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.

   Regards,
   Keiji Matsumoto

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" (http://www.princeton.edu/main/news/archive/S15/04/04O77/index.xml?section=topstories) 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

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