Zu Chongzhi

Zu Chongzhi: Seven Digits, and Nine Centuries

Some time in the fifth century, a Chinese official established that π lies between 3.1415926 and 3.1415927. The bounds were correct, and nobody anywhere on earth improved on them for roughly nine hundred years. The book in which Zu Chongzhi set out how he did it has been lost for the better part of a millennium, which means that his most celebrated achievement survives as a result without its proof — a fair emblem of everything we know and do not know about him.

A Family That Passed Down the Sky

He was born in 429 in Jiankang, the southern capital, on the site of modern Nanjing. His family's ancestry was traced to Fanyang, near present-day Baoding in Hebei; his grandfather Zu Chang had moved the household south to escape war during the Eastern Jin period, and served the Liu Song dynasty as Chief Minister for Palace Buildings. His father, Zu Shuozhi, was known as a scholar. A great-grandfather had served the Eastern Jin court.

This matters more than genealogy usually does. In early China, mathematical and astronomical competence was transmitted within families, from father to son, rather than through schools, and the Zu household had that tradition. His formal mathematical education came chiefly from Liu Hui's commentary on the *Nine Chapters on the Mathematical Art*, the foundational text of Chinese mathematics — and Liu Hui's method of inscribing polygons in a circle is precisely the technique Zu would later push to its limit.

Emperor Xiaowu of Liu Song heard of his ability and placed him at the Hualin Xuesheng academy, and afterwards in a research post at the imperial institution in Nanjing. By 461 he held an administrative position in Nanxu, modern Zhenjiang; he also served as an officer in Yangzhou and on the military staff at Jiankang.

The Daming Calendar and the Court

His major public work was the Daming, or Great Brightness, calendar. Sources differ on its date — 462 or 464 — and he compiled it around the time he moved to Louxian, in what is now Songjiang, Shanghai. It was built on a 391-year cycle containing 144 intercalary months, and it distinguished the sidereal from the tropical year, incorporating the precession of the equinoxes into a Chinese calendar.

Its accuracy was remarkable for the period. Zu put the length of the year at 365.24281481 days, against the modern 365.24219878. He gave the lunar month relative to the sun as 27.21223, where the modern value is 27.21222, and Jupiter's orbital period as 11.858 years against an actual 11.862. He was not right about everything: his figure for precession, one degree in about forty-five years and eleven months, is far off the true rate of roughly seventy years.

The court did not want it. A minister named Dai Faxing attacked the proposal, charging Zu with distorting the truth about heaven and violating the teaching of the classics. Zu replied that his calendar rested on careful observation and accurate calculation rather than authority. The emperor was persuaded; his successor was persuaded back, and implementation was cancelled. The Daming calendar came into use only around 510, roughly a decade after Zu's death. The documentary record of the dispute itself is thin, and much of what is repeated about it is later tradition.

The Number

Zu's calculation of π followed Liu Hui's algorithm: inscribe a regular polygon in a circle and keep doubling its sides, so that the polygon's perimeter converges on the circumference. Doing this by hand with counting rods, to the accuracy he reached, is an act of arithmetical endurance that is difficult to overstate. Sources disagree on how far he took it — some say a polygon of 12,288 sides, others 24,576 — which is itself a measure of how much has been lost.

He also gave two rational approximations. The Yuelü, 22/7, was already known to Archimedes. The Milü, 355/113, was not known to anyone else: it is the best rational approximation to π with a denominator of that size, accurate to six decimal places, and it was not obtained again outside China until Adriaan Anthoniszoon arrived at it in the Netherlands in 1585, more than a thousand years later. How Zu found it is unknown. Historians regard it as too good to be a lucky accident and assume a principled method, but the method is not recorded.

The Sphere, and a Principle Before Cavalieri

Working with his son Zu Gengzhi — himself an accomplished mathematician — Zu derived the correct formula for the volume of a sphere, πD³/6, equivalent to the familiar 4πr³/3. The derivation is the interesting part. It rests on the observation that two solids of equal height whose horizontal cross-sections have equal areas at every level must have equal volumes. That is the proposition Europe would later call Cavalieri's principle, after a seventeenth-century Italian. Chinese tradition names it after the Zus. They applied it to a Steinmetz solid — the intersection of two cylinders at right angles — and scaled the result by π/4.

Machines

Zu was also a builder. In 478 he reconstructed the south-pointing chariot, a direction-indicating vehicle that had fallen out of use after the Three Kingdoms period; the record credits him with making new bronze machinery that turned without a hitch and indicated direction consistently. He is credited with paddle boats capable of covering long distances without wind, which spread widely under the later Tang. In 488 he designed water-powered trip-hammer mills, inspected by Emperor Wu of Southern Qi in the early 490s. He wrote commentaries on the *Nine Chapters*, and a work of strange tales, the *Shu Yi Ji*, survives under his name.

Why Zu Is Called a Genius

Zu's reputation rests on results rather than on any contemporary verdict about his mind; no fifth-century assessment of his intelligence survives in a form worth quoting. What can be said precisely is what the results required.

Two distinct capacities are visible. The first is computational stamina of an extreme kind: doubling a polygon to twelve or twenty-four thousand sides, by hand, with counting rods, while controlling error tightly enough that the seventh decimal place can be bracketed rather than guessed. That is not a flash of insight but sustained, disciplined arithmetic. The second is genuinely conceptual — the cross-section argument behind the sphere's volume, which is a piece of reasoning about infinitesimals that Europe did not reach for another eleven centuries, and the derivation of 355/113, a result so well-chosen that it implies a theory of approximation nobody has been able to reconstruct.

The honest counter-case is that we cannot fully audit him. The *Zhui Shu*, where his methods were set out, became an imperial examination text in 656 after editing by Li Chunfeng, was dropped from the syllabus as too advanced for students, and was lost by the Song. Everything about how he worked is therefore inference from surviving numbers and later citation. The sphere result was joint work with his son, and the division of credit is unrecoverable. He was wrong about precession. And "genius" here is partly an artefact of survival: we celebrate the digits because the digits are what came through.

What Remains

Zu died in 500 or 501 — the sources differ. His calendar was adopted a decade later and his ratio bears his name in Chinese to this day. A crater on the Moon is called Tsu Chung-Chi, and asteroid 1888 carries his name. So do a modern stream cipher and a series of Chinese quantum computers. It is a peculiar afterlife for a man whose own book did not survive: honoured chiefly through machines that calculate, by a civilisation that lost the record of how he calculated.

Achievements

• Calculated π to seven decimal places, establishing that it lies between 3.1415926 and 3.1415927 — a precision unmatched worldwide for roughly 1,000 years.
• Discovered the fraction 355/113 (the Milü), an exceptionally accurate approximation of π not rediscovered in Europe until the 16th century.
• With his son Zu Gengzhi, derived a correct formula for the volume of a sphere using a cross-section principle anticipating Cavalieri.
• Compiled the Daming calendar, introducing a more accurate length of the solar year and accounting for precession of the equinoxes.
• Reconstructed mechanical devices including the south-pointing chariot.