1.
Domenico Fetti
–
Domenico Fetti was an Italian Baroque painter who had been active mainly in Rome, Mantua and Venice. Born in Rome to a painter, Pietro Fetti, Domenico is said to have apprenticed initially under Ludovico Cigoli. He then worked in Mantua from 1613 to 1622, patronized by the Cardinal, in the Ducal Palace, he painted the Miracle of the Loaves and Fishes. The series of representations of New Testament parables he carried out for his patrons studiolo gave rise to a popular specialty, and he and his studio often repeated his compositions. In August or September 1622, his feuds with some prominent Mantuans led him to move to Venice, into this mix, in the 1620s–30s, three foreigners—Fetti and his younger contemporaries Bernardo Strozzi and Jan Lys—breathed the first influences of Roman Baroque style. They adapted some of the coloration of Venice but adapted it to Caravaggio-influenced realism. In Venice, where he remained despite pleas from the Duke to return to Mantua, Fetti changed his style, in addition, he devoted attention to smaller cabinet pieces that adapt genre imaging to religious stories. His group of paintings entitled Parables, which represent New Testament scenes, are at the Dresden Gemäldegalerie and his painting style appears to have been influenced by Rubens. He would likely have continued to find excellent patronage in Venice had he not died there in 1623 or 1624, Jan Lys, eight years younger, but who had arrived in Venice nearly contemporaneously, died during the plague of 1629–30. Subsequently, Fettis style would influence the Venetians Pietro della Vecchia and his pupils in Mantua were Francesco Bernardi and Dionisio Guerri. Fettis works include, The Good Samaritan Melancholy Emperor Domitian Eve and Laboring Adam Angel in the Garden Jacobs Dream Portrait of an Actor Wittkower, pelican History of Art, Art and Architecture Italy, 1600–1750. Web Gallery of Art, paintings by Domenico Feti
Domenico Fetti
–
Melanconia (
Accademia, Venice)
Domenico Fetti
–
Portrait of a Man with a Sheet of Music (
Getty Museum)
Domenico Fetti
–
Portrait of an Actor (c. 1621–1622)
Domenico Fetti
–
Ideal Portrait of Gonzaga (c. 1620)
2.
Syracuse, Sicily
–
Syracuse is a historic city in Sicily, the capital of the province of Syracuse. The city is notable for its rich Greek history, culture, amphitheatres, architecture and this 2, 700-year-old city played a key role in ancient times, when it was one of the major powers of the Mediterranean world. Syracuse is located in the southeast corner of the island of Sicily, the city was founded by Ancient Greek Corinthians and Teneans and became a very powerful city-state. Syracuse was allied with Sparta and Corinth and exerted influence over the entirety of Magna Graecia, described by Cicero as the greatest Greek city and the most beautiful of them all, it equaled Athens in size during the fifth century BC. It later became part of the Roman Republic and Byzantine Empire, after this Palermo overtook it in importance, as the capital of the Kingdom of Sicily. Eventually the kingdom would be united with the Kingdom of Naples to form the Two Sicilies until the Italian unification of 1860, in the modern day, the city is listed by UNESCO as a World Heritage Site along with the Necropolis of Pantalica. In the central area, the city itself has a population of around 125,000 people, the inhabitants are known as Siracusans. Syracuse is mentioned in the Bible in the Acts of the Apostles book at 28,12 as Paul stayed there, the patron saint of the city is Saint Lucy, she was born in Syracuse and her feast day, Saint Lucys Day, is celebrated on 13 December. Syracuse was founded in 734 or 733 BC by Greek settlers from Corinth and Tenea, there are many attested variants of the name of the city including Συράκουσαι Syrakousai, Συράκοσαι Syrakosai and Συρακώ Syrako. The nucleus of the ancient city was the island of Ortygia. The settlers found the fertile and the native tribes to be reasonably well-disposed to their presence. The city grew and prospered, and for some time stood as the most powerful Greek city anywhere in the Mediterranean, colonies were founded at Akrai, Kasmenai, Akrillai, Helorus and Kamarina. The descendants of the first colonists, called Gamoroi, held power until they were expelled by the Killichiroi, the former, however, returned to power in 485 BC, thanks to the help of Gelo, ruler of Gela. Gelo himself became the despot of the city, and moved many inhabitants of Gela, Kamarina and Megera to Syracuse, building the new quarters of Tyche, the enlarged power of Syracuse made unavoidable the clash against the Carthaginians, who ruled western Sicily. In the Battle of Himera, Gelo, who had allied with Theron of Agrigento, a temple dedicated to Athena, was erected in the city to commemorate the event. Syracuse grew considerably during this time and its walls encircled 120 hectares in the fifth century, but as early as the 470s BC the inhabitants started building outside the walls. The complete population of its territory approximately numbered 250,000 in 415 BC, Gelo was succeeded by his brother Hiero, who fought against the Etruscans at Cumae in 474 BC. His rule was eulogized by poets like Simonides of Ceos, Bacchylides and Pindar, a democratic regime was introduced by Thrasybulos
Syracuse, Sicily
–
Ortygia island, where Syracuse was founded in
ancient Greek times.
