Sources in the Development of Mathematics pot

Sources in the Development of MathematicsThe discovery of infinite products by Wallis and infinite series by Newton marked thebeginning of the modern mathematical era.. Series and produc

Sources in the Development of Mathematics

The discovery of infinite products by Wallis and infinite series by Newton marked thebeginning of the modern mathematical era The use of series allowed Newton to findthe area under a curve defined by any algebraic equation, an achievement completelybeyond the earlier methods of Torricelli, Fermat, and Pascal The work of Newton andhis contemporaries, including Leibniz and the Bernoullis, was concentrated in math-ematical analysis and physics Euler’s prodigious mathematical accomplishmentsdramatically extended the scope of series and products to algebra, combinatorics, andnumber theory Series and products proved pivotal in the work of Gauss, Abel, andJacobi in elliptic functions; in Boole and Lagrange’s operator calculus; and in Cayley,Sylvester, and Hilbert’s invariant theory Series and products still play a critical role

in the mathematics of today Consider the conjectures of Langlands, including that ofShimura-Taniyama, leading to Wiles’s proof of Fermat’s last theorem

Drawing on the original work of mathematicians from Europe, Asia, and America,Ranjan Roy discusses many facets of the discovery and use of infinite series andproducts He gives context and motivation for these discoveries, including originalnotation and diagrams when practical He presents multiple derivations for manyimportant theorems and formulas and provides interesting exercises, supplementingthe results of each chapter

Roy deals with numerous results, theorems, and methods used by students,mathematicians, engineers, and physicists Moreover, since he presents original math-ematical insights often omitted from textbooks, his work may be very helpful tomathematics teachers and researchers

ranjan roy is the Ralph C Huffer Professor of Mathematics and Astronomy atBeloit College Roy has published papers and reviews in differential equations, fluidmechanics, Kleinian groups, and the development of mathematics He co-authored

Special Functions (2001) with George Andrews and Richard Askey, and authored chapters in the NIST Handbook of Mathematical Functions (2010) He has received

the Allendoerfer prize, the Wisconsin MAA teaching award, and the MAA Haimoaward for distinguished mathematics teaching

Cover image by NFN Kalyan; Cover design by David Levy

Sources in the Development

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Web sites is, or will remain, accurate or appropriate.

3.8 Sylvester: A Difference Equation and Euler’s Continued Fraction 45

v

4 The Binomial Theorem 51

4.4 Cauchy: Proof of the Binomial Theorem for Real Exponents 60

Contents vii

12 The Taylor Series 200

13.8 Johann Bernoulli: Integration of√

15.4 Johann Bernoulli’s Solution of a First-Order Equation 271

15.5 Euler on General Linear Equations with Constant Coefficients 272

Contents ix

16.6 Euler and Niklaus I Bernoulli: Partial Fractions Expansions of

17.5 Gauss: Cyclotomy, Lagrange Resolvents, and Gauss Sums 320

17.6 Kronecker: Irreducibility of the Cyclotomic Polynomial 324

18.7 Cauchy: Elementary Symmetric Functions as Rational

19.4 Clairaut: Exact Differentials and Line Integrals 354

20.10 Hamilton’s Algebra of Complex Numbers and Quaternions 393

21.3 Lagrange on the Longitudinal Motion of the Loaded

21.5 Fourier: Linear Equations in Infinitely Many Unknowns 412

22.3 Smith: Revision of Riemann and Discovery of the Cantor Set 431

Contents xi

24.5 Cauchy’s Proof of the Asymptotic Character of de Moivre’s Series 488

25.3 Maclaurin’s Derivation of the Euler–Maclaurin Formula 501

25.6 Euler on the Fourier Expansions of Bernoulli Polynomials 507

26.11 Euler’s Product for

27 The Hypergeometric Series 547

27.5 Gauss’s Proof of the Convergence ofF (a,b,c,x)

28.2 Legendre’s Proof of the Orthogonality of His Polynomials 578

28.10 Chebyshev’s Discrete Legendre and Jacobi Polynomials 594

29.7 Gauss’sq-Binomial Theorem and the Triple Product Identity 615

29.11 Cauchy and Ramanujan: The Extension of the Triple

Contents xiii

31.7 Feldheim and Lanzewizky: Orthogonality ofq-Ultraspherical

32.3 Dirichlet: Infinitude of Primes in an Arithmetic Progression 683

32.5 De la Vallée Poussin’s Complex Analytic Proof ofL χ (1) = 0 688

34.3 Differential Operators of Cayley and Sylvester 733

34.4 Cayley’s Generating Function for the Number of Invariants 736

37.4 Abel: Division of Elliptic Functions and Algebraic Equations 826

Contents xv

39.7 Nevanlinna’s Factorization of a Meromorphic Function 901

41.3 Gauss’s Proof thatZ×

But this is something very important; one can render our youthful students no greaterservice than to give them suitable guidance, so that the advances in science becomeknown to them through a study of the sources – Weierstrass to Casorati, December 21,1868

