''Mathematics in India: 500 BCE–1800 CE'' is a

monograph A monograph is a specialist work of writing (in contrast to reference works) or exhibition on a single subject or an aspect of a subject, often by a single author or artist, and usually on a scholarly subject. In library cataloging, ''monograph ...

about the history of

Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...

. It was written by American historian of mathematics

Kim Plofker Kim Leslie Plofker (born November 25, 1964) is an American historian of mathematics, specializing in Indian mathematics. Education and career Born in Chennai, India, Plofker received her bachelor's degree in mathematics from Haverford College. She ...

, and published in 2009 by the

Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial su ...

. The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has classified the book as essential for undergraduate mathematics libraries, their highest rating.

Topics

Plofker has organized ''Mathematics in India'' into nine chapters, roughly chronologically, according to the "mainstream narrative" of Indian chronology in a subject where accurate chronology is difficult and disputed. It covers the mathematics of the entire

Indian subcontinent The Indian subcontinent is a list of the physiographic regions of the world, physiographical region in United Nations geoscheme for Asia#Southern Asia, Southern Asia. It is situated on the Indian Plate, projecting southwards into the Indian O ...

, including the modern areas of

Afghanistan Afghanistan, officially the Islamic Emirate of Afghanistan,; prs, امارت اسلامی افغانستان is a landlocked country located at the crossroads of Central Asia and South Asia. Referred to as the Heart of Asia, it is bordere ...

India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...

, and

Pakistan Pakistan ( ur, ), officially the Islamic Republic of Pakistan ( ur, , label=none), is a country in South Asia. It is the world's List of countries and dependencies by population, fifth-most populous country, with a population of almost 24 ...

, but largely restricts itself to

Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...

-language sources. Unlike many previous works in this area, it views Indian mathematics as a coherent whole, strongly connected to Indian culture and religion, both influencing and being influenced by the other cultures of the world, rather than as a collection of milestones for measuring relative progress against other cultures. Much of the scholarly work on this subject has been contradictory and contentious, and Plofker is careful to provide evidence for the hypotheses she supports, discuss alternative hypotheses, and view the subject neutrally for itself rather than as a way to boost or put down Indian culture. Her book includes some speculative theories, but is well-grounded in recent scholarship, and focused on evidence from the source material. It carefully maintains a balance between the cultural and scientific context needed to understand the mathematics it describes, the major texts and oral traditions through which that mathematics has come down to us, and the cross-cultural transmission of mathematical knowledge with other cultures. The first introductory chapter provides an overview of Indian history of Indian mathematics and its scholarship, and of the religious and linguistic context of early Sanskrit texts, which leads to important differences from Indian mathematics to other ancient mathematical cultures developing from administrative or scientific works. Chapter two discusses the

Vedic period The Vedic period, or the Vedic age (), is the period in the late Bronze Age and early Iron Age of the history of India when the Vedic literature, including the Vedas (ca. 1300–900 BCE), was composed in the northern Indian subcontinent, betw ...

from 1500 to 500 BCE, and the

Shulba Sutras The ''Shulva Sutras'' or ''Śulbasūtras'' (Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction. Purpose and origins The ...

, religious instructional texts with significant mathematical content that are generally attributed to this period, although (as the book discusses) the absence of concrete astronomical observations within these texts has made it impossible to date them precisely. Topics from this period include its methods for reckoning time, its fascination with large numbers, the beginnings of decimal numbering and

integer factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

, geometric constructions using cords or ropes, the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

, and accurate approximations to pi and the

square root of two The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...

. This chapter also includes material on speculative links between Vedic India and ancient

Mesopotamia Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the F ...

, a pet theory of Plofker's advisor

David Pingree David Edwin Pingree (January 2, 1933, New Haven, Connecticut – November 11, 2005, Providence, Rhode Island) was an American historian of mathematics in the ancient world. He was a University Professor and Professor of History of Mathematics ...

, but it notes the weakness of evidence for these theories. The third chapter covers the next 500 years, the early classical period of India, including the Bhutasamkhya system for describing numbers in words and the invention of decimal place-value arithmetic (although Plofker suggests that the concept of

zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...

may be an import from China), connections between poetic meter and binary representations, early trigonometry, the works of

Pāṇini , era = ;;6th–5th century BCE , region = Indian philosophy , main_interests = Grammar, linguistics , notable_works = ' (Sanskrit#Classical Sanskrit, Classical Sanskrit) , influenced= , notable_ideas=Descript ...

and

Pingala Acharya Pingala ('; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody. The ' is a work of eight chapters in the late ...

(arguably including the invention of

recursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

), mathematics in

Jainism Jainism ( ), also known as Jain Dharma, is an Indian religions, Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current ...

and

Buddhism Buddhism ( , ), also known as Buddha Dharma and Dharmavinaya (), is an Indian religion or philosophical tradition based on teachings attributed to the Buddha. It originated in northern India as a -movement in the 5th century BCE, and gra ...

from this period, and possible Greek influences in trigonometry and

astrology Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of Celestial o ...

, which became one of the driving forces in later mathematics. Chapter four covers roughly the first millennium CE, and focuses mainly on Indian astronomy and

geocentrism In astronomy, the geocentric model (also known as geocentrism, often exemplified specifically by the Ptolemaic system) is a superseded description of the Universe with Earth at the center. Under most geocentric models, the Sun, Moon, stars, and ...

