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O pi12o2cq 0ai1q . /iv0i o. 2. 12oi0i. i. .ie.2ikq 000ie1q. v.3cp02.i 12i .piu vvoi0, -2i 010i, oae 2i ovou0i-, 12ou opi1uo, 12. .iou.2., 12qv .no/0iu2iuq /i./iu.1. k.i 12qv .n/0iq. K.1iko a.pu/0i. .ie.2ikq 000ie1q 0iv.i 2. "22oi0." 2ou vk0/q. O uk0i/q k.2cp.0 2i .pikc cvvoi0 q .piko opou 2ev 0e02pikev .v2ik0icvev, oae 0iv.i 2o 1q0io, q p.q ka. K.2cp.0 2i .1ikc 2ou i/io2q20 k.i k.0icpe10 2ou opi1o, 2. .i2q.2. k.i 2i koivc cvvoi0. qiopq10 c21i i. .0q.2ikq 00epi., c1. 12qv oaoi. ku00 apo2.1q aop0i v. .ao/0i00i 0 2q oq-00i. 2ev opi1ev, 2ev .i2qu2ev, 2ev koivev 0vvoiev k.i 2ev apo2u-10ev aou couv apoqoucve .ao/0i00i. .ie.2ikq 000ie1q k.i oi .ao/0ik2ikc /i./ik.1i0 aou 0i1q.0 0iv.i i1upc k.i .v.oie20 cpi 1q0p.. T. l2oi,0i. 0iv.i cv. .ao 2. 1q.v2iko20p. .0q.2iku cp. o-ev 2ev 0aoev. +i. 2300 povi. pq1i00i e 2o 12.00po k0i0vo i. 2q /i/.1k.i. 2ev .1ikev M.0q.2ikev k.i 0ai .pk02o .iev0 pq-1ioaoiq0qk0 e .1iko k0i0vo i. 2q /i/.1k.i. 2q +0e02pi. 10 aoc ep0 2ou ko1ou. L2qv ua0p/i1ii02q i12opi. 2q a.pq 2ou k.vcv. uo cpo 0ai12qoviko, 3io1o3iko, oo20viko q uq 31q k.i a0pi0ocvou /0v c0i .a.1oq10i 2ou 0k/o20 0 2qv i/i. 1uvo2q2. oae .u2o. 1 Theodoros G. Exarchakos, "The method of mathematical induction and the Pell equa-tion in ancient Greek Mathematics", Mentor, A Journal of Scientific and Educational Research, Hellenic Pedagogical Institute, Athens 1999, vol. 1, 10.4-16. L+O+ 9 0i 02.3p.100i 10 aup. aoc e110 k.i oi 0k/o10i 2ou 1q0p. uaooi02.i o2i 0a0pvov 2i 2000. K.2u 2o Loria1 2. l2oi,0i. 10 .-pi0o 0k/o10ev k.i 02.3pu10ev aopov v. .v2.evi12ov 2q ",0i. Kcuc/i." 2ou uv2q k.i i1e uao0iaov2.i ovo 2q i. +p.3q. ,. apca0i v. apo10c1ou0 0/e o2i q 0k/o2ikq /p.12qpio2q2. /0v c0i 0ai-k0v2pe10i 2qv apo1oq 2q ovo 12i 02.3pu10i k.i 2i 0k/o10i 2ev l2oi,0icv, .u a0pi.uv0i i. 0uq aoikii. .ao kpi2ikc 0c20, .ao /iop0e10i k.i .v.3opc, a0piq0i k.i uaovq.2i1o, 1oi. k.i 0i/ikc 0k/o10i i. 0i/iko oou. K.2u 2ov Ivor Thomas2 2. l2oi,0i. couv 0aqp0u10i 2o .v0peaivo u.o a0pi11o20po .ao ku00 uo cpo, 0k2o .ao 2q io. L2i 02.3pu10i, 0k/o10i, .v.00epqcv0 0k/o10i, 0i/ikc 0k-/o10i k.i 1oi.1o 2ev l2oi,0icv 0. .v.30p0o0 .v.u2iku 12qv a.pup.3o 5 .u2q 2q 0i1.eq. ,0epo0 oe .a.p.i2q2o v. a.-p.2qpq1ou0 0/e o2i q aio ookqpecvq 1povq ck/o1q 2ev l2oi-,0icv 0iv.i .u2q aou a0pi.uv02.i 12o cpo "Euclidis Opera Omnia" 2ou .vo 0qvi12q k.i i12opiko 2ev 0ai12qev J. L. Heiberg k.i 2ou +0p.vo oiou H. Menge. ck/o1q 2ev l2oi,0icv .ao 2ov Heiberg c0i iv0i 12. .p.i. qviku 0 a.pua0upq 02u3p.1q 12. A.2iviku. ao200i i. apo10cvq, 20kqpiecvq k.i aqpq kpt2tkq ck/o1q, aou .ao/i/0i ai12u 2o a0pi0o0vo 2ev l2oi,0icv k.i /i.2qp0i 2qv .u12qpo2q2. k.i 2q /oq aou .koou0o10 o uk0i/q. To cpo .u2o 2ou Heiberg 02.3pu12qk0 12. iku k.i 0k/o-0qk0 10 2p0i 2oou .ao 2ov T. L. Heath, o oaoio apo10010 i. 0i1.-eq k.i .pk02u 0v/i.3cpov2. k.i pq1i. 1oi.. L2ov 0qviko epo 2. l2oi,0i. a.pou1iuov2.i .ao 2ov vyy0-o L2.2q 10 cv. cpo 2ou aou 0k/o0qk0 k.2u 2qv a0pio/o 1952-1957 k.i .ao200i2.i .ao 2c110pi 2oou. To cpo 2ou . L2.u2q 0iv.i i. .v.a.p.eq 2ou cpou 2ou Heiberg. . 2. a.p.auve cp. 0iv.i .ioo. k.i 0.ip02iku pq1i., 2.u2opov. oe 0iv.i ai. 1auvi., q .v.ve1i. i. 2. 0qviku /0-/ocv. k.i ao /1pq12.. +i' .u2o 2o H.i/.eiko v12i2o2o 10 1u-v0p.1i. 0 2o Kcv2po p0uv. ai12qq k.i ka.i/0u1q (K..H.K), .ao3u1i1.v v. apoov 12qv ck/o1q 0vo vcou cpou, 0 a0pi0o0vo 2o oaoio 0. c0i e u1q 2. l2oi,0i. 2ou uk0i/q k.i 0. 1 G. Loria "12oi. 2cv M.0pu.2ikcv", k/o10i .M.. 1971, 2oo 1o, 10. 63. 2 Ivor Thomas Dictionary of Scientific Biography, vol. IV, Charles Scribner's sons, New York 1975, 10. 415. 10 YKA 'LTOX 0iv.i p.cvo 12q v0o0qvikq e11. 0 1povq opooi. k.i 1uoi1o. vo cpou aou v. .ao/i/0i ai12u 2o a0pi0o0vo k.i 2o av0. 2ou .p.iou 0qviko k0icvou 2ou uk0i/q, .u k.i aou 2.u2opov. v. aop0i v. /i..120i k.i v. k.2.voq00i .ao 2ov qv. .v.ve12q, .koq k.i .ao 2ov q 0i/iko. vo cpou pq1iou 12ov 0ka.i/0u2iko, 2o 3oi2q2q, 2o .0q2q, 2o ovio k.i 0viku 10 k.0cv.v aou 0v/i.3cp02.i v. 0vq0pe00i i. 2o a0pi0o0vo .u2o 2ou 2o1o 1q-.v2iko k.i /i.poviko cpou. ,0epo0 o2i 2. l2oi,0i., oae a.pou1iuov2.i .ao 2ov uk0i-/q, .ao20ov av0u.2ikq k.i aoi2i1ikq kqpovoiu oi ovo 2ev -qvev, .u ookqpou 2ou 0upea.ko k.i 2ou /i00vo epou. iv.i 0aocve uaopce1q .u k.i .vukq v. 2qv .v./0iou0, v. 2q /i.3uuou0 k.i v. 2qv .ioaoiq1ou0. 2. vk0/q: eq k.t q /p1q 2ov O uk0i/q 00ep0i2.i e o a0pi11o20po /iu1qo .0q.2iko oev 2ev 0aoev k.i 2o ovou 2ou c0i iv0i 1uvevuo 0 2q +0e02pi.. /q .ao 2qv .p.io2q2. .v.3cp02.i e o +0cu/2p q o l2oi,0ic2p. e, ao i. vepiou0 i. 2q eq k.i 2q /pu1q 2ou. ai1q a.p.-cv0i uve12o o 2oao 2q cvvq1q 2ou, k.0e k.i o .kpiq povo 2q cvvq1q k.i 2ou 0.vu2ou 2ou. ao 2. 12oi0i. aou cou0 12q /iu001q . k.i 2. oaoi. 0. a.-p.0c1ou0 12q 1uvc0i., 3.iv02.i o2i o uk0i/q 3oi2q10 12qv k.-/qi. 2ou Hu2ev., q2.v .0q2q 2ev .0q2ev 2ou k.i 0.20po .ao 2ov piq/q k.i 2ov p.2o10cvq 2ov Kupqv.io. q10 12qv 0-uv/p0i. k.2u 2qv 0aoq 2ou H2o0.iou 2ou 1ou 2ou Le2qp., oaou k.i 1uvc0010 2o aio a0pi3qo .ao 2. cp. 2ou, 2. l2oi,0i.. O Hpoko (410-485 .X.) 12. 1oiu 2ou auve 12o ape2o iio 2ev l2oi,0icv .v.3cp0i:1 ",i ao u02o .ao .o2o (2ov po2ivo 2ov Koo3evio k.i 2o -iiaao 2ov M0v/.io, /uo .0q2c 2ou Hu2ev. 12qv k./qi.) p2.v o ok0i/p, o oaoio 1oyk/v2c10 2. l2oi,0i., /.0 10 2op aoo .ao 2. 00cpu.2. 2oo /ooo, 1ouapc10 k.i 200ioaoip10 aoo .ao 2. 00-cpu.2. 2oo ,0.i2p2oo k.i i0o,p10 k.i 2oao0/2p10 10 ui. ./io,0o12p, 1 Proclus, "A Commentary on the First Book of Euclid's Elements", Glenn R. Morrow, Princeton University Press, Princeton New Jersey (1970), 10. 56-57. 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O pi12o2cq 0i0 .ao/0i0i 2qv apo-2.1q o2i oi evi0 2q u1q i1o1k0o 2pievou 0iv.i i101, aou 0iv.i q apo2.1q 5 2ou iiou 2ev l2oi,0icv. v 0aocve 2o20 uaqp.v 2. l2oi,0i., /0v uaqp0 .vukq o pi12o2cq v. .ao/0i0i 2qv apo2.1q .u2q. O 0uo +0p.vo 3iooo U. Von Wilamowitz-Moellendorff 02u .ao .kpopovi0 3iooikc 0c20 k.i .v.10i aou ck.v0 12. cp. 2ou Hu2ev. k.i 12. l2oi,0i., cp.0 2o 1929 o2i 2. l2oi,0i. 2ou uk0i/q apocpov2.i .ao 2q Loq 2ou Hu2ev.. +0p.vi/. i12opiko Eva Sachs, q oaoi. 0ai1q 0c2q10 2. l2oi,0i., k.0e k.i 2o .0q.2iko cpo 2ou Hu2ev., cp.0 2o 1917 o2i o uk0i/q 0iv.i u.0p2p-0yyovo 2ou Hu2ev.. q./q, .0q2q 2ev .0q2ev 2ou Hu2ev.. Tqv i/i. uaoq couv k.i oi Pe1oi .0q.-2ikoi V. Molodski k.i M. I. Vigodski. Yaupouv .pk02c .koq .v.3opc i. 2q eq k.i 2q /p.12qpio-2q2. 2ou uk0i/q. O Huaao, i. a.pu/0i., 12o cpo 2ou "lov.yc-yp" (VII 35), .v.3cp0i o2i o aoevio acp.10 .pk02o povo 12qv -0uv/p0i. 0 2ou .0q2c 2ou uk0i/q2. v uou0 uaoq o2i o aoevio cq10 k.i c/p.10 12qv 0uv/p0i. k.2u 2qv 0aoq 2ou H2o0.iou 2ou 3ou 2ou u0pc2q (246-221 a.X.) k.i 2ou H2o0.iou 1 pi12o2cou "v.o2iko Ho20." , 24, 41b 13-22. 2 Friedrich Hultsch, "Pappi Alexandrini Collectionis quae supersunt", 3 2ooi, Berlin 1876-1878, 2oo 1o, 10. 678. 12 YKA 'LTOX 2ou 4ou 2ou -ioau2op., aou .vc.0 2q .1i0i. 2o 221 a.X., uao-oiou0 o2i oi .0q2c 2ou uk0i/q cq1.v k.i 0pu12qk.v .u2q 2qv a0pio/o. O i/io o uk0i/q 12o cpo 2ou "-.ivou0v." a.p.0c20i i. apo2.1q .ao 2o cpo "H0i kivoou/vp 13.i." 2ou u2oukou .ao 2qv Hi2uvq. u2o 1q.iv0i o2i 2. -.ivou0v. 2ou uk0i/q, k.i 0aocve 2. l2oi,0i., couv p.30i 02u .ao 2o cpo .u2o 2ou u2oukou. Lq0ie-vou0 o2i 2o cpo 2ou u2oukou c0i p.30i 12o 2co 2ou 4ou .iev. a.X., 0iv.i 2o ov./iko .0q.2iko iio aou pu32qk0 apiv .ao 2. l2oi,0i. k.i 1e02.i 1q0p. ookqpo. v uou0 .koq uaoq o2i o u2ouko pi1ko2.v 12o .ao0io 2q /p.12qpio2q2u 2ou pe 12o 300 a.X. k.i q2.v 1povo k.i ve12o 2ou uk0i/q, aopo0 v. 1ua0puvou0 o2i k.i o uk0i/q /p.12qpioaoi0i2.i 2qv i/i. 0aoq q io .po20p.. O Friedrich Hultsch1 /iv0i e povo .kq 2ou uk0i/q 2o 295 a.X. O Peter Fraser2 1ua0p.iv0i o2i o uk0i/q 0vvq0qk0 ai0.vov 02. 330 k.i 320 k.i ac0.v0 pe 12o 270 a.X. ai1q, .pk02oi 0p0uvq2c uaooiouv o2i 2. c1. 2q eq 2ou uk0i/q (2o .ao0io 2q /pu1q 2ou) 0iv.i pe 12. 300 a.X. M02. .u2ev 2ev 0p0uvq2ev 0iv.i k.i oi Hankel, Gow, Zeuten, Cantor, Loria, Hoppe, Heath, Van der Waerden k.i uoi. O Clemens Thaer3 uaooi0i 2qv a0pio/o 2q .kq 2ou uk0i/q 02. 2ou 310 k.i 2ou 300 a.X. O Heinrich Vogt4 2oao0020i 2q cvvq1q 2ou uk0i/q 2o 365 a.X. k.i 2q 1v001q 2ev l2oi,0icv pe 12o 330 0 320 a.X. Ti i/i0 povo-oi0 /iv0i k.i o Max Steck5. e oi 1up.30i .u2oi /0v c..v u-aoq 2ou o2i o uk0i/q cq10 12qv 0uv/p0i. 2qv 0aoq 2ou H2o-0.iou 2ou 1ou 2ou Le2qp. k.i o2i o aoevio 1uv0pu12qk0 i. .pk02o poviko /iu12q. 0 2ou .0q2c 2ou uk0i/q. 1e oi 1u-p.30i .u2oi 1ucouv 2ov uk0i/q 0 2ov uk0i/q 2o M0.pc., 2o .0q.2iko k.i 3io1o3o aou q2.v 1povo 2ou Hu2ev.. 1 Friedrich Hultsch, "Eukleides", Pauli-Wissowa vi, 10. 104. 2 Peter Fraser, "Alexandria", vol.I, 10. 386-388 k.i vol. I Notes, kupie q note 82. 