Yigu yanduan

Yigu yanduan was based on Northern Song mathematician Jiang Zhou's (蒋周) Yigu Ji (益古集 Collection of Old Mathematics) which is not extant.

However, from fragments quoted in Yang Hui's work The Complete Algorithms of Acreage (田亩比类算法大全), this lost mathematical treatise Yigu Ji was about solving area problems with geometry.

Li Zhi used the examples of Yigu Ji to introduce the art of Tian yuan shu to newcomers to this field.

Although Li Zhi's previous monograph Ceyuan haijing also used Tian yuan shu, it is harder to understand than Yigu yanduan.

Yigu yanduan consists of three volumes with 64 problems solved using Tian yuan sh] in parallel with the geometrical method.

Li Zhi intended to introduce students to the art of Tian yuan shu through ancient geometry.

Yigu yanduan together with Ceyuan haijing are considered major contributions to Tian yuan shu by Li Zhi.

These two works are also considered as the earliest extant documents on Tian yuans shu.

All the 64 problems followed more or less the same format, starting with a question (问), followed by an answer (答曰), a diagram, then an algorithm (术), in which Li Zhi explained step by step how to set up algebra equation with Tian yuan shu, then followed by geometrical interpretation (Tiao duan shu).

This later arrangement conformed with contemporary convention of algebra equation( for instance, Qin Jiushao's Mathematical Treatise in Nine Sections), and later became a norm.

Yigu yanduan was first introduced to the English readers by the British Protestant Christian missionary to China, Alexander Wylie who wrote:Yi koo yen t'wan...written in 1282 consists of 64 geometrical problem, illustrated the principle of Plane Measurement, Evolution and other rules, the whole being developed by means of T'een yuen.

[1]In 1913 Van Hée translated all 64 problems in Yigu yanduan into French.

Example: problem 8 There is a square field, with a circular pool in the middle, given that the land is 13.75 mu, and the sum of the circumferences of the square field and the circular pool equals to 300 steps, what is the circumferences of the square and circle respective ?

Solve this equation to get diameter of circle = 20 steps, circumference of circle = 60 steps Problem 23 to 42, 20 problems in all solving geometry of rectangle embedded in circle with tian yuan shu Example, problem 35 Suppose we have a circular field with a rectangular water pool in the center, and the distance of a corner to the circumference is 17.5 steps, and the sum of length and width of the pool is 85 steps, what is the diameter of the circle, the length and width of the pool ?

equals to four times the area of the circle: subtracting four times the area of land to obtain: now The square of the sum of length and width of the pool =85*85 =7225 which is four times the pool area plus the square of the difference of its length and width (

=0 Solve this equation to get Problem 42 to 64, altogether 22 questions about the mathematics of more complex diagrams Q: fifty-fourth.

There is a square field, with a rectangular water pool lying on its diagonal.

The area outside the pool is one thousand one hundred fifty paces.

What is the area of the square field, what is the length and width of the pool?

Area of pool times 乘 1.96 ( the square root of 2) =1.4 one has