Symbole eis ten ereunan tes geometrikes algebras ton Pythagoreion

acini ER ESTAMATIS,E 5. — 145 — EYATIE NOY ETAMATII * wn Eva BÈ de <<< ita 12 BASTASSE D=2, P=} H=2, 241 «al STAMATIS E., Evupokh els thy Egeuvay ris yewperouis diytfoag tüv Mutayogelaoy [avec un résumé en angl.] : Platon VIII 1956 144-157. | La formule d’Archimède relative à l’approximation arithmétique de x se laisse intercepter par ”=2.9-1 172 =2.12-+1 409° = 2,29 —1 kim "H yeoperpixh andberkig 100 véuiou oynuatiouoù ty mAeupixbv xal Brapa» + XYMBOAH EIS THN EPEYNAN tpixby dpiduby uvnuovevetai bund 100 MpéxAou (') kal Exet dg EE : "Eotw terpáyovov nAeupas AB=a, ral diaywvlou Al=b,, lox. 1), Ste elvar () THE TEQMETPIKHE AATEBPAZ TON MYOATOPEION I. EISATATH 1. Ele sav [Siov tév Nußayspav dnoblSera: zard thy napébooiv À AnöberEis 100 nmepighpoo Suwvöpou Gempiinarog (EdüxAslbou 1, 47), bpropivai áxéparal Yoosıs the LErodaews z'==x"+y? nal À dvaxddupig av Gouppdtpoy (). El¢ rode Mufayopelour dv yéver dnobldera peroo GXAtov tò 11 BifAlov xal 16 nAgiotov tav dpisunuxdv tv Ztoryxelov 100 EdxAelSou xal spcaig tüv éxepalwy Abceuv the thodorog y=x Fl, (1), alrives xpnomedouan bid tov UnoAoyiopòdy the “ara mpootyyicoiv épilhnnnis tlic tig TR. Al éxépaiar Adoag tig téc ous (1) évonélovral, de yvwotév, wAcupixel Kal Siapetpixol &piduol, bide ol why tk rodtmv dvrrotorgoDor cle tag nAcupäg, of dè ele tdq biayovioug ta- "|tonioe [ = A , tpayoyvey (3). 1. diupergizoi pid poi 1 1+ 1= 2 2. 14 l= 3 3, 4, 2+ 5+ 5 3= 7= 12 2. 2, 2+ 7 3= 54+ 7=1 5. 12 4- 17 = 29 A pty (a, +5):=BEFEZ alvai wAcupa, à dèi (2a,-4-5,)=AB+BE+EZ slva: hayavıos tetpaydyvou,y BH. ‘Edv tal fig apoexiáceos tic BZ A&Awuev uña ZO=BZ xal 2.12+ 17 = 41 tv ouveyela ruñua OK=BH Kal tgappdoapev 10 eóxdelbriov Geopnua li, 10 dà AéBouev véov tatpàytovov, 100 dnoiou à ply mAcupa Dà elvan à 207-6K=3a,4-25,, y BE Biaybviog À ZA=DK=2BZ4-0K=4a,+4-30,. "Odev Agpfavopev 30 EGG oXiipa: +) Franyelos Stumalis, A coutcibution to the investigation of the ycometrical alge- Mievpizoi dgdpoi Gi 1) ‘Avene dv 31 "Axatmpita ‘A0nudv xazá thy cuvedplav sie Das; "Icuviou 1955 dik ” 2) Mizani Freparidov, Hlsuyoyi ety tiv loroglav Tüv Pycindy ‘Emamuiiv, a, 65-65 nat 1023, "AD ane IUUS. noxiog, Xydha alg ElxAsiZyyw I, 0,65 nai 498, ix, Priedicio, Teubner.