1.
Trigonometric function
–
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.
Uses of trigonometry
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The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics. In fine, it is the soul of science and it is an eternal truth, it contains the mathematical demonstration of which man speaks, and the extent of its uses are unknown. For the 25 years preceding the invention of the logarithm in 1614 and it used the identities for the trigonometric functions of sums and differences of angles in terms of the products of trigonometric functions of those angles. It does mean that things in these fields cannot be understood without trigonometry. For example, a professor of music may perhaps know nothing of mathematics, in some of the fields of endeavor listed above it is easy to imagine how trigonometry could be used. For example, in navigation and land surveying, the occasions for the use of trigonometry are in at least some cases simple enough that they can be described in a beginning trigonometry textbook. The resemblance between the shape of a string and the graph of the sine function is no mere coincidence. In oceanography, the resemblance between the shapes of some waves and the graph of the function is also not coincidental. In some other fields, among them climatology, biology, and economics, the study of these often involves the periodic nature of the sine and cosine function. Many fields make use of trigonometry in more advanced ways than can be discussed in a single article, often those involve what are called Fourier series, after the 18th- and 19th-century French mathematician and physicist Joseph Fourier. Fourier used these for studying heat flow and diffusion, Fourier series are also applicable to subjects whose connection with wave motion is far from obvious. Another example, mentioned above, is diffusion, among others are, the geometry of numbers, isoperimetric problems, recurrence of random walks, quadratic reciprocity, the central limit theorem, Heisenbergs inequality. A more abstract concept than Fourier series is the idea of Fourier transform, Fourier transforms involve integrals rather than sums, and are used in a similarly diverse array of scientific fields. Many natural laws are expressed by relating rates of change of quantities to the quantities themselves, for example, The rate of change of population is sometimes jointly proportional to the present population and the amount by which the present population falls short of the carrying capacity. This kind of relationship is called a differential equation, if, given this information, one tries to express population as a function of time, one is trying to solve the differential equation. Fourier transforms may be used to convert some differential equations to algebraic equations for which methods of solving them are known, in almost any scientific context in which the words spectrum, harmonic, or resonance are encountered, Fourier transforms or Fourier series are nearby. Intelligence quotients are sometimes held to be distributed according to the bell-shaped curve, about 40% of the area under the curve is in the interval from 100 to 120, correspondingly, about 40% of the population scores between 100 and 120 on IQ tests. Nearly 9% of the area under the curve is in the interval from 120 to 140, correspondingly, similarly many other things are distributed according to the bell-shaped curve, including measurement errors in many physical measurements
Uses of trigonometry
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The
Canadarm2 robotic manipulator on the
International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles.
4.
Trigonometric functions
–
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
–
Trigonometry
Trigonometric functions
Trigonometric functions
5.
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
–
Trigonometry
Inverse trigonometric functions
Inverse trigonometric functions
6.
List of trigonometric identities
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Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles and these identities are useful whenever expressions involving trigonometric functions need to be simplified. This article uses Greek letters such as alpha, beta, gamma, several different units of angle measure are widely used, including degrees, radians, and gradians,1 full circle =360 degrees = 2π radians =400 gons. The following table shows the conversions and values for some common angles, all angles in this article are re-assumed to be in radians, but angles ending in a degree symbol are in degrees. Per Nivens theorem multiples of 30° are the angles that are a rational multiple of one degree and also have a rational sine or cosine. The secondary trigonometric functions are the sine and cosine of an angle and these are sometimes abbreviated sin and cos, respectively, where θ is the angle, but the parentheses around the angle are often omitted, e. g. sin θ and cos θ. The sine of an angle is defined in the context of a right triangle, the tangent of an angle is the ratio of the sine to the cosine, tan θ = sin θ cos θ. These definitions are sometimes referred to as ratio identities, the inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the function for the sine, known as the inverse sine or arcsine, satisfies sin = x for | x | ≤1. This can be viewed as a version of the Pythagorean theorem, and follows from the equation x2 + y2 =1 for the unit circle. Dividing this identity by either cos2 θ or sin2 θ yields the other two Pythagorean identities,1 + tan 2 θ = sec 2 θ and 1 + cot 2 θ = csc 2 θ. For example, the formula was used to calculate the distance between two points on a sphere. By examining the unit circle, the properties of the trigonometric functions can be established. When the trigonometric functions are reflected from certain angles, the result is one of the other trigonometric functions. This leads to the identities, Note that the sign in front of the trig function does not necessarily indicate the sign of the value. For example, +cos θ does not always mean that cos θ is positive, in particular, if θ = π, then +cos θ = −1. By shifting the function round by certain angles, it is possible to find different trigonometric functions that express particular results more simply. Some examples of this are shown by shifting functions round by π/2, π, because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift
List of trigonometric identities
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Cosines and sines around the
unit circle
7.
Trigonometric tables
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In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science, the calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices. Modern computers and pocket calculators now generate trigonometric function values on demand, often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of functions is still used in computer graphics. In this case, calling generic library routines every time is unacceptably slow, one option is to call the library routines once, to build up a table of those trigonometric values that will be needed, but this requires significant memory to store the table. The other possibility, since a sequence of values is required, is to use a recurrence formula to compute the trigonometric values on the fly. Significant research has been devoted to finding accurate, stable recurrence schemes in order to preserve the accuracy of the FFT, modern computers and calculators use a variety of techniques to provide trigonometric function values on demand for arbitrary angles. On simpler devices that lack a hardware multiplier, there is an algorithm called CORDIC that is efficient, since it uses only shifts. All of these methods are implemented in hardware for performance reasons. The particular polynomial used to approximate a trig function is generated ahead of time using some approximation of an approximation algorithm. Trigonometric functions of angles that are multiples of 2π are algebraic numbers. The values for a/b·2π can be found by applying de Moivres identity for n = a to a bth root of unity, for this case, a root-finding algorithm such as Newtons method is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for transcendental trigonometric constants, however and this method was used by the ancient astronomer Ptolemy, who derived them in the Almagest, a treatise on astronomy. In modern form, the identities he derived are stated as follows, unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/N. For example, for N =256 the maximum error in the values is ~0.061. For N =1024, the error in the sine values is ~0.015. If the sine and cosine values obtained were to be plotted, N −1, where wr = cos and wi = sin. These two starting trigonometric values are computed using existing library functions
Trigonometric tables
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A page from a 1619 book of
mathematical tables.
Trigonometric tables
–
Trigonometry
8.
Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π
Unit circle
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Illustration of a unit circle. The variable t is an
angle measure.
9.
Law of sines
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In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. When the last of these equations is not used, the law is sometimes stated using the reciprocals, the law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. Numerical calculation using this technique may result in an error if an angle is close to 90 degrees. It can also be used when two sides and one of the angles are known. In some such cases, the triangle is not uniquely determined by this data, the law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant curvature, the area T of any triangle can be written as one half of its base times its height. Thus, depending on the selection of the base the area of the triangle can be written as any of, multiplying these by 2/abc gives 2 T a b c = sin A a = sin B b = sin C c. When using the law of sines to find a side of a triangle, in the case shown below they are triangles ABC and AB′C′. Given a general triangle the following conditions would need to be fulfilled for the case to be ambiguous, The only information known about the triangle is the angle A, the side a is shorter than the side c. The side a is longer than the altitude h from angle B, without further information it is impossible to decide which is the triangle being asked for. The following are examples of how to solve a problem using the law of sines, given, side a =20, side c =24, and angle C = 40°. Using the law of sines, we conclude that sin A20 = sin 40 ∘24, note that the potential solution A =147. 61° is excluded because that would necessarily give A + B + C > 180°. The second equality above readily simplifies to Herons formula for the area, the law of sines takes on a similar form in the presence of curvature. In the spherical case, the formula is, sin A sin α = sin B sin β = sin C sin γ. Here, α, β, and γ are the angles at the center of the sphere subtended by the three arcs of the spherical surface triangle a, b, and c, respectively, a, B, and C are the surface angles opposite their respective arcs. See also Spherical law of cosines and Half-side formula, in hyperbolic geometry when the curvature is −1, the law of sines becomes sin A sinh a = sin B sinh b = sin C sinh c. Define a generalized function, depending also on a real parameter K. The law of sines in constant curvature K reads as sin A sin K a = sin B sin K b = sin C sin K c
Law of sines
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A triangle labelled with the components of the law of sines. Capital A, B and C are the angles, and lower-case a, b, c are the sides opposite them. (a opposite A, etc.)
