The Egyptian Heritage In The Ancient Measurements Of The Earth

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

The Egyptian heritage in the ancient measurements of the earth GYULA PRISKIN The comparison of an Egyptian text recording the north-south extent of the country with the descriptions of Eratosthenes’ and Posidonius’ experiments to measure the earth’s circumference reveals that early Hellenic mathematical geography borrowed much information from Egyptian science. A further analysis of all the figures circulated for the length of the meridian in antiquity reinforces the case that Hellenic geographers were greatly influenced by Egyptian ideas when they formed their opinions on the size of the earth.

Introduction The literature on the measurements of the earth in antiquity – especially on Eratosthenes’ and Posidonius’ experiments – is so vast that it is almost impossible to compile a full bibliography on the subject, although hopefully a representative part of it will be cited along the present work. The obvious reason for this large number of publications, often appearing in lesser known or highly specialized journals, is the several moot points in the existing evidence that provide ample opportunity for different interpretations and scholarly debate. One such salient detail is for example the length of the stade the first earth-measurers used in their calculations, as the accuracy of their results really hinges on this particular point (see for instance Engels 1985; Gulbekian 1987; Dutka 1993). While opinions often widely differ on this and on many other aspects, one common feature clearly emerges: the authors – usually historians of science, astronomers, and classicists – committing themselves to contributing to the discussion have little or no knowledge of ancient Egypt in general, and ancient Egyptian science in particular. Their ignorance may be forgiven, as it coincides with the still widespread predilection of modern scholars – excepting of course orientalists – rarely to extend their cultural horizon beyond the ancient Greek civilization and Greek philosophers (as they in fact date the birth of science in the modern sense to this era). The neglect of Egypt, it must be admitted on the other hand, is also due to the scarcity of indigenous source material. Although various tomb-paintings illustrate that land surveying was quite developed in ancient Egypt as a result of the need to re-allocate farming plots annually (Lyons 1926; Butzer 1977, 525), and the Egyptians often described mythical landscapes in terms of measurements in their religious literature (Quirke 2003), there is comparably much less evidence to suggest that they made equally thorough efforts to map and measure the inhabitable world beyond their immediate surroundings. But that is not to say that evidence of this nature is completely lacking. Some fifteen years ago Christian Leitz claimed that the dimensions recorded for the different regions of the underworld in the Amduat, a funerary book depicting the sun god’s nocturnal journey on the walls of New Kingdom royal tombs (c. 1500-1000 BCE), may have derived from a prior measurement of the earth’s circumference (Leitz 1991, 101-104). His arguments, however, were not entirely convincing and indeed other explanations for the dimensions in the Amduat have since been put forward (Ferrari d’Occhieppo, Krauss & Schmidt-Kaler 1996; Priskin 2001, 110-113). On the other hand, his ideas also prompted some speculation about the shadow-measuring techniques the Egyptians might have possibly used to determine the length of the meridian (Zeidler 1997, 109-110; Wirsching 2002). All things considered, although Leitz might have wrongly identified the Amduat as the document proving the existence of mathematical geography in ancient Egypt, that does not necessarily mean that the quest for the proof of such a discipline should be abandoned once and for all. It seems wiser to explore other native textual traditions in order to seek out stronger evidence for a mathematic approach to geography by the Egyptians. In the initial part

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

of my paper I will do just that when I present and interpret an Egyptian text describing the north-south extent of the country that – given its content and context – naturally lends itself to an analysis from a geodetical point of view. Then I will turn my attention to the classical evidence and compare it with the Egyptian material, arguing that there are definite points of connection between the two pools of knowledge. These links reveal that the Greek geographers heavily relied on the feats of their Egyptian predecessors for their work. I must note here, however, that neither the investigation into the Egyptian sources nor the examination of the classical testimonies is an easy task. The first because of the extreme brevity of the evidence, so one must be careful not to read too much into the text and be carried away to see things that are simply not there, while the second because of the many layers of interpretation that have been overlapping the material since antiquity, so here one must be cautious to differentiate between facts and opinions. Still, with the right amount of wisdom and circumspection these pitfalls – I believe – can be avoided, and the whole issue can be put in an entirely new perspective. The traces of Egyptian mathematical geography True, if mathematical geography was in any form ever pursued in ancient Egypt, no systematic account of it has come down to us. However, there are some distinct – albeit faint – traces suggesting that Egyptian geography went beyond the simple description of the landscape and utilitarian cadastral surveys. Clement of Alexandria, a Christian author living in the Delta in the 2nd century CE, writes about the procession of the Egyptian priests carrying their sacred books on the occasion of a religious festival (Stromata 6.4). One of these priests, who is distinguished from his colleagues as the ‘scribe’, holds among other books a papyrus roll written about geography, and also some other papyri containing maps of the Nile valley. The closest authentic Egyptian source matching the description of Clement of Alexandria is undoubtedly the Tanis Geographical Papyrus, a badly damaged document written in hieroglyphics during the Graeco-Roman Period and found late in the 19th century (Griffith & Petrie 1889, 21-25). In the contents of the papyrus (various lists of festivals, nome capitals, sacred animals, place-names, gods, and numbers) there is only one piece of information that may be classified as having anything to do with mathematical geography. In one of the charred fragments the expression ‘106 iteru’ is clearly legible, and although its immediate context is lost, it is not far from a short series of hieroglyphic characters that refer to the distance between the Mediterranean coast and the apex of the Delta (Griffith & Petrie 1889, pl. 9). As a matter of fact, these fragments belong to a well-identifiable genre of Egyptian texts that give information on the entire north-south extent of the land and its subdivisions. The figure of 106 iteru first appears on a re-used building block originating from a destroyed temple of Amenemhet I (second half of 20th century BCE), while later testimonies include the reconstructed White Chapel of Senwosret I (end of 20th century BCE), various inscribed cubit rods (from 16th century BCE onwards), and the enclosure wall of the Graeco-Roman temple at Edfu (1st century BCE). All these sources (handily collected in Schlott-Schwab 1981, 3-5) unanimously equate the figure of 106 iteru with the overall distance from the northernmost region in the Delta to Elephantine, an island on the Nile opposite Aswan. The metric equivalent of 106 iteru can easily be found, as long as the length of the base unit of the Egyptian system of measurement, that of the royal cubit, is satisfactorily established, for it is generally accepted that one iteru comprised 20,000 royal cubits (Helck 1980, 1200; Clagett 1999, 7). In Egyptological literature customarily two values are cited for the length of the royal cubit: 0.5237 m, based on W. M. F. Petrie’s metrological investigations into the Giza pyramids (converted from inches; Petrie 1883, 179), and 0.525 m, following on

