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History Of Calculus

History of calculus

See also History of mathematics. Though the origins of integral calculus are generally regarded as going no farther back than to the ancient Greeks, there is evidence that the ancient Egyptians may have harbored such knowledge amongst themselves as well (see Moscow and Rhind Mathematical Papyri). Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See [http://mathpages.com/home/kmath343.htm Archimedes on Spheres & Cylinders].) Thus, Archimedes and others after used integral methods throughout history. Indian mathematician Bhaskara (1114-1185) was the first to conceive of differential calculus, using the "derivative" and inventing the basic idea of what is now known as Rolle's theorem. The 14th century Indian mathematician Madhava of Sangamagrama, along with other mathematicians of the Kerala school studied infinite series, convergence, differentiation, and iterative methods for solution of non-linear equations. Jyestadeva of the Kerala school wrote the first calculus text, the Yuktibhasa, which explores methods and ideas of calculus that were only repeated in seventeenth century Europe. In 17th century Europe, Isaac Barrow, Pierre de Fermat, Blaise Pascal, John Wallis and others are said to have discussed the idea of a derivative. René Descartes introduced the foundation for the methods of analytic geometry in 1637, providing the foundation for calculus later introduced by Isaac Newton and Gottfried Leibniz, independently of each other. Leibniz and Newton are usually credited with the invention, in the late 1600s, of differential and integral calculus as we know it today, but mainly developed the fundamental theorem of calculus and worked on notation. Lesser credit for the development of calculus is given to Barrow, Descartes, de Fermat, Huygens and Wallis. A Japanese mathematician, Kowa Seki, lived at the same time as Leibniz and Newton and also elaborated some of the fundamental principles of integral calculus independently. [http://www2.gol.com/users/coynerhm/0598rothman.html]

Invention of Calculus

Kowa Seki] Kowa Seki] Many of the results of Newton and Leibniz were known to mathematicians in Kerala, India almost 300 years previously. In 1835, Charles Whish published an article in the Transactions of the Royal Asiatic Society of Great Britain and Ireland, in which he claimed that the work of the Kerala school "laid the foundation for a complete system of fluxions." It was not until the 1940s however, that historians of mathematics verified Whish's claims, but their work is still underplayed in modern accounts of history of calculus, which holds Leibniz and Newton as its inventors. There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of the calculus in Europe. The truth of the matter is that the ideas of calculus were a part of the mathematical knowledge of their day, and they independently put those pieces together in different but coherent ways. Leibniz' notation of his calculus is clear and easy to understand, indicating he had a thorough understanding of the principles underlying calculus. notation, . Newton's terminology and notation was less flexible, and awkward to use compared to Leibniz's, yet it remained in British usage until the early 19th century, when the work of the Analytical Society successfully saw the introduction of Leibniz's notation in Great Britain. It is now generally thought that Newton had discovered several ideas related to calculus earlier than Leibniz had. However, Leibniz was the first to publish in Europe. Today, modern history credits both Leibniz and Newton as having discovered calculus independently. Newton provided a host of applications in physics, and his notation

\dot(x)

for the derivative of f with respect to x is still used in physics today, especially for derivatives with respect to time. Outside of physics it has mostly been displaced by the notation f'(x) for the derivative of f with respect to x. Also current is Leibniz's more flexible differential notation df/dx, again for the derivative of f with respect to x. Leibniz's notation is especially popular in the many situations when writing only f' would be ambiguous. Leibniz based his work on the concept of infinitesimals, as opposed to the fluxion of Isaac Newton, which is based upon the concept of the limit. Because infinitesimals were not put on a rigorous mathematical basis until the second half of the twentieth century, the delta-epsilon definition of limits and calculus became standard.

Controversy (Newton, Leibnitz... or Madhava?)

Madhava of Sangamagrama and the Kerala school were the first to come up with the important ideas of calculus in the 14th century and [http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_4.html some] [http://www.canisius.edu/topos/rajeev.asp ] propose these ideas may have been transmitted to Europe by the 17th century. There is no evidence by way of relevant manuscripts but the evidence of methodological similarities, communication routes and a suitable chronology for transmission is hard to dismiss. In the controversy between Newton and Leibniz, suggestions were made that the work of Leibniz was not independent, as he claimed, but influenced by reading copies of Newton's early manuscripts. That the Leibniz notation was original was common ground. There is evidence to show that Newton commenced work on the calculus about a decade before Leibniz did in 1676. Newton's work Method of Fluxions is presumed to be based on work carried out 1665-7, but it was not published until much later. Leibniz was in England in 1673 and again in 1676, and on the latter occasion did see some of Newton's manuscripts. In 1704 though, an anonymous pamphlet, later determined to have been written by Leibniz, accused Newton of having plagiarised Leibniz' work. Leibniz and Newton discussed their work on occasions when they met and by correspondence. It is quite possible that Newton got the ideas of the calculus from Leibniz. When comparing Leibniz's calculus with Newton's fluxion method, it is evident that Leibniz's terminology and notation indicate a depth of understanding of the principles of calculus that is absent in the fluxions method. A copy of one of Newton's very early manuscripts with annotations by Leibniz was found among Leibniz' papers after his death; the date when Leibniz first acquired this is unknown; it could have been after Leibniz had published his calculus. It is often stated that the controversy isolated English-speaking mathematicians from those in continental Europe for many years; and that this set back British analysis (i.e. calculus-based mathematics) for a very long time. Newton's terminology and notation was less flexible than that of Leibniz, yet it was retained in British university teaching usage until the early 19th century. At that point the Analytical Society successfully lobbied for the introduction of Leibniz's notation in Great Britain.

Rigorous foundations

Calculus was widely used, as it was a very powerful mathematical tool, but it was not until the mid-1800s that it was put on a rigorous foundation. For example, while the definition of the derivative itself has not changed since it was first introduced, it requires the notion of a limit. Newton, Leibniz, and their immediate successors interpreted limits intuitively instead of through precise definitions. This was standard practice at the time. Later, with the work of mathematicians like Augustin Louis Cauchy, Bernard Bolzano, and Karl Weierstrass, the foundations of calculus were clarified and made precise. The study of foundations eventually resulted in deep explorations of the concept of infinity by Georg Cantor and others.

