The Men of Mathematics

The Wikipedia Book of Men of Mathematics

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Contents Articles Men of Mathematics

1

Zeno of Elea

3

Zeno's paradoxes

5

Eudoxus of Cnidus

12

Archimedes

17

Archimedes' cattle problem

33

Book of Lemmas

36

Archimedes Palimpsest

39

René Descartes

45

Pierre de Fermat

58

List of things named after Pierre de Fermat

63

Blaise Pascal

64

Isaac Newton

75

Newton's laws of motion

98

Writing of Principia Mathematica

107

Method of Fluxions

114

Gottfried Wilhelm Leibniz

114

Bernoulli family

136

Jacob Bernoulli

137

Johann Bernoulli

140

Bernoulli differential equation

143

Bernoulli distribution

145

Bernoulli number

147

Bernoulli polynomials

169

Bernoulli process

176

Bernoulli trial

182

Bernoulli's principle

184

Leonhard Euler

198

Joseph Louis Lagrange

210

Pierre-Simon Laplace

220

Gaspard Monge

239

Joseph Fourier

243

Jean-Victor Poncelet

249

Poncelet Prize

252

Carl Friedrich Gauss

254

Augustin-Louis Cauchy

265

Nikolai Lobachevsky

275

Niels Henrik Abel

279

Carl Gustav Jacob Jacobi

286

William Rowan Hamilton

291

Évariste Galois

299

James Joseph Sylvester

306

Karl Weierstrass

310

Arthur Cayley

314

Sofia Kovalevskaya

319

George Boole

324

Charles Hermite

334

Leopold Kronecker

338

Bernhard Riemann

342

Ernst Kummer

346

Richard Dedekind

348

Henri Poincaré

352

Georg Cantor

370

References Article Sources and Contributors

384

Image Sources, Licenses and Contributors

395

Article Licenses License

399

Men of Mathematics

Men of Mathematics Men of Mathematics is a book on the history of mathematics written in 1937 by the mathematician E.T. Bell. After a brief chapter on three ancient mathematicians, the remainder of the book is devoted to the lives of about forty mathematicians who worked in the seventeenth, eighteenth and nineteenth centuries. The emphasis is on mainstream mathematics following on from the work. To keep the interest of readers, the book typically focuses on unusual or dramatic aspects of its subjects' lives. Men of Mathematics has inspired many young people, including the young John Forbes Nash Jr., to become mathematicians. It is not intended as a rigorous history, includes many anecdotal accounts, and presents a somewhat idealised picture of mathematicians, their personalities, research and controversies. In reviewing the faculty that served with Harry Bateman at Caltech, Clifford Truesdell wrote: ...[Bell] was admired for his science fiction and his Men of Mathematics. I was shocked when, just a few years later, Walter Pitts told me the latter was nothing but a string of Hollywood scenarios; my own subsequent study of the sources has shown me that Pitts was right, and I now find the contents of that still popular book to be little more than rehashes enlivened by nasty gossip and banal or indecent fancy..[1] An impression of the book was given by Rebecca Goldstein in her novel 36 Arguments for the Existence of God. Describing a character Cass Seltzer, she wrote on page 105: Right now he was reading E. T. Bell’s Men of Mathematics, which was the best yet, even though it had real mathematics in to slow him down. Some of these people sounded as if they had to be changelings, non-human visitors form some other sphere, with powers so prodigious they burst the boundaries of developmental psychology, lisping out profundities while other children were playing with their toes.[2]

Contents • • • • • • • • • • • • • • • • • • • •

Zeno (Fifth Century BC), Eudoxus (408–355 BC), Archimedes (287?–212 BC) Descartes (1596–1650) Fermat (1601–1665) Pascal (1623–1662) Newton (1642–1727) Leibniz (1646–1716) The Bernoullis (17th and 18th Century ) Euler (1707–1783) Lagrange (1736–1813) Laplace (1749 1827) Monge (1746–1818), Fourier (1768–1830) Poncelet (1788–1867) Gauss (1777–1855) Cauchy (1789–1857) Lobachevsky (1793–1856) Abel (1802–1829) Jacobi (1804–1851) Hamilton (1805–1865) Galois (1811–1832) Sylvester (1814–1897), Cayley (1821–1895)

• Weierstrass (1815–1897), Sonja Kowalewski [sic] (1850–1891) • Boole (1815–1864)

1

Men of Mathematics • • • • • •

Hermite (1822–1901) Kronecker (1823–1891) Riemann (1826–1866) Kummer (1810–1893), Dedekind (1831–1916) Poincaré (1854–1912) Cantor (1845–1918)

Notes and references [1] Truesdell, C. (1984). An idiot's fugitive essays on science: methods, criticism, training, circumstances. Berlin: Springer-Verlag. ISBN 0-387-90703-3. "Genius and the establishment at a polite standstill in the modern university: Bateman", pages 423 to 424 [2] Quoted in the College Mathematics Journal 43(3):231 (May 2010)

External links • Men of Mathematics (http://www.archive.org/details/MenOfMathematics) at the Internet Archive

2

Zeno of Elea

3

Zeno of Elea

Zeno shows the Doors to Truth and Falsity (Veritas et Falsitas). Fresco in the Library of El Escorial, Madrid. Born

ca. 490 BC

Died

ca. 430 BC (aged around 60)

Era

Pre-Socratic philosophy

Region

Western Philosophy

School

Eleatic school

Main interests

Metaphysics, Ontology

Notable ideas

Zeno's paradoxes

Zeno of Elea (pron.: /ˈziːnoʊəvˈɛliə/; Greek: Ζήνων ὁ Ἐλεάτης; ca. 490 BC – ca. 430 BC) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic.[1] He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".[2]

Life Little is known for certain about Zeno's life. Although written nearly a century after Zeno's death, the primary source of biographical information about Zeno is Plato's Parmenides dialogue.[3] In the dialogue, Plato describes a visit to Athens by Zeno and Parmenides, at a time when Parmenides is "about 65," Zeno is "nearly 40" and Socrates is "a very young man".[4] Assuming an age for Socrates of around 20, and taking the date of Socrates' birth as 469 BC gives an approximate date of birth for Zeno of 490 BC. Plato says that Zeno was "tall and fair to look upon" and was "in the days of his youth … reported to have been beloved by Parmenides".[4] Other perhaps less reliable details of Zeno's life are given by Diogenes Laërtius in his Lives and Opinions of Eminent Philosophers,[5] where it is reported that he was the son of Teleutagoras, but the adopted son of Parmenides, was "skilled to argue both sides of any question, the universal critic," and that he was arrested and perhaps killed at the hands of a tyrant of Elea. According to Plutarch, Zeno attempted to kill the tyrant Demylus, and failing to do so, "with his own teeth bit off his tongue, he spit it in the tyrant’s face."[6]

Works Although many ancient writers refer to the writings of Zeno, none of his writings survive intact. Plato says that Zeno's writings were "brought to Athens for the first time on the occasion of" the visit of Zeno and Parmenides.[4] Plato also has Zeno say that this work, "meant to protect the arguments of Parmenides",[4] was written in Zeno's youth, stolen, and published without his consent. Plato has Socrates paraphrase the "first thesis of the first argument" of Zeno's work as follows: "if being is many, it must be both like and unlike, and this is impossible, for neither can the like be unlike, nor the unlike like".[4]

Zeno of Elea According to Proclus in his Commentary on Plato's Parmenides, Zeno produced "not less than forty arguments revealing contradictions", [7] but only nine are now known. Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum, literally meaning to reduce to the absurd. Parmenides is said to be the first individual to implement this style of argument. This form of argument soon became known as the epicheirema (ἐπιχείρημα). In Book VII of his Topics, Aristotle says that an epicheirema is "a dialectical syllogism". It is a connected piece of reasoning which an opponent has put forward as true. The disputant sets out to break down the dialectical syllogism. This destructive method of argument was maintained by him to such a degree that Seneca the Younger commented a few centuries later, "If I accede to Parmenides there is nothing left but the One; if I accede to Zeno, not even the One is left."[8]

Zeno's paradoxes Zeno's paradoxes have puzzled, challenged, influenced, inspired, infuriated, and amused philosophers, mathematicians, and physicists for over two millennia. The most famous are the so-called "arguments against motion" described by Aristotle in his Physics.[9]

Notes [1] Diogenes Laërtius, 8.57, 9.25 [2] Russell, p. 347: "In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance..." [3] Plato (370 BC). Parmenides (http:/ / classics. mit. edu/ Plato/ parmenides. html), translated by Benjamin Jowett. Internet Classics Archive. [4] Plato, Parmenides 127b-e [5] Diogenes Laërtius. The Lives and Opinions of Eminent Philosophers, translated by C.D. Yonge. London: Henry G. Bohn, 1853. Scanned and edited for Peithô's Web. (http:/ / classicpersuasion. org/ pw/ diogenes/ dlzeno-eleatic. htm) [6] Plutarch, Against Colotes [7] Proclus, Commentary on Plato's Parmenides, p. 29 [8] Zeno in The Presocratics, Philip Wheelwright ed., The Odyssey Press, 1966, Pages 106-107. [9] Aristotle (350 BC). Physics (http:/ / classics. mit. edu/ Aristotle/ physics. html), translated by R.P. Hardie and R.K. Gaye. Internet Classics Archive.

References • Plato; Fowler, Harold North (1925) [1914]. Plato in twelve volumes. 8, The Statesman.(Philebus).(Ion). Loeb Classical Library. trans. W. R. M. Lamb. Cambridge, Mass.: Harvard U.P. ISBN 978-0-434-99164-8. OCLC 222336129. • Proclus; Morrow, Glenn R.; Dillon, John M. (1992) [1987]. Proclus' Commentary on Plato's Parmenides. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-02089-1. OCLC 27251522. • Russell, Bertrand (1996) [1903]. The Principles of Mathematics. New York, NY: Norton. ISBN 978-0-393-31404-5. OCLC 247299160. • Hornschemeier, Paul (2007). The Three Paradoxes. Seattle, WA: Fantagraphics Books.

4

Zeno of Elea

Further reading • • • • • •

Early Greek Philosophy Jonathan Barnes. (Harmondsworth, 1987). "Zeno and the Mathematicians" G. E. L. Owen. Proceedings of the Aristotelian Society (1957-8). Paradoxes Mark Sainsbury. (Cambridge, 1988). Zeno's Paradoxes Wesley C. Salmon, ed. (Indianapolis, 1970). Zeno of Elea Gregory Vlastos in The Encyclopedia of Philosophy (Paul Edwards, ed.), (New York, 1967). De compositie van de wereld Harry Mulisch. (Amsterdam, 1980).

External links • Zeno of Elea (http://plato.stanford.edu/entries/zeno-elea) entry by John Palmer in the Stanford Encyclopedia of Philosophy • Zeno of Elea (http://www-history.mcs.st-andrews.ac.uk/Biographies/Zeno_of_Elea.html) - MacTutor • Plato's Parmenides (http://classics.mit.edu/Plato/parmenides.html). • Aristotle's Physics (http://classics.mit.edu/Aristotle/physics.html). • Diogenes Laërtius, Life of Zeno, translated by Robert Drew Hicks (1925).

Zeno's paradoxes Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (ca. 490–430 BC) to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is usually assumed, based on Plato's Parmenides 128c-d, that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides's view. Thus Zeno can be interpreted as saying that to assume there is plurality is even more absurd than assuming there is only "the One". (Parmenides 128d). Plato makes Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point (Parmenides 128a-b). Some of Zeno's nine surviving paradoxes (preserved in Aristotle's Physics[1] and Simplicius's commentary thereon) are essentially equivalent to one another. Aristotle offered a refutation of some of them.[1] Three of the strongest and most famous—that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight—are presented in detail below. Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction. They are also credited as a source of the dialectic method used by Socrates.[2] Some mathematicians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution.[3] Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems.[4][5][6] The origins of the paradoxes are somewhat unclear. Diogenes Laertius, a fourth source for information about Zeno and his teachings, citing Favorinus, says that Zeno's teacher Parmenides was the first to introduce the Achilles and the Tortoise Argument. But in a later passage, Laertius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.[7]

5

Zeno's paradoxes

The Paradoxes of Motion Achilles and the tortoise In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. – as recounted by Aristotle, Physics VI:9, 239b15 In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.[8][9]

The dichotomy paradox That which is in locomotion must arrive at the half-way stage before it arrives at the goal.– as recounted by Aristotle, Physics VI:9, 239b10 Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

The resulting sequence can be represented as:

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox. Some, like Aristotle, regard the Dichotomy as really just another version of Achilles and the Tortoise.[10] There are two versions of the dichotomy paradox. In the other version, before Homer could reach the stationary bus, he must reach half of the distance to it. Before reaching the last half, he must complete the next quarter of the distance. Reaching the next quarter, he must then cover the next eighth of the distance, then the next sixteenth, and so on. There are thus an infinite number of steps that must first be accomplished before he could reach the bus, with no way to establish the size of any "last" step. Expressed this way, the dichotomy paradox is very much analogous to that of Achilles and the tortoise.

6

Zeno's paradoxes

The arrow paradox If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.[11] – as recounted by Aristotle, Physics VI:9, 239b5 In the arrow paradox (also known as the fletcher's paradox), Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not.[12] It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible. Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.[13]

Three other paradoxes as given by Aristotle Paradox of Place: "… if everything that exists has a place, place too will have a place, and so on ad infinitum."[14] Paradox of the Grain of Millet: "… there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially."[15] The Moving Rows (or Stadium): "… concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This...involves the conclusion that half a given time is equal to double that time."[16] For an expanded account of Zeno's arguments as presented by Aristotle, see Simplicius' commentary On Aristotle's Physics.

Proposed solutions According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes. Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.[17][18] Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities").[19] Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. (See: Geometric series, 1/4 + 1/16 + 1/64 + 1/256 + · · ·, The Quadrature of the Parabola.) Modern calculus achieves the same result, using more rigorous methods (see convergent series, where the "reciprocals of powers of 2" series, equivalent to the Dichotomy Paradox, is listed as convergent). These methods

7

Zeno's paradoxes allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.[3][20] Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."[21] Saint Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."[22] Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is a function of position with respect to time.[23][24] Nick Huggett argues that Zeno is begging the question when he says that objects that occupy the same space as they do at rest must be at rest.[13] Peter Lynds has argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.[25][26][27] Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. The ideas of Planck length and Planck time in modern physics place a limit on the measurement of time and space, if not on time and space themselves. According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem".[28][29] According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry. Jean Paul Van Bendegem has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.[3][30] Hans Reichenbach has proposed that the paradox may arise from considering space and time as separate entities. In a theory like general relativity, which presumes a single space-time continuum, the paradox may be blocked.[31]

The paradoxes in modern times Infinite processes remained theoretically troublesome in mathematics until the late 19th century. The epsilon-delta version of Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.[32] While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Brown and Moorcroft[4][5] claim that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise. Zeno's arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite. However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?[4][5][6][33] Today there is still a debate on the question of whether or not Zeno's paradoxes have been resolved. In The History of Mathematics, Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory

8

Zeno's paradoxes explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.'"[34] Bertrand Russell offered a "solution" to the paradoxes based on modern physics, but Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of 'Rorschach image' onto which people can project their most fundamental phenomenological concerns (if they have any)."[4]

The quantum Zeno effect In 1977,[35] physicists E. C. G. Sudarshan and B. Misra studying quantum mechanics discovered that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system.[36] This effect is usually called the "quantum Zeno effect" as it is strongly reminiscent of Zeno's arrow paradox. This effect was first theorized in 1958.[37]

Zeno behaviour In the field of verification and design of timed and hybrid systems, the system behaviour is called Zeno if it includes an infinite number of discrete steps in a finite amount of time.[38] Some formal verification techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.[39][40] In systems design these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.[41] A simple example of a system showing Zeno behaviour is a bouncing ball coming to rest. The physics of a bouncing ball can be mathematically analyzed in such a way, ignoring factors other than rebound, to predict an infinite number of bounces.

Writings about Zeno’s paradoxes Zeno’s paradoxes have inspired many writers • Leo Tolstoy in War and Peace (Part 11, Chapter I) discusses the race of Achilles and the tortoise when critiquing "historical science". • In the dialogue "What the Tortoise Said to Achilles", Lewis Carroll describes what happens at the end of the race. The tortoise discusses with Achilles a simple deductive argument. Achilles fails in demonstrating the argument because the tortoise leads him into an infinite regression. • In Gödel, Escher, Bach by Douglas Hofstadter, the various chapters are separated by dialogues between Achilles and the tortoise, inspired by Lewis Carroll’s works. • The Argentinian writer Jorge Luis Borges discusses Zeno’s paradoxes many times in his work, showing their relationship with infinity. Borges also used Zeno’s paradoxes as a metaphor for some situations described by Kafka. Borges traces, in an essay entitled "Avatars of the Tortoise", the many recurrences of this paradox in works of philosophy. The successive references he traces are Agrippa the Skeptic, Thomas Aquinas, Hermann Lotze, F.H. Bradley and William James.[42] • In Tom Stoppard's play Jumpers, the philosopher George Moore attempts a practical disproof with bow and arrow of the Dichotomy Paradox, with disastrous consequences for the hare and the tortoise. • Harry Mulisch's philosophical magnum opus, De compositie van de wereld (Amsterdam, 1980) is based on Zeno's Paradoxes mostly. Along with Herakleitos' thoughts and Cusanus' coincidentia oppositorum they constitute the foundation for his own system of the 'octave'. • In the novel Small Gods by Terry Pratchett the prophet Brutha encounters several Ephebian (Greek) philosophers in the country, attempting to disprove Zeno's paradox by shooting arrows at a succession of tortoises. So far, this has resulted only in a succession of "tortoise-kabobs."

9

Zeno's paradoxes

In popular culture • The Firesign Theatre's 1969 album How Can You Be in Two Places at Once When You're Not Anywhere at All contains a section originally titled "The Policemen's Brawl" but retitled "Zeno's Evil" when released on CD. In this segment, as the lead character is driving along in his new car, a series of audible highway signs reports that the distance to the Antelope Freeway is 1 mile, then 1⁄2 mile, then 1⁄4 mile, 1⁄8 mile, and so on. The signs' monolog is interrupted just after reaching the 1⁄512 mile mark. • The web comic xkcd makes reference to Zeno's paradoxes: the comic Advent Calendar [43] shows an advent calendar version of Achilles and the Tortoise paradox, and the comic Proof [44] shows a courtroom where Zeno claims to be able to prove that his client could not have killed anyone with an arrow, referencing the arrow paradox.

Notes [1] Aristotle's Physics (http:/ / classics. mit. edu/ Aristotle/ physics. html) "Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye [2] ([fragment 65], Diogenes Laertius. IX (http:/ / classicpersuasion. org/ pw/ diogenes/ dlzeno-eleatic. htm) 25ff and VIII 57). [3] Boyer, Carl (1959). The History of the Calculus and Its Conceptual Development (http:/ / books. google. com/ ?id=w3xKLt_da2UC& dq=zeno+ calculus& q=zeno#v=snippet& q=zeno). Dover Publications. p. 295. ISBN 978-0-486-60509-8. . Retrieved 2010-02-26. "If the paradoxes are thus stated in the precise mathematical terminology of continuous variables (...) the seeming contradictions resolve themselves." [4] Brown, Kevin. "Zeno and the Paradox of Motion" (http:/ / www. mathpages. com/ rr/ s3-07/ 3-07. htm). Reflections on Relativity. . Retrieved 2010-06-06. [5] Moorcroft, Francis. "Zeno's Paradox" (http:/ / web. archive. org/ web/ 20100418141459id_/ http:/ / www. philosophers. co. uk/ cafe/ paradox5. htm). Archived from the original (http:/ / www. philosophers. co. uk/ cafe/ paradox5. htm) on 2010-04-18. . [6] Papa-Grimaldi, Alba (1996). "Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides' Prohibition" (http:/ / philsci-archive. pitt. edu/ 2304/ 1/ zeno_maths_review_metaphysics_alba_papa_grimaldi. pdf) (PDF). The Review of Metaphysics 50: 299–314. . [7] Diogenes Laertius, Lives, 9.23 and 9.29. [8] "Math Forum" (http:/ / mathforum. org/ isaac/ problems/ zeno1. html). ., matchforum.org [9] Huggett, Nick (2010). "Zeno's Paradoxes: 3.2 Achilles and the Tortoise" (http:/ / plato. stanford. edu/ entries/ paradox-zeno/ #AchTor). Stanford Encyclopedia of Philosophy. . Retrieved 2011-03-07. [10] Huggett, Nick (2010). "Zeno's Paradoxes: 3.1 The Dichotomy" (http:/ / plato. stanford. edu/ entries/ paradox-zeno/ #Dic). Stanford Encyclopedia of Philosophy. . Retrieved 2011-03-07. [11] Aristotle. "Physics" (http:/ / classics. mit. edu/ Aristotle/ physics. 6. vi. html#752). The Internet Classics Archive. . "Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles." [12] Laertius, Diogenes (about 230 CE). "Pyrrho" (http:/ / en. wikisource. org/ wiki/ Lives_of_the_Eminent_Philosophers/ Book_IX#Pyrrho). Lives and Opinions of Eminent Philosophers. IX. passage 72. ISBN 1-116-71900-2. . [13] Huggett, Nick (2010). "Zeno's Paradoxes: 3.3 The Arrow" (http:/ / plato. stanford. edu/ entries/ paradox-zeno/ #Arr). Stanford Encyclopedia of Philosophy. . Retrieved 2011-03-07. [14] Aristotle Physics IV:1, 209a25 (http:/ / classics. mit. edu/ Aristotle/ physics. 4. iv. html) [15] Aristotle Physics VII:5, 250a20 (http:/ / classics. mit. edu/ Aristotle/ physics. 7. vii. html) [16] Aristotle Physics VI:9, 239b33 (http:/ / classics. mit. edu/ Aristotle/ physics. 6. vi. html) [17] Aristotle. Physics 6.9 [18] Aristotle's observation that the fractional times also get shorter does not guarantee, in every case, that the task can be completed. One case in which it does not hold is that in which the fractional times decrease in a harmonic series, while the distances decrease geometrically, such as: 1/2 s for 1/2 m gain, 1/3 s for next 1/4 m gain, 1/4 s for next 1/8 m gain, 1/5 s for next 1/16 m gain, 1/6 s for next 1/32 m gain, etc. In this case, the distances form a convergent series, but the times form a divergent series, the sum of which has no limit. Archimedes developed a more explicitly mathematical approach than Aristotle. [19] Aristotle. Physics 6.9; 6.2, 233a21-31 [20] George B. Thomas, Calculus and Analytic Geometry, Addison Wesley, 1951 [21] Aristotle. Physics (http:/ / classics. mit. edu/ Aristotle/ physics. 6. vi. html). VI. Part 9 verse: 239b5. ISBN 0-585-09205-2. . [22] Aquinas. Commentary on Aristotle's Physics, Book 6.861 [23] Huggett, Nick (1999). Space From Zeno to Einstein. ISBN 0-262-08271-3. [24] Salmon, Wesley C. (1998). Causality and Explanation (http:/ / books. google. com/ ?id=uPRbOOv1YxUC& pg=PA198& lpg=PA198& dq=at+ at+ theory+ of+ motion+ russell#v=onepage& q=at at theory of motion russell& f=false). p. 198. ISBN 978-0-19-510864-4. . [25] Lynds, Peter. Zeno's Paradoxes: a Timely Solution (http:/ / philsci-archive. pitt. edu/ 1197/ )

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Zeno's paradoxes [26] Lynds, Peter. Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity. Foundations of Physics Letter s (Vol. 16, Issue 4, 2003). doi:10.1023/A:1025361725408 [27] Time’s Up Einstein (http:/ / www. wired. com/ wired/ archive/ 13. 06/ physics. html), Josh McHugh, Wired Magazine, June 2005 [28] Van Bendegem, Jean Paul (17 March 2010). "Finitism in Geometry" (http:/ / plato. stanford. edu/ entries/ geometry-finitism/ #SomParSolProDea). Stanford Encyclopedia of Philosophy. . Retrieved 2012-01-03. [29] Cohen, Marc (11 December 2000). "ATOMISM" (https:/ / www. aarweb. org/ syllabus/ syllabi/ c/ cohen/ phil320/ atomism. htm). History of Ancient Philosophy, University of Washington. . Retrieved 2012-01-03. [30] van Bendegem, Jean Paul (1987). "Discussion:Zeno's Paradoxes and the Tile Argument". Philosophy of Science (Belgium) 54 (2): 295–302. doi:10.1086/289379. JSTOR 187807. [31] Hans Reichenbach (1958) The Philosophy of Space and Time. Dover [32] Lee, Harold (1965). "Are Zeno's Paradoxes Based on a Mistake?". Mind (Oxford University Press) 74 (296): 563–570. JSTOR 2251675. [33] Huggett, Nick (2010). "Zeno's Paradoxes: 5. Zeno's Influence on Philosophy" (http:/ / plato. stanford. edu/ entries/ paradox-zeno/ #ZenInf). Stanford Encyclopedia of Philosophy. . Retrieved 2011-03-07. [34] Burton, David, A History of Mathematics: An Introduction, McGraw Hill, 2010, ISBN 978-0-07-338315-6 [35] Sudarshan, E. C. G.; Misra, B. (1977). "The Zeno’s paradox in quantum theory". Journal of Mathematical Physics 18 (4): 756–763. Bibcode 1977JMP....18..756M. doi:10.1063/1.523304 [36] W.M.Itano; D.J.Heinsen, J.J.Bokkinger, D.J.Wineland (1990). "Quantum Zeno effect" (http:/ / www. boulder. nist. gov/ timefreq/ general/ pdf/ 858. pdf) (PDF). PRA 41 (5): 2295–2300. Bibcode 1990PhRvA..41.2295I. doi:10.1103/PhysRevA.41.2295. . [37] Khalfin, L.A. (1958). Soviet Phys. JETP 6: 1053. Bibcode 1958JETP....6.1053K [38] Paul A. Fishwick, ed. (1 June 2007). "15.6 "Pathological Behavior Classes" in chapter 15 "Hybrid Dynamic Systems: Modeling and Execution" by Pieter J. Mosterman, The Mathworks, Inc." (http:/ / books. google. com/ ?id=cM-eFv1m3BoC& pg=SA15-PA22). Handbook of dynamic system modeling. Chapman & Hall/CRC Computer and Information Science (hardcover ed.). Boca Raton, Florida, USA: CRC Press. pp. 15–22 to 15–23. ISBN 978-1-58488-565-8. . Retrieved 2010-03-05. [39] Lamport, Leslie (2002) (PDF). Specifying Systems (http:/ / research. microsoft. com/ en-us/ um/ people/ lamport/ tla/ book-02-08-08. pdf). Addison-Wesley. p. 128. ISBN 0-321-14306-X. . Retrieved 2010-03-06. [40] Zhang, Jun; Johansson, Karl; Lygeros, John; Sastry, Shankar (2001). "Zeno hybrid systems" (http:/ / aphrodite. s3. kth. se/ ~kallej/ papers/ zeno_ijnrc01. pdf). International Journal for Robust and Nonlinear control. . Retrieved 2010-02-28. [41] Franck, Cassez; Henzinger, Thomas; Raskin, Jean-Francois (2002). A Comparison of Control Problems for Timed and Hybrid Systems (http:/ / mtc. epfl. ch/ ~tah/ Publications/ a_comparison_of_control_problems_for_timed_and_hybrid_systems. html). . Retrieved 2010-03-02. [42] Borges, Jorge Luis (1964). Labyrinths. London: Penguin. pp. 237–243. ISBN 0-8112-0012-4. [43] http:/ / xkcd. com/ 994/ [44] http:/ / xkcd. com/ 1153/

References • Kirk, G. S., J. E. Raven, M. Schofield (1984) The Presocratic Philosophers: A Critical History with a Selection of Texts, 2nd ed. Cambridge University Press. ISBN 0-521-27455-9. • Huggett, Nick (2010). "Zeno's Paradoxes" (http://plato.stanford.edu/entries/paradox-zeno/). Stanford Encyclopedia of Philosophy. Retrieved 2011-03-07. • Plato (1926) Plato: Cratylus. Parmenides. Greater Hippias. Lesser Hippias, H. N. Fowler (Translator), Loeb Classical Library. ISBN 0-674-99185-0. • Sainsbury, R.M. (2003) Paradoxes, 2nd ed. Cambridge University Press. ISBN 0-521-48347-6.

External links • Silagadze, Z . K. " Zeno meets modern science, (http://uk.arxiv.org/abs/physics/0505042)" • Zeno's Paradox: Achilles and the Tortoise (http://demonstrations.wolfram.com/ ZenosParadoxAchillesAndTheTortoise/) by Jon McLoone, Wolfram Demonstrations Project. • Kevin Brown on Zeno and the Paradox of Motion (http://www.mathpages.com/rr/s3-07/3-07.htm) • Palmer, John (2008). "Zeno of Elea" (http://plato.stanford.edu/entries/zeno-elea/). Stanford Encyclopedia of Philosophy. This article incorporates material from Zeno's paradox on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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Eudoxus of Cnidus

Eudoxus of Cnidus Eudoxus of Cnidus (410 or 408 BC – 355 or 347 BC) was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy. Theodosius of Bithynia's important work, Sphaerics, may be based on a work of Eudoxus.

Life His name Eudoxus means "honored" or "of good repute" (in Greek Εὔδοξος, from eu "good" and doxa "opinion, belief, fame"). It is analogous to the Latin name Benedictus. Eudoxus's father Aeschines of Cnidus loved to watch stars at night. Eudoxus first travelled to Tarentum to study with Archytas, from whom he learned mathematics. While in Italy, Eudoxus visited Sicily, where he studied medicine with Philiston. Around 387 BC, at the age of 23, he traveled with the physician Theomedon, who according to Diogenes Laërtius some believed was his lover,[1] to Athens to study with the followers of Socrates. He eventually became the pupil of Plato, with whom he studied for several months, but due to a disagreement they had a falling out. Eudoxus was quite poor and could only afford an apartment at the Piraeus. To attend Plato's lectures, he walked the seven miles (11 km) each direction, each day. Due to his poverty, his friends raised funds sufficient to send him to Heliopolis, Egypt to pursue his study of astronomy and mathematics. He lived there for 16 months. From Egypt, he then traveled north to Cyzicus, located on the south shore of the Sea of Marmara, the Propontis. He traveled south to the court of Mausolus. During his travels he gathered many students of his own. Around 368 BC, Eudoxus returned to Athens with his students. According to some sources, around 367 he assumed headship of the Academy during Plato's period in Syracuse, and taught Aristotle. He eventually returned to his native Cnidus, where he served in the city assembly. While in Cnidus, he built an observatory and continued writing and lecturing on theology, astronomy and meteorology. He had one son, Aristagoras, and three daughters, Actis, Philtis and Delphis. 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 16th century, it became the basis for quantitative work in science for a century, until it was replaced by the algebraic methods of Descartes. Craters on Mars and the Moon are named in his honor. An algebraic curve (the Kampyle of Eudoxus) is also named after him a2x4 = b4(x2 + y2).

Mathematics Eudoxus is considered by some to be the greatest of classical Greek mathematicians, and in all antiquity, second only to Archimedes. He rigorously developed Antiphon's method of exhaustion, a precursor to the integral calculus which was also used in a masterly way by Archimedes in the following century. In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one another as the squares of their radii, volumes of spheres are to one another as the cubes of their radii, the volume of a pyramid is one-third the volume of a prism with the same base and altitude, and the volume of a cone is one-third that of the corresponding cylinder.[2] Eudoxus introduced the idea of non-quantified mathematical magnitude to describe and work with continuous geometrical entities such as lines, angles, areas and volumes, thereby avoiding the use of irrational numbers. In doing

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Eudoxus of Cnidus

13

so, he reversed a Pythagorean emphasis on number and arithmetic, focusing instead on geometrical concepts as the basis of rigorous mathematics. Some Pythagoreans, such as Eudoxus' teacher Archytas, had believed that only arithmetic could provide a basis for proofs. Induced by the need to understand and operate with incommensurable quantities, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit axioms. The change in focus by Eudoxus stimulated a divide in mathematics which lasted two thousand years. In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra.[3] The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the square root of 2 cannot be expressed as the ratio of two integers. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole. For example, Euclid provides an elaborate proof of the Pythagorean theorem (Elements I.47), by using addition of areas and only much later (Elements VI.31) a simpler proof from similar triangles, which relies on ratios of line segments. Ancient Greek mathematicians calculated not with quantities and equations as we do today, but instead they used proportionalities to express the relationship between quantities. Thus the ratio of two similar quantities was not just a numerical value, as we think of it today; the ratio of two similar quantities was a primitive relationship between them. Eudoxus was able to restore confidence in the use of proportionalities by providing an astounding definition for the meaning of the equality between two ratios. This definition of proportion forms the subject of Euclid's Book V. In Definition 5 of Euclid's Book V we read: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. Let us clarify it by using modern-day notation. If we take four quantities: a, b, c, and d, then the first and second have a ratio ; similarly the third and fourth have a ratio . Now to say that

we do the following: For any two arbitrary integers, m and n, form the equimultiples

m·a and m·c of the first and third; likewise form the equimultiples n·b and n·d of the second and fourth. If it happens that m·a > n·b, then we must also have m·c > n·d. If it happens that m·a = n·b, then we must also have m·c = n·d. Finally, if it happens that m·a < n·b, then we must also have m·c < n·d. Notice that the definition depends on comparing the similar quantities m·a and n·b, and the similar quantities m·c and n·d, and does not depend on the existence of a common unit of measuring these quantities. The complexity of the definition reflects the deep conceptual and methodological innovation involved. It brings to mind the famous fifth postulate of Euclid concerning parallels, which is more extensive and complicated in its wording than the other postulates. The Eudoxian definition of proportionality uses the quantifier, "for every ..." to harness the infinite and the infinitesimal, just as do the modern epsilon-delta definitions of limit and continuity. Additionally, the Archimedean property stated as definition 4 of Euclid's book V is originally due not to Archimedes but to Eudoxus.[4]

Eudoxus of Cnidus

Astronomy In ancient Greece, astronomy was a branch of mathematics; astronomers sought to create geometrical models that could imitate the appearances of celestial motions. Identifying the astronomical work of Eudoxus as a separate category is therefore a modern convenience. Some of Eudoxus' astronomical texts whose names have survived include: • Disappearances of the Sun, possibly on eclipses • Oktaeteris (Ὀκταετηρίς), on an eight-year lunisolar cycle of the calendar • Phaenomena (Φαινόμενα) and Entropon (Ἔντροπον), on spherical astronomy, probably based on observations made by Eudoxus in Egypt and Cnidus • On Speeds, on planetary motions We are fairly well informed about the contents of Phaenomena, for Eudoxus' prose text was the basis for a poem of the same name by Aratus. Hipparchus quoted from the text of Eudoxus in his commentary on Aratus.

Eudoxan planetary models A general idea of the content of On Speeds can be gleaned from Aristotle's Metaphysics XII, 8, and a commentary by Simplicius of Cilicia (6th century CE) on De caelo, another work by Aristotle. According to a story reported by Simplicius, Plato posed a question for Greek astronomers: "By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?" (quoted in Lloyd 1970, p. 84). Plato proposed that the seemingly chaotic wandering motions of the planets could be explained by combinations of uniform circular motions centered on a spherical Earth, apparently a novel idea in the 4th century. In most modern reconstructions of the Eudoxan model, the Moon is assigned three spheres: • The outermost rotates westward once in 24 hours, explaining rising and setting. • The second rotates eastward once in a month, explaining the monthly motion of the Moon through the zodiac. • The third also completes its revolution in a month, but its axis is tilted at a slightly different angle, explaining motion in latitude (deviation from the ecliptic), and the motion of the lunar nodes. The Sun is also assigned three spheres. The second completes its motion in a year instead of a month. The inclusion of a third sphere implies that Eudoxus mistakenly believed that the Sun had motion in latitude. The five visible planets (Venus, Mercury, Mars, Jupiter, and Saturn) are assigned four spheres each: • The outermost explains the daily motion. • The second explains the planet's motion through the zodiac. • The third and fourth together explain retrogradation, when a planet appears to slow down, then briefly reverse its motion through the zodiac. By inclining the axes of the two spheres with respect to each other, and rotating them in opposite directions but with equal periods, Eudoxus could make a point on the inner sphere trace out a figure-eight shape, or hippopede.

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Eudoxus of Cnidus

Importance of Eudoxan system Callippus, a Greek astronomer of the 4th century, added seven spheres to Eudoxus' original 27 (in addition to the planetary spheres, Eudoxus included a sphere for the fixed stars). Aristotle described both systems, but insisted on adding "unrolling" spheres between each set of spheres to cancel the motions of the outer set. Aristotle was concerned about the physical nature of the system; without unrollers, the outer motions would be transferred to the inner planets. A major flaw in the Eudoxan system is its inability to explain changes in the brightness of planets as seen from Earth. Because the spheres are concentric, planets will always remain at the same distance from Earth. This problem was pointed out in Antiquity by Autolycus of Pitane. Astronomers responded by introducing the deferent and epicycle, which caused a planet to vary its distance. However, Eudoxus' importance to Greek astronomy is considerable, as he was the first to attempt a mathematical explanation of the planets.

Ethics Aristotle, in The Nicomachean Ethics[5] attributes to Eudoxus an argument in favor of hedonism, that is, that pleasure is the ultimate good that activity strives for. According to Aristotle, Eudoxus put forward the following arguments for this position: 1. All things, rational and irrational, aim at pleasure; things aim at what they believe to be good; a good indication of what the chief good is would be the thing that most things aim at. 2. Similarly, pleasure's opposite − pain − is universally avoided, which provides additional support for the idea that pleasure is universally considered good. 3. People don't seek pleasure as a means to something else, but as an end in its own right. 4. Any other good that you can think of would be better if pleasure were added to it, and it is only by good that good can be increased. 5. Of all of the things that are good, happiness is peculiar for not being praised, which may show that it is the crowning good.[6]

References • Evans, James (1998). The History and Practice of Ancient Astronomy. Oxford University Press. ISBN 0-19-509539-1. OCLC 185509676. • Huxley, GL (1980). Eudoxus of Cnidus p. 465-7 in: the Dictionary of Scientific Biography, volume 4. • Lloyd, GER (1970). Early Greek Science: Thales to Aristotle. W.W. Norton.

Notes [1] [2] [3] [4] [5] [6]

Diogenes Laertius; VIII.87 Morris Kline, Mathematical Thought from Ancient to Modern Times Oxford University Press, 1972 pp. 48-50 ibid Knopp, Konrad (1951). Theory and Application of Infinite Series (English 2nd ed.). London and Glasgow: Blackie & Son, Ltd.. p. 7. largely in book ten this particular argument is referenced in book one

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Eudoxus of Cnidus

Further reading • De Santillana, G. (1968). "Eudoxus and Plato: A Study in Chronology". Reflections on Men and Ideas. Cambridge, MA: MIT Press. • Huxley, G. L. (1963). "Eudoxian Topics". Greek, Roman, and Byzantine Studies 4: 83–96. • Knorr, Wilbur Richard (1986). The Ancient tradition of geometric problems. Boston: Birkhäuser. ISBN 0-8176-3148-8. • Knorr, Wilbur Richard (1978). "Archimedes and the Pre-Euclidean Proportion Theory". Archives Intemationales d'histoire des Sciences 28: 183–244. • Neugebauer, O. (1975). A history of ancient mathematical astronomy. Berlin: Springer-Verlag. ISBN 0-387-06995-X. • Van der Waerden, B. L. (1988). Science Awakening (5th ed.). Leiden: Noordhoff.

External links • Working model and complete explanation of the Eudoxus's Spheres (http://www.youtube.com/ watch?v=_SFzDYSqR_4) • Dennis Duke, "Statistical dating of the Phaenomena of Eudoxus", DIO, volume 15 (http://www.dioi.org/vols/ wf0.pdf) see pages 7 to 23 • Diogenes Laërtius, Life of Eudoxus, translated by Robert Drew Hicks (1925). Wikisource • Eudoxus of Cnidus (http://www.britannica.com/EBchecked/topic/195005/Eudoxus-of-Cnidus) Britannica.com • Eudoxus of Cnidus (http://www.math.tamu.edu/~don.allen/history/eudoxus/eudoxus.html) Donald Allen, Professor, Texas A&M University • Eudoxos of Knidos (Eudoxus of Cnidus): astronomy and homocentric spheres (http://www.calstatela.edu/ faculty/hmendel/Ancient Mathematics/Eudoxus/Astronomy/EudoxusHomocentricSpheres.htm) Henry Mendell, Cal State U, LA • Herodotus Project: Extensive B+W photo essay of Cnidus (http://www.losttrails.com/pages/Hproject/Caria/ Cnidus/Cnidus.html) • Models of Planetary Motion—Eudoxus (http://faculty.fullerton.edu/cmcconnell/Planets.html#3), Craig McConnell, Ph.D., Cal State, Fullerton • O'Connor, John J.; Robertson, Edmund F., "Eudoxus of Cnidus" (http://www-history.mcs.st-andrews.ac.uk/ Biographies/Eudoxus.html), MacTutor History of Mathematics archive, University of St Andrews. • The Universe According to Eudoxus (http://hsci.cas.ou.edu/images/applets/hippopede.html) (Java applet)

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Archimedes

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Archimedes Archimedes of Syracuse (Greek: Ἀρχιμήδης)

Archimedes Thoughtful by Fetti (1620) Born

c. 287 BC Syracuse, Sicily Magna Graecia

Died

c. 212 BC (aged around 75) Syracuse

Residence

Syracuse, Sicily

Fields

Mathematics physics engineering astronomy invention

Known for

Archimedes' principle Archimedes' screw hydrostatics levers infinitesimals

Archimedes of Syracuse (Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer.[1] Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.[2] Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time.[3][4] He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.[5] He also defined the spiral bearing his name, formulae for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers.

Archimedes 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 the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements. Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus, while commentaries on the works of Archimedes written by Eutocius in the sixth century AD 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,[6] while the discovery in 1906 of previously unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.[7]

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 historian John Tzetzes that Archimedes lived for 75 years.[8] In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse.[9] A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure.[10] It is This bronze statue of Archimedes is at the unknown, for instance, whether he ever married or had children. Archenhold Observatory in Berlin. It was During his youth, Archimedes may have studied in Alexandria, Egypt, sculpted by Gerhard Thieme and unveiled in where Conon of Samos and Eratosthenes of Cyrene were 1972. contemporaries. He referred to Conon of Samos as his friend, while two of his works (The Method of Mechanical Theorems and the Cattle Problem) have introductions addressed to Eratosthenes.[a] Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he not be harmed.[11]

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Archimedes

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The last words attributed to Archimedes are "Do not disturb my circles" (Greek: μή μου τοὺς κύκλους τάραττε), a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in Latin as "Noli turbare circulos meos," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.[11] The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75 BC, 137 years after his death, the Roman orator Cicero A sphere has 2/3 the volume and surface area of was serving as quaestor in Sicily. He had heard stories about the tomb its circumscribing cylinder. A sphere and cylinder of Archimedes, but none of the locals was able to give him the were placed on the tomb of Archimedes at his location. Eventually he found the tomb near the Agrigentine gate in request. 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.[12] A tomb discovered in a hotel courtyard in Syracuse in the early 1960s was claimed to be that of Archimedes, but its location today is unknown.[13] The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by Polybius in his Universal History was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and Livy. 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.[14]

Discoveries and inventions 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 votive crown for a temple had been made for King Hiero II, who had supplied the pure gold to be used, and Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith.[15] Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density. While taking a bath, he noticed 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 volume of the crown. For practical purposes water is incompressible,[16] 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, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress,

Archimedes may have used his principle of buoyancy to determine whether the golden crown was less dense than solid gold.

Archimedes crying "Eureka!" (Greek: "εὕρηκα!," meaning "I have found it!"). The test was conducted successfully, proving that silver had indeed been mixed in.[17] The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement.[18] Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' principle, which he describes in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.[19] Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing the crown on a scale with a gold reference sample, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. Galileo 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."[20]

Archimedes' screw A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis described how King Hiero II commissioned Archimedes to design a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a naval warship. The Syracusia is said to have been the largest ship built in classical antiquity.[21] According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities. Since a ship of this size would The Archimedes screw can raise water efficiently. leak a considerable amount of water through the hull, the Archimedes screw was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The Archimedes screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.[22][23][24] 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.[25]

Claw of Archimedes The Claw of Archimedes is a weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker," 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.[26][27]

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21

Heat ray The 2nd century AD author Lucian wrote that during the Siege of Syracuse (c. 214–212 BC), Archimedes destroyed enemy ships with fire. Centuries later, Anthemius of Tralles mentions burning-glasses as Archimedes' weapon.[28] The device, sometimes called the "Archimedes heat ray", was used to focus sunlight onto approaching ships, causing them to catch fire. This purported weapon has been the subject of 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.[29] 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. This would have used the principle of the parabolic reflector in a manner similar to a solar furnace.

Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.

A test of the Archimedes heat ray was carried out in 1973 by the Greek scientist Ioannis Sakkas. The experiment took place at the Skaramagas naval base outside Athens. On this occasion 70 mirrors were used, each with a copper coating and a size of around five by three feet (1.5 by 1 m). The mirrors were pointed at a plywood mock-up of a Roman warship at a distance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burst into flames within a few seconds. The plywood ship had a coating of tar paint, which may have aided combustion.[30] A coating of tar would have been commonplace on ships in the classical era.[d] In October 2005 a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 one-foot (30 cm) square mirror tiles, focused on a mock-up wooden ship at a range of around 100 feet (30 m). Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was concluded that the device was a feasible weapon under these conditions. The MIT group repeated the experiment for the television show MythBusters, using a wooden fishing boat in San Francisco as the target. Again some charring occurred, along with a small amount of flame. In order to catch fire, wood needs to reach its autoignition temperature, which is around 300 °C (570 °F).[31][32] When MythBusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or failed) because of the length of time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. MythBusters also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances.[2] In December 2010, MythBusters again looked at the heat ray story in a special edition featuring Barack Obama, entitled President's Challenge. Several experiments were carried out, including a large scale test with 500 schoolchildren aiming mirrors at a mock-up of a Roman sailing ship 400 feet (120 m) away. In all of the experiments, the sail failed to reach the 210 °C (410 °F) required to catch fire, and the verdict was again "busted". The show concluded that a more likely effect of the mirrors would have been blinding, dazzling, or distracting the crew of the ship.[33]

Archimedes

Other discoveries and inventions While Archimedes did not invent the lever, he gave an explanation of the principle involved in his work On the Equilibrium of Planes. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas.[34][35] According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω)[36] 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.[37] 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.[38] Cicero (106–43 BC) mentions Archimedes briefly in his dialogue De re publica, which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse c. 212 BC, General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms, constructed by Archimedes and used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero 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.[39][40] This is a description of a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled On Sphere-Making. Modern research in this area has been focused on the Antikythera mechanism, another device from classical antiquity that was probably designed for the same 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.[41][42]

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Mathematics While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote: "He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life."[43] Archimedes was able to use 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 value of π. In Measurement of a Circle he did this by drawing a larger regular hexagon outside a circle and a smaller regular hexagon inside the circle, and progressively doubling the number of Archimedes used Pythagoras' Theorem to calculate the side of the 12-gon from that of the sides of each regular polygon, calculating the length of a side of each hexagon and for each subsequent doubling of the polygon at each step. As the number of sides increases, it becomes a sides of the regular polygon. 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 31⁄7 (approximately 3.1429) and 310⁄71 (approximately 3.1408), consistent with its actual value of approximately 3.1416.[44] He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle (πr2). In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. This is the Archimedean property of real numbers.[45] In Measurement of a Circle, Archimedes gives the value of the square root of 3 as lying between 265⁄153 (approximately 1.7320261) and 1351⁄780 (approximately 1.7320512). 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."[46] It is possible that he used an iterative procedure to calculate these values.[47]

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In The 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:

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 so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1⁄3. In The Sand Reckoner, Archimedes set out to calculate the number of grains of sand that the universe could contain. 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 (Gelo II, son of Hiero II), 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 As proven by Archimedes, the area of the region whether inhabited or uninhabited." To solve the problem, parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower Archimedes devised a system of counting based on the myriad. The figure. word is from the Greek μυριάς murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8×1063.[48]

Writings The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse.[49] The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to have existed only through references made to them 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.[b] During his lifetime, Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were collected by the Byzantine 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 Latin by Gerard of Cremona (c. 1114–1187 AD). 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.[50] Around the year 1586 Galileo Galilei invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.[51]

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Surviving works • On the Equilibrium of Planes (two volumes) The first book is in fifteen propositions with seven postulates, while the second book is in ten propositions. In this work Archimedes explains the Law of the Lever, stating, "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.[52] • On the Measurement of a Circle

Archimedes is said to have remarked of the lever: Give me a place to stand on, and I will move the Earth.

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 (π), showing that it is greater than 223⁄71 and less than 22⁄7. • 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 polar coordinates (r, θ) it can be described by the equation

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 the Sphere and the Cylinder (two volumes) In this 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 ⁄3πr3 for the sphere, and 2πr3 for the cylinder. The surface area is 4πr2 for the sphere, and 6πr2 for the cylinder (including its two bases), where r is the radius of the sphere and cylinder. The sphere has a volume two-thirds that of the circumscribed cylinder. Similarly, the sphere has an area two-thirds that of the cylinder (including the bases). A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request. • 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 (two volumes) In the first part of this treatise, 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. 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. Archimedes' principle of buoyancy is given in the work, stated as follows:

Archimedes

26 Any body wholly or partially immersed in a fluid experiences an upthrust equal to, but opposite in sense to, the weight of the fluid displaced.

• The 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 multiplied by the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio 1⁄4. • (O)stomachion 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. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Dr. Netz calculates that the pieces can be made into a square 17,152 ways.[53] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded.[54] 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).[55] Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of ὀστέον (osteon, bone) and μάχη (machē – fight). The puzzle is also known as the Loculus of Archimedes or Archimedes' Box.[56] • Archimedes' cattle problem This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a poem of 44 lines, 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. This version of the problem was first solved by A. Amthor[57] in 1880, and the answer is a very large number, approximately 7.760271×10206,544.[58] • The Sand Reckoner In this treatise, Archimedes counts the number of grains of sand that will fit inside 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 8×1063 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner or Psammites is the only surviving work in which Archimedes discusses his views on astronomy.[59] • The Method of Mechanical Theorems This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes 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.

Archimedes

Apocryphal works Archimedes' Book of Lemmas or Liber Assumptorum is a treatise with fifteen propositions on the nature of circles. The earliest known copy of the text is in Arabic. The scholars 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.[60] It has also been claimed that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes.[c] However, the first reliable reference to the formula is given by Heron of Alexandria in the 1st century AD.[61]

Archimedes Palimpsest The foremost document containing the work of Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople and examined a 174-page goatskin parchment of prayers written in the 13th century AD. He discovered that it was 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, which was a common practice in the Middle Ages as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th century AD copies of previously unknown treatises by Archimedes.[62] The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On Stomachion is a dissection puzzle in the Archimedes Palimpsest. October 29, 1998 it was sold at auction to an anonymous buyer for $2 [63] million at Christie's in New York. 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 is now stored at the Walters Art Museum in Baltimore, Maryland, where it has been subjected to a range of modern tests including the use of ultraviolet and x-ray light to read the overwritten text.[64] The treatises in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, Measurement of a Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems and Stomachion.

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Legacy • There is a crater on the Moon named Archimedes (29.7° N, 4.0° W) in his honor, as well as a lunar mountain range, the Montes Archimedes (25.3° N, 4.6° W).[65] • The asteroid 3600 Archimedes is named after him.[66] • 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 him which reads in Latin: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world).[67] • Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).[68]

The Fields Medal carries a portrait of Archimedes.

• 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.[69] • A movement for civic engagement targeting universal access to health care in the US state of Oregon has been named the "Archimedes Movement," headed by former Oregon Governor John Kitzhaber.[70]

Notes and references Notes a.  In the preface to On Spirals addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works. b.  The treatises by Archimedes known to exist only through references in the works of other authors are: On Sphere-Making and a work on polyhedra mentioned by Pappus of Alexandria; Catoptrica, a work on optics mentioned by Theon of Alexandria; Principles, addressed to Zeuxippus and explaining the number system used in The Sand Reckoner; On Balances and Levers; On Centers of Gravity; On the Calendar. Of the surviving works by Archimedes, T. L. Heath offers the following suggestion as to the order in which they were written: On the Equilibrium of Planes I, The Quadrature of the Parabola, On the Equilibrium of Planes II, On the Sphere and the Cylinder I, II, On Spirals, On Conoids and Spheroids, On Floating Bodies I, II, On the Measurement of a Circle, The Sand Reckoner. c.  Boyer, Carl Benjamin A History of Mathematics (1991) ISBN 0-471-54397-7 "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula — k = √(s(s − a)(s − b)(s − c)), where s is the semiperimeter — was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken chord' ... Archimedes is reported by the Arabs to have given several proofs of the theorem." d.   "It was usual to smear the seams or even the whole hull with pitch or with pitch and wax". In Νεκρικοὶ Διάλογοι (Dialogues of the Dead), Lucian refers to coating the seams of a skiff with wax, a reference to pitch (tar) or wax.[71]

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References [1] "Archimedes (c.287 - c.212 BC)" (http:/ / www. bbc. co. uk/ history/ historic_figures/ archimedes. shtml). BBC History. . Retrieved 2012-06-07. [2] "Archimedes Death Ray: Testing with MythBusters" (http:/ / web. mit. edu/ 2. 009/ www/ / experiments/ deathray/ 10_Mythbusters. html). MIT. . Retrieved 2007-07-23. [3] Calinger, Ronald (1999). A Contextual History of Mathematics. Prentice-Hall. p. 150. ISBN 0-02-318285-7. "Shortly after Euclid, compiler of the definitive textbook, came Archimedes of Syracuse (ca. 287 212 BC), the most original and profound mathematician of antiquity." [4] "Archimedes of Syracuse" (http:/ / www-history. mcs. st-and. ac. uk/ Biographies/ Archimedes. html). The MacTutor History of Mathematics archive. January 1999. . Retrieved 2008-06-09. [5] O'Connor, J.J. and Robertson, E.F. (February 1996). "A history of calculus" (http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ HistTopics/ The_rise_of_calculus. html). University of St Andrews. Archived (http:/ / web. archive. org/ web/ 20070715191704/ http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ HistTopics/ The_rise_of_calculus. html) from the original on 15 July 2007. . Retrieved 2007-08-07. [6] Bursill-Hall, Piers. "Galileo, Archimedes, and Renaissance engineers" (http:/ / www. sciencelive. org/ component/ option,com_mediadb/ task,view/ idstr,CU-MMP-PiersBursillHall/ Itemid,30). sciencelive with the University of Cambridge. . Retrieved 2007-08-07. [7] "Archimedes – The Palimpsest" (http:/ / web. archive. org/ web/ 20070928102802/ http:/ / www. archimedespalimpsest. org/ palimpsest_making1. html). Walters Art Museum. Archived from the original (http:/ / www. archimedespalimpsest. org/ palimpsest_making1. html) on 2007-09-28. . Retrieved 2007-10-14. [8] Heath, T. L., Works of Archimedes, 1897 [9] Plutarch. "Parallel Lives Complete e-text from Gutenberg.org" (http:/ / www. gutenberg. org/ etext/ 674). Project Gutenberg. Archived (http:/ / web. archive. org/ web/ 20070711045124/ http:/ / www. gutenberg. org/ etext/ 674) from the original on 11 July 2007. . Retrieved 2007-07-23. [10] O'Connor, J.J. and Robertson, E.F.. "Archimedes of Syracuse" (http:/ / www-history. mcs. st-andrews. ac. uk/ Biographies/ Archimedes. html). University of St Andrews. Archived (http:/ / web. archive. org/ web/ 20070206082010/ http:/ / www-history. mcs. st-andrews. ac. uk/ Biographies/ Archimedes. html) from the original on 6 February 2007. . Retrieved 2007-01-02. [11] Rorres, Chris. "Death of Archimedes: Sources" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Death/ Histories. html). Courant Institute of Mathematical Sciences. Archived (http:/ / web. archive. org/ web/ 20061210060235/ http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Death/ Histories. html) from the original on 10 December 2006. . Retrieved 2007-01-02. [12] Rorres, Chris. "Tomb of Archimedes: Sources" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Tomb/ Cicero. html). Courant Institute of Mathematical Sciences. Archived (http:/ / web. archive. org/ web/ 20061209201723/ http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Tomb/ Cicero. html) from the original on 9 December 2006. . Retrieved 2007-01-02. [13] Rorres, Chris. "Tomb of Archimedes – Illustrations" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Tomb/ TombIllus. html). Courant Institute of Mathematical Sciences. . Retrieved 2011-03-15. [14] Rorres, Chris. "Siege of Syracuse" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Siege/ Polybius. html). Courant Institute of Mathematical Sciences. Archived (http:/ / web. archive. org/ web/ 20070609013114/ http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Siege/ Polybius. html) from the original on 9 June 2007. . Retrieved 2007-07-23. [15] Vitruvius. "De Architectura, Book IX, paragraphs 9–12, text in English and Latin" (http:/ / penelope. uchicago. edu/ Thayer/ E/ Roman/ Texts/ Vitruvius/ 9*. html). University of Chicago. . Retrieved 2007-08-30. [16] "Incompressibility of Water" (http:/ / www. fas. harvard. edu/ ~scdiroff/ lds/ NewtonianMechanics/ IncompressibilityofWater/ IncompressibilityofWater. html). Harvard University. Archived (http:/ / web. archive. org/ web/ 20080317130651/ http:/ / www. fas. harvard. edu/ ~scdiroff/ lds/ NewtonianMechanics/ IncompressibilityofWater/ IncompressibilityofWater. html) from the original on 17 March 2008. . Retrieved 2008-02-27. [17] HyperPhysics. "Buoyancy" (http:/ / hyperphysics. phy-astr. gsu. edu/ Hbase/ pbuoy. html). Georgia State University. Archived (http:/ / web. archive. org/ web/ 20070714113647/ http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ pbuoy. html#c1) from the original on 14 July 2007. . Retrieved 2007-07-23. [18] Rorres, Chris. "The Golden Crown" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Crown/ CrownIntro. html). Drexel University. Archived (http:/ / web. archive. org/ web/ 20090311051318/ http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Crown/ CrownIntro. html) from the original on 11 March 2009. . Retrieved 2009-03-24. [19] Carroll, Bradley W. "Archimedes' Principle" (http:/ / www. physics. weber. edu/ carroll/ Archimedes/ principle. htm). Weber State University. Archived (http:/ / web. archive. org/ web/ 20070808132323/ http:/ / physics. weber. edu/ carroll/ Archimedes/ principle. htm) from the original on 8 August 2007. . Retrieved 2007-07-23. [20] Rorres, Chris. "The Golden Crown: Galileo's Balance" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Crown/ bilancetta. html). Drexel University. Archived (http:/ / web. archive. org/ web/ 20090224221137/ http:/ / math. nyu. edu/ ~crorres/ Archimedes/ Crown/ bilancetta. html) from the original on 24 February 2009. . Retrieved 2009-03-24. [21] Casson, Lionel (1971). Ships and Seamanship in the Ancient World. Princeton University Press. ISBN 0-691-03536-9. [22] Dalley, Stephanie. Oleson, John Peter. "Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World" (http:/ / muse. jhu. edu/ journals/ technology_and_culture/ toc/ tech44. 1. html). Technology and Culture Volume 44, Number 1, January 2003 (PDF). . Retrieved 2007-07-23. [23] Rorres, Chris. "Archimedes screw – Optimal Design" (http:/ / www. cs. drexel. edu/ ~crorres/ Archimedes/ Screw/ optimal/ optimal. html). Courant Institute of Mathematical Sciences. . Retrieved 2007-07-23.

