## Maths of Magellan

INTRODUCTION

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There is an archived document called Memorial attributed to Fernando de Magallanes in which the belonging of the Moluccas to Spain is justified AGI, PATRONATO, 34, R.13. -original here - in which different geographical knowledge of the time is declared that will allow us to infer the size assigned to the world by its author, as well as the location of both the demarcation meridian according to the Treaty of Tordesillas, and its antimeridian in the antipodes.

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The matter is of great interest in that it will allow us to mathematically demonstrate that Magellan already knew that he was going to face a vast ocean when he began to navigate the Pacific Ocean, contrary to the widely established opinion. Since in the primary sources there is no mention of it, possibly the root of this opinion, by which it is said that Magellan hoped to have found Asia shortly after leaving America behind, is the chronicle of Francisco López de Gómara, Historia General of the Indies:

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They [Magallanes and Rui Faleiro] also affirmed that the Malucas were not far from Panama and the Gulf of San Miguel, discovered by Vasco Núñez de Balboa. Francisco López de Gómara.

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The attribution of this unsigned Memorial would lead us to conclude that its author could not have been other than Fernando de Magallanes himself, or his colleague, the bachelor Rui Faleiro, because who else could write it? If we consider other plausible options, we will see that in all cases something fails us. Perhaps the main one is that it is cited at Cape San María, in present-day Uruguay, as a relevant geographical reference. It ceased to be after Magellan's voyage, but until then it was the last known cape on the coast of South America. Leaving this consideration aside, if we take Hernando de Colón as a candidate, his very lengthy writings on the problem of the size of the terrestrial sphere, and the problems to find empirically its size and the location of the meridians of demarcation, go in another sense. Nor does it seem logical that it was written by Elcano himself before leaving with Loaysa's army, since he does not address the King in the terms in which he does on other occasions, which by the way becomes an argument that would reinforce the hypothesis. that Magellan was its author.

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Memorial attributed to Fernando de Magallanes in which the belonging of the Moluccas to Spain is justified. AGI, PATRONATO, 34, R. 13.

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MEMORIAL ANALYSIS

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Very powerful sir

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Because it could be that the King of Portugal wanted at some time to say that the islands of Maluco are in his demarcation, and could order to change the defeats of the coasts and shorten the gulfs of the sea without anyone understanding it the way I understand it. , and I know how it could be done, I wanted by VA service to declare the heights of the lands and main capes and the heights in which they are, both in latitude and longitude, and with this VA will be notified so that, if what has been said, happens, I was deceased, know the truth.

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+ item the island of San Antón, which is one of those of Cape Verde on the coast of Guinea, where the distribution of these kingdoms with those of Portugal was made, the said island is twenty-two degrees east of the line of distribution.

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+ Item is the said island, it is convenient to know, the western tip at seventeen degrees of latitude.

Since the Treaty of Tordesillas established the demarcation line at 370 leagues west of Cape Verde, this information regarding the fact that those 370 leagues correspond to 22 degrees will help us to find the longitude of the circumference of the equator, or what is the same, the size of the Earth.

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On the one hand, we must find the length of the circumference at the parallel of Cape Verde. It is very simple, since, if 22 degrees measure 370 leagues, the 360 degrees of the entire circumference will measure:

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Longitude circumference of the parallel of Cape Verde = 360 °. 370 leagues / 22 ° = 6,054.55 leagues

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Having found this data, and knowing that Cape Verde is at a latitude of 17 ° N, we can obtain the terrestrial radius by applying the following formula:

Longitude of a terrestrial parallel in latitude α L (parallel) = 2πR cosα

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If we clear the radius in this expression we have:

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Terrestrial radius (R) = L (parallel) / (2π cosα) = 6,054.54 / (2π cos17) = 1,007.64 leagues

Once we have managed to determine the earth's radius, obtaining the length of the circumference of the equator is immediate using the formula:

L equator = 2πR = 2π. 1,007.64 leagues = 6,331.22 leagues

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If we divide these 6,331.22 leagues from the equator by the 360 degrees of the sphere, we also infer that Magellan attributed 17.58 leagues to each terrestrial degree, which will be a data that we are going to use. This data can also be compared with that obtained if we analyze Francisco Albo's Route: if we look at the days in which we are given the leagues traveled and the variation in latitude, choosing days with an important component to the North on the course to minimize errors , we will see that:

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If we discard the extreme values (in red) and obtain the average of the rest, we find that the mean of leagues per degree of the Route was 17.32, a value very close to the 17.58 that we have deduced from the Memorial and that comes to confirm this information.

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If we apply a relationship of 5.5 km per league (we will talk about it later) we obtain that the length of the circumference of the equator, or of any meridian, is 34,821.71 km, that is, 13% less than the 40,000 km real that today we know that it has -very approximately-.

