The Indian tradition in mathematics is long and glorious. It dates to the earliest times, and indeed many of the Indian discoveries from a period starting 5000 years ago correspond rather naturally to modern mathematical results. Celebration of Indian mathematics needs to consider the personalities among ancient mathematicians who laid a solid foundation for modern thinking. Our main purpose here is, by presenting very briefly some of the main contributions of ancient Indian mathematicians and astronomers, to argue and convince the reader that before the great Ramanujan, there have been thousands of years of rich mathematical discoveries in India and those personalities’ work also needs to be honored on Indian Mathematics Day.

The
government of India announced in 2012 that every year, Indian
mathematician Srinivasa Ramanujan’s birthday, December 22, will be
celebrated as national mathematics
day.^{1}Prime Minister’s speech at the 125th Birth Anniversary Celebrations of
Ramanujan at Chennai: https://archive.is/20120729041631/http://pmindia.nic.in/speech-details.php?nodeid=1117
(accessed on April 15, 2023). The year 2012 was the great
Ramanujan’s 125th birth anniversary. The government of India released a
commemorative stamp on that occasion as well. Brilliant contributions in
number theory and combinatorics by Ramanujan are well known [2
G. E. Andrews and B. C. Berndt,
Ramanujan’s lost
notebook. Part V. Springer, Cham (2018)
, 1
G. E. Andrews and B. C. Berndt, Ramanujan’s lost notebook. Part I.
Springer, New York (2005)
, 13
S. Ramanujan,
The lost notebook and other unpublished papers.
Springer-Verlag, Berlin; Narosa Publishing House, New Delhi (1988)
, 16
A. S. R. Srinivasa Rao,
Ramanujan and religion. Notices Amer. Math. Soc. 53, 1007–1008 (2006) ]. However, deep
astronomical and mathematical developments in India are several thousand
years older than Ramanujan. In this comment, we try to recollect a few
gems of the ancient Indian mathematics and its mathematicians who did
fundamental work in number systems, mathematics of astronomy, calculus,
etc., over more than 5000 years.

Our main purpose for writing this article is to
argue and convince that, while giving Ramanujan’s brilliant achievements
during the past 125 years their due place, reducing the Mathematics Day
in India to the celebration of Ramanujan’s birthday (who was born in the
19th century) is somewhat short-sighted. Our goal is to make sure Indian
Mathematics Day is seen as a celebration of thousands of years of
deep-rooted mathematical thought processes and discoveries
since the
times of *Shulba sutra*. Moreover, it should also be devoted to
celebrating many very strong mathematicians, such as, say, Harish
Chandra, who have come since Ramanujan’s time.

The origins of the mathematics that emerged in the Indian subcontinent
can be seen around the Shulba sutra period, around 1200 BCE to 500 BCE.
During this period the numbers up to $10_{12}$ were counted (in Vedic
Sanskrit this number was referred to as *Paradham*). The Vedic
period mathematics was confined to the geometry
of fire-altars and
astronomy, and these concepts were used to perform rituals by the
priests. Some of the famous names from that era are Baudhayana,
Apastamba, and Katyayana. In Table 1 we describe
Sanskrit sounds and their corresponding English numerals. Indian
mathematics also introduced the decimal number system that is in use
today and the concept of zero as a number. The concepts of sine (written
as *jaya* in Sanskrit) and cosine (*cojaya*), negative
numbers, arithmetic, and algebra were found in ancient Indian
mathematics [6
B. Datta and A. N. Singh, History of Hindu mathematics: A source
book. Part I: Numerical notation and arithmetic. Part II:
Algebra. Asia Publishing House, Bombay–Calcutta–New
Delhi–Madras–London-New York (1962)
].
The mathematics developed in India
was later translated and transmitted to China, East Asia, West Asia,
Europe, and Saudi Arabia. The classical period of Indian mathematics was
often attributed to the interval from
200 CE to 1400 CE, during which
works of several well-known mathematicians, like Varahamihira,
Aryabhata, Brahmagupta, Bhaskara, and Madhava have been translated into
other languages and transmitted outside the sub-continent.

