title: Bel Powley

text: Powley appeared as Maggie in "Tusk Tusk" at the Royal Court Theatre, London, in March 2009. On Broadway, she was Thomasina in the 2011 revival of "Arcadia" at the Ethel Barrymore Theatre. In October 2011, she once again appeared at the Royal Court as Tilly in "Jumpy", a role to which she returned in the West End transfer of "Jumpy" to the Duke of York's Theatre in August 2012. She appeared as Dawn in the March 2018 revival of "Lobby Hero" at the Hayes Theatre, alongside Chris Evans, Michael Cera and Bryan Tyree Henry.


title: Bel Powley

text: In an interview in 2013, Powley said she had a spot reserved to study history at the University of Manchester.


title: Bel Powley

text: It was announced in July 2021 via actor Douglas Booth's Instagram page that he and Powley are engaged to be married after meeting on the set of the film "Mary Shelley" in 2016.


title: History of mathematics

text: The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.


title: History of mathematics

text: The earliest mathematical texts available are from Mesopotamia and Egypt – "Plimpton 322" (Babylonian c. 2000 – 1900 BC), the "Rhind Mathematical Papyrus" (Egyptian c. 1800 BC) and the "Moscow Mathematical Papyrus" (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.


title: History of mathematics

text: The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek "μάθημα" ("mathema"), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.


title: History of mathematics

text: Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century.


title: History of mathematics

text: The origins of mathematical thought lie in the concepts of number, patterns in nature, magnitude, and form. Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.


title: History of mathematics

text: The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a "tally" of the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10." The Ishango bone, according to scholar Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.


title: History of mathematics

text: Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.


title: History of mathematics

text: Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC (Seleucid period). It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics.


title: History of mathematics

text: In contrast to the sparsity of sources in Egyptian mathematics, knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.


title: History of mathematics

text: The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around 2500 BC onward, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.


title: History of mathematics

text: Babylonian mathematics were written using a sexagesimal (base-60) numeral system. From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the decimal system. The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation. The notational system of the Babylonians was the best of any civilization until the Renaissance, and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet YBC 7289 gives an approximation of accurate to five decimal places. The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context. By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions. This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.


title: History of mathematics

text: Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs. The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time. Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem. However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles.


title: History of mathematics

text: Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa. Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.


title: History of mathematics

text: The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000–1800 BC. It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge, including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6). It also shows how to solve first order linear equations as well as arithmetic and geometric series.


title: History of mathematics

text: Another significant Egyptian mathematical text is the Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC. It consists of what are today called "word problems" or "story problems", which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum (truncated pyramid).


title: History of mathematics

text: Finally, the Berlin Papyrus 6619 (c. 1800 BC) shows that ancient Egyptians could solve a second-order algebraic equation.


title: History of mathematics

text: Greek mathematics refers to the mathematics written in the Greek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.


title: History of mathematics

text: Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.


title: History of mathematics

text: Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.


title: History of mathematics

text: Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers. Although he was preceded by the Babylonians, Indians and the Chinese, the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest Greco-Roman multiplication tables, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum). The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the "mensa Pythagorica".


title: History of mathematics

text: Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others. His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came. Plato also discussed the foundations of mathematics, clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions. The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.


title: History of mathematics

text: Eudoxus (408–c. 355 BC) developed the method of exhaustion, a precursor of modern integration and a theory of ratios that avoided the problem of incommensurable magnitudes. The former allowed the calculations of areas and volumes of curvilinear figures, while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, Aristotle (384–c. 322 BC) contributed significantly to the development of mathematics by laying the foundations of logic.


title: History of mathematics

text: In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria. It was there that Euclid (c. 300 BC) taught, and wrote the "Elements", widely considered the most successful and influential textbook of all time. The "Elements" introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the "Elements" were already known, Euclid arranged them into a single, coherent logical framework. The "Elements" was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today. In addition to the familiar theorems of Euclidean geometry, the "Elements" was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry, including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.


title: History of mathematics

text: Archimedes (c. 287–212 BC) of Syracuse, widely considered the greatest mathematician of antiquity, used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, . He also studied the spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid), and an ingenious method of exponentiation for expressing very large numbers. While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles. He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere.


title: History of mathematics

text: Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone. He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond"). His work "Conics" is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton. While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.


title: History of mathematics

text: Around the same time, Eratosthenes of Cyrene (c. 276–194 BC) devised the Sieve of Eratosthenes for finding prime numbers. The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline. Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers. Hipparchus of Nicaea (c. 190–120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle. Heron of Alexandria (c. 10–70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots. Menelaus of Alexandria (c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem. The most complete and influential trigonometric work of antiquity is the "Almagest" of Ptolemy (c. AD 90–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.


title: History of mathematics

text: Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics. During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis". The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the "Arithmetica", a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations. The "Arithmetica" had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the "Arithmetica" (that of dividing a square into two squares). Diophantus also made significant advances in notation, the "Arithmetica" being the first instance of algebraic symbolism and syncopation.


title: History of mathematics

text: Among the last great Greek mathematicians is Pappus of Alexandria (4th century AD). He is known for his hexagon theorem and centroid theorem, as well as the Pappus configuration and Pappus graph. His "Collection" is a major source of knowledge on Greek mathematics as most of it has survived. Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work.


title: History of mathematics

text: The first woman mathematician recorded by history was Hypatia of Alexandria (AD 350–415). She succeeded her father (Theon of Alexandria) as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed. Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as Proclus, Simplicius and Eutocius. Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in 529 AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus, the architects of the Hagia Sophia. Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.


title: History of mathematics

text: Although ethnic Greek mathematicians continued under the rule of the late Roman Republic and subsequent Roman Empire, there were no noteworthy native Latin mathematicians in comparison. Ancient Romans such as Cicero (106–43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman surveyors and calculators were far more interested in applied mathematics than the theoretical mathematics and geometry that were prized by the Greeks. It is unclear if the Romans first derived their numerical system directly from the Greek precedent or from Etruscan numerals used by the Etruscan civilization centered in what is now Tuscany, central Italy.


title: History of mathematics

text: Using calculation, Romans were adept at both instigating and detecting financial fraud, as well as managing taxes for the treasury. Siculus Flaccus, one of the Roman "gromatici" (i.e. land surveyor), wrote the "Categories of Fields", which aided Roman surveyors in measuring the surface areas of allotted lands and territories. Aside from managing trade and taxes, the Romans also regularly applied mathematics to solve problems in engineering, including the erection of architecture such as bridges, road-building, and preparation for military campaigns. Arts and crafts such as Roman mosaics, inspired by previous Greek designs, created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each tessera tile, the opus tessellatum pieces on average measuring eight millimeters square and the finer opus vermiculatum pieces having an average surface of four millimeters square.


title: History of mathematics

text: The creation of the Roman calendar also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the Roman Kingdom and included 356 days plus a leap year every other year. In contrast, the lunar calendar of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than the solar year, a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February. This calendar was supplanted by the Julian calendar, a solar calendar organized by Julius Caesar (100–44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in a 365-day cycle. This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by the Gregorian calendar organized by Pope Gregory XIII (), virtually the same solar calendar used in modern times as the international standard calendar.


title: History of mathematics

text: At roughly the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled, the Roman model first described by the Roman civil engineer and architect Vitruvius (c. 80 BC – c. 15 BC). The device was used at least until the reign of emperor Commodus (), but its design seems to have been lost until experiments were made during the 15th century in Western Europe. Perhaps relying on similar gear-work and technology found in the Antikythera mechanism, the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2 m) in diameter turning four-hundred times in one Roman mile (roughly 4590 ft/1400 m). With each revolution, a pin-and-axle device engaged a 400-tooth cogwheel that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.


title: History of mathematics

text: An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development. The oldest extant mathematical text from China is the "Zhoubi Suanjing", variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during the Warring States Period appears reasonable. However, the Tsinghua Bamboo Slips, containing the earliest known decimal multiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.


title: History of mathematics

text: Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten. Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system. Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the "suan pan", or Chinese abacus. The date of the invention of the "suan pan" is not certain, but the earliest written mention dates from AD 190, in Xu Yue's "Supplementary Notes on the Art of Figures".


title: History of mathematics

text: The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). The "Mo Jing" described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well. It also defined the concepts of circumference, diameter, radius, and volume.


title: History of mathematics

text: In 212 BC, the Emperor Qin Shi Huang commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the book burning of 212 BC, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is "The Nine Chapters on the Mathematical Art", the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles. It created mathematical proof for the Pythagorean theorem, and a mathematical formula for Gaussian elimination. The treatise also provides values of π, which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 AD) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724, as well as 3.162 by taking the square root of 10. Liu Hui commented on the "Nine Chapters" in the 3rd century AD and gave a value of π accurate to 5 decimal places (i.e. 3.14159). Though more of a matter of computational stamina than theoretical insight, in the 5th century AD Zu Chongzhi computed the value of π to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years. He also established a method which would later be called Cavalieri's principle to find the volume of a sphere.


title: History of mathematics

text: The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is the "Precious Mirror of the Four Elements" by Zhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method. The "Precious Mirror" also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100. The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (AD 1238–1298).


title: History of mathematics

text: Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.


title: History of mathematics

text: Japanese mathematics, Korean mathematics, and Vietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to the Confucian-based East Asian cultural sphere. Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's Ming dynasty (1368–1644). For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed the Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers. Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of mathematicians and astronomers, whereas in Japan it was more prevalent in the realm of private schools.


title: History of mathematics

text: The earliest civilization on the Indian subcontinent is the Indus Valley civilization (mature phase: 2600 to 1900 BC) that flourished in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.


title: History of mathematics

text: The oldest extant mathematical records from India are the Sulba Sutras (dated variously between the 8th century BC and the 2nd century AD), appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others. As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual. The Sulba Sutras give methods for constructing a circle with approximately the same area as a given square, which imply several different approximations of the value of π. In addition, they compute the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the Pythagorean theorem. All of these results are present in Babylonian mathematics, indicating Mesopotamian influence. It is not known to what extent the Sulba Sutras influenced later Indian mathematicians. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity.


title: History of mathematics

text: Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar. His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion. Pingala (roughly 3rd–1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system. His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem. Pingala's work also contains the basic ideas of Fibonacci numbers (called "mātrāmeru").


title: History of mathematics

text: The next significant mathematical documents from India after the "Sulba Sutras" are the "Siddhantas", astronomical treatises from the 4th and 5th centuries AD (Gupta period) showing strong Hellenistic influence. They are significant in that they contain the first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry. Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".


title: History of mathematics

text: Around 500 AD, Aryabhata wrote the "Aryabhatiya", a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology. Though about half of the entries are wrong, it is in the "Aryabhatiya" that the decimal place-value system first appears. Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the "Aryabhatiya" as a "mix of common pebbles and costly crystals".


title: History of mathematics

text: In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in "Brahma-sphuta-siddhanta", he lucidly explained the use of zero as both a placeholder and decimal digit, and explained the Hindu–Arabic numeral system. It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, all of which evolved from the Brahmi numerals. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix.


title: History of mathematics

text: In the 12th century, Bhāskara II lived in southern India and wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, derivatives, the mean value theorem and the derivative of the sine function. To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics.


title: History of mathematics

text: In the 14th century, Madhava of Sangamagrama, the founder of the Kerala School of Mathematics, found the Madhava–Leibniz series and obtained from it a transformed series, whose first 21 terms he used to compute the value of π as 3.14159265359. Madhava also found the Madhava-Gregory series to determine the arctangent, the Madhava-Newton power series to determine sine and cosine and the Taylor approximation for sine and cosine functions. In the 16th century, Jyesthadeva consolidated many of the Kerala School's developments and theorems in the "Yukti-bhāṣā".


title: History of mathematics

text: The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in Arabic, most of them were not written by Arabs, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Persians contributed to the world of Mathematics alongside Arabs.


title: History of mathematics

text: In the 9th century, the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote an important book on the Hindu–Arabic numerals and one on methods for solving equations. His book "On the Calculation with Hindu Numerals", written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word "algorithm" is derived from the Latinization of his name, Algoritmi, and the word "algebra" from the title of one of his works, "Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala" ("The Compendious Book on Calculation by Completion and Balancing"). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, and he was the first to teach algebra in an elementary form and for its own sake. He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as "al-jabr". His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."


title: History of mathematics

text: In Egypt, Abu Kamil extended algebra to the set of irrational numbers, accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions. His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci.


title: History of mathematics

text: Further developments in algebra were made by Al-Karaji in his treatise "al-Fakhri", where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes. The historian of mathematics, F. Woepcke, praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Also in the 10th century, Abul Wafa translated the works of Diophantus into Arabic. Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.


title: History of mathematics

text: In the late 11th century, Omar Khayyam wrote "Discussions of the Difficulties in Euclid", a book about what he perceived as flaws in Euclid's "Elements", especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations. He was also very influential in calendar reform.


