The Mathematical Treatise in Nine Sections is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247.

This book contains nine chapters:

1. Da Yan type (Indeterminate equations);
2. Heaven phenomena
3. Area of land and field
4. Surveying
5. Taxation
6. Storage of grains
7. Building construction
8. Military matters
9. Price and interest.

Each chapter contains nine problems, a total of 81 problems.
Apart from describing Chinese Remainder Theorem

Chinese remainder theorem

The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...

 for the first time and providing a constructive proof

Constructive proof

In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object...

 for it, the text investigated:

• Indeterminate equations
• "Lin Long method" for numerical solution of polynomial equations, 570 years before Horner's method
• Area
  Area
  Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
  
  s of geometric shapes
  Shape
  The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
  
   and
• Linear simultaneous equations

Like many traditional Chinese mathematical works, the text reflects a Confucian

Confucianism

Confucianism is a Chinese ethical and philosophical system developed from the teachings of the Chinese philosopher Confucius . Confucianism originated as an "ethical-sociopolitical teaching" during the Spring and Autumn Period, but later developed metaphysical and cosmological elements in the Han...

 administrator's concern with more practical mathematical problems, like calendrical

Calendar

A calendar is a system of organizing days for social, religious, commercial, or administrative purposes. This is done by giving names to periods of time, typically days, weeks, months, and years. The name given to each day is known as a date. Periods in a calendar are usually, though not...

, mensural

Mensural notation

Mensural notation is the musical notation system which was used in European music from the later part of the 13th century until about 1600."Mensural" refers to the ability of this system to notate complex rhythms with great exactness and flexibility...

, and fiscal

Finance

"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...

 problems.

The text existed in manuscript

Manuscript

A manuscript or handwrite is written information that has been manually created by someone or some people, such as a hand-written letter, as opposed to being printed or reproduced some other way...

 form in 1247, it was incorporated into The Yongle Encyclopedia

Yongle Encyclopedia

The Yongle Encyclopedia was a Chinese compilation of information commissioned by the Chinese Ming Dynasty emperor Yongle in 1403 and completed by 1408...

 in 1421; in 1787 the book was collected into Siku Quanshu

Siku Quanshu

The Siku Quanshu, variously translated as the Imperial Collection of Four, Emperor's Four Treasuries, Complete Library in Four Branches of Literature, or Complete Library of the Four Treasuries, is the largest collection of books in Chinese history and probably the most ambitious editorial...

, in 1842 appeared in woodblock printed edition. The 19th century British Protestant Christian

Christian

A Christian is a person who adheres to Christianity, an Abrahamic, monotheistic religion based on the life and teachings of Jesus of Nazareth as recorded in the Canonical gospels and the letters of the New Testament...

 missionary Alexander Wylie in his article Jottings on the Sciences of Chinese Mathematics published in North China Herald 1852, was the first person to introduce Mathematical Treatise in Nine Sections to the West. in 1971 Belgian sinologist Ulrich Libbrecht

Ulrich Libbrecht

Ulrich Libbrecht is a Belgian philosopher and active writer in the field of comparative philosophy. His magnum opus consists of four books in Dutch - called "Introduction to Comparative Philosophy"....

 published his doctorate dissertation, Chinese Mathematics in the Thirteenth Century, which earned him a degree cum laude at Leiden University

Leiden University

Leiden University , located in the city of Leiden, is the oldest university in the Netherlands. The university was founded in 1575 by William, Prince of Orange, leader of the Dutch Revolt in the Eighty Years' War. The royal Dutch House of Orange-Nassau and Leiden University still have a close...

.