The Southern Hemisphere is the half of Earth, south of the Equator. It contains parts of five continents, four oceans and most of the Pacific Islands in Oceania, its surface is 80.9% water, compared with 60.7% water in the case of the Northern Hemisphere, it contains 32.7% of Earth's land. Owing to the tilt of Earth's rotation relative to the Sun and the ecliptic plane, summer is from December to March and winter is from June to September. September 22 or 23 is the vernal equinox and March 20 or 21 is the autumnal equinox; the South Pole is in the center of the southern hemispherical region. Southern Hemisphere climates tend to be milder than those at similar latitudes in the Northern Hemisphere, except in the Antarctic, colder than the Arctic; this is because the Southern Hemisphere has more ocean and much less land. The differences are attributed to oceanic heat transfer and differing extents of greenhouse trapping. In the Southern Hemisphere the sun passes from east to west through the north, although north of the Tropic of Capricorn the mean sun can be directly overhead or due north at midday.
The Sun rotating through the north causes an apparent right-left trajectory through the sky unlike the left-right motion of the Sun when seen from the Northern Hemisphere as it passes through the southern sky. Sun-cast shadows turn anticlockwise throughout the day and sundials have the hours increasing in the anticlockwise direction. During solar eclipses viewed from a point to the south of the Tropic of Capricorn, the Moon moves from left to right on the disc of the Sun, while viewed from a point to the north of the Tropic of Cancer, the Moon moves from right to left during solar eclipses. Cyclones and tropical storms spin clockwise in the Southern Hemisphere due to the Coriolis effect; the southern temperate zone, a subsection of the Southern Hemisphere, is nearly all oceanic. This zone includes the southern tip of South Africa; the Sagittarius constellation that includes the galactic centre is a southern constellation and this, combined with clearer skies, makes for excellent viewing of the night sky from the Southern Hemisphere with brighter and more numerous stars.
Forests in the Southern Hemisphere have special features which set them apart from those in the Northern Hemisphere. Both Chile and Australia share, for example, unique beech species or Nothofagus, New Zealand has members of the related genera Lophozonia and Fuscospora; the eucalyptus is native to Australia but is now planted in Southern Africa and Latin America for pulp production and biofuel uses. 800 million humans live in the Southern Hemisphere, representing only 10–12% of the total global human population of 7.3 billion. Of those 800 million people, 200 million live in Brazil, the largest country by land area in the Southern Hemisphere, while 141 million live on the island of Java, the most populous island in the world; the most populous nation in the Southern Hemisphere is Indonesia, with 261 million people. Portuguese is the most spoken language in the Southern Hemisphere, followed by Javanese; the largest metropolitan areas in the Southern Hemisphere are São Paulo, Buenos Aires, Rio de Janeiro and Sydney.
The most important financial and commercial centers in the Southern Hemisphere are São Paulo, where the Bovespa Index is headquartered, along with Sydney, home to the Australian Securities Exchange, home to the Johannesburg Stock Exchange and Buenos Aires, headquarters of the Buenos Aires Stock Exchange, the oldest stock market in the Southern Hemisphere. Among the most developed nations in the Southern Hemisphere are Australia, with a nominal GDP per capita of US$51,850 and a Human Development Index of 0.939, the second highest in the world as of 2016. New Zealand is well developed, with a nominal GDP per capita of US$38,385 and a Human Development Index of 0.915, putting it at #13 in the world in 2016. The least developed nations in the Southern Hemisphere cluster in Africa and Oceania, with Burundi and Mozambique at the lowest ends of the Human Development Index, at 0.404 and 0.418 respectively. The nominal GDP per capitas of these two countries don't go above US$550 per capita, a tiny fraction of the incomes enjoyed by Australians and New Zealanders.
The most widespread religions in the Southern Hemisphere are Christianity in South America, southern Africa and Australia/New Zealand, followed by Islam in most of the islands of Indonesia and in parts of southeastern Africa, Hinduism, concentrated on the island of Bali and neighboring islands. The oldest continuously inhabited city in the Southern Hemisphere is Bogor, in western Java, founded in 669 CE. Ancient texts from the Hindu kingdoms prevalent in the area definitively record 669 CE as the year when Bogor was founded. However, there is some evidence that Zanzibar, an ancient port with around 200,000 inhabitants on
An analog computer or analogue computer is a type of computer that uses the continuously changeable aspects of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved. In contrast, digital computers represent varying quantities symbolically, as their numerical values change; as an analog computer does not use discrete values, but rather continuous values, processes cannot be reliably repeated with exact equivalence, as they can with Turing machines. Unlike machines used for digital signal processing, analog computers do not suffer from the discrete error caused by quantization noise. Instead, results from analog computers are subject to continuous error caused by electronic noise. Analog computers were used in scientific and industrial applications where digital computers of the time lacked sufficient performance. Analog computers can have a wide range of complexity. Slide rules and nomograms are the simplest, while naval gunfire control computers and large hybrid digital/analog computers were among the most complicated.
Systems for process control and protective relays used analog computation to perform control and protective functions. The advent of digital computing made simple analog computers obsolete as early as the 1950s and 1960s, although analog computers remained in use in some specific applications, like the flight computer in aircraft, for teaching control systems in universities. More complex applications, such as synthetic aperture radar, remained the domain of analog computing well into the 1980s, since digital computers were insufficient for the task. Setting up an analog computer required scale factors to be chosen, along with initial conditions—that is, starting values. Another essential was creating the required network of interconnections between computing elements. Sometimes it was necessary to re-think the structure of the problem so that the computer would function satisfactorily. No variables could be allowed to exceed the computer's limits, differentiation was to be avoided by rearranging the "network" of interconnects, using integrators in a different sense.
Running an electronic analog computer, assuming a satisfactory setup, started with the computer held with some variables fixed at their initial values. Moving a switch released the holds and permitted the problem to run. In some instances, the computer could, after a certain running time interval return to the initial-conditions state to reset the problem, run it again; this is a list of examples of early computation devices which are considered to be precursors of the modern computers. Some of them may have been dubbed as'computers' by the press, although they may fail to fit the modern definitions; the south-pointing chariot, invented in ancient China during the first millennium BC, can be considered the earliest analog computer. It was a mechanical-geared wheeled vehicle used to discern the southern cardinal direction; the Antikythera mechanism was an orrery and is claimed to be an early mechanical analog computer, according to Derek J. de Solla Price. It was designed to calculate astronomical positions.
It was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, has been dated to circa 100 BC. Devices of a level of complexity comparable to that of the Antikythera mechanism would not reappear until a thousand years later. Many mechanical aids to calculation and measurement were constructed for astronomical and navigation use; the planisphere was a star chart invented by Abū Rayḥān al-Bīrūnī in the early 11th century. The astrolabe was invented in the Hellenistic world in either the 1st or 2nd centuries BC and is attributed to Hipparchus. A combination of the planisphere and dioptra, the astrolabe was an analog computer capable of working out several different kinds of problems in spherical astronomy. An astrolabe incorporating a mechanical calendar computer and gear-wheels was invented by Abi Bakr of Isfahan, Persia in 1235. Abū Rayhān al-Bīrūnī invented the first mechanical geared lunisolar calendar astrolabe, an early fixed-wired knowledge processing machine with a gear train and gear-wheels, circa 1000 AD.
The castle clock, a hydropowered mechanical astronomical clock invented by Al-Jazari in 1206, was the first programmable analog computer. The sector, a calculating instrument used for solving problems in proportion, trigonometry and division, for various functions, such as squares and cube roots, was developed in the late 16th century and found application in gunnery and navigation; the planimeter was a manual instrument to calculate the area of a closed figure by tracing over it with a mechanical linkage. The slide rule was invented around 1620–1630, shortly after the publication of the concept of the logarithm, it is a hand-operated analog computer for doing division. As slide rule development progressed, added scales provided reciprocals and square roots and cube roots, as well as transcendental functions such as logarithms and exponentials and hyperbolic trigonometry and other functions. Aviation is one of the few fields where slide rules are still in widespread use for solving time–distance problems in light aircraft.
Mathematician and engineer Giovanni Plana devised a Perpetual Calendar machine which, though a system of pulleys and cylinders and over, could predict the perpetual calendar for every year from 0AD to 4000AD, keeping track of leap years and varying day length. The tide-predicting machine invented by Sir William Thomson in 1872 was of great utility to navigation in shallow waters, it used a system of pulleys and wires to automatically calculate predicted tide levels for a set period at a particular location. The di
Jakob Bartsch or Jacobus Bartschius was a German astronomer. Bartsch was born in Lauban in Lusatia, he was taught how to use the astrolabe by Sarcephalus, a librarian in Breslau. He studied astronomy and medicine at the University of Strassburg. In 1624 Bartsch published a book titled Usus astronomicus planisphaerii stellati containing star charts that depicted six new constellations introduced around 1613 by Petrus Plancius on a celestial globe published by Pieter van den Keere; these six new constellations were Camelopardalis, Jordanis, Monoceros and Vespa. He mentioned but did not depict Rhombus, a separate invention by Isaac Habrecht II. Bartsch was wrongly credited with having invented these figures. Only Camelopardalis and Monoceros survive today. Bartsch married Johannes Kepler's daughter Susanna on 12 March 1630 and helped Kepler with his calculations. After Kepler's death in 1630, Bartsch edited Kepler's posthumous work Somnium, he helped gather money from Kepler's estate for his widow. Bartsch died in Lauban in 1633.
Bartsch, Jacob. Usus Astronomicus Planisphaerii Stellati, 1624; the first cartographic use of the term planisphere
A calendar is a system of organizing days for social, commercial or administrative purposes. This is done by giving names to periods of time days, weeks and years. A date is the designation of a specific day within such a system. A calendar is a physical record of such a system. A calendar can mean a list of planned events, such as a court calendar or a or chronological list of documents, such as a calendar of wills. Periods in a calendar are though not synchronised with the cycle of the sun or the moon; the most common type of pre-modern calendar was the lunisolar calendar, a lunar calendar that adds one intercalary month to remain synchronised with the solar year over the long term. The term calendar is taken from calendae, the term for the first day of the month in the Roman calendar, related to the verb calare "to call out", referring to the "calling" of the new moon when it was first seen. Latin calendarium meant "account book, register"; the Latin term was adopted in Old French as calendier and from there in Middle English as calender by the 13th century.
A calendar can be on paper or electronic device. The course of the sun and the moon are the most salient natural recurring events useful for timekeeping, thus in pre-modern societies worldwide lunation and the year were most used as time units; the Roman calendar contained remnants of a ancient pre-Etruscan 10-month solar year. The first recorded physical calendars, dependent on the development of writing in the Ancient Near East, are the Bronze Age Egyptian and Sumerian calendars. A large number of Ancient Near East calendar systems based on the Babylonian calendar date from the Iron Age, among them the calendar system of the Persian Empire, which in turn gave rise to the Zoroastrian calendar and the Hebrew calendar. A great number of Hellenic calendars developed in Classical Greece, in the Hellenistic period gave rise to both the ancient Roman calendar and to various Hindu calendars. Calendars in antiquity were lunisolar, depending on the introduction of intercalary months to align the solar and the lunar years.
This was based on observation, but there may have been early attempts to model the pattern of intercalation algorithmically, as evidenced in the fragmentary 2nd-century Coligny calendar. The Roman calendar was reformed by Julius Caesar in 45 BC; the Julian calendar was no longer dependent on the observation of the new moon but followed an algorithm of introducing a leap day every four years. This created a dissociation of the calendar month from the lunation; the Islamic calendar is based on the prohibition of intercalation by Muhammad, in Islamic tradition dated to a sermon held on 9 Dhu al-Hijjah AH 10. This resulted in an observation-based lunar calendar that shifts relative to the seasons of the solar year; the first calendar reform of the early modern era was the Gregorian calendar, introduced in 1582 based on the observation of a long-term shift between the Julian calendar and the solar year. There have been a number of modern proposals for reform of the calendar, such as the World Calendar, International Fixed Calendar, Holocene calendar, the Hanke-Henry Permanent Calendar.