Integrals

Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential expressions can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. Frullani's theorem (1821), Bierens de Haan's work on the theory (1862) and his elaborate tables (1867), Dirichlet's lectures (1858) embodied in Meyer's treatise (1871), and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Schlömilch, Elliott, Leudesdorf, and Kronecker are among the noteworthy contributions. Eulerian integrals were first studied by Euler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows: :

\int_0^1 x^(1 - x)^dx

\int_0^\infty e^ x^dx

although these were not the exact forms of Euler's study. If n is an integer, it follows that

\int_0^\infty e^x^dx = (n-1)!

but if n is fractional it is a transcendent function. To it Legendre assigned the symbol

\Gamma

, and it is now called the gamma function. To the subject Dirichlet has contributed an important theorem Liouville, 1839), which has been elaborated by Liouville, Catalan, Leslie Ellis, and others. On the evaluation of

\Gamma x

and

\log \Gamma x

Raabe (1843-44), Bauer (1859), and Gudermann (1845) have written. Legendre's great table appeared in 1816.

Symbolic methods

Symbolic methods may be traced back to Taylor, and the analogy between successive differentiation and ordinary exponentials had been observed by numerous writers before the nineteenth century. Arbogast (1800) was the first, however, to separate the symbol of operation from that of quantity in a differential equation. François (1812) and Servois (1814) seem to have been the first to give correct rules on the subject. Hargreave (1848) applied these methods in his memoir on differential equations, and Boole freely employed them. Grassmann and Hermann Hankel made great use of the theory, the former in studying equations, the latter in his theory of complex numbers.

Calculus of variations

The calculus of variations may be said to begin with a problem of Johann Bernoulli's (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Euler first elaborated the subject. His contributions began in 1733, and his Elementa Calculi Variationum gave to the science its name. Lagrange contributed extensively to the theory, and Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. To this discrimination Brunacci (1810), Gauss (1829), Poisson (1831), Ostrogradsky (1834), and Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Hesse (1857), Clebsch (1858), and Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation.

Applications

The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the eighteenth century these applications were multiplied, until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of analysis. To Lagrange (1773) we owe the introduction of the theory of the potential into dynamics, although the name "potential function" and the fundamental memoir of the subject are due to Green (1827, printed in 1828). The name "potential" is due to Gauss (1840), and the distinction between potential and potential function to Clausius. With its development are connected the names of Dirichlet, Riemann, Neumann, Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami, and many of the leading physicists of the century. It is impossible in this place to enter into the great variety of other applications of analysis to physical problems. Among them are the investigations of Euler on vibrating chords; Sophie Germain on elastic membranes; Poisson, Lamé, Saint-Venant, and Clebsch on the elasticity of three-dimensional bodies; Fourier on heat diffusion; Fresnel on light; Maxwell, Helmholtz, and Hertz on electricity; Hansen, Hill, and Gyldén on astronomy; Maxwell on spherical harmonics; Lord Rayleigh on acoustics; and the contributions of Dirichlet, Weber, Kirchhoff, F. Neumann, Lord Kelvin, Clausius, Bjerknes, MacCullagh, and Fuhrmann to physics in general. The labors of Helmholtz should be especially mentioned, since he contributed to the theories of dynamics, electricity, etc., and brought his great analytical powers to bear on the fundamental axioms of mechanics as well as on those of pure mathematics. Category:Calculus Category:History of mathematics

History of mathematics

:See Timeline of mathematics for a timeline of events in mathematics. See list of mathematicians for a list of biographies of mathematicians. The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning". Today, the term refers to a specific body of knowledge - the rigorous, deductive study of numbers, shapes, patterns, and change. Every modern science depends on basic mathematics at the most fundamental level, including such operations as counting, addition and subtraction.

Fundamentals

In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. With counting established, then, the ideas of addition and subtraction naturally followed. See arithmetic. Mathematics undoubtedly could not have developed out of simple counting and arithmetic, however, without writing. Perhaps prehistoric peoples first expressed quantity by scratching lines on ground, rock or wood (see Numeral system: History). For example, paleontologists have discovered ochre rocks in a southern cave of South Africa adorned with scratched geometric patterns and dating to over 70,000 years ago [http://www.accessexcellence.org/WN/SU/caveart.html]. Also prehistoric artifacts discovered in Africa and France from between 35000 BC and 20000 BC indicate early attempts to quantify time (references: [http://www.tacomacc.edu/home/jkellerm/Papers/Menses/Menses.htm], [http://www.math.buffalo.edu/mad/Ancient-Africa/lebombo.html], [http://www.math.buffalo.edu/mad/Ancient-Africa/ishango.html]). Mathematics developed further, out of simple writing, with the development of pigments and paint. Predynastic Egyptians of the 5th millennium BC are the earliest known to have pictorially represented geometric spatial designs. Pigments and paints served other purposes in the historical development of mathematics. Pre-historic art and other early human inventions eventually led to
- Maps for expressing distance,
- Geometric shapes and other 3-dimensional figures for expressing basic form, and
- Graphs for expressing relations. Mathematics developed further, out of simple writing, with the development of other simple tools to record and communicate "quantity" among individuals and over periods of time. The Sumerians of the 4th millennium BC are the earliest known to have used numbers for complex calculations, using a base-60 mathematical system.

Developing the concept of "number" through equations

Many of the extensions of the concept of number can be seen as responses to equations that would otherwise have had no solution. In each of the extensions given below we start with an equation and then give the extension to the system which allows the equation to be solved. We start with the notion of natural numbers: positive integers and zero, although it should be noted that some ancient mathematics did not have the concept of zero. Also note that it was assumed that the normal algebraic operations

+\ -\ \times \ /

return only one value (division by zero is not defined).
-

5 \times X=3

requires the existence of fractional numbers for its solution. If we allow the solution of all equations of the form

m \times X=n

then we get the rational numbers (m and n are both integers).
-

X \times X-2=0

has no rational solution. Mathematicians responded by introducing radicals and real numbers, which allowed many polynomial equations to be solved.
-

X+1=0

requires the existence of negative numbers such as −1 for its solution.
-

X \times X+1=0

is the equation that introduces us to the complex numbers, which are discussed below.

