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102-658: The planisphere is the earliest surviving map showing Portuguese geographic discoveries in the east and west and is particularly notable for portraying a fragmentary record of the Brazilian coast, which the Portuguese explorer Pedro Álvares Cabral explored in 1500, the southern coast of Greenland , explored in the late 1490s, and the African coast of the Atlantic and Indian Oceans with remarkable accuracy and detail. It

204-406: A Riemann sphere , that is, as a projection of the sphere to the complex plane . In this case, loxodromes can be understood as certain classes of Möbius transformations . The formulation above can be easily extended to a spheroid . The course of the rhumb line is found merely by using the ellipsoidal isometric latitude . In formulas above on this page, substitute the conformal latitude on

306-541: A loxodrome line as: Loxodrom′ic Line is a curve which cuts every member of a system of lines of curvature of a given surface at the same angle. A ship sailing towards the same point of the compass describes such a line which cuts all the meridians at the same angle. In Mercator's Projection (q.v.) the Loxodromic lines are evidently straight. A misunderstanding could arise because the term "rhumb" had no precise meaning when it came into use. It applied equally well to

408-412: A rhumb line , rhumb ( / r ʌ m / ), or loxodrome is an arc crossing all meridians of longitude at the same angle , that is, a path with constant azimuth ( bearing as measured relative to true north ). Navigation on a fixed course (i.e., steering the vessel to follow a constant cardinal direction ) would result in a rhumb-line track . The effect of following a rhumb line course on

510-441: A royal monopoly of all profits from trading within the areas discovered. In 1418, two of Henry's captains, João Gonçalves Zarco and Tristão Vaz Teixeira were driven by a storm to Porto Santo an uninhabited island off the coast of Africa which may have been known to Europeans since the 14th century. In 1419 Zarco and Teixeira made a landfall on Madeira . They returned with Bartolomeu Perestrelo , and Portuguese settlement of

612-470: A compass rose are also called rhumbs . The expression "sailing on a rhumb" was used in the 16th–19th centuries to indicate a particular compass heading. Early navigators in the time before the invention of the marine chronometer used rhumb line courses on long ocean passages, because the ship's latitude could be established accurately by sightings of the Sun or stars but there was no accurate way to determine

714-410: A fast-sailing vessel which had better windward sailing ability than other vessels of the time. Portuguese navigators reached ever more southerly latitudes , advancing at an average rate of one degree a year. Senegal and Cape Verde Peninsula were reached in 1445. In the same year, the first overseas feitoria (trading post) was established under Henry's direction, on the island of Arguin off

816-549: A large wind-rose in central Africa, with a fleur-de-lis indicating North. This dense rhumb-line mesh was used in navigation as a reference, for reading and marking directions (courses) between places. Six scale bars graduated in Iberian leagues, with a variable number of sections (or logs ), are distributed over the chart's area. These were used to measure distances between places. Illustrations are few, but elaborate. Two cities are grandly depicted - Venice and Jerusalem . There

918-463: A loxodrome can extend beyond the right edge of the map, where it then continues at the left edge with the same slope (assuming that the map covers exactly 360 degrees of longitude). Rhumb lines which cut meridians at oblique angles are loxodromic curves which spiral towards the poles. On a Mercator projection the North Pole and South Pole occur at infinity and are therefore never shown. However

1020-427: A loxodrome, is simply the absolute value of the secant of the bearing (azimuth) times the north–south distance (except for circles of latitude for which the distance becomes infinite): where R is one of the earth average radii . Its use in navigation is directly linked to the style, or projection of certain navigational maps. A rhumb line appears as a straight line on a Mercator projection map. The name

1122-623: A permit to build a fort (Fort Manuel) and a trading post that was the first European settlement in India. There in 1503 they built the St. Francis Church . In 1502 Vasco da Gama took the island of Kilwa on the coast of Tanzania, where in 1505 the first fort of Portuguese East Africa was built to protect ships sailing in the East Indian trade. In 1505, king Manuel I of Portugal appointed Francisco de Almeida first Viceroy of Portuguese India for

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1224-526: A right angle from the north, or at some angle from the north which is some simple rational fraction of a right angle. These rhumb lines would be drawn so that they would converge at certain points of the map: lines going in every direction would converge at each of these points. See compass rose . Such maps would necessarily have been in the Mercator projection therefore not all old maps would have been capable of showing rhumb line markings. The radial lines on

1326-445: A slope Finding the loxodromes between two given points can be done graphically on a Mercator map, or by solving a nonlinear system of two equations in the two unknowns m = cot β and λ 0 . There are infinitely many solutions; the shortest one is that which covers the actual longitude difference, i.e. does not make extra revolutions, and does not go "the wrong way around". The distance between two points Δ s , measured along

1428-469: A three-year period, starting the Portuguese government in the east, headquartered at Kochi. That year the Portuguese conquered Kannur where they founded St. Angelo Fort . The Viceroy's son Lourenço de Almeida arrived in Ceylon (modern Sri Lanka), where he discovered the source of cinnamon . Finding it divided into seven rival kingdoms, he established a defense pact with the kingdom of Kotte and extended

