Appendix I: On the Early History of Pi Our earliest source on 'Ir is a problem from the Rhind mathematical papyrus, one of the few major sources we have for the mathematics of ancient Egypt. Although the material on the papyrus in its present form comes from ab out 1550 B.C. during the Middle Kingdom, scholars believe that the mathematics of the document originated in the Old Kingdom, which would date it perhaps to 1900 B.C. The Egyptian source does not state an explicit value for 'Ir, but teils, instead, how to compute the area of a circle as the square of eight-ninths of the diameter. (One is reminded of the 'classical' problem of squaring the circle.) Quite different is the passage in I Kings 7, 23 (repeated in 11 Chronicles 4, 21), which clearly implies that one should multiply the diameter of a circle by 3 to find its circumference. It is worth noting, however, that the Egyptian and the Biblical texts talk about what are , apriori, different concepts: the diameter being regarded as known in both cases, the first text gives a factor used to produce the area of the circle and the other gives (by implication) a factor used to produce the circumference.1 Also from the early second millenium are Babylonian texts from Susa. In the mathematical cuneiform texts the standard factor for multiplying the diameter to produce the circumference is 3, but, according to some interpre­ tations, the Susa texts give a value of 31.2 Another source of ancient values of'lr is the class of Indian works known as the Sulba Sütras, of which the four most important, and oldest are Baudhayana, Äpastamba, Katyayana and Manava (resp. BSS, ASS, KSS and MSS), com­ posed during the period from 800 to 200 B.C. The Sulba Sütras are concerned with mathematical rules for complying with the requirements of of Hindu rit­ ual, specifically with measuring and constructing sacrificial pits and altars. The requirements of religious ritualIed to the problems of squaring the circle and, as it has been called, 'circling the square'. For this latter problem all the Sulba Sütras state varying forms of a rule that pro duces , for a circle whose area is equal to that of a square of side a, the radius a(2 + J2) r= 6 (*), which gives for 'Ir in the context of areas a value nearly 3.088.3 For 'Ir as a IThe importance of making this distinction in historical investigations is pointed out in A. J. E. M. Smeur, "On the Value Equivalent to 11' in Ancient Mathematical Texts. A New Interpretation," Archive for History of Exact Sciences 6 (1970), 249 - 270. 20. Neugebauer, "Exact Sciences in Antiquity," (2nd Edition) Dover, New York (1969), p.47. 3This value results from circumscribing a circle around the given square and then in­ creasing one-half the side of the square by t of the difference between the radius of the circumscribed circle and half the side of the square. The resulting sum is alleged to be the radius of the circle equal to the given square. 677 factor for obtaining the circumference the MSS gives 3! - and advises the reader that 'not even a hair's length is extra,4. For squaring the circle BSS, KSS and ASS all state rules that give ell)2 as the area of a square equal to a circle, which they all admit is only a coarse approximation and which, indeed, gives 7r = 3.004. BSS also gives for the side, a, of a square equal in area to a circle of diameter d d d d d a=d--+--- + , 8 8 . 29 8 . 29 . 6 8 . 29 . 6 . 8 which was perhaps obtained by solving (*) above for a and using the Sulba's value of J2 = ~~~. 5 Given that the problem of expressing the area of a circle as that of a certain square appeared in so many ancient cultures it is hardly surprising that it appeared in ancient Greece as weIl. There is a jesting reference to it in Aristophanes's play The Birds (413 B.C.), and geometrical attacks on the problem from around that time, in the works of Hippocrates of Chios, Bryson and Antiphon, are known.6 In fact, Hippocrates was aware of the contents of what we know as Euclid's Elements XII, 2: 'Circles are to one another as the squares on their diameters,' a proposition which implies that the circle and the square on its diameter (and hence on its radius) bear a constant ratio to one another, but there is no evidence that any of this early work resulted in a numerical value for 7r. It was Archimedes (277 - 212 B.C.) who first established rigorously a connection