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A)\N DE\ H( PLEURA\ PRO\S €TH\N PLEURA\N E)/XH| ME/N TINA LO/GON, H(MIO/LION TUXO\N H)\ E)PI/TRITON €H)\ A)/LLON TINA\ TW=N E)PIMORI/WN H)\ TW=N E)PIMERW=N, TA\ ME\N A)P' €AU)TW=N GEGONO/TA TETRA/GWNA LO/GON E)/XOUSI PRO\S A)/LLHLA, O(\N €TETRA/GWNOS A)RIQMO\S PRO\S TETRA/GWNON A)RIQMO/N, OU) MH\N DE\ €TO\N TETRAPLASI/ONA, W(S E)PI\ TOU= KAI\ TOU= , W(=N AI( PLEURAI\ €LO/GON ME\N E)/XOUSIN, O(\N A)RIQMO\S PRO\S A)RIQMO/N, OU)X O(\N TETRA/-€GWNOS PRO\S TETRA/GWNON: TA\ GA\R DU/O KAI\ TRI/A, A(/PER EI)SI\ @1 €PLEURAI\ TOU= KAI\ TOU= , TO\N H(MIO/LION E)/XOUSI LO/GON: DIO\ €KAI\ OU) DU/NATAI EI)=NAI O( TOU= TETRAPLA/SIOS, W(S O( TOU= € KAI\ O( TOU= . šÑ¸³ôÿ{1&Ad lemma prop. 9$}1 ˜Ñ*OI(=ON O(/MOIOI E)PI/PEDOI/ EI)SIN O( KAI\ O( : A)NA/LOGON GA\R þï€µ°²²ÿï°°±ÿ¨Š˜Ñ‚E)/XOUSI TA\S PLEURA/S. W(S GA\R O( PRO\S TO\N , OU(/TWS O( PRO\S €TO\N . 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KAI/ E)STIN W(S H( *B*G PRO\S TH\N *Z*E, OU(/TWS TO\ U(PO\ €TW=N *A*B, *B*G PRO\S TO\ U(PO\ TW=N *A*B, *Z*E DIA\ TO\ A# TOU= #2#: €U(/FOS H( *A*B: SU/MMETROS DE\ H( *B*G TH=| *Z*E. í²³¹ôÿ{1&Ad prop. 34$}1 ›í*AI( TOIAU=TAI EU)QEI=AI MHTE/RES EI)SI\ TH=S R(HTO\N KAI\ ME/-€SON DUNAME/NHS A)LO/GOU. *(H *A*B , H( *B*G , TO\ H(/MISU TH=S *B*G €OU)DE\N , TO\ A)PO\ TH=S H(MISEI/AS TH=S *B*G . TO\ €H(/MISU TH=S *A*B , TO\ A)PO\ TH=S H(MISEI/AS TH=S *A*B <#2> € , H( PLEURA\ TH=S METACU\ TW=N TOMW=N , TO\ A)PO\ TH=S €METACU\ TW=N TOMW=N . €*(H *A*D , H( *D*B . €*TO\ A)PO\ TOU= H(MI/SEOS TH=S *B*G TETRA/GWNON E)A\N PARABLHQH=| €PRO\S TH\N H(MI/SEIAN TH=S *A*B, MA=LLON DE\ A)PO\ TOU= TETRAGW/-€NOU AU)TH=S TOU= <#2> , KATALIMPA/NETAI TO\ A)PO\ TH=S €METACU\ TW=N TOMW=N, O(/PER E)STI\N , H( PLEURA\ €AU)TOU= , H(/PER TH=| H(MISEI/A| PROSTEQEI=SA POIEI= TH\N *A*Z € , KAI\ H( *Z*B KATALIMPA/NETAI OU)DE\N . KAI\ TO\ U(PO\ €TW=N *A*Z, *Z*B I)/SON E)STI\ TW=| A)PO\ TH=S *B*E. 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A)LLA\ TO\ ME\N U(PO\ TW=N *B*A, *A*Z I)/SON E)STI\ TW=| A)PO\ TH=S €*A*D, TO\ DE\ U(PO\ TW=N *A*B, *B*Z I)/SON TW=| A)PO\ TH=S *B*D, W(S E)N €TW=| LH/MMATI E)DEI/XQH. W(/STE TO\ A)PO\ TH=S *A*D A)SU/MMETRO/N €E)STI TW=| A)PO\ TH=S *D*B. AI( *A*D, *D*B A)/RA DUNA/MEI A)SU/MMETROI/ €EI)SIN. õ²´¹ôÿ{1&Ad prop. 36$}1 ›õ*(H TW=N TOIOU/TWN R(HTW=N ME/SH A)NA/LOGON ME/SH E)STI/N. €OU)DEMI/A DE\ TOU/TWN OU)/TE SUNAMFO/TEROS ME/SH, H( DE\ SUGKEI-€ME/NH E)C AU)TW=N E)K DU/O O)NOMA/TWN KALEI=TAI. A)MFOTE/RWN TOI/-€NUN TW=N A)LO/GWN EI)SI\ PRO/GONOI KATA\ DIAFO/ROUS GENE/SEWS €TRO/POUS. *)/ESTW H( *A*B E(CA/POUS, H( DE\ *B*G PENTA/POUS. E)/STIN €OU)=N H( *A*B TH=S *B*G E)PI/PEMPTOS. TO\ DE\ U(PO\ TW=N *A*B, *B*G €E)STI\ : E(CA/KIS GA\R . TO\ DE\ A)PO\ TH=S *B*G : PENTA/KIS GA\R €PE/NTE . KAI\ O( A)/RA TOU= E)PI/PEMPTO/S E)STIN, W(S E)/XEI H( €*A*B PRO\S TH\N *B*G. 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TO\ MEI=ZON O)/NOMA &p. 76, 20]$ MEI=ZON O)/NOMA AU)TO\ TO\ €MEI=ZON TMH=MA KALEI=TAI. H( O(/LH &p. 76, 23]$ O(/LH DHLONO/TI H( E)K DU/O O)NOMA/TWN €KAI\ DIAIREQEI=SA, W(S U(PO/KEITAI. ‚¥³°±ôÿ{1&Ad prop. 48$}1 ›‚¥*)/ESTW O( *E*Z A)RIQMO\S MONA/DWN <#2>, O( DE\ *Z*H MONA/DWN € KAI\ LEPTW=N PRW/TWN , W(=N TESSA/RWN MONA/DWN KAI\ LE-€PTW=N PRW/TWN E)/STAI DU/NAMIS H)/TOI TETRA/GWNOS O( A)RIQ-€MO/S: TOU= GA\R EI)/KOSI PLEURA/ EI)SIN AI( TE/SSARES MONA/DES KAI\ € LEPTA/. TOU/TWN OU)=N E)XO/NTWN W(S O( PRO\S TO\N PE/NTE: €E)/XEI GA\R AU)TO\N KAI\ TE/SSARA AU)TOU= ME/RH: OU(/TWS O( TETRA/-€GWNOS O( A)PO\ TH=S *E*Z TH=S OU)/SHS <#2> MONA/DWN PRO\S TO\N EI)/KOSI €TETRA/GWNON TO\N A)PO\ TH=S *Z*H OU)/SHS MONA/DWN KAI\ LEPTW=N €PRW/TWN . E)/XEI TOI/NUN O( TO\N PE/NTE KAI\ TE/SSARA AU)TOU= €PE/MPTA: KAI\ O( OU)=N TO\N KAI\ TE/SSARA AU)TOU= PE/MPTA: €O( GA\R , W(=| U(PERE/XEI O( TOU= , O( OU)=N TE/SSARA PE/MPTA €E)STI\ TOU= . *)/ESTW O( *A*G O( , O( DE\ *G*B O( . O( OU)=N E)C AU)TW=N O( €PRO\S ME\N TO\N LO/GON E)/XEI, O(\N TETRA/GWNOS PRO\S TETRA/GWNON, €PRO\S DE\ TO\N OU)K E)/XEI. LO/GON DE\ E)/XEIN LE/GETAI A)RIQMO\S €PRO\S A)RIQMO/N, O(\N TETRA/GWNOS PRO\S TETRA/GWNON, O(/TAN ME-€TACU\ E)MPI/PTH| A)RIQMO\S A)NALOGI/AN SW/ZWN: DIO\ O( KAI\ O( €PRO\S TO\N LO/GON E)/XOUSIN, O(\N TETRA/GWNOS PRO\S TETRA/GWNON: €PI/PTEI GA\R METACU\ TOU= KAI\ O( <#2>, KAI/ E)STIN W(S O( PRO\S €TO\N <#2>, OU(/TWS O( <#2> PRO\S TO\N , METACU\ DE\ TOU= KAI\ O( . @1 *)/ESTW H( *D H( PLEURA\ TOU= OU)=SA MONA/DWN LEPTW=N €: E)/STW DH\ KAI\ H( *Z*H KAI\ AU)TH\ H( PLEURA\ TOU= : I)/SH €A)/RA H( *D TH=| *Z*H: SU/MMETROI A)/RA MH/KEI. H( DE\ *E*H OU)=SA €MONA/DWN LEPTW=N PRW/TWN E)/STW H( PLEURA\ TOU= . E)/STI €TOI/NUN W(S O( *G*A H)/TOI O( PRO\S TO\N *A*B [1E)/XETAI GA\R AU)TO\S €KAI\ TE/SSARA AU)TOU= PE/MPTA]1, OU(/TWS KAI\ O( A)PO\ TH=S *Z*H €TETRA/GWNOS O( PRO\S TO\N A)PO\ TH=S *E*Z TETRA/GWNON TO\N : €E)/XETAI GA\R KA)N TOU/TOIS O( U(PO\ TOU= KAI\ KAI\ TE/SSARA €AU)TOU= PE/MPTA: TA\ GA\R O)KTW/, OI(=S