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Optimization problems For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following:
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Linear programming When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular simplex algorit...
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By field of study Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric alg...
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Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields. By compl...
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Algorithms can be classified by the amount of time they need to complete compared to their input size: Constant time: if the time needed by the algorithm is the same, regardless of the input size. E.g. an access to an array element. Logarithmic time: if the time is a logarithmic function of the input size. E.g. binar...
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Some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms ...
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Continuous algorithms The adjective "continuous" when applied to the word "algorithm" can mean: An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in numerical analysis; or An algorithm in the form of a dif...
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Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), and hence algorithms are not patentable (as in Gottschalk v. Benson). However practical applications of a...
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Additionally, some cryptographic algorithms have export restrictions (see export of cryptography). History: Development of the notion of "algorithm" Ancient Near East The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia (modern Iraq). A Sumerian clay tablet found in Shurup...
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Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus circa 1550 BC. Algorithms were later used in ancient Hellenistic mathematics. Two examples are the Sieve of Eratosthenes, which was described in the Introduction to Arithmetic by Nicomachus, and the E...
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Discrete and distinguishable symbols Tally-marks: To keep track of their flocks, their sacks of grain and their money the ancients used tallying: accumulating stones or marks scratched on sticks or making discrete symbols in clay. Through the Babylonian and Egyptian use of marks and symbols, eventually Roman numerals ...
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Manipulation of symbols as "place holders" for numbers: algebra Muhammad ibn Mūsā al-Khwārizmī, a Persian mathematician, wrote the Al-jabr in the 9th century. The terms "algorism" and "algorithm" are derived from the name al-Khwārizmī, while the term "algebra" is derived from the book Al-jabr. In Europe, the word "alg...
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Cryptographic algorithms The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm.
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Mechanical contrivances with discrete states The clock: Bolter credits the invention of the weight-driven clock as "The key invention [of Europe in the Middle Ages]", in particular, the verge escapement that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "...
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Logical machines 1870 – Stanley Jevons' "logical abacus" and "logical machine": The technical problem was to reduce Boolean equations when presented in a form similar to what is now known as Karnaugh maps. Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part ...
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This machine he displayed in 1870 before the Fellows of the Royal Society. Another logician John Venn, however, in his 1881 Symbolic Logic, turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that a...
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Jacquard loom, Hollerith punch cards, telegraphy and telephony – the electromechanical relay: Bell and Newell (1971) indicate that the Jacquard loom (1801), precursor to Hollerith cards (punch cards, 1887), and "telephone switching technologies" were the roots of a tree leading to the development of the first computers...
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Telephone-switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. "He went home one evening in 1937 intending to test ...
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Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" open and closed): It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned."
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Mathematics during the 19th century up to the mid-20th century Symbols and rules: In rapid succession, the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by ...
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But Heijenoort gives Frege (1879) this kudos: Frege's is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed fro...
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The paradoxes: At the same time a number of disturbing paradoxes appeared in the literature, in particular, the Burali-Forti paradox (1897), the Russell paradox (1902–03), and the Richard Paradox. The resultant considerations led to Kurt Gödel's paper (1931)—he specifically cites the paradox of the liar—that completely...
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Effective calculability: In an effort to solve the Entscheidungsproblem defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" (i.e., a calculation that would succeed). In rapid succession the followi...
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Emil Post (1936) and Alan Turing (1936–37, 1939) Emil Post (1936) described the actions of a "computer" (human being) as follows: "...two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions.
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His symbol space would be "a two-way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single m...
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"One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e., INPUT] being marked with a stroke. Likewise, the answer [i.e., OUTPUT] is to be given in symbolic form by such a configuration of marked boxes...
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"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process terminates only when it comes to the direction of type (C ) [i.e., STOP]". See more at Post–Turing machine
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Alan Turing's work preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriter-like model derived from a youthful interest: "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter, and he could well ha...
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Turing—his model of computation is now called a Turing machine—begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers.
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"Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book...I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may ...
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"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observatio...
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Turing's reduction yields the following: "The simple operations must therefore include: "(a) Changes of the symbol on one of the observed squares "(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares. "It may be that some of these change necessarily inv...
