Ganitha

Introduction

Ganitha emerged in Bharat from 3000 BC until the end of the 18th century. In the classical period of Ganitha which is 500 BC to 1200 AD, important contributions were made by scholars like Aryabhata, Bhaskara, Brahmagupta, Bhaskaracharya, and Varahamira. The decimal number system in use today was first recorded in Bharathiya Ganitha. Ganithagnas (Bharathiya Mathematician) made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, Trigonometry and Spherical Trigonometry was further advanced in India and in particular, the modern definitions of Jya Sine, Ko-Jya Cosine, Utkrama-Jya Versine and Othkrama-Jya Inverse Sine were developed in Bharat. 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.

This workshop deals with the historical aspects of Ganitha. Anyhow if anybody is interested they could do further learning with some experts. 

Jyothishya not just Astrology

Jyothishya Shastra deals with the apparent movement of the 9 Grahas (heavenly bodies), through the 12 Rashis (not just zodiac constellations) and the 27 Nakshatras (asterisms and not stars). It is Cosmology, Astronomy and lastly Astrology.

Jyothishya: The general understanding of this Shastra especially amongst the so-called “Liberals” and “Scientific Thinkers” is that Jyothisha is superstition. Contrary to this Jyothisha Shastra deals with the apparent movement of the 9 Grahas (not planets), through the 12 Rashis (not just zodiac constellations) and the 27 Nakshatras (asterisms and not stars). Bharathiya Jyothishya (Indian astronomy) has a long history stretching from pre-historic to modern times. Some of the earliest roots of Bharathiya Jyothishya (Indian astronomy) can be dated to the period of Indus Valley Civilization or earlier. Jyothishya Astronomy later developed as a discipline of Vedanga or one of the “auxiliary disciplines” associated with the study of the Vedas, dating 1500 BC or even before that. The oldest known text is the Vedanga Jyotishya, dated to 1500–1300 BC.

Jyothisha Shastra deals with the apparent movement of the 9 Grahas (not planets), through the 12 Rashis (not just zodiac constellations) and the 27 Nakshatras (asterisms and not stars). It deals with these positions and their effects of these on our life. And it also suggests that if these can affect your life you too can counter-affect them through your Sadhana. Jyothishya Shastra is Astronomy rather than Astrology. Jyothishya deals with celestial events that repeat only once in a few thousand years. These events are represented by Jataka (Birth chart) of a person which depends on the Date of Birth, Time of Birth and the Place of Birth. 

Panchanga

The representation of Kaala (time) is always with respect to the Desha (place) in the Bharathiya Panchanga. Hence it is called Desha-Kala Panchanga. Each of the Janapadas of Bharath had a different Panchanga according to their time zones. The time zone was decided up on by the sunrise and the sunset which in turn depended on the latitude and longitude of the Janapada.

Panchanga gave the following details: 1) Kalpa, 2) Manvantara, 3) Yuga, 4) Shaka, 5) Samvatsara, 6) Ayayana, 7) Rithu, 8) Maasa, 9) Paksha, then the Panchanga 1) Thithi, 2) Vara, 3) Nakshatra 4) Yoga and 5) Karana so if you count you will find 14 parameters being given to each day but the Modren Gregorian Calandar gives only Year, Month, Date and Day.

After the advent of the Gregorian calendar we have forgotten the Bharathiya Panchanga but still the Panchanga is not what it is as usually called as “Hindu Calendar” or “Hindu Almanac” these terms are used only for convenience but they don’t convey the actual meaning. Just for starters the Bharathiya Panchanga is based on the movement of both the Surya and Chandra (sun and the moon, Solo-Lunar or Luno-Solar) through the 12 Raashis (Zodiac signs) and the 27 Nakshaktras (the 27 asterisms) in the sky. The Panchanga also indicates the seasons and seasonal changes we have to adopt in our life. It also gives the planning for agriculture and festivals. Other than all this it also gives us insights into the Astronomical events like Eclipses, Equinoxes, Unusual Planetary Confluences, etc. I shall attempt to bring about the understanding of the same and also why we need to follow our Panchanga.

Calendar

A calendar is a system of organizing days for social, religious, commercial or administrative purposes. This is done by giving names to periods of time which are limited to usually Year, Month, Date and Day of the Week. A date is the designation of a single, specific day within such a system. A calendar is also a physical record (often paper) of such a system. A calendar can also mean a list of planned events, such as a court calendar or a partly or fully chronological list of documents, such as a calendar of wills.

Almanac

An almanac is an annual publication listing a set of events like weather forecasts, farmers’ planting dates, tide tables, and other tabular data often arranged according to the calendar for the forthcoming year. It includes information such as the rising and setting times of the Sun and Moon, dates of eclipses (Sometimes), hours of high and low tides, and some Abrahamic religious festivals. The set of events noted in an almanac may be tailored for a specific group of readers, such as farmers, sailors, or astronomers.

Panchanga

The representation of Kaala (time) is always with respect to the Desha (place) in the Bharathiya Panchanga. Hence it is called Desha-Kala Panchanga. Each of the Janapadas of Bharath had a different Panchanga according to their time zones. The time zone was decided up on by the sunrise and the sunset which in turn depended on the latitude and longitude of the Janapada.

