Straightedge and compass construction

Straightedge and compass construction known as ruler-and-compass construction or classical construction, is the construction of lengths and other geometric figures using only an idealized ruler and compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, no markings on it; the compass is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates, it turns out to be the case that every point constructible using straightedge and compass may be constructed using compass alone. The ancient Greek mathematicians first conceived straightedge and compass constructions, a number of ancient problems in plane geometry impose this restriction; the ancient Greeks developed many constructions. Gauss showed that most are not; some of the most famous straightedge and compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields.

In spite of existing proofs of impossibility, some persist in trying to solve these problems. Many of these problems are solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone. In terms of algebra, a length is constructible if and only if it represents a constructible number, an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots; the "straightedge" and "compass" of straightedge and compass constructions are idealizations of rulers and compasses in the real world: The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to extend an existing segment; the compass can be opened arbitrarily wide.

Circles can only be drawn starting from two given points: a point on the circle. The compass may not collapse when it is not drawing a circle. Actual compasses do not collapse and modern geometric constructions use this feature. A'collapsing compass' would appear to be a less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a checkered history; each construction must be exact. "Eyeballing" it and getting close does not count as a solution. Each construction must terminate; that is, it must have a finite number of steps, not be the limit of closer approximations. Stated this way and compass constructions appear to be a parlour game, rather than a serious practical problem; the ancient Greek mathematicians first attempted straightedge and compass constructions, they discovered how to construct sums, products and square roots of given lengths.

They could construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, a regular polygon with 3, 4, or 5 sides. But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or a regular polygon with other numbers of sides. Nor could they construct the side of a cube whose volume would be twice the volume of a cube with a given side. Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle. No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed.

In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He showed that Gauss's sufficient constructibility condition for regular polygons is necessary. In 1882 Lindemann showed that π is a transcendental number, thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle. All straightedge and compass constructions consist of repeated application of five basic constructions using the points and circles that have been constructed; these are: Creating the line through two existing points Creating the circle through one point with centre another point Creating the point, th

Encyclopædia Britannica, Eleventh Edition

The Encyclopædia Britannica, Eleventh Edition is a 29-volume reference work, an edition of the Encyclopædia Britannica. It was developed during the encyclopaedia's transition from a British to an American publication; some of its articles were written by the best-known scholars of the time. This edition of the encyclopedia, containing 40,000 entries, is now in the public domain, many of its articles have been used as a basis for articles in Wikipedia. However, the outdated nature of some of its content makes its use as a source for modern scholarship problematic; some articles have special value and interest to modern scholars as cultural artifacts of the 19th and early 20th centuries. The 1911 eleventh edition was assembled with the management of American publisher Horace Everett Hooper. Hugh Chisholm, who had edited the previous edition, was appointed editor in chief, with Walter Alison Phillips as his principal assistant editor. Hooper bought the rights to the 25-volume 9th edition and persuaded the British newspaper The Times to issue its reprint, with eleven additional volumes as the tenth edition, published in 1902.

Hooper's association with The Times ceased in 1909, he negotiated with the Cambridge University Press to publish the 29-volume eleventh edition. Though it is perceived as a quintessentially British work, the eleventh edition had substantial American influences, not only in the increased amount of American and Canadian content, but in the efforts made to make it more popular. American marketing methods assisted sales; some 14% of the contributors were from North America, a New York office was established to coordinate their work. The initials of the encyclopedia's contributors appear at the end of selected articles or at the end of a section in the case of longer articles, such as that on China, a key is given in each volume to these initials; some articles were written by the best-known scholars of the time, such as Edmund Gosse, J. B. Bury, Algernon Charles Swinburne, John Muir, Peter Kropotkin, T. H. Huxley, James Hopwood Jeans and William Michael Rossetti. Among the lesser-known contributors were some who would become distinguished, such as Ernest Rutherford and Bertrand Russell.

