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        In order to try to discover the practical form in that scarce and very selected royal cosmographers of Portugal, trained by Behaim, could determine in so quickly as simple the exact geographical coordinates settled in the maps already commented and that now they are easily verifiable with the current maps, we will try to imagine some form in that the marine quadrant could be used for such effects. We estimate that this hypothetical form should not differ substantially from the one insinuated by Behaim in the locket in that appears under the pseudonym of Ptolemy in the map of 1507.


        It is known that the Portuguese Crown obtained the knowledge in order to know longitudes under the most absolute commitment to never transfer to any other country such a secret, for a short time before his death occurred in 1506, Behaim should have believed necessary to insinuate it by means of a drawing that for its total understanding, it would need to be supplemented with a verbal explanation that the Cosmographer exclusively gave to the cartographers authorized by the King.

        This verbal explanation would never have been registered in writing and if it was, it could have been among the files of burnt documents before the entry of the Spanish armies commanded by the Duke of Alba, so that Philipp II took up the Crown of Portugal.

        Let us try to imagine how the marine quadrant could be used for setting longitudes. The marine quadrant corresponds to the fourth part of a circle, because of that it contains an arch of 90º. If this instrument was taken from its upper ring, its geometrical ruler (alidade) or mobile viewfinder, that could be made in wood, naturally fell vertically attracted by the gravity force marking the degree 45 that could be considered as the meridian zero for these effects, wherever was the place where the observer was located.

        At noon the observer should have made pass by the center of the surface of the mobile finder the shade that casts a gnomon put in the inferior end of such finder. Then he could place the quadrant in horizontal position in order to lengthen the shade of the gnomon for more precision. Twelve hours later, he should have compared with absolute precision the stars that culminated at midnight with those that appeared for that date in the almanac or annual stellar calendar of 360 days commented previously. This date indicated to how many days or degrees the observer was from the 360 days or degrees in that the circumference was conventionally divided. The French pilot and cartographer Jacques de Vaulx seems to have modernized the above-mentioned method as we can see in the fig. 17 that illustrated his work “Cosmographie” in 1583.

17.- Possible method for setting longitudes.

17.- Possible method for setting longitudes.

        The problematic previous explanations with regard to the way how the longitudes could be set before the modern sextants were invented, are merely imaginative for what it would be of great historical interest that they were supplemented or discarded by seamen whose profession trained them in the science of discovering the position of their ship in the high sea, something difficult to understand for a lay person in such matters. I do not blush at all for having ventured to sense some accidental relatively valuable hypothesis; after all, the hypothesis is the mother of the science, if it has been lucky to be improved until reaching the quality a thesis universally accepted.


        The Portuguese Pedro Reinel should have been the assistant prepared by Behaim to follow him in the position of Royal Cosmographer, what happened at the death of him in 1506. Few years later, Reinel produced an excellent map of polar projection of the south hemisphere in which the Demarcation Line and its mentioned anti-meridian with dotted lines. Notice in this map an important political subterfuge, because he has displaced the Demarcation Line having it passed now through the Mouth of the Río de la Plata, knowingly that Portugal did not expose itself to The Moluccas leave outside of its hemisphere. See fig. 18.

18.- Reinel´s Circumpolar Map.

18.- Reinel´s Circumpolar Map.

        Very related to this latter is the planisphere that Diego de Ribeiro copied for the Emperor Charles V (Karl V of the Holy Roman Empire and Carlos I of Spain) before 1529, in which the Demarcation Line also appears going by the meridian of the estuary of the Río de la Plata, but with a deceiving anti-meridian because it presents a double demarcation. See fig. 19.

19.- Portuguese Map copied by Ribeiro.

19.- Portuguese Map copied by Ribeiro.

This way, on the left end of the map the anti-meridian appears located according to the Portuguese thesis, with The Moluccas inside its hemisphere, while in the right end appears, according to the Spanish thesis in which the anti-meridian is located over Sumatra and Malacca, as if it follows to Ptolemy.

        In this map, the nautical instruments used are clearly shown for its making. On the left end of the planisphere a marine quadrant appears, which is a testimony that such maps and demarcations were taken astronomically. Unfortunately, the small size of the letters in the referred notes to this instrument has not allowed us to know their content.

        In the center of the Pacific a great circle is observed in which appear the twelve months of the year corresponding to the twelve signs of the zodiac which insinuates that the quadrant is related to stellar angles.

This outstanding instrument was unnoticed professor Armando Cortesao in his important work on the planisphere by Ribeiro published in the Journal “Imago Mvndi", volume VIII, Leiden, Holland, 1951.

        A copy of the Ribeiro planisphere was given by the Emperor Charles V to the nuncio apostolic Cardinal Baltazar Castiglioni, and at the death of this in 1529, another similar one to his successor in the position, Cardinal Salvatti, maybe as expression of that agreed that same year, on April 22th, in the Treaty of Saragosa. By virtue of this treaty, the Spanish Crown sold to the king of Portugal his supposed rights in The Moluccas until a line traced to the east of them, traced from pole to pole, distant 300 leagues of 17.5 to the degree. For the first time Spain accepted this number in official form, without noticing that, it had recognized it as true the extension of the equatorial line expressed by Eratosthenes, what implied to leave aside all value to what mentioned by Ptolemy.

        When talking about the early American cartography it, becomes difficult to resist the temptation to include the letter of Piri Re`is of 1513, since in it, it is said concretely that it is based on the maps of the “infidel Portugueses", and “made at the engineers way". See fig. 20.

