Kerala School of Mathematics
Radhakrishna Warrier
I got a little time last week to devote to a quick study the contributions of the “Kerala School of Mathematics” to Mathematics and Astronomy. Given below are some information I could gather that may be of interest to this group.
First let me thank śrī Shankara, Bāndhavi ji, śrī Sunil (Swapna Sunil), and śrī Mēgh for their kind help. The links they provided helped me a great deal in understanding the contributions of the “Kerala School of Mathematics” and how this Kerala School came up with the fundamental concepts of calculus at least two hundred and fifty years before Newton. The very useful links they provided are:
- Printed copy of Yuktibhāṣā with explanations: Thanks śrī Shankara https://archive.org/details/Yukthibhasha1948/page/n107/mode/2up
- An interesting lecture (3 parts): Thanks Bāndhavi ji https://www.youtube.com/watch?v=yWZ15EKElH0
- An interesting presentation: Thanks śrī Sunil http://www.ircc.iitb.ac.in/IRCC-Webpage/rnd/PDF/IIT_Bombay_Research_Dissemination_Award_K_Ramasubramanian.pdf
- A palm leaf grantha of “Yuktibhāṣā”: Thanks śrī Sunil https://archive.org/details/raswhishNA-124/page/n1/mode/1up
- Vedāṅga Jyotiṣa of Lagadha:
Thanks
śrī Mēgh
https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol19_3_10_SupplementVedangjyotishaofLagdha.pdf
The people behind the “Kerala School of Mathematics” were:
- Mādhava (c. 1340–1420 CE) — pioneer of the Kerala School of Mathematics. https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama
- Paramēśvara (c. 1380–1460 CE) — a disciple of Mādhava, great observer and a prolific writer. https://en.wikipedia.org/wiki/Parameshvara
- Nīlakaṇṭha Sōmayājī (c. 1444–1550 CE) — monumental contributions Tantrasaṅgraha and Āryabhaṭiyabhāṣya. https://en.wikipedia.org/wiki/Nilakantha_Somayaji
- Jyēṣṭhadēva (c. 1530 CE) — author of the celebrated Yuktibhāṣā. https://en.wikipedia.org/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva
- Śaṅkara Vāriyar (c. 1500–1560 CE) — a brilliant mathematician and astronomer well known for his commentaries. https://en.wikipedia.org/wiki/Sankara_Variar
- Putumana Sōmayājī (c. 1532 CE) — author of several works, but well known for his Karaṇapaddhati. https://en.wikipedia.org/wiki/Puthumana_Somayaji
- Acyuta Piṣāraṭi (c. 1550–1621 CE) — a disciple of Jyēṣṭhadēva and a polymath – teacher of Melpattūr Nārāyaṇa Bhaṭṭatiri, the author of Nārāyaṇīyam. https://en.wikipedia.org/wiki/Achyutha_Pisharadi
Mādhava’s works have not survived to reach us in the present day. Tantrasaṅgraha of Nīlakaṇṭha Sōmayājī is based on Mādhava’s works and quotes Mādhava. Yuktibhāṣā in Malayalam prose by Jyēṣṭhadēva is based on Tantrasaṅgraha.
The Malayalam prose of the 1500’s of the Yuktibhāṣā is not difficult to understand. Author Jyēṣṭhadēva’s explanations are crisp and clear. See how he provides an example for trairāśikam (त्रैराशिकम्, the rule of three) which is finding the fourth quantity in a linear function when three values are known.
“ഇതിനുദാഹരണം. അഞ്ഞാഴിനെല്ലിന്നു് ഇരുനാഴി അരി എന്നിങ്ങനെ അറിഞ്ഞിട്ടിരിക്കുമ്പോൾ ഇതിന്റെ ശേഷം നെല്ലിന്നൊക്കെയ്ക്കും ഇങ്ങനെത്തൊരു അരിയോടുള്ള മാനസംബന്ധ നിയമമുണ്ടു് എന്നിരിക്കേണം. ആകയാൽ പന്തിരുനാഴി നെല്ലിന്നു് എത്ര അരിയുണ്ടെന്നു് അറിയേണ്ടുമ്പോൾ ഇത്ത്രൈരാശികമാകുന്ന ക്രിയ ഉപയോഗിക്കുന്നൂ.”
“An example for this [for trairāśikam]. If it is known that five nāẓi [a volumetric measure] of paddy gives two nāẓi of rice, and considering that this quantitative relationship between paddy and rice holds good hereafter, for finding the amount of rice from twelve nāẓi of paddy, this rule-of-three operation [trairāśikam kriya] is used.”
[Incidentally, this volumetric measure nāẓi was popular till relatively recently. I have heard my late mother and my late eldest sister speaking about rice, milk and oil measures in terms of nāẓi and uri.]
It is not the language, but the subject matter that needs careful attention. It is a mathematical and especially geometric treatise and yet it is all words. As per the premodern tradition, no diagrams, or symbolic notations are used. In other words, convenient figures, equations, and formulae are absent. Plenty of technical terms are used but Jyēṣṭhadēva clearly explains these in his work. However, for a newbie, there is need to frequently refer to the explanations when many of these technical terms are encountered.
Here are some technical terms that reflect the calculus fundamentals:
aṇu pariṇāmam = infinitesimal
asaṅkhyam ēṟiyāl = if increases limitless = if tends to infinity
saṃkalitam = integral; also sum of a series
saṃkalita saṃkalitam = integral of an integral
The following is a brief explanation as to why it is considered that Mādhava and the Kerala School of Mathematics knew of the concepts of calculus and used the techniques of calculus in their works.
I will not go deep into the technical matter.
The technical matter is very interesting, and the interested members can use the links provided to go into the mathematical discussions. I feel the following slide from Prof. P.P. Divakaran’s
presentation is sufficient for the present purpose.
Regards,
Radhakrishna Warrier
Shashi Joshi
Thanks,
Shashi
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Megh Kalyanasundaram
"But Plofker herself uses euphemisms like, you know, sort of pre-calculus, very brilliant but is it calculus, you know, that kind of very ambiguous terminology to describe what Madhava did. And part of my purpose is to remove such misunderstandings. That is the reason for the choice of my title.": P.P. Divakaran (almost verbatim) in https://www.youtube.com/watch?time_continue=1826&v=yWZ15EKElH0&feature=emb_logo (@~30:26)
Sivasenani Nori
L Srinivas
Radhakrishna Warrier
Sent: Monday, October 12, 2020 5:01 AM
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Subject: {भारतीयविद्वत्परिषत्} Re: Kerala School of Mathematics
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Bandhavi Brahma
Radhakrishna Warrier
Sent: Tuesday, October 13, 2020 8:56 AM
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Megh Kalyanasundaram
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Radhakrishna Warrier
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Subject: Re: {भारतीयविद्वत्परिषत्} Re: Kerala School of Mathematics