300 CE) offers an approximation of 3, in the ratios offered in Bhishma Parva verses: 6.12.40-45. In the 3rd century BCE, Archimedes proved the sharp inequalities < < , by means of regular 96-gons (accuracies of 2·10−4 and 4·10−4, respectively). In the 2nd century CE, Ptolemy used the value , the first known approximation accurate to three decimal places (accuracy 2·10−5). It is equal to which is accurate to two sexagesimal digits. The Chinese mathematician Liu Hui in 263 CE computed to between and by inscribing a 96-gon and 192-gon; the average of these two values is (accuracy 9·10−5). He also suggested that 3.14 was a good enough approximation for practical purposes. He has also frequently been credited with a later and more accurate result, π ≈ = 3.1416 (accuracy 2·10−6), although some scholars instead believe that this is due to the later (5th-century) Chinese mathematician Zu Chongzhi. Zu Chongzhi is known to have computed to be between 3.1415926 and 3.1415927, which was correct to seven decimal places. He also gave two other approximations of : π ≈ and π ≈ , which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation of using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's results surpass the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium. In Gupta-era India (6th century), mathematician Aryabhata, in his astronomical treatise Āryabhaṭīya stated: Approximating to four decimal places: π ≈ = 3.1416, Aryabhata stated that his result "approximately" ( "approaching") gave the circumference of a circle. His 15th-century commentator Nilakantha Somayaji (Kerala school of astronomy and mathematics) has argued that the word means not only that this is an approximation, but that the value is incommensurable (irrational). Middle Ages Further progress was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, found the Maclaurin series for arctangent, and then two infinite series for . One of them is now known as the Madhava–Leibniz series, based on The other was based on He used the first 21 terms to compute an approximation of correct to 11 decimal places as . He also improved the formula based on arctan(1) by including a correction: It is not known how he came up with