Development of Calculus

8/22/2019 Development of Calculus

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KERALA SCHOOL OF ASTRONOMY

AND

DEVELOPMENT OF CALCULUS

M.D.SRINIVAS

CENTRE FOR POLICY STUDIES

[email protected]


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BACKGROUND TO THE DEVELOPMENT

OF CALCULUS (c.500-1350)

Zero, infinitesimals and infinity

Background1)The concept ofpra - ntimantra ofopaniad.2)The concept of lopa in Pini, abhva in Nyya and nya in

Bauddha philosophy.3)nya as symbol in Chanda Stra (VIII.29) of Pigala (c.300BC)

Brahmagupta (c.628) on the mathematics of zero. The notion oftaccheda.

Bhskara II (c.1150) on the notion ofkhahara or ananta


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Bhskarcrya, while discussing the mathematics of zero inLlvat, notes

that when further operations are contemplated, the quantity beingmultiplied by zero should not be changed to zero, but kept as is. Further,when the quantity which is multiplied by zero is also divided by zero, then

it remains unchanged. He follows this up with an example and declares

that this kind of calculation has great relevance in astronomy.

What is the number which when multiplied by zero, being added to half ofitself multiplied by three and divided by zero, amounts to sixty-three?

[ - ]


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Bhskara, it seems, had not fully mastered this kind of "calculation withinfinitesimals" as is clear from some of the examples he considers in

Bjagaita, while solving quadratic equations by eliminating the middle

term (ekavara-madhyamharaa).

[ ]

[{0 (x + (x/2)}2

+ 2 {0(x + (x/2))}]/ 0 = 15.Bhskara in his Vsan just cancels out the zeroes and obtainsx = 2.


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Irrationals and iterative approximations

Background1)ulva-stra approximation for square-root of 2.2)ulva-stra approximation for .

Square-root algorithm in ryabhaya (Gaita 4) of ryabhaa(c.499). rdhara (c.850) on approximation of square-roots (Triatik

46).

Nryaa Paita (c.1356) on approximating square-roots by thesolutions ofvarga-prakti (Bjagaitvatasa 88).

Approximate value of (accurate to four decimal places) inryabhaya (Gaita 11). Successive approximations perhaps by

doubling of the circumscribing square, octagon etc. doubling, andcutting of corners (explained in Yuktibhand Kriykramakar).


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Second-order differences and interpolation in computation of

Rsines

Computation of Rsine-table (accurate to minutes in a circle ofcircumference 21,600 minutes) by the method of second-order

Rsine-differences in ryabhaya (Gtik 12, Gaita 12) of

ryabhaa (c.499).

Bj = Rsin (jh)

j = Bj+1 - Bj

j+1 - j = - Bj [(1 - 2)/ B1]

-Bj / B1

Second-order interpolation formula for finding arbitrary Rsinevalues in Khaakhdyaka (c. 665) of Brahmagupta.

Rsin (jh+

) =Bj + (/h) [(1/2)(

j +

j+1)(

/h)(

j-

j+1)/2]=Bj + (/h)(j+1 + j)/2 + (/h)2 (j+1 - j)/2

=Bj + (/h) j+1 + (/h) ((/h)-1)(j+1 - j)/2


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Summation of infinite geometric series

The geometric series 1 + 2 + ... 2n is summed in PigalasChanda-stra (c.300 BCE). Pigala also gives an algorithm forevaluating a positive integral power of a number in terms of anoptimal number of squaring and multiplication operations.

Mahvrcrya (c.850), in his Gaita-sra-sagraha gives thesum of a geometric series.

Vrasena (c. 816), in his Commentary Dhaval on theakhagama, has made use of the sum of the followinginfinite geometric series in his evaluation of the volume of thefrustum of a right circular cone:

1/4 + (1/4)2 + ... (1/4) n + .... = 1/3


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Ttklika-gati: Instantaneous velocity of a planet

Approximate formula for velocity (manda-gati) in terms ofRsine-differences was given by Bhskara I (c.630) and he alsocomments on its limitation (Laghu-bhskarya 2.14-15).

True velocity (sphua-manda-gati) in terms of Rcosine (as thederivative of Rsine) is given Laghu-mnasa (2.7) of Mujla (c.

932) andMah-siddhnta (3.15) ofryabhaa II (c. 950).

Bhskara II (c.1150) discusses the notion of instantaneousvelocity (ttklika-gati) and contrasts it with the so-called true

daily motion. He also evaluates the manda-gati and

ghra-gati(Vsan on Siddhnta-iromai 2.37-39).

