Mathematics of Seki Takakazu

Total Page：16

File Type：pdf, Size：1020Kb

13-4 Mathematics of Seki Takakazu Kenji Ueno The Seki Kowa Institute of Mathematics, Yokkaichi University Japanese mathematics of the Edo period started to develop into a new direction beyond Chinese classical mathematics when SEKI Takakazu (1645? - 1708) introduced a method to express algebraic equations of many unknowns (傍書法). Nevertheless, SEKI’s mathematics was based on Chinese classical mathematics. Indeed SEKI studied the Chinese mathematical book “Yang Hui Suanfa” (楊輝算法) carefully and he even composed its revised version. We can fi nd several evidences that SEKI was deeply infl uenced by the “Yang Hui Suanfa”. SEKI was the fi rst mathematician of the East Asian cultural sphere who developed general theories of mathematics. For example, he founded a theory of equations of one known formulating rigorously Horner’s method and more importantly the elimination theory of simultaneous equations of higher degree with many unknowns. Before him an equation had been regarded as only a tool to solve problems and was not the object of mathematical investigations. Although SEKI could not get any information of Western mathematics we can fi nd many similarities with SEKI’s mathematics and Western mathematics at that time. SEKI and the TAKEBE brothers, Katahiro and Kata-akira, who were the best students of SEKI planned to compose the mathematical books, which contain all the results they obtained and also all known results at that time. The fi rst version was completed at the end of the 18th century. But SEKI seemed not satisfactory of the books and he kept a distance from completing the fi nal version. The fi nal version was completed by TAKEBE Kata-akira after SEKI’s death and named “Taisei Sankei” (大成算経) consisting of 20 volumes. In my talk, I will explain about SEKI’s mathematics and its relationship with the “Yang Hui Suanfa” and “Suanxue Qimeng’’ (算学啓蒙). 152.

Copy Link

Recommended publications

Occasion of Receiving the Seki-Takakazu Prize
特集：日本数学会関孝和賞受賞 On the occasion of receiving the Seki-Takakazu Prize Jean-Pierre Bourguignon, the director of IHÉS A brief introduction to the Institut des Hautes Études Scientifiques The Institut des Hautes Études Scientifiques (IHÉS) was founded in 1958 by Léon MOTCHANE, an industrialist with a passion for mathematics, whose ambition was to create a research centre in Europe, counterpart to the renowned Institute for Advanced Study (IAS), Princeton, United States. IHÉS became a foundation acknowledged in the public interest in 1981. Like its model, IHÉS has a small number of Permanent Professors (5 presently), and hosts every year some 250 visitors coming from all around the world. A Scientific Council consisting of the Director, the Permanent Professors, the Léon Motchane professor and an equal number of external members is in charge of defining the scientific strategy of the Institute. The foundation is managed by an 18 member international Board of Directors selected for their expertise in science or in management. The French Minister of Research or their representative and the General Director of CNRS are members of the Board. IHÉS accounts are audited and certified by an international accountancy firm, Deloitte, Touche & Tomatsu. Its resources come from many different sources: half of its budget is provided by a contract with the French government, but institutions from some 10 countries, companies, foundations provide the other half, together with the income from the endowment of the Institute. Some 50 years after its creation, the high quality of its Permanent Professors and of its selected visiting researchers has established IHÉS as a research institute of world stature.
[Show full text]
Historical Development of the Chinese Remainder Theorem
Historical Development of the Chinese Remainder Theorem SHEN KANGSHENG Communicated by C. TRUESDELL 1. Source of the Problem Congruences of first degree were necessary to calculate calendars in ancient China as early as the 2 na century B.C. Subsequently, in making the Jingchu [a] calendar (237,A.D.), the astronomers defined shangyuan [b] 1 as the starting point of the calendar. If the Winter Solstice of a certain year occurred rl days after shangyuan and r2 days after the new moon, then that year was N years after shangyuan; hence arose the system of congruences aN ~ rl (mod 60) ~ r2 (mod b), . ~ ' where a is the number of days in a tropical year and b the number of days in a lunar month. 2. Sun Zi suanjing [c] (Master Sun's Mathematical Manual) Sun Zi suanjing (Problem 26' Volume 3) reads: "There are certain things whose number is unknown. A number is repeatedly divided by 3, the remainder is 2; divided by 5, the remainder is 3; and by 7, the remainder is 2. What will the num- ber be ?" The problem can be expressed as x --= 2 (mod 3) ~ 3 (mod 5) ~- 2 (rood 7). SUN ZI solved the problem as we do, giving x ~ 140 + 63 -k 30 ~=- 233 ~ 23 (rood 105). 1 Shangyuan is a supposed moment that occurred simultaneously with the midnight of jiazi [v] (the first day of the 60 day cycle), the Winter Solstice and the new moon. 286 SHEN KANGSHENG In speaking of the algorithm yielding these addends in the solution he continued: "In general, for G1 ~ 0(mod 5) ~ 0 (mod 7) = 1 (rood 3), take G1 = 70, G2 ~ 0(rood 3) ~ 0(mod 7) ~ 1 (mod 5), take G2 = 21, G3 ~ 0 (mod 3) ~ 0 (mod 5) ~ (1 rood 7), take G3 = 15.