U(PERE/XEI O( TOU= , TE/S-€SARA/ EI)SI TOU= DE/KA PE/MPTA. *KAT' A)/LLHN GRAFH\N TO\ A)PO\ TH=S *E*Z , TO\ A)PO\ TH=S €*Z*H , H( *Z*H H( PLEURA\ TOU= , TO\ A)PO\ TH=S *Q . *(O *A*G , O( *G*B , H( O(/LH *A*B , H( *D <#2>, H( *E*Z , TO\ €A)PO\ TH=S *Z*H O)KTW\ , H( *Z*H DU/O , TO\ A)PO\ TH=S *Q €<#2> , H( *Q , H( O(/LH *E*H <#2> . *KAT' A)/LLHN GRAFH\N O( *A*G , O( *G*B , O( *A*B , H( €*D , O( *Z*H <#2>, TO\ A)PO\ TH=S *Z*E , O( *Z*E <#2> , TO\ A)PO\ TH=S €*Q , H( *Q H( PLEURA\ TOU= . TH=| *D SU/MMETROS E)/STW MH/KEI H( *E*Z. &p. 77, 19]$ DU-€NA/MEQA TH=| E)KKEIME/NH| R(HTH=| SU/MMETRON LABEI=N, O(/TAN H)\ I)/SHN €H)\ DIPLASI/AN H)\ H(MI/SEIAN LA/BWMEN, OI(=ON EI)/ E)STIN H( E)KKEIME/NH €R(HTH\ E(CA/POUS, KAI\ LHYO/MEQA TH\N DWDEKA/PODA, SU/MMETROS €E)/STAI AU)TH=| MH/KEI: METREI= GA\R H( E(CA/POUS KAI\ E(AUTH/N: €PA=S GA\R A)RIQMO\S W(S E(AUTW=| E)FARMO/ZWN METRHTIKH/ E)STIN €E(AUTOU=. A)LLA\ KAI\ TH\N DWDEKA/PODA METREI= A)PARTIZO/NTWS H( €E(CA/POUS AU)TH\ KAI\ E(AUTH=S KAI\ TH=S DWDEKA/PODOS. O(MOI/WS KAI\ €E)PI\ TW=N A)/LLWN TA\ AU)TA\ R(HTE/ON TH=S TE H(MISEI/AS TH=S PRO-€KEIME/NHS R(HTH=S KAI\ TH=S I)/SHS KAI\ TH=S TRIPLASI/AS KAI\ E(CH=S. *DU/NATAI E)KTIQE/NAI EU)QEI=AN KAI\ POIEI=N H)\ DIA\ O(/RON €I)/SHN H)\ DIPLASI/AN H)\ H(MI/SEIAN DIA\ PO/RISMA #2# I# KAI\ E(CH=S. @1 W(/STE SU/MMETRO/N E)STI &p. 78, 5]$ TA\ GA\R TETRA/GWNA €TA\ PRO\S A)/LLHLA LO/GON E)/XONTA, O(\N A)RIQMO\S PRO\S A)RIQMO/N, €SU/MMETRA/ E)STIN. œ‚®ô{1&Ad prop. 49$}1 ›‚®*(O *A*G , O( *G*B , O( *A*B O(/LOS , H( *Z*H , H( *D <#2>, TO\ €A)PO\ TH=S *Z*E , H( *E*Z , TO\ A)PO\ TH=S *Q , €H( *Q , H( O(/LH *E*H . ‚¯³°´ôÿ{1&Ad prop. 50$}1 ›‚¯*(H *A*G KAI\ H( *G*B KAI\ H( O(/LH *A*B , O( *D , H( *E €<#2>, TO\ A)PO\ TH=S *Z*H , H( *Z*H H( PLEURA\ TOU= , TO\ €A)PO\ TH=S *H*Q , H( *H*Q , TO\ A)PO\ TH=S *K , H( *K €, H( O(/LH H( *Z*Q . *TOU= N# QEWRH/MATOS KAT' A)/LLHN GRAFH/N: O( *A*G , €O( *G*B , O( *A*B O(/LOS , H( *D , H( *E <#2>, TO\ A)PO\ TH=S *Z*H , H( €*Z*H , TO\ A)PO\ TH=S *Q*H , H( *Q*H , TO\ A)PO\ TH=S €*K , H( *K . ‚±³°¶ôÿ{1&Ad prop. 51$}1 þï€µ°²²ÿï°°±ÿ¨Š›‚±*(O *A*G , O( *G*B , H( *D <#2>, H( *E*Z , TO\ A)PO\ TH=S *Z*H €, H( *Z*H , TO\ A)PO\ TH=S *Q , H( *Q , TO\ A)PO\ €TH=S *E*Z , H( O(/LH *E*H . *TOU= NA#. O( *A*G , O( *G*B , O( *D <#2>, H( *E*Z , H( *Z*H € , TO\ A)PO\ TAU/THS <#3#2>, H( *Q PLEURA\ TOU= , H(/TIS €E)STI\N <#2> . @1 œ‚³ô{1&Ad prop. 52$}1 ›‚³*(O *A*G , O( *G*B , H( *D <#2>, H( *Z*H , TO\ A)PO\ TH=S *E*Z €, H( *E*Z D , TO\ A)PO\ TH=S *Q , H( *Q . H( *E*H O(/LH € . TO\ A)PO\ TH=S *Z*H . œ‚´ô{1&Ad prop. 53$}1 ›‚´*(O *A*G , O( *G*B , H( *E , TO\ A)PO\ TAU/THS , O( *D , €TO\ A)PO\ TH=S *Z*H , H( *Z*H , TO\ A)PO\ TH=S *H*Q , H( €*H*Q , TO\ A)PO\ TH=S *K DE/KA, H( *K , O(/LH H( *Z*Q € . ‚µ³±¶ôÿ{1&Ad prop. 54$}1 ›‚µ*(/EKTON KEFA/LAION DEIKNU/ON TA\S KATA\ SU/NQESIN E(\C €A)LO/GOUS XWRI/A POIOU/SAS PERIEXO/MENA U(PO\ R(HTH=S KAI\ MIA=S €TINOS TW=N E(\C E)K DU/O O)NOMA/TWN. *DIA\ TO\ MH# KAI\ DIA\ TO\ L#2# DUNATO\N TA\ EI)RHME/NA €PORI/SASQAI. *DEI= PRW=TON EU(REI=N TH\N E)K DU/O O)NOMA/TWN PRW/THN €KAI\ OU(/TWS DIAIREI=N EI)S TA\ O)NO/MATA DIA\ MB# I#. *TO\ A)PO\ TH=S *E*Z , TO\ *A*Q , TO\ €*H*K TE/SSARES, TO\ *E*L O)KTW\ , TO\ *Z*D O(MOI/WS TO\ U(PO\ €TW=N *A*B, *A*D , H( *A*D <#2> . *KAT' A)/LLHN GRAFH/N: O( *A*D , H( *A*B <#2>, H( *A*E €MONA/DWN <#2>, H( *E*D H( PLEURA\ TOU= , TO\ U(PO\ *A*B KAI\ *A*D € , H( *E*Z , TO\ A)PO\ TH=S *E*Z <#2> , H( *A*H , €H( *H*E , TO\ *A*Q , H( *M*N , TO\ *N*P , H( *N*C , €H( *M*C , TO\ *E*L . @1 [1*A*B]1#756, [1*A*H]1#753 #752#760, [1*H*E]1#760 #754#760, [1*E*Z]1#751 #752#759 #752#756 #753#760, [1*Z*D]1#751 #752#759 #752#756 #753#760 KAI\ H( *A*E TH=S *E*D &p. 90, 16$] EI) GA\R OU) DIAIREI=TAI €KATA\ TA\ EI)RHME/NA, OU)K E)/STIN E)K DU/O O)NOMA/TWN PRW/TH. PARABEBLH/SQW OU)=N &p. 91, 8$] KAI\ E)/STW LOIPO\N EI)/DEI €TETRAGW/NW| DIA\ LH=MMA TOU= IZ# I# KAI\ DIA\ IZ# I#, DIO/TI KAI\ EI)S €SU/MMETRA AU)TH\N DIAIREI= MH/KEI. ‚½³²°ôÿ{1&Ad prop. 55$}1 ›‚½*TO\ U(PO\ *A*H, *H*B XWRI/ON QE/LH|S E)NTO\S TOU= *A*G €XWRI/OU E)/GGRAYON QE/LH|S E)PI\ TA\ E(/TERA ME/RH TOU= PROBAI/NEI €TO\ QEW/RHMA TH=S DE\ *A*B E)C E(TE/RAS PARALLH/LOUS DIA\ TO\ *N*E, €*Z*D SHMEI=ON. *(H *A*B <#2>, H( *A*E PE/NTE , H( H(MI/SEIA TH=S *A*E € , TO\ A)PO\ TH=S H(MISEI/AS OU)DE\N , H( *E*D €, H( *E*Z , H( *Z*D , TO\ KATALEIPO/MENON META\ TH\N A)FAI/RE-€SIN TH=S KATAMETR..... , TO\ *A*G O(/LON , €H( MH\ PROSTIQEME/NHS TH=S METACU\ TW=N TOMW=N , H( €*H*E OU)DE\N , H( PROSTIQEME/NH PLEURA\ TH=| E(TE/RA| H(MI-€SEI/A| METACU\ TW=N TOMW=N, TO\ *A*Q H)/TOI TO\ *S*N , H( €AU)TW=N PLEURA\ , TO\ *H*K H)/TOI TO\ *N*P , H( AU)TW=N €PLEURA\ , TO\ *E*L , TO\ *Z*G , H( TO\ *A*G DUNAME/NH H( €*M*C . AI( *A*E, *E*D A)/RA &p. 93, 19$] EI) GA\R OU) DIAIREI=TAI OU(/-€TWS, OU)K A)/RA E)K DU/O O)NOMA/TWN E)STI\ DEUTE/RA DIA\ TO\N O(/RON €TW=N DEUTE/RWN, DIA\ MB# TOU= I#. @1 *KAI\ AI( *M*N, *N*C A)/RA ME/SAI &p. 95, 5$] GE/GRAPTAI €GA/R, O(/TI