Such ideas are mooted from time to time but have failed to gain traction because of the loss of continuity, massive upheaval in implementation, religious objections. A full calendar system has a different calendar date for every day, thus the week cycle is by itself not a full calendar system. The simplest calendar system just counts time periods from a reference date; this applies for Unix Time. The only possible variation is using a different reference date, in particular, one less distant in the past to make the numbers smaller. Computations in these systems are just a matter of subtraction. Other calendars have one larger units of time. Calendars that contain one level of cycles: week and weekday – this system is not common year and ordinal date within the year, e.g. the ISO 8601 ordinal date systemCalendars with two levels of cycles: year and day – most systems, including the Gregorian calendar, the Islamic calendar, the Solar Hijri calendar and the Hebrew calendar year and weekday – e.g. the ISO week dateCycles can be synchronized with periodic phenomena: Lunar calendars are synchronized to the motion of the Moon.
Solar calendars are based on perceived seasonal changes synchronized to the apparent motion of the Sun. Lunisolar calendars are based on a combination of both solar and lunar reckonings; the week cycle is an example of one, not synchronized to any external phenomenon. A calendar includes more than one type of cycle, or has both cyclic and non-cyclic elements. Most calendars incorporate more complex cycles. For example, the vast majority of them track years, months and days; the seven-day week is universal, though its use varies. It has run uninterrupted for millennia. Solar calendars assign a date to each solar day. A day may consist of the period between sunrise and sunset, with
Thābit ibn Qurra
Al-Ṣābiʾ Thābit ibn Qurrah al-Ḥarrānī was a Arab Sabian mathematician, physician and translator who lived in Baghdad in the second half of the ninth century during the time of Abbasid Caliphate. Thābit ibn Qurrah made important discoveries in algebra and astronomy. In astronomy, Thābit is considered one of the first reformers of the Ptolemaic system, in mechanics he was a founder of statics. Thābit was born in Harran in Upper Mesopotamia, which at the time was part of the Diyar Mudar subdivision of the al-Jazira region of the Abbasid Caliphate; the city of Harran was never Christianized. By the early Muslim conquests, the people of Harran were still adhering to the cult of Sin. Thābit and his pupils lived in the midst of the most intellectually vibrant, the largest, city of the time, Baghdad, he occupied himself with mathematics, astrology, mechanics and philosophy. In his life, Thābit's patron was the Abbasid Caliph al-Mu'tadid. Thābit became courtier. Thābit died in Baghdad. After him, the greatest Sabean name was al-Battani.
Thābit's native language was Syriac, the eastern Aramaic variety from Edessa, he was fluent in both Greek and Arabic. Thābit translated from Greek into Arabic works by Apollonius of Perga, Archimedes and Ptolemy, he revised the translation of Euclid's Elements of Hunayn ibn Ishaq. He rewrote Hunayn's translation of Ptolemy's Almagest and translated Ptolemy's Geography. Thābit's translation of a work by Archimedes which gave a construction of a regular heptagon was discovered in the 20th century, the original having been lost; the medieval astronomical theory of the trepidation of the equinoxes is attributed to Thābit. But it had been described by Theon of Alexandria in his comments of the Handy Tables of Ptolemy. According to Copernicus, Thābit determined the length of the sidereal year as 365 days, 6 hours, 9 minutes and 12 seconds. Copernicus based his claim on the Latin text attributed to Thābit. Thābit published his observations of the Sun. In mathematics, Thābit discovered an equation for determining amicable numbers.
He wrote on the theory of numbers, extended their use to describe the ratios between geometrical quantities, a step which the Greeks did not take. He is known for having calculated the solution to a chessboard problem involving an exponential series, he computed the volume of the paraboloid. He described a generalization of Pythagoras' theorem. In physics, Thābit rejected the Peripatetic and Aristotelian notions of a "natural place" for each element, he instead proposed a theory of motion in which both the upward and downward motions are caused by weight, that the order of the universe is a result of two competing attractions: one of these being "between the sublunar and celestial elements", the other being "between all parts of each element separately". and in mechanics he was a founder of statics. Only a few of Thābit's works are preserved in their original form. On the Sector-Figure which deals with Menelaus' theorem. On the Composition of Ratios Thabit number Thebit Roshdi Rashed, Thābit ibn Qurra.
