How is the heritage of the traditional Indian communities and their descendants abroad related in any way to the mathematical and astronomical traditions of India? Are there aspects that are still retained and practised today in the communities? Is there anything we can teach or tell our children in our community about it? If we or our children wanted to explore their own tradition, what would they need to do? Where should they go? How to go beyond the caricatured and misrepresented versions and translations of works in our tradition and look into the vast scientific texts of our ancestors? We explore these ideas and try to provide some directions in this article.

The *Taittireeya* branch of the *Yajurveda* and also *Rigveda* in the form of *Saamaveda* recitation is preserved by many communities in southern India, among others over the millennia. The oldest known reference and use of the positional decimal system is in the Rigveda.

द्वादश प्रथयश्चक्रमॆकं त्रीणि नाभ्यानि त उ तच्चिकॆत ।

तस्मिन् साकं त्रिंशता न शंङ्कवॊऽर्पिताः षष्टिर्न चलाचलासः ॥ 1.164.48 Rigveda

dvādaśa prathayaścakramkaṃ trīṇi nābhyāni ta u taccikta ।

tasmin sākaṃ triṃśatā na śaṃṅkav’rpitāḥ ṣaṣṭirna calācalāsaḥ ॥ 1.164.48 Rigveda

(There is a circle of year who’s 360 degrees are divided into twelve sections and three seasons.)

Did you ever ponder why there are 360 degrees of angles and not 400 for example? That is because the motion of the sun per day is roughly one degree, and 360 of them make a full circle! This is right in the Vedas. This is probably the oldest clear explanation of angles and why there are 360 degrees (called *amsha*). All this, obviously, is being described in the positional decimal system, as you may have noted.

“There is no separate branch for mathematics in India. The whole tradition of ancient Indian mathematics is basically a combination of the jyotisha (astronomical) tradition and Vedic tradition of India.”

*Nakshatra sukta* in the Vedas defines 27 sectors in the celestial sphere, each of 30 degrees, to cover the 360 degrees in the sky and not 460! Vedas clearly suggest the use of mathematics to compute the raising and setting of celestial bodies. Vedas cover not only arithmetic, geometry but also algebra. Thus, we can clearly see the Vedas preserved by our ancestors form the oldest foundations of mathematics. Yet more than 1000 branches of Vedas are lost. We do not know, therefore, what mathematics and other wisdom is lost with it.

Along with Vedas come the six auxiliary parts of Vedas called *Vedangas*: grammar (*vyakarana*), etymology (*nirukta*), framework for learning Vedas (*shiksha*), instructions (*kalpa*), prosody (*chandas*) and science of light (*jyotisha*/astronomy). While grammar, etymology, and learning form the basis for some of the earliest science on linguistics, prosody (chandas) lays the foundations for binary arithmetic as expounded in the work of *Pingala* *rishi* in his *Chandasutra*. The second most prominent work on Indian astronomy and mathematics is *Vedanga Jyotisha* by *Lagadha Muni*. This work provides mathematics for the creation of a *panchanga* or the calendar. The traditional Hindu calendar which has evolved through the works of *Aryabhata* and *Bhaskaracharya* and so on is based on observations and is still more accurate relative to the Gregorian calendar. It is very critical to know that the *Vedanga Jyotisha* does not mention astrology. It is a pure work of science. Therefore, a clear distinction should be made between Indian tradition of *jyotisha* which are astronomical and mathematical sciences and *phala jyotisha *which is astrology*.*

The *gruhyasutras* that we follow to perform all 16 *samskaras* including *upanayana (start of formal education)* and vivaaha (weddings) are developed by our ancient achaaryaas such as *Bhodhayana*, *Ashvalayana*, *Maanvya*, *Kaatyaayana* and so on. These also have mathematical calculations to construct the *yajna pits *and the like. Among these, *shulbasutra, *composed by *Aapasthamba, *the student of *Bhodhayana,* contains the famous theorem for right angle triangles.

दीर्घस्याक्ष्णयारज्जुः पार्श्वमानी तिर्यङ्मानी ।

यत्पृथग्भूतॆ कुरुतस्तदुभयं करॊति ॥ ( )

dīrghasyākṣṇayārajjuḥ pārśvamānī tiryaṅmānī ।

yatpṛthagbhūt kurutastadubhayaṃ karti ( )

It is interesting to note that the mathematical explanations in the *shulbasutra* are not rare, but are full of it. Commentaries of the *shulbasutra* by *Kapardi and Sundararaja *explain this in more detail. This is only to stress that the work is not something in passing, and that the tradition of commentaries continues all the way until recent centuries and is extensive. We may note before moving to discuss other works that geometry in India was called *rajju ganita* or thread calculus. It wasn’t simple planar geometry either, if works on spherical geometry by Aryabhata is any proof.

