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Economy of Alberta
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References
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References
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Economy of Alberta
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External links
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External links
CBC Digital Archives - Striking Oil in Alberta
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Economy of Alberta
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Table of Content
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Short description, Data, Current overview, Alberta's deficit, Alberta's credit rating, Alberta's real per capita GDP, Alberta's GDP compared to other provinces, Economic geography, Economic regions and cities, Calgary and Edmonton, Calgary-Edmonton Corridor, Calgary–Edmonton rivalry, Background, Employment, Extraction industries, Largest employers of Alberta, Sectors, Oil and gas extraction industries, Natural gas, Coal, Electricity, Mineral mining, Manufacturing, Biotechnology, Food processing, Transportation, Agriculture and forestry, Agriculture, Forestry, Services, Finance, Government, Technology, See also, References, External links
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Augustin-Louis Cauchy
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Short description
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Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real analysis), pioneered the field complex analysis, and the study of permutation groups in abstract algebra. Cauchy also contributed to a number of topics in mathematical physics, notably continuum mechanics.
A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated:
"More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)."
Cauchy was a prolific worker; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics.
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Augustin-Louis Cauchy
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Biography
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Biography
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Augustin-Louis Cauchy
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Youth and education
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Youth and education
Cauchy was the son of Louis François Cauchy (1760–1848) and Marie-Madeleine Desestre. Cauchy had two brothers: Alexandre Laurent Cauchy (1792–1857), who became a president of a division of the court of appeal in 1847 and a judge of the court of cassation in 1849, and Eugene François Cauchy (1802–1877), a publicist who also wrote several mathematical works. From his childhood he was good at math.
Cauchy married Aloise de Bure in 1818. She was a close relative of the publisher who published most of Cauchy's works. They had two daughters, Marie Françoise Alicia (1819) and Marie Mathilde (1823).
Cauchy's father was a highly ranked official in the Parisian police of the
Ancien Régime, but lost this position due to the French Revolution (14 July 1789), which broke out one month before Augustin-Louis was born. The Cauchy family survived the revolution and the following Reign of Terror during 1793–94 by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre in 1794, it was safe for the family to return to Paris. There, Louis-François Cauchy found a bureaucratic job in 1800, and quickly advanced his career. When Napoleon came to power in 1799, Louis-François Cauchy was further promoted, and became Secretary-General of the Senate, working directly under Laplace (who is now better known for his work on mathematical physics). The mathematician Lagrange was also a friend of the Cauchy family.
On Lagrange's advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, the best secondary school of Paris at that time, in the fall of 1802. Most of the curriculum consisted of classical languages; the ambitious Cauchy, being a brilliant student, won many prizes in Latin and the humanities. In spite of these successes, Cauchy chose an engineering career, and prepared himself for the entrance examination to the École Polytechnique.
In 1805, he placed second of 293 applicants on this exam and was admitted. One of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused Cauchy some problems in adapting. Nevertheless, he completed the course in 1807, at age 18, and went on to the École des Ponts et Chaussées (School for Bridges and Roads). He graduated in civil engineering, with the highest honors.
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Augustin-Louis Cauchy
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Engineering days
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Engineering days
After finishing school in 1810, Cauchy accepted a job as a junior engineer in Cherbourg, where Napoleon intended to build a naval base. Here Cauchy stayed for three years, and was assigned the Ourcq Canal project and the Saint-Cloud Bridge project, and worked at the Harbor of Cherbourg. Although he had an extremely busy managerial job, he still found time to prepare three mathematical manuscripts, which he submitted to the Première Classe (First Class) of the Institut de France. Cauchy's first two manuscripts (on polyhedra) were accepted; the third one (on directrices of conic sections) was rejected.
In September 1812, at 23 years old, Cauchy returned to Paris after becoming ill from overwork. Another reason for his return to the capital was that he was losing interest in his engineering job, being more and more attracted to the abstract beauty of mathematics; in Paris, he would have a much better chance to find a mathematics related position. When his health improved in 1813, Cauchy chose not to return to Cherbourg. Although he formally kept his engineering position, he was transferred from the payroll of the Ministry of the Marine to the Ministry of the Interior. The next three years Cauchy was mainly on unpaid sick leave; he spent his time fruitfully, working on mathematics (on the related topics of symmetric functions, the symmetric group and the theory of higher-order algebraic equations). He attempted admission to the First Class of the Institut de France but failed on three different occasions between 1813 and 1815. In 1815 Napoleon was defeated at Waterloo, and the newly installed king Louis XVIII took the restoration in hand. The Académie des Sciences was re-established in March 1816; Lazare Carnot and Gaspard Monge were removed from this academy for political reasons, and the king appointed Cauchy to take the place of one of them. The reaction of Cauchy's peers was harsh; they considered the acceptance of his membership in the academy an outrage, and Cauchy created many enemies in scientific circles.
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Augustin-Louis Cauchy
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Professor at École Polytechnique
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Professor at École Polytechnique
In November 1815, Louis Poinsot, who was an associate professor at the École Polytechnique, asked to be exempted from his teaching duties for health reasons. Cauchy was by then a rising mathematical star. One of his great successes at that time was the proof of Fermat's polygonal number theorem. He quit his engineering job, and received a one-year contract for teaching mathematics to second-year students of the École Polytechnique. In 1816, this Bonapartist, non-religious school was reorganized, and several liberal professors were fired; Cauchy was promoted to full professor.
When Cauchy was 28 years old, he was still living with his parents. His father found it time for his son to marry; he found him a suitable bride, Aloïse de Bure, five years his junior. The de Bure family were printers and booksellers, and published most of Cauchy's works. Aloïse and Augustin were married on April 4, 1818, with great Roman Catholic ceremony, in the Church of Saint-Sulpice. In 1819 the couple's first daughter, Marie Françoise Alicia, was born, and in 1823 the second and last daughter, Marie Mathilde.
The conservative political climate that lasted until 1830 suited Cauchy perfectly. In 1824 Louis XVIII died, and was succeeded by his even more conservative brother Charles X. During these years Cauchy was highly productive, and published one important mathematical treatise after another. He received cross-appointments at the Collège de France, and the .
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Augustin-Louis Cauchy
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In exile
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In exile
In July 1830, the July Revolution occurred in France. Charles X fled the country, and was succeeded by Louis-Philippe. Riots, in which uniformed students of the École Polytechnique took an active part, raged close to Cauchy's home in Paris.
These events marked a turning point in Cauchy's life, and a break in his mathematical productivity. Shaken by the fall of the government and moved by a deep hatred of the liberals who were taking power, Cauchy left France to go abroad, leaving his family behind. He spent a short time at Fribourg in Switzerland, where he had to decide whether he would swear a required oath of allegiance to the new regime. He refused to do this, and consequently lost all his positions in Paris, except his membership of the academy, for which an oath was not required. In 1831 Cauchy went to the Italian city of Turin, and after some time there, he accepted an offer from the King of Sardinia (who ruled Turin and the surrounding Piedmont region) for a chair of theoretical physics, which was created especially for him. He taught in Turin during 1832–1833. In 1831, he was elected a foreign member of the Royal Swedish Academy of Sciences, and the following year a Foreign Honorary Member of the American Academy of Arts and Sciences.
In August 1833 Cauchy left Turin for Prague to become the science tutor of the thirteen-year-old Duke of Bordeaux, Henri d'Artois (1820–1883), the exiled Crown Prince and grandson of Charles X. As a professor of the École Polytechnique, Cauchy had been a notoriously bad lecturer, assuming levels of understanding that only a few of his best students could reach, and cramming his allotted time with too much material. Henri d'Artois had neither taste nor talent for either mathematics or science. Although Cauchy took his mission very seriously, he did this with great clumsiness, and with surprising lack of authority over Henri d'Artois. During his civil engineering days, Cauchy once had been briefly in charge of repairing a few of the Parisian sewers, and he made the mistake of mentioning this to his pupil; with great malice, Henri d'Artois went about saying Cauchy started his career in the sewers of Paris. Cauchy's role as tutor lasted until Henri d'Artois became eighteen years old, in September 1838. Cauchy did hardly any research during those five years, while Henri d'Artois acquired a lifelong dislike of mathematics. Cauchy was named a baron, a title by which Cauchy set great store.
In 1834, his wife and two daughters moved to Prague, and Cauchy was reunited with his family after four years in exile.
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Augustin-Louis Cauchy
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Last years
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Last years
Cauchy returned to Paris and his position at the Academy of Sciences late in 1838. He could not regain his teaching positions, because he still refused to swear an oath of allegiance.
thumb|left|Cauchy in later life
In August 1839 a vacancy appeared in the Bureau des Longitudes. This Bureau bore some resemblance to the academy; for instance, it had the right to co-opt its members. Further, it was believed that members of the Bureau could "forget about" the oath of allegiance, although formally, unlike the Academicians, they were obliged to take it. The Bureau des Longitudes was an organization founded in 1795 to solve the problem of determining position at sea — mainly the longitudinal coordinate, since latitude is easily determined from the position of the sun. Since it was thought that position at sea was best determined by astronomical observations, the Bureau had developed into an organization resembling an academy of astronomical sciences.
In November 1839 Cauchy was elected to the Bureau, and discovered that the matter of the oath was not so easily dispensed with. Without his oath, the king refused to approve his election. For four years Cauchy was in the position of being elected but not approved; accordingly, he was not a formal member of the Bureau, did not receive payment, could not participate in meetings, and could not submit papers. Still Cauchy refused to take any oaths; however, he did feel loyal enough to direct his research to celestial mechanics. In 1840, he presented a dozen papers on this topic to the academy. He described and illustrated the signed-digit representation of numbers, an innovation presented in England in 1727 by John Colson. The confounded membership of the Bureau lasted until the end of 1843, when Cauchy was replaced by Poinsot.
Throughout the nineteenth century the French educational system struggled over the separation of church and state. After losing control of the public education system, the Catholic Church sought to establish its own branch of education and found in Cauchy a staunch and illustrious ally. He lent his prestige and knowledge to the École Normale Écclésiastique, a school in Paris run by Jesuits, for training teachers for their colleges. He took part in the founding of the Institut Catholique. The purpose of this institute was to counter the effects of the absence of Catholic university education in France. These activities did not make Cauchy popular with his colleagues, who, on the whole, supported the Enlightenment ideals of the French Revolution. When a chair of mathematics became vacant at the Collège de France in 1843, Cauchy applied for it, but received just three of 45 votes.
In 1848 King Louis-Philippe fled to England. The oath of allegiance was abolished, and the road to an academic appointment was clear for Cauchy. On March 1, 1849, he was reinstated at the Faculté de Sciences, as a professor of mathematical astronomy. After political turmoil all through 1848, France chose to become a Republic, under the Presidency of Napoleon III of France. Early 1852 the President made himself Emperor of France, and took the name Napoleon III.
The idea came up in bureaucratic circles that it would be useful to again require a loyalty oath from all state functionaries, including university professors. This time a cabinet minister was able to convince the Emperor to exempt Cauchy from the oath. In 1853, Cauchy was elected an International Member of the American Philosophical Society. Cauchy remained a professor at the university until his death at the age of 67. He received the Last Rites and died of a bronchial condition at 4 a.m. on 23 May 1857.
His name is one of the 72 names inscribed on the Eiffel Tower.
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Augustin-Louis Cauchy
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Work
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Work
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Augustin-Louis Cauchy
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Early work
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Early work
The genius of Cauchy was illustrated in his simple solution of the problem of Apollonius—describing a circle touching three given circles—which he discovered in 1805, his generalization of Euler's formula on polyhedra in 1811, and in several other elegant problems. More important is his memoir on wave propagation, which obtained the Grand Prix of the French Academy of Sciences in 1816. Cauchy's writings covered notable topics. In the theory of series he developed the notion of convergence and discovered many of the basic formulas for q-series. In the theory of numbers and complex quantities, he was the first to define complex numbers as pairs of real numbers. He also wrote on the theory of groups and substitutions, the theory of functions, differential equations and determinants.
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Augustin-Louis Cauchy
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Wave theory, mechanics, elasticity
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Wave theory, mechanics, elasticity
In the theory of light he worked on Fresnel's wave theory and on the dispersion and polarization of light. He also contributed research in mechanics, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. He wrote on the equilibrium of rods and elastic membranes and on waves in elastic media. He introduced a 3 × 3 symmetric matrix of numbers that is now known as the Cauchy stress tensor. In elasticity, he originated the theory of stress, and his results are nearly as valuable as those of Siméon Poisson.
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Augustin-Louis Cauchy
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Number theory
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Number theory
Other significant contributions include being the first to prove the Fermat polygonal number theorem.
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Augustin-Louis Cauchy
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Complex functions
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Complex functions
Cauchy is most famous for his single-handed development of complex function theory. The first pivotal theorem proved by Cauchy, now known as Cauchy's integral theorem, was the following:
where f(z) is a complex-valued function holomorphic on and within the non-self-intersecting closed curve C (contour) lying in the complex plane. The contour integral is taken along the contour C. The rudiments of this theorem can already be found in a paper that the 24-year-old Cauchy presented to the Académie des Sciences (then still called "First Class of the Institute") on August 11, 1814. In full form the theorem was given in 1825.
In 1826 Cauchy gave a formal definition of a residue of a function. This concept concerns functions that have poles—isolated singularities, i.e., points where a function goes to positive or negative infinity. If the complex-valued function f(z) can be expanded in the neighborhood of a singularity a as
where φ(z) is analytic (i.e., well-behaved without singularities), then f is said to have a pole of order n in the point a. If n = 1, the pole is called simple.
The coefficient B1 is called by Cauchy the residue of function f at a. If f is non-singular at a then the residue of f is zero at a. Clearly, the residue is in the case of a simple pole equal to
where we replaced B1 by the modern notation of the residue.
In 1831, while in Turin, Cauchy submitted two papers to the Academy of Sciences of Turin. In the first he proposed the formula now known as Cauchy's integral formula,
where f(z) is analytic on C and within the region bounded by the contour C and the complex number a is somewhere in this region. The contour integral is taken counter-clockwise. Clearly, the integrand has a simple pole at z = a. In the second paperCauchy, Mémoire sur les rapports qui existent entre le calcul des Résidus et le calcul des Limites, et sur les avantages qu'offrent ces deux calculs dans la résolution des équations algébriques ou transcendantes (Memorandum on the connections that exist between the residue calculus and the limit calculus, and on the advantages that these two calculi offer in solving algebraic and transcendental equations], presented to the Academy of Sciences of Turin, November 27, 1831. he presented the residue theorem,
where the sum is over all the n poles of f(z) on and within the contour C. These results of Cauchy's still form the core of complex function theory as it is taught today to physicists and electrical engineers. For quite some time, contemporaries of Cauchy ignored his theory, believing it to be too complicated. Only in the 1840s the theory started to get response, with Pierre Alphonse Laurent being the first mathematician besides Cauchy to make a substantial contribution (his work on what are now known as Laurent series, published in 1843).
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Augustin-Louis Cauchy
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''Cours d'analyse''
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Cours d'analyse
left|thumb|The title page of a textbook by Cauchy. In his book Cours d'analyse Cauchy stressed the importance of rigor in analysis. Rigor in this case meant the rejection of the principle of Generality of algebra (of earlier authors such as Euler and Lagrange) and its replacement by geometry and infinitesimals. Judith Grabiner wrote Cauchy was "the man who taught rigorous analysis to all of Europe". The book is frequently noted as being the first place that inequalities, and arguments were introduced into calculus. Here Cauchy defined continuity as follows: The function f(x) is continuous with respect to x between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself.
M. Barany claims that the École mandated the inclusion of infinitesimal methods against Cauchy's better judgement. Gilain notes that when the portion of the curriculum devoted to Analyse Algébrique was reduced in 1825, Cauchy insisted on placing the topic of continuous functions (and therefore also infinitesimals) at the beginning of the Differential Calculus. Laugwitz (1989) and Benis-Sinaceur (1973) point out that Cauchy continued to use infinitesimals in his own research as late as 1853.
Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero. There has been a vast body of literature written about Cauchy's notion of "infinitesimally small quantities", arguing that they lead from everything from the usual "epsilontic" definitions or to the notions of non-standard analysis. The consensus is that Cauchy omitted or left implicit the important ideas to make clear the precise meaning of the infinitely small quantities he used.
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Augustin-Louis Cauchy
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Taylor's theorem
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Taylor's theorem
He was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder. He wrote a textbook (see the illustration) for his students at the École Polytechnique in which he developed the basic theorems of mathematical analysis as rigorously as possible. In this book he gave the necessary and sufficient condition for the existence of a limit in the form that is still taught. Also Cauchy's well-known test for absolute convergence stems from this book: Cauchy condensation test. In 1829 he defined for the first time a complex function of a complex variable in another textbook. In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods; thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the introduction of the notion of uniform continuity.
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Augustin-Louis Cauchy
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Argument principle, stability
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Argument principle, stability
In a paper published in 1855, two years before Cauchy's death, he discussed some theorems, one of which is similar to the "Principle of the argument" in many modern textbooks on complex analysis. In modern control theory textbooks, the Cauchy argument principle is quite frequently used to derive the Nyquist stability criterion, which can be used to predict the stability of negative feedback amplifier and negative feedback control systems. Thus Cauchy's work has a strong impact on both pure mathematics and practical engineering.
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Augustin-Louis Cauchy
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Published works
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Published works
thumb|Leçons sur le calcul différentiel, 1829
Cauchy was very productive, in number of papers second only to Leonhard Euler. It took almost a century to collect all his writings into 27 large volumes:
(Paris : Gauthier-Villars et fils, 1882–1974)
His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced; these are mainly embodied in his three great treatises:
Le Calcul infinitésimal (1823)
Leçons sur les applications de calcul infinitésimal; La géométrie (1826–1828)
His other works include:
Exercices d'analyse et de physique mathematique (Volume 1)
Exercices d'analyse et de physique mathematique (Volume 2)
Exercices d'analyse et de physique mathematique (Volume 3)
Exercices d'analyse et de physique mathematique (Volume 4) (Paris: Bachelier, 1840–1847)
Analyse algèbrique (Imprimerie Royale, 1821)
Nouveaux exercices de mathématiques (Paris : Gauthier-Villars, 1895)
Courses of mechanics (for the École Polytechnique)
Higher algebra (for the )
Mathematical physics (for the Collège de France).
Mémoire sur l'emploi des equations symboliques dans le calcul infinitésimal et dans le calcul aux différences finis CR Ac ad. Sci. Paris, t. XVII, 449–458 (1843) credited as originating the operational calculus.
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Augustin-Louis Cauchy
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Politics and religious beliefs
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Politics and religious beliefs
Augustin-Louis Cauchy grew up in the house of a staunch royalist. This made his father flee with the family to Arcueil during the French Revolution. Their life there during that time was apparently hard; Augustin-Louis's father, Louis François, spoke of living on rice, bread, and crackers during the period. A paragraph from an undated letter from Louis François to his mother in Rouen says:
In any event, he inherited his father's staunch royalism and hence refused to take oaths to any government after the overthrow of Charles X.
He was an equally staunch Catholic and a member of the Society of Saint Vincent de Paul. He also had links to the Society of Jesus and defended them at the academy when it was politically unwise to do so. His zeal for his faith may have led to his caring for Charles Hermite during his illness and leading Hermite to become a faithful Catholic. It also inspired Cauchy to plead on behalf of the Irish during the Great Famine of Ireland.
His royalism and religious zeal made him contentious, which caused difficulties with his colleagues. He felt that he was mistreated for his beliefs, but his opponents felt he intentionally provoked people by berating them over religious matters or by defending the Jesuits after they had been suppressed. Niels Henrik Abel called him a "bigoted Catholic" and added he was "mad and there is nothing that can be done about him", but at the same time praised him as a mathematician. Cauchy's views were widely unpopular among mathematicians and when Guglielmo Libri Carucci dalla Sommaja was made chair in mathematics before him he, and many others, felt his views were the cause. When Libri was accused of stealing books he was replaced by Joseph Liouville rather than Cauchy, which caused a rift between Liouville and Cauchy. Another dispute with political overtones concerned Jean-Marie Constant Duhamel and a claim on inelastic shocks. Cauchy was later shown, by Jean-Victor Poncelet, to be wrong.
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Augustin-Louis Cauchy
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See also
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See also
List of topics named after Augustin-Louis Cauchy
Cauchy–Binet formula
Cauchy boundary condition
Cauchy's convergence test
Cauchy (crater)
Cauchy determinant
Cauchy distribution
Cauchy's equation
Cauchy–Euler equation
Cauchy's functional equation
Cauchy horizon
Cauchy formula for repeated integration
Cauchy–Frobenius lemma
Cauchy–Hadamard theorem
Cauchy–Kovalevskaya theorem
Cauchy momentum equation
Cauchy–Peano theorem
Cauchy principal value
Cauchy problem
Cauchy product
Cauchy's radical test
Cauchy–Rassias stability
Cauchy–Riemann equations
Cauchy–Schwarz inequality
Cauchy sequence
Cauchy surface
Cauchy's theorem (geometry)
Cauchy's theorem (group theory)
Maclaurin–Cauchy test
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Augustin-Louis Cauchy
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References
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References
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Augustin-Louis Cauchy
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Notes
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Notes
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Augustin-Louis Cauchy
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Citations
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Citations
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Augustin-Louis Cauchy
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Sources
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Sources
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Augustin-Louis Cauchy
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Further reading
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Further reading
Boyer, C.: The concepts of the calculus. Hafner Publishing Company, 1949.