The development of infinite series and products marked the beginning of the modern

mathematical era In his Arithmetica Infinitorum of 1656, Wallis made

groundbreak-ing discoveries in the use of such products and continued fractions This work had atremendous catalytic effect on the young Newton, leading him to the discovery of the

binomial theorem for noninteger exponents Newton explained in his De Methodis that

the central pillar of his work in algebra and calculus was the powerful new method ofinfinite series In letters written in 1670, James Gregory presented his discovery of sev-eral infinite series, most probably by means of finite difference interpolation formulas.Illustrating the very significant connections between series and finite difference meth-ods, in the 1670s Newton made use of such methods to transform slowly convergent

or even divergent series into rapidly convergent series, though he did not publish hisresults Illustrating the importance of this approach, Montmort and Euler soon used new

arguments to rediscover it Newton further wrote in the De Methodis that he conceived

of infinite series as analogues of infinite decimals, so that the four arithmetical tions and root extraction could be carried over to apply to variables Thus, he appliedinfinite series to discover the inverse function and implicit function theorems Newtonconcentrated largely on analysis and mathematical physics; Euler’s prodigious intellectbroadened Newton’s conception to apply infinite series and products to number theory,algebra, and combinatorics; this legacy continues unabated even today

opera-Infinite series have numerous manifestations, including power series, trigonometricseries,q-series, and Dirichlet series Their scope and power are evident in their piv-

otal role in many areas of mathematics, including algebra, analysis, combinatorics, andnumber theory As such, infinite series and products provide access to many mathe-matical questions and insights For example, Maclaurin, Euler, and MacMahon studiedsymmetric functions using infinite series; Euler, Dirichlet, Chebyshev, and Riemannemployed products and series to get deep insight into the distribution of primes Gauss

xvii

employedq-series to prove the law of quadratic reciprocity and Jacobi applied the

triple product identity, also discovered by Gauss, to determine the number of tations of integers as sums of squares Moreover, the correspondence between DanielBernoulli and Goldbach in the 1720s introduced the problem of determining whether agiven series of rational numbers was irrational or transcendental The 1843 publication

represen-of their letters prompted Liouville to lay the foundations represen-of the theory represen-of transcendentalnumbers

The detailed table of contents at the beginning of this book may prove even moreuseful than the index in locating particular topics or questions The preliminary remarks

in each chapter provide some background on the origins and motivations of the ideasdiscussed in the subsequent, more detailed, and substantial sections of the chapter.The exercises following these sections offer references so that the reader may perhapsconsult the original sources with a specific focus in mind Most works cited in the notes

at the end of each chapter should be readily accessible, especially since the number ofbooks and papers online is increasing steadily

Mathematics teachers and students may discover that the old sources, such as

Simpson’s books on algebra and calculus, Euler’s Introductio, or the correspondence

of Euler and Goldbach and the Bernoullis, are fruitful resources for calculus projects orundergraduate or graduate seminar topics Since early mathematicians often omitted tomention the conditions under which their results would hold, analysis students couldfind it very instructive to work out the range of validity of those results For example,Landen’s formula for the dilogarithm, while very insightful and significant, is incorrectfor a range of values, even where the series converge At an advanced level, importantresearch has arisen out of a study of old works Indeed, by studying Descartes andNewton, Laguerre revived a subject others had abandoned for two hundred years anddid his excellent work in numerical solutions of algebraic equations and extensions

of the rule of signs Again, André Weil recounted in his 1972 Ritt lectures on numbertheory that he arrived at the Weil conjectures through a study of Gauss’s two papers onbiquadratic residues

It is edifying and a lot of fun to read the noteworthy works of long ago; this iscommon practice in literature and is equally appropriate and beneficial in mathematics.For example, a calculus student might enjoy and learn from Cotes’s 1714 paper onlogarithms or Maria Agnesi’s 1748 treatment of the same topic in her work on analysis