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

Richard Lepsius’ statement in his work about existing cubit rods (Lepsius 1865, 5). Since in the Tanis Geographical Papyrus one passage expressly links the cubit with pharaoh Khufu, the builder of the largest pyramid at Giza (Griffith & Petrie 1889, pl. 14), in this paper priority will be given to the length of 0.5237 m, though it must be noted that the use of the other value would not lead to significantly different results in the calculations below, and therefore neither would it seriously affect the validity of the arguments in the foregoing reasoning. If then one cubit equals 0.5237 m, and one iteru is 20,000 cubits, 106 iteru will equal 1,110,244 m. This figure is very close to 1/36 of the earth’s circumference (Ce/36 = 1,111,330 m, if the polar and equatorial radii are taken as 6,356,752 m and 6,378,136 m respectively; Seidelmann 1992, 700). The difference is thus only 1086 m, but it is now difficult to decide whether the proximity of 106 iteru to a geodetically meaningful number is just a mere coincidence, or it reflects an earlier – quite precise – measurement of the earth’s circumference on the part of the Egyptians. Nonetheless, one thing is for certain: the figure of 106 iteru cannot have measured the latitudinal difference between the traditionally assigned extreme north and south points of Egypt (i.e. the north-south distance between the parallels running through these points). According to my own measurements, taken by a standard GPS device, the northernmost point of the Delta (literally, on the beach) lies at 31° 36.075' N, 31° 05.238' E, while the island of Elephantine stretches from 24° 05.94' N, 32° 53.629' E in the north to 24° 05.028' N, 32° 53.034' E in the south. The greatest possible distance in latitude between these coordinates is 31° 36.075' – 24° 05.028' = 7° 31.466', but certainly some margin for actual human settlements – as likely places of observations – must be allowed for to diminish this figure, especially by the sea in the north. Therefore, and also for the reason that no major changes affecting these locations have occured in the topography of the land since the 3rd millenium BC, it seems that the northern and southern end-points of Egypt, as conventionally conceived of by its native inhabitants, provided ancient geographers with a section of the globe that measured 7° 30', that is, 1/48 of the meridian (= 79.5 iteru). The 106 iteru – as it significantly exceeds this figure – has been thought to define the length of Egypt from Elephantine to the Mediterranean measured along the meandering riverbed of the Nile (Borchardt 1921, 120). While this proposition is certainly basically correct, it does not shed any light on how the Egyptians arrived at this value. The actual measurement of the Nile, given the enormous practical difficulties such an undertaking would involve, must be ruled out (Rawlins 1982b, 261). It is more likely that the 106 iteru is an authoritative figure and some alternatives must be scouted around to explain why from the infinite number of choices the Egyptians opted for 1/36 of the meridian to describe the winding course of the Nile, if indeed it was not chosen fortuitously. Simply enough, the actual latitudinal extent of Egypt (1/48 of the meridian) and the conventionally recorded length of the country (106 iteru = 1/36 of the meridian) were in the ratio of 3 : 4. So perhaps a possible clue is offered by the 2nd century CE Greek author Plutarch who writes (Isis and Osiris 56) that “the Egyptians liken the nature of the universe especially to this supremely beautiful of the triangles ... [which] has a vertical of three units of length, a base of four, and an hypotenuse of five.” In this case then it must be supposed that after the correct determination – by whatever means – of the difference in latitude between the edges of Egypt at the Mediterranean and Elephantine, the choice of 106 iteru for the length of the Nile was dictated by a simple numbers game that was based on the most elementary Pythagorean triangle. To sum up briefly what insight a modern reader might have into the possible geodetical knowledge of the ancient Egyptians by perusing their writings on the dimensions of their land, these are the plain facts: the figure of 106 iteru corresponds to 1/36 of the meridian, and incidentally this figure is to the actual latitudinal extent of Egypt (1/48 of the meridian) as 4 is