So far, we have considered a timeline in antiquity that is variously claimed to be anywhere from 6000 BCE to 2000 BCE. We have discussed only oral traditions and textual evidence and not evidence from *Sindhu-Saraswati* communities and archaeology (referred by colonialists as Indus Valley civilization). Let us only note that the tradition considers a continuity in civilization and does not look at the *Sindhu-Saraswati* communities as separate. While this aspect needs its own study and review, we list textual works possibly belonging to the same period or a bit after as our traditions tell us. In that vein, these are only some of the ancient works of Indian mathematics and astronomy known to us: *Suryasiddhantha, Pitamaha samhitha, Naarada samhitha, Atri samhitha, Brugu samhitha, Vasita samhitha, Kashyapa samhitha, Lomasha samhitha, Vysa samhitha, Shaukana samhitha, Garga samhitha, Parashara samhitha, Maya samhitha, Paulatsa samhitha Paulisha, Dhruvanaadi Granthas.*

Of these, *Varaahamihira *in his comparative study in 5th century CE of five systems of astronomy states clearly that the Indian *Suryasiddhantha* is the most accurate (सौरस्पष्टतरः) and others including that of *Romaka Siddhantha* as less reliable and inaccurate. Suryasiddhantha is the ancient work on Indian time keeping in mathematics, we do not clearly know the origin of. It is estimated variously to be from 1000 BCE to 500 CE, probably reflecting updates being made to it. We unfortunately do not have the same version of *Suryasiddhantha* that Varahamihira had. This is based on the quotes Varahamihira makes that do not exist in the copies that have survived. Suryasiddhantha is one of the most extensive in terms of mathematical and astronomical knowledge. It contains precise methods for prediction of eclipses, prediction of positions of planets, calculation of tilt of Earth’s axis, time Venus, Mercury, Jupiter, and Saturn take to come a full circle around the sun. It also calculates the size of planets, and most importantly to calculate the solar/topical year between two spring equinoxes. The table of sines it contains is more accurate and detailed than attributed to Greeks. There is a translation by Ebenezer Burgess of *Suryasiddhantha. *Ebenezer Burgess was a Christian pastor from the USA with a degree in theology who taught mathematics. The translation has errors and is biased. Those wanting to read should rely on the original Sanskrit. Most of the texts mentioned in this article are studied as part of study of *jyotisha* in Sanskrit *pathashalas*.

Here is an example of calculation of earth’s circumference stated in *Suryasiddhantha*.

यॊजनानि शतान्यष्टौ भूकर्णॊ द्विगुणानि तु ।

तद्वर्गतॊ दशगुणात् पदं भूपरिधिर्भवॆत् ॥

(yjanāni śatānyaṣṭau bhūkarṇ dviguṇāni tu ।

tadvargat daśaguṇāt padaṃ bhūparidhirbhavt ॥)

The radius of earth is 800 yojanas (determined separately). It further goes on to explain how circumference can be calculated for any given latitude (i.e size of earth at a place anywhere between the equator and the poles). The circumference of earth comes to 5059.64425627 yojanas or 40477 kms as opposed to the modern 40075 kms (value of pi 3.1435). The size of earth was measured using trigonometric functions, the table for which Suryasiddhantha provides.

*Aryabhata* is probably the most famous of Indian mathematicians and physicists. He is not so well known, however, for the theory of relative motion he proposed. Not so well known are his works on numerical differential equations and spherical geometry! *Aryabhatiya, *a major work on Indian astronomy, includes improvements to planetary position calculation among other things. *Bhaskara* *I*, *Somesvara* and *Prabhakara* aacharya’s commentaries on *Aryabhatia* were studied in traditional schools and were improved upon. The popular luni-solar calendar/panchanga in use among many communities in Karnataka and Andhra (*dharmika/baggona panchanga etc.*) is based on Aryabhata’s work and later improvements by *Bhaskara II*.

*Panchasiddhantika*, *Bruhat Samhita*, *Laghu Jataka and Bruhat Jataka *are five major works of *Varahamihira* in the 5th century CE. In his comparison of five schools of astronomy, *Varahamihira* is clear that “पौलिष रॊमकौ अस्पष्टौ” (The Paulisha and Roman schools are unclear and unreliable). He calls for the use of *Suryasiddhantha*, the Indian school of astronomy.

Area of a circle, as given in the *Ganitapaada* of *Aryabhateeya*: half of perimeter multiplied by circle’s radius!

*Brāhmasphuṭasiddhānta *written in 7th century CE by Brahmagupta is another famous work on mathematics. Brahmagupta is famous for solutions to indeterminate equations, among other things.

There are more works on what is called the Siddhanta part of *jyotisha* which is the primary source of Indian mathematics. This list might help give us a sense of how profound the tradition is. The works on mathematics and astronomy: *Phalita Grantha, Lata simha – Suryasiddhantha vyakhyana 6th century CE, Prathyusha – Shatpanchashika, Lallaacharya – Shishyadheevrudditantra, Padmanabha Bijaganita, Balabhadra – Gruhaganita, Vitteshvara – Karanasaara, Aryabhata II – Mahasiddhantha, Pratudhaka – Brahmasputa Siddhantha Teeka, Kamalaakara Bhatta – siddhantha Tatvaviveka.*

*Bhaskara I wrote Laghu Bhaskareeya, Mahābhāskarīya in the 7th century CE. *He followed the *Aryabhata* school.

The Lilavati Ganita written by Bhaskara II in the 12th century CE has been used to teach mathematics in our traditional *gurukulas* for more than 800 years. It continues to be used till today. It is the book, named after the daughter of *Bhaskara Acharya II*. Not only that, but it is the first part of *Siddhanthashiromani*, consisting of *Lilavati* (problems), *Bijaganitha* (Algebra), *Goladhyaya *(Spherical). Let us also note that one of the greatest mathematicians, Jyotishi, taught his daughter mathematics and named a book after her thousand years ago. Certainly some proof that women were not only taught but also taught mathematics. Among other things, Bhaskara II is famous for providing differentials for trigonometric functions. Many in traditional vedic communities used to study *Siddhanthashiromani* in the form of the book *Lilavati* (and its Kannada translation) at home. It is an excellent work full of problems that we should let our children have access to.

We now come to the last phase of Jyotisha, the ancient Indian mathematics, where major breakthroughs were made. *Achyuta* *Pisharati*, *Madhava* *Sangagrama*, *Nilakantha* *Somayaji* and *Parameshvara* are some names that standout among many more Jotishis from Kerala in the 14th century CE. *Nilakantha* *Somayaji* wrote eleven texts on astronomy, starting with the major work *Tantrasangraha*. He is famous for work on infinite series and for attributing the invention of what is today known as Taylor’s series to the Indian mathematician and Jotishi *Madhava* *Sangagrama*!

The beginning of decline of major *jyotisha siddhantha* or work on Indian mathematics can be seen starting 10th century CE in the north of India and around 16th century in the south of India, probably due to the decline of traditional Indian learning and destruction of libraries and killing of Brahmins by the invading colonizers. First the invading Muslim armies and then the British, who also burnt down Indian libraries. While the burning of Nalanda by Muslim invaders is rather famous, not so famous is the burning of the library of Jhansi Raani Lakshmibhai, *Manikarnika*. The British mathematician, Augustus De Morgan (born in *Madhurai*), famous for his “laws” in the 19th century, called the negative numbers evil! He also tried his luck in banning *panchangas*, the Indian calendars! C.K Raju, one of India’s mathematicians, has recorded many instances of undermining Indian *jyotisha* and mathematics by Europeans and glorification of Greek work using non-existent texts.

It is well known that the mathematics of India travelled to Europe through Arab and Persian translations. Gerbert of Aurillac, later known as Pope Sylvester II tried to introduce the Hindu positional decimal number system and later Fabinocci (Leonardo of Pisa) achieved some success. Matteo Ricci, a Catholic missionary, came to Goa in 1578 and according to his own letters he studied Indian astronomy/timekeeping in Kochi! One of the letters said, he was looking for “an honourable Moor or an intelligent Brahmin to tell him about the Indian methods of timekeeping”. It is important to note that Ricci was a student of Christopher Clavius who later proposed reform to the Gregorian calendar in 1582!

In closing, let’s take a look at the advanced notion of time in the *Tarkashastra*. The *Tarka-saṅgraha-deepika* of *Annam Bhatta* a commentary on the tradition of logic (*Nyaya Sutra* of Gautama) provides details on this. Indian *shastras* define time as ‘not direct (not pratyaksha). Time is ‘*upaadhi lakshana*’ or for instance as something used to measure the distance between two objects in the sky and that it is causal. We should not disregard these ancient texts as antiquated. There is more to learn by exploring these ideas found in our ancient texts.

In summary, we have a long and rich tradition of science and mathematics in the Vedic tradition that awaits our indulgence and understanding. We can directly refer to these texts and commentaries with the help of our traditional scholars to understand the tradition in its right context and scope. Works of our ancestors are much maligned by colonialists and their translations, and even the later translations along the same lines. It is critical that we do not take their word, but work with our traditional scholars to validate them. Ancient work on Indian mathematics and astronomy, still remains profound and hides much more wisdom than we are acquainted with. It is not an outdated set of findings that the current science has left behind. But, it is rich in approaches that will allow us to take science to the next level by understanding first principles. It allows us to examine science in a truly Indian context, devoid of dogmas of the other cultures. We should explore the Vedic tradition of science more and more. That journey can begin with as simple a step as taking a look at some of the texts mentioned in this article and talking to our traditional scholars about it.

**References:**

- Vedic verse quoted from: Vedic Sampattee By Pandit Raghunandan Sharma
- Apasthama sutra: https://archive.org/details/in.ernet.dli.2015.486875/page/20/mode/2up?view=theater
- C.K Raju – various lectures and books.
- Original mathematical and astronomical texts mentioned in the article.

Feature Image Credit: istockphoto.com

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