3 Clemens Thaer, "Die Elemente von Euklid", Leipzig, 1933-1937. 4 Heinrich Vogt, "Die Lebenzeit Euklids" Biblioteca Mathematica, 13 (1913), 10. 193-200. 5 Max Steck, "Proclus Diadochus", Halle, 1945. ca0 0ai1q k.i 2o up0po 2ou i/iou 1up.3c. 0 2i2o "Die geisting Tradition der frhen Euklid-Ausgaben", aou /qo1i-02qk0 12o a0pio/iko Forschungen und Fortschritte, 31 (1957), 10. 113-117. 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O Hpoko 12. 1oi. aou cp.0 auve 12qv "Hoi20i." 2ou Hu2ev. .v.3cp02.i 0 ao ko.k0u2iku oi. 12ov Hop3pio. O Ltnkto (2cq 5ou k.i .pc 6ou .iev. .X.) 0iv.i cv. .ao 2ou 1q.v2iko20pou 1oi.12c 2ou cpou 2ou pi12o2cq. L' .u2ov o30iou0 k.i 2q /iu1e1q aoev 1q.v2ikev .ao1a.1u2ev .ao 2qv i12opi. 2q +0e02pi. aou pu32qk0 .ao 2ov /qo i. 2qv a0pio/o .ao 2o ,.q cpi 2ov uk0i/q. Mi. a0piqq 2q i12opi. .u2q uaup-0i 12o cpo 2ou Hpokou. O Liaikio c0i pu0i 1oi. 12o ape2o iio 2ev l2oi,0icv k.i 0i/iku i. 2o 5o .i2q. 2ev a.p.qev. To cp-o 2ou /i.1e0qk0 0pike, 10 .p.ikq 02u3p.1q, .ao 2ov al-Nayrizi. Ti a0pi11o20p0 aqpo3opi0 i. 2o a0pi0o0vo 2ev l2oi,0icv 2i .v2o0 .ao 2ov Hpko 2o Ak0to1 (410-485 .X). O Hpoko cp.0 1oi. 12o ape2o iio 2ev l2oi,0icv 2ou uk0i/q. p.0 0ai1q 1oi. 10 .pk02u cp. 2ou Hu2ev.. Hooi i12opikoi 2ev 0ai-12qev, oae a.. oi J. L. Heiberg k.i T. L. Heath, ai120ouv o2i 2. 1o- 1 O Hpoko 0vvq0qk0 12qv Kev12.v2ivoaoq 2o 410 .X. k.i acp.10 2. a.i/iku 2ou povi. 12qv Euv0q 2q Auk0i., aou q2.v k.i o 2oao k.2.eq 2ev ovcev 2ou. Lao/.10 .piku 12qv 0uv/p0i. k.i 12q 1uvc0i. 12qv 0qv., oaou 0i0 /u1k.o 2o v0oa.2eviko 3io1o3o Ho2.po 2ov 0qv.io. i./c0qk0 2o Lupi.vo 12q /i0-0uv1q 2q k./qi., i .u2o ovou12qk0 "io/o,o" (iu/oo 2ou Hu2ev.). Hc-0.v0 2o 485 .X. 12qv 0qv.. 18 YKA 'LTOX i. aou uaupouv 12ou a0pi11o20pou ke/ik0 e 1oi. .vevev 0iv.i 1oi. 2ou Hpokou, 2. oaoi. kuaoioi uoi 1oi.12c 2. 0a.vc-..v 12. /iku 2ou k0i0v.. 0v uaupouv .ao/0i0i aou v. 0ai0.ievouv o2i o Hpoko c-p.0 1oi. k.i i. 2. uaooia. ii. 2ev l2oi,0icv. e uaupouv .pk02c 0v/0i0i o2i cp.0 2c2oi. 1oi., .3o .u2u pi1kov2.i /iu-1a.p2. 10 cp. uev 1oi.12ev. H.. 2. 1oi. 12o iio V aou .ao/i/ov2.i 10 .vevuo 1oi.12q, k.2u 2ov Heath .vqkouv 12ov Hpo-ko. Hooi uao12qpiouv o2i o Hpoko 12qpio0vo 12. cp. 2ou piq/q, 2ou aoeviou, k.0e k.i 12. cp. k.i 2. 1oi. 2ou pev. k.i 2ou Huaaou, c0i pu0i 1oi. 10 o. 2. ii. 2ev l2oi,0icv. u2o 0iv.i ao ai0.vov, .3o o Hpoko ao 1uvu .v.3cp02.i k.i 10 apo2u10i uev iiev. H.. apok0icvou v. /e10i a.pu/0i. aopi1.2o, .v.3cp02.i 12qv apo2.1q 1 2ou iiou , k.0e k.i 12qv apo2.1q 3 2ou iiou X. 3.3. To n0pt00vo 2ev 22oi0ov T. l2oi,0i., aou 0iv.i 2o 1q.v2iko20po .ao 2. cp. 2ou u-k0i/q, couv p.30i 10 13 ii.1. Hpiv .v.1ou0 2o a0pi0o0vo ku00 iiou, 0. apca0i 0/e v. 2ovi1ou0 o2i cp. 0 2ov 2i2o "l2oi-,0i." couv p.30i k.i .ao .pk02o .0q.2iko apiv .ao 2ov u-k0i/q. O aaokpu2q o Xio (470-410 a.X.), i. a.pu/0i., 0i0 pu-0i iio 0 2i2o "l2oi,0i.", oae .v.3cp02.i .ao 2ov /qo 12o cpo 2ou "12oi. 2p +0cu02i.", k.0e k.i .ao 2ov Hpoko. O Acev o 0qv.io (5o a.X. .iev.), .0q2q 2ou Hu2ev., cp.0 cv. cpo 0 2i2o "l2oi,0i.", 2o oaoio 00epq0qk0 aqpc120po .ao 2o .v2i12oio cp-o 2ou aaokpu2q, 2o1o e apo 2ov .pi0o 2ev apo2u10ev aou a0pi0i-0, o1o k.i e apo 2qv 0ai0c120pq .ao/0iq .u2ev 2ev apo2u10ev. ai1q ii. 0 2i2o "l2oi,0i." cp..v: O Kpi2ev o 0qv.io, .-0q2q 2ou Lekpu2q, ve12o .ao 2ov oevuo /iuoo, o Lii. o ,q.io, .0q2q 2ou -iouou, k.i o ,./io .ao 2q M.vq1i.. u12ue, .ao o. .u2u 2. 1upu.2. 2iao20 /0v c0i /i.-1e00i, 0k2o .ao 0pikc aqpo3opi0 aou uaupouv 10 uou 1u-p.30i k.i 12qv "12oi. 2p +0cu02i." 2ou /qou. To ov./tk 1 M0pikoi 00epov o2i 2. l2oi,0i. couv p.30i 10 15 ii., i.2i 1ua0pi.uvouv 1' .u2u k.i /o .ko. cp., aou pu32qk.v ao .po20p.. O iio XIV .v.3cp0-2.i cv. cpo 2ou Yikq, aou pu32qk0 2o 2o .X. .iev.. To iio XV c0i p.30i 2ov 6o .X. .iev. .ao cv. .0q2q 2ou 1i/epou 2ou Miq1iou. L+O+ 19 .0q.2tk to nov yp32qk0 nptv .n 2ov vk0/q k.t 2o o-noo 1o02.t okqpo 1q0p. 0v.t 2o to "H0p uivoouvq 13.p." 2ov v2vkov .n 2qv Ht2vq, 2o onoo c0t yp.30 k.2 2o 2co 2ov 4ov n.X. .tov.. O u2ouko oae 0ia.0, q2.v 1-povo k.i ve12o 2ou uk0i/q. O uk0i/q vepi0 ao k.u 2o cpo 2ou u2oukou. e, q /oq .u2o 2ou cpou k.i q /i./ik.1i. 2ev .ao/0i0ev 2ev apo2u10ev 0iv.i 0v20e /i.3op02ikq .ao 0k0ivq 2ev l2oi,0icv 2ou uk0i/q, .3o /0v uaupouv oi opi1oi o20 2. .i-2q.2. k.i oi koivc cvvoi0 aou ck..v 2. l2oi,0i. iio ov./iko 10 oq 2q .kp.ievq i12opi. 2ou. O Hpoko .v.3cp0i o2i o uk0i/q q2.v o 200u2.io aou cp.0 iio 0 2i2o l2oi,0i.. O oo 2q q /iu1e1q 2ev apoqo0vev q2.v, oae ai120ouv ooi 10/ov oi i12opikoi 2ev 0ai12qev, q 1uv2pi-a2ikq ua0poq 2ou cpou 2ou uk0i/q cv.v2i oev 2ev apoqo0vev cpev. u2q q ua0poq ck.v0 a0pi22q 2qv a.pq .u2ev 2ev cpev. L2. 13 ii. 2ev l2oi,0icv a0picov2.i 121 opi1oi, 5 .i2q.2., 9 koivc cvvoi0 (.ie.2.) k.i 465 apo2u10i (00epq.2.). Oi opi1uo 0iv.i .pk02u a0piopi1cvoi 10 .pi0o. O uk0i/q /0v .i10uv02.i 2qv .vukq v. opi10i .ko. k.i .1iku 0e02piku .-v2ik0i0v. q .1ikc .0q.2ikc cvvoi0. 0v uaup0i, i. a.pu/0i., opi1o i. 2o a.p.qop.o, 2o 2p.acio, 2o a0v2uevo ka. ai-1q /0v opi02.i q i1o2q2., a.poo aou pq1ioaoi0i2.i 12qv apo2.1q 4 2ou iiou . Oi opi1oi 2ev l2oi,0icv 0iv.i 0i20 0a0qq10i kuaoiev ap.u-2ev, 0i20 uao0c10i, 0i20 a.p./oc, /q./q apo2u10i aou 2i 00epo0 .q00i. .1iko .p.k2qpi12iko 2ev opi1ev 0iv.i o2i, 12ou ape2ou 2ouui12ov opi1o, o uk0i/q opi0i 2. 0e02piku .v2ik0i0v. apoao0c2ov2. 2qv a.pq 2ou. L' .u2o 1u3ev0i 0 2ov pi12o2cq, o oaoio 12. "v.o2iko 120.", oaou .v.a2110i i. .ao/0ik2ikq 00-epi., .v.3cp0i o2i "o oi1uo /i.3/0i .ao 2pv .ao/0ip, yi.2i o oi1uo /0v /,0i c 12o,o 2pv a.p 2oo .v2ik0iu/voo". v. a0ovck2q., k.2u 2q veq ., aou uaup0i 12. l2oi,0i. 0iv.i o2i o uk0i/q opi0i 2. 0e02piku .v2ik0i0v. epi apoqou-cve v. c0i k.0opi10i 2i .pikc cvvoi0. M0pikoi ui12. opi1oi .ao20ov oi i/ioi .pikc cvvoi0, oae 0iv.i q cvvoi. 2ou 1q0iou, 2q p.q ka. Oi opi1oi, 2. .i2q.2. k.i oi koivc cvvoi0 .ao20ov 2q e1-1. 2q 00epi. k.i 0iv.i 2. 00ci. ookqpou 2ou cpou. 20 YKA 'LTOX 3o opi10i 2o 1q0io, 2q p.q, 2ov kko ka. o uk0i/q a.p.0c20i 2. ncv20 .i2qu.2., 0 2. oaoi. apo1a.00i v. .ao/0i0i 2qv a.pq 2ev 0e02pikev 000ev k.i v. /i.2uae10i 2i i/io2q20 aou couv, k.0e k.i 2i 1c10i aou /icaouv .u2u 2. 0c0q. cvvoi. 2ou .i2q.2o uaup0i k.i 12. cp. 2ou pi12o2cq k.i 2ou Hu2ev., oe k.vcv. i12opiko 2ev 0ai12qev /0v .3i1q2q10 ao2c o2i 2. .i2q.2. k.i q 10ipu 0 2qv oaoi. a.pou1iuov2.i 0iv.i .aok0i12iku cpo 2ou i/iou 2ou uk0i/q. L2o cpo 2ou u2oukou /0v uaup0i /i.epi1o 02. opi1ev, .i2qu2ev k.i koivev 0vvoiev. 0 .u2c oi apo2u-10i ovouov2.i .p. u2o 3.iv02.i o2i .koou0o1.v k.i ooi oi .0q.2ikoi k.i 3io1o3oi aou apoqq0qk.v 2ou uk0i/q. To npo2o .2q. c0i o2i ".ao /o 1pu0i. uovo ui. 0o00i. y.u-up /i/,02.i". u2o 1q.iv0i o2i /o 0u00i0 aou couv 2. i/i. ukp. q /o 1q0i. koivu 2.u2iov2.i q, 2o 1q.v2iko20po, /o 0u00i0 /0v aopov v. a0pik0iouv kuaoi. 0ai3uv0i.. u2o 2o 0a.v..uv0i e cv.2q koivq cvvoi.. q./q, 2o .i2q. /0v 0iv.i i. .aq a.p./oq, .u 1uv0au02.i k.i 2qv a.p./oq 2q .q00i. uev apo2u10ev aou apoka2ouv e oikq 1uvca0i. .ao 2o .i2q.. K.i 2i apo2u10i .u2c o uk0i/q 2i pq1ioaoi0i epi a.p.acp. .ao/0iq. To /020po .2q., aou c0i o2i "ko00 a0a0.1u/vp 0o00i. ao0-k20iv02.i 1ov0,c k.i 0o0oyouuc k.i ao 2. /o ok. 2p", 1uv0au0-2.i o2i i. 0u00i. aop0i v. apo0k2.00i ovo k.2u i. /i00uv1q, 2q /i00uv1q 2ev ukpev 2q k.i .u2o aop0i v. iv0i i. ku00 0u00i.. u2o o/q0i 12o 1uacp.1. o2i q apock2.1q 10 ku00 k.200uv1q 0iv.i o-v./ikq. a0i/q /o 0u00i0 /0v aop0i v. couv koivo 2q. k.i ku00 0u00i. apo0k20iv02.i 0a` ua0ipov, q 0ai3uv0i. 12qv uk0i/0i. +0e0-2pi. 0iv.i ua0ipq 10 ku00 k.200uv1q. To 2p2o .2q., "u0 ko00 k/v2o k.i ko00 .k2iv. uao0i v. y.-30i kko", /qev0i o2i q .k2iv. aop0i v. 0iv.i .a0ipe ikpq q .a0i-pe 0uq. u2o 1q.iv0i o2i o epo 0iv.i 1uv0q k.i ua0ipo. q-./q .ao 2o .i2q. .u2o apoka20i q 1vvc0t. 2ov opov, .3o q .-k2iv. aop0i v. 0iv.i o1o 0cou0 ikpq k.t 2o n0tpo 2ov opov, .-3o q .k2iv. 0iv.i o1o 0cou0 0uq. ou0 c21i 2qv cvvoi. 2q 1u-vc0i. k.i 2ou .a0ipou 2ou epou, 2i oaoi0 o uk0i/q pq1ioaoi0i epi u0 0a0qq10i. To 2c2.p2o .2q., "o0 oi o0/ ycvi0 0iv.i i10 u02. 2oo", 0iv.i 0.3pe /i.3op02iko .ao 2. apoqo0v., i.2i .ao 2q i. apo-ao0c20i 2qv a.pq 2q op0q evi. epi apoqoucve v. 2qv c0i opi10i k.i .ao 2qv uq /c02.i o2i q op0q evi. 0iv.i apok.0opi1cvo L+O+ 21 c00o, .3o aop0i v. pq1ioaoiq00i e c2po 1kpi1q uev e-viev. ,0ep0i 0ai1q o2i oi evi0 0iv.i 0u0p.0, /i.3op02iku i. evi. i1q 0 i. op0q /0v 0iv.i oae1/qao20 op0q evi., oae .ac/0i0 o Huaao (ca0 a.pup.3o 5). koq, i. v. 1ukpi0ov /o i10 e-vi0 apca0i v. 02.kivq0ov, 2ouui12ov q i., v. 2oao002q0ov q i. auve 12qv uq k.i v. 1uac1ouv. u2o apoao0c20i o2i 2. 1q.2. /0 02.uov2.i k.2u 2q 02.kivq1q 2ou k.i o2i uaup0i ooiocv0i. 2ou epou. v2i v. opi0i q v. uao0c20i o. .u2u o uk0i/q, 0aic0i v. a.p./000i ovo o2i oi op0c evi0 0iv.i i10 02. 