H( DUNAME/NH A)/LOGON XWRI/ON A)/LOGO/S E)STIN. ‚Á³²²ôÿ{1&Ad prop. 56$}1 ›‚Á*(H *A*D , H( *A*E , TO\ A)PO\ TAU/THS KAI\ €TH=S R(HTH=S TH=S *A*B <#2> MONA/DWN OU)/SHS , H( *M*C H( €PLEURA\ TOU= *B*G , TO\ H(/MISU TH=S *A*E , H( *E*D € , H( *E*Z , TO\ A)PO\ TH=S *E*Z , TO\ €A)PO\ TH=S H(MISEI/AS TH=S *A*E <#2> , TO\ KATALEIFQE\N €A)PO\ TOU= A)PO\ TH=S H(MISEI/AS TH=S *A*E OU)DE\N , H( AU)TW=N €PLEURA\ , H( *A*H , H( *H*E OU)DE\N , TO\ *A*Q €H)/TOI TO\ *S*N , H( PLEURA\ AU)TW=N H( *M*N , TO\ €*H*K H)/TOI TO\ *N*P , H( PLEURA\ AU)TW=N H( *N*C , €TO\ *E*L <#2>, TO\ *Z*G <#2>. *KAT' A)/LLHN GRAFH\N EI)S TO\ N#2#: H( *A*B <#2>, H( *A*E € , H( *E*D , H( *A*D , TO\ *A*G <#3E> OU)DE\N <#2>, TO\ €*E*Z , H( *A*H <#2> , H( *H*E , H( PLEURA\ TOU= €*A*G , TO\ *A*Q , H( TOU/TWN PLEURA\ <#2> , TO\ €*H*K , H( TOU/TWN PLEURA\ . ‚Ã³²´ôÿ{1&Ad prop. 57$}1 ›‚Ã*(H *A*E , H( *E*D , H( *A*D O(/LH , TO\ *A*G €<#3H> , H( *A*B <#2> MONA/DWN, H( *E*Z , TO\ A)PO\ TAU/THS €H)/GOUN TO\ *E*L , H( *Z*D W(SAU/TWS I)/SH TH=| *E*Z, €O(MOI/WS KAI\ TO\ *Z*G I)/SON TW=| *E*L, TO\ A)PO\ TH=S H(MISEI/AS TH=S €*A*E , H( H(MI/SEIA TH=S *A*E , H( *A*H , H( *H*E , €TO\ *A*Q H)/TOI TO\ *S*N , H( AU)TW=N PLEURA\ H( *M*N <#2> @1 € , TO\ *H*K H)/TOI TO\ *N*P , H( AU)TW=N PLEURA\ H( €*N*C , H( O(/LH *M*C , TO\ *E*L , O(MOI/WS KAI\ €TO\ *Z*G. *TOU= NZ#. H( *A*B <#2>, H( *A*E , H( *E*D , H( *A*D € , TO\ *A*G , H( TOU/TWN PLEURA\ H( *E*Z € , TO\ A)PO\ TH=S *E*Z , H( *A*H , H( *H*E , TO\ *A*Q € OU)DE/N, H( TOU/TWN PLEURA\ , TO\ *H*K OU)DE/N, H( €TOU/TWN PLEURA\ . œ‚Åô{1&Ad prop. 58$}1 ›‚Å*(H *A*D ; H( *A*E , H( *E*D , H( *A*B <#2>, TO\ €*A*G O(/LON , H( AU)TW=N PLEURA\ H( *M*C , H( *E*Z , €TO\ A)PO\ TH=S *E*Z , H( *A*H , H( *H*E , TO\ *A*D €, H( AU)TW=N PLEURA\ H( *M*N , TO\ *H*K <#2> , H( AU)-€TW=N PLEURA\ H( *N*C , TO\ *E*L , TO\ *Z*G . œ‚Æô{1&Ad prop. 59$}1 ›‚Æ*(H *A*D O(/LH , H( *A*E , H( *E*D , €TO\ *A*G , H( AU)TW=N PLEURA\ H( *M*C , H( *E*Z € , H( *E*D O(MOI/WS, TO\ A)PO\ TH=S *E*Z , H( €*A*H , H( *H*E , TO\ *A*Q <#2>, H( AU)TW=N PLEURA\ €H( *M*N , TO\ *H*K <#2> , H( AU)TW=N PLEURA\ H( *N*C € , TO\ *E*L , O(MOI/WS KAI\ TO\ *Z*G. 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W(/STE TA\ A)PO\ TW=N *A*G, *G*D TOU= DI\S U(PO\ TW=N €*A*G, *G*D MEI/ZONA/ E)STI TW=| A)PO\ TH=S *A*D TETRAGW/NW|. I)/SA DE\ €TA\ ME\N A)PO\ TW=N *A*G, *G*D TOI=S A)PO\ TW=N *A*G, *G*B [1I)/SH GA\R @1 €E)TE/QH TH=| *G*B H( *G*D]1, TO\ DE\ DI\S U(PO\ TW=N *A*G, *G*D TW=| DI\S €U(PO\ TW=N *A*G, *G*B. TA\ A)/RA A)PO\ TW=N *A*G, *G*B MEI/ZONA/ €E)STI TOU= DI\S U(PO\ TW=N *A*G, *G*B TW=| A)PO\ TH=S *A*D: O(/PER €E)/DEI DEI=CAI. ‚É³³²ôÿ{1&Ad prop. 60$}1 ›‚É*)EK DU/O O)NOMA/TWN PRW/TH H)=N, O(/TAN TO\ MEI=ZON O)/NOMA €SU/MMETRON H)=N MH/KEI TH=| E)KKEIME/NH| R(HTH=|. *)/ESTW H( *A*B H( E)K DU/O O)NOMA/TWN , KAI\ DIH|RH/SQW €EI)S TA\ O)NO/MATA W(S EI)=NAI TO\ MEI=ZON O)/NOMA , TO\ DE\ E)/LAT-€TON . E)/STW DE\ KAI\ H( *D*E R(HTH/, H)/TOI KAI\ AU)TH\ , KAI\ €PARABEBLH/SQW H)/TOI MERISQH/TW TO\ A)PO\ TW=N GINO/MENON €TETRA/GWNON, TOUTE/STI TO\ A)PO\ TH=S *A*B TH=S E)K DU/O O)NOMA/-€TWN, A(/PER O)NO/MATA/ E)STIN, W(S EI)/RHTAI, O( KAI\ O( , MERI-€SQH/TW TOI/NUN TO\ A)PO\ TH=S E)K DU/O O)NOMA/TWN TETRA/GWNON €O)\N TRIW=N MURIA/DWN KAI\ DISXILI/WN TETRAKOSI/WN PARA\ TH\N €R(HTH\N TH\N *D*E OU)=SAN , KAI\ TO\ A)PO\ TOU= MERISMOU= EU(RE-€QE/N, O(/PER PLA/TOS PARABOLH=S KALEI=TAI, E)/STAI PA/NTWS AU)TH\ €H( E)K DU/O O)NOMA/TWN SUGKEIME/NH H)/TOI H( . *(H *A*B <#2> , TO\ A)PO\ TH=S *A*B , €H( *D*E , TO\ PLA/TOS TH=S PARABOLH=S H( *D*H , €TO\ A)PO\ TH=S *A*G H)/TOI TO\ *D*Q , H( *D*K , TO\ A)PO\ TH=S *G*B €H)/TOI TO\ *K*L , H( *K*M , TO\ A(/PAC €U(PO\ TW=N *A*G, *G*B , H( *M*N , O(MOI/WS KAI\ H( €*N*H KAI\ TO\ *N*Z. E(KA/TERON A)/RA TW=N &p. 104, 11]$ O(\ LE/GEI, E)STI/N, O(/TI €E(/KASTON PARALLHLO/GRAMMON TO\ PERIEXO/MENON A(/PAC U(PO\ €TW=N *A*G, *G*B OI(=ON TO\ *M*C E)STI TO\ A(/PAC U(PO\ TW=N *A*G, *G*B €KAI\ PA/LIN TO\ *N*Z TO\ A(/PAC U(PO\ TW=N *A*G, *G*B. E)PEI\ GA\R @1 €O(/LON TO\ *M*Z E)STI TO\ DI\S U(PO\ TW=N *A*G, *G*B, TE/TMHTAI DE\ €DI/XA H( *M*H, DH=LON, O(/TI TO\ *M*C H(/MISU/ E)STI TOU= *M*Z. W(/STE €TO\ A(/PAC E)STI\ TOU= U(PO\ TW=N *A*G, *G*B. ‚Í³³´ôÿ{1&Ad prop. 61$}1 ›‚Í*ME/SH H)=N H( DUNAME/NH XWRI/ON PERIEXO/MENON U(PO\ €R(HTW=N DUNA/MEI MO/NON SUMME/TRWN, OI(=ON H( EI)KOSITESSARA/-€POUS KAI\ TRIAKONTA/POUS MH/KEI ME/N EI)SIN A)SU/MMETROI, DU-€NA/MEI DE\ SU/MMETROI: TA\ GA\R A)P' AU)TW=N TETRA/GWNA TA/ TE € KAI\ TO\ E)NNAKO/SIOI KOINW=| XWRI/W| METROU=NTAI TW=| <#2>: €E(CA/KIS GA\R <#3#2> KAI\ E(CA/KIS E)NNAKO/SIOI. W(/STE H( EI)KOSI-€TESSARA/POUS KAI\ H( TRIAKONTA/POUS MH/KEI ME\N A)SU/MME-€TROI, DUNA/MEI DE\ SU/MMETROI/ EI)SI, PERIE/XOUSI DE\ XWRI/ON €PODW=N E(PTAKOSI/WN EI)/KOSI. H( OU)=N DUNAME/NH TO\ TOIOU=TON €XWRI/ON E)STI\ ME/SH. LHPTE/ON DH\ TH\N TOU= PLEURA\N TH\N DU-€NAME/NHN TO\N , KAI\ E)/STAI H( ME/SH. E)/STI DE\ H( PLEURA\ TOU= € . *(H *A*B , H( *D*E , TO\ A)PO\ TH=S *A*B €, H( *D*H , H( *A*G , H( *G*B , €TO\ A)PO\ TH=S *A*G ; H( *D*K <#2> , TO\ A)PO\ TH=S €*G*B , H( *K*M OU)DE\N , TO\ *M*C € ; H( *M*N , O(MOI/WS KAI\ TO\ *N*Z KAI\ H( *N*H. œ‚Ïô{1&Ad prop. 62$}1 ›‚Ï*(H *A*B O(/LH , TO\ A)PO\ TH=S *A*B , €H( *A*G , TO\ A)PO\ TH=S *A*G , H( *G*B , €TO\ A)PO\ TH=S *G*B OU)DE\N , H( *D*E , H( *D*H <#2> €OU)DE\N , H( *D*K , H( *K*M , TO\ *M*C €<#2> , TO\ DI\S U(PO\ TW=N *A*G, *G*B H)/TOI *M*Z € , H( *M*N , H( *M*H . @1 œ‚Ðô{1&Ad prop. 63$}1 ›‚Ð*(H *A*B , TO\ A)PO\ TH=S *A*B , H( €*A*G , TO\ A)PO\ TH=S *A*G , H( *G*B , €TO\ A)PO\ TAU/THS , H( *D*K , H( *D*E , H( €*K*M OU)DE\N , TO\ U(PO\ *A*B, *B*G H)/TOI TO\ *M*C € , TO\ DI\S , H( *M*H . œ‚Ñô{1&Ad prop. 64$}1 ›‚Ñ*(H *A*B , TO\ A)PO\ TH=S *A*B , H( *A*G € , TO\ A)PO\ TH=S *A*G , H( *G*B , TO\ A)PO\ €TH=S *G*B OU)DE\N , H( *D*E , H( *D*H , H( €*K*M OU)DE\N , O(MOI/WS KAI\ H( *M*N, H( *D*M € , H( *D*K , TO\ *M*C D , TO\ *M*Z € , H( *M*H, H(\N DI/XA TMHTE/ON EI)S TH\N *M*N KAI\ *N*H, <#2> € . œ‚Òô{1&Ad prop. 65$}1 ›‚Ò*(H *A*B , TO\ A)PO\ TAU/THS , H( *A*G € , TO\ A)PO\ TH=S *A*G , H( *G*B , TO\ A)PO\ €TH=S *G*B , H( *D*H , H( *D*E , €TO\ *D*L , H( *D*M <#2> , TO\ U(PO\ TW=N €*A*B, *B*G TO\ *M*C DU/O , H( *M*H . @1 ‚Ó³´³ôÿ{1&Ad prop. 66$}1 ›‚Ó*(H *A*E , H( *E*B , H( *A*B <#2> , H( *G*D DE/KA € , H( *G*Z , H( <*Z*D> . *(/EBDOMON KEFA/LAION, E)N W(=| PERI\ TH=S PRO\S TA\S KATA\ €SU/NQESIN <#2> A)LO/GOUS SUMMETRI/AS DIALE/GETAI DEIKNU/WN, O(/TI €H( E(KA/STH| SU/MMETROS O(MOEIDH/S E)STIN AU)TH=|, KAI\ E)/TI TA\S €DUNA/MEIS AU)TW=N PARA\ TA\S R(HTA\S PARABA/LLWN E)PISKE/PTETAI €TA\ PLA/TH TW=N XWRI/WN A)NTI/STROFON E(TE/RAN E(CA/DA TH=| E)N TW=| €<#2> KEFALAI/W| PARADOQEI/SH| TAU/THN EU(RW/N. MH/KEI &p. 113, 20]$ A)NAGKAI/WS TO\ MH/KEI PRO/SKEITAI, €E)PEI/, E)A/N EI)SI DUNA/MEI MO/NON SU/MMETROI, PROXWREI= E)K DU/O €O)NOMA/TWN EI)=NAI TH\N TH=| E)KKEIME/NH| E)K DU/O O)NOMA/TWN SU/M-€METRON DUNA/MEI MO/NON KAI\ AU)TH\N EI)=NAI E)K DU/O O)NOMA/TWN, TH=| €TA/CEI DE\ MH\ EI)=NAI TH\N AU)TH/N. GEGONE/TW W(S &p. 114, 4]$ PO/QEN DH=LON TOU=TO DUNATO\N €EI)=NAI, W(S TH\N *A*B PRO\S *G*D, OU(/TWS TH\N *A*E PRO\S E)LA/SSONA €TH=S *G*D; DIA\ TH=S A)DUNA/TOU. E)/STW H)\ PRO\S AU)TH\N H)\ PRO\S TH\N €MEI/ZONA TH=S *G*D: E)LE/GXETAI DIA\ ID# TOU= E#, O(/TI OU)/TE PRO\S €AU)TH\N TH\N *G*D OU)/TE PRO\S TH\N MEI/ZONA AU)TH=S. LEI/PETAI þï€µ°²²ÿï°°±ÿ¨Š›‚Ö†PRO\S TH\N E)LA/TTONA TH=S *G*D, TOUTE/STI TH\N *G*Z. KAI\ H( *G*Z TH=S *Z*D &p. 114, 18]$ DIA\ TOU= KQ# KAI\ L# €TOU= I# PORI/SASQAI TAU=TA DUNATO/N. ‚Ø³´µôÿ{1&Ad prop. 67$}1 ›‚Ø*(H *A*B , H( *G*D , H( *A*E , H( *E*B € , H( *G*Z , H( *Z*D . *DEI= PRW=TON EU(REI=N TH\N E)K TW=N DU/O ME/SWN PRW/THN €KAI\ DEUTE/RAN: KAI\ AU(=TAI DE\ EU(RI/SKONTAI DIA\ KH# KAI\ DIA\ KZ#: €KAI\ OU(/TWS DI/ELE EI)S TA\ O)NO/MATA, E)/XOUSI DE\ AI( DU/O KOINH=| @1 €DUNA/MEI MO/NON SU/MMETRON. A)/LLO E)STI\ NO/HMA TO\ LE/GEIN €EU)QEI=A EU)QEI/A| SU/MMETROS MH/KEI KAI\ A)/LLO EU)QEI=A EU)QEI/A| €SU/MMETROS DUNA/MEI MO/NON KAI\ A)/LLWS EU)QEI=A EU)QEI/A| SU/MMET-€ROS. TOU=TO GENIKW/TATON, TAUTI/ZETAI DE\ TO\ LE/GEIN EU)QEI=A EU)-€QEI/A| DUNA/MEI SU/MMETROS TW=| NOH/MATI TW=| LE/GEIN A(PLW=S EU)QEI=A €EU)QEI/A| SU/MMETROS. ‚Ú³´¹ôÿ{1&Ad prop. 68$}1 ›‚Ú*(H *A*B , H( *G*D , H( *A*E , H( *E*B € , H( *G*Z <#2> , H( *Z*D . 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SU/MMETRON DE\ TO\ A)PO\ TH=S €*E*B TW=| A)PO\ TH=S *Z*D: SU/MMETRON KAI\ TO\ SUGKEI/MENON TW=| €SUGKEIME/NW|: R(HTO\N E)KEI=NO KAI\ TOU=TO. *)EPEI/ E)STIN W(S H( *A*E PRO\S *E*B, H( *G*Z PRO\S TO\ *Z*D, €KAI/ E)STIN W(S TO\ A)PO\ TH=S *A*E PRO\S TO\ U(PO\ *A*E*B, OU(/TWS TO\ €A)PO\ TOU= *G*Z PRO\S TO\ U(PO\ *G*Z*D, E)NALLA/C E)STIN W(S TO\ A)PO\ €TH=S *A*E PRO\S TO\ A)PO\ TH=S *G*Z, OU(/TWS TO\ U(PO\ *A*E*B PRO\S TO\ €U(PO\ *G*Z*D. 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OU)K A)/RA SU/M-€METROI/ EI)SIN AI( *A*G KAI\ *A*H. ƒ¢´²±ôÿ{1&Ad prop. 93$}1 ›ƒ¢*)APOTOMH\ DE/ E)STI G#, O(/TAN MHDETE/RA SU/MMETROS H)=| €TH=| E)KKEIME/NH| R(HTH=| MH/KEI, H( DE\ O(/LH TH=S SUNARMOZOU/SHS €MEI=ZON DU/NATAI TW=| A)PO\ SUMME/TROU E(AUTH=|, KATA\ TOU\S G# €O(/ROUS. ƒ£ãïì ±ÿ*(H R(HTH\ *A*G €#754 @2 H( *A*D OU)DE/N €#752#756 €#751#755 @2 H( *A*H €#751#760 €#754#754 €#752#760 €#754#760 @2 H( AU)TH=S €H(MI/SEIA €#755 €#752#752 €#751#760 €#752#760 @1 TO\ U(PO\ R(HTH=S €KAI\ TH=S *A*D €#751 €#754#755 @2 H( *D*H €#751#760 €#751#758 €#755 €#754#760 @2 H( TAU/THS €H(MI/SEIA €#755 €#759 €#752 €#755#760 @2 TO\ A)PO\ €TAU/THS €#752#756 €#753#751 €#755#760 €#751#751 €#758 €#751 €#754#760 @2 TO\ A)PO\ TH=S €METACU\ TW=N €TOMW=N €#752 €#751#758 €#754 €#754#755 €#751#758 €#754#755 @2 ƒ£ãïì ±°ÿOU(= H( PLEURA/ €#751 €#753#751 €#751 €#751#754 @2 TO\ A)PO\ €TAU/THS €#752#758 €#754#759 €#755#754 €#755#756 €#752#756 €#754#756 €#754#760 @2 H( TO\ XWRI/ON €DUNAME/NH TO\ €*A*B €#751 €#751#759 €#752#751 @2 H( *A*Z €#756 €#755#753 €#751#751 €#753#754 @2 H( *Z*H €#753 €#755#751 €#759 €#756 @2 ›ƒ¤*)HPORH/QH TW=| PRO\S TH\N KATAGRAFH\N A)POBLEYAME/-€NW|, W(S, E)PEI\ PARA\ TH\N *A*H I)/SON TW=| A)PO\ TH=S H(MISEI/AS TH=S €E)LA/TTONOS, TOUTE/STI TW=| A)PO\ TH=S *E*H, PARALLHLO/GRAMMON €TO\ U(PO\ TW=N TMHMA/TWN TW=N *A*Z, *Z*H PERIEXO/MENON E)LLEI=PON €EI)/DEI TETRAGW/NW| TW=| U(PO\ TW=N *A*Z, *Z*I, PARA/LLHLOS DE\ H( €*Z*I TH=| *A*G, I)/SH A)/RA H( *A*Z TH=| *A*G: TO\ DE\ U(PO\ DU/O R(HTW=N €MH/KEI SUMME/TRWN PERIEXO/MENON R(HTO/N E)STI: W(/STE R(HTO/N €E)STI TO\ *A*I: A)LLA\ KAI\ ME/SON KATA\ TO\N GEWME/TRHN: H( GA\R €*A*Z R(HTH\ OU)=SA A)SU/MMETROS KAT' AU)TO\N TH=| *A*G R(HTH=| OU)/SH|: €W(/STE KAI\ ME/SON TO\ *A*I. E)/STI DE\ TOU=TO YEU=DOS. TO\ GA\R A)PO\ @1 €TH=S *A*Z A)NAGRAFO/MENON TETRA/GWNON I)/SAS E(/CEI TA\S PLEU-€RA/S, OU)K E)/STI DE\ H( *A*Z I)/SH TH=| *Z*I: H)= GA\R A)\N I)/SH H)=N KAI\ TH=| €*A*G: A)LLA\ TH=| *Z*I E)KBEBLHME/NH| KAI\ TH=| *A*G W(SAU/TWS E)K-€BEBLHME/NH|, W(S FE/RE EI)PEI=N E)PI\ €@3 TOU/TOU TOU= SXH/MATOS: TO\ GA\R €A)PO\ TH=S *A*Z A)NAGRAFO/MENON TE-€TRA/GWNON TO\ *Z*L E)STI KAI\ OU)XI\ €TO\ *Z*G. TO\ ME\N GA\R *Z*L R(HTO/N, €O(/TI KAI\ A)PO\ R(HTH=S TH=S *A*Z, TO\ €DE\ *Z*G ME/SON W(S U(PO\ DU/O R(HTW=N €DUNA/MEI MO/NON SUMME/TRWN PERI-€EXO/MENON. W(S OU)=N H( *L*C PRO\S *C*O: €SU/MMETROS DE/: OU(/TW TO\ *A*C PRO\S *C*H: SU/MMETRON A)/RA. KAI\ €W(S H( *L*C PRO\S *C*O, OU(/TW TO\ *G*C PRO\S *C*K: SU/MMETRON A)/RA. €E)A\N DE\ H)=| W(S O(/LON PRO\S O(/LON, OU(/TWS A)FAIREQE\N PRO\S A)FAIRE-€QE/N, KAI\ LOIPO\N A)/RA TO\ *A*I, *I*H E)/STAI W(S O(/LON PRO\S O(/LON. €E)PEI\ OU)=N H( AU)TH\ A)NALOGI/A SW/ZETAI, H( DEI=CIS PROBAI/NEI E)PI\ €TH=S *A*G DIA\ TO\ TAU/THN PROUPOTEQH=NAI R(HTH\N KAI\ MH\ TH\N *A*L. A)SU/MMETROS A)/RA E)STI\N H( *A*Z &p. 163, 9. 10$] E)PEI\ H( €*A*Z TH=| *H*D E)STIN A)SU/MMETROS, H( DE\ *H*D TH=| *E*H SU/M-€METROS, H( *A*Z A)/RA TH=| *E*H A)SU/MMETRO/S E)STIN. ƒ¦´²³ôÿ{1&Ad prop. 94$}1 ƒ¦ãïì ±ÿ*(H *A*D €#752 €#754#757 €#753#755 @2 H( *A*G €#754 @2 H( *A*H €#751#753 €#755 €#754#760 €#754#760 @2 H( AU)TH=S €H(MI/SEIA €#756 €#753#752 €#755#760 €#752#760 @2 TO\ A)PO\ €TAU/THS €#754#752 €#755#752 €#752 €#752#753 €#753#753 €#752#756 €#754#760 @2 O(\ ME/LLEI €PRO\S TO\ €A)PO\ TH=S €*Q*H PARA-€BLHQH=NAI @1 TO\ *A*B €XWRI/ON €#751#751 €#751#760 €#752#760 @2 H( AU)TOU= €PLEURA\ H( *A*G €#753 €#752#760 €#753#752 @2 H( PLEURA\ TOU= €A)PO\ TH=S METACU\ €TW=N TOMW=N €#754 €#752 €#753 €#754#754 @2 ƒ¦ãïì ±°ÿH( *Z*H €#752 €#753#760 €#751#759 €#753#756 @2 H( *A*Z €#751#760 €#753#755 €#752#751 €#754 @2 ›ƒ§*TAU/THN TH\N R(HTH\N LA/MBANE, H(\N E)CE/QOU E)N TH=| EU(RE/-€SEI TH=S D# A)POTOMH=S. ƒ¨´²µôÿ{1&Ad prop. 95$}1 ƒ¨ãïì ±ÿ*(H *D*H €#751#760 €#751#758 €#755 €#754#760 @2 H( *A*D €#753 €#754#757 €#755#752 @2 H( *A*G €#754 @2 H( *D*H €#751#760 €#751#758 €#755 €#754#760 @2 H( *A*H €#751#754 €#755 €#755#757 €#754#760 @2 H( AU)TH=S H(MI/SEIA €#757 €#752 €#755#758 €#755#760 @2 TO\ A)PO\ TH=S €H(MISEI/AS €#754#759 €#754#751 €#755#753 €#752#753 €#751 €#752#751 €#754#760 @2 TO\ *A*B €#751#754 €#752#753 €#753#756 @2 H( AU)TOU= €PLEURA\ H( *L*N €#754 €#757 €#753#757 @1 ƒ¨ãïì ±°ÿH( PLEURA\ TOU= A)PO\ TH=S €METACU\ TW=N TOMW=N €#754 €#754#758 €#754#758 €#753#756 @2 TO\ A)PO\ TH=S €METACU\ TW=N TOMW=N €#752#753 €#751#760 €#753 €#751#751 €#755#753 €#752#760 @2 H( *A*Z €#751#751 €#755#751 €#754#757 €#752#756 @2 H( *Z*H €#752 €#751#754 €#751#760 €#751#754 @2 ›ƒ©*)EPEI\ A)POTOMH/ E)STIN H( *A*B, E)/XEI TH\N PROSARMO/-€ZOUSAN AU)TH=|, KAI/ E)STIN H( O(/LH KAI\ H( PROSARMO/ZOUSA DUNA/-€MEI MO/NON SU/MMETROS. EI) DE\ OU)K EI)SI\N H( O(/LH KAI\ H( PROS-€ARMO/ZOUSA R(HTAI\ DUNA/MEI MO/NON SU/MMETROI, OU)DE\ A)POTOMH/ €E)STIN H( *A*B DIA\ OG# I#. œƒªô{1&Ad prop. 96$}1 ƒªãïì ±ÿ*(H *A*D €#753 €#752#757 €#755#760 @2 H( *A*G €#754 @2 H( *D*H €#751#760 €#751#758 €#755 €#754#760 @2 H( *A*H €#751#753 €#754#755 €#755#755 €#754#760 @2 H( H(MI/SEIA €TH=S *A*H €#756 €#755#752 €#755#758 €#755#760 @2 TO\ A)PO\ €TAU/THS €#754#757 €#752#752 €#751#759 €#751#760 €#752#754 @2 TO\ *A*B €XWRI/ON €#751#753 €#755#751 €#752#760 @2 H( *L*N H( €AU)TOU= PLEURA/ €#753 €#754#753 €#752#760 @2 TO\ A)PO\ TH=S METACU\ €TW=N TOMW=N €#752 €#755#760 €#752#758 €#755#759 @2 ƒªãïì ±°ÿH( AU)TOU= €PLEURA/ €#751 €#754#751 €#758 @2 H( *Z*H €#755 €#751#751 €#754#759 €#755#760 @1 ƒ«´²¹ôÿ{1&Ad prop. 97$}1 ƒ«ãïì ±ÿ*(H *A*B €#752#760 €TO\ A)PO\ €TH=S *A*B €#754#760#760 @2 H( *G*D €#754 €H( *G*Z €#751#760#760 @2 H( *H*B €#751#760 €#751#758 €#755 €#754#760 @2 H( *A*H €#753#760 €#751#752 €#755 €#754#760 @2 TO\ PLA/TOS TO\ *K*M €#752#756 €#753#751 €#755#760 €#751#751 €#758 €#751 €#754#760 @2 TO\ PLA/TOS TO\ *G*K €#752#752#759 €#753#752 €#754#756 €#755#751 €#310 €#751 €#754#760 @2 H( *G*M €#752#755#756 €#754 €#753#757 €#752 €#751#756 €#753 €#752#760 @2 H( *G*Z H( A)PO\ MONA/DWN . @2 ›ƒ¬*TO\ *N SHMEI=ON, O(/PER E)/TEME TH\N *Z*M DI/XA, OU) PESEI=-€TAI E)PI\ TH=S DIXOTOMI/AS TH=S MEI/ZONOS TH=S *G*M, E)PEI\ E)/STAI €H( *Z*M TH=| *G*M I)/SH. OU) MH\N OU)DE\ METACU\ TW=N *K, *M SHMEI/WN €PESEI=TAI TO\ *N: EI) GA\R PE/SH|, SUMBAI/NEI TO\ MEI=ZON TOU= þï€µ°²²ÿï°°±ÿ¨Š›ƒ¬…E)LA/TTONOS E)/LATTON EI)=NAI: O(/PER A)/TOPON. TA\ GA\R A)PO\ TW=N €*A*H, *H*B I)/SA E)STI\ TOI=S *G*Q, *K*L, TO\ DE\ A(/PAC U(PO\ TW=N €*A*H, *H*B I)/SON TW=| *N*L. KAI/ E)STI TO\ *N*L ME/SON A)NA/LOGON €TW=N *G*Q, *K*L: TW=N GA\R A)PO\ TW=N *A*H, *H*B ME/SON E)DEI/XQH €TO\ U(PO\ TW=N *A*H, *H*B. E)/STIN A)/RA, W(S TO\ *G*Q PRO\S TO\ *N*L, €TO\ *N*L PRO\S TO\ *K*L. MEI=ZON DE\ TO\ *G*Q TOU= *N*L: MEI=ZON @1 €A)/RA KAI\ TO\ *N*L TOU= *K*L: O(/PER A)/TOPON, TO\ MEI=ZON TOU= E)LA/S-€SONOS. OU)K A)/RA PESEI=TAI TO\ *N METACU\ TW=N *K, *M SHMEI/WN. LOIPO\N A)/RA TO\ *Z*L &p. 172, 24. 