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
–
Trigonometry
Trigonometric function
Trigonometric function
2.
Outline of trigonometry
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Trigonometry is a branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, geometry – mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is used extensively in trigonometry, angle – 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. D, ebook version, in PDF format, full text presented. Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company,1914, Trigonometry FAQ Trigonometry on Mathwords. com index of trigonometry entries on Mathwords. com Trigonometry on PlainMath. net Trigonometry Articles from PlainMath. Net
Outline of trigonometry
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∠, the angle symbol in
Unicode is
U+2220
3.
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.
4.
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
–
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.
5.
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
–
Trigonometric functions in the complex plane
Trigonometric functions
–
Trigonometry
Trigonometric functions
Trigonometric functions
6.
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
–
Inverse trigonometric functions in the
complex plane
Inverse trigonometric functions
–
Trigonometry
Inverse trigonometric functions
Inverse trigonometric functions
7.
List of 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
List of trigonometric identities
–
Cosines and sines around the
unit circle
8.
Exact trigonometric constants
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Exact algebraic expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. When they are, they are more specifically in terms of square roots. For an angle of a number of degrees, which is not a multiple 3°, the values of sine, cosine. Note that 1° = π/180 radians, according to Nivens theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2,1, −1/2, and −1. According to Bakers theorem, if the value of a sine and that is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values. The list in this article is incomplete in several senses, first, the trigonometric functions of all angles that are integer multiples of those given can also be expressed in radicals, but some are omitted here. Second, it is possible to apply the half-angle formula to find an expression in radicals for a trigonometric function of one-half of any angle on the list, then half of that angle. This article only gives the cases based on the Fermat primes 3 and 5, thus for example cos, given in the article 17-gon, is not given here. Fourth, this article deals with trigonometric function values when the expression in radicals is in real radicals—roots of real numbers. Many other trigonometric function values are expressible in, for example, in practice, all values of sines, cosines, and tangents not found in this article are approximated using the techniques described at Generating trigonometric tables. Several different units of measure are widely used, including degrees, radians. The following table shows the conversions and values for some common angles, Values outside the range are trivially derived from these values. This is because the sum of the angles of any n-gon is 180° ×, using cos 36 ∘ =5 +14, tan 36 ∘ =5 −25, this can be simplified to, V = a 34. The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructibility of right triangles, here right triangles made from symmetry sections of regular polygons are used to calculate fundamental trigonometric ratios. Each right triangle represents three points in a polygon, a vertex, an edge center containing that vertex. 2 sin θ =2 −2 cos 2 θ =2 −2 +2 cos 4 θ =2 −2 +2 +2 cos 8 θ and so on. If M =2 and N =2 then cos π17 = M −4 +28, crd is the chord function, crd θ =2 sin θ2. Thus sin 18 ∘ =11 +5 =5 −14, similarly crd 108 ∘ = crd = b a =1 +52, so sin 54 ∘ = cos 36 ∘ =1 +54
Exact trigonometric constants
–
The primary solution angles [
clarification needed] on the
unit circle are at multiples of 30 and 45 degrees.
9.
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
10.
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
–
Illustration of a unit circle. The variable t is an
angle measure.
11.
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
–
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.)
12.
Law of cosines
–
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. Though the notion of the cosine was not yet developed in his time, Euclids Elements, dating back to the 3rd century BC, the cases of obtuse triangles and acute triangles are treated separately, in Propositions 12 and 13 of Book 2. Using notation as in Fig.2, Euclids statement can be represented by the formula A B2 = C A2 + C B2 +2 and this formula may be transformed into the law of cosines by noting that CH = cos = − cos γ. Proposition 13 contains an analogous statement for acute triangles. In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi, the theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form, the theorem is used in triangulation, for solving a triangle or circle, i. e. These formulas produce high round-off errors in floating point calculations if the triangle is very acute and it is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for a the quadratic equation a2 − 2ab cos γ + b2 − c2 =0 and this equation can have 2,1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if b sin γ < c < b, only one positive solution if c = b sin γ and these different cases are also explained by the side-side-angle congruence ambiguity. Consider a triangle with sides of length a, b, c and this triangle can be placed on the Cartesian coordinate system by plotting the following points, as shown in Fig.4, A =, B =, and C =. By the distance formula, we have c =2 +2, an advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. right vs. obtuse. Drop the perpendicular onto the c to get c = a cos β + b cos α. Multiply through by c to get c 2 = a c cos β + b c cos α. By considering the other perpendiculars obtain a 2 = a c cos β + a b cos γ, b 2 = b c cos α + a b cos γ. Adding the latter two equations gives a 2 + b 2 = a c cos β + b c cos α +2 a b cos γ and this proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in any right triangle
Law of cosines
–
Figure 1 – A triangle. The angles α (or A), β (or B), and γ (or C) are respectively opposite the sides a, b, and c.
13.
Law of tangents
–
In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of tangents states that a − b a + b = tan tan , to prove the law of tangents we can start with the law of sines, a sin α = b sin β. Let d = a sin α, d = b sin β so that a = d sin α and b = d sin β. As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity tan = sin α ± sin β cos α + cos β. The law of tangents can be used to compute the missing side, on a sphere of unit radius, the sides of the triangle are arcs of great circles. Accordingly their lengths can be expressed in radians or any other units of angular measure, let A, B, C be the angles at the three vertices of the triangle and let a, b, c be the respective lengths of the opposite sides. The spherical law of tangents says tan tan = tan tan , Law of sines Law of cosines Law of cotangents Mollweides formula Half-side formula Tangent half-angle formula
Law of tangents
–
Figure 1 – A triangle. The angles α, β, and γ are respectively opposite the sides a, b, and c.
