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22.- STADIUM (FURLONG) : GEODESIC FORMULA THAT HARMONIZES SPACE AND TIME, OF FIXED AND STATIC CHARACTER, INDISPENSABLE FOR MAKING ASTRONOMICAL CARTOGRAPHY 


        Homer used the expression , “stadíe ysmíne” that the Latin experts interpreted as “stataria pugna” and that it passed to the English as “combat on firm feet”, to refer to a combat in which the fighter maintains a static, unalterable position, without giving a span of land (Il. 7, 241; 13, 314; 13, 514; 13, 713). In Greek “stadíe” it meant “fixed”, “firm”, “stable”, and “ysmíne”, “ combat.”


        The Greek adjective, “stadios, -a, -on” whose Ionic form was “stadíe”, is the past participle of the verb, “ístemi” that means “to fix”, “to immobilize”. Related with this concept, the word “static” also appears.


        With the centuries, the adjective “stadios” it was converted into its neuter form “stadion” to designate a measure itinerary “fixed” that was supposed it would be the distance that Heraklês ran with a single inspiration of air.

        Achilles Tatius of Alexandria, Greek philosopher of the IIIth century A.D., reproduced the work “The Phenomena” by Aratus of Soloi, very mentioned by Hipparchus. Surprisingly, Tatius added a brief comment that served as an introduction to this new copy, apparently very poorly edited:


(“Khaldaioi de periergotatoi genomenoi etolmesan tou heliou ton dromon kai tas horas

diorisasthai ten gar en tais isemeriais horan autou, kath hen isos dierkhetai ton polon, eis l’ orous merizousin, hoste to l’ meros tes en te isemerine hemera oron legesthai tou dromou tou heliou. Legousi de palin andros poreian mete trekhontos mete hemera badizontos, mete gerontos mete paidos, ten poreian einai tou heliou kai l’ stadion katharon einai”)
“The chaldaeans, very meticulous, had patience to determine the trip of the sun and the hours. Indeed, they settled down in thirty measures the time during which, in the equinoxes, it (the sun) passes by successively over the “poles”, so that the thirtieth part of the time in a equinox day, the measure of the trip of the sun was called. They (the chaldaeans) say, in turn that the march of a man, neither running neither very slow, neither old neither boy, is the march of the sun and it is of thirty pure furlongs.”

Hell in the southerm extreme of the Moskenes Island

Hell in the southerm extreme of the Moskenes Island

        In mathematical terms it could be expressed that the distance that an ordinary man can walk in one hour each one is equal to 30 furlongs of 125 human steps. This multiplication yields 3,750 human steps per hour, being each step of 5 feet and these of 3 spans, of the same person, that is to say twelve fingers. Because well, if we grant to each step the extension of 1,25 m., in function of recognizing that a person of 1,65 m of height commonly has a foot of 0,25 m., the furlong will have 156,25 m. and 30 furlongs will be 4,687,5 m.


        The best test that this it is the correct interpretation of the size of the itinerary furlong we obtain it by multiplying the 4,687.5 m. that they are walked in one hour by the 24 hours of the day and we will obtain a degree of 112,500 m. and an equatorial circumference of 40,500,000 m. that it exceeds by one percent to the 40,075,695 m. calculated today.
The theoretical number of 720 furlongs to the degree was not the followed one by Eratosthenes who seemingly obtained through the “poles” only 700 furlongs. By multiplying these 700 furlongs in the above form, yields a total of 109,375 m. for a degree and 39,375,000 m. for the equatorial circumference, lower by one percent to the real figure.

Tarim River

Tarim River

River basin of the Tarim River

River basin of the Tarim River

Argan (40°08’ N; 88°22’ E), Xinjiang, China, WNW horizon, April 8th. 1 A.D., 22h37m local time

 In this drawing, you can see the Pleiads situated at 15° over the western horizon, one hour before their occultation and half an hour after sunset.

 In this drawing, you can see the Pleiads situated at 15° over the western horizon, one hour before their occultation and half an hour after sunset.

Taken from "The sky astronomy software. The sky level I for celestron"

 

       These calculations show what the Chaldeans wanted to transmit, that the sun in one day travels, the same thing that a man would travel in one year if can walk in a continuous way. Would this be the reason of supposing that Heracles could only have made such a feat? It will have to notice that the calculation of the equatorial circumference corresponds to the total of 360°, but because of the fact that the Earth really turns around the Sun in 365 days, there are five days not numbered in the previous calculation, because of this fact, the ancient Egyptians called them “Epagomen-days”, that is “additional days”, and considered them holidays. These five days only were numbered for completing the Solar year, but not for geodesic purposes. Brilliant solution! This subterfuge appears oddly too in the Mayan Calendar called Haab, based in 18 months of 20 days each, more five years called Wayed, added to the end of the year.


Perhaps, it would be opportune to relate with the above-mentioned other measures of the Ancient times that have created many problems. As the Persian Parasanga as the Egyptian schoenus, or “schoinos”were worth 5000 steps each one, both measures equivalent to 40 furlongs as Eratosthenes said it. This quantity of 5,000 steps fits 17,5 times in a degree and it passed to the posterity with the name league , in circumstance that this is not the distance that a man can walk in one hour but in uncertain time, just as in the same way when an army when not in a hurry.


23.- THE NOMADS LEARNED HOW TO READ IN THE SKY LONG BEFORE THAN THE SEDENTARIES TO WRITE IN THE GROUND 

        A very old nomadic tradition remembered that the map of the Earth was shown in the sky.

    The necessity to discover a practical form to make maps to be guided in the deserts could begin by means of the identification of the stars that every night culminated on the zenith of a prepared observer that should allow him to familiarize intimately with the location of the fixed stars.

Nomads of the desert of Mauritania

Nomads of the desert of Mauritania

        In an effort to find a simple explanation to the eventual procedure that the old ones could have used to determine the stars that culminated over their heads at midnight, I think that they could have taken a hollow cane for one of their ends so that falling vertically reflected clearly in an mirror put under, the stars that crossed by the zenith at midnight.


If night to night, this procedure repeated in the same place during the 360 days in that its the Sun was supposed to move, they could make a star calendar very easy to identify for their dates and for the stars that were bounded and reflected in the observation mirror. As digression I would like to point out that to such a star catalog, they could also have called it “statistic” for the eventual relationship among “stadia” (furlongs) and big existent distances among different degrees of the Earth. Perhaps for it Sophocles in his work “Oedipus the King”, 880, he pointed out (“astrois ekmetroumenos khthona”) “measuring my steps in the earth by the stars.”


        For solving the problem of determining with accuracy the midnight, the Ancient people could use a sandglass or a clepsydra with duration of 12 hours, so that if they started to run at noon, their end had to coincide with the midnight. The necessary sand or water to mark 12 hours should be the half of the necessary one to cover from midday to midday.


        The happy observation of the fact that the day of the equinox the Pleiades was at 15º of the western horizon, that is to say at a hour of their decline, it should induce to a diligent guide of caravans to verify that during that same lapse of time reached to walk 3,750 steps. Of this ingenious observation, it could conclude that if it multiplied those 3,750 steps by the 8, 640 hours that a year of 360 days would have , we would obtain a result of 32, 400,000 steps for the equatorial circumference of the Earth. Well then, if each human step of five feet is equal to 1.25 meters, the 322,400,000 steps would be equal to 40,500,000 meters. Such a sequence of multiplications for those who managed the abacus skillfully from immemorial times would not exist.

 

        These extreme subtleties should be common in a population accustomed to knit with fine hands the low silk bud under imperious necessity to maintain a tiring concentration so that no thread was loose.


        Known the size of the Earth, it remained a second great problem to solve, this is how to determine longitudes on the base that to each day it corresponded to a degree. The Chinese tradition remembers in Yu the Great, VIIIth century B.C. or IVth century according to others, to an illustrious land surveyor and cartographer. Those who attribute him the “The Book of the Mounts and the Seas” or “Chan hai King” (“Shan Hai Jing”), they think that this work originally contained maps.


        The capacity to make maps should be a notable help for the Chinese authorities that from early, noticed the absolute necessity to build a very long wall of 5,000 km. of rammed ground, able to contain to the chivalry of the Mongol tribes of the North that subdued the calm peasants Chinese from millennia.

       We know now that between the eastern extreme of Manchuria and the Gulf of Peking, current Beijing, there is a geographical distance lower than one thousand kilometers of steppes which only when come to sea present low foothills which not accumulate ice during all the year.

         It would be unthinkable to suppose that it had been able to make such a work without the existence of indicative precise maps of the most suitable places where the wall continued along and that they registered the points up to where to take the necessary implements to supply to the thousands of men that were working on it.


        We suppose that Yu The Great could take advantage of an old artifice called in Greek “polos”, characterized to be a horizontal semi-sphere with a vertical projector of shade or “gnomon”, representative of the axis on which the Earth rotates. The word “Polos” in Greek means “axis” in its first meaning and it comes from the verb “poleo”, that means “to rotate”. See fig. 12.