20.- “Piri Re’is” Map.

20.- “Piri Re’is” Map.


The most prominent thing for our concern is the meridians that pass between the Canary Islands and those of Cabo Verde and for the mouths of the rivers Amazons and of Río de la Plata, indicating a falsified Demarcation Line. Both meridians are traced in parallels and highlighted by rose of the winds, what indicates it is a plane chart made on the basis that in the equatorial line 17.5 leagues fit at the degree, equivalent to 56.66 Arab miles. The Arab mile corresponds to a thousand camel steps according to Ibn al-Yayyab, mentioned by professor Joaquín Vallvé Bermejo in the Journal “Al-Andalus”, volume XLI, 1976.


        The old presumption that the Earth rotated around the sun the same as the planets and that allowed to make exact maps, it resurged with Copernicus' positions in 1543 and of Galileo in 1613, without none of them contributed some definitive evidence in its support. The astronomical studies advanced notably with Johannes Kepler's publication in 1621, the one that was improved by Isaac Newton in 1666 when outlining his Theory of the Universal Gravitation.


Toward 1670 the Italian-French Giovanni Domenico Cassini (Jean-Dominique Cassini) used the moons of Jupiter to find an unique hour with the one which to compare the local time of different places of the Earth and this way to determine longitudes.

        After the ascent in 1701 of the first Bourbon to the throne of Spain, Philipp V (Felipe V), French ships were authorized to look for protection in cases of urgency in the Spanish ports of the Pacific, that which was taken advantage of unduly to begin an incipient smuggling. England did not would like to lag behind and made an effort to get similar advantages taking advantage of the signature of the Treaty of Utrecht in 1713.


         The necessity to have precise maps in those which to indicate the places that could be discovered and with it to justify eventual rights of domain under first occupant's juridical title, it impelled the search of practical methods to determine longitudes to the margin of astronomical observations not always easy to carry out. To satisfy this urgency it was necessary to count on precision clocks to carry the Greenwich time on the sea, at least for a year, luckily knowing the meridian of a place whose location will interest to determine accurately, its longitude is obtained by comparing to how many hour-sun one will find regarding the meridian of Greenwich.

        For these effects the English Parliament proposed King George I in 1714 the creation of a prize of twenty thousand sterling pounds for who builds a clock without pendulum able to work without variations in the sea.

        After many vicissitudes, the watchmaker John Harrison obtained in 1764 the real recognition of the obtaining of the prize because his chronometer had only varied five seconds in a trip of six weeks.


        His invention based it on the use of a thin steel tape whose capacity to be uncoiled regularly produced a similar effect to which was made by means of strings that sustained weights able to make move the machinery of the wall clocks. Of here that the thin tape of steel of the new clocks will take the name of “spring”, although in nothing it resembles to a line.

Traslation movement Traslation movement

Surprisingly, the first scientific evidence of the movement of translation of the Earth around the Sun was only obtained in 1727 by James Bradley who discovered the phenomenon of the aberration or deviation of the light that did not have another explanation for him when trying to measure the parallax of the fixed stars that the movement of annual translation of the Earth. This scientific success was worth him the appointment of Director of the Observatory of Greenwich in 1742, after astronomer Edmond Halley's death.

Line of the Meridian of Greenwich


        The calculation of the measure of longitude of the equatorial circumference had variations through the years according to the improvement reached by the mensuration techniques. On March 19th, 1791, the Academy of Sciences of Paris proposed the creation of a new measure call “meter” that would be equal to the ten-thousand-millionth part of the quadrant of a terrestrial meridian.

        Of here that is considered in 10,000 km. the longitude of a meridian section from the Pole to the equatorial line.

        A more exact calculation had been able to obtain by measuring the advance of the first solar ray during a minute of time to at sunrise in the days of the equinox, on a near beach to the equatorial line as it could be on the coast of Nigeria. Curiously, the simplest formulas are those that take longer to appear.

Lagos.  Capital of Niger

Lagos.  Capital of Niger

        It would be absurd not to remember the method transmitted by Achilles Tatius, philosopher of Alexandria, already studied in the chapter 22 and according to which an ordinary man could walk during one hour 30 pure furlongs of 125 human steps each one and these of five feet, so that giving to each foot 0,25 m. a step of 1.25 m is obtained. and for it a furlong of 156.25 m. Of here that in one hour- man theoretically 4,687.5 m could advance. and in the 24 hours of the day 112,500 m., what would give for 360 days 40,500,000 m. This theoretical abstraction implies an error of 1% for excess regarding the 40,075,510 generally accepted m. In other words, a man could walk in one year the same distance that the Sun traveled in one day!

        The incredible previous comparison allows to risk the hypothesis that as well as in many occasions the man has obtained important successes through linking reasonable facts deduced starting from very simple observations, he could also find some solution principle to avoid the socio-economic apathy in that today the World is submerged.

This principle would have to originate in a very simple artificial formula to organize small companies that, for its acceptance social and easy administration, can give origin to bigger companies, those which in turn could multiply for the single effect of having given with a system that allows a good conducive understanding to a shared capitalization. The initial problem resides anything less in having a small capital of risk because ahead of time it could not make sure what degree of social acceptance it could generate this ideal. It is necessary to look for untiringly a reasonable formula that takes gradually to an economic “agreement”, possible seed of a new social pact.