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Ancient Measurements of the Circumference of the Earth


1. The reconstruction of the system of ancient lineal units allows to confirm the conclusion of French scholars of the eighteenth and early nineteenth centuries that the ancients had achieved a most accurate calculation of the circumference of the earth and that this calculation had been performed earlier than Greek times. The French scholars concluded that all the estimations reported by authors of the Hellenistic and Greek period repeat, according to different stadia, the statement of Aristotle that the mathematikoi had calculated the circumference of the earth as 400,000 stadia. Since the different stadia relate to each other by simple fractions, these scholars concluded that the ancient linear units were all modules of a single basic unit. Böckh, who tried to prove that all units of length, volume, and weight, could be calculated from a single unit of length, based himself on the studies of Gosselin and Letronne who in 1822 had tried to prove, each on his own way, what had become for almost a century a tenet among the members of the French Académie des Sciences and Académie des Inscriptions, namely, that all ancient reports are based on a single pre-Greek calculation of the circumference of the earth.

Through medieval times there had been preserved the memory of the Roman calculation of the meridian of degree as 75 Roman miles. Newton did not trust the accuracy of this figure and before publishing the Principia waited for the calculation of Picard, which actually proved less accurate because the toise used by Picard was slightly shorter than the toise used by other French scholars and by Picard himself in a second period.

The French scholars had not succeeded in calculating the length of the ancient Roman foot with a precision greater than 3 tenths of a millimeter. Since I have determined that the Roman foot (corresponding to a libra of 324 grams) has a length of 295.9454 mm., the ancient calculation proves to be even more accurate than had been believed. Seventy-five Roman miles indicate a degree of 110,979 m. Contemporary data based on a geoid of regular shape and calculated at sea level indicate a degree of 110,956 m. at latitude 36°. The ancient figure is perfect for latitude 37° where the degree is calculated as 110,975 m. The discrepancy between the ancient figure and the theoretical figure for latitude 36° is about 1/5000, less than a second of degree or 1/3600.

The common practice of Roman times was to divide the Roman mile not into 10 stadia of 600 Roman feet, but into 8 stadia of 600 artabic feet, equal to 625 Roman feet. Distances at sea were not calculated by Roman miles, but by artabic stadia. The artabic foot is particularly fitted to the calculation of geographic distances, since 100 such feet are equal to a second of degree. Hence a plethron of 100 artabic stadia fits exactly into the sexagesimal division of the degree. The Persian parasang, equal to an hour of march, is equal to 18,000 artabic feet, and is divided into the triple unit called milia in Roman times; since there are 20 parasangs to a degree, there are 60 milia to a degree. Hence, a man marching for eight hours each day in a year would make the circuit of the earth.

Aristotle (On the Heavens II, 298 B) reports that those mathematikoi who try “to calculate the circumference by proportional calculations” (analogizesthai peirontai), calculate the circumference as 400,000 stadia. By mathematikoi Aristotle refers either to Pythagoreans or to the followers of Chaldean mathematical science; it is not necessary to try to settle here the disputes about the meaning of the term mathematikoi in Aristotle.

The wording of Aristotle may refer to the fact that the ancients knew that the value of the degree varied from degree to degree, and hence could infer the circumference “by analogy” or “by proportional calculations” from the length of the degree at latitude 36°.

By stadion the Greeks meant either the distance covered in a minute of march or the distance covered in a double minute of march; generally they called stadion the double minute of march corresponding to the division of the day into 12 double hours, but the stadion of 300 feet or one minute of march was also used. The stadion mentioned by Aristotle is equal to 300 barley feet; if the feet are of the trimmed variety the circumference of the earth would be 39,952,644 m. The corresponding degree would be the one of 75 Roman miles of 20 parasangs.

The French scholars gathered evidence of the use of a stadion of about 100 m. by Greek writers, but today there is available a more unequivocal piece of evidence.

The Greek geographers of the Roman period report the figure of 240,000 and 180,000 stadia for the circumference. Both figures are based on artabic units. The second figure is based on a stadion of 720 artabic feet; the first on a stadion of 360 artabic cubits. The first figure could be also calculated by a stadion of 600 trimmed lesser feet.

Except for the figure of Aristotle, which is based on trimmed barley feet, all the other figures are based either on the trimmed basic foot or on the artabic foot computed as 25/24 of it; hence, all these figures express in different ways the value of 75 Roman miles or 20 parasangs to the degree.

Since the Greeks before the time of Aristotle did not possess the state organization necessary to proceed to the measurement of the degree, it follows that the degree had been calculated before Greek times. If the degree had been calculated before Greek times, it follows that it was not the Greeks that discovered that the earth is a sphere.

The French scholars thought that the calculation performed by Eratosthenes represents an independent figure, but this does not prove to be correct. Kleomedes reports that Eratosthenes calculated the latitude of Alexandria in Egypt and that of Syene at the First Cataract, and found that this distance, which is the entire length of Egypt, would be 1/50 of the circumference of the earth. Eratosthenes would have calculated the distance between Alexandria and Syene as 5000 stadia, so that the circumference is 250,000 stadia.

At the beginning of the nineteenth century it was determined that the Egyptian royal cubit is 525 mm. and hence it was concluded that Eratosthenes calculated by stadia of 300 Egyptian royal cubits. Newton too had tried quite successfully to ascertain the length of the Egyptian royal cubit from the dimensions of the Great Pyramid, in order to interpret Eratosthenes’ datum. But I have determined that the septenary cubit used in Egypt during the Hellenistic period is the Babylonian-Egyptian great cubit of 532.702 mm.

Ancient metrological tables state that the Philetairic or Ptolemaic royal cubit (which is the Babylonian-Egyptian royal cubit according to Boeckh’s terminology) is 9/5 of the Roman foot, so that the figure of Eratosthenes comes to be the usual figure of 75 Roman miles to the degree.

But several authors of the Roman period mention a degree of 700 stadia. This degree value should not be confused with that of Eratosthenes and is based on a stadion of 300 royal cubits of the Pharaonic period; these two points have been made by Letronne. I have reported that the correct Egyptian royal cubit was 525 mm., but it was at times computed as 524. mm. and at times as mm. Assuming a cubit of 525 mm. the degree would be 110,250, and assuming a cubit of 526.3 mm. It would be 110, It is easy to see why the figure of 700 stadia to the degree was chosen: it well fits the pattern of septenary reckoning in the Egyptian royal cubit. But the length of the degree at latitude 30°, the latitude of the pyramids and of the beginning of the Delta, is 110,849 mm. At the southern limit of the country, which is at latitude 24°, the degree 110,750 m. Hence, it seems that in Egypt, for the sake of a convenient reckoning there was adopted a value that it slightly in defect. Perhaps the figure was chosen because it was convenient and the methods of observation used did not allow to determine any error in it. This figure may have had great importance in convincing the ancients that the degree was shorter in Egypt than at latitude 36°.

Letronne, following remarks made by other scholars before him came to the conclusion that Eratosthenes did not rely on any actual survey on the ground in order to calculate the length of the degree, but merely calculated the latitude of Alexandria and of Syene and having found it to be 1/50 of the circumference of the earth, applied to it an established figure of the value of such an arc of circumference on the basis of a known value of the length of a degree. Since Alexandria and Syene are not on the same meridian, certainly Eratosthenes cannot have based himself on a measure on the ground. According to Kleomedes, Eratosthenes found that the difference in latitude between Alexandria and Syene is 1/50.

This very fact proves that Eratosthenes did not proceed to any actual measurement. Some of the French scholars had doubted that Eratosthenes had proceeded to an actual measurement of the degree. Letronne concluded that Eratosthenes calculated the latitude of Alexandria and the latitude of Syene and then calculated the length of the degree.

In my opinion Eratosthenes did try to apply to Egypt the figure about the length of the degree current in the Greek world, that of 20 parasangs or 75 Roman miles to the degree; but he proceeded in a manner different from that suggested by Letronne. He may have gathered the Egyptian information that the length of Egypt from Syene 1° 13’ 5” ; but the Egyptians probably calculated the end of country at the Canopic mouth which is at 31° 19’ and which is slightly to the north of Alexandria. Possibly the Egyptians used the round figure of 5000 stadia, but knew that this figure corresponded to the distance from Syene to a point near the latitude of Alexandria.

There is a score of articles trying to prove in detail that Eratosthenes proceeded to an independent evaluation of the circumference of the earth; but even though some of them show great ingenuity, none succeeds. The result ascribed to Eratosthenes, that is, a circumference of 250,000 stadia, repeats the traditional datum. In my opinion the purpose of Eratosthenes was simply to prove...

Archimedes (Sand Reckoner) speaks of the circumference of the earth as a datum that does not need any particular discussion and estimates it as 300,000 stadia. It is a case of a stadion of 300 trimmed basic cubits or Roman cubits of 443.9181 mm. The circumference is 135,000,000 Roman feet or 75 miles to the degree. Eratosthenes calculated that the distance from Alexandria to Syene was 5000 stadia, but calculated by stadia based on the Babylonian-Egyptian great cubit, which was the Egyptian royal cubit of his time. Hence, he assumed that between Alexandria and Syene there were 7° 12’. The latitude of Syene was of great importance to Eratosthenes because there was a long tradition that in Syene there was a well into which the sun...

Actually Syene had been under the Tropic around the year 2600 B.C., the time of the construction of the Great Pyramid of Gizah. This fact has been stressed by Jomard and Letronne. It may assumed that given the lack of precision in the methods of observation for a long period after this date, it was continued to assume that Syene was at the Tropic. But a new calculation of the situation of the Tropic was performed around the year 800 B.C. and this datum is reported by Eratosthenes.

