YBC 7289

YBC 7289 係一塊巴比倫泥板，出名在佢上面寫住六十進制入面開方2嘅約略數值。開方2係一個正方形邊長同對角線長度嘅比例，係數學入面基礎而重要嘅一個常數。刻住嘅數字換轉返做10進制嘅話準確度去到6個小數位，被譽為「現時已知古代世界最準嘅運算」。^[1]呢塊泥板相信係公元前1800至1600年間南美索不達米亞一個學生嘅作品。J. P. 摩根將佢捐咗畀耶魯巴比倫藏品系列。

內容

YBC 7289附上註解，正方形嘅邊上標上30，對角線標咗1 24 51 10，係六十進制數字，換轉十進制即係1.4142196...，係開方2嘅近似值；下邊嘅42 25 35，相當於42.4263888...，係邊長30嘅正方形對角線長度

泥板上面有一個正方形，連埋佢兩條對角線，其中一條邊標咗個六十進制嘅數字30，對角線就標咗兩個數字，第一個係1;24,51,10，相等於305470/216000 ≈ 1.414213，同開方2嘅誤差只係二百萬分之一；第二個係42;25,35 = 30547/720 ≈ 42.426，呢個數等於第一個數乘以30，係一個邊長為30嘅正方形嘅對角線長度近似值。^[2]

因為巴比倫用嘅60進制系統係無標示位值，所以泥板上邊嘅數字都有其他嘅演繹。例如邊長可以係代表0;30，即係30/60=1/2。喺呢個演繹之下，對角線上面第二個數就代表0;42,25,35=30547/43200 ≈ 0.70711，係 $1/{\sqrt {2}}$ 嘅一個近似值，而 $1/{\sqrt {2}}$ 就係邊長為1/2嘅正方形對角線嘅長度，呢個近似值嘅誤差比例同樣係二百萬分之一之內。David Fowler 同 Eleanor Robson 寫話「呢對係有幾何意義嘅倒數對」，但係同時佢哋亦都指出，雖然喺巴比倫數學入面倒數對好重要，因而令到呢個演繹好吸引，但係都有其他理據去質疑呢一個演繹。^[2]

泥板嘅背面被人擦咗一部分，不過Robson相信上面有一條相類似嘅長方形對角線長度問題，而邊長同對角線嘅比例係3:4:5。^[3]

詮繹

雖然YBC 7289嘅圖片成日都係好似上圖咁打斜展示出嚟，但係巴比倫人畫正方形嘅習慣係將四條邊沿住橫直方向畫嘅，而標咗數字嗰條邊喺頂。^[4]呢塊泥板圓形而且寫有大字，顯示佢應該係一塊「手上泥板」，係畀啲學生揸喺手上邊寫草稿用嘅。^[1]^[2]個學生好有可能係喺第二塊泥板到抄啲數字過嚟，不過喺另一塊泥板BM 96957 + VAT 6598到可以搵到計呢個數出嚟嘅迭代步驟。^[2]

1945年，Otto E. Neugebauer 同 Abraham Sachs 首先發現呢塊泥板嘅數學意義，^[2]^[5]「呢塊泥板展現咗古代世界最準確嘅運算」，準確度相當於十進制入邊六個小數位。^[1]其他巴比倫泥板上邊有六邊形同七邊形面積嘅運算，當中牽涉更加複雜嘅代數數，例如係 ${\sqrt {3}}$ 。^[2] ${\sqrt {3}}$ 呢個數字亦都係埃及一啲金字塔入邊嘅長度比例，不過YBC 7289上邊咁準確嘅數值顯示佢哋當年已經有能力去計呢啲數字出嚟，而唔係單憑實物比例估算。^[6]

托勒密喺佢本《天文學大成》入邊都有用到同一個近似值1;24,51,10，^[7]^[8]而佢無講呢個近似值係喺邊到嚟嘅，好可能呢個近似值喺當年已經廣為人知。^[7]

發掘同策展

現時唔知YBC 7289係喺美索不達米亞之中嘅邊到嚟嘅，不過佢嘅形狀同書寫風格顯可能係南部嚟嘅，製成時間係公元前1800年至1600年之間。^[1]^[2]

呢塊泥板係 J. P. 摩根 1909年捐畀耶魯大學嘅遺贈嘅一部份，成套遺贈組成咗耶魯巴比倫藏品系列。^[1]^[9]耶魯嘅文化遺產保育學院幫呢塊泥板整咗個電子模型，可以用嚟做3D打印。^[9]^[10]^[11]