10.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
Pythagorean theorem
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The
Plimpton 322 tablet records Pythagorean triples from Babylonian times.
Pythagorean theorem
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Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Pythagorean theorem
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Geometric proof of the Pythagorean theorem from the
Zhou Bi Suan Jing.
Pythagorean theorem
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Exhibit on the Pythagorean theorem at the
Universum museum in Mexico City
11.
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
–
Circle, centre O, radius r
Differentiation of trigonometric functions
–
Trigonometry
12.
Ancient Greek
–
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
–
Inscription about the construction of the statue of
Athena Parthenos in the
Parthenon, 440/439 BC
Ancient Greek
–
Ostracon bearing the name of
Cimon,
Stoa of Attalos
Ancient Greek
–
The words ΜΟΛΩΝ ΛΑΒΕ as they are inscribed on the marble of the 1955
Leonidas Monument at
Thermopylae
13.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
14.
Angle
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
Angle
–
An angle enclosed by rays emanating from a vertex.
15.
Triangle
–
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
–
The
Flatiron Building in New York is shaped like a
triangular prism
Triangle
–
A triangle
16.
Fourier transform
–
The Fourier transform decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform is called the frequency domain representation of the original signal, the term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, linear operations performed in one domain have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the domain corresponds to multiplication by the frequency. Also, convolution in the domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, after performing the desired operations, transformation of the result can be made back to the time domain. Functions that are localized in the domain have Fourier transforms that are spread out across the frequency domain and vice versa. The Fourier transform of a Gaussian function is another Gaussian function, Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can also be generalized to functions of variables on Euclidean space. In general, functions to which Fourier methods are applicable are complex-valued, the latter is routinely employed to handle periodic functions. The fast Fourier transform is an algorithm for computing the DFT, the Fourier transform of the function f is traditionally denoted by adding a circumflex, f ^. There are several conventions for defining the Fourier transform of an integrable function f, ℝ → ℂ. Here we will use the definition, f ^ = ∫ − ∞ ∞ f e −2 π i x ξ d x. When the independent variable x represents time, the transform variable ξ represents frequency. Under suitable conditions, f is determined by f ^ via the inverse transform, f = ∫ − ∞ ∞ f ^ e 2 π i ξ x d ξ, the functions f and f ^ often are referred to as a Fourier integral pair or Fourier transform pair. For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions, the Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. Many other characterizations of the Fourier transform exist, for example, one uses the Stone–von Neumann theorem, the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group. In 1822, Joseph Fourier showed that some functions could be written as an sum of harmonics
Fourier transform
17.
Mechanical engineering
–
Mechanical engineering is the discipline that applies the principles of engineering, physics, and materials science for the design, analysis, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the design, production and it is one of the oldest and broadest of the engineering disciplines. The mechanical engineering field requires an understanding of areas including mechanics, kinematics, thermodynamics, materials science, structural analysis. Mechanical engineering emerged as a field during the Industrial Revolution in Europe in the 18th century, however, Mechanical engineering science emerged in the 19th century as a result of developments in the field of physics. The field has evolved to incorporate advancements in technology, and mechanical engineers today are pursuing developments in such fields as composites, mechatronics. Mechanical engineers may work in the field of biomedical engineering, specifically with biomechanics, transport phenomena, biomechatronics, bionanotechnology. Mechanical engineering finds its application in the archives of various ancient, in ancient Greece, the works of Archimedes deeply influenced mechanics in the Western tradition and Heron of Alexandria created the first steam engine. In China, Zhang Heng improved a water clock and invented a seismometer, during the 7th to 15th century, the era called the Islamic Golden Age, there were remarkable contributions from Muslim inventors in the field of mechanical technology. Al-Jazari, who was one of them, wrote his famous Book of Knowledge of Ingenious Mechanical Devices in 1206 and he is also considered to be the inventor of such mechanical devices which now form the very basic of mechanisms, such as the crankshaft and camshaft. Newton was reluctant to publish his methods and laws for years, gottfried Wilhelm Leibniz is also credited with creating Calculus during the same time frame. On the European continent, Johann von Zimmermann founded the first factory for grinding machines in Chemnitz, education in mechanical engineering has historically been based on a strong foundation in mathematics and science. Degrees in mechanical engineering are offered at universities worldwide. In Spain, Portugal and most of South America, where neither B. Sc. nor B. Tech, programs have been adopted, the formal name for the degree is Mechanical Engineer, and the course work is based on five or six years of training. In Italy the course work is based on five years of education, and training, in Greece, the coursework is based on a five-year curriculum and the requirement of a Diploma Thesis, which upon completion a Diploma is awarded rather than a B. Sc. In Australia, mechanical engineering degrees are awarded as Bachelor of Engineering or similar nomenclature although there are a number of specialisations. The degree takes four years of study to achieve. To ensure quality in engineering degrees, Engineers Australia accredits engineering degrees awarded by Australian universities in accordance with the global Washington Accord, before the degree can be awarded, the student must complete at least 3 months of on the job work experience in an engineering firm. Similar systems are present in South Africa and are overseen by the Engineering Council of South Africa
Mechanical engineering
–
Mechanical engineers design and build
engines,
power plants, other machines...
Mechanical engineering
–
...
structures, and
vehicles of all sizes.
Mechanical engineering
–
An oblique view of a four-cylinder inline crankshaft with pistons
Mechanical engineering
–
Training FMS with learning robot
SCORBOT-ER 4u, workbench CNC Mill and CNC Lathe
18.
Electrical engineering
–
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
–
Electrical engineers design complex power systems...
Electrical engineering
–
... and electronic circuits.
Electrical engineering
–
The discoveries of
Michael Faraday formed the foundation of electric motor technology
Electrical engineering
–
Thomas Edison, electric light and (DC) power supply networks
19.