12.- “Polos”, according to a drawing of Vitruvius.

                             12.- “Polos”, according to a drawing of Vitruvius.

 

 


        As we had not obtained any information regarding the form that the poles was used to measure longitudes, it could supposed that in the edge of this instrument each one of the 360º of the circumference should be marked, representative of the meridians and in its interior 90 equivalent concentric circles to the parallels. The degree 270 and the degree 90 of the edge of the polos should be easily aligned in North-South direction by following the shade that casts the “gnomon” in the exact moment of the true noon. In this way, the degree 0 and the degree 180 would be pointing the East and the West respectively.

 
        Yu, the Great, should in his sharp imagination to find logical solutions to the pressing problems, although did not have on very rudimentary elements to get the searched objective. The very repeated successes in determining how much it lacked to arrive at the next oasis, confirmed him that, that was the correct way to know how to determine the locations.


        The intimate convincing that, through the polos, maps could be made, this gave origin to the empiric cartography in which certain some conclusions were certain, obtained through observations of perfect identifiable physical facts, although it didn't have an alternative proof to demonstrate the truth like it happens in the modern rationalistic method.

Third Part


POSSIBLE CARTOGRAPHIC TECHNIQUE IN THE MIDDLE AGE 

24.- DISTANT ANTECEDENTS OF THE MARINE QUADRANT

        Maybe we could find a distant antecedent regarding the use of the marine quadrant, in the news that the missionary William of Rubruk and Giovanni da Plan del Carpine, sent by the Franciscan Pope Innocentius IV the year 1245 before the Great Khan.

Pope Innocent IV

Pope Innocent IV

        They had to go by the valley of Ferghana, homeland of Alfraganus, the astronomer hired by Al Mamun, third caliph of Baghdad, fact that is hardly surprising because a very important center of astronomic studies existed in Ferghana since a long time ago.


        The valley of Ferghana, separated of the river Tarim basin by relatively low mountains, is important because the mean segment of the Silk Road passed by this region and because in different times it was under the influences of the Chinese, Mongolian, Persian and even Greek cultures, which explains how well located it appears in the Ancient Map mentioned by Eratosthenes. The valley of Ferghana is irrigate by the Syr Darya, river known by the Greeks with the name of “Jaxartes” (Yajartes). With respect to the river “Amu Darya”, the Greeks denominated it “Oxos” because their waters remembered the flavor of the vinegar, in Greek “Oxys”. We must bear in mind that both rivers flow parallely until their drainage in the Aral Sea. Alexander the Great founded the town of “Alexandreia Eschate”, “the Last Alexandria”, current “Kokand” (“Khujand”), in the western entrance of the Ferghana Valley. He did not advance further on because he was contained by the Scythian cavalry.

 "Mongol Horses, also called Takhi and Horses of Przewalski"

"Mongol Horses, also called Takhi and
Horses of Przewalski"; Photography taken from the chilean web site
"http://www.tierradecaballos.cl", key "Criaderos".


        It is possible that due to the extension of the valley of Ferghana for almost 300 km in Northeast-Southwest direction and flanked by high mountain ranges broken from the Pamir, have facilitated the astronomical observations by the experts made to build mathematical maps so necessary to assure the biggest success in the march of the commercial caravans along the deserts of the Central Asia in their trips to China.


        It would have been impossible not to distinguish in that place that a same group of stars moved quickly during every night and that however the same group took four months in disappearing completely for occident. This difference of speeds could only be explained by the movement of daily rotation of the Earth and for the movement of annual translation around the Sun the same as the other planets. It was always famous the valley of Ferghana as an astronomical observatory, may be due to it is surrounded by high cliffs, the stars can be seen at night, so clear that it gives the impression that they are very close to the Earth.

        It is the effect of depth that it could suggest the old astronomers that it was the Earth the one that rotated around the Sun and that for this reason the zodiac signs seem to be nearer to it than other constellations during the year.


        These missionaries could have brought some knowledge to make cartography, that they could pass especially to the Franciscans that governed Oxford, to Robert of Grosseteste (1168-1253), professor of John of Holywood and of Roger Bacon. John of Holywood (1220-1256) who changed into Latin its name for Sacroboscus, maintained the tradition that the true longitude of the equatorial circumference was the 252,000 furlongs found by Eratosthenes and transmitted by Pliny the Elder and Macrobios Theodosios, in contradiction to that affirmed by Ptolemy.