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"We may now construct a machine to do the work of this computer."
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A few years later, Turing expanded his analysis (thesis, definition) with this forceful expression of it: "A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Though it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have...
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J.B. Rosser (1939) and S.C. Kleene (1943) J. Barkley Rosser defined an 'effective [mathematical] method' in the following manner (italicization added): "'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a fin...
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Rosser's footnote No. 5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular, Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and Gödel and their use of recursion, in particular, Gödel's use in his famous paper On Formally Un...
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Stephen C. Kleene defined as his now-famous "Thesis I" known as the Church–Turing thesis. But he did this in the following context (boldface in original): "12. Algorithmic theories... In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independen...
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History after 1950 A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, foundations of mathematics (especially the Church–Turing thesis) and philosophy of mind (especially arguments about artificial int...
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Abstract machine Algorithm engineering Algorithm characterizations Algorithmic bias Algorithmic composition Algorithmic entities Algorithmic synthesis Algorithmic technique Algorithmic topology Garbage in, garbage out Introduction to Algorithms (textbook) List of algorithms List of algorithm general topics ...
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Bell, C. Gordon and Newell, Allen (1971), Computer Structures: Readings and Examples, McGraw–Hill Book Company, New York. . Includes an excellent bibliography of 56 references. , : cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively (mechanically) enumerable". Campagnolo, M...
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Further reading Knuth, Donald E. (2000). Selected Papers on Analysis of Algorithms. Stanford, California: Center for the Study of Language and Information. Knuth, Donald E. (2010). Selected Papers on Design of Algorithms. Stanford, California: Center for the Study of Language and Information. External links Dictionary...
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American Civil War
The American Civil War (April 12, 1861 – May 9, 1865; also known by other names) was a civil war in the United States between the Union (states that remained loyal to the federal union, or "the North") and the Confederacy (states that voted to secede, or "the South"). The central cause of the war was the status of slav...
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The practice of slavery in the United States was one of the key political issues of the 19th century. Decades of political unrest over slavery led up to the Civil War. Disunion came after Abraham Lincoln won the 1860 United States presidential election on an anti-slavery expansion platform. An initial seven southern sl...
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During 1861–1862 in the war's Western Theater, the Union made significant permanent gainsthough in the war's Eastern Theater the conflict was inconclusive. On January 1, 1863, Lincoln issued the Emancipation Proclamation, which made ending slavery a war goal, declaring all persons held as slaves in states in rebellion ...
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The Civil War effectively ended on April 9, 1865, when Confederate General Lee surrendered to Union General Grant at the Battle of Appomattox Court House, after Lee had abandoned Petersburg and Richmond. Confederate generals throughout the Confederate army followed suit. The conclusion of the American Civil War lacks a...
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The Civil War is one of the most studied and written about episodes in the history of the United States. It remains the subject of cultural and historiographical debate. Of particular interest is the persisting myth of the Lost Cause of the Confederacy. The American Civil War was among the earliest to use industrial wa...
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Causes of secession
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The causes of secession were complex and have been controversial since the war began, but most academic scholars identify slavery as the central cause of the war. The issue has been further complicated by historical revisionists, who have tried to offer a variety of reasons for the war. Slavery was the central source o...
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Slavery
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Slavery was the main cause of disunion. Slavery had been a controversial issue during the framing of the Constitution but had been left unsettled. The issue of slavery had confounded the nation since its inception, and increasingly separated the United States into a slaveholding South and a free North. The issue was ex...
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The motivations of the average person were not inherently those of their faction; some Northern soldiers were even indifferent on the subject of slavery, but a general pattern can be established. Confederate soldiers fought the war primarily to protect a Southern society of which slavery was an integral part. From the ...
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Abolitionists The abolitionists – those advocating the end of slavery – were very active in the decades leading up to the Civil War. They traced their philosophical roots back to the Puritans, who strongly believed that slavery was morally wrong. One of the early Puritan writings on this subject was The Selling of Jose...
