**Takebe** was a Japanese mathematician who wrote most of Seki's *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.
- Takebe served Tokugawa Ienobu during his three years as Shogun, and then served Tokugawa Ietsugu who was Shogun from 1712 to 1716.
- 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.
- Later in this biography we will describe work undertaken by Takebe specifically at Yoshimune's request.
- Takebe had made a careful study of Zhu Shijie's Chinese text Suanxue qimeng Ⓣ(Introduction to mathematical studies) 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 Ⓣ(Introduction to mathematical studies) which he intended as a text for students 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 Ⓣ(Art of assembling).
- In Chapter 2 of this work, Takebe explains the "method of celestial element" which Zhu Shijie had introduced in Suanxue qimeng Ⓣ(Introduction to mathematical studies) 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 Ⓣ(Introduction to mathematical studies).
- 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.
- Perhaps Takebe's greatest achievement was to devise a method to calculate a series expansion of a function.
- We have not yet mentioned the result for which many know Takebe's name, namely his calculation of π.
- 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.
- It allowed Takebe to find correct to 40 decimal places.
- The Tetsujutsu sankei Ⓣ(Introduction to mathematical studies) does more than describe Takebe's mathematical contributions, for in the work he also expounds his mathematical methodology.
- 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.
- 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.
- 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.

Born 1664, Edo (now Tokyo), Japan. Died 24 August 1739, Edo (now Tokyo), Japan.

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Astronomy, Origin Japan, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive