{"created":"2023-06-20T15:59:13.411747+00:00","id":4327,"links":{},"metadata":{"_buckets":{"deposit":"8d5d098e-be17-4c07-aa6c-2c88bcd6dc16"},"_deposit":{"created_by":17,"id":"4327","owners":[17],"pid":{"revision_id":0,"type":"depid","value":"4327"},"status":"published"},"_oai":{"id":"oai:minpaku.repo.nii.ac.jp:00004327","sets":["345:452"]},"author_link":["11337"],"item_9_biblio_info_7":{"attribute_name":"書誌情報","attribute_value_mlt":[{"bibliographicIssueDates":{"bibliographicIssueDate":"1989-01-27","bibliographicIssueDateType":"Issued"},"bibliographicIssueNumber":"3","bibliographicPageEnd":"719","bibliographicPageStart":"615","bibliographicVolumeNumber":"13","bibliographic_titles":[{"bibliographic_title":"国立民族学博物館研究報告"},{"bibliographic_title":"Bulletin of the National Museum of Ethnology","bibliographic_titleLang":"en"}]}]},"item_9_description_4":{"attribute_name":"抄録","attribute_value_mlt":[{"subitem_description":"A magic square is recorded in the Ta Tai Li Chi, compiled\nby Tai the Elder in the first century before or after the Christian\nera, in China. The book gives a sequence of the numbers 2, 9,\n4, 7, 5, 3, 6, 1, 8, which, when arranged in a square having three\nrows of three cells each, proves to be a magic square of order\nthree (Fig. 1.1). This is the first instance of magic squares so\nfar known to us. There are later occasional references to the\nsame magic square, or its variations, in Chinese literature, but\nit is not until the 13th century that magic squares of higher orders\nappear in China. Ting I-Tung discusses in his Ta Yen So Yin\n(ca. A.D. 1270) the relationship between numbers and I (divination),\nby using a number of diagrams made of numerical\nfigures and dots, among which occur magic squares of order\nthree and nine. Yang Hui records in his Hsii Ku Chai Ch'i Suan\nFa (written in A.D. 1274 but published in A.D. 1378) magic\nsquares of order three to ten.\nThe first example hitherto known of a magic square of\norder four occurs in the Brhatsanzhita (ca. A.D. 550), written by\nVarahamihira, an Indian authority on astronomy and divination\n(Fig. 1.3). The Kaksaputa, a work on magic ascribed to the\nfamous Buddhist philosopher Nagarjuna (2nd century A.D.),\ncontains a magic square of order four, but the authenticity of the\nwork is doubtful. Varahamihira's square is made of two sets of\nthe natural numbers 1 to 8. One of the four possible forms\n(Fig. 1.4a) of the original square reconstructed from Varahamihira's\nsquare, with a rotation of 90 degrees, coincides exactly with\nthe famous Islamic square (Fig. 1.5), that al-Bani (d. A.D. 1225)\nand al-Zinjani (ca. A.D. 1250) used frequently as a basic pattern\nfor talismans. This cannot be a mere coincidence because\n880 magic squares of order four are known to exist. This seems\nto indicate that magic squares were transmitted from India to\nthe Islamic world either directly, or, as in the case of chess\n(Indian caturariga), by way of Persia. It is also interesting that\nVarahamihira calls his square kacchaputa(=kaksaputa)or the\ncarapace of a turtle. This immediately reminds one of the\ntitle of the above-mentioned work ascribed to Nagarjuna, as well\nas of the old Chinese legend that, when the Emperor Yii visited\nthe river Lo, a miraculous turtle, on the back of which a diagram\ncalled the Lo Shu was written, came out of the river. The\ndiagram was interpreted as a magic square of order three by\nlater Chinese writers from the 10th century onward (Fig. 1.2a),\nalthough the original form of the Lo Shu itself can no longer be\nreconstructed on a well-documented basis.