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.

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

Some of the famous astronomers and mathematicians of this period are


 * Āryabhaṭa
 * Varāhamihira- a multi-faceted genius of the 6th century CE lived in Ujjain. He wrote Pañca-siddhāntikā and Bṛhatsaṃhitā. Pañca-siddhāntikā is a work on astronomy and Bṛhatsaṃhitā  is an encyclopedia on natural phenomena.
 * Brahmagupta
 * Bhāskara I
 * Bhāskara II

The Medieval Period (1200 CE to 1750 CE)
In this medieval period from 13th to 18th century CE, many commentaries were written on the earlier texts. A great school of mathematics and astronomy came to flourish in Kerala.


 * Nārāyaṇa Paṇḍita was a famous mathematician of the 14th century CE. His work, the Gaṇitakaumudī was composed in 1356 CE. It is nearly three times the size of Līlavatī with many more results and examples. For instance, it contains a separate chapter called Bhadragaṇita which deals with the mathematics of magic squares. The topic of combinatorics which deals with selection and arrangement of objects (permutations and combinations) is also extensively dealt with in this work.
 * Gaņeśa Daivajña who lived in the first half of 16th century CE was a distinguished astronomer who hailed from Nandigrāma in Konkan region. His work, the Buddhivilāsinīl is considered to be one of the finest commentaries on the Līlāvatīl as it gives detailed upapatti (proofs). He also composed a famous astronomical treatise, the Grahalāghava.
 * Krşņa Daivajña of the 16th century CE wrote the Bījapallava, a commentary on Bījagaņita which also contains many. upapattis (proofs).
 * Mādhava
 * Parameśvara
 * Nīlakaṇṭha Somasutvan
 * Śaṅkaravāriyar was a student Nīlakaṅṭha Somasutvan. He lived in the 16th century CE. His commentary on the Līlāvatī, the Kriyākramakarī is very famous.
 * Jyeṣṭadeva. a junior colleague of Nīlakaṅṭha Somasutvan, wrote the famous work Yuktibhāṣā in Malayalam language. Written around 1530 CE, this book presents detailed proofs of all the contributions of Mādhava and Nīlakaṅṭha in the fields of astronomy and mathematics. It is hailed as the first textbook of Calculus.
 * Putumaṇa somayājī authored an astronomical work, the Karaṇapaddhati in the sixteenth century C.E.
 * Śaṅkaravarman, the author of Sadratnamālā flourished in the first half of nineteenth century C.E.
 * Srinivasa Ramanujan