Chapter 49 Section Three

The calculation of pi and the approximation of irrational roots with decimal fractions (micro numbers) are actually the application of limit thinking in approximate calculations.Liu Hui also used this idea to calculate the area of ​​arc fields.He first proved that the formula for calculating the arc field area in "Nine Chapters of Arithmetic" was inaccurate, and then proposed a method for calculating the arc field density: he used the method of Pythagorean sawing round wood to find the diameter of the circle where the arc field is located, and then used Similar to the program of cutting a circle, the arc is divided into 2, 4, ..., "cutting and cutting to make it extremely thin", which uses the sum of the area of ​​a series of small triangles to approximate the area of ​​the arc field.He also used the Pythagorean theorem to find the chords and vectors of a series of small arc fields corresponding to the above-mentioned small triangles, that is, the base and height of this series of small triangles, "but the number of multiplication of the strings and vectors must be close to the density ratio. .” In this way, the arc field area can be accurate to the required level (as shown in Figure 36).

"Nine Chapters of Arithmetic" and later mathematicians only considered the area of ​​the arc field, and did not discuss the issue of arc length.Shen Kuo, a scientist in the Northern Song Dynasty, created the art of meeting circles in "Mengxi Bi Tan", and proposed an approximate formula for finding the arc length (see Figure 37): l=c+v/d Among them, d is the diameter of the circle where the arc is located, and c and v are still the chord and vector of the arc field.Later, Guo Shoujing and Wang Xun formulated the "Shoushi Calendar" and used Huiyuan many times.