{"id":5374,"date":"2025-08-31T22:19:13","date_gmt":"2025-08-31T22:19:13","guid":{"rendered":"https:\/\/flatarmal.is\/?p=5374"},"modified":"2025-09-01T14:48:48","modified_gmt":"2025-09-01T14:48:48","slug":"soguhornid-threttan-baekur-evklids-fra-alexandriu","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=5374","title":{"rendered":"S\u00d6GUHORNI\u00d0: \u00derett\u00e1n b\u00e6kur Evkl\u00ed\u00f0s fr\u00e1 Alexandr\u00edu"},"content":{"rendered":"\n

Krist\u00edn Bjarnad\u00f3ttir.<\/strong><\/p>\n\n\n\n

Ekki ver\u00f0ur lengi r\u00e6tt um \u00fe\u00e1tt st\u00e6r\u00f0fr\u00e6\u00f0innar \u00ed s\u00f6gu vestr\u00e6nnar menningar \u00e1n \u00feess a\u00f0 nafninu Evkl\u00ed\u00f0 breg\u00f0i fyrir. Hver var Evkl\u00ed\u00f0? \u00cd raun er l\u00edti\u00f0 vita\u00f0 um manninn. Hann bj\u00f3 \u00ed Alexandr\u00edu, gr\u00edskri n\u00fdlenduborg sem Alexander mikli stofna\u00f0i \u00e1ri\u00f0 325 f.Kr. vi\u00f0 \u00f3sa N\u00edlar \u00ed Egyptalandi. Tali\u00f0 er a\u00f0 Evkl\u00ed\u00f0 hafi veri\u00f0 uppi \u00e1 t\u00edmabilinu fr\u00e1 um 325\u2013320 fram til 280\u2013265 f.Kr.<\/p>\n\n\n\n

Evkl\u00ed\u00f0 safna\u00f0i saman allri st\u00e6r\u00f0fr\u00e6\u00f0i sem \u00feekkt var \u00e1 hans t\u00edmum \u00ed eina b\u00f3k, sem nefndist \u00e1 gr\u00edsku Stoikhe\u00eea<\/em> (\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03bf<\/em>, e. Elements<\/em>), en nefna m\u00e6tti Frum\u00fe\u00e6tti<\/em> \u00e1 \u00edslensku. Riti\u00f0, sem var fullb\u00fai\u00f0 um \u00e1ri\u00f0 300 f.Kr., skiptist \u00ed 13 b\u00e6kur. \u00dea\u00f0 var\u00f0 ein kunnasta st\u00e6r\u00f0fr\u00e6\u00f0ib\u00f3k s\u00f6gunnar, \u00fe\u00fddd \u00e1 m\u00f6rg tungum\u00e1l (ekki \u00fe\u00f3 \u00e1 \u00edslensku enn sem komi\u00f0 er), r\u00e6dd og r\u00fdnd, og kennslub\u00e6kur voru sni\u00f0nar eftir henni.<\/p>\n\n\n

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Mynd 1.<\/strong> Leifar af handriti Frum\u00fe\u00e1tta<\/em> fr\u00e1 1. \u00f6ld e.Kr.<\/figcaption><\/figure>\n<\/div>\n\n\n

Tali\u00f0 er l\u00edklegt a\u00f0 Evkl\u00ed\u00f0 hafi reynt a\u00f0 sn\u00ed\u00f0a b\u00f3k s\u00edna sem einfaldasta og sk\u00fdrasta, svo a\u00f0 textinn yr\u00f0i hverjum manni lj\u00f3s. \u00de\u00f3 er sagt a\u00f0 konungurinn Ptolemeus I, eftirma\u00f0ur Alexanders mikla, hafi kvarta\u00f0 vi\u00f0 Evkl\u00ed\u00f0 yfir a\u00f0 hann skildi ekki efni b\u00f3karinnar. Evkl\u00ed\u00f0 hafi \u00fe\u00e1 sagt hin fleygu or\u00f0: \u201e\u00dea\u00f0 er engin konungleg lei\u00f0 a\u00f0 r\u00famfr\u00e6\u00f0inni, herra minn\u201c.<\/p>\n\n\n\n

Efni Frum\u00fe\u00e1tta<\/em> vindur fram me\u00f0 aflei\u00f0slu (e. deduction), \u00feannig a\u00f0 sannanir og reglur eru leiddar af frumhugt\u00f6kum og frumforsendum. Reglur, sem \u00feannig hafa fengist, ver\u00f0a s\u00ed\u00f0an forsendur annarra reglna, einfaldra og fl\u00f3kinna.<\/p>\n\n\n\n