29

Archimedes [24] An animation of an Archimedes screw [25] "SS Archimedes" (http:/ / www. wrecksite. eu/ wreck. aspx?636). wrecksite.eu. . Retrieved 2011-01-22. [26] Rorres, Chris. "Archimedes' Claw – Illustrations and Animations – a range of possible designs for the claw" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Claw/ illustrations. html). Courant Institute of Mathematical Sciences. . Retrieved 2007-07-23. [27] Carroll, Bradley W. "Archimedes' Claw – watch an animation" (http:/ / physics. weber. edu/ carroll/ Archimedes/ claw. htm). Weber State University. Archived (http:/ / web. archive. org/ web/ 20070813202716/ http:/ / physics. weber. edu/ carroll/ Archimedes/ claw. htm) from the original on 13 August 2007. . Retrieved 2007-08-12. [28] Hippias, 2 (cf. Galen, On temperaments 3.2, who mentions pyreia, "torches"); Anthemius of Tralles, On miraculous engines 153 [Westerman]. [29] John Wesley. "A Compendium of Natural Philosophy (1810) Chapter XII, Burning Glasses" (http:/ / web. archive. org/ web/ 20071012154432/ http:/ / wesley. nnu. edu/ john_wesley/ wesley_natural_philosophy/ duten12. htm). Online text at Wesley Center for Applied Theology. Archived from the original (http:/ / wesley. nnu. edu/ john_wesley/ wesley_natural_philosophy/ duten12. htm) on 2007-10-12. . Retrieved 2007-09-14. [30] "Archimedes' Weapon" (http:/ / www. time. com/ time/ magazine/ article/ 0,9171,908175,00. html?promoid=googlep). Time Magazine. November 26, 1973. . Retrieved 2007-08-12. [31] Bonsor, Kevin. "How Wildfires Work" (http:/ / science. howstuffworks. com/ wildfire. htm). HowStuffWorks. Archived (http:/ / web. archive. org/ web/ 20070714174036/ http:/ / science. howstuffworks. com/ wildfire. htm) from the original on 14 July 2007. . Retrieved 2007-07-23. [32] Fuels and Chemicals – Auto Ignition Temperatures (http:/ / www. engineeringtoolbox. com/ fuels-ignition-temperatures-d_171. html) [33] "TV Review: MythBusters 8.27 – President's Challenge" (http:/ / fandomania. com/ tv-review-mythbusters-8-27-presidents-challenge/ ). . Retrieved 2010-12-18. [34] Rorres, Chris. "The Law of the Lever According to Archimedes" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Lever/ LeverLaw. html). Courant Institute of Mathematical Sciences. . Retrieved 2010-03-20. [35] Clagett, Marshall (2001). Greek Science in Antiquity (http:/ / books. google. com/ ?id=mweWMAlf-tEC& pg=PA72& lpg=PA72& dq=archytas+ lever& q=archytas lever). Dover Publications. ISBN 978-0-486-41973-2. . Retrieved 2010-03-20. [36] Quoted by Pappus of Alexandria in Synagoge, Book VIII [37] Dougherty, F. C.; Macari, J.; Okamoto, C.. "Pulleys" (http:/ / www. swe. org/ iac/ lp/ pulley_03. html). Society of Women Engineers. Archived (http:/ / web. archive. org/ web/ 20070718031943/ http:/ / www. swe. org/ iac/ LP/ pulley_03. html) from the original on 18 July 2007. . Retrieved 2007-07-23. [38] "Ancient Greek Scientists: Hero of Alexandria" (http:/ / www. tmth. edu. gr/ en/ aet/ 5/ 55. html). Technology Museum of Thessaloniki. Archived (http:/ / web. archive. org/ web/ 20070905125400/ http:/ / www. tmth. edu. gr/ en/ aet/ 5/ 55. html) from the original on 5 September 2007. . Retrieved 2007-09-14. [39] Cicero. "De re publica 1.xiv §21" (http:/ / www. thelatinlibrary. com/ cicero/ repub1. shtml#21). thelatinlibrary.com. . Retrieved 2007-07-23. [40] Cicero. "De re publica Complete e-text in English from Gutenberg.org" (http:/ / www. gutenberg. org/ etext/ 14988). Project Gutenberg. Archived (http:/ / web. archive. org/ web/ 20070929122153/ http:/ / www. gutenberg. org/ etext/ 14988) from the original on 29 September 2007. . Retrieved 2007-09-18. [41] Rorres, Chris. "Spheres and Planetaria" (http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Sphere/ SphereIntro. html). Courant Institute of Mathematical Sciences. . Retrieved 2007-07-23. [42] "Ancient Moon 'computer' revisited" (http:/ / news. bbc. co. uk/ 1/ hi/ sci/ tech/ 6191462. stm). BBC News. November 29, 2006. . Retrieved 2007-07-23. [43] Plutarch. "Extract from Parallel Lives" (http:/ / fulltextarchive. com/ pages/ Plutarch-s-Lives10. php#p35). fulltextarchive.com. . Retrieved 2009-08-10. [44] Heath, T.L.. "Archimedes on measuring the circle" (http:/ / www. math. ubc. ca/ ~cass/ archimedes/ circle. html). math.ubc.ca. . Retrieved 2012-10-30. [45] Kaye, R.W.. "Archimedean ordered fields" (http:/ / web. mat. bham. ac. uk/ R. W. Kaye/ seqser/ archfields). web.mat.bham.ac.uk. . Retrieved 2009-11-07. [46] Quoted in Heath, T. L. Works of Archimedes, Dover Publications, ISBN 0-486-42084-1. [47] McKeeman, Bill. "The Computation of Pi by Archimedes" (http:/ / www. mathworks. com/ matlabcentral/ fileexchange/ 29504-the-computation-of-pi-by-archimedes/ content/ html/ ComputationOfPiByArchimedes. html#37). Matlab Central. . Retrieved 2012-10-30. [48] Carroll, Bradley W. "The Sand Reckoner" (http:/ / physics. weber. edu/ carroll/ Archimedes/ sand. htm). Weber State University. Archived (http:/ / web. archive. org/ web/ 20070813215029/ http:/ / physics. weber. edu/ carroll/ Archimedes/ sand. htm) from the original on 13 August 2007. . Retrieved 2007-07-23. [49] Encyclopedia of ancient Greece By Wilson, Nigel Guy p. 77 (http:/ / books. google. com/ books?id=-aFtPdh6-2QC& pg=PA77) ISBN 0-7945-0225-3 (2006) [50] "Editions of Archimedes' Work" (http:/ / www. brown. edu/ Facilities/ University_Library/ exhibits/ math/ wholefr. html). Brown University Library. Archived (http:/ / web. archive. org/ web/ 20070808235638/ http:/ / www. brown. edu/ Facilities/ University_Library/ exhibits/ math/ wholefr. html) from the original on 8 August 2007. . Retrieved 2007-07-23.

30

Archimedes [51] Van Helden, Al. "The Galileo Project: Hydrostatic Balance" (http:/ / galileo. rice. edu/ sci/ instruments/ balance. html). Rice University. Archived (http:/ / web. archive. org/ web/ 20070905185039/ http:/ / galileo. rice. edu/ sci/ instruments/ balance. html) from the original on 5 September 2007. . Retrieved 2007-09-14. [52] Heath, T.L.. "The Works of Archimedes (1897). The unabridged work in PDF form (19 MB)" (http:/ / www. archive. org/ details/ worksofarchimede029517mbp). Archive.org. Archived (http:/ / web. archive. org/ web/ 20071006033058/ http:/ / www. archive. org/ details/ worksofarchimede029517mbp) from the original on 6 October 2007. . Retrieved 2007-10-14. [53] Kolata, Gina (December 14, 2003). "In Archimedes' Puzzle, a New Eureka Moment" (http:/ / query. nytimes. com/ gst/ fullpage. html?res=9D00E6DD133CF937A25751C1A9659C8B63& sec=& spon=& pagewanted=all). The New York Times. . Retrieved 2007-07-23. [54] Ed Pegg Jr. (November 17, 2003). "The Loculus of Archimedes, Solved" (http:/ / www. maa. org/ editorial/ mathgames/ mathgames_11_17_03. html). Mathematical Association of America. Archived (http:/ / web. archive. org/ web/ 20080519094951/ http:/ / www. maa. org/ editorial/ mathgames/ mathgames_11_17_03. html) from the original on 19 May 2008. . Retrieved 2008-05-18. [55] Rorres, Chris. "Archimedes' Stomachion" (http:/ / math. nyu. edu/ ~crorres/ Archimedes/ Stomachion/ intro. html). Courant Institute of Mathematical Sciences. Archived (http:/ / web. archive. org/ web/ 20071026005336/ http:/ / www. math. nyu. edu/ ~crorres/ Archimedes/ Stomachion/ intro. html) from the original on 26 October 2007. . Retrieved 2007-09-14. [56] "Graeco Roman Puzzles" (http:/ / www. archimedes-lab. org/ latin. html#archimede). Gianni A. Sarcone and Marie J. Waeber. Archived (http:/ / web. archive. org/ web/ 20080514130547/ http:/ / www. archimedes-lab. org/ latin. html) from the original on 14 May 2008. . Retrieved 2008-05-09. [57] Krumbiegel, B. and Amthor, A. Das Problema Bovinum des Archimedes, Historisch-literarische Abteilung der Zeitschrift Für Mathematik und Physik 25 (1880) pp. 121–136, 153–171. [58] Calkins, Keith G. "Archimedes' Problema Bovinum" (http:/ / web. archive. org/ web/ 20071012171254/ http:/ / andrews. edu/ ~calkins/ profess/ cattle. htm). Andrews University. Archived from the original (http:/ / www. andrews. edu/ ~calkins/ profess/ cattle. htm) on 2007-10-12. . Retrieved 2007-09-14. [59] "English translation of The Sand Reckoner" (http:/ / www. math. uwaterloo. ca/ navigation/ ideas/ reckoner. shtml). University of Waterloo. Archived (http:/ / web. archive. org/ web/ 20070811235335/ http:/ / www. math. uwaterloo. ca/ navigation/ ideas/ reckoner. shtml) from the original on 11 August 2007. . Retrieved 2007-07-23. [60] "Archimedes' Book of Lemmas" (http:/ / www. cut-the-knot. org/ Curriculum/ Geometry/ BookOfLemmas/ index. shtml). cut-the-knot. Archived (http:/ / web. archive. org/ web/ 20070711111858/ http:/ / www. cut-the-knot. org/ Curriculum/ Geometry/ BookOfLemmas/ index. shtml) from the original on 11 July 2007. . Retrieved 2007-08-07. [61] O'Connor, J.J. and Robertson, E.F. (April 1999). "Heron of Alexandria" (http:/ / www-history. mcs. st-and. ac. uk/ Biographies/ Heron. html). University of St Andrews. . Retrieved 2010-02-17. [62] Miller, Mary K. (March 2007). "Reading Between the Lines" (http:/ / www. smithsonianmag. com/ science-nature/ archimedes. html). Smithsonian Magazine. Archived (http:/ / web. archive. org/ web/ 20080119024939/ http:/ / www. smithsonianmag. com/ science-nature/ archimedes. html?) from the original on 19 January 2008. . Retrieved 2008-01-24. [63] "Rare work by Archimedes sells for $2 million" (http:/ / web. archive. org/ web/ 20080516000109/ http:/ / edition. cnn. com/ books/ news/ 9810/ 29/ archimedes/ ). CNN. October 29, 1998. Archived from the original (http:/ / edition. cnn. com/ books/ news/ 9810/ 29/ archimedes/ ) on May 16, 2008. . Retrieved 2008-01-15. [64] "X-rays reveal Archimedes' secrets" (http:/ / news. bbc. co. uk/ 1/ hi/ sci/ tech/ 5235894. stm). BBC News. August 2, 2006. Archived (http:/ / web. archive. org/ web/ 20070825091847/ http:/ / news. bbc. co. uk/ 1/ hi/ sci/ tech/ 5235894. stm) from the original on 25 August 2007. . Retrieved 2007-07-23. [65] Friedlander, Jay and Williams, Dave. "Oblique view of Archimedes crater on the Moon" (http:/ / nssdc. gsfc. nasa. gov/ imgcat/ html/ object_page/ a15_m_1541. html). NASA. Archived (http:/ / web. archive. org/ web/ 20070819054033/ http:/ / nssdc. gsfc. nasa. gov/ imgcat/ html/ object_page/ a15_m_1541. html) from the original on 19 August 2007. . Retrieved 2007-09-13. [66] "Planetary Data System" (http:/ / starbrite. jpl. nasa. gov/ pds-explorer/ index. jsp?selection=othertarget& targname=3600 ARCHIMEDES). NASA. Archived (http:/ / web. archive. org/ web/ 20071012171730/ http:/ / starbrite. jpl. nasa. gov/ pds-explorer/ index. jsp?selection=othertarget& targname=3600+ ARCHIMEDES) from the original on 12 October 2007. . Retrieved 2007-09-13. [67] "Fields Medal" (http:/ / web. archive. org/ web/ 20070701033751/ http:/ / www. mathunion. org/ medals/ Fields/ AboutPhotos. html). International Mathematical Union. Archived from the original (http:/ / www. mathunion. org/ medals/ Fields/ AboutPhotos. html) on July 1, 2007. . Retrieved 2007-07-23. [68] Rorres, Chris. "Stamps of Archimedes" (http:/ / math. nyu. edu/ ~crorres/ Archimedes/ Stamps/ stamps. html). Courant Institute of Mathematical Sciences. . Retrieved 2007-08-25. [69] "California Symbols" (http:/ / www. capitolmuseum. ca. gov/ VirtualTour. aspx?content1=1278& Content2=1374& Content3=1294). California State Capitol Museum. Archived (http:/ / web. archive. org/ web/ 20071012123245/ http:/ / capitolmuseum. ca. gov/ VirtualTour. aspx?content1=1278& Content2=1374& Content3=1294) from the original on 12 October 2007. . Retrieved 2007-09-14. [70] "The Archimedes Movement" (http:/ / www. archimedesmovement. org/ ). . [71] Casson, Lionel (1995). Ships and seamanship in the ancient world (http:/ / books. google. com/ books?id=sDpMh0gK2OUC& pg=PA18& dq=why+ were+ homer's+ ships+ black#v=onepage& q=why were homer's ships black& f=false). Baltimore: The Johns Hopkins University Press. pp. 211–212. ISBN 978-0-8018-5130-8. .

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Further reading • Boyer, Carl Benjamin (1991). A History of Mathematics. New York: Wiley. ISBN 0-471-54397-7. • Clagett, Marshall (1964-1984). Archimedes in the Middle Ages. 5 vols. Madison, WI: University of Wisconsin Press. • Dijksterhuis, E.J. (1987). Archimedes. Princeton University Press, Princeton. ISBN 0-691-08421-1. Republished translation of the 1938 study of Archimedes and his works by an historian of science. • Gow, Mary (2005). Archimedes: Mathematical Genius of the Ancient World. Enslow Publishers, Inc. ISBN 0-7660-2502-0. • Hasan, Heather (2005). Archimedes: The Father of Mathematics. Rosen Central. ISBN 978-1-4042-0774-5. • Heath, T.L. (1897). Works of Archimedes. Dover Publications. ISBN 0-486-42084-1. Complete works of Archimedes in English. • Netz, Reviel and Noel, William (2007). The Archimedes Codex. Orion Publishing Group. ISBN 0-297-64547-1. • Pickover, Clifford A. (2008). Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. ISBN 978-0-19-533611-5. • Simms, Dennis L. (1995). Archimedes the Engineer. Continuum International Publishing Group Ltd. ISBN 0-7201-2284-8. • Stein, Sherman (1999). Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 0-88385-718-9.

The Works of Archimedes online • Text in Classical Greek: PDF scans of Heiberg's edition of the Works of Archimedes, now in the public domain (http://www.wilbourhall.org) • In English translation: The Works of Archimedes (http://www.archive.org/details/ worksofarchimede029517mbp), trans. T.L. Heath; supplemented by The Method of Mechanical Theorems (http:// books.google.com/books?id=suYGAAAAYAAJ), trans. L.G. Robinson

External links • Archimedes (http://www.bbc.co.uk/programmes/b00773bv) on In Our Time at the BBC. ( listen now (http:// www.bbc.co.uk/iplayer/console/b00773bv/In_Our_Time_Archimedes)) • Archimedes (https://inpho.cogs.indiana.edu/thinker/2546) at the Indiana Philosophy Ontology Project • Archimedes (http://philpapers.org/s/archimedes) at PhilPapers • The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland (http://www. archimedespalimpsest.org/) • The Mathematical Achievements and Methodologies of Archimedes (http://mathdb.org/articles/archimedes/ e_archimedes.htm) • "Archimedes and the Square Root of 3" (http://www.mathpages.com/home/kmath038/kmath038.htm) at MathPages.com. • "Archimedes on Spheres and Cylinders" (http://www.mathpages.com/home/kmath343/kmath343.htm) at MathPages.com. • Photograph of the Sakkas experiment in 1973 (http://www.cs.drexel.edu/~crorres/bbc_archive/ mirrors_sailors_sakas.jpg) • Testing the Archimedes steam cannon (http://web.mit.edu/2.009/www/experiments/steamCannon/ ArchimedesSteamCannon.html) • Stamps of Archimedes (http://www.stampsbook.org/subject/Archimedes.html) • Eureka! 1,000-year-old text by Greek maths genius Archimedes goes on display (http://www.dailymail.co.uk/ sciencetech/article-2050631/Eureka-1-000-year-old-text-Greek-maths-genius-Archimedes-goes-display.html)

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Archimedes

33

Daily Mail, October 18, 2011.

Archimedes' cattle problem Archimedes' cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions. The problem was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of forty-four lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by A. Amthor. He gave the exact solution using exponentials and showed that it was about cattle, far more than could fit in the observable universe. The decimal form is too long for humans to calculate exactly, but multiple precision arithmetic packages on computers can easily write it out explicitly.

History In 1769, Gotthold Ephraim Lessing was appointed librarian of the Herzog August Library in Wolfenbüttel, Germany, which contained many Greek and Latin manuscripts.[1] A few years later, Lessing published translations of some of the manuscripts with commentaries. Among them was a Greek poem of forty-four lines, containing an arithmetical problem which asks the reader to find the number of cattle in the herd of the god of the sun. The name of Archimedes appears in the title of the poem, it being said that he sent it in a letter to Eratosthenes to be investigated by the mathematicians of Alexandria. The claim that Archimedes authored the poem is disputed, though, as no mention of the problem has been found in the writings of the Greek mathematicians.[2]

Problem The problem, from an abridgement of the German translations published by Georg Nesselmann in 1842, and by Krumbiegel in 1880, states: Compute, O friend, the number of the cattle of the sun which once grazed upon the plains of Sicily, divided according to color into four herds, one milk-white, one black, one dappled and one yellow. The number of bulls is greater than the number of cows, and the relations between them are as follows: White bulls

black bulls + yellow bulls,

Black bulls

dappled bulls + yellow bulls,

Dappled bulls

white bulls + yellow bulls,

White cows

black herd,

Black cows

dappled herd,

Dappled cows Yellow cows

yellow herd, white herd.

Archimedes' cattle problem

34

If thou canst give, O friend, the number of each kind of bulls and cows, thou art no novice in numbers, yet can not be regarded as of high skill. Consider, however, the following additional relations between the bulls of the sun: White bulls + black bulls = a square number, Dappled bulls + yellow bulls = a triangular number. If thou hast computed these also, O friend, and found the total number of cattle, then exult as a conqueror, for thou hast proved thyself most skilled in numbers.[2]

Solution The first part of the problem can be solved readily by setting up a system of equations. If the number of white, black, dappled, and yellow bulls are written as and , and the number of white, black, dappled, and yellow cows are written as

and

, the problem is simply to find a solution to:

which is a system of seven equations with eight unknowns. It is indeterminate, and has infinitely many solutions. The least positive integers satisfying the seven equations are:

which is a total of 50,389,082 cattle[2] and the other solutions are integral multiples of these. Note that the first four numbers are multiples of 4657, a value which will appear repeatedly below. The general solution to the second part of the problem was first found by A. Amthor[3] in 1880. The following version of it was described by H. W. Lenstra,[4] based on Pell's equation: the solution given above for the first part of the problem should be multiplied by

where

Archimedes' cattle problem and j is any positive integer. Equivalently, squaring w results in,

where {u,v} are the fundamental solutions of the Pell equation,

The size of the smallest herd that could satisfy both the first and second parts of the problem is then given by j = 1, and is about (first solved by Amthor). Modern computers can easily print out all digits of the answer. This was first done at the University of Waterloo, in 1965 by Hugh C. Williams, R. A. German, and Charles Robert Zarnke. They used a combination of the IBM 7040 and IBM 1620 computers.[5]

Pell Equation The constraints of the second part of the problem are straightforward and the actual Pell equation that needs to be solved can easily be given. First, it asks that B+W should be a square, or using the values given above, thus one should set k = (3)(11)(29)(4657)q2 for some integer q. That solves the first condition. For the second, it requires that D+Y should be a triangular number,

Solving for t,

Substituting the value of D+Y and k and finding a value of q2 such that the discriminant of this quadratic is a perfect square p2 entails solving the Pell equation,

Amthor's approach discussed in the previous section was essentially to find the smallest v such that it is integrally divisible by 2*4657. The fundamental solution of this equation has more than 100,000 digits.

References [1] Rorres, Chris. "Archimedes' Cattle Problem (Statement)" (http:/ / www. mcs. drexel. edu/ ~crorres/ Archimedes/ Cattle/ Statement. html). Archived (http:/ / web. archive. org/ web/ 20070124203443/ http:/ / www. mcs. drexel. edu/ ~crorres/ Archimedes/ Cattle/ Statement. html) from the original on 24 January 2007. . Retrieved 2007-01-24. [2] Merriman, Mansfield (1905). "The Cattle Problem of Archimedes". Popular Science Monthly 67: 660–665. [3] B. Krumbiegel, A. Amthor, Das Problema Bovinum des Archimedes, Historisch-literarische Abteilung der Zeitschrift Für Mathematik und Physik 25 (1880) 121-136, 153-171. [4] Lenstra, H. W. (2002). "Solving the Pell equation" (http:/ / www. ams. org/ notices/ 200202/ fea-lenstra. pdf) (PDF). Notices of the American Mathematical Society 29 (2): 182–192. . [5] Harold Alkema and Kenneth McLaughlin (2007). "Unbundling Computing at The University of Waterloo" (http:/ / www. cs. uwaterloo. ca/ 40th/ Chronology/ printable. shtml). University of Waterloo. Archived (http:/ / web. archive. org/ web/ 20110404172741/ http:/ / www. cs. uwaterloo. ca/ 40th/ Chronology/ printable. shtml) from the original on 4 April 2011. . Retrieved April 5, 2011. (includes pictures)

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Archimedes' cattle problem

36

Further reading • Dörrie, Heinrich (1965). "Archimedes' Problema Bovinum". 100 Great Problems of Elementary Mathematics. Dover Publications. pp. 3–7. • Williams, H. C.; German, R. A.; and Zarnke, C. R. (1965). "Solution of the Cattle Problem of Archimedes". Mathematics of Computation (American Mathematical Society) 19 (92): pp. 671–674. doi:10.2307/2003954. JSTOR 2003954. • Vardi, I. (1998). "Archimedes' Cattle Problem". American Mathematical Monthly (Mathematical Association of America) 105 (4): pp. 305–319. doi:10.2307/2589706.

Book of Lemmas The Book of Lemmas is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions on circles.[1]

History Translations The Book of Lemmas was first introduced in Arabic by Thābit ibn Qurra; he attributed the work to Archimedes. In 1661, the Arabic manuscript was translated into Latin by Abraham Ecchellensis and edited by Giovanni A. Borelli. The Latin version was published under the name Liber Assumptorum.[2] T. L. Heath translated Heiburg's Latin work into English in his The Works of Archimedes.[3][4] The first page of the Book of Lemmas as seen in The Works of Archimedes (1897).

Authorship The original authorship of the Book of Lemmas has been in question because in proposition four, the book refers to Archimedes in third person; however, it has been suggested that it may have been added by the translator.[5] Another possibility is that the Book of Lemmas may be a collection of propositions by Archimedes later collected by a Greek writer.[1]

New geometrical figures The Book of Lemmas introduces several new geometrical figures.

Book of Lemmas

37

Arbelos Archimedes' first introduced the arbelos in proposition four of his book:

The arbelos is the shaded region (grey).

“

If AB be the diameter of a semicircle and N any point on AB, and if semicircles be described within the first semicircle and having AN, BN as diameters respectively, the figure included between the circumferences of the three semicircles is "what Archimedes called αρβηλος"; and its [1] area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P.

”

The figure is used in propositions four through eight. In propositions five, Archimedes introduces the Archimedes' twin circles, and in proposition eight, he makes use what would be the Pappus chain, formally introduced by Pappus of Alexandria.

Salinon Archimedes' first introduced the salinon in proposition fourteen of his book:

The salinon is the blue shaded region.

“

Let ACB be a semicircle on AB as diameter, and let AD, BE be equal lengths measured along AB from A, B respectively. On AD, BE as diameters describe semicircles on the side towards C, and on DE as diameter a semicircle on the opposite side. Let the perpendicular to AB through O, the centre of the first semicircle, meet the opposite semicircles in C, F respectively. Then shall the area of the figure bounded by [1] the circumferences of all the semicircles be equal to the area of the circle on CF as diameter.

Archimedes proved that the salinon and the circle are equal in area.

”

Book of Lemmas

Propositions 1. If two circles touch at A, and if CD, EF be parallel diameters in them, ADF is a straight line. 2. Let AB be the diameter of a semicircle, and let the tangents to it at B and at any other point D on it meet in T. If now DE be drawn perpendicular to AB, and if AT, DE meet in F, then DF = FE. 3. Let P be any point on a segment of a circle whose base is AB, and let PN be perpendicular to AB. Take D on AB so that AN = ND. If now PQ be an arc equal to the arc PA, and BQ be joined, then BQ, BD shall be equal. 4. If AB be the diameter of a semicircle and N any point on AB, and if semicircles be described within the first semicircle and having AN, BN as diameters respectively, the figure included between the circumferences of the three semicircles is "what Archimedes called αρβηλος"; and its area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P. 5. Let AB be the diameter of a semicircle, C any point on AB, and CD perpendicular to it, and let semicircles be described within the first semicircle and having AC, CB as diameters. Then if two circles be drawn touching CD on different sides and each touching two of the semicircles, the circles so drawn will be equal. 6. Let AB, the diameter of a semicircle, be divided at C so that AC = 3/2 × CB [or in any ratio]. Describe semicircles within the first semicircle and on AC, CB as diameters, and suppose a circle drawn touching the all three semicircles. If GH be the diameter of this circle, to find relation between GH and AB. 7. If circles are circumscribed about and inscribed in a square, the circumscribed circle is double of the inscribed square. 8. If AB be any chord of a circle whose centre is O, and if AB be produced to C so that BC is equal to the radius; if further CO meets the circle in D and be produced to meet the circle the second time in E, the arc AE will be equal to three times the arc BD. 9. If in a circle two chords AB, CD which do not pass through the centre intersect at right angles, then (arc AD) + (arc CB) = (arc AC) + (arc DB). 10. Suppose that TA, TB are two tangents to a circle, while TC cuts it. Let BD be the chord through B parallel to TC, and let AD meet TC in E. Then, if EH be drawn perpendicular to BD, it will bisect it in H. 11. If two chords AB, CD in a circle intersect at right angles in a point O, not being the centre, then AO2 + BO2 + CO2 + DO2 = (diameter)2. 12. If AB be the diameter of a semicircle, and TP, TQ the tangents to it from any point T, and if AQ, BP be joined meeting in R, then TR is perpendicular to AB. 13. If a diameter AB of a circle meet any chord CD, not a diameter, in E, and if AM, BN be drawn perpendicular to CD, then CN = DM. 14. Let ACB be a semicircle on AB as diameter, and let AD, BE be equal lengths measured along AB from A, B respectively. On AD, BE as diameters describe semicircles on the side towards C, and on DE as diameter a semicircle on the opposite side. Let the perpendicular to AB through O, the centre of the first semicircle, meet the opposite semicircles in C, F respectively. Then shall the area of the figure bounded by the circumferences of all the semicircles be equal to the area of the circle on CF as diameter. 15. Let AB be the diameter of a circle., AC a side of an inscribed regular pentagon, D the middle point of the arc AC. Join CD and produce it to meet BA produced in E; join AC, DB meeting in F, and Draw FM perpendicular to AB. Then EM = (radius of circle).[1]

38

Book of Lemmas

References [1] Heath, Thomas Little (1897), The Works of Archimedes (http:/ / books. google. com/ ?id=bTEPAAAAIAAJ& printsec=titlepage), Cambridge University: University Press, pp. xxxii, 301–318, , retrieved 2008-06-15 [2] "From Euclid to Newton" (http:/ / www. brown. edu/ Facilities/ University_Library/ exhibits/ math/ nofr. html). Brown University. . Retrieved 2008-06-24. [3] Aaboe, Asger (1997), Episodes from the Early History of Mathematics (http:/ / books. google. com/ ?id=5wGzF0wPFYgC& printsec=frontcover), Washington, D.C.: Math. Assoc. of America, pp. 77, 85, ISBN 0-88385-613-1, , retrieved 2008-06-19 [4] Glick, Thomas F.; Livesey, Steven John; Wallis, Faith (2005), Medieval Science, Technology, and Medicine: An Encyclopedia (http:/ / books. google. com/ ?id=SaJlbWK_-FcC& printsec=frontcover#PPT9,M1), New York: Routledge, p. 41, ISBN 0-415-96930-1, , retrieved 2008-06-19 [5] Bogomolny, A. "Archimedes' Book of Lemmas" (http:/ / www. cut-the-knot. org/ Curriculum/ Geometry/ BookOfLemmas/ index. shtml). Cut-the-Knot. . Retrieved 2008-06-19.

Archimedes Palimpsest The Archimedes Palimpsest is a palimpsest (ancient overwritten manuscript) on parchment in the form of a codex (hand-written bound book, as opposed to a scroll). It originally was a 10th-century Byzantine copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes (c. 287 BC–c. 212 BC) of Syracuse and other authors, which was overwritten with a religious text. The manuscript currently belongs to an American private collector.

Overview Archimedes lived in the 3rd century BC, and a copy of his work was Ostomachion is a dissection puzzle in the Archimedes Palimpsest (shown after Suter from a made around 950 AD in the Byzantine Empire by an anonymous different source; this version must be stretched to scribe.[1] In 1229 the original Archimedes codex was unbound, scraped twice the width to conform to the Palimpsest) and washed, along with at least six other parchment manuscripts, including one with works of Hypereides. The parchment leaves were folded in half and reused for a Christian liturgical text of 177 pages; the older leaves folded so that each became two leaves of the liturgical book. The erasure was incomplete, and Archimedes' work is now readable after scientific and scholarly work from 1998 to 2008 using digital processing of images produced by ultraviolet, infrared, visible and raking light, and X-ray.[2][3] In 1906 it was briefly inspected in Istanbul by the Danish philologist Johan Ludvig Heiberg. With the aid of black-and-white photographs he arranged to have taken, he published a transcription of the Archimedes text. Shortly thereafter Archimedes' Greek text was translated into English by T. L. Heath. Before that it was not widely known among mathematicians, physicists or historians. It contains: • • • • • •

"On the Equilibrium of Planes" "Spiral Lines" "Measurement of a Circle" "On the Sphere and Cylinder" "On Floating Bodies" (only known copy in Greek) "The Method of Mechanical Theorems" (only known copy)

• "Stomachion" (only known copy).

39

Archimedes Palimpsest

40

The palimpsest also contains speeches by the 4th century BC politician Hypereides, a commentary on Aristotle's Categories by Alexander of Aphrodisias, and other works.[4]

Mathematical content The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy. In his other works, Archimedes often proves the equality of two areas or volumes with Eudoxus' method of exhaustion, an ancient Greek counterpart of the modern method of limits. Since the Greeks were aware that some numbers were irrational, their notion of a real number was a quantity Q approximated by two sequences, one providing an upper bound and the other a lower bound. If you find two sequences U and L, with U always bigger than Q, and L always smaller than Q, and if the two sequences eventually came closer together than any prespecified amount, then Q is found, or exhausted, by U and L. Archimedes used exhaustion to prove his theorems. This involved approximating the figure whose area he wanted to compute into sections of known area, which provide upper and lower bounds for the area of the figure. He then proved that the two bounds become equal when the subdivision becomes arbitrarily fine. These proofs, still considered to be rigorous and correct, used geometry with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in The Method.

A typical page from the Archimedes Palimpsest. The text of the prayer book is seen from top to bottom, the original Archimedes manuscript is seen as fainter text below it running from left to right

The method that Archimedes describes was based upon his investigations of physics, on the center of mass and the law of the lever. He compared the area or volume of a figure of which he knew the total mass and center of mass with the area or volume of another figure he did not know anything about. He divided both figures into infinitely many slices of infinitesimal width, and balanced each slice of one figure against a corresponding slice of the second figure on a lever. The essential point is that the two figures are oriented differently, so that the corresponding slices are at different distances from the fulcrum, and the condition that the slices balance is not the same as the condition that they are equal.

Archimedes Palimpsest

41

Once he shows that each slice of one figure balances each slice of the other figure, he concludes that the two figures balance each other. But the center of mass of one figure is known, and the total mass can be placed at this center and it still balances. The second figure has an unknown mass, but the position of its center of mass might be restricted to lie at a certain distance from the fulcrum by a geometrical argument, by symmetry. The condition that the two figures balance now allows him to calculate the total mass of the other figure. He considered this method as a useful heuristic but always made sure to prove the results he found using exhaustion, since the method did not provide upper and lower bounds. Using this method, Archimedes was able to solve several problems now treated by integral calculus, which was given its modern form in the seventeenth century by Isaac Newton and Gottfried Leibniz. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. (For explicit details, see Archimedes' use of infinitesimals.)

After imaging a page from the palimpsest, the original Archimedes text is now seen clearly

When rigorously proving theorems, Archimedes often used what are now called Riemann sums. In "On the Sphere and Cylinder," he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly. He adds the areas of the cones, which is a type of Riemann sum for the area of the sphere considered as a surface of revolution. But there are two essential differences between Archimedes' method and 19th-century methods: 1. Archimedes did not know about differentiation, so he could not calculate any integrals other than those that came from center-of-mass considerations, by symmetry. While he had a notion of linearity, to find the volume of a sphere he had to balance two figures at the same time; he never figured out how to change variables or integrate by parts. 2. When calculating approximating sums, he imposed the further constraint that the sums provide rigorous upper and lower bounds. This was required because the Greeks lacked algebraic methods that could establish that error terms in an approximation are small. A problem solved exclusively in the Method is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of Kepler's Stereometria. Some pages of the Method remained unused by the author of the palimpsest and thus they are still lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakanian have renamed n = 4 Archimedean globe (and the half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid.

Archimedes Palimpsest In Heiberg's time, much attention was paid to Archimedes' brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the palimpsest that appears to deal with a children's puzzle. Reviel Netz of Stanford University has argued that Archimedes discussed the number of ways to solve the puzzle, that is, to put the pieces back in their box. No pieces have been identified as such; the rules for placement, such as whether pieces are allowed to be turned over, are not known; and there is doubt about the board. The board illustrated here, as also by Netz, is one proposed by Heinrich Suter in translating an unpointed Arabic text in which twice and equals are easily confused; Suter makes at least a typographical error at the crucial point, equating the lengths of a side and diagonal, in which case the board cannot be a rectangle. But, as the diagonals of a square intersect at right angles, the presence of right triangles makes the first proposition of Archimedes' Stomachion immediate. Rather, the first proposition sets up a board consisting of two squares side by side (as in Tangram). A reconciliation of the Suter board with this Codex board was published by Richard Dixon Oldham, FRS, in Nature in March, 1926, sparking a Stomachion craze that year. Modern combinatorics reveals that the number of ways to place the pieces of the Suter board to reform their square, allowing them to be turned over, is 17,152; the number is considerably smaller – 64 – if pieces are not allowed to be turned over. The sharpness of some angles in the Suter board makes fabrication difficult, while play could be awkward if pieces with sharp points are turned over. For the Codex board (again as with Tangram) there are three ways to pack the pieces: as two unit squares side by side; as two unit squares one on top of the other; and as a single square of side the square root of two. But the key to these packings is forming isosceles right triangles, just as Socrates gets the slave boy to consider in Plato's Meno – Socrates was arguing for knowledge by recollection, and here pattern recognition and memory seem more pertinent than a count of solutions. The Codex board can be found as an extension of Socrates' argument in a seven-by-seven-square grid, suggesting an iterative construction of the side-diameter numbers that give rational approximations to the square root of two. The fragmentary state of the palimpsest leaves much in doubt. But it would certainly add to the mystery had Archimedes used the Suter board in preference to the Codex board. However, if Netz is right, this may have been the most sophisticated work in the field of combinatorics in Greek antiquity. Either Archimedes used the Suter board, the pieces of which were allowed to be turned over, or the statistics of the Suter board are irrelevant.

Modern history The Biblical scholar Constantin von Tischendorf visited Constantinople in the 1840s, and, intrigued by the Greek mathematics visible on the palimpsest, brought home a page of it. (This page is now in the Cambridge University Library.) It was Johan Heiberg who realized, when he studied the palimpsest in Constantinople in 1906, that the text was of Archimedes, and included works otherwise lost. Heiberg took photographs, from which he produced transcriptions, published between 1910 and 1915 in a complete works of Archimedes. It is not known how the palimpsest subsequently wound up in France.[5] From the 1920s, the manuscript lay unknown in the Paris apartment of a collector of manuscripts and his heirs. In 1998 the ownership of the palimpsest was disputed in federal court in New York in the case of the Greek Orthodox Patriarchate of Jerusalem v. Christie's, Inc. At some time in the distant past, the Archimedes manuscript had lain in the library of Mar Saba, near Jerusalem, a monastery bought by the Patriarchate in 1625. The plaintiff contended that the palimpsest had been stolen from one of its monasteries in the 1920s. Judge Kimba Wood decided in favor of Christie's Auction House on laches grounds, and the palimpsest was bought for $2 million by an anonymous buyer. Simon Finch, who represented the anonymous buyer, stated that the buyer was "a private American" who worked in "the high-tech industry", but was not Bill Gates.[6] (The German magazine Der Spiegel reported that the buyer is probably Jeff Bezos.)[6] At the Walters Art Museum in Baltimore, the palimpsest was the subject of an extensive imaging study from 1999 to 2008, and conservation (as it had suffered considerably from mold). This was directed by Dr. Will Noel, curator of manuscripts at the Walters Art Museum, and managed by Michael B. Toth of R.B. Toth Associates, with Dr. Abigail

42

Archimedes Palimpsest Quandt performing the conservation of the manuscript. A team of imaging scientists including Dr. Roger Easton from the Rochester Institute of Technology, Dr. Bill Christens-Barry from Equipoise Imaging, and Dr. Keith Knox with Boeing LTS used computer processing of digital images from various spectral bands, including ultraviolet and visible light, to reveal most of the underlying text, including of Archimedes. After imaging and digitally processing the entire palimpsest in three spectral bands prior to 2006, in 2007 they reimaged the entire palimpsest in 12 spectral bands, plus raking light: UV: 365 nanometers; Visible Light: 445, 470, 505, 530, 570, 617, and 625 nm; Infrared: 700, 735, and 870 nm; and Raking Light: 910 and 470 nm.[7] The team digitally processed these images to reveal more of the underlying text with pseudocolor. They also digitized the original Heiberg images. Dr. Reviel Netz[8] of Stanford University and Nigel Wilson have produced a diplomatic transcription of the text, filling in gaps in Heiberg's account with these images. All images are currently hosted on the website.[9] Sometime after 1938, one owner of the manuscript forged four Byzantine-style religious images in the manuscript in an effort to increase its value. It appeared that these had rendered the underlying text forever illegible. However, in May 2005, highly focused X-rays produced at the Stanford Linear Accelerator Center in Menlo Park, California, were used by Drs. Uwe Bergman and Bob Morton to begin deciphering the parts of the 174-page text that had not yet been revealed. The production of X-ray fluorescence was described by Keith Hodgson, director of SSRL. "Synchrotron light is created when electrons traveling near the speed of light take a curved path around a storage ring—emitting electromagnetic light in X-ray through infrared wavelengths. The resulting light beam has characteristics that make it ideal for revealing the intricate architecture and utility of many kinds of matter—in this case, the previously hidden work of one of the founding fathers of all science."[10] In April 2007, it was announced that a new text had been found in the palimpsest, which was a commentary on the work of Aristotle attributed to Alexander of Aphrodisias. Dr. Will Noel said in an interview: "You start thinking striking one palimpsest is gold, and striking two is utterly astonishing. But then something even more extraordinary happened." This referred to the previous discovery of a text by Hypereides, an Athenian politician from the fourth century BC, which has also been found within the palimpsest.[4] It is from his speech Against Diondas, and was published in 2008 in the German scholarly magazine Zeitschrift für Papyrologie und Epigraphik, vol. 165, becoming the first new text from the palimpsest to be published in a scholarly journal.[11] The transcriptions of the book were digitally encoded using the Text Encoding Initiative guidelines, and metadata for the images and transcriptions included identification and cataloging information based on Dublin Core Metadata Elements. The metadata and data were managed by Dr. Doug Emery of Emery IT. On October 29, 2008, (the tenth anniversary of the purchase of the palimpsest at auction) all data, including images and transcriptions, were hosted on the Digital Palimpsest Web Page for free use under a Creative Commons License, and processed images of the palimpsest in original page order were posted as a Google Book.[12] In late 2011 it was the subject of the Walters Art Museum exhibit "Lost and Found: The Secrets of Archimedes".

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Archimedes Palimpsest

Notes [1] Archimedes brought to light (http:/ / www. archimedespalimpsest. org/ pdf/ physicsworld-november2007. pdf) Physics World, November 2007. [2] "Reading Between the Lines, Smithsonian Magazine" (http:/ / www. smithsonianmag. com/ science-nature/ archimedes. html). . Retrieved 2009-03-31. [3] "The Archimedes Palimpsest Project" (http:/ / www. archimedespalimpsest. org/ digitalproduct1. html). Archived (http:/ / web. archive. org/ web/ 20090221153000/ http:/ / www. archimedespalimpsest. org/ digitalproduct1. html) from the original on 21 February 2009. . Retrieved 2009-03-31. [4] Morelle, Rebecca (2007-04-26). ""Text Reveals More Ancient Secrets", BBC News" (http:/ / news. bbc. co. uk/ 1/ hi/ technology/ 6591221. stm). Archived (http:/ / web. archive. org/ web/ 20090219230234/ http:/ / news. bbc. co. uk/ 1/ hi/ technology/ 6591221. stm) from the original on 19 February 2009. . Retrieved 2009-03-31. [5] "History of the Archimedes Manuscript" (http:/ / www. archimedespalimpsest. org/ palimpsest_history1. html). . Retrieved 2009-03-31. [6] Hisrhfield, Alan. Eureka Man, Walker & Co, NY, 2009; p. 187. [7] "File Naming Conventions" (http:/ / archimedespalimpsest. net/ Documents/ Internal/ FileNamingConventions. txt). . Retrieved 2009-03-31. [8] "The Scholarship of the Palimpsest" (http:/ / www. archimedespalimpsest. org/ scholarship_netz2. html). Archived (http:/ / web. archive. org/ web/ 20090515114709/ http:/ / www. archimedespalimpsest. org/ scholarship_netz2. html) from the original on 15 May 2009. . Retrieved 2009-03-31. [9] Archimedespalimpsest.net (http:/ / archimedespalimpsest. net/ ) [10] "Placed under X-ray gaze, Archimedes manuscript yields secrets lost to time" (http:/ / news-service. stanford. edu/ news/ 2005/ may25/ archimedes-052505. html). . Retrieved 2009-03-31. [11] Carey, C. et al., "Fragments of Hyperides’ Against Diondas from the Archimedes Palimpsest" (http:/ / www. uni-koeln. de/ phil-fak/ ifa/ zpe/ indices/ inhaltsverzeichnis_165. pdf), "Inhaltsverzeichnis", Zeitschrift für Papyrologie und Epigraphik, vol. 165, pp. 1-19. Retrieved 2009-10-11. [12] Archimedes Palimpsest (http:/ / books. google. com/ books?id=_zX8OG3QoF4C& printsec=frontcover& cad=0). . Retrieved 2009-03-31.

References • Reviel Netz and William Noel, The Archimedes Codex (http://www.orionbooks.co.uk/books/ the-archimedes-codex-paperback), Weidenfeld & Nicolson, 2007 • Dijksterhuis, E.J., Archimedes, Princeton U. Press, 1987, pages 129–133. copyright 1938, ISBN 0-691-08421-1, ISBN 0-691-02400-6 (paperback)

External links • • • • • • • • •

The Archimedes Palimpsest Project Web Page (http://www.archimedespalimpsest.org/) Digital Palimpsest on the Web (http://www.archimedespalimpsest.org/) The Nova Program outlined (http://www.pbs.org/wgbh/nova/archimedes/palimpsest.html) The Nova Program teacher's version (http://www.pbs.org/wgbh/nova/teachers/programs/3010_archimed. html) The Method: English translation (Heiberg's 1909 transcription) (http://books.google.com/ books?id=suYGAAAAYAAJ) Did Isaac Barrow read it? (http://dftuz.unizar.es/~rivero/research/isisletter.htm) May 2005 Stanford Report: Heather Rock Woods, "Archimedes manuscript yields secrets under X-ray gaze" (http://news-service.stanford.edu/news/2005/may25/archimedes-052505.html) May 19, 2005 Will Noel: Restoring The Archimedes Palimpsest (http://www.youtube.com/watch?v=t3IP_FmGams) (YouTube), Ignite (O'Reilly), August 2009 The Greek Orthodox Patriarchate of Jerusalem v. Christies’s Inc., 1999 U.S. Dist. LEXIS 13257 (S.D. N.Y. 1999) (http://web.archive.org/web/20060910145753/http://www.law.washington.edu/courses/andrews/ A503C_WiSp06/Documents/Greek_Orthodox_Patriarchate_of_Jerusalem_v.pdf) (via Archive.org)

• Eureka! 1,000-year-old text by Greek maths genius Archimedes goes on display (http://www.dailymail.co.uk/ sciencetech/article-2050631/Eureka-1-000-year-old-text-Greek-maths-genius-Archimedes-goes-display.html) Daily Mail, October 18, 2011.

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René Descartes

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René Descartes René Descartes

[1]

Portrait after Frans Hals, 1648 Born

31 March 1596 La Haye en Touraine, Touraine, France

Died

11 February 1650 (aged 53) Stockholm, Sweden

Nationality

French

Era

17th-century philosophy

Region

Western Philosophy

Religion

Roman Catholic

School

Cartesianism Rationalism Foundationalism Founder of Cartesianism

[2]

Main interests Metaphysics, Epistemology, Mathematics Notable ideas

Cogito ergo sum, method of doubt, Cartesian coordinate system, Cartesian dualism, ontological argument for the existence of Christian God, mathesis universalis; folium of Descartes

Signature

René Descartes (French: [ʁəne dekaʁt]; Latinized: Renatus Cartesius; adjectival form: "Cartesian";[3] 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings,[4][5] which are studied closely to this day. In particular, his Meditations on First Philosophy continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system — allowing reference to a point in space as a set of numbers, and allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system (and conversely, shapes to be described as equations) — was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution and has been described as an example of genius.

René Descartes

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Descartes frequently sets his views apart from those of his predecessors. In the opening section of the Passions of the Soul, a treatise on the Early Modern version of what are now commonly called emotions, Descartes goes so far as to assert that he will write on this topic "as if no one had written on these matters before". Many elements of his philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differs from the schools on two major points: First, he rejects the analysis of corporeal substance into matter and form; second, he rejects any appeal to ends—divine or natural—in explaining natural phenomena.[6] In his theology, he insists on the absolute freedom of God's act of creation. Descartes was a major figure in 17th-century continental rationalism, later advocated by Baruch Spinoza and Gottfried Leibniz, and opposed by the empiricist school of thought consisting of Hobbes, Locke, Berkeley, Jean-Jacques Rousseau, and Hume. Leibniz, Spinoza and Descartes were all well versed in mathematics as well as philosophy, and Descartes and Leibniz contributed greatly to science as well. He is perhaps best known for the philosophical statement "Cogito ergo sum" (French: Je pense, donc je suis; English: I think, therefore I am), found in part IV of Discourse on the Method (1637 – written in French but with inclusion of "Cogito ergo sum") and §7 of part I of Principles of Philosophy (1644 – written in Latin).

Biography Descartes was born in La Haye en Touraine (now Descartes), Indre-et-Loire, France. When he was one year old, his mother Jeanne Brochard died. His father Joachim was a member of the Parlement of Brittany at Rennes.[7] In 1606 or 1607 he entered the Jesuit Collège Royal Henry-Le-Grand at La Flèche[8] where he was introduced to mathematics and physics, including Galileo's work.[9] After graduation in December 1616, he studied at the University of Poitiers, earning a Baccalauréat and Licence in law, in accordance with his father's wishes that he should become a lawyer.[10] "I entirely abandoned the study of letters. Resolving to seek no knowledge other than that of which could be found in myself or else in the great book of the world, I spent the rest of my youth traveling, Graduation registry for Descartes at the Collège Royal Henry-Le-Grand, La Flèche, 1616 visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, and at all times reflecting upon whatever came my way so as to derive some profit from it." (Descartes, Discourse on the Method). In 1618, Descartes was engaged in the army of Maurice of Nassau in the Dutch Republic, but as a truce had been established between Holland and Spain, Descartes used his spare time to study mathematics.[11] In this way he became acquainted with Isaac Beeckman, principal of Dordrecht school. Beeckman had proposed a difficult mathematical problem, and to his astonishment, it was the young Descartes who found the solution. Both believed that it was necessary to create a method that thoroughly linked mathematics and physics.[12] While in the service of the Duke Maximilian of Bavaria, Descartes was present at the Battle of the White Mountain outside Prague, in November 1620.[13] On the night of 10–11 November 1619, while stationed in Neuburg an der Donau, Germany, Descartes experienced a series of three powerful dreams or visions that he later claimed profoundly influenced his life. He concluded from these visions that the pursuit of science would prove to be, for him, the pursuit of true wisdom and a central part of his life's work.[14] Descartes also saw very clearly that all truths were linked with one another, so that finding a fundamental truth and proceeding with logic would open the way to all science. This basic truth, Descartes found

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quite soon: his famous "I think".[12] In 1622 he returned to France, and during the next few years spent time in Paris and other parts of Europe. It was during a stay in Paris that he composed his first essay on method: Regulae ad Directionem Ingenii (Rules for the Direction of the Mind).[12] He arrived in La Haye in 1623, selling all of his property to invest in bonds, which provided a comfortable income for the rest of his life. Descartes was present at the siege of La Rochelle by Cardinal Richelieu in 1627. He returned to the Dutch Republic in 1628, where he lived until September 1649. In April 1629 he joined the University of Franeker, living at the Sjaerdemaslot, and the next year, under the name "Poitevin", he enrolled at the Leiden University to study mathematics with Jacob Golius and astronomy with Martin Hortensius.[15] In October 1630 he had a falling-out with Beeckman, whom he accused of plagiarizing some of his ideas. In Amsterdam, he had a relationship with a servant girl, Helena Jans van der Strom, with whom he had a daughter, Francine, who was born in 1635 in Deventer, at which time Descartes taught at the Utrecht University. Francine Descartes died in 1640 in Amersfoort, from Scarlet Fever. While in the Netherlands he changed his address frequently, living among other places in Dordrecht (1628), Franeker (1629), Amsterdam (1629–30), Leiden (1630), Amsterdam (1630–32), Deventer (1632–34), Amsterdam (1634–35), Utrecht (1635–36), Leiden (1636), Egmond (1636–38), Santpoort (1638–1640), Leiden (1640–41), Endegeest (a castle near Oegstgeest) (1641–43), and finally for an extended time in Egmond-Binnen (1643–49). Despite these frequent moves he wrote all his major work during his 20-plus years in the Netherlands, where he managed to revolutionize mathematics and philosophy. In 1633, Galileo was condemned by the Roman Catholic Church, and Descartes abandoned plans to publish Treatise on the World, his work of the previous four years. Nevertheless, in 1637 he published part of this work in three essays: Les Météores (The Meteors), La Dioptrique (Dioptrics) and La Géométrie (Geometry), preceded by an introduction, his famous Discours de la Métode (Discourse on the Method). In it Descartes lays out four rules of thought, meant to ensure that our knowledge rests upon a firm foundation. Descartes continued to publish works concerning both mathematics and philosophy for the rest of his life. In 1641 he published a metaphysics work, Meditationes de Prima Philosophia (Meditations on First Philosophy), written in Latin and thus addressed to the learned. It was followed, in 1644, by Principia Philosophiæ (Principles of Philosophy), a kind of synthesis of the Meditations and the Discourse. In 1643, Cartesian philosophy was condemned at the University of Utrecht, and Descartes began his long correspondence with Princess Elisabeth of Bohemia, devoted mainly to moral and psychological subjects. Connected with this correspondence, in 1649 he published Les Passions de l'âme (Passions of the Soul), that he dedicated to the Princess. In 1647, he was awarded a pension by the King of France. Descartes was interviewed by Frans Burman at Egmond-Binnen in 1648.

