1.
Trigonometric function
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
Trigonometric function
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Trigonometric functions in the complex plane
Trigonometric function
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Trigonometry
Trigonometric function
Trigonometric function
2.
History of trigonometry
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Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy, in Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata. During the Middle Ages, the study of continued in Islamic mathematics. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics, the term trigonometry was derived from Greek τρίγωνον trigōnon, triangle and μέτρον metron, measure. Our modern word sine is derived from the Latin word sinus, the Arabic term is in origin a corruption of Sanskrit jīvā, or chord. Sanskrit jīvā in learned usage was a synonym of jyā chord, Sanskrit jīvā was loaned into Arabic as jiba. Particularly Fibonaccis sinus rectus arcus proved influential in establishing the term sinus, the words minute and second are derived from the Latin phrases partes minutae primae and partes minutae secundae. These roughly translate to first small parts and second small parts, the ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, based on one interpretation of the Plimpton 322 cuneiform tablet, some have even asserted that the ancient Babylonians had a table of secants. There is, however, much debate as to whether it is a table of Pythagorean triples, the Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. Ahmes solution to the problem is the ratio of half the side of the base of the pyramid to its height, in other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face. Ancient Greek and Hellenistic mathematicians made use of the chord, given a circle and an arc on the circle, the chord is the line that subtends the arc. A chords perpendicular bisector passes through the center of the circle and bisects the angle. One half of the chord is the sine of one half the bisected angle, that is, c h o r d θ =2 sin θ2. Due to this relationship, a number of identities and theorems that are known today were also known to Hellenistic mathematicians. For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute angles, respectively, theorems on the lengths of chords are applications of the law of sines. And Archimedes theorem on broken chords is equivalent to formulas for sines of sums, the first trigonometric table was apparently compiled by Hipparchus of Nicaea, who is now consequently known as the father of trigonometry. Hipparchus was the first to tabulate the corresponding values of arc and it seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords
History of trigonometry
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Page from The Compendious Book on Calculation by Completion and Balancing by Muhammad ibn Mūsā al-Khwārizmī (c. AD 820)
History of trigonometry
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The chord of an angle subtends the arc of the angle.
History of trigonometry
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Guo Shoujing (1231–1316)
History of trigonometry
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Isaac Newton in a 1702 portrait by Godfrey Kneller.
3.
Trigonometric functions
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
Trigonometric functions
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Trigonometric functions in the complex plane
Trigonometric functions
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Trigonometry
Trigonometric functions
Trigonometric functions
4.
Inverse trigonometric functions
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In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. There are several notations used for the trigonometric functions. The most common convention is to name inverse trigonometric functions using a prefix, e. g. arcsin, arccos, arctan. This convention is used throughout the article, when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Similarly, in programming languages the inverse trigonometric functions are usually called asin, acos. The notations sin−1, cos−1, tan−1, etc, the confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, −1 = sec. Nevertheless, certain authors advise against using it for its ambiguity, since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. There are multiple numbers y such that sin = x, for example, sin =0, when only one value is desired, the function may be restricted to its principal branch. With this restriction, for x in the domain the expression arcsin will evaluate only to a single value. These properties apply to all the trigonometric functions. The principal inverses are listed in the following table, if x is allowed to be a complex number, then the range of y applies only to its real part. Trigonometric functions of trigonometric functions are tabulated below. This is derived from the tangent addition formula tan = tan + tan 1 − tan tan , like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,11 − z 2, as a binomial series, the series for arctangent can similarly be derived by expanding its derivative 11 + z 2 in a geometric series and applying the integral definition above. Arcsin = z + z 33 + z 55 + z 77 + ⋯ = ∑ n =0 ∞, for example, arccos x = π /2 − arcsin x, arccsc x = arcsin , and so on. Alternatively, this can be expressed, arctan z = ∑ n =0 ∞22 n 2. There are two cuts, from −i to the point at infinity, going down the imaginary axis and it works best for real numbers running from −1 to 1
Inverse trigonometric functions
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Inverse trigonometric functions in the complex plane
Inverse trigonometric functions
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Trigonometry
Inverse trigonometric functions
Inverse trigonometric functions
5.
Exact trigonometric constants
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Exact algebraic expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. When they are, they are more specifically in terms of square roots. For an angle of a number of degrees, which is not a multiple 3°, the values of sine, cosine. Note that 1° = π/180 radians, according to Nivens theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2,1, −1/2, and −1. According to Bakers theorem, if the value of a sine and that is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values. The list in this article is incomplete in several senses, first, the trigonometric functions of all angles that are integer multiples of those given can also be expressed in radicals, but some are omitted here. Second, it is possible to apply the half-angle formula to find an expression in radicals for a trigonometric function of one-half of any angle on the list, then half of that angle. This article only gives the cases based on the Fermat primes 3 and 5, thus for example cos, given in the article 17-gon, is not given here. Fourth, this article deals with trigonometric function values when the expression in radicals is in real radicals—roots of real numbers. Many other trigonometric function values are expressible in, for example, in practice, all values of sines, cosines, and tangents not found in this article are approximated using the techniques described at Generating trigonometric tables. Several different units of measure are widely used, including degrees, radians. The following table shows the conversions and values for some common angles, Values outside the range are trivially derived from these values. This is because the sum of the angles of any n-gon is 180° ×, using cos 36 ∘ =5 +14, tan 36 ∘ =5 −25, this can be simplified to, V = a 34. The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructibility of right triangles, here right triangles made from symmetry sections of regular polygons are used to calculate fundamental trigonometric ratios. Each right triangle represents three points in a polygon, a vertex, an edge center containing that vertex. 2 sin θ =2 −2 cos 2 θ =2 −2 +2 cos 4 θ =2 −2 +2 +2 cos 8 θ and so on. If M =2 and N =2 then cos π17 = M −4 +28, crd is the chord function, crd θ =2 sin θ2. Thus sin 18 ∘ =11 +5 =5 −14, similarly crd 108 ∘ = crd = b a =1 +52, so sin 54 ∘ = cos 36 ∘ =1 +54
Exact trigonometric constants
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The primary solution angles [clarification needed] on the unit circle are at multiples of 30 and 45 degrees.
6.
Law of tangents
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In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of tangents states that a − b a + b = tan tan , to prove the law of tangents we can start with the law of sines, a sin α = b sin β. Let d = a sin α, d = b sin β so that a = d sin α and b = d sin β. As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity tan = sin α ± sin β cos α + cos β. The law of tangents can be used to compute the missing side, on a sphere of unit radius, the sides of the triangle are arcs of great circles. Accordingly their lengths can be expressed in radians or any other units of angular measure, let A, B, C be the angles at the three vertices of the triangle and let a, b, c be the respective lengths of the opposite sides. The spherical law of tangents says tan tan = tan tan , Law of sines Law of cosines Law of cotangents Mollweides formula Half-side formula Tangent half-angle formula
Law of tangents
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Figure 1 – A triangle. The angles α, β, and γ are respectively opposite the sides a, b, and c.
7.
Law of cotangents
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In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. In the upper figure, the points of tangency of the incircle with the sides of the break the perimeter into 6 segments. In each pair the segments are of equal length, for example, the 2 segments adjacent to vertex A are equal. If we pick one segment from each pair, their sum will be the semiperimeter s, an example of this is the segments shown in color in the figure. The two segments making up the red line add up to a, so the blue segment must be of length s − a. Obviously, the five segments must also have lengths s − a, s − b, or s − c. By inspection of the figure, using the definition of the cotangent function, we have cot = s − a r, a number of other results can be derived from the law of cotangents. Note that the area of triangle ABC is also divided into 6 smaller triangles, also in 3 pairs, for example, the two triangles near vertex A, being right triangles of width s − a and height r, each have an area of 1/2r. From the addition formula and the law of cotangents we have sin sin = cot − cot cot + cot = a − b 2 s − a − b. This gives the result a − b c = sin cos as required, here, an extra step is required to transform a product into a sum, according to the sum/product formula. This gives the result b + a c = cos sin as required, the law of tangents can also be derived from this. Law of sines Law of cosines Law of tangents Mollweides formula Formula sheet database – law of cotangents, silvester, John R. Geometry, Ancient and Modern
Law of cotangents
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A triangle, showing the "incircle" and the partitioning of the sides. The angle bisectors meet at the incenter, which is the center of the incircle.
8.
Calculus
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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
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Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Calculus
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Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
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Maria Gaetana Agnesi
Calculus
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The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
9.
List of integrals of trigonometric functions
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The following is a list of integrals of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions, for a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral
List of integrals of trigonometric functions
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Trigonometry
10.
Differentiation of trigonometric functions
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The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. Common trigonometric functions include sin, cos and tan, for example, the derivative of f = sin is represented as f ′ = cos. F ′ is the rate of change of sin at a point a. All derivatives of trigonometric functions can be found using those of sin. The quotient rule is implemented to differentiate the resulting expression. Finding the derivatives of the trigonometric functions involves using implicit differentiation. Let θ be the angle at O made by the two radii OA and OB, since we are considering the limit as θ tends to zero, we may assume that θ is a very small positive number,0 < θ ≪1. Consider the following three regions of the diagram, R1 is the triangle OAB, R2 is the circular sector OAB, clearly, Area < Area < Area. Using basic trigonometric formulae, the area of the triangle OAB is 12 × | | O A | | × | | O B | | × sin θ =12 r 2 sin θ. Collecting together these three areas gives, Area < Area < Area ⟺12 r 2 sin θ <12 r 2 θ <12 r 2 tan θ, since r >0, we can divide through by ½·r2. This means that the construction and calculations are all independent of the circles radius, in the last step we simply took the reciprocal of each of the three terms. Since all three terms are positive this has the effect of reversing the inequities, e. g. if 2 <3 then ½ > ⅓. We have seen that if 0 < sin θ ≪1 then sin/θ is always less than 1 and, notice that as θ gets closer to 0, so cos θ gets closer to 1. Informally, as θ gets smaller, sin/θ is squeezed between 1 and cos θ, which itself it heading towards 1 and it follows that sin/θ tends to 1 as θ tends to 0 from the positive side. The last section enables us to calculate this new limit relatively easily and this is done by employing a simple trick. In this calculation, the sign of θ is unimportant, lim θ →0 = lim θ →0 = lim θ →0. The well-known identity sin2θ + cos2θ =1 tells us that cos2θ –1 = –sin2θ, to calculate the derivative of the sine function sin θ, we use first principles. By definition, d d θ sin θ = lim δ →0, using the well-known angle formula sin = sin α cos β + sin β cos α, we have, d d θ sin θ = lim δ →0 = lim δ →0
Differentiation of trigonometric functions
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Circle, centre O, radius r
Differentiation of trigonometric functions
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Trigonometry
11.
Ancient Greek
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Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
Ancient Greek
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Inscription about the construction of the statue of Athena Parthenos in the Parthenon, 440/439 BC
Ancient Greek
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Ostracon bearing the name of Cimon, Stoa of Attalos
Ancient Greek
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The words ΜΟΛΩΝ ΛΑΒΕ as they are inscribed on the marble of the 1955 Leonidas Monument at Thermopylae
12.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
Triangle
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The Flatiron Building in New York is shaped like a triangular prism
Triangle
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A triangle
13.
Geometry
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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
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Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Geometry
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An illustration of Desargues' theorem, an important result in Euclidean and projective geometry
Geometry
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Geometry lessons in the 20th century
Geometry
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A European and an Arab practicing geometry in the 15th century.
14.
Electrical engineering
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Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone. Subsequently, broadcasting and recording media made electronics part of daily life, the invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object. Electrical engineers typically hold a degree in engineering or electronic engineering. Practicing engineers may have professional certification and be members of a professional body, such bodies include the Institute of Electrical and Electronics Engineers and the Institution of Engineering and Technology. Electrical engineers work in a wide range of industries and the skills required are likewise variable. These range from basic circuit theory to the management skills required of a project manager, the tools and equipment that an individual engineer may need are similarly variable, ranging from a simple voltmeter to a top end analyzer to sophisticated design and manufacturing software. Electricity has been a subject of scientific interest since at least the early 17th century and he also designed the versorium, a device that detected the presence of statically charged objects. In the 19th century, research into the subject started to intensify, Electrical engineering became a profession in the later 19th century. Practitioners had created an electric telegraph network and the first professional electrical engineering institutions were founded in the UK. Over 50 years later, he joined the new Society of Telegraph Engineers where he was regarded by other members as the first of their cohort, Practical applications and advances in such fields created an increasing need for standardised units of measure. They led to the standardization of the units volt, ampere, coulomb, ohm, farad. This was achieved at a conference in Chicago in 1893. During these years, the study of electricity was considered to be a subfield of physics. Thats because early electrical technology was electromechanical in nature, the Technische Universität Darmstadt founded the worlds first department of electrical engineering in 1882. The first course in engineering was taught in 1883 in Cornell’s Sibley College of Mechanical Engineering. It was not until about 1885 that Cornell President Andrew Dickson White established the first Department of Electrical Engineering in the United States, in the same year, University College London founded the first chair of electrical engineering in Great Britain. Professor Mendell P. Weinbach at University of Missouri soon followed suit by establishing the engineering department in 1886
Electrical engineering
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Electrical engineers design complex power systems...
Electrical engineering
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... and electronic circuits.
Electrical engineering
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The discoveries of Michael Faraday formed the foundation of electric motor technology
Electrical engineering
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Thomas Edison, electric light and (DC) power supply networks
15.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
Plane (geometry)
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Vector description of a plane
Plane (geometry)
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Two intersecting planes in three-dimensional space
16.
Spherical trigonometry
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Spherical trigonometry is of great importance for calculations in astronomy, geodesy and navigation. The origins of spherical trigonometry in Greek mathematics and the developments in Islamic mathematics are discussed fully in History of trigonometry. This book is now available on the web. The only significant developments since then have been the application of methods for the derivation of the theorems. A spherical polygon is a polygon on the surface of the sphere defined by a number of great-circle arcs, such polygons may have any number of sides. Two planes define a lune, also called a digon or bi-angle, the analogue of the triangle. Three planes define a triangle, the principal subject of this article. Four planes define a spherical quadrilateral, such a figure, and higher sided polygons, from this point the article will be restricted to spherical triangles, denoted simply as triangles. Both vertices and angles at the vertices are denoted by the upper case letters A, B and C. The angles of spherical triangles are less than π so that π < A + B + C < 3π. The sides are denoted by letters a, b, c. On the unit sphere their lengths are equal to the radian measure of the angles that the great circle arcs subtend at the centre. The sides of proper spherical triangles are less than π so that 0 < a + b + c < 3π, the radius of the sphere is taken as unity. For specific practical problems on a sphere of radius R the measured lengths of the sides must be divided by R before using the identities given below, likewise, after a calculation on the unit sphere the sides a, b, c must be multiplied by R. The polar triangle associated with a triangle ABC is defined as follows, consider the great circle that contains the side BC. This great circle is defined by the intersection of a plane with the surface. The points B and C are defined similarly, the triangle ABC is the polar triangle corresponding to triangle ABC. Therefore, if any identity is proved for the triangle ABC then we can derive a second identity by applying the first identity to the polar triangle by making the above substitutions
Spherical trigonometry
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Eight spherical triangles defined by the intersection of three great circles.
17.
Curvature
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In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. This article deals primarily with extrinsic curvature and its canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius everywhere. Smaller circles bend more sharply, and hence have higher curvature, the curvature of a smooth curve is defined as the curvature of its osculating circle at each point. Curvature is normally a scalar quantity, but one may define a curvature vector that takes into account the direction of the bend in addition to its magnitude. The curvature of more objects is described by more complex objects from linear algebra. This article sketches the mathematical framework which describes the curvature of a curve embedded in a plane, the curvature of C at a point is a measure of how sensitive its tangent line is to moving the point to other nearby points. There are a number of equivalent ways that this idea can be made precise and it is natural to define the curvature of a straight line to be constantly zero. The curvature of a circle of radius R should be large if R is small and small if R is large, thus the curvature of a circle is defined to be the reciprocal of the radius, κ =1 R. Given any curve C and a point P on it, there is a circle or line which most closely approximates the curve near P. The curvature of C at P is then defined to be the curvature of that circle or line, the radius of curvature is defined as the reciprocal of the curvature. Another way to understand the curvature is physical, suppose that a particle moves along the curve with unit speed. Taking the time s as the parameter for C, this provides a natural parametrization for the curve, the unit tangent vector T also depends on time. The curvature is then the magnitude of the rate of change of T. Symbolically and this is the magnitude of the acceleration of the particle and the vector dT/ds is the acceleration vector. Geometrically, the curvature κ measures how fast the unit tangent vector to the curve rotates. If a curve close to the same direction, the unit tangent vector changes very little and the curvature is small, where the curve undergoes a tight turn. These two approaches to the curvature are related geometrically by the following observation, in the first definition, the curvature of a circle is equal to the ratio of the angle of an arc to its length. e. For such a curve, there exists a reparametrization with respect to arc length s. This is a parametrization of C such that ∥ γ ′ ∥2 = x ′2 + y ′2 =1, the velocity vector T is the unit tangent vector
Curvature
18.
Elliptic geometry
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Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a line must intersect. In fact, the perpendiculars on one side all intersect at the pole of the given line. There are no points in elliptic geometry. Every point corresponds to a polar line of which it is the absolute pole. Any point on this line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant, the distance between a pair of points is proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter The name elliptic is possibly misleading and it does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes, analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity. A simple way to picture elliptic geometry is to look at a globe, neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, with this identification of antipodal points, the model satisfies Euclids first postulate, which states that two points uniquely determine a line. Metaphorically, we can imagine geometers who are like living on the surface of a sphere. Even if the ants are unable to move off the surface, they can still construct lines, the existence of a third dimension is irrelevant to the ants ability to do geometry, and its existence is neither verifiable nor necessary from their point of view. Another way of putting this is that the language of the axioms is incapable of expressing the distinction between one model and another. In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the figures are similar, i. e. they have the same angles. In elliptic geometry this is not the case, for example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere. A line segment therefore cannot be scaled up indefinitely, a geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space
Elliptic geometry
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On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Elliptic geometry
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Projecting a sphere to a plane.
19.
Navigation
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Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another. The field of navigation includes four categories, land navigation, marine navigation, aeronautic navigation. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks, all navigational techniques involve locating the navigators position compared to known locations or patterns. Navigation, in a sense, can refer to any skill or study that involves the determination of position and direction. In this sense, navigation includes orienteering and pedestrian navigation, for information about different navigation strategies that people use, visit human navigation. In the European medieval period, navigation was considered part of the set of seven mechanical arts, early Pacific Polynesians used the motion of stars, weather, the position of certain wildlife species, or the size of waves to find the path from one island to another. Maritime navigation using scientific instruments such as the mariners astrolabe first occurred in the Mediterranean during the Middle Ages, the perfecting of this navigation instrument is attributed to Portuguese navigators during early Portuguese discoveries in the Age of Discovery. Open-seas navigation using the astrolabe and the compass started during the Age of Discovery in the 15th century, the Portuguese began systematically exploring the Atlantic coast of Africa from 1418, under the sponsorship of Prince Henry. In 1488 Bartolomeu Dias reached the Indian Ocean by this route, in 1492 the Spanish monarchs funded Christopher Columbuss expedition to sail west to reach the Indies by crossing the Atlantic, which resulted in the Discovery of America. In 1498, a Portuguese expedition commanded by Vasco da Gama reached India by sailing around Africa, soon, the Portuguese sailed further eastward, to the Spice Islands in 1512, landing in China one year later. The fleet of seven ships sailed from Sanlúcar de Barrameda in Southern Spain in 1519, crossed the Atlantic Ocean, some ships were lost, but the remaining fleet continued across the Pacific making a number of discoveries including Guam and the Philippines. By then, only two galleons were left from the original seven, the Victoria led by Elcano sailed across the Indian Ocean and north along the coast of Africa, to finally arrive in Spain in 1522, three years after its departure. The Trinidad sailed east from the Philippines, trying to find a path back to the Americas. He arrived in Acapulco on October 8,1565, the term stems from 1530s, from Latin navigationem, from navigatus, pp. of navigare to sail, sail over, go by sea, steer a ship, from navis ship and the root of agere to drive. Roughly, the latitude of a place on Earth is its angular distance north or south of the equator, latitude is usually expressed in degrees ranging from 0° at the Equator to 90° at the North and South poles. The height of Polaris in degrees above the horizon is the latitude of the observer, similar to latitude, the longitude of a place on Earth is the angular distance east or west of the prime meridian or Greenwich meridian. Longitude is usually expressed in degrees ranging from 0° at the Greenwich meridian to 180° east and west, sydney, for example, has a longitude of about 151° east. New York City has a longitude of 74° west, for most of history, mariners struggled to determine longitude
Navigation
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Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia
Navigation
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Dead reckoning or DR, in which one advances a prior position using the ship's course and speed. The new position is called a DR position. It is generally accepted that only course and speed determine the DR position. Correcting the DR position for leeway, current effects, and steering error result in an estimated position or EP. An inertial navigator develops an extremely accurate EP.
Navigation
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Pilotage involves navigating in restricted waters with frequent determination of position relative to geographic and hydrographic features.
Navigation
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Celestial navigation involves reducing celestial measurements to lines of position using tables, spherical trigonometry, and almanacs.
20.
Sumer
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Living along the valleys of the Tigris and Euphrates, Sumerian farmers were able to grow an abundance of grain and other crops, the surplus of which enabled them to settle in one place. Proto-writing in the dates back to c.3000 BC. The earliest texts come from the cities of Uruk and Jemdet Nasr and date back to 3300 BC, modern historians have suggested that Sumer was first permanently settled between c.5500 and 4000 BC by a West Asian people who spoke the Sumerian language, an agglutinative language isolate. These conjectured, prehistoric people are now called proto-Euphrateans or Ubaidians, some scholars contest the idea of a Proto-Euphratean language or one substrate language. Reliable historical records begin much later, there are none in Sumer of any kind that have dated before Enmebaragesi. Juris Zarins believes the Sumerians lived along the coast of Eastern Arabia, todays Persian Gulf region, Sumerian civilization took form in the Uruk period, continuing into the Jemdet Nasr and Early Dynastic periods. During the 3rd millennium BC, a cultural symbiosis developed between the Sumerians, who spoke a language isolate, and Akkadian-speakers, which included widespread bilingualism. The influence of Sumerian on Akkadian is evident in all areas, from lexical borrowing on a scale, to syntactic, morphological. This has prompted scholars to refer to Sumerian and Akkadian in the 3rd millennium BC as a Sprachbund, Sumer was conquered by the Semitic-speaking kings of the Akkadian Empire around 2270 BC, but Sumerian continued as a sacred language. Native Sumerian rule re-emerged for about a century in the Neo-Sumerian Empire or Third Dynasty of Ur approximately 2100-2000 BC, the term Sumerian is the common name given to the ancient non-Semitic-speaking inhabitants of Mesopotamia, Sumer, by the East Semitic-speaking Akkadians. The Sumerians referred to themselves as ùĝ saĝ gíg ga, phonetically /uŋ saŋ gi ga/, literally meaning the black-headed people, the Akkadian word Shumer may represent the geographical name in dialect, but the phonological development leading to the Akkadian term šumerû is uncertain. Hebrew Shinar, Egyptian Sngr, and Hittite Šanhar, all referring to southern Mesopotamia, in the late 4th millennium BC, Sumer was divided into many independent city-states, which were divided by canals and boundary stones. Each was centered on a dedicated to the particular patron god or goddess of the city. The Sumerian city-states rose to power during the prehistoric Ubaid and Uruk periods, classical Sumer ends with the rise of the Akkadian Empire in the 23rd century BC. Following the Gutian period, there is a brief Sumerian Renaissance in the 21st century BC, the Amorite dynasty of Isin persisted until c.1700 BC, when Mesopotamia was united under Babylonian rule. The Sumerians were eventually absorbed into the Akkadian population, 2500–2334 BC Akkadian Empire period, c. 2218–2047 BC Ur III period, c, 2047–1940 BC The Ubaid period is marked by a distinctive style of fine quality painted pottery which spread throughout Mesopotamia and the Persian Gulf. It appears that this culture was derived from the Samarran culture from northern Mesopotamia and it is not known whether or not these were the actual Sumerians who are identified with the later Uruk culture
Sumer
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Map of Sumer
Sumer
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The Samarra bowl, at the Pergamonmuseum, Berlin. The swastika in the center of the design is a reconstruction.
Sumer
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Fragment of Eannatum 's Stele of the Vultures
21.
Babylonians
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Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia. A small Amorite-ruled state emerged in 1894 BC, which contained at this time the city of Babylon. Babylon greatly expanded during the reign of Hammurabi in the first half of the 18th century BC, during the reign of Hammurabi and afterwards, Babylonia was called Māt Akkadī the country of Akkad in the Akkadian language. It was often involved in rivalry with its older fellow Akkadian-speaking state of Assyria in northern Mesopotamia and it retained the Sumerian language for religious use, but by the time Babylon was founded, this was no longer a spoken language, having been wholly subsumed by Akkadian. The earliest mention of the city of Babylon can be found in a tablet from the reign of Sargon of Akkad. During the 3rd millennium BC, a cultural symbiosis occurred between Sumerian and Akkadian-speakers, which included widespread bilingualism. The influence of Sumerian on Akkadian and vice versa is evident in all areas, from lexical borrowing on a scale, to syntactic, morphological. This has prompted scholars to refer to Sumerian and Akkadian in the millennium as a sprachbund. Traditionally, the religious center of all Mesopotamia was the city of Nippur. The empire eventually disintegrated due to decline, climate change and civil war. Sumer rose up again with the Third Dynasty of Ur in the late 22nd century BC and they also seem to have gained ascendancy over most of the territory of the Akkadian kings of Assyria in northern Mesopotamia for a time. The states of the south were unable to stem the Amorite advance, King Ilu-shuma of the Old Assyrian Empire in a known inscription describes his exploits to the south as follows, The freedom of the Akkadians and their children I established. I established their freedom from the border of the marshes and Ur and Nippur, Awal, past scholars originally extrapolated from this text that it means he defeated the invading Amorites to the south, but there is no explicit record of that. More recently, the text has been taken to mean that Asshur supplied the south with copper from Anatolia and these policies were continued by his successors Erishum I and Ikunum. During the first centuries of what is called the Amorite period and his reign was concerned with establishing statehood amongst a sea of other minor city states and kingdoms in the region. However Sumuabum appears never to have bothered to give himself the title of King of Babylon, suggesting that Babylon itself was only a minor town or city. He was followed by Sumu-la-El, Sabium, Apil-Sin, each of whom ruled in the same manner as Sumuabum. Sin-Muballit was the first of these Amorite rulers to be regarded officially as a king of Babylon, the Elamites occupied huge swathes of southern Mesopotamia, and the early Amorite rulers were largely held in vassalage to Elam
Babylonians
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Old Babylonian Cylinder Seal, hematite, The king makes an animal offering to Shamash. This seal was probably made in a workshop at Sippar.
Babylonians
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Geography
22.
Similarity (geometry)
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Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling and this means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other zoomed in or out at some level. For example, all circles are similar to other, all squares are similar to each other. On the other hand, ellipses are not all similar to other, rectangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure and it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem, due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of which is necessary and sufficient for two triangles to be similar,1, the triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is, If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′, all the corresponding sides have lengths in the same ratio, AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other, two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance, AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′ and this is known as the SAS Similarity Criterion. When two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′, there are several elementary results concerning similar triangles in Euclidean geometry, Any two equilateral triangles are similar. Two triangles, both similar to a triangle, are similar to each other. Corresponding altitudes of similar triangles have the ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, the statement that the point F satisfying this condition exists is Walliss Postulate and is logically equivalent to Euclids Parallel Postulate
Similarity (geometry)
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Sierpinski triangle. A space having self-similarity dimension ln 3 / ln 2 = log 2 3, which is approximately 1.58. (from Hausdorff dimension.)
Similarity (geometry)
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Figures shown in the same color are similar
23.
Nubia
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Nubia is a region along the Nile river located in what is today northern Sudan and southern Egypt. It was the seat of one of the earliest civilizations of ancient Africa, with a history that can be traced from at least 2000 B. C. onward, and was home to one of the African empires. Nubia was again united within Ottoman Egypt in the 19th century, the name Nubia is derived from that of the Noba people, nomads who settled the area in the 4th century following the collapse of the kingdom of Meroë. The Noba spoke a Nilo-Saharan language, ancestral to Old Nubian, Old Nubian was mostly used in religious texts dating from the 8th and 15th centuries AD. Before the 4th century, and throughout classical antiquity, Nubia was known as Kush, or, in Classical Greek usage, until at least 1970, the Birgid language was spoken north of Nyala in Darfur, but is now extinct. Nubia was divided into two regions, Upper and Lower Nubia, so called because of their location in the Nile river valley. Early settlements sprouted in both Upper and Lower Nubia, Egyptians referred to Nubia as Ta-Seti, or The Land of the Bow, since the Nubians were known to be expert archers. Modern scholars typically refer to the people from this area as the “A-Group” culture, fertile farmland just south of the Third Cataract is known as the “pre-Kerma” culture in Upper Nubia, as they are the ancestors. The Neolithic people in the Nile Valley likely came from Sudan, as well as the Sahara, by the 5th millennium BC, the people who inhabited what is now called Nubia participated in the Neolithic revolution. Saharan rock reliefs depict scenes that have been thought to be suggestive of a cult, typical of those seen throughout parts of Eastern Africa. Megaliths discovered at Nabta Playa are early examples of what seems to be one of the worlds first astronomical devices, around 3500 BC, the second Nubian culture, termed the A-Group, arose. It was a contemporary of, and ethnically and culturally similar to. The A-Group people were engaged in trade with the Egyptians and this trade is testified archaeologically by large amounts of Egyptian commodities deposited in the graves of the A-Group people. The imports consisted of gold objects, copper tools, faience amulets and beads, seals, slate palettes, stone vessels, and a variety of pots. Around 3300 BC, there is evidence of a kingdom, as shown by the finds at Qustul. The Nubian culture may have contributed to the unification of the Nile Valley. The earliest known depiction of the crown is on a ceremonial incense burner from Cemetery at Qustul in Lower Nubia. New evidence from Abydos, however, particularly the excavation of Cemetery U, around the turn of the protodynastic period, Naqada, in its bid to conquer and unify the whole Nile Valley, seems to have conquered Ta-Seti and harmonized it with the Egyptian state
Nubia
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Nubians in worship
Nubia
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Nubian woman circa 1900
Nubia
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Head of a Nubian Ruler
Nubia
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Ramesses II in his war chariot charging into battle against the Nubians
24.
Chord (geometry)
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A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just secant, is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse, a chord that passes through a circles center point is the circles diameter. Every diameter is a chord, but not every chord is a diameter, the word chord is from the Latin chorda meaning bowstring. Among properties of chords of a circle are the following, Chords are equidistant from the center if, a chord that passes through the center of a circle is called a diameter, and is the longest chord. If the line extensions of chords AB and CD intersect at a point P, the area that a circular chord cuts off is called a circular segment. The midpoints of a set of chords of an ellipse are collinear. Chords were used extensively in the development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the function for every 7.5 degrees. The circle was of diameter 120, and the lengths are accurate to two base-60 digits after the integer part. The chord function is defined geometrically as shown in the picture, the chord of an angle is the length of the chord between two points on a unit circle separated by that angle. The last step uses the half-angle formula, much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, all now lost, so presumably a great deal was known about them
Chord (geometry)
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The red segment BX is a chord (as is the diameter segment AB).
25.
Inscribed angle
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In geometry, an inscribed angle is the angle formed in the interior of a circle when two secant lines intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two points on the circle Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an angle to that of the central angle subtending the same arc. The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle, let O be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them V and A, draw line VO and extended past O so that it intersects the circle at point B which is diametrically opposite the point V. Draw an angle whose vertex is point V and whose sides pass through points A and B, Angle BOA is a central angle, call it θ. Lines OV and OA are both radii of the circle, so they have equal lengths, therefore, triangle VOA is isosceles, so angle BVA and angle VAO are equal, let each of them be denoted as ψ. Angles BOA and AOV are supplementary and they add up to 180°, since line VB passing through O is a straight line. Therefore, angle AOV measures 180° − θ and it is known that the three angles of a triangle add up to 180°, and the three angles of triangle VOA are, 180° − θ ψ ψ. Therefore,2 ψ +180 ∘ − θ =180 ∘, subtract 180° from both sides,2 ψ = θ, where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB. Given a circle whose center is point O, choose three points V, C, and D on the circle, draw lines VC and VD, angle DVC is an inscribed angle. Now draw line VO and extend it past point O so that it intersects the circle at point E. Angle DVC subtends arc DC on the circle, suppose this arc includes point E within it. Point E is diametrically opposite to point V, angles DVE and EVC are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them. Angle DOC is an angle, but so are angles DOE and EOC. Let θ0 = ∠ D O C, θ1 = ∠ D O E, θ2 = ∠ E O C, from Part One we know that θ1 =2 ψ1 and that θ2 =2 ψ2. Combining these results with equation yields θ0 =2 ψ1 +2 ψ2 therefore, by equation, θ0 =2 ψ0. The previous case can be extended to cover the case where the measure of the angle is the difference between two inscribed angles as discussed in the first part of this proof
Inscribed angle
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The inscribed angle θ is half of the central angle 2 θ that subtends the same arc on the circle (magenta). Thus, the angle θ does not change as its vertex is moved around on the circle (green, blue and gold angles).
26.
