Historical evidence
Three harvests, three unknowns
The first problem of the eighth chapter compares yields from three grades of grain. Its quantities correspond to the system 3x + 2y + z = 39, 2x + 3y + z = 34, and x + 2y + 3z = 26.
The cited historical account describes equations arranged as columns on a counting board. This page retains that orientation instead of silently rotating the board into a familiar school matrix.[NINE]
Reconstruction
Equations stand in columns
Each vertical stack is one complete equation: the three coefficients sit above its total. The modern row form printed beside it is a translation for comparison, not a claim about the source layout.
Rod colours historically helped distinguish signs in calculation. The Atlas also prints a plus or minus sign and an accessible label, so sign never depends on colour alone.[NINE]
Reconstruction
Reduce the board
Use Next operation to apply one whole-column replacement at a time. Undo returns to the preceding board and Reset restores the three original columns.
The guided sequence keeps exact integer entries until back-substitution. The last column becomes 36z = 99, from which the remaining grain values follow.[NINE]
Reconstruction · Fangcheng procedure
Reduce whole columns
| Place | Column 1 | Column 2 | Column 3 |
|---|---|---|---|
| Upper-grade coefficient x | |||
| Middle-grade coefficient y | |||
| Lower-grade coefficient z | |||
| Yield total |
Operation 0: Start with three equations written as columns.
Compare modern row notation
Here each historical board column is transcribed as one modern augmented row [x, y, z | total]. The orientation changes; the equations do not.
The current board remains exact; Undo and Reset never recompute from rounded decimals.
Modern translation
Read the triangular result
Substitution gives the exact values 37/4, 17/4 and 11/4. Putting those values back into the three original equations returns 39, 34 and 26.
“Matrix” and “Gaussian elimination” are useful modern comparisons. They are not labels supplied by the historical text, and they should not replace its column-based procedure.[NINE]
Open question
A text compiled over time
The Nine Chapters is a layered textual work. MacTutor reports proposed dates from around 200 BCE into the early centuries CE and describes later commentary and revision.
This exhibit therefore dates a procedure and textual tradition cautiously. It does not claim that one surviving physical counting board records this exact problem.[NINE-DATE]
Published accounts give date ranges rather than a single settled composition year. Liu Hui’s 263 CE commentary has its own date and should not be treated as the date of the anonymous underlying compilation.
Historical evidence
Procedure and explanation
A sequence of board operations can both solve a case and exhibit a reusable method. Historical work on the Nine Chapters cautions against treating a different style of justification as the absence of reasoning.
Compare this algorithm with the Atlas notebook studies: in both, a reproducible sequence matters, while the historical setting and representation remain specific.[NINE][NINE-DATE]
Open question
Why orientation matters
Rotating columns into rows preserves the modern system of equations, but it can hide how the historical procedure organized work. Representation shapes which operations feel natural and which invariants are easy to see.
Continue to Spectral for another use of arrays, or explore algorithms and equations through the historical Atlas.[NINE]