14.
Law of cotangents
–
In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. In the upper figure, the points of tangency of the incircle with the sides of the break the perimeter into 6 segments. In each pair the segments are of equal length, for example, the 2 segments adjacent to vertex A are equal. If we pick one segment from each pair, their sum will be the semiperimeter s, an example of this is the segments shown in color in the figure. The two segments making up the red line add up to a, so the blue segment must be of length s − a. Obviously, the five segments must also have lengths s − a, s − b, or s − c. By inspection of the figure, using the definition of the cotangent function, we have cot = s − a r, a number of other results can be derived from the law of cotangents. Note that the area of triangle ABC is also divided into 6 smaller triangles, also in 3 pairs, for example, the two triangles near vertex A, being right triangles of width s − a and height r, each have an area of 1/2r. From the addition formula and the law of cotangents we have sin sin = cot − cot cot + cot = a − b 2 s − a − b. This gives the result a − b c = sin cos as required, here, an extra step is required to transform a product into a sum, according to the sum/product formula. This gives the result b + a c = cos sin as required, the law of tangents can also be derived from this. Law of sines Law of cosines Law of tangents Mollweides formula Formula sheet database – law of cotangents, silvester, John R. Geometry, Ancient and Modern
Law of cotangents
–
A triangle, showing the "incircle" and the partitioning of the sides. The angle bisectors meet at the incenter, which is the center of the
incircle.
15.
Pythagorean theorem
–
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
–
The
Plimpton 322 tablet records Pythagorean triples from Babylonian times.
Pythagorean theorem
–
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
–
Geometric proof of the Pythagorean theorem from the
Zhou Bi Suan Jing.
Pythagorean theorem
–
Exhibit on the Pythagorean theorem at the
Universum museum in Mexico City
16.
Calculus
–
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
Calculus
–
Isaac Newton developed the use of calculus in his
laws of motion and
gravitation.
Calculus
–
Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
–
Maria Gaetana Agnesi
Calculus
–
The
logarithmic spiral of the
Nautilus shell is a classical image used to depict the growth and change related to calculus
17.
List of integrals of trigonometric functions
–
The following is a list of integrals of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions, for a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral
List of integrals of trigonometric functions
–
Trigonometry
18.
Differentiation of trigonometric functions
–
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
19.
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
20.
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
21.
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.
22.
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
23.
Hellenistic period
–
It is often considered a period of transition, sometimes even of decadence or degeneration, compared to the enlightenment of the Greek Classical era. The Hellenistic period saw the rise of New Comedy, Alexandrian poetry, the Septuagint, Greek science was advanced by the works of the mathematician Euclid and the polymath Archimedes. The religious sphere expanded to include new gods such as the Greco-Egyptian Serapis, eastern deities such as Attis and Cybele, the Hellenistic period was characterized by a new wave of Greek colonization which established Greek cities and kingdoms in Asia and Africa. This resulted in the export of Greek culture and language to new realms. Equally, however, these new kingdoms were influenced by the cultures, adopting local practices where beneficial, necessary. Hellenistic culture thus represents a fusion of the Ancient Greek world with that of the Near East, Middle East and this mixture gave rise to a common Attic-based Greek dialect, known as Koine Greek, which became the lingua franca through the Hellenistic world. Scholars and historians are divided as to what event signals the end of the Hellenistic era, Hellenistic is distinguished from Hellenic in that the first encompasses the entire sphere of direct ancient Greek influence, while the latter refers to Greece itself. The word originated from the German term hellenistisch, from Ancient Greek Ἑλληνιστής, from Ἑλλάς, Hellenistic is a modern word and a 19th-century concept, the idea of a Hellenistic period did not exist in Ancient Greece. Although words related in form or meaning, e. g, the major issue with the term Hellenistic lies in its convenience, as the spread of Greek culture was not the generalized phenomenon that the term implies. Some areas of the world were more affected by Greek influences than others. The Greek population and the population did not always mix, the Greeks moved and brought their own culture. While a few fragments exist, there is no surviving historical work which dates to the hundred years following Alexanders death. The works of the major Hellenistic historians Hieronymus of Cardia, Duris of Samos, the earliest and most credible surviving source for the Hellenistic period is Polybius of Megalopolis, a statesman of the Achaean League until 168 BC when he was forced to go to Rome as a hostage. His Histories eventually grew to a length of forty books, covering the years 220 to 167 BC, another important source, Plutarchs Parallel Lives though more preoccupied with issues of personal character and morality, outlines the history of important Hellenistic figures. Appian of Alexandria wrote a history of the Roman empire that includes information of some Hellenistic kingdoms, other sources include Justins epitome of Pompeius Trogus Historiae Philipicae and a summary of Arrians Events after Alexander, by Photios I of Constantinople. Lesser supplementary sources include Curtius Rufus, Pausanias, Pliny, in the field of philosophy, Diogenes Laertius Lives and Opinions of Eminent Philosophers is the main source. Ancient Greece had traditionally been a collection of fiercely independent city-states. After the Peloponnesian War, Greece had fallen under a Spartan hegemony, in which Sparta was pre-eminent but not all-powerful
Hellenistic period
–
The
Nike of Samothrace is considered one of the greatest masterpieces of
Hellenistic art.