René Descartes (right) with Queen Christina of Sweden (left).

A French translation of Principia Philosophiæ, prepared by Abbot Claude Picot, was published in 1647. This edition Descartes dedicated to Princess Elisabeth of Bohemia. In the preface Descartes praised true philosophy as a means to attain wisdom. He identifies four ordinary sources to reach wisdom, and finally says that there is a fifth, better and more secure, consisting in the search for first causes.[16] René Descartes died on 11 February 1650 in Stockholm, Sweden, where he had been invited as a tutor for Queen Christina of Sweden. The cause of death was said to be pneumonia; accustomed to working in bed until noon, he may have suffered damage to his health from Christina's demands for early morning study (the lack of sleep could

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have severely compromised his immune system). Descartes stayed at the French ambassador Pierre Chanut. In 1663, the Pope placed his works on the Index of Prohibited Books. As a Roman Catholic in a Protestant nation, he was interred in a graveyard used mainly for unbaptized infants in Adolf Fredriks kyrka in Stockholm. Later, his remains were taken to France and buried in the Abbey of Saint-Germain-des-Prés in Paris. Although the National Convention in 1792 had planned to transfer his remains to the Panthéon, they are, two centuries later, still resting between two other graves — those of the scholarly monks Jean Mabillon and Bernard de Montfaucon — in a chapel of the abbey. His memorial, erected in the 18th century, remains in the Swedish church.

Religious beliefs

The tomb of Descartes (middle, with detail of the inscription), in the Abbey of Saint-Germain-des-Prés, Paris

The religious beliefs of René Descartes have been rigorously debated within scholarly circles. He claimed to be a devout Roman Catholic, claiming that one of the purposes of the Meditations was to defend the Christian faith. However, in his own era, Descartes was accused of harboring secret deist or atheist beliefs. Contemporary Blaise Pascal said that "I cannot forgive Descartes; in all his philosophy, Descartes did his best to dispense with God. But Descartes could not avoid prodding God to set the world in motion with a snap of his lordly fingers; after that, he had no more use for God."[17] Stephen Gaukroger's biography of Descartes reports that "he had a deep religious faith as a Catholic, which he retained to his dying day, along with a resolute, passionate desire to discover the truth."[18] After Descartes died in Sweden, Queen Christina abdicated her throne to convert to Roman Catholicism (Swedish law required a Protestant ruler). The only Roman Catholic with whom she had prolonged contact was Descartes, who was her personal tutor.[19]

Philosophical work Descartes is often regarded as the first thinker to emphasize the use of reason to develop the natural sciences.[20] For him the philosophy was a thinking system that embodied all knowledge, and expressed it in this way:[21] Thus, all Philosophy is like a tree, of which Metaphysics is the root, Physics the trunk, and all the other sciences the branches that grow out of this trunk, which are reduced to three principal, namely, Medicine, Mechanics, and Ethics. By the science of Morals, I understand the highest and most perfect which, presupposing an entire knowledge of the other sciences, is the last degree of wisdom.

“

”

In his Discourse on the Method, he attempts to arrive at a fundamental set of principles that one can know as true without any doubt. To achieve this, he employs a method called hyperbolical/metaphysical doubt, also sometimes referred to as methodological skepticism: he rejects any ideas that can be doubted, and then reestablishes them in order to acquire a firm foundation for genuine knowledge.[22] Initially, Descartes arrives at only a single principle: thought exists. Thought cannot be separated from me, therefore, I exist (Discourse on the Method and Principles of Philosophy). Most famously, this is known as cogito ergo sum (English: "I think, therefore I am"). Therefore, Descartes concluded, if he doubted, then something or someone must be doing the doubting, therefore the very fact that he doubted proved his existence. "The simple meaning of the phrase is that if one is sceptical of existence, that is in and of itself proof that he does exist."[23]

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Descartes concludes that he can be certain that he exists because he thinks. But in what form? He perceives his body through the use of the senses; however, these have previously been unreliable. So Descartes determines that the only indubitable knowledge is that he is a thinking thing. Thinking is what he does, and his power must come from his essence. Descartes defines "thought" (cogitatio) as "what happens in me such that I am immediately conscious of it, insofar as I am conscious of it". Thinking is thus every activity of a person of which he is immediately conscious.[24] To further demonstrate the limitations of the senses, Descartes proceeds with what is known as the Wax Argument. He considers a piece of wax; his senses inform him that it has certain characteristics, such as shape, texture, size, color, smell, and so forth. When he brings the wax towards a flame, these characteristics change completely. However, it seems that it is still the same thing: it is still the same piece of wax, even though the data of the senses inform him that all of its characteristics are different. Therefore, in order to properly grasp the nature of the wax, he should put aside the senses. He must use his mind. Descartes concludes:

“

René Descartes at work

And so something which I thought I was seeing with my eyes is in fact grasped solely by the faculty of judgment which is in my mind.

”

In this manner, Descartes proceeds to construct a system of knowledge, discarding perception as unreliable and instead admitting only deduction as a method. In the third and fifth Meditation, he offers an ontological proof of a benevolent God (through both the ontological argument and trademark argument). Because God is benevolent, he can have some faith in the account of reality his senses provide him, for God has provided him with a working mind and sensory system and does not desire to deceive him. From this supposition, however, he finally establishes the possibility of acquiring knowledge about the world based on deduction and perception. In terms of epistemology therefore, he can be said to have contributed such ideas as a rigorous conception of foundationalism and the possibility that reason is the only reliable method of attaining knowledge. He, nevertheless, was very much aware that experimentation was necessary in order to verify and validate theories.[21] Descartes also wrote a response to scepticism about the existence of the external world. He argues that sensory perceptions come to him involuntarily, and are not willed by him. They are external to his senses, and according to Descartes, this is evidence of the existence of something outside of his mind, and thus, an external world. Descartes goes on to show that the things in the external world are material by arguing that God would not deceive him as to the ideas that are being transmitted, and that God has given him the "propensity" to believe that such ideas are caused by material things.

Dualism Descartes in his Passions of the Soul and The Description of the Human Body suggested that the body works like a machine, that it has material properties. The mind (or soul), on the other hand, was described as a nonmaterial and does not follow the laws of nature. Descartes argued that the mind interacts with the body at the pineal gland. This form of dualism or duality proposes that the mind controls the body, but that the body can also influence the otherwise rational mind, such as when people act out of passion. Most of the previous accounts of the relationship between mind and body had been uni-directional.

René Descartes Descartes suggested that the pineal gland is "the seat of the soul" for several reasons. First, the soul is unitary, and unlike many areas of the brain the pineal gland appeared to be unitary (though subsequent microscopic inspection has revealed it is formed of two hemispheres). Second, Descartes observed that the pineal gland was located near the ventricles. He believed the cerebrospinal fluid of the ventricles acted through the nerves to control the body, and that the pineal gland influenced this process. Cartesian dualism set the agenda for philosophical discussion of the mind–body problem for many years after Descartes's death.[25] In present day discussions on the practice of animal vivisection, it is normal to consider Descartes as an advocate of this practice, as a result of his dualistic philosophy. Some of the sources say that Descartes denied the animals could feel pain, and therefore could be used without concern.[26] Other sources consider that Descartes denied that animal had reason or intelligence, but did not lack sensations or perceptions, but these could be explained mechanistically.[27]

Descartes' moral philosophy For Descartes, morals was a science, the highest and most perfect of them, and like the rest of sciences had its roots in Metaphysics.[21] In this way he argues for the existence of God, investigates the place of men in nature, formulates the theory of mind-body dualism and defends free will. But, he being a convinced rationalist, clearly estates that reason suffices us in the search for the goods we should seek, and for him, virtue consists in the correct reasoning that should guide our actions. Nevertheless, the quality of this reasoning depends on knowledge, as a well informed mind will be more capable of making good choices, and also on mental condition. For this reason he said that a complete moral philosophy should include the study of the body. He discussed this subject in the correspondence with Princess Elisabeth of Bohemia, and as a result wrote his work The Passions of the Soul, that contains a study of the psychosomatic processes and reactions in man, with an emphasis on emotions or passions.[28] Men should seek the sovereign good that Descartes, following Zeno, identifies with virtue, as this produces a solid blessedness or pleasure. For Epicurus the sovereign good was pleasure, and Descartes says that in fact this is not in contradiction with Zeno's teaching, because virtue produces a spiritual pleasure, that is better than bodily pleasure. Regarding Aristotle opinion that happiness depends on the goods of fortune, Descartes does not deny that this goods contribute to happiness, but remarks that they are in great proportion outside our control, whereas our mind is under our complete control.[28] The moral writings of Descartes came at the last part of his life, but earlier, in his Discourse on Method he adopted three maxims to be able to act while he put all his ideas into doubt. This is known as his "Provisional Morals".

Historical impact Emancipation from Church doctrine Descartes has been often dubbed as the father of modern Western philosophy, the philosopher that with his sceptic approach has profoundly changed the course of Western philosophy and set the basis for modernity.[4][29] The first two of his Meditations on First Philosophy, those that formulate the famous methodic doubt, are the portion of Descartes writings that most influenced modern thinking.[30] It has been argued that Descartes himself didn't realize the extent of his revolutionary gesture.[31] In shifting the debate from "what is true" to "of what can I be certain?," Descartes shifted the authoritative guarantor of truth from God to Man (While traditional concept of "truth" implies an external authority, "certainty" instead relies on the judgement of the individual Man). In an anthropocentric revolution, Man is now raised to the level of a subject, an agent, an emancipated being equipped with autonomous reason. This is a revolutionary step which posed the basis of modernity (whose repercussion are still ongoing): the emancipation of man from Christian revelational truth and Church doctrine, a man that makes his own law and takes its own stand.[32][33][34] In modernity, the guarantor of truth is not God anymore but Man, a "self-conscious shaper and guarantor" of his reality.[35][36] Man in this way is turned into a reasoning adult, a subject and agent,[35] as

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René Descartes opposed to a child obedient to God. This change in perspective was characteristic of the shift from the Christian mediaval period to the modern period, it had been anticipated in other fields and now Descartes was giving it a formulation in the field of philosophy.[35][37] This anthropocentric perspective, establishing human reason as autonomous, posed the basis for the Enlightenment's emancipation from God and the Church. It also posed the basis for all subsequent anthropology.[38] Descartes philosophical revolution is sometimes said to have sparked modern anthropocentrism and subjectivism.[4][39][40][41]

Mathematical legacy One of Descartes' most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c". He also "pioneered the standard notation" that uses superscripts to show the powers or exponents; for example, the 4 used in x4 to indicate squaring of squaring.[42] He was first to assign a fundamental place for algebra in our system of knowledge, and believed that algebra was a method to automate or mechanize reasoning, particularly about abstract, unknown quantities. European mathematicians had previously viewed geometry as a more fundamental form of mathematics, serving as the foundation of algebra. Algebraic rules were given geometric proofs by mathematicians such as Pacioli, Cardan, Tartaglia and Ferrari. Equations of degree higher than the third were regarded as unreal, because a three dimensional form, such as a cube, occupied the largest dimension of reality. Descartes professed that the abstract quantity a2 could represent length as well as an area. This was in opposition to the teachings of mathematicians, such as Vieta, who argued that it could represent only area. Although Descartes did not pursue the subject, he preceded Leibniz in envisioning a more general science of algebra or "universal mathematics," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning.[43] Descartes' work provided the basis for the calculus developed by Newton and Leibniz, who applied infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics.[44] His rule of signs is also a commonly used method to determine the number of positive and negative roots of a polynomial. Descartes discovered an early form of the law of conservation of mechanical momentum (a measure of the motion of an object), and envisioned it as pertaining to motion in a straight line, as opposed to perfect circular motion, as Galileo had envisioned it. He outlined his views on the universe in his Principles of Philosophy. Descartes also made contributions to the field of optics. He showed by using geometric construction and the law of refraction (also known as Descartes's law or more commonly Snell's law, who discovered it 16 years earlier) that the angular radius of a rainbow is 42 degrees (i.e., the angle subtended at the eye by the edge of the rainbow and the ray passing from the sun through the rainbow's centre is 42°).[45] He also independently discovered the law of reflection, and his essay on optics was the first published mention of this law.[46]

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Contemporary reception Although Descartes was well known in academic circles towards the end of his life, the teaching of his works in schools was controversial. Henri de Roy (Henricus Regius, 1598–1679), Professor of Medicine at the University of Utrecht, was condemned by the Rector of the University, Gijsbert Voet (Voetius), for teaching Descartes's physics.[47]

Writings • 1618. Compendium Musicae. A treatise on music theory and the aesthetics of music written for Descartes's early collaborator, Isaac Beeckman. • 1626–1628. Regulae ad directionem ingenii (Rules for the Direction of the Mind). Incomplete. First published posthumously in Dutch translation in 1684 and in the original Latin at Amsterdam in 1701 (R. Des-Cartes Opuscula Posthuma Physica et Mathematica). The best critical edition, which includes the Dutch translation of 1684, is edited by Giovanni Crapulli (The Hague: Martinus Nijhoff, 1966). • 1630–1633. Le Monde (The World) and L'Homme (Man). Descartes's first systematic presentation of his natural philosophy. Man was published posthumously in Latin translation in 1662; and The World posthumously in 1664. • 1637. Discours de la méthode (Discourse on the Method). An introduction to the Essais, which include the Dioptrique, the Météores and the Géométrie. • 1637. La Géométrie (Geometry). Descartes's major work in mathematics. There is an English translation by Michael Mahoney (New York: Dover, 1979).

Handwritten letter by Descartes, December 1638.

• 1641. Meditationes de prima philosophia (Meditations on First Philosophy), also known as Metaphysical Meditations. In Latin; a French translation, probably done without Descartes's supervision, was published in 1647. Includes six Objections and Replies. A second edition, published the following year, included an additional objection and reply, and a Letter to Dinet. • 1644. Principia philosophiae (Principles of Philosophy), a Latin textbook at first intended by Descartes to replace the Aristotelian textbooks then used in universities. A French translation, Principes de philosophie by Claude Picot, under the supervision of Descartes, appeared in 1647 with a letter-preface to Princess Elisabeth of Bohemia. • 1647. Notae in programma (Comments on a Certain Broadsheet). A reply to Descartes's one-time disciple Henricus Regius. • 1647. La description du corps humaine (The Description of the Human Body). Published posthumously. • 1648. Responsiones Renati Des Cartes... (Conversation with Burman). Notes on a Q&A session between Descartes and Frans Burman on 16 April 1648. Rediscovered in 1895 and published for the first time in 1896. An annotated bilingual edition (Latin with French translation), edited by Jean-Marie Beyssade, was published in 1981 (Paris: PUF). • 1649. Les passions de l'âme (Passions of the Soul). Dedicated to Princess Elisabeth of the Palatinate. • 1656. Musicae Compendium (Instruction in Music). Posth. Publ.: Johannes Janssonius jun., Amsterdam. • 1657. Correspondance. Published by Descartes's literary executor Claude Clerselier. The third edition, in 1667, was the most complete; Clerselier omitted, however, much of the material pertaining to mathematics.

René Descartes In January 2010, a previously unknown letter from Descartes, dated 27 May 1641, was found by the Dutch philosopher Erik-Jan Bos when browsing through Google. Bos found the letter mentioned in a summary of autographs kept by Haverford College in Haverford, Pennsylvania. The College was unaware that the letter had never been published. This was the third letter by Descartes found in the last 25 years.[48][49]

Notes [1] Russell Shorto. Descartes' Bones. (Doubleday, 2008) p. 218; see also The Louvre, Atlas Database, http:/ / cartelen. louvre. fr [2] "René Descartes" (http:/ / www. newadvent. org/ cathen/ 04744b. htm). Newadvent.org. . Retrieved 30 May 2012. "...preferred to avoid all collision with ecclesiastical authority." [3] Colie, Rosalie L. (1957). Light and Enlightenment. Cambridge University Press. p. 58. [4] Bertrand Russell (2004) History of western philosophy (http:/ / books. google. com/ books?id=Ey94E3sOMA0C& pg=PA516) pp.511, 516-7 [5] Watson, Richard A. (31 March 2012). "René Descartes" (http:/ / www. britannica. com/ EBchecked/ topic/ 158787/ Rene-Descartes). Encyclopædia Britannica (Encyclopædia Britannica Online. Encyclopædia Britannica Inc). . Retrieved 31 March 2012. [6] Carlson, Neil R. (2001). Physiology of Behavior. Needham Heights, Massachusetts: Pearson: Allyn & Bacon. p. 8. ISBN 0-205-30840-6. [7] Rodis-Lewis, Geneviève (1992). "Descartes' life and the development of his philosophy" (http:/ / books. google. dk/ books?id=Prhr9FBdQ_MC). In Cottingham, John. The Cambridge Companion to Descartes. Cambridge University Press. p. 22. ISBN 978-0-521-36696-0. . [8] Desmond, p. 24 [9] Porter, Roy (1999) [1997]. "The New Science". The Greatest Benefit to Mankind: A Medical History of Humanity from Antiquity to the Present (paperback edition, 135798642 ed.). Great Britain: Harper Collins. p. 217. ISBN 0006374549. [10] Baird, Forrest E.; Walter Kaufmann (2008). From Plato to Derrida. Upper Saddle River, New Jersey: Pearson Prentice Hall. pp. 373–377. ISBN 0-13-158591-6. [11] "René Descartes" (http:/ / www. famousscientists. org/ rene-descartes). FamousScientists.org. . Retrieved 15 December 2011. [12] Guy Durandin, Les Principes de la Philosophie. Introduction et notes, Librairie Philosophique J. Vrin, Paris, 1970. [13] Battle of White Mountain (http:/ / www. britannica. com/ EBchecked/ topic/ 642395/ Battle-of-White-Mountain), Britannica Online Encyclopedia [14] Clarke, Desmond (2006). Descartes: A biography, pp. 58–59. Cambridge U. Press. http:/ / books. google. com/ books?id=W3D9KGVyz6sC [15] A.C. Grayling, Descartes: The Life of Rene Descartes and Its Place in His Times, Simon and Schuster, 2006, pp 151–152 [16] Blom, John J., Descartes. His Moral Philosophy and Psychology. New York University Press, 1978. ISBN 0-8147-0999-0 [17] Think Exist on Blaise Pascal (http:/ / thinkexist. com/ quotation/ i_cannot_forgive_descartes-in_all_his_philosophy/ 153298. html). Retrieved 12 February 2009. [18] The Religious Affiliation of philosopher and mathematician Rene Descartes (http:/ / www. adherents. com/ people/ pd/ Rene_Descartes. html). Webpage last modified 5 October 2005. [19] Smith, Kurt (Fall 2010). "Descartes' Life and Works" (http:/ / plato. stanford. edu/ entries/ descartes-works/ ). The Stanford Encyclopedia of Philosophy. . [20] Emily Grosholz (1991). Cartesian method and the problem of reduction (http:/ / books. google. com/ books?hl=en& lr=& id=2EtAVLU1eIAC& oi=fnd& pg=PA1). Oxford University Press. ISBN 0-19-824250-6. . "But contemporary debate has tended to...understand [Cartesian method] merely as the 'method of doubt'...I want to define Descartes's method in broader terms...to trace its impact on the domains of mathematics and physics as well as metaphysics." [21] René Descartes; Translator John Veitch. "Letter of the Author to the French Translator of the Principles of Philosophy serving for a preface" (http:/ / www. classicallibrary. org/ descartes/ principles/ preface. htm). . Retrieved December 2011. [22] Rebecca, Copenhaver. "Forms of skepticism" (http:/ / web. archive. org/ web/ 20050108095032/ http:/ / www. lclark. edu/ ~rebeccac/ forms. html). Archived from the original (http:/ / www. lclark. edu/ ~rebeccac/ forms. html) on 8 January 2005. . Retrieved 15 August 2007. [23] "Ten books: Chosen by Raj Persuade" (http:/ / bjp. rcpsych. org/ cgi/ content/ full/ 181/ 3/ 258). The British Journal of Psychiatry. . [24] Descartes, René (1644). The Principles of Philosophy (IX). [25] Stanford Encyclopedia of Philosophy (online): Descartes and the Pineal Gland. [26] Richard Dawkins (June 2012). "Richard Dawkins on vivisection: "But can they suffer?" (http:/ / boingboing. net/ 2011/ 06/ 30/ richard-dawkins-on-v. html). Boingboing. . Retrieved 2 July 2012. [27] "Animal Consciousness, #2. Historical background" (http:/ / plato. stanford. edu/ entries/ consciousness-animal/ #hist). Stanford Encyclopedia of Philosophy. Dec 1995/rev Oct 2010. . Retrieved 2 July 2012. [28] Blom, John J., Descartes. His moral philosophy and psychology. New York University Press. 1978. ISBN 0-8147-0999-0 [29] Heidegger [1938] (2002) p.76 quotation:

Descartes... that which he himself founded... modern (and that means, at the same time, Western) metaphysics. [30] Schmaltz, Tad M. Radical Cartesianism: The French Reception of Descartes (http:/ / books. google. com/ books?id=pIYcUBCOrNgC& pg=PA27) p.27 quotation:

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René Descartes The Descartes most familiar to twentieth-century philosophers is the Descartes of the first two Meditations, someone proccupied with hyperbolic doubt of the material world and the certainty of knowledge of the self that emerges from the famous cogito argument. [31] Roy Wood Sellars (1949) Philosophy for the future: the quest of modern materialism (http:/ / books. google. com/ books?id=y1wNAAAAIAAJ) quotation:

Husserl has taken Descartes very seriously in a historical as well as in a systematic sense [...] [in The Crisis of the European Sciences and Transcendental Phenomenology, Husserl] finds in the first two Meditations of Descartes a depth which it is difficult to fathom, and which Descartes himself was so little able to appreciate that he let go "the great discovery" he had in his hands. [32] Martin Heidegger [1938] (2002) The Age of the World Picture quotation:

For up to Descartes...a particular sub-iectum...lies at the foundation of its own fixed qualities and changing circumstances. The superiority of a sub-iectum...arises out of the claim of man to a...self-supported, unshakable foundation of truth, in the sense of certainty. Why and how does this claim acquire its decisive authority? The claim originates in that emancipation of man in which he frees himself from obligation to Christian revelational truth and Church doctrine to a legislating for himself that takes its stand upon itself. [33] Ingraffia, Brian D. (1995) Postmodern theory and biblical theology: vanquishing God's shadow (http:/ / books. google. com/ books?id=LHjZYbOLG8cC& pg=PA126) p.126 [34] Norman K. Swazo (2002) Crisis theory and world order: Heideggerian reflections (http:/ / books. google. com/ books?id=INP_cy6Mu7EC& pg=PA97) pp.97-9 [35] Lovitt, Tom (1977) introduction to Martin Heidegger's The question concerning technology, and other essays, pp.xxv-xxvi [36] Briton, Derek The modern practice of adult education: a postmodern critique (http:/ / books. google. com/ books?id=Hd_xwb6EolMC& pg=PA76) p.76 [37] Martin Heidegger The Word of Nietzsche: God is Dead pp.88-90 [38] Heidegger [1938] (2002) p.75 quotation:

With the interpretation of man as subiectum, Descartes creates the metaphysical presupposition for future anthropology of every kind and tendency. [39] Benjamin Isadore Schwart China and Other Matters (http:/ / books. google. com/ books?id=Wt4XDLEpjWYC& pg=PA95) p.95 quotation:

... the kind of anthropocentric subjectivism which has emerged from the Cartesian revolution. [40] Charles B. Guignon Heidegger and the problem of knowledge (http:/ / books. google. com/ books?id=5vFCfdWD5QEC& pg=PA23) p.23 [41] Husserl, Edmund (1931) Cartesian Meditations: An Introduction to Phenomenology quotation:

When, with the beginning of modern times, religious belief was becoming more and more externalized as a lifeless convention, men of intellect were lifted by a new belief, their great belief in an autonomous philosophy and science. [...] in philosophy, the Meditations were epoch-making in a quite unique sense, and precisely because of their going back to the pure ego cogito. Descartes, in fact, inaugurates an entirely new kind of philosophy. Changing its total style, philosophy takes a radical turn: from naïve Objectivism to transcendental subjectivism. [42] Tom Sorelli, Descartes: A Very Short Introduction, (2000). New York: Oxford University Press. p. 19. [43] Morris Kline, Mathematical Thought from Ancient to Modern Times, (1972). New York: Oxford University Press. pp. 280-281 [44] Gullberg, Jan (1997). Mathematics From The Birth Of Numbers. W. W. Norton. ISBN 0-393-04002-X. [45] Tipler, P. A. and G. Mosca (2004). Physics For Scientists And Engineers. W. H. Freeman. ISBN 0-7167-4389-2. [46] "René Descartes" (http:/ / encarta. msn. com/ encyclopedia_761555262/ Rene_Descartes. html#s3). Encarta. Microsoft. 2008. . Retrieved 15 August 2007. [47] Cottingham, John, Dugald Murdoch, and Robert Stoothof. The Philosophical Writings of Descartes.Cambridge: Cambridge University Press. 1985. 293. [48] Vlasblom, Dirk (25 February 2010). "Unknown letter from Descartes found" (http:/ / www. nrc. nl/ international/ article2492445. ece/ Unknown_letter_from_Descartes_found). Nrc.nl. . Retrieved 30 May 2012. [49] (Dutch) " Hoe Descartes in 1641 op andere gedachten kwam – Onbekende brief van Franse filosoof gevonden" (http:/ / www. nrc. nl/ wetenschap/ article2491995. ece/ Hoe_Descartes_in_1641_op_andere_gedachten_kwam)

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René Descartes

Bibliography References Collected works • Oeuvres de Descartes edited by Charles Adam and Paul Tannery, Paris: Léopold Cerf, 1897–1913, 13 volumes; new revised edition, Paris: Vrin-CNRS, 1964–1974, 11 vol. This work is traditionally cited with the initials AT (for Adam and Tannery) followed by a volume number in Roman numerals; thus AT VII refers to Oeuvres de Descartes volume 7. • Oeuvres de jeunesse (1616-1631) edited by Vincent Carraud, Paris: PUF, 2012.

Collected English translations • 1955. The Philosophical Works, E.S. Haldane and G.R.T. Ross, trans. Dover Publications. This work is traditionally cited with the initials HR (for Haldane and Ross) followed by a volume number in Roman numerals; thus HRII refers to volume 2 of this edition. • 1988. The Philosophical Writings of Descartes in 3 vols. Cottingham, J., Stoothoff, R., Kenny, A., and Murdoch, D., trans. Cambridge University Press.

Single works • 1618. Compendium Musicae. • 1628. Rules for the Direction of the Mind. • 1637. Discourse on the Method ("Discours de la Methode"). An introduction to Dioptrique, Des Météores and La Géométrie. Original in French, because intended for a wider public. • 1637. La Géométrie. Smith, David E., and Lantham, M. L., trans., 1954. The Geometry of René Descartes. Dover. • 1641. Meditations on First Philosophy. Cottingham, J., trans., 1996. Cambridge University Press. Latin original. Alternative English title: Metaphysical Meditations. Includes six Objections and Replies. A second edition published the following year, includes an additional Objection and Reply and a Letter to Dinet. HTML Online Latin-French-English Edition (http://www.wright.edu/cola/descartes/intro.html) • 1644. Les Principes de la philosophie. Miller, V. R. and R. P., trans., 1983. Principles of Philosophy. Reidel. • 1647. Comments on a Certain Broadsheet. • 1647. The Description of the Human Body. • 1648. Conversation with Burman. • 1649. Passions of the Soul. Voss, S. H., trans., 1989. Indianapolis: Hackett. Dedicated to Princess Elizabeth of Bohemia.

Secondary literature • Boyer, Carl (1985). A History of Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-02391-3. • Carriero, John (2008). Between Two Worlds. Princeton University Press. ISBN 978-0-691-13561-8. • Clarke, Desmond (2006). Descartes: A Biography. Cambridge: Cambridge University Press. ISBN 0-521-82301-3. • Costabel, Pierre (1987). René Descartes – Exercices pour les éléments des solides. Paris: Presses Universitaires de France. ISBN 2-13-040099-X. • Cottingham, John (1992). The Cambridge Companion to Descartes. Cambridge: Cambridge University Press. ISBN 0-521-36696-8. • Duncan, Steven M. (2008). The Proof of the External World: Cartesian Theism and the Possibility of Knowledge. Cambridge: James Clarke & Co. ISBN 978-02271-7267-4 http://www.lutterworth.com/jamesclarke/jc/titles/

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• • • • • •

•

proofew.htm. Farrell, John. "Demons of Descartes and Hobbes." Paranoia and Modernity: Cervantes to Rousseau (Cornell UP, 2006), chapter 7. Garber, Daniel (1992). Descartes' Metaphysical Physics. Chicago: University of Chicago Press. ISBN 0-226-28219-8. Garber, Daniel; Michael Ayers (1998). The Cambridge History of Seventeenth-Century Philosophy. Cambridge: Cambridge University Press. ISBN 0-521-53721-5. Gaukroger, Stephen (1995). Descartes: An Intellectual Biography. Oxford: Oxford University Press. ISBN 0-19-823994-7. Grayling, A.C. (2005). Descartes: The Life and times of a Genius. New York: Walker Publishing Co., Inc.. ISBN 0-8027-1501-X. Gillespie, A. (2006). Descartes' demon: A dialogical analysis of 'Meditations on First Philosophy.' (http://stir. academia.edu/documents/0011/0112/ Gillespie_Descartes_demon_a_dialogical_analysis_of_meditations_on_first_philosophy.pdf) Theory & Psychology, 16, 761–781. Martin Heidegger [1938] (2002) The Age of the World Picture in Off the beaten track (http://books.google. com/books?id=QImd2ARqQPMC&pg=PA66)

• Keeling, S. V. (1968). Descartes. Oxford: Oxford University Press. ISBN. • Melchert, Norman (2002). The Great Conversation: A Historical Introduction to Philosophy. New York: McGraw Hill. ISBN 0-19-517510-7. • Moreno Romo, Juan Carlos (Coord.), Descartes vivo. Ejercicios de hermenéutica cartesiana, Anthropos, Barcelona, 2007' • Ozaki, Makoto (1991). Kartenspiel, oder Kommentar zu den Meditationen des Herrn Descartes. Berlin: Klein Verlag.. ISBN 3-927199-01-X. • Moreno Romo, Juan Carlos, Vindicación del cartesianismo radical, Anthropos, Barcelona, 2010. • Schäfer, Rainer (2006). Zweifel und Sein – Der Ursprung des modernen Selbstbewusstseins in Descartes' cogito. Wuerzburg: Koenigshausen&Neumann. ISBN 3-8260-3202-0. • Serfati, M., 2005, "Geometria" in Ivor Grattan-Guinness, ed., Landmark Writings in Western Mathematics. Elsevier: 1–22. • Sorrell, Tom (1987). Descartes. Oxford: Oxford University Press.. ISBN 0-19-287636-8. • Vrooman, Jack Rochford (1970). René Descartes: A Biography. Putnam Press. • Watson, Richard A. (31 March 2012). "René Descartes" (http://www.britannica.com/EBchecked/topic/ 158787/Rene-Descartes). Encyclopædia Britannica (Encyclopædia Britannica Online. Encyclopædia Britannica Inc). Retrieved 31 March 2012. • Naaman-Zauderer, Noa (2010). Descartes' Deontological Turn: Reason, Will and Virtue in the Later Writings. Cambridge University Press. ISBN 978-0-521-76330-1.

External links Video • Bernard Williams interviewed about Descartes on "Men of ideas" (http://www.youtube.com/ watch?v=44h9QuWcJYk) • René Descartes (http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=8404) at Find a Grave General • Detailed biography of Descartes (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Descartes. html) • "René Descartes" in the 1913 Catholic Encyclopedia.

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René Descartes • Descartes featured on the 100 French Franc banknote from 1942. (http://www-personal.umich.edu/~jbourj/ money5.htm) • More easily readable versions of Meditations, Objections and Replies, Principles of Philosophy, Discourse on the Method, Correspondence with Princess Elisabeth, and Passions of the Soul. (http://www.earlymoderntexts.com) • 1984 John Cottingham translation of Meditations and Objections and Replies. (http://www.freewebs.com/ dqsdnlj/d.html) • René Descartes (1596–1650) (http://digitalcommons.unl.edu/modlangfrench/20/) Published in Encyclopedia of Rhetoric and Composition (1996) • Original texts of René Descartes in French (http://www.laphilosophie.fr/livres-de-Descartes-texte-integral. html) at La Philosophie • Descartes Philosophical Writings tr. by Norman Kemp Smith (http://www.archive.org/details/ descartesphiloso010838mbp) at archive.org • Studies in the Cartesian philosophy (1902) by Norman Kemp Smith (http://www.archive.org/details/ studiesincartes00smitgoog) at archive.org • The Philosophical Works Of Descartes Volume II (1934) (http://www.archive.org/details/ philosophicalwor005524mbp) at archive.org • Works by or about René Descartes (http://worldcat.org/identities/lccn-n79-61201) in libraries (WorldCat catalog) • Free scores by René Descartes at the International Music Score Library Project • René Descartes (1596—1650): Overview(IEP) (http://www.iep.utm.edu/descarte/) • René Descartes:The Mind-Body Distinction(IEP) (http://www.iep.utm.edu/descmind/) • Cartesian skepticism(DEP) (http://philosophy.uwaterloo.ca/MindDict/cartesianskepticism.html) Stanford Encyclopedia of Philosophy • • • • • • • • •

René Descartes (http://plato.stanford.edu/entries/descartes/) Descartes' Epistemology (http://plato.stanford.edu/entries/descartes-epistemology/) Descartes' Ethics (http://plato.stanford.edu/entries/descartes-ethics/) Descartes' Life and Works (http://plato.stanford.edu/entries/descartes-works/) Descartes' Modal Metaphysics (http://plato.stanford.edu/entries/descartes-modal/) Descartes' Ontological Argument (http://plato.stanford.edu/entries/descartes-ontological/) Descartes and the Pineal Gland (http://plato.stanford.edu/entries/pineal-gland/) Descartes' Physics (http://plato.stanford.edu/entries/descartes-physics/) Descartes' Theory of Ideas (http://plato.stanford.edu/entries/descartes-ideas/)

57

Pierre de Fermat

58

Pierre de Fermat Pierre de Fermat

Pierre de Fermat Born

1601 or 1607/8 Beaumont-de-Lomagne, France

Died

1665 Jan 12 Castres, France

Residence

France

Nationality

French

Fields

Mathematics and Law

Known for

Number theory Analytic geometry Fermat's principle Probability Fermat's Last Theorem Adequality

Influences

François Viète

Pierre de Fermat (French: [pjɛːʁ dəfɛʁma]; 17[1] August 1601 or 1607/8[2] – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Pierre de Fermat

Life and work Fermat was born in Beaumont-de-Lomagne, Tarn-et-Garonne, France; the late 15th century mansion where Fermat was born is now a museum. He was of Basque origin. Fermat's father was a wealthy leather merchant and second consul of Beaumont-de-Lomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. There is little evidence concerning his school education, but it may have been at the local Franciscan monastery. He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius's De Locis Planis to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of François Viète. From Bordeaux, Fermat went to Orléans where he studied law at the University. He received a degree in civil law before, in 1631, receiving Bust in the Salle des Illustres in Capitole de the title of councillor at the High Court of Judicature in Toulouse, Toulouse which he held for the rest of his life. Due to the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fluent in Latin, Basque, classical Greek, Italian, and Spanish, Fermat was praised for his written verse in several languages, and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often with little or no proof of his theorems. This allowed him to preserve his status as an "amateur" while gaining the recognition he desired. This naturally led to priority disputes with contemporaries such as Descartes and Wallis. He developed a close relationship with Blaise Pascal.[3] Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods."[4]

Work Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").[5] In Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation.[6] In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature.

59

Pierre de Fermat

60 Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series.[7] The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered the little theorem. He invented a factorization method—Fermat's factorization method—as well as the proof technique of infinite descent, which he used to prove Fermat's Last Theorem for the case n = 4. Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on.

Although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted Pierre de Fermat several of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat. His famous Last Theorem was first discovered by his son in the margin on his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. He had not bothered to inform even Marin Mersenne of it. It was not proved until 1994 by Sir Andrew Wiles, using techniques unavailable to Fermat. Although he carefully studied, and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, and he looked for all possible general solutions. He often proved that certain equations had no solution, which usually baffled his contemporaries. Through his correspondence with Pascal in 1654, Fermat and Pascal helped lay the fundamental groundwork for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory.[8] Fermat is credited with carrying out the first ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in his losing. Fermat subsequently proved why this was the case mathematically.[9] Fermat's principle of least time (which he used to derive Snell's law in 1657) was the first variational principle[10] enunciated in physics since Hero of Alexandria described a principle of least distance in the first century CE. In this way, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The terms Fermat's principle and Fermat functional were named in recognition of this role.[11]

Pierre de Fermat

61

Death Pierre de Fermat died at Castres, Tarn.[2] The oldest and most prestigious high school in Toulouse is named after him: the Lycée Pierre de Fermat. French sculptor Théophile Barrau made a marble statue named Hommage à Pierre Fermat as tribute to Fermat, now at the Capitole of Toulouse.

Place of burial of Pierre de Fermat in Place Jean Jaurés, Castres, France. Translation of the plaque: in this place was buried on January 13, 1665, Pierre de Fermat, councilor of the chamber of Edit and mathematician of great renown, celebrated for his theorem (sic), an + bn ≠ cn for n>2

Assessment of his work Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. According to Peter L. Bernstein, in his book Against the Gods, Fermat "was a mathematician of rare power. He was an independent inventor of analytic geometry, he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. In the course of what turned out to be an extended correspondence with Pascal, he made a significant contribution to the theory of probability. But Fermat's crowning achievement was in the theory of numbers."[12] Regarding Fermat's work in analysis, Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."[13] Of Fermat's number theoretic work, the great 20th-century mathematician André Holographic will handwritten by Weil wrote that "... what we possess of his methods for dealing with curves of Fermat on 4 March 1660 — kept at the Departmental Archives of genus 1 is remarkably coherent; it is still the foundation for the modern theory of Haute-Garonne, in Toulouse such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the descent which is rightly regarded as Fermat's own."[14] Regarding Fermat's use of ascent, Weil continued "The novelty consisted in the vastly extended use which Fermat made of it, giving him at least a partial equivalent of what we would obtain by the systematic use of the group theoretical properties of the rational points on a standard cubic."[15] With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers.

Pierre de Fermat

Notes [1] Křížek, M.; Luca, Florian; Somer, Lawrence (2001). 17 lectures on Fermat numbers: from number theory to geometry. CMS books in mathematics. Springer. p. v. ISBN 978-0-387-95332-8. [2] Klaus Barner (2001): How old did Fermat become? (http:/ / cat. inist. fr/ ?aModele=afficheN& cpsidt=14213014) Internationale Zeitschrift für Geschichte und Ethik der Naturwissenschaften, Technik und Medizin. ISSN 0036-6978. Vol 9, No 4, pp. 209-228. [3] Ball, Walter William Rouse (1888). A short account of the history of mathematics. General Books LLC. ISBN 978-1-4432-9487-4. [4] http:/ / www. ams. org/ notices/ 199507/ faltings. pdf [5] Gullberg, Jan. Mathematics from the birth of numbers, W. W. Norton & Company; p. 548. ISBN 0-393-04002-X ISBN 978-0393040029 [6] Pellegrino, Dana. "Pierre de Fermat" (http:/ / www. math. rutgers. edu/ ~cherlin/ History/ Papers2000/ pellegrino. html). . Retrieved 2008-02-24. [7] Paradís, Jaume; Pla, Josep; Viader, Pelagrí. "Fermat’s Treatise On Quadrature: A New Reading" (http:/ / papers. ssrn. com/ sol3/ Delivery. cfm/ SSRN_ID848544_code386779. pdf?abstractid=848544& mirid=5). . Retrieved 2008-02-24 [8] O'Connor, J. J.; Robertson, E. F.. "The MacTutor History of Mathematics archive: Pierre de Fermat" (http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ Biographies/ Fermat. html). . Retrieved 2008-02-24 [9] Eves, Howard. An Introduction to the History of Mathematics, Saunders College Publishing, Fort Worth, Texas, 1990. [10] "Fermat’s principle for light rays" (http:/ / relativity. livingreviews. org/ open?pubNo=lrr-2004-9& page=articlesu9. html). . Retrieved 2008-02-24. [11] Červený, V. (July 2002). "Fermat's Variational Principle for Anisotropic Inhomogeneous Media" (http:/ / www. ingentaconnect. com/ content/ klu/ sgeg/ 2002/ 00000046/ 00000003/ 00450806). Studia Geophysica et Geodaetica 46 (3): 567. doi:10.1023/A:1019599204028. . [12] Bernstein, Peter L. (1996). Against the Gods: The Remarkable Story of Risk. John Wiley & Sons. pp. 61–62. ISBN 978-0-471-12104-6. [13] Simmons, George F. (2007). Calculus Gems: Brief Lives and Memorable Mathematics. Mathematical Association of America. p. 98. ISBN 0-88385-561-5. [14] Weil 1984, p.104 [15] Weil 1984, p.105

Books referenced • Weil, André (1984). Number Theory: An approach through history From Hammurapi to Legendre. Birkhäuser. ISBN 0-8176-3141-0.

Further reading • Mahoney, Michael Sean (1994). The mathematical career of Pierre de Fermat, 1601 - 1665. Princeton Univ. Press. ISBN 0-691-03666-7. • Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate Ltd. ISBN 1-84115-791-0.

External links • • • •

Fermat's Achievements (http://fermatslasttheorem.blogspot.com/2005/05/fermats-achievements.html) Fermat's Fallibility (http://www.mathpages.com/home/kmath195/kmath195.htm) at MathPages History of Fermat's Last Theorem (French) (http://ns33717.ovh.net/spokus/default/EN/all/tpe_felix/) The Life and times of Pierre de Fermat (1601 - 1665) (http://www.maths.tcd.ie/pub/HistMath/People/ Fermat/RouseBall/RB_Fermat.html) from W. W. Rouse Ball's History of Mathematics • O'Connor, John J.; Robertson, Edmund F., "Pierre de Fermat" (http://www-history.mcs.st-andrews.ac.uk/ Biographies/Fermat.html), MacTutor History of Mathematics archive, University of St Andrews. zj:Pierre de Fermat

62

List of things named after Pierre de Fermat

List of things named after Pierre de Fermat This is a list of things named after Pierre de Fermat, a French amateur mathematician. This list is incomplete. • • • • • • • • • • • •

Fermat cubic Fermat number Fermat polygonal number theorem Fermat Prize Fermat pseudoprime Fermat quotient Fermat's factorization method Fermat's principle Fermat's spiral Fermat's last theorem Fermat's little theorem Fermat's theorem (stationary points)

• Fermat's theorem on sums of two squares

63

Blaise Pascal

64

Blaise Pascal Blaise Pascal

Born

19 June 1623 Clermont-Ferrand, Auvergne, France

Died

19 August 1662 (aged 39) Paris, France

Residence

France

Nationality

French

Era

17th-century philosophy

Region

Western philosophy

Religion

Roman Catholic

School

Jansenism, precursor to existentialism

Main interests Theology, mathematics, philosophy, physics Notable ideas

Pascal's Wager Pascal's triangle Pascal's law Pascal's theorem

Blaise Pascal (French: [blɛz paskal]; 19 June 1623 – 19 August 1662), was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli. Pascal also wrote in defense of the scientific method. In 1642, while still a teenager, he started some pioneering work on calculating machines, and after three years of effort and 50 prototypes[1] he invented the mechanical calculator.[2][3] He built 20 of these machines (called pascal's calculator and later pascaline) in the following ten years.[4] Pascal was an important mathematician, helping create two major new areas of research: he wrote a significant treatise on the subject of projective geometry at the age of 16, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science. Following Galileo and Torricelli, in 1646 he refuted Aristotle's followers who insisted that nature abhors a vacuum. Pascal's results caused many disputes before being accepted.

Blaise Pascal In 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism.[5] His father died in 1651. Following a mystical experience in late 1654, he had his "second conversion", abandoned his scientific work, and devoted himself to philosophy and theology. His two most famous works date from this period: the Lettres provinciales and the Pensées, the former set in the conflict between Jansenists and Jesuits. In this year, he also wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids. Pascal had poor health especially after his 18th year and his death came just two months after his 39th birthday.[6]

Early life and education Pascal was born in Clermont-Ferrand; he lost his mother, Antoinette Begon, at the age of three.[7] His father, Étienne Pascal (1588–1651), who also had an interest in science and mathematics, was a local judge and member of the "Noblesse de Robe". Pascal had two sisters, the younger Jacqueline and the elder Gilberte. In 1631, five years after the death of his wife,[8] Étienne Pascal moved with his children to Paris. The newly arrived family soon hired Louise Delfault, a maid who eventually became an instrumental member of the family. Étienne, who never remarried, decided that he alone would educate his children, for they all showed extraordinary intellectual ability, particularly his son Blaise. The young Pascal showed an amazing aptitude for mathematics and science. Particularly of interest to Pascal was a work of Desargues on conic sections. Following Desargues' thinking, the 16-year-old Pascal produced, as a means of proof, a short treatise on what was called the "Mystic Hexagram", Essai pour les coniques ("Essay on Conics") and sent it—his first serious work of mathematics—to Père Mersenne in Paris; it is known still today as Pascal's theorem. It states that if a hexagon is inscribed in a circle (or conic) then the three intersection points of opposite sides lie on a line (called the Pascal line). Pascal's work was so precocious that Descartes was convinced that Pascal's father had written it. When assured by Mersenne that it was, indeed, the product of the son not the father, Descartes dismissed it with a sniff: "I do not find it strange that he has offered demonstrations about conics more appropriate than those of the ancients," adding, "but other matters related to this subject can be proposed that would scarcely occur to a 16-year-old child."[9] In France at that time offices and positions could be—and were—bought and sold. In 1631 Étienne sold his position as second president of the Cour des Aides for 65,665 livres.[10] The money was invested in a government bond which provided if not a lavish then certainly a comfortable income which allowed the Pascal family to move to, and enjoy, Paris. But in 1638 Richelieu, desperate for money to carry on the Thirty Years' War, defaulted on the government's bonds. Suddenly Étienne Pascal's worth had dropped from nearly 66,000 livres to less than 7,300. Like so many others, Étienne was eventually forced to flee Paris because of his opposition to the fiscal policies of Cardinal Richelieu, leaving his three children in the care of his neighbor Madame Sainctot, a great beauty with an infamous past who kept one of the most glittering and intellectual salons in all France. It was only when Jacqueline performed well in a children's play with Richelieu in attendance that Étienne was pardoned. In time Étienne was back in An early Pascaline on display at the Musée des good graces with the cardinal, and in 1639 had been appointed the Arts et Métiers, Paris king's commissioner of taxes in the city of Rouen — a city whose tax records, thanks to uprisings, were in utter chaos. In 1642, in an effort to ease his father's endless, exhausting calculations, and recalculations, of taxes owed and paid, Pascal, not yet 19, constructed a mechanical calculator capable of addition and subtraction, called Pascal's calculator or the Pascaline. The Musée des Arts et Métiers in Paris and the Zwinger museum in Dresden, Germany, exhibit two of his original mechanical calculators. Though these machines are early forerunners to computer engineering, the calculator failed to be a great commercial success. Because it was extraordinarily expensive the Pascaline became

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little more than a toy, and status symbol, for the very rich both in France and throughout Europe. However, Pascal continued to make improvements to his design through the next decade and built 20 machines in total.

Contributions to mathematics Pascal continued to influence mathematics throughout his life. His Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle") of 1653 described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle. The triangle can also be represented:

Pascal's triangle. Each number is the sum of the two directly above it. The triangle demonstrates many mathematical properties in addition to showing binomial coefficients.

0 1

2

3

4

5 6

0 1 1 1

1

1

1 1

1 1 2 3

4

5

6

2 1 3 6

10 15

3 1 4 10 20 4 1 5 15 5 1 6 6 1

He defines the numbers in the triangle by recursion: Call the number in the (m+1)st row and (n+1)st column tmn. Then tmn = tm-1,n + tm,n-1, for m = 0, 1, 2... and n = 0, 1, 2... The boundary conditions are tm, −1 = 0, t−1, n for m = 1, 2, 3... and n = 1, 2, 3... The generator t00 = 1. Pascal concludes with the proof,

In 1654, prompted by a friend interested in gambling problems, he corresponded with Fermat on the subject, and from that collaboration was born the mathematical theory of probabilities. The friend was the Chevalier de Méré, and the specific problem was that of two players who want to finish a game early and, given the current circumstances of the game, want to divide the stakes fairly, based on the chance each has of winning the game from that point. From this discussion, the notion of expected value was introduced. Pascal later (in the Pensées) used a probabilistic argument, Pascal's Wager, to justify belief in God and a virtuous life. The work done by Fermat and Pascal into the calculus of probabilities laid important groundwork for Leibniz' formulation of the infinitesimal calculus.[11] After a religious experience in 1654, Pascal mostly gave up work in mathematics.

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Philosophy of mathematics Pascal's major contribution to the philosophy of mathematics came with his De l'Esprit géométrique ("Of the Geometrical Spirit"), originally written as a preface to a geometry textbook for one of the famous "Petites-Ecoles de Port-Royal" ("Little Schools of Port-Royal"). The work was unpublished until over a century after his death. Here, Pascal looked into the issue of discovering truths, arguing that the ideal of such a method would be to found all propositions on already established truths. At the same time, however, he claimed this was impossible because such established truths would require other truths to back them up—first principles, therefore, cannot be reached. Based on this, Pascal argued that the procedure used in geometry was as perfect as possible, with certain principles assumed and other propositions developed from them. Nevertheless, there was no way to know the assumed principles to be true. Pascal also used De l'Esprit géométrique to develop a theory of definition. He distinguished between definitions which are conventional labels defined by the writer and definitions which are within the language and understood by everyone because they naturally designate their referent. The second type would be characteristic of the philosophy of essentialism. Pascal claimed that only definitions of the first type were important to science and mathematics, arguing that those fields should adopt the philosophy of formalism as formulated by Descartes. In De l'Art de persuader ("On the Art of Persuasion"), Pascal looked deeper into geometry's axiomatic method, specifically the question of how people come to be convinced of the axioms upon which later conclusions are based. Pascal agreed with Montaigne that achieving certainty in these axioms and conclusions through human methods is impossible. He asserted that these principles can be grasped only through intuition, and that this fact underscored the necessity for submission to God in searching out truths.

Contributions to the physical sciences Pascal's work in the fields of the study of hydrodynamics and hydrostatics centered on the principles of hydraulic fluids. His inventions include the hydraulic press (using hydraulic pressure to multiply force) and the syringe. He proved that hydrostatic pressure depends not on the weight of the fluid but on the elevation difference. He demonstrated this principle by attaching a thin tube to a barrel full of water and filling the tube with water up to the level of the third floor of a building. This caused the barrel to leak, in what became known as Pascal's barrel experiment. By 1646, Pascal had learned of Evangelista Torricelli's experimentation with barometers. Having replicated an experiment that involved placing a tube filled with mercury upside down in a bowl of mercury, Pascal questioned what force kept some mercury in the tube and what filled the space above the mercury in the tube. At the time, most scientists contended that, rather than a vacuum, Portrait of Pascal some invisible matter was present. This was based on the Aristotelian notion that creation was a thing of substance, whether visible or invisible; and that this substance was forever in motion. Furthermore, "Everything that is in motion must be moved by something," Aristotle declared.[12] Therefore, to the Aristotelian trained scientists of Pascal's time, a vacuum was an impossibility. How so? As proof it was pointed out: • Light passed through the so-called "vacuum" in the glass tube.

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• Aristotle wrote how everything moved, and must be moved by something. • Therefore, since there had to be an invisible "something" to move the light through the glass tube, there was no vacuum in the tube. Not in the glass tube or anywhere else. Vacuums — the absence of any and everything — were simply an impossibility. Following more experimentation in this vein, in 1647 Pascal produced Experiences nouvelles touchant le vide ("New Experiments with the Vacuum"), which detailed basic rules describing to what degree various liquids could be supported by air pressure. It also provided reasons why it was indeed a vacuum above the column of liquid in a barometer tube. On 19 September 1648, after many months of Pascal's friendly but insistent prodding, Florin Périer, husband of Pascal's elder sister Gilberte, was finally able to carry out the fact-finding mission vital to Pascal's theory. The account, written by Périer, reads: "The weather was chancy last Saturday...[but] around five o'clock that morning...the Puy-de-Dôme was visible...so I decided to give it a try. Several important people of the city of Clermont had asked me to let them know when I would make the ascent...I was delighted to have them with me in this great work...