Complex numbers

When the complex numbers were introduced, there were many who argued that they were imaginary constructs to solve the cubic, and that they should not be considered 'real'. This is the origin of the terms imaginary and real for the numbers. However, mathematicians found the new world of complex numbers to be elegant and compelling. To represent a solution to the equation shown above (i.e.,

X - X+1=0

) mathematicians eventually settled on the letter i. However, in the early 19th century, one further extension of the real and complex numbers was found. All of the numbers described above are algebraic; but Liouville showed how to construct transcendental numbers, which could not be expressed as the roots of any algebraic equation. In order to construct these transcendental numbers one needs the "completeness axiom": Any set of real numbers with an upper bound has a least upper bound. This fills in the real line with all of the irrational numbers that cannot be derived merely from algebraic equations. It is worth noting that this is an entirely different type of extension. This is because of the cardinality of real or complex numbers is greater than that of the rationals. The Fundamental Theorem of Algebra shows that all polynomial equations over the complex numbers can be solved; thus there is no need for any further extension on algebraic grounds-nevertheless, many further extensions of the complex numbers do exist, such as the quaternions, or the surreal numbers. Mathematicians today rarely view the development of the complex numbers in this way (the preferred teaching method does not emphasize this stepwise development) but it demonstrates the tension in mathematics between the rigorous and the creative which is the main power behind much of modern mathematics.

Indian contributions

See: Indian mathematics Between 1500 BC and 1600 CE various treatises on mathematics were authored by Indian mathematicians in which were set forth for the first time, the concepts of zero, infinity, negative numbers, irrational numbers and binary numbers, the techniques of algebra and algorithm, square root and cube root, along with the methods of transformations, recursions, combinations and permutations, and the introductions of geometry, trigonometry, logarithms, quadratic equations, cubic equations, mathematical analysis and calculus, aswell as the discoveries of Pascal's triangle, Fibonacci numbers, Pythagorean theorem and Pythagorean triples. Vedic mathematics, as it is referred to today, is a separate field of study and courses are offered even in non-Indian universities. The concept of zero seems to have been a contribution of ancient Indian thought. Every ancient Indian language has multiple words to refer to the concept of 'void' or 'nothing' - 'Shunya' in Sanskrit. This word is used in early Sanskrit texts of the 4th century BC; the concept of zero is clearly explained in Pingala’s Sutra of the 2nd century BC. In Brahma-Phuta-Siddhanta of Brahmagupta (7th century), zero is lucidly explained; it was from a translation of this Indian text on mathematics (around 770 CE) that the Arab mathematicians perfected the decimal system and gave the world its current system of enumeration which we call the Hindu-Arabic numerals. From the Arabs the concept of zero was carried to Europe in the 8th century. Aryabhata in 499 CE computed the value of Pi to the fourth decimal place as 3.1416, while Madhava in the 14th century computed the value of Pi to the eleventh decimal place as 3.14159265359.

Miscellaneous historical notes

The Maya calendar utilized a base-20 number system which included the 'number' zero (also see Maya numerals). In China, Zu Chongzhi (5th century) of the Southern and Northern Dynasties was the first person to calculate the value of Pi to seven decimal places. The Mesopotamian cuneiform tablet Plimpton 232 records a number of Pythagorean triplets (3,4,5) (5,12,13). ..., dated 1900 BC, possibly millennia before Pythagoras.

External links

- [http://www-groups.dcs.st-andrews.ac.uk/~history/ The MacTutor History of Mathematics Archive] created by John J O'Connor and Edmund F Robertson, which contains biographies, timelines and historical articles about mathematical concepts.
- [http://members.aol.com/jeff570/mathsym.html Earliest uses of various mathematical symbols] by Jeff Miller
- [http://members.aol.com/jeff570/mathword.html Earliest known uses of some of the words of mathematics] by Jeff Miller
- [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/index.html History of Indian mathematics] by Ian Pearce
- [http://www.jewishencyclopedia.com/view.jsp?artid=259&letter=M&search=mathematics History of Mathematics, public domain article]
- Important publications in the history of mathematics
- [http://www.dean.usma.edu/math/people/rickey/hm/default.htm History of calculus] by Fred Rickey

References

- Boyer, C. B.: A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7)
- Hoffman, Paul, The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998 ISBN 0-7868-6362-5

Notes

Hoffman, p.187.
- ko:수학의 역사

Ancient Egypt

Ancient Egypt was a civilization along the Lower Nile extending from as far south as Jebel Barkal, Napata [http://www.newadvent.org/cathen/05329b.htm], and then northward to the Mediterranean Sea, though varying in size throughout its history between circa 3200 BC and 343 BC, ending with the conquest of Alexander the Great. As a civilization based on irrigation it is the quintessential example of a hydraulic empire.

Geography

Most of Egypt is in North Africa; though the Sinai Peninsula is in Southwest Asia. The country has shorelines on the Mediterranean Sea and the Red Sea; it borders Libya to the west, Sudan to the south, and the Gaza Strip, Palestine and Israel to the east. Ancient Egypt was divided into two kingdoms, known as Upper and Lower Egypt. Somewhat counter-intuitively, Upper Egypt was in the south and Lower Egypt in the north, named according to the flow of the Nile. The Nile river flows northward from a southerly point to the Mediterranean rather than southward from a northerly point. The Nile river, around which much of the population of the country clusters, has been the lifeline for Egyptian culture since the Stone Age and Naqada cultures. Two kingdoms formed Kemet ("the black land", in Ancient Egyptian Kmt), the name for the dark soil deposited by the Nile floodwaters. The desert was called Deshret ("the red land"), c.f. Herodotus: "Egypt is a land of black soil.... We know that Libya is a redder earth" (Histories, 2:12).

