World Mathematics in Africa

Mathematics has emerged in Africa in many varied forms over the millennia. Different civilisations have employed sophisticated arrangements of weights and measures, elaborate systems, and have played mathematical games. Former collection of French botanist Jean Baur. Donated to the Museum of Toulouse in 2010 Beyond that of the North African in various forms for the next 3,000  Akan gold-weights, and a Islamic and ancient Egyptian years. Comprehensive evidence of balance (18th–20th century CE); the weight on the left depicts civilisations, the history of Egyptian mathematics is rare, since a drummer, although the arms mathematics in Africa has been little- it would have been recorded on have been removed – possibly studied, and sources are sketchy. papyrus, which has not survived. to adjust the weight Further research will undoubtedly However, the very few remaining uncover more of the details of African sources allow us to reconstruct the mathematics that have been hidden sophisticated arithmetical techniques for centuries. that Egyptian scribes used in record- keeping. We also have evidence © The Trustees of the British Museum of geometrical understanding: for example, a standard procedure for calculating the area of a circle.

Alexandria in Egypt was also home, in the fourth century CE, to one of the first female mathematicians about whom we have any substantial gold trade, in the 14th century CE. The information: Hypatia, who Akan people in particular produced wrote commentaries on ancient brass weights, for weighing gold, that mathematical texts, and is said to were both elaborately and accurately Founded in 2003, the African have constructed astronomical crafted. Institute for Mathematical Sciences instruments. (AIMS) is a network of academic Mathematical Games institutions in Cameroon, Ghana, Ben2, CC BY-SA 3.0, via Wikimedia Commons Games of chance and strategy have Rwanda, Senegal, South Africa, been played throughout Africa and Tanzania. Its goal is to promote over the centuries. One example is post-graduate education, research, morabaraba, a version of the Roman and public engagement in the game of nine men’s morris, played in mathematical sciences in Africa.  The , South Africa and Botswana. Another controversial evidence of early game that is particularly widespread, in Africa with numerous variants and a host of Early arithmetic? different names, is mancala – alleged The Ishango bone is a tool made from to be one of the oldest games still to a baboon fibula, discovered in 1960 in be played anywhere. Indeed, mancala what is now the Democratic Republic  Hypatia of Alexandria tournaments now take place around One area of mathematics that has of the Congo. It dates to between (?–415 CE), as depicted by the world. seen particular growth in African Jules Maurice Gaspard in 1908 20,000 and 18,000 BCE. Notches institutions in recent years is that carved into the bone have been Weights in West Africa of machine learning, driven by the interpreted as , evidence The small hard seeds of the rosary foundation of such organisations of an arithmetical game, a record of pea were traditionally used as as Data Science Africa and Deep lunar cycles, or simply as grooves to standard weights by various peoples Learning Indaba. aid grip and therefore entirely non- in West Africa, owing to their

mathematical. consistency in size. However, metal © The Trustees of the British Museum weights became the norm following  A Kenyan mancala board –  The Rhind Mathematical Ancient Egypt the introduction of metal-working players take turns to move their Papyrus, the most complete playing pieces from one pit to and extensive surviving The ancient Egyptian civilisation techniques to present-day Ghana, and another, with the goal of capturing source on ancient Egyptian arose around 3,000 BCE and lasted the expansion of the trans-Saharan their opponent’s pieces mathematics

© The Trustees of the British Museum World history of mathematics Mathematics in the Americas

The countries of North, Central, and South America feature prominently in the modern international mathematical community.

