A History of Mathematics From Mesopotamia to Modernity

TheCalculus 167

A

y

C

R

O

B b

a

Fig. 1Indian calculation of the arc. Ifyis the arc AC,a=AB is the ‘Sine’ of the angle AOC, BO the ‘Cosine’, andRthe radius.a/bis

then the tangent, and the series givesyin terms ofaandb(ora/b) andR.

4. The Kerala connection

We may consider Madhava to have been the founder of modern analysis...( Joseph 1992, p. 293)

Joseph’s downright and still controversial statement calls attention to the ‘other priority question’—

whether the calculus was discovered (to put the claim at its strongest) in Kerala, south-west India,

in the late Middle Ages.^5 Here we have a contest which is not among mathematicians (there

is no record of an Indian seventeenth-century mathematician staking a claim for the calculus),

but among contemporary historians, whose interests are different. Specifically, Joseph and others

aim to attack the ‘Eurocentric’ story of mathematical discovery by drawing attention to parallel

discoveries in non-European contexts.

The problem is here, that the material is both unfamiliar and very inaccessible, although it has

been, in a very weak sense, known about since Charles Whish wrote about it in the 1830s. What

seems undeniable is that a number of astronomical texts from Kerala, whose dating is probably

between 1400 and 1600, give very sophisticatedinfinite seriesformulae for what we now call sinx,

cosx, and the inverse tangent ofx(the arc of the angle whose tangent isx, see Fig. 1). If the

radius of the circle isR, when the angle is 45◦the arc isπR/4 and the tangent is 1; and this gives

in particular a formula for what we know asπ. Series such as these were an important building

block in the calculus as Newton developed it, in fact his early papers give equal importance to the

calculus and the use of infinite series as ‘new’ methods. Similarly, the simplest of the formulae was

found by Leibniz early in his career (but, as he was disappointed to hear, had been found by others

before): it is commonly known as Gregory’s series. In Jyesthadeva’s sixteenth-century version it is

usually quoted as follows (see for example, Joseph 1992, p. 290):

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.

5. The subject of medieval Keralan mathematics is a very promising one for the enterprising researcher. The field is small, and
   the source material is not vast, even if much of it may be hard to locate (the Royal Asiatic Society library, near Paddington, may
   be helpful). The aspiring researcher needs to be prepared to learn Sanskrit and probably Mal ̄ayal ̄am, but the rewards could be
   substantial.