25]$ E)PEI\ TA\ A)PO\ €TW=N *A*H, *H*B DU/O TETRA/GWNA, W(S E)DEI/XQH, I)/SA EI)SI\ TW=| DI\S €U(PO\ TW=N *A*H, *H*B META\ TOU= A)PO\ TH=S *G*Z, H)\ KAI\ A)NA/PALIN €E)PEI\ TO\ DI\S U(PO\ TW=N *A*H, *H*B META\ TOU= A)PO\ TH=S *A*B €TETRAGW/NOU I)/SA EI)SI\ TOI=S DUSI\ TETRAGW/NOIS TW=| TE A)PO\ TH=S €*A*H KAI\ TW=| A)PO\ TH=S *H*B, E)/STI DE\ TO\ *G*E I)/SON TW=| A)PO\ TH=S €*A*B, LEI/PETAI TO\ DI\S U(PO\ TW=N *A*H, *H*B I)/SON EI)=NAI TW=| *Z*L. ƒ®´³²ôÿ{1&Ad prop. 98$}1 ƒ®ãïì ±ÿ*(H *A*B €#752 €#755#310 €#754#754 @2 H( *G*D €#754 @2 H( *B*H €#751 €#753#759 €#759 @2 H( *G*K €#755 €#752#751 €#754#754 €#754#757 €#751#752 @2 H( *K*M €OU)DE/N €#754#760 €#755#757 €#754#760 €#755#760 @2 TO\ A)PO\ TH=S *A*B €#310 €#755#752 €#752#755 €#753#756 €#751#756 @2 H( *G*Z €#752 €#751#753 €#756 €#752#754 €#754 @2 H( *A*H €#754 €#753#757 €#755#753 @2 ›ƒ¯LOIPO\N A)/RA &p. 175, 20$] E)PEI\ TO\ A)PO\ TH=S *A*B I)/SON €E)STI\ TW=| *G*E, TW=| DE\ A)PO\ TH=S *A*H TO\ *G*Q, KAI\ E)/TI TW=| A)PO\ €TH=S *B*H I)/SON TO\ *K*L, A)LLA\ TOI=S ME\N A)PO\ TW=N *A*H, *H*B €I)/SA E)STI\ TO/ TE DI\S U(PO\ TW=N *A*H, *H*B KAI\ TO\ A)PO\ TH=S *A*B: €TOU=TO GA\R DE/DEIKTAI E)N TW=| Z# QEWRH/MATI TOU= B# BIBLI/OU: @1 €W(=N TO\ A)PO\ TH=S *A*B I)/SON TW=| *G*E, LOIPO\N A)/RA TO\ DI\S U(PO\ €TW=N *A*H, *H*B I)/SON E)STI\ TW=| *Z*L. 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TH=| TA/CEI TH=| KAQ' AU(TH/N &p. 201, 13. 14$] H( ME/SH €A)POTOMH\ PRW/TH, ME/SH A)POTOMH\ DEUTE/RA, E)LA/TTWN, META\ €R(HTOU= ME/SON TO\ O(/LON POIOU=SA, META\ ME/SOU ME/SON TO\ O(/LON €POIOU=SA. €*)APOTOMH\N PRW/THN, DEUTE/RAN, TRI/THN, TETA/RTHN, PE/MPTHN, €E(/KTHN. @1 ƒÊ´¶·ôÿ{1&Ad prop. 112$}1 ›ƒÊ*)EK DU/O O)NOMA/TWN H)=N H( E)K DU/O R(HTW=N DUNA/MEI MO/-€NON SUMME/TRWN, O(/TAN DE\ A)PO\ R(HTH=S A)FAIREQEI=SA R(HTH\ DUNA/-€MEI MO/NON SU/MMETROS H)=N TH=| O(/LH|, H( LOIPH\ A)/LOGOS H)=N KAI\ €E)KALEI=TO A)POTOMH/. 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I)/SAI DE\ A)LLH/-€LAIS EI)SI\ KA)KEI=NAI DIA\ TO\ I)/SHN EI)=NAI TH\N *A*D TH=| *A*G €KAI\ I)SOSKELE\S KAQESTA/NAI TO\ TRI/GWNON. š‹±³ôÿ{1&Ad prop. 54 demonstr. alt.$}1 ˜‹*)EKKEI/SQW DOQEI=SA &p. 202, 12$] TW=| MEGE/QEI: OU(/TW €GA\R A)EI\ LAMBA/NEI A)ORISTW=S LE/GWN. *)/ESTIN A)/RA W(S TO\ *A PRO\S TO\ *B &p. 202, 21$] E)MA/-€QOMEN GA/R, O(/TI, E)A\N TE/SSARES EU)QEI=AI A)NA/LOGON W)=SIN, €KAI\ TA\ A)P' AU)TW=N EU)QU/GRAMMA O(/MOIA/ TE KAI\ O(MOI/WS €A)NAGEGRAMME/NA A)NA/LOGON E)/STAI. *KAI\ AI( LOIPAI\ A)/RA PLEURAI/ &p. 204, 8$] E)PEI\ €LO/GOS TH=S *G*D PRO\S TH\N *E*Z DOQEI/S, E)/STI DE\ KAI\ TO\ *A €O(/MOION TW=| *B, TW=N DE\ O(MOI/WN SXHMA/TWN AI( PLEURAI\ €A)NA/LOGO/N EI)SIN, PRO\S A(\S AU(=TAI A)NA/LOGO/N EI)SIN, KA)KEI=NAI €DEDOME/NAI E)/SONTAI. šŽ±µôÿ{1&Ad prop. 55 demonstr. alt.$}1 ˜Ž*DE/DOTAI A)/RA TW=| EI)/DEI &p. 204, 19$] E)MA/QOMEN GA\R €E)N TOI=S O(/ROIS, O(/TI EU)QU/GRAMMA SXH/MATA TW=| EI)/DEI DE-€DO/SQAI LE/GETAI, W(=N AI(/ TE GWNI/AI DEDOME/NAI EI)SI/ KTL. *DIA\ TA\ AU)TA\ DH/ &p. 204, 24$] W(S E)N TW=| SXOLI/W| €TOU= NB#: A)PO\ GA\R E(KA/STHS A)NAGRA/FONTES TETRA/GWNON €O(MOI/WS DEI/COMEN. @1 š±¹ôÿ{1&Ad prop. 67 demonstr. alt.$}1 ˜*(/WSTE KAI\ TOU= U(PO\ TW=N *E*G*D &p. 206, 13$] SXO/LION. €E)K TW=N LAMBANOME/NWN TH=| *G*D TH=| AU)TH=| A)PODEI/CEI TH=| @3 €E)PI\ TOU= CD# XRHSO/MEQA. €E)KQE/MENOI EU)QE/WS TH\N <20AB>20 €KAI\ TH=| ME\N *E*G I)/SHN €TH\N <20AG>20, TH=| DE\ *A*H €TH\N <20GB>20 KAI\ PRO\S O)R-€QA\S A)PO\ TOU= G TH\N <20GD>20 €I)/SHN OU)=SAN TH=| *G*D KAI\ €TA\ E(CH=S W(S E)N TW=| CD# QEWRH/MATI. *SXO/LION. W(S GA\R H( *E*G PRO\S *A*Z, OU(/TWS TO\ €U(PO\ TW=N *E*G*D PRO\S TO\ U(PO\ TW=N *A*Z, *G*D. *TOU= DE\ U(PO\ TW=N *A*Z, *G*D PRO\S TO\ *A*G*D €TRI/GWNON &p. 206, 15$] DIPLA/SION GA/R, FHSI/N, E)/STIN @3 €AU)TOU=. PW=S; E)KKEI/SQW TIS EU)QEI=A €H( <20HQ>20, KAI\ KEI/SQW TH=| ME\N *G*D I)/SH H( €<20HQ>20, TH=| DE\ *A*Z PRO\S O)RQA\S A)XQEI=SA €H( <20HK>20, KAI\ SUMPEPLHRW/SQW TO\ <20KQ>20 PAR-€ALLHLO/GRAMMON, KAI\ E)/STW DIAGW/NIOS €H( <20QK>20 A)NTI\ TH=S *A*D: TO\ A)/RA U(PO\ TW=N €<20QH, HK>20 E)STI TO\ <20KQ>20, KAI/ E)STI DI' AU)TOU= €H( <20QK>20: DIPLA/SION A)/RA E)STI\ TOU= <20KHQ>20 TRIGW/NOU: E)PI/ TE GA\R €TH=S AU)TH=S BA/SEW/S E)STI TH=S <20HQ>20 KAI\ E)N TAI=S AU)TAI=S PAR- @1€ALLH/LOIS TAI=S <20HQ, KL>20: KAI/ E)STI TO\ ME\N <20QK>20 PARALLHLO/-€GRAMMON I)/SON TW=| U(PO\ TW=N *A*Z, *G*D, TO\ DE\ <20KHQ>20 I)/SON €TW=| *A*G*D TRIGW/NW|: DIPLA/SION A)/RA TO\ U(PO\ TW=N *A*Z, *G*D €TOU= *A*G*D TRIGW/NOU. *PW=S TO\ U(PO\ TW=N *A*Z, *G*D TOU= *A*G*D TRIGW/NOU €DIPLA/SIO/N E)STIN; DEI/COMEN OU(/TWS. H)/XQW DIA\ TOU= *A @3 €TH=| *G*D PARA/LLHLOS H( *A*Z €KAI\ DIA\ TOU= *H TH=| *A*Z €PARA/LLHLOS H( *H*Q. DU/O þï€µ°²²ÿï°°²ÿ¯áðð óãèÿ˜“†A)/RA PARALLHLO/GRAMMA/ €E)STI TA\ *A*Q, *A*D [1U(PO/-€KEITAI GA\R KAI\ H( *A*G TH=| €*B*D PARA/LLHLOS]1 E)PI\ TH=S €AU)TH=S BA/SEWS O)/NTA TH=S €*A*H KAI\ E)N TAI=S AU)TAI=S €PARALLH/LOIS TAI=S *A*H, *Z*D: I)/SON A)/RA TO\ *A*Q PARALLHLO/-€GRAMMON TW=| *A*D PARALLHLOGRA/MMW|. KAI\ E)PEI\ TO\ U(PO\ €TW=N *A*Z, *A*H E)STI TO\ *A*Q, I)/SH DE\ H( *A*H TH=| *G*D, KAI\ €TO\ A)/RA U(PO\ TW=N *A*Z, *G*D E)STI TO\ *A*Q: DIPLA/SION €DE\ TO\ *A*Q TOU= *A*G*D TRIGW/NOU, E)PEI\ KAI\ TO\ *A*D: TO\ €A)/RA U(PO\ TW=N *A*Z, *G*D DIPLA/SIO/N E)STI TOU= *A*G*D €TRIGW/NOU. @1 š”³±ôÿ{1&Ad prop. 67 demonstr. tert.$}1 ˜”*KAI/ E)STI TOU= DI\S U(PO\ TW=N *B*A*G &p. 208, 7$] €DIA\ GA\R TO\ C#2# TO\ U(PO\ TW=N *B*A*G PRO\S TO\ TRI/GWNON €LO/GON E)/XEI DEDOME/NON: W(/STE KAI\ TO\ DI/S. *TA\ A)/RA A)PO\ TW=N *B*A*G &p. 208, 11_12$] E)N TW=| B# €TW=N STOIXEI/WN E)DEI/XQH TW=| IG# QEWRH/MATI. *TA\ A)/RA A)PO\ TW=N *B*A*G &p. 208, 14$] E)N TW=| B# €TW=N STOIXEI/WN E)DEI/XQH E)N TW=| D# QEWRH/MATI. *TOUTE/STI TW=| DI\S U(PO\ SUNAMFOTE/ROU TH=S *G*A*D €&p. 208, 17_18$] E)A\N GA\R LA/BWMEN TH\N <20BA>20 MI/AN EU)QEI=AN @3 €W(S A)/TMHTON, TH\N DE\ <20DAG>20 €MI/AN ME\N KAI\ AU)TH/N, TETMH-€ME/NHN DE\ KATA\ TO\ A, GI/NETAI €TO\ U(PO/ TE TH=S A)TMH/TOU €TH=S <20BA>20 KAI\ E(KA/STOU TW=N €TMHMA/TWN TW=N <20DA, AG>20 I)/SON €TW=| U(PO\ SUNAMFOTE/ROU TH=S <20DAG>20 KAI\ TH=S <20AB>20 DIA\ TO\ A# €TOU= DEUTE/ROU BIBLI/OU TW=N STOIXEI/WN: W(/STE KAI\ TO\ DI\S €U(PO\ TW=N <20BA, AD>20 META\ TOU= DI\S U(PO\ TW=N <20BA, AG>20 I)/SON €E)STI\ TW=| DI\S U(PO\ SUNAMFOTE/ROU TH=S <20GAD>20 KAI\ TH=S <20AB>20. *KAI\ TOU= U(PO\ SUNAMFOTE/ROU A)/RA TH=S *D*A*G €&p. 208, 26$] E)A\N GA\R POIH/SWMEN E)P' EU)QEI/AS TH\N *D*A TH=| €*A*G W(S TH\N *D*A*G KAI\ DIA\ TOU= *A TH=| *D*G PRO\S O)RQH\N €A)NASTH/SWMEN TH\N *A*B, DHLADH\ I)/SHS MENOU/SHS TH=S ME\N @1 €*D*A TH=| *D*A, TH=S DE\ *A*G TH=| *A*G, TH=S DE\ *B*A TH=| *B*A, €E)/STAI SAFE\S TO\ LEGO/MENON: W(S GA\R AI( BA/SEIS, OU(/TW €KAI\ TA\ PARALLHLO/GRAMMA TA\ U(PO\ TO\ AU)TO\ U(/YOS O)/NTA. *KAI\ TOU= DI\S U(PO\ SUNAMFOTE/ROU TH=S *D*A*G €&p. 210, 2$] E)/STW EU)QEI=A H( <20DE>20, KAI\ KEI/SQW TH=| ME\N *D*A I)/SH €H( <20DA>20, TH=| DE\ *A*G H( <20AG>20, KAI\ A)PO\ TOU= A TH=| <20DG>20 PRO\S @3 €O)RQA\S A)NESTA/TW H( <20AB>20, KAI\ €KEI/SQW H( <20AB>20 TH=| *A*B I)/SH. 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E)/STI DE\ KAI\ TOU= U(PO\ TW=N <20GA, AB>20 €PRO\S TO\ <20ABG>20 TRI/GWNON LO/GOS DOQEI\S DIA\ TO\ C#2# QEW/-€RHMA: KAI\ TO\ U(PO\ <20DAG, AB>20 A)/RA PRO\S TO\ <20ABG>20 TRI/GWNON €LO/GOS E)STI\ DOQEI\S DIA\ TO\ H# QEW/RHMA. *KAI\ TW=| DI\S U(PO\ TW=N *B*A, *G*Z &p. 210, 21_22$] €E)A\N GA\R SUMPLHRW/SWMEN TO\ U(PO\ TW=N <20BA, AG>20 PARALLHLO/-€GRAMMON W(S TO\ <20AH>20, KAI\ €@3 DIA\ TOU= Z PARA/LLHLON €A)GA/GWMEN TH=| <20AB>20, E)/PEITA €A)FE/LWMEN TO\ U(PO\ TW=N <20BAZ>20, €KATALEI/PETAI TO\ <20ZH>20 PAR-€ALLHLO/GRAMMON, O(/ E)STIN €U(PO\ TW=N <20BA, ZG>20: TH=| GA\R <20BA>20 I)/SH E)STI\N H( <20ZK>20. @1 *)EA\N GA\R A)PO\ TOU= DI\S U(PO\ TW=N *B*A*G A)FE/LW-€MEN TO\ DI\S U(PO\ TW=N *B*A*Z, TO\ KATALEIPO/MENO/N E)STI TO\ €DI\S U(PO\ TW=N *B*A, *Z*G. *(/WSTE KAI\ TOU= U(PO\ TW=N *E*G, *A*B &p. 212, 6$] €EI) GA\R TH\N *Z*G E)P' EU)QEI/AS TH=S *E*G NOH/SWMEN KAI\ €KOINO\N U(/YOS TH\N *B*A, E)/STAI TO\ LEGO/MENON DH=LON: W(S €GA\R H( *E*G BA/SIS PRO\S TH\N *G*Z BA/SIN, OU(/TWS TO\ *E*A €PARALLHLO/GRAMMON, TOUTE/STI TO\ U(PO\ TW=N *E*G, *B*A, €PRO\S TO\ *A*Z PARALLHLO/GRAMMON, TOUTE/STI TO\ U(PO\ TW=N €*Z*G, *A*B. *TOU= DE\ U(PO\ TW=N *A*B, *G*E PRO\S TO\ *A*B*G €&p. 212, 7_8$] DIA\ TO\ TH\N *G*E KA/QETON EI)=NAI E)PI\ TH\N €*B*A E)KBALLOME/NHN KAI\ GI/NESQAI DIPLA/SION TO\ U(PO\ TW=N €*B*A, *E*G TOU= *A*B*G TRIGW/NOU. *)EA\N GA\R DIA\ TOU= *G TH=| *E*B PARA/LLHLON A)GA/GW-€MEN KAI\ DIA\ TW=N *A, *B TH=| *E*G PARALLH/LOUS A)GA/GWMEN, €E)/STAI DH=LON. TO\ GA\R U(PO\ *E*G, *A*B E)STI TO\ *A*B, KAI\ €TO\ *A*B DIPLA/SIO/N E)STI TOU= *A*B*G TRIGW/NOU, KAI\ DIA\ €TOU=TO LO/GON E)/XEI PRO\S TO\ *A*B*G TRI/GWNON DEDOME/NON. *)EA\N GA\R DIA\ TOU= *G TH=| *E*B PARA/LLHLON A)GA/GW-€MEN KAI\ DIA\ TW=N *A, *B TH=| *E*G PARALLH/LOUS A)GA/GWMEN, €E)/STAI DH=LON: H( GA\R A)PO\ TOU= *A I)/SH E)STI\ TH=| *E*G, W(S €E)/XEI A)/NW TO\ SXO/LION. @1 š ³µôÿ{1&Ad prop. 67 demonstr. quart.