At a more advanced level, Euler gave not just one or two but at least eight derivations

of his famous formula

1/n2= π2/6 Reading these may serve to enlighten us on the

variety of approaches to the perennial problem of summing series, though most of theseapproaches are not mentioned in textbooks Students of literature routinely learn fromand enjoy reading the words of, say, Austen, Hawthorne, Turgenev, or Shakespeare Wemay likewise deepen our understanding and enjoyment of mathematics by reading andrereading the original works of mathematicians such as Barrow, Laplace, Chebyshev,

or Newton It might prove rewarding if mathematicians and students of mathematics

were to make such reading a regular practice In the introduction to his Development

of Mathematics in the 19th Century, Felix Klein wrote, “Thus, it is impossible to grasp even one mathematical concept without having assimilated all the concepts which led

up to its creation, and their connections.”

Preface xix

Wherever practical, I have tried to present a mathematician’s own notational ods Seeing an argument in its original form is often instructive and can give us insightinto its motivations and underlying rationale Because of the numerous notations forlogarithms, for simplicity I have denoted the logarithm of a real value by the familiarln; in the case of complex or non-e-based logarithms, I have used log.

meth-I am indebted to many persons who helped me in writing this book meth-I would first like

to thank my wife, Gretchen Roy, for her invaluable assistance in editing and preparingthe manuscript I am grateful to Kalyan for his beautiful cover art I thank my col-leagues: Paul Campbell for his expert and generous assistance with the indexes andtypesetting and Bruce Atwood for so cheerfully and accurately preparing the figures asthey now appear I am grateful to Ashish Thapa for his skillful typesetting and figureconstruction Many thanks to Doreen Dalman, who typeset the majority of the bookand did valuable troubleshooting I am obliged to Paul Campbell and David Heesenfor their meticulous work on the bibliography I benefited from the input of those whoread preliminary drafts of some chapters: Richard Askey, George Andrews, LonnieFairchild, Atar Mittal, Yu Shun, and Phil Straffin I was fortunate to receive assistancefrom very capable librarians: Cindy Cooley and Chris Nelson at Beloit College, TravisWarwick at the Kleene Mathematics Library at the University of Wisconsin, the effi-cient librarians at the University Library in Cambridge, and the kind librarians at St.Andrews University Library A W F Edwards, of Gonville and Caius College, alsogave me helpful guidance I am grateful for financial and other assistance from BeloitCollege; thanks to John Burris, Lynn Franken, and Ann Davies for their encourage-ment and support Heartfelt gratitude goes to Maitreyi Lagunas, Margaret Carey, MihirBanerjee, Sahib Ram Mandan, and Ramendra Bose Finally, I am deeply indebted to myparents for their intellectual, emotional, and practical support of my efforts to become

a mathematician I dedicate this book to their memory

1 Power Series in Fifteenth-Century Kerala

The mathematician-astronomers of medieval Kerala lived, worked, and taught inlarge family compounds called illams Madhava, believed to have been the founder

of the school, worked in the Bakulavihara illam in the town of Sangamagrama, a fewmiles north of Cochin He was an Emprantiri Brahmin, then considered socially inferior

to the dominant Namputiri (or Nambudri) Brahmin This position does not appear tohave curtailed his teaching activities; his most distinguished pupil was Paramesvara, aNamputiri Brahmin No mathematical works of Madhava have been found, though three

of his short treatises on astronomy are extant The most important of these describeshow to accurately determine the position of the moon at any time of the day Othersurviving mathematical works of the Kerala school attribute many very significantresults to Madhava Although his algebraic notation was almost primitive, Madhava’smathematical skill allowed him to carry out highly original and difficult research.Paramesvara (c.1380–c.1460), Madhava’s pupil, was from Asvattagram, aboutthirty-five miles northeast of Madhava’s home town He belonged to the Vatasreni illam,

a famous center for astronomy and mathematics He made a series of observations ofthe eclipses of the sun and the moon between 1395 and 1432 and composed severalastronomical texts, the last of which was written in the 1450s, near the end of his life.Sankara Variyar attributed to Paramesvara a formula for the radius of a circle in terms ofthe sides of an inscribed quadrilateral Paramesvara’s son, Damodara, was the teacher

of Jyesthadeva (c 1500–c 1570) whose works survive and give us all the survivingproofs of this school Damodara was also the teacher of Nilakantha (c 1450–c 1550)