1530-453: A trade monopoly for the newly discovered countries. The caravel , an existing ship type, was used in exploration from about 1440. It had a number of advantageous characteristics. These included shallow draft, which was suitable for approaching unknown coasts, and an efficient combination of hull shape (including a rudder attached to the sternpost, unlike some other contemporary types with side-mounted steering oars) and lateen rig, which gave

1632-547: A warehouse in the triangular trade between China, Japan and Europe. In 1570 the Portuguese bought a Japanese port where they founded the city of Nagasaki , thus creating a trading center that for many years was the port from Japan to the world. Portugal established trading ports at far-flung locations like Goa , Ormuz , Malacca , Kochi , the Maluku Islands , Macau , and Nagasaki . Guarding its trade from both European and Asian competitors, Portugal dominated not only

1734-686: Is also an elaborate depiction of the Portuguese castle of São Jorge da Mina ( Elmina Castle , on the Gold Coast of west Africa ), flanked by two African towns. Other illustrations include a lion-shaped mountain representing the Sierra Leone mountain range, the Tower of Babel (laid horizontal), the mythical Mountains of the Moon (legendary source of the Nile River ) in central Africa, and either

1836-457: Is derived from Old French or Spanish respectively: "rumb" or "rumbo", a line on the chart which intersects all meridians at the same angle. On a plane surface this would be the shortest distance between two points. Over the Earth's surface at low latitudes or over short distances it can be used for plotting the course of a vehicle, aircraft or ship. Over longer distances and/or at higher latitudes

1938-427: Is synonymous with the mathematically precise "loxodrome" because it has been made synonymous retrospectively. As Leo Bagrow states: the word ('Rhumbline') is wrongly applied to the sea-charts of this period, since a loxodrome gives an accurate course only when the chart is drawn on a suitable projection. Cartometric investigation has revealed that no projection was used in the early charts, for which we therefore retain

2040-489: Is the inverse hyperbolic sine . With this relationship between λ and φ , the radius vector becomes a parametric function of one variable, tracing out the loxodrome on the sphere: where is the isometric latitude . In the Rhumb line, as the latitude tends to the poles, φ → ± ⁠ π / 2 ⁠ , sin φ → ±1 , the isometric latitude arsinh(tan φ ) → ± ∞ , and longitude λ increases without bound, circling

2142-513: Is the air route between New York City and Hong Kong , for which the rhumb line path is 18,000 km (9,700 nmi). The great circle route over the North Pole is 13,000 km (7,000 nmi), or 5 + 1 ⁄ 2 hours less flying time at a typical cruising speed . Some old maps in the Mercator projection have grids composed of lines of latitude and longitude but also show rhumb lines which are oriented directly towards north, at

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2244-466: Is the earliest extant nautical chart that depicts places in Africa and parts of Brazil and India according to their actual latitudes. At the beginning of the 16th century, Lisbon was a buzzing metropolis where people from diverse backgrounds came in search of work, glory or fortune. There were also many undercover agents looking for the secrets brought by the Portuguese voyages to remote lands. Among them

2346-762: The Atlantic Coast of South America long before other nations even knew South America extended so far to the south. It also supplied great details of the Indian Ocean . The geographical information given on the Cantino map was copied into the Italian-made Canerio (or Caveri) map shortly after the Cantino map arrived in Italy and the Canerio, in turn, became the primary source for the design of

2448-596: The Azores were probably discovered in 1427 by Portuguese ships sailing under Henry's direction, and settled in 1432, suggesting that the Portuguese were able to navigate at least 745 miles (1,200 km) from the Portuguese coast. At around the same time as the unsuccessful attack on the Canary Islands, the Portuguese began to explore the North African coast. Sailors feared what lay beyond Cape Bojador at

2550-802: The Ragusan Republic (Dubrovnik). The Portuguese victory was critical for its strategy of control of the Indian Ocean: the Turks and Egyptians withdrew their navies from India, leaving the seas to the Portuguese, setting its trade dominance for almost a century, and greatly assisting the growth of the Portuguese Empire. It also marked the beginning of European colonial dominance in Asia. A second Battle of Diu in 1538 finally ended Ottoman ambitions in India, and confirmed Portuguese hegemony in

2652-650: The Serra do Mar . From the 15 original captainships, only two, Pernambuco and São Vicente, prospered. With permanent settlement came the establishment of the sugar cane industry and its intensive labor demands which were met with Native American and later African slaves. In 1534, Gujarat was occupied by the Mughals and the Sultan Bahadur Shah of Gujarat was forced to sign the Treaty of Bassein (1534) with

2754-606: The Table Mountain or Drakensberg range in South Africa. Along the central African coast are the various cross stone markers ( padrões ) erected by Diogo Cão and Bartolomeu Dias in the 1480s. In north Africa, there is the "Montes Claros" in the usual place of the Atlas Mountains , the legend below on the left reading that "this is the land of King Organo, whose king is very noble and very rich", and to

2856-637: The Zamorin of Calicut, leaving there some men to establish a trading post. Vasco da Gama's voyage to Calicut was the starting point for deployment of Portuguese feitoria posts along the east coast of Africa and in the Indian Ocean. Shortly after, the Casa da Índia was established in Lisbon to administer the royal monopoly of navigation and trade. Exploration soon lost private support, and took place under

2958-432: The great circle route is significantly shorter than the rhumb line between the same two points. However the inconvenience of having to continuously change bearings while travelling a great circle route makes rhumb line navigation appealing in certain instances. The point can be illustrated with an east–west passage over 90 degrees of longitude along the equator , for which the great circle and rhumb line distances are

3060-477: The scalar products λ̂ for constant φ traces out a parallel of latitude, while φ̂ for constant λ traces out a meridian of longitude, and together they generate a plane tangent to the sphere. The unit vector has a constant angle β with the unit vector φ̂ for any λ and φ , since their scalar product is A loxodrome is defined as a curve on the sphere that has a constant angle β with all meridians of longitude, and therefore must be parallel to

3162-483: The windrose lines as it did to loxodromes because the term only applied "locally" and only meant whatever a sailor did in order to sail with constant bearing , with all the imprecision that that implies. Therefore, "rhumb" was applicable to the straight lines on portolans when portolans were in use, as well as always applicable to straight lines on Mercator charts. For short distances portolan "rhumbs" do not meaningfully differ from Mercator rhumbs, but these days "rhumb"

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3264-411: The "A Famosa", where one of its gates still remains today. Learning of Siamese ambitions over Malacca, Albuquerque immediately sent Duarte Fernandes on a diplomatic mission to the kingdom of Siam (modern Thailand), where he was the first European to arrive, establishing amicable relations between the two kingdoms. In November that year, getting to know the location of the so-called " Spice Islands " in

3366-709: The Armada of the Islands protected ships from the Indies en route to Lisbon. In 1525, after Fernão de Magalhães 's expedition (1519–1522), Spain under Charles V sent an expedition to colonize the Moluccas islands , claiming that they were in his zone of the Treaty of Tordesillas , since there was not a set limit to the east. Led by García Jofre de Loaísa , the expedition reached the Moluccas, docking at Tidore . Conflict with

3468-610: The Cape Verde islands. However, as it was not possible at the time to correctly measure longitude , the exact boundary was disputed by the two countries until 1777. The completion of these negotiations with Spain is one of several reasons proposed by historians for why it took nine years for the Portuguese to follow up on Dias's voyage to the Cape of Good Hope, though it has also been speculated that other voyages were, in fact, taking place in secret during this time. Whether or not this

3570-672: The Corte-Real brothers, is labeled as Terra del Rey de Portuguall on the Cantino map. The map features a peculiar landmass in roughly the location of North America. Several theories offer potential explanations for this land that terminates in a peninsula, labeled " C. do fim do abrill " or "‘Cape of the end of April", pointing towards the Caribbean. It has been linked to Asia, the Yucatan, Florida, and Cuba. The area includes few defined cartographic details and names seemingly connected to

3672-564: The English expedition of 1498. But the depiction of the island on the map suggests it was based on the Portuguese mission of Labrador and Barcelos. Newfoundland was probably visited by an English expedition in 1497–98, and again, by the Portuguese explorer Gaspar Corte-Real in 1500 and 1501. The map makes clear that the land was discovered and charted for Portugal by Gaspar Corte-Real , for King Manuel I of Portugal . Portuguese discoveries Portuguese maritime exploration resulted in

3774-767: The Indian Ocean coast in Somalia , traveling secretly overland, as early as 1490; a diplomatic mission reached the ruler of that nation on October 19, 1520. In 1500, the second fleet to India (which also made landfall in Brazil) explored the East African coast in Southeast Africa , where Diogo Dias discovered the island that he named St. Lawrence, later known as Madagascar . This fleet, commanded by Pedro Álvares Cabral , arrived at Calicut in September, where

3876-613: The Indian Ocean. Under the government of Albuquerque, Goa was taken from the Bijapur sultanate in 1510 with the help of Hindu privateer Timoji . Coveted for being the best port in the region, mainly for the commerce in Arabian horses for the Deccan sultanates , it allowed the Portuguese to move on from their initial guest stay in Cochin. Despite constant attacks, Goa became the seat of

3978-831: The Island of Mozambique and Mombasa on the Kenyan coast. Madagascar was partly explored by Tristão da Cunha and in the same year Mauritius was discovered. In 1509, the Portuguese won the sea Battle of Diu against the combined forces of the Ottoman Sultan Beyazid II , the Sultan of Gujarat , the Mamlûk Sultan of Cairo , the Samoothiri Raja of Kozhikode , the Venetian Republic , and

