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The Forgotten Revolution

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The Forgotten Revolution - Lucio Russo The Forgotten Revolution How Science Was Born in 300 BC and Why It Had to Be Reborn With the Collaboration of the Translator, Silvio Levy ~ Springer 10.2 Eratosthenes' Measurement of the Meridian 273 10.2 Eratosthenes' Measurement of the Meridian Eratosthenes, with the method discussed in Section 3.2, obtained the value 252,000 stadia as the earth's circumference along a meridian. Estimating the accuracy of this measurement is not easy: there has been controversy on the value of a stadium in this context. The most likely reconstruction puts Eratosthenes' stadium in the range 155-160 m/ 6 implying an error of at most 2.4 °!.> below or 0.8% above the true value. Such remarkable ac­ curacy has often been seen with suspicion, especially because it is true only to a coarse approximation that Syene and Alexandria lie on the same meridian and that Syene lies on the tropic. Moreover, whereas modern measurements, first attempted by W. Snell in 1615,27 involved triangula­ tion over distances of a hundred kilometers or so,28 it is generally held that the distance between Syene and Alexandria was estimated by counting days of travel. The conclusion ordinarily accepted is that Eratosthenes did get an excellent value, but only as the result of a very lucky cancellation of errors.29 The size of the degree is not the only distance measurement that Eratos­ thenes is reported to have made. He in fact compiled a map of the whole known world. One of the data transmitted by Strabo is the distance from Alexandria to Rhodes, which Eratosthenes found to be 3750 stadia.Jt1 This value, too, is generally regarded as the result of a rough estimate31 that is u, A value of 157.5 m for the stadium used by Eratosthenes was determined by llultsch in his thorough investigation of Greek measurements ([Hultsch: GRM] , p. 61). Although different from the traditional value used in Greece, it has been accepted by most subsequent scholars as substan­ tially correct. The argument is based primarily on a passage of Pliny (Nal11ralis ilistoria, Xll §53), where the ratio between the stadium and the schoenus is reported to have two alternative values, one of them being called "Eratosthenes' ratio" (Emtostllenis ratio!lc). For the view that Er<ltosthenes used the traditional stildium, of about 185m, see [Rawlin s: ESNM ]. 27 Snell-best known for the sine law of refraction, abou t which more on page 348 - explicitly designed his measu rement as an attempt to duplicate Eratosthenes' feat. The vvor k, carried out in the Dutch fla tlands and described in Snell 's Eratosthe11es Hafm .l iiS {l ti 17), relied on his recovery of ancient methods of triangulation and of spherica l geometry, which culminated in the book lJoctrina triangolorum (Leiden, 1627}. On p. 62 of the latter we find a tantal izi ng tidbit of terminological infom1ation, noted in [Carnevale]: according to Snell, Hipparchus and Menelaus used the te rm tripleuron for spherical triangles. Because we possess no ancient testimonium on Hipparchus in connection with spherical geometry, this may mean that Snell had some source no longer extant. (In the case of Menelaus the source may be Pappus.) Note that Hippilrchus preceded Theodosius, the author of the oldest extant work on spherical geometry. 28 After Snell's and other attempts involving distances too small to be effective, in 1669 the French Academy undertook two careful measurements of distances over 100 krn, under Picard's d irection, and so obtained the first reliable values for the degree of the meridian, namely 57064.5 and 57057 Paris fathoms (toises de Pnris). Picard related this unit to a precisely defined pendulum, so we know it quite exactly (1949 mm); this gives an error of about 0.1 %. See [Picard ]. 29See, for example, [H eath: HCM], vol. 2, p. 107; compare [Neugebauer: HAMA ], p. 653. 30Strabo, Geography. II, v §24. 274 10. Lost Science close to right by accident. Do we know for sure that Eratosthenes assumed that Alexandria and Syene lay on the same meridian and that Syene was on the tropic? The primary source on the subject, Cleomedcs, actually wrote: Eratosthenes' method, being geometric, seems more difficult [than the previously explained method of Posidonius]. What he says will become clearer if we allow ourselves to make two assumptions. We assume first that Alexandria and Syene are on the same meridianY Cleomedes does not give a detailed account of the method- that would be pointless, Eratosthenes' work still being available-but a pedagogical precis meant for readers scared of the complex geometric arguments of the original work. He attains his goal of explaining Eratosthenes' method by taking an ideal case obtained by eliminating all technical difficulties; how else could he compress into three pages a work that occupied two books? Cleomedes also rounds off the numbers, evidently so as not to bother the reader with calculations inessential to an understanding of the method.33 We should not automatically attribute to Eratosthenes the si mplifications adopted by his popularizer. Elsewhere Cleomedes records a precious detail: that at noon on the day of the summer solstice sundials cast no shadow within a zone 300 stadia wide about the tropic line.34 Clearly, many measurements with sundials had been made, over a wide area, and the tropic was fixed as the midline of the shadow less zone.35 Thus it is reasonable to think that this line could be located precisely within a few tens of stadia, or a few minutes of arc.36 We must conclude that Eratosthenes, desirous of measuring the distance from Alexandria to the tropic, first took the trouble to find it precisely and didn't just assume on someone's say-so that it went through Syene. Cleomedes and other authors probably name Syenc because it was the 31 See, for exampl!o', [Neugebauer: HAMA], p. 653. 32Cleomedes, Carlestia, I §7, 35:49- 52 (ed. Todd). 1-' Cleomedes' value for the circumference (250,000 stadia, instead of the 252,000 reported by aU other sources) and for the difference in latitude between Syene and Alexand ria (1 / 50 of the whole circl e) are clearly obtained by rounding, an understandable liberty taken by someone whose aim is avowedly just to illustrate the method. 34 Cleomedes, Caelestia, I §7, 36:101-37:102 (ed. Todd). '"That this datum must have be€n determined by personnel sent on site for the purpose is said already in [Hultsch: PGES], p. 14. But J. Dutka objects that "it is questionilble whether in that era royal surveyors would be used for a purely scientific purpose" ([Dutka], p. 61). · 1"We also know that Era tosthenes could detect astronomie<1lly differences in latitude between spots more th <1 n 400 stadia apart along the same meridian (Strabo, Geoxraplly, II , i §35). The accu- . racy with which one can locate the tropic is much better than this margin of error, since it is easier to distinguish precisely between no shadow and some shadow· th an it is to distinguish between two approximately equ al nonzero magnitudes. The main source of subjective error in the shadow measurement is that the sun is not a point source. 10.2 Eratosthenes' Measurement of the Meridian 275 Egyptian town closest to the tropic and the most convenient base for an expedition to the tropic.37 As for the well whose bottom was lit by the sun at the solstice, Pliny says that it was dug out for a demonstration.:ll> Here a digression is warranted on an important aspect of experimen­ tal methodology. It is commonly thought that Hellenistic scientists were ignorant of the technique of averaging multiple measurements, because there is no direct documentary evidence for its use . 3 '~ But the placement of the tropic at the center of a shadowless zone, as logically implied by Cleomedes' statement just discussed, seems to be a case where the liter­ ature preserves an indirect trace of the method in question. The lack of direct testimonia about experimental averaging is hardly surprising, since the manuscript tradition preserved neither the works where the technique might have been used (such as Herophilus' research on heartbeats) nor the theoretical treatise by Eratosthenes titled On mean s, which might perhaps have cast some light on the issue.40 Next, the determination of the tropic through multiple simultaneous observations affords an accuracy that would be pointless if the distance to Alexandria were then estimated using days of journey. Is it possible that the distance was actually measured? Eratosthenes was the first person to make a map of Egypt. The degree of precision with which he managed to measure the distance from Alexandria to the tropic- that is, to the southern border of the kingdom- is equivalent to that with which the chart was made. A record of the work involved in this topographical survey can be found in the sources. Martianus Capella writes that the distance measurements on which Eratosthenes' estimate of the size of the earth relied were fur­ nished by the royal surveyors (mensores regii),41 and Strabo relays some data from Eratosthenes' map of Egypt.42 We know that already in the Pharaohs' time a detailed measuring of the land (yo:w~o: -r plrx) was made annually throughout Egypt. Under the Ptolemies the measurements were ~ 7 Strabo , Pliny and Arrian all say that Syene is on the tropic. The town lies near the first cataract of the Nile, which marked the boundary between Egypt and Ethiopia: therefore to get to the tropic one had to cross the border.
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