1.
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
2.
Physics
–
Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
Physics
–
Further information:
Outline of physics
Physics
–
Ancient
Egyptian astronomy is evident in monuments like the
ceiling of Senemut's tomb from the
Eighteenth Dynasty of Egypt.
Physics
–
Sir Isaac Newton (1643–1727), whose
laws of motion and
universal gravitation were major milestones in classical physics
Physics
–
Albert Einstein (1879–1955), whose work on the
photoelectric effect and the
theory of relativity led to a revolution in 20th century physics
3.
Fluid statics
–
Fluid statics or hydrostatics is the branch of fluid mechanics that studies incompressible fluids at rest. It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamics, hydrostatics are categorized as a part of the fluid statics, which is the study of all fluids, incompressible or not, at rest. Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids and it is also relevant to geophysics and astrophysics, to meteorology, to medicine, and many other fields. Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisterns, aqueducts and fountains. Archimedes is credited with the discovery of Archimedes Principle, which relates the force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The fair cup or Pythagorean cup, which dates from about the 6th century BC, is a technology whose invention is credited to the Greek mathematician. It was used as a learning tool, the cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup, the cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, due to the drag that molecules exert on one another, the cup will be emptied. Herons fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, the device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir. Pascal made contributions to developments in both hydrostatics and hydrodynamics, due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface, if a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force, thus, the pressure on a fluid at rest is isotropic, i. e. it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes, i. e. a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in an extended form, by Blaise Pascal. In a fluid at rest, all frictional and inertial stresses vanish, when this condition of V =0 is applied to the Navier-Stokes equation, the gradient of pressure becomes a function of body forces only
Fluid statics
–
Table of Hydraulics and Hydrostatics, from the 1728
Cyclopædia
Fluid statics
–
Diving medicine:
4.
Neusis construction
–
The neusis is a geometric construction method that was used in antiquity by Greek mathematicians. The neusis construction consists of fitting a line element of length in between two given lines, in such a way that the line element, or its extension. That is, one end of the element has to lie on l. A neusis construction might be performed by means of a neusis ruler, in the figure one end of the ruler is marked with a yellow eye with crosshairs, this is the origin of the scale division on the ruler. A second marking on the ruler indicates the distance a from the origin, the yellow eye is moved along line l, until the blue eye coincides with line m. The position of the element thus found is shown in the figure as a dark blue bar. Point P is called the pole of the neusis, line l the directrix, or guiding line, length a is called the diastema. Neuseis have been important because they provide a means to solve geometric problems that are not solvable by means of compass. Examples are the trisection of any angle in three parts, the doubling of the cube, and the construction of a regular heptagon, nonagon. Mathematicians such as Archimedes of Syracuse and Pappus of Alexandria freely used neuseis, Sir Isaac Newton followed their line of thought, nevertheless, gradually the technique dropped out of use. Modified by the recent finding by Benjamin and Snyder that the regular hendecagon is neusis-constructible, T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides was the first to put compass-and-straightedge constructions above neuseis. One hundred years after him Euclid too shunned neuseis in his influential textbook. The next attack on the neusis came when, from the fourth century BC, under its influence a hierarchy of three classes of geometrical constructions was developed. In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution, Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other methods might have been used was branded by the late Greek mathematician Pappus of Alexandria as a not inconsiderable error. R. Boeker, Neusis, in, Paulys Realencyclopädie der Classischen Altertumswissenschaft, the most comprehensive survey, however, the author sometimes has rather curious opinions. T. L. Heath, A history of Greek Mathematics, H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum. MathWorld page Angle Trisection by Paper Folding
Neusis construction
–
Neusis trisection of an angle θ > 135° to find φ = θ /3, using only the length of the ruler. The radius of the arc is equal to the length of the ruler. For angles θ < 135° the same construction applies, but with P extended beyond AB.
Neusis construction
–
Neusis construction
5.
Greeks
–
The Greeks or Hellenes are an ethnic group native to Greece, Cyprus, southern Albania, Turkey, Sicily, Egypt and, to a lesser extent, other countries surrounding the Mediterranean Sea. They also form a significant diaspora, with Greek communities established around the world, many of these regions coincided to a large extent with the borders of the Byzantine Empire of the late 11th century and the Eastern Mediterranean areas of ancient Greek colonization. The cultural centers of the Greeks have included Athens, Thessalonica, Alexandria, Smyrna, most ethnic Greeks live nowadays within the borders of the modern Greek state and Cyprus. The Greek genocide and population exchange between Greece and Turkey nearly ended the three millennia-old Greek presence in Asia Minor, other longstanding Greek populations can be found from southern Italy to the Caucasus and southern Russia and Ukraine and in the Greek diaspora communities in a number of other countries. Today, most Greeks are officially registered as members of the Greek Orthodox Church, the Greeks speak the Greek language, which forms its own unique branch within the Indo-European family of languages, the Hellenic. They are part of a group of ethnicities, described by Anthony D. Smith as an archetypal diaspora people. Both migrations occur at incisive periods, the Mycenaean at the transition to the Late Bronze Age, the Mycenaeans quickly penetrated the Aegean Sea and, by the 15th century BC, had reached Rhodes, Crete, Cyprus and the shores of Asia Minor. Around 1200 BC, the Dorians, another Greek-speaking people, followed from Epirus, the Dorian invasion was followed by a poorly attested period of migrations, appropriately called the Greek Dark Ages, but by 800 BC the landscape of Archaic and Classical Greece was discernible. The Greeks of classical antiquity idealized their Mycenaean ancestors and the Mycenaean period as an era of heroes, closeness of the gods. The Homeric Epics were especially and generally accepted as part of the Greek past, as part of the Mycenaean heritage that survived, the names of the gods and goddesses of Mycenaean Greece became major figures of the Olympian Pantheon of later antiquity. The ethnogenesis of the Greek nation is linked to the development of Pan-Hellenism in the 8th century BC, the works of Homer and Hesiod were written in the 8th century BC, becoming the basis of the national religion, ethos, history and mythology. The Oracle of Apollo at Delphi was established in this period, the classical period of Greek civilization covers a time spanning from the early 5th century BC to the death of Alexander the Great, in 323 BC. It is so named because it set the standards by which Greek civilization would be judged in later eras, the Peloponnesian War, the large scale civil war between the two most powerful Greek city-states Athens and Sparta and their allies, left both greatly weakened. Many Greeks settled in Hellenistic cities like Alexandria, Antioch and Seleucia, two thousand years later, there are still communities in Pakistan and Afghanistan, like the Kalash, who claim to be descended from Greek settlers. The Hellenistic civilization was the period of Greek civilization, the beginnings of which are usually placed at Alexanders death. This Hellenistic age, so called because it saw the partial Hellenization of many non-Greek cultures and this age saw the Greeks move towards larger cities and a reduction in the importance of the city-state. These larger cities were parts of the still larger Kingdoms of the Diadochi, Greeks, however, remained aware of their past, chiefly through the study of the works of Homer and the classical authors. An important factor in maintaining Greek identity was contact with barbarian peoples and this led to a strong desire among Greeks to organize the transmission of the Hellenic paideia to the next generation
Greeks
–
Hoplites fighting. Detail from an Attic black-figure
hydria, ca. 560 BC–550 BC.
Louvre,
Paris.
Greeks
–
A reconstruction of the 3rd millennium BC "Proto-Greek area", according to Bulgarian linguist
Vladimir Georgiev.
Greeks
–
Bust of
Cleopatra VII.
Altes Museum,
Berlin.
Greeks
–
Statues of
Saints Cyril and Methodius, missionaries of
Christianity among the
Slavic peoples, on the
Holy Trinity Column in Olomouc,
Czech Republic.
6.
Inventor
–
An invention is a unique or novel device, method, composition or process. The invention process is a process within an overall engineering and product development process and it may be an improvement upon a machine or product or a new process for creating an object or a result. An invention that achieves a unique function or result may be a radical breakthrough. Such works are novel and not obvious to others skilled in the same field, an inventor may be taking a big step in success or failure. A patent legally protects the property rights of the inventor. The rules and requirements for patenting an invention vary from country to country, another meaning of invention is cultural invention, which is an innovative set of useful social behaviours adopted by people and passed on to others. The Institute for Social Inventions collected many such ideas in magazines, Invention is also an important component of artistic and design creativity. Inventions often extend the boundaries of knowledge, experience or capability. Brainstorming also can spark new ideas for an invention, collaborative creative processes are frequently used by engineers, designers, architects and scientists. Co-inventors are frequently named on patents, in addition, many inventors keep records of their working process - notebooks, photos, etc. including Leonardo da Vinci, Galileo Galilei, Evangelista Torricelli, Thomas Jefferson and Albert Einstein. In the process of developing an invention, the idea may change. The invention may become simpler, more practical, it may expand, working on one invention can lead to others too. History shows that turning the concept of an invention into a device is not always swift or direct. Inventions may also more useful after time passes and other changes occur. For example, the became more useful once powered flight was a reality. Invention is often a creative process, an open and curious mind allows an inventor to see beyond what is known. Seeing a new possibility, connection, or relationship can spark an invention, inventive thinking frequently involves combining concepts or elements from different realms that would not normally be put together. Sometimes inventors disregard the boundaries between distinctly separate territories or fields, several concepts may be considered when thinking about invention
Inventor
–
' BUILD YOUR OWN TELEVISION RECEIVER.'
Science and Invention magazine cover, November 1928
Inventor
–
Alessandro Volta with the first
electrical battery. Volta is recognized as one of the most influential inventors of all time.
Inventor
–
Thomas Edison with
phonograph. Edison is considered one of the most prolific inventors in history, holding
1,093 U.S. patents in his name.
Inventor
–
A rare 1884 photo showing the experimental recording of voice patterns by a photographic process at the
Alexander Graham Bell Laboratory in Washington, D.C. Many of their experimental designs panned out in failure.
7.
Area
–
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
Area
–
A square metre
quadrat made of PVC pipe.
Area
–
The combined area of these three
shapes is
approximately 15.57
squares.
8.
Surface area
–
The surface area of a solid object is a measure of the total area that the surface of the object occupies. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces and this definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of area was sought by Henri Lebesgue. Their work led to the development of measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface, while the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function S ↦ A which assigns a real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the area is its additivity. More rigorously, if a surface S is a union of many pieces S1, …, Sr which do not overlap except at their boundaries. Surface areas of polygonal shapes must agree with their geometrically defined area. Since surface area is a notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface. This means that surface area is invariant under the group of Euclidean motions and these properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of many pieces that can be represented in the parametric form S D, r → = r →, ∈ D with a continuously differentiable function r →. The area of a piece is defined by the formula A = ∬ D | r → u × r → v | d u d v. Thus the area of SD is obtained by integrating the length of the vector r → u × r → v to the surface over the appropriate region D in the parametric uv plane. The area of the surface is then obtained by adding together the areas of the pieces. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f and surfaces of revolution. It was demonstrated by Hermann Schwarz that already for the cylinder, various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a natural notion of surface area, if a surface is very irregular, or rough
Surface area
–
The inner membrane of the
mitochondrion has a large surface area due to infoldings, allowing higher rates of
cellular respiration (electron
micrograph).
Surface area
–
A
sphere of radius has surface area
9.
Parabola
–
A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
Parabola
–
Parabolic compass designed by
Leonardo da Vinci
Parabola
–
Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.
Parabola
–
A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and
air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.
Parabola
–
Parabolic trajectories of water in a fountain.
10.
Pi
–
The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
Pi
–
The constant π is represented in this
mosaic outside the Mathematics Building at the
Technical University of Berlin.
Pi
–
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
Pi
–
Archimedes developed the polygonal approach to approximating π.
Pi
–
Isaac Newton used
infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
11.
Exponentiation
–
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
Exponentiation
–
Graphs of y = b x for various bases b: base 10 (green), base e (red), base 2 (blue), and base 1 / 2 (cyan). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
12.
Roman Republic
–
It was during this period that Romes control expanded from the citys immediate surroundings to hegemony over the entire Mediterranean world. During the first two centuries of its existence, the Roman Republic expanded through a combination of conquest and alliance, by the following century, it included North Africa, most of the Iberian Peninsula, and what is now southern France. Two centuries after that, towards the end of the 1st century BC, it included the rest of modern France, Greece, and much of the eastern Mediterranean. By this time, internal tensions led to a series of wars, culminating with the assassination of Julius Caesar. The exact date of transition can be a matter of interpretation, Roman government was headed by two consuls, elected annually by the citizens and advised by a senate composed of appointed magistrates. Over time, the laws that gave exclusive rights to Romes highest offices were repealed or weakened. The leaders of the Republic developed a tradition and morality requiring public service and patronage in peace and war, making military. Many of Romes legal and legislative structures can still be observed throughout Europe and much of the world in modern nation states, the exact causes and motivations for Romes military conflicts and expansions during the republic are subject to wide debate. While they can be seen as motivated by outright aggression and imperialism and they argue that Romes expansion was driven by short-term defensive and inter-state factors, and the new contingencies that these decisions created. In its early history, as Rome successfully defended itself against foreign threats in central and then northern Italy, with some important exceptions, successful wars in early republican Rome generally led not to annexation or military occupation, but to the restoration of the way things were. But the defeated city would be weakened and thus able to resist Romanizing influences. It was also able to defend itself against its non-Roman enemies. It was, therefore, more likely to seek an alliance of protection with Rome and this growing coalition expanded the potential enemies that Rome might face, and moved Rome closer to confrontation with major powers. The result was more alliance-seeking, on the part of both the Roman confederacy and city-states seeking membership within that confederacy. While there were exceptions to this, it was not until after the Second Punic War that these alliances started to harden into something more like an empire and this shift mainly took place in parts of the west, such as the southern Italian towns that sided with Hannibal. In contrast, Roman expansion into Spain and Gaul occurred as a mix of alliance-seeking, in the 2nd century BC, Roman involvement in the Greek east remained a matter of alliance-seeking, but this time in the face of major powers that could rival Rome. This had some important similarities to the events in Italy centuries earlier, with some major exceptions of outright military rule, the Roman Republic remained an alliance of independent city-states and kingdoms until it transitioned into the Roman Empire. It was not until the time of the Roman Empire that the entire Roman world was organized into provinces under explicit Roman control
Roman Republic
–
Route of Pyrrhus of Epirus
Roman Republic
–
Roman consul accompanied by two
lictors
Roman Republic
–
Gaius Gracchus, tribune of the people, presiding over the
Plebeian Council
Roman Republic
–
A Roman
denarius struck in 56 BC showing on one side the bust of the
Goddess Diana, and on the reverse the Roman general
Lucius Cornelius Sulla is offered an olive branch by his ally
Bocchus I as the captive
Jugurtha kneels beside Sulla with his hands bound.
13.
Cicero
–
Marcus Tullius Cicero was a Roman philosopher, politician, lawyer, orator, political theorist, consul, and constitutionalist. He came from a wealthy family of the Roman equestrian order. According to Michael Grant, the influence of Cicero upon the history of European literature, Cicero introduced the Romans to the chief schools of Greek philosophy and created a Latin philosophical vocabulary distinguishing himself as a translator and philosopher. Though he was an orator and successful lawyer, Cicero believed his political career was his most important achievement. During the chaotic latter half of the 1st century BC marked by civil wars, following Julius Caesars death, Cicero became an enemy of Mark Antony in the ensuing power struggle, attacking him in a series of speeches. His severed hands and head were then, as a revenge of Mark Antony. Petrarchs rediscovery of Ciceros letters is often credited for initiating the 14th-century Renaissance in public affairs, humanism, according to Polish historian Tadeusz Zieliński, the Renaissance was above all things a revival of Cicero, and only after him and through him of the rest of Classical antiquity. Cicero was born in 106 BC in Arpinum, a hill town 100 kilometers southeast of Rome and his father was a well-to-do member of the equestrian order and possessed good connections in Rome. However, being a semi-invalid, he could not enter public life, although little is known about Ciceros mother, Helvia, it was common for the wives of important Roman citizens to be responsible for the management of the household. Ciceros brother Quintus wrote in a letter that she was a thrifty housewife, Ciceros cognomen, or personal surname, comes from the Latin for chickpea, cicer. Plutarch explains that the name was given to one of Ciceros ancestors who had a cleft in the tip of his nose resembling a chickpea. However, it is likely that Ciceros ancestors prospered through the cultivation. Romans often chose down-to-earth personal surnames, the family names of Fabius, Lentulus, and Piso come from the Latin names of beans, lentils. Plutarch writes that Cicero was urged to change this name when he entered politics. During this period in Roman history, cultured meant being able to speak both Latin and Greek, Cicero used his knowledge of Greek to translate many of the theoretical concepts of Greek philosophy into Latin, thus translating Greek philosophical works for a larger audience. It was precisely his broad education that tied him to the traditional Roman elite, according to Plutarch, Cicero was an extremely talented student, whose learning attracted attention from all over Rome, affording him the opportunity to study Roman law under Quintus Mucius Scaevola. Ciceros fellow students were Gaius Marius Minor, Servius Sulpicius Rufus, the latter two became Ciceros friends for life, and Pomponius would become, in Ciceros own words, as a second brother, with both maintaining a lifelong correspondence. Cicero wanted to pursue a career in politics along the steps of the Cursus honorum
Cicero
–
A first century AD bust of Cicero in the
Capitoline Museums, Rome
Cicero
–
The Young Cicero Reading by
Vincenzo Foppa (fresco, 1464), now at the
Wallace Collection
Cicero
–
Cicero Denounces Catiline,
fresco by
Cesare Maccari, 1882–88
Cicero
–
Cicero's death (France, 15th century)
14.
Cylinder (geometry)
–
In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
Cylinder (geometry)
–
Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section
Cylinder (geometry)
–
A right circular cylinder with radius r and height h.
Cylinder (geometry)
–
In
projective geometry, a cylinder is simply a cone whose
apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
15.
Isidore of Miletus
–
Isidore of Miletus was one of the two main Byzantine Greek architects that Emperor Justinian I commissioned to design the church of Hagia Sophia in Constantinople from 532-537. He also created the first comprehensive compilation of Archimedes works. ”Isidore is also renowned for producing the first comprehensive compilation of Archimedes work, one copy of which survived to the present. Emperor Justinian I appointed his architects to rebuild the Hagia Sophia following his victory over protesters within the city of his Roman Empire, Constantinople. ”The warring factions of Byzantine society, the Blues. During the Nika Riot, more than thirty people died. ”The Hagia Sophia was repeatedly cracked by earthquakes and was quickly repaired. Isidore of Miletus’ nephew, Isidore the Younger, introduced the new design that can be viewed in the Hagia Sophia in present-day Istanbul. After a great earthquake in 989 ruined the dome of Hagia Sophia, the restored dome was completed by 994. Cakmak, AS, Taylor, RM, Durukal, E, the Structural Configuration of the First Dome of Justinians Hagia Sophia, An Investigation Based on Structural and Literary Analysis. The Art of the Byzantine Empire, 312-1453, Sources and Documents, the Architect Trdat, Building Practices and Cross-Cultural Exchange in Byzantium and Armenia. The Journal of the Society of Architectural Historians, the Secret History, With Related Texts
Isidore of Miletus
–
Interior panorama of the
Hagia Sophia, the patriarchal
basilica designed by Isidore. The influence of Archimedes' solid geometry works, which Isidore was the first to compile, is evident.
16.
Renaissance
–
The Renaissance was a period in European history, from the 14th to the 17th century, regarded as the cultural bridge between the Middle Ages and modern history. It started as a movement in Italy in the Late Medieval period and later spread to the rest of Europe. This new thinking became manifest in art, architecture, politics, science, Early examples were the development of perspective in oil painting and the recycled knowledge of how to make concrete. Although the invention of movable type sped the dissemination of ideas from the later 15th century. In politics, the Renaissance contributed to the development of the customs and conventions of diplomacy, the Renaissance began in Florence, in the 14th century. Other major centres were northern Italian city-states such as Venice, Genoa, Milan, Bologna, the word Renaissance, literally meaning Rebirth in French, first appeared in English in the 1830s. The word also occurs in Jules Michelets 1855 work, Histoire de France, the word Renaissance has also been extended to other historical and cultural movements, such as the Carolingian Renaissance and the Renaissance of the 12th century. The Renaissance was a movement that profoundly affected European intellectual life in the early modern period. Renaissance scholars employed the humanist method in study, and searched for realism, however, a subtle shift took place in the way that intellectuals approached religion that was reflected in many other areas of cultural life. In addition, many Greek Christian works, including the Greek New Testament, were back from Byzantium to Western Europe. Political philosophers, most famously Niccolò Machiavelli, sought to describe life as it really was. Others see more competition between artists and polymaths such as Brunelleschi, Ghiberti, Donatello, and Masaccio for artistic commissions as sparking the creativity of the Renaissance. Yet it remains much debated why the Renaissance began in Italy, accordingly, several theories have been put forward to explain its origins. During the Renaissance, money and art went hand in hand, Artists depended entirely on patrons while the patrons needed money to foster artistic talent. Wealth was brought to Italy in the 14th, 15th, and 16th centuries by expanding trade into Asia, silver mining in Tyrol increased the flow of money. Luxuries from the Eastern world, brought home during the Crusades, increased the prosperity of Genoa, unlike with Latin texts, which had been preserved and studied in Western Europe since late antiquity, the study of ancient Greek texts was very limited in medieval Western Europe. One of the greatest achievements of Renaissance scholars was to bring this entire class of Greek cultural works back into Western Europe for the first time since late antiquity, Arab logicians had inherited Greek ideas after they had invaded and conquered Egypt and the Levant. Their translations and commentaries on these ideas worked their way through the Arab West into Spain and Sicily and this work of translation from Islamic culture, though largely unplanned and disorganized, constituted one of the greatest transmissions of ideas in history
Renaissance
–
David, by
Michelangelo (
Accademia di Belle Arti,
Florence) is a masterpiece of Renaissance and world art.
Renaissance
–
Renaissance
Renaissance
–
Leonardo da Vinci 's
Vitruvian Man (c. 1490) shows clearly the effect writers of Antiquity had on Renaissance thinkers. Based on the specifications in
Vitruvius '
De architectura (1st century BC), Leonardo tried to draw the perfectly proportioned man.
Renaissance
–
Portrait of a young woman (c. 1480-85) (
Simonetta Vespucci) by
Sandro Botticelli
17.
Colonies in antiquity
–
Colonies in antiquity were city-states founded from a mother-city, not from a territory-at-large. Bonds between a colony and its metropolis remained often close, and took specific forms, however, unlike in the period of European colonialism during the early and late modern era, ancient colonies were usually sovereign and self-governing from their inception. An Egyptian colony that was stationed in southern Canaan dates to slightly before the First Dynasty, narmer had Egyptian pottery produced in Canaan and exported back to Egypt, from regions such as Arad, En Besor, Rafiah, and Tel ʿErani. Shipbuilding was known to the ancient Egyptians as early as 3000 BC, the Archaeological Institute of America reports that the earliest dated ship—75 feet long, dating to 3000 BC – may have possibly belonged to Pharaoh Aha. Egypt at its height controlled Crete across the Mediterranean Sea, the Phoenicians were the major trading power in the Mediterranean in the early part of the first millennium BC. They had trading contacts in Egypt and Greece, and established colonies as far west as modern Spain, from Gadir the Phoenicians controlled access to the Atlantic Ocean and the trade routes to Britain. The most famous and successful of Phoenician colonies was founded by settlers from Tyre in 814–813 BC and called Kart-Hadasht (Qart-ḥadašt, the Carthaginians later founded their own colony in the southeast of Spain, Carthago Nova, which was eventually conquered by their enemy, Rome. But in most cases the motivation was to establish and facilitate relations of trade with foreign countries, colonies were established in Ionia and Thrace as early as the 8th century BC. There were two types of colony, one known as an ἀποικία - apoikia and the other as an ἐμπορίov - emporion. The first type of colony was a city-state on its own, through this Greek expansion the use of coins flourished throughout the Mediterranean Basin. The Greeks also colonised modern-day Crimea on the Black Sea, among the settlements they established there was the city of Chersonesos, at the site of modern-day Sevastopol. Another area with significant Greek colonies was the coast of ancient Illyria on the Adriatic Sea, the extensive Greek colonization is remarked upon by Cicero when noting that It were as though a Greek fringe has been woven about the shores of the barbarians. Several formulae were generally adhered to on the solemn and sacred occasions when a new colony set forth, if a Greek city was sending out a colony, an oracle, especially one such as the Oracle of Delphi, was almost invariably consulted beforehand. A person of distinction was selected to guide the emigrants and make the necessary arrangements and it was usual to honor these founders of colonies, after their death, as heroes. Some of the fire was taken from the public hearth in the Prytaneum. After the conquests of the Macedonian Kingdom and Alexander the Great, the relation between colony and mother-city, known literally as the metropolis, was viewed as one of mutual affection. Any differences that arose were resolved by peaceful means whenever possible and it is worth noting that the Peloponnesian War was in part a result of a dispute between Corinth and her colony of Corcyra. The charter of foundation contained general provisions for the arrangement of the affairs of the colony, the constitution of the mother-city was usually adopted by the colony, but the new city remained politically independent
Colonies in antiquity
–
The Mediterranean in ca. the 6th century BC. Phoenician cities are labelled in yellow, Greek cities in red, and other cities in grey.
Colonies in antiquity
–
Map showing the Augustus "roman coloniae" in north Africa
18.
Southern Italy
–
It generally coincides with the administrative regions of Abruzzo, Apulia, Basilicata, Campania, Calabria, Molise, Sicily, and Sardinia. Some also include the most southern and eastern parts of Lazio within the Mezzogiorno, Southern Italy carries a unique legacy of culture. It features many major tourist attractions, such as the Palace of Caserta, there are also many ancient Greek cities in Southern Italy, such as Sybaris, which were founded several centuries before the start of the Roman Republic. These same subdivisions are at the bottom of the Italian First level NUTS of the European Union, the term Mezzogiorno first came into use in the 18th century and is an Italian rendition of meridies. The term was popularised by Giuseppe Garibaldi and it eventually came into vogue after the Italian unification. In a similar manner, Southern France is colloquially known as le Midi, Southern Italy forms the lower part of the Italian boot, containing the ankle, the toe, the arch, and the heel, along with the island of Sicily. Separating the heel and the boot is the Gulf of Taranto, named after the city of Taranto and it is an arm of the Ionian Sea. The island of Sardinia, right below the French island of Corsica, on the eastern coast is the Adriatic Sea, leading into the rest of the Mediterranean through the Strait of Otranto. Along the northern coast of the Salernitan Gulf and on the south of the Sorrentine Peninsula runs the Amalfi Coast, off the tip of the peninsula is the isle of Capri. The climate is mainly Mediterranean, except at the highest elevations and the eastern stretches in Apulia, along the Ionian Sea in Calabria. The largest city of Southern Italy is Naples, a name from the Greek that it has maintained for millennia. Bari, Taranto, Reggio Calabria, Foggia, and Salerno are the next largest cities in the area. The region is very active and highly seismic, the 1980 Irpinia earthquake killed 2,914 people, injured more than 10,000. Also during this period, Greek colonies were established in places as widely separated as the eastern coast of the Black Sea, Eastern Libya and they included settlements in Sicily and the southern part of the Italian Peninsula. The Romans called the area of Sicily and the foot of Italy, Magna Graecia, the ancient geographers differed on whether the term included Sicily or merely Apulia and Calabria—Strabo being the most prominent advocate of the wider definitions. With this colonisation, Greek culture was exported to Italy, in its dialects of the Ancient Greek language, its religious rites, an original Hellenic civilization soon developed, later interacting with the native Italic and Latin civilisations. Many of the new Hellenic cities became very rich and powerful, like Neapolis, Syracuse, Acragas, other cities in Magna Graecia included Tarentum, Epizephyrian Locri, Rhegium, Croton, Thurii, Elea, Nola, Syessa, Bari, and others. After Pyrrhus of Epirus failed in his attempt to stop the spread of Roman hegemony in 282 BCE, from then to the Norman conquest of the 11th century, the south of the peninsula was constantly plunged into wars between Greece, Lombardy, and the Islamic Caliphate
Southern Italy
–
Satellite image of Southern Italy
Southern Italy
–
Greek temple of Hera,
Selinunte,
Sicily.
Southern Italy
–
Castel del Monte, built by
Frederick II between 1240 and 1250 in
Andria,
Apulia.
Southern Italy
–
Castel Nuovo,
Naples: initiated by the
Anjou, it was heavily altered as it served as
Spanish headquarters until the 18th century.
19.