[Show full text]
ON MATHEMATICAL TERMINOLOGY: CULTURE CROSSING in NINETEENTH-CENTURY CHINA the History of Mathematical Terminologies in Nineteent
ANDREA BRÉARD ON MATHEMATICAL TERMINOLOGY: CULTURE CROSSING IN NINETEENTH-CENTURY CHINA INTRODUCTORY REMARKS: ON MATHEMATICAL SYMBOLISM The history of mathematical terminologies in nineteenth-century China is a multi-layered issue. Their days of popularity and eventual decline are bound up with the complexities of the disputes between the Qian-Jia School who were concerned with textual criticism and the revival of ancient native Chinese mathematical texts, and those scholars that were interested in mathematical studies per se1 (and which we will discuss in a more detailed study elsewhere). The purpose of the following article is to portray the ‘introduction’ of Western mathematical notations into the Chinese consciousness of the late nineteenth century as a major signifying event, both (i) in its own right within the context of the translation of mathematical writ- ings from the West; and (ii) with respect to the way in which the tradi- tional mathematical symbolic system was interpreted by Qing commentators familiar with Western symbolical algebra. Although one might assume to ﬁnd parallel movements in the commentatorial and translatory practices of one and the same person, nineteenth-cen- tury mathematical activities will be shown to be clearly compartmen- talized. In fact, for the period under consideration here, it would be more appropriate to speak of the ‘reintroduction’ of Western mathematical notations. Already in 1712 the French Jesuit Jean-François Foucquet S. J. (1665–1741) had tried to introduce algebraic symbols in his Aer- rebala xinfa ̅ (New method of algebra). This treatise, which was written for the Kangxi emperor, introduced symbolical 1 In particular, the ‘three friends discussing astronomy and mathematics’ (Tan tian san you ǆ ) Wang Lai (1768–1813), Li Rui (1763–1820) and Jiao Xun ǌ (1765–1814).
[Show full text]
Muramatsu's Method Aryabhata (499 AD) Brahmagupta (640 AD
π in Other Eastern Countries Japan India In the Edo Period of Japan (1603 to 1868), 3.16 was used for π. But Aryabhata (499 AD) as people recognized that this value was not accurate, different values for π were calculated. Wasan scholars such as Muramatsu Shigekiyo, Seki Takakazu, Kamata Toshikiyo, Takebe Katahiro, and Matsunaga Aryabhata usedp the perimeter of a 384-sided poly- Yoshisuke all made calculations of π, and accomplished results gon to find π ≈ 9:8684. Later Aryabhata would comparable to their European mathematician counterparts. publish an “approximation” for his square root value, 3.1416. Note that his approximation is Muramatsu’s Method more accurate than his official value! Shigekiyo Muramatsu published Sanso in 1663. Muramatsu served the Asano family and possibly had a math insti- Brahmagupta (640 AD) tute in Edo (present day Tokyo). In his Brahmagupta used a pattern of inscribed polygons that work, Sanso, Muramatsu arranged prob- increased in the number of sides to calculate the perime- lems published earlier in Japanese math- ter of a circle.p Using these data points, he came to believe ematics texts without answers. Mura- that π = 10: matsu classified these problems into dif- ferent levels according to how difficult he thought they’d be to learn. Asano Crest It is in this book that he also showed the calculation of π from the regular inscribed polygon of 32,768 sides. He correctly obtained the Ramanujan (1887-1920) value 3.1415926. Thus, this book was the first to contain a mathematical calculation of π in Japan. 1 Srinivasa Ramanujan found several formulas for π.