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Augustin-Louis Cauchy
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External links
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External links
Augustin-Louis Cauchy – Œuvres complètes (in 2 series) Gallica-Math
Augustin-Louis Cauchy – Cauchy's Life by Robin Hartshorne
Category:1789 births
Category:1857 deaths
Category:19th-century French mathematicians
Category:Corps des ponts
Category:École des Ponts ParisTech alumni
Category:École Polytechnique alumni
Category:Fellows of the American Academy of Arts and Sciences
Category:Foreign members of the Royal Society
Category:French Roman Catholics
Category:French geometers
Category:History of calculus
Category:French mathematical analysts
Category:Linear algebraists
Category:Members of the French Academy of Sciences
Category:Members of the Royal Swedish Academy of Sciences
Category:Recipients of the Pour le Mérite (civil class)
Category:French textbook writers
Category:Academic staff of the University of Turin
Category:International members of the American Philosophical Society
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Augustin-Louis Cauchy
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Table of Content
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Short description, Biography, Youth and education, Engineering days, Professor at École Polytechnique, In exile, Last years, Work, Early work, Wave theory, mechanics, elasticity, Number theory, Complex functions, ''Cours d'analyse'', Taylor's theorem, Argument principle, stability, Published works, Politics and religious beliefs, See also, References, Notes, Citations, Sources, Further reading, External links
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Archimedes
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Short description
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Archimedes of Syracuse ( ; ) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in classical antiquity, and one of the greatest mathematicians of all time. Archimedes anticipated modern calculus and analysis by applying the concept of the infinitesimals and the method of exhaustion to derive and rigorously prove many geometrical theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.
Archimedes' other mathematical achievements include deriving an approximation of pi (), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever, the widespread use of the concept of center of gravity, and the enunciation of the law of buoyancy known as Archimedes' principle. In astronomy, he made measurements of the apparent diameter of the Sun and the size of the universe. He is also said to have built a planetarium device that demonstrated the movements of the known celestial bodies, and may have been a precursor to the Antikythera mechanism. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.
Archimedes died during the siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting Archimedes' tomb, which was surmounted by a sphere and a cylinder that Archimedes requested be placed there to represent his most valued mathematical discovery.
Unlike his inventions, Archimedes' mathematical writings were little known in antiquity. Alexandrian mathematicians read and quoted him, but the first comprehensive compilation was not made until by Isidore of Miletus in Byzantine Constantinople, while Eutocius' commentaries on Archimedes' works in the same century opened them to wider readership for the first time. In the Middle ages, Archimedes' work was translated into Arabic in the 9th century and then into Latin in the 12th century, and were an influential source of ideas for scientists during the Renaissance and in the Scientific Revolution. The recent discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest has also provided new insights into how he obtained mathematical results.
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Archimedes
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Biography
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Biography
thumb|right|Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West
The details of Archimedes life are obscure; a biography of Archimedes mentioned by Eutocius was allegedly written by his friend Heraclides Lembus, but this work has been lost, and modern scholarship is doubtful that it was written by Heraclides to begin with.Commentarius in dimensionem circuli (Archimedis opera omnia ed. Heiberg-Stamatis (1915), vol. 3, p. 228); Commentaria in conica (Apollonii Pergaei quae Graece exstant, ed. Heiberg (1893) vol. 2, p. 168: "Hērakleios"
Based on a statement by the Byzantine Greek scholar John Tzetzes that Archimedes lived for 75 years before his death in 212 BC, Archimedes is estimated to have been born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. In the Sand-Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known; Plutarch wrote in his Parallel LivesPlutarch, Life of Marcellus that Archimedes was related to King Hiero II, the ruler of Syracuse, although Cicero and Silius Italicus suggest he was of humble origin. It is also unknown whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth; though his surviving written works, addressed to Dositheus of Pelusium, a student of the Alexandrian astronomer Conon of Samos, and to the head librarian Eratosthenes of Cyrene, suggested that he maintained collegial relations with scholars based there. In the preface to On Spirals addressed to Dositheus, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works.
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Archimedes
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Golden wreath
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Golden wreath
thumb|Measurement of volume (a) before and (b) after an object has been submerged, with (∆V) indicating the rising amount of liquid is equal to the volume of the object
Another story of a problem that Archimedes is credited solving with in service of Hiero II is the "wreath problem." According to Vitruvius, writing about two centuries after Archimedes' death, King Hiero II of Syracuse had commissioned a golden wreath for a temple to the immortal gods, and had supplied pure gold to be used by the goldsmith.Vitruvius, De Architectura, Book IX, 3 However, the king had begun to suspect that the goldsmith had substituted some cheaper silver and kept some of the pure gold for himself, and, unable to make the smith confess, asked Archimedes to investigate. Later, while stepping into a bath, Archimedes allegedly noticed that the level of the water in the tub rose more the lower he sank in the tub and, realizing that this effect could be used to determine the golden crown's volume, was so excited that he took to the streets naked, having forgotten to dress, crying "Eureka!, meaning "I have found [it]!" According to Vitruvius, Archimedes then took a lump of gold and a lump of silver that were each equal in weight to the wreath, and, placing each in the bathtub, showed that the wreath displaced more water than the gold and less than the silver, demonstrating that the wreath was gold mixed with silver
A different account is given in the Carmen de Ponderibus,Metrologicorum Scriptorum reliquiae, ed. F. Hultsch (Leipzig 1864), II, 88 an anonymous 5th century Latin didactic poem on weights and measures once attributed to the grammarian Priscian. In this poem, the lumps of gold and silver were placed on the scales of a balance, and then the entire apparatus was immersed in water; the difference in density between the gold and the silver, or between the gold and the crown, causes the scale to tip accordingly.Carmen de Ponderibus, lines 124-162 Unlike the more famous bathtub account given by Vitruvius, this poetic account uses the hydrostatics principle now known as Archimedes' principle that is found in his treatise On Floating Bodies, where a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Galileo Galilei, who invented a hydrostatic balance in 1586 inspired by Archimedes' work, considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."
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Archimedes
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Launching the ''Syracusia''
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Launching the Syracusia
A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of Syracuse. Athenaeus of Naucratis in his Deipnosophistae quotes a certain Moschion for a description on how King Hiero II commissioned the design of a huge ship, the Syracusia, which is said to have been the largest ship built in classical antiquity and, according to Moschion's account, it was launched by Archimedes.Athenaeus, Deipnosophistae, V.40-45 Plutarch tells a slightly different account,Plutarch, Life of Marcellus 7-8 relating that Archimedes boasted to Hiero that he was able to move any large weight, at which point Hiero challenged him to move a ship. These accounts contain many fantastic details that are historically implausible, and the authors of these stories provide conflicting about how this task was accomplished: Plutarch states that Archimedes constructed a block-and-tackle pulley system, while Hero of Alexandria attributed the same boast to Archimedes' invention of the baroulkos, a kind of windlass. Heronis Opera Vol II, 1, 256, III 306 Pappus of Alexandria attributed this feat, instead, to Archimedes' use of mechanical advantage, the principle of leverage to lift objects that would otherwise have been too heavy to move, attributing to him the oft-quoted remark: "Give me a place to stand on, and I will move the Earth."Pappus of Alexandria, Synagoge Book VIII
Athenaeus, likely garbling the details of Hero's account of the baroulkos, also mentions that Archimedes used a "screw" in order to remove any potential water leaking through the hull of the Syracusia. Although this device is sometimes referred to as Archimedes' screw, it likely predates him by a significant amount, and none of his closest contemporaries who describe its use (Philo of Byzantium, Strabo, and Vitruvius) credit him with its use.
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Archimedes
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War machines
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War machines
thumb|Mirrors placed as a parabolic reflector to attack upcoming ships
The greatest reputation Archimedes earned during antiquity was for the defense of his city from the Romans during the Siege of Syracuse. According to Plutarch,Life of Marcellus, 25-27 Archimedes had constructed war machines for Hiero II, but had never been given an opportunity to use them during Hiero's lifetime. In 214 BC, however, during the Second Punic War, when Syracuse switched allegiances from Rome to Carthage, the Roman army under Marcus Claudius Marcellus attempted to take the city, Archimedes allegedly personally oversaw the use of these war machines in the defense of the city, greatly delaying the Romans, who were only able to capture the city after a long siege. Three different historians, Plutarch, Livy, and Polybius provide testimony about these war machines, describing improved catapults, cranes that dropped heavy pieces of lead on the Roman ships or which used an iron claw to lift them out of the water, dropping the back in so that they sank.
A much more improbable account, not found in any of the three earliest accounts (Plutarch, Polybius, or Livy) describes how Archimedes used "burning mirrors" to focus the sun's rays onto the attacking Roman ships, setting them on fire. The earliest account to mention ships being set on fire, by the 2nd century CE satirist Lucian of Samosata,Lucian, Hippias, ¶ 2, in Lucian, vol. 1, ed. A. M. Harmon, Harvard, 1913, does not mention mirrors, and only says the ships were set on fire by artificial means, which may imply that burning projectiles were used. The first author to mention mirrors is Galen, writing later in the same century.Galen, On temperaments 3.2 Nearly four hundred years after Lucian and Galen, Anthemius, despite skepticism, tried to reconstruct Archimedes' hypothetical reflector geometry.Anthemius of Tralles, On miraculous engines 153. The purported device, sometimes called "Archimedes' heat ray", has been the subject of an ongoing debate about its credibility since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes, with mixed results. See p. 144.
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Archimedes
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Death
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Death
thumb|The Death of Archimedes (1815) by Thomas Degeorge
There are several divergent accounts of Archimedes' death during the sack of Syracuse after it fell to the Romans: The oldest account, from Livy,Livy, Ab Urbe Condita Book XXV, 31 says that, while drawing figures in the dust, Archimedes was killed by a Roman soldier who did not know he was Archimedes. According to Plutarch,Life of Marcellus, XIX, 1 the soldier demanded that Archimedes come with him, but Archimedes declined, saying that he had to finish working on the problem, and the soldier killed Archimedes with his sword. Another story from Plutarch has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items. Another Roman writer, Valerius Maximus (fl. 30 AD), wrote in Memorable Doings and Sayings that Archimedes' last words as the soldier killed him were "... but protecting the dust with his hands, said 'I beg of you, do not disturb this." which is similar to the last words now commonly attributed to him, "Do not disturb my circles," which otherwise do not appear in any ancient sources.
Marcellus was reportedly angered by Archimedes' death, as he considered him a valuable scientific asset (he called Archimedes "a geometrical Briareus") and had ordered that he should not be harmed.Plutarch, Parallel LivesJaeger, Mary. Archimedes and the Roman Imagination. p. 113. Cicero (106–43 BC) mentions that Marcellus brought to Rome two planetariums Archimedes built, which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets, one of which he donated to the Temple of Virtue in Rome, and the other he allegedly kept as his only personal loot from Syracuse."Cicero, De republica Pappus of Alexandria reports on a now lost treatise by Archimedes On Sphere-Making, which may have dealt with the construction of these mechanisms. Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing, which was once thought to have been beyond the range of the technology available in ancient times, but the discovery in 1902 of the Antikythera mechanism, another device built BC designed with a similar purpose, has confirmed that devices of this kind were known to the ancient Greeks, with some scholars regarding Archimedes' device as a precursor.
While serving as a quaestor in Sicily, Cicero himself found what was presumed to be Archimedes' tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up and was able to see the carving and read some of the verses that had been added as an inscription. The tomb carried a sculpture illustrating Archimedes' favorite mathematical proof, that the volume and surface area of the sphere are two-thirds that of an enclosing cylinder including its bases.
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Archimedes
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Mathematics
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Mathematics
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics, both in applying the techniques of his predecessors to obtain new results, and developing new methods of his own.