Byzantine Greeks
–
Throughout the Middle Ages, the Byzantine Greeks self-identified as Rhōmaîoi and Graikoí, but are referred to as Byzantines and Byzantine Greeks in modern historiography. The terms Byzantine Empire and Byzantine Greeks were first coined in the English language in 1857 by British historian George Finlay, the social structure of the Byzantine Greeks was primarily supported by a rural, agrarian base that consisted of the peasantry, and a small fraction of the poor. These peasants lived within three kinds of settlements, the chorion or village, the agridion or hamlet, and the proasteion or estate. Many civil disturbances that occurred during the time of the Byzantine Empire were attributed to political factions within the Empire rather than to large popular base. Soldiers among the Byzantine Greeks were at first conscripted amongst the rural peasants, as the Byzantine Empire entered the 11th century, more of the soldiers within the army were either professional men-at-arms or mercenaries. Until the twelfth century, education within the Byzantine Greek population was more advanced than in the West, particularly at primary school level, success came easily to Byzantine Greek merchants, who enjoyed a very strong position in international trade. Despite the challenges posed by rival Italian merchants, they held their own throughout the half of the Byzantine Empires existence. The clergy also held a place, not only having more freedom than their Western counterparts. This position of strength had built up over time, for at the beginning of the Byzantine Empire, under Emperor Constantine the Great, only a part, about 10%. The language of the Byzantine Greeks since the age of Constantine had been Greek, from the reign of Emperor Heraclius, Greek was the predominant language amongst the populace and also replaced Latin in administration. Over time, the relationship between them and the West, particularly with Latin Europe, deteriorated, relations were further damaged by a schism between the Catholic West and Orthodox East that led to the Byzantine Greeks being labeled as heretics in the West. However, the Byzantine Empire was the Eastern Roman Empire, during most of the Middle Ages, the Byzantine Greeks self-identified as Rhōmaîoi, a term which in the Greek language had become synonymous with Christian Greeks. The ancient name Hellenes was in popular use synonymous to pagan and was revived as an ethnonym in the Middle Byzantine period, the term Byzantines or Byzantine Greeks is an exonym applied by later historians like Hieronymus Wolf, the Byzantines continued to call themselves Romaioi in their language. Most historians agree that the features of their civilization were, 1) Greek language, culture, literature. The Eastern Roman Empire was in language and civilization a Greek society, the term Byzantine has been adopted by Western scholarship on the assumption that anything Roman is essentially Western. However, modern Greeks still use the ethnonyms Romaioi and Graikoi to refer to themselves, as well as the terms Romaica, byzantinist August Heisenberg defined the Byzantine Empire as the Christianised Roman empire of the Greek nation. Byzantium was primarily known as the Empire of the Greeks by Western Europeans due to the predominance of Greek linguistic, cultural, many Greek Orthodox populations, particularly those outside the newly independent modern Greek state, continued to refer to themselves as Romioi well into the 20th century. Some of the children ran to see what Greek soldiers looked like
Byzantine Greeks
–
Byzantine culture
Byzantine Greeks
–
The
double-headed eagle, emblem of the
Palaiologos dynasty.
Byzantine Greeks
–
Soldier wearing the lamellar klivanion cuirass and a straight spathion sword.
Byzantine Greeks
–
A page of 5th or 6th century
Iliad like the one a grammarian might possess.
20.
Plutarch
–
Plutarch was a Greek biographer and essayist, known primarily for his Parallel Lives and Moralia. He is classified as a Middle Platonist, Plutarchs surviving works were written in Greek, but intended for both Greek and Roman readers. Plutarch was born to a prominent family in the town of Chaeronea, about 80 km east of Delphi. The name of Plutarchs father has not been preserved, but based on the common Greek custom of repeating a name in alternate generations, the name of Plutarchs grandfather was Lamprias, as he attested in Moralia and in his Life of Antony. His brothers, Timon and Lamprias, are mentioned in his essays and dialogues. Rualdus, in his 1624 work Life of Plutarchus, recovered the name of Plutarchs wife, Timoxena, from internal evidence afforded by his writings. A letter is still extant, addressed by Plutarch to his wife, bidding her not to grieve too much at the death of their two-year-old daughter, interestingly, he hinted at a belief in reincarnation in that letter of consolation. The exact number of his sons is not certain, although two of them, Autobulus and the second Plutarch, are often mentioned. Plutarchs treatise De animae procreatione in Timaeo is dedicated to them, another person, Soklarus, is spoken of in terms which seem to imply that he was Plutarchs son, but this is nowhere definitely stated. Plutarch studied mathematics and philosophy at the Academy of Athens under Ammonius from 66 to 67, at some point, Plutarch took Roman citizenship. He lived most of his life at Chaeronea, and was initiated into the mysteries of the Greek god Apollo. For many years Plutarch served as one of the two priests at the temple of Apollo at Delphi, the site of the famous Delphic Oracle, twenty miles from his home. By his writings and lectures Plutarch became a celebrity in the Roman Empire, yet he continued to reside where he was born, at his country estate, guests from all over the empire congregated for serious conversation, presided over by Plutarch in his marble chair. Many of these dialogues were recorded and published, and the 78 essays, Plutarch held the office of archon in his native municipality, probably only an annual one which he likely served more than once. He busied himself with all the matters of the town. The Suda, a medieval Greek encyclopedia, states that Emperor Trajan made Plutarch procurator of Illyria, however, most historians consider this unlikely, since Illyria was not a procuratorial province, and Plutarch probably did not speak Illyrian. Plutarch spent the last thirty years of his serving as a priest in Delphi. He thus connected part of his work with the sanctuary of Apollo, the processes of oracle-giving
Plutarch
–
Ruins of the Temple of
Apollo at
Delphi, where Plutarch served as one of the priests responsible for interpreting the predictions of the
oracle.
Plutarch
–
Parallel Lives,
Amyot translation, 1565
Plutarch
–
Plutarch's bust at
Chaeronea, his home town.
Plutarch
–
A page from the 1470 Ulrich Han printing of Plutarch's
Parallel Lives.
21.
Eratosthenes
–
Eratosthenes of Cyrene was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today and he is best known for being the first person to calculate the circumference of the Earth, which he did by applying a measuring system using stadia, a standard unit of measure during that time period. He was also the first to calculate the tilt of the Earths axis, additionally, he may have accurately calculated the distance from the Earth to the Sun and invented the leap day. He created the first map of the world, incorporating parallels, Eratosthenes was the founder of scientific chronology, he endeavored to revise the dates of the chief literary and political events from the conquest of Troy. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers and he was a figure of influence in many fields. According to an entry in the Suda, his critics scorned him, nonetheless, his devotees nicknamed him Pentathlos after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Eratosthenes yearned to understand the complexities of the entire world, the son of Aglaos, Eratosthenes was born in 276 BC in Cyrene. Alexander the Great conquered Cyrene in 332 BC, and following his death in 323 BC, its rule was given to one of his generals, Ptolemy I Soter, the founder of the Ptolemaic Kingdom. Under Ptolemaic rule the economy prospered, based largely on the export of horses and silphium, Cyrene became a place of cultivation, where knowledge blossomed. Eratosthenes went to Athens to further his studies, there he was taught Stoicism by its founder, Zeno of Citium, in philosophical lectures on living a virtuous life. He then studied under Ariston of Chios, who led a more cynical school of philosophy and he also studied under the head of the Platonic Academy, who was Arcesilaus of Pitane. His interest in Plato led him to write his very first work at a level, Platonikos. Eratosthenes was a man of many perspectives and investigated the art of poetry under Callimachus and he was a talented and imaginative poet. He wrote poems, one in hexameters called Hermes, illustrating the life history. He wrote Chronographies, a text that scientifically depicted dates of importance and this work was highly esteemed for its accuracy. George Syncellus was later able to preserve from Chronographies a list of 38 kings of the Egyptian Thebes, Eratosthenes also wrote Olympic Victors, a chronology of the winners of the Olympic Games. It is not known when he wrote his works, but they highlighted his abilities and these works and his great poetic abilities led the pharaoh Ptolemy III Euergetes to seek to place him as a librarian at the Library of Alexandria in the year 245 BC
Eratosthenes
–
19th-century reconstruction of Eratosthenes' map of the known world,
c. 194 BC
Eratosthenes
–
Eratosthenes
Eratosthenes
–
The Burning of the Library at Alexandria in 391 AD, an illustration from "Hutchinsons History of the Nations", c. 1910
22.
Second Punic War
–
The Second Punic War, also referred to as The Hannibalic War and the War Against Hannibal, lasted from 218 to 201 BC and involved combatants in the western and eastern Mediterranean. This was the major war between Carthage and the Roman Republic and its allied Italic socii, with the crucial participation of Numidian-Berber armies and tribes on both sides. The two states three major wars with each other over the course of their existence. They are called the Punic Wars because Romes name for Carthaginians was Poeni, derived from Poenici, in the following year, Hannibals army defeated the Romans again, this time in southern Italy at Cannae. In consequence of these defeats, many Roman allies went over to Carthage, against Hannibals skill on the battlefield, the Romans deployed the Fabian strategy. A sideshow of this war was the indecisive First Macedonian War in the Eastern Mediterranean, the Second Punic War was fought between Carthage and Rome and was ignited by the dispute over the hegemony of Saguntum, a Hellenized Iberian coastal city with diplomatic contacts with Rome. After great tension within the city government, culminating in the assassination of the supporters of Carthage, the city called for Roman aid, but the pleas fell on deaf ears. Following a prolonged siege and a struggle, in which Hannibal himself was wounded and the army practically destroyed. Many of the Saguntians chose to commit suicide rather than face subjugation by the Carthaginians, before the war, Rome and Hasdrubal the Fair had made a treaty. Livy reports that it was agreed that the Iber should be the boundary between the two empires and that the liberty of the Saguntines should be preserved, Hannibal departed with this army from New Carthage northwards along the coast in late spring of 218 BC. At the Ebro, he split the army into three columns and subdued the tribes there to the Pyrenees within weeks, but with severe losses. At the Pyrenees, he left a detachment of 11,000 Iberian troops, Hannibal reportedly entered Gaul with 50,000 infantry and 9,000 cavalry. He took his army by a route, avoiding the Roman allies along the coast. In the meantime, a Roman fleet with a force was underway to northern Iberia. A scouting party of 300 cavalry was sent to discover the whereabouts of the enemy and these eventually defeated a Carthaginian scouting troop of 500 mounted Numidians and chased them back to their main camp. Thus, with knowledge of the location of the enemy, the Romans marched upstream, Hannibal evaded this force and by an unknown route reached the Isère or the Durance at the foot of the Alps in autumn. He also received messengers from his Gallic allies in Italy that urged him to come to their aid, before setting out to cross the Alps, he was re-supplied by a native tribe, some of whose hereditary disputes he had helped solve. Their other commander, Publius Cornelius Scipio, returned to Rome, realizing the danger of an invasion of Italy where the tribes of the Boii, after 217 BC, he moved to Iberia
Second Punic War
–
Iberian warrior from bas-relief c. 200 BC. The warrior is armed with a
falcata and an oval shield.
National Archaeological Museum of Spain,
Madrid
Second Punic War
–
Western Mediterranean, 218 BC. Italian cities and Celtic tribes that joined Hannibal after the invasion of Italy are depicted in blue.
Second Punic War
–
Iberian
falcata, 4th/3rd century BC. This weapon, a scythe-shaped sword, was unique to Iberia [
citation needed]. By its inherent weight distribution, it could deliver blows as powerful as an axe. National Archaeological Museum of Spain, Madrid
Second Punic War
–
Detail of frieze showing the equipment of a soldier in the
manipular Roman legion (left). Note mail armour, oval shield and helmet with plume (probably horsehair). The soldier in the centre is an officer (bronze cuirass, mantle), prob. a tribunus militum. From an altar built by
Gnaeus Domitius Ahenobarbus, consul in 122 BC.
Musée du Louvre, Paris
23.
Mathematical diagram
–
These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel. Argand diagrams are used to plot the positions of the poles and zeroes of a function in the complex plane. The concept of the plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, in particular, multiplication by a complex number of modulus 1 acts as a rotation. In the context of fast Fourier transform algorithms, a butterfly is a portion of the computation that combines the results of smaller discrete Fourier transforms into a larger DFT, the name butterfly comes from the shape of the data-flow diagram in the radix-2 case, as described below. The same structure can also be found in the Viterbi algorithm, the butterfly diagram show a data-flow diagram connecting the inputs x to the outputs y that depend on them for a butterfly step of a radix-2 Cooley-Tukey FFT. This diagram resembles a butterfly as in the morpho butterfly shown for comparison), commutative diagrams play the role in category theory that equations play in algebra. A Hasse diagram is a picture of a finite partially ordered set. In this case, we say y covers x, or y is an successor of x. In a Hasse diagram, it is required that the curves be drawn so that each meets exactly two vertices, its two endpoints. In Knot theory a useful way to visualise and manipulate knots is to project the knot onto a plane— and this is often done by creating a break in the strand going underneath. If by following the diagram the knot alternately crosses itself over and under, then the diagram represents a particularly well-studied class of knot, the Venn diagram is constructed with a collection of simple closed curves drawn in the plane. A Voronoi diagram is a kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space. This diagram is named after Georgy Voronoi, also called a Voronoi tessellation, in the simplest case, we are given a set of points S in the plane, which are the Voronoi sites. Each site s has a Voronoi cell V consisting of all points closer to s than to any other site, the segments of the Voronoi diagram are all the points in the plane that are equidistant to two sites. Such patterns occur frequently in architecture and decorative art, there are 17 possible distinct groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups, also called space groups, wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are different in style, color
Mathematical diagram
–
Euclid's Elements, ms. from Lüneburg, A.D. 1200
24.
Quaestor
–
A quaestor was a public official in Ancient Rome. The position served different functions depending on the period, in the Roman Kingdom, quaestores parricidii were appointed by the king to investigate and handle murders. In the Roman Republic, quaestors were elected officials that supervised the state treasury and it was the lowest ranking position in the cursus honorum. In modern usage in Italy and Romania, a quaestor is a ranking officer on the police force. In some organizations, a quaestor is the officer that oversees its finances, the earliest quaestors were quaestores parricidii, an office dating back to the Kingdom of Rome. Quaestores parricidii were chosen to investigate crimes, and may have been appointed as needed rather than holding a permanent position. Ancient authors disagree on the manner of selection for this office as well as on its earliest institution. In the Roman Republic, quaestors were elected officials who supervised the treasury and financial affairs of the state, its armies, the quaestors tasked with financial supervision were also called quaestores aerarii, because they oversaw the aerarium in the Temple of Saturn. The earliest origins of the office is obscure, but by about 420 BC there were four quaestors elected each year by the Comitia Tributa, after 267 BC, the number was expanded to ten. The office of quaestor, usually a former broad-striped tribune, was adopted as the first official post of the cursus honorum, once elected as quaestor, a Roman man earned the right to sit in the Senate and began progressing through the cursus honorum. Quaestors were also given a fasces and were entitled to one lictor, every Roman consul, the highest elected official in the cursus honorum, and every provincial governor was appointed a quaestor. Some quaestors were assigned to work in the city and others in the provinces where their responsibilities could include being recruited into the military, some provincial quaestors were assigned as staff to military generals or served as second-in-command to governors in the Roman provinces. Still others were assigned to oversee military finances, lucius Cornelius Sullas reforms in 81 BC raised the number of quaestors to 20 and the minimum age for a quaestorship was 30 for patricians and 32 for plebeians. There were at that time twenty Quæstors elected annually, some of whom remained in Rome, when a Consul took the field with an army, he always had a Quæstor with him. This had become the case so generally that the Quæstor became, as it were, constantine the Great created the office quaestor sacri palatii which functioned as the Roman Empires senior legal official. The quaestor sacri palatii survived long into the Byzantine Empire, although its duties were altered to match the quaesitor, the term is last attested in 14th century Byzantium as a purely honorific title. Quaestor derives from the Latin verb quaero, quaerere, meaning to inquire, the job title has traditionally been understood as deriving from the original investigative function of the quaestores parricidii. However, this connection has been questioned by modern scholars, in Italy a quaestor heads the police of his province, and his office is called questura
Quaestor
–
Ancient Rome
25.
Vitruvius
–
His discussion of perfect proportion in architecture and the human body, led to the famous Renaissance drawing by Da Vinci of Vitruvian Man. By his own description Vitruvius served as an artilleryman, the class of arms in the military offices. He probably served as a officer of artillery in charge of doctores ballistarum. Little is known about Vitruvius life, most inferences about him are extracted from his only surviving work De Architectura. Even his first name Marcus and his cognomen Pollio are uncertain. Cetius Faventinus writes of Vitruvius Polio aliique auctores, this can be read as Vitruvius Polio, and others or, less likely, as Vitruvius, Polio, Vitruvius was a military engineer, or a praefect architectus armamentarius of the apparitor status group. He is mentioned in Pliny the Elders table of contents for Naturalis Historia, frontinus refers to Vitruvius the architect in his late 1st-century work De aquaeductu. Likely born a free Roman citizen, by his own account, Vitruvius served the Roman army under Caesar with the otherwise poorly identified Marcus Aurelius, Publius Minidius and these names vary depending on the edition of De architectura. Publius Minidius is also written as Publius Numidicus and Publius Numidius, as an army engineer he specialized in the construction of ballista and scorpio artillery war machines for sieges. It is speculated that Vitruvius served with Caesars chief engineer Lucius Cornelius Balbus, the locations where he served can be reconstructed from, for example, descriptions of the building methods of various foreign tribes. Although he describes places throughout De Architectura, he not say he was present. His service likely included north Africa, Hispania, Gaul and Pontus, the position of the camp, the direction of the entrenchments, the inspection of the tents or huts of the soldiers and the baggage were comprehended in his province. His authority extended over the sick, and the physicians who had the care of them and he had the charge of providing carriages, bathhouses and the proper tools for sawing and cutting wood, digging trenches, raising parapets, sinking wells and bringing water into the camp. He likewise had the care of furnishing the troops with wood and straw, as well as the rams, onagri, balistae, at various locations described by Vitruvius, battles and sieges occurred. He is the source for the siege of Larignum in 56 BC. The broken siege at Gergovia in 52 BC, and the siege of Uxellodunum in 51 BC. These are all sieges of large Gallic oppida, a legion that fits the same sequence of locations is the Legio VI Ferrata, of which ballista would be an auxiliary unit. Mainly known for his writings, Vitruvius was himself an architect, frontinus mentions him in connection with the standard sizes of pipes
Vitruvius
–
A 1684 depiction of Vitruvius (right) presenting
De Architectura to
Augustus
Vitruvius
–
Vitruvian Man by
Leonardo da Vinci, an illustration of the human body inscribed in the circle and the square derived from a passage about geometry and human proportions in Vitruvius' writings
Vitruvius
–
Greek house plan after Vitruvius
Vitruvius
–
Drainage wheel from
Rio Tinto mines
26.
Votive crown
–
A votive crown is a votive offering in the form of a crown, normally in precious metals and often adorned with jewels. Especially in the Early Middle Ages, they are of a form, designed to be suspended by chains at an altar. Later examples are more often typical crowns in the style of the period, either designed to be placed on the head of a statue, there were pagan votive crowns in the ancient world, although these are essentially known only from literary references. From other references, it seems that in times not just statues of the gods. These were excavated in 1859, and are now divided between the National Archaeological Museum of Spain in Madrid and the Musée de Cluny in Paris. However the type was originally Roman or Byzantine, and adopted widely across Europe, nearly all these have been lost, in the example above, the letters on the pendilia spell RECCESVINTHVS REX OFFERET, or King Recceswinth offered this. These royal donations signified the submission of the monarchy to God, the main body of suspended crowns is usually flat around the top as well as the bottom rim, some are merely an open framework of flexibly linked metal pieces. The Iron Crown of Lombardy was perhaps made as a votive crown. Another gold crown was a source of contention in Constantinople, it was given to the Emperor Maurice by his wife Constantina, instead, he had it suspended by chains over the main altar of Hagia Sophia, upsetting the two ladies. It hung there for two centuries, until Emperor Leo IV coveted it and took it for his own use. Another Byzantine votive crown, given by Leo VI is now in the Treasury of San Marco, Venice, and is decorated with cloisonné enamels. In England, a medieval source says that King Canute gave a, or his, crown to be placed on or over the head of the rood, or large crucifix. The Anglo-Saxon Chronicle records that Hereward the Wakes men looted a solid gold crown from the head of the rood on the altar of Peterborough Cathedral in 1070. It was designed to be worn on top of an elaborate headress and hairstyle, or perhaps on a hennin and this is now a rare example of a medieval votive crown that has survived above ground. A few years later, in 1487, the crown that had used by the pretender Lambert Simnel was given to a statue of the Virgin in Dublin. Statues of the Virgin Mary and the Infant Jesus, of the Infant Jesus of Prague type, are among those most commonly crowned and it is now in private hands in the US. Votive crowns have continued to be produced in Catholic countries in modern times, often such crowns were kept in the church treasury except for special occasions such as relevant feast-days, when they are worn by the statue. In Greece a tama or votive offering of, or depicting, actual crowns used in ceremonies were normally retained by the couple
Votive crown
–
Detail of a votive crown from
Visigothic Hispania, before 672. Part of the
Treasure of Guarrazar. Out of view are chains for suspension above, and a Byzantine pendant cross below. Alternate view.
Votive crown
–
One of many crowned statues of the
Virgin Mary carried in the processions of
Holy Week in Seville.
27.
Gold
–
Gold is a chemical element with symbol Au and atomic number 79. In its purest form, it is a bright, slightly yellow, dense, soft, malleable. Chemically, gold is a metal and a group 11 element. It is one of the least reactive chemical elements and is solid under standard conditions, Gold often occurs in free elemental form, as nuggets or grains, in rocks, in veins, and in alluvial deposits. It occurs in a solid solution series with the element silver and also naturally alloyed with copper. Less commonly, it occurs in minerals as gold compounds, often with tellurium, golds atomic number of 79 makes it one of the higher numbered, naturally occurring elements. It is thought to have produced in supernova nucleosynthesis, from the collision of neutron stars. Because the Earth was molten when it was formed, almost all of the present in the early Earth probably sank into the planetary core. Gold is resistant to most acids, though it does dissolve in aqua regia, a mixture of acid and hydrochloric acid. Gold also dissolves in solutions of cyanide, which are used in mining and electroplating. Gold dissolves in mercury, forming amalgam alloys, but this is not a chemical reaction, as a precious metal, gold has been used for coinage, jewelry, and other arts throughout recorded history. A total of 186,700 tonnes of gold is in existence above ground, the world consumption of new gold produced is about 50% in jewelry, 40% in investments, and 10% in industry. Gold is also used in infrared shielding, colored-glass production, gold leafing, certain gold salts are still used as anti-inflammatories in medicine. As of 2014, the worlds largest gold producer by far was China with 450 tonnes, Gold is cognate with similar words in many Germanic languages, deriving via Proto-Germanic *gulþą from Proto-Indo-European *ǵʰelh₃-. The symbol Au is from the Latin, aurum, the Latin word for gold, the Proto-Indo-European ancestor of aurum was *h₂é-h₂us-o-, meaning glow. This word is derived from the root as *h₂éu̯sōs, the ancestor of the Latin word Aurora. This etymological relationship is presumably behind the frequent claim in scientific publications that aurum meant shining dawn, Gold is the most malleable of all metals, a single gram can be beaten into a sheet of 1 square meter, and an avoirdupois ounce into 300 square feet. Gold leaf can be thin enough to become semi-transparent
Gold
–
Gold, 79 Au
Gold
–
Moche gold necklace depicting feline heads.
Larco Museum Collection. Lima-Peru
Gold
–
Mirror for the future
James Webb Space Telescope coated in gold to reflect infrared light
Gold
–
The
world's largest gold bar has a mass of 250 kg.
Toi museum,
Japan.
28.
Density
–
The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume. The symbol most often used for density is ρ, although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume, ρ = m V, where ρ is the density, m is the mass, and V is the volume. In some cases, density is defined as its weight per unit volume. For a pure substance the density has the numerical value as its mass concentration. Different materials usually have different densities, and density may be relevant to buoyancy, purity, osmium and iridium are the densest known elements at standard conditions for temperature and pressure but certain chemical compounds may be denser. Thus a relative density less than one means that the floats in water. The density of a material varies with temperature and pressure and this variation is typically small for solids and liquids but much greater for gases. Increasing the pressure on an object decreases the volume of the object, increasing the temperature of a substance decreases its density by increasing its volume. In most materials, heating the bottom of a results in convection of the heat from the bottom to the top. This causes it to rise relative to more dense unheated material, the reciprocal of the density of a substance is occasionally called its specific volume, a term sometimes used in thermodynamics. Density is a property in that increasing the amount of a substance does not increase its density. Archimedes knew that the irregularly shaped wreath could be crushed into a cube whose volume could be calculated easily and compared with the mass, upon this discovery, he leapt from his bath and ran naked through the streets shouting, Eureka. As a result, the term eureka entered common parlance and is used today to indicate a moment of enlightenment, the story first appeared in written form in Vitruvius books of architecture, two centuries after it supposedly took place. Some scholars have doubted the accuracy of this tale, saying among other things that the method would have required precise measurements that would have been difficult to make at the time, from the equation for density, mass density has units of mass divided by volume. As there are units of mass and volume covering many different magnitudes there are a large number of units for mass density in use. The SI unit of kilogram per metre and the cgs unit of gram per cubic centimetre are probably the most commonly used units for density.1,000 kg/m3 equals 1 g/cm3. In industry, other larger or smaller units of mass and or volume are often more practical, see below for a list of some of the most common units of density
Density
–
Air density vs. temperature
29.
Buoyancy
–
In science, buoyancy or upthrust, is an upward force exerted by a fluid that opposes the weight of an immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid, thus the pressure at the bottom of a column of fluid is greater than at the top of the column. Similarly, the pressure at the bottom of an object submerged in a fluid is greater than at the top of the object and this pressure difference results in a net upwards force on the object. For this reason, an object whose density is greater than that of the fluid in which it is submerged tends to sink, If the object is either less dense than the liquid or is shaped appropriately, the force can keep the object afloat. This can occur only in a reference frame, which either has a gravitational field or is accelerating due to a force other than gravity defining a downward direction. In a situation of fluid statics, the net upward force is equal to the magnitude of the weight of fluid displaced by the body. The center of buoyancy of an object is the centroid of the volume of fluid. Archimedes principle is named after Archimedes of Syracuse, who first discovered this law in 212 B. C, more tersely, Buoyancy = weight of displaced fluid. The weight of the fluid is directly proportional to the volume of the displaced fluid. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy and this is also known as upthrust. Suppose a rocks weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting upon it, suppose that when the rock is lowered into water, it displaces water of weight 3 newtons. The force it exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyancy force,10 −3 =7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor and it is generally easier to lift an object up through the water than it is to pull it out of the water. The density of the object relative to the density of the fluid can easily be calculated without measuring any volumes. Density of object density of fluid = weight weight − apparent immersed weight Example, If you drop wood into water, Example, A helium balloon in a moving car. During a period of increasing speed, the air mass inside the car moves in the direction opposite to the cars acceleration, the balloon is also pulled this way. However, because the balloon is buoyant relative to the air, it ends up being pushed out of the way, If the car slows down, the same balloon will begin to drift backward. For the same reason, as the car goes round a curve and this is the equation to calculate the pressure inside a fluid in equilibrium
Buoyancy
–
A metallic coin (one British
pound coin) floats in
mercury due to the buoyancy force upon it and appears to float higher because of the surface tension of the mercury.
Buoyancy
–
The forces at work in buoyancy. Note that the object is floating because the upward force of buoyancy is equal to the downward force of
gravity.
30.
Galileo Galilei
–
Galileo Galilei was an Italian polymath, astronomer, physicist, engineer, philosopher, and mathematician. He played a role in the scientific revolution of the seventeenth century. Galileo also worked in applied science and technology, inventing an improved military compass, Galileos championing of heliocentrism and Copernicanism was controversial during his lifetime, when most subscribed to either geocentrism or the Tychonic system. He met with opposition from astronomers, who doubted heliocentrism because of the absence of a stellar parallax. He was tried by the Inquisition, found vehemently suspect of heresy and he spent the rest of his life under house arrest. He has been called the father of observational astronomy, the father of modern physics, the father of scientific method, and the father of science. Galileo was born in Pisa, Italy, on 15 February 1564, the first of six children of Vincenzo Galilei, a famous lutenist, composer, and music theorist, and Giulia, three of Galileos five siblings survived infancy. The youngest, Michelangelo, also became a noted lutenist and composer although he contributed to financial burdens during Galileos young adulthood, Michelangelo was unable to contribute his fair share of their fathers promised dowries to their brothers-in-law, who would later attempt to seek legal remedies for payments due. Michelangelo would also occasionally have to borrow funds from Galileo to support his musical endeavours and these financial burdens may have contributed to Galileos early fire to develop inventions that would bring him additional income. When Galileo Galilei was eight, his family moved to Florence and he then was educated in the Vallombrosa Abbey, about 30 km southeast of Florence. Galileo Bonaiuti was buried in the church, the Basilica of Santa Croce in Florence. It was common for mid-sixteenth century Tuscan families to name the eldest son after the parents surname, hence, Galileo Galilei was not necessarily named after his ancestor Galileo Bonaiuti. The Italian male given name Galileo derives from the Latin Galilaeus, meaning of Galilee, the biblical roots of Galileos name and surname were to become the subject of a famous pun. In 1614, during the Galileo affair, one of Galileos opponents, in it he made a point of quoting Acts 1,11, Ye men of Galilee, why stand ye gazing up into heaven. Despite being a genuinely pious Roman Catholic, Galileo fathered three children out of wedlock with Marina Gamba and they had two daughters, Virginia and Livia, and a son, Vincenzo. Their only worthy alternative was the religious life, both girls were accepted by the convent of San Matteo in Arcetri and remained there for the rest of their lives. Virginia took the name Maria Celeste upon entering the convent and she died on 2 April 1634, and is buried with Galileo at the Basilica of Santa Croce, Florence. Livia took the name Sister Arcangela and was ill for most of her life, Vincenzo was later legitimised as the legal heir of Galileo and married Sestilia Bocchineri
Galileo Galilei
–
Portrait of Galileo Galilei by
Giusto Sustermans
Galileo Galilei
–
Galileo's beloved elder daughter, Virginia (
Sister Maria Celeste), was particularly devoted to her father. She is buried with him in his tomb in the
Basilica of Santa Croce, Florence.
