Chapter 42 The third section stacking technique and recruiting technique
Yang Hui's second-order arithmetic series summation method is usually called stacking technique, and Zhu Shijie pushed the research of stacking technique to the highest peak. In the "Siyuan Yujian" volume, "Jiao Cao Shaped Section", "Ruxiang Tricks" and "Fruit Pile Stacking" in the volume 33 questions, all of which are the sum of known high-order arithmetic progressions to find the number of items The problem.In order to solve these problems, it is necessary to list a higher-order equation according to the respective summation formulas, and then use the "positive and negative formulas" to find its roots.Among these problems, Zhu Shijie proposed a series of triangular stack formulas:
Zizania haystack (or Zizania grass plot):
Triangular stack (or one-shaped stack):
Scattering star-shaped stack (or triangular-shaped stack):
Triangular star-shaped stack (or star-studded one-shaped stack):
The triangular scattering of stars is more like a stack:
These formulas seem to be incoherent in Zhu Shijie's book, but from among them, the latter one is called the first one, that is, the sum of the first n terms of the former one is the nth term of the latter one, They formed a complete system in Zhu Shijie's mind.Let's look at their relationship with the Jiaxian triangle again: the above-mentioned numbers are the numbers on the 2nd, 3rd, 4th, 5th, and 6th diagonal lines of the Jiaxian triangle in turn, and their sum is exactly the 3rd, 4th, 5th, and 6th. , the nth number on the 7 slanted lines, which is why Zhu Shijie used two sets of parallel lines parallel to the left and right slanted lines to connect the numbers of Jia Xian's triangle.It can be seen that Zhu Shijie has mastered the general formula of triangular stack:
Obviously, when p=1, 2, 3, 4, 5, it is the above triangular stack formula.Zhu Shijie also solved a series of high-order arithmetic series summation problems with the product of four-corner stacks as a general term, as well as more complex series summation problems such as Lanfeng-shaped stacks.
Guo Shoujing (1231-1316 A.D.), Wang Xun (1235-1281 A.D.) and other astrologers in the Yuan Dynasty used the trick to calculate the diurnal degrees of the day and month.Zhu Shijie also developed to a very complete level the use of tricks to solve the summation problem of high-order arithmetic series. In the appendix to the fifth question of "Ruxiang tricks": "(There is a lawsuit now) to recruit soldiers according to the cube, the first move is three feet, and the second move is one foot more, and the number is called soldiers. Now recruit 15 squares, ask about recruiting troops Geometry?" "The art said: Find the top difference of twenty-seven, the second difference of thirty-seven, the third difference of twenty-four, and the bottom difference of six. The recruiter: this move is for the upper accumulation, and now the move is reduced by one for the bottom of the bamboo grass. The accumulation is two accumulations, and now subtract two to form the triangular foundation accumulation to form three accumulations, and now subtract three to form the three corners and one accumulation to form the lower accumulation. Multiply each accumulation by each difference, and combine the four, that is the number of soldiers recruited." Suppose the number of days is x, and f(x) is the total number of recruits on the xth day, then the number of recruits per day is (2+x). When x=1, 2, 3, 4..., the value of f(x) and The grades are as follows:
Upper difference △=27, second difference △=37, third difference △=24, lower difference △=6.And the upper product is n, and the second product is the product of bamboo shoots based on (n-1)
This formula is exactly the same as the modern general form.In Europe, it was only in the writings of J. Gregory (AD 1670) that the recruiting technique was first explained, while the general formula appeared in the writings of Newton (AD 1676).Zhu Shijie pointed out that the coefficients of the recruiting formula are exactly the products of the triangular piles in turn, which is his outstanding contribution.In the above formula, n=15, then
That is, the total number of recruits on the 15th.
Triangular stack (or one-shaped stack):
Scattering star-shaped stack (or triangular-shaped stack):
Triangular star-shaped stack (or star-studded one-shaped stack):
The triangular scattering of stars is more like a stack:
These formulas seem to be incoherent in Zhu Shijie's book, but from among them, the latter one is called the first one, that is, the sum of the first n terms of the former one is the nth term of the latter one, They formed a complete system in Zhu Shijie's mind.Let's look at their relationship with the Jiaxian triangle again: the above-mentioned numbers are the numbers on the 2nd, 3rd, 4th, 5th, and 6th diagonal lines of the Jiaxian triangle in turn, and their sum is exactly the 3rd, 4th, 5th, and 6th. , the nth number on the 7 slanted lines, which is why Zhu Shijie used two sets of parallel lines parallel to the left and right slanted lines to connect the numbers of Jia Xian's triangle.It can be seen that Zhu Shijie has mastered the general formula of triangular stack:
Obviously, when p=1, 2, 3, 4, 5, it is the above triangular stack formula.Zhu Shijie also solved a series of high-order arithmetic series summation problems with the product of four-corner stacks as a general term, as well as more complex series summation problems such as Lanfeng-shaped stacks.
Guo Shoujing (1231-1316 A.D.), Wang Xun (1235-1281 A.D.) and other astrologers in the Yuan Dynasty used the trick to calculate the diurnal degrees of the day and month.Zhu Shijie also developed to a very complete level the use of tricks to solve the summation problem of high-order arithmetic series. In the appendix to the fifth question of "Ruxiang tricks": "(There is a lawsuit now) to recruit soldiers according to the cube, the first move is three feet, and the second move is one foot more, and the number is called soldiers. Now recruit 15 squares, ask about recruiting troops Geometry?" "The art said: Find the top difference of twenty-seven, the second difference of thirty-seven, the third difference of twenty-four, and the bottom difference of six. The recruiter: this move is for the upper accumulation, and now the move is reduced by one for the bottom of the bamboo grass. The accumulation is two accumulations, and now subtract two to form the triangular foundation accumulation to form three accumulations, and now subtract three to form the three corners and one accumulation to form the lower accumulation. Multiply each accumulation by each difference, and combine the four, that is the number of soldiers recruited." Suppose the number of days is x, and f(x) is the total number of recruits on the xth day, then the number of recruits per day is (2+x). When x=1, 2, 3, 4..., the value of f(x) and The grades are as follows: