - npHMon. TI03TOMY
y"QZ - npHM:aH JIHHHH, BCTpeqalOm;aHCH C AB B Z, II, CJIeJJ;oBaTeJIb-
JlO. ./-lB HCTpeQaeT CD AO Toro, l\aK OHa -BCTpeTllT XZ. 31'0 If eCTb Tpe-
6ye 10e, )TO - TO, "'ITO MbI XOTeJIH» [14, c. 86-87].
· C.,lPL\YCT oTMeTlIiJ , "'ITO, npJiIBeAH «<PIIJIOCO CRoe 060cHOBaHlIe»
eYHl.ec.TBOBaHHH pannOOTCTOHID;IIX npHMbIX, aH-I-IanpII3H OTHOCHT
YTBepjl AeIJMe 0 cyIi ecTBOBaHHH paBHOOTCTOHmHX rrpHl\1bIX K «rrepBII"'I-
IIhL'l YT t pfH,UeIllIfll\f», T. e. K aKCIIOl\Ial\I. 81'0 rrepBaH B HCTOpllH TeopMH
na pa.:"1..TJ(\ LHbIX JIHHIIH OTl\pbITaH 3aMeHa aKCHOl\lbI 0 rrapaJIJJeJIbHbIX
,lJ.py['oi1, 1'13 KOTOpOH 3TY aKCIIOMY l\fOiJ\HO BbIBeCTII, OAHaKO ,lJ;eJIaeT
aTO aII- II aHplI3H e e P06RO, corrponOiK.n;aH HOBYIO aRClIo?tIY rronI)ITKOH
"<pn.JIocolllcKoro 060cHoBaHIIH. AORa3aTeJIbCTBO aH-HaiipII3I1, TaR iue,
Hal\ BTopoe J];ORa3aTeJILC1I60 116H l\OppJ)I, OCHonaHO Ha J];ORa3aTeJIbCTBe
cyu eCTB()BaIlHH npHl\foyrOJIbHIJRa B npeAJIOil\eHHH 5 (nepBble 'tIeTbIpe
npe,n rJOr"eHMH TpaKTYIOT 06 o6mlIx nepneHJ];III\YJIHpaX paBHOOTCTOH-
JLI.UX npHI\JJ)IX). fI pH AOKa3a TeJIbCTBe V nOCTYJIa Ta B npe]l:JIOmeHIIH 7
RH-IIanpIl3H, l\aK II Y I1GH I\OPPbI, HCIIOJIb3yeTcH al\CHOMa EBAoRca -
A pxn -IcJJ;a.
TpaI TaT aH-I-IaHpH3R ynOl\fHHaeTCH B 1\0 fMeHTapHHX XaH.H1\Ia
BMeCTC C TpaKTaTa:rv11I A6y ,n:iHa' apa aJI-Xa3HHa (YM. OK. 965) II
A6y'AGAaJIJIaxa arn-IIlaHHH (X-XI BB.), ynoMHHaHIIR 0 1\0TOpbIX
B npyrMx HCTO"'IHHl\aX HaM HeH3BeCTHbI. 063THX TpaKTaTax Xaii.HM
nlICaJI: « tl TO KacaeTCH TaKHX rr03AHeHIIIHX Y"'IeHbIX, RaK aJI-Xa3HH,
arn-lllaHHM., aH-HaftpH3J1, KOTopbIe nbITaJIHCb JJ;OKa3aTb 3TO, TO HH-
RO}lY H3 HHX He YJ];aJIOCb npe.n;CTanHTb CTpororo JJ;ORaaaTeJIbCTBa, KaiI -
lJ;bIH II3 HHX OCHOBbIBaJICH Da TOM, "£ITO HBJI.HeTCH He 60JIee JIerKIIl\1 .no-
ny eHHeM, "'IeM AOKa3yeMoe» [32, c. 114-115].
,IJ,0 Hac He AomeJI H Tpal\TaT XpHCTJlaHCKOro cnH eHHHKa IOxaHHbI
1I6H IOcy<pa aJI-Xapnca H6H aJI-BaTpHKa aJI-Racca (YM. OK. 980)
«0 nepeCeqeHJlH ABYX np.Hl\fLIX JIHHH:U, DhIxoAH HX H3 KOHD;OB npHMoii
JIHHlIll no.n; yrJIa?tIH, MeHbIllIIl\1H .n;nyx npHMbIX» (<DR HJITHi58 aJI-!a -
,!aHH aJI-MycTa RMaHII aJI- apHAiI aHH ltlHH ,!,apa aH a 1 ?tfycTa-
l fil\1 'aJIa a \aJIJI MHH 33.BHiiaTaHH 1 8Hl\laTaiiH), nocBHu eHHbIii CYJIT8HY
CaH<p aJ];-).::(aYJIe, YMepmel\{y B 950 r. STOT TpaKTaT yrroMIIHaeTCH IOxaH-
HOH 1I6H IOcy ol\{ B ero «!\HHre 0 pan;HOHaJIbHblX H nppan;IIOHaJIbHbIX
BeJIHqHIIaX» (Ma1 aJIa i1:-JI-Ma 3.AHp aJI-MYHTal a Ba- - ,aM?tf) [17,
c. 215], a TaKme aJI-Xanacpli D ero IJ;HTHpOBaBmeMC.H yme Hal\11I nlICb-
le K aT- TYCH.
45
Te n e OC06eHHOCTH 1I3JIO}HeHIIH
TeOpHII napaJIJIeJIbHbIX, 1ITO y Ara-
HHCa, M6H I\oppbl H aH-HaiipII3M,
BCTpe1IaIOTCH H OAHOM 113 TpaKTa-
TOB 3Hal\leHHTOrO Y1IeHOrO-9H HK-
JIOrreAHCTa M6H CIIHbl (ABIIIJ;eHHbI).
A6y <AJIM aJI-XycaiiH H6H(A6)l.aJIJIax
U6H CHHa (980-1037), ypO}I\eHe
ByxapbI, pa60TaBIlIMH B Byxape, Xope3Me M pa3JIII1IHLIX ropoA8.x
11 paHa, rrpoc.rraBHJICH CBOHM «l\aHoHol\f Me]J:II IIHbD), no ROTOPOMY Y1IH-
JIHCb Bpa1IH l\aK CTpaH nCJIaMa, TaR H 3anaAHOH EBponbI B TeqeHlIe
HeCROJIbKHX CTOJIeTHH. HapHAY C 3THM OH 6blJI aBTOpo .{ Tpex 9IL( HRJIO-
rrenH1IeCI\HX TpaKTaTOB - «HHHrH HC eJIeHlIfI []J:YlIIH OT HeBemeCTBa]»
(RHTa6 am-illH«pa) II ee COKpa eHHH - «I\HHrH cnaCeHHH» (1\IITa6
an-BaAiRa) Ha apa6CKOM H3bIKe II «}\HHrlI 3HaHHH» (;:::(aHllIII- HaM:e)
Ha «papcn, a TaKme pHJJ;a MaTeMaTII1IeCI\I1:X 11 aCTpOHOMlf1-1eCI\HX TpaK-
Ta TOB. 8H III\JIOne]J:H1IeCKHe TPYAbI M6H CJiIHbl cO]J:epmaJIH «I\BaApH-
BIIYM», T. e. reOMeTplIlO, ap:u«pMeTHKY, aCTpOHOMHIO H TeopHIO MY3hI-
RH, rr pU1IeM: reOM:eTpHH H3JIaraJIaCb no EBKJIII]J:y C ,n;onOJIHenlf.fIM:1I aB-
Topa. K TaKIIl\I ]J:onOJIHeHHHM OTHOCIITCH pa3AeJI 0 napaJIJIeJIbHbIX JIH-
HHHX reOl\IeTplI"tJeCROH rJIaBbl «}\HHrH 3HaHHH», ony6JllIROBaHHoiI BO
«ppalln;Y3CKOM: nepeBOne M. AlileHa II A. Maoce [38] II B PYCCRO-M nepe-
BO]J:e A. M. BoroYT]J:HHoBa [11] (.pHJIOCO.pIIH II .pH3Hl\a) H B.A. Po-
3eH«peJIbAa II H. A. Ca.noBcKoro [12] (MaTeMaTlIl\a, racTpoHoMIIH II l\IY-
3bll\a). B HaqaJIe 3Toro pa3AeJIa 116H CHHa TIHIIleT:
«JIIIHIIH, OT,n;eJIeHHble APyr OT ,n;pyra, MorYT 6bITb pacnOJIOmeHhI
TaR, 1ITO 1\0Hen; o.n;Boii HaKJIOHeH K ]J:pyroiI; eCJIH IIX TIpo.n;OJIil\aTb
B 3TY CTOPOHY, TO OHH nepeceKYTCH; OHH He nepeCel\YTCH. eCJIJI IIX
rrpO]J:OJI}I\HTb B ApyrylO CTOPOHY; TaROBbI, HarrpHMep, JIHHHH AB H CD»
pH c. 31) [12, c. 21]. HaK MbI BHAHM, H3JIOmeHHe I16H CIIHhI HaqHHaeT-
CH C YTBepmAeHHH, BeCbMa 6JIH3KOrO K «npHH HTIY», rrpHTIIICbIBaeMO-
MY XaHHMOM ApHCTOTeJIIO.
;:::(aJIee M6H CnBa npoJJ:OJIa aeT:
«OHH MorYT 6hlTb OTAeJIeHbl APyr OT .n;Pyra Tal\, qTO paCCTOHHIIH l\fe-
mAY .n;BYl\fH HX KOHn;aMH paBHbI MelliAY co6oiI (pIIC. 32). ECJIIf rrpo-
AOJIiI\HTb nepneHAIIRYJIHp, BOCCTaBJIeHHbIH OT OAHOH JIHHIfH ,n;o npy.
roii, TO 3TOT neprreHAHKYJIHp 6YAeT nepneHAHKYJIHpOM H R APyroH JIH-
HHH. B CaMOl\1 ]J:eJIe, eCJIH 6bI OH He 6:blJI neprteHnlfl\YJI.HpeH 1\ 3TOH
APyroH JIHHIIII, TO O]J:IIH H3 yrJIOB 6bIJI 6bI OCTpbIM, a npyroH - TY-
TII)Il\I, n I\OHII;bl co CTOpOHbI Tynoro yrJIa 6hIJIII 6bI OOJIee Y]J:aJIeHLI,
a KOHII;bI CO CTOpOHbl OCTporo yrJIa 6blJIH 6bl 60JIee npn6JIlIiUeHbl.
TaI\OBbl, HanpHAlep, 3TII ABe JIHHHH, Ha3bIBae Ihle napaJI:IeJIbHhIl\flI»
[12, c. 21-22]. 3]J:ecb 116H CHHa onpeneJIHeT napa.JIJIeJIbHhIe JIIIHHII
l\aK 3KBII1J;IICTaHTHbIe, T. e. AeJIaeT npe]J:nOJIOmeHHe, paBHOClfJIbHOe
YCHJIeHHOH aKCIIOMe 0 uapaJIJIeJIbHbIX. OTclOAa 116H CHHa ]J:OKa3bIBaeT
c..TJenYIOIIJ;lfe rrpe JIOmeHHH: «ECJIU JIHBHH naAaeT Ha ABe rra paJIJIeJI b-
HbIe JIHHIIJiI, 6YAY1IH nepneHAHI\YJIHpHOH K oAHon H3 HIIX, TO OHa
Bcer]J:a rtepIIeHAlIl\YJIHpHa If K npyroii. TaKoBa, HanplIMep, JIHHHH EG,
uaAalO aH Ha JIHHHH AB H CD (pHC. 33).
8.
B033pe1tUR
11 61t C U1tbl
46
B CJIyqae, eCJlu JIIIHIIH He nepneH,IJ;MKYJlHpHa HH K O,IJ;Hon1I3 napaJl-
:[e ' bHblX .TIIIHlln, a H8KJIOHeH8 B O,IJ;HY CTOPOHY, npllQel\l OJJ;IIH ee ROHen;
naxo.n;nTCH C OJJ;HOH CTOpOHbI, a BTOpOH - C ,IJ;Pyro:u, TO ,IJ;Ba yrJI8 6y-
.n;YT yn-\e He rrpHMbIl\111, a OCTpbIMII (pIlC. 34) C OJJ;HOH CTOpOHbI - Ta-
KOB yrOJr I "X.J{ Ha JIIIHHII HF II yrOJI KON Ha JIHHIIII KL - II TyrrbI-
1\111 C ,n;pyroH - TaKOBbI ,IJ;Ba yrJla, Cl\lemHbIe C HIIMI1. TIOCKOJlbRY
JIH lIlt H J.ll1\' HMeeT BcerJJ;a O,IJ;HH II TOT me HaKJlOH, TO YBeJlIIQeHlle II
Y1\feIIbIIJeHMe BeJlIIQJIHbI yrJloB B3allMHO KOMneHcMpyeTCH. n OSTOMY
yrO TI l/XN paneH yrJlY NOL, a yrOJI HXM paBeH yrJIY KOM, 1I6o
HaRJIOH OJJ;IIHaKOB ,IJ;JlH BIIX o6olIx. TI03TOMY 3TH ,IJ;aa COOTBeTCTBeH-
HbIX yr.aa paBHbI. BHeIlIHIIH yrOJI rrpll TOQKe X II BHYTpeHHIIH yrOJI npH
3TOH TOQKe paBHbI KaK BepTIIKaJIbHbIe; n09TOl\IY yrO TJ [ X F
panCH yrJlY NOK; n03TOMY BHYTpeHHIIH yroJl paBeH CBoeMY co-
OTBeTCTBeHHOMY BHeIlIHeM:y yrJIY. "LITO KacaeTCH nBYx yrJlOB, CMeil -
HI>IX npll TOQRaX X II 0, TO OHII paBHbI JJ;BYM rrpHMbIl\1 yrJlaM:, n03TOMY
A B B A B E A
I
D c .D fi C
C
Puc.31 Puc.32 Puc.33
Af
H
c
F
K
l
A
A
Puc.34
Puc.35
B
B C A .D C
I
& H E & E
Puc.37 Puc.38
c
Puc.36
A
47
nBa yrJI3, BHYTpeHHIIX rrplI TOqKe X, p3BHbI ABYAI rrpHMbIl'v1 yr"Tl31\I.
YrOJI XOL p3BeH yrJIY OXlI, TaR l\aK eCJIII yrOJI FXO rrplI6aBHTb
K TOl'vIY IIJIII npyrOMY H3 3THX ,lI;BYX yrJIOB, TO C Ka}HAbIl'vl H3 HHX OH 6y-
AeT paBeH }];BYM rrpHMbIM yrJIaM.
ECJIU OAH3 JIIIHIIH rrepeceKaeT }];Be }];pyrlle JlIIHIIII II rrpH aTOM ABa
BHYTpeHHIIX OL(HOCTOpOHHIIX yrJI3 p3BHbI !(BYM npHl\lbIM yrJI3M, TO 9TH
ABe JIHHHH nap3JIJIeJIbHbI, eCJIH 6bI 3TII ]J;Be JIIIHIIII He 6bIJIH nap3JIJIeJII>-
HbI, TO O,!J;H3 H3 HHX HaKJIOHHJI3CI> 6bI K }];pyroM H BCJIeACTBHe 3Toro OHII
6bI nepeCeKJIHCb. 3TO Heo6xo}];IIMO npHBeJIO 6bI 1\ ,IJ;BYM HeJIenOC'l'HM::
npeiH,IJ;e Beero npH rrepeCeqeHHII, HarrpHM:ep, rrpHl\1bIX AB II CD B ro'q-
Ke B 06p330B3JIIICb 6bI yrJIbI E, GlIB, HO MbI rOBOpIIJIII, 'tITO OAHOCTO-
pOHHlIe yrJIbI paBHbI ]J;BYl\1 npHMbIM yrJIaM, 3 eCJIII K HHM rrpn6aBHTI>
TpeT1I1 yrOJI TpeyrOJII>HHK3 - B, TO TPII yrJIa 3Toro TpeyrOJIbIlMKa,
T. e. yrJIbI E, GlIB, 6bIJIH 6bI 60JII>IIIe ABYX npHMbIX yrJIOB. lllIn e 6y-
]J;eT ,IJ;OKaaaHo, qTO 8TO HeJIerro (pIIC. 35).
BTopaH HeJIenOCTI>: TorAa 6bI 6bIJIO Heo6xoAHMO, -qTo6bI BHemHnM
yrOJI AEG TpeyrOJIbHIIKa 6bIJI ObI paBeH BHYTpeHHel\1Y yrJIY 3Toru Tpe-
yrOJII>HHKa» [12, c. 22-23]. 3JJ;ecb M6H CHHa 1I3 CBoero HeHBHoro upe]J;-
nOJIOmeHIIH BbIBeJI n;eJIbIM PH}]; TeopeM, nMelO JlX l\leCTO TOJIbKO 1\ reo-
MeTp1I1I EBI\JIII}];a. If, HaKOHen;, 113 3THX Teope:M M6H CHHa BbIUO}];IIT
V rrOCTYJIaT:
«MbI TaKme YTBepmAaeM: eCJIH OJ];Ha JIHHIfH nepeCel\aeT ,IJ;Be APyrlIe
n HHM II np" 3TOM ABa BHYTpeHHHX O]J;HOCTepOHHHX yrJIa Mellbllie
ABYX rrpHl\lbIX yrJIOB, TO 3TII }];Be JIIIHIlH, eCJIII HX npo,n;OJImHTb C aTOM
me CTOpOHbl, rrepeceRYTcH. Tal\oBbI, HanplIMep, JIIIHIIH A K II CL
Ha -qepTe}Re (pIIC. 36). 8TO nOTOMY, 'lITO O]J;Ha Hal\JIOHHeTCH K npyroii,
II BCJIe}];CTBHe 3Toro HaKJIOHa olin rrepeceKYTcH. B caMOM AeJIe, eCJIII
6bI oJ];na 1t3 HHX He H3KJIOHHJIaCb K npyroii, TO OHH 6bIJIII 6bI napaJIJIeJIb-
IIbIl\lI1, a eCJIH 6bI OHII 6bIJIII napaJIJIeJIbHbIMII, TO !(Ba YKaaaHHbIX yrJIa
6hIJIII 6bI paBllbl ]J;BYM rrpHMbIM yrJIal\l, K3K 3TO 6bIJIO y Ke ,lI;OKaaaHO Ha-
Mil paHI>IlIe» [12, c. 24].
.D:aJIee M6H CHHa ]J;OKa3bIBaeT e e HeCKOJIbl\O TeopeM reOM:eTpHII
EnI\JlH,IIa:
«ECJlH }lbI XOTIIj\J1 npOI3eCTI1 :IHHIlIO, napaJIJI.eJIbHYIO rrpH loM AB
(pIIC. 37), TO OTMeTlIM Ha Heii TO"({KY C II BOCCTaBll 1 B HeM nepneHJ];II-
I\YJIHp ClI. B TOqRe // DOCCTaBlIl\1 3aTe 1 neprreHAJIRYJIHp HE. OTclOAa
CJIenyeT, "liTO ,nBa O,II;HOCTOpOIlHHX yrJIa panHbI ;a;nYl\1 npHMbIl\1 yrJIaM II
"({TO J{Ba COOTBeTCTBYIOII(IIX yrJIa paBHbI l\Iemny co6oii:; CJIeL\oBaTeJIb-
no, nBe JIIIHHU n3paJIJIeJIbHbI.
ECJIU lbI rOBoplI l, qTO O,II;Ha JIIIHHH napaJIJIeJIbUa }];pyroM, ROTopaH
n CBOIO Oqepe;a;b napaJIJIeJIbHa TpeTbeii: (pIlC. 38), TO 113 3Toro CJleJJ;yeT,
qTO nepBaH JIIIHIIH rrapaJIJIeJII>H3. TpeTl>eii. B rrpOTIIBHOAI CJIyqae KOHen;
nepBOH JIIIHIIH H31\JIOHHJICH 6bI 1\ TpeTbeii JIIIHHII II, CTaJIO 6bITb, K JIH.-
HIII1, 1\ KOTOpOH rrepBaH JIIIHIIH napaJIJIeJIbHa, 3 3TO HeJIenO» [12,
c. 25].
48
Teopuefl na paJIJIeJIblIT IX JIHIIIIH 3a-
HHMaJICH II 31IaIVleHHTI)TH «OTel on-
THKH» 116H aLTI-X aiICal\I, c.pe He-
HeKOBOH JIaTIIHCROH JIIiTepaType II3-
BeCTHLIH nOn Hl\IeHel\1 j\LTJbXa3eH..
npOHCIIIeJJ;mH I B pe3YJIbTaTe 06bIq-
HOH B TO Bpe lH (QIaTHHn3aJ nll» apa6-
CKUX JIl\leH. lI6H aJI-XaUc.al\,[ 6bIJI
ypOllieHl e .[ BacpbI B MpaKe. fIep-
BOHaqaJIbHO OH 3aHIIl\IaJICH Ha pO-
nHlIe nOJIHTHqeCKOH JJ;eHTeJIbHOCTbID
H JJ;OCTHr Ta 1 name nOJI1HHOCTII Ba-
3Hpa, HO 3aTel\1 OCTaBIIJI I\H3Hb npH JJ;Bope II 3aHHJIC H HaYKoi1:. noc.rre
Toro RaK OH pa3pa60TaJI npoeKT perYJIlIpOBaHHH BOJJ; HHJIa C nOMOII];blO
nJIOTHHbI, 01-1 6bIJI npHrJIameH B HaHp, rL{e H npOmHJI JJ;O c IepTn. Y6e-
nHBlliHCb, 1]:1'0 TeXHIIKa TOro BpeMeHH He n03BOJIHeT nOCTpOllTb
TaKYIO nJIOTMHY (31'0 6bIJIO CneJIaHO B AcyaHe TOJIbKO B XX B. C no-
MOIIJ;blO COBeTCKoro COI03a), M6H aJI-XaHCaM, onacaHCb HaKa3aHHH,
B TeqeHHe pH,n;a JIeT, BnJIOTb JJ;0 cMepTII nplIrJIaCHBJUerO ero npaBHTeJIH
ErHnTa., CHMYJIHpOBaJI CYl\laCmecTBHe. M6H aJI-XaMCaM 6bIJI O,n;HHM H3
OCHOBOnOJIOrl-\HIIKOB 3KcnepHMeHT8JIbHO:R H3HKH. Ero TPY,n; no onTH-
Ke, nepeBe,n;eHHblM Ha JIaTbIHb nOli: Ha3BaHHe I «Opticae thesaurus»
(<COKpOBHm;e OnTIIKJ.i» r: 6J \, B ;reqeHHe MHorllX BeKOB CJIYj-RHJI OCHOBHblM
PYKOBO,n;CTBOM KaK B apa6cKHx CTpaHax, TaK If B EBpone. M6H all-Xaii-
caM: 6bIJI aBTopOM em;e MHorHX COqHHeHH:R no onTIIKe, M:aTe:M:aTIIRe II
aCTpOHOl\IHH, B qaCTHOCTH nBYX KOMMeHTapHeB K «HaqaJIaM» EBKJIH,n;a.
TeopHH napaJIJIeJIbHbIX paCCMaTpHBaeTCH B 060HX 3TIIX COqHHeHHHX -
II B «HOMMeHTapHHX K BBeJJ;eHHHM KHHrn EBKJIlIna "I-IaqaJIa") (III p
My anapaT KHTa6 YJ JIllnHC iI-JI-YrYJI), nOCBHrn;eHHOH KOM1\IeHTHpo-
BaHll10 TOJI.bKO BBO]j;HhIX qaCTeH KHHr «HaqaJI», H B «HHlIre 0 pa3peme-
HHH COMHeHHH B KHHre EBKJIl1na "H aqa.TIa"» (KHTa6 ii aJILTJ IIIYKYK
KHTa6 Y 1.\JIllnHC «piI-JI- Y cy JI), Cone"pn-\am;eii KOM leIlTa PUH H Cal\-IIIM
npe,n;JIOmeHHHM 3Toro COqHHeHHH. Pa3neJI 0 napaJIJIe:IbllbIX .JII.IJ:HIIHX
nepBoro COqHHeHHH 116H aJl-XaHCaMa 6I)IJI H3JJ;aH B PYCCKOM. nepeBo,n;e
[20, c. 743-762].. B aTOM: TpaKTaTe, HaK If BO BTOpO 1 TpaKTaTe 116H
HOppI I, u;eHTpaJIbHbIM nYHKTO 1 nOKa3RTeJIbCTBa CJIymlIT cymecT-
BOBaHlfe npHMoyrOJIbHHKa, BbITeKalOrn;ee H3 npennO:IOa eHHH 0 cym;e-
CTBOBaHHH paBHOOTCTO.Hrn;HX npH 'lbIX, KOTopoe 060CHOBbIB!}eTCH C no.
1\10IIJ;blO KHHeMaTHqeCKHX co06paj-KeHHii, aHaJIOfHqHbIX co06pameUIfHM
116H KOppbI.
116H aJI- X a:ncal\I HaqHHaeT C KPUTHKU onpeneL eHIfH na paLrr.TJeL en
Y EBRJlH]j;a: «EBKJIII,n; CKa3aJI: napaJIJIeJIbHbIe npH}[bIe JIHHUII - Ta-
RHe, KOTopbIe JIemaT B onHon nJIOCKOCTII II eCJIU npo,n;OJImaTb I1X B o6e
CTOpOHbI 6eCKOHeqHO, OHII He nepeceKYTCH HH C onHOH 1I3 CTOpOH. IIpH
aTOM JIBHHH npeJJ;CTaBJIHIOTCH paCIIOJIOmeHHblMH B OJ(HOH nlIOCROCTII
II He nepeceKalOm;HMHCH, eCJIII npOLtOJImaTb HX B Kaj-R]j;YIO !13 ,n;BYX
CTOpOH, npHqeAI aTH JIHHIIH }IOlliHO npo,n;OJImaTb nOCTOHHHO B 06e CTO-
pOHbI o,n;HOBpe}leHHO, 11 npH aT01\1 OHH He nepeceKYTCH, HO HeB03MOil\HO
npe]j;CTaBBTL nOCTOHHHoro npH6aBJIeHHH TaRoro po,n;a. He nOCTHralO-
9.
K U1leMamU1.£eCKan
meopUR
nap aJlJl eJl'b1l'blX
1161l aJl- X auca.Ma
49
IJJ;ero KOUIJ;a. ,1].JIH npe,nCTaRJIeHIIH 3TOrO HeT nYTU, TaR RaK Bce, qTO
npe,nCTaBJIHeTCH, KOJIe"tJHO, H JIHHHH, 0 KOTOpblX 3neCb rOBOpI1.TCH,
npe,IJ;CTaBJIHIOTCH JIHJIHHMH KOHeqHOH BeJIHqHHhI. ECJIII fHe aTH BeJIU-
qHHbI YBeJIHqHBalOTCH 6eCKOJIeqHO, HeT cnocooa nOHHTb II H306pa3H Tb
HX KOHII;bI nOCJIe YBeJIHqeHHH. ECJIH aTH KOHIJ;bI He COBna,naIOT Ha KO-
He"tJHoM! paCCTOHHHH, OJIH TaKOBbI, KaR CKa3aHO, HO nOJIHTb cy ecT-
BOBaHHe ,IJ;BYX JIMHHii, npO]J;OJImalOm;HXCH 6eCKOHeqHO B 06e CTOpOJIbI
II He nepeceKalOm;UXCH, :MOiRJIO, TOJIbKO c,neJIaB HCJIbI f cym;ecTBoBaHHe
npHl\lbIX JIHJIIiH aToro BH,na. l\10rRHO npe]J;CTaBHTb ce6e KOJIeqHOe npo-
1(OJI KeHHe, HO 6eCKOHeqHOe npOI(OJImeJIHe, He06xo,nHMOe ,IJ;JIfI HHX,
n peACTaBI1Tb ce6e HeJJb3H. n YTb, II03BO.,TIHJOIll.Hll [nOHHTb] c.y ecTBo-
B8HUe npHMbIX JJI1HIIH 3TOrO BI1]J;a,- 3TO B3HTHe OrpaHUqeHHOH npH-
:MOM .TIlIHlIlI, paOIIOJIOmeHHOiI Ha IIJfOCKOCTII, 1\ paTHOI1: npeACTaBIIM
ce6e OrpaHI1"tJeHHYIO npH IYIO JIHHIIIO, paCnOJIOiI eHHYIO JIa aToii nJIO-
CKOCTIf, JI BTOpylO OrpaHH"tJeHJIYIO npHl\fYIO JIHHHIO, lI ferom;yro C nep-
BOll JIHHHeH OJJ;HY 06myro TO"tJKY, If npeJJ;CTaBI1AI ce6e, 'ITO HanpaBJIeHHe
BTOpoii JIHHHI1 COBna,naeT C HanpaBJIeHHeM: nepBOll JIMHHH, CMhICJI
aTOrO MOiI I10 ce6e npe,nCTaBHTb 6e3 COMHeHllH. TaKIf 1 06pa301\I, CO-
r JIaCHO 3TOl\IY npe,nCTaBJIeHHIO cym;eCTBYIOT JJ;Be npH1\IWe JIHJIHH, npH-
MhIKalOlII;ne B O,nJIOH TOqKe, OrpaHMqeHllbIe C 06e:ux CTOpOH. ,r:(aJIee
npeJJ;CTaBHM ce6e ern;e OJJ;I1Y JIllHJlIO, npHl\fbIKalOm;ylO KO BTOpOH TaK
me, Kal\ BTopaH K nepBoii, ,nao;TJee "tJeTBepTYIO JIHHHIO, npHMhIKalOm;YIO
K TpeTbeii no npHl\fepy nepBbIX JIHHHii, II TO iI\e npeJJ;CTaBIIM ce6e npH
,IJ;pyroM KOHl(e nepBoH JIHHHH. ECJIH npeLtCTaBHTb ce6e OrpaHHqeHJIYIO
npHMYIO JIHHHIO B3HTOii TaKH I o6pa30M KpaTHoii, olla 6y,neT YBeJIH-
qHBaTbCH B o6e CTOpOHbI 6ecKoHe"tJJIO, T. e. ee l\IOiRHO HeOrpaHHqeHHO
YBeJIHqHBaTb B ee HanpaBJIeHHH» [20, c. 743-744].
BCJIeJJ; 3aTeM If6H aJI-XaHCaM nepeXO]J;HT K onpe,neJIeHHIO napaJl-
JIeJIeii, no cym;eCTBY pa3BHBaH onpeneJIeJIHe Ca6HTa H9H KOppbI, JIa
KOToporo, BnpOqe l, OH He CCbIJIaeTCH.
« aJIee rrpe,nCTaBIIM ce6e ,IJ;pyrylO OrpaHHqeJIHYIO npHMYIO JIHHHIO,
CTOHm;ylO JIa nepBoH no,n npHMbIM yrJIOl\l B Ton iRe nJIOCKOCTH, B KO-
TOpOll HaXO,IJ;HTCH nepBaH JIHJIHH. aJIee npe,nCTaBHM ce6e aTY JIHJIHIO
,IJ;BHmymeHcH O,nHI1M H3 ee KOJIn;OB no nepBOH npHMOH JIHHIIH B O,IJ;HY
H3 CTOpOH. Ee ,nBHmeJIHe eCTb O,nHO npOCToe ,nBHmeJIHe, T. e. 6e3 113-
MeHeHIIH ]J;BHmeHHll, He COCTOH ee H3 ,IJ;BlImeHHH If nOKOH, O,IJ;JIO ,nBH-
meHHe 6e3 1I3ru6oB. ECJIII npH aTOl\f nepeJIOce B TeqeJIlle Bcero BpeMe-
HH ,nBHmeJIHH [nepeHOCIIMaH] JIHHHH nepneH,nIIKYJIHpHa K JIHJIHH Ha
nJIOCKOCTH, T. e. K nepBOH JIHHHH, pacnOJIOmeJIJIOH JIa JIeH, TO KOJIen;
aTOll nepneJI,IJ;HKYJIHpHoii JIJIJIHH 3a Bpe fH ,nBHmeHHH OIIHllieT npHMYIO
JIHJIHIO, nepneJI,nUKY JIHPHYIO K Hen, H npe,IJ;CTaBJIeJIUe aTOll JIHHHII,
KaR H npe,nCTaBJIeHHe ,nBH ReHHH aToro BHLta, B03MOmJIO» [20, c. 744].
oKa3aTeJIbCTBO M6H aJI-XaHCaMa TaKOBO: «B CaMOl\1 ,neJIe, npH
,IJ;BHmeJIHH BCHKOH JIHJIHH ,IJ;BHmYTCH H Bce ee TOqKH, u npH 3T01\I ,IJ;BH-
meHHH Kam,IJ;aH TOqKa onHCbIBaeT JIHJIHIO: npu ,nBHmeHHH BCHKaH
TOlIKa C]J;BHraeTCH Ha HeKOTopoe paCCTOHHHe, T. e. ,nJIHHY, n03TOMY
BCHKoe paCCTOHHHe, Ha KOTopoe C,IJ;BHraeTCH TOqKa, eCTb ]J;JIHHa 6e3
lliHPHHhI, TaK KaK TOqKa He HMeeT pa3MepOB II nplI ee ,IJ;BumeHHH He
MomeT nOJlYlIHTbCH lliHpHHa, JIO BCHKaH ,nJIHJIa 6e3 illHpHHhI eCTb
50
.TJI1HIIH. n03TO IY BCHKBH ADMIl,ymaHcH TOIJJ{a OnnCbIHaCT o!lJ1HIIJO»
[20, C. 744-745].
fIpMBeAeHHoe paCCY;'I-\JJ;eHI1:e M6H aJI-Xaiica la IIHTepeCHO Te I, "tJTO
3JJ;eCb BnepBblc B 1I3BeCTHoii Ha:M 1\laTe}IaTMQeCKOii JIMTepaType IICI10JIb-
ayeTcH onpeJJ:e.iJeHI1e 2 I\HMrM I «HaqaJI» EBKJIJI;n:3: «JlnHHH - J{oilMHa
oea HIMPH:HhI» [10, T. 1, c. 12]. I3b1HoJJ; 116H aJI-Xaiica [a 0 1'01\1, qTO
«BCHHaH ABl1n-\ym;aHcH TOqHa onMCbIBaeT JIMHHIO», COBnanaeT C YTnop-
il\JJ;eHMej\1 nH<paropeiiQeB, npOTH"B KOToporo 60pOJICH ApnCTOTeJIb.
BnOCJIeJJ;cTBIfI1, nOJIe IM3I1pYH C 31'111\1 paccya,n;eIIJfe1\I 116H aJI- Xo iica Ia,
XaHHM: I1l1caJI: «}\aRoe OTHOlJICHI1e MM:eeTCH le;I-\AY re01\IeTpneM H ABH-
llieHMe1\[ If 4:1'0 CJIeJJ;yeT nOHH1\IBTb nOJJ; ABI1 +\eJIlJe}I? COrJIaCHO yqeHhlM,
HeCO:MHeHHO, '11'0 JIUHIIH Mon-\eT cym;ecTHoB8Tb TO,JlbHO HB nOBepXHO-
CTI1, a nOBepXHOCTb - B TeJIe, T. e. JII1HHH 1\IO,I,eT OhJTb TOJIhKO Ii TeJIe
M He 1\IOmeT npelUlICCTBOBaTb nonepXIIOCTII. I-iaR ,J-\e aHa lO;I eT ;n:BU-
raTbCH OTBJIeqeHHO 01' ee npeA1\IeTa? l\aR JIIfl1I1H l\IOiIieT 6blTb o6pa30-
HaHa ABI1;l\eHl1Cl\I TOqHI1, B TO Bpel\'UI HaH OHa npenulecTByeT TO'IKC
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n3 HHX - npHMoii yrOJI. ECJIH J-I-\e JIHHHH E}i' Ha.ITOmnTCH H8 JIHHHIO
BA, TOQRa E 6y,n;eT BHellIHeH no OTHOIlleHlI1O K JI:HHUH AB, npIlqe 1
npeBbIllieHue 6YAeT co CTOpOHbI TOqRH A, TaR l\aK JlHHHH EF, RaK
6bIJIO ,D;ORa3aHO, BCJIe;l(CTBlIe paBeHCTBa yrJIOB EFB H ABD 60JIbIlle
JIHHHH AB. TI03TOMY [JIIIHHH] Elt"' npH ee HaJIOn eHHH Ha JIHHHIO BA
6y,n;eT JIHHUeH BH. JIHHHH BH nOCJIe 31'01'0 6YAeT ABHraTbCH B CTO-
POHY BD, r)];e OHa 6YAeT paBHa CBOeltlY IIepBOHaqaJIbHOMY IIOJlOme-
54
A
c
H
E
D
B
Puc.39
F
HIfIO. .ECJIH TOqKa B rrpIf J{BlfiI,eHHH JIIfHIfH BH COBnal1aeT C TOQKOH D,
JIIIIIHH BH HaJIOmllTCH Ha JIIIHHIO BC, TaR :KaR yrJIbI HBF II C DB
paBHbI :Ka:K ABa npHMbIX [yrJIa]. ECJIH JIIIHHH BH HaJIOfnlITCH Ha JIII-
HIlIO DC, TOq:Ka H COBlIaAeT C TOqKOH C, TaR RaR JIIIHIIH HB eCTb
JIIIHHH EF, a JIHHHH EF paBHa JIHHHII CD, T. e. COBnaAaeT C Heft, :Kor-
]J;a npH ABHH,eHHH JIHHHH EF BAOJIb JIHHHII f""D TOqR8 F COBnaAaeT
C TOqKOH D II TOqRa E COBlIa]J;aeT C TOq:KOH ('. Ho npH onpeAeJIeHIIH
napaJIJIeJIbHbIX JIHHHH l\1:bI ]J;OKa3aJIII, qTO eCJIH BCHKaH [npHMaH]
JIIIHIIH ABHmeTCH TaKHM 06pa30M, TO ee KOH bI onHCbIBalOT npHMYIO
JIHHIIIO. IIoaTOMY TOqKa E nplI ABHllieHlI1I JIIIHIiH EF BJJ;OJIb JIHHHII F B
onHCbIBaeT npHMYIO JIHHHIO. Ho JIHHHH, onHCbIBaeM:an: TOq:KOJI E,
eCTb JIHHHH EHC. TaKIIl\I 06pa30M, JIHHHH EHC - npHMaH JIHHIfH.
Ho JIIIHHH EAC, IIO IIpe]J;nOJIO,ReHHIO, IIpHMaH JIHHHH, coe]J;HHHIO aH
TOqRH E H C, II JIHHHH EHC, OTJIHqHaH 01' JIIIHIIH EAC, If MeeT C HeB
]J;Be 06 He TOq:KH E If C. TaK KaK o6e aTH JIIIHHH - npHMbIe, ABe npH-
MbIe JIHHHH OrpaHHqHBalOT IIOBepXHocTb, qTO HeJIerro. OTCIO]J;a BbITeKaeT,
qTO HeJIelIO II Harne npeAnOJIOmeHHe 0 TOM, qTO JIHHHH CD 60JIbIIIe
JIRHHH AB, 1'. e. JIHHHH CD He 60Jlbme JIHHHII AB» [20, c. 749-750].
AHaJlOrHQHO JJ;OKa3bIBaeTCH, qTO JlHHHH CD He MeHbme JIHHHH AB
II qTO BCHKHH IIeplIeH]J;H:KY JIHp, onytu;eHHbIii H3 TOqKH JIHHHH A C HR.
JIHHHIO BD, paBeH JIHHHH AB: «.H YTBepm]J;alO, q1'O yrOJI DCA npHl\1:oii.
M3BecTHo, qTO yrOJI EAB - rrpHMoH. IIpoBeJJ;eM JIHHHIO AD. llYCTb
OHa nepeceRaeT JIHHHIO BC B TOqRe K. JIHHHH AB paBHa JIHHHH CD,
[cTopoHa] BD [TpeyrOJIbHHKOB ABD H CDB] o6 aH, T. e. CTOpOHbI
AB II BD paBHbI CTopOHaM CD H DB, a yroJl ABD paBeH yrJIY CDB,
RaR ABa rrpHl\fbIX yrJIa. II03TOl\fY OCHOBaHHe AD paBHO OCHOBaHHIO
CB II TpeyrOJIbHHK ABD paBeH TpeyrOJIbHHKY CDB. l103TOMY OCTaJIb-
HbIe ,nBa yrJIa [TpeyrOJIbHHKa ABD] paBHbI OCTaJIbHbIM ]J;BY 1 yrJIaM
[TpeyrOJIbHHKa CBD], T. e. yrOJI DAB paBeH yrJIY BCD, yrOJI ADB
paBeH yrJIY CBD. lIo CYl\fMa BYX yrJIOB CDS paBHa CYl\:fMe ]J;BYX yrJIOB
55
ABD, II03TOl\IY yrOJI ABK paneH yrJIY KDC. 110 H3BeCTHO, "tJTO yrOJI
KAB paBeH yrJIY KG"'D, a JIHHHH AB paBlla JIlIHIIH CD. 1103Tol\:IY
TpeyrOJIbHHK AKB paBeH TpeyrOJIbHHKY CKD. n03TOMY JIIIH.HH AK
paBHa JIHHHH KC, TaK .RaK OHM JIea"aT npOTIIB paBHbIX yrJIon ABK
H CDK. ECJII1 fJ e JIIIHHH AK paBHa JIIIHllH KC, TO yrOJI KAC paBeH
yrJIY KCA. 110 I13BeCTHO, "tJTO yrOJI KAB paBeH yrJIY KCD, n03TO IY
CY I la ;n:BYX yrJIOB BA C paBHa CYM le ,Il.BYX yrJIOB DCA, II TaR KaK
yrOJI BAC npHM:OH, TO yroJI DCA TaR,Re npHl\:loH. TaRHM 06pa301\I,
BCHRHH nepneH,Il.HRYJIHp, ony eHHbIH H3 [TOqeR] JlIIHIIII AC Iia JIHHIIIO
BD, paBeH JlHHI111 AB H 06pa3yeT C JlIIHlfeH AC npHMoH yrOJI.
TaRHM: fI-\e o6pa30M [,Il.ORa3bIBaeTCH, "liTO] BCHKHH nepneH,Il.HRYJIHp,
onYlI.\elIHbIH 113 [TO"tJeR] JIHHIIH BD Ha JIIIHIlIO AC, paBeH JIHHIIH AB
H aaRJllOqaeT C JlHHHeii BD npHl\IOH yrOJI, TaR KaR ,Il.ORa3aTeJlbCTBO
1];JIH 3THX nepneH]];HKYJlHpOB TaRoe fI e, RaR ,Il.JlH nepneH,Il.IIRYJIHpOB,
ony eHHbIX H3 [TO"tJeK] JlIIHIIII A C Ha JIHHHIO B D. 3TO II eCTb TO, "tJTO
MbI XOTeJlH ,r(OKa3aTb.
113 3Toro HCHO II ,n;OKaaaTe-JlbCTBO Toro, "tJTO JlHHIIH CA paBHa JlH-
HI1I1 DB: eCJIIf H3 ROHlJ;OB npHMoB JIHHIIII BD rrpOBe]];eHbI ABe np HMble
JIHHHH BA II DC, 3aRJIIO"tJalO He C JIHHHeH BD npHMble yrJIbI ABlJ
II CVB, H JIHHHH CA nepneHAHRYJIHpHa R JIHHHH AB, TO JIHHHH CA
paBHa JlllHlI1f DB» [20, c. 751-7521.
HaRoHe , 116H aJl-XaHCaM nepeXO)J;HT K caMOMY ,Il.ORa3aTeJIbCTBY
V rrOCTYJIaTa: «ECJIII 8TO ,Il.OKa3aHO, TO nYCTb npHMhIe JIIIHHH AC H
EDG nepeceRalOTCH JIHHHeH BD, H nYCTb yrJlbI DBC H BDG M:eHbIIIe
,Il.BYX npHMbIX [yrJIOB] (pHC. 40). .H YTBepm]];alO, qTO npHl\:lble AC H DG,
eCJIIf 6eCKOlIe"tJHO npO,Il.OJImaTb HX B CTOPOHY CG, nepeceRaIOTCH»
[20, c. 7521. CJIHmROM CJlOFI\HbIH qepTeFI-\ 116H aJI-XaHCaMa 06'bHCHH-
eTCH TeM, "tJTO RaR CJIyqaH V nOCTYJlaTa, ROr]];a ceRy aH nepeceKaeT
,Il.Be npHIvlble 110]]; npHMblIv[ H OCTpbIM yr JlaMII, TaR U CJIyqaH, Ror.n;a. ce-
RYll\aH nepeceRaeT )J;Be npHl\:lbIe no]]; TynhIM II OCTphIM II rro,II. )J;BYIvIH OCT-
pbIMH yrJlal\IH, 116H aJl-XaHCaM ,Il.ORa3bIBaeT Ha O)J;H01\1 H TOM me qep-
Teme, Ha ROTOpOM: B 3aBHCHMOCTH OT 06CTOHTeJlbCTB OAHH H TOT me yrOJI
H306pamaeT OCTpbIH HJlII npHMOH JIH60 me np"HMOH IIJlH TynoH yroJI.
« OKa3aTeJIbCTBO 3Toro. nYCTb yrJlhI DBC II BDG B cYl\:IMe l\feHb-
me )J;BYX npH lbIX, TOT,Il.a O,Il.HH H3 3THX yrJloB BO Bcex CJlY'tJaHx MeHbme
npHMoro, TaR RaR, eCJlII 6bI HH O]];HH 113 HHX He 6bIJI MeHbIIIe npHl\IOrO,
IIX CYMMa He 6bIJla 6bI feHbIIIe .,Il.BYX npHl\fbIX, a HX CYM la, no rrpen;rro-
JIO,I-\eHH O, l\leHbme )J;BYX npHMbIX. II03TOM:y O]];IlH H3 HHX BO Rcex CJlY-
'tJaHX l\feHbIIIe npHl\:IOro. nYCTb'yrOJI DBC l\feHbIIIe npH loro. EC.ill1 yrooJ'I
DBC MeHbme npHMoro'- yrOJi BDG l\fO,I\eT 6bITb MeHbIIIe npHMoro,
M:OFI eT 6bITb 60JlbIIIe npHMoro II IOFI\eT 6bITb npHl\:lhIM. nYCTb CHaqaJIa
yrOJI BDG npH foH. TIpoBe)J;eM H3 TO"tJRH B npHl\:IYIO BK, 3aKoillOQalO-
m;ylO C IIpH fOii BD npH loH yrOJI, T. e. yrOJI IJBK. I1pe)J;nO"T{Oa Il I
Ha JIHHHII BC npOH3BOJIbHYlO TOqRY, nYC1'b 8TO 6YJJ;eT To"tJKa C, II ony-
CTHl\I H3 TO'tJRH C nepneH)J;HRYJlHP Ha [JIHHlllO] BD; OH yna)J;eT 118 JlH-
HIlJO ED IIJlH Bue ee co CTOpOHbI B IIJIH co CTOPOHhI./). Ou He yna]];eT
BHe JIHHHH B D CO CTOpOHhI B, TaK HaR, eCJIU npO)J;OJIffiHTb JIIIHIlIO B D
B ee HanpaBJIeHHH B CTOPOHY B, OHa 3aKJIIO'tJaeT C JIHHUen BC TynoH
yrOJI BHe JIIIHIIH BD, II eCJIH nepneH]];HHYJIHp ynaJJ;eT H8 JIUHUIO I)B
56
BHe CTOpOHbI B, CliB - TpeyrOJIbHHK, OAHU 113 yrJIOB KOTOpOrO -
npHMoiI, TOT, KOTOpI>IH npH KOHu.e nepneH,nllKYJIHpa, OIIYlT eHHOrO Ha
JIHHfIIO BI) H3 TOl.JKI1 C, a ,npyroii yrOJI - TynoiI, TOT, KOTOphIH nplI
TOl.JKe B, a 3TO HeJIenO, TaR RaR R npe,nJIOjJ eHHI1 17 RHHI'H I [«HaqaJI» J'
,nOR33aHO, "tJTO BCHRHe ,nBa yrJIa TpeyrOJIbHI1Ka :MeHbllIe ,nBYX npHl\fbIX
[yrJIon]. TIepneH,nlIKYJIHP, onYII(eHHbIH H3 TO"tJRM C Ha JIHHI110 Bl),
yuaJJ;eT co CTOpOHbI D, HO He yna,neT Ha Ca}IOe TO"lIKY 1J, TaR RaR eCJIH
OJ)I OH ynaJI Ha cal\loe TO"tJRY 1), npHMoii yrOJI ORa3aJICH ObI MeHbllie
npHl\IOrO yrJIa, nOTOMY "tJTO JIHHHH, nponeJJ;eHHaH H3 TOl.JRM C H TO"tJRY
[), nepeCeqeT JII1HHIO B J), II, eCJIH TO"tJRa C He HaXO!(I1TCH Ha JIlIHHI1
DG, ABa npHl\fbIX yr JIa 61)IJI" 61>1 He paBHbI APyr JJ;Pyry, qTO HeJIeno.
TI03TOl\fY nepneH,nHRYJIHp He yna,n;eT Ha TOqKY D. rr03TO fY nepneH-
,nHRYJIHp, na}J;alO HH 113 TO"tJRH C Ha JIHHHIO BD, nepeceRaeT JIIIHI1IO
BD B TOqRe, HaxoAHm;eMCH l\felliAY TOqKaMH B H D, HJIM B TO"tJRe BHe
8TOM JIHHIfII co CTOpOHbI TOqRH D. ECJIH 6bl 8TOT nepneHJJ;I1RYJIHp nepe-
ceR JIHHHIO BD BO BHellIHeii TOqRe CO CTOpOHbI TOqKH D, nOJIy"tJHJICH 6b!
npHMoyrOJIbHbIH TpeyrOJIbHHR, npHqeM TOqKa D ObIJIa 6bI OAHon 113
CTOpOH 3Toro TpeyrOJIbHHRa, HO 113 8TOH TO"tJRH K 3TOll CTopOHe npo-
Be1(eHa [JIHHHH] nOJJ; npHl\fbIM yr JIOl\-I. T. e. non yrJIO I BJJG; n03TOl\fY
JII1HHH DC nonaJJ;aeT BHYTPb 3Toro TpeyrOJIbHI1Ra, H, eCJIH oeCROHel.J-
HO npO)..(OJIiHaTb JIHHHIO J)G B ee HanpaBJIellI1M, OHa nepeCel.JeT OJJ;HY
H3 CTOpOH aToro npHMoyrOJIbHOrO TpeyrOJIbHHRa "JIH ,He npOHJJ;eT
"tJepe3 BepIlIHHY TpeyrOJIbHHRa» [20, c. 752-754].
