Seki, Founder of Modern Mathematics in Japan

Seki Takakazu is a Japanese mathematician in the early period of the Edo era. He is known as the first in the world to study the so-called “resultants and determinants.” However, one had not known so much about his life before the author’s investigation started about two years ago. On the occasion of the 300th posthumous anniversary of Seki, the author has found some significant records and facts concerning his family, especially his adoptive father and his first career. Now it is possible to write a “Curriculum Vitae of Seki Takakazu.”

Moreover, his professional career in the Kōfu fief is now clarified by a document added in proof at the end of the paper.

There are two major activities of mathematics—theorem-proving and algorithm-creating. While theorem-proving, which originated in ancient Greece, had been being the backbone of the deductive tradition in the history of mathematics, the algorithm-creating, which flourished in ancient and medieval East, formed a strong algorithmic trend in the evolution of mathematics. The main purpose of this paper is to argue the indispensable role in advancing the development of mathematics played by the algorithmic tradition as one of so-called

main lines of mathematical development.

Description of the algorithmic characters of the Oriental mathematics constitutes the first part of this paper. Some representative Chinese and Wasan algorithms in ancient and medieval times are observed and their modern implication is discussed. The second part of the paper analyzes the algorithmic tendency in the origin of the modern Western mathematics by taking Descartes’ geometry as a case study. Finally the author proposes some questions about the relation of ancient Oriental mathematics with main lines of mathematical development for further discussion.

The mathematical cuneiform tablet Plimpton 322 is one of the most important source materials in history of mathematics. It lists fifteen Pythagorean triples together with a certain table which suggests that one of the angles of a right-angled triangle decreases from

$$ 45^\circ\ \text{to}\ 31^\circ, $$

as O. Neugebauer explained in detail. However, we do not know the principle of constructing the fifteen triples and the true purpose of the table.

In this paper I shall clarify the mathematical meanings of a few technical terms which occur in the headings of the four columns of the tablet, and also the constructing principle of the listed numbers by analyzing Babylonian calculation methods. As a result, we can conclude that the Babylonian scribe of our tablet calculated the fifteen Pythagorean triples using the trigonometric table of the first column which was made by a kind of linear-interpolation.

Some parallels can be drawn between Archimedes and Liu Hui, Seki Takakazu, among other oriental mathematicians in ancient and medieval times, especially in terms of the concepts and methods concerning infinitesimal. It would be fantastic to make a comparative study along this direction, but this is not my task in this short talk. Instead, a focus will be put on Archimedes and his works as portrayed in Chinese literature of the Ming and Qing Dynasties.

The talk is based on a holistic investigation of the literature of the Ming and Qing Dynasties, with a timeline focused on two phases: first, the late Ming and early Qing period, i.e. the first half of the 17th century, when interest in calendar reform afforded an opportunity for the introduction into China of classical Western astronomical and mathematical knowledge; and second, following the conclusion of the Second Opium War in the late Qing Dynasty, that is, from 1861 to the onset of the 20th century, contemporary Western science was widely introduced into China. This period was also for Chinese mathematics a transition from the traditional to the modern phase.

Therefore the exact content of the talk should be “Archimedes and his Works in Chinese Literature of the Ming and Qing Dynasties.”

I will begin with mathematics, and then discuss mechanics and legends, before making a brief conclusion.

When discussing ancient mathematical theories, scholars often limit themselves to Greek mathematics and, especially to its axiomatic system, which they use as the standard to evaluate traditional mathematics in other cultures: whichever failed to form an axiomatic system is considered to be without theory. Therefore, even those scholars who highly praise the achievements in ancient Chinese mathematics consider that “the greatest deficiency in old Chinese mathematical thought was the absence of the idea of rigorous proofs” and that there is no formal logic in ancient Chinese mathematics; in particular it did not have deductive logic. They further contend that, “in the flight from practice into the realm of pure intellect, Chinese mathematics did not participate,” [5, p. 151] and conclude that Chinese mathematics has no theory.