2ou. O Hpoko c0i o2i .v a.p./00o0 o2i q apo2.1q .u2q 0iv.i .ao ovq 2q 3.v0pq k.i /0 p0iu02.i .ao/0iq, 2o20 /0v 0iv.i .i2q., .u .ie. (Hpo-ko, 188.1-3) To ncn2o .2q., ".v ui. 0o00i. 2/uv0i /o o0 k.i 1,pu.2i0i 2i 0v2o k.i 0ai 2. .o2o u/p ycvi0 v. /,oov o0oi1u. uiko20o .ao /o o0/, 2o20 oi 0o00i0 o2.v ao0k2.0ov 0a oa0iov 0. 1ov.v2p0ov ao 2o u/o aoo 1,pu.2iov2.i oi uiko200 2cv /o o0cv ycvi0", 0iv.i 2o .2qu. 2ov n.p.zzqzov, aou e ve12ov .a.1oq10 2o1o 2ou .p.iou qv0 .0q.2iko, o1o k.i 2ou v0o20pou k.i 2ou 1povou. v.3cp02.i 12i a.puq0, 0u00i0 0 2qv cvvoi. aou k.0opi10 o uk0i/q 12ov opi1o 23. .1ikq cvvoi. 2q a.p.qi-. i. 2ov uk0i/q 0iv.i o2i oi 0u00i0 0iv.i 1uv0aia0/0 k.i /0v 2cvov2.i. oi oe 0ai12qov0 k.0opiouv 2i a.puq0 0u00i0 0 2qv cvvoi. 2q "i1p .ao12.1p" q 2q "/i00ov1p". ae .v.3cp0i o Hpoko1, o Ho10i/evio k.i o +civo o Po/io 00epo1.v o2i /o 0u00i0 0iv.i a.puq0, o2.v oi .ao12u10i ku00 1q0iou 2q i. .ao 2qv uq 0iv.i i10. Tqv i/i. cvvoi. i. 2i a.pu-q0 0u00i0 0i0 k.i o Liaikio, oae .v.3cp0i o al-Nayrizi2. O -io.o 12. 1oi. aou cp.0 i. 2o cpo 2ou pi12o2cq ("v.o2iko Ho20." , 16.65 . 4) ovou0i a.puq0 /o 0u00i0 aou couv 2qv i/i. /i00uv1q. To aca2o .i2q. c0i .a.1oq10i aoo 0ai12qov0 .ao 2qv .p.io2q2. cpi 2ov a0p.1cvo .iev.. Hooi ai120u.v o2i 2o .i2q. .u2o 0. aopo10 v. .ao/0i00i k.i 0aocve q uk0i/0i. +0e02pi. 0. aopo10 v. 000ie00i .ie.2iku epi .u2o. O H2o0.io q2.v 1 G. Friedlein, "Procli Diadochi in Primum Euclidis Elementorum Librum Commentarii", Leipzig 1883, 10i/. 176.7-10. 2 M. Curtze, "Das angebliche werkdes Eukleides berdie Waage", Zeitschrifte fr Ma-thematik und Physik, 19 (1874) 10. 25.8-27.14. 22 YKA 'LTOX o ape2o aou apo1au0q10 v. 2o .ao/0i0i, .u epi .ao2c01.1. O Hpoko, oae .v.3cp0i2, apo1au0q10 o i/io v. .ao/0i0i 2o ok0i-/0io .i2pu., epi oe 0ai2ui.. Oi apo1au00i0 q2.v aoc k.i .-kpopov0 0 2o i/io auv2. .pvq2iko .ao2c01.. O Saccheri 12o cpo 2ou "Euclides ab omni naevo Vindicatus" (1773) apo1a.00i v. /i0p0u-vq10i aoi. 0. q2.v q +0e02pi. epi 2o ok0i/0io .i2pu.. -.iv02.i v. 0i0 1uv0i/q2oaoiq10i o2i /0v aopo10 v. uaup0i uq +0e02pi. /i-a. 10 .u2q aou 000ie10 o uk0i/q. ae 0iv.i ve12o, oi apo1au-00i0 1uv0i12qk.v kupie k.2u 2o 18o k.i 19o .iev. .ao /iu3opou .iooou .0q.2iko 2q 0aoq, k.i o/qq1.v 12q /qioupi. u-ev up-0ok0i/0icv +0cu02icv. O Lobachevski (1826), o Bolyai (1832) k.i o Riemann (1854) q2.v 0pikoi .ao 2ou /qioupo 2ev up-0ok0i/0icv +0cu02icv. Oi uoiv vvoi0 couv 0viko20pq 03.poq. 0v 03.poov2.i /q./q ovo 12q +0e02pi., .u k.i 10 u0 0ai12q0, oae a.. 12i 3u1ikc 0ai12q0. Oi koivc cvvoi0 i. a.pu/0i., "o2.v /o u0-y/0p 0iv.i i1. u0 2i2o u/y00o, 2o20 0iv.i k.i u02. 2oo i1." q "o2.v 10 i1. ao10/1oou0 i1. aoka2oov i1." ka. 0iv.i cvvoi0 aou i1ouv 0viku acp.v 2q +0e02pi.. 1e .u2o 0iv.i k.i o oo aou ovou12qk.v "koiv/ /vvoi0". L2. l2oi,0i. 0v2oaiou0 koivc cvvoi0 oi oaoi0 0iv.i 1uvca0i. kuaoiev .i2qu2ev. H.. q koivq cvvoi. "/o 0o00i0 /0v a0ik0ioov 0ai3ov0i.", 0iv.i 1uvca0i. 2ou .i2q.2o 1, oae .v.3cp.0 a.p.auve. O Hpoko, o oaoio .koou00i ai12u 2qv opooi. 2ou uk0i/q, /i.-3opoaoi0i2.i ovo e apo 2o 1q0io .u2o. q./q, .v2i 2ou opou "uoi-v vvoi0" pq1ioaoi0i 2ov opo ".iou.2." k.i kuv0i /iukpi1q 0-2. .i2q.2o k.i .ie.2o. K.2u 2ov Hpoko, 2o koivo .p.-k2qpi12iko 02. .i2q.2o k.i .ie.2o 0iv.i o2i k.i 2. /o /0 p0i-uov2.i .ao/0iq, .u 00epov2.i e ve12u k.i pq1ioaoiov2.i e 1q0i. 0kkivq1q 12q /i./ik.1i. .ao/0iq. i.3cpouv 02. 2ou oae 2. 00epq.2. 0 2. apoq.2.. T. .ie.2. 00epov e /0/o-cv. apu.2. aou 0iv.i u01. apo3.vq k.i 0ko. k.2.voq2u k.i .ao 2ov q 0i/iko. ve 12o .i2q. .ao/0o.120 ku2i 2o oaoio aop0i 0ko-. v. .ao/0i00i, v. 0aivoq00i q v. k.2.1k0u.120i, epi v. .a.i20i2.i i/i.i20po koao 1kcq i' .u2q 2qv .ao/oq. 1 G. Friedlein, "Procli Diadochi in Primum Euclidis Elementorum Librum Commenta-rii", Leipzig 1883, 10i/0 361. 5 367. 27. 2 G. Friedlein, o.a. 10i/0 368. 24 373. 2 L+O+ 23 K.2u 2ov pi12o2cq ".2qu." 0iv.i i. apo2.1q aou 0ai/c02.i .ao/0iq, .u 00i 2q 00epo0 e /0/ocvq epi .ao/0iq ("v.o-2iko 120.", 76b 32-34). O Hpoko 12o cpo 2ou "l,oi. 12o ac2o iio 2cv l2oi,0icv" 1ua0pi.uv0i ovo acv20 .ie.2. (Hpoko, 10. 193.10-14): Hoyu.2. aoo 0iv.i i1. u0 2o i/io aoyu. 0iv.i k.i u02. 2oo i1.. v i1. ao1200ov 10 i1. aoka2oov i1.. v i1. .3.i00ov .ao i1., 2. oaooia. 0iv.i i1.. To oo 0iv.i u0y.20o .ao 2o u/o 2oo k.i Hoyu.2. aoo 2.o2iov2.i 0iv.i i1. u02. 2oo L2i /iu3op0 0k/o10i k.i 02.3pu10i 2ev l2oi,0icv o .pi0o 2ev koivev 0vvoiev (.ieu2ev) /i.3cp0i. H.. 12q 02u3p.1q 2ou Barocius a0picov2.i 10 .ie.2., 0ve 12o cpo 2ou Heiberg uaupouv 9 .ie.2., 2. oaoi. ovou0i koiv/ /vvoi0. e o Heiberg 2. 2c110-p. .ao 2. 9 .ie.2. 2. 00ep0i o2i 0iv.i v0o20p0 apo10qk0 k.i 0aocve 00ep0i e vq1i. .ie.2. 2ou uk0i/q ovo 2. acv20 aou a0picov2.i 12o iio 2ou Hpokou. Oi npo210i 0iv.i k.2.v0qcv0 10 o. 2. ii., .vuo. 0 2o a0pi0o0vo 2ou. Oi a0pi11o20p0 apo2u10i, 115 10 .pi0o, a0pic-ov2.i 12o iio X, aou .v.3cp02.i 12ou .102pou .pi0o. 0-20po 10 .pi0o apo2u10ev 0iv.i 2o iio , 2o oaoio a0pic0i 48 apo2u-10i, k.i .koou0ov 2o iio VII (ape2o iio 2q 00epi. .pi0ev), 2o oaoio a0pic0i 39 apo2u10i k.i 2o iio X (ape2o iio 2q L20-p0o02pi.), 2o oaoio a0pic0i 0ai1q 39 apo2u10i. Ku00 apo2.1q aou 0iv.i aqpq, /q./q aou c0i o. 2. cpq 2q, .koou00i 2. 0q q.2.: Tq /i.2ac1p (0k3cvp1p) 2q apo2.1q. L2q /i.2ae1q /0v uaup-ouv pu.2. q 1o.. Tqv /k001p. q./q 2qv .vuu1q 2ev /0/ocvev k.i 2qv .v.a.pu-12.1q 2q apo2.1q 0 1uk0kpicvo 1q., 2o oaoio c0i pu.2. 2ou 0qviko .3uq2ou. Tou .a.p.i2q2ou ao1/ioi1uo. Tqv k.2.1k0op, oaou .u2q 0iv.i .v.k.i.. k.2.1k0uq a0pic0i k.i 2i .a.p.i2q20 apo10qk0 12o 1q.. Tqv .ao/0ip q 0ai/0ip k.i To 1oua/.1u.. L0 kuaoi0 apo2u10i aop0i v. qv uaupouv o. .u2u 2. q.-2., 0a0i/q /0 p0iuov2.i, oae a.. oi opi1oi q q k.2.1k0uq. e q /i.2ae1q 2q apo2.1q, q .vuu1q 2ev /0/ocvev, q .ao/0iq k.i 2o 24 YKA 'LTOX 1uacp.1. uaupouv auv2o20 10 ku00 apo2.1q. L2o 2co 2ou 1ua0-pu1.2o, 2.v npk0t2.t yt. .n/0tq, .v.pu302.i q apo2.1q ]B (.f./.), aou 1q.iv0i o2i 0k0ivo aou cap0a0 v. .ao/0i00i .ao/0i0qk0, 0ve 2.v npk0t2.t yt. k.2.1k0vq, .v.pu302.i q cv/0i-q ] B (.f.n.), /q./q 0k0ivo aou cap0a0 v. k.2u-1k0u.120i k.2.1k0uu12qk0. L2. l2oi,0i. 0. 1uv.v2q1ou0 k.i o0 10/ov 2i 00o/ou .ao-/0iq aou pq1ioaoiov2.i 1q0p.. ,. po0 a.. 2q u0o/o 2q 1ov0n.,o,q, 2q 1ov002iuq u0o/o, 2qv .v.zo2iuq u0o/o, 2q c0o/o 2q 0i 2onov .n.,o,q, k.0e k.i 2q c0o/o 2q 2z0i. 0n.,o,q. u/0o/o 2p 1ov0a.ycyp pq1ioaoi0i2.i ao 1uvu 12qv .-ao/0iq 2ev apo2u10ev. O uk0i/q pq1ioaoi0i 1uvq0e 2q 1ov002ikp u/0o/o. E0kivu .ao 2. /0/ocv., 2. ve12u, k.i 0 2q oq00i. 2ev opi1ev, 2ev .i2qu2ev k.i 2ev .ieu2ev, k.0e k.i 2ev apo2u10ev 2ev oaoiev q .ao/0iq c0i apoqq00i, .koou00i i. 0a.eikq /i./ik.1i., e1aou v. k.2.q0i 12o q2o0vo .ao2c01.. Tqv .v.o2ikp u00o/o 2q pq1ioaoio10 1a.vio20p.. e, 13ev. 0 2ov Huaao (lov.ycyp, VII, 0i1.eq 1-3), o uk0i/q q2.v ao 0ai/cio 12q pq1ioaoiq1q 2q .v.o2ikp u00o/oo, .3o 2q pq1i-oaoi0i ao 1uvu 12. u. cp. 2ou, 2. "0/ou/v.", 2. "Hoi1u.2." k.i 12o "Toaoi ao 0ai3ov0i0". u2u 1ua0pi.uvov2.i 12q 1uoq .0q.2ikev cpev aou 0iv.i ve12q e ",p1.ooi 2p voo1p", oae 0. /o0 a.p.ku2e. L2. l2oi,0i. iv02.i 1uvq pq1q 2q 00o/ou 2q "0i o2oaov .-a.ycyp", k.0e k.i pq1q 2q 00o/ou 2q 2/0i. 0a.ycyp. c0o-/o 2q 2/0i. 0a.ycyp ape2opq1ioaoiq0qk0 .ao 2ou Hu0.op0i-ou1 k.i 2q pi1kou0 12. cp. 2ou pi12o2cq k.i 2ou piq/q. L2. l2oi,0i. pq1ioaoi0i2.i i. 2qv .ao/0iq .pk02ev apo2u10ev, oae i. a.pu/0i. 2q apo2.1q 14 2ou iiou VII, 2q apo2.1q 13 2ou i-iou VIII k.i 2q apo2.1q 20 2ou iiou IX. To iio a0pic0i 23 opi1o, 5 .i2q.2., 9 koivc cvvoi0 k.i 48 apo2u10i (00epq.2.). t2q.2. k.t kotvc cvvot0 vnpovv -vo 12o npo2o to k.t .no20ov 2t 10t 2q .te.2tkq 000-e1q 2q vk0/0t. +0e02p.. 1 T. G. Exarchakos, "The Method of Mathematical Induction and the Pell Equation in ancient Greek Mathematics", Mentor 1 (1999), 10. 4-16. L+O+ 25 Oi 48 apo2u10i aopov v. epi12ov 12i 0q k.2qopi0: Oi apo2u10i 1 cpi 26 .ao20ov 2q +0e02pi. 2ev 1q0iev, 2ev p.ev, 2ev eviev k.i 2ev 2pievev. Oi apo2u10i .u2c, .i 0 2i apo2u10i 27 k.i 28 .ao20ov 2q 0o0vq oo//20p (aoo2p) +0cu02i.. q./q 2q +0e02pi. aou opiouv 2. 2c1-10p. ape2. .i2q.2. 2ev l2oi,0icv. apo2.1q 4 aou .ao/i/02.i .ao 2ov /qo 12o ,.q, 12. a0pi11o20p. 1pov. .ie.2i-ku 1u12q.2. 00ep0i2.i .ie.. L2o .ie.2iko 112q. 2ou Hilbert q apo2.1q 4 03.vi02.i e .ie. .5 k.i o Hilbert .-ao/0ikv0i o2i 2o .ie. .u2o 0iv.i .v0up2q2o .ao 2. u.. L2qv .ao/0iq 2q apo2.1q 6 iv02.i i. ape2q 3opu 12. l2oi-,0i. q pq1q 2q 00o/ou 2q 0i o2oaov .a.ycyp. Oi apo2u10i 27 cpi 34 .ao20ov 2q +0cu02i. 2cv a..-pcv. i0i v. 0ai1quvou0 0/e o2i i. 2qv .ao/0iq 2q apo-2.1q 29 0iv.i .a.p.i2q2q q pq1ioaoiq1q 2ou 5ou .i2q.2o (2ou .i2q.2o 2ev a.p.qev 2ou uk0i/q) k.i o2i q apo2.1q .u2q 0iv.i q ape2q apo2.1q 2ev l2oi,0icv aou i. 2qv .ao/0iq 2q iv02.i pq1q 2ou 5ou .i2q.2o. +i. 2qv .ao/0iq 2q apo2.