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''Yigu yanduan'' (益古演段 old mathematics in expanded sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi. Yigu yanduan was based on North Song mathematician Jiang Zhou (蒋周) Yigu Ji (益古集 Collection of Old Mathematics)which was extinct. However from fragments quoted in Yang Hui's work ''The Complete Algorithms of Acreage(田亩比类算法大全)'', we know that 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 new comers to this field. Although Li Zhi's previous monograph ''Ceyuan haijing'' also used tian yuan shu, however it is harder to understand than Yigu yanduan. Yigu yanduan was later collected into Siku Quanshu. Yigu yanduan consists of three volumes with 64 problems solved with Tian yuan shu in parallel with geometrical method. Li Zhi intended to introduced students to the art of Tian yuan shu thru ancient geometry. Yigu yanduan together with Ceyuan haijing are considered major contribution 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, it started 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). The order of arrangement of Tian yuan shu equation in Yigu yanduan is the reverse of that in Ceyuan haijing, i.e., here with the constant term at top, followed by first order tian yuan, second order tian yuan, third order tian yuan etc. 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 western world by the British Protestant Christian missionary to China, Alexander Wylie who wrote: In 1913 Van Hée translated all 64 problems in Yigu yanduan into French.〔van Hée ''Li Yeh, Mathématicien Chinois du XIIIe siècle'', TP,1913,14,537〕 ==Volume I== Problem 1 to 22, all about the mathematics of a circle embedded in a square. 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 ? Anwwer: The circumference of the square is 240 steps, the circumference of the circle is 60 steps.'' Method: set up tian yuan one (celetial element 1) as the diameter of the circle, x :::::::File:Counting rod 0.png TAI :::::::File:Counting rod v1.png multiply it by 3 to get the circumference of the circle 3x (pi ~~3) :::::::File:Counting rod 0.png TAI :::::::File:Counting rod v3.png subtract this from the sum of circumferences to obtain the circumference of the square :::::::File:Counting rod v3.pngFile:Counting rod 0.pngFile:Counting rod 0.png TAI :::::::::File:Counting rod v-3.png The square of it equals to 16 times the area of the square :::::::File:Counting rod v9.pngFile:Counting rod 0.pngFile:Counting rod 0.pngFile:Counting rod 0.pngFile:Counting rod 0.png TAI :::::::File:Counting rod h1.pngFile:Counting rod v-8.pngFile:Counting rod 0.pngFile:Counting rod 0.png ::::::::::File:Counting rod v9.png Again set up tian yuan 1 as the diameter of circle, square it up and multiplied by 12 to get 16 times the area of circle as ::::::::File:Counting rod 0.png TAI ::::::::File:Counting rod 0.png :::::::File:Counting rod h1.pngFile:Counting rod v2.png subtract from 16 time square area we have 16 times area of land :::::::File:Counting rod v9.pngFile:Counting rod 0.pngFile:Counting rod 0.pngFile:Counting rod 0.pngFile:Counting rod 0.png TAI :::::::File:Counting rod h1.pngFile:Counting rod v-8.pngFile:Counting rod 0.pngFile:Counting rod 0.png ::::::::::File:Counting rod v-3.png put it at right hand side and put 16 times 13.75 mu = 16 * 13.75 *240 =52800 steps at left, after cancellation, we get =0: :::::::File:Counting rod v3.pngFile:Counting rod h7.pngFile:Counting rod v2.pngFile:Counting rod 0.pngFile:Counting rod 0.png TAI :::::::File:Counting rod h1.pngFile:Counting rod v-8.pngFile:Counting rod 0.pngFile:Counting rod 0.png ::::::::::File:Counting rod v-3.png Solve this equation to get diameter of circle = 20 steps, circumference of circle = 60 steps 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Yigu yanduan」の詳細全文を読む スポンサード リンク
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