Mount Etna is visible in the distance.
Syracuse, Sicily
–
A Syracusan
tetradrachm (c. 415–405 BC), sporting
Arethusa and a
quadriga.
Syracuse, Sicily
–
Decadrachme from Sicile struck at Syracuse and sign d'Évainète
Syracuse, Sicily
–
The siege of Syracuse in a 17th-century engraving.
3.
Magna Graecia
–
The settlers who began arriving in the 8th century BC brought with them their Hellenic civilization, which was to leave a lasting imprint in Italy, such as in the culture of ancient Rome. Most notably the Roman poet Ovid referred to the south of Italy as Magna Graecia in his poem Fasti, according to Strabo, Magna Graecias colonization started already at the time of the Trojan War and lasted for several centuries. Also during that period, Greek colonies were established in places as widely separated as the eastern coast of the Black Sea, Eastern Libya and they included settlements in Sicily and the southern part of the Italian Peninsula. The Romans called the area of Sicily and the foot of Italy Magna Graecia since it was so densely inhabited by the Greeks, the ancient geographers differed on whether the term included Sicily or merely Apulia and Calabria, Strabo being the most prominent advocate of the wider definitions. With colonization, Greek culture was exported to Italy, in its dialects of the Ancient Greek language, its religious rites, an original Hellenic civilization soon developed, later interacting with the native Italic civilisations. Many of the new Hellenic cities became very rich and powerful, like Neapolis, Syracuse, Acragas Paestum, other cities in Magna Graecia included Tarentum, Epizephyrian Locri, Rhegium, Croton, Thurii, Elea, Nola, Ancona, Syessa, Bari and others. Following the Pyrrhic War in the 3rd century BC, Magna Graecia was absorbed into the Roman Republic, a remarkable example of the influence is the Griko-speaking minority that still exists today in the Italian regions of Calabria and Apulia. Griko is the name of a language combining ancient Doric, Byzantine Greek, there is a rich oral tradition and Griko folklore, limited now but once numerous, to around 30,000 people, most of them having become absorbed into the surrounding Italian element. Some scholars, such as Gerhard Rohlfs, argue that the origins of Griko may ultimately be traced to the colonies of Magna Graecia, one example is the Griko people, some of whom still maintain their Greek language and customs. For example, Greeks re-entered the region in the 16th and 17th century in reaction to the conquest of the Peloponnese by the Ottoman Empire, especially after the end of the Siege of Coron, large numbers of Greeks took refuge in the areas of Calabria, Salento and Sicily. Greeks from Coroni, the so-called Coronians, were nobles, who brought with them substantial movable property and they were granted special privileges and tax exemptions. Other Greeks who moved to Italy came from the Mani Peninsula of the Peloponnese, the Maniots were known for their proud military traditions and for their bloody vendettas, many of which still continue today. Another group of Maniot Greeks moved to Corsica, Ancient Greek dialects Greeks in Italy Italiotes Graia Graïke Graecus Griko people Griko language Hellenic civilization Names of the Greeks Cerchiai L. Jannelli L. Longo F. The Greek Cities of Magna Graecia and Sicily, in Dictionary of Greek and Roman Geography. 21 June,2005,17,19 GMT18,19 UK, salentinian Peninsula, Greece and Greater Greece. Traditional Griko song performed by Ghetonia, traditional Griko song performed by amateur local group. Second Interdisciplinary Symposium on the Hellenic Heritage of Southern Italy, the Greeks in the West, genetic signatures of the Hellenic colonisation in southern Italy and Sicily
Magna Graecia
–
Cities of Magna Graecia and other Greek settlements in Italy (in red)
Magna Graecia
–
Northwestern
Magna Graecia
–
Greek temples of
Paestum,
Campania
Magna Graecia
–
Mosaic from
Caulonia,
Calabria
4.
Fluid statics
–
Fluid statics or hydrostatics is the branch of fluid mechanics that studies incompressible fluids at rest. It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamics, hydrostatics are categorized as a part of the fluid statics, which is the study of all fluids, incompressible or not, at rest. Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids and it is also relevant to geophysics and astrophysics, to meteorology, to medicine, and many other fields. Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisterns, aqueducts and fountains. Archimedes is credited with the discovery of Archimedes Principle, which relates the force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The fair cup or Pythagorean cup, which dates from about the 6th century BC, is a technology whose invention is credited to the Greek mathematician. It was used as a learning tool, the cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup, the cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, due to the drag that molecules exert on one another, the cup will be emptied. Herons fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, the device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir. Pascal made contributions to developments in both hydrostatics and hydrodynamics, due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface, if a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force, thus, the pressure on a fluid at rest is isotropic, i. e. it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes, i. e. a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in an extended form, by Blaise Pascal. In a fluid at rest, all frictional and inertial stresses vanish, when this condition of V =0 is applied to the Navier-Stokes equation, the gradient of pressure becomes a function of body forces only
Fluid statics
–
Table of Hydraulics and Hydrostatics, from the 1728
Cyclopædia
Fluid statics
–
Diving medicine:
5.
Lever
–
A lever is a machine consisting of a beam or rigid rod pivoted at a fixed hinge, or fulcrum. A lever is a body capable of rotating on a point on itself. On the basis of the location of fulcrum, load and effort and it is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide an output force. The ratio of the force to the input force is the mechanical advantage of the lever. The word lever entered English about 1300 from Old French, in which the word was levier and this sprang from the stem of the verb lever, meaning to raise. The verb, in turn, goes back to the Latin levare, itself from the adjective levis, the words primary origin is the Proto-Indo-European stem legwh-, meaning light, easy or nimble, among other things. The PIE stem also gave rise to the English word light, the earliest remaining writings regarding levers date from the 3rd century BC and were provided by Archimedes. Give me a place to stand, and I shall move the Earth with it is a remark of Archimedes who formally stated the correct mathematical principle of levers. The distance required to do this might be exemplified in astronomical terms as the distance to the Circinus galaxy - about 9 million light years. It is assumed that in ancient Egypt, constructors used the lever to move, a lever is a beam connected to ground by a hinge, or pivot, called a fulcrum. The ideal lever does not dissipate or store energy, which there is no friction in the hinge or bending in the beam. This is known as the law of the lever, the mechanical advantage of a lever can be determined by considering the balance of moments or torque, T, about the fulcrum. T1 = F1 a, T2 = F2 b where F1 is the force to the lever. The distances a and b are the distances between the forces and the fulcrum. Since the moments of torque must be balanced, T1 = T2, the mechanical advantage of the lever is the ratio of output force to input force, M A = F2 F1 = a b. Levers are classified by the positions of the fulcrum, effort. It is common to call the force the effort and the output force the load or the resistance
Lever
–
Levers can be used to exert a large force over a small distance at one end by exerting only a small force over a greater distance at the other.
Lever
–
A lever in balance
Lever
–
This is an engraving from Mechanics Magazine published in London in 1824.
6.
Neusis construction
–
The neusis is a geometric construction method that was used in antiquity by Greek mathematicians. The neusis construction consists of fitting a line element of length in between two given lines, in such a way that the line element, or its extension. That is, one end of the element has to lie on l. A neusis construction might be performed by means of a neusis ruler, in the figure one end of the ruler is marked with a yellow eye with crosshairs, this is the origin of the scale division on the ruler. A second marking on the ruler indicates the distance a from the origin, the yellow eye is moved along line l, until the blue eye coincides with line m. The position of the element thus found is shown in the figure as a dark blue bar. Point P is called the pole of the neusis, line l the directrix, or guiding line, length a is called the diastema. Neuseis have been important because they provide a means to solve geometric problems that are not solvable by means of compass. Examples are the trisection of any angle in three parts, the doubling of the cube, and the construction of a regular heptagon, nonagon. Mathematicians such as Archimedes of Syracuse and Pappus of Alexandria freely used neuseis, Sir Isaac Newton followed their line of thought, nevertheless, gradually the technique dropped out of use. Modified by the recent finding by Benjamin and Snyder that the regular hendecagon is neusis-constructible, T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides was the first to put compass-and-straightedge constructions above neuseis. One hundred years after him Euclid too shunned neuseis in his influential textbook. The next attack on the neusis came when, from the fourth century BC, under its influence a hierarchy of three classes of geometrical constructions was developed. In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution, Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other methods might have been used was branded by the late Greek mathematician Pappus of Alexandria as a not inconsiderable error. R. Boeker, Neusis, in, Paulys Realencyclopädie der Classischen Altertumswissenschaft, the most comprehensive survey, however, the author sometimes has rather curious opinions. T. L. Heath, A history of Greek Mathematics, H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum. MathWorld page Angle Trisection by Paper Folding
Neusis construction
–
Neusis trisection of an angle θ > 135° to find φ = θ /3, using only the length of the ruler. The radius of the arc is equal to the length of the ruler. For angles θ < 135° the same construction applies, but with P extended beyond AB.
Neusis construction
–
Neusis construction
7.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
8.
Physics
–
Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
Physics
–
Further information:
Outline of physics
Physics
–
Ancient
Egyptian astronomy is evident in monuments like the
ceiling of Senemut's tomb from the
Eighteenth Dynasty of Egypt.
Physics
–
Sir Isaac Newton (1643–1727), whose
laws of motion and
universal gravitation were major milestones in classical physics
Physics
–
Albert Einstein (1879–1955), whose work on the
photoelectric effect and the
theory of relativity led to a revolution in 20th century physics
9.
Engineering
–
The term Engineering is derived from the Latin ingenium, meaning cleverness and ingeniare, meaning to contrive, devise. Engineering has existed since ancient times as humans devised fundamental inventions such as the wedge, lever, wheel, each of these inventions is essentially consistent with the modern definition of engineering. The term engineering is derived from the engineer, which itself dates back to 1390 when an engineer originally referred to a constructor of military engines. In this context, now obsolete, a referred to a military machine. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, the word engine itself is of even older origin, ultimately deriving from the Latin ingenium, meaning innate quality, especially mental power, hence a clever invention. The earliest civil engineer known by name is Imhotep, as one of the officials of the Pharaoh, Djosèr, he probably designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both civilian and military domains, the Antikythera mechanism, the first known mechanical computer, and the mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed, the first steam engine was built in 1698 by Thomas Savery. The development of this gave rise to the Industrial Revolution in the coming decades. With the rise of engineering as a profession in the 18th century, similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of Thomas Newcomen and the Scottish engineer James Watt gave rise to mechanical engineering. The development of specialized machines and machine tools during the revolution led to the rapid growth of mechanical engineering both in its birthplace Britain and abroad. John Smeaton was the first self-proclaimed civil engineer and is regarded as the father of civil engineering. He was an English civil engineer responsible for the design of bridges, canals, harbours and he was also a capable mechanical engineer and an eminent physicist. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of hydraulic lime and his lighthouse remained in use until 1877 and was dismantled and partially rebuilt at Plymouth Hoe where it is known as Smeatons Tower. The United States census of 1850 listed the occupation of engineer for the first time with a count of 2,000, there were fewer than 50 engineering graduates in the U. S. before 1865. In 1870 there were a dozen U. S. mechanical engineering graduates, in 1890 there were 6,000 engineers in civil, mining, mechanical and electrical. There was no chair of applied mechanism and applied mechanics established at Cambridge until 1875, the theoretical work of James Maxwell and Heinrich Hertz in the late 19th century gave rise to the field of electronics
Engineering
–
The
steam engine, a major driver in the
Industrial Revolution, underscores the importance of engineering in modern history. This
beam engine is on display in the
Technical University of Madrid.
Engineering
–
Relief map of the
Citadel of Lille, designed in 1668 by
Vauban, the foremost military engineer of his age.
Engineering
–
The Ancient Romans built
aqueducts to bring a steady supply of clean fresh water to cities and towns in the empire.
Engineering
–
The
International Space Station represents a modern engineering challenge from many disciplines.
10.
Astronomy
–
Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
Astronomy
–
A
star -forming region in the
Large Magellanic Cloud, an
irregular galaxy.
Astronomy
–
A giant
Hubble mosaic of the
Crab Nebula, a
supernova remnant
Astronomy
–
19th century
Sydney Observatory,
Australia (1873)
Astronomy
–
19th century
Quito Astronomical Observatory is located 12 minutes south of the
Equator in
Quito,
Ecuador.
11.
Invention
–
An invention is a unique or novel device, method, composition or process. The invention process is a process within an overall engineering and product development process and it may be an improvement upon a machine or product or a new process for creating an object or a result. An invention that achieves a unique function or result may be a radical breakthrough. Such works are novel and not obvious to others skilled in the same field, an inventor may be taking a big step in success or failure. A patent legally protects the property rights of the inventor. The rules and requirements for patenting an invention vary from country to country, another meaning of invention is cultural invention, which is an innovative set of useful social behaviours adopted by people and passed on to others. The Institute for Social Inventions collected many such ideas in magazines, Invention is also an important component of artistic and design creativity. Inventions often extend the boundaries of knowledge, experience or capability. Brainstorming also can spark new ideas for an invention, collaborative creative processes are frequently used by engineers, designers, architects and scientists. Co-inventors are frequently named on patents, in addition, many inventors keep records of their working process - notebooks, photos, etc. including Leonardo da Vinci, Galileo Galilei, Evangelista Torricelli, Thomas Jefferson and Albert Einstein. In the process of developing an invention, the idea may change. The invention may become simpler, more practical, it may expand, working on one invention can lead to others too. History shows that turning the concept of an invention into a device is not always swift or direct. Inventions may also more useful after time passes and other changes occur. For example, the became more useful once powered flight was a reality. Invention is often a creative process, an open and curious mind allows an inventor to see beyond what is known. Seeing a new possibility, connection, or relationship can spark an invention, inventive thinking frequently involves combining concepts or elements from different realms that would not normally be put together. Sometimes inventors disregard the boundaries between distinctly separate territories or fields, several concepts may be considered when thinking about invention
Invention
–
' BUILD YOUR OWN TELEVISION RECEIVER.'
Science and Invention magazine cover, November 1928
Invention
–
Alessandro Volta with the first
electrical battery. Volta is recognized as one of the most influential inventors of all time.
