Ancient Indian Astronomy, and the Concept of Zero

 

Deepali Gaskell and Dr Peter Ford MBE 

 

30 November 2018

 

Georgian Bath and Commerce, Industry, Astronomy and Mathematics 

The second half of the eighteenth century was a time of great economic and social change in the world.  This was partly brought about by the rapid rise in trade such as with the increased activity of the British East India Company which had been granted a charter by Queen Elizabeth I in 1600 to break the Dutch stranglehold in sea trade to the East.

 

The Georgian era saw developments in engineering and technology that, in turn, produced the Industrial Revolution and also transformed the City of Bath into a fashionable social milieu for people from all over the country.   Many came to the city to “take to the waters” and to get the latest gossip, as well as to exchange ideas on science, the arts and current events.  Architecture in Bath flourished during this period with the building of gems such as the Royal Crescent, the Circus, Lansdown Crescent and Camden Crescent.  Bath was a melting pot for ideas from all the diverse people present in Bath at the time. The Bath Chronicle founded in 1760 carries interesting snippets on the many personalities visiting Bath. They ranged from senior members of the British East India Company and scholars such as William Jones who translated from the Sanskrit texts for the East India Company Researches, to the aristocracy and people connected with the army and navy. Some also came to Bath to progress their careers and interests, such as Edmund Rack and people from Hanover such as William Herschel.

 

It is of interest that William Jones of the East India Company arrived in Bath in 1777, and wrote of his experience of Bath in a lively and revealing letter to Viscount Althorp.  Francis Wilford, the other principal contributor to the Researches, was born in Hanover.  He had joined the East India Company initially in 1781, and became a member of the Sanskrit Scholars and Orientalists Circle that worked with William Jones and the Asiatick Society of Bengal.

 

William Herschel was also from Hanover (like George 1st of England who came to England in 1714 to ascend the British throne). Herschel had taken up observing the night sky after he met Dr William Watson, a physician and one-time mayor of Bath, whose interests included looking at the night sky through a telescope. Herschel developed a fascination for this and set about building his own telescope, producing some of the finest examples of the day.  It was through his observations of the northern hemisphere that he discovered Uranus -the first planet to be discovered since antiquity - on March 13th 1781.

 

Banking was a growing industry at this time, with the creation of the Hoare’s, Childs and Barings banks, all supporting the East India Company in its drive for trade and industry.  The Hoare family was based at Stourhead in Wiltshire.  Francis Baring the founder of Barings Bank, was also friend and mentor to the Marquis of Lansdowne who lived in Bowood House in nearby Calne.  It was here in Calne that a new gas which was found to support life now called Oxygen was discovered in 1775 by Joseph Priestley, tutor to the sons of the first Marquis.  He was a member of the Bath Philosophical Society, which in 1824 evolved into the Bath Royal Literary and Scientific Institution (BRLSI) when the third Marquis of Lansdowne became its first director (See: BRLSI publication, ‘Bath and the Rise of Science’).

 

These bankers would have mingled with the up-and-coming in commerce and others who all shared ideas here in Bath.  The presence in Bath of these influential people in the world of Banking and Commerce and from the East India Company all went to create an ambience for discussion and exchange.

 

Into the Melting Pot of Bath

Some of the people who visited Bath in the second half of the 18th C would have had a connection with India at the same time as the East India Company was entering a period of intensified activity in India. This would have brought people who came to Bath in contact with the ideas generated by the activities of the East India Company.  This included an awareness of the Astronomy practised in Ancient India and the invention there of the system of numbers using the Zero.

 

Amongst those present in Bath at the time was Robert Clive, ‘Clive of India’, who had laid the foundation of British presence in India when he won the Battle of Plassey in 1757.  He lived in Bath at the Circus from 1767.  Warren Hastings, appointed the first Governor General of India by the British East India Company in 1773, had a great respect for the ancient Indian scriptures.  He also had a connection with Bath, and was mentioned in the Bath Chronicle of June 8, 1790.  Hastings helped establish the East India Company more firmly ‘by looking back to the earliest precedents possible’ and promoting the foundation of the Asiatik ResearchesIn 1784 by the Orientalist Scholar William Jones.  This became a storehouse of knowledge about India.  The volumes are in the collection at the Stourhead Library where we came across them.  Henry Hoare the Magnificent who created the garden at Stourhead had married Susan Colt, who was born in Calcutta and was the daughter of an East India Company man.