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

to 3. These traces on their own, however, do not furnish sufficient evidence to claim that the Egyptians were capable of measuring the earth, and would perhaps be no more than curious coincidences, were it not for the works of the first classical geographers in which the same figures appear and the same proportional relation is encountered. Cleomedes There may have been several attempts to evaluate the size of the earth by a scientific method in antiquity, but only two of them were described in detail in classical literature. Unfortunately, the original reports of those who had been personally responsible for these two experiments, the writings of Eratosthenes (3rd century BCE) and Posidonius (end of 2nd century BCE), have not survived. An account of what they did in order to measure the earth is given in a book written in the 2nd century CE by a Stoic author, Cleomedes (The Heavens 1.7). In his work, which can best be described as a collection of lectures aimed at the students of his philosophical school, he singles out Eratosthenes and Posidonius as the two ablest scientists whose methods had surpassed the attempts of others. It must be noticed, however, that Cleomedes here makes a value judgement the veracity of which cannot be determined in the absence of further sources. Either there were other serious attempts to measure the earth in the Hellenic world that simply eluded the attention – or intellect – of Cleomedes, or he was right in that Eratosthenes’ and Posidonius’ results stood out from a series of more speculative experiments. In other words, the lack of other sources on the subject means that it cannot be determined whether he – perhaps due to his own stance in the philosophical debates of his day (see Bowen 2003) – distorted the truth, or he did pass on posterity the descriptions of those experiments that were in fact worth mentioning from a scientific point of view. Certainly, from the works of some other highly reputable scholars it is clear that there were a couple of more figures for the earth’s circumference circulated in antiquity (Archimedes, Sand Reckoner 1.8; Aristotle, On the Heavens 2.14); however, these authors did not comment on the procedures by which these figures had been found. Whatever the case is, though, Cleomedes’ treatise on the one hand does offer enough information for a collation with the Egyptian sources, and on the other does highlight the strong similarity to them. While the question on the number of scientifically relevant measurements of the earth will probably remain unresolved until perhaps new evidence comes to light, another of Cleomedes’ distortions can easily be rectified. Not only does Cleomedes say that Eratosthenes and Posidonius were ahead of their contemporaries in the field of geodesy, but he also sets up an order of preference between the two scholars, dismissing the view that Eratosthenes’ method was more obscure and repeatedly quoting his figure for the earth’s circumference in other passages of his book (Cleomedes, The Heavens 1.5.72 & 2.1.294-295). This judgement of Cleomedes’ is still widely reflected in the modern literature and is usually summarized by the statement that the most accurate measurement of the earth in antiquity was given by Eratosthenes (Fischer 1975, 152-153). If, however, the account of Cleomedes is truly closely read, it becomes obvious that there is no outstanding difference in methodology – and in quality – between Eratosthenes and Posidonius, as they both use the principles of proportionality and equality of angles to determine the size of the earth (Taisbak 1974, 256259). Of course they rely on different natural phenomena in their experiments – Eratosthenes on the sun and Posidonius on a bright star – but the earlier’s solar method and the latter’s stellar one are still essentially the same, and after all there is no way of telling whether the figures they establish for their needed terrestrial distances really represent different levels of accuracy. It is only a pity that this simple truth has only been recognized recently. It has, however, a very obvious consequence: when the classical measurements of the earth are analysed, Eratosthenes and Posidonius must be treated on the same level and must be

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

recognized as sources with equal value, and this is precisely what will be done in the following two chapters. Eratosthenes According to Cleomedes (The Heavens 1.7.49-110), Eratosthenes reckoned that the meridional arc corresponding to a given celestial arc could be measured by observing the shadows cast by the sun at two sufficiently distant locations lying on the same meridian. Eratosthenes reportedly chose Alexandria and Aswan as the two places befitting his experiment. Alexandria seems to be a natural choice because Eratosthenes worked in that city as the head of the famous library, while as regards shadows, the other place, Aswan, was a special one, he believed, because it lay straight on the tropic, so at noon on the day of summer solstice the sun cast no shadows there, hovering exactly overhead. On the same day at noon he also gauged the length of the shadow at Alexandria by a hemispherical bowl known as the skaphe, and found that it represented 1/50 part of a circle. Applying the principle of equal angles, he now determined that the meridional arc between Alexandria and Aswan had to be of the same length (Figure 1). He also knew that the terrestrial distance between these places equalled 5000 stades. Now applying the principle of proportionality he could reckon that the great circle must have measured 50 times 5000 stades, that is, 250,000 stades.

Figure 1 Eratosthenes’ measurement of the angular distance between Alexandria (A) and Aswan (S). Angle BFA, observed to be 1/50 of the circle, equals angle AOS.