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Archimedes
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Method of exhaustion
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Method of exhaustion
thumb|Archimedes calculates the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon
In Quadrature of the Parabola, Archimedes states that a certain proposition in Euclid's Elements demonstrating that the area of a circle is proportional to its diameter was proven using a lemma now known as the Archimedean property, that “the excess by which the greater of two unequal regions exceed the lesser, if added to itself, can exceed any given bounded region.” Prior to Archimedes, Eudoxus of Cnidus and other earlier mathematicians applied this lemma, a technique now referred to as the "method of exhaustion," to find the volume of a tetrahedron, cylinder, cone, and sphere, for which proofs are given in book XII of Euclid's Elements.
In Measurement of a Circle, Archimedes employed this method to show that the area of a circle is the same as a right triangle whose base and height are equal to its radius and circumference. He then approximated the ratio between the radius and the circumference, the value of , by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of lay between 3 (approx. 3.1429) and 3 (approx. 3.1408), consistent with its actual value of approximately 3.1416. In the same treatise, he also asserts that the value of the square root of 3 as lying between (approximately 1.7320261) and (approximately 1.7320512), which he may have derived from a similar method.
thumb|upright=.8|A proof that the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola
In Quadrature of the Parabola, Archimedes used this technique to prove that the area enclosed by a parabola and a straight line is times the area of a corresponding inscribed triangle as shown in the figure at right, expressing the solution to the problem as an infinite geometric series with the common ratio :
If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and whose third vertex is where the line that is parallel to the parabola's axis and that passes through the midpoint of the base intersects the parabola, and so on. This proof uses a variation of the series which sums to .
He also used this technique in order to measure the surface areas of a sphere and cone,On the Sphere and Cylinder 13-14, 33-34, 42, 44 to calculate the area of an ellipse,On Conoids and Spheroids 4 and to find the area contained within an Archimedean spiral.On Spirals, 24-25
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Archimedes
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Mechanical method
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Mechanical method
In addition to developing on the works of earlier mathematicians with the method of exhaustion, Archimedes also pioneered a novel technique using the law of the lever in order to measure the area and volume of shapes using physical means. He first gives an outline of this proof in Quadrature of the Parabola alongside the geometric proof, but he gives a fuller explanation in The Method of Mechanical Theorems. According to Archimedes, he proved the results in his mathematical treatises first using this method, and then worked backwards, applying the method of exhaustion only after he had already calculated an approximate value for the answer.
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Archimedes
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Large numbers
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Large numbers
Archimedes also developed methods for representing large numbers.
In The Sand Reckoner, Archimedes devised a system of counting based on the myriad, the Greek term for the number 10,000, in order to calculate a number that was greater than the grains of sand needed to fill the universe. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8. In doing so, he demonstrated that mathematics could represent arbitrarily large numbers.
In the Cattle Problem, Archimedes challenges the mathematicians at the Library of Alexandria to count the numbers of cattle in the Herd of the Sun, which involves solving a number of simultaneous Diophantine equations. A more difficult version of the problem in which some of the answers are required to be square numbers, and the answer is a very large number, approximately 7.760271.
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Archimedes
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Archimedean solids
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Archimedean solids
In a lost work described by Pappus of Alexandria, Archimedes also proved that there are exactly thirteen semiregular polyhedra.
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Archimedes
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Writings
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Writings
thumb|upright=.8|Front page of Archimedes' Opera, in Greek and Latin, edited by David Rivault (1615)
Archimedes made his work known through correspondence with mathematicians in Alexandria, which were originally written in Doric Greek, the dialect of ancient Syracuse.
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Archimedes
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Surviving works
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Surviving works
The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).
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Archimedes
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''Measurement of a Circle''
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Measurement of a Circle
This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos. In Proposition II, Archimedes gives an approximation of the value of pi (), showing that it is greater than (3.1408...) and less than (3.1428...).
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Archimedes
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''The Sand Reckoner''
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The Sand Reckoner
In this treatise, also known as Psammites, Archimedes finds a number that is greater than the grains of sand needed to fill the universe. This book mentions the heliocentric theory of the Solar System proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies, and attempts to measure the apparent diameter of the Sun. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner is the only surviving work in which Archimedes discusses his views on astronomy. Adapted from
Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus' heliocentric model of the universe, in the Sand-Reckoner. Without the use of either trigonometry or a table of chords, Archimedes determines the Sun's apparent diameter by first describing the procedure and instrument used to make observations (a straight rod with pegs or grooves), applying correction factors to these measurements, and finally giving the result in the form of upper and lower bounds to account for observational error.
Ptolemy, quoting Hipparchus, also references Archimedes' solstice observations in the Almagest. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years.
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Archimedes
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''On the Equilibrium of Planes''
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On the Equilibrium of Planes
There are two books to On the Equilibrium of Planes: the first contains seven postulates and fifteen propositions, while the second book contains ten propositions. In the first book, Archimedes proves the law of the lever, which states that:
Earlier descriptions of the principle of the lever are found in a work by Euclid and in the Mechanical Problems, belonging to the Peripatetic school of the followers of Aristotle, the authorship of which has been attributed by some to Archytas.
Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas.
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Archimedes
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''Quadrature of the Parabola''
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Quadrature of the Parabola
In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height. He achieves this by two different methods: first by applying the law of the lever, and by calculating the value of a geometric series that sums to infinity with the ratio 1/4.
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Archimedes
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''On the Sphere and Cylinder''
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On the Sphere and Cylinder
thumb|A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases|209x209px
In this two-volume treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. The volume is 3 for the sphere, and 23 for the cylinder. The surface area is 42 for the sphere, and 62 for the cylinder (including its two bases), where is the radius of the sphere and cylinder.
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Archimedes
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''On Spirals''
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On Spirals
This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in modern polar coordinates (, ), it can be described by the equation with real numbers and .
This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.
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Archimedes
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''On Conoids and Spheroids''
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On Conoids and Spheroids
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.
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Archimedes
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''On Floating Bodies''
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On Floating Bodies
There are two books of On Floating Bodies. In the first book, 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 since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.
Archimedes' principle of buoyancy is given in this work, stated as follows:
Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in direction to, the weight of the fluid displaced.
In the second part, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, similar to the way that icebergs float.
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Archimedes
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''Ostomachion''
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Ostomachion
thumb|Ostomachion is a dissection puzzle found in the Archimedes Palimpsest|200x200px
Also known as Loculus of Archimedes or Archimedes' Box, this is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Reviel Netz of Stanford University argued in 2003 that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Netz calculates that the pieces can be made into a square 17,152 ways. The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded. The puzzle represents an example of an early problem in combinatorics.
The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the Ancient Greek word for "throat" or "gullet", stomachos (). Ausonius calls the puzzle , a Greek compound word formed from the roots of () and ().
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Archimedes
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The cattle problem
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The cattle problem
In this work, addressed to Eratosthenes and the mathematicians in Alexandria, Archimedes challenges them to count the numbers of cattle in the Herd of the Sun, which involves solving a number of simultaneous Diophantine equations. Gotthold Ephraim Lessing discovered this work in a Greek manuscript consisting of a 44-line poem in the Herzog August Library in Wolfenbüttel, Germany in 1773. There is a more difficult version of the problem in which some of the answers are required to be square numbers. A. Amthor first solved this version of the problemKrumbiegel, B. and Amthor, A. Das Problema Bovinum des Archimedes, Historisch-literarische Abteilung der Zeitschrift für Mathematik und Physik 25 (1880) pp. 121–136, 153–171. in 1880, and the answer is a very large number, approximately 7.760271.
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Archimedes
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''The Method of Mechanical Theorems''
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The Method of Mechanical Theorems
As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.
In this work Archimedes uses a novel method, an early form of Cavalieri's principle,; ; ; to rederive the results from the treatises sent to Dositheus (Quadrature of the Parabola, On the Sphere and Cylinder, On Spirals, On Conoids and Spheroids) that he had previously used the method of exhaustion to prove, using the law of the lever he applied in On the Equilbrium of Planes in order to find the center of gravity of an object first, and reasoning geometrically from there in order to more easily derive the volume of an object. Archimedes states that he used this method to derive the results in the treatises sent to Dositheus before he proved them more rigorously with the method of exhaustion, stating that it is useful to know that a result is true before proving it rigorously, much as Eudoxus of Cnidus was aided in proving that the volume of a cone is one-third the volume of cylinder by knowing that Democritus had already asserted it to be true on the argument that this is true by the fact that the pyramid has one-third the rectangular prism of the same base.
This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906.
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Archimedes
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Apocryphal works
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Apocryphal works
Archimedes' Book of Lemmas or Liber Assumptorum is a treatise with 15 propositions on the nature of circles. The earliest known copy of the text is in Arabic. T. L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.
Other questionable attributions to Archimedes' work include the Latin poem Carmen de ponderibus et mensuris (4th or 5th century), which describes the use of a hydrostatic balance, to solve the problem of the crown, and the 12th-century text Mappae clavicula, which contains instructions on how to perform assaying of metals by calculating their specific gravities.Dilke, Oswald A. W. 1990. [Untitled]. Gnomon 62(8):697–99. .Berthelot, Marcel. 1891. "Sur l histoire de la balance hydrostatique et de quelques autres appareils et procédés scientifiques." Annales de Chimie et de Physique 6(23):475–85.
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Archimedes
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Lost works
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Lost works
Many written works by Archimedes have not survived or are only extant in heavily edited fragments: Pappus of Alexandria mentions On Sphere-Making, as well as a work on semiregular polyhedra, and another work on spirals, while Theon of Alexandria quotes a remark about refraction from the Catoptrica. Principles, addressed to Zeuxippus, explained the number system used in The Sand Reckoner; there are also On Balances; On Centers of Gravity.
Scholars in the medieval Islamic world also attribute to Archimedes a formula for calculating the area of a triangle from the length of its sides, which today is known as Heron's formula due to its first known appearance in the work of Heron of Alexandria in the 1st century AD, and may have been proven in a lost work of Archimedes that is no longer extant.
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Archimedes
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Archimedes Palimpsest
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Archimedes Palimpsest
thumb|In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost
In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople to examine a 174-page goatskin parchment of prayers, written in the 13th century, after reading a short transcription published seven years earlier by Papadopoulos-Kerameus. He confirmed that it was indeed a palimpsest, a document with text that had been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, a common practice in the Middle Ages, as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th-century copies of previously lost treatises by Archimedes. The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. It is the only known source of The Method of Mechanical Theorems, referred to by Suidas and thought to have been lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts.
The treatises in the Archimedes Palimpsest include:
On the Equilibrium of Planes
On Spirals
Measurement of a Circle
On the Sphere and Cylinder
On Floating Bodies
The Method of Mechanical Theorems
Stomachion
Speeches by the 4th century BC politician Hypereides
A commentary on Aristotle's Categories
Other works
The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On 29 October 1998, it was sold at auction to an anonymous buyer for a total of $2.2 million.Christie's (n.d). Auction results The palimpsest was stored at the Walters Art Museum in Baltimore, Maryland, where it was subjected to a range of modern tests including the use of ultraviolet and light to read the overwritten text. It has since returned to its anonymous owner.