People

Libya]] A recent genetic study links the maternal lineage of a traditional population from Upper Egypt to Eastern Africa . A separate study further narrows the genetic lineage to Northeast Africa () and reveals also that modern day Egyptians "reflect a mixture of European, Middle Eastern, and African." Champollion the Younger, who deciphered the Rosetta Stone, claimed in Expressions et Termes Particuliers ("Expression of Particular Terms") that Kmt did not actually refer to the soil but to a negroid population in the sense of "Black Nation." Modern day professional Egyptologists, linguists and historians, however, overwhelmingly agree that the term referred to the soil rather than the people. Herodotus wrote, "the Colchians are Egyptians... on the fact that they are black-skinned (melanchrôs) and wooly-haired (oulothrix)" (Histories Book 2:104). Later authors, including Aristotle and Diodorus Siculus, repeated Herodotus' description of "black-skinned". Melanchros is also used of the sunburnt complexion of Odysseus (Od. 16.176). Although analyzing the hair of ancient Egyptian mummies from the Late Middle Kingdom has revealed evidence of a stable diet , mummies from circa 3200 BC show signs of severe anemia and hemolitic disorders . A few teams of European scientists reported that cocaine, hashish and nicotine have been found in the skin and hair of Egyptian mummies . The results of these studies have been harshly criticized (e.g., ref. ) by mainstream scientists and Egyptologists as flawed and inaccurate.

History

:Main article: History of ancient Egypt The ancient Egyptians themselves traced their origin to a land they called Punt, or "Ta Nteru" ("Land of the Gods"). Once commonly thought to be located on what is today the Somali coast, Punt now is thought to have been in either southern Sudan or Eritrea. The history of ancient Egypt proper starts with Egypt as a unified state, which occurred sometime around 3000 BC. Though archaeological evidence indicates a developed Egyptian society may have existed for a much longer period (see Predynastic Egypt). Along the Nile, in 10th millennium BC, a grain-grinding culture using the earliest type of sickle blades had been replaced by another culture of hunters, fishers, and gathering peoples using stone tools. Evidence also indicates human habitation in the southwestern corner of Egypt, near the Sudan border, before 8000 BC. Climate changes and/or overgrazing around 8000 BC began to desiccate the pastoral lands of Egypt, eventually forming the Sahara (c.2500 BC), and early tribes naturally migrated to the Nile river where they developed a settled agricultural economy and more centralized society (see Nile: History). There is evidence of pastoralism and cultivation of cereals in the East Sahara in the 7th millennium BC. By 6000 BC ancient Egyptians in the southwestern corner of Egypt were herding cattle and constructing large buildings. Mortar (masonry) was in use by 4000 BC. The Predynastic Period continues through this time, variously held to begin with the Naqada culture. Some authorities however begin the Predynastic Period earlier, in the Lower Paleolithic (see Predynastic Egypt). Egypt unified as a single state circa 3000 BC. Egyptian chronology involves assigning beginnings and endings to various dynasties beginning around this time. The conventional Egyptian chronology is the accepted developments during the 20th century, but do not include any of the major revision proposals that have also been made in that time. Even within a single work, often archeologists will offer several possible dates or even several whole chronologies as possibilities. Consequently, there may be discrepancies between dates shown here and in articles on particular rulers. Often there are also several possible spellings of the names. Typically, Egyptologists divide the history of pharaonic civilization using a schedule laid out first by Manetho's Aegyptaica.
- List of pharaohs: The pharaohs stretch from before 3000 BC to around 30 BC.
- Dynasties (see also: List of Egyptian dynasties):
- Early Dynastic Period of Egypt (1st to 2nd Dynasties; until ca. 27th century BC)
- Old Kingdom (3rd to 6th Dynasties; 27th to 22nd centuries BC)
- First Intermediate Period (7th to 11th Dynasties)
- Middle Kingdom of Egypt (11th to 14th Dynasties; 20th to 17th centuries BC)
- Second Intermediate Period (14th to 17th Dynasties)
- Hyksos (15th to 16th Dynasties)
- New Kingdom of Egypt (18th to 20th Dynasties; 16th to 11th centuries BC)
- Third Intermediate Period (21st to 25th Dynasties; 11th to 7th centuries BC)
- Late Period of Ancient Egypt (26th to 31st Dynasties; 7th century BC to 332 BC)
- Achaemenid Dynasty
- Graeco-Roman Egypt (332 BC to AD 639)
- Ptolemaic Dynasty
- Roman Empire

Government

Nomes were the subnational administrative divisions of Upper and Lower Egypt. The pharaoh was the ruler of these two kingdoms and headed the ancient Egyptian state structure. The pharaoh served as monarch, spiritual leader and commander-in-chief of both the army and navy. The pharaoh was supposed to be divine, a connection between men and gods. Below him in the government, were the viziers (one for Upper Egypt and one for Lower Egypt) and various officials. Under him on the religious side were the high priest and various other priests. Generally, the position was handed down from father to eldest son. Sometimes this rule was broken, and occasionally a woman assumed power.

Language

The ancient Egyptians spoke an Afro-Asiatic language related to Chadic, Berber and Semitic languages. Records of the ancient Egyptian language have been dated to about 3200 BC. Scholars group the Egyptian language into six major chronological divisions:
- Archaic Egyptian (before 2600 BC)
- Old Egyptian (2600–2000 BC)
- Middle Egyptian (2000–1300 BC)
- Late Egyptian (1300–700 BC)
- Demotic Egyptian (7th century BC–4th century AD)
- Coptic (3rd–12th century AD)

Writing

For many years, the earliest known hieroglyphic inscription was the Narmer Palette, found during excavations at Hierakonpolis (modern Kawm al-Ahmar) in the 1890s, which has been dated to c.3200 BC. However recent archaeological findings reveal that symbols on Gerzean pottery, c.4000 BC, resemble the traditional hieroglyph forms . Also in 1998 a German archeological team under Gunter Dreyer excavating at Abydos (modern Umm el-Qa'ab) uncovered tomb U-j, which belonged to a Predynastic ruler, and they recovered three hundred clay labels inscribed with proto-hieroglyphics dating to the Naqada IIIA period, circa 33rd century BC , . Egyptologists refer to Egyptian writing as hieroglyphs, today standing as the world's earliest known writing system. The hieroglyphic script was partly syllabic, partly ideographic. Hieratic is a cursive form of Egyptian hieroglyphs and was first used during the First Dynasty (c. 2925 BC – c. 2775 BC). The term Demotic, in the context of Egypt, came to refer to both the script and the language that followed the Late Ancient Egyptian stage, i.e. from the Nubian 25th dynasty until its marginalization by the Greek Koine in the early centuries AD. After the conquest of Umar ibn al-Khattab, the Coptic language survived into the Middle Ages as the liturgical language of the Christian minority. Beginning from around 2700 BC, Egyptians used pictograms to represent vocal sounds -- both vowel and consonant vocalizations (see Hieroglyph: Script). By 2000 BC, 26 pictograms were being used to represent 24 (known) main vocal sounds. The world's oldest known alphabet (c. 1800 BC) is only an abjad system and was derived from these uniliteral signs as well as other Egyptian hieroglyphs. The hieroplyphic script finally fell out of use around the 4th century and began to be rediscovered after the 15th century (see Hieroglyphica).