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

Eratosthenes’ procedure seems simple enough, but at a closer look it is fraught with a number of mistakes and leads to a few problems that have not yet been satisfactorily explained. The inconsistencies in the description of his method may even suggest that Eratosthenes did not actually carry out any experiment himself, but in the fashion of a true desk scientist he based his opinions on the teachings of previous authorities or commonly known assumptions, or perhaps plain hearsay (Newton 1980, 387). For one thing, Alexandria and Aswan do not lie on the same meridian, and the difference of about three degrees was perhaps noticable for a serious observer as early as the 3rd century BCE, even if the determination of longitudes – in the absence of reliable clocks – was an inherently imprecise undertaking in antiquity. Nor was Aswan situated below the tropic in the 3rd century BCE, even if allowance is made for the sun’s apparent diameter, because due to precession the line defining the northernmost position of the sun had moved a few kilometres southward by the time of Eratosthenes (Ball 1942, 40). From a practical point of view, it is easier to imagine a hemispherical bowl with a pointer casting a distinct shadow on it than actually build such a device or observe where the tip of the shadow really ends on the curving surface (Newton 1980, 381; Gratwick 1995, 202). It is also a nagging question why some classical authors augmented Eratosthenes’ figure for the earth’s circumference to 252,000 stades (for a list see Goldstein 1984, 412). Or quite on the contrary, did Cleomedes slightly adjust the 252,000 stades to suit the calculations he recorded? The claim that this second scenario is more likely, and that the 252,000 stades is not based on precise geodetic measurements, may find support in the remarks that the number 252,000 might have been the outcome of a Pythagorean numbers game, as it is the lowest common denominator of the integers from 1 to 10 (Rawlins 1982a, 216), and that the same figure is cited by Pliny (Natural History 2.83) as the one Pythagoras had established for the distance between the sun and the moon. The accuracy of Eratosthenes’ figure, if it was indeed arrived at by some way of measurement and not by some mental arithmetic, is also difficult to judge. Since it is easily conceivable that the angular distance between the two places around Alexandria and Aswan where he actually made his readings of the skaphe was in fact 1/50 part of the circle (although some even question the precision and experimental background of this figure, Newton 1980, 384; Goldstein 1984, 412), the clue to this matter lies with the length of the stade he used in his experiment, and eventually with the accuracy of his figure for the terrestrial distance between his two observation points. As for the origins of the 5000 stades, which almost all researchers believe in any case to be a rounded figure, two theories have so far been put forward by modern researchers. One of them claims, prompted by a remark from the 5th century CE Roman author Martianus Capella (The Marriage 598), that this figure was provided to Eratosthenes by the bematists of the Ptolemaic rulers of Egypt, that is, royal pacers who measured long distances by counting the number of their steady steps, each representing the same length (see Dutka 1993, 61-62). The other theory conjectures that the 5000 stades is a projection of an astronomical distance onto the terrestrial arc between Alexandria and Aswan (Rawlins 1982a, 215; Thurston 2002, 66). A closer examination of classical and Egyptian sources reveals, however, that neither of these assumptions is correct. Cleomedes not only relates that Eratosthenes took the way from Alexandria to Aswan as equalling 5000 stades, but he also says that the area where the sun’s rays were perpendicular to the earth on summer solstice spanned a circle with a diameter of 300 stades (The Heavens 1.7.75-76). Whether this area centred on Aswan or stretched south from it, he does not specify. Nevertheless, the two figures added up equal 5300 stades, and interestingly enough Strabo also writes (Geography 17.1.2) that according to Eratosthenes Egypt measured from Aswan to the sea in the north 5300 stades. Thus it seems that Cleomedes’ 5000 stades for the

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

Alexandria-Aswan distance is only a later development, and the original figure for the extent of Egypt – certainly for Eratosthenes and Strabo, but possibly for other Hellenic geographers as well – was 5300 stades. All the more so because the connection of this figure with the Egyptian sources is not difficult to see. The number 5300 equals precisely 50 times 106, so it must be surmised that the length of 5300 stades is a direct conversion from 106 iteru, the length the Egyptians traditionally ascribed to their land since at least the 20th century BCE (Priskin 2004b, 60-61). From another perspective, this identity of the two figures also implies that Eratosthenes took over an Egyptian tradition of converting different units of measurement according to which one iteru equalled 50 stades. Without advancing too far into the treacherous minefield of ancient metrology, two pieces of evidence can be cited here to back up this claim. One is the name Tachompso referring to the southern border of the Dodekaschoinos, a 12-iteru long region south of Elephantine. In Egyptian this name read ¦i-kmi-600 (Locher 1999, 260), suggesting that for the Egyptians 12 iteru equalled 600 smaller units; consequently one iteru consisted of 50 of these unspecified smaller units of length which the Greeks must have identified with the stade. The other is again provided by Strabo’s description of the Nile, as he says that the distance from the first waterfall at Aswan to the great cataract was 1200 stades (Geography 17.1.2). If converted with a factor of 50, this figure equals 24 iteru, which must also be a corrupt form of an Egyptian distance, because hieroglyphic inscriptions occasionally described the Dodekaschoinos as 12 iteru on the eastern side of the river and 12 iteru on the western side, totalling 24 iteru altogether (Locher 1999, 341). Furthermore, it must be noted that all the distances in Strabo’s map of the Nile can be divided by 50 (Priskin 2004b, 62). Therefore, beside the immediately obvious fact that his whole enterprise was set in Egypt, it is now entirely clear that one of the most crucial pieces of information in Eratosthenes’ experiment, the figure for the terrestrial distance between Alexandria and Aswan, was a direct borrowing from the Egyptians. Eratosthenes certainly did not stand alone as the only Greek scholar relying on indigenous information for the length of Egypt, because Herodotus in the 5th century BCE also quoted the same Egyptian figures for this distance, although in a corrupt form (Priskin 2004a). It may well have been that Eratosthenes misunderstood his sources and did not realize that the 5300 stades (= 106 iteru) originally could not refer to the straight-line distance between the extreme northern and southern limits of Egypt. It seems, however, that he did adjust this figure to his needs when he truncated it to 5000 stades, perhaps having been aware of the fact that Alexandria lay a bit further to the south than the northernmost part of the Delta. At the same time, he may also have had some information that the tropic no longer exactly fell upon Aswan, so he shifted 300 stades from the north to the south, as intimated by Cleomedes. And there is of course the possibility that it is Eratosthenes’ commentator who is responsible for the truncation of the 5300 stades. Posidonius Cleomedes says (The Heavens 1.7.7-48) that Posidonius was able to measure the earth’s circumference by determining a given terrestrial arc with the help of a bright star. Posidonius believed that the Greek city on the island of Rhodes lay on the same meridian as Alexandria. He is said to have observed that the star Canopus (a Carinae) culminated just above the horizon at Rhodes. Then he divided the celestial circle into twelve equal parts in accordance with the twelve signs of the zodiac. Since in Alexandria Canopus reportedly reached its highest elevation on the night sky one quarter of a zodiacal segment above the horizon, Posidonius surmised – applying the principle of equal angles – that the arc between Rhodes and Alexandria was 1/48 of the whole circle (Figure 2). As for the terrestrial distance between Rhodes and Alexandria, Posidonius reckoned it to be 5000 stades, although in Cleomedes it is