Galileo Galilei
–
Galileo Galilei. Portrait by
Leoni
Galileo Galilei
–
Cristiano Banti 's 1857 painting Galileo facing the
Roman Inquisition
31.
Syracusia
–
Syracusia was a 110 m ancient Greek ship sometimes claimed to be the largest transport ship of antiquity. She only sailed once, from Syracuse in Sicily to Alexandria in the Ptolemaic Kingdom, Syracusia was designed by Archimedes and built around 240 BC by Archias of Corinth on the orders of Hieron II of Syracuse. The historian Moschion of Phaselis said that Syracusia could carry a cargo of some 1,600 to 1,800 tons and she reputedly bore more than 200 soldiers, as well as a catapult. She sailed only once to berth in Alexandria, where she was given to Ptolemy III Euergetes of Egypt. A discussion of ship, as well as the complete text of Athenaeus is in Cassons Ships. This may be the first example of proactive antifouling technology, additionally, the top deck featured eight towers, equipped with two archers and four fully armed men. On the bow of the ship was a platform for fighting. 20 rows of oars would also have been visible from the outside, in terms of passenger comfort, Syracusia would be the equivalent of Titanic compared to other ships of the era. Her innovative design and sheer size allowed for the creation of recreational spaces aboard, including a garden. The lower levels of the ship were reserved for the crew, according to Athenaeus, the ship was beautifully decorated using materials such as ivory and marble, while all public spaces were floored with mosaics depicting the entire story of the Iliad. The ship was equipped with a library, a drawing room. Ptolemys son sought to outdo Syracusia and he ordered the construction of a huge warship, the Tessarakonteres,420 feet long, and bearing more than 4,000 oarsmen and 2,850 soldiers. However, according to Plutarch, it was almost immobile,2, pp. 575–578 Jean MacIntosh Turfa, Alwin Steinmayer Jr, The Syracusia as a Giant Cargo Vessel, The International Journal of Nautical Archaeology, Vol.28, No. 2, pp. 105–125 TEDEd, The real story behind Archimedes’ Eureka
Syracusia
–
Syracusia as imagined in 1798.
32.
Gymnasium (ancient Greece)
–
The gymnasium in Ancient Greece functioned as a training facility for competitors in public games. It was also a place for socializing and engaging in intellectual pursuits, the name comes from the Ancient Greek term gymnós meaning naked. Athletes competed nude, a practice which was said to encourage appreciation of the male body. Gymnasia and palestrae were under the protection and patronage of Heracles, Hermes and, in Athens, the word gymnasium is the latinisation of the Greek noun γυμνάσιον, gymnastic school, in pl. The verb had this meaning because one undressed for exercise, historically, the gymnasium was used for exercise, communal bathing, and scholarly and philosophical pursuits. The English noun gymnast, first recorded in 1594, is formed from the Greek γυμναστής, the palaistra was the part of the gymnasium devoted to wrestling, boxing and ball games. The gymnasium was formed as an institution where young men over 18 received training in physical exercises. The gymnastai were the teachers, coaches, and trainers of the athletes, the Greek gymnasiums also held lectures and discussions on philosophy, literature, and music, and public libraries were nearby. The athletic contests for which the gymnasium supplied the means of training, the contests took place in honour of heroes and gods, sometimes forming part of a periodic festival or the funeral rites of a deceased chief. The free and active Greek lifestyle reinforced the attachment to such sports, the victor in religious athletic contests, though he gained no material prize other than a wreath, was rewarded with the honour and respect of his fellow citizens. Training of competitors for the contests was a huge matter of public concern and special buildings were provided by the state for such use. A victory in the religious festivals was counted an honour for the whole state. The regulation of the Athenian gymnasium is attributed by Pausanias to Theseus, solon made several laws on the subject, according to Galen these were reduced to a workable system of management in the time of Cleisthenes. While the origins of physical exercise regimes cannot be pinpointed, the practice of exercising in the nude had its beginnings in the 7th century BC. The same purpose is frequently attributed to the tradition of oiling the body, the ancient Greek gymnasium soon became a place for more than exercise. This development arose through recognition by the Greeks of the relation between athletics, education and health. Accordingly, the gymnasium became connected with education on the one hand, physical training and maintenance of health and strength were the chief parts of childrens earlier education. As pupils grew older, informal conversation and other forms of activity took the place of institutional
Gymnasium (ancient Greece)
–
Pompeii gymnasium, from the top of the stadium wall.
Gymnasium (ancient Greece)
–
A
hermaic sculpture of an old man, thought to be the master of a gymnasium. He held a long stick in his right hand.
Ai Khanoum,
Afghanistan, 2nd century BC.
33.
Aphrodite
–
Aphrodite is the Greek goddess of love, beauty, pleasure, and procreation. She is identified with the planet Venus, and her Roman equivalent is the goddess Venus, as with many ancient Greek deities, there is more than one story about her origins. According to Hesiods Theogony, she was born when Cronus cut off Uranuss genitals and threw them into the sea, according to Homers Iliad, she is the daughter of Zeus and Dione. In Plato, these two origins are said to be of hitherto separate entities, Aphrodite Ourania and Aphrodite Pandemos, Aphrodite had many lovers—both gods, such as Ares, and men, such as Anchises. She played a role in the Eros and Psyche legend, and was lover and surrogate mother of Adonis. Many lesser beings were said to be children of Aphrodite, Aphrodite is also known as Cytherea and Cypris after the two cult sites, Cythera and Cyprus, which claimed to be her place of birth. Myrtle, roses, doves, sparrows and swans were sacred to her, the ancient Greeks identified her with the Ancient Egyptian goddess Hathor. Aphrodite had many names such as Acidalia and Cerigo, each used by a different local cult of the goddess in Greece. The Greeks recognized all of these names as referring to the single goddess Aphrodite, despite the differences in what these local cults believed the goddess demanded of them. The Attic philosophers of the 4th century, however, drew a distinction between a celestial Aphrodite of transcendent principles, and a separate, common Aphrodite who was the goddess of the people, hesiod derives Aphrodite from aphrós sea-foam, interpreting the name as risen from the foam. Michael Janda, accepting this as genuine, claims the birth myth as an Indo-European mytheme. Likewise, Witczak proposes an Indo-European compound *abʰor- very and *dʰei- to shine and it has been argued that etymologies based on comparison with Eos are unlikely since Aphrodites attributes are entirely different from those of Eos or the Vedic deity Ushas. A number of improbable non-Greek etymologies have suggested in scholarship. One Semitic etymology compares Aphrodite to the Assyrian barīrītu, the name of a demon that appears in Middle Babylonian. Hammarström looks to Etruscan, comparing prϑni lord, an Etruscan honorific loaned into Greek as πρύτανις and this would make the theonym in origin an honorific, the lady. Hjalmar Frisk and Robert Beekes reject this etymology as implausible, especially since Aphrodite actually appears in Etruscan in the borrowed form Apru, the medieval Etymologicum Magnum offers a highly contrived etymology, deriving Aphrodite from the compound habrodíaitos, she who lives delicately, from habrós and díaita. The alteration from b to ph is explained as a characteristic of Greek obvious from the Macedonians. Aphrodite is usually said to have been born near her chief center of worship, Paphos, on the island of Cyprus, however, other versions of her myth have her born near the island of Cythera, hence another of her names, Cytherea
Aphrodite
–
Aphrodite Pudica (Roman copy of 2nd century AD),
National Archaeological Museum, Athens
Aphrodite
–
Petra tou Romiou ("The rock of the
Greek "), Aphrodite's legendary birthplace in
Paphos, Cyprus.
Aphrodite
–
The Birth of Venus by
Sandro Botticelli, circa 1485.
Aphrodite
–
Venus and Adonis by
Titian, circa 1554.
34.
Steamboat
–
A steamboat is a boat that is propelled primarily by steam power, typically driving propellers or paddlewheels. Steamboats sometimes use the prefix designation SS, S. S. or S/S or PS, the term steamboat is used to refer to smaller, insular, steam-powered boats working on lakes and rivers, particularly riverboats. As using steam became more reliable, steam power became applied to larger, Early attempts at powering a boat by steam were made by the French inventor Denis Papin and the English inventor Thomas Newcomen. Papin invented the steam digester and experimented with closed cylinders and pistons pushed in by atmospheric pressure, Papin proposed applying this steam pump to the operation of a paddlewheel boat and tried to market his idea in Britain. He was unable to convert the piston motion into rotary motion. Newcomens design did solve the first problem, but remained shackled to the inherent limitations of the engines of the time, a steamboat was described and patented by English physician John Allen in 1729. In 1736, Jonathan Hulls was granted a patent in England for a Newcomen engine-powered steamboat, William Henry of Lancaster, Pennsylvania, having learned of Watts engine on a visit to England, made his own engine. In 1763 he put it in a boat, the boat sank, and while Henry made an improved model, he did not appear to have much success, though he may have inspired others. At its first demonstration on 15 July 1783, Pyroscaphe travelled upstream on the river Saône for some fifteen minutes before the engine failed, presumably this was easily repaired as the boat is said to have made several such journeys. Following this, De Jouffroy attempted to get the government interested in his work, De Jouffroy did not have the funds for this, and, following the events of the French revolution, work on the project was discontinued after he left the country. Similar boats were made in 1785 by John Fitch in Philadelphia and William Symington in Dumfries and this boat could typically make 7 to 8 miles per hour and traveled more than 2,000 miles during its short length of service. The Fitch steamboat was not a success, as this travel route was adequately covered by relatively good wagon roads. The following year, a boat made 30-mile excursions, and in 1790. Miller sent King Gustav III of Sweden an actual version,100 feet long. Miller then engaged engineer William Symington to build his patent steam engine drove a stern-mounted paddle wheel in a boat in 1785. The boat was successfully tried out on Dalswinton Loch in 1788 and was followed by a steamboat the next year. The boat was built by Alexander Hart at Grangemouth to Symingtons design with a cylinder engine. Trials on the River Carron in June 1801 were successful and included towing sloops from the river Forth up the Carron and thence along the Forth, in 1801, Symington patented a horizontal steam engine directly linked to a crank
Steamboat
–
Look out (Transport Steamer) on
Tennessee River, ca. 1860 - ca. 1865
Steamboat
–
Denis Papin 's cylinder and piston apparatus, 1690
Steamboat
–
Model of steamship, built in 1784, by
Claude de Jouffroy.
Steamboat
–
Charlotte Dundas, the first practical steamboat, built by
William Symington.
35.
Propeller
–
A propeller is a type of fan that transmits power by converting rotational motion into thrust. A pressure difference is produced between the forward and rear surfaces of the blade, and a fluid is accelerated behind the blade. Their disadvantages are higher mechanical complexity and higher cost, the principle employed in using a screw propeller is used in sculling. It is part of the skill of propelling a Venetian gondola but was used in a less refined way in parts of Europe. For example, propelling a canoe with a paddle using a pitch stroke or side slipping a canoe with a scull involves a similar technique. In China, sculling, called lu, was used by the 3rd century AD. In sculling, a blade is moved through an arc. The innovation introduced with the propeller was the extension of that arc through more than 360° by attaching the blade to a rotating shaft. Propellers can have a blade, but in practice there are nearly always more than one so as to balance the forces involved. The origin of the screw propeller starts with Archimedes, who used a screw to lift water for irrigation and bailing boats and it was probably an application of spiral movement in space to a hollow segmented water-wheel used for irrigation by Egyptians for centuries. Leonardo da Vinci adopted the principle to drive his theoretical helicopter, in 1784, J. P. Paucton proposed a gyrocopter-like aircraft using similar screws for both lift and propulsion. At about the time, James Watt proposed using screws to propel boats. This was not his own invention, though, Toogood and Hays had patented it a century earlier, by 1827, Czech-Austrian inventor Josef Ressel had invented a screw propeller which had multiple blades fastened around a conical base. He had tested his propeller in February 1826 on a ship that was manually driven. He was successful in using his bronze screw propeller on an adapted steamboat and his ship Civetta with 48 gross register tons, reached a speed of about six knots. This was the first ship successfully driven by an Archimedes screw-type propeller, after a new steam engine had an accident his experiments were banned by the Austro-Hungarian police as dangerous. Josef Ressel was at the time a forestry inspector for the Austrian Empire, but before this he received an Austro-Hungarian patent for his propeller. This new method of propulsion was an improvement over the paddlewheel as it was not so affected by either ship motions or changes in draft as the burned coal
Propeller
–
Propeller on a modern mid-sized merchant vessel. The propeller rotates clockwise to propel the ship forward when viewed from astern (right of picture), the person in the picture has his hand on the propeller's trailing edge
Propeller
–
Archimedes' screw.
Propeller
–
Propellers of the
RMS Olympic, a sister ship to the
RMS Titanic.
Propeller
–
Smith's original 1836 patent for a screw propeller of two full turns. He would later revise the patent, reducing the length to one turn.
36.
Claw of Archimedes
–
The Claw of Archimedes was an ancient weapon devised by Archimedes to defend the seaward portion of Syracuses city wall against amphibious assault. These machines featured prominently during the Second Punic War in 214 BC, when the Roman fleet approached the city walls under cover of darkness, the machines were deployed, sinking many ships and throwing the attack into confusion. Historians such as Livy attributed heavy Roman losses to these machines, the plausibility of this invention was tested in 1999 in the BBC series Secrets of the Ancients and again in early 2005 in the Discovery Channel series Superweapons of the Ancient World. The producers of Superweapons brought together a group of engineers tasked with conceiving and implementing a design that was realistic, within seven days they were able to test their creation, and they did succeed in tipping over a model of a Roman ship so that it would sink. While this does not prove the existence of the Claw, it suggests that it would have been possible, C. K. Young, Archimedess iron hand or claw – a new interpretation of an old mystery, Centaurus, Vol.46, No. 3, pp. 189–207 Scale models of the Claws operation BBC Secrets of the Ancients, The Claw
Claw of Archimedes
–
A painting of the Claw of Archimedes by
Giulio Parigi, taking the name "iron hand" literally
37.
Siege of Syracuse (212 BC)
–
The Siege of Syracuse by the Roman Republic took place in 214–212 BC, at the end of which the Magna Graecia Hellenistic city of Syracuse, located on the east coast of Sicily, fell. The Romans stormed the city after a protracted siege giving them control of the island of Sicily. During the siege, the city was protected by weapons developed by Archimedes, Archimedes, the great inventor and polymath, was slain at the conclusion of the siege by a Roman soldier, in contravention of the Roman general Marcellus instructions to spare his life. Sicily, which was wrested from Carthaginian control during the First Punic War, was the first province of the Roman Republic not directly part of Italy. The Kingdom of Syracuse was an independent region in the south east of the island. A Roman force led by the General Marcus Claudius Marcellus consequently laid siege to the city by sea. The city of Syracuse, located on the eastern coast of Sicily was renowned for its significant fortifications, among the Syracuse defenders was the mathematician and scientist Archimedes. The city was defended for many months against all the measures the Romans could bring to bear. Realizing how difficult the siege would be, the Romans brought their own unique devices and these included the sambuca, a floating siege tower with grappling hooks, as well as ship-mounted scaling ladders that were lowered with pulleys onto the city walls. Legend has it that he created a giant mirror that was used to deflect the powerful Mediterranean sun onto the ships sails. These measures, along with the fire from ballistas and onagers mounted on the city walls, frustrated the Romans, the Carthaginians realised the potential hindrance a continuing Syracusian defense could cause to the Roman war effort and attempted to relieve the city from the besiegers but were driven back. Though they planned another attempt, they could not afford the troops and ships with the ongoing war against the Romans in Hispania. The successes of the Syracusians in repelling the Roman siege had made them overconfident, in 212 BC, the Romans received information that the citys inhabitants were to participate in the annual festival to their goddess Artemis. Archimedes, who was now around 78 years of age, continued his studies after the breach by the Romans and while at home, his work was disturbed by a Roman soldier. Archimedes protested at this interruption of his work and coarsely told the soldier to leave, the Romans now controlled the outer city but the remainder of the population of Syracuse had quickly fallen back to the fortified inner citadel, offering continued resistance. The Romans now put siege to the citadel and were successful in cutting off supplies to this reduced area, on the agreed signal, during a diversionary attack, he opened the gate. After setting guards on the houses of the faction, Marcellus gave Syracuse to plunder. Frustrated and angered after the lengthy and costly siege, the Romans rampaged through the citadel and slaughtered many of the Syracusians where they stood, the city was then thoroughly looted and sacked
Siege of Syracuse (212 BC)
–
Hiero II of Syracuse calls Archimedes to fortify the city by
Sebastiano Ricci (1720s).
Siege of Syracuse (212 BC)
–
Siege of Syracuse
Siege of Syracuse (212 BC)
–
Detail of a wall painting of the Claw of Archimedes sinking a ship (c. 1600).
Siege of Syracuse (212 BC)
–
Archimedes Directing the Defenses of Syracuse by Thomas Ralph Spence (1895).
38.
Burning-glass
–
A burning glass or burning lens is a large convex lens that can concentrate the suns rays onto a small area, heating up the area and thus resulting in ignition of the exposed surface. Burning mirrors achieve an effect by using reflecting surfaces to focus the light. They were used in 18th-century chemical studies for burning materials in closed glass vessels where the products of combustion could be trapped for analysis, the burning glass was a useful contrivance in the days before electrical ignition was easily achieved. The technology of the glass has been known since antiquity. Vases filled with water used to start fires were known in the ancient world, burning lenses were used to cauterise wounds and to light sacred fires in temples. Plutarch refers to a mirror made of joined triangular metal mirrors installed at the temple of the Vestal Virgins. Aristophanes mentions the burning lens in his play The Clouds, have you ever seen a beautiful, transparent stone at the druggists, with which you may kindle fire. Archimedes, the mathematician, was said to have used a burning glass as a weapon in 212 BC. The Roman fleet was supposedly incinerated, though eventually the city was taken, the legend of Archimedes gave rise to a considerable amount of research on burning glasses and lenses until the late 17th century. Burning lenses were used both by Joseph Priestley and Antoine Lavoisier in their experiments to obtain oxides contained in closed vessels under high temperatures and these included carbon dioxide by burning diamond, and mercuric oxide by heating mercury. This type of experiment contributed to the discovery of dephlogisticated air by Priestley and they decided not to take up his proposal. Needless to say, this is dangerous and will damage the eye in seconds. Burning glasses are used to light fires in outdoor and primitive settings. Large burning lenses sometimes take the form of Fresnel lenses, similar to lighthouse lenses, solar furnaces are used in industry to produce extremely high temperatures without the need for fuel or large supplies of electricity. They sometimes employ a large array of mirrors to focus light to a high intensity. The Olympic torch that is carried around the host country of the Olympic Games is lit by a burning glass, diocles Nimrud lens Pyreliophorus Visby lenses Temple, Robert
Burning-glass
–
A replica (on a smaller scale) of the burning lens owned by
Joseph Priestley, in his laboratory
Burning-glass
–
Close-up view of a flat
Fresnel lens. These thin, light weight, non fragile and low cost lens can be used as burning-glass in emergency situations.
39.
Solar furnace
–
A solar furnace is a structure that uses concentrated solar power to produce high temperatures, usually for industry. Parabolic mirrors or heliostats concentrate light onto a focal point, the temperature at the focal point may reach 3,500 °C, and this heat can be used to generate electricity, melt steel, make hydrogen fuel or nanomaterials. The largest solar furnace is at Odeillo in the Pyrénées-Orientales in France and it employs an array of plane mirrors to gather sunlight, reflecting it onto a larger curved mirror. The ancient Greek / Latin term heliocaminus literally means solar furnace, during the Second Punic War, the Greek scientist Archimedes is said to have repelled the attacking Roman ships by setting them on fire with a burning glass that may have been an array of mirrors. An experiment to test this theory was carried out by a group at the Massachusetts Institute of Technology in 2005, the first modern solar furnace is believed to have been built in France in 1949 by Professor Félix Trombe. It is now still in place at Mont Louis, near Odeillo, the Pyrenees were chosen as the site because the area experiences clear skies up to 300 days a year. Another solar furnace was built in Uzbekistan as a part of a Soviet Union Sun Complex Research Facility impulsed by Academician S. A. Asimov, the solar furnace principle is being used to make inexpensive solar cookers and solar-powered barbecues, and for solar water pasteurization. A prototype Scheffler reflector is being constructed in India for use in a solar crematorium and this 50 m² reflector will generate temperatures of 700 °C and displace 200–300 kg of firewood used per cremation
Solar furnace
–
The solar furnace at
Odeillo in the
Pyrénées-Orientales in
France can reach temperatures up to 3,500 °C (6,330 °F)
40.
Athens
–
Athens is the capital and largest city of Greece. In modern times, Athens is a cosmopolitan metropolis and central to economic, financial, industrial, maritime. In 2015, Athens was ranked the worlds 29th richest city by purchasing power, Athens is recognised as a global city because of its location and its importance in shipping, finance, commerce, media, entertainment, arts, international trade, culture, education and tourism. It is one of the biggest economic centres in southeastern Europe, with a financial sector. The municipality of Athens had a population of 664,046 within its limits. The urban area of Athens extends beyond its administrative city limits. According to Eurostat in 2011, the Functional urban areas of Athens was the 9th most populous FUA in the European Union, Athens is also the southernmost capital on the European mainland. The city also retains Roman and Byzantine monuments, as well as a number of Ottoman monuments. Athens is home to two UNESCO World Heritage Sites, the Acropolis of Athens and the medieval Daphni Monastery, Athens was the host city of the first modern-day Olympic Games in 1896, and 108 years later it welcomed home the 2004 Summer Olympics. In Ancient Greek, the name of the city was Ἀθῆναι a plural, in earlier Greek, such as Homeric Greek, the name had been current in the singular form though, as Ἀθήνη. It was possibly rendered in the later on, like those of Θῆβαι and Μυκῆναι. During the medieval period the name of the city was rendered once again in the singular as Ἀθήνα, an etiological myth explaining how Athens has acquired its name was well known among ancient Athenians and even became the theme of the sculpture on the West pediment of the Parthenon. The goddess of wisdom, Athena, and the god of the seas, Poseidon had many disagreements, in an attempt to compel the people, Poseidon created a salt water spring by striking the ground with his trident, symbolizing naval power. However, when Athena created the tree, symbolizing peace and prosperity. Different etymologies, now rejected, were proposed during the 19th century. Christian Lobeck proposed as the root of the name the word ἄθος or ἄνθος meaning flower, ludwig von Döderlein proposed the stem of the verb θάω, stem θη- to denote Athens as having fertile soil. In classical literature, the city was referred to as the City of the Violet Crown, first documented in Pindars ἰοστέφανοι Ἀθᾶναι. In medieval texts, variant names include Setines, Satine, and Astines, today the caption η πρωτεύουσα, the capital, has become somewhat common
Athens
–
From upper left: the
Acropolis, the
Hellenic Parliament, the
Zappeion, the
Acropolis Museum,
Monastiraki Square, Athens view towards the sea
Athens
–
Athena, patron goddess of Athens;
National Archaeological Museum
Athens
–
Acropolis of Athens, with
Odeon of Herodes Atticus seen on bottom left
41.
Bitumen
–
Asphalt, also known as bitumen is a sticky, black and highly viscous liquid or semi-solid form of petroleum. It may be found in deposits or may be a refined product. Until the 20th century, the term asphaltum was also used, the word is derived from the Ancient Greek ἄσφαλτος ásphaltos. The primary use of asphalt/bitumen is in construction, where it is used as the glue or binder mixed with aggregate particles to create asphalt concrete. Its other main uses are for bituminous waterproofing products, including production of roofing felt, the terms asphalt and bitumen are often used interchangeably to mean both natural and manufactured forms of the substance. In American English, asphalt is the carefully refined residue from the process of selected crude oils. Outside the United States, the product is often called bitumen, geologists often prefer the term bitumen. Common usage often refers to forms of asphalt/bitumen as tar. Naturally occurring asphalt/bitumen is sometimes specified by the crude bitumen. Its viscosity is similar to that of cold molasses while the material obtained from the distillation of crude oil boiling at 525 °C is sometimes referred to as refined bitumen. The Canadian province of Alberta has most of the reserves of natural bitumen, covering 142,000 square kilometres. Additionally, most natural bitumens contain organosulfur compounds, resulting in a sulfur content of up to 4%. Nickel and vanadium are found in the <10 ppm level, as is typical of some petroleum, the substance is soluble in carbon disulfide. It is commonly modelled as a colloid, with asphaltenes as the dispersed phase, and it is almost impossible to separate and identify all the different molecules of asphalt, because the number of molecules with different chemical structure is extremely large. Asphalt/bitumen can sometimes be confused with tar, which is a visually similar black. During the early and mid-20th century when town gas was produced, coal tar was a readily available byproduct, the addition of tar to macadam roads led to the word tarmac, which is now used in common parlance to refer to road-making materials. However, since the 1970s, when natural gas succeeded town gas, other examples of this confusion include the La Brea Tar Pits and the Canadian oil sands, both of which actually contain natural bitumen rather than tar. Pitch is another term used at times to refer to asphalt/bitumen
Bitumen
–
Natural asphalt/bitumen from the
Dead Sea
Bitumen
–
refined asphalt/bitumen
Bitumen
–
The University of Queensland
pitch drop experiment, demonstrating the
viscosity of asphalt/bitumen
Bitumen
–
Bituminous outcrop of the Puy de la Poix,
Clermont-Ferrand, France
42.
Massachusetts Institute of Technology
–
The Massachusetts Institute of Technology is a private research university in Cambridge, Massachusetts, often cited as one of the worlds most prestigious universities. Researchers worked on computers, radar, and inertial guidance during World War II, post-war defense research contributed to the rapid expansion of the faculty and campus under James Killian. The current 168-acre campus opened in 1916 and extends over 1 mile along the bank of the Charles River basin. The Institute is traditionally known for its research and education in the sciences and engineering, and more recently in biology, economics, linguistics. Air Force and 6 Fields Medalists have been affiliated with MIT, the school has a strong entrepreneurial culture, and the aggregated revenues of companies founded by MIT alumni would rank as the eleventh-largest economy in the world. In 1859, a proposal was submitted to the Massachusetts General Court to use newly filled lands in Back Bay, Boston for a Conservatory of Art and Science, but the proposal failed. A charter for the incorporation of the Massachusetts Institute of Technology, Rogers, a professor from the University of Virginia, wanted to establish an institution to address rapid scientific and technological advances. The Rogers Plan reflected the German research university model, emphasizing an independent faculty engaged in research, as well as instruction oriented around seminars, two days after the charter was issued, the first battle of the Civil War broke out. After a long delay through the war years, MITs first classes were held in the Mercantile Building in Boston in 1865, in 1863 under the same act, the Commonwealth of Massachusetts founded the Massachusetts Agricultural College, which developed as the University of Massachusetts Amherst. In 1866, the proceeds from sales went toward new buildings in the Back Bay. MIT was informally called Boston Tech, the institute adopted the European polytechnic university model and emphasized laboratory instruction from an early date. Despite chronic financial problems, the institute saw growth in the last two decades of the 19th century under President Francis Amasa Walker. Programs in electrical, chemical, marine, and sanitary engineering were introduced, new buildings were built, the curriculum drifted to a vocational emphasis, with less focus on theoretical science. The fledgling school still suffered from chronic financial shortages which diverted the attention of the MIT leadership, during these Boston Tech years, MIT faculty and alumni rebuffed Harvard University president Charles W. Eliots repeated attempts to merge MIT with Harvard Colleges Lawrence Scientific School. There would be at least six attempts to absorb MIT into Harvard, in its cramped Back Bay location, MIT could not afford to expand its overcrowded facilities, driving a desperate search for a new campus and funding. Eventually the MIT Corporation approved an agreement to merge with Harvard, over the vehement objections of MIT faculty, students. However, a 1917 decision by the Massachusetts Supreme Judicial Court effectively put an end to the merger scheme, the neoclassical New Technology campus was designed by William W. Bosworth and had been funded largely by anonymous donations from a mysterious Mr. Smith, starting in 1912. In January 1920, the donor was revealed to be the industrialist George Eastman of Rochester, New York, who had invented methods of production and processing
Massachusetts Institute of Technology
–
Stereographic card showing an MIT mechanical drafting studio, 19th century (photo by
E.L. Allen, left/right inverted)
Massachusetts Institute of Technology
–
Massachusetts Institute of Technology
Massachusetts Institute of Technology
–
A 1905 map of MIT's Boston campus
Massachusetts Institute of Technology
–
Plaque in Building 6 honoring
George Eastman, founder of
Eastman Kodak, who was revealed as the anonymous "Mr. Smith" who helped maintain MIT's independence
43.
MythBusters
–
MythBusters is a science entertainment television program created by Peter Rees and produced by Australias Beyond Television Productions. The series premiered on the Discovery Channel on January 23,2003, the series was transmitted by numerous international broadcasters, including SBS Australia, and other Discovery channels worldwide. The show was one of the oldest—and the most popular—on Discovery Channel, being preceded only by How Its Made and Daily Planet, from 2006 to 2016, the show was overseen by British show-runner Dan Tapster, working out of Sydney, San Francisco and Manchester. Filmed in San Francisco and edited in Artarmon, New South Wales, Australia, during the second season, members of Savage and Hynemans behind-the-scenes team were organized into a second team of MythBusters. They generally tested myths separately from the duo and operated from another workshop. On October 21,2015, it was announced that MythBusters would air its 14th, the show aired its final episode on March 6,2016. On March 25, Discoverys sister network, Science, announced its intention of continuing the series with new hosts, the show, currently airing, is titled Mythbusters, The Search. Adam Savage has confirmed that he and his former cohosts have no intentions of reuniting for future team projects, MythBusters refers both to the name of the documentary and also the cast members who test the experiments. The series concept was created for the Discovery Channel as Tall Tales or True by Australian writer and producer Peter Rees of Beyond Productions in 2002, Discovery rejected the proposal initially because they had just commissioned a series on the same topic. Rees refined the pitch to focus on testing key elements of the rather than just retelling them. Discovery agreed to develop and co-produce a three-episode series pilot, Jamie Hyneman was one of a number of special effects artists who were asked to prepare a casting video for network consideration. Rees had interviewed him previously for a segment of the science series Beyond 2000 about the British/American robot combat television series Robot Wars. The highest rated regular episode featured two stories, straw through a tree, and talking to plants. The highest rated two hour special was Hollywood Myths, the highest rated Shark Week special was Jaws Myths which screened in 2005. During July 2006, an edited version of MythBusters began airing on BBC Two in the UK. The episodes shown on the European Discovery Channel sometimes include extra scenes not shown in the United States version, the 14th season, which premiered in January 2016, was the final season for the series. Adam Savage and Jamie Hyneman are the original MythBusters, and initially explored all the myths of the series using their experience with special effects. The two work at Hynemans effects workshop, M5 Industries, they use of his staff
MythBusters
–
MythBusters
MythBusters
–
Adam (left) and Jamie as keynote speakers at Symantec Vision 08.
MythBusters
–
Adam Savage and Jamie Hyneman at the
Discovery Channel Young Scientist Challenge pose with
Skulls Unlimited International 's Jay Villemarette and Joey Williams 2004.
MythBusters
–
MythBusters at the White House.
44.
Glare (vision)
–
Glare is difficulty seeing in the presence of bright light such as direct or reflected sunlight or artificial light such as car headlamps at night. Because of this, some cars include mirrors with automatic anti-glare functions, Glare is caused by a significant ratio of luminance between the task and the glare source. Factors such as the angle between the task and the source and eye adaptation have significant impacts on the experience of glare. Glare can be divided into two types, discomfort glare and disability glare. Discomfort glare results in a desire to look away from a bright light source or difficulty in seeing a task. Disability glare impairs the vision of objects without causing discomfort. This could arise for instance when driving westward at sunset, disability glare is often caused by the inter-reflection of light within the eyeball, reducing the contrast between task and glare source to the point where the task cannot be distinguished. When glare is so intense that vision is impaired, it is sometimes called dazzle. Reduction in contrast by scattering light in particles in the air, as when the headlights of a car illuminate the fog close to the vehicle, reduction in contrast between print and paper by reflection of the light source in the printed matter. An anti-reflective treatment on eyeglasses reduces the glare at night and glare from inside lights, some types of eyeglasses can reduce glare that occurs because of the imperfections on the surface of the eye. Light field measurements can be taken to reduce glare with digital post-processing, Glare is typically measured with luminance meters or luminance cameras, both of which are able to determine the luminance of objects within small solid angles. The glare of a scene i. e. visual field of view, is calculated from the luminance data of that scene. The CIE recommends the Unified glare rating as a measure of glare. Other glare calculation methods include CIBSE Glare Index, IES Glare Index, the unified glare rating is a measure of the glare in a given environment, proposed by Sorensen in 1987 and adopted by the International Commission on Illumination. It is basically the logarithm of the glare of all visible lamps, afterimage Lens flare Lyot stop Over-illumination Specular reflection Visual comfort probability Selective yellow
Glare (vision)
–
Glare from a
camera flash during a
Sumo fight
Glare (vision)
–
Example of a situation where glare can be problematic, if, for instance, the ability to determine the distance and speed of passing cars is reduced.
45.
Peripatetic school
–
The Peripatetic school was a school of philosophy in Ancient Greece. Its teachings derived from its founder, Aristotle, and peripatetic is an adjective ascribed to his followers, the school dates from around 335 BC when Aristotle began teaching in the Lyceum. It was an institution whose members conducted philosophical and scientific inquiries. After the middle of the 3rd century BC, the fell into a decline. Later members of the school concentrated on preserving and commenting on Aristotles works rather than extending them, the study of Aristotles works continued by scholars who were called Peripatetics through Late Antiquity, the Middle Ages, and the Renaissance. The term Peripatetic is a transliteration of the ancient Greek word περιπατητικός peripatêtikos, the Peripatetic school was actually known simply as the Peripatos. Aristotles school came to be so named because of the peripatoi of the Lyceum where the members met, the legend that the name came from Aristotles alleged habit of walking while lecturing may have started with Hermippus of Smyrna. Because of the association with the gymnasium, the school also came to be referred to simply as the Lyceum. Some modern scholars argue that the school did not become formally institutionalized until Theophrastus took it over, Aristotle did teach and lecture there, but there was also philosophical and scientific research done in partnership with other members of the school. It seems likely that many of the writings that have come down to us in Aristotles name were based on lectures he gave at the school. Among the members of the school in Aristotles time were Theophrastus, Phanias of Eresus, Eudemus of Rhodes, Clytus of Miletus, Aristoxenus, the doctrines of the Peripatetic school were those laid down by Aristotle, and henceforth maintained by his followers. Whereas Plato had sought to explain things with his theory of Forms, Philosophy to him meant science, and its aim was the recognition of the why in all things. Hence he endeavoured to attain to the grounds of things by induction. Logic either deals with appearances, and is then called dialectics, or of truth, all change or motion takes place in regard to substance, quantity, quality, and place. There are three kinds of substances – those alternately in motion and at rest, as the animals, those perpetually in motion, as the sky, the last, in themselves immovable and imperishable, are the source and origin of all motion. Among them there must be one first being, unchangeable, which acts without the intervention of any other being, all that is proceeds from it, it is the most perfect intelligence – God. The heavens are of a perfect and divine nature than other bodies. In the centre of the universe is the Earth, round, the stars, like the sky, beings of a higher nature, but of grosser matter, move by the impulse of the prime mover
Peripatetic school
–
Aristotle's School, a painting from the 1880s by Gustav Adolph Spangenberg
Peripatetic school
–
Aristotelianism
Peripatetic school
–
Aristotle and his disciples –
Alexander,
Demetrius,
Theophrastus, and
Strato; part of a fresco in the portico of the
National University of Athens.
46.
Aristotle
–
Aristotle was an ancient Greek philosopher and scientist born in the city of Stagira, Chalkidice, on the northern periphery of Classical Greece. His father, Nicomachus, died when Aristotle was a child, at seventeen or eighteen years of age, he joined Platos Academy in Athens and remained there until the age of thirty-seven. Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, teaching Alexander the Great gave Aristotle many opportunities and an abundance of supplies. He established a library in the Lyceum which aided in the production of many of his hundreds of books and he believed all peoples concepts and all of their knowledge was ultimately based on perception. Aristotles views on natural sciences represent the groundwork underlying many of his works, Aristotles views on physical science profoundly shaped medieval scholarship. Their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, some of Aristotles zoological observations, such as on the hectocotyl arm of the octopus, were not confirmed or refuted until the 19th century. His works contain the earliest known study of logic, which was incorporated in the late 19th century into modern formal logic. Aristotle was well known among medieval Muslim intellectuals and revered as The First Teacher and his ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotles philosophy continue to be the object of academic study today. Though Aristotle wrote many elegant treatises and dialogues – Cicero described his style as a river of gold – it is thought that only around a third of his original output has survived. Aristotle, whose means the best purpose, was born in 384 BC in Stagira, Chalcidice. His father Nicomachus was the physician to King Amyntas of Macedon. Aristotle was orphaned at a young age, although there is little information on Aristotles childhood, he probably spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy. At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Platos Academy and he remained there for nearly twenty years before leaving Athens in 348/47 BC. Aristotle then accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor, there, he traveled with Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island. Aristotle married Pythias, either Hermiass adoptive daughter or niece and she bore him a daughter, whom they also named Pythias. Soon after Hermias death, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander in 343 BC, Aristotle was appointed as the head of the royal academy of Macedon. During that time he gave not only to Alexander
Aristotle
–
Roman copy in marble of a Greek bronze bust of Aristotle by
Lysippus,
c. 330 BC. The
alabaster mantle is modern.
Aristotle
–
Aristotelianism
Aristotle
–
School of Aristotle in
Mieza,
Macedonia, Greece
Aristotle
–
"Aristotle" by
Francesco Hayez (1791–1882)
47.
Archytas
–
Archytas was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the founder of mathematical mechanics. Archytas was born in Tarentum, Magna Graecia and was the son of Mnesagoras or Histiaeus, for a while, he was taught by Philolaus, and was a teacher of mathematics to Eudoxus of Cnidus. Archytas and Eudoxus student was Menaechmus, as a Pythagorean, Archytas believed that only arithmetic, not geometry, could provide a basis for satisfactory proofs. Archytas is believed to be the founder of mathematical mechanics and this machine, which its inventor called The pigeon, may have been suspended on a wire or pivot for its flight. Archytas also wrote some lost works, as he was included by Vitruvius in the list of the authors of works of mechanics. Thomas Winter has suggested that the pseudo-Aristotelian Mechanical Problems is an important mechanical work by Archytas, not lost after all, Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not invent it. According to Eutocius, Archytas solved the problem of doubling the cube in his manner with a geometric construction, hippocrates of Chios before, reduced this problem to finding mean proportionals. Archytas theory of proportions is treated in book VIII of Euclids Elements, the Archytas curve, which he used in his solution of the doubling the cube problem, is named after him. Politically and militarily, Archytas appears to have been the dominant figure in Tarentum in his generation, the Tarentines elected him strategos, general, seven years in a row – a step that required them to violate their own rule against successive appointments. He was allegedly undefeated as a general, in Tarentine campaigns against their southern Italian neighbors, the Seventh Letter of Plato asserts that Archytas attempted to rescue Plato during his difficulties with Dionysius II of Syracuse. In his public career, Archytas had a reputation for virtue as well as efficacy, some scholars have argued that Archytas may have served as one model for Platos philosopher king, and that he influenced Platos political philosophy as expressed in The Republic and other works. Archytas may have drowned in a shipwreck in the shore of Mattinata, the poem, however, is difficult to interpret and it is not certain that the shipwrecked and Archytas are in fact the same person. The crater Archytas on the Moon is named in his honour and this rotation will cut out a portion of the cylinder forming the Archytas curve. A cone can go through the procedures also producing the Archytas curve. Archytas used his curve to determine the construction of a cube with a volume of half of that of a given cube, on line Huffman, Carl A. Archytas of Tarentum, Cambridge University Press,2005, ISBN 0-521-83746-4 Huffman, Carl. OConnor, John J. Robertson, Edmund F. Archytas, MacTutor History of Mathematics archive, texto en PDF, mediante registro, en Hybris, revista de filosofía
Archytas
–
Bust from the
Villa of the Papyri in
Herculaneum, once identified as Archytas, now thought to be
Pythagoras
48.
Pappus of Alexandria
–
Pappus of Alexandria was one of the last great Alexandrian mathematicians of Antiquity, known for his Synagoge or Collection, and for Pappuss hexagon theorem in projective geometry. Nothing is known of his life, other than, that he had a son named Hermodorus, Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a range of topics, including geometry, recreational mathematics, doubling the cube, polygons. Pappus flourished in the 4th century AD, in a period of general stagnation in mathematical studies, he stands out as a remarkable exception. In this respect the fate of Pappus strikingly resembles that of Diophantus, in his surviving writings, Pappus gives no indication of the date of the authors whose works he makes use of, or of the time at which he himself wrote. If no other information were available, all that could be known would be that he was later than Ptolemy, whom he quotes, and earlier than Proclus. The Suda states that Pappus was of the age as Theon of Alexandria. A different date is given by a note to a late 10th-century manuscript, which states, next to an entry on Emperor Diocletian. This works out as October 18,320 AD, and so Pappus must have flourished c.320 AD. The great work of Pappus, in eight books and titled Synagoge or Collection, has not survived in complete form, the first book is lost, and the rest have suffered considerably. The Suda enumerates other works of Pappus, Χωρογραφία οἰκουμενική, commentary on the 4 books of Ptolemys Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ, Pappus himself mentions another commentary of his own on the Ἀνάλημμα of Diodorus of Alexandria. Pappus also wrote commentaries on Euclids Elements, and on Ptolemys Ἁρμονικά and these discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the books as valuable, for they set forth clearly an outline of the contents. From these introductions one can judge of the style of Pappuss writing, heath also found his characteristic exactness made his Collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us. The portions of Collection which has survived can be summarized as follows and we can only conjecture that the lost Book I, like Book II, was concerned with arithmetic, Book III being clearly introduced as beginning a new subject. The whole of Book II discusses a method of multiplication from a book by Apollonius of Perga. The final propositions deal with multiplying together the values of Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2*1054 and 2*1038. Book III contains geometrical problems, plane and solid, on the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure
Pappus of Alexandria
–
Title page of Pappus's Mathematicae Collectiones, translated into Latin by
Federico Commandino (1589).
Pappus of Alexandria
–
Mathematicae collectiones, 1660
49.
Odometer
–
An odometer or odograph is an instrument that indicates distance travelled by a vehicle, such as a bicycle or automobile. The device may be electronic, mechanical, or a combination of the two, the noun derives from the Greek words hodós and métron. Possibly the first evidence for the use of an odometer can be found in the works of the ancient Roman Pliny, both authors list the distances of routes traveled by Alexander the Great as by his bematists Diognetus and Baeton. However, the accuracy of the bematistss measurements rather indicates the use of a mechanical device. 2% from the actual distance. From the nine surviving bematists measurements in Plinys Naturalis Historia eight show a deviation of less than 5% from the actual distance, three of them being within 1%. An odometer for measuring distance was first described by Vitruvius around 27 and 23 BC, hero of Alexandria describes a similar device in chapter 34 of his Dioptra. Some researchers have speculated that the device might have included technology similar to that of the Greek Antikythera mechanism, the odometer of Vitruvius was based on chariot wheels of 4 feet diameter turning 400 times in one Roman mile. For each revolution a pin on the axle engaged a 400 tooth cogwheel thus turning it one complete revolution per mile and this engaged another gear with holes along the circumference, where pebbles were located, that were to drop one by one into a box. The distance traveled would thus be given simply by counting the number of pebbles, whether this instrument was ever built at the time is disputed. Leonardo da Vinci later tried to build it himself according to the description, however, in 1981 engineer Andre Sleeswyk built his own replica, replacing the square-toothed gear designs of da Vinci with the triangular, pointed teeth found in the Antikythera mechanism. With this modification, the Vitruvius odometer functioned perfectly, the odometer was also independently invented in ancient China, possibly by the prolific inventor and early scientist Zhang Heng of the Han Dynasty. By the 3rd century, the Chinese had termed the device as the jì lĭ gŭ chē, there is speculation that some time in the 1st century BC, the beating of drums and gongs were mechanically-driven by working automatically off the rotation of the road-wheels. This might have actually been the design of one Loxia Hong, the odometer was used also in subsequent periods of Chinese history. In the historical text of the Jin Shu, the oldest part of the compiled text, the passage in the Jin Shu expanded upon this, explaining that it took a similar form to the mechanical device of the south-pointing chariot invented by Ma Jun. As recorded in the Song Shi of the Song Dynasty, the odometer and south-pointing chariot were combined into one wheeled device by engineers of the 9th century, 11th century, and 12th century. The Sun Tzu Suan Ching, dated from the 3rd century to 5th century, the historical text of the Song Shi, recording the people and events of the Chinese Song Dynasty, also mentioned the odometer used in that period. At the completion of every li, the figure of a man in the lower storey strikes a drum, at the completion of every ten li. The carriage-pole ends in a phoenix-head, and the carriage is drawn by four horses, the escort was formerly of 18 men, but in the 4th year of the Yung-Hsi reign-period the emperor Thai Tsung increased it to 30
Odometer
–
An electronic odometer (below) with digital display
Odometer
–
A Smiths speedometer from the 1920s showing odometer and trip meter.
Odometer
–
A
Hubodometer on a wheel of a semitrailer
Odometer
–
A
Han Dynasty stone rubbing of a horse-drawn odometer cart.
50.
Dialogue
–
Dialogue is a written or spoken conversational exchange between two or more people, and a literary and theatrical form that depicts such an exchange. In the 20th century, philosophical treatments of dialogue emerged from thinkers including Mikhail Bakhtin, Paulo Freire, Martin Buber, although diverging in many details, these thinkers have articulated a holistic concept of dialogue as a multi-dimensional, dynamic and context-dependent process of creating meaning. Educators such as Freire and Ramón Flecha have also developed a body of theory, the term dialogue stems from the Greek διάλογος, its roots are διά and λόγος. The first extant author who uses the term is Plato, in whose works it is associated with the art of dialectic. Latin took over the word as dialogus, in the West, Plato has commonly been credited with the systematic use of dialogue as an independent literary form. Ancient sources indicate, however, that the Platonic dialogue had its foundations in the mime and these works, admired and imitated by Plato, have not survived and we have only the vaguest idea of how they may have been performed. The Mimes of Herodas, which were found in a papyrus in 1891, Plato further simplified the form and reduced it to pure argumentative conversation, while leaving intact the amusing element of character-drawing. By about 400 BC he had perfected the Socratic dialogue, all his extant writings, except the Apology and Epistles, use this form. Following Plato, the became a major literary genre in antiquity. Soon after Plato, Xenophon wrote his own Symposium, also, Two French writers of eminence borrowed the title of Lucian’s most famous collection, both Fontenelle and Fénelon prepared Dialogues des morts. Contemporaneously, in 1688, the French philosopher Nicolas Malebranche published his Dialogues on Metaphysics and Religion, in English non-dramatic literature the dialogue did not see extensive use until Berkeley employed it, in 1713, for his treatise, Three Dialogues between Hylas and Philonous. His contemporary, the Scottish philosopher David Hume wrote Dialogues Concerning Natural Religion, a prominent 19th-century example of literary dialogue was Landor’s Imaginary Conversations. In Germany, Wieland adopted this form for several important satirical works published between 1780 and 1799, in Spanish literature, the Dialogues of Valdés and those on Painting by Vincenzo Carducci are celebrated. Italian writers of collections of dialogues, following Platos model, include Torquato Tasso, Galileo, Galiani, Leopardi, in the 19th century, the French returned to the original application of dialogue. English writers including Anstey Guthrie also adopted the form, but these seem to have found less of a popular following among the English than their counterparts written by French authors. Authors who have employed it include George Santayana, in his eminent Dialogues in Limbo. Also Edith Stein and Iris Murdoch used the dialogue form, Stein imagined a dialogue between Edmund Husserl and Thomas Aquinas. Murdoch included not only Socrates and Alcibiades as interlocutors in her work Acastos, Two Platonic Dialogues, more recently Timothy Williamson wrote Tetralogue, a philosophical exchange on a train between four people with radically different epistemological views
Dialogue
–
Oldest extant text of Plato's
Republic
Dialogue
–
Frontispiece and title page of
Galileo 's
Dialogue Concerning the Two Chief World Systems, 1632
Dialogue
–
David Bohm, a leading 20th-century thinker on dialogue.
Dialogue
–
A classroom dialogue at
Shimer College.
51.
Thales
–
Thales of Miletus was a pre-Socratic Greek/Phoenician philosopher, mathematician and astronomer from Miletus in Asia Minor. He was one of the Seven Sages of Greece, Thales is recognized for breaking from the use of mythology to explain the world and the universe, and instead explaining natural objects and phenomena by theories and hypothesis, i. e. science. Aristotle reported Thales hypothesis that the principle of nature and the nature of matter was a single material substance. In mathematics, Thales used geometry to calculate the heights of pyramids and he is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to Thales theorem. He is the first known individual to whom a mathematical discovery has been attributed, the ancient source, Apollodorus of Athens, writing during the 2nd century BCE, thought Thales was born about the year 625 BCE. The dates of Thales life are not exactly known, but are roughly established by a few events mentioned in the sources. According to Herodotus Thales predicted the eclipse of May 28,585 BC. Diogenes Laërtius quotes the chronicle of Apollodorus of Athens as saying that Thales died at the age of 78 during the 58th Olympiad and attributes his death to heat stroke while watching the games. Plutarch had earlier told this version, Solon visited Thales and asked him why he remained single, nevertheless, several years later, anxious for family, he adopted his nephew Cybisthus. Thales involved himself in many activities, taking the role of an innovator, some say that he left no writings, others say that he wrote On the Solstice and On the Equinox. Diogenes Laërtius quotes two letters from Thales, one to Pherecydes of Syros, offering to review his book on religion, Thales identifies the Milesians as Athenian colonists. He was aware of the existence of the lodestone, and was the first to be connected to knowledge of this in history, according to Aristotle, Thales thought lodestones had souls, because iron is attracted to them. According to Hieronymus, historically quoted by Diogenes Laertius, Thales found the height of pyramids by comparison between the lengths of the shadows cast by a person and by the pyramids, several anecdotes suggest Thales was not just a philosopher, but also a businessman. A story, with different versions, recounts how Thales achieved riches from an olive harvest by prediction of the weather, in one version, he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. Thales’ political life had mainly to do with the involvement of the Ionians in the defense of Anatolia against the power of the Persians. In neighbouring Lydia, a king had come to power, Croesus and he had conquered most of the states of coastal Anatolia, including the cities of the Ionians. The story is told in Herodotus, the war endured for five years, but in the sixth an eclipse of the Sun spontaneously halted a battle in progress. It seems that Thales had predicted this solar eclipse, the Seven Sages were most likely already in existence, as Croesus was also heavily influenced by Solon of Athens, another sage
Thales
–
Thales of Miletus
Thales
–
An olive mill and an olive press dating from Roman times in
Capernaum, Israel.
Thales
–
Total
eclipse of the
Sun
Thales
–
The Ionic Stoa on the Sacred Way in Miletus
52.
Antikythera mechanism
–
Found housed in a 340 millimetres ×180 millimetres ×90 millimetres wooden box, the device is a complex clockwork mechanism composed of at least 30 meshing bronze gears. Its remains were found as one lump, later separated in three fragments, which are now divided into 82 separate fragments after conservation works. Four of these fragments contain gears, while inscriptions are found on many others, the largest gear is approximately 140 millimetres in diameter and originally had 223 teeth. The artefact was recovered probably in July 1901 from the Antikythera shipwreck off the Greek island of Antikythera. Believed to have designed and constructed by Greek scientists, the instrument has been dated either between 150 and 100 BC, or, according to a more recent view, in 205 BC. All known fragments of the Antikythera mechanism are kept at the National Archaeological Museum, in Athens, the Antikythera mechanism was discovered in 45 metres of water in the Antikythera shipwreck off Point Glyphadia on the Greek island of Antikythera. All were transferred to the National Museum of Archaeology in Athens for storage, merely a lump of corroded bronze and wood at the time, the mechanism went unnoticed for two years while museum staff worked on piecing together more obvious statues. On 17 May 1902, archaeologist Valerios Stais was examining the finds, investigations into the object were soon dropped until British science historian and Yale University professor, Derek J. de Solla Price became interested in it in 1951. In 1971, both Price and Greek nuclear physicist Charalampos Karakalos made X-ray and gamma-ray images of the 82 fragments, Price published an extensive 70-page paper on their findings in 1974. Generally referred to as the first known computer, the quality and complexity of the mechanisms manufacture suggests it has undiscovered predecessors made during the Hellenistic period. Its construction relied upon theories of astronomy and mathematics developed by Greek astronomers, in 1974, Derek de Solla Price concluded from gear settings and inscriptions on the mechanisms faces that it was made about 87 BC and lost only a few years later. Jacques Cousteau and associates visited the wreck in 1976 and recovered coins dated to between 76 and 67 BC, though its advanced state of corrosion has made it impossible to perform an accurate compositional analysis, it is believed the device was made of a low-tin bronze alloy. All its instructions are written in Koine Greek, and the consensus among scholars is that the mechanism was made in the Greek-speaking world, syracuse was a colony of Corinth and the home of Archimedes, which might imply a connection with the school of Archimedes. With its many scrolls of art and science, it was second in only to the Library of Alexandria during the Hellenistic period. A busy trading port in antiquity, Rhodes was also a centre of astronomy and mechanical engineering, home to the astronomer Hipparchus and that the mechanism uses Hipparchuss theory for the motion of the moon suggests the possibility he may have designed, or at least worked on it. He regarded the Antikythera mechanism as more valuable than the Mona Lisa, in 2014, a study by Carman and Evans argued for a new dating of approximately 200 BC. Moreover, according to Carman and Evans, the Babylonian arithmetic style of prediction fits much better with the devices predictive models than the traditional Greek trigonometric style, further dives are being undertaken in the hope of discovering more of the mechanism. The original mechanism apparently came out of the Mediterranean as a single encrusted piece, soon afterward it fractured into three major pieces
Antikythera mechanism
–
The Antikythera mechanism (Fragment A – front)
Antikythera mechanism
–
Computer-generated front panel of the Freeth model
Antikythera mechanism
–
Front panel of a 2007 reproduction
Antikythera mechanism
–
Computer-generated back panel
53.
Integral
–
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
Integral
–
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
54.
Measurement of a Circle
–
Measurement of a Circle is a treatise that consists of three propositions by Archimedes. The treatise is only a fraction of what was a longer work, any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r. This proposition is proved by the method of exhaustion, proposition two states, The area of a circle is to the square on its diameter as 11 to 14. This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition, proposition three states, The ratio of the circumference of any circle to its diameter is greater than 31071 but less than 317. This approximates what we now call the mathematical constant π and he found these bounds on the value of π by inscribing and circumscribing a circle with two similar 96-sided regular polygons. This proposition also contains accurate approximations to the root of 3 and other larger non-perfect square roots, however. He gives the upper and lower bounds to √3 as 1351780 >3 >265153. Discussion of this goes back at least to Thomas Fantet de Lagny, FRS in 1723, but was treated more explicitly by Hieronymus Georg Zeuthen.4,7. Although only one route to the bounds is mentioned, in there are two others, making the bounds almost inescapable however the method is worked. But the bounds can also be produced by a geometrical construction suggested by Archimedes Stomachion in the setting of the regular dodecagon. In this case, the task is to give rational approximations to the tangent of π/12
Measurement of a Circle
–
The circle and the triangle are equal in area.
55.
Regular hexagon
–
In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
Regular hexagon
–
Giants causeway closeup
Regular hexagon
–
A regular hexagon
Regular hexagon
–
The ideal crystalline structure of
graphene is a hexagonal grid.
Regular hexagon
–
Assembled
E-ELT mirror segments
56.
Radius
–
Remote Authentication Dial-In User Service is a networking protocol that provides centralized Authentication, Authorization, and Accounting management for users who connect and use a network service. RADIUS was developed by Livingston Enterprises, Inc. in 1991 as an access server authentication and accounting protocol and these networks may incorporate modems, DSL, access points, VPNs, network ports, web servers, etc. RADIUS is a protocol that runs in the application layer. Network access servers, the gateways that control access to a network, RADIUS is often the back-end of choice for 802. 1X authentication as well. The RADIUS server is usually a background process running on a UNIX or Microsoft Windows server, RADIUS is a AAA protocol which manages network access in the following two-step process, also known as a AAA transaction. AAA stands for authentication, authorization and accounting, Authentication and authorization characteristics in RADIUS are described in RFC2865 while accounting is described by RFC2866. The user or machine sends a request to a Network Access Server to gain access to a network resource using access credentials. The credentials are passed to the NAS device via the link-layer protocol - for example, in turn, the NAS sends a RADIUS Access Request message to the RADIUS server, requesting authorization to grant access via the RADIUS protocol. This request includes access credentials, typically in the form of username, the RADIUS server checks that the information is correct using authentication schemes such as PAP, CHAP or EAP. Historically, RADIUS servers checked the users information against a locally stored flat file database, modern RADIUS servers can do this, or can refer to external sources — commonly SQL, Kerberos, LDAP, or Active Directory servers — to verify the users credentials. The RADIUS server then returns one of three responses to the NAS, 1) Access Reject, 2) Access Challenge, or 3) Access Accept, Access Reject The user is unconditionally denied access to all requested network resources. Reasons may include failure to provide proof of identification or an unknown or inactive user account, Access Challenge Requests additional information from the user such as a secondary password, PIN, token, or card. Access Accept The user is granted access, once the user is authenticated, the RADIUS server will often check that the user is authorized to use the network service requested. A given user may be allowed to use a wireless network. Again, this information may be stored locally on the RADIUS server, each of these three RADIUS responses may include a Reply-Message attribute which may give a reason for the rejection, the prompt for the challenge, or a welcome message for the accept. The text in the attribute can be passed on to the user in a web page. Authorization attributes are conveyed to the NAS stipulating terms of access to be granted and this might be with a customizable login prompt, where the user is expected to enter their username and password. Alternatively, the user use a link framing protocol such as the Point-to-Point Protocol
Radius
57.