[Show full text]
Mathematical Philosophy of Takebe Katahiro
数理解析研究所講究録第 1831 巻 2013 年 57-65 57 Mathematical Philosophy of Takebe Katahiro Morimoto, Mitsuo 1 Seki Kowa Institute of Mathematics, Yokkaichi University Professor Emeritus, Sophia University Abstract Takebe Katahiro (1664-1739) was a Japanese mathematician, who exposed his philosophy on Mathematics itself and on Mathematical Research. In the Taisei Sankei (1711), he wrote Chapter 4 entitled the “Three Essentials” describing four classes of mathematical problems, the status of parameters in a problem and the classffica- tion of numbers. Note that his thought was based on Chinese traditional philosophy and on the achievement of Japanese mathematics in the early Edo period. Note also he recognized some numbers are not algebraic. In the Tetsujutsu Sankei (1722) he recommended the inductive heuristic method in mathe- matical research and recognized a mathematical research would be successful once the character of a mathematical object and that of a mathematician were accommodated to each other, com- paring his method of calculation of the circular $co$ efficient with that of his master Seki Takakazu and citing his own discovery of the infinite power series expansion formula of an inverse trigono- metric function. The Meiji Restoration (1868) was a tuming point in the history of Japan from the feudalism to the constitutional monarchism. The “ordinance on school system” (1872) of the new gov- emment defined mathematics in Japanese schools to be of “European style” thus abandoning the Japanese traditional mathematics. This policy was proved to be efficient at least for a half century and Takebe’s philosophy on Mathematics was buried in the complete oblivion. 1 Takebe Katahiro Takebe Katahiro 建部賢弘 (1664-1739) was one of great mathematicians in the Edo Period (1603- 1868) of Japan.
[Show full text]
A Review of the History of Japanese Mathematics
Revue d’histoire des mathématiques, 7 (2001), p. 137–155. A REVIEW OF THE HISTORY OF JAPANESE MATHEMATICS Tsukane OGAWA (*) ABSTRACT. — This review aims to introduce Japanese mathematics to a non- expert and a non-Japanese readership. It briefly characterizes mathematics in Japan, surveys its history, as it developed over the last century, and provides a large (if not exhaustive) bibliography of works in the primary European languages. RESUM´ E´.—APERÇ U SUR L’HISTOIRE DES MATHEMATIQUES´ JAPONAISES. Le but de cette note est de présenter les mathématiques japonaises à un public non spécialisé dans le domaine. Les mathématiques au Japon sont brièvement caractérisées, leur histoire, telle qu’elle s’est développée durant le dernier siècle, est passée en revue et finalement une importante bibliographie dans les principales langues européennes est proposée, même si elle ne peut prétendre à l’exhaustivité. 1. INTRODUCTION – NOT ONLY SANGAKU The custom of hanging sangaku ( ), wooden plates on which are inscribed mathematical problems and their answers, under the roofs of shrines in the Edo period (1603–1867) in Japan is familiar enough to have been illustrated and described in Scientific American, May 1998 [Rothman 1998]. It is, however, obvious that sangaku is not all there is to Japanese mathematics. It would be fallacious to consider that the essence of Japanese mathematics reveals itself in sangaku. As a beginning, this essay attempts briefly to introduce Japanese mathematics to a non-Japanese readership. It will answer the following questions : 1) What kind of mathematical disciplines developed in Japan ? 2) How were mathematical expressions written in vertical typesetting ? 3) How did the Japanese calculate ? 4) Is it true that, lacking proofs, Japanese mathematics was not logical ? (*) Texte re¸cu le 7 mars 1999, révisé le 11 novembre 2000.
[Show full text]
Cambridge University Press 978-1-108-70945-3 — Series and Products in the Development of Mathematics Vol 1, 2Nd Ed
Cambridge University Press 978-1-108-70945-3 — Series and Products in the Development of Mathematics Vol 1, 2nd ed. Ranjan Roy Index More Information Index Abel, Niels Henrik, 88, 90, 93, 95–99, 102–104, 137, Arithmetica Universalis, 118, 140, 142 220, 242, 253, 367, 368, 385, 387, 388, 394, 402, Ars Conjectandi, 26, 27 448, 449, 471, 543, 556, 562, 563, 569, 584, 600 Artin, Emil, 443, 486, 488, 496 Abel mean, 569, 579 Artis Analyticae Praxis, 117 Abhyankar, Shreeram, 172 Aryabhatyabhasya, 2, 3 Acosta, D. J., 142 Ash, J. Marshall, 656 Acta Eruditorum, 145, 150, 244, 271, 303, 305, 342 Askey, Richard, 458, 482, 498, 628, 659, 709, 710, Aepinus, Franz, 91 720 Ahlfors, Lars, 664 asymptotic series, 503, 506 Aiyar, T. V. V., 22 Aubrey, John, 58 Akhiezer, N. I., 712, 720 al-Biruni, 188 Babbage, Charles, 597, 599, 600, 616, 638 al-Haytham, 8, 20, 21, 24, 30, 241 Bag, Amulya Kumar, 20, 22 al-Karaji, 81 Baillaud, B., 76 al-Kashi, 81, 166 Barnes, E. W., 498, 664, 665 al-Samawal, 81 Barnes’s integral, 498, 666 d’Alembert, Jean, 87, 250, 251, 262, 265, 266, 307, 322, 337, 524–526, 536, 598, 599 Barrow, Isaac, 144, 167, 173, 301, 328, 329 algebraic analysis, 103, 254, 366, 596, 602 de Beaune, Florimond, 151, 249, 260, 328 Almagest, 21, 186 Beery, Janet, 212 Altmann, Simon, 720 Berkeley, George, 115 Analyse alg´ebrique, 88, 101–103, 181, 202, 374, 394 Berndt, Bruce, 48, 686 Analytical Society, 368, 599, 600, 638 Bernoulli, Daniel, 26, 218, 284, 307, 319–321, 327, Anderson, Alexander, 178 331, 332, 334, 335, 361–364, 366, 399, 400, 437, Anderson, G.
