# The Bakhshali Manuscript and the Indian Zero

### The Issues

The announcement in September last year,1 along with various media releases, of the first ever radiocarbon dating of the Indian mathematical text known as the Bakhshali Manuscript2 (referred to as BM or, simply, ‘the manuscript’, from now on) has given rise to a flurry of interest in this important document. There has already been an article published by Kim Plofker et al. [10], of which Hayashi (see footnote 2) is one of the co-authors, which calls into question some of the publicly stated inferences—and the underlying assumptions—of the person who seems to have been the moving spirit behind the radiocarbon measurements, Marcus du Sautoy of Oxford University. Perhaps more comments and interpretations are on the way. A great deal of the interest around this manuscript is focussed on the prolific occurrence of a symbol for zero in the form of a dot, especially highlighted by du Sautoy. An accurate date is of obvious importance for the history of the written zero and of the mathematical zero more generally. (The two histories are not the same, as we shall see presently.) Nevertheless, this narrow focus has been to the detriment both of other interesting mathematics in it and to the general background of the numerical culture of Indian mathematics, against which it seems appropriate to place what is actually present in the BM. In particular, the history of the zero itself cannot be properly interpreted without paying attention to the evolution of decimal place-value enumeration.

The earliest evidence for a mastery of the principles underlying decimal enumeration comes from far older sources than historians usually give credit to, a lapse that the statements of du Sautoy as well as the article of Plofker et al. are not free from. The main reason for the oversight is probably that these sources are not primarily mathematical in intent or content. The earliest such are in fact Vedic saṃhitā texts, especially the Ṛgveda and the Taittirīya Saṃhitā recension of the Yajurveda (1200–1000 bce). They have plenty to tell us about the beginnings—as far as we can tell now—of the ‘decimal culture’. The Ṛgveda, in particular, is rich in number names which reflect faithfully the principles governing the formation of based numbers with $10$ as the base [1].3

A second reason for the oversight has been the tendency to think of place-value enumeration as inseparably tied to a written, symbolic, representation of numbers or something visually evocative of it (like tokens placed in compartments). This is a misunderstanding and part of the task of retracing the roots of based enumeration is to correct this misunderstanding. The earliest Vedic texts show in an absolutely unambiguous manner that one can implement the place-value paradigm just as effectively with spoken number names—which we may call the nominal representation of decimal numbers—as with number symbols, a fact whose significance cannot be emphasised enough. As is now widely accepted, Vedic seers had no writing; there was no choice but to rely on the spoken sound. The rules for forming numbers above 9—serving the same purpose as the now-universal sequence of numerals in the written symbolic notation—had to be realised by applying rules of nominal composition—in other words, rules of grammar.4 This basic distinction in the way based numbers are formed in a written and an oral tradition has consequences which we are not always conscious of.

One of these consequences—this is an elementary observation, briefly recapitulated in a later section nevertheless—is that there is no functional need in an oral treatment of numbers for a zero, either as a counting number or as a placeholder. We shall also see that, logically and mathematically, there is no distinction between the two.

Strictly speaking, there is also no need for it in carrying out the elementary operations of arithmetic,5 though it helps in simplifying them. This is the explanation of why there is such a long gap, about a millennium, between the perfected decimal number nomenclature of the Ṛgveda and the first appearance of what is generally thought to be a signature of place-value numeration, the mathematical zero, śūnya (in the Chandaḥsūtra of Pi$\dot{\mathrm n}$gala, 3rd–2nd century bce). The centuries following, say till about 3rd–4th century ce, saw a somewhat confused evolution of the symbolic—but not positional—representations of the decimal place-value principle and, along with it, the idea of the mathematical zero. What we see in the BM is the end result of this evolution. Clearly, an understanding of how the earlier experimentation led up to the strictly positional numbers of the BM, based on evidence and free from unsupported speculation to the extent possible, will be helpful not only in contextualising the manuscript itself and its zero but also in throwing light on the history of the general idea of the zero, much written about though not always with the necessary discipline. Given that the announced datings of the BM folios vary widely over the samples—in a range of about 800 years—and so are not very useful in arriving at a chronology, there is a good case for a fresh look at the internal evidence.

The BM was written in ink on the bark of the Himalayan birch tree. The Oxford measurements were carried out on samples of the bark from three different folios (avoiding the inked portions). One supposes that adequate care was taken in preparing the samples and so it is extremely surprising that the age measurements returned such widely separated ranges, 224–383, 680–779 and 885–993, all in years ce.

From the available information, it is unclear what kind of uncertainties these ranges represent, in particular how estimated systematic and statistical errors contribute to them. It seems unlikely that the errors are wholly statistical in origin as corrected for the known standard uncertainties in the calibration of radiocarbon counts, for two related reasons: (i) they are too large for samples as ‘young’ as, at most, 2000 years and the range for the first (the oldest) sample is much larger than for the other two; (ii) even if we accept them, the mean ages are too far apart (in terms of the errors) to be explained by statistical and other known uncertainties.