Surveying
–
Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional position of points and the distances and angles between them. A land surveying professional is called a land surveyor, Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages and the law. Surveying has been an element in the development of the environment since the beginning of recorded history. The planning and execution of most forms of construction require it and it is also used in transport, communications, mapping, and the definition of legal boundaries for land ownership. It is an important tool for research in other scientific disciplines. Basic surveyance has occurred since humans built the first large structures, the prehistoric monument at Stonehenge was set out by prehistoric surveyors using peg and rope geometry. In ancient Egypt, a rope stretcher would use simple geometry to re-establish boundaries after the floods of the Nile River. The almost perfect squareness and north-south orientation of the Great Pyramid of Giza, built c.2700 BC, the Groma instrument originated in Mesopotamia. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao Suanjing or The Sea Island Mathematical Manual, the Romans recognized land surveyors as a profession. They established the basic measurements under which the Roman Empire was divided, Roman surveyors were known as Gromatici. In medieval Europe, beating the bounds maintained the boundaries of a village or parish and this was the practice of gathering a group of residents and walking around the parish or village to establish a communal memory of the boundaries. Young boys were included to ensure the memory lasted as long as possible, in England, William the Conqueror commissioned the Domesday Book in 1086. It recorded the names of all the owners, the area of land they owned, the quality of the land. It did not include maps showing exact locations, abel Foullon described a plane table in 1551, but it is thought that the instrument was in use earlier as his description is of a developed instrument. Gunters chain was introduced in 1620 by English mathematician Edmund Gunter and it enabled plots of land to be accurately surveyed and plotted for legal and commercial purposes. Leonard Digges described a Theodolite that measured horizontal angles in his book A geometric practice named Pantometria, joshua Habermel created a theodolite with a compass and tripod in 1576. Johnathon Sission was the first to incorporate a telescope on a theodolite in 1725, in the 18th century, modern techniques and instruments for surveying began to be used. Jesse Ramsden introduced the first precision theodolite in 1787 and it was an instrument for measuring angles in the horizontal and vertical planes
Surveying
–
A surveyor at work with an infrared reflector used for distance measurement.
Surveying
–
Table of Surveying, 1728
Cyclopaedia
Surveying
–
A map of India showing the Great Trigonometrical Survey, produced in 1870
Surveying
–
A German engineer surveying during the
First World War, 1918
20.
Plane (geometry)
–
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)
–
Vector description of a plane
Plane (geometry)
–
Two intersecting planes in three-dimensional space
21.
Curvature
–
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
22.
Hyperbolic geometry
–
In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry, Hyperbolic geometry can be extended to three and more dimensions, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry, For any line R and any point P which does not lie on R. In the plane containing line R and point P there are at least two lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
Hyperbolic geometry
–
A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the
Institute For Figuring
Hyperbolic geometry
–
Lines through a given point P and asymptotic to line R
Hyperbolic geometry
–
A coral with similar geometry on the
Great Barrier Reef
Hyperbolic geometry
–
M.C. Escher 's
Circle Limit III, 1959
23.
Babylonians
–
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
–
Old Babylonian
Cylinder Seal,
hematite, The king makes an animal offering to
Shamash. This seal was probably made in a workshop at
Sippar.
Babylonians
–
Geography
24.
Greek mathematics
–
Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
Greek mathematics
–
Statue of Euclid in the
Oxford University Museum of Natural History
Greek mathematics
–
An illustration of
Euclid 's proof of the
Pythagorean Theorem
Greek mathematics
–
The
Antikythera mechanism, an ancient mechanical calculator.
25.
Euclid
–
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
Euclid
–
Euclid by
Justus van Gent, 15th century
Euclid
–
One of the oldest surviving fragments of Euclid's Elements, found at
Oxyrhynchus and dated to circa AD 100 (
P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid
–
Statue in honor of Euclid in the
Oxford University Museum of Natural History
26.
Chord (geometry)
–
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)
–
The red segment BX is a chord (as is the diameter segment AB).
27.
Nicaea
–
The ancient city is located within the modern Turkish city of İznik, and is situated in a fertile basin at the eastern end of Lake Ascanius, bounded by ranges of hills to the north and south. It is situated with its west wall rising from the lake itself, the lake is large enough that it could not be blockaded from the land easily, and the city was large enough to make any attempt to reach the harbour from shore-based siege weapons very difficult. The ancient city is surrounded on all sides by 5 kilometres of walls about 10 metres high and these are in turn surrounded by a double ditch on the land portions, and also included over 100 towers in various locations. Large gates on the three sides of the walls provided the only entrance to the city. Today the walls have been pierced in places for roads. The later version however was not widespread even in Antiquity, Antigonus is also known to have established Bottiaean soldiers in the vicinity, lending credence to the tradition about the citys founding by Bottiaeans. Following Antigonus defeat and death at the Battle of Ipsus in 301 BC, the city was captured by Lysimachus, who renamed it Nicaea, in tribute to his wife Nicaea, who had recently died. Sometime before 280 BC, the city came under the control of the dynasty of the kings of Bithynia. This marks the beginning of its rise to prominence as a seat of the royal court, the two cities dispute over which one was the pre-eminent city of Bithynia continued for centuries, and the 38th oration of Dio Chrysostom was expressly composed to settle the dispute. Along with the rest of Bithynia, Nicaea came under the rule of the Roman Republic in 72 BC. The geographer Strabo described the city as built in the typical Hellenistic fashion with great regularity, in the form of a square, measuring 16 stadia in circumference, i. e. approx. This monument stood in the gymnasium, which was destroyed by fire but was restored with increased magnificence by Pliny the Younger, in his writings Pliny makes frequent mention of Nicaea and its public buildings. Emperor Hadrian visited the city in 123 AD after it had been damaged by an earthquake. The new city was enclosed by a wall of some 5 kilometres in length. Reconstruction was not completed until the 3rd century, and the new set of walls failed to save Nicaea from being sacked by the Goths in 258 AD, by the 4th century, Nicaea was a large and prosperous city, and a major military and administrative centre. Emperor Constantine the Great convened the First Ecumenical Council there, the city remained important in the 4th century, seeing the proclamation of Emperor Valens and the failed rebellion of Procopius. During the same period, the See of Nicaea became independent of Nicomedia and was raised to the status of a metropolitan bishopric, many of its grand civic buildings began to fall into ruin, and had to be restored in the 6th century by Emperor Justinian I. Nicaea became the capital of the Opsician Theme in the 8th century and remained a center of administration, a Jewish community is attested in the city in the 10th century
Nicaea
–
The Lefke Gate, part of Nicaea's city walls.
Nicaea
–
The theatre, restored by
Pliny the Younger.
28.
Ptolemy
–
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
–
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
–
Early
Baroque artist's rendition
Ptolemy
–
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
–
Prima Europe tabula. A C15th copy of Ptolemy's map of Britain
29.
Almagest
–
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
–
Ptolemy's Almagest became an authoritative work for many centuries.
Almagest
Almagest
–
Picture of George Trebizond's Latin translation of Almagest
30.
Indian mathematics
–
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
–
The design of the domestic fire altar in the Śulba Sūtra
31.
Astronomy in medieval Islam
–
Islamic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age, and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa and these included Greek, Sassanid, and Indian works in particular, which were translated and built upon. Islamic astronomy also had an influence on Chinese astronomy and Malian astronomy, a significant number of stars in the sky, such as Aldebaran, Altair and Deneb, and astronomical terms such as alidade, azimuth, and nadir, are still referred to by their Arabic names. A large corpus of literature from Islamic astronomy remains today, numbering approximately 10,000 manuscripts scattered throughout the world, even so, a reasonably accurate picture of Islamic activity in the field of astronomy can be reconstructed. These observations were based on the rising and setting of stars. Anwa continued to be developed after Islamization by the Arabs, where Islamic astronomers added mathematical methods to their empirical observations, according to David King, after the rise of Islam, the religious obligation to determine the qibla and prayer times inspired more progress in astronomy for centuries. The first astronomical texts that were translated into Arabic were of Indian and Persian origin, another text translated was the Zij al-Shah, a collection of astronomical tables compiled in Sasanid Persia over two centuries. Fragments of texts during this period indicate that Arabs adopted the function in place of the chords of arc used in Greek trigonometry. The House of Wisdom was an established in Baghdad under Abbasid caliph Al-Mamun in the early 9th century. From this time, independent investigation into the Ptolemaic system became possible, Astronomical research was greatly supported by the Abbasid caliph al-Mamun through The House of Wisdom. Baghdad and Damascus became the centers of such activity, the caliphs not only supported this work financially, but endowed the work with formal prestige. The first major Muslim work of astronomy was Zij al-Sindh by al-Khwarizmi in 830, the work contains tables for the movements of the sun, the moon and the five planets known at the time. The work is significant as it introduced Ptolemaic concepts into Islamic sciences and this work also marks the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others, al-Khwarizmis work marked the beginning of nontraditional methods of study and calculations. In 850, al-Farghani wrote Kitab fi Jawani, the book primarily gave a summary of Ptolemic cosmography. However, it also corrected Ptolemy based on findings of earlier Arab astronomers, al-Farghani gave revised values for the obliquity of the ecliptic, the precessional movement of the apogees of the sun and the moon, and the circumference of the earth. The book was circulated through the Muslim world, and even translated into Latin. The period when a distinctive Islamic system of astronomy flourished, the period began as the Muslim astronomers began questioning the framework of the Ptolemaic system of astronomy
Astronomy in medieval Islam
–
An 18th-century Persian astrolabe, kept at the
Whipple Museum of the History of Science in
Cambridge, England.
