The Bakhshali manuscript
The Bakhshali manuscript is an ancient mathematical treatise that was found in 1881 in the village of Bakhshali, approximately 80 kilometers northeast of Peshawar (then in India, now in Pakistan). Among the topics covered in this document, at least in the fragments that have been recovered, are solutions of systems of linear equations, indeterminate (Diophantine) equations of the second degree, arithmetic progressions of various types, and rational approximations of square roots (more on this below).
The manuscript features an extensive usage of decimal arithmetic — the same full-fledged positional decimal arithmetic with zero system that we use today (although the symbols for the digits are a bit different).
The manuscript appears to be a copy of an even earlier work. As Japanese scholar Takao Hayashi has noted, the manuscript includes the statement “sutra bhrantim asti” (“there is a corruption in the numbering of this sutra”), indicating that the work is a commentary on an earlier work.
Ever since its discovery in 1881, scholars have debated its age. Some, like British scholar G. R. Kaye, assigned the manuscript to the 12th century, in part because he believed that its mathematical content was derivative from Greek sources. In contrast, Rudolf Hoernle assigned the underlying manuscript to the “3rd or 4th century CE.” Similarly, Bibhutibhusan Datta concluded that the older document was dated “towards the beginning of the Christian era.” Gurjar placed it between the second century BCE and the second century CE. In a more recent analysis, Japanese scholar Takao Hayashi assigned the commentary to the seventh century, with the underlying original not much older. (See this paper for references.)
Recent tests by the Bodelian Library
Recently the Bodelian Library in London, where the Bakhshali manuscript has been housed for decades, commissioned a radiocarbon dating study on the manuscript. The test results, which were announced on 14 September 2017, are quite surprising.
These tests found that the samples they examined dated from three different time periods: one from 885-993 CE, one from 680-779 CE and a third from 224-383 CE. The latter date means that at least some of the manuscript is hundreds of years older than Hayashi’s consensus date of seventh century. Indeed, the Bakhshali manuscript’s numerous usages of zero (represented by a centered dot) now means that the manuscript is the oldest known ancient artifact with zero.
Square roots in the Bakhshali manuscript
One particularly intriguing item in the Bakhshali manuscript is the following algorithm for computing square roots:
In the case of a number whose square root is to be found, divide it by the by the approximate root [the root of the nearest square number]; multiply the denominator of the resulting [ratio of the remainder to the divisor] by two; square it [the fraction just obtained]; halve it; divide it by composite fraction [the first approximation]; subtract [from the composite fraction]; [the result is] the refined root. [Translation due to M. N. Channabasappa]
In modern notation, this algorithm is as follows. To obtain the square root of a number q, start with an approximation x0 and then calculate, for n >= 0,
an = (q – xn2) / (2 xn)
xn+1 = xn + an – an2 / (2 (xn + an))
In the examples presented in the Bakhshali manuscript, this formula is used to obtain rational approximations to square roots only for integer arguments q, only for integer-valued starting values x0, and is only applied once in case (even though the result after one iteration is described as the “refined root,” possibly suggesting it could be repeated). But from a modern perspective, the scheme clearly can be repeated, and in fact converges very rapidly to sqrt(q), as we shall see below.
Several explicit applications of this scheme are presented in the Bakshshali manuscript. One example is to find an accurate rational approximation to the solution of the quadratic equation 3 x2 / 4 + 3 x / 4 = 7000. The manuscript notes that x = (sqrt(336009) – 3) / 6, and then calculates an accurate value for sqrt(336009), starting with the approximation 579. The result obtained is
579 + 515225088 / 777307500 = 450576267588 / 777307500
This is 579.66283303325903841…, which agrees with sqrt(336009) = 579.66283303313487498… to 12-significant-digit accuracy. From a modern perspective, this happens because the Bakhshali square root algorithm is quartically convergent — each iteration approximately quadruples the number of correct digits in the result, provided that either exact rational arithmetic or sufficiently high precision floating-point arithmetic is used.
For additional details see the paper Ancient Indian square roots: An exercise in forensic paleo-mathematics.
A Eurocentric bias in historical mathematics studies?
Discoveries such as these underscore the regrettable legacy of a Eurocentric bias in traditional studies on the history of mathematics and science. Western scholars such as G. R. Kaye (mentioned above) quickly convinced themselves that artifacts such as the Bakhshali manuscript, which clearly contain sophisticated mathematical work, must have been derivative of western sources, e.g., Greek mathematics, and were unwilling to accept that groundbreaking work could have arisen elsewhere.
Most likely the redating of the Bakhshali manuscript is just the first step in the rectifying these errors and granting full recognition to early mathematical and scientific work in India, China and the Middle East. It’s about time.