Bhskara II notes the relation between maximum equation ofcentre (correction to displacement) and the vanishing of velocity

correction (Vsan on Siddhnta-iromai, Gola 4.3).


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Ttklika-gati: Instantaneous velocity of a planet

[ .- ]

[ . ]


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Surface area and volume of a sphere

In ryabhaya (Gaitapda 7), the volume of a sphere wasincorrectly estimated as the product of the area of a great circleby its square-root.

Bhskarcrya II (c.1150) has given the correct relation betweenthe diameter, the surface area and the volume of a sphere in his

Llvat.

In his V

san

commentary on Siddh

nta-

iromai

Bhskara hasalso presented justifications for these results. The surface area is

obtained by adding the areas of the sectors into which thesurface of the sphere is divided by a number of great circlesdrawn at equal distance. Volume of the sphere evaluated by

summing the volumes of pyramids with apex at the centre.


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NRYAA PAITA ON VRASAKALITA (c.1350)

ryabhaya, gives the sum of the sequence of natural numbers

1 + 2 + ... + n = n(n+1)/2

As also the result of first order repeated summation:

1.2/2 + 2.3/2 + ... + n(n+1)/2 = n(n+1)(n+2)/6

ryabhaas result for repeated summation was generalised to arbitraryorder by Nrayaa Paita (c.1350):


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THE KERALA SCHOOL OF ASTRONOMY (c.1350-1825)

Kerala traces its ancient mathematical traditions to Vararuci. There are

speculations that

ryabhaa hailed from Kerala. In the classical period,there were many great Astronomer-Mathematicians in Kerala such as

Haridatta (c.650-700), Devcrya (c.700), Govindasvmin (c.800),akaranryaa (c.850) and Udayadivkara (c.1100). However it was

Mdhava of Sagamagrma (near Ernakulam) who pioneered a new

School of Astronomy and Mathematics

Mdhava (c.1340-1425): Vevroha, Sphuacandrpti, and a few tracts

are all that is available apart from citations in later works.

Paramevara of Vaasseri (c.1360-1455), a disciple of Mdhava:

Dggaita, Goladpik, and commentaries on Sryasiddhnta,

ryabhaya, Mahbhskarya, Laghubhskarya, Laghumnasa and

Llvat and Siddhntadpik on Govindasvmins commentary onMhabhskarya.


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THE KERALA SCHOOL OF ASTRONOMY (c.1350-1825)

Nlakaha Somayjof Tkaiyr (c.1444-1555), student of Dmodara son

of Paramevara: Tantrasagraha, ryabhayabhya, Golasra,

Candracchygaita, Siddhntadarpaa, Jyotirmms and Grahasphu-

nayane Vikepavsan.

Jyehadeva of Parakroa (c.1500 - 1610), student of Dmodara: Gaita-

Yukti-Bh (c. 1530 in Malaylam),Dkkaraa.

Citrabhnu (c.1475-1550), student of Nlakaha: Karamta,

Ekaviatipranottara.

akara Vriyr of Tkuaveli (c.1500-1560), student of Citrabhnu:

Karaasra, commentaries Kriykramakar (c.1535) onLlvat, Yukti-dpik,

Kriykalpa (in Malaylam) andLaghuvivti on Tantrasagraha.Acyuta Pirai (c.1550-1621), student of Jyehadeva and teacher of

Nryana Bhatiri: Sphuanirayatantra, Karaottama, Rigola-sphuanti,

Malaylam commentary on Vevroha and a few tracts.

The school continued to flourish till early nineteenth century. Some of theimportant works are Karaapaddhati (c.1700?) of Putumana Somayj and

Sadratnamlofakaravarman (c.1774-1839).


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NLAKAHA ON THE IRRATIONALITY OF

One of the main motivations of the mathematical work of the Kerala

school is paridhi-vysa sambandha, obtaining accurately the relationbetween the circumference of a circle. ryabhaa (c.499) had given the

following approximate value for:

[ ]One hundred plus four multiplied by eight and added to sixty-two

thousand: This is the approximate measure of the circumference of a circle

whose diameter is twenty-thousand.