Science and Philosophy in Ninth-Century Baghdad, Walter de Gruyter, 2009. Francis J. Carmody: The astronomical works of Thābit b. Qurra. 262 pp. Berkeley and Los Angeles: University of California Press, 1960. Rashed, Roshdi. Les Mathématiques Infinitésimales du IXe au XIe Siècle 1: Fondateurs et commentateurs: Banū Mūsā, Ibn Qurra, Ibn Sīnān, al-Khāzin, al-Qūhī, Ibn al-Samḥ, Ibn Hūd. London. Reviews: Seyyed Hossein Nasr in Isis 89 pp. 112-113. Churton, Tobias; the Golden Builders: Alchemists and the First Freemasons. Barnes and Noble Publishing, 2006. Hakim S Ayub Ali. Zakhira-i Thābit ibn Qurra, India, 1987. Palmeri, JoAnn. "Thābit ibn Qurra". In Thomas Hockey; the Biographical Encyclopedia of Astronomers. New York: Springer. Pp. 1129–30. ISBN 978-0-387-31022-0. O'Connor, John J.. Rosenfeld, B. A.. T.. "Thābit Ibn Qurra, Al-Ṣābiʾ Al-Ḥarrānī". Complete Dictionary of Scientific Biography. Encyclopedia.com. Thabit ibn Qurra on Astrology & Magic
Muhammad ibn Musa al-Khwarizmi
Muḥammad ibn Mūsā al-Khwārizmī Latinized as Algorithmi, was a Persian scholar who produced works in mathematics and geography under the patronage of the Caliph Al-Ma'mun of the Abbasid Caliphate. Around 820 AD he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad. Al-Khwarizmi's popularizing treatise on algebra presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications; because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing", he has been described as the father or founder of algebra. The term algebra itself comes from the title of his book, his name gave rise to the terms algorithm. His name is the origin of guarismo and of algarismo, both meaning digit. In the 12th century, Latin translations of his textbook on arithmetic which codified the various Indian numerals, introduced the decimal positional number system to the Western world.
The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical text-book of European universities. In addition to his best-known works, he revised Ptolemy's Geography, listing the longitudes and latitudes of various cities and localities, he further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial. Few details of al-Khwārizmī's life are known with certainty, he was born into a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan. Muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy; the epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul, a viticulture district near Baghdad. However, Rashed suggests: There is no need to be an expert on the period or a philologist to see that al-Tabari's second citation should read "Muhammad ibn Mūsa al-Khwārizmī and al-Majūsi al-Qutrubbulli," and that there are two people between whom the letter wa has been omitted in an early copy.
This would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī even the origins of his knowledge, had not been made. G. J. Toomer... with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmī's religion, Toomer writes: Another epithet given to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was an adherent of the old Zoroastrian religion; this would still have been possible at that time for a man of Iranian origin, but the pious preface to al-Khwārizmī's Algebra shows that he was an orthodox Muslim, so al-Ṭabarī's epithet could mean no more than that his forebears, he in his youth, had been Zoroastrians. Ibn al-Nadīm's Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833. After the Muslim conquest of Persia, Baghdad became the centre of scientific studies and trade, many merchants and scientists from as far as China and India traveled to this city, as did al-Khwārizmī.
He worked in Baghdad as a scholar at the House of Wisdom established by Caliph al-Ma’mūn, where he studied the sciences and mathematics, which included the translation of Greek and Sanskrit scientific manuscripts. Douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmī's contributions to mathematics, geography and cartography established the basis for innovation in algebra and trigonometry, his systematic approach to solving linear and quadratic equations led to algebra, a word derived from the title of his book on the subject, "The Compendious Book on Calculation by Completion and Balancing". On the Calculation with Hindu Numerals written about 820, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe, it was translated into Latin as Algoritmi de numero Indorum. Al-Khwārizmī, rendered as Algoritmi, led to the term "algorithm".
Some of his work was based on Persian and Babylonian astronomy, Indian numbers, Greek mathematics. Al-Khwārizmī corrected Ptolemy's data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea and Africa, he wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a world map for al-Ma'mun, the caliph, overseeing 70 geographers. When, in the 12th century, his works spread to Europe through Latin translations, it had a profound impact on the advance of mathematics in
Ancient Greek astronomy
Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the ancient Greek, Greco-Roman, Late Antiquity eras, it is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the language of scholarship throughout the Hellenistic world following the conquests of Alexander. This phase of Greek astronomy is known as Hellenistic astronomy, while the pre-Hellenistic phase is known as Classical Greek astronomy. During the Hellenistic and Roman periods, much of the Greek and non-Greek astronomers working in the Greek tradition studied at the Musaeum and the Library of Alexandria in Ptolemaic Egypt; the development of astronomy by the Greek and Hellenistic astronomers is considered, by historians, to be a major phase in the history of astronomy. Greek astronomy is characterized from the start by seeking a rational, physical explanation for celestial phenomena. Most of the constellations of the northern hemisphere derive from Greek astronomy, as are the names of many stars and planets.
It was influenced by Egyptian and Babylonian astronomy. References to identifiable stars and constellations appear in the writings of Homer and Hesiod, the earliest surviving examples of Greek literature. In the oldest European texts, the Iliad and the Odyssey, Homer has several astronomical phenomena including solar eclipses. Eclipses that can permit the dating of these events as the place is known and the calculation of the time is possible if other celestial phenomena are described at the same time. In the Iliad and the Odyssey, Homer refers to the following celestial objects: the constellation Boötes the star cluster Hyades the constellation Orion the star cluster Pleiades Sirius, the Dog Star the constellation Ursa Major Hesiod, who wrote in the early 7th century BC, adds the star Arcturus to this list in his poetic calendar Works and Days. Though neither Homer nor Hesiod set out to write a scientific work, they hint at a rudimentary cosmology of a flat Earth surrounded by an "Ocean River."
Some stars set. At certain times of the year, certain stars will set at sunrise or sunset. Speculation about the cosmos was common in Pre-Socratic philosophy in the 6th and 5th centuries BC. Anaximander described a cyclical earth suspended in the center of the cosmos, surrounded by rings of fire. Philolaus the Pythagorean described a cosmos with the stars, Sun, Earth, a counter-Earth —ten bodies in all—circling an unseen central fire; such reports show that Greeks of the 6th and 5th centuries BC were aware of the planets and speculated about the structure of the cosmos. A more detailed description about the cosmos, Sun and the Earth can be found in the Orphism, which dates back to the end of the 5th century BC, it is even older. Within the lyrics of the Orphic poems we can find remarkable information such as that the Earth is round, it has an axis and it moves around it in one day, it has three climate zones and that the Sun magnetizes the Stars and planets; the name "planet" comes from the Greek term πλανήτης, meaning "wanderer", as ancient astronomers noted how certain lights moved across the sky in relation to the other stars.
Five planets can be seen with the naked eye: Mercury, Mars and Saturn, the Greek names being Hermes, Ares and Cronus. Sometimes the luminaries, the Sun and Moon, are added to the list of naked eye planets to make a total of seven. Since the planets disappear from time to time when they approach the Sun, careful attention is required to identify all five. Observations of Venus are not straightforward. Early Greeks thought that the evening and morning appearances of Venus represented two different objects, calling it Hesperus when it appeared in the western evening sky and Phosphorus when it appeared in the eastern morning sky, they came to recognize that both objects were the same planet. Pythagoras is given credit for this realization. In classical Greece, astronomy was a branch of mathematics; this tradition began with the Pythagoreans. The study of number comprising the four arts was called the Quadrivium. Although he was not a creative mathematician, Plato included the quadrivium as the basis for philosophical education in the Republic.
He encouraged Eudoxus of Cnidus, to develop a system of Greek astronomy. According to a modern historian of science, David Lindberg: "In their work we find a shift from stellar to planetary concerns, the creation of a geometrical model, the "two-sphere model," for the representation of stellar and planetary phenomena, the establishment of criteria governing theories designed to account for planetary observations"; the two-sphere model is a geocentric model that divides the cosmos into two regions, a spherical Earth and motionless and a spherical heavenly realm centered on the Earth, which may contain multiple rotating spheres made of aether. Plato's main books on cosmology are the Republic. In them he described the two-sphere model and said there were eight circles or spheres carrying the seven planets and the fixed stars. According to the "Myth of Er" in the Republic, the cosmos is the Spindle of Nec