^{page 464 note 1} Daśagītikasūtra, verse 5.

^{page 464 note 2} Gaṇitapāda, verse 10.

^{page 464 note 3} Śishyadhīvṛiddhida, p. 10, verse 56; for π see p. 28, verse 3.

^{page 464 note 4} Brāhma-Siddhānta, p. 10, verse 36; for π see p. 198, verse 40.

^{page 464 note 5} The latest refinements seem to be those given by Young, , from Clarke, in his General Astronomy (1904), p. 601Google Scholar:—

equatorial semidiameter, 3963'296 miles;

polar semidiameter, 3949–790 miles.

^{page 465 note 1} Commentary on the Bṛihat-Saṁhitā, vol. 1, p. 48.

^{page 465 note 2} Siddhāntaśirōmaṇi, ed. Bapu Deva Sastri, p. 52, verse I: in verse 52 on p. 261 he has given the diameter more precisely as 1581.

^{page 465 note 3} Līlāvatī, verses 5,6: he has referred expressly to this in his statement under Siddhīntaśirōmain, p. 52, verse 1.

^{page 466 note 1} Lewis, , Astronomy of the Ancients (1862), p. 198Google Scholar; Bunbury, , Ancient Geography (2nd ed., 1883), vol. 1, p. 623Google Scholar. The equivalent is 28,959 or 27,800 miles, according to the two valuations of the stadium. But we are concerned here with the actual figures, not with the values of them.

On the assumption that Eratosthenes had practically the true circumference, proposals have been made on the one side to determine the value of the stadium from his figures, and on the other side to decide which one he used out of various stadia. But it is reasonable to hold, with Bunbury (p. 624), that, writing for Greeks, he used “the customary Greek stade, the length of which was familiar to them all”: and a perusal of details fully justifies the same writer's decision that “his conclusion was erroneous, because his data were inaccurate, and his observations defective.” His process was the proper one, of arcmeasurement; and we know that similar attempts were made in other countries also in early times: but we have no good reason for believing that any early people could perform the operation with any real approach to accuracy; they could not determine with sufficient exactness either the distance between any two points or the latitudes of them.

Eratosthenes may be treated reasonably thus. According to the two valuations of the stadium, his original estimate for the circumference, 250,000 stadia, conies to28,728 or 27,579 miles, and with π = 3·14159 these give as the diameter 9144 or 8778 miles; all without fractions. These results may be regarded as creditable enough for so early a time, though they were, of course, useless for any really scientific purposes: even the higher of them is not very much more faulty in excess than was, in the opposite direction, the estimate (see farther on) which prevailed in Europe till nearly the end of the seventeenth century.

For an idea of the delicacy and difficulty of the operation of arcmeasurement, reference may be made to Airy's, Popular Astronomy, revised issue of 1891, pp. 50–71Google Scholar. See also Proctor's, Old and New Astronomy (1892), pp. 87–91Google Scholar.

The modern measurements which have given us the true dimensions began in 1528; see Airy's, “Figure of the Earth” in the Encyclopædia Metropolitana, vol. 5, Mixed Sciences, vol. 3 (1845), pp. 165–240Google Scholar: I am indebted to Dr. Burgess for drawing my attention to this article. In that year, Fernel found the value of one degree on the north of Paris to be 56,746 toises: with the toise taken at 2·1315 yards, this gives 1° = 68·724 miles; and from this we should have circumference 24,740 miles and diameter 7875 miles, without fractions. Other measurements were made in or about 1617 and 1637. In 1669, Picard obtained the result of 57,060 toises, = 69·104 miles, as the mean value of one degree between Sourdon and Malvoisine, which would give circumference 24,877 miles and diameter 7912 miles. And this last result, becoming known to Newton, enabled him to establish in 1687 his theory of gravitation, in respect of which he had previously failed in consequence of following the then still usual estimate of 60 miles as the length of a degree, with the result of 21,600 miles circumference and 6875 miles diameter.

^{page 467 note 1} Bunbury, op. cit., vol. 2, pp. 9, 228–9.

^{page 467 note 2} Natural History, 2. 112. The story was that, after his death, there was found in his tomb a letter from him, stating that he had descended to the lowest part [the centre] of the earth, and that the distance [the radius] was 42,000 stadia.

^{page 467 note 3} His value for the circumference of course works out to more than that of Eratosthenes; because it was the figures for the diameter that he took over, and he applied to them the practically correct value π = 3·1416, instead of the rough value, 3, by which they had been obtained.

^{page 467 note 4} Bunbury, op. cit., 1. 396. The equivalent is 46,080 or 44,127 miles, according to the valuations of the stadium. This value of the circumference perhaps was (but perhaps was not) associated by the mathematicians with the idea of the earth being a flat disc; a view which Aristotle rejected: he held, with the Pythagoreans, that the earth is a sphere. But it does not follow that the idea would reach the Hindūs with the measure: and any circular plan, with dimensions marked on it, would show the earth as a flat surface, even though it was known to be a sphere.

^{page 468 note 1} Another Greek value was that ot Posidonius (about is.c. 135–51), who reduced the circumference first to 240,000 and then to 180,000 stadia Lewis, p. 215; Bunbury, vol. 2, pp. 95, 539. This latter figure, 180,000, was taken over by Marinus of Tyre (second century), and was adopted from him by Ptolemy (A.D. 139 and 161): Bunbury, vol. 2, pp. 539, 564. The equivalent is 20,685 or 19,857 miles. The Hindūs would denote the 180,000 stadia by 2250 yōjana: but such a value does not seem to be found in their books.

^{page 468 note 2} Pañchasiddhāntikā, trans., p. 71, verse 18; and compare p. 16, verse 14; p. 57, verse 10.

^{page 469 note 1} Brāhma-Siddhānta, p. 414, vevse 10.

^{page 469 note 2} Trans., vol. i, p. 312.

^{page 469 note 3} Brāhma-Siddhānta, p. 198, verse 40.

^{page 469 note 4} Primitive Universal Standard of Weights and Measures (1835), p. 73. It may be noted, as a curiosity, that on p. 53 he made practical use of the long value of π quoted by me in this Journal, 1911. 793, and took the decimal even two places farther, ending with 32384.