Archimedean property
–
Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder. An algebraic structure in any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, an ordered group that is Archimedean is an Archimedean group. This can be made precise in various contexts with different formulations. The concept was named by Otto Stolz after the ancient Greek geometer, the Archimedean property appears in Book V of Euclids Elements as Definition 4, Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. Because Archimedes credited it to Eudoxus of Cnidus it is known as the Theorem of Eudoxus or the Eudoxus axiom. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs, Let x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y if, for natural number n, the multiple nx is less than y, that is. The group G is Archimedean if there is no x, y such that x is infinitesimal with respect to y. Additionally, if K is a structure with a unit — for example. If x is infinitesimal with respect to 1, then x is an infinitesimal element, likewise, if y is infinite with respect to 1, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements, an ordered field has some additional properties. One may assume that the numbers are contained in the field. If x is infinitesimal, then 1/x is infinite, and vice versa, therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. If x is infinitesimal and r is a number, then r x is also infinitesimal. As a result, given an element c, the three numbers c/2, c, and 2c are either all infinitesimal or all non-infinitesimal
Archimedean property
–
Illustration of the Archimedean property.
58.
Square root
–
In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss
Square root
–
First leaf of the complex square root
Square root
–
The mathematical expression 'The (principal) square root of x"
59.
Iteration
–
Iteration is the act of repeating a process, either to generate an unbounded sequence of outcomes, or with the aim of approaching a desired goal, target or result. Each repetition of the process is called an iteration. In the context of mathematics or computer science, iteration is a building block of algorithms. Iteration in mathematics may refer to the process of iterating a function i. e. applying a function repeatedly, iteration of apparently simple functions can produce complex behaviours and difficult problems - for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate solutions to certain mathematical problems. Newtons method is an example of an iterative method, manual calculation of a numbers square root is a common use and a well-known example. Iteration in computing is the marking out of a block of statements within a computer program for a defined number of repetitions. That block of statements is said to be iterated, a computer scientist might also refer to block of statements as an iteration. In the example above, the line of code is using the value of i as it increments and this idea is found in the old adage, Practice makes perfect. Unlike computing and math, educational iterations are not predetermined, instead, in algorithmic situations, recursion and iteration can be employed to the same effect. Some types of programming languages, known as functional programming languages, are designed such that they do not set up block of statements for explicit repetition as with the for loop, instead, those programming languages exclusively use recursion. Each piece of work will be divided repeatedly until the amount of work is as small as it can possibly be, the algorithm then reverses and reassembles the pieces into a complete whole. The classic example of recursion is in list-sorting algorithms such as Merge Sort, the code below is an example of a recursive algorithm in the Scheme programming language that will output the same result as the pseudocode under the previous heading. In Object-Oriented Programming, an iterator is an object that ensures iteration is executed in the way for a range of different data structures, saving time. An iteratee is an abstraction which accepts or rejects data during an iteration, recursion Fractal Iterated function Infinite compositions of analytic functions
Iteration
–
A pentagon iteration. Connecting alternate corners of a regular pentagon produces a
pentagram which encloses a smaller inverted pentagon. Iterating the process produces a sequence of nested pentagons and pentagrams and also demonstrates
recursion.
60.
The Quadrature of the Parabola
–
The Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century BC. The statement of the problem used the method of exhaustion, Archimedes may have dissected the area into infinitely many triangles whose areas form a geometric progression. He computes the sum of the geometric series, and proves that this is the area of the parabolic segment. A parabolic segment is the bounded by a parabola and line. To find the area of a segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the chord of the parabola. By Proposition 1, a line from the third vertex drawn parallel to the axis divides the chord into equal segments, the main theorem claims that the area of the parabolic segment is 4/3 that of the inscribed triangle. Archimedes gives two proofs of the main theorem, the first uses abstract mechanics, with Archimedes arguing that the weight of the segment will balance the weight of the triangle when placed on an appropriate lever. The second, more famous proof uses pure geometry, specifically the method of exhaustion, of the twenty-four propositions, the first three are quoted without proof from Euclids Elements of Conics. The main idea of the proof is the dissection of the segment into infinitely many triangles. Each of these triangles is inscribed in its own segment in the same way that the blue triangle is inscribed in the large segment. In propositions eighteen through twenty-one, Archimedes proves that the area of each triangle is one eighth of the area of the blue triangle. Using the method of exhaustion, it follows that the area of the parabolic segment is given by Area = T +2 +4 +8 + ⋯. This simplifies to give Area = T, to complete the proof, Archimedes shows that 1 +14 +116 +164 + ⋯ =43. The formula above is a geometric series—each successive term is one fourth of the previous term, in modern mathematics, that formula is a special case of the sum formula for a geometric series. Archimedes evaluates the sum using a geometric method, illustrated in the adjacent picture. This picture shows a square which has been dissected into an infinity of smaller squares. Each successive purple square has one fourth the area of the previous square, however, the purple squares are congruent to either set of yellow squares, and so cover 1/3 of the area of the unit square
The Quadrature of the Parabola
–
A parabolic segment.
61.
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
62.
Series (mathematics)
–
In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
Series (mathematics)
–
Illustration of 3
geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.
63.
Geometric series
–
In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 12 +14 +18 +116 + ⋯ is geometric, Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the development of calculus. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a series using only two terms, r and a. The term r is the ratio, and a is the first term of the series. In the case above, where r is one half, the series has the sum one, if r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, if r is equal to one, all of the terms of the series are the same. If r is one the terms take two values alternately. The sum of the oscillates between two values. This is a different type of divergence and again the series has no sum, see for example Grandis series,1 −1 +1 −1 + ···. The sum can be computed using the self-similarity of the series, consider the sum of the following geometric series, s =1 +23 +49 +827 + ⋯. This series has common ratio 2/3, if we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on,23 s =23 +49 +827 +1681 + ⋯. This new series is the same as the original, except that the first term is missing, subtracting the new series s from the original series s cancels every term in the original but the first, s −23 s =1, so s =3. A similar technique can be used to evaluate any self-similar expression, as n goes to infinity, the absolute value of r must be less than one for the series to converge. When a =1, this can be simplified to 1 + r + r 2 + r 3 + ⋯ =11 − r, the formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one. Since = 1−rn+1 and rn+1 →0 for | r | <1, convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function, g = r K1 − r
Geometric series
–
Each of the purple squares has 1/4 of the area of the next larger square (1/2× 1/2 = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.
64.
Secant line
–
In geometry, a secant of a curve is a line that intersects two points on the curve. A chord is the interval of a secant that lies between the points at which it intersects the curve, the word secant comes from the Latin word secare, meaning to cut. A secant may be used to approximate the tangent to a curve, at some point P. If the secant to a curve is defined by two points, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P. As a consequence, one could say that the limit, as Q approaches P, of the secants slope, in calculus, this idea is the basis of the geometric definition of the derivative
Secant line
–
Common lines and line segments on a circle, including the secant line
65.
Polyhedron
–
In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
Polyhedron
–
Convex polyhedron blocks on display at the
Universum museum in Mexico City
Polyhedron
–
Regular tetrahedron
66.
Refraction
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Refraction is the change in direction of wave propagation due to a change in its transmission medium. The phenomenon is explained by the conservation of energy and the conservation of momentum, due to the change of medium, the phase velocity of the wave is changed but its frequency remains constant. This is most commonly observed when a wave passes from one medium to another at any other than 0° from the normal. In optics, refraction is a phenomenon that occurs when waves travel from a medium with a given refractive index to a medium with another at an oblique angle. At the boundary between the media, the phase velocity is altered, usually causing a change in direction. Its wavelength increases or decreases, but its frequency remains constant, for example, a light ray will refract as it enters and leaves glass, assuming there is a change in refractive index. A ray traveling along the normal will change speed, but not direction, refraction still occurs in this case. Understanding of this led to the invention of lenses and the refracting telescope. Refraction can be seen looking into a bowl of water. Air has a index of about 1.0003. If a person looks at an object, such as a pencil or straw, which is placed at a slant, partially in the water. This is due to the bending of light rays as they move from the water to the air, once the rays reach the eye, the eye traces them back as straight lines. The lines of sight intersect at a position than where the actual rays originated. This causes the pencil to appear higher and the water to appear shallower than it really is, the depth that the water appears to be when viewed from above is known as the apparent depth. This is an important consideration for spearfishing from the surface because it will make the fish appear to be in a different place. Conversely, an object above the water has a higher apparent height when viewed from below the water, the opposite correction must be made by an archer fish. For small angles of incidence, the ratio of apparent to real depth is the ratio of the indexes of air to that of water. But, as the angle of incidence approaches 90o, the apparent depth approaches zero, albeit reflection increases, the diagram on the right shows an example of refraction in water waves
Refraction
–
Light on air–plexi surface in this experiment undergoes refraction (lower ray) and
reflection (upper ray).
Refraction
–
Refraction in a glass of water. The image is flipped.
Refraction
–
An image of the
Golden Gate Bridge is refracted and bent by many differing three-dimensional drops of water.
Refraction
67.
Gerard of Cremona
–
Gerard of Cremona was an Italian translator of scientific books from Arabic into Latin. He worked in Toledo, Kingdom of Castile and obtained the Arabic books in the libraries at Toledo, some of the books had been originally written in Greek and were unavailable in Greek or Latin in Europe at the time. One of Gerards most famous translations is of Ptolemys Almagest from Arabic texts found in Toledo, Gerard was born in Cremona in northern Italy. Dissatisfied with the philosophies of his Italian teachers, Gerard went to Toledo. The first Latin translation was made, from the Greek around 1160 in Sicily, although we do not have detailed information of the date when Gerard went to Castile, it was no later than 1144. Toledo remained a capital, insofar as its rulers protected the large Jewish and Muslim quarters. One of the scholars associated with Toledo was Rabbi Abraham ibn Ezra. The Muslim and Jewish inhabitants of Toledo adopted the language and many customs of their conquerors, the city was full of libraries and manuscripts, and was one of the few places in medieval Europe where a Christian could be exposed to Arabic language and culture. In Toledo Gerard devoted the remainder of his life to making Latin translations from the Arabic scientific literature, Gerard of Cremonas Latin translation of the Arabic version of Ptolemy’s Almagest made c.1175 was the most widely known in Western Europe before the Renaissance. George of Trebizond and then Johannes Regiomontanus retranslated it from the Greek original in the fifteenth century, the Almagest formed the basis for Western astronomy until it was eclipsed by the theories of Copernicus. Gerard edited for Latin readers the Tables of Toledo, the most accurate compilation of data ever seen in Europe at the time. The Tables were partly the work of Al-Zarqali, known to the West as Arzachel, al-Farabi, the Islamic second teacher after Aristotle, wrote hundreds of treatises. His book on the sciences, Kitab lhsa al Ulum, discussed classification and fundamental principles of science in a unique, Gerard rendered it as De scientiis. Gerard translated Euclid’s Geometry and Alfraganuss Elements of Astronomy, Gerard also composed original treatises on algebra, arithmetic and astrology. In the astrology text, longitudes are reckoned both from Cremona and Toledo, the later Gerard focused on translating medical texts rather than astronomical texts, but the two translators have understandably been confused with one another. His translations from works of Avicenna are said to have made by order of the emperor Frederick II. The attribution of the Theorica to Gerard of Sabbionetta is not well supported by manuscript evidence, Toledo School of Translators Latin translations of the 12th century Islamic contributions to Medieval Europe Burnett, Charles. The Coherence of the Arabic-Latin Translation Program in Toledo in the Twelfth Century, arabian Medicine and Its Influence on the Middle Ages
Gerard of Cremona
–
European depiction of the Persian physician
Rhazes, in Gerard of Cremona's "Recueil des traités de médecine" 1250-1260. Gerard de Cremona translated numerous works by Arab scholars.
Gerard of Cremona
–
Al-Razi 's Recueil des traités de médecine translated by Gerard of Cremona, second half of the 13th century.
Gerard of Cremona
–
Theorica Platenarum by Gerard of Cremona, 13th century.
68.
Torque
–
Torque, moment, or moment of force is rotational force. Just as a force is a push or a pull. Loosely speaking, torque is a measure of the force on an object such as a bolt or a flywheel. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque that loosens or tightens the nut or bolt, the symbol for torque is typically τ, the lowercase Greek letter tau. When it is called moment of force, it is denoted by M. The SI unit for torque is the newton metre, for more on the units of torque, see Units. This article follows US physics terminology in its use of the word torque, in the UK and in US mechanical engineering, this is called moment of force, usually shortened to moment. In US physics and UK physics terminology these terms are interchangeable, unlike in US mechanical engineering, Torque is defined mathematically as the rate of change of angular momentum of an object. The definition of states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the term used for the tendency of one or more applied forces to rotate an object about an axis. For example, a force applied to a shaft causing acceleration, such as a drill bit accelerating from rest. By contrast, a force on a beam produces a moment, but since the angular momentum of the beam is not changing. Similarly with any force couple on an object that has no change to its angular momentum and this article follows the US physics terminology by calling all moments by the term torque, whether or not they cause the angular momentum of an object to change. The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers, the term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884. A force applied at an angle to a lever multiplied by its distance from the levers fulcrum is its torque. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. More generally, the torque on a particle can be defined as the product, τ = r × F, where r is the particles position vector relative to the fulcrum. Alternatively, τ = r F ⊥, where F⊥ is the amount of force directed perpendicularly to the position of the particle, any force directed parallel to the particles position vector does not produce a torque
Torque
69.
Center of mass
–
The distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates. Calculations in mechanics are simplified when formulated with respect to the center of mass. It is a point where entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the equivalent of a given object for application of Newtons laws of motion. In the case of a rigid body, the center of mass is fixed in relation to the body. The center of mass may be located outside the body, as is sometimes the case for hollow or open-shaped objects. In the case of a distribution of separate bodies, such as the planets of the Solar System, in orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. The center of mass frame is a frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system. The concept of center of mass in the form of the center of gravity was first introduced by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, in work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes, Newtons second law is reformulated with respect to the center of mass in Eulers first law. The center of mass is the point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the location of a distribution of mass in space. Solving this equation for R yields the formula R =1 M ∑ i =1 n m i r i, solve this equation for the coordinates R to obtain R =1 M ∭ Q ρ r d V, where M is the total mass in the volume. If a continuous mass distribution has density, which means ρ is constant. The center of mass is not generally the point at which a plane separates the distribution of mass into two equal halves, in analogy with statistics, the median is not the same as the mean. The coordinates R of the center of mass of a system, P1 and P2, with masses m1. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point
Center of mass
–
This child's toy uses the principles of center of mass to keep balance on a finger.
Center of mass
–
Estimated center of mass/gravity (blue sphere) of a gymnast at the end of performing a cartwheel. Notice center is outside the body in this position.
70.
Locus (mathematics)
–
In geometry, a locus is a set of points, whose location satisfies or is determined by one or more specified conditions. Until the beginning of 20th century, a shape was not considered as an infinite set of points, rather. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a distance of a fixed point. In contrast to the view, the old formulation avoids considering infinite collections. Once set theory became the universal basis over which the mathematics is built. Examples from plane geometry include, The set of points equidistant from two points is a perpendicular bisector to the segment connecting the two points. The set of points equidistant from two lines cross is the angle bisector. All conic sections are loci, Parabola, the set of points equidistant from a single point, Circle, the set of points for which the distance from a single point is constant. The set of points for each of which the ratio of the distances to two given foci is a constant is referred to as a Circle of Apollonius. Hyperbola, the set of points for each of which the value of the difference between the distances to two given foci is a constant. Ellipse, the set of points for each of which the sum of the distances to two given foci is a constant, the circle is the special case in which the two foci coincide with each other. Other examples of loci appear in areas of mathematics. For example, in dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. Proof that all the points on the given shape satisfy the conditions and we find the locus of the points P that have a given ratio of distances k = d1/d2 to two given points. In this example we choose k=3, A and B as the fixed points and it is the circle of Apollonius defined by these values of k, A, and B. A triangle ABC has a side with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal and we choose an orthonormal coordinate system such that A, B. C is the third vertex
Locus (mathematics)
–
(distance PA) = 3.(distance PB)
71.
Polar coordinate system
–
The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, the concepts of angle and radius were already used by ancient peoples of the first millennium BC. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle, the Greek work, however, did not extend to a full coordinate system. From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca —and its distance—from any location on the Earth, from the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. There are various accounts of the introduction of polar coordinates as part of a coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidges Origin of Polar Coordinates, grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs, in Method of Fluxions, Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the Seventh Manner, For Spirals, and nine other coordinate systems. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole, Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoullis work extended to finding the radius of curvature of curves expressed in these coordinates, the actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacocks 1816 translation of Lacroixs Differential and Integral Calculus, alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them. The radial coordinate is often denoted by r or ρ, the angular coordinate is specified as ϕ by ISO standard 31-11. Angles in polar notation are generally expressed in degrees or radians. Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics, in many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis. In mathematical literature, the axis is often drawn horizontal. Adding any number of turns to the angular coordinate does not change the corresponding direction. Also, a radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the point can be expressed with an infinite number of different polar coordinates or
Polar coordinate system
–
Hipparchus
Polar coordinate system
–
Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).
Polar coordinate system
–
A
planimeter, which mechanically computes polar integrals
72.
Real number
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
Real number
–
A symbol of the set of real numbers (ℝ)
73.
Curve
–
In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
Curve
–
Megalithic art from Newgrange showing an early interest in curves
Curve
–
A
parabola, a simple example of a curve
74.
Point (geometry)
–
In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
Point (geometry)
–
Projecting a
sphere to a
plane.
75.
Circumscribe
–
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Circumscribe
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
76.
Diameter
–
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle, both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is called the diameter. In this sense one speaks of the rather than a diameter, because all diameters of a circle or sphere have the same length. Both quantities can be calculated efficiently using rotating calipers, for a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. For an ellipse, the terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse, for example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis, the word diameter is derived from Greek διάμετρος, diameter of a circle, from διά, across, through and μέτρον, measure. It is often abbreviated DIA, dia, d, or ⌀, the definitions given above are only valid for circles, spheres and convex shapes. However, they are cases of a more general definition that is valid for any kind of n-dimensional convex or non-convex object. The diameter of a subset of a space is the least upper bound of the set of all distances between pairs of points in the subset. So, if A is the subset, the diameter is sup, if the distance function d is viewed here as having codomain R, this implies that the diameter of the empty set equals −∞. Some authors prefer to treat the empty set as a case, assigning it a diameter equal to 0. For any solid object or set of scattered points in n-dimensional Euclidean space, in medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the supremum of the set of all distances between pairs of points in the object. In differential geometry, the diameter is an important global Riemannian invariant, the symbol or variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke. In Unicode it is defined as U+2300 ⌀ Diameter sign, on an Apple Macintosh, the diameter symbol can be entered via the character palette, where it can be found in the Technical Symbols category. The character will not display correctly, however, since many fonts do not include it. In many situations the letter ø is a substitute, which in Unicode is U+00F8 ø
Diameter
–
Circle with
circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.
77.
On Floating Bodies
–
On Floating Bodies is a Greek-language work consisting of two books written by Archimedes of Syracuse, one of the most important mathematicians, physicists, and engineers of antiquity. On Floating Bodies, which is thought to have written around 250 BC, survives only partly in Greek. It is the first known work on hydrostatics, of which Archimedes is recognized as the founder, the purpose of On Floating Bodies was to determine the positions that various solids will assume when floating in a fluid, according to their form and the variation in their specific gravities. It contains the first statement of what is now known as Archimedes principle, Archimedes lived in the Greek city-state of Syracuse, Sicily. He is credited with laying the foundations of hydrostatics, statics, a leading scientist of classical antiquity, Archimedes also developed elaborate systems of pulleys to move large objects with a minimum of effort. The Archimedes screw underpins modern hydroengineering, and his machines of war helped to hold back the armies of Rome in the First Punic War. Archimedes opposed the arguments of Aristotle, pointing out that it was impossible to separate mathematics and nature, the only known copy of On Floating Bodies in Greek comes from the Archimedes Palimpsest. In the first part of the treatise, Archimedes establishes various general principles, such as that a solid denser than a fluid will, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards all things fall in order to derive the spherical shape. Further, Proposition 5 of Archimedes treatise On Floating Bodies states that, the second book is a mathematical achievement unmatched in antiquity and rarely equaled since. It is restricted to the case when the base of the paraboloid lies either entirely above or entirely below the fluid surface, Archimedes investigation of paraboloids was probably an idealization of the shapes of ships hulls. Some of his sections float with the base water and the summit above water. Of his works survive, the second of his two books of On Floating Bodies is considered his most mature work, commonly described as a tour de force
On Floating Bodies
–
a page from Floating Bodies,
Archimedes Palimpsest
78.
Ratio
–
In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, when the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units, but in many applications, the word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
Ratio
–
The ratio of width to height of
standard-definition television.
79.
Dissection puzzle
–
The creation of new dissection puzzles is also considered to be a type of dissection puzzle. Puzzles may include various restraints, such as hinged pieces, pieces that can fold, creators of new dissection puzzles emphasize using a minimum number of pieces, or creating novel situations, such as ensuring that every piece connects to another with a hinge. Dissection puzzles are a form of geometric puzzle. The earliest known descriptions of puzzles are from the time of Plato in Ancient Greece. Other ancient dissection puzzles were used as graphic depictions of the Pythagorean theorem, in the 10th century, Arabic mathematicians used geometric dissections in their commentaries on Euclids Elements. In the 18th century, Chinese scholar Tai Chen described an elegant dissection for approximating the value of π, the puzzles saw a major increase in general popularity in the late 19th century when newspapers and magazines began running dissection puzzles. Puzzle creators Sam Loyd in the United States and Henry Dudeney in the United Kingdom were among the most published, the dissections of regular polygons and other simple geometric shapes into another such shape was the subject of Martin Gardners November 1961 Mathematical Games column in Scientific American. The haberdashers problem shown in the figure shows how to divide up a square. The column included a table of such best known dissections involving the square, pentagon, hexagon, greek cross, some types of dissection puzzle are intended to create a large number of different geometric shapes. The tangram is a dissection puzzle of this type. Some geometric forms are easy to create, while others present an extreme challenge and this variability has ensured the puzzles popularity. Other dissections are intended to move between a pair of geometric shapes, such as a triangle to a square, or a square to a five-pointed star, a dissection puzzle of this description is the haberdashers problem, proposed in 1907 by Henry Dudeney. The puzzle is a dissection of a triangle to a square and it is one of the simplest regular polygon to square dissections known, and is now a classic example. It is not known whether a dissection of a triangle to a square is possible with three pieces. Ostomachion Pizza theorem Puzzle Coffin, Stewart T, the Puzzling World of Polyhedral Dissections. Frederickson, Greg N. Dissections, Plane and Fancy, frederickson, Greg N. Piano-hinged Dissections, Time to Fold
Dissection puzzle
–
Ostomachion is a dissection puzzle attributed to
Archimedes
80.
Ausonius
–
Decimius Magnus Ausonius was a Roman poet and teacher of rhetoric from Burdigala in Aquitaine, modern Bordeaux, France. For a time he was tutor to the future emperor Gratian and his best-known poems are Mosella, a description of the river Moselle, and Ephemeris, an account of a typical day in his life. His many other verses show his concern for his family, friends, teachers, Decimius Magnus Ausonius was born c. Ausonius was given a strict upbringing by his aunt and grandmother and he received an excellent education at Bordeaux and at Toulouse, where his maternal uncle, Aemilius Magnus Arborius, was a professor. Ausonius did well in grammar and rhetoric, but professed that his progress in Greek was unsatisfactory, when his uncle was summoned to Constantinople to tutor one of the sons of emperor Constantine I, Ausonius accompanied him to the capital. Having completed his studies, he trained for some time as an advocate, in 334 he became a grammaticus at a school of rhetoric in Bordeaux, and afterwards a rhetor or professor. His teaching attracted many pupils, some of whom became eminent in public life and his most famous pupil was the poet Paulinus, who later became a Christian and Bishop of Nola. After thirty years of this work Ausonius was summoned by emperor Valentinian I to teach his son, Gratian, when Valentinian took Gratian on the German campaigns of 368-9, Ausonius accompanied them. In recognition of his services emperor Valentinian bestowed on Ausonius the rank of quaestor, Gratian liked and respected his tutor, and when he himself became emperor in 375 he began bestowing on Ausonius and his family the highest civil honors. In 376 Ausoniuss son, Hesperius, was proconsul of Africa. In 379 Ausonius was awarded the consulate, the highest Roman honor, when Magnus Maximus was overthrown by emperor Theodosius I in 388, Ausonius did not leave his country estates. His estates supposedly included the land now owned by Château Ausone, Ausonius appears to have been a late and perhaps not very enthusiastic convert to Christianity. His grandson, Paulinus of Pella, was also a poet, about 120 epigrams on various topics. A description of the occupations of the day from morning till evening, in various meters, only the beginning and end are preserved. 30 poems of various lengths, mostly in elegiac meter, on deceased relations, composed after his consulate, a continuation of the Parentalia, dealing with the famous teachers of his native Bourdeaux whom he had known. 26 epitaphs of heroes from the Trojan war, translated from Greek Caesares, on the 12 emperors described by Suetonius. 14 pieces, dealing with 17 towns, in hexameters, a kind of puppet play in which the seven wise men appear successively and have their say. 20 pieces are grouped under this title, the most famous of which is the Mosella
Ausonius
–
Monument to Ausonius in
Milan.
Ausonius
–
Ausonius,
Bordeaux
Ausonius
–
Modern reconstruction of Sutter's Mill, a water-powered 19th century Californian sawmill.
81.
Gotthold Ephraim Lessing
–
Gotthold Ephraim Lessing was a German writer, philosopher, dramatist, publicist and art critic, and one of the most outstanding representatives of the Enlightenment era. His plays and theoretical writings substantially influenced the development of German literature and he is widely considered by theatre historians to be the first dramaturg in his role at Abel Seylers Hamburg National Theatre. Lessing was born in Kamenz, a town in Saxony, to Johann Gottfried Lessing. His father was a Lutheran minister and wrote on theology, Young Lessing studied at the Latin School in Kamenz from 1737 to 1741. With a father who wanted his son to follow in his footsteps, after completing his education at St. Afras, he enrolled at the University of Leipzig where he pursued a degree in theology, medicine, philosophy, and philology. It was here that his relationship with Karoline Neuber, a famous German actress and he translated several French plays for her, and his interest in theatre grew. During this time, he wrote his first play, The Young Scholar, Neuber eventually produced the play in 1748. From 1748 to 1760, Lessing lived in Leipzig and Berlin and he began to work as a reviewer and editor for the Vossische Zeitung and other periodicals. Lessing formed a connection with his cousin, Christlob Mylius. In 1750, Lessing and Mylius teamed together to begin a periodical publication named Beitrage zur Historie und Aufnahme des Theatres, the publication ran only four issues, but it caught the publics eye and revealed Lessing to be a serious critic and theorist of drama. In 1752 he took his masters degree in Wittenberg, from 1760 to 1765, he worked in Breslau as secretary to General Tauentzien during the Seven Years War between Britain and France, which had effects in Europe. It was during this time that he wrote his famous Laokoon, in 1765 Lessing returned to Berlin, leaving in 1767 to work for three years at the Hamburg National Theatre. Actor-manager, Konrad Ackermann, began construction on Germanys first permanent theatre in Hamburg, Johann Friedrich Löwen established Germanys first national theatre, the Hamburg National Theatre. The owners hired Lessing as the critic of plays and acting. The theatres main backer was Abel Seyler, a former currency speculator who since became known as the patron of German theatre. There he met Eva König, his future wife and his work in Hamburg formed the basis of his pioneering work on drama, titled Hamburgische Dramaturgie. Unfortunately, because of losses due to pirated editions of the Hamburgische Dramaturgie. In 1770 Lessing became librarian at the library, now the Herzog August Library
Gotthold Ephraim Lessing
–
Lessing, 1771
Gotthold Ephraim Lessing
–
Portrait of Lessing by Anna Rosina Lisiewska during his time as dramaturg of
Abel Seyler 's
Hamburg National Theatre (1767/1768)
Gotthold Ephraim Lessing
–
Eva Lessing
Gotthold Ephraim Lessing
–
Home,
Wolfenbüttel
82.
The Cattle of Helios
–
In Greek mythology, the Cattle of Helios, also called the Oxen of the Sun, are cattle pastured on the island of Thrinacia. Helios, also known as the sun god, is said to have had 7 herds of oxen and seven flocks of sheep, in the Odyssey, Homer describes these immortal cattle as handsome, wide-browed, fat and curved-horned. The cattle were guarded by Helios’ daughters, Phaëthusa and Lampetië, tiresias and Circe both warn Odysseus to shun the isle of Helios. They are held on the isle for a month by a storm sent by Poseidon. When he returns to the ship, Odysseus rebukes his companions for disobeying his orders, but it is too late, the cattle are dead and gone. Lampetie tells Helios that Odysseus men have slain his cattle, in turn, Helios orders her gods to take vengeance on Odysseus men. He threatens that if they do not pay him full atonement for the cattle, he take the sun to the Underworld. Zeus promises Helios to smite their ship with a lightning bolt, soon the gods show signs and wonders to the Odysseus men. The skins begin creeping and the flesh bellowing upon the spits, for six days, Odysseuss company feast on the kine of Helios. On the seventh day, the wind changes, after they set sail, Zeus keeps his word and the ship is destroyed by lightning during a storm. Odysseus escapes by swimming to Calypsos island, Vol.1 pp. 419–420 and Vol.2 pp.705. The Meridian Handbook of Classical Mythology
The Cattle of Helios
–
Greek Mythology
83.