Ptolemy
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Claudius Ptolemy was a Greek writer, known as a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in the city of Alexandria in the Roman province of Egypt, wrote in Koine Greek, beyond that, few reliable details of his life are known. His birthplace has been given as Ptolemais Hermiou in the Thebaid in a statement by the 14th-century astronomer Theodore Meliteniotes. This is a very late attestation, however, and there is no reason to suppose that he ever lived elsewhere than Alexandria. Ptolemy wrote several treatises, three of which were of importance to later Byzantine, Islamic and European science. The first is the astronomical treatise now known as the Almagest, although it was entitled the Mathematical Treatise. The second is the Geography, which is a discussion of the geographic knowledge of the Greco-Roman world. The third is the treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatika but more known as the Tetrabiblos from the Greek meaning Four Books or by the Latin Quadripartitum. The name Claudius is a Roman nomen, the fact that Ptolemy bore it indicates he lived under the Roman rule of Egypt with the privileges and political rights of Roman citizenship. It would have suited custom if the first of Ptolemys family to become a citizen took the nomen from a Roman called Claudius who was responsible for granting citizenship, if, as was common, this was the emperor, citizenship would have been granted between AD41 and 68. The astronomer would also have had a praenomen, which remains unknown and it occurs once in Greek mythology, and is of Homeric form. All the kings after him, until Egypt became a Roman province in 30 BC, were also Ptolemies, abu Mashar recorded a belief that a different member of this royal line composed the book on astrology and attributed it to Ptolemy. The correct answer is not known”, Ptolemy wrote in Greek and can be shown to have utilized Babylonian astronomical data. He was a Roman citizen, but most scholars conclude that Ptolemy was ethnically Greek and he was often known in later Arabic sources as the Upper Egyptian, suggesting he may have had origins in southern Egypt. Later Arabic astronomers, geographers and physicists referred to him by his name in Arabic, Ptolemys Almagest is the only surviving comprehensive ancient treatise on astronomy. Ptolemy presented his models in convenient tables, which could be used to compute the future or past position of the planets. The Almagest also contains a catalogue, which is a version of a catalogue created by Hipparchus
Ptolemy
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Engraving of a crowned Ptolemy being guided by the muse Astronomy, from Margarita Philosophica by Gregor Reisch, 1508. Although Abu Ma'shar believed Ptolemy to be one of the Ptolemies who ruled Egypt after the conquest of Alexander the title ‘King Ptolemy’ is generally viewed as a mark of respect for Ptolemy's elevated standing in science.
Ptolemy
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Early Baroque artist's rendition
Ptolemy
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A 15th-century manuscript copy of the Ptolemy world map, reconstituted from Ptolemy's Geography (circa 150), indicating the countries of " Serica " and "Sinae" (China) at the extreme east, beyond the island of "Taprobane" (Sri Lanka, oversized) and the "Aurea Chersonesus" (Malay Peninsula).
Ptolemy
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Prima Europe tabula. A C15th copy of Ptolemy's map of Britain
27.
Almagest
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The Almagest is the critical source of information on ancient Greek astronomy. It has also been valuable to students of mathematics because it documents the ancient Greek mathematician Hipparchuss work, Hipparchus wrote about trigonometry, but because his works appear to have been lost, mathematicians use Ptolemys book as their source for Hipparchuss work and ancient Greek trigonometry in general. The treatise was later titled Hē Megalē Syntaxis, and the form of this lies behind the Arabic name al-majisṭī. Ptolemy set up a public inscription at Canopus, Egypt, in 147 or 148, the late N. T. Hamilton found that the version of Ptolemys models set out in the Canopic Inscription was earlier than the version in the Almagest. Hence it cannot have been completed before about 150, a century after Ptolemy began observing. The Syntaxis Mathematica consists of thirteen sections, called books, an example illustrating how the Syntaxis was organized is given below. It is a 152-page Latin edition printed in 1515 at Venice by Petrus Lichtenstein, then follows an explanation of chords with table of chords, observations of the obliquity of the ecliptic, and an introduction to spherical trigonometry. There is also a study of the angles made by the ecliptic with the vertical, Book III covers the length of the year, and the motion of the Sun. Ptolemy explains Hipparchus discovery of the precession of the equinoxes and begins explaining the theory of epicycles. Books IV and V cover the motion of the Moon, lunar parallax, the motion of the apogee. Book VI covers solar and lunar eclipses, books VII and VIII cover the motions of the fixed stars, including precession of the equinoxes. They also contain a catalogue of 1022 stars, described by their positions in the constellations. The brightest stars were marked first magnitude, while the faintest visible to the eye were sixth magnitude. Each numerical magnitude was twice the brightness of the following one and this system is believed to have originated with Hipparchus. The stellar positions too are of Hipparchan origin, despite Ptolemys claim to the contrary, Book IX addresses general issues associated with creating models for the five naked eye planets, and the motion of Mercury. Book X covers the motions of Venus and Mars, Book XI covers the motions of Jupiter and Saturn. Book XII covers stations and retrograde motion, which occurs when planets appear to pause, Ptolemy understood these terms to apply to Mercury and Venus as well as the outer planets. Book XIII covers motion in latitude, that is, the deviation of planets from the ecliptic, the cosmology of the Syntaxis includes five main points, each of which is the subject of a chapter in Book I
Almagest
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Ptolemy's Almagest became an authoritative work for many centuries.
Almagest
Almagest
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Picture of George Trebizond's Latin translation of Almagest
28.
Byzantine
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It survived the fragmentation and fall of the Western Roman Empire in the 5th century AD and continued to exist for an additional thousand years until it fell to the Ottoman Turks in 1453. During most of its existence, the empire was the most powerful economic, cultural, several signal events from the 4th to 6th centuries mark the period of transition during which the Roman Empires Greek East and Latin West divided. Constantine I reorganised the empire, made Constantinople the new capital, under Theodosius I, Christianity became the Empires official state religion and other religious practices were proscribed. Finally, under the reign of Heraclius, the Empires military, the borders of the Empire evolved significantly over its existence, as it went through several cycles of decline and recovery. During the reign of Maurice, the Empires eastern frontier was expanded, in a matter of years the Empire lost its richest provinces, Egypt and Syria, to the Arabs. This battle opened the way for the Turks to settle in Anatolia, the Empire recovered again during the Komnenian restoration, such that by the 12th century Constantinople was the largest and wealthiest European city. Despite the eventual recovery of Constantinople in 1261, the Byzantine Empire remained only one of several small states in the area for the final two centuries of its existence. Its remaining territories were annexed by the Ottomans over the 15th century. The Fall of Constantinople to the Ottoman Empire in 1453 finally ended the Byzantine Empire, the term comes from Byzantium, the name of the city of Constantinople before it became Constantines capital. This older name of the city would rarely be used from this point onward except in historical or poetic contexts. The publication in 1648 of the Byzantine du Louvre, and in 1680 of Du Canges Historia Byzantina further popularised the use of Byzantine among French authors, however, it was not until the mid-19th century that the term came into general use in the Western world. The Byzantine Empire was known to its inhabitants as the Roman Empire, the Empire of the Romans, Romania, the Roman Republic, Graikia, and also as Rhōmais. The inhabitants called themselves Romaioi and Graikoi, and even as late as the 19th century Greeks typically referred to modern Greek as Romaika and Graikika. The authority of the Byzantine emperor as the legitimate Roman emperor was challenged by the coronation of Charlemagne as Imperator Augustus by Pope Leo III in the year 800. No such distinction existed in the Islamic and Slavic worlds, where the Empire was more seen as the continuation of the Roman Empire. In the Islamic world, the Roman Empire was known primarily as Rûm, the Roman army succeeded in conquering many territories covering the entire Mediterranean region and coastal regions in southwestern Europe and north Africa. These territories were home to different cultural groups, both urban populations and rural populations. The West also suffered heavily from the instability of the 3rd century AD
Byzantine
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Tremissis with the image of Justinian the Great (r. 527–565) (see Byzantine insignia)
Byzantine
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Byzantine lamellar armour klivanium (Κλιβάνιον) - a predecessor of Ottoman krug mirror armour
Byzantine
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The Baptism of Constantine painted by Raphael 's pupils (1520–1524, fresco, Vatican City, Apostolic Palace); Eusebius of Caesarea records that (as was common among converts of early Christianity) Constantine delayed receiving baptism until shortly before his death
Byzantine
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Restored section of the Theodosian Walls.
29.
Indian mathematics
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Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama, the decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, in addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China and this was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved, all mathematical works were orally transmitted until approximately 500 BCE, thereafter, they were transmitted both orally and in manuscript form. A later landmark in Indian mathematics was the development of the series expansions for functions by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a theory of differentiation and integration. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of practical mathematics. The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4,2,1, considered favourable for the stability of a brick structure. They used a system of weights based on the ratios, 1/20, 1/10, 1/5, 1/2,1,2,5,10,20,50,100,200. They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, the inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts, bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. The religious texts of the Vedic Period provide evidence for the use of large numbers, by the time of the Yajurvedasaṃhitā-, numbers as high as 1012 were being included in the texts. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta, With three-fourths Puruṣa went up, the Satapatha Brahmana contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Śulba Sūtras list rules for the construction of fire altars. Most mathematical problems considered in the Śulba Sūtras spring from a single theological requirement, according to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. The diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately and they contain lists of Pythagorean triples, which are particular cases of Diophantine equations
Indian mathematics
Indian mathematics
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The design of the domestic fire altar in the Śulba Sūtra
30.
Mathematics in medieval Islam
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Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries, the study of algebra, the name of which is derived from the Arabic word meaning completion or reunion of broken parts, flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a scholar in the House of Wisdom in Baghdad, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi deals with ways to solve for the roots of first. He also introduces the method of reduction, and unlike Diophantus, Al-Khwarizmis algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the work of Diophantus, which was syncopated. The transition to symbolic algebra, where symbols are used, can be seen in the work of Ibn al-Banna al-Marrakushi. It is important to understand just how significant this new idea was and it was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a theory which allowed rational numbers, irrational numbers, geometrical magnitudes. It gave mathematics a whole new development path so much broader in concept to that which had existed before, another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. Several other mathematicians during this time expanded on the algebra of Al-Khwarizmi. Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation, omar Khayyam found the general geometric solution of a cubic equation. Omar Khayyám wrote the Treatise on Demonstration of Problems of Algebra containing the solution of cubic or third-order equations. Khayyám obtained the solutions of equations by finding the intersection points of two conic sections. This method had used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al-Dīn al-Ṭūsī developed an approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. His surviving works give no indication of how he discovered his formulae for the maxima of these curves, various conjectures have been proposed to account for his discovery of them. The earliest implicit traces of mathematical induction can be found in Euclids proof that the number of primes is infinite, the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique
Mathematics in medieval Islam
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A page from the The Compendious Book on Calculation by Completion and Balancing by Al-Khwarizmi.
Mathematics in medieval Islam
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Engraving of Abū Sahl al-Qūhī 's perfect compass to draw conic sections.
Mathematics in medieval Islam
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The theorem of Ibn Haytham.
31.
Spherical geometry
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Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean, two practical applications of the principles of spherical geometry are navigation and astronomy. In plane geometry, the concepts are points and lines. On a sphere, points are defined in the usual sense, the equivalents of lines are not defined in the usual sense of straight line in Euclidean geometry, but in the sense of the shortest paths between points, which are called geodesics. On a sphere, the geodesics are the circles, other geometric concepts are defined as in plane geometry. Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry, for example, it shares with that geometry the property that a line has no parallels through a given point. An important geometry related to that of the sphere is that of the projective plane. Locally, the plane has all the properties of spherical geometry. In particular, it is non-orientable, or one-sided, Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher-dimensional spherical geometries exist, see elliptic geometry, the earliest mathematical work of antiquity to come down to our time is On the rotating sphere by Autolycus of Pitane, who lived at the end of the fourth century BC. The book of unknown arcs of a written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the law of sines. The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe, however, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15,1771, pp. 195–216, Opera Omnia, Series 1, Volume 28, pp. 142–160. L. Euler, De mensura angulorum solidorum, Acta academiae scientarum imperialis Petropolitinae 2,1781, p. 31–54, Opera Omnia, Series 1, vol. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientarum imperialis Petropolitinae 4,1783, p. 91–96, Opera Omnia, Series 1, vol. L. Euler, Geometrica et sphaerica quaedam, Mémoires de lAcademie des Sciences de Saint-Petersbourg 5,1815, p. 96–114, Opera Omnia, Series 1, vol. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientarum imperialis Petropolitinae 3,1782, p. 72–86, Opera Omnia, Series 1, vol
Spherical geometry
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On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore, it is a two dimensional manifold.
32.
Western Europe
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Western Europe, or West Europe, is the region comprising the western part of Europe. Below, some different geographic and geopolitical definitions of the term are outlined, prior to the Roman conquest, a large part of Western Europe had adopted the newly developed La Tène culture. This cultural and linguistic division was reinforced by the later political east-west division of the Roman Empire. The division between these two was enhanced during Late Antiquity and the Middle Ages by a number of events, the Western Roman Empire collapsed, starting the Early Middle Ages. By contrast, the Eastern Roman Empire, mostly known as the Greek or Byzantine Empire, survived, in East Asia, Western Europe was historically known as taixi in China and taisei in Japan, which literally translates as the Far West. The term Far West became synonymous with Western Europe in China during the Ming dynasty, the Italian Jesuit priest Matteo Ricci was one of the first writers in China to use the Far West as an Asian counterpart to the European concept of the Far East. In his writings, Ricci referred to himself as Matteo of the Far West, the term was still in use in the late 19th and early 20th centuries. Post-war Europe would be divided into two spheres, the West, influenced by the United States, and the Eastern Bloc. With the onset of the Cold War, Europe was divided by the Iron Curtain, behind that line lie all the capitals of the ancient states of Central and Eastern Europe. Although some countries were neutral, they were classified according to the nature of their political. This division largely defined the popular perception and understanding of Western Europe, the world changed dramatically with the fall of the Iron Curtain in 1989. The Federal Republic of Germany peacefully absorbed the German Democratic Republic, COMECON and the Warsaw Pact were dissolved, and in 1991, the Soviet Union ceased to exist. Several countries which had part of the Soviet Union regained full independence. Although the term Western Europe was more prominent during the Cold War, it remains much in use, in 1948 the Treaty of Brussels was signed between Belgium, France, Luxembourg, the Netherlands and the United Kingdom. It was further revisited in 1954 at the Paris Conference, when the Western European Union was established and it was declared defunct in 2011, after the Treaty of Lisbon, and the Treaty of Brussels was terminated. When the Western European Union was dissolved, it had 10 member countries, six member countries, five observer countries. The CIA divides Western Europe into two smaller subregions, regional voting blocs were formed in 1961 to encourage voting to various UN bodies from different regional groups. The European Union is an economic and political union of 28 member states that are located primarily in Europe, some Western and Northern European countries of Iceland, Norway, Switzerland and Liechtenstein are members of EFTA, though cooperating to varying degree with the European Union
Western Europe
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The Great Schism in Christianity, the predominant religion in Western Europe at the time.
Western Europe
Western Europe
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Geopolitical Occident of Europe
33.
Regiomontanus
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Johannes Müller von Königsberg, better known as Regiomontanus, was a mathematician and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrumental in the development of Copernican heliocentrism in the following his death. Regiomontanus wrote under the name of Ioannes de Monteregio, the adjectival Regiomontanus was first used by Philipp Melanchthon in 1534. He is named for Königsberg in Lower Franconia, not after the larger Königsberg in Prussia, at eleven years of age, Regiomontanus became a student at the university in Leipzig, Saxony. In 1451 he continued his studies at Alma Mater Rudolfina, the university in Vienna, there he became a pupil and friend of Georg von Peuerbach. In 1452 he was awarded his “magister artium” at the age of 21 in 1457 and it is known that he held lectures in optics and ancient literature. Regiomontanus continued to work with Peuerbach learning and extending the known areas of astronomy, mathematics. In 1460 the papal legate Basilios Bessarion came to Vienna on a diplomatic mission, being a humanist scholar and great fan of the mathematical sciences, Bessarion sought out Peuerbachs company. Peuerbachs Greek was not good enough to do a translation but he knew the Almagest intimately so instead he started work on a modernised, improved abridgement of the work. Bessarion also invited Peuerbach to become part of his household and to him back to Italy when his work in Vienna was finished. Peuerbach accepted the invitation on the condition that Regiomontanus could also accompany them, however Peuerbach fell ill in 1461 and died only having completed the first six books of his abridgement of the Almagest. On his death bed Peuerbach made Regiomontanus promise to finish the book and he went to work for János Vitéz, archbishop of Esztergom. There he calculated extensive astronomical tables and built astronomical instruments, in 1467 he went to Buda, and the court of Matthias Corvinus of Hungary, for whom he built an astrolabe, and where he collated Greek manuscripts for a handsome salary. The tables that he created while living in Hungary, his Tabulae directionum, were designed for astrology, here he founded the worlds first scientific printing press, and in 1472 he published the first printed astronomical textbook, the Theoricae novae Planetarum of his teacher Georg von Peurbach. Regiomontanus and Bernhard Walther observed the comet of 1472, Regiomontanus tried to estimate its distance from Earth, using the angle of parallax. These values, of course, fail by orders of magnitude, the 1472 comet was visible from Christmas Day 1471 to 1 March 1472, a total of 59 days. In 1475, Regiomontanus was called to Rome by Pope Sixtus IV on to work on the calendar reform. Sixtus promised substantial rewards, including the title of bishop of Regensburg, on his way to Rome, stopping in Venice, he commissioned the publication of his Calendarium with Erhard Ratdolt
Regiomontanus
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Regiomontanus
Regiomontanus
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Plaque at Regiomontanus' birthplace
Regiomontanus
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De triangulis planis et sphaericis libri
Regiomontanus
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Title page for Qvesta opra da ogni parte e un libro doro, 1476
34.