Aflei\u00f0slukerfi Evkl\u00ed\u00f0s hefst \u00e1 10 frumforsendum, forsendum sem eru \u00f3sanna\u00f0ar en \u00f3umdeildar. \u00de\u00e6r eru oft nefndar frumsendur<\/em>. Allar a\u00f0rar setningar eru leiddar af \u00feeim. Fimm fyrstu frumsendurnar eru almennar og var\u00f0a alla st\u00e6r\u00f0fr\u00e6\u00f0i en hinar fimm seinni fjalla um r\u00famfr\u00e6\u00f0ina s\u00e9rstaklega.<\/p>\n\n\n\n

Almennu frumsendurnar eru:<\/p>\n\n\n\n

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1. S\u00e9u tveir hlutir b\u00e1\u00f0ir jafnir \u00feri\u00f0ja hlut \u00fe\u00e1 eru \u00feeir l\u00edka jafnir hvor \u00f6\u00f0rum.
2. S\u00e9 jafnmiklu b\u00e6tt vi\u00f0 tvo jafna hluti ver\u00f0a \u00fatkomurnar jafnar.
3. S\u00e9 jafnmiki\u00f0 teki\u00f0 af tveim j\u00f6fnum hlutum ver\u00f0a \u00fatkomurnar jafnar.
4. Hlutir sem fylla sama r\u00fam eru jafnir.
5. Heild er st\u00e6rri en hluti hennar.<\/p>\n<\/div>\n<\/div>\n\n\n\n

R\u00famfr\u00e6\u00f0ilegu frumsendurnar eru:<\/p>\n\n\n\n

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1. Fr\u00e1 hva\u00f0a punkti sem er m\u00e1 draga beina l\u00ednu \u00ed hva\u00f0a punkt sem er.
2. Strik af endanlegri lengd m\u00e1 framlengja \u00ed beina l\u00ednu.
3. H\u00e6gt er a\u00f0 draga hring me\u00f0 hva\u00f0a fjarl\u00e6g\u00f0 sem er fyrir rad\u00edus og hva\u00f0a punkt sem er fyrir mi\u00f0ju.
4. \u00d6ll r\u00e9tt horn eru jafn st\u00f3r.
5. Ef strik sker tv\u00e6r l\u00ednur og innri hornin \u00e1 a\u00f0ra hli\u00f0 \u00feess eru samtals minni en tv\u00f6 r\u00e9tt horn \u00fe\u00e1 skerast l\u00ednurnar tv\u00e6r \u00feeim megin vi\u00f0 striki\u00f0 ef \u00fe\u00e6r eru framlengdar \u00f3takmarka\u00f0.<\/p>\n<\/div>\n<\/div>\n\n\n\n

(\u00de\u00fd\u00f0andi Atli Har\u00f0arson, 2001).<\/p>\n\n\n\n

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Mynd 2. <\/strong>\u00dealesar regla.  <\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n
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Evkl\u00ed\u00f0 var \u00fe\u00f3 ekki fyrstur manna til a\u00f0 byggja upp aflei\u00f0slukerfi, enda er rit hans a\u00f0 mestu safn \u00feess sem \u00e1\u00f0ur var \u00feekkt. L\u00f6ngu \u00e1\u00f0ur haf\u00f0i \u00deales fr\u00e1 M\u00edletos (624 \u2013 547 f.Kr.) sett r\u00famfr\u00e6\u00f0i fram \u00ed aflei\u00f0slukerfi og s\u00fdnt fram \u00e1 a\u00f0 s\u00e9 \u00fer\u00edhyrningur innrita\u00f0ur \u00ed hring \u00feannig a\u00f0 eitt horn hans s\u00e9 \u00e1 hringferlinum og m\u00f3tl\u00e6g hli\u00f0 hornsins mi\u00f0strengur hringsins, \u00fe\u00e1 er horni\u00f0 \u00e1 hringferlinum r\u00e9tt horn. Reglan er nefnd \u00dealesar regla<\/em>.<\/p>\n<\/div>\n<\/div>\n\n\n\n

Kr\u00f3kalei\u00f0 Frum\u00fe\u00e1tta<\/em> til vorra t\u00edma<\/strong><\/p>\n\n\n\n

