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Archimedes

Archimedes of Syracuse (c. 287) 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, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,

Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include 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. 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 mathematical discoveries.

Unlike his inventions, Archimedes' mathematical writings were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530AD by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes by Eutocius in the 6th century opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance and again in the 17th century, while the discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.

Biography
Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek scholar John Tzetzes that Archimedes lived for 75 years before his death in 212 BC. In the Sand-Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known. A biography of Archimedes was written by his friend Heracleides, but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth. From his surviving written works, it is clear that he maintained collegial relations with scholars based there, including his friend Conon of Samos and the head librarian Eratosthenes of Cyrene.

The standard versions of Archimedes' life were written long after his death by Greek and Roman historians. The earliest reference to Archimedes occurs in The Histories by Polybius (c. 200–118 BC), written about 70 years after his death. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city from the Romans. Polybius remarks how, during the Second Punic War, Syracuse switched allegiances from Rome to Carthage, resulting in a military campaign under the command of Marcus Claudius Marcellus and Appius Claudius Pulcher, who besieged the city from 213 to 212 BC. He notes that the Romans underestimated Syracuse's defenses, and mentions several machines Archimedes designed, including improved catapults, crane-like machines that could be swung around in an arc, and other stone-throwers. Although the Romans ultimately captured the city, they suffered considerable losses due to Archimedes' inventiveness.

Cicero (106–43 BC) mentions Archimedes in some of his works. While serving as a quaestor in Sicily, Cicero 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. He also mentions that Marcellus brought to Rome two planetariums Archimedes built. The Roman historian Livy (59 BC–17 AD) retells Polybius' story of the capture of Syracuse and Archimedes' role in it. Plutarch (45–119 AD) wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse, although Cicero suggests he was of humble origin.. Plutarch also provides at least two accounts on how Archimedes died after the city was taken. According to the most popular account, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet Marcellus, but he declined, saying that he had to finish working on the problem. This enraged the soldier, who killed Archimedes with his sword. Another story has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items. 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.

The last words attributed to Archimedes are "Do not disturb my circles" (Latin, "Noli turbare circulos meos"; Katharevousa Greek, "μὴ μου τοὺς κύκλους τάραττε"), a reference to the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier.. There is no reliable evidence that Archimedes uttered these words and they do not appear in Plutarch's account. A similar quotation is found in the work of Valerius Maximus (fl. 30 AD), who wrote in Memorable Doings and Sayings, "... sed protecto manibus puluere 'noli' inquit, 'obsecro, istum disturbare'" ("... but protecting the dust with his hands, said 'I beg of you, do not disturb this).

Archimedes' principle
The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a crown for a temple had been made for King Hiero II of Syracuse, who supplied the pure gold to be used. The crown was likely made in the shape of a votive wreath. Archimedes was asked to determine whether some silver had been substituted by the goldsmith without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density.

In this account, Archimedes noticed while taking a bath that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the golden crown's volume. Archimedes was so excited by this discovery that he took to the streets naked, having forgotten to dress, crying "Eureka!" ("εὕρηκα, heúrēka!, lit. 'I have found [it]!'). For practical purposes water is incompressible, so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, its density could be obtained; if cheaper and less dense metals had been added, the density would be lower than that of gold. Archimedes found that this is what had happened, proving that silver had been mixed in.

The story of the golden crown does not appear anywhere in Archimedes' known works. The practicality of the method described has been called into question due to the extreme accuracy that would be required to measure water displacement. Archimedes may have instead sought a solution that applied the hydrostatics principle known as Archimedes' principle, found in his treatise On Floating Bodies: a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Using this principle, it would have been possible to compare the density of the crown to that of pure gold by balancing it on a scale with a pure gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. 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."

Law of the lever
While Archimedes did not invent the lever, he gave a mathematical proof of the principle involved in his work On the Equilibrium of Planes. 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.

There are several, often conflicting, reports regarding Archimedes' feats using the lever to lift very heavy objects. Plutarch describes how Archimedes designed block-and-tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move. According to Pappus of Alexandria, Archimedes' work on levers and his understanding of mechanical advantage caused him to remark: "Give me a place to stand on, and I will move the Earth" (δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω). Olympiodorus later attributed the same boast to Archimedes' invention of the baroulkos, a kind of windlass, rather than the lever.