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Finally, we will be able to locate the demarcation antimeridian at 180º from the one already found 370 leagues from Cape Verde, which with the data obtained would be:

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180º x 17.58 leagues per degree x 5.5 km per league = 17,404 km

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In the map that we have created as support, we will show it reflected by measuring this distance towards the west or Castilian side of the demarcation meridian.

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+ Item the Cape of San Agustín, which is in the land of Brazil in the demarcation of Portugal, at eight degrees of latitude and twenty degrees of longitude from the distribution line [actually, ten. Irrelevant data in what we are dealing with here].

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+ Item the Cape of Santa Maria, which is the same land as Brazil of Portugal, is at thirty-five degrees of latitude and at six and a quarter degrees of longitude from said island [actually, eleven. Irrelevant data in what we are dealing with here].

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+ Item the Cape of Good Hope with the Cape of Santa Maria runs east west and there is the Cape of Good Hope at thirty-five degrees of latitude and seventy-five degrees of longitude to the east of the line.

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With this information, Magellan begins to help us locate the antimeridian demarcation of Tordesillas in the eastern half of the world. It will do so by referencing three sections, of which this is the first. By giving us the geographical longitude of the Cape of Good Hope referenced from the demarcation meridian, we can find that in kilometers this distance is the following:

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75 ° x 17.58 leagues per degree x 5.5 km per league = 7,252 km

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This will therefore be the distance to be measured on the equator to the east of the demarcation meridian, to obtain the first coordinate of the cape. By adding your latitude, we will locate on the map a point that will virtually correspond to the Cape of Good Hope.

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The error obtained with respect to the real position of the cape we can verify that it is very low, only 200 km, which confirms that the transformation from leagues to kilometers that we are driving is the same or very close to the correct one.

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The accumulated error found in the location of the Cape of Good Hope is less than 200 km compared to the real one.

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+ Item, the said Cape of Good Hope is in defeat with Malacca to the north-east or south-west and there are one thousand and six hundred leagues on the way from the said Cape of Good Hope to the port of Malacca.

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From the point previously found with the virtual position of the Cape of Good Hope, where Magellan located it, he now tells us that there are 1,600 leagues up to Malacca, which multiplied by 5.5 km per league gives a value of 8,800 km.

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We do not know whether it refers to measurements on the equator from the length of the cape to the length of Malacca, or whether it is a distance measured directly from one point to another. We are going to consider this last option because it is on the safety side (the value it will give us will imply a smaller land size than if we take the first option).

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This leads us to create a virtual point on the map with the location where Magellan claimed to be the city of Malacca. In this case, his estimate in leagues makes him fall short at 625 km. Malacca is a bit further from the Cape of Good Hope than he said.

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We have been curious to find the real orthodromic distance between the positions described in the Memorial between the Cape of Good Hope and Malacca, in leagues, and we have been able to verify that the theoretical distance between both points on a sphere with the aforementioned 1,007 radius, 64 leagues is 1,556.88 leagues, that is, Magellan had an error of only 2.7% in his calculation: something truly impressive.

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The accumulated error found in the location of Malacca is 625 km, in this case less, compared to its actual position.

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3D model created in AutoCAD of the world described in the Memorial, taking the league as a unit of measurement, and with a radius of 1,007.64 leagues. On the model we can measure the orthodromic distance between the Cape of Good Hope and Malacca after positioning both points by their coordinates in degrees. The orthodromic distance found is 1,556.88 leagues, only 2.7% less than what is indicated in the Memorial.

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+ Item the said port of Malaca is to the north of the conuncial one degree and there is from it to the other line of demarcation that is to the east seventeen and a half degrees.

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Finally, with this third indication we come to cover the pending distance to get to the demarcation antimeridian, which tells us to be 17.5 degrees East of Malacca. So, we have that this is equivalent to:

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17.5 degrees x 17.58 leagues per degree x 5.5 km = 1,692 km

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That we must measure on the equator to locate the antimeridian.

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If we advance 17.5 degrees or 1,692 km towards the West from the virtual position of Malacca, measured on the equator, we can reach the Antimeridian on the Portuguese side according to the description of the Memorial.

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+ Item the islands of Maluco are five, it is convenient to know, the three that are closest to the second line of the demarcation that are all north and south at two and a half degrees of longitude and the island in the middle is below the conuncial.

+ item the other two islands are in the way of the first two. which is north south and four degrees east of the second line, it is convenient to know, two north of the equinoctial and two south of the equinoctial, settled by the Portuguese pilots who discovered them.

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Finally, the distance at which Magellan located the Moluccas from the Antimeridian is 4 degrees to the East, and therefore within the Castilian hemisphere.

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These four degrees, transformed into kilometers, correspond to 387 km.

