The Pinnacle of Algorithms and the Source of Logic: When the Nine Chapters on the Mathematical Art Meets Euclid's Elements

At the dawn of human civilization, two mathematical works emerged in almost the same era. Like two rivers flowing in different directions, they quietly shaped the completely different intellectual traditions of the East and West.

One came from China,
the other from ancient Greece.

They are:

• China’s Nine Chapters on the Mathematical Art
• Ancient Greece’s Elements (from Euclid’s work)

To this day, our understanding of science, technology, and even artificial intelligence still carries the DNA of these two books.

I. Historical Context: A Philosopher’s Truth vs. an Administrator’s Handbook

In the history of Western science, the earliest and most influential mathematical work is Euclid’s Elements (the source of what is often called ‘Euclidean geometry’);
whereas in the history of ancient Chinese science, the earliest, most widespread, and most influential mathematical classic is the Nine Chapters on the Mathematical Art, compiled during the Han Dynasty.

Comparing these two “most important mathematical classics of their respective civilizations” is not about determining a winner, but about understanding a more practical question:

Why did ancient mathematics of a similarly high level
ultimately contribute to modern natural science via two completely different paths?

1️⃣ Euclid’s Elements: Mathematics for the Sake of ‘Truth Itself’

• Date of Compilation: Circa 300 BCE
• Author: Euclid
• Role: A ‘Logical Ladder’ to the Laws of the Universe

Euclid’s Elements is not concerned with harvests, taxes, or engineering costs.
It is concerned with a more fundamental question:

How should mathematics itself be defined?

It establishes a “deducible world”: starting from the fewest possible premises to deduce as many necessary conclusions as possible.
This is why many later Western disciplines adopted it as a “model for theoretical writing and scientific argumentation.”

Some have even pointed out that the structure of Newton’s Principia Mathematica clearly borrows from the Euclidean system’s “definition-axiom-proposition-proof” style.

2️⃣ Nine Chapters on the Mathematical Art: Mathematics in Service of the Real World

• Date of Compilation: Circa 2nd century BCE – 1st century CE (Western Han to Eastern Han Dynasty)
• Author: Compiled and expanded by mathematicians over generations
• Role: An ‘Administrative Handbook’ for a unified empire

It solves the most practical problems:

• Land surveying
• Tax calculation
• Construction projects (building dikes, granaries, digging canals)

This is a book written for officials and engineers.
It does not pursue the “pure truth of the universe”; it pursues:

Solving problems correctly, quickly, and practically.

Precisely because of this strong practicality, the study and application of the Nine Chapters continued in China for a long time, even extending into the Ming and Qing dynasties—it became a “standard algorithm library” for the ancient system of governance and engineering.

II. A Difference in Starting Points: Beginning with ‘Definitions’ vs. Beginning with ‘Problems’

📐 Euclid’s Elements: Building a Universe from Nothing

Euclid’s opening does not calculate areas or solve equations.
Instead, it first defines the world itself.

Original Text:

• Definition 1: A point is that which has no part.
• Definition 2: A line is breadthless length.
• Postulate 1: To draw a straight line from any point to any point.

Here, Euclid is not performing calculations; he is engaging in abstract modeling.
The entire book is built on an extremely rigorous deductive chain:

Definitions → Axioms/Postulates → Propositions → Proofs

This is a form of mathematics that starts from the question ‘Why is this necessarily true?’

🌾 Nine Chapters on the Mathematical Art: Wisdom Grown from the Land

The opening of the Nine Chapters is extremely “down-to-earth.”
Almost every chapter begins with a single phrase:

“Now given…”
(Meaning: Now there is a practical problem)

Original Text:

Now given a field 15 paces wide and 16 paces long. Question: what is the area of the field?
Method: Multiply the width and length to get the area in square paces.

Its structure is very consistent:

Problem → Answer → Method (Algorithm)

It does not explain why “length × width = area.”
It directly gives you a set of reusable algorithms called “methods” (术)—which, from today’s perspective, are almost like pseudocode.