Hellenistic period
–
Alexander fighting the Persian king
Darius III. From the
Alexander Mosaic,
Naples National Archaeological Museum.
Hellenistic period
–
Alexander's empire at the time of its maximum expansion.
24.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
Geometry
–
Visual checking of the
Pythagorean theorem for the (3, 4, 5)
triangle as in the
Chou Pei Suan Ching 500–200 BC.
Geometry
–
An illustration of
Desargues' theorem, an important result in
Euclidean and
projective geometry
Geometry
–
Geometry lessons in the 20th century
Geometry
–
A
European and an
Arab practicing geometry in the 15th century.
25.
Astronomy
–
Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
Astronomy
–
A
star -forming region in the
Large Magellanic Cloud, an
irregular galaxy.
Astronomy
–
A giant
Hubble mosaic of the
Crab Nebula, a
supernova remnant
Astronomy
–
19th century
Sydney Observatory,
Australia (1873)
Astronomy
–
19th century
Quito Astronomical Observatory is located 12 minutes south of the
Equator in
Quito,
Ecuador.
26.
Astronomers
–
An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
Astronomers
–
The Astronomer by
Johannes Vermeer
Astronomers
–
Galileo is often referred to as the Father of
modern astronomy
Astronomers
–
Guy Consolmagno (Vatikan observatory), analyzing a meteorite, 2014
Astronomers
–
Emily Lakdawalla at the Planetary Conference 2013
27.
Pure mathematics
–
Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in there is much overlap in the activity of pure. To develop accurate models for describing the world, many applied mathematicians draw on tools. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research, ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between arithmetic, now called number theory, and logistic, now called arithmetic. Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, the term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, in the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, in fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved, Pure mathematician became a recognized vocation, achievable through training. One central concept in mathematics is the idea of generality. One can use generality to avoid duplication of effort, proving a general instead of having to prove separate cases independently. Generality can facilitate connections between different branches of mathematics, category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generalitys impact on intuition is both dependent on the subject and a matter of preference or learning style. Often generality is seen as a hindrance to intuition, although it can function as an aid to it. Each of these branches of abstract mathematics have many sub-specialties. A steep rise in abstraction was seen mid 20th century, in practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, the point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central
Pure mathematics
–
An illustration of the
Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.
28.
Applied mathematics
–
Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of science and specialized knowledge. The term applied mathematics also describes the professional specialty in which work on practical problems by formulating and studying mathematical models. The activity of applied mathematics is thus connected with research in pure mathematics. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory, quantitative finance is now taught in mathematics departments across universities and mathematical finance is considered a full branch of applied mathematics. Engineering and computer science departments have made use of applied mathematics. Today, the applied mathematics is used in a broader sense. It includes the areas noted above as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of mathematics are now important in applications. There is no consensus as to what the various branches of applied mathematics are, such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. Many mathematicians distinguish between applied mathematics, which is concerned with methods, and the applications of mathematics within science. Mathematicians such as Poincaré and Arnold deny the existence of applied mathematics, similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to industrial problems is also called industrial mathematics. Historically, mathematics was most important in the sciences and engineering. Academic institutions are not consistent in the way they group and label courses, programs, at some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and Mathematics. It is very common for Statistics departments to be separated at schools with graduate programs, many applied mathematics programs consist of primarily cross-listed courses and jointly appointed faculty in departments representing applications. Some Ph. D. programs in applied mathematics require little or no coursework outside of mathematics, in some respects this difference reflects the distinction between application of mathematics and applied mathematics. Research universities dividing their mathematics department into pure and applied sections include MIT, brigham Young University also has an Applied and Computational Emphasis, a program that allows student to graduate with a Mathematics degree, with an emphasis in Applied Math
Applied mathematics
–
Efficient solutions to the
vehicle routing problem require tools from
combinatorial optimization and
integer programming.
29.
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
30.
Wave equation
–
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in physics—such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics, historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The wave equation is a partial differential equation. It typically concerns a time t, one or more spatial variables x1, x2, …, xn, and a scalar function u = u, whose values could model, for example. The wave equation for u is ∂2 u ∂ t 2 = c 2 ∇2 u where ∇2 is the Laplacian, therefore, the sum of any two solutions is again a solution, in physics this property is called the superposition principle. The wave equation, and modifications of it, are found in elasticity, quantum mechanics, plasma physics. The wave equation in one dimension can be written as follows. This equation is described as having only one space dimension x. The wave equation in one dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string that is vibrating in a two-dimensional plane, another physical setting for derivation of the wave equation in one space dimension utilizes Hookes Law. The wave equation in the case can be derived from Hookes Law in the following way. In the case of a stress pulse propagating through a beam the beam acts much like a number of springs in series. A beam of constant cross section made from an elastic material has a stiffness K given by K = E A L Where A is the cross sectional area. Traveling means that the shape of these individual arbitrary functions with respect to x stays constant, however the functions are translated left and this was derived by Jean le Rond dAlembert. However, the waveforms F and G may also be generalized functions, in that case, the solution may be interpreted as an impulse that travels to the right or the left. The basic wave equation is a differential equation and so it will adhere to the superposition principle. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually
Wave equation
–
French scientist
Jean-Baptiste le Rond d'Alembert (b. 1717) discovered the wave equation in one space dimension.
