How the Ancient Egyptians had Calculated the Earth's Circumference between 3750-1500 BC: a revision of the method used by Eratosthenes (2020 Update) (original) (raw)

commentary on How to Measure the Earth' Circumference.pdf

2018

In 'How to Measure the Earth's circumference' Michael Baizerman mentions the modern equatorial measure in conjunction with the meridian degree length. Confusing. The Greek methodologies invariably were conducted in a north/south direction. They were attempting to measures a distance along a meridian. This distance was then expanded to a count of diverse stade values applied to the meridian circumference. At no time was an equatorial circumference given in the ancient world. What emerges from an assessment of these values when evaluated with knowledge of the measures of the ancient world is that these diverse stadia values all ultimately gave the same dimension for the circumference of Earth in a meridian direction. It was John Michell who extending and adapting the works of previous scholars including Berryman, Stechini and Petrie to name but three, revealed the ancient measuring system. While I disagree with some of his conclusions, his measures were correct. Michell claimed that various differences in values were due to the difference in degree lengths around the meridian quadrant whereas I argue that the same values can be found via counts of time such as days of the month in its various formats. In fact the measures applied greatly predate the interpretation of the Earth's shape from a pure globe to a flattened version. It was not until well into the Christian era that reasonably accurate knowledge of this flattening became available. The unit measures, whether cubits, reeds or stadia, among other denominations are related to the meridian circumference of Earth as accepted before the French determinations for the metric system. Some of these divide accurately into that meridian while others work via specific factors. The Earth's diameter was also derived from this via a pi value of 3.1418181818. This diameter of course was 7920 miles. The circumference measure can be seen as 24883.2 British miles. It was Michell who calculated this value and it and its component parts are verified via ancient cubit rods, measures of buildings etc. Here is an extract from my work Measurements of the Gods showing the values applied by Michell, the French calculation for the metric values and NASA [early 1990s] :

ERATOSTHENESʼ MEASUREMENTS OF THE EARTH: ASTRONOMICAL AND GEOGRAPHICAL SOLUTIONS

Orbis Terrarum, 16 (2018, ), pp. 221--254, 2018

Nowadays, Eratosthenes of Cyrene (276-194 BC) is mainly known for measuring the circumference of the Earth, quoted in the ancient sources as either 250000 or 252000 stades. Whereas the former figure is related to the astronomical observation reported by Cleomedes, the latter could have been derived from the geographical information available to Eratosthenes. We demonstrate that the greater of the two figures is interrelated with the number of other figures, including the equinoctial gnomon ratio in Alexandria and the width of the tropic zone, which have been attributed to Eratosthenes. In addition, it is shown how the arc of the tropic zone could be expressed as a fraction of a full circle in pre-trigonometrical times. Furthermore , we discuss a recent attempt by CARMAN and EVANS to explain the figure of 252000 stades as a result of taking into account the finite distance to the Sun. We argue that this contains some historical inconsistencies and astronomical problems.

Measurement of the Earth's radius based on historical evidence of its curvature

Physics Education, 2005

Probably the most direct observation of the Earth's curvature is how objects appear from over the horizon when we approach them and disappear as we get further away from them. Similarly, the portion of a high object (a building or a mountain) that is visible depends on the height of the site where the observation is made. Based upon these very obvious facts, a simple method to estimate the Earth's radius R has been applied. The method does not need either sophisticated instrumentation or complex mathematics. In our application of the method presented here, the result is R = 6600 ± 600 km in the best case. A discussion is presented about the possible use of this method in ancient times. Surprisingly enough, we have not found any reference to the use of this method despite its being simpler than, for example, the classical approach of Eratosthenes.