Invention
–
Thomas Edison with
phonograph. Edison is considered one of the most prolific inventors in history, holding
1,093 U.S. patents in his name.
Invention
–
A rare 1884 photo showing the experimental recording of voice patterns by a photographic process at the
Alexander Graham Bell Laboratory in Washington, D.C. Many of their experimental designs panned out in failure.
12.
Greek language
–
Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
Greek language
–
Idealized portrayal of
Homer
Greek language
–
regions where Greek is the official language
Greek language
–
Greek language road sign, A27 Motorway, Greece
13.
Greeks
–
The Greeks or Hellenes are an ethnic group native to Greece, Cyprus, southern Albania, Turkey, Sicily, Egypt and, to a lesser extent, other countries surrounding the Mediterranean Sea. They also form a significant diaspora, with Greek communities established around the world, many of these regions coincided to a large extent with the borders of the Byzantine Empire of the late 11th century and the Eastern Mediterranean areas of ancient Greek colonization. The cultural centers of the Greeks have included Athens, Thessalonica, Alexandria, Smyrna, most ethnic Greeks live nowadays within the borders of the modern Greek state and Cyprus. The Greek genocide and population exchange between Greece and Turkey nearly ended the three millennia-old Greek presence in Asia Minor, other longstanding Greek populations can be found from southern Italy to the Caucasus and southern Russia and Ukraine and in the Greek diaspora communities in a number of other countries. Today, most Greeks are officially registered as members of the Greek Orthodox Church, the Greeks speak the Greek language, which forms its own unique branch within the Indo-European family of languages, the Hellenic. They are part of a group of ethnicities, described by Anthony D. Smith as an archetypal diaspora people. Both migrations occur at incisive periods, the Mycenaean at the transition to the Late Bronze Age, the Mycenaeans quickly penetrated the Aegean Sea and, by the 15th century BC, had reached Rhodes, Crete, Cyprus and the shores of Asia Minor. Around 1200 BC, the Dorians, another Greek-speaking people, followed from Epirus, the Dorian invasion was followed by a poorly attested period of migrations, appropriately called the Greek Dark Ages, but by 800 BC the landscape of Archaic and Classical Greece was discernible. The Greeks of classical antiquity idealized their Mycenaean ancestors and the Mycenaean period as an era of heroes, closeness of the gods. The Homeric Epics were especially and generally accepted as part of the Greek past, as part of the Mycenaean heritage that survived, the names of the gods and goddesses of Mycenaean Greece became major figures of the Olympian Pantheon of later antiquity. The ethnogenesis of the Greek nation is linked to the development of Pan-Hellenism in the 8th century BC, the works of Homer and Hesiod were written in the 8th century BC, becoming the basis of the national religion, ethos, history and mythology. The Oracle of Apollo at Delphi was established in this period, the classical period of Greek civilization covers a time spanning from the early 5th century BC to the death of Alexander the Great, in 323 BC. It is so named because it set the standards by which Greek civilization would be judged in later eras, the Peloponnesian War, the large scale civil war between the two most powerful Greek city-states Athens and Sparta and their allies, left both greatly weakened. Many Greeks settled in Hellenistic cities like Alexandria, Antioch and Seleucia, two thousand years later, there are still communities in Pakistan and Afghanistan, like the Kalash, who claim to be descended from Greek settlers. The Hellenistic civilization was the period of Greek civilization, the beginnings of which are usually placed at Alexanders death. This Hellenistic age, so called because it saw the partial Hellenization of many non-Greek cultures and this age saw the Greeks move towards larger cities and a reduction in the importance of the city-state. These larger cities were parts of the still larger Kingdoms of the Diadochi, Greeks, however, remained aware of their past, chiefly through the study of the works of Homer and the classical authors. An important factor in maintaining Greek identity was contact with barbarian peoples and this led to a strong desire among Greeks to organize the transmission of the Hellenic paideia to the next generation
Greeks
–
Hoplites fighting. Detail from an Attic black-figure
hydria, ca. 560 BC–550 BC.
Louvre,
Paris.
Greeks
–
A reconstruction of the 3rd millennium BC "Proto-Greek area", according to Bulgarian linguist
Vladimir Georgiev.
Greeks
–
Bust of
Cleopatra VII.
Altes Museum,
Berlin.
Greeks
–
Statues of
Saints Cyril and Methodius, missionaries of
Christianity among the
Slavic peoples, on the
Holy Trinity Column in Olomouc,
Czech Republic.
14.
Greek mathematics
–
Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
Greek mathematics
–
Statue of Euclid in the
Oxford University Museum of Natural History
Greek mathematics
–
An illustration of
Euclid 's proof of the
Pythagorean Theorem
Greek mathematics
–
The
Antikythera mechanism, an ancient mechanical calculator.
15.