I believe that none of the commentators has called attention to an important qualification contained in Kleomedes’ report. After having stated that a gnomon does not cast any shadow at high noon on the solstice, Kleomedes states “This phenomenon takes place for an extension of 300 stadia”. This indicates that the method of observation was accurate within the range of about 25’ or roughly half a degree. If this is the accuracy of observation the operation cannot have been performed by Eratosthenes, whose data are most imprecise. According to Kleomedes, Eratosthenes performed his observation with a scaphe, which is a hemisphere with an upright pointer at the counter, that is, an instrument that would hardly allow an observation more precise than a degree. But the figures suggest that when it was observed that the Tropic is at 23° 51’ 20”, a figure that Eratosthenes reports and that was true around 800 B.C., it was also noted that Syene was considered to be at the Tropic. Since actually Syene is about 14’ North of this point, somebody assumed that the phenomenon of the sun falling vertically proved correct over a band of 26’ by doubling the figure, assuming a possibility of error in either direction. All this indicates that the method of observation used in determining the latitude of Syene could not be in error more than a couple of minutes.

For Eratosthenes Syene is at 23° 51’ 20”, so that Alexandria should be at 31° 03’ 20”. Letronne by analyzing the figures given by Ptolemy about the latitudes of Egypt arrives at the conclusion that there is a constant error in defect of 14’ to 15’, which corresponds to the semidiameter of the sun. In observing the elevation of the sun it was believed that the instruments indicated the center of the sun, whereas they indicated the Northern margin. It can be assumed that the same type of error occurred in Eratosthenes’ datum. Hence, Alexandria should have been at 31° 18’. The latitude of Alexandria at the Pharos is 31° 13’ 5” ; according to Ptolemy it is 30° 58’. It is possible to assume that Eratosthenes was less accurate than Ptolemy; but I would make another suggestion: The calculation of Eratosthenes is based on some traditional datum about the length of Egypt; the Egyptians may have calculated the length of the country from Syene to the Canopic mouth of the Nile. Ptolemy reports for Canopos a latitude 31° 05’, which becomes the correct one by adding 15’ or 14’.

In substance, Eratosthenes did not intend to calculate the length of the degree or the circumference of the earth, which were well known current data. He was only interested in explaining how this calculation could be performed and in proving that it was correct. To this effect he quoted Egyptian data to the effect that the extension of the country is 1/50 of the circumference of the earth. He made use of the two Egyptian traditions: one that Syene is at the Tropic and the other the Tropic is at 23° 51’ 20”. As a Greek of the Hellenistic age he assumed Alexandria was the Northern end of the country, whereas probably Canopos had been it for the Egyptians. All that Eratosthenes had to do was to convert the usual figure of 75 Roman inches to the degree or a similar one, into one of stadium of 300 royal cubits equal to 9/5 of Roman foot, so that the total length of Egypt would be 5000 stadia.

Several ancient writers, among them Strabo, mention a calculation of the degree as equal to 700 stadia. Several modern scholars have confused this calculation with that of Eratosthenes, but ...

...and that there a gnomon at high noon at the solstice would not cast any shadow. This had not been true since about the year 2600 B.C., the age of the Pyramids of Gizah, but that the Tropic was at Syene was a well established tradition since Pliny states that at Syene there was a well into which the sun fell perpendicularly.

The French scholars believed that the calculation of Eratosthenes by which the circumference of the earth is 252,000 stadia represents a value differennt from those mentioned up to now. They correctly calculated the stadion of Eratosthenes as 300 Egyptian cubits, but they did not know that in the Hellenistic age the royal cubit of Egypt is the one called Babylonian-Egyptian great cubit. I have calculated this unit as 532.702 mm; ancient metrological tables compute it, quite exactly, as 9/5 of the Roman foot, so that the datum of Eratosthenes can be easily converted into that of 75 Roman miles to the degree. The degree is equal to 6 according to Eratosthenes.

A number of texts of the Roman period mention a value of 700 stadia to the degree and some of them ascribe it to Eratosthenes; but it is quite clear that the figure of 700 stadia is merely a rounding of Eratosthenes’ figure.

When as a result of the Napoleonic Expedition to Egypt modern geographical data became available, it became clear that Eratosthenes had not performed any direct test of the length of the degree. Kleomedes, a writer of the second century A.D., reports that Eratosthenes calculated that Alexandria and Syene, at the First Cataract, the Southern border of Egypt, were on the same meridian and were at a distance of 1/50 of the circumference of the earth (that is, 7° 12”). Eratosthenes would have calculated that the distance between Alexandria and Syene is 5000 stadia. But Alexandria and Syene are not on the same meridian (there is a difference of about 3° in longitude) and their distance is not 5000 stadia.

Letronne thought that Eratosthenes calculated the difference between the latitude of Syene and that of Alexandria and then multiplied this difference for a known length of the degree in stadia and arrived at 5000 stadia. But the difference of latitude between Alexandria and Syene is less than 7° 12’.

I would suggest a possible different explanation of Eratosthenes’s procedure. He may have based himself on a traditional report that the total length of Egypt was 1/50 of the circumference of the earth. The length of Egypt was calculated from to Syene to some point in the Delta.

 
 

Ptolemy gives the following coordinates of longitude and latitude for Syene: 62° 23° 50’

For the Island of Elephantine they are 61° 30’ 23°55’

Ptolemy also gives the following coordinates for four mouths of the Nile

 
 

Bolbitic mouth 61° 15’ 31° 5’

Sebenytic mouth 61° 30’ 31° 5’

Pinept 61° 45’ 31° 5’

Canopic mouth 60° 50’ 31° 5’

He gives the same latitude for Canopos which is at 60° 45’ 31° 5’ and amazingly also for the Pharos of Alexandria 60° 20’ 31° 5’ and for Alexandria itself 60° 45’ 31° 5’ But another point of this geography he places Alexandria at 31° and in the Almagest at 30° 58’.

From these data one can gather that 31° 5’ was traditionally the northem limit of Egypt. Since Alexandria was for the Greeks at the northern extremity of Egypt, it was placed at latitude 31° 5’, on the line 31° 5’ across the other margin of the Delta. There was some position that was on the meridian of Syene. Eratosthenes transposed to Alexandria what was the traditional terminal point of Egypt in the Delta.

According to Kleomedes, Eratosthenes assumed that Syene was at the Tropic...

...been considered the northern limit of Egypt.

There is an indication that Eratosthenes deliberately followed traditional Egyptian data, since Kleomedes after having reported that the gnomon does not cast any shadow at Syene at the solstice adds “this phenomenon takes place for a distance of 300 stadia.” The probable meaning of this sentence is that somebody had found that the absence of shadow would be found almost 26’ south of Syene where it was traditionally supposed to occur. The Tropic changes ...stadia.” The probable meaning of this sentence is that somebody had found that the absence of a shadow could be found almost 26’ south of Syene where it was traditionally supposed to occur. The Tropic changes according to an angle of aobut 25’ in 3000 years.

If 14’ are added to Eratosthenes’ if 14’ are added to Eratosthenes’ figure for the Tropic or to Ptolemy’s figure for Syene, there results the correct latitude for Syene which is 24° 5’.

By adding 7° 12’ to the alleged latitude of Syene, there results 31° 03’ 20” or 31°, which well corresponds to the line 31° 5’, which seems to have...

...at high noon at the solstice. It can be proved that Eratosthenes calculates the Tropic on the assumption that the Tropic passes through Syene, since he states that the ecliptic is 11/126 of meridian, so that the Tropic would be at 23° 51’ 20”. All calculatioins of latitude obtained by a gnomon erred in defect by 14’ or 15’, which...

...1½26 of the meridian. Probably this figure wass obtained by deducting 7° 12’ from the latitude of Alexandria, which should be at 31° 3’ 20”. The actual latitude of Alexandria at the Pharos is 31° and in the city. According to Ptolemy the latitude is 30° 58’; Letronne notes that Ptolemy’s latitudes...

stadia.” the probable meaning of this sentence it thuel probably somebody had formed that the obsence of scholar could be found almost 26’ south of Syenen here it the was triditionally supposed to occur, teh Tropic changes according to an angle 251 about 25’ in 3000 years . Dince to some point in the Delta

 
 

Ptolemy gives the following coordin notes of longitud a and latude for Since; 62 0 32 0 50’

For the Inland of Elephantine they are 61 0 30 23 0 55 ``

Kleomedes

Our contemporary scholars are committed to the dogma that Hellenistic science was superior to pre-Aristotelian science. They ignore the fact that Plato was the last great representive of the mythical-mathematical interpretation of the universe, a conception that results in a method similar to the intuitional-deductive method of modern physical science. They gloss over the fact that in the generation of Plato there began to be advocated a common-sense antimathematical interpretation of the universe. This is the reason why Aristotle, who was the first to try to build a science based on this method, championed by his teacher Isocrates, speaks with a certain scorn of the opinion of the mathematikoi who declared that the Earth has a circumference of 400,000 stadia.

After him only Archimedes, who was an anomaly in his own time in that he remained faithful to the tradition of mathematical science, is the only one who indicates that the circumference of the Earth was a positively known datum: 300,000 stadia.