I think that Liu Hui’s commentary (263 A.D.) to the Nine Chapters on the Mathematical Procedures, hereafter Nine Chapters, completely proved the formulas and solutions in Nine Chapters. It, mainly based on deductive logics, elucidated deep mathematical theories. Even though Nine Chapters itself does not contain mathematical reasoning and proofs, which is a major flaw in the pursuit of mathematical theory in the history of Chinese mathematics, there are certain correct abstract procedures that possess a general applicability which should be considered as mathematical theories in Chinese mathematics. Sir Geoffrey Lloyd, after explaining the difference between Liu Hui’s and Euclid’s mathematics, said “Mais cela ne signifie pas une absence d’intérêt pour la validation des résultats ou pour la recherche d’une systématisation” [3, préface, p. xi]. Based on the Nine Chapters, this article will discuss Liu Hui’s contribution to the mathematical theory in order to stimulate more fruitful discussions.

The alternative algorithm of the 7th problem in Liu Hui’s Sea Island Mathematical Canon is mathematically incorrect. In this short note we try to restore it, based on a characteristic of mathematical formulas by Liu Hui.

Mathematics and astronomical system of ancient Japan had been influenced by those of ancient Korea since the early 5th century, Ōjin period of Japan. It was also clear that the fundamental languages, especially the numeral and arithmetic terms of both Korea and Japan were based on the common linguistic ancestors. These strongly indicate that both nations possessed the very close way of thinking and the value in mathematics. However, after the 17th century, there was a major difference in their way of thinking of mathematics.

Comparative researches between eastern and western mathematics of ancient and mediaeval times revealed that there are two major activities of mathematics, that is, theorem-proving and equation-solving. Theorem-proving mainly originated from Greek mathematics, meanwhile, equation-solving was an important content of Chinese mathematics. The fact seems to lead to the conclusion that Chinese mathematics was basically based on the practical problems, and in the mean time European one on geometrical problems. Based on analyses of Chinese mathematical works, the author of the present paper argues that Chinese mathematicians paid more attentions to the design and the construction of their geometrical models different from those introduced in living practices, in particular, in the 13–14th centuries. Using their geometrical models, they constructed their mathematical contexts and problems to meet their needs in displaying their mathematical ideas and in breaking through the limitation of practical problems. The research style of Chinese mathematics, therefore, changed in some way and strengthened its theoretical aspect.

The

Suanxue Qimeng

(the Introduction of Mathematics) was written by Zhu Shijie in 1299 during the

Yuan

dynasty. The book had been lost in the

Ming

dynasty but was conserved in Korea and reprinted several times as mathematical textbook for the education of mathematical experts of the Korean dynasty. In the last decade of the 16th century a copy of the book was transferred to Japan. In the

Edo

period in Japan, the book was reprinted several times with annotation. The last chapter of the book explained the procedure of celestial element (

tianyuanshu

or

tengen jutsu

), a method for representing a polynomial of one variable with numerical coefficients by a column vector of its coefficients. Having mastered this procedure, Seki Takakazu generalized it in such a way that polynomials of several variables could be handled with.

Mādhava of San

$$ \dot{\rm n}$$

gamagrāma was a mathematical astronomer who flourished around 1400 A.D. in South India. Not a few mathematical formulas attributed to him have been transmitted by scholars of his school to these days. The most important among them are power series expansions of trigonometrical functions sine, cosine, arctangent, and so on. Because these series are not found in his extant astronomical treatises it is not always clear what parts of them were found by Mādhava himself, though some, the expansion of the circumference of a circle for example, can be attributed to him with a certainty.

After taking a glance at Mādhava himself and his school, I will explain in this paper howMādhava derived the power series of the circumference C of a circle with diameter d together with the last corrective term:

$$ C= \frac{4d}{1}-\frac{4d}{3}+\frac{4d}{5}-\frac{4d}{7}+\cdots+(-1)^{n-1} \frac{4d}{2n-1}+(-1)^n\cdot 4d \cdot\frac{n}{(2n)^2+1}, $$

according to the commentary

Kriyākramakarī

(ca. AD 1550), composed by Śa

$$ \dot{\rm n}$$

kara, who was a scholar situated near the end of the Mādhava school.

