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Juan Sebastián Elcano

Juan Sebastián Elcano (Elkano in modern Basque; sometimes given as del Cano; 1486/1487 – 4 August 1526) was a Spanish navigator, ship-owner and explorer of Basque origin from Getaria, part of the Crown of Castile when he was born, best known for having completed the first circumnavigation of the Earth in the Spanish ship Victoria on the Magellan expedition to the Spice Islands. He received recognition for his achievement by Charles I of Spain with a coat of arms bearing a globe and the Latin motto Primus circumdedisti me (You were the first to circumnavigate me).

Despite his achievements, information on Elcano is scarce and he is the subject of great historiographical controversy, because of the scarcity of original sources which illuminate his private life and personality. Even in Spain, for example, the first biographies about him were written in the second half of the 19th century, after three centuries of neglect by historians.

Following his success, the king entrusted him with another large expedition to the Spice Islands, headed by the nobleman García Jofre de Loaisa, which was not completed. Elcano died of scurvy in the Pacific Ocean during this venture.

Name
Elcano's name has been written in different ways by historians. Although today in the Basque language the form "Elkano" is used, his signature seems to read Delcano, or possibly Del Cano, although it is difficult to be sure: Enrique Santamaría shows that his lack of dexterity when signing his name leads him to leave gaps between the letters, so sometimes he signed "del ca no". In Spanish historiography it most commonly has been interpreted as "Del Cano" (or de Cano), but also as simply "Cano".

However, near Getaria (today between Zarautz and Aia) is the neighborhood of Elkano, where the surname is "Elkano" or "Elcano". For this reason, "Elcano" has been used as a surname in Spanish and "Elkano" in Basque to indicate that it was from the Elkano lineage.

Mitxelena, in his book Apellidos vascos, interprets the surname Elkano from Basque. Between the suffixes -ano and -no, he provides arguments for the latter, the diminutive suffix. He proposes that the first part of the surname is elge. Mitxelena reconstructs the previous form of this elge, and argues that the surname Elkano developed from the association of these two morphemes and that it is also a toponym. It exists in different areas of the Basque Country as a minor place name, but also as the name of a village in Navarre.

Family
Juan Sebastián Elcano's mother was Catalina del Puerto or Catalina Portu, and his father was Domingo Sebastián Elcano. The Portu or Puerto were a powerful family of clergy and scribes. Elcano's grandmother, Catalina's mother, was Domenja Olazabal. Formerly it was believed that she belonged to a noble family of the Tolosa area. It is known that Catalina, who was alive at the time of Elcano's death, asked the king upon her arrival in Getaria 10 years later for the pension that had been promised to Elcano, legally due her according to his will.

Domingo Sebastián Elcano and Catalina Portu had eight children, their first son being born in 1481, and Juan Sebastián being the fourth born son. Next came Domingo Elcano, who was given the name of his father and became a priest in Getaria. The four other sons in the family were Martín Pérez, Antón Martín, Juan Martín, and Ochoa Martín. Martín Pérez, Antón Martín and Ochoa Martín were also mariners, and sailed with Juan Sebastián Elcano in the second expedition to the Moluccas. The daughters, i.e., Juan Sebastián's sisters, were Sebastiana de Elcano and Inés de Elcano. Apparently Elcano also had a half-sister, María, Domingo's illegitimate daughter.

Elcano had two children: a son, Domingo Elcano, with María Hernández de Hernialde in Getaria, and a daughter, María, with a woman called María Bidaurreta in Valladolid. In his will, he bequeathed 100 ducats to his son's mother, María Hernández de Hernialde. To his daughter he left 40 ducats, conditional on her coming to live in Getaria before she was 4 years old.

Family social position and economic status
A number of historians have said that the Elcanos were a family of maritime transporters who operated in the Mediterranean. Their ownership of a vessel is suggested by the amount of taxes the family paid the Crown in 1500.

Some 19th century sources say that the Elcano family belonged to the nobility, but this is questionable. Elcano asked the king for the right to bear arms, a privilege of the nobles, and for his brothers to accompany him on his next expedition, indicating that the Elcano brothers, on their father's side at least, were neither nobles nor hidalgos, and that the Portus, on their mother's side, were probably not nobles either. However, the Olazabal family on his grandmother's side may have been hidalgos, but since nobility was not inherited in the matrilineal line, the Portus, through Catalina the daughter of Domenja, would not be hidalgos nor would be the Elcanos, the grandchildren of Domenja.

Fernandez de Navarrete states that, in addition to being a fisherman and a mariner, Elcano acted as a smuggler on board a French ship, but no original sources are given to confirm this. On the other hand, some biographies point out that the Elcano family experienced economic hardship because Elcano's father died young and his mother had to support eight siblings, but this assertion is unfounded.

As attested in documentation, in 1500, when Elcano was about 14 years old, a large tax was levied in Getaria, with his father Domingo Elcano appearing in the thirteenth position on the tax rolls, having paid 23½ maravedís. It seems, therefore, that they were quite solvent, since they are named among the richest families of the town. Additionally, the royal document later granting him a pardon states that when Elcano was young and acting as a merchant in the Mediterranean, he was the owner of a 200-ton vessel. This would also indicate that his family had been financially solvent.

Elcano got rich from the circumnavigation voyage, earning 613,250 maravedis. This was an immense fortune, an amount equivalent to the salary of a sea pilot for 20 years, as compared to the 23½ maravedis paid by his father in municipal taxes. Of that fortune, 104,526 maravedis was his shipmaster's and captain's wage, while the rest was earned by selling the cloves imported from the Moluccas.

Birth
The date of Elcano's birth is unknown, but it can be inferred with a fair amount of certainty that his year of birth was 1486 or 1487. Spanish historiographers have written that he was 42 years old when he sailed with Magellan in 1518, which would place his birth year in 1476. However, before setting sail Elcano himself had confirmed that he was "approximately" 32 years old, as recorded in a document of August 1519; therefore, it is reasonable to believe that he was born in 1486 or 1487.

There is little doubt about Elcano's place of birth because in his will, drawn by Elcano himself, he mentions that his birthplace is Getaria. It is usually said that he was born in the house located in San Roque Street in the municipality of Getaria, today called the "Birthplace of Juan Sebastián Elcano". There is a plaque commemorating the event next to the house. Even if he was born there, the house did not belong to the Elcano family, but to his maternal grandfather's family, the Portus.

In the chronicles of the time, Elcano is also presented as a "Getarian". In 1601, the chronicler Juan de Mariana, after writing that Elcano was from Getaria, adds: "from Biscay by nationality or Guipuscoan". At that time the Basques were called 'Vizcaínos' or 'Biscayans'. It has been deduced from indirect information that the Basque language was his mother tongue, but it is unequivocable that he also communicated in Spanish, as can be seen in the letters he wrote to the king, and in the interrogations he underwent in Seville and in Valladolid. It appears that he could read Latin as well, because the two books referred to in his will were written in that language.

Early life
Little is known about Elcano's youth. As a young man he owned a large ship and sailed in the Mediterranean. He sold the ship to the Savoyards to resolve legal problems he had incurred because the king had not paid him a salary for 'services rendered' on the kings's orders. It is often repeated, for example, that in 1509 he participated in the conquest of Oran, in the Mediterranean, under the direction of Cardinal Francisco Jiménez de Cisneros, commanding his own his two-hundred-ton ship. However, in the register of ships participating in the conquest of Oran, no captain Elcano, or anything similar, is recorded. According to Spanish historiography, Elcano participated with his ship in the Italian wars under the leadership of Gonzalo Fernandez de Cordova, known as The Great Captain (El Gran Capitán) (1495–1504). This is also doubtful because there are no primary sources to prove it, and, above all because Elcano was then 8 years old (17 according to the Spanish historiography), and it is impossible that an 8-year-old boy was already a ship owner or fought in the war.

The certificate of pardon that the king issued to Elcano has been preserved. It mentions that he acted in the service of the king "in the Levante and Africa", but these actions are not further specified. It also states that since the king did not pay him the promised salaries, he had been forced to sell his ship to the House of Savoy. Presumably when he had asked for money from the merchants of Savoy he had to offer them the ship as a guarantee. Therefore, in the case of his participating in a military campaign in the Mediterranean 'in the service of the king', perhaps the events of the catastrophe of Gelves (1510) was a factor, since that military defeat would explain Elcano's debt. Forced to pay for the ship and the crew's wages, but with no war prizes, this could have been the cause of his getting into debt. At the age of twenty-three, unable to pay off his financial obligation and still on board the ship in Italy, he had to hand it over to the Savoyards to satisfy the debt. Another hypothesis is that Elcano performed the 'royal services' in the autumn of 1516, during the battle to take Algiers, also a military defeat, and that he then had to relinquish his ship.

It is known that in the summer of 1515 Elcano joined the local militia. The Royal Corregidor asked 500 Gipuzkoans to go to Hondarribia and San Sebastian to face the threat of the French. Elcano went there, along with 11 compatriots, charging 30 maravedís per day. It would appear that if he had to enroll in the local militia in 1515 and he did not own a ship at the time, his engaging in the battle to take Algiers is unlikely. Being in his twenties during the 1510s, it is unusual that Elcano already owned a ship of 200 tons, and for such a young man to have this responsibility, but it is clear that Elcano was a precocious sailor who rose very quickly as a professional mariner. Nonetheless, he committed a serious infraction by selling his ship to the Savoyards, an illegal action at a time when wars were fought with ships. This infringement would cause him numerous problems in the following years.

After returning from the Mediterranean campaign, as he had legal problems it is likely that he remained in the Mediterranean, in Catalonia or in the Valencian Country, or perhaps also in Alicante. (In his will he left 24 ducats to the Alicante church of Santa Veronica.) However, at the end of June 1517 he appeared again in Getaria, as is known since he signed as a witness to the debt letter of a compatriot. It is possible his son with the young Getarian María Hernández Hernialde was conceived at that time, between 1517 and 1518, although they were not married, probably because Elcano did not have a stable residence. At the end of 1518 Elcano left Getaria and traveled to Seville to join Magellan's expedition. However, after returning from the expedition, at the age of 35 or 36, the king would forgive him that legal debt on February 13, 1523, and he was able to stabilize his situation in Valladolid.