The number systems present since the Vedic days, especially since the
Sukla Yajurveda and their Sanskrit sounds, were as follows: $1$ (*Eka*), $10$ (*Dasa*),
$100$ (*Sata*), $1000$ (*Sahasra*), $10_{4}$ (*Aayuta*),
$10_{5}$ (*Laksa* or *Niyuta*), $10_{7}$ (*Koti*), $10_{12}$ (*Sanku* or *Paraardha*),
$10_{17}$ (*Maha Sanku*), $10_{22}$ (*Vrnda*), $10_{52}$ (*Samudra*),
$10_{62}$ (*Maha-ogha*).

In addition to the number systems of ancient India, still today in India
are heard the popular *“Vishnu Sahasra Nama Stotra,”* which dates
back to the Mahabharata epic. In this, there is a verse that sounds like
Sahasra *“Koti Yugadharine Namah.”* If we translate this verse,
then, as we saw above, *Sahasra* means $1000$, and *Koti*
means $10_{7}$, so a simple translation of the phrase *Sahasra Koti*
could mean $10_{10}$. The entire phrase has been interpreted in
different ways. We do not list here all possible interpretations and
confine ourselves to number systems.

The deep investigations in astronomy and the solar system, geometry, and ground-breaking mathematical calculations by ancient and medieval great scholars in India, for example, Baudhayana, Varahamihira, Aryabhata, Bhaskara I & II, Pingala, Madhava, and many more, are well known (see [10 G. G. Joseph, The crest of the peacock: Non-European roots of mathematics. 2nd ed., Princeton University Press, Princeton, NJ (2000) , 7 B. R. Evans, The development of mathematics throughout the centuries: A brief history in a cultural context. John Wiley & Sons, New York (2014) , 5 B. Dâjî, Brief notes on the age and authenticity of the works of Âryabhaṭa, Varâhamihira, Brahmagupta, Bhaṭṭ otpala, and Bhâskarâchârya. Journal of the Royal Asiatic Society 1, 392–418 (1865) ] and [17 A. S. R. Srinivasa Rao and S. G. Krantz, Dynamical systems: From classical mechanics and astronomy to modern methods. J. Indian Inst. Sci. 101, 419–429 (2021) , p. 423]).

It seems that celebrating national mathematics day in India only as part of Ramanujan’s birthday is confining the glory and celebration of Indian mathematics to a little over 100 years of the past. Schools and colleges across India have celebrated Ramanujan’s birthday for many decades, but that is different from exclusively limiting national day only to the great Ramanujan.

A good deal can be written on ancient scholar’s work from India; material on this can be found, for example in [8 R. C. Gupta, T. A. Sarasvati Amma (c. 1920–2000): A great scholar of Indian geometry. Bhāvanā 3 (2019) , 12 K. Plofker, Mathematics in India. Princeton University Press, Princeton, NJ (2009) , 15 T. A. Sarasvati Amma, Geometry in ancient and medieval India. Motilal Banarsidass, Delhi (1979) , 6 B. Datta and A. N. Singh, History of Hindu mathematics: A source book. Part I: Numerical notation and arithmetic. Part II: Algebra. Asia Publishing House, Bombay–Calcutta–New Delhi–Madras–London-New York (1962) , 3 A. Bürk, Das Āpastamba-Śulba-Sūtra. Zeitschrift der Deutschen Morgenländischen Gesellschaft 55, 543–591 (2001) ]. In this opinion piece, we highlight only a few of them.

Shulba sutras were believed to have started in India around 2000 BCE through verbal usage. Their compilation in Sanskrit started perhaps 1000 years later by Baudhayana then by Manava, Apastamba, Katyayana and consisted of geometric-shaped fire-altars for performing ancient Indian rituals [3 A. Bürk, Das Āpastamba-Śulba-Sūtra. Zeitschrift der Deutschen Morgenländischen Gesellschaft 55, 543–591 (2001) , 12 K. Plofker, Mathematics in India. Princeton University Press, Princeton, NJ (2009) ]. Some of these sutras also contain the statements of Pythagorean theorems and triples. For example, Apastamba provided the following triples:

for constructing fire-altars [15 T. A. Sarasvati Amma, Geometry in ancient and medieval India. Motilal Banarsidass, Delhi (1979) ].

These sutras can be used to find the approximate value of $2 $ [15 T. A. Sarasvati Amma, Geometry in ancient and medieval India. Motilal Banarsidass, Delhi (1979) , 12 K. Plofker, Mathematics in India. Princeton University Press, Princeton, NJ (2009) ], using the expression

In
the Vedic period astrology (*Jyotisha*) of India, the magic squares
(*anka-yantra*) were used to please and worship nine planets of the solar
system [9
R. C. Gupta,
Mystical mathematics of ancient planets. Indian J. Hist.
Sci. 40, 31–53 (2005)
, 18
G. P. H. Styan and K. L. Chu,
An illustrated
introduction to some old magic squares from India. In Combinatorial
matrix theory and generalized inverses of matrices, pp. 227–252, Springer,
New Delhi (2013)
]. Figure 1 is about ancient magic
squares for the Sun and the other eight planets in our solar system.

In the 5th century CE, Aryabhata calculated, among many other things, that the moon orbit takes 27.396 days, the value of $π=3.1416$, etc. He is believed to have started the study of properties of sine and cosine in trigonometry.

According to [12 K. Plofker, Mathematics in India. Princeton University Press, Princeton, NJ (2009) ], the Leibniz infinite series

was known in the works of Indian mathematician Madhava, who lived three centuries before Leibniz.

In
the 12th century CE, Bhaskara described in his famous book
*Bijaganita* the rules of algebraic operations on positive,
negative signs, rules of zero (*shunya*), and infinity
(*anantam*). His book also shows how to obtain solutions to
intermediate equations of the first degree [14
K. Ramasubramanian, T. Hayashi and C. Montelle, (eds.),
Bhāskara-prabhā.
Sources Stud. Hist. Math. Phys. Sci.,
Springer, Singapore (2019)
].
Bhaskara’s book titled *Siddhanta siromani*
provided a detailed account of Indian astronomy and its development.
Computations of the planetary movements, shapes of planets, rotation
axis, lunar month days, etc., were explained in detail. See Figure 2.
Pavuluri Mallana translated Mahavira’s *Ganitasarasamgraha* from *Sanskrit* in the 11th century to another ancient indian
language, *Telugu*; Joseph [11
G. G. Joseph,
Indian
mathematics: Engaging with the world from ancient to modern times. World Scientific Publishing, Hackensack, NJ (2016)
] thinks that this stood as a role model for other subsequent translations. Bhaskara’s
*Lilavati Ganitam* was for the first time translated from *Sanskrit* to *Telugu* in the 12th century by Eluganti Peddana [4
P. Chenchiah and R. M. Bhujanga, A history of Telugu literature.
The Association Press, Calcutta; Oxford University Press, London (1928)
],
and into English first in 1816 by John Taylor, then in 1817 by Henry Thomas Holbrooke, who was considered as the
first European Sanskrit scholar.

What we advocate in this piece is for an exposition of deep-rooted
mathematical knowledge in India, and not an exhaustive account of all
possible results and conclusions. Several of the ancient texts in the
language *Sanskrit* are either lost or preserved in museums.

Srinivasa Ramanujan’s work undoubtedly shines as part of modern Indian mathematics but thousands of years ancient mathematical discoveries, the introduction of various branches of pure and applied mathematics needs a proper representation in any celebration of India’s contribution to world mathematics.

We hope that this short list of significant examples will convince the readers as well as the decision makers of the need to incorporate and celebrate all the rich past and contemporary history of Indian mathematics during the Indian Mathematics Day.

*Acknowledgements. *
We are thankful to the reviewer for the
comments that helped us to make
the point in the article clearer. We are
also thankful to Prof. Ralf Krömer, editor of the
EMS Magazine, and the editorial staff and copyeditor of EMS Press for their edits to
our article.

^{1}Prime Minister’s speech at the 125th Birth Anniversary Celebrations of Ramanujan at Chennai: https://archive.is/20120729041631/http://pmindia.nic.in/speech-details.php?nodeid=1117 (accessed on April 15, 2023).

^{2}This book is available for free at https://upload.wikimedia.org/wikipedia/commons/7/75/Lilavatiganitamu00bhassher.pdf. © Sundarayya Vignana Kendram, Bagh Lingampally, Hyderabad, India

## References

- G. E. Andrews and B. C. Berndt, Ramanujan’s lost notebook. Part I. Springer, New York (2005)
- G. E. Andrews and B. C. Berndt, Ramanujan’s lost notebook. Part V. Springer, Cham (2018)
- A. Bürk, Das Āpastamba-Śulba-Sūtra. Zeitschrift der Deutschen Morgenländischen Gesellschaft 55, 543–591 (2001)
- P. Chenchiah and R. M. Bhujanga, A history of Telugu literature. The Association Press, Calcutta; Oxford University Press, London (1928)
- B. Dâjî, Brief notes on the age and authenticity of the works of Âryabhaṭa, Varâhamihira, Brahmagupta, Bhaṭṭ otpala, and Bhâskarâchârya. Journal of the Royal Asiatic Society 1, 392–418 (1865)
- B. Datta and A. N. Singh, History of Hindu mathematics: A source book. Part I: Numerical notation and arithmetic. Part II: Algebra. Asia Publishing House, Bombay–Calcutta–New Delhi–Madras–London-New York (1962)
- B. R. Evans, The development of mathematics throughout the centuries: A brief history in a cultural context. John Wiley & Sons, New York (2014)
- R. C. Gupta, T. A. Sarasvati Amma (c. 1920–2000): A great scholar of Indian geometry. Bhāvanā 3 (2019)
- R. C. Gupta, Mystical mathematics of ancient planets. Indian J. Hist. Sci. 40, 31–53 (2005)
- G. G. Joseph, The crest of the peacock: Non-European roots of mathematics. 2nd ed., Princeton University Press, Princeton, NJ (2000)
- G. G. Joseph, Indian mathematics: Engaging with the world from ancient to modern times. World Scientific Publishing, Hackensack, NJ (2016)
- K. Plofker, Mathematics in India. Princeton University Press, Princeton, NJ (2009)
- S. Ramanujan, The lost notebook and other unpublished papers. Springer-Verlag, Berlin; Narosa Publishing House, New Delhi (1988)
- K. Ramasubramanian, T. Hayashi and C. Montelle, (eds.), Bhāskara-prabhā. Sources Stud. Hist. Math. Phys. Sci., Springer, Singapore (2019)
- T. A. Sarasvati Amma, Geometry in ancient and medieval India. Motilal Banarsidass, Delhi (1979)
- A. S. R. Srinivasa Rao, Ramanujan and religion. Notices Amer. Math. Soc. 53, 1007–1008 (2006)
- A. S. R. Srinivasa Rao and S. G. Krantz, Dynamical systems: From classical mechanics and astronomy to modern methods. J. Indian Inst. Sci. 101, 419–429 (2021)
- G. P. H. Styan and K. L. Chu, An illustrated introduction to some old magic squares from India. In Combinatorial matrix theory and generalized inverses of matrices, pp. 227–252, Springer, New Delhi (2013)

## Cite this article

Steven G. Krantz, Arni S. R. Srinivasa Rao, Ancient Indian mathematics needs an honorific place in modern mathematics celebration. Eur. Math. Soc. Mag. (2023), published online first

DOI 10.4171/MAG/139