Astronomy in medieval Islam
–
Muhammad Salih Tahtawi of
Sindh headed the task of creating a massive, seamless
celestial globe using a secret
wax casting method in the
Mughal Empire, the famous celestial globe of Muhammad Salih Tahtawi is inscribed with
Arabic and
Persian inscriptions and was completed in the year 1631.
Astronomy in medieval Islam
–
The
Tusi-couple is a mathematical device invented by
Nasir al-Din al-Tusi in which a small
circle rotates inside a larger circle twice the
diameter of the smaller
circle. Rotations of the circles cause a point on the
circumference of the smaller circle to
oscillate back and forth in
linear motion along a diameter of the larger circle.
Astronomy in medieval Islam
–
An illustration from al-Biruni's astronomical works, explains the different phases of the moon.
32.
Germany
–
Germany, officially the Federal Republic of Germany, is a federal parliamentary republic in central-western Europe. It includes 16 constituent states, covers an area of 357,021 square kilometres, with about 82 million inhabitants, Germany is the most populous member state of the European Union. After the United States, it is the second most popular destination in the world. Germanys capital and largest metropolis is Berlin, while its largest conurbation is the Ruhr, other major cities include Hamburg, Munich, Cologne, Frankfurt, Stuttgart, Düsseldorf and Leipzig. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity, a region named Germania was documented before 100 AD. During the Migration Period the Germanic tribes expanded southward, beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation, in 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the German Revolution of 1918–1919, the Empire was replaced by the parliamentary Weimar Republic, the establishment of the national socialist dictatorship in 1933 led to World War II and the Holocaust. After a period of Allied occupation, two German states were founded, the Federal Republic of Germany and the German Democratic Republic, in 1990, the country was reunified. In the 21st century, Germany is a power and has the worlds fourth-largest economy by nominal GDP. As a global leader in industrial and technological sectors, it is both the worlds third-largest exporter and importer of goods. Germany is a country with a very high standard of living sustained by a skilled. It upholds a social security and universal health system, environmental protection. Germany was a member of the European Economic Community in 1957. It is part of the Schengen Area, and became a co-founder of the Eurozone in 1999, Germany is a member of the United Nations, NATO, the G8, the G20, and the OECD. The national military expenditure is the 9th highest in the world, the English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine. This in turn descends from Proto-Germanic *þiudiskaz popular, derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- people, the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a mine in Schöningen where three 380, 000-year-old wooden javelins were unearthed
Germany
–
The
Nebra sky disk is dated to c. 1600 BC.
Germany
–
Flag
Germany
–
Martin Luther (1483–1546) initiated the
Protestant Reformation.
Germany
–
Foundation of the
German Empire in
Versailles, 1871.
Bismarck is at the center in a white uniform.
33.
Regiomontanus
–
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
–
Regiomontanus
Regiomontanus
–
Plaque at Regiomontanus' birthplace
Regiomontanus
–
De triangulis planis et sphaericis libri
Regiomontanus
–
Title page for Qvesta opra da ogni parte e un libro doro, 1476
34.
George of Trebizond
–
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
–
George of Trebizond.
George of Trebizond
–
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
–
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
–
1580 portrait (artist unknown) in the Old Town City Hall,
Toruń
Nicolaus Copernicus
–
Toruń birthplace (ul. Kopernika 15, left). Together with the house at no. 17 (right), it forms the Muzeum Mikołaja Kopernika.
Nicolaus Copernicus
–
Copernicus' maternal uncle,
Lucas Watzenrode the Younger
Nicolaus Copernicus
–
Collegium Maius,
Kraków
36.
Bartholomaeus Pitiscus
–
Bartholomaeus Pitiscus was a 16th-century German trigonometrist, astronomer and theologian who first coined the word trigonometry. Pitiscus was born to parents in Grünberg in Lower Silesia, nowadays in Poland. He studied theology in Zerbst and Heidelberg, a Calvinist, he was appointed to teach the ten-year-old Frederick IV, Elector Palatine of the Rhine, by Fredericks Calvinist uncle Johann Casimir of Simmern, as Fredericks father had died in 1583. Pitiscus was subsequently appointed court chaplain at Breslau and court preacher to Frederick, Pitiscus supported Fredericks subsequent measures against the Roman Catholic Church. It consists of five books on plane and spherical trigonometry, Pitiscus edited Thesaurus mathematicus in which he improved the trigonometric tables of Georg Joachim Rheticus and also corrected Rheticus’s Magnus Canon doctrinæ triangulorum. The lunar crater Pitiscus is named after him, the classical scholar Samuel Pitiscus was his nephew. Verlag Harri Thun, Frankfurt a. M.1990 ISBN 3-8171-1164-9 OConnor, John J. Robertson, Edmund F. Bartholomaeus Pitiscus, MacTutor History of Mathematics archive, University of St Andrews
Bartholomaeus Pitiscus
–
Literature [edit]
37.
Triangulation
–
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to it from known points. Optical 3D measuring systems use this principle as well in order to determine the spatial dimensions, basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital device. The projection centers of the sensors and the point on the object’s surface define a triangle. Within this triangle, the distance between the sensors is the base b and must be known, by determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3D coordinate, is calculated from the triangular relations. Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry, the use of triangles to estimate distances dates to antiquity. He measured the length of the shadows and that of his own at the same moment. Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope, i. e. the reciprocal of gradients as measured today. The slopes and angles were measured using a rod that the Greeks called a dioptra
Triangulation
–
Triangulation of
Kodiak Island in 1929.
Triangulation
–
Liu Hui (c. 263), How to measure the height of a sea island. Illustration from an edition of 1726
Triangulation
–
Gemma Frisius 's 1533 proposal to use triangulation for mapmaking
38.
Leonhard Euler
–
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
–
Portrait by
Jakob Emanuel Handmann (1756)
Leonhard Euler
–
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
–
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
–
Euler's grave at the
Alexander Nevsky Monastery
39.
Taylor series
–
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
–
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.
40.
Complementary angles
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
Complementary angles
–
An angle enclosed by rays emanating from a vertex.
41.
Ratio
–
In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, when the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units, but in many applications, the word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
Ratio
–
The ratio of width to height of
standard-definition television.
42.
Hypotenuse
–
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
–
A right-angled triangle and its hypotenuse.
43.