Archimedes' screw

A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of Syracuse. Athenaeus of Naucratis quotes a certain Moschion in a description on how King Hiero II commissioned the design of a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a display of naval power. The Syracusia is said to have been the largest ship built in classical antiquity and, according to Moschion's account, it was launched by Archimedes. The ship presumably was capable of carrying 600 people and included garden decorations, a gymnasium, and a temple dedicated to the goddess Aphrodite among its facilities. The account also mentions that, in order to remove any potential water leaking through the hull, a device with a revolving screw-shaped blade inside a cylinder was designed by Archimedes.

Archimedes' screw was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The screw is still in use today for pumping liquids and granulated solids such as coal and grain. Described by Vitruvius, Archimedes' device may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon. 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' claw
Archimedes is said to have designed a claw as a weapon to defend the city of Syracuse. Also known as "", the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it.

There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled Superweapons of the Ancient World built a version of the claw and concluded that it was a workable device. Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.

Heat ray

As legend has it, Archimedes arranged mirrors as a parabolic reflector to burn ships attacking Syracuse using focused sunlight. While there is no extant contemporary evidence of this feat and modern scholars believe it did not happen, Archimedes may have written a work on mirrors entitled Catoptrica, and Lucian and Galen, writing in the second century AD, mentioned that during the siege of Syracuse Archimedes had burned enemy ships. Nearly four hundred years later, Anthemius, despite skepticism, tried to reconstruct Archimedes' hypothetical reflector geometry.

The purported device, sometimes called "", 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, mostly with negative results. It has been suggested that a large array of highly polished bronze or copper shields acting as mirrors could have been employed to focus sunlight onto a ship, but the overall effect would have been blinding, dazzling, or distracting the crew of the ship rather than fire. Using modern materials and larger scale, sunlight-concentrating solar furnaces can reach very high temperatures, and are sometimes used for generating electricity.

Astronomical instruments
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.

Cicero's De re publica portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse in the Second Punic War, Marcellus is said to have taken back to Rome two mechanisms which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets. Cicero also mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus:

"Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione. When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth when the Sun was in line."

This is a description of a small planetarium. Pappus of Alexandria reports on a now lost treatise by Archimedes dealing with the construction of these mechanisms entitled On Sphere-Making. Modern research in this area has been focused on the Antikythera mechanism, another device built c. 100 BC probably designed with a similar purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.

Mathematics
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote that Archimedes "placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life", though some scholars believe this may be a mischaracterization.

Method of exhaustion
Archimedes was able to use indivisibles (a precursor to infinitesimals) in a way that is similar to modern integral calculus. Through proof by contradiction (reductio ad absurdum), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the areas of figures and the value of π.

In Measurement of a Circle, he did this 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 π lay between 3$1⁄7$ (approx. 3.1429) and 3$10⁄71$ (approx. 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle ($\pi r^2$ ).

Archimedean property
In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as the Archimedean property of real numbers.

Archimedes gives the value of the square root of 3 as lying between $265⁄153$ (approximately 1.7320261) and $1351⁄780$ (approximately 1.7320512) in Measurement of a Circle. The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of how he had obtained it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results." It is possible that he used an iterative procedure to calculate these values.

The infinite series
In Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is $4⁄3$ times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio $1⁄4$:

 * $$\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. \;$$

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 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to $1⁄3$.

Myriad of myriads
In The Sand Reckoner, Archimedes set out to calculate a number that was greater than the grains of sand needed to fill the universe. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote:"There are some, King Gelo, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited."To solve the problem, Archimedes devised a system of counting based on the myriad. The word itself derives from the Greek μυριάς, for the number 10,000. 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.

Writings
The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse. Many written works by Archimedes have not survived or are only extant in heavily edited fragments; at least seven of his treatises are known to have existed due to references made by other authors. Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the now-lost Catoptrica.

Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were first collected by the Byzantine Greek architect Isidore of Miletus (c. 530 AD), while commentaries on the works of Archimedes written by Eutocius in the sixth century AD helped to bring his work a wider audience. 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).

