The history of Mathematics

The history of Mathematics

Every culture on earth has developed some mathematics. In some cases, this mathematics has spread from one culture to another. Now there is one predominant international mathematics, and this mathematics has quite a history. It has roots in ancient Egypt and Babylonia, then grew rapidly in ancient Greece. Mathematics written in ancient Greek was translated into Arabic. About the same time some mathematics of India was translated into Arabic. Later some of this mathematics was translated into Latin and became the mathematics of Western Europe. Over a period of several hundred years, it became the mathematics of the world.
There are other places in the world that developed significant mathematics, such as China, southern India, and Japan, and they are interesting to study, but the mathematics of the other regions have not had much influence on current international mathematics. There is, of course, much mathematics being done these and other regions, but it is not the traditional math of the regions, but international mathematics.
By far, the most significant development in mathematics was giving it firm logical foundations. This took place in ancient Greece in the centuries preceding Euclid. See Euclid's Elements. Logical foundations give mathematics more than just certainty-they are a tool to investigate the unknown.
By the 20th century the edge of that unknown had receded to where only a few could see. One was David Hilbert, a leading mathematician of the turn of the century. In 1900 he addressed the International Congress of Mathematicians in Paris, and described 23 important mathematical problems.
Mathematics continues to grow at a phenomenal rate. There is no end in sight, and the application of mathematics to science becomes greater all the time.
The History of Mathematics
Instructor: G. Donald Allen
Spring, 1997
The Origins of Mathematics
{The origins of mathematics accompanied the evolution of social systems. Many, many social needs require
· counting,
· calculations,
· measurement
· The worth of a herdsman cannot be known unless some basics facts of counting are known.
· An temple cannot be built unless certain facts about triangles, squares, and volumes are known.
· An inheritance cannot be distributed unless certain facts about division (fractions) are known.
From practical needs such as these, mathematics was born.
One view is that the core of early mathematics is based upon two simple questions.
· How many?
· How much?
This is the cardinal number viewpoint.
Ordinals
Another view is that mathematics may have an even earlier basis on ordinals used perhaps for rituals in religious practices or simply the pecking order for eating the fresh game. Such basic questions are thus:
· Who is first, etc?
· What comes first, etc?
We will take the cardinal numbers viewpoint in the following.
HOW MANY?
As indicated earlier, as society formed and organized, the need to express quantity emerged. Even at this early level, perhaps as early as 250,000 years ago, there must have begun a transition from sameness to similarity of numbers.
one wolf one sheep
two dogs wolf two rabbits
five warriors five spears
This abstraction of the concept of number was a major step toward modern mathematics.
HOW MANY?
From artifacts even more than 5,000 years old, notches on bones have been noted. Were these to count seasons, kills, children? We don't know. But the need to denote quantity must have been significant.
The English language, as others, has quantifier to indicate plurality
school of fish
pack of wolves
flock of geese
HOW MANY?
Other examples of counting and enumerations reveal just how enumeration began and proceeded.
1. The Indians of the Tamanaca on the Orinoco River.
HOW MANY?
2. The Dammara tribe in Africa (19th century). Trading of tobacco sticks for sheep. The tradesman knew the equivalence:
2 sticks = 1 sheep
However, he was unable to cipher correctly the formula:
4 sticks = 2 sheep
So, at the very early stage of counting numerical equivalences there is no such fact as two times two equals four.
3. Certain Australian aboriginal trives counted to two only, with any number larger than two called simply as much or many.
4. Other South American Indians on the tributaries of the Amazon were equally lacking in number words. They count count to six, but had no words for three, four, five, or six. For example, three was expressed as two-one.
5. The bushmen of Africa could count to ten with just two words.
ten=2+2+2+2+2+2
For larger numbers the descriptive phrases became too long.
Even earlier records
6. The earliest records of counting do not come from words but from physical evidence -- scratches on sticks or stones. Old stone age peoples had devised a system of tallyign by groups as early as 30,000 B.C. There is an example of the shinbone from a young wolf found in Czechoslovakia in 1937. It is about 7 inches long, and is engraved with 55 deeply cut notches, of about equal length, arranged in groups of 5. (Modern systems!!!)
7. There is other evidence dating from 8500 B.C on the shores of Lake Edward (in the Queen Elizabeth National Park in Uganda). An incised bone fossil contains groups of notches in three definite columns. Odd and unbalanced, it does not appear decorative. One set of is arranged in groups of 11, 21, 19 and 9 notches. Another is arranged in groups of 3, 6, 4, 8, 10, 5, 5, 7 notches. Many have conjectured on the meaning of these groups. (Lunar months, doubling, halving, ...)