睇埋

參考資料

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 Beery, Janet L.; Swetz, Frank J. (July 2012), "The best known old Babylonian tablet?", Convergence, Mathematical Association of America, doi:10.4169/loci003889
↑ ^2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 Fowler, David; Robson, Eleanor (1998), "Square root approximations in old Babylonian mathematics: YBC 7289 in context", Historia Mathematica, 25 (4): 366–378, doi:10.1006/hmat.1998.2209, MR 1662496
↑ Robson, Eleanor (2007), "Mesopotamian Mathematics", 出自 Katz, Victor J. (編), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, p. 143, ISBN 978-0-691-11485-9
↑ Friberg, Jöran (2007), Friberg, Jöran (編), A remarkable collection of Babylonian mathematical texts, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, p. 211, doi:10.1007/978-0-387-48977-3, ISBN 978-0-387-34543-7, MR 2333050
↑ Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series, American Oriental Society and the American Schools of Oriental Research, New Haven, Conn., p. 43, MR 0016320
↑ Rudman, Peter S. (2007), How mathematics happened: the first 50,000 years, Prometheus Books, Amherst, NY, p. 241, ISBN 978-1-59102-477-4, MR 2329364
↑ ^7.0 ^7.1 Neugebauer, O. (1975), A History of Ancient Mathematical Astronomy, Part One, Springer-Verlag, New York-Heidelberg, pp. 22–23, ISBN 978-3-642-61910-6, MR 0465672
↑ Pedersen, Olaf (2011), Jones, Alexander (編), A Survey of the Almagest, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, p. 57, ISBN 978-0-387-84826-6
↑ ^9.0 ^9.1 Lynch, Patrick (April 11, 2016), "A 3,800-year journey from classroom to classroom", Yale News, 喺2017-10-25搵到
↑ A 3D-print of ancient history: one of the most famous mathematical texts from Mesopotamia, Yale Institute for the Preservation of Cultural Heritage, January 16, 2016, 喺2017-10-25搵到
↑ Kwan, Alistair (April 20, 2019), Mesopotamian tablet YBC 7289, University of Auckland, doi:10.17608/k6.auckland.6114425.v1

[bs-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 Beery, Janet L.; Swetz, Frank J. (July 2012), "The best known old Babylonian tablet?", Convergence, Mathematical Association of America, doi:10.4169/loci003889

[fr-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 Fowler, David; Robson, Eleanor (1998), "Square root approximations in old Babylonian mathematics: YBC 7289 in context", Historia Mathematica, 25 (4): 366–378, doi:10.1006/hmat.1998.2209, MR 1662496

[robson-3] Robson, Eleanor (2007), "Mesopotamian Mathematics", 出自 Katz, Victor J. (編), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, p. 143, ISBN 978-0-691-11485-9

[friberg-4] Friberg, Jöran (2007), Friberg, Jöran (編), A remarkable collection of Babylonian mathematical texts, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, p. 211, doi:10.1007/978-0-387-48977-3, ISBN 978-0-387-34543-7, MR 2333050

[ns-5] Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series, American Oriental Society and the American Schools of Oriental Research, New Haven, Conn., p. 43, MR 0016320

[rudman-6] Rudman, Peter S. (2007), How mathematics happened: the first 50,000 years, Prometheus Books, Amherst, NY, p. 241, ISBN 978-1-59102-477-4, MR 2329364

[neuhist-7] 7.0 ^7.1 Neugebauer, O. (1975), A History of Ancient Mathematical Astronomy, Part One, Springer-Verlag, New York-Heidelberg, pp. 22–23, ISBN 978-3-642-61910-6, MR 0465672

[ped-8] Pedersen, Olaf (2011), Jones, Alexander (編), A Survey of the Almagest, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, p. 57, ISBN 978-0-387-84826-6

[y1-9] 9.0 ^9.1 Lynch, Patrick (April 11, 2016), "A 3,800-year journey from classroom to classroom", Yale News, 喺2017-10-25搵到

[y2-10] A 3D-print of ancient history: one of the most famous mathematical texts from Mesopotamia, Yale Institute for the Preservation of Cultural Heritage, January 16, 2016, 喺2017-10-25搵到

[renders-11] Kwan, Alistair (April 20, 2019), Mesopotamian tablet YBC 7289, University of Auckland, doi:10.17608/k6.auckland.6114425.v1

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