-1q 28 /0v 0iv.i .v.k.i. q pq1q 2ou 5ou .i2q.2o. iv.i oe .v.k.i. q pq1q 2ou 1ou .i2q.2o, o2i /q./q "/o 1pu0i. oioov 2p 0/1p ui. uovo 0o00i.". aocve, q apo2.1q .u2q aop0i v. 0v2.00i k.i 12qv .aou2q +0e02pi. k.i .ko. i10i k.i i. 2i q-0uk0i/0i0 +0e02pi0 2ou Lobatshevski k.i 2ou Riemann. koq apca0i v. 0ai1quvou0 o2i 12qv apo2.1q 32, 2o cpo aou .v.3cp0i o2i "2o o0oi1u. 2cv ycvicv 2iycvoo 0iv.i /o o0/" 0iv.i 0aivoq1q 2ev Hu0.op0iev, oi oaoioi, 13ev. 0 2ov Hpoko, q2.v oi ape2oi aou 2o .ac/0i.v. Oi apo2u10i 35 cpi 48 .v.3cpov2.i 12q 00epi. 2ev 0./ev, aou 0. ookqpe00i 12. 0ao0v. ii.. O uk0i/q /iv0i 2qv cvvoi. 2ou 0./o pq1ioaoiev2. 2ov opo "a0i0,ou0vo 1,p-u.2o". Oi apo2u10i .u2c .v.3cpov2.i 12. 0./u 2pievev k.i a.p.qopuev. apo2.1q 47 0iv.i 2o Hu0.op0io 00e-pq., 0ve q apo2.1q 48 0iv.i q .v2i12po3q apo2.1q 2q 47. To iio 0iv.i 2o ikpo20po iio 2ev l2oi,0icv, .u 0iv.i ao 1q.v2iko, i.2i 0i1u0i 2q 0e02pikq u0p. 2ev qvev k.i 26 YKA 'LTOX opi0i cv. vco 0e02piko 1q., 2o ,vouov., aou 0i0 0uq 03.po-q i. 2q 1q 0e02pikev apo2u10ev 12qv .p.io2q2.. H0pic0i 2 opi1o k.i 14 apo2u10i. Oi /o opi1oi k.0opiouv 2qv cvvoi. 2ou op0oeviou a.p.qopuou k.i 2ou veov.. O opi-1o 2ou veov., k.2u 2o iocvq 2o A.cp2io, o30i02.i 12ov v.i-.v/po. Oi apo2u10i 2ou iiou 0iv.i 03.poc 12i apo2u10i 44 k.i 45 2ou iiou k.i .pk02c .v.3cpov2.i 10 02.1q.2i1o 0ai3.v0iev. M0pikc .ao 2i apo2u10i .u2c .v.3cpov2.i 12q 0e02pikq 1q 0i1e10ev /020pou .0o, 12q 1q 0i1e10ev aou .po20p. ovou12qk.v 0i1c10i Pell, 12ou a0upiko k.i /i.-02piko .pi0o, 12qv 0ai0.ie1q .0pikev 1c10ev, 12q 0vik0u1q 2ou Hu0.op0iou 00epq.2o. Oi apo2u10i 5, 6, 11 k.i 14 i. a.pu/0i., .v.3cpov2.i ou1i.12iku 12q 1q 2ev 0i1e10ev .x - x2 2, .x + x2 2, x2 .x .2 k.i x2 . .v2i12oi.. i/iku q apo2.1q 11 .3opu 12o npzqu. 2q po1q 2ouq, 2o oaoio .ao/i/02.i 12ou Hu0.op0iou. To apoq. .u2o .a.i20i "v. /i.ip000 0o0,p.uuo 2uqu. 10 u1o u.i upo z,o". v . 0iv.i 2o /o0cv 0u0p.o 2q. k.i x, . x, 2. 2q.2. 12. oaoi. /i.ipc0qk0, 2o20 0. i10i q 1c1q x - .xx.= q x2 .(. - x) q x2 .x .2. aocve, 2o apoq. ou1i.12iku .v.q2u 2q 1q 2q /0u20-pou0i. 0i1e1q x2 .x .2. Oi apo2u10i 9 k.i 10 0iv.i ao 1q.v2ikc, i.2i couv aoc 03.poc k.i kupie i.2i aopov v. pq1ioaoiq0ov 12q 1q io-3.v2ikev 0i1e10ev 2q 0vikq op3q x2 .y2 1 (1) aou 0iv.i ve12c e 0i1o10i 2oo Pell. apo2.1q 9 c0i o2i ".v + 0v.i 2o u1o 0o0o,puuoo 2uqu.2o u.i 1qu0o 2oo 2uqu.2o (L. 1), 220 i10i q 11q 2 2 2(+2 +2) (2) L+O+ 27 l,pu. 1 v 0c1ou0 + x k.i y, 0. cou0 + = + = + + = x + y. + + x + y + x = 2x + y k.i 0aocve q 1c1q (2) iv02.i (2x + y)2 + y2 = 2x2 + 2(x+y)2 (3) Tqv apo2.1q 3, k.2u 2ov Hpoko, 2qv 0i.v .ao/0i0i oi Hu0.-op0ioi, 0 2qv 0q /i.2ae1q: "v . u.i / 0v.i q nz0op u.i q /i.,ovio 0v 202p.,o-voo, 220 . + / u.i 2. + / 0v.i q nz0op u.i q /i.,ovio 0v z-zoo 202p.,ovoo". q./q, oi Hu0.op0ioi 0i.v .ao/0i0i 2q 1c1q1 (2. /)2 /2 2.2 2(./)2 aou 0iv.i q 1c1q (3). M0 2q pq1q 2q 1c1q .u2q oi Hu0.op0ioi uaooi1.v 2ou a0upiko k.i 2ou /i.02piko .pi0o, pq1io-aoiev2. i. ape2q 3opu 12qv i12opi. 2ev M.0q.2ikev 2q c0o/o 2q 2/0i. 0a.ycyp2. ao 2q 1c1q (3) apoka20i q 1c1q (2x + y)2 2(x + y)2 = 2x2 y2 ao 2qv 200u2.i. .u2q 1c1q a.p.2qpo0 o2i: v 2x2 y2 1, 2o20 2(x+y)2 (2x+y)2 = 1 k.i .v 2x2 y2= 1, 2o20 2(x+y)2 (2x+y)2 = 1. q./q, .v 2. x, y 0iv.i pi0 i. .ao 2i 0i1e10i y2 = 2x2 1 2o20, 2. x + y k.i 2x + y 0iv.i pi0 i. .ao 2i 0i1e10i (2x+y)2 = 2(x+y)2 1 aou 0iv.i ou1i.12iku q apo2.1q 2ev Hu0.op0iev. 1 T. G. Exarchakos, "The method of mathematical induction and the Pell equation in .ncient Greek Mathematics", Mentor 1 (1999), 10. 4-16. 2 B. L. Van der Waerden, "Geometry and Algebra in ancient civilizations", Springer Verlag, Berlin, Heidelberg, New York, Tokyo (1983), 10.136. 28 YKA 'LTOX ao 2qv apo2.1q 10 aopo0 v. cou0 .vuo. 1ua0pu1.2.. Hp..2iku, q apo2.1q 10 c0i: v + 0v.i 2o u1o 0o0,p.uuoo 2uqu.2o u.i 0v.i 1qu0o 12qv npou2.1q 2oo (L. 2), 220 0. i10i q 11q 2 2 2(+2 +2) l,pu. 2 v 0c1ou0 + x k.i y, 2o20 + + + x y k.i 2(xy) + y = 2xy. aocve, q a.p.auve 1c1q /iv0i (2xy)2 + y2 = 2x2 + 2(xy)2 aou 0iv.i .vuoq 2q 1c1q (3). ao 2q 1c1q .u2q aopo0 v. 1ua0puvou0 o2i .v 2. x, y 0iv.i pi0 i. .ao 2i 0i1e10i y2 = 2x2 1, 2o20 2. x y k.i 2x y 0iv.i pi0 i. .ao 2i 0i1e10i (2x y)2 = 2(x y)2 1 aou 0iv.i 0ai1q q apo2.1q 2ev Hu0.op0iev. ao o. 2. a.p.auve apoka20i o2i oi apo2u10i 9 k.i 10 0iv.i i1o/v.0 0 2q 1q io3.v2ikev 0i1e10ev 2q op3q y2 = 2x2 1 k.i aopo0 v. .ao/e1ou0 2i apo2u10i .u2c 12ou Hu0.op0iou. Oi apo2u10i 12 k.i 13 0iv.i k.i oi /o i1o/v.0 0 2qv apo2.1q: 20 u00 2p,ovo + u0 nz0op ., , , .v212oi. i10i q 11q: .2 2 2 2 1uv To iio .v.3cp02.i 12ov kko k.i 12. 12oi0i. 2ou, 12i 1c10i 0u00i. k.i kkou k.i 12i 1c10i aou aopov v. couv oi kkoi 02. 2ou. H0pic0i 11 opi1o k.i 37 apo2u10i. v/i.3cpov L+O+ 29 a.pou1iu0i q apo2.1q 16, 12qv oaoi. 0i1u02.i i. ape2q 3opu q cv-voi. 2q k0.2o0i/o ycvi., /q./q i. 0ik2q evi., aou c0i 2q i. a0upu 2q 0u00i. k.i 2q /020pq 2oo kkou. IV To iio .u2o .ao200i2.i .ao 7 opi1o k.i 16 apo2u10i, oi oaoi0 .v.3cpov2.i 12qv 0p.3q k.i a0pip.3q 10 kko 0u0upuev 1qu2ev, k.0e k.i 12qv k.2.1k0uq k.vovikev aouevev 00p.-cvev q a0pi0p.cvev 10 kko 0 a0upc 3, 4, 5, 6 k.i 15 .v2i12oi.. L3ev. 0 2ov Hpoko (10i/. 269.8-21), o uk0i/q .ac/0i0 2qv apo2.1q 16 2ou iiou .u2o 0 1koao v. 2q pq1ioaoiq10i 12qv 12povoi.. K.2u 2ov Hpoko, .v aupou0 cv.v kko aou a0pvu .ao 2ou aoou k.i 1' .u2ov 0puou0 k.voviko /0k.a0v2uevo (apo2.-1q 16), 2o20 0. vepiou0 2i .ao12u10i 2ev aoev .ao 2ov 1q0pivo k.i .ao 2ou e/i.ko kkou, .3o vepio0 o2i oi .ao12u10i 02. 1q0pivo k.i e/i.kev kkev 0iv.i i1q 0 2qv a0upu k.vo-viko /0k.a0v2.evou. 2.v cv. k.voviko aoevo 0 v a0upc (v 3) 0iv.i k.2u-1k0uu1io, 2o20 0 2q oq00i. 2q apo2.1q 30 2ou iiou aopo-0 v. k.2.1k0uu1ou0 k.voviko aoevo 0 2v, 4v, 8v k.i 0vike 0 2k v a0upc 0 kN. ai1q, .v cv. v-evo 0iv.i k.2.1k0uu1io (v 5) k.i 0iv.i cv. 3u1iko .pi0o /i.ipc2q 2ou v, 2o20 0 2q oq00i. 2q apo2.1q 10 2ou iiou VI aopo0 v. k.2.1k0uu1ou0 k.voviko aoevo 0 a0upc, i. ku00 3u1iko .pi0o , 0 3. ao 2qv 200u2.i. .u2q 1c1q k.i 0 2q oq00i. 2q apo2.1q .v ./, u.i /, u.i (.,) 1, 220 ./,, aopo0 v. k.2.1k0uu1ou0 k.voviku aoev. 0 v a0upc, o2.v 2. k.voviku aoev. 0 k.i v a0upc 0iv.i k.2.1k0uu1i. k.i (, v) 1. L2o iio .u2o couv k.2.1k0u.120i 2. k.voviku v-ev. 0 v 3, 4, 5, 6, 15. p. aopo0 v. k.2.1k0uu1ou0 k.i o. 2. k.vovi-ku -ev. 0 2v, 32v, 5 2v, 3 52v. O Heiberg 12o cpo 2ou "Euclidis Opera Omnia" (Loi. 12o iio IV, 1oio 4 (2oo V, 10.273.13-15)), .ao/i/0i k.i 2i 16 apo-2u10i .u2o 2ou iiou 12ou Hu0.op0iou. 30 YKA 'LTOX V iv.i 2o aio 1q.v2iko k.i 2o a0pi11o20po pq1ioaoiqcvo iio .ao o. 2. ii. 2ev l2oi,0icv. H0pic0i 18 opi1o k.i 25 apo2u10i. oop3iu k.i q .i. 2ou iiou .u2o pi1k02.i 12ou opi-1o. O opi1o 3 2ou oou /o 000ev 0iv.i ao 0u3uq k.i /0i-v0i ao1q /u1koi. uaup0i v. opi10i cv. oo, 0ve 0iv.i 0koo v. 2ov k.2.voq10i. O opi1o 4 aou c0i o2i "o u0y/0p /y02.i o2i /,oov oyo u02. 2oo, o2.v ao.a.1i.ou0v. k.2op. uaoou0 v. a02,oou0 c120 2o /v. v. oa0/,0i 2oo ooo" .aok0i0i 2o .a0ipe 0uo k.i 2o .a0ipe ikpo k.i 0iv.i ou1i.12iku i1o/v.o 0 0k0ivo aou 0iv.i 1q0p. ve-12o e .ou. 2oo piuq/q. ai1q, o opi1o 5 k.2u 2ov oaoio "T/110. u0y/0p /u0 o2i /-,oov 2ov i/io oyo 2o ac2o ao 2o /020o k.i 2o 2i2o ao 2o 2/2.2o, o2.v i10 3o/ ao.ao1i. 2oo ac2oo k.i 2oo 2i2oo 10 1,/1p u0 2. i10 3o/ ao.ao1i. 2oo /020oo k.i 2oo 2/2.2oo, u0 oaoiov/pao20 ao.a.1i.1uo, 0iv.i u0y.20. p i1. p uiko20. o2.v .o2o p30ov 10 .v2i12oi,p 2op", c0i k.2.aq0i 2q .0q.2ikq 1kcq. u2o o opi1o 0iv.i k.2.aqk2iko 12q 1qq 2ou k.i i10i i. 102p. k.i .102p. 0c0q. M0 1povo 1uoi1o o opi1o .u2o /i.2uaev02.i e 0q: "v ., , y, / 0iv.i 2/110. u0y/0p, 2o20 =., .v k.i uovo .v yi. oaoioo1/pao20 3o1iko .i0uo u k.i v i1,0i ui. .ao 2i 1,/10i. . > v k.i 1oy,ovc > v/ . v k.i 1oy,ovc v/ . v k.i 1oy,ovc v/". O opi1o .u2o 0iv.i 3.v0po o2i 0iv.i .v.k.i. k.i ik.vq 1uv0q-kq, e120 2c110p. 0c0q v. couv 2ov i/io oo. O De Morgan1 0q0i 2ov opi1o 5 e 0q: "T/110. u0y/0p, k.i 2oo i/ioo 0i/oo k.i C k.i D 2oo i/ioo p ooo 0i/oo, /y02.i o2i /,oov 2ov i/io oyo, o2.v o. 2. ao.ao1i. 2oo uaoov v. k.2.v/-uov2.i u02. 2cv ao.a.1icv 2oo u0 2o i/io /io12pu., oac 2. .-v2i12oi,. ao.ao1i. 2oo C u02. .o2cv 2oo D". q./q, .v k.i v 0iv.i 3u1ikoi .pi0oi k.i .v pi1k02.i 02. v k.i (v1), 2o20 2o C pi1k02.i 02. vD k.i (v1)D. 1 A. De Morgan, "Proportion", The Penny Cyclopaedia XIX, p.51 L+O+ 31 O K. Weierstrass1 pq1ioaoiq10 .u2ov 2ov opi1o i. v. opi10i 2qv i1o2q2. 02. .pi0ev. O Heath2 c/0i0 ae o opi1o .u2o o/q0i 12o /i.0pi1o 2ev .pi0ev 10 ku10i, aou 0iv.i i/i0 0 2i 2oc Dedekind. koou0ov aooi .ko. opi1oi, oi oaoioi .v.3cpov2.i 12. /iu3op. 0i/q .v.oiev k.i 12i i/io2q2c 2ou. Oi 25 apo2u10i aou .koou0ov 0iv.i o0 03.poq 2ev opi-1ev 10 1uk0kpicv. .v2ik0i0v.. O Barrow3 .v.30po0vo 12o iio V 2ev l2oi,0icv c0i o2i /0v uaup0i aio 0u3uq .v.kuuq. Tiao2. aio 12.00po, aio .kpic k.i aio k.2.aqk2iko /0v c0i cpi 2ep. a.pou1i.120i .ao 2q 00epi. 2ev .v.