Thus, according to ryabhaa, 62832/20000=3.1416Nlakaha Somayaji in his ryabhaya-bhya, while discussing square-roots, explains that only the approximate value is given for as thetraditional methods for its evaluation of involve square-roots:


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NLAKAHA ON THE IRRATIONALITY OF

Later, Nlakaha states that the ratio of the circumference to the diameter

of a circle cannot be expressed as the ratio of two integers exactly.

?

...

"Why then has an approximate value been mentioned here instead of the

actual value? This is the explanation. Because the actual value cannot be

expressed. Why? Given a certain unit of measurement in which thediameter has no fractional part, the same measure when applied to

measure the circumference will certainly have a fractional part. ...Thus

when both are measured by the same unit they cannot both without

fractional parts. Even if you go a long way (by choosing smaller andsmaller units of measure) a small fractional part will remain. The import

[ofsanna] is that there will never be a situation where both are integral."


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NLAKAHA ON THE SUM OF INFINITE GEOMETRIC SERIES

Vrasena (c. 816), had made use of the sum of the following infinite

geometric series1/4+(1/4)

2+ (1/4)

n+. =1/3

This is proved in the ryabhaya-bhya by Nlakaha Somayj, who

makes use of this series for deriving an approximate expression for a small

arc in terms of the corresponding chord in a circle. Nlakaha begins hisdiscussion of the sum of the infinite geometric series by posing the issue

as follows:

?

"The entire series of powers of 1/4 adds up to just 1/3. How is it

known that [the sum of the series] increases only up to that [limiting

value] and that it actually does increase up to that [limiting value]?"


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NLAKAHA ON THE SUM OF INFINITE GEOMETRIC SERIES

Nlakaha obtains the sequence of results

1/3 = 1/4 + 1/(4.3)1/(4.3) = 1/(4.4) + 1/(4.4.3)1/(4.4.3) = 1/(4.4.4) + 1/(4.4.4.3)

and so on, from which he derives the general result

1/3 - [1/4 + (1/4)2 + ... + (1/4)n] = (1/4n)(1/3)Nlakaha then goes on to present the following crucial argument toderive the sum of the infinite geometric series: As we sum more terms, the

difference between 1/3 and sum of powers of 1/4 (as given by the right

hand side of the above equation), becomes extremely small, but neverzero. Only when we take all the terms of the infinite series together do we

obtain the equality

1/4 + (1/4)2 + ... + (1/4)n + ... = 1/3

Nlakaha uses the above series to prove the following relationbetween the cpa (arc), jy (Rasine and ara (Rversine) for small arc:

Cpa [(1+1/3) ara2 + Jy2 ]1/2


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BINOMIAL SERIES EXPANSION

In obtaining the accurate relation between circumference and diameter, the

binomial series expansion plays an important role.

Given three positive numbers a, b, c, with b > c. we have the identity

Now substituting for (b-c)/b on the right from the equation and iteratingwe get

If we set [(b-c)/c] =x, the above is the well-known binomial series


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BINOMIAL SERIES EXPANSION

The derivation of the binomial series as well as the other results that we

discuss are found in Gaita-Yukti-Bha and Kriykramakar.

As regards the binomial series they note that there is no logical end to the

process of iterations and that one may stop after having obtained results tothe desired accuracy when the later terms will only get smaller and

smaller. They also note that the latter will happen only when (b-c) < c

(which is the condition for the convergence of the binomial expansion).


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SUM OF INTEGRAL POWERS OF NATURAL NUMBERS

The derivation of the Mdhava series for also involves estimating, for

large n, the value of the sama-ghta-sakalita

Sn(k)=1k + 2k + ... n k

Explicit formulae were given by ryabhaa for k = 1, 2 and 3.

Sn(1)=1+2+...n

=n(n+1)/2

Sn(2)=1

2+2

2+... +n

2=n(n+1)(2n+1)/6

Sn(3)=1

3+2

3+... +n

3=[1

+2+... +n]

2 =[n(n+1)/2]

2

Gaita-Yukti-Bh and Kriykramakarderive the following estimate forthe general sama-ghta-sakalita:

Sn(k)= 1

k+2

k++n

k nk+1/(k+1) forlargen

They also give an estimate for the repeated summation (vra-sakalita)

Vn(1)=1+2+3+...+n =n(n+1)/2

Vn(k)=V1

(k1)+V2

(k1)++Vn

(k1) n

k+1/[(k+1)!] forlargen


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MDHAVA SERIES FOR

The following verses of Mdhava are cited in Yuktibh and

Kriykramar

, which also present a detailed derivation of the relationbetween diameter and the circumference:

