The ''Mathematical Treatise in Nine Sections'' () is a mathematical text written by Chinese Southern Song 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, mathematical structure, structure, space, Mathematica ...

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 number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers. In modern times indeterminate equati ...

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

Surveying Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Plane (mathematics), two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of Point (geom ...

Taxation A tax is a mandatory financial charge or levy imposed on an individual or legal person, legal entity by a governmental organization to support government spending and public expenditures collectively or to Pigouvian tax, regulate and reduce nega ...

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

interest In finance and economics, interest is payment from a debtor 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 f ...

. 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 example, x^5-3x+1=0 is an algebraic equati ...

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 measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

s and

volume Volume is a measure of regions in 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) ...

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 abstractio ...

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, and is variously described as a tradition, philosophy, religion, theory of government, or way of life. Founded by Confucius ...

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

mensural Mensural notation is the musical notation system used for polyphony, polyphonic European vocal music from the late 13th century until the early 17th century. The term "mensural" refers to the ability of this system to describe precisely measur ...

, and fiscal problems. The text existed in

manuscript A manuscript (abbreviated MS for singular and MSS for plural) was, traditionally, any document written by hand or typewritten, as opposed to mechanically printed or reproduced in some indirect or automated way. More recently, the term has ...

form in 1247, it was incorporated into The

Yongle Encyclopedia The ''Yongle Encyclopedia'' () or ''Yongle Dadian'' () is a Chinese ''leishu'' encyclopedia commissioned by the Yongle Emperor (1402–1424) of the Ming dynasty in 1403 and completed by 1408. It comprised 22,937 manuscript rolls in 11,095 vol ...

in 1421; in 1787 the book was collected into the Complete Library of the Four Treasuries, in 1842 appeared in woodblock printed edition. The 19th century British

Protestant Protestantism is a branch of Christianity that emphasizes Justification (theology), justification of sinners Sola fide, through faith alone, the teaching that Salvation in Christianity, salvation comes by unmerited Grace in Christianity, divin ...

Christian A Christian () is a person who follows or adheres to Christianity, a Monotheism, monotheistic Abrahamic religion based on the life and teachings of Jesus in Christianity, Jesus Christ. Christians form the largest religious community in the wo ...

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, also referred to as China studies, is a subfield of area studies or East Asian studies involved in social sciences and humanities research on China. It is an academic discipline that focuses on the study of the Chinese civilizatio ...

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''; ) is a Public university, public research university in Leiden, Netherlands. Established in 1575 by William the Silent, William, Prince of Orange as a Protestantism, Protestant institution, it holds the d ...

.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, be ...

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