The ''Mathematical Treatise in Nine Sections'' () is a mathematical text written by Chinese

Southern Song The Song dynasty (; ; 960–1279) was an Dynasties in Chinese history, imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. ...

dynasty

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Qin Jiushao Qin Jiushao (, ca. 1202–1261), courtesy name Daogu (道古), was a Chinese mathematician, meteorologist, inventor, politician, and writer. He is credited for discovering Horner's method as well as inventing Tianchi basins, a type of rain gau ...

in the year 1247. The mathematical text has a wide range of topics and is taken from all aspects of the society of that time, including agriculture, astronomy, water conservancy, urban layout, construction engineering, surveying, taxation, armament, military and so on. This book contains nine chapters: #Da Yan type (

Indeterminate equation In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation ax + by =c is a simple indeterminate equation, as is x^2=1. Indeterminate equations cannot be solv ...

s); #Heaven phenomena #Area of land and field #

Surveying Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...

Taxation A tax is a compulsory financial charge or some other type of levy imposed on a taxpayer (an individual or legal person, legal entity) by a governmental organization in order to fund government spending and various public expenditures (regiona ...

#Storage of grains #Building construction #Military matters #Price and

interest In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distinct ...

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

Chinese Remainder Theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...

for the first time and providing a

constructive proof In mathematics, a constructive proof is a method of mathematical proof, proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also ...

for it, the text investigated: *

s *"Linglong method" (, roughly "method of harmoniously alternating evolution") for numerical solution of

algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...

s, 570 years before

Horner's method In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Hor ...

Area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

s and

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

s of geometric objects and *

Linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction o ...

Like many traditional Chinese mathematical works, the text reflects a

Confucian Confucianism, also known as Ruism or Ru classicism, is a system of thought and behavior originating in ancient China. Variously described as tradition, a philosophy, a Religious Confucianism, religion, a humanistic or rationalistic religion, ...

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

mensural Mensural notation is the musical notation system used for European vocal polyphonic music from the later part of the 13th century until about 1600. The term "mensural" refers to the ability of this system to describe precisely measured rhythmi ...

, and

fiscal Fiscal usually refers to government finance. In this context, it may refer to: Economics * Fiscal policy, use of government expenditure to influence economic development * Fiscal policy debate * Fiscal adjustment, a reduction in the government ...

problems. The text existed in

manuscript A manuscript (abbreviated MS for singular and MSS for plural) was, traditionally, any document written by hand – or, once practical typewriters became available, typewritten – as opposed to mechanically printing, printed or repr ...

form in 1247, it was incorporated into The

Yongle Encyclopedia The ''Yongle Encyclopedia'' () or ''Yongle Dadian'' () is a largely-lost Chinese ''leishu'' encyclopedia commissioned by the Yongle Emperor of the Ming dynasty in 1403 and completed by 1408. It comprised 22,937 manuscript rolls or chapters, in 1 ...

in 1421; in 1787 the book was collected into Siku Quanshu, in 1842 appeared in woodblock printed edition. The 19th century British

Protestant Protestantism is a Christian denomination, branch of Christianity that follows the theological tenets of the Reformation, Protestant Reformation, a movement that began seeking to reform the Catholic Church from within in the 16th century agai ...

Christian Christians () are people who follow or adhere to Christianity, a monotheistic Abrahamic religion based on the life and teachings of Jesus Christ. The words ''Christ'' and ''Christian'' derive from the Koine Greek title ''Christós'' (Χρι ...

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 Sinology, or Chinese studies, is an academic discipline that focuses on the study of China primarily through Chinese philosophy, language, literature, culture and history and often refers to Western scholarship. Its origin "may be traced to the ex ...

Ulrich Libbrecht published his doctorate dissertation, ''Chinese Mathematics in the Thirteenth Century'', which earned him a degree cum laude at

Leiden University Leiden University (abbreviated as ''LEI''; nl, Universiteit Leiden) is a Public university, public research university in Leiden, Netherlands. The university was founded as a Protestant university in 1575 by William the Silent, William, Prince o ...

.Ulrich Libbrecht: ''Chinese Mathematics in the Thirteenth Century'': "Shu-shu Chiu-chang" of Ch'in Chiu-shao, Dover Publications Inc.,

Notes

References

*Guo, Shuchun
"Qin Jiushao"
''

Encyclopedia of China The ''Encyclopedia of China'' () is the first large-entry modern encyclopedia in the Chinese language. The compilation began in 1978. Published by the Encyclopedia of China Publishing House, the encyclopedia was issued one volume at a time, begin ...

'' (Mathematics Edition), 1st ed.
Siku Quanshu edition of the book - 數學九章 (四庫全書本)
{{Song dynasty topics Song dynasty literature Chinese mathematics Mathematics manuscripts 1247 works Treatises 13th-century Chinese books