-oiev aou uaup0i 12o iio V 2ev l2oi,0icv. O Cayley c0i o2i "ao /1ko. 0. 000i ko2i 12. M.0pu.2iko aio ouo3o .ao .o2o 2o yop20o2iko 5o iio 2cv l2oi,0icv". L3ev. 0 cv.v .vevuo 1oi.12q 2ou iiou V, 0pikoi cv0 o2i 2o iio .u2o 0iv.i cpo 2ou i/iou 2ou uk0i/q4. u uo .ve-vuo 0ai1q 1oi.12q c0i o2i 2o iio cap0a0 v. .ao/o00i 12ov -/oo5 k.i apo10c20i o2i q .ao/o1q 2ou cpou 12ov uk0i/q /0v 0iv.i u0o, 0 2qv cvvoi. o2i q 2.k2oaoiq1q 2ev apo2u10ev k.i q i0pupq1q 2ou .v.vepi02.i .ao oou o2i 0iv.i /ou0iu 2ou uk0i/q. VI To a0pi0o0vo .u2o 2ou iiou 0iv.i 0ai1q ao 1q.v2iko. H0-pic0i 5 opi1o k.i 33 apo2u10i. M0pikc .ao 2i apo2u10i .u2c 0iv.i .v.k.i0 i. 2q 0c2q 2ev .102pev .pi0ev aou 002uov2.i 12o i-io X. M0pikc .ao 2i apo2u10i, k.i kupie oi apo2u10i 27-29, couv 0uq 1q.1i., i.2i .ao20ov 2q u1q i. 2q 0c2q 2ev kevikev 2o-ev. apo2.1q 1 k.i q apo2.1q 33 /0ivouv 2o ao1o 1q.v2iko 0iv.i o opi1o 5 2ou iiou V, .3o 12qv apo2.1q 1 "2. 2iycv. k.i 2. a..p-oy.uu. aoo /,oov i1. ,p /,oov oyo 0u./cv i1o u0 2o oyo 2cv o10-cv", iv02.i oo i. 2qv i1o2q2. 2ou oou 2ev 0./ev 0 oo qkev 1 H. G. Zeuthen, "Lehre von den Kegelschnitten im Altertum", 10.2 2 T. L. Heath, "The Thirteen Books of Euclids Elements", I, 124-126. 3 Isaac Barrow, "Lectiones habitae in scholiis publicis academia cantabrigiensis". Cam-bridge (1684), 10.336. 4 J. L. Heiberg and H. Menge, "Euclidis Opera Omnia", 1oio 1 12o iio V, 10i/. 280.7-9. 5 J. L. Heiberg and H. Menge o.a., 1oio 3 12o iio V, 10i/. 282.13-20. 32 YKA 'LTOX (u10ev), aou epi 2ov opi1o /0 0. 0i0 cvvoi.. ai1q 12qv apo2.1q 33, "10 i1oo kkoo oi 0aik0v20 p oi 0yy0y.uu/v0 ycvi0 0iv.i oac 2. .-v2i12oi,. 2o. 12. oaoi. .ivoov", iv02.i oo i. 2qv i1o2q2. 2ou oou eviev 0 2o oo 2ev .v2i12oiev 2oev. L2qv apo2.1q 27 a0pic02.i i. ape2q 3opu 12qv i12opi. 2ev M.0q.2ikev q cvvoi. 2ou u,i12oo. L2q 1povq p.3q q apo2.1q .u2q 0iv.i i1o/v.q 0 2qv apo2.1q: N. 000i p u/yi12p 2iup 2oo yivou/voo x(.-x). a0i/q x . x . 12.00po, 2o ivo0vo iv02.i ci12o o2.v x . x q x = 21.. apo2.1q 31, "12. o0oycvi. 2iycv., 2o aoycvo aoo yo302.i u0 a0oo 2pv oao-20ivoo1., /,0i 0u./ov i1o ao 2o o0oi1u. 2cv 0u./cv 2cv aooycvcv aoo yo3ov2.i aovc 12i ko0020 a0o/ 2oo 2iycvoo k.i 0iv.i ouoi. k.i ouoic k0iu0v. u0 2o aoycvo aoo yo302.i aovc 12pv oao20ivoo1.", 0iv.i i. 0.uu1i. 0vik0u1q 2ou Hu0.op0iou 00epq.2o. L3ev. 0 2ov Hpoko1, 2o Hu0.op0io 00epq. 2o .v.-kuu.v oi Hu0.op0ioi, .u o uk0i/q c/e10 i. y0vtkq k.t 0k-0.pq .n/0tq, /i.2ae10 2o .v2i12po3o 00epq. k.i 2. 0vik0ucv. 00epq.2. 2ou Hu0.op0iou 00epq.2o k.i c/e10 i. vc. .ao/0iq 2ou Hu0.op0iou 00epq.2o. O uk0i/q 0. aopo10 v. .ao/0i0i 0ko. 2o Hu0.op0io 00epq. pq1ioaoiev2. 2qv apo2.1q 8. e apo2iq10 v. 2o .ao-/0i0i 0 12oi0i. 2ou iiou , epi v. 0ac0i 12qv .ao/0iq .v.o-i0 k.i ooio2q20 0u0upuev 1qu2ev. VII To iio VII .pi0i 0 23 opi1o k.i 1uv0i0i 0 2q /i.2ae1q k.i .ao/0iq 39 apo2u10ev. /e .v.a21102.i q 00epi. 2ev .v.oiev 0 .pi0o, q oaoi. 1uv0i02.i k.i 12. 0ao0v. .pi0q2iku ii.. 00epi. 2ev .v.oiev aou a.pou1iu12qk0 12o iio V, 03.po02.i 10 0e02piku 0c0q 12o iio VI, k.i 10 .pi0o 12o iio VII. O opi1o 21, "2/110-i .i0uoi .ao20ov .v.oyi. o2.v o ac2o 0iv.i 2o i/io ao.ao1io p 2o i/io u/o p 2. i/i. u/p 2oo /020oo, oac 0iv.i o 2i2o 2oo 202o2oo" 3.i-v02.i v. c0i 03.poq ovo i. 102p. 0c0q k.i 0aocve /0v 03.po- 1 Hpoko, "l,oi. aovc 12o ac2o iio 2cv l2oi,0icv", 10. 426. L+O+ 33 02.i q 0vikq 00epi. 2ev .v.oiev oae .u2q .v.a20qk0 12o iio V. e .u2o /0v i10i, i.2i o opi1o 21 10 1uv/u.1o k.i 0 2qv apo2.1q 19 0iv.i i1o/v.o 0 2ov opi1o 5 2ou iiou V, o2.v 03.po02.i 10 .-pi0o. aocve k.i 12. .pi0q2iku ii. 2ev l2oi,0icv aopov v. pq1ioaoiq0ov o0 oi apo2u10i 2ou iiou V epi i/i.i20po apoq.. apo2.1q 1 0iv.i ao 1q.v2ikq k.i 0iv.i 1q0p. ve12q k.i e o .yoi0uo 2oo ok0i/p. Luv0i02.i 0 2i apo2u10i 2 k.i 3, aou .v.3cpov2.i 12qv 0p01q 2ou M.K.. /o q 2piev .pi0ev .v2i12oi.. Oi 0ao0v0 apo2u10i 4 cpi k.i 20 .ao20ov 2qv .pi0q2ikq 00epi. 2ev apo2u10ev 2ou iiou .u2o. Mi. 0ao0vq ou/. apo2u10ev (21 cpi 32) .v.3cpov2.i 12ou ape2ou .pi0o k.i 12ou ape2ou apo .qou, k.0e k.i 12i i/io2q20 2ev .pi0ev .u2ev. 200u2.i. apo2.1q .u2q 2q ou/. . c0i o2i "ko00 .i0uo p 0iv.i ac2o p 0iv.i yivou0vo ac2oo", aou 0iv.i 000ie/q apo2.1q 2q 00epi. .-pi0ev. Oi uaooia0 apo2u10i (33 cpi 39) .v.3cpov2.i 12qv 00epi. 2ev ao.a.1iev, 2ev koivev ao.a.1iev, 12qv 0p01q 2ou .K.H. /o .pi0ev (apo2.1q 34) k.i 12qv 0p01q 2ou .K.H. 2piev .pi0ev (apo2.1q 36). Oi opi1oi 2ou iiou .u2o 0iv.i oi ovoi opi1oi aou uaupouv 12. ii. VII, VIII k.i IX. O 200u2.io opi1o 0i1u0i i. ape2q 3o-pu 2qv cvvoi. 2ou 2z0ioo .pi0uo: "T/0io .i0uo 0iv.i o .i0uo aoo 0iv.i i1o ao 2. u/p 2oo". q./q, .v cv. .pi0o . i1o2.i 0 2o u0poi1. oev 2ev /u-v.2ev /i.ip02ev 2ou (0k2o .ao 2ov 0.u2o 2ou), 2o20 0iv.i 2c0io. H.. o .pi0o 6 0iv.i 2c0io i.2i 1 2 3 6. ai1q o .pi0o 28 0iv.i 2c0io, i.2i 1 2 4 7 14 28. 0p01q 2c0iev .pi0ev 002u-02.i 12qv apo2.1q 36 2ou iiou X. VIII To iio VIII a0pic0i 27 apo2u10i k.i .v.3cp02.i 10 10ipc .-pi0ev, 1uv00i .v.oi0 k.i 12q +0e02pi. 2ev .pi0ev. pk02c apo2u10i .v.3cpov2.i 10 0e02pikc apoo/ou k.i 12ov 0e02piko c1o .pi0ev. Oi apo2u10i 1 k.i 3 0iv.i .v2i12po30. L2qv apo2.1q 1 .ao/0ikv02.i o2i ".v 10 ui. y0cu02ikp aoo/o o ac2o k.i o 200o2.i-o oo 2p 0iv.i ac2oi u02. 2oo, 2o20 o .i0uo 2cv ocv 2p aoo-/oo 0iv.i o 0o,i12o /ov.2o aoo uaoov v. 1,pu.2i1oov 2pv aoo/o". ve 12qv apo2.1q 3 .ao/0ikv02.i 2o .v2i12po3o. 34 YKA 'LTOX pk02c .ao 2i apo2u10i .u2c .vqkouv 10 .0q.2iko apo0vc-120pou 2ou uk0i/q. M0pikc apo2u10i, oae oi apo2u10i 11, 12, 26 k.i 27 .v.3cpov2.i 12ov a.2eviko /iuoo "Tiu.io". ai1q, oi apo2u10i 8, 9 k.i 10 apca0i v. q2.v ve12c 12qv k./qi. 2ou Hu2ev.. To iio X a0pic0i 36 apo2u10i. Oi ape20 apo2u10i .u2o 2ou iiou 0iv.i ao .ac k.i .v.3cpov2.i 12i 1uv00i .v.oi0. q./q 10 .pi0o aou pi1kov2.i 10 0e02pikq apoo/o. ae 3.i-v02.i k.i 12o iio .u2o, oae k.i 12o iio VIII, k0v2piko 0c. 0iv.i oi 1uv00i .v.oi0. iv.i .p.k2qpi12iko o2i o opo "1ov0,p .v.oyi." /0v opi02.i 10 k.vcv. 1q0io 2ev l2oi,0icv. L2o iio .u2o uaupouv k.i 0.ip02iku 1q.v2ikc apo2u10i. -ae i. a.pu/0i. q apo2.1q 14, /q./q q apo2.1q ".v 0o,i12o .i0-uo u02i/2.i .ao ac2oo .i0uo, 2o20 /0 0. u02i/2.i .ao k.v/v.v oo ac2o .i0uo, 0k2o .ao 2oo .,iko". apo2.1q .u2q 0iv.i 2o 000ie-/0 00epq. 2q pi0q2ikq, k.2u 2o oaoio ku00 .pi0o 2pca02.i 10 i-vo0vo ape2ev a.p.ov2ev k.2u ov./iko 2poao, 0k2o c.i. .ao 2qv .v2i02u001q 2ev a.p.ov2ev. apo2.1q 20 o2i "oi ac2oi .i0uoi 0iv.i a0i11o20oi .ao ko00 ap0o ac2cv .i0ucv", /q./q o2i oi ape2oi .-pi0oi 0iv.i ua0ipoi 12o aq0o, 0iv.i i. .ao 2i .1iko20p0 apo2u10i 2q 00epi. .pi0ev. +i. 2qv .ao/0iq 2q apo2.1q .u2q o uk0i/q pq-1ioaoi0i 2q c0o/o 2q 0i o2oaov .a.ycyp .u k.i 2q c0o/o 2q 2/-0i. 0a.ycyp. ai1q, 12qv .ao/0iq 2ev apo2u10ev 8 k.i 9 iv02.i pq-1q 2q 00o/ou 2q 2/0i. 0a.ycyp. Oi apo2u10i 21 cpi 34 .v.-3cpov2.i 12ou up2iou k.i a0pi22o .pi0o, 2i i/io2q20 2ou k.i 2i 1c10i aou couv 02. 2ou. Oi apo2u10i 18 k.i 19 0iv.i, ou1i.12iku, kpi2qpi. a.pq 2q 2pi2q k.i 2q 2c2.p2q .v.oou .v2i12oi.. apo2.1q 35 12qv ou1i. . /iv0i 2o u0poi1. 2ev v ape2ev opev i. 0e02pikq apoo/ou. M0 1povq opooi. q apo2.1q .u-2q aop0i v. /i.2uae00i e 0q: "l0 ko00 y0cu02ikp aoo/o, o oyo 2p /i.3oo 2oo ac2oo .ao 2o /020o ao 2ov ac2o, 0iv.i i1o u0 2o oyo 2p /i.3oo 2oo ac2oo .ao 2ov 200o2.io ao 2o o0oi1u. ocv 2cv ocv 2p aoo/oo apv 2oo 200o2.ioo". q./q, .v ., .e, .e2, . . . , .ev 0iv.i oi opoi 2q apoo/ou, 2o20 13ev. 0 2qv apo2.1q .u2q, 0. cou-0: L+O+ 35 = . .e.. .ev, oaou Lv . .e .e2 . . . .ev-1 ao 2q 1c1q .u2q apoka20i o ve12o 2ao 2ou .0poi1.2o Lv 2ev v ape2ev opev 2q apoo/ou, .3o Lv = ( )1 e. .e. .e. .e .v v= . apo2.1q 36 0iv.i 0ai1q .pk02u 1q.v2ikq, .3o .ao/0ikv0i cv.v 2ao aou . 0ai2pca0i v. uaooi1ou0 2c0iou .pi0o. apo-2.1q .u2q 0 1povq opooi. /i.2uaev02.i e 0q: v 10 ui. y0cu02ikp aoo/o u0 ac2o oo 2p uovo/. k.i oyo 2 aooou0 o1oo1/pao20 ooo, /c o2oo 2o o0oi1u. ocv /iv0i .i0uo ac2o, 2o20 2o o0oi1u. .o2o .v ao.a.1i.100i u0 2ov 200o2.io oo 2p aoo/oo /iv0i yivou0vo 2/0io .i0uo. q./q, .v aupou0 2qv apoo/o 1, 2, 22, . . ., 2v-1 k.i 2o u0poi1. Lv = 1+2+22+. . . +2v-1 = 2v-1 0iv.i .pi0o ape2o, 2o20 o .pi0o 2v = 2v-1 (2v-1) 0iv.i .pi0o 2c0io. M0 2q pq1ioaoiq1q 2ou 2aou .u2o k.i i. /iu3op0 2ic 2ou v aopo0 v. po0 2c0iou .pi0o. H.. M0 v 2, o 22 1 3 0iv.i ape2o. p. o 2 3 6 0iv.i 2c0io. M0 v 3, o 23 1 7 0iv.i ape2o. aocve o 22 (23 -1) 28 0iv.i 2c0io. M0 v 5, o 25 1 31 0iv.i ape2o. aocve o 24 (25 1) =16 31 496 0iv.i 2c0io. M0 v 13, o 213 1 8191 0iv.i .pi0o ape2o. aocve o .pi0o 212 (213 1) = 4096 8191 = 33 550 336 0iv.i 2c0io. O Euler .ac/0i0 o2i o 2ao .u2o 2ou uk0i/q /iv0i ovo up-2iou 2c0iou .pi0o k.i oi a0pi22oi, .v 2c2oioi .pi0oi uaupouv, /ivov2.i .ao 2ov 2ao: p4v+1 x2, 36 YKA 'LTOX oaou p 4 1 k.i x a0pi22o q /i.ipo0vo /i. 2ou p. Mcpi 1q0p. oe /0v c0i p000i k.vcv. a0pi22o 2c0io .pi0o. To iio .u2o 0iv.i 2o 0k20vc120po iio 2ev l2oi,0icv k.i a0-pic0i 115 apo2u10i aou .v.3cpov2.i 12. .102p. 0c0q. apo2.1q 1, "o2.v /o0ov /o ovi1. u0y/0p k.i .ao 2o u0y.20o .3.i000i 2upu. u0y.20o .ao 2o ui1o 2oo k.i .ao .o2o aoo u/v0i .3.i000i 2upu. u0y.20o .ao 2o ui1o 2oo k.i .o2o yiv02.i 1ov0,c k.i ./i.0ia2c, 0. u0iv0i /v. u/y00o uiko20o 2oo /o0/v2o uiko20oo u0y/0oo", 0iv.i ou1i.12iku q u0o/o 2q 0v2zq1q, oae pq1ioaoiq0qk0 .po20p. .ao 2ov uk0i/q 12o iio X. apo2.1q .u2q 0iv.i aup. ao 1q.v2ikq i. 2ou uppq2ou .pi0o, i.2i ou1i.12iku 0i1u0i 2qv cvvoi. 