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The American Revolution and the cause of liberty added tremendous impetus to the abolitionist cause. Slavery, which had been around for thousands of years, was considered normal and was not a significant issue of public debate prior to the Revolution. The Revolution changed that and made it into an issue that had to be...
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The establishment of the Northwest Territory as "free soil" – no slavery – by Manasseh Cutler and Rufus Putnam (who both came from Puritan New England) would also prove crucial. This territory (which became the states of Ohio, Michigan, Indiana, Illinois, Wisconsin and part of Minnesota) doubled the size of the United ...
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In the decades leading up to the Civil War, abolitionists, such as Theodore Parker, Ralph Waldo Emerson, Henry David Thoreau and Frederick Douglass, repeatedly used the Puritan heritage of the country to bolster their cause. The most radical anti-slavery newspaper, The Liberator, invoked the Puritans and Puritan values...
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By 1840 more than 15,000 people were members of abolitionist societies in the United States. Abolitionism in the United States became a popular expression of moralism, and led directly to the Civil War. In churches, conventions and newspapers, reformers promoted an absolute and immediate rejection of slavery. Support f...
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Abolitionist sentiment was not strictly religious or moral in origin. The Whig Party became increasingly opposed to slavery because they saw it as inherently against the ideals of capitalism and the free market. Whig leader William H. Seward (who would serve in Lincoln's cabinet) proclaimed that there was an "irrepress...
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Manifest destiny heightened the conflict over slavery, as each new territory acquired had to face the thorny question of whether to allow or disallow the "peculiar institution". Between 1803 and 1854, the United States achieved a vast expansion of territory through purchase, negotiation, and conquest. At first, the new...
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The Mexican–American War and its aftermath was a key territorial event in the leadup to the war. As the Treaty of Guadalupe Hidalgo finalized the conquest of northern Mexico west to California in 1848, slaveholding interests looked forward to expanding into these lands and perhaps Cuba and Central America as well. Prop...
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By 1860, four doctrines had emerged to answer the question of federal control in the territories, and they all claimed they were sanctioned by the Constitution, implicitly or explicitly. The first of these "conservative" theories, represented by the Constitutional Union Party, argued that the Missouri Compromise apport...
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The second doctrine of Congressional preeminence, championed by Abraham Lincoln and the Republican Party, insisted that the Constitution did not bind legislators to a policy of balance—that slavery could be excluded in a territory as it was done in the Northwest Ordinance of 1787 at the discretion of Congress; thus Con...
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Senator Stephen A. Douglas proclaimed the third doctrine: territorial or "popular" sovereignty, which asserted that the settlers in a territory had the same rights as states in the Union to establish or disestablish slavery as a purely local matter. The Kansas–Nebraska Act of 1854 legislated this doctrine. In the Kansa...
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The fourth doctrine was advocated by Mississippi Senator Jefferson Davis, one of state sovereignty ("states' rights"), also known as the "Calhoun doctrine", named after the South Carolinian political theorist and statesman John C. Calhoun. Rejecting the arguments for federal authority or self-government, state sovereig...
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States' rights A long running dispute over the origin of the Civil War is to what extent states' rights triggered the conflict. The consensus among historians is that the Civil War was fought about states' rights. But the issue is frequently referenced in popular accounts of the war and has much traction among Souther...
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Historian James McPherson points out that even if Confederates genuinely fought over states' rights, it boiled down to states' right to slavery. McPherson writes concerning states' rights and other non-slavery explanations:
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Before the Civil War, the Southern states used federal powers in enforcing and extending slavery at the national level, with the Fugitive Slave Act of 1850 and Dred Scott v. Sandford decision. The faction that pushed for secession often infringed on states' rights. Because of the overrepresentation of pro-slavery facti...
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Sectionalism Sectionalism resulted from the different economies, social structure, customs, and political values of the North and South. Regional tensions came to a head during the War of 1812, resulting in the Hartford Convention, which manifested Northern dissatisfaction with a foreign trade embargo that affected the...
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Historians have debated whether economic differences between the mainly industrial North and the mainly agricultural South helped cause the war. Most historians now disagree with the economic determinism of historian Charles A. Beard in the 1920s, and emphasize that Northern and Southern economies were largely compleme...