[Show full text]
Indigenous Japanese Mathematics, Wasan Murata Tamotsu
1 Indigenous Japanese Mathematics, Wasan Murata Tamotsu Historical Background and Prehistory Every civilization in history has its own fate: awakening, growing up, flourishing, and declining. Indeed, even the Greek civilization had once declined, and, after having sunk into oblivion, the Renaissance literally \revived" it. A tradition which started in antiquity and has persisted up to the present, if any, would be the Chinese tradition, whose most noteworthy features include several philosophical ideas and the invention of paper and Chinese characters (Kanji). The tradition of Japanese civilization, even if it is not so long and profound as that of the Chinese, has similar phases at any rate. It started fairly early, even though it does not possess any original religion or philosophy. As a matter of fact, the Japanese civilization developed as a satellite of China, but this does not mean that the Japanese simply imitated the Chinese. After having adopted and assimilated it, the Japanese began to advance and refine it in their own way, of which one characteristic is their particular sense of beauty, and another is perhaps a lack of original thinkers of the first order. Today the opinion seems to be widely accepted that a driving force of the rapid progress in modern Japan is found in the Meiji Restoration (1868) or, at the lat- est, in the Americanization after the defeat (1945). However, this opinion is not appropriate, because what supported that progress is a large amount of the cul- tural accumulations in history, particularly in the Edo period (under the Tokugawa shogunate; 1603{1867). In fact, at the beginning of the 6th century, books including Buddhist sutras (written in Kanji) were officially imported into Japan via Korea.
[Show full text]
PDF(Final Proof)
79-05.tex : 2018/8/21 (14:34) page: 83 Advanced Studies in Pure Mathematics 79, 2018 Mathematics of Takebe Katahiro and History of Mathematics in East Asia pp. 83–99 Determinants by Seki Takakazu from the group-theoretic viewpoint Naoki Osada Abstract. In the Kai Fukudai no Ho¯ (Methods of Solving Concealed Problems)re- revised in 1683, Seki Takakazu gave two eliminating procedures for a system of n polynomial equations of degree n−1. Both procedures are derived from a formula in which the determinant of the coefficient matrix of the system of equations vanishes. Seki called the first procedure the successive multiplica- tion of equations by coefficients of other equations (chikushiki koj¯ o¯). Because the first procedure was so complicated, he invented another procedure con- sisting of shuffles (koshiki¯ ) and oblique multiplications (shajo¯), an extension of the rule of Sarrus. Although there are some errors when n ≥ 5, we can prove that Seki’s conception of the second procedure is essentially correct. We can summarize that Seki’s second procedure is based on a coset decomposition of the sym- metric group Sn with respect to the dihedral subgroup Dn. We also clarify Seki’s terms forward (jun)andbackward (gyaku) which have not yet been explained by historians of mathematics. §1. Introduction The Kai Fukudai no Ho¯ [解伏題之法](Methods of Solving Concealed Problems) re-revised by Seki Takakazu [関孝和] (1640s–1708) in 1683 was passed down among students of his school in the form of manuscripts such as [15, 16]. A concealed problem (fukudai [伏題]) means the problem which is solved using a system of equations with several unknowns.