-" Zuuffhigee Ve P. 246, €42, L. Denbner, Teubner, 1) Tennis Suyenaci, Philosophi Piatonici, %% E, Hiller, 0, 43, Tenbner—A Zrapis, Vigatto) l'aunassta Alva 3153) BE=AB=a, kal év ouvexela uña EZ=AT=5,. Kata tov EoxAetBnv 11, 10 Où elvan: (La, + Bu)? +5, = 20,’ -|-2 (a, + 5,). Kalenedh B,=2a,, Où Exopev Bi dgaiptgews toùrcu rata pin ix tic uponyovulyns thioducas, (La,-+5,)=2(a0,-+5,)*."H oxtois Suos adtq onualve Bu bra of the Pythagorrans. sod dxutypatxed x. Mzayl Seepavidov. $ 1 B,=20,7, ‘Enl tig nporkiáoeos ric AB Aaubdávonev tuna 1 2, 5 >> ‘O véuos oynpationod tv mAsupiKGy xal bicpetpinGy ápi0uOv beso0n Unò 100 Otwvog roû Euvpvalou xal Eye: bc xáro0r : TPcvgizoi dordpoi 2 Grugia dip, són, IL, o, 8 ('Opyav "End, Syedimdv Bibdlov, Atapergixot dgidpol ë, Gy = di m 5, da ni 2a, ai= +5, 5, = 2a, + di + 5 a = Ci + 5, è, = 2% -F di i H ay = ay—1-}- Bv1 By = 2av14- dv "Edy Ofoajtey &,=1 Kal B,=31, \auf&vopzy toda xat& tòv Ghova tov Euva- —Puul= Henri Michel, De Pytlagore A Euctide, p. 133, Paris 1950 (Soc. d* 1) Zyóma slg Moderslav Madswvog, iéj IL 0. 21 4.8, 393 xd, nd AL Lullsch, 11). Gl. Les Belles Lettres. Cantor, Vorlesungen über Geschichte der Mathematik unter ‘Theor von Siuyena,— 7 Meath, A history of Greek mathematics I, p. 91, Oxford 1921, At the Clarendon Press — di, Moris Cohen + 3. E, Drahkin, A source book in Greek science, Kroll, Teubner, n. 13, MeGraw - Hill book company inc, New York, Toronto, London, 1919,

Puoùg xal té¢ tudo 1@v peyadutépev Biaywvluv, tv ouvex@v xard tov dvotepw véuov xataoxsualoptvev pPopfav, Biapetpixoùg dpiGpode, dk Exopev: vatoy nAsuprxeòg xal Bioustpixoùc dpiluovc, Aror tag dxepalag Adoziq 150 bE " oGocos y'=2x Fl, fi bv ay 2-+-(—1)v (v=1, 2, 3,...). "Ex 1 Tloktzelag 100 Miárcovos nAinpogopevpeta, Bri ol tAcupixol xal Bia. Hieugizoi dgrdpoi petpinol dpiCpol Foav ywwotol ele adtév. “Exel as dgduör dad diopérecwr fiv acunddos, deoptrwy dvayiyv@oxopev <Íxarór piv Iris Indorwr, domjror di dSvoive (545 0). "Evrat8a & NAGtov Onawlogerar tobe nAsumixodc Kal Siapertpr 40058 &piBpots ral Bi xal plav dkepaiav Ado tic dvatipa sfiodoews, tv ?=2,5°—], Tobto ouvéyerai Ex tod Mpdxdou, Bor yodpet «ónov di rò córeyyve darts, olor eiplvizs dv yewpargig rergdywror rergaydrov dindmor, iv dgsduole di iis Entddos Cinidoror ¿vos diorros>. Kal d\Aaxod cod yap dor retpdywvog dps0nós rerpa” ew dirkdarog af pa) Adyes vis nov súveyyus. d ydg dad rod L* rod dad 100 a” dixldosds tony hoz dlorsoss (*). (Elvai Bnd. 7*=2. 51). «KbxXou Mätpnaig? xpnoipoon3 (A) 283 CYT II, sal 265923. . 15392,y 1351%=3. È 78071 153 760" Feoperpiriv Snd5eEw tOv oxtotav rodteav Ömeßäronev ele tiv 'Axabnulav "ASnvbv (1). 7 7 y «BP, vá elva ton npdg tiv (B) tEctepiniv ywviav loonAcdpcu tpiy@vou. Kata tav EbnAeldnv 11, 12, tàv xa» Aédwuev sv nAzupàv AB=a, xal tiv AF=5,, Mug Pepalos elvar À ueyaÂuté- | 25, + 25, a, =2a a, = 2a, +5 +5, & = Ja, 5, = Ja, + 25, + 25, : î ' a, = 2ay—1-+ By By = Jav-1t 2by—1 IMevgizoi dyrdpoi Aupergizoi dgidpoi a= 3 5, = 5 a, = 41 5, — 71 a, ra 153 5, = 265 E i IlZeugizoi Ggidpoi Ainpergizoi águd pol a, = 1 5, = dy = 4 Es = 7 aw 15 = 26 56 5, = 97 "y pa Biayóvios 100 pôufou ABA, Ga elvar Zy. 2 ô, = Da, di = 3a, Ol dpidpol oùtoi napéxoua tag dxapalag Abasıs tig LErodorar y?