An illustration of Pascal's barrel experiment of 1646

"...at eight o'clock we met in the gardens of the Minim Fathers, which has the lowest elevation in town....First I poured 16 pounds of quicksilver...into a vessel...then took several glass tubes...each four feet long and hermetically sealed at one end and opened at the other...then placed them in the vessel [of quicksilver]...I found the quick silver stood at 26" and 3½ lines above the quicksilver in the vessel...I repeated the experiment two more times while standing in the same spot...[they] produced the same result each time... "I attached one of the tubes to the vessel and marked the height of the quicksilver and...asked Father Chastin, one of the Minim Brothers...to watch if any changes should occur through the day...Taking the other tube and a portion of the quick silver...I walked to the top of Puy-de-Dôme, about 500 fathoms higher than the monastery, where upon experiment...found that the quicksilver reached a height of only 23" and 2 lines...I repeated the experiment five times with care...each at different points on the summit...found the same height of quicksilver...in each case..."[13] Pascal replicated the experiment in Paris by carrying a barometer up to the top of the bell tower at the church of Saint-Jacques-de-la-Boucherie, a height of about fifty meters. The mercury dropped two lines. In the face of criticism that some invisible matter must exist in Pascal's empty space, Pascal, in his reply to Estienne Noel, gave one of the 17th century's major statements on the scientific method, which is a striking anticipation of the idea popularised by Karl Popper that scientific theories are characterised by their falsifiability: "In order to show that a hypothesis is evident, it does not suffice that all the phenomena follow from it; instead, if it leads to something contrary to a single one of the phenomena, that suffices to establish its falsity."[14] His insistence on the existence of the vacuum also led to conflict with other prominent scientists, including Descartes. Pascal introduced a primitive form of roulette and the roulette wheel in the 17th century in his search for a perpetual motion machine.[15]

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Adult life, religion, philosophy, and literature For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either. The ends of things and their beginnings are impregnably concealed from him in an impenetrable secret. He is equally incapable of seeing the nothingness out of which he was drawn and the infinite in which he is engulfed. Blaise Pascal, Pensées #72

Religious conversion In the winter of 1646, Pascal's 58 year-old father broke his hip when he slipped and fell on an icy street of Rouen; given the man's age and the state of medicine in the 17th century, a broken hip could be a very serious condition, perhaps even fatal. Rouen was home to two of the finest doctors in France: Monsieur Doctor Deslandes and Monsieur Doctor de La Bouteillerie. The elder Pascal "would not let anyone other than these men attend him...It was a good choice, for the old man survived and was able to walk again..."[16] But treatment and rehabilitation took three months, during which time La Bouteillerie and Deslandes had become household guests. Both men were followers of Jean Guillebert, proponent of a splinter group from the main body of Catholic teaching known as Jansenism. This still fairly small sect was making surprising inroads into the French Catholic community at that time. It espoused rigorous Augustinism. Blaise spoke with the doctors frequently, Pascal studying the cycloid, by and upon his successful treatment of Étienne, borrowed works by Jansenist Augustin Pajou, 1785, Louvre authors from them. In this period, Pascal experienced a sort of "first conversion" and began to write on theological subjects in the course of the following year. Pascal fell away from this initial religious engagement and experienced a few years of what some biographers have called his "worldly period" (1648–54). His father died in 1651 and left his inheritance to Pascal and Jacqueline, of which Pascal acted as her conservator. Jacqueline announced that she would soon become a postulant in the Jansenist convent of Port-Royal. Pascal was deeply affected and very sad, not because of her choice, but because of his chronic poor health; he too needed her. "Suddenly there was war in the Pascal household. Blaise pleaded with Jacqueline not to leave, but she was adamant. He commanded her to stay, but that didn't work, either. At the heart of this was...Blaise's fear of abandonment...if Jacqueline entered Port-Royal, she would have to leave her inheritance behind...[but] nothing would change her mind."[17] By the end of October in 1651, a truce had been reached between brother and sister. In return for a healthy annual stipend, Jacqueline signed over her part of the inheritance to her brother. Gilberte had already been given her inheritance in the form of a dowry. In early January, Jacqueline left for Port-Royal. On that day, according to Gilberte concerning her brother, "He retired very sadly to his rooms without seeing Jacqueline, who was waiting in the little parlor..."[18] In early June 1653, after what must have seemed like endless badgering from Jacqueline, Pascal formally signed over the whole of his sister's inheritance to Port-Royal, which, to him, "had begun to smell like a cult."[19] With two-thirds of his father's estate now gone, the 29 year old Pascal was now consigned to genteel poverty. For a while, Pascal pursued the life of a bachelor. During visits to his sister at Port-Royal in 1654, he displayed contempt for affairs of the world but was not drawn to God.[20]

Blaise Pascal

Brush with death On 23 November 1654, between 10:30 and 12:30 at night, Pascal had an intense religious vision and immediately recorded the experience in a brief note to himself which began: "Fire. God of Abraham, God of Isaac, God of Jacob, not of the philosophers and the scholars..." and concluded by quoting Psalm 119:16: "I will not forget thy word. Amen." He seems to have carefully sewn this document into his coat and always transferred it when he changed clothes; a servant discovered it only by chance after his death.[21] This piece is now known as the Memorial. The story of the carriage accident as having led to the experience described in the Memorial is disputed by some scholars.[22] His belief and religious commitment revitalized, Pascal visited the older of two convents at Port-Royal for a two-week retreat in January 1655. For the next four years, he regularly travelled between Port-Royal and Paris. It was at this point immediately after his conversion when he began writing his first major literary work on religion, the Provincial Letters.

The Provincial Letters Beginning in 1656, Pascal published his memorable attack on casuistry, a popular ethical method used by Catholic thinkers in the early modern period (especially the Jesuits, and in particular Antonio Escobar). Pascal denounced casuistry as the mere use of complex reasoning to justify moral laxity and all sorts of sins. The 18-letter series was published between 1656 and 1657 under the pseudonym Louis de Montalte and incensed Louis XIV. The king ordered that the book be shredded and burnt in 1660. In 1661, in the midsts of the formulary controversy, the Jansenist school at Port-Royal was condemned and closed down; those involved with the school had to sign a 1656 papal bull condemning the teachings of Jansen as heretical. The final letter from Pascal, in 1657, had defied Alexander VII himself. Even Pope Alexander, while publicly opposing them, nonetheless was persuaded by Pascal's arguments. Aside from their religious influence, the Provincial Letters were popular as a literary work. Pascal's use of humor, mockery, and vicious satire in his arguments made the letters ripe for public consumption, and influenced the prose of later French writers like Voltaire and Jean-Jacques Rousseau. Wide praise has been given to the Provincial Letters.

The Pensées Pascal's most influential theological work, referred to posthumously as the Pensées ("Thoughts"), was not completed before his death. It was to have been a sustained and coherent examination and defense of the Christian faith, with the original title Apologie de la religion Chrétienne ("Defense of the Christian Religion"). The first version of the numerous scraps of paper found after his death appeared in print as a book in 1669 titled Pensées de M. Pascal sur la religion, et sur quelques autres sujets ("Thoughts of M. Pascal on religion, and on some other subjects") and soon thereafter became a classic. One of the Apologie's main strategies was to use the contradictory philosophies of skepticism and stoicism, personalized by Montaigne on one hand, and Epictetus on the other, in order to bring the unbeliever to such despair and confusion that he would embrace God. Pascal's Pensées is widely considered to be a masterpiece, and a landmark in French prose. When commenting on one particular section (Thought #72), Sainte-Beuve praised it as the finest pages in the French language.[23] Will Durant hailed it as "the most eloquent book in French prose."[24] In Pensées, Pascal surveys several philosophical paradoxes: infinity and nothing, faith and reason, soul and matter, death and life, meaning and vanity—seemingly arriving at no definitive conclusions besides humility, ignorance, and grace. Rolling these into one he develops Pascal's Wager.

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Last works and death T. S. Eliot described him during this phase of his life as "a man of the world among ascetics, and an ascetic among men of the world." Pascal's ascetic lifestyle derived from a belief that it was natural and necessary for a person to suffer. In 1659, Pascal fell seriously ill. During his last years, he frequently tried to reject the ministrations of his doctors, saying, "Sickness is the natural state of Christians."[25] Louis XIV suppressed the Jansenist movement at Port-Royal in 1661. In response, Pascal wrote one of his final works, Écrit sur la signature du formulaire ("Writ on the Signing of the Form"), exhorting the Jansenists not to give in. Later that year, his sister Jacqueline died, which convinced Pascal to cease his polemics on Jansenism. Pascal's last major achievement, returning to his mechanical genius, was inaugurating perhaps the first bus line, moving passengers within Paris in a carriage with many seats. In 1662, Pascal's illness became more violent, and his emotional condition had severely worsened since his sister's death, which Pascal's epitaph in Saint-Étienne-du-Mont, where he was buried happened the previous year. Aware that his health was fading quickly, he sought a move to the hospital for incurable diseases, but his doctors declared that he was too unstable to be carried. In Paris on 18 August 1662, Pascal went into convulsions and received extreme unction. He died the next morning, his last words being "May God never abandon me," and was buried in the cemetery of Saint-Étienne-du-Mont.[25] An autopsy performed after his death revealed grave problems with his stomach and other organs of his abdomen, along with damage to his brain. Despite the autopsy, the cause of his poor health was never precisely determined, though speculation focuses on tuberculosis, stomach cancer, or a combination of the two.[26] The headaches which afflicted Pascal are generally attributed to his brain lesion.

Legacy In honor of his scientific contributions, the name Pascal has been given to the SI unit of pressure, to a programming language, and Pascal's law (an important principle of hydrostatics), and as mentioned above, Pascal's triangle and Pascal's wager still bear his name. Pascal's development of probability theory was his most influential contribution to mathematics.[27] Originally applied to gambling, today it is extremely important in economics, especially in actuarial science. John Ross writes, "Probability theory and the discoveries following it changed the way we regard uncertainty, risk, decision-making, and an individual's and society's ability to influence the course of future events."[28] However, it should be noted that Pascal and Fermat, though doing important early work in probability theory, did not Death mask of Blaise Pascal. develop the field very far. Christiaan Huygens, learning of the subject from the correspondence of Pascal and Fermat, wrote the first book on the subject. Later figures who continued the development of the theory include Abraham de Moivre and Pierre-Simon Laplace.

Blaise Pascal In literature, Pascal is regarded as one of the most important authors of the French Classical Period and is read today as one of the greatest masters of French prose. His use of satire and wit influenced later polemicists. The content of his literary work is best remembered for its strong opposition to the rationalism of René Descartes and simultaneous assertion that the main countervailing philosophy, empiricism, was also insufficient for determining major truths. In France, prestigious annual awards, Blaise Pascal Chairs are given to outstanding international scientists to conduct their research in the Ile de France region.[29] One of the Universities of Clermont-Ferrand, France – Université Blaise Pascal – is named after him. The University of Waterloo, Ontario, Canada, holds an annual math contest named in his honour.[30] Roberto Rossellini directed a filmed biopic (entitled Blaise Pascal) which originally aired on Italian television in 1971. Pascal was a subject of the first edition of the 1984 BBC Two documentary, Sea of Faith, presented by Don Cupitt.

Works • Essai pour les coniques (1639) • Experiences nouvelles touchant le vide (1647) • Traité du triangle arithmétique (1653) • • • •

Lettres provinciales (1656–57) De l'Esprit géométrique (1657 or 1658) Écrit sur la signature du formulaire (1661) Pensées (incomplete at death)

References [1] (fr) La Machine d’arithmétique, Blaise Pascal (http:/ / fr. wikisource. org/ wiki/ La_Machine_d’arithmétique), Wikisource [2] Marguin, Jean (1994) (in fr). Histoire des instruments et machines à calculer, trois siècles de mécanique pensante 1642–1942. Hermann. p. 48. ISBN 978-2-7056-6166-3. [3] d'Ocagne, Maurice (1893) (in fr). Le calcul simplifié (http:/ / cnum. cnam. fr/ CGI/ fpage. cgi?8KU54-2. 5/ 248/ 150/ 369/ 363/ 369). Gauthier-Villars et fils. p. 245. . [4] Mourlevat, Guy (1988) (in fr). Les machines arithmétiques de Blaise Pascal. Clermont-Ferrand: La Française d'Edition et d'Imprimerie. p. 12. [5] "Blaise Pascal" (http:/ / www. newadvent. org/ cathen/ 11511a. htm). Catholic Encyclopedia. . Retrieved 2009-02-23. [6] Hald, Anders A History of Probability and Statistics and Its Applications before 1750, (Wiley Publications, 1990) pp.44 [7] Devlin, Keith, The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, Basic Books; 1 edition (2008), ISBN 978-0-465-00910-7, p. 20. [8] O'Connor, J.J.; Robertson, E.F. (August 2006). "Étienne Pascal" (http:/ / www-history. mcs. st-andrews. ac. uk/ Biographies/ Pascal_Etienne. html). University of St. Andrews, Scotland. . Retrieved 5 February 2010. [9] The Story of Civilization: Volume 8, "The Age of Louis XIV" by Will & Ariel Durant; chapter II, subsection 4.1 p.56) [10] Connor, James A., Pascal's wager: the man who played dice with God (HarperCollins, NY, 2006) ISBN 0-06-076691-3 p. 42 [11] "The Mathematical Leibniz" (http:/ / www. math. rutgers. edu/ courses/ 436/ Honors02/ leibniz. html). Math.rutgers.edu. . Retrieved 2009-08-16. [12] Aristotle, Physics, VII, 1. [13] Périer to Pascal, 22 September 1648, Pascal, Blaise. Oeuvres complètes. (Paris: Seuil, 1960), 2:682. [14] Pour faire qu'une hypothèse soit évidente, il ne suffit pas que tous les phénomènes s'en ensuivent, au lieu que, s'il s'ensuit quelque chose de contraire à un seul des phénomènes, cela suffit pour assurer de sa fausseté, in Les Lettres de Blaise Pascal: Accompagnées de Lettres de ses Correspondants Publiées, ed. Maurice Beaufreton, 6th edition (Paris: G. Crès, 1922), 25–26, available at http:/ / gallica. bnf. fr and translated in Saul Fisher, Pierre Gassendi's Philosophy and Science: Atomism for Empiricists Brill's Studies in Intellectual History 131 (Leiden: E. J. Brill, 2005), 126 n.7 [15] MIT, "Inventor of the Week Archive: Pascal : Mechanical Calculator" (http:/ / web. mit. edu/ invent/ iow/ pascal. html), May 2003. "Pascal worked on many versions of the devices, leading to his attempt to create a perpetual motion machine. He has been credited with introducing the roulette machine, which was a by-product of these experiments." [16] Connor, James A., Pascal's wager: the man who played dice with God (HarperCollins, NY, 2006) ISBN 0-06-076691-3 p. 70 [17] Miel, Jan. Pascal and Theology. (Baltimore: Johns Hopkins University Press, 1969), p. 122 [18] Jacqueline Pascal, "Memoir" p. 87 [19] Miel, Jan. Pascal and Theology. (Baltimore: Johns Hopkins University Press, 1969), p. 124

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Blaise Pascal [20] Richard H. Popkin, Paul Edwards (ed.), Encyclopedia of Philosophy, 1967 edition, s.v. "Pascal, Blaise.", vol. 6, p. 52–55, New York: Macmillan [21] Pascal, Blaise. Oeuvres complètes. (Paris: Seuil, 1960), p. 618 [22] MathPages, Hold Your Horses (http:/ / www. mathpages. com/ home/ kmath558/ kmath558. htm) [23] Sainte-Beuve, Seventeenth Century (http:/ / books. google. com/ books?id=I0P0A8XK29QC& pg=PA167) ISBN 1-113-16675-4 p. 174 (2009 reprint). [24] The Story of Civilization: Volume 8, "The Age of Louis XIV" by Will & Ariel Durant, chapter II, Subsection 4.4, p. 66 ISBN 1-56731-019-2 [25] Muir, Jane. Of Men and Numbers (http:/ / books. google. com/ books?id=uV3rJkmnQhsC& printsec=frontcover). (New York: Dover Publications, Inc, 1996). ISBN 0-486-28973-7, p. 104. [26] Muir, Jane. Of Men and Numbers (http:/ / books. google. com/ books?id=uV3rJkmnQhsC& printsec=frontcover). (New York: Dover Publications, Inc, 1996). ISBN 0-486-28973-7, p. 103. [27] "Blaise Pascal" (http:/ / www. famousscientists. org/ blaise-pascal). FamousScientists.org. . Retrieved 2011-12-15. [28] Ross, John F. (2004). "Pascal's legacy". EMBO Reports 5 (Suppl 1): S7–S10. doi:10.1038/sj.embor.7400229. PMC 1299210. PMID 15459727. [29] "Chaires Blaise Pascal" (http:/ / www. chaires-blaise-pascal. org/ uk/ index. html). Chaires Blaise Pascal. . Retrieved 2009-08-16. [30] "CEMC – Pascal, Cayley and Fermat – Mathematics Contests – University of Waterloo" (http:/ / www. cemc. uwaterloo. ca/ contests/ pcf. html). Cemc.uwaterloo.ca. 2008-06-23. . Retrieved 2009-08-16.

Further reading • Adamson, Donald. Blaise Pascal: Mathematician, Physicist, and Thinker about God (1995) ISBN 0-333-55036-6 • Adamson, Donald. "Pascal's Views on Mathematics and the Divine," (http://books.google.com/ books?id=AMOQZfrZq-EC&pg=PA405) Mathematics and the Divine: A Historical Study (eds. T. Koetsier and L. Bergmans. Amsterdam: Elsevier 2005), pp. 407–21. • Broome, J.H. Pascal. (London: E. Arnold, 1965). ISBN 0-7131-5021-1 • Davidson, Hugh M. Blaise Pascal. (Boston: Twayne Publishers), 1983. • Farrell, John. "Pascal and Power". Chapter seven of Paranoia and Modernity: Cervantes to Rousseau (Cornell UP, 2006). • Goldmann, Lucien, The hidden God; a study of tragic vision in the Pensees of Pascal and the tragedies of Racine (original ed. 1955, Trans. Philip Thody. London: Routledge, 1964). • Jordan, Jeff. Pascal's Wager: Pragmatic Arguments and Belief in God. (Oxford: Clarendon Press, 2006). • Landkildehus, Søren. "Kierkegaard and Pascal as kindred spirits in the Fight against Christendom" in Kierkegaard and the Renaissance and Modern Traditions (ed. Jon Stewart. Farnham: Ashgate Publishing, 2009). • Mackie, John Leslie. The Miracle of Theism: Arguments for and against the Existence of God. (Oxford: Oxford University Press, 1982). • Saka, Paul (2001). "Pascal's Wager and the Many Gods Objection". Religious Studies 37 (3): 321–41. doi:10.1017/S0034412501005686. • Stephen, Leslie. "Pascal" (in English). Studies of a Biographer. 2. London: Duckworth and Co.. p. 241–284. • Tobin, Paul. "The Rejection of Pascal's Wager: A Skeptic's Guide to the Bible and the Historical Jesus". authorsonline.co.uk, 2009. • Yves Morvan, Pascal à Mirefleurs ? Les dessins de la maison de Domat, Impr. Blandin, 1985.(FRBNF40378895)

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External links • Pascal's Memorial (http://www.users.csbsju.edu/~eknuth/pascal.html) in orig. French/Latin and modern English, trans. Elizabeth T. Knuth. • Biography, Bibliography. (http://www.biblioweb.org/-PASCAL-Blaise-.html) (in French) • Works by Blaise Pascal (http://www.gutenberg.org/author/Pascal+Blaise) at Project Gutenberg • Works by Blaise Pascal on Open Library at the Internet Archive • Blaise Pascal featured on the 500 French Franc banknote in 1977. (http://www-personal.umich.edu/~jbourj/ money5.htm) • Blaise Pascal's works (http://www.intratext.com/Catalogo/Autori/Aut852.htm): text, concordances and frequency lists •  "Blaise Pascal". Catholic Encyclopedia. New York: Robert Appleton Company. 1913. • Etext of Pascal's Pensées (http://www.ccel.org/ccel/pascal/pensees.html) (English, in various formats) • Etext of Pascal's Lettres Provinciales (http://oregonstate.edu/instruct/phl302/texts/pascal/letters-a.html) (English) • Etext of a number of Pascal's minor works (http://www.bartleby.com/48/3/) (English translation) including, De l'Esprit géométrique and De l'Art de persuader. • O'Connor, John J.; Robertson, Edmund F., "Blaise Pascal" (http://www-history.mcs.st-andrews.ac.uk/ Biographies/Pascal.html), MacTutor History of Mathematics archive, University of St Andrews.

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Isaac Newton Sir Isaac Newton

Godfrey Kneller's 1689 portrait of Isaac Newton (age 46). Born

25 December 1642

[NS: 4 January 1643]

[1]

Woolsthorpe-by-Colsterworth, Lincolnshire, England Died

20 March 1726 (aged 83)

[OS: 20 March 1726 [1] NS: 31 March 1727] Kensington, Middlesex, England Resting place

Westminster Abbey

Residence

England

Nationality

English (later British)

Fields

• •

Physics Natural philosophy

• •

Mathematics Astronomy

• •

Alchemy Christian theology

Institutions

• • •

University of Cambridge Royal Society Royal Mint

Alma mater

Trinity College, Cambridge

Academic advisors • •

[2] Isaac Barrow [3][4] Benjamin Pulleyn

Notable students

Roger Cotes William Whiston

• •

Isaac Newton

76 Known for

Influences

Influenced

• • •

•

Newtonian mechanics Universal gravitation Infinitesimal calculus • •

Optics Binomial series

• •

Principia Newton's method

[5] Henry More

• •

[6] Polish Brethren [7] Robert Boyle

• •

Nicolas Fatio de Duillier John Keill Signature

Sir Isaac Newton PRS MP (25 December 1642 – 20 March 1726) was an English physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, who has been considered by many to be the greatest and most influential scientist who ever lived.[8][9] His monograph Philosophiæ Naturalis Principia Mathematica, published in 1687, laid the foundations for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion, which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motion of objects on Earth and that of celestial bodies is governed by the same set of natural laws: by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation he removed the last doubts about heliocentrism and advanced the scientific revolution. The Principia is generally considered to be one of the most important scientific books ever written, both due to the specific physical laws the work successfully described, and for its style, which assisted in setting standards for scientific publication down to the present time. Newton built the first practical reflecting telescope[10] and developed a theory of colour based on the observation that a prism decomposes white light into the many colours that form the visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound. In mathematics, Newton shares the credit with Gottfried Leibniz for the development of differential and integral calculus. He generalised the binomial theorem to non-integer exponents, developed Newton's method for approximating the roots of a function, and contributed to the study of power series. Although an unorthodox Christian, Newton was deeply religious and his occult studies took up a substantial part of his life. He secretly rejected Trinitarianism and refused holy orders.[11]

Life Early life Isaac Newton was born (according to the Julian calendar in use in England at the time) on Christmas Day, 25 December 1642, (NS 4 January 1643.[1]) at Woolsthorpe Manor in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. He was born three months after the death of his father, a prosperous farmer also named Isaac Newton. Born prematurely, he was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug (≈ 1.1 litres). When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabus Smith, leaving her son in the care of his maternal grandmother, Margery Ayscough. The young Isaac disliked his stepfather and maintained some enmity towards his mother for marrying him, as revealed by this entry in a list of sins committed up to the age of 19: "Threatening my father and mother Smith to

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burn them and the house over them."[12] Although it was claimed that he was once engaged,[13] Newton never married. From the age of about twelve until he was seventeen, Newton was educated at The King's School, Grantham. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, where his mother, widowed by now for a second time, attempted to make a farmer of him. He hated farming.[14] Henry Stokes, master at the King's School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a schoolyard bully, he became the top-ranked student.[15] The Cambridge psychologist Simon Baron-Cohen considers it "fairly certain" that Newton had Asperger syndrome.[16] In June 1661, he was admitted to Trinity College, Cambridge as a sizar – a sort of work-study role.[17] At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers, such as Descartes, and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student,[18] Newton's private studies at his home in Woolsthorpe over the subsequent two years saw the development of his theories on calculus,[19] optics and the law of gravitation. In 1667, he returned to Cambridge as a fellow of Trinity.[20] Fellows were required to become ordained priests, something Newton desired to avoid due to his unorthodox views. Luckily for Newton, there was no specific deadline for ordination, and it could be postponed indefinitely. The problem became more severe later when Newton was elected for the prestigious Lucasian Chair. For such a significant appointment, ordaining normally could not be dodged. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II (see "Middle years" section below).

Newton in a 1702 portrait by Godfrey Kneller

Isaac Newton (Bolton, Sarah K. Famous Men of Science. NY: Thomas Y. Crowell & Co., 1889)

Middle years Mathematics Newton's work has been said "to distinctly advance every branch of mathematics then studied".[21] His work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newton's mathematical papers.[22] The author of the manuscript De analysi per aequationes numero terminorum infinitas, sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as:[23] Mr Newton, a fellow of our College, and very young ... but of an extraordinary genius and proficiency in these things. Newton later became involved in a dispute with Leibniz over priority in the development of infinitesimal calculus (the Leibniz–Newton calculus controversy). Most modern historians believe that Newton and Leibniz developed infinitesimal calculus independently, although with very different notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did not give a full account until 1704, while Leibniz began publishing a full account of his methods in 1684. (Leibniz's notation and "differential Method", nowadays recognised

Isaac Newton as much more convenient notations, were adopted by continental European mathematicians, and after 1820 or so, also by British mathematicians.) Such a suggestion, however, fails to notice the content of calculus which critics of Newton's time and modern times have pointed out in Book 1 of Newton's Principia itself (published 1687) and in its forerunner manuscripts, such as De motu corporum in gyrum ("On the motion of bodies in orbit"), of 1684. The Principia is not written in the language of calculus either as we know it or as Newton's (later) 'dot' notation would write it. But his work extensively uses an infinitesimal calculus in geometric form, based on limiting values of the ratios of vanishing small quantities: in the Principia itself Newton gave demonstration of this under the name of 'the method of first and last ratios'[24] and explained why he put his expositions in this form,[25] remarking also that 'hereby the same thing is performed as by the method of indivisibles'. Because of this, the Principia has been called "a book dense with the theory and application of the infinitesimal calculus" in modern times[26] and "lequel est presque tout de ce calcul" ('nearly all of it is of this calculus') in Newton's time.[27] His use of methods involving "one or more orders of the infinitesimally small" is present in his De motu corporum in gyrum of 1684[28] and in his papers on motion "during the two decades preceding 1684".[29] Newton had been reluctant to publish his calculus because he feared controversy and criticism.[30] He was close to the Swiss mathematician Nicolas Fatio de Duillier. In 1691, Duillier started to write a new version of Newton's Principia, and corresponded with Leibniz.[31] In 1693 the relationship between Duillier and Newton deteriorated, and the book was never completed. Starting in 1699, other members of the Royal Society (of which Newton was a member) accused Leibniz of plagiarism, and the dispute broke out in full force in 1711. The Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud. This study was cast into doubt when it was later found that Newton himself wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both Newton and Leibniz until the latter's death in 1716.[32] Newton is generally credited with the generalised binomial theorem, valid for any exponent. He discovered Newton's identities, Newton's method, classified cubic plane curves (polynomials of degree three in two variables), made substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic series by logarithms (a precursor to Euler's summation formula), and was the first to use power series with confidence and to revert power series. Newton's work on infinite series was inspired by Simon Stevin's decimals.[33] He was appointed Lucasian Professor of Mathematics in 1669 on Barrow's recommendation. In that day, any fellow of Cambridge or Oxford was required to become an ordained Anglican priest. However, the terms of the Lucasian professorship required that the holder not be active in the church (presumably so as to have more time for science). Newton argued that this should exempt him from the ordination requirement, and Charles II, whose permission was needed, accepted this argument. Thus a conflict between Newton's religious views and Anglican orthodoxy was averted.[34]

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Optics From 1670 to 1672, Newton lectured on optics.[36] During this period he investigated the refraction of light, demonstrating that a prism could decompose white light into a spectrum of colours, and that a lens and a second prism could recompose the multicoloured spectrum into white light.[37] Modern scholarship has revealed that Newton's analysis and resynthesis of white light owes a debt to corpuscular alchemy.[38] He also showed that the coloured light does not change its properties by separating out a coloured beam and shining it on various objects. Newton noted that regardless of whether it was reflected or scattered or transmitted, it stayed the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as Newton's theory of colour.[39]

A replica of Newton's second Reflecting telescope that he presented to the Royal Society [35] in 1672

From this work, he concluded that the lens of any refracting telescope would suffer from the dispersion of light into colours (chromatic aberration). As a proof of the concept, he constructed a telescope using a mirror as the objective to bypass that problem.[40] Building the design, the first known functional reflecting telescope, today known as a Newtonian telescope,[40] involved solving the problem of a suitable mirror material and shaping technique. Newton ground his own mirrors out of a custom composition of highly reflective speculum metal, using Newton's rings to judge the quality of the optics for his telescopes. In late 1668[41] he was able to produce this first reflecting telescope. In 1671, the Illustration of a dispersive prism decomposing white light into the Royal Society asked for a demonstration of his colours of the spectrum, as discovered by Newton reflecting telescope.[42] Their interest encouraged him to publish his notes On Colour, which he later expanded into his Opticks. When Robert Hooke criticised some of Newton's ideas, Newton was so offended that he withdrew from public debate. Newton and Hooke had brief exchanges in 1679–80, when Hooke, appointed to manage the Royal Society's correspondence, opened up a correspondence intended to elicit contributions from Newton to Royal Society transactions,[43] which had the effect of stimulating Newton to work out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector (see Newton's law of universal gravitation – History and De motu corporum in gyrum). But the two men remained generally on poor terms until Hooke's death.[44]

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80

Newton argued that light is composed of particles or corpuscles, which were refracted by accelerating into a denser medium. He verged on soundlike waves to explain the repeated pattern of reflection and transmission by thin films (Opticks Bk.II, Props. 12), but still retained his theory of 'fits' that disposed corpuscles to be reflected or transmitted (Props.13). Later physicists instead favoured a purely wavelike explanation of light to account for the interference patterns, and the general phenomenon of diffraction. Today's quantum mechanics, photons and the idea of wave–particle duality bear only a minor resemblance to Newton's understanding of light. In his Hypothesis of Light of 1675, Newton posited the existence of the ether to transmit forces between particles. The contact with the theosophist Henry More, revived his interest in alchemy. He replaced the ether with occult forces based on Hermetic ideas of attraction and repulsion between particles. John Maynard Keynes, who acquired Facsimile of a 1682 letter from Isaac Newton to many of Newton's writings on alchemy, stated that "Newton was not [45] Dr William Briggs, commenting on Briggs' "A the first of the age of reason: He was the last of the magicians." New Theory of Vision". Newton's interest in alchemy cannot be isolated from his contributions to science.[5] This was at a time when there was no clear distinction between alchemy and science. Had he not relied on the occult idea of action at a distance, across a vacuum, he might not have developed his theory of gravity. (See also Isaac Newton's occult studies.) In 1704, Newton published Opticks, in which he expounded his corpuscular theory of light. He considered light to be made up of extremely subtle corpuscles, that ordinary matter was made of grosser corpuscles and speculated that through a kind of alchemical transmutation "Are not gross Bodies and Light convertible into one another, ...and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?"[46] Newton also constructed a primitive form of a frictional electrostatic generator, using a glass globe (Optics, 8th Query). In an article entitled "Newton, prisms, and the 'opticks' of tunable lasers[47] it is indicated that Newton in his book Opticks was the first to show a diagram using a prism as a beam expander. In the same book he describes, via diagrams, the use of multiple-prism arrays. Some 278 years after Newton's discussion, multiple-prism beam expanders became central to the development of narrow-linewidth tunable lasers. Also, the use of these prismatic beam expanders led to the multiple-prism dispersion theory.[47] Mechanics and gravitation In 1679, Newton returned to his work on (celestial) mechanics, i.e., gravitation and its effect on the orbits of planets, with reference to Kepler's laws of planetary motion. This followed stimulation by a brief exchange of letters in 1679–80 with Hooke, who had been appointed to manage the Royal Society's correspondence, and who opened a correspondence intended to elicit contributions from Newton to Royal Society transactions.[43] Newton's reawakening interest in astronomical matters received further stimulus by the appearance of a comet in the winter of 1680–1681, on which he corresponded with John Flamsteed.[48] After the exchanges with Hooke, Newton worked out a proof that the elliptical form of planetary orbits would result from a

Newton's own copy of his Principia, with hand-written corrections for the second edition

Isaac Newton centripetal force inversely proportional to the square of the radius vector (see Newton's law of universal gravitation – History and De motu corporum in gyrum). Newton communicated his results to Edmond Halley and to the Royal Society in De motu corporum in gyrum, a tract written on about 9 sheets which was copied into the Royal Society's Register Book in December 1684.[49] This tract contained the nucleus that Newton developed and expanded to form the Principia. The Principia was published on 5 July 1687 with encouragement and financial help from Edmond Halley. In this work, Newton stated the three universal laws of motion that enabled many of the advances of the Industrial Revolution which soon followed and were not to be improved upon for more than 200 years, and are still the underpinnings of the non-relativistic technologies of the modern world. He used the Latin word gravitas (weight) for the effect that would become known as gravity, and defined the law of universal gravitation. In the same work, Newton presented a calculus-like method of geometrical analysis by 'first and last ratios', gave the first analytical determination (based on Boyle's law) of the speed of sound in air, inferred the oblateness of the spheroidal figure of the Earth, accounted for the precession of the equinoxes as a result of the Moon's gravitational attraction on the Earth's oblateness, initiated the gravitational study of the irregularities in the motion of the moon, provided a theory for the determination of the orbits of comets, and much more. Newton made clear his heliocentric view of the solar system – developed in a somewhat modern way, because already in the mid-1680s he recognised the "deviation of the Sun" from the centre of gravity of the solar system.[50] For Newton, it was not precisely the centre of the Sun or any other body that could be considered at rest, but rather "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and this centre of gravity "either is at rest or moves uniformly forward in a right line" (Newton adopted the "at rest" alternative in view of common consent that the centre, wherever it was, was at rest).[51] Newton's postulate of an invisible force able to act over vast distances led to him being criticised for introducing "occult agencies" into science.[52] Later, in the second edition of the Principia (1713), Newton firmly rejected such criticisms in a concluding General Scholium, writing that it was enough that the phenomena implied a gravitational attraction, as they did; but they did not so far indicate its cause, and it was both unnecessary and improper to frame hypotheses of things that were not implied by the phenomena. (Here Newton used what became his famous expression "hypotheses non fingo"[53]). With the Principia, Newton became internationally recognised.[54] He acquired a circle of admirers, including the Swiss-born mathematician Nicolas Fatio de Duillier, with whom he formed an intense relationship. This abruptly ended in 1693, and at the same time Newton suffered a nervous breakdown.[55]

Classification of cubics Besides the work of Newton and others on calculus, the first important demonstration of the power of analytic geometry was Newton's classification of cubic curves in the Euclidean plane in the late 1600s. He divided them into four types, satisfying different equations, and in 1717 Stirling, probably with Newton's help, proved that every cubic was one of these four. Newton also claimed that the four types could be obtained by plane projection from one of them, and this was proved in 1731.[56]

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Later life In the 1690s, Newton wrote a number of religious tracts dealing with the literal interpretation of the Bible. Henry More's belief in the Universe and rejection of Cartesian dualism may have influenced Newton's religious ideas. A manuscript he sent to John Locke in which he disputed the existence of the Trinity was never published. Later works – The Chronology of Ancient Kingdoms Amended (1728) and Observations Upon the Prophecies of Daniel and the Apocalypse of St. John (1733) – were published after his death. He also devoted a great deal of time to alchemy (see above). Newton was also a member of the Parliament of England from 1689 to 1690 and in 1701, but according to some accounts his only comments were to complain about a cold draught in the chamber and request that the window be closed.[57] Newton moved to London to take up the post of warden of the Royal Mint in 1696, a position that he had obtained through the patronage of Charles Montagu, 1st Earl of Halifax, then Chancellor of the Exchequer. He took charge of England's great recoining, somewhat treading on the toes of Lord Lucas, Governor of the Tower (and securing the job of deputy comptroller of the temporary Chester branch for Edmond Halley). Newton became perhaps the best-known Master of the Mint upon the death of Thomas Neale in 1699, a position Newton held for the last 30 years of his life.[58][59] These appointments were intended as sinecures, but Newton took them seriously, retiring from his Cambridge duties in 1701, and exercising his power to reform the currency and punish clippers and counterfeiters. As Master of the Mint in 1717 in the "Law of Queen Anne" Newton moved the Pound Sterling de facto from the silver standard to the gold standard by setting the bimetallic relationship between gold coins and the silver penny in favour of gold. This caused silver sterling coin to be melted and shipped out of Britain. Newton was made President of the Royal Society in 1703 and an associate of the French Académie des Sciences. In his position at the Royal Society, Newton made an enemy of John Flamsteed, the Astronomer Royal, by prematurely publishing Flamsteed's Historia Coelestis Britannica, which Newton had used in his studies.[60] Isaac Newton in old age in 1712, portrait by Sir James Thornhill

In April 1705, Queen Anne knighted Newton during a royal visit to Trinity College, Cambridge. The knighthood is likely to have been motivated by political considerations connected with the Parliamentary election in May 1705, rather than any recognition of Newton's scientific work or services as Master of the Mint.[62] Newton was the second scientist to be knighted, after Sir Francis Bacon. Towards the end of his life, Newton took up residence at Cranbury Park, near Winchester with his niece and her husband, until his death in 1726.[63] His half-niece, Catherine Barton Conduitt,[64] served as his hostess in social affairs at his house on Jermyn Street in London; he was her "very loving Uncle,"[65] according to his letter to her when she was recovering from smallpox.

Personal coat of arms of Sir Isaac [61] Newton

Newton died in his sleep in London on 20 March 1726 (OS 20 March 1726; NS 31 March 1727)[1] and was buried in Westminster Abbey. A bachelor, he had divested much of his estate to relatives during his last years, and died intestate. After his death, Newton's hair was examined and found to contain mercury, probably resulting from his alchemical pursuits. Mercury poisoning could explain Newton's eccentricity in late life.[66]

Isaac Newton

After death Fame French mathematician Joseph-Louis Lagrange often said that Newton was the greatest genius who ever lived, and once added that Newton was also "the most fortunate, for we cannot find more than once a system of the world to establish."[67] English poet Alexander Pope was moved by Newton's accomplishments to write the famous epitaph: Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light. Newton himself had been rather more modest of his own achievements, famously writing in a letter to Robert Hooke in February 1676: If I have seen further it is by standing on the shoulders of giants.[68] Two writers think that the above quote, written at a time when Newton and Hooke were in dispute over optical discoveries, was an oblique attack on Hooke (said to have been short and hunchbacked), rather than – or in addition to – a statement of modesty.[69][70] On the other hand, the widely known proverb about standing on the shoulders of giants published among others by 17th-century poet George Herbert (a former orator of the University of Cambridge and fellow of Trinity College) in his Jacula Prudentum (1651), had as its main point that "a dwarf on a giant's shoulders sees farther of the two", and so its effect as an analogy would place Newton himself rather than Hooke as the 'dwarf'. In a later memoir, Newton wrote: I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.[71] Albert Einstein kept a picture of Newton on his study wall alongside ones of Michael Faraday and James Clerk Maxwell.[72] Newton remains influential to today's scientists, as demonstrated by a 2005 survey of members of Britain's Royal Society (formerly headed by Newton) asking who had the greater effect on the history of science, Newton or Einstein. Royal Society scientists deemed Newton to have made the greater overall contribution.[73] In 1999, an opinion poll of 100 of today's leading physicists voted Einstein the "greatest physicist ever;" with Newton the runner-up, while a parallel survey of rank-and-file physicists by the site PhysicsWeb gave the top spot to Newton.[74] Commemorations

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Newton's monument (1731) can be seen in Westminster Abbey, at the north of the entrance to the choir against the choir screen, near his tomb. It was executed by the sculptor Michael Rysbrack (1694–1770) in white and grey marble with design by the architect William Kent. The monument features a figure of Newton reclining on top of a sarcophagus, his right elbow resting on several of his great books and his left hand pointing to a scroll with a mathematical design. Above him is a pyramid and a celestial globe showing the signs of the Zodiac and the path of the comet of 1680. A relief panel depicts putti using instruments such as a telescope and prism.[75] The Latin inscription on the base translates as: Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, Newton statue on display at the explored the course and figures of the planets, the paths of comets, Oxford University Museum of the tides of the sea, the dissimilarities in rays of light, and, what no Natural History other scholar has previously imagined, the properties of the colours thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25 December 1642, and died on 20 March 1726/7. — Translation from G.L. Smyth, The Monuments and Genii of St. Paul's Cathedral, and of Westminster Abbey (1826), ii, 703–4.[75] From 1978 until 1988, an image of Newton designed by Harry Ecclestone appeared on Series D £1 banknotes issued by the Bank of England (the last £1 notes to be issued by the Bank of England). Newton was shown on the reverse of the notes holding a book and accompanied by a telescope, a prism and a map of the Solar System.[76] A statue of Isaac Newton, looking at an apple at his feet, can be seen at the Oxford University Museum of Natural History. A large bronze statue, Newton, after William Blake, by Eduardo Paolozzi, dated 1995 and inspired by Blake's etching, dominates the piazza of the British Library in London.

Personal life Newton never married, and no evidence has been uncovered that he had any romantic relationship. Although it is impossible to verify, it is commonly believed that he died a virgin, as has been commented on by such figures as mathematician Charles Hutton,[77] economist John Maynard Keynes,[78] and physicist Carl Sagan.[79] Eduardo Paolozzi's Newton, after William Blake (1995), outside the British Library

French writer and philosopher Voltaire, who was in London at the time of Newton's funeral, claimed to have verified the fact, writing that "I have had that confirmed by the doctor and the surgeon who were with him when he died"[80] (allegedly he stated on his deathbed that he was a virgin[81][82]). In 1733, Voltaire publicly stated that Newton "had neither passion nor weakness; he never went near any woman".[83][84] Newton did have a close friendship with the Swiss mathematician Nicolas Fatio de Duillier, whom he met in London around 1690.[85] Their friendship came to an unexplained end in 1693. Some of their correspondence has survived.

Isaac Newton

Religious views In a minority view, T.C. Pfizenmaier argues that Newton held the Eastern Orthodox view on the Trinity rather than the Western one held by Roman Catholics, Anglicans and most Protestants.[86] However, this type of view 'has lost support of late with the availability of Newton's theological papers',[87] and now most scholars identify Newton as an Antitrinitarian monotheist.[6][88] 'In Newton's eyes, worshipping Christ as God was idolatry, to him the fundamental sin'.[89] Historian Stephen D. Snobelen says of Newton, "Isaac Newton was a heretic. But ... he never made a public declaration of his private faith—which the orthodox would have deemed extremely radical. He hid his faith so well that scholars are still unravelling his personal beliefs."[6] Snobelen concludes that Newton was at least a Socinian sympathiser (he owned and had thoroughly read at least eight Socinian books), possibly an Arian and almost certainly an Newton's tomb in Westminster anti-trinitarian.[6] In an age notable for its religious intolerance, there are few Abbey public expressions of Newton's radical views, most notably his refusal to receive holy orders and his refusal, on his death bed, to receive the sacrament when it was offered to him.[6] Although the laws of motion and universal gravitation became Newton's best-known discoveries, he warned against using them to view the Universe as a mere machine, as if akin to a great clock. He said, "Gravity explains the motions of the planets, but it cannot explain who set the planets in motion. God governs all things and knows all that is or can be done."[90] Along with his scientific fame, Newton's studies of the Bible and of the early Church Fathers were also noteworthy. Newton wrote works on textual criticism, most notably An Historical Account of Two Notable Corruptions of Scripture. He placed the crucifixion of Jesus Christ at 3 April, AD 33, which agrees with one traditionally accepted date.[91] He also tried unsuccessfully to find hidden messages within the Bible. Newton wrote more on religion than he did on natural science. He believed in a rationally immanent world, but he rejected the hylozoism implicit in Leibniz and Baruch Spinoza. The ordered and dynamically informed Universe could be understood, and must be understood, by an active reason. In his correspondence, Newton claimed that in writing the Principia "I had an eye upon such Principles as might work with considering men for the belief of a Deity".[92] He saw evidence of design in the system of the world: "Such a wonderful uniformity in the planetary system must be allowed the effect of choice". But Newton insisted that divine intervention would eventually be required to reform the system, due to the slow growth of instabilities.[93] For this, Leibniz lampooned him: "God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion."[94] Newton's position was vigorously defended by his follower Samuel Clarke in a famous correspondence. A century later, Pierre-Simon Laplace's work "Celestial Mechanics" had a natural explanation for why the planet orbits don't require periodic divine intervention.[95]

Effect on religious thought Newton and Robert Boyle's mechanical philosophy was promoted by rationalist pamphleteers as a viable alternative to the pantheists and enthusiasts, and was accepted hesitantly by orthodox preachers as well as dissident preachers like the latitudinarians.[96] The clarity and simplicity of science was seen as a way to combat the emotional and metaphysical superlatives of both superstitious enthusiasm and the threat of atheism,[97] and at the same time, the second wave of English deists used Newton's discoveries to demonstrate the possibility of a "Natural Religion".

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Isaac Newton

Newton, by William Blake; here, Newton is depicted critically as a "divine geometer".

86 The attacks made against pre-Enlightenment "magical thinking", and the mystical elements of Christianity, were given their foundation with Boyle's mechanical conception of the Universe. Newton gave Boyle's ideas their completion through mathematical proofs and, perhaps more importantly, was very successful in popularising them.[98] Newton refashioned the world governed by an interventionist God into a world crafted by a God that designs along rational and universal principles.[99] These principles were available for all people to discover, allowed people to pursue their own aims fruitfully in this life, not the next, and to perfect themselves with their own rational powers.[100]

Newton saw God as the master creator whose existence could not be denied in the face of the grandeur of all creation.[101][102][103] His spokesman, Clarke, rejected Leibniz' theodicy which cleared God from the responsibility for l'origine du mal by making God removed from participation in his creation, since as Clarke pointed out, such a deity would be a king in name only, and but one step away from atheism.[104] But the unforeseen theological consequence of the success of Newton's system over the next century was to reinforce the deist position advocated by Leibniz.[105] The understanding of the world was now brought down to the level of simple human reason, and humans, as Odo Marquard argued, became responsible for the correction and elimination of evil.[106]

End of the world In a manuscript he wrote in 1704 in which he describes his attempts to extract scientific information from the Bible, he estimated that the world would end no earlier than 2060. In predicting this he said, "This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fanciful men who are frequently predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions fail."[107]

Enlightenment philosophers Enlightenment philosophers chose a short history of scientific predecessors – Galileo, Boyle, and Newton principally – as the guides and guarantors of their applications of the singular concept of Nature and Natural Law to every physical and social field of the day. In this respect, the lessons of history and the social structures built upon it could be discarded.[108] It was Newton's conception of the Universe based upon Natural and rationally understandable laws that became one of the seeds for Enlightenment ideology.[109] Locke and Voltaire applied concepts of Natural Law to political systems advocating intrinsic rights; the physiocrats and Adam Smith applied Natural conceptions of psychology and self-interest to economic systems; and sociologists criticised the current social order for trying to fit history into Natural models of progress. Monboddo and Samuel Clarke resisted elements of Newton's work, but eventually rationalised it to conform with their strong religious views of nature.

Counterfeiters As warden of the Royal Mint, Newton estimated that 20 percent of the coins taken in during The Great Recoinage of 1696 were counterfeit. Counterfeiting was high treason, punishable by the felon's being hanged, drawn and quartered. Despite this, convicting the most flagrant criminals could be extremely difficult. However, Newton proved to be equal to the task.[110] Disguised as a habitué of bars and taverns, he gathered much of that evidence himself.[111] For all the barriers placed to prosecution, and separating the branches of government, English law still had ancient and formidable customs of authority. Newton had himself made a justice of the peace in all the home

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counties—there is a draft of a letter regarding this matter stuck into Newton's personal first edition of his Philosophiæ Naturalis Principia Mathematica which he must have been amending at the time.[112] Then he conducted more than 100 cross-examinations of witnesses, informers, and suspects between June 1698 and Christmas 1699. Newton successfully prosecuted 28 coiners.[113] One of Newton's cases as the King's attorney was against William Chaloner.[114] Chaloner's schemes included setting up phony conspiracies of Catholics and then turning in the hapless conspirators whom he had entrapped. Chaloner made himself rich enough to posture as a gentleman. Petitioning Parliament, Chaloner accused the Mint of providing tools to counterfeiters (a charge also made by others). He proposed that he be allowed to inspect the Mint's processes in order to improve them. He petitioned Parliament to adopt his plans for a coinage that could not be counterfeited, while at the same time striking false coins.[115] Newton put Chaloner on trial for counterfeiting and had him sent to Newgate Prison in September 1697. But Chaloner had friends in high places, who helped him secure an acquittal and his release.[114] Newton put him on trial a second time with conclusive evidence. Chaloner was convicted of high treason and hanged, drawn and quartered on 23 March 1699 at Tyburn gallows.[116]

Laws of motion In the Principia, Newton gives the famous three laws of motion, stated here in modern form. Newton's First Law (also known as the Law of Inertia) states that an object at rest tends to stay at rest and that an object in uniform motion tends to stay in uniform motion unless acted upon by a net external force. The meaning of this law is the existence of reference frames (called inertial frames) where objects not acted upon by forces move in uniform motion (in particular, they may be at rest). Newton's Second Law states that an applied force,

, on an object equals the rate of change of its momentum,

,

with time. Mathematically, this is expressed as

Since the law applies only to systems of constant mass,[117] m can be brought out of the derivative operator. By substitution using the definition of acceleration, the equation can be written in the iconic form

The first and second laws represent a break with the physics of Aristotle, in which it was believed that a force was necessary in order to maintain motion. They state that a force is only needed in order to change an object's state of motion. The SI unit of force is the newton, named in Newton's honour. Newton's Third Law states that for every action there is an equal and opposite reaction. This means that any force exerted onto an object has a counterpart force that is exerted in the opposite direction back onto the first object. A common example is of two ice skaters pushing against each other and sliding apart in opposite directions. Another example is the recoil of a firearm, in which the force propelling the bullet is exerted equally back onto the gun and is felt by the shooter. Since the objects in question do not necessarily have the same mass, the resulting acceleration of the two objects can be different (as in the case of firearm recoil). Unlike Aristotle's, Newton's physics is meant to be universal. For example, the second law applies both to a planet and to a falling stone. The vector nature of the second law addresses the geometrical relationship between the direction of the force and the manner in which the object's momentum changes. Before Newton, it had typically been assumed that a planet orbiting the Sun would need a forward force to keep it moving. Newton showed instead that all that was needed was an inward attraction from the Sun. Even many decades after the publication of the Principia, this counterintuitive idea was not universally accepted, and many scientists preferred Descartes' theory of vortices.[118]

Isaac Newton

Apple incident

Reputed descendants of Newton's apple tree, at the Cambridge University Botanic Garden and the Instituto Balseiro library garden

Newton himself often told the story that he was inspired to formulate his theory of gravitation by watching the fall of an apple from a tree.[119] Although it has been said that the apple story is a myth and that he did not arrive at his theory of gravity in any single moment,[120] acquaintances of Newton (such as William Stukeley, whose manuscript account of 1752 has been made available by the Royal Society)[121] do in fact confirm the incident, though not the cartoon version that the apple actually hit Newton's head. Stukeley recorded in his Memoirs of Sir Isaac Newton's Life a conversation with Newton in Kensington on 15 April 1726:[122] ... We went into the garden, & drank tea under the shade of some appletrees, only he, & myself. amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. "why should that apple always descend perpendicularly to the ground," thought he to him self: occasion'd by the fall of an apple, as he sat in a comtemplative mood: "why should it not go sideways, or upwards? but constantly to the earths centre? assuredly, the reason is, that the earth draws it. there must be a drawing power in matter. & the sum of the drawing power in the matter of the earth must be in the earths centre, not in any side of the earth. therefore dos this apple fall perpendicularly, or toward the centre. if matter thus draws matter; it must be in proportion of its quantity. therefore the apple draws the earth, as well as the earth draws the apple." John Conduitt, Newton's assistant at the Royal Mint and husband of Newton's niece, also described the event when he wrote about Newton's life:[123] In the year 1666 he retired again from Cambridge to his mother in Lincolnshire. Whilst he was pensively meandering in a garden it came into his thought that the power of gravity (which brought an apple from a tree to the ground) was not limited to a certain distance from earth, but that this power must extend much further than was usually thought. Why not as high as the Moon said he to himself & if so, that must influence her motion & perhaps retain her in her orbit, whereupon he fell a calculating what would be the effect of that supposition. In similar terms, Voltaire wrote in his Essay on Epic Poetry (1727), "Sir Isaac Newton walking in his gardens, had the first thought of his system of gravitation, upon seeing an apple falling from a tree." It is known from his notebooks that Newton was grappling in the late 1660s with the idea that terrestrial gravity extends, in an inverse-square proportion, to the Moon; however it took him two decades to develop the full-fledged theory.[124] The question was not whether gravity existed, but whether it extended so far from Earth that it could also be the force holding the Moon to its orbit. Newton showed that if the force decreased as the inverse square of the

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Isaac Newton distance, one could indeed calculate the Moon's orbital period, and get good agreement. He guessed the same force was responsible for other orbital motions, and hence named it "universal gravitation". Various trees are claimed to be "the" apple tree which Newton describes. The King's School, Grantham, claims that the tree was purchased by the school, uprooted and transported to the headmaster's garden some years later. The staff of the [now] National Trust-owned Woolsthorpe Manor dispute this, and claim that a tree present in their gardens is the one described by Newton. A descendant of the original tree[125] can be seen growing outside the main gate of Trinity College, Cambridge, below the room Newton lived in when he studied there. The National Fruit Collection at Brogdale[126] can supply grafts from their tree, which appears identical to Flower of Kent, a coarse-fleshed cooking variety.[127]

Writings • • • • • •

Method of Fluxions (1671) Of Natures Obvious Laws & Processes in Vegetation (unpublished, c. 1671–75)[128] De motu corporum in gyrum (1684) Philosophiæ Naturalis Principia Mathematica (1687) Opticks (1704) Reports as Master of the Mint [129] (1701–25)

• Arithmetica Universalis (1707) • The System of the World, Optical Lectures, The Chronology of Ancient Kingdoms, (Amended) and De mundi systemate (published posthumously in 1728) • Observations on Daniel and The Apocalypse of St. John (1733) • An Historical Account of Two Notable Corruptions of Scripture (1754)

References [1] During Newton's lifetime, two calendars were in use in Europe: the Julian ("Old Style") calendar in protestant and Orthodox regions, including Britain; and the Gregorian ("New Style") calendar in Roman Catholic Europe. At Newton's birth, Gregorian dates were ten days ahead of Julian dates: thus his birth is recorded as taking place on 25 December 1642 Old Style, but can be converted to a New Style (modern) date of 4 January 1643. By the time of his death, the difference between the calendars had increased to eleven days: moreover, he died in the period after the start of the New Style year on 1 January, but before that of the Old Style new year on 25 March. His death occurred on 20 March 1726 according to the Old Style calendar, but the year is usually adjusted to 1727. A full conversion to New Style gives the date 31 March 1727. [2] Mordechai Feingold, Barrow, Isaac (1630–1677) (http:/ / www. oxforddnb. com/ view/ article/ 1541), Oxford Dictionary of National Biography, Oxford University Press, September 2004; online edn, May 2007. Retrieved 24 February 2009; explained further in Mordechai Feingold's " Newton, Leibniz, and Barrow Too: An Attempt at a Reinterpretation (http:/ / www. jstor. org/ stable/ 236236)" in Isis, Vol. 84, No. 2 (June 1993), pp. 310–338. [3] "Newton, Isaac" (http:/ / www. chlt. org/ sandbox/ lhl/ dsb/ page. 50. a. php) in the Dictionary of Scientific Biography, n.4. [4] Gjersten, Derek (1986). The Newton Handbook. London: Routledge & Kegan Paul. [5] Westfall, Richard S. (1983) [1980]. Never at Rest: A Biography of Isaac Newton. Cambridge: Cambridge University Press. pp. 530–1. ISBN 978-0-521-27435-7. [6] Snobelen, Stephen D. (1999). "Isaac Newton, heretic: the strategies of a Nicodemite" (http:/ / www. isaac-newton. org/ heretic. pdf) (PDF). British Journal for the History of Science 32 (4): 381–419. doi:10.1017/S0007087499003751. . [7] Stokes, Mitch (2010). Isaac Newton (http:/ / books. google. gr/ books?id=zpsoSXCeg5gC& pg=PA97& lpg=PA97& dq=#v=onepage& q="Boyle influenced Newton"& f=false). Thomas Nelson. p. 97. ISBN 1595553037. . Retrieved 17 October 2012. [8] See below, under Fame. [9] Burt, Daniel S. (2001). The biography book: a reader's guide to nonfiction, fictional, and film biographies of more than 500 of the most fascinating individuals of all time (http:/ / books. google. com/ books?id=jpFrgSAaKAUC). Greenwood Publishing Group. p. 315. ISBN 1-57356-256-4. ., Extract of page 315 (http:/ / books. google. com/ books?id=jpFrgSAaKAUC& pg=PA315) [10] "The Early Period (1608–1672)" (http:/ / etoile. berkeley. edu/ ~jrg/ TelescopeHistory/ Early_Period. html). James R. Graham's Home Page. . Retrieved 3 February 2009. [11] Christianson, Gale E. (1996). Isaac Newton and the scientific revolution (http:/ / books. google. com/ books?id=O61ypNXvNkUC& pg=PA74). Oxford University Press. p. 74. ISBN 0-19-509224-4. . [12] Cohen, I.B. (1970). Dictionary of Scientific Biography, Vol. 11, p.43. New York: Charles Scribner's Sons

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Isaac Newton [13] This claim was made Dr. Stukeley in 1727, in a letter about Newton written to Dr. Richard Mead. Charles Hutton, who in the late 18th century collected oral traditions about earlier scientists, declares that there "do not appear to be any sufficient reason for his never marrying, if he had an inclination so to do. It is much more likely that he had a constitutional indifference to the state, and even to the sex in general." Charles Hutton "A Mathematical and Philosophical Dictionary" (1795/6) II p.100. [14] Westfall 1994, pp 16–19 [15] White 1997, p. 22 [16] James, Ioan (January 2003). "Singular scientists". Journal of the Royal Society of Medicine 96 (1): 36–39. doi:10.1258/jrsm.96.1.36. PMC 539373. PMID 12519805. [17] Michael White, Isaac Newton (1999) page 46 (http:/ / books. google. com/ books?id=l2C3NV38tM0C& pg=PA24& dq=storer+ intitle:isaac+ intitle:newton& lr=& num=30& as_brr=0& as_pt=ALLTYPES#PPA46,M1) [18] ed. Michael Hoskins (1997). Cambridge Illustrated History of Astronomy, p. 159. Cambridge University Press [19] Newton, Isaac. "Waste Book" (http:/ / cudl. lib. cam. ac. uk/ view/ MS-ADD-04004/ ). Cambridge University Digital Library. . Retrieved 10 January 2012. [20] Venn, J.; Venn, J. A., eds. (1922–1958). " Newton, Isaac (http:/ / venn. lib. cam. ac. uk/ cgi-bin/ search. pl?sur=& suro=c& fir=& firo=c& cit=& cito=c& c=all& tex=RY644J& sye=& eye=& col=all& maxcount=50)". Alumni Cantabrigienses (10 vols) (online ed.). Cambridge University Press. [21] W W Rouse Ball (1908), "A short account of the history of mathematics", at page 319. [22] D T Whiteside (ed.), The Mathematical Papers of Isaac Newton (Volume 1), (Cambridge University Press, 1967), part 7 "The October 1666 Tract on Fluxions", at page 400, in 2008 reprint (http:/ / books. google. com/ books?id=1ZcYsNBptfYC& pg=PA400). [23] D Gjertsen (1986), "The Newton handbook", (London (Routledge & Kegan Paul) 1986), at page 149. [24] Newton, 'Principia', 1729 English translation, at page 41 (http:/ / books. google. com/ books?id=Tm0FAAAAQAAJ& pg=PA41). [25] Newton, 'Principia', 1729 English translation, at page 54 (http:/ / books. google. com/ books?id=Tm0FAAAAQAAJ& pg=PA54). [26] Clifford Truesdell, Essays in the History of Mechanics (Berlin, 1968), at p.99. [27] In the preface to the Marquis de L'Hospital's Analyse des Infiniment Petits (Paris, 1696). [28] Starting with De motu corporum in gyrum, see also (Latin) Theorem 1 (http:/ / books. google. com/ books?id=uvMGAAAAcAAJ& pg=RA1-PA2). [29] D T Whiteside (1970), "The Mathematical principles underlying Newton's Principia Mathematica" in Journal for the History of Astronomy, vol.1, pages 116–138, especially at pages 119–120. [30] Stewart 2009, p.107 [31] Westfall 1980, pp 538–539 [32] Ball 1908, p. 356ff [33] Błaszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), "Ten misconceptions from the history of analysis and their debunking", Foundations of Science, arXiv:1202.4153, doi:10.1007/s10699-012-9285-8 [34] White 1997, p. 151 [35] King, Henry C (2003). ''The History of the Telescope'' By Henry C. King, Page 74 (http:/ / books. google. com/ ?id=KAWwzHlDVksC& dq=history+ of+ the+ telescope& printsec=frontcover). Google Books. ISBN 978-0-486-43265-6. . Retrieved 16 January 2010. [36] Newton, Isaac. "Hydrostatics, Optics, Sound and Heat" (http:/ / cudl. lib. cam. ac. uk/ view/ MS-ADD-03970/ ). Cambridge University Digital Library. . Retrieved 10 January 2012. [37] Ball 1908, p. 324 [38] William R. Newman, "Newton's Early Optical Theory and its Debt to Chymistry," in Danielle Jacquart and Michel Hochmann, eds., Lumière et vision dans les sciences et dans les arts (Geneva: Droz, 2010), pp. 283-307. A free access online version of this article can be found at the Chymistry of Isaac Newton project (http:/ / webapp1. dlib. indiana. edu/ newton/ html/ Newton_optics-alchemy_Jacquart_paper. pdf) [39] Ball 1908, p. 325 [40] White 1997, p170 [41] Hall, Alfred Rupert (1996). '''Isaac Newton: adventurer in thought''', by Alfred Rupert Hall, page 67 (http:/ / books. google. com/ ?id=32IDpTdthm4C& pg=PA67& lpg=PA67& dq=newton+ reflecting+ telescope+ + 1668+ letter+ 1669& q=newton reflecting telescope 1668 letter 1669). Google Books. ISBN 978-0-521-56669-8. . Retrieved 16 January 2010. [42] White 1997, p168 [43] See 'Correspondence of Isaac Newton, vol.2, 1676–1687' ed. H W Turnbull, Cambridge University Press 1960; at page 297, document No. 235, letter from Hooke to Newton dated 24 November 1679. [44] Iliffe, Robert (2007) Newton. A very short introduction, Oxford University Press 2007 [45] Keynes, John Maynard (1972). "Newton, The Man". The Collected Writings of John Maynard Keynes Volume X. MacMillan St. Martin's Press. pp. 363–4. [46] Dobbs, J.T. (December 1982). "Newton's Alchemy and His Theory of Matter". Isis 73 (4): 523. doi:10.1086/353114. quoting Opticks [47] Duarte, F. J. (2000). "Newton, prisms, and the 'opticks' of tunable lasers" (http:/ / www. opticsjournal. com/ F. J. DuarteOPN(2000). pdf). Optics and Photonics News 11 (5): 24–25. Bibcode 2000OptPN..11...24D. doi:10.1364/OPN.11.5.000024. . [48] R S Westfall, 'Never at Rest', 1980, at pages 391–2. [49] D T Whiteside (ed.), 'Mathematical Papers of Isaac Newton', vol.6, 1684–1691, Cambridge University Press 1974, at page 30.

