Napoléon Bonaparte was a French statesman and military leader who rose to prominence during the French Revolution and led several successful campaigns during the French Revolutionary Wars. He was Emperor of the French as Napoleon I from 1804 until 1814 and again in 1815 during the Hundred Days. Napoleon dominated European and global affairs for more than a decade while leading France against a series of coalitions in the Napoleonic Wars, he won most of these wars and the vast majority of his battles, building a large empire that ruled over much of continental Europe before its final collapse in 1815. He is considered one of the greatest commanders in history, his wars and campaigns are studied at military schools worldwide. Napoleon's political and cultural legacy has endured as one of the most celebrated and controversial leaders in human history, he was born in Corsica to a modest family of Italian origin from minor nobility. He was serving as an artillery officer in the French army when the French Revolution erupted in 1789.
He rose through the ranks of the military, seizing the new opportunities presented by the Revolution and becoming a general at age 24. The French Directory gave him command of the Army of Italy after he suppressed a revolt against the government from royalist insurgents. At age 26, he began his first military campaign against the Austrians and the Italian monarchs aligned with the Habsburgs—winning every battle, conquering the Italian Peninsula in a year while establishing "sister republics" with local support, becoming a war hero in France. In 1798, he led a military expedition to Egypt, he became First Consul of the Republic. Napoleon's ambition and public approval inspired him to go further, he became the first Emperor of the French in 1804. Intractable differences with the British meant that the French were facing a Third Coalition by 1805. Napoleon shattered this coalition with decisive victories in the Ulm Campaign and a historic triumph over the Russian Empire and Austrian Empire at the Battle of Austerlitz which led to the dissolution of the Holy Roman Empire.
In 1806, the Fourth Coalition took up arms against him because Prussia became worried about growing French influence on the continent. Napoleon defeated Prussia at the battles of Jena and Auerstedt marched his Grande Armée deep into Eastern Europe and annihilated the Russians in June 1807 at the Battle of Friedland. France forced the defeated nations of the Fourth Coalition to sign the Treaties of Tilsit in July 1807, bringing an uneasy peace to the continent. Tilsit signified the high-water mark of the French Empire. In 1809, the Austrians and the British challenged the French again during the War of the Fifth Coalition, but Napoleon solidified his grip over Europe after triumphing at the Battle of Wagram in July. Napoleon invaded the Iberian Peninsula, hoping to extend the Continental System and choke off British trade with the European mainland, declared his brother Joseph Bonaparte the King of Spain in 1808; the Spanish and the Portuguese revolted with British support. The Peninsular War lasted six years, featured extensive guerrilla warfare, ended in victory for the Allies against Napoleon.
The Continental System caused recurring diplomatic conflicts between France and its client states Russia. The Russians were unwilling to bear the economic consequences of reduced trade and violated the Continental System, enticing Napoleon into another war; the French launched a major invasion of Russia in the summer of 1812. The campaign did not yield the decisive victory Napoleon wanted, it resulted in the collapse of the Grande Armée and inspired a renewed push against Napoleon by his enemies. In 1813, Prussia and Austria joined Russian forces in the War of the Sixth Coalition against France. A lengthy military campaign culminated in a large Allied army defeating Napoleon at the Battle of Leipzig in October 1813, but his tactical victory at the minor Battle of Hanau allowed retreat onto French soil; the Allies invaded France and captured Paris in the spring of 1814, forcing Napoleon to abdicate in April. He was exiled to the island of Elba off the coast of Tuscany, the Bourbon dynasty was restored to power.
Napoleon took control of France once again. The Allies responded by forming a Seventh Coalition which defeated him at the Battle of Waterloo in June; the British exiled him to the remote island of Saint Helena in the South Atlantic, where he died six years at the age of 51. Napoleon's influence on the modern world brought liberal reforms to the numerous territories that he conquered and controlled, such as the Low Countries and large parts of modern Italy and Germany, he implemented fundamental liberal policies throughout Western Europe. His Napoleonic Code has influenced the legal systems of more than 70 nations around the world. British historian Andrew Roberts states: "The ideas that underpin our modern world—meritocracy, equality before the law, property rights, religious toleration, modern secular education, sound finances, so on—were championed, consolidated and geographically extended by Napoleon. To them he added a rational and efficient local administration, an end to rural banditry, the encouragement of science and the arts, the abolition of feudalism and the greatest codification of laws since the fall of the Roman Empire".
The ancestors of Napoleon descended from minor Italian nobility of Tuscan origin who had come to Corsica fr
Lorenzo Mascheroni was an Italian mathematician. He was born near Lombardy. At first interested in the humanities, he became professor of mathematics at Pavia. In his Geometria del Compasso, he proved that any geometrical construction which can be done with compass and straightedge, can be done with compasses alone. However, the priority for this result belongs to the Dane Georg Mohr, who had published a proof in 1672 in an obscure book, Euclides Danicus; this problem was the source of a musical composition called "Mascheroni Circles", performed by David Stutz on the album Iolet. In his Adnotationes ad calculum integrale Euleri he published a calculation of what is now known as the Euler–Mascheroni constant denoted as γ, he died in Paris. O'Connor, John J..
