The Gougu Theorem and Pythagoras: Same Truth, Different Paths

Different Modes of Thought and Their Legacy Through Two Giants of Eastern and Western Mathematics

• The fundamental differences between the Eastern ‘Gougu theorem’ and the Western ‘Pythagorean theorem’
• A deep understanding of how mathematics reflects each civilization’s worldview
• Discovering mathematical principles applied in Korean history from Silla’s Seokguram Grotto to Joseon’s Angbuilgu sundial

The Gougu Theorem: Truth Proven Through Practicality

The relationship among the three sides of a right triangle ($a^2+b^2=c^2$) was explored differently in East and West. In the East, it was known as the Gougu theorem, discovered and applied through practical problem-solving. This theorem was systematized in the ancient Chinese mathematical classic 『Nine Chapters on the Mathematical Art』.

『Nine Chapters on the Mathematical Art』: A Guidebook for Problem Solving

『Nine Chapters on the Mathematical Art (九章算術)』 is a compendium of ancient Chinese mathematical knowledge, organized around thoroughly practical problem solving rather than abstract theory.

As seen from chapter titles like ‘Land Measurement’ and ‘Engineering Works,’ this book served as a manual for officials and technicians facing real-world problems. Each problem follows a structure presenting a specific situation (“Now there is…”), followed by a solution procedure or algorithm (“The method says…”). This highlights China’s pragmatic mathematical tradition that valued establishing repeatable solution procedures over abstract proofs.

Liu Hui’s Proof: Wisdom Through Cutting and Rearranging

The earliest logical explanation of the Gougu theorem appears in the 3rd century mathematician Liu Hui’s commentary. He visually and perfectly proved the theorem’s validity using an original principle called ‘cutting and complementing’ (出入相補).

This method involves cutting a figure into several pieces and rearranging them like a puzzle to intuitively show equal areas. By cutting the squares on the legs (勾, a and 股, b) and fitting them perfectly into the square on the hypotenuse (弦, c), it demonstrates that $a^2+b^2=c^2$. This is not a deductive proof following logical necessity but rather an algorithmic demonstration based on area preservation. The goal was not to establish a universal truth but to empirically verify a reliable rule for calculation.

The Pythagorean Theorem: A Tower Built on Logic

In the West, this theorem is known as the Pythagorean theorem, and its development followed a completely different path. Greek mathematics began as a philosophical inquiry into understanding the cosmos through ’numbers.’

Discovery of Irrational Numbers and Philosophical Crisis

The Pythagorean school believed “all things are numbers,” but their belief system was shaken by the discovery of incommensurability, i.e., irrational numbers.

The fact that the hypotenuse length ($\sqrt{2}$) of a right isosceles triangle with legs of length 1 cannot be expressed as a ratio of integers was a profound shock. This contradiction led Greek mathematics to move beyond observation and calculation, embracing deductive reasoning that starts from self-evident truths (axioms) and derives conclusions through logical steps.

Euclid’s Elements: The Essence of Deductive Proof

This axiomatic methodology culminated in Euclid’s 『Elements』. The proof of the Pythagorean theorem appears as Proposition 47 in Book 1, a highly sophisticated logical construction fundamentally different from Liu Hui’s visual rearrangement.

Euclid links congruence conditions and area relations through a chain of logical lemmas to prove that $a^2+b^2=c^2$ necessarily follows. Instead of asking readers to manipulate figures, it forces acceptance of the conclusion based on established axioms and theorems. The concluding “Q.E.D.” declares that this proposition has attained the status of a universal truth within a closed logical system, independent of reality.

Comparison: Pragmatism vs. Idealism

These two proofs vividly illustrate the contrasting modes of thought in approaching mathematical truth between two civilizations. China’s approach was not a ‘failed’ attempt at Greek-style proof but a successful implementation of a different mathematical reasoning.

The Gougu Principle in Korean History

Korea adopted the Chinese pragmatic mathematical tradition and used it as a core tool for state governance.

Architecture: Realizing Divine Proportions

Have you ever wondered about the mathematical principles hidden in the elegant curves of traditional Korean roofs? The Gougu method was essential for determining the lengths of rafters and eaves and for establishing right angles in buildings.

• Bulguksa and Seokguram: These masterpieces of Silla architecture exemplify mathematical precision. The stable proportions of the Seokgatap and Dabotap pagodas and the perfect geometric balance inside Seokguram were all realized through the Gougu method. Here, mathematics transcended mere technique to become a sacred language embodying the ideal of the Buddhist land on earth.

State Governance: Techniques for Measuring Land and Sky

The Gougu method was a fundamental technology underpinning national administration.

• Land Surveying and Taxation: During the Joseon era’s land survey project called ‘Yangjeon (量田),’ irregular lands were divided into right triangles to calculate area using the Gougu method. This was the core of fair taxation and national fiscal administration.
• Astronomical Instrument Design: Precise angle calculations and component design for instruments like the sundial Angbuilgu and the celestial globe Honcheonui were based on Gougu principles.
• Hwaseong Seongyeok Uigwe: The construction report of Hwaseong Fortress includes designs of cranes and other machinery, demonstrating that the Gougu method was part of systematic engineering technology for national projects.

Conclusion

The Gougu theorem and the Pythagorean theorem clearly demonstrate how a single universal truth can be understood differently through the prism of culture.

• Key Summary

  1. Same truth, different approaches: The Gougu theorem was proven through practical, algorithmic demonstration, while the Pythagorean theorem was proven through abstract, deductive argument.
  2. Reflection of worldviews: This reflects the fundamental difference between the East’s focus on solving real problems and the West’s pursuit of ideal truths.
  3. Foundation of Korean history: Korea adopted the practical knowledge of the Gougu theorem and developed it into core technologies for architecture, astronomy, and administration.

Next time you visit a museum or a historic palace, why not look for the mathematical principles hidden in the beautiful heritage? You may discover a new joy in uncovering the wisdom and pragmatic spirit of our ancestors.