Cosine
–
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
Cosine
–
Trigonometric functions in the complex plane
Cosine
–
Trigonometry
Cosine
Cosine
44.
Adjacent side (right triangle)
–
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
Adjacent side (right triangle)
–
The
Flatiron Building in New York is shaped like a
triangular prism
Adjacent side (right triangle)
–
A triangle
45.
Tangent (trigonometric function)
–
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)
–
Trigonometric functions in the complex plane
Tangent (trigonometric function)
–
Trigonometry
Tangent (trigonometric function)
Tangent (trigonometric function)
46.
Trigonometric identities
–
Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles and these identities are useful whenever expressions involving trigonometric functions need to be simplified. This article uses Greek letters such as alpha, beta, gamma, several different units of angle measure are widely used, including degrees, radians, and gradians,1 full circle =360 degrees = 2π radians =400 gons. The following table shows the conversions and values for some common angles, all angles in this article are re-assumed to be in radians, but angles ending in a degree symbol are in degrees. Per Nivens theorem multiples of 30° are the angles that are a rational multiple of one degree and also have a rational sine or cosine. The secondary trigonometric functions are the sine and cosine of an angle and these are sometimes abbreviated sin and cos, respectively, where θ is the angle, but the parentheses around the angle are often omitted, e. g. sin θ and cos θ. The sine of an angle is defined in the context of a right triangle, the tangent of an angle is the ratio of the sine to the cosine, tan θ = sin θ cos θ. These definitions are sometimes referred to as ratio identities, the inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the function for the sine, known as the inverse sine or arcsine, satisfies sin = x for | x | ≤1. This can be viewed as a version of the Pythagorean theorem, and follows from the equation x2 + y2 =1 for the unit circle. Dividing this identity by either cos2 θ or sin2 θ yields the other two Pythagorean identities,1 + tan 2 θ = sec 2 θ and 1 + cot 2 θ = csc 2 θ. For example, the formula was used to calculate the distance between two points on a sphere. By examining the unit circle, the properties of the trigonometric functions can be established. When the trigonometric functions are reflected from certain angles, the result is one of the other trigonometric functions. This leads to the identities, Note that the sign in front of the trig function does not necessarily indicate the sign of the value. For example, +cos θ does not always mean that cos θ is positive, in particular, if θ = π, then +cos θ = −1. By shifting the function round by certain angles, it is possible to find different trigonometric functions that express particular results more simply. Some examples of this are shown by shifting functions round by π/2, π, because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift
Trigonometric identities
–
Cosines and sines around the
unit circle
47.
Polygon
–
In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
Polygon
–
Historical image of polygons (1699)
Polygon
–
Some different types of polygon
Polygon
–
The
Giant's Causeway, in
Northern Ireland
48.
Radian
–
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
–
A chart to convert between degrees and radians
Radian
–
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.
49.
Infinite series
–
In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
Infinite series
–
Illustration of 3
geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.
50.
Mnemonics in trigonometry
–
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
–
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.
51.
Interpolate
–
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
–
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.
52.
Programming language
–
A programming language is a formal computer language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to programs to control the behavior of a machine or to express algorithms. From the early 1800s, programs were used to direct the behavior of such as Jacquard looms. Thousands of different programming languages have created, mainly in the computer field. Many programming languages require computation to be specified in an imperative form while other languages use forms of program specification such as the declarative form. The description of a language is usually split into the two components of syntax and semantics. Some languages are defined by a document while other languages have a dominant implementation that is treated as a reference. Some languages have both, with the language defined by a standard and extensions taken from the dominant implementation being common. A programming language is a notation for writing programs, which are specifications of a computation or algorithm, some, but not all, authors restrict the term programming language to those languages that can express all possible algorithms. For example, PostScript programs are created by another program to control a computer printer or display. More generally, a language may describe computation on some, possibly abstract. It is generally accepted that a specification for a programming language includes a description, possibly idealized. In most practical contexts, a programming language involves a computer, consequently, abstractions Programming languages usually contain abstractions for defining and manipulating data structures or controlling the flow of execution. Expressive power The theory of computation classifies languages by the computations they are capable of expressing, all Turing complete languages can implement the same set of algorithms. ANSI/ISO SQL-92 and Charity are examples of languages that are not Turing complete, markup languages like XML, HTML, or troff, which define structured data, are not usually considered programming languages. Programming languages may, however, share the syntax with markup languages if a computational semantics is defined, XSLT, for example, is a Turing complete XML dialect. Moreover, LaTeX, which is used for structuring documents. The term computer language is used interchangeably with programming language
Programming language
–
The
Manchester Mark 1 ran programs written in
Autocode from 1952.
Programming language
–
A selection of textbooks that teach programming, in languages both popular and obscure. These are only a few of the thousands of programming languages and dialects that have been designed in history.
53.
Floating point unit
–
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
–
An Intel 80287
54.
Marine chronometer
–
A marine chronometer is a timepiece that is precise and accurate enough to be used as a portable time standard, it can therefore be used to determine longitude by means of celestial navigation. Timepieces made in Switzerland may display the word chronometer only if certified by the COSC, to determine a position on the Earths surface, it is necessary and sufficient to know the latitude, longitude, and altitude. Altitude considerations can, of course, be ignored for vessels operating at sea level, until the mid-1750s, accurate navigation at sea out of sight of land was an unsolved problem due to the difficulty in calculating longitude. Navigators could determine their latitude by measuring the angle at noon or, in the Northern Hemisphere. To find their longitude, however, they needed a standard that would work aboard a ship. Observation of regular celestial motions, such as Galileos method based on observing Jupiters natural satellites, was not possible at sea due to the ships motion. The lunar distances method, initially proposed by Johannes Werner in 1514, was developed in parallel with the marine chronometer, the Dutch scientist Gemma Frisius was the first to propose the use of a chronometer to determine longitude in 1530. The purpose of a chronometer is to measure accurately the time of a fixed location. This is particularly important for navigation, knowing GMT at local noon allows a navigator to use the time difference between the ships position and the Greenwich Meridian to determine the ships longitude. The creation of a timepiece which would work reliably at sea was difficult, christiaan Huygens, following his invention of the pendulum clock in 1656, made the first attempt at a marine chronometer in 1673 in France, under the sponsorship of Jean-Baptiste Colbert. He obtained a patent for his invention from Colbert, but his clock remained imprecise at sea, the first published use of the term was in 1684 in Arcanum Navarchicum, a theoretical work by Kiel professor Matthias Wasmuth. This was followed by a theoretical description of a chronometer in works published by English scientist William Derham in 1713. Attempts to construct a marine chronometer were begun by Jeremy Thacker in England in 1714. In 1714, the British government offered a prize for a method of determining longitude at sea. His first two sea timepieces H1 and H2 used this system, but he realised that they had a sensitivity to centrifugal force. However, H3s circular balances still proved too inaccurate and he abandoned the large machines. Harrison solved the problems with his much smaller H4 chronometer design in 1761. H4 looked much like a large five-inch diameter pocket watch, in 1761, Harrison submitted H4 for the £20,000 longitude prize
Marine chronometer
–
Breguet twin barrel box chronometer.
Marine chronometer
–
The marine "Chronometer" of
Jeremy Thacker used
gimbals and a
vacuum in a bell jar.
Marine chronometer
–
Henry Sully (1680-1729) presented a first marine chronometer in 1716.
Marine chronometer
–
John Harrison 's H1 marine chronometer of 1735.
55.
Light
–
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
–
An example of refraction of light. The straw appears bent, because of refraction of light as it enters liquid from air.