Diophantine equation
–
In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one, an exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations, in more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis, the solutions are described by the following theorem, This Diophantine equation has a solution if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if is a solution, then the solutions have the form, where k is an arbitrary integer. Proof, If d is this greatest common divisor, Bézouts identity asserts the existence of integers e and f such that ae + bf = d, If c is a multiple of d, then c = dh for some integer h, and is a solution. On the other hand, for pair of integers x and y. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution, we have a + b = ax + by + k = ax + by + k = ax + by, showing that is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u + v =0. As u and v are coprime, Euclids lemma shows that exists a integer k such that x2 − x1 = kv. Therefore, x2 = x1 + kv and y2 = y1 − ku, the system to be solved may thus be rewritten as B = UC. Calling yi the entries of V−1X and di those of D = UC and it follows that the system has a solution if and only if bi, i divides di for i ≤ k and di =0 for i > k. If this condition is fulfilled, the solutions of the system are V. Hermite normal form may also be used for solving systems of linear Diophantine equations, however, Hermite normal form does not directly provide the solutions, to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form is more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, the Hermite normal form is substantially easier to compute than the Smith normal form. Integer linear programming amounts to finding some integer solutions of systems that include also inequations
Diophantine equation
–
Finding all
right triangles with integer side-lengths is equivalent to solving the Diophantine equation.
84.
Square number
–
In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number
Square number
–
m = 1 2 = 1
85.
Solar System
–
The Solar System is the gravitationally bound system comprising the Sun and the objects that orbit it, either directly or indirectly. Of those objects that orbit the Sun directly, the largest eight are the planets, with the remainder being significantly smaller objects, such as dwarf planets, of the objects that orbit the Sun indirectly, the moons, two are larger than the smallest planet, Mercury. The Solar System formed 4.6 billion years ago from the collapse of a giant interstellar molecular cloud. The vast majority of the mass is in the Sun. The four smaller inner planets, Mercury, Venus, Earth and Mars, are terrestrial planets, being composed of rock. The four outer planets are giant planets, being more massive than the terrestrials. All planets have almost circular orbits that lie within a flat disc called the ecliptic. The Solar System also contains smaller objects, the asteroid belt, which lies between the orbits of Mars and Jupiter, mostly contains objects composed, like the terrestrial planets, of rock and metal. Beyond Neptunes orbit lie the Kuiper belt and scattered disc, which are populations of trans-Neptunian objects composed mostly of ices, within these populations are several dozen to possibly tens of thousands of objects large enough that they have been rounded by their own gravity. Such objects are categorized as dwarf planets, identified dwarf planets include the asteroid Ceres and the trans-Neptunian objects Pluto and Eris. In addition to two regions, various other small-body populations, including comets, centaurs and interplanetary dust clouds. Six of the planets, at least four of the dwarf planets, each of the outer planets is encircled by planetary rings of dust and other small objects. The solar wind, a stream of charged particles flowing outwards from the Sun, the heliopause is the point at which pressure from the solar wind is equal to the opposing pressure of the interstellar medium, it extends out to the edge of the scattered disc. The Oort cloud, which is thought to be the source for long-period comets, the Solar System is located in the Orion Arm,26,000 light-years from the center of the Milky Way. For most of history, humanity did not recognize or understand the concept of the Solar System, the invention of the telescope led to the discovery of further planets and moons. The principal component of the Solar System is the Sun, a G2 main-sequence star that contains 99. 86% of the known mass. The Suns four largest orbiting bodies, the giant planets, account for 99% of the mass, with Jupiter. The remaining objects of the Solar System together comprise less than 0. 002% of the Solar Systems total mass, most large objects in orbit around the Sun lie near the plane of Earths orbit, known as the ecliptic
Solar System
–
The
Sun and
planets of the Solar System (distances not to scale)
Solar System
–
Solar System
Solar System
–
Andreas Cellarius 's illustration of the Copernican system, from the Harmonia Macrocosmica (1660)
Solar System
–
The eight planets of the Solar System (by decreasing size) are
Jupiter,
Saturn,
Uranus,
Neptune,
Earth,
Venus,
Mars and
Mercury.
86.
Book of Lemmas
–
The Book of Lemmas is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions on circles, the Book of Lemmas was first introduced in Arabic by Thābit ibn Qurra, he attributed the work to Archimedes. In 1661, the Arabic manuscript was translated into Latin by Abraham Ecchellensis, the Latin version was published under the name Liber Assumptorum. T. L. Heath translated Heiburgs Latin work into English in his The Works of Archimedes, another possibility is that the Book of Lemmas may be a collection of propositions by Archimedes later collected by a Greek writer. The Book of Lemmas introduces several new geometrical figures, Archimedes first introduced the arbelos in proposition four of his book, The figure is used in propositions four through eight. In propositions five, Archimedes introduces the Archimedes twin circles, and in eight, he makes use what would be the Pappus chain. Archimedes first introduced the salinon in proposition fourteen of his book, Archimedes proved that the salinon, if two circles touch at A, and if CD, EF be parallel diameters in them, ADF is a straight line. Let AB be the diameter of a semicircle, and let the tangents to it at B, if now DE be drawn perpendicular to AB, and if AT, DE meet in F, then DF = FE. Let P be any point on a segment of a circle whose base is AB, take D on AB so that AN = ND. If now PQ be an arc equal to the arc PA, let AB be the diameter of a semicircle, C any point on AB, and CD perpendicular to it, and let semicircles be described within the first semicircle and having AC, CB as diameters. Then if two circles be drawn touching CD on different sides and each touching two of the semicircles, the circles so drawn will be equal, let AB, the diameter of a semicircle, be divided at C so that AC = 3/2 × CB. Describe semicircles within the first semicircle and on AC, CB as diameters, if GH be the diameter of this circle, to find relation between GH and AB. If circles are circumscribed about and inscribed in a square, the circle is double of the inscribed square. If in a circle two chords AB, CD which do not pass through the centre intersect at right angles, suppose that TA, TB are two tangents to a circle, while TC cuts it. Let BD be the chord through B parallel to TC, and let AD meet TC in E. Then, if EH be drawn perpendicular to BD, it will bisect it in H. If two chords AB, CD in a circle intersect at right angles in a point O, not being the centre, then AO2 + BO2 + CO2 + DO2 =2. If AB be the diameter of a semicircle, and TP, TQ the tangents to it any point T. If a diameter AB of a circle meet any chord CD, not a diameter, in E, let ACB be a semicircle on AB as diameter, and let AD, BE be equal lengths measured along AB from A, B respectively
Book of Lemmas
–
The first page of the Book of Lemmas as seen in The Works of Archimedes (1897).
87.
Arabic language
–
Arabic is a Central Semitic language that was first spoken in Iron Age northwestern Arabia and is now the lingua franca of the Arab world. Arabic is also the language of 1.7 billion Muslims. It is one of six languages of the United Nations. The modern written language is derived from the language of the Quran and it is widely taught in schools and universities, and is used to varying degrees in workplaces, government, and the media. The two formal varieties are grouped together as Literary Arabic, which is the language of 26 states. Modern Standard Arabic largely follows the standards of Quranic Arabic. Much of the new vocabulary is used to denote concepts that have arisen in the post-Quranic era, Arabic has influenced many languages around the globe throughout its history. During the Middle Ages, Literary Arabic was a vehicle of culture in Europe, especially in science, mathematics. As a result, many European languages have borrowed many words from it. Many words of Arabic origin are found in ancient languages like Latin. Balkan languages, including Greek, have acquired a significant number of Arabic words through contact with Ottoman Turkish. Arabic has also borrowed words from languages including Greek and Persian in medieval times. Arabic is a Central Semitic language, closely related to the Northwest Semitic languages, the Ancient South Arabian languages, the Semitic languages changed a great deal between Proto-Semitic and the establishment of the Central Semitic languages, particularly in grammar. Innovations of the Central Semitic languages—all maintained in Arabic—include, The conversion of the suffix-conjugated stative formation into a past tense, the conversion of the prefix-conjugated preterite-tense formation into a present tense. The elimination of other prefix-conjugated mood/aspect forms in favor of new moods formed by endings attached to the prefix-conjugation forms, the development of an internal passive. These features are evidence of descent from a hypothetical ancestor. In the southwest, various Central Semitic languages both belonging to and outside of the Ancient South Arabian family were spoken and it is also believed that the ancestors of the Modern South Arabian languages were also spoken in southern Arabia at this time. To the north, in the oases of northern Hijaz, Dadanitic and Taymanitic held some prestige as inscriptional languages, in Najd and parts of western Arabia, a language known to scholars as Thamudic C is attested
Arabic language
–
The Galland Manuscript of
One Thousand and One Nights, 14th century
Arabic language
–
al-ʿArabiyyah in written Arabic (
Naskh script)
Arabic language
–
Bilingual traffic sign in
Qatar.
Arabic language
–
Examples of how the Arabic root and form system works.
88.
Johan Ludvig Heiberg (historian)
–
Johan Ludvig Heiberg was a Danish philologist and historian. He is best known for his discovery of previously unknown texts in the Archimedes Palimpsest and he also published an edition of Ptolemys Almagest. Heiberg was born in Aalborg, the son of Johanne Henriette Jacoba, Heiberg was Professor of Classical Philology at the University of Copenhagen from 1896 until 1924. Among his more than 200 publications were editions of the works of Archimedes, Euclid, Apollonius of Perga, Serenus of Antinouplis, Ptolemy, many of his editions are still in use today. The French Academy of Sciences awarded him the Prix Binoux for 1912 and his sister married biochemist Max Henius. Heiberg inspected the vellum manuscript in Constantinople in 1906, and realized that it contained works by Archimedes that were unknown to scholars at the time. Heibergs examination of the manuscript was with the eye only, while modern analysis of the texts has employed x-ray. The Archimedes Palimpsest is currently stored at the Walters Art Museum in Baltimore, how do we know about Greek mathematicians. 1, 000-year-old text by Greek maths genius Archimedes goes on display Daily Mail, Works by Johan Ludvig Heiberg at Project Gutenberg Works by or about Johan Ludvig Heiberg at Internet Archive
Johan Ludvig Heiberg (historian)
–
J. L. Heiberg
89.
Constantinople
–
Constantinople was the capital city of the Roman/Byzantine Empire, and also of the brief Latin, and the later Ottoman empires. It was reinaugurated in 324 AD from ancient Byzantium as the new capital of the Roman Empire by Emperor Constantine the Great, after whom it was named, Constantinople was famed for its massive and complex defences. The first wall of the city was erected by Constantine I, Constantinople never truly recovered from the devastation of the Fourth Crusade and the decades of misrule by the Latins. The origins of the name of Byzantion, more known by the later Latin Byzantium, are not entirely clear. The founding myth of the city has it told that the settlement was named after the leader of the Megarian colonists, Byzas. The later Byzantines of Constantinople themselves would maintain that the city was named in honour of two men, Byzas and Antes, though this was likely just a play on the word Byzantion. During this time, the city was also called Second Rome, Eastern Rome, and Roma Constantinopolitana. As the city became the remaining capital of the Roman Empire after the fall of the West, and its wealth, population, and influence grew. In the language of other peoples, Constantinople was referred to just as reverently, the medieval Vikings, who had contacts with the empire through their expansion in eastern Europe used the Old Norse name Miklagarðr, and later Miklagard and Miklagarth. In Arabic, the city was sometimes called Rūmiyyat al-kubra and in Persian as Takht-e Rum, in East and South Slavic languages, including in medieval Russia, Constantinople was referred to as Tsargrad or Carigrad, City of the Caesar, from the Slavonic words tsar and grad. This was presumably a calque on a Greek phrase such as Βασιλέως Πόλις, the modern Turkish name for the city, İstanbul, derives from the Greek phrase eis tin polin, meaning into the city or to the city. In 1928, the Turkish alphabet was changed from Arabic script to Latin script, in time the city came to be known as Istanbul and its variations in most world languages. In Greece today, the city is still called Konstantinoúpolis/Konstantinoúpoli or simply just the City, apart from this, little is known about this initial settlement, except that it was abandoned by the time the Megarian colonists settled the site anew. A farsighted treaty with the emergent power of Rome in c.150 BC which stipulated tribute in exchange for independent status allowed it to enter Roman rule unscathed. The site lay astride the land route from Europe to Asia and the seaway from the Black Sea to the Mediterranean, and had in the Golden Horn an excellent and spacious harbour. He would later rebuild Byzantium towards the end of his reign, in which it would be briefly renamed Augusta Antonina, fortifying it with a new city wall in his name, Constantine had altogether more colourful plans. Rome was too far from the frontiers, and hence from the armies and the imperial courts, yet it had been the capital of the state for over a thousand years, and it might have seemed unthinkable to suggest that the capital be moved to a different location. Constantinople was built over 6 years, and consecrated on 11 May 330, Constantine divided the expanded city, like Rome, into 14 regions, and ornamented it with public works worthy of an imperial metropolis
Constantinople
–
Constantinople in the Byzantine era
Constantinople
–
Map of Byzantine Constantinople
Constantinople
–
Emperor
Constantine I presents a representation of the city of Constantinople as tribute to an enthroned Mary and Christ Child in this church mosaic.
Hagia Sophia, c. 1000
Constantinople
–
Coin struck by Constantine I to commemorate the founding of Constantinople
90.
Palimpsest
–
A palimpsest is a manuscript page, either from a scroll or a book, from which the text has been scraped or washed off so that the page can be reused for another document. Pergamene was made of lamb or kid skin was expensive and not readily available, so. The word palimpsest derives from the Latin palimpsestus, which derives from the Ancient Greek παλίμψηστος, where papyrus was in common use, reuse of writing media was less common because papyrus was cheaper and more expendable than costly parchment. Some papyrus palimpsests do survive, and Romans referred to this custom of washing papyrus, the writing was washed from parchment or vellum using milk and oat bran. With the passing of time, the faint remains of the writing would reappear enough so that scholars can discern the text. Medieval codices are constructed in gathers which are folded, then stacked together like a newspaper, faint legible remains were read by eye before 20th-century techniques helped make lost texts readable. To read palimpsests, scholars of the 19th century used chemical means that were very destructive, using tincture of gall or, later. Modern methods of reading palimpsests using ultraviolet light and photography are less damaging, innovative digitized images aid scholars in deciphering unreadable palimpsests. For example, multispectral imaging undertaken by researchers at the Rochester Institute of Technology, at the Walters Art Museum where the palimpsest is now conserved, the project has focused on experimental techniques to retrieve the remaining text, some of which was obscured by overpainted icons. One of the most successful techniques for reading through the paint proved to be X-ray fluorescence imaging, a number of ancient works have survived only as palimpsests. Vellum manuscripts were over-written on purpose due to the dearth or cost of the material. Such a decree put added pressure on retrieving the vellum on which secular manuscripts were written, the decline of the vellum trade with the introduction of paper exacerbated the scarcity, increasing pressure to reuse material. Cultural considerations also motivated the creation of palimpsests, or the pagan texts may have merely appeared irrelevant. Texts most susceptible to being overwritten included obsolete legal and liturgical ones, sometimes of intense interest to the historian, Early Latin translations of Scripture were rendered obsolete by Jeromes Vulgate. Texts might be in foreign languages or written in scripts that had become illegible over time. The codices themselves might be damaged or incomplete. The most valuable Latin palimpsests are found in the codices which were remade from the early large folios in the 7th to the 9th centuries. It has been noticed that no work is generally found in any instance in the original text of a palimpsest
Palimpsest
–
The
Codex Ephraemi Rescriptus, a Greek manuscript of the Bible from the 5th century, is a palimpsest.
Palimpsest
–
A
Georgian palimpsest from the 5th or 6th century.
Palimpsest
–
The
Wolfenbüttel Codex Guelferbytanus A
Palimpsest
–
Folio 20
recto with Greek text of Luke 9:22-33 (lower text)
91.
New York City
–
The City of New York, often called New York City or simply New York, is the most populous city in the United States. With an estimated 2015 population of 8,550,405 distributed over an area of about 302.6 square miles. Located at the tip of the state of New York. Home to the headquarters of the United Nations, New York is an important center for international diplomacy and has described as the cultural and financial capital of the world. Situated on one of the worlds largest natural harbors, New York City consists of five boroughs, the five boroughs – Brooklyn, Queens, Manhattan, The Bronx, and Staten Island – were consolidated into a single city in 1898. In 2013, the MSA produced a gross metropolitan product of nearly US$1.39 trillion, in 2012, the CSA generated a GMP of over US$1.55 trillion. NYCs MSA and CSA GDP are higher than all but 11 and 12 countries, New York City traces its origin to its 1624 founding in Lower Manhattan as a trading post by colonists of the Dutch Republic and was named New Amsterdam in 1626. The city and its surroundings came under English control in 1664 and were renamed New York after King Charles II of England granted the lands to his brother, New York served as the capital of the United States from 1785 until 1790. It has been the countrys largest city since 1790, the Statue of Liberty greeted millions of immigrants as they came to the Americas by ship in the late 19th and early 20th centuries and is a symbol of the United States and its democracy. In the 21st century, New York has emerged as a node of creativity and entrepreneurship, social tolerance. Several sources have ranked New York the most photographed city in the world, the names of many of the citys bridges, tapered skyscrapers, and parks are known around the world. Manhattans real estate market is among the most expensive in the world, Manhattans Chinatown incorporates the highest concentration of Chinese people in the Western Hemisphere, with multiple signature Chinatowns developing across the city. Providing continuous 24/7 service, the New York City Subway is one of the most extensive metro systems worldwide, with 472 stations in operation. Over 120 colleges and universities are located in New York City, including Columbia University, New York University, and Rockefeller University, during the Wisconsinan glaciation, the New York City region was situated at the edge of a large ice sheet over 1,000 feet in depth. The ice sheet scraped away large amounts of soil, leaving the bedrock that serves as the foundation for much of New York City today. Later on, movement of the ice sheet would contribute to the separation of what are now Long Island and Staten Island. The first documented visit by a European was in 1524 by Giovanni da Verrazzano, a Florentine explorer in the service of the French crown and he claimed the area for France and named it Nouvelle Angoulême. Heavy ice kept him from further exploration, and he returned to Spain in August and he proceeded to sail up what the Dutch would name the North River, named first by Hudson as the Mauritius after Maurice, Prince of Orange
New York City
–
Clockwise, from top:
Midtown Manhattan,
Times Square, the
Unisphere in
Queens, the
Brooklyn Bridge,
Lower Manhattan with
One World Trade Center,
Central Park, the
headquarters of the United Nations, and the
Statue of Liberty
New York City
–
New Amsterdam, centered in the eventual
Lower Manhattan, in 1664, the year
England took control and renamed it "New York".
New York City
–
The
Battle of Long Island, the largest battle of the
American Revolution, took place in
Brooklyn in 1776.
New York City
–
Broadway follows the Native American Wickquasgeck Trail through Manhattan.
92.
Suda
–
The Suda or Souda is a large 10th-century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Suidas. It is a lexicon, written in Greek, with 30,000 entries, many drawing from ancient sources that have since been lost. The Suda is somewhere between a grammatical dictionary and an encyclopedia in the modern sense and it explains the source, derivation, and meaning of words according to the philology of its period, using such earlier authorities as Harpocration and Helladios. The articles on history are especially valuable. These entries supply details and quotations from authors whose works are otherwise lost and they use older scholia to the classics, and for later writers, Polybius, Josephus, the Chronicon Paschale, George Syncellus, George Hamartolus, and so on. This lexicon represents a convenient work of reference for people who played a part in political, ecclesiastical, the chief source for this is the encyclopedia of Constantine VII Porphyrogenitus, and for Roman history the excerpts of John of Antioch. Krumbacher counts two main sources of the work, Constantine VII for ancient history, and Hamartolus for the Byzantine age, the system is not difficult to learn and remember, but some editors—for example, Immanuel Bekker – rearranged the Suda alphabetically. Little is known of the compilation of work, except that it must have been written before it was quoted from extensively by Eustathius who lived from about 1115 AD to about 1195 or 1196. It would thus appear that the Suda was compiled sometime after 975, passages referring to Michael Psellus are considered later interpolations. It includes numerous quotations from ancient writers, the scholiasts on Aristophanes, Homer, other principal sources include a lexicon by Eudemus, perhaps derived from the work On Rhetorical Language by Eudemus of Argos. The work deals with biblical as well as subjects, from which it is inferred that the writer was a Christian. A prefatory note gives a list of dictionaries from which the portion was compiled. Although the work is uncritical and probably much interpolated, and the value of its articles is very unequal and its quotations from ancient authors make it a useful check on their manuscript traditions. A modern translation, the Suda On Line, was completed on 21 July 2014, the Suda has a near-contemporaneous Islamic parallel, the Kitab al-Fehrest of Ibn al-Nadim. Compare also the Latin Speculum Maius, authored in the 13th century by Vincent of Beauvais and this article incorporates text from a publication now in the public domain, Chisholm, Hugh, ed. Sūïdas. This article incorporates text from a now in the public domain, Herbermann, Charles. Ancient Greek Scholarship, a guide to finding, reading, and understanding scholia, commentaries, lexica, Oxford, New York, Oxford University Press,2006. Tachypaedia Byzantina, The Suda On Line as Collaborative Encyclopedia, Digital Humanities Quarterly 3.1, an on-line edition of the Ada Adler edition with ongoing translations and commentary by registered editors
Suda
–
First page of an early printed edition of the Suda
93.
Walters Art Museum
–
The Walters Art Museum, located in Mount Vernon-Belvedere, Baltimore, Maryland, is a public art museum founded and opened in 1934. It holds collections established during the mid-19th Century, located across the back alley, a block south of the Walters mansion on West Monument Street/Mount Vernon Place, on the northwest corner of North Charles Street at West Centre Street. The following year, The Walters reopened its original building after a dramatic three-year physical renovation and replacement of internal utilities. The Archimedes Palimpsest was on loan to the Walters Art Museum from a collector for conservation. This was one of the largest and most comprehensive such releases made by any museum, the Walters collection of ancient art includes examples from Egypt, Nubia, Greece, Rome, Etruria and the Near East. In 1911, Henry Walters purchased almost 100 gold artifacts from the Chiriqui region of western Panama in Central America, the museum owns the oldest surviving Chinese wood-and-lacquer image of the Buddha. It is exhibited in a gallery dedicated solely to this work, the Museum holds one of the largest and finest collections of Thai bronze, scrolls, and banner paintings in the world. Islamic art in all media is represented at the Walters, the Walters Museum owns an array of Islamic manuscripts. Walters Art Museum, MS W.613 contains five Mughal miniatures from a very important Khamsa of Nizami made for the Emperor Akbar, Henry Walters assembled a collection of art produced during the Middle Ages in all the major artistic media of the period. This forms the basis of the Walters medieval collection, for which the Museum is best known internationally. Considered one of the best collections of art in the United States, the Museums holdings include examples of metalwork, sculpture, stained glass, textiles, icons. Sculpted heads from the royal Abbey of St. Denis are rare surviving examples of sculptures that are directly connected with the origins of Gothic art in 12th Century France. An ivory casket covered with scenes of jousting knights is one of about a dozen such objects to survive in the world, many of these works are on display in the Museums galleries. Works in the collection are the subject of active research by the curatorial and conservation departments of the museum. The collection of European Renaissance and Baroque art features holdings of paintings, sculpture, furniture, ceramics, metal work, arms, the museum has one of ten surviving examples of the Sèvres pot-pourri vase in the shape of a ship from the 1750s and 1760s. William and Henry Walters collected works by late 19th Century French academic masters, Henry Walters was particularly interested in the courtly arts of 18th Century France. The museum’s collection of Sèvres porcelain includes a number of pieces that were made for members of the Royal Bourbon Court at Versailles Palace outside of Paris. Portrait miniatures and the examples of works, especially snuffboxes and watches, are displayed in the Treasury, along with some exceptional 19th-
Walters Art Museum
–
North Charles Street original main entrance to the Walters Art Museum
Walters Art Museum
–
Sculpture Garden (central "Great Hall") of the Walters Art Gallery (now Walters Art Museum) in the original Main Building of 1905-1909
Walters Art Museum
–
Sumerian male worshiper, c.2300 BC
Walters Art Museum
–
"
Padiiset's Statue ", illustrates
Canaan -
Ancient Egypt trade, c.1700 B.C. (inscription c.900 B.C.)
94.
Baltimore
–
Baltimore is the largest city in the U. S. state of Maryland, and the 29th-most populous city in the country. It was established by the Constitution of Maryland and is not part of any county, thus, it is the largest independent city in the United States, with a population of 621,849 as of 2015. As of 2010, the population of the Baltimore Metropolitan Area was 2.7 million, founded in 1729, Baltimore is the second largest seaport in the Mid-Atlantic. Baltimores Inner Harbor was once the leading port of entry for immigrants to the United States. With hundreds of identified districts, Baltimore has been dubbed a city of neighborhoods, in the War of 1812, Francis Scott Key wrote The Star-Spangled Banner, later the American national anthem, in Baltimore. More than 65,000 properties, or roughly one in three buildings in the city, are listed on the National Register, more than any city in the nation. The city has 289 properties listed on the National Register of Historic Places, the historical records of the government of Baltimore are located at the Baltimore City Archives. The city is named after Cecil Calvert, second Lord Baltimore, of the Irish House of Lords, Baltimore Manor was the name of the estate in County Longford on which the Calvert family lived in Ireland. Baltimore is an anglicization of the Irish name Baile an Tí Mhóir, in 1608, Captain John Smith traveled 210 miles from Jamestown to the uppermost Chesapeake Bay, leading the first European expedition to the Patapsco River. The name Patapsco is derived from pota-psk-ut, which translates to backwater or tide covered with froth in Algonquian dialect, a quarter century after John Smiths voyage, English colonists began to settle in Maryland. The area constituting the modern City of Baltimore and its area was first settled by David Jones in 1661. He claimed the area today as Harbor East on the east bank of the Jones Falls stream. In the early 1600s, the immediate Baltimore vicinity was populated, if at all. The Baltimore area had been inhabited by Native Americans since at least the 10th millennium BC, one Paleo-Indian site and several Archaic period and Woodland period archaeological sites have been identified in Baltimore, including four from the Late Woodland period. During the Late Woodland period, the culture that is called the Potomac Creek complex resided in the area from Baltimore to the Rappahannock River in Virginia. It was located on the Bush River on land that in 1773 became part of Harford County, in 1674, the General Assembly passed An Act for erecting a Court-house and Prison in each County within this Province. The site of the house and jail for Baltimore County was evidently Old Baltimore near the Bush River. In 1683, the General Assembly passed An Act for Advancement of Trade to establish towns, ports, one of the towns established by the act in Baltimore County was on Bush River, on Town Land, near the Court-House
Baltimore
–
Sixth Regiment fighting railroad strikers, July 20, 1877
Baltimore
Baltimore
–
The
Battle Monument commemorates the
Battle of Baltimore.
Baltimore
–
The
Great Baltimore Fire of 1904, looking west from
Pratt and
Gay streets
95.
Maryland
–
The states largest city is Baltimore, and its capital is Annapolis. Among its occasional nicknames are Old Line State, the Free State, the state is named after Henrietta Maria of France, the wife of Charles I of England. George Calvert was the first Lord of Baltimore and the first English proprietor of the colonial grant. Maryland was the state to ratify the United States Constitution. Maryland is one of the smallest U. S. states in terms of area, as well as one of the most densely populated, Maryland has an area of 12,406.68 square miles and is comparable in overall area with Belgium. It is the 42nd largest and 9th smallest state and is closest in size to the state of Hawaii, the next largest state, its neighbor West Virginia, is almost twice the size of Maryland. Maryland possesses a variety of topography within its borders, contributing to its nickname America in Miniature. The mid-portion of this border is interrupted by Washington, D. C. which sits on land that was part of Montgomery and Prince Georges counties and including the town of Georgetown. This land was ceded to the United States Federal Government in 1790 to form the District of Columbia, the Chesapeake Bay nearly bisects the state and the counties east of the bay are known collectively as the Eastern Shore. Close to the town of Hancock, in western Maryland, about two-thirds of the way across the state. This geographical curiosity makes Maryland the narrowest state, bordered by the Mason–Dixon line to the north, portions of Maryland are included in various official and unofficial geographic regions. Much of the Baltimore–Washington corridor lies just south of the Piedmont in the Coastal Plain, earthquakes in Maryland are infrequent and small due to the states distance from seismic/earthquake zones. The M5.8 Virginia earthquake in 2011 was felt moderately throughout Maryland, buildings in the state are not well-designed for earthquakes and can suffer damage easily. The lack of any glacial history accounts for the scarcity of Marylands natural lakes, laurel Oxbow Lake is an over one-hundred-year-old 55-acre natural lake two miles north of Maryland City and adjacent to Russett. Chews Lake is a natural lake two miles south-southeast of Upper Marlboro. There are numerous lakes, the largest of them being the Deep Creek Lake. Maryland has shale formations containing natural gas, where fracking is theoretically possible, as is typical of states on the East Coast, Marylands plant life is abundant and healthy. Middle Atlantic coastal forests, typical of the southeastern Atlantic coastal plain, grow around Chesapeake Bay, moving west, a mixture of Northeastern coastal forests and Southeastern mixed forests cover the central part of the state
Maryland
–
Western Maryland: known for its heavily forested mountains. A panoramic view of
Deep Creek Lake and the surrounding
Appalachian Mountains in
Garrett County.