Wave equation
Wave equation
–
Swiss Mathematician and Physicist
Leonhard Euler (b. 1707) discovered the wave equation in three space dimensions.
31.
Periodic function
–
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, any function which is not periodic is called aperiodic. A function f is said to be periodic with period P if we have f = f for all values of x in the domain, geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P and this definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane. A function that is not periodic is called aperiodic, for example, the sine function is periodic with period 2 π, since sin = sin x for all values of x. This function repeats on intervals of length 2 π, everyday examples are seen when the variable is time, for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position of the system are expressible as periodic functions, for a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a function is the function f that gives the fractional part of its argument. In particular, f = f = f =, =0.5 The graph of the function f is the sawtooth wave. The trigonometric functions sine and cosine are periodic functions, with period 2π. The subject of Fourier series investigates the idea that a periodic function is a sum of trigonometric functions with matching periods. According to the definition above, some functions, for example the Dirichlet function, are also periodic, in the case of Dirichlet function. For example, f = sin has period 2 π therefore sin will have period 2 π5, a function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions, if L is the period of the function then, L =2 π / k One common generalization of periodic functions is that of antiperiodic functions. This is a function f such that f = −f for all x, for example, the sine or cosine function is π-antiperiodic and 2π-periodic. A further generalization appears in the context of Bloch waves and Floquet theory, in this context, the solution is typically a function of the form, f = e i k P f where k is a real or complex number. Functions of this form are sometimes called Bloch-periodic in this context, a periodic function is the special case k =0, and an antiperiodic function is the special case k = π/P
Periodic function
32.
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
33.
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
34.
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
35.
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
36.
Right angle
–
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two adjacent parts of a straight line. More precisely, if a ray is placed so that its endpoint is on a line, as a rotation, a right angle corresponds to a quarter turn. The presence of an angle in a triangle is the defining factor for right triangles. The term is a calque of Latin angulus rectus, here rectus means upright, in Unicode, the symbol for a right angle is U+221F ∟ Right angle. It should not be confused with the similarly shaped symbol U+231E ⌞ Bottom left corner, related symbols are U+22BE ⊾ Right angle with arc, U+299C ⦜ Right angle variant with square, and U+299D ⦝ Measured right angle with dot. The symbol for an angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland. Right angles are fundamental in Euclids Elements and they are defined in Book 1, definition 10, which also defines perpendicular lines. Euclid uses right angles in definitions 11 and 12 to define acute angles, two angles are called complementary if their sum is a right angle. Book 1 Postulate 4 states that all angles are equal. Euclids commentator Proclus gave a proof of this using the previous postulates. Saccheri gave a proof as well but using a more explicit assumption, in Hilberts axiomatization of geometry this statement is given as a theorem, but only after much groundwork. A right angle may be expressed in different units, 1/4 turn, 90° π/2 radians 100 grad 8 points 6 hours Throughout history carpenters and masons have known a quick way to confirm if an angle is a true right angle. It is based on the most widely known Pythagorean triple and so called the Rule of 3-4-5 and this measurement can be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem, Thales theorem states that an angle inscribed in a semicircle is a right angle. Two application examples in which the angle and the Thales theorem are included. Cartesian coordinate system Orthogonality Perpendicular Rectangle Types of angles Wentworth, G. A, Euclid, commentary and trans. by T. L. Heath Elements Vol.1 Google Books
Right angle
–
A right angle is equal to 90 degrees.
37.
Spherical trigonometry
–
Spherical trigonometry is of great importance for calculations in astronomy, geodesy and navigation. The origins of spherical trigonometry in Greek mathematics and the developments in Islamic mathematics are discussed fully in History of trigonometry. This book is now available on the web. The only significant developments since then have been the application of methods for the derivation of the theorems. A spherical polygon is a polygon on the surface of the sphere defined by a number of great-circle arcs, such polygons may have any number of sides. Two planes define a lune, also called a digon or bi-angle, the analogue of the triangle. Three planes define a triangle, the principal subject of this article. Four planes define a spherical quadrilateral, such a figure, and higher sided polygons, from this point the article will be restricted to spherical triangles, denoted simply as triangles. Both vertices and angles at the vertices are denoted by the upper case letters A, B and C. The angles of spherical triangles are less than π so that π < A + B + C < 3π. The sides are denoted by letters a, b, c. On the unit sphere their lengths are equal to the radian measure of the angles that the great circle arcs subtend at the centre. The sides of proper spherical triangles are less than π so that 0 < a + b + c < 3π, the radius of the sphere is taken as unity. For specific practical problems on a sphere of radius R the measured lengths of the sides must be divided by R before using the identities given below, likewise, after a calculation on the unit sphere the sides a, b, c must be multiplied by R. The polar triangle associated with a triangle ABC is defined as follows, consider the great circle that contains the side BC. This great circle is defined by the intersection of a plane with the surface. The points B and C are defined similarly, the triangle ABC is the polar triangle corresponding to triangle ABC. Therefore, if any identity is proved for the triangle ABC then we can derive a second identity by applying the first identity to the polar triangle by making the above substitutions
Spherical trigonometry
–
Eight spherical triangles defined by the intersection of three great circles.
