Center of the Earth in the Geocentric Model

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Center of the Earth in the Geocentric Model

Postby The Golden Turd » Fri Feb 05, 2016 10:33 pm

It's going to take me a few posts to finish this OP, as I have to hit multiple sources, the larger scope was on the development of the concepts of creating instruments for the use of determining time and space in the ancient world, that MAY be related to Diocles' insights. I will have to, our of concern for quality preserverance, ask Peter Kropotkin and James Saint to stay out if the discussion, it's aimed at a more scientific minded crowd, with interests in Greek and Roman philosophy and science.

I'll make a note when the OP is finished a few posts down, I'll have multiple premises going in different directions, such as the building of analog gears for primitive clocks, the use of gnomons for exploration and mapping, and the discovery of the procession of the equinox.

I apologize in advance to anyone reading this post interested in the history of science appalled by the trolling, I usually make such posts on history sites. I'm going to give it a shot here, to see if the forum is mature enough to handle it. Please ignore any discussions of Affectance or debates about people engaging in homosexual intrigue with objects in their house, animals, or other humans, or spontaneous rants about conservatives.

I'm taking this quote from Diocles "On Burning Mirrors" p. 38-42

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http://smile.amazon.com/DIOCLES-Burning ... ng+mirrors

We discuss first of all an assumption constantly made by the astronomers, namely that every point on earth can be treated as the center of the earth. Sometimes people who try to discredit the mathematical scientists and say that they construct their subject on a weak foundation scoff (at this): for some of them (the scientists) assert that the(size of the) radii of the spheres in known and that each one is greater than the one (next to it) by more than 30 million stades, while others asset it (is greater by) more than 50 million stades. People were inclined to this second opinion, because they trusted the doctrines of the ancients in this matter; but they say that if the way can be found to avoid using this principle, and we are not not forced to use it by the requirements of the subject of time measuring instruments which use the shadow, then it is best not to use it. We shall proceed with our discussion in order to explain what they were doubtful about. We say, then, that the above mentioned point (i. e. the point on the earth's surface) can be treated as the center of the earth and if the universe. In this we must state the cogent analogy which we use in this matter and in others, and we state that the phenomena occurring in the case of gnomons are similar to what would occur if they really lay at the center (of the earth). But those time measuring instruments using the shadow which indicates the hours without having a gnomon in them reach a degree of minute accuracy such as cannot be attained in this matter by any other kind (of instruments). It may be possible for you to examine the way these instruments are used, if you would care to do so, because (firstly) there is in them something to evoke your astonishment: for we are able to do something which others have (merely) talked about in connection with the subject, and (to decide) whether they hit the mark or not in what they wrote about it; and because, secondly to sum up, none of the elements needed in the above-mentioned time measuring instruments is lacking in your operation. I believe that the operation we must expound to you is something you may understand easily and briefly, so it does not require that you give it close attention and study. As for the matter of the gnomons used by the astronomers, they achieve great accuracy when they are made according to the old methods,which used to be employed in making time measuring instruments in which the shadow is used. But many of the surfaces in which it (the shadow) is used is impossible to make, and many of them are very difficult to make. In short, you must realize that the knowledge of this is difficult and requires care and preserverance; whoever has spent pains on it will will attest the truth of what we say.
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Re: Center of the Earth in the Geocentric Model

Postby The Golden Turd » Fri Feb 05, 2016 10:58 pm

It is believed Diocles was a contemporary of Apollonius of Perga.

The dates given by Wikipedia for Apollonius is 262-190 BC

https://en.m.wikipedia.org/wiki/Apollonius_of_Perga

However, Pytheas was a Greek explorer, who lived, again, according to Wikipedia, in 350-285 BC.

Pytheas had been dead for 100 years. Yet, he is credited with exploring the coasts of England, the Baltic, and first Westerner to enter above the Arctic Circle, using Gnomons to mark his latitude.

Latitude by the altitude of the sun Edit
In discussing the work of Pytheas, Strabo typically uses direct discourse: "Pytheas says ..." In presenting his astronomical observations, he changes to indirect discourse: "Hipparchus says that Pytheas says ..." either because he never read Pytheas' manuscript (because it was not available to him) or in deference to Hipparchus, who appears to have been the first to apply the Babylonian system of representing the sphere of the earth by 360°.[53]

Strabo uses the degrees, based on Hipparchus.[54] Neither say that Pytheas did. Nevertheless, Pytheas did obtain latitudes, which, according to Strabo, he expressed in proportions of the gnōmōn ("index"), or trigonometric tangents of angles of elevation to celestial bodies. They were measured on the gnōmōn, the vertical leg of a right triangle, and the flat leg of the triangle. The imaginary hypotenuse looked along the line of sight to the celestial body or marked the edge of a shadow cast by the vertical leg on the horizontal leg.