CJIe,nyeT 3aMeTHTb, qTO B npHBeneHHOM paCCYiI\neHHH lI6H aJI-Xaw-
caM, TaR ;Re KaR MHOfIIe ero npe,IJ;UleCTBeHHI1RH, nOJIb3yeTcH «aK-
CHOl\IOH IIaIII'a», <POPMYJIHPYH ee B OTJIUQlle OT 60JIee paHHHX MaTe-
MaTHKOB (CM., HanpMMep, c. 30) KaK Crrel(Ua "JbHOe OqeBH,IJ;Hoe YT-
B epmJJ;eHHe. 8TO nepBbIii H HCTOplIU reOMeTpHH CJIyqaH HBHOH <pOpl\IY-
JIHpOBRH aKCHOMbI, BBeJJ;eHHe ROTopon B 1882 r. MopHIJ;el\1 II allie 2\1
[57 a] nOJI01HHJIO HaqaJIO BbIJJ;eJIeHHJO OTCYTCTBOBaBIIIHX y EBR.JIU-
,IJ;a «aRCHOM rropH]J;Ra», KOTopbIe 6bIJIH BKJIIOqeHbI B BH]J;e oT]J;eJlbHOii
rpynnbI B CUCTeMY aRCHOM, npeAJIOiHeHHYIO ,il,. fHJlb6epTOM B ero
«OcHO,BaHIIHX reOMeTpIlH».
C<POpl\fYJIHpOBaB «aKCHOMY nama», M6H aJI-XaHCal\I npOAOJliHaeT:
«1-10 JIHHIIH DG He MomeT nepeCeqb nepneHJJ;IIKYJIHpa, na]J;alO ero 1I3
TOQKI1 C Ha JIHHHIO BD, II He MomeT rrponTH Qepe3 BepIIIHHY TpeyrOJlb-
HHKa, Tal( Ra , eCJlu 6bI OHa nepeCeKJIa nepneHJJ;HKYJIHp HJlH npoxo-
}J;HJIa 6bI qepea BepIIIIIHY TpeyrOJlbHHRa, nOJlyqHJlCfI 6bI TpeyrO:.IbHUR
c ,nBY:MH npHMbll\fH yrJlal\IH, "liTO HeJleno. n03TOMY JlHHMH DG, eCJlH
npOJJ;OJIiI\IITb ee B ee HanpaBJleHHH, M01ReT rrepeCeqb TOJIbKO JIHHUIO
BC, 110 eCJII1 OHa nepeceRaeT JIHHHIO BC, JIIIHIfH DG H BC nepece-
KaIOTCH.
ECJIH n\e nepneH]J;HRYJlHp, ony:w;eHHbIH 113 TOqKH C Ha JIHHfIlO BD,
He nepeceRaeT ee BO BHemHen TOqRe co CTOpOHbI B H BO BHeIllHeH
TO"({Re CO CTOpOHbI D, a TaRme He nepeceKaeT ee HIl B TOqRe B, 1111
B TOQRe D., OH nepeceRaeT ee B TOqRe MemJJ;Y TOqRaMH B H D. IIYCTb
nepneHAIIKYJIHp eCTb JIIIHHH CH If TOqRa H HaXOJJ;HTCH MemAY TOqRa-
Mil D II B. [TIp0JJ;OJImHM JIHHBIO] CH B ee HanpaBJleHHJ.l B CTOPOHY C
11 npOAOJlII\UM ee TaJ(iI\e B ee HarrpaBJleHHH B CTOPOHY H, OTJIO;I IIl\l r 113
57
npO,IJ;OJI}IieHHH JIHHIIH C II JIHHHIO C .F'], paBHYJO r JIIIHHII 1 C H, OTJIOJ-I\IIl\1
[Ha npOJJ;OJIfI eHHH JIHHHH BC] JIHHIIIO CI ., paBHYIO [JJllHJllI] BC, -
coe,IJ;HHHM FL. TaK KaK JIHHHU BC H CL paBHbI, a TaR}He JIHHHH CH
II CF, H BepTllKaJIbHbIe yrJIbI LCp' II BCH paBHhI, OCHOBaHHe LF
paBHO OCHOBaHHIO BH, II ,IJ;Ba ,IJ;pyrlIx yrJIa paBHbI Kan\,IJ;blH CBOe .fY
COOTBeTCTBYIOll.(el\IY. noaTo IY yrOJI CFL paBeH npHMOMY yrJIY CHB,
T. e. yrOJI Cp'L - npHMOH. OnYCTIIM H3 TOqKH F nepneHIU1KYJIHp
lIa JIHHIIIO BK, nYCTb 3TO 6YJJ;eT FK. JIHHIIH FK paBHa JIIIHUn HB,
a yrOJI K F H - npH IOH, KaR 8TO 6bIJIO ,IJ;OKa3aHO B rrpe,IJ;nOCbI TIKe.
Ho yrOJI CF L TaKfJie npHMOH, n03TO}IY JIHH;HH LF K - npH}faH. TaR
KaK }J;oRa38Ho, qTO JIHHHH LF paBHa JIHHIIH BH, JIHHHH LK paBHa
YJJ:BoeHHOH JII1HUn B H. ECJIH JIHHIIH Lf( He 60JILllie JJIIHHIf BD't
npO,IJ;OJIil\lIl\f JIUHIIIO BL B ee HanpaBJIeHMII B CTOPOHY J II npOJJ;OJIa\H I
JIHHUIO K L B ee HanpaBJIeHMH TaRII-\e B CTOPOHY L, OTJIOa\Ifl\1 JIHHMH
LM M LN II coeJJ;I1HIlJ\11 N M, npH"tJeM JIIlHHH LlV/ H LN [COOTBeTCTBeH-
HO] paBHbI JIHHHHl\'1 LK M LB, 11 BepTIIKaJIbHbIe yrJIbI N LM H BLK
pa BHbJ. 1103ToMY OCHOBaHlfe JlJ N paBHO OCHOBaHHIO B K, M TpeyrOJIb-
HHR N Lit! paBeH [TpeyrO TJbHIIRY] BLN, H OCTaJIbHble yr:IhI [Tpe-
yrOJIbHHROB 1 paHHbI OCTaJIbHbll\l yrJIaM, RarH,IJ;blH paBeH COOTBeTCTBYIO-
m;eMyeM:Y. TI03TOl\IY yrOJI NML paBeH yrJIY BKL. Ho yrOJI BKL -
npH IoiL n03TO IY yrOJI N M L [TaRiHe] npHMOH» [20.. c. 754-756l.
3aMeTIIJ\1J, qTO Ha qepTen e J16u aJI-X.aHcaMa (HanI plIC. 40) J(JIHIIH
LM paBHaH JIUHMH KI-" U300paJI\eHa HeCKOJIbRO 60JIbllIeH, "tJe)f KL,
a npHMoH yrOJI LM N Jla06pai)\eH B BM,IJ;e OCTporo yrJIa. n"pO,n:OJIiHIIM
ll3JIOn eHHe 116H aJI-XaHca la:
«JIHHMH LM paBHa JIHHIIH LK, T. e. ynBoeHHoH JIMIII1II HB.
n03TOMY JIMHHH M K - yqeTBepeHllaH JIHHHH H B. OnYCTHl\1 113 TO"tJ-
KH N nepneHnHRYJIHp Ha JIHHIfIO BK U ee npO,IJ;OJI)HeHlIe. Torna NQ -
yqeTBepeHHaH JIIfHHH liB. ECJJH npOJI;OJI}HHM JIIIHIIII BN II QN B HX
HanpaBJIeHHHX B CTOPOHY TOqKII N, OTJIOrHHM Ha HIIX JIHHIII1., paBHbIe
COOTBeTCTBeHHO TJJIHIIH I BN II QN, II coenMHIIM IIX KOH bI, nOJIy-qIIT-
CH TpeyrOJIbIIHR, paBHT)IH TpeyrOJIbHHRY BQN, II, eCJIII onYCTHTb H3
BepIIIHHI)I 3Toro TpeyrOJIbHIIRa nepneH}J;MKYJIHp Ha JIHHIIIO BQ n ee
npOnOJIiI,eHlIe, 3TOT nepneH,IJ;I1RYJIHp - Y,IJ;BOeHHbI:M nepneHJJ;HRY.ll HP
NQ. lIo nepneHJJ;IIRYJIHp NQ - yqeTBepeHHaH JIHHHH BH, n03TOMY
3TOT nepneIIJJ;IIRYJIHp - YBOCbMepeHHaH JIHHHH BH, 8TOT nepneHJJ;U-
KYJIHp ony eH H3 TOqKH JIHHIIH BN IIJIH ee npO,IJ;OJImeHI1H H3 nHHIIIO
BK II ee npOJJ;OJIrHeHHe. TaKIIM o6pa30M, MomHO onYCRaTb nepneH,IJ;II-
KY"TJHpbI U3 [TOqeK] JIHHHH BL Ha JIHHIIIO BK H ee npOnOJIllieHlIe 6ec-
KOHeqHO. f\aiJ-\.IJ;bIH 1I3 3THX nepneH,IJ;HKYJIHpOB nOJIY"tJeH ynBoeHlleM:
JIHHIIIf BH H Bcex [JIHHMB], npeJJ;IIIeCTBYIOID-HX eMY» [20, c. 756].
aJIee M6u aJI-XaHca:M, TaR me, KaR ero npe,IJ;IIIeCTBeHHHKJI.. nOJIb-
30BaBIIIHHCH aRCIfO [OH EBJJ;ORca - ApXIIMe,IJ;a, B OTJIlIqUe OT HUX
HBHO <pOP:MYJIlIpyeT ee, CCbIJIaHCb Ha EBKJIHna, y KOToporo ona <pnry-
pnpyeT B BM,IJ;e onpeJJ;eJIeHHH 4 KHlIru V «Ha"tJaJI». «,l(JIH BC:HRIIX ABYX
pa3JIHQHbIX JIUHHH,- nHIueT M6H aJI-XaHCal\I,- eCJIH ynBaHBaTb
:MeHbIUYIO 6eCROHeqHO MHoro pa3, ee BeJIHqlIHa CTaHeT 60JIbJIle OO"TJb-
IueH BeJIIIQHHbI. 8Ta npeJJ;nOCI)IJIR3 paHbIIIe He 6blJIa HymHa ]],JlH 3Toro
J{ORa3aTeJIbCTBa, HO I BKJIlfn npH IeHHJI ee B CBoeH KHlfre B JJ;pyrJIX
58
CBOIIX JJ;OKa3aTeJIbCTBaX. O'lleBH)J}IO, 'liTO YKa3aHHOe HarvIJI )]:OKa3bIBaeT-
CfI TOJIbKO C ee nOMO blO. II03TOIVIY MOmHO OnYCTI1Tb nepneHJJ;JIKYJIflp
H3 TO'llKJI JIJIHJIJI BN HJIJI ee npoJJ;OJI)UeHJIfI, KOTOpbIH ynaJJ;eT Ha
JIJIHJIIO BK II ee npOJJ;OJIrReHJIe JI 6YJJ;eT 60JIbIIIe JIIIHJIM DB. IIycTb
TaKoif neprreH,n;HKYJIflp - NQ. Ho [JIIIHJIfI] NQ paBHa lVI K, n03TOMY
JIJIHJIfI M K 60JIbllIe JIJIHJIJI BD. OTJIOmJIM [JIJIHMIO] KO, paBHYIO
BD. II YTBepm,n;aIo, 'liTO JIHHHH DG, eCJIJI npOJJ;OJDHIlTb ee B ee HanpaB-
JIeHIlJI JJ;O JIJIHHH KM, rrepeCe'lleT ee B TO"tlKe 0, T. e. OHa nepeCe'lleT
JIIfHIIIO KM, II HeB03MO)1\1I0, 'IIT06bI OHa He nepeCeKJIa JIJIHHH KM,
eCJIIl npOJJ;OJI)KaTb ee 6eCKOHe'llHO. OnYCTJIM JI3 TO'llKJI K rrepneHJJ;JI-
KYJIflp Ha JIJIHIlIO DG II ee rrpoJJ;OJI» eHJIe, nYCTb 3TO KI. [JIJIHMfI]
KI OTJIJIqHa OT KM: eCJIJI JIHHHfI KI eCTb JIJIHJIfI KM, TO'llKa I Ha-
XOJJ;JITCfI Ha JIHHJIH KM HJIU ee npOJJ;OJIsl eHJIIl, HO TO'llKa I HaXOJJ;JITCH
Ha JIJIHIIJI DGI, II eCJIJI JIHHJIfl KI eCTb JIJIHIIH KAI, TO'llKa I - 06II1-aH
TO'llKa JIJIHJIH DI H KM, T. e. DI nepeCeKae'fCfI C JIHHJIeii KM, COB-
rra,n;alO eii C KI, H, 3Ha'llJIT, DG nepeceKaeTCfI C JIJIHJIeM KM. TaKJIM
06pa30M, JIMHHH KI - lIe JIJIHHH KM» [20, c. 756-757].
Ha qepTe)l\e H6H aJI-XaMCaMa (HallI pHC. 40) 1I306pameHo onpOBep-
rae:M:oe Hi\{ rrOJIO)l\eHHe, n03TOMY JIOMaHafl DGI Ha 3TOM 'IIepTerRe JI30-
6pai-KaeT npflMYIO JIJIHIfIO, a TynoH yrOJI K I]) 1I306pan aeT npflMoM
yrOJI .
1I6H aJI-XaMcarvI npOJJ;OJIH\aeT: «EcJIU JIJIlIJIfI KI - He JIJIHJIfI K M,
OHH nepeceKalOTCH B 06UJ;eH TO'llKe K. fI03TOl\IY yrOJI I KB OTJII1QeH
OT yrJIa M KB. JIMHJIII BK II DI npone.u;eHbI M3 KOIIl\OB JIJIHI1JI BD
H BMeCTe C JIJIHMeH BD 33KJIIO'IIaIOT rrpflMbIe yrJIbI DBK 11 BDI,
..TIJIHIfH KI nepneHJJ;UKYJIflpHa K JIMIIJIJI DI, n03TorvIY yrOJI I KB -
rrpflM:oM no JJ;OKa3aHHOMY B npeJJ;nOCblJIKe. TaK KaK yrOJI M KB Tome
npflMOH, 3TH JJ;Ba yrJIa paBlIbI, HO 3TO HeJIenO, II, TaKIfM 06pa301\f,
HeJIerrO Harne npeJJ;nOJIOrReHHe 0 T01\I, QTO JIIIHJIM DG II KM He
nepeceKaIOTCfI. II03TOMY JIHHJIH DG nepeceKaeT JIJIHJIIO KJVl. H
YTBep)l\JJ;alO, 'liTO 01I3 nepeceKaeT ee KaK pa3 B TO'llKe O. llYCTb OHM
nepeceKalOTCH B TO'llKe, OTJIJI'IIHOH OT TOqKJI 0, a IIMeHHO B TO'llKe ]
Tor.u;a TOqKa P Haxo.u;MTCfI MetI\.u;y 0 JI K JIJIH BHe JIMHIIJI OK co CTO-
pOHbI TO'llKJI O. IIycTb CHa'llaJIa OHa lIaXOJJ;UTCH BO BHellIHeM TOqKe
co CTOpOHbI TO'llKII 0, KaK Ha 'IIepTeme. TorJJ;a JIIlHlffl P K 60JIblUe
JIJIHJIH KO, a JIJIHMfI KO paBHa JIMHMII BD. fI03ToMY JIJIHJIfI J K
60JIbllIe JlMHJIH BD. Ho JIMHMH B K JI D jJ npOBeneHbI JI3 KOII OB JIJIHJIM
BD, II. Ka)l\JJ;bIH JI3 yrJIOB KBD H PDB - npflMoM. ll03TOMY JIHHJIfI
PK neprreHJJ;HKYJIflpHa K JIHHIfJI BK M JIJIHHH PK paBHa JIHHJII:I BD,
1I0 paHbIIIe OHa 6bIJIa 60JIbIIIe ee, 'liTO HeJIeno. II03TOl\lY JIJIHIIH DG
He nepeceKaeT JIIIHJIIO KO BO BHellIHeM TO'llKe co CTOpOHbI TOqKIl O.
ECJIJI )I\e TO'llKa P HaXOJJ;IITCfI Meil JJ;y TOqKaMH K M 0, JIMHIlfl K})
MeHbllIe JII:IHMH BD, HO C rrOl\IOIIJ;blO npeJJ;noCbIJIKH 6bIJIO ,n;OKa3aHO,
'liTO JIJIHJIH K P paBHa JII1HMJI B D, n03TOMY TO, 'liTO K] l\leHblIle BD,
HeJIeno. IIoJTOMY JIUHJIfI DG He nepeceKaeT JIJIHMIO KAf TaKII I 00-
pa30M, 'liTO OHM nepeceKalOTCfI Mea ,n;y TO'llKal\IJI KilO M BO BHeIIIHeH
TO'llKe co CTOpOlIbI TOQKH 0, Ii:, CJIeJJ;OBaTeJIbHO, OHI1 nepeceKalOTCH
B TO'llKe 0 M JIMeeTCH JIJIHJIH DGO. TOQKa 0 HaXOJJ;HTCfI Ie»,.u;y TOqKa-
MU K H M, II, CJIeJJ;OBa TeJIbHO, TOQKa 0 JIJIM naxo.u;UTCH 1\fe»,.u;y TO"lIKa-
59
1\111 L H AI, HJIH OHa eCTb TO'llRa L, llJIII OHa HaXOJJ;IITCfI MeiU.n;y TOqRa-
Mil [( II L. ECJIII TO'llRa 0 eCTb TO'llRa L, JIIIHllfi K L paBHa JIlIHUn BD,
JIHHllfI DC JJ;OCTllraeT TO'llRH L, ROTopafl HaXOJJ;UTCfI Ha JIIIHIIII BL.
TaR RaK JIIIHHH DC ,n;OCTJIraeT TO'llRII L, ORa nepeceKaeTCfI c BL.
ECJIII T01:IRa 0 Haxo,n;HTCH 1\leil AY TO'llK8?tIH K II L, JIIIHllfI LK OOJIbIIIe
JIIIHIIII BD II JIHUIIH DO JJ;OCTllraeT TO'llRII 0, Haxo.n;fI eBcfI Meil\JJ;y
TOqR31\IH KilL. TaR RaR JIHHHH DO nepeceRaeTcfI C JJHHUeii LK,
ona ncpeceRaeTcH C JIHHlIeM .BL B TO'llKe MeiHAY TO'llKaMH L If B»
[20, c. 757 -758].
To.. 1:ITO JIIIHI.JfI DO DC nepeceKacTcH C JIHHlleH BL == BC
n CJIyqac, HorJJ;a /.JK > BD, JJ;ORa3hIBaeTCfI aHaJIOrll'llHO npIlBeJJ;eH-
nOMY BbJllle CJIyqalO.
«EC.,1:M iI\e JIHHHH J)O,- npOJJ;OJIsRaeT 116H 8JI-XaMCaM,- nepece-
KaeT JIMHHIO LK B TOqI{e 0 l\,fei)\;O;Y TO'llKa1\111 L II M, yrOJI KOD -
npflMoii, TaR RaR JII1HHH BK H DO npOBeJ(eHbI H3 KOBIlOB JIliBHII BD
H R8iHAI)]M 113 yrJlon KBD 1:1 ODB - npflMOM, a JIHHIIH KO nepneH-
AIIRYJIHpna R oITJIIHlln KB. n03TQMY B CIIJIY npe;u.noCLJJII\JI yrOJI
KOD npHMoll II JIHHllfI DO nepeceKaeTCfI C JI.HHl1eH LM. l c:In npo-
,n;OJIiI,MTb JIIIHI1IO DO B ee HanpanJIeHHIJ BHYTPL TpeyrOJIbHIIR8 £111 N,
T. e. npoJJ;OJJiHIITb B ee HanpaB.,TJeHRH Ha TaKoe paccToHHlle, 1:.J:TO OHa
nona;o;eT BHYTPb TpeyrOJlbHI1Ra LAJN, ona nepeCeqeT OAHY H3 CTOpOH
LN, 1VM HJIII npOMJ\eT Qepe3 neplUIIHY N. 110 OHa He 1\10iReT nepe--
Ce1Jb CTOPOHY M N, TaR RaR, eC1Ju 6bI Olla nepeCCKJI8 CTOPOHY 111 N,
IIOJIyqllJICfI 6bI TpeyrOJIbHHR C ,lJ,BY1\IH npH1\fI)I1\IItJ yrJIal\-1l1, a 3TO HCJleno.
'ro ,he ObIJIO 6bI B CJIY1:lae, eCJIH 61)1 OHa npOllJJIa 'IIepe3 TO IRY N.
n03TOlVIY JIHHHfI DO He nepeceRaeT JIIIHI1H M 1V H He npOXO]J,IIT IIPpe3
TOQI\Y N, XOTfI If nonaJJ;aeT nnYTPb TpeyrOJIbHIIRa LMN. ECJIU OHa
HaXO]UITCfI BHYTPH TpeyrOJIbHIIKa LlVI N 1:1 eCJIH 6eCKOHeQHO npOAOJI-
iRI1TL ee B ee H8npaBJIeHIIII, OHa He nepeCeqeT JIIIHHH 11-1 N H He npoi1:-
.n;eT qepe3 TOQRY N 1103T01\fY OHa nepeCe'lleT JIIIHIlIO LN B T01:.J:Ke S
JIHHIIH DGOS, 1"aK qTO TO'llRa S HaXOJJ;IITCfI Ha npflMoH JIIIHIIII BCLN
H Ha npflMoH JIIIIIIIM DGOS, Ii JIIIHIIH BC II DG nepeceRalOTC.R B TO"IJ:-
Re S. TaRIIl\f OOpa301\1, eCJIII yroJI DBC MeHI)rne npflMoro, 8 yrOJI
BDG - npfl?tl0M, JIIIH1III BC II DG, eCJIH HX npo.nOJIiJ-\IITb B CTOPOHY
TOqeR C II G II npOJJ;OJIiI\aTb IIX 6eCKOHeQHO, nepeceKYTCfI» [20, e. 758].
HaK Mbl 1l1:l.n;IIM, I16H aJI-XaUCaM npOBeJI CBoe JJ;ORa3aTeJIb.CTBO JJ;JIH
CJIYQafl, RorJJ;a 2BH < BD < /jBIl, HO COBepmeHHO TaR me npOBO-
JJ;IITCH JJ;OKa3aTeJIbCTBO H B 06ll\eM cJIY'llae, Kor.n;a 2 1l BH < I D <
< 2 n + I BH. I-Ia 'IIepTeiJ\e (CAt. pllC. 40) BD == 4BH, HO, RaR YK83bI-
naJIOCb BbIIlle, 3JJ;eCb npeJJ;nOJIaraeTCH, '111'0 KL L}1tf, T. e. KJ.'I
== 4BH II, CJIe,n;onaTeJIbHO, BD < 4BH.
M6H aol -X aMca 1 JJ;OKa3hIBaeT Ha TOM iRe 'IIepTeme V nOCTY:18T A.TIH
CJIY'IIafl, KO JJ;.a ceKy afl nepeceRaeT JJ;Be npflMhIe no.n; .n;BYMH OCTpbI-
1\11:1 yrJIaMH, npH e!\I Ha epTeiI\e 116H aJI-Xa:Hca la (C1\1. pllC. 40) npH-
1\10M yrOJI BDG H306paa-\aeT OCTpbIH yrOJI, TaR qTO ceRYlI\aH BD
COCTaBJIfleT C npflMblMII AB II ED .n;Ba OCTphIX yrJIa: «Ee.:'III yrOJI
BDG - OCTpblM II onYCTIITb 113 TO'llKH B nepneHAHRYJIflp lIa .11IHIlIO
DG, OH yna.n;eT CO CTOpOHLI G, TaK KaR, eCJII:I 6hI OR ynaJl CO CTOpO-
HbI E, nOJIyqllJICfI 6bl TpeyrOJIbHIIK, yr JIbl KOToporo OhJJIII 61)( UOJIhIIIe
60
nyx npHl\lbIX I yr tlYJOB], 1I0TOl\IY qTO yrOJI JTOrO TpeyrOJIbIIlfl\a npH
OCHOBaHUH nepneHJJ;IIRYJIHpa - npHlVIoM, a yrOJI BDE - TynoH
B CIIJIY Toro, qTO yrOJI BDG - OCTpbIH II yrJIJ,I TpeyrOJIbHIfROB OIJTJIII
ObI 60JlblIle fJ;BY.x npH1\IbIX r yrJIOB 1, a HTO He TIeno. rI03TO IY neplIPHp;n-
RY TIHp, onym;eHHbIM 1I3 TO'lRH B Ha TIIIHlIIO DG. ynaJ(eT TOJIbRO eo
CTOpOHbI G. ECJIH 3'rOT nepneH,I(JIRYJIHp ynaJJ;eT co CTOpOHbI C. 3TOT
nepneHJJ;JIRYJIHp paCCe'lleT yrOJI DBC 1I CCRYII aH 6YJJ;eT 3aI{JIIot.laTb
C JIJIHJIM BC OCTpLIH yroJI. TaR KHR 3TOT nepIIeHJJ;IIRY TJHp aaR.TJIOtlaeT
C JIJIHJIeM DG npHl\fOH yrOJI, C JIYQai1 I1oJ(06eH TOl\IY CJlYl.laJo. (\orJJ.a
JIHHHH BD 3aRJIIO'llaeT C JIJIHJIeM BC OCTpblH yroJI, a C .ITUHlIeM DC -
npHMoH yrOJI. rI03TO lY, TaR iI e R3R n npe,n:J)J)xYIII.eM JJ;0I\a3aTe TJJ)CTBe,
)XORa3bIBaeTCH, "£ITO JIllHJIU BC II DG nepeceRYTCH» [20, c. 758-75H).
3Ha'llJITeJIbHO CJIOJJ\Hee JJ;ORa3aTeJILCTBO 116H a.rr-X ancaj\fa V IIOCTY-
JIaTa JJ;JIH CtlTJY"tJaH, Ror)Xa ceRYII aH nepeceRaCT J].Be np.fI lble nOL( OCT-
pbIM II TynbIM yrJIaMM. B HTOl\l CJIy-qae Ha "tIepTeiI\e J10H aJI- X aiic a Ia
npHMoH yro TI BDG JI306paniaeT TY110M yrOJI, a OCTpblii yro TI P'XD
JI306paiHaeT npHl\lOll -yrOJI. M6H a.rr-Xai1cal\f !IpnOeracT 3;I.eCh eII e
R O,l];IIOH YCJIOBHOCTM: BBMJJ;Y Toro, "tJTO 3JJ;eCJ) OH IIO.:Ib3yeTcH lIe nce l
-qepTeiI\Ol\f, a TOJIbRO O)XHOH ero 'tJaCTbIO, on n03ll0JIH€T ceoe 00031Ia-
'IIHTb HOBbIe TO"tIRII TCl\lJI rHe 6YRBal\fM, ROTOphIl\'II1 Ha JI.pyrnx. Tenepb
yme HeIlYiI"HbIX el\-IY "tJ:aCTHX "tIepTejUa 011 0003Ha'llaJI JJ:pyrne 'fOTIRH:
« ECJJH yrO:l J DG - Tynoii, TO yroJI E DJ ' - oc:rphlii. P3 AP: HM
JIHHJIIO BD nOnOJIaM n TO'llRe 1/ II onYCTHl\I 1I3 TO"tJRlI F nepIle]IJJ.nKy ;rIHp
Ha JIJIHJIIO EDG. OH yna)XeT co CTOpOHbI E, TaR RaR, eC TJH 61)1 011
ynaJI co CTOpOllbl G, nOJIY'IIJIJICH 6bI TpeyrOJIbllJIR, yr TIw ROToporo
6LIJIII 6bI 60JIbIIIe )];BYX npHMDlX yrJIOB, nOTOl\IY 'tJTO r yro.;T{ JTOFO
TpeyrOJIbHHRa npH OCHOBaHHJI nepneH)J;HRYJIHpa npHMoftJ, a yrOJI
BDG - TynoH; HO :3TO BeJIeno. TI03TOMY OIl yna)XeT Ha JIIIHHIO DE.
nYCT:b nepneH)J;BRYJIHp 6YJJ;eT FX. TaR RaR yrJH>I DBC II BDG 1\IeHblIIe
)J;BYX npHMhIx [yrJIOB], a yrJIbl XDB II BDG paBlIbI )XBYM npHl\lbIM,
yroJI XDB 60JI:bIIIe yrJIa DBC. nOCTpOI1M yroJI F1JN, paBHhIH
FBC, II npo)J;OJImHM DN B ce HanpaBJIeHHI1 AO P, II, TaR EaR yrJIbI
DNX II DXN [yrJIbT1 B TpeyrOJIbHHRe. T. e. l\fCHbllle ,l!BYX npH n)TX
[yrJIOB], a yrOJI DXN - npHMoii, yrOJI DNX MeHblIle npH!\IOrO yr TIa,
a TaRiI e yrOJI F N j BepTMRaJIbHI>lM no OTlIOIIIeHIIIO R He IY Ii n03TOl\lY
paBHLIH el\fY. TI03TOMY yrOJI FlV 1) leHbIne npH Ioro, 11'1 eC jH :Mh[
onYCTJIl\f M3 T01JRH }l nepnCH)XURYJIHp Ha JIHHHIO N l . OH ynaneT co
CTOpOHbI P, TaR RaR yrOJI FNP OCTphIU, a l/ND - Tynon. I1YCTb
3TOT nepneHJJ;JII(YJIHp - "JIIIIIUH F P OnycTII1vI H3 TO'llRJI }' nepncHJJ;H-
RYJIHp Ha JIJIHJIIO BC, nYCTb 3TO 6YJJ;eT F8. YroJI }/8B TpeyrOJII>Hn-
Ra F8B paBeH yrJIY TpeyrOJIbHHRa l/jJD, TaR RaR ooa onM - npH n)le
[yrJIbI], II yrOJI FB8 paBen yrJJY FDP, JIHHJIH B}l paRIIa JIIIHIIII p'D,
n03TOl\IY TpeyrO:IbHJIR F B 8 paBeH TpeyrOJIbHJIKY j'D JJ, OeTa.,TJhHble
yrJIbI [TpeyrOJIbHIIROB] paBHbl OCTaJII>HbIl\-1 yrJIaM, lJ yrO TI SF S
paBeH D F}.J. JIIIHIIH 8 J1"'JJ npOJJ;OJIllieHa B O)XHOl\f nanpan.aeHHn,
a JIJIHMH S X nepcceRaeT JIIIHJIIO /) P II pacceRaeT yrOJI D F J) rIo-
3TO IY JIUHMH X It"' 3aRJ1IO'IIaeT C JIBHJIeM 1/8 yrOJI, CTHrnnae Ihlii &lYJUlIlIeH
8 X, JIRHIIH S X 3aKJIIO'IIaeT yr JIbl RaK C J1IIHJleH X f"', TaR]I c .,jI1Hueii
FS. Yro.,rr FXD - npHl\IOH, n03TOl\IY yroJI SXD - OCTphIiL Yro..TJ
61
p'se - npHM:oii, n03TO IY yrOJI xse - OCTphIii. EC.,TJII OnYCTIITb 1I3
TOqKII S nepneHJJ;IIRYJIHp na JIIIHIIIO XD, OH ynan;eT co CTOpOHbl D,
TaR RaR yrOJI SXD - OCTpbIii, M eCJIU nepneHJJ;IIKYJIHp, onym;eHHbIM
H3 TOqRII S Ha JIHHlllO D X, ynaJJ;eT co CTOpOHbl D, OR pacceKaeT yrOJI
X se, a eCJIII OH paCCe'lleT yrOJI X se, OIl 6YAeT 3aKJIIO'IIaTb C [oJ'IU-
HHeM] se OCTpbIM yrOJI; 3TOT nepneHAHKYJIHp 3aKoJTJIO'IIaeT C JII1HlIeii
XDG npHMOH yrOJI, n03TOl\IY, eCJIU yrOJI BDG TynoiL cym;ecTByeT
JIHHIIH, nepeCeKalOIII.aH JIIIHIIII Be H DG II 3aKJIlOllalOm;aH OCTpbIM
yrOJI KaR C JIIIHlIe:n Be, TaK M C JIIIHHeM l.)G, OTKYAa CJIe.n;yeT, 'liTO
3Tl1 JIIIHUU IomHO COe)];IIHHTb 11 JIIIHlleii no)]; npH.MbIl\1 yrJIOl\I K O)];HOH
1I3 HMX. Onpe)];eJIMB 3TO B TOM n-\e nopHJJ;Ke, liTO 11 npll npe)];bI)];ym;eM
)];OKa3aTeJIbCTBe, )];OKa l-\eM, 'liTO B CJIyqae JIIIHHII A e II EG, eCJIII npo-
)];OJIrRIITb MX B CTOPOHY e II G II npO)];OJlmaTb IIX 6eCKOHe'llHO, OHM
nepeceRYTCH. I103TOl\fY eCJIII npHMaH JIIIHHH na)];aeT Ha ,n;Be npHMbIe
JIIIHIIII II BHYTpeHHlIe yrJIbI no O)];HY 113 .n;BYX CTOpOH l\IeHbme )];BYX
npHMbIX [yrJIOB], TO 3TII np!IMb1e JIIIHIIM npn npO)];OJIHieHHII B 3TY
cTopony nepeceKYTCH, a 3TO II eCTb TO, "(ITO l\fhI XOTBJIII )];0 Ka3aTI»)
[20, c. 759-762].
I1pIIBeJJ;eHHoe HaMH JJ;OKa3aTeJIbCTBO V nOCTYJIaTa, IIpMHa,n;JIe;'J a-
m;ee 116H aJI-XaiicaMY, pa3BIIBaH B OCHOBHOM II)];eM AOKa3aTeJIbCTBa
I1UH ROPPbI, npenJIOmeHHbIe IIM B ero BTOpOlvl TpaKTaTe, 06JIanaeT
PHJJ;O I cym;ecTBeHHbIX oco6eHHocTe:n: nplI )];OKa3aTe.JIbCTBe cym;ecTBO...
BaHIIH npHMoyrOJIbHIIKa IICnOJIb3yeTcH npHel\f paCCMOTpeHMH Tpex
rl1IIOTe3 0 'IIeTBepTOM yrJIe qeTbIpexyrOJIbHIIKa 116H aJI-XaHCal\la -
JlaM:6epTa, a B naJIbHeiirnllx 3Tanax )];OKa3aTeJIbCTBa YTBepmJJ;eHI1H
aKCHOM Ilaula II EBJJ;OKca - ApXIIMe,D;a, KOTOpbIMH M6H aJI-XaMCaM
nOJIb3yeTcH HCoJ'Ie)]; 3a 116H Roppoii II Apyrlll\III CBOIIl\-IM npe.n;IlleCTBen-
HIIKaMII, Tm;aTeJIbHO orOBapllBalOTCH nepBoe KaK OqeBII)];HOe YTBepil\-
)];eHlle, a BTopoe - KaK YTBep \neHlle, C.pOp1vIYJIM pOBaHHoe EBRJIIif-
)];OM, HO npllMeIlHBmeeCH lIl\f n03)];Hee.
B )];pyro:n CBoeM KHllre - ll;IITHpOBaBmeMCH B TpaKTaTe Hacnp
aJJ;-,1];HHa aT- TyclI [28] «I- HMre 0 pa3pemeHHII COMHeHHH» - JI6H
aJI-XaMCaM, COCJIaBmlICb Ha TO, 'liTO OH )];ORa3aJI V nOCTYJIaT B.. «l\HI1:re
KOl\-Il\feHTa pMeB RO BBe)];eHIIHM», YKa3bIBaeT, "({TO 3TOMY nOCTY JIa TY
paBHOCHJIbHO YTBepm)];eHHe: ,n;Be nepeceKalOm;lIeCH npHMhle He napaJI-
JIeJIbHbI OJJ;HOH JIHHHM (T. e. 'IIepe3 TO'llKY HeJIb3H npOBeCTII )];BYX pa3-
JIMqHhIX napaJIJleJIeii R O)];HOH npHMOM), npH'IIeM «3TO YTBepiI )];eHlle
60JIee Hal' JIH)];HO )];JIH 'IIYBCTBa II 60JIee npOHIIKaeT B Ayrny, 'IIeM TO»
[28, c. 486-487].
62
O HeJ1,aBHerO Bpe IeHU 0 pa-
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HbIe u;eHTpaJIbHhIe yrJIhI BbIceRalOT Ha oKpymHocTHX paBHhIX KpyroB
paBHbIe )];yrll. XaHH1\1 OTBepraeT PH)]; nOTIbITOR ero npe)];IIIeCTBeHHHROB
)];OKa3aTb 9TOT nOCTYJIaT, KaK JIOrlf1JeCKII HeCOCTOHTeJIbHhIe, npHqeM
Ha3bIBaeT nonbITRn )];OKa3aTeJIbCTBa 3TOH aRCHOMhI, npIlHa)];JIema He
fepOHY, EBTORHIO, aJI-Xa3I1HY, am-llIaHHII If aH-HaiIpII3H. AOKa3a-
1'eJIbCTBO M6H aJI-XaHCaMa XaHHl\f KplITHRyeT 3a nplIMeHeHlle )J;BH-
meHHH, TaR RaK caM OB BCJIe)]; 3a ApHCTOTeJIeM CqlfTaeT HenpaBOl\lep-
HbIM nplIl\feHeHlIe )J;BlImeHIIH K reOMeTpHII.
aJIee XaBJIM .pOpl\IYJlHpyeT nHTb nplIBe)];eHHbIX HaMil BbIme
.( c. 11) «npHHD;HnOB, 3aHMCTBOBaHHhIX y .pHJIOco.pa» (A pn:cTOTeJlH),
'IeTBepTbIH 1I3 KOTOpbIX, KaK YKa3bIBaJIOCb, 3RBIIBaJIeHTeH V nOCTYJIa-
7Y EBRJIII)J;a.
BHa1JaJIe XaHJIM )];OKa3bIBaeT, qTO )];Ba neprreH)];IIRYJIHpa R o)];Hoi
npHMOH He 1\IorYT nepeceKaTbCH, TaK KaK B 9TOM CJIyqae OHB .n;O.rriHHbl
nepeceKaTbCH B )J;BYX TOqKaX no 06e CTOpOHbI OT 9TOH npHMoH. OTCIO-
p;a H 1I3 nepBoro YTBep}K)];eHHH nplIBl\Hna 4 CJle)];yeT, "({TO .n;Ba nepneH-
JJ;HRYJIHpa K O)];HOU npHMOH He MorYT CXO)];IITbCH.
M3 BToporo YTBepil,JJ;eHIIH npHHl\lfna 4 CJIeJJ;yeT, QTO 9TII JJ:Ba nep-
neH)];IIKYJIHpa He MorYT II pacxo.n;IITbCH, TaK RaK OHI:I )];OJIJRHhI 6bIJIR
6hI paCXO)];lfTbCH JJ;Pyr OT )J;pyra no 06e CTOpOHbI OT 3TOH npHl\tIoH.
TI09T01\:IY ABa nepneH)];HKYJIHpa K O.n;HOH npHMOn )];OJIiI,HbI 6bITb 9RBH-
IICTaHTHbIMH.
aJIee XaHHM ]];OKa3bIBaeT BOCel\Ib npe)];JIOmeHHH, KOTopbIe .n;OJIiHHhI
QbITb, no ero MHeHHIO, BCTaBJleHhI B I KHHry «HaqaJI» BMeCTO npe)];JIO-
;fieHHH 29 9TOB RHllrH, c KOT poro EBKJIB)J; HaqHHaeT H3JIOmeHHe
TeoplIU napaJIJleJlbHhIX JIHHHH, OCHOBaHHOB Ha ero V nOCTYJIaTe.
IIepe)]; 9THMH npe)J;JIOmeHHHMH XaHHM numeT:
«Tenepb MhI )J;OJImHhI npHHHTb 28 npe.n;JIOmeHBB KHHrH «HaqaJIa)),
TaK RaK OHH He Hym)J;3IOTCH B 9TOB npe)J;nOCLIJIRe. Ho B HeB Hym)];aeT-
oCH npe)];JIOmeHHe 29, BLIpa)l\aIOmee 3aKOHOMepHOCTb napaJIJIeJIbHhIX
JIHHHH. IIo9TOMY TOT, RTO X01JeT, nYCTb nOCTaBHT nepBoe npe)J;JIome-
Hlle 3Ton KHHrH BMeCTO npe)J;JIOmeHHH 29 KH TH I, BKJIIOqaH ero, eCJIH
3aX01JeT AJJJIax, B co.n;epmaHHe KHHrH.
3)];eCb "Thl YBH)J;Hmb HCTHHHoe «)J;OKaaaTeJIbCTBO Toro, n01JeMY 9TO
67
3 'c
TaR», npH 6JIarOCRJIOHHOCTH H nOMO H AJIJIaxa; RTO nplI6eraeT
K HeMY, OH PYROBO,lJ;IIT HM H YJJ;OBJIeTBOpHeT ero» [32, c. 120]. «,I{oRa-
aa TeJIbCTBO Toro, n01JeMY 3TO TaR» - H3BeCTHLIH TepMHH a pHC1'OTeJIeB-
CROH JIOrHRII, npOTIIBOnOJIOmHbIH aplICTOTeJIeBCROMY TepMBHY «JJ;ORa-
3aTeJIbCTBO Toro, 1JTO 31'0 TaR». Bo BTOpOM CJIyqae HMeeTCH B BII,lJ;Y
):(ORaaaTeJIbCTBO, y6emJJ;3lOmee B npaBIIJIbHOCTH HeROToporo nO.7Ioa e-
HIIH, HO He BbIHCHHIOm;ee ero npHqHHLI, a B nepBOM - JJ;OKaaaTeJIbCT-
BO, y6emJJ;alOm;ee B np3BHJIbHOCTII qero-JIH60 C nO:MoIIJ;blO BbIHCHeHIIH
ero npnqBHbI.
XallHM, RaK 11 MHorHe ero npeJJ;IIIeCTBeHHHRII, HaqHHaeT C IJ;oRa-
3aTeJIbCTBa cymecTBoBaHIIH npHMoyrOJIbHHRa. ,I:{JIH 3Toro OH CTpOIIT
1JeTLIpexyrOJIbHIIR, 06pa30BaHHblll ,lJ;BYM.H nepneHJJ;HRYJIHpaMII A C II
BDj paBHoH JJ;JIHHLI, BOCCTaBJIeHHbIMII R .OJJ;HOH npHMoH AB, II 01'-
pe3KaMH AB H CD. 01'01' 1JeTbIpexyrOJIbHHK, TaR me RaK qeTbIpex:..
yrOJIbHIIR M6H aJI-XaiiCaMa, CbIrpaJI Ba}I\HYIO pOJIb B IICTOpHH
HeeBRJIHJJ;OBOH reOMeTpHII; ero qaCTO HaaLIBalOT 1JeTbIp xyrOJIbHIIROM
CaRRep:o no IIMeHH MaTeMaTHRa XVIII B., BHOBb paCCl\IaTpHBaBIIIero
TaROH qeTLIpexyrOJIbHHK. 3aMeTHl\f, qTO qeTblpexyrOJIbHHR XaHHl\Ia -
CaKKepH ):(eJIIITCH cBoeii OCblO CHMMeTpHH Ha JJ;Ba 1JeTbIpexyrOJIbHIIRa
H6H aJI- XaiiC3Ma -JIaM6epTa.
B npeJJ;JIo» eH1IH 1, ROTopoe XaHHM npeJJ;JIaraeT C1JIITaTb npeJJ;JIO-
meHHeM 29. RHHrII I «Ha-qaJI», OR, 3a,lJ;aBmHCb npHMOM JIIIHlIeii AB,
npOBOJJ;IIT JJ;Be paBHbIe JIHHH1I A C II BD, nepneH,lJ;HKYJIHpHbIe AB,
II rOBoplIT:
«OHB napaJIJIeJIbHbI, RaR nORaaaHO EBRJIHJJ;OM B npe,lJ;JIOmeHIIH 26
[I RHlIrlIJ. CoeJJ;HHIIM CD (pRC. 44). H YTBepiUJJ;alO, 1JTO yrOJI ACD
paBeH yrJIY BDC» [32, c. 120J.
IIpeJJ;JIomeHHe 2 XanHMa (<30-e "Ha1JaJI u ») rJIaCHT: «PaCCMOTpHM
CHOBa .pHrypy ABCD, pa3JJ;eJIHM AB nOnOJIaM B E H npOBe,lJ;eM EG
nepneH,lJ;HRYJIHpHO K AB (pnc. 45). .R YTBepmJJ;alO, 1JTO CG paBHa GD
II 1JTO EG nepneHJJ;IIRYJlHpHa DC» [32, c. 121 J.
QeHTpaJIbHbIl\f nYHKTOM JJ;OKa3aTeJIbCTBa XaHHMa HBJIHeTCH ero
npeJJ;JIOmeHHe 3, B ROTOpOM onpOBepralOTCH rHnOTe3LI OCTporo II TY-
norD yrJIa ):(JIH nOCTpoeHHoro HM 1JeTbIpexyrOJILHIIRa.
«TpeTbe npeJJ;JIOmeHHe, T. e. 31-e "Ha1JaJI". PaCCMOTpHl\I CHOBa
«pHrypy ABCD (pIIC. 46). .H YTBepmJJ;alO, 1JTO yrJIbI ACD H BDC
np Hl\lbIe.
,1];oRa3aTeJIbCTBo. Pa3JJ;eJIIIM AB nonOJIal\l BE, BOCCTaBHM rrepneH-
):(IIRYJIHp EG, npoJJ;OJImHM ero B ero HanpaBJIeHIIH, OTJIOjRII 1 GK,
paBHYIO GE, II rrpoBe,n;eM HKF nepneHJJ;IIRYJIHpHO R EK.
,1];aJIee npOJJ;OJIJI\HM A C H BD. OHH nepeceRYT H KF B H II F, TaR
RaR A C H EK napaJIJIeJIbHbI, a paCCTOHHlIe l\-IemJJ;Y JJ;BYMH napaJIJIeJIb-
HbIMH He H3MeHHeTCH, II eCJIH M.bl 6YJJ;el\1 npOJJ;OJIiJiaTb JJ;O 6eCROHe1JHO-
CTH A C, napaJIJIeJIbHYIO JIHHIIH EK, H 6YJJ;e?-I npOJJ;OJImaTb JJ;O 6ecRo-
HeqHOCTH H K, na paJIJIeJIbHYIO JIHHMH GC, OHII, O'lleBH):(HO, He06xoIJ;lI-
.MO nepeceRYTCH. Coe):(HHHJtf CK II DK. TorJJ;a, TaK RaR JIHHIIH DG
paBHa GC, a GK 06m;aH II B TO }He BpeMH nepneH):(IIKYJIHpHa [K DG
II GC], TO OCHOB3HIIH DK II KC paBHbI H yrJIbI GCK II GDK paBHLI.
ll03TOMY yrJIbI HCK H KDF TaRiHe paBHLI II JJ;onOJIHHTeJIbHbIe yrJIbI
68
])
c
F
K
H
B
A
Puc.44
.D & C
S 8 M E l A N
Puc.46
.
B E A
Puc.45
DKG II CKG pRBHbI, II OCTaBmHeCH yrJIbI H K'C JlI. F KD TaKme paBHbI.
TIo9TOMY, TaK KaK JlJlI.HJlI.H DK paBHa KC, TO CH paBHa DF II HK
paBHa KF. ECJIII yrJlbI ACD'JlI. BDC npHMbIe, 9TO HCTHHHO nOHeBOJIe.
EcnD me OHB He npHMbIe, TO RamJJ;bIH H3 HHX llJIH MeH:bIIIe npHMoro,
JlI.JIH 6oJI:bme ero.
TI YCT:b CHa"llaJIa OHH MeH:bme npHJ\1IOro. ECJIH MbI HBJIOmJlI.M nJIOCKYIO
<pHrypy CF Ha IIJIOCKYIO <pJlI.rypy CB, TO GK HaJIOmHTCH Ha GE TaR
me, KaR H F HB AB, npJlI."IIeM H F 6YJJ;eT paBHa JIIIHHJlI. N S, TaK KRK
yrOJI HCG 6oJI:bme yrJIa ACG H JIHHJlI.H HF 6oJI:bme AB. TO"llHO TaR
I\e, eCJIH 8TH JJ;Be JIHHJlI.JlI. [CH II DF] npOJJ;OJImaT:b JJ;O 6eCKOHe"llHOCTll t
TO KamJI;aH JlI.3 coeJJ;HHHIOIII;:nx [JlI.x] JIHHHH B nopHJJ;Ke nOCJIeJJ;OBaTeJI:b-
HOCTJlI. 6YJJ;eT 60JI:bme, qeM JJ;pyraH. TIo3TOMY JIHHHH AC II BD 6YJJ;YT
pacxoJJ;HT:bCH. TO"tlHO TaR me JIIIHHJlI. A C H BD npH HaJIOrKeHHH B JJ;Py-
rOM HanpaBJIeHHU 6YJJ;YT paCXOJJ;HT:bCH, "liTO JJ;OKa3IiIBaeTCH COBep-
meHHO TaR rue, TaR KaR nOJIOmeHHfI no o6e CTOpOHhI npH HaJIOiI e-
HIIII He06xoJI;HMO COBnaJJ;aIOT. TIo9TOMY JI;Be IIpHMhIe JIHHHH nepeceRa-
lOT nOJJ; npH IIiIMH yrJIBMH npHMYIO [JIHHHIO], a 3aTeM no o6e CTOpO-
HbI OT 3TOH JIHHHH paCCTOHHHe MemJJ;y HHMH YBeJIH"IIHBaeTCH. Ho 9TO
B cnJIY aRCHOMbI HeJIenO, eCJIII npeJJ;CTBBIITb ce6e IIpHl\fH3HY. 110-
3TOMY MemJI;y 9THl'tIH JI;BYMH JIHHHHMH IIl\feeTCH onpeJJ;eJIeHHOe paCCTOH-
Hue. 8TO JlI.3 Toro, "tlTO paccMaTpHBaJIOCL <pHJIOCO.pOl\1.