According to the mathematical tradition, the basic meridian of the Earth was 31°14’, the meridian that passed through the Apex of the Nile Delta and the mouth of the Boristhenes or Don and was considered the axis indicated by the Nile and the Boristhenes. Beginning with Diakaiarkhos of Messana, a pupil of Aristotle, this meridian was reinterpreted in common-sense terms. It was conceived as passing through Syene ( ), Alexandria ( ), Rhodes ( ), the Dardanelles ( ), the Bosphorus ( ), and the mouth of the Don ( ). In this way a mathematical line was identified with the course that could be followed by a navigator going from the southern limit of Egypt proper to the mouth of the Don. All Greek geographers adopted this practical meridian as their basic meridian.

Our contemporary scholars of ancient history and ancient science assume as proven the superiority of the method introduced by Aristotle. For this method logos is logic, the proper interrelation of verbal statements; verbal statements are considered grounded if they correspond to simple immediate sense experiences, if they conform to common sense. Leaving out of the question what Aristotle may have thought or advocated, it is a fact that as a result Hellenistic science inclined more and more to stress the importance of verbal statements and to accept them as valid if uttered by a source invested with authority.

Eratosthenes was not the first to measure the circumference of the Earth, but the first to argue, contrary to the opinion of Aristotle, that the calculations about the circumference of the Earth could be accepted as proven in terms of the new scientific style.

A series of ancient authors credits Eratosthenes as having introduced the calculation of the degree as equal to 700 stadia, but there is not a single writer who indicates that he based himself on an empirical survey of the ground. Contemporary scholars exalt Eratosthenes as a great scientist and as a pioneer in mathematical geography, but none of the ancient writers who were acquainted with his works indicate this. If Eratosthenes had been such an innovator, Ptolemy who discusses at length the problem of the dimensions of the Earth in the Prolegomena to his Geography would have said at least some words to this effect. Theon of Smyrna and Proklos, who lived in Alexandria do not make any reference to the alleged discovery of Eratosthenes in their extensive commentaries on ancient mathematical science. Strabo, who had before his eyes the writings of Eratosthenes and discusses them at length, does not ascribe to Eratosthenes any specific achievement in the field of empirical geodesy or of theoretical geography. Strabo mentions repeatedly the figure of 700 stadia to the degree, but justifies it only in these words: “We suppose as Hipparchos, that the size of the Earth is 252,000 stadia, a figure given also by Eratosthenes.” He would not have spoken in these terms if Eratosthenes had provided a complete mathematical demonstration.

In a memoir of 1817, Letronne proved conclusively that Eratosthenes did not proceed to any geodetic operation; in this he developed an argument advanced earlier by the astronomer Delambre. But contemporary scholars prefer to eulogize the greateness of Eratosthenes without answering to the evidence submitted by Letronne. Great value is placed upon some dramatic words of Pliny.

The commonsense approach of Eratosthenes is clearly indicated by the method he followed in subdividing the distance between the Pole and the Equator, once he had accepted that the circumference of the Earth is 252,000 stadia. He counted by the practical basic meridian that I have mentioned earlier. For reasons that I shall explain below he placed the Tropic at latitude 23° 51’ 30“N; hence, he placed the Polar Circle at the same distance from the Pole, that is, 16,700 stadia. He identified the latitude of the mythical land of Thule with the Polar Circle (66° 08’N). Then he counted 10,500 stadia or 15° 00’ from Thule to the latitude of the mouth of the Boristhenes or Don and the southern limit of the Morass Maiotis or Sea of Azov; thereby he was referring to a pre-Greek fundamental meridian which passes through the Strait of Kerch, entrance to the Sea of Azov, and is at 45° 12’N. He treats the Bosphoros and the Dardanelles as a single latitude and places the 5,000 stadia or 7°8’30” to the south of the mouth of the Don. He was referring to a fundamental meridian of pre-Greek geography which was 41°12’N (5°00’ the north of the basic meridian of Rhodes) and passes through the northern limit of the Bosphorus. The difference of latitude between Lysimacheia and Alexandria (Base of the Nile Delta, 31°12’N) is 10° 00’ or 7,000 stadia, but Eratosthenes made it 8,100 stadia. He should have placed Rhodes at the middle point between the Bosphorus and Alexandria, whereas he placed it 3750 stadia north of Alexandria. He claims that the latitude of Rhodes was calculated by observing the shadow of the Sun with a gnomon; this indicates that the instrument of which he disposed was a crude one indeed: a real difference of 5°00’ was read as 5° 28’. In reading the instrument he was inclined to force Rhodes to the north, because he had not broken completely with the erroneous belief (based on misread textual evidence) that Athens was at the latitude of Rhodes; he placed Athens 400 stadia or 34’ north of Rhodes. He counted 5,000 stadia from Alexandria to Syene, identified with the Tropic, and counted 5,000 stadia more to Meroe. He placed the limit of inhabited Earth 3,000 stadia further south, leaving 8,700 stadia for the distance from this point to the Equator.

These very approximate reckonings presuppose as given that the degree is 700 stadia.

There is only one text that describes the method by which Eratosthenes tried to prove that the figure of 700 stadia is correct.

The argument reported by Kleomedes concludes that the distance between Syene and Alexandria which is 5,000 stadia is 1/50 of the circumference of the Earth, so that the circumference is 250,000 stadia. Since Eratosthenes adopted as correct the figure of 252,000 stadia, it follows that in this demonstration he was interested only in proving that the figure was about right and not against common sense.

The most important datum of the entire calculation, the actual distance between Syene and Alexandria, is accepted as given. There is no indication of any sort that Eratosthenes proceeded in any way to measure the distance on the ground. A geodetic survey over the distance of more than 7° would have been an impossibility even for the Egyptians. Most likely the Egyptians had proceeded, as Calif al-Mamun, by measuring on the ground the length of a degree or two, and had obtained the total figure about the length of their country by multiplying the value of a degree by the difference of latitude obtained astronomically. For the Egyptians the distance between latitude 24°00’N, the southern limit of Egypt proper, which they identified with the Tropic and to which they referred by the name of Syene, even though Syene was at 24°00’N, and the base of the Delta, was a matter of common knowledge which they recorded in their monuments.

The Egyptians had two positions for the base of the Delta, a conventional line, one at 31°12’N and one at 31°06’N.

In Pharaonic Egypt the stadion was computed by the Egyptian royal cubit of 525 mm., whereas in later times the stadion was computed by the Babylonian/Egyptian great cubit of 532.207 mm. By the first unit, 5,000 stadia are 7°06’N, whereas by the second unit they are 7°12’. For this reason Eratosthenes may have found himself confronted by the information that the length of Egypt was 5,000 stadia, but stadia that made the circumference of the Earth 252,000 and 250,000 stadia respectively. Apparently Eratosthenes was not concerned with this difference and used one figure in one calculation and another figure in another. He used the second figure in his demonstration that the calculation is acceptable, and used the second figure in his geographical studies, since it provided the convenient figure of 700 stadia to the degree. However, Hipparchus observed that the figure of 700 stadia, assuming a distance of 5,000 stadia between Alexandria and Syene, is slightly too short.

For Eratosthenes, since there was an anciently established tradition that the length of Egypt was 5,000 stadia, this could be accepted as fact. It was common sense, it was something on which there was general agreement (sensus communis, koine epinoia), it was what the Epikureans and Stoics called a prolepsis; it was not against the daily experience of those who lived in Egypt. For him this was as true as the belief that Alexandria and Syene were on the same mediridian, a belief based on the fact that in Hellenistic Egypt, Alexandria was the northern end of Egypt. This was just as true as the belief that Rhodes was to the north of Alexandria, because a ship leaving Alexandria in a northerly direction would head for Rhodes. If Eratosthenes had proceeded to any deductive verification, he would have found that the proposition that the distance between Alexandria and Syene is 5,000 stadia is not correct.

Ptolemy reports that according to Eratosthenes and Hipparchos the angle of the Ecliptic was and indicates that he found this datum himself empirically. Actually the Tropic was at 23°45’ in the age of Eratosthenes and had moved to 23° 41’ by the age of Ptolemy. The datum was an old Egyptian one which had been correct around 1080 B.C. (middle of the zodiacal period of Aries). It is possible that Eratosthenes tried to test the angle of the Ecliptic by the instruments at his disposal and found that the Egyptian figure was not contradicted.

Eratosthenes accepted the Egyptian tradition that at high noon at the summer solstice the Sun does not cast any shadow at Syene. This was true when the Tropic was at 23°51’N because the position of the shadow of the Sun is determined by the upper margin of the solar disk. Eratosthenes no longer possessed the information that calculations of latitude based on the shadow of the Sun must be corrected by the semidiameter of the solar disk. Ptolemy too did not possess this information, since in the Almagest ( ) he places Alexandria at latitude 30°58’N, that is, 14’ or half a solav diameter south of the true position; in his Geography he rounds the figure, placing Alexandria at 31°00’N. However, Eratosthenes had some inkling of the problem, since Kleomedes reports that the area without any shadow extends for 300 stadia (half degree by a geographic stadion of 600 to the degree). Apparently Eratosthenes misunderstood the information that the apparent disk of the Sun extends 300 stadia to the south of the point without shadow; since the instrument he used to measure the shadow of the Sun could not be more precise than half a degree, he concluded that the phenomenon of the lack of shadow is observable for about this angle.