=3x'-2, voi elvar: 1*==3,1*—2, 523,32, 19%=3.11'-2, xAm. "Eáv BLo@pev a,=1, 5,=2, AapPdvopev: “è ABI (ov. 2), 100 brolou h peyadurtpa yovla, + 5, | 8, "Edy Béowpev a=!, b=1, Acuf&vouev: mot Gveu Anobeldewg ag oxéoeis AGtn ouvilerat npòg toùc mieupixoùg xal Biaperpixode Kpıdyodc. Oswpoa Ju HEV loooxails dubluyóviov tplywvov, 5, a, = 2a, a= 2a, 4 obx Forces dvds Slovtog qupiv ülkor Eliov dimldawr baiozar, Gonep rod dirò «e nsriddos d dad 2.°O "Apxınhöng ele tiv apayuatelav abtod .eye Aupetgizoi dg poi 2 5,1=3x,?, kal ouvenós 5, :a,=Y3. 'EpapuóZopev topa áxpipOe tiv und tod Mpó- Ce = 730 6, = 1351 KAou Onoberxvuonévnav ¡pidobov Bid rv i i Enòbe:fiv tv ix tetpayovev oxnudrtoy Ol ápibuol oGrot napéyouor tag dxepalac AdoEIC tic ¿Eidos y'=3x?-4-1, upoxustóvicov nAeupiköv Kal SiaperpinGy GppGyv. ‘Enl 1g npoextacews tig AB Aapfßävouev tufipa BE=AB=a, xal tv cuvexelg tuñua EZ=Al=b,. Kare tov Ecxdelónv 11, 10 Bà Exapev: (La,+5,)+5,*=20,4-2(0,-45,)", xal tx radtng: (2a,-4-8,)*=4a,'+ 40,8 pd. rot elvai: 23=3, 341, 7°*=3,4%41, 261=3,15"-41, xAm. Ol A6yot è t@v (A) xal (8) ánoredodor dúo dxodovblag, Ex tdv Snolwv_ñ (1). uèv tOv (A) elvai ad&avopzévn, À di tv {B) o0lvouoz, Tè xonòv ppayua todtev elvas à di, bg elvaı Elvas ipa xal 3(2c,-!-5,)'==120,"-4-12a,8,-+-35,%, “AAAG 6,%==3a,*, ‘Enopfvws 3{20,+5,)’=9a,?--12a,5,-445,?=(da,-+25,)". ‘H axdoıs Suws adın onnalva Bu A piv Raurdı) calva wAsupd, A BEI (24,201) clvar À peyadutipa biayómos duolou AbuBov pdc tov ABA 100 AZHO. 'Eav inl tiie npoextaceog ins AZ Adbopev tpijua loov npèç AZ xal tv ouveyela rufina loov upós AH, tote Exouev sarà tov alıdv vouov tiv nAeupov N 783Henne. Ti. < 965 tota 57ich 263 à ¢ 133! Kal 2) 265%==3.153*—2, 13513 780°4-1, Keme, Oo yvooté, è "Apxiuñbns. kal tiv peyakuttpav biaydviov véou buolou AduBou npdg tòv Apxırdv, toi nrzun& piv elvat h 2AZ-+-AH, SiayOvog Bi peyadutépa Y 3AZ-+2AH fl 70,440, «al 120:4-75; ávuworolyos. KakoOvteg tag inde tüv nheupov nAeupixods dpi 362 , 97 O¢ xpnoimonoiet radra &veu únobel- . IL. "Ex tv divotepo ExkteBEvrov ouyváyopev tó cupnépaopa dr ol Mudaydperot Éyvopiüov Kal dc dxepalag Adoeic 1fc tftodoenue 1) IfadvZos eis Ebro 1, o, 01 xal 427, 142. G. Friedicin, Teubner, Sy 2) Bh, Tipuwwak ‘Axctyutag AQyvay, 2.6.1955, a, 255 x. di. nal nponyodpevey 160406 P= day "+(A —4)v (— 1)" (v=1, 2, 3... xal ADS, dxtparog pù terpywvog), kal Bri À TR elvas 10 xouvèv clic orzo.