Voyage of circumnavigation
The Magellan-Elcano expedition (1519 - 1522), which completed the first circumnavigation of the Earth, started with 5 ships (Trinidad, San Antonio, Concepción, Victoria and Santiago) along with 234 crewmen (some sources raise the number of sailors to 247.) Although Elcano had departed Spain aboard the Concepción, the return voyage from the Moluccas to Seville was made by the single surviving ship, Victoria, captained by Elcano. Having sailed under extreme conditions, only 21 people arrived in Seville, the 18 Europeans and 3 Moluccans.

To complete the round-the-world voyage, they had to sail 69,918 km After three years of rough passages, most of the sailors had died. A few made it back alive, but 147 men lost their lives, three-fifths of those who had set sail from Seville. This figure is a reflection of the number of unforeseen events, difficulties and vicissitudes they encountered during the voyage. Fifty-five of those who sailed on the return voyage were deserters coming from South America on the first part of the voyage aboard the San Antonio. They had not circumnavigated the world because they decided to retreat to the Strait of Magellan. Others who returned would spend some time in Asia or Cape Verde, although they later managed to get to Europe; thus they also could claim, once they disembarked in Spain, that they had sailed around the world. In other words, in addition to the 17 sailors who arrived in Seville along with Elcano, more sailors would sail around the world, albeit later, and not in the same vessels.

In addition to Elcano, the expedition included 34 other Basques, the largest representation after the Andalusians, along with 28 Portuguese, 19 Genoese and 21 Castilians. Nine of the Basques were with Elcano aboard the Concepción – Elcano and Juan de Acurio, Antonio Bermejo's boatswain, wanted trustworthy people around them.

Spice Island
Elcano participated in a fierce mutiny against Ferdinand Magellan before the convoy discovered the Strait of Magellan, the passage between mainland South America and Tierra del Fuego. His life was spared by Magellan and after five months of hard labour in chains he was made captain of the galleon. Santiago was later destroyed in a storm. The fleet sailed across the Atlantic Ocean to the eastern coast of Brazil and into what is now Puerto San Julián in Argentina. Several months later they discovered a passage now known as the Strait of Magellan located in the southern tip of South America and sailed through the strait. The crew of San Antonio mutinied and returned to Spain. On 28 November 1520, three ships set sail for the Pacific Ocean and about 19 men died before they reached Guam on 6 March 1521. Conflicts with the nearby island of Rota prevented Magellan and Elcano from resupplying their ships with food and water. They eventually gathered enough supplies and continued their journey to the Philippines and remained there for several weeks. Close relationships developed between the islanders and the Spaniards, who began to evangelize and convert the Cebuano tribes to Christianity; they also became involved in tribal warfare between rival Filipino groups on Mactan Island.

On 27 April 1521, Magellan was killed and the Spaniards defeated by natives in the Battle of Mactan in the Philippines. The surviving members of the expedition could not decide who should succeed Magellan. The men finally chose a joint command with the leadership divided between Duarte Barbosa and João Serrão. Within four days these two were also dead, killed after being betrayed by the host at a feast given by Rajah Humabon. The mission was now teetering on the edge of disaster, and João Lopes de Carvalho took command of the fleet, leading it on a meandering journey through the Philippine archipelago.

During the six-month listless journey after Magellan died, and before reaching the Moluccas, Elcano's stature grew as the men became disillusioned with the weak leadership of Carvalho. The two ships, Victoria and Trinidad, finally reached their destination, the Moluccas, on 6 November. The men rested and reprovisioned the ships in this haven, then filled their holds with the precious cargo of cloves and nutmeg. On 18 December, the ships were ready to leave, but Trinidad sprang a leak, and could not be repaired. Carvalho stayed with the ship along with 52 others hoping to return later.

Victoria, commanded by Elcano along with 17 other European survivors of the 240-man expedition and 4 survivors out of 13 of the Timorese men, continued its westward voyage to Spain traversing the Indian and Atlantic Oceans. They reached Sanlúcar de Barrameda on 6 September 1522.

Antonio Pigafetta, an Italian scholar, was a crew member of the Magellan and Elcano expedition, and later wrote several documents concerning its events. According to Pigafetta the voyage covered 14,460 leagues – about 81,449 km.

Elcano under Magellan's leadership
Ferdinand Magellan was the expedition leader, its captain-general (capitán-general). Being Portuguese, he had traveled in his youth through South Asia with the Portuguese army, getting to know those islands, finding safe harbors and places to stay and establishing his mastery of the maritime routes for trade. As a result of those experiences, Magellan knew the exact location of the Moluccas islands, then called the "Spice Islands", or at least he made King Charles V believe this. He claimed, wrongly, that they were in the Castilian hemisphere, following the Treaty of Tordesillas. He was appointed captain-general because he had this information, and this is also the reason he was responsible for planning the expedition's route to the East Indies. The relationship between Magellan and Elcano quickly became strained, precisely because Magellan did not want to show anyone the route and did not want to reveal exactly where the Moluccas were. Magellan condemned Elcano in San Julian (Patagonia) for his participation in the San Julian revolt.

Elcano confessed his part in the mutinies against Magellan during his interrogation at Valladolid after the circumnavigation, adding that he had not written anything while Magellan was alive because he feared him. Magellan had been killed in the Philippines on April 27, 1521, by warriors from the island of Mactan.

Although the objective set by the king was to open the route to the Spice Islands, when they reached Asia, Magellan began to pursue his personal objectives as a captain. The king had promised to make him governor of those islands in the Castilian hemisphere and that he should enjoy commercial rights to the trade of the two main islands. This is probably why the expedition did not sail directly to the Moluccas islands to acquire spices, but further north. They also traveled widely in the region, getting involved in the internal conflicts of the resident peoples of the various islands.

Voyage after Magellan's death
After Magellan's death, the Portuguese Duarte Barbosa, a relative of Magellan, was appointed captain-general, but he was also killed in Cebu together with Captain João Serrão of the Trinidad, in an ambush at a dinner organized by the Hindu leader of the island, the rajah called Humabon. In that ambush on May 1 in Mactan, about 35 sailors lost their lives. In this situation, on May 2, 1521, it was decided to burn the ship Concepción because there were not enough sailors, only 116 or 117, to crew three ships. Thus the expedition was reduced from five to two vessels.

Now the expedition had only two vessels to return to Seville, Victoria and Trinidad. However, Elcano was not immediately appointed captain. First another Portuguese, Juan Lopez de Carvalho, was appointed in May 1521. Disagreeing with Carvalho's manner of command, the sailors dismissed Carvalho and elected Elcano as captain of the ship on September 17, 1521. Clarifying what happened between May and September is complicated because there are several versions of the events. How and why the unusual decision was made to remove Carvalho and place Elcano in command has not yet been determined, but it is certain that Elcano was chosen as captain by the crew to replace Carvalho.

After arriving in the Moluccas and loading the cargo of cloves, once in South Asia Captain Elcano changed his original plan. He proposed to continue westward returning to Europe around the Cape of Good Hope in Africa, without turning back or rounding Cape Horn in South America. This change of plans would culminate with the first round-the-world voyage.

From Tidore to Cape Verde
The expedition finally reached Tidore in the Molucca Islands, in what is now Indonesia. There they found the precious spice they were looking for, i.e., cloves. They made an agreement with the local rajah, whom they called the Almansur of Tidore Island, and who brought them tons of cloves. (The Victoria returned to Seville with 27 tons of cloves.) As there were not so many cloves on Tidore Island, the rajah brought some from the neighboring islands as well. In the meantime, the Trinidad was disabled. Having heard of the approach of the Portuguese, and with the danger of tarrying, they decided to return alone on the Victoria, with Elcano as captain. They had specific orders from the Crown to return by the route they had gone, but they did not comply with them, and took the westward course with the aim of sailing around the world. Elcano proposed to do this because, as he wrote, "They were going to do what could be narrated", indicating that he was aware of the historical import of their voyage, as is clear in the letter he wrote to the newly enthroned king:

"Your Majesty will know that we should have the highest esteem for having discovered and encircled the roundness of the world, for we have gone to the West and returned by the East." Elcano allowed the sailors to choose their ship, since they were to circumnavigate in waters belonging to Portugal. Forty seven sailors chose to return with Elcano aboard the Victoria, and 13 members decided to stay in the Moluccas. At that time there were twelve Basques left in the expedition, of whom eight decided to return with Elcano, the other three remaining aboard the Trinidad. The ship Victoria left Tidore Island on December 21, 1521, bound for Seville. They sailed into a strong storm which damaged the ship. On the nearby island of Mallua (today called Pulau Wetar) they had to remain fifteen days for repairs. From Tidore they sailed to the island of Timor, and after spending a few days there, they set sail on February 7, 1522. From that day until July 9, when they reached Cape Verde, they would not set foot on land again. The expedition planned to sail from Timor to Seville, a distance of 27,000 km kilometers, without a stopover, to avoid meeting the Portuguese on the way. They did not succeed, because after sailing over 20,000 km, their supplies, health and the condition of Victoria had become impossible; the sailors decided by vote to stop and recover at Cape Verde.

From Timor, the Victoria traveled west, on a line around 10° 32' South – probably within 500 km of the Australian mainland and passing close to Christmas Island and the Cocos Islands – before heading southwest, to avoid Portuguese India, and crossing the equator into the southern hemisphere. By going so far south, they also managed to avoid the opposing monsoon winds blowing from Africa that time of year. On March 18, the crew sighted an island that Elcano named Desesperanza ("Despair"; later known as Ile Amsterdam), because he was unable to find a safe place to land, and his crew was desperate for fresh water, 40 days out from Timor.

While they were still in the Indian Ocean, the crew of Victoria began to run out of food. They had only rice cooked in seawater left to eat, and scurvy began to make the sailors seriously ill. Under these circumstances, the idea of making landfall in Mozambique spread on the ship. It was dangerous to land there, however, where the Portuguese might capture them. Elcano consulted the crew, who decided by vote to continue sailing, without stopping in Mozambique.

