Liu Hui's Secant Circle Method and Infinitesimal Method to Solve Physical Problems
"Round, one is the same length." A circle has a center. From the center, all points on the circle are equal in length.
Mathematical significance: A circle has a center, and the distance (i.e. radius) from the center to all points on the circle is equal.
About the definition of "artificial intelligence". Mozi said, "Hey, a China person is as long." The word "er" here is a circle. Mozi pointed out that the circle can be drawn with compasses or checked with compasses. Compasses have been widely used before Mozi, but it is Mozi's contribution to give a precise definition of circle. Mozi's definition of circle is exactly the same as that in Euclid's Geometry.
When it comes to finding the circumference of a circle, the calculation of pi involves a pi, and the ancients always used "three circumference and one diameter" according to experience. In practice, it is found that the three-dimensional diameter is not the relationship between the circumference and diameter of a circle, but the ratio of the circumference and diameter of a hexagon inscribed in a circle. In the third century AD, China mathematician Liu Hui made an in-depth study on this issue. In his words, "Wednesday's work is from the ring of six shackles." Commenting on Nine Chapters of Arithmetic, he said: "Scholars learn from the past and from its mistakes. Without clear evidence, it is difficult to argue. The image of all things is not round, and Fiona Fang's rate is sincere in the near, although it is far away. "
After a long period of searching and studying hard, Liu Hui finally realized the true meaning and created Out of the Circle, which shocked several circles at home and abroad.
He started from a regular hexagon and doubled the number of sides to 12, 24, 48, 96, 192. Calculate the side lengths of hexagon, dodecagon and 24 one by one, and then multiply them by the number of sides to get the perimeter, which gradually approaches the perimeter of the circle, so the ratio of the perimeter of the regular polygon to the diameter of the circle will gradually approach the pi.
What kind of figure evolved from the circle? "Zhou Kuai Shu Jing" wrote: "The number method comes from the square, and the circle comes from the square." "The moment of the ring is round and the moment of the ring is square." "The number of squares is code, and squares are circles."
So Liu Hui saw the relationship between the circle and the square, and proved the area calculation rules of the circle in "Nine Chapters of Arithmetic" by the following methods:
A circle is inscribed with a regular N-polygon, and its area and perimeter are marked as Sn, PN and An respectively.
Let AB be one side of a circle inscribed with a regular hexagon and AC be one side inscribed with a regular hexagon 12.
Sobc =1/2db * DC =1/4a6 * r =1/2p6 * r, similarly, s24 =1/2p12 * rs2n =1/.
In order to determine the upper limit of the circular area, he also proposed that
S2 n & lt； S<s n+2 (S2 n-S2 n) = S2 n+(S2 n-S2 n), then we get: 314× 64/625 < s <: 3 14× 169/625,
From S = 1/2L r, L≈2 S2 n/r= 628, so π=628/200= 3. 14.
Generally speaking, there are
Liu Hui skillfully used the characteristics of "square is easy to calculate, but circle is difficult to calculate" in secant, and regarded a circle as a regular polygon with infinite sides, which is unknown, while a regular polygon with finite sides is available and known. In fact, the method of finite approximation to infinity is summed up as "if you cut it carefully, you will lose little, and if you cut it again, you will lose nothing!" This sentence. Liu Hui's secant method also shows that if the number of inscribed sides of a regular polygon of a circle is infinitely increased, then the perimeter of the regular polygon is infinitely close to the perimeter of the circle.
Although Liu Hui's Circlesis written more than 700 years ago, it still has profound enlightening significance for studying and studying modern mathematics.