Symbole eis ten ereunan tes geometrikes algebras ton pythagoreion

AMBAS ES. | n Y = MPAKTIKA THX AKAAHMIAZ AOHNQN — 1351 Va und 862 97 26 Cs (a Cao € 56 C6 TT 1351°=3.780° +1. | Wir haben. also. auf Grund der von Theon von Smyrna und Proklos überlieferten Methode, die Beziehungen die Archi-— medes, ohne Erklärung, als bekannt, angibt. - ie, {x or ad. TEQMETPIKH AATEBPA.— EvufoAh eis tiv Eoevvav tic yewuetolxñc di veßoas tov Iludayooeiov, $106 Evayy. Zraudın* ‘Avexowody xo tod x, Muay) Erepavidov. : I. EIEATOTH 1. Els row iótow zóv lludayópav ircdiderar zati thy mapadoow À Amödeıkız Too repoñuou óuwvópos Demprparos (Edxdeidos I, 47), dprspévar dxéparar Adoers sg ¿Ercóczws 2° =x" + y" ual i Avanadudıs mov douppérowv? Eis sobs Mudayopelous du viver Arodiderar petal) x\wv vo II BiBMiov xal +ù mIelorov +6Gv Apıdunrınav rov Ezoryzciowv tod Edxreldou, xal à eúpeois av duepatwy Adoewy 7% ¿En omsews y =2x*F 1, (1), atrives yonotpedouar Sik tov broloyıspöv 7% zark rpooëyyuow ¿burns tiie Ti Y 2, A! dntparar dices THe ÉErcoceuws (1) dvopalovean de yywmatéy, rheuprzot nal Bixperpixol dpidpot. Side: of wév ix toútwv Avristorgooaw ele tac mhevone, of DE els Tis Bexytovious rterpxydveay ”. ‘O vôuos synparıspod 20 Thevomdy zal Sraperoiumdiv ¿pur treomdy Ind 09 Oéwvos tod Epupvaiou ual Eyer by xdtwi: T.evorxoi dpıdjeoi diapetoizoi agud poi 1 2. 1 + 3. 24 t= 2 3 5 = 1 2 2+ 1= 3 2+ 3 = 7 * EVANGFLOS STAMATIS. A contribution to the investigation of the geometrical algebra of the Pythagoreans. ! MIXAHA ZSTE@ANIAOY, Eiszywyr, dd; shy ictopiay sv ‘Pusmdv "Erusrrnuv. +. 53-69 ani 102-3, *Adivar 1938.— Ipoxho; sl; oydhiz Elxhcidos I, o. 65 zai 438. “Ezò. Friedlein, Teubner. —‘Ixpfiizou V. P. 246, 2x5. L. Deubner, Teubner. 2 Theonis Smyrnaei, Philosophi Platoniki, £xô. E. Hiller, 3. 43, Teubner. E. XTA- MATH, Eöxkelöoy Vewperziz - Decía 20 uüv, top. 11,0 8. ("Oor. "Exd. Nyokx@r Bifiicor, 1953 "Adzvaı). — PAUL-HENRI MICHEL, De Pythagore à Euclide. p 438, Paris 1950.— (Sue. d'éd. Les Belles Lettres) M. CANTOR Vorlesungen über Geschichte der Mathematik u. Theon von Smyrna.— T. HEATH A history of Greek mathematics I, p. 91, Oxford 1921 at the Clarendon Press —R Moris Conex and J. E. Draskın, A source book in Greek science, p.43, MeGraw-Hill book company ine. New York, Toronto, London, 1984: Stamatis, Evangelos SYMBOLE EIS TEN EREUNAN TES GEOMETRIKES ALGEBRAS TON PYTHAGOREION In: — Prektika tes Akademias Athenon, 30 - 1955-p.262-282

SYNEAPIA DIE 2 10YNIOY de 12 4. 5 + ir 12 +17 = 28 =2 Tae = i — 1 1° mie P +1 2 — 1 5° 7 = ba 263 41 =7 2-12-1 Elvas sei PV ( + „al 1055 17" ee Le Y & 3 41° =2: 2 — 1 xx. ‘H yewperpian drédatie Tod vópoo oyypaticpod Ty Thevowdiy vol drappo. roy dordunv pynuovevstor bd rod lpóxkov? nal Exe: be Eins: i "Eozw tetpdywvov rheupäs AB=4 xal Sizywviov AT =A,, (ox. 1), Gre siva A Sy: 1. = 2,7. "Ext +75 mposxzácews zig AB lauBivopev ip BE-=AB= za iv ouveyels suina EZ—AT=6,. Kara zóv EdxdelSyv II, 10 da elvas (2a,+6,) + +2 (1 +51)". +6 =20 Kal ineidi, 5,2017, Da Eywpev 3: Apatpéseus ToûTou xatà palm in 373 = ant, ). 'H oyésts Gums adri onuaxlver da: rpunyouuévns ÉErcwseus, (2a, +8) + EZ elvor xheves, à dì (24 45)=AB=+BE+EZ elvar dix— $ pèy (a, +6,)=BE 775 BZ ABuoye suina vd rezpayovou, % BH. "Eàv ¿ni zis mpoextacews 1 Lydia eli Dodireizv Madvivog, top. II 0. 24 «i 393 x È. tno F. HULTSCH, ixö. Kroll, Teubner.