Rounding the perilous Cape of Good Hope at the southern tip of Africa was very difficult for them. The Portuguese called it the "Cape of Storms". At first they headed south to take advantage of the wind, but the ship could not continue because of bad weather, usual in those parts. For nine weeks they stayed there, freezing, with their sails lowered. At last Elcano made the sailors a dangerous proposition: to round the cape close to the coast. On the one hand there was the danger that the storms would drive them into the shore, and on the other, if they avoided that fate, that they might encounter the Portuguese on the eastern side of the cape. They pressed on, sailing westward in the Indian Ocean, close to the African coast, and at last managed to round the cape and head northward off the continent's western coast.

Landing in Cape Verde
The expedition could not sail directly to Seville because the situation on board was unsustainable, and they needed to land somewhere. In desperation, and in search of food, they first reconnoitered the African coast off what is now Guinea Bissau and Senegal, but could not find a suitable place to land. They decided by vote to call at Cape Verde where they could obtain food and repair the ship, although the Portuguese were in charge in Cape Verde. This was the final leg of the voyage, and Elcano's men had not made landfall for months, suffering two or three deaths a week. Seawater was entering the vessel's wooden hull, due to naval shipworms tunneling into and damaging it, and as the crew were weak from hunger, they did not have the strength to pump the water out of the bilges, and there were fewer and fewer of them left alive.

To make matters worse, they had no money and no goods except cloves to exchange for provisions. Acknowledging that they had cloves would reveal that the expedition was returning from Asia rather than the Americas, and the Portuguese would come out against them. By unknown means they were able to pay for the first two loads of food, but they used part of their cargo of cloves to pay for the third and thus revealed their identity to the Portuguese. (This assumption has been dismissed by scholars like Enrique Santamaría, who argues that such "errors" were impossible). Consequently, they were pursued by the Portuguese and forced to flee, leaving 13 sailors prisoners of the Portuguese in Cape Verde.

It is not entirely clear whether their third risky purchase at Cape Verde was to obtain provisions or slaves. Most of the original accounts, including Elcano's, speak only of food, but Bustamante, the doctor-barber on board, says that they went down to look for slaves because of the imperative need for labor to pump out the water filling the bilges. This is a possibility, as the African slave trade was common and legal in Portugal.

Arrival
The ship didn't make the passage from Cape Verde to Seville in a straight line because the wind was driving them off course. They turned around by 'Volta do mar largo', taking a wide course to the west, sailing almost as far northward as Galicia and from there southward to return to Seville. They landed at Sanlúcar de Barrameda on September 6, 1522, and two days later, they entered Seville on September 8, after almost three years of voyaging. Of the 234 (or 247) sailors who set sail, only 18 arrived.

As soon as the ship Victoria arrived in Sanlúcar de Barrameda, Elcano began writing a 700-word letter addressed to Charles V, in which he never mentions himself, emphasising that the expedition had achieved its goal of bringing back the desired spices, had "brought peace" to those islands, and had obtained the friendship of their kings and lords, attested by their signatures on documents. He went on to highlight the extreme hardships undergone during the expedition. Elcano did not forget the members of the crew captured in Cabo Verde by the Portuguese, begging the king to initiate all necessary actions leading to their release. He ends the letter with commentary about their discoveries, the roundness of the world, setting sail to the west and coming back from the east.

Court of Valladolid
Once the voyage was over, upon arriving in Seville, Elcano and a few selected men took the road to Valladolid, which at the time was the residence of Charles V and his court. The king wanted Elcano to personally tell him about the expedition. In his letter of invitation, the king offered him horses to make the trip, although the road from Seville to Valladolid was traveled more often by carriage than on horseback. The invitation was intended only for Elcano and two others, 'Those who have better sense', but Elcano brought more crew members with him. The little group arrived in Valladolid, including the three sailors from the Moluccas who wanted to meet the king.

King Charles V soon received Elcano, at the latest one month after the circumnavigation. Elcano appeared at the court in Valladolid, and spoke in the presence of the king, giving his account of the voyage, possibly in three conversations: first with the king, perhaps in private; then with the court experts, to clarify technical and financial matters and also to describe the events of the voyage, including the mutiny and deaths that occurred; and finally, with a group of humanist learned men more interested in the various cultures that the expedition encountered. It is not known exactly how these meetings went. He would spend three years in Valladolid, near the Court. There he joined María Bidaurreta (in some sources De Vida Urreta) and had another daughter.

Charles V granted Elcano an augmentation of his coat of arms featuring a world globe with the words Primus circumdedisti me (Latin: "You first encircled me"). However, this augmentation has been contested, as Elcano was not previously a member of the nobility. Elcano claimed three gifts from the king as a reward for his achievement, namely an annual pension of 500 ducats which he never received, two armed men to escort him and an official statement pardoning him for the sale of his ship to the Savoyard bankers. Following Elcano's death and after lengthy lawsuits, his mother Catalina del Puerto never managed to obtain any of his pension. By 1567, after she too had died, Elcano's heirs and other relatives continued to demand that the pension be honored. His male heirs were given the hereditary title of Marquis of Buglas, i.e., Negros Island in the Philippines. In the modern era, the country with the most people surnamed "Elcano" is currently the Philippines.

Relations with Portugal
The king of Portugal, John III, filed a piracy complaint against Elcano for theft of the cargo of cloves from the Moluccas, as they were part of his domains, and for his escape from Cape Verde with the precious cargo. He petitioned the King of Castile to return the cloves and to arrest and punish Elcano. King Charles V disregarded the request and protected Elcano.

Portugal and Castile both claimed the Moluccas Islands. The need to revise the Treaty of Tordesillas was not easy to resolve. Without knowing the exact size of the world, it was difficult to place those islands in a certain area. The key was to fix the location of the anti-meridian relative to the meridian established in the Treaty of Tordesillas, the two meridians forming a great circle that divides the earth into two hemispheres, to determine on which side of the antimeridian the Moluccas Islands were located. Meetings to resolve the matter were held in 1524 in the towns of Elvas and Badajoz along the border between Spain and Portugal. Five of the six Basques who circumnavigated the world participated in these assemblies.

To accurately establish this antimeridian, each of the Crowns assembled the best cosmographers of the time. Each delegation appointed three astronomers or cartographers, three sea pilots and three mathematicians. The Spanish delegation also appointed Fernando Colón, son of Christopher Columbus, with the objective of determining the location of the Moluccas. The negotiating team included Sebastian Cabot (Sebastián Caboto), Juan Vespucio, Diogo Ribeiro, Estêvão Gomezs, Simón Alcazaba and Diego López de Sigueiro.

In the Badajoz-Elvas assemblies, the most authoritative voice of the Castilian delegation was Elcano himself, who devoted himself to his work as a cosmographer. He brought to these meetings a sphere of the world he made himself, on which some accounts say he had marked the trajectory of the round-the-world voyage. These meetings were unsuccessful because the attendant parties were unable to agree on the exact location of the Moluccas Islands, as it could not be deduced from the ships' logs, and in any case the Portuguese did not recognize Elcano's authority. It is not known exactly why, but suddenly Elcano and the pilot Estêvão Gomes were absent from those meetings on March 15, 1524. Shortly thereafter, on May 20, 1524, the king pronounced his support of Elcano, as it was said that there were those who wanted to harm him (the royal charter speaks of "wounded, dead or crippled"), and three days later, on May 23, 1524, the assembly met again in Badajoz, but without Elcano. The conjecture is that he was threatened by the Portuguese in the case of the alleged piracy in the Moluccas. By October 1524 Elcano was in the Basque Country preparing for a second expedition to the Moluccas. Two years later, in 1529, John III of Portugal and Charles V of Castile signed the Treaty of Zaragoza, Castile recognizing the Moluccas as being on the Portuguese side of the antimeridian of the line of demarcation previously specified in the Treaty of Tordesillas.

Last expedition
In 1525, Elcano was back at sea as a member of the Loaísa expedition to the Moluccas with García Jofre de Loaísa, who had been appointed by Charles V as captain-general of a fleet composed of seven ships. Loaísa was in command of the carrack (Spanish: nao) Santa María de la Victoria, the fleet's flagship, while Elcano was named pilot-major of the expedition, in command of Sancti Spiritus, with his brother Martín Pérez serving as its pilot. They were sent by Charles V to claim the Indies in his name, a venture that was irksome to Portugal. The expedition, financed by German international banking and merchant families, the Fuggers and the Welsers, sailed from A Coruña in July 1525. It consisted of four carracks, two caravels and a patache, with 450 sailors to crew them. Its purpose was securing a military foothold in the Maluccas, and forming alliances with the native rulers to gain control of the local spice trade with the establishment of a base for the Spanish Crown's mercantile and colonial agents. Although Elcano had more experience, the king chose García Jofre de Loaísa as commander of the expedition because he was a nobleman, his status in line with the royal objective of establishing a Castilian authority in the Moluccas. Elcano was accompanied by his brothers Martín Pérez, Ochoa Martín and Antón Martín, his nephew Esteban Mutio, and his brother-in-law Santiago de Guevara, captain of Santiago. Elcano's influence was evident in the preparations for the expedition. Of the seven ships, four had been built in Basque territory and the number of natives of the Gipuzkoan coast in the crew had increased considerably, including those in positions of responsibility, compared to the first expedition. Many of them were close to Elcano. The expedition faced numerous misfortunes. Before reaching the strait, two vessels were lost because they couldn't find its entrance. The Sancti Spiritus, piloted by Elcano, ran aground in a storm and was abandoned, with its men being distributed among the other ships, and the San Gabriel deserted the expedition, returning to Castile after sailing up the Brazilian coast. The remaining ships completed their passage through the Strait of Magellan on 26 May 1526, and set out for the Pacific crossing. A storm scattered the ships on June 2 and only one, Victoria, remained. Another of the ships ended up in Zihuatanejo, Mexico.

Death
The ship continued to the Moluccas in poor condition, and the crew suffered the ravages of scurvy. The accountant Alonso de Tejada, the pilot Antonio Bermejo and thirty-two other crew members died. Finally, General Loaisa himself died on 30 July. Elcano then took command of the navigation, and was named captain-general, a position he had been promised by the king from the beginning. After a short time he died of scurvy, probably on 6 August 1526. All his relatives who accompanied him also died, except, perhaps, his brother Ochoa Martín, who may have been aboard the ship that landed at Zihuatanejo. In one of the documents that appeared later in Laurgain, dated January 29, 1529, the king calls "Johan Ochoa Martínez del Cano" to go on the expedition of Simón de Alcazaba to the Moluccas. If this referred to Elcano's brother, then he was still alive in 1529. Before dying, Juan Sebastián had made a will – one of those signing as a witness was Andrés de Urdaneta.