1

who composed the famous treatise called the Tantrasangraha (c 1500), a digest of the

mathematical and astronomical knowledge of his time His works allow us determine

his approximate dates since in his Aryabhatyabhasya, Nilakantha refers to his

observa-tion of solar eclipses in 1467 and 1501 Nilakantha made several efforts to establish newparameters for the mean motions of the planets and vigorously defended the necessity

of continually correcting astronomical parameters on the basis of observation SankaraVariyar (c 1500–1560) was his student

The surviving texts containing results on infinite series are Nilakantha’s graha, a commentary on it by Sankara Variyar called Yuktidipika, the Yuktibhasa by Jyesthadeva and the Kriyakramakari, started by Variyar and completed by his student Mahisamangalam Narayana All these works are in Sanskrit except the Yuktibhasa, writ-

Tantrasan-ten in Malayalam, the language of Kerala These works provide a summary of majorresults on series discovered by these original mathematicians of the indistinct past:

A Series expansions for arctangent, sine, and cosine:

3 cosθ = 1 − θ2

2!+θ4 4! − ··· ,

4 sin2θ = θ2− θ4

(22−2/2)+ θ6

(22−2/2)(32−3/2)− θ8

(22−2/2)(32−3/2)(42−4/2) + ···

In the proofs of these formulas, the range ofθ for the first series was 0 ≤ θ ≤ π/4

and for the second and third was 0≤ θ ≤ π/2 Although the series for sine and

cosine converge for all real values, the concept of periodicity of the trigonometricfunctions was discovered much later

1.1 Preliminary Remarks 3

and any last term is to be subtracted from the next above, the remainder from the term then next above, and so on, to obtain the jya (sine) of the arc.

So ifr is the radius and s the arc, then the successive terms of the repeated operations

mentioned in the description are given by

s · s2(22+ 2)r2, s · s2

arctangent series, and series B.4; note that B.4 can be derived from the arctangent

by takingθ = π/6 The extant manuscripts do not appear to attribute the other series

to a particular person The Yuktidipika gives series B.6, including the remainder; it

is possible that this series is due to Sankara Variyar, the author of the work We cansafely conclude that the power series for arctangent, sine, and cosine were obtained byMadhava; he is, thus, the first person to express the trigonometric functions as series

In the 1660s, Newton rediscovered the sine and cosine series; in 1671, James Gregoryrediscovered the series for arctangent

The series for sin2θ follows directly from the series for cos θ by an application of

the double angle formula, sin2θ =1

2(1 − cos2θ) The series for π/4 (B.1) has several

points of interest Whenn → ∞, it is simply the series discovered by Leibniz in 1673.

However, this series is not useful for computational purposes because it convergesextremely slowly To make it more effective in this respect, the Madhava school added

a rational approximation for the remainder aftern terms They did not explain how they

arrived at the three expressionsf i (n) in B.1 However, if we set

n2+ 4

n(n2+ 5) = f3(n). (1.4)

Although this continued fraction is not mentioned in any extant works of the Keralaschool, their approximants indicate that they must have known it, at least implic-

itly In fact, continued fractions appear in much earlier Indian works The Lilavati of

Bhaskara (c 1150) used continued fractions to solve first-order Diophantine equations

and Variyar’s Kriyakramakari was a commentary on Bhaskara’s book.

The approximation in equation B.6 is similar to that in B.1 and gives further dence that the Kerala mathematicians saw a connection between series and continuedfractions If we write

Newton, who was very interested in the numerical aspects of series, also found the

f1(n) = 1/(2n) approximation when he saw Leibniz’s series He wrote in a letter of

1676 to Henry Oldenburg:

By the series of Leibniz also if half the term in the last place be added and some other like device

be employed, the computation can be carried to many figures.

Though the accomplishments of Madhava and his followers are quite impressive, themembers of the school do not appear to have had any interaction with people outside ofthe very small region where they lived and worked By the end of the sixteenth century,the school ceased to produce any further original works Thus, there appears to be nocontinuity between the ideas of the Kerala scholars and those outside India or evenfrom other parts of India

3− f1(2) − f1(4)

+

1

5− f1(4) − f1(6)



− ··· (1.8)The (n + 1)th term in this series is

1.3 Jyesthadeva on Sums of Powers 5

Thus, we arrive at equation B.2 Equation B.3 is similarly obtained:

π

4 = (1 − f2(2)) −

1

3− f2(2) − f2(4)

+

1

5− f2(4) − f2(6)



− ··· , (1.10)and here the(n + 1)th term is

1

2n + 1−

n (2n)2+ 1−

n + 1 (2n + 2)2+ 1=

Thus, taking fifty terms of 1−1

3 +1

5 − ··· and using the approximation f2(n), the

last inequality shows that the error in the value of π becomes less than 4 × 10−10.The Leibniz series with fifty terms is normally accurate in computingπ up to only

one decimal place; by contrast, the Keralese method of rational approximation of theremainder produces numerically useful results

The Sanskrit texts of the Kerala school with few exceptions contain merely thestatements of results without derivations It is therefore extremely fortunate that

Jyesthadeva’s Malayalam text Yuktibhasa, containing the methods for obtaining the formulas, has survived Sankara Variyar’s Yuktidipika is a modified Sanskrit version of the Yuktibhasa It seems that the Yuktibhasa was the text used by Jyesthadeva’s stu-

dents at his illam From this, one may surmise that Variyar, a student of Nilakantha,also studied with Jyesthadeva whose illam was very close to that of Nilakantha

A basic result used by the Kerala school in the derivation of their series is that

This relation has a long history; sums of powers of integers have been used in the study

of area and volume problems at least since Archimedes Archimedes summed

forp = 1 and p = 2 For p = 2, he proved the more general result: If A1,A2, ,A nare

n lines (we may take them to be numbers) forming an ascending arithmetical

progression in which the common difference is equal toA1(the least term), then

to find the volume of revolution of segment of a parabola about its base The lation involved sums of cubes and fourth powers of integers Al-Haytham proved hisgeneralization by means of a diagram; it can be expressed in modern notation by

Whenever we wish to obtain the sum (sankalitam) of any given powers [say thepth powers of

natural numbers, up to an assigned limitn], we multiply the sankalitam of the next lower powers

[that is,(p − 1)th powers, up to the given limit n] by the limit [n] The result will contain the

required sankalitam and also the sankalitam of all the sankalitams of all lower powers up to various limits.

Jyesthadeva’s next lemma stated:

Multiply the lower [power] sankalitam [up to the limit ofn] by the limit [n] Subtract from this

product the result of dividing the product by an integer one more than the given power[p] The

result will be [asymptotically equal to] the desired sankalitam.

Thus

nS (p−1) n

com-S (p−1)

1.4 Arctangent Series in the Yuktibhasa 7

which is certainly true forp = 1 From this it can be deduced that

S1(p−1) + S2(p−1) + ··· + S (p−1)

n ∼ 1p+ 2p + ··· + n p

S (p) n

p as n → ∞.

Jyesthadeva asserted this but verified it only forp = 2 and 3 But once we fill in the

gap by proving this for allp, equation (1.18) implies that

This was Jyesthadeva’s argument for(1.15)

1.4 Arctangent Series in the Yuktibhasa

The following derivation of the arctangent series, attributed to Madhava, boils down

to the integration of 1/(1 + x2), as do the methods of Gregory and Leibniz.

In Figure1.1,AC is a quarter circle of radius one with center O; OABC is a square.

The sideAB is divided into n equal parts of length δ so that nδ = 1 and P k−1 P k = δ.

EF and P k−1 D are perpendicular to OP k Now, the trianglesOEF and OP k−1 D are

similar, implying that

1.5 Derivation of the Sine Series in the Yuktibhasa

Once again, Madhava’s derivation of the sine series has similarities with Leibniz’sderivation of the cosine series In Figure1.2,suppose that OP = θ,OP = R, P is

the midpoint of the arcP−1P1, andP Q is perpendicular to OA, where O is the origin

of the coordinate system LetP = (x,y),P1= (x1,y1), and P−1= (x−1,y−1) From the

similarity of the trianglesP−1Q1P1andOP Q, we have

1.5 Derivation of the Sine Series in the Yuktibhasa 9

Figure 1.2 Derivation of the sine series.

In fact, Bhaskara earlier stated this last relation and proved it in the same way; heapplied it to the discussion of the instantaneous motion of planets Interestingly, inthe 1650s, Pascal used a very similar argument to show that

cosθ dθ = sin θ and

 t0sinududt. (1.24)

We also note that Leibniz found the series for cosine using a similar method of repeatedintegration In Jyesthadeva, the integrals are replaced by sums and double integrals bysums of sums The series is then obtained by using successive polynomial approxi-mations for sinθ For example, when the first approximation sin u ≈ u is used in the

right-hand side of(1.24), the result is

sinθ − θ ∼ − θ3

3! or sinθ ∼ θ −

θ3

3!.When this approximation is employed in(1.24), we obtain

sinθ − θ ∼ − θ3

3! +

θ5

5!.Briefly, Jyesthadeva arrived at the sums approximating(1.24) by first dividingAP into

n equal parts using division points P1,P2, ,P n−1 Denote the midpoint of the arc

Now start withk = n−1 and multiply the equations by 1,2, ,n−1 respectively and

sum up the resulting equations We then have

(y1 + (y1 + y2) + ··· + (y1 + y2 + ···y n−1 )). (1.28)