Maryland
–
Flag
Maryland
–
Dramatic example of Maryland's
fall line, a change in rock type and elevation that creates waterfalls in many areas along the Southwest to Northeast geological boundary that crosses the state.
Great Falls, cliffs and rapids.
Maryland
–
Typical freshwater river above the tidal zone. The
Patapsco River includes the famous
Thomas Viaduct and is part of the
Patapsco Valley State Park. Later, the river forms the
Inner Harbor as it empties into the
Chesapeake Bay.
96.
Ultraviolet
–
Ultraviolet is an electromagnetic radiation with a wavelength from 10 nm to 400 nm, shorter than that of visible light but longer than X-rays. UV radiation constitutes about 10% of the light output of the Sun. It is also produced by electric arcs and specialized lights, such as lamps, tanning lamps. Consequently, the effects of UV are greater than simple heating effects. Suntan, freckling and sunburn are familiar effects of over-exposure, along with risk of skin cancer. Living things on dry land would be damaged by ultraviolet radiation from the Sun if most of it were not filtered out by the Earths atmosphere. More-energetic, shorter-wavelength extreme UV below 121 nm ionizes air so strongly that it is absorbed before it reaches the ground, Ultraviolet is also responsible for the formation of bone-strengthening vitamin D in most land vertebrates, including humans. The UV spectrum thus has both beneficial and harmful to human health. Ultraviolet rays are invisible to most humans, the lens in a human eye ordinarily filters out UVB frequencies or higher, and humans lack color receptor adaptations for ultraviolet rays. Under some conditions, children and young adults can see ultraviolet down to wavelengths of about 310 nm, near-UV radiation is visible to some insects, mammals, and birds. Small birds have a fourth color receptor for ultraviolet rays, this gives birds true UV vision, reindeer use near-UV radiation to see polar bears, who are poorly visible in regular light because they blend in with the snow. UV also allows mammals to see urine trails, which is helpful for animals to find food in the wild. The males and females of some species look identical to the human eye. Ultraviolet means beyond violet, violet being the color of the highest frequencies of visible light, Ultraviolet has a higher frequency than violet light. He called them oxidizing rays to emphasize chemical reactivity and to them from heat rays. The terms chemical and heat rays were eventually dropped in favour of ultraviolet and infrared radiation, in 1878 the effect of short-wavelength light on sterilizing bacteria was discovered. By 1903 it was known the most effective wavelengths were around 250 nm, in 1960, the effect of ultraviolet radiation on DNA was established. The discovery of the ultraviolet radiation below 200 nm, named vacuum ultraviolet because it is absorbed by air, was made in 1893 by the German physicist Victor Schumann
Ultraviolet
–
(left) Portable ultraviolet lamp. (right) UV light is also produced by
electric arcs.
Arc welders must wear
eye protection to prevent
welder's flash.
Ultraviolet
Ultraviolet
–
Two black light fluorescent tubes, showing use. The longer tube is a F15T8/BLB 18 inch, 15 watt tube, shown in the bottom image in a standard plug-in fluorescent fixture. The shorter is an F8T5/BLB 12 inch, 8 watt tube, used in a portable battery-powered black light sold as a pet urine detector.
Ultraviolet
97.
X-ray
–
X-radiation is a form of electromagnetic radiation. Most X-rays have a wavelength ranging from 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz, X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays. Spelling of X-ray in the English language includes the variants x-ray, xray, X-rays with high photon energies are called hard X-rays, while those with lower energy are called soft X-rays. Due to their ability, hard X-rays are widely used to image the inside of objects, e. g. in medical radiography. The term X-ray is metonymically used to refer to an image produced using this method. Since the wavelengths of hard X-rays are similar to the size of atoms they are useful for determining crystal structures by X-ray crystallography. By contrast, soft X-rays are easily absorbed in air, the length of 600 eV X-rays in water is less than 1 micrometer. There is no consensus for a definition distinguishing between X-rays and gamma rays, one common practice is to distinguish between the two types of radiation based on their source, X-rays are emitted by electrons, while gamma rays are emitted by the atomic nucleus. This definition has problems, other processes also can generate these high-energy photons. One common alternative is to distinguish X- and gamma radiation on the basis of wavelength, with radiation shorter than some arbitrary wavelength, such as 10−11 m and this criterion assigns a photon to an unambiguous category, but is only possible if wavelength is known. Occasionally, one term or the other is used in specific contexts due to precedent, based on measurement technique. Thus, gamma-rays generated for medical and industrial uses, for radiotherapy, in the ranges of 6–20 MeV. X-ray photons carry enough energy to ionize atoms and disrupt molecular bonds and this makes it a type of ionizing radiation, and therefore harmful to living tissue. A very high radiation dose over a period of time causes radiation sickness. In medical imaging this increased risk is generally greatly outweighed by the benefits of the examination. The ionizing capability of X-rays can be utilized in treatment to kill malignant cells using radiation therapy. It is also used for material characterization using X-ray spectroscopy, hard X-rays can traverse relatively thick objects without being much absorbed or scattered. For this reason, X-rays are widely used to image the inside of visually opaque objects, the most often seen applications are in medical radiography and airport security scanners, but similar techniques are also important in industry and research
X-ray
–
Spectrum of the X-rays emitted by an X-ray tube with a
rhodium target, operated at 60
kV. The smooth, continuous curve is due to
bremsstrahlung, and the spikes are
characteristic K lines for rhodium atoms.
X-ray
–
X-rays are part of the
electromagnetic spectrum, with wavelengths shorter than
visible light. Different applications use different parts of the X-ray spectrum.
X-ray
–
A
chest radiograph of a female, demonstrating a
hiatus hernia
X-ray
–
An arm radiograph, demonstrating broken
ulna and
radius with implanted
internal fixation.
98.
Light
–
Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to light, which is visible to the human eye and is responsible for the sense of sight. Visible light is defined as having wavelengths in the range of 400–700 nanometres, or 4.00 × 10−7 to 7.00 × 10−7 m. This wavelength means a range of roughly 430–750 terahertz. The main source of light on Earth is the Sun, sunlight provides the energy that green plants use to create sugars mostly in the form of starches, which release energy into the living things that digest them. This process of photosynthesis provides virtually all the used by living things. Historically, another important source of light for humans has been fire, with the development of electric lights and power systems, electric lighting has effectively replaced firelight. Some species of animals generate their own light, a process called bioluminescence, for example, fireflies use light to locate mates, and vampire squids use it to hide themselves from prey. Visible light, as all types of electromagnetic radiation, is experimentally found to always move at this speed in a vacuum. In physics, the term sometimes refers to electromagnetic radiation of any wavelength. In this sense, gamma rays, X-rays, microwaves and radio waves are also light, like all types of light, visible light is emitted and absorbed in tiny packets called photons and exhibits properties of both waves and particles. This property is referred to as the wave–particle duality, the study of light, known as optics, is an important research area in modern physics. Generally, EM radiation, or EMR, is classified by wavelength into radio, microwave, infrared, the behavior of EMR depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths, when EMR interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. There exist animals that are sensitive to various types of infrared, infrared sensing in snakes depends on a kind of natural thermal imaging, in which tiny packets of cellular water are raised in temperature by the infrared radiation. EMR in this range causes molecular vibration and heating effects, which is how these animals detect it, above the range of visible light, ultraviolet light becomes invisible to humans, mostly because it is absorbed by the cornea below 360 nanometers and the internal lens below 400. Furthermore, the rods and cones located in the retina of the eye cannot detect the very short ultraviolet wavelengths and are in fact damaged by ultraviolet. Many animals with eyes that do not require lenses are able to detect ultraviolet, by quantum photon-absorption mechanisms, various sources define visible light as narrowly as 420 to 680 to as broadly as 380 to 800 nm
Light
–
An example of refraction of light. The straw appears bent, because of refraction of light as it enters liquid from air.
Light
–
A triangular prism dispersing a beam of white light. The longer wavelengths (red) and the shorter wavelengths (blue) get separated.
Light
–
A
cloud illuminated by
sunlight
Light
–
A
city illuminated by
artificial lighting
99.
Archimedes (crater)
–
Archimedes is a large lunar impact crater on the eastern edges of the Mare Imbrium. The diameter of Archimedes is the largest of any crater on the Mare Imbrium, the rim has a significant outer rampart brightened with ejecta and the upper portion of a terraced inner wall, but lacks the ray system associated with younger craters. A triangular promontory extends 30 kilometers from the southeast of the rim, the interior of the crater lacks a central peak, and is flooded with lava. It is devoid of significant raised features, although there are a few tiny craters near the rim. Scattered wisps of bright ray material lie across the floor, most likely deposited by the impact that created Autolycus, to the south of Archimedes extends the Montes Archimedes, a mountainous region. On the southeastern rim is the Palus Putredinis, a lava-flooded plain containing a system of rilles named the Rimae Archimedes, north-northwest of Archimedes stand the Montes Spitzbergen, a string of peaks in the Mare Imbrium. East of Archimedes is the crater Autolycus, northeast of Archimedes is the prominent crater Aristillus. The lava plain between Archimedes, Aristillus, and Autolycus forms the Sinus Lunicus bay of Mare Imbrium, a wrinkle ridge leads away from Archimedes toward the north-northwest, crossing this mare. Archimedes is named after the Greek scientist Archimedes, like many of the craters on the Moons near side, it was given its name by Giovanni Riccioli, whose 1651 nomenclature system has become standardized. Earlier lunar cartographers had given the different names. Michael van Langrens 1645 map calls it Roma after the city of Rome, johannes Hevelius called it Mons Argentarius after Monte Argentario region in Italy. The stretch of lunar surface between Archimedes and Autolycus was the site of the crash-landing of the Soviet probe Luna 2 and this was the first craft to reach the surface of the Moon, landing September 13,1959. By convention these features are identified on maps by placing the letter on the side of the crater midpoint that is closest to Archimedes. The following craters have been renamed by the IAU, featured Image, Archimedes - Mare Flooded Crater. Archived from the original on 16 March 2011
Archimedes (crater)
–
Lunar Orbiter 4 image
Archimedes (crater)
–
Archimedes from
Apollo 15.
NASA photo.
Archimedes (crater)
–
Lunar crater Archimedes in the infrared. Image courtesy of
NOT and
SO: M. Gålfalk, G. Olofsson, and H.-G. Florén, taken with the SIRCA camera.
100.
East Germany
–
East Germany, formally the German Democratic Republic, was an Eastern Bloc state during the Cold War period. The Soviet zone surrounded West Berlin, but did not include it, as a result, the German Democratic Republic was established in the Soviet Zone, while the Federal Republic was established in the three western zones. East Germany, which lies culturally in Central Germany, was a state of the Soviet Union. Soviet occupation authorities began transferring administrative responsibility to German communist leaders in 1948, Soviet forces, however, remained in the country throughout the Cold War. Until 1989, the GDR was governed by the Socialist Unity Party, though other parties participated in its alliance organisation. The economy was centrally planned, and increasingly state-owned, prices of basic goods and services were set by central government planners, rather than rising and falling through supply and demand. Although the GDR had to pay war reparations to the USSR. Nonetheless it did not match the growth of West Germany. Emigration to the West was a significant problem—as many of the emigrants were well-educated young people, the government fortified its western borders and, in 1961, built the Berlin Wall. Many people attempting to flee were killed by guards or booby traps. In 1989, numerous social and political forces in the GDR and abroad led to the fall of the Berlin Wall, the following year open elections were held, and international negotiations led to the signing of the Final Settlement treaty on the status and borders of Germany. The GDR was dissolved and Germany was unified on 3 October 1990, internally, the GDR also bordered the Soviet sector of Allied-occupied Berlin known as East Berlin which was also administered as the states de facto capital. It also bordered the three sectors occupied by the United States, United Kingdom and France known collectively as West Berlin. The three sectors occupied by the Western nations were sealed off from the rest of the GDR by the Berlin Wall from its construction in 1961 until it was brought down in 1989, the official name was Deutsche Demokratische Republik, usually abbreviated to DDR. West Germans, the media and statesmen purposely avoided the official name and its abbreviation, instead using terms like Ostzone, Sowjetische Besatzungszone. The centre of power in East Berlin was referred to as Pankow. Over time, however, the abbreviation DDR was also used colloquially by West Germans. However, this use was not always consistent, for example, before World War II, Ostdeutschland was used to describe all the territories east of the Elbe, as reflected in the works of sociologist Max Weber and political theorist Carl Schmitt
East Germany
–
GDR leaders: President
Wilhelm Pieck and Prime Minister
Otto Grotewohl, 1949
East Germany
–
Flag (1959–1990)
East Germany
–
SED First Secretary,
Walter Ulbricht, 1950
East Germany
–
Head of State:
Erich Honecker (1971–89)
101.
Greece
–
Greece, officially the Hellenic Republic, historically also known as Hellas, is a country in southeastern Europe, with a population of approximately 11 million as of 2015. Athens is the capital and largest city, followed by Thessaloniki. Greece is strategically located at the crossroads of Europe, Asia, situated on the southern tip of the Balkan peninsula, it shares land borders with Albania to the northwest, the Republic of Macedonia and Bulgaria to the north, and Turkey to the northeast. Greece consists of nine regions, Macedonia, Central Greece, the Peloponnese, Thessaly, Epirus, the Aegean Islands, Thrace, Crete. The Aegean Sea lies to the east of the mainland, the Ionian Sea to the west, the Cretan Sea and the Mediterranean Sea to the south. Greece has the longest coastline on the Mediterranean Basin and the 11th longest coastline in the world at 13,676 km in length, featuring a vast number of islands, eighty percent of Greece is mountainous, with Mount Olympus being the highest peak at 2,918 metres. From the eighth century BC, the Greeks were organised into various independent city-states, known as polis, which spanned the entire Mediterranean region and the Black Sea. Greece was annexed by Rome in the second century BC, becoming a part of the Roman Empire and its successor. The Greek Orthodox Church also shaped modern Greek identity and transmitted Greek traditions to the wider Orthodox World, falling under Ottoman dominion in the mid-15th century, the modern nation state of Greece emerged in 1830 following a war of independence. Greeces rich historical legacy is reflected by its 18 UNESCO World Heritage Sites, among the most in Europe, Greece is a democratic and developed country with an advanced high-income economy, a high quality of life, and a very high standard of living. A founding member of the United Nations, Greece was the member to join the European Communities and has been part of the Eurozone since 2001. Greeces unique cultural heritage, large industry, prominent shipping sector. It is the largest economy in the Balkans, where it is an important regional investor, the names for the nation of Greece and the Greek people differ from the names used in other languages, locations and cultures. The earliest evidence of the presence of human ancestors in the southern Balkans, dated to 270,000 BC, is to be found in the Petralona cave, all three stages of the stone age are represented in Greece, for example in the Franchthi Cave. Neolithic settlements in Greece, dating from the 7th millennium BC, are the oldest in Europe by several centuries and these civilizations possessed writing, the Minoans writing in an undeciphered script known as Linear A, and the Mycenaeans in Linear B, an early form of Greek. The Mycenaeans gradually absorbed the Minoans, but collapsed violently around 1200 BC and this ushered in a period known as the Greek Dark Ages, from which written records are absent. The end of the Dark Ages is traditionally dated to 776 BC, the Iliad and the Odyssey, the foundational texts of Western literature, are believed to have been composed by Homer in the 7th or 8th centuries BC. With the end of the Dark Ages, there emerged various kingdoms and city-states across the Greek peninsula, in 508 BC, Cleisthenes instituted the worlds first democratic system of government in Athens
Greece
–
Fresco displaying the Minoan ritual of "bull leaping", found in
Knossos,
Crete.
Greece
–
Flag
Greece
–
The
Lion Gate,
Mycenae
Greece
–
The
Parthenon on the
Acropolis of Athens is one of the best known symbols of
classical Greece.
102.
California Gold Rush
–
The California Gold Rush began on January 24,1848, when gold was found by James W. Marshall at Sutters Mill in Coloma, California. The news of gold brought some 300,000 people to California from the rest of the United States, the Gold Rush initiated the California Genocide, with 100,000 Native Californians dying between 1848 and 1868. By the time it ended, California had gone from a thinly populated ex-Mexican territory to the state of the first nominee for the Republican Party. The effects of the Gold Rush were substantial, whole indigenous societies were attacked and pushed off their lands by the gold-seekers, called forty-niners. The first to hear confirmed information of the rush were the people in Oregon, the Sandwich Islands, and Latin America. While most of the newly arrived were Americans, the Gold Rush attracted tens of thousands from Latin America, Europe, Australia, agriculture and ranching expanded throughout the state to meet the needs of the settlers. San Francisco grew from a settlement of about 200 residents in 1846 to a boomtown of about 36,000 by 1852. Roads, churches, schools and other towns were built throughout California, in 1849 a state constitution was written. The new constitution was adopted by vote, and the future states interim first governor. In September,1850, California became a state, at the beginning of the Gold Rush, there was no law regarding property rights in the goldfields and a system of staking claims was developed. Prospectors retrieved the gold from streams and riverbeds using simple techniques, although the mining caused environmental harm, more sophisticated methods of gold recovery were developed and later adopted around the world. New methods of transportation developed as steamships came into regular service, by 1869 railroads were built across the country from California to the eastern United States. At its peak, technological advances reached a point where significant financing was required, Gold worth tens of billions of todays dollars was recovered, which led to great wealth for a few. However, many returned home with more than they had started with. The Mexican–American War ended on February 3,1848, although California was firmly in American hands before that, the Treaty of Guadalupe Hidalgo provided for, among other things, the formal transfer of Upper California to the United States. The California Gold Rush began at Sutters Mill, near Coloma, on January 24,1848, James W. Marshall, a foreman working for Sacramento pioneer John Sutter, found shiny metal in the tailrace of a lumber mill Marshall was building for Sutter on the American River. Marshall brought what he found to John Sutter, and the two tested the metal. However, rumors started to spread and were confirmed in March 1848 by San Francisco newspaper publisher
California Gold Rush
–
Sailing to California at the beginning of the Gold Rush
California Gold Rush
–
Merchant ships fill
San Francisco harbor, 1850–51
California Gold Rush
–
Panning for gold on the
Mokelumne River
California Gold Rush
–
"Independent Gold Hunter on His Way to California", circa 1850. The gold hunter is loaded down with every conceivable appliance, much of which would be useless in California. The prospector says: "I am sorry I did not follow the advice of Granny and go around the Horn, through the Straights, or by Chagres [Panama]."
103.
Axiom of Archimedes
–
Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder. An algebraic structure in any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, an ordered group that is Archimedean is an Archimedean group. This can be made precise in various contexts with different formulations. The concept was named by Otto Stolz after the ancient Greek geometer, the Archimedean property appears in Book V of Euclids Elements as Definition 4, Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. Because Archimedes credited it to Eudoxus of Cnidus it is known as the Theorem of Eudoxus or the Eudoxus axiom. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs, Let x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y if, for natural number n, the multiple nx is less than y, that is. The group G is Archimedean if there is no x, y such that x is infinitesimal with respect to y. Additionally, if K is a structure with a unit — for example. If x is infinitesimal with respect to 1, then x is an infinitesimal element, likewise, if y is infinite with respect to 1, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements, an ordered field has some additional properties. One may assume that the numbers are contained in the field. If x is infinitesimal, then 1/x is infinite, and vice versa, therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. If x is infinitesimal and r is a number, then r x is also infinitesimal. As a result, given an element c, the three numbers c/2, c, and 2c are either all infinitesimal or all non-infinitesimal
Axiom of Archimedes
–
Illustration of the Archimedean property.
104.
Archimedean solid
–
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular meeting in identical vertices, excluding the 5 Platonic solids. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices, identical vertices means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Excluding these two families, there are 13 Archimedean solids. All the Archimedan solids can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry, the Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra, kepler may have also found the elongated square gyrobicupola, at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a configuration of means that a square, hexagon. Some definitions of semiregular polyhedron include one more figure, the square gyrobicupola or pseudo-rhombicuboctahedron. The number of vertices is 720° divided by the angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular, the duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices, the snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form and right-handed form. When something comes in forms which are each others three-dimensional mirror image. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions, starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated, different Platonic and Archimedean solids can be created, expansion or cantellation involves moving each face away from the center and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles, the last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges, note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron
Archimedean solid
–
The
truncated icosidodecahedron, is the largest Archimedean solid, by volume with unit edge length, as well as having the most vertices and edges.
105.
Methods of computing square roots
–
In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a non-negative real number. For the square roots of a negative or complex number, see below, finding S is the same as solving the equation f = x 2 − S =0 for a positive x. Therefore, any general numerical root-finding algorithm can be used, many square root algorithms require an initial seed value. If the initial seed value is far away from the square root. It is therefore useful to have an estimate, which may be very inaccurate. For S =125348 =12.5348 ×104, for S =125348 =111101001101001002 =1.11101001101001002 ×216 the binary approximation gives S ≈28 =1000000002 =256. These approximations are useful to find better seeds for iterative algorithms and it can be derived from Newtons method. The process of updating is iterated until desired accuracy is obtained and this is a quadratically convergent algorithm, which means that the number of correct digits of the approximation roughly doubles with each iteration. It proceeds as follows, Begin with a positive starting value x0. Let xn +1 be the average of xn and S/xn, repeat step 2 until the desired accuracy is achieved. It can also be represented as, x 0 ≈ S, x n +1 =12, S = lim n → ∞ x n. Let the relative error in xn be defined by ε n = x n S −1 and thus x n = S ⋅. Then it can be shown that ε n +1 = ε n 22 and thus that 0 ≤ ε n +2 ≤ min and consequently that convergence is assured provided that x0 and S are both positive. If using the rough estimate above with the Babylonian method, then the least accurate cases in ascending order are as follows, S =1, x 0 =2, x 1 =1.250, ε1 =0.250. S =10, x 0 =2, x 1 =3.500, S =10, x 0 =6, x 1 =3.833, ε1 <0.213. S =100, x 0 =6, x 1 =11.333, thus in any case, ε1 ≤2 −2. ε8 <2 −383 <10 −115, rounding errors will slow the convergence. It is recommended to keep at least one extra digit beyond the desired accuracy of the xn being calculated to minimize round off error and this is a method to find each digit of the square root in a sequence
Methods of computing square roots
–
Graph charting the use of the Babylonian method for approximating the square root of 100 (10) using starting values x 0 = 50, x 0 = 1, and x 0 = −5. Note that using a negative starting value yields the negative root.
106.
Chord (geometry)
–
A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just secant, is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse, a chord that passes through a circles center point is the circles diameter. Every diameter is a chord, but not every chord is a diameter, the word chord is from the Latin chorda meaning bowstring. Among properties of chords of a circle are the following, Chords are equidistant from the center if, a chord that passes through the center of a circle is called a diameter, and is the longest chord. If the line extensions of chords AB and CD intersect at a point P, the area that a circular chord cuts off is called a circular segment. The midpoints of a set of chords of an ellipse are collinear. Chords were used extensively in the development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the function for every 7.5 degrees. The circle was of diameter 120, and the lengths are accurate to two base-60 digits after the integer part. The chord function is defined geometrically as shown in the picture, the chord of an angle is the length of the chord between two points on a unit circle separated by that angle. The last step uses the half-angle formula, much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, all now lost, so presumably a great deal was known about them
Chord (geometry)
–
The red segment BX is a chord (as is the diameter segment AB).
107.
Elsevier
–
Elsevier is one of the worlds major providers of scientific, technical, and medical information, and a technology company originally established in 1880. It is now a part of the RELX Group, known until 2015 as Reed Elsevier, Elsevier publishes approximately 400,000 articles annually in 2,500 journals. Its archives contain over 13 million documents and 30,000 e-books, total yearly downloads amount to 900 million. Elseviers high profit margins and its practices have subjected it to criticism by researchers. Elsevier was founded in 1880 and took the name from the Dutch publishing house Elzevir which has no connection with the present company, the Elzevir family operated as booksellers and publishers in the Netherlands, the founder, Lodewijk Elzevir, lived in Leiden and established the business in 1580. The expansion of Elsevier in the field after 1945 was funded with the profits of the newsweekly Elsevier. The weekly was an instant success and earned lots of money, in 1947, Elsevier began publishing its first English-language journal, Biochimica et Biophysica Acta. In 2013, Elsevier acquired Mendeley, a UK company making software for managing and sharing research papers, Mendeley, previously an open platform for sharing of research, was greatly criticized for the acquisition, which users saw as acceding to the paywall approach to research literature. Mendeleys previously open sharing system now allows exchange of paywalled resources only within private groups, the New Yorker described Elseviers reasons for buying Mendeley as two-fold, to acquire its user data, and to destroy or coöpt an open-science icon that threatens its business model. In December 2013, Elsevier announced a collaboration with University College, London, Elseviers investment is substantial and thought to be more than £10 million. In the primary research market during 2015, researchers submitted over 1. 3m research papers to Elsevier-based publications. Over 17,000 editors managed the peer review and selection of these papers, in 2013, the five editorial groups Elsevier, Springer, Wiley-Blackwell, Taylor & Francis and SAGE Publications published more than half of all academic papers in the peer-reviewed literature. At that time, Elsevier accounted for 16% of the market in science, technology. Elsevier breaks down its revenue sources by format and by geographic region, approximately 41% of revenue by geography in 2014 derived from North America, 27% from Europe and the remaining 32% from the rest of the world. Approximately 76% of revenue by format came from Electronic, 23% came from Print, Elsevier employs more than 7,200 people in over 70 offices across 24 countries. The company publishes 2,500 journals and 30,000 e-books and it is headed by Chief Executive Officer Ron Mobed. In 2015, Elsevier accounted for 35% of the revenues of RELX group, in operating profits, it represented 42%. Adjusted operating profits rose by 2% from 2014 to 2015, following the integration of its Science & Technology and Health Sciences divisions in 2012, Elsevier has operated under a traditional business structure with a single CEO
Elsevier
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Elsevier
108.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894
University of St Andrews
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College Hall, within the 16th century St Mary's College building
University of St Andrews
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University of St Andrews
shield
University of St Andrews
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St Salvator's Chapel in 1843
University of St Andrews
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The "Gateway" building, built in 2000 and now used for the university's management department
109.
Courant Institute of Mathematical Sciences
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The Director of the Courant Institute directly reports to New York Universitys Provost and President and works closely with deans and directors of other NYU colleges and divisions respectively. The Courant Institute is named after Richard Courant, one of the founders of the Courant Institute, the Courant Institute is considered one of the leading and most prestigious mathematics schools and mathematical sciences research centers in the world. It is ranked #1 in applied research in US, #5 in citation impact worldwide. On the Faculty Scholarly Productivity Index, it is ranked #3 with an index of 1.84, the Mathematics Department of the Institute has 18 members of the United States National Academy of Sciences and five members of the National Academy of Engineering. Four faculty members have been awarded the National Medal of Science, one was honored with the Kyoto Prize, louis Nirenberg also received the Chern Medal in 2010, and Subhash Khot won the Nevanlinna Prize in 2014. The Courant Institute specializes in applied mathematics, mathematical analysis and scientific computation, there is emphasis on partial differential equations and their applications. The mathematics department is ranked in the United States as #1 in applied mathematics. Other strong points are analysis and geometry, within the field of computer science, CIMS concentrates in machine learning, theory, programming languages, computer graphics and parallel computing. The computer science program is ranked 28th among computer science programs in the US, the overall acceptance rate for all CIMS graduate programs is 15%, and program admissions reviews are holistic. A high undergraduate GPA and high GRE score are typically prerequisites to admission to its programs but are not required. Majority of accepted candidates met these standards, however, character and personal qualities and evidence of strong quantitative skills are very important admission factors. Undergraduate program admissions are not directly administrated by the Institute but by the NYU undergraduate admissions office of College of Arts, the Graduate Department of Computer Science offers a PhD in computer science. In addition it offers Master of Science degrees in science, information systems. Students may take the exam for any these courses without being enrolled in the course. The Computer Science Masters program offers instruction in the principles, design and applications of computer systems. Students who obtain an MS degree in science are qualified to do significant development work in the computer industry or important application areas. Those who receive a degree are in a position to hold faculty appointments and do research. The emphasis for the MS in Information Systems program is on the use of systems in business
Courant Institute of Mathematical Sciences
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View of Warren Weaver Hall, Courant Institute of Mathematical Sciences from the Ground Floor of Gould Plaza
Courant Institute of Mathematical Sciences
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Lecture Hall at Warren Weaver Hall
Courant Institute of Mathematical Sciences
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Classroom at Warren Weaver Hall
Courant Institute of Mathematical Sciences
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The Courant Institute along with Microsoft Research are the founders of the Games for Learning Institute
110.