George of Trebizond
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George of Trebizond was a Greek philosopher, scholar and humanist. He was born on the Greek island of Crete, and derived his surname Trapezuntius from the fact that his ancestors were from the Byzantine Greek Trapezuntine Empire. He learned Latin from Vittorino da Feltre, and made rapid progress that in three years he was able to teach Latin literature and rhetoric. His reputation as a teacher and a translator of Aristotle was very great, and he was selected as secretary by Pope Nicholas V, an ardent Aristotelian. He subsequently returned to Rome, where in 1471 he published a very successful Latin grammar based on the work of another Greek grammarian of Latin, additionally an earlier work on rhetoric Greek principles garnered him wide recognition, even from his former critics who admitted his brilliance and scholarship. He died in poverty in 1486 in Rome. G. Voigt, Die Wiederbelebung des klassischen Altertums, article by C. F. Behr in Ersch, for a complete list of his numerous works, consisting of translations from Greek into Latin and original essays in Greek and Latin, see Fabricius, Bibliotheca Graeca, xii. Byzantine scholars in Renaissance Harris, Jonathan, Byzantines in Renaissance Italy, in Online Reference Book for Medieval Studies – http, //the-orb. net/encyclop/late/laterbyz/harris-ren. T. M. Izbicki, G. Christianson and P. Krey, letter no.61. Encyclopædia Britannica,2007 ed. Attribution This article incorporates text from a now in the public domain, Chisholm, Hugh. Jonathan Harris, Greek Émigrés in the West, 1400–1520, ISBN 1-871328-11-X John Monfasani, George of Trebizond. A biography and a study of his rhetoric and logic, Leiden, texts, Documents, and Bibliographies of George of Trebizond, Binghamton, NY, RSA,1984. Lucia Calboli Montefusco, Ciceronian and Hermogenean Influences on George of Trebizonds Rhetoricorum Libri V, Rhetorica 26.2, Greek Studies in the Italian Renaissance, London,1992
George of Trebizond
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George of Trebizond.
George of Trebizond
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Page from Book X of George of Trebizond's Commentary on the Almagest. On the left, is a model of the planet Mercury, showing its closest approach to the earth; on the right, is information about Mercury and the beginning of his commentary on the planet Venus.
35.
Nicolaus Copernicus
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Copernicus was born and died in Royal Prussia, a region that had been part of the Kingdom of Poland since 1466. A polyglot and polymath, he obtained a doctorate in law and was also a mathematician, astronomer, physician, classics scholar, translator, governor, diplomat. In 1517 he derived a quantity theory of money – a key concept in economics –, Nicolaus Copernicus was born on 19 February 1473 in the city of Toruń, in the province of Royal Prussia, in the Crown of the Kingdom of Poland. His father was a merchant from Kraków and his mother was the daughter of a wealthy Toruń merchant, Nicolaus was the youngest of four children. His brother Andreas became an Augustinian canon at Frombork and his sister Barbara, named after her mother, became a Benedictine nun and, in her final years, prioress of a convent in Chełmno, she died after 1517. His sister Katharina married the businessman and Toruń city councilor Barthel Gertner and left five children, Copernicus fathers family can be traced to a village in Silesia near Nysa. The villages name has been variously spelled Kopernik, Copernik, Copernic, Kopernic, Coprirnik, in the 14th century, members of the family began moving to various other Silesian cities, to the Polish capital, Kraków, and to Toruń. The father, Mikołaj the Elder, likely the son of Jan, Nicolaus was named after his father, who appears in records for the first time as a well-to-do merchant who dealt in copper, selling it mostly in Danzig. He moved from Kraków to Toruń around 1458, Nicolaus father was actively engaged in the politics of the day and supported Poland and the cities against the Teutonic Order. In 1454 he mediated negotiations between Polands Cardinal Zbigniew Oleśnicki and the Prussian cities for repayment of war loans, Copernicuss father married Barbara Watzenrode, the astronomers mother, between 1461 and 1464. The Modlibógs were a prominent Polish family who had been known in Polands history since 1271. The Watzenrode family, like the Kopernik family, had come from Silesia from near Świdnica and they soon became one of the wealthiest and most influential patrician families. Lucas Watzenrode the Elder, a merchant and in 1439–62 president of the judicial bench, was a decided opponent of the Teutonic Knights. In 1453 he was the delegate from Toruń at the Grudziądz conference that planned the uprising against them, Lucas Watzenrode the Younger, the astronomers maternal uncle and patron, was educated at the University of Kraków and at the universities of Cologne and Bologna. He was an opponent of the Teutonic Order, and its Grand Master once referred to him as the devil incarnate. In 1489 Watzenrode was elected Bishop of Warmia against the preference of King Casimir IV, as a result, Watzenrode quarreled with the king until Casimir IVs death three years later. Watzenrode was then able to close relations with three successive Polish monarchs, John I Albert, Alexander Jagiellon, and Sigismund I the Old. He was a friend and key advisor to each ruler, Watzenrode came to be considered the most powerful man in Warmia, and his wealth, connections and influence allowed him to secure Copernicus education and career as a canon at Frombork Cathedral
Nicolaus Copernicus
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1580 portrait (artist unknown) in the Old Town City Hall, Toruń
Nicolaus Copernicus
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Toruń birthplace (ul. Kopernika 15, left). Together with the house at no. 17 (right), it forms the Muzeum Mikołaja Kopernika.
Nicolaus Copernicus
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Copernicus' maternal uncle, Lucas Watzenrode the Younger
Nicolaus Copernicus
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Collegium Maius, Kraków
36.
Gemma Frisius
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Gemma Frisius, was a physician, mathematician, cartographer, philosopher, and instrument maker. He created important globes, improved the mathematical instruments of his day and applied mathematics in new ways to surveying, gemmas rings are named after him. Frisius was born in Dokkum, Friesland, of parents who died when he was young. He moved to Groningen and later studied abroad at the University of Leuven, Belgium and his oldest son, Cornelius Gemma, edited a posthumous volume of his work and continued to work with Ptolemaic astronomical models. Of particular fame were the terrestrial globe of 1536 and the globe of 1537. On the first of these Frisius is described as the author with technical assistance from Van der Heyden, on the second globe Mercator is promoted to co-author. In 1533, he described for the first time the method of still used today in surveying. This was only a theoretical presentation of the concept — due to topographical restrictions, nevertheless, the figure soon became well known all across Europe. Twenty years later, in ~1553, he was the first to describe how an accurate clock could be used to determine longitude, Frisius created or improved many instruments, including the cross-staff, the astrolabe, and the astronomical rings. His students included Gerardus Mercator, Johannes Stadius, John Dee, Andreas Vesalius, Frisius died in Leuven at the age of 46. According to an account by his son, Cornelius, Gemma died from kidney stones, a lunar crater has been named after him. Gualterus Arsenius, the 16th century scientific instrument maker, was his nephew, haasbroek, Gemma Frisius, Tycho Brahe and Snellius and their triangulations. Robert Haardt, The globe of Gemma Frisius, W. Karrow, Mapmakers of the Sixteenth Century and Their Maps. G. Kish, Medicina, mensura, mathematica, The Life, minneapolis 1967, sowie sein Artikel in Dictionary of Scientific Biography A. Pogo, Gemma Frisius, his method of determining longitude
Gemma Frisius
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Gemma Frisius, (Maarten van Heemskerck, c. 1540-1545)
Gemma Frisius
Gemma Frisius
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Page from Cosmographia
Gemma Frisius
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Frontispiece of Arithmeticae practicae methodus facilis
37.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
Leonhard Euler
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Portrait by Jakob Emanuel Handmann (1756)
Leonhard Euler
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1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Leonhard Euler
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Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as " v − e + f = 2".
Leonhard Euler
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Euler's grave at the Alexander Nevsky Monastery
38.
Colin Maclaurin
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Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a case of the Taylor series, is named after him. Owing to changes in orthography since that time, his surname is alternatively written MacLaurin, Maclaurin was born in Kilmodan, Argyll. His father, Reverend and Minister of Glendaruel John Maclaurin, died when Maclaurin was in infancy and he was then educated under the care of his uncle, the Reverend Daniel Maclaurin, minister of Kilfinan. At eleven, Maclaurin entered the University of Glasgow and this record as the worlds youngest professor endured until March 2008, when the record was officially given to Alia Sabur. In the vacations of 1719 and 1721, Maclaurin went to London, where he acquainted with Sir Isaac Newton, Dr Benjamin Hoadly, Samuel Clarke, Martin Folkes. He was admitted a member of the Royal Society, in 1722, having provided a substitute for his class at Aberdeen, he traveled on the Continent as tutor to George Hume, the son of Alexander Hume, 2nd Earl of Marchmont. During their time in Lorraine, he wrote his essay on the percussion of bodies, upon the death of his pupil at Montpellier, Maclaurin returned to Aberdeen. In 1725 Maclaurin was appointed deputy to the professor at Edinburgh, James Gregory. On 3 November of that year Maclaurin succeeded Gregory, and went on to raise the character of that university as a school of science, Newton was so impressed with Maclaurin that he had offered to pay his salary himself. Maclaurin used Taylor series to characterize maxima, minima, and points of inflection for infinitely differentiable functions in his Treatise of Fluxions. Maclaurin attributed the series to Taylor, though the series was known before to Newton and Gregory, nevertheless, Maclaurin received credit for his use of the series, and the Taylor series expanded around 0 is sometimes known as the Maclaurin series. Maclaurin also made significant contributions to the attraction of ellipsoids. Clairaut, Euler, Laplace, Legendre, Poisson and Gauss, Maclaurin showed that an oblate spheroid was a possible equilibrium in Newtons theory of gravity. The subject continues to be of scientific interest, and Nobel Laureate Subramanyan Chandrasekhar dedicated a chapter of his book Ellipsoidal Figures of Equilibrium to Maclaurin spheroids, independently from Euler and using the same methods, Maclaurin discovered the Euler–Maclaurin formula. He used it to sum powers of arithmetic progressions, derive Stirlings formula, Maclaurin contributed to the study of elliptic integrals, reducing many intractable integrals to problems of finding arcs for hyperbolas. His work was continued by dAlembert and Euler, who gave a more concise approach and this publication preceded by two years Cramers publication of a generalization of the rule to n unknowns, now commonly known as Cramers rule. In 1733, Maclaurin married Anne Stewart, the daughter of Walter Stewart, Maclaurin actively opposed the Jacobite Rebellion of 1745 and superintended the operations necessary for the defence of Edinburgh against the Highland army
Colin Maclaurin
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Colin Maclaurin (1698–1746)
Colin Maclaurin
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Memorial, Greyfriars Kirkyard, Edinburgh
39.
Brook Taylor
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Brook Taylor FRS was an English mathematician who is best known for Taylors theorem and the Taylor series. Brook Taylor was born in Edmonton to John Taylor of Bifrons House in Patrixbourne, Kent and he entered St Johns College, Cambridge, as a fellow-commoner in 1701, and took degrees of LL. B. and LL. D. in 1709 and 1714, respectively. Taylors Methodus Incrementorum Directa et Inversa added a new branch to higher mathematics, among other ingenious applications, he used it to determine the form of movement of a vibrating string, by him first successfully reduced to mechanical principles. From 1715 his studies took a philosophical and religious bent and he corresponded in that year with the Comte de Montmort on the subject of Nicolas Malebranches tenets. Unfinished treatises, On the Jewish Sacrifices and On the Lawfulness of Eating Blood, written on his return from Aix-la-Chapelle in 1719, were afterwards found among his papers. His marriage in 1721 with Miss Brydges of Wallington, Surrey, led to an estrangement from his father, which ended in 1723 after her death in giving birth to a son, by the date of his fathers death in 1729 he had inherited the Bifrons estate. Taylors fragile health gave way, he fell into a decline and he was buried in London on 2 December 1731, near his first wife, in the churchyard of St Annes, Soho. A posthumous work entitled Contemplatio Philosophica was printed for private circulation in 1793 by Taylors grandson, Sir William Young, prefaced by a life of the author, and with an appendix containing letters addressed to him by Bolingbroke, Bossuet, and others. Several short papers by Taylor were published in Phil, vols. xxvii to xxxii, including accounts of some interesting experiments in magnetism and capillary attraction. A French translation was published in 1757, in Methodus Incrementorum, Taylor gave the first satisfactory investigation of astronomical refraction. Taylor, Brook, Methodus Incrementorum Directa et Inversa, London, Taylor is an impact crater located on the Moon, named in honour of Brook Taylor. Brook Taylor’s Work on Linear Perspective, anderson, Marlow, Katz, Victor, Wilson, Robin. Sherlock Holmes in Babylon, And Other Tales of Mathematical History, Brook Taylor and the Method of Increments. Archive for History of Exact Sciences, oConnor, John J. Robertson, Edmund F. Brook Taylor, MacTutor History of Mathematics archive, University of St Andrews, beningbrough Hall has a painting by John Closterman of Taylor aged about 12 with his brothers and sisters. See also NPG5320, The Children of John Taylor of Bifrons Park Brook Taylors pedigree Taylor, a crater on the Moon named after Brook Taylor
Brook Taylor
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Brook Taylor (1685-1731)
Brook Taylor
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Methodus incrementorum directa et inversa, 1715
Brook Taylor
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Brook Taylor
40.
Taylor series
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In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the functions derivatives at a single point. The concept of a Taylor series was formulated by the Scottish mathematician James Gregory, a function can be approximated by using a finite number of terms of its Taylor series. Taylors theorem gives quantitative estimates on the error introduced by the use of such an approximation, the polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that functions Taylor polynomials as the degree increases, a function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an interval is known as an analytic function in that interval. The Taylor series of a real or complex-valued function f that is differentiable at a real or complex number a is the power series f + f ′1. Which can be written in the more compact sigma notation as ∑ n =0 ∞ f n, N where n. denotes the factorial of n and f denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and 0 and 0. are both defined to be 1, when a =0, the series is also called a Maclaurin series. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for 1/1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ so the Taylor series for 1/x at a =1 is 1 − +2 −3 + ⋯. The Taylor series for the exponential function ex at a =0 is x 00, + ⋯ =1 + x + x 22 + x 36 + x 424 + x 5120 + ⋯ = ∑ n =0 ∞ x n n. The above expansion holds because the derivative of ex with respect to x is also ex and this leaves the terms n in the numerator and n. in the denominator for each term in the infinite sum. The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a result, but rejected it as an impossibility. It was through Archimedess method of exhaustion that a number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later, in the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. The Kerala school of astronomy and mathematics further expanded his works with various series expansions, in the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a method for constructing these series for all functions for which they exist was finally provided by Brook Taylor. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, if f is given by a convergent power series in an open disc centered at b in the complex plane, it is said to be analytic in this disc
Taylor series
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As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin(x) and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
41.
Hypotenuse
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In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. For example, if one of the sides has a length of 3. The length of the hypotenuse is the root of 25. The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timaeus 54d, a folk etymology says that tenuse means side, so hypotenuse means a support like a prop or buttress, but this is inaccurate. The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle are a and b and that of the hypotenuse is c, many computer languages support the ISO C standard function hypot, which returns the value above. The function is designed not to fail where the straightforward calculation might overflow or underflow, some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. This gives both the length of the hypotenuse and the angle the hypotenuse makes with the line at the same time when given x and y. The angle returned will normally be given by atan2. Orthographic projections, The length of the hypotenuse equals the sum of the lengths of the projections of both catheti. And The square of the length of a cathetus equals the product of the lengths of its projection on the hypotenuse times the length of this. Given the length of the c and of a cathetus b. The adjacent angle of the b, will be α = 90° – β One may also obtain the value of the angle β by the equation. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. Hypotenuse
Hypotenuse
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A right-angled triangle and its hypotenuse.
42.
Tangent (trigonometric function)
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
Tangent (trigonometric function)
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Trigonometric functions in the complex plane
Tangent (trigonometric function)
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Trigonometry
Tangent (trigonometric function)
Tangent (trigonometric function)
43.
Multiplicative inverse
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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity,1. The multiplicative inverse of a fraction a/b is b/a, for the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, the reciprocal function, the function f that maps x to 1/x, is one of the simplest examples of a function which is its own inverse. In the phrase multiplicative inverse, the qualifier multiplicative is often omitted, multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba, then inverse typically implies that an element is both a left and right inverse. The notation f −1 is sometimes used for the inverse function of the function f. For example, the multiplicative inverse 1/ = −1 is the cosecant of x, only for linear maps are they strongly related. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, in the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, the property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no other than 1 and −1 has an integer reciprocal. In modular arithmetic, the multiplicative inverse of a is also defined. This multiplicative inverse exists if and only if a and n are coprime, for example, the inverse of 3 modulo 11 is 4 because 4 ·3 ≡1. The extended Euclidean algorithm may be used to compute it, the sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i. e. nonzero elements x, y such that xy =0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring, the linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case. A ring in which every element has a multiplicative inverse is a division ring. As mentioned above, the reciprocal of every complex number z = a + bi is complex. In particular, if ||z||=1, then 1 / z = z ¯, consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property
Multiplicative inverse
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The reciprocal function: y = 1/ x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.
44.
Inverse trigonometric function
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In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. There are several notations used for the trigonometric functions. The most common convention is to name inverse trigonometric functions using a prefix, e. g. arcsin, arccos, arctan. This convention is used throughout the article, when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Similarly, in programming languages the inverse trigonometric functions are usually called asin, acos. The notations sin−1, cos−1, tan−1, etc, the confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, −1 = sec. Nevertheless, certain authors advise against using it for its ambiguity, since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. There are multiple numbers y such that sin = x, for example, sin =0, when only one value is desired, the function may be restricted to its principal branch. With this restriction, for x in the domain the expression arcsin will evaluate only to a single value. These properties apply to all the trigonometric functions. The principal inverses are listed in the following table, if x is allowed to be a complex number, then the range of y applies only to its real part. Trigonometric functions of trigonometric functions are tabulated below. This is derived from the tangent addition formula tan = tan + tan 1 − tan tan , like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,11 − z 2, as a binomial series, the series for arctangent can similarly be derived by expanding its derivative 11 + z 2 in a geometric series and applying the integral definition above. Arcsin = z + z 33 + z 55 + z 77 + ⋯ = ∑ n =0 ∞, for example, arccos x = π /2 − arcsin x, arccsc x = arcsin , and so on. Alternatively, this can be expressed, arctan z = ∑ n =0 ∞22 n 2. There are two cuts, from −i to the point at infinity, going down the imaginary axis and it works best for real numbers running from −1 to 1
Inverse trigonometric function
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Inverse trigonometric functions in the complex plane
Inverse trigonometric function
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Trigonometry
Inverse trigonometric function
Inverse trigonometric function
45.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
Radian
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A chart to convert between degrees and radians
Radian
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An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to an angle of 2 π radians.
46.
Mnemonics in trigonometry
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In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions. Another method is to expand the letters into a sentence, such as Some Old Houses Can Always Hide Their Old Age, communities exposed to Chinese dialect may choose to remember it as TOA-CAH-SOH, which also means big-footed woman in Hokkien. Azals Mnemonic is a mnemonic to SOH-CAH-TOA for people who have different names for the legs of a triangle, i. e. Perpendicular for Opposite. Azals Mnemonic goes like this, Some People Have Curly Black Hairs Through Proper Brushing, here, Some People Have is for Sine=Perpendicular/Hypotenuse, Curly Black Hairs is for Cosine=Base/Hypotenuse, and Through Proper Brushing is for Tangent=Perpendicular/Base. Another mnemonic permits all of the basic identities to be read off quickly, although the word part of the mnemonic used to build the chart does not hold in English, the chart itself is fairly easy to reconstruct with a little thought
Mnemonics in trigonometry
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Signs of trigonometric functions in each quadrant. The mnemonic " All S cience T eachers (are) C razy" lists the functions which are positive from quadrants I to IV.
47.
Mnemonic
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A mnemonic device, or memory device is any learning technique that aids information retention in the human memory. Mnemonics make use of encoding, retrieval cues, and imagery as specific tools to encode any given information in a way that allows for efficient storage. Mnemonics aid original information in becoming associated with something more meaningful—which, in turn, the word mnemonic is derived from the Ancient Greek word μνημονικός, meaning of memory, or relating to memory and is related to Mnemosyne, the name of the goddess of memory in Greek mythology. Both of these words are derived from μνήμη, remembrance, memory, mnemonics in antiquity were most often considered in the context of what is today known as the art of memory. Ancient Greeks and Romans distinguished between two types of memory, the memory and the artificial memory. The former is inborn, and is the one that everyone uses instinctively, the latter in contrast has to be trained and developed through the learning and practice of a variety of mnemonic techniques. Mnemonic systems are techniques or strategies consciously used to improve memory and they help use information already stored in long-term memory to make memorisation an easier task. Mnemonic devices were much cultivated by Greek sophists and philosophers and are referred to by Plato. In later times the poet Simonides was credited for development of these techniques, the Romans valued such helps in order to support facility in public speaking. The Greek and the Roman system of mnemonics was founded on the use of mental places and signs or pictures, to recall these, an individual had only to search over the apartments of the house until discovering the places where images had been placed by the imagination. Except that the rules of mnemonics are referred to by Martianus Capella, among the voluminous writings of Roger Bacon is a tractate De arte memorativa. Ramon Llull devoted special attention to mnemonics in connection with his ars generalis, about the end of the 15th century, Petrus de Ravenna provoked such astonishment in Italy by his mnemonic feats that he was believed by many to be a necromancer. His Phoenix artis memoriae went through as many as nine editions, about the end of the 16th century, Lambert Schenkel, who taught mnemonics in France, Italy and Germany, similarly surprised people with his memory. He was denounced as a sorcerer by the University of Louvain, the most complete account of his system is given in two works by his pupil Martin Sommer, published in Venice in 1619. In 1618 John Willis published Mnemonica, sive ars reminiscendi, containing a statement of the principles of topical or local mnemonics. Giordano Bruno included a memoria technica in his treatise De umbris idearum, other writers of this period are the Florentine Publicius, Johannes Romberch, Hieronimo Morafiot, Ars memoriae, and B. The philosopher Gottfried Wilhelm Leibniz adopted a very similar to that of Wennsshein for his scheme of a form of writing common to all languages. Wennssheins method was adopted with slight changes afterward by the majority of subsequent original systems and it was modified and supplemented by Richard Grey, a priest who published a Memoria technica in 1730
Mnemonic
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Detail of Giordano Bruno 's statue in Rome. Bruno was famous for his mnemonics, some of which he included in his treatises De umbris idearum and Ars Memoriae.
Mnemonic
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Knuckle mnemonic for the number of days in each month of the Gregorian Calendar. Each knuckle represents a 31-day month.
48.
Interpolate
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In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. It is often required to interpolate the value of that function for a value of the independent variable. A different problem which is related to interpolation is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complex to evaluate efficiently, a few known data points from the original function can be used to create an interpolation based on a simpler function. In the examples below if we consider x as a topological space, the classical results about interpolation of operators are the Riesz–Thorin theorem and the Marcinkiewicz theorem. There are also many other subsequent results, for example, suppose we have a table like this, which gives some values of an unknown function f. Interpolation provides a means of estimating the function at intermediate points, there are many different interpolation methods, some of which are described below. Some of the concerns to take into account when choosing an appropriate algorithm are, how many data points are needed. The simplest interpolation method is to locate the nearest data value, one of the simplest methods is linear interpolation. Consider the above example of estimating f, since 2.5 is midway between 2 and 3, it is reasonable to take f midway between f =0.9093 and f =0.1411, which yields 0.5252. Another disadvantage is that the interpolant is not differentiable at the point xk, the following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by g, then the linear interpolation error is | f − g | ≤ C2 where C =18 max r ∈ | g ″ |. In words, the error is proportional to the square of the distance between the data points, the error in some other methods, including polynomial interpolation and spline interpolation, is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants, polynomial interpolation is a generalization of linear interpolation. Note that the interpolant is a linear function. We now replace this interpolant with a polynomial of higher degree, consider again the problem given above. The following sixth degree polynomial goes through all the seven points, substituting x =2.5, we find that f =0.5965. Generally, if we have n points, there is exactly one polynomial of degree at most n−1 going through all the data points
Interpolate
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An interpolation of a finite set of points on an epitrochoid. Points through which curve is splined are red; the blue curve connecting them is interpolation.
49.
Slide rule
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The slide rule, also known colloquially in the United States as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as exponents, roots, logarithms and trigonometry, though similar in name and appearance to a standard ruler, the slide rule is not ordinarily used for measuring length or drawing straight lines. Slide rules exist in a range of styles and generally appear in a linear or circular form with a standardized set of markings essential to performing mathematical computations. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in calculations common to those fields, at its simplest, each number to be multiplied is represented by a length on a sliding ruler. As the rulers each have a scale, it is possible to align them to read the sum of the logarithms. The Reverend William Oughtred and others developed the rule in the 17th century based on the emerging work on logarithms by John Napier. Before the advent of the calculator, it was the most commonly used calculation tool in science. In its most basic form, the slide rule uses two logarithmic scales to allow rapid multiplication and division of numbers and these common operations can be time-consuming and error-prone when done on paper. More elaborate slide rules allow other calculations, such as roots, exponentials, logarithms. Scales may be grouped in decades, which are numbers ranging from 1 to 10. Thus single decade scales C and D range from 1 to 10 across the width of the slide rule while double decade scales A and B range from 1 to 100 over the width of the slide rule. Numbers aligned with the marks give the value of the product, quotient. The user determines the location of the point in the result. Scientific notation is used to track the decimal point in more formal calculations, addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule. Most slide rules consist of three strips of the same length, aligned in parallel and interlocked so that the central strip can be moved lengthwise relative to the other two. The outer two strips are fixed so that their relative positions do not change. Some slide rules have scales on both sides of the rule and slide strip, others on one side of the outer strips and both sides of the slide strip, still others on one side only. A sliding cursor with a vertical alignment line is used to find corresponding points on scales that are not adjacent to other or
Slide rule
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A typical ten-inch student slide rule (Pickett N902-T simplex trig).
Slide rule
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Cursor on a slide rule.
Slide rule
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This slide rule is positioned to yield several values: From C scale to D scale (multiply by 2), from D scale to C scale (divide by 2), A and B scales (multiply and divide by 4), A and D scales (squares and square roots).
Slide rule
50.
Scientific calculator
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A scientific calculator is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science, engineering, and mathematics. They have almost completely replaced slide rules in almost all traditional applications, there is also some overlap with the financial calculator market. A few have multi-line displays, with recent models from Hewlett-Packard, Texas Instruments, Casio, Sharp. By providing a method to enter an entire problem in as it is written on the page using simple formatting tools, the HP-35, introduced on February 1,1972, was Hewlett-Packards first pocket calculator and the worlds first handheld scientific calculator. Like some of HPs desktop calculators it used RPN, introduced at US$395, the HP-35 was available from 1972 to 1975. Texas Instruments, after the introduction of units with scientific notation, came out with a handheld scientific calculator on January 15,1974. TI continues to be a player in the calculator market. Casio and Sharp have also been major players, with Casios fx series being a common brand. Casio is also a player in the graphing calculator market
Scientific calculator
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Casio FX-77, a solar-powered scientific calculator from the 1980s using a single-line display
Scientific calculator
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The TI-84 Plus: A typical graphing calculator by Texas Instruments
Scientific calculator
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A modern scientific calculator with a dot matrix LCD display
51.
Floating point unit
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A floating-point unit is a part of a computer system specially designed to carry out operations on floating point numbers. Typical operations are addition, subtraction, multiplication, division, square root, some systems can also perform various transcendental functions such as exponential or trigonometric calculations, though in most modern processors these are done with software library routines. This could be an integrated circuit, an entire circuit board or a cabinet. Where floating-point calculation hardware has not been provided, floating point calculations are done in software, emulation can be implemented on any of several levels, in the CPU as microcode, as an operating system function, or in user space code. When only integer functionality is available the CORDIC floating point emulation methods are most commonly used, in most modern computer architectures, there is some division of floating-point operations from integer operations. This division varies significantly by architecture, some, like the Intel x86 have dedicated floating-point registers, in earlier superscalar architectures without general out-of-order execution, floating-point operations were sometimes pipelined separately from integer operations. Since the early 1990s, many microprocessors for desktops and servers have more than one FPU, the modular architecture of Bulldozer microarchitecture uses a special FPU named FlexFPU, which uses simultaneous multithreading. Each physical integer core, two per module, is threaded, in contrast with Intels Hyperthreading, where two virtual simultaneous threads share the resources of a single physical core. Some floating-point hardware only supports the simplest operations - addition, subtraction, but even the most complex floating-point hardware has a finite number of operations it can support - for example, none of them directly support arbitrary-precision arithmetic. When a CPU is executing a program calls for a floating-point operation that is not directly supported by the hardware. In systems without any floating-point hardware, the CPU emulates it using a series of simpler fixed-point arithmetic operations that run on the arithmetic logic unit. The software that lists the series of operations to emulate floating-point operations is often packaged in a floating-point library. In some cases, FPUs may be specialized, and divided between simpler floating-point operations and more complicated operations, like division, in some cases, only the simple operations may be implemented in hardware or microcode, while the more complex operations are implemented as software. In the 1980s, it was common in IBM PC/compatible microcomputers for the FPU to be separate from the CPU. It would only be purchased if needed to speed up or enable math-intensive programs, the IBM PC, XT, and most compatibles based on the 8088 or 8086 had a socket for the optional 8087 coprocessor. Other companies manufactured co-processors for the Intel x86 series, coprocessors were available for the Motorola 68000 family, the 68881 and 68882. These were common in Motorola 68020/68030-based workstations like the Sun 3 series, there are also add-on FPUs coprocessor units for microcontroller units /single-board computer, which serve to provide floating-point arithmetic capability. These add-on FPUs are host-processor-independent, possess their own programming requirements and are provided with their own integrated development environments
Floating point unit
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An Intel 80287
52.
Satellite navigation system
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A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning. It allows small electronic receivers to determine their location to high precision using time signals transmitted along a line of sight by radio from satellites, the system can be used for providing position, navigation or for tracking the position of something fitted with a receiver. The signals also allow the receiver to calculate the current local time to high precision. Satnav systems operate independently of any telephonic or internet reception, though these technologies can enhance the usefulness of the information generated. A satellite navigation system with global coverage may be termed a global satellite system. As of December 2016 only the United States NAVSTAR Global Positioning System, the Russian GLONASS, the European Unions Galileo GNSS is scheduled to be fully operational by 2020. China is in the process of expanding its regional BeiDou Navigation Satellite System into the global BeiDou-2 GNSS by 2020, India currently has satellite-based augmentation system, GPS Aided GEO Augmented Navigation, which enhances the accuracy of NAVSTAR GPS and GLONASS positions. India has already launched the IRNSS, with an operational name NAVIC and it is expected to be fully operational by June 2016. France and Japan are in the process of developing regional navigation systems as well, Global coverage for each system is generally achieved by a satellite constellation of 18–30 medium Earth orbit satellites spread between several orbital planes. The actual systems vary, but use orbital inclinations of >50°, Ground based augmentation is provided by systems like the Local Area Augmentation System. GNSS-2 is the generation of systems that independently provides a full civilian satellite navigation system. These systems will provide the accuracy and integrity monitoring necessary for civil navigation and this system consists of L1 and L2 frequencies for civil use and L5 for system integrity. Development is also in progress to provide GPS with civil use L2 and L5 frequencies, making it a GNSS-2 system. ¹ Core Satellite navigation systems, currently GPS, GLONASS, Galileo, Global Satellite Based Augmentation Systems such as Omnistar and StarFire. Regional SBAS including WAAS, EGNOS, MSAS and GAGAN, Regional Satellite Navigation Systems such as Chinas Beidou, Indias NAVIC, and Japans proposed QZSS. Continental scale Ground Based Augmentation Systems for example the Australian GRAS, Regional scale GBAS such as CORS networks. Local GBAS typified by a single GPS reference station operating Real Time Kinematic corrections, early predecessors were the ground based DECCA, LORAN, GEE and Omega radio navigation systems, which used terrestrial longwave radio transmitters instead of satellites. These positioning systems broadcast a radio pulse from a known master location, the delay between the reception of the master signal and the slave signals allowed the receiver to deduce the distance to each of the slaves, providing a fix. The first satellite system was Transit, a system deployed by the US military in the 1960s
Satellite navigation system
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North Surveying Stealth GNSS Double Frequency Receiver board.
Satellite navigation system
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Satellite navigation using a laptop and a GPS receiver
Satellite navigation system
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launched GNSS satellites 1978 to 2014
Satellite navigation system
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General
53.
Light
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Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to light, which is visible to the human eye and is responsible for the sense of sight. Visible light is defined as having wavelengths in the range of 400–700 nanometres, or 4.00 × 10−7 to 7.00 × 10−7 m. This wavelength means a range of roughly 430–750 terahertz. The main source of light on Earth is the Sun, sunlight provides the energy that green plants use to create sugars mostly in the form of starches, which release energy into the living things that digest them. This process of photosynthesis provides virtually all the used by living things. Historically, another important source of light for humans has been fire, with the development of electric lights and power systems, electric lighting has effectively replaced firelight. Some species of animals generate their own light, a process called bioluminescence, for example, fireflies use light to locate mates, and vampire squids use it to hide themselves from prey. Visible light, as all types of electromagnetic radiation, is experimentally found to always move at this speed in a vacuum. In physics, the term sometimes refers to electromagnetic radiation of any wavelength. In this sense, gamma rays, X-rays, microwaves and radio waves are also light, like all types of light, visible light is emitted and absorbed in tiny packets called photons and exhibits properties of both waves and particles. This property is referred to as the wave–particle duality, the study of light, known as optics, is an important research area in modern physics. Generally, EM radiation, or EMR, is classified by wavelength into radio, microwave, infrared, the behavior of EMR depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths, when EMR interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. There exist animals that are sensitive to various types of infrared, infrared sensing in snakes depends on a kind of natural thermal imaging, in which tiny packets of cellular water are raised in temperature by the infrared radiation. EMR in this range causes molecular vibration and heating effects, which is how these animals detect it, above the range of visible light, ultraviolet light becomes invisible to humans, mostly because it is absorbed by the cornea below 360 nanometers and the internal lens below 400. Furthermore, the rods and cones located in the retina of the eye cannot detect the very short ultraviolet wavelengths and are in fact damaged by ultraviolet. Many animals with eyes that do not require lenses are able to detect ultraviolet, by quantum photon-absorption mechanisms, various sources define visible light as narrowly as 420 to 680 to as broadly as 380 to 800 nm
Light
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An example of refraction of light. The straw appears bent, because of refraction of light as it enters liquid from air.
