A playful side to twelfth-century mathematics

As editors of the book Lilavati's Daughters: The Women Scientists of India, reviewed by Asha Gopinathan (Nature 460, 1082; 2009), we would like to elaborate on the background to its title. Lilavati was a mathematical treatise of the twelfth century, composed by the mathematician and astronomer Bhaskaracharya (1114–85) — also known as Bhaskara II — who was a teacher of repute and author of several other texts. The name Lilavati, which literally means 'playful', is a surprising title for an early scientific book. Some of the mathematical problems posed in the book are in verse form, and are addressed to a girl, the eponymous Lilavati. However, there is little real evidence concerning Lilavati's historicity. Tradition holds that she was Bhaskaracharya's daughter and that he wrote the treatise to console her after an accident that left her unable to marry. But this could be a later interpolation, as the idea was first mentioned in a Persian commentary. An alternative view has it that Lilavati was married at an inauspicious time and was widowed shortly afterwards. Other sources have implied that Lilavati was Bhaskaracharya's wife, or even one of his students — raising the possibility that women in parts of the Indian subcontinent could have participated in higher education as early as eight centuries ago. However, given that Bhaskara was a poet and pedagogue, it is also possible that he chose to address his mathematical problems to a doe-eyed girl simply as a whimsical and charming literary device.

The Archaeometallurgical Implications of New Findings of Traditional Crafts of Making High Tin ‘Delta’ Bronze Mirrors and ‘Beta’ Bronze Vessels in Kerala State of South India

New metallurgical and ethnographic observations of the traditional manufacture of specular high-tin bronze mirrors in Kerala state of southern India are discussed, which is an exceptional example of a surviving craft practice of metal mirror-making in the world. The manufacturing process has been reconstructed from analytical investigations made by Srinivasan following a visit late in 1991 to a mirror making workshop and from her technical studies of equipment acquired by Glover in March 1992 from another group of mirror makers from Pathanamthita at an exhibition held at Crafts Museum, Delhi. Finished and unfinished mirror from two workshops were of a binary, copper-tin alloy of 33% tin which is close to the composition of pure delta phase, so that these mirrors are referred to here as ‘delta’ bronzes. For the first time, metallurgical and field observations were made by Srinivasan in 1991 of the manufacture of high-tin ‘beta’ bonze vessels from Palghat district, Kerala, i‥e of wrought and quenched 23% tin bronze. This has provided the first metallurgical record for a surviving craft of high-tin bronze bowl making which can be directly related to archaeological finds of high-tin bronze vessels from the Indian subcontinent and Southeast Asia. New analytical investigations are presented of high-tin beta bronzes from the Indian subcontinent which are some of the earliest reported worldwide. These coupled with the archaeometallurgical evidence suggests that these high-tin bronze techniques are part of a long, continuing, and probably indigenous tradition of the use of high-tin bronzes in the Indian subcontinent with finds reported even from Indus Valley sites. While the source of tin has been problematic, new evidence on bronze smelting slags and literary evidence suggests there may have been some sources of tin in South India.