III. The Same Mathematics, Different Thought Processes

1️⃣ The Pythagorean Theorem: Proving Necessity vs. Applying Effectiveness

• Euclid’s Elements
  In Proposition I.47, Euclid uses complex geometric constructions to rigorously prove the logical necessity of the relationship between the three sides of a right-angled triangle.
• Nine Chapters on the Mathematical Art
  Directly provides the formula:

  Multiply the gou and gu (the two shorter sides) by themselves, add them, and take the square root. This is the hypotenuse.
  It then immediately applies it to practical problems: measuring water depth, tree height, and determining engineering dimensions.

One is proving ‘why it is always correct,’
the other is verifying ‘if it is useful enough.’

2️⃣ Engineering Problem: A ‘Showdown’ for Calculating Officials

In ancient mathematical examinations (like the Mingsuan ke of the Tang Dynasty), there was a highly representative engineering problem:

Original Problem (from the ‘Shang Gong’ chapter):

Now given a dike to be built, with a bottom width of 2 zhang, a top width of 8 chi, a height of 4 chi, and a length of 12 zhang and 7 chi.
Question: What is the volume?

Method (Algorithm):

Add the top and bottom widths, halve the sum, multiply by the height, and then multiply by the length.

This is the standard algorithm for the volume of a trapezoidal prism.

If Euclid were to solve this:
He would require you to prove the formula for the area of a trapezoid and derive it by decomposing the trapezoid into triangles.
A Chinese examination candidate, however, would simply: move the counting rods and execute the algorithm.

This highly algorithmic educational system gave ancient China an extremely powerful capacity for engineering execution—it could turn “governance” into something “calculable” and “engineering” into something “operable.”

3️⃣ Positive and Negative Numbers: The Art of Red and Black

This is one of Chinese mathematics’ most important contributions to humanity.

• Nine Chapters on the Mathematical Art · ‘Fangcheng’ (Equations) Chapter
  To handle the cancellation of coefficients, it introduced positive and negative numbers:

  Red counting rods represent positive, black counting rods represent negative.
  It also systematically provided the rules of addition and subtraction for the “Method of Positive and Negative.”

• In contrast
  Ancient Greek mathematics long held that “length cannot be negative” and thus considered negative numbers absurd.

The attitude of Chinese mathematicians was more like that of an engineer:

As long as the algorithm is logically self-consistent, it doesn’t need to be intuitively visible.

IV. If You Only Have Time for a Glance, This Table Is Enough

V. The Key Divergence: Why Was the Nine Chapters ‘Long-Lived’ but Less ‘Foundational’ Than Elements?

The practicality of the Nine Chapters did indeed ensure its study and application in China continued into the Ming and Qing dynasties. Many later mathematicians often deepened and extended their work based on the problems posed in the Nine Chapters.

But it also had an unavoidable structural shortcoming:

It provided almost no clear mathematical concepts, theorem systems, or systematic proofs.

In contrast, Euclid’s Elements was the opposite.
It might not directly help build a dike, but it provided the “scaffolding for the edifice of science”:

• How to define concepts
• How to establish axioms and postulates
• How to prove propositions
• How to build a self-consistent system

This structure had a profound impact on later natural sciences—modern science requires not only the ability to “calculate” but also the ability to “build theories.”

So, the final result is:
The Nine Chapters is more like an “algorithm library for governance and engineering,”
while Elements is more like a “building code for scientific theory.”

VI. How Does This Divergence Continue Today?

This difference between the “algorithmic gene” and the “logical gene” has re-emerged in the 21st century.

• The West (the Elements gene) excels at going from 0 to 1:
  Neural network theory, large model frameworks, computational paradigm innovation
  It solves: Why the model is viable
• The East (the Nine Chapters gene) excels at going from 1 to 100:
  Algorithm compression, inference acceleration, engineering implementation, cost optimization
  It solves: How to perfect the model

This is not about which is superior, but about different choices of path.

Conclusion

The Nine Chapters on the Mathematical Art is the wisdom of the earth, teaching us how to solve problems pragmatically;
Euclid’s Elements is a contemplation under the starry sky, teaching us how to construct a rational world.

To this day, we still stand at the intersection of these two traditions.

True breakthroughs often come from:

Using logic to verify algorithms,
and using algorithms to implement logic.