Taking the median values as the most probable dates (for the individual samples) and half the ranges as—given the lack of information—standard deviations leads to a statistically improbable picture: the dates of the first and the third samples differ by about 5 (combined) standard deviations. It will therefore be meaningless to combine the results from the three samples to calculate a most probable date as around 700 ce for the manuscript as a whole (with somewhat smaller uncertainties). That leaves us with the following possible explanations:

(1) The surviving folios (70 in number out of an unknown total) were prepared and written upon at different times; relying on the median ages, the three samples would then date from ca. 304, 730 and 939 ce respectively. The paper of Plofker et al. cited above has given strong reasons, based on the continuity of the mathematical material in some of the folios and of the writing style, why that is extremely unlikely.

(2) The writing took place at one time, but on bark removed from the living tree at different times varying from the 4th to the 10th century ce. 6 That would require a hoard of birch bark, periodically replenished by freshly harvested bark, and kept in stock for very long periods (and drawn upon as the need arose). Is that any more plausible than alternative 1?

Discounting the possibility of the measurements (and the prior preparation of the samples) being mishandled—the technique is a mature one now, after undergoing many refinements over several decades—there seems to be only one viable option left.

(3) The folios have been contaminated by recently ingested (atmospheric) carbon. As is recorded by A.F.R. Hoernle to whose care in Calcutta the manuscript passed after its discovery, its unearthing was accidental and carelessly handled. Also, we do not know the conditions under which the folios were stored in Calcutta for about two decades, before they were sandwiched between mica sheets and sent to England.7 We do know that they deteriorated quite badly over time, to the point where they were made inaccessible to scholars by the Bodleian Library.

Whatever the correct explanation, it will be imprudent at this stage to speculate about which of the three dates, if any, is likely to be closer to the truth. Given the lack of authentic information about the measurements themselves and the likelihood of contamination,8 we cannot assign even an approximate date to the BM; the datings do not help us in any way to place the manuscript in relation to the evolution of mathematical thought in India. While waiting for more informative experiments—taking samples from as many folios as is practicable and subjecting the results to sophisticated statistical analysis, which may take a long time—what can still be done is to compare the mathematics in the manuscript (and the mathematics not in the manuscript) to material in other dated texts covering any candidate date for the BM. The story of the zero is one, in fact the most productive, such point of comparison.

There has never been a shortage of suggestions regarding the date of the BM, beginning with Hoernle and continuing upto the present day. The majority of them rely on extraneous, non-mathematical considerations, mainly to do with the language (recognisably a variant of Sanskrit) and the script (a variant of Nāgari), some based on what we may call ideological preferences. The suggested dates range from the last centuries bce to the 12th century ce. The extremes of this quite impossibly wide time-span were put forward in the days when the study of Indian mathematical history was influenced by the colonial experience (on both sides) and are not taken seriously since a long time; the very late dates were unviable already at the time they were suggested and the very early ones are in any case ruled out by the Nāgari script of the text. What survives the cuts is roughly the same range as is covered—coincidentally it may be stressed—by the radiocarbon measurements. Hayashi [7], who has looked carefully at the work done prior to his own, is of the view that the text is very likely of the 7th century ce.

Folio 16 of the Bakhshali manuscript, recto and verso ((Click on the images to see larger high-resolution images): The Oxford measurement gives its radiocarbon date as 224–383 CE. Bodleian Libraries, University  of  Oxford

### The Framework: Based Numbers and their Representations

The role the zero plays in the number system is closely linked to the concept of numbers in a base (or based numbers), even though it is independent of any particular base. The earliest recorded evidence we have for based numbers (with $60$ as the base) is from Old Babylonia (ca. 1800 bce); the earliest such from India (with the now universal $10$ as the base) can be dated, in view of the fully finished form in which they appear in the Ṛgveda, perhaps 1400–1300 bce or even slightly earlier. The base (our default choice from now on will be $10$) serves as a unit by which to measure the magnitude of a number precisely, by repeated applications of the operation of division with remainder, exactly as in any physical measurement. The result of the measurement is expressed commonly in writing as the ordered sequence of numerals

$$n_{k}n_{k-1}\ldots n_{1}n_{0}.$$

What it stands for is the assertion that the sequence identifies a number $N$ as the value of a polynomial in which the variable is fixed at $10$:

$$N=n_{k}10^{k}+n_{k-1}10^{k-1}+\cdots +n_{1}10^{1}+n_{0}10^{0}$$

and the coefficients $\{n_{j}\}$ take values in the set $\{0,1,\ldots,9\}$. One essential difference between the full polynomial expression and its representation by the sequence of coefficients is that the absence of any particular power of $10$, say $10^{j}$, can be taken care of by simply dropping the corresponding term in the polynomial:

$$N=n_{k}10^{k}+\cdots n_{j+1}10^{j+1}+n_{j-1}10^{j-1}+\cdots +n_{0}$$

while in its representation by coefficients the absence has to be explicitly indicated by setting $n_{j}=0$:

$$N=n_{k}\ldots n_{j+1}\,0\,n_{j-1}\ldots n_{0}$$

or by some other equivalent convention such as enclosing the coefficients in a linear array of compartments of which the $j$th compartment is left empty. (The Babylonians did not compartmentalise the coefficients of the powers of $60$ in the beginning and that makes the reading of their numbers ambiguous, especially when zero(s) occurs on the extreme right as the coefficient(s) of the lowest power(s); eventually (around the 3rd century bce) at about the time when Pi$\dot{\mathrm n}$gala introduced śūnya in India as a number by itself (see below), they did get around to using a symbol for zero.)

The description above of the place-value principle is tailored to the written representation of numbers as a model, partly because of the demands of writing as our mode of communication (as in this article). But the principle itself is abstract: given a number $N$ and another number $b < N$ ($b$ is the base), divide $N$ by $b$: $N=q_{0}b+n_{0}$ with $n_{0}. Next, divide the quotient $q_{0}$ by $b$: $q_{0}=q_{1}b+n_{1}$ and repeat the operation until $q_{k}$ for some $k$ becomes less than $b$. Put that $q_{k}$ equal to $n_{k}$ and we get the polynomial representation written down earlier (with $b=10$). The point of course is not that ancient civilisations which tamed numbers by means of a base really thought in such abstract terms but that we, who wish to understand the relationship between the written/symbolic and the nominal representations (or any other representation for that matter), have to go beyond the specifics to the abstract structure underlying them to do so.9

In a culture that does not have (or has renounced) writing but is highly literate otherwise, as the Vedic culture was, the step of replacing the polynomial expression by its sequence of coefficients is unavailable or, at least, highly unnatural. The entirely natural Vedic solution to the problem was to give a spoken identity to every number. This involved the following steps: first give names to the numerals upto $9$ (the oral analogue of the familiar symbolic numerals): eka, dvi, …, nava; then give names to the powers of $10$: daśa, śata, sahasra, ayuta, …; then combine the names of the coefficients ($n_{j}$) and the powers of $10$ ($10^{j}$) to arrive at the names of each term $n_{j}10^{j}$ in the polynomial, following the rules of Vedic grammar for composing the two names to implement the mathematical operation of multiplication; then invoke another grammatical rule to implement the operation of addition of such individual terms, to finally arrive at a name for the number $N$.10

Thus, the hallmarks of the nominal representation can be summarised in two points: (i) the need to invent new names11 for higher and higher powers of $10$; and (ii) the paramount importance of the rules of nominal composition in giving every number a unique identity. (There is no grammar involved in point (i), though the names employed are generally words which have other meanings.) Once these points are understood, it becomes obvious—though it may come as a surprise to some—that the nominal representation is as unambiguous, complete and rigorous as the symbolic, as far as enumeration is concerned.

Point (i) above, the need to invent endless new names, accounts for the extraordinary fascination lists of names of powers of $10$ held for Indian texts of all sorts, starting with the Taittirīya Saṃhitā and continuing well into the common era. Point (ii) is the more relevant one for our concerns here because our main source of information on the genesis of decimal counting is also the oldest (readable) text India can lay claim to, namely, the Ṛgveda. The Ṛgveda has an enormous number, close to 3000, of number names, a few hundred of them being names of compound numbers—those which require the application of grammar to form and analyse. A very thorough analytic study of these names [1], which also reviews the general framework of based numbers in greater detail than given here) reveals that all of them, with two or three irresoluble exceptions, conform to the ideal rule-based nominal representation outlined above.

It is thus no longer justified to link a mastery of decimal enumeration to the skill of writing; the Ṛgveda already exhibits such a mastery.12 Given the consensus date of the Ṛgveda (1200–1100 bce), and given also the time it must have taken for the decimal paradigm to evolve to maturity, it seems safe to date its birth to $\sim$ 1400–1300 bce at the latest.

The nominal representation differs from the symbolic most crucially in the fact that it is a faithful transcription of the full polynomial expression as distinct from the listing of its coefficients. In particular, just as the vanishing terms of the polynomial do not have to be explicitly displayed, the nominal representation has no functional need for a zero in the expression of any positive integer, i.e., as a ‘placeholder’; that role of the zero is subsumed in the names of the (positive integral) powers of $10$. (One might say that the zero is the price to be paid for a greater degree of abstraction and economy.) As an illustration, the number $60,099$ occurs in the Ṛgveda and its name, formed by the application of the rules of multiplicative and additive composition, is ṣaṣṭiṃ sahasrā navatiṃnava (the separation of the words matters) having the literal translation ‘sixty thousand ninety-and-nine’.13

It also has no need for zero as a number by itself, i.e., the one whose successor is $1$ in the Jyeṣṭhadeva–Peano14 view of numbers: for instance, no atheist will say ‘there is 0 god’, they would rather say ‘there is no god’. Given the weight exerted by the oral tradition on all sciences (in fact on all literary compositions) and over such a long period, it then becomes easy to understand why the first occurrence of śūnya as a number in Pi$\dot{\mathrm n}$gala’s Chandaḥsūtra took so long after the decimal system was perfected.15 The real advance was the construction of the latter; the zero was a contingent notion, introduced when it became useful. A final remark is that the placeholder $0$ of the symbolic representation simply reflects the vanishing of a coefficient in the polynomial for a number, just as $1$ or $2$ or … $9$ as a placeholder identifies it as a coefficient in the polynomial; it is not a different kind of zero just as there are not different kinds of $1$ or $2$.

The surprise in all this is that, in our attachment to writing as an indispensable artefact of civilisational progress, we have overlooked for so long the overwhelming evidence for the true antiquity of the decimal system, in one of the most intensely studied texts from that antiquity, the Ṛgveda.

### The Early History of the Zero; Brāhmi Numbers

Many scholars are of the view that the Indian zero has a grammatical ancestor, lopa, as expressed in the famous sūtra of Pāṇini ($\sim$ 500 bce, to remind ourselves of the overall chronology): adarśanaṃ lopaḥ (“that which does not appear is lopa”, or even “that which disappears is lopa”).16 What lopa signals in grammar is a rule for the dropping of an affix which should otherwise be present in the formation of certain compounds. The affix is absent, with nothing left behind to indicate that a phonetic unit has been removed. That is not the same as the role of the zero; the zero is either not needed (the oral representation) or its symbol must be present to show the vanishing of a coefficient in the polynomial (or be ‘actively’ absent, as when a compartment in an array of coefficients is left empty).

In contrast with lopa, there is no ambiguity at all about the next episode in the history of zero. This too occurs in a text connected to language—not on grammar but on prosody, by Pi$\dot{\mathrm n}$gala (referred to earlier)—but it occurs in a problem posed and solved mathematically. In a certain combinatorial question, Pi$\dot{\mathrm n}$gala is required to separate numbers arising at an intermediate stage into subsets of even and odd numbers. This he does in true Pāṇinian manner (and as we do nowadays) by giving the two subsets labels and the labels he chooses are—fortunately for the history of the zero and unlike the meaningless metalinguistic markers of Pāṇini—dvi and śūnya. Since dvi is undoubtedly the number $2$, one will have to be perverse to suggest that śūnya is not a number; that number can only be the cardinality of the empty set, $0$, the number preceding $1$.

Is it a chance coincidence that the first reference to the zero came at roughly the same time as the first recorded occurrence—setting aside the unread script of the Harappans from two millennia earlier—of writing, in the edicts of Ashoka, around 3rd century bce? There is no evidence that Pi$\dot{\mathrm n}$gala’s reference is to a written śūnya. One may be tempted to think, nevertheless, that there might have been a causal connection between the imperative need for a symbolic zero in written numbers and the emergence of a specific name for it. Ashoka’s edicts were inscribed dominantly in the Brāhmi script and it is natural then to ask what the proliferation of Brāhmi inscriptions in the following five or six centuries, in different durable media including a large number of coins, have to tell us about symbolic numbers. The remarkable fact is that though there are many numbers in them, all presented in a symbolic notation, there is no zero in the numbers.

Brāhmi numbers have been the cause of much mystification for historians (and numismatists) over a long time. Many proposals were made about their origins and possible affiliations to other ancient number signs from outside India, mostly on morphological grounds, none of which met with general scholarly approval.17 The key to clearing the mystification is in fact not in their morphology but in their syntactic features, as seen in the mature form Brāhmi number notation acquired after a somewhat confused and experimental early phase [6]. First of all, though they are symbolic (not, apparently, nominal), they do not follow a positional system. The symbols for the numerals up to $9$ are arbitrary, as they necessarily are in any decimal system, since all that matters is that their numerical values were universally agreed upon. Next, $10$, $100$ and $1000$ are also single symbols, again arbitrary and for the same reason. Next, a multiple of a power of $10$ is expressed by attaching a suffix corresponding to the multiplier to the symbol for that power of $10$. (For the morphological implementation of these structural modifications, occasionally somewhat irregular, see the reproductions in [6].) Finally, additive composition is implemented by juxtaposing the symbols for the appropriate powers of $10$.

It is blindingly obvious in the light of these features where the Brāhmi numbers came from, conceptually. They are a faithful symbolic version of the rules for forming decimal number names that had held sway from the time of the Ṛgveda, i.e., of the polynomial representation, and they result from the slavish transcription of names to symbols. They are not positional because they have not reduced a number to a list of its (polynomial) coefficients.

In particular, the mystery of the missing Brāhmi zero is solved. Pi$\dot{\mathrm n}$gala’s śūnya, just like his dvi, refers to a number in its own right, not to a written place-value entry, because the place-value notation had not yet been invented.

The historically important conclusion from all this is that Brāhmi numerals represented the continuing hold of the oral tradition on the newly emergent written systems of enumeration, that they served in their time as the bridge connecting nominal numbers to modern symbolic place-value numbers. Without understanding their structure—especially the missing zero—we cannot understand what came after. Evidence of absence is sometimes very useful.18

### Dating Bakhshali: Āryabhaṭīya as a Benchmark

One lesson we can take from the discussion in the previous section is that the really important question is: what is the earliest date at which written place-value numbers appear (rather than the date of the first written zero, which is just one of its several characteristic features)? The dates of the cave shrines in which Brāhmi numbers occur as part of inscriptions are reasonably reliably known (and there are also, of course, the dated coins) mainly from the history of their sponsoring dynasties. On that basis, the hybrid Brāhmi system was in current use through the 3rd century ce, perhaps a little later. We can therefore feel secure in dating the transition to a fully positional system to after, say, $\sim$ 350 ce. How much after? The earliest date for which firm inscriptional evidence can be adduced is 595 ce (according to [6]; I am not aware of any new evidence pushing this date further back). These facts by themselves do not put any upper limit on the date of the BM but they do make it highly unlikely that it was originally composed (not just written down or copied) before the 3rd century ce. All three radiocarbon dates are still in play.

To restrict the date to a narrower range, there seem to be only two options available: the usual analysis of language and script, or a comparative study of its mathematical content in relation to other, ideally firmly dated, texts. The first has been the option generally followed by most historians but it has not led to anything like a consensus view (as is evident from the debates that are still going on);19 variations in the linguistic characteristics with time and place have turned out to be frustratingly hard to pin down.

Mathematical comparisons are, in contrast, rather thin on the ground, apart from some incidental and casual remarks. There are several very well studied and dated texts covering the relevant period, from the Āryabhaṭīya (end of the 5th century ce) to Varāhamihira’s Pañcasiddhāntikā to the books of Bhāskara I and Brahmagupta ($\sim$ 630 ce) and of still later authors. A number of mathematical results and themes lend themselves to such comparisons, especially with the Āryabhaṭīya, of which the most immediately useful are listed below. But before that, it is necessary to remind ourselves of the essential criteria, logically obvious but not always observed in practice, that must govern any such exercise.

In order to establish a plausible upper limit date for an undated text A, it is not enough to look at another (dated) text B with which it has some material in common. To take only one relevant instance out of many, it is not enough that negative numbers and arithmetic with them, not to mention zero itself, are found both in the BM and in Brahmagupta’s work, to hypothesise that they might be roughly contemporaneous. The comparison becomes meaningful if it can be shown that a topic in common is better developed in B than in A. Even more useful is evidence of absence, the converse situation in which a topic addressed in B is absent or poorly treated in A or, more generally and decisively, if A has nothing that is more advanced than in B. The earliest such text B can then be credibly considered to have been written after A. In relation to the BM, the Āryabhaṭīya lends itself to the role of such a benchmark admirably, partly because the ambient arithmetical and algebraic culture is roughly of the same level of accomplishment. Here are some of the more specific points of comparison, in summary:

(1) Fractions and negative numbers occur routinely in the BM. Among the rules written down in the sūtra portions of the manuscript are $a\pm (-b)=a\mp b$ and $(a/b)\times(c/d)=(ac/bd)$ (and the corresponding rule for division). But the rule for adding fractions by the use of the common denominator is absent. The Āryabhaṭīya describes the reduction to a common denominator in the same verse as the multiplication rules. It is a well-recognised historical fact—seen even today in the classroom—that the manipulations involved in the addition of fractions have generally posed a bigger challenge to beginners than their multiplication, presumably because of the greater arithmetical sophistication of the underlying logic.

(2) The next two verses of the Āryabhaṭīya then lay down the procedure for solving systems of linear equations in complete generality, by inverting the order of operations, transposition of terms, etc. The BM has numerical examples illustrating such procedures but nothing of this generality.

(3) The formula for the sum of the general arithmetic series is given in both texts. The Āryabhaṭīya looks at the series from several different angles and, furthermore, assumes that the special case of the sum of the first $n$ numbers, $S_{1}(n)=\sum_{i=1}^{n}i$, is already known and goes on to give the formulae for the more difficult sums of squares and cubes, $S_{2}(n)=\sum_{i=1}^{n}i^{2}$ and $S_{3}(n)=\sum_{i=1}^{n}i^{3}$ as well as the ‘sum of sums’ $S^{2}(n)=\sum_{i=1}^{n}S_{1}(i)$. The BM does not have them.

(4) The BM is familiar with Diophantine problems. Among the linear equations treated are indeterminate equations (systems with more unknowns than equations; there is also a quadratic equation in two unknowns). The Āryabhaṭīya has the general solution of the linear Diophantine equation in two unknowns, arrived at by the algorithm known famously as kuṭṭaka. Nothing remotely approaching it in generality and completeness is to be found in the BM.

I have chosen these few examples with the specific intention of highlighting topics which the BM is not unfamiliar with but whose satisfactory elucidation had to wait for Āryabhaṭa. Each one of them may not be as decisive by itself as one could wish—especially in view of the missing portions of the manuscript—but together, in my view, they are a strong argument in favour of a date for the BM earlier to that of the Āryabhaṭīya, namely 499 ce. The reasoning and the conclusion they lead to owe nothing to the new interest in the question of dating and actually predate it by some years (as explained in [5]) as do some more general considerations to which we can now turn.

### Dating Bakhshali: General Considerations

First is the question, once again, of whether there is anything in the BM more advanced or sophisticated than its counterpart in the Āryabhaṭīya. Aside from the points made in the last section, there is an immediate candidate: the square root algorithms in the two texts. They are strikingly different in spirit and execution. The Āryabhaṭan method is more or less what is taught in schools today, an adaptation of long division; the steps are specific to based numbers with $10$ as the base. The method of the BM is algebraic and consists of a series of iterated (recursive) steps, each step being an application of what is called Heron’s approximation to the square root.20 It is base-independent and structural and, so, by modern standards, more abstract and hence ‘advanced’. But the fact is that the Heron formula was known already to the authors of the Śulbasūtra (at least in its equivalent geometric form) as was, very likely, the idea of recursive computations, for example in the rational approximation to $\sqrt{2}$ (see [3]). Such techniques remained one of the staples of the Indian recursive approach to mathematics throughout its history. It is not implausible that Āryabhaṭa, who had to work out many numerical square roots in his astronomy and trigonometry, preferred a method of his own invention.

There are other pointers, not linked to the Āryabhaṭīya, to a relatively early date. The least decisive of them is the morphology of the numerals up to $9$. Several of the Bakhshali numerals have a close resemblance to their Brāhmi cousins but not to what they became in later Nāgari writing (see the tables in [6]). The problem is that there are also numerals (e.g., $1$, $2$, $3$ and possibly $4$) that hardly changed, from their first appearance until modern times, except for accommodating slightly more stylised cursive forms. Among other, more decisive, archaic features of the BM are the fact that compound number symbols (those which would occupy more than one place in a true place-value notation) are often accompanied by their names and sometimes these names are also broken up into subgroups, the name of each subgroup being written as though it was a stand-alone number. This is a survival from the nominal representation; we often read symbolically written numbers by breaking them up, e.g., 1947 as nineteen forty-seven. (A spectacularly long number written in this manner in the BM will be found in [6], see also [7].) Related to this is the practice of stating problems and their solutions both in symbolic and narrative styles. The impression is of a certain discomfort in relying solely on the new-fangled symbolic methods, including for the numbers themselves; it is as though mathematical writing was passing through a transitional phase, with one foot still in the familiar oral tradition whose last remnant in the form of Brāhmi numbers was about to become extinct. These archaic features disappeared quite rapidly where numbers actually were written down, as in inscriptions, while mathematical (and astronomical) texts either continued to express numbers through their names or turned increasingly to the use of bhūtasaṃkhyā (‘metaphoric numbers’). And that gives us yet another indicator of an early date for the BM: bhūtasaṃkhyā are present in the BM, but sparsely, while the Pañcasiddhāntikā (mid-6th century ce) bristles with them.

One originality in the BM that has provoked varying interpretations is the use of the dot, bindu, to stand for both zero and the (temporarily) unknown solution of a numerically posed problem, the ancestor of the modern $x$, the unknown in short. Plofker et al. ([10]) have something to say about it. They write that the use of the same symbol for both zero and the unknown points to an “algebra-like” role for the written $0$, in that it can be subjected to the same arithmetical operations as any other number including $x$; since $x$ is a number that will be determined once the problem is solved. This is of course trivially true but it is not the whole story. If that was the intention, any other symbol would have done as well, preferably one without a fixed numerical connotation—or even, to stretch a point, $1$ or $2$ or … $9$. There is a subtler reason for the double role of the bindu and that goes to the heart of Indian place-value philosophy itself. The Indian view has always (at least from the time of the Āryabhaṭīya) been that the places (sthāna) are conceived as locations, vacant (śūnya) to begin with, to be occupied by any of the numerals $1$ to $9$ and, if not so occupied, to remain empty. (As I have already noted, the distinction between an ‘algebraic’ $0$ and a place-value $0$ is a false distinction, one that was not made in India in the beginning.) It is this emptiness that the zero-dot (śūnya-bindu) signifies, as well as the potential of its being displaced by any numeral, unknown at the outset. Many texts, not only mathematical (an outstandingly clear example is the reference to place-value by the linguist-philosopher Bhartṛhari, very possibly a rough contemporary of Āryabhaṭa, in his Vākyapadīya, see [1]), testify to the correctness of this interpretation. Indeed, as late a text as Yuktibhāṣā (1520–1525 ce) describes this process of occupation by a Malayalam word whose literal sense is ‘to invade’. The fact that $0$ became in time a number like any other—special only by virtue of its being the neutral element in the operation of addition (its “algebra-like” role if one wishes)—is extraneous to the Bakhshali use of the same symbol for $0$ and $x$; it is not a sign of a late concern with algebraic abstraction. It is true that the natural home of $0$, as of negative numbers, is in the algebraic properties of the set of all integers (more generally, of all rationals)—that they form a group under addition21—but that recognition belongs to a very modern perspective, not historically relevant here. In any case, such considerations are not an argument for a relatively late date for the BM.

### Conclusions

‘Conclusions’ is probably too strong a word to describe the more or less plausible inferences we have drawn from the mathematics in the surviving folios of the BM; the ghost of the lost folios—very likely lost forever—will always hover over them. Setting aside such futile regrets, it would seem that the material we do have, on the basis of comparisons with other dated texts and in the general context of the growth of mathematics in India, determines a rather narrow window of time for its composition. Critical to this determination are two factors: the (non-positional) organising principle of the Brāhmi numbers, itself derived from the much older decimal number names, and the relatively greater mathematical sophistication of the routine arithmetic and algebra of the Āryabhaṭīya (we are not speaking here of Āryabhaṭa’s ‘new’ mathematics, his trigonometry and finite calculus), the first for what it does not have and the second for what it does. All in all, the conclusion has to be that the date of the BM lies somewhere between the end of the era of Brāhmi numbers and the composition of the Āryabhaṭīya, say 350–500 ce. That will contradict neither the fundamental principle of comparative history—what comes later cannot influence what came earlier—nor, in their present state of uncertainty, manuscriptological considerations like the material, language, script, etc. It is also consistent with the date of the first mention of a śūnya bindu, not just śūnya, in other writing, namely in Subandhu’s romantic fiction Vāsavadattā, 4th century ce.

Overriding all these details, however, is the general historical context. The great material, spiritual and intellectual status that Gandhāra had enjoyed for many centuries as a cultural crossroads of antiquity came to an end in the 5th century ce as a result of the invasion of the Huns. By about the middle of the century, the region and all its centres of learning, dominantly Buddhist by then, had turned to ashes and rubble under their onslaught. The monks and the professors of universities like Takṣaśila, just over 100 km southeast of Bakhshali—those who remained alive—travelled to the peripheral parts of the Gupta empire (for a summary of the evidence, dominantly art-historical, see [5]) and northwest India turned into an intellectual desert, never to regain its past glory. Who in that wilderness would have written a finely crafted text in mathematics and for whom would he have written it? Not for Buddhist merchants, there were none left. This for me is just as important a reason for placing the BM before mid-5th century ce as the mathematical arguments that we have looked at.

acknowledgements I give warm thanks to Vidyanand Nanjundaiah for his insightful comments on an earlier version of this article. My thanks are also due to the reviewers for their helpful suggestions.

### References

• [1] Bhagyashree Bavare and P.P. Divakaran. 2013. Genesis and Early Evolution of Decimal Enumeration: Evidence from Number Names in the Ṛgveda. Ind. J. History of Science. 48.4: 535.
• [2] M.N. Channabasappa. 1976. On the Square Root Formula in the Bakhshali Manuscript. Ind. J. History of Science. 11: 112.
• [3] P.P. Divakaran. “Notes on Yuktibhāṣā: Recursive Methods in Indian Mathematics” in Studies in the History of Indian Mathematics (C.S. Seshadri, Ed., 2010: Hindustan Book Agency, New Delhi).
• [4] P.P. Divakaran. 2016. What is Indian about Indian Mathematics? Ind. J. History of Science. 51.1: 56.
• [5] P.P. Divakaran. The Mathematics of India: Concepts, Methods, Connections (2018: Hindustan Book Agency, New Delhi).
• [6] Bibhutibhushan Datta and Avadhesh Narayan Singh. History of Hindu Mathematics, Vol. I (reprinted 2001: Bharatiya Kala Prakashan, Delhi).
• [7] Takao Hayashi. The Bakhshali Manuscript (1995: Egbert Forsten, Groningen).
• [8] David Mumford. “What is so Baffling about Negative Numbers?” in Studies in the History of Indian Mathematics (C.S. Seshadri, Ed., 2010: Hindustan Book Agency, New Delhi).
• [9] M.D. Pandit. “Reflections on Pāṇinian Zero” in The Concept of Śūnya (A.K. Bag and S.R. Sarma, Eds., 2003: Indira Gandhi National Centre for the Arts and Indian National Science Academy, New Delhi).
• [10] Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk. 2017. The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating. Hist. Science in South Asia. 5.1: 134.
• [11] K.V. Sarma. Gaṇitayuktibhāṣā of Jyeṣṭhadeva, Vol. I (English translation from Malayalam, 2008: Hindustan Book Agency, New Delhi).
• [12] Frits Staal. “On the Origins of Zero” in Studies in the History of Indian Mathematics (C.S. Seshadri, Ed., 2010: Hindustan Book Agency, New Delhi).
• [13] D.T. Whiteside, Ed., The Mathematical Papers of Isaac Newton, Vol. III (1969: The University Press, Cambridge).