Inventor
–
An invention is a unique or novel device, method, composition or process. The invention process is a process within an overall engineering and product development process and it may be an improvement upon a machine or product or a new process for creating an object or a result. An invention that achieves a unique function or result may be a radical breakthrough. Such works are novel and not obvious to others skilled in the same field, an inventor may be taking a big step in success or failure. A patent legally protects the property rights of the inventor. The rules and requirements for patenting an invention vary from country to country, another meaning of invention is cultural invention, which is an innovative set of useful social behaviours adopted by people and passed on to others. The Institute for Social Inventions collected many such ideas in magazines, Invention is also an important component of artistic and design creativity. Inventions often extend the boundaries of knowledge, experience or capability. Brainstorming also can spark new ideas for an invention, collaborative creative processes are frequently used by engineers, designers, architects and scientists. Co-inventors are frequently named on patents, in addition, many inventors keep records of their working process - notebooks, photos, etc. including Leonardo da Vinci, Galileo Galilei, Evangelista Torricelli, Thomas Jefferson and Albert Einstein. In the process of developing an invention, the idea may change. The invention may become simpler, more practical, it may expand, working on one invention can lead to others too. History shows that turning the concept of an invention into a device is not always swift or direct. Inventions may also more useful after time passes and other changes occur. For example, the became more useful once powered flight was a reality. Invention is often a creative process, an open and curious mind allows an inventor to see beyond what is known. Seeing a new possibility, connection, or relationship can spark an invention, inventive thinking frequently involves combining concepts or elements from different realms that would not normally be put together. Sometimes inventors disregard the boundaries between distinctly separate territories or fields, several concepts may be considered when thinking about invention
Inventor
–
' BUILD YOUR OWN TELEVISION RECEIVER.'
Science and Invention magazine cover, November 1928
Inventor
–
Alessandro Volta with the first
electrical battery. Volta is recognized as one of the most influential inventors of all time.
Inventor
–
Thomas Edison with
phonograph. Edison is considered one of the most prolific inventors in history, holding
1,093 U.S. patents in his name.
Inventor
–
A rare 1884 photo showing the experimental recording of voice patterns by a photographic process at the
Alexander Graham Bell Laboratory in Washington, D.C. Many of their experimental designs panned out in failure.
16.
Scientist
–
A scientist is a person engaging in a systematic activity to acquire knowledge that describes and predicts the natural world. In a more restricted sense, a scientist may refer to an individual who uses the scientific method, the person may be an expert in one or more areas of science. The term scientist was coined by the theologian, philosopher and historian of science William Whewell and this article focuses on the more restricted use of the word. Scientists perform research toward a comprehensive understanding of nature, including physical, mathematical and social realms. Philosophers aim to provide an understanding of fundamental aspects of reality and experience, often pursuing inquiries with conceptual, rather than empirical. When science is done with a goal toward practical utility, it is called applied science, an applied scientist may not be designing something in particular, but rather is conducting research with the aim of developing new technologies and practical methods. When science seeks to answer questions about aspects of reality it is sometimes called natural philosophy. Science and technology have continually modified human existence through the engineering process, as a profession the scientist of today is widely recognized. Jurisprudence and mathematics are often grouped with the sciences, some of the greatest physicists have also been creative mathematicians and lawyers. There is a continuum from the most theoretical to the most empirical scientists with no distinct boundaries, in terms of personality, interests, training and professional activity, there is little difference between applied mathematicians and theoretical physicists. Scientists can be motivated in several ways, many have a desire to understand why the world is as we see it and how it came to be. They exhibit a strong curiosity about reality, other motivations are recognition by their peers and prestige, or the desire to apply scientific knowledge for the benefit of peoples health, the nations, the world, nature or industries. Scientists tend to be motivated by direct financial reward for their work than other careers. As a result, scientific researchers often accept lower average salaries when compared with other professions which require a similar amount of training. The number of scientists is vastly different from country to country, for instance, there are only 4 full-time scientists per 10,000 workers in India while this number is 79 for the United Kingdom and the United States. According to the US National Science Foundation 4.7 million people with science degrees worked in the United States in 2015, across all disciplines, the figure included twice as many men as women. Of that total, 17% worked in academia, that is, at universities and undergraduate institutions, 5% of scientists worked for the federal government and about 3. 5% were self-employed. Of the latter two groups, two-thirds were men, 59% of US scientists were employed in industry or business, and another 6% worked in non-profit positions
Scientist
Scientist
–
Chemical scientists in a laboratory of the
University of La Rioja
Scientist
–
"No one in the history of civilization has shaped our understanding of science and natural philosophy more than the great Greek philosopher and scientist
Aristotle (384-322 BC), who exerted a profound and pervasive influence for more than two thousand years" —Gary B. Ferngren
Scientist
–
Alessandro Volta, the inventor of the
electrical battery and discoverer of
methane, is widely regarded as one of the greatest scientists in history.
17.