Eratosthenes observed as proven that the Sun does not cast any shadow at Syene at the summer solstice; this was true by observations that were not too accurate. From this he concluded that Syene is at the Tropic, at latitude 23°51’N

Having accepted this figure as demonstrated, he tried to verify the position of Alexandria. He mentions the use of the skophe which is a hemisphere with a vertical point at the center. Keeping the margin of the sphere horizontal and the pointer perpendicular to the Earth, the angle of the shadow cast by the pointer can be read on an arc inside the hemisphere. Letronne argued that Kleomedes must have been misquoting Eratosthenes because the latter would hardly have used the skophe, a crude instrument that cannot provide a reading much more accurate than half a degree. But Eratosthenes was interested in providing a demonstration that conforms to common sense experience: the skophe had the advantage of having a shape that mirrors the shape of the vault of heaven. With the use of the skophe it was shown that, on the day in which the Sun supposedly does not cast any shadow at Syene, at Alexandria the shadow in the skophe is 1/50 of circumference. The effort of Eratosthenes, as quoted by Kleomedes, is not directed at proving the accuracy of the angle of 1/50 of circumference, but at proving that the angle of the shadow with the skophe corresponds to the angle in the sky between the perependicular passing through Syene and the perpendicular passing through Alexandria.

Eratosthenes concluded that it was proved that Alexandria is 1/50 of circumference north of the Tropic. This was not true, but there was a compensation of errors, since the Tropic was identified with the latitude of Syene and the latitude of Alexandria was placed half a solar diameter south of its true position.

Possibly the two figures about the latitutude of Alexandria that occur in Ptolemy was derived from Eratosthenes. In the calculation reported by Kleomedes, Eratosthenes placed Alexandria at latitude 31°00’N, 1/50 of circumference, or 7°8’30” north of the supposed position of the Tropic and Syene. By a more accurate observation with instruments superior to the skophe, he may have placed Alexandria at latitude 30°58’N or 7°06’ north of Syene and of the Tropic; he may have found this result confirmed by Egyptian data that gave correctly the difference of latitude between Syene (24°06’N) and the latitude of Alexandria (31°12’N). Hence, Eratosthenes adopted the figure of 252,000 stadia for the circumference of the Earth.

For writers who lived in the age that followed Aristotle, the problem was not that of providing an accurate calculation of the circumference of the Earth, but of proving that the circumference could be measured and that the calculations of the mathematikoi were not mere fantastications. Just before quoting the arguments used by Eratosthenes, Kleomedes presents an argument ascribed to...

15,000 stadia between Lysimacheia and Syene. The interval between gamma Draconis and Cancer cannot be 1/15 of circumference or 24°.

Since Kleomedes ascribes to Eratosthenes a hodgepodge of notions that are not certainly his, it follows that he can well have ascribed to him a measurement of the Earth that nobody else mentions.

I suspect Poseidonios to be the author of this argument. The author cannot be Kleomedes himself, because in other parts he proves to be acquainted with the position of gamma Draconis and with the distance between Alexandria and Lysimacheia. Probably Poseidonios was interested in presenting a purely hypothetical argument. He found that Eratosthenes computed an angle of 20,000 stadia between Cancer, at the Tropic, and the head of Draco. He found an estimate of about 20,000 stadia between Syene and Lysimacheia, based on rounding the figure of 17° at 1111.1 stadia to the degree. He interpreted 20,000 stadia as 24° at the rate of 833.3 stadia to the degree (24 x 833.3 = 20,000). The calculation by a stadion of 833.3 to the degree is implied in the assumption that the circumference of the Earth is 300,000 stadia....

Delimited the Arctic circle for those who lived in Greece. Since gamma Draconis had a declination of it did not set for those who lived at latitude . Eratosthenes may have said that there was an angle of 20,000 stadia or 28°34’ between Cancer at the Tropic and gamma Draconis, placed at about 52° 34’N. But Eratosthenes...

To prove that the Earth was not flat became a difficult problem for the Stoics, because immediate sense experience suggested that it was flat. Having granted that “the judgment of the eyes” indicates that the Earth is “flat and two-dimensional,” Kleomedes proceeds to observe that there is some sense experience that indicates that it is curved (I, 8, 40-41). To this effect he observes that “It is said that the Persians, who live towards sunrise, meet the rays of the Sun four hours earlier than the Iberians who live towards the sunset.” The marvellous calculation by which the holy area of Persepolis had been placed exactly 60°00’ from the two ends of the Oikoumene becomes here a commonly accepted opinion to the effect that the Sun rises 4 hours earlier for the Persians than for the Iberians. Kleomedes speaks of Persians and Iberians, rather than of Persia and Iberia, because he is concerned with sensory experience, something to which the people of these two countries can be witness.

A further proof that the Earth is not flat is that if it were so, “It would happen that days would have the same length for all men, which is completely contrary to what can be seen.” The Stoics do not reject outright the pre-Aristotelian concept of the logos, but empty it of all mathematical meaning and reduce it to crude sensory experience. If we consider the mathematical foundations of Newton’s theory of gravitation and compare it with its image in the mind of the common men who came to accept it as reasonable, we can have a scale of what the Stoics did with the logos.

Kleomedes proceeds to provide a more elaborate demonstration of the sphericity of the Earth, which runs as follows:

If the Earth were flat and twoâmensional in shape, the whole diameter of the kosmos would be 100,000 stadia.

In fact, for those who live at Lysimacheia, the head of Draco is at the zenith, whereas Cancer is above those who live in the neighborhood of Syene. The arc between Draco and Cancer is 1/15 of the meridian drawn through Lysimacheia and Syene, as is indicated by the shadows cast by gnomons; 1/15 of the whole circle is 1/5 of the diameter.

Now, if, supposing that the Earth is twoâmensional, we lower two perpendiculars from the extremities of the arc that goes from Draco to Cancer, they will touch the two extremities of the diameter that goes from Lysimacheia to Syene on the meridian drawn through these two places. Now, there will be 20,000 staida between the two perpendiculars, because there are 20,000 stadia between Lysimacheia and Syene. Since this interval is 1/5 of the entire diameter, the diameter of the entire meridian will be 100,000 stadia. If the diameter of the kosmos is 100,000, the great circle will be 300,000.

The Earth, which is only a point in relation to the kosmos, is 250,000 stadia, and the Sun, which is several times larger than the Earth, belongs to only a small fraction of the sky. How could it not be evident from all this that the Earth cannot be two-dimensional?

The argument presupposes as proved that the Earth has a circumference of 250,000 stadia; but this is accepted as proved because there was an authoritative opinion to this effect. The question whether Lysimcheia and Syene are on the same meridian is not even discussed, because it was current opinion. The datum that there are 20,000 stadia between Lysimacheia and Syene must have been derived from a text that reckoned 18° by a stadion of 1111.1 to the degree, between the Tropic or Syene and the parallel of the Bosphoros (42°12’N); Eratosthenes identifies the latitude of Lysimacheia at the Dardanelles with that of the Bosphoros. Another source must have provided the information that the star gamma Draconis was at the zenith 20,000 stadia north of the Tropic, which are 28°34’ by a stadion of 700 to the degree. According to Hipparchos’ commentary on Aratos’ Phainomena (I. 4, 8), gamma Draconis is at the zenith at about 53°N. There is further introduced into the argument the assumed conclusion that the Earth has a circumference of 300,000 stadia, which is true by a stadion of 833.3 to the degree. Hence, both Lysimacheia and Draco are placed 24° North of Syene, because by this stadion 20,000 stadia are 24°.

This hodgepodge of numerical data is comprehensible from the point of view of philological, anti-mathematical geography. What counts are the written authorities that give specific figures; it does not matter that the figures are based on different stadia. Strabo who begins his work by expressing his condemnation of mathematical geography, steadily treats as equivalent figures based on different stadia.

Precision in measurement does not count, as indicated by the computation pi=3.<127>

We should not be surprised at the procedure, since it is similar to that followed by most of our contemporary scholars: they interpret ancient geographical texts by assuming that there was only one kind of stadion and when the figures do not fit they resort to the argument that the ancient value of the stadion was not precisely defined. To the names of those who hold this view I may add that of D. R. Dicks who in a recent (London, 1968?) work on The Geographical Fragments of Hipparchus dedicates a special chapter to “The Value of the Stade,” in which he claims that there was only one stadion, the geographic stadion of 8 to a Roman mile, but adds (p. 45):

Despite the fact that there seem to have been local variations in its length, it was regarded as a constant standard—everyone knew roughly what distance was meant when one city was said to be 50 stades or 500 stades from another. The ancients certainly did not trouble themselves about exact measurements based on carefully worked out equivalents for the various systems in the modern manner.

Here there is represented the usual demagogic argument that the ancients were not precise in measurement because this is a modern privilege. But how much credence can be given to this boast is indicated by the inaccuracies of this author in quoting modern data about latitudes, a carelessness in measurement that culminates in the amazing statement (p. 168) that the Ecliptic and the Tropic reached the maximum angle, which would be about 23°53’ in 2000 B.C.! By data of this sort any rational discussion of ancient astronomy and mathematical geography becomes impossible. It is no wonder that scholars who hold opinions such as these praise Hellenistic geographers as superior in scientific accuracy to their predecessors.

Kleomedes then proceeds to argue that not only is the Earth round, but that it can be measured. The exact measurement does not interest him, because as long as it can be measured, the existence in the world of a logos, in the Stoic sense, is proved. According to him the two best demonstrations are those of Poseidonios and of Eratosthenes. He considers that of Eratosthenes less desirable because it is not simple, since it makes recourse to a geometric argument. Both demonstrations “make certain assumptions and arrive to the demonstrations from these assumptions.”

The argument of Poseidonios runs as follows, according to Kleomedes:

Rhodes and Alexandria lie under the same meridian, and the distance between the two cities is considered to be 5,000 stadia. Let us suppose that it is so.