The first verse gives the Mdhava (Leibniz) series


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MDHAVA SERIES FOR

Mdhava also gave the cp-karaa series (Gregory Series) giving the cpa

(arc) associated with any jy

(Rsine) [It is also noted that we must ensurethat numerator < denominator in each term]:


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MDHAVA SERIES FOR

By using the cpkaraa series for an arc equal to one-twelfth of the

circumference (30), Mdhava gets a different series (later discovered byAbraham Sharp) for the ratio of the circumference to the diameter:

For an arc s which is one-twelfth of the diameter


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END CORRECTION TERMS

The Mdhava series for the circumference of a circle (in terms of odd

numbersp = 1, 3, 5, ...)C=4d[11/3+....+ (1)

(p1)/21/p+...]

is an extremely slowly convergent series. In order to facilitate

computation, Mdhava has given a procedure of using end-correction

terms (antya-saskra), of the form

In fact, the famous verses of Mdhava, which give the relation between

the circumference and diameter, also include the end-correction term

C=4d[11/3+....+...+(1)(p1)/2

1/p

+ (1)(p+1)/2{(p+1)/2}/{(p+1)2+1}]


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END CORRECTION TERMS

Mdhava has also given a finer end-correction term

C=4d[11/3+....+...+(1)(p1)/2

1/p

+(1)(p+1)/2[{(p+1)/2}2+1]/[{(p+1)2+5}{(p+1)/2}]

To Mdhava is attributed a value of accurate to eleven decimal places

which is obtained by just computing fifty terms with the above correction.


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MDHAVA CONTINUED FRACTION FOR

Both Yuktibh and Kriykramargive a derivation of the end correction

terms given by Mdhava, which involve a careful estimate of the errorinvolved in terms of inverse powers of comparison of the odd numberp.

By carrying this process further, we find that the end-correction term 1/apcan be expressed as a continued fraction:

Using the above correction term forp = 1, we get what may be called the

Mdhava continued fraction for :

2/(4 )=2+12/2+ 2

2/2+ 3

2/2+...

Brouncker Continued Fraction (1656):


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TRANSFORMED SERIES FOR

Adding and subtracting the end-correction terms, we can write the

Mdhava series for in the form:

By choosing different correction terms we get different transformed series

many of which are also converge faster than the Mdhava (Leibniz) series.

If we choose what may be termed the zero-th order correction divisor,

ap = 2p+2, we get the series


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TRANSFORMED SERIES FOR

If we choose the first-order correction divisor given by Mdhava,

then we get the series

:

Yuktibh and Kriykramardo not discuss the transformed series when

we use the more accurate correction divisor given by Mdhava. We caneasily see that it involves the seventh powers of the odd numbers.


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A HISTORY OF APPROXIMATIONS TO

Approximation to Accuracy

(Decimalplaces)

Method Adopted

Rhind Papyrus - Egypt

(Prior to 2000 BCE)

256/81 = 3.1604 1 Geometrical

Babylon (2000 BCE) 25/8 = 3.125 1 Geometrical

ulva Stras (Prior to

800 BCE)

3.0883 1 Geometrical

Jaina Texts (500 BCE) (10) = 3.1623 1 Geometrical

Archimedes (250 BCE) 3 10/71 < < 3 1/7 2 Polygon doubling(6.2

4= 96 sides)

Ptolemy (150 CE) 3 17/120 = 3. 141666 3 Polygon doubling(6.2

6= 384 sides)

Lui Hui (263) 3.14159 5 Polygon doubling

(6.29

= 3072 sides)

Tsu Chhung-Chih

(480?)

355/113 = 3.1415929

3.1415927

6

7

Polygon doubling

(6.29

= 12288

sides)

ryabhaa (499) 62832/20000 = 3.1416 4 Polygon doubling(4.2

8= 1024 sides)


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A HISTORY OF APPROXIMATIONS TO

Approximation to Accuracy

(Decimalplaces)

Method Adopted

Mdhava (1375) 2827433388233/9.1011

= 3.141592653592

11 Infinite series with

end corrections

Al Kashi (1430) 3.1415926535897932 16 Polygon doubling (6.227

sides)

Francois Viete (1579) 3.1415926536 9 Polygon doubling (6. 216

sides)

Romanus (1593) 3.1415926535..... 15 Polygon doubling

Ludolph Van Ceulen

(1615)

3.1415926535..... 32 Polygon doubling (262

sides)

Wildebrod Snell

(1621)

