The brothers did their utmost to spread Seki's work, to make it easier to understand, and to defend it against detractors. They were the main craftsmen of Seki's project (launched 1683) to record mathematical knowledge in an encyclopaedia. The 'Taisei sankei' (Comprehensive Classic of Mathematics), in 20 volumes, was finally completed by Takebe Kataaki in 1710. It gives a good picture of Seki's skill at reformulating problems, as well as Takebe Katahiro's ability to correct, perfect, and extend his master's intuitions.Seki came from the samurai class and this meant that he became a public servant. He held the position of examiner of accounts to Tokugawa Ienobu, the Lord of Kofu. When his lord became heir to the Shogun, the hereditary military dictator of Japan, Seki became Shogunate samurai and in 1704 was given a position of honour as master of ceremonies in the Shogun's household. This brought Takebe into contact with the leading members of the Tokugawa family. He was an officer to Tokugawa Bakuhu, and then he was close to Tokugawa Ienobu, the Lord of Kofu, as he rose to become Shogun in 1709. Takebe served Tokugawa Ienobu during his three years as Shogun, and then served Tokugawa Ietsugu who was Shogun from 1712 to 1716. From 1695 when he began his close association with members of the Tokugawa family, Takebe spent less time on his study of mathematics. However he next served Tokugawa Yoshimune, who became the eighth Tokugawa Shogun in 1716. Yoshimune was significantly different from the earlier Tokugawa rulers, having an inquiring mind. Takebe's enthusiasm for the study of mathematics and astronomy was invigorated again from 1716. Encouraged by Takebe, Yoshimune relaxed the edict forbidding the introduction of foreign books, including scientific books, which led to the growth of interest in Western science in Japan. Yoshimune was fascinated by astronomy and had a large globe made. He also had a telescope which had been in the Netherlands. Later in this biography we will describe work undertaken by Takebe specifically at Yoshimune's request.
Tatebe was only nineteen years old when he published his first mathematics book the Kenki Sanpo Ⓣ (1683). He followed this in 1685 with the book Hatsubi Sanpo Endan Genkai Ⓣ. These early works were based on the methods Seki had developed for handling polynomials. These methods were extended to handle polynomials in a single variable but with variable coefficients. Tatebe improved and extended the methods of his teacher and applied them to a wide range of problems in these books. Takebe had made a careful study of Zhu Shijie's Chinese text Suanxue qimeng Ⓣ published in 1299 and the work had been a great help to him in developing his theory of polynomials. In 1690 Takebe published an annotated Japanese translation of the Suanxue qimeng Ⓣ which he intended as a text for students of mathematics.
In 1683 Seki started a project to compile an encyclopaedia of mathematics. This 20 volume work appeared in 1710 with the title Taisei sankei (Comprehensive Classic of Mathematics). None of the work was written by Seki himself and it is clear that the first twelve volumes were written by Takebe. Takebe Kataaki also played a major role in compiling the Taisei sankei and the final eight volumes are due to him. One of the most significant ideas introduced in Takebe's part of the text is the method of Enri, which is definite integration. For many years historians believed that this idea was due to Seki and only the writing was due to Takebe. However, modern research leads most historians to claim that the method of Enri was in fact due to Takebe.
The most important of Takebe's work is Tetsujutsu Sankei Ⓣ. Morimoto writes in :-
In 1722, Takebe wrote the 'Tetsujutsu Sankei' (Mathematical Treatise on the Technique of Linkage) to explain how mathematical research could be done in accordance of one's inclination, based on 12 examples of mathematical investigation.In Chapter 2 of this work, Takebe explains the "method of celestial element" which Zhu Shijie had introduced in Suanxue qimeng Ⓣ as a method of representing a polynomial in one variable on a counting board. The extension of this method to polynomials with variable coefficients, due to Seki and Takebe, was presented in Chapter 6 of the Tetsujutsu Sankei Ⓣ. This effectively allowed Takebe to handle polynomials in several variables. Although Takebe did not explicitly have the operation of differentiation, nevertheless, he stated a result in Chapter 6 which is equivalent to the statement that if a cubic polynomial takes an extreme value at a point the derivative vanishes at that point. We should note, however, that Ogawa writes in :-
The purpose of this paper is to consider the essence of mathematics of Takebe Katahiro (1664-1739) by investigating his method of finding the maximum in 'Tetsujutsu Sankei' (1722). It has been said that he had first done a calculation of a derivation for finding the maximum in Chapter 6 of the treatise, but that is not strictly true. A close look at the chapter will reveal that his method has nothing to do with the theory of differential.Perhaps Takebe's greatest achievement was to devise a method to calculate a series expansion of a function. Ancient Greek mathematicians had been perplexed by the problem of squaring the circle and in the 17th century Japanese mathematicians looked at a similar problem, namely the problem of finding a polynomial which expressed the length s of an arc of a circle subtended by a chord with sagitta k. The sagitta is the line from the midpoint of the chord to the midpoint of the arc of the circle it subtends. Now we know that no such polynomial exists but Takebe found an infinite series expressing s in terms of k, namely
(s/2)2 = k + (1/3)k2 + (1.8) / (3.9)k3 + (1.8.9) / (3.15.14)k4 + (22.214.171.124) / (126.96.36.199)k5 + (188.8.131.52.25) / (184.108.40.206.33)k6 + ...In fact, looked at in modern terms, what Takebe was calculating was the Taylor expansion of arcsin(√k)2 about k = 0.