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Protectionism Owners of slaves preferred low-cost manual labor with no mechanization. Northern manufacturing interests supported tariffs and protectionism while Southern planters demanded free trade. The Democrats in Congress, controlled by Southerners, wrote the tariff laws in the 1830s, 1840s, and 1850s, and kept red...
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Nationalism and honor Nationalism was a powerful force in the early 19th century, with famous spokesmen such as Andrew Jackson and Daniel Webster. While practically all Northerners supported the Union, Southerners were split between those loyal to the entirety of the United States (called "Southern Unionists") and thos...
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While the South moved towards a Southern nationalism, leaders in the North were also becoming more nationally minded, and they rejected any notion of splitting the Union. The Republican national electoral platform of 1860 warned that Republicans regarded disunion as treason and would not tolerate it. The South ignored ...
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The election of Abraham Lincoln in November 1860 was the final trigger for secession. Efforts at compromise, including the Corwin Amendment and the Crittenden Compromise, failed. Southern leaders feared that Lincoln would stop the expansion of slavery and put it on a course toward extinction. When Lincoln won the presi...
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According to Lincoln, the American people had shown that they had been successful in establishing and administering a republic, but a third challenge faced the nation: maintaining a republic based on the people's vote, in the face of an attempt to destroy it. Outbreak of the war
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Secession crisis The election of Lincoln provoked the legislature of South Carolina to call a state convention to consider secession. Before the war, South Carolina did more than any other Southern state to advance the notion that a state had the right to nullify federal laws, and even to secede from the United States....
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Among the ordinances of secession passed by the individual states, those of three—Texas, Alabama, and Virginia—specifically mentioned the plight of the "slaveholding states" at the hands of Northern abolitionists. The rest make no mention of the slavery issue and are often brief announcements of the dissolution of ties...
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As Southerners resigned their seats in the Senate and the House, Republicans were able to pass projects that had been blocked by Southern senators before the war. These included the Morrill Tariff, land grant colleges (the Morrill Act), a Homestead Act, a transcontinental railroad (the Pacific Railroad Acts), the Natio...
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In December 1860, the Crittenden Compromise was proposed to re-establish the Missouri Compromise line by constitutionally banning slavery in territories to the north of the line while guaranteeing it to the south. The adoption of this compromise likely would have prevented the secession of the Southern states, but Linc...
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On March 4, 1861, Abraham Lincoln was sworn in as president. In his inaugural address, he argued that the Constitution was a more perfect union than the earlier Articles of Confederation and Perpetual Union, that it was a binding contract, and called any secession "legally void". He had no intent to invade Southern sta...
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The Davis government of the new Confederacy sent three delegates to Washington to negotiate a peace treaty with the United States of America. Lincoln rejected any negotiations with Confederate agents because he claimed the Confederacy was not a legitimate government, and that making any treaty with it would be tantamou...
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Complicating Lincoln's attempts to defuse the crisis were the actions of the new Secretary of State, William Seward. Seward had been Lincoln's main rival for the Republican presidential nomination. Shocked and deeply embittered by this defeat, Seward only agreed to support Lincoln's candidacy after he was guaranteed th...
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Battle of Fort Sumter Fort Sumter is located in the middle of the harbor of Charleston, South Carolina. Its garrison had recently moved there to avoid incidents with local militias in the streets of the city. Lincoln told its commander, Major Robert Anderson, to hold on until fired upon. Confederate president Jefferson...
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The attack on Fort Sumter enormously invigorated the North to the defense of American nationalism.
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On April 15, 1861, Lincoln called on all the states to send forces to recapture the fort and other federal properties. The scale of the rebellion appeared to be small, so he called for only 75,000 volunteers for 90 days. In western Missouri, local secessionists seized Liberty Arsenal. On May 3, 1861, Lincoln called for...
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Maryland, Delaware, Missouri, and Kentucky were slave states that had divided loyalties to Northern and Southern businesses and family members. Some men enlisted in the Union Army and others in the Confederate Army. West Virginia separated from Virginia and was admitted to the Union on June 20, 1863.
