Yuktibhāṣā

Yuktibhasa (Malayalam:യുക്തിഭാഷ ; meaning — rationale language ) also known as Ganita Yuktibhasa (compendium of astronomical rationale), is a major treatise on Mathematics and Astronomy, written by Indian astronomer Jyesthadeva of the Kerala School of Mathematics in AD 1530. The treatise is a consolidation of the discoveries of the works of Madhava of Sangamagrama, Nilakantha Somayaji, Parameswara, Jyeshtadeva, Achyuta Panikkar 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. The work predates those of European mathematicians by over three centuries. However, the treatise was largely unnoticed beyond Kerala, as the book was written in the local language of Malayalam.

The work was unique for its time for containing exacting proofs of the theorems it presented. 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, diamater 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, fractions, Rule of Three, Kuttakaram, circle and disquisition on R-Sine; and the latter about astronomy.

Mathematics

Explanation of the sine rule in Yuktibhasa

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:

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.

This yields $r\theta ={\frac {r\ sin\theta }{cos\theta }}-(1/3)\,r\,{\frac {\left(sin\theta \right)^{3}}{\left(cos\theta \right)^{3}}}+(1/5)\,r\,{\frac {\left(sin\theta \right)^{5}}{\left(cos\theta \right)^{5}}}-(1/7)\,r\,{\frac {\left(sin\theta \right)^{7}}{\left(cos\theta \right)^{7}}}+...$

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

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

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 achieved centuries before their European counterparts, to whom the credit is often misattributed.

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 triangles, ellipses and parallax correction. The planetary theory described in the book is similar to that later adopted by Danish astronomer Tycho Brahe.

See also

• Indian Mathematics
• Kerala School
• Possible transmission of Kerala mathematics to Europe
• Eurocentrism

External links

• Biography of Jyesthadeva — School of Mathematics and Statistics University of St Andrews, Scotland

References

• K V Sharma & S Hariharan (1990). Yuktibhasa of Jyesthadeva — A book on rationales in Indian Mathematics and Astronomy - an analytic appraisal. Indian Journal of History of Science.