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.

Surviving works
The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).

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 ($\pi$), showing that it is greater than $223⁄71$ (3.1408...) and less than $22⁄7$ (3.1428...).

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. 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.

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:

"Magnitudes are in equilibrium at distances reciprocally proportional to their weights."

Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas.

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 in one of his proofs by calculating the value of a geometric series that sums to infinity with the ratio 1/4.

On the Sphere and Cylinder
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 $4⁄3$π$r$3 for the sphere, and 2π$r$3 for the cylinder. The surface area is 4π$r$2 for the sphere, and 6π$r$2 for the cylinder (including its two bases), where $r$ is the radius of the sphere and cylinder.

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 ($r$, $θ$), it can be described by the equation $$\, r=a+b\theta$$ with real numbers $a$ and $b$.

This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.

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.

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 self-gravitating 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.

Ostomachion
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 (στόμαχος). Ausonius calls the puzzle Ostomachion, a Greek compound word formed from the roots of osteon (ὀστέον) and machē (μάχη).

The cattle problem
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. It is addressed to Eratosthenes and the mathematicians in Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations. 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 problem in 1880, and the answer is a very large number, approximately 7.760271.

The Method of Mechanical Theorems
This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses indivisibles, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. He may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.

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.

It has also been claimed that the formula for calculating the area of a triangle from the length of its sides was known to Archimedes, though its first appearance is in the work of Heron of Alexandria in the 1st century AD. 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.

Archimedes Palimpsest

The foremost document containing Archimedes' work is the Archimedes Palimpsest. 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 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.

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 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 X-ray light to read the overwritten text. It has since returned to its anonymous owner.

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

Legacy
Sometimes called the father of mathematics and mathematical physics, Archimedes had a wide influence on mathematics and science.

Mathematics and physics
Historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity. Eric Temple Bell, for instance, wrote: "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."

Likewise, Alfred North Whitehead and George F. Simmons said of Archimedes: "... in the year 1500 Europe knew less than Archimedes who died in the year 212 BC ...""If we consider what all other men accomplished in mathematics and physics, on every continent and in every civilization, from the beginning of time down to the seventeenth century in Western Europe, the achievements of Archimedes outweighs it all. He was a great civilization all by himself." Reviel Netz, Suppes Professor in Greek Mathematics and Astronomy at Stanford University and an expert in Archimedes notes: "And so, since Archimedes led more than anyone else to the formation of the calculus and since he was the pioneer of the application of mathematics to the physical world, it turns out that Western science is but a series of footnotes to Archimedes. Thus, it turns out that Archimedes is the most important scientist who ever lived."

Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes. Galileo called him "superhuman" and "my master", while Huygens said, "I think Archimedes is comparable to no one", consciously emulating him in his early work. Leibniz said, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times". Gauss's heroes were Archimedes and Newton, 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".

The inventor Nikola Tesla praised him, saying:

"Archimedes was my ideal. I admired the works of artists, but to my mind, they were only shadows and semblances. The inventor, I thought, gives to the world creations which are palpable, which live and work."

Honors and commemorations
There is a crater on the Moon named Archimedes (29.7°N, -4°W) in his honor, as well as a lunar mountain range, the Montes Archimedes (25.3°N, -4.6°W).

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").

Archimedes has 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.

Concepts

 * Arbelos
 * Archimedean point
 * Archimedes' axiom
 * Archimedes number
 * Archimedes paradox
 * Archimedean solid
 * Archimedes' twin circles
 * Methods of computing square roots
 * Salinon
 * Steam cannon

People

 * Diocles
 * Pseudo-Archimedes
 * Zhang Heng

title: Archimedes


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Alternative medicine

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.

Definitions and terminology
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. 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).

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.

Complementary or integrative medicine
Complementary medicine (CM) or integrative medicine (IM) is when alternative medicine is used together with mainstream functional 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" 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."

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.

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.

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."

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.

Unscientific belief systems
Alternative medicine, such as using naturopathy or homeopathy in place of conventional medicine, is based on belief systems not grounded in science.

Traditional ethnic systems
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.

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.

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.

NCCIH classification
The US agency 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.