Many articles were carried over from some with minimal updating. Some of the book-length articles were divided into smaller parts for easier reference, yet others much abridged; the best-known authors contributed only a single article or part of an article. Most of the work was done by British Museum scholars and other scholars; the 1911 edition was the first edition of the encyclopædia to include more than just a handful of female contributors, with 34 women contributing articles to the edition. The eleventh edition introduced a number of changes of the format of the Britannica, it was the first to be published complete, instead of the previous method of volumes being released as they were ready. The print type was subject to continual updating until publication, it was the first edition of Britannica to be issued with a comprehensive index volume in, added a categorical index, where like topics were listed. It was the first not to include long treatise-length articles. Though the overall length of the work was about the same as that of its predecessor, the number of articles had increased from 17,000 to 40,000.

It was the first edition of Britannica to include biographies of living people. Sixteen maps of the famous 9th edition of Stielers Handatlas were translated to English, converted to Imperial units, printed in Gotha, Germany by Justus Perthes and became part this edition. Editions only included Perthes' great maps as low quality reproductions. According to Coleman and Simmons, the content of the encyclopedia was distributed as follows: Hooper sold the rights to Sears Roebuck of Chicago in 1920, completing the Britannica's transition to becoming a American publication. In 1922, an additional three volumes, were published, covering the events of the intervening years, including World War I. These, together with a reprint of the eleventh edition, formed the twelfth edition of the work. A similar thirteenth edition, consisting of three volumes plus a reprint of the twelfth edition, was published in 1926, so the twelfth and thirteenth editions were related to the eleventh edition and shared much of the same content.

However, it became apparent that a more thorough update of the work was required. The fourteenth edition, published in 1929, was revised, with much text eliminated or abridged to make room for new topics; the eleventh edition was the basis of every version of the Encyclopædia Britannica until the new fifteenth edition was published in 1974, using modern information presentation. The eleventh edition's articles are still of value and interest to modern readers and scholars as a cultural artifact: the British Empire was at its maximum, imperialism was unchallenged, much of the world was still ruled by monarchs, the tragedy of the modern world wars was still in the future, they are an invaluable resource for topics omitted from modern encyclopedias for biography and the history of science and technology. As a literary text, the encyclopedia has value as an example of early 20th-century prose. For example, it employs literary devices, such as pathetic fallacy, which are not as common in modern reference texts.

In 1917, using the pseudonym of S. S. Van Dine, the US art critic and author Willard Huntington Wright published Misinforming a Nation, a 200+

Public domain

The public domain consists of all the creative works to which no exclusive intellectual property rights apply. Those rights may have been forfeited, expressly waived, or may be inapplicable; the works of William Shakespeare and Beethoven, most early silent films, are in the public domain either by virtue of their having been created before copyright existed, or by their copyright term having expired. Some works are not covered by copyright, are therefore in the public domain—among them the formulae of Newtonian physics, cooking recipes, all computer software created prior to 1974. Other works are dedicated by their authors to the public domain; the term public domain is not applied to situations where the creator of a work retains residual rights, in which case use of the work is referred to as "under license" or "with permission". As rights vary by country and jurisdiction, a work may be subject to rights in one country and be in the public domain in another; some rights depend on registrations on a country-by-country basis, the absence of registration in a particular country, if required, gives rise to public-domain status for a work in that country.

The term public domain may be interchangeably used with other imprecise or undefined terms such as the "public sphere" or "commons", including concepts such as the "commons of the mind", the "intellectual commons", the "information commons". Although the term "domain" did not come into use until the mid-18th century, the concept "can be traced back to the ancient Roman Law, as a preset system included in the property right system." The Romans had a large proprietary rights system where they defined "many things that cannot be owned" as res nullius, res communes, res publicae and res universitatis. The term res nullius was defined as things not yet appropriated; the term res communes was defined as "things that could be enjoyed by mankind, such as air and ocean." The term res publicae referred to things that were shared by all citizens, the term res universitatis meant things that were owned by the municipalities of Rome. When looking at it from a historical perspective, one could say the construction of the idea of "public domain" sprouted from the concepts of res communes, res publicae, res universitatis in early Roman law.