In 1934 Yoshio Mikami (1875–1950) published a paper devoted to the Vietnamese mathematical treatise Guide [towards] Understanding of Calculational Methods

Chi Minh Toan Phap

. His analysis of several topics discussed in the treatise (representation of numbers with counting rods, format of multiplication table, generic problems of various categories, etc.) allowed him to advance hypotheses concerning the origin and the time of compilation of the treatise. The book studied by Mikami nowadays is not available. In the present paper the author examines Mikami’s work and provides a description of the Vietnamese mathematical treatise

Chi Minh Lap Thanh Toan Phap

by Phan Huy Khuong (preface 1820) textually close to that investigated by Mikami.

Yoshida Mitsuyoshi (1598–1672) published the Jinkōki first in 1627. This was a problem book of elementary mathematics for everyday use but it also contained many interesting problems which attracted readers. This book became so popular that there have been more than 300 versions published during the

Edo

era (1603–1868) in Japan. In these notes, we shall survey the first edition of the Jinkōoki, and the problems which were added in later editions.

We shall describe briefly the contents of the Résumé of Works on Mathematics (

Katsuyō Sanpō

) of Seki Takakazu (ca.1640–1708). This monograph is the posthumous publication of the great mathematician of the

Edo

period and contains most of his representative works on mathematics.

In this paper, I shall explain a crucial point of Seki Takakazu’s argument which claims that there is a close relationship between the solid of revolution of an arc-figure (called here a finger ring) and an obliquely cut segment of a cylinder (called an

onglet

). I think his argument is the most brilliant; no one has ever attained his level. But his explanation is too concise; no one has ever found its proper interpretation. Several years ago I succeeded in deciphering his arguments in seeking the volume of solids of revolution for the first time and published it in a series of papers in Japanese. Here I shall show some of my interpretations with many figures for illustration.

It is known that a system of simultaneous congruences of first degree which will be henceforth called a remainder problem, first appeared in the Sunzi’s Arithmetical Canon (c.AD400). Later in China, remainder problems were discussed in many books, and some of these Chinese books were introduced into Japan by the seventeenth century. The eminent Japanese mathematician Seki Takakazu (c.1642–1708) investigated remainder problems adopting the term the art of cutting bamboo [jianguan shu] which is found in the Chinese book, Yang Hui’s Arts on Arithmetic [Yanghui Suanfa] (1275) by Yang Hui. Seki is supposed to have consulted the Chinese book, but Seki’s method is much more advanced than Yang Hui’s. Seki generalized the theory on the remainder problem and showed the procedure for the solution systematically. The aim of this paper is to analyze Seki’s method on the remainder problem in comparison with Chinese books, especially with the Mathematical Treatise in Nine Chapters [Shushu Jiuzhang] (1247) by Qin Jiushao.

It is usually said that Seki Takakazu (ca. 1642 – 1708) calculated the volume of a solid of revolution using the Pappus-Guldinus theorem. However, it is difficult to say that Seki was influenced by Mechanics. In fact, Seki calculated the volumes of all parts which constitute a figure and added them. We may claim that Seki found the Pappus-Guldinus theorem from these calculations and will explain Seki’s method of construction for mathematical objects.

As late as 1938, people doubted whether Gottfried Wilhelm Leibniz ever dealt with determinants. Thus, Gerhard Kowalewski wrote in 1938: “Strangely, nothing relating to determinants and their application has been found in his (viz. Leibniz’s) manuscripts until now” [19, p. 125]. Later on, Morris Kline, hinting at Leibniz’s often cited letter dating from 1693 to L’Hospital, erroneously wrote in 1972: “The solutions of simultaneous linear equations in two, three, and four unknowns by the method of determinants was created by Maclaurin, probably in 1729, and published in his posthumous Treatise of Algebra (1748)” [8, p. 606].

Seki Takakazu (1642?–1708) is a mathematician in the Edo Period (1603– 1868) of Japan. He was distinguished far from the other mathematicians in Japan at that time. We have so far failed to find any name of person who taught him mathematics in spite of all our efforts of investigations at this occasion of 300 years after his death. His disciples are few and his monumental treatise

Complete Book of Mathematics

(1683–1711) of approximately 1800 pages has practically been ignored by mathematicians who claimed themselves to be in Seki’s school of mathematicians and also by later historians of mathematics until these days. Yet he was not isolated not only in Japan but also in the world. We will show the evidence in what follows.