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Isaac Newton [50] See Curtis Wilson, "The Newtonian achievement in astronomy", pages 233–274 in R Taton & C Wilson (eds) (1989) The General History of Astronomy, Volume, 2A', at page 233 (http:/ / books. google. com/ books?id=rkQKU-wfPYMC& pg=PA233). [51] Text quotations are from 1729 translation of Newton's Principia, Book 3 (1729 vol.2) at pages 232–233 (http:/ / books. google. com/ books?id=6EqxPav3vIsC& pg=PA233). [52] Edelglass et al., Matter and Mind, ISBN 0-940262-45-2. p. 54 [53] On the meaning and origins of this expression, see Kirsten Walsh, Does Newton feign an hypothesis? (https:/ / blogs. otago. ac. nz/ emxphi/ 2010/ 10/ does-newton-feign-an-hypothesis/ ), Early Modern Experimental Philosophy (https:/ / blogs. otago. ac. nz/ emxphi/ ), 18 October 2010. [54] Westfall 1980. Chapter 11. [55] Westfall 1980. pp 493–497 on the friendship with Fatio, pp 531–540 on Newton's breakdown. [56] Conics and Cubics, Robert Bix, Springer Undergraduate Texts in Mathematics, 2nd edition, 2006, Springer Verlag. [57] White 1997, p. 232 [58] "[ Newton: Physicist And ... Crime Fighter? (http:/ / www. npr. org/ templates/ story/ story. php?storyId=105012144|Isaac)]". Science Friday. 5 June 2009. NPR. [59] Thomas Levenson (2009). Newton and the counterfeiter : the unknown detective career of the world's greatest scientist. Houghton Mifflin Harcourt. ISBN 978-0-15-101278-7. OCLC 276340857. [60] White 1997, p.317 [61] Gerard Michon. "Coat of arms of Isaac Newton" (http:/ / www. numericana. com/ arms/ index. htm#newton). Numericana.com. . Retrieved 16 January 2010. [62] "The Queen's 'great Assistance' to Newton's election was his knighting, an honor bestowed not for his contributions to science, nor for his service at the Mint, but for the greater glory of party politics in the election of 1705." Westfall 1994 p.245 [63] Yonge, Charlotte M. (1898). "Cranbury and Brambridge" (http:/ / www. online-literature. com/ charlotte-yonge/ john-keble/ 6/ ). John Keble's Parishes – Chapter 6. www.online-literature.com. . Retrieved 23 September 2009. [64] Westfall 1980, p. 44. [65] Westfall 1980, p. 595 [66] "Newton, Isaac (1642–1726)" (http:/ / scienceworld. wolfram. com/ biography/ Newton. html). Eric Weisstein's World of Biography. . Retrieved 30 August 2006. [67] Fred L. Wilson, History of Science: Newton citing: Delambre, M. "Notice sur la vie et les ouvrages de M. le comte J. L. Lagrange," Oeuvres de Lagrange I. Paris, 1867, p. xx. [68] Letter from Isaac Newton to Robert Hooke, 5 February 1676, as transcribed in Jean-Pierre Maury (1992) Newton: Understanding the Cosmos, New Horizons [69] John Gribbin (2002) Science: A History 1543–2001, p 164. [70] White 1997, p187. [71] Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton (1855) by Sir David Brewster (Volume II. Ch. 27) [72] "Einstein's Heroes: Imagining the World through the Language of Mathematics", by Robyn Arianrhod UQP, reviewed by Jane Gleeson-White, 10 November 2003, The Sydney Morning Herald [73] "Newton beats Einstein in polls of Royal Society scientists and the public" (http:/ / royalsociety. org/ News. aspx?id=1324& terms=Newton+ beats+ Einstein+ in+ polls+ of+ scientists+ and+ the+ public). The Royal Society. . [74] "Opinion poll. Einstein voted "greatest physicist ever" by leading physicists; Newton runner-up" (http:/ / news. bbc. co. uk/ 2/ hi/ science/ nature/ 541840. stm). BBC News. 29 November 1999. . Retrieved 17 January 2012. [75] "Famous People & the Abbey: Sir Isaac Newton" (http:/ / www. westminster-abbey. org/ our-history/ people/ sir-isaac-newton). Westminster Abbey. . Retrieved 13 November 2009. [76] "Withdrawn banknotes reference guide" (http:/ / www. bankofengland. co. uk/ banknotes/ denom_guide/ nonflash/ 1-SeriesD-Revised. htm). Bank of England. . Retrieved 27 August 2009. [77] Hutton, Charles (1815). A Philosophical and Mathematical Dictionary Containing... Memoirs of the Lives and Writings of the Most Eminent Authors, Volume 2 (http:/ / books. google. ca/ books?id=_xk2AAAAQAAJ& pg=PA100& lpg=PA100& dq=Charles+ Hutton+ Isaac+ Newton+ constitutional+ indifference& source=bl& ots=gxI1T-5UzL& sig=NJHnmCqkPwNalnOSrUXZZgkfODs& hl=en#v=onepage& q=Charles Hutton Isaac Newton constitutional indifference& f=false). p. 100. . Retrieved 11 September 2012. [78] John Maynard Keynes. "Newton: the Man" (http:/ / www-history. mcs. st-and. ac. uk/ Extras/ Keynes_Newton. html). University of St Andrews School of Mathematics and Statistics. . Retrieved 11 September 2012. [79] Carl, Sagan (1980). Cosmos (http:/ / books. google. ca/ books?id=_-XhL6_xsVkC& pg=PA55& lpg=PA55& dq=Isaac+ Newton+ virgin& source=bl& ots=pfxDt6lG8I& sig=u4GtOW8G0jCFdrppKL2o0j9ZAKU& hl=en& sa=X& ei=jrJJULeTIYnDigLs14Fo& ved=0CEMQ6AEwAzge#v=onepage& q=Isaac Newton virgin& f=false). New York: Random House. ISBN 0394502949. . Retrieved 11 September 2012. [80] Letters on England, 14, pp. 68-70, as referenced in the footnote for the quote in p. 6 of James Gleick's biography, Isaac Newton [81] Stokes, Mitch (2010). Isaac Newton (http:/ / books. google. ca/ books?id=zpsoSXCeg5gC& pg=PA154& lpg=PA154& dq=Isaac+ Newton+ virgin+ confess& source=bl& ots=jL4JIVcIJe& sig=JYyHgrFXKVc_fQrc_Xr3FXjJYkw& hl=en#v=onepage& q=Isaac Newton virgin confess& f=false). Thomas Nelson. p. 154. ISBN 1595553037. . Retrieved 11 September 2012.

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Isaac Newton [82] Foster, Jacob (2005). "Everybody Loves Einstein" (http:/ / www. oxonianreview. org/ issues/ 5-1/ 5-1foster. html). The Oxonian Review 5 (1). . [83] Gjertsen, Derek (1986). The Newton Handbook (http:/ / books. google. ca/ books?id=cqIOAAAAQAAJ& pg=PA105& lpg=PA105& dq=Isaac+ Newton+ virgin& source=bl& ots=Sf2QL1yV2J& sig=0m7VW3Ca0_jKFl-k-P8FNAATuaY& hl=en#v=onepage& q=Isaac Newton virgin& f=false). Taylor & Francis. p. 105. ISBN 0710202792. . Retrieved 11 September 2012. [84] Fara, Patricia (2011). Newton: The Making of Genius. Pan Macmillan. ISBN 1447204530. [85] Professor Robert A. Hatch, University of Florida. "Newton Timeline" (http:/ / web. clas. ufl. edu/ users/ ufhatch/ pages/ 13-NDFE/ newton/ 05-newton-timeline-m. htm). . Retrieved 13 August 2012. [86] Pfizenmaier, T.C. (1997). "Was Isaac Newton an Arian?". Journal of the History of Ideas 58 (1): 57–80. [87] Snobelen, Stephen D. (1999). "Isaac Newton, heretic: the strategies of a Nicodemite" (http:/ / www. isaac-newton. org/ heretic. pdf) (PDF). British Journal for the History of Science 32 (4): 381–419. doi:10.1017/S0007087499003751. . [88] Avery Cardinal Dulles. The Deist Minimum (http:/ / www. firstthings. com/ print. php?type=article& year=2008& month=08& title_link=the-deist-minimum--28). January 2005. [89] Westfall, Richard S. (1994). The Life of Isaac Newton. Cambridge: Cambridge University Press. ISBN 0-521-47737-9. [90] Tiner, J.H. (1975). Isaac Newton: Inventor, Scientist and Teacher. Milford, Michigan, U.S.: Mott Media. ISBN 0-915134-95-0. [91] John P. Meier, A Marginal Jew, v. 1, pp. 382–402 after narrowing the years to 30 or 33, provisionally judges 30 most likely. [92] Newton to Richard Bentley 10 December 1692, in Turnbull et al. (1959–77), vol 3, p. 233. [93] Opticks, 2nd Ed 1706. Query 31. [94] H. G. Alexander (ed) The Leibniz-Clarke correspondence, Manchester University Press, 1998, p. 11. [95] Neil Degrasse Tyson (November 2005). "The Perimeter of Ignorance" (http:/ / www. haydenplanetarium. org/ tyson/ read/ 2005/ 11/ 01/ the-perimeter-of-ignorance). Natural History Magazine. . [96] Jacob, Margaret C. (1976). The Newtonians and the English Revolution: 1689–1720. Cornell University Press. pp. 37, 44. ISBN 0-85527-066-7. [97] Westfall, Richard S. (1958). Science and Religion in Seventeenth-Century England. New Haven: Yale University Press. p. 200. ISBN 0-208-00843-8. [98] Haakonssen, Knud. "The Enlightenment, politics and providence: some Scottish and English comparisons". In Martin Fitzpatrick ed.. Enlightenment and Religion: Rational Dissent in eighteenth-century Britain. Cambridge: Cambridge University Press. p. 64. ISBN 0-521-56060-8. [99] Frankel, Charles (1948). The Faith of Reason: The Idea of Progress in the French Enlightenment. New York: King's Crown Press. p. 1. [100] Germain, Gilbert G.. A Discourse on Disenchantment: Reflections on Politics and Technology. p. 28. ISBN 0-7914-1319-5. [101] Principia, Book III; cited in; Newton's Philosophy of Nature: Selections from his writings, p. 42, ed. H.S. Thayer, Hafner Library of Classics, NY, 1953. [102] A Short Scheme of the True Religion, manuscript quoted in Memoirs of the Life, Writings and Discoveries of Sir Isaac Newton by Sir David Brewster, Edinburgh, 1850; cited in; ibid, p. 65. [103] Webb, R.K. ed. Knud Haakonssen. "The emergence of Rational Dissent." Enlightenment and Religion: Rational Dissent in eighteenth-century Britain. Cambridge University Press, Cambridge: 1996. p19. [104] H. G. Alexander (ed) The Leibniz-Clarke correspondence, Manchester University Press, 1998, p. 14. [105] Westfall, 1958 p201. [106] Marquard, Odo. "Burdened and Disemburdened Man and the Flight into Unindictability," in Farewell to Matters of Principle. Robert M. Wallace trans. London: Oxford UP, 1989. [107] "Papers Show Isaac Newton's Religious Side, Predict Date of Apocalypse" (http:/ / web. archive. org/ web/ 20070813033620/ http:/ / www. christianpost. com/ article/ 20070619/ 28049_Papers_Show_Isaac_Newton's_Religious_Side,_Predict_Date_of_Apocalypse. htm). Associated Press. 19 June 2007. Archived from the original (http:/ / www. christianpost. com/ article/ 20070619/ 28049_Papers_Show_Isaac_Newton's_Religious_Side,_Predict_Date_of_Apocalypse. htm) on 13 August 2007. . Retrieved 1 August 2007. [108] Cassels, Alan. Ideology and International Relations in the Modern World. p2. [109] "Although it was just one of the many factors in the Enlightment, the success of Newtonian physics in providing a mathematical description of an ordered world clearly played a big part in the flowering of this movement in the eighteenth century" John Gribbin (2002) Science: A History 1543–2001, p 241 [110] White 1997, p. 259 [111] White 1997, p. 267 [112] Newton, Isaac. "Philosophiæ Naturalis Principia Mathematica" (http:/ / cudl. lib. cam. ac. uk/ view/ PR-ADV-B-00039-00001/ ). Cambridge University Digital Library. pp. 265–266. . Retrieved 10 January 2012. [113] Westfall 2007, p.73 [114] White 1997, p 269 [115] Westfall 1994, p 229 [116] Westfall 1980, pp. 571–5 [117] Halliday; Resnick. Physics. 1. pp. 199. ISBN 0-471-03710-9. "It is important to note that we cannot derive a general expression for Newton's second law for variable mass systems by treating the mass in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass."

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Isaac Newton [Emphasis as in the original] [118] Ball 1908, p. 337 [119] White 1997, p. 86 [120] Scott Berkun (27 August 2010). The Myths of Innovation (http:/ / books. google. com/ books?id=kPCgnc70MSgC& pg=PA4). O'Reilly Media, Inc.. p. 4. ISBN 978-1-4493-8962-8. . Retrieved 7 September 2011. [121] Newton's apple: The real story (http:/ / www. newscientist. com/ blogs/ culturelab/ 2010/ 01/ newtons-apple-the-real-story. php). New Scientist. 18 January 2010. . Retrieved 10 May 2010 [122] Hamblyn, Richard (2011). " Newtonian Apples: William Stukeley (http:/ / books. google. co. uk/ books?id=1xKFSqsDj0MC& pg=PT57)". The Art of Science. Pan Macmillan. ISBN 978-1-4472-0415-2. [123] Conduitt, John. "Keynes Ms. 130.4:Conduitt's account of Newton's life at Cambridge" (http:/ / www. newtonproject. sussex. ac. uk/ view/ texts/ normalized/ THEM00167). Newtonproject. Imperial College London. . Retrieved 30 August 2006. [124] I. Bernard Cohen and George E. Smith, eds. The Cambridge Companion to Newton (2002) p. 6 [125] Alberto A. Martinez Science Secrets: The Truth about Darwin's Finches, Einstein's Wife, and Other Myths, page 69 (University of Pittsburgh Press, 2011). ISBN 978-0-8229-4407-2 [126] "Brogdale — Home of the National Fruit Collection" (http:/ / www. brogdale. org/ ). Brogdale.org. . Retrieved 20 December 2008. [127] "From the National Fruit Collection: Isaac Newton's Tree" (http:/ / www. brogdale. org. uk/ image1. php?varietyid=1089). . Retrieved 10 January 2009. [128] Newton's alchemical works (http:/ / webapp1. dlib. indiana. edu/ newton/ index. jsp) transcribed and online at Indiana University. Retrieved 11 January 2007. [129] http:/ / www. pierre-marteau. com/ editions/ 1701-25-mint-reports. html

Bibliography • Ball, W.W. Rouse (1908). A Short Account of the History of Mathematics. New York: Dover. ISBN 0-486-20630-0. • Christianson, Gale (1984). In the Presence of the Creator: Isaac Newton & His Times. New York: Free Press. ISBN 0-02-905190-8. This well documented work provides, in particular, valuable information regarding Newton's knowledge of Patristics • Craig, John (1958). "Isaac Newton – Crime Investigator". Nature 182 (4629): 149–152. Bibcode 1958Natur.182..149C. doi:10.1038/182149a0. • Craig, John (1963). "Isaac Newton and the Counterfeiters". Notes and Records of the Royal Society of London 18 (2): 136–145. doi:10.1098/rsnr.1963.0017. • Levenson, Thomas (2010). Newton and the Counterfeiter: The Unknown Detective Career of the World's Greatest Scientist. Mariner Books. ISBN 978-0-547-33604-6. • Stewart, James (2009). Calculus: Concepts and Contexts. Cengage Learning. ISBN 978-0-495-55742-5. • Westfall, Richard S. (1980, 1998). Never at Rest. Cambridge University Press. ISBN 0-521-27435-4. • Westfall, Richard S. (2007). Isaac Newton. Cambridge University Press. ISBN 978-0-19-921355-9. • Westfall, Richard S. (1994). The Life of Isaac Newton. Cambridge University Press. ISBN 0-521-47737-9. • White, Michael (1997). Isaac Newton: The Last Sorcerer. Fourth Estate Limited. ISBN 1-85702-416-8.

Further reading • Andrade, E. N. De C. (1950). Isaac Newton. New York: Chanticleer Press. ISBN 0-8414-3014-4. • Bardi, Jason Socrates. The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time. 2006. 277 pp. excerpt and text search (http://www.amazon.com/dp/1560259922) • Bechler, Zev (1991). Newton's Physics and the Conceptual Structure of the Scientific Revolution. Springer. ISBN 0-7923-1054-3.. • Berlinski, David. Newton's Gift: How Sir Isaac Newton Unlocked the System of the World. (2000). 256 pages. excerpt and text search (http://www.amazon.com/dp/0743217764) ISBN 0-684-84392-7 • Buchwald, Jed Z. and Cohen, I. Bernard, eds. Isaac Newton's Natural Philosophy. MIT Press, 2001. 354 pages. excerpt and text search (http://www.amazon.com/dp/0262524252)

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Isaac Newton • Casini, P (1988). "Newton's Principia and the Philosophers of the Enlightenment". Notes and Records of the Royal Society of London 42 (1): 35–52. doi:10.1098/rsnr.1988.0006. ISSN 0035–9149. JSTOR 531368. • Christianson, Gale E (1996). Isaac Newton and the Scientific Revolution. Oxford University Press. ISBN 0-19-530070-X. See this site (http://www.amazon.com/dp/019530070X) for excerpt and text search. • Christianson, Gale (1984). In the Presence of the Creator: Isaac Newton & His Times. New York: Free Press. ISBN 0-02-905190-8. • Cohen, I. Bernard and Smith, George E., ed. The Cambridge Companion to Newton. (2002). 500 pp. focuses on philosophical issues only; excerpt and text search (http://www.amazon.com/dp/0521656966); complete edition online (http://www.questia.com/read/105054986) • Cohen, I. B (1980). The Newtonian Revolution. Cambridge: Cambridge University Press. ISBN 0-521-22964-2. • Craig, John (1946). Newton at the Mint. Cambridge, England: Cambridge University Press. • Dampier, William C; Dampier, M. (1959). Readings in the Literature of Science. New York: Harper & Row. ISBN 0-486-42805-2. • de Villamil, Richard (1931). Newton, the Man. London: G.D. Knox. – Preface by Albert Einstein. Reprinted by Johnson Reprint Corporation, New York (1972). • Dobbs, B. J. T (1975). The Foundations of Newton's Alchemy or "The Hunting of the Greene Lyon". Cambridge: Cambridge University Press. • • • • • • • • • • • • • • •

Gjertsen, Derek (1986). The Newton Handbook. London: Routledge & Kegan Paul. ISBN 0-7102-0279-2. Gleick, James (2003). Isaac Newton. Alfred A. Knopf. ISBN 0-375-42233-1. Halley, E (1687). "Review of Newton's Principia". Philosophical Transactions 186: 291–297. Hawking, Stephen, ed. On the Shoulders of Giants. ISBN 0-7624-1348-4 Places selections from Newton's Principia in the context of selected writings by Copernicus, Kepler, Galileo and Einstein Herivel, J. W. (1965). The Background to Newton's Principia. A Study of Newton's Dynamical Researches in the Years 1664–84. Oxford: Clarendon Press. Keynes, John Maynard (1963). Essays in Biography. W. W. Norton & Co. ISBN 0-393-00189-X. Keynes took a close interest in Newton and owned many of Newton's private papers. Koyré, A (1965). Newtonian Studies. Chicago: University of Chicago Press. Newton, Isaac. Papers and Letters in Natural Philosophy, edited by I. Bernard Cohen. Harvard University Press, 1958,1978. ISBN 0-674-46853-8. Newton, Isaac (1642–1726). The Principia: a new Translation, Guide by I. Bernard Cohen ISBN 0-520-08817-4 University of California (1999) Pemberton, H (1728). A View of Sir Isaac Newton's Philosophy. London: S. Palmer. Shamos, Morris H. (1959). Great Experiments in Physics. New York: Henry Holt and Company, Inc.. ISBN 0-486-25346-5. Shapley, Harlow, S. Rapport, and H. Wright. A Treasury of Science; "Newtonia" pp. 147–9; "Discoveries" pp. 150–4. Harper & Bros., New York, (1946). Simmons, J (1996). The Giant Book of Scientists – The 100 Greatest Minds of all Time. Sydney: The Book Company. Stukeley, W. (1936). Memoirs of Sir Isaac Newton's Life. London: Taylor and Francis. (edited by A. H. White; originally published in 1752) Westfall, R. S (1971). Force in Newton's Physics: The Science of Dynamics in the Seventeenth Century. London: Macdonald. ISBN 0-444-19611-0.

Religion • Dobbs, Betty Jo Tetter. The Janus Faces of Genius: The Role of Alchemy in Newton's Thought. (1991), links the alchemy to Arianism • Force, James E., and Richard H. Popkin, eds. Newton and Religion: Context, Nature, and Influence. (1999), 342pp . Pp. xvii + 325. 13 papers by scholars using newly opened manuscripts

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Isaac Newton • Ramati, Ayval. "The Hidden Truth of Creation: Newton's Method of Fluxions" British Journal for the History of Science 34: 417–438. in JSTOR (http://www.jstor.org/stable/4028372), argues that his calculus had a theological basis • Snobelen, Stephen "'God of Gods, and Lord of Lords': The Theology of Isaac Newton's General Scholium to the Principia," Osiris, 2nd Series, Vol. 16, (2001), pp. 169–208 in JSTOR (http://www.jstor.org/stable/301985) • Snobelen, Stephen D. (1999). "Isaac Newton, Heretic: The Strategies of a Nicodemite". British Journal for the History of Science 32 (4): 381–419. doi:10.1017/S0007087499003751. JSTOR 4027945. • Pfizenmaier, Thomas C. (January 1997). "Was Isaac Newton an Arian?". Journal of the History of Ideas 58 (1): 57–80. JSTOR 3653988. • Wiles, Maurice. Archetypal Heresy. Arianism through the Centuries. (1996) 214 pages, with chapter 4 on 18th century England; pp. 77–93 on Newton, excerpt and text search (http://books.google.com/ books?id=DGksMzk37hMC&printsec=frontcover&dq="Arianism+through+the+Centuries"). Primary sources • Newton, Isaac. The Principia: Mathematical Principles of Natural Philosophy. University of California Press, (1999). 974 pp. • Brackenridge, J. Bruce. The Key to Newton's Dynamics: The Kepler Problem and the Principia: Containing an English Translation of Sections 1, 2, and 3 of Book One from the First (1687) Edition of Newton's Mathematical Principles of Natural Philosophy. University of California Press, 1996. 299 pp. • Newton, Isaac. The Optical Papers of Isaac Newton. Vol. 1: The Optical Lectures, 1670–1672. Cambridge U. Press, 1984. 627 pp.

• • • • •

• •

• Newton, Isaac. Opticks (4th ed. 1730) online edition (http://books.google.com/ books?id=GnAFAAAAQAAJ&dq=newton+opticks&pg=PP1&ots=Nnl345oqo_& sig=0mBTaXUI_K6w-JDEu_RvVq5TNqc&prev=http://www.google.com/search?q=newton+opticks& rls=com.microsoft:en-us:IE-SearchBox&ie=UTF-8&oe=UTF-8&sourceid=ie7&rlz=1I7GGLJ&sa=X& oi=print&ct=title&cad=one-book-with-thumbnail) • Newton, I. (1952). Opticks, or A Treatise of the Reflections, Refractions, Inflections & Colours of Light. New York: Dover Publications. Newton, I. Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World, tr. A. Motte, rev. Florian Cajori. Berkeley: University of California Press. (1934). Whiteside, D. T (1967–82). The Mathematical Papers of Isaac Newton. Cambridge: Cambridge University Press. ISBN 0-521-07740-0. – 8 volumes. Newton, Isaac. The correspondence of Isaac Newton, ed. H. W. Turnbull and others, 7 vols. (1959–77). Newton's Philosophy of Nature: Selections from His Writings edited by H. S. Thayer, (1953), online edition (http:/ /www.questia.com/read/5876270). Isaac Newton, Sir; J Edleston; Roger Cotes, Correspondence of Sir Isaac Newton and Professor Cotes, including letters of other eminent men (http://books.google.com/books?as_brr=1&id=OVPJ6c9_kKgC& vid=OCLC14437781&dq="isaac+newton"&jtp=I), London, John W. Parker, West Strand; Cambridge, John Deighton, 1850 (Google Books). Maclaurin, C. (1748). An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books. London: A. Millar and J. Nourse. Newton, I. (1958). Isaac Newton's Papers and Letters on Natural Philosophy and Related Documents, eds. I. B. Cohen and R. E. Schofield. Cambridge: Harvard University Press.

• Newton, I. (1962). The Unpublished Scientific Papers of Isaac Newton: A Selection from the Portsmouth Collection in the University Library, Cambridge, ed. A. R. Hall and M. B. Hall. Cambridge: Cambridge University Press. • Newton, I. (1975). Isaac Newton's 'Theory of the Moon's Motion' (1702). London: Dawson.

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External links • • • • • • • • • •

Newton's Scholar Google profile (http://scholar.google.com.au/citations?user=xJaxiEEAAAAJ&hl=en) ScienceWorld biography (http://scienceworld.wolfram.com/biography/Newton.html) by Eric Weisstein Dictionary of Scientific Biography (http://www.chlt.org/sandbox/lhl/dsb/page.50.a.php) "The Newton Project" (http://www.newtonproject.sussex.ac.uk/prism.php?id=1) "The Newton Project – Canada" (http://www.isaacnewton.ca/) "Rebuttal of Newton's astrology" (http://web.archive.org/web/20080629021908/http://www.skepticreport. com/predictions/newton.htm) (via archive.org) "Newton's Religious Views Reconsidered" (http://www.galilean-library.org/snobelen.html) "Newton's Royal Mint Reports" (http://www.pierre-marteau.com/editions/1701-25-mint-reports.html) "Newton's Dark Secrets" (http://www.pbs.org/wgbh/nova/newton/) - NOVA TV programme from The Stanford Encyclopedia of Philosophy: • "Isaac Newton" (http://plato.stanford.edu/entries/newton/), by George Smith • "Newton's Philosophiae Naturalis Principia Mathematica" (http://plato.stanford.edu/entries/ newton-principia/), by George Smith • "Newton's Philosophy" (http://plato.stanford.edu/entries/newton-philosophy/), by Andrew Janiak

• • • • • • • • • • • • • •

• "Newton's views on space, time, and motion" (http://plato.stanford.edu/entries/newton-stm/), by Robert Rynasiewicz "Newton's Castle" (http://www.tqnyc.org/NYC051308/index.htm) - educational material "The Chymistry of Isaac Newton" (http://www.dlib.indiana.edu/collections/newton), research on his alchemical writings "FMA Live!" (http://www.fmalive.com/) - program for teaching Newton's laws to kids Newton's religious position (http://www.adherents.com/people/pn/Isaac_Newton.html) The "General Scholium" to Newton's Principia (http://hss.fullerton.edu/philosophy/GeneralScholium.htm) Kandaswamy, Anand M. "The Newton/Leibniz Conflict in Context" (http://www.math.rutgers.edu/courses/ 436/Honors02/newton.html) Newton's First ODE (http://www.phaser.com/modules/historic/newton/index.html) – A study by on how Newton approximated the solutions of a first-order ODE using infinite series O'Connor, John J.; Robertson, Edmund F., "Isaac Newton" (http://www-history.mcs.st-andrews.ac.uk/ Biographies/Newton.html), MacTutor History of Mathematics archive, University of St Andrews. Isaac Newton (http://genealogy.math.ndsu.nodak.edu/id.php?id=74313) at the Mathematics Genealogy Project "The Mind of Isaac Newton" (http://www.ltrc.mcmaster.ca/newton/) - images, audio, animations and interactive segments Enlightening Science (http://www.enlighteningscience.sussex.ac.uk/home) Videos on Newton's biography, optics, physics, reception, and on his views on science and religion Newton biography (University of St Andrews) (http://www-history.mcs.st-andrews.ac.uk/Mathematicians/ Newton.html) Chisholm, Hugh, ed. (1911). "Newton, Sir Isaac". Encyclopædia Britannica (11th ed.). Cambridge University Press. and see at s:Author:Isaac Newton for the following works about him: • "Newton, Sir Isaac" in A Short Biographical Dictionary of English Literature by John William Cousin, London: J. M. Dent & Sons, 1910. • "Newton, Isaac," in Dictionary of National Biography, London: Smith, Elder, & Co., (1885–1900) • Memoirs of Sir Isaac Newton's life by William Stukeley, 1752

Writings by Newton

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• Newton's works – full texts, at the Newton Project (http://www.newtonproject.sussex.ac.uk/prism. php?id=43) • The Newton Manuscripts at the National Library of Israel - the collection of all his religious writings (http://web. nli.org.il/sites/NLI/English/collections/Humanities/Pages/newton.aspx) • Works by Isaac Newton (http://www.gutenberg.org/author/Isaac_Newton) at Project Gutenberg • "Newton's Principia" (http://rack1.ul.cs.cmu.edu/is/newton/) – read and search • Descartes, Space, and Body and A New Theory of Light and Colour (http://www.earlymoderntexts.com/), modernised readable versions by Jonathan Bennett • Opticks, or a Treatise of the Reflections, Refractions, Inflexions and Colours of Light (http://www.archive.org/ stream/opticksoratreat00newtgoog#page/n6/mode/2up), full text on archive.org • "Newton Papers" (http://cudl.lib.cam.ac.uk/collections/newton) - Cambridge Digital Library • See Wikisource at s:Author:Isaac Newton for the following works by him: • • • • •

Philosophiae Naturalis Principia Mathematica Opticks: or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light Observations upon the Prophecies of Daniel and the Apocalypse of St. John New Theory About Light and Colour An Historical Account of Two Notable Corruptions of Scripture Related navpages: • • • • • •

Lucasian Professors of Mathematics (over 20 topics) Royal Society presidents 1700s (over 15 topics) Age of Enlightenment (over 60 topics) Metaphysics (over 130 topics) Philosophy of science (over 130 topics) Scientists whose names are used as SI units (over 20 topics)

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Newton's laws of motion Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces. They have been expressed in several different ways over nearly three centuries,[1] and can be summarized as follows: 1. First law: If an object experiences no net force, then its velocity is constant: the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero).[2][3][3] 2. Second law: The acceleration a of a body is parallel and directly proportional to the net force F acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body, i.e., F = ma. 3. Third law: When a first body exerts a force F1 on a second body, the second body simultaneously exerts a force F2 = −F1 on the first body. This means that F1 and F2 are equal in magnitude and opposite in direction. The three laws of motion were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first Newton's First and Second laws, in Latin, from the published in 1687.[4] Newton used them to explain and investigate original 1687 Principia Mathematica. [5] the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.

Overview Newton's laws are applied to bodies (objects) which are considered or idealized as a particle,[6] in the sense that the extent of the body is neglected in the evaluation of its motion, i.e., the object is small compared to the distances involved in the analysis, or the deformation and rotation of the body is of no importance in the analysis. Therefore, a planet can be idealized as a particle for analysis of its orbital motion around a star.

Isaac Newton (1643-1727), the physicist who formulated the laws

In their original form, Newton's laws of motion are not adequate to characterize the motion of rigid bodies and deformable bodies. Leonard Euler in 1750 introduced a generalization of Newton's laws of motion for rigid bodies called the Euler's laws of motion, later applied as well for deformable bodies assumed as a continuum. If a body is represented as an assemblage of discrete particles, each governed by Newton’s laws of motion, then Euler’s laws can be derived from Newton’s laws. Euler’s laws can, however, be taken as axioms describing the laws of motion for extended bodies, independently

of any particle structure.[7] Newton's laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Some authors interpret the first law as defining what an inertial reference frame is; from this point of view,

Newton's laws of motion the second law only holds when the observation is made from an inertial reference frame, and therefore the first law cannot be proved as a special case of the second. Other authors do treat the first law as a corollary of the second.[8][9] The explicit concept of an inertial frame of reference was not developed until long after Newton's death. In the given interpretation mass, acceleration, momentum, and (most importantly) force are assumed to be externally defined quantities. This is the most common, but not the only interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is still useful as an approximation when the speeds involved are much slower than the speed of light.[10]

Newton's first law The first law law states that if the net force (the vector sum of all forces acting on an object) is zero, then the velocity of the object is constant. Velocity is a vector quantity which expresses both the object's speed and the direction of its motion; therefore, the statement that the object's velocity is constant is a statement that both its speed and the direction of its motion are constant. The first law can be stated mathematically as

Consequently, • An object that is at rest will stay at rest unless an unbalanced force acts upon it. • An object that is in motion will not change its velocity unless an unbalanced force acts upon it. This is known as uniform motion. An object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest (demonstrated when a tablecloth is skillfully whipped from under dishes on a tabletop and the dishes remain in their initial state of rest). If an object is moving, it continues to move without turning or changing its speed. This is evident in space probes that continually move in outer space. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely. Newton placed the first law of motion to establish frames of reference for which the other laws are applicable. The first law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to forces is a straight line at a constant speed.[8][11] Newton's first law is often referred to as the law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an inertial reference frame is that the total net force acting on it is zero. In this sense, the first law can be restated as: In every material universe, the motion of a particle in a preferential reference frame Φ is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in Φ. That is, a particle initially at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it.[12] Newton's laws are valid only in an inertial reference frame. Any reference frame that is in uniform motion with respect to an inertial frame is also an inertial frame, i.e. Galilean invariance or the principle of Newtonian relativity.[13]

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History From the original Latin of Newton's Principia:

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Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.

Translated to English, this reads:

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Law I: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its [14] state by force impressed.

”

Aristotle had the view that all objects have a natural place in the universe: that heavy objects (such as rocks) wanted to be at rest on the Earth and that light objects like smoke wanted to be at rest in the sky and the stars wanted to remain in the heavens. He thought that a body was in its natural state when it was at rest, and for the body to move in a straight line at a constant speed an external agent was needed to continually propel it, otherwise it would stop moving. Galileo Galilei, however, realized that a force is necessary to change the velocity of a body, i.e., acceleration, but no force is needed to maintain its velocity. In other words, Galileo stated that, in the absence of a force, a moving object will continue moving. The tendency of objects to resist changes in motion was what Galileo called inertia. This insight was refined by Newton, who made it into his first law, also known as the "law of inertia"—no force means no acceleration, and hence the body will maintain its velocity. As Newton's first law is a restatement of the law of inertia which Galileo had already described, Newton appropriately gave credit to Galileo. The law of inertia apparently occurred to several different natural philosophers and scientists independently, including Thomas Hobbes in his Leviathan.[15] The 17th century philosopher René Descartes also formulated the law, although he did not perform any experiments to confirm it.

Newton's second law Explanation The second law states that the net force on an object is equal to the rate of change (that is, the derivative) of its linear momentum p in an inertial reference frame:

The second law can also be stated in terms of an object's acceleration. Since the law is valid only for constant-mass systems,[16][17][18] the mass can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,

where F is the net force applied, m is the mass of the body, and a is the body's acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it. Consistent with the first law, the time derivative of the momentum is non-zero when the momentum changes direction, even if there is no change in its magnitude; such is the case with uniform circular motion. The relationship also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the momentum. Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems (see below).

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Newton's second law requires modification if the effects of special relativity are to be taken into account, because at high speeds the approximation that momentum is the product of rest mass and velocity is not accurate.

Impulse An impulse J occurs when a force F acts over an interval of time Δt, and it is given by[19][20]

Since force is the time derivative of momentum, it follows that This relation between impulse and momentum is closer to Newton's wording of the second law.[21] Impulse is a concept frequently used in the analysis of collisions and impacts.[22]

Variable-mass systems Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot be directly treated by making mass a function of time in the second law;[17] that is, the following formula is wrong:[18]

The falsehood of this formula can be seen by noting that it does not respect Galilean invariance: a variable-mass object with F = 0 in one frame will be seen to have F ≠ 0 in another frame.[16] The correct equation of motion for a body whose mass m varies with time by either ejecting or accreting mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; the result is[16]

where u is the relative velocity of the escaping or incoming mass as seen by the body. From this equation one can derive the Tsiolkovsky rocket equation. Under some conventions, the quantity u dm/dt on the left-hand side, known as the thrust, is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of acceleration, the equation becomes F = ma.

History Newton's original Latin reads: Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

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This was translated quite closely in Motte's 1729 translation as: Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.

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According to modern ideas of how Newton was using his terminology,[23] this is understood, in modern terms, as an equivalent of: The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed. Motte's 1729 translation of Newton's Latin continued with Newton's commentary on the second law of motion, reading: If a force generates a motion, a double force will generate double the motion, a triple force triple the

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motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both. The sense or senses in which Newton used his terminology, and how he understood the second law and intended it to be understood, have been extensively discussed by historians of science, along with the relations between Newton's formulation and modern formulations.[24]

Newton's third law The third law states that all forces exist in pairs: if one object A exerts a force FA on a second object B, then B simultaneously exerts a force FB on A, and the two forces are equal and opposite: FA = −FB.[25] The third law means that all forces are interactions between different bodies,[26][27] and thus that there is no such thing as a unidirectional force or a force that acts on only one body. This law is sometimes referred to as the action-reaction law, with FA called the "action" and FB the "reaction". The action and the reaction are simultaneous, and it does not matter which is called the action and which is called reaction; both forces are part of a single interaction, and neither force exists without the other.[25] The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road).

An illustration of Newton's third law in which two skaters push against each other. The skater on the left exerts a force F on the skater on the right, and the skater on the right exerts a force −F on the skater on the right. Although the forces are equal, the accelerations are not: the less massive skater will have a greater acceleration due to Newton's second law.

From a conceptual standpoint, Newton's third law is seen when a person walks: they push against the floor, and the floor pushes against the person. Similarly, the tires of a car push against the road while the road pushes back on the tires—the tires and road simultaneously push against each other. In swimming, a person interacts with the water, pushing the water backward, while the water simultaneously pushes the person forward—both the person and the water push against each other. The reaction forces account for the motion in these examples. These forces depend on friction; a person or car on ice, for example, may be unable to exert the action force to produce the needed reaction force.[28]

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History Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi.

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Law III: To every action there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions.

A more direct translation than the one just given above is: LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. — Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another, and by its force changes the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, as the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.[29] In the above, as usual, motion is Newton's name for momentum, hence his careful distinction between motion and velocity. Newton used the third law to derive the law of conservation of momentum;[30] however from a deeper perspective, conservation of momentum is the more fundamental idea (derived via Noether's theorem from Galilean invariance), and holds in cases where Newton's third law appears to fail, for instance when force fields as well as particles carry momentum, and in quantum mechanics.

Importance and range of validity Newton's laws were verified by experiment and observation for over 200 years, and they are excellent approximations at the scales and speeds of everyday life. Newton's laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena. These three laws hold to a good approximation for macroscopic objects under everyday conditions. However, Newton's laws (combined with universal gravitation and classical electrodynamics) are inappropriate for use in certain circumstances, most notably at very small scales, very high speeds (in special relativity, the Lorentz factor must be included in the expression for momentum along with rest mass and velocity) or very strong gravitational fields. Therefore, the laws cannot be used to explain phenomena such as conduction of electricity in a semiconductor, optical properties of substances, errors in non-relativistically corrected GPS systems and superconductivity. Explanation of these phenomena requires more sophisticated physical theories, including general relativity and quantum field theory. In quantum mechanics concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state; at speeds that are much lower than the speed of light, Newton's laws are just as exact for these operators as they are for classical objects. At speeds comparable to the speed of light, the second law holds in the original form F = dp/dt, where F and p are four-vectors.

Newton's laws of motion

Relationship to the conservation laws In modern physics, the laws of conservation of momentum, energy, and angular momentum are of more general validity than Newton's laws, since they apply to both light and matter, and to both classical and non-classical physics. This can be stated simply, "Momentum, energy and angular momentum cannot be created or destroyed." Because force is the time derivative of momentum, the concept of force is redundant and subordinate to the conservation of momentum, and is not used in fundamental theories (e.g., quantum mechanics, quantum electrodynamics, general relativity, etc.). The standard model explains in detail how the three fundamental forces known as gauge forces originate out of exchange by virtual particles. Other forces such as gravity and fermionic degeneracy pressure also arise from the momentum conservation. Indeed, the conservation of 4-momentum in inertial motion via curved space-time results in what we call gravitational force in general relativity theory. Application of space derivative (which is a momentum operator in quantum mechanics) to overlapping wave functions of pair of fermions (particles with half-integer spin) results in shifts of maxima of compound wavefunction away from each other, which is observable as "repulsion" of fermions. Newton stated the third law within a world-view that assumed instantaneous action at a distance between material particles. However, he was prepared for philosophical criticism of this action at a distance, and it was in this context that he stated the famous phrase "I feign no hypotheses". In modern physics, action at a distance has been completely eliminated, except for subtle effects involving quantum entanglement. However in modern engineering in all practical applications involving the motion of vehicles and satellites, the concept of action at a distance is used extensively. The discovery of the Second Law of Thermodynamics by Carnot in the 19th century showed that every physical quantity is not conserved over time, thus disproving the validity of inducing the opposite metaphysical view from Newton's laws. Hence, a "steady-state" worldview based solely on Newton's laws and the conservation laws does not take entropy into account.

References and notes [1] For explanations of Newton's laws of motion by Newton in the early 18th century, by the physicist William Thomson (Lord Kelvin) in the mid-19th century, and by a modern text of the early 21st century, see:•

[2] [3]

[4] [5] [6]

[7]

Newton's "Axioms or Laws of Motion" starting on page 19 of volume 1 of the 1729 translation (http:/ / books. google. com/ books?id=Tm0FAAAAQAAJ& pg=PA19#v=onepage& q=& f=false) of the "Principia"; • Section 242, Newton's laws of motion (http:/ / books. google. com/ books?id=wwO9X3RPt5kC& pg=PA178) in Thomson, W (Lord Kelvin), and Tait, P G, (1867), Treatise on natural philosophy, volume 1; and • Benjamin Crowell (2000), Newtonian Physics. Halliday Browne, Michael E. (1999-07) (Series: Schaum's Outline Series). Schaum's outline of theory and problems of physics for engineering and science (http:/ / books. google. com/ ?id=5gURYN4vFx4C& pg=PA58& dq=newton's+ first+ law+ of+ motion& q=newton's first law of motion). McGraw-Hill Companies. pp. 58. ISBN 978-0-07-008498-8. . See the Principia on line at Andrew Motte Translation (http:/ / ia310114. us. archive. org/ 2/ items/ newtonspmathema00newtrich/ newtonspmathema00newtrich. pdf) Andrew Motte translation of Newton's Principia (1687) Axioms or Laws of Motion (http:/ / members. tripod. com/ ~gravitee/ axioms. htm) [...]while Newton had used the word 'body' vaguely and in at least three different meanings, Euler realized that the statements of Newton are generally correct only when applied to masses concentrated at isolated points;Truesdell, Clifford A.; Becchi, Antonio; Benvenuto, Edoardo (2003). Essays on the history of mechanics: in memory of Clifford Ambrose Truesdell and Edoardo Benvenuto (http:/ / books. google. com/ ?id=6LO_U6T-HvsC& printsec=frontcover& dq=essays+ in+ the+ History& cd=9#v=snippet& q="isolated points"). New York: Birkhäuser. p. 207. ISBN 3-7643-1476-1. . Lubliner, Jacob (2008). Plasticity Theory (Revised Edition) (http:/ / www. ce. berkeley. edu/ ~coby/ plas/ pdf/ book. pdf). Dover Publications. ISBN 0-486-46290-0. .

[8] Galili, I.; Tseitlin, M. (2003). "Newton's First Law: Text, Translations, Interpretations and Physics Education" (http:/ / www. springerlink. com/ content/ j42866672t863506/ ). Science & Education 12 (1): 45–73. Bibcode 2003Sc&Ed..12...45G. doi:10.1023/A:1022632600805. .

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Newton's laws of motion [9] Benjamin Crowell. "4. Force and Motion" (http:/ / www. lightandmatter. com/ html_books/ 1np/ ch04/ ch04. html). Newtonian Physics. ISBN 0-9704670-1-X. . [10] In making a modern adjustment of the second law for (some of) the effects of relativity, m would be treated as the relativistic mass, producing the relativistic expression for momentum, and the third law might be modified if possible to allow for the finite signal propagation speed between distant interacting particles. [11] NMJ Woodhouse (2003). Special relativity (http:/ / books. google. com/ ?id=ggPXQAeeRLgC& printsec=frontcover& dq=isbn=1852334266#PPA6,M1). London/Berlin: Springer. p. 6. ISBN 1-85233-426-6. . [12] Beatty, Millard F. (2006). Principles of engineering mechanics Volume 2 of Principles of Engineering Mechanics: Dynamics-The Analysis of Motion, (http:/ / books. google. com/ ?id=wr2QOBqOBakC& lpg=PP1& pg=PA24#v=onepage& q). Springer. p. 24. ISBN 0-387-23704-6. . [13] Thornton, Marion (2004). Classical dynamics of particles and systems (http:/ / books. google. com/ ?id=HOqLQgAACAAJ& dq=classical dynamics of particles and systems) (5th ed.). Brooks/Cole. p. 53. ISBN 0-534-40896-6. . [14] Isaac Newton, The Principia, A new translation by I.B. Cohen and A. Whitman, University of California press, Berkeley 1999. [15] Thomas Hobbes wrote in Leviathan:

That when a thing lies still, unless somewhat else stir it, it will lie still forever, is a truth that no man doubts. But [the proposition] that when a thing is in motion it will eternally be in motion unless somewhat else stay it, though the reason be the same (namely that nothing can change itself), is not so easily assented to. For men measure not only other men but all other things by themselves. And because they find themselves subject after motion to pain and lassitude, [they] think every thing else grows weary of motion and seeks repose of its own accord, little considering whether it be not some other motion wherein that desire of rest they find in themselves, consists. [16] Plastino, Angel R.; Muzzio, Juan C. (1992). "On the use and abuse of Newton's second law for variable mass problems". Celestial Mechanics and Dynamical Astronomy (Netherlands: Kluwer Academic Publishers) 53 (3): 227–232. Bibcode 1992CeMDA..53..227P. doi:10.1007/BF00052611. ISSN 0923-2958. "We may conclude emphasizing that Newton's second law is valid for constant mass only. When the mass varies due to accretion or ablation, [an alternate equation explicitly accounting for the changing mass] should be used." [17] Halliday; Resnick. Physics. 1. pp. 199. ISBN 0-471-03710-9. "It is important to note that we cannot derive a general expression for Newton's second law for variable mass systems by treating the mass in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass." [Emphasis as in the original] [18] Kleppner, Daniel; Robert Kolenkow (1973). An Introduction to Mechanics. McGraw-Hill. pp. 133–134. ISBN 0-07-035048-5. "Recall that F = dP/dt was established for a system composed of a certain set of particles[. ... I]t is essential to deal with the same set of particles throughout the time interval[. ...] Consequently, the mass of the system can not change during the time of interest." [19] Hannah, J, Hillier, M J, Applied Mechanics, p221, Pitman Paperbacks, 1971 [20] Raymond A. Serway, Jerry S. Faughn (2006). College Physics (http:/ / books. google. com/ ?id=wDKD4IggBJ4C& pg=PA247& dq=impulse+ momentum+ "rate+ of+ change"). Pacific Grove CA: Thompson-Brooks/Cole. p. 161. ISBN 0-534-99724-4. . [21] I Bernard Cohen (Peter M. Harman & Alan E. Shapiro, Eds) (2002). The investigation of difficult things: essays on Newton and the history of the exact sciences in honour of D.T. Whiteside (http:/ / books. google. com/ ?id=oYZ-0PUrjBcC& pg=PA353& dq=impulse+ momentum+ "rate+ of+ change"+ -angular+ date:2000-2009). Cambridge UK: Cambridge University Press. p. 353. ISBN 0-521-89266-X. . [22] WJ Stronge (2004). Impact mechanics (http:/ / books. google. com/ ?id=nHgcS0bfZ28C& pg=PA12& dq=impulse+ momentum+ "rate+ of+ change"+ -angular+ date:2000-2009). Cambridge UK: Cambridge University Press. p. 12 ff. ISBN 0-521-60289-0. . [23] According to Maxwell in Matter and Motion, Newton meant by motion "the quantity of matter moved as well as the rate at which it travels" and by impressed force he meant "the time during which the force acts as well as the intensity of the force". See Harman and Shapiro, cited below. [24] See for example (1) I Bernard Cohen, "Newton’s Second Law and the Concept of Force in the Principia", in "The Annus Mirabilis of Sir Isaac Newton 1666–1966" (Cambridge, Massachusetts: The MIT Press, 1967), pages 143–185; (2) Stuart Pierson, "'Corpore cadente. . .': Historians Discuss Newton’s Second Law", Perspectives on Science, 1 (1993), pages 627–658; and (3) Bruce Pourciau, "Newton's Interpretation of Newton's Second Law", Archive for History of Exact Sciences, vol.60 (2006), pages 157–207; also an online discussion by G E Smith, in 5. Newton's Laws of Motion (http:/ / plato. stanford. edu/ entries/ newton-principia/ index. html#NewLawMot), s.5 of "Newton's Philosophiae Naturalis Principia Mathematica" in (online) Stanford Encyclopedia of Philosophy, 2007. [25] Resnick; Halliday; Krane (1992). Physics, Volume 1 (4th ed.). p. 83. [26] C Hellingman (1992). "Newton’s third law revisited". Phys. Educ. 27 (2): 112–115. Bibcode 1992PhyEd..27..112H. doi:10.1088/0031-9120/27/2/011. "Quoting Newton in the Principia: It is not one action by which the Sun attracts Jupiter, and another by which Jupiter attracts the Sun; but it is one action by which the Sun and Jupiter mutually endeavour to come nearer together." [27] Resnick and Halliday (1977). "Physics". John Wiley & Sons. pp. 78–79. "Any single force is only one aspect of a mutual interaction between two bodies." [28] Hewitt (2006), p. 75

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Newton's laws of motion [29] This translation of the third law and the commentary following it can be found in the "Principia" on page 20 of volume 1 of the 1729 translation (http:/ / books. google. com/ books?id=Tm0FAAAAQAAJ& pg=PA20#v=onepage& q=& f=false). [30] Newton, Principia, Corollary III to the laws of motion

Further reading and works referred to • Crowell, Benjamin, (2011), Light and Matter (http://www.lightandmatter.com/lm/), (2011, Light and Matter), especially at Section 4.2, Newton's First Law (http://www.lightandmatter.com/html_books/lm/ch04/ch04. html#Section4.2), Section 4.3, Newton's Second Law (http://www.lightandmatter.com/html_books/lm/ ch04/ch04.html#Section4.3), and Section 5.1, Newton's Third Law (http://www.lightandmatter.com/ html_books/lm/ch05/ch05.html#Section5.1). • Feynman, R. P.; Leighton, R. B.; Sands, M. (2005). The Feynman Lectures on Physics. Vol. 1 (2nd ed.). Pearson/Addison-Wesley. ISBN 0-8053-9049-9. • Fowles, G. R.; Cassiday, G. L. (1999). Analytical Mechanics (6th ed.). Saunders College Publishing. ISBN 0-03-022317-2. • Likins, Peter W. (1973). Elements of Engineering Mechanics. McGraw-Hill Book Company. ISBN 0-07-037852-5. • Marion, Jerry; Thornton, Stephen (1995). Classical Dynamics of Particles and Systems. Harcourt College Publishers. ISBN 0-03-097302-3. • Newton, Isaac, "Mathematical Principles of Natural Philosophy", 1729 English translation based on 3rd Latin edition (1726), volume 1, containing Book 1 (http://books.google.com/books?id=Tm0FAAAAQAAJ), especially at the section Axioms or Laws of Motion starting page 19 (http://books.google.com/ books?id=Tm0FAAAAQAAJ&pg=PA19). • Newton, Isaac, "Mathematical Principles of Natural Philosophy", 1729 English translation based on 3rd Latin edition (1726), volume 2, containing Books 2 & 3 (http://books.google.com/books?id=6EqxPav3vIsC). • Thomson, W (Lord Kelvin), and Tait, P G, (1867), Treatise on natural philosophy (http://books.google.com/ books?id=wwO9X3RPt5kC), volume 1, especially at Section 242, Newton's laws of motion (http://books. google.com/books?id=wwO9X3RPt5kC&pg=PA178). • NMJ Woodhouse (2003). Special relativity (http://books.google.com/?id=ggPXQAeeRLgC& printsec=frontcover&dq=isbn=1852334266#PPA6,M1). London/Berlin: Springer. p. 6. ISBN 1-85233-426-6.

