Critique of My View and a Response

Keiji Matsumoto from Japan has kindly communicated to me his critique of the view points exposed in my introduction to the Sangaku pages. Below I included a slightly edited copy of his message followed by my response.

I read your article on Sangaku, which was nice. but I found several misunderstandings in your writings.

  1. 'Kanbun=Latin' is not quite correct. I think Sangaku was, and is, readable by relatively many, if one knows technical terms of math at that time. (At least, I can read them though I have not studied Kanbun. Of course, however, I don't think I can write one properly.) Japanese writing uses, in daily basis,. a lot of Kanji, or Chinese characters, so people know its meanings. How about grammar? I have to say grammar of Kanbun, or ancient Chinese writing, is quite simple. (On the other hand, the number of characters is tremendous.) In addition, Kanbun of Sangaku is quite simple. I believe that at least a few persons in a village could read them (only if they are interested ...)

  2. Just like western mathematics, focus of Japanese math at that period was also calculus and algebra. Even geometrical problems are often treated as application of algebra. Especially, in the time of Seki Kowa, they simply applied Pythagorean theorem, and solved everything by brute-force algebra. I think most interesting finding by Japanese mathematics at that time are:

    1. A necessary and sufficient condition for a two polynomials to have a common root. R(f, g) = 0, where R is the resultant. The result is independently found by Seki Kowa (Tokyo) and Iseki (Osaka) . They used determinant to represent them. (It is often pointed out that Seki, in his first writing on the topic, wrote incorrect the definition of determinant for more than 5×5. However, he presented correct formula later. Some believe his first mistake was of typo-nature; with small correction, they claim they can recover the correct one.)
    2. numeric acceleration method, like Aitken's Δ-process (Seki), Romberg's extrapolation (Takebe). (Both of them anticipates results obtained in western mathematics.)
    3. numeric methods to solve linear/nonlinear multivariable equations.
    4. study of the number of positive/negative roots, conditions for having degeneracy. (These and (a) are obviously connected to (c)).
    5. Infinite series expansion and continued fraction expansion of arctan², motivated to compute π (Takebe).
    6. similar results for various (inverse) trigonometric functions.
    7. various formulas for integration.
    8. discovery of inversion. motivated to simplify Sangaku-like problem etc.

    But, for the sake of visual effect, geometrical problems are preferred for Sangaku. It is also said that Sangaku was not (at least, not mainly) for demonstration of the most recent result: Many of them are by students, to advertise how far they had developed.

  3. First, the development of Rangaku was only after the middle of the Edo period. Before that, the import of books from the West was strictly prohibited, due to anti-chirstian policy. (Some points out that, anyway up to that time, Japanese were busy with learning from traditional Chinese civilizations, and there were not so much demand for more advanced science/technology.) Hence, most of researchers sees negligible western inference on Japanese mathematics. In fact, there is no reason to assume impact from the west -- the development from traditional Chinese math to Japanese math was so continuous. Also, almost all Japanese mathematicians could not read dutch. (I only know two exceptions.) On the other hand, the influence of traditional Chinese math and astronomy was essential. while they were almost replaced by Western math and astronomy at that time. This gap was, I think, due to the fact that China had been occupied by 'barbarians' (=Manchurian), and Japanese/Korean at that time secretly looked down on Ching dynasty. For example, Japanese astronomers first did not pay attention to contemporary Chinese astronomical documents, and were interested in older Chinese calenders. Change occurred after the death of Seki. Nakane, a leader of Seki school, proposed to Shogun to relax the ban on import of books from west. Especially, he correctly pointed out the development of western astronomy. As a result, Chinese translation of western astronomical works were imported. trigonometry, spherical trigonometry, and log entered into Japan at that time. (Log was used (1) to compute n-th root, (2) theory of tuning, or to compute equal temperament.) Even after that, we didn't learn a lot. One reason was that they (mathematicians/astronomers) couldn't read Dutch. Those who could read did not know math. For example, a leading Japanese astronomer (Yoshitoki Takahashi) 'translated' a book on astronomy by Lalande. But it is said that his 'translation' was almost like Description, helped by equations and pictures. (So, very hard to see sentence by sentence correspondence.) Indeed, even at the early Meiji era, many important projects (construction, voyage, etc ...) relied on the traditional Japanese mathematics, since leading engineers at that time were educated in the Edo era, and this was the only math they could property use.

    I think impact of Rangaku was more on medical/biological sciences. As for technologies like clock, I think influence was more via real machinery itself.

  4. The question of proportion and class of people who contributed/enjoyed mathematics. Surely it was limited to relatively wealthy people, I think. But the social class, in view of these four division (I don't think this division, and the ordering among the four, correctly reflects the reality of the society at that time), was really 'from all the social classes'.

    I read a book on the life of math teacher who travels all over the country. He mainly taught to farmers and merchants. the problems they treated were not necessarily for practical use. Some of them were mainly for entertainment (not so deep, though). In general, culture/science of Japan at that time very much relies on support from wealthy merchant and farmers. For example, astronomy in the latter half of Edo period was mainly pushed forward by a school which was supported by merchants in Osaka, (some important scholarly are from among them, e.g., Shigetomi Hazama). Finally, Shogun hired some of them, and made them samurai.