On August 7, Elcano's body was wrapped in a shroud and tied to a board with ropes. Afterwards, it was placed on the deck of the ship while the surviving men recited the "Lord's Prayer" and the Ave Maria (Hail Mary). When they finished, a weight was tied to the shroud, and Alonso de Salazar, the new captain general of the Army, nodded his head. The four sailors tilted the plank over the gunwale until the weight of the corpse caused it to drop into the sea.

Writings by Elcano
Two letters addressed to King Charles V and a will are the only documents in Elcano's own handwriting known to have survived to the present day. In the interrogations at Seville before the expedition and at Valladolid after its completion, the answers given by Elcano were also recorded in writing and have also survived. Elcano's voice can be heard in them.

While Magellan was alive Elcano wrote nothing because he feared his wrath, he addressed this point in the Valladolid interrogations. After Magellan's death, however, following Elcano's appointment as captain, he began to write down (as he himself said in the Valladolid investigation) what had happened and what he had seen. This chronicle of the first world circumnavigation, written by his own hand, has been lost. Along with Elcano's account, the other important written chronicles of the voyage, including Magellan's or Pigaffeta's originals, are also lost. In the case of Magellan and Elcano a hypothesis has been proposed that both their writings could be translated in the Latin chronicle of the royal secretary Maximilianus Transilvanus ,and that the main passages of the chronicle written by Elcano have survived in the contemporary chronicles rewritten by Maximiliano Transilvano and Gonzalo Fernández de Oviedo.

First Letter to Charles V
The first letter was written in Sanlúcar de Barrameda to let the king know that he has returned from his circumnavigation. The text is precise and correct, as merchants, unlike diplomats, usually write. He promises the king documents, letters of submission from the monarchs of the remote island and samples of spices to account for the success of the voyage. He also brings with him five parrots, which were appreciated at the Court. He explains in 10 lines what happened up to the death of Magellan and, subsequently, what happened in his charge in the other 40–45. It is at that point that he asks the king to make an effort to free the 13 who have been detained in Cape Verde. He asks for a financial reward from Charles V, but mentions that the feat was accomplished by collective decision.

Second Letter to Charles V
The purpose of the second letter was to ask for bonuses. Approximately 40 days after returning from the voyage around the world, Elcano arrived at the Court of Valladolid where the king received his request to name him captain-general. In it he also asked the king for the commercial rights of trade in the Moluccas or the habit of the Order of Santiago, which were entrusted by the king to Magellan. The king's secretary (Francisco de los Cobos) denied all of these requests. Apparently he was not accorded respect at the Court of Castile, and they did not want to give Elcano the orders that had been promised to Magellan. Santamaría argues that this animadversion was caused by Elcano's lack of noble status. It took him months to obtain the other three bonuses he had been promised – a pension, a pardon and the right to an armed guard – probably as a result of the rapprochement with the Court.

Elcano's will
The will is also a source of information about Elcano's personal life. In addition to citing all his claimed assets in great detail, it speaks of his family. It shows that although he never married, he had sexual relationships with at least two women, both of them Basque or of Basque origin, and each of whom bore his child. His son Domingo was born before the world circumnavigation and his daughter María afterwards.

The will was drawn up on July 26, 1526, and opened by the president of the Council of the Indies upon his arrival in Seville, ten years after it was made. All seven witnesses to the will were Basques, which is surprising, given the situation and the location in the middle of the Pacific, a clear indication of trust and solidarity among them. The witnesses were Martín García Karkizano, Andrés Gorostiaga, Hernando Gebara, Andrés Urdaneta, Juanes Zabala, Martín Uriarte and Andrés Aletxe.

In his will Elcano left money to churches such as San Salvador de Getaria, Itziar, Sasiola (Deba), Our Lady of Arantzazu, San Pelayo in Zarautz and Nuestra Señora de Guadalupe in Hondarribia. However, he stipulated that these bequests should be paid out of the money owed him by the king according to the terms of their agreement, funds which he never received. This leaving of money to the church only if he were paid by the King has been used by Santamaría to dismiss the idea of Elcano being a very religious man.

He also left portions of those funds to his heirs, but if they died, as happened, his mother would receive the benefit of his properties. He treated the four of them generously in the distribution of goods: one hundred ducats of gold to Domingo's mother (37,500 maravedís), and another 40 ducats (15,000 maravedís) to the daughter, if she married, as a dowry. On the other hand, he requested that his daughter be taken from Valladolid to Getaria when she turned 4 years old, because Getaria was closer to his heart than the Court of Valladolid.

Thanks to the will, it is known that Elcano owned two books written in Latin, indicating that he knew how to read that language. Both books refer to astronomy, one being the Almanac of Regiomontanus, which allowed navigators to determine longitude at sea with observations of the moon. These two books were left to Andres San Martin, his pilot cosmographer who disappeared on the circumnavigation, if he should be found alive. This suggests that in the second expedition Elcano was still hoping to find a companion lost in the first expedition.

In 1533, seven years after Elcano's death, his mother was still in court with the royal treasury seeking the salaries corresponding to the rank of captain due to her son and the other payments he never received, and his pension of 500 ducats per year. The will also shows that Elcano did not own slaves, otherwise he would have had to cite them.

Transilvanus' text
The humanist Maximilianus Transylvanus was present at the royal court, listening to Elcano. From what he heard he wrote his own chronicle of the voyage, in the form of a letter, which he sent to his patron, Matteo Lang von Wellenburg. The book was printed in Cologne in January 1523 and in November of that year another printing was made in Rome by Gian Matteo Giberti, assistant to Pope Clement VII. Giovanni Battista Ramusio years later assembled a collection of travels and introduced the text of Transilvanus. The text was written in Latin, for political purposes and according to the tastes of Chancellor Gattinara. It is possible that this text by Transilvano is a compilation of collected testimonies: the account of Magellan's part of the voyage is very different from that of Elcano's part, as if they had two different authors. Another chronicler of the time who read these two texts, Fernandez de Oviedo, alluded to the parity between the texts of Elcano and Transilvano: "It is almost him", he wrote.

If the second part of the Transylvanus text is a direct translation of Elcano's report, he appears as an exponent of the humanist and utopian intellectual movements then current in Europe, and the text, which says that the societies of the Moluccas were peaceful, treated their neighbors well and were hospitable to foreigners can be seen as one of the first explications of the myth of the noble savage and a critique of the corruption of European civilization: "All of them show great respect and care not to cause harm or discomfort to neighboring towns, even more so to neighbors from neighboring islands, and even more so to foreigners or pilgrims."

Basque philosopher Ekai Txapartegi defends the proposition that Transilvanus's text was written by Elcano, asserting that its description of the island of Borneo as a utopia and its depiction of the customs of the inhabitants reveals the humanist aspect of his political thought, as well as his opposition to the expansionist ambitions of empire and warlike kings, and the imposition of the Christian faith on peaceful pagan societies. Not much is said in the text about the events that transpired when the expedition was in Borneo, only that they talked to the local king, made some exchanges and moved on. What they were doing there is not explained in the passage, although it is known that Borneo was the scene of conflict for the travelers, indicating that the writer is speaking more of his personal utopian vision than of a literal description of Borneo.

Historiography
Elcano's achievement has been eclipsed in traditional historiography by that of Magellan, who planned and led the famous expedition until he died before it reached the Spice Islands. More recently, Portugal's solo candidacy to UNESCO to get Magellan's expedition and the resulting circumnavigation (without mentioning Elcano) recognised as a Portuguese Intangible World Heritage has provoked a major controversy with Spain, thereafter seemingly settled by the submission by said countries of a new joint application to honour the circumnavigation route. According to the Basque historian Xabier Alberdi Lonbide, Elcano was relegated to a secondary role in French and Spanish historiography of the 19th century and there remains a minor figure. Lonbide writes that the maritime heritage of the Basque Country generally has been consigned to oblivion in British, French, and Spanish historiography, and that Basque historiography has not managed to overcome this situation, although in the Basque Country Elcano is regarded as the most universal representative of Basque culture, and he has greater stature.

Elcano, deleted from History
Throughout the world Elcano has been a marginal character because he was almost forgotten for three centuries. In the first accounts of the voyage, the cold reception that the successful circumnavigation of the world got at the Court of Castile is noticeable; in the long seventeen-page account written by Peter Martyr d'Anghiera, for example, Elcano was not mentioned even once. The reason for this neglect is perhaps that those who sailed around the world were not hidalgos, but common sailors, a fact that contravened late medieval social attitudes among the upper classes.

Nor did the book, Relazione del primo viaggio intorno al mondo, by Antonio Pigafetta, the Venetian scholar who sailed with Magellan and who was among the 18 men that survived the expedition and returned to Spain, mention Elcano. In its received form, his chronicle exaggerates his own role, perhaps explaining why in Italian historiography Elcano disappears and Capitano Pigafetta himself appears as an important character.

In recent years, however, another point of view has spread: Pigafetta quoted Elcano, but as the works of Pigafetta that have survived to modern times were modified in transmission, perhaps mentions of Elcano were suppressed. The original Pigafetta chronicle is lost, and the account in its present form would be a retranslation of a translation made in France, written later, and suppressing or modifying many sections. The hypothesis is that, since France was at war with Castile, the French eliminated all the "Spaniards" that Pigafetta had praised in his telling of the voyage. This 'eliminatory' view was further extended when it was written by Philip II's chronicler, embellishing the roles of the hidalgos Magellan and Pigafetta and minimising that of the plebeians. Spanish historiography transformed Magellan into a leader who favored the unity of Portugal and Spain, although he had betrayed Portugal by becoming Castilian. Thus, he became a hero for the Castilians whose political position was consequently enhanced.