Literature

- c. 2500 BC: Westcar Papyrus
- c. 1800 BC: Story of Sinuhe
- c. 1800 BC: Ipuwer papyrus
- c. 1800 BC: Papyrus Harris I
- c. 1000 BC: Story of Wenamun

Culture

The Egyptian religions, embodied in Egyptian mythology, were the succession of beliefs held by the people of Egypt, until the coming of Christianity and Islam. These were conducted by Egyptian priests or magicians, but the use of magic and spells is questioned. The religious nature of ancient Egyptian civilization influenced its contribution to the arts of the ancient world. Many of the great works of ancient Egypt depict gods, goddesses, and pharaohs, who were also considered divine. Ancient Egyptian art in general is characterized by the idea of order. Evidence of mummies and pyramids outside ancient Egypt indicate reflections of ancient Egyptian belief values on other prehistoric cultures, transmitted in one way over the Silk Road. Some scholars have speculated that Egypt's art pieces are sexually symbolic.

Ancient achievements

symbolic See Predynastic Egypt for inventions and other significant achievements in the Sahara region before the Protodynastic Period. The art and science of engineering was present in Egypt, such as accurately determining the position of points and the distances between them (known as surveying). These skills were used to outline pyramid bases. The Egyptian pyramids took the geometric shape formed from a polygonal base and a point, called the apex, by triangular faces. Hydraulic Cement was first invented by the Egyptians. The Al Fayyum Irrigation (water works) was one of the main agricultural breadbaskets of the ancient world. There is evidence of ancient Egyptian pharaohs of the twelfth dynasty using the natural lake of the Fayyum as a reservoir to store surpluses of water for use during the dry seasons. From the time of the First dynasty or before, the Egyptians mined turquoise in Sinai Peninsula. The earliest evidence (circa 1600 BC) of traditional empiricism is credited to Egypt, as evidenced by the Edwin Smith and Ebers papyri. The roots of the Scientific method may be traced back to the ancient Egyptians. The ancient Egyptians are also credited with devising the world's earliest known alphabet, decimal system and complex mathematical formularizations, in the form of the Moscow and Rhind Mathematical Papyri. An awareness of the golden ratio seems to be reflected in many constructions, such as the Egyptian pyramids.

Timeline

(All dates are approximate.)

Predynastic

See main article and timeline: Predynastic Egypt.
- 3500 BC: Senet, world's oldest (confirmed) board game
- 3500 BC: Faience, world's earliest known earthenware

Dynastic

- 3300 BC: Bronze works (see Bronze Age)
- 3200 BC: Egyptian hieroglyphs fully developed (see First dynasty of Egypt)
- 3200 BC: Narmer Palette, world's earliest known historical document
- 3100 BC: Decimal system, , world's earliest (confirmed) use
- 3100 BC: Wine cellars, world's earliest known
- 3100 BC: Mining, Sinai Peninsula
- 3050 BC: Shipbuilding in Abydos,
- 3000 BC: Exports from Nile to Israel: wine (see Narmer)
- 3000 BC: Copper plumbing (see Copper: History)
- 3000 BC: Papyrus, world's earliest known paper
- 3000 BC: Medical Institutions
- 2900 BC: possible steel: carbon-containing iron,
- 2700 BC: Surgery, world's earliest known
- 2700 BC: precision Surveying
- 2700 BC: Uniliteral signs, forming basis of world's earliest known alphabet
- 2600 BC: Sphinx, still today the world's largest single-stone statue
- 2600s–2500 BC: Shipping expeditions: King Sneferu and Pharaoh Sahure. See also , .
- 2600 BC: Barge transportation, stone blocks (see Egyptian pyramids: Construction)
- 2600 BC: Pyramid of Djoser, world's earliest known large-scale stone building
- 2600 BC: Menkaure's Pyramid & Red Pyramid, world's earliest known works of carved granite
- 2600 BC: Red Pyramid, world's earliest known "true" smooth-sided pyramid; solid granite work granite]
- 2580 BC: Great Pyramid of Giza, the world's tallest structure until AD 1300
- 2500 BC: Beekeeping,
- 2400 BC: Astronomical Calendar, used even in the Middle Ages for its mathematical regularity
- 2200 BC: Beer,
- 1860 BC: possible Nile-Red Sea Canal (Twelfth dynasty of Egypt)
- 1800 BC: Alphabet, world's oldest known
- 1800 BC: Berlin Mathematical Papyrus, , 2nd order algebraic equations
- 1800 BC: Moscow Mathematical Papyrus, generalized formula for volume of frustum
- 1650 BC: Rhind Mathematical Papyrus: geometry, cotangent analogue, algebraic equations, arithmetic series, geometric series
- 1600 BC: Edwin Smith papyrus, medical tradition traces as far back as c. 3000 BC
- 1550 BC: Ebers Medical Papyrus, traditional empiricism; world's earliest known documented tumors (see History of medicine)
- 1500 BC: Glass-making, world's earliest known
- 1258 BC: Peace treaty, world's earliest known (see Ramesses II, )
- 1160 BC: Turin papyrus, world's earliest known geologic and topographic map
- 5th–4th century BC (or perhaps earlier): battle games petteia and seega; possible precursors to Chess (see Origins of chess)

Other

- c.2500 BC: Westcar Papyrus
- c.1800 BC: Ipuwer papyrus
- c.1800 BC: Papyrus Harris I
- c.1400 BC: Tulli Papyrus
- c.1300 BC: Brugsch Papyrus
- Unknown date: Rollin Papyrus