TIYCT:b Tenep:b KamJJ;hIH H3 HHX [yrJIOB A CD H BDC] 6oJI:bme npH-
Ioro. TorJJ;a npH HaJIOmeHHH JIIIHIIH H F 6YJJ;eT paBHa LM, ROTopaH
6YJJ:eT ?tfeH:bme AB, TaR me KaR Bce coeJJ:HHHlOmHe JIHHHH, JlI. 9TH JJ:Be
TJIIHIIH 6YJJ;YT CXOJJ;HTbCH. C JJ;pyroH CTOpOHhI TaR}Iie 6YJJ:eT cxomJJ;eHlIe,
TaK Kal\ nOJIOmeHHH no o6e CTOpOHhI npH HaJIOmeHmI connaJJ;aIOT.
69
EC,JlH TLI HeMHOrO nonYMaem:b, TLI 3TO nOHMem:b. Ho 3TO CorJIaCHO
CKpaaHHOMY BbIme onHTb HeJIeno.
(J09TOMY Be JIBHBB [AB B FH] He MorYT 6LITb paaJIB'lHLIMB, T. e.
OHYI paBHLI. TaR KaR OHB paBHbI, }J;Ba yrJla TaKme paBHLI, BCJleJJ:cTBBe
'lero OHB HBJIHIOTCH npHMLIMB. TLI nOHMem:b 3TO IIPB He6oJlbmO 1
pa MhIIIIJleJlBB. IIo3ToMY, 'lTo6LI Ba6emaTb MHOrOCJIOBBH, Mid OCTa-
Bufl 3TOT Bonpoc. TOT, KTO aaXO'leT npoBecTH nOJJ;p06Hoe }J;oKaaaTeJI:b-
CTWO, CMomeT 3TO CJJ:eJlaT:b, He Hym aHcb B Hamen nOMO H.
(?mH6Ka noaJJ:HeHmHX [Y'leHbIX] B JJ;ORaaaTeJI:bCTBe 9TOH npeJJ:IIOCldJI-
RB npo:UCXOJJ;HT OT Toro, 'lTO OHD He yqHTbIBaJIB 9ry aKCBOMY, aiRe
ecdB ee nOJJ;Jlema ee II cRaayeMoe npeJJ;CTaBJIHJIOC:b npaBHJI:bHO. II Te,
or{opLIe 06J1a aIOT rJIy60KoH HHTYH Hei II npOHH aTeJI:bHbIM YMOM,
MOCYT He Y'lHThIBaT:b MHorBX aKCHOM Ha-aa Toro, "tJTO OHB He npeJJ;-
CTa BJIHIOT UX nOJJ;JIema HX II cKaayeMldX. Ho nepBHqHOCT:b II IICTHH-
HOc;T:b YTBepiRJ];eHHH He TOJI:bRO B npe CTaBJleHHH ero nOJJ;JIeiRa ero H
CKwayeMoro, TaR RaR cnpaBe}J;JIHBOCT:b HJIH HecnpaBe}J;JIHBOCTb YTBepiR-
eIlBH aaBHCBT He OT caMHX nOJJ;JIeiRa ero II cRaayeMoro, a TOJl:bRO
OT BHalI Mea\JJ;Y HHMII. B 3TOM COCTOBT npHqHHa, no KOTOPOH aRCHO-
Ma MOiReT He yqHTbIBaT:bCH. IIOHMH 9TO» [32, c. 121-122].
yfTaR, }J;onYCRan rHnOTeahI OCTporo II Tynoro yrJIon, XaHHM CHaqaJIa
neI1erlI6aeT "tJepTem no npHMoH CD D IIORaahIBaeT, 'lTO npH 9TOM
IIpJ1 r!lnoTeae OCTporo yrJIa OTpeaOR H F COBIIaJJ;aeT C OTpeaKOM N S ,
a nPH rHIIOTeae Tynoro yrJIa - C OTpeaROM LM, T. e. B nepBOM-CJIY-
'Iae BepXHee OCHOBaHHe 'IeTliIpexyrOJIbHBRa ORaaLIBaeTCH 6oJI:bme
HII} Hero B ero 60RoBhIe CTOpOHbI Y}J;aJIHIOTCH JJ;pyr OT JJ;pyra, a BO
BTOPOM - BepXHee OCHOBaHBe MeH:bme HHiRHerO D 6oKoBbIe CTOpOHhI
'IeT,PIpexyrOJl:bHHRa npH6JIHiRalOTCH JJ;pyr R JJ;pyry. ,IJ:aJlee, neperH6aH
'lepTeiR no npHMoH AB, XaHHM BHJJ;HT, "tJTO B nepBOM CJIY'lae }J;Ba
nepneHJ];HRYJIHpa K OJJ;HOH npHMoH pacxoJJ;HTCH no 06e CTOpOHbI OT
3T0;1 npHMoH, a BO BTOpOM 9TH nepneHJJ;HKYJlHPhI CXOJJ;HTCH no 06e
CT01>0HhI OT 9TOH npHMoH. Ho 9TO nOJIOiReHHe II,poTHBOpeqlIT IIPHH-
BdY 4, T. e. JJ;Ba nepneHJJ;HRYJlHpa K OJJ:HOH npHMoH HaXOJJ;HTCH Ha
noc OHHHOM paCCTOHHHB B, CJIe OBaTeJIbHO, RaK rHIIOTeaa OCTporo
yrna, TaR H rHnOTeaa Tynoro yrJla nplIBeJJ:eHhI K npOTBBOpeqHIO
C 31 HM IIpBH HnOM. TeM CaMhIM ORaaaHO cy ecTBoBaHHe npHMO-
yro}lbHBKa.
;j.aJIee XaHHM onpeJJ;eJIHeT «paCCTOHHHe Mem y JJ:BYMH npOH3BOJIb-
HhI?t D [npHMLIMB] JIHHHHMH» RaK «JIHHHIO, coeJJ;HHHIOID:YIO HX TaKIIM
06p8.aOM, 'lTO BHYTpeHHHe yrJIhI paBHbI». OH pacCYiRJJ;aeT TaR:
«(J1.Be JIHHBII AB H AC nepeceRalOTCH B TOqRe A (pHC. 47). II
YTB pmJJ;alO, 'lTO OHH paCRpbIBalOTCH B paCXO HTCfI o 6eCROHe'IHOcTH.
JlP 3Toro onHmeM Ha n:eHTpa A Rpyr ABC Ha paCCTOHHHII AB.
PacCToHHBe MeiRJJ;y JJ;BYMH JIHHHHMH npB HX nepece"tJeHHH C RpyroM
eCT}? JlHHHH BC. IIpoJJ;oJIiRBM AB B ee HanpaBJIeHHH H onHmeM Rpyr
AD . ,lJ:aJIee npO]:(OJImHM A C B ee HanpaBJIeHHB o ee nepeCeqeHIIH
C KJ'yroM [ADE] B TO"tJRe E D coe;n;HHBM DE. TorJJ;a paCCTOHHBe l\lem-
y BYMH JIHHHHMB eCTb DE, npH"tJeM JIHHHH DE 60JI:bme BC, H eCJIH
npe CTaBBT:b ce6e CMIiICJI Kpyra, yrJIa II npHMoH JIHHHH, TO, 6ea COM:-
HeH H, 3TO aKCHOMa» [32, c. 133].
70
3aMeTIIM, 'liTO npOBeJJ;eHlle yr C IJ;eHTpOM B BepmHHe npHMOnJI-
HeiiHoro yrJI8 II XOPJJ; 9THX yr HanOMHHaeT paccym eHHH, C nOMO blO
KOTOpbIX JI;ORa3bIBaJIH aRCHOMY 0 napaJIneJIbHIiIX CIIMnnHR:nH H
aJI-,n>l\ayxapH. B03MOiRHO, qTO XaHHM 3aHMCTBOBan CBOII BIiIBOJJ;hI 113
aTHX HJIH JJ;pyrlIx JJ;ORa3aTeJIbCTB YRaaaHHOH aRCHOMbI, npnHaJJ;JIema-
ID;IIX ero npeJJ;IIIeCTBeHHHRaM. MhI yme ynoMHHaJIH, 'liTO paC l\IaTpHBae-
Mo.e 3JJ;eCb YTBepiRJJ;eHlle - npHHIJ;Hn 3, «3aHMCTBOBaHHhIH y <pHJIOCO-
.pa», HBJIHeTCH CJIeJJ;CTBHeM aRC:UOM reOMeTpHH EBKJIHJI;a, He 3aBHCH-
IIJ;1Il\1 OT aRCIIOMhI 0 na paJIJIeJIbHbIX.
OnpeJI;eJIHB paCCTOHHHe MemJJ;y npHMbIMH JIHHBHMH, XaHHM cnpa-
IIIIIBaeT: «ECJIH }J;aHbI JJ;Be npHMhIe JIIIHHII AB II AC Ha nJIOCRQCTB
II npeJJ;nOJIOiRHM Ha AB TO"tJRY E, TO paCCTOHHHe MemJI;Y TO"tJROH E
II JIHlIlIeH DC eCTb JIHHHH EG :n yrOJI E paBeH yrJIY G (pHC. 48). Ho
KaR npOBeCTII IIa TOqRH E JIHHHIO R CD, 'ITo6bI BHYTpeHHHe yrJILI
ObIJIH paBHIiI? MCnpaBJIeHHe OCHOB reOMeTpHII - }J;eJIO reOMeTpa,
a He <pHJIOCO<pa. MomHo JIH npoBecTII JIHHIIIO, 06JIa}J;aIO YIO 9THM
CBOHCTBOM? 8TOT Bonpoc OTHOCHTCH R :nCRYCCTBY aBTopa [.pHJIOCO -
CRIIX] npHHl\lInoB. Pa3'DHCHHM 9TO CJIeJI;YIOIIJ;HM o6pa30M. lI3 E
MOil,HO npoBoJJ;HTb K CD 6eCqHCJIeHHliIe JIBHHH, o6pa3YIO He Ha CBO:nX
ROHI ax 6eC'IHCJIeHHliIe yrJIbI, OTJIllqaIOIII;HeCH JJ;pyr OT Pyra TeM, 'liTO
OJJ;IIH 60JIbIIIe :nJIH MeHbme }J;pyroro. Ho TaR KaR Ha }J;BYX ROHl\aX
[coeJJ;HHHTeJIbHOH npHMoii] :nMeIOTCfI pa3JIII'IIHbIe [yrJIIiI], OAHH 113 KO-
TOpLIX 60JIbIIIe HJIH MeHbme JI;pyroro, TO B CIIJIY Toro, 'ITO BeJIlIqHBLI
JJ;eJIHMIiI O 6eCROHelIHOCTH, Heo6xoAHMO B03MOiRBO II paBeHCTBO ,Il;BYX
yrJIOB [EGF II GEH].
OTJIOmlIM EH II GF, paBHliIe APyr APyry, :n CoeJI;HHHM HF. TorJI;a
yrOJI H paBeH [yrJIY] F, KaR nORa3aHO B nepBOM CJIyqae, TaR '{TO
HF eCTb paCCTOHH:ne. II09TOMY, eCJIH HF 60JIbme EG, }J;Be JIHHBB
pacxoJJ;HTCH.
aJIee OTJIOmHM H K H F L, paBHliIe }J;pyr APyry, II coeJJ;:nHHM KL.
TorJJ;a KL eCTL paccToHH:ne. IIo eCJIH KL MeHbme HF, }J;Be JIHII:n:n
CXOJJ;HTCH B CHJIY aRcHoMIiI, TaK RaR OHH CHaqaJIa paCXO}J;HJIHCb. To
sRe Heo6xoJJ;HMO 6YJJ;e1' II B TOM CJIYlIae, eCJIH OHH paBHIiI.
ECJIH HF MeHbme EG, JJ;Be JIHHIIH CXOJJ;HTCH. B CHJIY nORa3aHHoro
Ha1-IH KL Heo6xoJJ;HMO MeHbIIIe HF, TaR RaR B npOTBBHOM CJIyqae
MhI B CIIJIY aKCIIOMIiI nOJIyqHM HeJIenOCTb.
IIo3TOM:y HCHO, "tlTO, eCJIH JJ;Be npHMliIe nHHHH Ha OJJ;HOH nJIOCRo.CTH
B O)J;IIOM HanpaBJIeHHH CXOJI;HTCH, HeB03MOiRHO, lITo6bI OHH paCXOJJ;H-
JIHCb B aTOM HanpaBJIeHHH. To me HMeeT MecTo, eCJIII OHH pacxoJJ;HTCH.
8TO OO'DHCHeHHe HBJIHeTCH <piIJIOCO<pCKHM, a He reOMeTpHqeCRHM. O-
6aBJIeHHbIH Ha IH npHMep npeJJ;Ba3HaqeH )J:JIH Toro, qTOOIiI CJJ;eJIaTb
113JIOn eHHoe 60JIee Har JIHJI;HIiIl\l II 60JIee 01IeBHJI;HbIl\1 JI;JIH Tex, RTO He
o6JIaJI;aeT OCTpOH IIHTYH lIeH» [32, c. 121-124].
HaR BHJJ;HO, DPH JJ;ORa3aTeJIbCTBe B03MOiRHOCTH npOBeJJ;eHHH npH-
}IOH, COCTaBJIHIOm;eii C JJ;BYMH )J:aHHbIMH npHMbIMH paBHDIe yrJIhI,
XaHHM CCbIJIaeTCH Ha npHHIJ;Hn 1, «3aHMcTBoBaHHDIH y .pllJIOco.pa»,
IIrpalOIII;HH y Hero pOJIb np:nHl(Hna HenpepbIBHOCTII, a «aRCHOMOih)
IIa3bIBaeT npHHIJ;lIn 4, ROTOpbIM OH 3aMeHIIJI V nOCTYJIaT.
,[(aJIee XanHM IIIIIIIeT: «TaR RaR JJ;ORa3aHO, 'liTO eCJIH AaHa npH laH
71
c
A
1J
A
Puc.47
E
.c
8
Puc.49
.0
B
c
A
Puc.48
c
A
H
&
E
H
G
Puc.50
Puc.51
8
JJ
8
II
F
JIHHHH, B 060HX KOHu;aX KOTOpOH BOCCTaBJIeHhI rreprreH,IJ;HKYJIHpbI II
aa 8TUX rreprreH,IJ;HKYJIHpaX OTJIOmeHM rrpOH3BOJIbHMe paBHMe H-
HHH, TO paCCTOHHHH Mem,IJ;Y 8THMH JIHHHHMH neprreH,IJ;HKYJIHpHhI
K HHM H paBHhI Me}H,IJ;Y co6on, a ,IJ;Be JIHHHH He CXOJI;HTCH II He pac-
XO,IJ;HTCH, 6Y,IJ;eM Ha3liIBaTb TaKHe ,IJ;Ba nepneH,IJ;HKYJIHpa 8KBH,IJ;HCTaHT-
HbIMH» [32, c. 125]. B rrpeJJ;JIOmeHHH 4 (<32- 1 "HaqaJI"») Xa:HHM L{O-
Ka3bIB3eT, qTO B rrpHMoyrOJIbHHKe rrpOTHBOnOJIOmItliIe CTOpOHbI paB-
RbI: « aHa nJIOCKaH <pHrypa ABCD C rrpHMhIMH yrJIaMH (pHC. 49).
H YTBepm,IJ;alO, qTO AB paBHa CD, a AD paBHa BC» [32, c. 125J.
B rrpe,IJ;JIOmeHHH 5 «(33-M "HalIaJI"») ,IJ;OK83hIBaeTCH, liTO ,IJ;Ba rreprreH-
)];HKYJIHpa K O,IJ;HO:H rrpHMOH 06JIaAalOT Te:M CBOHCTBOM, liTO Kam,IJ;bIH
nepneHAHKYJIHp K O,IJ;HOMY H3 HUX HBJIHeTCH IIX 06 HM nepneH,IJ;HKY-
JIHpOM: «JIUBHH AB II CD 8KBH,IJ;HCTaHTHhI (puc. 50). JI YTBepm,IJ;alO,
'ITO BCHKaH JIBHHH, rrepneH,IJ;HKYJIHpHaH K O,IJ;HOH H3 HIIX, rrepneH,IJ;II-
KYJIHpHa K ,IJ;pyrOH» [32, c. 125].
B npe,IJ;JIOmeHHH 6 «<34-M "HaqaJI"») ,IJ;OKa3hIBaeTCH, liTO eCJIH ,IJ;Be
rrpHMliIe rrapaJIJIeJIbHhIe B CMIiICJIe EBKJIH,IJ;a, T. e. He nepeceKalOTCH
rrpH npO,IJ;OJImeHHH, TO OHH HBJIHIOTCH )];BYMH rrepneH,IJ;HKYJIHpaMII
K OJJ;HOH npHMoii: «BCHKHe JJ;Be JIHHUH, rrapaJIJIeJIbHliIe COrJIaCHO onpe-
J];eJIeHHIO EBKJIHJI;a, T. e. He nepeceRalO HeCH, 6e3 BCHKoro JJ;pyroro
YCJIOBHH 8KBII,IJ;HCTaHTHhD) [32, c. 126].
B rrpe]];JIOmeHHII 7 (T. e. no ClIeTY XaHHMa 35-M «HaqaJI») YC7aHaB-
JIHBaeTCH, 'lTG, eCJIU JJ;Be napaJIJIeJIbHhIe npHMhIe nepeceK310TCH
TpeTben rrp MOH, HaKpeCTJIema He II COOTBeTCTBeHHhIe yrJIhI paBHbI,
72
a BHYTpeHHHe OJJ;HOCTopOHHHe yrJIbI COCTaBJIHIOT B CYMMe ,n;Ba rrpflMhIX
yrJIa. 3TO rrpe,n;JIOmeHHe COBna,n;aeT C npeJJ;JIOmeHHeM 29 KHHrII I EB-
KJIH,n;a, HO npH ero JJ;ORaaaTeJIbCTBe XaHHM orrHpaeTCH yme He Ha
aKCHOMY EBKJIHJJ;a 0 napaJIJIeJIbHbIX, a Ha CBOH npeAJIOmeHIIH.
11, HaKOHeD;, B npeJJ;JIOmeHHH 8 XallHM ,n;OKaabIBaeT V nOCTYJIaT
B OPMYJIHpOBKe EBKJIII)];a:
«llpeJJ;JIOmeHHe 8, T. e. 36-e "HalIaJI". JIHHIIH EG - npHMaH. OT
Hee npOBeJJ;eHbI )];Be JIHHIIH EA 11 GC, rrp:nlIeM yrJIbI AEG II CGE
[BMecTe] MeHbme ;a:BYX npHMIiIX (pIIC. 51). fl YTBepmnalO, lITO OHB
nepeceKalOTCH CO CTOpOHhI A.
,r(oKaaaTeJIbCTBo. TIpo,n;OJImHM 8TH ,n;Be JIHHIIII B IIX HarrpaBJIeHHH.
TIYCTb yrOJI AEG MeHbme [yrJIa] EGD; nOCTpOHM yroJIHEG, paBHbIH
[yrJIY] EGD. Tor,n;a )];Be JIIIHIIH HEF II DGC napaJIJIeJIbHLI, KaR
.n;OKaaaJI EBKJIHn B npe)];JIOmeHIIH 27 KHlIr:n I, II JIHHHH A E, nepece-
KalOm;aH [JJHHHIO] HF, nepeCeqeT JIHHHIO CD co CTOpOHhI A. 3TO
II eCTb TO, lITO MhI XOTeJIH JJ;oKaaaTb.
BOT HCTHHHoe )];OKa3aTeJIbCTBO YTBepm)];eHlIll 0 napaJIJIeJIbHbIX
B COOTBeTCTBIIH C ero Cl\IIiICJIOM H l\eJlblO. CJIe)];OBaJIO 6hI )];06aBIITb 8TH
npeJJ;JIOiI\eHIIH B "HaqaJIa" B TaKOM rropHJJ;Ke, RaK l\lhI H3JIOmHJIH IIX
B 8TOH KH:nre. OH:n BbITeKalOT lIS npHBD;HrrOB TIepBoH HJIOCO H:n [T. e.
IIJIOCO HH ApHCTOTeJIH.- Aem.]. Mhl BKJIlOlIHJIH :nx CIOAa, XOTH
OHH II BbIXO,n;HT 3a npeJJ;eJIhI CYIII;HOCTH 8Toro IICKYCCTBa, TaK KaK MhI
He CMOrJIH Ha6e)I\aTb 8Toro, BCJIeJJ;CTBlle Toro, lITO 8TOT Borrpoc TpYJJ;eH
II 06cym)];aJICH MHorHMH JIIO}J;bMH. II08To1\fY MhI ,n;06aBIIJIH BO BBe,n;eHH:n
yrroMHHYTbIe npHHD;HnbI, TaK KaK 8TO IICKYCC1'BO HymJJ;aeTcH B HIIX
JJ;JIH Toro, lIT06bI IIMeTb npOlIHYIO HJIOCO CKYIO OCHOBY H He BbI3liIBaTb
nOJJ;oapeHHH H COMHeHHll y Tex, KTO pa3MbIlliJIHeT Ha,n; HBM» [32,
c. 127].
TIplIBeJJ;eHHoe pacCY)I\)];eHHe XallHMa 6JIIIaKO K JJ;OKaSaTeJIbCTBY
TI pOKJIa, C TOll paaHHD;ell, lITO nOCTOHHCTBO paCCTOHHIIH :p..lem,n;y He-
nepeceKalOIII;HMHCH rrpHMIiIMH EH II CD y XaHHMa 060CHoBbIBaeTcH,
a y IIpoKJIa OrpaHHqeHHOCTb 8Toro paCCTOHHIIH MOJIqaJIIIBO rrpeJJ;no-
JIaraeTCH. Ho BamHellmell aaCJIyroH XaHHMa B TeopHII rrapaJIJIeJIb-
HbIX JIIIHHH, KOHeqHO, HBJIHeTCH He YCOBepmeHCTBOBaHHe AOKaaaTeJIb-
CTBa IIpoRJIa, a nepBaH B HCTOpHB reoMeTplIH - BO BCHKOM cJIyqae
nepBaJl HaBeCTHaH HaM - OTKpLITaH aaMeHa V rrOCTYJIaTa 9KBHBaJIeHT-
HhIl\-1 (H 60JIee HarJIH)];HbIM C TOlIKII apeHHH aBTopa) nOCTYJIaTOM II pac-
Cl\-IOTpeHlIe qeTbIpexyrOJIbHIIKa XaHHMa-CaKKep:n. B xo,u;e 8TBX
paccymJJ;eHHH OPMYJIIIPYIOTCH nepBhIe TeopeMbI r:nnep60JIHlIeCKOH B
8JIJIIInTlllIeCKOH HeeBKJIII)];OBLIX reoMeTp:nH. TaKIIMII TeopeMaMB HB-
JIHIOTCH nOJIyqaeMbIe B xO}J;e JJ;OKaaaTeJIbCTBa rrpe,n;JIOmeHIIH 3 XaHHMa
CJIeJJ;CTBHH, lITO B CJIyqae Tynoro H OCTporo yrJIOB BepXHBe OCHOBaHIIH
"tJeTbIpexyrOJIbHIIKa XallHMa-CaKKepll COOTBeTCTBeHHO KOpOqe IIJIU
,nJIIIHHee HHlliHero II 'tlTO )];Ba nepneH)];HKYJIHpa R OAHOH npHMOH B CJIY-
lIae rlIrrOTeabI Tynoro yrJIa npH6JIHmalOTCH 1J:pyr K APyry rro o6e
CTOpOHbI OT OCHOBaHIIH II nepeceKalOTcH C 06eux CTOpOH, a B cJIyqae
rlIrrOTe3hI oCTporo yrJIa 8TH rreprreH)];HKYJIHpbI paCXOJJ;HTCH no 06e
CTOpOHIiI OT HX OCHOBaHIIH.
79
TIonhITKa YJIyqmlITb n;OKa3aTeJIb-
CTBO XaHHMa 6hIJIa rrpeJJ;npllHHTa B
HaqaJIe XIII B. XycaM a - IIHOM
<1>aJJ;JIaJIJIaXOM ac-CaJIapOM (YA-1. OR.
1262), 3aHIIMaBmlIM nOCT rJIaBHO-
ro aCTpOJIOra "PH JJ:Bope rrOCJIeJJ;-
HIIX xope3MmaXOB nepeJJ; MOHrOJIb-
CRlIl\f 3aBoeBaHIIeM U nplI JJ;Bope
MOHrOJIbCROrO 3aBoeBaTeJIH XYJIary-
xaHa. Ac-CaJIap 6bIJI aBTopOM He O-
me mlIx JJ;O Hac «IDaXCRIIX aCTpOHOMlIqeCKIIX Ta6JIII » II TpaKTaTa 0
nOJIHOM "1JeTbIpeXCTopOHHIIKe, IITlIpOBaBmerOCH }-IaclIp aJJ;-,[(HHoM
8T-TYCII B ero 1I3BeCTHOM TpYJJ;e no TOMY me Bonpocy [27, c. 23-24];
KaK ac-CaJIap, TaK II aT':'TyclI paCCMaTpllBaJIII nJIOCKUH I{ c.peplIqe-
CKHH CJIyqalI TeopeMhI MeHeJIaH 0 nOJIHOM qeThIpeXCTopOHHIIKe,
a TaKm JJ;PyrlIe TeopeMbI c<pepIIqeCKOH TpHrOHOMeTpIIU II HX npIIM:e-
HeHHe K pemeHHIO c«pepHqeCKHX TpeyrOJIbHHROB.
TIonbITKa ac-CaJIapa JJ;ORa3aT V nOCTYJIaT, CKame1\1 3apaHee, 6hIJIa
COBceM HeYJJ;aqHoH. TpaKTaT ac-CaJIapa, B KOTOpOM npOH3BOJJ;IITCH
9Ta no IlbITK a , Ha3hIBaeTCH «TI peJJ;nOChIJIKH JJ;JIH JJ;OKa3a TeJIbCTBa nOCTY-
JIaTa, rrpHBeJJ;eHHoro EBKJIHJJ;OM B HaqaJIe nepBoH KHHrH, OTHOCH ero-
CH K napaJIJIeJIbHhIM JIHHHHM» (MYJ:\aJJ;JJ;aM3.T JIH Ta6HHH aJI-l\fY98JJ;apa
aJIJIaTH aKapaxa Y15JIHJJ;[HC] <pH 9aJJ;P aJI-Ma 8JIa aJI-YJI8 «pH M8
iiaTa(aJIaf:\y 6H-JI- YTYT aJI-MYTaB83IIHHa).
<DaRcIIMIIJIe PYRonHcH TpaRTaTa ac-CaJIapa 6hIJIO nOMemeHo
iI\. XYMaH B ero «XaHHMH-HaMe» ["33, c. 285-294]; HMeeTCH ero
PYCCRIIH nepeBOJJ; [24].
,[(oKa3aTeJIbCTBO ac-CaJIapa COCTOHT 113 meCTll «npe nOChIJIOK» II
JJ;OKa3aTeJIbCTBa caMOH aKCHOMM 0 napaJIJIeJIbHbIX.
TIepBaH H3 «npeJJ;nOCbIJIOR» ac-CaJIapa COBnaJJ;aeT C npeJJ;JIOiReHHel\1 1
.n;OKa3aTeJIbCTBa XaHHMa. Ac-CaJIap, KaK H XaHHM, npeJJ;nOJIaraeT,
qTO BepXHHe yrJIbI nOCTpoeHHoro 11M qeThIpexyrOJIbHIIRa XaHHMa
MorYT 6bITb TynbIMB H OCTpbIMII. aJIee ac-CaJIap nHmeT:
«3TO COBna aeT C TeM, KaR onpeJJ;eJIHeTCH paCCTOHHHe MeiHJJ;Y JJ;BY-
IH JIHHHHMH IIJIH paCCTOHHlIe MemJJ;y JJ;BYMH TOqRaMH Ha HHX: OHO
orrpe.n;eJIHeTCH no eJIHqllHe JIHHHH, coe HHHIO eH 9TH .n;Be JIHHIIH II
06pa3YIO ei C HHMB JJ;Ba paBHbIX yrJIa» [24, c. 285].
TaRHM o6pa30M, ac-CaJIap nOHHMaeT TepMHH «paCCTOHHHe Melli.IJ;y
.n;BYMH JIHHHHMH» B TOM me CMhICJIe, qTO II XaHHM.
aJIee CJIeJJ;yeT «rrpeJJ;nOCbIJIKa» 2 ac-CaJIapa, HBJIHlOmaHCH B ero
.IJ;OKaaaTeJIbCTBe OCHOBHOH:
«TIpe.IJ;noChIJIKa 2. JIH Ram oH npHMoH JIHHIIII, 113 KOHD;OB ROTOpoii
rrpOBeJJ;eHhI .n;Be npHMhIe JIHHUH, CTOH He Ha HeR npHMo, T. e. He Ha-
KJIOHHhIe HB R OJJ;HOH 113 JJ;BYX CTOpOH, RaK JIHHHH AC II BD, npOBe-
}J;eHHhIe 113 KOHD;OB JIHHHH AB YRa3aHHhIM cnoco60M (pIIC. 52), T. e.
HBJIHIO HeCH neprreHJJ;BRYJIHpHhIMH K Hen, RaK 6hI HB YJJ;aJIHJIHCb OHH
OT TOqeR IIX BbIxo.n;a, OHH He CKJIOHHhI HH K rrpH6JIHiH.eHHIO, HH
K YJJ;aJIeHHIO.
XOTH 8TO OqeBII.IJ;HO II 6JIB3KO .n;JIH rrOHHMaHHH, ,D;06aBIIl\1 JJ;OKa3aTeJIb-
12.
n Onbtm1Ca
XycaM ao-Auua
ac-CaJlapa
74
CTBO, a 1IMeHHO, npOBeJJ;eM Ha TO"llKH E, HaXOJJ;HII\eUCH MemJJ;Y T01JKaMH
A lIB, JI1IHHIO EG, KOTopaH TaK}Re CTOIIT Ha aTOH nHHIIH, nOJJ;o6Ho
JJ;BYM nepBhIM JIHHHHM. ECJIH JJ;Ba nepneHJJ;HKYJIHpa, BbIXOJJ;H He 113
KOH OB .n;aHHOH JIHH II, CKJIOHHbI K npH6JIlImeHHIO, TO nOJI01ReHIIe
JIIIHIIH EG C Kaa JJ;oH 1I3 JIHHHH AC II BD TaKOBO a e, KaK nOJIOmeHHe-
HX caMHX, H n03TOMY JIHHHH EG Heo6xoJJ;HMO CKJIOHH8 K npH6JIHiRe-
HHIO K KamJJ;oH H3 HHX. ECJIII iRe OHH CKJIOHHhI K YJJ;aJIeHHIO, TO II OHa
CKJIOHHa K YJJ;aJIeHHIO OT KamAou 113 HHX. Ho TaKoe nOJIOmeHlIe,
O"lleBHJJ;HO, He HMeeT feCTa, TaK KaK eCJIH JIHHHH nona aeT MeiRJJ;Y
JJ;BYMH JIHHHlIMII, TO HeB03MomHO, "IITo6bI OHa 6LIJIa CKJIOHHa K npll-
6JIHiI\eHHIO K OJJ;HOH H3 HHX 6ea Toro, "IITo6LI OHa 6liIJIa CKJIOHHa
K YJJ;aJIeHHIO OT APyroH, H 6hIJIa CKJIOHHa K y aJIeHHIO OT OJJ;HOH '1I3
HIIX 6e3 Toro, "IITo6hI OHa 6liIJIa CKJIOHHa K npH6JIH1ReHHIO K APyrOH.
113 3Toro Y3HaeTCH OKa3aTeJIbCTBO Toro, "liTO JIHHHH, CO eJJ;HHH 10-
II(aH KOH LI BYX p8BHliIX nepneHJJ;IIKYJIHpOB, BbIXOJJ;HIII;HX 1I3 KOH OB
JJ;8HHOH JIHHHH, Heo6xoJJ;HMO paBHa JJ;aHHOH JIHHIIH, nOJJ;o6Ho TOMY,
HaK JIHHHH CD, coeJJ;HHHIOIII; H [KOH bI] paBHbIX neprreHJJ;HKynHpoB
A C H ED, BLIXOJJ;lIIII;HX H3 HOH OB JIHHHH AB, Heo6xoJJ;HMO paBH8 AB:
eCJIH CD He paBHa AB, TO OHa 60JIbIIle HJIH MeHbme ee. ECJIH OHa
60JIbme, TO JJ;Be JIHHHH CKJIOHHhI K YJJ;aJIeHHIO, a eCJIH OHa MeHbme,
TO OHII CKJIOHHbI K npH6JIHmeHHIO. Ho YiRe 1I3BeCTHo, "liTO B aTOM rro-
JIOmeHHH paCCTOHHHe MemJJ;Y HHMH nOCTOllHHO: He npH6aBJIHeTCH II
He y6aBJIHeTCH» [24, c. 286].
YTBepm eHHe, 1JTO JJ;Ba neprreHJJ;HKYJIHpa K npHMOH He rrplI6JIu-
malOTCH H He YJJ;aJIHIOTCH, 6LIJIO JJ;ORa3aHO XaUHMOM B ero rrpeJJ;JI01Re-
HHII 3 nYTeM onpOBepmeHHH npeJJ;nOJIOmeHHH 0 TOM, "liTO Ba BepXHBX
yrJIa "IIeTLIpexyrOJIbHIIKa XaHHMa HBJIHIOTCH TYDLIMB IIJIII OCTpbIMIf,
C rrOMOIII;blO nplIH una 4, «3alIl\fCTBOBaHHoro Y «pBJIOcoq.a».
Ac-CaJIap 3aMeHHeT TOHKIIH BhIBOJJ; XaHHMa rpy60 OIIlH6olIHhIM
paccym)];eHHeM, B KOTOpOM CHa"ll8JIa JJ;onYCKaeTCH npH6JIHmeHIIe llJIII
YJJ;aJIeHlle )];BYX o6 :nx neprreH)];HKYJIHpOB, 3aTeM paccMaTplIBaeTCJI
TpeTIIH nepneHJJ;HKYJIHp, HaxoJJ;HIII;HHCH MemJJ;Y HIIMII, II aBTOp YCMaT-
pHBaeT npOTHBOpeqHe B TOM, "liTO TpeTIIH nepneHABKYJIHp B aTOM
CJIyqae TaKme JJ;OJImeH OJJ;HOBpeMeHHO nplI6JIHmaThCH K KpaHHBM nep-
.D
8
D
IT
c
F
G
B E A
Puc.52 Puc.53
C
75
neH.n;IIKYJIHpaM HJIH y.n;aJIHTbCH OT I1HX,- no ero MHeHHIO, eCJIH npH-
MaH, Haxo.n;H aHCH Mem.n;y .n;BYMH rrepneH.n;HKYJIHpaMH, npH6JIHmaeT-
CH K O.n;HOMY H3 HIIX, OHa .n;OJliI\Ha y.n;aJlHTbCH OT .n;Pyroro. <DaKTHqeCKH,
B nOCJIeLJ;HeM YTBepmLJ;eHHH HeHBHO npeLJ;nOJIaraeTCH, "liTO KpaHHIIe
rreprreH.n;UKY JIHphI Haxo,n;HTCH Ha rrOCTOHHHOM paCCTOHHHH .n;pyr OT
.n;pyra.
aJIee ac-CaJIap rrlImeT:
«ECJIH YCTaHOBJIeHa 8Ta npeAnocM Ka, OHa HaT8JIKHBaeT Hac Ha
npe.n;nOChIJIKY 3: eCJlH LJ;Ba yrJIa A H B caMH He npH IhIe, a TOJIbKO
paBHhI HM [B CYMMe], TO rrOJIOmeHUe 8TIIX JJ;B YX JIIIHUH TaKOBO iHe,
KaK nOJIOmeHHe yrroMHHYThIX paHee [npHMbIX], T. e. OHH HIIKor.n;a He
rrpU6JIHmalOTCH If He y.n;aJIHIOTCH. TOT, "IIbH IIHTYIID;UH He nOMoraeT
eMY B rrOCTlImeHIIH 9TOH npeJJ:nOChIJIKH, ,n;o6beTCH 9Toro C rrOMom;bIO
pa3MhIillJIeHHH, rOBopH: JJ;JlH BCHKHX .n;BYX JIHHUi, nYCTb, [HanpHMep],
AB II CD, rrpHMaH JIHHHH, npOBe.n;eHHaH OT o.n;Hon U3 HHX K ,n;pyroi,
3 HMeHHO EG, o6pa3yeT .n;Ba O.n;HOCTOpOHHHX yrJIa, a HMeHHO AEG
II CGE, paBHhIe [B CYMMe] .n;BYM rrpHMhIM, Tor.n;a MeiHJJ;Y HUMH IfM:eeTCH
npHMaH JIHHIIH, HBJIHIOIT(aHCH rrepneH.n;HKYJlHpOM K 06eHM JlHHHHl\{
BMeCTe. 8TO nOTOMY, "liTO, eCJIH H3 cepe.n;IIHhI JlHHHH EG, a HMeHHO
TO"llRH H, onYCTHTb rrepneHJJ;HRYJIHp HF Ha JIHHHIO Cl) (pHC. 53),
TO OH Heo6xoJJ;Hl\-IO BCTpeTHT JIHHIIIO CD He B TOqKe G, rrYCTb 8TO -
TO"llKa F. OTJIOiRHM Ha JIHHIIH EB JIHHHIO EK, paBHYIO JIHHHH FG,
II coe.n;HHHM TO"llRH K II H npHMOH JlUHllei, 3TO KH. TorJJ;a JIHHHH EH
II EK paBHbI JIHHHHM HG H GF, H HaRpeCTJlema Ue yrJIbI E II G
paBHhI. IT08TOMY paBHhI H yrJIbI K H F, a TaKme paBHhI II yrJlbI rrpH
H. Ho yroJl F - npHMoH, n08TOMY npHMoH II [yroJI] K. .H YTBep»tJJ;alO,
"liTO JIHHHH KH - B H3rrpaBJIeHHII [JlIIHHII] F H, TaR KaR yrJIbI npH
TO"llKe H paBHhI, II eCJIH npH6aBHTb K 060HM 06 Hii yrOJI EHF,
yrJILI EHK II EHF .[BMeCTe] paBHhI yrJIaM EHF II GHF, qTO paBHO
.n;BYM npHMhIM. II09TOMY yrJIhI EHK II EHF [BMeCTe] TaRme paBHbI
.n;BYM npHMLIM. IT03TOMY JIIIHHH KH - B HanpaBJIeHHR JIUHUH FH.
ECJIII Haxo;n:IITCH JIHHIIH, coe;n:HHHIO aH AB H CD no.n; npHMhIMH
yrJIaMH, TO JIBHHH AB II CD He rrpH6JIHmalOTCH H He y.n;aJIHIOTCH,
H IIX MomHO npo.n;OJImaTb 6e3 KOHD;a [6e3 BCTpe"llHJ. 8TO H eCTb TO, qTO
MbI XOTeJlII .n;OKa3aTb» [24, c. 286-287J.
«ITpe.n;noCbIJIKa» 3 ac-CaJlapa COBIIaAaeT C O HHM H3 YTBepm eHH:ii
rrpe.n;nOJIOmeHHH 7 XaHHMa. Y TBepm,n;eHHe «npeJJ;IIOChIJIKH» 4 ac-Ca-
Jlapa COBnaJJ;aeT C YTBepm.n;eHHeM npe.n;JlOmeHHH 3 XaiiHMa, HO .n;OKa-
3hIBaeTCH yme HCXOJJ;H H3 Toro, "liTO 60KOBhIe CTOpOHbI qeThIpexyrOJIb-
HHKa XaHHMa HBJIHIOTCH paBHOOTCTOHIIJ;HMH npHMhIMH. «fIpe;n:noChIJIKa»
5 ac-CaJIapa COBIIa;n:aeT C npe.n;JIOmeHHeM 4 XaiHMa. ,I:(JIH rrOHIIM:a-
HHH xo.n;a paccymJJ;eHHH ac-CaJIapa cy ecTBeHHa ero «npe.n;nOChI TlKa»,
YTBepm;n:eHlle ROTOpOH COBrra.n;aeT C YTBepmJJ;eHHeM nplIHD;lIna 3,
«3aIIMCTBOBaHHoro y «pHJIOco«pa», KOTOphIH XaHHM npHHIIMaJI 6es
nORa3aTeJIbCTBa. MbI yme rOBOpIIJIII, "liTO 8TOT npHHD;IIII npHHa.n;JIelliHT
K .a6COJIIOTHO:U reOMeTpHH, T. e. CIIpaBe.n;JlHB KaR B eBRJlH.n;OBO:U, TaR
II B rHnep6oJlllqeCKOH reOMeTplIB. He npIlBOJJ;H COOTBeTCTBYlOmero
;n:OBOJIbHO JJ;JIHHHOrO BbIBo.n;a ac-CaJla pa, MhI OTMeTHM, "liTO B Hel\f nplI-
MeHHeTCH aKCUOMa EB;n:OKca-ApXHM:eJJ;a.
76
«TIpeIJ;IIOCbIJIKa» 6 ac-CaJIapa IIOKaahIBaeT, QTO, OTIIpaBJIHHCb OT
JJ;OKaaaTeJIbCTBa XaHHMa II IIbITaHCb YJIyqmHTb ero; ac-CaJIap CTpe-
MHJICH .n;OKaaaTb «npHHl\HnhI», IIpHHHMaBillHeCH XaHHMoM 6ea JJ;OKaaa-
TeJIbCTBa. OJJ;HaKO, He IIOHHB JJ;ORaaaTeJIbCTBa OCHOBHoro IIpeJJ;JIO-
meHHH 3 XaHHMa, ac-CaJIap aaMeHHJI ero oillI1601IHhIM paccymJJ;eHHeM,
HeHBHO JJ;onYCTHB TO, lITO OH IIbITa.rrCH JJ;OKaaaTb.
OKaaaTeJIbCTBO V nOCTYJIaTa Y ac-CaJIapa, no cy eCTBY, COBIIa-
JJ;aeT C JJ;OKaaaTeJIbCTBOM IIpe,n;JIOmeHHH 8 XaHHMa.
TpaKTaT ac-CaJIapa IIOKaabIBaeT, qTO IIOIIhITKH COBepilleHCTBOBaHHH
;a:OKaaaTeJIbCTBa V IIOCTYJIaTa rrYTeM IJ;OKaaaTeJIbCTBa nOJIOiHeHHH,
IIpI1HHMaeMhIX reOMeTpaMH 6ea AOKaaaTeJIbCTB, IIpOJJ;OJImaJII1Cb" H,
XOTH caMO JJ;ORaaaTeJIbCTBO V IIOCTYJIaTa y ac-CaJIapa H OKaaaJIOCb
HeYJJ;aqHhIM, OH Bce me HCIIpaBHJI OJJ;HH ,n;e.peKT JJ;OKaaaTeJIbCTBa XaHH-
Ma - IIOKaaaJI, qTO npHH HII 3, «aaHMCTBOBaHHhIH Y <p 1I JIOCO <Pa»,
I{OTOPhIH XaHHM CqHTaJI aKcHoMoH, Ha caMOM IJ;eJIe MomeT 6hITb
JJ;OKaaaH Ha OCHOBaHHH JJ;pyrHx IIOJIOmeHHB reOMeTpHH EBKJIH.n;a.
MhI l\fHOrO paa l\HTHpOBaJIH co-
lIHHeHHe no TeOp1I11 IIapaJIJIeJIbHbIX
HacHp aJJ;- HHa aT-TyclI. HaclIp
aJJ;- HH A6y iRa(.pap Myxal\IMaJJ;
H6H MyxaMMaJJ; aT-TyclI, ypomeHeD;
Tyca B XopacaHe HJIH CaBe B
mH6aJIe II YlIeHHK RaMaJI aJJ;- IIHa
n6H IOHHca, 6hIJI OJJ;HHM lIa Kpyrr-
He:nmHX yqeHhIX-SHD;HKJIOlleJJ;HCTOB
rOCYAapCTB IICJIaMa II HeKOTopoe
BpeMlI aaHHMaJICH nOJIHTHqeCKOH
JJ;eHTeJIbHOCTblO. AT- TYCH - a3TOp
COqHHeHHH 110 MaTeMaTHKe II aCTpo-
HOMIIH, II B qaCTHOCTH I{OMM:eHTa-
pHeB RO MHorHM COlIHHeHHHM EBKJIHJJ;a, ApxHMeJJ;a, AnOJIJIOHHH, TITO-
."IeMeH H JJ;pyrHx rpeqeCKHX RJIaCCHKOB, a TaKme yrroMHHaBillerOCH
HaMH «TpaRTaTa 0 1I0JIHOM qeTbIpeXCTopOHHHRe» [27]. OH ROMMeH-
TlIpOBaJI TaKme <PHJIOco<pcRHe H JIOrHlIeCKHe TPYJJ;hI M6H CIIHb!
II caM HaIIHCaJI HeCKOJIbRO .pIIJIOCO.pCRHX TPYJJ;OB, Ha ROTOpbIX
HaH60JIbillYlO IIonYJIHpHOCTb nplI06peJIa «HaclIpoBa 3TIIKa» (A JIaI:\-1I
Ha HpH). AT-TyclI IIpHHaJJ;JIemaT TaRme COlIHHeHHH no .pHallKe H
l\IHHepaJIOrHH H, HaKOHen;, lIeTBepOCTHillHlI, qaCTb ROTOpbIX npUIIHCbI-
BaeTClI XaHHMY. B Te-qeHHe MHorHX JIeT aT-TycH pa60TaJI aCTpOHO-
ltfOM H aCTpOJIOrOM B «rocy)];apCTBe accaCHHOB), Coa.D;aHHOM IDIIIITaMH-
nCMaHJIHTaMH B ropax CeBepHoro MpaHa. 3JJ;ecb OH HanHCaJI pH.D;
CBOHX TPYJJ;OB, B lIaCTHOCTH TpaKTaT 0 napaJIJIeJIbHhIX If 06pa60TKY
«HalIaJI» EBRJIH.D;a, ROTopbIe MbI 6y,n;el\1 paccMaTpllBaTb HHme. Acca-
CIIHbI, HaaBaHHe KOTOpbIX BO MHorHX eBpoIIeHCKHx HahIKax CTaJIO
CHHOHHMOM y6HHU;, paCChIJIaJIH CBOHX areHTOB-«<peJJ;aH» ,n;JIH YHHqTOme-
HUH BparOB HpaHCRHX lliHHTOB - 6arJJ;a,n;CKHX XaJIH<pOB, eBponeiI-
CRHX Rpe OHocu;eB II MOHrOJIbCKHX aaBOeBaTeJIeii. EBponeHU;hI npo-
aBaJIH STHX TeppopHCTOB «accaCHHaMH» - OT nepCH,n;CKoro CJIOBa
«xamHmHH» - «OIIbHHHIOID:HeCH ramHmeM». RorJJ;a B 1256 r. MOHrOJIhI
13.
T eopuR
napaJlJleJl'b1f/btX
,
.llUUUU
Hacup ao-Auua
am- T ycu
77
OBJIa eJIH rocy apcTBOM aCCaCHHOB, aT-TYCH nepemeJI Ha c.JIym6y
R MOHrOJIbCKOMY xaHY XYJIary H BCKope CMeHHJI npH ero JJ;Bope yme
BCTpeqaBllierOCJI HaM XycaM a - HHa ac-CaJIapa Ha nOCTY rJIaBHOro
aCTpOJIOra. B 1258 r. aT-TycH yqaCTBOBaJI B noxoJJ;e XYJIary-xaH8
Ha EarJJ:aJJ:, nOJIOil\HBmeM KOHen; 6arJJ:a CKOMY XaJIHq.aTY, Ii BeJI
neperOBopbl C nOCJIeJJ;HHM XaJIII.pOM 0 cJJ;alle. B 1259 r. aT-TycH oc-
HOBaJI B CTOJIH e XYJIary Mapare (IOmHhIR A3ep6aHJJ;iRaH) o6cepBa-
TOpHIO H MO HYIO HayqHYIO mKOJIY, rJJ;e 6bIJIH c06paHLI JIY'tImlIe yqe-
Hble, nonaBmHe Ha TeppHTopHIO, 3aBoeBaHHYlO MOHrOJIaMII. OH 6bIJI
TaRme COBeTHHROM XYJIary Hero npeeMHHKOB «IIJILXaHOB» no <pH-
HaHCOBLIM BonpocaM H pa3pa60TaJI npoeKT HaJIOrOBOH peq.opMbl, ocy-
eCTBJ1eHHOii OJJ;HHM H3 «HJIbXaHOB».