All meridians are formed by great circles of the kosmos, such that they cut the kosmos into two parts and are drawn through the two Poles of the kosmos.

Having established as hypothesis that things are so, Poseidonios next divides the zodiac into 48 parts by cutting each of the 12 signs into 4 parts; the zodiac is similar to the meridians, since it cuts the kosmos into two equal parts.

Now, if the meridian that passes through Alexandria and Syene is divided into 48 parts like the zodiac, each fraction shall be equal to the fractions of the zodiac that we have mentioned.

Given that things are so, Poseidonios says next that Canopus is the name of a brilliant star placed at the south near the Rudder of the ship Argo. This is not seen at all in Greece, since Aratos does not mention it in the Phainomena, but when we advance towards the south it begins to be seen at Rhodes where, after having been seen at the horizon, it set immediately because of the rotation of the kosmos. When, having navigated from Rhodes for 5,000 stadia, we arrive at Alexandria, we find that this star, when it is exactly at the middle of the sky, has an elevation above the horizon of a fourth of a sign, that is, 1/48 of zodiac...

...world.

When, having navigated from Rhodes for 5,000 stadia, we arrive at Alexandria, we find that this star, when it is exactly at the middle of the sky, has an elevation above the horizon of a fourth of a sign, that is, 1/48 of zodiac. It follows of necessity that the arc of celestial meridian corresponding to the distance between the two cities is also the forty-eighth part of the same meridian. The horizon of Rhodes and the horizon of Alexandria are distant 1/48 of zodiacal circle

Since the part of Earth that lies below this segment is considered to be five thousand stadia, the great circle of the Earth is found to be two hundred forty thousand stadia, if the distance from Rhodes is five thousand stadia. If not, the distance shall be in proportion to the distance.

The argument of Poseidonios appeals as much as possible to immediate sense experience. It assumes that “we” could start from Rhodes by ship and tell directly when we have navigated 5,000 stadia. In order to avoid a mathematical division of the meridian, it refers in a rather cumbersome way to the zodiacal circle which is visibly divided into parts. If it is accepted that the zodiacal circle is divided into parts, the great circle passing through the Pole can also be assumed to be divided into parts, because both divide the kosmos into two halves. This argument sounds unnecessary and childish, unless the polemics against the mathematikoi are kept in mind.

For Hellenistic philology, immediate sensible experience and textual authority are treated as equivalent, because they are both grounds for which propositions are accepted as evident truth. It was true that the star Canopus had an elevation of about 1°30’ Alexandria; but given this it should have been visible at the horizon at Athens, and much more clearly visible at Rhodes. In fact, Hipparchos in his commentary on Aratos (I. 12, 7) states that Canopus (alpha Argus) is about 38°30’ from the southern Pole, whereas the circle of the invisible stars begins at about 37° for Athens and at about 36° for Rhodes, so that the star can be seen passing above the horizon at Athens and even more certainly can be observed in the zone of Rhodes. ApparentlyAratos had not mentioned at all the star Canopus, so that it could be presumed that it could not be seen at the latitude of Athens. But Geminus (Isagoge 13D) said “This star is difficult to observe in Rhodes and can be seen only from elevated points.”

That Canopus has an elevation of 7°30’ at the latitude of Alexandria is accepted as a fact, because this is something that can be seen with the eyes; that Rhodes and Alexandria are on the same meridian is presented as an hypothesis, since it is something that we cannot sense directly, even though it is not contrary to sensory experience.

Since Strabo declares that Poseidonios computed the circumference of the Earth as 180,000 stadia, and since Eratosthenes referred to three estimates of the distance between Alexandria and Rhodes (sailors estimated the distance as 5,000 and as 4,000 stadia, whereas the shadow of gnomons indicates a distance of 3,750 stadia), it follows that Poseidonios based himself on the figures of Eratosthenes and concluded that the Earth could be either 240,000 (=5,000 x 48) or 180,000 (3750 x 48) stadia.

It is clear that Poseidonios did not claim to have submitted a new calculation of the circumference of the Earth. Modern scholars who speak of a computation of Poseidonios are just as careless as Strabo. Probably Strabo read the figures of 240,000 and of 180,000 stadia in Poseidonios and since the second corresponded to a stadion of 500 to the degree which was in use, he concluded that the second figure of Poseidonios was intended to be a value for the circumference of the Earth. Berger, in his study of Eratosthenes, has recognized that Poseidonios did not proceed to any calculation of the circumference, but he does not see that for the same reasons it must be concluded the same about Eratosthenes.

He says that Syene and Alexandria are under the same meridian. Since meridians are great circles of the kosmos, the circles on the Earth that are below them must also be great circles. Hence, the great circle on the Earth will have the same size as that which will be proved by this procedure to be that of the (celestial) great circle passing through Syene and Alexandria. Then he says, and it is really so, that Syene is under the summer Tropic. When the Sun enters Cancer and makes the solstitial turn and is exactly at the middle of the sky, of necessity the pointers (gnomones) of the dials become without shadow, and it is logos that this should happen over a diameter of 300 stadia.

In Alexandria at the same time the pointers of the dial do not cast a shadow because this city is more to the north of Syene. Since these cities are on the same meridian and great circle, if we draw an arc from the end of the shadow cast by the pointer to the foot of the pointer, this arc will be a segment of the great circle that is drawn on the skophe, because the skophe of the dial is below the great circle. If, next, we imagine perpendiculars to the Earth lowered from each of the two pointers, they will meet at the center of the Earth. Since the dial is perpendicularly below the sun, if we imagine a straight line drawn from the Sun to the tip of the pointer, it will be a single straight line from the Sun to the center of the Earth. If we imagine, in relation to the skophe that is in Alexandria, another straight line drawn from the end of the shadow cast by the point, through the tip of the pointer, to the sun, this line will be parallel to the one mentioned before, no matter from which part of the Sun the line is drawn and to which part of the Earth.

It is clear that Eratosthenes never proceeded to any survey on the ground in order to ascertain whether Syene and Alexandria are 5,000 stadia apart. If he had proceeded to this survey, he would have found that Syene and Alexandria are not on the same meridian, their difference of longitude being about 3°. For Eratosthenes the distance of 5,000 stadia is something that he accepts as given, as he accepts as given that Alexandria and Syene are on the same meridian.

What interests Eratosthenes is the geometric demonstration and the empirical demonstration by the skophe.

The main difficulty for Eratosthenes was that he did not know what was known to the Egyptians, that the position of the shadow of the Sun is determined by the upper margin of the solar disk. The Egyptians knew that when the point without shadow at the Summer Solstice was at the latitude of Syene (24°06’N), the Tropic, being calculated by the center of the solar disk, was at latitude 23°51’N.

Ptolemy reports that according to Eratosthenes and Hipparchos the angle of the Ecliptic was . He accepts this figure himself and suggests that he obtained it by testing; possibly he tried to verify it by the means at his disposal and found it not contradicted.

The datum was an old Egyptian datum which had been correct around 1080 B.C. (middle of the zodiacal period of Aries); the Tropic was at 23°45’N in the age of Eratosthenes and at 23° 41’N in the age of Ptolemy. Eratosthenes found himself confronted also with the tradition that the Sun does not cast any shadow at Syene, which was true when the Tropic was at 23°51’N. From this Eratosthenes concluded that Syene is at the Tropic. The problem of the shadow of the Sun was solved by assuming that the rays of the Sun come parallel to the Earth from any point of the surface of the Sun. Hence, the point without shadow extends on the surface of the Earth for half a diameter of the Sun. For this reason Kleomedes says that the area without shadow has a diameter of 300 stadia. The figure was probably obtained by counting by stadia of 600 to the degree, so that it corresponds to 30’, the usual value of the semidiameter of the Sun. But it is possible that Eratosthenes computed the semidiameter of the Sun as 25’. This indicates that the instruments used by Eratosthenes could not measure more precisely than half a degree, so that Eratosthenes could conceive of the zodiac as a zone 25’ wide.

Letronne thought that Kleomedes had misquoted Eratosthenes by ascribing to the latter a calculation by the skophe which is a crude instrument that does not allow a precision much greater than half a degree. But the limit of precision that was achieved by Eratosthenes is indicated by his statement that the shadow of gnomones (pointers of dials) prove that Rhodes is 3,750 stadia north of Alexandria, that is, 5°22’. According to Eratosthenes Athens is 400 stadia north of Rhodes. This permits to determine which observation by the dial Eratosthenes had in mind. The latitude of Athens was determined by assuming that it was the latitude at which the shadow cast by a sundial varies as 4:3 from the Solstice to the Equinox. Computing exactly this is true for latitude 36°52’N. Assuming that Athens was at this latitude, Eratosthenes placed Athens 30’ to the south, rounding the figure of 350 stadia to 400 or 34’. Traditions foreced Hellenistic geographers to assume that Athens and Rhodes were on the same meridian; hence, Eratosthenes set them apart about as much as the limit of his instrumental precision.

It can be believed that Eratosthenes based his astronomical calculations on a sundial in the shape of a skophe. The skophe is a hemisphere...

Eratosthenes must have taken the figure of 5,000 stadia from Egyptian tradition: it was somehting that probably was inscribed on monuments and which was not contradicted by experience. The Egyptians counted the length of their country from latitude 24°00’ which they referred in practice under the name of Syene, to the latitude of the Base of the Delta which could be reckoned as either 31°06’ or at 31°12’N. Probably when they used to reckon by Egyptian royal cubits they said that the length of Egypt was 5,000 stadia counting to latitude 31°06’N. When they introduced the Babylonian-Egyptian great cubit which is 72/71 longer, they may have counted the length of Egypt up to latitude 31°12’N, which is the latitude of Alexandria. Eratosthenes may have found...

