Suanfa Tong Zong

In 1592, Chinese mathematician Cheng Dawei published Suanfa Tong Zongtranslated in English as either General Source of Computational Methods or Systematic Treatise on Mathematics. The text became a reference book for both professional and amateur mathematicians, with a great influence on the development of Japanese mathematics as well. 

Cheng Dawei

A portrait of Cheng Dawei 

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The problems below appeared in chapters 3, 7, and 12,  respectively. In the first six we are asked to show that each given relationship between the area of a circle and that of a regular polygon is true.

Suanfa Tong Zong

Sources: Sacred mathematics: Japanese Temple Geometry by Fukagawa Hidetoshi and Tony Rothman (2008); A History of Chinese Mathematics by Jean-Claude Martzloff (2006).

Triangle in Circle

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The area of the equilateral triangle is about seven sixteenths the area of the circle.

Square in Circle

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The area of the square is about two thirds the area of the circle.

Hexagon in Circle

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The area of the regular hexagon is about six sevenths the area of the circle.

Circle in Triangle

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The area of the circle is about four sevenths the area of the equilateral triangle.

Circle in Square

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The area of the circle is about three fourths the area of the square.

Circle in Hexagon

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The area of the circle is about six sevenths the area of the regular hexagon.

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"How Deep is the Pond?" problem as illustrated in Suanfa Tong Zong

How Deep is the Pond?

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Two reeds of equal length project 3 syaku above the pond's surface. One reed is pulled 9 syaku towards the shore such that it is now just touching the surface. How deep is the pond?

An Equilateral Triangle

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An equilateral triangle of side 20 is divided into three equal parts. What are the area and the side lengths of each quadrilateral?

Right Triangle and Circle

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What is the radius of the circle?

Right Triangle and Square

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What is the side of the square?