Yuktibhasa

01The reasons behind the rules

Most Sanskrit works on ganita state their rules in verse and leave the reasons to the teacher. Jyeshthadeva wrote the reasons down. Yukti means the reasoning that establishes a rule, and bhasha the spoken language, here Malayalam. He says he follows Nilakantha Somayaji’s Tantrasangraha and sets out the upapattis, the demonstrations, as he received them from his teacher Damodara. The prose is plain and patient, built for a student who wants to see each step, and it often stops to say: now we will show why this is so.

02Jyeshthadeva of the Nila

Jyeshthadeva was a Namboodiri of the Parangngottu family, whose house stood near Alathiyur on the banks of the Nila, the Bharathapuzha, in the heart of the Kerala school’s country. A note in a later astronomical manuscript, found by K. V. Sarma, places his birth around 1500 and says he lived to a great age. He studied under Damodara, son of the astronomer Parameshvara, and was a younger contemporary of Nilakantha, whose Tantrasangraha dates from 1500. His own pupil Achyuta Pisharati in turn taught Melpathur Narayana Bhattathiri, the poet of the Narayaniyam. The Yuktibhasa is usually placed around 1530, though some put it a few decades later.

03From Madhava to Nilakantha

The Kerala school begins with Madhava of Sangamagrama, near today’s Irinjalakuda, who lived around 1340 to 1425. Almost none of his own writing on series survives; his results come down through his students. Parameshvara of Vatasseri spent fifty-five years observing eclipses and planets and wrote the Drigganita of 1431. His son Damodara taught both Nilakantha and Jyeshthadeva. Nilakantha Somayaji of Kundapura, Trikkantiyur, wrote the Tantrasangraha in 432 verses and a great commentary on the Aryabhatiya. The teaching went on for three more centuries, in the illams along the Nila, from guru to shishya, in Sanskrit verse and Malayalam explanation.

04Two parts, fifteen chapters

The first part is mathematics in seven chapters. It begins gently with the operations, ten problems on two numbers, fractions and the rule of three, then the kuttaka. Chapter six, on the circle, is the longest in the book and holds the infinite series. Chapter seven builds sines and cosines and gives their series, with the cyclic quadrilateral. The second part, eight chapters, applies all this to the sky: the true planets, the spheres of earth and heaven, fifteen problems of spherical astronomy, the gnomon’s shadow, eclipses, vyatipata, the visibility correction, and the horns of the moon.

05The series for π

Chapter six takes a quarter circle inside a square, divides one side into ever smaller equal pieces, and adds up the tiny arcs cut off by lines drawn to the centre. To sum them it needs the sums of the powers of the first n numbers, and it shows that as n grows, the sum of the k-th powers approaches n raised to k+1, divided by k+1. From this comes Madhava’s series: the circumference is four times the diameter, minus four times a third of it, plus a fifth, minus a seventh, and so on without end. Because the plain series closes in slowly, the text derives correction terms for the remainder, which make a few terms give many correct decimals.

06Sine, cosine and arctangent

The same method, applied to an arc of any size, gives the series for the arc in terms of its tangent, which Europe later called Gregory’s series; set the arc to one-eighth of the circle and the π series falls out. Chapter seven then takes a small arc, finds how sine and cosine change across it, and by summing again reaches the series for sine and cosine: the sine is the arc, less the arc cubed over six, plus the arc to the fifth over one hundred and twenty, and on. Madhava used such series to compute a sine table correct to many places, and a value of π correct to eleven decimals is credited to him.

07The astronomy part

The second half explains Nilakantha’s revised model of the planets. In the Tantrasangraha Nilakantha had changed the old rule for Mercury and Venus, so that all five planets move round the mean sun, which itself goes round the earth. Jyeshthadeva gives the geometry behind it, then the reasons for the rules of shadows, latitudes and eclipses, with the earth’s shadow and the moon’s parallax worked out on the sphere. He explains the visibility correction, drik-karma, and how to compute the height of the moon’s horns. The aim throughout is to show why the numbers in the panchanga come out as they do.

08Brought to light

Charles Whish of the Madras civil service described the Kerala series in a paper to the Royal Asiatic Society published in 1834, naming the Tantrasangraha and the Yuktibhasha among his sources, but little followed for a century. In 1948 Rama Varma Maru Thampuran and A. R. Akhileswara Aiyer published the mathematics part in Malayalam with notes. K. V. Sarma edited the whole text, and his English translation with explanatory notes by K. Ramasubramanian, M. D. Srinivas and M. S. Sriram appeared in two volumes in 2008. A Sanskrit version, the Ganita-yukti-bhasha, also survives, though who made it is not known.

09Kerala and Europe

The series for π, sine, cosine and arctangent reach Europe in the later seventeenth century, with James Gregory, Isaac Newton and Gottfried Leibniz. Jesuit missionaries were in Kerala from the sixteenth century, learning Malayalam and studying the local calendar, and some historians have asked whether Kerala mathematics travelled west with them. No document has yet been found that proves it, and the question stays open. What is certain is the date: the proofs in the Yuktibhasa were written down more than a century before Europe had them, and historians of mathematics now cite it among the earliest texts anywhere to argue step by step towards the ideas of the calculus.

What it contains

1. 1
   1. Parikarma

   The basic operations: addition, subtraction, multiplication, division, squares and square roots. Each is explained with the reason it works, which sets the tone for the whole book.

2. 2
   2. Dashaprashna

   Ten problems in which two quantities are found from any two of their sum, difference, product, sum of squares and difference of squares. The proofs are drawn as rectangles and squares.

3. 3
   3. Bhinnaganita

   Fractions and the operations on them. This prepares for the long sums of fractions in the chapter on the circle.

4. 4
   4. Trairashika

   The rule of three and why proportion works. Jyeshthadeva treats it as the root of nearly every computation in astronomy.

5. 5
   5. Kuttakara

   Indeterminate equations of the first degree, solved by the pulveriser. The reasoning shows why the chain of divisions must end in a whole-number answer.

6. 6
   6. Paridhi and vyasa

   The circle and its circumference: the infinite series for π, the sums of powers of integers, the arctangent series, and correction terms for faster results. It is the longest and most cited chapter.

7. 7
   7. Jya

   Sines and cosines: how to derive them, their infinite series, and the sine of a sum of arcs. It also gives the diagonals, area and circumradius of a cyclic quadrilateral.

8. 8
   8. Grahagati

   Mean and true planets, with the reasons for the manda and shighra corrections. Nilakantha’s revised model of the planets is explained here.

9. 9
   9. Bhugola and bhagola

   The sphere of the earth and the sphere of the stars, and the circles drawn on them. It prepares the ground for spherical trigonometry.

10. 10
    10. Fifteen problems

    Fifteen problems relating the sun’s longitude, declination, right ascension and other quantities on the sphere. Each is solved from two given quantities.

11. 11
    11. Chhaya

    Directions, latitude and time from the shadow of the gnomon. The reasons are given for the rules used in daily reckoning.

12. 12
    12. Grahana

    Eclipses of the sun and moon, with the earth’s shadow and the moon’s parallax. It explains why eclipse timings need the corrections they do.

13. 13
    13. Vyatipata

    The moments when the declinations of the sun and moon are equal, which the panchanga marks as vyatipata. The text shows how to find them.

14. 14
    14. Drikkarma

    The visibility correction: how a planet’s place on the ecliptic is turned into the moment it rises or sets for an observer. It links calculation to what the eye sees.

15. 15
    15. Shringonnati

    The elevation of the moon’s horns: which tip of the crescent stands higher on a given evening. It closes the book.