$}1 ˜ *PW=S ME\N TH\N U(PO\ *D*E*G DU/NAMAI SUSTH/SASQAI €I)/SHN TH=| U(PO\ *A*D*G XWRI\S TW=N *)APOLLWNI/OU; OU(/TWS. €E)PEI\ GA\R I)/SH E)STI\N H( U(PO\ *A*G*D TH=| U(PO\ *A*D*G, MEI/ZWN €E)STI\N H( U(PO\ *B*G*D TH=S U(PO\ *A*D*G. KEI/SQW OU)=N I)/SH €TH=| U(PO\ *B*G*D H( U(PO\ *B*D*E, KAI\ E)KBEBLH/SQW H( *B*G. €E)/STI DE\ KOINH\ H( PRO\S TW=| *B GWNI/A TOU= TE *D*B*G TRI-€GW/NOU KAI\ TOU= *D*B*E. 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E)K-€BLHQEI/SHS A)/RA TH=S <20ZE>20 O)CEI=A E)/STAI H( U(PO\ <20DEL>20. KA/QETOS €OU)=N H)/XQW H( <20DL>20, KAI\ TH=| U(PO\ <20LDE>20 I)/SH KEI/SQW H( U(PO\ <20HAK>20. €LOIPH\ A)/RA H( U(PO\ <20DEL>20 I)/SH E)STI\ TH=| U(PO\ <20AKH>20, W(/STE KAI\ €H( E)FECH=S H( U(PO\ <20DEZ>20 TH=| E)FECH=S TH=| U(PO\ <20AKG>20 I)/SH E)STI/N. *TOUTE/STI TO\ U(PO\ TW=N *E*G*B &p. 214, 6_7$] E)A\N €GA\R EU)QEI=A GRAMMH\ TMHQH=|, W(S E)/TUXEN, TO\ U(PO\ TH=S O(/LHS €KAI\ E(NO\S TW=N TMHMA/TWN PERIEXO/MENON O)RQOGW/NION I)/SON €E)STI\ TW=| TE U(PO\ TW=N TMHMA/TWN KAI\ TW=| A)PO\ TOU= PRO-€EIRHME/NOU TETRAGW/NW|. *)/ISON E)STI\ TW=| A)PO\ *B*D &p. 214, 7_8$] E)A\N GA\R €TREI=S EU)QEI=AI A)NA/LOGON, TO\ U(PO\ PRW/THS KAI\ TRI/THS I)/SON €E)STI\ TW=| A)PO\ TH=S DEUTE/RAS. š¤³¹ôÿ{1&Ad prop. 68 demonstr. alt.$}1 ˜¤*PW=S DUNATO\N POIH=SAI, W(S TO\ *A PARALLHLO/GRAM-€MON PRO\S TO\ *B PARALLHLO/GRAMMON, OU(/TWS TH\N *K PRO\S *L; €EI)LH/FQW TW=N *G*D, *E*Z TRI/TH A)NA/LOGON. 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E)A\N GA\R METASTRE/YWMEN TO\N ZW|DIAKO\N KAI\ POIH/SWMEN TO\ €*A*G U(PO\ GH=N, W(S E)/XEI H( KATAGRAFH/, H( AU)TH\ A)PO/DEICIS A(RMO/SEI. &Ad finem prop.$ TH=S GA\R AU)TH=S MENOU/SHS KATA-€GRAFH=S E)A\N METASTRE/YWMEN TO\N ZW|DIAKO\N KAI\ POIH/SW-€MEN TO\ *A*G H(MIKU/KLION TOU= ZW|DIAKOU= U(PO\ GH=N, H( AU)TH\ €A)PO/DEICIS PROBH/SETAI KAI\ DEIXQH/SONTAI AI( I)/SON A)PE/-€XOUSAI TOU= I)SHMERINOU= E)N I)/SW| XRO/NW| KAI\ A)NATE/LLOUSAI. @1 šû±²¹ôÿ{1&Ad prop. 14 demonstr. alt.$}1 ˜û*(H *T*U A)/RA PERIFE/REIA &p. 124, 4$] DIA\ GA\R TOU= *U €KAI\ TOU= *M PO/LOU TOU= *A*B*G KU/KLOU ME/GISTOS KU/KLOS €GEGRA/FQW O( *M*UW: O( *M*UW A)/RA O)RQO/S E)STI PRO\S TO\N €*A*B*G: DIA\ GA\R TW=N PO/LWN AU)TOU= E)STIN. 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KOI-€NO\S PROSKEI/SQW O( XRO/NOS, E)N W(=| H( *E*K E)CALLA/TTEI TO\ €FANERO\N H(MISFAI/RION: O( A)/RA XRO/NOS, E)N W(=| H( *K*L DU/NEI, €META\ TOU= XRO/NOU, E)N W(=| H( *E*K E)CALLA/TTEI TO\ FANERO\N €H(MISFAI/RION, I)/SOS E)STI\ TW=| XRO/NW|, E)N W(=| H( *E*Z A)NATE/LLEI €KAI\ H( *E*K E)CALLA/TTEI TO\ FANERO\N H(MISFAI/RION. A)LL' O( @3 €ME\N XRO/NOS, E)N W(=| H( €*L*K DU/NEI KAI\ H( *K*E €E)CALLA/TTEI TO\ FANERO\N €H(MISFAI/RION, O( XRO/NOS €E)STI/N, E)N W(=| H( *E*L E)C-€ALLA/SSEI TO\ FANERO\N €H(MISFAI/RION: O( DE\ XRO/-€NOS, E)N W(=| H( *E*Z A)NA-€TE/LLEI, META\ TOU= XRO/-€NOU, E)N W(=| H( *E*K E)CAL-€LA/TTEI TO\ FANERO\N H(MI-€SFAI/RION, O( XRO/NOS €E)STI/N, E)N W(=| H( *Z*E*K E)CALLA/TTEI TO\ FANERO\N H(MISFAI/RION: €AI( *Z*E*K, *E*K*L A)/RA E)N I)/SW| XRO/NW| E)CALLA/SSOUSI TO\ €FANERO\N H(MISFAI/RION. €A)LL' E)/STW H( *E*Z PERIFE/REIA E)LA/SSWN TETARTHMORI/OU: €KAI\ H( *E*K A)/RA E)LA/SSWN E)STI\ TETARTHMORI/OU. KEI/SQW TETAR-€THMORI/OU H( *E*M KAI\ KEI/SQW TH=| *M*K I)/SH H( *K*N. LOIPH\ €A)/RA H( *E*N LOIPH=| TH=| *M*L E)STIN I)/SH. KAI\ H( *E*N E)/GGION €TH=S SUNAFH=S TOU= QERINOU= TROPIKOU= H)/PER H( *M*L: E)N €PLEI/ONI A)/RA XRO/NW| H( *E*N DU/NEI H)/PER H( *M*L &[prop. XII].$ €DIA\ TA\ AU)TA\ DH\ KAI\ H( *N*K E)N PLEI/ONI XRO/NW| DU/NEI H)/PER €H( *K*M: KAI\ H( *E*K A)/RA TH=S *K*L E)N PLEI/ONI XRO/NW| DU/NEI. €E)N W(=| DE\ XRO/NW| H( *E*K DU/NEI, H( *E*Z A)NATE/LLEI &[schol. €&nr. 107]$ E)N PLEI/ONI A)/RA XRO/NW| H( *E*Z A)NATE/LLEI H)/PER H( @1 €*K*L DU/NEI. KOINO\S PROSKEI/SQW O( XRO/NOS, E)N W(=| H( *E*K €E)CALLA/TTEI TO\ FANERO\N H(MISFAI/RION: E)N PLEI/ONI A)/RA €XRO/NW| H( *Z*E*K E)CALLA/TTEI TO\ FANERO\N H(MISFAI/RION H)/PER €H( *E*L. *)EK PERISSOU=. 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KOINO\S PROSKEI/SQW O( XRO/NOS, E)N W(=| H( *M*E*H E)C-€ALLA/SSEI TO\ FANERO\N H(MISFAI/RION: E)N PLEI/ONI A)/RA XRO/NW| þï€µ°²²ÿï°°¶ÿ›€ˆ˜H( *K*E*H E)CALLA/TTEI TO\ FANERO\N H(MISFAI/RION H)/PER H( €*M*E*Z. *)EK PERISSOU=. TW=N AU)TW=N U(POKEIME/NWN A)PEI- @1€LH/FQWSAN I)/SAI TE KAI\ A)PENANTI/ON PERIFE/REIAI AI( *Z*H, €*Q*K, KAI\ E)/STW H( *Z*H E)/GGION TH=S SUNAFH=S TOU= QERINOU= €TROPIKOU= H)/PER H( *Q*K: LE/GW, O(/TI E)N PLEI/ONI XRO/NW| H( €*Z*H E)CALLA/SSEI TO\ FANERO\N H(MISFAI/RION H)/PER H( *Q*K. €E)PEI\ GA\R H( *Z*H E)/GGIO/N E)STI TH=S SUNAFH=S TOU= QERINOU= @3 €TROPIKOU= H)/PER H( *Q*K, €MEI/ZWN E)STI\N H( *Q*E TH=S €*E*Z. KEI/SQW TH=| ME\N *Z*E €I)/SH H( *E*L, TH=| DE\ *Z*H €I)/SH H( *L*M. E)PEI\ OU)=N AI( €*L*M, *Z*H I)/SON A)PE/XOUSI €TH=S SUNAFH=S TOU= QERI-€NOU=, E)N W(=| XRO/NW| H( *L*M €E)CALLA/TTEI TO\ FANERO\N €H(MISFAI/RION, E)N TOU/TW| €KAI\ H( *Z*H &[prop. XIV].$ €E)N PLEI/ONI DE\ XRO/NW| H( €*L*M E)CALLA/TTEI TO\ FA-€NERO\N H(MISFAI/RION H)/PER H( *Q*K. 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