2016-04 - Blog Posts

Sangaku Saturday #8

Having established that sangaku were, in part, a form of advertisement for the local mathematicians, we can look at the target demographic. Who were the mathematicians of the Edo period? What did they work on and how?

The obvious answer is that the people in the Edo period who used mathematics were the ones who needed mathematics. As far back as the time when the capital was in Kashihara, in the early 8th century, evidence of mathematical references has been uncovered (link to a Mainichi Shinbun article, with thanks to @todayintokyo for the hat tip). All kinds of government jobs - accounting, such as determining taxes, customs, or engineering... - needed some form of mathematics. Examples above: 8th-century luggage labels and coins at the Heijô-kyô Museum in Nara, and an Edo-period ruler used for surveying shown at Matsue's local history museum.

As such, reference books for practical mathematics have existed for a long time, and continued to be published to pass on knowledge to the next generation. But sangaku are different: they are problems, not handbooks.

More on that soon. Below the cut is the solution to our latest puzzle.

Recall that SON is a right triangle with SO = 1 and ON = b. These are set values, and our unknowns are the radii p, q and r of the circles with centres A, B and C. While these are unknown, we assume that this configuration is possible to get equations, which we can then solve.

1: The two circles with centres B and C are tangent to a same line, so we can just re-use the very first result from this series, so

2: Also recalling what we said in that first problem about tangent circles, we know that

Moreover, PA = AO - OP = AO - CQ = (p+2*q) - r. Thus, using Pythagoras's theorem in the right triangle APC, we get a new expression for PC:

since 2(p+q)=1 (the first relation). Equating the two expressions we now have of PC², we solve the equation for r:

again using the first relation to write 2q-1 = -2p.

It only remains to find a third equation for p to solve the problem.