Eratosthenes began with an Egyptian datum that gives to Egypt the length of 7°12’ and 5,000 stadia. This figure is correct computing by Babylonian-Egyptian great cubits; it gives the same values to the degree.

But in his geographical works Eratosthenes seems to have computed the length of Egypt differently. From Ptolemy we learn that Eratosthenes and Hipparchos assumed that the Ecliptic had an angle of 11/83 of circle or 23°51’20” ; this last pharaonic value was accurate around the year 1080 B.C. Ptolemy himself adopted this figure even though the angle of the Ecliptic was 23°45’ in the age of Eratosthenes and 23°41’ in the age of Ptolemy. Ptolemy places Alexandria at 31°00’N in his Geography ( ) and at 28°58’N in the Almagest. It is possible that Eratosthenes assumed that Alexandria was at latitude 31°00’N, with the result that the length of Egypt from the Tropic to Alexandria would be 7°08’40”. Assuming that this length was 5,000 stadia, Eratosthenes arrived at a circumference of 252,000 stadia equal to 700 stadia to the degree.<M^>

Probably the figure of 700 stadia was an old Egyptian datum based on stadia of 300 Egyptian royal cubits. In relation to this value of the stadion the length of Egypt may have been interprented as 7°06’. Hipparchos may have noticed this fact because even though he found convenient to use the value of 700 stadia to the degree, he considered it a little too short. The latitude 30°58’N for Alexandria may have been derived from Hipparchos who placed this city 7°06’ north of the Tropic in his fine calculations, even though in his geographical reckonings he seems to place it at 31°07’.

A passage of Pliny that states how much Hipparchos added to the figure of Eratosthenes is unfortunately corrupt:

Hipparchus et in coarguendo eo et in reliqua omnia diligentia minus adicit stadiorum paulo minux XXVI M.

The last figure is uncertain in the manuscripts. I read it as XXIII. It is impossible that Hipparchos added 26,000 stadia because this would not have been a minor addition (more than 10%) and because he too reckoned by 700 stadia to the degree. I suggest that Hipparchos made the degree equal to 700.5 stadia, since 700:700.5 = 7°06’:7°8’40”. He added 180 stadia to the circumference of 252,000 stadia. Pliny, who converts stadia of any sort into Roman miles at the rate of 8 to the mile, said “a little less than 23 miles.” 180:8=22.5).

 
 


STADION

Itinerary distances were measured by a unit called stadion in Greek; it is the equivalent of the English furlong. The stadion is the sexagesimal multiple of the measuring rod. The measuring cane could be 5 or 6 feet or cubits or, being double, 10 or 12 feet or cubits. In the English system since, for special reasons, there is a rod of 11 feet, the furlong is 660 feet. The measuring cane of 5 or 6 feet is equal to the single step (passus), movement forward of the foot, whereas the measuring cane of 10 or 12 feet is equal to the full step (gressus), movement of the two feet. By stretching the movements the steps can become 5 or 6 cubits and 10 or 12 cubits. Hence, the stadion is equal to 60 steps. The single step corresponds to a second of time.

The stadion of 300 or 360 feet or cubits was considered equal to a minute of march, whereas the stadion of 600 or 720 feet or cubits was considered equal to the double minute of march, 1/60 of double hour obtained by dividing the full day into 12 hours. The measuring cane of 10 feet or cubits was preferred in Egypt, Greece and Rome, where measures were computed in part decimally, whereas the measuring can of 12 cubits is typical of cuneiform mathematical texts which aim at a consistent sexagesimal reckoning. The stadion of 720 cubits occurs in cuneiform mathematical texts; Egyptians preferred to count by stadia of 300 cubits, whereas the Greeks and the Romans preferred the stadion of 600 feet.

There existed a great variety of stadia, but only a few were used in geographical computations.

The stadion most commonly used to calculate itinerary and geographical distances was the stadion of 600 artabic feet, which is often called Greek stadion by metrologists and which I call geograpic stadion. This stadion of 600 artabic feet is equal to 625 Roman feet. The roads of the Roman Republic and the Empire were measured by miles (milia passus) , equal to 1000 passus of 5 Roman feet, but the mile was subdivided into 8 artabic stadia marked by stones smaller than the mile stones. In documents of the Roman period, distances at sea are always computed by geographic stadia. The stadion may be subdivided into 6 plethra of 100 feet.

The artabic stadion was preferred for the calculation of itinerary and geographic distances because a plethron of 100 artabic feet is exactly a minute of degree and 600 stadia are a degree; 10 stadia are a minute of degree.

The official units of measure of the Persian Empire were based on the artabic foot (edge of the cube containing an artaba). Thirty geographic stadia of 600 artabic feet formed a Persian parasang, considered an hour of march. There are 20 parasangs in a degree. Since it was assumed that a man could march 10 hours a day, a day of march is half a degree.

Counting by Roman miles, the degree is 75 miles. But at the basic latitude of 36°N, the axis of the Mediterranean, the degree of longitude was converted as 4/5 of the equatorial degree, so that a degree of longitude could be reckoned as 60 miles, a mile to a minute of degree.

2. An artabic foot is 308.2765 mm., being 25/24 of Roman foot of 295.9454 mm. It is worth stressing that this value was obtained empirically by measuring ancient buildings, before the principles that determine the volume of the ancient artaba had been discovered by Segrè, and before I determined the mathematical interrelation of ancient units of length.

All scholars agree that this stadion is the typical stadion of ancient geography; those scholars who try to argue that the ancients had only approximate notions about the size of the circumference of the earth assert that this was the only stadion used by geographers. But since this last group of scholars has not paid any attention to the exact value of the artabic foot and the geographic stadion, they have not come to realize that calculations by this stadion imply a degree of 110,979 m. and a circumference of the earth computed as 39,952,629 m.

This implies an astounding achievement by ancient scientists, but the achievement becomes even more astounding when one takes into consideration that 110,979 m. is the value of the degree at latitude 37°N by our theoretical geoid. If the length of the degree had been measured in Mesopotamia at latitutde 36°N, the result would have been that because of the elevation about sea level. Latitude 36°N was the basic latitude for Mesopotamian astronomers and for Greek geographers; latitude 36°N is a basic latitude also in Chinese, Indian, and Persian calculations. To this evidence there must be added the striking datum that when Calif al-Mamun ordered a geodetic survey in order to test the traditional value of the degree, the operations wee performed in Mesopotamia at latitude 36°N.

I would not draw the inference that the length of the artabic foot and hence the value of all units of length, volume, and weight, was established knowing the length of the degree at latitude 36° but that one of the reasons why latitude 36° was chosen as the basic one was that it had been found out that counting the degree as 360,000 artabic feet, the resulting value was perfect for latitude 36°. This would imply that the degrees could be measured with such precision that it could be noticed that the degrees of latitude become longer as one moves to the north. This is not absolutely impossible, because my conclusion is that the ancients could locate geographical positions with a precision greater than a minute of degree. By our theoretical geoid a degree of latitude is 110,567 m. at the Equator and 111,052 m. at latitude 41°; to notice a difference of about 500 m. in the length of the degree was not beyond the technical possibilities of ancient metrics. However, it is necessary to study in situ the geographical position of relevant ancient monuments in order to ascertain in a positive way the limit of precision that could be achieved by ancient geodetic surveys.

The artabic foot had two values, one of 307.796 mm., which is exactly the edge of the artaba and hence is (cube root of 9)/cube root of 8) of the Roman foot, and one of 308.276mm., computed by the practical relation 25/24 of Roman foot. Whereas buildings appear to have been planned by either form of the artabic foot, in all texts concerning land surveying, itinerary distances, and geographical distances, the stadion is always 625 Roman feet, indicating that the second value of the artabic foot was found preferable for geographic purposes.

There are texts that mention a stadion of 500 to the degree, such that the circumference is 180,000 stadia. These are stadia of 720 artabic feet. A stadion of 500 to the degree has the advantage that the corresponding length of the degree of longitude at latitude 36° is 400 stadia.

Aristotle ( ) reports that according to the mathematikoi the circumference of the earth is 400,000 stadia. Archimedes (Sand Reckoner ), who lived a century later, mentions as an accepted fact that the circumference of the earth was computed as 300,000 stadia. Since Gosselin, concluding the researches of scholars of the eighteenth century, poiointed out that stadia of 1111.1 and 833.3 to the degree are most common in ancient geographical calculations, it follows that Aristotle and Archimedes were quoting standard measurements of the circumference of the earth. Gossellin and before him D’Anville, spent years of their lives testing against actual distances on the ground the figures of ancient geographers based on these two stadia, and concluded that the dimensions based on stadia of 1111.1 and 833.3 to the degree are equivalent in value to dimensions calculated by geographic stadia of 600 to the degree.

Modern scholars protest, without submittinng any specific argument, that it would have been impossible for Aristotle to know the exact circumference of the earth. According to them it was just in the age of Aristotle that it was first realized that the earth is a sphere and not a disk. If Aristotle was acquainted with an exact measurement of the circumference of the earth, it follows that the calculation had been performed by non-Greeks, because we can rest assured that up to that time there had never been a Greek state organization that had the means to proceed to a geodetic survey.