This informal paper, an expanded version of the short talk given by the second author at the conference on the occasion of the 300th anniversary of Seki’s death, is slightly non-standard in nature and perhaps requires a short explanatory preamble. The authors are not professional historians of mathematics, and no attempt has been made to interpret the material discussed from a historical viewpoint. Instead, the first section contains several specific mathematical comments, from the point of view of a contemporary professional mathematician (D.Z.), on a few of the problems and solutions of Seki and of his predecessors in China and Japan, pointing out places where the mathematical content is unexpectedly naive or unexpectedly sophisticated, or where particular mathematical features of the problems permit deductions about their authors’ methods or views.

Ming Antu (1692?–1763?), of Mongolian nationality, is a famous Chinese astronomer, mathematician and topographist. He began to work in the Imperial Observatory in 1713. He participated in the work of compiling and editing three very important books in astronomy and joined the team of China’s area measurement.

Seki Takakazu (ca. 1642–1708) and his pupils develop the Wasan [Japanese mathematics] into the most advanced mathematics in the world outside Europe in the Edo period (1603–1867), and mathematical achievements of Seki Takakazu and Takebe Katahiro (1664–1739) laid foundation for the Wasan. In order to commemorate this outstanding mathematician, this paper surveys Seki’s influence on Takebe Katahiro, who was his excellent pupil.

In the 17th century, Japanese mathematicians could calculate the arc length numerically at any accuracy once the diameter and the sagitta were given numerically. But they could not find any formulas to express the arc length in terms of the diameter and the sagitta; only polynomials or fractions of polynomials were considered as formulas. Finally in 1722, Takebe Katahiro (1664–1739) succeeded in expressing the arc length in terms of the diameter and the sagitta in the form of an infinite series expression.

The method of successive divisions developed by Wasan mathematicians was considered as a kind of Diophantine approximations. Nevertheless their main aim is not to find the close approximating fractions of irrational numbers but to obtain the simple fractions which enter the assigned range. We shall here reproduce their crucial algorithm. About fifty years later another Wasan mathematician made an alternate treatment for the same problem which is rather troublesome but gives systematical and complete solutions.

Hatsubi-Sanpō written by Seki Takakazu (ca.1642–1708) was published in 1674 in the form of woodblock printing. The rest of his works are known only in the form of hand-copied books. His case was not exceptional, however. A famous Confucianist Arai Hakuseki (1657–1725) was Seki’s colleague as a government official. Arai also wrote a lot of works as manuscripts but did not publish them in woodblock printings; his works spread widely as hand-copied books. Why they did not publish printed books? We can see some suggestion through recent studies on publications in the Edo period.

The mathematics developed properly in Japan during the Edo period (1603–1867) is called Wasan (Japanese Mathematics). Its roots are in the ancient Chinese mathematics which flourished in the Song and Yuan dynasties (962–1368), in particular, but the later developments in Japan have been thought to be independent of China or Europe. The author claims, however, that Wasan was influenced by Europe first through Chinese translations of European books starting with the Fundamentals for Astronomy (1629), which was purchased by a son of Tokugawa Ieyasu as early as 1632, and from the end of the eighteenth century on directly through books in Dutch.

We give a brief survey of the development of mathematics in Vietnam since 1947, when the first mathematical research paper written by a Vietnamese mathematician was published in an international journal. We describe how mathematics in Vietnam developed under very special conditions: the anti-French resistance, the struggle for the reunification of the country, the American war, the economic crisis, and the change toward a market economy.

The Complete Book of Mathematics is the most comprehensive treatise of mathematics in the Edo Period of Japan. The 20 volume book is about 900 sheets or 1800 pages long. Seki Takakazu (1642?–1708), Takebe Kataakira (1661–1721) and Takebe Katahiro (1664–1739) spent 28 years (1683–1711) in writing it. Unfortunately, the Book has never been published as a whole. We reproduce here Volume 4 Three Essentials, which is a rare exposition of mathematical philosophy during the Edo period.