Open problems

There is a question as to the sophistication of ancient Egyptian technology, and there are several open problems concerning real and alleged ancient Egyptian achievements. Certain artifacts and records do not fit with conventional technological development systems. It is not known why there is no neat progression to an Egyptian Iron Age nor why the historical record shows the Egyptians taking so long to begin using iron. It is unknown how the Egyptians shaped and worked granite. The exact date the Egyptians started producing glass is debated. Some question whether the Egyptians were capable of long distance navigation in their boats and when they became knowledgeable sailors. It is contentiously disputed as to whether or not the Egyptians had some understanding of electricity and if the Egyptians used engines or batteries. The relief at Dendera is interpreted in various ways by scholars. The topic of the Saqqara Bird is controversial, as is the extent of the Egyptians' understanding of aerodynamics. It is unknown for certain if the Egyptians had kites or gliders. Beekeeping is known to have been particularly well developed in Egypt, as accounts are given by several Roman writers — Virgil, Gaius Julius Hyginus, Varro and Columella. It is unknown whether Egyptian beekeeping developed independently or as an import from Southern Asia.

External links

- [http://www.ancientegypt.co.uk/ Ancient Egypt] - maintained by the British Museum, this site provides a useful introduction to Ancient Egypt for older children and young adolescents
- [http://archaeology.about.com/od/ancientegypt/ Ancient Egypt and Egyptians] articles and resources from About Archaeology
- [http://www.bbc.co.uk/history/ancient/egyptians/ BBC History: Egyptians] - provides a reliable general overview and further links
- [http://www.ancientneareast.net/egypt.html Ancientneareast.net: Ancient Egypt] - provides a comprehensive listing of resources relating to the archaeology of Ancient Egypt
- [http://www.newton.cam.ac.uk/egypt/ Egyptology Resources] - maintained by Dr Nigel Strudwick, offers one reliable guide to online documentation of Ancient Egypt
- [http://www.kv5.com/ The Theban Mapping Project] - although focusing on the Theban region (modern Luxor), this site holds much of general interest relating to Ancient Egypt

Notes

# # # # # # # # # # # # # # # # # # # # # # Category:Ancient Egypt ja:古代エジプト ms:Mesir purba

Eudoxus of Cnidus

Eudoxus of Cnidus (Greek Εύδοξος) (410 or 408 BC - 355 or 347 BC) was a Greek astronomer, mathematician, physician, scholar and friend of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy. Theodosius of Bithynia's Sphaerics may be based on a work of Eudoxus. He was a pupil in mathematics of Archytas in Athens. In mathematical astronomy his fame is due to the introduction of the astronomical globe, and his early contributions to understanding the movement of the planets. His work on proportions shows tremendous insight into numbers; it allows rigorous treatment of continuous quantities and not just whole numbers or even rational numbers. When it was revived by Tartaglia and others in the 1500s, it became the basis for quantitative work in science for a century, until it was replaced by the algebraic methods of Descartes. Eudoxus invented the method of exhaustion, which was used in a masterly way by Archimedes. The work of Eudoxus and Archimedes as precursors of calculus was only exceeded in mathematical sophistication and rigour by Indian Mathematician Bhaskara and later by Newton. An algebraic curve (the Kampyle of Eudoxus) is named after him :a²x⁴ = b⁴(x² + y²). Also, craters on Mars and the Moon are named in his honor.

External link

- http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Eudoxus.html
- http://www.mathwright.com/book_pgs/book680.html Category:410 BC births Category:408 BC births Category:355 BC deaths Category:347 BC deaths Category:Greek and Roman astronomers Category:Ancient Greek mathematicians

Method of exhaustion

The method of exhaustion is a way of finding the area or volume of a shape that is not easily defined in terms of traditional shapes. The process, the creation of which is credited to Eudoxus, is performed by estimating the area or volume using a number of known shapes which, when put together, provide a reasonable estimation of the volume or area. Using more, smaller shapes will, of course, improve this estimation. Using an infinite number of such shapes to find the area or volume exactly is the purpose of integral calculus, which has made this process obsolete. The method of exhaustion is often associated with Archimedes. Category:Volume Category:Euclidean geometry Category:Integral calculus Category:History of mathematics

Archimedes

:For other senses of this word, see Archimedes (disambiguation). Archimedes (disambiguation) Archimedes (Greek: Αρχιμηδης ) (287 BC–212 BC) was an ancient mathematician, physicist, engineer, astronomer and philosopher born in the Greek seaport colony of Syracuse. He is considered by some math historians to be one of history's greatest mathematicians, along with possibly Newton, Gauss and Euler.