title: History of mathematics

text: In the 13th century, Nasir al-Din Tusi (Nasireddin) made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate. In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating "n"th roots, which was a special case of the methods given many centuries later by Ruffini and Horner.


title: History of mathematics

text: Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, al-Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam and the development of an algebraic notation by al-Qalasādī.


title: History of mathematics

text: During the time of the Ottoman Empire and Safavid Empire from the 15th century, the development of Islamic mathematics became stagnant.


title: History of mathematics

text: In the Pre-Columbian Americas, the Maya civilization that flourished in Mexico and Central America during the 1st millennium AD developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics. Maya numerals used a base of twenty, the vigesimal system, instead of a base of ten that forms the basis of the decimal system used by most modern cultures. The Maya used mathematics to create the Maya calendar as well as to predict astronomical phenomena in their native Maya astronomy. While the concept of zero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.


title: History of mathematics

text: In precolumbian americas the Inca empire had made various accomplishment in mathematical field.The Incas used their own numeral system which has the base ten.


title: History of mathematics

text: It is also confirmed that the Incas use of the decimal system by the interpretation of the quipus, which are organized in such a way that the knots — according to their location — can represent: units, tens, hundreds, etc.


title: History of mathematics

text: However, the main confirmation of the use of this system is expressed in the denomination of the numbers in Quechua, in which the numbers are developed in decimal form. This can be appreciated in the following table:


title: History of mathematics

text: Most of the mathematics in Japan had either imported from China although many of the mathematical accomplishment was also made by Japanese mathematicans.


title: History of mathematics

text: One prominent mathematician during Edo period was Seki Takakazu. He created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations.


title: History of mathematics

text: He had a significant impact on calculus. He discovered the Newton or Newton-Raphson method for solving equations (independent from Newton) and also had a version of the Newton interpolation formula.


title: History of mathematics

text: In 1674, Seki published Hatsubi Sanpō (発微算法), giving solutions to all 15 problems. The method he used is called "bōsho-hō". He introduced the use of kanji to represent unknowns and variables in equations.


title: History of mathematics

text: He also developed Bernoulli numbers independently from Jacob Bernoulli. They are also called as seki numbers. He studied equations treating both positive and negative roots.


title: History of mathematics

text: His manuscript and other treatises in 1683 show that the Japanese theory of determinate was fully applied to elimination method of arbitrary constant.It is assumed that he had invented the determinants. Seki pushed ahead with elimination theory, based on resultants, in the Kaifukudai no Hō (解伏題之法). To express the resultant, he developed the notion of the determinant.


title: History of mathematics

text: Mathematicans like Takebe Katahiro played and important role in developing Enri (" circle principle"), a crude analog to the Western calculus. He obtained power series expansion of formula_1 in 1722, 15 years earlier than Euler. He used Richardson extrapolation in 1695, about 200 years earlier than Richardson. He also computed 41 digits of π, based on polygon approximation and Richardson extrapolation.


title: History of mathematics

text: Ajima Naonobu was credited for introducing calculus into Japanese mathematics.Ajima also posed the question of inscribing three mutually tangent circles in a triangle; circles are now known as Malfatti circles after the later work of Gian Francesco Malfatti, but two triangle centers derived from them, the Ajima–Malfatti points, are named after Ajima.


title: History of mathematics

text: ππMedieval European interest in mathematiRichardson.Hey concerns quite different froπe icians. One driving element was the belief that mathematics .vided the key to understanding the created order of nature, frequently justified by Plato's "Timaeus" and the biblical passage (in the "Book of Wisdom") that God had "ordered all things in measure, and number, and weight".


title: History of mathematics

text: Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the term "quadrivium" to describe the study of arithmetic, geometry, astronomy, and music. He wrote "De institutione arithmetica", a free translation from the Greek of Nicomachus's "Introduction to Arithmetic"; "De institutione musica", also derived from Greek sources; and a series of excerpts from Euclid's "Elements". His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.


title: History of mathematics

text: In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's "The Compendious Book on Calculation by Completion and Balancing", translated into Latin by Robert of Chester, and the complete text of Euclid's "Elements", translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona. These and other new sources sparked a renewal of mathematics.


title: History of mathematics

text: Leonardo of Pisa, now known as Fibonacci, serendipitously learned about the Hindu–Arabic numerals on a trip to what is now Béjaïa, Algeria with his merchant father. (Europe was still using Roman numerals.) There, he observed a system of arithmetic (specifically algorism) which due to the positional notation of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. Leonardo wrote "Liber Abaci" in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the Fibonacci sequence (known to Indian mathematicians for hundreds of years before that) which was used as an unremarkable example within the text.


title: History of mathematics

text: The 14th century saw the development of new mathematical concepts to investigate a wide range of problems. One important contribution was development of mathematics of local motion.


title: History of mathematics

text: Thomas Bradwardine proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing:


title: History of mathematics

text: V = log (F/R). Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.


title: History of mathematics

text: One of the 14th-century Oxford Calculators, William Heytesbury, lacking differential calculus and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by [a body] if... it were moved uniformly at the same degree of speed with which it is moved in that given instant".


title: History of mathematics

text: Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".


title: History of mathematics

text: Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled. In a later mathematical commentary on Euclid's "Elements", Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.


title: History of mathematics

text: The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. They used it with base 2 rather than base 10 which most of the culture had adopted.


title: History of mathematics

text: During the Renaissance, the development of mathematics and of accounting were intertwined. While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in Flanders and Germany) or abacus schools (known as "abbaco" in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing bookkeeping operations, but for complex bartering operations or the calculation of compound interest, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.


title: History of mathematics

text: Piero della Francesca (c. 1415–1492) wrote books on solid geometry and linear perspective, including "De Prospectiva Pingendi (On Perspective for Painting)", "Trattato d’Abaco (Abacus Treatise)", and "De quinque corporibus regularibus (On the Five Regular Solids)".


title: History of mathematics

text: Luca Pacioli's "Summa de Arithmetica, Geometria, Proportioni et Proportionalità" (Italian: "Review of Arithmetic, Geometry, Ratio and Proportion") was first printed and published in Venice in 1494. It included a 27-page treatise on bookkeeping, ""Particularis de Computis et Scripturis"" (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the mathematical puzzles it contained, and to aid the education of their sons. In "Summa Arithmetica", Pacioli introduced symbols for plus and minus for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. "Summa Arithmetica" was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized.


title: History of mathematics

text: In Italy, during the first half of the 16th century, Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations. Gerolamo Cardano published them in his 1545 book "Ars Magna", together with a solution for the quartic equations, discovered by his student Lodovico Ferrari. In 1572 Rafael Bombelli published his "L'Algebra" in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations.


title: History of mathematics

text: Simon Stevin's book "De Thiende" ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of decimal notation, which influenced all later work on the real number system.


title: History of mathematics

text: Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his "Trigonometria" in 1595. Regiomontanus's table of sines and cosines was published in 1533.


title: History of mathematics

text: During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that involved, were studied intensely.


title: History of mathematics

text: The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe. Galileo observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant, Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of logarithms by John Napier and Jost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion.


title: History of mathematics

text: The analytic geometry developed by René Descartes (1596–1650) allowed those orbits to be plotted on a graph, in Cartesian coordinates.


title: History of mathematics

text: Building on earlier work by many predecessors, Isaac Newton discovered the laws of physics explaining Kepler's Laws, and brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world.


title: History of mathematics

text: In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of utility theory in the 18th–19th century.


title: History of mathematics

text: The most influential mathematician of the 18th century was arguably Leonhard Euler (1707–1783). His contributions range from founding the study of graph theory with the Seven Bridges of Königsberg problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol "i", and he popularized the use of the Greek letter formula_2 to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.


title: History of mathematics

text: Other important European mathematicians of the 18th century included Joseph Louis Lagrange, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics.


title: History of mathematics

text: Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.


title: History of mathematics

text: This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds.


title: History of mathematics

text: The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalizes the ideas of curves and surfaces.


title: History of mathematics

text: The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1. Boolean algebra is the starting point of mathematical logic and has important applications in electrical engineering and computer science.


title: History of mathematics

text: Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion.


title: History of mathematics

text: Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem). Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks. On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers.


title: History of mathematics

text: Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.


title: History of mathematics

text: In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L.E.J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics.


title: History of mathematics

text: The 19th century saw the founding of a number of national mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Matematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883, and the American Mathematical Society in 1888. The first international, special-interest society, the Quaternion Society, was formed in 1899, in the context of a vector controversy.


title: History of mathematics

text: The 20th century saw mathematics become a major profession. Every year, thousands of new Ph.D.s in mathematics were awarded, and jobs were available in both teaching and industry. An effort to catalogue the areas and applications of mathematics was undertaken in Klein's encyclopedia.


title: History of mathematics

text: In a 1900 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics. These problems, spanning many areas of mathematics, formed a central focus for much of 20th-century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.


title: History of mathematics

text: Notable historical conjectures were finally proven. In 1976, Wolfgang Haken and Kenneth Appel proved the four color theorem, controversial at the time for the use of a computer to do so. Andrew Wiles, building on the work of others, proved Fermat's Last Theorem in 1995. Paul Cohen and Kurt Gödel proved that the continuum hypothesis is independent of (could neither be proved nor disproved from) the standard axioms of set theory. In 1998 Thomas Callister Hales proved the Kepler conjecture.


title: History of mathematics

text: Mathematical collaborations of unprecedented size and scope took place. An example is the classification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages. A group of French mathematicians, including Jean Dieudonné and André Weil, publishing under the pseudonym "Nicolas Bourbaki", attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.


title: History of mathematics

text: Differential geometry came into its own when Albert Einstein used it in general relativity. Entirely new areas of mathematics such as mathematical logic, topology, and John von Neumann's game theory changed the kinds of questions that could be answered by mathematical methods. All kinds of structures were abstracted using axioms and given names like metric spaces, topological spaces etc. As mathematicians do, the concept of an abstract structure was itself abstracted and led to category theory. Grothendieck and Serre recast algebraic geometry using sheaf theory. Large advances were made in the qualitative study of dynamical systems that Poincaré had begun in the 1890s.


title: History of mathematics

text: Measure theory was developed in the late 19th and early 20th centuries. Applications of measures include the Lebesgue integral, Kolmogorov's axiomatisation of probability theory, and ergodic theory. Knot theory greatly expanded. Quantum mechanics led to the development of functional analysis. Other new areas include Laurent Schwartz's distribution theory, fixed point theory, singularity theory and René Thom's catastrophe theory, model theory, and Mandelbrot's fractals. Lie theory with its Lie groups and Lie algebras became one of the major areas of study.


title: History of mathematics

text: Non-standard analysis, introduced by Abraham Robinson, rehabilitated the infinitesimal approach to calculus, which had fallen into disrepute in favour of the theory of limits, by extending the field of real numbers to the Hyperreal numbers which include infinitesimal and infinite quantities. An even larger number system, the surreal numbers were discovered by John Horton Conway in connection with combinatorial games.


title: History of mathematics

text: The development and continual improvement of computers, at first mechanical analog machines and then digital electronic machines, allowed industry to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: Alan Turing's computability theory; complexity theory; Derrick Henry Lehmer's use of ENIAC to further number theory and the Lucas-Lehmer test; Rózsa Péter's recursive function theory; Claude Shannon's information theory; signal processing; data analysis; optimization and other areas of operations research. In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of discrete concepts and the expansion of combinatorics including graph theory. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too time-consuming to deal with by pencil and paper calculations, leading to areas such as numerical analysis and symbolic computation. Some of the most important methods and algorithms of the 20th century are: the simplex algorithm, the fast Fourier transform, error-correcting codes, the Kalman filter from control theory and the RSA algorithm of public-key cryptography.


title: History of mathematics

text: At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the natural numbers plus either addition or multiplication (but not both), was decidable, i.e. could be determined by some algorithm. In 1931, Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as Peano arithmetic, was in fact incompletable. (Peano arithmetic is adequate for a good deal of number theory, including the notion of prime number.) A consequence of Gödel's two incompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all of analysis and geometry), truth necessarily outruns proof, i.e. there are true statements that cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated.


title: History of mathematics

text: One of the more colorful figures in 20th-century mathematics was Srinivasa Aiyangar Ramanujan (1887–1920), an Indian autodidact who conjectured or proved over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also made major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.


title: History of mathematics

text: Paul Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. Mathematicians have a game equivalent to the Kevin Bacon Game, which leads to the Erdős number of a mathematician. This describes the "collaborative distance" between a person and Erdős, as measured by joint authorship of mathematical papers.


title: History of mathematics

text: Emmy Noether has been described by many as the most important woman in the history of mathematics. She studied the theories of rings, fields, and algebras.


title: History of mathematics

text: As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century there were hundreds of specialized areas in mathematics and the Mathematics Subject Classification was dozens of pages long. More and more mathematical journals were published and, by the end of the century, the development of the World Wide Web led to online publishing.


title: History of mathematics

text: In 2000, the Clay Mathematics Institute announced the seven Millennium Prize Problems, and in 2003 the Poincaré conjecture was solved by Grigori Perelman (who declined to accept an award, as he was critical of the mathematics establishment).


title: History of mathematics

text: Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched. There is an increasing drive toward open access publishing, first popularized by arXiv.


title: History of mathematics

text: There are many observable trends in mathematics, the most notable being that the subject is growing ever larger, computers are ever more important and powerful, the application of mathematics to bioinformatics is rapidly expanding, and the volume of data being produced by science and industry, facilitated by computers, is expanding exponentially.


title: Frances Conroy

text: Frances Hardman Conroy is an American actress. She is best known for playing Ruth Fisher on the television series "Six Feet Under" (2001–2005), for which she won a Golden Globe and three Screen Actors Guild Awards, and received four Primetime Emmy Awards nominations for Outstanding Lead Actress in a Drama Series. She is also known for playing the older version of Moira O'Hara in of the television anthology series "American Horror Story", which garnered Conroy her first Saturn Award for Best Supporting Actress on Television nomination, and as well a Primetime Emmy Awards nomination for Outstanding Supporting Actress in a Miniseries or a Movie. Conroy subsequently portrayed The Angel of Death, Myrtle Snow, Gloria Mott, Mama Polk, Bebe Babbitt, and Belle Noir on seven further seasons of the show: "", "", "", "", "", "", and , respectively. Conroy is the fourth actor who has appeared in most seasons of the show. For her performance in "Coven", she was nominated again for a Primetime Emmy Award for Outstanding Supporting Actress in a Miniseries or a Movie.


title: Frances Conroy

text: She starred as Dawn in the first season of the Hulu original series "Casual", which was nominated for a Golden Globe Award for its first season. In 2017, she starred as Nathalie Raven in "The Mist". In 2019, she starred in "Joker" as the titular character's mother. In 2021, she had a supporting role in "The Power of the Dog".


title: Frances Conroy

text: Conroy was born in Monroe, Georgia, the daughter of Ossie Hardman (née Ray) and Vincent Paul Conroy. Her father, who was of Irish descent, was a business executive, and her mother also worked in business. During the 1971–72 school year, she was a student at Dickinson College in Carlisle, Pennsylvania, where she was a member of the Mermaid Players and appeared in college theatrical productions. She moved to New York City to study drama at the Neighborhood Playhouse and the Juilliard School. She was a member of Juilliard's Drama Division "Group 6" (1973–1977) which also included Kevin Conroy (no relation), Kelsey Grammer, Harriet Sansom Harris, and Robin Williams.


title: Frances Conroy

text: During the 1970s, she performed regularly with regional and touring theatrical companies (most notably The Acting Company), and appeared as Desdemona at the Delacorte Theatre in a production of "Othello" with Richard Dreyfuss and Raul Julia. She returned to the Delacorte in the summer of 1998 as, "Mrs. Antrobus" in Thorton Wilder's "The Skin Of Our Teeth". One of her first film appearances was as a Shakespearean actress in Woody Allen's 1979 classic, "Manhattan". In 1980, she made a very well received Broadway debut in Edward Albee's "The Lady From Dubuque". She focused primarily on her stage career for the next two decades, appearing in such productions as "Our Town", "The Little Foxes", and "The Ride Down Mt. Morgan", receiving one Tony and four Drama Desk Award nominations (including a Drama Desk win for "The Secret Rapture"). Conroy had a small role in the 1984 movie "Falling in Love", as a waitress in a swanky restaurant. In 1988, she appeared as the elder daughter of Burt Lancaster's dying patriarch in "Rocket Gibraltar". That same year, she appeared in supporting roles in "Dirty Rotten Scoundrels" and "Another Woman". In 1992, she played the political science teacher Christine Downes in the film "Scent of a Woman".


title: Frances Conroy

text: She is best known for her critically acclaimed work on HBO's original drama series "Six Feet Under", which she starred in from 2001 until the series' end in 2005, playing the volatile family matriarch Ruth Fisher. During the series' run, Conroy won the Golden Globe Award for Best Actress – Television Series Drama in 2004. She was also nominated for four Primetime Emmy Award for Outstanding Lead Actress in a Drama Series awards, without winning. Along with the cast of the show, they were awarded two times in 2003 and 2004 the Screen Actors Guild Award for Outstanding Performance by an Ensemble in a Drama Series, as well as Conroy winning the individual 2004 award for Outstanding Performance by a Female Actor in a Drama Series. The series ended in 2005, having aired 63 original episodes.


title: Frances Conroy

text: In 2008, she landed a guest role in ABC's "Desperate Housewives" as Virginia Hildebrand, a rich woman who tries to buy the Solis family's love.


title: Frances Conroy

text: Along with many guest appearances on television in the mid-2000s, she was cast in a recurring role on "How I Met Your Mother", as Barney Stinson's mother Loretta Stinson. And in 2010, Conroy portrayed the character Angie Dinkley in the animated show "Scooby-Doo! Mystery Incorporated".


title: Frances Conroy

text: In 2010, Conroy played Madylyn, the wife of Robert De Niro's character Jack Mabrey, in the John Curran thriller "Stone". In the same year, Conroy played the recurring role of Peggy Haplin in the short-lived ABC drama series "Happy Town".


title: Frances Conroy

text: In 2011, she was cast in a new drama series for FX titled "". Conroy was one of two actresses, the other being Alexandra Breckenridge, playing a woman named Moira O'Hara. When the season came to an end in 2011, it was announced that the show was intended to be an anthology series, in which each season would be completely different with new characters each season. During this time, Conroy was nominated for a fifth Primetime Emmy Award in the Outstanding Supporting Actress in a Miniseries or a Movie category.


title: Frances Conroy

text: For the second season, dubbed "", Conroy re-joined the cast playing Shachath, The Angel of Death, alongside an unnamed prison transfer, in a multi-episode arc. And in 2013, she guest starred on "Royal Pains"; portraying Blythe Ballard. She also co-starred in the television movie "Ring of Fire", where she played Maybelle Carter. Conroy was once more cast for "American Horror Story"'s third season during this time, "", being promoted to series regular, playing Myrtle Snow, a member of the Witches Council in New Orleans, Louisiana. She was subsequently nominated for a sixth Primetime Emmy Award and her second nomination for Outstanding Supporting Actress in a Miniseries or a Movie.