TpaRTaT aT-TYCII 0 napaJIJIeJILHLIX HOCHT HaaBaHHe «TpaRTaT, IIC-
n;eJIJlIO HH COMHeHlIe no nOBOJJ;Y napaJIJIeJIbHbIX JIHHHH» (ap-PncaJIa
am-maq.HHHa (aH maKK q.H-JI- YrY T aJI-MYTaB3.aHHiia). Apa6cRIIH
TeRCT TpaRTaTa 6blJI 113JJ;aH B C60pHHRe [30] BMeCTe C o6pa6oTKaMH
aT- TYCH TaK Ha3blBaeMbIX «npOMemYTOqHLIX KHHr» (KYTy6 aJI-MYTa-
BacHTa) - COqHHeH:nu rpeqeCRIIX KJIacc:nKOB, H3yqaBIDIIxcJI MemJJ;Y
«HaqaJIaMH» EBKJIHJJ;a 11 «AJIMareCTOM» IIToJIeMeJl, a TaRiRe BMeCTe
C 06pa60TRaMH aT-Tyc:n MaTeMaTHqeCKHX TpaRTaTOB 6paTLeB BaHY
Myca (IX B.) II Ca6IITa :n6H ROPPbI. ROMMeHTHpoBaHHbIH pyecKHH ne-
peBOJJ; TpaKTaTa [28] 6bIJI ony6JIHKOBaH aBTopaMH. B TpaKTaTe BHa-
qaJIe 1I3JIaralOTCJI II KpnTHRYIOTCH TeoplIH napaJIJIeJIbHbIX JIIIHlIii M6H
8JI-XaiicaMa, aJI- mayxapll II XaHHMa. AT-TYCH HanHCaJI eBoe co-
1I1IHeH:ne o 1258 r.: 9TO CJIe;lJ;yeT 1I3 Toro, 'tITO B 9TOM ro y YMep MaTe-
MaTHK AJIaM a - HH aJI-XaHaq.lI, C KOTOp:bIM OH BeJI nepen:nCKY 06
aTOM TpYJJ;e (06 aJI-XaHacp:n II ynoMHHYTOH nepenlICRe HaM npHJJ;eTCH
e e rOBopIITb). RorJJ;a aT- TyclI pa60TaJI B rOCYJJ;apCTBe aecaCHHOB
HaA TeopHeH napaJIJIeJIbHbIX, eMY 6bIJIH He ocTynHbl B nOAJIHHHHHe MHO-
rHe pa60ThI no 9TOMY Bonpocy. B qaCTHOCTH, B 9TO BpeMJI aT-TYCH He
pacnOJIaraJI «ROMMeHTapHJlMH K BBeJJ;eHHJlM RHHrlI EBKJIIIJJ;a "Haqa-
JIa"» M6H aJI-XaiiC8Ma, r;lJ;e npOBO;lJ;HJIOCb ero OKaaaTeJIbCTBO V nOCTY-
JIaTa, H 6blJI 3HaKOM TOJIbKO C ero «RHHrOH 0 paapemeHH:n COMHeHIIH
B KHHre EBRJIHJJ;a "HaqaJIa"». IIoaTo y lIS CJIOB H6H aJI-XaHCaMa 0
paBHOCBJIbHOCTH V IIOCTYJIaTa 0 HeB03MOiRHOCTH cy ecTBoBaHIIH BYX
nepeceKalO HXCJI npJlMhIX, napaJIJIeJILHbIX TpeTbeii npHMO:H, :n 0 TOM,
qTO nOCJIeJJ;Hee YTBepil\JJ;eHHe HarJIJlJJ;Hee V nOCTYJIaTa, aT-TYCB 3a-
RJIlOqHJI, 6YJI;TO M6H aJI-XaHCaM He IIbITaJICH ;lJ;OKa3b1BaTb V nOCTYJIaT,
a TOJIbKO XOTeJI 3aMeHHTb ero 60JIee HarJIJI;lJ;HbIM YTBepmJJ;eHHeM. AT-
TYCH I(IITHpyeT lIe BCIO qaCTb reOMeTpHqeCKOrO TpaRTaTa XaiiJlMa, OT-
HOCJlm;YIOCJI K napaJIJIeJIbHbIM JIHH:nJlM, a TOJILKO BoceMb npe;lJ;JIome-
HHH, B KOTOpbIX npHBOJJ:HTCH JJ:ORa3aTeJIbCTBO V nOCTYJIaTa, II He 3a-
IeqaeT, qTO JJ;ORa3aTeJIbCTBO XaHHMa OCHOBaHO Ha npIlHI HIIe 4,
«3aHMCTBOBaHHOM Y <pHJIOcocpa), eOCTOHm;eM lI3 JI;BYX YTBepmJJ;eHIf:U,
paBHOCIIJIbHhIX V nOCTYJIaTY; MomeT 6bITb, 9TO 06'LHCHJleTCH TeM,
qro aT- TyclI He pacnOJIaraJI :n nOJIHhlM TeRCTOM TpaRTaTa XaiiHMa. AT-
TYCH CqlITaeT, qTO XaHHM COBepmaeT B 9TOM JI;ORa3aTeJIbCTBe JIor:n-
qeCRYIO OllIH6RY, nOJIbSYJlCb YTBepmJJ;eHHeM, 'tITO paeCTOHHHe femJJ:Y
;lJ;BYMJI nepeceKalO BMHCH npJlMIiIMH HeOrpaHHqeHHO YBeJIII'tIHBaeT-
78
CH, T. e. npIIH HnOM 3 XaHHMa, ROTOphIH AORaShIBaJI ac-CaJIap. MhI
YiRe rOBOpIlJIII, lITO 8TO YTBepm,n;eHlIe He saBHCHT OT V nOCTYJIaTa, II,
CJIe,n;OBaTeJIbHO, ynpeR aT-TYCII B a,n;pec XaHHMa COBeprneHHO HeOCHO-
BaTeJIeH. B HalIaJIe CBoero TpaRTaTa aT-TYCH nHCaJI:
«MHorlIe BhI,n;alO HeCH JIIO,n;H 8TOH HaYRH He CMOTpeJIII rJIaSaMII
cnpaBe,n;JIHBOCTII B CBOHX nonhITRax AORasaTb 8TY [npennOCLIJIKY].
B TOM, lITO H BH,n;eJI, JI He HameJI ,n;OCTaTOlIHOrO ,n;OKaSaTeJIbCTBa. Ha-
npOTHB, Ram,n;LIH lIS 8THX IICCJIe,n;OBaTeJIeH, npH,n;epmHBaBmHXCH B
CBoeM ,n;OKaaaTeJIbCTBe paSJIHlIHLIX nYTeii, OrpaHHlIHBaJICH nOJIOBH-
HOH [,n;OKaSaTeJIbCTBa]. Cpe,n;1I HHX 6LIJID Te, RTO aaMeHHJI ee ,n;pyrHM
nOCTYJIaTOM, 60JIee 01IeBH,n;HLIM, TaKOB 6LIJI A6y AJIH 1I6H aJI-XaHCaM,
60JIblliOH SHaTOK HaYKII MaTeMaTHKH. Cpe,n;H HIiX 6LIJIH Te, RTO ,n;aBaJI
en JIOmHOe ,n;OKaSaTeJIbCTBO, IICXO,n;H HS npe,n;nOCbIJIKH, He npe,n;me-
CTBYIO eH eH no 01IeBII,n;HOCTH H HCHOCTII, TaROB 6:bIJI MHpOBOH YlIeHhIH
A6Y-JI-<DaTx OMap aJI-XaWIMH. Cpe]:(H HHX 6LIJIH II Te, KTO IICKa}KaJI
npe,n;noCLIJIKY, HX paccym,n;eHHH He y,n;OBJIeTBOpHIOT npOHH aTeJIbHO-
ro II paSYl\fHOrO lIeJIOBeRa, TaKOB 6LIJI ,n;OCTOCJIaBHLIH aJI- A66ac H6H
CalI,n; aJI- mayxaplI. H He HameJI CJIOB 0 Tex, RTO B 8TOM Bonpoce ,n;0-
meJI ,n;o KOH a. TI08TOMY ,n;JIH HC eJIeHHJI OT 8Toro He,n;yra H ClIeJI Hym-
HbIM npenCTaBHTb nOJIHOCTbIO Bce TO, qTO JI HameJI B KHHrax, II YKa-
saTb Ha Te He,n;OCTaTKII B HeJIenOCTH, KOTopLIe JI 06HapY}KHJI B HIIX.
H ,n;eJIalO 8TO nYTeBo,n;HTeJIeM H YKasaTeJIeM sa6JIym,n;eHIIH ,n;JIJI yqa-
HXCJl II npe,n;CTaBJIJlIO 8TO BO Bceii nOJIHOTe, ,n;OCTOHHOH paaYl\fHLIX.
qTO KacaeTCH M6H aJI-XaHCaMa, !l;a nOMHJIyeT ero AJIJIax, TO OH
B CBoeH KHHre, OSarJIaBJIeHHOH "PaapemeHHe COMHeHHH B KHHre EB"
KJIH,n;a", nOJIbaOBaJICJI BMeCTO 8TOH npe,n;nOCLIJIKH ,n;PyroH npe,n;nOCLIJI-
KOH, nOJIaraJl, qTO OHa 60JIee HarJIJI,n;Ha ,n;JIJI qYBCTBa II 60JIee npOHH-
KaeT B ,n;ymy, ,n;OKaaaTeJIbCTBO me npaBIIJIbHOCTH 8Toro nOCTYJIaTa
BMeCTe C po,n;CTBeHHLIMH eMY [YTBepiI(,n;eHIIJlMH] nOMe eHo B npyroH
ero KHHre, OaarJIaBJIeHHOH "KOMMeHTapHH K BBe,n;eHHJlM". MIi[ He
HamJIII HII o,n;Horo 8KSeMnJIJlpa ee H SHaKOMIiI C Heii TOJIbKO no 8TOH
KHHre, T. e. "PaapemeHHIO COMHeHHH". Cy,n;JI no TOMY, qTO HSJIOmeHO
B 8Toii KHlIre, B ero CJIOBaX npOJlBHJIaCb nYTaHII a BCJIenCTBHe ero
HeHCKY.CHOCTII B HCCJIe,n;yeMoii Hl\f s,n;eCb HaYKe - reOMeTpHH H He,n;o-
CTaTOqHOH nOCJIe,n;OBaTeJIbHOCTII npH HayqeHlI1I OCHOB HaYKH. B CBO-
HX CJIOBaX OH He y6epercJI OT KpyroBoro ,n;OKaSaTeJIbCTBa: npH onpe-
,n;eJIeHHH na paJIJIeJIbHOCTH JIIIHlIii OH ,n;onYCKaeT ,n;BHmeHHe nepneB,n;H-
KYJIJlpa, BOCCTaBJIeHHoro R npJlMOH JIHHHH, B,n;OJIb JIHHHH, K KOTOpOH
BOCCTaBJIeH nepneH,n;IIKYJIJlp, npHqeM ,n;BHmeHneM ero ,n;pyroro ROH a
06pasyeTcJI JIHHIIJI, napaJIJIeJIbHaJl nepBoH. aJIee OH nepexo,n;HT K
,n;OKaSaTeJIbCTBY cnopHoi npe,n;nOCLIJIKH, r,n;e OH SaMeHJleT TO, qTO
He06xonHMO, Ha TO, lITO Tpe6yeTcH, T. e. Ha 8TO YTBepm,n;eHHe, XOTJI
OH H rOBopHT, qTO ero ,n;OKaSaTeJIbCTBO npaBHJIbHoe. Ero ,n;OKaaaTeJIb-
CTBO OCHOBaHO Ha HCnOJIbSOBaHHH ,n;BHmeHHJI, qTO IIPH eCTeCTBeHHOM
no,n;pas,n;eJIeHHII 06'beRTOB MaTeMaTHKH npHBo,n;HT K nYTaHH e; 8TO
nOKaabIBaeT OTCYTCTBHe ero HCKYCHOCTH B 8THX Be ax. B ero ROMMeH-
TapHJlX 11 BII,n;eJI, qTO B ,n;OKaaaTeJIbCTBe HeROTopLIX nOCTYJIaTOB OB ,n;0-
BOJIbCTByeTcH BOCCTaBJIeHHeM nepneH,n;HRYJIJlpOB K Ram,n;OH JIHHHH,
.n;pyrHe me HaYlIHlile sa,n;alIH OH 6ea He06xo,n;HMOCTB CBO,n;HT K npHH-
.'19
HnaM; BCe 8TO YRa3:bIBaeT Ha OTCYTCTBlIe ero IICRYCHOCTH B HaYRe
,n;ORa3aTeJI:bCTBa II OCHOBHBIX HaYRax» [28, c. 485-487].
KaR BH,n;HO, He pacnOJIaraH TeRCTOM «KOMMeHTaplIeB R BBe,n;eHlI.R I»
M6H aJI-XaHCaMa, aT-TYCD Bce me IIMeJI HeROTOp:bIe CBe,n;eHIIH 0 ero
,n;OR83aTeJI:bCTBe V nOCTYJIaTa, H3JIOmeHHOM B 8TOl\f TpaRTaTe. fOBOpH
o «eCTeCTBeHHO 1 no,n;pa3,n;eJIeBIIH 06'beRTOB M8TeMaTIIRH», aT- Tycl1
RpHTHROBaJI npHMeHeHlIe M6H:aJI-XaHCaMOM ,n;BlImeHIIH R reOMeTplIII
C Tex me apHCTOTeJIeBCRHX n03H HH, lITO II XaHHM.
« q TO RacaeTCH npe!J;nOC:bIJIRII, - npo,n;OJImaeT ilT- TYCD 1I3JIOa\e-
HHe lI,n;eH M6H aJI-XaiiCaMa, - ROTopaH, no ero MHeHHIO, 60JIee Ha-
rJIH,n;Ha ,n;JIJI lIYBcTBa II 60JIee npoHHRaeT B ,n;ymy, qeM 8TOT nOCTYJIaT,
TO OH nOJIb30BaJICJI elO B Tex MeCTax, r,n;e HYiKHO nOJIb30BaTbCH 8THM
nOCTYJIaTOM. OHa COCTOHT B TOM, qTO, eCJIII ,n;Be npJlM:bIe JIIIHHH nepe-
ceRaIOTCJI, OHH He l\IorYT 6:bITb napaJIJIeJIbH:bI O,n;HOH npJlMOH JIIIHIIH»
[28, c. 487].
TIepe,n; 1I3JIOmeHIIeM ,n;ORa3aTeJIbCTBa XaHJlMa aT- TYCD nlIcaJI:
«qTO RacaeTCH aJI-XallJlMH, ,n;a nOMHJIyeT ero AJIJIax, TO OH B nep-
BOll RHHre TpaRTaTa, 03arJIaBJIeHHOrO "ROMMeHTaplIlI R TPy,n;HOCTJlM
BO BBe,n;eHIIJlX RHHrlI EBRJIH,n;a", nplIBeJI ,n;ORa3aTeJIbCTBO Tpe6yel\IOrO
nOCTYJIaTa B BOCbMII npe,n;JIOiKeHIIJlX II YRa3aJI, qTO OH nOMeCTHJI 8TO
B RHHry "HaqaJI" nOCJIe npe,n;JIOmeHIIJI 28. M:bI ,n;ORa}KeM 8TO ero CJIO-
BaMH, a 3aTeM JI YRaiKY MeCTa 1I3'bJlHOB, IIMelO HXCJI y HCCJIe,n;Ol\aTeJIJI,
eCJID 8Toro 3aXOqeT BCeB:bIIDHIIH AJIJIax» [28, c. 489].
aJIee npHBo,n;HTCJI BoceMb npe,n;JIOiKeHHH XallJlMa, nOCJIe qero
aT-TycH nHmeT:
«M:bI nplIBeJIH CJIOBa aJI-Xa:HJlMII. H rOBOplO: He CRp:bITO OT npOHH-
n;aTeJIbHOrO H B,n;YM:bIBalO erocJI B 8TII CJIOBa, qTO Bce, qTO OH rOBopUT
,n;o ROHn;a npe,n;JIOiKeHHJI 5, npaBIIJIbHO H HeCOMHeHHO, 3a IICRJIlOqeHlIeM
ero CJIOB B npe,n;JIOiKeHHH 3: "TIPO,n;OJIiKIIM AC H BD, Tor,n;a OHII
nepeceRYT H KF B H II F": 8TO He ,n;ORa3aHO B TOM, qTO OH H3JIOfHIIJI.
HpoMe Toro, B ROHn;e TpeTbero npe,D;JIOmeHHJI OB npHBo,n;IIT ,D;JIH 60JIb-
meft JlCHOCTII CJIOBa, ,D;oRa3:bIBalO He B03MomHOCTb npoBeJ]:eHHH 1I3
TOqRH O,n;HOH 1I3 ,n;BYX ,n;aHHhIx JIHBIIH R ,n;pyro:H TaRoH JIIIHHII, qTo6:bI
BHYTpeHHlIe yrJIIiI 6:bIJIII paBH:bI, cnoco6oM 4>HJIOCO 8, a He reOMeTpa.
no ero CJIOBaM, eCJIII MomHO rrpOBeCTII H3 E R CD 6eCROHeqHO MHoro
nIIHH:H C 6eCROHeqHO MHorHMH yrJIa IH C o6elIx CTOpOH 8THX ]IBYX
JIHHHll, TO Heo6xo,n;HMO B031\fOmHO H p8BeHCTBO. Ho qTO 1I3BeCTHO e:l\IY
o cy ecTBoBaHHH Henpep:bIBH:bIX yrJIOB MemJJ;y HepaBH:bIMII He C <pH-
JIoco.pcRoii, 8 C reOMeTpHqeCROn TOqRIl 3peHIIJI? ECJIII ,n;onYCTHTb, "tJTO
H3BeCTHO cy ecTBoBaHHe HeROTOp:bIX yrJIOB, MeHbillHX, H HeROTopbIX
yrJIOB, 60JIbIIIHX yrJIa nplI TOqRe E, C reOMeTpHqeCROH TOqRD 3peHIIJI,
TO OTRYJJ;a H3BeCTHO, qTO MeiKJ]:y 8THl\fH J]:BYMH yrJIaMH, MeHbillnl\l II
66JIbillIlM:, cy ecTByeT paBeHCTBO? TIpe,n;nOJIOmeHHe 0 TaROM yrJIe
JJ;OJIiRHO 61i1Tb J]:ORa3aHO. Ha ,n;pyrlIx qepTemax TaRoe YTBepmJJ;eHHe
nORa3:bIBaeTCJI nocpeJJ;CTBOM nOR83aTeJIbCTB8 Heo6xoAIIMOCTH CYIT(e-
CTBOBaHHH HeROToporo yrJIa. TaR npoHcxo,n;HT, eCJIII npoBoJ]:IITb He-
paBH:bIe JIHHHH 113 OJ]:HOn TOqRU ORPY}KHOCTII Rpyra R JJ;pyrHM TO"tJRaM
Ton me oRpymHocTH, Tor,n;a Rpyr J]:eJIHTCJI R8mJ]:on H3 8TDX JIIlHDH Ha
ABa CerMerJIIiI. YrJIbI, OrpaHHqeHH:bIe AyroH oRpymHocTH II 8TIIMU npH
80
MbIMH JlIIHII.HMH, MeHbille np.HMOrO, KOr,n;a CerMeHTbI MeHbIIIe nOJlY-
Rpyra, H 60Jlbme np.HMOrO, ROr,n;a CerMeHTbI 60Jlbme nOJlYRpyra,
a Mem,n;y 8THMH 60JlbllIHMH H MeHbillHMH IIMeIOTC.H paBHLIe, Kor,n;a [yroJl]
np.HMoii, KaK ,n;ORa3aHO B TpIIn aTOM npenJlOmeHHII TpeTben RHHrn
"Ha'tlaJI". TO'tlHO TaK me II a,n;eCb Mem,n;y [60JlbllIIIMH II] M:eHbmlI IH HMe-
IOTC.H paBH:bIe [yrJlbI]. llOJlY1J:eHHOe ,n;ORa3aTeJlbCTBO He06xo,n;H IOCTIf
[cy ecTBOBaHH.H] OTHOCHTC.H K HeKOTOp:bIM np.HMOJIHHeHHbIM yr Jla?tl,.
HO He ro,n;IITC.H ,n;JI.H HeKOTopLIX yrJlOB, orpaHH'tJeHHIiIX np.HMLIMH II
RpyrJlIiIMH JlHHH.HMH...
B npe,n;JlOmeHHII 6 OH nplIMeHHJI npe,n;nOCbIJlKY 0 TOM, lITO, eCJlH ,IJ;B&
JlIIHHII nepeceKaIOTC.H, paCCTO.HHHe MeiR,n;y nepeCeRalOll1;HMHC.H JIHHII.H-
MH YBeJlII'tIHBaeTC.H ,n;o 6eCROHe'tJHOCTII, II, TaK KaK paCCTO.HHlIe Mem,n;y
,n;BYM.H 8RBH,n;HCTaHTHIiIMH [JIIIHII.HMH] oiHo H TO me, paCCTO.HHHe Mem-
JJ;y nepeceRalO HMHC.H JIIIHII.HMII CTaHeT 60JIbme 8Toro o,n;lIoro paCCTO.H-
HH.H, II JIHHH.H, nepeceKalO a.H O,n;HY 1I3 ,n;BYX 8KBII,n;IICTaHTHbIX JIIIHIIH,
nepeceKaeT 06e- 8TH JIHHHH. OT pa3YMHoro He CKp:bITO, 'tITO 8TO Ta ca-
Ma.H npe,n;nOCLIJIRa, ROTOPYIO M6H aJI-XaHCaM nOCTaBIIJI BMeCTO COM-
HHTeJIbHOrO nOCTYJIaTa; Tenepb MLI SHa-eM ee CY HOCTb. ECJIH 6bI OR
npHBeJI "IITO-HII6y,n;b no,n;06Hoe B Ha'tJaJIe ,n;ORaaaTeJIbCTBa nOCTYJIaTa,
,n;eJIO QbIJIO 6bI ,n;JI.H Hero ropas,n;o JIer'tle H OH He HY}K,n;aJIC.H 6bI B IHO-
rOCJIOBHH» [28, c. 496-499J.
AT- TYCH nOJIeMH3HpyeT C XaH.H IOM TaRme no nOBO,n;y onpe,n;eJle-
BH.H paCCTO.HHH.H Mem,n;y JJ;BYM.H np.HMbIMH: aT- TYCH npennO'tlHTaeT-
IISMep.HTb 8TO paCCTO.HHHe nepneH,n;IIKYJI.HpO?tI, ony eHHIiIM lIS TO'tlRH
O,n;HOH H3 np.HMbIX Ha ,n;pyrylO, II, RaK MbI YBH,n;IIM, BOCnOJIbayeTC.H-
HMeHHO 8THM onpe,n;eJIeHHeM B CBoeM ,n;ORaSaTeJIbCTBe V nOCTYJIaTa._
MaJIOmHB ,n;aJIee npHBe,n;eHHoe HaMOBIilIIIe ,n;ORaaaTeJIbCTBO aJI-
mayxaplI, aT- TYCH no,n;Bepr ero cnpaBe,n;JIHBOH RpIITIIKe:
«fl rOBOplO: OH He ,n;OKaSaJI 6:bI CBOHX npe,n;JIOmeHIIH, eCJIH 6bI He-
npHMell.HJI paCCMaTpHBaeMYIO npe,n;nOCbIJIKY. TIpH ,n;OKa3aTeJIbCTBe.
nepBoro YTBepm,n;eHHH B rrepBOM lIa 8THX npe,n;JIOmeHIIH 6bIJlO nOJIY'tIe-
HO, 'tITO, eCJlH JIIIHH.H nanaeT Ha )J;Be JIHHHII H npH 8TOM 06paaYIOTC.H'
paBHbIe HaKpeCTJIema lIe [yrJILI], 8TH ,n;Be JIIIHHH napaJIJleJIbHLI. Ho
113 3Toro YTBepm,n;eHII.H He CJIeAyeT, 'tITO H Bce OCTaJIbHbIe JIIIHIIII, na-
,n;alO He H3 HHX, 06JIa,n;aIOT, RaK nepBaH JIIIHII.H, paBeHCTBOM HaKpeCT-
JIema HX yrJIOB. Bo BTOpOl\f YTBepm,n;eHHH, npIlCoe,n;HHeHHOM K nep-
BOMY YTBepm,n;eHHIO, ,n;OKaabIBaeTCH, lITO eCJIH npe,n;nOJIOmHTb 'tIeTbIpe
TO'tlKH Ha ,n;BYX napaJIJIeJIbHLIX JIIIHH.HX, Ha KOTopbIe na,n;aeT e e o,n;Ha
npHMaH, npH'tIeM paCCTO.HHHH OT ,n;BYX To'tJeK ,n;o MeCTa na,n;eHH.H Ha
GABOn JIIIHIIH paBHbI paCCTOHBIIHM OT ABYX TOlIeK AO MeCTa n3AeHIIH Da
,n;Pyroii JIHHHH, TO paCCTOHHHe Mem,n;y )J;BYM.H TO'tlKaMH paBHO paCCTO.H-
HHIO Me}KJJ;y )J;pyrHMH )J;BYl\I.H TO'tlKaMH, a paCCTO.HHHe Kam,n;OH TO'tlKH
OT MeCTa na,n;eHHH Ha ee JIHHHII paBHO paCCTOHHHIO COOTBeTCTBeHHOH
TO'tlRH OT ,n;pyroro MeCTa na,n;eHH.H. HanpH lep, eCJIH Ha JIHHHH AB II
CD nanaeT JlHHH.H EG YKa3aHHbIM 06paaoM H TbI npennOJIOmHIDb 'tJeTbI-
pe TO'tlKH H, F, I, K no o6e CTOpOHbI OT EG (pHC. 54), npH'tIeM paCCTO.H-
Hpe OT H ]];0 E paBHO paCCTOHHHIO OT K ]];0 G, a paCCTOHHHe OT F ]:(0
E paBHO paCCTOHHHIO OT I ]];0 G, TO paCCTOHHIIe OT H,IJ;O G paBHO pac-
CTO.HHHID OT K ]];0 E, paCCTOHHHe OT F ]];0 G paBHO paCCTO.HHHIO OT I
81
..
A
A
B
F
E
H A
IJ C
.D K G I C C E II
Pue 54 Pue.55 Puc.56
C F (; A C H E A
JJ
H E B
.D
6'
F r; 0
a
Puc.57
;n;0 E, a paCCTOJlHlle OT H ]]:0 I paBHO paCCTOJlHll1O OT F ;D;O K. O;n;H3KO
He He06xo]]:HMO, 1J:T06hI paCCTOJlHIIJI OT ]]:aHHhIX T01J:eK Ha O]]:HOH lIa
JIIIHHH ]]:0 COOTBeTCTBYIOIIJ:IIX [DM TOqeK Ha ]]:PyroH JIHHIIH] 6LIJIH 6LI
paBHLI, HanpllMep, lIT06LI paCCTOJlHHe OT H ;n;o I 6bIJIO 6LI paDHO pac-
CTOJlHHIO OT A ]]:0 cooTBeTcTBYIOIIJ:eH eH TOqKH H paCCTOHHHIO OT O]]:HO:H
Ha ]]:BYX JI106L1x COOTBeTCTBeHHLlX T01J:eK ;D;O ]]:PyroH, II Bo06m;e He He-
06XO]]:HMO, lIT06bI paCCTOJlHIIJI Mem]]:y TOqKaMH 6bIJIH 6L1 paBHbI B Co-
OTBeTCTBHH C YKaa8HHLlM CBOHCTBOM, TaK KaK ]]:OKaaaTeJIbCTBO He
co]]:epmlIT 06ru;eii HCTHHLI ;D;JIJI OCTaJIbHLlX TOlIeK, a lIa paBeHCTB p3C-
CTOJlHHH YKaaaHHLlX TOqeK He CJIe]]:yeT C He06xo]]:HMOCTblO paBeHCTBO
paCCTOJlHIIH [Bcex] TOqeR, 06JIa;D;alOIIJ:HX YR8saHHbIM CBOHCTBOM, XOTJI
TaKoe paBeHCTBO II BOSMOiKHO. TOlIHO TaK iRe HS paBeHcTBa ]]:BYX xop]]:
B Kpyre, Haxo]]:Jlm;IIXCJI Ha paBHLlx paCCTOJlHHJlX OT n;eHTpa, He CJIe-
;n;yeT C He06xo;D;IIMOCTblO paBeHCTBO ABYX J(pyrHx XOP;D; B aTOM Kpyre.
AaJIee, BO BTOpOM npe;D;JIOiKeHHH Tpe6yeTcH ;n;OKaaaTb paBeHCTBO
JIHHHH ED II BC, OJ(Ha 113 KOTOpLIX - OCHOBaHHe OJ(Horo lIa [Tpe-
yrOJIbHHKOB], a ;D;pyraJl - YABoeHHaJl CTopOHa [BToporo] (pHC. 55).
3TO paBeHCTBO OCHOBaHO Ha ynOMJlHYTOM J(OKaSaTeJIbCTBe npe;D;@;D;Y-
m;ero npe;D;JIOmeHHH, H OHO He BLlnOJIHJleTCJI, eCJIH TOlIRH E, B, C, D
He 06na]]:alOT CBoicTBOM, YKaaaHHLlM B ;D;OKaaaTeJIbCTBe, TaK KaK JIII-
HHJI, na]]:alOm;aJl Ha JIHHHD AB H FC TaKHM 06paaOM, lITO H3KpeCTJIe-
}Kam;He yrJILI paBHLI, MOiKeT 6L1Tb II JIHHHeH ED, II Tor]]:3 E H D - )J;Be
COOTBeTCTBeHHLIe TOlIKH, a MomeT 6LITb JIHHlIeii BC, H Tor]]:3 C II B -
;D;Be COOTBeTCTBeHHLIe TOqRH. PaBeHcTBo paccToHHIIH Mem;D;y 8TH?\IH
TOQKaMH He CJIe;n;yeT Ha aToro ]]:OKa3aTeJIbCTBa C He06xo)J;HMOCTblO,
TaR KaK paBeHCTBO paCCTOJlHHH OT KaiK]]:oi TOqKH ]]:0 COOTBeTCTBYIOm;eii
ei He ]]:OKaaaHO, paccym]]:eHHe He JlBJIJleTCH 06m;HM, He p3cnpOCTpa-
82
HHeTCH Ha Bce TO'tlKH H He cnpaBe;D;JIHBO ;D;JIH JII06LIX ;D;BYX TO'tleK. 31:Y
OIIIH6KY paCCMaTpHBaJI aBTOp "JIorHRH" B CBoeH RHHre, 03arJIaBJIeH-
HOH "CO<pHCTHKa", B rJIaBe 0 npHHHTIIH BO BHHMaHHe 06CTOJlTeJIbCTB,
r,n;e H3JIaralOTCH npH'tIHHLI OillH60K H3-3a HenpHJlTHH BO BHIIl\laHlle
YCJIOBHH» [28, c. 508-5101. ABTOp «JIorHKH» - 8TO ApIICTOTeJIb.
«JIorHKOH» (MaHTHK) apa6bI Ha3bIBaJIH c06paHHe JIOrH'tIeCKHX TpaK-
TaTOB ApHCTOTeJIH, 06LI'tIHO Ha3bIBaeMoe HaMH «OpraHOH», «C04>IIC-
THRa» - O)J;HO 1I3 8THX JIorlIlIeCKIIX CO'tlHHeHHH «0 COq.HcTII'tIeCRHX on-
pOBepiReHIIHX» [2, T. 2, c. 533-593]. «TI09TOMY,- npO;D;OJImaeT
aT-TycH,- BTopoe YTBepiR,n;eHHe [XaHJlMa] HeBepHO. HeBepHo II
YTBepmneHlle npenJIomeHHH, II TO, 'qTO CJIe,n;yeT 3a HHM, TaR RaR Bce 9TO
onnpaeTCH Ha Hero. qTO me RacaeTCH npe,n;nOCLIJIKH, Ha ROTOpOH
OCHOBaHO npe,n;JIOmeHHe 5, B KOTOpOH YTBepm,n;aeTcH He06xO;D;IIMOCTb
Toro, 'tITO MeHblliaH H3 ,n;BYX COOTHOCH IIXCJl BeJIII'qHH o,n;Horo po](a
npeB30H,n;eT nOJIOBHHY 60JIbllieH, TO 8TO - Ta npeAnOCbIJIRa, KOTOpylO
OIl BRJIIO'tIIIJI BO BBe,n;eHlIe R I KHHre ,n;JIH YBeJIH'tIeHHH HCHOCTH H YCIIJIe-
HHH ,n;OKa3a TeJIbHOCTIU) {28, c. 510]. 3,n;ecb HMeeTCH B BH,n;y aKCHOMa
EBJJ;oKca - ApxHMe,n;a, KOTopaH, RaK MbI YiRe rOBOpHJIH, 6bIJIa c4>op-
MYJIHpOBaHa EBKJIH,n;OM B HallaJIe V KHllrH «Ha'tlaJI» II ROTOpylO XaHHM
BRJIIO'tIaJ1 BO BBeneHHe R I RHHre.
,I.{aJIee aT- TYCH npllBo,n;HT co6cTBeHHoe ,n;OKa3aTeJIbCTBO aRCDOMbI
o napaJIJIeJIbHLIX:
«q TO RacaeTCH nYTeH, KOTOpbIMII 11 HCCJIe;D;OBaJI 8TO nOCJIe H3Ylle-
HUH CJIOB 8THX Y'tIeHLIX, TO MLI 1I3JIOmHM Harny pe'qb B ceMU npe,IJ;JIO-
meHHJlX, )J;Ba H3 ROTOpLIX B3HTbI H3 npe,IJ;JIOmeHHH aJI-XaHJlMII - 8TO
2-e H 4-e 113 aTHX npe,IJ;JIOmeHHH, JlBJIJlIOm;IIXC1I 1-M II 4-M H3 ero npe,IJ;-
JIOmeHHH. TIYCTb HallaJIO KHlIrH "Ha'tlaJIa" - 28 npe,IJ;JIOmeHUH
I KHurH, B KOTOpLIX HeT COMHIITeJIbBOrO nOCTYJIaTa, OCTalOTCH 6e3
H3MeHeHHH, a 3aTeM ,IJ;06aBIIM 8TH npeJ{JIOmeHHH.
TIpeJ{JIOiI\eHHe 1. KpaT'tIaHmaJl H3 JIIIHHH, npOBeJ{eHHhIX 113 TO'tlKH
RO HCHKon JIHHHD, KOH LI KOTOpOH He OrpaHHlleHbI, Ha3LIBaeMaH pac-
CTOHHueM OT aTOH TO'qKH J{O 8TOH JIIIHHH, eCTb nepneHJ{HRYJIJlp, ony-
m;eHHhI:U H3 TOllKB Ha JIIIHHIO.
llpHMep. JIHHIIH AB - nepneH,IJ;HKYJIHp, onym;eHHbIH 113 TO'tlKII A
aa JIHHHIO GD (pHC. 56). .H YTBepmJ{alO, 'qTO OHa - KpaT'tIaHmaH 1I3
nIlHII:U, KOTophIe MomHO npOBeCTH 1I3 TO'qRH K J1HHIIH.
OKa3aTeJIbCTBO. llpOBeJ{eM JIIIHHIO AE 03 TO'tlKH K JIHHHU. TIOJIY-
'tI1Il\f TpeyrOJIbHHK AEB. 'YroJI B B aTOM TpeyrOJIbHHKe npJlMOH, no-
aTOMY yrOJI E MeHbrne npHMoro, TaR KaK ,n;Ba yrJIa TpeyrOJIbHUKa
l\leHbille ,n;BYX npJlMLIX, KaK ;D;OKa3aHO B npe;D;JIOmeHHH 17. TI03TOMY
AB, CTHrIIBalOm;aH MeHblliHH yrOJI AEB, KOpO'tle, 'qeM AE, CTHrHBaro-
aH 60JIbillHH yrOJI, KaK ,n;OKa3aHO B npeJ{JIOmeHHH 19. 3TO OTHOCIITCJI
KO BCel\f JIHHHHM, KOTopbIe MomHO npOBeCTII H3 TOllKII A K JIHHUH GE.
lloaToMY A"B - KpaT'tIaiIillaH H3 JIHHHH, npOBe,n;eHHLIX 113 TO"tJKH K
;D;aHHOH JIIIHHH. OHa Ha3LIBaeTCrH paCCTOHHHeM Mem;D;y HHl\IH B COOT-
BeTCTBIIH C TeM, 'tITO rOBOpHJI aBTOp "Ha'tlaJI" BO BBe;n;eHHII KIll KHH-
re. 3TO II eCTb TO, "tJTO MbI XOTeJIH ;D;ORa3aTb» [28, c. 511-512]..
llpenJIO l\eHHe 2 aT- TyclI: «ECJIH BOCCTaBHTb ;D;Ba paBHhIX nepneH-
,D;IIKYJIHpa K npH?tloH JIHHHII II coeJ];IIHHTb IIX KOH hI npHMoH JIHHUeH"
89
TO 06paayeMLIe IIMII yrJIbI paBBLI» [28, c. 512], KaK YKa3bIBaJI BLIme aT-
TYCII, COBIIa,n;aeT C npe,n;JIOmeBlleM 1 ,n;OKa3aTeJILCTBa XaHHMa.
OCBOBHLIM B J]:OKa3aTeJIbCTBe aT-TyclI HBJIHeTCH npeJ]:JIOmeHlle 3,
COBnaJ{aIO ee C npeJ]:JIOmeHlleM 3 XaHHMa no .pOPMYJIllpOBKe, HO OT-
JIllqaIO eeCH OT Hero cnoc060M AOKaaaTeJILCTBa:
«II peJ]:JIOmeHlle 3. ECJIII BOCCTaBIITL J]:Ba paBHLIX nepneHJJ;HKYJIHpa
II coeJ]:HHIITL HX KOH LI npHMoH JIHHlleH, TO 06paayeMLIe IIMH yrJIbI
npHMLIe.
IIpHMep. PaBHLIe nepneH,n;HKYJIHpLI AB II CD BOCCTaBJIeHbI K JIH-
HHH BD, HX KOHI{LI coeJ]:HHeHbI JIHHlleH AC (pHC. 57, a, 6). .H YT-
Bep}KJ]:aIO, lITO paBHLIe yrJIIiI BAC H DCA npHMLIe.
,IJ:oKaaaTeJIbCTBo. ECJIH OHU He npHl\fbIe, TO OBIl TynLIe IIJIH OCT-
pbIe. IIpeAnoJIomHM CHaqaJIa, lITO OHII TynLIe, II BOCCTaBHM Ha nep-
BOM lIepTeiRe (pHC. 57, a) B TOlIKe A nepneHJ]:IIKYJIHp AE K JIHHHH
A C, KaK AOKaaaHO B npe,n;JIOmeHIIII 11. OH Heo6xo,n;HMO nona,n;eT Mem-
,n;y JIHHHHl\fH AB II CD, H yrOJI AED, BHemHHH yrOJI npHMoyrOJIbHO-
ro TpeyrOJIbHHK8 ABE, 60JIbme BHYTpeHHero npHMoro yrJIa, HaK
J];OKa3aHO B npe,n;JIOmeHHH 16. IIoaToMY OB TaKme TynoH. ,IJ:aJIee BOC-
CTaBIIM B TOlIKe E nepneBJ]:IIKYJIHp EG K JIHHIIII BD. OH nona,n;eT Mem-
l1.y JIIIHHHMH AE H CD, H yrOJI EGC, BHemHHH yrOJI TpeyrOJIbHIIKa
EAG, 60JIbme yrJIa A - BHYTpeHHero npHMoro yrJIa. IIoaToMY OH
TaK}Ke Tynou. ,I:("aJIee BOCCTaBHM B TOqKe G nepneH,n;IlKYJIHp GH CHO-
Ba K JIIIHHH A C H B aTOM me nopHAKe 6YJ]:eM BOCCTaBJIHTb nepneHJ];H-
KYJIHpLI H ,n;aJIee ,n;o 6eCKOHeqHOCTII. TorAa nepneH,n;HKYJIHpLI, npOBe-
,n;eHHbIe H3 TOqeK, pacnOJIOmeHHhIX Ha JIHHHH AC nOJ]; npHMLIM yrJIOM
K JIHHHIl BD,- nepneHJ]:HKYJIHpLI AB, GE, FH nOCJIeJ]:OBaTeJIbHO
YBeJIHqHBaIOTCH no ,IJ;JIIIHe, II caMbIH KOpOTKHH 113 HIIX - nepneHJ]:H-
KYJIHp AB, CTHrIlBaIO HH OCTpbIH yrOJI AEB B TpeyrOJIbHHKe ABE,
noaTOMY OH KOpOlIe AE, CTHrIlBaIO eH npHMOH yrOJI ABE, KaK ,n;o-
Ka3aHO B npeJ]:JIOmeHHH 19. AE, CTHrHBaIO aH OCTpbIH yrOJI AGE
B TpeyrOJIbHHKe AEG, KOpOlIe GE, cTHrHBaIO eH npHMoH yrOJI EAG.
n03TOMY AB TaKme KOpOqe GE. TaKHl\f iRe 06pa30M AOKa3LIBaeTCH,
lITO GE KOpOlIe H F H, a F H - Toro, lITO CJIe,n;yeT 3 HeH. TaKHl\f 06-
pa30M, Te H3 8THX nepneHJ{HKYJIHpOB, KOTopbIe 6JIHme K AB, 6YAYT
KOpOqe, H paCCTOHHHH MemJ]:y TOqKaMH, HBJIHIO HMHCH OCHOBaHIIH-
MH nepneBJ]:HKYJIHpOB, ony eHHLIx 1I3 TOlIeK JIHBHH AC Ha JIIIHIlIO
BD, nOCJIeJ];OBaTeJIbHO YBeJIllqHBaIOTCH B CTOPOHY C, II, CJIe,n;OBaTeJIb-
HO, JIHHIIH AC H BD pacxo,n;HTCH B CTOPOHY C H CXOJ]:HTCH B CTOPOHY A.
Ho yrOJI DCA Tome TynoH, TaK KaK OH paBeH yrJIY BAC B CIIJIY
npeJ]:LIJ];y ero npe,n;JIOmeHIIH. IIoaToMY TaK me, KaK paHbme, ,n;OKa3bI-
BaeTCH, qTO JIHHHII CA II DB pacxo,n;HTCH B CTOPOHY A II CXO,IJ;HTCH
B CTOPOHY C. Ho aTO HeJIeno. CJIe,n;oBaTeJIbHO, yrJIbI BAC II DCA He
TynhIe.
ECJIII me aTII yrJILI OCTpbIe, onYCTIIM Ha BTOpOM qepTeme (pIlC. 57, 6)
113 TOqKII B nepneHJ];IIKYJIHp BE K JIIIHIIII AC, KaK )J;OHaaaHO B IIpeJ];JIO-
meBlI1I 12. OH Heo6xo)J;HMO nona,n;eT MemJ];y JIIIBHHMII AB II CD, TaR
HaK yrOJI A OCTphIH II HeB03MomHO, qT06LI OH nonaJI BHe aTIlX JIIIHIIH.
B npHMoyrOJIbHOM TpeyrOJIbHIIKe AEB yrOJI ABE OCTPLIH, noaTOMY
yrOJI EBD, COCTaBJIHIO HH BMeCTe C yrJIOM ABE npHMoH yrOJI ABD,
84
TaRme OCTpbIH. aJIee OrrYCTHM 1f3 TOqRH E rreprreHAHKYJIHp EG R
JlHHHH BD. OH IIOrralJ;eT MemJJ;Y JIHHHHMH AB H CD, II yroJl GEC OCT-
phIi. aJIee, orrYCTHM H3 TOqRH G neprreHlJ;IIKYJlHp GH CHOBa R JIHHHH
AC II B aTOM me rropHJJ;Re 6y,n;eM orrYCRaTb rreprreHlJ;HRYJlHphI H JJ;aJIee
)];0 6eCROHeqHOCTH. TorJJ;a rreprreHL(IIRYJlHphI, rrpOBeL(eHHhIe H3 TO-
qeR, pacnOJIOmeHHhlX Ha JIHHHH AC nOJJ; npHMbIl\f yrJIOM K JIIIHHH
BD, - nepneHlJ;IIKYJIHpbI AB, EG, H F nOCJIe OBaTeJIbHO YMeHbmalOT-
CH no lJ;JIHHe, II caMhIii JJ;JIHHHbIH H3 HRX - nepneHJJ;HKYJIHp AB. Ta-
KHM 06pa30M JJ;OKa3bIBaeTCH, qTO JIHHHH AC II BD CXOlJ;HTCH B CTOPC-
HY C H paCXO]]:HTCH B CTOPOHY A. Ho yrOJI DCA Tome OCTpbIH, TaK
RaK OH pa-BeH yrJIY BAC B CHJIY npeJJ;bIJJ;y ero npeJJ;JIOmeHIIH. TIoaTo-
MY TaR me, KaK paHbme, JJ;OKa3bIBaeTCH, qTO JIHHIIH CA II DB CXOlJ;HT-
CH B CTOPOHY A II paCXOlJ;HTCH B CTOPOHY C. Ho aTO BeJIeno. CJIelJ;o-
BaTeJIbHO, yrJlbI BAC II DCA He OCTpbIe, a TaK KaK OHH II He TYIIbIe,.
OHH rrpHMbIe. 3TO H eCTb TO, qTO MhI XOTeJIH JJ;OKa3aTb» [28, c. 512-
514].
3]]:ecb aT- TYCII, TaR me KaK Xa HM, paCCMaTpHBaeT qeTbIpex-
yrOJIbHHKH XaHHl\fa H Te tRe TPH rHIIOTe3hI 06 IIX BepXHHX yrJlax. On-
pOBepraH rHnOTe3hI Tynoro II OCTporo yrJIa, aT-TycH JJ;OKaSLIBaeT, 1J:TO
B rrepBOM cJlyqae BOCCTaBJIHeMhle HM nepneHJJ;HKYJlHPLI YBeJlHqIIBaIOT-
CH OT KpaH OCHOBaHHH K ero cepeJJ;HHe, a BO BTOpOM cJlyqae onYCKae-
MbIe HM nepneHJJ;HRYJIHpLI YMeHbmalOTCH OT KpaH OCHOBaHIIH K ero
cepe]]:IIHe, H CqHTaeT aTO IIpOTHBOpeqa HM CHMMeTpHII qeTbIpexyrOJlb-
HHKOB OTHOCHTeJlbHO nepneHJJ;HKYJIHpOB, coenHHHIO HX cepe,n;HHbI
BepXHHX II HHmHHX OCHOBaHHH. Ha caMOM me ]]:eJle Bce nepneHJJ;HKY-
JlHphI, nOCTpoeHHhIe no aT-TyclI, JIemaT no OJJ;HY CTOPOHY OT OCH CHM-
MeTp1I1I qeTLIpexyroJlbHHKa; aTO JJ;ORa3LIBaeT, qTO OCHOBaHHH 1J:eTLI-
pexyrOJlbHHR8 XaHHMa npH6JIHmalOTCH APyr K ,n;pyry no 06e CTOpo-
RhI OT OCH CHMMeTpHH B cJlyqae rHnOTe3bI Tynoro yrJIa II ynaJlHIOTCH
J];pyr OT npyra B cJlyqae rHrrOTe3LI OCTporo yrJla, 1J:TO ]]:OKa3hIBaJI II
XaHHM 0 ]]:BYX nepneH]]:HKYJlHpaX R OJJ;HOH npHMoii.
IIpeJJ;JIOmeHlIe 4 aT- TYCH: «BCHKHe ABe npOTHBOIIOJIOmHLIe CTOpO-
HLI npHMoyroJlbHoro qeTLlpexyroJlbHHKa paBHbI» [28, c. 515], KaR
YRasLlBAJI BhIme aT-TycH, COBnaJJ;aeT C npenJlomeHHeM 4 XaHHMa.
IIpe]]:JlomeHHe 5 aT-TycH: «ECJIH npHMaH JlHHHH na]]:aeT Ha ABa
nepneH]]:HKYJlHpa, BOCCTaBJleHHbIe K ,n;pyroH npHMOH JlHHHH npOHS-
BOJlbHbIM 06pa30M, TO HaKpeCTJlema He yrJILI paBHLI, BHemHHH yrOJl
paBeH COOTBeTCTBeHHOMY BHYTpeHHeMY, a BHYTpeHHlIe O]]:HOCTOpOH-
HHe yrJlLl paBHhI ]]:BYM npHl\fbIM» [28, c. 515] - COBnaAaeT C npeJJ;Jlo-
iReHHeM 7 XaHHMa.
B npe]]:JIOmeHHH 6 aT- TYCH ]]:OKa3hIBaeT V 1I0CTYJlaT ]]:JlH 1J:aCTHOro
cJlyqaH nepneH]]:BKYJlHpa H HaKJIOHHOH: «IIpenJlomeHHe 6. ECJIH ]]:Be
rrpHMbIe JIHHIIH, ROHIJ;hI KOTOpbIX He OrpaHIIQeHbI, IIepeceKaIOTCH He
no]]: npHMLlM yrJIOM H K onHOH 113 HHX BOCCTaBJIeH nepneH]]:HKYJIHp,
TO, eCJIH npOJJ;OJImBTb ero B O]]:HY H3 CTOpOH, OH nepeCeqeT ,n;pyrylO
JIUHHIO CO CTOpOHLI yrJla MeiR]]:Y nepneH]]:HRYJIHpOM H JIHHHeR, nepe-
ceKalO eii rrepneHJJ;BRYJIHp. ·
IIpHMep. JlHHBH AB II CD nepeceKaIOTCH B TOqRe E He no,n; npH-
MLIM yrJIOM. K JIHHHH CD BOCCTaBJIeH nepneH]]:HKYJIHp HG (pHC. 58).
85
.H YTBepmnalO, "liTO, eCJIII npOnOJIffilITb ero B O;D;BY lIS CTOpOB, OB ne-
peCelleT JIIIHIIIO A B .
,[(oRa3aTeJIbCTBo. YroJI AEB - 1I3 ;D;BYX paSJIlIlIHLlX yrJIOB,
AEC II CEB, BMeCTe paBHLlX nBYM npHMLlM B CIIJIY npe;D;JIOmeBIIH 13.
M3 ;D;aBHoii npOlI3BOJIbBOH T01lRII F JIIIBIIII AEB MomHO onYCTIITb nep-
neH;D;JiIRYJIHp F K Ha JIIIBIIIO CD, RaR AORasano B npeAJIOmeHlI1I 12.
