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Sangaku Sunday #2

As the tags in a reblog by @todayintokyo indicated, I waffled about what we'll do in this series in the first post without really defining its main object!

Sangaku are wooden tablets on display at Shintô shrines or Buddhist temples in Japan, featuring geometry problems and their solutions, usually without proof. They started appearing in the Edo period, a particular time for the Japanese people and Japanese scientists. The votive role of these tablets has been debated as far back as the Edo period, as indicated by Meijizen who wrote in 1673:

"There appears to be a trend these days, of mathematical problems on display at shrines. If they were true votive tablets (ema), they should contain a prayer of some sort. Lacking that, one wonders what they are for, other than to celebrate the mathematical genius of their authors. Their meaning eludes me."

I feel the debate on their religious role is overrated. If you look at some food offerings at shrines today, I don't think you'll find a prayer on the bottle of tea or pack of rice, as the prayer is made at the time of offering. It likely is the same for sangaku tablets, which went on display with other offerings. But, as Meijizen hinted, they did have another purpose.

Until we expand on that, below the cut is the solution of last weekend's problem.

Place the point H on the line between A and C1 so that the distance between A and C1 is equal to r2. As the lines (AC1) and (BC2) are both perpendicular to the line (AB), they are parallel, and since AH=BC2=r2, HABC2 is a parallelogram with two right angles: it's a rectangle.

So the length we want, AB, is equal to HC2. The triangle HC1C2 has a right angle at the vertex H, so we can use Pythagoras's theorem:

HC1² + HC2² = C1C2²

In this equality, two lengths are known: C1C2=r1+r2, and

HC1 = AC1-AH = r1-r2 (assuming r1>r2, if not just switch the roles of r1 and r2)

Thus, HC2² = (r1+r2)²-(r1-r2)² = 4 r1 r2 after expanding both expressions (e.g. (r1+r2)² = (r1+r2)x(r1+r2) = r1² + 2 r1 r2 + r2²).

Taking the square root yields the result.