The Conversation recently posted an article with the title, *Five ways ancient India changed the world – with maths*, which to be honest left much to be desired as a piece of mathematical history. First off, if you are going to write about #histSTM then a piece of good advice is avoid BuzzFeed style lists, history should never be presented as a collection of bullet points; such an approach is bound to produce dubious and inaccurate claims and statements, as in this case.

The first major problem with this piece is the title; in reality it should read four contributions that Brahmagupta made to the history of mathematics with his *Brāhmasphuṭasiddhānta* and one development in Indian mathematics, which failed to transfer outside of India.

The first four elements of the list are the number system, zero, solutions of quadratic equations and rules for negative numbers, which are all, as I said above, taken from Brahmagupta’s *Brāhmasphuṭasiddhānta*, which was written in the seventh century CE. Both zero and negative numbers are parts of the number system so we really only have one item not three but I will return in detail to this and the quadratic formula later. First I want to deal with the fifth item on the list, basis for calculus.

This is something I blogged about several years ago in a brief outline of the history of calculus. What we have here is the so-called Kerala School of mathematics, which flourished in the 14^{th} to 16^{th} centuries and did some quite remarkable work on infinite series, anticipating work that was first done in Europe in the 17^{th} century. This work is indeed the basis on which calculus stand, however there are various caveats that need to be made here about any potential influence on the world. First the extent to which the Kerala School anticipated calculus is debatable. George Gheverghese Joseph from whose book *The Crest of the Peacock*: *Non-European Roots of Mathematics* (Penguin) I first learnt of the Kerala School is convinced that what they had is a full blown calculus, whereas Kim Plofker in her excellent *Mathematics in India* (Princeton UP) is far less convinced. However the real problem is that although Joseph sets up a plausible route of cultural transfer from Kerala to Europe, all investigations have drawn a blank and there is absolutely no evidence for such a transfer. As far as we know the Kerala School flourished and died without influencing the history of mathematics outside of their own circle. This is not an uncommon phenomenon in the history of science.

Let us return to Brahmagupta. His text is indeed the text that introduced the so-called Hindu-Arabic decimal place value number system to the world outside of India, first to the Islamic Empire and then through them to medieval Europe. However this wasn’t the only place value number system from antiquity and not even the only decimal one. The Chinese also had a decimal place value number system and historians of mathematics still don’t know if the Chinese influenced the Indians or the Indians the Chinese or whether the two systems developed totally independently of each other. Of course the Babylonians also had, much earlier than the Indians, a place value number system but a base sixty (sexagesimal) one not a base ten (decimal) one. There was certainly knowledge transfer between Babylon and India did the Indians get the idea of a place value number system from the Babylonians? We do know that the Indians took over a lot of their astronomy from the Greeks and Greek astronomers used the Babylonian sexagesimal place value numbers system in their astronomical texts, did a knowledge transfer take place here? A lot of unanswered questions but although we do have the decimal place value numbers system from Brahmagupta there are still a lot of open questions as to where he got it from.

With zero as a number we are on safer ground, although the Babylonians did develop and use a place holder zero, as did the Greeks in their astronomical texts, it really does appear that zero as a number, and not just a place holder, is a genuine unique India invention. There is however even here an important caveat; Brahmagupta thought one could divide by zero, which as every modern school kid knows is not on.

Turning to negative numbers, whilst Brahmagupta does indeed correctly describe their use in his *Brāhmasphuṭasiddhānta* he wasn’t the first to do so. In this case the Chinese beat him to it in *The Nine Chapters on the Mathematical Art, *which dates from 202-186 BCE, so some eight hundred years before Brahmagupta. The author of the article write that “European mathematicians were reluctant to accept negative numbers as meaningful” but so were Islamic mathematicians and also some prominent later Indian mathematicians.

In his piece the author write:

*In his seminal text, the astronomer Brahmagupta introduced rules for solving quadratic equations (so beloved of secondary school mathematics students)*…

Whilst it is true that Brahmagupta presents what is now know as the quadratic formula the Babylonians knew how to solve them at least two thousand years earlier. They however used two formulas for the two solutions based on the so-called reduced quadratic (where the parameter for x^{2} is reduced to 1 by division). The Babylonians of course rejected negative and imaginary solutions. Euclid solves quadratic equations geometrically, which is why we call them quadratic, meaning square). So there were methods for solving quadratic equations long before Brahmagupta.

Whilst by no means whishing to diminish the undoubted Indian contributions to the history of mathematics, what I am trying to make clear here is that any aspect of the history of mathematics or science has a context, a pre-history and a post-history and to ignore those aspect when presenting any given aspect automatically produces a distorted and misleading picture.

Couple of questions.

We do know that the Indians took over a lot of their astronomy from the Greeks and Greek astronomers used the Babylonian sexagesimal place value numbers system in their astronomical texts, did a knowledge transfer take place here?This use of sexagesimal notation also extended to place-value fractions, though AFAIK, the Babylonians (and the Seleucids) didn’t have a “sexagesimal point”; they relied on context. Now, the invention of the decimal point is usually attributed to Simon Stevin. Is there any evidence that the Indians used decimal notation for fractions?

the so-called reduced quadratic (where the parameter for x2 is reduced to 1 by division)Hmm, haven’t heard “reduced” being using that way before. A reduced

cubicis one missing an x-squared term; this can always be achieved by a substitution y=x+c. By analogy, a reduced quadratic would be one missing a linear term (I would think). (Then there are reduced binary quadratic forms, but that’s another kettle of fish.)Michael: I think what you call “reduced” is usually called “depressed”.

As far as I know the Indian’s didn’t use decimal fractions but the Chinese did. In fact there is, as always, quite a long history of the use of decimal fractions before Simon Stevin came on the scene with his

De Thiendein 1585. He didn’t in fact introduce the decimal point and his system of fractions is rather unwieldy as a result. The decimal point seems to have been first used by Christoph Clavius.Ah, thanks. Always nice to learn something new.

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