Yuktibhasa ( ; meaning —
rationale language) also known as Ganita
Yuktibhasa (Mathematical Rationale Language), is a
major treatise on Mathematics and Astronomy, written by Indian
astronomer
Jyesthadeva of the Kerala School of Mathematics in about AD
1530. The treatise is a consolidation of the discoveries by
Madhava of Sangamagrama,
Nilakantha Somayaji,
Parameswara,
Jyeshtadeva,
Achyuta Pisharati and other
astronomer-mathematicians at the Kerala School.
Yuktibhasa
is mainly based on Nilakantha's
Tantra Samgraha. It is
considered as the first text on
calculus
and predates those of
European mathematicians
such as
James Gregory
by over a century.
However, the treatise was largely unnoticed
beyond Kerala
, as the book
was written in the local language of Malayalam. However,
some have argued that mathematics from Kerala were transmitted to
Europe (see
Possible transmission of Keralese mathematics to Europe).
The work was unique for its time, since it contained
proof and derivations of the
theorems that it presented; something that was not
usually done by any
mathematicians of
that era. Some of its important developments in analysis include:
the
infinite series expansion of a
function, the
power series, the
Taylor series, the
trigonometric series of
sine,
cosine,
tangent and
arctangent, the second and third order Taylor
series approximations of
sine and
cosine, the power series of
π,
π/4,
θ, the radius, diameter and
circumference, and
tests
of convergence.
Contents
Yuktibhasa contains most of the developments of earlier
Kerala School mathematicians, particularly
Madhava and
Nilakantha. The text is divided into two
parts — the former deals with
mathematical analysis of
arithmetic,
algebra,
trigonometry and
geometry,
Logistics,
algebraic problems,
fraction,
Rule of Three,
Kuttakaram,
circle and disquisition
on R-Sine; and the latter about astronomy.
Mathematics
As per the old Indian tradition of starting off new chapters with
elementary content, the first four chapters of the
Yuktibhasa contain elementary mathematics, such as
division, proof of
Pythagorean
theorem,
square root determination,
etc. The radical ideas are not discussed until the sixth chapter on
circumference of a
circle.
Yuktibhasa contains the derivation
and proof of the
power series for
inverse tangent,
discovered by Madhava. In the text, Jyesthadeva describes Madhava's
series in the following manner:
This yields r\theta={\frac {r \sin \theta }{\cos \theta
}}-(1/3)\,r\,{\frac { \left(\sin \theta \right) ^
{3}}{ \left(\cos \theta \right) ^{3}}}+(1/5)\,r\,{\frac {
\left(\sin \theta \right) ^{5}}{ \left(\cos
\theta \right) ^{5}}}-(1/7)\,r\,{\frac { \left(\sin \theta
\right) ^{7}}{ \left(\cos \theta \right) ^{
7}}} + \ldots
which further yields the theorem
- \theta = \tan \theta - (1/3) \tan^3 \theta + (1/5) \tan^5
\theta - \ldots
popularly attributed to
James Gregory,
who discovered it three centuries after Madhava. This series was
traditionally known as the Gregory series but scholars have
recently begun referring to it as the Madhava-Gregory series, in
recognition of Madhava's work.
The text also elucidates Madhava's
infinite series expansion of
π:
- \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} +
\cdots + \frac{(-1)^n}{2n + 1} + \cdots
which he obtained from the power series expansion of the
arc-tangent function.
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. These
were the most accurate approximations of π after almost a thousand
years.
The text describes that he gave two methods for computing the value
of π.
- One of these methods is to obtain a rapidly converging series
by transforming the original infinite series of π. By doing so, he
obtained the infinite series
- \pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot
3^2}-{1\over7\cdot 3^3}+\cdots\right)
and used the first 21 terms to compute an approximation of π
correct to 11 decimal places as 3.14159265359.
- The other method was to add a remainder term to the original
series of π. The remainder term was used
- \frac{n^2 + 1}{4n^3 + 5n}
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
Yukthibhasa contains many
elementary, and complex mathematics,
including,
- Proof for the expansion of the sine and
cosine functions.
- Integer solutions of systems of first degree equations (solved
using a system known as kuttakaram)
- Rules for finding the sines and the cosines of the sum and
difference of two angles.
- The earliest statement of Wallis
product and the Taylor
series.
- Geometric derivations of series.
- Tests of convergence (often
attributed to Cauchy)
- Fundamentals of calculus: differentiation, term by term integration, iterative
methods for solutions of non-linear
equations, and the theory that the area under a curve is its
integral.
Most of these results were long before their European counterparts,
to whom credit was traditionally attributed.
Astronomy
Chapters seven to seventeen of the text deals essentially with
subjects of astronomy, viz.
Planetary
orbit,
Celestial sphere,
ascension,
declination, directions and shadows,
spherical triangle,
ellipses and
parallax
correction. The planetary theory described in the book is similar
to that later adopted by
Danish
astronomer
Tycho Brahe.
See also
Notes
References
External links
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