38.
Sphere
–
A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
Sphere
–
Circumscribed cylinder to a sphere
Sphere
–
A two-dimensional
perspective projection of a sphere
Sphere
Sphere
–
Deck of playing cards illustrating engineering instruments, England, 1702.
King of spades: Spheres
39.
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
40.
Elliptic geometry
–
Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a line must intersect. In fact, the perpendiculars on one side all intersect at the pole of the given line. There are no points in elliptic geometry. Every point corresponds to a polar line of which it is the absolute pole. Any point on this line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant, the distance between a pair of points is proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter The name elliptic is possibly misleading and it does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes, analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity. A simple way to picture elliptic geometry is to look at a globe, neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, with this identification of antipodal points, the model satisfies Euclids first postulate, which states that two points uniquely determine a line. Metaphorically, we can imagine geometers who are like living on the surface of a sphere. Even if the ants are unable to move off the surface, they can still construct lines, the existence of a third dimension is irrelevant to the ants ability to do geometry, and its existence is neither verifiable nor necessary from their point of view. Another way of putting this is that the language of the axioms is incapable of expressing the distinction between one model and another. In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the figures are similar, i. e. they have the same angles. In elliptic geometry this is not the case, for example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere. A line segment therefore cannot be scaled up indefinitely, a geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space
Elliptic geometry
–
On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Elliptic geometry
–
Projecting a
sphere to a
plane.
41.
Navigation
–
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another. The field of navigation includes four categories, land navigation, marine navigation, aeronautic navigation. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks, all navigational techniques involve locating the navigators position compared to known locations or patterns. Navigation, in a sense, can refer to any skill or study that involves the determination of position and direction. In this sense, navigation includes orienteering and pedestrian navigation, for information about different navigation strategies that people use, visit human navigation. In the European medieval period, navigation was considered part of the set of seven mechanical arts, early Pacific Polynesians used the motion of stars, weather, the position of certain wildlife species, or the size of waves to find the path from one island to another. Maritime navigation using scientific instruments such as the mariners astrolabe first occurred in the Mediterranean during the Middle Ages, the perfecting of this navigation instrument is attributed to Portuguese navigators during early Portuguese discoveries in the Age of Discovery. Open-seas navigation using the astrolabe and the compass started during the Age of Discovery in the 15th century, the Portuguese began systematically exploring the Atlantic coast of Africa from 1418, under the sponsorship of Prince Henry. In 1488 Bartolomeu Dias reached the Indian Ocean by this route, in 1492 the Spanish monarchs funded Christopher Columbuss expedition to sail west to reach the Indies by crossing the Atlantic, which resulted in the Discovery of America. In 1498, a Portuguese expedition commanded by Vasco da Gama reached India by sailing around Africa, soon, the Portuguese sailed further eastward, to the Spice Islands in 1512, landing in China one year later. The fleet of seven ships sailed from Sanlúcar de Barrameda in Southern Spain in 1519, crossed the Atlantic Ocean, some ships were lost, but the remaining fleet continued across the Pacific making a number of discoveries including Guam and the Philippines. By then, only two galleons were left from the original seven, the Victoria led by Elcano sailed across the Indian Ocean and north along the coast of Africa, to finally arrive in Spain in 1522, three years after its departure. The Trinidad sailed east from the Philippines, trying to find a path back to the Americas. He arrived in Acapulco on October 8,1565, the term stems from 1530s, from Latin navigationem, from navigatus, pp. of navigare to sail, sail over, go by sea, steer a ship, from navis ship and the root of agere to drive. Roughly, the latitude of a place on Earth is its angular distance north or south of the equator, latitude is usually expressed in degrees ranging from 0° at the Equator to 90° at the North and South poles. The height of Polaris in degrees above the horizon is the latitude of the observer, similar to latitude, the longitude of a place on Earth is the angular distance east or west of the prime meridian or Greenwich meridian. Longitude is usually expressed in degrees ranging from 0° at the Greenwich meridian to 180° east and west, sydney, for example, has a longitude of about 151° east. New York City has a longitude of 74° west, for most of history, mariners struggled to determine longitude
Navigation
–
Table of geography, hydrography, and navigation, from the 1728
Cyclopaedia
Navigation
–
Dead reckoning or DR, in which one advances a prior position using the ship's course and speed. The new position is called a DR position. It is generally accepted that only course and speed determine the DR position. Correcting the DR position for
leeway, current effects, and steering error result in an estimated position or EP. An
inertial navigator develops an extremely accurate EP.
Navigation
–
Pilotage involves navigating in restricted waters with frequent determination of position relative to geographic and hydrographic features.
Navigation
–
Celestial navigation involves reducing celestial measurements to lines of position using tables,
spherical trigonometry, and
almanacs.
42.
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
43.