opéyua Suo d&xchoutiGy, jnäg ad avouivas kal pis pRivodans, Bidts À ánóberEie roúrov Elva AxpiBSs À adrh npds tag dvorépo txtedeloag. Mapéyouev riv AróbeiEiv Did tés Éfiodueic: by? == Gay? + (5-47 (—1)v Bv? = Pay? -+217-—4)Y (—Yr if a = 4, ss = 17, 5; = 9 5, = o, = 72, = 161 6,. 161 eso 161 j,99 38 Fr;72 <7 Cu V5 ow € 5 <a" dv? = Say? + (8—4) (—1)y 4 Elvas bè 3 by > =1ay * -+(17—4)v (—1)v 2 xal thy VS, Ve, 17,...777. Elve BE yvwotóv Ex tod Ocatitou 100 Mi&tw vos, dr 6 Gtdbmpocg("} (8 Kupnvatog Satie Dewpeiron Mudaydpeioc) dnéSercEe 18 Golppuerpov tig PS, Vo... Yi: (Qealtntos 147 D—148 B). Il, i. BV? = Say 14 (54) (1) xal YT. 6 BP=2a, xal ñ biayóvios Al=b, Elvar dpa Kal B,1=5u 1 mposxtfoeog tig AB Aaufávopev tuñua EZ=AT==5,. Kata tdv (2a,+5,)'45,*=20,14-24, +51) (2a, +8,)'==4a,?-}-4a,8,+-5," Elvai dpa xal / | /\ 5(20,-}-8,)*=200,7-}-202,5,-+-351". "AM à vu È 8,3=5a;3. "Exopévac By += Way "4 (10—4)¥(—1)¥, xat YIO. 1.2 1 "AAAG B,*=100,”, ‘Exoutvag 10(2a,-}-d,)*=40c,*4-600,*-+-40a,5,-4-45,*==(10&1-|-28,)?. ‘H ayéoic Sus abın onpalver, Br À piv (20,-+5,) elvur nitupà, A si (100,-+28,) biayóvios Spolou dèploywvicu naparinkoypänpou npdg tó Aapyixbv. "O vbuoc tic rataoxeuñc tv éuolov ly ouvexela napa\inAoyp&puwv elvas = "Odev 64 elvar Ey. 3 "H oytaig Spog aütn onpalver, Su À piy (20,-+-5,)=AZ elvas nheupé, 52 (Sur -25)=AH elvat bimyóvios Biiolou mpdg td cpyiKkdv dp3oyavlou nanakknkoypduuou 100 AZHO, Katk rdv npopavii véuov tho xataokeuñs ly guvexela duolov dpBoyovlov rapaX\nAoypappwov dà Aoperquaoi dgiOpot Mievpizoi dgi0poi Gi Exupev: Araperpıxoi dosOpol ö, a, = 2a, +- Bi a= 10a, + 25, dim doi +3 5, = 10a, +25 3 a Cy = 2a, db, + 5, 5; = 5a, ob 25, Qa = 2a, + di 5, == Sa, + 20, = = 2% + 5, 5, = 5a, + 25, + Bei by = Sav—-: 1) Panty + Wissowa, Realenzyklopidie unter AL Cuutor, E, Frank, F, Uultsch, G, Junge, IL Vogt, Theodoros “+ 25y—-1. Dort Literaturangabe : IL Q. Zenthen, Era Sachs, T, Bonne» sen, I Hasse-Al, Scholz, Y, Meath. —2, Und IM L, van der Waerden, Die Arithinetik der Pythagorcer II, Die Thearie des Irrationalen, Mathem, Annalen, 120, 6/6 Meft, 1949, Elvar' 54 Kal Springer Verlag, Berlin, Göttingen, Heidelberg —3, A Reidemeister, Die Arithmetik der Griechen, Leipzig, 1910.-4. J. Æ Hofmann, Geschichte der Mathematik 1, S, 26-27, berlin, 193 (Sammlung Göschen, 226, $ ay = 2ay=1 + By ‘Edy BEoopev : Gv = Zav-1 Bl=3a,, Sudte "Eoapnößovies tiv nponyoujévny Kataoxevay (Il. 1) Aayfávopev (20,-+-8,)?=4,"-+-42,5,4-D,* 10(2a,-}-6,)?=40x,°-+-40a,8,-1-105,* Elvas dipa xal npogavig. 5S(2a,4-5,)*= 200, 9+-5ex,?