title: Frances Conroy

text: In 2014, she returned for the fourth season of the series, titled "", as Gloria Mott, a wealthy woman whose psychotic, spoiled son wants to join the freak show. Conroy later returned in the 2016 season, , as a cannibal.


title: Frances Conroy

text: As of 2015, Conroy starred as Dawn on the Hulu original series "Casual". In July 2017, it was announced via Twitter that Conroy would appear in "", marking her sixth appearance on the show..


title: Frances Conroy

text: In 2018, Conroy reprised her respective roles as Myrtle Snow and Moira O'Hara from "Coven" and "Murder House" in Ryan Murphy's "", her seventh appearance on the show. Conroy co-starred in the 2019 film ""Joker"", an origin story film of the Batman supervillain the Joker, as the character's mother.


title: Frances Conroy

text: She returned to the "American Horror Story" franchise in 2021 with "". The same year she acted as Mrs. Burbank in the critically acclaimed film "The Power of the Dog", winning her first Satellite Award and receiving a Critics' Choice Awards nomination as a member of the cast.


title: Frances Conroy

text: In 1980, she married Jonathan Furst; they divorced in the late 1980s. Since 1992, Conroy has been married to the actor Jan Munroe.


title: Frances Conroy

text: According to reports, her right eye was injured in a car accident, for which she underwent surgery. The damage to the cornea left the eye discolored.


title: Voyager 2

text: Voyager 2 is a space probe launched by NASA on August 20, 1977, to study the outer planets and interstellar space beyond the Sun's heliosphere. As a part of the Voyager program, it was launched 16 days before its twin, "Voyager 1", on a trajectory that took longer to reach gas giants Jupiter and Saturn but enabled further encounters with ice giants Uranus and Neptune. "Voyager 2" remains the only spacecraft to have visited either of the ice giant planets. "Voyager 2" was the fourth of five spacecraft to achieve Solar escape velocity, which allowed it to leave the Solar System.


title: Voyager 2

text: "Voyager 2" successfully fulfilled its primary mission of visiting the Jovian system in 1979, the Saturnian system in 1981, Uranian system in 1986, and the Neptunian system in 1989. The spacecraft is now in its extended mission of studying interstellar space. It has been operating for as of ; , it has reached a distance of from Earth.


title: Voyager 2

text: The probe entered interstellar space on November 5, 2018, at a distance of (about 16:58 light-hours) from the Sun and moving at a velocity of relative to the Sun. "Voyager 2" has left the Sun's heliosphere and is traveling through the interstellar medium, a region of outer space beyond the influence of the Solar System, joining "Voyager 1", which had reached the interstellar medium in 2012. "Voyager 2" has begun to provide the first direct measurements of the density and temperature of the interstellar plasma.


title: Voyager 2

text: "Voyager 2" remains in contact with Earth through the NASA Deep Space Network. In 2020, maintenance to the Deep Space Network cut outbound contact with the probe for eight months. Contact was reestablished on November 2, 2020, when a series of instructions was transmitted, subsequently executed, and relayed back with a successful communication message. On February 12, 2021, full communications with the probe were restored after a major antenna upgrade that took a year to complete. The DSS 43 communication antenna, which is solely responsible for communications with the probe, is located near Canberra, Australia.


title: Voyager 2

text: In the early space age, it was realized that a periodic alignment of the outer planets would occur in the late 1970s and enable a single probe to visit Jupiter, Saturn, Uranus, and Neptune by taking advantage of the then-new technique of gravity assists. NASA began work on a Grand Tour, which evolved into a massive project involving two groups of two probes each, with one group visiting Jupiter, Saturn, and Pluto and the other Jupiter, Uranus, and Neptune. The spacecraft would be designed with redundant systems to ensure survival through the entire tour. By 1972 the mission was scaled back and replaced with two Mariner program-derived spacecraft, the Mariner Jupiter-Saturn probes. To keep apparent lifetime program costs low, the mission would include only flybys of Jupiter and Saturn, but keep the Grand Tour option open. As the program progressed, the name was changed to Voyager.


title: Voyager 2

text: The primary mission of "Voyager 1" was to explore Jupiter, Saturn, and Saturn's moon, Titan. "Voyager 2" was also to explore Jupiter and Saturn, but on a trajectory that would have the option of continuing on to Uranus and Neptune, or being redirected to Titan as a backup for "Voyager 1". Upon successful completion of "Voyager 1"<nowiki>'</nowiki>s objectives, "Voyager 2" would get a mission extension to send the probe on towards Uranus and Neptune.


title: Voyager 2

text: Constructed by the Jet Propulsion Laboratory (JPL), "Voyager 2" included 16 hydrazine thrusters, three-axis stabilization, gyroscopes and celestial referencing instruments (Sun sensor/Canopus Star Tracker) to maintain pointing of the high-gain antenna toward Earth. Collectively these instruments are part of the Attitude and Articulation Control Subsystem (AACS) along with redundant units of most instruments and 8 backup thrusters. The spacecraft also included 11 scientific instruments to study celestial objects as it traveled through space.


title: Voyager 2

text: Built with the intent for eventual interstellar travel, "Voyager 2" included a large, parabolic, high-gain antenna () to transceive data via the Deep Space Network on the Earth. Communications are conducted over the S-band (about 13 cm wavelength) and X-band (about 3.6 cm wavelength) providing data rates as high as 115.2 kilobits per second at the distance of Jupiter, and then ever-decreasing as distance increases, because of the inverse-square law. When the spacecraft is unable to communicate with Earth, the Digital Tape Recorder (DTR) can record about 64 megabytes of data for transmission at another time.


title: Voyager 2

text: "Voyager 2" is equipped with three Multihundred-Watt radioisotope thermoelectric generators (MHW RTG). Each RTG includes 24 pressed plutonium oxide spheres, and provided enough heat to generate approximately 157 W of electrical power at launch. Collectively, the RTGs supplied the spacecraft with 470 watts at launch (halving every 87.7 years). They were predicted to allow operations to continue until at least 2020 and have already done so.


title: Voyager 2

text: Because of the energy required to achieve a Jupiter trajectory boost with an payload, the spacecraft included a propulsion module made of a solid-rocket motor and eight hydrazine monopropellant rocket engines, four providing pitch and yaw attitude control, and four for roll control. The propulsion module was jettisoned shortly after the successful Jupiter burn.