Light
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A triangular prism dispersing a beam of white light. The longer wavelengths (red) and the shorter wavelengths (blue) get separated.
Light
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A cloud illuminated by sunlight
Light
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A city illuminated by artificial lighting
54.
Acoustics
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Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of society with the most obvious being the audio. Hearing is one of the most crucial means of survival in the animal world, accordingly, the science of acoustics spreads across many facets of human society—music, medicine, architecture, industrial production, warfare and more. Likewise, animal species such as songbirds and frogs use sound, art, craft, science and technology have provoked one another to advance the whole, as in many other fields of knowledge. Robert Bruce Lindsays Wheel of Acoustics is a well accepted overview of the fields in acoustics. The word acoustic is derived from the Greek word ἀκουστικός, meaning of or for hearing, ready to hear and that from ἀκουστός, heard, audible, which in turn derives from the verb ἀκούω, I hear. The Latin synonym is sonic, after which the term used to be a synonym for acoustics. Frequencies above and below the range are called ultrasonic and infrasonic. If, for example, a string of a length would sound particularly harmonious with a string of twice the length. In modern parlance, if a string sounds the note C when plucked, a string twice as long will sound a C an octave lower. In one system of tuning, the tones in between are then given by 16,9 for D,8,5 for E,3,2 for F,4,3 for G,6,5 for A. Aristotle understood that sound consisted of compressions and rarefactions of air which falls upon, a very good expression of the nature of wave motion. The physical understanding of acoustical processes advanced rapidly during and after the Scientific Revolution, mainly Galileo Galilei but also Marin Mersenne, independently, discovered the complete laws of vibrating strings. Experimental measurements of the speed of sound in air were carried out successfully between 1630 and 1680 by a number of investigators, prominently Mersenne, meanwhile, Newton derived the relationship for wave velocity in solids, a cornerstone of physical acoustics. The eighteenth century saw advances in acoustics as mathematicians applied the new techniques of calculus to elaborate theories of sound wave propagation. Also in the 19th century, Wheatstone, Ohm, and Henry developed the analogy between electricity and acoustics, the twentieth century saw a burgeoning of technological applications of the large body of scientific knowledge that was by then in place. The first such application was Sabine’s groundbreaking work in architectural acoustics, Underwater acoustics was used for detecting submarines in the first World War
Acoustics
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Principles of acoustics were applied since ancient times: Roman theatre in the city of Amman.
Acoustics
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Artificial omni-directional sound source in an anechoic chamber
Acoustics
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Jay Pritzker Pavilion
Acoustics
55.
Optics
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Optics is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light, because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties. Most optical phenomena can be accounted for using the classical description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice, practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines, physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically, the model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that waves were in fact electromagnetic radiation. Some phenomena depend on the fact that light has both wave-like and particle-like properties, explanation of these effects requires quantum mechanics. When considering lights particle-like properties, the light is modelled as a collection of particles called photons, quantum optics deals with the application of quantum mechanics to optical systems. Optical science is relevant to and studied in many related disciplines including astronomy, various engineering fields, photography, practical applications of optics are found in a variety of technologies and everyday objects, including mirrors, lenses, telescopes, microscopes, lasers, and fibre optics. Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, the earliest known lenses, made from polished crystal, often quartz, date from as early as 700 BC for Assyrian lenses such as the Layard/Nimrud lens. The ancient Romans and Greeks filled glass spheres with water to make lenses, the word optics comes from the ancient Greek word ὀπτική, meaning appearance, look. Greek philosophy on optics broke down into two opposing theories on how vision worked, the theory and the emission theory. The intro-mission approach saw vision as coming from objects casting off copies of themselves that were captured by the eye, plato first articulated the emission theory, the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in Timaeus, some hundred years later, Euclid wrote a treatise entitled Optics where he linked vision to geometry, creating geometrical optics. Ptolemy, in his treatise Optics, held a theory of vision, the rays from the eye formed a cone, the vertex being within the eye. The rays were sensitive, and conveyed back to the observer’s intellect about the distance. He summarised much of Euclid and went on to describe a way to measure the angle of refraction, during the Middle Ages, Greek ideas about optics were resurrected and extended by writers in the Muslim world
Optics
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Optics includes study of dispersion of light.
Optics
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The Nimrud lens
Optics
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Reproduction of a page of Ibn Sahl 's manuscript showing his knowledge of the law of refraction, now known as Snell's law
Optics
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Cover of the first edition of Newton's Opticks
56.
Biology
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Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, evolution, distribution, identification and taxonomy. Modern biology is a vast and eclectic field, composed of branches and subdisciplines. However, despite the broad scope of biology, there are certain unifying concepts within it that consolidate it into single, coherent field. In general, biology recognizes the cell as the unit of life, genes as the basic unit of heredity. It is also understood today that all organisms survive by consuming and transforming energy and by regulating their internal environment to maintain a stable, the term biology is derived from the Greek word βίος, bios, life and the suffix -λογία, -logia, study of. The Latin-language form of the term first appeared in 1736 when Swedish scientist Carl Linnaeus used biologi in his Bibliotheca botanica, the first German use, Biologie, was in a 1771 translation of Linnaeus work. In 1797, Theodor Georg August Roose used the term in the preface of a book, karl Friedrich Burdach used the term in 1800 in a more restricted sense of the study of human beings from a morphological, physiological and psychological perspective. The science that concerns itself with these objects we will indicate by the biology or the doctrine of life. Although modern biology is a recent development, sciences related to. Natural philosophy was studied as early as the ancient civilizations of Mesopotamia, Egypt, the Indian subcontinent, however, the origins of modern biology and its approach to the study of nature are most often traced back to ancient Greece. While the formal study of medicine back to Hippocrates, it was Aristotle who contributed most extensively to the development of biology. Especially important are his History of Animals and other works where he showed naturalist leanings, and later more empirical works that focused on biological causation and the diversity of life. Aristotles successor at the Lyceum, Theophrastus, wrote a series of books on botany that survived as the most important contribution of antiquity to the plant sciences, even into the Middle Ages. Scholars of the medieval Islamic world who wrote on biology included al-Jahiz, Al-Dīnawarī, who wrote on botany, biology began to quickly develop and grow with Anton van Leeuwenhoeks dramatic improvement of the microscope. It was then that scholars discovered spermatozoa, bacteria, infusoria, investigations by Jan Swammerdam led to new interest in entomology and helped to develop the basic techniques of microscopic dissection and staining. Advances in microscopy also had a impact on biological thinking. In the early 19th century, a number of biologists pointed to the importance of the cell. Thanks to the work of Robert Remak and Rudolf Virchow, however, meanwhile, taxonomy and classification became the focus of natural historians
Biology
Biology
Biology
Biology
57.
Ultrasound
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Ultrasound is sound waves with frequencies higher than the upper audible limit of human hearing. Ultrasound is no different from normal sound in its physical properties and this limit varies from person to person and is approximately 20 kilohertz in healthy, young adults. Ultrasound devices operate with frequencies from 20 kHz up to several gigahertz, Ultrasound is used in many different fields. Ultrasonic devices are used to detect objects and measure distances, Ultrasound imaging or sonography is often used in medicine. In the nondestructive testing of products and structures, ultrasound is used to detect invisible flaws, industrially, ultrasound is used for cleaning, mixing, and to accelerate chemical processes. Animals such as bats and porpoises use ultrasound for locating prey, scientist are also studying ultrasound using graphene diaphragms as a method of communication. Acoustics, the science of sound, starts as far back as Pythagoras in the 6th century BC, echolocation in bats was discovered by Lazzaro Spallanzani in 1794, when he demonstrated that bats hunted and navigated by inaudible sound and not vision. The first technological application of ultrasound was an attempt to detect submarines by Paul Langevin in 1917, the piezoelectric effect, discovered by Jacques and Pierre Curie in 1880, was useful in transducers to generate and detect ultrasonic waves in air and water. Ultrasound is defined by the American National Standards Institute as sound at frequencies greater than 20 kHz, in air at atmospheric pressure ultrasonic waves have wavelengths of 1.9 cm or less. The upper frequency limit in humans is due to limitations of the middle ear, auditory sensation can occur if high‐intensity ultrasound is fed directly into the human skull and reaches the cochlea through bone conduction, without passing through the middle ear. Children can hear some high-pitched sounds that older adults cannot hear, the Mosquito is an electronic device that uses a high pitched frequency to deter loitering by young people. Bats use a variety of ultrasonic ranging techniques to detect their prey and they can detect frequencies beyond 100 kHz, possibly up to 200 kHz. Many insects have good hearing and most of these are nocturnal insects listening for echolocating bats. This includes many groups of moths, beetles, praying mantids, upon hearing a bat, some insects will make evasive manoeuvres to escape being caught. Ultrasonic frequencies trigger an action in the noctuid moth that cause it to drop slightly in its flight to evade attack. Tiger moths also emit clicks which may disturb bats echolocation, dogs and cats hearing range extends into the ultrasound, the top end of a dogs hearing range is about 45 kHz, while a cats is 64 kHz. The wild ancestors of cats and dogs evolved this higher hearing range to hear sounds made by their preferred prey. A dog whistle is a whistle that emits ultrasound, used for training and calling dogs, porpoises have the highest known upper hearing limit, at around 160 kHz
Ultrasound
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Ultrasound image of a fetus in the womb, viewed at 12 weeks of pregnancy (bidimensional-scan)
Ultrasound
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An ultrasonic examination
Ultrasound
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Bats use ultrasounds to navigate in the darkness.
Ultrasound
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Sonogram of a fetus at 14 weeks (profile)
58.
Chemistry
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Chemistry is a branch of physical science that studies the composition, structure, properties and change of matter. Chemistry is sometimes called the science because it bridges other natural sciences, including physics. For the differences between chemistry and physics see comparison of chemistry and physics, the history of chemistry can be traced to alchemy, which had been practiced for several millennia in various parts of the world. The word chemistry comes from alchemy, which referred to a set of practices that encompassed elements of chemistry, metallurgy, philosophy, astrology, astronomy, mysticism. An alchemist was called a chemist in popular speech, and later the suffix -ry was added to this to describe the art of the chemist as chemistry, the modern word alchemy in turn is derived from the Arabic word al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία and this may have Egyptian origins since al-kīmīā is derived from the Greek χημία, which is in turn derived from the word Chemi or Kimi, which is the ancient name of Egypt in Egyptian. Alternately, al-kīmīā may derive from χημεία, meaning cast together, in retrospect, the definition of chemistry has changed over time, as new discoveries and theories add to the functionality of the science. The term chymistry, in the view of noted scientist Robert Boyle in 1661, in 1837, Jean-Baptiste Dumas considered the word chemistry to refer to the science concerned with the laws and effects of molecular forces. More recently, in 1998, Professor Raymond Chang broadened the definition of chemistry to mean the study of matter, early civilizations, such as the Egyptians Babylonians, Indians amassed practical knowledge concerning the arts of metallurgy, pottery and dyes, but didnt develop a systematic theory. Greek atomism dates back to 440 BC, arising in works by such as Democritus and Epicurus. In 50 BC, the Roman philosopher Lucretius expanded upon the theory in his book De rerum natura, unlike modern concepts of science, Greek atomism was purely philosophical in nature, with little concern for empirical observations and no concern for chemical experiments. Work, particularly the development of distillation, continued in the early Byzantine period with the most famous practitioner being the 4th century Greek-Egyptian Zosimos of Panopolis. He formulated Boyles law, rejected the four elements and proposed a mechanistic alternative of atoms. Before his work, though, many important discoveries had been made, the Scottish chemist Joseph Black and the Dutchman J. B. English scientist John Dalton proposed the theory of atoms, that all substances are composed of indivisible atoms of matter. Davy discovered nine new elements including the alkali metals by extracting them from their oxides with electric current, british William Prout first proposed ordering all the elements by their atomic weight as all atoms had a weight that was an exact multiple of the atomic weight of hydrogen. The inert gases, later called the noble gases were discovered by William Ramsay in collaboration with Lord Rayleigh at the end of the century, thereby filling in the basic structure of the table. Organic chemistry was developed by Justus von Liebig and others, following Friedrich Wöhlers synthesis of urea which proved that organisms were, in theory
Chemistry
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Solutions of substances in reagent bottles, including ammonium hydroxide and nitric acid, illuminated in different colors
Chemistry
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Democritus ' atomist philosophy was later adopted by Epicurus (341–270 BCE).
Chemistry
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Antoine-Laurent de Lavoisier is considered the "Father of Modern Chemistry".
Chemistry
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Laboratory, Institute of Biochemistry, University of Cologne.
59.
Meteorology
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Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not occur until the 18th century, the 19th century saw modest progress in the field after weather observation networks were formed across broad regions. Prior attempts at prediction of weather depended on historical data, Meteorological phenomena are observable weather events that are explained by the science of meteorology. Different spatial scales are used to describe and predict weather on local, regional, Meteorology, climatology, atmospheric physics, and atmospheric chemistry are sub-disciplines of the atmospheric sciences. Meteorology and hydrology compose the interdisciplinary field of hydrometeorology, the interactions between Earths atmosphere and its oceans are part of a coupled ocean-atmosphere system. Meteorology has application in diverse fields such as the military, energy production, transport, agriculture. The word meteorology is from Greek μετέωρος metéōros lofty, high and -λογία -logia -logy, varāhamihiras classical work Brihatsamhita, written about 500 AD, provides clear evidence that a deep knowledge of atmospheric processes existed even in those times. In 350 BC, Aristotle wrote Meteorology, Aristotle is considered the founder of meteorology. One of the most impressive achievements described in the Meteorology is the description of what is now known as the hydrologic cycle and they are all called swooping bolts because they swoop down upon the Earth. Lightning is sometimes smoky, and is then called smoldering lightning, sometimes it darts quickly along, at other times, it travels in crooked lines, and is called forked lightning. When it swoops down upon some object it is called swooping lightning, the Greek scientist Theophrastus compiled a book on weather forecasting, called the Book of Signs. The work of Theophrastus remained a dominant influence in the study of weather, in 25 AD, Pomponius Mela, a geographer for the Roman Empire, formalized the climatic zone system. According to Toufic Fahd, around the 9th century, Al-Dinawari wrote the Kitab al-Nabat, ptolemy wrote on the atmospheric refraction of light in the context of astronomical observations. St. Roger Bacon was the first to calculate the size of the rainbow. He stated that a rainbow summit can not appear higher than 42 degrees above the horizon, in the late 13th century and early 14th century, Kamāl al-Dīn al-Fārisī and Theodoric of Freiberg were the first to give the correct explanations for the primary rainbow phenomenon. Theoderic went further and also explained the secondary rainbow, in 1716, Edmund Halley suggested that aurorae are caused by magnetic effluvia moving along the Earths magnetic field lines. In 1441, King Sejongs son, Prince Munjong, invented the first standardized rain gauge and these were sent throughout the Joseon Dynasty of Korea as an official tool to assess land taxes based upon a farmers potential harvest. In 1450, Leone Battista Alberti developed a swinging-plate anemometer, and was known as the first anemometer, in 1607, Galileo Galilei constructed a thermoscope
Meteorology
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Atmospheric sciences
Meteorology
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Parhelion (sundog) at Savoie
Meteorology
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Twilight at Baker Beach
Meteorology
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A hemispherical cup anemometer
60.
Oceanography
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Oceanography, also known as oceanology, is the study of the physical and the biological aspects of the ocean. Paleoceanography studies the history of the oceans in the geologic past, humans first acquired knowledge of the waves and currents of the seas and oceans in pre-historic times. Observations on tides were recorded by Aristotle and Strabo, early exploration of the oceans was primarily for cartography and mainly limited to its surfaces and of the animals that fishermen brought up in nets, though depth soundings by lead line were taken. Although Juan Ponce de León in 1513 first identified the Gulf Stream, Franklin measured water temperatures during several Atlantic crossings and correctly explained the Gulf Streams cause. Franklin and Timothy Folger printed the first map of the Gulf Stream in 1769-1770, information on the currents of the Pacific Ocean was gathered by explorers of the late 18th century, including James Cook and Louis Antoine de Bougainville. James Rennell wrote the first scientific textbooks on oceanography, detailing the current flows of the Atlantic, during a voyage around the Cape of Good Hope in 1777, he mapped the banks and currents at the Lagullas. He was also the first to understand the nature of the intermittent current near the Isles of Scilly, Robert FitzRoy published a four-volume report of the Beagles three voyages. In 1841–1842 Edward Forbes undertook dredging in the Aegean Sea that founded marine ecology, the first superintendent of the United States Naval Observatory, Matthew Fontaine Maury devoted his time to the study of marine meteorology, navigation, and charting prevailing winds and currents. His 1855 textbook Physical Geography of the Sea was one of the first comprehensive oceanography studies, many nations sent oceanographic observations to Maury at the Naval Observatory, where he and his colleagues evaluated the information and distributed the results worldwide. Despite all this, human knowledge of the oceans remained confined to the topmost few fathoms of the water, almost nothing was known of the ocean depths. The Royal Navys efforts to all of the worlds coastlines in the mid-19th century reinforced the vague idea that most of the ocean was very deep. As exploration ignited both popular and scientific interest in the regions and Africa, so too did the mysteries of the unexplored oceans. The seminal event in the founding of the science of oceanography was the 1872-76 Challenger expedition. As the first true oceanographic cruise, this laid the groundwork for an entire academic. In response to a recommendation from the Royal Society, The British Government announced in 1871 an expedition to explore worlds oceans, charles Wyville Thompson and Sir John Murray launched the Challenger expedition. The Challenger, leased from the Royal Navy, was modified for scientific work, under the scientific supervision of Thomson, Challenger travelled nearly 70,000 nautical miles surveying and exploring. On her journey circumnavigating the globe,492 deep sea soundings,133 bottom dredges,151 open water trawls and 263 serial water temperature observations were taken, around 4,700 new species of marine life were discovered. The result was the Report Of The Scientific Results of the Exploring Voyage of H. M. S, Murray, who supervised the publication, described the report as the greatest advance in the knowledge of our planet since the celebrated discoveries of the fifteenth and sixteenth centuries
Oceanography
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HMS Challenger undertook the first global marine research expedition in 1872.