-+-20cr, 8 ,-}-45, "= (Sa,4-25,)* Micvpixol dgıdpoi — 1 3 ba Kai Le 110, Eöxkelönv Il, 10, 6G Exwpuev : Kal tx rane +1 by = 5.av? -|- (5—4)¥ (11. Gi ‘Ent 4 Ele 1d uponyotuevoy oyfina 3 Aaufévouev AD=o, di std 3E=AB=a, ral iv ouveyela tufñua xal sr 1 = 5,179 39 : Oewpodpev dp0oydviov napadAn: + Adypappnov, tó ABTA (oy. 3), Evüa Este AB=a,, 5, 1? GS=5 AapPavouev 5, = Kal ‘Edv Ofcopev a, = 1, dvi = Say? + (5—4)¥ Li” xal By = 10av-14 28v1 . AapPd&vopev b, = 2 a 1 a, 4 5, == 14 a, = 112 Ba == 356 2 : + È <.--Y]0.. = is 2 = 10.1% — 6 14 = 10,4 +6 68% = 10 22 — 6 356’ == 10,112? + 6* i by 1210, av 104 (= Walter de Gruyter und Co),—5, Robert $, Drumhangh, Matos Mathematical Imagination, p, 140, Indiana Univ, Press, Moomington, 1954, By 3=170,2-4(17—4)" (1), xal YI7, Ele td aúro oyfja 3 Aauf&vopev AD=a, Br=ia, A=, énéte alvar

b'=17a; ral ai = VIT, 'Enappdfouev néAiv thy KataoKeviy (Il. Avipincorzoî dni pol a; ë, a, = 2a, + 6, 6, == 7a, + 25, a= 2a, |. 5, di I 7a, + 25; 17(2a,-+-6,)?=63a,?-{-2210,3-4-680,8,-j-45,’=(170,-4-25,)?, a, = 20, + LA = 7a, + 25, "H oxto Bug abtn (17a,-|-25,) BiayGviog Spolou onualve, Sri ñ uv (2a,-+5,) elvor nAgupé, à bè Sdpdoyivlou napurAnioypGpiou upde td úpyxixóv. ‘O vôuoc oxnuariojiod 1öv duoluv dpdoy, naparindoyoduuov elvas npogavás. “OSev Où Excoyuev ID.eugwmoi égibpoi Srapergizol “gil jiol a; 'Eáv Gloopev IMeugixoi Quoi 8,°:=170,3. (2c,-$-5,)"=4a,*+4-4a,5,-+-6,7. Elva: Epa Kal 17(2c,-+6,)'=68a,?--682,5,4-375,". ‘AMG . “Enopévas Möndre Exouev 151 — 8 : 3 Gy = 2av-1 + bv1 ‘Edv GEowmpev a,= Dv == Tari + 25y—1 1 «al & = a, == 68 6, a= 20, + 5, ba = Ma, “+ 26, a,=2u, a= 2a, +5 6, = la, +5 &,= 17a, +23; +23. ; 2 50 = xal i i by = 17ay-1 + 25v-1 S$, 2 2, Anußkvayev 6, =: 233 233 , 11 ST: >< Saale de <a Elvas 5è Gv == 2av—1 + Bei a = 1 xal : : 2=7,19 —3 1l=7,4 23° 30= 7,19 — 3 Aaußkvonev 233= 7 , 63° -+- 3° i a, = 168 Elvas Be By tee 7. ay th (74) (1). 5, = 713 ER. LA, = 7.9 —13 21° = 17.4 + 13 110? == 17.29 — 13 vat 11. 5. Ele 16 nponyodnevov axfiua 4 Aauf&vopev AB=a,, BF=3a,, Af:=s.. Kata tov EoxAelBnv Il, 12 elvat 6,"==13a,", Kal à. =Y]3. M&Aw tgapudfopev 4 thy adtiv Kataoxeuiy 6 Kal nponyouptvac, rot Anufikvonev énéte Kata tó Il, 10 100 EuxAclBou Ba elvan > 713* = 17, 168? 4- 13¢ La +55 45) =20,"4-2(a,+5,)”, : Sy *= 17.av* ++ (17-4)" (-1)v Hi, 4. Gewpodpev tò poufosidic raparAnAbypanpuov ABTA (ox. 4) Evda yovla 8 ZA ABI=120', AB=a, Br==2a, Kal A peyariuitpa Biaybviog Alb, Kard rov Ebxielónv 1, 12 04 elvai 6%=7a,? Sabre Se =V7, Gi MeAw ZoapusLlonev tiv adrhv xa: taoxeuy (11. 1), ónóre Aaufüvoyrec BE=a,, 13(2a,4-5,)1-=520,"4-1170,14-520,5,-1-45,1==(13x%,-1-2b,)*, 'H oxtoig duws adın onpalver, Sui $ puèv (2a,--5,) elvas nicupá, À dI (13a,-+25,) biayóvios peyaruripa duolou poufosibobs nmapadAnroypadypou npd¢ 10 dpyixdy, Kata rdv npopavñ vópov xataoreufic tor byolav houßosböv napar: Anioyp&upuwy Où Eyopev TTievgixol dordpot (Qu,+5,) ==, -4a,5,4-8,1. 