T01lRa K IlJID nonaneT Memny E II G, IlJIII Ha T01lRY G, IInlI BHe [OT-
pe3Ka EG] co CTOpOHLI E llJIII C. ECJIII OHa nonaneT Memny E II G,
npennOJIOa\IIM npRMYIO nIIHUIO, paBUYIO EK, 8TO JIIIBIIH QZ, npo-
]J;OJI}KIiM ee B CTOPOHY Z II OTJIOmHM ZS, paBBYIO eH, RaR 8TO nORaaa-
HO B npenJIomeHIIH 3, HeCRonbRO pa3 AO Tex nop, DORa CY:hIMa 8THX
nOBTOpeHJiI:H nHHlI1I QZ He npeBaOH eT JIHHIIIO EG. 3TO QV, nYCTb 8TH
nOBTopeHHH - OTpe3RII QZ, ZS, ST, TV; Ra}K]J;LlH 113 HIIX paBeB JIH-
HIIH EK. 3aTe I OTJIOllilIM Ha JIHBIIII.AE TaRoe me 1IIICJIO nlIBIIH EF,
FX, XO, OP B nOCJIe;D;OBaTeJIbHOM nopHJ(Re, paBHLlX no BenllllUHe
FE. ,l],aJIee onYCTII 11I3 T01leK X, 0 II P nepneH]J;IIKYJIHpLl XL, O f
II P N Ha JIHHHIO CD, KaR JI:O:KaSaHO B npe]J;JIOmeHIIII 12. OnycTBM H3
T01lKII F nepne'H;D;IIKYJIHP FI Ha JIIIHIlIO XL. Tor]J;a B TpeyrOJILHIIKaX
E F K II FIX yr JILl E F K II EX I, BBYTpeHHIIH II BHemBIIH, paBHLI B CH-
JIY npe;D;LI;D;y ero npe]J;JIOmeHBH, TaR KaR nepneH;D;IIRYJIHpLI F K. If
XL BOCCTaBJIeHLI K npHMoH EK II Ha HIIX na;D;aeT npHMaH XE, yrJIbJ
EKF H FIX - npHMLle, a CTOpOBLlEF II FX paBHLI. Cne]J;oBaTeJIbHO,.
TpeyrOJIbHHRB XIF II EFK paBHLI, RaR ]J;OKasaHO B npe;D;JIome-
BIIB 26. TI08TOMY CTopOHa IF paBBa CTopOHe EK. Ho lIeTLIpexyrOJILBHK
IF LK npHMoyrOJILHLlH, TaR RaR Kam]J;aH nIIBIIH, nepneHJ(IIRYJIHp-
HaH R O;D;HOMY 1I3 nepneH;D;HRYJIHpOB, nepneB]J;IIRYJIHpHa II R OCTaJIb-
HbI I, KaR ORasaBO B npe]J;LI]J;y eM rrpeJ(JIOmeHHII. ll08TOMY npOTH-
BOnOJIOmHbIe CTOpOHbI IF II LK paBHbI, RaR ]J;OKasaHO B 1JeTBepTOM H3.
8THX npe]J;JIOmeHIIH. TI08TOMY JIIIHIIII EK II KL paBBbI. T01lHO TaR me
,nORa3L1BaeTCH, "liTO JIIIHBII KL II LM, a TaRme LM II MN paBBbI, H
TaKIIM 06pa30 1, Bce JIHBHH EK, KL, LM II MN paBBLI. IToBTOMY
CYMl\Ia 8THX JIHHIIH paBHa CYMMe OTpe3KOB QZ; ZS, ST II TV, T. e.
JIHBIIII QV, TaK KaR Bce 9TB JIIIBHH paBHLI II Kam]J;aH 113 HBX paBHa JIII-
HHH EK. Ho JIHHIIH QV ]J;JIIIHHee JIHBHB EG, n09TOMY II EN TaKme.
lJ;JIHBHee ee. n03TOMY T01lKa N He06xolJ;HMO HaXO]J;IITCH BBe JIIIBIIII EG
co CTOpOHbI C II nepneH]J;IIKYJI1.IP GH Haxo;D;BTCH BHYTPII TpeyrOJIb-
HHKa PNE. TI08TO IY, eCJIB npO]J;OJI}KIITL nepneH]J;IIRYJIHp GH napaJI-
.JIeJIbHO nepneBJ{HRYJIHPY P N ;0:0 BbIxo;D;a 113 TpeyrOJIbHHKa P N E, OH
He06xo;D;IIMO nepeCe1leT CTOPOHY AB. ECJIII T01lRa K nona;D;eT Ha T01l-
RY G, nepneH HKYJIHpLl COBna]J;YT, II, eCJIII npO;D;OJImHTL GH, nOJIY-
"IIHTClI BLlCOTa GH F Tpe-yrOJIbHIIKa F KE. B 8TOM CJIY1lae YTBepmlJ;e-
Hlle 01leBH;D;HO. 3TO II eCTb TO, 1ITO MLI XOTeJIII ,nOKa3aTb. BLlJIO lJ;OKa-
saHO, "tJTO BCTpe1la npolI30ii1J:eT CO CTOpOHLI OCTporo yrJIa, T. e. yrJI8
AEG. B 9TO?\1 npeJ{JIOmeHHH IICnOJILSOBaJIOCL YTBepmAeBlle 0 TOM,
lITO eCJIH B3HTh KpaTHOH 60JIee KOPOTKYIO 113 ;D;BYX JIIIHIIH C orpaBll-
"qeHHLlMH KOB aMII, TO OHa npeB30H]J;eT 60JIee J{JIIIBBYIO liS BIIX,-
H "
8TO YTBepiRneHlle lbI ynoMHHaJIH, ero IICnOJIL80BaJI aBTOp" &lIaJI
B npelJ;JIOffieHIIU 1 KHHrlI X» [28, c. 5-16-518].
QUTHpoBaHHoe nOKa3aTeJILCTBO 6JIII3KO KO nTopOMY AOKa3aTeJIb-
86
A
I c
]J E
J( l /HC N C
8 6
V T S Z [J B
Puc.58 Puc.59
H
A
CTBY M6H ROppLI II B J]:OKaaaTeJIbCTBY M6H aJI-XaHCaMa II, TaK me
RaK OHH, onlIpaeTCH Ha aKCHOMY llama II aKCHOMY EBJ]:oKca - Ap-
XHMeJ]:a, KOTopLIe aT-TyclI, nOJ]:o6Ho 1I6H-aJI-XaHcaMY, cpOPMYJIHpyeT
B HBHOM BHJ];e. rOBOpH 0 BTOpOH lIa 9THX aKCHOM, aT-TycH ynoMHHaeT,
"ITO OHa IIcnOJIbaOBaJIaCb EBKJIHJ]:OM B npe JIOmeHHII 1 KHHrlI X «Ha-
qaJI», aa KOTopoe OnHpaJICH «MeTOJ]; HC"IeprrLIBaHHH», IlrpaBillHH B
JJ;peBHOCTII pOJIb HameH TeopHII npeJ]:eJIOB II npHMeHHBmlIHcH J]:peBHII-
?tIH rpeKaMH npH ,n;OKaaaTeJIbCTBe Tex TeopeM 0 ]];JIHHaX KpHBLIX JIII-
HHU, nJIO a;g;Hx nJIOCKHX cpHryp II KpHBLIX nOBepxHocTeH H 06'beMaX
Ten, KOTopbIe B HaCTOHIIJ:ee BpeMfI JJ;OKaaLIBaIOTCH C nOMOI.IJ;bIO IIHTer-
paJIbHOrO HC"IHCneHHH.
B npeJ]:JIOmeHHII 7 aT- TYCH J]:aeT opHrlIHaJILHOe J]:OKaaaTeJIbCTBO
o6mero CJIY"IaH V nOCTY'JIaTa:
«IIpeJJ;JIomeHHe 7, co,n;epma ee )J;OKaaaTeJIbCTBO nOCTYJIaTa: eCJIH
npHMaH JIHHHH naJ]:aeT Ha J]:Be npHMbIe JIHHHH II o6paayeT C HHMH
BHYTpeHHlIe O]];HOCTOpOHHlIe yrJILI, MeHbmHe ,n;BYX npHMLIX, TO 9TH
JJ;Be npHl\fLIe, eCJIH npoJ]:OJImHTb HX B 9TY CTOPOHY, nepeceKYTCH.
IIplIMep. JIHHHH AB naJ]:aeT Ha JIHHHII CD II EG II o6paayeT yrnLI
CHP II EFH, MeHbme ,n;BYX npHMhIX (pHC. 59). H YTBepmJ];alO, "ITO
JIIIHHII CD II EG, eCJIH rrpo,n;OJIiKHTb IIX B CTOPOHY C, nepeceKYTCH.
AOKaaaTeJIbCTBo. ECJIH OJ];HB 1I3 yrJIOB CHF II EFH npJlMOU, TO
JJ:pyroH Heo6xo)J;HMO OCTphIH. l108TOMY OnHa 113 JIHHHH CD II EF lIe-
p ceKaeT JIHHHIO AB nOJ]: yrJIOM, He HBJIHIO HMCH npHMLIM, a ,n;pyraJl
nepneHJ]:HKYJIHpH8 K HeH, rr09TOMY, eCJIH npOJ]:OJImHTb HX, OHH IIepe-
ceKYTCH CO CTOpOHLI OCTporo [yrJIa], KaK ,n;OKasaHO B. npeJ]:LI,n;y eM
npeJ]:JIOme-HIIH.
87
ECJIII OJJ;IIB 1I3 yrJIOB Tynoii, nYCTb BTO yrOJI CHF, BOCCTaBIIMB TO'll-
Re H nepnelInlIKYJIHp HI R JJIIH:HII CD, RaK ,IJ;OKaaaHO B npe;nJIO-
meBIIH 11, a 113 TO'llKII F OnYCTIIM H8 Bee nepneHJ]:IIRYJIHp F K, R8K
)J;OKa3aHO B npe](JIOmeHIIII 12. TorJ]:a MLI YTBepmJ]:aeM, 'liTO TaR RaK
yrJILI CHF II EFH BMeCTe MeHbme J]:BYX npHMLIX, a yrOJI CHI npn-
IOH, TO yrJIbl IHF II HFI BMeCTe MeHbme O,IJ;Horo npllMOro. Ho yrJIhI
I H F II H F K, HaR HaKpeCTJIema lIe, 06pa8YIO lIecH npll na](eHIIII
JIIIHIIII AB Ba ](Ba nepneHJ]:IIKYJIHpa I H II F K, paBB:bI, KaR J]:ORa3aHO
B nHTOM 113 3TIIX npe](JIOiReHIIH. CJIeJ]:OBaTeJIbHO, BeC.b yrOJI KF I MeBb-
me OJ{Horo npH:Moro, T. e. OB OCTp:bIii. llOBTOMY JIIIHIIII KF II EF ne-
peceRalOTCH He non npHMLIM yr JIOM, a JIIIHIIH H K nepneH](IIRYJIHpHa
R OJJ;HOH liS BIIX, a II:M:eHHO R JIIIBIIII KF. TI03TOMY JIIIHIIII CK 11 EFt
eCJIII npOJJ;OJIllilITb IIX TaRme B CTOPOHY C II E, nepe.ceRYTCH, RaR ](0-
Ra3aHO B npe](:bI,D;ym;eM npeJJ;JIO)I eHIIII.
ECJIII 06a yrJIa OCTp:ble, BOCCTaBIIM B TO'llKe F nepneHJ]:IIKYJIHp F K
H JIIIHIIII EG, RaK JJ;OKa3aBO B npelJ;JIOmeHIIII 11, II onYCTIIM liS TO"llKII
H nepneHlJ;IIRYJIHp HI Ha Hee, KaK ,D;OKaSaBO B npeJ]:JIOmeHIIII 12.
TorJ]:a yrOJI EFK npHMoii, a yrJIId KFH II FHI, KaK HaKpeCTJIema-
lIe, 06paaYlOlUnecH npll nalJ;eHIIH JIIIHIIII AB Ha ,n;Ba nepneHJJ;IIKY-
JIHpa HI II KF, paBH:bI, KaK lJ;ORa3aHO B nHTOM H8 BTHX npeJ]:JIome-
HHn. CJIelJ;oBaTeJIbHO, yrJILI FHI II HFI BMeCTe paBHLI OlJ;HOMY npH-
MO IY, a, TaR RaR yrJI:b1 EFH II CHF no npeJ]:nOJIOmeHHIO MeBbme
,D;BYX npHM:bIX, yrOJI I HC MeHbIIIe npHMoro, T. e. OB OCTp:bIii. TI03TO-
MY JIIIHIIII I H n CH nepeceKalOTCH Be non npHMLIM yrJIOM, a EI nep-
neHJ]:IIKYJIHpHa K OlJ;HOU 1I3 HIIX, 8 IIMeBHO K I H. TIOBTOMY [JIIIBIIU)
CD II EG nepeceKYTCH, eCJIII npOJ]:OJImlITb IIX B CTOPOHY C II E, KaK
JJ;OKa3aHO B npe,D;LIJJ;y eM npe,IJ;JIOmeHIIH. 3TO II eCTb TO, 'liTO Tpe60-
BaJIOCb JJ;ORaSaTb» [28, c. 519-520].
BCJIen sa TeM aT- TyclI npe](JIaraeT J];pyroH BaplIaHT J]:ORaaaTeJIbCTBa
aRCHOM:bI 0 napaJIJIeJIbH:bIX, COCTonITl;IIU lIS npeJ]:JIOmeHHH, 3aMeHHIO-
IIX npelJ;JIOmeHIIH 6 II 7 nepBoro J]:ORaSaTeJIbCTBa, II H3 npe,IJ;JIome-
HIIH 8:
«BMecTo npenJIomeHIIH 6. Bo BCHKOM OCTpOM npHMOJIHHeiiBO 1
yrJIe, eCJIII OTJIOmlITb Ha o,IJ;Hen D8 ero CTOpOH paBH:ble JIIIHIIH II onyc-
TIITb liS To"tJeK JJ;eJIeHIIH nepneBlJ;IIKYJIHp:b1 Ha JJ:pyrylO CTOPOHY, TO
JIIIBHH, OTCeKaeM:bIe OCHOBaHIIH 11I nepneH,IJ;IIKYJIHpOB Ba aTOH CTOpO-
He, TaKme paBH:b!.
npIIMe"p. B OCTpOM yrJIe BAC Ha AB OTKJIa,D;LIBaIOTCH paBHLIe JIII-
HIID AD, DE II EG, II 113 IIX [KOHqOB] onYCRalOTCH nepneH,IJ;HKYJIH-
p:bI DR, EF II GI Ha JIIIHIlIO AC (pIlC. 60). .H YTBepm,n;alO, "liTO JUIHHII
AH, HF II FI, OTCeKae {LIe OCHOBaHIIHMH nepneH](IIRYJIHpOB, TaRiRe
paBHLI.
,n:OKa3aTeJIbCTBO. TIOCTpOIl I npH TO'llKe D Ha JIIIHIIH ED yrOJI
EDK, paBBLIn yrJIY A, KaK nOKaaaHO B npe,IJ;JIOmeHIIH 23. Tor](a
B TpeyrOJIbHIIKaX AHD H DKE yrJI:bI A H D paBH:bI, yrJILI D II E,
BBemHIIH II BHYTpeHHH:U, 06pa30BaHHLle npll naJ]:eHIIII JIIIHIIH AE Ha
,D;Ba nepneH]J;HKYJIHpa DR lIEF, paBH:bI, RaK J]:ORa3aHO B nHTO 1 n3
8TIIX npelJ;JIOmeBlln, II CTOpOB:bI AD H DE paBHLI. CJIelJ;OBaTeJIbHO,
3TH lJ;Ba TpeyrOJIbHIIRa paBHLI, II CTopOHa AH paBHa CTopOHe DK,
88
8
A
8
c
o
x
PUC. 60
PUC. 61
a IIpHMOH yrOJI H paBeH yrJIY K, KaK nOKaaaHO B npe.n;JIOmeHHH 26.
IIoaToMY nJIOCKaH cpHrypa D H F K - npHMoyrOJIbHbIH qeTbIpexyrOJIb-
BHK, H ee npOTHBOIIOJIOmH:bIe CTOpOH:bI D K H H F paBHbI, KaK nORasa-
HO B lIeTBepTOM HS 3THX npenJIomeHHii. IIoaToMY JIIIHHH AH, paBHaH
DK, paBHa D HF. Ta HM me 06pasoM nOKaSbIBaeTCH, 'lITO HF paBHa
Fl. 3TO H eCTb TO, liTO MbI XOTeJIII nOKaSaTb) [28, c. 520-521].
,[(aHHoe YTBepm eHHe aT-TycH paBHOCHJIbHO YTBepmneHHIO 0
cy ecTBoBaHHIl IIon06HbIX TpeyrOJIbHHKOB JII06bIX paSJ\;IepOB H, CJIe-
nOBaTeJIbHO, YCHJIeHHOii aKCHOMe 0 napaJIJIeJIbHbIX.
A T- TYCH nponoJImaeT:
«BMecTo npenJIOmeHHH 7. ECJIH B npHMOJIHHeiiHO 1 yrJIe npenno-
JIOmHTb TOqKY Memny ero CTopOHaMH, TO HX MomHO coenHHHTb npH-
Moii JIHHHeii, npoxO):(HIT(eu qepe3 aTY TOllKY.
IIpHMep. B npHMOJIHHeiiHOM yrJIe ABC Memny CTopOHaMH AB H
BC npennOJIOmeH8 TOqKa D (pHC. 61). .H YTBepm.n;alO, 'lITO CTOpOHbI
AB H BC MomHO coenllHHTb npHMOH JIHHHeii, npoxonHIT(eH qepes
TOqKY D.
,[(OKaSaTeJIbCTBO. IIpoBeneM HS n;eHTpa B Ha paCCTOHHHH BD ny-
ry EDG, npOXonHIT(YIO lIepea TOqKY D. IIpoBeneM xopny EG H pa3ne-
JIHM nonOJIaM yrOJI EBG lIHHHeH BI, KaK nOKasaHO B npenJIomeHHII 9.
Torna TpeyrOJIbHHKH EBH H GBH paBHbI, TaK KaR CTOpOH:bI EB D
BH paBHbI CTopOHaM GB H BH, a yrJIhl B paBHbI. II03TOJ\;IY paBH:bI II
CTOpOHbI EH H HG H yrJIbI H, KaK nOKaaaHO B npenJIomeHHB 4. IIoa-
TOMY EH nepneHnHRYJIHpHa R BH. IIponoJImHJ\;I BH no nepeCeqeHHH
c nyroH EDG B TOqKe F. IIoBTopHM JIIIHHIO BH no Tex nop, IIORa CYJ\;I-
ria 3TIlX JIHHIIH He npeBsoiineT JIIIHHIO BF, nYCTb 3Ta CYMMa - JIH-
HUH OX. OTJIOmHM Ha BA JIHHHH, KamnaH HS KOTOpbIX paBHa JIHHHH
BE, B TOM me qHCJIe, 'lITO II qHCJIO, BO CKOJIbKO OX KpaTHO BH, 3TO
89
BE, EK. OnYCTHM HS KOBD;OB 3THX JIHHUH nepneH HKYJIHpbl B8 nH-
BHIO BH, 8TO nepneH HKYJIHpbI EH, KL. 3TH nepneB HKYJIHpLI OT-
CeKYT H8 JIHHHH BH p8BHbIe JIHHHH BH, HL, KaK OK83aBO B npe hI-
YIIJ;eM npeJJ;JIOmeHHH. Ho HX CYMMa, paBBaH JIHHHH OX, . JIHHHee
JIHBHH B F. II08TOJ\;IY KOHe nepneH HKYJIHp8 K L B8 JIHBHH BI,
T. e. TOqKR L, B8XO HTCH BRe JIHHHII BF. ,I:(8JIee OTJIOmllM H8 BC JIH-,
BllIO BM, paBBYIO BK, II COe HHHM ML. Tor):(a TpeyrOJIbHHKH BKL
H BML paBHbl, TaK KaK Y HHX 06 aH CTOpOaa BL, paBHble CTOpOHLI
BK 11 BM II yrJIbl B, K8K OKa38Ho B npe,n;JIOiKeBHH 4. II08ToMY yrOJI
MLB paBeH npHMOMY yrJIbI KLB, H npHMble KL H LM coe HHHIOTCH
B CBoeM BanpaBJIeBHH B OJJ;HY JIHBHIO B CHJIY npe JIOiKeHHH 14. ,I:(a-
JIee coe HBHM BD, npO OJIiKHM ee):(o N H nOCTpOHM npH TOqKe D
Ha JIHBHH DN yrOJI PDN, paBBblH yrJIY DNL, K8K ,n;OKa38HO B npe -
JIOmeBHH 23. Tor a JIHBHH PD 11 NM n8p8JIJIeJIbBbl, K8K paBbme,
B CIIJIY paBeBCTB8 BaKpeCTJIema HX [yrJIOB], T. e. yrJIOB PDN II
DNM, KaK nOKa3aBO B npe JIOmeHHH 27. IIpo,n;oJIiKHM PD o Tex
nop, nOKa OHa He BbIii eT H3 TpeyrOJIbBHK8 BKM B TOllK8X P H Z.
JIHHHH PZ coe HBHeT CTOpOHbl AB H BC H npoxo,n;HT lIepes aBHYIO
TOllKY D. 3TO H eCTb TO, 'lITO Mbl XOTeJIH ):(OK838Tb) [28, c. 521-523].
OTO YTBepm,n;eBHe COBna,n;aeT C TeM, KOTOpbIM nOJIb30BaJIHCL o
aT-TycH CUMnJIHKHii II aJI-,I:(mayxapH, 8 nOCJIe Hero - JIemaH,D;p.
O,n;HaKO aT- TycII He CqllTaeT 8TO YTBepm,n;eHHe OqeBH HbIM, 8 npHBO.
,n;HT ero ,n;OKa3aTeJIbCTBO. Mbl ya-\e YKa3bIB8JIH, 'lITO 3TO YTBepm eBHe
paBBOCHJIbHO V nOCTYJIaTY. IIpe JIomeBHe 8 8T-TYCH COBna,n;8eT C no-
CJIeJJ;HHM npe JIOmeHlIeM OKa3aTeJIbCTB8 8JI-,I:(iK8YX8pH. TeopHH na-
paJIJIeJIbHbIX JIHBHii 6bIJIa 1I3JIOmeH8 HacHp aA-,I:(HHoM 8T-TYCH H B
ero 06pa60TKe «HaqaJI) EBKJIH a - TaR Ha3bIB8eMOM «M3JIOmeHHII
KBllrH HaqaJI reOMeTpHII EBKJIH,n;a) (T8 pHp KIITa6 8JI-Y YJI <pH-JI-
xaH aC8 JIH Y E5JIH HC), 06bIqBO KpaTKO Ha3blB8eMOM «M3JIOmeHHeAI
EBHJIHna) (Ta pfip Y JIHJJ;HC). 3TO OnBO 113 B8H60JIee nonYJIJlpBblX
CHCTeMaTUqeCKHX COqHHeHHM H8 apa6CKOM H3b1Ke, HJ\;IeIO eeCH nOqT}I
B Kam oii 6H6JIHOTeKe apa6CHHX PYKonHceii. KBHr8 6blJI8 113 aHa Ha
apa6CKOM H3blHe B 1881 r. B TerepaHe. Pa3 eJI 29, nocBH HHbIH
TeopHH napaJIJIeJIbHblX JIHBHii, 6bIJI ony6JIHKOBaB B8 8pa6cKo 1 H3blKe
A. H. Ca6poii [61], a ero PYCCKHii nepeno - r. ,I(. MaMe 6eiiJIH
[16, c. 13-22].
OTMeTHM, 'lITO B CBoeAI «H3JIOmeHHH EBKJIH a) 8T-TYCH npe,n;nOCbl-
JIaJI nOCTYJI aTal\1 EBKJIH a CJIe,n;YIO lIe CJIOB8: «.fl rOBOplO nepBblM
H3 He06xo H loro, 'lITO nOJIaraeTCH, 'lITO TOqKa, JIHBIIH, nOBepXBOCTb,
npHMaH JIHHHH, nJIOCHaH nOBepXHOCTb H Kpyr cy eCTBYIOT H 'lITO MhI
BbI eJIHeM TOqKY Ha JII060ii JIHHHII HJIII nOBepxHocTH H npe,n;noJIarae1\1
JIHHIIIO Ha JII060ii nOBepxHocTH IIJIII npoxo,n;H eii qepe3 KaKYIO yroA-
HO TOqKY) [29, c. 3] 1, H TeM Cal\lblM BnepBble B HCTOpHH MaTeMaTUHH
aT- TycII C«POPMYJIHpOBaJI TO, 'lITO 1\lbl B HaCTOH ee BpeMH Ha3blBae1\1
«aKCHO laMH cy ecTBOBaHHH) H «aKCHOMaMII BbI6opa).
1 Harne BHRMaHRe Ha OPMYJIRpOBRH aRCROM CYlI\eCTBOBaHHJI H BbI60pa B «HaJIo.
meHRH EBRJIR)];a» aT- TycR 06paTRJIa BHHMaHRe H. r. E epOaarnBIIJIIl, llaY'lJaBmaH
3TO C01JHHeHlIe RaR HJIoJIor-apa6HcT.
90
B «II3JIomeBHH KBHrll HallaJI» BOCrrpOIl3Bep;eHLI o6a BapnaBTa ,n;O-
Ka3aTeJlbCTBa V rrOCTYJIaTa :03 «TpaKTaTa, UCn;eJIHIOII:\erO COMHeHUe»,
HO B OTJIHlIHe OT 3TOrO TpaRTaTa nOCJIe CPOPMYJIUpOBKH V rrOCTYJIaTa
BO BBe)J;eHHH R I RHHre aT- TYCH 3p;eCb nHCaJI:
«H rOBopIO, lITO nOCJIeABee YTBepm,n;eBlIe Be HBJIHeTCH aKCHOMoi
II MomeT 6LITb p;OKa3aHO TOJIbKO B reOMeTpHlIeCKOH HaYKe. 06 aTOM:
JIYllme rOBopHTL He BO BBep;eHHH, H H ,n;OKamy ero B Ha,n;JIemall\el\1
l\leCTe. BMeCTO Hero H CTaBJIIO ;n;Pyroe YTBepmAeIIlIe: eCJIU rrpHl\fLIe
JIHHHU, JIema IIe B O,n;HOH nJIOCROCTII, CXOAHTCH B O)J;HOM HarrpaBJIe-
HUH, OHH He MorYT B 3TOM HarrpaBJIeHHII paCXOJJ;HTLCH, eCJIH TOJlbKO
OHU He rrepeceKaIOTCH» [29, c. 4].
B IIOCJIep;HeM npe,n;JIOmeHHH aT-TycH 4>OPMYJIUpyeT nOCTYJIaT, 3a-
l\feHHIO IIH V nOCTYJIaT. 3TOT nOCTYJIaT BeCLMa noxom Ha «4-i rrplIH-
n;Hrr <pIIJIOcocpa», KOTOpLIM nOJIL30BaJICH XaHHl\I, o,n;HaKO B OTJIIIlIIIe
OT Hero rrOCTYJIaT aT-TYCH paBHOCHJIeH He V nOCTYJIaTY, a YCHJIeHHOH
aKCIIOMe rrapaJIJIeJIbHOCTH.
YrrOl\IIIHaBmaHCH Hal\fH rrepenHCKa aT-TYCII C aJI-XaHa4> nOKa3LI-
BaeT, lITO aT-TYCH o6cym,n;aJI CBoi TpaKTaT 0 napaJIJIeJIbHLIX JIIIHIIHX
C ,n;PyrHMH YlIeHLIMII. IIO-BII,n;IIl\10MY, KTO-TO 113 aTHX YlIeBLIX YKa3aJI
aT-TYCfI Ha TO, lITO B «TpaKTaTe, IIC-U;eJIHIO eM COMHeHUe» OH COBep-
mHJI JIOrHlIeCKYIO omH6KY nOCTYJIHpOBaHIIH OCHOBaHHH, II peKOMeH,n;o-
BaJI eMY ,n;onOJIHUTL ero U3JIOmeHHe HBHLIl\f nOCTYJIaTOM, paBHOCIIJIL-
HLIM V nOCTYJIaTY HJIII cop;epH am;eMY YTBepm,n;eHHe, paBHOCHJIbHOe
eMY, Ral\ nocTynUJI paHee XaHHl\1. Bo BCHROM CJIyqae, aT-Tycu nepe-
CMOTpeJI CBOIO nepBOBallaJIbHYIO TOtJKY 3peHHH II 1);onOJIHHJI CBoe U3-
JIOmeHHe TeopHII napaJIJIeJIbHLIX JIIIHHH npHBep;eHHLIM rrOCTYJIaTOl\I.
I13JIOmeHHe Teopuu napaJIJIeJIb-
HbIX, Ha KOTopoe CCbIJIaJIIICb Brro-
CJIe,n;CTBHH B XV I lB. ,[(m. B aJIJIUC
11 B XVIII B. ,[(iI . CaKKepH II KOTO-
poe OHII ClIIITaJIH npHHaAJIema Hl\1
HacHp a,n;- uHY aT-TYCH, He COBna-
]];aeT C npUBep;eBH:bIl\1 BbIme JJ;OKa3a-
TeJILCTBOM, nplIBe,n;eHHbIM B HerrOJI-
HOl\1 BH)];e, 6e3 aKCUOl\lbI, 3aM:eHHIO-
m;ei V nOCTYJIaT EBKJIH,n;a, B «TpaK-
TaTe, HCn;eJIHIO el\1 COl\fHeHHe no nOBO,n;y rrapaJIJIeJIbH:bIX JIUHIIH»
11 B nOJIHOM BHp;e - B «M3JIOmeHIIH KHlIrH HatJaJI reOMeTpHH EBKJIII-
,n;a». BaJIJIHC II CaKKepH HMeJIII B BU,n;y «RHury lI3JIOmeHUH HallaJI
EBKJIH,n;a, COllHHeHBYIO xo,n;mOH HaclIp a,n;- HHOl\f aT-T)TCU») (KUTa6
Ta pHp Y YJI JIB Y JIII,n;UC l\IHHTa< JIii4> Ba,n;ma Ha Hp aJJ;- HH ar-1'y-
CH), 1I3,n;aHHYIO no-apa6CKH C JIaTHHCKHl\1 TUTYJILHLIl\1 JIIICTOM (Eucli-
dis Elementorum geometricorum libri tredecim ex traditione doctis-
simi Nasiridini Tusini) B PUMe n 1594 r. [48].
PBMcKoe H3,n;aHHe 3HallHTeJIbHO OTJIHlIaeTCH OT paccMoTpeHHoro
BbIme «M3JIOmeHIIH KHurll HallaJI»). 8TO H3;n;aBoe COAepiRlIT 1f3JIO-
meHlIe TOJIbKO TpHHa.lI.n;aTII KHHr, HaUHcaHHbIX caMH!.1 EBKJIHJJ;Ol\I,
a paCCMOTpeHBoe HaMH BLIme COllUHeHUe aT-Tycu - H3JIOmeHHe e e
TaR Ha3LIBae1\lbIX X IV II XV RHHf, Hepe,n;KO rrpIIUIICbIBael\lhIX EBKJIU-
14.
A01Ca3ameJl/bCm(30,
nptlnUC'bl(3aeMOe
am-Tycu:
91
AY. ,I{aJIee, B pnl\ICKOM H3,n;aHIIH TeKCT EBKJIH,n;a nOJ];BeprHYT 3Ha'tJlI-
TeJIbHO 60JIbmeii nepepa60TKe, 'tJeM B COlIHHeHHH aT-TYCII. CYIII-eCT-
BeHHbIl\f pa3JIHqlIeM Mem,n;y 06eHMH TeKCTaMH HBJIHeTCH TaKaie TO, 'tJTO
CO'tJIIHeHHe aT-TYCII COXpaHHJIOCb B OrpOMHOM 'tJHCJIe PYKOnHCeii, B TO
BpeMH KaK COXpaHHJlHCb .TOJIbKO J};Be PYKOnHCH TeKCTa pHMCKOrO H3-
)J;aHIIH, O)J;Ha nOJIHaH H o,n;Ha HenOJIHaH, XpaHH lIeCH B 6H6JIII.OTeKe
Me,n;lIqH BO <DJIOpeHn;IIH, npH'tJeM pHMCKoe H3)J;aHHe CJ];eJIaHO C IIOJIHOH
PYKonuCH. TOT «paRT, 'tJTO 3HaqUTeJIbHO JIyqmlIH BapHaHT H3JIOmeHHH
EBKJIH,n;a nOJIyqHJI 3Ha'tJHTeJIbHO MeHbmee pacnpOCTpaHeHHe, 'tJe 1 XY)J;-
mHH, BhI3BaJI COl\IHeHHe B npHHa,n;JIemHOCTU plIMCKoro H3,n;aHIIH «113-
JIOmeHIIH Ha'tJaJI EBKJIIf,n;a» aT- TYCII [21]. 3TO COl\IHeHHe IIOJIyqHJIO
)J;aJIbHeiimee no,n;TBepmJ];eHHe nOCJIe Toro, KaK A. II. Ca6pa [60, c. 15]
YCTaHOBHJI, 'tJTO, Ral\ YKa3aHO BO «pJIOpeHTHHcKOii pYKonIICH, opHrlI-
HaJJ pHMCKoro 1I3,n;aHIIH 6bIJI COlIHHeH B 1298 r., T. e. nOCJIe CMepTH
aT-TycH. Ha aTOM OCHOBaHHH ,[(m. Mep,n;OK Ha3bIBaeT aBTopa «II3JIo-
meHHH "HaqaJI" EBKJIHAa» «IIceBAo-TycH» [57, c. 455]. O)J;HaKo, KaK
06HapymlIJI X. TJIJIameB, B cnHCKe COqHHeHHH aT-TycH, COCTaBJIeH-
HOM ero yqeHUKOM HU3aM: aJ];- HHoM aH-HaHca6YPH, xpaHHm;eMcH
B TamKeHTCKOM IIHcTIITYTe BOCTOKOBe,n;eHHH [26], YKa3aHhI TPH CO"lH-
HeHHH aT-TycH, nOCBH eHHbIe KOMMeHTupOBaHHIO EBKJIII,n;a: no,n;
M 9 «II3JIomeBHe EBKJIH,n;a», no,n; M 24 «TpaKTaT 0 EBKJIH,n;e» (PH-
caJIa q.H-JI-Y JIiI)J;lIc) (HeCOl\fHeHHO, aTO COKp8lIJ;eHHOe Ha3BaHHe
«TpaRTaTa, HCn;eJIHIO erO COMHeHHH no nOBO,n;y napaJIJIeJIbH:bIX JIH-
HHH») II IIO M 27 «IIpHMeqaHHH K EBKJIH,n;y» (8aBami1 'aJIa YJ)JIH-
AHC). IIocJIe,n;Hee COqlIHeHHe O Hac He ,n;OIDJIO. B03MomHO, liTO «113-
JIOmeHHe "HalIaJI" EBKJIH,n;a» 6hIJIO COCTaBJIeHO nOCJIe CJ\;IepTII aT- TYCH
C YlIeTOM «TIpHMe'tJaHuH K EBKJIH)J;Y» KeM:-HH6y,n;b H3 ero YlIeHHKOB,
HanpuMep ero CbIHOl\f Ca,n;p a,n;- HHOM H6H xo,n;mOH HaCHp a)J;-,[(HHoM,
6hIBIDHM BCJIe)J; 3a OTD;oM AHpeKTOpOl\f MaparHHCKoii 06cepBaTopuH.
3TO H3JIOmeHHe «HaqaJD) EBKJIH,n;a OrJIHlIaeTCH OT paccMoTpeHHoro
Hal\IH BhIme «M3JIOmeHIIH EBKJIH)J;a» aT- TYCH 60JIee cy ecTBeHHoii ne-
pepa60TKOH CIICTeMhI nocTYJIaTOB EBKJIH,n;a. Pa3BHBaH ,n;aJIbwe. H)J;eD
aT-TyclI 06 aHCHOMax cy ecTBoBaHHH H BbI60pa, IIceB,n;o-TYCH B
CBoeM «M3JIOmeHHH EBKJIII)J;a» cpOpl\fYJIHpyeT aKCHOMhI cym;ecTBoBa-
HUH H BhI60pa yme B KalIeCTBe HOBbIX nOCTYJIaTOB. I1epBhIH CPOPMY-
.JIHpyeTcH TaK: «Ram,n;aH TOlIKa, npHl\laH IIJIH RpyrJIaH JIHHHH, nJIOCKaH
HJIH KpyrJIaH nOBepXHOCTb cym;ecTByeT B CHJIY He06xo,n;UMOCTH cy-
eCTBOBaHHH .n;BHmy eiicH C«pePhI, Bce CTOpOHhI CYlI(eCTBOBaHHH 1\0-
TOpOH orpaHH'tJeHhI». ,[(aJIee nOCTYJIlIpyeTCH: «IIepece'tJeHHe BCHKHX
JJ;BYX JIIIHHH eCTb TOlIl\a, HBJIHIO aHCH KOHD;Ol\1 Kam,n;OH H3 HHX» U «IIe-
peCelIeHHe BCHKHX ,n;BYX nOBepXHOCTeii eCTb JIHHHH, HBJIHIO aHCH
KpaeM Kam,IJ:OH H3 HHX. ..» ,[(aJIee «POPMYJIHpyeTcH «nOCTYJIaT BhI60pa»:
«MhI MOmel\1 npe,n;nOJIOmlITb Ha KaiI(,n;OH JIHHIIH H nOBepXHOCTH TOqHY,
HBJIHIO YIOCH npe,n;eJIOl\f lIYBCTBeHHoro YKa3aHIIH». 3aTel\1 CJIe}J;yeT
nOCTYJIaT, 0606 aIOW;lIii I nOCTYJIaT EBKJIIIAa: «MhI MOa(el\1 coe)J;HHHTb
Kam,n;bIe ABe TO'tJKII npHl\fOH HJIH APyroH JIUHHeH». 3TOT nOCTYJI aT 06oc-
HOBhIBaeTCH Te l, 'tJTO «MemAY Ram,n;hIl\IH ABYl\IH TOlIKaMII l\lhI l\fOmeM
npe,n;nOJIOmHTb TOqKH B IIX HanpaBJIeHHII II npe,n;IIOJIOiHHTb, lITO TOqKa,
cOBnaAaIOm;aH C o,n;Hon H3 3THX ,n;BYX TOqeK, ,n;BHmeTCH no HanpaBJIe-
92
HHIO K )];pyroH TOlIKe H npOXO,D;HT lIepe3 [Karl\)];YIO] TOlIKY, npe,IJ;nOJIO-
meHHYIO B HanpaBJIeH1IH ,l];aHHbIX TOlIeR, B TelIeHHe BCerO BpeMeHII ee
,IJ;BHmeHIIH, 3aKaHlIHBalOm;erOCH BO BTOpOH TOlIKe» [48, c. 7].
B HaCTOHm;ee BpeMH IVILI CKa3aJIH 6bl, lITO «nOCTYJIaT npOBe)];eHHH
JI1IHIIH lIepe3 TOlIKY» HBJIHeTCH ]J;BoHcTBeHHbIl\f K nOCTYJIaTY BbI60pa
TOlIKH H3 JIHHHII; OTl\feTHl\f, lITO nceB)J;o-TYCII He TOJIbKO He 1I36eraeT
npHI\feHeHHH ,IJ;BIImeHIIH K reOMeTpHH, KaK 3TO ,IJ;eJI aJiH EBKJI1I)]; H CJIe-
)J;OBaBmHe YtJeHHIO ApHCTOTeJIH M6H CHHa II XaiiHl\f, HO «pOpl\fYJIHpyeT
3TO nplIl\feHeHlle yme B CBOHX nOCTYJI8Tax. aJIee CJIe)];yeT nOCTYJIaT
npO,D;OJImeHHH npHl\fOH JIHHHII B ee HanpaBJIeHIIH, T. e. II nOCTYJIaT
EBKJIH,n;a, HeO,IJ;HORpaTBO ynoMHBaBmaHCH HaMil aKCHOMa EB,D;OKCa -
ApXHl\'le)J;a, BnepBbIe nOMem;eHBaH B CHCTeJ\;Ie aKCHOM I KHHrH «Ha-
lIaJI»; nOCTYJIaT 0 T01\f, 'lITO KOHelIHaH BeJIHqHHa He MomeT 6blTL pa3,IJ;e-
JIeHa Ha 6eCKOHetJHO MHoro paBH:bIX qacTeii, II, HaKOHen;, V nOCTYJIaT
EBKJIH,D;a. HeKOTopble H3 npHBeneHH:bIX aneCL «nOCTYJIaTOB» nCeB,D;O-
TYCH ,IJ;OKaabIBaeT: HanpHMep, KaK MLI BH]J;eJIH, nOCTYJIaT 0 COe)];HHH-
MOCTH ,IJ;BYX TOlIeK JIHHHeii OB ,n;OKaa:bIBaeT C nOMoID;LIO paccym,n;eHIIH
o ,l];BHmeHHII TOllKH, COBna,IJ;alOm;eH nepBOHaqaJIbHO C nepBOH TOtJKOii,
KO BTOpOH TOlIKe. HHme npHBO,IJ;HTCH nonbITKa )J;OKa3aTeJIbCTBa V no-
CTYJIaTa. OTM:eTIIJ\;I TaKme cym;ecTBeHHO OTJIHtJarom;HiicH OT CTHJIH EB-
KJIH,IJ;a H Bcex ero npemHHX 06pa60TOK CTHJIb nceB,IJ;o-TYCII, rOBopH-
m;ero 0 «lIYBCTBeHHOM YKa3aHIIH» II «cym;ecTBoBaHHH ;l];BHmym;eicH
C epLI».
,[(OKaaaTeJILCTBO V nOCTYJIaTa nceB,IJ;o-TYCH, nOMem;eHHoe B ero
H3JIOmeHIIH I KHHrH «HatJaJI» EBKJIH,n;a, 6LIJIO n3BeCTHO B 3ana,IJ;HOii
EBpone yme B XVII B., 1\0r,IJ;a ,[(m. BaJIJIHC BKJIlOtJHJI B CBOIO pa60TY
o V nOCTYJIaTe JIaTHHCKHii nepeBO,IJ; COOTBeTCTBeHHoro pa3,IJ;eJIa COlIH-
HeHHH nceB,IJ;o-TYCII, BblnOJIHeHHblii 3. IIOKOKOl\f [68, c. 664-673],
lIl\felOTCH TaKme «ppaHIJ;Y3cKHU nepeBO,IJ; [41] H PYCCl\lIe nepeBO,D;bI 3Toro
pa3,n;eJI3 [13, c. 119-121; 16, c. 22-32]. ,[{oKa3aTeJIbCTBO nCeB,IJ;O-
TYCH COCTOIIT 113 Tpex «npe)J;IIOCbIJIOK» II Tpex npe,IJ;JIOmeHlIii:
«TIpe)J;noCblJIl\a 1. ECJIU Ha BCHKHe ,IJ;Be npHMble JIHHHH, pacnoJIo-
meHHble B nJIOCKOCTH, l\aK JIHHHII AB H CD, na,IJ;alOT npHl\fble JIHHHH,
l\aK JIIIHHH EG, HF, KL, MN H XO, Kam,n;aH H3 KOTOpbIX nepneH,IJ;H-
I\YJIHpH3 K JIHHHH CD H nepeceKaeT JIHHHIO AB no)]; OCTpDIl\f H TynbIM
yrJlaMH, npnlIeM: Bce yrJIbI B CTOPOHY BD - OCTpble, a B CTOPOHY
AC - Tyrrble (pHC. 62), TO H YTBepm,IJ;alO, lITO JIHHHH AB 11 CD no-
CTOHHHO rrp:n6JIHmaIOTCH ,D;pyr K ,IJ;pyry B CTOPOHY BD AO Tex nop, nOKa
OHM He rrepeceKYTCH, H nOCTOHHHO y,n;aJIHIOTCJI )J;pyr OT ,D;pyra B CTO-
POHY AC, T. e. nepneH,IJ;HKYJIHp EG 60JIbme rrepneH)J;HKYJIHpa HF,
TOT - [60JIbme] rrepneH)J;HKYJIHpa KL, TOT [60JIbme] neprreH)];HKY-
JIHpa MN, 3 TOT - [60JIbme] nepneH,IJ;HKYJlHpa XO, a nepneH,D;IIKYJIHp
X 0 l\feHbme neprreH,D;HKYJIJIpa M N, TOT [MeHbme] nepneH,IJ;IIKYJIHpa
KL [II T. ,n;.] )J;O nOCJIe,IJ;Hero. TOqHO TaK me, eCJIH Kam,IJ;aH 113 npHl\fbIX
JlIIHHH, na)J;alOm;1IX Ha ,D;Be npHMbIe JlHHHH nepneH)J;HKYJIHpHO K O,IJ;Hoii
113 HHX, YBeJIlItJHBaeTCH, eCJIII Mid 6epeM HX B O,D;HY H3 )J;BYX CTOpOH
,D;BYX JIIIH1I:U, 11 YMeHbmaeTCH, eCJIH MDI 6epeM HX B ,IJ;pyrylO H3 ,D;BYX
CTOpOH 3TIIX ,IJ;BYX JIIIHHH, T. e. ,D;Be npHMbIe JIHHIIH nOCTOHHHO Y,IJ;aJIHIOT-
CH B CTOPOHY YBeJIHtJeHHH IIeprreH,IJ;HKYJIHpOB II npH6JIHmalOTCH B ,IJ;PY-
99
E A
JJ
c
B
JJ 0
N l F &
PUC. 62
c
B
A
PUC. 63
rYIO CTOPOHY, T. e. B CTOPOHY Yl\leHbmeHHH nepneHAHRYJIHpOB AO
Tex nop, nORa 3TH ,n;Be JIHHHH He nepeceRYTCH, TO Ram,n;aH H3 npHl\lhIX
JIHHIIH, nepnen]J;HKYJIHpHLIX K O,n;HOH U3 3TUX ,n;BYX npHMJ?IX, nepece-
qeT 3TY npHl\lYro no,n; npHMhIM yrJIOM, II 3Ta JIUHUH He 6y,n;eT HaKJIOHeHf.\
HH R 0,IJ;HOM:YII3 nepneH]J;UKYJIHpOB, HO Kam,n;hIH H33THX nepneH,IJ;HKYJIH-
pOB nepeCeqeT BTOPYro U3 3THX ,II;BYX npHl\lbIX no,IJ; ,IJ;BYl\IH yrJI3l\11I,
O,n;HH 113 KOTOphIX OCTphIii, a APyroH Tynoii, npUqeM Bce OCTpbIe
yrJIbI - B CTOPOHY npll6JIumeHUH ,n;BYX npHMbIx, a Bce TynbIe yrJIbI -
B CTOPOHY HX y,n;aJIeHHH, 11 3Ta JIHHUH HaRJIOHeHa R RarR,n;OMY 113 nep-
neH HKYJIHpOB B CTOPOHY npH6JIHmeHHH 11 HaKJIOHeH3 OT Kam,n;OrO
H3 HHX B CTOPOHY y,n;aJIeHIIH. 3TH ,n;B3 YTBepil\,n;eHHH OqeBH,n;HhI, Hl\lII
060HMII B CHHY UX OqeBH,n;HOCTH nOJIb30BaJIIICb HeKOTOpbIe ,n;peBHHe
II n03]J;HeiimHe reOl\leTphJ» [48, c. 28-29].
TI pe,n;nOCLIJIKa 1 nceB]J;O- TYCH HrpaeT pOJIb aKCHOl\ILI. Car.lo YTBepH\-
.,neHHe 3TOii npe,u;nOCLIJIKH He 3aBHCHT OT V nOCTYJIaTa, O,IJ;HaKO <!>aKTII-
qeCKII TaM, r,n;e aBTOp CChIJIaeTCH Ha CBOIO npe,n;nOCLIJIKY 1, OH Hl\leeT
B BII,n;y He 3TO YTBepa\,n;eHue, a HeKOTopoe YTBepm,n;eHHe, 6JIH3KOe K aK-
CIIOl\le «1I3JIOmeHHH KHHrH HaqaJD) Hacup a,n;- UHa aT-TYCII. CJIe,n;YIO-
II lIe .D;Be npe,n;nOChIJIKH HBJIHIOTCH TeopeMaMU:
«TI pe,n;nOCLIJIK8 2. ECJIII ,n;Be npHMhIe JIHHUH, BOCCTaBJIeHHLIe
B KOH ax npHM:oii JIUHHII nepneH,n;IIKYJIHpHO K Heft, paBHbI H IIX KOH-
IJ;bI coe)J;HHeHLI npHM:oii JIIIHHeii, TO Kam,n;hIH H3 yrJIOB, 06pa30BaHHhIX
neprreH,n;HKYJIHpaMH H npHMOH JIuHueii, coe,n;HHHIo eii HX ROH bI,-
npHl\loii. I1YCTL npHMaH JIHHHH - AB, nepneH,n;IIKYJIHphI AC II BD
II IIX KOH hI - TOqKH C II D coe,n;uHeHhI npHl\lOii JIHHlIeii (pHC. 63).
Tor,n;a H YTBepm,n;aro, 'lITO Kam,n;hIii H3 yrJIOB ACD H BDC - rrpHl\loH.
oKa3aTeJIbCTBO 3Toro. ECJIH yrOJI ACD - He npHMOH, TO OH
H.JIII OCTphIii HJIH Tynoii. ECJIH OH OCTphIii, TO JIIIHUD AB II CD IIOCTOHH-
HO nplI6JIUiHaIOTcH B CTOPOHY D, n03TOl\IY nepneH,n;IIKYJIHp A C 60JIb-
me nepneH,IJ;HKYJIHpa BD B CHJIY npe,n;nOCbIJIKH 1. lIo OHH paBHhI,
a 3TO HeJIenO» [48, c. 29].
TaRHl\1 06pa30M, CChIJIaHCb Ha npe,n;noCbIJIRY 1, nceB,n;o-TYCH Ul\leeT
BII,n;y nOCTOHHHoe npU6JIllmeHHe npHMhIX AB II CD ,n;pyr R APyry.
)]:aJIee OH rOBopHT 0 nepneHAIIKYJIlIpaX AC H BD:
«EcJIII me yrOJI Tynoii, TO OHH YAaJIHIOTCH B CTOPOHY D, n03TOl\I'Y"
neprreH)J;HRYJIHp AC l\leHbme nepneH,n;URYJIHpa BD B CUJIY npe,n;no-
94
CbIJIKII 1, HO OHH paBHbI, a 3TO HeJIeno. IIoaToMY yroJl ACD - npJl-
J\;Ioii:. TOqHO TaK me }J;OKaa:bIBaeTCH II lITO yroJl ADC - npJlMoii.