All that I can do is to add to the more than adequate demonstrations of D’Anville and Gossellin by submitting some specific examples of pre-Greek geodetic calculations based on the stadia of 1111.1 and 833.3 to the degree, proving further that these calculations were exact and gave the same values as the calculations by the geographic stadion of 600 to the degree.

Since I have established the principles by which the ancient lineal units were related to each other and hence can quote precise values for all the lineal units, I can add, in support of the conclusions of the French scholars of the eighteenth and early nineteenth centuries that the stadion of 833.3 to the degree is a stadion of 300 Roman cubits (cubit of 443.918 mm.; stadion of 133.175 m.) whereas the stadion of 1111.1 to the degree is composed of 300 trimmed barley feet (18/16 of Roman foot or 332.9387 mm.), a stadion of 99.882 m., which can be also interpreted as 360 trimmed lesser feet (15/16 of Roman foot or 277.4489 mm.)

The stadia mentioned up to this point are in simple relation to each other and express a single estimate of the degree and of the circumference of the earth:

Units

Relation to
Stadia of

Roman foot

Trimmed lesser feet

15/16
300 360 600 720

Roman cubits

1.5
833.3
1111.1

 

(300,000)
(400,000)

Artabic feet

25/24
600
500

 

(216,000)
(180,000)

Trimmed barley feet

18/16
1111.1
(400,000)

Degrees of 1111.1 and 833.3 stadia have particular advantages in reckonings that apply to the original earthly Oikoumene, the Oikoumene that corresponds to the heavenly Oikoumene and extends from the Equator to the northern limit of Egypt at latitude 31°. In Egyptian reckoning the important latitudes were the latitude of Thebes (2/7 of the distance from the Equator to the Pole or 25°42.8’N) where the degree of longitude is 9/10 of equatorial degree (cos 25°50’=0.900) and latitude 31°00 where the degree of longitude is 6/7 of equatorial degree (6/7=0.8571=cos 31°00). By a stadion of 1111.1 to the degree a degree of longitude at the latitude of Thebes is 1000 stadia. By a stadion of 833.3 to the degree a degree of longitude at latitude 31°00, is 700 stadia. Just below it will appear why this figure of 700 stadia was particularly desirable for the Egyptians.

Above latitude 31°, the fundamental parallels were 36°00’ or 36°12’N (latitude of Rhodes) and the parallel 5°00’ farther to the north, parallel 41°12’ (passing through the northern limit of the Bosphorus). The degree of longitude was reckoned as 4/5 of equatorial degree at the latitude of Rhodes (cos36°52’= 0.8000) and 3/4 of equatorial degree at latitude 41°12’N (cos 41°24’=0.7501). By an equatorial degree of 1111.1 stadia, the degree of longitude was 833.3 at latitude 41°12’N. Another fundamental parallel was 45°12’N, where the degree could be counted as 7/10 of equatorial degree (cos 45°34’N=0.7000); but the reckonings were adjusted to a figure of about 7.12.

Greek and Roman writers associate the name of Eratosthenes, who was the head of the Library of Alexandria in the second half of the third century B.C., with the introduction of a stadion of 700 to the degree, by which the circumference of the earth is 252,000 stadia.

Since it is known that the Egyptian royal cubit had a value of 525 mm., a large number of scholars has concluded that this stadion was equal to 300 Egyptian royal cubits. Reckoning by this stadion of 157.50 m, the degree is 110,250 m. and the circumference of the earth is 39,690,000 m.

But there is another group of contemporary scholars who object that such an achievement would have been impossible in ancient times, whether in the pre-Greek or in the Greek period. Typical is the opinion of J. Oliver Thomson who observes that a calculation of the circumference as 39,690 km, which is 24,662 English miles, or only 200 miles short of the actual figure, must be rejected a priori as beyond the possibilities of ancient science. He exclaims: “Even with some discount for happy cancelling out of various errors, it is very wonderful, if true, and should be cried from the housetops as the best thing in ancient science, a thing never really improved on till Picard in Newton’s time. But it seems too good to be true.” Thomson follows the method of substituting rhetoric for mathematical analysis, a metod considered inherent to the humanistic character of ancient studies. He also follows the strange epistemological principle by which if an ancient calculation appears to be correct, it must be presumed to have been interpreted erroneously; it is a peculiar principle by which interpretations that make the figures of a Greek inscription or a cuneiform text appear correct, are assumed to be suspect.

Thomson argues that ancient geographers used only one kind of stadion, that which I call geographic, so that by a calculation of 700 stadia to the degree the estimate would have been about 1/7 in excess (he says 11-12%, but quotes other scholars who reckon the excess as slightly less than 1/7). Thomson and those he follows do not realize that they are contradicting themselves because if there was a stadion such as the one that I call geographical stadion, since it is beyond question that most commonly the degree is computed as 600 stadia, it must be concluded that the length of the degree had been computed accurately. Thomson grants that 8 such stadia make a Roman mile, and the length of the Roman mile is something that can be determined with good accuracy by direct inspection of Roman roads.

The scholars who have recognized that Eratosthenes was reckoning by stadia of 300 Egyptian royal cubits do not know that this unit was no longer the official unit of Egypt in Hellenistic times. During the period of Persian domination or earlier in the period of the Assyrian conquest, the Babylonian-Egyptian great cubit of 532.702 mm. (28/16 English feet) replaced the royal cubit of Pharaonic Egypt. Hence, when Eratosthenes mentioned the figure of 700 stadia to the degree, he was quoting a computation of Pharaonic Egypt.

The Egyptian calculated the degree by a stadion of 700 to the degree not only because they preferred computations based on the factor 7 (the Egyptian royal cubit is 28 fingers, and the Babylonian-Egyptian great cubit that used to be 32 fingers in Mesopotamia came to be divided into 28 fingers when it was introduced into Egypt), but also because a stadion of 700 to the degree gives a length of 600 stadia for the degree of longitude at latitude 31°.

The relation between the two types of cubit is almost exactly 71:72 (71/72 * 525.303=532.702). Hence, the Egyptians that calculated the length of their country either as 7°06’ or as 7°12’ did not have to change the value of 5000 stadia for the length of Egypt: it could be reckoned as 7°12’ by the shorter cubit and 7°06’ by the longer cubit.

The exact value of the Egyptian royal cubit waas 525 mm., so that the degree waas 110.250m., a value that is slightly too short even for the latitude of Egypt. By our theoretical geoid the degree is 110,567 m. at the Equator, 110,753 m. at the Tropic, and 110,866 m. at latitude 31°.

The Egyptian royal cubit occurred also with an increased value of 526.323 mm. (edge of cube containing 16,000 basic sheqel or qedet of 9.1125 grams), so that by it the degree would be 110,527 m. This degree is correct at the Equator; it is a fact that the Egyptians began their calculations from the Equator.

The calculation of the degree by a stadion of 300 Egyptian royal cubits gives a value of the degree that is slightly inferior to the value of 600 geographic stadia; the latter value, however, is in excess for the latitude of Egypt. Apparently Hipparchos in his comments on Eratosthenes’ Geography noticed this fact, since Pliny states:

The last figure is uncertain. I read it as XXIII.

Scholars have not succeeded in explaining this text, because it would be incorrect if it meant that Hipparchos added 26,000 stadia to the 252,000 of Eratosthenes as the circumference of the Earth; it is beyond dispute that Hipparchos too reckoned 700 stadia to the degree. Pliny states that Hipparchos added the mentioned amount as a matter of nice reckoning, whereas more than 10% would not be a minor correction. In my opinion, Hipparchos noticed that 700 stadia of 300 Egyptian royal feet being 700 x 300 x 526.323 mm = 110,525 m., are less than 600 geographic stadia or 110,979 m., and hence suggested that a half stadion be added to each degree (700.5 stadia = 110,648 m.). He added 180 stadia to the circumference; Pliny who always converts stadia of any sort into Roman miles at the rate of 8 stadia to the mile, declared that Hipparchos added “little less than 23 Roman miles” (180/8=22.5).

For people who lived when the measures were in use, it was obvious that 700 stadia of 300 Egyptian royal cubits were less than 600 geographic stadia. This explains why, whereas a series of ancient sources states that Eratosthenes reckoned the circumference of the Earth as 252,000 stadia, there is a passage of Kleomedes (probably based on Poseidonios) asserting that Eratosthenes reckoned the circumference as 250,000 stadia. The second reckoning does not reduce the value of 252,000 stadia but increases it, because it implies a shift to the reckoning by the then more common Babylonian-Egyptian great cubits: since it was a matter of general knowledge that a cubit of this variety is 8/5 of Roman cubit, it took only a moment to realize that 250,000 stadia of 300 Babylonian-Egyptian great cubits are just as much as 216,000 geographic stadia and 300,000 stadia of 300 Roman cubits.

Eratosthenes himself probably cited the figure of 700 stadia to the degree with the circumferencce of 252,000 stadia, but reckoned the quadrant as 62,500 stadia rather than 63,000. This would explain a discrepancy of 500 stadia in Eratosthenes’ figures for the distance between the Pole and the Equator as reported by Strabo.

...that 5000 stadia of 700 to the degree (by the Egyptian royal cubit) are equal to the distance from latitude 31°12’, latitude of Alexandria, to the latitude of Syene, which is 24°06’. Hence, Eratosthenes...

Ptolemy in his Geography computes the latitude of Alexandria as 31°00’N, whereas in the Almagest he computes it as 23°58’N. It may be that these figures are based on Eratosthenes. Eratosthenes began with the datum that the Tropic is 23°51’20“N, and dividing 5000 stadia by 700 stadia to the degree concluded that Alexandria was 7°8’40” (exactly 7°8.57’) north of the Tropic and hence at latitude 31°00’N. By this reckoning the circumference of the earth is 252,000 stadia.