It was in the interest of Castile to elevate Magellan, and to push Elcano aside, but also of the Catholic Church. The tug of war between the empire of Charles V and the Roman Catholic Church was constant. The Holy Roman Empire presented itself as a Catholic Empire, but at the same time it did not control the center of religion, that is, Rome. The Church did not want Charles V to have too much power. Therefore, the Church wanted to mark a contrast between the bloody evangelisation that Spain carried out in the Americas and the supposedly peaceful evangelisation that Portugal carried out in Asia. Since it was convenient for it to emphasize that Spain's attempt at evangelisation had been drenched with blood, while Portugal's was a builder of civilization, the Church found it advantageous to praise Magellan. Thus, Magellan was praised by Rome and England, who were allied with Portugal against Spain. Magellan was taken as a model of civilization, and Francis Drake as the executor of his dream.

Although many technical and economic documents have survived, most of the originals have disappeared; their existence is known because these documents are cited by those who received the records of the times. Neither the actual text of the chronicle of Elcano's voyage, records of the proceedings of the trial of Carvalho, the documentation provided by Elcano at the Badajoz-Elvas meetings, nor the ship's logbook have been found. The Trinidad ship's book has also been lost, although it was taken by Portugal. Historian Enrique Santamaría believes these documents have been destroyed, probably in the 19th century; he finds it difficult to credit that they were lost, they being of immense historical value, and observes that documents with many other details of the voyage have survived.

Navarrete's account
In the context of the Napoleonic Wars, feelings of nationalism arose in most European countries. In Spain this sentiment had a conservative component supported by the monarchy, that in its retrospective view, held that the Spanish people needed to construct a new national historiography; this task fell to Martín Fernández de Navarrete, director of the Royal Academy of History. In 1825 Fernández de Navarrete wrote a modern 'official' account of the first circumnavigation of the earth, Colección de los viajes y descubrimientos que hicieron por mar los españoles, in which Magellan is praised, and the figure of Elcano passed down by tradition was remade.

At a time when Spain was experiencing the decolonization of America, the Spanish government desired to present a national historical narrative that legitimized the power of the ruling elites, a message that inevitably would diminish the position of the lower classes. In Navarrete's telling Magellan became an exemplar of those elevated persons who must struggle against the incapacity of common men to understand the complexities of governance, portraying him as of noble character, an upholder of the Christian faith, and a leader in the advancement of civilization. In contrast to these virtuous qualities he attributed to Magellan, Elcano was a man in the habit of deciding important matters democratically by vote, one who had never conquered another race of people that could be evangelized and civilized, nor had he ever even tried. Thus, in Navarrete's account, crediting Elcano with the completion of the first circumnavigation is understood as an injustice to the memory of Magellan's achievements, and Navarette's history is an attempt to rectify this error. The historical conception of a Spanish nation as presented by Navarrete has subsequently been propagated by historians of other nationalities, and remained a dominant view internationally for many years.

Later constructions
In 1861 the soldier Juan Cotarelo Garastazu, a field marshal (mariscal de campo) of the Spanish army, wrote the first biography of Elcano. He writes that Elcano was "humble and obedient" and that he had managed to sail around the world in obedience to his king, submitting as proof that in the rigorous Valladolid inquiry concerning the circumnavigation, Elcano responded in a humble, modest way to the thirteen questions he was asked.

Eustaquio Fernández de Navarrete, grandson of Martín Fernández de Navarrete, wrote a more complete biography of Elcano in 1872. This biography has a more neutral, analytical approach to its subject and has been the main reference for historians. As it was problematic for some people that Elcano lacked a noble pedigree, in order to vindicate his reputation, Eustaquio Fernández de Navarrete suggested, without providing any proof, that Elcano had taken part in the Siege of Oran and in the Italian wars.

From Cánovas to the Francoism
Elcano was a posthumous victim of political machinations in the 19th century when Antonio Cánovas del Castillo faced the creation of peripheral Basque nationalist movements and fuerismo concerning the laws (fueros) of the different provinces of the Basque region; this occurred in a chaotic scenario after the overthrow of the First Spanish Republic. The so-called foral laws set up a legal framework in which authority was shared between the monarchy, whose agencies governed local affairs of interest to the Crown, and provincial institutions, which managed their own affairs concerning local traditions and customs. Cánovas believed that the foralists used a distorted history to defend their politics and, for this reason he wanted to minimize the role of the Basques and the Navarrese in Iberian history. Thus, he tried to disappear not only Elcano, but also Blas de Lezo, Churruca, Andrés de Urdaneta, and Legazpi from this history, establishing a damnatio memoriae. To eradicate Elcano's place as a heroic figure, Cánovas reinforced the use of the Portuguese Magellan as a symbol of the political unity of the Iberian Peninsula, averring that Elcano was no more than the "humble master" (modesto maestre) of the ship Victoria that completed the first circumnavigation of the Earth.

Prime Minister of Spain Primo de Rivera, however, wanted to restore Elcano's reputation and named a training ship of the Spanish Navy, Juan Sebastián de Elcano (A-71), after him. Franco's regime tried to make Elcano a national mythical figure, using the narrative written by Eustaquio Fernández de Navarrete (1872).

Amado Melón Ruiz de Gordejuela in his book Magallanes o la Primera Vuelta al Mundo, published in 1940 in the collection La España Imperial, affirmed that Elcano participated in the siege of Oran and was an assistant to "The Great Captain", Gonzalo Fernandez de Cordova, who campaigned in Italy between 1495 and 1504. Melón Ruiz wanted to place Elcano in Oran, and added 10 years to his age (otherwise he would have been only 8 years old), and promulgated the fiction that Elcano was born in 1476 rather than 1487. With the acceptance of this fabricated birth date by historians, the false information that Elcano participated in the Oran campaign has been widely published in history books.

500th anniversary
The 500th anniversary of the first circumnavigation was celebrated in Spain on September 6, 2022. In view of this, several initiatives such as the Elkano Foundation have arisen, both to complete the history and perpetuate its memory and to plan the celebration of the event.

As part of the commemoration of the 500th anniversary of the first circumnavigation of the Earth on September 6, 1522, the Basque Maritime Museum, with the support of the Elkano Foundation, has published a new book: ''Elcano y el País Vasco. Cómo se hizo posible la primera vuelta al mundo'' (June 2022).

Elcano in art
There is no description or contemporary artwork depicting Elcano, and all artworks depicting him were made centuries later, often with ahistorical elements in their composition and featuring anachronistic clothing.

Cinema
In 2019 an animated movie was made in the Basque language by Ángel Alonso with the title Elkano, lehen mundu bira. In 2020 another animation movie by Manuel H. Martin called El viaje más largo was presented in Sevilla. In 2022 Amazon Prime broadcast the series Boundless, with Álvaro Morte as Elcano. In 2022 the movie Uncharted mentioned him. The movie is based on his journey with the Magellan Expedition.

title: Juan Sebastián Elcano


text:
Transfinite number

In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined in 1895 by Georg Cantor,   who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers. Nevertheless, the term transfinite also remains in use.

Notable work on transfinite numbers was done by Wacław Sierpiński: Leçons sur les nombres transfinis (1928 book) much expanded into Cardinal and Ordinal Numbers (1958, 2nd ed. 1965 ).

Definition
Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of marbles), whereas ordinal numbers specify the order of a member within an ordered set (e.g., "the  man from the left" or "the  day of January"). When extended to transfinite numbers, these two concepts are no longer in one-to-one correspondence. A transfinite cardinal number is used to describe the size of an infinitely large set, while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered. The most notable ordinal and cardinal numbers are, respectively:

 * $$\omega$$ (Omega): the lowest transfinite ordinal number. It is also the order type of the natural numbers under their usual linear ordering.
 * $$\aleph_0 $$ (Aleph-null): the first transfinite cardinal number. It is also the cardinality of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, $$\aleph_1.$$ If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-null. Either way, there are no cardinals between aleph-null and aleph-one.

The continuum hypothesis is the proposition that there are no intermediate cardinal numbers between $$\aleph_0$$ and the cardinality of the continuum (the cardinality of the set of real numbers): or equivalently that $$\aleph_1$$ is the cardinality of the set of real numbers. In Zermelo–Fraenkel set theory, neither the continuum hypothesis nor its negation can be proved.

Some authors, including P. Suppes and J. Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition, the following are all equivalent:
 * $$\mathfrak{m}$$ is a transfinite cardinal. That is, there is a Dedekind infinite set $$A$$ such that the cardinality of $$A$$ is $$\mathfrak {m}.$$
 * $$\mathfrak{m} + 1 = \mathfrak{m}.$$
 * $$\aleph_0 \leq \mathfrak{m}.$$
 * There is a cardinal $$\mathfrak{n}$$ such that $$\aleph_0 + \mathfrak{n} = \mathfrak{m}.$$

Although transfinite ordinals and cardinals both generalize only the natural numbers, other systems of numbers, including the hyperreal numbers and surreal numbers, provide generalizations of the real numbers.

Examples
In Cantor's theory of ordinal numbers, every integer number must have a successor. The next integer after all the regular ones, that is the first infinite integer, is named $$\omega$$. In this context, $$\omega+1$$ is larger than $$\omega$$, and $$\omega\cdot2$$, $$\omega^{2}$$ and $$\omega^{\omega}$$ are larger still. Arithmetic expressions containing $$\omega$$ specify an ordinal number, and can be thought of as the set of all integers up to that number. A given number generally has multiple expressions that represent it, however, there is a unique Cantor normal form that represents it, essentially a finite sequence of digits that give coefficients of descending powers of $$\omega$$.

