Madhava series

In mathematics, a Madhava series or Leibniz series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics and later by Gottfried Wilhelm Leibniz, among others. These expressions are the Maclaurin series expansions of the trigonometric sine, cosine and arctangent functions, and the special case of the power series expansion of the arctangent function yielding a formula for computing π. The power series expansions of sine and cosine functions are respectively called Madhava's sine series and Madhava's cosine series. The power series expansion of the arctangent function is sometimes called Madhava–Gregory series or Gregory–Madhava series. These power series are also collectively called Taylor–Madhava series. The formula for π is referred to as Madhava–Newton series or Madhava–Leibniz series or Leibniz formula for pi or Leibnitz–Gregory–Madhava series. These further names for the various series are reflective of the names of the Western discoverers or popularizers of the respective series.

The derivations use many calculus related concepts such as summation, rate of change, and interpolation, which suggests that Indian mathematicians had a solid understanding of the concept of limit and the basics of calculus long before they were developed in Europe. Other evidence from Indian mathematics up to this point such as interest in infinite series and the use of a base ten decimal system also suggest that it was possible for calculus to have developed in India almost 300 years before its recognized birth in Europe.

No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava series. However, in the writing of later members of the Kerala school of astronomy and mathematics like Nilakantha Somayaji and Jyeshthadeva one can find unambiguous attributions of these series to Madhava. It is also in the works of these later astronomers and mathematicians one can trace the Indian proofs of these series expansions. These proofs provide enough indications about the approach Madhava had adopted to arrive at his series expansions.

Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although the number 1 can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc., (as even the ancient Egyptians and Greeks had known), the exact total of 1 can only be achieved by adding up infinitely many fractions. But Madhava went further and linked the idea of an infinite series with geometry and trigonometry. He realized that, by successively adding and subtracting different odd number fractions to infinity, he could home in on an exact formula for pi (this was two centuries before Leibniz was to come to the same conclusion in Europe).

Madhava's series in modern notations

In the writings of the mathematicians and astronomers of the Kerala school, Madhava's series are described couched in the terminology and concepts fashionable at that time. When we translate these ideas into the notations and concepts of modern-day mathematics, we obtain the current equivalents of Madhava's series. These present-day counterparts of the infinite series expressions discovered by Madhava are the following:

No.	Series	Name	Western discoverers of the series and approximate dates of discovery
1	sin x = x − ⁠x3/3!⁠ + ⁠x5/5!⁠ − ⁠x7/7!⁠ + ...	Madhava's sine series	Isaac Newton (1670) and Wilhelm Leibniz (1676)
2	cos x = 1 − ⁠x2/2!⁠ + ⁠x4/4!⁠ − ⁠x6/6!⁠ + ...	Madhava's cosine series	Isaac Newton (1670) and Wilhelm Leibniz (1676)
3	arctan x = x − ⁠x3/3⁠ + ⁠x5/5⁠ − ⁠x7/7⁠ + ...	Madhava's series for arctangent	James Gregory (1671) and Wilhelm Leibniz (1676)
4	⁠π/4⁠ = 1 − ⁠1/3⁠ + ⁠1/5⁠ − ⁠1/7⁠ + ...	Madhava's formula for π	James Gregory (1671) and Wilhelm Leibniz (1676)

Madhava series in "Madhava's own words"

None of Madhava's works, containing any of the series expressions attributed to him, have survived. These series expressions are found in the writings of the followers of Madhava in the Kerala school. At many places these authors have clearly stated that these are "as told by Madhava". Thus the enunciations of the various series found in Tantrasamgraha and its commentaries can be safely assumed to be in "Madhava's own words". The translations of the relevant verses as given in the Yuktidipika commentary of Tantrasamgraha (also known as Tantrasamgraha-vyakhya) by Sankara Variar (circa. 1500 - 1560 CE) are reproduced below. These are then rendered in current mathematical notations.