2ou opiou .koou0i. .ao i. apeiq povikq 12iq k.i /iv0i 2q /uv.2o2q2. v. p00ov apo10i12ikc pq2c 2ic ku00 uppq2ou .pi0o. Hp..2iku, .v . k.i 0 0iv.i 2. /0/ocv. 0c0q 0 0 ., 2o20 .v 1uoi1ou0 0 11, 12, . ,1v, 2. 0c0q aou ku00 3opu .3.ipo0 k.2u 2qv a.p.auve /i./ik.1i. k.i Lk = 11 12 . 1k 0 k 1, 2, 3, . . . 2o20: M02u 2qv ape2q .3.ip01q 0. c0i .3.ip000i q ao1o2q2. L1 11. M02u 2q /020pq .3.ip01q 0. c0i .3.ip000i 1uvoiku q ao1o2q2. L2 1112 k.i 0viku, 02u 2q vio12q .3.ip01q 0. c0i .3.ip000i 1uvo-iku q ao1o2q2. Lv 1112. . .1v. L3ev. 0 2qv apo2.1q 1, 0. uaup0i kuaoio v 2c2oio, e120 q /i.3opu . Lv v. 0iv.i ikpo20pq .ao 2o 0, i. oaoio/qao20 0 o1o/q-ao20 ikpo. u2o 1q.iv0i o2i 0 1q0pivq opooi. k.i 1uoi1o q apo2.1q .u2q aop0i v. /i.2uae00i e 0q: +i. ku00 0 > 0, o1o/qao20 ikpo, 0. uaup0i cv. /0ik2q v0, o oaoio 0. 0.p2u2.i .ao 2o 0 k.i 2c2oio, e120 v. i10i > < 0 .

aou 1q.iv0i o2i . OLP

= . L+O+ 37 apo2.1q .u2q pq1ioaoi0i2.i 1u12q.2iku 12o iio X i. 2ov uaooi1o 2ev 0./ev k.i 2ev okev. ai1q pq1ioaoi0i2.i 1u12q.2iku .ao 2ov piq/q. O piq/q 12qv 0i1.eq 2ou cpou 2ou "H0i 13.i. k.i koiv/oo" .v.3cp0i o2i q apo2.1q .u2q o30i0-2.i 12ov /oo. apo2.1q 2, ".v /o0ov /o ovi1. u0y/0p k.i .v0o3.i0i2.i ao-v2o20 2o uiko20o .ao 2o u0y.20o, 2o oaooiao aoo ko00 3oo u/v0i /0v k.2.u02o 2o aopyou0vo 2oo, 2o20 2. u0y/0p 0iv.i .1uu02.", ou-1i.12iku k.0opi0i ao20 /o 0c0q 0iv.i .102p.. Oi apo2u10i 3 k.i 4 .v.3cpov2.i 12qv 0p01q 2ou M.K.. /o k.i 2piev .pi0ev .v2i12oi.. K.2u 2ov uk0i/q, q cvvoi. 2ou 102pou 0c0ou 0iv.i 0up-20pq .ao 2q 1q0pivq. O uk0i/q a.ipv0i i. 0u00i., c12e 0, 2qv oaoi. 00ep0i pq2q. To20 ku00 0u00i. aou c0i koivo c2po 0 2qv 0 00ep0i2.i pq2q, .u k.i ku00 0u00i. aou 2o 202puevo 2q c0i koivo c2po 0 2qv 0 00ep0i2.i 0ai1q pq2q. q./q k.2u 2ov uk0i/q, .v i. 0u00i. /0v 0iv.i pq2q, .u 2o 202puevo 2q 0iv.i pq2o, 00ep0i2.i k.i .u2q pq2q. 0 oi 0u00i0 aou /0v 0iv.i pq2c o20 e apo 2o qko 2ou, o20 e apo 2o 202puevo 2ou 2i ovou0i op20. apo2.1q 9 c0i o2i "oi a0o/ 202.ycvoo 0iv.i 1uu02. p .1uu02. u0y/0p 10 upko, .v k.i uovo .v 2. 202oycv. /,oov p /0v /-,oov oyo 202oycvo .i0uo". q./q, .v . k.i 0iv.i oi a0upc 202p.evev k.i , v /o .pi0oi, 2o20: . : : v .2 : 2 2 : v2 apo2.1q .u2q .ao/i/02.i 12o ,0.i2q2o1. O ,0.i2q2o 0i0 0ai-/0i0i 0uo 0v/i.3cpov i. 2ou .102pou .pi0o k.i .u2o apo-ka20i .ao 2o ve12o /iuoo 2ou Hu2ev., ",0.i2p2o" (1470 ff). ai-1q, cv. 1oio auve 12o iio X aou c0i /i.1e00i 12. p.iku k.i .ao/i/02.i .ao 2ov Heiberg 12ov Huaao, .v.3cp0i o2i o ,0.i2q2o 0i0 .v.k.0i k.i /iu3op. 0i/q .102pev 000ev, aou 0iv.i ve12u e /icvou. k.i .ao2oup.2 iv.i 0ovo o2i .pk02c .ao 2i apo2u10i 2ev .pi0q2ikev i-iev 2ev l2oi,0icv, k.0e k.i 2ou iiou X, o30iov2.i 10 .0q.2i- 1 J. L. Heiberg and H. Menge, "Euclidis Opera Omnia", 2oo V, 1oio 62 12o X i-io 2ev l2oi,0icv. 2 W. Thomson, "Commentary of Pappus on Book X of Euclid s Elements". p.iko k0i0vo 0 02u3p.1q, 1q0ie10i, a.p.2qpq10i k.i 0ioio .ao 2ou G. Junge and W. Thomson. Cambridge, Mass (1930), 0a.v0k2ae1q 1968. 38 YKA 'LTOX ko aou cq1.v apiv 2ov uk0i/q k.i kupie 12ov /oo, 2o ,0.i-2q2o k.i 2ou .0q.2iko 2q k./qi. 2ou Hu2ev.. Oi .1ikc .v.k.0i i. 2ou .102pou kupie .pi0o civ.v .u2q 2qv 0aoq. e q 2.k2oaoiq1q .u2ev 2ev apo2u10ev, q 1uaqpe1q 2ou, q 200ioaoiq1q 2ou k.i q i0pupq1q 2ou civ.v .ao 2ov uk0i/q. v. 1oio, 2o oaoio o Heath .ao/i/0i 12ov Huaao, c0i 2. 0q.1 "O ok0i/p k.0oi10 .o12po k.vov0 yc .ao 2. 1uu02. k.i .1uu02. u0y/0p y0viko. /c10 1oyk0kiu/voo oi1uo k.i /i/ki-v0 u02. 2oo 2. p2o .ao 2. op2. u0y/0p. 0010 k.i /v. u0yoo .i0-uo k.vovcv yi. 2. .1uu02. u0y/0p k.i 20iko //0i0 k.0.o 2p 1pu.1i. 2oo k.i 2pv op 0a/k2.1p 2oo". T. 2pi. 200u2.i. ii., 2. ii. X, X k.i X, .ao20ov 2q L20p0o02pi. 2ev l2oi,0icv. ao20ov2.i .ao apo2u10i aou .v.3c-pov2.i 12q +0e02pi. 2ou epou k.i uaupouv k.i 0ui120 aou .v.-3cpov2.i 12qv aia0/o02pi.. To iio X a0pic0i 28 opi1o k.i 39 apo2u10i. iv.i 2o ovo .ao 2. 2pi. ii. aou a0pic0i opi1o, oi oaoioi pq1ioaoiov2.i k.i i. 2. ii. X k.i X. Oi opi1oi .v.3cpov2.i 12i evi0 2ou epou, 12qv a.p.qi. 12o epo, k.0e k.i 12. .1iku 120p0u 1e.2., 2. ao0/p., 2ov kiv/po, 2ov kevo k.i 2q 13.ip.. Oi apo2u10i 2ou ii-ou .u2o 002uouv 2i 1c10i 0u00i. k.i 0aiac/ou, k.0e k.i 2i 1c-10i 0aiac/ev 02. 2ou. ai1q, 2i i/io2q20 2ev 120p0ev 0e02pi-kev 1qu2ev k.i 2i 1c10i a.p.q0aiac/ev k.i api1.2o. To i-io .u2o c0i aoc ooio2q20 0 2o iio 2ev l2oi,0icv. H.. oi apo2u10i 1 cpi 5 2ou iiou .ao20ov 2i .1ikc apo2u10i, 2. 00ci. 2q aia0/o02pi.. ai1q, oi apo2u10i 1 - 5 2ou iiou X .ao20ov 2. 00ci. 2q L20p0o02pi.. Oi apo2u10i 11 k.i 12 2ou i-iou .v.3cpov2.i 12q up.q k.0c2ou 10 /o00i1. 0u00i.. K.i oi apo-2u10i 11 k.i 12 2ou iiou X .v.3cpov2.i 12qv k.2.1k0uq k.0c2ou 10 0aia0/o. apo2.1q 20 2ou iiou .v.3cp02.i 12qv 2pievikq .vi1o2q-2. 0aia0/ev eviev k.i q apo2.1q 20 2ou iiou X .v.3cp02.i 12qv 2pievikq .vi1o2q2. i. 2i evi0 2q kopu3q ka. 1 T. L. Heath, "The Thirteen Books of Euclids Elements", 2oo . L+O+ 39 To iio X a0pi.uv0i 18 apo2u10i, oi oaoi0 .v.3cpov2.i kupie 12i 1c10i 2ev 120p0ev 0e02pikev 1qu2ev. Yaupouv k.i kuaoi0 apo2u10i 2q aia0/o02pi., oae a.. oi apo2u10i 1 k.i 2 oi oaoi0 .v.3cpov2.i 12o 0./ov kukiko /i1kou. Oi apo2u10i 2q L20p0o02pi. 002ov 2qv 0ai3uv0i. k.i 2ov oko 120p0ev 1qu2ev epi v. /ivouv kuaoi. 1uk0kpicvq .pi0q2ikq c2pq1q. L2i .ao/0i-0i 2ev apo2u10ev iv02.i 1uvu pq1q 2q 00o/ou 2q 0uv2q1q (apo2.1q 1 2ou iiou X). Xpq1ioaoi0i2.i, i. a.pu/0i., q c0o/o 2q 0uv2q1q 12qv .ao/0iq 2ev apo2u10ev: "o kkoi 0iv.i .vooyoi 2cv /i.u/2cv 2oo", "Ho.ui/0 u0 2iycvik/ o10i k.i 2o i/io ,o 0iv.i .vooy0 2cv o10cv 2oo", "O oyko 2oo kcvoo 0iv.i 2o /v. 2i2o 2oo oykoo 2oo koiv/oo u0 2ov oaoiov /,0i 2pv i/i. o1p k.i i1. ,p", "Kcvo k.i kiv/o aoo /,oov i1. ,p /,oov oyo i1o u0 2o 2i-ao1io oyo 2cv o10cv 2oo". ,. apca0i v. 0ai1q.v00i 0/e o2i o uk0i/q o2.v .v.3cp02.i 10 oo 120p0ev, 0vvo0i 2o oo 2ev okev .u2ev 2ev 120p0ev. To iio X a0pic0i 18 apo2u10i. L2i ape20 apo2u10i iv0-2.i 0p.3q 10 kko 2ev k.vovikev aouevev aou /0v 0i.v ookq-pe00i 12o iio IV. Oi uaooia0 apo2u10i .1oov2.i 0 2qv 0p.-3q 10 13.ip. k.vovikev aouc/pev. L2i apo2u10i 13, 14, 15, 16 k.i 17 iv02.i q 0p.3q 10 13.ip. 2ev k.vovikev 120p0ev, aup.i/., ok2.c/pou, kou, 0iko1.c/pou k.i /e/0k.c/pou .v2i12oi.. T. acv20 .u2u k.voviku 120p0u 0i.v iv0i .v2ik0i0vo cp0uv. .ao 2ou .p.iou qv0 .0q.2iko apiv .ao 2ov uk0i/q. O Hu2ev kuv0i pq1q .u2ev 2ev 120p0ev 12o /iuoo "Tiu.io", i' .u2o 2. 1uk0kpicv. 120p0u 0iv.i ve12u e nz.2oviu 120p0. To k0v2piko .0q.2iko apoq. 12i apo2u10i .u2c 0iv.i v. uaooi120i q a0upu ku00 k.voviko 120p0o 10 1uvup2q1q 0 2qv .-k2iv. 2q 13.ip. c1. 12qv oaoi. c0i 0p.30i. O uk0i/q uaooi-10 2qv a0upu 2q aup.i/., 2ou kou k.i 2ou ok2.c/pou, 1uv.p2q10i 2q .k2iv. 2q 13.ip.. +i. 2o /e/0ku0/po k.i 2o 0iko1u0/po c/0i0 o2i oi a0upc 2ou 0iv.i .102p. 0c0q. L2qv apo2.1q 18 o uk0i/q 2oao0020i 2i a0upc 2ev acv20 k.-vovikev 120p0ev 12o i/io 1q., kuv0i 1kpi1q 02. 2ev a0upev 40 YKA 'LTOX .u2ev k.i 20iku .ao/0ikv0i o2i /0v uaupouv u. k.voviku 120p0u aou v. aopov v. 0p.3ov 10 13.ip.. L3ev. 0 cv. 1oio1 .vevuou 1oi.12q, o ,0.i2q2o q2.v o ape2o aou k.2.1k0.10 q a.pou1i.10 2. a.p.auve acv20 k.voviku 120p0u. i/iku oi apo2u10i 14 k.i 16 o30iov2.i 12o ,0.i2q2o. 4. T. . cpy. 2ov vk0/q 4.1. +0vtk k2o .ao 2. l2oi,0i. o uk0i/q c0i pu0i k.i u. 1q.v2iku cp., .ao 2. oaoi., /u12ue, 1eov2.i 0ui12.. Hoc aqpo3opi0 k.i .v.3opc 12. cp. .u2u 2ou uk0i/q cou0 .ao aoo .p.iou qv0 .0q.2iko, 1oi.12c, 02.3p.12c k.i 0k/o20 2ev .p.i-ev 0qvikev cpev. Oi a0pi11o20p0 aqpo3opi0 apocpov2.i .ao 2ov Huaao k.i 2ov Hpoko. L2o .p.iko 0iko Fihrist a.pou1iu02.i i. i12. 0 2. ovo.2. .0q.2ikev k.i 3io1o3ev. k0i 12o ovo. 2ou uk0i/q .v.3cpov2.i e cp. 2ou 2. 0q (.2ivikq 02u3p.1q . Suter2): "-.ivou0v.", "0/ou/v.", "l2oi,0i. Moo1ikp" (.u2o .v.3cp02.i e "iio 2cv 2ovcv"), "H0i /i.i/10cv", "Hoi1u.2.", "H0i ./cv k.i 0.3cv 1cuo2cv" k.i "iio K.vovcv". Kuaoi. .ao 2. cp. 2ou uk0i/q a0picov2.i (ookqp. q .ao-1au1.2u 2ou 10 cp. uev .p.iev qvev .0q.2ikev, 10 .p.