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Maryland's territory surrounded the United States' capital of Washington, D.C., and could cut it off from the North. It had numerous anti-Lincoln officials who tolerated anti-army rioting in Baltimore and the burning of bridges, both aimed at hindering the passage of troops to the South. Maryland's legislature voted ov...
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In Missouri, an elected convention on secession voted decisively to remain within the Union. When pro-Confederate Governor Claiborne F. Jackson called out the state militia, it was attacked by federal forces under General Nathaniel Lyon, who chased the governor and the rest of the State Guard to the southwestern corner...
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Kentucky did not secede; for a time, it declared itself neutral. When Confederate forces entered the state in September 1861, neutrality ended and the state reaffirmed its Union status while maintaining slavery. During a brief invasion by Confederate forces in 1861, Confederate sympathizers organized a secession conven...
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After Virginia's secession, a Unionist government in Wheeling asked 48 counties to vote on an ordinance to create a new state on October 24, 1861. A voter turnout of 34 percent approved the statehood bill (96 percent approving). Twenty-four secessionist counties were included in the new state, and the ensuing guerrilla...
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A Unionist secession attempt occurred in East Tennessee, but was suppressed by the Confederacy, which arrested over 3,000 men suspected of being loyal to the Union. They were held without trial.
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General features of the war The Civil War was a contest marked by the ferocity and frequency of battle. Over four years, 237 named battles were fought, as were many more minor actions and skirmishes, which were often characterized by their bitter intensity and high casualties. In his book The American Civil War, John K...
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As the first seven states began organizing a Confederacy in Montgomery, the entire U.S. army numbered 16,000. However, Northern governors had begun to mobilize their militias. The Confederate Congress authorized the new nation up to 100,000 troops sent by governors as early as February. By May, Jefferson Davis was push...
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In the first year of the war, both sides had far more volunteers than they could effectively train and equip. After the initial enthusiasm faded, reliance on the cohort of young men who came of age every year and wanted to join was not enough. Both sides used a draft law—conscription—as a device to encourage or force v...
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When the Emancipation Proclamation went into effect in January 1863, ex-slaves were energetically recruited by the states and used to meet the state quotas. States and local communities offered higher and higher cash bonuses for white volunteers. Congress tightened the law in March 1863. Men selected in the draft could...
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In both the North and South, the draft laws were highly unpopular. In the North, some 120,000 men evaded conscription, many of them fleeing to Canada, and another 280,000 soldiers deserted during the war. At least 100,000 Southerners deserted, or about 10 percent; Southern desertion was high because, according to one h...
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From a tiny frontier force in 1860, the Union and Confederate armies had grown into the "largest and most efficient armies in the world" within a few years. Some European observers at the time dismissed them as amateur and unprofessional, but British historian John Keegan concluded that each outmatched the French, Prus...
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At the start of the Civil War, a system of paroles operated. Captives agreed not to fight until they were officially exchanged. Meanwhile, they were held in camps run by their army. They were paid, but they were not allowed to perform any military duties. The system of exchanges collapsed in 1863 when the Confederacy r...
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Historian Elizabeth D. Leonard writes that, according to various estimates, between five hundred and one thousand women enlisted as soldiers on both sides of the war, disguised as men. Women also served as spies, resistance activists, nurses, and hospital personnel. Women served on the Union hospital ship Red Rover and...
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Mary Edwards Walker, the only woman ever to receive the Medal of Honor, served in the Union Army and was given the medal for her efforts to treat the wounded during the war. Her name was deleted from the Army Medal of Honor Roll in 1917 (along with over 900 other, male MOH recipients); however, it was restored in 1977.
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Naval tactics The small U.S. Navy of 1861 was rapidly enlarged to 6,000 officers and 45,000 men in 1865, with 671 vessels, having a tonnage of 510,396. Its mission was to blockade Confederate ports, take control of the river system, defend against Confederate raiders on the high seas, and be ready for a possible war wi...
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American Civil War
Modern navy evolves The Civil War occurred during the early stages of the industrial revolution. Many naval innovations emerged during this time, most notably the advent of the ironclad warship. It began when the Confederacy, knowing they had to meet or match the Union's naval superiority, responded to the Union blocka...
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