Not all infinite integers can be represented by a Cantor normal form however, and the first one that cannot is given by the limit $$\omega^{\omega^{\omega^{...}}}$$ and is termed $$\varepsilon_{0}$$. $$\varepsilon_{0}$$ is the smallest solution to $$\omega^{\varepsilon}=\varepsilon$$, and the following solutions $$\varepsilon_{1}, ...,\varepsilon_{\omega}, ...,\varepsilon_{\varepsilon_{0}}, ...$$ give larger ordinals still, and can be followed until one reaches the limit $$\varepsilon_{\varepsilon_{\varepsilon_{...}}}$$, which is the first solution to $$\varepsilon_{\alpha}=\alpha$$. This means that in order to be able to specify all transfinite integers, one must think up an infinite sequence of names: because if one were to specify a single largest integer, one would then always be able to mention its larger successor. But as noted by Cantor, even this only allows one to reach the lowest class of transfinite numbers: those whose size of sets correspond to the cardinal number $$\aleph_{0}$$.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Transfinite number


text:
Talk:Fermi–Dirac statistics

Miscellaneous discussion
Awesome plots, Unc.hbar. My plot was thrown together quickly, with the intent of spurring someone else to make a better one. I see the plan worked in record time. Ed Sanville 14:27, 14 May 2005 (UTC)

Is the word "statistics" plural? I think so, but Florian Marquardt introduced some grammar suggesting it's singular. He also left in some of my plural forms, leaving the article sounding rather confused. I've changed it to be consistently plural. If anyone knows better please say so.

Another grammer matter: was quantum mechanics really invented  or was it perhaps  discovered  or maybe  introduced ?... just a thought

Also, F-D statistics are used for electrons in solids, not just electrons generally. In free space, you don't usually need QM, and inside atoms or molecules, you need non-statistical QM. -- Tim Starling
 * [I've inserted this comment out of chronological order.]
 * Statistics as a subject field is used with a singular verb.   --Bob K31416 (talk) 18:25, 11 January 2009 (UTC)

 * Electrons can be be found in other forms than 'in solids' or 'in free space', there are in every (non-solid/non gas) molecule also.

 * Oh really? Perhaps that's why I wrote above "inside atoms or molecules, you need non-statistical QM". I guess there could be applications for F-D statistics for electrons in molecular or atomic gases – perhaps at high temperatures – but informing me smugly that there are electrons in liquids and gases is probably not the best way to start such a discussion.
 * Sorry, I had not intention of being sarcastic or offending, I just wanted to state that solid-state physics]] is not the only field where F-D apply; i.e. F-D seems to be used in dense-matter physics, white-dwarf & neutron star are also (but not so good) example. - looxix 22:36 Mar 3, 2003 (UTC)

 * I didn't mean to suggest that F-D statistics can never be used for liquids or gases. I thought the statement "F-D statistics applies in particular to the electrons and therefore are often used in the study of solids" was misleading. It implies that anything containing electrons is suitable for study with F-D statistics. This is certainly untrue. -- Tim

 * F-D statistics is not only used for electrons in solids, in fact, a metal conductor is better interpreted as a 'gas' of electrons embedded in a solid mesh/lattice (see Fermi gas).

 * The "gas" analogy may be a useful one, but it does miss some of the more curious properties of solids. Although a small fraction of the electrons in a metal are free to move and respond to external forces (like the particles of a classical gas), not all of them are. Why not? -- Tim
 * I didn't say that all electrons are part of the "gas". - looxix 22:36 Mar 3, 2003 (UTC)

 * also F-D (as B-E) statistics apply to indistinguishable particle, if the electrons are bound in a solid, then they are distinguishable.
 * of course I may be totaly wrong. - looxix 22:14 Mar 2, 2003 (UTC)

 * About that last point, yes unfortunately you are "totaly wrong". Electrons are always indistinguishable. See identical particles for a sketchy explanation. -- Tim Starling 04:29 Mar 3, 2003 (UTC)
 * Indeed, I confounded with nuclei in a crystal. - looxix 22:36 Mar 3, 2003 (UTC)

Derivation
The new "derivation" is NOT a derivation of Fermi-Dirac statistics. The "another derivation" is much closer to a derivation, since it starts from statistical assumptions and statistical relationships. Any derivation that does not introduce the statistics of how indistinguishable particles are distributed among various energy levels while assuming each distribution is equally likely, is not a derivation. The present "derivation" introduces the partition function as a given! The partition function follows naturally from a statistical analysis of Fermions and is not a given!. The present "derivation" introduces the probability of an electron having a particular energy as a given, but this is just what a derivation is supposed to derive! This is not a derivation. It needs to go somewhere else, under a different title. By removing it I don't mean to seem like I am throwing it out, I just don't know quite where it fits. PAR 01:31, 15 May 2005 (UTC)

 * Yes, I am taking partition function for granted because in science we prove something once and then use it again and again. One can look up derivation of the partition function if (s)he so desires.  Also, someone looking up FD statistics in Wikipedia rather than SM text would more easily follow my logic than yours.  Feel free to change the title, please don't erase my work. Unc.hbar 02:08, 15 May 2005 (UTC)

 * Do you like my plots? Unc.hbar 02:21, 15 May 2005 (UTC)

I am reading and rereading what you wrote, and I cannot follow it. You say energy is 0 or epsilon, and use this, then later bring in the chemical potential from I don't know where, to fix the fact that energy is not zero. You say gi is unity, but then later say you don't need to put it in because its "put in later", implying it is not unity. When you say n-bar is a probability, you have to have a level subscript, or else its an integral, and then its not a probability. It just needs tons of work before it makes any sense. This is not to say that the second derivation is clear as a bell. It needs work too. PAR 03:14, 15 May 2005 (UTC)

 * It is a well-known fact that energy can be offset without significant implications to the problem, I don't see why I can't slip $$\mu$$ in at the last moment :). I don't like carrying the degeneracy around since it only multiplies the answer at the end.  I am not a fan of subscripts, which is odd because I study GR.

 * Statistical Mechanics is a bunch of hand-waving. There are 2 kinds of hand-waving: (1) rigorous; (2) non-rigorous.  Title my stuff as "non-rigorous hand-waving" if you will.  We could just cut and paste the book.  I think the idea if this website is to provide something books can't/don't.

 * I agree, the "derivation" is not crystal clear, I'll keep working on it.

 * I think my stuff is more undergraduate, yours is graduate-level. Maybe we should title them so?  How's FD for Dummies and FD for Hard-Core Physicists?

Also, I think the plots are excellent. I fixed the captions for mu and epsilon. I tried to do something similar with the Bose gas article. What did you use to make the plots?

 * Thanks. I used SM (Supermongo). hbar 03:47, 15 May 2005 (UTC)

-- I think that the "Derivation using Lagrange multipliers" should be changed. In particular using the Stirling's approximation for the factorial of "ni" when the quantity "ni" is a very small number (between 0 and 1 if you assume it to be a continuous variable) doesn't seem to make much sense as Stirling's approximation is good only if ni is large. 190.188.251.111 (talk) 05:55, 24 July 2009 (UTC)Muad'Dib

Dead link report
Dead link: http://www.boutichesaid.cv.dz/FermiDirac/FermiDirac.htm —Preceding unsigned comment added by Joey11 (talk • contribs) 11:27, 19 January 2009 (UTC)

Equation positioning
In the derivation that I just added, when I use a Windows based computer with non-Safari browsers, the alignment is Ok for all the equations, i.e. they are aligned by keeping the equals sign in the same place on each line. The Windows/non-Safari computer was the one that I used when I was aligning. However, when I use a MAC and Safari browser, some of the equations become misaligned. There seems to be a shifting of the lines which might be related to the use of ::: and/or hard spaces at the beginning of the line and its interaction with the TeX markup, but I'm not sure. Does anyone have any suggestions on how to get consistent alignment on both types of computers/browsers? Thanks. --Bob K31416 (talk) 11:07, 7 March 2009 (UTC)

 * I suspect you're the one responsible for the things I commented on below. If you're using something external to TeX to make it align correctly, then you're doing it wrong.  All those preceeding colons for spacing is absolutely wrong; you should be working WITHIN TeX to do that.  The "align" environment is used in TeX for alignment of "="s on different lines.  They way you (if it was you) did it is lunacy; it's a horrible mess. Michael Hardy (talk) 22:30, 23 October 2009 (UTC)

What is this mess supposed to say??

 * Rearranging the summations,
 * $$ \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \ \   \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$\bar{n}_i \ = \ \frac {\qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \quad}  { }  $$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$\bar{n}_i \ = \ \frac {\qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \quad}  { }  $$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$

The above is in this article and looks like something written by a lunatic. If I knew what it is supposed to say I have no doubt that I could write it correctly with much simpler TeX code than what appears here. If someone's using all those "\qquad"s in this kind of manner, that's a sure sign that they don't know how to use TeX properly. Michael Hardy (talk) 22:27, 23 October 2009 (UTC)

 * The problem was introduced with your recent edits. Please look at the version here before your recent edits and you will see that it doesn't have the problem that you introduced. Also, note that the version before your edits correctly corresponds to the way the equation appears in the cited Reif reference and is eq 9.3.2 there.

 * Re "If I knew what it is supposed to say I have no doubt that I could write it correctly with much simpler TeX code than what appears here."

 * By all means do that if you can. As I mentioned before, what it is suppose to mean is what was in the article before you edited it and introduced the problem. I would suggest that you revert the article back to what it was before you started editing and try again. --Bob K31416 (talk) 23:23, 23 October 2009 (UTC)

 * One thing I should add, in your efforts to simplify the code, do not change the appearance of the equations without consensus. --Bob K31416 (talk) 23:55, 23 October 2009 (UTC)

The problem was very much there before my edits. It said this:
 * $$ \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \ \   \sideset{ }{^{(i)}}\sum_{n_1,n_2,...} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$\bar{n}_i \ = \ \frac {\qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \quad}  { }  $$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,...} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,...} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$

I changed it to this: The former form does not just flout Wikipedia's indentation conventions (see WP:MOSMATH) but it tortures the software and other Wikipedians. It uses complicated formatting, relying both on TeX and colons, etc., outside of TeX, making it hard to edit and getting an inferior result (notice among other things how your horizontal line doesn't go as far to the right as the numerator and denominator do). That weird "\sideset" thing is nearly impossible to deal with while editing.
 * $$ \bar{n}_i = \frac{\displaystyle \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} }{\displaystyle \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} } $$
 * $$ \bar{n}_i = \frac{\displaystyle \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} }{\displaystyle \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} } $$

TeX is sophisticated (even the stripped-down version of it used on Wikipedia). It can do all this stuff simply, as you see above, rather than using the complicated workarounds you used. Michael Hardy (talk) 00:07, 24 October 2009 (UTC) .... The article still needs a lot of cleanup. But it's looking much better than it did six hours ago. Michael Hardy (talk) 00:09, 24 October 2009 (UTC)

 * The above latest version that you introduced contains an error. It does not include the superscript "(i)" in the two summations on the right, which is in the previous version before you started editing and the cited reference. (You may want to see the text of the article for the meaning of the superscript.) --Bob K31416 (talk) 00:22, 24 October 2009 (UTC)

OK, I've reinstated the (i).