Classical antiquity
–
It is the period in which Greek and Roman society flourished and wielded great influence throughout Europe, North Africa and Southwestern Asia. Conventionally, it is taken to begin with the earliest-recorded Epic Greek poetry of Homer, and continues through the emergence of Christianity and it ends with the dissolution of classical culture at the close of Late Antiquity, blending into the Early Middle Ages. Such a wide sampling of history and territory covers many disparate cultures, Classical antiquity may refer also to an idealised vision among later people of what was, in Edgar Allan Poes words, the glory that was Greece, and the grandeur that was Rome. The culture of the ancient Greeks, together with influences from the ancient Near East, was the basis of art, philosophy, society. The earliest period of classical antiquity takes place before the background of gradual re-appearance of historical sources following the Bronze Age collapse, the 8th and 7th centuries BC are still largely proto-historical, with the earliest Greek alphabetic inscriptions appearing in the first half of the 8th century. Homer is usually assumed to have lived in the 8th or 7th century BC, in the same period falls the traditional date for the establishment of the Ancient Olympic Games, in 776 BC. The Phoenicians originally expanded from Canaan ports, by the 8th century dominating trade in the Mediterranean, carthage was founded in 814 BC, and the Carthaginians by 700 BC had firmly established strongholds in Sicily, Italy and Sardinia, which created conflicts of interest with Etruria. The Etruscans had established control in the region by the late 7th century BC, forming the aristocratic. According to legend, Rome was founded on April 21,753 BC by twin descendants of the Trojan prince Aeneas, Romulus and Remus. As the city was bereft of women, legend says that the Latins invited the Sabines to a festival and stole their unmarried maidens, leading to the integration of the Latins and the Sabines. Archaeological evidence indeed shows first traces of settlement at the Roman Forum in the mid-8th century BC, the seventh and final king of Rome was Tarquinius Superbus. As the son of Tarquinius Priscus and the son-in-law of Servius Tullius, Superbus was of Etruscan birth and it was during his reign that the Etruscans reached their apex of power. Superbus removed and destroyed all the Sabine shrines and altars from the Tarpeian Rock, the people came to object to his rule when he failed to recognize the rape of Lucretia, a patrician Roman, at the hands of his own son. Lucretias kinsman, Lucius Junius Brutus, summoned the Senate and had Superbus, after Superbus expulsion, the Senate voted to never again allow the rule of a king and reformed Rome into a republican government in 509 BC. In fact the Latin word Rex meaning King became a dirty and hated throughout the Republic. In 510, Spartan troops helped the Athenians overthrow the tyrant Hippias, cleomenes I, king of Sparta, put in place a pro-Spartan oligarchy conducted by Isagoras. Greece entered the 4th century under Spartan hegemony, but by 395 BC the Spartan rulers removed Lysander from office, and Sparta lost her naval supremacy. Athens, Argos, Thebes and Corinth, the two of which were formerly Spartan allies, challenged Spartan dominance in the Corinthian War, which ended inconclusively in 387 BC
Classical antiquity
–
The
Parthenon is one of the most iconic symbols of the classical era, exemplifying ancient Greek culture
18.
Calculus
–
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
Calculus
–
Isaac Newton developed the use of calculus in his
laws of motion and
gravitation.
Calculus
–
Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
–
Maria Gaetana Agnesi
Calculus
–
The
logarithmic spiral of the
Nautilus shell is a classical image used to depict the growth and change related to calculus
19.
Mathematical analysis
–
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, in Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere in the 5th century, the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones, in the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
Mathematical analysis
–
A
strange attractor arising from a
differential equation. Differential equations are an important area of mathematical analysis with many applications to
science and
engineering.
20.
Method of exhaustion
–
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes small, the possible values for the area of the shape are systematically exhausted by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction and this amounts to finding an area of a region by first comparing it to the area of a second region. The idea originated in the late 5th century BC with Antiphon, the theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate areas and volumes. It was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle, the first use of the term was in 1647 by Grégoire de Saint-Vincent in Opus geometricum quadraturae circuli et sectionum. The method of exhaustion is seen as a precursor to the methods of calculus, the development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. Euclid used the method of exhaustion to prove the following six propositions in the book 12 of his Elements, proposition 2 The area of a circle is proportional to the square of its radius. Proposition 5 The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases, proposition 10 The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. Proposition 11 The volume of a cone of the height is proportional to the area of the base. Proposition 12 The volume of a cone that is the similar to another is proportional to the cube of the ratio of the diameters of the bases, proposition 18 The volume of a sphere is proportional to the cube of its diameter. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater area and greater number of sides. He also provided the bounds 3 + 10/71 < π <3 + 10/70, the Method of Mechanical Theorems The Quadrature of the Parabola Trapezoidal rule
Method of exhaustion
–
Grégoire de Saint-Vincent
21.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
Geometry
–
Visual checking of the
Pythagorean theorem for the (3, 4, 5)
triangle as in the
Chou Pei Suan Ching 500–200 BC.
Geometry
–
An illustration of
Desargues' theorem, an important result in
Euclidean and
projective geometry
Geometry
–
Geometry lessons in the 20th century
Geometry
–
A
European and an
Arab practicing geometry in the 15th century.
22.
Theorem
–
In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. In some cases, a picture alone may be sufficient to prove a theorem, because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being trivial, or difficult, or deep and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
Theorem
–
A
planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The
four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
23.
Area
–
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
Area
–
A square metre
quadrat made of PVC pipe.
Area
–
The combined area of these three
shapes is
approximately 15.57
squares.
24.
Circle
–
A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
Circle
–
The
compass in this 13th-century manuscript is a symbol of God's act of
Creation. Notice also the circular shape of the
halo
Circle
–
A circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre (O) in magenta.
Circle
–
Circular piece of silk with Mongol images
Circle
–
Circles in an old
Arabic astronomical drawing.
25.
Surface area
–
The surface area of a solid object is a measure of the total area that the surface of the object occupies. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces and this definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of area was sought by Henri Lebesgue. Their work led to the development of measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface, while the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function S ↦ A which assigns a real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the area is its additivity. More rigorously, if a surface S is a union of many pieces S1, …, Sr which do not overlap except at their boundaries. Surface areas of polygonal shapes must agree with their geometrically defined area. Since surface area is a notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface. This means that surface area is invariant under the group of Euclidean motions and these properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of many pieces that can be represented in the parametric form S D, r → = r →, ∈ D with a continuously differentiable function r →. The area of a piece is defined by the formula A = ∬ D | r → u × r → v | d u d v. Thus the area of SD is obtained by integrating the length of the vector r → u × r → v to the surface over the appropriate region D in the parametric uv plane. The area of the surface is then obtained by adding together the areas of the pieces. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f and surfaces of revolution. It was demonstrated by Hermann Schwarz that already for the cylinder, various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a natural notion of surface area, if a surface is very irregular, or rough
Surface area
–
The inner membrane of the
mitochondrion has a large surface area due to infoldings, allowing higher rates of
cellular respiration (electron
micrograph).
Surface area
–
A
sphere of radius has surface area
26.
Volume
–
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
Volume
–
A
measuring cup can be used to measure volumes of
liquids. This cup measures volume in units of
cups,
fluid ounces, and
millilitres.
27.
Sphere
–
A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
Sphere
–
Circumscribed cylinder to a sphere
Sphere
–
A two-dimensional
perspective projection of a sphere
Sphere
Sphere
–
Deck of playing cards illustrating engineering instruments, England, 1702.
King of spades: Spheres
28.
Parabola
–
A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
Parabola
–
Parabolic compass designed by
Leonardo da Vinci
Parabola
–
Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.
Parabola
–
A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and
air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.
Parabola
–
Parabolic trajectories of water in a fountain.
29.
Pi
–
The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
Pi
–
The constant π is represented in this
mosaic outside the Mathematics Building at the
Technical University of Berlin.
Pi
–
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
Pi
–
Archimedes developed the polygonal approach to approximating π.
Pi
–
Isaac Newton used
infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
30.
Archimedes spiral
–
The Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates it can be described by the equation r = a + b θ with real numbers a and b, changing the parameter a will turn the spiral, while b controls the distance between successive turnings. Archimedes described such a spiral in his book On Spirals, the Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance, hence the name arithmetic spiral. In contrast to this, in a logarithmic spiral these distances, as well as the distances of the intersection points measured from the origin, the Archimedean spiral has two arms, one for θ >0 and one for θ <0. The two arms are connected at the origin. Only one arm is shown on the accompanying graph, taking the mirror image of this arm across the y-axis will yield the other arm. Some sources describe the Archimedean spiral as a spiral with a constant separation distance between successive turns, there is a curve slightly different from the Archimedean spiral, the involute of a circle, whose turns have constant separation distance in the latter sense of parallel curves. Sometimes the term Archimedean spiral is used for the general group of spirals r = a + b θ1 / c. The normal Archimedean spiral occurs when c =1, other spirals falling into this group include the hyperbolic spiral, Fermats spiral, and the lituus. Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones, one method of squaring the circle, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to trisect an angle, both approaches relax the traditional limitation on the use of straightedge and compass. The Archimedean spiral has a variety of real-world applications, scroll compressors, made from two interleaved involutes of a circle of the same size that almost resemble Archimedean spirals, are used for compressing liquids and gases. Asking for a patient to draw an Archimedean spiral is a way of quantifying human tremor, additionally, Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter. They are also used to model the pattern occurs in a roll of paper or tape of constant thickness wrapped around a cylinder. Page with Java application to interactively explore the Archimedean spiral and its related curves Online exploration using JSXGraph Archimedean spiral at mathcurve
Archimedes spiral
–
Three 360° turnings of one arm of an Archimedean spiral