Madhava's sine series

In Madhava's own words

Madhava's sine series is stated in verses 2.440 and 2.441 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows.

Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even numbers increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva, as collected together in the verse beginning with "vidvan" etc.

Rendering in modern notations

Let r denote the radius of the circle and s the arc-length.

• The following numerators are formed first:

  $s\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2}\cdot s^{2},\qquad \cdots$

• These are then divided by quantities specified in the verse.

  $s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}},\qquad s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}},\qquad s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}+6)r^{2}}},\qquad \cdots$

• Place the arc and the successive results so obtained one below the other, and subtract each from the one above to get jiva:

  ${\text{jiva}}=s-\left[s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}-\left[s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}-\left[s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}+6)r^{2}}}-\cdots \right]\right]\right]$

Transformation to current notation

Let θ be the angle subtended by the arc s at the centre of the circle. Then s = r θ and jiva = r sin θ. Substituting these in the last expression and simplifying we get

$\sin \theta =\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\quad \cdots$

which is the infinite power series expansion of the sine function.

Madhava's reformulation for numerical computation

The last line in the verse ′as collected together in the verse beginning with "vidvan" etc.′ is a reference to a reformulation of the series introduced by Madhava himself to make it convenient for easy computations for specified values of the arc and the radius. For such a reformulation, Madhava considers a circle one quarter of which measures 5400 minutes (say C minutes) and develops a scheme for the easy computations of the jiva′s of the various arcs of such a circle. Let R be the radius of a circle one quarter of which measures C. Madhava had already computed the value of π using his series formula for π. Using this value of π, namely 3.1415926535922, the radius R is computed as follows: Then

R = 2 × 5400 / π = 3437.74677078493925 = 3437 arcminutes 44 arcseconds 48 sixtieths of an arcsecond = 3437′ 44′′ 48′′′.

Madhava's expression for jiva corresponding to any arc s of a circle of radius R is equivalent to the following:

${\begin{aligned}{\text{jiva }}&=s-{\frac {s^{3}}{R^{2}(2^{2}+2)}}+{\frac {s^{5}}{R^{4}(2^{2}+2)(4^{2}+4)}}-\cdots \\[6pt]&=s-\left({\frac {s}{C}}\right)^{3}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{3}}{3!}}-\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{5}}{5!}}-\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{7}}{7!}}-\cdots \right]\right]\right].\end{aligned}}$

Madhava now computes the following values:

No.	Expression	Value	Value in Katapayadi system
1	R × (π / 2)3 / 3!	2220′   39′′   40′′′	ni-rvi-ddhā-nga-na-rē-ndra-rung
2	R × (π / 2)5 / 5!	273′   57′′   47′′′	sa-rvā-rtha-śī-la-sthi-ro
3	R × (π / 2)7 / 7!	16′   05′′   41′′′	ka-vī-śa-ni-ca-ya
4	R × (π / 2)9 / 9!	33′′   06′′′	tu-nna-ba-la
5	R × (π / 2)11 / 11!	44′′′	vi-dvān

The jiva can now be computed using the following scheme:

jiva = s − (s / C)³ [ (2220′ 39′′ 40′′′) − (s / C)² [ (273′ 57′′ 47′′′) − (s / C)² [ (16′ 05′′ 41′′′) − (s / C)²[ (33′′ 06′′′) − (s / C)² (44′′′ ) ] ] ] ].

This gives an approximation of jiva by its Taylor polynomial of the 11'th order. It involves one division, six multiplications and five subtractions only. Madhava prescribes this numerically efficient computational scheme in the following words (translation of verse 2.437 in Yukti-dipika):

vi-dvān, tu-nna-ba-la, ka-vī-śa-ni-ca-ya, sa-rvā-rtha-śī-la-sthi-ro, ni-rvi-ddhā-nga-na-rē-ndra-rung . Successively multiply these five numbers in order by the square of the arc divided by the quarter of the circumference (5400′), and subtract from the next number. (Continue this process with the result so obtained and the next number.) Multiply the final result by the cube of the arc divided by quarter of the circumference and subtract from the arc.