IIPAKTIKA AKAAHMIAY A@HNOQN ZO=BZ zal iv cuveyela surux OK=BH xai ipappiowuev o ebudciderov Decdeno II, 10, 0% AckBwyev véov zerpéywvoy, Tod drofov uèv rheupx BX eivar % ZO+0K=3a4 +281, 4 dì dtxyonos 4 ZA=BK=2BZ+ OK =4aq, + 38). "Oder LapSavopey 76 5575 oyfua IDevquaoi agudpoi (ty - a | Ataperoizoi Gord poi "7er à SO e =a +4 d = 24 + ù ts = Us + da ds = 2a, + 0 ci = ds + à è, = 203 + à Uy = W-ı + Óy-1 dy = 20v-1 + Ov: 'Exv diowu:sy a; =1 xal 6,1, AnpBavopey robs zatà TOV Mimva zóv Zpupvatov rieupizods xal dtaperpizods Aprbuoós, tor Tas Axepaías Quoss THe ÉEtowaeux y7=2x?F1, 4 dv=20°+(—-1)", (v=1, 2, 3, ...). "Ex +75 Modireias 700 IM.atovos TArnpopopoúyzda dr: of =heuprxol zat drapetormoì dofuol Foxy yrwazoi els adrov. “Exel dvayıyvaszopey sixatév piv derby aro Stauérowy ento reur#doc, Seouéveny Evög ixzatov, Apphrwv Sì Suvoive (546 c). 'Evzavda 6 IMarwy Sramicazia: tobe zhevomobs xxl Biapetoizodbe terduode E at di È vai play Frepaiav Low 77% p 15 ¿voip ifromasws, THY T° =2°5°—1. Toëro ouvayeì Y var gx tod [lodu).ov, Garis5 yoape: «Grz0v Di TO oúveyyuz Ayaro u er, oloy ebodvrec YPX? p 5 èv vemuerpiz tetpayovoy Dir). horow, dy Apılluuis Di oùx Zyovres ¿vos05 Biowpz TETOAYWVOY Y 8 4 tas papiv po Zidov dos Sir)dotoy Úrdoy em, Gorep TOD 2770 7% mevrados 6 drrò 7% i émtabos Doro ¿vos Diovtos>. Kal dXayoù «od yao tort tetpaywvos pros +eparco dirdatos sì pr Ayer tig Tov oûveyyuc. 6 yxo dro 105 Ü Tod and cod = Simdaords ¿ori ¿vos> Géovroc!. (lv dv). 71=2-5*— 1) 6 ' 2. 'O "Apy:uhins ele civ roxyuarelas adrod «Kiko Mirenmis> yprso- FOUT dvev Arodettews ras syste 265 — , 1351 TEE (Va (Cu i é nad 265 =3.1537—2, 1351%=3.780"4-1. Vewpetpixyy drrodertiv Tv 0yéczww Tourwv ¿rmeBídlopev eis shy "Axadyyulav "Aluvav’. 7 cuvdierar > mo co . siae nai Ouxuereixoùs % pero LA 43 Alta mod: robe= mievorzod: xD ous. do:tuose. * IIpoxAos slg EdxAcidyy I, o. Gl xai 427, 2x0, G. FRIEDLEIN, Teubner, 2 BA. lpaxtiez, ävur., o. 255 x. EE © EWSOULLEV

