The Bakhshali Manuscript: Ancient Indian Math

Bakhshkali manuscrtipt
The Bakhshkali manuscrtipt.

This interesting artifact is an ancient Indian mathematical manuscript written on birch bark. It was found near the village of Bakhshali in 1881, in the North-West Frontier Province of British India, as was known back then (now Khyber Pakhtunkhwa province, in Pakistan).

The Bakhshali Manuscript is written in Śāradā script and Gatha dialect which is a combination of the ancient Indian languages of Sanskrit and Prakrit. The manuscript is incomplete – there are only 70 leaves of birch bark, many of which are mere scraps. The Bakhshali manuscript is currently stored in the Bodleian Library at the University of Oxford, but is too fragile to be examined by scholars.

It does not appear to belong to any specific period. That being said, some researchers classify it as a work from the early “Classical period”, while others suggest that it may be the work of Jaina mathematicians. This is chronologically plausible but there is no proof it was composed by Jain scholars.

One scientist dates the ancient manuscript back between 2nd century BC and 2nd century AD. The evidence he offers is based on detailed analysis of the contents of the manuscript. It includes the language in which it was written which died out around 300 AD, discussion of currency found in several math problems, and the lack of techniques known to have been developed in the 5th century.

The controversy and debate surrounding the date of the Bakhshali manuscript was particularly intense when it was first discovered and highlights the resistance of European historians to accept new discoveries and evidence of the origins of various mathematical results.

Clearly establishing a date for the composition of the manuscript is extremely important as it has a vast bearing on the significance of its mathematical content.


There are eight principal topics observed in the Bakhshali manuscript, including examples of the “rule of three”, solution of linear equations with as many as five unknowns, solution of the quadratic equation  which was developed with incredible precision, arithmetic and geometric progressions, compound equations, quadratic indeterminate equations (origin of type ax/c = y), simultaneous equations, fractions and other advances in notation including use of zero and negative sign, and improved method for calculating square root and hence approximations for irrational numbers.

By the end of the 2nd century AD mathematics in India had attained a considerable stature, and had become detached from purely practical and religious requirements, although it is worth mentioning that over the next 1000 years the majority of mathematical developments occurred within works on astronomy.