Madhava's cosine series

In Madhava's own words

Madhava's cosine series is stated in verses 2.442 and 2.443 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows.

Multiply the square of the arc by the unit (i.e. the radius) and take the result of repeating that (any number of times). Divide (each of the above numerators) by the square of the successive even numbers decreased by that number and multiplied by the square of the radius. But the first term is (now)(the one which is) divided by twice the radius. Place the successive results so obtained one below the other and subtract each from the one above. These together give the śara as collected together in the verse beginning with stena, stri, etc.

Rendering in modern notations

Let r denote the radius of the circle and s the arc-length.

• The following numerators are formed first:

$r\cdot s^{2},\qquad r\cdot s^{2}\cdot s^{2},\qquad r\cdot s^{2}\cdot s^{2}\cdot s^{2},\qquad \cdots$

• These are then divided by quantities specified in the verse.

$r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}},\qquad r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}-4)r^{2}}},\qquad r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}-4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}-6)r^{2}}},\qquad \cdots$

• Place the arc and the successive results so obtained one below the other, and subtract each from the one above to get śara:

${\text{sara}}=r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}-\left[r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}-4)r^{2}}}-\left[r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}-4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}-6)r^{2}}}-\cdots \right]\right]$

Transformation to current notation

Let θ be the angle subtended by the arc s at the centre of the circle. Then s = rθ and śara = r(1 − cos θ). Substituting these in the last expression and simplifying we get

$1-\cos \theta ={\frac {\theta ^{2}}{2!}}-{\frac {\theta ^{4}}{4!}}+{\frac {\theta ^{6}}{6!}}+\quad \cdots$

which gives the infinite power series expansion of the cosine function.

Madhava's reformulation for numerical computation

The last line in the verse ′as collected together in the verse beginning with stena, stri, etc.′ is a reference to a reformulation introduced by Madhava himself to make the series convenient for easy computations for specified values of the arc and the radius. As in the case of the sine series, Madhava considers a circle one quarter of which measures 5400 minutes (say C minutes) and develops a scheme for the easy computations of the śara′s of the various arcs of such a circle. Let R be the radius of a circle one quarter of which measures C. Then, as in the case of the sine series, Madhava gets R = 3437′ 44′′ 48′′′.

Madhava's expression for śara corresponding to any arc s of a circle of radius R is equivalent to the following:

${\begin{aligned}{\text{jiva }}&=R\cdot {\frac {s^{2}}{R^{2}(2^{2}-2)}}-R\cdot {\frac {s^{4}}{R^{4}(2^{2}-2)(4^{2}-4)}}-\cdots \\&=\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{2}}{2!}}-\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{4}}{4!}}-\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{6}}{6!}}-\cdots \right]\right]\right]\end{aligned}}$

Madhava now computes the following values:

No.	Expression	Value	Value in Katapayadi system
1	R × (π / 2)2 / 2!	4241′   09′′   00′′′	u-na-dha-na-krt-bhu-re-va
2	R × (π / 2)4 / 4!	872′   03′′   05 ′′′	mī-nā-ngo-na-ra-sim-ha
3	R × (π / 2)6 / 6!	071′   43′′   24′′′	bha-drā-nga-bha-vyā-sa-na
4	R × (π / 2)8 / 8!	03′   09′′   37′′′	su-ga-ndhi-na-ga-nud
5	R × (π / 2)10 / 10!	05′′   12′′′	strī-pi-śu-na
6	R × (π / 2)12 / 12!	06′′′	ste-na

The śara can now be computed using the following scheme:

śara = (s / C)² [ (4241′ 09′′ 00′′′) − (s / C)² [ (872′ 03′′ 05 ′′′) − (s / C)² [ (071′ 43′′ 24′′′) − (s / C)²[ (03′ 09′′ 37′′′) − (s / C)² [(05′′ 12′′′) − (s / C)² (06′′′) ] ] ] ] ]

This gives an approximation of śara by its Taylor polynomial of the 12'th order. This also involves one division, six multiplications and five subtractions only. Madhava prescribes this numerically efficient computational scheme in the following words (translation of verse 2.438 in Yukti-dipika):

The six stena, strīpiśuna, sugandhinaganud, bhadrāngabhavyāsana, mīnāngonarasimha, unadhanakrtbhureva. Multiply by the square of the arc divided by the quarter of the circumference and subtract from the next number. (Continue with the result and the next number.) Final result will be utkrama-jya (R versed sign).

Madhava's arctangent series

In Madhava's own words

Madhava's arctangent series is stated in verses 2.206 – 2.209 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses is given below. Jyesthadeva has also given a description of this series in Yuktibhasa.

Now, by just the same argument, the determination of the arc of a desired sine can be (made). That is as follows: The first result is the product of the desired sine and the radius divided by the cosine of the arc. When one has made the square of the sine the multiplier and the square of the cosine the divisor, now a group of results is to be determined from the (previous) results beginning from the first. When these are divided in order by the odd numbers 1, 3, and so forth, and when one has subtracted the sum of the even(-numbered) results from the sum of the odd (ones), that should be the arc. Here the smaller of the sine and cosine is required to be considered as the desired (sine). Otherwise, there would be no termination of results even if repeatedly (computed).

By means of the same argument, the circumference can be computed in another way too. That is as (follows): The first result should by the square root of the square of the diameter multiplied by twelve. From then on, the result should be divided by three (in) each successive (case). When these are divided in order by the odd numbers, beginning with 1, and when one has subtracted the (even) results from the sum of the odd, (that) should be the circumference.

Rendering in modern notations

Let s be the arc of the desired sine (jya or jiva) y. Let r be the radius and x be the cosine (kotijya).

• The first result is ${\tfrac {y\cdot r}{x}}$.
• Form the multiplier and divisor ${\tfrac {y^{2}}{x^{2}}}$.
• Form the group of results:

${\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}},\qquad {\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}},\qquad \cdots$

• These are divided in order by the numbers 1, 3, and so forth:

${\frac {1}{1}}{\frac {y\cdot r}{x}},\qquad {\frac {1}{3}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}},\qquad {\frac {1}{5}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}},\qquad \cdots$

• Sum of odd-numbered results:

${\frac {1}{1}}{\frac {y\cdot r}{x}}+{\frac {1}{5}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}+\cdots$

• Sum of even-numbered results:

${\frac {1}{3}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}+{\frac {1}{7}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}+\cdots$

• The arc is now given by

$s=\left({\frac {1}{1}}{\frac {y\cdot r}{x}}+{\frac {1}{5}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}+\cdots \right)-\left({\frac {1}{3}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}+{\frac {1}{7}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}+\cdots \right)$

Transformation to current notation

Let θ be the angle subtended by the arc s at the centre of the circle. Then s = rθ, x = kotijya = r cos θ and y = jya = r sin θ. Then y / x = tan θ. Substituting these in the last expression and simplifying we get

• $\theta =\tan \theta -{\frac {\tan ^{3}\theta }{3}}+{\frac {\tan ^{5}\theta }{5}}-{\frac {\tan ^{7}\theta }{7}}+\quad \cdots$.

Letting tan θ = q we finally have

• $\tan ^{-1}q=q-{\frac {q^{3}}{3}}+{\frac {q^{5}}{5}}-{\frac {q^{7}}{7}}+\quad \cdots$

Another formula for the circumference of a circle

The second part of the quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows.