SYNEAPIA THX 2 IOYNIOY 265 taosxeric dpBiuyoviov sptywvor, 70 ABI (sy. 2), sod Groiou % pevadutipa yuvía, + ABI, và eivar ton pos #hv Efwrepıinhv Yavizv lsoxketcou sprymvov. Katz cov Edxdeldyy 11, 12, ¿dv xaréowyev + mhevexy AB=@ xal av AD=0,, ru u ff a i te a —— . « = sa x es à x depuis shy bro tod lipdxdAou irodermvyo- / 7 Be # ouverts di: =\ 3. Noopuóloper topa N Tu / Pd ye 7 / / LI A / / AE pévyy pédodoy Dix Thv Anode Tv in TE / wae / BeBalos clvar + peyadutipa Drayuvoz tot o ' "a eouBou ABTA, 0% eivar 0,341”. xal = f 3 TPRYOVWY CLAUXTWY TPOMUTTÓVTWV TAEUe zy. 2. piney al Giausrorüv dprdpov. “Ext ris mpoextasews is AB hauSévousv =urux BE = AB = EZ=AT'=5,. xal dy ouveyelx Tur px Kazx zóv Ebxdeiünv II, 10 dx Eywpev. La +5) + =2%04 + 2 (a, + da, nat ix aura (1). (2a, + 8) =40, +4, +8, Elvar ¿pa xxl (2a, + 8) = 120’ + 120,0, + 30,2 AMA Ô, = 34°. ‘Eropévws 3(241 + 5) =90 + 120,5, + 48,7 =(8a, + 28,)°. "H oyéon Gums alien onuxiver örı % wey (Qa, +5,) elvas zhevpx, + Sè (3a, + 20,) elvar i, ueyahuréox dtayıhvros bolos póp Bos mpüs zöv ABTA +05 AZHO. "Ev ¿ni #75 mpoexvésews ig AZ AdSwpev zutun tsov pds AZ xual dv auvsgelz tuijua toos mobs AH, wore Eyouev zatà civ abröv vöuov Thy mAeucdy mal Thy peyadutipas dizyomov véou éuolou péuSou reds zöy Apyixdv, Fro mAcuex piv chat 4 2AZ + AH, Gtxywvios Sì ueyakuzépx, % 3AZ+2AH,% Ta, + Ad, ai 124, + 70, dvrısrolgog. Kxl.oövres tke Tinks THY mheupy mheupmobs Apıiluobg xal Tas Tias Tüv peyakuTéswy Staywviey, Tv cuverdiv zatà Tov Avariow VOUOY rataozevatopivwy copBwy, dixperetxods aprtwods, DA Exywyev Tievorzoi aqubpoi. Aranerowoi quoi. a, ö, @ = 2a, + 0, ds = 34 + 9, ts = 2 + è ds = Juy + 202 a = la + 4; 4 = Sas + 2 a = a + & "Se = 34 + 2 Gv = va + dy = Savi +

IIPAKTIKA AKAAHMIAZ AOHNON "Edy Décwpev a=1 , b=1 lauBévouev (A) [evqiroi aed pol. a Atapetoixol dordpoi. =1 “ 6,=1 - do = 5 o = 3 f@3=11 d¿=19 - ay =41 1, =71 a=155 35 =265 O! Zpıdpoi obror rapéyouat tag dmepatas Niger tHe ifloacws y°=3x"—2, Timor elvar: 17=3-1°-2 5°=3-3°-2 19°=3-11°-2, “MT. 'Exv décopev x, =1, 5,=2 AawBevouev: Misupizoi domi. (B) Atoperotxot aqui poi. cx, = 1 à, = 2 a = | de = Y x, = 15 Ge = 26 a = 56 dy = 97 a, == 209 ca = TRO =a I| 362 1351 Ot aprthwol oùrot rapéyouor Tac txspatxg users ne ttiawosms y =3x?+ 1, rar elvas hd D? , Sy 2 = 26° - = 1? + dI + 3°15? -r , non , a”, Sa , Ol oyo a "I (A) xal (B) drrovehodo: S40 dxohouviac bx 20v ónolwv % A pev zov (A) elvas adZousx, Y BE taHv (B) pUivovra. Tó xowòv ppéyux robruv elvas + V3, rot elvat vi Org zat 7 19 , 41 11 r 263 / LS / 1351 a ET Res ESS » 780 ¿ 362 , 97 , 26 * 209 Ÿ 56 Y 15 : 48 2) 265°=3-153°—2, 1351°=3- 780°’ +1, Ws yonoiporoet TAÜTX dvev kATO- SelEcwe, wg yvwaord, 6 'Apyiuòdns.

SYNEAPIA THE 2 IOYNIOY 1955 "Ex tv dvurépo ixteftiviov auyvayopev TO gupripaapua ott ol Fuixydpeto: dyvapılov nai tas ixepatas Mugen tHe ÉEtowaeuwc N, (v=1,2,3... et RDS, Axégaros ah TeTEXywvoc), nal drt À Li siva: 70 zomòv pexypx Bio dxokoudrdv, puts adEodans nat puis obwovane, Bids: À irás sorte siva droits 7 abit pds Tas AVWTÉPU) ixtettzica:. Iazéyopev shy Amödsıkım dix Tas 2Esauoets , 5 = se + (5-4) ley 5 = 6 + (6—4)" (-1) 5 = Tar + (7-4) (-—1)” 5? = Sar + (8-4) (-1) 2° = tia" + (17-44) (A, nal Thy V5, Vs, V Lea 17. Elvas Si yvwozóv ix Tod Oearrítos Tod ID.<=wvos dm: è Oeddwpos! (6 Kusnvatos, doris Dempeirar [ludat te yopews) AméSerke To doúpperpov zis V3, V5 ... \17, (Oexirnros 147 D— 148 B). 1.1. + (5-4) (—1) nod V5 Sy =50 Quwmpoduey dpdoyivor zaxpxl)}nAdyeaupoy, 76 ABTA AB=u, BP=2u4 xual 4 Gixywvios (cy. 3), ¿vda toro AD =x3,". Elvai dpe 8,*=50,”, xal += VE. "Eri rie mpoentäoews ig AB VapBavopey spa BE= = è =AB=a, val tv ouvegein suina EZ=AT'=8,. Kat tov EùxXet8v 11,10 Ba Eympev. La, + di + 6,7 20 + 2{a + da)”, mal in Tadeng Sy. 3. 11, Pavıy - Wissowa, Realenzyklopádie unter ‘Theodoros. Dort Literaturangabe: M. Cantor, E. FRANK, F. HtLTscH, G, June, H. Voot, H, G. ZEUTHEN, EVA — 2. Und W. L. WAN DER Sachs, T. BoxxEsEN, H. HaSSE-H. SemoLz, T. HEATH, WAERDEN, Die Arithmetik der Pythagoreer II. Die Theorie des Irrationalen, father. 3. K. “Annalen, 120, 3./6. Heft, 1940, Springer Verlag, Berlin, Göttingen, Heidelber—g. HOFMAXX, E. J. 4. Leipzig.— 1940, Griechen, der Arithmetik REIDEMEISTER, Die Geschichte der Mathematik I, S. 26-27, Berlin, 1953. (Sammlung Göschen, 226. Walter de Gruyter und C°,) 5. ROBERT s. BRUMBAUGH, Plato's Mathematical Imagination, p. 146, Indiana University Press, Bloomington, 1954.