title: Voyager 2

text: Sixteen hydrazine MR-103 thrusters on the mission module provide attitude control. Four are used to execute trajectory correction maneuvers; the others in two redundant six-thruster branches, to stabilize the spacecraft on its three axes. Only one branch of attitude control thrusters is needed at any time.


title: Voyager 2

text: Thrusters are supplied by a single diameter spherical titanium tank. It contained of hydrazine at launch, providing enough fuel until 2034.


title: Voyager 2

text: "For more details on the Voyager space probes' identical instrument packages, see the separate article on the overall Voyager Program."


title: Voyager 2

text: The "Voyager 2" probe was launched on August 20, 1977, by NASA from Space Launch Complex 41 at Cape Canaveral, Florida, aboard a Titan IIIE/Centaur launch vehicle. Two weeks later, the twin "Voyager 1" probe was launched on September 5, 1977. However, "Voyager 1" reached both Jupiter and Saturn sooner, as "Voyager 2" had been launched into a longer, more circular trajectory.


title: Voyager 2

text: "Voyager 1"s initial orbit had an aphelion of , just a little short of Saturn's orbit of . "Voyager 2"s initial orbit had an aphelion of , well short of Saturn's orbit.


title: Voyager 2

text: In April 1978, a complication arose when no commands were transmitted to "Voyager 2" for a period of time, causing the spacecraft to switch from its primary radio receiver to its backup receiver. Sometime afterwards, the primary receiver failed altogether. The backup receiver was functional, but a failed capacitor in the receiver meant that it could only receive transmissions that were sent at a precise frequency, and this frequency would be affected by the Earth's rotation (due to the Doppler effect) and the onboard receiver's temperature, among other things. For each subsequent transmission to "Voyager 2", it was necessary for engineers to calculate the specific frequency for the signal so that it could be received by the spacecraft.


title: Voyager 2

text: "Voyager 2"s closest approach to Jupiter occurred at 22:29 UT on July 9, 1979. It came within of the planet's cloud tops.


title: Voyager 2

text: Jupiter's Great Red Spot was revealed as a complex storm moving in a counterclockwise direction. Other smaller storms and eddies were found throughout the banded clouds.


title: Voyager 2

text: "Voyager 2" returned images of Jupiter, as well as its moons Amalthea, Io, Callisto, Ganymede, and Europa. During a 10-hour "volcano watch", it confirmed "Voyager 1"s observations of active volcanism on the moon Io, and revealed how the moon's surface had changed in the four months since the previous visit. Together, the Voyagers observed the eruption of nine volcanoes on Io, and there is evidence that other eruptions occurred between the two Voyager fly-bys.


title: Voyager 2

text: Jupiter's moon Europa displayed a large number of intersecting linear features in the low-resolution photos from "Voyager 1". At first, scientists believed the features might be deep cracks, caused by crustal rifting or tectonic processes. Closer high-resolution photos from "Voyager 2", however, were puzzling: the features lacked topographic relief, and one scientist said they "might have been painted on with a felt marker". Europa is internally active due to tidal heating at a level about one-tenth that of Io. Europa is thought to have a thin crust (less than thick) of water ice, possibly floating on a -deep ocean.


title: Voyager 2

text: Two new, small satellites, Adrastea and Metis, were found orbiting just outside the ring. A third new satellite, Thebe, was discovered between the orbits of Amalthea and Io.


title: Voyager 2

text: While passing behind Saturn (as viewed from Earth), "Voyager 2" probed Saturn's upper atmosphere with its radio link to gather information on atmospheric temperature and density profiles. "Voyager 2" found that at the uppermost pressure levels (seven kilopascals of pressure), Saturn's temperature was , while at the deepest levels (120 kilopascals), the temperature increased to . The north pole was found to be cooler, although this may be seasonal ("see also Saturn Oppositions").


title: Voyager 2

text: After the fly-by of Saturn, the camera platform of "Voyager 2" locked up briefly, putting plans to officially extend the mission to Uranus and Neptune in jeopardy. The mission's engineers were able to fix the problem (caused by an overuse that temporarily depleted its lubricant), and the "Voyager 2" probe was given the go-ahead to explore the Uranian system.


title: Voyager 2

text: The closest approach to Uranus occurred on January 24, 1986, when "Voyager 2" came within of the planet's cloudtops. "Voyager 2" also discovered 11 previously unknown moons: Cordelia, Ophelia, Bianca, Cressida, Desdemona, Juliet, Portia, Rosalind, Belinda, Puck and Perdita. The mission also studied the planet's unique atmosphere, caused by its axial tilt of 97.8°; and examined the Uranian ring system. The length of a day on Uranus as measured by "Voyager 2" is 17 hours, 14 minutes. Uranus was shown to have a magnetic field that was misaligned with its rotational axis, unlike other planets that had been visited to that point, and a helix-shaped magnetic tail stretching 10 million kilometers (6 million miles) away from the Sun.


title: Voyager 2

text: When "Voyager 2" visited Uranus, much of its cloud features were hidden by a layer of haze; however, false-color and contrast-enhanced images show bands of concentric clouds around its south pole. This area was also found to radiate large amounts of ultraviolet light, a phenomenon that is called "dayglow". The average atmospheric temperature is about . Surprisingly, the illuminated and dark poles, and most of the planet, exhibit nearly the same temperatures at the cloud tops.


title: Voyager 2

text: Detailed images from "Voyager 2"s flyby of the Uranian moon Miranda showed huge canyons made from geological faults. One hypothesis suggests that Miranda might consist of a reaggregation of material following an earlier event when Miranda was shattered into pieces by a violent impact.


title: Voyager 2

text: "Voyager 2" discovered two previously unknown Uranian rings. Measurements showed that the Uranian rings are distinctly different from those at Jupiter and Saturn. The Uranian ring system might be relatively young, and it did not form at the same time that Uranus did. The particles that make up the rings might be the remnants of a moon that was broken up by either a high-velocity impact or torn up by tidal effects.


title: Voyager 2

text: In March 2020, NASA astronomers reported the detection of a large atmospheric magnetic bubble, also known as a plasmoid, released into outer space from the planet Uranus, after reevaluating old data recorded during the flyby.


title: Voyager 2

text: Following a mid-course correction in 1987, "Voyager 2"s closest approach to Neptune occurred on August 25, 1989. Through repeated computerized test simulations of trajectories through the Neptunian system conducted in advance, flight controllers determined the best way to route "Voyager 2" through the Neptune-Triton system. Since the plane of the orbit of Triton is tilted significantly with respect to the plane of the ecliptic, through mid-course corrections, "Voyager 2" was directed into a path about above the north pole of Neptune. Five hours after "Voyager 2" made its closest approach to Neptune, it performed a close fly-by of Triton, the larger of Neptune's two originally known moons, passing within about .