Some etymology.
· The word tally comes from the French verb tailler, ``to cut", like the English word tailor. The root Latin word is tailler, talea, ``to cut". Note also the English word write comes from the Anglo-saxon writan, ``to scratch".
· Our word calculate comes from the Latin calculus, pebble.
· The English thrice, just like the Latic ter, has the double meaning: three times, and many. There is a plausible connection between the Latin tres, three, and trans, beyond. The same can be said regarding the French trés, very, and trois, three.
· Some primitive languages have words for every color but have no word for color. Others have all the number words but no word for number. The same is true for other words.
· English is very rich in native expressions for types of collections: flock, herd, set, lot, bunch, to name a few. Yet the words collection and aggregate are of foreign extraction. From Bertrand Russell we have the quote, ``It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number two." Today we have many terms to describe the idea of two: couple, set, team, twin, brace, etc.
Tallying
Tally sticks have been used since the beginning of counting. But it was not limited to ``primitive" peoples. The acceptance of tally sticks as promissory notes or bills of exchange reached all levels of development in the British Exchequer tallies. (12 century onwards.) It took an act of parliament in 1846 to abolish the practice.
An anecdote: The double tally stick was used by the Bank of England. If someone lent the Bank money, the amount was cut on a stick and the stick was then cut in half. The piece retained by the Bank was called the foil, and the other half was called the stock. It was the receipt issued by the Bank. The holder of said became a ``stockholder" and owned ``bank stock". When the holder would return the stock was carefully checked agained the foil; if they agreed, the owner would be paid the correct amount in kind or currency. A written certificate that was presented for remittance and checked against its security later became a ``check".
Tallying on a bone or stick is both ancient and modern. A more ancient form of counting was done by means of knots tied in a cord -- though counting is carried out to this day by knots or beads. Both objects and days were so tallied. From King Darius of Persia, we have this command given to the Ionians:
The King took a leather thong and tying sixty knots in it called together the Ionian tyrants and spoke thus to them: ``Untie every day one of the knots; if I do not return before the last day to which the knots will hold out, then leave your station and return to your several homes."
Knotted cords, called quipus were also used by the Incas of Peru. The conquering Spaniards noted that each village and an official of the knots, who maintained complex accounts on knotted cords of several colors and thicknesses, and performed a function similar to today's city treasurer.
HOW MANY?
Systems of enumeration.
Primitive:
notches, sticks, stones
Egyptians:
symbols for 1, 10, 100, 1,000, ... 1,0000,000.
Babylonians:
two symbols only--cunieform
Greeks:
alphabetical denotations, plus special symbols
Roman:
Roman numerals, I,V,X,L,C,D,M..
Arabs:
Ten special symbols for numbers.
Modern:
Ten special symbols for numbers.
Methods of ciphering.
Devices:
Abacus, counting boards.
Symbolic:
Arithmetic.
HOW MANY?
Bases for numbering systems
· binary -- early
· ternary -- early
· quinary -- early
· decimal
· vigesimal
· sexigesimal
· combinations of several
A study among American Indians showed that about one third used a decimal scheme; one third used a quinary/decimal scheme; fewer than a third used a binary scheme; and about one fifth used a vegesimal system. and a ternary scheme was used by only one percent.
HOW MUCH?
When counting or asking how many, we can limit discussions to whole positive integers. When asking how much, integers no longer suffice. Examples:
Given 17 seedlings, how can they be planted in five rows?
Given 20 talons of gold, how can they be distribution to three persons?
Given 12 pounds of salt, how can it be divided into five equal containers?
When asking how much we are led directly to the need of fractions.
HOW MUCH?
Another how much question is connected with measurement.
Where?
· Construction. To build graneries, or ovens to bake bread, or pyramids, or temples we need formulas for quantity, or area or volume.
· Planting. To divide arable plots we need formulas for plane area and those for seasons.
· Astronomy. To study the motions of stars we need angular and temporal measurment.
· Taxes and commerce. To properly assess taxes, we need ways to compute percentages (fractions).
To consider questions of how much we need more advanced numbers and arithmetic; we also need concepts of geometry.

