Yigu yanduan

Last updated
Preface to Yigu yanduan, Siku Quanshu Yi Gu Yan Duan .jpg
Preface to Yigu yanduan, Siku Quanshu

Yigu yanduan (益古演段 Old Mathematics in Expanded Sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi. Yigu yanduan was based on Northern Song mathematician Jiang Zhou (蒋周)'s Yigu Ji (益古集 Collection of Old Mathematics) which was extinct. However, from fragments quoted in Yang Hui's work The Complete Algorithms of Acreage (田亩比类算法大全), we know that this lost mathematical treatise Yigu Ji was about solving area problems with geometry. Li Zhi used the examples of Yigu Ji to introduce the art of Tian yuan shu to newcomers to this field. Although Li Zhi's previous monograph Ceyuan haijing also used Tian yuan shu, it is harder to understand than Yigu yanduan.

Contents

Yigu yanduan was later collected into Siku Quanshu .

Yigu yanduan consists of three volumes with 64 problems solved using Tian yuan sh] in parallel with the geometrical method. Li Zhi intended to introduce students to the art of Tian yuan shu through ancient geometry. Yigu yanduan together with Ceyuan haijing are considered major contributions to Tian yuan shu by Li Zhi. These two works are also considered as the earliest extant documents on Tian yuans shu.

All the 64 problems followed more or less the same format, starting with a question (问), followed by an answer (答曰), a diagram, then an algorithm (术), in which Li Zhi explained step by step how to set up algebra equation with Tian yuan shu, then followed by geometrical interpretation (Tiao duan shu). The order of arrangement of Tian yuan shu equation in Yigu yanduan is the reverse of that in Ceyuan haijing, i.e., here with the constant term at top, followed by first order tian yuan, second order tian yuan, third order tian yuan etc. This later arrangement conformed with contemporary convention of algebra equation( for instance, Qin Jiushao's Mathematical Treatise in Nine Sections ), and later became a norm.

Yigu yanduan was first introduced to the English readers by the British Protestant Christian missionary to China, Alexander Wylie who wrote:

Yi koo yen t'wan...written in 1282 consists of 64 geometrical problem, illustrated the principle of Plane Measurement, Evolution and other rules, the whole being developed by means of T'een yuen. [1]

In 1913 Van Hée translated all 64 problems in Yigu yanduan into French. [2]

Volume I

Problem 8 in Yigu yanduan solved by in line Tian yuan shu Yi Gu Di Ba Wen .jpg
Problem 8 in Yigu yanduan solved by in line Tian yuan shu

Problem 1 to 22, all about the mathematics of a circle embedded in a square.

Example: problem 8

There is a square field, with a circular pool in the middle, given that the land is 13.75 mu, and the sum of the circumferences of the square field and the circular pool equals to 300 steps, what is the circumferences of the square and circle respective ?

Anwwer: The circumference of the square is 240 steps, the circumference of the circle is 60 steps.

Method: set up tian yuan one (celestial element 1) as the diameter of the circle, x

Counting rod 0.png TAI
Counting rod v1.png

multiply it by 3 to get the circumference of the circle 3x (pi ~~3)

Counting rod 0.png TAI
Counting rod v3.png

subtract this from the sum of circumferences to obtain the circumference of the square

Counting rod v3.png Counting rod 0.png Counting rod 0.png TAI
Counting rod v-3.png

The square of it equals to 16 times the area of the square

Counting rod v9.png Counting rod 0.png Counting rod 0.png Counting rod 0.png Counting rod 0.png TAI
Counting rod h1.png Counting rod v-8.png Counting rod 0.png Counting rod 0.png
Counting rod v9.png

Again set up tian yuan 1 as the diameter of circle, square it up and multiplied by 12 to get 16 times the area of circle as

Counting rod 0.png TAI
Counting rod 0.png
Counting rod h1.png Counting rod v2.png

subtract from 16 time square area we have 16 times area of land

Counting rod v9.png Counting rod 0.png Counting rod 0.png Counting rod 0.png Counting rod 0.png TAI
Counting rod h1.png Counting rod v-8.png Counting rod 0.png Counting rod 0.png
Counting rod v-3.png

put it at right hand side and put 16 times 13.75 mu = 16 * 13.75 *240 =52800 steps at left, after cancellation, we get

