Sanskrit-Prakrit interaction in elementary mathematics as reflected in Arabic and Italian formulations of the rule of three – and something more on the rule elsewhere

Sanskrit sources from A¯ ryabhata to Bha¯skara II have a standard formulation of the rule of three. However, it is clear that mathematics must also have been spoken of and performed during this period (and before) in vernacular environments,
and that the two levels must have interacted – not least because the erudite astronomer-mathematicians use commercial arithmetic as the introduction to mathematics. But we have no surviving vernacular texts. From Brahmagupta onward, however, the standard Sanskrit formulation is supplemented by the observation that two of the known magnitudes are similar in kind, and the third dissimilar. This could be an innovation made within the Sanskrit tradition, but comparison with Arabic and Italian medieval sources seems to rule this out. Instead, it must have been current in the commercial community spanning the Indian Ocean and the mediterranean – but since the Sanskrit scholars are not likely to have borrowed from Arabic traders, also in vernacular commercial
arithmetic as practised within India. So far, the story seems simple and coherent. However, if Latin twelfththirteenth-
century writings and sources from the late medieval Ibero-Provencal area are taken into account, loose ends turn up that show the simple story not to be the whole story.