 

Fig 1: Bath Chronicle and Weekly Gazette 24th Jan 1799
(Courtesy of the Bath Record Office: Archives and local studies)

 

Publication of the researches and the notices in the Bath Chronicle of the time also contributed to the ambience for discussion and exchange in Bath.  There would have been, among the participants of the discussions at the Bath Philosophical Society awareness of the Researches Hastings commissioned, and these would have been discussed at the meetings.  The melting pot of Bath therefore provided an ideal venue for enquiring minds to exchange ideas generated by the East India Company’s researches, including an interest in Astronomy.  This helped to propel the Enlightenment Movement leading to discoveries and inventions that revolutionised the world.

 

Beginnings of Modern Mathematics Revealed

 

  

 

Fig 2: Extracts from the Researches 
© Reproduced by kind permission of National Trust

 

Hastings had been prompted to institute the Researches when the East India Company as it deepened its commercial expansion into Bengal came across the system of accounting used by tradesmen there.  They found the system used there to be much simpler and more logical than that which they had been using.  It included the accounting for debts, which was something new.   This discovery had led to the adoption in the 1770s by the Company of the Indian system of accounting.  This is what is in use today in modern accounting practice.

 

Beginnings of Modern Mathematics in Ancient Indian Astronomy

Modern mathematics as we know it today originated in ancient India, as was revealed in these Researches conducted in the 1770s. The translations from the ancient Indian texts, the Vedas, showed that modern mathematics developed from the number system invented by astronomers in ancient India based on the Zero, a new the concept at the time, to facilitate the calculation of Time which involved both very large and very small numbers.

 

The astronomers had observed that all movements by the astral bodies that were found to be present in the heavens took place in cycles.  This then suggested that all things moving in circular or elliptical orbits had their own independent focal points.  The heavenly bodies - the sun, moon and stars and the earth, our habitat - were all observed to move in periodic motions. These observations were made high up in the Himalayan Mountains where Indian civilisation has been based since antiquity.  This led to the sages or astronomers asking deep questions about the nature of existence and what constituted and was at the core of all observable matter.  These questions led to the invention of sophisticated systems of measuring and computing Time – the time that it took for all matter to become observable from the beginning of existence. This system of measurement used the symbol we call the Zero, denoted by a space within a circle.

 

[Some other ancient civilisations have been credited with having invented the zero, such as the Mesopotamians in the 7th century BC.  There would have been some cultural interchange at different time periods between adjacent cultural groups, each with its own distinct linguistic and cultural features. However the concept of the zero was not developed by the other cultures. The zero was not used as a number or in any calculations by these cultures.  There is evidence that Pythagoras visited India in the 3rd C BC where he came across the principles of geometry being used by the astronomers there.]

 

This system used the 10 digits (fingers), with the numbers 1-9 assigned to the first 9 digits and for the tenth digit it introduced the Shunya(now called zero) which was used with the number one to indicate the end of the first nine and the beginning of the next series of ten numbers.

 

The use of the Shunya or Zero made it possible to perform calculations using:

  1. the 10 digits in repeated sequences of 10, which allowed the calculation of very large and very small numbers in powers of 10 in what we now know as the Decimal system (based on the Sanskrit word ‘Dus’ meaning ten).
  2. using this place holder Zero or Nothing with 1, thereby creating the basis of the binary system which is at the heart of modern Information Technology.

 

The origin of the Zero from the concept of the origin of matter

In considering the concept of the origin of matter, the investigations the sages made indicated that matter, when broken into smaller and smaller particles, ultimately approached a state of there being ‘an empty space’ - Shunya or ‘Nothing’.  This was denoted by a circle - a space within a boundary.