 * 1) Whole medical systems: Cut across more than one of the other groups; examples include traditional Chinese medicine, naturopathy, homeopathy, and ayurveda.
 * 2) 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.
 * 3) "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.)
 * 4) Manipulative and body-based practices: feature manipulation or movement of body parts, such as is done in bodywork, chiropractic, and osteopathic manipulation.
 * 5) Energy medicine: is a domain that deals with putative and verifiable energy fields:
 * 6) * 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.
 * 7) * Bioelectromagnetic-based therapies use verifiable electromagnetic fields, such as pulsed fields, alternating-current, or direct-current fields in a non-scientific manner.

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".

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).

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).

Efficacy
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:

"Contrary to much popular and scientific writing, many alternative cancer treatments have been investigated in good-quality clinical trials, and they have been shown to be ineffective. The label 'unproven' is inappropriate for such therapies; it is time to assert that many alternative cancer therapies have been 'disproven'."

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. [[Image:Altmed function 2.svg|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.]]

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.

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.

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.

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."

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:

"Why is it so popular, then? Ernst blames the providers, customers and the doctors whose neglect, he says, has created the opening into which alternative therapists have stepped. 'People are told lies. There are 40 million websites and 39.9 million tell lies, sometimes outrageous lies. They mislead cancer patients, who are encouraged not only to pay their last penny but to be treated with something that shortens their lives.' At the same time, people are gullible. It needs gullibility for the industry to succeed. It doesn't make me popular with the public, but it's the truth."

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.

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.

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%.

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.

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.

In palliative care
Complementary therapies are often used in palliative care or by practitioners attempting to manage chronic pain in patients. Integrative medicine is considered more acceptable in the interdisciplinary approach used in palliative care than in other areas of medicine. "From its early experiences of care for the dying, palliative care took for granted the necessity of placing patient values and lifestyle habits at the core of any design and delivery of quality care at the end of life. If the patient desired complementary therapies, and as long as such treatments provided additional support and did not endanger the patient, they were considered acceptable." The non-pharmacologic interventions of complementary medicine can employ mind-body interventions designed to "reduce pain and concomitant mood disturbance and increase quality of life."

Regulation

The alternative medicine lobby has successfully pushed for alternative therapies to be subject to far less regulation than conventional medicine. Some professions of complementary/traditional/alternative medicine, such as chiropractic, have achieved full regulation in North America and other parts of the world and are regulated in a manner similar to that governing science-based medicine. In contrast, other approaches may be partially recognized and others have no regulation at all. In some cases, promotion of alternative therapies is allowed when there is demonstrably no effect, only a tradition of use. Despite laws making it illegal to market or promote alternative therapies for use in cancer treatment, many practitioners promote them.

Regulation and licensing of alternative medicine ranges widely from country to country, and state to state. In Austria and Germany complementary and alternative medicine is mainly in the hands of doctors with MDs, and half or more of the American alternative practitioners are licensed MDs. In Germany herbs are tightly regulated: half are prescribed by doctors and covered by health insurance.

Government bodies in the US and elsewhere have published information or guidance about alternative medicine. The U.S. Food and Drug Administration (FDA), has issued online warnings for consumers about medication health fraud. This includes a section on Alternative Medicine Fraud, such as a warning that Ayurvedic products generally have not been approved by the FDA before marketing.

Risks and problems
The National Science Foundation has studied the problematic side of the public's attitudes and understandings of science fiction, pseudoscience, and belief in alternative medicine. They use a quote from Robert L. Park to describe some issues with alternative medicine:

"Alternative medicine is another concern. As used here, alternative medicine refers to all treatments that have not been proven effective using scientific methods. A scientist's view of the situation appeared in a recent book (Park 2000b)':

Between homeopathy and herbal therapy lies a bewildering array of untested and unregulated treatments, all labeled alternative by their proponents. Alternative seems to define a culture rather than a field of medicine—a culture that is not scientifically demanding. It is a culture in which ancient traditions are given more weight than biological science, and anecdotes are preferred over clinical trials. Alternative therapies steadfastly resist change, often for centuries or even millennia, unaffected by scientific advances in the understanding of physiology or disease. Incredible explanations invoking modern physics are sometimes offered for how alternative therapies might work, but there seems to be little interest in testing these speculations scientifically."

