Shinto Shrine - Blog Posts
Trip to Japan 2014
Day 1 - Tokyo Tour
(Click Photos for Hi Res)
Sangaku Saturday #4
In the previous info post, we went over the debate on the religious aspect of sangaku, and the fact that the absence of prayers on these tablets was more puzzling to some than the mathematics. As such, the tablets are not ema prayer tablets, but donations, which usually don't feature prayers on them. Case in point, some consecrated sake and French wine seen at Meiji-jingû in 2016.
Beyond wishing for good fortune and health, such donations serve two very worldly purposes: to contribute to the life and prestige of the shrine or temple (having a famous contributor makes the shrine famous by association), and to advertise the donor in return, as their name is on display. See this large torii at Fushimi Inari Taisha paid for by TV Asahi (テレビ朝日).
With that in mind, Meijizen's cynical comment from 1673 that sangaku aim "to celebrate the mathematical genius of their authors" may not far from the truth. The authors of sangaku are looking to gain notoriety through the publicity that the shrine or temple provides. But was the bemused Meijizen the target audience?
More on that in a couple of weeks. Below the cut is the solution to last week's problem.
The solution to the first problem (below the cut in this post) is the key. Name K, L and M the intersections of the three circles with the horizontal line. Then, by using that previous result,
Indeed, as in that problem, we can construct three right triangles, ABH, ACI and BCJ and apply Pythagoras's theorem in each.
Now, it suffices to note that KL = KM + LM, so
or, dividing by 2*squareroot(pqr), we get the desired result:
Inverting and squaring this yields the formula for r:
This gives us the means to construct this figure on paper using a compass and a marked ruler. Having chosen two radii p and q and constructed the two large circles (remember that AB=p+q) and a line tangent to both, placing M and C is done after calculating the lengths IK=CM=r and IC=KM=2*sqrt(pr).