The Panchanga gave the following details: 1) Kalpa, 2) Manvantara, 3) Yuga, 4) Shaka, 5) Samvatsara, 6) Ayayana, 7) Rithu, 8) Maasa, 9) Paksha, then the Panchanga 1) Thithi, 2) Vara, 3) Nakshatra 4) Yoga and 5) Karana so if you count you will find 14 parameters being given to each day. This was given and detailed with information like Rithu Purvanumana (weather forecasts), Krishi Prayojana (farmers’ planting and probable harvest dates), Taranga Anuman (tide tables hours of high and low tides), and other tabular data like the Suryodaya, Suryastha, Chandodaya Chandrastha (rising and setting times of the Sun and Moon), dates of Grahana (eclipses were always marked) along with the Nakshatras that got affected by the Eclipse, all 33 Parvas (festivals), Jatras (Temple Fares), Santha (Shandy) and many other things.

Shulaba Sutras Geometry and Trigonometry

They contain details of making various altars for rituals using various geometrical shapes. Hence they contain the whole knowledge of Geometry and Trigonometry.

Shulaba Sutras are texts which belong to the Shrouta tradition containing the procedures of various rituals. They are the first sources of knowledge of Bharathiya Ganitha (Indian mathematics) from the Vedic period.

The Shulaba Sutras are texts which belong to the Shrouta tradition containing the procedures of various rituals. They contain details of making various altars for rituals using various geometrical shapes. Hence they contain the whole knowledge of Geometry. The Shrauta Sutras, considered to be appendices to the Vedas, are the source for Shulba Sutras. They are the first sources of knowledge of Bharathiya Ganitha (Indian mathematics) from the Vedic period. The “Shrauthis” used to create unique Homa/Yagna Kundas (unique fire-altar shapes) which were associated with begetting unique gifts from the Devas Gods.

The most significant of these are 1) Baudhayana, 2) Apasthambha, 3) Manava, 4) Katyayana, the others like 5) Maitrayana, 6) Varaha, 7) Vadhula and 8) Hiranyakshin are some of the minor ones. Most of these contain the “Pythagoras Theorem” in various different sutras. These sutras also discuss about finding squares and square roots. The also contain theorems of Trigonometry.

The Shulaba Sutras from Shulaba meaning “string, cord, rope” are sutra texts belonging to the Shrauta Shastras which contains details about the Homa Kunda and the Yagjna Kunda making (Fire Altar) and containing geometry related to making them. The Baudhayana Sulba Sutra, the dates of which are given variously as between the 800 BC to 500 BC, contains a list of Pythagorean triples and a statement of the Pythagorean Theorem, both in the special case of the isosceles right triangle and in the general case, as does the Apastamba Sulba Sutra (c. 600 BC).

Bharathiya Ganitagjna

Baudhyana 600 BC in his Shulaba Sutras says “drighachatursrasyakshnaya rajjuh parshvamani triyagmani cha yatpthagbhute kuruthstadhubhuayam karoti” which means ‘A rope stretched along the length of the diagonal produces an area which the vertical and the horizontal sides make’. This is nothing but the Pythagoras Theorem.

Pingala 300 BC is credited with Chandass Shastra which presents the first known description of binary numerical system; he uses Laghu (Light) and Guru (Heavy) and later uses Shoonya 0 and Ekam 1. Here he ranks binary patterns starting at 1 (four 0000) as the first one and the nth pattern to n-1 with increasing positional value.

Baudhyana 600 BC in his Shulaba Sutras says “drighachatursrasyakshnaya rajjuh parshvamani triyagmani cha yatpthagbhute kuruthstadhubhuayam karoti” which means ‘A rope stretched along the length of the diagonal produces an area which the vertical and the horizontal sides make’. This is nothing but the Pythagoras Theorem. Pingala 300 BC is credited with Chandass Shastra which presents the first known description of binary numerical system; he uses Laghu (Light) and Guru (Heavy) and later uses Shoonya 0 and Ekam 1. Here he ranks binary patterns starting at 1 (four 0000) as the first one and the nth pattern to n-1 with increasing positional value. This is a workshop just to introduce the lives and works of the Bharathiya Ganitahgnas who lived from about 800 BC to the present. Here we get a peek into the life and works of greats like the following:

Vedic Period

1. Baudhayana 1800 BC
2. Katyayana 1500 BC
3. Pingala 1500 BC from the ancient period,

Classical Peroid  

1. Aryabhatta 476–550 AD,
2. Bhaskara-1 600–680 AD,
3. Brahmagupta 598–670 AD,
4. Bhaskaracharya 1114–1185 AD,
5. Varahamira 505–587 AD
6. Shridhara 650–850 AD
7. Mahavira 9th century AD

Medieval Period

1. Narayana Pandit
2. Madhava of Sangamagrama
3. Parameshvara (1360–1455), discovered drk-ganita, a mode of astronomy based on observations
4. Nilakantha Somayaji (1444–1545), mathematician and astronomer
5. Raghunatha Siromani (1475–1550), logician
6. Mahendra Suri (14th century)
7. Shankara Variyar (c. 1530)
8. Jyeshtadeva (1500–1610), author of Yuktibhāṣā
9. Achyuta Pisharati (1550–1621), mathematician and astronomer
10. Munishvara (17th century)
11. Kamalakara (1657)
12. Jagannatha Samrat (1730)