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Archimedes
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Legacy
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Legacy
thumb| Bronze statue of Archimedes in Berlin
Sometimes called the father of mathematicsFather of mathematics: Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, p 19. and mathematical physics,James H. Williams Jr., Fundamentals of Applied Dynamics, p 30., Carl B. Boyer, Uta C. Merzbach, A History of Mathematics, p 111., Stuart Hollingdale, Makers of Mathematics, p 67., Igor Ushakov, In the Beginning, Was the Number (2), p 114. historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity.;
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Archimedes
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Classical antiquity
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Classical antiquity
The reputation that Archimedes had for mechanical inventions in classical antiquity is well-documented; AthenaeusDeipnosophistae, V, 206d) recounts in his Deipnosophistae how Archimedes supervised the construction of the largest known ship in antiquity, the Syracusia, while ApuleiusApologia, 16 talks about his work in catoptrics. PlutarchPlutarch, Parallel lives had claimed that Archimedes disdained mechanics and focused primarily on pure geometry, but this is generally considered to be a mischaracterization by modern scholarship, fabricated to bolster Plutarch's own Platonist values rather than to an accurate presentation of Archimedes, and, unlike his inventions, Archimedes' mathematical writings were little known in antiquity outside of the works of Alexandrian mathematicians. The first comprehensive compilation was not made until by Isidore of Miletus in Byzantine Constantinople, while Eutocius' commentaries on Archimedes' works earlier in the same century opened them to wider readership for the first time.
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Archimedes
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Middle ages
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Middle ages
Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD), and into Latin via Arabic by Gerard of Cremona (c. 1114–1187). Direct Greek to Latin translations were later done by William of Moerbeke (c. 1215–1286) and Iacobus Cremonensis (c. 1400–1453).
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Archimedes
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Renaissance and early modern Europe
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Renaissance and early modern Europe
thumb|1612 drawing of a now-lost bronze coin depicting Archimedes
During the Renaissance, the Editio princeps (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin, which were an influential source of ideas for scientists during the Renaissance and again in the 17th century. Reprinted in
Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes. Galileo Galilei called him "superhuman" and "my master",Matthews, Michael. Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion Can Contribute to Science Literacy. p. 96. while Christiaan Huygens said, "I think Archimedes is comparable to no one", consciously emulating him in his early work. Gottfried Wilhelm Leibniz said, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times".Boyer, Carl B., and Uta C. Merzbach. 1968. A History of Mathematics. ch. 7.
Italian numismatist and archaeologist Filippo Paruta (1552–1629) and Leonardo Agostini (1593–1676) reported on a bronze coin in Sicily with the portrait of Archimedes on the obverse and a cylinder and sphere with the monogram ARMD in Latin on the reverse. Although the coin is now lost and its date is not precisely known, Ivo Schneider described the reverse as "a sphere resting on a base – probably a rough image of one of the planetaria created by Archimedes," and suggested it might have been minted in Rome for Marcellus who "according to ancient reports, brought two spheres of Archimedes with him to Rome".
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Archimedes
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In modern mathematics
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In modern mathematics
thumb|The Fields Medal carries a portrait of Archimedes|190x190px
Gauss's heroes were Archimedes and Newton,Jay Goldman, The Queen of Mathematics: A Historically Motivated Guide to Number Theory, p 88. and Moritz Cantor, who studied under Gauss in the University of Göttingen, reported that he once remarked in conversation that "there had been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein".E.T. Bell, Men of Mathematics, p 237 Likewise, Alfred North Whitehead said that "in the year 1500 Europe knew less than Archimedes who died in the year 212 BC." The historian of mathematics Reviel Netz,Reviel Netz, William Noel, The Archimedes Codex: Revealing The Secrets of the World's Greatest Palimpsest echoing Whitehead's proclamation on Plato and philosophy, said that "Western science is but a series of footnotes to Archimedes," calling him "the most important scientist who ever lived." and Eric Temple Bell,E.T. Bell, Men of Mathematics, p 20. wrote that "Any list of the three "greatest" mathematicians of all history would include the name of Archimedes. The other two usually associated with him are Newton and Gauss. Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first."
The discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.
The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with a carving illustrating his proof on the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri ("Rise above oneself and grasp the world").
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Archimedes
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Cultural influence
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Cultural influence
The world's first seagoing steamship with a screw propeller was the SS Archimedes, which was launched in 1839 and named in honor of Archimedes and his work on the screw.
Archimedes has also appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).
The exclamation of Eureka! attributed to Archimedes is the state motto of California. In this instance, the word refers to the discovery of gold near Sutter's Mill in 1848 which sparked the California gold rush.
There is a crater on the Moon named Archimedes () in his honor, as well as a lunar mountain range, the Montes Archimedes ().
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Archimedes
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See also
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See also
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Archimedes
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Concepts
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Concepts
Arbelos
Archimedean point
Archimedes' axiom
Archimedes number
Archimedes paradox
Archimedean solid
Archimedes' twin circles
Methods of computing square roots
Salinon
Steam cannon
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Archimedes
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People
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People
Zhang Heng
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Archimedes
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Notes
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Notes
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Archimedes
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Footnotes
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Footnotes
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Archimedes
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Citations
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Citations
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Archimedes
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References
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References
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Archimedes
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Ancient testimony
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Ancient testimony
Plutarch, Life of Marcellus
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Archimedes
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Modern sources
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Modern sources
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Archimedes
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Further reading
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Further reading
Clagett, Marshall. 1964–1984. Archimedes in the Middle Ages 1–5. Madison, WI: University of Wisconsin Press.
Clagett, Marshall. 1970. "Archimedes". In Charles Coulston Gillispie, ed. Dictionary of Scientific Biography. Vol. 1 (Abailard–Berg). New York: Charles Scribner's Sons. .
Gow, Mary. 2005. Archimedes: Mathematical Genius of the Ancient World. Enslow Publishing. .
Hasan, Heather. 2005. Archimedes: The Father of Mathematics. Rosen Central. .
Netz, Reviel. 2004–2017. The Works of Archimedes: Translation and Commentary. 1–2. Cambridge University Press. Vol. 1: "The Two Books on the Sphere and the Cylinder". . Vol. 2: "On Spirals". .
Netz, Reviel, and William Noel. 2007. The Archimedes Codex. Orion Publishing Group. .
Pickover, Clifford A. 2008. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. .
Simms, Dennis L. 1995. Archimedes the Engineer. Continuum International Publishing Group. .
Stein, Sherman. 1999. Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. .
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Archimedes
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External links
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External links
Heiberg's Edition of Archimedes. Texts in Classical Greek, with some in English.
The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
Testing the Archimedes steam cannon
Category:3rd-century BC Greek writers
Category:People from Syracuse, Sicily
Category:Ancient Greek engineers
Category:Ancient Greek inventors
Category:Ancient Greek geometers
Category:Ancient Greek physicists
Category:Hellenistic-era philosophers
Category:Doric Greek writers
Category:Sicilian Greeks
Category:Mathematicians from Sicily
Category:Scientists from Sicily
Category:Ancient Greek murder victims
Category:Ancient Syracusans
Category:Fluid dynamicists
Category:Buoyancy
Category:280s BC births
Category:210s BC deaths
Category:Year of birth uncertain
Category:Year of death uncertain
Category:3rd-century BC Greek mathematicians
Category:3rd-century BC Syracusans
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Archimedes
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Table of Content
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Short description, Biography, Golden wreath, Launching the ''Syracusia'', War machines, Death, Mathematics, Method of exhaustion, Mechanical method, Large numbers, Archimedean solids, Writings, Surviving works, ''Measurement of a Circle'', ''The Sand Reckoner'', ''On the Equilibrium of Planes'', ''Quadrature of the Parabola'', ''On the Sphere and Cylinder'', ''On Spirals'', ''On Conoids and Spheroids'', ''On Floating Bodies'', ''Ostomachion'', The cattle problem, ''The Method of Mechanical Theorems'', Apocryphal works, Lost works, Archimedes Palimpsest, Legacy, Classical antiquity, Middle ages, Renaissance and early modern Europe, In modern mathematics, Cultural influence, See also, Concepts, People, Notes, Footnotes, Citations, References, Ancient testimony, Modern sources, Further reading, External links
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Alternative medicine
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Short description
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Alternative medicine is any practice that aims to achieve the healing effects of medicine despite lacking biological plausibility, testability, repeatability or evidence of effectiveness. Unlike modern medicine, which employs the scientific method to test plausible therapies by way of responsible and ethical clinical trials, producing repeatable evidence of either effect or of no effect, alternative therapies reside outside of mainstream medicine and do not originate from using the scientific method, but instead rely on testimonials, anecdotes, religion, tradition, superstition, belief in supernatural "energies", pseudoscience, errors in reasoning, propaganda, fraud, or other unscientific sources. Frequently used terms for relevant practices are New Age medicine, pseudo-medicine, unorthodox medicine, holistic medicine, fringe medicine, and unconventional medicine, with little distinction from quackery.
Some alternative practices are based on theories that contradict the established science of how the human body works; others appeal to the supernatural or superstitious to explain their effect or lack thereof. In others, the practice has plausibility but lacks a positive risk–benefit outcome probability. Research into alternative therapies often fails to follow proper research protocols (such as placebo-controlled trials, blind experiments and calculation of prior probability), providing invalid results. History has shown that if a method is proven to work, it eventually ceases to be alternative and becomes mainstream medicine.
Much of the perceived effect of an alternative practice arises from a belief that it will be effective, the placebo effect, or from the treated condition resolving on its own (the natural course of disease). This is further exacerbated by the tendency to turn to alternative therapies upon the failure of medicine, at which point the condition will be at its worst and most likely to spontaneously improve. In the absence of this bias, especially for diseases that are not expected to get better by themselves such as cancer or HIV infection, multiple studies have shown significantly worse outcomes if patients turn to alternative therapies. While this may be because these patients avoid effective treatment, some alternative therapies are actively harmful (e.g. cyanide poisoning from amygdalin, or the intentional ingestion of hydrogen peroxide) or actively interfere with effective treatments.
The alternative medicine sector is a highly profitable industry with a strong lobby, and faces far less regulation over the use and marketing of unproven treatments. Complementary medicine (CM), complementary and alternative medicine (CAM), integrated medicine or integrative medicine (IM), and holistic medicine attempt to combine alternative practices with those of mainstream medicine. Traditional medicine practices become "alternative" when used outside their original settings and without proper scientific explanation and evidence. Alternative methods are often marketed as more "natural" or "holistic" than methods offered by medical science, that is sometimes derogatorily called "Big Pharma" by supporters of alternative medicine. Billions of dollars have been spent studying alternative medicine, with few or no positive results and many methods thoroughly disproven.
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Alternative medicine
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Definitions and terminology
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Definitions and terminology
thumb|upright|Marcia Angell: "There cannot be two kinds of medicine – conventional and alternative."
The terms alternative medicine, complementary medicine, integrative medicine, holistic medicine, natural medicine, unorthodox medicine, fringe medicine, unconventional medicine, and new age medicine are used interchangeably as having the same meaning and are almost synonymous in most contexts. Terminology has shifted over time, reflecting the preferred branding of practitioners.Gorski, David (August 15, 2011). "Integrative medicine": A brand, not a specialty. sciencebasedmedicine.org. (Retrieved March 25, 2022). For example, the United States National Institutes of Health department studying alternative medicine, currently named the National Center for Complementary and Integrative Health (NCCIH), was established as the Office of Alternative Medicine (OAM) and was renamed the National Center for Complementary and Alternative Medicine (NCCAM) before obtaining its current name. Therapies are often framed as "natural" or "holistic", implicitly and intentionally suggesting that conventional medicine is "artificial" and "narrow in scope".
The meaning of the term "alternative" in the expression "alternative medicine", is not that it is an effective alternative to medical science (though some alternative medicine promoters may use the loose terminology to give the appearance of effectiveness). Loose terminology may also be used to suggest meaning that a dichotomy exists when it does not (e.g., the use of the expressions "Western medicine" and "Eastern medicine" to suggest that the difference is a cultural difference between the Asian east and the European west, rather than that the difference is between evidence-based medicine and treatments that do not work).
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Alternative medicine
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Alternative medicine
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Alternative medicine
Alternative medicine is defined loosely as a set of products, practices, and theories that are believed or perceived by their users to have the healing effects of medicine, but whose effectiveness has not been established using scientific methods, or whose theory and practice is not part of biomedicine, or whose theories or practices are directly contradicted by scientific evidence or scientific principles used in biomedicine. "Biomedicine" or "medicine" is that part of medical science that applies principles of biology, physiology, molecular biology, biophysics, and other natural sciences to clinical practice, using scientific methods to establish the effectiveness of that practice. Unlike medicine, an alternative product or practice does not originate from using scientific methods, but may instead be based on hearsay, religion, tradition, superstition, belief in supernatural energies, pseudoscience, errors in reasoning, propaganda, fraud, or other unscientific sources.
Some other definitions seek to specify alternative medicine in terms of its social and political marginality to mainstream healthcare. This can refer to the lack of support that alternative therapies receive from medical scientists regarding access to research funding, sympathetic coverage in the medical press, or inclusion in the standard medical curriculum. For example, a widely used definition devised by the US NCCIH calls it "a group of diverse medical and health care systems, practices, and products that are not generally considered part of conventional medicine". However, these descriptive definitions are inadequate in the present-day when some conventional doctors offer alternative medical treatments and introductory courses or modules can be offered as part of standard undergraduate medical training; alternative medicine is taught in more than half of US medical schools and US health insurers are increasingly willing to provide reimbursement for alternative therapies.
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Alternative medicine
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Complementary or integrative medicine
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Complementary or integrative medicine
Complementary medicine (CM) or integrative medicine (IM) is when alternative medicine is used together with mainstream medical treatment in a belief that it improves the effect of treatments. For example, acupuncture (piercing the body with needles to influence the flow of a supernatural energy) might be believed to increase the effectiveness or "complement" science-based medicine when used at the same time. Significant drug interactions caused by alternative therapies may make treatments less effective, notably in cancer therapy.
Several medical organizations differentiate between complementary and alternative medicine including the UK National Health Service (NHS), Cancer Research UK, and the US Center for Disease Control and Prevention (CDC), the latter of which states that "Complementary medicine is used in addition to standard treatments" whereas "Alternative medicine is used instead of standard treatments."
Complementary and integrative interventions are used to improve fatigue in adult cancer patients.
David Gorski has described integrative medicine as an attempt to bring pseudoscience into academic science-based medicine with skeptics such as Gorski and David Colquhoun referring to this with the pejorative term "quackademia". Robert Todd Carroll described Integrative medicine as "a synonym for 'alternative' medicine that, at its worst, integrates sense with nonsense. At its best, integrative medicine supports both consensus treatments of science-based medicine and treatments that the science, while promising perhaps, does not justify"Robert Todd Carroll. Integrative medicine. Skeptic's Dictionary Rose Shapiro has criticized the field of alternative medicine for rebranding the same practices as integrative medicine.
CAM is an abbreviation of the phrase complementary and alternative medicine. The 2019 World Health Organization (WHO) Global Report on Traditional and Complementary Medicine states that the terms complementary and alternative medicine "refer to a broad set of health care practices that are not part of that country's own traditional or conventional medicine and are not fully integrated into the dominant health care system. They are used interchangeably with traditional medicine in some countries."
In the 1990s, integrative medicine started to be marketed by a new term, functional medicine.
The Integrative Medicine Exam by the American Board of Physician Specialties includes the following subjects: Manual Therapies, Biofield Therapies, Acupuncture, Movement Therapies, Expressive Arts, Traditional Chinese Medicine, Ayurveda, Indigenous Medical Systems, Homeopathic Medicine, Naturopathic Medicine, Osteopathic Medicine, Chiropractic, and Functional Medicine.
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Alternative medicine
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Other terms
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Other terms
Traditional medicine (TM) refers to certain practices within a culture which have existed since before the advent of medical science, Many TM practices are based on "holistic" approaches to disease and health, versus the scientific evidence-based methods in conventional medicine. The 2019 WHO report defines traditional medicine as "the sum total of the knowledge, skill and practices based on the theories, beliefs and experiences indigenous to different cultures, whether explicable or not, used in the maintenance of health as well as in the prevention, diagnosis, improvement or treatment of physical and mental illness." When used outside the original setting and in the absence of scientific evidence, TM practices are typically referred to as "alternative medicine".
is another rebranding of alternative medicine. In this case, the words balance and holism are often used alongside complementary or integrative, claiming to take into fuller account the "whole" person, in contrast to the supposed reductionism of medicine.
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Alternative medicine
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Challenges in defining alternative medicine
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Challenges in defining alternative medicine
Prominent members of the science and biomedical science community say that it is not meaningful to define an alternative medicine that is separate from a conventional medicine because the expressions "conventional medicine", "alternative medicine", "complementary medicine", "integrative medicine", and "holistic medicine" do not refer to any medicine at all. Others say that alternative medicine cannot be precisely defined because of the diversity of theories and practices it includes, and because the boundaries between alternative and conventional medicine overlap, are porous, and change. Healthcare practices categorized as alternative may differ in their historical origin, theoretical basis, diagnostic technique, therapeutic practice and in their relationship to the medical mainstream. Under a definition of alternative medicine as "non-mainstream", treatments considered alternative in one location may be considered conventional in another.
Critics say the expression is deceptive because it implies there is an effective alternative to science-based medicine, and that complementary is deceptive because it implies that the treatment increases the effectiveness of (complements) science-based medicine, while alternative medicines that have been tested nearly always have no measurable positive effect compared to a placebo. Journalist John Diamond wrote that "there is really no such thing as alternative medicine, just medicine that works and medicine that doesn't", a notion later echoed by Paul Offit: "The truth is there's no such thing as conventional or alternative or complementary or integrative or holistic medicine. There's only medicine that works and medicine that doesn't. And the best way to sort it out is by carefully evaluating scientific studies—not by visiting Internet chat rooms, reading magazine articles, or talking to friends."
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Alternative medicine
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Types
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Types
Alternative medicine consists of a wide range of health care practices, products, and therapies. The shared feature is a claim to heal that is not based on the scientific method. Alternative medicine practices are diverse in their foundations and methodologies. Alternative medicine practices may be classified by their cultural origins or by the types of beliefs upon which they are based. Methods may incorporate or be based on traditional medicinal practices of a particular culture, folk knowledge, superstition, spiritual beliefs, belief in supernatural energies (antiscience), pseudoscience, errors in reasoning, propaganda, fraud, new or different concepts of health and disease, and any bases other than being proven by scientific methods. Different cultures may have their own unique traditional or belief based practices developed recently or over thousands of years, and specific practices or entire systems of practices.
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Alternative medicine
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Unscientific belief systems
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Unscientific belief systems
thumb|"They told me if I took 1000 pills at night I should be quite another thing in the morning", an early 19th-century satire on Morison's Vegetable Pills, an alternative medicine supplement
Alternative medicine, such as using naturopathy or homeopathy in place of conventional medicine, is based on belief systems not grounded in science.
Proposed mechanismIssuesNaturopathyNaturopathic medicine is based on a belief that the body heals itself using a supernatural vital energy that guides bodily processes.In conflict with the paradigm of evidence-based medicine. Many naturopaths have opposed vaccination, and "scientific evidence does not support claims that naturopathic medicine can cure cancer or any other disease".HomeopathyA belief that a substance that causes the symptoms of a disease in healthy people cures similar symptoms in sick people.Developed before knowledge of atoms and molecules, or of basic chemistry, which shows that repeated dilution as practiced in homeopathy produces only water, and that homeopathy is not scientifically valid.
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Alternative medicine
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Traditional ethnic systems
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Traditional ethnic systems
thumb|Ready-to-drink traditional Chinese medicine mixture
thumb|Acupuncture involves insertion of needles in the body.
Alternative medical systems may be based on traditional medicine practices, such as traditional Chinese medicine (TCM), Ayurveda in India, or practices of other cultures around the world. Some useful applications of traditional medicines have been researched and accepted within ordinary medicine, however the underlying belief systems are seldom scientific and are not accepted.
Traditional medicine is considered alternative when it is used outside its home region; or when it is used together with or instead of known functional treatment; or when it can be reasonably expected that the patient or practitioner knows or should know that it will not work – such as knowing that the practice is based on superstition.
ClaimsIssuesTraditional Chinese medicineTraditional practices and beliefs from China, together with modifications made by the Communist party make up TCM. Common practices include herbal medicine, acupuncture (insertion of needles in the body at specified points), massage (Tui na), exercise (qigong), and dietary therapy.The practices are based on belief in a supernatural energy called qi, considerations of Chinese astrology and Chinese numerology, traditional use of herbs and other substances found in China, a belief that the tongue contains a map of the body that reflects changes in the body, and an incorrect model of the anatomy and physiology of internal organs.AyurvedaTraditional medicine of India. Ayurveda believes in the existence of three elemental substances, the doshas (called Vata, Pitta and Kapha), and states that a balance of the doshas results in health, while imbalance results in disease. Such disease-inducing imbalances can be adjusted and balanced using traditional herbs, minerals and heavy metals. Ayurveda stresses the use of plant-based medicines and treatments, with some animal products, and added minerals, including sulfur, arsenic, lead and copper(II) sulfate.Safety concerns have been raised about Ayurveda, with two U.S. studies finding about 20 percent of Ayurvedic Indian-manufactured patent medicines contained toxic levels of heavy metals such as lead, mercury and arsenic. A 2015 study of users in the United States also found elevated blood lead levels in 40 percent of those tested. Other concerns include the use of herbs containing toxic compounds and the lack of quality control in Ayurvedic facilities. Incidents of heavy metal poisoning have been attributed to the use of these compounds in the United States.
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Alternative medicine
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Supernatural energies
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Supernatural energies
Bases of belief may include belief in existence of supernatural energies undetected by the science of physics, as in biofields, or in belief in properties of the energies of physics that are inconsistent with the laws of physics, as in energy medicine.