-ikc 02.3pu10i aou .v.k.30qk.v .po20p., 10 0qviku 0ipop.-3. (ve12o ke/ik0), 10 apei0 0k/o10i 2ev l2oi,0icv, k.0e k.i 10 v0o20p0 0k/o10i k.i 0c20. H.. 2. cp. "0/ou/v.", "Hoi1u.2." k.i "Toaoi ao ai3ov0i0" a0pi.uvov2.i 10 i. 1uoq .0q.-2ikev cpev 2ou Huaaou, aou 0iv.i ve12q e ",p1.oo 2p voo-1p". T. "0/ou/v." k.i 2. 1oi. 2ou M.pivou auve 12o cpo .u2o uaupouv 12o .2ik.vo 0qviko ke/ik. 190. T. "Oa2iko" k.i 2. "-.i-vou0v." pc0qk.v 10 0ipop.3. 2ou 10ou .iev. 12q A.up0v2i.vq i-io0qkq 2q -ep0v2i. (Ms3 XXVIII, 3 k.i Ms Philos. Gr. No 103). To "H0i ./cv k.i 0.3cv 1cuo2cv" a0pic02.i 10 .2ivikq 02u- 1 J. L. Heiberg and H. Menge, "Euclidis Opera Omnia", 2oo V, 1oio 1 12o iio X, 10i/. 654.5-6. 2 H. Suter,"Das Mathematiker Verzeichniss im Fihrist in Abhandlungen zur Geschichte der Mathematik", vi,1892, 10.1-87 (kupie 10. 17). 3 Ms = manuscript, /q./q 0ipop.3o. L+O+ 41 3p.1q 12qv ck/o1q 2q .1i0i. 2ou 1537, k.0e k.i 12o cpo 2ou Gregory 2o 1703 (ca0 a.pup.3o 5). T. "0/ou/v." a0picov2.i k.i 10 0ipop.3o 2ou 11ou .iev., aou .ao200i2.i .ao /o 2oou k.i pi-1k02.i 12q qo1i. iio0qkq 2q Maoovi. 0 .pi0o 18-19. L2o cpo 2ev Heiberg k.i Menge "Euclidis Opera Omnia" uaup-ouv k.i .v.ov2.i 2. cp. "0/ou/v.", "-.ivou0v.", "l2oi,0i. Moo-1ikp", "Oa2iko" k.i "K.2oa2iko" (ca0 k.i a.pup.3o 5). ai1q, .pk02u .ao 2. u. cp. 2ou uk0i/q a.pou1iuov2.i, c12e k.i 1uvo-a2iku, .ao 2ov T. L. Heath 12o cpo 2ou "Euclid: The Thirteen Books of the Elements" (10i/0 7-28). T. cp. aou .ao/i/ov2.i 12ov uk0i/q 0iv.i: 1. "0/ou/v.", 2. "-.ivou0v." q "l3.iiko" 3. "Oa2iko" 4. "K.2oa2iko" 5. "l2oi,0i. Moo1ikp" 6. "iio a0i /i.i/10cv". 7. "Hoi1u.2." 8. "Kcviko" 9. "Toaoi ao 0ai3ov0i0" 10. "T0o/oi." 11. "Mp,.vikp" 12. "H0i ./cv k.i 0.3cv 1cuo2cv" ,. .v.30p0o0 a0piqa2iku 10 k.0cv. .ao .u2u. 4.2. To cpyo "0/ouv." ae q/q .v.3cp.0, 2o cpo .u2o, k.0e k.i 2. 1oi. 2ou M.-pivou 2q Nuaoq (.0q2q k.i iopu3ou 2ou Hpokou) auve 1' .u2o, a0picov2.i 12o .2ik.vo 0qviko ke/ik. 190, k.i .ao20ov 2ov 2oo VI 2ou cpou"Euclidis Opera Omnia", aou 0k/o0qk0 .ao 2ov H. Menge1 2o 1896. ai1q, 2. "0/ou/v." a0picov2.i k.i 12o 0ipop.3o 18-19 2q qo1i. iio0qkq 2q Maoovi.. 1 J. L. Heiberg and H. Menge "Euclidi's Opera Omnia", Leipzig 2oo VI (1896), 10.254. 42 YKA 'LTOX iv.i 2o ov./iko 0k2o 2ev l2oi,0icv cpo 2ou uk0i/q 2o o-aoio 1e02.i ookqpo 12. qviku. H0pic0i 15 opi1o k.i 94 apo-2u10i. Oi opi1oi .v.3cpov2.i 12ou /iu3opou 2poaou 0 2ou oaoiou kuaoio .v2i.uv02.i kuaoi. apu.2. e /0/ocv.. H.. oi 0ai3u-v0i0, oi 0u00i0, oi evi0 k.i oi ooi 00epov2.i o2i 0iv.i "/0/ocv." e apo 2o c00o, o2.v aopo0 v. 2. 1ukpivou0 e apo u. 0c0q. T. 0u0p.. 1q.2. c02.i o2i 0iv.i "/0/ocv." 12o epo q "/0/ocv." 12q op3q, o2.v oi evi0 k.i oi ooi 2ev a0upev 2ou 0iv.i /0/ocv.1. Lq0i., p.c k.i evi0 0iv.i "/0/ocv. e apo 2q 0c1q", o2.v k.2couv 12.00pq 0c1q ka. L2. 1oiu 2ou o M.pivo .-v.3cp02.i 12ou /iu3opou 2poaou 0 2ou oaoiou o opo "/0/ou/v." iv02.i .v2iqa2o .ao 2ou /iu3opou qv0 0ec2p0. L2. "0/ou/v." pi1kou0 12oi0i. aou aopov v. 00epq0ov 03.poc 12. ii. I cpi VI 2ev l2oi,0icv. Oi 94 apo2u10i aou .koou0ov .v.3cpov2.i 10 /iu3op. 0c.2. k.i .1iov2.i 12qv i/c. o2i ".v oi1u/v. 12oi,0i. 0vo 1,pu.2o 0iv.i /0/ou/v., 2o20 v. /0i,00i o2i o. 12oi,0i. 2oo i/ioo 1,pu.2o 0iv.i /0/ou/v.". H.: np2.1q 4 c0t: v .ao /0/ou/vo u/y00o .3.i000i /0/ou/vo u/y00o, 2o20 .o2o aoo u/v0i 0iv.i /0/ou/vo u/y00o. np2.1q 22 c0t: v /oo u0y/0p /,oov /0/ou/vo oyo ao 2i2o u/y00o, 2o20 k.i 2o o0oi1uo 2oo /,0i /0/ou/vo oyo c ao 2o 2i2o u/y00o. np2.1q 84 c0t: v /oo 0o00i0 y.uu/ a0i/,oov ui. /0/ou/-vp 0ai3ov0i. 10 ui. /0/ou/vp ycvi. k.i .v p ui. .ao .o2/ 0iv.i u0y.-20p .ao 2pv op k.2o /0/ou/vp ao1o2p2., 2o20 ko00 ui. .ao 2i y.uu/ .o2/ 0iv.i /0/ou/vp. apo2.1q .u2q, oae 0iv.i 3.v0po o/q0i 12q 1q 1u12q.2o /0u2cpou .0o. v x,y 0iv.i oi p.c, . k.i 2 q /0/ocvq /i.3o-pu k.i q 0ai3uv0i. .v2i12oi., 2o20 0. cou0 v. 1ou0 2o 112q. x y = . x y = 2 np2.1q 85 0iv.i i/i. 0 2qv apo2.1q 84, ovo aou 00ep0i e /0/ocvo 2o u0poi1. 2ev p.ev k.i oi 2q /i.3opu 2ou. apo2.1q .u2q o/q0i 12q 1q 1u12q.2o 2q op3q 1 Dictionary of Scientific Biography, Charles Scribnre's sons, New York, 1975, vol. IV, 10.425. L+O+ 43 x + y = . x y = 2 np2.1q 92 .v.3cp02.i 12ov kko k.i c0i e 0q: v .ao /0/ou/vo 1pu0io 12o 01c20iko 0vo /0/ou/voo kkoo 3/oou0 0o00i. aoo 2/uv0i 2ov kko, 2o20 2o o0oycvio a..poy.uuo aoo /,0i a0o-/ 2i 0o00i0 .ao 2o 1pu0io u/,i 2ov kko 0iv.i /0/ou/vo. ae vepiou0, o uk0i/q o2.v .v.3cp02.i 10 0u00i0 0vvo0i 0u0p.. 2q.2.. ao 2o a.p.ku2e 1q. cou0 o2i +.

()(+) R2 .2, oaou R 0iv.i /0/ocvo k.i . 0iv.i /0/ocvo e .ao12.1q 2ou kcv2pou 2ou kkou .ao 2o /0/ocvo 1q0io .u2o. O Huaao 12q "lov.ycyp" a0pipu30i 2o cpo "0/ou/v." 2ou uk0i/q 0 .pk02c 0a2ocp0i01. 4.3. To cpyo "-.ivu0v." q "23.ipiu" To cpo .u2o c0i .12povoiko a0pi0o0vo. To k0i0vo 2ou uk0i/q /0 1e02.i. Yaupouv oe kuaoi. .v.00epqcv. k0i0v., cv. 2ou 10ou .i-ev., aou pc0qk0 12q A.up0v2i.vq iio0qkq 2q -ep0v2i. k.i cv. 2ou 12ou .iev., aou pc0qk0 12q icvvq. To /020po k0i0vo a0pic02.i 10 i. 1uoq .12povoikev cpev, 2qv oaoi. o Huaao 0i0 ovou10i ",p1.oo 2p 12ovoui." k.i 0iv.i ve12q 10 . e "Mikp 12ovoui.", 10 .v2i-/i.12oq apo 2q "M0yop 12ovoui." 2ou H2o0.iou. T. k0i0v. .u2u, aou 00ep0i2.i o2i 0iv.i ao kov2u 12o .piko k0i0vo 2ou uk0i/q, a0picov2.i 12ov 2oo VIII 2ou cpou "Euclides Opera Omnia". O 2oo .u2o 0k/o0qk0 .ao 2o Heiberg 2o 1916. 1 F. Hultsch, "Pappi Alexandrini Collectionis quae supersunt", 2oo , 10i/0 636 k.i 638-640. 44 YKA 'LTOX To cpo "-.ivou0v." a0pic0i 2qv 0i1.eq k.i 16 apo2u10i. L2qv 0i1.eq 0q0i2.i i.2i 2o La.v 0iv.i 13.ipiko k.i uaupouv k.i /o 0pikoi opi1oi 20vikev opev. k0i o uk0i/q /iv0i 2ov opo "oiov2." k.i 0i1u0i i. ape2q 3opu 2ov opo "u01puivo kko". L2i apo2u10i .ao/0ikv02.i o2i 2. u12p. couv 1uk0kpicv0 0c10i k.i o2i .v.2couv k.i /ouv 10 k.0opi1cvou povou. Yaupouv apo2u10i aou .v.3cpov2.i 12qv a0pi12po3q 2ev oup.viev 1eu2ev k.i 2q oupuvi. 13.ip.. O Huaao (lov.ycyp VI, 104-130) 1oiu0i 0a2o0pe .pk02c .ao 2i apo2u10i .u2o 2ou cpou. ,. apca0i v. a.p.2qpq1ou0 0/e o2i i. .pk02c apo2u10i /0 /i-vov2.i .ao/0i0i k.i 0ai1q o2i oi apo2u10i .u2c uaupouv k.i 12o cpo 2ou u2oukou. +i' .u2o uaqp0 q 0k/oq o2i apocpov2.i .ao ku-aoio a..io20po cpo, 2o oaoio q2.v ve12o 2o1o 12ov uk0i/q o1o k.i 12ov u2ouko. O Paul Tannery1 i1upi12qk0 o2i 2. "-.ivou0v." q2.v cpo 2ou /oou. 4.4. To cpyo "On2iu" O Hpoko2 k.i o Huaao3 .ao/i/ouv 2o cpo "Oa2iko" 12ov u-k0i/q. O Huaao ui12. 1ua0pi.uv0i 2o cpo .u2o 12qv .12po-voikq 1uoq "Mikp 12ovoui.". L2o k0i0vo .u2o, aou 3.iv02.i o2i c0i p.30i .ao 2ov uk0i/q apiv .ao 2o cpo 2ou "-.ivou0v.", o Huaao c0i apo10c10i .koq 12 apo2u10i. Lq0p. 1eov2.i /uo op3c 2ou cpou "Oa2iko". ape2q .v.-00epqcvq op3q, aou civ0 .ao 2o ,cev. 2ov 0.v/pivo, q2.v 2o ovo ve12o k0i0vo cpi 2o 2co 2ou 19ou .iev.. L2o k0i0vo .u2o uaup0i i. 0i1.eq, q oaoi. 3.iv02.i v. c0i apo12000i .ao 2o ,cev.. /020pq op3q o30i02.i 12o Heiberg. K.2u 2o 2co 2ou 19ou .iev. o Heiberg .v.kuu0 /uo k0i0v., 2o cv. 2ou 10ou .iev. 12q A.u-p0v2i.vq iio0qkq 2q -ep0v2i. k.i 2o uo 2ou 12ou .iev. 12q icvvq. M0 u1q 2. k0i0v. .u2u 1uvc0010 2q vc. op3q 2ou cpou "O-a2iko", aou 00ep0i2.i o2i .ao/i/0i 2o .u00v2iko k0i0vo 2ou uk0i/q k.i a0pic02.i 12o cpo 2ev Heiberg k.i Henge "Euclidis Opera Omnia4". 1 Paul Tannery,"La Geometriae Grecque" , H.pi1i, 1887. 2 G. Friedlein, 'Procli Diadochi in Primum Euclidis Elementorum Librum Commentarii", Leipzig 1883 k.i 0a.vck/o1q 1967 (10.69. 2). 3 F. Hultsch, "Pappi Alexandrini Collectionis quae supersunt" 3 2ooi, Berlin 1876-1878 (2oo , 10. 568-594). 4 J. L. Heiberg and H. Menge "Euclidis Opera Omnia", 2oo VII (1895). L+O+ 45 T. "Oa2iko", oae a.pou1iuov2.i 12qv ck/o1q 2ou Heiberg, .p-iouv 0 opi1o auve 10 0c.2. oa2ikq k.i a0picouv .koq 58 apo2u10i. Oi opi1oi aopo0 v. ao0 o2i /0v .v2.aokpivov2.i 12q 1q0pivq cvvoi. 