4080-487: The Moluccas, Albuquerque sent an expedition to find them. Led by António de Abreu , the expedition arrived in early 1512. Abreu went by Ambon, while his deputy commander Francisco Serrão advanced to Ternate , where a Portuguese fort was allowed. That same year, in Indonesia, the Portuguese took Makassar , reaching Timor in 1514. Departing from Malacca, Jorge Álvares came to southern China in 1513. This visit followed

4182-509: The Portuguese already established in nearby Ternate was inevitable, starting nearly a decade of skirmishes. An agreement was reached only with the Treaty of Zaragoza (1529) , which gave the Moluccas to Portugal and the Philippines to Spain. In 1530, John III organized the colonization of Brazil around 15 capitanias hereditárias ("hereditary captainships"), that were given to anyone who wanted to administer and explore them, to overcome

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4284-468: The Portuguese coat of arms marking their claims, and built forts and trading posts. From these bases, the Portuguese engaged profitably in the slave and gold trades. Portugal enjoyed a virtual monopoly of the Atlantic slave trade for over a century, exporting around 800 slaves annually. Most were brought to the Portuguese capital Lisbon, where it is estimated black Africans came to constitute 10 percent of

4386-472: The Portuguese government, under the name of Estado da India (State of India) , with the conquest triggering compliance of neighbor kingdoms: Gujarat and Calicut sent embassies, offering alliances and grants to fortify. Albuquerque began that year in Goa the first Portuguese mint in India, taking the opportunity to announce the achievement. In April 1511 Albuquerque sailed to Malacca in modern-day Malaysia,

4488-705: The Portuguese naval efforts. The first victims of slave raids by Portuguese and Spanish were the Guanches of the Canary Islands, a people of Berber origin, who put up fierce resistance but were reduced to near extinction by pillaging and enslavement. In 1415, the Portuguese occupied the North African city of Ceuta to gain a foothold in Morocco and control shipping through the Strait of Gibraltar. They also hoped to extend Christianity and provide an outlet for Portuguese nobles looking to gain riches and honor in war. Among

4590-667: The Portuguese nobleman Pedro Álvares Cabral became the first European to discover Brazil . In 1297, King Dinis of Portugal took personal interest in the development of exports and organized the export of surplus production to European countries. On May 10, 1293, he instituted a maritime insurance fund for Portuguese traders living in the County of Flanders , which were to pay certain sums according to tonnage, accrued to them when necessary. Wine and dried fruits from Algarve were sold in Flanders and England, salt from Setúbal and Aveiro

4692-443: The Portuguese, establishing an alliance to regain the country, giving in exchange Daman , Diu , Mumbai , and Bassein . In 1538 the fortress of Diu was again surrounded by Ottoman ships. Another siege failed in 1547, putting an end to Ottoman ambitions and confirming Portuguese hegemony. In 1542 Jesuit missionary Francis Xavier arrived in Goa at the service of King John III of Portugal , in charge of an Apostolic Nunciature . At

4794-536: The Tordesillas line. Cabral recommended to the Portuguese King that the land be settled, and two follow-up voyages were sent in 1501 and 1503. The land was found to be abundant in pau-brasil , or brazilwood , from which it later inherited its name, but the failure to find gold or silver meant that for the time being Portuguese efforts were concentrated on India. The aim of Portugal in the Indian Ocean

4896-778: The arrival in Guangzhou , where trade was established. Later a trading post at Macau would be established. The Portuguese empire expanded into the Persian Gulf as Portugal contested control of the spice trade with the Ottoman Empire . In 1515, Afonso de Albuquerque conquered the Huwala state of Hormuz at the head of the Persian Gulf, establishing it as a vassal state. Aden , however, resisted Albuquerque's expedition in that same year, and another attempt by Albuquerque's successor Lopo Soares de Albergaria in 1516. Bahrain

4998-545: The coast of Mauritania . It was created to attract Muslim traders and monopolize the business in the routes traveled in North Africa, starting the chain of Portuguese feitorias along the coast. In 1446, Álvaro Fernandes pushed on almost as far as present-day Sierra Leone , and the Gulf of Guinea was reached in the 1460s. As a result of the first meager returns of the African explorations, in 1469 king Afonso V granted

5100-501: The coast. Between 1325 and 1357 Afonso IV of Portugal granted public funding to raise a proper commercial fleet and ordered the first maritime explorations, with the help of Genoese , under command of admiral Manuel Pessanha. In 1341 the Canary Islands , already known to Genoese seafarers, were officially rediscovered under the patronage of the Portuguese king, but in 1344 Castile disputed ownership of them, further propelling

5202-462: The control in coastal areas, where in 1517 was founded the fortress of Colombo. In 1506, a Portuguese fleet under the command of Tristão da Cunha and Afonso de Albuquerque , conquered Socotra at the entrance of the Red Sea and Muscat in 1507, having failed to conquer Ormuz , following a strategy intended to close those entrances into the Indian Ocean. That same year, fortresses were built in

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5304-477: The defeat at Tangier, Henry retired to Sagres on the southern tip of Portugal where he continued to direct Portuguese exploration until his death in 1460. In 1443, Prince Pedro , Henry's brother, granted him the monopoly of navigation, war, and trade in the lands south of Cape Bojador . Later this monopoly would be backed by the Papal bulls Dum Diversas (1452) and Romanus Pontifex (1455), granting Portugal

5406-465: The destination point does not remain constant. If one were to drive a car along a great circle one would hold the steering wheel fixed, but to follow a rhumb line one would have to turn the wheel, turning it more sharply as the poles are approached. In other words, a great circle is locally "straight" with zero geodesic curvature , whereas a rhumb line has non-zero geodesic curvature. Meridians of longitude and parallels of latitude provide special cases of

5508-508: The enterprise and financial experience of these rivals of the Republic of Venice . In the second half of the fourteenth century outbreaks of bubonic plague led to severe depopulation: the economy was extremely localized in a few towns, and migration from the country led to the abandonment of agricultural land and an increase in rural unemployment. Only the sea offered opportunities, with most people settling in fishing and trading areas along

5610-625: The equator into the Southern Hemisphere and found the islands in the Gulf of Guinea, including São Tomé and Príncipe . In 1471, Gomes' explorers reached Elmina on the Gold Coast (present day Ghana ), and discovered a thriving overland gold trade between the natives and visiting Arab and Berber traders. Gomes established his own trading post there, which became known as “A Mina” ("The Mine"). Trade between Elmina and Portugal grew in

5712-655: The exclusive patronage of the Portuguese Crown. The second voyage to India was dispatched in 1500 under Pedro Álvares Cabral . While following the same south-westerly route across the Atlantic Ocean as da Gama (to take advantage of the most favorable winds), Cabral made landfall on the Brazilian coast. This was probably an accidental discovery, but it has been speculated that the Portuguese secretly knew of Brazil's existence and that it lay on their side of

5814-451: The first trade agreement in India was signed. For a short time a Portuguese factory was installed there, but it was attacked by Muslims on December 16 and several Portuguese, including the scribe Pêro Vaz de Caminha , died. After bombarding Calicut as a retaliation, Cabral went to rival Kochi . Profiting from the rivalry between the Maharaja of Kochi and the Zamorin of Calicut, the Portuguese were well received and seen as allies, getting

5916-406: The following years Portuguese mariners discovered and settled the rest of the Azores. Henry suffered a serious setback in 1437 after the failure of an expedition to capture Tangier , having encouraged his brother, King Edward , to mount an overland attack from Ceuta. The Portuguese army was defeated and only escaped destruction by surrendering Prince Ferdinand, the king's youngest brother . After

6018-484: The full loxodrome on an infinitely high map would consist of infinitely many line segments between the two edges. On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the north or south pole. All loxodromes spiral from one pole to the other. Near the poles, they are close to being logarithmic spirals (which they are exactly on a stereographic projection , see below), so they wind around each pole an infinite number of times but reach

6120-486: The goal of defending the country against Muslim pirate raids, thus laying the basis for the Portuguese Navy and establishment of a Genoese merchant community in Portugal. Forced to reduce their activities in the Black Sea, the Republic of Genoa had turned to North Africa for trade in wheat and olive oil and a search for gold – navigating also into the ports of Bruges (Flanders) and England. Genoese and Florentine communities were established in Portugal, which profited from

6222-410: The information brought by the fleet of João da Nova , who arrived in Lisbon in September 1502. In the third phase the island named quaresma , together with some names written in cursive would have been added. The European rediscovery of Greenland is thought to have been by João Fernandes Labrador and Pedro de Barcelos between 1495 and 1498, and also visited by Giovanni Caboto (John Cabot), in

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6324-462: The islands began. There, wheat and later sugarcane were cultivated, as in Algarve, by the Genoese, becoming profitable activities. This helped both them and Prince Henry become wealthier. A Portuguese attempt to capture Grand Canary , one of the nearby Canary Islands , which had been partially settled by Spaniards in 1402 was unsuccessful and met with protestations from Castile. Although the exact details are uncertain, cartographic evidence suggests

6426-442: The islands recently [discovered]... in part of the Indies: from Alberto Cantino to Duke Hercole". While it enlightened the Italians to many new territories yet unknown to them, it was made obsolete within months by subsequent Portuguese mapping voyages. Nevertheless, its importance to the Portuguese–Italian trade relations should not be understated; this map provided the Italians with knowledge of Brazil's coastline and that of much of

6528-418: The longitude. The ship would sail north or south until the latitude of the destination was reached, and the ship would then sail east or west along the rhumb line (actually a parallel , which is a special case of the rhumb line), maintaining a constant latitude and recording regular estimates of the distance sailed until evidence of land was sighted. The surface of the Earth can be understood mathematically as

6630-422: The monopoly of trade in part of the Gulf of Guinea to merchant Fernão Gomes , for an annual payment of 200,000 reals . Gomes was also required to explore 100 leagues (480 km) of the coast each year for five years. He employed explorers João de Santarém , Pedro Escobar , Lopo Gonçalves , Fernão do Pó , and Pedro de Sintra , and exceeded the requirement. Under his sponsorship, Portuguese explorers crossed

6732-415: The most important eastern point in the trade network, where Malay met Gujarati, Chinese, Japanese, Javanese, Bengali, Persian and Arabic traders, described by Tomé Pires as invaluable. The port of Malacca became then the strategic base for Portuguese trade expansion with China and Southeast Asia, under the Portuguese rule in India with its capital at Goa . To defend the city a strong fort was erected, called

6834-407: The name 'portolan'. For a sphere of radius 1, the azimuthal angle λ , the polar angle − ⁠ π / 2 ⁠ ≤ φ ≤ ⁠ π / 2 ⁠ (defined here to correspond to latitude), and Cartesian unit vectors i , j , and k can be used to write the radius vector r as Orthogonal unit vectors in the azimuthal and polar directions of the sphere can be written which have

6936-426: The need to defend the territory, since an expedition under the command of Gonçalo Coelho in 1503 had found the French making incursions on the land. That same year, there was a new expedition from Martim Afonso de Sousa with orders to patrol the whole Brazilian coast, banish the French, and create the first colonial towns: São Vicente on the coast, and São Paulo near the edge of the inland plateau ( planalto ) and

7038-406: The newly discovered western lands on the highly influential wall map of the world produced by Martin Waldseemüller in 1507 under the auspices of Rene, Duke of Lorraine. This old map, made of six glued parchment sheets, was kept in the Ducal Library, Ferrara , for about 90 years, until Pope Clement VIII transferred it to another palace in Modena , Italy. More than two centuries later, in 1859,

7140-427: The next decade. In 1481, the recently crowned João II decided to build São Jorge da Mina fort (Elmina Castle) and factory to protect this trade, which was then held again as a royal monopoly. In 1482, Diogo Cão discovered the mouth of the Congo River . In 1486, Cão continued to Cape Cross , in present-day Namibia , near the Tropic of Capricorn . In 1488, Bartolomeu Dias rounded the Cape of Good Hope near

7242-478: The numerous territories and maritime routes recorded by the Portuguese as a result of their intensive maritime journeys during the 15th and 16th centuries. Portuguese sailors were at the vanguard of European exploration, chronicling and mapping the coasts of Africa and Asia, then known as the East Indies , and Canada and Brazil (the West Indies ), in what came to be known as the Age of Discovery . Methodical expeditions started in 1419 along West Africa's coast under

7344-738: The palace was ransacked and the Cantino Map lost. It was found by Giuseppe Boni, Director of the Biblioteca Estense, in that same year, in a butcher's store in Modena. The Cantino world map can currently be found in Modena , Italy, at the Biblioteca Estense . The Cantino planisphere is the earliest extant example of the so-called latitude chart , which was developed following the introduction of astronomical navigation, during

7446-687: The participants of the action was the young Prince Henry the Navigator . Appointed governor of the Order of Christ in 1420, while personally holding profitable monopolies on resources in Algarve, he took the lead role in encouraging Portuguese maritime exploration until his death in 1460. He invested in sponsoring voyages down the coast of Mauritania , gathering a group of merchants, shipowners, and other stakeholders interested in new opportunities for maritime trade. Later his brother Prince Pedro granted him

7548-600: The pole in a finite distance. The pole-to-pole length of a loxodrome (assuming a perfect sphere ) is the length of the meridian divided by the cosine of the bearing away from true north. Loxodromes are not defined at the poles. The word loxodrome comes from Ancient Greek λοξός loxós : "oblique" + δρόμος drómos : "running" (from δραμεῖν drameîn : "to run"). The word rhumb may come from Spanish or Portuguese rumbo/rumo ("course" or "direction") and Greek ῥόμβος rhómbos , from rhémbein . The 1878 edition of The Globe Encyclopaedia of Universal Information describes

7650-545: The population. In 1492 Christopher Columbus 's discovery for Spain of the New World , which he believed to be Asia, led to disputes between the Spanish and Portuguese. These were eventually settled by the Treaty of Tordesillas in 1494 which divided the world outside of Europe in an exclusive duopoly between the Portuguese and the Spanish, along a north–south meridian 370 leagues , or 970 miles (1,560 km), west of

7752-573: The portolan-chart model ). The construction of the rhumb line system in the Cantino planisphere uses two circles (some charts use only one, others use as many as three, depending on size): the western circle is centered on the Cape Vert islands, the eastern circle is centered in India. The circumference of each circle is marked with sixteen equally spaced points, from which radiate the 32 classic rhumbs: 0°, 11 1/4°, 22 1/2°, 33 3/4°, etc. The western and eastern outer circles are tangent to each other at

7854-411: The rhumb line, where their angles of intersection are respectively 0° and 90°. On a north–south passage the rhumb line course coincides with a great circle, as it does on an east–west passage along the equator . On a Mercator projection map, any rhumb line is a straight line; a rhumb line can be drawn on such a map between any two points on Earth without going off the edge of the map. But theoretically

7956-519: The right that this is the "land of the King of Nubia, the king of which is continuously making war on Prester John and is a moor and a great enemy of Christians". By the time of the map's creation, European voyages had made landfall across the Atlantic Ocean. Christopher Columbus had completed his first three voyages to a land that he called both Cuba and Asia. John Cabot had completed three voyages from Bristol under Henry VII of England . Very little

8058-655: The same time Francisco Zeimoto, António Mota , and other traders arrived in Japan for the first time. According to Fernão Mendes Pinto , who claimed to be in this journey, they arrived at Tanegashima , where the locals were impressed by European firearms , which would be immediately made by the Japanese on a large scale. In 1557 the Chinese authorities allowed the Portuguese to settle in Macau through an annual payment, creating

8160-401: The same, at 10,000 kilometres (5,400 nautical miles). At 20 degrees north the great circle distance is 9,254 km (4,997 nmi) while the rhumb line distance is 9,397 km (5,074 nmi), about 1.5% further. But at 60 degrees north the great circle distance is 4,602 km (2,485 nmi) while the rhumb line is 5,000 km (2,700 nmi), a difference of 8.5%. A more extreme case

8262-574: The second half of the fifteenth century. Contrary to the portolan charts of the Mediterranean, which were constructed on the basis of magnetic courses and estimated distances between places, in the latitude chart, places were represented according to their latitudes. In the Cantino planisphere, latitudes were incorporated only in the coasts of Africa, Brazil and India, while Europe and the Caribbean Sea continued to be represented according to

8364-524: The so-called Padrão Real , kept in the Armazéns da Índia . However, there is no historical evidence that such order was ever made and the theory is weakened by the presence of numerous mistakes. One would expect a carefully made copy of an official standard, if it existed in Portugal at that time, would be accurate. A more plausible explanation is that the map was surreptitiously acquired shortly after it

8466-415: The southern tip of Africa, disproving the view that had existed since Ptolemy that the Indian Ocean was separate from the Atlantic. Also at this time, Pêro da Covilhã reached India via Egypt and Yemen , and visited Madagascar. He recommended further exploration of the southern route. As the Portuguese explored the coastlines of Africa, they left behind a series of padrões , stone crosses inscribed with

8568-454: The sphere ever so fast in a spiral towards the pole, while tending to a finite total arc length Δ s given by Let λ be the longitude of a point on the sphere, and φ its latitude. Then, if we define the map coordinates of the Mercator projection as a loxodrome with constant bearing β from true north will be a straight line, since (using the expression in the previous section) with

8670-636: The sponsorship of prince Henry the Navigator , with Bartolomeu Dias reaching the Cape of Good Hope and entering the Indian Ocean in 1488. Ten years later, in 1498, Vasco da Gama led the first fleet around Africa to the Indian subcontinent , arriving in Calicut and starting a maritime route from Portugal to India. Portuguese explorations then proceeded to southeast Asia, where they reached Japan in 1542, forty-four years after their first arrival in India. In 1500,

8772-573: The surface of a globe was first discussed by the Portuguese mathematician Pedro Nunes in 1537, in his Treatise in Defense of the Marine Chart , with further mathematical development by Thomas Harriot in the 1590s. A rhumb line can be contrasted with a great circle , which is the path of shortest distance between two points on the surface of a sphere. On a great circle, the bearing to

8874-453: The time, as Europeans did not know what lay beyond on the African coast, and did not know whether it was possible to return once it was passed. Henry wished to know how far the Muslim territories in Africa extended, and whether it was possible to reach the source of the lucrative tran-Saharan caravan gold trade and perhaps to join forces with the long-lost Christian kingdom of Prester John that

8976-570: The trade between Asia and Europe, but also much of the trade between different regions of Asia, such as India, Indonesia, China, and Japan. Jesuit missionaries, such as the Francis Xavier , followed the Portuguese to spread Christianity to Asia with mixed success. 15th century 16th century 15th century 16th century 17th century 18th century 19th century 16th century 17th century Rhumb line In navigation ,

9078-916: The unit vector β̂ . As a result, a differential length ds along the loxodrome will produce a differential displacement where gd {\displaystyle \operatorname {gd} } and gd − 1 {\displaystyle \operatorname {gd} ^{-1}} are the Gudermannian function and its inverse, gd ⁡ ψ = arctan ⁡ ( sinh ⁡ ψ ) , {\displaystyle \operatorname {gd} \psi =\arctan(\sinh \psi ),} gd − 1 ⁡ φ = arsinh ⁡ ( tan ⁡ φ ) , {\displaystyle \operatorname {gd} ^{-1}\varphi =\operatorname {arsinh} (\tan \varphi ),} and arsinh {\displaystyle \operatorname {arsinh} }