Volume 4 is named Three Essentials [三要, san’yō] and composed of three sections. Each section starts with a general statement followed by problems, which serve as examples for the general statements. Section 1 is named Symbols and Figures [象形, shō-kei] and divided into (abstract) symbols [(抽) 象, (chū)shō] (Problems 1 – 6), (concrete) symbols [(表) 象, (hyō)shō] (Problems 7 – 11), planar figures [平形, heikei] (Problems 12 – 16), and solid figures [立形, ryūkei] (Problems 17 – 21). Section 2 is named Flow and Ebb [満干, man-kan] (Problems 22 – 37). Section 3 is named Numbers [数, sū] and divided into two subsections: the first subsection is named Provisional and Stationary [動静] numbers (Problems 38 – 47). The second subsection deals with two kinds of finitely presented numbers [整数], i.e., Whole [全] numbers (≒ finite decimal fractions) (Problems 48 – 52) and Complicated [繁] numbers (≒ fractions) (Problems 53 – 57), and two kinds of Inexhaustible numbers [不尽], i.e., Residual [畸] numbers (≒ algebraic numbers) (Problems 58 – 62) and Degraded [零] numbers (≒ transcendentals and observed constants) (Problems 63 – 67).

Seki Takakazu (1642?–1708) classified mathematical problems into three categories: explicit problems, which can be solved by arithmetic, implicit problems, which can be solved by an algebraic equation of one unknown, and concealed problems, which needs simultaneous algebraic equations with more than one unknowns. He is believed to have written for each category of problems a book of their solutions. The three books: Methods of Solving Explicit Problems, Methods of Solving Implicit Problems and Methods of Solving Concealed Problems are called Seki’s Trilogy and used as the standard textbooks in the later Seki School of Mathematics. If a student masters a book of them, he is given a license which certificates his degree of understanding in mathematics.

Each part of these books is without doubt Seki’s writing but we don’t think that the books as they are now are the same as what Seki wrote. The Methods of Solving Implicit Problems may be his original but the others seem to be the results of edition by Yamaji Nushizumi (1704–1772), who instituted Seki’s School of Mathematics. The following are our restorations of the Yamaji editions around 1726 or later.

Abstract This is a collated text of the first of Seki’s Trilogy. The book is composed of four chapters: Chapter 1 Addition and Subtraction [加減kagen], Chapter 2 Partition and Joining [分合bungō], Chapter 3 Whole Products [全乘zenjō] and Chapter 4 Partial Products [折乘setsujō]. In Chapter 1 easiest examples are given to show how a mathematical problem is interpreted as an algorithm on a counting board. In Chapter 2 the byscript method [傍書法bōsho hō] of Seki is introduced by which we can express a polynomial as a sum of monomials represented by a numerical coefficient of counting rods and literal factors written on its right side. The heading means the distributive law of sums and products. Chapters 3 and 4 deal with the area and the volume of geometric figures. Formulas are not given by the algebraic byscript method but as algorithms on a counting board. The Pythagorean theorem and the Eudoxos formula of the volume of a pyramid are proved by figures. The volume formula of sphere segments is correct but its proof by an illustration is hard to understand.

This is a collated text of the second of Seki’s Trilogy. This is a very concise monograph of Celestial Element Method [天元術 tengen jutsu] and Root Extraction Method [開方術 kaihō jutsu]. In Chapter 1: Setting Element [立元], Celestial Element [天元 tengen], that is, the unknown is represented by a counting rod of 1 in the modulus [方hō] class of a counting board. Polynomials [式 shiki] in Celestial Element with numerical coefficients and their operations [演段 endan] are discussed in Chapter 2: Addition and Subtraction [加減 kagen], and Chapter 3: Mutual Multiplications [相乗 sōjō] as manipulations of counting rods on a counting board. Two polynomials representing the same quantity define an Equation [開方式 kaihōsiki] by subtracting one from the other as shown in Chapter 4: Mutual Cancellation [相 消]. The last Chapter 5: Root Extraction [開方] is devoted to the Numerical Extraction of Roots [開方術], that is the same as the so called Horner method which is erroneously attributed to W. G. Horner (1819). Japanese learnt these methods from Zhu Shijie [朱世傑] by his Introduction to Mathematics [算学啓蒙] (1299) but the methods described here are more refined. At the end of the book, Newton’s method of approximation is explained very succinctly.