    Keiji Matsumoto

Here is my response.

Dear Keiji (if I may):

many thanks for writing. I appreciate your effort very much. I also appreciate the Sangaku phenomenon and the peculiarity of the Edo period.

The only thing that I object to is an unintended (by Rothman, Pedoe and Fukagawa) side effect of creating a Sangaku myth. That this is what might be happening is demonstrated by an article in "News at Princeton" (Rothman helps reveal intricacies of ancient math phenomenon) where the popularity of sangaku is compared to that of sudoku. I just do not believe the comparison holds water. Sangaku was a terrific phenomenon but could not have been as popular as it may appear from Rothman and Fukagawa's work. The numbers just do not support that - I think you are not disagreeing with that.

Also, I did not mean to denigrate the Wasan. As you rightly observe, in several developments in Calculus and Algebra the Japanese discoveries preceded those in the West - I have no issue with that. However, Wasan is a very broad term. In my view, so little has been done in geometry and this little has been based mostly on the well known facts, like theorems of Pythagoras and Ptolemy, that it is safe to assume that the political circumstances of seclusion had very little effect on the development of geometry. Calculus is of course a different matter altogether.

I also have no doubt that Sangaku touched one way or another members of all social strata. However to my ear, to claim in English that members of "all social classes" have been involved in a certain development is to imply large numbers from all classes. Indeed, in a recent book [Fukagawa and Rothman, p. 9-10] we read

... the inscriptions on the tablets make it clear that whole classes of students, children, and occasionally women dedicated sangaku. So the best answer to the question "Who created them?" seems to be "everybody."

For you to understand my take on Rothman's and subsequent claims, let's go to extreme. If only 1 member of every societal group had an interest in Sangaku, Rothman would never mention "all social classes". For him to do so he had to believe that substantial numbers from every class have been involved with Sangaku. And, as I already mentioned, I simply do not believe that the numbers were at all large. This was the point of my argument. Drawn by an admirable enthusiasm for the delightful phenomenon of sangaku, Rothman takes a huge literary license.

Finally, I am asking for your permission to append your letter and my response to the Sangaku page at my site. This would make the story more interesting and the case more convincing. Please let me know.

All the best,
Alexander Bogomolny


  1. H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008


  1. Sangaku: Reflections on the Phenomenon
  2. Critique of My View and a Response
  3. 1 + 27 = 12 + 16 Sangaku
  4. 3-4-5 Triangle by a Kid
  5. 7 = 2 + 5 Sangaku
  6. A 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored Sangaku Problem
  10. A Sangaku: Two Unrelated Circles
  11. A Sangaku by a Teen
  12. A Sangaku Follow-Up on an Archimedes' Lemma
  13. A Sangaku with an Egyptian Attachment
  14. A Sangaku with Many Circles and Some
  15. A Sushi Morsel
  16. An Old Japanese Theorem
  17. Archimedes Twins in the Edo Period
  18. Arithmetic Mean Sangaku
  19. Bottema Shatters Japan's Seclusion
  20. Chain of Circles on a Chord
  21. Circles and Semicircles in Rectangle
  22. Circles in a Circular Segment
  23. Circles Lined on the Legs of a Right Triangle
  24. Equal Incircles Theorem
  25. Equilateral Triangle, Straight Line and Tangent Circles
  26. Equilateral Triangles and Incircles in a Square
  27. Five Incircles in a Square
  28. Four Hinged Squares
  29. Four Incircles in Equilateral Triangle
  30. Gion Shrine Problem
  31. Harmonic Mean Sangaku
  32. Heron's Problem
  33. In the Wasan Spirit
  34. Incenters in Cyclic Quadrilateral
  35. Japanese Art and Mathematics
  36. Malfatti's Problem
  37. Maximal Properties of the Pythagorean Relation
  38. Neuberg Sangaku
  39. Out of Pentagon Sangaku
  40. Peacock Tail Sangaku
  41. Pentagon Proportions Sangaku
  42. Proportions in Square
  43. Pythagoras and Vecten Break Japan's Isolation
  44. Radius of a Circle by Paper Folding
  45. Review of Sacred Mathematics
  46. Sangaku à la V. Thebault
  47. Sangaku and The Egyptian Triangle
  48. Sangaku in a Square
  49. Sangaku Iterations, Is it Wasan?
  50. Sangaku with 8 Circles
  51. Sangaku with Angle between a Tangent and a Chord
  52. Sangaku with Quadratic Optimization
  53. Sangaku with Three Mixtilinear Circles
  54. Sangaku with Versines
  55. Sangakus with a Mixtilinear Circle
  56. Sequences of Touching Circles
  57. Square and Circle in a Gothic Cupola
  58. Steiner's Sangaku
  59. Tangent Circles and an Isosceles Triangle
  60. The Squinting Eyes Theorem
  61. Three Incircles In a Right Triangle
  62. Three Squares and Two Ellipses
  63. Three Tangent Circles Sangaku
  64. Triangles, Squares and Areas from Temple Geometry
  65. Two Arbelos, Two Chains
  66. Two Circles in an Angle
  67. Two Sangaku with Equal Incircles
  68. Another Sangaku in Square
  69. Sangaku via Peru
  70. FJG Capitan's Sangaku

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