ClaimsIssuesBiofield therapyIntended to influence energy fields that, it is purported, surround and penetrate the body.Advocates of scientific skepticism such as Carl Sagan have criticized the lack of empirical evidence to support the existence of the putative energy fields on which these therapies are predicated.Bioelectromagnetic therapyUse verifiable electromagnetic fields, such as pulsed fields, alternating-current, or direct-current fields in an unconventional manner.Asserts that magnets can be used to defy the laws of physics to influence health and disease.ChiropracticSpinal manipulation aims to treat "vertebral subluxations" which are claimed to put pressure on nerves.Chiropractic was based on the belief that manipulating the spine unblocks the flow of a supernatural vital energy called Innate Intelligence, thereby affecting health and disease. Vertebral subluxation is a pseudoscientific entity not proven to exist.ReikiPractitioners place their palms on the patient near Chakras that they believe are centers of supernatural energies in the belief that these supernatural energies can transfer from the practitioner's palms to heal the patient.Lacks credible scientific evidence.
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Alternative medicine
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Herbal remedies and other substances
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Herbal remedies and other substances
Substance based practices use substances found in nature such as herbs, foods, non-vitamin supplements and megavitamins, animal and fungal products, and minerals, including use of these products in traditional medical practices that may also incorporate other methods. Examples include healing claims for non-vitamin supplements, fish oil, Omega-3 fatty acid, glucosamine, echinacea, flaxseed oil, and ginseng. Herbal medicine, or phytotherapy, includes not just the use of plant products, but may also include the use of animal and mineral products. It is among the most commercially successful branches of alternative medicine, and includes the tablets, powders and elixirs that are sold as "nutritional supplements". Only a very small percentage of these have been shown to have any efficacy, and there is little regulation as to standards and safety of their contents.
thumb|right|A chiropractor "adjusting" the spine
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Alternative medicine
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Religion, faith healing, and prayer
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Religion, faith healing, and prayer
ClaimsIssuesChristian faith healingThere is a divine or spiritual intervention in healing.Lack of evidence for effectiveness. Unwanted outcomes, such as death and disability, "have occurred when faith healing was elected instead of medical care for serious injuries or illnesses". A 2001 double-blind study of 799 discharged coronary surgery patients found that "intercessory prayer had no significant effect on medical outcomes after hospitalization in a coronary care unit."
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Alternative medicine
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NCCIH classification
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NCCIH classification
The United States agency National Center for Complementary and Integrative Health (NCCIH) has created a classification system for branches of complementary and alternative medicine that divides them into five major groups. These groups have some overlap, and distinguish two types of energy medicine: veritable which involves scientifically observable energy (including magnet therapy, colorpuncture and light therapy) and putative, which invokes physically undetectable or unverifiable energy. None of these energies have any evidence to support that they affect the body in any positive or health promoting way.
Whole medical systems: Cut across more than one of the other groups; examples include traditional Chinese medicine, naturopathy, homeopathy, and ayurveda.
Mind-body interventions: Explore the interconnection between the mind, body, and spirit, under the premise that they affect "bodily functions and symptoms". A connection between mind and body is conventional medical fact, and this classification does not include therapies with proven function such as cognitive behavioral therapy.
"Biology"-based practices: Use substances found in nature such as herbs, foods, vitamins, and other natural substances. (As used here, "biology" does not refer to the science of biology, but is a usage newly coined by NCCIH in the primary source used for this article. "Biology-based" as coined by NCCIH may refer to chemicals from a nonbiological source, such as use of the poison lead in traditional Chinese medicine, and to other nonbiological substances.)
Manipulative and body-based practices: feature manipulation or movement of body parts, such as is done in bodywork, chiropractic, and osteopathic manipulation.
Energy medicine: is a domain that deals with putative and verifiable energy fields:
Biofield therapies are intended to influence energy fields that are purported to surround and penetrate the body. The existence of such energy fields have been disproven.
Bioelectromagnetic-based therapies use verifiable electromagnetic fields, such as pulsed fields, alternating-current, or direct-current fields in a non-scientific manner.
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Alternative medicine
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History
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History
The history of alternative medicine may refer to the history of a group of diverse medical practices that were collectively promoted as "alternative medicine" beginning in the 1970s, to the collection of individual histories of members of that group, or to the history of western medical practices that were labeled "irregular practices" by the western medical establishment. It includes the histories of complementary medicine and of integrative medicine. Before the 1970s, western practitioners that were not part of the increasingly science-based medical establishment were referred to "irregular practitioners", and were dismissed by the medical establishment as unscientific and as practicing quackery. Until the 1970s, irregular practice became increasingly marginalized as quackery and fraud, as western medicine increasingly incorporated scientific methods and discoveries, and had a corresponding increase in success of its treatments. In the 1970s, irregular practices were grouped with traditional practices of nonwestern cultures and with other unproven or disproven practices that were not part of biomedicine, with the entire group collectively marketed and promoted under the single expression "alternative medicine".Quack Medicine: A History of Combating Health Fraud in Twentieth-Century America, Eric W. Boyle,
Use of alternative medicine in the west began to rise following the counterculture movement of the 1960s, as part of the rising new age movement of the 1970s. This was due to misleading mass marketing of "alternative medicine" being an effective "alternative" to biomedicine, changing social attitudes about not using chemicals and challenging the establishment and authority of any kind, sensitivity to giving equal measure to beliefs and practices of other cultures (cultural relativism), and growing frustration and desperation by patients about limitations and side effects of science-based medicine. At the same time, in 1975, the American Medical Association, which played the central role in fighting quackery in the United States, abolished its quackery committee and closed down its Department of Investigation. By the early to mid 1970s the expression "alternative medicine" came into widespread use, and the expression became mass marketed as a collection of "natural" and effective treatment "alternatives" to science-based biomedicine. By 1983, mass marketing of "alternative medicine" was so pervasive that the British Medical Journal (BMJ) pointed to "an apparently endless stream of books, articles, and radio and television programmes urge on the public the virtues of (alternative medicine) treatments ranging from meditation to drilling a hole in the skull to let in more oxygen".
An analysis of trends in the criticism of complementary and alternative medicine (CAM) in five prestigious American medical journals during the period of reorganization within medicine (1965–1999) was reported as showing that the medical profession had responded to the growth of CAM in three phases, and that in each phase, changes in the medical marketplace had influenced the type of response in the journals. Changes included relaxed medical licensing, the development of managed care, rising consumerism, and the establishment of the USA Office of Alternative Medicine (later National Center for Complementary and Alternative Medicine, currently National Center for Complementary and Integrative Health).
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Alternative medicine
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Medical education
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Medical education
Mainly as a result of reforms following the Flexner Report of 1910 medical education in established medical schools in the US has generally not included alternative medicine as a teaching topic. Typically, their teaching is based on current practice and scientific knowledge about: anatomy, physiology, histology, embryology, neuroanatomy, pathology, pharmacology, microbiology and immunology. Medical schools' teaching includes such topics as doctor-patient communication, ethics, the art of medicine, and engaging in complex clinical reasoning (medical decision-making). Writing in 2002, Snyderman and Weil remarked that by the early twentieth century the Flexner model had helped to create the 20th-century academic health center, in which education, research, and practice were inseparable. While this had much improved medical practice by defining with increasing certainty the pathophysiological basis of disease, a single-minded focus on the pathophysiological had diverted much of mainstream American medicine from clinical conditions that were not well understood in mechanistic terms, and were not effectively treated by conventional therapies.
By 2001 some form of CAM training was being offered by at least 75 out of 125 medical schools in the US. Exceptionally, the School of Medicine of the University of Maryland, Baltimore, includes a research institute for integrative medicine (a member entity of the Cochrane Collaboration). Medical schools are responsible for conferring medical degrees, but a physician typically may not legally practice medicine until licensed by the local government authority. Licensed physicians in the US who have attended one of the established medical schools there have usually graduated Doctor of Medicine (MD). All states require that applicants for MD licensure be graduates of an approved medical school and complete the United States Medical Licensing Examination (USMLE).
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Alternative medicine
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Efficacy
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Efficacy
thumb|Edzard Ernst, an authority on scientific study of alternative therapies and diagnoses and the first university professor of CAM, in 2012
There is a general scientific consensus that alternative therapies lack the requisite scientific validation, and their effectiveness is either unproved or disproved. Many of the claims regarding the efficacy of alternative medicines are controversial, since research on them is frequently of low quality and methodologically flawed. Selective publication bias, marked differences in product quality and standardisation, and some companies making unsubstantiated claims call into question the claims of efficacy of isolated examples where there is evidence for alternative therapies.
The Scientific Review of Alternative Medicine points to confusions in the general population – a person may attribute symptomatic relief to an otherwise-ineffective therapy just because they are taking something (the placebo effect); the natural recovery from or the cyclical nature of an illness (the regression fallacy) gets misattributed to an alternative medicine being taken; a person not diagnosed with science-based medicine may never originally have had a true illness diagnosed as an alternative disease category.
Edzard Ernst, the first university professor of Complementary and Alternative Medicine, characterized the evidence for many alternative techniques as weak, nonexistent, or negative and in 2011 published his estimate that about 7.4% were based on "sound evidence", although he believes that may be an overestimate. Ernst has concluded that 95% of the alternative therapies he and his team studied, including acupuncture, herbal medicine, homeopathy, and reflexology, are "statistically indistinguishable from placebo treatments", but he also believes there is something that conventional doctors can usefully learn from the chiropractors and homeopath: this is the therapeutic value of the placebo effect, one of the strangest phenomena in medicine.
In 2003, a project funded by the CDC identified 208 condition-treatment pairs, of which 58% had been studied by at least one randomized controlled trial (RCT), and 23% had been assessed with a meta-analysis. According to a 2005 book by a US Institute of Medicine panel, the number of RCTs focused on CAM has risen dramatically.
, the Cochrane Library had 145 CAM-related Cochrane systematic reviews and 340 non-Cochrane systematic reviews. An analysis of the conclusions of only the 145 Cochrane reviews was done by two readers. In 83% of the cases, the readers agreed. In the 17% in which they disagreed, a third reader agreed with one of the initial readers to set a rating. These studies found that, for CAM, 38.4% concluded positive effect or possibly positive (12.4%), 4.8% concluded no effect, 0.7% concluded harmful effect, and 56.6% concluded insufficient evidence. An assessment of conventional treatments found that 41.3% concluded positive or possibly positive effect, 20% concluded no effect, 8.1% concluded net harmful effects, and 21.3% concluded insufficient evidence. However, the CAM review used the more developed 2004 Cochrane database, while the conventional review used the initial 1998 Cochrane database.
Alternative therapies do not "complement" (improve the effect of, or mitigate the side effects of) functional medical treatment. Significant drug interactions caused by alternative therapies may instead negatively impact functional treatment by making prescription drugs less effective, such as interference by herbal preparations with warfarin.
In the same way as for conventional therapies, drugs, and interventions, it can be difficult to test the efficacy of alternative medicine in clinical trials. In instances where an established, effective, treatment for a condition is already available, the Helsinki Declaration states that withholding such treatment is unethical in most circumstances. Use of standard-of-care treatment in addition to an alternative technique being tested may produce confounded or difficult-to-interpret results.