This is the result corresponding to (1.24) To obtain the successive polynomial mations, Jyesthadeva had to work with sums of powers of integers; in order to deal withthese sums, he applied the same lemma(1.15)he had used for the arctangent series

The noted twelfth-century Indian mathematician Bhaskara, who lived and worked in

the area now known as Karnataka, used continued fractions in his c 1150 Lilavati The

Kerala school was certainly familiar with Bhaskara’s work, since they commented on

it It is therefore possible that they were aware of the specific continued fractions(1.2)and (1.6) for the error terms, even though they mentioned only the first few convergents

of these fractions They did not indicate how they obtained these convergents Somehistorians have suggested that Madhava may have found the approximations for theerror term, without knowing the continued fractions, by comparing the first few partialsums of the series with a known rational approximation ofπ Others speculate that

Madhava may have used a method of Wallis

Whether or not Madhava knew it, Wallis’s technique can be used to derive thecontinued fractions of which the Kerala school gave the convergents; this may be ofinterest Start with the functional equation(1.4) forf (n),

2r (0) n+1 − n = n + 2 + 2

r n+1 (1) ,

1.6 Continued Fractions 11

and a similar relation holds forr (0)

n−1 When these values are substituted in (1.31), some

calculation gives us



2r (1) n+1 − (n − 2)2r (1)

n−1 − (n + 2)= n2. (1.32)Once again,r (1)

n−1 − (n + 6)= n2. (1.34)Inductively, it can be shown that if

It may be instructive to consider another method for finding the continued fractions

of the Kerala school, also a method for obtaining the successive convergents There iscertainly no clear indication that this method was discovered before Gauss did so in hiswork on approximate quadrature, published in 1813 Start with a series of the form

Note that it is always possible to associate a continued fraction with (1.35) by applyingsuccessive division Write(1.35) as

a1= 1/2, a2= 0, a3= −1/2, a4= 0.

From these values, we obtain the second convergent of the continued fraction forf (n)

by applying the process described in (1.36):

By also usinga5= 5/2 and a6= 0, we obtain the third convergent: (n2+4)/(2n(n2+5)).

One problem that arises in the computation of a1, a2, a3, etc., is finding the seriesexpansions of 1/(n + 1)2, 1/(n − 1)2, 1/(n + 1)3, etc Although this may appear torequire knowledge of the binomial theorem for negative integer powers, observe thatthe series may be obtained by repeatedly multiplying the geometric series by itself Inour chapter on the binomial theorem, we see that Newton verified the correctness ofhis binomial theorem by multiplying series

This result (equivalent to B.5) is easily derived from series B.2; it is

con-tained in the Karanapaddhati, by an unknown author from the Putumana illam in

Sivapur, Kerala The result is described: “Six times the diameter is divided rately by the square of twice the squares of even integers minus one, diminished

sepa-by the squares of the even integers themselves The sum of the resulting tients increased by thrice the diameter is the circumference.” See Bag (1966).Also see Srinivasienger (1967), p 149

1.7 Exercises 13

where f3 is defined in (1.4) This gives π correct to eleven decimal places.

In one of his astronomical works, Madhava gave a value ofπ: “For a circle of

diameter 9×1011units, the circumference is 2,827,433,38,233 units.” This givesthe approximate value ofπ as 3.14159265359, correct to eleven decimal places.

The Sadratnamala by Sankara Verman of unknown date gives π to seventeen

decimal places See Parameswaran (1983), p 194, and Srinivasiengar (1967)

3 Prove al-Haytham’s formula (1.18)

4 This exercise outlines the proof of Paramesvara’s formula for the radius of the

circle circumscribing a cyclic quadrilateral, as given in the Kriyakramakari.

First, prove that the product of the flank sides of any triangle divided by thediameter of its circumscribed circle is equal to the altitude of the triangle Thisresult follows from a rule given by Brahmagupta (c 628) in an astronomical

work, the Brahmasphutasiddhanta.