Hipparchus
–
Hipparchus of Nicaea was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry but is most famous for his discovery of precession of the equinoxes. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes and he is known to have been a working astronomer at least from 162 to 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some and he was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Babylonians. He developed trigonometry and constructed trigonometric tables, and he solved problems of spherical trigonometry. With his solar and lunar theories and his trigonometry, he may have been the first to develop a method to predict solar eclipses. Relatively little of Hipparchuss direct work survives into modern times, although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus was preserved by later copyists. There is a tradition that Hipparchus was born in Nicaea, in the ancient district of Bithynia. His birth date was calculated by Delambre based on clues in his work, Hipparchus must have lived some time after 127 BC because he analyzed and published his observations from that year. Hipparchus obtained information from Alexandria as well as Babylon, but it is not known when or if he visited these places and he is believed to have died on the island of Rhodes, where he seems to have spent most of his later life. It is not known what Hipparchuss economic means were nor how he supported his scientific activities and his appearance is likewise unknown, there are no contemporary portraits. In the 2nd and 3rd centuries coins were made in his honour in Bithynia that bear his name and show him with a globe, this supports the tradition that he was born there. As an astronomer of antiquity his influence, supported by ideas from Aristotle, held sway for nearly 2000 years, Hipparchuss only preserved work is Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις. This is a critical commentary in the form of two books on a popular poem by Aratus based on the work by Eudoxus. Hipparchus also made a list of his works, which apparently mentioned about fourteen books. His famous star catalog was incorporated into the one by Ptolemy, the first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as the father of trigonometry. There are a variety of mis-steps in the more ambitious 2005 paper, According to one book review, both of these claims have been rejected by other scholars
Hipparchus
–
Hipparchus as he appears in "
The School of Athens " by
Raphael.
44.
Indus Valley Civilization
–
The Indus Valley Civilisation was a Bronze Age civilisation mainly in the northwestern regions of South Asia, extending from what today is northeast Afghanistan to Pakistan and northwest India. Along with ancient Egypt and Mesopotamia it was one of three early civilisations of the Old World, and of the three, the most widespread, at its peak, the Indus Civilisation may have had a population of over five million. Inhabitants of the ancient Indus river valley developed new techniques in handicraft, the Indus cities are noted for their urban planning, baked brick houses, elaborate drainage systems, water supply systems, and clusters of large non-residential buildings. The discovery of Harappa, and soon afterwards, Mohenjo-Daro, was the culmination of work beginning in 1861 with the founding of the Archaeological Survey of India in the British Raj, excavation of Harappan sites has been ongoing since 1920, with important breakthroughs occurring as recently as 1999. This Harappan civilisation is called the Mature Harappan culture to distinguish it from the cultures immediately preceding and following it. The early Harappan cultures were preceded by local Neolithic agricultural villages, as of 1999, over 1,056 cities and settlements had been found, of which 96 have been excavated, mainly in the general region of the Indus and Ghaggar-Hakra Rivers and their tributaries. Among the settlements were the urban centres of Harappa, Mohenjo-daro, Dholavira, Ganeriwala in Cholistan. The Harappan language is not directly attested and its affiliation is uncertain since the Indus script is still undeciphered, a relationship with the Dravidian or Elamo-Dravidian language family is favoured by a section of scholars. Recently, Indus sites have been discovered in Pakistans northwestern Frontier Province as well, other IVC colonies can be found in Afghanistan while smaller isolated colonies can be found as far away as Turkmenistan and in Maharashtra. The largest number of colonies are in the Punjab, Sindh, Rajasthan, Haryana, Indus Valley sites have been found most often on rivers, but also on the ancient seacoast, for example, Balakot, and on islands, for example, Dholavira. There is evidence of dry river beds overlapping with the Hakra channel in Pakistan, many Indus Valley sites have been discovered along the Ghaggar-Hakra beds. Among them are, Rupar, Rakhigarhi, Sothi, Kalibangan, Harappan Civilisation remains the correct one, according to the common archaeological usage of naming a civilisation after its first findspot. John wrote, I was much exercised in my mind how we were to get ballast for the line of the railway and they were told of an ancient ruined city near the lines, called Brahminabad. Visiting the city, he found it full of hard well-burnt bricks, and, convinced there was a grand quarry for the ballast I wanted. These bricks now provided ballast along 93 miles of the track running from Karachi to Lahore. In 1872–75, Alexander Cunningham published the first Harappan seal and it was half a century later, in 1912, that more Harappan seals were discovered by J. J. H. MacKay, and Marshall. By 1931, much of Mohenjo-Daro had been excavated, but excavations continued, such as that led by Sir Mortimer Wheeler, director of the Archaeological Survey of India in 1944. Among other archaeologists who worked on IVC sites before the independence in 1947 were Ahmad Hasan Dani, Brij Basi Lal, Nani Gopal Majumdar, and Sir Marc Aurel Stein
Indus Valley Civilization
–
Indus Valley Civilization
Indus Valley Civilization
–
Ceremonial vessel, Harappan, 2600–2450 BCE.
LACMA
Indus Valley Civilization
–
Excavated ruins of
Mohenjo-daro, with the
Great Bath in the front
Indus Valley Civilization
–
Indus Valley pottery, 2500–1900 BCE
45.
Lothal
–
Lothal is one of the most prominent cities of the ancient Indus valley civilisation, located in the Bhāl region of the modern state of Gujarāt and dating from 3700 BCE. Discovered in 1954, Lothal was excavated from 13 February 1955 to 19 May 1960 by the Archaeological Survey of India, the official Indian government agency for the preservation of ancient monuments. It was a vital and thriving trade centre in ancient times, with its trade of beads, gems and valuable ornaments reaching the far corners of West Asia, the techniques and tools they pioneered for bead-making and in metallurgy have stood the test of time for over 4000 years. Lothal is situated near the village of Saragwala in the Dholka Taluka of Ahmedabad district and it is six kilometres south-east of the Lothal-Bhurkhi railway station on the Ahmedabad-Bhavnagar railway line. It is also connected by roads to the cities of Ahmedabad, Bhavnagar, Rajkot. The nearest cities are Dholka and Bagodara, the findings consist of a mound, a township, a marketplace, and the dock. Adjacent to the excavated areas stands the Archaeological Museum, where some of the most prominent collections of Indus-era antiquities in India are displayed, when British India was partitioned in 1947, most Indus sites, including Mohenjo-daro and Harappa, became part of Pakistan. The Archaeological Survey of India undertook a new program of exploration, many sites were discovered across northwestern India. Lothal stands 670 kilometers from Mohenjo-daro, which is in Sindh, the meaning of Lothal in Gujarati to be the mound of the dead is not unusual, as the name of the city of Mohenjo-daro in Sindhi means the same. People in villages neighbouring to Lothal had known of the presence of an ancient town, as recently as 1850, boats could sail up to the mound. In 1942, timber was shipped from Broach to Saragwala via the mound, a silted creek connecting modern Bholad with Lothal and Saragwala represents the ancient flow channel of a river or creek. Speculation suggests that owing to the small dimensions of the main city, Lothal was not a large settlement at all. However, the ASI and other contemporary archaeologists assert that the city was a part of a river system on the trade route of the ancient peoples from Sindh to Saurashtra in Gujarat. Lothal provides with the largest collection of antiquities in the archaeology of modern India and it is essentially a single culture site—the Harappan culture in all its variances is evidenced. An indigenous micaceous Red Ware culture also existed, which is believed to be autochthonous, two sub-periods of Harappan culture are distinguished, the same period is identical to the exuberant culture of Harappa and Mohenjo-daro. After the core of the Indus civilisation had decayed in Mohenjo-daro and Harappa, Lothal seems not only to have survived and its constant threats - tropical storms and floods - caused immense destruction, which destabilised the culture and ultimately caused its end. Topographical analysis also shows signs that at about the time of its demise, thus the cause for the abandonment of the city may have been changes in the climate as well as natural disasters, as suggested by environmental magnetic records. Lothal is based upon a mound that was a salt marsh inundated by tide, small channel widths when compared to the lower reaches suggest the presence of a strong tidal influence upon the city—tidal waters ingressed up to and beyond the city
Lothal
–
Archaeological remains at the lower town of Lothal
Lothal
–
To the northwest of Lothal lies the
Kutch (see also
Dholavira) peninsula, which was a part of the
Arabian Sea until very recently in history. Owing to this, and the proximity of the
Gulf of Khambhat, Lothal's river provided direct access to sea routes. Although now sealed off from the sea, Lothal's topography and geology reflects its maritime past.
Lothal
–
An ancient well, and the city drainage canals
Lothal
–
The bathroom-toilet structure of houses in Lothal
46.
Sumer
–
Living along the valleys of the Tigris and Euphrates, Sumerian farmers were able to grow an abundance of grain and other crops, the surplus of which enabled them to settle in one place. Proto-writing in the dates back to c.3000 BC. The earliest texts come from the cities of Uruk and Jemdet Nasr and date back to 3300 BC, modern historians have suggested that Sumer was first permanently settled between c.5500 and 4000 BC by a West Asian people who spoke the Sumerian language, an agglutinative language isolate. These conjectured, prehistoric people are now called proto-Euphrateans or Ubaidians, some scholars contest the idea of a Proto-Euphratean language or one substrate language. Reliable historical records begin much later, there are none in Sumer of any kind that have dated before Enmebaragesi. Juris Zarins believes the Sumerians lived along the coast of Eastern Arabia, todays Persian Gulf region, Sumerian civilization took form in the Uruk period, continuing into the Jemdet Nasr and Early Dynastic periods. During the 3rd millennium BC, a cultural symbiosis developed between the Sumerians, who spoke a language isolate, and Akkadian-speakers, which included widespread bilingualism. The influence of Sumerian on Akkadian is evident in all areas, from lexical borrowing on a scale, to syntactic, morphological. This has prompted scholars to refer to Sumerian and Akkadian in the 3rd millennium BC as a Sprachbund, Sumer was conquered by the Semitic-speaking kings of the Akkadian Empire around 2270 BC, but Sumerian continued as a sacred language. Native Sumerian rule re-emerged for about a century in the Neo-Sumerian Empire or Third Dynasty of Ur approximately 2100-2000 BC, the term Sumerian is the common name given to the ancient non-Semitic-speaking inhabitants of Mesopotamia, Sumer, by the East Semitic-speaking Akkadians. The Sumerians referred to themselves as ùĝ saĝ gíg ga, phonetically /uŋ saŋ gi ga/, literally meaning the black-headed people, the Akkadian word Shumer may represent the geographical name in dialect, but the phonological development leading to the Akkadian term šumerû is uncertain. Hebrew Shinar, Egyptian Sngr, and Hittite Šanhar, all referring to southern Mesopotamia, in the late 4th millennium BC, Sumer was divided into many independent city-states, which were divided by canals and boundary stones. Each was centered on a dedicated to the particular patron god or goddess of the city. The Sumerian city-states rose to power during the prehistoric Ubaid and Uruk periods, classical Sumer ends with the rise of the Akkadian Empire in the 23rd century BC. Following the Gutian period, there is a brief Sumerian Renaissance in the 21st century BC, the Amorite dynasty of Isin persisted until c.1700 BC, when Mesopotamia was united under Babylonian rule. The Sumerians were eventually absorbed into the Akkadian population, 2500–2334 BC Akkadian Empire period, c. 2218–2047 BC Ur III period, c, 2047–1940 BC The Ubaid period is marked by a distinctive style of fine quality painted pottery which spread throughout Mesopotamia and the Persian Gulf. It appears that this culture was derived from the Samarran culture from northern Mesopotamia and it is not known whether or not these were the actual Sumerians who are identified with the later Uruk culture
Sumer
–
Map of Sumer
Sumer
–
The
Samarra bowl, at the
Pergamonmuseum, Berlin. The
swastika in the center of the design is a reconstruction.