title: Voyager 2

text: "Voyager 2" discovered previously unknown Neptunian rings, and confirmed six new moons: Despina, Galatea, Larissa, Proteus, Naiad and Thalassa. While in the neighborhood of Neptune, "Voyager 2" discovered the "Great Dark Spot", which has since disappeared, according to observations by the Hubble Space Telescope. The Great Dark Spot was later hypothesized to be a region of clear gas, forming a window in the planet's high-altitude methane cloud deck.


title: Voyager 2

text: With the decision of the International Astronomical Union to reclassify Pluto as a dwarf planet in 2006, the flyby of Neptune by "Voyager 2" in 1989 retroactively became the point when every known planet in the Solar System had been visited at least once by a space probe.


title: Voyager 2

text: Once its planetary mission was over, "Voyager 2" was described as working on an interstellar mission, which NASA is using to find out what the Solar System is like beyond the heliosphere. "Voyager 2" is currently transmitting scientific data at about 160 bits per second. Information about continuing telemetry exchanges with "Voyager 2" is available from Voyager Weekly Reports.


title: Voyager 2

text: In July 1994, an attempt was made to observe the impacts from fragments of the comet Comet Shoemaker–Levy 9 with Jupiter. The craft's position meant it had a direct line of sight to the impacts and observations were made in the ultraviolet and radio spectrum. "Voyager 2" failed to detect anything, with calculations showing that the fireballs were just below the craft's limit of detection.


title: Voyager 2

text: On November 29, 2006, a telemetered command to "Voyager 2" was incorrectly decoded by its on-board computer—in a random error—as a command to turn on the electrical heaters of the spacecraft's magnetometer. These heaters remained turned on until December 4, 2006, and during that time, there was a resulting high temperature above , significantly higher than the magnetometers were designed to endure, and a sensor rotated away from the correct orientation. As of this date it had not been possible to fully diagnose and correct for the damage caused to "Voyager 2"s magnetometer, although efforts to do so were proceeding.


title: Voyager 2

text: On August 30, 2007, "Voyager 2" passed the termination shock and then entered into the heliosheath, approximately closer to the Sun than "Voyager 1" did. This is due to the interstellar magnetic field of deep space. The southern hemisphere of the Solar System's heliosphere is being pushed in.


title: Voyager 2

text: On April 22, 2010, "Voyager 2" encountered scientific data format problems. On May 17, 2010, JPL engineers revealed that a flipped bit in an on-board computer had caused the problem, and scheduled a bit reset for May 19. On May 23, 2010, "Voyager 2" resumed sending science data from deep space after engineers fixed the flipped bit. Currently research is being done regarding marking the area of memory with the flipped bit off limits or disallowing its use. The Low-Energy Charged Particle Instrument is currently operational, and data from this instrument concerning charged particles is being transmitted to Earth. This data permits measurements of the heliosheath and termination shock. There has also been a modification to the on-board flight software to delay turning off the AP Branch 2 backup heater for one year. It was scheduled to go off February 2, 2011 (DOY 033, 2011–033).


title: Voyager 2

text: On July 25, 2012, "Voyager 2" was traveling at relative to the Sun at about from the Sun, at −55.29° declination and 19.888 h right ascension, and also at an ecliptic latitude of −34.0 degrees, placing it in the constellation Telescopium as observed from Earth. This location places it deep in the scattered disc, and traveling outward at roughly per year. It is more than twice as far from the Sun as Pluto, and far beyond the perihelion of 90377 Sedna, but not yet beyond the outer limits of the orbit of the dwarf planet Eris.


title: Voyager 2

text: On September 9, 2012, "Voyager 2" was from the Earth and from the Sun; and traveling at (relative to the Sun) and traveling outward at about per year. Sunlight takes 13.73 hours to get to "Voyager 2". The brightness of the Sun from the spacecraft is magnitude −16.7. "Voyager 2" is heading in the direction of the constellation Telescopium. To compare, Proxima Centauri, the closest star to the Sun, is about 4.2 light-years (or ) distant. "Voyager 2"s current relative velocity to the Sun is . This calculates as per year, about 10% slower than "Voyager 1". At this velocity, 81,438 years would pass before "Voyager 2" reaches the nearest star, Proxima Centauri, were the spacecraft traveling in the direction of that star. "Voyager 2" will need about 19,390 years at its current velocity to travel a complete light year.


title: Voyager 2

text: On November 7, 2012, "Voyager 2" reached from the Sun, making it the third human-made object to reach that distance and the second that was still sending data to Earth at that distance. "Voyager 1" was from the Sun, and "Pioneer 10" is presumed to be at . While Pioneer has ceased communications, both the Voyager spacecraft are performing well and are still communicating.


title: Voyager 2

text: In 2013, "Voyager 1" was escaping the Solar System at a speed of about per year, while "Voyager 2" was escaping at per year.


title: Voyager 2

text: By February 25, 2019, "Voyager 2" was at a distance of from the Sun. There is a variation in distance from Earth caused by the Earth's revolution around the Sun relative to "Voyager 2".


title: Voyager 2

text: It was originally thought that "Voyager 2" would enter interstellar space in early 2016, with its plasma spectrometer providing the first direct measurements of the density and temperature of the interstellar plasma. In December 2018, the Voyager project scientist, Edward C. Stone, announced that "Voyager 2" reached interstellar space on November 5, 2018.


title: Voyager 2

text: In October 2020, astronomers reported a significant unexpected increase in density in the space beyond the Solar System as detected by the "Voyager 1" and "Voyager 2" space probes. According to the researchers, this implies that "the density gradient is a large-scale feature of the VLISM (very local interstellar medium) in the general direction of the heliospheric nose".


title: Voyager 2

text: As the power from the RTG slowly reduces, various items of equipment have been turned off on the spacecraft. The first science equipment turned off on "Voyager 2" was the PPS in 1991, which saved 1.2 watts.


title: Voyager 2

text: In 2023, "Voyager 2" is expected to pass "Pioneer 10" to become the second furthest spacecraft from the Sun at a distance of around 12.4 billion miles.


title: Voyager 2

text: The probe is expected to keep transmitting weak radio messages until at least the mid-2020s, more than 48 years after it was launched.