I came to this article simply to bypass an unprintable redirect. I found all the nonstandard formatting. Usually that's easy to fix. Then I came to the place where you're using all those preceeding colons to do formatting usually done within the math tags, and they're interacting with the stuff inside the tags in an odd way. It takes a lot of close attention to things that normally wouldn't be there to figure out how to do even minor edits on a thing like that. Michael Hardy (talk) 01:33, 24 October 2009 (UTC)

 * Re "Then I came to the place where you're using all those preceeding colons to do formatting..."
 * I looked at WP:MOSMATH and I found "When displaying formulae on their own line, one should indent the line with one or more colons ..." Perhaps you had a different part in mind, since this quote doesn't prohibit what I did. Could you give the excerpt here that you were referring to? Thanks.

 * Also, I used the colons to approximately center equations and/or line up their equal signs, as is usual in most physics texts and journal articles that I recall reading. Perhaps Wikipedia's software is not sophisticated enough to do this automatically? If it can do this automatically, please implement that feature. Thanks. --Bob K31416 (talk) 04:01, 24 October 2009 (UTC)

TeX is sophisticated enough to format complex fractions. Doing it by such strange means as you used just makes things difficult to edit.

Centering is not the usage convention. It's normally indented by one colon, but if the paragraph is already intended, as by a bullet or otherwise, then two colons may be needed. Possibly more than two could be needed in some cases. Michael Hardy (talk) 01:47, 29 October 2009 (UTC)

 * I have previously and carefully simplified the equation code using \displaystyle, as indicated in the article's history. Also, see my recent additions  to Help:Displaying_a_formula which adds two examples on the subject of "Sum in a fraction".

 * Re "Centering is not the usage convention." - It is the usage convention in the scientific community outside of Wikipedia. However, left-align is the predominant usage in Wikipedia, so I have just changed the alignment in the article to left-align.
 * --Bob K31416 (talk) 14:12, 29 October 2009 (UTC)

Recent edits by Michael Hardy
Recently there were a considerable number of changes to the article. Some of these changes introduced outright physics errors, most did not improve the article, and they were accompanied by rude and untruthful remarks. These included unsubstantiated accusations of violation of Wikipedia style guidelines. If the editor insists on including these changes, consensus will be needed on this talk page through civil discussion. --Bob K31416 (talk) 12:23, 25 October 2009 (UTC)

FD distribution
The article says n_i is the "average number of fermions in the state i", which is wrong. Either an N (total number of fermions in the system) is missing in the numerator, or the given n_i is the probability that a fermion is in the state i.--190.188.10.179 (talk) 15:16, 3 October 2010 (UTC)
 * See p. 339 of the source Reif, where $$ \bar{n}_s $$ is defined for quantum distribution functions. "Let us now consider the case of arbitrary temperature T and calculate, for the several cases of interest, the mean number of particles $$ \bar{n}_s $$ in a particular state s." --Bob K31416 (talk) 13:37, 15 September 2011 (UTC)

Can someone stop Fat&Happy to revert my changes?
I made some changes at the beginning of this page. The sources are 1,2 and 5. Fermi published on March, Dirac on October of 1926 (In my country in the same year October follows March). Moreover Dirac mentioned the fact it in his book,ref. 5. Fat&Happy is continuously reverting my changed, without discussing it, and I have the opinion, without reading them. This modus operandi is everything except than collaborative... — Preceding unsigned comment added by Antonio.napoli (talk • contribs) 04:25, 22 February 2012 (UTC)

Moreover, in the Italian publication (ref. 1), the number 3 indicates the month (i.e. March). Fermi first published it in Italian, BUT some English scientists, in the same period when Dirac was going to publish his work, were already aware of Fermi work. Search for "Thomas-Fermi". Furthermore, the particles of this statistics are called Fermi's particles (by Dirac himself), the energy level, Fermi's levels, and so on and so forth....Why didn't they call Dirac's particles, or Dirac's energy if Fermi wasn't the first? Last but not the least, in science whenever a double name is assigned to a law, the common practice is to follow the alphabetic order (if the two scientists published the work contemporaneously) or the first name is usually the name of the one who defined and published the work as first.

I didn't want to change the name of the statistics, (I just wanted to specify who was the first to understand its principles), and since Dirac himself pointed this out in his book, I don't understand why this person is continuously reverting what I write.

 * If you have a reliable source that states that Fermi's work predates Dirac's, please quote it here on the talk page. Otherwise people have to dig up the source themselves and its much harder to "stop Fat&Happy". I agree with Fat&Happy that the article dates often but do not necessarily determine the order of the names. PAR (talk) 07:45, 22 February 2012 (UTC)

I mentioned the sources. Did you read what I wrote? Ref. 1, 2 and 5 of the current wikipedia page on this topic. Another valuable source, "Whereas according to Dirac, it was first studied by Fermi, and Dirac called it Fermi statistics and the corresponding particles fermions.[5]", from the same wiki page...

 * Did you read what I wrote? I asked you to quote the sources, not to just mention them. I doubt if the above quote is from "Principles of Quantum Mechanics". Find the book "Principles of Quantum mechanics" by Dirac, and quote from the book where it says that the subject was first studied by Fermi, and how Dirac called it Fermi statistics and the corresponding particles fermions. In other words, write down the sentences, here on the talk page, taken from the book, that support the above statement. PAR (talk) 14:12, 22 February 2012 (UTC)

ok, I will do, if you can explain me why I need to do this to support my points, while Fat&Happy can just delete without proving, or even mentioning anything. Where is the difference between me and this person? Which are his credits concerning the knowledge about Fermi's achievements and life to have the right to delete whatever he wants?? Following what you wrote, Wikipedia would be empty, because NOBODY reports the sentence (this article for example)...but since you desire so much to have it I will find to make you happy...You should have a look at the Italian page of Fermi, mainly written by me (Antonio Napoli), and see how many time I really reported exactly Fermi or other scientist words...exactly what you ask me, the opposite of what the large majority does... This is one example. Original letter written (but in the italian translation) from Fermi to Dirac....Use any translator to understand it. I will find also the original...because Fat&Happy can delete whatever he wants, while myself has to prove any single line... « Caro Signore! Nel suo interessante lavoro, On the theory of quantum mechanics, ha proposto una teoria del gas ideale basata sul principio di esclusione di Pauli. Ora una teoria sul gas ideale che è praticamente identica alla sua è stata pubblicata da me all'inizio del 1926 (Zs. f. Phys. 36, p. 902, Lincei Rend. Febbraio 1926). Poiché immagino lei non ha visto il mio articolo mi permetto di attrarre la sua attenzione su di esso. Sinceramente suo Enrico Fermi. »

Fermi defined the statistics before Dirac:

http://books.google.com.tw/books?id=-pL56OcVubgC&pg=PA767&lpg=PA767&dq=fermi+letter+to+Dirac+1926&source=bl&ots=jABE5WeLTD&sig=4ESQy3pPYjKXmcALr1uaPjta6eE&hl=en&sa=X&ei=YQJFT9fwG--XiAeaqcWqAw&redir_esc=y#v=onepage&q=fermi%20letter%20to%20Dirac%201926&f=false

The Creation of Wave Mechanics; Early Response and Applications 1925-1926, page 767. http://www.springer.com/physics/book/978-0-387-95180-5?cm_mmc=Google-_-Book%20Search-_-Springer-_-0

In this same page you find confirmation of what Dirac, wrote, that Fermi was the first to discover the statistics...

 * Why can't you simply find the quote and state it?

 * Looking at your first reference From Mehra and Rechenberg "The Historical Develpment of Quantum Theory" Volume 5, Part 2, I find the following quote:

 * "Paul Dirac, like Heisenberg, made use of Schroedinger's wave mechanical approach in treating quantum systems of several identical particles; without taking notice of Fermi's papers, he rediscovered Fermi (-Dirac) statistics a few months later."

 * All you have to do is state this quote, and then change the article the way you originally had it, with a reference to the above book. I am fairly sure that this will satisfy Fat&Happy. It's that simple.

 * Fat&Happy cannot delete whatever he wants. He cannot delete properly referenced statements.

 * The strict rules are that you should not enter something into Wikipedia without referencing a reliable source. If you enter something without a reliable source, then anyone can delete it. The person who deleted it does not have to prove that your unreferenced statement is wrong.

 * People who delete unreferenced statements without explanation or discussion are not being cooperative or friendly, I agree, but they are not wrong. If you know something to be true, then simply quoting the reference that proves it on the talk page will fix this. If the person now deletes it, that person is wrong.PAR (talk) 16:15, 22 February 2012 (UTC)

I guess you know you cannot copy text from google books....