Sumer
–
Fragment of
Eannatum 's
Stele of the Vultures
47.
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
48.
Similarity (geometry)
–
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling and this means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other zoomed in or out at some level. For example, all circles are similar to other, all squares are similar to each other. On the other hand, ellipses are not all similar to other, rectangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure and it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem, due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of which is necessary and sufficient for two triangles to be similar,1, the triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is, If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′, all the corresponding sides have lengths in the same ratio, AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other, two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance, AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′ and this is known as the SAS Similarity Criterion. When two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′, there are several elementary results concerning similar triangles in Euclidean geometry, Any two equilateral triangles are similar. Two triangles, both similar to a triangle, are similar to each other. Corresponding altitudes of similar triangles have the ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, the statement that the point F satisfying this condition exists is Walliss Postulate and is logically equivalent to Euclids Parallel Postulate
Similarity (geometry)
–
Sierpinski triangle. A space having self-similarity dimension ln 3 / ln 2 = log 2 3, which is approximately 1.58. (from
Hausdorff dimension.)
Similarity (geometry)
–
Figures shown in the same color are similar
49.
Nubia
–
Nubia is a region along the Nile river located in what is today northern Sudan and southern Egypt. It was the seat of one of the earliest civilizations of ancient Africa, with a history that can be traced from at least 2000 B. C. onward, and was home to one of the African empires. Nubia was again united within Ottoman Egypt in the 19th century, the name Nubia is derived from that of the Noba people, nomads who settled the area in the 4th century following the collapse of the kingdom of Meroë. The Noba spoke a Nilo-Saharan language, ancestral to Old Nubian, Old Nubian was mostly used in religious texts dating from the 8th and 15th centuries AD. Before the 4th century, and throughout classical antiquity, Nubia was known as Kush, or, in Classical Greek usage, until at least 1970, the Birgid language was spoken north of Nyala in Darfur, but is now extinct. Nubia was divided into two regions, Upper and Lower Nubia, so called because of their location in the Nile river valley. Early settlements sprouted in both Upper and Lower Nubia, Egyptians referred to Nubia as Ta-Seti, or The Land of the Bow, since the Nubians were known to be expert archers. Modern scholars typically refer to the people from this area as the “A-Group” culture, fertile farmland just south of the Third Cataract is known as the “pre-Kerma” culture in Upper Nubia, as they are the ancestors. The Neolithic people in the Nile Valley likely came from Sudan, as well as the Sahara, by the 5th millennium BC, the people who inhabited what is now called Nubia participated in the Neolithic revolution. Saharan rock reliefs depict scenes that have been thought to be suggestive of a cult, typical of those seen throughout parts of Eastern Africa. Megaliths discovered at Nabta Playa are early examples of what seems to be one of the worlds first astronomical devices, around 3500 BC, the second Nubian culture, termed the A-Group, arose. It was a contemporary of, and ethnically and culturally similar to. The A-Group people were engaged in trade with the Egyptians and this trade is testified archaeologically by large amounts of Egyptian commodities deposited in the graves of the A-Group people. The imports consisted of gold objects, copper tools, faience amulets and beads, seals, slate palettes, stone vessels, and a variety of pots. Around 3300 BC, there is evidence of a kingdom, as shown by the finds at Qustul. The Nubian culture may have contributed to the unification of the Nile Valley. The earliest known depiction of the crown is on a ceremonial incense burner from Cemetery at Qustul in Lower Nubia. New evidence from Abydos, however, particularly the excavation of Cemetery U, around the turn of the protodynastic period, Naqada, in its bid to conquer and unify the whole Nile Valley, seems to have conquered Ta-Seti and harmonized it with the Egyptian state
Nubia
–
Nubians in worship
Nubia
–
Nubian woman circa 1900
Nubia
–
Head of a Nubian Ruler
Nubia
–
Ramesses II in his war chariot charging into battle against the Nubians
50.
Greek mathematics
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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
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Statue of Euclid in the
Oxford University Museum of Natural History
Greek mathematics
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An illustration of
Euclid 's proof of the
Pythagorean Theorem
Greek mathematics
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The
Antikythera mechanism, an ancient mechanical calculator.