Jyeshtadeva
Encyclopedia
Jyeṣṭhadeva (c. 1500 – c. 1610) was an astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

-mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 of the Kerala school of astronomy and mathematics founded by Sangamagrama Madhava (c.1350 – c.1425) . He is best known as the author of Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

,
a commentary in Malayalam of Tantrasamgraha
Tantrasamgraha
Tantrasamgraha is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics....

 by Nilakantha Somayaji
Nilakantha Somayaji
Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...

 (1444–1544). In Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

, Jyeṣṭhadeva had given complete proofs
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

 and rationale of the statements in Tantrasamgraha
Tantrasamgraha
Tantrasamgraha is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics....

. This was unusual for traditional Indian mathematicians of the time. An analysis of the mathematics content of Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

 has prompted some scholars to call it "the first textbook of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

". Jyeṣṭhadeva also authored Drk-karana a treatise on astronomical observations.

Life period of Jyeṣṭhadeva

There are a few references to Jyeṣṭhadeva scattered across several old manuscripts. From these manuscripts, one can deduce a few bare facts about the life of Jyeṣṭhadeva. He was a Namputiri belonging to the Parangngottu family (Sanskrtised as Parakroda) born about the year 1500 CE. He was a pupil of Damodara
Damodara
Vatasseri Damodara Nambudiri was an astronomer-mathematician of the Kerala school of astronomy and mathematics who flourished during the fifteenth century CE. He was a son of Vatasseri Paramesvara who developed the drigganita system of astronomical computations...

 and a younger contemporary of Nilakantha Somayaji
Nilakantha Somayaji
Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...

. Achyuta Pisharati
Achyuta Pisharati
Acyuta Piṣāraṭi was an Sanskrit grammarian, astrologer, astronomer and mathematician who studied under Jyeṣṭhadeva and was a member of Mādhava of Sañgamāgrama's Kerala school of astronomy and mathematics...

 was a pupil of Jyeṣṭhadeva. In the concluding verse of his work titled Uparagakriyakrama, completed in 1592, Achyuta Pisharati
Achyuta Pisharati
Acyuta Piṣāraṭi was an Sanskrit grammarian, astrologer, astronomer and mathematician who studied under Jyeṣṭhadeva and was a member of Mādhava of Sañgamāgrama's Kerala school of astronomy and mathematics...

 has referred to Jyeṣṭhadeva as his aged benign teacher. From a few references in Drkkarana, a work believed to be of Jyeṣṭhadeva, one may conclude that Jyeṣṭhadeva lived up to about 1610 CE.

Parangngottu, the family house of Jyeṣṭhadeva, still exists in the vicinity of Trikkandiyur and Alathiyur. There are also several legends
Kottarathil Sankunni
Kottarathil Sankunni is a well known author of Malayalam literature, was born on 23 March 1855 in Kottayam, Travancore. He made huge contributions in both poetry and prose. He died on 22 July 1937. He started compiling the legends of Kerala in 1909 and completed the work in eight volumes over a...

 connected with members of Parangngottu family.

Mathematical lineage

Little is known about the mathematical traditions in Kerala
Kerala
or Keralam is an Indian state located on the Malabar coast of south-west India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....

 prior to Sangamagrama Madhava.
Vatasseri Paramesvara was a direct disciple of Madhava. Damodara
Damodara
Vatasseri Damodara Nambudiri was an astronomer-mathematician of the Kerala school of astronomy and mathematics who flourished during the fifteenth century CE. He was a son of Vatasseri Paramesvara who developed the drigganita system of astronomical computations...

 was a son of Paramesvara. Nilakantha Somayaji
Nilakantha Somayaji
Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...

 and Jyeshthadeva were pupils of Damodara. Jyeṣṭhadeva's pupil was Achyuta Pisharati
Achyuta Pisharati
Acyuta Piṣāraṭi was an Sanskrit grammarian, astrologer, astronomer and mathematician who studied under Jyeṣṭhadeva and was a member of Mādhava of Sañgamāgrama's Kerala school of astronomy and mathematics...

 and Melpathur Narayana Bhattathiri
Melpathur Narayana Bhattathiri
Melpathur Narayana Bhattathiri , third student of Achyuta Pisharati, was of Madhava of Sangamagrama's Kerala school of astronomy and mathematics. He was a mathematical linguist . His most important scholarly work, Prkriya-sarvawom, sets forth an axiomatic system elaborating on the classical system...

 was Achyuta Pisharati
Achyuta Pisharati
Acyuta Piṣāraṭi was an Sanskrit grammarian, astrologer, astronomer and mathematician who studied under Jyeṣṭhadeva and was a member of Mādhava of Sañgamāgrama's Kerala school of astronomy and mathematics...

's student.

Jyeshthadeva's works

Main articles : Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

, Ganita-yukti-bhasa
Ganita-yukti-bhasa
Ganita-yukti-bhasa is either the title or a part of the title of three different books:*Ganita-yukti-bhasa of Jyesthadeva, published by Springer, is the first critical edition with an English translation of Yuktibhasa, a seminal treatise in the Malayalam language composed in c.1530 CE by...



Jyeṣṭhadeva is known to have composed only two works, namely, Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

and Drkkarana.
The former is commentary with rationales of Tantrasamgraha
Tantrasamgraha
Tantrasamgraha is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics....

 of Nilakantha Somayaji
Nilakantha Somayaji
Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...

 and the latter is a treatise on astronomical computations.

Three factors make Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

 unique in the history of the development of mathematical thinking in the Indian subcontinent:
  • It is composed in the spoken language of the local people, namely, the Malayalam language. This is in contrast to the centuries old Indian tradition of composing scholarly works in the Sanskrit
    Sanskrit
    Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...

     language which was the language of the learned.
  • The work is in prose, again in contrast to the prevailing style of writing even technical manuals in verse. All the other notable works of the Kerala school
    Kerala school
    Kerala school may refer to*Kerala school of astronomy and mathematics, a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which flourished between the 14th and 16th centuries CE....

     are in verse.
  • Most importantly, Yuktibhāṣā
    Yuktibhasa
    Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

     was composed intentionally as a manual of proofs. The very purpose of writing the book was to record in full detail the rationales of the various results discovered by mathematicians-astronomers of the Kerala school, especially of Nilakantha Somayaji
    Nilakantha Somayaji
    Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...

    . This book is proof enough to establish that the concept of proof was not unknown to Indian mathematical traditions.

Further references

  • Details on the English translation of Yuktibhāṣā
    Yuktibhasa
    Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

     by K. V. Sarma
    K. V. Sarma
    K. V. Sarma was an Indian historian of science, particularly the astronomy and mathematics of the Kerala school.His doctoral thesis was at the Panjab University in 1977....

    : (This is a critical translation of the original Malayalam text by K.V. Sarma with explanatory notes by K. Ramasubramanian, M.D. Srinivas and M.S. Sriram.)
  • For a review of the English translation of Yuktibhāṣā
    Yuktibhasa
    Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

    :
  • For a modern explanation of Jyeṣṭhadeva's proof of the power series expansion of the arctangent function:
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