Having established as hypothesis that things are so, Poseidonios next divides the zodiac into 48 parts by cutting each of the 12 signs into 4 parts; the zodiac is similar to the meridians, since it cuts the world into 2 equal parts.

Now if the meridian that passes through Alexandria and Syene is divided into 48 parts, like the zodiac, each fraction shall be equal to the fractions of the zodiac that we have mentioned.

Touton ouJtw” egountoon

Poseidonios says next that Canopus is the name of a very brilliant star placed at the south, near the rudder of the ship Argo. This is not seen at all in Greece, since Aratos does not mention it in his Phainomena. When we advance towards the south it begins to be seen at Rhodes, and being seen at the horizon immediately it sets because of the rotation of the...

In Alexandria at the same time the pointers of horologia do cast a shadow because this city is more to the north than Syene. Since these cities are on the same meridian and great circles, if we draw a circumference from the end of the shadow of the gnomon up to the base of the gnomon of this horologion, this circumference will be a segment of the great circles that are drawn through a scophe, because the scophe of the horologion is subjected to a greatest circle. If next [noesaimen] perpendiculars to the earth, through each gnomon, they shall meet at the center of the earth. Since the horologion is perpendicularly below the sun, if we imagine a straight line drawn from the sun to the ... gnomon of the horologion, there will be a single straight line drawn from the sun to the center of the earth. If we imagine another straight line drawn from [from the scophe that is at Alexandria] the extreme of the shadow of the gnomon through the tip of the gnomon of the sun, this line and the line mentioned before will be parallel no matter from which oar to fhte sun they are drawn to whichever part of the earth. (I,10<197>ends at 55)

Kleomedes remarks that the argument of Eratosthenes is less manifest or obvious (saphes), because it resorts to geometry:

Things said by him will appear evident if we assume as preliminary the following. Let it be supposed here, first, that Syene and Alexandria are under the same meridian, second, that the interval between the two cities is 5,000 stadia, third, that the rays emitted from any point of the Sun to any point of the Earth are parallel. The geometers suppose that these things are so...

The fourth supposition is the geometic principle by which, given that the rays of the Sun are parallel, if the rays of the Sun fall perpendicularly at Syene and fall at an angle at Alexandreia, the angle according to which the rays fall at Alexandreia is equal to the angle formed at the center of the Earth by the perpendiculars lowered from Syene and from Alexandria. After referring to this point “demonstrated by geometers,” Kleomedes continues (I.10; 52<197>)

He who has mastered these points, will not have any difficulty in comprehending the procedure of Eratosthenes which is the following:

He says that Syene and Alexandria are under the same meridian. Since the merdians are the greatest circles of the kosmos, the circles of the earth subject to them must also be greatest circles. Hence, according to this procedure, the circle passing through Syene and Alexandria shall prove to be as great as the greatest circle of the earth. Then he says, and it is really so, that Syene is under the Summer Tropic. When the Sun enters Cancer and makes the Summer turn, being exactly at the middle of the sky, of necessity the pointers of the horologia become without a shadow, and it is logos that this happens over a diameter of 300 stadia.

And such is the procedure of Poseidonios about the size of the Earth, whereas that of Eratosthenes follows a geometrical procedure, and seems somewhat less clear. Let it be supposed first that Syene and Alexandria are under the same meridian, and second that the interval between the two cities is 5,000 stadia, and third that the rays emitted from any part of the Sun to any part of the Earth are parallel; the geometers suppose that these things are so; let us suppose what is demonstrated by the geometers.

...Poseidonios in which this tries to prove how the earth could be measured.

The text reads:

Fieret etiam, ut dies apud omnes aequales essent; quod totum contrarium astim eis, quae apparent.

Geminus—Isagoge, ed Pet. 13D: “Dieser Stern [Canopus] is in Rhodus schwer zu beobachten” or visible only from elevated points.

Hipparchos —Commentarii on Aratos & Eudoxos Phainomena .

I,4,8 gamma Draconis 37° südliche Schëufe.

Neighbordhood of Athens Polaris was visible up to about 37° where the gnomon indicates that the shadows of day and night arethe same are in the relation 4:3 alpha Argus, or Canopus I, 12, 7 about 38°30’ from the Polar circle, always invisible is distant from the Pole about 37° in Athens and about 36° in Rhodes. this star is north of the circle of invisible stars of Greece, and is observed in the zone of Rhodes.

He says that Rhodes and Alexandria are on the same meridian... Anmd the interval between the two cities seems to be five thousand stadia; let us suppose that it is so. All meridians are formed by great circles of the kosmos, such that they cut it into two parts and are drawn through the two poles of it.

The calculation is based on the usual false assumption that Syene, Alexandria and Rhodes are on the same meridian.

The star Canopus did have at that time an elevation of about 7°30’ at the latitude of Alexandria, but it was not at the horizon of Rhodes. The argument was probably derived from the assumptioon that Rhodes was at the latitude of Athens (37° 58’N) which is almost 7° north of Alexandria. Hipparchos in his commentary on Aratos states that Canopos has an elevatioon of 38°30’ and is well visible at Athens and even more at Rhodes. In the argument of Poseidonios the mathematical premises are presented as hypothetical. Poseidonios does not claim that the circumference of the Earth is 240,000 stadia. He merely says that if the difference of latitude between Alexandria and rhodes is found to be 7°30’ and if the distance between these two places is 5,000 stadia, the circumference of the Earth is 240,000 stadia. The assumption which is false is that Rhodes and Alexandria are on the same meridian.

Letronne pointed out that Poseidonios never claimed to have measured the circumference of the earth, but a number of scholars continue to repeat that Eratosthenes’ calculation was followed by a novel one by Poseidonios.

The argument ascribed by Kleomedes to Poseidonios appears to have been based on some statement of Eratosthenes. Eratosthenes reports that in the estimate of sailors Rhodes was either 5,000 or 4,000 stadia from Alexandria, whereas by the observation of the shadow of the sun the distance is found to be 3750 stadia. Since Strabo quotes Poseidonios (II 95) as having said that the circumference of the earth is 180,000 stadia, it follows that in hypothetical reckoning Poseidonios considered also the figure of 3750 stadia for distance between Rhodes and Alexandria and arived at 180,000 (48 x 3750 = 18,000).

It appears that the sailors had preserved better than the literati the tradition of mathematical geography. The distance between Alexandria and Rhodes is 5° or something less, if one counts to the southern limit of Rhodes (35°52’N). Hence, reckoning by round figures, the distance was assumed to be 5000 stadia by the stadion of 1111.1 to the degree and 4000 stadia by the stadion of 833.3 to the degree.

Whereas for a mathematikos the exact distance between the latitude of Rhodes and that of Alexandria would have been the crux of the matter, for Poseidonios this was a point that could be left vague. In the same spirit Eratosthenes thought that this was a matter to be decided by opinion: for this reason he quoted the calculation base don the angle of the shadow of the Sun together with the current beliefs of sailors.

In the scientific style developed in the Hellenistic age logos becomes logic, that is, the proper interrelation of verbal statements; logos is no longer the discovery of interrelations within the physis. This is made most clear by an argument used by Kleomedes to prove that the earth is not flat. The argument may have been used by Poseidonios. Kleomedes speaks as follows:

If the earth were flat in shape, forming a plane, the theoretical diameter of the kosmos would be 100,000 stadia.

For those who live at Lysimacheia the head of Draco is at the zenith, whereas Cancer is above those that live at Syene. The meridian drawn through Lysimacheia and Syene is 1/5 of the circumference from Draco to Cancer, as is indicated by the shadow of gnomons.

One fifth of the diameter is about 1/15 of the circumference. Supposing that the earth is a plane, let us draw perpendiculars at the two extremities of the ard that ends at Draco and Cancer; they will touch the diameter by which there is measured the meridian that passes through Syene and Lysimacheia. Between the two perpendiculars there will be two myriads of stadia, because there are 20,000 stadia from Syene to Lysimacheia. Since this interval is one fifth of the entire diameter, the diameter of the entire meridian will be ten myriads. If the diameter of the kosmos is ten myriads, the greatest circle of the kosmoos will have thirty myriads.

Now, the earth, which is only a point in relation to the kosmos, is 250,000 [stadia]. The Sun which is several times larger than the earth, represents only a small part of the sky. How could it not be evident from these that it is impossible for the earth to be a plane? (I 8, 42-43).

The argument is not concerned with mathematical precision, as indicated by the computatioin by pi=3. In the new scientific style exact correspondence of measurements (akribeia) is not the essence of evidence. A starting poioint must have been a text that said that parallel 42°12’N (the parallel of the Bosphorus) was 20,000 stadia north of Syene; by a stadion of 1111.1 to the degree 20,000 stadia are 18°. Following the method adopted by Eratosthenes, the latitude of the Bosphorus is identified with that of Lysimacheia. Another source must have provided the information tthat the star gamma Draconis was at the zenith 20,000 stadia north of Syene or 28°34’ by a stadioin of 700 to the degree; according to Hipparchos ( ) this star is at the zenith at about latitude 53°N. But by the calculation that makes the circumference of the earth 300,000 stadia, a degree is 833.3 stadia and 20,000 stadia are 24°00’. Hence, Lysimacheia and the latitude where gamma Draconis is at the zenith, are placed 24° to the north of Syene. In the final conclusion, the statement...


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