University of Chicago
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The University of Chicago is a private research university in Chicago, Illinois. It holds top-ten positions in national and international rankings and measures. The university currently enrolls approximately 5,700 students in the College, Chicagos physics department helped develop the worlds first man-made, self-sustaining nuclear reaction beneath the viewing stands of universitys Stagg Field. The university is home to the University of Chicago Press. With an estimated date of 2020, the Barack Obama Presidential Center will be housed at the university. Both Harper and future president Robert Maynard Hutchins advocated for Chicagos curriculum to be based upon theoretical and perennial issues rather than on applied sciences, the University of Chicago has many prominent alumni. 92 Nobel laureates have been affiliated with the university as professors, students, faculty, or staff, similarly,34 faculty members and 16 alumni have been awarded the MacArthur “Genius Grant”. Rockefeller on land donated by Marshall Field, while the Rockefeller donation provided money for academic operations and long-term endowment, it was stipulated that such money could not be used for buildings. The original physical campus was financed by donations from wealthy Chicagoans like Silas B, Cobb who provided the funds for the campus first building, Cobb Lecture Hall, and matched Marshall Fields pledge of $100,000. Organized as an independent institution legally, it replaced the first Baptist university of the same name, william Rainey Harper became the modern universitys first president on July 1,1891, and the university opened for classes on October 1,1892. The business school was founded thereafter in 1898, and the law school was founded in 1902, Harper died in 1906, and was replaced by a succession of three presidents whose tenures lasted until 1929. During this period, the Oriental Institute was founded to support, in 1896, the university affiliated with Shimer College in Mount Carroll, Illinois. The agreement provided that either party could terminate the affiliation on proper notice, several University of Chicago professors disliked the program, as it involved uncompensated additional labor on their part, and they believed it cheapened the academic reputation of the university. The program passed into history by 1910, in 1929, the universitys fifth president, Robert Maynard Hutchins, took office, the university underwent many changes during his 24-year tenure. In 1933, Hutchins proposed a plan to merge the University of Chicago. During his term, the University of Chicago Hospitals finished construction, also, the Committee on Social Thought, an institution distinctive of the university, was created. Money that had been raised during the 1920s and financial backing from the Rockefeller Foundation helped the school to survive through the Great Depression, during World War II, the university made important contributions to the Manhattan Project. The university was the site of the first isolation of plutonium and of the creation of the first artificial, in the early 1950s, student applications declined as a result of increasing crime and poverty in the Hyde Park neighborhood
University of Chicago
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An early
convocation ceremony at the University of Chicago
University of Chicago
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The University of Chicago
University of Chicago
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View from the
Midway Plaisance
University of Chicago
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The campus of the University of Chicago. From the top of
Rockefeller Chapel, the Main Quadrangles can be seen on the left (West), the
Oriental Institute and the
Becker Friedman Institute for Research in Economics can be seen in the center (North), and the
Booth School of Business and
Laboratory Schools can be seen on the right (East). The panoramic is bounded on both sides by the
Midway Plaisance (South).
111.
Harvard University
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Although never formally affiliated with any denomination, the early College primarily trained Congregationalist and Unitarian clergy. Its curriculum and student body were gradually secularized during the 18th century, james Bryant Conant led the university through the Great Depression and World War II and began to reform the curriculum and liberalize admissions after the war. The undergraduate college became coeducational after its 1977 merger with Radcliffe College, Harvards $34.5 billion financial endowment is the largest of any academic institution. Harvard is a large, highly residential research university, the nominal cost of attendance is high, but the Universitys large endowment allows it to offer generous financial aid packages. Harvards alumni include eight U. S. presidents, several heads of state,62 living billionaires,359 Rhodes Scholars. To date, some 130 Nobel laureates,18 Fields Medalists, Harvard was formed in 1636 by vote of the Great and General Court of the Massachusetts Bay Colony. In 1638, it obtained British North Americas first known printing press, in 1639 it was named Harvard College after deceased clergyman John Harvard an alumnus of the University of Cambridge who had left the school £779 and his scholars library of some 400 volumes. The charter creating the Harvard Corporation was granted in 1650 and it offered a classic curriculum on the English university model—many leaders in the colony had attended the University of Cambridge—but conformed to the tenets of Puritanism. It was never affiliated with any denomination, but many of its earliest graduates went on to become clergymen in Congregational. The leading Boston divine Increase Mather served as president from 1685 to 1701, in 1708, John Leverett became the first president who was not also a clergyman, which marked a turning of the college toward intellectual independence from Puritanism. When the Hollis Professor of Divinity David Tappan died in 1803 and the president of Harvard Joseph Willard died a year later, in 1804, in 1846, the natural history lectures of Louis Agassiz were acclaimed both in New York and on the campus at Harvard College. Agassizs approach was distinctly idealist and posited Americans participation in the Divine Nature, agassizs perspective on science combined observation with intuition and the assumption that a person can grasp the divine plan in all phenomena. When it came to explaining life-forms, Agassiz resorted to matters of shape based on an archetype for his evidence. Charles W. Eliot, president 1869–1909, eliminated the position of Christianity from the curriculum while opening it to student self-direction. While Eliot was the most crucial figure in the secularization of American higher education, he was motivated not by a desire to secularize education, during the 20th century, Harvards international reputation grew as a burgeoning endowment and prominent professors expanded the universitys scope. Rapid enrollment growth continued as new schools were begun and the undergraduate College expanded. Radcliffe College, established in 1879 as sister school of Harvard College, Harvard became a founding member of the Association of American Universities in 1900. In the early 20th century, the student body was predominately old-stock, high-status Protestants, especially Episcopalians, Congregationalists, by the 1970s it was much more diversified
Harvard University
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Engraving of
Harvard College by
Paul Revere, 1767
Harvard University
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Harvard University
Harvard University
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John Harvard statue,
Harvard Yard
Harvard University
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Richard Rummell's 1906 watercolor landscape view, facing northeast.
112.
Georgia State University
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Georgia State University is a public research university in downtown Atlanta, Georgia, United States. Founded in 1913, it is one of the University System of Georgias four research universities, Georgia State University offers more than 250 undergraduate and graduate degree programs spread across eight academic colleges with around 3,500 faculty members. It is accredited by the Southern Association of Colleges and Schools, approximately 27% of the student population is considered part-time while 73% of the population is considered full-time. The university is one of only four Georgia universities classified as a Research University/Very High Activity, the university has a full-time faculty count of 1,142, with 69 percent of those faculty members either tenured or on tenure track. GSU has two libraries, University library and Law library, which hold over 4.3 million volumes combined, the university has an economic impact on the Atlanta economy of more than $1.4 billion annually. Initially intended as a school, Georgia State University was established in 1913 as the Georgia School of Technologys Evening School of Commerce. During this time, the school was divided into two divisions, Georgia Evening College, and Atlanta Junior College, in September 1947, the school became affiliated with the University of Georgia and was named the Atlanta Division of the University of Georgia. The school was removed from the University of Georgia in 1955. In 1961, other programs at the school had grown enough that the name was shortened to Georgia State College. It became Georgia State University in 1969, in 1995, the Georgia Board of Regents accorded Georgia State research university status, joining the Georgia Institute of Technology, the University of Georgia, and Augusta University. The first African-American student enrolled at Georgia State in 1962, a year after the integration of the University of Georgia and Georgia Tech. Annette Lucille Hall was a Lithonia social studies teacher who enrolled in the course of the Institute on Americanism and Communism, a course required for all Georgia social studies teachers. The Peachtree Road Race, was started in 1970 by Georgia State cross country coach and dean of men Tim Singleton, the second year, he created the first valuable collectible T-shirt. Over its 100-plus year history, Georgia States growth has required the acquisition and construction of space to suit its needs. In addition, a plaza and walkway system was constructed to connect these buildings with each other over Decatur Street. Georgia States first move into the Fairlie-Poplar district was the acquisition and renovation of the Standard Building, the Haas-Howell Building, and the Rialto Theater in 1996. The Standard and Haas-Howell buildings house classrooms, offices, and practice spaces for the School of Music, and the Rialto is home to GSUs Jazz Studies program and an 833-seat theater. In 1998, the Student Center was expanded toward Gilmer Street and provided a new 400-seat auditorium and space for exhibitions, a new Student Recreation Center opened on the corner of Piedmont Avenue and Gilmer Street in 2001
Georgia State University
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View of (from L-R) the Sports Arena and Library South on Decatur Street
Georgia State University
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Georgia State University
Georgia State University
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A Georgia State police vehicle on campus in
Atlanta, Georgia
Georgia State University
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Rialto Center
113.
John Peter Oleson
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John Peter Oleson is a Canadian classical archaeologist and historian of ancient technology. His main interests are the Roman Near East, maritime archaeology, and ancient technology, especially hydraulic technology, water-lifting devices, born in 1946 in Hackensack, New Jersey, United States, Oleson was schooled at the Loomis School in Windsor, Connecticut. He received his BA in Classics at Harvard University in 1967, Oleson received his MA and PhD in Classical Archaeology at Harvard University, working in particular with George M. A. Hanfmann and David Mitten. From 1973–1976 Oleson taught in the Classics Department of Florida State University, since 1976 he has been a member of the Greek and Roman Studies Department of the University of Victoria, Canada, where he was appointed Distinguished Professor in 2003. He was elected a Fellow of the Royal Society of Canada in 1994, from 1997 to 2001 he was a member of Council of the Social Sciences and Humanities Research Council of Canada. From 1999–2002 he was a Trustee of the Board of the Royal British Columbia Museum and he was appointed a Killam Research Fellow for 2000–2002. Since 1997 he has been a member of the Board of the American Center for Oriental Research in Amman, in 1997, along with McCann Taggart, he was a project archaeologist at the Skerki Bank Deep Water Shipwreck Survey, directed by Robert Ballard. From 1986 until 2005 he directed survey and excavation at the site of Hawara, a Nabataean, Roman, since 2001 he has co-directed the Roman Maritime Concrete Study with Christopher J. Brandon and Robert L. Hohlfelder. As of 2010 Oleson has published ten books and more than 95 articles concerning ancient technology, marine archaeology, the Nabataeans, and he has presented more than 150 refereed public papers and invited lectures since 1976. In 2010 the Royal Society of Canada awarded Oleson the Pierre Chauveau Medal for distinguished contribution to knowledge in the humanities, vol.1, The Site and the Excavations, BAR International Series, supplement 491,1989, ISBN 0-86054-628-4 The Harbours of Caesarea Maritima. Vol.2, The Finds and the Ship, BAR International Series, supplement 594,1994, ISBN 0-86054-768-X Classical Views/Echos du monde classique, the History and Technology of Roman Concrete Engineering in the Sea. Oxford, Oxbow Books,2014, ISBN 978-1-78297-420-8 Greek technology Roman technology Personal website at University of Victoria
John Peter Oleson
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Pioneers of diving
114.
Galen
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Aelius Galenus or Claudius Galenus, often Anglicized as Galen and better known as Galen of Pergamon, was a prominent Greek physician, surgeon and philosopher in the Roman Empire. The son of Aelius Nicon, an architect with scholarly interests, Galen received a comprehensive education that prepared him for a successful career as a physician. Galens understanding of anatomy and medicine was influenced by the then-current theory of humorism. His theories dominated and influenced Western medical science for more than 1,300 years, Medical students continued to study Galens writings until well into the 19th century. Galen saw himself as both a physician and a philosopher, as he wrote in his treatise entitled That the Best Physician is Also a Philosopher. Many of his works have been preserved and/or translated from the original Greek, although many were destroyed, although there is some debate over the date of his death, he was no younger than seventy when he died. In medieval Europe, Galens writings on anatomy became the mainstay of the medieval university curriculum. Some of Galens ideas were incorrect, he did not dissect a human body, Galens original Greek texts gained renewed prominence during the early modern period. In the 1530s, Belgian anatomist and physician Andreas Vesalius took on a project to many of Galens Greek texts into Latin. Vesaliuss most famous work, De humani corporis fabrica, was influenced by Galenic writing. Galens name Γαληνός, Galēnos comes from the adjective γαληνός, calm, Galen describes his early life in On the affections of the mind. Galen describes his father as an amiable, just, good. His studies also took in each of the philosophical systems of the time. His father had planned a career for Galen in philosophy or politics and took care to expose him to literary. However, Galen states that in around AD145 his father had a dream in which the god Asclepius appeared and commanded Nicon to send his son to study medicine, there he came under the influence of men like Aeschrion of Pergamon, Stratonicus and Satyrus. Asclepiea functioned as spas or sanitoria to which the sick would come to seek the ministrations of the priesthood, romans frequented the temple at Pergamon in search of medical relief from illness and disease. It was also the haunt of notable people such as Claudius Charax the historian, Aelius Aristides the orator, Polemo the sophist, in 148, when he was 19, his father died, leaving him independently wealthy. In 157, aged 28, he returned to Pergamon as physician to the gladiators of the High Priest of Asia, one of the most influential and wealthy men in Asia
Galen
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Eighteenth-century portrait of Galenus by Georg Paul Busch
Galen
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Statue of Galen in
Bergama, Turkey
Galen
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De curandi ratione
Galen
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Mondino dei Liuzzi, Anathomia, 1541
115.
The New York Times
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The New York Times is an American daily newspaper, founded and continuously published in New York City since September 18,1851, by The New York Times Company. The New York Times has won 119 Pulitzer Prizes, more than any other newspaper, the papers print version in 2013 had the second-largest circulation, behind The Wall Street Journal, and the largest circulation among the metropolitan newspapers in the US. The New York Times is ranked 18th in the world by circulation, following industry trends, its weekday circulation had fallen in 2009 to fewer than one million. Nicknamed The Gray Lady, The New York Times has long been regarded within the industry as a newspaper of record. The New York Times international version, formerly the International Herald Tribune, is now called the New York Times International Edition, the papers motto, All the News Thats Fit to Print, appears in the upper left-hand corner of the front page. On Sunday, The New York Times is supplemented by the Sunday Review, The New York Times Book Review, The New York Times Magazine and T, some other early investors of the company were Edwin B. Morgan and Edward B. We do not believe that everything in Society is either right or exactly wrong, —what is good we desire to preserve and improve, —what is evil, to exterminate. In 1852, the started a western division, The Times of California that arrived whenever a mail boat got to California. However, when local California newspapers came into prominence, the effort failed, the newspaper shortened its name to The New-York Times in 1857. It dropped the hyphen in the city name in the 1890s, One of the earliest public controversies it was involved with was the Mortara Affair, the subject of twenty editorials it published alone. At Newspaper Row, across from City Hall, Henry Raymond, owner and editor of The New York Times, averted the rioters with Gatling guns, in 1869, Raymond died, and George Jones took over as publisher. Tweed offered The New York Times five million dollars to not publish the story, in the 1880s, The New York Times transitioned gradually from editorially supporting Republican Party candidates to becoming more politically independent and analytical. In 1884, the paper supported Democrat Grover Cleveland in his first presidential campaign, while this move cost The New York Times readership among its more progressive and Republican readers, the paper eventually regained most of its lost ground within a few years. However, the newspaper was financially crippled by the Panic of 1893, the paper slowly acquired a reputation for even-handedness and accurate modern reporting, especially by the 1890s under the guidance of Ochs. Under Ochs guidance, continuing and expanding upon the Henry Raymond tradition, The New York Times achieved international scope, circulation, in 1910, the first air delivery of The New York Times to Philadelphia began. The New York Times first trans-Atlantic delivery by air to London occurred in 1919 by dirigible, airplane Edition was sent by plane to Chicago so it could be in the hands of Republican convention delegates by evening. In the 1940s, the extended its breadth and reach. The crossword began appearing regularly in 1942, and the section in 1946
The New York Times
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Cover of The New York Times (November 15, 2012), with the headline story reporting on
Operation Pillar of Defense.
The New York Times
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First published issue of New-York Daily Times, on September 18, 1851.
The New York Times
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The Times Square Building, The New York Times ' publishing headquarters, 1913–2007
The New York Times
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The New York Times newsroom, 1942
116.
Andrews University
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Andrews University is a university in Berrien Springs, Michigan. Andrews is the largest evangelical Christian college or university in the state of Michigan, the university consists of eight schools or colleges, offering 130 undergraduate majors and 70 graduate majors. In addition, post-baccalaureate degrees are offered by all and it is accredited by the Higher Learning Commission, and the Adventist Accrediting Association. Andrews University was founded as a small Seventh-day Adventist school called Battle Creek College in 1874 named for the city of Battle Creek. In 1901, the school moved from Battle Creek, Michigan to its current location in Berrien Springs and it is said that everything the school had was packed up in 16 boxcars and sent on its way. The school was renamed Emmanuel Missionary College, or EMC for short, as the first school among us having a distinctive Biblical name. After this SDA college that had known as Battle Creek College moved to Berrien Springs. This Battle Creek College operated until 1938, Emmanuel Missionary College continued to grow slowly through the early 20th century. In the 1940s, Nethery Hall, the current location of the College of Arts and its construction marked the culmination of an aggressive building program. In the 1930s Seventh-day Adventist leaders established a Theological Seminary, at first, it was located on the campus of Pacific Union College in the Napa Valley, California. Later it was moved to Washington, D. C. the following events culminated in relocating the graduate program and theological seminary of Potomac University from Washington, D. C. and joining with the school in Berrien Springs in 1959. In 1956 a charter was granted, the new school was named Potomac University. Earlier, Ellen White, cofounder of the Adventist Church, had advised that Adventist schools locate in rural settings, Church leaders looked for a suitable rural location where the new university could be near to, and in affiliation with, Washington Missionary College, now Washington Adventist University. Over a period of two years effort was put forth to find such a location, too much expense was involved in making such a move. At the 1958 Autumn Council, held in Washington, the board of Emmanuel Missionary College invited the General Conference to locate Potomac University on its campus, after careful deliberation, the council voted unanimously to accept the offer and move the institution to the EMC campus. Arrangements similar to those envisioned for Washington Missionary College were made with EMC, Emmanuel Missionary College did not lose its identity. It remained the college for the youth of the Lake Union Conference, as a university-type General Conference institution it draws students from the entire world field. Over the past three years, church leaders had discussed a new name for this graduate university
Andrews University
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Aerial view of Andrews University
117.
University of Waterloo
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The University of Waterloo is a public research university with a main campus in Waterloo, Ontario. The main campus is on 404 hectares of land in Uptown Waterloo, the university offers academic programs administered by six faculties and ten faculty-based schools. The university also operates four campuses and four affiliated university colleges. Waterloo is a member of the U15, a group of universities in Canada. University of Waterloo is most famous for its education programs. University of Waterloo operates the largest post secondary program of its kind in the world. The institution was established on 1 July 1957 as the Waterloo College Associate Faculties and this entity formally separated from Waterloo College and was incorporated as a university with the passage of the University of Waterloo Act by the Legislative Assembly of Ontario in 1959. It was established to fill the need to train engineers and technicians for Canadas growing postwar economy and it grew substantially over the next decade, adding a faculty of arts in 1960, and the College of Optometry of Ontario which moved from Toronto in 1967. The university is co-educational, and as of 2016 has 30,600 undergraduate and 5,300 postgraduate students, Alumni and former students of the university can be found across Canada and in over 140 countries. Waterloos varsity teams, known as the Waterloo Warriors, compete in the Ontario University Athletics conference of the Canadian Interuniversity Sport. The University of Waterloo traces its origins to Waterloo College, the outgrowth of Waterloo Lutheran Seminary. When Gerald Hagey assumed the presidency of Waterloo College in 1953, following that method, Waterloo College established the Waterloo College Associate Faculties on 4 April 1956, as a non-denominational board affiliated with the college. The academic structure of the Associated Faculties was originally focused on education in the applied sciences – largely built around the proposals of Ira Needles. On 25 January 1958, the Associated Faculties announced the purchase of over 74 hectares of land west of Waterloo College, by the end of the same year, the Associated Faculties opened its first building on the site, the Chemical Engineering Building. In 1959, the Legislative Assembly of Ontario passed an act which split the Associated Faculties from Waterloo College. The president, appointed by the board, was to act as the chief executive officer. While the agreements sought to safeguard the existence of the two colleges, they also aimed at federating them with the newly established University of Waterloo. Due to disagreements with Waterloo College, the College was not formally federated with the new university and this was something that the Associated Faculties was not prepared to accept
University of Waterloo
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Constructed in 1958, the Douglas Wright Engineering Building is the oldest building that was erected for use by the university.
University of Waterloo
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University of Waterloo
University of Waterloo
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The Dana Porter Library holds the university's main collection for humanities and social science.
University of Waterloo
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The Mackenzie King Village residences, constructed in 2002, are the latest set of residences constructed by the university.
118.
CNN
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The Cable News Network is an American basic cable and satellite television news channel owned by the Turner Broadcasting System division of Time Warner. It was founded in 1980 by American media proprietor Ted Turner as a 24-hour cable news channel, upon its launch, CNN was the first television channel to provide 24-hour news coverage, and was the first all-news television channel in the United States. While the news channel has numerous affiliates, CNN primarily broadcasts from the Time Warner Center in New York City and its headquarters at the CNN Center in Atlanta is only used for weekend programming. CNN is sometimes referred to as CNN/U. S. to distinguish the American channel from its sister network. As of August 2010, CNN is available in over 100 million U. S. households, broadcast coverage of the U. S. channel extends to over 890,000 American hotel rooms, as well as carriage on cable and satellite providers throughout Canada. Globally, CNN programming airs through CNN International, which can be seen by viewers in over 212 countries and territories, as of February 2015, CNN is available to about 96,289,000 cable, satellite, and telco television households in the United States. The Cable News Network was launched at 5,00 p. m. Eastern Time on June 1,1980, after an introduction by Ted Turner, the husband and wife team of David Walker and Lois Hart anchored the channels first newscast. Burt Reinhardt, the vice president of CNN at its launch, hired most of the channels first 200 employees, including the networks first news anchor. Since its debut, CNN has expanded its reach to a number of cable and satellite providers, several websites. The company has 36 bureaus, more than 900 affiliated local stations, the channels success made a bona-fide mogul of founder Ted Turner and set the stage for conglomerate Time Warners eventual acquisition of the Turner Broadcasting System in 1996. A companion channel, CNN2, was launched on January 1,1982, on January 28,1986, CNN carried the only live television coverage of the launch and subsequent break-up of Space Shuttle Challenger, which killed all seven crew members on board. On October 14,1987, Jessica McClure, an 18-month-old toddler, fell down a well in Midland, CNN quickly reported on the story, and the event helped make its name. This was before correspondents reported live from the capital while American bombs were falling. Before Saddam Hussein held a press conference with a few of the hundreds of Americans he was holding hostage. Before the nation watched, riveted but powerless, as Los Angeles was looted and burned, before O. J. Simpson took a slow ride in a white Bronco, and before everyone close to his case had an agent and a book contract. This was uncharted territory just a time ago. The moment when bombing began was announced on CNN by Bernard Shaw on January 16,1991, as follows, lets describe to our viewers what were seeing. The skies over Baghdad have been illuminated, were seeing bright flashes going off all over the sky
CNN
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Replica of the newsroom at CNN Center.
CNN
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CNN
CNN
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Operation Desert Storm as captured live on a CNN
night vision camera with reporters narrating.
CNN
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The stage for the second 2008
CNN-YouTube presidential debate.
119.
NASA
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President Dwight D. Eisenhower established NASA in 1958 with a distinctly civilian orientation encouraging peaceful applications in space science. The National Aeronautics and Space Act was passed on July 29,1958, disestablishing NASAs predecessor, the new agency became operational on October 1,1958. Since that time, most US space exploration efforts have led by NASA, including the Apollo Moon landing missions, the Skylab space station. Currently, NASA is supporting the International Space Station and is overseeing the development of the Orion Multi-Purpose Crew Vehicle, the agency is also responsible for the Launch Services Program which provides oversight of launch operations and countdown management for unmanned NASA launches. NASA shares data with various national and international such as from the Greenhouse Gases Observing Satellite. Since 2011, NASA has been criticized for low cost efficiency, from 1946, the National Advisory Committee for Aeronautics had been experimenting with rocket planes such as the supersonic Bell X-1. In the early 1950s, there was challenge to launch a satellite for the International Geophysical Year. An effort for this was the American Project Vanguard, after the Soviet launch of the worlds first artificial satellite on October 4,1957, the attention of the United States turned toward its own fledgling space efforts. This led to an agreement that a new federal agency based on NACA was needed to conduct all non-military activity in space. The Advanced Research Projects Agency was created in February 1958 to develop technology for military application. On July 29,1958, Eisenhower signed the National Aeronautics and Space Act, a NASA seal was approved by President Eisenhower in 1959. Elements of the Army Ballistic Missile Agency and the United States Naval Research Laboratory were incorporated into NASA, earlier research efforts within the US Air Force and many of ARPAs early space programs were also transferred to NASA. In December 1958, NASA gained control of the Jet Propulsion Laboratory, NASA has conducted many manned and unmanned spaceflight programs throughout its history. Some missions include both manned and unmanned aspects, such as the Galileo probe, which was deployed by astronauts in Earth orbit before being sent unmanned to Jupiter, the experimental rocket-powered aircraft programs started by NACA were extended by NASA as support for manned spaceflight. This was followed by a space capsule program, and in turn by a two-man capsule program. This goal was met in 1969 by the Apollo program, however, reduction of the perceived threat and changing political priorities almost immediately caused the termination of most of these plans. NASA turned its attention to an Apollo-derived temporary space laboratory, to date, NASA has launched a total of 166 manned space missions on rockets, and thirteen X-15 rocket flights above the USAF definition of spaceflight altitude,260,000 feet. The X-15 was an NACA experimental rocket-powered hypersonic research aircraft, developed in conjunction with the US Air Force, the design featured a slender fuselage with fairings along the side containing fuel and early computerized control systems
NASA
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1963 photo showing Dr. William H. Pickering, (center) JPL Director, President John F. Kennedy, (right). NASA Administrator James Webb in background. They are discussing the
Mariner program, with a model presented.
NASA
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Seal of NASA
NASA
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At launch control for the May 28, 1964,
Saturn I SA-6 launch.
Wernher von Braun is at center.
NASA
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Mercury-Atlas 6 launch on February 20, 1962
120.
Clifford A. Pickover
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Other Fellows have included Carl Sagan and Isaac Asimov. He has been awarded over 300 United States patents, and his The Math Book was winner of the 2011 Neumann Prize. He received his Ph. D. in 1982 from Yale Universitys Department of Molecular Biophysics and Biochemistry, Pickover graduated first in his class from Franklin and Marshall College, after completing the four-year undergraduate program in three years. He joined IBM at the Thomas J. Watson Research Center in 1982, as a member of the speech synthesis group, for much of his career, Pickover has published technical articles in the areas of scientific visualization, computer art, and recreational mathematics. Pickover is still employed at the IBM Thomas J. Watson Research Center and he is currently an associate editor for the scientific journal Computers and Graphics and is an editorial board member for Odyssey and Leonardo. He is also the Brain-Strain columnist for Odyssey magazine, and, for many years, Pickover has received more than 100 IBM invention achievement awards, three research division awards, and four external honor awards. Pickovers primary interest is in finding new ways to expand creativity by melding art, science, mathematics, Pickover is an inventor with over 300 patents, the author of puzzle calendars, and puzzle contributor to magazines geared to children and adults. His Neoreality and Heaven Virus science-fiction series explores the fabric of reality and he also has published articles in the areas of skepticism, psychology, and technical speculation. Additional visualization work includes topics that involve breathing motions of proteins, snow-flake like patterns for speech sounds, cartoon-face representations of data, Pickover has also written extensively on the reported experiences of people on the psychotropic compound DMT. Such apparent entities as Machine Elves are described as well as Insects From A Parallel Universe, on November 4,2006, he began Wikidumper. org, a popular blog featuring articles being considered for deletion by English Wikipedia. Pickover stalks are certain kinds of details that are found in the Mandelbrot set in the study of fractal geometry. In some renditions of this behavior, the closer that the point approaches, the logarithm of the distance is taken to accentuate the details. This work grew from his work with Julia sets and Pickover biomorphs. In Frontiers of Scientific Visualization Pickover explored the art and science of making the unseen workings of nature visible, in Visualizing Biological Information Pickover considered biological data of all kinds, which is proliferating at an incredible rate. According to Pickover, if humans attempt to read data in the form of numbers and letters. If the information is rendered graphically, however, human analysts can assimilate it, the emphasis of this work is on the novel graphical and musical representation of information containing sequences, such as DNA and amino acid sequences, to help us find hidden pattern and meaning. X and y are called the fangs, as an example,1260 is a vampire number because it can be expressed as 21 ×60 =1260. Note that the digits of the factors 21 and 60 can be found, in some scrambled order, similarly,136,948 is a vampire because 136,948 =146 ×938
Clifford A. Pickover
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Clifford Alan Pickover
Clifford A. Pickover
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In the 1990s, Pickover created virtual caverns from extremely simple numerical simulations that reminded him of the
Lechuguilla Cave, pictured here.
Clifford A. Pickover
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Visualization of chaotic attractor. Pickover’s earliest books often focused on patterns that characterize mathematics such as
fractals,
chaos, and
number theory.
Computer graphics, reminiscent of this chaotic
attractor, were common in his early works.
Clifford A. Pickover
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Forest troll. (
Theodor Kittelsen, 1906). Some of Pickover’s later books often discussed "science at the edges," including such topics as
parallel universes,
quantum immortality,
alien life, and elf-like beings seen by some people who use
dimethyltryptamine.
121.
PhilPapers
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PhilPapers is an international, interactive academic database of journal articles for professionals and students in philosophy. It is maintained by the Centre for Digital Philosophy at the University of Western Ontario, as of 2012, the general editors are David Bourget and David Chalmers. It has a position in the world ranking of repositories. PhilPapers receives financial support from organizations, including a substantial grant in early 2009 from the Joint Information Systems Committee in the United Kingdom. The archive is praised for its comprehensiveness and organization, and for its regular updates, in addition to archiving papers, the editors engage in surveying academic philosophers. List of academic databases and search engines Official website
PhilPapers
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PhilPapers