.R YTBepiR)J;alO TaKme, 'lITO JIIIHIIJI CD paBHa JIIIHIIH AB.
oKaaaTeJIbCTBO 3Toro. ECJIII CD He paBHa AB, TO OHa :MeHbme IIJIH
60JIbme ee. ECJIH OHa M:eHLme, TO Heo6xo}J;IIMO, 1J:To6bI JIHHHH AC
II ED nOCTOHHHO nplI6JIIImaJIIICL 6:bI B CTOPOHY C II nOCTOHHHO y,n;a-
JIHJIIICL 6bI B CTOPOHY B. Tor,n;a yrOJI ABD HJIH BAC 6bIJI 6:bI OCTpbIl\I,
a yrOJI CDB IIJIII yrOJI ACD 6bIJI 6bI Tyn IM B CHJIY BToporo YTBepm-
,n;eHIIH npe)J;nOCbIJIKII 1, HO OHII npHMbIe, a aTO HeJIeno. ECJIH me CD
60JILIIle AB, TO JIIIHIIII AC II BD nOCTOHHHO nplI6JIllmaIOTCJI B CTOpO-
HY B II y,n;aJIHIOTCH B CTOPOHY C. Ho Tor)J;a yrOJI CDB llJIH yrOJI A CD" -
OCTpbIH, a yrOJI ABD IIJIII BAC - Tynoii B CHJIY BToporo YTBepm,n;e-
HHH npe,D;nOCbIJIKH 1, HO OHII npJlMbIe, a aTO HeJIenO) [48, c. 29].
3)J;ecL TaKme nCeB,D;O- TYCH CC:bIJIaeTCH Ha CBOIO npe,n;nOC:bIJIKY 1
B YHaaaHHOM: Hal\fH Cl\f:bICJIe. IITaK, B npennOCbIJIKe 2 nceBno-TYCH
}J;OKa3LIBaeTCJI cy ecTBoBaHHe npJlMoyrOJILHIJKa. 3}J;ecL cPaKTII1J:eCKH
paCCl\laTpUBaeTCJI 1J:eTLIpexyrOJILHUK 1I6H aJI-XaiiCaMa II TPII runOTe-
3:bI 0 ero yrJIe D, npH1J:eM rHnOTe3LI OCTporo H Tynoro yrJIa onpoBep-
raIOTCH.C nOMO LIO «nepBoH npe)J;nOCLIJIKH), nOHHMaeMOH B YKa3aH-
HOl\1 CM:bICJIe. MomHo, O)J;HaKO, paCCMaTplIBaTL qeTLIpexyrOJILHHK
npe)J;nOCLIJIKII 2 11 KaK qeTbIpexyrOJILHHK XaiiJll\fa, KaK 3TO, nO-BII-
,D;IIMOMY, AeJIaJI CaKKepu.
II pe)J;nOCbIJIK3 3 nCeB,D;O- TYCH IIrpaeT OC06yIO POJIL. OHa 6y,n;eT
npHl\feHeHa HM TOJILKO B Cal\fOM KOHr(e ero }J;OKa3aTeJILCTBa V nOCTYJIa-
Ta, H 6e3 Hee JIerKO o6oiiTHCL; B. <1>. RaraHY, He pacnOJIaraBmeM:y
nOJIHLIM TeKCTOM )J;OKa3aTeJILCTBa nceB,n;o-TYCH, 6LIJIO HeJlCHO, aaqel\1
HymHa aTa npe)J;nOCbIJIKa [13, c. 123]. Ha3Ha1J:eHlIe me ee B CaMOl\-1
)J;eJIe COCTOHT B TOM, 1J:TO B Hei BnepBLIe H3 aKTa cy ecTBoBaHHH
npHMoyrOJILHIIK8 B:bIBe)J;eHa TeopeMa 0 paBeHCTBe CYMMbI yrJIOB npH-
MOJIHHeHHOro TpeyrOJILHHKa ,D;BYM npJlMLIM yrJIaM. o6aBIIM, 'lITO B
OPMYJIlIpOBKe aTOll npe)J;nOCMJIKH no,n;qepKIIBaeTCH, 'lITO 3)J;eCL IIMe-
IOTCH B BII,n;y npJll\IOJIHHeiHMe TpeyrOJILHIIKII,- aBTopy, HeCOMHeHHO,
6:bIJIO 113BeCTHO, 'lITO B C epH1J:eCKIIX TpeyrOJILHIIKaX cYl\Il\la yrJIOB
60JILme )J;BYX npHMbIx.
«TIpe)J;nOC:bIJIKa 3. B Kam)J;OM TpeyrOJILHIIKe C npHMOJIIIHe:HHbIl\IH
CTopOHaMII Tpll yrJIa [BMecTe COCTaBJIHIOT] ,D;Ba npJll\fLIX' [yrJIa]. I1YCTb
yrOJI ABC TpeyrOJILHHKa ABC - npHMoii. Tor)J;a H YTBepm,n;alO,
'lITO [yrJIM] BAC H BCA paBH:bI ABYM npJlMMM (pIIC. 64»)) [48, c. 29].
BHaqaJIe npe,n;nOCbIJIKa )J;OKa3bIBaeTCH }J;JIJI npJll\foyrOJILHOrO Tpe-
yrOJI LHIIKa:
«,[(oKa3aTeJILCTBO 3Toro. BOCCTaBIIM B T01J:Ke C nepneH,n;IIKYJIHp CD
K cTopoHe BC no npeJJ;JI omeHlI1O 11, OTJIOmlIl\1 Ha CD [JIHHHIO], paBHYIO
AB no npe)J;JIOme 1I1O 3, II coe)J;IIHHM T01J:KII A II D npHl\fOii JIIIHlIeii.
Tor)J;a JIIIHHH AD paBHa JIIIHIIH BC, 8 yroJl ADC - npHMOH B CIIJIY
iIpe,n;noCMJIKII 2. Ho CTOpOHbI AB II BC II yrOJI ABC TpeyrOJILHIIKa
ABC COOTBeTCTBeHHO paBHLI CTOpOH8M AD 11 DC D yrJIY ADC.
TI03TOl\IY B CHJIY npe)J;JIOa-\eHIIH 4 yrOJI ACD paBeH yrJIY BAC, a npH-
AIOi yrOJI BCD paBeH yrJIaM BCA II DCA. I103TOMY yrJI:bI BAC II BCA
[Bl\IecTe] p aBH:bI npHl\10MY [yrJIY]. Tel\1 Ca fbIl\1 YTBepm,n;eHlIe YCTaHOB-
95
JIeHO. 3TO II eCTL TO, qTO MLI XOTeJIH ,n;ORa3aTb») [48, c. 29-30]. 3,n;ecb
H ,n;aJIee, 3a I:ICKJIlOqeHlIel\f orOBopeHHoro CJIyqaH, no,n; npe,n;JIOmeHHH-
lUll Hl\feIOTCH B BH,n;y npe,n;JIOmeHHH I KHlIrH «HaqaJI» EBRJIII,n;a.
3aTeM npe,n;noc nRa ,n;ORa3MBaeTCH ,n;JIH TynoyrOJIbHOrO TpeyrOJIb-
HHKa: « aJIee, nYCTL [B TpeyrOJIbHIIRe HMeeTcH] TynoH yrOJI. TorAa
II YTBepm,n;alO, qTO TPH yrJIa TpeyrOJIblJlIKa ABC [BI\leCTe] paBHLI ,n;BYl\1
lip Hl\lLIM yr JI aM.
AORa3aTeJIbCTBO 3Toro. llYCTb yrOJI ABC - TynoH (pIIC. 65).
Ba yrna BCHRoro TpeyrOJIbHHRa B CHJIY npe,n;JIOmeHHH 17 l\feHbIIle
,n;BYX npHl\fLIX. llo3Tol\fY yrOJI ACB - OCTphli. ECJIII npHMaH JIHHIIH
na,n;aeT [Ha ,n;pyrylO], TO ,n;Ba 06pa3YIO HeCH nplI 3TO f yrJIa B CHJIY
npe,n;JIOmeHIIH 13 paBHbI ,n;BYl\f npHMbIM II yrOJI ACB - OCTphI:U:,
a Cl\leiKHhIH C HHl\f yrOJI - Tyno:u:. ECJIH onYCTHTb H3 TOqKH A neprreH-
,n;IIRYJIHp AD Ha CTOPOHY BC no npe,n;JIOmeHIIIO 12, TO HeB03MOil\HO,
lIT06LI OH ynan Ha O,l];HY 113 TOqeR B II C, TaR KaK yrOJI ABC IIJIH yrOJI
ACB He MomeT 6 TL npHMhIM; OH He MomeT ynacTb II Mem,n;y TOqKaMH
B II C IInH Ha CTOPOHY BC nOCJIe ee npo,n;OJImeHHH B CTOPOHY C, TaR
RaK ,n;BYl\fH yrJIaMH TpeyrOJIbHHRa He MorYT 6bITL yrJIhI ABD H ADF
IIJIH ABa yrJIa, O,n;IIH 113 KOTOpLIX ACD - CMemHhIH C yrJIOl\f AC B,
a BTOpOH yrOJI ACD - 60JILillUH npHl\fOrO, TaR KaE o6a OHII B CIIJIY
npe)J;JIOmeHHH 17 MeHbme npHMoro. llo3Tol\fY OH yna,n;eT Ha CTOPOHY
BC nOCJIe ee npo,n;OJImeHHH B CTOPOHY B. IIo3Tol\fY Kam,n;aH napa yr-
JIOB DAB, ABD H DCA, ACD paBHa npHl\IOMY [yrJIY]. TI03TOMY eCJIH
oT6poCHTL 06 HH yrOJI DAB, OCTaHeTCH yrOJI ABD, paBHhIH yrJIaM
BAC II ACD. Ho yrJILI ABD If ABC B CHJIY npe,n;JIOmeHIIH 13 [BMecTe]
paBHLI npHMOMY yrJIY. IIo3ToMY yrOJI ABC BMeCTe C yrnal\-III BAC
II ACB paBeH ,n;BYl\1 npHMhIl\f. 3TO II eCTb TO, 'lITO l\fhI XOTeJIH ,n;OKa-
3aTL» [48, c. 30].
M, HaROHen;, npe,n;nOChIJIRa ]J;OKa3bIBaeTCH ,n;JIH oCTpoyrOJIbHOrO
TpeyrOJILHHRa: «)]:anee nYCTb Kam,n;LIH yrOJI TpeyrOJIbHHKa ABC -
OCTpbIH (pHC. 66). Tor,n;a H YTBepm,n;alO, qTO yrJIbI 3Toro TpeyrOJIbHH-
I\a [BMecTe] paBHbI ,n;BYl\1 npHMLIM yrJIaM.
,n:oRa3aTenLcTBo 3Toro. OnYCTIIl\f 113 TOqRII A nepneHJJ;IIKYJIHp AD
Ha CTOPOHY BC no npe,n;JIOmeHIIIO 12. OH He yna,n;eT HH Ha OAHY II3 TO-
tIeK B II C, TaK RaK Tor,n;a npHl\fOH yrOJI 6hIJI 6hl OCTpbIM, II He Ha BC
nOCJIe ee npo,n;OJImeHHH B O,IJ;HY H3 ,IJ;BYX CTOpOH, TaR RaR Tor,n;a ,IJ;Ba
yrJIa TpeyrOJILHIIKa 6LIJIH 6bI 60JIbme ,IJ;BYX npHMbIx, a 3TO yrJIbI ABD
II ADB IIJIH yrJILI ACD II ADC, KOTopbIe B CIIJIY npe,n;JIOrReHHH 17
MeHbme ,n;BYX npHMMX. IIo3Tol\fY OH yna,n;eT 1\fem,n;y TOqRaMII B II C.
I103TOl\fY yrJIbI ABD H BAD [BMecTe] paBHhI npHMOl\IY yrJIY, H yrJIhI
ACD H CAD TaKiI\e [BMecTe] paBHhI npHMOl\fY yrJIY B CHJIY nepBoro
npe,n;JIOmeHUH 3TOH npe,n;noCbIJIKH. llo3Tol\fY Bce yrJIbI TpeyrOJIbHIIRa
ABC [oMecTe] paBHbI )];BYM npHl\fbIl\1. 8TO II eCTb TO, qTO l\lbI XOTeJlH
]];ORa3aTb» [48, c. 30). )]:aJIee nceB)];o-TyclI ,n;OKa3bIBaeT V nOCTYJIaT,
onHTb,KaR II ero npe,n;mecTBeHHIIKH, CHaqana )];JIH CJIyqaH nepneH,n;H-
KYJIHpa H HaRJlOHHOH, a 3aTe1\{ ,n;JIH ,n;BYX )];pyrHx CJIyqaeB.
«IIoCKOJIbKY 3TH npe,n;nOCbIJIKH YCTaHOBJIeHbI, l\lbI rOBOpHl\f: nYCTL
,D;Be npHMbIe JIHHHII, Ha KOTopbIe na,n;aeT npHMaH JIIIHHH,- JIHHHH
AB H CD, a JIHHHH, na)];alOm;aH Ha HHX, - JIHHHH EG, nepeceRalOm;aH
96
D
A
c
8
Puc.64
B
A
6
B
JJ
&
c
6
A
c
B
Puc.65
JJ
A
c
]}
'1
B
A
Puc.66
B
6
c
FC
ti
8
])
A
Puc.68
Puc.67
G
c
A
z
HX B TOllnaX E II G, II rrYCTL npH aTOM 06paaYIOTCH yrJIbI BEG II DCE,
.MeHbmDe ,n;BYX npHMhIX, nplIlIeM Heo6xo,n;HMO OAHU D3 HIIX rrpHMoi,
a APyroii OCTpbIi, llJID 06a - OCTphIe, HJIH OAHH 113 HHX Tyrroii:,
a ,n;pyroH OCTpLIii:. Tor,n;a 3TH ]J;Be npHMhIe BO Bcex Tpex CJIyqaHx, eCJIH
npOAOJliRlITb HX B HX HarrpaBJIeHDHX B CTOPOHY B ,n;o 6eCKOHellHOCTH,
BCTpeTHTCH.
,gOKa3aTeJILCTBO aToro. B nepBOM CJIyqae nYCTL yrOJI BEG OCTphli,
a yrOJI DGE - npHMoi. OTMeTHl\f Ha JIHHHH BE rrpOH3BOJIbHYlO
TOqKY H II orrYCTHM H3 Hee no npeAJIOmeHHIO 12 JIHHHIO HF, rreprreH-
)!I;HRYJlSpHO JIHHHH EG. TorAa OHa HaJIOiRHTCH Ha JIHHlllO DC HJIII
4 E. A. P03CHcJ>C:Ib]l;, A. II. IOIIIMCBIPI 97
ynaAeT B TOqRY J\;IemAY TOlIKal\fH C II G, IIJIK 1\fem y TOqKal\fH E II C,
IIJIH BO BHemHIOIO no OTHomeHllIO R HHl\f TOqRY co CTOpOHbI E
(pHC. 67, a - e). qeTBepTbIu cJIyqau HeB03M:omeH, nOCROJIbRY B 3TOM
CJIyqae 6bIJIO 6bI B03l\fOiRHO, tIT06bI )J;Ba yrJIa HFE H HEF TpeyrOJIb-
HHKa H F E 6bIJIH 6bI 60JILme BYX npHMbIX, TaR RaR yrOJI Ii EF TY-
noi, HO B CHJIY npe JIOmeHHH 3 3TO HeJIeno» [48, c. 30-31]. IIo,n;
«npe,n;JIOmeHHel\f 3» 3JJ;eCb nOHHMaeTCH npe]];JIOmeHlle 3 nCeB,IJ;O- TyclI.
«,IJ:aJIee, eCJIII JIHHHH CD npO,l];OJImeHa B ee HanpaBJIeHIIH, OHa BCTpe-
THTCH C JIHHlleH AB: B nepBOM CJIyqae 3TO OqeBII,n;HO. Bo BTOpOM CJIY-
qae HeB03MOiKHO, lIT06bI JIHHHH CD BCTpeTIIJIaCb C nepneH,l];HI<YJIHpOl\f
H F, TaK RaR eCJIH 6bI 3TO 6bIJIO TaR, TO B rro 1IyqlIBmeMCH TpeyrOJIb-
HilKe [DCF] ABa yrJIa DCF II DFC 6bIJIH 6bI paBHbI ]J;BYM npHl\IbIl\f,
HO B CHJIY rrpe]J;JIOmeHIIH 17 OHH l\feHbme HX, a 3TO HeJIeno. OHa He
l\fOmeT BCTpeTIITbCH C JIHHHeH EC, TaR KaR ABe rrpHl\fbIe JIHHHH He 1\10-
rYT OrpaHHqHBaTb nJIOCRYIO <plIrypy, n03TOMY OHa BCTpeTIIT JIIIHIIIO
AB. B TpeTbel\f CJIyqae B03bl\feM EF KpaTHoii CTOJIbRO. pa3, 1J:To6bI OHa
CTaJIa 60JIbme JIIIBHD EC, 3TO JIHBHH EF, FK, KL, LM, II OTJIOmlll\f
no npe JIOmeHHIO 3 KHHrH I «HaqaJI» EBKJI1I]J;a Ha JIHHIIH BH JIIIHHH,
Kam,l];aH 113 KOTOpbIX paBBa" JIHHHH EH, 3TO .T.IHHIIH H N, N X, XO,
1J:HCJIO ROTOpbIX Bl\feCTe C JIHHHei EH paBHO qlICJIY e:IeHHH JIIIHHII
EM (pHC. 68). BOCCTaBH 1 no npeAJIOmeHHIO 11 B TOqRe E nepneH II-
KYJIHp EP II OTJIOiKHl\f Ha Hel\f no npe,l];JIOmeHHIO 3 PE, paBHYIO HF.
Coe HHIIM TO"tJKH P H H npHl\fOH JIHHHeH. Tor,n;a KaiK blii H3 yrJIOB
EPH 11 PHF npHMoH II CTopOHa PH paBHa CTopOHe EF B CHJIY rrpe,l];-
nOCbIJIRII 2. OnycTHl\f no npe;n:JIOmeBHIO 12 113 TOqRII N rreprreH,l];HRY-
JIHp N K Ha EG. TaK RaR JIHHHH EB II EG nOCTOHHHO Y,l];aJIHIOTCH B
CTOPOHY B, nepneH IIRYJIHp N K 60JIbllIe nepneH IIRYJIHpa HF BBH,l];Y
rrpe,l];rrOCbIJIRH 1. TI03TOMY OTJIOllilll\l Ha HeM no npe,n;JIOi-I\eHHIO 3 JIII-
HI-1I0 KQ, paBHYIO nepneH,n;1IRYJIHPY H F, II coe,n;UHHM TOqRII H II Q
npHl\fOi JIIIHlIei. Ram]];bIH 113 yrJIOB F HQ 11 KQH - rrpHMoii, CTOpO-
Ha FK paBHa CTopOHe HQ B CHJIY npe,n;nOCbIJIRII 2, a yrOJI FHP -
npHMoi. TI03TOl\IY JIIIHIIH PH - B HanpaBJIeHUn JIHHIIH HQ, TaR 'tITO
B CIIJIY npe,n;JIOmeHHH 14 JIIIHHH PQ - o,n;Ha JIHHIIH. TI03TOl\fY yrOJI
HQ/( - npHIHoii. YroJI HQN - npHMoH B CHJIY npeLJ:JIOmeHHH 13,
a yrJIbI PHE H QHN - BepTHRaJIbHbIe, paBHbIe B CIIJIY npe,n;JIome-
HIIH 15. CTopoHa EH TpeyrOJIbHHR8 HEP paBHa CTopOHe HN Tpe-
yrOJI:bHIII\a HQN. TI031'Ol\IY B CIIJIY npe,n;JIOil eHnH 26 CTopOHa PH
paBlIa CTopOHe HQ. Ho cTopoHa F K paBHa cTopoHe HQ.. II03TO IY
cTopoHa F K paBHa cTopoHe PH. Ho cTopoHa EF paBHa CTopOHe PH,
n03TOl\fY CTopOHa EF p8BHa CTopOHe F K II neprreH,n;IIRYJIHp N K na-
,n;aeT B TO"tJKY K JIHHIIH EG. OnycTIIM no npe]J;JIOmeHIIIO 12 H3 TOqfiH X
nepneH HRYJIHp XL Ha CTOPOHY EG, OTJIOmHl\f IT.o npe,n;JIOmeHIIIO 3
JIHHHIO ZL, paBHYIO JIHHHH N K, TaR KaK JIIIHIIH XL 60JI:bme JIHHHH
N K BBH,n;y npen;rrOCbIJIKII 1, H coe,n;HHHl\f TOllRII N H Z rrpHMoH JIIIHHeH.
Tor,n;a Ham,n;bIi 1I3 yrJIOB KNZ II LZN - npH Ioii, a CTopOH.a KL
paBH8 CTopOHe NZ B CHJIY npe;J;noChIJIKH 2. TIPO,n;OJIiI,IIM nepneH,L];H-
J\YJIHp HF B CTOPOHY H B ero HanpaBJIeHHH AO 6eCKOHelIHOCTH, OT-
.iOiI\HM na Hel\f no rrpe]:(JlOmeHHIO 3 F R, paBHYIO N K, Ii coe,n;HHIIM TOq-
I{H R H N rrpHMoi JIHHHeH. Tor,n;a KaiK,D;LIH 113 yrJloB F RN H K N R -
98
B
c
8
&
Puc.69
PUC. 70
npHMoi If CTopOHa F K paBHa CTopOHe RN B ClIJIY npe,n;nOCbIJIKH 2.
IIo9ToMY, TaR Kal\ yrOJI LZN - npHMoii, yrOJI NZX - rrpHMOR II
CHJIY npe,n;.nOmeHHH 13. IIo9ToMY Ram,n;bIii H3 yrJIOB H RN H NZX -
npH1\fOii, yrJlhI HNR H ZNX paBHbI B CHJIY npeJJ;JIOmeHHH 15, a CTO-
pOHbI NH H ...VX paBHhJ. l109TOMY B CHJIY npe,n;JIOmeHlIH 26 CTopOHa
NZ Tpeyro.1IbHlIKa NZX paBHa CTopOHe NR TpeyrOJIbHHKa NRZ.
II09TOMY FK paBHa NZ. Ho KL paBHa NZ, n03TOMY FK paBHa KL.
l109TOMY nepneH,n;HKYJIHp XL na,n;aeT B TOllKY L JIHHHH EG. IIpo,n;OJI-
mUM KN B CTOPOHY N B ee HanpaBJIeHHH ,n;o 6eCKOHellHOCTH, OTJlOmHM
Ha Hei no npe,n;JIOmeHHIO 3 KS, paBHyro LX, H coe)J;HHHM TOqKH S
H X npHMoH JIlIBHeR. Tor,n;a Kam,n;bIB HS yrJIOB KS X H LX S - IIPH-
MOR, CTopOHa KL paBHa CTopOHe SX B CHJIY npe,n;nOCbIJIKH 2. Onyc-
THM 1I3 TO"1J H 0 no npe)J;JIOmeHHIO 12 nepneH,n;HKYJIHp OM Ha JIHHHIO
GE. TaR RaR OM B CHJIY npe,n;nOCbIJIKH 1 60JIbme LM, OTJIO}KHM
Ha HeM no npe,n;JIOmeHHIO 3 T M, paBHYro CTopOHe LX, coe,n;HHHM X
II T npHMoH .1:HHHei. Tor,n;a Kam,n;bIi H3 yrJIOB LXT H MTX - npll-
)IOB. n09TOl\fY B CIIJIY npe,n;JIOmeHHH 14 JIHHHH ST - npHMaH JIHHHH,
CTopOHa L1t1 paBHa CTopOHe XT B CHJIY npe,n;nOCbIJIKH 2 H yrOJI
fTX - npHMoi. l103TOMY yrOJI XTO npHMoii, yrJIbI SXN H OXT
paBHhl B CII. y npe,n;JIOmeHHH 15, H CTOpOBhI NX H XO paBHbL IIo3To-
AfY B CHJIY npe.n;JIOmeHHH 26 CTopOBa XT paBHa CTopOHe XS. IIo3To-
IY CTOpOH8 KL paBHa CTopOHe LM B CHJIY nO,l(06HOrO TOl\IY, lITO 6bIJIO
paHbme. TIo3TOMY nepneB,n;HKYJIHp OM rra,n;aeT B TOllKY M 'npHMoii
EC. H03TOMY npHMaH DC 3aRJI l( qeHa Mem,n;y nepneH)J;HKYJIHpal\fH
XL If OM. l109TO IY, eCJIO MLI npOjJ;OJImHM ee B CTOPOHY D B ee Ha-
npaBJIeHHIf.. HeB03 IO;RHO, lIT06L1 OBa BCTpeTHJIaCb C OJ];HHM H3 nep-
rreH.n;HRYJIHpOB XL H OM, H60 eCJlH 9TO He TaK, TO nYCTb 3TO npo..
H30H.n;eT B TOqKe D, Tor,n;a B TpeyrOJIbHHKe DC M HJIH DC L 6y,n;YT ,n;Ba
npHMbIx yr.lIa - yrJIbI DLC H DCL HJIH DCM H DMC, HO Kan,.n;bIe
.n;Ba yrJIa TpeyrOJIbHHl\a B CHJIY npe,l(JIOllieHH-H 17 MeBbme ,n;BYX npH-
l\.JbIX, a 3TO HeJIeno. l109TOMY JIHHHH CD BCTpeTHTCH C JIHHlleii AB»
[48, c. 31-33].
99
4*
B npHBeAeHHOM 1(OKa3aTeJILCTBe nCeBAO-TYCH, RaR H y ero npeA-
meCTBeHBHKOB, HCnOJIL3YIOTCH aRCHOMhI nama II EB)];ORCa - APXH-
Me)];a.
OAHaRO B OTJIHlIHe OT 1I6H aJI-XaUCaMa H aT-TYCH nceBAo-TYCH,
cRpynYJIe3HO OTMellaH Bce npHMeHHeMhIe HM npe)];JIOmeHHH «HallaJI»
EBRJIH)];a, He cpOPMYJIHpyeT 8TII aRCHOMhI HBBO. ,I(aJIee, nceB)];o-TYCH
AaeT BeCLMa opHrHHaJILHOe H H3H Hoe AORa3aTeJI:bCTBO V nOCTYJIaTa
B BYX ,l(pyrHx CJIyqallx:
«qTO RaCaeTClI 2-ro CJIyqall, RorAa Ram,l(bIH H3 yrJIOB BEC H DCE -
OCTpbIH (pHC. 69), TO, TaR RaK yrOJI DCE - OCTpbIH, yrOJI DCG -
TynoH B CHJIY npe)];JIOmeBHlI 13. BoccTaBHM B TOllRe C no npe)];JIome-
HHIO 11 rrepneH)];HRYJIHp HC R JIHHHH EG B CTOPOHY D, OH nOna)];eT
Mem)];y CTopOHaMH DC H CG. TI08TOMY, eCJIH MbI npO)];OJImHM ero
B CTOPOHY H B ero HanpaBJIeHHH, OB BCTpeTHT JIHHHIO AB B CHJIY
npe)];bI)];ymero npeAJIOmeHHH, OH BCTpeTHT ee B T01:J:Ke H. II08ToMY,
eCJIH MbI npO)];OJImHM JIHHHIO DC B ee HanpaBJIeHHH B CTOPOHY D,
OBa BCTpeTHT AB Mem)];y TOllRaMH E H H. 3TO 01leBH)];HO BBH)];y He-
B03MomHOCTH Toro, lIT06hI ABe npllMbIe OrpaHHlIHBaJIH nJIOcKYIO
cpHrypy.
qTO RacaeTCH 3-ro CJIY1:J:aH, ROr)];a yroJI BEC - OCTpbIH, a yroJI
DCE - TynoH (pHC. 70), TO, TaK RaR yrJIbI BEC H DCE MeHLme )];BYX
npHMbIX, a yrOJI DCE BMeCTe CO CMemHbIM K HeMY B CHJIY npeJ];JIome-
HHH 13 paBeH )];BYM npHMbIM, yrOJI C, CMelliHbIH C yrJIOM DCE, 60JIL-
me yrJIa BEC. OTMeTHl\1 Ha JIHHHH EC npOH3BOJILHYlO TOllKY H
II onYCTHM H3 flee no npe)];JIOmeHHIO 12 nepneH)];HRYJIHp HF Ha JIH-
HIIIO EB. OqeBH)];HO, lITO OH He yna)];eT B TOllRY E. OH He yna)];eT H Ha
JlIIHHIO AE, Tal\ RaK eCJIH 6bI 8TO 6bIJIO TaK, TO ABa yrJIa TpeyrOJIbHHKa
6bIJIH 6bI 60JILme )];BYX npHMbIX, a B npe)];JIOmeHHH 17 6bIJIO AORa3aHO,
1:J:TO OHH MeHLme ,D;BYX npHMbIX, lITO HeJIeno. IIycTL OB yna)];eT B TOllKY
F 11 pO)];OJImHl\I JIIIHHIO F H B ee HanpaBJIeHHH B CTOPOHY H ,1];0 K.
TaR KaK npHMoH yroJI HFE BMeCTe C yrJIOM EHF B CHJIY rrpe)];JIome-
HHH 17 MeHLme )];BYX npHMbIx, a OCTpbIH yrOJI EH F B CHJIY npeJJ;Jlo-
meHHH 15 paBeH yrJIY CHK, yrOJI C, CMemHbIH C yrJIOl\{ DCE, l\-IeHb-
me npHl\fOro. IIo8ToMY l\am)];bIH H3 ABYX yrJIOB CH K II C, CMemHbIH
C yrJIOM DCE,- OCTpbIH. II08ToMY JIHHHH HK H DC, eC"TIH npO)];OJI-
lliHTL I1X B CTOPOHY K, BCTpeTHTCH B CHJIY BToporo H3 JJ;BYX npe)];bI)];y-
HX npe)];JIOmeHHii. IIycTL OHH BCTpeTHTCH B T01:J:I<e K. TaR KaK yrJIbI
EHF H CHK paBBbI B CHJIY rrpe,l(JIOmeBHlI 15, a yrOJI C 60JILme yrJIa
HEF, npHMoH yrOJI EFH 60JILme yrJIa HKC, nOCROJIbRY TPH yrJIa
Ram)];OrO TpeyrOJILHHKa C npHMOJIHHeHHi>IMH CTopOHaMH paBHbI )];BYl\1
npHMbIM B CHJIY npeAnOCbIJIKH 3, n03TOMY H KC - OCTpbIH yrOJI.
Ho yrOJI BF K - npHMoii B CHJIY npe)];JIOmeBHH 13. II09ToMY, eC:IH
MbI npO)];OJImHM JIHHHH AB H CD B CTOPOHY BD, OHH BCTpeTHTCH B-
CHJIY 1-ro HS 8THX npeAJIOmeBHH B TOR me CTopoBe OT JIHBHH, "na)];aIO-
eii no)]; [yrJIaMH MeHLmHMH] ,II;BYX npllMbIx. 3TO II eCTb TO, MbI
XOTeJIH )];OKa3aTL» [48, c. 33].
HeKoppeRTHall 4>oPMYJIHpOBRa npeAnOCbIJIRH 1 9Toro AOKaS8TeJIb-
CTBa AeJIaeT e e MeHee BepOllTHLIM ero npHHa)];JIeiRHOCTb C8MOMY
8T-TYCII, KaK H Bcero opUrHHaJIa pHMCRoro H3AaHHH. Bnpo'tJ:eM, KaK
100
rOBOpHJIOCb, MOmHO ]J;YMaTb, 1J:TO HaH60JIee Opll:rIlHaJIbHhIe 1J:aCTU 3TO....
ro ]J;OKa3aTeJIbCTBa - npe]J;nOCLIJIKa 3 II )J;OKa3aTeJIbCTBO V nOCTYJIa-
Ta B ]J;BYX nOCJIe)J;HIIX CJIYtIaHX - 3aHMCTBOBaHLI H3 «IIpHMeqaHIIB:
K EBKJIlI)J;Y» HJIH )J;Pyroro COqIIHeHHH, npHHa)J;JIeiHamero aT- Tyc-n, KO-
Topoe 6LIJIO HanHcaHO IIM B KOHIJ;e lliH3HH.
KpoMe Tpex H3JIOmeHHLIX Hal\IU
,n;OKa3aTeJILCTB V nOCTYJIaTa, npu-
Ha)J;JIemalI\lIxaC-CaJIapy.aT- TYCH
H nceB)J;O- TYCH, B X I I lB. 6LIJI 0 Ha-
llHCaHO e e HeCROJILKO CO'tJ:HHeHU:U,
nOCBH eHHbIX IIp06"T(e ie rrapaJIJIeJIL-
HbIX JIHHnH. 1\lbl Y}l e D;HTnpOBaJIH
H3 nllCbl\la (AJIa}! a;IJ;- HHa HaHCapa
aJI-XaHa$H K HaCHp aA- IIHY
aT- TYCH no nOBO;IJ;Y ero TpaKTaTa 0 na-
paJIJIeJIbHbIX JIlIHHHX ]J;allHoe aJI-XaHa.pH H3JIOmeHIIe HaqaJIa nOllbITKU
.n;OKa3aTeJIbCTBa CHMnJII1KlIH. BeCbl\!a HHTcpecna KpI1THKa 9TOH IIO-
IIbITKlI, KOTOpyro aJI-XaHa.pH npIlBO)J;IIT nOCJIe CBoero 113JIOiUeHHH:
«O;IJ;HaRO 1\10iI\HO npe;IJ;nOJIOmHTb, lITO JIHHHH BD C caMoro HaqaJIa
OTKJIOHHeTCH OT HanpaBJIeHHH JIHUHH BG, a KaiR)J;aH Xop;IJ;a, CTHrIIBaro-
m;aH yrOJI GBD, nonaJ];aeT Mem)J;Y TOqKaMH A lIB, eCJIII AB ,a;eJIHM:a
)];0 6eCKOHeqHOCTH» [60 c. 8-9 H 19].
3,n:ecb aJl-XaHa.pH YRa3bIBaeT H1\IeHHO TOT CJIY'tJ:an, ROTOpbIH IIl\feeT
MeCTO B rHnep60JIIIQeCKOH reOl\feTpHH - Kor]J;a Bce «XOP)J;bI», CTHrH-
Barom;lIe yrOJI GBD, npOXO;IJ;HT HHiRe HeKOTopOH BHYTpeHlIeii: TOQKH
9Toro yrJIa; OH HMeeT B BH)J;Y, tITO npe]J;llOJIOmeHHe 0 HCB03MOiHHOCTU
TaKoro C YtIaH, na KOTOpOM OCHOBaHO )];OKa3aTeJIbCTBO CHMII IIKHH,
He BbITeKaeT 113 aKCHOM, Ha KOTOpbIX OCHOBbIBaeTCH ero AORa3aTeJIbCT-
BO. OTl\IeTHM, QTO aJI-XaHa.pII, nO)J;06HO aJI-KIIII;IJ;H, aJI-EHpYHH
II XaiHMY, CBH3bIBaeT np06J1el\-IY V IIOCTYJIaTa C )J;eJIlIMOCTblO reOM:eT-
pnqeCKHX BeJInqHH AO 6eCKOHetIHOCTH.
HaH60JIee IIonYJIHpHOH KaK B XIII, TaK II B IIOCJIeAYIO IIX BeRax,
6bIJIa nOIIbITKa ,n;OKa3aTe:IbCTBa aKCIIOMLI 0 napaJIJIeJIbHbIX, IIpHHa)J;-
JIemaII(aH Acnp a,n;-)J:HHY aJI-MYJ a.p aJIY H6H Ol\lapy aJI-A6xapu,
na3LIBaeMOMY TaKiKe aJl-A6axpH (YM. 1263), ypOiKeHD;Y A6xapa
B mIl6aJIe, yqeUIIKY KaMa.rr a)];- HHa H6H IOHHca, KOTOpbJM 6LIJI
Y1JIITeJIeM aT-TyclI. Pa60Ta.rr aJI-.A6xapH B Mocy.rre H Mp611JIe
(MpaK). Ero nonLITRa )J;OKa3aTeJI:bCTBa aKCHOMLI 0 napaJI.rreJIbHLIX
H3JIO)ReUa B COCTaBJIeHHOH HM 06pa60TKe «HallaJI» EBKJIHAa, coxpa-
HHBmeHCH nO)J; Ba3BaHHHMII «¥ COBepmeUCTBOBaHlle "CTHXHH"» (McJIa
aJI-MCTH aT) H «¥ COBepmeHCTBOBanne "Ha1J:aJI" EBKJIH)J;a» (McJIa;
Y YJI 'Y JIH]J;HC). OT.n;eJI:bBLIe pa3AeJIhI 8Ton RBHrH, B TOM 'tJ:HCJIe II pa3-
]J;eJI 0 napaJIJIeJI:bHLIX JIHHIIHX, BocnpOB3Be.n;eHLI B BeCLMa pacnpOCTpa-
BeHHbIX HeKor,IJ;a «Ilpe;IJ;JIomeHHHx 060cHOBaHHH» (AmK8JI aT-Ta'cHc)
illaMc a;IJ;-,l(IIHa MyxaMMa;IJ;a 116H Ampa a aJI-XyceiiHII ac-CaM:ap-
RaH,IJ;H, ypomeuD;a CaMapRan.n;a, iKHBmero BO BTOpOH nOJIOBHHe XIII B.
B t)LIBmero COTPY,IJ;HHROM MaparlIHcRoH 06cepBaTopBH RT-TYCB.
Hall60JIee 1I3BeCTHLIl\1 C-TaJI TeRCT 3TOH KHHrH C KOMMeHTa
8uaHeHHToro caMapKaH;IJ;CKOrO l\laTeMaTHKa D aCTpOHOMa XV B. Ka-
15.
T eopuu
a.,-rz - X auacjJu
u a.n-A6xalJu
101
31I-3a e ap-PYl\IU II 1\lyxal\Il\la)];a aJI-Xa )];1I, HeO}J;HOKpaTHO H3,[(aBaB-
UlIIHCH B CTa 16y.,TJ:e [25]. qacTb KHlIrlI ac-Ca fapRaH,[(II, CO,rJ;epH all\aH
nonMTKY an-A6xapll OKaaaTb aRClIOMY 0 napa neJIbHMX JIHHHHX,
6bIJIa 1I3f];aHa Xal\fIITO I HJIbraHOl\1 nO-TypeU;nII [44, c. 113-118]
H no-.ppaHu;yacKII [45] H aBTOpal\flI [22, c. 599-602]. O)];HaKO B 3THX
H3)J;aHHHX aBTopCTBO aJI-A6xaplI He YKa3ano, ero YCTaHOBHJI
A. II. Ca6pa (Cl\I. [57, c.455]).
II pHBe)];eM )];OKaaaTeJIbCTBO aJI-A6xapu:
«3}J;ecb l\leCTO )];JlH 06eIJ:(aHHoro )];OKa3aTenbCTBa lIaBeCTHoro IIOCTY-
JIaTa. <1>HJIOCO.p AClIp a)];- IIH aJI-A6xapn CKa3a.rr: eCJIH yrOJI ABC
paaJ]:eJIeH JIIIHlIeH BH IIOIIOJIaM, TO B 3TOAI yr.rre fOiRHO rrpOBeCTH 6ec-
KOHeQUO }IHOrO Xop,n; TaKHl\f 06pa30M, QTO OHII IlaXO)];HTCH J];pyr. no)];
)];pyroM H I,aiR)];aH ua HHX HBJIHeTCH OCHOBaHlIeM: paBH06e,n;peHHoro
TpeyrOJIhnHI(a (puc. 71). AeHCTBHTeJIbHO, OTnOiKlIl\f BE, paBHYIO
BG, If COe HHII\I EG. Tor)];a EB II BH paBHbI BG II BH, a yrJIhI [EBH
II HBG] paBHhI, n03TOl\IY 06a yrJIa H paBIIhI, II BH neprreH,rJ;UKy.i1Hp-
Ha I EG. OTJIOmHM BF, paBHYIO BK, II coeJ;IIHn I FK. TorJJ;a JlMHHH
F/( He npOXO,D;HT qepea TOqKY H, TaK KaK TorJ];a yrJIhI BHF H BHK
6bI:III 6h1 paBH!>I )J;BYl\I rrpHl\fhlM:, HO BHE II BI1G paIlblIIe 6hIJIH paBHbI
111\1; OHa He rrepeceRaeT EG, TaK KaK TOr)];a JJ;Be rrpHl\lbIe TIHHHH orpaHH-
'IIIBaJIH 6bI rrJIOCH:OCTb. IIoaTOMY F K IIpOXOJJ;lIT qepea TOqKY, HaXO H-
YIOCH no)]; TO'IKOH H,- qepe3 TOqKY L. ITo 3TO IY o6paa y l\IOiRHO
npOBeCTH 6eCKOHe'IHO MHoro XOP)];» [22, c. 598J. AJI-A6xapH, no,[(06-
HO CHMnJIIIKIIIO, Ha'IHHaeT C ,[(OKaaaTeJIbCTBa Toro, 'ITO MOiKHO IIpO-
BeCTO 6eCKOHe'IHO l\fHOrO npHMhlX, nepeCeKaIOIIJ;UX 06e CTOpOHhI yrJIa.
,[(aJIee OH nepeXO)];HT K )];OKaaaTeJIbCTBY V nOCTy.,TIaTa, Ha'IHHaH, KaK
PI ero npe)];meCTBeHRHKH, co CJIyqaH nepneH)];nRYJIHpa II HaRJIOHHOU:
«ECJIH 3TO YCTaHOBJIeHO, MhI rOBopHM: eCJIH nlIHIIH rra)J;aeT Ha Be
JIHHHH TaKIIM 06paao l, 'ITO 06paaYlOTcH )];Ba RHYTpeHHHe O,rJ;HOCTOPOH-
HHe yrJIa, l\feHbllIIIe ,[(BYX rrpHl\f:bIX, TO OHII BCTpeTHTCH C aTOH CTOpOHbI,
eCJIH npO)];OJIiI HTb HX. 8TH yrJIhI o6a OCTpbIe, n:n:n O)];HH Ha HHX OCT-
pbI:ii, a )];pyroi - npH1\IOH HJIH Tynon.
IIycTb o,l];nH Ha HHX OCTPLIH, a )];pyroH - npH IOH. IIarrpM Iep, Ha
JIUHIIH AC H BD na,J:aeT JIHHHH AB TaKHM oopaaOM, 'ITO 06paayeTcH
npHMOH yroJI ABD II OCTphlH yrOJI BAC (pHC. 72). I10CTpOHl\1 yrOJI
BAE, paBHhlH BA C, II IIpO)];OJIiKHM A B B ee HanpaBJIeHHH )];0 G. Tor-
,n;a yrOJI EAC ,n;eJIIITCH nOnOJIaM JIHHHeH AG. rroaTo IY B Hel\f l\IOiKHO
npOBeCTH XOp,D;bI, Haxo,n;HIJ:(lIeCH ,[(pyr rro)]; JJ;pyro I, KaK 6h1JIO [,D;OKa-
saRo] paHbme. IlpoBe,[(el\1 Xop,[(y rro)]; TO'IKOH B, nYCTb 3TO 6YJJ;eT EF:
ORa naXO)];HTCH rroJ]; TO'IKOH B. TaR KaK AG rreprreH)];HKYJIHpHa R EF,
GF He BCTpeTllTCH C BD, TaR KaK Torl];a rrOJIYQII.7ICH 6h1 TpeyrOJIbHHR
C JJ;ByMH rrpHMLIl\fll yr.ila IH, a aTO OTBepraeTCH B ClI l:Y rrpe)];JIOmeHHH
17 RHI1:rH I «lIaqaJD) II HeJIelIO TaK;Re B CII.rry ee rrpeJ.(."10meHHH 32, T. e.
[npeJ];JIOmeHHH] 20 aTOH MoeH KHHrH, 6ea Toro, 'IT06h1 rrpH ,[(oKa3a-
TeJILCTBe 3Toro 6bI "1 6bI HCnOJlbaOBaH 3TOT nOCTYJlaT; HCIIOJlbaOBaTb
ero DPH ,[(oKaaaTeJIbCTBe 6LIJIO 6bI HenpaBHJIbHO; eCJ.IH nO;KeJlaeT Bce-
BbIillHHH AJIJIax. l\lbI HaJIOiKH f aTO npeJJ;JlOifieHHe I10CJIe OI\OI{qaHHH
aTOll pe'Iu. MOiRHO J(oKaaLIBaTL OTCYTcTBHe BCTpeqH TaR }Ke, RaK 3TO
.6bI:I0 JJ;01\a3aIlO B [rrpeJJ;JIOmeHHHJ 18 aToM RHllrll, T. e. [npeJJ;JIO/RA-
102
8
A
r
&
E
PUC. 72
l
Puc.71
A
G
A
{,'
A
F
G
E
8
Puc.74
JJ
Puc.73
HHn] 28 [KHHrlI] I «HaqaJI». B 3TOM Hym,n;aIOTCH II npH BYX ,n;pyrllx
npe,n;nOJIOFI eHHHX. ll03TOl\IY, eCJIH npo,n;OJIllilITb ED B ee HanpaBJle-
HIIH, OHa nepeCeqeT JIHHIIIO AF» [22, c. 597].
B npIlBeJJ;eHHOl\I paCCYiI .neHHn, TaRiHe 6JIHaKO 1 R CHM:IIJIBKIIIO,
aJI-A6xapll, CCLIJIaHCb Ha ,n;OKaaaHHYIO Hl\1 JIel\ll\fY, aKTHqeCRH nOJIb-
ayeTcH He ee YTBepm,n;eHne:M, OTHOCHIII.nMCH R a6COJIIOTIIOH reOMeT-pHII,
a HeHBHO npe,n;nOJIaraeT, qTO qepea Kam,n;YIO T01:J:KY yrJIa lomHO npo-
BeCTII npHMYIO, nepeCeKaIOIl\YIO 06e ero CTOpOHbI, qTO 3KBBBaJIeHTHO
V nOCTYJIaTY. CJIe,n;yeT OTAleTHTb, 1:J:TO aJI-A6xaplI nOJIbayeTcH aJJ;eCL
II aKCHOMOH nama, He CPOpAIYJIHPYH ee HBHO.
,[(aJIee aJI-A6xapH nepeXo,IJ;HT K ,n;oKaaaTeJIbCTBY ,n;BYX APyrllx CJIY-
qaeB V nocTYJIaTa:
«TIYCTh 06a yrJIa oCTpble. BepHeMcH K TOM:Y me "tIepTemy C Toi
paaHHu;ei, 'tJ:TO yroJI ABD TaKme OCTpbIi (pIlC. 73). TaR KaK OH OCT-
PhIH, yroJI GBD - TynoH, a AGF - rrpHMoii. TI03TOMY JIHHHH GF
He BCTpe1:J:aeT BD, TaR RaK Tor.n;a nOJIYQHJICH 6bI TpeyrOJIbHIIK C npH-
109
MbIM n. TynblM yrJIaMII, a 3TO HeJIenO B CIIJIY Toro me npe,IJ;JIOmeHIIH.
IIo3ToMY BD, eCJIII ee npOJ];OJImIITL, nepeCeqeT AC.
n Y3Tb O,IJ;HH lIa yrJIOB OCTpbIi, a ));pyroH - Tynoi. HailplIl\1ep,
Ha .iJMHMM AB II CD na));aeT JIIIHIIH EG TaKIIM 06pa30M, 'tJ:TO 06paaYIOT-
CR yrJIbI BEG H DGE, MeHhilllle J];BYX npHMLIX, lIa ROTOpDIX yroJl
DGE Tynon, a YI:OJI BEG - OCTpbIH (pIlC. 74). Paa));eJIHM JIHHHIO
EG IIOnOJIaM B TOqRe H. IIpoBe));eM Ha TOqKII H JIHHHIO HF, nepneH-
nMHYJIHpHYIO K CD, II npO));OJIiKHM ee B ee HanpaBJIeBHH [,n;o ML
Tor,n;a yrOJI HFG - npHMou, a FHG - OCTpbIi. DoaToMY [yroJI]
EHM - OCTpbIi If BEH - OCTpbIii. II03ToMY JIMBHII AE II HM
BCTpeqaIOTCH, IIyCTh IIX BCTpeqa - B To"tJ:Ke K. Tor));a yrOJI EKH -
TynoH, TaR KaK IIHaqe OB 6bIJI 6bI IIpHMbIM IIJIH OCTpbIM. ECJIII 6bI OH
6bIJJ IIpR 1bIM, TO, TaR KaR yrJIbI EKH II EHK paBHbI yrJIaM HFG
II FHG, a Ell paBHa HG, yrOJI KEH 6bIJI 6bI paBeH HGF. Tor));a,
npM6aBHB 06 Hii yrOJI DGE, MLI nOJlyqMJIH 6bI, qTO J];Ba [npHMhlx]
yrJIa paBHbI yrJIaM DGE II KEH, HO aTII ,IJ;Ba yrJIa MeHhille ,IJ;BYX npH-
MbIX, a aTO BeJIeno. ECJIII me "3TOT yrOJI OCTpbIi, TO, TaR RaR yrOJI
KFG - npHMOU, JIIIHIIH AB II CD BCTpeTHJIIICh 6bI; rrYCTh HX BCTpe-
qa - B TOqRe L. TorJ];a yrJIbI BEG II DGE 6bIJIH 6bI MeHhille J];BYX
npR lbIX, a yrJIbI AEG II KEG paBHhI ));BYM npHMbIM; n09TOMY yrOJI
DGE 6I)IJI 6bI MeHhme yrJIa AEG, TaR qTO BHemHHi yrOJI 6bIJI 6DI
l\feHbllIe BH YTpeHHero, "liTO BeJIeno.
ECJII1 YCTaHOBJIeHO, qTO yrOJI EKH - Tynoi, TO yroJI BKF -
OCTphIH, a yroJI GF K - npHMoi. II08TOMY JIHBIIH AB II CD BCTpe-
qaIOTCH. 8TO II eCTh TO, "liTO MbI XOTeJIII ));ORaaaTL)) [22, c. 600-601J.
3aCJIymnBaeT BHIIMaBIIH, "liTO )J;OKaaaTeJIhCTBO V nOCTYJIaTa B IIO-
CJIe,IJ;HeM CJIyqae COBnaJJ:aeT C ,n;ORaaaTeJIhCTBOM nCeB}];O- TYCH.
RaK rrOKaaaJI A. AXMeJJ:oB [3], MomBO npOCJIe}];HTh cy ecTBeHHoe
CXO)J;CTBO MemJ];y «II peJJ:J10meBIIHMH 060cHOBaBIIH» ac-CaMapKaH)J;H
H KHMron nceBJ];O- TYCH II B J];pyrllx MeCTax.