Pytheas took the altitude of the sun at Massalia at noon on the longest day of the year and found that the tangent was the proportion of 120 (the length of the gnōmōn) to 1/5 less than 42 (the length of the shadow).[55] Hipparchus, relying on the authority of Pytheas (says Strabo[56]), states that the ratio is the same as for Byzantium and that the two therefore are on the same parallel. Nansen and others prefer to give the cotangent 209/600,[57] which is the inverse of the tangent, but the angle is greater than 45° and it is the tangent that Strabo states. His number system did not permit him to express it as a decimal but the tangent is about 2.87.

It is unlikely that any of the geographers could compute the arctangent, or angle of that tangent. Moderns look it up in a table. Hipparchos is said to have had a table of some angles. The altitude, or angle of elevation, is 70° 47’ 50″[57] but that is not the latitude.

At noon on the longest day the plane of longitude passing through Marseilles is exactly on edge to the sun. If the Earth's axis were not tilted toward the sun, a vertical rod at the equator would have no shadow. A rod further north would have a north-south shadow, and as an elevation of 90° would be a zero latitude, the complement of the elevation gives the latitude. The sun is even higher in the sky due to the tilt. The angle added to the elevation by the tilt is known as the obliquity of the ecliptic and at that time was 23° 44′ 40″.[57] The complement of the elevation less the obliquity is 43° 13′, only 5′ in error from Marseilles's latitude, 43° 18′.[58]

Latitude by the elevation of the north pole Edit
A second method of determining the latitude of the observer measures the angle of elevation of a celestial pole, north in the northern hemisphere. Seen from zero latitude the north pole's elevation is zero; that is, it is a point on the horizon. The declination of the observer's zenith also is zero and therefore so is his latitude.

As the observer's latitude increases (he travels north) so does the declination. The pole rises over the horizon by an angle of the same amount. The elevation at the terrestrial North Pole is 90° (straight up) and the celestial pole has a declination of the same value. The latitude also is 90.[59]

Moderns have Polaris to mark the approximate location of the North celestial pole, which it does nearly exactly, but this position of Polaris was not available in Pytheas' time, due to changes in the positions of the stars. Pytheas reported that the pole was an empty space at the corner of a quadrangle, the other three sides of which were marked by stars.[60] Their identity has not survived but based on calculations these are believed to have been α and κ in Draco and β in Ursa Minor.[61]

Pytheas sailed northward with the intent of locating the Arctic Circle and exploring the "frigid zone" to the north of it at the extreme of the earth. He did not know the latitude of the circle in degrees. All he had to go by was the definition of the frigid zone as the latitudes north of the line where the celestial arctic circle was equal to the celestial Tropic of Cancer, the tropikos kuklos (refer to the next subsection). Strabo's angular report of this line as being at 24° may well be based on a tangent known to Pytheas, but he does not say that. In whatever mathematical form Pytheas knew the location, he could only have determined when he was there by taking periodic readings of the elevation of the pole (eksarma tou polou in Strabo and others).

Today the elevation can be obtained easily on ship with a quadrant. Electronic navigational systems have made even this simple measure unnecessary. Longitude was beyond Pytheas and his peers, but it was not of as great a consequence, because ships seldom strayed out of sight of land. East-west distance was a matter of contention to the geographers; they are one of Strabo's most frequent topics. Because of the gnōmōn north-south distances were accurate often to within a degree.

It is unlikely that any gnōmōn could be read accurately on the pitching deck of a small vessel at night. Pytheas must have made frequent overnight stops to use his gnōmōn and talk to the natives, which would have required interpreters, probably acquired along the way. The few fragments that have survived indicate that this material was a significant part of the periplus, possibly kept as the ship's log. There is little hint of native hostility; the Celts and the Germans appear to have helped him, which suggests that the expedition was put forward as purely scientific. In any case all voyages required stops for food, water and repairs; the treatment of voyagers fell under the special "guest" ethic for visitors.