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

suggested that it was just a hypothetical value accepted only for the sake of demonstrating the procedure by which the earth’s circumference could be effectively measured. Nonetheless, relying now on the principle of proportionality, Posidonius calculated the whole meridian to be of 48 × 5000 stades = 240,000 stades.

Figure 2 Posidonius’ measurement of the angular distance between Rhodes (R) and Alexandria (A). Angle CHA, observed to be 1/48 of the circle, equals angle ROA.

Just as with Eratosthenes, Posidonius’ method seems simple enough, but it is also fraught with grave errors and major inconsistencies. Again, Rhodes and Alexandria do not lie on the same meridian, although the longitudinal difference between these two places, about two degrees, is smaller than the one between Alexandria and Syene. What is perhaps more important is the fact that his figure for the latitudinal difference between Rhodes and Alexandria – 1/48 of the circle – is in error with a sizeable margin, as in reality this figure equals approximately 1/68 of the meridian. This probably once more indicates that Posidonius may not have made observations of the star Canopus himself, but leaned on earlier sources to fabricate a plausible method for measuring the earth’s circumference, without being too much concerned about the exactitude of the particular details. As a matter of fact, it has been pointed out that in the time of Posidonius (end of 2nd century BCE) Canopus did not culminate on the horizon at Rhodes but a significant one degree above it, even in the northern part of the island (Drabkin 1943, 510). Another similarity between Eratosthenes and Posidonius is that the latter’s figure for the circumference of the earth was also replaced by another one in sources other than Cleomedes; in this case, however, not a minor adjustment took place, but

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

the hugely different figure of 180,000 stades is cited by Strabo (Geography 2.2.2) as Posidonius’ estimate for the length of the whole meridian. Perhaps in conjunction with this Strabo also relates (Geography 2.5.24) that the terrestrial distance between Rhodes and Alexandria was 3750 stades according to Eratosthenes’ measurement with the sundial. The contradictions that abound in the description of Posidonius’ method for measuring the earth can again find their explanations if the data associated with it are pitted against the Egyptian evidence. Thus as far as the arc equalling 1/48 of the meridian is concerned, the earlier sources on which Posidonius relied are not difficult to establish, because – as it was shown in an earlier chapter – it was the land of Egypt that provided an earthly observer with the ideal conditions to determine a terrestrial arc with the length of 1/48 of the whole circle. The different figures for the terrestrial distance between Rhodes and Alexandria also constitute a telltale sign in the direction of Egyptian origins. In the first place, Cleomedes cites 5000 stades for this distance, which is the same figure as Eratosthenes used to describe Egypt following on – it was argued in the previous chapter – native sources. Even more revealing is the proportional relation between the figure of 5000 stades and the one that Strabo ascribes to the Rhodes-Alexandria distance, 3750 stades. This ratio is simply but perhaps not surprisingly 4 : 3, the same that was demonstrated to have existed between the canonized figure of 106 iteru and that of 79.5 iteru, the actual terrestrial distance between the parallels running through the traditionally conceived boundaries of Egypt. All these fine details perceived in connection with Posidonius’ determination of the earth’s circumference suggest that the description of his method originally applied to an experiment that must have been carried out in Egypt by its native inhabitants. Canopus – which anyway is one of the few star names that derive from the ancient Egyptian language (Kunitzsch & Smart 1986, 25) – was visible in the Nile valley all throughout antiquity, and it culminated not much above the horizon from the vantage point of an observer standing in the northern part of the Delta at the beginning of the Egyptian civilization, around 3000 BCE (about two degrees, as opposed to the roughly one degree at Rhodes in 100 BCE, StarCalc 5.72 2002). It seems therefore that Posidonius, acting on the relevant observation that in his time the culmination of Canopus approximately aligned with the horizon at the location of Rhodes, supplanted the original Egyptian description and moved it to a territory that was more familiar to him, yet he did not bother about modifying the other factual data he had gained from his primary sources. These original sources – as the arc amounting to 1/48 of the circle and the relation between 5000 and 3750 stades indicate – were of course Egyptian. Other figures for the earth’s circumference in antiquity Although it is only with Eratosthenes and Posidonius that their methods to measure the circumference of the earth have been preserved (albeit in secondary sources), some further much esteemed ancient scholars report a few other figures that had apparently gained a certain degree of recognition among the members of contemporary scientific communities. Thus Archimedes relates the figure of 300,000 stades (Sand Reckoner 1.8), Aristotle speaks of 400,000 stades (On the Heavens 2.14), whereas Claudius Ptolemy – by saying that every degree of the great circle equals 500 stades – settles down for 180,000 stades in his highly influential Geography (1.7) that was to influence opinions on the size of the earth for many generations to come. This latest figure incidentally coincides with the one Strabo credits with Posidonius, yet there is no hint in Claudius Ptolemy’s Geography that he chose this value as the result of the work of his Stoic predecessor. Altogether therefore five different figures for the earth’s circumference crop up in Hellenic sources that can be judged textually coherent, each reported by or associated with a reputable authority of their time: 180,000 stades, 240,000 stades, 252,000 stades (variant 250,000 stades), 300,000 stades and 400,000 stades.

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

a

A 180,000

B 240,000

Posidonius, Claudius Ptolemy

Posidonius

252,000

b

Eratosthenes

c

300,000

400,000

Archimedes

Aristotle

Table 1 Different figures (in stades) for the earth’s circumference current in antiquity. The ratio between columns A and B is 3 to 4. Lines a and c are, incidentally, in the ratio of 3 to 5.

The proliferation of numbers for the earth’s circumference may perhaps be partly explained if they are arranged in a grid with two columns and three lines (Table 1). Out of the five figures, four can be put in such pairs that the members are in the ratio of 3 : 4 to each other (columns A and B). This seems to indicate that the choice of the different figures arose from a simple numbers game the origins of which in all probability go back to ancient Egypt, where the actual latitudinal extent of the land was in the same relation of 3 to 4 with the traditionally acclaimed figure for the country’s length. Ultimately, of course, this numbers game – as intimated by Putarch – found its roots in a play with the sides of the simplest Pythagorean triangle of 3-4-5. It must be pointed out here, however, that out of the five numbers in Table 1 four can also be paired vertically, in which case the ratio between the members is 3 : 5 (lines a and c). That this relation is very likely fortuitous, and that the numbers game intended is in fact based on the ratio of 3 : 4, is suggested by Eratosthenes’ figure, which both in the horizontal and vertical classifications remains the odd one out, initially perhaps coming from a different numbers game (see above). The possible matches for 252,000 stades would be either 189,000 or 336,000 stades in the 3 : 4 scheme, and 151,200 or 420,000 stades in the 3 : 5 scheme. Although there is no direct evidence for any of these numbers in classical sources, it is quite probably not without significance that if the individual figures in Strabo’s passage on the length of the Nile (Geography 17.1.2) are added up, it turns out that Eratosthenes determined the distance from Meroe to the Mediterranean as 18,900 stades, exactly 1/10 of the figure (189,000) that should occupy slot A-b in Table 1. There are some further hints in classical literature suggesting that the 3 : 4 ratio was somehow part of the reckonings about the size of the earth. In an Armenian text that preserved the lost geography of Pappus, another Alexandrian scholar working in the 3rd century CE, a rather obscure passage says that for different purposes of geometric and ‘aerometric’ measurements there were two different kinds of the stade (Ananias of Širak, Geography 1.6), whose lengths – it can be reconstructed from the slightly corrupt numbers – were in the ratio of 3 to 4. Indeed, it has been conjectured (although, to my knowledge, not on the basis of this particular Armenian treatise, but rather following on from the existence of two figures for the Rhodes-Alexandria distance) that the two values for the earth’s circumference associated with Posidonius, those of 180,000 and 240,000 stades, do not denote different measurements of the meridian, but the same one expressed in stades with differing lengths (see Drabkin 1943, 510). Quite obviously, no definitive judgement on this contention can be formed here, or perhaps not even after a laborious study of all the relevant metrological evidence that has come down to us, but the gist of the matter seems to be quite clear: the 3 : 4 ratio did feature in one way or another in classical sources on geography. Speaking of the possible points of linkage between Egypt and Hellenic geography, finally it must be noted that Claudius Ptolemy may have arrived at 180,000 stades for the length of the meridian, the same figure as is credited with Posidonius, by an alternative interpretation of the Egyptian evidence at his disposal – or at least this different look at the evidence may have

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

reinforced him that he should adopt no other figure. By the time of Ptolemy the division of the great circle into 360 degrees had replaced the earlier ad hoc astronomical gradation techniques, such as, for example, the one used by Posidonius (Neugebauer 1975, 590-594). Ptolemy, or a previous authority he was relying on, may have had a better understanding of the Egyptian sources than Eratosthenes, and may have realized that the 5000 (in place of 5300) stades did not originally refer to the latitudinal extent of the country, but was a putative number for 1/36 of the meridian (the 106 iteru recorded in the Egyptian inscriptions). If it was so, quite clearly 5000 stades were equal to 360 ÷ 36 = 10 degrees, and consequently one degree equalled 500 stades. In the absence of more evidence this is of course largely conjecture, but may explain why Ptolemy does not refer to Posidonius and why he only indirectly specifies the length of the meridian by saying that it is common knowledge that one degree of latitude corresponds to 500 stades. Conclusion The ancient Egyptians seem to have been, perhaps decidedly, very laconic about the art of geodesy they possessed. Practically the only piece of information they have made public in this regard, and the only one that has made it to the present through the ravages of time, is a number referring to a distance that strongly presupposes the knowledge of the earth’s circumference. This singular textual tradition, when compared with the actual extent of the land between its canonical boundaries, suggests that the discipline of mathematical geography in Egypt was a mixture of genuine scientific observations and a numbers game based on the triangle of 3-4-5. The ultimate reasons for the introduction of this numbers game into the Egyptian view of the world, as of now, remain elusive (they may have simply come from a cultic practice or may in fact have originated from some metrological considerations), but the analysis of the figures attributed to the length of the meridian by Hellenic scientists has shown that the same numbers game was taken over and perpetuated by the Greeks. What makes sure that simultaneously with this seemingly irrational play with numbers there was a rigorous scientific side to Egyptian geodetic thought, which at some point led to a fairly correct determination of the earth’s circumference, is firstly the trust that the earliest Greek geographers placed in their Egyptian sources when they collected the factual information needed for their work. This has been demonstrated during the careful examination of the accounts on Eratosthenes’ and Posidonius’ experiments. It is of course, given the nature of the evidence, also a distinct possibility that the methodological or instrumental foundations of these undertakings were also rooted in Egyptian science. Secondly, the firm scientific background of Egyptian geodesy is also revealed by the surprising accuracy of its findings, irrespective of whether they were preserved in hieroglyphic documents (as, for example, the figure of 106 iteru for Ce/36), or in the writings of Hellenic philosophers (say, the 1/48 arc of the meridian). The sole objective of this paper was to show the definite areas of connection between the little information currently available on Egyptian mathematical geography and its Hellenic counterpart, so any speculation on how this precision had been achieved would have led it astray. I leave this matter, at least for the moment, to others to ponder over.

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

References Ananias of Širak, 7th c. CE. – The Geography of Ananias of Širak (AŠXARHACcOYCc). The Long and Short Recensions. Introduction, translation and commentary by R. H. Hewsen. Wiesbaden: Dr. Ludwig Reichert, 1992. Archimedes, 3rd c. BCE. – The Works of Archimedes, ed. by T. L. Heath. New York: Dover, 1912. Aristotle, 4th c. BCE. – On the Heavens, transl. by W. K. C. Guthrie. Cambridge & London: Harvard University Press & Heinemann, 1939. Ball, J. B., 1942. Egypt in the Classical Geographers. Cairo: Government Press. Borchardt, L., 1921. Ein weiterer Versuch zur Längenbestimmung der ägyptischen Meilen (ì-trw), in Festschrift zu C. F. Lehmann-Haupts sechzigstem Geburtstage, eds., K. Regling & H. Reich. Wien & Leipzig: Wilhelm Braumüller, 119-123. Bowen, A. C., 2003. Cleomedes and the measurement of the earth: A question of procedures. Centaurus 45, 59-68. Butzer, K. W., 1977. Geographie, in Lexikon der Ägyptologie vol 2. eds., W. Helck, E. Otto & W. Westendorf. Wiesbaden: Harrassowitz, cc. 525-530. Clagett, M., 1999. Ancient Egyptian Science: A Source Book vol. 3. Ancient Egyptian Mathematics. Philadelphia: American Philosophical Society.

Claudius Ptolemy, 2nd c. CE. – The Geography, transl. and ed. by E. L. Stevenson. New York: Dover, 1991. Clement of Alexandria, 2nd c. CE. – The Stromata or Miscellanies, no translator. N. p.: Kessinger, n. d. Cleomedes, 2nd c. CE. – Cleomedes’ Lectures on Astronomy. A Translation of The Heavens. Introduction and commentary by A. C. Bowen & R. B. Todd. Berkeley, Los Angeles & London: University of California Press, 2004. Drabkin, I. E., 1943. Posidonius and the circumference of the earth. Isis 34, 509-512. Dutka, J., 1993. Eratosthenes’ measurement of the earth reconsidered. Archive for History of Exact Sciences 46, 55-66. Engels, D., 1985. The length of Eratosthenes’ stade. American Journal of Philology 106, 298-311. Ferrari d’Ochieppo, K., Krauss, R. & Schmidt-Kaler, T., 1996. Die Gefilde der altägyptischen Unterwelt: Spiegelbild der Sonnenbahnen im Jahreslauf. Zeitschrift für Ägyptische Sprache und Altertumskunde 123, 104-109. Fischer, I., 1975. Another look at Eratosthenes’ and Posidonius’ determinations of the earth’s circumference. Quarterly Journal of the Royal Astronomical Society 16, 152-167. Goldstein, B. R., 1984. Eratosthenes on the “measurement” of the earth. Historia Mathematica 11, 411-416.

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

Gratwick, A. S., 1995. Alexandria, Syene, Meroe: Symmetry in Eratosthenes’ Measurement of the World, in The Passionate Intellect. Essays on the Transformation of Classical Traditions, ed., L. Ayres. New Brunswick & London: Transaction Publishers. Griffith, F. Ll., & Petrie, W. M. F., 1889. Two Hieroglyphic Papyri from Tanis. London: Trübner. Gulbekian, E., 1987. The origin and value of the stadion unit used by Eratosthenes in the third century B.C. Archive for History of Exact Sciences 37, 359-363. Helck, W., 1980. Maße und Gewichte (pharaonische Zeit), in Lexikon der Ägyptologie vol. 3. eds., W. Helck, E. Otto & W. Westendorf. Wiesbaden: Harrassowitz, cc. 1199-1209. Kunitzsch, P. & Smart, T., 1985. Short Guide to Modern Star Names and Their Derivations. Wiesbaden: Harrassowitz. Leitz, C., 1991. Studien zur ägyptischen Astronomie. Wiesbaden: Harrassowitz. Lepsius, R., 1865. Die alt-aegyptische Elle und ihre Eintheilung. Reprinted as The Ancient Egyptian Cubit and Its Subdivision, English translation by J. Degreef, ed., Michael St. John. London: Museum Bookshop, 2001. Locher, J., 1999. Topographie und Geschichte der Region am ersten Nilkatarakt in griechischrömischen Zeit. Stuttgart & Leipzig: B. G. Teubner. Lyons, H., 1926. Two notes on land-measurement in Egypt. Journal of Egyptian Archaeology 12, 242244. Martianus Capella, 5th c. CE. – The Marriage of Philology and Mercury, transl. by W. H. Stahl, R. Johnson & E. L. Burge. New York: Columbia University Press, 1977. Neugebauer, O., 1975. A History of Ancient Mathematical Astronomy. Berlin, Heidelberg & New York: Springer-Verlag. Newton, R. R., 1980. The sources of Eratosthenes’ measurement of the earth. Quarterly Journal of the Royal Astronomical Society 21, 379-387. Petrie, W. M. F., 1883. The Pyramids and Temples of Gizeh. London: Field & Tuer. Pliny, 1st c. CE. – Natural History, transl. by H. Rackham, W. H. S. Jones & D. E. Eichholz. Cambridge & London: Harvard University Press & Heinemann, 1938-1962. Plutarch, 2nd c. CE. – Isis and Osiris, transl. by F. C. Babbitt. Cambridge & London: Harvard University Press & Heinemann, 1969. Priskin, G., 2001. The philosopher’s stones: the Great Pyramid. Discussions in Egyptology 51, 109120. Priskin, G. 2004a. Herodotus on the extent of Egypt. Göttinger Miszellen 201, 63-67. Priskin, G. 2004b. Reconstructing the length and subdivision of the iteru from late Egyptian and Graeco-Roman texts. Discussions in Egyptology 60, 57-71. Quirke, S., 2003. Measuring the Underworld, in Mysterious Lands, eds., D. O’Connor & S. Quirke. London: UCL Press, 161-181.

Published in Göttinger Miszellen 208 (2006), pp. 75-88.

Rawlins, D., 1982a. The Eratosthenes-Strabo Nile map. Is it the earliest surviving instance of spherical cartography? Did it supply the 5000 stades arc for Eratosthenes’ experiment? Archive for History of Exact Sciences 26, 211-219. Rawlins, D., 1982b. Eratosthenes’ geodesy unraveled: Was there a high-accuracy Hellenistic astronomy? Isis 73, 259-265. Schlott-Schwab, A., 1981. Die Ausmasse Ägyptens nach altägyptischen Texten. Wiesbaden: Harrassowitz. Seidelmann, P. K., ed., 1992. Explanatory Supplement to the Astronomical Almanac. Mill Valley: University Science Books. StarCalc 5.72, 2002. Planetarium software by A. E. Zavalishin. Strabo, 1st c. CE. – The Geography of Strabo, transl. by H. Jones. Cambridge & London: Harvard University Press & Heinemann, 1917-1932. Taisbak, C. M., 1974. Posidonius vindicated at all costs? Modern scholarship versus the Stoic earth measurer. Centaurus 18, 253-269. Thurston, H., 2002. Greek mathematical astronomy reconsidered. Isis 93, 58-69. Wirsching, A., 2002. Mit Schattenmessungen die Erdkrümmung erkennen und den Erdumfang berechnen. Göttinger Miszellen 191, 89-99. Zeidler, J., 1997. Die Länge der Unterwelt nach ägyptischer Vorstellung. Göttinger Miszellen 156, 101-112.

e-mail: [email protected]