Oceanography
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Thermohaline circulation
Oceanography
–
Ocean currents (1911)
Oceanography
–
Oceanographic Museum Monaco
61.
Physical science
–
Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a physical science, in natural science, hypotheses must be verified scientifically to be regarded as scientific theory. Validity, accuracy, and social mechanisms ensuring quality control, such as review and repeatability of findings, are amongst the criteria. Natural science can be broken into two branches, life science, for example biology and physical science. Each of these branches, and all of their sub-branches, are referred to as natural sciences, physics – natural and physical science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. More broadly, it is the analysis of nature, conducted in order to understand how the universe behaves. Branches of astronomy Chemistry – studies the composition, structure, properties, branches of chemistry Earth science – all-embracing term referring to the fields of science dealing with planet Earth. Earth science is the study of how the natural environment works and it includes the study of the atmosphere, hydrosphere, lithosphere, and biosphere. Branches of Earth science History of physical science – history of the branch of science that studies non-living systems. It in turn has many branches, each referred to as a physical science, however, the term physical creates an unintended, somewhat arbitrary distinction, since many branches of physical science also study biological phenomena. History of astrodynamics – history of the application of ballistics and celestial mechanics to the problems concerning the motion of rockets. History of astrometry – history of the branch of astronomy that involves precise measurements of the positions and movements of stars, History of cosmology – history of the discipline that deals with the nature of the Universe as a whole. History of physical cosmology – history of the study of the largest-scale structures, History of planetary science – history of the scientific study of planets, moons, and planetary systems, in particular those of the Solar System and the processes that form them. History of neurophysics – history of the branch of biophysics dealing with the nervous system, History of chemical physics – history of the branch of physics that studies chemical processes from the point of view of physics. History of computational physics – history of the study and implementation of algorithms to solve problems in physics for which a quantitative theory already exists. History of condensed matter physics – history of the study of the properties of condensed phases of matter. History of cryogenics – history of the cryogenics is the study of the production of low temperature. History of biomechanics – history of the study of the structure and function of biological systems such as humans, animals, plants, organs, History of fluid mechanics – history of the study of fluids and the forces on them
Physical science
–
Chemistry, the central science, partial ordering of the sciences proposed by Balaban and Klein.
62.
Geodesy
–
Geodesists also study geodynamical phenomena such as crustal motion, tides, and polar motion. For this they design global and national networks, using space and terrestrial techniques while relying on datums. Geodesy — from the Ancient Greek word γεωδαισία geodaisia — is primarily concerned with positioning within the temporally varying gravity field, such geodetic operations are also applied to other astronomical bodies in the solar system. It is also the science of measuring and understanding the earths geometric shape, orientation in space and this applies to the solid surface, the liquid surface and the Earths atmosphere. For this reason, the study of the Earths gravity field is called physical geodesy by some, the geoid is essentially the figure of the Earth abstracted from its topographical features. It is an idealized surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. The geoid, unlike the ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between the geoid and the ellipsoid is called the geoidal undulation. It varies globally between ±110 m, when referred to the GRS80 ellipsoid, a reference ellipsoid, customarily chosen to be the same size as the geoid, is described by its semi-major axis a and flattening f. The quantity f = a − b/a, where b is the axis, is a purely geometrical one. The mechanical ellipticity of the Earth can be determined to high precision by observation of satellite orbit perturbations and its relationship with the geometrical flattening is indirect. The relationship depends on the density distribution, or, in simplest terms. The 1980 Geodetic Reference System posited a 6,378,137 m semi-major axis and this system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics. It is essentially the basis for geodetic positioning by the Global Positioning System and is also in widespread use outside the geodetic community. The locations of points in space are most conveniently described by three cartesian or rectangular coordinates, X, Y and Z. Since the advent of satellite positioning, such systems are typically geocentric. The X-axis lies within the Greenwich observatorys meridian plane, the coordinate transformation between these two systems is described to good approximation by sidereal time, which takes into account variations in the Earths axial rotation. A more accurate description also takes polar motion into account, a closely monitored by geodesists
Geodesy
–
An old geodetic pillar (1855) at Ostend, Belgium
Geodesy
–
Geodesy
Geodesy
–
A Munich archive with lithography plates of maps of Bavaria
Geodesy
–
Geodetic Control Mark (example of a deep benchmark)
63.
Architecture
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Architecture is both the process and the product of planning, designing, and constructing buildings and other physical structures. Architectural works, in the form of buildings, are often perceived as cultural symbols. Historical civilizations are often identified with their surviving architectural achievements, Architecture can mean, A general term to describe buildings and other physical structures. The art and science of designing buildings and nonbuilding structures, the style of design and method of construction of buildings and other physical structures. A unifying or coherent form or structure Knowledge of art, science, technology, the design activity of the architect, from the macro-level to the micro-level. The practice of the architect, where architecture means offering or rendering services in connection with the design and construction of buildings. The earliest surviving work on the subject of architecture is De architectura. According to Vitruvius, a building should satisfy the three principles of firmitas, utilitas, venustas, commonly known by the original translation – firmness, commodity. An equivalent in modern English would be, Durability – a building should stand up robustly, utility – it should be suitable for the purposes for which it is used. Beauty – it should be aesthetically pleasing, according to Vitruvius, the architect should strive to fulfill each of these three attributes as well as possible. Leon Battista Alberti, who elaborates on the ideas of Vitruvius in his treatise, De Re Aedificatoria, saw beauty primarily as a matter of proportion, for Alberti, the rules of proportion were those that governed the idealised human figure, the Golden mean. The most important aspect of beauty was, therefore, an inherent part of an object, rather than something applied superficially, Gothic architecture, Pugin believed, was the only true Christian form of architecture. The 19th-century English art critic, John Ruskin, in his Seven Lamps of Architecture, Architecture was the art which so disposes and adorns the edifices raised by men. That the sight of them contributes to his health, power. For Ruskin, the aesthetic was of overriding significance and his work goes on to state that a building is not truly a work of architecture unless it is in some way adorned. For Ruskin, a well-constructed, well-proportioned, functional building needed string courses or rustication, but suddenly you touch my heart, you do me good. I am happy and I say, This is beautiful, le Corbusiers contemporary Ludwig Mies van der Rohe said Architecture starts when you carefully put two bricks together. The notable 19th-century architect of skyscrapers, Louis Sullivan, promoted an overriding precept to architectural design, function came to be seen as encompassing all criteria of the use, perception and enjoyment of a building, not only practical but also aesthetic, psychological and cultural
Architecture
–
Brunelleschi, in the building of the dome of Florence Cathedral in the early 15th-century, not only transformed the building and the city, but also the role and status of the architect.
Architecture
–
Section of Brunelleschi 's dome drawn by the architect Cigoli (c. 1600)
Architecture
–
The Parthenon, Athens, Greece, "the supreme example among architectural sites." (Fletcher).
Architecture
–
The Houses of Parliament, Westminster, master-planned by Charles Barry, with interiors and details by A.W.N. Pugin
64.
Computer graphics
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Computer graphics are pictures and films created using computers. Usually, the term refers to computer-generated image data created with help from specialized hardware and software. It is a vast and recent area in computer science, the phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes referred to as CGI. The overall methodology depends heavily on the sciences of geometry, optics. Computer graphics is responsible for displaying art and image data effectively and meaningfully to the user and it is also used for processing image data received from the physical world. Computer graphic development has had a significant impact on many types of media and has revolutionized animation, movies, advertising, video games, the term computer graphics has been used a broad sense to describe almost everything on computers that is not text or sound. Such imagery is found in and on television, newspapers, weather reports, a well-constructed graph can present complex statistics in a form that is easier to understand and interpret. In the media such graphs are used to illustrate papers, reports, thesis, many tools have been developed to visualize data. Computer generated imagery can be categorized into different types, two dimensional, three dimensional, and animated graphics. As technology has improved, 3D computer graphics have become more common, Computer graphics has emerged as a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. Screens could display art since the Lumiere brothers use of mattes to create effects for the earliest films dating from 1895. New kinds of displays were needed to process the wealth of information resulting from such projects, early projects like the Whirlwind and SAGE Projects introduced the CRT as a viable display and interaction interface and introduced the light pen as an input device. Douglas T. Ross of the Whirlwind SAGE system performed an experiment in 1954 in which a small program he wrote captured the movement of his finger. Electronics pioneer Hewlett-Packard went public in 1957 after incorporating the decade prior, and established ties with Stanford University through its founders. This began the transformation of the southern San Francisco Bay Area into the worlds leading computer technology hub - now known as Silicon Valley. The field of computer graphics developed with the emergence of computer graphics hardware, further advances in computing led to greater advancements in interactive computer graphics. In 1959, the TX-2 computer was developed at MITs Lincoln Laboratory, the TX-2 integrated a number of new man-machine interfaces
Computer graphics
–
A Blender 2.45 screenshot, displaying the 3D test model Suzanne.
Computer graphics
–
Spacewar! running on the Computer History Museum 's PDP-1
Computer graphics
–
Dire Straits ' 1985 music video for their hit song Money For Nothing - the "I Want My MTV " song – became known as an early example of fully three-dimensional, animated computer-generated imagery.
Computer graphics
–
Quarxs, series poster, Maurice Benayoun, François Schuiten, 1992
65.
Circumcircle
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In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Circumcircle
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
66.
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.
67.
Small-angle approximation
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The small-angle approximation is a useful simplification of the basic trigonometric functions which is approximately true in the limit where the angle approaches zero. They are truncations of the Taylor series for the trigonometric functions to a second-order approximation. This truncation gives, sin θ ≈ θ cos θ ≈1 − θ22 tan θ ≈ θ, where θ is the angle in radians. The small angle approximation is useful in areas of engineering and physics, including mechanics, electromagnetics, optics, cartography, astronomy. The accuracy of the approximations can be seen below in Figure 1, as the angle approaches zero, it is clear that the gap between the approximation and the original function quickly vanishes. The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A. As is shown, H and A are almost the length, meaning cos θ is close to 1. Cos θ ≈1 − θ22 The opposite leg, O, is equal to the length of the blue arc. Simplifying leaves, sin θ ≈ tan θ ≈ θ, the Maclaurin expansion of the relevant trigonometric function is sin θ = ∑ n =0 ∞ n. + ⋯ where θ is the angle in radians.01, Figure 3 shows the relative errors of the small angle approximations. The angles at which the error exceeds 1% are as follows. Sin θ ≈ θ at about 0.244 radians, cos θ ≈1 − θ2/2 at about 0.664 radians. In astronomy, the angle subtended by the image of a distant object is only a few arcseconds. The linear size is related to the size and the distance from the observer by the simple formula D = X d 206265 where X is measured in arcseconds. The number 7005206265000000000♠206265 is approximately equal to the number of arcseconds in a circle, the exact formula is D = d tan and the above approximation follows when tan X is replaced by X. The second-order cosine approximation is useful in calculating the potential energy of a pendulum. The small-angle approximation also appears in structural mechanics, especially in stability and this leads to significant simplifications, though at a cost in accuracy and insight into the true behavior. The 1 in 60 rule used in air navigation has its basis in the small-angle approximation, skinny triangle Infinitesimal oscillations of a pendulum Versine and haversine Exsecant and excosecant
Small-angle approximation
–
Approximately equal behavior of some (trigonometric) functions for x → 0
68.
Trigonometry in Galois fields
–
In mathematics, trigonometry analogies are supported by the theory of quadratic extensions of finite fields, also known as Galois fields. The main motivation to deal with a finite field trigonometry is the power of the discrete transforms, in the real DTTs, inevitably, rounding is necessary, because the elements of its transformation matrices are derived from the calculation of sines and cosines. This is the motivation to define the cosine transform over prime finite fields. In this case, all the calculation is done using integer arithmetic, the set GI of Gaussian integers over the finite field GF plays an important role in the trigonometry over finite fields. If q = pr is a power such that −1 is a quadratic non-residue in GF, then GI is defined as GI =. Thus GI is an isomorphic to GF. Trigonometric functions over the elements of a Galois field can be defined as follows, Let ζ be an element of multiplicative order N in GI, q = pr, p an odd prime such that p ≡3. The GI-valued k-trigonometric functions of in GI are defined as cos k = ⋅, sin k = ⋅ and we write cosk and sink as cosk and sink, respectively. The trigonometric functions above introduced satisfy properties P1-P12 below, in GI, unit circle, sin k 2 + cos k 2 ≡1. Even/Odd, cos k ≡ cos k , euler formula, ζ k i ≡ cos k + j sin k . Addition of arcs, cos k ≡ cos k cos k − sin k sin k , double arc, cos k 2 ≡ ⋅, sin k 2 ≡ ⋅ has modulus one and belongs to GI. The complex Z plane in GF can be constructed from the set of GI, The supra-unimodular set of GI, denoted Gs, is the set of elements ζ = ∈ GI. The structure <Gs, *>, is a group of order 2. The elements ζ = a + jb of the supra-unimodular group Gs satisfy 2 ≡1, Gs is precisely the group of phases G θ. If p is a Mersenne prime, the elements ζ = a + jb such that a2 + b2 ≡ −1 are the generators of Gs, Let p =31, a Mersenne prime, and ζ =6 + j16. Then r = | | ≡ | |13 | | ≡7, so that ϵ = ζ /r =23 + j20, therefore ε has order 2 =64. A unimodular element β of order N, such that N |25, therefore ε has order 2 =16, so it is a generator of the group Gs. A generator ε of the group is used to construct the Z plane over GF
Trigonometry in Galois fields
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Figure 1. Roots of unity in GF(11 2) expressed as elements of GI(11).
69.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
70.
Clark University
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Clark University is an American private research university located in Worcester, Massachusetts, the second largest city in New England. It is adjacent to University Park about 50 miles west of Boston, founded in 1887 with a large endowment from its namesake Jonas Gilman Clark, a prominent businessman, Clark was one of the first modern research universities in the United States. Originally an all-graduate institution, Clarks first undergraduates entered in 1902. S, News & World Report and as one of 40 Colleges That Change Lives. The university competes intercollegiately in 17 NCAA Division III varsity sports as the Clark Cougars and is a part of the New England Womens and Mens Athletic Conference, intramural and club sports are also offered in a wide range of activities. Clark was ranked no.27 on the U. S. News list of Best Value Schools, the university is also the alma mater of at least three living billionaires, in addition to its alumni having won two Pulitzer Prizes and an Emmy Award. An Act of Incorporation was duly enacted by the legislature and signed by the governor on March 31 of that same year. Opening on October 2,1889, Clark was the first all-graduate university in the United States, with departments in mathematics, physics, chemistry, biology, G. Stanley Hall was appointed the first president of Clark University in 1888. He had been a professor of psychology and pedagogy at Johns Hopkins University, Hall spent seven months in Europe visiting other universities and recruiting faculty. He became the founder of the American Psychological Association and earned the first Ph. D. in psychology in the United States at Harvard, Clark has played a prominent role in the development of psychology as a distinguished discipline in the United States ever since. This had been opposed by President Hall in years past but Clark College opened in 1902. Clark College and Clark University had different presidents until Halls retirement in 1920, Clark University began admitting women after Clarks death, and the first female Ph. D. in psychology was awarded in 1908. Early Ph. D. students in psychology were ethnically diverse, in 1920, Francis Sumner became the first African American to earn a Ph. D. in psychology. Clark withdrew its membership in 1999, citing a conflict with its mission, in order to celebrate the 20th anniversary of Clarks opening, President Hall invited a number of leading thinkers to the University. This was Freuds only set of lectures in the United States, in the 1920s Robert Goddard, a pioneer of rocketry, considered one of the founders of space and missile technology, served as a professor and chairman of the Physics Department. On November 23,1929, noted aviator Charles Lindbergh visited campus, the Robert H. Goddard Library, a distinctive modern building in the brutalist style by architect John M. Johansen, was completed in 1969. In 1963, student DArmy Bailey invited Malcolm X to campus to speak and he delivered a speech in Atwood Hall. On March 15,1968, The Jimi Hendrix Experience performed at Clark University as part of the bands American tour in support of Axis, the Experience played in the Atwood Hall, which could accommodate more than six hundred students. Tickets for the concerts, which sold out, were priced, with seats priced at $3.00, $3.50
Clark University
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Group photo 1909 in front of Clark University. Front row: Sigmund Freud, G. Stanley Hall, Carl Jung; back row: Abraham A. Brill, Ernest Jones, Sándor Ferenczi.