Next, prove that the area of the cyclic quadrilateral is given by

A =s(s − a)(s − b)(s − c) where s = (a + b + c + d)/2

and a,b,c,d are the lengths of the sides of the quadrilateral This was also stated by Brahmagupta The Yuktibhasa contains a complete proof See also

Kichenassamy (2010), who convincingly argues that Brahmagupta had a proof

and reconstructs it from indications in Brahmasphutasiddhanta.

Then, letABCDbe the quadrilateral obtained fromABCD by interchanging

the sidesAD and CD, so that AD= CD = c and CD= AD = d Show that

ifx,y,z denote the three diagonals AC,BD,BD, respectively, then

yz = ab + cd,zx = bc + da,xy = ca + bd.

This is, of course, Ptolemy’s theorem Ptolemy’s formula is equivalent to the

addition formula for the sine function; his Almagest, containing this relation,

is heavily indebted to the Chords in a Circle of Hipparchus Bhaskara defined the three diagonals in his Lilavati See Boyer and Merzbach (1991) and Maor(1998), pp 87–94 Finally, prove that the radius of the circle circumscribing thecyclic quadrilateral is

6 Use al-Haytham’s formula(1.19)to obtain

This proves the missing step in the Yuktibhasa.

1.8 Notes on the Literature

The Weierstrass quotation at the beginning of the preface is a translation by the author

of a passage in a letter to Casorati; see Neuenschwander (1978b), p 73 It seems thatthe work of Madhava and his followers on series became known outside India onlywhen a British civil servant and Indologist, Charles M Whish, wrote a paper on the

subject, posthumously published in the Transactions of the Royal Asiatic Society of

Great Britain and Ireland in 1835 This journal was founded by British Indologists in

the early 1830s, though Sir William Jones had first conceived the idea about fifty yearsearlier Unfortunately, Whish’s paper had little impact Interest in the Kerala schoolwas renewed in the twentieth century by the efforts of C Rajagopal and his associates,who published several papers on the topic See Rajagopal (1949), Rajagopal and Aiyar(1951), Rajagopal and Venkataraman (1949), Rajagopal and Rangachari (1977), andRajagopal and Rangachari (1986) The translation of the verse enunciating the seriesfor sine is taken from Rajagopal and Rangachari (1977), p 96 The translations of

verses in the Yuktibhasa concerning sums of powers are contained in Rajagopal and

Aiyar (1951), p 70 Newton’s letter is quoted from Newton (1959–60), p 140 Wediscussed two methods for deriving continued fractions for the remainder term; forfurther details, see Srinivasiengar (1967), pp 149–151 and Rajagopal and Rangachari(1977) The latter paper makes use of Whiteside’s (1961) reconstruction of Wallis’sincomplete derivation of Brouncker’s continued fraction forπ.

The Yuktibhasa of Jyesthadeva and the Tantrasangraha of Nilakantha have recently

been published with commentaries in English by Sarma (1977) and (2008) Sarma(2008) was posthumously published with additional notes by Ramasubramanian,Srinivasa, and Sriram This two-volume translation with extensive and informativecommentary contains both the mathematical and astronomical portions; the originalMalayalam text extends to 300 pages Sarma (1972) also discusses the Kerala school,but from the astronomical point of view Biographical information on the members ofthe Kerala school, as well as numerous other ancient and medieval Indian astronomersand mathematicians, can be found in David Pingree’s five-volume work (1970–1994)

1.8 Notes on the Literature 15

Readers who wish to read more on the Indian work on series, but with modern tion, may consult Roy (1990), Katz (1995), and Bressoud (2002) These papers areconveniently available in Anderson, Katz, and Wilson (2004) Also see the papers byParameswaran (1983) on Madhava, Bag (1966) on the Karanapaddhati, Gupta (1977)

nota-on Paramesvara’s rule for radius of the cyclic quadrilateral, and Sarma and Hariharan

(1991) on the Yuktibhasa Plofker (2009) presents a scholarly, detailed, and readablediscussion of Kerala mathematics, with several excerpts onπ translated from Sankara Variyar’s Kriyakramakari She also presents the derivation of the sine series with trans- lations from the Yuktidipika and describes Takao Hayashi’s suggested reconstruction of

Madhava’s remainder term results In order to derive the continued fraction, Hayashiand his collaborators have compared the values of partial sums of Madhava’s seriesforπ with the then-known rational approximations for π Van Brummelen (2009), onthe history of trigonometry, discusses the contributions of the Kerala school and relatesthem to the astronomical work of medieval India In the context of the development

of astronomy, Van Brummelen (2009) presents the Yuktibhasa derivation of the sine

series This accessible presentation is very helpful, since the mathematics of the Keralaschool was largely motivated by an interest in astronomy

2 Sums of Powers of Integers

polynomial inn; the asymptotic value simply yields the term with the highest power.