BeCbMa 6JIIIaRO K )J;ORa3aTeJIhCT-
BY aJI-A6xapH ));ORaaaTeJIhCTBO V
IIOcTYJIaTa MYXbH a));-AIIHa aJI-Mar-
p1l6H; ero He));aBHO pacCMOTpeJI
A. M. Ca6pa [60].
MYXhM a));- IIH JIXhH H6B My-
XaMMa)); H6B A611-m-illYRp aJl-Mar-
pn6H, HaabIBaeMLIH TaKn e aJI-AB-
));aJIYCH, HaK YKaahIBalOT ero BHC-
6hI, 6bIn ypomeHn;eM MycYJIhMaHCRoi MClIaHHH (aJI-AHAaJIYc) IIJIH Ce-
Bepo-3ana,IJ;HoH A4>PHKH (MarpIl6a). B 1260 r., 6Y));y"tJ:H B CIlPHH, OB
nOlIaJI B nJIelI K MOHrOJIaM, npenpOBO));IIBillllM ero B MaparHHcKYIO
aCTpOHOMHqeCKYIO 06cepBaTopuIO HaCHp-a));- HHa aT- TYCII, C ROTO.'
poi II 6hIJIH CBHaaHbI nOCJIeJ{Hlle ro,n;bI ero mIl3HH.
IIepBble Ba H3 qeTLIpeX npe));JIOiKeHHH YKaaaHHoi PYKonHcH MbI
pacCMOTpeJlH BbIme, qeTBepToe C TOqHOCTblO ));0 060aHaqeHlli COBna-
;naeT C npeJJ:JIOmeHHeM 4 aJI-A6xapn. IIpeAJIomeHHe 3 COBnanaeT
C npe,n;nomeHueM 3 aJI-A6xapu no OPMYJIHpOBKe, HO OTJIH'tJ:aeTCH
CBOMM ));OKaaaTeJIhCTBOM, 60JIee 6JIII3KHM R npeAnOCJIe,n;HeMY npeJ];'JIO-
16.
o a3a e bC 80
a -Ma'tpu6u
104
E
E A
Z K
F
B Puc. 75 Puc.76
A
a
meHHIO ,nORaaaTeJIbCTBa rrCeBIJ;O- TYCH. Qlop IY.jIHpOBRa rrpe!))IOJl-\eHIIH
3 YRaaaHlIoro COQHHeHIIH 8JI-l\:larpH6u TaRona:
«TIYCTb KaiJ\,nbIH IfR yrJIOB 4 II B 0CTpl)]ii (p:uc. 75). Tor,na
JIHHlIH A C II BD npH npOJJ;OJImeHI1:JJ nepeceKYTCH co CTOpOHbI C H D.
OnycTIIM neprreH,nHRYJIHp AE Ha BD. Tor,na yrOJI E - npJIMOH,
a yrOJI EA C - OCTpbIH H JIIIHIIII A C H BD nepeceRYTCJI IIPII rrpo-
,nOJImeHHH» [60, c. 11 H20).
AJI-MarpH6H HanHCaJI e e JJ.pyroe «l1aJIomenue "HaqaJI" EBRJIH-
)];a» (Ta pllp YCYJI YI JIH,nH(, , 6JIHaROe K HaaBaHHbIM COQHHeHIIJIM
aT- TyclI H IIceB,no- TYCH. B 3TOl\f BTOpOM «MaJIOi-ReHHH» aJI-MarpII611
BMeeTCH e e O,nIlH BapnaHT AORaaaTeJIbCTBa \T nocTYJIaTa. Bo BBe]J;eHHH
aJI-Marpll6H onpe.neJIHeT napaJIJIeJIbHbIe npHl\IbIe CJIe!J;YIOrn;IIl\1 06pa30 I:
«llapaJIJIeJIbHbIe npHMbIe JIHHIIII - TaRHe, KOTopbIe JIemaT B O,nHOH
IIJIOCHOCTH U TaKoBbI, qTO npOU3BOJIbHaH npHMaH JIUHIIH, na,nalOmaJI
Ha HHX, o6paayeT O,nHOCTopoHHHe yrJIbI, paBHbIe ,nBYM npJIMbIM» [60,
c. 17]. OTclO,na CJIe,nyeT, qTO BCHRHH nepneH,nURYJIHp R O,nHOH lIa ,nBYX
TaRHX npHMbIX HBJIHeTCH HX 06 I1:l\I rrepneH,nHRYJIJIpOM, T. e., B qaCT-
HOCTH, BhITeRaeT cy ecTBoBaHue npHMoyrOJILHHRa, a nepeceRaH .nBe
TaRDe npHMbIe ,nBYMH npHMbIl\111. no,n paBHbIMH OCTpbIMH IIJIH TynbIMII
yrJIaMII, MbI nOJIyqlIM CYll\eCTBOBaHue napaJIJIeJIOrpaMMa. Ma cy e-
CTBOBaHIIH npHMoyrOJIbHHRa JIerRO BbIBO.nHTCH, 1JTO JIHHHII, napaJI-
JIeJIbHbIe B C:MbICJIe aJI-MarpH6H, HBJIHIOTCH paBHOOTCTOH nMH. HCHo,
1JTO OIIpe,neJIeHHe aJI-MarpH6n 3RBHBaJIeHTHO YCHJIeHHOH aKCBOMe 0
na paJIJIeJIbHhIX II HCRJII01JaeT RaR rHnep60JIu1JeCRYro, TaR H 3JIJIBnTH-
qeCRYIO reOMeTpHIO.
TIpBBeAeM 8TO AOKaaaTeJIbCTBO aJI-MarpH6H:
«TIpeAnOCbIJIRa. ECJIH JIIIHHH na,naeT Ha ,nBe npHMlde JIHHUB II 06-
paayeT ,nBa O,nHOCTopo HHe yrJIa, MeHbIDHe ABYX npHMbIX, TO, eCJIII
npO)];OJImUTL 3TU JIUHHH B 3TY CTOPOHY, OHB BCTpeTJITCH.
TIpBMep 3Toro. Ha JIBHBH AB H CD na,naeT JIUHIIH AC II 06paayeT
ABa yrJIa BAC II DCA, MeHblline ABYX npJIMbIX (puc. 76). JI. YTBepm-
1);alO, qTO eCJIU HX npOAOJIiRHTL AO 6eCROHe1:JHOCTII, OHII BCTpeTJITCJI.
105
ORaaaTeJIbCTBO 3TOrO. ECJIH OLUtH 113 yrJIOB - npHMOH, {bI aa-
KOHqHM ,n;ORaaa TeJIbCTBO, RaR 6y,n;eT YRaaaHO. ECJIH 8TO He TaR, npo-
]::(OJImIIM DC no 6eCROHeqHOCTH H onYCTHl\1 Ha Hee rrepneHtJ;HRYJIHp
A E ([ B CIIJIY] [npetJ;JIOiKeHHH] 12). IT pOnOJIlliHM ero ,n;o 6eCKOHeQHOCTIf
B CTOPOHY F. ITOCTpOHM rrpH TOqKe A JIHHHH AE yrOJI KAE, paBHLIH
yrJIY BAE ([B CII "1Y] [npenJlomeHHH] .23). rrpOL(OJIiRH I JIHHIIH AB II
AK B CTOPOHY B H K no 6eCKOHeQHOCTH ([B CIIJIY] BBe,n;eHIIH). OTMe-
THM Ha AB TOQKY L, OTJIOmlIM AM, paBHYJO AL ([B CHJIY] [npenJIo-
iKeHHH] 3), II CoeLI:HHlI I lltIL. Tor,a;a, TaK KaK yrJIM KAE II BAE -
OCTpbIe ([B CIIJIY] BBeneHHH), JIHHIIH ML nepeceqeTAF B N TaK KaK
nBe CTOpOHLI AL II AM paBHbI, CTopOHa A1V - 06 aH, a 06a yrJIa
A paBHbl, 06a yrJIa N TpeyrOJlbHIIROB ANL II ANM, rrpHJ\1IbIe
([B CIIJIY] [npenJIOi-ReIlHH] 4 II BBeAeHIIH). EC.TJU 1'O"[IRa N -Mem,n;y TOQ-
KaMII E II F, MbI aaROHqlIJIII nOCTpoeHlle. ECJlH a e HeT, TO OTJlOllilIM
JIIIHIIIJ MO II LX, paBHble AM II AL ([B c:UJlY] [npe,n;JIOmeHIIH] 3),
H coe,n;HHIIM XO, OHa nepeCeqeT AF B P TaK me, KaK Mbl ,lJ;ORaabIBaJIII,
nOKai-KeM, qTO 06a yr,,"1a P TpeyrOJIbHHROB APO II APX - npHMLIe
([ B CHJIY] [npe,n;JlOmeHHH] 4). ECJIII TOqKa P HaXOnHTCH MemAY TOqRa-
MH E II F, ,n;JIH Hac JJ;OCTaTOqHO. ECJIH me HeT, npoJJ;OJlllilll\{ JIIfHIIIO
N L no 6eCKOHeqHOCTlI II onycTIIM Ha Hee Ha TOqKH X nepneHJJ;HRy..TIHp
X"Q ([B CIIJIY] [npenJIOmeHHH] 12). Tor,n;a TaR KaK yrOJI ANL-
npHMOH yrOJI II yrOJI LQX - rrpHMoii, yrJIbI ALN II XLQ paBHLI
([B CIIJIY] [npe,n;JIOmeHIIH] 15), a JIIIHIIH AL JI LX paBHbl, TO JIIIHHH AN
II QX paBHbl ([B CHJIY] [npeJJ;JIOmeHHH] 26). TOQHO TaR fRe, TaR KaK
yrOJI QN P - npHMOH II yroJl N P X TaKi-Re JIpH IOii, KaK l\lbl nOKaaa-
JIll, yrOJI Q - npHMoii. IToaTOMY llJIOCKaH cpHrypa - napaJIJIeJIO-
rpaMM. IToaTOMY JIIIHIIII QX II NP paBHLI ([B CHJIY] [npetJ;JIOmeHIIH]
34). ECJIH [9Toro] HeJJ;OCTaTo'tJ:HO, nocTynaeM no o6paau;y 9Toro JJ;eHcT-
BIlH: OTKJIanLIBaeM Ha AF RpaTHble JIHHHM AN no Tex nop.. nORa He
6YJJ;eT nOCTHrHYTa TaKaH KpaTHaH, KOTopaH 60Jlbme AE, nycTb 8TO
AZ ([B CIIJIY] BBeneHIIH). JIIIHHH, KOTopaH OTceRaeT AZ HaAF - KZ.
Tor)];a yrJILI Z -'- npHMbIe, KaK MbI .lJ;ORaaaJIH ,n;JIH yrJla AN L. YrJlbI
E - TaKme npHl\lbIe. IToaToMY JlHHHH ED He BCTpeTIIT ZB ([B CMJlY]
[npe)];JIOH\eHHHJ 28) II He BCTpeTHT AE ([B CHJlY] BB eHJIR). IIo3TOMY
OBa He06xonHMo BCTpeTHT AB.
OTcro,na 11 Ha [rrpe)];JIOmeHHH] 28 CJIeJJ;yeT, QTO BCHKHe JJ;Be JIHHIIH
Ha nJIOCKOCTII IIJIH BCTpeqaIOTCH, HJlH napaJIJIeJlbHbl, TaR KaR, eCJIH
Ha HHX na)];aeT npHMaH JIHHHH, 06paaYIOm;aH OnHOCTOpOHHHe yrJIbl,
MeHLmHe )];BYX npHMbIX, OHH BCTpeTHTCH, a eCJIII 06paaYIOTCH nBa yrJIa,
paBHLIe ,nBYM I1pHMbIM, npHMLIe rrapaJIJIeJIbHLI ([B CIIJIY] BBe):(eHHH).
TaRIlM 06paaOM, COMHeHHe, HMeBmeeCH B aTOM Borrpoce, YCTpaHeHO
6JIarO)];apH HameMY ycoBepmeHcTBoBaHllIO BBeJJ;eHHH» [60, c. 15-17
H 21-24].
CCblJIKH aJI-MarpH6H Ha BBe,neHHe OTHOCHTCH COOTBeTCTBeHHO R:
1) II nocTynaTY (BoaMomHocTH .HeorpaHHqeHHOrO npO)];OJImeHIIH npH-
MOB), 2) onpe)];eJIeHHIO OCTporo yrJIa KaK yrJIa, MeHLrnero npHMoro,
3) 'oupe)];eJIeHu"IO rrpHMoro yrJIa RaR yrJIa, paBHoro CMemHOMY C HHM,
4) aKcBoMe EBAoKca - ApXHMe)];a, ROTOpylO OH nOMe aeT rrepen I RHH-
ron, iI, HaKoli , 5) aKCUOMe 0 TO I, qTO ,n;Be npHMbIe He MOr"VT Qr-
106
paHH1.JI:IBaTb nJIOCKOH .pIIrYPbI. OKa3aTeJIbCTB.o «rrpe,IJ;IIOChI:IKH»
3JI-l\tIarpII6H, C OJ].Hoii CTOpOHhI, HOCHT CJIe,n;bI ,n;OHa3aTe bCTB, BOCXOAH-
IUJ1X R CIIl\frrJIHRIIIO, 3 C ,n;pyroH CTOpOHbI, aH3JIOrlI'IHO ,n;OKa3aTeJIb-
CTHa1-I, OCHOBaHHbIM Ha onpeJJ;eJIeHIIH napaJIJIeJIbHbIX JIHHIIH, 8RBHBa-
JIeHTHOM YCHJIeHHOH aKCIIO fe 0 IIapaJIJIeJIbHbIX, C TIOMOIJJ;blO qerO
,n;ORa3bIBaeTCH CYIJJ;eCTBOBaHHe npHMoyrOJIbHIIRa. TaR me, RaE II6H
aJI-XaUCaM, aT-rrYCH II TIceB,n;o-TYCII, aJl-Marpn6II nOJIb3yeTcH 3RCHO-
M:OH EBnOKca - ApXH:Mena, a TaKn e MCIIOJIb3yeT YTBepiR,AeHHe, RB-
JJHIOU eeCH, paBHOCUJlbHbI I aKCMOMe llama.
1\ ROH:U;y XIII JIJIIf Haqa.!IY XIV B.
OTHOCHTCH e e OJ].Ha IIonhITKa
,n;OKa3aTeJIbCTB3 V nOCTY:Ja Ta, npll-
Ha,n;JIemam;aH y IeHI1RY HacHp
a,n;-,D,IIHa aT-TyclI RYT6 an- IIHY
am IIIHpa3H.
KYT6 a,n;-,[(IIH IaxMY,n; n6H l\1ac'YA
arn-IIIHpa3H (1236-1311), ypome-
HeI IIIHpa3a, BHaqaJle Y"IlHJICH Me-
AHU;IIHe, .plIJIOCO.pUII H IOplIcrrpy,n;eH-
U;1I11 y CBoero OTn;a Mac'yna aJl-Ka-
3apYHH, a 3aTeM MaTeNIaTHKe II
aCTpOH01.fIIH Y HacHp an- IIHa aT-Tycli B Mapare. TIoccopHBIDncb C
aT-rrYCH", aIII-IIIlIpa3II nORIIHYJI Mapary II pa60TaJI Bpa'IOl\I II cynbeii
B pa3JIIIqHbIX ropo,nax IIpaHa II MaJIOll A3IIH. IIocJIe y,na'IHOrO BbI-
nOJIHeHHH o,n;Horo ,n;rrTIJlOMaTlIqeCKOrO rropyqeHIIH B ErlInTe, ,n;aHHoro
MOHrOJIbCKIIM xaHOM, arn- IIIIIpa31I B 1297 r. B03Bpa TIIJICH B CTOJIIIU;y
xaHOB, ROTopaH K 9TOM:y Bpe IeHII 6bIJIa nepeHeceHa B Te6pu3, H CTaJI
rJIaBOH Te6plI3cROH aCTpOHOl\flIqeCROH 06cepBaToplIH, HBJIHBIIIeHCH
npeeMHn:u;eii MaparlIHcRoll 06cepBaTopIIII aT-Tycn. B CBOHX Tpy,n;ax
«BbICIlIIIH npe,n;eJI nOCTlImeHIIH He6eCHbIX c.pep» M «IIIaxcRIfH no,n;apOR
no aCTpOH01.fHII» a III- IIInpa3H, pa3BIIBaH aCTpOHO}llIqeCKHe n,n;elI
aT- TyclI.. npe,n;JIomlIJI CBOll BaplIaHT TeopHIf ,n;BlIFI eHIIH IIJIaHeT. l\;{aTe-
MaTIIKe nOCBHIIJ;eHbI ero ROM1.feHTaplIlI R «TpaRTaTY 0 ,nBJImeHHII Ra
qeHIIH II 06 OTHomeHIIH Iem,n;y nJIOCKIIM II RpIIBbIl\J), nanlIcaHHOMY,
nO-BII,n;IIMOMY, aT-TyclI; B 9TIIX ROl\fMeHTapHHX paCCl\laTpHBalOTcH Ta-
Rile BonpOCbI, RaK RBa,npaTypa Rpyra, CnpHMJIeHlIe KpHBbIX, pa3Bep-
TbIBaHHe KplIBbIX nOBepXHOCTeii Ha nJIOCROCTb, yrJIbI RacaHHH, npo6-
JIeMbI MaTeMaTHqeCROro aT01.fH3 Ia H T. n. BeCb la norrYJIHpHLI 6LIJIB
ROMMeHTapllH am-IIIllpa3H R TpaKTaTY ac-CyxpaBapJJ;H «<1>IIJIOCo«l»BH
03a peHHH».
TeoplIH napaJIJIeJIbHbIX JIIIH:nH 1I3JIOmeHa am-III:npa31I B ero Ha-
nHcaHHOM H8 nepCH,n;CKOM H3bIRe 8H:U;HRJIOneJJ;lIqeCROM TpaKTaTe
«iReMQymlIHa ROpOBbI JJ;JIH YKpameHHH ,I:(y6a,n;ma» ( yppa aT-Tanm
JIll yppa aJJ;-,I:(y6a,n;m) [34], rrOCBHIJJ;eHHOM rll:IHHCKOMY 8MHPY ,I:(y-.
6aJJ;my. 8TO COllllHeHHe HanllcaHO no nJIaHY aBn;IIKJIOneJJ;HlIeCR X
«I{HlIrn IIC eJIeJlIIH», «!\HHrH crraCeHHH» II «HHHrII 3UaBHH» H6u
CUHId, 0 ROTOpLIX MLI rOBOplIJIlI BLIIIIe, II COCTOHT II3 qacTeA, 9CBJI-
eHHbIX JIOrHKe, 4»HJIOC04»IIH, eCTeCTB03BaBllIO II MaTeMaTUqeCKItK
HaYKa:p.1 - reOMeTpHH, aCTpOHOMHH, apH MeTHRe II ?fY3LIKe. reo MeT-
17
T pa'Km061Ca
np06JleMbt
o napaJlJleflb1-lbtX
y am-lllupa3u
107
pHlIeCKaH tIaCTh «iReMtIymHBbl KOpOHbI» B 3BallIf'l'eJIhHOH tIaCTII HB-
JIHeTCH nepeBoAoM C apa6CRoro Ha nepCH CKHH«M3nOmeHHH KHHrH
"HallaJI") aT-TYCH, OnHaKO B OTJIIPiHe OT aT-TYCII am-lliupa3H, no-
BH HMOMY, no,n; BJIHHBHeM M6H CHBhI, npe JIOiRHJI )J;ORa3aTe,J'l bCTBa
Bcex nOCTYJIaTOB EBKJIHna.
feOMeTplltIeCKHH H apH4>MeTHtIeCKUH pa3)J;eJIhI «/ReMtIymHBhI KO-
POHhI» H3YQaJIa X. P. MY3a4>apoBa [18], ony6JIHKOBaBmaH PYCCKHH
nepeBO)J; nonLITKH nOKa3aTeJILCTBa am-illHpa3H V nOCTYJIaTa [18,
c. 89-90).
XOTH B OCH llHO I reOMeTpHtIeCKaH KBHra «iKeMlIymHBa ROPOHLI»
6JIII3Ka K KBHre aT-TycH, BO B HeROTopblX MeCTax OHa 6JIH3Ka H K
«M3JIOmeHIIIO EBKJIHna) nceB)J;o-TYCH. B lIaCTHoCTH, B HatIaJIe 9TOH
RHJfrH npHBe)J;eHbl 14 «npe)J;nOChIJIOK», 3HatIHTeJILHO 60JIee 6JIH3RHX
R nOCTYJIaTa:M nceBno-TycH, tIeM R nOCTYJIaTal\1 EBRJIH)J;a, BocnpOlf3Be-
neHHblM B RHMre aT- TyclI. IIepBble meCTh «npe)J;nOChIJIOK» am- IIIHpa3H
COBnaJ]:3IOT C nepBbIMH meCThIO nOCTYJIaTaMH nceBAo- TYCII, )J;aJIee CJIe-
nVIOT: 7) 110CTYJIaT 0 TOM, "lJTO yrOJI, paBHblH npHMOMY, npHMon;
b) 1l0CTYJIaT 0 TOM, qTO npHMaH B ee HanpaBJIeHHH MomeT coe)J;IIHHThCH
TOJIhRO C O,n;HOH enpHMo:ii; 9-12) I-IV nOCTYJIaThI EBRJIHJJ;a; 13) noc-
TYJIaT 0 TOM, tITO nBe npHMLIe He MorYT OrpaHlItIHBaTh nJIOCKOH .pH-
ryphI, H 14) V nOCTYJI3T EBRJIH)J;3.
TaK RaK TerepaHCRoe II3n;aHHe «JReMqymHH:bI KOPOHhI» [34] 6hIJIO
HaM He,n;ocTynHo, MbI npHBe)J;eM pa3)J;eJI 9TOH RHHrH, nOCBHlI\eHHbIH no-
Ra3aTeJIhCTBY V nOCTYJIaTa, no PYKOllIIC.u M 816IIHcTIITYTa BOCTORO-
Be)J;eHHH AH Y3CCP, ROTOpylO H3YtIaJIa X. po. MY3a.papOBa (PYC-
CRHll: nepeBo)J; [18] HenOJIOH H He BnOJIHe TOtIeH):
«M3 nOHHTIIH napaJIJIeJIhHblX JIHHHH H3BeCTHO, tITO O;O;Ha M3 ,D;BYX
napaJIJIeJIhHMX HaXOnlITCH no O)J;HY CTOPOHY OT APyroH napaJI eJIh-
HOH II OnHa 11:3 HIIX He MOiReT HaXO)J;HTLCH no pa3HhIe CTOpOHbI OT ;o;PY-
rOB, HHatIe OHII BCTpeTHTcH II He 6Y)J;YT napaJIJIeJIhHbIMII II paCCTOHHHH
Mem)J;y HIIMH 6Y)J;YT pa3JIHtIBhI, TaK CtIMTaeT KaiR)J;LIH 3)J;paBOMbICJIHII\HH
yqeHhIH. II03TOMY BCHKHe Be npH lble JIIIHHM, MemAY ROTOpbIMH JIe-
mHT npHMaH oi1JHHMH, napaJIJIeJIbHaH HM 06eHM, napaJIJleJIhHbI Memny
co60ii, TaK KaK eCJIII 6bI OHH BCTpeTHJIHCh, TO He06xo,D;HMO O)J;Ha H3
ABYX napaJIJIeJlhHhIX HaXOnHJIaCh 6hI no pa3HLIe CTOpOHbI OT npyroH-
II )J;Be napaJIJIeJIbHhIe JIHHHH BCTpeTHJIHCh 6bl, a 9TO HeJIenO, TaR KaR
paCCTOHHIIH Mem,IJ;y HHMII 6blJIH 6bI pa3JIHtIH:bI. II09TOMY )J;JIH BCHKHX
ABYX BCTpetIalO HXCH npHMhIX JIHHHH npHMaH JIHHIIH, BaXO)J;HlI\aHCH
MemAY HHMII, eCJlH IIpO OJImHTh HX no 6eC-KOHetIHOCTH, He06xonHMO
BCTpeTHTCH C O HOH H3 HHX, BHatIe BCTpetIalOlI\HeCH [JIHHIIH] 6hIJIH
6LI napaJIJIeJIhHbI. II09TOMY, eCJIH npHMaH JIHHHH JIe>KHT B nJIOCROCTH
H B 8TOH nJIOCKOCTH HaXO)J;HTCH ABe BCTpetIalOII:(HeCH [JIHHHH], OHa He-
06XO)J;H 10 nepeCeqeT O)J;BY H3 8THX JIHHHH, eCJIH IIpOnOJImHTh HX no
DeCKOHetIHOCTII, r)J;e 6bI MbI ee HIl npennO.T.IOmllJIH, Memny HHMH HJIH
DHe HX, H npHlIHHa 8Toro - B TOM, tITO HIIKaKaH tIaCTh nJIOCKOCTH,
HaXOnH aHCH Mem;o;y nBYMH nepeCeKalOII:tHMHCH [npHMhI IH] JIHHHH-
MH, He BbIXO,IJ;HT 113 8TOB nJIOCKOH <plIrYPhI.
II08TOMY BCHKlIe ,IJ;Be npHMLIe JIHHHH, HanpHMep AB H CD, Ha
ROTopLIe rranaeT npHMaH JIHHHH, HanpHMep EG, 06pa3yrom;aH BHYT-
108
B
E
A
.D
A
F
Puc.77
l
.
6
K
.D
H
PUC. 78
Puc.79
c
peHHHe O;n;HOCTOpOHHHe yrJIbI BEG II DGE, MeHbmHe ;n;BYX npHMbIX yr-
:lOB (pHC. 77), eCJIH npO;n;OJIiKHTb HX C aTOU CTOpOHbI - BCTpeTHTCH.
TIpnqHH3 3Toro - TO, qTO yrOJI BEG BMeCTe c [yrJIOM) EGD no
npe;n;rrOJIOiKeHHIO l\leHbme ;n;BYX rrpHMbIX, B TO BpeMH KaK HSBeCTHO,
'ITO C yrJIOM AEG OH COCTaBJIHeT ;n;Ba npHMbIx. TI03TOMY [yrOJI) AEG
60JIbme [yrJIa) EGD. TIpe;n;CTaBHM ce6e, lITO CTopOHa GEA HaJIOmeHa
Ha [CTOPOHY) EGD, Tor;n;a [JIHHHH) EA yna;n;eT Ha [JIHHHIO] HGF. MbI
rOBopHM, lITO HF nJIH napaJIJIeJIbHa AB, HJIH He rrapaJIJIeJIbHa. ECJIH
OHH rrapaJIJIeJIbHbI, TO no npHqHHe Toro, qTO AB nona;n;eT Mem;n;y
HF H CD, eCJIH nx rrpO;n;OJIiKHTb, OHa ;n;OJIiRHa BCTpeTHTb OAHY H3
HIIX, HO BCTpeqa C HF HeB03MOiKHa, noaTOMY OHa BCTpeTHT CD,
a 3TO H eCTb Tpe6yeMOe» [JI. 6 06. TamKeHTCKOU pYKonHcH). KaK MbI
BR;n;HM, XO;n; MbICJIeU am-illHpa3H S;n;eCb 6JIII30K K paCCYiK;n;eHHHM
TIpOKJIa H HanOMHHaeT TaKme paccym;n;eHHH XaUHMa B nOCJIe;n;HeM
npeJtJIOmeHHH pa3;n;eJIa ero KOMMeHTapHeB K EBKJIH;n;y, nOCBH eHHoro
TeoplIH napaJIJIeJIbHLIX, C TOU pasHIIn:eu, lITO B OTJIHqHe OT XaUHMa
alIl- illHpa3H He rrOJIb3yeTcH nOCTYJIaTOM, 8KBHBaJIeHTHLIM V nOCTY-
.-J"'IaTY, HO, TaR me KaK IIpOKJI, HeHBHO npe;n;nOJIaraeT, qTO paCCTOHHHe
fefHn;y napaJIJIeJIbHLIMH npHMLIMII OrpaHlIqeHO; lIS npeJtJIOmeHHH am-
illHpa3H, npe;n;meCTBYIO IIX ero ;n;OKaSaTeJIbCTBY V nOCTYJIaT.a, BH;n;HO
;n;ame, qTO OH CqHTaeT, qTO paCCTOHHHH Mem;n;y napaJIJIeJIbHLIMII npH-
MLIMH He MorYT 6bITb paSJIHqHLlMH. Ho am-illHpaSII He OrpaHH'IU-
BaeTCH 8THM H paCCMaTplIBaeT CJIyqau, Kor;n;a npHMLIe AB II HF He
napaJIJIeJIbHLI, T. e. CJIyqau, Kor;n;a ;n;Be npHMLle, 06JIa;n;alO He paBHbI-
MH HaKpeCTJIeiKa HMH yrJIaMII, nepeceKaIOTCH:
109
«ECJIII OHII He napaJIJIeJIbHbI, TO OHH BCTpeTHTCH .HJIII CO CTOpOHbI
H A, IlJIII co CTOpOHbI B F. llepBoe HeB03MomHO, no TOM npH'tJHHe, qTO
yrOJI AEG paBeH [yrJIY] EGF II yrOJI BEG He06xo II 10 paBeH yrJIY
EGH, TaR RaR Kam LIe Ba yrJIa IIplI E II, no 06HO 3TOMY, :Ka;+\,I(ble
,l);Ba yrJIa nplI G paBHLI BYl\f npHMbIM yrJIal\f. ECJIH EA H GH BCTpe-
qaIOTCH, HanpHMep, B [TO'tJRe] K (pIIC. 78) H eCJIH HaJIOFRliTb AEG Ha
EGE, TO EGH He06xo HMO IlaJIOmHTCH Ha BEG H EB Ii GF BCTpe-
THTCH co CTOpOHLI B F, HO OHH BCTpe'tJaIOTCH co CTOpOHbI H A , n03TO IY
He06xo HMO H F H AB - JJ:Be npHMbIe [JIIIHHII], orpaHII"tJHBaIOIIJ;lIe
IIJIOCRYIO <pHrypy, a BTO HeJIeno. BTopoe nplIBOAIIT J\ Tpe60BaHHIO,
'tJT06bI [BTH nBe nVHMbIe JIHHHII] 6bIJIH TaRHMII me co CTOPOHbI BF
fIpe.n;cTaBHM ce6e, 1JTO OlIlI nepeceRYTCH B [1'o't{Re] L (pur. 79). JIIIHIIH
CD no TOll npn'tJHHe, "tJTO OHa pacceRaeT yrOJI E'GL, eC.,1JI ee npoJJ:OJI-
mHTb, nepece'tJeT OCHOBaHlIe EL, TaR RaR, eCJIH 6bI OHa nepeCel\JI3
[CTOPOHY] GL, Be IIpH lble JIHHIIH He06xoAIIMO orpaHII'tJIIBaJIII 6bI
nJIOCRYIO .plIrypy, a 3TO HeB01MOmHO» [JI. 6 06. TamReHTCROH PYRO-
fiIICH]. ·
3 ecb am-lliHpa3H onYCRaeT nepece't{eHHe npHMbIX C paBHbIMII
Ha:KpeCTJIema HMH yrJIaMII, 'tJTO IIMeeT MeCTO TOJIbRO B BJIJIHnTH"tJec-
ROB reoMeTpHH. BCJIe CTBHe 3Toro am-llITHpa3H JIerRO onpoBepraeT
C eJIaHHOe npe nOJIOmeHHe, CCbIJIaHCb Ha TO nOJIOmeHHe a6COJIIOTHOH
reOMeTpHH, 't{TO Be npHMLIe He MorYT orpaHH'tJHBaTb nJIOCKYIO «pHrypy.
IIonYTHo am-lliHpaSH .paRTHqeCKII J];OKaSLIBaeT 't{aCTHbIH cJIY'tJai YT-
Bepm eHHH aKCHOMbI llama JIH npHMoi, npoxo H eM 'tJepes O HY H3
BepmHH TpeyrOJIbHHKa. IIoBTOMY paSMLIIIIJIeHIIH am-IDHpa311 0 Teo-
pHH napaJIJIeJIbHLIX JIHHHH B ero «iReWIymHHe ROPOHbD) BeCbMa HH-
TepeCHLI JIH IICTOpHII 8Toro pas eJIa reOMeTpHH.
IIepBoe IISBeCTHoe HaM ORa3a-
TeJIbCTBO " nOCTYJIaTa B cpe HeBe-
KOBOH EBpone MLI BCTpe'tJaeM B on-
TH'tJeCKOM TpaKTaTe nOJIbCKoro Y"tJe-
Horo XIII B. BHTeJIO (OR. 1230-
1235, YM. nOCJIe 1275). CaM BIITeJIO
Ha3LIBaJI ce6H «CbIHOM TlOpHHrOB II
IIOJIHKOB», oTKy a MomllO C,D;eJIaTb
BLIBO , 't{TO O HH H3 ero pO H TeJIeH
6bIJI nOJIHKOM, a pyroii - BbIXO -
n;eM H3 TropHHrHH. CBoei pO,D;HHOH
BHTeJIO C'tJHTaJI llOJIbIll y. TpaRTaT
BIITeJIO J3 O HIIX PYKOIIHCHX IIMeHO-
BaJICH «OnTIIK », B pyrHx - «IIep-
cneKTHBa», OH 6LIJI HanHcaH no-
JIaTLIHII II IIs aH B 1572 r. <D. PHC-
BepO I BMeCTe co 3Ha IeHIIToi «RHHroH OIITHRH» 116H aJI-XaHCaMa
[36]. Co'tJIIHeHlle BHTeJIO COCTOHT 1I3 eCHTH KHHr, cOJ{epmaHlle KO-
TOpbIX 6JIHSKO K ceMH RHlIraM co'tJHHeHHH 116H aJI-XaHCaMa, 6L1BmerO
OCHOBHLlM IICTO'tJHIIKOM co'tJHHeHHH BHTeJIO, KOTOpbIU, Bnpo'tJeM, HC-
IIOJIb30BaJI TRKme TPY,lIbI EBRJIH a, IIToJIe leH, faJIeHa, aJI-RHH JiI,.
M6H-CHHLI H pyrHx Y'tJeHbIX.
18.
B.nURHue
apa6c ux mpY006
1-la meopu1O
nap a.n.n e.nb1tblX
l3 EGpone
XIII-XIV Ge'K06
110
ORa3aTeJIbCTBO BI1TeJIO V nOCTYJIaTa 113JIOmeHO B npe JIOmeHHH
14 RHHrH I «ITepcneRTHBM». 3TO npeAnomeHHe HeAaBHO H3yqaJIOCb
C. YHryPY [67], npHBe mero naTHHCKHH TeKCT BToro npeJJ;JIOmeHHH
[67, c. 205-206] Hero aHrnHHCKHH nepeBO [67, c. 207]. rrpe JIo-
jJ,eHue 14 KHHrH I BHTeJIo no CBoeH 4>oPMYJIHpOBRe HeCROJIbKO mHpe
\7" nOCTYJIaTa EBRJIH,na:
«ECJIH npHMaH JIIIHHH, na alOII{aH Ha ,lJ;Be npHMMe nHHHH, 06pa3yeT
HepaBHbIe HaKpeCTJIema He yrJIbI, HJIH Ba BHYTpeHHHX [O HOCTO-
pOHHHX], MeHbillHe BYX rrpHMMx [yrJIOB], HJIH BHemHHH yrOJI, HepaB-
HbIH BHYTpeHHeMY [c TOH me CTOpOHM], TO 8TH Be npHMble He06xo H-
fO BCTpeTHTCH CO CTOpOHhI MeHbillHX yrnoB II He MorYT BCTpeTHTbCH C
,a:pyroH CTOpOHbI; eCJIH me JIHHHII BCTpeqaIOTCH, TO Heo6xo HMO, qT06bI
YRa3aHHbIe yrJIbI BeJIM ce6H O HHM 1I3 ynoMHHYTbIX cnOC060B».
Xo,n paccym,neHUH BHTeJIO TaKOB: OH rrpe nOJIaraeT, qTO Be
npHMble IIPO. YRa3aHHbIX ycnoBHHX He nepeceRalOTCH, H AeJIaeT BbIBO,I(,
"lITO, nOCKOJIbKY npH Ible He nepeceRalOTCH, «OqeBH HO», OHH napaJI-
JIeJIbHhI, HO B 3TO I CJIyqae B CHJIY rrpe JIOmeHHH 29 KHlIrlI I «HaqaJI»
EBRJIH,na H3 napaJIJIeJIbHOCTH npHMbIX BLITeKaeT paBeHCTBO HaRpeCT-
JIeif\a IIX yrJIOB, paBeHCTBO CYMMhI BHYTpeHHHX O HOCTOpOHHHX yr-
JIOB ,llBYM rrpHMbI?v1 II paBeHCTBO BHeIIIHero yrJIa BHYTpeHHeMY, qTO
npoTHBOpeqHT YCJIOBHIO TeopeMM. Ha caMOM me ene rrpe IIOJIOmeHHe,
qTO JIH HenepeceKalO HXCH npHMbIX HMeeT MeCTO npe JIOmeHHe 29
RHIIrH I «HaqaJI», HaR 1I3BeCTHO, onHpalO eeCH Ha V nOCTYJIaT, caMO
paBHOCIIJIbHO V rrOCTYJIaTY. BecbMa BepOHTHO, qTO IIHTepeC BHTeJIo
R V rrOCTYJIaTY 06'bHCHHeTcH ero 3HaKOMCTBOM C HeRoTopbIMH apa6-
CRH IH OKa3aTeJIbCTBaMH V nOCTYJIaTa. O HaKo HCHO, qTO YPOBeHb
JIOrH"IleCKOH CTporOCTH BHTeJIo e e CHJIbHO OTCTaBaJI OT ypOBHH, AOC-
THrHYToro K 3TOMY BpeMeHH B CTpaHax IICJIaMa.
,[(Ba 3arra HoeBporreHcRHx AOKa3aTeJIbCTBa V nOCTYJIaTa, 60JIee
6JIH3RHX C TO"llRH 3peHIIH HX HayqHbIX AOCTOHHCTB K ,nORa3aTeJIbCTBaM
B apa6cKoH JIIITepaType Bonpoca, MbI BCTpe"llaeM B XIV B. Y BYX Ma-
TeMaTHKOB, nHcaBmnx Ha peBHeeBpeHcRoM H3bIRe. O HO 113 8THX
OKa3aTeJIbCTB npHHa,nJIemHT repCOHH Y (JIeBH 6eH repmoHY, 1288-
1344), H3BeCTHOMY Y"IleHoMy-«pHJIOco4>Y, pa60TaBmeMY B IOmHoH
<DpaHIJ;HH. 3TO ,lJ;OKa3aTeJIbCTBO II3JIOmeHO B ero «KOMMeHTapIIHx K
BBe eHHHM KHHrH EBKJIH a» (BeHYP nTH aT ce4>ep l1 nH Yc). COOT-
BeTcTBYIO HH TeRCT HMeeTCH B PYCCKOM nepeBo e M. r. ITOJIbCROrO
[20, c. 763-776].
B OTJIH"IlHe OT M6H aJI-XaHCaMa, KOTOpLlH COBepmHJI «nOCTYJIHpO-
BaHHe OCHOBaHHH», repcoHH , KaK H XaHHM, npe nOChIJIaeT CBoeMY
ORa3aTeJIbCTBY « Be 113BeCTHLle IIpeAnOCbIJIKH». O Ha 1I3 HIIX - 3TO
{<aRCHOMa EB oRca-ApXHMe a», npHMeHHBmaHCH 0 M6H aJI-XaHCa-
IO)I, BTopaH rJIaCHT: «JIIIHHH, KOTopaH HaKnOHeHa, npH6JIllmaeTCH
R TOH CTopOHe, C KOTOpOH 06pa3yeTcH npHMoH yrOJI» [20, c. 794].
«JIIIHHH, ROTopaH HaKJIOHeHa»,- 3TO TaKaH npHMaH, JIH KOTOpOH no
OTHomeHHIO K HeRoTopoH PyroH npHMoH II HeKoTopoB npHMoH, na,:",
.[(alO e:D Ha 8TH Be rrpHMLle, BLlnOJIHHIOTCH ycnoBHH aKCIIOMLI EBKnll-
.[(a 0 rrapaJIJIeJIbHMX. ITepe 4>opMynHpoBKOH npe nOCMJIOK repco-
HOA 3aMe"llaeT, "lITO, «eCJIH npHMaH uaAaeT Ha ABe npHMLle II BHYTpeH-
111
Hue OAHOCTopOHHHe yrJIhI MeHbme ABYX npH?vILIX, BCHRaH npHMan
JIUHHH, naAalOID;aH Ha HHX, 06paayeT C TOH CTOpOHbI BHYT.peHHHe yr-
JIbl MeHbme ABYX npHMhIX» [20, c. 763]. TaRHM 06paaOl\f, BO BTOpOH
npe;lJ;IIOCbIJIRe IIl\feeTCH B BHAY, liTO «JIHHHH, ROTopaH HaRJIOHeHa»,
npH6JIHmaeTCH R aHHOH rrpHMOH Ha BceM npOTHmeHHH 06eHX JIHHHH.
8TO YTBepm eHHe paBHOCHJIbHO V nOCTYJIaTY EBKJIHAa, HO B HeRO TO-
pOM CMLICJIe 60JIee HarJIHAHO.
B npe JIOmeHHII 1 repcoBB JJ;OKaSLIBaeT, liTO «HeBOaMomeH rrpH-
MOJIHHeHHLIH lIeTLIpexyrOJIbHIIK, Bce yrJILI KOTopor 6bIJIH 6LI TynbI?vID
HJIH OCTPLIMH» [20, c. 765]. 3 eCb, TaR me RaR y M6H aJI-XaHCaMa II
XaHHMa, paccMaTpllBalOTcH rllnOTeabI oCTporo, Tyrroro Ii rrpHl\IOrO
yrJIa H onpoBepraIOTCH Be nepBLIe liS HII . ECJIH lIeTLIpexyrOJIbHHI\
XaHHMa MomeT 6bITb nOJIY1IeH lIS lIeTLIpexyrOJIbHHKa M6H aJI-Xa:nca-
Ma oTpameHHeM OT OAHOB liS ero CTOpOH, npHMLlKalO Hx K BYM
npHMLIM yrJIaM, TO, OTpamaH rrOCJIe HHH lIeTLIpexyrOJIbHHK OT ero
OCHOBaHHH, npHMLIKalO ero R ABYM npHMbIM yrJIal\I, ?vILI nOJJyqHM
lIeTbIpexyrOJIbHHR repCOHIIAa. l1epBLIe ABe rHnOTeSbI orrpOBepraIOTCH
npH rroMOID;H BTOpOH npeAnOCbIJIKII.
Ma CJIeACTBHH rllnOTe3LI npHMoro yrJIa, HaH eHHbIX repCOHH Ol\f,
OTMeTHM TeopeMY: «ECJIH OAHY 113 60KOBbIX CTOpOH paBH06eApeHHoro
TpeyrOJIbHHRa rrpOAOJImHTb liS TOllRII HX nepeCelleHHH B ee BanpaB-
JIeHHII Ha TaKYIO me BeJIHq}{HY II npOBeCTII OCHOBaHHe, TO OBa COCTaB-
JIHeT C nepBLIM OCHOBaHHeM npHMOH yrOJI» [20, c. 766]. fepcoHHJJ; npo-
AOJImaeT CTOPOHY AB paBH06eApeHHoro TpeyrOJIbHHKa ABC o G Ha
paCCTOHHHe BG = BA (pHC. 80) II nOKaSbIBaeT, qTO JIHHHH CG nep-
neH URYJIHpHa AC. OH CTpOHT BbICOTY BD TpeyrOJIbHHRa ABC II
npO OJImaeT ee no E Ha paCCTOHHlle BE == BD II nORa3bIBaeT, 1.JTO
lIeTLIpexyrOJIbHHK CDEG - npHMoyrOJIbHHR. B xo e ORaaaTeJIb-
CTBa onpOBepraIOTCH npennOJIOmeHHH, liTO CG == KL > ED H CG ==
= MN < ED, COOTBeTCTBYIO He rllnOTeaaM OCTporo H Tyrroro yrJIa.
B lIHCJIe npyrHx CJIeACTBIIH rllnOTe3L1 npHMoro yrJIa fepcoHHA
AORaabIBaeT, liTO CYMMa yrJIOB TpeyrOJIbHHRa paBHa BYM npHMbIM
yrJIaM.
V nOCTYJIaT OKaSLIBaeTCH CHallaJIa nJIH CJIyqaH, RorJJ;3 npHMbIe
AB H CD nepeceKaIOTCH JIHHHeH AG, ncpneHAHRYJIHpHOH JIHHHH CD.
repcoHH onYCKaeT liS npOHSBOJIbHOH TOllKH H JIHHHH AB rrepneH-
AHKYJIHp Ha AG II YABaHBaeT OTpeSKH AH II AF AO Tex nop, rrORa OT-
peSOK AL, nOJIyqeHHLIH nOCJIe OBaTeJIbHLIM y BoeHHeM OTpeaRa AF,
He npeB30HAeT OTpeSOK AG. TorAa npRMaH CD nepeCel\aeT CTOPOHY
AL nOJIyqeHHOrO TpeyrOJIbHIIKa ALN, II, TaR KaR OHa He MomeT ne-
peCeqb CTOPOHY LN IlJIII BTOpOH pas nepeCeqb CTOPOHY AL, OHa He-
06XO HMO nepeceKaeT CTOPOHY AN, T. e. JIHHHIO AB. B aTOM BLIBo e
HCnOJIbayeTcH nepBas npennOCLIJIR8 repCOHII a, T. e. al\CHOMa EB OR-
ca-ApXHMeJJ;a, H HeHBHO npHMeHHeTCH aKCHOMa llama. Ha npHBe-
AeHHOM HaMil lIepTeme (pHC. 81) onpOBepraeTCH IIpe nOJIOmeHHe, qTO
npSMaH BTOpOH pas nepeceKaeT CTOPOHY AL TpeyrOJIbHHRa ALN
B qeJIOM AOKaaaTeJIbCTBO 01IeHb 6JIH3KO K JJ;ORaaaTeJIbCTBY H6I1 aJI-
XaicaMa. ,I(aJIee repcoHH , TaK me KaK M6H aJI-XaHCaM, ORaSbIBaeT
aKCHOMY 0 napaJIJIeJIbHbIX AJIH CJIyqaH, KorJJ;a Be npHMbIe nepece-
112
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II TynbIM yr JIOM.
BTopoe OKaSaTeJIbCTBO al\CHOMbI 0 napaJIJIeJIbHbIX, HaIIHcaHHoe
H3 lIaJIeKTe ApeBHeeBpeHcKoro HSMKa, KOTOpbIM nOJIb30BaJIHCb HC-
naHCKHe eBpeH, IIpHHa JIemHT HeKoeMY AJIb«pOHCO. TaK I\al\ aBTOp,
nHcaBillHH Ha peBHeeBpeiicKoM H3MKe, HOCHJI xpHCTHaHCI\Oe H1\IH,
OH, HeCOMHeHHO, HBJIHJICH Kpe eHLIM eBpeeM. I1s eJIoro pH a I\pe-
eHLIX eBpeeB, mHBillHX B I1cIIaHHH II HOCHBillHX HMH AJIb«pOHCO,
HaH60JIee BepOHTHhIM aBTopOM 8Toro Tpal\TaTa HBJIHeTCH AJIb«pOHCO
e BaJIbH OJIH (1270-1346), IISBeCTHLlH rJIaBHLIM 06pasoM Kal\ Bpaq
H aBTOp nOJIeMHqeCI\HX COqHHeHHH Ha peJIHrH03HbIe TeMLI, a TaKiKe
l\aK 3HaTOK KaJIeHAapeii. TpaKTaT AJIb«pOHCO HaSLIBaeTCH «BLInpH-
l\fHTeJIb I\pHBOrO» (Meiiammep (a1506). Ha PYI\OnHCb 3Toro TpaKTaTa,
xpaHH YIOCH B BpHTaHCI\OM MY3ee (Hebr. 86894/6), BnepBLIe 06pa-
THJI BHHMaHHe C. H. JIYPbe [15, c. 128, 141, 148, 183]; PYKOIIHCb
6hIJIa onHcaHa f. M. fJIYCKHHOH [8, 9] II ony6JIHKOBaHa B PYCCKOM
IIepeBo e [1 J. TpaKTaT HanHcaH 60JIee «pHJIOCO«pOM, -qeM MaTeMaTHI\OM.
B OCHOBHOM OH nOCBH eH I\Ba paType Kpyra H pyrHM np06JIeMaM,
CBH3aHHLIM C HH«pHHIITe3HMaJIbHLIMH paccymAeHHHMH. Bonpoc 06
al\CHOMe 0 napaJIJIeJIbHLIX 3 eCb CJIymHT npHMepOM nJIOAOTBOpHoro
npHMeHeHHH ABllmeHHH K reOMeTpHH. BHaqaJIe AJIb«pOHCO KpHTHKyeT
OKaSaTeJIbCTBO AraHllca, KOTopoe OH HaSIiIBaeT OKa3aTeJIbCTBOM aH-
HaiipHsH, a saTeM nplIBOAHT CBoe c06cTBeHHoe, COCTOH ee lIB meCTII
Dpe JIO ReHHii.
B I1pe JIOmeHIIII 1 nOK8SLIBaeTCH, 'tJTO npH npO OJImeHHH Me,n;HaHLI
.wTJ1060ro TpeyrOJIbHHKa Ha paccToHHHe, paBHoe 3TOH MeJJ;HaHe, IIOJIY-
qaeTCH 'tJeTbIpexyrOJIbHHK, Y KOToporo npOTlfBOnOJIOmHhIe CTOpOHLI H
npoTHBOnOJIOmHLle yrJIhI paBHDI; :3TO npe,n;JIOmeHHe OTHOCHTCH K a6-
COLTJIOTHOH reoMeTpHH. BO:.BTOPOM npe JIOmeHHH npe bI y ee nOCTpoe-
Hue aBTOp npHMeHHeT K npHMoyrOJIbHOMY TpeyrOJIbHHKY II, HCnOJIb-
3)'H TO me «npOCToe BHiReHlle», 'tJTO II M6H Roppa H M6H aJI-XaUCaM,
OKa3LIBaeT, 'ITO nOJIY'IeHHDIH 'tJeTDIpexyrOJIbHHK - npHMoyrOJIbHHK.
