Secant is to replace the circle with a circle inscribed with a regular polygon. Liu Hui thinks that when the number of regular polygons inscribed in a circle increases infinitely, its perimeter will be closer and closer to that of the circle. "

When the number of regular polygons inscribed in a circle is infinite, the limit of its perimeter is the perimeter of the circle and the limit of its area is the area of the circle. It contains the earliest concept of limit and the idea of linear curve transformation, which is of great significance to the later development of limit theory in higher mathematics.

According to secant technique, Liu Hui calculated from the regular hexagon inscribed in the circle, and the number of sides gradually doubled, and then calculated the regular 12 polygon and the regular 24 polygon successively, so that the regular polygon inscribed in the circle gradually approached the circle, thus verifying the calculation formula of the circle area and calculating the more accurate pi value.

The value of π=3. 14 124 is obtained. Not only that, he continued to calculate until he figured out the area of the regular 3072 polygon inscribed with a circle, and worked out a more accurate value π=3. 14 16.

Extended data

A short story about cyclotomy.

Liu Hui in ancient China has been trying to study the calculation of pi. Once, Liu Hui saw a stonemason working on a stone, which made him feel very interesting, so he observed it carefully. "wow! A square stone turned out to be an octagonal stone after four corners were chiseled off by a stonemason. After going to eight corners, it became a hexagon again. " A square stone is processed into a smooth cylinder and cut off with an axe at a time.

Who would have thought that what seemed ordinary to ordinary people triggered the spark of Liu Hui's wisdom? He thought, "Can the stonemason's method of processing stones be used in the study of pi?"

So, Liu Hui used this method to divide circles gradually and gave it a try, and the effect was good. He invented the unprecedented "cyclotomy". He followed the idea of cyclotomy, counting from the regular hexagon inscribed in the circle, doubling the number of sides in turn, and calculating the area of the regular 12 polygon and the regular 24 polygon in turn ... until the regular 192 polygon.

The approximate value of pi is 157/50 (3. 14). Later, he calculated the area of the inscribed circle of the regular 3072 polygon, thus obtaining a more accurate approximation of pi: π ≈ 3927/1250 (3.1416).

Baidu encyclopedia-cyclotomy