The so-called secant method is a method of calculating pi by using the circumference of a regular polygon inscribed in a circle to approach the circumference infinitely. This method is a brand-new method created by Liu Hui after criticizing and summarizing various old calculation methods in the history of mathematics.
In ancient China, since the pre-Qin period, the calculation of a circle has always been based on "the diameter of three weeks is one". However, the results calculated with this value often have great errors. As Liu Hui said, the circumference of a circle calculated by "Three Circumferences Diameter One" is actually not the circumference of a circle, but the circumference of a circle inscribed by a regular hexagon (see figure 1-5- 1), and its value is far less than the actual circumference. Zhang Heng of the Eastern Han Dynasty was not satisfied with this result. He began to study the relationship between a circle and its circumscribed circle (see figure 1-5-2) and got pi. This value is better than "three-circumference diameter one", but Liu Hui thinks that the calculated circumference must be longer than the actual circumference, which is inaccurate. Under the guidance of limit thought, Liu Hui put forward the secant method for calculating pi, which is not only bold and innovative, but also rigorous, and points out a scientific way for calculating pi.
In Liu Hui's view, because the circumference calculated by "three-circumference diameter one" is actually the circumference of a circle inscribed by a regular hexagon, it is quite different from the circumference; Then we can divide the circumference into six arcs on the basis of inscribed regular hexagon, and then continue to divide each arc into two, thus making a regular dodecagon inscribed in the circle. Isn't the circumference of this regular dodecagon closer to the circumference than that of a regular hexagon? If the circumference is further divided into circles inscribed with a regular quadrangle, then the circumference of a regular quadrangle must be closer to the circumference than that of a regular dodecagon. (see figure 1-5-3). This shows that the finer the circumference is divided, the smaller the error is, and the closer the circumference of inscribed regular polygon is to the circumference. This continuous division continues until the circumference can no longer be divided, that is, when the number of sides inscribed with a regular polygon in a circle is infinite, its circumference is "combined" with the circumference, which is exactly the same.
According to this idea, Liu Hui calculated the area of a circle inscribed by a regular polygon and a regular polygon with 3072 sides, thus obtaining two approximate values of pi, 3. 14 and 3. 14 16. This result was the most accurate data for calculating pi in the world at that time. Liu Hui has great confidence in this new method of secant circle, and has extended it to all aspects of circle calculation, thus greatly promoting the development of mathematics since the Han Dynasty.
Later, in the Northern and Southern Dynasties, Zu Chongzhi continued to work hard on the basis of Liu Hui, and finally got pi: accurate to the seventh place after the decimal point. In the west, this achievement was achieved by the French mathematician Veda in 1593, which was more than 1 100 years later than Zu Chongzhi. Zu Chongzhi also got two fractional values of pi, one is "approximate rate" and the other is "density rate". In the west, this value was obtained by Otto in Germany and Antonitz in the Netherlands at the end of 16, which was later than Zu Chongzhi 1 100. History will never forget the great contribution of Liu Hui's new method of secant circle to the development of ancient mathematics in China.