Discoveries and inventions

Euler Archimedes became a popular figure as a result of his involvement in the defense of Syracuse against the Roman siege in the First and Second Punic Wars. He is reputed to have held the Romans at bay with war machines of his own design; to have been able to move a full-size ship complete with crew and cargo by pulling a single rope[http://www.smith.edu/hsc/museum/ancient_inventions/shipshaker2.html]; to have discovered the principles of density and buoyancy, also known as Archimedes' principle, while taking a bath (thereupon taking to the streets naked calling "Eureka" - "I have found it!"); and to have invented the irrigation device known as Archimedes' screw. He has also been credited with the possible invention of the odometer during the First Punic War. One of his inventions used for military defense of Syracuse against the invading Romans was the claw of Archimedes. It is said that he prevented one Roman attack on Syracuse by using a large array of mirrors (speculated to have been highly polished shields) to reflect sunlight onto the attacking ships causing them to catch fire. This popular legend was tested on the Discovery Channel's MythBusters program. After a number of experiments, whereby the hosts of the program tried burning a model wooden ship with a variety of mirrors, they concluded that the enemy ships would have had to have been virtually motionless and very close to shore for them to ignite, an unlikely scenario during a battle. A group at MIT subsequently performed their own tests and concluded that the mirror weapon was a possibility [http://web.mit.edu/2.009/www/lectures/10_ArchimedesResult.html], although later tests of their system showed it to be ineffective in conditions that more closely matched the described siege [http://www.sfgate.com/cgi-bin/article.cgi?f=/n/a/2005/10/22/state/n121443D54.DTL]. Archimedes was killed by a Roman soldier in the sack of Syracuse during the Second Punic War, despite orders from the Roman general, Marcellus, that he was not to be harmed. The Greeks said that he was killed while drawing an equation in the sand; engrossed in his diagram and impatient with being interrupted, he is said to have muttered his famous last words before being slain by an enraged Roman soldier: Μὴ μοὺ τους κύκλους τάραττε ("Don't disturb my circles"). This story was sometimes told to contrast the Greek high-mindedness with Roman ham-handedness; however, it should be noted that Archimedes designed the siege engines that devastated a substantial Roman invasion force, so his death may have been out of retribution. In creativity and insight, he exceeded any other mathematician prior to the European Renaissance. In a civilization with an awkward numeral system and a language in which "a myriad" (literally "ten thousand") meant "infinity", he invented a positional numeral system and used it to write numbers up to 10⁶⁴. He devised a heuristic method based on statistics to do private calculation that we would classify today as integral calculus, but then presented rigorous geometric proofs for his results. To what extent he actually had a correct version of integral calculus is debatable. He proved that the ratio of a circle's perimeter to its diameter is the same as the ratio of the circle's area to the square of the radius. He did not call this ratio π but he gave a procedure to approximate it to arbitrary accuracy and gave an approximation of it as between 3 + 1/7 and 3 + 10/71. He was the first, and possibly the only, Greek mathematician to introduce mechanical curves (those traced by a moving point) as legitimate objects of study. He proved that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height. (See the illustration below. The "base" is any secant line, not necessarily orthogonal to the parabola's axis; "the same base" means the same "horizontal" component of the length of the base; "horizontal" means orthogonal to the axis. "Height" means the length of the segment parallel to the axis from the vertex to the base. The vertex must be so placed that the two horizontal distances mentioned in the illustration are equal.)

Image:Parabola.png

In the process, he calculated the oldest known example of a geometric series with the ratio 1/4: :

\sum_^\infty 4^ = 1 + 4^ + 4^ + 4^ + \cdots =  \; .

If the first term in this series is the area of the triangle in the illustration then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines in the illustration. Essentially, this paragraph summarizes the proof. Archimedes also gave a quite different proof of nearly the same proposition by a method using infinitesimals (see "How Archimedes used infinitesimals"). He proved that the area and volume of the sphere are in the same ratio to the area and volume of a circumscribed straight cylinder, a result he was so proud of that he made it his epitaph. Archimedes is probably also the first mathematical physicist on record, and the best before Galileo and Newton. He invented the field of statics, enunciated the law of the lever, the law of equilibrium of fluids and the law of buoyancy. (He famously discovered the latter when he was asked to determine whether a crown had been made of pure gold, or gold adulterated with silver; he realized that the rise in the water level when it was immersed would be equal to the volume of the crown, and the decrease in the weight of the crown would be in proportion; he could then compare those with the values of an equal weight of pure gold). He was the first to identify the concept of center of gravity, and he found the centers of gravity of various geometric figures, assuming uniform density in their interiors, including triangles, paraboloids, and hemispheres. Using only ancient Greek geometry, he also gave the equilibrium positions of floating sections of paraboloids as a function of their height, a feat that would be taxing to a modern physicist using calculus. Apart from general physics he was an astronomer, and Cicero writes that the Roman consul Marcellus brought two devices back to Rome from the sacked city of Syracuse. One device mapped the sky on a sphere and the other predicted the motions of the sun and the moon and the planets (i.e., an orrery). He credits Thales and Eudoxus for constructing these devices. For some time this was assumed to be a legend of doubtful nature, but the discovery of the Antikythera mechanism has changed the view of this issue, and it is indeed probable that Archimedes possessed and constructed such devices. Pappus of Alexandria writes that Archimedes had written a practical book on the construction of such spheres entitled On Sphere-Making. Archimedes' works were not widely recognized, even in antiquity. He and his contemporaries probably constitute the peak of Greek mathematical rigour. During the Middle Ages the mathematicians who could understand Archimedes' work were few and far between. Many of his works were lost when the library of Alexandria was burnt (twice actually) and survived only in Latin or Arabic translations. As a result, his mechanical method was lost until around 1900, after the arithmetization of analysis had been carried out successfully. We can only speculate about the effect that the "method" would have had on the development of calculus had it been known in the 16th and 17th centuries.

Writings by Archimedes

- On the Equilibrium of Planes (2 volumes) :This scroll explains the law of the lever and uses it to calculate the areas and centers of gravity of various geometric figures.
- On Spirals :In this scroll, Archimedes defines what is now called Archimedes' spiral. This is the first mechanical curve (i.e., traced by a moving point) ever considered by a Greek mathematician. Using this curve, he was able to square the circle.
- On the Sphere and The Cylinder :In this scroll Archimedes obtains the result he was most proud of: that the area and volume of a sphere are in the same relationship to the area and volume of the circumscribed straight cylinder.
- On Conoids and Spheroids :In this scroll Archimedes calculates the areas and volumes of sections of cones, spheres and paraboloids.
- On Floating Bodies (2 volumes) :In the first part of this scroll, Archimedes spells out the law of equilibrium of fluids, and proves that water around a center of gravity will adopt a spherical form. This is probably an attempt at explaining the observation made by Greek astronomers that the Earth is round. Note that his fluids are not self-gravitating: he assumes the existence of a point towards which all things fall and derives the spherical shape. One is led to wonder what he would have done had he struck upon the idea of universal gravitation. :In the second part, a veritable tour-de-force, 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, which is reminiscent of the way icebergs float, although Archimedes probably was not thinking of this application.
- The Quadrature of the Parabola :In this scroll, Archimedes calculates the area of a segment of a parabola (the figure delimited by a parabola and a secant line not necessarily perpendicular to the axis). The final answer is obtained by triangulating the area and summing the geometric series with ratio 1/4.
- Stomachion :This is a Greek puzzle similar to Tangram. In this scroll, Archimedes calculates the areas of the various pieces. This may be the first reference we have to this game. Recent discoveries indicate that Archimedes was attempting to determine how many ways the strips of paper could be assembled into the shape of a square. This is possibly the first use of combinatorics to solve a problem.
- Archimedes' Cattle Problem :Archimedes wrote a letter to the scholars in the Library of Alexandria, who apparently had downplayed the importance of Archimedes' works. In these letters, he dares them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations, some of them quadratic (in the more complicated version). This problem is one of the famous problems solved with the aid of a computer. The solution is a very large number, approximately (See the external links to the Cattle Problem.)
- The Sand Reckoner :In this scroll, Archimedes counts the number of grains of sand fitting inside the universe. This book mentions Aristarchus' theory of the solar system, contemporary ideas about the size of the Earth and the distance between various celestial bodies. From the introductory letter we also learn that Archimedes' father was an astronomer.
- "The Method" :In this work, which was unknown in the Middle Ages, but the importance of which was realised after its discovery, Archimedes pioneered the use of infinitesimals, showing how breaking up a figure in an infinite number of infinitely small parts could be used to determine its area or volume. Archimedes did probably consider these methods not mathematically precise, and he used these methods to find at least some of the areas or volumes he sought, and then used the more traditional method of exhaustion to prove them. This particular work is found in what is called the Archimedes Palimpsest. Some details can be found at how Archimedes used infinitesimals.