2ou opi1o, .u 0iv.i uao0c10i (a.p./oc). +i. a.pu/0i., o opi1o 1 a.p./c02.i o2i q op.1q apok.0i2.i .ao 2i oa2ikc .k2iv0 aou 0kivov .ao 2o u2i 2ou a.p.2qpq2q k.i k.2u-qouv 12o a.p.2qpo0vo .v2ik0i0vo. O opi1o 2 c0i o2i 2o 1q. aou a0pik0i02.i .ao 2i oa2ikc .k2iv0 0iv.i cv. kevo, o oaoio c0i 2qv kopu3q 12o u2i 2ou a.p.2qpq2q k.i u1q 2ou 0iv.i 2o cpo 2ou .v2ik0icvou aou ca0i o a.p.2qpq2q. L0 uo opi1o uao0c20i o2i .v2ik0i0v. aou 3.ivov2.i uao i10 evi0 0iv.i i1., 0ve 0k0iv. aou 3.ivov2.i uao uvi10 evi0 0iv.i ooie uvi1. ka. 4.5. To cpyo "K.2on2piu" O Hpoko .ao/i/0i 12ov uk0i/q cv. cpo 0 2ov 2i2o "K.2o-a2iko". +pu30i 1uk0kpicv.: "Yao,oov aoo o. u.0pu.2iko /y. 2oo ok0i/p, oac 0iv.i 2. Oa2iko 2oo, 2. K.2oa2iko 2oo, 2. l2oi,0i. Moo1ikp k.i 2o uiko 2oo iio H0i /i.i/10cv". T. "K.2oa2iko" a0-picov2.i 12ov 2oo VII 2ou cpou "Euclidis Opera Omnia" 2ev Heiberg k.i Menge. O12o1o, 00ep0i2.i o2i 2o cpo aou .v.3cp0i o Hpoko /0v 2o c-0i pu0i o uk0i/q, .u .vqk0i 10 1up.3c. ao 02.0vc120po. Yaupouv 0v/0i0i o2i 1up.3c. 2ou 0iv.i o ,cev k.i o2i o Hpoko .ao a.p./poq 2o .ac/e10 12ov uk0i/q. 4.6. To cpyo "22oi0. Moo1iuq" ae .v.3cp.0 apoqoucve, o Hpoko .ao/i/0i 12ov u-k0i/q k.i cv. cpo 0 2ov 2i2o "l2oi,0i. Moo1ikp". To i/io cpo .ao-/i/0i 12ov uk0i/q k.i o M.pivo. L2. "l2oi,0i. Moo1ikp" a0picov2.i ou1i.12iku /o ou1ikc 0p.1i0. ape2q ovou02.i "K.2.2oup K.vovo", (/q./q /i.ip01q 2q ki.k.) k.i 0pqv00i 2q 00epi. a0i uoo1ikp 2ev Hu0.op0iev. /020pq ovou02.i "i1.ycyp uovikp" k.i 0iv.i .1i1cvq 12q 00epi. 2ou pi12o0vou. 00epi. .u2q 0iv.i .v2i002q 0 2q 00epi. 2ev Hu0.op0iev. u2q q .v2i3.1q 1q.iv0i o2i 0iv.i uov /1koo v. c0i p.30i ookqpo 2o cpo .ao 2ov uk0i/q. 46 YKA 'LTOX Lq0p., q 3iooikq kpi2ikq c0i /020i o2i q "i1.ycyp -uovikp" 0iv.i p.cvq .ao cv.v .0q2q 2ou pi12o0vou, 2ov K0ovi-/q. ve q "K.2.2oup K.vovo" .ao/i/02.i 0 0.io2q2. 12ov uk0i-/q, 0a0i/q, 0k2o 2ev uev, 0iv.i .0q.2ikq ou1ikq k.i 2o 3o, q 0aioq 2ev c0ev, q op3q 2ev apo2u10ev k.i 0viku q .vua2uq 2ou k0icvou 1u3evov 0 .u2u 2ev l2oi,0icv. O Hop3pio, aou cp.0 1oi. 12o cpo "uoviko" 2ou H2o-0.iou, .v.3cp02.i .pk02c 3opc 12qv "K.2.2oup K.vovo", 00epe-v2. o2i 0iv.i cpo 2ou uk0i/q. H.. 10 kuaoio 1q0io pu30i1: "K.i .o2o o l2oi,0ic2p ok0i/p 12o /yo 2oo K.vovo K.2.2oup.". K.i a.p.ku2e:"l2ov K.vov. K.2.2oup 2oo ok0i/p.". 4.7. To cpyo "izo /i.ip10ov" To cpo .u2o /0 1e02.i ookqpo 12. qviku. v. 0uo cpo 2ou a0pic02.i 12ov 2oo VIII 2ou cpou "Euclides Opera Omnia", aou 0k-/o0qk0 .ao 2ov Heiberg 2o 1916. To k0i0vo .u2o 0iv.i .1i1cvo 10 /o kupie aqc: 1. L0 i. .p.ikq 02u3p.1q 2ou 10ou .iev., 2qv oaoi. .v.ku-u0 o Franz Woepcke 12qv 0vikq iio0qkq 2ou H.pi1io 2o 1849. To k0i0vo .u2o 2o 02c3p.10 o Woepcke 12. +.i-ku k.i 2o /qo1i0u102 12o 0ai12qoviko a0pio/iko Journal Asiatique 2o 1851. 2. L2o cpo Practica Geometriae, aou pu32qk0 2o 1220 .ao 2o Leonardo Fibonacci. L2o cpo .u2o uaup0i cv. 0uo cpo .ao 2o iio "H0i /i.i/10cv" 2ou uk0i/q. Yaupouv 1q0p. i1upc 0v/0i0i o2i 2o k0i0vo .u2o 2ou Fibonacci apocp02.i .ao k0i0v. 2ou uk0i/q aou 1q0p. /0 1eov2.i. To 1915 o R. C. Archibald .v.k.2.1k0.10 ookqpo 2o cpo "H0i /i.i/10cv", .1i1cvo 12. cp. 2ou L. Fibonacci k.i 2ou F. Woepcke 3. 1 Porphyry "Commentary on Ptolemy's Harmonica", Wallis, Opera Mathematica III, Oxford, 1669, 10. 267. 31-32 k.i 272. 26-27. 2 F. Woepcke, "Notice sur des traductions arabes de deux auvrages perdus d' Euclide", Journal Asiatique, 1851. 3 R. C. Archibald, "Euclid's Book on Divisions of Figures with a Restoration Based on Woepcke's Text and the Practica Geometriae of Leonardo Pisano", Cambridge, 1915. L+O+ 47 O Hpoko .v.3cp02.i 12o "iio /i.i/10cv" k.i c0i o2i o u-k0i/q /i.ip0i /0/ocv. 0u0p.. 1q.2. 10 ooi. q .vooi. 0u0-p.. 1q.2.1. 0 oi apo2u10i 2ou cpou .v.3cpov2.i 10 0u0p.. 1q.2., 2. oaoi. /i.ipov2.i 0 0u00i0 p.c 10 u. 0u0p.. 1q.2., aou couv /0/ocvq 1c1q. H.. /i.ipov2.i 2piev., a.p.qop.., 2p.-aci., 202pua0up., kkoi, kukikoi 2o0i 10 /o q a0pi11o20p. i1. cpq q 10 /o q a0pi11o20p. cpq 2. oaoi. couv /o0cv2. oo. Oi apo2u10i 0iv.i 36 1uvoiku, .u .ao/0i0i uaupouv ovo i. 2c110pi, 2i apo2u10i 19, 20, 28 k.i 29. K.2.pu3ou0 0v/0ik2iku 0pikc .ao .u2c 2i apo2u10i, oae uaupouv 12o cpo 2ou Heiberg: N. /i.i000i /0/ou/vo 2iycvo 10 /o i1. u/p .ao 0o00i. aoo 0iv.i a.opp ao 2p o1p 2oo 2iycvoo (apo2.1q 1). N. /i.i000i 2iycvo 10 2i. i1. u/p .ao /o 0o00i0 a.o-p0 ao 2p o1p 2oo 2iycvoo (apo2.1q 3). N. /i.i000i 2.a/io 10 /o i1. u/p .ao ui. 0o00i. aoo 0iv.i a.opp ao 2i o10i 2oo 2.a0ioo (apo2.1q 4). N. /i.i000i a..poy.uuo 10 /o i1. u/p .ao ui. 0o00i. y.uup, aoo /i/,02.i .ao /v. /0/ou/vo 1pu0io 2o oaoio i1k02.i aovc 10 ui. a0oo 2oo a..poyouuoo (apo2.1q 6). N. /i.i000i 2.a/io 10 /o i1. u/p .ao ui. 0o00i. aoo a0-vo .ao /0/ou/vo 1pu0io, 2o oaoio .vpk0i 10 ui. a0oo 2oo 2.a0ioo (apo2.1q 8). N. /i.i000i /0/ou/vo 2iycvo 10 /o i1. u/p .ao ui. 0o00i. y.uup aoo a0vo .ao /v. /0/ou/vo 1pu0io, 2o oaoio i1k02.i u/1. 12o 2iycvo. (apo2.1q 19). l0 /0/ou/vo 2iycvo v. 3/oou0 0o00i., p oaoi. v. a0vo .ao /v. /0/ou/vo 1pu0io aoo i1k02.i 12o 01c20iko 2oo 2iycvoo k.i v. .aokoa20i .ao 2o 2iycvo /v. 2upu. 2oo, aoo v. /,0i /0-/ou/vo oyo ao 2o 2iycvo. (apo2.1q 20). L2qv .ao/0iq aou uaup0i 12o k0i0vo, .aokoa202.i .ao 2o 2pievo 2q. i1o 0 2o 312ou 2pievou. 1 G. Friedlein, "Procli Diadochi in primum Euclidis Elementorum librum commentarii", Leipzig 1883, 10. 68. 23-27. 48 YKA 'LTOX l0 /0/ou/vo kko v. 3/oou0 /oo a.op0 0o00i0, oi o-aoi0 v. 2/uvoov 2ov kko k.i v. .aokoa2oov .ao .o2ov /v. /0/ou/vo ko1u. 2oo. (apo2.1q 29). O uk0i/q 12qv .ao/0iq 2q apo2.1q 29 k.2.1k0uu0i cv. 2q. 2ou kkou i1o 0 2o cv. 2pi2o 2ou kkou. v.3cp0i oe o2i 0 ooio 2poao k.2.1k0uu02.i 2o cv. 2c2.p2o 2ou kkou, 2o cv. aca2o 2ou k.i ku00 2q. oaoiou/qao20 ku1.2o 2ou kkou. O pev 12. "M02iko" (2oo , 10. 8) v0i 2qv apo2.1q 29 o2.v 2o ku1. 0iv.i 2o cv. 2pi2o 2ou kkou. K.2.q0i 12o 1uac-p.1. o2i 2o apoq. /0v c0i 1q 0 pq2o .pi0o k.i apo1a.00i v. p0i apo10i12ikc 2ic i. 2q 1q .u2o 2ou apoq.2o. ,0epo0 pq1io v. apo10c1ou0 0/e o2i o John Dee pqk0 i. 0p.1i. 0 2i2o "De divisionibus" 0 a0pi0o0vo 12. .p.iku kuaoiou Muhammad Bagdadinus, 2q 02c3p.10 12. A.2iviku k.i 2o 1563 c/e10 .v2ip.3o 2q .2ivikq 02u3p.1q 12ov Commandinus, o oaoio 2o /qo1i0u10 2o 1570 0 2o ovou 2ou k.i 2o ovo. 2ou Dee. To k0i0vo .u2o a0pic02.i 12o cpo 2ou Gregory 2ou 1703. 4.8. To cpyo "Hop1u.2." O Hpoko1 k.i o Huaao2 .ao/i/ouv 2o cpo "Hoi1u.2." 12ov u-k0i/q. ao200i2.i .ao 2pi. ii. k.i, 13ev. 0 2ov Huaao, a0pic0i 171 apo2u10i k.i 38 q.2.. Oi apo2u10i .v.3cpov2.i kupie 10 0e-02piko 2oaou 1q0iev q p.ev. O Huaao 2o 1ua0pi.uv0i 12q 1uoq .0q.2ikev cpev ",p1.oo 2p voo1p". ao 2. 12oi0i. aou a.pou1iuouv o Hpoko k.i o Huaao, o Robert Simson3 k.i o Michel Chasles4 ck..v 1q.v2ikc apo1au00i0 v. 2o .v.k.2.1k0uu1ouv. To cpo "Hoi1u.2." aop0i v. p000i 12. cp. "Euklid-Studien" 2ou Heiberg (10i/0 56-79) k.i "Le scienze essate nell' antica Grecia" (1914, 10i/0 245-268) 2ou G. Loria. L2o cpo "Euclidis the thirteen books of Elements", 2ou Heath, (2oo , 10i/0 10-15), uaupouv 0k202.cv. .ao1au1.2. 2ou cpou .u2o. 1 G. Friedlein, 'Procli Diadochi in primum Euclidis Elementorum librum commentarii", Leipzig 1883 k.i 0a.vck/o1q 1967 (10.69. 2) 2 F. Hultsch, "Pappi Alexandrini Collectionis quae supersunt" 3 2ooi, Berlin 1876-1878 (2oo , 10. 568-594) 3 Robert Simson, "De porismatibus tractatus", Opera quaedam reliqua, Glasgow 1776, 10.315-594. 4 Michel Chasles, "Le trois livres de porismes d' Euclide", Paris, 1860. L+O+ 49 4.9. To cpyo "Koviu" ao 2ov Huaao ("lov.ycyp", V, 10. 672) vepiou0 o2i o u-k0i/q 0i0 pu0i cv. cpo 0 2ov 2i2o "Kcviko". To cpo c0i .00i, e12o1o o Huaao .v.3cp02.i 1uvu 1' .u2o k.i .ao 2i .v.3opc .u2c vepiou0 .pk02u i. 2o a0pi0o0vo 2ou. Mui12., 0a0i/q .v.3cp02.i 2o1o 1' .u2o o1o k.i 10 .v2i12oio cpo 2ou pi12.iou, e v. uaqp.v k.i v. kuko3opo1.v 12qv 0aoq 2ou, ai12002.i o2i 2o cpo .u2o 2ou uk0i/q u0qk0 02u 2qv 0aoq 2ou Huaaou. O Huaao 12q "lov.ycyp" (iio V, 10. 672, 10ipu 18 k.0.) .v.3cp0i o2i 2. "Kcviko" 2ou uk0i/q .ao20ov2.i .ao 2c110p. i-i. k.i a0picouv k.i 2i kevikc 2oc. po20p., o aoevio .v.-00epq10 k.i 1uaqpe10 2. 2c110p. .u2u ii., apo10010 u. 2c1-10p. /iku 2ou k.i c21i /qiopq10 2o a0pi3qo cpo 2ou "Kcvik/ 2o-u/", 2o oaoio .ao200i2.i 1uvoiku .ao ok2e ii. ("20,p"). Luvc .v.3opc 12. "Kcviko" 2ou uk0i/q kuv0i k.i o pi-q/q, o oaoio ui12. .v.3cp02.i k.i 10 1uk0kpicv0 apo2u10i 2ou cpou .u2o. 13.e, o uk0i/q vepi0 o2i i. kevikq 2oq 0vvic2.i .ao 2qv 2oq 0vo kevou .ao cv. 0aia0/o. ,0epo10 2p0i op0o kevou, oi oaoioi 0i.v 2q evi. 2q kopu3q o0i., op0q q .0i. .v2i12oi.. Oi 2oc 2ev kevev .u2ev 0 0aia0/o ku002o 12q 0vc20ip. 2ou kevou 1q.2iouv 2i kevikc 2oc, 2i oaoi0 ovo.0 2oup ooycvioo kcvoo, 2oup o0oycvioo kcvoo k.i 2oup .uoycvioo kcvoo. +i. 2i kevikc .u2c 2oc o ao-evio k.0opi10 2ou opou /0i,p, a..op k.i oa0op .v2i12oi.. e o uk0i/q vepi0 o2i q ua0poq aop0i v. 1q.2i100i .ao 2qv 2oq 0vo kevou q 0vo kuiv/pou .ao 0aia0/o aou /0v 0iv.i a.puqo apo 2q u1q. L2o cpo 2ou "-.ivou0v." c0i: "v /v. kcvo p kiv/o 2up00i .ao /v. 0aia0/o aoo /0v 0iv.i a.opo ao 2p o1p 2oo kcvoo p 2oo koiv-/oo, p 2oup 0iv.i ui. 2oup ooycvioo kcvoo, p oaoi. uoio0i u0 0o0o". 4.10. To cpyo "1noi npo 0ni3v0i0" To cpo .u2o /0v c0i /i.1e00i. To a0pi0o0vo 2ou aopo0 v. 2o uao0c1ou0 .ao /iu3op0 .v.3opc aou pi1kou0 12. cp. 2ou Hpokou k.i 2ou Huaaou. 21i a.. vepiou0 o2i .ao20ov2.v .ao /uo ii.. O Huaao .ao/i/0i 2o cpo .u2o 12ov uk0i/q (lov.ycyp, VII, 636, 23-24), k.i 2o 0v2u110i 12q .0q.2ikq 1uoq ",p1.oo 2p voo1p". ai1q, .ao/i/0i 12ov uk0i/q /o q.2. auve 12ou 50 YKA 'LTOX y0cu02iko 2oaoo 0ai3ov0i0, 2. oaoi. k.2.pu30i 12ov 2oo