Counting rod v3.png Counting rod h7.png Counting rod v2.png Counting rod 0.png Counting rod 0.png TAI
Counting rod h1.png Counting rod v-8.png Counting rod 0.png Counting rod 0.png
Counting rod v-3.png

Solve this equation to get diameter of circle = 20 steps, circumference of circle = 60 steps

Volume II

Yi Gu Di 36Wen .jpg

Problem 23 to 42, 20 problems in all solving geometry of rectangle embedded in circle with tian yuan shu

Example, problem 35

Suppose we have a circular field with a rectangular water pool in the center, and the distance of a corner to the circumference is 17.5 steps, and the sum of length and width of the pool is 85 steps, what is the diameter of the circle, the length and width of the pool ?

Answer: The diameter of the circle is one hundred steps, the length of pool is 60 steps, and the width 25 steps. Method: Let tian yuan one as the diagonal of rectangle, then the diameter of circle is tian yuan one plus 17.5*2

multiply the square of diameter with equals to four times the area of the circle:

subtracting four times the area of land to obtain:

four times the area of pool = =

now

The square of the sum of length and width of the pool =85*85 =7225 which is four times the pool area plus the square of the difference of its length and width ()

Further double the pool area plus equals to = the square of the diagonal of the pool thus

( four time pool area + the square of its dimension difference ) - (twice the pool area + square if its dimension difference) equals = twice the pool area

so four times the area of pool =

equate this with the four times pool area obtained above

=

we get a quadratic equation =0 Solve this equation to get

Volume III

Yi Gu Yan Duan Xia 54.jpg

Problem 42 to 64, altogether 22 questions about the mathematics of more complex diagrams

Q: fifty-fourth. There is a square field, with a rectangular water pool lying on its diagonal. The area outside the pool is one thousand one hundred fifty paces. Given that from the corners of the field to the straight sides of the pool are fourteen paces and nineteen paces. What is the area of the square field, what is the length and width of the pool?

Answer: The area of the square field is 40 square paces, the length of the pool is thirty five paces, and the width is twenty five paces.

Let the width of the pool be Tianyuan 1.

Counting rod 0.png TAI
Counting rod v1.png

Add the width of the pool to twice the distance from field corner to short long side of pool equals to the length of diagonal of the field x+38

Counting rod h3.png Counting rod v8.png
Counting rod v1.png TAI

Square it to obtain the area of square with the length of the pool diagonal as its sides

Counting rod h1.png Counting rod v4.png Counting rod h4.png Counting rod v4.png
Counting rod h7.png Counting rod v6.png TAI
Counting rod v1.png
The length of pool minus the width of pool multiplied by 2 = 2 (19-14) = 10

Pool length = pool width +10: x+10

Counting rod h1.png Counting rod 0.png TAI
Counting rod v1.png

Pool area = pool with times pool length : x(x+10) =

Counting rod 0.png TAI
Counting rod h1.png Counting rod 0.png
Counting rod v1.png

Area of pool times 乘 1.96 ( the square root of 2) =1.4

one has

Counting rod h1.png Counting rod v9.png Counting rod h6.png tai
Counting rod v1.png Counting rod h9.png Counting rod v6.png

Area of diagonal square subtract area of pool multiplied 1.96 equals to area of land times 1.96:

-
Counting rod h1.png Counting rod v4.png Counting rod h4.png Counting rod v4.png
Counting rod h5.png Counting rod v6.png Counting rod h4.png TAI
Counting rod 0.png Counting rod h9.png Counting rod v-6.png

Occupied plot times 1.96 =1150 * 1.96 =2254=

hence =:

Counting rod v8.png Counting rod v-1.png Counting rod 0.png
Counting rod h5.png Counting rod v6.png Counting rod h4.png TAI
Counting rod 0.png Counting rod h9.png Counting rod v-6.png

Solve this equation and we obtain

width of pooll 25 paces therefore pool length =pool width +10 =35 paces length of pool =45 paces

Related Research Articles

Area Extent of a two-dimensional surface

Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

Circumference linear distance around the outside of a closed curve or circular object

In geometry, the circumference is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length around any closed figure. Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk.

