安島直円
安島 直円（あじま なおのぶ、享保17年（1732年） - 寛政10年4月5日（1798年5月20日））は江戸時代中期の財政家・和算家[1]。新庄藩士。本姓は藤原氏。家系は藤原秀郷流の安島氏。仮名は万蔵[2]。字は伯規、号は南山。諱は直円。名は資料によっては安島万蔵とも載せ、新庄藩の資料では安嶋直円ともある。父は新庄藩御勘定頭・安島庄右衛門清英。兄弟に安島弥惣次清茂、伊東平蔵直休がいる。妻は於なを。家禄は80石。寛政10年（1798年）江戸藩邸にて没。江戸（現在の東京都港区三田）の曹洞宗常林寺に葬られた他、国元の菩提寺である出羽国最上郡新庄町（山形県新庄市）は桂嶽寺に分骨された。戒名は祖眞院智算量空居士。位階は贈従五位。和算に長け、同門の藤田定資をして「当代の名人」と言わしめ、江戸時代の数学の発展に寄与、後世の人は関孝和と並んで和算の二大焦点と評した[3]。
家系[編集]
『郷土資料叢書第10輯』の「戸沢家中分限帳（二）」によれば、安島直円の生家は代々新庄藩士で、元々は常陸守護・佐竹氏の家臣で陸奥国白河郡棚倉の城代・安島丹後守久成を祖とする安島氏の分家とされる。安島久成の子・安島隼人が棚倉を没落し、常陸国宍戸において宍戸藩主として入封した戸沢氏に200石で召し抱えられ、家臣となった。直円は隼人から数えて四代・安島五左衛門の次男・安島庄右衛門清英が分家し、その清英の子として生まれた[3]。
系譜[編集]
新庄藩士 安島氏の系譜は以下のとおり[4]。
系譜 安島丹後守久成 ― 隼人 ― 五左衛門 ― 甚内 - 五左衛門 ― 〇庄右衛門清英 ― ◎贈従五位萬蔵直円 ― 萬蔵広茂 ― 銀之助 ― 安島操 ― 鋼三郎 ― 釛三郎 ― かつ
- 安島直円には、弟に弥惣次直茂（元文6年（1741年）生まれ、同腹、宝暦6年（1756年）6月召出し）と、平蔵直休（延享2年（1745年）生まれ、同腹、母方の伊東家断絶につき御家再興のため伊東姓に改める）がいる。
- 直円の母は、脇坂氏の家臣・熊谷仁左衛門の女と記録されている。
生涯[編集]
享保17年(1732年)、江戸の新庄藩邸において生まれ、当初は万蔵と名付けられた。父は安島五左衛門の次男・庄右衛門清英で、本来ならば部屋住みとなるところ、才能を見出され80石で藩の召し抱えとなり、別家を建てた。江戸常府御会所勤めを経て勘定方となり、江戸藩邸における会計責任者を務めた。直円も幼年より和算に親しみ、和算中西流の大家・入江広忠が主催していた江戸の和算塾・入江塾に通っていた。寛保3年（1743年）12歳で元服し、当時和算において大いに成長していたため、父が数学者としての大成を願って諱を直円と命名した。後に関流の家元・山路主住の門下となり、さらなる数学の道を究めた。宝暦4年（1754年）父が亡くなり家督を継いで家禄80石を相続、同年、宝暦暦制定に協力する。同6年（1756年）には吟味役兼金元方を命ぜられ、12年（1762年）に御勘定頭に昇進、3人扶持を加増された。天明5年（1785年）10月に本締手代、11月には郡奉行へと昇進を重ね、同6年（1786年）には本締役を命ぜられ、20石を加増された。同年中にはさらに10石の加増があり、120石の禄を賜るまでになった。こうした一連の加増は藩財政の建て直しに貢献した功績に基づくといわれている[5]。
直円の研究は独創的なものが多い。特に円理については、円柱の相貫体の体積を二重級数で表す、円弧の長さを求めるのに弦を等分する方法を完成させるなどの結果を与えた。また幾何学においても三斜三円術（安島‐マルファッティの定理）や四円六斜術（ケーシーの定理）など、円と多角形とが接する際にその大きさを求める問題の解法を、ヨーロッパに先駆けて発見している。後期の和算の特徴の一つであるこのような幾何的図形の研究について、安島は極めて基礎的な分野で貢献している。また整数方程式・対数・循環小数についても優れた研究を残している。世間では同門の藤田定資の方が知名度が高く「名人」と呼ばれたが、彼自身は直円を「名人」と呼んだと伝えられる。師匠筋の山路家が天文方だった影響で、暦に関する著書もあり、『授時暦便蒙』『安子西洋暦考草』『安島先生便蒙之術』『交食蒙求俗解』の4編が残されている。内容は研究というよりは編暦計算の実務に携わる人のための教科書と見られ、そのための工夫が随所に見られる。
これらの和算に対する功績について、日本学士院院長・菊池大麓は直円を高く評価した。こうしたことから安島直円は和算の歴史において関孝和とともに和算史上の二大焦点といわれた。大正4年（1925年）従五位を贈位される[3]。
遺功・業績・顕彰[編集]
- 明和5年（1768年）『授時暦便蒙』完成
- 天明6年（1786年）新庄藩財政回復に功、20石加増の後さらに10石加増
- 大正4年（1925年）贈従五位
- 平成10年（1998年）安島直円没後200年に際し、その遺功を讃えるために有志（安島直円顕彰会）が新庄市の西山の丘に顕彰碑を建立
その他の著書[編集]
- 『円柱穿空円術』
- 『祇園算学解』
その他、エピソード[編集]
- 月面には安島の名に由来するクレーターナオノブ (Naonobu) が存在する。
- 安島直円の業績を讃えるため、今日ではその郷里の新庄市において安島直円顕彰会が結成され、顕彰がなされている。
- 墓所の常林寺は、東京都港区三田4丁目に所在する（最寄駅は都営浅草線三田駅、下車5分）。https://ja.wikipedia.org/wiki/%E5%AE%89%E5%B3%B6%E7%9B%B4%E5%86%86ゼロ除算の発見は日本です：∞？？？∞は定まった数ではない・・・・人工知能はゼロ除算ができるでしょうか：とても興味深く読みました：ゼロ除算の発見と重要性を指摘した：日本、再生核研究所ゼロ除算関係論文・本\documentclass[12pt]{article}\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}\numberwithin{equation}{section}\begin{document}\title{\bf Announcement 412: The 4th birthday of the division by zero $z/0=0$ \\(2018.2.2)}\author{{\it Institute of Reproducing Kernels}\\Kawauchi-cho, 5-1648-16,\\Kiryu 376-0041, Japan\\}\date{\today}\maketitleThe Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotelēs (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).In particular, Brahmagupta defined as $0/0=0$ in Brāhmasphuṭasiddhānta (628), however, our world history stated that his definition $0/0=0$ is wrong over 1300 years, but, we showed that his definition is suitable.For the details, see the references and the site: http://okmr.yamatoblog.net/We wrote a global book manuscript \cite{s18} with 154 pagesand stated in the preface and last section of the manuscript as follows:\bigskip{\bf Preface}\medskipThe division by zero has a long and mysterious story over the world (see, for example, H. G. Romig \cite{romig} and Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628. In particular, note that Brahmagupta (598 -668 ?) established the four arithmetic operations by introducing $0$ and at the same time he defined as $0/0=0$ inBrhmasphuasiddhnta. Our