Cancer researcher Andrew J. Vickers has stated:
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Alternative medicine
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Perceived mechanism of effect
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Perceived mechanism of effect
Anything classified as alternative medicine by definition does not have a proven healing or medical effect. However, there are different mechanisms through which it can be perceived to "work". The common denominator of these mechanisms is that effects are mis-attributed to the alternative treatment. thumb|right|350px|How alternative therapies "work":
a) Misinterpreted natural course – the individual gets better without treatment.
b) Placebo effect or false treatment effect – an individual receives "alternative therapy" and is convinced it will help. The conviction makes them more likely to get better.
c) Nocebo effect – an individual is convinced that standard treatment will not work, and that alternative therapies will work. This decreases the likelihood standard treatment will work, while the placebo effect of the "alternative" remains.
d) No adverse effects – Standard treatment is replaced with "alternative" treatment, getting rid of adverse effects, but also of improvement.
e) Interference – Standard treatment is "complemented" with something that interferes with its effect. This can both cause worse effect, but also decreased (or even increased) side effects, which may be interpreted as "helping".
Researchers, such as epidemiologists, clinical statisticians and pharmacologists, use clinical trials to reveal such effects, allowing physicians to offer a therapeutic solution best known to work. "Alternative treatments" often refuse to use trials or make it deliberately hard to do so.
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Alternative medicine
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Placebo effect
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Placebo effect
A placebo is a treatment with no intended therapeutic value. An example of a placebo is an inert pill, but it can include more dramatic interventions like sham surgery. The placebo effect is the concept that patients will perceive an improvement after being treated with an inert treatment. The opposite of the placebo effect is the nocebo effect, when patients who expect a treatment to be harmful will perceive harmful effects after taking it.
Placebos do not have a physical effect on diseases or improve overall outcomes, but patients may report improvements in subjective outcomes such as pain and nausea. A 1955 study suggested that a substantial part of a medicine's impact was due to the placebo effect. However, reassessments found the study to have flawed methodology. This and other modern reviews suggest that other factors like natural recovery and reporting bias should also be considered.
All of these are reasons why alternative therapies may be credited for improving a patient's condition even though the objective effect is non-existent, or even harmful. David Gorski argues that alternative treatments should be treated as a placebo, rather than as medicine. Almost none have performed significantly better than a placebo in clinical trials. Furthermore, distrust of conventional medicine may lead to patients experiencing the nocebo effect when taking effective medication.
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Alternative medicine
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Regression to the mean
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Regression to the mean
A patient who receives an inert treatment may report improvements afterwards that it did not cause. Assuming it was the cause without evidence is an example of the regression fallacy. This may be due to a natural recovery from the illness, or a fluctuation in the symptoms of a long-term condition. The concept of regression toward the mean implies that an extreme result is more likely to be followed by a less extreme result.
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Alternative medicine
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Other factors
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Other factors
There are also reasons why a placebo treatment group may outperform a "no-treatment" group in a test which are not related to a patient's experience. These include patients reporting more favourable results than they really felt due to politeness or "experimental subordination", observer bias, and misleading wording of questions. In their 2010 systematic review of studies into placebos, Asbjørn Hróbjartsson and Peter C. Gøtzsche write that "even if there were no true effect of placebo, one would expect to record differences between placebo and no-treatment groups due to bias associated with lack of blinding." Alternative therapies may also be credited for perceived improvement through decreased use or effect of medical treatment, and therefore either decreased side effects or nocebo effects towards standard treatment.
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Alternative medicine
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Use and regulation
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Use and regulation
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Alternative medicine
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Appeal
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Appeal
Practitioners of complementary medicine usually discuss and advise patients as to available alternative therapies. Patients often express interest in mind-body complementary therapies because they offer a non-drug approach to treating some health conditions.
In addition to the social-cultural underpinnings of the popularity of alternative medicine, there are several psychological issues that are critical to its growth, notably psychological effects, such as the will to believe, cognitive biases that help maintain self-esteem and promote harmonious social functioning, and the post hoc, ergo propter hoc fallacy.
In a 2018 interview with The BMJ, Edzard Ernst stated: "The present popularity of complementary and alternative medicine is also inviting criticism of what we are doing in mainstream medicine. It shows that we aren't fulfilling a certain need-we are not giving patients enough time, compassion, or empathy. These are things that complementary practitioners are very good at. Mainstream medicine could learn something from complementary medicine."
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Alternative medicine
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Marketing
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Marketing
Alternative medicine is a profitable industry with large media advertising expenditures. Accordingly, alternative practices are often portrayed positively and compared favorably to "big pharma".
The popularity of complementary & alternative medicine (CAM) may be related to other factors that Ernst mentioned in a 2008 interview in The Independent:
Paul Offit proposed that "alternative medicine becomes quackery" in four ways: by recommending against conventional therapies that are helpful, promoting potentially harmful therapies without adequate warning, draining patients' bank accounts, or by promoting "magical thinking". Promoting alternative medicine has been called dangerous and unethical.
thumb|Friendly and colorful images of herbal treatments may look less threatening or dangerous when compared to conventional medicine. This is an intentional marketing strategy.
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Alternative medicine
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Social factors
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Social factors
Authors have speculated on the socio-cultural and psychological reasons for the appeal of alternative medicines among the minority using them in lieu of conventional medicine. There are several socio-cultural reasons for the interest in these treatments centered on the low level of scientific literacy among the public at large and a concomitant increase in antiscientific attitudes and new age mysticism. Related to this are vigorous marketing of extravagant claims by the alternative medical community combined with inadequate media scrutiny and attacks on critics. Alternative medicine is criticized for taking advantage of the least fortunate members of society.
There is also an increase in conspiracy theories toward conventional medicine and pharmaceutical companies, mistrust of traditional authority figures, such as the physician, and a dislike of the current delivery methods of scientific biomedicine, all of which have led patients to seek out alternative medicine to treat a variety of ailments. Many patients lack access to contemporary medicine, due to a lack of private or public health insurance, which leads them to seek out lower-cost alternative medicine. Medical doctors are also aggressively marketing alternative medicine to profit from this market.
Patients can be averse to the painful, unpleasant, and sometimes-dangerous side effects of biomedical treatments. Treatments for severe diseases such as cancer and HIV infection have well-known, significant side-effects. Even low-risk medications such as antibiotics can have potential to cause life-threatening anaphylactic reactions in a very few individuals. Many medications may cause minor but bothersome symptoms such as cough or upset stomach. In all of these cases, patients may be seeking out alternative therapies to avoid the adverse effects of conventional treatments.
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Alternative medicine
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Prevalence of use
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Prevalence of use
According to research published in 2015, the increasing popularity of CAM needs to be explained by moral convictions or lifestyle choices rather than by economic reasoning.
In developing nations, access to essential medicines is severely restricted by lack of resources and poverty. Traditional remedies, often closely resembling or forming the basis for alternative remedies, may comprise primary healthcare or be integrated into the healthcare system. In Africa, traditional medicine is used for 80% of primary healthcare, and in developing nations as a whole over one-third of the population lack access to essential medicines.
In Latin America, inequities against BIPOC communities keep them tied to their traditional practices and therefore, it is often these communities that constitute the majority of users of alternative medicine. Racist attitudes towards certain communities disable them from accessing more urbanized modes of care. In a study that assessed access to care in rural communities of Latin America, it was found that discrimination is a huge barrier to the ability of citizens to access care; more specifically, women of Indigenous and African descent, and lower-income families were especially hurt. Such exclusion exacerbates the inequities that minorities in Latin America already face. Consistently excluded from many systems of westernized care for socioeconomic and other reasons, low-income communities of color often turn to traditional medicine for care as it has proved reliable to them across generations.
Commentators including David Horrobin have proposed adopting a prize system to reward medical research. This stands in opposition to the current mechanism for funding research proposals in most countries around the world. In the US, the NCCIH provides public research funding for alternative medicine. The NCCIH has spent more than US$2.5 billion on such research since 1992 and this research has not demonstrated the efficacy of alternative therapies. As of 2011, the NCCIH's sister organization in the NIC Office of Cancer Complementary and Alternative Medicine had given out grants of around $105 million each year for several years. Testing alternative medicine that has no scientific basis (as in the aforementioned grants) has been called a waste of scarce research resources.
That alternative medicine has been on the rise "in countries where Western science and scientific method generally are accepted as the major foundations for healthcare, and 'evidence-based' practice is the dominant paradigm" was described as an "enigma" in the Medical Journal of Australia. A 15-year systematic review published in 2022 on the global acceptance and use of CAM among medical specialists found the overall acceptance of CAM at 52% and the overall use at 45%.
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Alternative medicine
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In the United States
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In the United States
In the United States, the 1974 Child Abuse Prevention and Treatment Act (CAPTA) required that for states to receive federal money, they had to grant religious exemptions to child neglect and abuse laws regarding religion-based healing practices. Thirty-one states have child-abuse religious exemptions.
The use of alternative medicine in the US has increased, with a 50 percent increase in expenditures and a 25 percent increase in the use of alternative therapies between 1990 and 1997 in America. According to a national survey conducted in 2002, "36 percent of U.S. adults aged 18 years and over use some form of complementary and alternative medicine." Americans spend many billions on the therapies annually. Most Americans used CAM to treat and/or prevent musculoskeletal conditions or other conditions associated with chronic or recurring pain. In America, women were more likely than men to use CAM, with the biggest difference in use of mind-body therapies including prayer specifically for health reasons". In 2008, more than 37% of American hospitals offered alternative therapies, up from 27 percent in 2005, and 25% in 2004. More than 70% of the hospitals offering CAM were in urban areas.
A survey of Americans found that 88 percent thought that "there are some good ways of treating sickness that medical science does not recognize". Use of magnets was the most common tool in energy medicine in America, and among users of it, 58 percent described it as at least "sort of scientific", when it is not at all scientific. In 2002, at least 60 percent of US medical schools have at least some class time spent teaching alternative therapies. "Therapeutic touch" was taught at more than 100 colleges and universities in 75 countries before the practice was debunked by a nine-year-old child for a school science project.
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Alternative medicine
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Prevalence of use of specific therapies
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Prevalence of use of specific therapies
The most common CAM therapies used in the US in 2002 were prayer (45%), herbalism (19%), breathing meditation (12%), meditation (8%), chiropractic medicine (8%), yoga (5–6%), body work (5%), diet-based therapy (4%), progressive relaxation (3%), mega-vitamin therapy (3%) and visualization (2%).
In Britain, the most often used alternative therapies were Alexander technique, aromatherapy, Bach and other flower remedies, body work therapies including massage, Counseling stress therapies, hypnotherapy, meditation, reflexology, Shiatsu, Ayurvedic medicine, nutritional medicine, and yoga. Ayurvedic medicine remedies are mainly plant based with some use of animal materials. Safety concerns include the use of herbs containing toxic compounds and the lack of quality control in Ayurvedic facilities.
According to the National Health Service (England), the most commonly used complementary and alternative medicines (CAM) supported by the NHS in the UK are: acupuncture, aromatherapy, chiropractic, homeopathy, massage, osteopathy and clinical hypnotherapy.
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