Circle Simple curve of Euclidean geometry

A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

A perimeter is a path that encompasses/surrounds a two-dimensional shape. The term may be used either for the path, or its length—in one dimension. It can be thought of as the length of the outline of a shape. The perimeter of a circle or ellipse is called its circumference.

Parallelogram quadrilateral with two pairs of parallel sides

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

Rhombus quadrilateral in which all sides have the same length

In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

Thaless theorem The angle formed by a point on a circle and the 2 ends of a diameter is a right angle

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, then the angle ∠ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem, and is mentioned and proved as part of the 31st proposition, in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, who is said to have offered an ox as a sacrifice of thanksgiving for the discovery, but sometimes it is attributed to Pythagoras.

Octagon polygon shape with eight sides

In geometry, an octagon is an eight-sided polygon or 8-gon.

Curve of constant width Convex planar shape whose width is the same regardless of the orientation of the curve

In geometry, a curve of constant width is a convex planar shape whose width is the same regardless of the orientation of the curve.

Square Regular quadrilateral

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.

Arbelos

In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line that contains their diameters.

Area of a circle area enclosed by a circle

In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents a constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter.

Arc (geometry) Closed segment of a differentiable curve

In Euclidean geometry, an arc is a connected subset of a differentiable curve. Arcs of lines are called segments or rays, depending whether they are bounded or not. A common curved example is an arc of a circle, called a circular arc. In a sphere, an arc of a great circle is called a great arc.

Indiana Pi Bill 1897 attempt to define by legislation a method of squaring the circle

The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the Indiana General Assembly, one of the most notorious attempts to establish mathematical truth by legislative fiat. Despite its name, the main result claimed by the bill is a method to square the circle, rather than to establish a certain value for the mathematical constant π, the ratio of the circumference of a circle to its diameter. The bill, written by the crank Edward J. Goodwin, does imply various incorrect values of π, such as 3.2. The bill never became law, due to the intervention of Professor C. A. Waldo of Purdue University, who happened to be present in the legislature on the day it went up for a vote.

Non-integer base of numeration

A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numeral system. For a non-integer radix β > 1, the value of

Liu Huis <span class="texhtml mvar" style="font-style:italic;">π</span> algorithm

Liu Hui's π algorithm was invented by Liu Hui, a mathematician of the Cao Wei Kingdom. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 or as . Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wan Fan (219–257) provided π ≈ 142/45 ≈ 3.156. All these empirical π values were accurate to two digits. Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of π to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy of five digits: π ≈ 3.1416.

Li Ye, born Li Zhi, courtesy name Li Jingzhai, was a Chinese mathematician and writer who published and improved the tian yuan shu method for solving polynomial equations of one variable. Along with the 4th-century Chinese astronomer Yu Xi, Li Ye is one of the few Chinese mathematicians to propose the idea of a spherical Earth instead of a flat one before the arrival of European science in China during the 17th century.

<i>Ceyuan haijing</i>

Ceyuan haijing is a treatise on solving geometry problems with the algebra of Tian yuan shu written by the mathematician Li Zhi in 1248 in the time of the Mongol Empire. It is a collection of 692 formula and 170 problems, all derived from the same master diagram of a round town inscribed in a right triangle and a square. They often involve two people who walk on straight lines until they can see each other, meet or reach a tree or pagoda in a certain spot. It is an algebraic geometry book, the purpose of book is to study intricated geometrical relations by algebra.

Tian yuan shu Chinese system of algebra

Tian yuan shu is a Chinese system of algebra for polynomial equations created in the 13th century. It is first known from the writing of Li Zhi, though it was created earlier.

<i>Jade Mirror of the Four Unknowns</i>

Jade Mirror of the Four Unknowns, Siyuan yujian (四元玉鉴), also referred to as Jade Mirror of the Four Origins, is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie. With this masterpiece, Zhu brought Chinese algebra to its highest level.

References

  1. Alexander Wylie, Notes on Chinese Literature, p117, Shanghai 1902, reprinted by kessinger Publishing
  2. van Hée Li Yeh, Mathématicien Chinois du XIIIe siècle, TP,1913,14,537