3.1415926535..... 34 Modified polygon doubling

(2

30

sides)Grienberger (1630) 3.1415926535..... 39 Modified polygon doubling

Isaac Newton (1665) 3.1415926535..... 15 Infinite series

Abraham Sharp (1699) 3.1415926535..... 71 Infinite series for tan-1

(1/ 3)

John Machin (1706) 3.1415926535..... 100 Infinite series relation

/4 = 4 tan-1 (1/5)-

tan-1 (1/239)Ramanujan (1910,

1914) Gosper (1985)

17

Million

Modular Equation

Kondo, Yee (2010) 5 Trillion Modular Equation


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A HISTORY OF EXACT RESULTS FOR

Mdhava (1375) /4 = 1 1/3 + 1/5 1/7 + ...

/

12 = 1- 1/3.3 +1/3

2

.5 1/3

3

.7 +.../4 = 3/4 + 1/(33-3) 1/(53-5) + 1/(73-7) - .../16 = 1/(15+4.1) -1/(35+4.3) +1/(55+4.5) - ...

Francois Viete (1593) 2/ = [1/2] [1/2 + 1/2(1/2)] [1/2 + 1/2(1/2+1/2(1/2))]...(Infinite product)

John Wallis (1655) 4/ = (3/2)(3/4) (5/4)(5/6)(7/6)(7/8)... (Infinite product)

William Brouncker(1658)

4/ = 1+ 12/2+ 32/2+ 52/2+ ... (Continued fraction)

Isaac Newton (1665) = 3 3 /4 + 24 [1/3.8 1/5.32 1/7.128 1/9.512...]

James Gregory (1671) tan-1

(x) = 1x/3 +x2/5 - ...

Gottfried Leibniz

(1674)

/4 = 1 1/3 + 1/5 1/7 + ...

Abraham Sharp

(1699)

/ 12 = 1- 1/3.3 +1/32.5 1/33.7 +...

John Machin (1706) /4 = 4 tan-1 (1/5) - tan-1 (1/239)

Ramanujan (1910, 1914)


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RAMANUJANS SERIES FOR

One of Ramanujan's early papers is on the Modular equations and

approximations to

. Though published later from London in 1914 (QJM1914, 350-372), it is said to embody so much of Ramanujans early Indian

work. Here is a sample of his results:

Ramanujan also notes that the last series "is extremely rapidly convergent".

Indeed in late 1980s, it blazed a new trail in the saga of computation of.

MDHAVA SERIES FOR RSINE AND RVERSINE


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MDHAVA SERIES FOR RSINE AND RVERSINE

The jy or bhuj-jy of an arc of a circle is actually the half the chord

(ardha-jy

or jy

rdha) of double the arc. In the figure below, if r is theradius of the circle, jy (Rsine), koi or koi-jy (Rcosine) and ara

(Rversine) of the cpa (arc) EC = s= r, are given by:

jy(arcEC) =Rsin(s)=CD=r sin

koi(arcEC)=Rcos(s)=OD=rcos

ara(arcEC)=Rvers(s)=ED=rrcos

NLAKAHAS DERIVATION OF RYABHAA RELATION FOR


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NLAKAHAS DERIVATION OF RYABHAA RELATION FOR

SECOND ORDER SINE DIFFERENCES

We consider a given arc of arc-length s, which is divided into n equal arc-

bits. Ifs = r, then the j-th pia-jyBj and the corresponding koi-jy

Kj, and the ara Sj, are

Bj=Rsin(js/n)=rsin(j/n)=rsin(js/rn) [Cj Pj in the Figure]

Kj=Rcos(js/n)=rcos(j/n)=rcos(js/rn) [Cj Tj in the Figure]

Sj=Rvers(js/n)=r[1cos(j/n)]=r[1 cos(js/rn)] [Pj E in the Figure]

RYABHAA RELATION FOR SECOND ORDER SINE DIFFERENCES


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RYABHAA RELATION FOR SECOND-ORDER SINE-DIFFERENCES

Let M j+1 be the mid-point of the arc-bit CjCj+1 and similarly M j the mid-

point of the previous (j-th) arc-bit. We shall denote the pia-jy

of the arcEMj+1 as Bj+1/2 and clearly Bj+1/2 = Mj+1Qj+1. The corresponding

Kj+1/2 = Mj+1Uj+1 and Sj+1/2 = EQj+1. Similarly, Bj-1/2 = MjQj, Kj-1/2 = MjUj and

