La Cartografia Antica dell India. Out of these maps, two have been reproduced using a manuscript of Lokaprakasa, originally compiled by the polymath Ksemendra (Kashmir, 11th century CE), as a source. The other manuscript, used as a source by Francesco I, is titled Samgraha'.Samarangana Sutradhara, a Sanskrit treatise by Bhoja (11th century), includes a chapter about the construction of mechanical contrivances (automata), including mechanical bees and birds, fountains shaped like humans and animals, and male and female dolls that refilled oil lamps, danced, played instruments, and re-enacted scenes from Hindu mythology. Late Medieval and Early Modern periods (1206–1858 CE) Madhava of Sangamagrama (c. 1340 – 1425) and his Kerala school of astronomy and mathematics developed and founded mathematical analysis. The infinite series for π was stated by him, and he made use of the series expansion of to obtain an infinite series expression, now known as the Madhava-Gregory series, for . Their rational approximation of the error for the finite sum of their series are of particular interest. They manipulated the error term to derive a faster converging series for . They used the improved series to derive a rational expression, for correct up to nine decimal places, i.e. (of 3.1415926535897...). The development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) was carried out by mathematicians of the Kerala School in the 15th century CE. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). Shēr Shāh of northern India issued silver currency bearing Islamic motifs, later imitated by the Mughal Empire. The Chinese merchant Ma Huan (1413–51) noted that gold coins, known as fanam, were issued in Cochin and weighed a total of one fen and one li according to the Chinese standards. They were of fine quality and could be exchanged in China for 15 silver coins of four-li weight each. In 1500, Nilakantha Somayaji of the Kerala school of astronomy and mathematics, in his Tantrasangraha, revised Aryabhata's elliptical model for the planets Mercury and Venus. His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century. The seamless celestial globe was invented in Kashmir by Ali Kashmiri ibn Luqman in 998 AH (1589–90 CE), and twenty other such globes were later produced in Lahore