Location of the Arctic Circle Edit
The ancient Greek view of the heavenly bodies on which their navigation was imported from Babylonia by the Ionian Greeks, who used it to become a seafaring nation of merchants and colonists during the Archaic period in Greece. Massalia was an Ionian colony. The first Ionian philosopher, Thales, was known for his ability to measure the distance of a ship at sea from a cliff by the very method Pytheas used to determine the latitude of Massalia, the trigonometric ratios.

The astronomic model on which ancient Greek navigation was based, which is still in place today, was already extant in the time of Pytheas, the concept of the degrees only being missing. The model[62] divided the universe into a celestial and an earthly sphere pierced by the same poles. Each of the spheres were divided into zones (zonai) by circles (kukloi) in planes at right angles to the poles. The zones of the celestial sphere repeated on a larger scale those of the terrestrial sphere.

The basis for division into zones was the two distinct paths of the heavenly bodies: that of the stars and that of the sun and moon. Astronomers know today that the Earth revolving around the sun is tilted on its axis, bringing each hemisphere now closer to the sun, now further away. The Greeks had the opposite model, that the stars and the sun rotated around the earth. The stars moved in fixed circles around the poles. The sun moved at an oblique angle to the circles, which obliquity brought it now to the north, now to the south. The circle of the sun was the ecliptic. It was the center of a band called the zodiac on which various constellations were located.

The shadow cast by a vertical rod at noon was the basis for defining zonation. The intersection of the northernmost or southernmost points of the ecliptic defined the axial circles passing through those points as the two tropics (tropikoi kukloi, "circles at the turning points") later named for the zodiacal constellations found there, Cancer and Capricorn. During noon of the summer solstice (therinē tropē) rods there cast no shadow.[63] The latitudes between the tropics were called the torrid zone (diakekaumenē, "burned up").

Based on their experience of the Torrid Zone south of Egypt and Libya, the Greek geographers judged it uninhabitable. Symmetry requires that there be an uninhabitable Frigid Zone (katepsugmenē, "frozen") to the north and reports from there since the time of Homer seemed to confirm it. The edge of the Frigid Zone ought to be as far south from the North Pole in latitude as the Summer Tropic is from the Equator. Strabo gives it as 24°, which may be based on a previous tangent of Pytheas, but he does not say. The Arctic Circle would then be at 66°, accurate to within a degree.[64]

Seen from the equator the celestial North Pole (boreios polos) is a point on the horizon. As the observer moves northward the pole rises and the circumpolar stars appear, now unblocked by the Earth. At the Tropic of Cancer the radius of the circumpolar stars reaches 24°. The edge stands on the horizon. The constellation of mikra arktos (Ursa Minor, "little bear") was entirely contained within the circumpolar region. The latitude was therefore called the arktikos kuklos, "circle of the bear". The terrestrial Arctic Circle was regarded as fixed at this latitude. The celestial Arctic Circle was regarded as identical to the circumference of the circumpolar stars and therefore a variable.

When the observer is on the terrestrial Arctic Circle and the radius of the circumpolar stars is 66° the celestial Arctic Circle is identical to the celestial Tropic of Cancer.[65] That is what Pytheas means when he says that Thule is located at the place where the Arctic Circle is identical to the Tropic of Cancer.[27] At that point, on the day of the Summer Solstice, the vertical rod of the gnōmōn casts a shadow extending in theory to the horizon over 360° as the sun does not set. Under the pole the Arctic Circle is identical to the Equator and the sun never sets but rises and falls on the horizon. The shadow of the gnōmōn winds perpetually around it.

Latitude by longest day and shortest solar elevation Edit
Strabo uses the astronomical cubit (pēchus, the length of the forearm from the elbow to the tip of the little finger) as a measure of the elevation of the sun. The term "cubit" in this context is obscure; it has nothing to do with distance along either a straight line or an arc, does not apply to celestial distances, and has nothing to do with the gnōmōn. Hipparchus borrowed this term from Babylonia, where it meant 2°. They in turn took it from ancient Sumer so long ago that if the connection between cubits and degrees was known in either Babylonia or Ionia it did not survive. Strabo states degrees in either cubits or as a proportion of a great circle. The Greeks also used the length of day at the summer solstice as a measure of latitude. It is stated in equinoctial hours (hōrai isēmerinai), one being 1/12 of the time between sunrise and sunset on an equinox.

Based partly on data taken from Pytheas, Hipparchus correlated cubits of the sun's elevation at noon on the winter solstice, latitudes in hours of a day on the summer solstice, and distances between latitudes in stadia for some locations.[66] Pytheas had proved that Marseilles and Byzantium were on the same parallel (see above). Hipparchus, through Strabo,[67] adds that Byzantium and the mouth of the Borysthenes, today's Dnepr river, were on the same meridian and were separated by 3700 stadia, 5.3° at Strabo's 700 stadia per a degree of meridian arc. As the parallel through the river-mouth also crossed the coast of "Celtica", the distance due north from Marseilles to Celtica was 3700 stadia, a baseline from which Pytheas seems to have calculated latitude and distance.[68]

Strabo says that Ierne (Ireland) is under 5000 stadia (7.1°) north of this line. These figures place Celtica around the mouth of the Loire river, an emporium for the trading of British tin. The part of Ireland referenced is the vicinity of Belfast. Pytheas then would either have crossed the Bay of Biscay from the coast of Spain to the mouth of the Loire, or reached it along the coast, crossed the English channel from the vicinity of Brest, France to Cornwall, and traversed the Irish Sea to reach the Orkney Islands. A statement of Eratosthenes attributed by Strabo to Pytheas, that the north of the Iberian Peninsula was an easier passage to Celtica than across the Ocean,[69] is somewhat ambiguous: apparently he knew or knew of both routes, but he does not say which he took.

At noon on the winter solstice the sun stands at 9 cubits and the longest day on the summer solstice is 16 hours at the baseline through Celtica.[70] At 2500 stadia, approximately 283 miles, or 3.6°, north of Celtica, are a people Hipparchus called Celtic, but whom Strabo thinks are the British, a discrepancy he might not have noted if he had known that the British were also Celtic. The location is Cornwall. The sun stands at 6 cubits and the longest day is 17 hours. At 9100 stadia, approximately 1032 miles, north of Marseilles, 5400 or 7.7° north of Celtica, the elevation is 4 cubits and the longest day is 18 hours. This location is in the vicinity of the Firth of Clyde.

Here Strabo launches another quibble. Hipparchus, relying on Pytheas, according to Strabo, places this area south of Britain, but he, Strabo, calculates that it is north of Ierne. Pytheas, however, rightly knows what is now Scotland as part of Britain, land of the Picts, even though north of Ierne. North of southern Scotland the longest day is 19 hours. Strabo, based on theory alone, states that Ierne is so cold[27] that any lands north of it must be uninhabited. In the hindsight given to moderns Pytheas, in relying on observation in the field, appears more scientific than Strabo, who discounted the findings of others merely because of their strangeness to him. The ultimate cause of his skepticism is simply that he did not believe Scandinavia could exist. This disbelief may also be the cause of alteration of Pytheas' data.


https://en.m.wikipedia.org/wiki/Apollonius_of_Perga
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Re: Center of the Earth in the Geocentric Model

Postby The Golden Turd » Fri Feb 05, 2016 11:14 pm

This came out in the news a week or two back, prior to me reading Diocles (still reading him).

Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph

http://science.sciencemag.org/content/351/6272/482.full


I will explain later on what interests me regarding this, I believe it had a influence (or was influenced by) Greek science, especially some of Zenodorus' presumptions regarding Circles.

https://en.m.wikipedia.org/wiki/Zenodor ... hematician)


Of all regular polygons of equal perimeter, that is the greatest in area which has the most angles.

A circle is greater than any regular polygon of equal contour.

Of all polygons of the same number of sides and equal perimeter the equilateral and equiangular polygon is the greatest in area.

Of all solid figures the surfaces of which are equal, the sphere is the greatest in solid content.


The way the time graph was used via sections to determine a trajectory can also be used to calculate a sphere from it's center. This would produce paradoxes we still struggle with to this day regarding Pi, as we view a circle as a perfect, abstract object, and simultaneously a real, concrete thing to be measured..... despite all circular shaped objects being a irregular polygon in physical form (and thus can't be measured by Zenodorus' measurements from the center).
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Re: Center of the Earth in the Geocentric Model

Postby The Golden Turd » Mon Feb 08, 2016 4:59 am

Sorry, I just finished watching a Korean series called "The Slave Hunters", and was researching most of the characters biographies the last few days.... got a little distracted with Joseon era biographies.

Ephorus

I have to make note of him as he wrote on History, navies, and inventions. His works are largely list, but is a variable in the mix that can't be looked over.

https://en.m.wikipedia.org/wiki/Ephorus
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