 * Bye PAR (talk) 16:44, 22 February 2012 (UTC)

Average number
The first graph in this article does not show an average, but a probability. Of course, the two are linked but obviously not the same. The Physics-writer D.C. Giancoli talks about "an occupation-probability" and Hyperphysics mentions "a probability that a particle has energy E". It makes me even more suspicious about the use of an average number in this article. But correct me if I'm wrong. — Preceding unsigned comment added by Koitus~nlwiki (talk • contribs) 22:11, 11 June 2021 (UTC)
 * See Talk:Fermi–Dirac_statistics. The given source Reif derives the Fermi-Dirac Distribution as an average occupation, not a probability of occupation. However, in this case of the Fermi-Dirac Distribution  the maximum number of particles in a state is one and the average equals the probability. Compare with the Bose-Einstein Distribution where the  average occupation can be greater than 1 and thus not equal to the probability. Bob K31416 (talk) 22:55, 19 June 2021 (UTC)

Lead
The following message was moved here from my talk page.
 * "I don't understand, what's the problem with my changes? I explained the reasons in the edit summaries. Do you believe the new version is worse (in which respect)? Evgeny (talk) 15:05, 19 August 2021 (UTC)"

You started out by adding info re non-interacting particles that was already in the lead. Later you removed what was already in the lead, saying it was repetitious, and also you rewrote the lead. As I wrote in my edit summary, "please explain on the talk page what the deficiencies were that your last two edits were trying to improve and how your edits improved them...". Bob K31416 (talk) 18:01, 19 August 2021 (UTC)
 * What do you mean by "already in the lead"? A correct statement (that the FD distribution follows from the FD stats only for the case of non-interacting particles) must be made upfront. Evgeny (talk) 19:05, 19 August 2021 (UTC)
 * We could change
 * A result is the Fermi–Dirac distribution of particles over energy states.
 * to
 * A result for the case of non-interacting particles is the Fermi–Dirac distribution of those particles over energy states.
 * Bob K31416 (talk) 19:35, 19 August 2021 (UTC)
 * Fine with me, but I really see no principal difference from my version that you undid with the comment "already discussed in second paragraph". Why doesn't the second paragraph bother you now?  Evgeny (talk) 21:23, 19 August 2021 (UTC)
 * I took a second look at your version. How does this look for you.
 * Fermi-Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many identical particles that are in thermodynamic equilibrium and obey the Pauli exclusion principle, where no two particles can occupy the same state. A result for the case of non-interacting particles is the Fermi–Dirac (F-D) distribution over energy states. F-D statistics and the F-D distribution are named after Enrico Fermi and Paul Dirac, each of whom derived the F-D distribution independently in 1926 (although Fermi derived it before Dirac).[1][2]

 * The particles of F-D statistics have half-integer spin (1/2, 3/2, etc.) and are called fermions. F–D statistics is most commonly applied to electrons, which are spin-1/2 fermions. A counterpart to F–D statistics is Bose–Einstein (B–E) statistics, which applies to identical and indistinguishable particles with integer spin (0, 1, 2, etc.) called bosons. In classical physics, Maxwell–Boltzmann (M–B) statistics applies to particles that are identical but treated as distinguishable. For both B–E and M–B statistics, more than one particle can occupy the same state, unlike F–D statistics where a state's occupancy is limited to one particle.

 * Bob K31416 (talk) 01:05, 20 August 2021 (UTC)

 * I don't know if what we're doing is right. 99% of the article's contents is about the distribution. Not much is said (and not much can be said, in fact) about the statistics. It might be better to have a short stand-alone article on quantum statistics, describing both FD and BE, and rename this article to "FD distribution". Evgeny (talk) 16:52, 20 August 2021 (UTC)
 * Maybe. My inclination is to work with what we have. Bob K31416 (talk) 20:57, 20 August 2021 (UTC)

title: Talk:Fermi–Dirac statistics


text:
Rocksteady

Rocksteady is a music genre that originated in Jamaica around 1966. A successor of ska and a precursor to reggae, rocksteady was the dominant style of music in Jamaica for nearly two years, performed by many of the artists who helped establish reggae, including harmony groups such as the Techniques, the Paragons, the Heptones and the Gaylads; soulful singers such as Alton Ellis, Delroy Wilson, Bob Andy, Ken Boothe and Phyllis Dillon; musicians such as Jackie Mittoo, Lynn Taitt and Tommy McCook. The term rocksteady comes from a popular (slower) dance style mentioned in the Alton Ellis song "Rocksteady", that matched the new sound. Some rocksteady songs became hits outside Jamaica, as with ska, helping to secure the international base reggae music has today.

Characteristics
The Jamaican musicians and producers who developed the rocksteady term and sound from 1966 to 1968 had grown up jazz and R&B, had played through ska and were influenced by other genres, most notably rhythm and blues, mento, calypso and US Soul music, and by Caribbean and African music.

The tempo became slower with the development of rocksteady than it had been in ska. The guitar and piano players began to experiment with occasional accents around the basic offbeat pattern.

The slowing that occurred with rocksteady allowed bass players to explore more fat, dark, loose, slow tones than ska bass. The slower tempo and smaller band-sizes in turn led to a much larger focus on the bass line in general, which eventually became one of the recognizable characteristics of Jamaican music. In rocksteady, the lead guitar often doubles the bass line, in the muted picking style created by Lynn Taitt (as on "Run for Cover" by Lee "Scratch" Perry).

Lyrics
Due in part to the heavy borrowing from US soul songs, many rocksteady songs are love songs; e.g. "Sharing You" by Prince Buster, which is a cover of a soul singer Mitty Collier's original, and "Queen Majesty" by the Techniques, which is a cover of "Minstrel and Queen" by the Impressions.

There are rocksteady songs about religion and the Rastafari movement, though not to the same extent as in reggae. Rocksteady coincided with the rise of rude boys and some rocksteady songs reflect this (usually negatively) such as "Rude Boy Gone A Jail" by the Clarendonians and, most famously, "Judge Dread" by Prince Buster.

Alton Ellis was anti-rudie, and Alton Ellis and the Flames' "Cry Tough", released before the term rocksteady was in vogue, urged Jamaicans in the ghettos to stay tough through the hard times.

History
As a popular musical style, rocksteady was short-lived - the genre's heyday only lasted about two years, from around summer 1966 until spring 1968. However, timeless hits like "Stir It Up" and "Bend Down Low" by the original Wailers that had vocals by Bob Marley and rhythms by the Soul Brothers were created during this era, among many others, launching careers of the biggest reggae singers ever.

Also, in the middle to later part of the decade, as ska began to fade in popularity and the optimism that accompanied Independence in 1962 dwindled, young people from the Jamaican countryside were flooding into the urban ghettos of Kingston—in neighborhoods such as Riverton City, Greenwich Town and Trenchtown. Many of them became delinquents who exuded a certain coolness and style. These unruly youths became known as rude boys.

Alton Ellis is sometimes said to be the father of rocksteady for his hit "Rocksteady"; however, other candidates for the first rocksteady single include "Take It Easy" by Hopeton Lewis, "Tougher Than Tough" by Derrick Morgan and "Hold Them" by Roy Shirley.

One account of rocksteady's inception comes from the film "Studio One Drummie and the history of rocksteady music": rock steady music started at Studio One in 1966 because when Joe Isaacs at age 15 replaced Skatalites drummer Lloyd Knibbs, Isaacs could not play drums fast enough to keep up with the pace of ska - music director Jackie Mittoo (The Mozart of Jamaica), slowed down the tempo.

Another account comes from a Jamaican radio interview, pianist Gladstone Anderson said that bandleader Lynn Taitt made a suggestion to slow the music down whilst recording "Take It Easy". Taitt backed this up in a 2002 interview, stating: "I told 'Gladdy to slow the tempo and that's how Take It Easy and rocksteady came about. Rocksteady is really slow ska."

Rocksteady's dominance meant that all record labels of the time released music in the genre; Studio One, Bunny Lee and Prince Buster were predominant, with Reid. The record producer Duke Reid released Alton Ellis' "Girl I've Got a Date" on his Treasure Isle label, as well as recordings by the Techniques, the Silvertones, the Jamaicans and the Paragons. Reid's work with these groups helped establish the vocal sound of rocksteady. Some would consider the rocksteady years to be Treasure Isle's best.

Notable solo artists include Delroy Wilson, Ken Boothe and Phyllis Dillon (known as the "Queen of Rocksteady"). Other musicians who were crucial in creating rocksteady included keyboard player Jackie Mittoo, drummers Joe Isaacs and Winston Grennan, bassist Jackie Jackson and saxophonist Tommy McCook. When ska band the Skatalites disbanded (64/65—accounts vary) McCook went to work at the Treasure Isle label and Jackie Mittoo went to the Studio One label—these two artists/arrangers became instrumental in the way these two labels became dominant and helped to form the sound of Rocksteady.

Despite its short lifespan, rocksteady's influence is great. Many reggae artists began in rocksteady (and/or ska)—most commonly reggae singers grew out of rocksteady groups, e.g., Junior Byles came from the Versatiles, John Holt was in the Paragons, both Pat Kelly and Slim Smith sang with the Techniques (Pat Kelly sings lead on "You Don't Care") and Ronnie Davis was in the Tennors while Winston Jarrett was in the Righteous Flames. The Wailing Wailers were similarly a vocal harmony trio (modelled on the Impressions) who came from ska, through rocksteady and became a reggae band with just the one main vocalist.

Derrick Harriott noted, "Ask any Jamaican musician and they'll tell you the rocksteady days were the best days of Jamaican music."

Transformation into reggae
Several factors contributed to the evolution of rocksteady into reggae in the late 1960s. The emigration to Canada of key musical arrangers Jackie Mittoo and Lynn Taitt—and the upgrading of Jamaican studio technology—had a marked effect on the sound and style of the recordings. Bass patterns became more complex and increasingly dominated the arrangements, and the piano gave way to the electric organ. Other developments included horns fading farther into the background; the introduction of a scratchier, more percussive rhythm guitar; the addition of African-style hand drumming, and a more precise, intricate and aggressive drumming style.

Also around this time (1969–70) the use of a vocal-free or lead instrument-free dub or B-side "version" became popular in Jamaica; at the beginning this involved the use of rocksteady tracks, most notably with U-Roy deejaying over Treasure Isle rhythms (made by a young Osbourne Ruddock, later known as King Tubby, beginning with "Wake the Town"). Indeed, this collaboration provided rocksteady with an afterlife as U-Roy rocksteady-based songs rode high in the charts (1970–71), even as reggae began to establish itself as the new sound.

By the late 1960s, the Rastafari movement became more popular in Jamaica and rocksteady became less popular. Many reggae songs became focused less on romance and more on black consciousness, politics and protest. The release of the 1972 film The Harder They Come and the rise of Jamaican superstar Bob Marley brought reggae to an international level that rocksteady never reached.

Although rocksteady was a short-lived phase of Jamaican popular music, its influence on what came after: reggae, dub and dancehall is significant. Many bass lines originally created for rocksteady songs continue to be used in contemporary Jamaican music. Such as the rhythm from "Never Let Go" by Slim Smith (sometimes known as the 'answer rhythm') and "Real Rock" both from the Studio One label; "My Conversation" also sung by Slim Smith, produced by Bunny Lee; "Queen Majesty" sung by the Techniques and "Lonely Street" by the Conquerors, both for Treasure Isle label.

title: Rocksteady