R\u00edki Alexanders mikla st\u00f3\u00f0 ekki lengi. Um \u00e1ri\u00f0 270 f.Kr. haf\u00f0i \u00fea\u00f0 li\u00f0ast \u00ed sundur. Um s\u00ed\u00f0ir hnigna\u00f0i einnig n\u00fdsk\u00f6pun \u00e1 svi\u00f0i st\u00e6r\u00f0fr\u00e6\u00f0i \u00ed gr\u00edska menningarheiminum. \u00cdb\u00faar R\u00f3mar, R\u00f3mverjar, t\u00f3ku a\u00f0 \u00feenja \u00fat r\u00edki sitt yfir \u00cdtal\u00eduskagann. \u00deeir h\u00f6f\u00f0u hafi\u00f0 frekari \u00fat\u00feenslu til su\u00f0urs \u00feegar st\u00e6r\u00f0fr\u00e6\u00f0ingurinn Arkimedes f\u00e9ll fyrir hendi r\u00f3mversks hermanns \u00ed S\u00fdrak\u00fasu \u00e1 Sikiley \u00e1ri\u00f0 212 f.Kr. \u00dea\u00f0 ger\u00f0ist \u00ed annarri p\u00fanversku styrj\u00f6ldinni milli R\u00f3mverja og \u00edb\u00faa Kar\u00feag\u00f3, borgar vi\u00f0 sunnanvert Mi\u00f0jar\u00f0arhaf \u00fear sem n\u00fa er T\u00fanis.<\/p>\n\n\n\n

R\u00f3mverjar r\u00e9\u00f0u l\u00f6gum og lofum vi\u00f0 Mi\u00f0jar\u00f0arhaf \u00fear til R\u00f3maveldi klofna\u00f0i \u00ed austr\u00f3mverskan og vestr\u00f3mverskan hluta. Vestr\u00f3mverska r\u00edki\u00f0 f\u00e9ll endanlega \u00e1ri\u00f0 476 e.Kr. \u00d3l\u00e6sir germanskir \u00fej\u00f3\u00f0flokkar l\u00f6g\u00f0u undir sig meginhluta Evr\u00f3pu. Forn st\u00e6r\u00f0fr\u00e6\u00f0irit t\u00fdndust og gleymdust \u00ed Vestur-Evr\u00f3pu en voru a\u00f0 nokkru var\u00f0veitt \u00ed austr\u00f3mverska r\u00edkinu, B\u00fdsans, og \u00ed Mi\u00f0austurl\u00f6ndum.<\/p>\n\n\n\n

B\u00f6\u00fe\u00edus, sem uppi var um 500 e.Kr., er talinn hafa \u00fe\u00fdtt \u00e1 lat\u00ednu b\u00f3k eftir N\u00edk\u00f3makkos fr\u00e1 Gerasa sem var uppi um 100 e.Kr. Borgin Gerasa var \u00fe\u00e1 \u00ed S\u00fdrlandi en er n\u00fa \u00ed J\u00f3rdan\u00edu. B\u00f3k N\u00edk\u00f3makkosar var ritu\u00f0 \u00e1 gr\u00edsku og nefndist Frumatri\u00f0i talnafr\u00e6\u00f0innar<\/em> (gr. Arithm\u0113tik\u0113 eisag\u014dg\u0113<\/em>). \u00de\u00fd\u00f0ing B\u00f6\u00fe\u00edusar var notu\u00f0 til kennslu \u00ed d\u00f3mkirkjusk\u00f3lum eftir a\u00f0 menntastarfi r\u00f3mversk-ka\u00fe\u00f3lsku kirkjunnar \u00f3x \u00e1smegin.<\/p>\n\n\n\n

Fr\u00e1 n\u00edundu til t\u00f3lftu aldar var texti B\u00f6\u00fe\u00edusar meginkennsluefni\u00f0 \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i. \u00dear var fjalla\u00f0 um talnaspeki, sl\u00e9ttar t\u00f6lur og oddat\u00f6lur, pr\u00edmt\u00f6lur og samsettar t\u00f6lur, \u00edtarlega flokkun \u00e1 hlutf\u00f6llum og mismunandi me\u00f0alt\u00f6l: hreint me\u00f0altal (e. arithmetic mean), faldme\u00f0altal (e. geometric mean) og \u00fe\u00fdtt me\u00f0altal (e. harmonic mean). N\u00e1msefni\u00f0 \u00ed r\u00famfr\u00e6\u00f0i f\u00f3lst \u00ed skilgreiningum, frumsendum og setningum \u00far I. og II. b\u00f3k Frum\u00fe\u00e1tta<\/em>, nokkrum s\u00f6nnunum og flatarm\u00e1lsreglum. \u00deannig b\u00e1rust Frum\u00fe\u00e6ttir<\/em> sm\u00e1m saman til baka til Evr\u00f3pu, fyrst \u00ed brotum \u00far gr\u00edsku en s\u00ed\u00f0ar \u00ed heilsteyptri \u00fe\u00fd\u00f0ingu \u00far arab\u00edsku.<\/p>\n\n\n\n