7(2a4-4-5} Bbi'e7a;?, ‘Enopulvoz Aiapergexol digaQpioi N ay Raro) 45 =20 40,45). "AMG tE iis Elva &pa ral 13(20,-+-6,) =52a,7+520,6,-f- 138,9, > tropivoc 6,%=:13a,! ' ‘AAG EZ==5,, Elvat dipa «al M201,-}-6,}?==200,3-j-200, 51-4-76,*. a,= 2a, oo 5, Bi = 13a, + 28, a, = 2a, + 5, LA = 13a, + 26, i i Gy = iv + By ‘Edy Séowuev a, by = 13av-ı + 2bv—1 1 xal - & = 2 2,85 + 55 < me 497 , 1 113 Som tag <a XapuP&vopev 28, 7-4-2 a, 4-260, 5,---46,*== =(70,425,)". "H oxtow uns córm onnalver, Bri À pèv (20,45) elvan zAcupd, h St (Ja, +-28,) Biayfytog ueyaruripa dpotoy npôc 146 dpxixóv pouPocibodg napaddAn: Loypepyzou, ¡Card toy mpopavi) véuov xaraoxeuñs duolwv Pophoebav wapadrAnAoypdppov Od Eyopev: EZ=5, (La,+45,)1=4a,"-440,5,+5,?. Où Eymuev xará tóv Eörkelönv 11, 10: BE he BE=a,, i Elvar 5d

252,+oteYiIal,aa=5,5,5ö,+Z2a,%=a=,aEYdrS.,&pxia1dSp6lo,vtatwSaairvapGedrtiopabiexyGlevboeuKuearvlpniaxwloAdcx,ucproiaùxxoGlsvevxdabvlyriÜtonxtooneloSyycoiivoodnyoodxdnxuoaarltaippodnodgg10Gè%Ayr"eoEnBvépnitxoogd KLe12,4=169202a,6l+7Ba:v=—1&<=,2Éb5y.+-PE6,N8i.07SA=)B19y<6,%-|+a=vi,=52GdyaElva 25,++Ilday,==55,,d5i+24a,=aa,,aa==,yAapPavopev7xa5lLéat2Koadlo=e=iCcIB.b(,Do1vav,nag(xVea6Etl-s4c)rua+gixGaaly?V1iB521y,0t.e=.=s..aBa,i,7E7Ol8kv,oamtp1u2a,y"Eùv GAanpavonev2loapev"Edv25,(+—1}6(%-—4=)9r°b6+H4i;-r136e"13n,==t4d69i,7+=a,2=%2caa,,3 54Elva: 4=Cy .2258,,++1R2a0,,—7—11.5238—=749Biowpev‘Edy28y22—58,1,(-+1++)Gva-Gv8+2a-a7i,(,29==—44==4)—5d5v—+5ya,,8=8.=.a898y5.?1,7=1=0?25&57E°2v?—B56a1,+o+ablQ241ar=%,8,2==aGaa,C%vs,y +aB2M=5yrv:—ai1 81.8.AEulvar=aOS,s5èv=*|122=0a0,,1l<a++yC"6L5&+oP,yn0(3=1VIT14l2a)u,y2(?4i+(gr—NàSb5is-,<&a==)Krg4a0rl1(,—ljYv«T.al. I , 10 a"Yu25127==8.4+di24=1%6a,512+25,aa.,==224178404518s,)"*=—1n2e,.1402v8*?440+6=&15t,78,8(=121-5412)58"(IrAaufpévouew a*Eáv1B.é7o.mp8u8evay=r1unterx(al—1}xal18."Edy1G.l9o.wpAeivkASG8aGaavi,v,?====]B22a1ua,y41tyv4—s*1-++M1a52y,d-v4*-)i+r„a(Nlrl,axa(5dddl5ii-y,,1-V===),2l5l2a4r,8al+VIA2A5a,uBévouev 1 uOùro04elvaı =a12.,1—78120,+28, $ “ 80 16 av=2av-1+Dy=1by=1lay-ı+2601 xal 48 E, +125avu-1 IA6r.a a; 1 i 5, +(B,1—3*1—a4)=vy"®v î =8913—5,2°5" 14=73%. 213=.?1 Xapuf&vopev