Sj-1/2 = EQj. The full-chord of the arc-bit s/n may be denoted . Then a

simple argument based on trair

ika (similar triangles) leads to therelations for Rsine and Rcosine differences

j=Bj+1Bj=( /R)Kj+1/2

Kj1/2Kj+1/2=(Sj+1/2Sj1/2)=( /R)Bj

Thus, we obtain the relation for second-order sine differences:

j+1 j = (Bj+1Bj)(BjBj1)= ( /R)2Bj=(2 1)Bj/B1

With n =24, ryabhaa used the approximation: (1 2) 1,B1225'

Nlakaha (Tantrasagraha): B1224'50",(1 2)/B11/233'30"

akara Vriyar (Laghuvivti):B1224'50"22"',(1 2)/B11/233'32"

MDHAVA SERIES FOR RSINE


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MDHAVA SERIES FOR RSINE

Yuktibh and Yuktidpik give the derivation of this and the Rversine

series by dividing the arc s into a large number n of equal parts and using

the relations between second-order Rsine differences (khaa-jyntara)

and the Rsines of arcs ns/j (piajys)-

MDHAVA SERIES FOR RVERSINE


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MDHAVA SERIES FOR RVERSINE

The verses giving the Rsine and Rversine series also note that the methodof obtaining accurate approximations to Rsine and Rversine values as

encoded in the mnemonics (also due to Mdhava) Vidvn etc and Stena

etc, indeed follow from these series.

NLAKAHAS FORMULA FOR INSTANTANEOUS VELOCITY


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NLAKAHA S FORMULA FOR INSTANTANEOUS VELOCITY

(c.1500)

Instead of basing the calculation of instantaneous velocity on theapproximate form of manda-correction, Nlakaha Somayj uses the

exact form of the manda correction

=m+Sin1[(r0/R)(1/R)Rsin(m)]

Nlakaha gives the correct formula for the correction to the meanvelocity in his treatise Tantrasagraha.

Nlakaha gives the derivative of the second term above in the form

[{(r0/R)Rcos(m)}/{R2(r0/R)

2Rsin

2(m)}

1/2][(d/dt)(m)]

ACYUTAS FORMULA FOR INSTANTANEOUS VELOCITY (c 1600)


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ACYUTA S FORMULA FOR INSTANTANEOUS VELOCITY (c.1600)

Acyuta Pirai in his Sphuaniraya-tantra gives the Nlakaha formula

for the instantaneous velocity. He also discusses an alternative prescription

for manda-correction due to Mujla (c.932) given by

=m+[(r/R)Rsin(m)]/[R(r/R)Rcos(m)]

Acyuta notes that in this model the manda-correction also depends on the

hypotenuse and hence the correction to the mean velocity is given by:

This gives the derivative of the second term above as

[{(r/R)Rcos(m)}{(r/R)Rsin(m)}2/{R(r/R)Rcos(m)}]

x [1/{R(r/R)Rcos(m)}] [(d/dt) (m)]

REFERENCES


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REFERENCES

1.Gaitayuktibh (c.1530) of Jyehadeva (in Malayalam):Gaitdhyya, Ed., with Notes in Malayalam, by Ramavarma

Thampuran and A. R. Akhileswara Aiyer, Trichur 1948. Ed. with Tr. byK.V.Sarma with Explanatory Notes by K Ramasubramanian, M DSrinivas and M.S.Sriram, 2 Volumes, Hindustan Book Agency, Delhi2008.

2.Kriykramakar (c.1535) of akara Vriyar on Llvat ofBhskarcrya II: Ed. by K.V. Sarma, Hoshiarpur 1975.

3.K.V.Sarma, A History of the Kerala School of Hindu Astronomy,Hoshiarpur (1972)

4.S. Parameswaran, The Golden Age of Indian Mathematics, SwadeshiScience Movement, Kochi 1998.

5.C.K.Raju, Cultural Foundations of Mathematics: The Nature ofMathematical Proof and the Transmission of the Calculus from India to

Europe in the 16thc.CE, Pearson Education, Delhi 2007.6.G.G.Joseph,A Passage to Infinity, Sage, Delhi (2009)7. K.Ramasubramanian and M.D.Srinivas, Development of Calculus in

India, in C.S.Seshadri (ed) , Studies in History of Indian Mathematics,Hindustan Book Agency, Delhi (2010), pp.201-286.

Development of Calculus

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