Hinn hluti R\u00f3mar\u00edkis var\u00f0 austr\u00f3mverska r\u00edki\u00f0, B\u00fdsansr\u00edki\u00f0 e\u00f0a Miklagar\u00f0sr\u00edki\u00f0, \u00fear sem r\u00e9tttr\u00fana\u00f0arkirkjan r\u00edkti. B\u00fdsansr\u00edki\u00f0 st\u00f3\u00f0 fram til 1453. Mi\u00f0st\u00f6\u00f0 \u00feess var \u00ed Konstant\u00edn\u00f3pel, Miklagar\u00f0i. \u00dear geymdist gr\u00edski st\u00e6r\u00f0fr\u00e6\u00f0iarfurinn og um s\u00ed\u00f0ir barst hann til H\u00fass viskunnar \u00ed kal\u00edfar\u00edki m\u00faslima \u00ed Bagdad. H\u00fas viskunnar, Bayt Ul-Hikma, var b\u00f3kasafn og \u00fe\u00fd\u00f0ingarsetur \u00ed Bagdad, stofna\u00f0 \u00e1 t\u00edmum Abbassida kal\u00edfaveldisins. B\u00f3kasafni\u00f0 geymdi me\u00f0al annars \u00fe\u00fd\u00f0ingar \u00e1 fr\u00e6\u00f0iritum sem voru upphaflega ritu\u00f0 \u00e1 gr\u00edsku og sanskr\u00edt. B\u00f3kasafni\u00f0 skapa\u00f0i \u00fev\u00ed hagst\u00e6\u00f0 skilyr\u00f0i fyrir samruna austr\u00e6nnar og vestr\u00e6nnar menningar.<\/p>\n\n\n\n

\u00c1 t\u00edmabilinu 700 \u2013 900 e.Kr. h\u00f6f\u00f0u \u00fej\u00f3\u00f0ir Mi\u00f0austurlanda \u00fe\u00fdtt gr\u00edsk rit \u00e1 arab\u00edsku, me\u00f0al annars Frum\u00fe\u00e6tti<\/em> Evkl\u00ed\u00f0s, en einnig rit Arkimedesar og fleiri gr\u00edskra h\u00f6funda, og b\u00e6tt vi\u00f0 \u00feau. Framhald s\u00f6gunnar er s\u00ed\u00f0an a\u00f0 forngr\u00edsku ritin \u00ed arab\u00edskum b\u00faningi t\u00f3ku a\u00f0 berast til Evr\u00f3pu \u00ed gegnum Sp\u00e1n eftir a\u00f0 kal\u00edfaveldinu t\u00f3k a\u00f0 hnigna \u00fear \u00e1 12 \u00f6ld. Latneskar \u00fe\u00fd\u00f0ingar \u00e1 gr\u00edsku og arab\u00edsku ritunum ollu mikilli vakningu Evr\u00f3pu \u00ed st\u00e6r\u00f0fr\u00e6\u00f0ilegum efnum, og h\u00fan hefur sta\u00f0i\u00f0 allar g\u00f6tur s\u00ed\u00f0an.<\/p>\n\n\n\n

Um efni Frum\u00fe\u00e1tta<\/em> var fjalla\u00f0 \u00e1 \u00f3l\u00edka vegu eftir menningarsamhengi. Mynd 3 s\u00fdnir hluta \u00far handriti \u00e1 gr\u00edsku fr\u00e1 9. \u00f6ld um setningu 2 \u00ed b\u00f3k VIII, sem segir fr\u00e1 hlutfallinu 3\/2 milli talnanna 8 \u2013 12 \u2013 18 \u2013 27.<\/p>\n\n\n

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Mynd 3.<\/strong> \u00dar handritinu Vat. Gr. 190 (codex P), fol. 115v. fr\u00e1 9. \u00f6ld. \u00dea\u00f0 er rita\u00f0 \u00e1 gr\u00edsku og var\u00f0veitt \u00ed Vat\u00edkaninu. Ofan vi\u00f0 efstu r\u00f6\u00f0 l\u00f3\u00f0r\u00e9ttu strikanna standa 2 og 3, ofan vi\u00f0 mi\u00f0r\u00f6\u00f0ina 4,  6, og 9, og ofan vi\u00f0 ne\u00f0stu r\u00f6\u00f0ina 8, 12, 18, og 27. <\/figcaption><\/figure>\n<\/div>\n\n\n

Umfj\u00f6llun um setningu 2 \u00ed b\u00f3k VIII er einnig a\u00f0 finna \u00ed ritger\u00f0inni Algorismus<\/em>, sem talin er \u00fe\u00fddd \u00e1 \u00edslensku \u00far lat\u00ednu um mi\u00f0ja 13. \u00f6ld, en handriti\u00f0 AM 736 III 4to er tali\u00f0 rita\u00f0 um 1550. \u00dear birtast s\u00f6mu t\u00f6lur \u00ed talnaspekilegum b\u00faningi, l\u00edtt tengdar aflei\u00f0slu. T\u00f6lurnar 8 \u2013 12 \u2013 18 \u2013 27 eru tengdar vi\u00f0 h\u00f6fu\u00f0skepnurnar eld \u2013 loft \u2013 vatn \u2013 j\u00f6r\u00f0 (lat. ignis \u2013 aer \u2013 aqua \u2013 terra).<\/p>\n\n\n

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Mynd 4.<\/strong> Tilv\u00edsun \u00ed 8. b\u00f3k Frum\u00fe\u00e1tta<\/em> \u00ed ritger\u00f0inni Algorismus <\/em>sem \u00fe\u00fddd var \u00e1 \u00edslensku \u00e1 13. \u00f6ld (skr\u00e1\u00f0 \u00ed AM 736 III 4to fr\u00e1 um 1550).<\/figcaption><\/figure>\n<\/div>\n\n\n

Ofan vi\u00f0 hringinn \u00e1 Mynd 4, \u00fear sem stendur ignis <\/em>(eldur) stendur \u201etres trium tres (\u00ferisvar \u00fer\u00edr \u00ferisvar)<\/em>\u201c (27), vi\u00f0 hringinn me\u00f0 aer <\/em>(loft) stendur \u201etres trium bis xviii (\u00ferisvar \u00fer\u00edr tvisvar)<\/em>\u201c (18), vi\u00f0 aqua<\/em> (vatn) m\u00e1 sj\u00e1 \u201ebis bini tres xii (tvisvar tveir \u00ferisvar)<\/em>\u201c (12). \u00cd hringjunum sex \u00e1 mi\u00f0ri mynd m\u00e1 sj\u00e1 tilv\u00edsanir \u00ed ge\u00f0slag f\u00f3lks: acutus, subtilis, mobilis, obtusus, corpulus, immobilis <\/em>(lauslega \u00fe\u00fdtt: \u00e1kafur, vi\u00f0kv\u00e6mur, hviklyndur, deigur\/daufur, \u00feunglamalegur, \u00f3bifanlegur). Lesandanum er l\u00e1ti\u00f0 eftir a\u00f0 huglei\u00f0a merkingu \u00feessara or\u00f0a um mannlega eiginleika og sko\u00f0a tengsl \u00feeirra \u00e1 myndinni vi\u00f0 frumefnin eld, loft, vatn og j\u00f6r\u00f0.<\/p>\n\n\n\n

Kennslub\u00e6kur \u00ed r\u00famfr\u00e6\u00f0i fyrir framhaldssk\u00f3la voru gjarnan sni\u00f0nar eftir Frum\u00fe\u00e1ttum<\/em> langt fram \u00e1 tuttugustu \u00f6ld. Nefna m\u00e1 R\u00famfr\u00e6\u00f0i <\/em>eftir Julius Petersen sem kennd var vi\u00f0 Menntask\u00f3lann \u00ed Reykjav\u00edk og forvera hans, L\u00e6r\u00f0a sk\u00f3lann, \u00ed um eitt hundra\u00f0 \u00e1r (um 1870 \u2013 um 1970). \u00de\u00f3tt b\u00f3kin hafi \u00fe\u00f3tt nokku\u00f0 stirfin, m\u00e1 segja a\u00f0 uppbygging textans me\u00f0 aflei\u00f0slua\u00f0fer\u00f0 og aflei\u00f0slukerfi hafi lagt grundv\u00f6ll a\u00f0 \u00fej\u00e1lfun \u00ed r\u00f6klei\u00f0slu sem er eitt af meginvi\u00f0fangsefnum st\u00e6r\u00f0fr\u00e6\u00f0in\u00e1ms.<\/p>\n\n\n\n