Because Madhava and his school were primarily interested in integration, and thus inthe highest power, they failed to note the full significance of the polynomial itself

In the seventeenth century, Fermat was very interested in asymptotic values, since

he too wished to evaluate
a

0 x p dx While the Indians followed the geometric approach

of al-Haytham, Fermat arrived at his results through figurate numbers

By contrast, in the early seventeenth century, Johann Faulhaber (1580–1635) tiated an approach to the topic of sums of powers, taking an algebraic and numbertheoretic point of view Thus, he gave the expression of sums of powers as a polyno-mial, of which the asymptotic value was just the first term Faulhaber’s approach wasalso motivated by his fascination with figurate numbers

ini-The figurate numbers have been studied since ancient times ini-The one-dimensionalfigurate numbers are merely the consecutive positive integers 1,2,3, ,n The two-

dimensional figurate numbers are the triangular numbers, where the nth triangular

16

There exists an ancient Egyptian papyrus (c 300 BC) containing the formula for the

nth triangular and tetrahedral numbers These formulas in modern notation can be

When written this way, it is clear that the figurate numbers are related to the number

of combinations ofk things chosen from m different things, for appropriate m and k It

appears that the connection between figurate numbers and combinations was recognized

by Narayana Pandita whose Ganita Kaumudi of 1356 makes this explicit.

Narayana Pandita also algebraically extended the figurate numbers by taking sums

of sums of sequences So the sequence after the tetrahedral numbers would be

1, 1 + 4 = 5, 1 + 4 + 10 = 15, 1 + 4 + 10 + 20 = 34,

Some of the earlier mathematicians may have refrained from doing this because theydid not conceive of dimensionality beyond three as meaningful In effect, Narayanahad the formula

Fermat rediscovered (2.4) around 1635, though he apparently never wrote down

a proof In the margin of his copy of Diophantus’s Book on Polygonal Numbers, he

wrote that he had discovered this proposition and called it beautiful and wonderful Healso noted that the margin was too small to contain his proof, though we may surmise

it to have been an inductive proof It was the common practice of mathematicians

up to the nineteenth century to work out a number of special cases as evidence for

the correctness of the general result But in his work Plane Loci Fermat proved a

proposition by induction and included the crucial n to n + 1 step; one may surmise

that Fermat accomplished this around 1630 It is possible that Fermat had learned of

the need to supply this step from F Maurolico’s Arithmeticorum Libri Duo, written

in 1557 and published in 1575 Maurolico proved by complete induction that the sum

of the first n odd integers was n2 Also, in 1654 Pascal gave a lucid exposition of

mathematical induction in his treatise on the arithmetical triangle; he too was familiarwith Maurolico’s work.

In fact, Fermat used (2.4) to determine the asymptotic values of sums of powers Tosee this, observe that since



n + p − 1 p

prob-an algorithm for expressingS (p)

n as a polynomial inn; though he worked with Bernoulli

numbers, he failed to note their significance It was not the practice in Faulhaber’s time

to give proofs of algorithms Two centuries later, in a paper on the Euler–Maclaurinformula, Jacobi provided proofs of some of Faulhaber’s formulas

Around 1700, Jakob Bernoulli gave a simple method for computing the polynomial

inn for S (p)

n Bernoulli numbers, a sequence of rational numbers, play a significant role

in the determination of this polynomial Bernoulli’s interest in the summation of finiteand infinite series was connected with his study of probability theory Jakob Bernoulli(1654–1705) was the eldest in an illustrious scientific and mathematical family, includ-ing his brother Johann, nephews Niklaus I, Niklaus II, Daniel, and Johann II In 1676,Bernoulli received a degree in theology from the University of Basel, intending to gointo the ministry He then traveled in Europe, coming into contact with the Dutch math-ematician Hudde and members of the Royal Society These experiences aroused hisinterest in science and mathematics In the 1680s, he taught himself mathematics byreading short treatments by Leibniz on differentiation and integration; he then taughtthis subject to his younger brother Johann One of the first mathematicians to graspLeibniz’s calculus, Jakob Bernoulli proceeded to apply it to fundamental problems inmechanics and to differential equations The study of Huygens’s treatise on games ofchance led Bernoulli to a study of probability theory, on which he wrote the first known

full-length text, Ars Conjectandi From 1687 until his death, Bernoulli happily served