Mathematical Treatise In Nine Sections

The Mathematical Treatise in Nine Sections, a rather dry title for something that clearly held Qin Jiushao’s attention for a good while. Don’t confuse it with The Nine Chapters on the Mathematical Art, though I suppose one could be forgiven for that. This one, the Shu Shu Jiu Zhang as it’s known in its more elegant, traditional Chinese form (數書九章), or its simplified counterpart (数书九章), was penned by none other than Qin Jiushao, a Southern Song dynasty mathematician who clearly had a flair for the dramatic, or at least for meticulously documenting his findings in 1247. It’s the kind of work that makes you wonder what else was happening in the world that year. Probably not much as interesting as this, if you ask me.

This treatise isn’t some slim pamphlet; it’s a comprehensive survey of mathematical applications, drawn from every conceivable corner of society back then. Think agriculture, the grand designs of astronomy, the intricate systems of water conservancy, the very layout of their cities, the brute force of construction engineering, the meticulous art of surveying, the ever-present burden of taxation, the chilling efficiency of armament, and the grim necessities of military strategy. It’s a snapshot, really, of a world grappling with numbers and trying to make them make sense.

The book itself is structured into nine distinct chapters, each a deep dive into a particular facet of this numerical universe:

• The Da Yan type: This delves into Indeterminate equations , which are precisely as fun as they sound.
• Heaven phenomena: Because even the stars needed a good dose of mathematical scrutiny.
• Area of land and field: Practical, I’ll give them that. Someone had to figure out who owned what.
• Surveying : Essential for everything from land disputes to city planning. Imagine trying to survey a round city from afar with just the tools they had.
• Taxation : The universal constant, even in the 13th century.
• Storage of grains: Feeding a population is a mathematical problem in itself.
• Building construction: Structures don’t just appear; they require careful calculation.
• Military matters: Strategy, logistics, the numbers behind warfare.
• Price and interest : The bedrock of any economy, then as now.

Each of these chapters presents nine problems, bringing the total to a neat, if somewhat daunting, eighty-one problems. It’s a testament to Qin Jiushao’s thoroughness, or perhaps his peculiar sense of humor.

But this isn’t just a collection of problems. It’s where the Chinese Remainder Theorem makes its first appearance, complete with a constructive proof . That’s no small feat. Beyond that, Qin Jiushao explored:

• More Indeterminate equations , because one chapter wasn’t enough.
• The “Linglong method” (玲瓏開方; línglóng kāifāng), which translates to something like “method of harmoniously alternating evolution.” It was a technique for numerically solving algebraic equations . And here’s the kicker: it predates Horner’s method by a staggering 570 years. Yes, you read that right. Five centuries. Apparently, some ideas are just ahead of their time, or perhaps just ahead of the West.
• The calculation of Areas and volumes for all sorts of geometric objects . Geometry, the foundation of so much.
• And the elegant, if sometimes frustrating, world of Linear system .

Like many of its contemporaries, this text reflects the mindset of a Confucian administrator. Their focus was squarely on practical matters: the precise timing of calendrical events, the measurements needed for various tasks (mensural ), and the complex realities of fiscal policies. It’s less about abstract theory and more about making the world work, one equation at a time.

The journey of this text is almost as interesting as its content. It began as a manuscript in 1247, a tangible object passed from hand to hand. Then, in 1421, it found its way into the monumental Yongle Encyclopedia . By 1787, it was deemed worthy of inclusion in the Complete Library of the Four Treasuries , a collection of immense cultural significance. The year 1842 saw its appearance in a woodblock printed edition, making it more accessible, though still requiring dedicated effort to obtain.

The West, however, remained largely oblivious until 1852. That’s when Alexander Wylie, a British Protestant Christian missionary with a keen interest in Chinese mathematics, published “Jottings on the Sciences of Chinese Mathematics” in the North China Herald. He was the first to really introduce the Mathematical Treatise in Nine Sections to Western scholars. It took another century and a bit for it to receive more in-depth academic attention. In 1971, Ulrich Libbrecht, a Belgian sinologist , published his doctoral dissertation, Chinese Mathematics in the Thirteenth Century, focusing on Qin Jiushao’s work. He even earned a degree cum laude from Leiden University for it. It’s a shame it took so long for these ideas to cross continents, but at least they made the journey eventually.

It’s fascinating, really, how much mathematical sophistication was brewing in China during the Song dynasty , a period often overlooked in Western historical narratives. This book, Shu Shu Jiu Zhang, is a testament to that. It’s not just a collection of problems; it’s a window into a mind that saw the world through the lens of numbers and found patterns where others saw chaos. And frankly, it’s about time more people recognized that.