contains most of the developments of the earlier Kerala school, particularly those of
Madhava and
Nilakantha Somayaji. The text is divided into two parts – the former deals with
mathematical analysis and the latter with astronomy. The first four chapters of the section contain elementary mathematics, such as division, the
Pythagorean theorem,
square roots, etc. Novel ideas are not discussed until the sixth chapter on the
circumference of a
circle. contains a derivation and proof for the
power series of
inverse tangent discovered by Madhava. In the text, Jyeṣṭhadeva describes Madhava's series in the following manner: In modern mathematical notation, : 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, :\theta = \tan\theta - \frac13 \tan^3\theta + \frac15 \tan^5\theta - \cdots \ , which in Europe was conventionally called ''
Gregory's series'' after
James Gregory, who independently discovered 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. :\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 independently discovered it in 1673. Using a rational approximation of this series, Jyeṣṭhadeva gave values of 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 :\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\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\frac{n^2 + 1}{4n^3 + 5n} was used in the infinite series expansion of \frac{\pi}{4} to improve the approximation of π to 13 decimal places of accuracy when
n=76. Apart from these, the 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 • 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. 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. Specifically,
grahagati: planetary motion,
bhagola: sphere of the zodiac,
madhyagraha: mean planets,
sūryasphuṭa: true sun,
grahasphuṭa: true planets
bhū-vāyu-bhagola: spheres of the earth, atmosphere, and asterisms,
ayanacalana:
precession of the equinoxes pañcadaśa-praśna: fifteen problems relating to
spherical triangles dig-jñāna: orientation,
chāyā-gaṇita: shadow computations,
lagna: rising point of the
ecliptic,
nati-lambana: parallaxes of latitude and longitude
grahaṇa: eclipse
vyatīpāta visibility correction of planets moon's cusps and phases of the moon ==Modern editions==