物理学への道標【科学史から】

建部賢弘は日本の数学者で、和算を大成した人物です。
江戸時代1664年に生まれています。

関ヶ原の合戦が1600年で江戸太平の世の中が200年ほど
だったことを思い返せば建部はまさに江戸時代の中期
に活躍したと言えますね。

時は享保の時代で8代将軍の暴れん坊将軍「徳川吉宗」
の信頼を得て享保四年（1719年）「日本総図」
を作成します。また、
師である関孝和の業績に関する著作を多数残しました。
その内容は歴史的な記述というよりも
数学の側面からの解説書であったようです。

そもそも、関孝和は沢口一之が残した『古今算法記』での未解決問題を関さん独自の点竄術を使って解決していました。そこで「関さんの悪い所」なのですが、省略し過ぎで難しい本だったのです。ここで面白いのは関西系の数学者からツッコミ食らっていた事情です。頑固な江戸のおじいちゃんが関西人からツッコまれていたのですが、建部さんは丁寧な解説で「正しいでしょう？」って感じの話し方が出来たのです。きっと関西人たちも納得したはずです。関西人であれスッキリした瞬間です。

そして、師匠の関孝和と建部賢弘と建部建部賢明の三人で全20巻の
「大成算経」は当時の和算をまとめ上げた秀作として評価されました。

建部賢弘の大きな業績として円に対しての定量的な追及があります。物凄い精度で円について考えていったのです。そもそも、精度の高い真円が描けたとしてもその円での半径とこの長さの関係は自明ではありません。今でこそ、子供たちも3.14…と記憶していけるのですが理論的に真円が描けたと考えた時の弧の長さは「三角関数を使って級数を作り極限」を求めていくしかありません。この三角関数、級数、極限といった概念を和算の中で正確に使っていくデリケートさが求められるのです。建部賢弘は丁寧に言葉を選んで誰でもわかる表現をして未知の世界に挑んでいったのです。建部以前の時代から使われていた正多角形を円が囲む近似から考えていって、逆に正多角形に円が囲まれた部分を想像して、円の面積がA以上B以下であると証明していくのです。そして円弧の長さがα以上β以下であると証明していったのです。そして建部賢弘は円周率を41桁まで正確に出したのです。世界的に考えても数値的な解法として優れた業績でした。

その他にも建部賢弘は多くの業績を日本に残しましたが、以下備忘録的に羅列します。

・指数1/2の二項級数の禁じ解法を紹介
・ディオファントス方程式の近似解法を紹介
・帰納法に基づいた数値解析の方法論を紹介
・無限の概念を「不尽」として導入
・三角関数の内容を表の形で明示

そして今、日本数学会では建部賢弘特別賞や建部賢弘奨励賞
という形で若手数学者を奨励する賞を設けています。
建部賢弘のように若かりし人が
新しい分野を開いていく姿を期待しています。

〆

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2022/10/06_初稿投稿
2022/10/13_ 改訂投稿

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（2022年10月時点での対応英訳）

Kenhiro Tatebe was a Japanese mathematician and a great exponent of Japanese arithmetic.
He was born in 1664 during the Edo period.

The Battle of Sekigahara took place in 1600, and the Edo period lasted about 200 years.
If we recall that the Battle of Sekigahara took place in 1600 and the peaceful Edo period lasted for about 200 years
The time was the Kyoho period, and he was active in the 8th generation.

The time was the Kyoho period, and he gained the trust of the 8th shogun, "Tokugawa Yoshimune," the ruffian and tyrant shogun.
In 1719, he gained the trust of the 8th shogun, the ruffian Tokugawa Yoshimune, and produced the "General Map of Japan" in 1719.
in 1719, under the trust of the 8th shogun, Yoshimune Tokugawa. In addition
He also wrote many works on the achievements of his mentor, Seki Takakazu.
The contents of these works are not so much historical descriptions as commentaries on the
The contents of these works seem to have been commentaries from a mathematical point of view rather than historical descriptions.

To begin with, Seki Takakazu solved the unsolved problems in Sawaguchi Kazuyuki's "Kokin Keiken" by using Seki's original point-falsification technique. However, the book was difficult to read because of the excessive omissions as "Seki's bad point. What is interesting here is the fact that he was criticized by a mathematician from the Kansai region. The stubborn old man from Edo was getting flack from the Kansai people, but Mr. Tatebe was able to give a polite explanation and say, "Isn't that right? He was able to speak in a way that made the Kansai people understand. I am sure the Kansai people were convinced. It was a moment of great clarity, even for Kansai people.

And then, his master Seki Takakazu, Tatebe Masahiro, and Tatebe Tatebe Kenmei together produced a 20-volume book, "The Great Calculation Sutra," which was published in 1949.
The "Taisei Keikyo" was highly regarded as an excellent work that summarized the Japanese mathematics of the time.

One of Kembe's major achievements was his quantitative pursuit of the circle. He thought about the circle with tremendous precision. Even if a highly accurate circle could be drawn, the relationship between the radius and the length of the circle would not be self-evident. Nowadays, children can memorize 3.14..., but theoretically, when a perfect circle is drawn, the length of the arc can only be obtained by using trigonometric functions to create a series and finding the limit. The concepts of trigonometric functions, series, and limits must be used with delicacy and precision in Japanese arithmetic. Kenhiro Tatebe carefully chose his words to express them in a way that anyone could understand, challenging the unknown. He began by considering the approximation of a circle enclosing a regular polygon, which had been used since the pre-Tatebe era, and then, conversely, imagined the area of a circle enclosed by a regular polygon, proving that the area of the circle is greater than A and less than or equal to B. He then used a circle with an arc length of at least α and less than or equal to B. He then proved that the length of the arc is greater than or equal to α and less than or equal to β. Then, Kenhiro Tatebe obtained pi to exactly 41 digits. This was an outstanding achievement in numerical solving, even when considered on a global scale.

Kenhiro Tatebe also left many other achievements in Japan, which are listed below as a reminder.

Introduced a forbidden solution method for binomial series with exponent 1/2.
Introduced an approximate solution method for Diophantine equation.
Introduction of a methodology for numerical analysis based on induction
Introduces the concept of infinity as "inexhaustibility
・Contents of trigonometric functions are clearly stated in the form of tables.

Now, the Mathematical Society of Japan is honoring young mathematicians in the form of the Kenhiro Tatebe Special Award and the Kenhiro Tatebe Encouragement Award.
The Mathematical Society of Japan now offers prizes to encourage young mathematicians in the form of the Ken Hiroshi Tatebe Special Prize and the Ken Hiroshi Tatebe Encouragement Prize.
We hope to see young people like Kenhiro Tatebe
to open up new fields of study.

Translated with www.DeepL.com/Translator (free version)