Mathematics

Explanation of the sine rule in Yuktibhāṣā

This subjects treated in the mathematics part of the Yuktibhāṣā can be divided into seven chapters:^{[1]: xxxvii}

1. parikarma: logistics (the eight mathematical operations)
2. daśapraśna: ten problems involving logistics
3. bhinnagaṇita: arithmetic of fractions
4. trairāśika: rule of three
5. kuṭṭakāra: pulverisation (linear indeterminate equations)
6. paridhi-vyāsa: relation between circumference and diameter: infinite series and approximations for the ratio of the circumference and diameter of a circle
7. jyānayana: derivation of Rsines: infinite series and approximations for sines.^[10]

The first four chapters of the contain elementary mathematics, such as division, the Pythagorean theorem, square roots, etc.^[11] Novel ideas are not discussed until the sixth chapter on circumference of a circle. Yuktibhāṣā contains a derivation and proof for the power series of inverse tangent, discovered by Madhava.^[5] In the text, Jyesthadeva describes Madhava's series in the following manner:

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.

In modern mathematical notation,

${\displaystyle r\theta ={r{\frac {\sin \theta }{\cos \theta }}}-{\frac {r}{3}}{\frac {\sin ^{3}\theta }{\cos ^{3}\theta }}+{\frac {r}{5}}{\frac {\sin ^{5}\theta }{\cos ^{5}\theta }}-{\frac {r}{7}}{\frac {\sin ^{7}\theta }{\cos ^{7}\theta }}+\cdots }$

or, expressed in terms of tangents,

${\displaystyle \theta =\tan \theta -{\frac {1}{3}}\tan ^{3}\theta +{\frac {1}{5}}\tan ^{5}\theta -\cdots \ ,}$

which in Europe was conventionally called Gregory's series after James Gregory, who rediscovered it in 1671.

The text also contains Madhava's infinite series expansion of π which he obtained from the expansion of the arc-tangent function.

${\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n}}{2n+1}}+\cdots \ ,}$

which in Europe was conventionally called Leibniz's series, after Gottfried Leibniz who rediscovered it in 1673.

Using a rational approximation of this series, he gave values of the number π as 3.14159265359, correct to 11 decimals, and as 3.1415926535898, correct to 13 decimals.

The text describes two methods for computing the value of π. First, obtain a rapidly converging series by transforming the original infinite series of π. By doing so, the first 21 terms of the infinite series

${\displaystyle \pi ={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)}$

was used to compute the approximation to 11 decimal places. The other method was to add a remainder term to the original series of π. The remainder term${\textstyle {\frac {n^{2}+1}{4n^{3}+5n}}}$ was used in the infinite series expansion of ${\displaystyle {\frac {\pi }{4}}}$ to improve the approximation of π to 13 decimal places of accuracy when n=76.

Apart from these, the Yuktibhāṣā contains many elementary and complex mathematical topics, including,

• Proofs for the expansion of the sine and cosine functions
• The sum and difference formulae for sine and cosine
• Integer solutions of systems of linear equations (solved using a system known as kuttakaram)
• Geometric derivations of series
• Early statements of Taylor series for some functions

Astronomy

Chapters eight to seventeen deal with subjects of astronomy: planetary orbits, celestial spheres, ascension, declination, directions and shadows, spherical triangles, ellipses, and parallax correction. The planetary theory described in the book is similar to that later adopted by Danish astronomer Tycho Brahe.^[12] The topics covered in the eight chapters are computation of mean and true longitudes of planets, Earth and celestial spheres, fifteen problems relating to ascension, declination, longitude, etc., determination of time, place, direction, etc., from gnomonic shadow, eclipses, Vyatipata (when the sun and moon have the same declination), visibility correction for planets and phases of the moon.^[10]

Specifically,^{[1]: xxxviii}

8. grahagati: planetary motion, bhagola: sphere of the zodiac, madhyagraha: mean planets, sūryasphuṭa: true sun, grahasphuṭa: true planets
9. bhū-vāyu-bhagola: spheres of the earth, atmosphere, and asterisms, ayanacalana: precession of the equinoxes
10. pañcadaśa-praśna: fifteen problems relating to spherical triangles
11. dig-jñāna: orientation, chāyā-gaṇita: shadow computations, lagna: rising point of the ecliptic, nati-lambana: parallaxes of latitude and longitude
12. grahaṇa: eclipse
13. vyatīpāta
14. visibility correction of planets
15. moon's cusps and phases of the moon