Heimildir og frekara lesefni<\/strong><\/p>\n\n\n\n

Atli Har\u00f0arson. (2001). Fimm kaflar um s\u00f6gu r\u00famfr\u00e6\u00f0innar. https:\/\/atlivh.com\/textar\/stae\/fimm_kaflar_um_sogu_rumfraedinnar.pdf<\/a><\/p>\n\n\n\n

J\u00f3n \u00deorvar\u00f0arson (2005). Og \u00e9g skal hreyfa j\u00f6r\u00f0ina<\/em>. Reykjav\u00edk, ST\u00c6 ehf.<\/p>\n\n\n\n

Krist\u00edn Bjarnad\u00f3ttir. (2016, 29. apr\u00edl). Hvernig barst \u00feekking um st\u00e6r\u00f0fr\u00e6\u00f0i \u00e1 milli menningar\u00fej\u00f3\u00f0a \u00e1 mi\u00f0\u00f6ldum og hver var \u00fe\u00e1ttur Araba \u00ed \u00fev\u00ed?<\/em> V\u00edsindavefurinn. https:\/\/visindavefur.is\/svar.php?id=72148<\/a><\/p>\n\n\n\n

Krist\u00edn Bjarnad\u00f3ttir & Bjarni V. Halld\u00f3rsson (2021). Algorismus: Hindu-Arabic Arithmetic in GKS 1812 4to. \u00cd Gunnar Har\u00f0arson, Christian Etheridge, Gu\u00f0r\u00fan Nordal & Svanhildur \u00d3skarsd\u00f3ttir (ritstj.), A World in Fragments, Studies on the Encyclopedic Manuscript GKS 1812 4to<\/em>. Reykjav\u00edk, \u00c1rnastofnun.<\/p>\n\n\n\n

Krist\u00edn Bjarnad\u00f3ttir, Dirk De Bock, & Jo\u00e3o Bosco Pitombeira de Carvalho (2024). Chapter 23. The origin and evolution of the concept of mathematics curriculum. \u00cd M. A. (Ken)Clements, Berinderjeet Kaur, Thomas Lowrie, Vilma Mesa, & Johan Prytz (ritstj.), Fourth International Handbook of Mathematics Education<\/em>. Springer.<\/p>\n\n\n\n

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Krist\u00edn Bjarnad\u00f3ttir, fyrrum pr\u00f3fessor emerita<\/em> vi\u00f0 H\u00e1sk\u00f3la \u00cdslands<\/em><\/strong><\/p>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

Krist\u00edn Bjarnad\u00f3ttir. Ekki ver\u00f0ur lengi r\u00e6tt um \u00fe\u00e1tt st\u00e6r\u00f0fr\u00e6\u00f0innar \u00ed s\u00f6gu vestr\u00e6nnar menningar \u00e1n \u00feess a\u00f0 nafninu Evkl\u00ed\u00f0 breg\u00f0i fyrir. Hver var Evkl\u00ed\u00f0? \u00cd raun er l\u00edti\u00f0 vita\u00f0 um manninn. Hann bj\u00f3 \u00ed Alexandr\u00edu, gr\u00edskri n\u00fdlenduborg sem Alexander mikli stofna\u00f0i \u00e1ri\u00f0 325 f.Kr. vi\u00f0 \u00f3sa N\u00edlar \u00ed Egyptalandi. Tali\u00f0 er a\u00f0 Evkl\u00ed\u00f0 hafi veri\u00f0 uppi…<\/p>\n","protected":false},"author":1,"featured_media":5385,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[138,139,137,21,140],"class_list":["post-5374","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","tag-evklid","tag-frumthaettir","tag-soguhornid","tag-staerdfraedi","tag-thalesar-regla"],"_links":{"self":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/5374","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5374"}],"version-history":[{"count":14,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/5374\/revisions"}],"predecessor-version":[{"id":5415,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/5374\/revisions\/5415"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/media\/5385"}],"wp:attachment":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5374"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5374"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}