### Footnotes

1. “Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol ‘zero’”. 14 September 2017. Bodleian Libraries, University of Oxford.
2. The indispensable source book for the contents of the manuscript (and much other related information) is Takao Hayashi’s critical edition consisting of an English translation accompanied by detailed annotations [7]. For purposes of orientation, it is to be noted from the outset that the work is undoubtedly in the mainstream of Indian mathematical tradition. Given the historical and geographical continuity of this tradition, there does not seem to be much point in replacing ‘Indian’ in the above sentence by expressions like ‘Indic’, ‘Subcontinental’, etc., as is the current scholarly custom, even though the manuscript was dug up (in 1881) in the region of Gandhāra, in Bakhshali village in what is now Pakistan, about 140 km northwest of Takṣaśila, on the southern border of Swat.
3. It remains something of a mystery as to why this fact, obvious even in a casual reading, was overlooked for so long.
4. For a detailed analysis of how this is accomplished, see [1].
5. For illustrative examples, see [4]. It may help the reader to remember that the multiplication tables we memorised in childhood were for the names of numbers and that we learned them by reciting them; there is no other way.
6. The science of carbon dating is based on the fact that living plant tissue has a fraction of its total carbon content in the form of radioactive C$^{14}$ which is continuously renewed, mainly from atmospheric CO$_{2}$ by photosynthesis; once removed from the living plant, the tissue can no longer absorb atmospheric carbon by a chemical process.}
7. It will be rash to try to pinpoint, in our state of ignorance, the mechanism of contamination. A possibility is the growth of chlorophyll-containing organisms on the dead bark, followed by diffusion. There are other instances of carbon dates disagreeing with the results of different, supposedly reliable, dating techniques.
8. Plofker et al. say that accepting that all the folios were written by one scribe (option 2 above) will imply that the third (the most recent) date is the correct one. On the other hand, contamination by ‘modern carbon’ causes an underestimation of the true age. So, one way to accommodate the discrepancy is to invoke selective contamination of the samples, in which case the earliest date will be favoured as it would be the least contaminated, and it will in fact give a lower bound on the true age if that sample is uncontaminated.
9. Historically, the notion of a polynomial in the abstract was introduced for the first time and explained in a long and revelatory passage in the Malayalam text Yuktibhāṣā of Jyeṣṭhadeva (ca. 1520 ce; it has an English translation [11]. Its model was the set of decimal numbers in the polynomial representation. The passage includes a schematic notation for a polynomial in one variable as a listing of its coefficients (including zero and negative numbers), very close to its modern definition as a sequence of coefficients (which are elements of a ring) of which all but a finite number vanish. When Isaac Newton (in the 1660s) needed to make sense of infinite (power) series, he too modelled them on decimal numbers (which he invoked as the “doctrine recently established” [13]).
10. These grammatical rules are not different in principle from what is practised in daily speech today, though greatly more precise; unlike us, the Vedic people had no symbolic representation to fall back upon.
11. This could have been obviated by inventing a grammatical rule for the operation of exponentiation but exponentiation, unlike root extraction, was not given the status of a fundamental arithmetical operation in India.
12. The Vedic people also seem to have used other representations when the occasion called for them. In the repeated chanting of the gāyatrī, which is a mantra from one of the earlier Books of the Ṛgveda, recited on certain occasions by the orthodox to this day, the count is kept by assigning a set of $9$ finger joints as the numerals, with the right hand as the place of units and the left as the place of tens. See [1].
13. This is not as trivial a matter as may appear. As is well known, the Āryabhaṭīya was sometimes referred to as Āryāṣṭaśata by some later mathematicians, for the 108 verses in its three substantive chapters. Both Henry Thomas Colebrooke (who translated Brahmagupta and Bhāskara II) and Rev. Ebenezer Burgess (the translator of the Sūryasiddhānta), independently, read aṣṭaśata as $800$, at a time, of course, when George Thibaut’s edition of the Āryabhaṭīya had not been published, an error they would not have made if they had paid attention to their Pāṇini.
14. Yuktibhāṣā, Chapter 1, where the notion of the successor of a number was first introduced and identified with the result of adding $1$.
15. One can also do elementary arithmetical operations involving positive integers and rationals quite easily without having to have a zero, once one has a multiplication table for the powers $10$ by their name; those who would like to be convinced can consult [4].
16. Two strong advocates of the view are Frits Staal [12] and M.D. Pandit [9]. Pandit suggests specifically that lopa is the place-value zero.
17. For the early theories, see Datta and Singh [6]. The situation is no better today.
18. For more on the questions addressed in this section and the next two, see [5].
19. The paper [10] is the latest episode in this debate.
20. Descriptions in modern notation can be found in many places, [2], [7], [3] and [5] among others.
21. For a historically and mathematically perceptive account, see Mumford’s article [8]. Mumford’s primary focus is on negative numbers but his approach to the question naturally subsumes the zero.
After his retirement from TIFR Mumbai as Professor of Theoretical Physics, P.P. Divakaran’s main interest has been the history of the mathematical tradition and culture of India.