The Italians are a Romance ethnic group and nation native to the Italian peninsula and its neighbouring insular territories. Most Italians share a common culture, ancestry or language. All Italian nationals are citizens of the Italian Republic, regardless of ancestry or nation of residence and may be distinguished from people of Italian descent without Italian citizenship and from ethnic Italians living in territories adjacent to the Italian Peninsula without Italian citizenship; the majority of Italian nationals are speakers of a regional variety thereof. However, many of them speak another regional or minority language native to Italy. In 2017, in addition to about 55 million Italians in Italy, Italian-speaking autonomous groups are found in neighbouring nations: a quarter million are in Switzerland, a large population is in France, the entire population of San Marino, there are smaller groups in Slovenia and Croatia in Istria and Dalmatia; because of the wide-ranging diaspora, about 5 million Italian citizens and nearly 80 million people of full or partial Italian ancestry live outside their own homeland, which include the 62.5% of Argentina's population, 1/3 of Uruguayans, 40% of Paraguayans, 15% of Brazilians, people in other parts of Europe bordering Italy, the Americas and the Middle East.
Italians have influenced and contributed to diverse fields, notably the arts and music and technology, cuisine, jurisprudence and business both abroad and worldwide. Furthermore, Italian people are known for their localism, both regionalist and municipalist; the Latin name Italia according to Strabo's Geographica was used by Greeks to indicate the southwestern tip of the Italian peninsula, corresponding to the current region of Calabria, from the strait of Messina to the line connecting the gulf of Salerno and gulf of Taranto. It most originates with Oscan Víteliú, meaning "land of young cattle"; the bull was a symbol of the southern Italic tribes and was depicted goring the Roman wolf as a defiant symbol of free Italy during the Social War. The name was extended to include all the Italian peninsula south of the Rubicon, still by the end of the 1st century BC, to all of the peninsula and beyond. Latin Italicus as a substantive meaning "a man of Italy" is first recorded in Pliny the Elder, Letters 9.23.
The adjective italianus, from which are derived the Italian name of the Italians is medieval. The Italian peninsula was divided into a multitude of tribal or ethnic territory prior to the Roman conquest of Italy in the 3rd century BC. After a series of wars between Greeks and Etruscans, the Latins, with Rome as their capital, gained the ascendancy by 272 BC, completed the conquest of the Italian peninsula by 218 BC; this period of unification was followed by one of conquest in the Mediterranean, beginning with the First Punic War against Carthage. In the course of the century-long struggle against Carthage, the Romans conquered Sicily and Corsica. In 146 BC, at the conclusion of the Third Punic War, with Carthage destroyed and its inhabitants enslaved, Rome became the dominant power in the Mediterranean; the process of Italian unification, the associated Romanization, culminated in 88 BC, when, in the aftermath of the Social War, Rome granted its Italian allies full rights in Roman society, extending Roman citizenship to all Italic peoples.
From its inception, Rome was a republican city-state, but four famous civil conflicts destroyed the republic: Lucius Cornelius Sulla against Gaius Marius and his son, Julius Caesar against Pompey, Marcus Junius Brutus and Gaius Cassius Longinus against Mark Antony and Octavian, Mark Antony against Octavian. Octavian, the final victor, was accorded the title of Augustus by the Senate and thereby became the first Roman emperor. Augustus created for the first time an administrative region called Italia with inhabitants called "Italicus populus", stretching from the Alps to Sicily: for this reason historians like Emilio Gentile called him Father of Italians. In the 1st century BC, Italia was still a collection of territories with different political statuses; some cities, called municipia, had some independence from Rome, while others, the coloniae, were founded by the Romans themselves. Around 7 BC, Augustus divided Italy into eleven regiones. During the Crisis of the Third Century the Roman Empire nearly collapsed under the combined pressures of invasions, military anarchy and civil wars, hyperinflation.
In 284, emperor Diocletian restored political stability. The importance of Rome declined; the seats of the Caesars were Augusta Treverorum for Constantius Chlorus and Sirmium (on the Riv
A circle is a simple closed shape. It is the set of all points in a plane; the distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior. A circle may be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. A circle is a plane figure bounded by one line, such that all right lines drawn from a certain point within it to the bounding line, are equal; the bounding line is called the point, its centre. Annulus: a ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle. Centre: the point equidistant from all points on the circle. Chord: a line segment whose endpoints lie on the circle, thus dividing a circle in two sements. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; this is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, its length is twice the length of a radius. Disc: the region of the plane bounded by a circle. Lens: the region common to two overlapping discs. Passant: a coplanar straight line that has no point in common with the circle. Radius: a line segment joining the centre of a circle with any single point on the circle itself. Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.
Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to. Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Tangent: a coplanar straight line that has one single point in common with a circle. All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries; the word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, meaning "hoop" or "ring".
