Sangaku with Quadratic Optimization
[Fukagawa and Rothman, p. 117, problem 42] refer to Fujita Kagen's book Zoku Shinpeki Sanpo for the following problem
Take a point on the hypotenuse AB of the right triangle ABC. From that point draw perpendiculars to the legs of the triangles. The perpendiculars form a rectangle. Find the location of the point on the hypotenuse that maximizes the area of the rectangle.
According to Fukagawa and Rothman, the problem was originally proposed in 1806 by Hotta Sensuke, a student of Fujita school, and written on a tablet hung in the Gikyosha shrine of Niikappugun, Hokkaido.
References
 H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008
Sangaku
 Sangaku: Reflections on the Phenomenon
 Critique of My View and a Response
 1 + 27 = 12 + 16 Sangaku
 345 Triangle by a Kid
 7 = 2 + 5 Sangaku
 A 49^{th} Degree Challenge
 A Geometric Mean Sangaku
 A Hard but Important Sangaku
 A Restored Sangaku Problem
 A Sangaku: Two Unrelated Circles
 A Sangaku by a Teen
 A Sangaku FollowUp on an Archimedes' Lemma
 A Sangaku with an Egyptian Attachment
 A Sangaku with Many Circles and Some
 A Sushi Morsel
 An Old Japanese Theorem
 Archimedes Twins in the Edo Period
 Arithmetic Mean Sangaku
 Bottema Shatters Japan's Seclusion
 Chain of Circles on a Chord
 Circles and Semicircles in Rectangle
 Circles in a Circular Segment
 Circles Lined on the Legs of a Right Triangle
 Equal Incircles Theorem
 Equilateral Triangle, Straight Line and Tangent Circles
 Equilateral Triangles and Incircles in a Square
 Five Incircles in a Square
 Four Hinged Squares
 Four Incircles in Equilateral Triangle
 Gion Shrine Problem
 Harmonic Mean Sangaku
 Heron's Problem
 In the Wasan Spirit
 Incenters in Cyclic Quadrilateral
 Japanese Art and Mathematics
 Malfatti's Problem
 Maximal Properties of the Pythagorean Relation
 Neuberg Sangaku
 Out of Pentagon Sangaku
 Peacock Tail Sangaku
 Pentagon Proportions Sangaku
 Proportions in Square
 Pythagoras and Vecten Break Japan's Isolation
 Radius of a Circle by Paper Folding
 Review of Sacred Mathematics
 Sangaku à la V. Thebault
 Sangaku and The Egyptian Triangle
 Sangaku in a Square
 Sangaku Iterations, Is it Wasan?
 Sangaku with 8 Circles
 Sangaku with Angle between a Tangent and a Chord
 Sangaku with Quadratic Optimization
 Sangaku with Three Mixtilinear Circles
 Sangaku with Versines
 Sangakus with a Mixtilinear Circle
 Sequences of Touching Circles
 Square and Circle in a Gothic Cupola
 Steiner's Sangaku
 Tangent Circles and an Isosceles Triangle
 The Squinting Eyes Theorem
 Three Incircles In a Right Triangle
 Three Squares and Two Ellipses
 Three Tangent Circles Sangaku
 Triangles, Squares and Areas from Temple Geometry
 Two Arbelos, Two Chains
 Two Circles in an Angle
 Two Sangaku with Equal Incircles
 Another Sangaku in Square
 Sangaku via Peru
 FJG Capitan's Sangaku
Contact Front page Contents Geometry Up
Copyright © 19962018 Alexander Bogomolny
Using the notations from the diagram, the task is to minimize the are
The configuration features several pairs of similar triangles that may be used to derive a relationship between x and y. In particular,
 = 

from which S = xy = (b/a)(ax  x²) and the problem reduces to finding maximum of the quadratic function
I quote from [Fukagawa and Rothman, p. 139]:
Here is a traditional solution from the manuscript Solutions to Problems of Zoku Shinpeki Sanpo by Kitagawa Moko (17631833). In this case the traditional solution is pretty much what any calculus student would do (but see chapter 9.)
... Setting the derivative df/dx to 0 gives
As I am sure most of the calculus students will indeed find the maximum of function
The question of differentiation in some sense is more difficult than that of integration because, although traditional Japanese mathematicians wrote volumes on integration techniques, they were virtually silent about how they took derivatives. One thing is certain: the Japanese did not have the concept of differentiation as we know it. In no traditional manuscript do we the fundamental formula for the derivative:


Without this concept it is difficult or impossible to develop a formal theory of differentiation. Perhaps for this reason in the wasan differentiation was confined to finding the maximum and minimum of functions.
How did the traditional geometers do this? For quadratic equations, parabolas, one knows without calculus that the maximum or minimum is at the vertex of the parabola, and so for functions of the form
My interpretation of the above is that, faced with the task of finding the maximum of the function obtained in the sangaku at hand, the traditional Japanese mathematicians would not, perhaps for the wrong reasons, calculate and subsequently equate to zero the derivative of quadratic functions. Unlike a majority of the present day calculus students, they would simply point out to a solution which is especially easy to surmise (and remember) for the function
Contact Front page Contents Geometry Up
Copyright © 19962018 Alexander Bogomolny
68354727