Mathematics Used by American
Indians North of Mexico
For the American Indians north of Mexico, we may say that although their bonds of superstition and lack of an adequate number symbolism limited their mathematical progress, number still played an important role in their religious beliefs. In addition, they used many geometric figures in ornamentation and construction.


Sacred Numbers.
Specific reference to the use of three, four five, seven, and thirteen in religious ceremonies is extant, with four being the more prevalent. The may be due to the four points of the compass. Here are examples.
· Five was the mystical number of some of the Pacific Coast Indians. Three and five were sacred to the Iroquois.
· Seven was used by the Zuñi, Cherokee, Creeks, and most of the Plains tribes.
Thirteen was adopted by the Hopi Pawnee, and the Zuñi. It was also widely used in Central America.
· In the Pueblo Snake Dance the Snake Men prepared eight days for the ceremony; the snakes used were of four kinds obtained from a four days' hunt in the four directions.
· An Apache prayed to his gods at least once every four days, and if possible every day, for four times a day. Apache medicine men used this number in their remedies, e.g. four roots of one herb, roots of four varieties, ...
· Suppose a member of the Potawatomi tribe was accused of murder, but that the tribal chief thought he was not guilty, a pipe bearer would bring flint and steel and attempt to light the chief's pipe. If he was successful within four strokes of the steel the man went free, otherwise the man was executed. An influential man might get away with three murders, but for four murders, nothing could save him.
· The Iroquois when smoking would take three puffs from a pipe. Only three trials were allowed in physical contests. Five days or multiple thereof must elapse between the announcement of and the beginning of a celebration.
Counting
Counting on fingers was nearly universal among Indian tribes. Sometimes the fingers were bent in during counting, other times the fingers were extended from a fist during counting.
· Usually, both hands, beginning with the left hand were used to count ten. To get the next ten, some tribes used the toes; others used the fingers again. The Zuñi counted the second ten on their knuckles.
· Tally marks (vertical strokes) were used to denote one. Grouping was not generally evident. The Dakotas used only the vertical stroke. The Creeks also used this but a cross was used for ten.
· Evidence of subtraction has been found. For example in the Bellacoola language of British Columbia we have:
16 one man less four
18 = one man less two
26 = one man and two hands less four
36 = two men less four
· Traces of multiplication can be found in the number words of the Zuñi. We have
10 all the fingers
20 = two times all the fingers
100 = the fingers all the fingers
1000 = the fingers all the fingers times all the fingers
Mounds and Other Earthworks.
Most Indian mounds have been found in the eastern United States. Most were conical. The typical pyramidal mound was a truncated quadrilateral pyramid. The largest is located in Illinois. It is one hundred feet high and has a seven hundred foot base. Mounds have also been discovered and some excavated in Minnisota, Wisconsin, Ohio, Iowa, Georgia, and Mississippi. Many of the mounds were constructed well before Europeans arrived in the new world, some as early as 1000 BCE. The practice seems to have ending in about 1300 CE.


One group of mounds found in Ohio have bases in the shapes of circles, squares, and octagons --- all being quite accurate to true figures. The angles of one measuring more than 900 feet on its base made angles differing from right angles by less than one degree. Could this have been achieved by line of sight? It is an interesting exercise to explain how a people, living in the wilderness, with no tradition of geometry and little tradition of contruction could have made not just one but four nearly right angles on so gigantic a scale. Below is the Adena Serpent mound near Locust Grove, Ohio. Constructed in the second century BCE, it measures 1336 ft (405 meters) long by about 3-6 feet high. Note the use of spirals and semicircles as a part of a quite regular wavy line.

One mound in Georgia constructed by the Etowah Indians is the tallest structure in the area (about 61 feet), and furnishes an impressive view of the Etowah River Valley. The top covers about an acre of land. Another quadrilateral pyramid was a near accurate square with sides pointing in the compass directions. Below is a picture of the terraced-pyramidal Monk's mound rising 100 feet above the surrounding lansdscape, which is a part of the Cahokia mounds in Illinois, just across the Mississippi river from St. Louis. Cahokia is the largest pyramds construction north of Mexico. At its peak it was home to 30,000-50,000 people. Depending on the elevation used it measures 954 ft in the north-south direction and 775 ft in the east-west direction.

Two purposes of these mounds emerge. They were a location of temples and dwellings for the upper crust of the tribes. These sites were clearly more defensible in the event of rather common attacks from neighboring tribes. The other purpose was for burial, and some mounds were exclusively for this purpose. Though these mounds have supplied archeologists with a steady supply of artifacts about the the customs and daily life of the tribes, there has been little by way of mathematical abilities uncovered.

Ornamentation.
Anyone who has visited New Mexico has seen the beautiful blankets still woven by the local Indians. These blankets and other objects of Indian art throughout North America have an assortment of geometrical patterns and themes. Navajo pottery, for example, exhibits opposed sets of isosceles triangles, line bordering dots, hooked spirals, double spirals, vertical and horizontal lines, and stepped figures.

Mathematics in China


Primary sources are Mikami's The Development of Mathematics in China and Japan and Li Yan and Du Shiran's Chinese Mathematics, a Concise History. See the bibliography below.

1. Numerical notation, arithmetical computations, counting rods
o Traditional decimal notation -- one symbol for each of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, 1000, and 10000. Ex. 2034 would be written with symbols for 2,1000,3,10,4, meaning 2 times 1000 plus 3 times 10 plus 4. Goes back to origins of Chinese writing.
o Calculations performed using small bamboo counting rods. The positions of the rods gave a decimal place-value system, also written for long-term records. 0 digit was a space. Arranged left to right like Arabic numerals. Back to 400 B.C.E. or earlier.
o Addition: the counting rods for the two numbers placed down, one number above the other. The digits added (merged) left to right with carries where needed. Subtraction similar.
o Multiplication: multiplication table to 9 times 9 memorized. Long multiplication similar to ours with advantages due to physical rods. Long division analogous to current algorithms, but closer to "galley method."
2. Zhoubi suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (c. 100 B.C.E.-c. 100 C.E.)
o Describes one of the theories of the heavens. Early Han dynasty (206 B.C.E -220 C.E.) or earlier. Book burning of 213 B.C.E..
o States and uses the Pythagorean theorem for surveying, astronomy, etc. Proof of the Pythagorean theorem.
o Calculations including with common fractions.
3. The Nine Chapters on the Mathematical Art (Jiuzhang Suanshu) (c. 100 B.C.E.-50 C.E.)
Collects mathematics to beginning of Han dynasty. 246 problems in 9 chapters. Longest surviving and most influential Chinese math book. Many commentaries.
o Ch 1, Field measurement: systematic discussion of algorithms using counting rods for common fractions including alg. for GCD, LCM; areas of plane figures, square, rectangle, triangle, trapezoid, circle, circle segment, sphere segment, annulus -- some accurate, some approximations.
o Ch 2,3,6 on proportions, Cereals, Proportional distribution, Fair taxes.
o Ch 4, What width?: given area or volume find sides. Describes usual algorithms for square and cube roots but takes advantage of computations with counting rods
o Ch 5, Construction consultations: volumes of cube, rectangular parallelepiped, prism frustums, pyramid, triangular pyramid, tetrahedron, cylinder, cone, and conic frustum, sphere -- some approximations, some use pi=3
o Ch 7, Excess and deficients: false position and double false position
o Ch 8, Rectangular arrays: Gives elimination algorithm for solving systems of three or more simultaneous linear equations. Involves use of negative numbers (red reds for pos numbers, black for neg numbers). Rules for signed numbers.
o Ch 9, Right triangles: applications of Pythagorean theorem and similar triangles, solves quadratic equations with modification of square root algorithm, only equations of the form x^2 + a x = b, with a and b positive.
4. Sun Zi (c. 250? C.E.)
Wrote his mathematical manual. Includes "Chinese remainder problem" or "problem of the Master Sun": find n so that upon division by 3 you get a remainder of 2, upon division by 5 you get a remainder of 3, and upon division by 7 you get a remainder of 2. His solution: Take 140, 63, 30, add to get 233, subtract 210 to get 23.
5. Liu Hui (c. 263 C.E.)
o Commentary on the Nine Chapters
Approximates pi by approximating circles polygons, doubling the number of sides to get better approximations. From 96 and 192 sided polygons, he approximates pi as 3.141014 and suggested 3.14 as a practical approx.
States principle of exhaustion for circles
Suggests Calvalieri's principle to find accurate volume of cylinder
o Haidao suanjing (Sea Island Mathematical Manual). Originally appendix to commentary on Ch 9 of the Nine Chapters. Includes nine surveying problems involving indirect observations.
6. Zhang Qiujian (c. 450?)
Wrote his mathematical manual. Includes formula for summing an arithmetic sequence. Also an undetermined system of two linear equations in three unknowns, the "hundred fowls problem"
7. Zu Chongzhi (429-500) Astronomer, mathematician, engineer.
o Collected together earlier astronomical writings. Made own astronomical observations. Recommended new calendar.
o Determined pi to 7 digits: 3.1415926. Recommended use 355/113 for close approx. and 22/7 for rough approx.
o With father carried out Liu Hui's suggestion for volume of sphere to get accurate formula for volume of a sphere.
8. Liu Zhuo (544-610) Astronomer
Introduced quadratic interpolation (second order difference method).
9. Wang Xiaotong (fl. 625) Mathematician and astronomer.
Wrote Xugu suanjing (Continuation of Ancient Mathematics) of 22 problems. Solved cubic equations by generalization of algorithm for cube root.
10. Translations of Indian mathematical works.
By 600 C.E., 3 works, since lost. Levensita, Indian astronomer working at State Observatory, translated two more texts, one of which described angle measurement (360 degrees) and a table of sines for angles from 0 to 90 degrees in 24 steps (3 3/4 degree) increments.
Hindu decimal numerals also introduced, but not adopted.
11. Yi Xing (683-727) tangent table.
12. Jia Xian (c. 1050)
Written work lost. Streamlined extraction of square and cube roots, extended method to higher-degree roots using binomial coefficients.
13. Qin Jiushao (c. 1202 - c. 1261)
Shiushu jiuzhang (Mathemtaical Treatise in Nine Sections), 81 problems of applied math similar to the Nine Chapters. Solution of some higher-degree (up to 10th) equations. Systematic treatment of indeterminate simultaneous linear congruences (Chinese remainder theorem). Euclidean algorithm for GCD.
14. Li Chih (a.k.a. Li Yeh) (1192-1279)
Ceyuan haijing (Sea Mirror of Circle Measurements), 12 chapters, 170 problems on right triangles and circles inscribed within or circumscribed about them. Yigu yanduan (New Steps in Computation), geometric problems solved by algebra.
15. Yang Hui (fl. c. 1261-1275)
Wrote sevral books. Explains Jiu Xian's methods for solving higher-degree root extractions. Magic squares of order up through 10.
16. Guo Shoujing (1231-1316).
Shou shi li (Works and Days Calendar). Higher-order differences (i.e., higher-order interpolation).
17. Zhu Shijie (fl. 1280-1303)
Suan xue qi meng (Introduction to Mathematical Studies), and Siyuan yujian (Precious Mirror of the Four Elements). Solves some higher degree polynomial equations in several unknowns. Sums some finite series including (1) the sum of n^2 and (2) the sum of n(n+1)(n+2)/6. Discusses binomial coefficients. Uses zero digit.
Chronology of Mathematicians and Mathematical Works

Early traditional texts
These developed in a gradual accumulation of material over centuries. The dates given are roughly when they reached their final form.
· Suan shu shu (A Book on Arithmetic) (c. 180 B.C.E.). A book of bamboo strips found in 1984 near Jiangling in Hubei province.
· Zhoubi suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (c. 100 B.C.E.-c. 100 C.E.)
· Jiuzhang suanshu (Nine Chapters of the Mathematical Art) (c. 100 B.C.E.-50 C.E.)
The development of mathematics
· Zhang Heng (78-139)
o Ling xian (Spiritual Constitution of the Universe)
· Liu Hong (fl. 178-187)
o Qian xiang li (Calendrical Science Based on the Celestial Appearances) (178-187)
· Wang Fan (217-257)
· Sun Zi (c. 250?)
o Sunzi suanjing (Master Sun's Mathematical Manual)
· Zhao Shuang (Jun Qing) (c. 260)
o Zhoubi suanjing zhu (Commentary on the `Zhoubi Suanjing')
· Liu Hui (c. 263)
o Jiushang suanshu zhu (Commentary on the `Nine Chapters of the Mathematical Art')
o Haidao suanjing (Sea Island Mathematical Manual)
· Xiahou Yang (c. 350?)
o Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual)
· Zhang Qiujian (c. 450?)
o Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual)
· Zu Chongzhi (Wenyuan) (429-500)
o Da ming li (Da Ming Calendar) (462)
o Zhui shu (Method of Interpolation)
o Jiuzhang shu yi zhu (Commentary on the Methods and Essence of the `Nine Chapters')
o Chong cha zhu (Commentary on Double Differences)
· Zu Geng
o Zhui shu (Method of Interpolation)
· Zhen Luan (Shuzun) (fl. 566)
o Tian he li (Tian He Calendar) (462)
o Wucao suanjing (Mathematical Manual of the Five Government Departments)
o Wujing suanshu (Arithmetic in Five Classics)
o Shushu juji (Memoir on some Traditions of Mathematical Art)
· Liu Zhuo (544-610)
o Huang ji li (Imperial Standard Calendar) (600)
· Wang Xiaotong (fl. 625)
o Xugu suanjing (Continuation of Ancient Mathematics)
· Li Chunfeng (fl. 664)
o Edited the Shibu suanjing (Ten Books of Mathematical Classics) .
This collection included the Jiuzhang suanshu (Nine Chapters of the Mathematical Art), Haidao suanjing (Sea Island Mathematical Manual), Sunzi suanjing (Master Sun's Mathematical Manual), Wucao suanjing (Mathematical Manual of the Five Government Departments), Wujing suanshu (Arithmetic in Five Classics), Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual), Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual), Zhui shu (Method of Interpolation), and Xugu suanjing (Continuation of Ancient Mathematics).
· Yi Xing (683-727)
o Da yan li (Da Yan Calendar) (727)
· Levensita (fl. 718)
o Jiu zhi li (Catching Nines Calendar) (718) translated from an Indian work
Three century interlude
followed by
The zenith of mathematical development
· Jia Xian (c. 1050)
o Jia Xian suanjing (Jia Xian's Mathematical Manual)
· Shen Kuo (1031-1095)
o Meng qu bi tan (Dream Pool Essays)
· Li Zhi (Li Ye) (Jingzhai) (1192-1279)
o Ceyuan haijing (Sea Mirror of Circle Measurements) (1248)
o Yigu yanduan (New Steps in Computation) (1259)
· Liu Yi (fl. c. 1225)
o Yigu genyuan (Discussion of the Old Sources)
· Qin Jiushao (Daogu) (c. 1202-c. 1261)
o Shiushu jiuzhang (Mathemtaical Treatise in Nine Sections) (1247)
· Guo Shoujing (1231-1316)
· Yang Hui (Qianguang) (fl. 1261-1275)
o Xiangjie jiuzhang suanfa (A Detailed Analysis of the Mathematical Methods in the `Nine Chapters') (1261)
o Riyong suan fa (Computing Methods for Daily Use) (1262)
o Yang Hui suan fa (Yang Hui's Methods of Computation) (1274)
o Fasuan qu yong ben mo (Alpha and Omega of a Selection on the Applications of Arithmetical Methods (1274)
o Xugu zhaiqi suan fa (Continuation of Ancient Mathematical Methods for Elucidating the Strange [Properties of Numbers]) (1275)
o Jiuzhang suan fa zuan lei (Reclassification of the Mathematical Methods in the `Nine Chapters')
o Tian mu bi lei cheng chu jie fa (Practical Rules of Arithmetic for Surveying) (1275)
o Cheng chu tong bian suan bao (Precious Reckoner for Variations of Multiplication and Division)
· Wang Xun (1235-1281)
· Shou shi li (Works and Days Calendar), Guo Shoujing, Wang Xun, and others. (1280).
· Zhu Shijie (Hanqing, Songting) (fl. 1280-1303)
o Suan xue qi meng (Introduction to Mathematical Studies) (1299) There is a Japanese edition of 1658.
o Siyuan yujian (Precious Mirror of the Four Elements) (1303)
The decline of mathematics
· Sha keshi (fl. 1321)
o He fang tong yi (On the Prevention of River Flooding) (1321)
· Ding Ju (fl. 1355)
o Ding ju suan fa (Ding Ju's Arithmetical Methods) (1355)
· He Pingzi (fl. 1373)
o Xiangming suan fa (Explanations of Arithmetic) (1373)
· Liu Shilong (fl. 1424)
o Jiu zhang tong ming suanfa (Methods of Calculation in the `Nine Chapters') (1424)
· Wu Jing (fl. 1450)
o Jiu zhang suan fa bi lei da quan (Complete Description of the `Nine Chapters' on Arithmetical Techniques) (1450)
· Wang Wensu (fl. 1524)
o Suan xue baojian (Precious Mirror of Mathematics) (1524)
· Xu Xinlu
o Pan zhu suan fa (Method of Calculating on an Abacus) (1573)
· Ke Sangquin (fl. 1578)
o Shu xue tong gui (Rules of mathematics) (1578)
· Matteo Ricci (1552-1610)
· Niccolo Longobardi (1559-1654)
· Xu Guangqi (Zixian, Xuanhu) (1562-1633)
· Li Zhizao (Zhenzhi) (1565-1630)
Introduction of Western Mathematics
· Cheng Dawei (Rusi, Binqu)(fl. 1592)
o Suan fa tong zong (Systematic Treatise on Arithmetic) (1592) Reprinted in Japan in 1675.
o Zhi zhi uan fa tong zong (Postscript to the Systematic Treatise on Arithmetic) (1592)
o Suan fa zuan yao (Highlights of Calculation Methods (1598)
· Huang Longyi (fl. 1604)
o Suan fa ji nan (Directory of Calculation Methods) (1604)
· Johann Terrenz Schreck (1576-1630)
o Da ce (Complete Surveying)
o Ge tu ba xian biao (Tables of Trigonometric Functions) (1631)
o Ce tian yue shuo (Brief Description of the Measurement of the Heavens)
· Li Tianjing (1579-1659)
· Translation of euclid.html">Euclid's Elements, first six books, Matteo Ricci and Xu Guangqi (1607)
· Giulio Aleni (1582-1649)
o Ji he yao fa (Essentials of Geometry)
· Johann Adam Schall von Bell (1591-1666)
· Giacomo Rho (1593-1638)
o Chou suan (Napier's Bones) (1628)
o Ce liang quan yi (Complete Theory of Surveying) (1631)
o Bi li gui jie (Manual for proportional dividers) (1631)
· Tong wen suan zhi (Treatise on European Arithmetic) an edited translation of Clavius's Epitome of Practical Arithmetic, Matteo Ricci and Li Zhizao (1631)
· Chong zhen li shu (Chong Zhen Reign Treatise on Astronomy and Calendrical Science). (1631-1634).
A collection of 137 books in five submissions edited by Xu Guanqi and Li Tianjing with support of many others. It included Aleni's Ji he yao fa (Essentials of Geometry); Terrenz's Da ce (Complete Surveying), Ge tu ba xian biao (Tables of Trigonometric Functions), and Ce tian yue shuo (Brief Description of the Measurement of the Heavens); and Rho's Ce liang quan yi (Complete Theory of Surveying), Bi li gui jie (Manual for proportional dividers), and Chou suan (Napier's Bones).
· Jean Nicolas Smogulecki (1611-1656)
· Xi yang xin fa li shu (Treatise on Astronomy and Calendrical Science According to the New Western Methods).
A collection of 100 books in 17 volumes emended by Schall von Bell from the Chong zhen li shu (Chong Zhen Reign Treatise on Astronomy and Calendrical Science) (1645)
· Xue Fengzuo (d. 1680)
· Fang Zhongtong (1633-1698)
· Li xue hui tong (Understanding Calendar Making) (1652-1654)
A collection of books published by Smogulecki and Xue Fengzuo. Included are
o Bi li si xian xin biao (New Tables for Four Logarithmic Trigonometric Functions)
o Bi li dui shu biao (Logarithm Tables with Explanations)
o San jiaofa (Essentials of Trigonometry) (1653)
· Tian bu zhen yuan (True Course of Celestial Motions) (1653)
A collection of books written by Smogulecki and Xue Fengzuo. Includes
o San jiao suan fa (Method of Trigonometrical Calculations)
· Mei Wending (Dingjiu, Wu'an) (1633-1721)
o See Mei Juecheng, Mei shi congshu jiyao (Collected Works of the Mei Family) for publication of Mei Wending's written comments on mathematics
· Shu li jing yun (Collected Basic Principles of Mathematics) (1723).
Supervised by Emperor Kang Xi (Aixinjueluo) (1654-1722), edited by Mei Juecheng, Chen Houyao, He Guozong, Ming Antu, Mei Wending, and others.
· Mei Juecheng
o In 1761, Mei Juecheng complied Mei Wending's written commentaries into the Mei shi congshu jiyao (Collected Works of the Mei Family). It included several works on mathematics: Bisuan (Pen Calculations), Chou suan (Napier's bones), Du suan shi li (Proportional Dividers), Shao guang shi yi (Supplement to `What Width'), Fang cheng lun (Theory of Rectangular Arrays), Gougu ju yu (Right-angled Triangles), Jihe tong jie (Explanations in Geometry), Ping san jiao ju yao (Elements of Plane Trigonometry), Fang yuan mi ji (Squares and Circles, Cubes and Spheres), Jihe bu bian (Supplement to Geometry), Hu san jiao ju yao (Elements of Spherical Trigonometry), Huan zhong shu chi (Geodesy), and Qiandu celiang (Surveying Solids).
Mathematics under the "Closed Door" Policy
· Chen Shiren (1676-1722)
o Shao guang bu yi (Supplement to `What Width')
· Ming Antu (d. 1765)
o Suanjing shishu (Ten Mathematical Manuals) (1773)
o Ge yuan mi lu jie fa (Quick Method for Determining Close Ratios in Circle Division) (1774)
· Jiao Xun (1763-1820)
o Da yan qiu yi shu (Technique for Finding 1 by the Great Extension)
· Ruan Yuan (1764-1849)
· Wang Lai (Xiaoying, Hengzhai) (1768-1813)
o Hengzhai suanxue (Hengzhai's Mathematics)
o Hengahai yi shu (Unpublished Works of Hengzhai) (1834, edited by Xia Xie)
· Chou ren zhuan (Biographies of Mathematicians and Astronomers) (1795-1799). Edited by Ruan Yuan.
· Li Huang (d. 1811)
o Jiuzhang suanshu xi cao tu shuo (Careful Explanation of the `Nine Chapters on the Mathematical Art')
o Haidao suanjing xi cao tu shuo (Careful Explanation of the `Sea Island Mathematical Manual')
o Xu gu suanjing kao zhu (Commentary on the `Continuation of Ancient Mathematical Methods for Elucidating the Strange [Properties of Numbers]')
· Li Rui (Shangzhi, Sixiang) (1773-1817)
o Li shi suan xue yi shu (Collected Mathematical Works of Li Rui)
· Luo Tengfeng
o Yi you lu (Records of the Art of Learning) (1815)
· Xiang Mingda (1789-1850)
o Xiang shu yi yuan (The Source of Series) (1888, edited by Dai Xu)
· Luo Shilin (1789-1853)
o Siyuan yujian xicao (Commentary on the `Precious Mirror of the Four Elements') (1836)
· Dong Youcheng (Fangli) (1791-1823)
o Ge yuan mi lu tu jie (Explanation for the `Determination of Close Ratios in Circle Division')
· Gu Guanjuang (1799-1862)
o Zhoubi suanjing xiao kan ji (A Textual Criticism of the `Zhoubi Suanjing')
· Shen Qinpei (fl. 1829)
o Siyuan yujian xicao (Commentary on the `Precious Mirror of the Four Elements') (1829)
· Zhang Dunren (fl. 1831)
o Qiu yi suan shu (Techniques of Finding 1) (1831)
· Dai Xu (1805-1860)
o Dai shu jian fa (Concise Technique of Logarithms) (1846)
International Mathematics in China
· Li Shanlan (Renshu, Qiuren) (1811-1882)
o Duo ji bi lei (Sums of Piles of Various Types)
o Fang yuan chan you (Explanation of the Square and the Circle)
o Hu shi qi mi (Unveiling the Secrets of Arc and Sagitta)
o Dui shu tan yuan (Seeking the Source of Logarithms)
o Several translations of Western mathematics (1852-1866)
· Hua Hengfan (Ruo Ting) (1833-1902)
o Xingsu xuan suan (Mathematical Papers form the Xing Su Study)
o Several translations of Western mathematics (1868-1886)
· Shi Richun
o Qiu yi shu zhi (Path to the Technique of Finding 1) (1873)
· Huang Zongxian
o Qiu yi shu tong jie (Explanation of the Technique of Finding 1) (1873)
The Origins of Mathematics.
Like every other aspect of human invention, mathematics has its origin, and like every technology, and mathematics is at least partly that, its origin is based upon needs of mankind. The particular needs are those arising from the wants of society. The more complex the society, the more complex the needs. The primitive tribe has little mathematical needs beyond counting. The complex society intent on building great temples, mustering conquering armies, or managing large capital assets has logistical problems that demand mathematics to solve.
Long before Pythagoras considered proving the famous theorem named after him, others tackled the just-as-complex operation of counting. You will see that for some the concept of two times two equals fours is advanced beyond comprehension, while for others counting past three is very complex. That counting began more than 50,000 years ago and many peoples even today and even in complex societies have trouble counting suggests that its creation was not as simple as we may.