It should also be pointed out to the student that for many practical purposes one of the limits of [pi] stated by Archimedes, namely, 3 1/7, is sufficient. For more accurate work 3.1416 is usually a satisfactory approximation. Indeed, the late Professor Newcomb stated that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope." Probably the earliest approximation of the value of [pi] was 3. This appears very commonly in antiquity, as in I Kings vii, 23, and 2 Chronicles iv, 2. In the Ahmes papyrus (_ca._ 1700 B.C.) there is a rule for finding the area of the circle, expressed in modern symbols as (8/9)^2_d_^2, which makes [pi] = 256/81 or 3.1604.... Archimedes, using a plan somewhat similar to ours, found that [pi] lay between 3 1/7 and 3 10/71. Ptolemy, the great Greek astronomer, expressed the value as 3 17/120, or 3.14166.... The fact that Ptolemy divided his diameter into 120 units and his circumference into 360 units probably shows, however, the influence of the ancient value 3. In India an approximate value appears in a certain poem written before the Christian era, but the date is uncertain. About 500 A.D. Aryabhatta (or possibly a later writer of the same name) gave the value 62832/20000, or 3.1416. Brahmagupta, another Hindu (born 598 A.D.), gave [sqrt](10), and this also appears in the writings of the Chinese mathematician Chang Heng (78-139 A.D.). A little later in China, Wang Fan (229-267) gave 142 / 45, or 3.1555...; and one of his contemporaries, Lui Hui, gave 157 / 50, or 3.14. In the fifth century Ch'ung-chih gave as the limits of [pi], 3.1415927 and 3.1415926, from which he inferred that 22/7 and 355/113 were good approximations, although he does not state how he came to this conclusion. In the Middle Ages the greatest mathematician of Italy, Leonardo Fibonacci, or Leonardo of Pisa (about 1200 A.D.), found as limits 3.1427... and 3.1410.... About 1600 the Chinese value 355/113 was rediscovered by Adriaen Anthonisz (1527-1607), being published by his son, who is known as Metius (1571-1635), in the year 1625. About the same period the French mathematician Vieta (1540-1603) found the value of [pi] to 9 decimal places, and Adriaen van Rooman (1561-1615) carried it to 17 decimal places, and Ludolph van Ceulen (1540-1610) to 35 decimal places. It was carried to 140 decimal places by Georg Vega (died in 1793), to 200 by Zacharias Dase (died in 1844), to 500 by Richter (died in 1854), and more recently by Shanks to 707 decimal places. There have been many interesting formulas for [pi], among them being the following: [pi]/2 = 2/1 . 2/3 . 4/3 . 4/5 . 6/5 . 6/7 . 8/7 . 8/9 . .... (Wallis, 1616-1703)