How to Measure the Earth's Circumference

The chapter of my book, "The Enchanting Encounter with the East", compares the estimations of the Earth's circumference made by Greek, Arab, and European scholars. It sets the 'consensus' value of ancient measures to facilitate its conversion to modern equivalents. Tags: Stadium, Arab mile, Roman mile, Eratosthenes, Posidonius, Pierre d'Ailly, Ahmad Al-Farghani, Christopher Columbus

Ptolemy's Circumference of the Earth

MPIWG Preprint N.464, 2014

The relationship between the determination of the circumference of the Earth and the geographical mapping performed by Ptolemy in his Geography is discussed. A simple transformation of the Ptolemaic coordinates to the circumference of the Earth measured by Eratosthenes, based on the assumption that the metrical values of the stadion used by both Ptolemy and Eratosthenes are equivalent, drastically improves the positions of the locations given in Ptolemy's catalogue at least for a great part of the oikoumenē. Comparing the recalculated positions of the identified localities with their actual positions, it turns out that the distances extracted by Ptolemy from ancient sources are remarkably precise. This in turn confirms the high precision of Eratosthenes's result for the circumference of the Earth. It is shown that many distortions of Ptolemy's world map can be explained as pure mathematical consequences of a mapping onto the surface of a sphere of wrong size.

The Circumference of the Earth and Ptolemy´s World Map (Tupikova/Geus)

Abstract. The interlink between the determination of the circumference of the Earth and the geographical mapping performed by Ptolemy in his Geography (c. 150 AD) is discussed. As Ptolemy himself stated, he used the value of 180,000 stades for the circumference of the Earth which is in stark contrast to the famous result of Eratosthenes, i.e. 252,000 stades. Many scholars see these different values as a result of the diverse definitions of a stade used by both ancient authors. Such a view cannot, nevertheless, explain the excessive distortion of Ptolemy's world map along the east-west direction. We have treated the problem with the methods of spherical trigonometry and have shown that many features of Ptolemy’s map can be easily explained and corrected under the presupposition that the length of the stade used by Ptolemy coincides with that of Eratosthenes. The latitudinal distortion in Ptolemaic map is caused by his underestimating of the size of the Earth in combination with his efforts to keep the known latitudes of the localities. Another mathematical consequence is a significant displacement of the localities lying approximately on the same meridian (so-called antikemenoi poleis), as in the case Carthage in relation to Rome. It is shown that a simple transformation of the Ptolemaic coordinates to the circumference of the Earth measured by Eratosthenes drastically improves the positions of the locations given in Ptolemy´s catalogue. The comparison of the recalculated positions of the identified localities with their actual positions confirmed a very high precision of Eratosthenes' result for the circumference of the Earth. Zusammenfassung Untersucht wird der Zusammenhang zwischen der Bestimmung des Erdumfangs und dem kartographischen Verfahren, das Ptolemaios in seiner Geographie (ca. 150 n. Chr.) anwendet. Wie Ptolemaios selbst behauptet, verwendet er statt des Resultats der berühmten Erd-umfangsmessung des Eratosthenes (252 000 Stadien) den Wert von 180 000 Stadien für die Erdgröße. Die meisten Forscher heute führen diesen Unter-schied auf unterschiedliche Stadien-Längen zurück. Eine solche Ansicht kann aber z. B. nicht die starke Ost-West-Überdehnung der Oikumene erklären, wie sie aus der Erdkarte des Ptolemaios bekannt ist. Wir haben daher dieses Problem mit den Methoden der Sphärischen Trigonometrie behandelt und können nun zeigen, dass sich viele Verzerrungen leicht erklären und korrigieren lassen, wenn wir annehmen, dass das Ptolemäische Stadion mit dem des Eratosthenes identisch ist. Vielmehr wird die Über-dehnung der Oikumene durch den zu geringen Erdumfang bei Ptolemaios in Verbindung mit seinem Bestreben, die überlieferten Breitenangaben für bekann¬te Orte zu bewahren, verursacht. Eine andere mathematische Konsequenz ist die signifikante Verlagerung von Orten, die auf dem gleichen Meridian liegen (so genannte antikemenoi poleis), wie die von Karthago gegenüber Rom. Wir können nachweisen, dass durch eine Rekalkulation der Ptolemaischen Koordinaten in Bezug auf die “richtige” Erdgröße des Eratosthenes die Positionen der Orte in seinem Katalog dramatisch verbessert werden. Ein Vergleich zwischen den rekalkulierten und modernen Koordinaten bestätigt außerdem die Genauigkeit der Eratosthenischen Erdmessung.