, including the use of verse forms and interpolation to make memorization of trigonometric tables possible. Chapters five and six concern the medieval period of India. Chapter five overlaps in time with the later parts of chapter four, and concerns the works of

Aryabhata Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...

Bhāskara I Bhāskara () (commonly called Bhāskara I to avoid confusion with the 12th-century mathematician Bhāskara II) was a 7th-century Indian mathematician and astronomer who was the first to write numbers in the Hindu–Arabic decimal system with a ...

, and

Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...

, and

Mahāvīra Mahavira (Sanskrit: महावीर) also known as Vardhaman, was the 24th ''tirthankara'' (supreme preacher) of Jainism. He was the spiritual successor of the 23rd ''tirthankara'' Parshvanatha. Mahavira was born in the early part of the 6 ...

, and the Bakhshali manuscript, including the invention of

negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...

s and

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

Brahmagupta's formula In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides; its generalized version (Bretschneider's formula) can be used with non-cyclic ...

for the area of

cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...

s, and the solution of

Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...

. Chapter six covers later mathematicians

Bhāskara II Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiroman ...

and Narayana Pandita, Bhāskara's works on

geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...

, and the development of ideas related to

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

(although not really calculus itself). It also discusses the position of mathematicians in society, and the nature of mathematical canon, commentary, and proof in those times. The

Kerala school of astronomy and mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, K ...

founded by

Madhava of Sangamagrama Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinjalakuda in Thrissur District, Kerala, India. He is ...

is the topic of the seventh chapter, which includes Madhava's works on

series expansion In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division) ...

s of trigonometric functions and the calculation of pi, and developments by

Nilakantha Somayaji Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensi ...

in the theory of astronomy. Chapter eight covers the interactions between India and

mathematics in medieval Islam Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius of Perga, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress wa ...

, including the transmission of decimal notation to the west and an increased awareness of mathematical rigor in India. Chapter nine concerns colonial and early modern times in India, the influence of European mathematics, and ongoing developments within Indian mathematics from the 16th through 18th centuries. Unfortunately, it stops just before the time of

Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...

. The book concludes with a collection of still-unresolved major research questions in the area of Indian mathematics. Two appendices cover aspects of Sanskrit grammar and prosody that are important for understanding Indian mathematics, a glossary of technical terms, and a collection of biographies of Indian mathematicians. Throughout, many images of documents and artifacts of mathematical interest are included.

Audience and reception

''Mathematics in India'' does not require that its readers have any background in mathematics or the history of mathematics. It makes scholarship in this area accessible to a general audience, for instance by replacing many Sanskrit technical terms by English phrases, although it is "more of a research monograph than a popular book". Its readers are likely to come from many different audiences, including mathematicians, historians, indologists, philosophers, linguists, and philologists, and it succeeds in navigating the different expectations of these audiences. Reviewer James Rauff recommends ''Mathematics in India'' to all students or teachers of the history of mathematics, calling it "meticulously researched, carefully argued, and beautifully written", and Benno van Dalen goes further, calling it required reading for all future students of this topic. Dominik Wujastyk calls it "path-breaking", "a classic work that should be owned and read by any scholar interested in the history of science in South Asia". Although calling it difficult reading for non-specialists, Ward Stewart suggests that it could also be valuable to high school teachers and that some of its material could be incorporated into their lessons, and although A. K. Bag calls it "mainly meant for the foreign audience", B. Ramanujam writes that it deserves to be better known among Indian schoolteachers in particular. Dominik Wujastyk suggests using it as the basis for university-level courses, and Toke Knudsen highlights its value as reference material for researchers in this area. Both van Dalen and Agathe Keller write that the comprehensive English-language history of Indian mathematics in ''Mathematics in India'' was long-awaited, and several reviewers point to the '' History of Hindu Mathematics'' by

Bibhutibhushan Datta Bibhutibhushan Datta (also Bibhuti Bhusan Datta; Bengali : বিভূতিভূষণ দত্ত, Bibhūtibhūṣaṇ Datta) (28 June 1888 – 6 October 1958) was a historian of Indian mathematics. Datta came from a poor Bengali family. He ...

and Awadhesh Narayan Singh from the 1930s as the only previous work that filled that role, albeit one organized by topic rather than by time. Reviewers also noted the novelty of the book's focus on mathematical astronomy, with Alexander Jones calling it "the best general introduction to the history of astronomy in India that we currently have". Despite some quibbles, Keller and Clemency Montelle both call the book "destined to be a classic". A rare negative review is given by Satyanad Kichenassamy, who takes issue with the book's consideration of social context rather than purely of the mathematical content of the works it discusses, with its emphasis on astronomy as a force for mathematical development, with its omission of

Malayalam Malayalam (; , ) is a Dravidian language spoken in the Indian state of Kerala and the union territories of Lakshadweep and Puducherry (Mahé district) by the Malayali people. It is one of 22 scheduled languages of India. Malayalam was des ...

-language works, with "a tendency to conflate ancient mathematical concepts with modern ones", and with many details of its conclusions.

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, pages = 182–184 , title = Review of ''Mathematics in India'' , volume = 25 Books about the history of mathematics 2009 non-fiction books Indian mathematics