We have not yet mentioned the result for which many know Takebe's name, namely his calculation of π. He describes in Chapter 9 of the Tetsujutsu Sankei Ⓣ the right way to proceed if one wants to calculate the circumference of a circle of a given diameter. He writes (in Morimoto's translation from ):-
If he who decomposes the circumference of a circle cuts the diameter equally and horizontally into thin slices, seeks the [length of the] right and left oblique chords cut by the horizontal lines and adds the oblique chords to seek the [approximate] circular circumference, then the parts of circumference are not equal even if he cuts the diameter equally. Therefore, if he seeks the circumference doubling the sections of the diameter, these numbers being disobedient to the attribute, he stagnates in determining the extreme number and never obtain a basis to understand the attribute of circle. On the other hand, when he cuts the circumference into the four angular forms [i.e., by an inscribed square] and further doubling angles [i.e., forming an inscribed octagon, etc.], the circumference is cut into equal length and the numbers are obedient to the attribute of circumference. Therefore, doubling the number of angles and seeking the angular circumferences at each step, by the repeated application of the procedure of incremental divisor he can determine the extreme number rapidly and obtain a basis to understand the attribute of a circle.Now of course this gives a method to approximate π but the method converges very slowly. Takebe invented a method for the acceleration of convergence using a technique for the successive removal of various powers of the argument, in this case the error term. This idea is essentially equivalent to the Romberg algorithm. It allowed Takebe to find correct to 40 decimal places. See  for further details.
The Tetsujutsu sankei Ⓣ does more than describe Takebe's mathematical contributions, for in the work he also expounds his mathematical methodology. Let us give two quotations dealing with this part of his thought; first the description by Annick Horiuchi in :-
He distinguished two ways of solving a mathematical problem (and two corresponding types of mathematicians): an "investigation based on numbers," an inductive approach that involves scrutinizing and manipulating data until one finds a general law; and an "investigation based on principle," a reasoned approach that involves directly utilizing rules and procedures, as in algebra. The two approaches are often complementary, as he demonstrated by showing that an infinite series that he had obtained inductively could also be derived algebraically. His procedure for calculating the infinite series played a key role in the development of analysis in Japan in the following decades.The article  by Chang Zhou and Jian Ke Zhang on Takebe's mathematical thought and methodology is summarised by the authors as follows:-
In view of mathematical methodology and the Neo-Confucianism of the Song and Yuan dynasties, this paper discusses the essential character of 'Tetsujutsu' which runs through 'Tetsujutsu Sankei', and Takebe's mathematical thought and methodology reflected from the 'Jishitsu Setsu' at the end of the book. The paper comes to the conclusion that Takebe's 'Tetsujutsu' is, actually, the colligation of the methods of deduction and induction, in which Takebe pays more attention to the method of induction, but he ignores the proof of precisely relying on the method of deduction. Based on our analysis, we conclude that the mathematical thought behind the author's preface and the 'Jishitsu Setsu' of the 'Tetsujutsu Sankei' is connected to the mainstream philosophy of Zhu Xi (1130-1200) and Wang Shouren (1472-1528) of the time, rather than having nothing to do with the philosophical thought, which is the view of some scholars. Furthermore, the origin of Takebe's mathematical thought may be traced back to the Neo-Confucianism of the Song and Yuan dynasties in China.We promised earlier in this article to return to the work which Takebe undertook at Shogun Yoshimune's request. This is studied in detail in , which is reviewed by Brunetto Piochi who explains that :-
... presents the contribution in astronomy and calendar science of Japanese mathematician Takebe, who worked around 1720-1730 on those subjects. The author explains in particular the work of Takebe on two topics: (i) the variation of the tropic year, i.e. the time between two consecutive winter solstices; (ii) the way of calculating the position of the polar star in the sky. By this study, the author draws the conclusion that Takebe (whose work is still unpublished, and partly lost) had a role in the history of scientific thought, as his aim was to improve, by the assistance of geometry and mathematics, calculation techniques so that astronomy and calendar science could become, in Japan too, exact sciences.
Article by: J J O'Connor and E F Robertson