\nIn the Islamic world discovery of magic squares is often\nconnected with the ancient Greeks. According to al-Bani, for\nexample, the above-mentioned Islamic square of order four was\ninvented by the philosopher Plato. None of those Islamic\nlegends, however, is verified in the Greek literature. It is certain\nthat Theon of Smyrna (2nd century A.D.) made use of a natural\nsquare of nine cells in order to illustrate the significant position\nthat the number five occupies among the natural numbers from\none to nine (Fig. 1.6), but he seems not to have been aware of the\nconcept of magic squares.\nIn India a magic square of order three appears for the first\ntime in Vrnda's medical work, Siddhayoga (ca. A.D. 900), although\na legend asserts that Garga (in the first century before or after\nthe Christian era?), a legendary authority on divination, recommended\nthe use of magic squares of order three in order to\npacify the nine planets (navagrahas). Vrnda recommends his\nmagic square (Fig. 1.7) to women in labor for an easy delivery.\nThe same usage of magic squares was recorded also in the\nIslamic world from the 9th century A.D. onward. Al-Tabari,\nfor example, describes it in his medical work, Firdaus al-hikma\n(A.D. 850), and adds that it was his father's prescription.\nMagic squares of order five and higher appear for the first\ntime in the Rasa'il of the Ikhwan al-Safa' (ca. A.D. 983), an\nencyclopaedic work on Islamic theology. The book illustrates\nmagic squares of order three to nine. It seems that there was no\ngeneral method for constructing those squares, but the Muslims\nseem to have begun to investigate general construction methods\nat about the same time. In fact, several rules of Ibn al-Haytham\n(ca. A.D. 965-1039) and al-Isfard'ini (d. ca. A.D. 1120) have\nbeen handed down to us in an Arabic manuscript. It is, however,\nin the works of al-Buni and al-Zinjani that fully generalized\nmethods are stated. Interestingly, they flourished in the same\ncentury as Ting I-Tung and Yang Hui. About the same time\na magic square of order six, which had been constructed according\nto the framing method of the Muslims and incised on\nan iron plate with Arabic numerals, was transmitted to China.\nA little later in India Thakkura Pherii and Narayana gave\ngeneral methods in their mathematical works, Ganitasara (ca.\nA.D. 1315) and Ganitakaurnudi (A.D. 1356), respectively.\nNarayana, in particular, discusses magic squares in a systematic\nfashion, and correctly gives the number 384 of pandiagonal\nmagic squares of order four.\nTheir elder contemporary, Moschopoulos (ca. A.D. 1300),\nof Byzantin, also gave general methods, in which Islamic influences\nare evident. For example, he uses the reversed form of\nthe above-mentioned Islamic square of order four as a basic\npattern for constructing his evenly-even magic squares. Similarities\nwith Indian methods are also found in his methods. It is\nprobable that he transmitted magic squares from the Orient to\nEurope, but his exact role has yet to be investigated.\nAs has been mentioned above, Indian and Islamic peoples\nused magic squares of order three for magical effects in obstetrics,\nand in China magic squares were studied in connection with I\n(divination). Magic squares certainly had \"magical\" significances\nin those days, and it is highly probable that knowledge\nof magic squares, and especially their construction methods, were\ntransmitted only orally. That generalized construction methods\nbegan to be written down and published in the 13th and the 14th\ncenturies in the Islamic world, China, Byzantin, and India may\nimply that magic squares were losing their secrecy almost\nsimultaneously throughout the Old World.\nIn Europe \"planetary\" squares, which too have their roots in\nthe Islamic world, are mentioned even in the 15th and the 16th\ncenturies, but from the 17th century onward magic squares began\nto attract purely mathematical minds, such as Fermat, Frenicle\nand Euler. It is in the same centuries that Japanese mathematicians,\nTakakazu Seki (A.D. 1642-1708) and others, began to study\nthem under the influence of the Yang Hui Suan Fa and Ch'eng\nTa-Wei's Suan Fa T'ung Tsung (A.D. 1593).\nI have restricted this study to the periods before the 17th\ncentury. It should be noted also that, except for Indian literature,\nI have relied mainly on secondary sources. This is\nespecially so in the case of Arabic literature, for which I owe\nmuch to Ahrens, Bergstrasser, Hermelink, Saidan, Cammann,\nSchuster, and Sesiano, through their articles.\nChapter 1 is an introduction, and gives a brief sketch of the\nhistory of magic squares before the 17th century; Chapter 2 is\na chronological table of authors of magic squares; Chapter 3 is\nan alphabetical list of the authors referred to in Chapter 2\n(under each item the following information is given: 1) original\nsources, 2) descriptions of the magic squares, 3) secondary\nsources, and 4) references to the list in Chapter 5). In Chapter\n4 I classify the construction methods actually prescribed by the\nancient and medieval authors, and describe each of them with\nillustrations; and Chapter 5 is an annotated list of all the magic\nsquares, as far as I know, that belong to the periods before the\n17th century, according to their dimensions and constant sums.","subitem_description_type":"Abstract"}]},"item_9_identifier_registration":{"attribute_name":"ID登録","attribute_value_mlt":[{"subitem_identifier_reg_text":"10.15021/00004319","subitem_identifier_reg_type":"JaLC"}]},"item_9_publisher_33":{"attribute_name":"出版者","attribute_value_mlt":[{"subitem_publisher":"国立民族学博物館"}]},"item_9_publisher_34":{"attribute_name":"出版者（英）","attribute_value_mlt":[{"subitem_publisher":"National Museum of Ethnology"}]},"item_9_source_id_10":{"attribute_name":"書誌レコードID","attribute_value_mlt":[{"subitem_source_identifier":"AN00091943","subitem_source_identifier_type":"NCID"}]},"item_9_source_id_8":{"attribute_name":"ISSN","attribute_value_mlt":[{"subitem_source_identifier":"0385-180X","subitem_source_identifier_type":"ISSN"}]},"item_9_version_type_16":{"attribute_name":"著者版フラグ","attribute_value_mlt":[{"subitem_version_resource":"http://purl.org/coar/version/c_970fb48d4fbd8a85","subitem_version_type":"VoR"}]},"item_creator":{"attribute_name":"著者","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"林, 隆夫"},{"creatorName":"ハヤシ, タカオ","creatorNameLang":"ja-Kana"},{"creatorName":"Hayashi, Takao","creatorNameLang":"en"}],"nameIdentifiers":[{}]}]},"item_files":{"attribute_name":"ファイル情報","attribute_type":"file","attribute_value_mlt":[{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2015-11-19"}],"displaytype":"detail","filename":"KH013_3_004.pdf","filesize":[{"value":"4.3 MB"}],"format":"application/pdf","licensetype":"license_note","mimetype":"application/pdf","url":{"label":"KH013_3_004.pdf","url":"https://minpaku.repo.nii.ac.jp/record/4327/files/KH013_3_004.pdf"},"version_id":"72692e03-5d81-463a-bcda-225dfe4e7308"}]},"item_language":{"attribute_name":"言語","attribute_value_mlt":[{"subitem_language":"jpn"}]},"item_resource_type":{"attribute_name":"資源タイプ","attribute_value_mlt":[{"resourcetype":"departmental bulletin paper","resourceuri":"http://purl.org/coar/resource_type/c_6501"}]},"item_title":"方陣の歴史 : 16世紀以前に関する基礎研究","item_titles":{"attribute_name":"タイトル","attribute_value_mlt":[{"subitem_title":"方陣の歴史 : 16世紀以前に関する基礎研究"},{"subitem_title":"A Preliminary Study in the History of Magic Squares before the Seventeenth Century","subitem_title_language":"en"}]},"item_type_id":"9","owner":"17","path":["452"],"pubdate":{"attribute_name":"公開日","attribute_value":"2010-02-16"},"publish_date":"2010-02-16","publish_status":"0","recid":"4327","relation_version_is_last":true,"title":["方陣の歴史 : 16世紀以前に関する基礎研究"],"weko_creator_id":"17","weko_shared_id":-1},"updated":"2023-06-20T19:27:54.670115+00:00"}