between the area of a circle and its circumference in Prop. 1 of his treatise Measurement oE the Circle, which we include in this volume. It is in this treatise as weIl where his famous upper and lower bounds for 7r occur, in the form of Proposition 3: the circumference of any circle is greater than 3 ~~ times the diameter and less than 3 ~g times the diameter. Heron of Alexandria (fl. ca. 60 A.D.) writes, in his Metrica (i, 26), that Archimedes in a (now lost) work Plinthides and Cylinders made a better ap­ proximation to 7r. The figures are garbled, but it seems the denominators were on the orders of tens of thousands and the numerators on the orders of hundreds of thousands. The French historian, P. Tannery, has made two possible emendations of the text, either of which gives as the mean of the two limits, the approximation 7r = 3.141596. In his Almagest (vi, 7) Ptolemy (150 A.D.) gives for the circumference of a circle the value 3 + :0 +~, i.e. 3.1416, times its diameter. Since this exactly corresponds to the perimeter of a regular 360-gon inscribed in a circle, calculated according to his table of chords in a 4For other values from the MSS and references to the literature on the Sulba Sutras, see R. C. Gupta, "New Indian Values ofpi from the Mänava Sulba Sutra." Centaurus 31 (1988), 114 - 125. 5Datta B., "Science of the Sulba," University of Calcutta, Calcutta (1932), 143-4. 6See pp. 220-235 in T.L. Heath, A History of Greek Mathematics, vol. 1, Clarendon Press, Oxford, 1921, for details on this and the other Greek material cited below. 678 circle, there is no doubt that Ptolemy used his ehord table (Almagest i,ll) as the basis for ealeulating 'Ir. In the period from the late fifth to the early seventh eentury of our era in India two astronomers stated rules for measuring circles that implied values of 'Ir. The earlier of these, .Ä.ryabhata (b. 476), in his work known as the Äryabhatiya gives correctly the area of a circle as that of the rectangle whose si des are half the circumference and half the diameter, and then gives the volume of a sphere as the area of the [great] circle multiplied by its square root. 7 Then in Rule 10 we are told 'Add 4 to 100, multiply by 8, and add 62,000. The result is ap­ proximately the cireumference of a circle of which the diameter is 20,000.'8 This value of 'Ir, ~~~~~, yields the decimal equivalent 3.1416.9 In his notes on Rule 10. Kaye states that the second of the two writers we referred to above, the astronomer Brahmagupta (b. 598), gives for 'Ir the values 3 and VW, and that Brahmagupta finds fault with .Ä.rhybhata for using in one place 3393 and in another 3393 10 1080 1050' Both Greek and Indian science came to the court of the 'Abbasid califs in the latter part of the eighth and early ninth centuries. One of the most im­ portant writers of this latter period was Mul,unmad ibn Müsa al-Khwarizml, whose Algebra was eomposed during the first third of the ninth eentury and dedicated to the Calif al-Ma'mün. Between its opening section on algebra and its third (and final) section on the calculation of inheritances, al-Khwarizml's work contains, a seetion on mensuration, 11 with the following rules: The 'prac­ tical man' takes 3+ as the value whieh, multiplied by the diameter, produces the cireumferenee, 'though it is not quite exact.' Geometers, on the other hand, take the cireumference either as VlO· d· d or, if they are astronomers, as ~~03~Od. He next gives the same rule as that found in .Ä.ryabhata and the Banü Müsa ( see below), Le. that any circle is equal to the reet angle whose one side is half the eireumferenee and whose perpendicular side is the radius. Finally, al-KhwarizmI states that 'If you multiply the diameter of any circle by itself, and subtract from the produet one-seventh and half of one-seventh of the same, then the remainder is equal to the area of the circle, , 7«The Äryabhatiya of Äryabhata," (W.E. Clark, tr.), Chicago, U. of Chicago Press (1930). See Rule 7, p. 27. 8The use of large values for the diameter was one way of avoiding fractions. 9 An earlier work than the Äryabhatiya is the Suryasiddhanta, although its present form incorporates later material. The value of 11' in an early part of the Suryasiddhanta is v'iö( = 3.1623), but in the section on the Sine table the value used is 10,8000: 3438(= 3.14136). lOKaye, G.R., "Notes on Indian Mathematics, No.