Enri - Enri

In Japanese mathematics, this refers to the relationship that holds between the diameter, chord, height, etc., of a circle or an arc. The first appearance of the principle of circles was in the preface of the Kokin Sanpo-ki (1671) by Sawaguchi Kazuyuki, which states that it is easy to find a formula that holds for polygons, but it is difficult to know the principle of circles. A famous problem on the principle of circles is the 10th question in the Jingoki, which asks the reader to cut out an arc of a certain area from a given circle. At the time, 3.16 was used for the value of pi, but thanks to the efforts of mathematicians, by the late 17th century, pi, the product of a circle, and the product of a ball could be calculated accurately. At the beginning of the 18th century, Takebe Katahiro showed a formula equivalent to the Maclaurin expansion of (sin ^-1 x) ² in his book Kohaijutsu. Later, Matsunaga Yoshisuke and Kurushima Yoshihiro created formulas for the infinite series expansion of inverse trigonometric functions and trigonometric functions, which are also called Enri. Pi was also calculated to more than 50 significant digits. Ajima Naonobu calculated the volume of the intersecting part of two cylinders by double integrals. Wada Yasushi, a disciple of Ajima, completed numerous tables of definite integrals and made achievements in the fields of multiple integrals and differentiation. These definite integral tables and differentiations are also called Enri tables or Enri. At the end of the Edo period, various curves such as catenaries and cycloids, figures enclosed by these curves, and the rotational bodies of these figures were studied, and problems were posed on finding the length, area, volume, or surface area of ​​curves, and the center of gravity of plane and solid figures. Wada Yasushi's Enri table was used to solve these problems, and these types of problems were also called Enri.

[Shimohira Kazuo]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

和算において、円や弧について、直径、弦、高さなどの間に成り立つ関係をいう。円理は、沢口一之(さわぐちかずゆき)著の『古今算法記』（1671）の序文にあるのが初見で、多角形について成り立つ公式を求めるのはやさしいが、円理を知るのは困難だとある。円理の問題として有名なのは『塵劫記(じんごうき)』の遺題第10問で、与えられた円から、ある面積の弓形を切り取れという問題である。当時、円周率は3.16が使われていたが、数学者の努力により、17世紀後半には、円周率、円積率、玉率が正確に求められるようになった。18世紀の初め建部賢弘(たけべかたひろ)は(sin^-1x)²のマクローリン展開に相当する公式をその著『円理弧背術(こはいじゅつ)』で示した。その後、松永良弼(よしすけ)と久留島義太(くるしまよしひろ)により、逆三角関数や三角関数の無限級数展開の公式がつくられ、これらも円理とよばれる。円周率も有効数字50桁(けた)以上求められた。安島直円(あじまなおのぶ)は、円柱と円柱の相貫部の体積を二重積分で求めた。安島の孫弟子の和田寧(やすし)は数多くの定積分表を完成し、多重積分や微分についても業績を残した。これらの定積分表や微分も円理表あるいは円理とよばれる。幕末になると、懸垂線（カテナリー）やサイクロイドなど種々の曲線や、それらの曲線によって囲まれた図形、その図形の回転体が研究され、曲線の長さ、面積、体積、あるいは表面積、平面図形や立体図形の重心を求める問題が提出された。これらの問題を解くのに、和田寧の円理表が利用されるとともに、このような問題も円理とよばれた。

［下平和夫］

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