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方陣の歴史 : 16世紀以前に関する基礎研究
https://doi.org/10.15021/00004319
https://doi.org/10.15021/00004319e49a0a8e-0c9a-4819-b361-570a06ec7f02
| 名前 / ファイル | ライセンス | アクション |
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KH013_3_004.pdf (4.3 MB)
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| Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||||||||
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| 公開日 | 2010-02-16 | |||||||||||
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| タイトル | 方陣の歴史 : 16世紀以前に関する基礎研究 | |||||||||||
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| タイトル | A Preliminary Study in the History of Magic Squares before the Seventeenth Century | |||||||||||
| 言語 | en | |||||||||||
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| 言語 | jpn | |||||||||||
| 資源タイプ | ||||||||||||
| 資源タイプ識別子 | http://purl.org/coar/resource_type/c_6501 | |||||||||||
| 資源タイプ | departmental bulletin paper | |||||||||||
| ID登録 | ||||||||||||
| ID登録 | 10.15021/00004319 | |||||||||||
| ID登録タイプ | JaLC | |||||||||||
| 著者 |
林, 隆夫
× 林, 隆夫
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| 内容記述タイプ | Abstract | |||||||||||
| 内容記述 | A magic square is recorded in the Ta Tai Li Chi, compiled by Tai the Elder in the first century before or after the Christian era, in China. The book gives a sequence of the numbers 2, 9, 4, 7, 5, 3, 6, 1, 8, which, when arranged in a square having three rows of three cells each, proves to be a magic square of order three (Fig. 1.1). This is the first instance of magic squares so far known to us. There are later occasional references to the same magic square, or its variations, in Chinese literature, but it is not until the 13th century that magic squares of higher orders appear in China. Ting I-Tung discusses in his Ta Yen So Yin (ca. A.D. 1270) the relationship between numbers and I (divination), by using a number of diagrams made of numerical figures and dots, among which occur magic squares of order three and nine. Yang Hui records in his Hsii Ku Chai Ch'i Suan Fa (written in A.D. 1274 but published in A.D. 1378) magic squares of order three to ten. The first example hitherto known of a magic square of order four occurs in the Brhatsanzhita (ca. A.D. 550), written by Varahamihira, an Indian authority on astronomy and divination (Fig. 1.3). The Kaksaputa, a work on magic ascribed to the famous Buddhist philosopher Nagarjuna (2nd century A.D.), contains a magic square of order four, but the authenticity of the work is doubtful. Varahamihira's square is made of two sets of the natural numbers 1 to 8. One of the four possible forms (Fig. 1.4a) of the original square reconstructed from Varahamihira's square, with a rotation of 90 degrees, coincides exactly with the famous Islamic square (Fig. 1.5), that al-Bani (d. A.D. 1225) and al-Zinjani (ca. A.D. 1250) used frequently as a basic pattern for talismans. This cannot be a mere coincidence because 880 magic squares of order four are known to exist. This seems to indicate that magic squares were transmitted from India to the Islamic world either directly, or, as in the case of chess (Indian caturariga), by way of Persia. It is also interesting that Varahamihira calls his square kacchaputa(=kaksaputa)or the carapace of a turtle. This immediately reminds one of the title of the above-mentioned work ascribed to Nagarjuna, as well as of the old Chinese legend that, when the Emperor Yii visited the river Lo, a miraculous turtle, on the back of which a diagram called the Lo Shu was written, came out of the river. The diagram was interpreted as a magic square of order three by later Chinese writers from the 10th century onward (Fig. 1.2a), although the original form of the Lo Shu itself can no longer be reconstructed on a well-documented basis. In the Islamic world discovery of magic squares is often connected with the ancient Greeks. According to al-Bani, for example, the above-mentioned Islamic square of order four was invented by the philosopher Plato. None of those Islamic legends, however, is verified in the Greek literature. It is certain that Theon of Smyrna (2nd century A.D.) made use of a natural square of nine cells in order to illustrate the significant position that the number five occupies among the natural numbers from one to nine (Fig. 1.6), but he seems not to have been aware of the concept of magic squares. In India a magic square of order three appears for the first time in Vrnda's medical work, Siddhayoga (ca. A.D. 900), although a legend asserts that Garga (in the first century before or after the Christian era?), a legendary authority on divination, recommended the use of magic squares of order three in order to pacify the nine planets (navagrahas). Vrnda recommends his magic square (Fig. 1.7) to women in labor for an easy delivery. The same usage of magic squares was recorded also in the Islamic world from the 9th century A.D. onward. Al-Tabari, for example, describes it in his medical work, Firdaus al-hikma (A.D. 850), and adds that it was his father's prescription. Magic squares of order five and higher appear for the first time in the Rasa'il of the Ikhwan al-Safa' (ca. A.D. 983), an encyclopaedic work on Islamic theology. The book illustrates magic squares of order three to nine. It seems that there was no general method for constructing those squares, but the Muslims seem to have begun to investigate general construction methods at about the same time. In fact, several rules of Ibn al-Haytham (ca. A.D. 965-1039) and al-Isfard'ini (d. ca. A.D. 1120) have been handed down to us in an Arabic manuscript. It is, however, in the works of al-Buni and al-Zinjani that fully generalized methods are stated. Interestingly, they flourished in the same century as Ting I-Tung and Yang Hui. About the same time a magic square of order six, which had been constructed according to the framing method of the Muslims and incised on an iron plate with Arabic numerals, was transmitted to China. A little later in India Thakkura Pherii and Narayana gave general methods in their mathematical works, Ganitasara (ca. A.D. 1315) and Ganitakaurnudi (A.D. 1356), respectively. Narayana, in particular, discusses magic squares in a systematic fashion, and correctly gives the number 384 of pandiagonal magic squares of order four. Their elder contemporary, Moschopoulos (ca. A.D. 1300), of Byzantin, also gave general methods, in which Islamic influences are evident. For example, he uses the reversed form of the above-mentioned Islamic square of order four as a basic pattern for constructing his evenly-even magic squares. Similarities with Indian methods are also found in his methods. It is probable that he transmitted magic squares from the Orient to Europe, but his exact role has yet to be investigated. As has been mentioned above, Indian and Islamic peoples used magic squares of order three for magical effects in obstetrics, and in China magic squares were studied in connection with I (divination). Magic squares certainly had "magical" significances in those days, and it is highly probable that knowledge of magic squares, and especially their construction methods, were transmitted only orally. That generalized construction methods began to be written down and published in the 13th and the 14th centuries in the Islamic world, China, Byzantin, and India may imply that magic squares were losing their secrecy almost simultaneously throughout the Old World. In Europe "planetary" squares, which too have their roots in the Islamic world, are mentioned even in the 15th and the 16th centuries, but from the 17th century onward magic squares began to attract purely mathematical minds, such as Fermat, Frenicle and Euler. It is in the same centuries that Japanese mathematicians, Takakazu Seki (A.D. 1642-1708) and others, began to study them under the influence of the Yang Hui Suan Fa and Ch'eng Ta-Wei's Suan Fa T'ung Tsung (A.D. 1593). I have restricted this study to the periods before the 17th century. It should be noted also that, except for Indian literature, I have relied mainly on secondary sources. This is especially so in the case of Arabic literature, for which I owe much to Ahrens, Bergstrasser, Hermelink, Saidan, Cammann, Schuster, and Sesiano, through their articles. Chapter 1 is an introduction, and gives a brief sketch of the history of magic squares before the 17th century; Chapter 2 is a chronological table of authors of magic squares; Chapter 3 is an alphabetical list of the authors referred to in Chapter 2 (under each item the following information is given: 1) original sources, 2) descriptions of the magic squares, 3) secondary sources, and 4) references to the list in Chapter 5). In Chapter 4 I classify the construction methods actually prescribed by the ancient and medieval authors, and describe each of them with illustrations; and Chapter 5 is an annotated list of all the magic squares, as far as I know, that belong to the periods before the 17th century, according to their dimensions and constant sums. |
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| 書誌情報 |
国立民族学博物館研究報告 en : Bulletin of the National Museum of Ethnology 巻 13, 号 3, p. 615-719, 発行日 1989-01-27 |
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| 出版者 | ||||||||||||
| 出版者 | 国立民族学博物館 | |||||||||||
| 出版者（英） | ||||||||||||
| 出版者 | National Museum of Ethnology | |||||||||||
| ISSN | ||||||||||||
| 収録物識別子タイプ | ISSN | |||||||||||
| 収録物識別子 | 0385-180X | |||||||||||
| 書誌レコードID | ||||||||||||
| 収録物識別子タイプ | NCID | |||||||||||
| 収録物識別子 | AN00091943 | |||||||||||
| 著者版フラグ | ||||||||||||
| 出版タイプ | VoR | |||||||||||
| 出版タイプResource | http://purl.org/coar/version/c_970fb48d4fbd8a85 | |||||||||||
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Cite as
Hayashi, Takao, 1989, A Preliminary Study in the History of Magic Squares before the Seventeenth Century: 国立民族学博物館, 615–719 p.
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