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South and West Asia consists of a wide region extending from the present-day country of Turkey in the west to Bangladesh and India in the east.

**3rd millennium BCE Sexagesimal system of the Sumerians**:**2nd millennium BCE Babylonian Pythagorean triples**. According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 1850 BCE^{ [1] }"contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,^{ [2] }indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia.**1st millennium BCE Baudhayana Śulba Sūtras Earliest statement of Pythagorean Theorem**: According to ( Hayashi 2005 , p. 363), the*Śulba Sūtras*contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."

Since the statement is aThe diagonal rope (

) of an oblong (rectangle) produces both which the flank (*akṣṇayā-rajju**pārśvamāni*) and the horizontal () <ropes> produce separately."*tiryaṇmānī*^{ [3] }*sūtra*, it is necessarily compressed and what the ropes*produce*is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.^{ [3] }

- ↑ Mathematics Department, University of British Columbia,
*The Babylonian tablet Plimpton 322*. - ↑ Three positive integers form a
*primitive*Pythagorean triple if and if the highest common factor of is 1. In the particular Plimpton322 example, this means that and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see Plimpton 322 for details. - 1 2 ( Hayashi 2005 , p. 363)

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

The **history of mathematics** deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.

**Hindu astrology**, also called **Indian astrology**, **Jyotisha** or **Jyotishya**, and more recently **Vedic astrology**, is the traditional Hindu system of astrology. It is one of the six auxiliary disciplines in Hinduism that is connected with the study of the Vedas.

* Vedic Mathematics* is a book written by the Indian monk Bharati Krishna Tirtha, and first published in 1965. It contains a list of mathematical techniques, which were falsely claimed to have been retrieved from the Vedas and to contain advanced mathematical knowledge.

Acharya **Pingala** was an ancient Indian poet and mathematician, and the author of the * Chandaḥśāstra*, the earliest known treatise on Sanskrit prosody.

**Mahāvīra** was a 9th-century Jain mathematician possibly born in Mysore, in India. He authored *Gaṇita-sāra-saṅgraha* or the Compendium on the gist of Mathematics in 850 AD. He was patronised by the Rashtrakuta king Amoghavarsha. He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics. He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems. He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle. Mahāvīra's eminence spread throughout South India and his books proved inspirational to other mathematicians in Southern India. It was translated into the Telugu language by Pavuluri Mallana as *Saara Sangraha Ganitamu*.

The ** Baudhāyana sūtras** are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. They belong to the

**Greek mathematics** refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly attested from the late 7th century BC to the 6th century AD, around the shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire region, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language. The development of mathematics as a theoretical discipline and the use of proofs is an important difference between Greek mathematics and those of preceding civilizations.

**Indian mathematics** emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, and Varāhamihira. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made significant early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.

The * Shulva Sutras* or

**Johan Frederik** "**Frits**" **Staal** was the department founder and Emeritus Professor of Philosophy and South/Southeast Asian Studies at the University of California, Berkeley. Staal specialized in the study of Vedic ritual and mantras, and the scientific exploration of ritual and mysticism. He was also a scholar of Greek and Indian logic and philosophy and Sanskrit grammar.

**Śrauta** is a Sanskrit word that means "belonging to śruti", that is, anything based on the Vedas of Hinduism. It is an adjective and prefix for texts, ceremonies or person associated with śruti. The term, for example, refers to Brahmins who specialise in the *śruti* corpus of texts, and Śrauta Brahmin traditions in modern times can be seen in Kerala and Coastal Andhra.

The **Kerala school of astronomy and mathematics** or the **Kerala school** was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called *Tantrasangraha*, and again in a commentary on this work, called *Tantrasangraha-vakhya*, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work *Yuktibhasa*, written in Malayalam, by Jyesthadeva, and also in a commentary on *Tantrasangraha*.

**Kalpa** means "proper, fit" and is one of the six disciplines of the Vedānga, or ancillary science connected with the Vedas – the scriptures of Hinduism. This field of study is focused on the procedures and ceremonies associated with Vedic ritual practice.

**Pāṇini** was a Sanskrit philologist, grammarian, and revered scholar in ancient India, variously dated between the 6th and 4th century BCE.

**Geometry** is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a *geometer*. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

The **Bhūtasaṃkhyā system** is a method of recording numbers in Sanskrit using common nouns having connotations of numerical values. The method was introduced already in astronomical texts in antiquity, but it was expanded and developed during the medieval period. A kind of rebus system, bhūtasaṃkhyā has also been called the "concrete number notation".

**T. A. Sarasvati Amma** was a scholar born in Cherpulassery, Palakkad District, Kerala, India. She has contributed to the fields of history of Mathematics and Sanskrit, through her work on *Geometry of ancient and medieval India*.

**Bibhutibhushan Datta** was a historian of Indian mathematics.

* Mathematics in India: 500 BCE–1800 CE* is a monograph about the history of Indian mathematics. It was written by American historian of mathematics Kim Plofker, and published in 2009 by the Princeton University Press. The Basic Library List Committee of the Mathematical Association of America has classified the book as essential for undergraduate mathematics libraries, their highest rating.

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