. 2 92 — 543 , 18 2= 14, 1% — 10 185 = 14. 4 + 92 = 14. 25° — i¢ Mpdg obyxptow napadironev tà ¿Eayóneva Kal tv 660 nedödwv. by = 20v-1-}-bv-1. A'. MéGoboc elbizà, av = ay—14-By—1, xal ar T < 36 < Pla Cia Elva: &È 155 — a,= 10 5:53 = 14.144 + 10* è | b.= 1 a,= 2 œ = 3 o, = 12 = 3 = 7 = 17 5, Pd 1. TI. 5, a; a = 2a, Cr = 2a; + A 5,: 6, B,= 15a, 5, = 150, + + 25, 25, = 25, + ds è, = 15 + 28, ar = 2av-1 + Br: "Esv Bfomuev = {Oia «a=1 = 2 By * = 2av*? +(—1}". 7=2.5-1, 39=2.241, P=2.1-1, 5=2 Sv = mn + 25,1 1 de Be E CT <a ¿ds Kine ¿51 bi Air Alá By? = 13ay* + (15-4v(-1)", xal VIS 5, = 24 a, = 17 AcuS&vopey e : Y = «= 27 a, = 152 1 Elva: Sè | < > <a VI5 € E < 2 2= 15, xel 1 — 1 4 + 312 19 = 15. 58 = 15. 27: — 13» 601? = 15.152 + 11° prua 1. 10 yewperzpixh kataokeuñ, tiv ónolav pynuoveder è MaékXoc, &yer ele (1), Thy énoBrixvuonévny xovae n=2. +2 Aik a; = 4, — 2 +2 róxdelberov Sebpnua Il, 4. "Edv 5,%=Aa,?, av =2ar-ıtrdr-r, Eva Sv =20r-14-25r-1. 5, = 1, a == 1 dy = 3 (2a, +5, )?=4o,*-+40,5,--5,', 16 22.3 10=2,7 — 2 24 = 2,17 3°. MéBoboc yevixh, 2. "Ex Gv nponyouptvay zacpatnpodpev, bu à xard td coxAelBeiov Bro» rAv taurbinta, By hy de by = 20 2 + (2—4) 1)" by '= 15.av? + (15-v (1. HI. rr. hy hy > Ga > STE «> Ce > Ga > Ca hoy CA => |—®corso 4 = 601 b,= 1 + .-B = 4 À 5, dxéoaiog pi) tetpdywvos. Bü elva kal (A—4)5,=(A—4)Aa,”, Onéte Ex tic (1) tyopey M2a,+8,)'=4A.0,7+40,5,-+(A—4)Ac,?-+-45,2=(Aa,-4-25,), "Enoutvog Sk elvai, 1) Tilevgizoi derdpoi Ci Gi = 22, 5, a 5, cg, = 20; e Gi = 20, “> 6; 5, 5. = ia: cL 25, ds = Aa, — 25, 19=2,1 —1 2=2.7 —2 14 = 2, 10° — 2 — > 48 = 2.34 & = las + 25, By = Nava + 2641 ay = 2av-1 + dv ca DBS SS CV A Re 3) 5, 5; 5, EW? “>: = Cee Anjerpixoi ági pol Ge CE By t= 2.00 * HR CR, ar Ge È (ut, de . dà Sy * = Am ? «+ RAN (-1)v a ==] , diEu; =2, Dr SAGv—3+28v—1, Srav A=2 Tay uÉ9obov tadınv xakodpev yevixhv npòc Sidxpiow Gud tc ueûbbou tic Siaorméslons Gnd r00 Otovac roi Zuupvalou ev eidicfiv. xel tod MpSxAou, tiv srolav xadod- 3 a= 4 Bb, = 6 u, = 48 5, = 68, î 3 111. 2. Elvas Buvatèv À V2 ve ÖroAoyıcdj rai Lx rGv nAcupixSyv xal biapetaxady Apıbuöv 196 nopgäis av = 2av—1-j-By—-1,

@ = 2.4 +2 is tiv Ind tod "Apxiuñbouc napsyopévn? Tipay the 0% eiyouev Bià tv 12 LMa i Tr BE l AS 22 tata Sme + 22,)*. ,2,15,2 =(102,-+10( 2, -$-2,)? = 1002,?-$-102 Side numbers ay =2av—1-+2y—1, diagonal numbers cri” ” by = 2.0, ++ (2—4v (1). Kor dvuoteylav apèç 2 + Cité Sag Sta Elva Gavepôy, Bu alvar nponmoripa h Ind tod Diwvoc 35 63 2 < is VCH Ce — by 2102, 2410-0 (Mr. When el, 4=2, Co 3. Inthe same figure 3 we take AB=2,, PT=ia,, AT=5, Then 2,°=172,", 172,7. Because 6Sabi 2) 08e 2,:2=/T7, and 12 , =(1 70H oe 4-452 4682,2 ia, is 17(22,42,)=2892,7- Side numbers a ST of the related identity is proved book 11, proposition 10, Side numbers ay = wer tiv, diagonal numbers 22, Fl, For 2,1, 3 =1 we have +<i A 2. Archimedes for the arithmetical dv = 2ay—1-}-3y—1, 2,9 == CA <1 <4. | approximation to # starts from a greater and a lesser limit to the value of Y3, which without remark as known: 263 36 15 <T5< 2 265°=S.1537—2, of tke side—and disgonal—rumbers. In the figure 2 is AB=x, the side ATZ?, the greater diayonal of the rhomS ABFA, and the greater angle ABT—120%, Thea, 3,3=32x,3, 2,: ==? 