168 A History ofMathematics
Here ‘Sine’ and ‘Cosine’ mean the lines AB, BO in the diagram (Fig. 1.)—sine and cosine multiplied
by the radius. In some respects this is like Leibniz’s series:arc=Rx−Rx^3
3+R
x^5
5−R
x^7
7+··· (1)
which becomes our ‘modern’ series for tan−^1 (x)by settingR=1. Like the Leibnizian series, it is
forthright about the use of an infinite number of terms, a novelty in India it would seem as it was
to be in Europe. The equation (1) is, though, rather different from the form in which it was given by
the Keralans, since (as the quotation above indicates) they wrote the formula in verse, and in words,
without any use of symbols at all. One could add that since there is on the whole no explanation
of how the series were arrived at, we can only guess at the methods; but we shall see that the
seventeenth-century European mathematicians were often silent about how they had found their
series and integrals, for their own reasons. In Keralan sources the original discovery is ascribed to
Madhava, a famous fourteenth-century writer whose mathematical works are mostly lost. On the
credit side, the practical Keralans realized that the series (1) is useless for computation; you can add
50 terms and you will still be making mistakes in the second figure ofπ/4, because your next term
will be 1011 , roughly 0.01. They accordingly refined the series to give a number of others, more
useful, but still without explanation. In all this, they can rightly claim 100 years’ priority, at least.
We have here a prime example of two traditions whose aims were completely different. The
Euclidean ideology of proof which was so influential in the Islamic world had no apparent influence
in India (as al-B ̄ir ̄un ̄i had complained long before), even if there is a possibility that the Greek tables
of ‘trigonometric functions’ had been transmitted and refined. To suppose that some version of
‘calculus’ underlay the derivation of the series must be a matter of conjecture.
The single exception to this generalization is a long work, much admired in Kerala, which was
known asYukti-bhasaby Jyesthadeva; this contains something more like proofs—but again, given
the different paradigm, we should be cautious about assuming that they are meant to serve the same
function. Both the authorship and date of this work are hard to establish exactly, (the date usually
claimed is the sixteenth century), but it does give explanations of how the formulae are arrived at
whichcouldbe taken as a version of the calculus.
As I have stressed before, in dealing with al-Samaw‘al’s algebra (for example), or the use of series
by Oresme, this ‘anticipation’ as such (who was first?) is not a sensible object of history. Even if we
could establish the existence of a ‘transmission line’ from sixteenth-century Kerala to seventeenth-
century Europe (see Donald F. Lach 1965 for some evidence), we must recognize the existence of
what Kuhn called incommensurability; different research efforts and a different language of science
directed to different ends. What we know of Keralan society in the period is rather unlike the Europe
of the Scientific Revolution, in particular there is no evidence of interest in the use of mathematics
for warfare, mining, book-keeping, and so on.^6 The series of Madhava and his followers are—when
translated into algebraic notation—the same as those found in Western Europe in the seventeenth
century, but the translation has the effect of changing the aim and context of the work. It seems
of more value historically to study the Kerala tradition in itself, with reference to its own society
(how and where it was used, for example) than to argue about its role as a precursor. We can use
the Keralan material to attack ideas of the uniqueness of Western discoveries, and for that matter
to point out alternatives to the Western way of doing mathematics; but these projects ask us to see- This is not to say that evidence may not come to light in future.