Roger Bacon

Roger Bacon


        It would be necessary to suppose that Sacroboscus found the way of using the Marine Quadrant and the Compass, as it could be deduced from his knowledge of the works of Alfraganus and Albategnius and that he titled to one of his books “De Compositione Cuadrantis” that I have found mentioned only in the Encyclopedia Traccani.


        On the other hand, Roger Bacon (1220-1292) sent in 1267 to the Franciscan Pope Clement IV his work “Opus Maius", accompanied by mathematical instruments. In his work, he insisted in the necessity of verifying the theoretical knowledge by practical means, so he is considered the founder of the inductive method or modern rationalism that it induces of well-known facts an unknown truth.

        The Pope encouraged him to continue ahead with his studies, although for the dedication of Bacon to invent instruments to demonstrate the validity of his theories, there were some people who disqualified him to suppose merely empiric knowledge.


        According to Leo Bagrow, toward the year 1290 the Byzantine monk Maximos Planudes would have copied in the monastery of Vatopédi some version of a group of Ancient Maps to accompany to the “Geography” of Ptolemy.


        It is also of that time the Catalonian world map of 1375, drawn by the Jewish Abraham Cresques and given to the king Charles V of France, which presents the same deviation of the parallel one of Alexandria that is appreciated in the “Ancient Map”. Maybe if for these same years, it began to appear a remarkable series of denominated “portolani” maps in Italy, very well made and beautifully colored, decorated in countless spiders or roses of the winds that induced the commentators of later centuries to suppose that have been made based on of the use of the compass, in circumstances that such a supposition lacks all foundation since the compass only indicates direction and never distances that it is the essential element to make maps. Moreover, it would seem to be an extension of Muslim maps whose main objective was to guide to the faithful people for praying toward the holy city of Mecca.

25.- ORIGIN OF THE AMERICAN ASTRONOMIC CARTOGRAPHY

    It is also possible that the use of the marine quadrant and of the pair of compasses have arrived until the famous mathematical and astronomer Johann Müller (1436–1476) born in Königsberg or “Mountain of the King", origin of the Latin "Regiomontanus".

Johan Müller

Johan Müller

His student was Martin of Bohemian or Behaim who also signed a work under the pseudonym of Henricus Martellus. It would seem to be that who affirmed that Regiomontanus would have introduced the cross-staff or staff of Jacob as a nautical instrument, fell in the serious error of confusing this instrument with the marine quadrant, that does not have anything in common and that of him Behaim took it.


        The nautical chart that is preserved in the National Library of Paris, badly attributed to Sebastian Cabot (Sebastiano Gaboto) because he never had true cartographer knowledge, at most it could only be a copy made by himself. See fig. 13.

13.- Map of America mistakenly attributed to John Cabot.

13.- Map of America mistakenly attributed to John Cabot.

 

This chart is doubtless made in Portugal, both for the gradation of the meridians and also because it corresponds to the geographical discoveries made in 1500 by Gaspar de Corte Real, who was considered as missing because had separated from the rest of the expedition until one year later returned to his house on the island of Terceira of the Azores Islands.


        We can affirm the above-mentioned facts since the immense quantity of toponymes that appear in such a chart , which indicates that it had to be a very detailed exploration, which it is impossible to achieve in so brief time as used by John Cabot (Giovanni Gaboto) and his son Sebastian. The Portuguese cartographers also surely knew around 1501 that to the north of the Caribbean there was not any path The Moluccas, as it attests Saint Christopher's image in the planisphere of 1501 smartly attributed to the Spanish Juan de la Cosa.


        In the Portuguese planisphere sold to the Ambassador Cantino and sent to the Duke of Ferrara in 1502, an identical deviation appears to that of the parallel one of Alexandria that we had already seen in the Ancient Map and in the planisphere of Cresques, only that in the “Cantino” map this deviation is presented in the western end, where Cuba was located at 6º on the Tropic. This anomaly demonstrates that all these maps were made with the same type of instrument that we suppose it would be the marine quadrant.


        The most important in the “Cantino” map rests on that according to the Treaty of Tordesillas, the Demarcation Line, located at 370 leagues from the islands of Cape Verde --of those that fit 17.5 times in a degree--it should go by through 46º30´ L.O.G., next to the mouth of the river Amazon and to be prolonged toward the south until leaving to the Atlantic next to the island of “Buen Abrigo” (“Good Shelter”), immediate to Sao Vicente and Santos cities.


        Given these facts, its antemeridian, located at 180º toward east, left The Moluccas at 3º inside the hemisphere of Portugal according to strict astronomical calculations already carried out previously by Lusitanian astronomers. For a true proof, it is enough to verify that the distance “equatorial-tropic line”--24º-- fits exactly 7.5 times between the Demarcation Line, very well traced in Brazil, and its antemeridian that should be 133º30 ' L.E.G. What a transcendent Portuguese message to determine the exact measures of the Earth! See fig. 14.

 

14.- “Cantino” Map.

14.- “Cantino” Map.


        Have it in mind that the deviation of the Caribbean appears in the world map that Portugal sent to the Duke René of Lorraine and that it was reproduced by Waldssemüller in 1507. The intellectual and material author of both maps it could not be but Martin Behaim, Royal Cosmographer and Cartographer to the Portuguese at the service of king Joao II, with residence on Faial island of the archipelago of the Azores Islands. The capacity that Behaim had to measure longitudes confirms the fact that in Nüremberg he said that his father-in-law was at 700 equatorial leagues from that city. In fact, Nüremberg is located 11º L.E.G. and Faial island at 29º L.O.G., this is at 40º equatorial of distance that multiplied by 17.5 leagues by degree, it gives the exact quantity of 700 leagues.


The cartographic skills of Behaim can be seen after a quick observation of his world map in two hemispheres, presided by its sphinx and dated in Nüremberg in 1492. In the western hemisphere of this map Africa appears quite well designed, with its southeast extreme in the same meridian that Alexandria and with the current Cape Good Hope in its real figure and located at 36º of latitude S., identical characteristics to those presented in the map that he signed under the pseudonym of Henricus Martellus.


        The longitudes of the mentioned world map by Behaim are counted from the meridian of the Canary Islands toward East up to the 180º that go by it would seem to be the peninsula of Malacca, just as Ptolemy erroneously had supposed it in his time. However, Behaim simulated to believe in this error, because he knew well that Eratosthenes had pointed out that this distance was equal to something more than a third of the terrestrial globe (123º). Only as from 1494, the Portuguese charts began to be graduated from Cape Verde (Cabo Verde) in order to be related with that agreed in the Treaty of Tordesillas.


        In relation to the important map of 1507 absurdly attributed to Waldseemüller and in that for the first time the name “America” appears, we would like to demonstrate that his author is Behaim who incorporated in it, the recent discoveries and it located them among concordant meridians with the 252,000 furlongs that Eratosthenes found that they fit in the equinox line, with in fact he recognized the obsolescence of the 180,000 furlongs that Ptolemy believed that existed in it.


In the world map of 1507 already referred, there are two small lockets in the superior part. On the right locket, drawn next to Amerigo Vespucci's figure, Central and South America appear notably well configured, so much in their Atlantic costs as in those of the Pacific, what demonstrates that America had already been very well mapped. See fig. 15.

15.- Medallion in map attributed to Waldssemüller, with the first geographic representation of America.

15.- Medallion in map attributed to Waldssemüller, with the first geographic representation of America.

        It is impressive the fidelity of the design, perfectly located inside meridians that today is enough well-known. The gradation of this small map finishes its 360º in the islands of Cabo Verde, in connection with the Treaty of Tordesillas of 1494.


Observe how the meridian 320 of the map of this locket goes by the eastern end of “La Española Island” (“Hispaniola”), the for the eastern coast of the mouth of the Maracaibo Lake and it finishes in the western mouth of the Strait of Magellan, all that corresponds accurately to the current meridian 70 L.O.G.


        On the left locket, relative to the old world, Behaim appears hiding under the pseudonym of Ptolemy, with a great marine quadrant that has used for the calculation of longitudes, as long as the geometric compass places it in hands of Amerigo Vespucci. In this engraving Behaim shows an incipient smile in his eyes and lips, as reminding the Infante (Prince) Dom Henrique's motto, “Talant de bie faire” or “the happiness of making the things right”. See fig. 16.

16.- Effigy of Behaim with used marine quadrant.

16.- Effigy of Behaim with used marine quadrant.

 

        It should be observed that Behaim did not use for the roses of the winds at all because his objective was to make exact maps and not indication of unnecessary directions.

 


        This map of 1507 should have been the one that impressed Hernando de Magellan (Fernao de Magalhaes) when he saw it in the secretariat of the king of Portugal. Magellan did not doubt to attribute it to Martín Behaim and his columnist Pigaffeta attested this way it. In this map it appeared to insinuate a strait in the South which through you could arrive to The Moluccas by navigating along West, although it was not clear to what height such a path would be.

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