B npe,n;JIOmeHHHX 3-5 ,n;OKasaHDI pa3JIHQHhle 4>aKTbI, HBJIHIO HeCH
CJIe,n;CTBHeM aKCHOMbI 0 napaJIJIeJIbH.bIX, BbITeKaIO He II3 «,n;OKa3aHHO-
ro» AJIb«pOHCO cy ecTBoBaHIIH npHMoyrOJIbHHKa: paBeHCTBO CY:Mi\IbI
OCTp.bIX yrJIOB npHMoyrOJIbHIIKa BYM npHMbIM; BLIIIOJIHeHHe «rHl1OTe-
3bI npHMoro yrJIa» ,n;JIH'tJeTDIpexyrOJIbHHKa XaHHMa H paBeHCTBO Bepx-
Hero OCHOBaHHH 3Toro lIeTLIpexyrOJIbHHKa HHJRHe1\IY; Teope la 0 TO I,
QTO neplIeH HKYJIHp, ony eHHLIii H3 cepe HHDI npHMoyrOLTJbHOrO
TpeyrOJIbHHKa Ha ero KaTeT, ,n;eJIHT ero nOnOJIaM. B npeJJ;JIOmeHHH 6
AJIb«pOHCO BhlBO HT ,r IIoCTy aT, nOJIb3YHCb aKcHoMoH EB,IJ;oKCa-
ApxHMe,n;a. .
If repcoIlH , II AJIb<pOHCO HaXO HJIIICb nOJJ: CHJIbHhIM BJIIIHHHe}I
I16H aJI-XaHCaMa II LJ;pyrHx apa6cKHx l\laTeMaTlfKOB, nepeBOJJ;bI COllH-
HeHllH i KOTOpbIX Ha npeBHeeBpeHcKHH B 3TO BpeMH 6DIJIH ,n;OBOJIbHO
mHpoKo pacIIpocTpaHeHLI B IOmHoH EBpone.
CJIeJJ;YIOIIJ;HM aa «OIITHHOH» BH-
TeJIO H3BeCTHLIM HaM CO'tJHHeHHel\1 Ha
JIaTHHCKOl\1 H3LIKe, B KOTOpOM 06-
.
cym aJIaCb TeoplIH napaJIJIeJIbHhIX,
6bIJIO «ACTpoHoMH'tJecKoe 3epKaJIO,
SaKJIIO'tJalO ee 'tJeJIOBe"tJeCKHM pa3Yl\-1
B KamJJ;oH HaYKe» (Speculum ast-
ronomicum terminans intellectum
humanum in omni scientia)
<1>. B. fplIcoroHo. <1>e epIlK Bap
TOJIaH"tJ fpHcoroHo (1472-1538),
ypomeHen; IOrOCJIaBCKOro ropOJJ;3
3a apa, pa60TaJI B IITaJIIIII, Herb TpaKTaT 6hIJI ony6JIHKOBaH B
BeHen;HH B 1507 r. 3TO co'tJHHeHHe HeJJ;aBHO H3Y'tJHJI 3. CTHIIaHH'tJ [641.
B rJIaBe 9 3Toro co'tJHHeHHH, ROTopaH Ha3hIBaeTCH «0 napaJIJIeJIbHbIX
JIHHHHX», fpHcoroBo KpHTHKyeT «MHorIIX 1\laTeMaTHKOB, aHTlIlIHbIX,
apa6cKHx II JIaTHHCKHX», KOTopLIe nblTaJIHCb ,IJ;OKaaaTb V IIOCTYJIaT,
I1CXO H liS onpe,n;eJIeHHH napaJIJIeJIbHhIX JIHHHi, KaK paBHOOTCTOH HX.
YKasaB, 'tJTO o cero nHH OH He HameJI perneHHH, fpHcorOHO BbI-
CKasaR HHTepeCHYID MhICJIb: «H Bo06pa3HJI ce6e cnoc06 IIpOBe,n;eHHH
Ha. OJ];HOH nOBepxHOCTH BYX HepaBHOOTCTOH HX IIpHMhIX JIHHU:U,
KOTopLle MomHO npo,n;OJImHTb ,n;o 6eCKOHe'tJHOCTH II KOTopLIe, TeM He
l\leHee, MorYT HIIRor,n;a He BCTpeTHTbCH» [64, c. 371]. 3. CTHnaHH'tJ
neveBeJI JIaTIIHCKoe CJIOBO «superficie» - «llOBepXHOCTb» cep60xop-
BaTCKHM CJIOBOM «ravan» - «IIJIOCKOCTb» If OTMeTHJI, 'tJTO npHMhIe,
o KOTOpDIX rOBOplIJI fpllcoroHo, IIMeIDT MeCTO Ha rHnep60JIIIlIeCKOH
nJIOCKOCTH. O HaKo fpucoroHo Hlfr e He rOBopHT, 'tJTO nOBepXHOCTb,
19.
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napaJtJteJtb1-tbtX
l3 El3pone
XV I-XV I I I l3e1<:O(3
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o KOTOpOH IIJ];eT pe"tJb, HBJIHeTCH IIJIOCKOCTbIO. T01iHO CXaSaTb, lITO OH
1I le:I B BHAY, MbI He }'IOfI\eM; ero paccym eHIIH 3aCJIymHBaIOT J];aJIb-
He:UIlIero aHaJIH3a.
MbI ynoMHHaJIH pH aBTopOB, nlIcaBIDlIX Ha rpe'tJeCKOM II apa6CKO I
H3bIKax, nbITaBIlIHXCH ,UOKa3aTb V nOCTYJIaT, IICXO H DS YKasaHHoro
fplIcorOHO onpeJJ;eJIeHHH na paJIJIeJIbHhIX JIIIHHH. B03MOJI\HO, lITO, ro-
BOpH 0 JIaTHHCKHX aBTopax, fpHcorOHO IIMeJI B- BHJJ;y BIITeJIo,
8 B03:\fOiRHO, "tJTO lI?vIeJIIfCb II J];pyrHe, HeH3BeCTHbIe HaM JIaTHHCKHe CO-
"tJlIlIeHIIH, B KOTOphIX JJ;OKaabIBaeTCH V nOCTYJIaT.
B 1572 r. Ha JIaTIIHCKOM H3hIRe 6bIJIH HaI1e1JaTaHLI cpa3Y .n;Ba .n;OKa-
38TeJIbCTB"a V nOCTYJIaTa EBKJIHJJ;a - B H3 aHIIH PllcHepa «OnTHKII»
BlITe:IO H AOKaaaTeJIbCTBO IIpoKJIa B JIaTHHCKOM nepeBo.n;e «Ha"tJa.rr»
EBKJIIIAa, BLInOJIHeHHOM <De epllro ROMMaH HHo (1509-1575) [49].
qepea Ba ro a nOHBJIHeTCH opHrlIHaJIbHOe nOKa3aTeJIbCTBO V no-
CTYJIaTa, npIIHaJJ;JIema ee ypomeHn;y fa!\I6ypra XpHCTO.pOpy RJIaBIIIO
(IlIJIlOcceJIIO, 1537-1612), pa60TaBmeMY B PIIMe H yqaCTBOBaBme IY
TaM B pa3pa60TKe rperoplIaHCKoro KaJIeH,napH. 8TO «JJ;OKaSaTeJIbCTBO»
npHBe CHO B 113JIOmeHHH RJIaBHH «HaqaJI» EBKJIH a, ony6JIHKOBaH-
HOM: BnepBbIe B 1574 r. [43].
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Kam aH TO'tJKa KOTOpOll OTCTOHT OT npHMOll :IlIHHH, HaXO H eHCH B
rOll me llJIOCKOCTH, Ha paBHOM paCCTOHHIIH, HBJIHeTCH npH1\IOH» [43,
c. 50 J. RJIaBHH IICXO HT lIa eBKJIH}J;oBa onpe,neJIeHHH npHMoi KaK T{H-
HHII, «paRHO pacnOJIOmeHHOH no OTHomeHMIO KO BceM To"tJKaM Ha Hell».
TaK KaR Bce TO"tJF"tI paCC?vIaTpHBaeMOH H?vI .illIHHII OTCTOHT OT npHMO:U
Ha paBHOM paCCTOHHHH, TO RJIaBlIH C'tJIITaeT, 'tJTO 3T8 JIIIHHH «paBHO
pacnOJIOmeHa» 110 OTHomeHlI1O K TO'tJKaM Ha Hei, T. e. eCTb npHMaH. OH
onpOBepraeT npe}J;nOJIOme.HHe, 'tJTO 3Ta JIHHHH HBJIHeTCH oKpymHocTbIO,
TaK RaE JIHHHH, paBHOOTCTOH aH OT oKpymHocTH, HBJIHeTCH oKpym-
HOCTblO II, eCJIH 6LI paCC?vIaTpHBaeMaH JIIIHHH 6LIJIa oKpymHocTblO,
oKpymnocTblO }J;OJIif\Ha 6bIJIa 6LITb II }J;aHHaH JIIIHHH. OH C'tJHTaeT oqe-
BH}J;HLIM, tJ:TO, KpOMe npHMOH II oKpymHocTII, pyrHx JIIIHHH, «paBHO
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cy eCTBY, }J;onYCKalO HX HaJIOmeHHe Ha ce6H, He MomeT 6bITb. B rn-
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HOCTH, HBJIHeTCH 3KBH HcTaHTa, KOTopaH II HBJIHeTCH B 3TOM cJIyqae
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Ioii.
RJIaBHH He ynoMHHaeT HMen apa6cKIIx Y'tJeHLIx, no rOBoplIT: «J!
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IHe HHKor}J;8 He npeJ];CTaBJIHJIOCb B03MomnOCTII npOlleCTL 9TO }J;OKa3a-
TeJIbCTBO, XOTH H ynopHo HeCKOJIbKO pas npOCHJI 8TO Y Toro, HTO BJIa-
neeT 8TIIM apa6CKIIM EBKJIH}J;OM» [43, c. 503]. nO-BHAHMOMY, RJIaBIIH
6bIJI 3HaHOM C JJ;OKaSaTeJIbCTBOM M6H ROpPhI IIJIII H6H aJI-XaUCaMa
113 BTOphIX PYK. OC06eHHO] 6JIHSKII K M6H Roppe II .lI6n an-XaiiCaMY
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meTCH IIO npyroH npHMOH, o6pa;3YH CBOHJ.I ROH:U;OM C Hei Bce Bpel\IH
npH1\IOH yrOJI, ee pyroii KOHen; onllChIBaeT npHMYIO JIBHHIO»; 2) «Ec-
H R npllMOH JIHHHI:I BOCCTaBJIeHhI }J;Ba p8BHb1X nepneH}J;IIKYJIHpa, KO-
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npJIMYIO, paBeH nepBOMY nepneH,IJ;HRYJIJIpy» [43, c. 51-52]. 3aTeM
RJIaBHH paCCMaTpHB8eT qeTLIpexyrOJIbHHK XaHHMa II ,IJ;OKa3MBaeT,
qTO ero BepXHHe ,IJ;Ba yrJIa npHMhIe. 3JJ;ecb, O,IJ;HaKO, He paCCMaTpI:IBaIOT-
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yrOJIbHHRa ABCD (pRC. 82). COrJIaCHO npe,IJ;LI,IJ;YmeMY npe,IJ;rrOJIome-
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yrOJIbHIIKaM AEFC B EBDF, TaR qTO Bce BepXHHe yrJIbI 3THX -qeTLI-
pexyrOJIbHHKOB TaKme paBHbI ,IJ;pyr JJ;pyry. Ho yrJILI CFE II DFE
CMemHbIe, n03TOMY Bce 3TH yrJIhI npHMhIe.
,n:ORa3aB cy ecTBoBaHlIe rrpHMoyrOJIbHHKa, RJIaBIIH n;OK8shIBaeT
V nOCTYJIaT crrepBa, KaR II MHorHe apa6CKHe yqeHLIe, ,IJ;JIJI CJIyqaH
rrepIIeH,IJ;HKYJIJIpa K HaRJIOHHOH, a 3aTeM ,IJ;JIJI o6 ero CJIyqaH.
B caMOM HaqaJIe XVII B. B BOJIOHbe 6bIJIH ony6JIHKOBaHhI «He6oJIL-
moe COQIIHeHHe 0 paBHOOTCTOH HX II HepaBHOOTCTOH HX npJIMLIX
JIHHHJIX» [42, 11. 1] II « OIIOJIHeHHe R He6olIbillOMY COqHHeHHIO 0 paB-
HOOTCTOJI HX H H.epaBHOOTCTOHII\HX rrpJIMMX JIHHHJIX» [42, q. 2] I1beT-
po AHTOHHO KaTaJIb,IJ;1I (1548-1626), oco6eHHo H3BeCTHoro OTKp:bITl:IeM
D;enHhIX ,IJ;po6ei. IIepBoe COqHHeHHe HaTaJIb,IJ;H COJJ;epmHT ,IJ;Ba onpe-
,IJ;eJIeHHJI H IIpeJJ;JIOmeHHJI 1-15, BTopoe - npeJJ;JIOmeHI:IJI 16-33 II
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rrapaJIJIeJIbHhIX. B onpeJJ;eJIeHHH 1 orrpe,IJ;eJIHeTCJI paCCTOHHlle OT TOq-
KH ,IJ;O npJIMoiI, B onpe,n;eJIeHHH 2 - paBHooTcToH He II HepaBHOOT-
CTOJI lIe npHMLIe. lIs oIiy eHIIH 0 cy ecTBoBaHHIJ paBHOOTCTOH HX
npJIMLIX yme B rrepBOM COqHHeHIII:I RaTaJIb,IJ;H BhIBO,IJ;HT pJI,n; YTBepiRJJ;e-
Hili, 3RBHBaJIeUTHMX YCHJIeHHOH aKCHOMe 0 rrapaJIJIeJIbHhIx. TaKOBbI,
116
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ABe npHMhIe JIHHHH - paBHOOTCTOH He, TO JIHHHH, npOBeneHHLIe
OT nepBOH nepneHAHKYJIHpHO KO BTOpOH, 6YAYT rrepneHJJ;HKYJIHpHhI K
nepBOH JIHHHH» [42, q. 1, c. 3] IIJIII OJJ;HO H3 CJIe}];CTBHH H3 rrpeAJIOme-
HUH 10: «ECJIH CJIOiKIITL TPH yrJIa npOH3BOJIbHOrO TpeyrOJIbHIIKa, TO
B CYMMe nOJIyqaTCH }];Ba npHMhIX yrJIa» [42, q. 1, c. 16]. oKa3aTeJIbCTBO
V nOCTYJIaTa y KaTaJIb]];H OCHOBhIBaeTCH Ha npeJJ;JIOmeHHH 30: < ECJIH
.nBe HepaBHOOTCTOHII:(Me npHMhIe JIHHHH CXOJ]JITCH C O]];HOH CTOpOHbI, TO
He06xo]];HMO, "lIT06bI npH npOJJ;OJImeHHH B TY CTOPOHY, B KOTOpyIO OHII
npH6JIHmaIOTCH, B KOH e KOHIJ;OB OHH nepeCeKJIHCb» [42, q. 2, c. 32].
<l>oPMYJIHpOBKa 8Toro YTBepmJJ;eHHH COBIIaJ];aeT C «IIpIIHIJ;HrrOl\O),
npHnHChIBaeMhI { XanHMOM ApHCTOTeJIIO.
R XVII B. OTHOCHTCH TaKme TeopHH napaJIJIeJIbHLIX moBaHHH
AJIb«pOHCO BOpeJIJIB (1608-1679) 11 BHTaJIe mOp]];aHO (1633-1711).
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meHHIO K ]];PyroH IIpH IOH T3KIIM 06pa30M, qTO BCer]];a npHMbIKaeT K
nocJIenHeii CBOHM KOHD;OM, TO JJ;pyraH 1\0HeQHaH TO"IIKa OTpe3Ka B
CBOCM JJ;BHiHeHHII onHilleT npHMYIO JIMHIIIO» [40, c. 32]. lIo}]; CJIOBaMH
«rrepeHOCHTCH rrorrepeK» IIMeeTCH B BH]];y nepeHOC no}]; nOCTOHHHhIM
yr TIOM K AaHHoH npHMoH, II03TOMY YTBepmJJ;eHlIe BOpeJIJIH, no cYII:(e-
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KaK II 3TH apa6CKIIe IaTeMaTHKII, BopeJIJlII nbITaeTCH 060CHORaTh CBOB
npllH Hrr C nOMO hIO I\IIHeMaTHq.eCKHX co06pameHHH. 3aTeM OH JJ;O-
Ka3bIBaeT, "lITO ABa nepneH;lJ;HKY JIflpa H oJJ;Hoii npHMOR paBHOOTCTOHT
JJ;pyr OT JJ;pyra, orrpe;lJ;eJIHeT rrapH.JIJIeJIhHhIe npHMble KaK P3BHOOTCTOH-
II:(He II BhIBO]];HT V rrOCTYJIaT 113 yu ecTBOBaHHH npHMoyrOJlhHMK3.
AHaJIOrlI"lIHhIH BhInOJJ; V nOCTY.Hhld H leeTCH II B «BOCCTaHOBJleHHOM
EBKJIIIJJ;e, IIJlH BOCCTaHOBJIeHHbIX H 06JIerqeHHbIX aHTMqHhIX f-Iaqa-
JIaX reOMeTplI1I B 15 I\HlIraX» [49a] BHT3011P I{mopJJ;aHo. TIpeJJ;JIOmeHlIe
o TOM, qTO reOMeTpHqeCKOe MeCTO npHMbIX, paBHOOTCTOHru;HX OT npH-
l\IOH, eCTb rrpHMaH, 060CHOBhIBaeTCH 3J{eCh C no \1 0 LF T}!C .,1 eM {bI: «ECJIH
"lIepe3 ;lJ;Be TOQKM l\aKon-HH6YJJ;b KplIBon, 06paII:(eHHon cBoeR BorHY-
TOCThlO 1\ D, npOBe]];eHa npHMaH A C II eCJIH II3 6eCI\OHeQHO 60JlbIIIOrO
qlICJJ.a TOqeK }];yrH A C OIIYCTHTh nepneH]];HKYJIHpbI Ha npHMYJO, TO H
YTBep;I\AalO, "lITO HeB03 10mHO, qT06bI 3TH nepneH;lJ;MKYJIHphI 6hIJIH
p3BHhI JJ;pyr JJ;Pyry» [49a, c. 41]. .n:mopJJ;aHo paCCl\IaTpUBaeT npHMYIO
Gl", COe]];HHHeT 1\0HD;bI paBHbIX nepneHJJ;MI\YJlHPOB AG II DBF npHMoH
AC (pHC. 83). JIHHHH ABC, pa3YMeeTcH, He HBJIHeTCH paBHOOTCTOH-
II:(eH OT rrpHMOH GF, OJJ;HaKO rrpH JJ;OK33aTeJIbCTBe Toro'l qTO reOMeTpH-
qeCKOe MeCTO TO"IIel\, paBHOOTCTOHII:(HX OT npHMOH, eCTb npH?laH,
,[(a\opJJ;aHO npHMeHHeT 8TY JIel\IMY K 'tJepTemaM, Ha 1\0TOpbIX He HMelOT
MeCTa COOTHomeHHH MemJJ;Y npH IOH GF H ]];yron ABC. ,n:aJIee ,IJ;mop-
]];aHO paCCMaTpHBaeT qeTblpexyrOJIbHHK XaHHMa II JJ;OHa3hIBaeT, qTO
ero BepXHHe yrJIhI rrpHMble, 3 H3 cy ecTBoBaHIIH npHMoyroJlhHHHa
BhIBOJJ;IIT V nOCTYJIaT.
R XVII B. OTHOCHTCH He paa ynoMHHaBillHHCH TpaKTaT mOHa
BaJIJIHCa «0 nHTOM nocTYJIaTeHnHToM onpeJJ;eJIeHHH B HHMre VI EBKJIH-
117
]];a, reO leTplI eCROe pacCYn\;n;eHHe» [68], orry6JIIIHOBaHHLIH B 1693 r.
B co6paHMII COqHHeHIIH BaJIJIHCa, HO HarrHcaHHbIM 3HaqMTeJIbHO paHb-
me. TpaHTaT COCTOIIT 113 Tpex qaCTeH: B rrepBOH paCC laTpHBaeTCH TaR
Ha3LIBael\IOe onpeneJIeHlIe 5 HHHrlI VI «Ha aJI» - onpe;n;eJIeHHe co-
CTaBHoro OTHomeIIIIH, HnJIHIO eeCH nCTaBROH rro3.n;HeHillHx pe,naKTO-
pOB. Bo BTOpOH qaCTII rrplIBeAeH rrepeBO,n 8. TIoRoRa C apa6cRoro
.n;Ol\a3aTeJIbCTBa V nOCTYJIa1'a 113 yrroMIIHaBIIIerOCH HaJ\1I1 COqIlHeHHH,
rrpHrrlICbIBaeMoro a1'- Tycn. B 1'peTbeH aC1'H ;o;aHO AOHaaaTeJIbCTBO V
rrOCTYJIa1'a, rrpIIHa;O;JIea-\aIII.ee caMOMY BaJIJIlIcy. BaJI '1MC nORa3bIBaeT
V nOCTYJlaT C rrOMO bIO CJIe;O;YIOID;ero nOC1'YJIaTa: «HaKOHeu;, 11 rrpH IY
(CqlITaH yme H3BeC1'HbIMH rrpHpoAY OTHOIlIeHHH II yqeHlIe 0 rroAo6HbIX
.pHrypax) B RaqeCTBe 06 ero nOHHTIIH: ;o;JlH BCHROM .pnrYPLI B03l\IOmHa
;o;pyraH no,n;06HaH en .pHrypa npOH3BOJlbHOH BeJII1:qIIHLI [68, c. 676J.
BaJIJIHC 060CHoBLIBaeT eCTeCTBeHHOCTb BToro «06mero rrOHlITIIH», T. e.
aRCIIOMLI, TeM, qTO ero qaCTHbIM CJIyqaeM lIBJlHe1'CH III IIOCTYJIaT EBK-
JIHAa «M3 BClIKOrO D;eHTpa II BCHRHM paCTBopOM MO l\eT 6bITb onHcaH
Rpyr» [10, T. 1, c. 14]. ,I:(JIlI Toro qT06hI AORa3a1'b, q1'O ABe npHMLIe
AB H CD, 06pa3YIOID;He C npHMoH AC yrJILI, B CYM Ie MeHbillHe ABYX
rrpHMhIX, nepeceRaIOTCH (pHC. 84), BaJIJIHC nepe;n;BHraeT 3TY npHMYIO
AB B;n;OJIb npHMoH A C TaR, lITO yrOJI CAB OCTaeTCH rrOCTOHHHLIl\tJ.
I1pH ,nOCTaTOqHOM rrpH6J1Hfl\eHHH K TO Ke C npHl\IaH AB 3anMeT rro-
JlOmeHlle a , B ROTOpO:M OHa nepeCeqeTCn C rrpHMoH CD B TOqRe :TI.
no rrpe.n;nOJIOmeHHIO BaJIJIlIca, ;n;JIH nOJlyqe:iIHoro TpeyrOJIbHHRa aCn
IIMee1'CH rro;n;06HLIH TpeyroJlbHHK ACP'; TOqRa P II HBJIHeTCH HCROMon
TO ROii nepeCeqeHHH. OTMeTIIM, qTO rrepeABHmeHne npHMoH AB
B.n;OJlb npHMoH A C CHOBa HanOMHHaeT HaM KHHeMaTlIqeCKHe paccym-
.n;eHIIH M6H HOppbI 11 M6H aJI-XaHcaMa.
TIo pa60Te BaJIJIHCa rr03HaROMHJICH C COqHHeHHeM, npHnHCLIBae-
?-IbIM aT-TYCH, H ,I:(mHpOJIal\IO CaRKepn, ROTOpLIH B CBoeM «EBRJIHAe,
OqH eHHOM OT Bcex nHTeH, HJIH feOMeTplIqeCROH nOrrbITl\e YCTaHoBHTb
ca:MbIe rrepBbIe HaqaJIa BceH reOMeTpHH» [62] nO;n;Bepr RpHTHRe ;n;OKa-
aaTeJlbCTBa BaJlJIllca II nceBAo-Tycn. M. r. JIaM6epT [55J B CBOIO oqe-
peAb 6b1JI 3HaKOM C COqHHeHHeM CaRKeplI II ;n;pyrHx 60JIee paHHHX
aB1'opOB eCJIH He Henocpe,n;CTBeHHO, TO, no RpallHeiI Mepe, no yrro-
MHHYTOH BhIllie AHccepTaD;HH f. C. HJIIOreJlH [54], ony6JIHROBaBwero
B 1763 r. RpllTHqeCKHB aHaJIIIS Bcex BamHeHmHX npe,n;bIAym;HX AOKa-
aaTeJIbCTB nOCTYJIaTa 0 napaJIJIeJIbHhIX H rrpHmenmero K SaKJIIO-
qeHHIO, qTO Bce OHH HeYAOBJIeTBOpIITeJIbHhI.
MLI yme HeCKOJIbKO pas ynoMHHaJIII COqllHeHHH CaKKepll II JIaM-
6epTa, KOTopbIe lIayqaJIH CBOHCTBa Tex me qeTbIpexyrOJIbHHKOB,
qTO II apa6cKlle aBTopbI. TpYAhI 060HX 3TIIX MaTeMaTHKOB B HeKO-
TOpOM CMLICJIe HenocpenCTBeHHO npeAmeCTBOBaJIH c03AaHllIO nepBLIX
HeeBKJIH;n;OBbIX CHCTeM H n08TOMY aacnymHBaIOT" OTJ];eJJbHoro pac-
CMOTpeHHH.
CaKKepH no,n;Bepr KpHTHKe ,n;OKa3aTenbCTBO V nocTynaTa y BaJI-
JIHCa II rrorrblTKY J];OKa3aTeJIbCTBa IIceB,n;o- TYCH II IIpenJIOmHJI c06-
C1'BeHHoe nOKaaaTeJIbCTBO nOCTYJIaTa. OH paCCMOTpee!"1 TOT me paB-
H06e,npeHHbIH ;n;ByrrpllMoyronbHHK, liTO H XaHHM H 81'-Tycn, II BbI-
CKaa8JI Te me TPH runOTe3bI 0 ero BepXHHX yrJlax.
118
rllnOTe3Y Tynoro YI'JIa CaKKeplI onpOBepI'aeT Te I <paKTOM, "IlTO
B aTOM CJIyqae, KaK II B CJIyqae I'llITOTe3bI rrpHMOI'O YI'JIa, BepeH V
nOCTYJIaT EBKJIHJ];a. OTCIOJ];a CaKKepH J];eJIaeT BbIBOJ];, qTO rrpll I'Hno-
Te3e Tynoro yr JIa IIMeeT MeCTO 061>IqHaH I'eOMeTpIIH, BCJIeJ];CTBlIe
"Ilero I'HnOTe3a Tynoro YI'JIa «BCen;eJIO JIOmHa», 1I60 caMa ce6H pa3-
pYiliaeT. CMbICJI 3TOI'O paccymn;eHHH COCTOIIT B TOA:I, lITO I'HrfoTe3a
Tynoro YI'JIa, nplI KOTOPOH BblnOJIHHeTCH V nOCTYJIaT, rrpOTIIBOpeqllT
OCTaJIbHbIM aKCHOMaM I'eOMeTpHII EBKJIII;r(a.
IIocJIe 3TOI'O CaKKeplI nepeXOJ];HT K onpOBepmeHHIO I'HnOTe3bI
OCTpoI'O YI'JIa. 3J];eCb OH <paKTHqeCKH rrpOHHKaeT I'opa3;r(0 ;r(aJIbme,
qeA:I ero rrpeJ];illeCTBeHHHKII, B HaqaJIa I'eOMeTpHH JI06a"tJeBCKOI'O.
OH ;r(OKa3bIBaeT, "tJTO B 3TOM CJIyqae J];Be npHMble IIJIII nepeceKaroTcH,
lIJIII IIMelOT 06u lIli nepneHJ];HKYJIHp, no 06e CTOpOHbl OT KOTOpoI'O
OHH yJ];aJIHIOTCH J];pyr OT J];pYI'a, nJIII yJ];aJIHIOTCH J];pyr OT ;r(pyra C O;r(-
HOH CTOpOHbI, IIJIII aCIIMnTOTlIqeCKII C6JIIImalOTCH C J];pYI'oH CTOpOHbl.
B nOCJIeJ];HeM CJIyqae CaKKeplI npIIXOJ];IIT K 3aKJIIO"tJeHHIO, qTO 3TlI
npH lhIe J];OJImHbI HMeTb 06 ylO TOqKY II 06mlIii nepneH;r(JIKYJIHp
B 6eCKOHe"llHOCTH. PeaJIbHbIH CMbIcn 3TOI'O BbIBOJ];a COCTOHT B TOM,
QTO eCJIH orrYCKaTb H3 TOqeK OJ];HOH 1I33THX npHMbIX nepneHJ];HKYJIHpbI
Ha ,n;pYI'YIO npHMYro, TO npH CTpeMJIeHHII TOqKH B 6eCKOHe"tJHOCTb
J];JIIIHa 3TOI'O rreprreHJ];HKYJIJlpa CTpeMHTCJI K HYJIIO, a YI'OJI, COCTaB-
JIHeMhIH HM C nepBoii npHMOH,- K npHMOMY YI'JIY. OJ];HaKO CaKKeplI
npe,n;CTaBJIHJI ce6e 06 ylO TOqKY H nepBbIll nepneH;r(HKYJIHp B 6ec-
KOHeQHOCTII KaK 06blqHbIe 06 ylO TOqKY II 06 Hll nepneHJ];IIRYJIHp
II ,n;eJIaJI OTCIOJ];a BbIBO;r(, lITO B TpeTbeM CJIyqae J];Be npHMble KacalOTCH
J];pyr J];pyra B 6eCKOHeqHOCTH; OTKYJ];a 3aKJIlOqaJI, qTO I'lIrrOTe3a OCT pO-
1'0 YI'JIa JIOmHa, KaK npOTHBOpeq aH nplIpO;r(e rrpHMOH JIIIHIIII.
He yJ];OBJIeTBOpHBillIICb 3THM J];OKa3aTeJIbCTBOM, CaKKeplI pac-
CMaTplIBaeT I'eOMeTplIqeCKOe MeCTO TOqeK nJIOCKOCTII, paBHOOTCTOH-
IIX OT rrpHMoll. B OTJIlIqlIe OT CBOIIX npeJ];IIIeCTBeHHIIKOB OH 3HaeT,
qTQ 3Ta JIIIHHJI B CJIyqae I'lIrrOTe3bI OCTpoI'O yr JIa He rrpHMaH II He
oKpymHoCTb. BbIqHCJIHH J];JIHHY J];YI'H 3TOH JIHHHII, CaKKepn ,n;eJIaeT
OIIIu6KY II IIaXOJ];HT, qTO 3Ta J];JIHHa paBHa paCCTOHHHIO MeiR;r(y OCHO-
BaHlIHMH rrepne J];HKYJIHpOB, ony eHHbIX 1I3 3TlIX TOqeK Ha npHMYro.
C pyroll CTOpOHbI, paHee OH nOKa3aJI, qTO 3TH nepneHJ];HKYJIHpbI
yJ];aJIHIOTCH J];pyr OT J];pyra II paCCTOHHHH Meil\;r(y KOHn;aMH J];YI'H, He
I'OBopH yme 0 J];JIIIHe 3TOll J];YI'II, J];OJImHbI 6bITb 60JIbIIIe paCCTOHHHH
M:elli,n;y OCHOBaHHHMH nepneHJ];HKYJIHpOB. IIOJIyqlIB 3TO npOTHBope-
que, OH BHOBb 06 HBHJI I'HnOTe3Y OCTpoI'O YI'JIa onpoBepI'HYTon.
PaccYll\:J];eHHH JIaA:I6epTa cOJ];epmaJIII 60JIee npHMeqaTeJIbHbIe MbICJIII.
fipHBeJ];H <POPMYJIllpOBKY «11-H aKCHOMbI» (T. e. V nocTYJIaTa), JIaA:l-
6epT I'OBopHT, qTO 3TO OCHOBHoe YTBepmJ];eHHe J];aJIeKO He TaK HCHO
II OqeBHJ];HO, KaK OCTaJIbHble.
,IJ:aJIee OH paCCA:IaTpHBaeT TOT me TpHnpJlMoYI'OJILHHK, "IlTO 116H
aJI-XaHCaM, H Te me I'HrrOTe3bI npHMOI'O, Tynoro II OCTpoI'O YI'JIOB.
JIaM6epT, KaK H CaKKepH, 6e3 TpYJ];a onpOBepI'aeT I'HnoTeay
Tynoro YI'JIa, nOKa3bIBaH, '1ITO B ee CJIY"llae J];Ba rreprreHJ];HKYJIHpa
K OnHOH npJlMOH nepeceKalOTCH, qTO He rrpOTHBOpeqHT V nOCTYJIaTY,
HO rrpOTIIBOpeqHT OCTaJIbHbIM aKCHOMaM I'eOMeTpHH EBKJIII.[{a. OR
119
OTMeqaeT, qTO rHIIOTeaa TYIIOrO yrJIa HMeeT MeCTO Ha c<pepe, eCJIII
B KaqeCTBe IIpHMbIX paCCMaTpllBaTb ee 60JIbllilIe KpyrlI.
lIepexoJI;H K rlIIIOTeae OCTpOrO yr JIa, JIaM6epT JI;OKaabIBaeT e e
60JIbme YTBepmJI;eHIIH, HMeIO HX MeCTO B reOMeTpBB JI06aqeBCKOrO,
'1IeM CaKKeplI.. B qaCTHOCTH, OH HaXO,nIIT, qTO CYMMa yrJIOB TpeyrOJIb-
HHl\a IIPII rlIIIOTeae OCTpOrO yrJIa MeHbme nBYX npJlMbIX, II, cpaBHH-
BaH 3TOT <paKT C TeopeMOH 0 TOM, '1ITO B CJIyqae rHnOTeabI TYIIoro
yrJIa CYMMa yrJIOB TpeyrOJIbHHKa 60JIbme JI;BYX npHMbIX, aaHBJIHeT,
QTO TaKIIM 06paaOM npll TpeTbell rHnOTeae MomHO HJI;TH e e JI;aJIbme.
OH CTpeMHTCH nOJIyqHTb TaKHM 06paaol\1 pHJI; napaJI;OKCaJIbHbIX CJIen-
CTBIIH C n;eJIbIO BbIHBIITb, He o6HapymaTcH JIB IIPII 3TO}1 npOTHBO-
peqlIH. OH IIpHaHaeT, qTO OIIpOBeprHYTb 3TY rHIIoTeay COBceM He-
JIerKO.
H aH60JIee HHTepeCHbIM H3 3TIIX CJIe,nCTBHll HBJIHeTCJI TO, qTO eCJIII
6bI HMeJIa MeCTO TpeTbH rlIllOTeaa, TO cy ecTBOBaJIa 6LI a6COJIIOTHall
Mepa JI;JIIIHbI KamJI;Oll JIIIHHH, nJIo aJI;II HCHKOH nOBepXHOCTII II 06'beMa
JII060ro TeJIa. KOHeqHO, TaKOH BbIBOJI; YJI;HBHTeJIeH, H60 HHKTO paHee
He COMHeBaJICH, '1ITO HeT HIIKaKOH a6COJIIOTHOll MepbI npOCTpaHCTBa.
OTOT BbIBO JIaM6epT Ha3bIBaeT «BOCXHTIITeJIbHbIl\I» H BbIabIBalO H}1
meJIaHlle, qT06bI TpeTbH rlIHOTeaa 6bIJIa CnpaBeJI;JIIIBa. BIIpo'tJe l,
JI;06aBJIHeT JIaM6epT, IIPII .3TOM BOaHHKJIII 6bI cepbe3HbIe HeYJI;06cTBa:
TplIrOHOMeTplIqeCKHe Ta6JIII LI CTaJIII 6bI CJIIIIIIKOM 06mlIpHbIMH,
HCqe3JIII 6bI 1I0JI;061Ie II llponop HOHaJIbHOCTb <pHryp - BeAb HH
OJI;HY <pHrypy HeJIbaH 6bIJIO 6LI Ha06paaHTb HHa'tJe, qeM B HaTypaJIb-
HYIO BeJIHqIIHY.
BbIHCHHB, qTO IIpll rHIIOTeae OCTporo yrJIa CYMMa yrJIOB TpeyrOJIb-
HHKa MeHbIIIe JI;BYX npHMbIX, JIaM6epT nOKaaLIBaeT, '1ITO HenOCTaTOK
3TOH CYMMbI yrJIOB JI;O JI;BYX npHMbIX IIpOnOpn;IIOHaJIeH nJIom.ann
TpeyrOJIbHHKa. JIaM6epT COIIOCTaBJIHeT 3TO C TeM, qTO Ha c<pepe, rJI;e
HMeeT MeCTO rHIIOTeaa TYIIoro yrJIa, CYMMa yrJIOB TpeyrOJIbHHRa
60JIbIIIe JI;BYX IIpHMbIX II Ha6bITOK 8TO:H CYl\IMbI HaJI; JI;BYMH npHMbIl\fH
npOIIOpn;HOHaJIeH IIJIO aJI;H TpeyrOJIbHHKa; OH OTMeqaeT, qTO CKaaaH
Hoe HM 0 C<pepHqeCKHX TpeyrOJIbHHKaX JI;OKaaMBaeTCH HeaaBHCHMO
«OT TPYJJ:HOCTII IIa paJIJIeJIbHbIX JIHHHH» II B rrpeJI;IIOJIOmeHHH TOJIbKO
TOH aKCHOMbI, '1ITO BCHKaH nJIOCKOCTb, IIpOXOJI;Hm.aJl '1Iepea eHTp
C<pePbI, JI;eJIHT ee Ha JI;Be paBHLIe 'tJaCTH.
EbITb MomeT, HaH60JIee aaMeqaTeJIbHbIM HBJIHeTCH npOaOpJIHBOe
YKaaaHHe JIaM6epTa 0 TOM, qTO OTCIO):(a OH «):(OJImeH 6LIJI 6LI nO'1lTII
CneJIaTb BbIBon, '1ITO TpeTbH rHnOTeaa IIpOHBJIHeTCH Ha nOBepXHOCTJI
MHHMOH C<pepLI. lIo KpaHHeH Mepe, JI;OJImHO 6bITb He'1lTO, nOqeMY Ha
llJIOCKOCTH ee He ynaeTcH onpOBeprHYTb TaK npOCTO, KaK BTOpyro
rHnoTeay» [14a, c. 259]. II JI;eHCTBHTeJIbHO, B KOH e XIX B. AHp1t
IIyaHKape nOKaaaJI, qTO, non06HO TOMY, KaK 3JIJIHnTH'1IeCKaH
reOMeTpHH ocymeCTBJIHeTCH Ha 06bIqHOH c<pepe, eCJIH OTom,neCTBHTb
ee JI;HaMeTpaJIbHO npOTHBOnOJIOmHbIe TOqKH, rHnep6oJIH'1IeCKaH
reOMeTpHH aHaJIOrHqHbIM 06paaoM ocym.eCTBJIHeTCH Ha ,nBynOJIOCTHOM
rHnep6oJIOHJI;e, eCJIH aaMeHHTb B npOCTpaHCTBe 06LPIHYlO eBKJIHJI;OBY
MeTpHRY «IIceBJI;oeBKJIHnOBOH», B KOTOpOH 3TOT rHnep60JIOHA 6yn eT
c epoH MHHMoro panHyca. ECJIH CIIpoeKTlIpOBaTb 3TY «MHHMYIO
120
c<pepy» H3 ee n;eHTpa Ha KaCaTeJIbHYIO IIJIOCKOCTb, MbI 1I0JIY"IlHM IIH-
TepnpeTan;lilO BeJIbTpaMH - RJIeHHa rlIllep60JIH"IleCKOH IIJIOCKOCTII Ha
06:bI"IlHOH llJIOCKOCTII, a eCJIII cnpoeKTlIpOBaTb TY me c«J)epy 1I3
OJJ;Horo ee nOJIlOca Ha KaCaTeJIbHYIO llJIOCKOCTb B JJ;PyroM 1I0JIIOCe,
MbI nOJIY"IlllM HHTeplIpeTan;HIO IIyaHKa pe rlIllep60JIlIQeCKOH llJIOCKO-
CTH B Kpyre eBKJIIIJJ;OBOH nJIOCKOCTII.
Cpe):(H JJ;pyrllx BbIBOJJ;OB H3 «TpeTbell rll OTe3bI», c):(eJIaHHbIX JIaM-
6epTOM, OTMeTIIM JJ;OKa3aTeJIbCTBO Toro <paKTa, 'tJTO B 9TOM CJIY"Ilae
J:\Ba rrepneHJJ;IIKYJIHpa K OJJ;HOH IIpHMOll paCXOJJ;HTCH no 06e CTOpOH:bI
OT 3Toro rreplIeHJJ;IIKYJIHpa. Bblme MbI BIlJJ;eJIH, 'tJTO 3TO CJIeJJ;CTBlIe
Ha «rllnOTeahI OCTporo yrJIa» 6hIJIO yCTaHOBJIeHO e e XaHHMOM.
RaK HII IlHTepeCHbI nplIBeJJ;eHH:bIe MbICJIH JIaM6epTa, Bce me npH
J:\aJIbHeHmeM HCCJIeJJ;OBaHHII, JJ;OIlYCTIIB cepbeaHYIO orn1l6KY, OH 110-
JIar aeT, "IlTO eMY YJJ;aJIOCb OllponeprHYTb «TpeTblO rIIIIOTe3Y»: OT JIOiIiHB
Ha npHMOH PHJJ; paBH:bIX OTpeaKOB II BOCCTaBIlB B HX KOH ax nepneH-
JJ;HKYJIHp:bI paBHOH AJIIIH:bI, OH nOJIY"IlaeT IIpaBIlJIbHYIO B:bIllYKJIYIO
JIOMaHYIO. JIaM6epT C"IlHTaeT, 'tJTO "Ilepe3 BepmHHbI 9TOH JIOMaHOll
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121
JIMTEPATYPA
1. A AbfPOHCO. BhIIIpHMHTeJIb RpHBOrO (MeiIammep 'aK06)/IIep r. M. r nYCRII-
lion, rrpHMeq. r. M. rJIYCKHHOii, C. .H. JIypbe HE. A. PoaeHcpeJILAa. M.:
HaYRa, 1983.
2. A p ucmome.rtb. CoqHHeHHH/IIep. no,n; pe . J;3. <I>. ACMyca, 3. H. MHKena,n;3e,
M. ,1];. POiRaHCKoro. B 3-x T. M.: MhICJIL, 1974-1981.
3. AXJrte8fJ8 A. ABTOp pHMcRoro H3AaHHH «HaqaJI» EBKJIH,n;a.- H3B. AH
Y3CCP. Cep. cpH3.-MaT. HaYR, 1971, T. 15, .M 5, c. 9-12.
4. A.rt-Btlp[j1-£u A6[j-p-Paufii1-£. PaC3HJI. Xaii apa6aA, 1369 x. (1948).
5. BOHo.rta P HeeBKJIH,n;oBa reoMeTpHH: RpHTHKO-HCTOpHqeCKOe HCCJIeAOBaHHe
ee pa3BHTHH/IIep. A. RYJIHmepa. CII6.: 06m;ecTB. nOJILSa, 1910.
6. By.rteaw,08 II. F., AXMe808 A. Bepy:iIH H aJI-RHHAH 0 TeopHH napaJIJIeJIbHhlx.-
06m;eCTn-. HaYKH B Y36eRHcTaHe, 1977, 8, c. 30-36.
7. rU.rtb6epm 1(. OCHoBaHHH reoMeTpHH/IIep. M. C. rpa}J;llITenHa. M.; JI.:
rOcTexH3,n;aT, 1948.
8. F AYCw,U1-£a F M. 0 HeH3AaHHOM cpe HeBeRoBoM TpaRTaTe «Meyasser 'aqob»,
xpaHHm;C)fCH B EpHTaHcKoM Myaee.- IIaJIecT. c6., 1974, BhIn. 5(88),
c. 152-158.
9. rAYCw,uHa F. M. 06 aBTope MaTeMaTHqeCROrO TpaKTaTa «Meyas;er 'aqob».-
IIaJIecT. c6., 1978, Bhln. 26(89), c. 66-=----75.
to. E8W,.aUfJ. HaqaJIa /llep. ,1];. ,1];. Mop yxaiI-EoJITOBCRoro. B 3-x T. M.; JI.:
rOcTexH3AaT, 1948-1950.
11. H61-£ CU1-£a. ,1];aHHm-HaM3 - RHHra 3HaHHH/IIep. A. M. EoroYT,D;IIHoBa.
,1]; yrnaH6e: T a,n;iRrOCH3}J;a T , 1957.
12. H61-£ CU1-£a. MaTeMaTHqeCRHe rJIaBhI «RHHrH SHaHHH» (AOHHllIHoMa)/IIep.
B. A. P03eH«peJIL a H H. A. Ca oBcRoro. ,1];ymaH6e: IiIPCPOH, 1967.
13. H aea1-£ B. (/J. OCHoBaHHH reOMeTpHH. YqeHHe 06 060cHoBaHHH reOMeTpHH
B xOJJ;e ero HCTOpHqeCKOrO paSBHTHH. M.: focTexHs aT, 1949, T. 1.
14. I.fyp6iiHll A6t/-.rt-{(iicUM. PHiia H 3Hi-'iH-H HpaHM a3 B pHSMH Ta M6u CfiHa.
TerepaH, 1350 c. x. (1971).
14a. JIanmee B. JI. JIaM6epT: - reOMeTp.- MCT.-MaT. H:CCJIe ., 1980, Bblii. 25,
c. 248-260.
15. JIypbe C. H. TeopHH 6eCKOHeqHO MaJIhIX y peBHHX aTO:\UICTOB. M.; JI.:
M3 -Bo AH CCCP, 1935.
16. MaMerJ6eu.au Fl(. MyxaMMe HacHpa ,n;HH TYCH 0 TeopHH napaJIJIeJIbHhIX
JIHHHll H TeopHH oTHollIeHHn. BaRY: H3 -Bo AH A3ep6CCP, 1959.
17. M am8Ue8Cw,aJ£ F. II. Y1JeHHe 0 qHCJIe Ha Cpe)];HeBeROBoM EJIHiRHeM If CpeAHe1.1
BocToKe. TallIKeHT: <I>aH, 1967.
18. MyaaifJapo8a X. P. 0 MaTeMaTHqeCRHX rJIaBaX 3Hn;HKJIOne HqeCKOrO npo-
H3BeACHHH RYT6a ,n;HHa allI-IIIHpasH «,1];yppa aT-Ta)];iR JIH ryppa aA-,1];y6a m»
(meMtJymHHa KOpOHhI JIH YKpallIeHHH ,n;y6aAiRa).- Yq. 3an. Tap;m. ,roc.
YH-Ta (TpYAhI MeX.-MaT. cpaR.), 1970, T. 1, .M 1, c. 85-93.
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JIaTa EBKJIHAa.- MaB. AH ApMCCP. Cep. cpH3.-)IaT. HaYK, 1960, T. 13,
.M 1, c. 153-164.
20. Poae1-£fPe.rtbfJ B. A. ,1];oHa3aTCJIbCTBa nHToro nOCTYJIaTa EBKJIH)];a cpe}J;He-
BeROBhlX MaTeMaTHROB XacaHa H6H aJI-XaHCaMa H JIbBa fepcoHHAa.-
RCT.-MaT. HCCJIe,n;., 1958, BbIII. 11, c. 733-782.
20a. Poael-lfpeltbO B. A. HCTOPUH HeeBKJII1AoBOH reOMeTpHH. M.: HaYRa, 1976.
21. POae1-£fPeAb8 B. A., Ky6eco8 A. H., Co6upoe r C. l\TO 6hIJI aBTopoM pHMCKO-
ro H3p;aHHH «M3JIoiReHHH EBKJIlIJJ;a HacHp a,n;- HHa. 8T-TycH»? - Bonp.
HCTOpIIH eCTeCTB03H. H TCXH., 1966, BblII. 20, c. 51-53.
122
2ta. Poae1ufie.tttO B. A., IOwKeeu'I{ A. II. 0 TpaKTaTe HacHp aA-AuHa aT-Ty-
cn 0 napaJIJIeJIbIlI,IX JII1HHHX.- MCT.-MaT. HccneA., 1960, BbIn. 13,
c. 475-482.
22. PoaeHifje.abo B. A., 10 ULKeeUIl A. II.o oKa3aTCJIbCTBa nHToro nOCTYJIaTa EB-
RJIIJ a y Ca6HTa H6H ROppLI H IIIaMc aA- I1Ha ac-CaMapRaH H.- MCT.-MaT.
HCCJIeA., 1961, BLIn. 14, c. 587-602.
23. Ca6um U6H Hoppa. RHlIra 0 TOM, qTO ABe JUIHHH, npOBe)];eHlllde no,[( yrJIaMIf,
MeHLillHMH )];BYX npHMLIX, BCTpeTHTcH/IIep. E. A. PoaeHcI»eJIb)];a.- MCT.-
MaT. HCCJIeA., 1963, BLIn. 15, c. 363-380.
4. A c-Ca.aap X yca.M a8-J( UH. IIpeAIIOChIJIKH )];JIH ;r;OKa3aTeJILCTBa nocTYJIaTa
o napaJIJIeJIbHLIX JIHHHHX, rrpI1BeAeHHoro EBKJIHJJ;OM B HaqaJIC nepBoH RHHrH/
[Jep. E. A. P03eHcI»eJIbAa H H. r. XaftpcT)J;HHOBOii.- I1cT.-MaT. HCCJIe .,
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26. T JI,.aaUle8 X. HOBLIe )];aHHLIe 06 HCTOpHH MaTeMaTJIKH B Cpe)];Heii A3HH
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