 

The formation of matter was understood to take place over Time which allowed for the particles to coalesce.  Matter was therefore seen to be a function of time. In effect, the state of complete Shunya or ‘Nothing’ (Zero) could be approached but never quite reached.  The circle also represented the motions that were observed, which brought into focus that there would be a core or centre. The sages calculated the time taken by particles to coalesce into matter and assume more complex forms including the heavenly bodies. The material world, or form (Prakriti or Nature in Sanskrit), was seen to be in fact part of a cycle of creation, destruction and balance (the trinity of forces); and that it was minute particles spinning round a centre, as were all the celestial bodies, that was at the core of all form.   It was thought that all perceptible material form was ephemeral, illusion or Maya- and was in fact ultimately a collection of much smaller entities, which were in turn made up of a Force or Shakti or which is now understood as Energy.  Each anu (or atom in today’s language), was in turn made up of sub-particles, and eventually of the param anuor the ‘ultimate small particle’ all held together by a Force, which was at the core of all existence.  The concept of these cosmic cycles is central to the philosophy of the Vedas and hence the word chakra (wheel or cycle) came into use to describe it.

 

Fig 3: This Chakra is seen at the centre of the Indian flag.

 

 

The Concept of the Zero, and Measurement of Time in Powers of 10

The use of the Zero and the 10-based system (or the Decimal system based on the Sanskrit word Dus or ten) enabled calculating in positive powers of ten up towards infinity and in negative powers of ten towards zero. This had been until the time of Fibonacci at the beginning of the 13th century a mystery to the Arabs.

 

Fig 4: Time in ascending and descending powers of 10 for large calculations

Just to understand the scale of the calculations, in these charts in fig 4, the measurements start with the formation of matter with a Truti, roughly 1/1600 of a second.  This leads on to several ages and cosmic periods, called the cycles of Bramha (the Creator), relating to the Four ages or Yugas which make a Kalpa, Kalpa being equal to 4.32 billion solar years.

The cycle then reverts to the beginning and starts again.

Division of time came down to measuring the formation of matter from the state of ‘Nothing’ or Shunya; 2 elementary particles making one double particle; 3 double particles making one hexaparticle, the time taken being a Truti (10 to the power minus 7 solar years) the time taken for the integration of 18 elementary particles. 

This has resonance with modern physics in understanding the nature of matter.

 

 

Passage to the West

Further research by us confirmed that the accounting system that the East India Company had adopted and that which is in use today came from India, as did the system of numbers and the concept of the zero which was originally only brought to the West as recently as the 13th C.

 

In both cases, it was the need to help calculate the value of wealth being generated through trade that gave rise to the requirement for a systematic method of calculating.  This is what happened with the East India Company, which was a trading operation, as was the case in the 13th C when another trader - a Venetian from a merchant family, Leonardo of Pisa, better known as Fibonacci, looked into the Persian scholar Al Khwarizmi’s Arabic translation from the Sanskrit, called Kitab al Fuzul F’il Hisab al Hindi(literally translated, ‘Book of Paraphernalia of Accounting of Hindi (or the Hindus)’.  The concept first came to the West via various translations into Arabic with slightly different titles from the original Sanskrit texts.  They were then subsequently translated into Latin.

 

The Arab world that was also engaged largely in Trade, had picked up the Indian system of numbers 1-9 and the zero, but had not been using the zero in accounting as they had not understood the function of the zero.  The system in place until the 13th C was a tally system using the digits 1-9, which was known in the Mediterranean region and the West as the Hindu-Arab numbers because the Arabs had taken the numbers from India.   These numbers were helpful in tallying up, using the Abacus, to add and subtract, and the values were then represented back in Roman letters. The Roman system of notation was descriptive but difficult to use in complex calculations.

 

Commerce, Banking and Trade

So it is that, just as the modern accounting system was a result of Warren Hastings encountering the sophisticated one followed in Bengal, so also in Fibonacci’s time it was the commercial connection that had led to the discovery in 1202 by Fibonacci, of the Indian system of numbers using the Zero.

 

Fibonacci’s deciphering in 1202 of the use of the Zero as a number helped keep account of the growing trade in Italy, which at the time consisted of several principalities engaged in trade and in close contact with the Arabs who dominated the trade in the Mediterranean region. Fibonacci’s work was subsequently overwritten by successors who used the new technology of printing to further the work under their own names, omitting to mention Fibonacci, and describing the numbers as Hindu-Arabic.  Hence few records exist of Fibonacci’s life and how he deciphered the Zero before the arrival of printing in Europe.  With the advent of printing hand-written manuscript texts were superseded by printed ones, and Fibonacci’s contributions were passed over.  The first printed text in Italy was the Aritmetica di Treviso in December 1478.  In Felipo Calandri’s Pitagora Aritmetice Introductor, printed in Florence in 1491, references to Fibonacci’s book Liber Abacci were omitted, although it would have been the initial source for most of the material.