In the past, the various cultures  Artur Ávila, awarded of the continent, particularly of Rather than recording information the Fields Medal in 2014 for ‘contributions South America, developed a range in a written form, some Andean to dynamical systems of systems, which were civilisations, including the Incas of theory, which have employed most extensively in ancient Peru, developed a system changed the face of the field, using the powerful connection with time-keeping. of record-keeping using knots in idea of renormalization as a unifying principle’

bundles of wool and cotton cords © MFO The Aztecs called quipu (‘talking knots’). The use of base-20 number systems was widespread across South The colours of the cords and the However, the mathematical America, both linguistically and  in a spacing of the knots were crucial to communities of other countries are notationally. The Aztecs, for example, rare surviving text of the the encoding of information on the now growing. In 1995, the Unión 13th or 14th century CE employed a non-positional base-20 quipu. Although much of the content Matemática de América Latina y el number system with separate signs of the surviving quipu remains to Caribe was created, with the initial for 1, 20, 400, and 8000. These Dates in the long count were written be deciphered, we know that one of involvement of Argentina, Brazil, numerals may be found in surviving using a base-20 positional numeral their major uses was the recording of Chile, Colombia, Cuba, Mexico, survey documents, which record system, where a dot and a bar stood numerical information. Uruguay, Venezuela, and Peru; in areas and perimeters of fields – but for ‘1’ and ‘5’, respectively; a picture 2018, Rio de Janeiro became the first we do not know how these were of a shell often indicated an empty The Americas in the present day city in the southern hemisphere to measured or calculated. position. During the past two centuries, the host an International Congress of mathematical focus of the Americas Mathematicians. The study of the has been on Canada and the United recent history of mathematics in the

These images are obtained through the courtesy of the World Digital Library. The codex itself is held by the National Library of States. US-based mathematicians Americas is also changing: it has hold the largest number of Fields usually concentrated on white male Medals; so far (2020) only one mathematicians, but the contributions South American has won the award: of other, previously-overlooked groups Artur Ávila of Brazil. are now being recognised.

 Illustration from El  The 2010 book primer nueva corónica Hidden Figures: y buen gobierno The American Dream (The First New and the Untold Story Chronicle and Good of the Black Women Government), 1615, Who Helped Win the by Peruvian Felipe Space Race Guaman Poma de Ayala; this is one of the few surviving accounts of the use of quipu

 An early 16th-century land survey, annotated in both Spanish and the Aztec language Nahuatl

The Maya The Maya culture emerged in Mesoamerica around 2,000 BCE, and elements of it survive to the present day. From around the fifth century BCE, the Maya employed an elaborate calendar, consisting of cycles within cycles. In the calendar, a 260-day count was combined with a year of 365 days; the two came back into step every © Paul Johnson 52 years, known as a Calendar Round. For dates on monuments, the Long Count was used; this was constructed  An inscription displaying a date in the  An Incan quipu consisting of around much larger units of time, Maya Long Count; numerals appear above coloured and knotted cords, some spanning centuries. the respective signs for units of time encoding numerical information World history of mathematics Mathematics in China

Mathematics has a long pedigree in China, both as an accomplishment expected of Confucian literati, and as a subject that was key to training as a civil servant.

The Nine Chapters on the Mathematical Art One of the most influential of the traditional Chinese mathematical texts was the Jiǔzhāng suànshù 九章算術 (The Nine Chapters on the Mathematical Art), probably dating originally from c. 300 BCE, but in use for over 1,000 years as a manual for the training of administrators. TheJi ǔzhāng suànshù consists of 246 arithmetical problems and their methods of solution. Its seventh and eighth chapters, for example, deal with the solution of systems of up to five simultaneous equations in five unknowns. The method used, fāngchéng 方程, is equivalent to the process known in the West  Page from an edition of the Sūnzĭ suànjīng  The Yang Hui triangle 杨辉三角形 (a.k.a. as Gaussian elimination, and may printed more than 1,000 years after the Pascal’s triangle), as found in the Sìyuán yùjiàn original text was written; beyond his name, we 四元玉鉴 (Jade Mirror of the Four Unknowns, have been influenced by the use of know nothing about the alleged author, Sun Zi 1303) by Zhū Shìjié 朱世傑 (1249–1314) counting boards in China. Wang Zhenyi Although mathematical learning in Jean-Claude Martzloff and StephenJean-Claude Martzloff S. Wilson, A history Springer, of Chinese mathematics, Berlin, 1997 China was confined largely to men, some women did also engage in scientific studies. Most notable was

https://math-physics-problems.wikia.org/wiki/Liu_Hui CC-BY-SA https://math-physics-problems.wikia.org/wiki/Liu_Hui Wang Zhenyi 王貞儀 (1768–97), who wrote treatises on both astronomy (the procession of the equinoxes) and mathematics (trigonometry).

Because traditional Chinese mathematics focused on methods rather than proofs, it has often

 The Jiǔzhāng suànshù was transmitted  The handwritten numerals shown been compared unfavourably with  Chen Jingrun 陳景潤 (1933–96), who down the centuries via commentaries that above derive from Chinese counting the ancient Greek mathematics made progress towards the proof of often added substantially to its content. rods: bamboo rods were used to that has traditionally provided the the Twin Prime Conjecture by proving A particularly famous commentary was represent within spaces in 1966 that there are infinitely many that supplied by Liu Hui 劉徽, perhaps on a grid, with red rods for positive benchmark for how mathematics primes p for which p + 2 is either prime in the third century CE numbers, and black for negative ‘should’ be done. Nowadays, however, or a product of two primes historians of mathematics strive to  Liu Hui’s method for the calculation of the area Master Sun’s Mathematical Manual treat ancient Chinese mathematics,  Wang Zhenyi 王貞儀 of a circle, given its diameter and circumference A later text that also became and other non-Western mathematical (1768–97), astronomer implicitly takes 3.14024 as the value of π and mathematician required reading for civil traditions, purely on their own terms. servants was the Sūnzĭ suànjīng 孫子算經 (Master Sun’s China occupies a prominent position Mathematical Manual), probably in international mathematics; written in the third century CE. in August 2002, Beijing became only the second Asian city to host Motivated by calendrical problems, the International Congress of the Sūnzĭ suànjīng deals with Mathematicians. congruences as well as ordinary arithmetic, and is the earliest source for what we now call the Chinese Remainder Theorem on the solution of simultaneous congruences. World history of mathematics Mathematics in India

India has a mathematical tradition that goes back many thousands of years.

The Indus Valley Civilisation, Ramanujan which emerged around 3,000 India’s most famous mathematician BCE, had a centralised system of of the twentieth century was Srinivasa weights and measures, along with Ramanujan சீனிவாச இராமானுசன் measuring instruments. Although (1887–1920). Born in present-day the Indus script has not yet been Tamil Nadu, Ramanujan was largely deciphered, India nevertheless boasts self-taught in mathematics. mathematical texts whose originals date from 800–500 BCE (but are now In 1913, he began a correspondence known only from later copies) in  The Tantrasaṅgrahaḥ with G. H. Hardy in Cambridge, the form of the Śulbasūtras शुल्वसूत्रम, तंत्रसंग्रह (1501), an which led to his being invited there ् astronomical treatise by which concern geometry and the the following year. Ramanujan Nīlakaṇṭha Somayājī नीलकण्ठ construction of sacrificial altars. सोमयाजि (1444–1544), remained in Cambridge until shortly containing power series before his early death in 1920, aged expansions for trigonometric functions, written on palm just 32; in 1918, he was elected both leaves in Sanskrit verse a Fellow of the Royal Society and of Trinity College, Cambridge.

deals mostly with plane and spherical  The oldest surviving trigonometry, but also covers the undisputed occurrence of the symbol 0 for zero appears in  The Bakhshali manuscript is one of the solution of various types of equations, an inscription in Chaturbhuj oldest surviving original Indian mathematical and contains an approximation of Temple in Gwalior, Madhya texts (estimates place it anywhere between π as 3.1416. Other figures from the Pradesh, India the second and the tenth centuries CE). It is a compendium of mathematical problems and classical period are Brahmagupta rules for their solution. Depending on its date, ब्रह्मगुप्त (c. 598–668 CE), who was Numerals it may contain the oldest known example of the first scholar to set out rules for Indian scholars employed a the symbol 0 for zero. performing arithmetic with negative positional number system from early numbers and zero, and Bhāskara I in the first millennium CE. Initially, भास्कर प्रथम (c. 600–680 CE), author of empty columns were indicated with three astronomical treatises. a word meaning ‘emptiness’, but eventually a dot, and then a circle,  Srinivasa Ramanujan, who Kerala School came to be used. This system of made profound contributions to analysis and number theory; During the 14th–16th centuries CE, numerals was adopted by Middle with Hardy, he obtained an a school of mathematics and Eastern scholars in the ninth century, asymptotic formula for the astronomy flourished in Kerala in and was transmitted to in the partition function southern India. Often motivated by following centuries – both by Italian astronomical problems, the school merchants and via Islamic Iberia – Indian Women and Mathematics produced treatises on arithmetic, to become the number system that Like many countries, India has algebra, geometry, approximation we use today, often termed ‘Hindu- an association devoted to promoting of roots of equations, and magic ’. the involvement of women in squares, and also developed power mathematics: Indian Women and series expansions for trigonometric Mathematics, founded in 2009.