3. According to Euclid II Prop. 10 we have (2-2, 2e a, It is also 0e, 20.) @2,+5,P=42,"+-12,5,+2,%, “a, P2)=12%0,1+122,5,-253,”, and because (1) 3,%=32,?, Sao,'=92,24122,0,348,=132,-+20,)0 ER When Uy ler ten, a= 4 = 1 ace 3 = 6 a,= 11 M = a= 41 e, * = 163 diagonal numbers When 1 è; = Za 4 = 7 19 = 16 y= 26 7 „= 66 A= Y dy == 265 “= à, = 362 a == 150 ds = 1361 dy = Jay * — 2 1 5 19 11 | 285 TFR — and 1351 Sy "== Jar th 1, 862 ,97 <a ‘wa _.% 1, 7 1 TT In the figure 3 we take ADma, Dax, Al, Then 9,=b4,?, 3, ta) 6 » is Sa py 20 Side numbers è,1=0,?, 40a EE, = (5,28, av =20y-1--3y-1, diagonal numbers è, —Say—3-4-22y—3, by = Say Hr (-1)v. itv" D. N., bv ECTS D PATES by "= Tay 34-(7-—4)v (Ir . ES <q <7 7 Sg 3 . Because melde, is 15(22,-+2,)?=1602,3-522,2,4-18,)=(13x,4-22,)? 6 à ih y 5. In the same figure 4 we take AB, br=3x,, AP=0,. +2, 198,?. 4-522,8,+5a , +2.2 jr-=52a,313, 13 =/13 Thea 8,=18a,9, S,:a,=} ch. cSt vi 3 136r 2-8 + os Eee = 23 In the same way we take the side—and the diagonal--numbers for the Ys, TE: Yu. Yi, GE Y id. (We mention Theseteins of Plato 147 DI III. If %>5, integer no square number and 2 =da,, then AIR, and according to the identity (i) Ada, +2, 0 RN HB 0,8 4402, + en) Side numbers Diagonal numbers Ci è, a, = Da, + è, = da, + 2, 2 11, tr the following we start from the identity (1). and E(2a,-+3,)3:=202;3--202,83,-103,%, Because AB=d,, Pr=2x,, Al=ty The angle ABD=120», When a,.=1, d=2, dv —Say—1-4+-22y—3. «= | ‘ and 713 , 21 Sas, 122,45) = dx,34-234,2,4-1%,3=(70,-4-22,)". SN, ay = The law of formation of the correspondig side—and diagonsl—aumbers is evidently. Side numbers 3 110 $72, Because =284,%4= 232,2, +72, i %,î=7x,3, bia =}i7, and 72,4%,)* u When a,=1, 2,=2, _ 195133, 780.1, We give the following interpretation on the archimedeza formula, with pytkagoscan method IA. 2 Then the side - and diameter- (dingonal - ) numbers is explained by Theon of Smyrna. According to Proclua hy Euclid By = Bay HI 4. In the figure 4 we take SUMMARY dv=17ay-1422v1, ay=2ay1-+8v-1, D. N. tod Zuupvalou Kal 105 Mpóxiov biaowBriva pédobog Sie rd OnoAcyioudy tic Ÿ 2. 1. 1, The law of formation 9 "Ar € I <>» Y 5 .. 4 & 72 < q 2. In the same figure 3 we take AB=2,, BO =3x,, AT =}, Then §2=102,', 2=2.+ 20° = 2.14 + 2° 659° = 2,48 +2 T3 161 — 38 2 When a =1, $,=2, 2 î ay = 2av-14- dvi 5 ty = mer Der di ges = AI x Me n g (1). We take here always =), 2,—2, X=2 and A=3 and %y *=22y *+(2—1)v ' are special cases,