Light
–
A triangular prism dispersing a beam of white light. The longer wavelengths (red) and the shorter wavelengths (blue) get separated.
Light
–
A
cloud illuminated by
sunlight
Light
–
A
city illuminated by
artificial lighting
56.
Music theory
–
Music theory is the study of the practices and possibilities of music. The term is used in three ways in music, though all three are interrelated. The first is what is otherwise called rudiments, currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, Theory in this sense is treated as the necessary preliminary to the study of harmony, counterpoint, and form. The second is the study of writings about music from ancient times onwards, Music theory is frequently concerned with describing how musicians and composers make music, including tuning systems and composition methods among other topics. However, this medieval discipline became the basis for tuning systems in later centuries, Music theory as a practical discipline encompasses the methods and concepts composers and other musicians use in creating music. The development, preservation, and transmission of music theory in this sense may be found in oral and written music-making traditions, musical instruments, and other artifacts. In ancient and living cultures around the world, the deep and long roots of music theory are clearly visible in instruments, oral traditions, and current music making. Many cultures, at least as far back as ancient Mesopotamia and ancient China, have also considered music theory in more formal ways such as written treatises, in modern academia, music theory is a subfield of musicology, the wider study of musical cultures and history. Etymologically, music theory is an act of contemplation of music, from the Greek θεωρία, a looking at, viewing, contemplation, speculation, theory, also a sight, a person who researches, teaches, or writes articles about music theory is a music theorist. University study, typically to the M. A. or Ph. D level, is required to teach as a music theorist in a US or Canadian university. Methods of analysis include mathematics, graphic analysis, and especially analysis enabled by Western music notation, comparative, descriptive, statistical, and other methods are also used. See for instance Paleolithic flutes, Gǔdí, and Anasazi flute, several surviving Sumerian and Akkadian clay tablets include musical information of a theoretical nature, mainly lists of intervals and tunings. The scholar Sam Mirelman reports that the earliest of these dates from before 1500 BCE. Further, All the Mesopotamian texts are united by the use of a terminology for music that, much of Chinese music history and theory remains unclear. The earliest texts about Chinese music theory are inscribed on the stone and they include more than 2800 words describing theories and practices of music pitches of the time. The bells produce two intertwined pentatonic scales three tones apart with additional pitches completing the chromatic scale, Chinese theory starts from numbers, the main musical numbers being twelve, five and eight. Twelve refers to the number of pitches on which the scales can be constructed, the Lüshi chunqiu from about 239 BCE recalls the legend of Ling Lun. On order of the Yellow Emperor, Ling Lun collected twelve bamboo lengths with thick, blowing on one of these like a pipe, he found its sound agreeable and named it huangzhong, the Yellow Bell
Music theory
–
Ancient Egyptian musicians playing lutes in an ensemble.
Music theory
–
Pythagoras and
Philolaus engaged in theoretical investigations, in a woodcut from
Franchinus Gaffurius, Theorica musicæ (1492)
Music theory
–
A set of bells from China, 5th Century BCE.
Music theory
–
Barbershop quartets, such as this US Navy group, sing 4-part pieces, made up of a melody line (normally the second-highest voice, called the "lead") and 3 harmony parts.
57.
Acoustics
–
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
–
Principles of acoustics were applied since ancient times:
Roman theatre in the city of
Amman.
Acoustics
–
Artificial omni-directional sound source in an
anechoic chamber
Acoustics
–
Jay Pritzker Pavilion
Acoustics
58.
Optics
–
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
–
Optics includes study of
dispersion of light.
Optics
–
The Nimrud lens
Optics
–
Reproduction of a page of
Ibn Sahl 's manuscript showing his knowledge of the law of refraction, now known as
Snell's law
Optics
–
Cover of the first edition of Newton's Opticks
59.
Ultrasound
–
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
–
Ultrasound image of a fetus in the womb, viewed at 12 weeks of pregnancy (bidimensional-scan)
Ultrasound
–
An ultrasonic examination
Ultrasound
–
Bats use ultrasounds to navigate in the darkness.
Ultrasound
–
Sonogram of a fetus at 14 weeks (profile)
60.
Oceanography
–
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
–
HMS Challenger undertook the first global marine research expedition in 1872.
Oceanography
–
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.
Architecture
–
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
63.
Civil engineering
–
Civil engineering is traditionally broken into a number of sub-disciplines. It is the second-oldest engineering discipline after military engineering, and it is defined to distinguish non-military engineering from military engineering, Civil engineering takes place in the public sector from municipal through to national governments, and in the private sector from individual homeowners through to international companies. Engineering has been an aspect of life since the beginnings of human existence, during this time, transportation became increasingly important leading to the development of the wheel and sailing. The construction of pyramids in Egypt were some of the first instances of large structure constructions, the Romans developed civil structures throughout their empire, including especially aqueducts, insulae, harbors, bridges, dams and roads. In the 18th century, the civil engineering was coined to incorporate all things civilian as opposed to military engineering. The first self-proclaimed civil engineer was John Smeaton, who constructed the Eddystone Lighthouse, in 1771 Smeaton and some of his colleagues formed the Smeatonian Society of Civil Engineers, a group of leaders of the profession who met informally over dinner. Though there was evidence of some meetings, it was little more than a social society. In 1818 the Institution of Civil Engineers was founded in London, the institution received a Royal Charter in 1828, formally recognising civil engineering as a profession. The first private college to teach engineering in the United States was Norwich University. The first degree in engineering in the United States was awarded by Rensselaer Polytechnic Institute in 1835. The first such degree to be awarded to a woman was granted by Cornell University to Nora Stanton Blatch in 1905, throughout ancient and medieval history most architectural design and construction was carried out by artisans, such as stonemasons and carpenters, rising to the role of master builder. Knowledge was retained in guilds and seldom supplanted by advances, structures, roads and infrastructure that existed were repetitive, and increases in scale were incremental. Brahmagupta, an Indian mathematician, used arithmetic in the 7th century AD, based on Hindu-Arabic numerals, Civil engineers typically possess an academic degree in civil engineering. The length of study is three to five years, and the degree is designated as a bachelor of engineering. The curriculum generally includes classes in physics, mathematics, project management, design, after taking basic courses in most sub-disciplines of civil engineering, they move onto specialize in one or more sub-disciplines at advanced levels. In most countries, a degree in engineering represents the first step towards professional certification. After completing a degree program, the engineer must satisfy a range of requirements before being certified. Once certified, the engineer is designated as a engineer, a chartered engineer
Civil engineering
–
A multi-level stack interchange, buildings, houses, and park in
Shanghai, China.
Civil engineering
–
Philadelphia City Hall in the United States is still the world's tallest masonry load bearing structure.
Civil engineering
–
Leonhard Euler developed the theory explaining the
buckling of columns
Civil engineering
–
John Smeaton, the "father of civil engineering"
64.
Computer graphics
–
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.
Cartography
–
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively, the fundamental problems of traditional cartography are to, Set the maps agenda and select traits of the object to be mapped. This is the concern of map editing, traits may be physical, such as roads or land masses, or may be abstract, such as toponyms or political boundaries. Represent the terrain of the object on flat media. This is the concern of map projections, eliminate characteristics of the mapped object that are not relevant to the maps purpose. This is the concern of generalization, reduce the complexity of the characteristics that will be mapped. This is also the concern of generalization, orchestrate the elements of the map to best convey its message to its audience. This is the concern of map design, modern cartography constitutes many theoretical and practical foundations of geographic information systems. The earliest known map is a matter of debate, both because the term map isnt well-defined and because some artifacts that might be maps might actually be something else. A wall painting that might depict the ancient Anatolian city of Çatalhöyük has been dated to the late 7th millennium BCE, the oldest surviving world maps are from 9th century BCE Babylonia. One shows Babylon on the Euphrates, surrounded by Assyria, Urartu and several cities, all, in turn, another depicts Babylon as being north of the world center. The ancient Greeks and Romans created maps since Anaximander in the 6th century BCE, in the 2nd century AD, Ptolemy wrote his treatise on cartography, Geographia. This contained Ptolemys world map – the world known to Western society. As early as the 8th century, Arab scholars were translating the works of the Greek geographers into Arabic, in ancient China, geographical literature dates to the 5th century BCE. The oldest extant Chinese maps come from the State of Qin, dated back to the 4th century BCE, in the book of the Xin Yi Xiang Fa Yao, published in 1092 by the Chinese scientist Su Song, a star map on the equidistant cylindrical projection. Early forms of cartography of India included depictions of the pole star and these charts may have been used for navigation. Mappa mundi are the Medieval European maps of the world, approximately 1,100 mappae mundi are known to have survived from the Middle Ages. Of these, some 900 are found illustrating manuscripts and the remainder exist as stand-alone documents, the Arab geographer Muhammad al-Idrisi produced his medieval atlas Tabula Rogeriana in 1154
Cartography
–
A medieval depiction of the
Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy's
Geography and using his second map projection. The translation into Latin and dissemination of Geography in Europe, in the beginning of the 15th century, marked the rebirth of scientific cartography, after more than a millennium of stagnation.
Cartography
–
Valcamonica rock art (I), Paspardo r. 29, topographic composition, 4th millennium BC
Cartography
–
The
Bedolina Map and its tracing, 6th–4th century BC
Cartography
–
Copy (1472) of
St. Isidore's TO map of the world.
66.
Crystallography
–
Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. The word crystallography derives from the Greek words crystallon cold drop, frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein to write. In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography, X-ray crystallography is used to determine the structure of large biomolecules such as proteins. Before the development of X-ray diffraction crystallography, the study of crystals was based on measurements of their geometry. This involved measuring the angles of crystal faces relative to other and to theoretical reference axes. This physical measurement is carried out using a goniometer, the position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to face is plotted on the net. Each point is labelled with its Miller index, the final plot allows the symmetry of the crystal to be established. Crystallographic methods now depend on analysis of the patterns of a sample targeted by a beam of some type. X-rays are most commonly used, other beams used include electrons or neutrons and this is facilitated by the wave properties of the particles. Crystallographers often explicitly state the type of beam used, as in the terms X-ray crystallography and these three types of radiation interact with the specimen in different ways. X-rays interact with the distribution of electrons in the sample. Electrons are charged particles and therefore interact with the charge distribution of both the atomic nuclei and the electrons of the sample. Neutrons are scattered by the atomic nuclei through the nuclear forces, but in addition. They are therefore also scattered by magnetic fields, when neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute deuterium for hydrogen, because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies. An image of an object is made using a lens to focus the beam. However, the wavelength of light is three orders of magnitude longer than the length of typical atomic bonds and atoms themselves
Crystallography
–
A crystalline solid: atomic resolution image of
strontium titanate. Brighter atoms are
strontium and darker ones are titanium.
67.
Game development
–
Video game development is the process of creating a video game. Development is undertaken by a developer, which may range from one person to a large business. Traditional commercial PC and console games are funded by a publisher. Indie games can take time and can be produced cheaply by individuals. The indie game industry has seen a rise in recent years with the growth of new distribution systems. The first video games were developed in the 1960s, but required mainframe computers and were not available to the general public, commercial game development began in the 1970s with the advent of first-generation video game consoles and home computers. Due to low costs and low capabilities of computers, a programmer could develop a full game. The average cost of producing a video game rose from US$1–4 million in 2000 to over $5 million in 2006. Mainstream PC and console games are developed in phases. First, in pre-production, pitches, prototypes, and game design documents are written, if the idea is approved and the developer receives funding, a full-scale development begins. This usually involves a team of 20–100 individuals with various responsibilities, including designers, artists, programmers, Game development is the software development process by which a video game is produced. Games are developed as an outlet and to generate profit. Development is normally funded by a publisher, well-made games bring profit more readily. However, it is important to estimate a games financial requirements, failing to provide clear implications of games expectations may result in exceeding allocated budget. In fact, the majority of games do not produce profit. Most developers cannot afford changing development schedule and require estimating their capabilities with available resources before production, the game industry requires innovations, as publishers cannot profit from constant release of repetitive sequels and imitations. Every year new independent development companies open and some manage to develop hit titles, similarly, many developers close down because they cannot find a publishing contract or their production is not profitable. It is difficult to start a new due to high initial investment required
Game development
–
The XGS PIC 16-Bit game development board, a game development tool similar to those used in the 1990s.
68.
Circumscribed circle
–
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
Circumscribed circle
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
69.
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.
70.
Skinny triangle
–
A skinny triangle in trigonometry is a triangle whose height is much greater than its base. The solution of triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to the angle in radians. The solution is simple for skinny triangles that are also isosceles or right triangles. The skinny triangle finds uses in surveying, astronomy and shooting, the proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Applying the small angle approximations to the law of sines above results in and this result is equivalent to assuming that the length of the base of the triangle is equal to the length of the arc of circle of radius r subtended by angle θ. This approximation becomes more accurate for smaller and smaller θ. The error is 10% or less for angles less than about 43°, the error of this approximation is less than 10% for angles 31° or less. Applications of the skinny triangle occur in any situation where the distance to a far object is to be determined and this can occur in surveying, astronomy, and also has military applications. The skinny triangle is used in astronomy to measure the distance to solar system objects. The base of the triangle is formed by the distance between two measuring stations and the angle θ is the angle formed by the object as seen by the two stations. This baseline is usually very long for best accuracy, in principle the stations could be on opposite sides of the Earth. However, this distance is short compared to the distance to the object being measured. The alternative method of measuring the angles is theoretically possible. The base angles are very nearly right angles and would need to be measured with much greater precision than the angle in order to get the same accuracy. The same method of measuring angles and applying the skinny triangle can be used to measure the distances to stars. In the case of stars however, a longer baseline than the diameter of the Earth is usually required, instead of using two stations on the baseline, two measurements are made from the same station at different times of year. During the intervening period, the orbit of the Earth around the Sun moves the station a great distance. This baseline can be as long as the axis of the Earths orbit or, equivalently
Skinny triangle
–
Fig.2 Length of
arc l approaches length of
chord b as angle θ decreases
71.
Small-angle approximation
–
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
72.
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).
73.
Michiel Hazewinkel
–
Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam. After graduation Hazewinkel started his career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, and Gerard van der Hoek. From 1973 to 1975 he was also Professor at the Universitaire Instelling Antwerpen, were Marcel van de Vel was his PhD student. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra, Analysis, in 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited books, and numerous articles. With Michel Demazure and Pierre Gabriel, on invariants, canonical forms and moduli for linear, constant, finite dimensional, dynamical systems. Moduli and canonical forms for linear dynamical systems II, The topological case, on Lie algebras and finite dimensional filtering. Stochastics, a journal of probability and stochastic processes 7. 1–2. Nonexistence of finite-dimensional filters for conditional statistics of the sensor problem. Systems & control letters 3.6, 331–340, the algebra of quasi-symmetric functions is free over the integers
Michiel Hazewinkel
–
Michiel Hazewinkel, 1987
74.
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
–
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
–
Clark University
Clark University
–
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.