This is the last of Seki’s Trilogy. Here he gives a general procedure to eliminate a common unknown from two polynomial equations, which is the same as the elimination theory of Étienne Bézout. The book has five chapters. In Chapter 1: Real and Fictitious [真虚 shinkyo] and Chapter 2: Two Equations [兩式 ryōshiki] we learn how to formulate a system of algebraic equations as a succession of two equations in an unknown to be eliminated with polynomials of the other unknowns as coefficients. Chapter 3: Estimates of Degrees [定乘 teijō] gives an estimate of the degree of eliminated equations. The elimination is carried out in two steps. Chapter 4: Transformed Equations [換式 kanshiki] shows how to construct

n

equations of degree less than

n

out of a system of two equations of degree

≤ n

. Then the eliminated equation is obtained as the determinant of their coefficients equated to 0 as shown in Chapter 5: Creative and Annihative Terms [生剋 seikoku], where Seki gives the expansion of determinants of order up to 4. The disorder of table happens in the case of order 4, for which we append an amendment for disorder and alterations of sheets 14 and 15

1

. If Seki had stopped here, he would have been praised for having written the most concise and complete book on ellimination. His errors occurred in the expansion of determinant of order 5, for which we refer the reader to Goto–Komatsu [1].

The Complete Book of Mathematics is the most comprehensive treatise of mathematics in the Edo Period of Japan. The 20 volume book is about 900 sheets or 1800 pages long. Seki Takakazu (1642?–1708), Takebe Kataakira (1661–1721) and Takebe Katahiro (1664–1739) spent 28 years (1683–1711) in writing it. Unfortunately, the Book has never been published as a whole. Except for Volume 4 we reproduce here only one volume, Volume 10 Geometry, which is the first volume on geometry in the treatise. They develop here the basics of the plain geometry as algebraic relations among line segments which specify the geometric objects in question. This is exactly the same standpoint as René Descartes’ (1596–1650) in his

Géométrie

(1637). At the end of the volume they discuss algebraic relations among the sides and diagonals in a general pentagon and hexagon by making use of Seki’s theory of resultants.

Volume 10 is named Algorithms for Appearances [形法 keihō] as the first of five volumes on geometry. It consists of four chapters. Chapter 1 Algorithms for Squares [方法 hōhō] deals with squares. Chapter 2 is entitled Algorithms for Rectangles [直法 chokuhō]. The longer side of a rectangle [直 choku] is called [縦 tate] or length [長 chō] and the shorter side [横 yoko] or [平 hei] or width [闊 katsu]. Their product is the area [積 seki]. In the following Chapter 3 Algorithms for Right Triangles [勾股法 kōko hō] this is used to prove the Pythagorean theorem. The shorter leg hook [勾 kō] and the longer leg leg [股 ko] of a right triangle stand for the right triangle itself. The hypothenuse is called chord [弦 gen]. There are no formal treatments of proportions as in Euclid’s Elements but it is remarked that rectangles or right triangle with a fixed ratio of heights to widths make a straight line. There are two appendices to Chapter 3. The first one deals with Pythagorean triplets, i. e. the integral solutions of

x

2

+

y

2

=

z

2

. The second appendix is a brief introduction to the methods of survey described in Sea Islands Mathematics [海島算經 Haidao Suanjing] by Liu Hui. In the last Chapter 4 Algorothms for Polygons [斜法 shahō] the authors develop a general theory of triangles [三斜 sansha], quadrilaterals [四 斜 shisha], pentagons [五斜] and hexagons [六斜] with the use of their elimination theory

This is an enlarged version of the authors’ papers published in Journal of Northwest University (Natural Science Edition), 33 Nos.3 and 4 (2003). Added is Section 3, where proofs are given of the main properties of determinants and resultants as originally introduced by Seki Takakazu (1642?–1708), Étienne Bézout (1739–83), James J. Sylvester (1814–97) and Arthur Cayley (1821–95).