THE HEROES BEHIND THE ZERO

 The Vedas are having six angas. Jyothisha or mathematics is one important anga of the Vedas. An old sloka says that ‘like the crown for the peacock and the diamond for the cobra, mathematics adorn the Vedangas”. Our forefather attained proficiency in different branches of mathematics like arithmetic, beejaganita (algebra) geometry, trigonometry, spherical trigonometry and astronomy.

The important contribution of the ancient Indians is the invention of zero-the very basis for modern computer algorithms. In the words of G.B.Halsted “The importance of the creation of the zero mark can never be exaggerated. No single mathematical creation has been more potent for the general on go of intelligence and power”. The concept of zero, the concept of positional value system in numeration, the concept of Pi, the techniques of algebra square and cube roots quadratic equations have been clearly explained in ancient Indian texts.

The very idea of “Sunya” of ‘Nil” as symbolized by zero is a great leap in mathematics. The first reference to zero comes in Chanda Sastra of Pingala 200 BC and in Panini’s work. Later Brahmagupta in ‘Brahma Sputa Siddhanta’ mentions about the operations of Zero.

Zero is not just a number. It is a fundamental concept like the “Shunyata’ of Acharya Nagarjuna. It is a giant leap into abstraction. This opened up a new vista of negative numbers. It also made possible the representation of high value numbers with limited characters. Thus the system of abstract numbers, as distinct from representation of quantities, and the concept of infinity and ‘nil’ developed.

Pingala’s Chanda-Shastra describes the concept of Zero.




“In Gayatri Chandas one pada has six letters. When the number is halved it becomes three. Remove one from three and make it half to get one. Remove one from it, thus Zero is gotten”.

The concept of Infinity is also indicated in the famous verse of Rigveda.


Likewise with the place of Zero, the decimal system with number representing an absolute value and also a place value became a very important factor in mathematical development.

The Sindu-Saraswati civilization dating back third millenieum BC, scales and standardization of weights and measures were noteworthy features. The lengths were measured in scales representing 1.32 inches to the Inch Numerical System. During the Vedic Age, Vedics or altars for yajnas were constructed based on different norms. Baudhayana, Apastambha and Katyayana were outstanding mathematicians of this era. Sulba Sutras-mathematical texts explain the relationship between the sides and hypotenuse with right angle triangle, which is known as ‘Pythagoras Theorm”.

Brahmagupta made great contribution to the cyclic quadrilateral theories. In fact, it is know as ‘Brahmagupta’s quadrilateral’. He has also contributed to the solution of indeterminate equations of the second degree.

Aryabhatiya-bhasya of Bhaskara is a great mathematical treatise. Bhaskara was a maths teacher at Valabhi in Saurashitra in 9th century AD. Apart from this he also wrote Mahabhaskariya, Laghu Bhaskariya and Aryabhatiya Bhasya Bhaskara’s work dominated the field of mathematics. And were popular, especially in South India. In AD 1930, Bibhuti Bhusan Datta discovered the existence of two Bhaskaras in Indian Mathematical Tradition.

Arya Bhatiya Bhasya is a full exposition of the works of Aryabhata. In the commentary, bhaskara also refers to other mathematicians like Maskari, Mudgala, Purana and Putana. He gives an exposition “In the ganita Pada the Acharya has dealt with the subject to ganita by indications only, whereas in the Kalakriya pada and golagula pada, he has dealt with reckoning of time and spherical astronomy in greater detail. So by the word Ganita used by Aryabhata I, one must understand a ‘bit of mathematics . Otherwise, the subject of mathematics is vast.

There are eight Vyavaharas (determinations) viz, mistaka (mixture), Ksertra (plan figures) kkata (excavations) Citi (pile of bricks etc) Krakacika (saw problems) rasi (heaps of grain and Chaya (shadow). Of the Vyavaharika Ganita (practical or Commercial mathematics: pati Ganita which is in eight classes, there are four bijas (methods of analysis) namely Yavattavat (theory of simple equations), VargaVarga (theory of quadratic equations), Ghanaghana (theory of cubic equations) and Vishama (theory of equations)involving several unknowns). Rules and examples pertaining to each one of them have been compiled (in independent works) by the masters Maskari, Purana, Mudgala and others. How can that be stated by the Acarya (Aryabhata I) in a small works (the Aryabhatiya)?. So, we have rightly said a bit of mathematics”. The above statement by Bhaskara I goes to prove that there have been a number of textbooks in mathematics.

The most important achievement of Indian Mathematics is the development of decimal numeration and place value system. This can be called one of the most significant developments in the history of mankind.
In the words of Pierre Simpon De Laplace, “ It is India that gave the ingenious method of expressing all numbers by ten symbols, each symbol receiving a place value position, as well as an absolute value. We shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Appollinius”.

The yoga sutra Bhashya ( 150 AD )describes this.




“In the unit place the digit has the same value, in the 10th place 10 times the value and in 100th place 100 times the value, as a woman is called mother, daughter and sister”.


THE CONCEPT OF ROOTS:

Somewhere around 6th century BC the great Indian mathematicians Apastamba authored the Sulba Sutras. Here he gives the value of the square root to the sixth decimal places Apastamba’s subla sutra is divided into 6 sections, 21 chapters and 223 sutra. It gives the geometrical propositions for the construction of altars.

His mathematics was not merely theoretical; it was for practical applications. The general linear equation was also solved in the Apastamba Sutras.

While the plane geometry was developed through the necessity of designing sacrificial altars, alter algebra or Bija-ganitam developed Bija-Ganitam measures ‘other mathematics’. Thus it has been developed as a parallel system of computation. Another interpretation of ‘Bija” means root, implying that is the origin of mathematics. It is feasible due to the fact that Vedic literatures gives, in places, short ways of computation while Aryabhata is also credited with the first treatise on algebra Bhaskaras Siddhanta Siromani elaborated this.

In the area of Plane geometry Bhaskara gives several solutions. We can see some of them here:

1. PROBLEM OF AREA OF A SCALENE TRIANGLE.

What is the area of a scalene triangle whose sides are 15 and 13 units and base to 14 units? Bhaskara gives an algorithm which is, similar to the modern flow chart. Bhaskara’s method. Difference of the squares of the sides is to be obtained by multiplying their sum and differences. The difference thus obtained is to be divided by the base. The quotient to be added to or subtracted from the base. Half of these values are the parts of the base(made by the altitude) from these value the altitude is to be found and then the area of the given triangle (1/2 bh).

2. Value of p is first given in Aryabhatiya of Aryabhata I (5th century AD) Ganita-pada of Dasagatika in Aryabhatiya says.


(Hundred plus four multiplied by eight and added to sixty two thousand: this is the most approximate measure of the circumference of a circle whose diameter is twenty thousand). Aryabhata has used a complex algorithm to arrive at this.


3. Parameswara’s formula for the circum radius of a cyclic quadrilateral.

4. Trigonometric series for the tanX values. In modern mathematics, inverse tangent function is called Gregorian series after the 17th century Scottish mathematician James Gregory. In fact, the solution had been given nearly 2 centers earlier by Madhava in his algorithm.

5. The approximation formula for the third order Taylor series is given by Parameswara in 14th century in his commentary called Siddhanta depika on Govindaswamin’s Commentary on Mahabhaskariya.

RIG VEDIC ASTRONOMY

There are great references to the astronomical facts in the Rig Veda Sathapate Brahmana gives the names of 27 stars and 27 Upa-nakshatras. At the same time other texts like Taitriya, Kakthaka and Maitrayana. Samhitas give the name of 28 stars. Names of other constellations are also mentioned. Early reference is to a six days week called ‘Sadaha’. Later seven days week came into being; while some samhitas speak about six seasons, other speak of five, clubbing hemante and Sisira.

(Vasanta, Grishma, Varsha, Sarada, Hemanta, Sisira)

The seminal text here is Vedanga Jyotisa of Logadha around 1300 BC.

It was followed by Siddanta Astronomy broader in scope. Varahamih describes five systems of Siddhanta astronomy, Partamaha, Vasisha, Romaka, Pauhka and Saura-Shere are common points between Indian and other astronomical system. We
· The division of the Zodiac into 27 or 28 asterisms, commonly found among Indian, Chinese and Arab systems.
· The twelve fold division of the ecliptic circle into zodiacal signs
· The theory of epicycles
· Parallel systems of astrology
· Common names of planets in Indian and Greek System

Surya Siddhanta also talks about the spherical shape of the earth.

“As the earth is round every person considers himself at the top of the earth where he or she is standing. So downward direction is towards the centre of the earth for everyone”.

Bhugholadhyaya, (Surya Siddhanta) . The idea of spherical shape of the earth, was accepted in the west only after the 14th century.

The Panchavamsa Brahmana (6.8.6) states that the heavens are 1000 earth diameters away from the earth. The sun was taken to be halfway to the heavens. This suggests a distance to the Sun to be about 500 earth diameters from the earth, which is about 4357 million yojanas , yajurveda (hymn 17) dealing with the nature of the Universe, counts numbers in power of ten upto 1012.
We can see here the observation of Ebenezer Burgess, on the translation of Surya Siddhanta .

“In reference to most (of the above points), the evidence of originality I regard as clearly in favour of the Hindus; and in regard to some, and those the more important, this evidence appears to me nearly or quite conclusive… As to the lunar division of the Zodiac… the undoubted antiquity of this division among the Hindus, in connection with the absence or paucity of such evidence among any other people, incline me decidedly to the opinion that the division is of a purely Hindu origin.
As to the solar division… this was known to the Hindus centuries before any traces can be found in existence among any other people.
The theory of epicycles. The difference in the development of this theory in the Greek and Hindu systems of astronomy precludes the idea that one of these people derived more than a hint respecting it from the other. And so far as this point alone is concerned, we have as much reasons to suppose the Greeks to have been the borrowers as the contrary; but other considerations seem to favour the supposition that the Hindus were the original inventors of this theory.
As regards astrology, there is not much honor, in any estimation, connected with its invention and culture. But the honor of original invention, such as it is, lies, I think between the Hindus and the Chaldeans. The evidence of priority of invention and culture seems, on the whole to be in favour of the former…There is abundant testimony to the fact that the division of the day into twenty-four hours existed in the East, if not actually in India, before it did in Greece. In reference, further, to the so-called Greek words found in Hindu astronomical treatises, I would remark that we may with entire propriety refer them to that numerous class of words common to the Greek and Sanskrit languages which either came to both from a common source, or passed from the Sanskrit to the Greek at a period of high antiquity.

As to the names of the planets, I remark that the identity of all of them in the Hindu and the Greek systems is not to my mind clearly made out.

And in regard to…data and results-as for instance, the amount of the annual precession of the equinoxes, the relative size of the sun and the moon as compared with the earth, the greatest equation of the centre for the sun- the Hindus are more nearly correct than the Greeks, and in regard to the times of the revolutions of the planets they are very nearly as correct: it appearing from a comparative view of the sidereal revolutions of the planets, that the Hindus are most nearly correct in four items, and Ptolemy in six. There has evidently been very little astronomical borrowing between the Hindus and the Greeks. And in relation to the points that prove a communication from one people to the other… I am inclined to think that the course of derivation was from east to west rather than from west to east”.
The reference to the five planets in the Vedic literature is brought out as 34 lights in Rig Veda 10.5.3 (27 stars, the sun, the moon and the five planets).

[Brahaspati : RV 4.50.4
Vena (Venus) RV 10.123
Sukra(Venus) RV 3.32.2
Cosmology was also discussed in these texts.
INFINITESIMAL CALCULUS
The Infinitesimal Calculus in India is a product of the attempts of the earlier astronomers to chase the instantaneous motion of the planets. Bramha Gupta calculated the average velocity of the Planet Mars based on its instantaneous motion. The concept of the differentiation also appears in Siddhanta Siromani of Bhaskaracharya II. It was studied in connection with the instantaneous motion of the planet and the position angle of the ecliptic of secondary to the equator. The ideas of different co-efficient has been clearly stated by Bhaskaracharya in the same book. The theorem that for a maximum value for a function, its first differential co-efficient should be zero is stated by Bhaskaracharya in his Siddhanta Siromani.

The idea of integration and its development also has been indicated in the same book in connection with the surface area and volume of a sphere.

The idea of summation of an infinite series is found in Yukthi Basha nearly a century earlier than Newton or Leibnicz.
Conclusion:
What I have presented so far is only on the nature of “a trial” and appetizer in the field of ancient Indian Mathematics – It is really vast in scope and profound in depth. There has been a continuous stream of Indian mathematicians from Apastamba to Ramanujam. With such abundance of mathematical heritage, India can rise to be a World power!

P.S.
This lecture was delivered at the University of Pondicherry on 27.12.2003 during International WAVES Conference.