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+ and this membership that I give to VA very well to keep, that time may come that is necessary and will excuse differences, and this I say with a healthy conscience, not having respect for anything other than to tell the truth.

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With all the above, we have theoretically obtained the location for Magellan of the antimeridian demarcation, both for the Portuguese and for the Castilian hemisphere, obtaining that between them there is a stretch of the Earth of width 5,253 km, with which Magellan did not have.

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This value also confirms the validity of all the exposed calculations, since it coincides exactly with the 5,178 km of difference existing up to the 40,000 km considered of circumference at the equator, also adding the extra 75 km that today we know that the equator has as a result of the flattening at the poles, and the 34,822 km found in the first point of this exhibition.

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It could be interpreted that the weak point of this study is the correlation that we have considered valid of 5.5 km per league. There is no strict way to establish it, but we can gauge its validity: we previously found that, with this correlation, the error in the position of the Cape of Good Hope measured from the demarcation meridian was only 200 km (a positive 2%, put that Magellan located it 200 km further), while the error between the cape and Malacca was 600 km, negative for falling short (-6%).

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If we made the calculation of the actual km per league between the cape and Malacca, we would have a higher value, 5.84 km per league, with which the error in the dimension of the Earth would be even smaller. On the other hand, if we start from the error in the location of the Cape of Good Hope, this relationship would be 5.49 km per league. In any case, these are checks that help us to confirm that the value of 5.5 km per league is valid.

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As support, access the map created in Google Maps.

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The position of the antimeridian of the Treaty of Tordesillas turns out to be located in two different places depending on whether we find it from the demarcation meridian towards the Castilian or Portuguese side, being between both a strip of the Earth equivalent to the error in the dimension of the world , of 5,253 km measured on the equator, or 13% with respect to its complete circumference.

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ABOUT THE DISTANCES IN LEGUAS

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When trying to find out what was the size of the world considered in the Memorial, we have two factors that determine it: the error in the calculation of the size of the sphere contained in its dimensioning in leagues, and on the other hand, the transformation from leagues to kilometers.

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If one repeats all the operations carried out here, attributing the value of 6,318 km to each league, the demarcation meridian will be located a little further west, and the antimeridian will coincide exactly at 20,000 km, just on the other side of the equator, and therefore at 180 degrees of longitude at the opposite end of the sphere. There would be no error in the size of the sphere, more than the small one due to its flatness. However, the positions described in the Memorial for the Cape of Good Hope and Malacca would be meaningless, because they would have very bulky errors.

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This is due to the fact that the distances in that smallest imaginary balloon that is handled in the Memorial are the same as in the real-size balloon, that is, the distance from one point to another is what it is and not another. Therefore, if one uses the concept that the terrestrial sphere is smaller, the distance between two points covers more degrees of longitude than it should.

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If we use the factor of 5.5 km per league, we have seen that the position of the Cape of Good Hope and Malacca is very close to the real one, measuring on a sphere also of real size, like the one offered to us in Google Maps. In addition, since with this value the two theoretical positions of the antimeridian would fit, between which the proper distance of the error in the dimension of the world would be included, we can say with greater conviction that this is the correct value.

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If we used a value of km per league between 5.5 and 6,318 maximum (if we go beyond this value we would speak of a larger Earth than it is) the positions of the Cape of Good Hope and Malacca would begin to move too far. what corresponds to them, and we would be assuming that the error in the previous dimensioning of the sphere was less. Therefore, the distances measured in degrees should be reduced with respect to those indicated in the Memorial.

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For all these reasons, only with leagues of 5.5 km does this world described in the Memorial block.

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CONCLUSIONS

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Thanks to this Memorial, the antimeridian of the Treaty of Tordesillas can be found from the demarcation meridian on both the east and west sides, yielding two theoretical positions, among which the 5,253 km wide section of the Earth will be included. that Magellan did not expect to find.

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The antimeridian found by the Portuguese hemisphere would be located cutting the island of Borneo while, if it is found by the Castilian hemisphere, it would pass between Australia and New Zealand.

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As we have found, for the author of this Memorial the sphere of the Earth had a circumference length at the equator, or at any meridian, of 34,822 km.

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The error that this supposes with respect to the actual 40,075 km is 5,253 km. In percentage, the size of the Earth obtained is 13% smaller than the real one.

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For all this, it is also demonstrated why Magellan took more than 10,000 km to reach the latitude of the Moluccas when he sailed through the Pacific, although it ended up being even greater than he had expected.

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Thus, the quote that the chronicler Antonio de Herrera put in the mouth of the pilot Esteban Gómez, who, when consulted by Magellan about what decision to make when they began to enter the Strait, claimed that, with supplies for three months, they might not be enough to reach the Moluccas because there was a great gulf to pass, and if it took them a few days of calm or storms they would all perish .