75.
Linear algebra
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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, the set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns, such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics, for instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces, combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Because linear algebra is such a theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramers Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, the study of matrix algebra first emerged in England in the mid-1800s. In 1844 Hermann Grassmann published his Theory of Extension which included foundational new topics of what is called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb, while studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses. Crucially, Cayley used a letter to denote a matrix. In 1882, Hüseyin Tevfik Pasha wrote the book titled Linear Algebra, the first modern and more precise definition of a vector space was introduced by Peano in 1888, by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its form in the first half of the twentieth century. The use of matrices in quantum mechanics, special relativity, the origin of many of these ideas is discussed in the articles on determinants and Gaussian elimination. Linear algebra first appeared in American graduate textbooks in the 1940s, following work by the School Mathematics Study Group, U. S. high schools asked 12th grade students to do matrix algebra, formerly reserved for college in the 1960s. In France during the 1960s, educators attempted to teach linear algebra through finite-dimensional vector spaces in the first year of secondary school and this was met with a backlash in the 1980s that removed linear algebra from the curriculum. To better suit 21st century applications, such as mining and uncertainty analysis
Linear algebra
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The three-dimensional
Euclidean space R 3 is a vector space, and lines and planes passing through the
origin are vector subspaces in R 3.
76.
Arithmetic
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Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
Arithmetic
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Arithmetic tables for children, Lausanne, 1835
Arithmetic
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A scale calibrated in imperial units with an associated cost display.
77.
Category theory
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Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows. A category has two properties, the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, several terms used in category theory, including the term morphism, are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself, Category theory has practical applications in programming language theory, in particular for the study of monads in functional programming. Categories represent abstraction of other mathematical concepts, many areas of mathematics can be formalised by category theory as categories. Hence category theory uses abstraction to make it possible to state and prove many intricate, a basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it. The arrows of category theory are said to represent a process connecting two objects, or in many cases a structure-preserving transformation connecting two objects. There are, however, many applications where more abstract concepts are represented by objects. The most important property of the arrows is that they can be composed, in other words, linear algebra can also be expressed in terms of categories of matrices. A systematic study of category theory allows us to prove general results about any of these types of mathematical structures from the axioms of a category. The class Grp of groups consists of all objects having a group structure, one can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proven from the axioms that the identity element of a group is unique, in the case of groups, the morphisms are the group homomorphisms. The study of group homomorphisms then provides a tool for studying properties of groups. Not all categories arise as structure preserving functions, however, the example is the category of homotopies between pointed topological spaces. If one axiomatizes relations instead of functions, one obtains the theory of allegories, a category is itself a type of mathematical structure, so we can look for processes which preserve this structure in some sense, such a process is called a functor. Diagram chasing is a method of arguing with abstract arrows joined in diagrams. Functors are represented by arrows between categories, subject to specific defining commutativity conditions, functors can define categorical diagrams and sequences
Category theory
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Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ∘ f. (The category's three identity morphisms 1 X, 1 Y and 1 Z, if explicitly represented, would appear as three arrows, next to the letters X, Y, and Z, respectively, each having as its "shaft" a circular arc measuring almost 360 degrees.)
78.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
Group theory
–
Water molecule with symmetry axis
Group theory
–
The popular puzzle
Rubik's cube invented in 1974 by
Ernő Rubik has been used as an illustration of
permutation groups.
79.
Differential equation
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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.
80.
Game theory
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Game theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory is used in economics, political science, and psychology, as well as logic, computer science. Originally, it addressed zero-sum games, in one persons gains result in losses for the other participants. Today, game theory applies to a range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals. Modern game theory began with the idea regarding the existence of equilibria in two-person zero-sum games. Von Neumanns original proof used Brouwer fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this provided an axiomatic theory of expected utility. This theory was developed extensively in the 1950s by many scholars, Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2014, John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Early discussions of examples of two-person games occurred long before the rise of modern, the first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, and uncle to James Waldegrave, a British diplomat, in 1713. In this letter, Waldegrave provides a mixed strategy solution to a two-person version of the card game le Her. James Madison made what we now recognize as an analysis of the ways states can be expected to behave under different systems of taxation. In 1913 Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels and it proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems, the Danish mathematician Zeuthen proved that the mathematical model had a winning strategy by using Brouwers fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false. Game theory did not really exist as a field until John von Neumann published a paper in 1928. Von Neumanns original proof used Brouwers fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern
Game theory
–
An extensive form game
81.
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
–
This
Togliatti surface is an
algebraic surface of degree five. The picture represents a portion of its real
locus.
82.
Analytic geometry
–
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
–
Cartesian coordinates
83.
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
–
A triangle immersed in a saddle-shape plane (a
hyperbolic paraboloid), as well as two diverging
ultraparallel lines.
84.
Finite geometry
–
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
–
Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are "parallel".
85.
History of mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind Mathematical Papyrus, All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period
History of mathematics
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A proof from
Euclid 's
Elements, widely considered the most influential textbook of all time.
History of mathematics
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The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
History of mathematics
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Image of Problem 14 from the
Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
History of mathematics
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One of the oldest surviving fragments of Euclid's Elements, found at
Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
86.
Mathematical physics
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Mathematical physics refers to development of mathematical methods for application to problems in physics. It is a branch of applied mathematics, but deals with physical problems, there are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics, both formulations are embodied in analytical mechanics. These approaches and ideas can be and, in fact, have extended to other areas of physics as statistical mechanics, continuum mechanics, classical field theory. Moreover, they have provided several examples and basic ideas in differential geometry, the theory of partial differential equations are perhaps most closely associated with mathematical physics. These were developed intensively from the half of the eighteenth century until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics. The theory of atomic spectra developed almost concurrently with the fields of linear algebra. Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic, Quantum information theory is another subspecialty. The special and general theories of relativity require a different type of mathematics. This was group theory, which played an important role in quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the description of cosmological as well as quantum field theory phenomena. In this area both homological algebra and category theory are important nowadays, statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics and it is related with the more mathematical ergodic theory. There are increasing interactions between combinatorics and physics, in statistical physics. The usage of the mathematical physics is sometimes idiosyncratic. Certain parts of mathematics that arose from the development of physics are not, in fact, considered parts of mathematical physics. The term mathematical physics is sometimes used to research aimed at studying and solving problems inspired by physics or thought experiments within a mathematically rigorous framework
Mathematical physics
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An example of mathematical physics: solutions of
Schrödinger's equation for
quantum harmonic oscillators (left) with their
amplitudes (right).
87.
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.
88.
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
89.
Probability theory
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events, two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, a great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory and this culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of space, introduced by Richard von Mises. This became the mostly undisputed axiomatic basis for modern probability theory, most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment. The power set of the space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results, one collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the set of the sample space of die rolls. In this case, is the event that the die falls on some odd number, If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one, the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6 and this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, discrete probability theory deals with events that occur in countable sample spaces. Modern definition, The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by Ω
Probability theory
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The
normal distribution, a continuous probability distribution.
Probability theory
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The
Poisson distribution, a discrete probability distribution.
90.
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.
91.
Statistics
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
Statistics
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Scatter plots are used in descriptive statistics to show the observed relationships between different variables.
Statistics
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More
probability density is found as one gets closer to the expected (mean) value in a
normal distribution. Statistics used in
standardized testing assessment are shown. The scales include
standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines, and percentages in standard nines.
Statistics
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Gerolamo Cardano, the earliest pioneer on the mathematics of probability.
Statistics
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Karl Pearson, a founder of mathematical statistics.