Arrange coefficients, then eliminate unknowns.
In modern notation: elementary operations on an augmented matrix.
From counting rods to unknowns · three conceptual leaps
Before writing x,
they already made
unknowns computable.
Classical Chinese mathematics organized calculation through counting rods, position, and procedure: quantities were placed on a counting board, then transformed by repeatable operations until the unknowns were found.
Begin with a counting board↓unknown元yuan
方程number positions
天元one unknown
四元elimination
c. 1st century CEThe Nine Chapters13th centuryHeavenly Unknown1303Jade Mirror
I · Fangcheng procedure
Two kinds of grain:
how much does each yield?
Chapter Eight of the Nine Chapters, “Fangcheng,” solves simultaneous linear problems. Each condition occupied a column on the counting board; repeated scaling and subtraction eliminated unknowns one by one.
Relation to modern Gaussian elimination
The same computational structure
Try it
How do two rows reveal the answer?
01 / 04
Simplified example
Two bundles of upper-grade grain plus one bundle of lower-grade grain yield five dou; one upper plus three lower yield seven. Find the yield of each bundle.
Set the unknowns
x = yield per upper bundle
y = yield per lower bundle
Modern equations
2x + y = 5x + 3y = 7
Modern augmented matrix
215137
How to read it: each row is one condition; the first two columns are coefficients of x and y; the right side contains the totals. Row operations change the representation without changing the solution.
215
137
EncodePlace both equations in one table
The modern row layout is the transpose of the traditional counting-board orientation.
One action, two vocabularies
Counting-board operations ⇄ linear algebra
各置Place coefficients by positionModern: form the augmented matrix
2 1 │ 51 3 │ 7encode遍乘Scale a whole columnModern: scale one row
R₂ ← 2R₂1 3 │ 7 → 2 6 │ 14scale直除Subtract and eliminate one termModern: row replacement
R₂ ← R₂ − R₁2 6 │ 14 − 2 1 │ 5= 0 5 │ 9eliminate以法除之Normalize and back-substituteModern: pivot and solve
R₂ ← R₂ ÷ 5y = 9/5, x = 8/5solveThe four cards continue the same problem. Modern notation uses rows R₁ and R₂; the classical counting board used the transposed orientation.
II · The Heavenly Unknown
One yuan
gave the unknown a place.
“Establish one Heavenly Unknown as…”— a standard setting phrase in thirteenth-century texts
The method represented the coefficients of successive powers through position. It did not write the letter x, but it could organize and operate on a polynomial in one unknown.
A small problem
One side of a rectangular field is one unit longer than the other; its area is twelve.
Let the shorter side be the Heavenly Unknown. Move the slider.
12square units
unknown = 3unknown + 1 = 4Reduce the story to a polynomial
x(x + 1) = 12→x² + x − 12 = 0
quadratic place1x²
unknown place1x
constant place−121
The letter and exponent are modern translations; the classical method encoded them through position.
III · The Four Unknowns
Heaven, Earth, Human, Matter:
four directions for four unknowns.
In the 1303 Jade Mirror of the Four Unknowns, Zhu Shijie used Heaven, Earth, Human, and Matter for as many as four unknown quantities, then eliminated them successively.
Fourth introductory example · “Four Forms Meet at the Origin”Heaven · x
In this problem, the Heavenly Unknown is the shorter leg.
Original Classical Chinese
今有股乘五较与弦幂加句乘弦等。只云句除五和与股幂减句弦较同。问黄方带句股弦共几何？答曰：一十四步。草曰：立天元一为句，地元一为股，人元一为弦，物元一为开数，四象和会求之。
English: The answer is fourteen. Set the Heavenly Unknown as the shorter leg, Earth as the longer leg, Human as the hypotenuse, and Matter as the sought total; combine the four forms to solve it.
Read the original text ↗Modern notation
- Heaven
- x · shorter leg
- Earth
- y · longer leg
- Human
- z · hypotenuse
- Matter
- u · requested total
The “yellow square” is the incircle’s diameter: d = x + y − z. Thus u = x + y + z + d = 2x + 2y.
Step 1 · Set four unknowns
Translate the problem into four relations
Heaven xEarth yHuman zMatter u
x² + y² = z²
y · five differences = z² + xz
five sums / x = y² − (z − x)
u = x + y + z + (x + y − z)
The first three relations determine the triangle; the fourth records the requested total.
The animation reconstructs the original elimination with modern symbols. The transmitted book represents its operations through multidirectional counting-rod diagrams.
From counting board to algebra
One method,
step-for-step correspondence.
Represent the coefficients of successive powers by position.
In modern notation: a polynomial in x.
Organize Heaven, Earth, Human, and Matter.
Here they are transcribed as x, y, z, and u.
Reduce four unknowns to one.
Modern algebra performs the same reduction by substitution and elimination.
Placed on a world timeline
Different notations,
connected mathematical acts.
- The Nine ChaptersFangcheng problems use tabulation and elimination.
- Heavenly UnknownExtant texts systematically represent one-variable polynomials by position.
- Jade MirrorZhu Shijie extends polynomial elimination to four unknowns.
- European symbolic algebraViète generalizes literal notation; Descartes popularizes x, y, z and modern exponents.
Sources
Follow the text
back to the page.
Dates and terminology follow conservative historical descriptions. Modern letters and row notation are translations used to expose the algorithms.