Development of Mathematics

Indian mathematics has a hoary past and the history of ancient Indian mathematics dates back to several millennia. We shall discuss the history of Indian mathematics in terms of the following broad periodization:

1. The Ancient period (Prior to 600 BCE)
2. The Early Classical period (600 BCE to 400 CE)
3. The Later Classical period (400 CE to 1200 CE)
4. The Medieval period (1200 CE to 1750 CE)

There have been significant contributions to mathematics by Indians in the modern period (post 1750 CE) also. The legendary Indian mathematician Srinivasa Ramanujan (1887-1920 CE) and many other stalwarts have contributed significantly to the world of mathematics in the 20th and 21st century.

The Ancient Period (Prior to 600 BCE)

The oldest available work of mankind is the ṛg-veda. It contains 1,028 sūktas with 10,462 mantras. These mantras were compiled in the millennia prior to 2000 BCE. Historians call it the Vedic period. According to historians, the ancient period is the period prior to 600 BCE. In this period, the Vedas and the canonical texts of Vedāngas were composed.

There are four Vedas - ṛg, Yajur, Sāma and Atharva. The Vedas are composed of mantras. There are several mathematical aspects contained in these Vedic mantras. Some of them are listed below.

• Enumeration of numbers in powers of 10 till 10¹⁹ (Taīttirīya-saṃhitā, 7.2.20)
• Decimal place value nomenclature for numbers.
• Odd number series (Taīttirīya-saṃhitā, 7.2.11)
• Even number series (Taīttirīya-saṃhitā, 7.2.13)
• Arithmetic Progression with common difference 4, 5, 10, 20 and 100 (Taīttirīya-saṃhitā, 7.2.15-19)
• Factors and non-factors (śatapatha-brāhmaṇa, 10.24.2.1-20)
• Sum of series (śatapatha-brāhmaṇa, 10.5.4)
• Multiplication operation (ṛg-veda, 8.19.37).
• Geometric progression (Pañcaviṃśati-brāhmaṇa, 18.3)

To understand the import of the Vedic mantras, six supporting disciplines evolved. They are

1. Śikṣā which deals with classification and pronunciation of sounds (phonetics)
2. Vyākaraṇa which deals with grammar.
3. Chandas which discuss prosody or the study of metres.
4. Kalpa which discuss performance of yajñas and construction of altars and other accessories.
5. Nirukta which deals with etymology of words and their meanings.
6. Jyotiṣa which is the science of astronomy.

These six are called as Vedāngas.

The body of literature called śulbasūtras was composed in this period. They form a part of the Kalpa Vedānga. The Samskrit word śulba means 'a rope'. The word sūtra denotes a short pithy rule or statement. The śulbasūtras deal with various aspects of geometry which are involved in the construction of altars. Using rope (śulba or rajju) and stick or gnomon (śaṅku), many exact and approximate constructions are stated in these texts. Currently, we know eight śulbasūtras named after their authors. Among them, the four popular ones are Baudhāyana-śulbasūtra , āpastamba-śulbasūtra , Kātyāyana-śulbasūtra and Mānava-śulbasūtra . Historians uer their period to be before 800 BCE. śulbasūtras have been regarded as the most ancient texts of geometry. We find a precise formulation of what came to be later known as the Pythogoras theorem already in the śulbasūtras.

The Early Classical Period (600 BCE to 400 CE)

The Early Classical period starts from 600 BCE. The period when the doctrines of Buddhism and Jainism originated is generally dated by historians to be around 500 BCE. The science of mathematics is also popular in the Buddhist and Jain traditions The Buddhists consider mathematics to be a noble art. They call it saṅkhyāna- the science of numbers. The Jains consider the art of counting to be an essential part of their philosophical education. They classify their sacred literature into four divisions.They are Dravyānuyoga, Caraṇa-karaṇānuyoga , Gaṇitānuyoga and Dharmakathānuyoga. Gaṇitānuyoga consists of arithmetic and astronomy. Some of the Jain works, important from the view of mathematics, are Sūrya-prajñapti, Candra prajñapti, Sthānāṅga-sūtra, Bhagavatī-sūtra, Tattvārthādhigama-sūtra and Anuyogadvāra-sūtra.

Piṅgala who composed Chandassūtra lived in the 3rd century BCE. In this seminal text dealing with chandas (metres of Samskrit poetry), he developed various as algorithms related to permutations and combinations and binary representation of numbers. His Meru-Prastāra is the same as what is currently referred to as the Pascal's triangle.

The ancient astronomical siddhāntas, including the original Vaśiṣṭha, Paitāmaha and Sūrya-siddhānta belong to this period. Another important work which is attributed to this period is the Bakhshālī Manuscript. The following is the story of its discovery in the 19th century. Bakhshālī is the name of a village in what was then the North-West Frontier Province of British India. It is near Peshawar in Khyber Pakhtunkhwa province, in present-day Pakistan. A manuscript of a mathematical work was discovered in this village in 1881 CE. It was accidentally found by a farmer both while digging a ruined stone enclosure of his house. Since its author is not known, it is called the Bakhshāli manuscript. Historians are unable to come to a definitive understanding of its exact period. The estimates of the period of the manuscripts based on different dating methods (even based on carbon dating) vary from 1st century CE to 7th century CE. The Bakhshālī Manuscript has a large number of illustrative problems with solutions covering arithmetic, commercial mathematics and some algebra as well as geometry.

The Later Classical Period (400 CE to 1200 CE)

Later Classical period is considered by scholars to be the 'Golden Age of Indian Mathematics'. Many great mathematicians flourished during this period. Indian mathematical contributions and discoveries were transmitted to many other regions of the world during this period. This golden period starts with the famous astronomer Āryabhaṭa and culminates in Bhāskara II, the author of the famous Līlāvatī.

in traditions -Khåna - the sential part of our divisions.

thel

and

omy. Some of

apti, Candra2-sútra and

vesical period is normally considered to be 600 BCE to 500 CE since that is

iod of ancient Greek and Roman civilization. For Europeans, this is indeed lassical period. For them, the medieval period (500 CE to 1200 CE) was not a

ous period. On the other hand, in India, there is continuity from 500 BCE to 1200 CE in political, social, philosophical and cultural domain. Hence we refer to the period (400 CE to 1200 CE) as the later classical period.

We shall briefly discuss the works of some famous astronomers and mathematicians of this period.

BCE. In this oped various esentation of ed to as the

al Vasistha, twork which

the story of hat was then ar in Khyber nathematical buta farmer

wn, it finitive

Āryabhata was born in 476 CE