*The Indian influence on early Islamic mathematics*.

For well over 2500 years, Sanskrit texts have recorded the mathematical interests and achievements of Indian scholars, scientists, priests and merchants. Literally millions of manuscripts attest to this tradition, and a few of its highlights (e.g., decimal place-value numerals, use of negative numbers, solution of indeterminate equations, power series in the Kerala school) are standard fare in general histories of mathematics. However, the historiography of Indian mathematics istill comparatively new. The Europeans who first encountered Indian mathematical texts in the eighteenth and nineteenth centuries were often completely at sea concerning their age, their development, and even their identity. Even when the background and content of the best-known treatises were sorted out in the early nineteenth century, historians still had many vexing problems to contend with.

Much mathematical material was embedded in the very unfamiliar context of medieval Indian astronomy and astrology; its presentation in highly compressed Sanskrit verse and commentarial prose was even more unfamiliar; and yet it bore many similarities, from its decimal numerals to its trigonometric formulas, to certain features of western mathematics.

Into this new territory came the early authors of general histories of mathematics, foraging for grand narratives. They incorporated into their overviews of world mathematics many of the newly-gleaned facts about the Indian tradition, establishing a standard, but seriously incomplete, picture of Indian mathematics that still serves as the basic framework of its treatment in most modern histories.

Its historiography was from the beginning co-opted for political purposes, beginning at least with nineteenth-century colonial disparagements of native learning in order to justify westernized education for future colonial functionaries. Many Indian nationalists responded in kind by promoting various separatist or Hindu nationalist historiographies, often including extravagant claims for the autonomy and/or antiquity of their scientific traditions.

The impact of such positions is especially noticeable in the widespread neglect and ignorance (sometimes even denial) of the remarkable exchanges in the second millennium between Hindu or Jaina and Muslim mathematicians in India.

Transmissions between India and the Islamic world were also crucial for the development of early Islamic mathematics. However, many of the most basic questions concerning them, which demand a familiarity with both Sanskrit and Arabic, have yet to be thoroughly investigated.

This research project seeks to reform the antiquated standard narrative of Indian mathematics by drawing on underutilized sources as well as familiar classics to present an up-to-date, detailed and reliable account of its history. It will also serve as a guide to more specialized studies, briefly discussing the major historiographic trends in this field and describing its chief vehicles for published research in India and elsewhere. Attention will be given to astronomical texts, in which mathematical results are often embedded, and to cultural contexts and regional schools, which are largely neglected in existing surveys.

The joint research by Dr Plofker and Dr Hogendijk on the circulation of mathematical knowledge between India and the Islamic world will strengthen our understanding of some of the roots of Islamic mathematics, and of modern European mathematics as well.