Clark University
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Clark University
Clark University
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Main façade of Jonas Clark Hall, the main academic facility for undergraduate students.
Clark University
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The Traina Center for the Arts is located in the former Downing Street School.
71.
Elementary algebra
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Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values and this use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real, the use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations, algebraic notation describes how algebra is written. It follows certain rules and conventions, and has its own terminology, a term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators. By convention, letters at the beginning of the alphabet are used to represent constants. They are usually written in italics, algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation. and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example,3 × x 2 is written as 3 x 2, usually terms with the highest power, are written on the left, for example, x 2 is written to the left of x. When a coefficient is one, it is usually omitted, likewise when the exponent is one. When the exponent is zero, the result is always 1, however 00, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents. Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters, for example, exponents are usually formatted using superscripts, e. g. x 2. In plain text, and in the TeX mark-up language, the symbol ^ represents exponents. In programming languages such as Ada, Fortran, Perl, Python and Ruby, many programming languages and calculators use a single asterisk to represent the multiplication symbol, and it must be explicitly used, for example,3 x is written 3*x. Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general numbers and this is useful for several reasons. Variables may represent numbers whose values are not yet known, for example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as C = P +20. Variables allow one to describe general problems, without specifying the values of the quantities that are involved, for example, it can be stated specifically that 5 minutes is equivalent to 60 ×5 =300 seconds
Elementary algebra
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A typical algebra problem.
Elementary algebra
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Two-dimensional plot (magenta curve) of the algebraic equation
72.
Differential equation
–
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
Differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
73.
Dynamical systems theory
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Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems, when difference equations are employed, the theory is called discrete dynamical systems. Some situations may also be modeled by mixed operators, such as differential-difference equations, much of modern research is focused on the study of chaotic systems. This field of study is called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems. Dynamical systems theory and chaos theory deal with the qualitative behavior of dynamical systems. Or Does the long-term behavior of the system depend on its initial condition, an important goal is to describe the fixed points, or steady states of a given dynamical system, these are values of the variable that dont change over time. Some of these points are attractive, meaning that if the system starts out in a nearby state. Similarly, one is interested in points, states of the system that repeat after several timesteps. Periodic points can also be attractive, sharkovskiis theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system. Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos, the branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory. The concept of systems theory has its origins in Newtonian mechanics. Before the advent of fast computing machines, solving a system required sophisticated mathematical techniques. Some excellent presentations of mathematical dynamic system theory include, and, the dynamical system concept is a mathematical formalization for any fixed rule that describes the time dependence of a points position in its ambient space. Examples include the models that describe the swinging of a clock pendulum, the flow of water in a pipe. A dynamical system has a state determined by a collection of real numbers, small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold, the evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule may be deterministic or stochastic and it argues that differential equations are more suited to modelling cognition than more traditional computer models. In mathematics, a system is a system that is not linear—i. e
Dynamical systems theory
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The Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.
74.
Algebraic geometry
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Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. A point of the plane belongs to a curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the points, the inflection points. More advanced questions involve the topology of the curve and relations between the curves given by different equations, Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. In the 20th century, algebraic geometry split into several subareas, the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field. The study of the points of a variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry. The study of the points of an algebraic variety is the subject of real algebraic geometry. A large part of singularity theory is devoted to the singularities of algebraic varieties, with the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties and this means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of algebraic geometry, mainly concerned with complex points. Wiless proof of the longstanding conjecture called Fermats last theorem is an example of the power of this approach. For instance, the sphere in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 −1 =0. A slanted circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 −1 =0, x + y + z =0, first we start with a field k. In classical algebraic geometry, this field was always the complex numbers C and we consider the affine space of dimension n over k, denoted An. When one fixes a system, one may identify An with kn. The purpose of not working with kn is to emphasize that one forgets the vector space structure that kn carries, the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k
Algebraic geometry
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This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.
75.
Analytic geometry
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In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete, usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane and Euclidean space, the numerical output, however, might also be a vector or a shape. That the algebra of the numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is thought to have anticipated the work of Descartes by some 1800 years. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves and that is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation, analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the term used for analytic geometry, is named after Descartes. This work, written in his native French tongue, and its philosophical principles, initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartess masterpiece receive due recognition, Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a form of Ad locos planos et solidos isagoge was circulating in Paris in 1637. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry, as a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonard Euler who first applied the method in a systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the point of origin. These are typically written as an ordered pair and this system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates. In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ from the polar axis
Analytic geometry
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Cartesian coordinates
76.
Differential geometry
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Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century, since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas, Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. These unanswered questions indicated greater, hidden relationships, initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric and this is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Various concepts based on length, such as the arc length of curves, area of plane regions, the notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds, a distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, i. e. for small neighborhoods of points, any two regular curves are locally isometric. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat, an important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant. These are the closest analogues to the plane and space considered in Euclidean and non-Euclidean geometry. Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite, a special case of this is a Lorentzian manifold, which is the mathematical basis of Einsteins general relativity theory of gravity. Finsler geometry has the Finsler manifold as the object of study. This is a manifold with a Finsler metric, i. e. a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M is a function F, TM → [0, ∞) such that, F = |m|F for all x, y in TM, F is infinitely differentiable in TM −, symplectic geometry is the study of symplectic manifolds. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed, a diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, in dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism
Differential geometry
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A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
77.
Finite geometry
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A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points, a geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are systems that could be called finite geometries, attention is mostly paid to the finite projective. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field, Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of three or greater is isomorphic to a projective space over a finite field. However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes, similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes, There are two main kinds of finite plane geometry, affine and projective. In an affine plane, the sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, For every two distinct points, there is exactly one line that contains both points. Playfairs axiom, Given a line ℓ and a point p not on ℓ, There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry, the simplest affine plane contains only four points, it is called the affine plane of order 2. Since no three are collinear, any pair of points determines a line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered parallel, or a square where not only opposite sides, but also diagonals are considered parallel. More generally, an affine plane of order n has n2 points and n2 + n lines, each line contains n points. The affine plane of order 3 is known as the Hesse configuration. A projective plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, the intersection of any two distinct lines contains exactly one point
Finite geometry
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Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are "parallel".
78.
Graph theory
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In mathematics graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, Graphs are one of the prime objects of study in discrete mathematics. Refer to the glossary of graph theory for basic definitions in graph theory, the following are some of the more basic ways of defining graphs and related mathematical structures. To avoid ambiguity, this type of graph may be described precisely as undirected, other senses of graph stem from different conceptions of the edge set. In one more generalized notion, V is a set together with a relation of incidence that associates with each two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices, Many authors call this type of object a multigraph or pseudograph. All of these variants and others are described more fully below, the vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge, V and E are usually taken to be finite, and many of the well-known results are not true for infinite graphs because many of the arguments fail in the infinite case. The order of a graph is |V|, its number of vertices, the size of a graph is |E|, its number of edges. The degree or valency of a vertex is the number of edges that connect to it, for an edge, graph theorists usually use the somewhat shorter notation xy. Graphs can be used to model many types of relations and processes in physical, biological, social, Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the network is sometimes defined to mean a graph in which attributes are associated with the nodes and/or edges. In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the structure of a website can be represented by a directed graph, in which the vertices represent web pages. A similar approach can be taken to problems in media, travel, biology, computer chip design. The development of algorithms to handle graphs is therefore of major interest in computer science, the transformation of graphs is often formalized and represented by graph rewrite systems. Graph-theoretic methods, in forms, have proven particularly useful in linguistics. Traditionally, syntax and compositional semantics follow tree-based structures, whose power lies in the principle of compositionality
Graph theory
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A drawing of a graph.
79.
Mathematical statistics
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Mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. Statistical science is concerned with the planning of studies, especially with the design of randomized experiments, the initial analysis of the data from properly randomized studies often follows the study protocol. Of course, the data from a study can be analyzed to consider secondary hypotheses or to suggest new ideas. A secondary analysis of the data from a planned study uses tools from data analysis, data analysis is divided into, descriptive statistics - the part of statistics that describes data, i. e. summarises the data and their typical properties. Mathematical statistics has been inspired by and has extended many options in applied statistics, more complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures. A probability distribution can either be univariate or multivariate, important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution. g, inferential statistics are used to test hypotheses and make estimations using sample data. Whereas descriptive statistics describe a sample, inferential statistics infer predictions about a population that the sample represents. The outcome of statistical inference may be an answer to the question what should be done next, where this might be a decision about making further experiments or surveys, or about drawing a conclusion before implementing some organizational or governmental policy. For the most part, statistical inference makes propositions about populations, more generally, data about a random process is obtained from its observed behavior during a finite period of time. e. In statistics, regression analysis is a process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. Less commonly, the focus is on a quantile, or other parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the function which can be described by a probability distribution. Many techniques for carrying out regression analysis have been developed, nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite-dimensional. Nonparametric statistics are not based on parameterized families of probability distributions. They include both descriptive and inferential statistics, the typical parameters are the mean, variance, etc
Mathematical statistics
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Illustration of linear regression on a data set. Regression analysis is an important part of mathematical statistics.
80.
Numerical analysis
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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Being able to compute the sides of a triangle is important, for instance, in astronomy, carpentry. Numerical analysis continues this tradition of practical mathematical calculations. Much like the Babylonian approximation of the root of 2, modern numerical analysis does not seek exact answers. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors, before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead and these same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of differential equations. Car companies can improve the safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving differential equations numerically. Hedge funds use tools from all fields of analysis to attempt to calculate the value of stocks. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments, historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use programs for actuarial analysis. The rest of this section outlines several important themes of numerical analysis, the field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago, to facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. The function values are no very useful when a computer is available. The mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of analysis, since now longer
Numerical analysis
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Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296...
Numerical analysis
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Direct method
Numerical analysis
81.
Mathematical optimization
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In mathematics, computer science and operations research, mathematical optimization, also spelled mathematical optimisation, is the selection of a best element from some set of available alternatives. The generalization of optimization theory and techniques to other formulations comprises an area of applied mathematics. Such a formulation is called a problem or a mathematical programming problem. Many real-world and theoretical problems may be modeled in this general framework, typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain A of f is called the space or the choice set. The function f is called, variously, a function, a loss function or cost function, a utility function or fitness function, or, in certain fields. A feasible solution that minimizes the objective function is called an optimal solution, in mathematics, conventional optimization problems are usually stated in terms of minimization. Generally, unless both the function and the feasible region are convex in a minimization problem, there may be several local minima. While a local minimum is at least as good as any nearby points, a global minimum is at least as good as every feasible point. In a convex problem, if there is a minimum that is interior, it is also the global minimum. Optimization problems are often expressed with special notation, consider the following notation, min x ∈ R This denotes the minimum value of the objective function x 2 +1, when choosing x from the set of real numbers R. The minimum value in case is 1, occurring at x =0. Similarly, the notation max x ∈ R2 x asks for the value of the objective function 2x. In this case, there is no such maximum as the function is unbounded. This represents the value of the argument x in the interval, John Wiley & Sons, Ltd. pp. xxviii+489. (2008 Second ed. in French, Programmation mathématique, Théorie et algorithmes, Editions Tec & Doc, Paris,2008. Nemhauser, G. L. Rinnooy Kan, A. H. G. Todd, handbooks in Operations Research and Management Science. Amsterdam, North-Holland Publishing Co. pp. xiv+709, J. E. Dennis, Jr. and Robert B
Mathematical optimization
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Graph of a paraboloid given by f(x, y) = −(x ² + y ²) + 4. The global maximum at (0, 0, 4) is indicated by a red dot.
82.
Order theory
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Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a framework for describing statements such as this is less than that or this precedes that. This article introduces the field and provides basic definitions, a list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the order on the natural numbers e. g.2 is less than 3,10 is greater than 5. This intuitive concept can be extended to orders on sets of numbers, such as the integers. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general, other familiar examples of orderings are the alphabetical order of words in a dictionary and the genealogical property of lineal descent within a group of people. The notion of order is very general, extending beyond contexts that have an immediate, in other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the relation, e. g. Pediatricians are physicians. However, many other orders do not and those orders like the subset-of relation for which there exist incomparable elements are called partial orders, orders for which every pair of elements is comparable are total orders. Order theory captures the intuition of orders that arises from such examples in a general setting and this is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes sense, because one can derive numerous theorems in the general setting. These insights can then be transferred to many less abstract applications. Driven by the wide usage of orders, numerous special kinds of ordered sets have been defined. In addition, order theory does not restrict itself to the classes of ordering relations. A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found and this section introduces ordered sets by building upon the concepts of set theory, arithmetic, and binary relations. Suppose that P is a set and that ≤ is a relation on P, a set with a partial order on it is called a partially ordered set, poset, or just an ordered set if the intended meaning is clear. By checking these properties, one sees that the well-known orders on natural numbers, integers, rational numbers
Order theory
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Hasse diagram of the set of all divisors of 60, partially ordered by divisibility
83.
Philosophy of mathematics
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The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics, the logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably, the latter, however, may be used to refer to several other areas of study. Another refers to the philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Recurrent themes include, What is the role of Mankind in developing mathematics, what are the sources of mathematical subject matter. What is the status of mathematical entities. What does it mean to refer to a mathematical object, what is the character of a mathematical proposition. What is the relation between logic and mathematics, what is the role of hermeneutics in mathematics. What kinds of play a role in mathematics. What are the objectives of mathematical inquiry, what gives mathematics its hold on experience. What are the human traits behind mathematics, what is the source and nature of mathematical truth. What is the relationship between the world of mathematics and the material universe. The origin of mathematics is subject to argument, whether the birth of mathematics was a random happening or induced by necessity duly contingent upon other subjects, say for example physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics, there are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Greek philosophy on mathematics was strongly influenced by their study of geometry, for example, at one time, the Greeks held the opinion that 1 was not a number, but rather a unit of arbitrary length. A number was defined as a multitude, therefore,3, for example, represented a certain multitude of units, and was thus not truly a number. At another point, an argument was made that 2 was not a number. These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the root of two
Philosophy of mathematics
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David Hilbert
84.
Representation theory
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The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system. Representation theory is pervasive across fields of mathematics, for two reasons, secondly, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics, the success of representation theory has led to numerous generalizations. One of the most general is in category theory, let V be a vector space over a field F. For instance, suppose V is Rn or Cn, the standard space of column vectors over the real or complex numbers respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using n × n matrices of real or complex numbers, there are three main sorts of algebraic objects for which this can be done, groups, associative algebras and Lie algebras. The set of all invertible n × n matrices is a group under matrix multiplication, matrix addition and multiplication make the set of all n × n matrices into an associative algebra and hence there is a corresponding representation theory of associative algebras. If we replace matrix multiplication MN by the matrix commutator MN − NM, then the n × n matrices become instead a Lie algebra, there are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group G or algebra A on a vector space V is a map Φ, G × V → V or Φ, A × V → V with two properties. First, for any g in G, the map φ, V → V v ↦ Φ is linear, the requirement for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation is ignored. Equation is an expression of the associativity of matrix multiplication. This doesnt hold for the commutator and also there is no identity element for the commutator. This approach is more concise and more abstract. The vector space V is called the space of φ. It is also common practice to refer to V itself as the representation when the homomorphism φ is clear from the context, otherwise the notation can be used to denote a representation. When V is of dimension n, one can choose a basis for V to identify V with Fn
Representation theory
85.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
Set theory
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Georg Cantor
Set theory
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A Venn diagram illustrating the intersection of two sets.
86.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
Topology
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Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.
87.
Discrete mathematics
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Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Discrete mathematics therefore excludes topics in mathematics such as calculus. Discrete objects can often be enumerated by integers, more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no definition of the term discrete mathematics. Indeed, discrete mathematics is described less by what is included than by what is excluded, continuously varying quantities, the set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of mathematics that deals with finite sets. Conversely, computer implementations are significant in applying ideas from mathematics to real-world problems. Although the main objects of study in mathematics are discrete objects. In university curricula, Discrete Mathematics appeared in the 1980s, initially as a computer science support course, some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is seen as a preparatory course. The Fulkerson Prize is awarded for outstanding papers in discrete mathematics, the history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, in logic, the second problem on David Hilberts list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Gödels second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself, Hilberts tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done, at the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades, operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in mathematics, particularly in graph theory. Formal verification of statements in logic has been necessary for development of safety-critical systems. Computational geometry has been an important part of the computer graphics incorporated into modern video games, currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP
Discrete mathematics
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Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.
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National Diet Library
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The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. Hani envisioned the new body as both a citadel of popular sovereignty, and the means of realizing a peaceful revolution, the National Diet Library opened in June 1948 in the present-day State Guest-House with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori, the philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo