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Aryabhata의 제곱근 풀이법 -Aryabhatiya를 중심으로-
차유만 ( You Man Cha ),정승석 ( Seung Suk Jung ) 인도철학회 2013 印度哲學 Vol.0 No.39
The paper is to explore the square roots computation in the Aryabhatiya written by Aryabhata(476-550 CE), the first mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. The difficulty of his explanation on the square roots computation is not from mathematical complications but from implicated descriptions of Sanskrit. In addition, his computations have been in- included cluded numerous exceptional rules. Aryabhata`s regulations implicated in the verse of the Aryabhatiya 2.4 can be basically applied under the square root of five digits. Thorough the application of his calcu-lations, lations, however, it is possible to be calculated the square root of higher oder. For calculating of the square roots Aryabha.a accurately provides the method that the answer can be obtained by repeatedly calculations of its approximations. For example, in order to calculation more than five dig-its of the square the process of root seventeen steps calcu-lations lations is necessary stages. the application of these methods can solved the approximation of irrational number square roots. Thus, it would be understood that the exceptional rules for the necessity of practical application are due to narrow down the approximate estimate for calculating the square roots. This calculated methods do not appeared in any liter-atures atures before the Aryabhatiya of Aryabhataya. Accordingly, it can not be determinated whether the methods is Aryabhata`s thought or is existed previous Aryabhata. Nevertheless, it is noticeably evaluated the worthy of mathematical performance that the calculating of the square roots of single figures can be extended to be solved the higher digits numbers` one. Like identifying in this paper, the rules of calculating appeared in Indian mathematical literatures would be indis-pensably needed for more accurate examinations. Especially it has to be needed that for applied strong implicated de-scriptions scriptions in Sanskrit literatures has to be preceded by careful and diversified methods investigation.
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<특집논문> 불교문화와 공간성의 문제 : 법현(法顯)이 적용한 Yojana에 대한 재검토
차유만 ( You Man Cha ),정승석 ( Seung Suk Jung ) 동아시아불교문화학회 2014 동아시아불교문화 Vol.0 No.17
유순(yojana)은 거리를 계량하는 기본 단위로서 인도에서 고대로부터 통용되어 왔다. 그러나 1유순의 거리는 학자들 사이에서 4km로부터 23km로까지 산정될 만큼 이견이 분분하여 아직까지도 정설로 확정되어 있지 않다. 그러나 비교적 최근에 일본에서는 모리 쇼지(森 章司)와 모토자와 츠 나오(本澤 綱夫)의 공동 연구로 1유순의 거리를 면밀하게 고찰하고 현 지 답사로 그 거리를 계량했다. 이들은 11km대의 거리가 1유순일 것이 라는 결론을 도출해 낸 동시에, 절대 단위로서 1유순의 거리가 13km일 개연성이 있다는 견해를 제시했다. 이처럼 다소 납득하기 어려운 견해 는 자료 분석의 방식에서 기인한 것으로 생각된다. 본 논문은 이 같은 선행 연구에서 제공하는 자료들을 충분히 활용하 면서 1유순의 거리에 대한 계량을 독자적인 방식으로 시도했다. 이로써 과거 인도에서 보편적으로 통용되었던 1유순의 거리를 재확인하고자 한 다. 이를 위해 우선 6세기 전후에 인도의 수학자로 활동했던 아리야바 타(Aryabhata)가 정의한 1유순의 거리를 파악했다. 이어서 현장(玄奬) 과 의정(義淨)의 기술에 따른 1유순의 거리를 검토했다. 이후 1유순의 계량에 가장 신뢰할 만한 정보를 제공하는 법현(法顯)의 기록 중 비교 적 명료한 구간을 선별하여 선행 연구의 결과를 재검토했다. 선행 연구에서는 체감에 따라 편차가 적지 않은 계량을 일괄하여 평 균한 수치로 1유순의 거리를 산정했다. 그러나 본 논문에서는 1유순으 로 계량된 거리의 구간별 분포를 고려하여, 체감도가 가장 유사한 거리 들에서 개연성이 큰 수치를 1유순으로 도출하였다. 이 결과, 과거 인도 에서는 약 13.5km가 1유순의 보편적인 거리였을 것으로 추정된다. 이 거리는 아리야바타가 정의한 1유순의 거리(13.54km)와 거의 동일할 뿐 만 아니라, 현장과 의정의 기술에 의거하여 도출한 1유순의 거리 (13.67km)와도 근사하다. Yojana had been used in India from ancient times as a measure of length. It, in particular, was the basic unit of measuring distance. But there are so many views about the length of one yojana insomuch as one yojana is reckoned to be 4km or up to 23km among scholars. So there is not the established theory about the length of one yojana up to now. On the other hand, two Japanese scholars named Shoji Mori(森 章 司) and Tsunao Motozawa(本澤 綱夫) collaborated in researching this issue a little recently. They researched the length of one yojana in all its aspects, and measured it by doing fieldwork. Here they drew a conclusion that the length of one yojana would be more than 11km not to 12km, and at the same time they suggested that the length of one yojana as an absolute unit was probable to be 13km. Such a somewhat unacceptable view seems to be caused by the method of data analysis. The present research tried to reckon the length of one yojana by our own methods, while making full use of data given by the preceding research. The purpose here is to reconfirm the distance of one yojana employed in the past India. For the purpose of this paper, in the beginning, we grasped the length of one yojana defined by Aryabhata who lived before and after the sixth century as a mathematician of India. And then we examined the length of one yojana according to the records of Xuanzang(玄奬) and Yijing(義淨). Lastly, we selected the relatively clear sections between two places through the records of Faxian(法顯) who provided information deserves the most trust to reckon one yojana, and we re-examined them together with the results of the preceding research. Two scholars of the preceding research calculated the average of all yojana-reckonings which are likely to be much different according to sensory factors. As a result, the average was reckoned to be one yojana by them. But we, in the present research, considered the distribution of distances which were reckoned as one yojana. As a result, we drew a numerical value which is highly probable to be one yojana from the most similar sensory distances. In the end, we come to the conclusion that the length of one yojana was very likely to be about 13.5km in the past India. This length is almost equal to 13.54km defined as one yojana by Aryabhata. Furthermore, it is approximately equal to the length of one yojana(i.e. 13.67km) drawn by the records of Xuanzang and Yijing.
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