Travel Diary of Mathematician Yamaguchi Kanzan
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18th of September: There was a great statue of Buddha in the Hokoji temple of Kyoto. Yamaguchi wanted to see it, but the statue had been destroyed. “I have started in the Hokoji temple at the place of the great statue of Buddha and visited the Kiyomizu temple. There I recorded a sangaku hung by Ritouken in 1822. On the tablet, the names of lovers of mathematics were written.” 21st of September: “In mathematician Takeda’s house, I saw one sangaku problem that is to be hung in the near future and another problem which was already written on the tablet of the Tenman shrine. I have written them in my diary.” We give Takeda’s problem “to be hung in the near future” as problem 17. The other problem on the Tenman shrine sangaku was proposed by Morikawa Jihei and is presented as problem 18.
Problem 17 Here is Takeda’s problem “to be hung in the near future”: We are given a segment of a circle of radius R. Draw two smaller circles of radius r inside the segment such that they touch each other; both are tangent to the chord AB; one touches the arc AB and one touches the tangent line AP. If AB = k and AP = m, find r in terms of k, d, and m. d
P
The answer is on page 281. m
r
r
B
A k
Problem 18 This is Morikawa’s problem at the Tenman shrine, proposed at an unknown date. Two circles with radii a and b sit on the line l and touch each other. Between them is an inscribed a square with side x. Find the minimum of x in terms of a and b. The tablet contained no solution, but Morikawa had written, “I will be very happy if someone can solve this problem.” And so, says Yamaguchi, “I went to Morikawa’s home with my friend Takeda and asked him what the answer b is. He said that he could not solve the problem a yet.” Neither does Yamaguchi’s diary contain a x solution and, like Morikawa, we would be very l happy if someone solves this problem.