When the first early copyright law was first established in Britain with the Statute of Anne in 1710, public domain did not appear. However, similar concepts were developed by French jurists in the 18th century. Instead of "public domain", they used terms such as publici juris or propriété publique to describe works that were not covered by copyright law; the phrase "fall in the public domain" can be traced to mid-19th century France to describe the end of copyright term. The French poet Alfred de Vigny equated the expiration of copyright with a work falling "into the sink hole of public domain" and if the public domain receives any attention from intellectual property lawyers it is still treated as little more than that, left when intellectual property rights, such as copyright and trademarks, expire or are abandoned. In this historical context Paul Torremans describes copyright as a, "little coral reef of private right jutting up from the ocean of the public domain." Copyright law differs by country, the American legal scholar Pamela Samuelson has described the public domain as being "different sizes at different times in different countries".

Definitions of the boundaries of the public domain in relation to copyright, or intellectual property more regard the public domain as a negative space. According to James Boyle this definition underlines common usage of the term public domain and equates the public domain to public property and works in copyright to private property. However, the usage of the term public domain can be more granular, including for example uses of works in copyright permitted by copyright exceptions; such a definition regards work in copyright as private property subject to fair-use rights and limitation on ownership. A conceptual definition comes from Lange, who focused on what the public domain should be: "it should be a place of sanctuary for individual creative expression, a sanctuary conferring affirmative protection against the forces of private appropriation that threatened such expression". Patterson and Lindberg described the public domain not as a "territory", but rather as a concept: "here are certain materials – the air we breathe, rain, life, thoughts, ideas, numbers – not subject to private ownership.

The materials that compose our cultural heritage must be free for all living to use no less than matter necessary for biological survival." The term public domain may be interchangeably used with other imprecise or undefined terms such as the "public sphere" or "commons", including concepts such as the "commons of the mind", the "intellectual commons", the "information commons". A public-domain book is a book with no copyright, a book, created without a license, or a book where its copyrights expired or have been forfeited. In most countries the term of protection of copyright lasts until January first, 70 years after the death of the latest living author; the longest copyright term is in Mexico, which has life plus 100 years for all deaths since July 1928. A notable exception is the United States, where every book and tale published prior to 1924 is in the public domain.

Parabola

In mathematics, a parabola is a plane curve, mirror-symmetrical and is U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define the same curves. One description of a parabola involves 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. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane, parallel to another plane, tangential to the conical surface; 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", is the point where the parabola is most 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, parallel to the directrix and passes through the focus.

Parabolas can open up, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light light which travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. 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 other forms of energy. This reflective property is the basis of many practical uses of parabolas; the parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are used in physics 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 area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes via the method of exhaustion in the third century BC, in his The Quadrature of the Parabola; the name "parabola" is due to Apollonius. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved; the focus–directrix property of the parabola and other conic sections 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 parabolic reflector could produce an image was 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, James Gregory; 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 and radar receivers. A parabola can be defined geometrically as a set of points in the Euclidean plane: A parabola is a set of points, such that for any point P of the set the distance | P F | to a fixed point F, the focus, is equal to the distance | P l | to a fixed line l, the directrix: The midpoint V of the perpendicular from the focus F onto the directrix l is called vertex and the line F V the axis of symmetry of the parabola. If one introduces cartesian coordinates, such that F =, f > 0, the directrix has the equation y = − f one obtains for a point P = from | P F | 2 = | P l | 2 the equation x 2 + 2 = 2. Solving for y yields y = 1 4 f x 2; the parabola is U-shaped. The horizontal chord through the focus is called the latus rectum; the latus rectum is parallel to the directrix. The semi-latus

Geometry

Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.

He studied the sp