world history, however, stated that his definition $0/0=0$ is wrong over 1300 years, but, we will see that his definition is right and suitable.The division by zero $1/0=0/0=z/0$ itself will be quite clear and trivial with several natural extensions of the fractions against the mysterously long history, as we can see from the concepts of the Moore-Penrose generalized inverses or the Tikhonov regularization method to the fundamental equation $az=b$, whose solution leads to the definition $z =b/a$.However, the result (definition) will show thatfor the elementary mapping\begin{equation}W = \frac{1}{z},\end{equation}the image of $z=0$ is $W=0$ ({\bf should be defined from the form}). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere (\cite{ahlfors}). �As the representation of the point at infinity of the Riemann sphere by thezero $z = 0$, we will see some delicate relations between $0$ and $\infty$ which show a strongdiscontinuity at the point of infinity on the Riemann sphere. We did not consider any value of the elementary function $W =1/ z $ at the origin $z = 0$, because we did not consider the division by zero$1/ 0$ in a good way. Many and many people consider its value by the limiting like $+\infty $ and $- \infty$ or thepoint at infinity as $\infty$. However, their basic idea comes from {\bf continuity} with the common sense orbased on the basic idea of Aristotle. --For the related Greece philosophy, see \cite{a,b,c}. However, as the division by zero we will consider its value ofthe function $W =1 /z$ as zero at $z = 0$. We will see that this new definition is valid widely inmathematics and mathematical sciences, see (\cite{mos,osm}) for example. Therefore, the division by zero will give great impacts to calculus, Euclidean geometry, analytic geometry, differential equations, complex analysis in the undergraduate level and to our basic ideas for the space and universe.We have to arrange globally our modern mathematics in our undergraduate level. Our common sense on the division by zero will be wrong, with our basic idea on the space and the universe since Aristotle and Euclid. We would like to show clearly these facts in this book. The content is in the undergraduate level.\bigskip\bigskip{\bf Conclusion}\medskipApparently, the common sense on the division by zero with a long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on derivatives we have a great missing since $\tan (\pi/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero.This book is an elementary mathematics on our division by zero as the first publication of books for the topics. The contents have wide connections to various fields beyond mathematics. The author expects the readers write some philosophy, papers and essays on the division by zero from this simple source book.The division by zero theory may be developed and expanded greatly as in the author's conjecture whose break theory was recently given surprisingly and deeply by Professor Qi'an Guan \cite{guan} since 30 years proposed in \cite{s88} (the original is in \cite {s79}).We have to arrange globally our modern mathematics with our division by zero in our undergraduate level.We have to change our basic ideas for our space and world.We have to change globally our textbooks and scientific books on the division by zero.\bibliographystyle{plain}\begin{thebibliography}{10}\bibitem{ahlfors}L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.\bibitem{cs}L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.\bibitem{guan}Q. Guan, A proof of Saitoh's conjecture for conjugate Hardy H2 kernels, arXiv:1712.04207.\bibitem{kmsy}M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.\bibitem{ms16}T. Matsuura and S. Saitoh,Matrices and division by zero z/0=0,Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt\\ http://dx.doi.org/10.4236/alamt.2016.62007.\bibitem{ms18}T. Matsuura and S. Saitoh,Division by zero calculus and singular integrals. (Submitted for publication)\bibitem{mms18}T. Matsuura, H. Michiwaki and S. Saitoh,$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.\bibitem{msy}H. Michiwaki, S. Saitoh and M.Yamada,Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html\bibitem{mos}H. Michiwaki, H. Okumura and S. Saitoh,Division by Zero $z/0 = 0$ in Euclidean Spaces,International Journal of Mathematics and Computation, {\bf 2}8(2017); Issue 1, 2017), 1-16.\bibitem{osm}H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.\bibitem{os}H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).\bibitem{o}H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.\bibitem{os18}H. Okumura and S. Saitoh,Applications of the division by zero calculus to Wasan geometry.(Submitted for publication).\bibitem{ps18}S. Pinelas and S. Saitoh,Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.\bibitem{romig}H. G. Romig, Discussions: Early History of Division by Zero,American Mathematical Monthly, Vol. {\bf 3}1, No. 8. (Oct., 1924), pp. 387-389.\bibitem{s79}S. Saitoh, The Bergman norm and the Szeg$\ddot{o}$ norm, Trans. Amer. Math. Soc. {\bf 249} (1979), no. 2, 261--279.\bibitem{s88}S. Saitoh, Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, {\bf 189}. Longman Scientific \& Technical, Harlow; copublished in the United States with John Wiley \& Sons, Inc., New York, 1988. x+157 pp. ISBN: 0-582-03564-3\bibitem{s14}S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/\bibitem{s16}S. Saitoh, A reproducing kernel theory with some general applications,Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .\bibitem{s17}S. Saitoh, Mysterious Properties of the Point at Infinity、arXiv:1712.09467 [math.GM](2017.12.17).\bibitem{s18}S. Saitoh, Division by zero calculus (154 pages: draft): （http://okmr.yamatoblog.net/）\bibitem{ttk}S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.\bibitem{a}https://philosophy.kent.edu/OPA2/sites/default/files/012001.pdf\bibitem{b}http://publish.uwo.ca/~jbell/The 20Continuous.pdf\bibitem{c}http://www.mathpages.com/home/kmath526/kmath526.htm\bibitem{ann179}Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.\bibitem{ann185}Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.\bibitem{ann237}Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.\bibitem{ann246}Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.\bibitem{ann247}Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.\bibitem{ann250}Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.\bibitem{ann252}Announcement 252 (2015.11.1): Circles andcurvature - an interpretation by Mr.Hiroshi Michiwaki of the division byzero $r/0 = 0$.\bibitem{ann281}Announcement 281 (2016.2.1): The importance of the division by zero $z/0=0$.\bibitem{ann282}Announcement 282 (2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.\bibitem{ann293}Announcement 293 (2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.\bibitem{ann300}Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.\bibitem{ann326}Announcement 326 (2016.10.17): The division by zero z/0=0 - its impact to human beings through education and research.\bibitem{ann352}Announcement 352(2017.2.2): On the third birthday of the division by zero z/0=0.\bibitem{ann354}Announcement 354(2017.2.8): What are $n = 2,1,0$ regular polygons inscribed in a disc? -- relations of $0$ and infinity.\bibitem{362}Announcement 362(2017.5.5): Discovery of the division by zero as $0/0=1/0=z/0=0$\bibitem{380}Announcement 380 (2017.8.21): What is the zero?\bibitem{388}Announcement 388(2017.10.29): Information and ideas on zero and division by zero (a project).\bibitem{409}Announcement 409 (2018.1.29.): Various Publication Projects on the Division by Zero.\bibitem{410}Announcement 410 (2018.1 30.): What is mathematics? -- beyond logic; for great challengers on the division by zero.\end{thebibliography}\end{document}List of division by zero:\bibitem{os18}H. Okumura and S. Saitoh,Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum.Saburou Saitoh, Mysterious Properties of the Point at Infinity、