functions, two centuries before these  Participants of the 2012 It holds annual conferences and same ideas appeared in Europe. IWM Annual Conference regional workshops throughout India.

 Statue of Āryabhaṭa outside the Inter-University Centre for Astronomy and Astrophysics in Pune, Maharashtra, India

Classical Period The so-called ‘classical period’ of Indian mathematics began in the middle of the first millennium CE with the work of Āryabhaṭa I आर्यभट (476–550 CE). His Sanskrit treatise Āryabhaṭīya आर्यभटीयम् (c. 510 CE) World history of mathematics Mathematics in Japan

During the seventeenth to nineteenth centuries, Japanese mathematicians developed a unique form of mathematics, known as wasan.

During the early 20th century, many Japanese students travelled to Europe – Germany, in particular – to study mathematics; in the second half of the century, however, the focus shifted to the USA and to Japanese institutions. In 1990, Kyoto became the first Asian city to host an International Congress  A soroban, the Japanese problems, and in reaction to them, of Mathematics. that the more systematic wasan After Japan opened up to the West gradually developed.  A page from Seki’s textbook in the mid-nineteenth century, Kaikendai no hō 解見題之法 (Methods of Solving Explicit its government decreed that Seki Takakazu, the Japanese Problems), featuring techniques Western mathematical techniques Newton for calculating areas of plane (洋算 yōsan) should be adopted. The Seki Takakazu 関 孝和 (?–1708) figures and volumes of some solids exception to this was that children is considered the father of wasan. should still be taught to use the Moving away from a view of Sangaku Japanese abacus, the soroban 算盤. mathematics as a collection of Alongside the more systematic wasan, isolated problems, he sought general the tradition of posing isolated To date (2020), In earlier centuries, Japanese methods. In contrast to Western problems remained strong, driven Japan boasts three Fields Medalists: mathematics had been heavily mathematics, wasan was inductive by mathematical hobbyists. Rival (clockwise from above) influenced by texts from China, and rather than deductive: mathematical mathematical sects developed across Kodaira Kunihiko European missionaries had taught enthusiasts (Seki himself was an Japan, with little interaction between 小平 邦彦 (1915–97), Hironaka Heisuke Western mathematics in Japan in accountant by day) carried out different groups. These schools often 広中 平祐 (b. 1931), the early seventeenth century, but arithmetical experimentations to had religious links, and displayed and Mori Shigefumi during Japan’s period of isolation establish patterns and results. their mathematical achievements via 森 重文 (b. 1951) from the rest of the world, from the votive tablets, sangaku 算額, hung mid-seventeenth century onwards, outside Shinto shrines or Buddhist a Japanese form of mathematics had temples. These showed arithmetical flourished:wasan 和算. problems or geometrical figures with numerical solutions, but never In the 17th century there arose in indicated how the problem had Japan a tradition of posing ever been solved. harder and more intricate, and often rather artificial, mathematical problems, often taken from Chinese textbooks, whose goal was to challenge the reader rather than to develop a comprehensive understanding of the underlying  Seki Takakazu 関 孝和, mathematics. It was out of such sometimes referred to as  Page from a mid-seventeenth century ‘the Japanese Newton’ arithmetic textbook Shinpen Jinkoki 新編塵 劫記, compiled by Yoshida Mitsuyoshi 吉田 Using such an approach, Seki was 光由 (1598–1673). The problem illustrated is equivalent to the so-called ‘Josephus problem’ one of the first mathematicians to that was posed in Mediaeval Europe discover the Bernoulli numbers,  Example of a and to develop the idea of a sangaku (c.1846) from Io Shrine in Osaka determinant. He is also credited Prefecture with the foundation of the branch  Sangaku (1861) from of mathematics known as enri 円理, Sōzume shrine in Okayama City, showing or circle principles, which sought women and children general methods for calculating areas, performing calculations volumes, and lengths of curves, and on soroban therefore paralleled the development of calculus in Europe. World history of mathematics Mathematics in the Middle East

Some of the world’s earliest mathematics was developed in the Middle East by the Sumerian and Babylonian cultures after c. 3,000BCE , much of which was eventually transmitted to ancient Greece.

A later Middle Eastern mathematical  13th-century depiction of solving equations were presented as  Omar Khayyam’s solution and astronomical school flourished scholars in the Abbasid Library step-by-step procedures; a Latinised of a cubic equation of the at the Bayt al-Ḥikmah form x³ + cx + d = bx2 under Islamic influence, particularly version of his name gives us the word (for b, c, d positive) during the 9th and 10th centuries CE. ‘algorithm’. The pursuit of knowledge was seen as having religious merit, and certain verses in the Quran were interpreted as encouraging scientific enquiry.

Babylonian mathematics The scribes of ancient Mesopotamia employed a base-60 positional numerical notation, many examples of which appear on surviving clay tablets. Using this notation, the scribes were able to solve a range As well as producing new ideas, of arithmetical and geometrical In his Compendious Mediaeval Islamic scholars also problems, as well as to develop a Book on Calculation by helped to preserve the knowledge sophisticated astronomy – their use of Completion and Balancing of the ancient world, via their systematic translation of writings الكتاب املخترص يف حساب الجرب واملقابلة base-60 is preserved in our division of an hour in 60 minutes, a minute (c. 820 CE), al-Khwārizmī set out whose originals are now lost.

into 60 seconds, and a circle into methods for the systematic solution  Page from al-Khwārizmī’s Certain texts, such as parts of 360 degrees. of quadratic equations, which for him Compendious Book giving Diophantus’ Arithmetica, a Greek fell into six distinct cases because a geometrical justification for his compendium of algebraic problems rule for the solution of a quadratic he did not use negative numbers or equation of the form x² + bx = c from the third century CE, are known zero. The ‘al-jabr’ (‘balancing’) of (for b and c both positive) only through surviving Arabic the text’s Arabic title was taken over translations. into European languages as our word Omar Khayyam مریم میرزاخانی algebra’. Al-Khwārizmī’s rules for Two hundred years later, al- Maryam Mirzakhani‘  Statue of al-Khwārizmī that stands Khwārizmī’s methods for solving (1977–2017) was born in Iran, but in front of the Faculty of Mathematics quadratic equations were extended to subsequently moved to the United of Amirkabir University of Technology the case of cubics by Omar Khayyam States. In 2014, she became the first in Tehran who lived in woman to win a Fields Medal for ,(1131–1048) عمر خیّام  Mathematical clay tablet, modern-day Iran. Famous in the “her outstanding contributions to the dating from 1900–1600 BCE, West as a poet, Khayyam also studied dynamics and geometry of Riemann showing an area calculation astronomy and mathematics. As an surfaces and their moduli spaces.” House of Wisdom astronomer, he devised a new solar calendar, while his contributions بيت الحكمةThe Bayt al-Ḥikmah (House of Wisdom) was founded to mathematics covered both in Baghdad at the end of the geometry and algebra. In particular, eighth century CE. It was created he provided solutions of cubic to produce Arabic translations of equations, on a case-by-case basis, as ancient Greek texts, but soon became intersections of conic sections. a centre for original scholarship as well. Developments were made in particular in both mathematics (especially algebra) and astronomy (especially spherical trigonometry).

Al-Khwārizmī One of the Bayt al-Ḥikmah’s most famous figures during the ninth century CE was the Persian scholar Muḥammad ibn Mūsā  Omar Khayyam, Persian  Maryam Mirzakhani mathematician, astronomer, (1977–2017), the first woman محمد بن موىس خوارزمی al-Khwārizmī (c. 780–850 CE). and poet to win a Fields Medal World history of mathematics Mathematics in Oceania

The various cultures of Oceania provide us with many examples of mathematics appearing in unexpected places.

The same mathematical ideas have Te Rangiua and Te Urikore playing Mu Torere. Original photographic prints and postcards from file print Box collection, 3. Ref: Alexander PAColl-5671-07. Turnbull Library, Wellington, New Zealand. /records/23218316 often appeared implicitly in different places around the globe, but tend to be articulated and conceptualised in distinct ways. As scholars, we must therefore be flexible in what we interpret as ‘mathematics’.

We can analyse the latent mathematical notions of different cultures using Western ideas, but  A graph showing the  Each mū tōrere player this doesn’t necessarily mean that an interrelations between the begins with four counters sections of Warlpiri society; each, arranged around the awareness of these same concepts in unlabelled nodes indicate outer points of the playing their Western formulation is present permissible marriages, and grid. There are 86 attainable in the original ideas: an appreciation arrows indicate the subsection configurations on the game to which a child of that board, including 3 winning of symmetry, for example, does not marriage will belong positions for each player  Kōwhaiwhai in a house automatically signal a knowledge of of the Ngāti Porou at Waiomatatini, East Cape, the notion of a group. Warlpiri population is divided into points with a view to blocking their New Zealand eight named subsections, and the opponent. A combinatorial analysis Number systems subsections to which parents belong of versions of the game played on The many cultures and languages determine those of their children in grids having more than or fewer than of Oceania provide a variety of a systematic way. If we label these eight points suggests that the eight- examples of different counting interrelationships in a certain way, pointed version provides just the systems, constructed on a range of we find the structure of the dihedral right level of complexity to make for bases, including 2, 4, 5, 6, and 10. group of order 8. an interesting game. The Oksapmin of Papua New Guinea Marcia Ascher, A Multicultural View of Mathematical Ethnomathematics: Ideas, Taylor and Francis, 1991, p. 78 employ a base-27 counting system  There is a formal structure which involves the enumeration to kōwhaiwhai, whose carving must be both ritually and of different parts of the body. technically correct

In 2018, the University of Sydney launched its global Mathematical  The enumeration of parts of the  The drawing of continuous figures  A specially made mū tōrere Research Institute, with the goals body, according to the Oksapmin in sand has an important ritualistic board. The game is equally well of drawing leading mathematical base-27 counting system purpose for the Malekula people played with pebbles on a grid of Vanuatu. In particular, diagrams marked out in sand, for example scientists from around the world to The logic of kinship such as that pictured here were work with Australian collaborators, Systems of kinship have particular used to illustrate Malekula kinship Māori rafter patterns and of increasing public engagement relationships for early Western visitors. importance amongst many Australian With appropriate labelling, we may Kōwhaiwhai are traditional painted with mathematics. Aboriginal peoples. Such systems, identify the structure of a dihedral motifs that appear on the rafters of group of order 6 whereby each person is assigned Māori meeting houses. Created by a  The University of Sydney to a traditional subgroup of the master carpenter, these are geometric Main Quadrangle population, encode the rights and Mū tōrere patterns that tell stories about the obligations of individuals within Games of strategy are played in people who built the meeting house, society, and determine who may many cultures around the world, represent ancestors, heroes or gods, marry whom. and the Māori of New Zealand or reflect aspects of the natural are no exception. The game of mū environment. Notions of symmetry A kinship system that has received tōrere is played with counters on a and balance are often central to these a lot of attention is that of the grid shaped like an eight-pointed designs, which lend themselves to Warlpiri people of Australia’s star. Players take it in turns to move analysis via Western mathematical Northern Territory. The entire their counters to adjacent empty techniques.