$c={\sqrt {12d^{2}}}-{\frac {\sqrt {12d^{2}}}{3\cdot 3}}+{\frac {\sqrt {12d^{2}}}{3^{2}\cdot 5}}-{\frac {\sqrt {12d^{2}}}{3^{3}\cdot 7}}+\quad \cdots$

Since c = π d this can be reformulated as a formula to compute π as follows.

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

This is obtained by substituting q = $1/{\sqrt {3}}$ (therefore θ = π / 6) in the power series expansion for tan⁻¹ q above.

Comparison of the convergence of two Madhava series (the one with √12 in dark blue) and several historical infinite series for π. S_n is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail)

See also

• Madhava of Sangamagrama
• Madhava's sine table
• Padé approximant
• Taylor series
• Laurent series
• Puiseux series

Further reading

• Joseph, George Gheverghese (October 2010) [1991]. The Crest of the Peacock: Non-European Roots of Mathematics (3rd ed.). Princeton University Press. ISBN 978-0-691-13526-7.
• K. V. Sarma, A History of the Kerala School of Hindu Astronomy (Hoshiarpur, 1972).
• A. K. Bag, Madhava's sine and cosine series, Indian J. History Sci. 11 (1) (1976), 54–57.
• D. Gold and D Pingree, A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine, Historia Sci. No. 42 (1991), 49–65.
• R. C. Gupta, Madhava's and other medieval Indian values of pi, Math. Education 9 (3) (1975), B45–B48.
• R. C. Gupta, Madhava's power series computation of the sine, Ganita 27 (1–2) (1976), 19–24.
• R. C. Gupta, On the remainder term in the Madhava–Leibniz's series, Ganita Bharati 14 (1–4) (1992), 68–71.
• R. C. Gupta, The Madhava–Gregory series, Math. Education 7 (1973), B67–B70.
• T. Hayashi, T. Kusuba and M. Yano, The correction of the Madhava series for the circumference of a circle, Centaurus 33 (2–3) (1990), 149–174.
• R. C. Gupta, The Madhava–Gregory series for tan⁻¹x, Indian Journal of Mathematics Education, 11(3), 107–110, 1991.
• Kim Plofker (2009). Mathematics in India. Princeton: Princeton University Press. pp. 217–254. ISBN 978-0-691-12067-6.
• "The discovery of the series formula for π by Leibniz, Gregory, and Nilakantha" by Ranjan Roy in :  Marlow Anderson; Victor Katz; Robin Wilson, eds. (2004). Sherlock Holmes in Babylon and other tales of mathematical history. The Mathematical Association of America. pp. 111–121. ISBN 0-88385-546-1.
• "Ideas of calculus in Islam and India" by Victor J Katz in :  Marlow Anderson; Victor Katz; Robin Wilson, eds. (2004). Sherlock Holmes in Babylon and other tales of mathematical history. The Mathematical Association of America. pp. 122–130. ISBN 0-88385-546-1.
• "Was calculus invented in India?" by David Bressoud in :  Marlow Anderson; Victor Katz; Robin Wilson, eds. (2004). Sherlock Holmes in Babylon and other tales of mathematical history. The Mathematical Association of America. pp. 131–137. ISBN 0-88385-546-1.
• Victor J Katz, ed. (2007). "Chapter 4 : Mathematics in India IV. Kerala School". The mathematics of Egypt, Mesopotemia, China, India and Islam: A source book. Princeton: Princeton University Press. pp. 480–495. ISBN 978-0-691-11485-9.
• Glen Van Brummelen (2009). The mathematics of the heavens and the earth : the early history of trigonometry. Princeton: Princeton University Press. pp. 113–120. ISBN 978-0-691-12973-0.
• D. Pouvreau, Trigonométrie et "développements en séries" en Inde médiévale, I.R.E.M. de l'Université de Toulouse III (2003), 162 pages. OCLC 758823300
• D. Pouvreau, "Sur l'accélération de la convergence de la série de Madhava-Leibniz", Quadrature, n°97 (2015), pp. 17–25. ISBN 978-2-7598-0528-0