The origins of the words circus and circuit are related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand; the circle is the basis for the wheel, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Early science geometry and astrology and astronomy, was connected to the divine for most medieval scholars, many believed that there was something intrinsically "divine" or "perfect" that could be found in circles; some highlights in the history of the circle are: 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π. 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed explanation of the circle.
Plato explains the perfect circle, how it is different from any drawing, definition or explanation. 1880 CE – Lindemann proves that π is transcendental settling the millennia-old problem of squaring the circle. The ratio of a circle's circumference to its diameter is π, an irrational constant equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: C = 2 π r = π d; as proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2. Equivalently, denoting diameter by d, A r e
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
In geometry, a line segment is a part of a line, bounded by two distinct end points, contains every point on the line between its endpoints. A closed line segment includes both endpoints. Examples of line segments include the sides of a square. More when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices, or otherwise a diagonal; when the end points both lie on a curve such as a circle, a line segment is called a chord. If V is a vector space over R or C, L is a subset of V L is a line segment if L can be parameterized as L = for some vectors u, v ∈ V, in which case the vectors u and u + v are called the end points of L. Sometimes one needs to distinguish between "open" and "closed" line segments. One defines a closed line segment as above, an open line segment as a subset L that can be parametrized as L = for some vectors u, v ∈ V. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.
In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R 2 the line segment with endpoints A = and C = is the following collection of points:. A line segment is a non-empty set. If V is a topological vector space a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, skew, or none of these; the last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane they must cross each other, but that need not be true of segments. In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line.
Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set; this is important because it transforms some of the analysis of convex sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and substitute other segments into another statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant. A complete orbit of this ellipse traverses the line segment twice; as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Some frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, the internal angle bisectors. In each case there are various equalities relating these segment lengths to others as well as various inequalities. Other segment
Compass (drawing tool)
A pair of compasses known as a compass, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can be used as tools to measure distances, in particular on maps. Compasses can be used for mathematics, drafting and other purposes. Compasses are made of metal or plastic, consist of two parts connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. One part has a spike at its end, the other part a pencil, or sometimes a pen. Prior to computerization and other tools for manual drafting were packaged as a "bow set" with interchangeable parts. By the mid-twentieth century, circle templates supplemented the use of compasses. Today these facilities are more provided by computer-aided design programs, so the physical tools serve a didactic purpose in teaching geometry, technical drawing, etc. Compasses are made of metal or plastic, consist of two parts connected by a hinge that can be adjusted to change the radius of the circle one wishes to draw.
One part has a spike at its end, the other part a pencil, or sometimes a pen. The handle is about half an inch long. Users can grip it between their pointer thumb. There are two types of legs in a pair of compasses: the straight or the steady leg and the adjustable one; each has a separate purpose. The screw on the hinge holds the two legs in its position; the tighter the screw, the better the compass’ performance. The needle point is located on the steady leg, serves as the center point of circles that are drawn; the pencil lead draws the circle on a particular material. Alternatively, an ink nib or attachment with a technical pen may be used; this holds the pencil pen in place. Circles can be made by fastening one leg of the compasses into the paper with the spike, putting the pencil on the paper, moving the pencil around while keeping the hinge on the same angle; the radius of the circle can be adjusted by changing the angle of the hinge. Distances can be measured on a map using compasses with two spikes called a dividing compass.
The hinge is set in such a way that the distance between the spikes on the map represents a certain distance in reality, by measuring how many times the compasses fit between two points on the map the distance between those points can be calculated. Compasses-and-straightedge constructions are used to illustrate principles of plane geometry. Although a real pair of compasses is used to draft visible illustrations, the ideal compass used in proofs is an abstract creator of perfect circles; the most rigorous definition of this abstract tool is the "collapsing compass". Euclid showed in his second proposition that such a collapsing compass could be used to transfer a distance, proving that a collapsing compass could do anything a real compass can do. A beam compass is an instrument with a wooden or brass beam and sliding sockets, or cursors, for drawing and dividing circles larger than those made by a regular pair of compasses. Scribe-compasses is an instrument used by other tradesmen; some compasses can be used in this case to trace a line.
It is the compass in the most simple form. Both branches are crimped metal. One branch has a pencil sleeve while the other branch is crimped with a fine point protruding from the end. A wing nut on the hinge serves two purposes: first it tightens the pencil and secondly it locks in the desired distance when the wing nut is turned clockwise. Loose leg wing dividers are made of all forged steel; the pencil holder, thumb screws, brass pivot and branches are all well built. They are used for stepping off repetitive measurements with some accuracy. A proportional compass known as a military compass or sector, was an instrument used for calculation from the end of the sixteenth century until the nineteenth century, it consists of two rulers of equal length joined by a hinge. Different types of scales are inscribed on the rulers. A reduction compass is used to enlarge patterns while conserving angles. A pair of compasses is used as a symbol of precision and discernment; as such it finds a place in logos and symbols such as the Freemasons' Square and Compasses and in various computer icons.
English poet John Donne used the compass as a conceit in "A Valediction: Forbidding Mourning". Dividers Circle Geometrography Masonic Square and Compasses Technical drawing tools Beam or trammel compass