IIPAKTIKA THE AKAAHMIAZ AOHNQN (2a, + d,)'=4a,°4+ 40,8, + 5,* Elva ¿ox nat 5(2a, + è,)° =20a,* + 200,8, + 55,2. ANG 8,5 =5a,’. Exouévos * E 5Qa, + 5,)'=200,* + da, + 200,0, + 45,*=(50, + 26,)’. 'H oyéots Gums ar onpalva dot % uèv (20, + 8,)—AZ siva rheupé, + Se (5u,+26,) = AH elvas diaydivios ópolos poc TO dpyindv dpdoyuwvioy mxpxAdnoypeppou 00 AZHO. Katà +ùv roopuv vopov tHe axraoneuñe dv cuverela dpotwy dpdoywviow maparinioypeppwy Da Zywmpev: IM evorzor dow noi. Ataperorxot dgWuot. it, ô, Œ = Bur + 0, gs = Dax + 20; (tg = to + dy Ss = Dos + 26. a = Its + Ox dy — Das + 25, ay 5 dv Ô; = Diva + (4 = 1, ly = "Fav Jécwuer Un Bu Livar Se 2 4 = 2 = 4, D, = 9 cs = 17, dg = 38 38 sq 2° 1. 161 hapSavopev ,9 VO 4 AN NES E — | 9 = 35:44 = D + 1 35S* = 5°17? — 1 d = ray IT. 2. Dd 20v_1 | ax + (5—4)' (— 1)". Sv? = 10ay" + (10—4)"(—1)", ai VIO - Kis 70 mpoqyouusvoy oyux 3 hapßavouev AB=a,, BE =54t1, éxore 81*=100,°, nel 2 =VI0 . "EqgappoCovtss thy mponyoupevyy zarasxeuty (II. 1) Xap Bevopev

IOYNIOY SYNEAPIA THX 2 La, + da) =4a,’ + dad, + 8," Elva dpa zat 10(2a, + 8.) =40a,’ + 40u,5, + 105,* "AN 3,7 =10a,’. 'Exoyévos 10(2a, + 5, =40a,* + 60a,? + 400,0, + 45,7 = 10, + 28,)". H oygore Gums ar ampalver Ore 4 pv (2x4 + 8) elvar rheupt, + dì (100, + 28,) Brayivios Gpotou datoyuvio» mapakiqhoypxumov mods TO Apzıziv. "O vopos Ht AxTacreute Tv byolwy iv auveyelz napa)imioypippwv siva: mpogavis. "Odev OX siva: Iigugizoi dpiiitoi, Aanetgiroi aod poi. & di us = a + 8 Ds = 10a, + 28, ag = a + è ò. = Was + 20. a = e + ds dà, = Wey + 28, cin = aut dv & = Ma + 21 Exv Biowpev Elva se a, = 1 wet $ = 2 Us = 4 ds = ld a = 22 da a Si = 356 112 = US Coe CO (3507 2° = 10-1° xal JapSdvoucv — 6 14° = 10:44 + 6° 68° = 10-227 — 6° 856° = 10-112" + 6* (- 1) de 10-44 + (10 — 4) II. 3 By? = 170" + (17-4)" (—1)", xed VTT. Els <6 aUtò oy7ua 3 )xpfavopey AB=a,, BF =4da,, AD=8,, éxdve elvar 'Eoxouólouev wahw Thy xxcaoxeuty (II. 1) ömdre Zyousy 5,¿=170,* nai Y (2a, + 5,)'=4a,* + 4a,0, + d,

MPAKTIKA Elva dea ‘Eropivog xal THE AKAAHMIAZ AOHNQN 17(2a, + 8,)?’ —68a,? + 680,5, + 175, Ada 8, —17a;. 17(2a, + 8,)? =68a,? + 221a,? + 680,5, + 45,7 =(17a, + 28,)’. ‘H ooygotg ópos aÿrn onualver dr % piv (2a, +5,) elvar wAsupa, % Si. (170, + 25,) Suxyovios duotov SpPoywviou rapalimiovedppov mpds TO doytxdy. ‘O vópos oxnuariouou Tüv "Odev $a Gyotwy doboy maparknroyecmumy slvat mpopaviis. Zywpev M.everzol apıduot. Araperoizoi dard noi. di 5, a = 2a, + 9 ds = 7a, + 28, da = Ya + % Ss = lía + 2% ai = 2a, + à & = lTa + 23; dv = 2ava + bya Sv = l70v1 + a, = 1 d = 4 ‘Edy Décwpev Elva 5 wat 8, = 2 da = a; = 29 da = 110 = 168 & = 713 VIT CRC 2? = 17-1! — 13 21° = 1774" +13 110? — 135 = 17:29" dapBivopay 2) a Cae ra. ral 113° = 17-168" + 13° By? = 17m? / + (17-4) (-1)" "na 2 Oswpoöuev TO popBozidis rxpx)).nÀ0- vpauuov ABTA (ay, 4) Evita ywvlx ABT=120°, AB=a,, za À peyadutipa Guxyovos BI =24, AU=d, Karà zóv EdxdetSyy 11, 12 0% elvar 5, =70,* äE ÓMOTE =a =\'7, uw ipappolojnev THY A adri xxtacusuny (II. 1), óróre VapBavoy- & E 2 res BE=a,, EZ=5,, dà Exwpev nave +èv Eixisióny IT, 10,

ho ~! ons FYNEAPIA THX 2 IOYNIOY BE Fs (2a, +51) + 8,7 =2u, + Aa, + da), La, + 5,’ =4a,’ + 4u,5, +. = 28a,° + 28a,5, + 28: Elvat Zea xal = 7(2u + 6.) 5 =70 0. Exopévos ‘Ada 7(2u, + 8,) =28a,? +210, + 230,5, +40,* =(Ta, + 26.) . "H oyeaıs dpws aber ompalvar dir 7 pèv (2a, +5,) eivar rheupd, % di ho(Ta, + 25,) dtayehviog peyaæhuréo® duotou roûs TO Apyınöv poubosidode mapa) voduuov. Kara rdv moopavi vouov AATADAENTE buciwy copBosdiy mapakAndoypauuwv Da Eywpev: Aruuetpixoi dorduol TlAeverxoi agıduot. d, a, + 25, = Ta, + 28. dy = Te + 28 dv = Tavi + = a, + à, ds us = 2% + di dy a = Ya, + % da = ai + dea ‘Edy Séowuey 5 Bit Où nal ja, a 2e + «4 = 1 xal e 5, = 2, hauldvopev ll a, = 4 dò. = da = 15 ds = 90 a, = $85 $, = 233 ex 2 (VT gg pi DE, Ca Ce ari - 3 1° = 7-4 + 8° 50° = 71:19 -- 3° 233° = 7.88’ + 3 = di = Tea + (1-4) (1). II. 5. Ele 20 roonyobuevor aymua 4 haußävopev AB =a,, BI = 3a,, AT =5,. Kara zöv Edxdeióno 11, 12 elvar 5, = 13a,*, za rs =V13 . [law ipasuolopev iv

[IPAKTIKA THX AKAAHMIAS "ABOHNON abi natagzeuiy Ws zal mponyoupsvwes, Got Aaußavousv BE = a, EZ = 6, ónóre AATA +0 II, 10 +où Edxdefdou Da sivar (2a, + 8) + 6? =2e,? + 2(a, + da, de Ts (2a, +8,)?=4a,’ + 4a,5, +8,’ . Elvar Sox ai 13(2a, + 5,520, + 52058, + 135," . di ld‘: "ADD Éropévuws 13(2a, + 6,) =52a,? + 11T,” +52,’ + 48, = " ‘H oxeoıs Guws alti onuaivez (13a,+25,) (13, + 25,)*. Gr % piv (2u,+6,) elvar mheupd, 4 dè Brayeivies peyahurica, ópolou popßosdods raoadrnhoypauuou rpôs o dpyiziv. Kara sóv popa” vôuov xxtaaxsu%s Tüv duolwy pop.ßosößv apre loypiuuoyv dà Eywusyv Tievorzoi dod noi. AMauetgixor ded pol. a; di to = 2a, + $, bs = 13a, + 26, a = e + D di = Ike + 235 u = 2a, + 0 Sy = Du + 2 de = avi + dea dv = 13uv1 + "Lay Biowusv a, = | nat 5, = 2 da = 4 d = 17 ug = 25 1 = 56 «y = 136 5; = 497 % = 13-13 — 9 1354 +9 86° = 13-257 — $ 407" = 13-136? + Y dy = Ida,’ + 13-4 (— DY. day Sdvopsy

EYNEAPIA THE 2 TOYNIOY KE "Ex av duuwrése ixtederaGv xataozeviov zul dnobsilewy xothetara: aùrovonTos Ó SANUXTIGUÔS Avrıstolywv cup nal Grapetouxdv Gorter dr dhyeSorx00 zabapis brohoyıspod xxl odyt yewpstpizod, did THY VID. \6.NS.NII.VI2.\14, nai tac suvapete Erowaeus. Oisw fà siva: IL. 6. At Sy? = Gay? + (G—4) (- 1)" “y — xa Va d, to = Ba + 4, ds = te, + 2, us = a. + À, & = be, + 26, aa = Ze, + 4; & = bas + 2, Gy = 2ayva + Svar dv = Gara + Eav fiowpey a, = | nai ò, = 2 18 dy = 44 a; = 80 di Elva: Se + << “VG ci 2° = G-13 — 2 10° = G'4 + 2? 44° — le] .. dap Savopey ò, = 10 a, = 4 ag = 20,1 G18? LEZ Il = Ie 196 al 2° Cc ‘80° + 21 S = Ga + (6-4) (—1). II. 7. Aux =! + (8—4)"(—1)" xai VS . ty à, | du = 2a, + 3, dy = Sa + 2, ag = 2, + 5, & = Sa, + 26. ag = Pu, + 5s & = Sa + 26, dv = dvi + dy = durar + "Eàv Décwpev dvi -~- oa, = | è, = 2 a dy = 4 = 2 laphivonev

TIPAKTIKA THE AKAAHMIAS AOHNON = 96 9 dy = 272 a ne 12 7% = (Co. yg ATC, nah Elva Sì f= | 5 — 4 12° = 8-47 + 4 9 = 8-20° — 4 272 = 8.96 + 4! by? = 8-ay? + (8-4) (—1)". IL 8 A 5 = 11a" + (11 — 4)" (—1), xat VII . dy à, co = Ya + bd ds = lla + 2, a = 24, + 8. dy = lla, + 3, a = lu + À, & = lla + 2; ty = Bava + da dv = Ils + ‘Edy découuss Liar dì = a, = 1 nai à, = 2 la = 4 ds = 15 ln = 23 ds = 74 cu = 120 à, = 401, REIT a ¢ =. 7 Pp =F 15° = 11-4” FT 14 = 11-237 — 73 401? = 11-120 + 7! de = la? nal + (1 — 4)" Lap Savousy

SYNEAPIA THE 2 IOYNIOY 1955 (— 1)", moi y 12. — 4)2 Ta Ack Se?= lou + (1 5, fy 28, {fs = tte + da ds = 12. + 20; it, = Dax + D di er 1 Ya + 25; ily = Pin + dvi dy = l2ar 1 SJ "Fiv Désoper , dì Elvas a, = | 5, = 2 je 5, = 16 Zvi LapSavopey da = 448. a, = 128 i HS , 16 2,8,\ B=ET nai T PES 3S = 19. 5 1G? = 12-47 + 8 so? = 12-247 — 85 448° = 12-128? + 5! Il. 10. A ? — 4)" (NN, nat VIE. + (14 SE= May 4, & = Ma, + ds = da, + a; = 143 + thy a = Da, + 5, as = Dita + & Cs aus Pas + ds au 'Eáv Siowpev Sy = bes Boat = 5, = 2 = a = | de = 18 ay. = 4 à 9 92 oe | = dy = 92 $, = 548 ‚518 , 18 nar u a; = 26 a, = 144 Elvas dì lduva + 7 Vida Sa LapBavopey

IIPAKTIKA THE AKAAHMIAZ AOHNQN — 10 18? + 10° 14-43 92° = 14-26? -— 10° 5487 = 14-144? + 10* u IL 11. A& 8)? =15ey? + (15-4) (—1)", xal V15. (ty 5, a = a + au” = Qty + 'Exv Béswusy a, = 1 à 95 Elva: 52 2 à & = Ilda; dvi dv = lday-ı + 6, = 2 —- 601 2 15-1° — 11 19° 15-44 + 11° 9$ = 15:27 — 115 601 + 11° e 37 ¢ ie i Il 19 Sii fat 15-152 + 28 28y1 au avoue nah Sv? = 15-ay? + (15—4)" (—1)". III. 1. "Ex av roonyoupéves raparnpobuev rt mark +O ebxdctScrov Demcypa 11, 10 yemperorr zxtxsxeuy, thy drrotav pvqmoveder è [Tpdxdoc, dyer els tiv <avzôznræ tiv (1), (2a, +5) =4a + 40,5,45,7, amodexvopevey zati sò cdzdsidciov Dewpyyx II, 4. 5, =—da,?, Evda

SYNEAPIA THX 2 IOYNIOY 1955 15D, dnéparos più, Terpaywvos, Dà sivar zul (,—A)5,* = (A—A)Aa,?, rote Ex 778 (1) Exopev Ia, + da = au, + Had, + O. — Uña +40, = (zu, +25). 'Exouévos Di siva Arugiergixot deri pot. TlAeverzoi dgid poi. 3, my 1) + & 5 = 14 + 20% + ds di = hats + 255 dv dv = hays + da (14 = = 2a 2113 Gy = Jay + AN ra A Boe . st di à ende) di CCE, di = ha’ + rs (— n 3) 2 HI. AR 29.1 ’ Di mal în TÜV meupindiv zal Dixperpreiy Elvas Suvaròv +, V2 vàx úmodoyio By = Aas + 20v.1,Grav À = 2. Av = Qaver+Sy1, dorduov re uoppiis Tv uédodoy sadryy xxdodpev yevixhy mpög Bikzpsow azo zis uedódos #75 Si:aceteions inò où Oéwvos 700 Éuupvxiou ual rod llgóxdou, tiv órolav zadovpev eidirrr. Tlode odyxprow rapaléroper tà ¿fayópeva uxt tiv Sio usdodwy. A’. MéDodos etdix4, a, = Days Fir. Uy = ys + Ov , Sy = 1 à, = 1 = 2 & = 3 as = 5 ds = 7 a, = 12 u = 17, a“ Te EEE Te E DI |, B= 2-2" +1, 7 =2. 5’ —1,--+- 5, = 2° +(—1)". Papo [Au a=1, 5,=2. a = 3

MPAKTIKA AKAAHMIAX 5, AOHNON By à Sa "a > o 24 2 2 2 Sy* B'. Aw Metode< a,=1, AENA, 2-4)(-1)]. ily = Privy +0: ’ dy = 241 +21 . 1 y ug EU . + à, = 1. . ty Day? 2 = $ = 10 = 34 ds “a 14 di ‘ty \ Dite! (Q-4)"3.(-1)7. uz à, = 2 do = & ds = 20 d =

SYNEAPIA THE 2 JOYNIOY 1055 2.1? + 2 Gert + 2 20? = 2-147 + 2 G8? = 248 + 9 de? = dq? + (2 — Se. (— 1)" , * , a - RS, > Kaz avrtazorziav mpös thy tnd tod > Ù , ‘ 'Agyiwidous mapezopevay TIMAY 375 V3 Da etyopev, dik THY v2, I 4 14 48 164 sr : 192 232 , GS eee Cae Care ae side Sr an ti 9 OG SE SG Elvat pavepöv, Ott elvat meoriporipa % Sad tod Oéwvos +05 Syuevatou zai 705 [odxrdou Sixswetsx pidodos did tov brokoyısuav Tic Fa. SUMMARY I 1. The law of formation of the side- and diameter- (diagonal -) numbers is explained by Theon of Smyrna. According to Proclus the relateil identity is proved by Euclid book II, proposition ro. Side numbers ay = av + 6, diagonal numbers % = 2ity_1 + dv. ‚VW 5,*=%0* 371. For a, =1, 6, =1 we have + (+ (o 1/2 --- D | rolus 9. Archimedes for the arithmetical approximation to x starts from a greater and a lesser limit to the value of 13, which without remark as known, 265 — y 1351 ae < YS C5 : =9 =3:153°—2, 965? ont? =53:780" won? +1. .s 1551 We give the following interpretation on the archimedean formula, with pythagorean method of the side—and diagonal—numbers. In the figure 2 is AB =a; the side, AT =, the greater diagonal of the rhomb ABTA, and the greater angle ABT = 120°, Then 5,’ =3a,’, 5,:4, = 173 According to Euclid II Prop. 10 we have (2a, +8, +8, =24,° + 2a, + d,)* (Qu, + Spe da + -+a,d, + è", - It is also (1) 3(2a, + 5,)* = 12a,?+ 124,5, + 35,°, and because 0,30, a, Thelaw 3(2a,+8,)" =9u,” + 124,5, + 45,7,=(3 of formation of the corresponding side— and diagonal —numbers is evidently. Side numbers ay = 2av1+ v1, diagonal numbers dy = Fav: + 2x...

MPAKTIKA THE AKAAHMIAE AOHNDN When a = 1 ò, = 1 a = 3 & = 5 When a = 4 u = Tl a, 7 | ~~ and Sv = 3a —2 I ò, = 2 a = 4 ds = 7 a, = 56 à = 97 % = 780 de = 1351 and re Cr a Ge NT à, =3ay’+1. (E E Cr (+ IT. In the following we start from the identity (1). 1. In the figure 3 we take ein, AB=a,, Br=2a,, Fica AT =8,. d:ia=V5, and 5(2a,+5,)? =200,* + 201,3, +55, . Then Because d, = Da,’ is 5(2a, * 5,) = 204,’ + da,” + 204,0, + 45, = Ga, + 25,)”. Side numbers ay = 2ay1+68y.1, diagonal numbers, 8) =5ay1+ 2615 Sv? = dey? + (5—4)" (—1). 2. 4, =2, a,=1, When -|uo In the same figure 3 we take AB=a,, BT =3a,, AT=ò,. Then = 100, cause S,:a,= V10, and 10(2« ,+ 8,)° = 40a,? + 400,5,+ 105," . Be- §,°=10a,? Side numbers is 10(2a, + 5,) = 100u,? +400,8, + 48,’ = (10a, + 25,)” . ay =2ay1+8.1, diagonal numbers bv? = 10ay? + (10—4)¥ (—1) 3. In the same When figure 3 we take Then 8,°=17a,?, 5,:0,=M17, and Because § = 101 + 281» u=1, AB=«4, 6, =2, BU=4a,, AP=06, 17(2c,+5,)? = 6S8u,? + 68a,5,+ 175, . è,’ = 178,° = 2894 + 68a,5, + 46,7 = (17a, + 25,)”.. Side numbers ay =2avit+8y4, D.N. 8¢=17 ey +254, be? = 1 Tay? +(17— 4)" [-1)". x When 4. a,=1 ¿ <2, à, 2 =< 110 In the figure y we take AB=u,, ABT' = 120°. Then 8,7 =7ie,?, te = — ro Cees] 39 BI =2a,, 713 Ce 21 ai AI =8, The angle 7, and 1%0,+ 8,)? = 28a,?+-28a,5,+ 701. Because §,°=7e,?, Tu, +5) = Ha, + 280,8, + 45,7 = (Ta, + 28,)*. SN. tv = Fava +501, DIN. De = Titor + 2e, By? = Tay’? + (7 — 4)” (— 1)" re When 5. «,=1, Jn the same Then 5,=2, 2 + DD figure y we take 5=1347, 5 — SUR 7 AB=a«,, Sie, =) 13, 1320, +8,) ; 293 11 IT: = 520," + 520,8, + 135, .

SYNEAPIA THE 2 JOYNIOY 1951 Because dè, = 134,7 is 1820, +0) = 1604 + 52400, + 48,7 = (Sa, 26,7. SN. ty = Dirt + dv A D. N. dy = Ladys + 21 à à,” — 13a,’ + (18 —4)*(— 1}. , RG 2 APTE 1 (= a | iho take In the same way we 497 , 17 € 136\ 4 the side—und the diagonal— numbers Mar, Vs, Mio. (We mention Theactetus of Plato Ls for the Web A 147 D). HI. #3 S 5, (1. 4), = (A—4)ha,*, (2a, integer no square number and and according to the identity (1) 8,°=4a,', then 25, = {ia +42 ad + 25,7, = Aha,’ + M— Aa? + Hd, +46, = (Lay + 25,). Diagonal numbers Side numbers d, u, = 2a, + di dy = de + mi = RITA -- ds ds = Las + 20: D jé 29: 20.1 = (ty = Puy fly == Pity + DI a ds fees (JE à. ds Da; di = Darts dvi by = hir at eee F4 TE eee !» —de #5 à, den (N. We take here always ra + (141 St ì=2 and 20, te A=5 DA — @=1, - ,=2. are special cases. BIBAIOTPASIA PAULY-WiIssOWA: Real Encrelopädie der classischen Altertumswissenschaft: unter Arithmetica, Theodoros. J. E. HoFMaxx: Geschichte der Mathematik I, Walter de Gruyter und Co, B 226, Berlin, 1053. MIXAHA STE@ANIAHE : Eloayoyh ets vir iacogiay riv peor 'Emormu&r, "Adijvea, 1935. M. R. Conex-J. E. Dragkix: A source book in Greek science. Me Graw- Hill book company, inc. New York, Toronto, London, 1948. A. Reum-K,. VOGEL: Einleitung in die Altertumswissenschaft II, 3: Exakte Naturwissenschaften, Leipzig - Berlin, 1933. ROBERT S. BRUMBAUGH: Plato’s mathematical imagination. Indiana, University Press. Bloomington, 1954. “pavL-HENRI MICHEL: De Pythagore à Euclide, Soc. d' édit. Les Belles Lettres, Paris,1050. B. L. van der WAERDEN: Die Arithmetik der Pythagoreer II, Die Theorie des Irrationalen. Matñem. Annalen B. 120, Heit 5/6, 1939, Springer Verlag, Berlin, Got tingen, Heidelberg.

HPAKTIKA THX AKAAHMIAY AOHNDN K. REIDEMEISTER: O. NEUGEBAUER: M. Castor: . J ". HEATH: Die Arithmetik der Griechen. Leipzig- Berlin, 1940. The exact sciences in antiquity. Kopenhagen, 1951. Vorlesungen über Geschichte der Mathematik I, 1907, Teubner. A history of Greek mathematics, I, II. Oxford at the Clarendon Press 1921. K. A. lM'EQPTOYAHE: | W. A. HErDEL: "H eiinvian émormun, "Eyz. Astızöv «"H}ıos>, Tónos: “EAs 0.; The heroic age of science. Baltimore 1933. F. ExRIQUES-G, SANTILLANA: Storia del pensiero scientifico. Bologna 1932. G. Lorta: A. REY: Le scienze esatte nell’ Antica Grecia. Milano, 1895, 1914. La science grecque, Paris, 1933. P.,TANNERY: Mémoires scientifiques I, II, III. Toulouse, Paris. H. Hasse-H. Scnorz: Die Grundlagenkrisis der griechischen Mathematik, Berlin- Charlottenburg 1928.