9180-673: The voyages of Columbus, Cabot, and Corte Real. Other maps depicting the same land include the Caverio map , the maps of Martin Waldseemüller , and the Johannes Schöner globes . These describe the land variously as Terra ultra incognita ("Land beyond unknown"), Vlterius incognita terra ("Land further beyond unknown"), Terra de Cvba ("Land of Cuba"), Parias (a native place name from Four Voyages of Amerigo Vespucci ), and Asie partis ("Part of Asia"). The Brazilian coast

9282-472: Was Alberto Cantino, who was sent to Portugal by Ercole I d'Este, Duke of Ferrara , with the formal intention of horse trading , while secretly collecting information on Portuguese discoveries . A popular theory, introduced in the earliest studies of the map, suggests that the Cantino Planisphere was ordered to an official Portuguese mapmaker, who made a copy of the royal cartographic pattern,

9384-435: Was a profitable export to northern Europe, and leather and kermes , a scarlet dye, were also exported. Portugal imported armor and munitions, fine clothes, and several manufactured products from Flanders and Italy. In 1317, King Dinis made an agreement with Genoese merchant sailor Manuel Pessanha (Pessagno), appointing him first Admiral with trade privileges with his homeland in return for twenty warships and crews, with

9486-519: Was captured in 1521, when a force led by António Correia defeated the Jabrid King, Muqrin ibn Zamil . In a shifting series of alliances, the Portuguese dominated much of the southern Persian Gulf for the next hundred years. The island of Mozambique became a strategic port on the regular maritime route linking Lisbon to Goa, and Fort São Sebastião and a hospital were built there. In the Azores,

9588-448: Was certainly the last to be added and reached its present form in three phases: to the first belong an initial coastline running to southeast from Golfo fremosso to Cabo Sam Jorge , and from there, to north of Porto Seguro , continuing further south to the tip of the landmass. An inscription off Porto Seguro records the discovery and naming of Vera Cruz, as Brazil was initially called: Porto Seguro. Vera Cruz, so called by this name,

9690-450: Was discovered in 1500. Only a relatively small portion of the coast, between the flag near the Vera cruz inscription and the northern side of the baia de todos os santos would have been surveyed, in 1500, by the fleet of Pedro Álvares Cabral . To the second phase belongs the pasting of the strip of parchment between Rio de sã franc° and Golfo fremosso , which would have been based on

9792-481: Was found by Pedro Alvares Cabral, a gentleman of the household of the King of Portugal, which he discovered in going as commander of fourteen ships that the King sent to Calicut and, on the way to India, he came across this land here, which he thought to be mainland [terra firma], in which there are many people who are observed, men and women, to walk about as naked as their mothers bore them: they are rather fair-skinned than reddish brown and have very slick hair. This land

9894-440: Was known about Cabot's third voyage, including whether Cabot ever returned to England. While sailing to Greenland, known but little understood by contemporary Europeans, João Fernandes Lavrador made landfall on a nearby coast. Gaspar Corte-Real and his brother Miguel , members of the Portuguese royal household, sailed West under Manuel I of Portugal to find a Northwest Passage to Asia. Newfoundland , visited in 1500 and 1501 by

9996-483: Was made for some nobleman or official client. From a letter sent by Cantino to his patron, the Duke of Ferrara, on 19 November 1502, we know that he paid 12 golden ducats for it, which was a considerable amount for the time. An Italian inscription in the back of the map reads: Carta de navigar per le Isole nouam trovate in le parte de India: dono Alberto Cantino al S. Duca Hercole , which translates as "Navigational chart of

10098-489: Was rumoured to exist somewhere to the east. In 1434, one of Prince Henry's captains, Gil Eanes , passed this obstacle. Once this psychological barrier had been crossed, it became easier to probe further along the coast. Within two decades of exploration, Portuguese ships had bypassed the Sahara . Westward exploration continued over the same period: Diogo de Silves discovered the Azores island of Santa Maria in 1427 and in

10200-582: Was the case, the long-standing Portuguese goal of finding a sea route to Asia was finally achieved in a ground-breaking voyage commanded by Vasco da Gama . Vasco da Gama's squadron left Portugal on 8 July 1497, consisting of four ships and a crew of 170 men. It rounded the Cape and continued along the coast of Southeast Africa, where a local pilot was brought on board who guided them across the Indian Ocean, reaching Calicut in western India in May 1498. After some conflict, da Gama got an ambiguous letter for trade with

10302-426: Was to ensure the monopoly of the spice trade . Taking advantage of the rivalries that pitted Hindus against Muslims, the Portuguese established several forts and trading posts between 1500 and 1510. In East Africa , small Islamic states along the coast of Mozambique , Kilwa , Brava , Sofala and Mombasa were destroyed, or became either subjects or allies of Portugal. Pêro da Covilhã had reached Ethiopia (via

10404-455: Was valuable at the beginning of the sixteenth century because it showed detailed and up-to-date strategic information in a time when geographic knowledge of the world was growing at a fast pace. It is important in the modern era because it contains unique historical information about the maritime exploration and the evolution of nautical cartography during the Age of Discovery . The Cantino planisphere

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