 

Until Fibonacci was able to apply the Indian system of numbers using the Zero and the base of 10 in an organised way for commercial use by traders, traders in the West recorded their numerical data using Roman numerals and performed calculations in a very elaborate way with the widely used tally systems using ten fingers or a mechanical abacus.  It was only when Fibonacci understood the Hindu system of using 10 numerals including the Zero that traders were able to value their goods to calculate the increasing trade using the decimal system from India.    It was also for the first time that debts (negative figures) were properly accounted for using the Indian system.  Fibonacci also introduced the term Abacus, to denote the method of calculating using what was then referred to as the Hindu-Arabic number system, 1-9.   Eventually in the 16th Century, 400 years after Fibonacci, the mathematical systems from India came into use in Europe introduced by the Jesuits in their syllabuses, around 1570, due mainly to the work of Christoph Clavius.

 

Fig 5: The Mediterranean region, sphere of Arab trade and influence

 

Fibonacci’s merchant father would have encouraged him to learn about the mathematics that the Arab tradesmen were using so as to help in his career development.  This would have brought him into contact with the Hindu number system being used by the Arabs, one source having been identified as mentioned above, as the ‘Book of Chapters on Hindu Arithmetic’, or Kitab al Fusul fi-il Hisab al-Hindi (roughly translated as Book of use of Hindi calculations) written by Abu al Hasan Ahmed in Ibrahim al Uqlidisi in Damascus in 952 AD.  This was translated as Algebra into Latin by Robert of Chester in 1145, and may also have been by Adelard of Bath as Arithmetic around the same time.  Fibonacci’s book Liber Abacci would have been based on his reading of all the above.  It was divided into 25 chapters, starting with the recognition of the Indian figures and how they were written, through to the concept of whole numbers, integers and functions that were performed by them, to applying them in valuing merchandise and to finding square and cubic roots, geometry and algebra to arrive at solutions to mathematical problems.

 

There are some interesting observations here about the translations from the Sanskrit.  The trigonometrical sine function that originated in India was translated into Arabic without the use of vowels.  When Robert of Chester translated this into Latin, not knowing the Sanskrit origin, he supplied the vowels which resulted in a word meaning bay or inlet, which in Latin is sinus, hence the word sine now used for that function.  Fibonacci was helped in his deciphering the system based on the zero by his connecting with series prevalent in Nature, such as in the arrangement of petals in wild roses.  His number series, generated by each successive number being the sum of the two previous numbers, had been first mentioned in the Chandahshastra (The Art of Prosody) written by the Sanskrit grammarian Pingala around the 4th BC.

 

Although the number system is often credited to Al Khwarizmi, he had, in the 9th C, translated into Arabic from the Sanskrit a method of accounting for the use of traders, and had not made use of the Zero to enable calculations in powers of 10 as the Indian astronomers had been doing for a very long time. Hence his description Al Jabr.  He called his translation Kitab Al Jabr Wa I Mukablaor ‘Book on (matters of) Business’, which had led to ‘Algebra’, featuring unknowns (Referred to today as Al Khwarizmi’s Algebra).

 

The Church in Europe had texts preserved in the archives of Spain and Italy of translations from the Arabic into Latin. Adelard of Bath had also attempted to understand the system in the 12th C when he translated into Latin a work he called Algorithmi Numero Indorum, or, ‘Al Khwarizmi on the ‘Hindu Art of Reckoning’.

 

Fig 6: Extract from the first chapter of Dixit Algorizmi or ‘So Said Al Khwarizmi’

 

Dixit Algorizmi

Adelard of Bath called his translation from Al Khwarizmi’s Arabic into Latin, ‘Algoritmi de numero indorum’  (or, ‘Al Khwarizmi on the Hindu art of Reckoning’).  This was also referred to as ‘Dixit Algoritmi de numero Indorum’, or ‘So said Al Khwarizmi on Hindu Calculations’.

 

There was also a translation by an Italian Gherardo Cremona around 1150, who called it Liber maumeti filii moysi alchoarismide algebra et al muchabala, which was his understanding of the aforementioned translation by Al Khwarizmi.  This was probably used by Fibonacci along with Al Khwarizmi’s Arabic translations when he finally ‘cracked’ the Zero and brought it to the West.