arXiv:1712.09467 [math.GM]Hiroshi Okumura and Saburou SaitohThe Descartes circles theorem and division by zero calculus. ２０１７．１１．１４L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$, Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.T. Matsuura and S. Saitoh,Matrices and division by zero z/0=0,Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt\\ http://dx.doi.org/10.4236/alamt.2016.62007.T. Matsuura and S. Saitoh,Division by zero calculus and singular integrals. (Submitted for publication).T. Matsuura, H. Michiwaki and S. Saitoh,$\log 0= \log \infty =0$ and applications. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.)H. Michiwaki, S. Saitoh and M.Yamada,Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.htmlH. Michiwaki, H. Okumura and S. Saitoh,Division by Zero $z/0 = 0$ in Euclidean Spaces,International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16.H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,Journal of Technology and Social Science (JTSS), 1(2017), 70-77.S. Pinelas and S. Saitoh,Division by zero calculus and differential equations. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics).S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/S. Saitoh, A reproducing kernel theory with some general applications,Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .再生核研究所声明371（2017.6.27）ゼロ除算の講演― 国際会議 https://sites.google.com/site/sandrapinelas/icddea-2017 報告1/0=0、0/0=0、z/0=0http://ameblo.jp/syoshinoris/entry-12276045402.html1/0=0、0/0=0、z/0=0http://ameblo.jp/syoshinoris/entry-12263708422.html1/0=0、0/0=0、z/0=0ソクラテス・プラトン・アリストテレス その他
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。ドキュメンタリー 2017: 神の数式 第２回 宇宙はなぜ生まれたのか
〔NHKスペシャル〕神の数式 完全版 第3回 宇宙はなぜ始まったのか
〔NHKスペシャル〕神の数式 完全版 第1回 この世は何からできているのか
NHKスペシャル 神の数式 完全版 第4回 異次元宇宙は存在するか
再生核研究所声明 411(2018.02.02): ゼロ除算発見4周年を迎えてゼロ除算の論文
Mysterious Properties of the Point at InfinityAlgebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-1972018.3.18．午前中 最後の講演： 日本数学会 東大駒場、函数方程式論分科会 講演書画カメラ用 原稿
The Japanese Mathematical Society, Annual Meeting at the University of Tokyo. 2018.3.18.
https://ameblo.jp/syoshinoris/entry-12361744016.html より*057 Pinelas,S./Caraballo,T./Kloeden,P./Graef,J.(eds.):Differential and Difference Equations with Applications:ICDDEA, Amadora, 2017.(Springer Proceedings in Mathematics and Statistics, Vol. 230)May 2018 587 pp.ゼロ除算の論文が２編、出版になりました：
ICDDEA: International Conference on Differential & Difference Equations and Applications
Differential and Difference Equations with Applications
ICDDEA, Amadora, Portugal, June 2017
• Editors
• (view affiliations)
• Sandra Pinelas
• Tomás Caraballo
• Peter Kloeden
• John R. Graef
Conference proceedingsICDDEA 2017
log0=log∞=0log0=log∞=0 and Applications
Hiroshi Michiwaki, Tsutomu Matuura, Saburou Saitoh
Pages 293-305
Division by Zero Calculus and Differential Equations
Sandra Pinelas, Saburou Saitoh
Pages 399-418ゼロ除算（division by zero）
1/0=0、0/0=0、z/0=0
テーマ：数学これは最も簡単な 典型的なゼロ除算の結果と言えます。 ユークリッド以来の驚嘆する、誰にも分る結果では ないでしょうか？Hiroshi O. Is It Really Impossible To Divide By Zero?. Biostat Biometrics Open Acc J. 2018; 7(1): 555703. DOI: 10.19080/BBOJ.2018.07.555703ゼロで分裂するのは本当に不可能ですか？ - Juniper Publishers
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