Negative outcomes
According to the Institute of Medicine, use of alternative medical techniques may result in several types of harm:
 * "Direct harm, which results in adverse patient outcome."
 * "Economic harm, which results in monetary loss but presents no health hazard;"
 * "Indirect harm, which results in a delay of appropriate treatment, or in unreasonable expectations that discourage patients and their families from accepting and dealing effectively with their medical conditions;"

Interactions with conventional pharmaceuticals
Forms of alternative medicine that are biologically active can be dangerous even when used in conjunction with conventional medicine. Examples include immuno-augmentation therapy, shark cartilage, bioresonance therapy, oxygen and ozone therapies, and insulin potentiation therapy. Some herbal remedies can cause dangerous interactions with chemotherapy drugs, radiation therapy, or anesthetics during surgery, among other problems. An example of these dangers was reported by Associate Professor Alastair MacLennan of Adelaide University, Australia regarding a patient who almost bled to death on the operating table after neglecting to mention that she had been taking "natural" potions to "build up her strength" before the operation, including a powerful anticoagulant that nearly caused her death.

To ABC Online, MacLennan also gives another possible mechanism:

"And lastly there's the cynicism and disappointment and depression that some patients get from going on from one alternative medicine to the next, and they find after three months the placebo effect wears off, and they're disappointed and they move on to the next one, and they're disappointed and disillusioned, and that can create depression and make the eventual treatment of the patient with anything effective difficult, because you may not get compliance, because they've seen the failure so often in the past."

Side-effects
Conventional treatments are subjected to testing for undesired side-effects, whereas alternative therapies, in general, are not subjected to such testing at all. Any treatment – whether conventional or alternative – that has a biological or psychological effect on a patient may also have potential to possess dangerous biological or psychological side-effects. Attempts to refute this fact with regard to alternative therapies sometimes use the appeal to nature fallacy, i.e., "That which is natural cannot be harmful." Specific groups of patients such as patients with impaired hepatic or renal function are more susceptible to side effects of alternative remedies.

An exception to the normal thinking regarding side-effects is homeopathy. Since 1938, the FDA has regulated homeopathic products in "several significantly different ways from other drugs." Homeopathic preparations, termed "remedies", are extremely dilute, often far beyond the point where a single molecule of the original active (and possibly toxic) ingredient is likely to remain. They are, thus, considered safe on that count, but "their products are exempt from good manufacturing practice requirements related to expiration dating and from finished product testing for identity and strength", and their alcohol concentration may be much higher than allowed in conventional drugs.

Treatment delay
Alternative medicine may discourage people from getting the best possible treatment. Those having experienced or perceived success with one alternative therapy for a minor ailment may be convinced of its efficacy and persuaded to extrapolate that success to some other alternative therapy for a more serious, possibly life-threatening illness. For this reason, critics argue that therapies that rely on the placebo effect to define success are very dangerous. According to mental health journalist Scott Lilienfeld in 2002, "unvalidated or scientifically unsupported mental health practices can lead individuals to forgo effective treatments" and refers to this as opportunity cost. Individuals who spend large amounts of time and money on ineffective treatments may be left with precious little of either, and may forfeit the opportunity to obtain treatments that could be more helpful. In short, even innocuous treatments can indirectly produce negative outcomes. Between 2001 and 2003, four children died in Australia because their parents chose ineffective naturopathic, homeopathic, or other alternative medicines and diets rather than conventional therapies.

Unconventional cancer "cures"
There have always been "many therapies offered outside of conventional cancer treatment centers and based on theories not found in biomedicine. These alternative cancer cures have often been described as 'unproven,' suggesting that appropriate clinical trials have not been conducted and that the therapeutic value of the treatment is unknown." However, "many alternative cancer treatments have been investigated in good-quality clinical trials, and they have been shown to be ineffective.... The label 'unproven' is inappropriate for such therapies; it is time to assert that many alternative cancer therapies have been 'disproven'."