 

In Fibonacci’s time what is now Italy was a collection of independent states each with its own hierarchy and ruled by the Holy Roman Emperor.  The currency was denarii(coins), 12 denaris made a solidus(shilling), and twenty solidi made a Libra(pound), as in Britain until the decimalisation system came in.  Adelard of Bath’s translation in 1150s, Venice and Pisa, trade, as well as Fibonacci’s translation in 1202, all led to the adoption of the Indian system of numbers based on the Zero

 

Appendix
Indian Mathematicians in more modern times – Brahmagupta

The brilliant conceptual leap to include zero as a number in its own right, rather than merely as a placeholder a blank or empty space within a number as it had been treated until that time, was made by the Indian mathematician and astronomer Brahmagupta (598–668 AD) who lived most of his life in Bhillamala (modern Bhinmal) and later headed the astronomical observatory at Ujjain in central India.

 

Brahmagupta lived in more modern times as compared to earlier Vedic periods prior to 6,000 BC. He is considered to be the father of algebra, geometry and trigonometry as he was the first to formulate the solving of quadratic equations.   He also described gravity as a universal force, one millennium before Newton expounded his theories on gravitation, and he explained how in the universe bodies do not travel in a straight line but in curved circular motion, and he headed the most famous university in the world at the time, in Ujjain.  Brahamgupta’s main work was the ‘Brāhma-sphuṭa-siddhānta’ ("Doctrine from Brahma") which he wrote aged 30 in 628 AD. Much of the book is about astronomy, but it is also the first book to provide rules for arithmetic operations that involve zero and negative numbers.  This was a theoretical treatise which was followed by another more practical manual which he called Khaṇḍakhādyaka ("edible bite") in 665 AD for the use of his students.  Bramhagupta’s work can be seen as an extension of the original work found in the Vedas, which needed to be made more understandable by the less scholarly and available for application in everyday use. Traditionally, the sacrosanct use of mathematics in earlier times was to understand and calculate accurately the times for and locations of astral bodies for the performance of ceremonies.  It gave rise to the highly developed astronomy and geometry, algebra and trigonometry found in the Vedas.  

 

Vedangashad developed as ancillary studies for the Vedas from around the end of the 1st millennium BC, a few millennia after the Vedic texts were composed.   This was because by then the language of the Vedas, which were composed in the formal language Sanskrit, was no longer a spoken language and the knowledge in these texts needed to be made more easily understood and interpreted for students.  Many scholars and astronomers, over the following thousand years carried on this work, amongst the most notable being Brahmagupta.  He was a somewhat controversial figure because he simplified the mathematics which was considered to be the preserve of the Brahmins or the priestly class.  In chapter 18 of the Brāhma sphuṭa siddhāntahe recorded the arithmetic rules for handling positive numbers (“properties”) and negative numbers (“debts”).  During the 7th century AD negative numbers were used in India to represent debts:

 

  • A debt minus zero is a debt.
  • A property minus zero is a property.
  • Zero minus zero is a zero.
  • A debt subtracted from zero is a property. 
  • A property subtracted from zero is a debt.
  • Zero multiplied by a debt or a property or a zero is zero.
  • The product of a debt and a property is a debt, of two debts a property, and of two properties a property.
  • The Brāhma-phuṭa-siddhāntacontains many contributions to algebra and geometry and trigonometry.  For example Brahmagupta found the sum of the squares of the first n natural numbers to be n(n+1)(2n+1)⁄6 and the sum of the cubes of the first n natural numbers as (n(n + 1)⁄2)².

 

Bibliography

We found the material in the aforementioned journals in the Stourhead library very helpful, and have made use of both collections in writing this article, as well as from sources in the archives at the Bath Guildhall, and the BRLSI. We would like to thank Tony Symes, convener for the lecture, for all his help in the preparation of this summary.

  1. The Researches with translations from the Vedas
  2. Bath and the Rise of Science (BRLSI publication) 
  3. Guidebook, Bowood House
  4. Edmund Rack (BRLSI publication)
  5. Adelard of Bath (BRLSI publication) 
  6. The British East India Company Journals  (in BRLSI archival collection)
  7. ‘The Man of Numbers’ by Keith Devlin
  8. The Bath Records office, Guildhall Bath for editions of the Bath Chronicle from the 1770’s