Quotes about Archimedes

- "Perhaps the best indication of what Archimedes truly loved most is his request that his tombstone include a cylinder circumscribing a sphere, accompanied by the inscription of his amazing theorem that the sphere is exactly two-thirds of the circumscribing cylinder in both surface area and volume!" (Laubenbacher and Pengelley, p. 95)¹
- "...but regarding the work of an engineer and every art that ministers the needs of life as ignoble and vulgar, he devoted his earnest efforts only to those studies the subtlety and charm of which are not affected by the claims of necessity." Plutarch, possibly explaining why Archimedes produced no writings that describe precisely the design of his inventions. It has also been suggested that this statement merely reflects the prejudices of Plutarch and his peers, influenced by Platonic beliefs in pure reasoning and deduction over experimentation and inductive processes. Given Archimedes's prodigious output as an engineer, Plutarch's often quoted comments on him seem hard to believe by modern historians.

Named after Archimedes

- Archimedes crater on the Moon.
- Asteroid 3600 Archimedes, named in his honour
- The Acorn Archimedes
- Archimedean property
- Archimedean solid
- Archimedean point
- Archimedean tiling
- Archimedean spiral
- Archimedean field
- Trammal of Archimedes
- Claw of Archimedes
- Archimedean screw
- Archimedean copula

External links

- [http://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/index.shtml Archimedes' Book of Lemmas] at cut-the-knot
- [http://agutie.homestead.com/files/rhombicubocta.html Archimedes and the Rhombicuboctahedron] by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
- [http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html Archimedes Home Page]
-
- [http://www.thewalters.org/archimedes/frame.html The Archimedes Palimpsest] web pages at the Walters Art Museum.
- [http://www.pbs.org/wgbh/nova/archimedes/palimpsest.html NOVA program on Archimedes Palimpsest]
- [http://www.mcs.drexel.edu/~crorres/Archimedes/Crown/CrownIntro.html Archimedes - The Golden Crown] points out that in reality Archimedes may well have used a more subtle method than the one in the classic version of the story.
- [http://www.math.ubc.ca/~cass/archimedes/parabola.html Archimedes' Quadrature Of The Parabola] Translated by Thomas Heath.
- [http://www.math.ubc.ca/~cass/archimedes/circle.html Archimedes' On The Measurement Of The Circle] Translated by Thomas Heath.
- [http://www.mcs.drexel.edu/~crorres/Archimedes/Cattle/Statement.html Archimedes' Cattle Problem]
- [http://mathworld.wolfram.com/ArchimedesCattleProblem.html Archimedes' Cattle Problem]
-
- [http://www.cut-the-knot.org/pythagoras/archi.shtml Angle Trisection by Archimedes of Syracuse (Java)]
- [http://www.cut-the-knot.org/ctk/Parabola.shtml#ArchimedesTriangle Archimedes'Triangle (Java)]
- [http://www.cut-the-knot.org/pythagoras/Archimedes.shtml An ancient extra-geometric proof]
- [http://www.cut-the-knot.org/ctk/Parabola.shtml#SqParabola Archimedes' Squaring of Parabola (Java)] at cut-the-knot
- [http://www.mlahanas.de/Greeks/Mirrors.htm Archimedes and his Burning Mirrors, Reality or Fantasy?]

Notes

Note 1: p. 95, Mathematical Expeditions: Chronicles by the Explorers by Laubenbacher and Pengelley (1999) ISBN 0387984348 (Hardcover) ISBN 038798433X (Paperback) Category:287 BC births Category:212 BC deaths Category:Ancient Greek engineers Category:Ancient Greek inventors Category:Ancient Greek mathematicians Category:Ancient Greek physicists Category:Hellenistic philosophers Category:Sicilian Greeks Category:Murdered scientists Category:History of physics ko:아르키메데스 ja:アルキメデス simple:Archimedes th:อาร์คิมิดีส

Archimedes

Discoveries and inventions

Image:Parabola.png

In the process, he calculated the oldest known example of a geometric series with the ratio 1/4: :

\sum_^\infty 4^ = 1 + 4^ + 4^ + 4^ + \cdots =  \; .

History of calculus

Invention of Calculus

Controversy (Newton, Leibnitz... or Madhava?)

Rigorous foundations

Integrals

Symbolic methods

Calculus of variations

Applications

History of mathematics

Fundamentals

Developing the concept of "number" through equations

Complex numbers

Indian contributions

Miscellaneous historical notes

External links

References

Notes

Ancient Egypt

Geography

People

History

Government

Language

Writing

Literature

Culture

Ancient achievements

Timeline

Predynastic

Dynastic

Other

Open problems

See also

Further reading

External links

Notes

Eudoxus of Cnidus

External link

Method of exhaustion

Archimedes

Discoveries and inventions

Writings by Archimedes

Quotes about Archimedes

Named after Archimedes

See also

External links

Notes

Archimedes

Discoveries and inventions

Writings by Archimedes