Edzard Ernst has stated:

"any alternative cancer cure is bogus by definition. There will never be an alternative cancer cure. Why? Because if something looked halfway promising, then mainstream oncology would scrutinize it, and if there is anything to it, it would become mainstream almost automatically and very quickly. All curative 'alternative cancer cures' are based on false claims, are bogus, and, I would say, even criminal."

Rejection of science
Complementary and alternative medicine (CAM) is not as well researched as conventional medicine, which undergoes intense research before release to the public. Practitioners of science-based medicine also discard practices and treatments when they are shown ineffective, while alternative practitioners do not. Funding for research is also sparse making it difficult to do further research for effectiveness of CAM. Most funding for CAM is funded by government agencies. Proposed research for CAM are rejected by most private funding agencies because the results of research are not reliable. The research for CAM has to meet certain standards from research ethics committees, which most CAM researchers find almost impossible to meet. Even with the little research done on it, CAM has not been proven to be effective. Studies that have been done will be cited by CAM practitioners in an attempt to claim a basis in science. These studies tend to have a variety of problems, such as small samples, various biases, poor research design, lack of controls, negative results, etc. Even those with positive results can be better explained as resulting in false positives due to bias and noisy data.

Alternative medicine may lead to a false understanding of the body and of the process of science. Steven Novella, a neurologist at Yale School of Medicine, wrote that government-funded studies of integrating alternative medicine techniques into the mainstream are "used to lend an appearance of legitimacy to treatments that are not legitimate." Marcia Angell considered that critics felt that healthcare practices should be classified based solely on scientific evidence, and if a treatment had been rigorously tested and found safe and effective, science-based medicine will adopt it regardless of whether it was considered "alternative" to begin with. It is possible for a method to change categories (proven vs. unproven), based on increased knowledge of its effectiveness or lack thereof. Prominent supporters of this position are George D. Lundberg, former editor of the Journal of the American Medical Association (JAMA) and the journal's interim editor-in-chief Phil Fontanarosa.

Writing in 1999 in CA: A Cancer Journal for Clinicians Barrie R. Cassileth mentioned a 1997 letter to the United States Senate's Subcommittee on Public Health and Safety, which had deplored the lack of critical thinking and scientific rigor in OAM-supported research, had been signed by four Nobel Laureates and other prominent scientists. (This was supported by the National Institutes of Health (NIH).)

In March 2009, a staff writer for The Washington Post reported that the impending national discussion about broadening access to health care, improving medical practice and saving money was giving a group of scientists an opening to propose shutting down the National Center for Complementary and Alternative Medicine. They quoted one of these scientists, Steven Salzberg, a genome researcher and computational biologist at the University of Maryland, as saying "One of our concerns is that NIH is funding pseudoscience." They noted that the vast majority of studies were based on fundamental misunderstandings of physiology and disease, and had shown little or no effect.

Writers such as Carl Sagan, a noted astrophysicist, advocate of scientific skepticism and the author of The Demon-Haunted World: Science as a Candle in the Dark (1996), have lambasted the lack of empirical evidence to support the existence of the putative energy fields on which these therapies are predicated.

Sampson has also pointed out that CAM tolerated contradiction without thorough reason and experiment. Barrett has pointed out that there is a policy at the NIH of never saying something does not work, only that a different version or dose might give different results. Barrett also expressed concern that, just because some "alternatives" have merit, there is the impression that the rest deserve equal consideration and respect even though most are worthless, since they are all classified under the one heading of alternative medicine.

Some critics of alternative medicine are focused upon health fraud, misinformation, and quackery as public health problems, notably Wallace Sampson and Paul Kurtz founders of Scientific Review of Alternative Medicine and Stephen Barrett, co-founder of The National Council Against Health Fraud and webmaster of Quackwatch. Grounds for opposing alternative medicine include that:
 * Alternative therapies typically lack any scientific validation, and their effectiveness either is unproven or has been disproved.
 * It is usually based on religion, tradition, superstition, belief in supernatural energies, pseudoscience, errors in reasoning, propaganda, or fraud.
 * Methods may incorporate or base themselves on traditional medicine, folk knowledge, spiritual beliefs, ignorance or misunderstanding of scientific principles, errors in reasoning, or newly conceived approaches claiming to heal.
 * Research on alternative medicine is frequently of low quality and methodologically flawed.
 * Treatments are not part of the conventional, science-based healthcare system.
 * Where alternative therapies have replaced conventional science-based medicine, even with the safest alternative medicines, failure to use or delay in using conventional science-based medicine has caused deaths.

Many alternative medical treatments are not patentable, which may lead to less research funding from the private sector. In addition, in most countries, alternative therapies (in contrast to pharmaceuticals) can be marketed without any proof of efficacy – also a disincentive for manufacturers to fund scientific research.

English evolutionary biologist Richard Dawkins, in his 2003 book A Devil's Chaplain, defined alternative medicine as a "set of practices that cannot be tested, refuse to be tested, or consistently fail tests." Dawkins argued that if a technique is demonstrated effective in properly performed trials then it ceases to be alternative and simply becomes medicine.

CAM is also often less regulated than conventional medicine. There are ethical concerns about whether people who perform CAM have the proper knowledge to treat patients. CAM is often done by non-physicians who do not operate with the same medical licensing laws which govern conventional medicine, and it is often described as an issue of non-maleficence.

According to two writers, Wallace Sampson and K. Butler, marketing is part of the training required in alternative medicine, and propaganda methods in alternative medicine have been traced back to those used by Hitler and Goebels in their promotion of pseudoscience in medicine.

In November 2011 Edzard Ernst stated that the "level of misinformation about alternative medicine has now reached the point where it has become dangerous and unethical. So far, alternative medicine has remained an ethics-free zone. It is time to change this."

Harriet Hall criticized the low standard of evidence accepted by the alternative medicine community:

"Science-based medicine has one rigorous standard of evidence, the kind [used for pharmaceuticals] .... CAM has a double standard. They gladly accept a lower standard of evidence for treatments they believe in. However, I suspect they would reject a pharmaceutical if it were approved for marketing on the kind of evidence they accept for CAM."

Conflicts of interest
Some commentators have said that special consideration must be given to the issue of conflicts of interest in alternative medicine. Edzard Ernst has said that most researchers into alternative medicine are at risk of "unidirectional bias" because of a generally uncritical belief in their chosen subject. Ernst cites as evidence the phenomenon whereby 100% of a sample of acupuncture trials originating in China had positive conclusions. David Gorski contrasts evidence-based medicine, in which researchers try to disprove hyphotheses, with what he says is the frequent practice in pseudoscience-based research, of striving to confirm pre-existing notions. Harriet Hall writes that there is a contrast between the circumstances of alternative medicine practitioners and disinterested scientists: in the case of acupuncture, for example, an acupuncturist would have "a great deal to lose" if acupuncture were rejected by research; but the disinterested skeptic would not lose anything if its effects were confirmed; rather their change of mind would enhance their skeptical credentials.

Use of health and research resources
Research into alternative therapies has been criticized for "diverting research time, money, and other resources from more fruitful lines of investigation in order to pursue a theory that has no basis in biology." Research methods expert and author of Snake Oil Science, R. Barker Bausell, has stated that "it's become politically correct to investigate nonsense." A commonly cited statistic is that the US National Institute of Health had spent $2.5 billion on investigating alternative therapies prior to 2009, with none being found to be effective.

World Health Organization

 * Benchmarks for training in traditional / complementary and alternative medicine
 * Summary.

Journals

 * Alternative Medicine Review: A Journal of Clinical Therapeutics. Sandpoint, Idaho : Thorne Research, c. 1996 NLM ID: 9705340
 * Alternative Therapies in Health and Medicine. Aliso Viejo, California : InnoVision Communications, c1995- NLM ID: 9502013
 * BMC Complementary and Alternative Medicine . London: BioMed Central, 2001 NLM ID: 101088661
 * Complementary Therapies in Medicine. Edinburgh; New York : Churchill Livingstone, c. 1993 NLM ID: 9308777
 * Evidence Based Complementary and Alternative Medicine: eCAM. New York: Hindawi, c. 2004 NLM ID: 101215021
 * Forschende Komplementärmedizin / Research in Complementary Medicine
 * Journal for Alternative and Complementary Medicine New York : Mary Ann Liebert, c. 1995
 * Scientific Review of Alternative Medicine (SRAM)

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