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Another conclusion that we could draw from all this is that, when considering a smaller Earth, when these pioneering navigators traveled to the Moluccas through the Portuguese hemisphere, they found that they were beyond the Antimeridian and therefore in the Castilian hemisphere, as stated by Magellan, while when navigating in the opposite direction, as they did in this expedition, the same thing happened to them, that is, they also believed that they had crossed the antimeridian before reaching the Moluccas. That strip of the "excess" Earth of more than 5,000 km in which the Moluccas remained was no-man's land under a conception of the Earth smaller than its real size.

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This opens up another series of questions: did Elcano realize that the Earth was larger, and that is why he ended up affirming to Carlos V that the Moluccas were in the Castilian hemisphere? Did he choose the path back through the Portuguese hemisphere conscious of that the geographical longitude of the Moluccas brought him closer to Spain by navigating that side of the world? Does his anticipated turn to the Northwest, believing he had exceeded the Cape of Good Hope, gives us an indication that it was then that he could really realize that the Earth Was he older than they had previously believed? This is certainly an exciting topic, but I think that for the moment we have already gone a long way with our deductions, and we cannot solve these questions without going too much into the speculative field.

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Under a smaller conception of the Earth, the Moluccas were beyond the antimeridian demarcation of the Treaty of Tordesillas, whether one traveled to them from the Portuguese hemisphere or from the Castilian one.

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A LAST HYPOTHESIS

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After discussing these results with two experts and friends, Braulio Vázquez Campos , from the General Archive of the Indies, and Luis Robles Macías , author of the blog History and Maps , both gave me new ideas to continue investigating and deepening the issue.

Among other things, Braulio spoke to me about the allocation of 5.5 km to each league: why did Gonzalo Gómez de Espinosa say in his letter to the King that Darien, Panama, was only 1,800 leagues from Maluco, or 2,000 leagues according to the letter from the pilots Juan Bautista and León Pancaldo? It would only be a valid statement with a greater km per league allocation, but that would make some data from the Memorial not add up, and could lead us to the conclusion that the Earth was considered even larger, something more than doubtful.

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This island, sir, I left on the sixth day of the month of April in the year 1522, and we made our way to go to demand the mainland where Andrés Niño made the caravels, which is in the South Sea; where, sir, I found that from Maluco to the first land there were but one thousand and eight hundred leagues. (Gonzalo Gómez de Espinosa, AGI, INDIFFERENT, 1528, N.2)

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We left the said island of Tidori on the sixth day of the month of April in the year 1522, and we made our way to demand the mainland of the southern part, where sir, your honor will know that from the said mainland to Maluco is no further than two thousand leagues. (Juan Bautista de Punzorol and León Pancaldo, As Gavetas da Torre do Tombo, XV, 10-34)

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For his part, Luis told me about the world map known as Kunstmann IV , the facsimile that is preserved of the lost map that is believed to be the work of Jorge and Pedro Reinel, from the year 1519, and that some experts affirm that it is the one that Magellan commissioned before his departure, as even indicated in the card of this map of the National Geographic Institute of Spain, and that for this reason it could perhaps capture all that is collected in the Memorial that we have just analyzed.

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Initially, I did not link both information with each other, but after thinking about it, I realized that perhaps our expedition members knew this map, or carried a similar one from among those that Nuño García de Toreno prepared for them at the Casa de Contratación before their They were based on it to calculate the distance between the Maluco and the Darien. After studying it and doing the proper geometric checks, that is exactly what I now believe.

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On the drawing, the line between the Cape of Good Hope and Malacca must be curved because it is the great circle distance. However, the difference in length between this and the straight line we have calculated to be only 1%, which leads us to consider it a negligible error.

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As it also seems quite clear, the distance of 2,000 leagues between the Maluco and the Darien mentioned by Juan Bautista and León Pancaldo would correspond to that existing up to the ism of Panama itself. Furthermore, Espinosa was very exact, because he had always asked me what he meant when he said that "to the first land." Now I have an answer: to the first land reflected on this map.

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Kustmann IV World Map attributed to Jorge and Pedro Reinel, from the year 1519, and which could be a map commissioned by Magellan before leaving for the Spice. If we make the pertinent verifications, we verify that the distance between the Maluco and the Darién corresponds to the 1,800 leagues that Captain Gonzalo Gómez de Espinosa estimated that he had to travel from the Moluccas to reach the Darién.

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PS One last note that I find fascinating: rereading the chronicle of López de Castanheda (year 1554) I have found this quote, which I had not considered important until now. When referring to the capture of the nao Trinidad by the Portuguese Antonio de Brito, Castanheda says:

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... and on the ship were found books by the astrologer San Martín, who was with Fernando de Magallanes and died on the trip, and also two planispheres of Fernando de Magallanes made by Pedro Reynel.

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