[Show full text]
Elementary Soroban Arithmetic Techniques in Edo Period Japan
Elementary Soroban Arithmetic Techniques in Edo Period Japan Rosalie Joan Hosking Tsukane Ogawa Mitsuo Morimoto Seki Kowa Institute of Seki Kowa Institute of Seki Kowa Institute of Mathematics Mathematics Mathematics Yokkaichi University Yokkaichi University Yokkaichi University In this article, we introduce methods for doing basic arithmetic on the Japanese abacus – known as the soroban – as used during the Edo period (1603 – 1868 CE). This introduction is informed by books one and two of the Taisei Sankei 大成算経 ‘Great Accomplishment of Mathematics’, a mathematical work compiled by Seki Takakazu, Takebe Katahiro, and Takebe Kataakira between 1683 and 1711 CE. I Introduction During the Japanese Edo period (1603 – 1868 CE), the most common tool for cal- culation was the Japanese bead abacus known as the soroban 算盤. This device was used by merchants, farmers, and mathematicians alike to calculate everything from the cost of exchanging silver into gold to calculating the value of π. In this article, we introduce the common techniques for doing basic arithmetic on the soroban in the traditional style, so modern students, mathematicians, and historians alike can see how this device was used for over 250 years in Japan. We also provide a series of exercises in Section III for those who desire to try applying the techniques of the soroban themselves to traditional Edo period problems. Before introducing the techniques for basic calculation on the soroban, we first briefly detail the history and layout of this device, as well as the nature of the number system used in the Edo period. I.1 Soroban I.1.1 History Since ancient times, the Japanese have performed arithmetic using calculation tools originating from China.
[Show full text]
Birth, Growth and Computation of Pi to Ten Trillion Digits Ravi P Agarwal1*, Hans Agarwal2 and Syamal K Sen3
Agarwal et al. Advances in Difference Equations 2013, 2013:100 http://www.advancesindifferenceequations.com/content/2013/1/100 R E S E A R C H Open Access Birth, growth and computation of pi to ten trillion digits Ravi P Agarwal1*, Hans Agarwal2 and Syamal K Sen3 *Correspondence: [email protected] Abstract 1Department of Mathematics, Texas A&M University-Kingsville, Kingsville, The universal real constant pi, the ratio of the circumference of any circle and its TX, 78363, USA diameter, has no exact numerical representation in a finite number of digits in any Full list of author information is number/radix system. It has conjured up tremendous interest in mathematicians and available at the end of the article non-mathematicians alike, who spent countless hours over millennia to explore its beauty and varied applications in science and engineering. The article attempts to record the pi exploration over centuries including its successive computation to ever increasing number of digits and its remarkable usages, the list of which is not yet closed. Keywords: circle; error-free; history of pi; Matlab; random sequence; stability of a computer; trillion digits All circles have the same shape, and traditionally represent the infinite, immeasurable and even spiritual world. Some circles may be large and some small, but their ‘circleness’,their perfect roundness, is immediately evident. Mathematicians say that all circles are similar. Before dismissing this as an utterly trivial observation, we note by way of contrast that not all triangles have the same shape, nor all rectangles, nor all people. We can easily imag- ine tall narrow rectangles or tall narrow people, but a tall narrow circle is not a circle at all.
[Show full text]
A Traditional Solution to a Nineteenth Century Japanese Temple Problem
Journal for History of Mathematics http://dx.doi.org/10.14477/jhm.2017.30.2.053 Vol. 30 No. 2 (Apr. 2017), 53–69 Solving Sangaku: A Traditional Solution to a Nineteenth Century Japanese Temple Problem Rosalie Joan Hosking This paper demonstrates how a nineteenth century Japanese votive temple prob- lem known as sangaku from Okayama prefecture can be solved using traditional mathematical methods of the Japanese Edo (1603–1868 CE). We compare a mod- ern solution to a sangaku problem from Sacred Geometry: Japanese Temple Problems of Tony Rothman and Hidetoshi Fukagawa with a traditional solution of Ōhara Toshi- aki (?–1828). Our investigation into the solution of Ōhara provides an example of traditional Edo period mathematics using the tenzan jutsu symbolic manipulation method, as well as producing new insights regarding the contextual nature of the rules of this technique. Keywords: sangaku, wasan, tenzan jutsu, boshō hō, Katayamahiko, Ōhara Toshiaki, Sanpō Tenzan Shinan. MSC: A01,A27 1 Introduction During the Japanese Edo period (1603–1868 CE), a unique practice of creating vo- tive tablets with mathematical content emerged. These tablets—known as sangaku 算額—contained a variety of geometrical problems and were dedicated to Shinto shrines and Buddhist temples throughout the country. This tradition remained largely hidden to historians until the publishing of Japanese Temple Problems: San Gaku by David Pedoe and Fukagawa Hidetoshi in 1989 CE [5]. This work provided a collection of problems taken directly from various sangaku, along with a selection of solutions using modern mathematical methods and termi- nology. This was followed up with the more extensive Sacred Mathematics—Japanese Rosalie Joan Hosking: School of Math.
[Show full text]