External links • MIT Physics video lecture (http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/ detail/Video-Segment-Index-for-L-6.htm) on Newton's three laws • Light and Matter (http://www.lightandmatter.com/lm/) – an on-line textbook • Motion Mountain (http://www.motionmountain.net) – an on-line textbook • Simulation on Newton's first law of motion (http://phy.hk/wiki/englishhtm/firstlaw.htm) • " Newton's Second Law (http://demonstrations.wolfram.com/NewtonsSecondLaw/)" by Enrique Zeleny, Wolfram Demonstrations Project. • Newton's 3rd Law demonstrated in a vacuum (http://www.youtube.com/watch?v=9gFMObYCccU)

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Writing of Principia Mathematica

Writing of Principia Mathematica Isaac Newton composed Principia Mathematica during 1685 and 1686,[1] and it was published in a first edition on July 5, 1687 and began changing the world. Widely regarded as one of the most important works in both the science of physics and in applied mathematics during the Scientific revolution, the work underlies much of the technological and scientific advances from the Industrial Revolution (usually dated from 1750) which its tools helped to create.

Authoring Principia Work begins In the other letters written in 1685 and 1686, he asks Flamsteed for information about the orbits of the moons of Jupiter and Saturn, the rise and fall of the spring and neap tides at the solstices and the equinoxes, about the flattening of Jupiter at the poles (which, if certain, he says, would conduce much to the stating the reasons of the precession of the equinoxes), and about the universal application of Kepler's third law. "Your information for Jupiter and Saturn has eased me of several scruples. I was apt to suspect there might be some cause or other unknown to me which might Newton's own copy of his Principia, with hand written corrections disturb the sesquialtera proportion. For the influences for the second edition. of the planets one upon another seemed not great enough, though I imagined Jupiter's influence greater than your numbers determine it. It would add to my satisfaction if you would be pleased to let me know the long diameters of the orbits of Jupiter and Saturn, assigned by yourself and Mr Halley in your new tables, that I may see how the sesquiplicate proportion fills the heavens, together with another small proportion which must be allowed for." [2] Upon Newton's return from Lincolnshire in the beginning of April 1685, he seems to have devoted himself to the preparation of his work. In the spring he had determined the attractions of masses, and thus completed the law of universal gravitation. In the summer he had finished the second book of the Principia, the first book being the treatise De motu corporum in gyrum, which he had enlarged and completed. Except for correspondence with Flamsteed we hear nothing more of the preparation of the Principia until April 21, 1686, when Halley read to the Royal Society his Discourse concerning Gravity and its Properties, in which he states "that his worthy countryman Mr Isaac Newton has an incomparable treatise of motion almost ready for the press," and that the law of the inverse square "is the principle on which Mr Newton has made out all the phenomena of the celestial motions so easily and naturally, that its truth is past dispute." At the next meeting of the Society, on April 28, 1686, "Dr Vincent presented to the Society a manuscript treatise entitled Philosophiae Naturalis Principia Mathematica, and dedicated to the Society by Mr Isaac Newton." Although this manuscript contained only the first book, yet such was the confidence the Society placed in the author that an order was given "that a letter of thanks be written to Mr Newton; and that the printing of his book be referred to the consideration of the council; and that in the meantime the book be put into the hands of Mr Halley, to make a report thereof to the council." Although there could be no doubt as to the intention of this report, no step was taken towards the publication of the work. At the next meeting of the Society, on May 19, 1686, some dissatisfaction seems to have been expressed at the delay, as it was ordered "that Mr Newton's work should be printed forthwith in quarto, and that a letter should be

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Writing of Principia Mathematica written to him to signify the Society's resolutions, and to desire his opinion as to the print, volume, cuts and so forth." Three days afterwards Halley communicated the resolution to Newton, and stated to him that the printing was to be at the charge of the Society. At the next meeting of the council, on June 2, 1686, it was again ordered "that Mr Newton's book be printed," but, instead of sanctioning the resolution of the general meeting to print it at their charge, they added "that Mr Halley undertake the business of looking after it, and printing it at his own charge, which he engaged to do." In order to explain to Newton the cause of the delay, Halley in his letter of May 22, 1686 alleges that it arose from "the president's attendance on the king, and the absence of the vice-president's, whom the good weather had drawn out of town"; but there is reason to believe that this was not the true cause, and that the unwillingness of the council to undertake the publication arose from the state of the finances of the Society. Halley certainly deserves the gratitude of posterity for undertaking the publication of the work at a very considerable financial risk to himself. In the same letter Halley found it necessary to inform Newton of Hooke's conduct when the manuscript of the Principia was presented to the Society. Sir John Hoskyns was in the chair when Dr Vincent presented the manuscript, and praised the novelty and dignity of the subject. Hooke was offended because Sir John did not mention what he had told him of his own discovery. Halley only communicated to Newton the fact "that Hooke had some pretensions to the invention of the rule for the decrease of gravity being reciprocally as the squares of the distances from the centre," acknowledging at the same time that, though Newton had the notion from him, "yet the demonstration of the curves generated thereby belonged wholly to Newton." "How much of this," Halley adds, "is so, you know best, so likewise what you have to do in this matter; only Mr Hooke seems to expect you should make some mention of him in the preface, which 'tis possible you may see reason to prefix. I must beg your pardon that 'tis I that send you this ungrateful account; but I thought it my duty to let you know it, so that you might act accordingly, being in myself fully satisfied that nothing but the greatest candour imaginable is to be expected from a person who has of all men the least need to borrow reputation." In thus appealing to Newton's honesty, Halley obviously wished that Newton should acknowledge Hooke in some way. Indeed, he knew that before Newton had announced the inverse law, Hooke and Wren and himself had spoken of it and discussed it, and therefore justice demanded that Hooke especially should receive credit for having maintained it as a truth of which he was seeking the demonstration, even though none of them had given a demonstration of the law. On June 20, 1686 Newton wrote to Halley the following letter: "Sir, In order to let you know the case between Mr Hooke and me, I give you an account of what passed between us in our letters, so far as I could remember; for 'tis long since they were writ, and I do not know that I have seen them since. I am almost confident by circumstances, that Sir Chr. Wren knew the duplicate proportion when I gave him a visit; and then Mr Hooke (by his book Cometa written afterwards) will prove the last of us three that knew it. I A page from the Principia intended in this letter to let you understand the case fully; but it being a frivolous business, I shall content myself to give you, the heads of it in short, viz, that I never extended the duplicate proportion lower than to the superficies of the earth, and before a certain demonstration I found the last year, have suspected it did not reach accurately enough down so low; and therefore in the doctrine of projectiles never used it nor considered the motions of the heavens; and consequently Mr Hooke could not from my letters, which were about projectiles and the regions descending hence to the centre, conclude me ignorant of the

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Writing of Principia Mathematica theory of the heavens. That what he told me of the duplicate proportion was erroneous, namely, that it reached down from hence to the centre of the earth. "That it is not candid to require me now to confess myself, in print, then ignorant of the duplicate proportion in the heavens; for no other reason but because he had told it me in the case of projectiles, and so upon mistaken grounds, accused me of that ignorance. That in my answer to his first letter I refused his correspondence, told him I had laid philosophy aside, sent him, only the experiment of projectiles (rather shortly hinted than carefully described), in compliment to sweeten my answer, expected to hear no further from him; could scarce persuade myself to answer his second letter; did not answer his third, was upon other things; thought no further of philosophical matters than, his letters put me upon it, and therefore may be allowed not to have had my thoughts of that kind about me so well at that time. That by the same reason he concludes me then ignorant of the rest of the duplicate proportion, he may as well conclude me ignorant of the rest of that theory I had read before in his books. That in one of my papers writ (I cannot say in what year, but I am sure some time before I had any correspondence with Mr Oldenburg, and that's above fifteen years ago), the proportion of the forces of the planets from the sun, reciprocally duplicate of their distances from him, is expressed, and the proportion of our gravit to the moon's conatus recedendi a centro terrae is calculated, though not accurately enough. That when Hugenius put out his Horol. Oscill., a copy being presented to me, in my letter of thanks to him I gave those rules in the end thereof a particular commendation for their usefulness in Philosophy, and added out of my aforesaid paper an instance of their usefulness, in comparing the forces of the moon from the earth, and earth from the sun; in determining a problem about the moon's phase, and putting a limit to the sun's parallax, which shows that I had then my eye upon comparing the forces of the planets arising from their circular motion, and understood it; so that a while after, when Mr Hooke propounded the problem solemnly, in the end of his attempt to prove the motion of the earth, if I had not known the duplicate proportion before, I could not but have found it now. Between ten and eleven years ago there was an hypothesis of mine registered in your books, wherein I hinted a cause of gravity towards the earth, sun and planets, with the dependence of the celestial motions thereon; in which the proportion of the decrease of gravity from the superficies of the planet (though for brevity's sake not there expressed) can be no other than reciprocally duplicate of the distance from the centre. And I hope I shall not be urged to declare, in print, that I understood not the obvious mathematical condition of my own hypothesis. But, grant I received it afterwards from Mr Hooke, yet have I as great a right to it as to the ellipse. For as Kepler knew the orb to be not circular but oval, and guessed it to be elliptical, so Mr Hooke, without knowing what I have found out since his letters to me, can know no more, but that the proportion was duplicate quam proximè at great distances from the centre, and only guessed it to be so accurately, and guessed amiss in extending that proportion down to the very centre, whereas Kepler guessed right at the ellipse. And so, Mr Hooke found less of the proportion than Kepler of the ellipse. "There is so strong an objection against the accurateness of this proportion, that without my demonstrations, to which Mr Hooke is yet a stranger, it cannot be believed by a judicious philosopher to be any where accurate. And so, in stating this business, I do pretend to have done as much for the proportion as for the ellipsis, and to have as much right to the one from Mr Hooke and all men, as to the other from Kepler; and therefore on this account also he must at least moderate his pretences. "The proof you sent me I like very well. I designed the whole to consist of three books; the second was finished last summer being short, and only wants transcribing, and drawing the cuts fairly. Some new propositions I have since thought on, which I can as well let alone. The third wants the theory of comets. In autumn last I spent two months in calculations to no purpose for want of a good method, which made

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Writing of Principia Mathematica me afterwards return to the first book, and enlarge it with diverse propositions some relating to comets others to other things, found out last winter. The third I now design to suppress. Philosophy is such an impertinently litigious lady, that a man has as good be engaged in lawsuits, as have to do, with her. I found it so formerly, and now I am no sooner come near her again, but she gives me warning. The two first books, without the third, will not so well bear the title of Philosophiae Naturalis Principia Mathematica; and therefore I had altered it to this, De Motu Corporum libri duo. "But, upon second thoughts, I retain the former title. It will help the sale of the book, which I ought not to diminish now it's yours. The articles are with the largest to be called by that name. "If you please you may change the word to sections, though it be not material. In the first page, I have struck out the words uti posthac docebitur as referring to the third book; which is all at present, from your affectionate friend, and humble servant, "Is. NEWTON." On June 20, 1686, Halley wrote to Newton: "I am heartily sorry that in this matter, wherein all mankind ought to acknowledge their obligations to you, you should meet with anything that should give you unquiet"; and then, after an account of Hooke's claim to the discovery as made at a meeting of the Royal Society, he concludes: "But I found that they were all of opinion that nothing thereof appearing in print, nor on the books of the Society, you ought to be considered as the inventor. And if in truth he knew it before you, he ought not to blame any but himself for having taken no more care to secure a discovery, which he puts so much value on. What application he has made in private, I know not; but I am sure that the Society have a very great satisfaction, in the honour you do them, by the dedication of so worthy a treatise. Sir, I must now again beg you, not to let your resentments run so high, as to deprive us of your third book, wherein the application of your mathematical doctrine to the theory of comets and several curious experiments, which, as I guess by what you write, ought to compose it, will undoubtedly render it acceptable to those, who will call themselves Philosophers without Mathematics, which are much the greater number. Now you approve of the character and paper, I will push on the edition vigorously. I have sometimes had thoughts of having the cuts neatly done in wood, so as to stand in the page with the demonstrations. It will be more convenient, and not much more charge. If it please you to have it so, I will try how well it can be done; otherwise I will have them in somewhat a larger size than those you have sent up. I am, Sir, your most affectionate humble servant, E. HALLEY." On June 30, 1686 the council resolved to license Newton's book, entitled Philosophiae Naturalis Principia Mathematica. On July 14, 1686, Newton wrote to Halley approving of his proposal to introduce woodcuts among the letterpress, stating clearly the differences which he had from Hooke, and adding, "And now having sincerely told you the case between Mr Hooke and me, I hope I shall be free for the future from the prejudice of his letters. I have considered how best to compose the present dispute, and I think it may be done by the inclosed scholium to the fourth proposition." This scholium was "The inverse law of gravity holds in all the celestial motions, as was discovered also independently by my countrymen Wren, Hooke and Halley." After this letter of Newton's the printing of the Principia was begun. The second book, though ready for the press in the autumn of 1686, was not sent to the printers until March 1687. The third book was presented to the Society, on April 6, and the whole work published about midsummer in that year, July 5, 1687.[3] It was dedicated to the Royal Society, and to it was prefixed a set of Latin hexameters addressed by Halley to the author. The work, as might have been expected, caused a great deal of excitement throughout Europe, and the whole of the impression was very soon sold. In 1691 a copy of the Principia was hard to obtain.

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Conflict between the University and James II While Newton was writing the second and third books of the Principia, an event occurred at Cambridge which had the effect of bringing him before the public. James II had in 1686 conferred the deanery of Christ Church at Oxford on John Massey, a person whose sole qualification was that he was a member of the Church of Rome; and the king had boasted to the pope's legate that "what he had done at Oxford would very soon be done at Cambridge." In February 1687 James issued a mandate directing that Father Alban Francis, a Benedictine monk, should be admitted a master of arts of the University of Cambridge, without taking the oaths of allegiance and supremacy. Upon receiving the mandamus John Pechell, the master of Magdalene College, who was vice-chancellor, sent a messenger to the Duke of Albemarle, the chancellor, to request him to get the mandamus recalled; and the registrary and the bedell waited upon Francis to offer him instant admission to the degree if only he would take the necessary oaths. A menacing letter was despatched by Sunderland—respectful explanations were returned, but the university showed no sign of compliance, nor suggested a compromise. The vice-chancellor and deputies from the senate were summoned to appear before the High commission court at Westminster. Newton was one of the eight deputies appointed by the senate for this purpose. The deputies, before starting for London, held a meeting to prepare their case for the court. A compromise which was put forward by one of them was resisted by Newton. On April 21 the deputation, with their case carefully prepared, appeared before the court. Lord Jeffreys presided at the board. The deputation appeared as a matter of course before the commissioners, and was dismissed. On April 27 they gave their plea. On May 7 it was discussed, and feebly defended by the vice-chancellor. The deputies maintained that in the late reign several royal mandates had been withdrawn, and that no degree had ever been conferred without the oaths having been previously taken. Jeffreys spoke with his accustomed insolence to the vice-chancellor, silenced the other deputies when they offered to speak, and ordered them out of court. When recalled the deputies were reprimanded, and Pechell was deprived of his office as vice-chancellor, and of his salary as master of Magdalene. Newton returned to Trinity College to complete the Principia. While thus occupied he had an extensive correspondence with Halley, a very great part of which is extant. The following letter from Halley, dated London, July 5, 1687, announcing the completion of the Principia, is of particular interest: "I have at length brought your book to an end, and hope it will please you. The last errata came just in time to be inserted. I will present from you the book you desire to the Royal Society, Mr Boyle, Mr Paget, Mr Flamsteed, and if there be any else in town that you design to gratify that way; and I have sent you to bestow on your friends in the University 20 copies, which I entreat you to accept. In the same parcel you will receive 40 more, which having no acquaintance in Cambridge, I must entreat you to put into the hands of one or more of your ablest booksellers to dispose of them. I intend the price of them, bound in calves' leather, and lettered, to be [OCR error] shillings here. Those I send you I value in quires at 6 shillings, to take my money as they are sold, or at 5 sh. for ready, or else at some short time; for I am satisfied there is no dealing in books without interesting the booksellers; and I am contented to let them go halves with me, rather than have your excellent work smothered by their combinations. I hope you will not repent you of the pains you have taken in so laudable a piece, so much to your own and the nation's credit, but rather, after you shall have a little diverted yourself with other studies, that you will resume those contemplations wherein you had so great success, and attempt the perfection of the lunar theory, which will be of prodigious use in navigation, as well as of profound and public speculation. You will receive a box from me on Thursday next by the wagon, that starts from town tomorrow."

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Illness in 1693 In 1692 and 1693 Newton seems to have had a serious illness, the nature of which has given rise to very considerable dispute. In a letter dated the September 13, 1693, addressed to Samuel Pepys, he writes: "Some time after Mr Millington had delivered your message, he pressed me to see you the next time I went to London. I was averse, but upon his pressing consented, before I considered what I did, for I am extremely troubled at the embroilment I am in, and have neither ate nor slept well this twelvemonth, nor have my former consistency of mind. I never designed to get any thing by your interest, nor by icing James's favour, but am now sensible that I must withdraw from your acquaintance, and see neither you nor the rest of my friends any more, if I may but have them quietly. I beg your pardon for saying I would see you again, and rest your most humble and obedient servant." And in a letter written to John Locke in reply to one of his about the second edition of his book, and dated the 15th of October 1693, Newton wrote: "The last, winter, by sleeping too often by my fire, I got an ill habit of sleeping; and a distemper, which this summer has been epidemical, put me farther out of order, so that when I wrote to you, I had not slept an hour a night for a fortnight together, and for five days together not a wink. I remember I wrote to you, but what I said of your book I remember not. If you please to send me a transcript of that passage, I will give you an account of it if I can." The loss of sleep to a person of Newton's temperament, whose mind was never at rest, and at times so wholly engrossed in his scientific pursuits that he even neglected to take food, must necessarily have led to a very great deal of nervous excitability. It is not astonishing that rumours got abroad that there was a danger of his mind giving way, or, according to a report which was believed at the time, that it had actually done so. Pepys must have heard such rumours, as in a letter to his friend Millington, the tutor of Magdalene College at Cambridge, dated September 26, 1693, he wrote: "I must acknowledge myself not at the ease I would be glad to be at in reference to excellent Mr Newton; concerning whom (methinks) your answer labours under the same kind of restraint which (to tell you the truth) my asking did. For I was 10th at first dash to tell you that I had lately received a letter from him so surprising to me for the inconsistency of every part of it, as to be put into great disorder by it, from the concern I have for him, lest it should arise from that which of all mankind I should least dread from him and most lament for I mean a discomposure in head, or mind, or both. Let me, therefore, beg you, Sir, having now told you the true ground of the trouble I lately gave you, to let me know the very truth of the matter, as far at least as comes within your knowledge." On September 20, 1693, Millington wrote to Pepys that he had been to look for Newton some time before, but that "he was out of town, and since," he says, "I have not seen him, till upon the 28th I met him at Huntingdon, where, upon his own accord, and before I had time to ask him any question, he told me that he had written to you a very odd letter, at which he was much concerned; added, that it was in a distemper that much seized his head, and that kept him awake for above five nights together, which upon occasion he desired I would represent to you, and beg your pardon, he being very much ashamed he should be so rude to a person for whom he hath so great an honour. He is now very well, and though I fear he is under some small degree of melancholy, yet I think there is no reason to suspect it hath at all touched his understanding, and I hope never will; and so I am sure all ought to wish that love learning or the honour of our nation, which it is a sign how much it is looked after, when such a person as Mr Newton lies so neglected by those in power." The illness of Newton was very much exaggerated by foreign contemporary writers. Christiaan Huygens, in a letter dated June 8, 1694, wrote to Leibniz, "I do not know if you are acquainted with the accident which has happened to the good Mr Newton, namely, that he has had an attack of phrenitis, which lasted eighteen months, and of which they say his friends have cured him by means of remedies, and keeping him shut up." To which Leibniz, in a letter dated the 22nd of June, replied, "I am very glad that I received information of the cure of Mr Newton at the same time that I first heard of his illness, which doubtless must have been very alarming."

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Initial election to Parliament The active part which Newton had taken in defending the legal privileges of the university against the encroachments of the crown had probably at least equal weight with his scientific reputation when his friends chose him as a candidate for a seat in parliament as one of the representatives of the university. The other candidates were Sir Robert Sawyer and Mr Finch. Sir Robert headed the poll with 125 votes, Newton next with 122 and Mr Finch was last with 117 votes. Newton retained his seat only about a year, from January 1689 till the dissolution of the Coventry Parliament in February 1690. During this time Newton does not appear to have taken part in any of the debates in the House, but he was not neglectful of his duties as a member. On April 30, 1689 he moved for leave to bring in a bill to settle the charters and privileges of the University of Cambridge, just as Sir Thomas Clarges did for Oxford at the same time, and he wrote a series of letters to Dr Lovel, the vice-chancellor of the university, on points which affected the interests of the university and its members. Some of the members of the university who had sworn allegiance to James had some difficulty in swearing allegiance to his successor. On February 12, 1689, the day of the coronation of William and Mary, Newton intimated to the vice-chancellor that he would soon receive an order to proclaim them at Cambridge. He enclosed a form of the proclamation, and expressed a hearty "wish that the university would so compose themselves as to perform the solemnity with a reasonable decorum."

References [1] For information on Newton's later life and post-Principia work, see Isaac Newton's later life. [2] (Letter of mid-January (before 14th) 1684|1685 (Old Style), published as #537 in Vol.2 of "The Correspondence of John Flamsteed", ed. E.G. Forbes et al., 1997. (This reference was supplied after original compilation of the present article, and gives original spellings; but most spellings and punctuations in the text above have been modernised. The words 'sesquialtera' and 'sesquiplicate', now archaic, refer to the relation between a given number and the same multiplied by its own square root: or to the square root of its cube, which comes to the same thing: the 'one-and-a-half-th' power, as it were.) [3] Richard S. Westfall, Never at Rest, ISBN 0-521-27435-4 (paperback) Cambridge 1980..1998.

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Method of Fluxions

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Method of Fluxions Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus (fluents was his term for integral calculus). He originally developed the method at Woolsthorpe Manor during the closing of Cambridge during the Great Plague of London from 1665 to 1667, but did not choose to make his findings known (similarly, his findings which eventually became the Philosophiae Naturalis Principia Mathematica were developed at this time and hidden from the world in Newton's notes for many years). Gottfried Leibniz developed his calculus around 1673, and published it in 1684, fifty years before Newton. The calculus notation we use today is mostly that of Leibniz, although Newton's dot notation for differentiation for denoting derivatives with respect to time is still in current use throughout mechanics and circuit analysis. Newton's Method of Fluxions was formally published posthumously, but following Leibniz's publication of the calculus a bitter rivalry erupted between the two mathematicians over who had developed the calculus first and so Newton no longer hid his knowledge of fluxions.

External links • Method of Fluxions [1] at the Internet Archive

References [1] http:/ / www. archive. org/ details/ methodoffluxions00newt

Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz Born

July 1, 1646 Leipzig, Electorate of Saxony, Holy Roman Empire

Died

November 14, 1716 (aged 70) Hanover, Electorate of Hanover, Holy Roman Empire

Gottfried Wilhelm Leibniz

115 Nationality

German

Era

17th-/18th-century philosophy

Region

Western Philosophy

Main interests Mathematics, metaphysics, logic, theodicy, universal language Notable ideas

Infinitesimal calculus Monads Best of all possible worlds Leibniz formula for π Leibniz harmonic triangle Leibniz formula for determinants Leibniz integral rule Principle of sufficient reason Diagrammatic reasoning Notation for differentiation Proof of Fermat's little theorem Kinetic energy Entscheidungsproblem AST Law of Continuity Transcendental Law of Homogeneity Characteristica universalis Ars combinatoria Calculus ratiocinator [2] Universalwissenschaft

Signature

Gottfried Wilhelm von Leibniz (German: [ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪbnɪts][3] or [ˈlaɪpnɪts][4]) (July 1, 1646 – November 14, 1716) was a German mathematician and philosopher. He occupies a prominent place in the history of mathematics and the history of philosophy. Leibniz developed the infinitesimal calculus independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published. His visionary Law of Continuity and Transcendental Law of Homogeneity only found mathematical implementation in the 20th century. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685[5] and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is at the foundation of virtually all digital computers. In philosophy, Leibniz is mostly noted for his optimism, e.g., his conclusion that our Universe is, in a restricted sense, the best possible one that God could have created. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th century advocates of rationalism. The work of Leibniz anticipated modern logic and analytic philosophy, but his philosophy also looks back to the scholastic tradition, in which conclusions are produced by applying reason to first principles or prior definitions rather than to empirical evidence. Leibniz made major contributions to physics and technology, and anticipated notions that surfaced much later in philosophy, probability theory, biology, medicine, geology, psychology, linguistics, and information science. He wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, and in unpublished manuscripts. He wrote in several languages, but primarily in Latin, French, and German.[6] As of 2013, there is no complete gathering of the writings of Leibniz.[7]

Gottfried Wilhelm Leibniz

Biography Early life Gottfried Leibniz was born on July 1, 1646 in Leipzig, Saxony (at the end of the Thirty Years' War), to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his family journal: "On Sunday 21 June [NS: 1 July] 1646, my son Gottfried Wilhelm is born into the world after six in the evening, ¾ to seven [ein Viertel uff sieben], Aquarius rising."[8] His father (a German of Sorbian ancestry[9]) died when Leibniz was six years old, and from that point on he was raised by his mother. Her teachings influenced Leibniz's philosophical thoughts in his later life. Leibniz's father had been a Professor of Moral Philosophy at the University of Leipzig and Leibniz inherited his father's personal library. He was given free access to this from the age of seven. While Leibniz's schoolwork focused on a small canon of authorities, his father's library enabled him to study a wide variety of advanced philosophical and theological works – ones that he would not have otherwise been able to read until his college years.[10] Access to his father's library, largely written in Latin, also led to his proficiency in the Latin language. Leibniz was proficient in Latin by the age of 12, and he composed three hundred hexameters of Latin verse in a single morning for a special event at school at the age of 13.[11] He enrolled in his father's former university at age 15,[12] and he completed his bachelor's degree in philosophy in December 1662. He defended his Disputatio Metaphysica de Principio Individui, which addressed the principle of individuation, on June 9, 1663. Leibniz earned his master's degree in philosophy on February 7, 1664. He published and defended a dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum, arguing for both a theoretical and a pedagogical relationship between philosophy and law, in December 1664. After one year of legal studies, he was awarded his bachelor's degree in Law on September 28, 1665. In 1666, at age 20, Leibniz published his first book, On the Art of Combinations, the first part of which was also his habilitation thesis in philosophy. His next goal was to earn his license and doctorate in Law, which normally required three years of study then. In 1666, the University of Leipzig turned down Leibniz's doctoral application and refused to grant him a doctorate in law, most likely due to his relative youth (he was 21 years old at the time).[13] Leibniz subsequently left Leipzig.[14] Leibniz then enrolled in the University of Altdorf, and almost immediately he submitted a thesis, which he had probably been working on earlier in Leipzig.[15] The title of his thesis was Disputatio Inauguralis De Casibus Perplexis In Jure. Leibniz earned his license to practice law and his Doctorate in Law in November 1666. He next declined the offer of an academic appointment at Altdorf, saying that "my thoughts were turned in an entirely different direction.[16] As an adult, Leibniz often introduced himself as "Gottfried von Leibniz". Also many posthumously published editions of his writings presented his name on the title page as "Freiherr G. W. von Leibniz." However, no document has ever been found from any contemporary government that stated his appointment to any form of nobility.[17]

1666–74 Leibniz's first position was as a salaried alchemist in Nuremberg, though he may have only known fairly little about the subject at that time.[18] He soon met Johann Christian von Boyneburg (1622–1672), the dismissed chief minister of the Elector of Mainz, Johann Philipp von Schönborn.[19] Von Boyneburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for his Electorate.[20] In 1669, Leibniz was appointed Assessor in the Court of Appeal. Although von Boyneburg died late in 1672, Leibniz remained under the employment of his widow until she dismissed him in 1674.

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Von Boyneburg did much to promote Leibniz's reputation, and the latter's memoranda and letters began to attract favorable notice. Leibniz's service to the Elector soon followed a diplomatic role. He published an essay, under the pseudonym of a fictitious Polish nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. The main force in European geopolitics during Leibniz's adult life was the ambition of Louis XIV of France, backed by French military and economic might. Meanwhile, the Thirty Years' War had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows. France would be invited to take Egypt as a stepping stone towards an eventual conquest of the Dutch East Indies. In return, France would agree to leave Germany and the Netherlands undisturbed. This plan obtained the Elector's cautious support. In 1672, the French government invited Leibniz to Paris for discussion,[21] but the plan was soon overtaken by the outbreak of the Franco-Dutch War and became irrelevant. Napoleon's failed invasion of Egypt in 1798 can be seen as an unwitting implementation of Leibniz's plan. Thus Leibniz began several years in Paris. Soon after arriving, he met Dutch physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was patchy. With Huygens as mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including inventing his version of the differential and integral calculus. He met Nicolas Malebranche and Antoine Arnauld, the leading French philosophers of the day, and studied the writings of Descartes and Pascal, unpublished as well as published. He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus; they corresponded for the rest of their lives. In 1675 he was admitted by the French Academy of Sciences as a foreign honorary member, despite his lack of attention to the academy. When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on a related mission to the English government in London, early in 1673.[22] There Leibniz came into acquaintance of Henry Oldenburg and John Collins. He met with the Royal Society where he demonstrated a calculating machine that he had designed and had been building since 1670. The machine was able to execute all four basic operations (adding, subtracting, multiplying, and dividing), and the Society quickly made him an external member. The mission ended abruptly when news reached it of the Elector's death, whereupon Leibniz promptly returned to Paris and not, as had been planned, to Mainz.[23]

Stepped Reckoner

The sudden deaths of Leibniz's two patrons in the same winter meant that Leibniz had to find a new basis for his career. In this regard, a 1669 invitation from the Duke of Brunswick to visit Hanover proved fateful. Leibniz declined the invitation, but began corresponding with the Duke in 1671. In 1673, the Duke offered him the post of Counsellor which Leibniz very reluctantly accepted two years later, only after it became clear that no employment in Paris, whose intellectual stimulation he relished, or with the Habsburg imperial court was forthcoming.

House of Hanover, 1676–1716 Leibniz managed to delay his arrival in Hanover until the end of 1676 after making one more short journey to London, where he was later accused by Newton of being shown some of Newton's unpublished work on the calculus.[24] This fact was deemed evidence supporting the accusation, made decades later, that he had stolen the calculus from Newton. On the journey from London to Hanover, Leibniz stopped in The Hague where he met Leeuwenhoek, the discoverer of microorganisms. He also spent several days in intense discussion with Spinoza, who

Gottfried Wilhelm Leibniz had just completed his masterwork, the Ethics.[25] Leibniz respected Spinoza's powerful intellect, but was dismayed by his conclusions that contradicted both Christian and Jewish orthodoxy. In 1677, he was promoted, at his request, to Privy Counselor of Justice, a post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for the period. Among the few people in north Germany to accept Leibniz were the Electress Sophia of Hanover (1630–1714), her daughter Sophia Charlotte of Hanover (1668–1705), the Queen of Prussia and his avowed disciple, and Caroline of Ansbach, the consort of her grandson, the future George II. To each of these women he was correspondent, adviser, and friend. In turn, they all approved of Leibniz more than did their spouses and the future king George I of Great Britain.[26] The population of Hanover was only about 10,000, and its provinciality eventually grated on Leibniz. Nevertheless, to be a major courtier to the House of Brunswick was quite an honor, especially in light of the meteoric rise in the prestige of that House during Leibniz's association with it. In 1692, the Duke of Brunswick became a hereditary Elector of the Holy Roman Empire. The British Act of Settlement 1701 designated the Electress Sophia and her descent as the royal family of England, once both King William III and his sister-in-law and successor, Queen Anne, were dead. Leibniz played a role in the initiatives and negotiations leading up to that Act, but not always an effective one. For example, something he published anonymously in England, thinking to promote the Brunswick cause, was formally censured by the British Parliament. The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting the calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence. He began working on the calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675. By 1677 he had a coherent system in hand, but did not publish it until 1684. Leibniz's most important mathematical papers were published between 1682 and 1692, usually in a journal which he and Otto Mencke founded in 1682, the Acta Eruditorum. That journal played a key role in advancing his mathematical and scientific reputation, which in turn enhanced his eminence in diplomacy, history, theology, and philosophy. The Elector Ernest Augustus commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared; the next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less. They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes. In 1708, John Keill, writing in the journal of the Royal Society and with Newton's presumed blessing, accused Leibniz of having plagiarized Newton's calculus.[27] Thus began the calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so have tended to acquit Leibniz, pointing to important differences between Leibniz's and Newton's versions of the calculus.

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In 1711, while traveling in northern Europe, the Russian Tsar Peter the Great stopped in Hanover and met Leibniz, who then took some interest in Russian matters for the rest of his life. In 1712, Leibniz began a two-year residence in Vienna, where he was appointed Imperial Court Councillor to the Habsburgs. On the death of Queen Anne in 1714, Elector George Louis became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he Leibniz's correspondence, papers and notes from 1669-1704, completed at least one volume of the history of the National Library of Poland. Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the Dowager Electress Sophia, died in 1714.

Death Leibniz died in Hanover in 1716: at the time, he was so out of favor that neither George I (who happened to be near Hanover at the time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz was a life member of the Royal Society and the Berlin Academy of Sciences, neither organization saw fit to honor his passing. His grave went unmarked for more than 50 years. Leibniz was eulogized by Fontenelle, before the Academie des Sciences in Paris, which had admitted him as a foreign member in 1700. The eulogy was composed at the behest of the Duchess of Orleans, a niece of the Electress Sophia.

Personal life Leibniz never married. He complained on occasion about money, but the fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had, by and large, paid him well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was all too often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which put him in a bad light during the calculus controversy. On the other hand, he was charming, well-mannered, and not without humor and imagination.[28] He had many friends and admirers all over Europe. On Leibniz's religious views, although he is considered by some biographers as a deist since he did not believe in miracles and believed that Jesus Christ has no real role in the universe, he was nonetheless a theist.[29][30][31][32]

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Philosopher Leibniz's philosophical thinking appears fragmented, because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and many letters to many correspondents. He wrote only two book-length philosophical treatises, of which only the Théodicée of 1710 was published in his lifetime. Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics, which he composed in 1686 as a commentary on a running dispute between Nicolas Malebranche and Antoine Arnauld. This led to an extensive and valuable correspondence with Arnauld;[33] it and the Discourse were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy with a journal article titled "New System of the Nature and Communication of Substances".[34] Between 1695 and 1705, he composed his New Essays on Human Understanding, a lengthy commentary on John Locke's 1690 An Essay Concerning Human Understanding, but upon learning of Locke's 1704 death, lost the desire to publish it, so that the New Essays were not published until 1765. The Monadologie, composed in 1714 and published posthumously, consists of 90 aphorisms. Leibniz met Spinoza in 1676, read some of his unpublished writings, and has since been suspected of appropriating some of Spinoza's ideas. While Leibniz admired Spinoza's powerful intellect, he was also forthrightly dismayed by Spinoza's conclusions,[35] especially when these were inconsistent with Christian orthodoxy. Unlike Descartes and Spinoza, Leibniz had a thorough university education in philosophy. He was influenced by his Leipzig professor Jakob Thomasius, who also supervised his BA thesis in philosophy. Leibniz also eagerly read Francisco Suárez, a Spanish Jesuit respected even in Lutheran universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle, but viewed their work through a lens heavily tinted by scholastic notions. Yet it remains the case that Leibniz's methods and concerns often anticipate the logic, and analytic and linguistic philosophy of the 20th century.

The Principles Leibniz variously invoked one or another of seven fundamental philosophical Principles:[36] • Identity/contradiction. If a proposition is true, then its negation is false and vice versa. • Identity of indiscernibles. Two distinct things cannot have all their properties in common. If every predicate possessed by x is also possessed by y and vice versa, then entities x and y are identical; to suppose two things indiscernible is to suppose the same thing under two names. Frequently invoked in modern logic and philosophy. The "identity of indiscernibles" is often referred to as Leibniz's Law. It has attracted the most controversy and criticism, especially from corpuscular philosophy and quantum mechanics. • Sufficient reason. "There must be a sufficient reason [often known only to God] for anything to exist, for any event to occur, for any truth to obtain."[37] • Pre-established harmony.[38] "[T]he appropriate nature of each substance brings it about that what happens to one corresponds to what happens to all the others, without, however, their acting upon one another directly." (Discourse on Metaphysics, XIV) A dropped glass shatters because it "knows" it has hit the ground, and not because the impact with the ground "compels" the glass to split. • Law of Continuity. Natura non saltum facit. • Optimism. "God assuredly always chooses the best."[39] • Plenitude. "Leibniz believed that the best of all possible worlds would actualize every genuine possibility, and argued in Théodicée that this best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature's perfection." Leibniz would on occasion give a rational defense of a specific principle, but more often took them for granted.[40]

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The monads Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in Monadologie. According to Leibniz, monads are elementary particles with blurred perception of each other. Monads can also be compared to the corpuscles of the Mechanical Philosophy of René Descartes and others. Monads are the ultimate elements of the universe. The monads are "substantial forms of being" with the following properties: they are eternal, indecomposable, individual, subject to their own laws, un-interacting, and each reflecting the entire universe in a pre-established harmony (a historically important example of panpsychism). Monads are centers of force; substance is force, while space, matter, and motion are merely phenomenal. The ontological essence of a monad is its irreducible simplicity. Unlike atoms, monads possess no material or spatial character. They also differ from atoms by their complete mutual independence, so that interactions among monads are only apparent. Instead, by virtue of the principle of pre-established harmony, each monad follows a preprogrammed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment. (These "instructions" may be seen as analogs of the scientific laws governing subatomic particles.) By virtue of these intrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.g., each human being constitutes a monad, in which case free will is problematic. God, too, is a monad, and the existence of God can be inferred from the harmony prevailing among all other monads; God wills the pre-established harmony. Monads are purported to have gotten rid of the problematic: • Interaction between mind and matter arising in the system of Descartes; • Lack of individuation inherent to the system of Spinoza, which represents individual creatures as merely accidental.

Theodicy and optimism (Note that the word "optimism" here is used in the classic sense of optimal, not in the mood-related sense, as being positively hopeful.) The Theodicy[41] tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. It must be the best possible and most balanced world, because it was created by an all powerful and all knowing God, who would not choose to create an imperfect world if a better world could be known to him or possible to exist. In effect, apparent flaws that can be identified in this world must exist in every possible world, because otherwise God would have chosen to create the world that excluded those flaws. Leibniz asserted that the truths of theology (religion) and philosophy cannot contradict each other, since reason and faith are both "gifts of God" so that their conflict would imply God contending against himself. The Theodicy is Leibniz's attempt to reconcile his personal philosophical system with his interpretation of the tenets of Christianity.[42] This project was motivated in part by Leibniz's belief, shared by many conservative philosophers and theologians during the Enlightenment, in the rational and enlightened nature of the Christian religion, at least as this was defined in tendentious comparisons between Christian and non Western or "primitive" religious practices and beliefs. It was also shaped by Leibniz's belief in the perfectibility of human nature (if humanity relied on correct philosophy and religion as a guide), and by his belief that metaphysical necessity must have a rational or logical foundation, even if this metaphysical causality seemed inexplicable in terms of physical necessity (the natural laws identified by science). Because reason and faith must be entirely reconciled, any tenet of faith which could not be defended by reason must be rejected. Leibniz then approached one of the central criticisms of Christian theism:[43] if God is all good, all wise and all powerful, how did evil come into the world? The answer (according to Leibniz) is that, while God is indeed unlimited in wisdom and power, his human creations, as creations, are limited both in their wisdom and in their will (power to act). This predisposes humans to false beliefs, wrong decisions and ineffective actions in the exercise of their free will. God does not arbitrarily inflict pain and suffering on humans; rather he permits both moral evil (sin)

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Gottfried Wilhelm Leibniz and physical evil (pain and suffering) as the necessary consequences of metaphysical evil (imperfection), as a means by which humans can identify and correct their erroneous decisions, and as a contrast to true good. Further, although human actions flow from prior causes that ultimately arise in God, and therefore are known as a metaphysical certainty to God, an individual's free will is exercised within natural laws, where choices are merely contingently necessary, to be decided in the event by a "wonderful spontaneity" that provides individuals an escape from rigorous predestination. This theory drew controversy and refutations, that are collected in the article Best of all possible worlds.

Symbolic thought Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion: The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right.[44] Leibniz's calculus ratiocinator, which resembles symbolic logic, can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda[45] that can now be read as groping attempts to get symbolic logic—and thus his calculus—off the ground. But Gerhard and Couturat did not publish these writings until modern formal logic had emerged in Frege's Begriffsschrift and in writings by Charles Sanders Peirce and his students in the 1880s, and hence well after Boole and De Morgan began that logic in 1847. Leibniz thought symbols were important for human understanding. He attached so much importance to the invention of good notations that he attributed all his discoveries in mathematics to this. His notation for the infinitesimal calculus is an example of his skill in this regard. C.S. Peirce, a 19th-century pioneer of semiotics, shared Leibniz's passion for symbols and notation, and his belief that these are essential to a well-running logic and mathematics. But Leibniz took his speculations much further. Defining a character as any written sign, he then defined a "real" character as one that represents an idea directly and not simply as the word embodying the idea. Some real characters, such as the notation of logic, serve only to facilitate reasoning. Many characters well known in his day, including Egyptian hieroglyphics, Chinese characters, and the symbols of astronomy and chemistry, he deemed not real.[46] Instead, he proposed the creation of a characteristica universalis or "universal characteristic", built on an alphabet of human thought in which each fundamental concept would be represented by a unique "real" character: It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by transposing these characters and by a species of calculus.[47] Complex thoughts would be represented by combining characters for simpler thoughts. Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers in the universal characteristic, a striking anticipation of Gödel numbering. Granted, there is no intuitive or mnemonic way to number any set of elementary concepts using the prime numbers. Leibniz's idea of reasoning through a universal language of symbols and calculations however remarkably foreshadows great 20th century developments in formal systems, such as Turing completeness, where computation was used to define equivalent universal languages (see Turing degree). Because Leibniz was a mathematical novice when he first wrote about the characteristic, at first he did not conceive it as an algebra but rather as a universal language or script. Only in 1676 did he conceive of a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting characteristic included a logical calculus, some combinatorics, algebra, his analysis situs (geometry of situation), a universal concept language, and more.

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Gottfried Wilhelm Leibniz What Leibniz actually intended by his characteristica universalis and calculus ratiocinator, and the extent to which modern formal logic does justice to the calculus, may never be established.[48]

Formal logic Leibniz is the most important logician between Aristotle and 1847, when George Boole and Augustus De Morgan each published books that began modern formal logic. Leibniz enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion, and the empty set. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two: 1. All our ideas are compounded from a very small number of simple ideas, which form the alphabet of human thought. 2. Complex ideas proceed from these simple ideas by a uniform and symmetrical combination, analogous to arithmetical multiplication. The formal logic that emerged early in the 20th century also requires, at minimum, unary negation and quantified variables ranging over some universe of discourse. Leibniz published nothing on formal logic in his lifetime; most of what he wrote on the subject consists of working drafts. In his book History of Western Philosophy, Bertrand Russell went so far as to claim that Leibniz had developed logic in his unpublished writings to a level which was reached only 200 years later.

Mathematician Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular.[49] In the 18th century, "function" lost these geometrical associations. Leibniz was the first to see that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination. Leibniz's discoveries of Boolean algebra and of symbolic logic, also relevant to mathematics, are discussed in the preceding section. The best overview of Leibniz's writings on the calculus may be found in Bos (1974).[50]

Calculus Leibniz is credited, along with Sir Isaac Newton, with the invention of infinitesimal calculus (that comprises differential and integral calculus). According to Leibniz's notebooks, a critical breakthrough occurred on November 11, 1675, when he employed integral calculus for the first time to find the area under the graph of a function y = ƒ(x). He introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia. This cleverly suggestive notation for the calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus until 1684.[51] The product rule of differential calculus is still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule. Leibniz exploited infinitesimals in developing the calculus, manipulating them in ways suggesting that they had paradoxical algebraic properties. George Berkeley, in a tract called The Analyst and also in De Motu, criticized these. A recent study argues that Leibnizian calculus was free of contradictions, and was better grounded than Berkeley's empiricist criticisms.[52] From 1711 until his death, Leibniz was engaged in a dispute with John Keill, Newton and others, over whether Leibniz had invented the calculus independently of Newton. This subject is treated at length in the article Leibniz-Newton controversy.

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Gottfried Wilhelm Leibniz Infinitesimals were officially banned from mathematics by the followers of Karl Weierstrass, but survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential. Beginning in 1960, Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory, in the context of a field of hyperreal numbers. The resulting non-standard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning. Robinson's transfer principle is a mathematical implementation of Leibniz's heuristic law of continuity, while the standard part function implements the Leibnizian transcendental law of homogeneity.

Topology Leibniz was the first to use the term analysis situs,[53] later used in the 19th century to refer to what is now known as topology. There are two takes on this situation. On the one hand, Mates, citing a 1954 paper in German by Jacob Freudenthal, argues: Although for Leibniz the situs of a sequence of points is completely determined by the distance between them and is altered if those distances are altered, his admirer Euler, in the famous 1736 paper solving the Königsberg Bridge Problem and its generalizations, used the term geometria situs in such a sense that the situs remains unchanged under topological deformations. He mistakenly credits Leibniz with originating this concept. ...it is sometimes not realized that Leibniz used the term in an entirely different sense and hence can hardly be considered the founder of that part of mathematics.[54] But Hideaki Hirano argues differently, quoting Mandelbrot:[55] To sample Leibniz' scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in 'packing,'... My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling. In "Euclidis Prota"..., which is an attempt to tighten Euclid's axioms, he states,...: 'I have diverse definitions for the straight line. The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets.' This claim can be proved today.[56] Thus the fractal geometry promoted by Mandelbrot drew on Leibniz's notions of self-similarity and the principle of continuity: natura non facit saltus. We also see that when Leibniz wrote, in a metaphysical vein, that "the straight line is a curve, any part of which is similar to the whole", he was anticipating topology by more than two centuries. As for "packing", Leibniz told to his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled with three even smaller circles by the same procedure. This process can be continued infinitely, from which arises a good idea of self-similarity. Leibniz's improvement of Euclid's axiom contains the same concept.

Scientist and engineer Leibniz's writings are currently discussed, not only for their anticipations and possible discoveries not yet recognized, but as ways of advancing present knowledge. Much of his writing on physics is included in Gerhardt's Mathematical Writings.

Physics Leibniz contributed a fair amount to the statics and dynamics emerging about him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton was thoroughly convinced that space was absolute. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695.[57]

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Gottfried Wilhelm Leibniz Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense. For instance, he anticipated Albert Einstein by arguing, against Newton, that space, time and motion are relative, not absolute. Leibniz's rule is an important, if often overlooked, step in many proofs in diverse fields of physics. The principle of sufficient reason has been invoked in recent cosmology, and his identity of indiscernibles in quantum mechanics, a field some even credit him with having anticipated in some sense. Those who advocate digital philosophy, a recent direction in cosmology, claim Leibniz as a precursor. The vis viva Leibniz's vis viva (Latin for living force) is mv2, twice the modern kinetic energy. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter.[58] Here too his thinking gave rise to another regrettable nationalistic dispute. His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes in France; hence academics in those countries tended to neglect Leibniz's idea. In reality, both energy and momentum are conserved, so the two approaches are equally valid.

Other natural science By proposing that the earth has a molten core, he anticipated modern geology. In embryology, he was a preformationist, but also proposed that organisms are the outcome of a combination of an infinite number of possible microstructures and of their powers. In the life sciences and paleontology, he revealed an amazing transformist intuition, fueled by his study of comparative anatomy and fossils. One of his principal works on this subject, Protogaea, unpublished in his lifetime, has recently been published in English for the first time. He worked out a primal organismic theory.[59] In medicine, he exhorted the physicians of his time—with some results—to ground their theories in detailed comparative observations and verified experiments, and to distinguish firmly scientific and metaphysical points of view.

Social science In psychology,[60] he anticipated the distinction between conscious and unconscious states. In public health, he advocated establishing a medical administrative authority, with powers over epidemiology and veterinary medicine. He worked to set up a coherent medical training programme, oriented towards public health and preventive measures. In economic policy, he proposed tax reforms and a national insurance program, and discussed the balance of trade. He even proposed something akin to what much later emerged as game theory. In sociology he laid the ground for communication theory.

Technology In 1906, Garland published a volume of Leibniz's writings bearing on his many practical inventions and engineering work. To date, few of these writings have been translated into English. Nevertheless, it is well understood that Leibniz was a serious inventor, engineer, and applied scientist, with great respect for practical life. Following the motto theoria cum praxis, he urged that theory be combined with practical application, and thus has been claimed as the father of applied science. He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, clocks, etc. With Denis Papin, he invented a steam engine. He even proposed a method for desalinating water. From 1680 to 1685, he struggled to overcome the chronic flooding that afflicted the ducal silver mines in the Harz Mountains, but did not succeed.[61]

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Gottfried Wilhelm Leibniz Computation Leibniz may have been the first computer scientist and information theorist.[62] Early in life, he documented the binary numeral system (base 2), then revisited that system throughout his career.[63] He anticipated Lagrangian interpolation and algorithmic information theory. His calculus ratiocinator anticipated aspects of the universal Turing machine. In 1934, Norbert Wiener claimed to have found in Leibniz's writings a mention of the concept of feedback, central to Wiener's later cybernetic theory. In 1671, Leibniz began to invent a machine that could execute all four arithmetical operations, gradually improving it over a number of years. This "Stepped Reckoner" attracted fair attention and was the basis of his election to the Royal Society in 1673. A number of such machines were made during his years in Hanover, by a craftsman working under Leibniz's supervision. It was not an unambiguous success because it did not fully mechanize the operation of carrying. Couturat reported finding an unpublished note by Leibniz, dated 1674, describing a machine capable of performing some algebraic operations.[64] Leibniz also devised a (now reproduced) cipher machine, recovered by Nicholas Rescher in 2010.[65] Leibniz was groping towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace. In 1679, while mulling over his binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched cards.[66] Modern electronic digital computers replace Leibniz's marbles moving by gravity with shift registers, voltage gradients, and pulses of electrons, but otherwise they run roughly as Leibniz envisioned in 1679.

Librarian While serving as librarian of the ducal libraries in Hanover and Wolfenbuettel, Leibniz effectively became one of the founders of library science. The latter library was enormous for its day, as it contained more than 100,000 volumes, and Leibniz helped design a new building for it, believed to be the first building explicitly designed to be a library. He also designed a book indexing system in ignorance of the only other such system then extant, that of the Bodleian Library at Oxford University. He also called on publishers to distribute abstracts of all new titles they produced each year, in a standard form that would facilitate indexing. He hoped that this abstracting project would eventually include everything printed from his day back to Gutenberg. Neither proposal met with success at the time, but something like them became standard practice among English language publishers during the 20th century, under the aegis of the Library of Congress and the British Library. He called for the creation of an empirical database as a way to further all sciences. His characteristica universalis, calculus ratiocinator, and a "community of minds"—intended, among other things, to bring political and religious unity to Europe—can be seen as distant unwitting anticipations of artificial languages (e.g., Esperanto and its rivals), symbolic logic, even the World Wide Web.

Advocate of scientific societies Leibniz emphasized that research was a collaborative endeavor. Hence he warmly advocated the formation of national scientific societies along the lines of the British Royal Society and the French Academie Royale des Sciences. More specifically, in his correspondence and travels he urged the creation of such societies in Dresden, Saint Petersburg, Vienna, and Berlin. Only one such project came to fruition; in 1700, the Berlin Academy of Sciences was created. Leibniz drew up its first statutes, and served as its first President for the remainder of his life. That Academy evolved into the German Academy of Sciences, the publisher of the ongoing critical edition of his works.[67]

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Lawyer, moralist With the possible exception of Marcus Aurelius, no philosopher has ever had as much experience with practical affairs of state as Leibniz. Leibniz's writings on law, ethics, and politics[68] were long overlooked by English-speaking scholars, but this has changed of late.[69] While Leibniz was no apologist for absolute monarchy like Hobbes, or for tyranny in any form, neither did he echo the political and constitutional views of his contemporary John Locke, views invoked in support of democracy, in 18th-century America and later elsewhere. The following excerpt from a 1695 letter to Baron J. C. Boyneburg's son Philipp is very revealing of Leibniz's political sentiments: As for.. the great question of the power of sovereigns and the obedience their peoples owe them, I usually say that it would be good for princes to be persuaded that their people have the right to resist them, and for the people, on the other hand, to be persuaded to obey them passively. I am, however, quite of the opinion of Grotius, that one ought to obey as a rule, the evil of revolution being greater beyond comparison than the evils causing it. Yet I recognize that a prince can go to such excess, and place the well-being of the state in such danger, that the obligation to endure ceases. This is most rare, however, and the theologian who authorizes violence under this pretext should take care against excess; excess being infinitely more dangerous than deficiency.[70] In 1677, Leibniz called for a European confederation, governed by a council or senate, whose members would represent entire nations and would be free to vote their consciences;[71] this is sometimes tendentiously considered an anticipation of the European Union. He believed that Europe would adopt a uniform religion. He reiterated these proposals in 1715.

Ecumenism Leibniz devoted considerable intellectual and diplomatic effort to what would now be called ecumenical endeavor, seeking to reconcile first the Roman Catholic and Lutheran churches, later the Lutheran and Reformed churches. In this respect, he followed the example of his early patrons, Baron von Boyneburg and the Duke John Frederick—both cradle Lutherans who converted to Catholicism as adults—who did what they could to encourage the reunion of the two faiths, and who warmly welcomed such endeavors by others. (The House of Brunswick remained Lutheran because the Duke's children did not follow their father.) These efforts included corresponding with the French bishop Jacques-Bénigne Bossuet, and involved Leibniz in a fair bit of theological controversy. He evidently thought that the thoroughgoing application of reason would suffice to heal the breach caused by the Reformation.

Philologist Leibniz the philologist was an avid student of languages, eagerly latching on to any information about vocabulary and grammar that came his way. He refuted the belief, widely held by Christian scholars in his day, that Hebrew was the primeval language of the human race. He also refuted the argument, advanced by Swedish scholars in his day, that a form of proto-Swedish was the ancestor of the Germanic languages. He puzzled over the origins of the Slavic languages, was aware of the existence of Sanskrit, and was fascinated by classical Chinese. He published the princeps editio (first modern edition) of the late medieval Chronicon Holtzatiae, a Latin chronicle of the County of Holstein.

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Sinophile Leibniz was perhaps the first major European intellect to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other works by, European Christian missionaries posted in China. Having read Confucius Sinicus Philosophus on the first year of its publication,[72] he concluded that Europeans could learn much from the Confucian ethical tradition. He mulled over the possibility that the Chinese characters were an unwitting form of his universal characteristic. He noted with fascination how the I Ching hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.[73] Leibniz's attraction to Chinese philosophy originates from his perception that Chinese philosophy was similar to his own.[72] The historian E.R. Hughes suggests that Leibniz's ideas of "simple substance" and "pre-established harmony" were directly influenced by Confucianism, pointing to the fact that they were conceived during the period that he was reading Confucius Sinicus Philosophus.[72]

As polymath While making his grand tour of European archives to research the Brunswick family history that he never completed, Leibniz stopped in Vienna between May 1688 and February 1689, where he did much legal and diplomatic work for the Brunswicks. He visited mines, talked with mine engineers, and tried to negotiate export contracts for lead from the ducal mines in the Harz mountains. His proposal that the streets of Vienna be lit with lamps burning rapeseed oil was implemented. During a formal audience with the Austrian Emperor and in subsequent memoranda, he advocated reorganizing the Austrian economy, reforming the coinage of much of central Europe, negotiating a Concordat between the Habsburgs and the Vatican, and creating an imperial research library, official archive, and public insurance fund. He wrote and published an important paper on mechanics. Leibniz also wrote a short paper, first published by Louis Couturat in 1903,[74] summarizing his views on metaphysics. The paper is undated; that he wrote it while in Vienna was determined only in 1999, when the ongoing critical edition finally published Leibniz's philosophical writings for the period 1677–90. Couturat's reading of this paper was the launching point for much 20th-century thinking about Leibniz, especially among analytic philosophers. But after a meticulous study of all of Leibniz's philosophical writings up to 1688—a study the 1999 additions to the critical edition made possible—Mercer (2001) begged to differ with Couturat's reading; the jury is still out.

Posthumous reputation As a mathematician and philosopher When Leibniz died, his reputation was in decline. He was remembered for only one book, the Théodicée, whose supposed central argument Voltaire lampooned in his Candide. Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description. Thus Voltaire and his Candide bear some of the blame for the lingering failure to appreciate and understand Leibniz's ideas. Leibniz had an ardent disciple, Christian Wolff, whose dogmatic and facile outlook did Leibniz's reputation much harm. He also influenced David Hume who read his Théodicée and used some of his ideas.[75] In any event, philosophical fashion was moving away from the rationalism and system building of the 17th century, of which Leibniz had been such an ardent proponent. His work on law, diplomacy, and history was seen as of ephemeral interest. The vastness and richness of his correspondence went unrecognized. Much of Europe came to doubt that Leibniz had discovered the calculus independently of Newton, and hence his whole work in mathematics and physics was neglected. Voltaire, an admirer of Newton, also wrote Candide at least in part to discredit Leibniz's claim to having discovered the calculus and Leibniz's charge that Newton's theory of

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Gottfried Wilhelm Leibniz universal gravitation was incorrect. The rise of relativity and subsequent work in the history of mathematics has put Leibniz's stance in a more favorable light. Leibniz's long march to his present glory began with the 1765 publication of the Nouveaux Essais, which Kant read closely. In 1768, Dutens edited the first multi-volume edition of Leibniz's writings, followed in the 19th century by a number of editions, including those edited by Erdmann, Foucher de Careil, Gerhardt, Gerland, Klopp, and Mollat. Publication of Leibniz's correspondence with notables such as Antoine Arnauld, Samuel Clarke, Sophia of Hanover, and her daughter Sophia Charlotte of Hanover, began. In 1900, Bertrand Russell published a critical study of Leibniz's metaphysics.[76] Shortly thereafter, Louis Couturat published an important study of Leibniz, and edited a volume of Leibniz's heretofore unpublished writings, mainly on logic. They made Leibniz somewhat respectable among 20th-century analytical and linguistic philosophers in the English-speaking world (Leibniz had already been of great influence to many Germans such as Bernhard Riemann). For example, Leibniz's phrase salva veritate, meaning interchangeability without loss of or compromising the truth, recurs in Willard Quine's writings. Nevertheless, the secondary English-language literature on Leibniz did not really blossom until after World War II. This is especially true of English speaking countries; in Gregory Brown's bibliography fewer than 30 of the English language entries were published before 1946. American Leibniz studies owe much to Leroy Loemker (1904–85) through his translations and his interpretive essays in LeClerc (1973). Nicholas Jolley has surmised that Leibniz's reputation as a philosopher is now perhaps higher than at any time since he was alive.[77] Analytic and contemporary philosophy continue to invoke his notions of identity, individuation, and possible worlds, while the doctrinaire contempt for metaphysics, characteristic of analytic and linguistic philosophy, has faded. Work in the history of 17th- and 18th-century ideas has revealed more clearly the 17th-century "Intellectual Revolution" that preceded the better-known Industrial and commercial revolutions of the 18th and 19th centuries. The 17th- and 18th-century belief that natural science, especially physics, differs from philosophy mainly in degree and not in kind, is no longer dismissed out of hand. That modern science includes a "scholastic" as well as a "radical empiricist" element is more accepted now than in the early 20th century. Leibniz's thought is now seen as a major prolongation of the mighty endeavor begun by Plato and Aristotle: the universe and man's place in it are amenable to human reason. In 1985, the German government created the Leibniz Prize, offering an annual award of 1.55 million euros for experimental results and 770,000 euros for theoretical ones. It is the world's largest prize for scientific achievement. The collection of manuscript papers of Leibniz at the Gottfried Wilhelm Leibniz Bibliothek – Niedersächische Landesbibliothek were inscribed on UNESCO's Memory of the World Register in 2007.[78]

Leibniz biscuits Leibniz-Keks, a popular brand of biscuits, are named after Gottfried Leibniz. These biscuits honour Leibniz because he was a resident of Hanover, where the company is based.[79]

Writings and edition Leibniz mainly wrote in three languages: scholastic Latin, French and German. During his lifetime, he published many pamphlets and scholarly articles, but only two "philosophical" books, the Combinatorial Art and the Théodicée. (He published numerous pamphlets, often anonymous, on behalf of the House of Brunswick-Lüneburg, most notably the "De jure suprematum" a major consideration of the nature of sovereignty.) One substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain, which Leibniz had withheld from publication after the death of John Locke. Only in 1895, when Bodemann completed his catalogues of Leibniz's manuscripts and correspondence, did the enormous extent of Leibniz's Nachlass become clear: about 15,000 letters to more than 1000 recipients plus more than 40,000 other items. Moreover, quite a few of these letters are of essay length. Much of his vast correspondence, especially the letters dated after 1685, remains unpublished, and much of

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Gottfried Wilhelm Leibniz what is published has been so only in recent decades. The amount, variety, and disorder of Leibniz's writings are a predictable result of a situation he described in a letter as follows: I cannot tell you how extraordinarily distracted and spread out I am. I am trying to find various things in the archives; I look at old papers and hunt up unpublished documents. From these I hope to shed some light on the history of the [House of] Brunswick. I receive and answer a huge number of letters. At the same time, I have so many mathematical results, philosophical thoughts, and other literary innovations that should not be allowed to vanish that I often do not know where to begin.[80] The extant parts of the critical edition[81] of Leibniz's writings are organized as follows: • • • • • • • •

Series 1. Political, Historical, and General Correspondence. 21 vols., 1666–1701. Series 2. Philosophical Correspondence. 1 vol., 1663–85. Series 3. Mathematical, Scientific, and Technical Correspondence. 6 vols., 1672–96. Series 4. Political Writings. 6 vols., 1667–98. Series 5. Historical and Linguistic Writings. Inactive. Series 6. Philosophical Writings. 7 vols., 1663–90, and Nouveaux essais sur l'entendement humain. Series 7. Mathematical Writings. 3 vols., 1672–76. Series 8. Scientific, Medical, and Technical Writings. In preparation.

The systematic cataloguing of all of Leibniz's Nachlass began in 1901. It was hampered by two world wars, the Nazi dictatorship (with the Holocaust, which affected a Jewish employee of the project, and other personal consequences), and decades of German division (two states with the cold war's "iron curtain" in between, separating scholars and also scattering portions of his literary estates). The ambitious project has had to deal with seven languages contained in some 200,000 pages of written and printed paper. In 1985 it was reorganized and included in a joint program of German federal and state (Länder) academies. Since then the branches in Potsdam, Münster, Hanover and Berlin have jointly published 25 volumes of the critical edition, with an average of 870 pages, and prepared index and concordance works.

Selected works The year given is usually that in which the work was completed, not of its eventual publication. • 1666. De Arte Combinatoria (On the Art of Combination); partially translated in Loemker §1 and Parkinson (1966). • 1671. Hypothesis Physica Nova (New Physical Hypothesis); Loemker §8.I (partial). • 1673 Confessio philosophi (A Philosopher's Creed); an English translation is available. • 1684. Nova methodus pro maximis et minimis (New method for maximums and minimums); translated in Struik, D. J., 1969. A Source Book in Mathematics, 1200–1800. Harvard University Press: 271–81. • 1686. Discours de métaphysique; Martin and Brown (1988), Ariew and Garber 35, Loemker §35, Wiener III.3, Woolhouse and Francks 1. An online translation [82] by Jonathan Bennett is available. • 1703. Explication de l'Arithmétique Binaire (Explanation of Binary Arithmetic); Gerhardt, Mathematical Writings VII.223. An online translation [83] by Lloyd Strickland is available. • 1710. Théodicée; Farrer, A.M., and Huggard, E.M., trans., 1985 (1952). Wiener III.11 (part). An online translation [84] is available at Project Gutenberg. • 1714. Monadologie; translated by Nicholas Rescher, 1991. The Monadology: An Edition for Students. University of Pittsburg Press. Ariew and Garber 213, Loemker §67, Wiener III.13, Woolhouse and Francks 19. Online translations: Jonathan Bennett's translation [82]; Latta's translation [85]; French, Latin and Spanish edition, with facsimile of Leibniz's manuscript. [86] • 1765. Nouveaux essais sur l'entendement humain; completed in 1704. Remnant, Peter, and Bennett, Jonathan, trans., 1996. New Essays on Human Understanding. Cambridge University Press. Wiener III.6 (part). An online translation [87] by Jonathan Bennett is available.

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Collections Five important collections of English translations are Wiener (1951), Loemker (1969), Ariew and Garber (1989), Woolhouse and Francks (1998), and Strickland (2006). The ongoing critical edition of all of Leibniz's writings is Sämtliche Schriften und Briefe.[81]

Notes [1] The History of Philosophy, Vol. IV: Modern Philosophy: From Descartes to Leibniz by Frederick C. Copleston (1958) [2] Franz Exner, "Über Leibnitz'ens Universal-Wissenschaft", 1843; "Universalwissenschaft" (http:/ / www. zeno. org/ Meyers-1905/ A/ Universalwissenschaft) in the Meyers Großes Konversations-Lexikon; Stanley Burris, "Leibniz's Influence on 19th Century Logic" (http:/ / plato. stanford. edu/ entries/ leibniz-logic-influence/ ), Stanford Encyclopedia of Philosophy [3] Max Mangold (ed.), ed. (2005) (in German). Duden-Aussprachewörterbuch (Duden Pronunciation Dictionary) (7th ed.). Mannheim: Bibliographisches Institut GmbH. ISBN 978-3-411-04066-7. [4] Eva-Maria Krech et al. (ed.), ed. (2010) (in German). Deutsches Aussprachewörterbuch (German Pronunciation Dictionary) (1st ed.). Berlin: Walter de Gruyter GmbH & Co. KG. ISBN 978-3-11-018203-3. [5] David Smith, p.173-181 (1929) [6] Roughly 40%, 30%, and 15%, respectively. www.gwlb.de (http:/ / www. gwlb. de/ Leibniz/ Leibniz-Nachlass/ index. htm). Leibniz-Nachlass (i.e. Legacy of Leibniz), Gottfried Wilhelm Leibniz Bibliothek (one of the three Official Libraries of the German state Lower Saxony). [7] Baird, Forrest E.; Walter Kaufmann (2008). From Plato to Derrida. Upper Saddle River, New Jersey: Pearson Prentice Hall. ISBN 0-13-158591-6. [8] Leibnitiana (http:/ / www. gwleibniz. com/ friedrich_leibniz/ friedrich_leibniz. html) [9] Johann Amos Comenius, Comenius in England, Oxford University Press, 1932, p. 6 [10] Mackie (1845), 21 [11] Mackie (1845), 22 [12] Mackie (1845), 26 [13] Jolley, Nicholas (1995). The Cambridge Companion to Leibniz. Cambridge University Press.:20 [14] Mackie (1845), 38 [15] Mackie (1845), 39 [16] Mackie (1845), 40 [17] Aiton 1985: 312 [18] Mackie (1845), 41-42 [19] Mackie (1845), 43 [20] Mackie (1845), 44-45 [21] Mackie (1845), 58-61 [22] Mackie (1845), 69-70 [23] Mackie (1845), 73-74 [24] On the encounter between Newton and Leibniz and a review of the evidence, see Alfred Rupert Hall, Philosophers at War: The Quarrel Between Newton and Leibniz (Cambridge, 2002), pp. 44–69. [25] Mackie (1845), 117-118 [26] For a recent study of Leibniz's correspondence with Sophia Charlotte, see MacDonald Ross (http:/ / www. philosophy. leeds. ac. uk/ GMR/ homepage/ sophiec. html) (1998). [27] Mackie (1845), 109 [28] See Wiener IV.6 and Loemker § 40. Also see a curious passage titled "Leibniz's Philosophical Dream," first published by Bodemann in 1895 and translated on p. 253 of Morris, Mary, ed. and trans., 1934. Philosophical Writings. Dent & Sons Ltd. [29] Gottfried Wilhelm Leibniz (2012). Peter Loptson. ed. Discourse on Metaphysics and Other Writings. Broadview Press. pp. 23–24. ISBN 9781554810116. "The answer is unknowable, but it may not be unreasonable to see him, at least in theological terms, as essentially a deist. He is a determinist: there are no miracles (the events so called being merely instances of infrequently occurring natural laws); Christ has no real role in the system; we live forever, and hence we carry on after our deaths, but then everything — every individual substance — carries on forever. Nonetheless, Leibniz is a theist. His system is generated from, and needs, the postulate of a creative god. In fact, though, despite Leibniz's protestations, his God is more the architect and engineer of the vast complex world-system than the embodiment of love of Christian orthodoxy." [30] Christopher Ernest Cosans (2009). Owen's Ape & Darwin's Bulldog: Beyond Darwinism and Creationism. Indiana University Press. pp. 102–103. ISBN 9780253220516. "In advancing his system of mechanics, Newton claimed that collisions of celestial objects would cause a loss of energy that would require God to intervene from time to time to maintain order in the solar system (Vailati 1997, 37–42). In criticizing this implication, Leibniz remarks: "Sir Isaac Newton and his followers have also a very odd opinion concerning the work of God. According to their doctrine, God Almighty wants to wind up his watch from time to time; otherwise it would cease to move." (Leibniz 1715, 675) Leibniz argues that any scientific theory that relies on God to perform miracles after He had first made the universe indicates that God lacked sufficient foresight or power to establish adequate natural laws in the first place. In defense of Newton's theism, Clarke is unapologetic: "'tis

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Gottfried Wilhelm Leibniz not a diminution but the true glory of his workmanship that nothing is done without his continual government and inspection"' (Leibniz 1715, 676–677). Clarke is believed to have consulted closely with Newton on how to respond to Leibniz. He asserts that Leibniz's deism leads to "the notion of materialism and fate" (1715, 677), because it excludes God from the daily workings of nature." [31] Andreas Sofroniou (2007). Moral Philosophy, from Hippocrates to the 21st Aeon. Lulu.com. ISBN 9781847534637. "In a commentary on Shaftesbury published in 1720, Gottfried Wilhelm Leibniz, a Rationalist philosopher and mathematician, accepted the Deist conception of God as an intelligent Creator but refused the contention that a god who metes out punishments is evil." [32] Shelby D. Hunt (2003). Controversy in Marketing Theory: For Reason, Realism, Truth, and Objectivity. M.E. Sharpe. p. 33. ISBN 9780765609311. "Consistent with the liberal views of the Enlightenment, Leibniz was an optimist with respect to human reasoning and scientific progress (Popper 1963, p.69). Although he was a great reader and admirer of Spinoza, Leibniz, being a confirmed deist, rejected emphatically Spinoza's pantheism: God and nature, for Leibniz, were not simply two different "labels" for the same "thing"." [33] Ariew & Garber, 69; Loemker, §§36, 38 [34] Ariew & Garber, 138; Loemker, §47; Wiener, II.4 [35] Ariew & Garber, 272–84; Loemker, §§14, 20, 21; Wiener, III.8 [36] Mates (1986), chpts. 7.3, 9 [37] Loemker 717 [38] See Jolley (1995: 129–31), Woolhouse and Francks (1998), and Mercer (2001). [39] Loemker 311 [40] For a precis of what Leibniz meant by these and other Principles, see Mercer (2001: 473–84). For a classic discussion of Sufficient Reason and Plenitude, see Lovejoy (1957). [41] Rutherford (1998) is a detailed scholarly study of Leibniz's theodicy. [42] Magill, Frank (ed.). Masterpieces of World Philosophy. New York: Harper Collins (1990). [43] Magill, Frank (ed.) (1990) [44] The Art of Discovery 1685, Wiener 51 [45] Many of his memoranda are translated in Parkinson 1966. [46] Loemker, however, who translated some of Leibniz's works into English, said that the symbols of chemistry were real characters, so there is disagreement among Leibniz scholars on this point. [47] Preface to the General Science, 1677. Revision of Rutherford's translation in Jolley 1995: 234. Also Wiener I.4 [48] A good introductory discussion of the "characteristic" is Jolley (1995: 226–40). An early, yet still classic, discussion of the "characteristic" and "calculus" is Couturat (1901: chpts. 3,4). [49] Struik (1969), 367 [50] Jesseph, Douglas M. (1998). "Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes" (http:/ / muse. jhu. edu/ journals/ perspectives_on_science/ v006/ 6. 1jesseph. html). Perspectives on Science 6.1&2: 6–40. . Retrieved 31 December 2011. [51] For an English translation of this paper, see Struik (1969: 271–84), who also translates parts of two other key papers by Leibniz on the calculus. [52] Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y [53] Loemker §27 [54] Mates (1986), 240 [55] HIRANO, Hideaki. "Leibniz's Cultural Pluralism And Natural Law" (http:/ / www. t. hosei. ac. jp/ ~hhirano/ academia/ leibniz. htm). . Retrieved March 10, 2010. [56] Mandelbrot (1977), 419. Quoted in Hirano (1997). [57] Ariew and Garber 117, Loemker §46, W II.5. On Leibniz and physics, see the chapter by Garber in Jolley (1995) and Wilson (1989). [58] See Ariew and Garber 155–86, Loemker §§53–55, W II.6–7a [59] On Leibniz and biology, see Loemker (1969a: VIII). [60] On Leibniz and psychology, see Loemker (1969a: IX). [61] Aiton (1985), 107–114, 136 [62] Davis (2000) discusses Leibniz's prophetic role in the emergence of calculating machines and of formal languages. [63] See Couturat (1901): 473–78. [64] Couturat (1901), 115 [65] See N. Rescher, Leibniz and Cryptography (Pittsburgh, University Library Systems, University of Pittsburgh, 2012). [66] The Reality Club: Wake Up Call for Europe Tech (http:/ / www. edge. org/ discourse/ schirrmacher_eurotech. html) [67] On Leibniz's projects for scientific societies, see Couturat (1901), App. IV. [68] See, for example, Ariew and Garber 19, 94, 111, 193; Riley 1988; Loemker §§2, 7, 20, 29, 44, 59, 62, 65; W I.1, IV.1–3 [69] See (in order of difficulty) Jolley (2005: chpt. 7), Gregory Brown's chapter in Jolley (1995), Hostler (1975), and Riley (1996). [70] Loemker: 59, fn 16. Translation revised. [71] Loemker: 58, fn 9 [72] Mungello, David E. (1971). "Leibniz's Interpretation of Neo-Confucianism". Philosophy East and West 21 (1): 3–22. doi:10.2307/1397760.

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Gottfried Wilhelm Leibniz [73] On Leibniz, the I Ching, and binary numbers, see Aiton (1985: 245–48). Leibniz's writings on Chinese civilization are collected and translated in Cook and Rosemont (1994), and discussed in Perkins (2004). [74] Later translated as Loemker 267 and Woolhouse and Francks 30 [75] Vasilyev, 1993 (http:/ / www. humesociety. org/ hs/ issues/ v19n1/ vasilyeu/ vasilyeu-v19n1. pdf) [76] Russell, 1900 [77] Jolley, 217–19 [78] "Letters from and to Gottfried Wilhelm Leibniz within the collection of manuscript papers of Gottfried Wilhelm Leibniz" (http:/ / portal. unesco. org/ ci/ en/ ev. php-URL_ID=22464& URL_DO=DO_TOPIC& URL_SECTION=201. html). UNESCO Memory of the World Programme. 2008-05-16. . Retrieved 2009-12-15. [79] "Bahlsen products FAQ" (http:/ / www. bahlsen. de/ root_bahlsen_anim/ index. php). . [80] 1695 letter to Vincent Placcius in Gerhardt. [81] www.leibniz-edition.de (http:/ / www. leibniz-edition. de/ ). See photograph there. [82] http:/ / www. earlymoderntexts. com [83] http:/ / www. leibniz-translations. com/ binary. htm [84] http:/ / www. gutenberg. org/ etext/ 17147 [85] http:/ / www. rbjones. com/ rbjpub/ philos/ classics/ leibniz/ monad. htm [86] http:/ / www. helicon. es/ dig/ 8542205. pdf [87] http:/ / www. earlymoderntexts. com/ f_leibniz. html

References Primary literature • Alexander, H G (ed) The Leibniz-Clarke Correspondence. Manchester: Manchester University Press, 1956. • Ariew, R & D Garber, 1989. Leibniz: Philosophical Essays. Hackett. • Arthur, Richard, 2001. The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672–1686. Yale University Press. • Cohen, Claudine and Wakefield, Andre, 2008. Protogaea. University of Chicago Press. • Cook, Daniel, and Rosemont, Henry Jr., 1994. Leibniz: Writings on China. Open Court. • Loemker, Leroy, 1969 (1956). Leibniz: Philosophical Papers and Letters. Reidel. • Remnant, Peter, and Bennett, Jonathan, 1996 (1981). Leibniz: New Essays on Human Understanding. Cambridge University Press. • Riley, Patrick, 1988. Leibniz: Political Writings. Cambridge University Press. • Sleigh, Robert C., Look, Brandon, and Stam, James, 2005. Confessio Philosophi: Papers Concerning the Problem of Evil, 1671–1678. Yale University Press. • Strickland, Lloyd, 2006. The Shorter Leibniz Texts: A Collection of New Translations. Continuum. • Ward, A. W. Leibniz as a Politician (lecture, 1911) • Wiener, Philip, 1951. Leibniz: Selections. Scribner. • Woolhouse, R.S., and Francks, R., 1998. Leibniz: Philosophical Texts. Oxford University Press.

Secondary literature • Adams, Robert Merrihew. Lebniz: Determinist, Theist, Idealist. New York: Oxford, Oxford University Press, 1994. • Aiton, Eric J., 1985. Leibniz: A Biography. Hilger (UK). • Antognazza, M.R.(2008) Leibniz: An Intellectual Biography. Cambridge Univ. Press. • [ Edit this reference (http://en.wikipedia.org/w/index.php?title=Template:BarrowTipler1986&action=edit)] Barrow, John D.; Tipler, Frank J. (19 May 1988). The Anthropic Cosmological Principle (http:/ / books. google. com/ books?id=uSykSbXklWEC& printsec=frontcover). foreword by John A. Wheeler. Oxford: Oxford University Press. ISBN 9780192821478. LC 87-28148 (http://lccn.loc.gov/87028148). Retrieved 31 December 2009. • Albeck-Gidron, Rachel, The Century of the Monads: Leibniz's Metaphysics and 20th-Century Modernity, Bar-Ilan University Press.

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Gottfried Wilhelm Leibniz • Bos, H. J. M. (1974) "Differentials, higher-order differentials and the derivative in the Leibnizian calculus," Arch. History Exact Sci. 14: 1—90. • Couturat, Louis, 1901. La Logique de Leibniz. Paris: Felix Alcan. • Davis, Martin, 2000. The Universal Computer: The Road from Leibniz to Turing. WW Norton. • Deleuze, Gilles, 1993. The Fold: Leibniz and the Baroque. University of Minnesota Press. • Du Bois-Reymond, Paul, 18nn. "Leibnizian Thoughts in Modern Science". • Finster, Reinhard & Gerd van den Heuvel. Gottfried Wilhelm Leibniz. Mit Selbstzeugnissen und Bilddokumenten. 4. Auflage. Rowohlt, Reinbek bei Hamburg 2000 (Rowohlts Monographien, 50481), ISBN 3-499-50481-2. • Grattan-Guinness, Ivor, 1997. The Norton History of the Mathematical Sciences. W W Norton. • Hall, A. R., 1980. Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge University Press. • Heidegger, Martin, 1983. The Metaphysical Foundations of Logic. Indiana University Press. • Hirano, Hideaki, 1997. "Cultural Pluralism And Natural Law." Unpublished. • Hostler, J., 1975. Leibniz's Moral Philosophy. UK: Duckworth. • Jolley, Nicholas, ed., 1995. The Cambridge Companion to Leibniz. Cambridge University Press. • LeClerc, Ivor, ed., 1973. The Philosophy of Leibniz and the Modern World. Vanderbilt University Press. • Lovejoy, Arthur O., 1957 (1936) "Plenitude and Sufficient Reason in Leibniz and Spinoza" in his The Great Chain of Being. Harvard University Press: 144–82. Reprinted in Frankfurt, H. G., ed., 1972. Leibniz: A Collection of Critical Essays. Anchor Books. • Mandelbrot, Benoît, 1977. The Fractal Geometry of Nature. Freeman. • Mackie, John Milton; Guhrauer, Gottschalk Eduard, 1845. Life of Godfrey William von Leibnitz. Gould, Kendall and Lincoln. • Mates, Benson, 1986. The Philosophy of Leibniz: Metaphysics and Language. Oxford University Press. • Mercer, Christia, 2001. Leibniz's metaphysics: Its Origins and Development. Cambridge University Press. • Morris, Simon Conway, 2003. Life's Solution: Inevitable Humans in a Lonely Universe. Cambridge University Press. • Perkins, Franklin, 2004. Leibniz and China: A Commerce of Light. Cambridge University Press. • Rensoli, Lourdes, 2002. El problema antropologico en la concepcion filosofica de G. W. Leibniz. Leibnitius Politechnicus. Universidad Politecnica de Valencia. • Riley, Patrick, 1996. Leibniz's Universal Jurisprudence: Justice as the Charity of the Wise. Harvard University Press. • Rutherford, Donald, 1998. Leibniz and the Rational Order of Nature. Cambridge University Press. • Struik, D. J., 1969. A Source Book in Mathematics, 1200–1800. Harvard University Press. • Ward, P. D., and Brownlee, D., 2000. Rare Earth: Why Complex Life is Uncommon in the Universe. Springer Verlag. • Wilson, Catherine, 1989. 'Leibniz's Metaphysics. Princeton University Press. • Zalta, E. N., 2000. " A (Leibnizian) Theory of Concepts (http://mally.stanford.edu/Papers/leibniz.pdf)", Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy 3: 137–183. • Smith, David Eugene (1929). A Source Book in Mathematics. New York and London: McGraw-Hill Book Company, Inc..

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External links • An extensive bibliography (http://www.worldcat.org/profiles/mciocchi/lists/1786513) • Internet Encyclopedia of Philosophy: " Leibniz (http://www.utm.edu/research/iep/l/leib-met.htm)" – Douglas Burnham. • Stanford Encyclopedia of Philosophy. Articles on Leibniz (http://plato.stanford.edu/search/searcher. py?query=Leibniz). • O'Connor, John J.; Robertson, Edmund F., "Gottfried Wilhelm Leibniz" (http://www-history.mcs.st-andrews. ac.uk/Biographies/Leibniz.html), MacTutor History of Mathematics archive, University of St Andrews. • George MacDonald Ross, Leibniz (http://etext.leeds.ac.uk/leibniz/leibniz.htm), Originally published: Oxford University Press (Past Masters) 1984; Electronic edition: Leeds Electronic Text Centre July 2000 • Gottfried Wilhelm Leibniz (http://genealogy.math.ndsu.nodak.edu/id.php?id=60985) at the Mathematics Genealogy Project • Works by Gottfried Leibniz (http://www.gutenberg.org/author/Leibniz+Gottfried+Wilhelm+Freiherr+von) at Project Gutenberg • Gottfried Wilhelm Leibniz (http://www.dmoz.org/Society/Philosophy/Philosophers/L/ Leibniz,_Gottfried_Wilhelm/) at the Open Directory Project • translations (http://www.earlymoderntexts.com) by Jonathan Bennett, of the New Essays, the exchanges with Bayle, Arnauld and Clarke, and about 15 shorter works. • Leibnitiana (http://www.gwleibniz.com/) – Gregory Brown. • Gottfried Wilhelm Leibniz: Texts and Translations (http://philosophyfaculty.ucsd.edu/faculty/rutherford/ Leibniz/index.html), compiled by Donald Rutherford, UCSD • Leibniz-translations.com (http://www.leibniz-translations.com/) Scroll down for many Leibniz links. • Leibniz Prize. (http://www.dfg.de/en/news/scientific_prizes/leibniz_preis/index.html) • Philosophical Works of Leibniz translated by G.M. Duncan (http://www.archive.org/details/ philosophicalwor00leibuoft) • Leibnitiana (http://www.gwleibniz.com/), links and resources compiled by Gregory Brown, University of Houston. • Leibnizian Resources (http://www.helsinki.fi/~mroinila/leibniz1.htm), many links organized by Markku Roinila, University of Helsinki. • Leibniz Bibliography (http://www.leibniz-bibliographie.de/DB=1.95/LNG=EN/ ?COOKIE=U8000,K8000,I0,B1999++++++,SY,NVZG,D1.95,E0ed05df2-2e89,A,H,R194.95.154.1,FY) at the Gottfried Wilhelm Leibniz Library.

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Bernoulli family

136

Bernoulli family The Bernoullis (German: [bɛʁˈnʊli][1]; (English: pron.: /bərˈnuːli/)) were a patrician family of merchants and scholars, originally from Antwerp, who settled in Basel, Switzerland. The name is sometimes misspelled Bernou-ill-i and mispronounced accordingly.[2] Leon Bernoulli was a doctor in Antwerp, which at that time was in the Spanish Netherlands. He died in 1561 and in 1570 his son, Jacob, emigrated to Frankfurt am Main to escape from the Spanish persecution of the Huguenots.[3] Jacob’s grandson, a spice trader also named Jacob, moved in 1620 to Basel, Switzerland, and was granted Swiss citizenship. His son Niklaus (1623-1708), Leon’s great-great-grandson, married Margarethe Schönauer. Niklaus had three sons: • Jacob Bernoulli (1654–1705; also known as James or Jacques) Mathematician after whom Bernoulli numbers are named. • Nicolaus Bernoulli (1662–1716) Painter and alderman of Basel. • Johann Bernoulli (1667–1748; also known as Jean) Mathematician and early adopter of infinitesimal calculus. In addition to those mentioned above, the Bernoulli family produced many notable artists and scientists, in particular a number of famous mathematicians in the 18th century: • • • • • •

Nicolaus I Bernoulli (1687–1759) Mathematician. Nicolaus II Bernoulli (1695–1726) Mathematician; worked on curves, differential equations, and probability. Daniel Bernoulli (1700–1782) Developer of Bernoulli's principle and St. Petersburg paradox. Johann II Bernoulli (1710–1790; also known as Jean) Mathematician and physicist. Johann III Bernoulli (1744–1807; also known as Jean) Astronomer, geographer, and mathematician. Jacob II Bernoulli (1759–1789; also known as Jacques) Physicist and mathematician.

Devices and ideas named for members of the family • • •

Bernoulli differential equation • Bernoulli distribution • Bernoulli number • •

Bernoulli polynomials Bernoulli process Bernoulli trial Bernoulli's principle

References [1] Mangold, Max (1990) Duden - Das Aussprachewörterbuch. 3. Auflage. Mannheim/Wien/Zürich, Dudenverlag. [2] Talk page, section Pronunciation [3] Historic Lexicon of the Swiss, Bernoulli (http:/ / www. hls-dhs-dss. ch/ textes/ f/ F20951. php)

• Family tree (http://www-groups.dcs.st-and.ac.uk/~history/Diagrams/Bernoulli_family.gif) at the MacTutor History of Mathematics archive. • Bernoulli family in German (http://www.hls-dhs-dss.ch/textes/d/D20951.php), French (http://www. hls-dhs-dss.ch/textes/f/F20951.php) and Italian (http://www.hls-dhs-dss.ch/textes/i/I20951.php) in the online Historical Dictionary of Switzerland.

Jacob Bernoulli

137

Jacob Bernoulli For other family members named Jacob, see Bernoulli family.

Jacob Bernoulli

Jakob Bernoulli Born

6 January 1655 Basel, Switzerland

Died

16 August 1705 (aged 50) Basel, Switzerland

Residence

Switzerland

Nationality

Swiss

Fields

Mathematician

Institutions

University of Basel

Alma mater

University of Basel

Doctoral students Johann Bernoulli Jacob Hermann Nicolaus I Bernoulli Known for

Bernoulli differential equation Bernoulli numbers (Bernoulli's formula Bernoulli polynomials Bernoulli map) Bernoulli trial (Bernoulli process Bernoulli scheme Bernoulli operator Hidden Bernoulli model Bernoulli sampling Bernoulli distribution Bernoulli random variable Bernoulli's Golden Theorem) Bernoulli's inequality Lemniscate of Bernoulli

Notes Brother of Johann Bernoulli.

Jacob Bernoulli (also known as James or Jacques) (27 December 1654/6 January 1655 – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family.

Jacob Bernoulli

138

Jacob Bernoulli was born in Basel, Switzerland. Following his father's wish, he studied theology and entered the ministry. But contrary to the desires of his parents, he also studied mathematics and astronomy. He traveled throughout Europe from 1676 to 1682, learning about the latest discoveries in mathematics and the sciences. This included the work of Robert Boyle and Robert Hooke. He became familiar with calculus through a correspondence with Gottfried Leibniz, then collaborated with his brother Johann on various applications, notably publishing papers on transcendental curves (1696) and isoperimetry (1700, 1701). In 1690, Jacob Bernoulli became the first person to develop the technique for solving separable differential equations. Upon returning to Basel in 1682, he founded a school for mathematics and the sciences. He was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life.

Important works Jacob Bernoulli is best known for the work Ars Conjectandi (The Art of Conjecture), published eight years after his death in 1713 by his Jacob Bernoulli's grave. nephew Nicholas. In this work, he described the known results in probability theory and in enumeration, often providing alternative proofs of known results. This work also includes the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers. The terms Bernoulli trial and Bernoulli numbers result from this work. The lunar crater Bernoulli is also named after him jointly with his brother Johann.

Discovery of the mathematical constant e Bernoulli discovered the constant e by studying a question about compound interest which required him to find the value of the following expression (which is in fact e):

One example is an account that starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25. Compounding quarterly yields $1.00×1.254 = $2.4414..., and compounding monthly yields $1.00×(1.0833...)12 = $2.613035.... Bernoulli noticed that this sequence approaches a limit (the force of interest) for more and smaller compounding intervals. Compounding weekly yields $2.692597..., while compounding daily yields $2.714567..., just two cents more. Using n as the number of compounding intervals, with interest of 100%/n in each interval, the limit for large n is the number that came to be known as e; with continuous compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1, and yields (1+R) dollars at simple interest, will yield eR dollars with continuous compounding.

Jacob Bernoulli

Personal life Bernoulli chose a figure of a logarithmic spiral and the motto Eadem mutata resurgo ("Changed and yet the same, I rise again") for his gravestone; the spiral executed by the stonemasons was, however, an Archimedean spiral.,[1] “[Jacques Bernoulli] wrote that the logarithmic spiral ‘may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self’.” (Livio 2002: 116). Jacob had a daughter and a son.

References [1] Jacob (Jacques) Bernoulli (http:/ / www-gap. dcs. st-and. ac. uk/ ~history/ Biographies/ Bernoulli_Jacob. html), The MacTutor History of Mathematics archive (http:/ / www-gap. dcs. st-and. ac. uk/ ~history/ ), School of Mathematics and Statistics, University of St Andrews, UK.

Further reading • Hoffman, J.E. (1970–80). "Bernoulli, Jakob (Jacques) I". Dictionary of Scientific Biography. 2. New York: Charles Scribner's Sons. pp. 46–51. ISBN 0684101149. • Schneider, I., 2005, "Ars conjectandi" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 88–104. • Livio, Mario, 2002, The golden ratio: the story of Phi, the extraordinary number of nature, art, and beauty. London.

External links • Jacob Bernoulli (http://genealogy.math.ndsu.nodak.edu/id.php?id=54440) at the Mathematics Genealogy Project • O'Connor, John J.; Robertson, Edmund F., "Jacob Bernoulli" (http://www-history.mcs.st-andrews.ac.uk/ Biographies/Bernoulli_Jacob.html), MacTutor History of Mathematics archive, University of St Andrews. • Jakob Bernoulli: Tractatus de Seriebus Infinitis (http://www.kubkou.se/pdf/mh/jacobB.pdf) (pdf) • Weisstein, Eric W., Bernoulli, Jakob (1654–1705) (http://scienceworld.wolfram.com/biography/ BernoulliJakob.html) from ScienceWorld.

139

Johann Bernoulli

140

Johann Bernoulli Johann Bernoulli

Johann Bernoulli (portrait by Johann Rudolf Huber, circa 1740) Born

27 July 1667 Basel, Switzerland

Died

1 January 1748 (aged 80) Basel, Switzerland

Residence

Switzerland

Nationality

Swiss

Fields

Mathematician

Institutions

University of Groningen University of Basel

Alma mater

University of Basel

Doctoral advisor

Jacob Bernoulli

Other academic advisors Nikolaus Eglinger Doctoral students

Daniel Bernoulli Leonhard Euler Johann Samuel König Pierre Louis Maupertuis

Other notable students

Guillaume de l'Hôpital

Known for

Development of infinitesimal calculus Catenary solution Bernoulli's rule Bernoulli's identity

Notes Brother of Jakob Bernoulli, and the father of Daniel Bernoulli.

Johann Bernoulli (27 July 1667 – 1 January 1748; also known as Jean or John) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educated Leonhard Euler in his youth.

Johann Bernoulli

Early life and education Johann was born in Basel, the son of Nikolaus Bernoulli, an apothecary, and his wife, Margaretha Schonauer and began studying medicine at Basel University. His father desired that he study business so that he might take over the family spice trade, but Johann Bernoulli disliked business and convinced his father to allow him to study medicine instead. However, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob.[1] Throughout Johann Bernoulli’s education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus. They were among the first mathematicians to not only study and understand calculus but to apply it to various problems.[2]

Adult life After graduating from Basel University Johann Bernoulli moved to teach differential equations. Later, in 1694, Johann Bernoulli married Dorothea Falkner and soon after accepted a position as the professor of mathematics at the University of Groningen. At the request of Johann Bernoulli’s father-in-law, Johann Bernoulli began the voyage back to his home town of Basel in 1705. Just after setting out on the journey he learned of his brother’s death to tuberculosis. Johann Bernoulli had planned on becoming the professor of Greek at Basel University upon returning but instead was able to take over as professor of mathematics, his older brother’s former position. As a student of Leibniz’s calculus, Johann Bernoulli sided with him in 1713 in the Newton–Leibniz debate over who deserved credit for the discovery of calculus. Johann Bernoulli defended Leibniz by showing that he had solved certain problems with his methods that Newton had failed to solve. However, due to his opposition to Newton and the study that vortex theory over Newton’s theory of gravitation which ultimately delayed acceptance of Newton’s theory in continental Europe.[3] In 1724 he entered a competition sponsored by the French Académie Royale des Sciences, which posed the question: What are the laws according to which a perfectly hard body, put into motion, moves another body of the same nature either at rest or in motion, and which it encounters either in a vacuum or in a plenum? In defending a view previously espoused by Leibniz he found himself postulating an infinite external force required to make the body elastic by overcoming the infinite internal force making the body hard. In consequence he was disqualified for the prize, which was won by Maclaurin. However, Bernoulli's paper was subsequently accepted in 1726 when the Académie considered papers regarding elastic bodies, for which the prize was awarded to Mazière. Bernoulli received an honourable mention in both competitions.

Private life Although Jakob and Johann worked together before Johann graduated from Basel University, shortly after this, the two developed a jealous and competitive relationship. Johann was jealous of Jakob's position and the two often attempted to outdo each other. After Jakob's death Johann's jealousy shifted toward his own talented son, Daniel. In 1738 the father–son duo nearly simultaneously published separate works on hydrodynamics. Johann Bernoulli attempted to take precedence over his son by purposely predating his work two years prior to his son’s. Johann married Dorothea Falkner, daughter of an Alderman of Basel. He was the father of Nicolaus II Bernoulli, Daniel Bernoulli and Johann II Bernoulli and uncle of Nicolaus I Bernoulli. The Bernoulli brothers often worked on the same problems, but not without friction. Their most bitter dispute concerned finding the equation for the path followed by a particle from one point to another in the shortest time, if the particle is acted upon by gravity alone, a problem originally discussed by Galileo. In 1697 Jakob offered a reward for its solution. Accepting the challenge, Johann proposed the cycloid, the path of a point on a moving wheel, pointing out at the same time the relation this curve bears to the path described by a ray of light passing through strata of variable density. A protracted, bitter dispute then arose when Jakob challenged the solution and proposed his own. The dispute marked the origin of a new discipline, the calculus of variations.

141

Johann Bernoulli

L'Hôpital controversy Bernoulli was hired by Guillaume de L'Hôpital to tutor him in mathematics. Bernoulli and L'Hôpital signed a contract which gave l'Hôpital the right to use Bernoulli’s discoveries as he pleased. L'Hôpital authored the first textbook on infinitesimal calculus, "Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes" in 1696, which mainly consisted of the work of Bernoulli, including what is now known as L'Hôpital's rule.[4][5][6] Subsequently, in letters to Leibniz, Varignon and others, Bernoulli complained that he had not received enough credit for his contributions, in spite of the fact that l'Hôpital acknowledged fully his debt in the preface of his book: "Je reconnais devoir beaucoup aux lumières de MM. Bernoulli, surtout à celles du jeune (Jean) présentement professeur à Groningue. Je me suis servi sans façon de leurs découvertes et de celles de M. Leibniz. C'est pourquoi je consens qu'ils en revendiquent tout ce qu'il leur plaira, me contentant de ce qu'ils voudront bien me laisser." "I recognize I owe much to Messrs. Bernoulli's insights, above all to the young (John), currently a professor in Groningue. I did unceremoniously use their discoveries, as well as those of Mr. Leibniz. For this reason I consent that they claim as much credit as they please, and will content myself with what they will agree to leave me."

References [1] [2] [3] [4] [5] [6]

A Short History of Mathematics, by V. Sanford, Houghton, Mifflin Company, (1958) The Bernoulli Family, by H. Bernhard, Doubleday, Page & Company, (1938) Johann and Jacob Bernoulli, by J.O. Fleckstein, Mathematical Association of America, (1949) The Story of a Number, by Eli Maor, Princeton University Press, Princeton, (1998) p. 116, ISBN 0-691-05854-7 The Mathematics of Great Amateurs, by Julian Lowell Coolidge, Dover, New York, (1963), pp. 154–163 A Source Book in Mathematics, 1200–1800, ed. D. J. Struck, Harvard University Press, Cambridge, MA, (1969), pp.312–316

External links • Johann Bernoulli (http://genealogy.math.ndsu.nodak.edu/id.php?id=53410) at the Mathematics Genealogy Project • O'Connor, John J.; Robertson, Edmund F., "Johann Bernoulli" (http://www-history.mcs.st-andrews.ac.uk/ Biographies/Bernoulli_Johann.html), MacTutor History of Mathematics archive, University of St Andrews. • Golba, Paul, " Bernoulli, Johan (http://www.shu.edu/projects/reals/history/bernoull.html)'" • " Johann Bernoulli (http://www.bernoulli.ag.vu/)" • Weisstein, Eric W., Bernoulli, Johann (1667–1748) (http://scienceworld.wolfram.com/biography/ BernoulliJohann.html) from ScienceWorld. • C. Truesdell The New Bernoulli Edition (http://links.jstor.org/sici?sici=0021-1753(195803)49:12. 0.CO;2-1) Isis, Vol. 49, No. 1. (Mar., 1958), pp. 54–62, discusses the strange agreement between Bernoulli and de l'Hôpital on pages 59–62.

142

Bernoulli differential equation

143

Bernoulli differential equation In mathematics, an ordinary differential equation of the form

is called a Bernoulli equation when n≠1, 0, which is named after Jakob Bernoulli, who discussed it in 1695 (Bernoulli 1695). Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.

Solution Dividing by

yields

A change of variables is made to transform into a linear first-order differential equation.

The substituted equation can be solved using the integrating factor

Example Consider the Bernoulli equation

We first notice that

is a solution. Division by

yields

Changing variables gives the equations

which can be solved using the integrating factor

Multiplying by

,

Note that left side is the derivative of

. Integrating both sides results in the equations

Bernoulli differential equation

The solution for

as well as

144

is

.

Verifying using MATLAB symbolic toolbox by running x = dsolve('Dy-2*y/x=-x^2*y^2','x') gives both solutions: 0 x^2/(x^5/5 + C1) also see a solution [1] by WolframAlpha, where the trivial solution

is missing.

References • Bernoulli, Jacob (1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. anni de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis", Acta Eruditorum. Cited in Hairer, Nørsett & Wanner (1993). • Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0.

External links • Bernoulli equation [2], PlanetMath.org. • Differential equation [3], PlanetMath.org. • Index of differential equations [4], PlanetMath.org.

References [1] [2] [3] [4]

http:/ / www. wolframalpha. com/ input/ ?i=y%27-2*y%2Fx%3D-x^2*y^2 http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=7032 http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=2629 http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=7023

Bernoulli distribution

145

Bernoulli distribution Bernoulli Parameters Support PMF

CDF

Mean Median

Mode

Variance Skewness Ex. kurtosis Entropy MGF CF PGF Fisher information

In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . So if X is a random variable with this distribution, we have:

A classical example of a Bernoulli experiment is a single toss of a coin. The coin might come up heads with probability p and tails with probability 1-p. The experiment is called fair if p=0.5, indicating the origin of the terminology in betting (the bet is fair if both possible outcomes have the same probability). The probability mass function f of this distribution is

This can also be expressed as

Bernoulli distribution

146

The expected value of a Bernoulli random variable X is

, and its variance is

Bernoulli distribution is a special case of the Binomial distribution with n = 1.[1] The kurtosis goes to infinity for high and low values of p, but for

the Bernoulli distribution has a lower

kurtosis than any other probability distribution, namely -2. The Bernoulli distributions for 0≤p≤1 form an exponential family. The maximum likelihood estimator of p based on a random sample is the sample mean.

Related distributions • If

are independent, identically distributed (i.i.d.) random variables, all Bernoulli distributed with

success probability p, then

(binomial distribution). The Bernoulli

distribution is simply . • The categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values. • The Beta distribution is the conjugate prior of the Bernoulli distribution. • The geometric distribution is the number of Bernoulli trials needed to get one success.

Notes [1] McCullagh and Nelder (1989), Section 4.2.2.

References • McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5. • Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9

External links • Hazewinkel, Michiel, ed. (2001), "Binomial distribution" (http://www.encyclopediaofmath.org/index. php?title=p/b016420), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Weisstein, Eric W., " Bernoulli Distribution (http://mathworld.wolfram.com/BernoulliDistribution.html)" from MathWorld.

Bernoulli number

147

Bernoulli number In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are B0 = 1, B1 = ±1⁄2, B2 = 1⁄6, B3 = 0, B4 = −1⁄30, B5 = 0, B6 = 1⁄42, B7 = 0, B8 = −1⁄30. If the convention B1=−1⁄2 is used, this sequence is also known as the first Bernoulli numbers (A027641 / A027642 in OEIS); with the convention B1=+1⁄2 is known as the second Bernoulli numbers (A164555 / A027642 in OEIS). Except for this one difference, the first and second Bernoulli numbers agree. Since Bn=0 for all odd n>1, and many formulas only involve even-index Bernoulli numbers, some authors write Bn instead of B2n. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712[1][2] in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the analytical engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine.[3] As a result, the Bernoulli numbers have the distinction of being the subject of the first computer program.

Sum of powers Bernoulli numbers feature prominently in the closed form expression of the sum of the m-th powers of the first n positive integers. For m, n ≥ 0 define

This expression can always be rewritten as a polynomial in n of degree m + 1. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:

where the convention B1 = +1/2 is used. (

denotes the binomial coefficient, m+1 choose k.)

For example, taking m to be 1 gives the triangular numbers 0, 1, 3, 6, ... (sequence A000217 in OEIS).

Taking m to be 2 gives the square pyramidal numbers 0, 1, 5, 14, ... (sequence A000330 in OEIS).

Some authors use the convention B1 = −1/2 and state Bernoulli's formula in this way:

Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sum of powers. Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005).

Bernoulli number

Definitions Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of the most useful ones are mentioned: • • • •

a recursive equation, an explicit formula, a generating function, an algorithmic description.

For the proof of the equivalence of the four approaches the reader is referred to mathematical expositions like (Ireland & Rosen 1990) or (Conway & Guy 1996). Unfortunately in the literature the definition is given in two variants: Despite the fact that Bernoulli defined B1 = 1/2 (now known as "second Bernoulli numbers"), some authors set B1 = −1/2 ("first Bernoulli numbers"). In order to prevent potential confusions both variants will be described here, side by side. Because these two definitions can be transformed simply by into the other, some formulae have this alternatingly (-1)n-term and others not depending on the context, but it is not possible to decide in favor of one of these definitions to be the correct or appropriate or natural one (for the abstract Bernoulli numbers).

Recursive definition The recursive equation is best introduced in a slightly more general form

This defines polynomials Bm in the variable n known as the Bernoulli polynomials. The recursion can also be viewed as defining rational numbers Bm(n) for all integers n ≥ 0, m ≥ 0. The expression 00 has to be interpreted as 1. The first and second Bernoulli numbers now follow by setting n = 0 (resulting in B1=−1⁄2, "first Bernoulli numbers") respectively n = 1 (resulting in B1=+1⁄2, "second Bernoulli numbers").

Here the expression [m = 0] has the value 1 if m = 0 and 0 otherwise (Iverson bracket). Whenever a confusion between the two kinds of definitions might arise it can be avoided by referring to the more general definition and by reintroducing the erased parameter: writing Bm(0) in the first case and Bm(1) in the second will unambiguously denote the value in question.

148

Bernoulli number

Explicit definition Starting again with a slightly more general formula

the choices n = 0 and n = 1 lead to

There is a widespread misinformation that no simple closed formulas for the Bernoulli numbers exist (Gould 1972). The last two equations show that this is not true. Moreover, already in 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers (Saalschütz 1893), usually giving some reference in the older literature.

Generating function The general formula for the generating function is

The choices n = 0 and n = 1 lead to

Algorithmic description Although the above recursive formula can be used for computation it is mainly used to establish the connection with the sum of powers because it is computationally expensive. However, both simple and high-end algorithms for computing Bernoulli numbers exist. Pointers to high-end algorithms are given the next section. A simple one is given in pseudocode below. Algorithm Akiyama–Tanigawa algorithm for second Bernoulli numbers Bn Input: Integer n≥0. Output: Second Bernoulli number Bn. for m from 0 by 1 to n do A[m] ← 1/(m+1) for j from m by -1 to 1 do A[j-1] ← j×(A[j-1] - A[j]) return A[0] (which is Bn) •

"←" is a shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item.

•

"return" terminates the algorithm and outputs the value that follows.

149

Bernoulli number

150

Efficient computation of Bernoulli numbers In some applications it is useful to be able to compute the Bernoulli numbers B0 through Bp − 3 modulo p, where p is a prime; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime. It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) p2 arithmetic operations would be required. Fortunately, faster methods have been developed (Buhler et al. 2001) which require only O(p (log p)2) operations (see big-O notation). David Harvey (Harvey 2008) describes an algorithm for computing Bernoulli numbers by computing Bn modulo p for many small primes p, and then reconstructing Bn via the Chinese Remainder Theorem. Harvey writes that the asymptotic time complexity of this algorithm is O(n2 log(n)2+eps) and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed Bn for n = 108. Harvey's implementation is included in Sage since version 3.1. Pavel Holoborodko (Holoborodko 2012) computed Bn for n = 2*108 using Harvey's implementation, which is a new record. Prior to that Bernd Kellner (Kellner 2002) computed Bn to full precision for n = 106 in December 2002 and Oleksandr Pavlyk (Pavlyk 2008) for n = 107 with 'Mathematica' in April 2008. Computer

Year

n

Digits*

J. Bernoulli

~1689 10

1

L. Euler

1748

30

8

J.C. Adams

1878

62

36

D.E. Knuth, T.J. Buckholtz 1967

1672

3330

G. Fee, S. Plouffe

1996

10000

27677

G. Fee, S. Plouffe

1996

100000

376755

B.C. Kellner

2002

1000000

4767529

O. Pavlyk

2008

10000000

57675260

D. Harvey

2008

100000000 676752569

P. Holoborodko

2012

200000000

• Digits is to be understood as the exponent of 10 when B(n) is written as a real in normalized scientific notation.

Different viewpoints and conventions The Bernoulli numbers can be regarded from four main viewpoints: • • • •

as standalone arithmetical objects, as combinatorial objects, as values of a sequence of certain polynomials, as values of the Riemann zeta function.

Each of these viewpoints leads to a set of more or less different conventions. Bernoulli numbers as standalone arithmetical objects. Associated sequence: 1/6, −1/30, 1/42, −1/30, … This is the viewpoint of Jakob Bernoulli. (See the cutout from his Ars Conjectandi, first edition, 1713). The Bernoulli numbers are understood as numbers, recursive in nature, invented to solve a certain arithmetical problem, the summation of powers, which is the paradigmatic application of the Bernoulli numbers. These are also the numbers appearing in the Taylor series expansion of tan(x) and tanh(x). It is misleading to call this viewpoint 'archaic'. For example Jean-Pierre Serre uses it in his highly acclaimed book A Course in Arithmetic which is a standard textbook used at many universities today.

Bernoulli number

151

Bernoulli numbers as combinatorial objects. Associated sequence: 1, +1/2, 1/6, 0, … This view focuses on the connection between Stirling numbers and Bernoulli numbers and arises naturally in the calculus of finite differences. In its most general and compact form this connection is summarized by the definition of the Stirling polynomials σn(x), formula (6.52) in Concrete Mathematics by Graham, Knuth and Patashnik.

In consequence Bn = n! σn(1) for n ≥ 0. Bernoulli numbers as values of a sequence of certain polynomials. Assuming the Bernoulli polynomials as already introduced the Bernoulli numbers can be defined in two different ways: • Bn = Bn(0). Associated sequence: 1, −1/2, 1/6, 0, … • Bn = Bn(1). Associated sequence: 1, +1/2, 1/6, 0, … The two definitions differ only in the sign of B1. The choice Bn = Bn(0) is the convention used in the Handbook of Mathematical Functions. Bernoulli numbers as values of the Riemann zeta function. Associated sequence: 1, +1/2, 1/6, 0, … Using this convention, the values of the Riemann zeta function satisfy nζ(1 − n) = −Bn for all integers n≥0. (See the paper of S. C. Woon; the expression nζ(1 − n) for n = 0 is to be understood as limx → 0 xζ(1 − x).)

Applications of the Bernoulli numbers The Bernoulli numbers as given by the Riemann zeta function.

Asymptotic analysis Arguably the most important application of the Bernoulli number in mathematics is their use in the Euler–MacLaurin formula. Assuming that ƒ is a sufficiently often differentiable function the Euler–MacLaurin formula can be written as [4]

This formulation assumes the convention B1 = −1/2. Using the convention B1 = 1/2 the formula becomes

Here ƒ(0) = ƒ which is a commonly used notation identifying the zero-th derivative of ƒ with ƒ. Moreover, let ƒ(−1) denote an antiderivative of ƒ. By the fundamental theorem of calculus,

Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula

Bernoulli number This form is for example the source for the important Euler–MacLaurin expansion of the zeta function (B1 = 1⁄2)

Here

denotes the rising factorial power.[5]

Bernoulli numbers are also frequently used in other kinds of asymptotic expansions. The following example is the classical Poincaré-type asymptotic expansion of the digamma function (again B1 = 1⁄2).

Taylor series of tan and tanh The Bernoulli numbers appear in the Taylor series expansion of the tangent and the hyperbolic tangent functions:

Use in topology The Kervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes of exotic (4n − 1)-spheres which bound parallelizable manifolds involves Bernoulli numbers. Let ESn be the number of such exotic spheres for n ≥ 2, then

The Hirzebruch signature theorem for the L genus of a smooth oriented closed manifold of dimension 4n also involves Bernoulli numbers.

Combinatorial definitions The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the inclusion-exclusion principle.

Connection with Worpitzky numbers The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function n! and the power function km is employed. The signless Worpitzky numbers are defined as

They can also be expressed through the Stirling numbers of the second kind

152

Bernoulli number

153

A Bernoulli number is then introduced as an inclusion-exclusion sum of Worpitzky numbers weighted by the sequence 1, 1/2, 1/3, …

This representation has B1 = 1/2. Worpitzky's representation of the Bernoulli number B0 =

1/1

B1 =

1/1 − 1/2

B2 =

1/1 − 3/2 + 2/3

B3 =

1/1 − 7/2 + 12/3 − 6/4

B4 =

1/1 − 15/2 + 50/3 − 60/4 + 24/5

B5 =

1/1 − 31/2 + 180/3 − 390/4 + 360/5 − 120/6

B6 = 1/1 − 63/2 + 602/3 − 2100/4 + 3360/5 − 2520/6 + 720/7

A second formula representing the Bernoulli numbers by the Worpitzky numbers is for n ≥ 1

Connection with Stirling numbers of the second kind If

where

denotes Stirling numbers of the second kind[6] then one has:

denotes the falling factorial.

If one defines the Bernoulli polynomials

where

for

as[7]:

are the Bernoulli numbers.

Then after the following property of binomial coefficient:

one has,

One also has following for Bernoulli polynomials,[7]

The coefficient of j in

is

Comparing the coefficient of j in the two expressions of Bernoulli polynomials, one has:

Bernoulli number

154

(resulting in B1=1/2) which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.[8][9][10]

Connection with Stirling numbers of the first kind The two main formulas relating the unsigned Stirling numbers of the first kind

to the Bernoulli numbers (with

B1 = 1/2) are

and the inversion of this sum (for n ≥ 0, m ≥ 0)

Here the number An,m are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table. Akiyama–Tanigawa number n\m

0

1

2

3

4

0

1

1/2

1/3

1/4 1/5

1

1/2

1/3

1/4

1/5

...

2

1/6

1/6

3/20

...

...

3

0

1/30

...

...

...

4

−1/30

...

...

...

...

The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above.

Connection with Eulerian numbers There are formulas connecting Eulerian numbers

to Bernoulli numbers:

Both formulas are valid for n ≥ 0 if B1 is set to ½. If B1 is set to −½ they are valid only for n ≥ 1 and n ≥ 2 respectively.

Bernoulli number

155

Connection with Balmer series A link between Bernoulli numbers and Balmer series could be seen in sequence A191567 in OEIS.

Representation of the second Bernoulli numbers See A191302 in OEIS. The number are not reduced. Then the columns are easy to find, the denominators being A190339. Representation of the second Bernoulli numbers B0

=

1 = 2/2

B1

=

1/2

B2

=

1/2 − 2/6

B3

=

1/2 − 3/6

B4

=

1/2 − 4/6 +  2/15

B5

=

1/2 − 5/6 +  5/15

B6

=

1/2 − 6/6 +  9/15 −   8/105

B7

=

1/2 − 7/6 + 14/15 − 28/105

A binary tree representation The Stirling polynomials σn(x) are related to the Bernoulli numbers by Bn = n!σn(1). S. C. Woon (Woon 1997) described an algorithm to compute σn(1) as a binary tree. Woon's tree for σn(1)

Woon's recursive algorithm (for n ≥ 1) starts by assigning to the root node N = [1,2]. Given a node N = [a1,a2,..., ak] of the tree, the left child of the node is L(N) = [−a1,a2 + 1, a3, ..., ak] and the right child R(N) = [a1,2, a2, ..., ak]. A node N = [a1,a2,..., ak] is written as ±[a2,..., ak] in the initial part of the tree represented above with ± denoting the sign of a1. Given a node N the factorial of N is defined as

Restricted to the nodes N of a fixed tree-level n the sum of 1/N! is σ

n(1), thus

Bernoulli number

For example B1 = 1!(1/2!), B2 = 2!(−1/3! + 1/(2!2!)), B3 = 3!(1/4! − 1/(2!3!) − 1/(3!2!) + 1/(2!2!2!)).

Asymptotic approximation Leonhard Euler expressed the Bernoulli numbers in terms of the Riemann zeta function as

It then follows from the Stirling formula that, as n goes to infinity,

Including more terms from the zeta series yields a better approximation, as does factoring in the asymptotic series in Stirling's approximation.

Integral representation and continuation The integral

has as special values b(2n) = B2n for n > 0. The integral might be considered as a continuation of the Bernoulli numbers to the complex plane and this was indeed suggested by Peter Luschny in 2004. For example b(3) = (3/2)ζ(3)Π−3Ι and b(5) = −(15/2) ζ(5) Π −5Ι. Here ζ(n) denotes the Riemann zeta function and Ι the imaginary unit. It is remarkable that already Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated

Euler's values are unsigned and real, but obviously his aim was to find a meaningful way to define the Bernoulli numbers at the odd integers n > 1.

The relation to the Euler numbers and π The Euler numbers are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers E2n are in magnitude approximately (2/π)(42n − 22n) times larger than the Bernoulli numbers B2n. In consequence:

This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers. In fact π could be computed from these rational approximations. Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since for n odd Bn = En = 0 (with the exception B1), it suffices to consider the case when n is even.

156

Bernoulli number

These conversion formulas express an inverse relation between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to π. These numbers are defined for n > 1 as

and S1 = 1 by convention (Elkies 2003). The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by Leonhard Euler in a landmark paper (Euler 1735) ‘De summis serierum reciprocarum’ (On the sums of series of reciprocals) and has fascinated mathematicians ever since. The first few of these numbers are

The Bernoulli numbers and Euler numbers are best understood as special views of these numbers, selected from the sequence Sn and scaled for use in special applications.

The expression [n even] has the value 1 if n is even and 0 otherwise (Iverson bracket). These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of Rn = 2Sn / Sn+1 when n is even. The Rn are rational approximations to π and two successive terms always enclose the true value of π. Beginning with n = 1 the sequence starts

These rational numbers also appear in the last paragraph of Euler's paper cited above.

An algorithmic view: the Seidel triangle The sequence Sn has another unexpected yet important property: The denominators of Sn divide the factorial (n − 1)!. In other words: the numbers Tn = Sn(n − 1)! are integers. Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as

These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers En are given immediately by T2n + 1 and the Bernoulli numbers B2n are obtained from T2n by some easy shifting, avoiding rational arithmetic. What remains is to find a convenient way to compute the numbers Tn. However, already in 1877 Philipp Ludwig von Seidel (Seidel 1877) published an ingenious algorithm which makes it extremely simple to calculate Tn.

157

Bernoulli number

158

Seidel's algorithm for Tn

[begin] Start by putting 1 in row 0 and let k denote the number of the row currently being filled. If k is odd, then put the number on the left end of the row k − 1 in the first position of the row k, and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper. At the end of the row duplicate the last number. If k is even, proceed similar in the other direction. [end] Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont (Dumont 1981)) and was rediscovered several times thereafter. Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz (Knuth & Buckholtz 1967) gave a recurrence equation for the numbers T2n and recommended this method for computing B2n and E2n ‘on electronic computers using only simple operations on integers’. V. I. Arnold rediscovered Seidel's algorithm in (Arnold 1991) and later Millar, Sloane and Young popularized Seidel's algorithm under the name boustrophedon transform.

A combinatorial view: alternating permutations Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis (André 1879) & (André 1881). Looking at the first terms of the Taylor expansion of the trigonometric functions tan x and sec x André made a startling discovery.

The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansion of tan x + sec x has as coefficients the rational numbers Sn.

André then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers).

Bernoulli number

159

Related sequences The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers: B0 = 1, B1 = 0, B2 = 1/6, B3 = 0, B4 = -1/30, A176327 / A027642 in OEIS. Via the second row of its inverse Akiyama-Tanigawa transform (sequence A177427 in OEIS), they lead to Balmer series A061037 / A061038.

A companion to the second Bernoulli numbers See A190339. These numbers are the eigensequence of the first kind. A191754 / A192366 = 0, 1/2, 1/2, 1/3, 1/6, 1/15, 1/30, 1/35, 1/70, -1/105, -1/210, 41/1155, 41/2310, -589/5005, -589/10010 ...

Arithmetical properties of the Bernoulli numbers The Bernoulli numbers can be expressed in terms of the Riemann zeta function as Bn = − nζ(1 − n) for integers n ≥ 0 provided for n = 0 and n = 1 the expression − nζ(1 − n) is understood as the limiting value and the convention B1 = 1/2 is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates that p is a prime number if and only if pBp−1 is congruent to −1 modulo p. Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.

The Kummer theorems The Bernoulli numbers are related to Fermat's last theorem (FLT) by Kummer's theorem (Kummer 1850), which says: If the odd prime p does not divide any of the numerators of the Bernoulli numbers B2, B4, ..., Bp−3 then xp + yp + zp = 0 has no solutions in non-zero integers. Prime numbers with this property are called regular primes. Another classical result of Kummer (Kummer 1851) are the following congruences. Let p be an odd prime and b an even number such that p − 1 does not divide b. Then for any non-negative integer k

A generalization of these congruences goes by the name of p-adic continuity.

p-adic continuity If b, m and n are positive integers such that m and n are not divisible by p − 1 and

, then

Since Bn = —n ζ(1 — n), this can also be written where u = 1 − m and v = 1 − n, so that u and v are nonpositive and not congruent to 1 modulo p − 1. This tells us that the Riemann zeta function, with 1 − p−s taken out of the Euler product formula, is continuous in the p-adic numbers on odd negative integers congruent modulo p − 1 to a particular , and so can be extended to a continuous function ζp(s) for all p-adic integers

, the p-adic zeta function.

Bernoulli number

Ramanujan's congruences The following relations, due to Ramanujan, provide a method for calculating Bernoulli numbers that is more efficient than the one given by their original recursive definition:

Von Staudt–Clausen theorem The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt (von Staudt 1840) and Thomas Clausen (Clausen 1840) independently in 1840. The theorem states that for every n > 0,

is an integer. The sum extends over all primes p for which p − 1 divides 2n. A consequence of this is that the denominator of B2n is given by the product of all primes p for which p − 1 divides 2n. In particular, these denominators are square-free and divisible by 6.

Why do the odd Bernoulli numbers vanish? The sum

can be evaluated for negative values of the index n. Doing so will show that it is an odd function for even values of k, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that B2k+1−m is 0 for m odd and greater than 1; and that the term for B1 is cancelled by the subtraction. The von Staudt Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question (valid for n > 1). From the von Staudt Clausen theorem it is known that for odd n > 1 the number 2Bn is an integer. This seems trivial if one knows beforehand that in this case Bn = 0. However, by applying Worpitzky's representation one gets

as a sum of integers, which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let Sn,m be the number of surjective maps from {1, 2, ..., n} to {1, 2, ..., m}, then . The last equation can only hold if

This equation can be proved by induction. The first two examples of this equation are n = 4:  2 + 8 = 7 + 3, n = 6:  2 + 120 + 144 = 31 + 195 + 40. Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers.

160

Bernoulli number

161

A restatement of the Riemann hypothesis The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the Riemann hypothesis (RH) which uses only the Bernoulli number. In fact Marcel Riesz (Riesz 1916) proved that the RH is equivalent to the following assertion: For every ε > 1/4 there exists a constant Cε > 0 (depending on ε) such that |R(x)|  3 bodies by Qiudong Wang in the 1990s.

Work on relativity Local time Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "luminiferous aether"), could be synchronised. At the same time Dutch theorist Hendrik Lorentz was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time"

Marie Curie and Poincaré talk at the 1911 Solvay [18] Conference.

and

introduced the hypothesis of length contraction to explain the failure of optical and electrical experiments to detect motion relative to the aether (see Michelson–Morley experiment).[19] Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In The Measure of Time (1898), Poincaré said, " A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that scientists have to set the

Henri Poincaré

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constancy of the speed of light as a postulate to give physical theories the simplest form.[20] Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame.[21] Principle of relativity and Lorentz transformations He discussed the "principle of relative motion" in two papers in 1900[21][22] and named it the principle of relativity in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest.[23] In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance." In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz.[24] In a second letter to Lorentz, Poincaré gave his own reason why Lorentz's time dilation factor was indeed correct after all: it was necessary to make the Lorentz transformation form a group and gave what is now known as the relativistic velocity-addition law.[25] Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote:[26]

“

The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which I will call by the name of Lorentz) of the form:

and showed that the arbitrary function

”

must be unity for all

(Lorentz had set

by a different

argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination is invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing [27]

imaginary coordinate, and he used an early form of four-vectors.

as a fourth

Poincaré expressed a disinterest in a

four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit.[28] So it was Hermann Minkowski who worked out the consequences of this notion in 1907. Mass–energy relation Like others before, Poincaré (1900) discovered a relation between mass and electromagnetic energy. While studying the conflict between the action/reaction principle and Lorentz ether theory, he tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included.[21] He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. Poincaré concluded that the electromagnetic field energy of an electromagnetic wave behaves like a fictitious fluid ("fluide fictif") with a mass density of E/c2. If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible—it's neither created or destroyed—then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions. However, Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid. Poincaré performed a Lorentz boost (to order v/c) to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow perpetual motion, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore he argued that also in this case there has to be another compensating mechanism in the ether.

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Poincaré himself came back to this topic in his St. Louis lecture (1904).[23] This time (and later also in 1908) he rejected[29] the possibility that energy carries mass and criticized the ether solution to compensate the above mentioned problems: The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy. [..] Shall we say that the space which separates the oscillator from the receiver and which the disturbance must traverse in passing from one to the other, is not empty, but is filled not only with ether, but with air, or even in inter-planetary space with some subtile, yet ponderable fluid; that this matter receives the shock, as does the receiver, at the moment the energy reaches it, and recoils, when the disturbance leaves it? That would save Newton's principle, but it is not true. If the energy during its propagation remained always attached to some material substratum, this matter would carry the light along with it and Fizeau has shown, at least for the air, that there is nothing of the kind. Michelson and Morley have since confirmed this. We might also suppose that the motions of matter proper were exactly compensated by those of the ether; but that would lead us to the same considerations as those made a moment ago. The principle, if thus interpreted, could explain anything, since whatever the visible motions we could imagine hypothetical motions to compensate them. But if it can explain anything, it will allow us to foretell nothing; it will not allow us to choose between the various possible hypotheses, since it explains everything in advance. It therefore becomes useless. He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass , Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of Madame Curie. It was Albert Einstein's concept of mass–energy equivalence (1905) that a body losing energy as radiation or heat was losing mass of amount m = E/c2 that resolved[30] Poincaré's paradox, without using any compensating mechanism within the ether.[31] The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. However, concerning Poincaré's solution of the Center of Gravity problem, Einstein noted that Poincaré's formulation and his own from 1906 were mathematically equivalent.[32] Algebra and number theory Poincaré first introduced the group theory in physics, in particular, he first studied the group of Lorentz transformations. He also made major contributions to the theory of discrete groups and their representations. He applied the Poincare group a theoretical approach, which became a major tool in many future studies – from topology to the theory of relativity. Poincaré first introduced the group theory in physics, in particular, he first studied the group of Lorentz transformations.[33] Topology The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of high-quality geometry. The term "topology" (instead of previously used Analysis situs). Some important concepts were introduced by Enrico Betti and Bernhard Riemann. But the foundation of this science, and quite elaborate for a space of any dimension was created by Poincare. His first article on this topic appeared in 1894.[34] His research in geometry led to abstract topological definition of homotopy and homology. He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers, the fundamental group, has

Topological transformation of the torus into a mug

Henri Poincaré proved a formula relating the number of edges, vertices and faces of n-dimensional polyhedron (the Euler–Poincaré), gave the first precise formulation of the intuitive notion of dimension.[35]

Astronomy and celestial mechanics Poincare published two classic monograph, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They entered the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions. Generalize the theory of Bruns (1887), Poincaré showed that the three-body principle is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area were the first major achievements in celestial mechanics since Isaac Newton.[36].. These include the idea of Poincaré, who later became the base for mathematical "chaos theory" (see, in particular, the Poincare recurrence theorem) and the general theory of dynamical systems. Poincare authored important works on astronomy for the equilibrium figures gravitating rotating fluid. He introduced the important concept of bifurcation points, proved the existence of equilibrium figures of non-ellipsoid, including ring-shaped and pear-shaped figures, their stability. For this discovery, the Poincaré received the Gold Medal of the Royal Astronomical Society (1900).[37] Poincaré and Einstein Einstein's first paper on relativity was published three months after Poincaré's short paper,[26] but before Poincaré's longer version.[27] It relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure (Einstein synchronisation) that Poincaré (1900) had described, but was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on special relativity. Einstein acknowledged Poincaré in the text of a lecture in 1921 called Geometrie und Erfahrung in connection with non-Euclidean geometry, but not in connection with special relativity. A few years before his death Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...."[38] Differential equations and mathematical physics After defending his doctoral thesis on the study of singular points of the system of differential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). In these articles, he built a new branch of mathematics, called "qualitative theory of differential equations." Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (saddle, focus, center, node), introduced the concept of a limit cycle and the loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the finite-difference equations, he created a new direction – the asymptotic analysis of the solutions. He applied all these achievements to study practical problems of mathematical physics and celestial mechanics, and the methods used were the basis of its topological works.[39][40]

360

Henri Poincaré

361

The singular points of the integral curves

Sadle

Focus

Node

Assessments Poincaré's work in the development of special relativity is well recognised,[30] though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.[41] Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time.[42][43][44][45][46] While this is the view of most historians, a minority go much further, such as E. T. Whittaker, who held that Poincaré and Lorentz were the true discoverers of Relativity.[47]

Character Poincaré's work habits have been compared to a bee flying from flower to flower. Poincaré was interested in the way his mind worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris. He linked his way of thinking to how he made several discoveries. The mathematician Darboux claimed he was un intuitif (intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. He did not care about being rigorous and disliked logic. He believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.

Toulouse's characterisation Poincaré's mental organisation was not only interesting to Poincaré himself but also to Édouard Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled Henri Poincaré (1910).[48][49] In it, he discussed Poincaré's regular schedule:

Photographic portrait of H. Poincaré by Henri Manuel

• He worked during the same times each day in short periods of time. He undertook mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7 p.m.. He would read articles in journals later in the evening.

Henri Poincaré • His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper. • He was ambidextrous and nearsighted. • His ability to visualise what he heard proved particularly useful when he attended lectures, since his eyesight was so poor that he could not see properly what the lecturer wrote on the blackboard. These abilities were offset to some extent by his shortcomings: • He was physically clumsy and artistically inept. • He was always in a rush and disliked going back for changes or corrections. • He never spent a long time on a problem since he believed that the subconscious would continue working on the problem while he consciously worked on another problem. In addition, Toulouse stated that most mathematicians worked from principles already established while Poincaré started from basic principles each time (O'Connor et al., 2002). His method of thinking is well summarised as: Habitué à négliger les détails et à ne regarder que les cimes, il passait de l'une à l'autre avec une promptitude surprenante et les faits qu'il découvrait se groupant d'eux-mêmes autour de leur centre étaient instantanément et automatiquement classés dans sa mémoire. (Accustomed to neglecting details and to looking only at mountain tops, he went from one peak to another with surprising rapidity, and the facts he discovered, clustering around their center, were instantly and automatically pigeonholed in his memory.) —Belliver (1956)

Attitude towards transfinite numbers Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured.[50] Poincaré said, "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."[51]

View on economics Poincaré saw mathematical work in economics and finance as peripheral. In 1900 Poincaré commented on Louis Bachelier's thesis "The Theory of Speculation", saying: "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." (Bernstein, 1996, pp. 199–200) Bachelier's work explained what was then the French government's pricing options on French Bonds and anticipated many of the pricing theories in financial markets used today.[52]

Views on education The choice can be explained only by the memory of the intuitive notion of what this combination took place, and if this memory is lacking, the choice will seems unjustified. However, for understand a theory, it is sufficient to note that the path is not followed cut by an obstacle, it is necessary to understand the reasons that made him be chosen. —POINCARÉ, Poincare and Euclides Roxo – The History of Relations between Philosophy of Mathemathics and Education, page 53 Therefore, the main goal of education, according to Poincare, is to develop some mathematical faculties of the mind, including intuition. If intuition is abolished how we would develop some faculties of the mind? That is the question with which Poincaré seeks to convince educators of the time, prompting them to care about an education focused on the development of reasoning student considering intuition primary factor for this to occur. Poincaré argues that some people who study Mathematics, fundamentally need their practical applications, for example, engineers. In this

362

Henri Poincaré case, you must learn to see and correct rapidly. The intuition becomes an essential requirement. Others become teachers. They should also cultivate intuition, since otherwise they would have a misconception of science, seeing by only one way and, furthermore, how they could develop in its students a quality that they themselves do not have? [53]

Honours Awards • • • • • • • •

Oscar II, King of Sweden's mathematical competition (1887) American Philosophical Society 1899 Gold Medal of the Royal Astronomical Society of London (1900) Bolyai Prize in 1905 Matteucci Medal 1905 French Academy of Sciences 1906 Académie Française 1909 Bruce Medal (1911)

Named after him • • • • • •

Institut Henri Poincaré (mathematics and theoretical physics center) Poincaré Prize (Mathematical Physics International Prize) Annales Henri Poincaré (Scientific Journal) Poincaré Seminar (nicknamed "Bourbaphy") The crater Poincaré on the Moon Asteroid 2021 Poincaré

Philosophy Poincaré had philosophical views opposite to those of Bertrand Russell and Gottlob Frege, who believed that mathematics was a branch of logic. Poincaré strongly disagreed, claiming that intuition was the life of mathematics. Poincaré gives an interesting point of view in his book Science and Hypothesis: For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rule. Poincaré believed that arithmetic is a synthetic science. He argued that Peano's axioms cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is a priori synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of Immanuel Kant (Kolak, 2001, Folina 1992). He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions. However, Poincaré did not share Kantian views in all branches of philosophy and mathematics. For example, in geometry, Poincaré believed that the structure of non-Euclidean space can be known analytically. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "conventionalism". Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics. He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.[54]

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Free will Poincaré's famous lectures before the Société de Psychologie in Paris (published as Science and Hypothesis, The Value of Science, and Science and Method) were cited by Jacques Hadamard as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation.[55] Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves chance. It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.[56] Poincaré's two stages—random combinations followed by selection—became the basis for Daniel Dennett's two-stage model of free will.[57]

References This article incorporates material from Jules Henri Poincaré on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Footnotes and primary sources [1] [2] [3] [4] [5]

"Poincaré’s Philosophy of Mathematics" (http:/ / www. iep. utm. edu/ poi-math/ #H3): entry in the Internet Encyclopedia of Philosophy. Poincaré pronunciation examples at Forvo (http:/ / www. forvo. com/ word/ poincaré/ ) Belliver, 1956 Sagaret, 1911 The Internet Encyclopedia of Philosophy (http:/ / www. utm. edu/ research/ iep/ p/ poincare. htm) Jules Henri Poincaré article by Mauro Murzi — Retrieved November 2006. [6] Joseph McCabe (1945). A Biographical Dictionary of Ancient, Medieval, and Modern Freethinkers (http:/ / www. infidels. org/ library/ historical/ joseph_mccabe/ dictionary. html). Haldeman-Julius Publications. . Retrieved 10 April 2012. "In his last words (published as Last Thoughts, 1913) he entirely rejects Christianity and believes in God only in the sense that he is the moral ideal. In effect he was an atheist." [7] Poincaré, Henri (January 1, 1913). Dernières Pensées (http:/ / www. ac-nancy-metz. fr/ enseign/ philo/ textesph/ Dernierespensees. pdf). p. 138. . Retrieved 10 April 2012. "Les dogmes des religions révélées ne sont pas les seuls à craindre. L'empreinte que le catholicisme a imprimée sur l'âme occidentale a été si profonde que bien des esprits à peine affranchis ont eu la nostalgie de la servitude et se sont efforcés de reconstituer des Eglises ; c'est ainsi que certaines écoles positivistes ne sont qu'un catholicisme sans Dieu. Auguste Comte lui-même rêvait de discipliner les âmes et certains de ses disciples, exagérant la pensée du maître, deviendraient bien vite des ennemis de la science s'ils étaient les plus forts." [8] O'Connor et al., 2002 [9] Carl, 1968 [10] D. Stillwell, Mathematics and its history. pages = 432–435 [11] Sageret, 1911 [12] see Galison 2003 [13] Lorentz, Poincaré et Einstein (http:/ / www. lexpress. fr/ idees/ tribunes/ dossier/ allegre/ dossier. asp?ida=430274) [14] Mathematics Genealogy Project (http:/ / www. genealogy. ams. org/ id. php?id=34227) North Dakota State University. Retrieved April 2008. [15] McCormmach, Russell (Spring, 1967), "Henri Poincaré and the Quantum Theory", Isis 58 (1): 37–55, doi:10.1086/350182 [16] Irons, F. E. (August, 2001), "Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms", American Journal of Physics 69 (8): 879–884, Bibcode 2001AmJPh..69..879I, doi:10.1119/1.1356056 [17] Diacu, F. (1996), "The solution of the n-body Problem", The Mathematical Intelligencer 18 (3): 66–70, doi:10.1007/BF03024313

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Henri Poincaré [18] Hsu, Jong-Ping; Hsu, Leonardo (2006), A broader view of relativity: general implications of Lorentz and Poincaré invariance (http:/ / books. google. com/ books?id=amLqckyrvUwC), 10, World Scientific, p. 37, ISBN 981-256-651-1, , Section A5a, p 37 (http:/ / books. google. com/ books?id=amLqckyrvUwC& pg=PA37) [19] Lorentz, H.A. (1895), Versuch einer theorie der electrischen und optischen erscheinungen in bewegten Kõrpern, Leiden: E.J. Brill [20] Poincaré, H. (1898), "The Measure of Time", Revue de métaphysique et de morale 6: 1–13 [21] Poincaré, H. (1900), "La théorie de Lorentz et le principe de réaction", Archives néerlandaises des sciences exactes et naturelles 5: 252–278. See also the English translation (http:/ / www. physicsinsights. org/ poincare-1900. pdf) [22] Poincaré, H. (1900), "Les relations entre la physique expérimentale et la physique mathématique" (http:/ / gallica. bnf. fr/ ark:/ 12148/ bpt6k17075r/ f1167. table), Revue générale des sciences pures et appliquées 11: 1163–1175, . Reprinted in "Science and Hypothesis", Ch. 9–10. [23] Poincaré, Henri (1904/6), "The Principles of Mathematical Physics", The Foundations of Science (The Value of Science), New York: Science Press, pp. 297–320 [24] Letter from Poincaré to Lorentz, Mai 1905 (http:/ / www. univ-nancy2. fr/ poincare/ chp/ text/ lorentz3. xml) [25] Letter from Poincaré to Lorentz, Mai 1905 (http:/ / www. univ-nancy2. fr/ poincare/ chp/ text/ lorentz4. xml) [26] Poincaré, H. (1905), "On the Dynamics of the Electron", Comptes Rendus 140: 1504–1508 (Wikisource translation) [27] Poincaré, H. (1906), "On the Dynamics of the Electron", Rendiconti del Circolo matematico Rendiconti del Circolo di Palermo 21: 129–176, doi:10.1007/BF03013466 (Wikisource translation) [28] Walter (2007), Secondary sources on relativity [29] Miller 1981, Secondary sources on relativity [30] Darrigol 2005, Secondary sources on relativity [31] Einstein, A. (1905b), "Ist die Trägheit eines Körpers von dessen Energieinhalt abhängig?" (http:/ / www. physik. uni-augsburg. de/ annalen/ history/ papers/ 1905_18_639-641. pdf), Annalen der Physik 18: 639–643, Bibcode 1905AnP...323..639E, doi:10.1002/andp.19053231314, . See also English translation (http:/ / www. fourmilab. ch/ etexts/ einstein/ specrel/ www). [32] Einstein, A. (1906), "Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie" (http:/ / www. physik. uni-augsburg. de/ annalen/ history/ papers/ 1906_20_627-633. pdf), Annalen der Physik 20 (8): 627–633, Bibcode 1906AnP...325..627E, doi:10.1002/andp.19063250814, [33] Poincare, Selected works in three volumes. page = 682 [34] D. Stillwell, Mathematics and its history. pages = 419–435 [35] PS Aleksandrov, Poincare and topology. pages = 27–81 [36] D. Stillwell, Mathematics and its history. pages = 434 [37] A. Kozenko , The theory of planetary figures, pages = 25–26 [38] Darrigol 2004, Secondary sources on relativity [39] Kolmogorov, AP Yushkevich, Mathematics of the 19th century Vol = 3. page = 283 ISBN 978-3764358457 [40] Kolmogorov, AP Yushkevich, Mathematics of the 19th century. pages = 162–174 [41] Galison 2003 and Kragh 1999, Secondary sources on relativity [42] Holton (1988), 196–206 [43] Hentschel (1990), 3–13 [44] Miller (1981), 216–217 [45] Darrigol (2005), 15–18 [46] Katzir (2005), 286–288 [47] Whittaker 1953, Secondary sources on relativity [48] Toulouse, E.,1910. Henri Poincaré [49] http:/ / books. google. com. mx/ books/ about/ Henri_Poincar%C3%A9_par_le_Dr_Toulouse. html?id=mpjWPQAACAAJ [50] Dauben 1979, p. 266. [51] Van Heijenoort, Jean (1967), From Frege to Gödel: a source book in mathematical logic, 1879–1931 (http:/ / books. google. com/ ?id=v4tBTBlU05sC& pg=PA190), Harvard University Press, p. 190, ISBN 0-674-32449-8, , p 190 (http:/ / books. google. com/ books?id=v4tBTBlU05sC& pg=PA190) [52] Dunbar, Nicholas (2000), Inventing money, JOHN WILEY & SONS, LTD, ISBN 0-471-49811-4 [53] Poincare and Euclides Roxo – The History of Relations between Philosophy of Mathematics and Education, pages = 52, 53, and 54 [54] Poincaré, Henri (2007), Science and Hypothesis (http:/ / books. google. com/ books?id=2QXqHaVbkgoC), Cosimo,Inc. Press, p. 50, ISBN 978-1-60206-505-5, , Extract of page 50 (http:/ / books. google. com/ books?id=2QXqHaVbkgoC& pg=PA50#v=onepage& q& f=false) [55] Hadamard, Jacques. An Essay On The Psychology Of Invention In The Mathematical Field. Princeton Univ Press (1949) [56] Science and Method, Chapter 3, Mathematical Discovery, 1914, pp.58 [57] Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on Mind and Psychology. The MIT Press, p.293

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Poincaré's writings in English translation Popular writings on the philosophy of science: • Poincaré, Henri (1902–1908), The Foundations of Science (http://www.archive.org/details/ foundationsscie01poingoog), New York: Science Press; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908). • 1913. Last Essays. (http://www.archive.org/details/mathematicsandsc001861mbp), New York: Dover reprint, 1963 On algebraic topology: • 1895. Analysis Situs (http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf). The first systematic study of topology. On celestial mechanics: • 1892–99. New Methods of Celestial Mechanics, 3 vols. English trans., 1967. ISBN 1-56396-117-2. • 1905–10. Lessons of Celestial Mechanics. On the philosophy of mathematics: • Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Univ. Press. Contains the following works by Poincaré: • 1894, "On the nature of mathematical reasoning," 972–81. • 1898, "On the foundations of geometry," 982–1011. • 1900, "Intuition and Logic in mathematics," 1012–20. • 1905–06, "Mathematics and Logic, I–III," 1021–70. • 1910, "On transfinite numbers," 1071–74.

General references • • • • • •

• • • • • • • •

Bell, Eric Temple, 1986. Men of Mathematics (reissue edition). Touchstone Books. ISBN 0-671-62818-6. Belliver, André, 1956. Henri Poincaré ou la vocation souveraine. Paris: Gallimard. Bernstein, Peter L, 1996. "Against the Gods: A Remarkable Story of Risk". (p. 199–200). John Wiley & Sons. Boyer, B. Carl, 1968. A History of Mathematics: Henri Poincaré, John Wiley & Sons. Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton Uni. Press. Dauben, Joseph (1993, 2004), "Georg Cantor and the Battle for Transfinite Set Theory" (http://www. acmsonline.org/journal/2004/Dauben-Cantor.pdf), Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA), pp. 1–22. Internet version published in Journal of the ACMS 2004. Folina, Janet, 1992. Poincare and the Philosophy of Mathematics. Macmillan, New York. Gray, Jeremy, 1986. Linear differential equations and group theory from Riemann to Poincaré, Birkhauser Jean Mawhin (October 2005), "Henri Poincaré. A Life in the Service of Science" (http://www.ams.org/notices/ 200509/comm-mawhin.pdf) (PDF), Notices of the AMS 52 (9): 1036–1044 Kolak, Daniel, 2001. Lovers of Wisdom, 2nd ed. Wadsworth. Murzi, 1998. "Henri Poincaré" (http://www.iep.utm.edu/p/poincare.htm). O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré" (http://www-history.mcs. st-andrews.ac.uk/Mathematicians/Poincare.html). University of St. Andrews, Scotland. Peterson, Ivars, 1995. Newton's Clock: Chaos in the Solar System (reissue edition). W H Freeman & Co. ISBN 0-7167-2724-2. Sageret, Jules, 1911. Henri Poincaré. Paris: Mercure de France.

• Toulouse, E.,1910. Henri Poincaré (http://quod.lib.umich.edu/cgi/t/text/ text-idx?c=umhistmath;idno=AAS9989.0001.001).—(Source biography in French) at University of Michigan Historic Math Collection.

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Secondary sources to work on relativity • Cuvaj, Camillo (1969), "Henri Poincaré's Mathematical Contributions to Relativity and the Poincaré Stresses", American Journal of Physics 36 (12): 1102–1113, Bibcode 1968AmJPh..36.1102C, doi:10.1119/1.1974373 • Darrigol, O. (1995), "Henri Poincaré's criticism of Fin De Siècle electrodynamics", Studies in History and Philosophy of Science 26 (1): 1–44, doi:10.1016/1355-2198(95)00003-C • Darrigol, O. (2000), Electrodynamics from Ampére to Einstein, Oxford: Clarendon Press, ISBN 0-19-850594-9 • Darrigol, O. (2004), "The Mystery of the Einstein–Poincaré Connection" (http://www.journals.uchicago.edu/ doi/full/10.1086/430652), Isis 95 (4): 614–626, doi:10.1086/430652, PMID 16011297 • Darrigol, O. (2005), "The Genesis of the theory of relativity" (http://www.bourbaphy.fr/darrigol2.pdf) (PDF), Séminaire Poincaré 1: 1–22 • Galison, P. (2003), Einstein's Clocks, Poincaré's Maps: Empires of Time, New York: W.W. Norton, ISBN 0-393-32604-7 • Giannetto, E. (1998), "The Rise of Special Relativity: Henri Poincaré's Works Before Einstein", Atti del XVIII congresso di storia della fisica e dell'astronomia: 171–207 • Giedymin, J. (1982), Science and Convention: Essays on Henri Poincaré's Philosophy of Science and the Conventionalist Tradition, Oxford: Pergamon Press, ISBN 0-08-025790-9 • Goldberg, S. (1967), "Henri Poincaré and Einstein's Theory of Relativity", American Journal of Physics 35 (10): 934–944, Bibcode 1967AmJPh..35..934G, doi:10.1119/1.1973643 • Goldberg, S. (1970), "Poincaré's silence and Einstein's relativity", British journal for the history of science 5: 73–84, doi:10.1017/S0007087400010633 • Holton, G. (1973/1988), "Poincaré and Relativity", Thematic Origins of Scientific Thought: Kepler to Einstein, Harvard University Press, ISBN 0-674-87747-0 • Katzir, S. (2005), "Poincaré's Relativistic Physics: Its Origins and Nature", Phys. Perspect. 7 (3): 268–292, Bibcode 2005PhP.....7..268K, doi:10.1007/s00016-004-0234-y • Keswani, G.H., Kilmister, C.W. (1983), "Intimations Of Relativity: Relativity Before Einstein" (http://osiris. sunderland.ac.uk/webedit/allweb/news/Philosophy_of_Science/PIRT2002/Intimations of Relativity.doc), Brit. J. Phil. Sci. 34 (4): 343–354, doi:10.1093/bjps/34.4.343 • Kragh, H. (1999), Quantum Generations: A History of Physics in the Twentieth Century, Princeton University Press, ISBN 0-691-09552-3 • Langevin, P. (1913), "L'œuvre d'Henri Poincaré: le physicien" (http://gallica.bnf.fr/ark:/12148/bpt6k111418/ f93.chemindefer), Revue de métaphysique et de morale 21: 703 • Macrossan, M. N. (1986), "A Note on Relativity Before Einstein" (http://espace.library.uq.edu.au/view. php?pid=UQ:9560), Brit. J. Phil. Sci. 37: 232–234 • Miller, A.I. (1973), "A study of Henri Poincaré's "Sur la Dynamique de l'Electron", Arch. Hist. Exact. Scis. 10 (3–5): 207–328, doi:10.1007/BF00412332 • Miller, A.I. (1981), Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2 • Miller, A.I. (1996), "Why did Poincaré not formulate special relativity in 1905?", in Jean-Louis Greffe, Gerhard Heinzmann, Kuno Lorenz, Henri Poincaré : science et philosophie, Berlin, pp. 69–100 • Schwartz, H. M. (1971), "Poincaré's Rendiconti Paper on Relativity. Part I", American Journal of Physics 39 (7): 1287–1294, Bibcode 1971AmJPh..39.1287S, doi:10.1119/1.1976641 • Schwartz, H. M. (1972), "Poincaré's Rendiconti Paper on Relativity. Part II", American Journal of Physics 40 (6): 862–872, Bibcode 1972AmJPh..40..862S, doi:10.1119/1.1986684

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Henri Poincaré • Schwartz, H. M. (1972), "Poincaré's Rendiconti Paper on Relativity. Part III", American Journal of Physics 40 (9): 1282–1287, Bibcode 1972AmJPh..40.1282S, doi:10.1119/1.1986815 • Scribner, C. (1964), "Henri Poincaré and the principle of relativity", American Journal of Physics 32 (9): 672–678, Bibcode 1964AmJPh..32..672S, doi:10.1119/1.1970936 • Walter, S. (2005), Henri Poincaré and the theory of relativity (http://www.univ-nancy2.fr/DepPhilo/walter/ papers/hpeinstein2005.htm), in Renn, J., , Albert Einstein, Chief Engineer of the Universe: 100 Authors for Einstein (Berlin: Wiley-VCH): 162–165 • Walter, S. (2007), Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910 (http:// www.univ-nancy2.fr/DepPhilo/walter/), in Renn, J., , The Genesis of General Relativity (Berlin: Springer) 3: 193–252 • Zahar, E. (2001), Poincare's Philosophy: From Conventionalism to Phenomenology, Chicago: Open Court Pub Co, ISBN 0-8126-9435-X Non-mainstream • Keswani, G.H., (1965), "Origin and Concept of Relativity, Part I", Brit. J. Phil. Sci. 15 (60): 286–306, doi:10.1093/bjps/XV.60.286 • Keswani, G.H., (1965), "Origin and Concept of Relativity, Part II", Brit. J. Phil. Sci. 16 (61): 19–32, doi:10.1093/bjps/XVI.61.19 • Keswani, G.H., (11966), "Origin and Concept of Relativity, Part III", Brit. J. Phil. Sci. 16 (64): 273–294, doi:10.1093/bjps/XVI.64.273 • Leveugle, J. (2004), La Relativité et Einstein, Planck, Hilbert—Histoire véridique de la Théorie de la Relativitén, Pars: L'Harmattan • Logunov, A.A. (2004), Henri Poincaré and relativity theory, Moscow: Nauka, arXiv:physics/0408077, Bibcode 2004physics...8077L, ISBN 5-02-033964-4 • Whittaker, E.T. (1953), "The Relativity Theory of Poincaré and Lorentz", A History of the Theories of Aether and Electricity: The Modern Theories 1900–1926, London: Nelson

External links • Works by Henri Poincaré (http://www.gutenberg.org/author/Henri+Poincaré) at Project Gutenberg • Free audio download of Poincaré's Science and Hypothesis (http://librivox.org/ science-and-hypothesis-by-henri-poincare/), from LibriVox. • Internet Encyclopedia of Philosophy: " Henri Poincare (http://www.utm.edu/research/iep/p/poincare. htm)"—by Mauro Murzi. • Henri Poincaré (http://genealogy.math.ndsu.nodak.edu/id.php?id=34227) at the Mathematics Genealogy Project • Henri Poincaré on Information Philosopher (http://www.informationphilosopher.com/solutions/scientists/ poincare/) • O'Connor, John J.; Robertson, Edmund F., "Henri Poincaré" (http://www-history.mcs.st-andrews.ac.uk/ Biographies/Poincare.html), MacTutor History of Mathematics archive, University of St Andrews. • A timeline of Poincaré's life (http://www.univ-nancy2.fr/ACERHP/documents/kronowww.html) University of Nancy (in French). • Bruce Medal page (http://phys-astro.sonoma.edu/brucemedalists/Poincare/index.html) • Collins, Graham P., " Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions, (http://www.sciam. com/print_version.cfm?articleID=0003848D-1C61-10C7-9C6183414B7F0000)" Scientific American, 9 June 2004.

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Henri Poincaré • BBC In Our Time, " Discussion of the Poincaré conjecture, (http://www.bbc.co.uk/radio4/history/inourtime/ inourtime.shtml)" 2 November 2006, hosted by Melvynn Bragg. See Internet Archive (http://web.archive.org/ web/*/http://www.bbc.co.uk/radio4/history/inourtime/inourtime.shtml) • Poincare Contemplates Copernicus (http://www.mathpages.com/home/kmath305/kmath305.htm) at MathPages • High Anxieties – The Mathematics of Chaos (http://www.youtube.com/user/ thedebtgeneration?feature=mhum#p/u/8/5pKrKdNclYs0) (2008) BBC documentary directed by David Malone looking at the influence of Poincaré's discoveries on 20th Century mathematics.

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Georg Cantor

370

Georg Cantor Georg Cantor

Born

Georg Ferdinand Ludwig Philipp Cantor March 3, 1845 Saint Petersburg, Russian Empire

Died

January 6, 1918 (aged 72) Halle, Province of Saxony, German Empire

Residence

Russian Empire (1845–1856), German Empire (1856–1918)

Nationality

German

Fields

Mathematics

Institutions

University of Halle

Alma mater

ETH Zurich, University of Berlin

Doctoral advisor

Ernst Kummer Karl Weierstrass

Doctoral students Alfred Barneck Known for

Set theory

Georg Ferdinand Ludwig Philipp Cantor (pron.: /ˈkæntɔr/ KAN-tor; German: [ɡeˈɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.[2] Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God [4] — on one occasion equating the theory of transfinite numbers with pantheism[5] — a proposition which Cantor vigorously rejected. The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."[7] Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing

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decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong".[8] Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries,[9] though some have explained these episodes as probable manifestations of a bipolar disorder.[10] The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.[11] It has been suggested that Cantor believed his theory of transfinite numbers had been communicated to him by God.[12] David Hilbert defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created."[13]

Life Youth and studies Cantor was born in 1845 in the western merchant colony in Saint Petersburg, Russia, and brought up in the city until he was eleven. Georg, the oldest of six children, was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm's brother) was the well-known musician and the soloist in the Russian empire in an imperial orchestra.[14] Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, Cantor entered the University of Zürich. After receiving a substantial inheritance upon his father's death in 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research.

Teacher and researcher In 1867, Cantor completed his dissertation, on number theory, at the University of Berlin. After teaching briefly in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle.[15] In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday. Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible.[16] Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,[17] perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.[18] Worse yet, Kronecker, a well-established figure

Georg Cantor within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle. In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor. In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.[19] Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta.[20] He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party: Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about Acta Mathematica.[21] Cantor suffered his first known bout of depression in 1884.[22] Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence: ... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness.[23] This crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of Elizabethan literature thinking there might be evidence that Francis Bacon wrote the plays attributed to Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.[24] Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84. He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung and chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time.

Late years After Cantor's 1884 hospitalization, there is no record that he was in any sanatorium again until 1899.[22] Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly (while Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics.[25] Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. (Konig is now remembered as having only pointed out that some sets cannot be well-ordered, in disagreement with Cantor.) Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated.[26] Although Ernst Zermelo

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Georg Cantor demonstrated less than a day later that König's proof had failed, Cantor remained shaken, even momentarily questioning God.[11] Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years.[27] He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the Deutsche Mathematiker–Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904. In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but this did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person. Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I.[28] The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.

Mathematical work Cantor's work between 1874 and 1884 is the origin of set theory.[29] Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle.[30] No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics. In one of his earliest papers, Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and uncountable sets (nondenumerable infinite sets).[31] Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter (aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). This notation is still in use today. The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his famous address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium.[32] The US philosopher Charles Sanders Peirce praised Cantor's set theory, and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor

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corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works.

Number theory, trigonometric series and ordinals Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the nth derived set Sn of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S1 as its set of zeros, where S1 is the set of limit points of S. If Sk+1 is the set of limit points of Sk, then he could construct a trigonometric series whose zeros are Sk+1. Because the sets Sk were closed, they contained their Limit points, and the intersection of the infinite decreasing sequence of sets S, S1, S2, S3,... formed a limit set, which we would now call Sω, and then he noticed that Sω would also have to have a set of limit points Sω+1, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ω, ω+1, ω+2, ... [33] Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics".[34] Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.[35]

Set theory The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article,[29] "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").[37] This article was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements).[38] Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof is more complex than the more elegant diagonal argument that he gave in 1891.[39] Cantor's article also contains a new method of constructing transcendental numbers. Transcendental numbers were first constructed by Joseph Liouville in 1844.[40]

An illustration of Cantor's diagonal argument for the existence of [36] uncountable sets. The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.

Cantor established these results using two constructions. His first construction shows how to write the real algebraic numbers[41] as a sequence a1, a2, a3, .... In other words, the real algebraic numbers are countable. Cantor starts his

Georg Cantor second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence — that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.[42] Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.[43] Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole.[44] Cantor also introduced the Cantor set during this period. The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types. In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem. Cantor wrote on the Goldbach conjecture in 1894. In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory.[45] The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schroeder theorem.

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Georg Cantor One-to-one correspondence Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor famously wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!")[46] The result that he found so astonishing has implications for geometry and the notion of dimension. A bijective function. In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence, and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.

This paper displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Weierstrass supported its publication.[47] Nevertheless, Cantor never again submitted anything to Crelle. Continuum hypothesis Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.[9] The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").[48] Paradoxes of set theory Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program.[49] In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.[11] In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set A, the cardinal number of the power set of A is strictly larger than the cardinal number of A (this fact is now known as Cantor's theorem). This paradox, together with Burali-Forti's, led Cantor to formulate a concept called limitation of size,[50] according to which the collection of all ordinals, or of all

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Georg Cantor sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as proper classes. One common view among mathematicians is that these paradoxes, together with Russell's paradox, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.[51]

Philosophy, religion, and Cantor's mathematics The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.[52] He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.[53] To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications — he identified the Absolute Infinite with God,[54] and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.[12] Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence.[55] Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.[6] Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set.[56] Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."[6] Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.[8] Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[4] In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".[57] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:[58] ... the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers.[59] Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism — and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs.[60]

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In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim,[61] as well as theologians such as Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with pantheism.[5] Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.[58] Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his famous assertion that "the essence of mathematics is its freedom."[62] These ideas parallel those of Edmund Husserl.[63] Meanwhile, Cantor himself was fiercely opposed to infinitesimals, describing them as both an "abomination" and "the cholera bacillus of mathematics". Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering: ... I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.[64] Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity.

Cantor's ancestry "Very little is known for sure about the origin and education of George Woldemar Cantor."[65] Cantor's paternal grandparents were from Copenhagen, and fled to Russia from the disruption of the Napoleonic Wars. There is very little direct information on his grandparents.[66] Cantor was sometimes called Jewish in his lifetime,[67] but has also variously been called Russian, German, and Danish as well. Jakob Cantor, Cantor's grandfather, gave his children Christian saints' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they converted to Christianity. Cantor's father, Georg Waldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. His mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating:

The title on the memorial plaque (in Russian): "In this building was born and lived from 1845 till 1854 the great mathematician and creator of set theory Georg Cantor", Vasilievsky Island, Saint-Petersburg.

Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigung der Hebräer sein, im socialen Leben sind mir Christen lieber ...[65] ("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians...") which could be read to imply that she was of Jewish ancestry.[68] There were documented statements, during the 1930s, that called this Jewish ancestry into question:

Georg Cantor More often [i.e., than the ancestry of the mother] the question has been discussed of whether Georg Cantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical Institute in Copenhagen from the year 1937 concerning his father: "It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completely without doubt was not a Jew ..."[65] It is also later said in the same document: Also efforts for a long time by the librarian Josef Fischer, one of the best experts on Jewish genealogy in Denmark, charged with identifying Jewish professors, that Georg Cantor was of Jewish descent, finished without result. [Something seems to be wrong with this sentence, but the meaning seems clear enough.] In Cantor's published works and also in his Nachlass there are no statements by himself which relate to a Jewish origin of his ancestors. There is to be sure in the Nachlass a copy of a letter of his brother Ludwig from 18 November 1869 to their mother with some unpleasant antisemitic statements, in which it is said among other things: ...[65] (the rest of the quote is finished by the very first quote above). In Men of Mathematics, Eric Temple Bell described Cantor as being "of pure Jewish descent on both sides," although both parents were baptized. In a 1971 article entitled "Towards a Biography of Georg Cantor," the British historian of mathematics Ivor Grattan-Guinness mentions (Annals of Science 27, pp. 345–391, 1971) that he was unable to find evidence of Jewish ancestry. (He also states that Cantor's wife, Vally Guttmann, was Jewish). In a letter written by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p. 306), Cantor states that his paternal grandparents were members of the Sephardic Jewish community of Copenhagen. Specifically, Cantor states in describing his father: "Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde..." ("He was born in Copenhagen of Jewish (lit: "Israelite") parents from the local Portuguese-Jewish community.")[69] In addition, Cantor's maternal great uncle,[70] a Hungarian violinist Josef Böhm, has been described as Jewish,[71] which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community.[72] In a letter to Bertrand Russell, Cantor described his ancestry and self-perception as follows: Neither my father nor my mother were of german blood, the first being a Dane, borne in Kopenhagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not a regular just Germain, for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with my father and mother and brothers and sister, eleven years old in the year 1856, into Germany.[73]

Historiography Until the 1970s, the chief academic publications on Cantor were two short monographs by Schönflies (1927) — largely the correspondence with Mittag-Leffler — and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by Eric Temple Bell's Men of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics"; and as "one of the worst".[74] Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics, and fills the picture with stereotypes. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell — including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents.[75] A critique of Bell's book is contained in Joseph Dauben's biography.[76]

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Notes [1] Grattan-Guinness 2000, p. 351 [2] The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources. [3] Dauben 2004, p. 1. [4] Dauben, 1977, p. 86; Dauben, 1979, pp. 120, 143. [5] Dauben, 1977, p. 102. [6] Dauben 1979, p. 266. [7] Dauben 2004, p. 1. See also Dauben 1977, p. 89 15n. [8] Rodych 2007 [9] Dauben 1979, p. 280:"...the tradition made popular by Arthur Moritz Schönflies blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression. [10] Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness as "cyclic manic-depression". [11] Dauben 1979, p. 248 [12] Dauben 2004, pp. 8, 11, 12-13. [13] Hilbert 1926, p. 170; see Reid 1996, p. 177 [14] ru: The musical encyclopedia (Музыкальная энциклопедия) (http:/ / dic. academic. ru/ dic. nsf/ enc_music/ 924/ Бём) [15] O'Connor, John J, and Robertson, Edmund F (1998). "Georg Ferdinand Ludwig Philipp Cantor" (http:/ / www-history. mcs. st-andrews. ac. uk/ Biographies/ Cantor. html). MacTutor History of Mathematics. . [16] Dauben 1979, p. 163. [17] Dauben 1979, p. 34. [18] Dauben 1977, p. 89 15n. [19] Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355. [20] Dauben 1979, p. 138. [21] Dauben 1979, p. 139. [22] Dauben 1979, p. 282. [23] Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884. [24] Dauben 1979, pp. 281–283. [25] Dauben 1979, p. 283. [26] For a discussion of König's paper see Dauben 1979, 248–250. For Cantor's reaction, see Dauben 1979, p. 248; 283. [27] Dauben 1979, p. 283–284. [28] Dauben 1979, p. 284. [29] Johnson 1972, p. 55. [30] This paragraph is a highly abbreviated summary of the impact of Cantor's lifetime of work. More details and references can be found later. [31] A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable". [32] Reid 1996, p. 177. [33] Cooke, Roger (1993), "Uniqueness of trigonometric series and descriptive set theory, 1870–1985", Archive for History of Exact Sciences 45 (4): 281, doi:10.1007/BF01886630. [34] Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science. doi:10.1007/s10699-011-9223-1 See link (http:/ / www. springerlink. com/ content/ tj7j2810n8223p43/ ) [35] Ehrlich, P. (2006) The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci. 60, no. 1, 1–121. [36] This follows closely the first part of Cantor's 1891 paper. [37] Cantor 1874. English translation: Ewald 1996, p. 840–843. [38] For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous—see Moore 1995, p. 114. [39] For this, and more information on the mathematical importance of Cantor's work on set theory, see e.g., Suppes 1972. [40] Education.fr (http:/ / www. bibnum. education. fr/ mathematiques/ propos-de-lexistence-des-nombres-transcendants) [41] The real algebraic numbers are the real roots of polynomial equations with integer coefficients. [42] For more details on Cantor's article, see Cantor's first uncountability proof and Gray 1995 (http:/ / mathdl. maa. org/ mathDL/ 22/ ?pa=content& sa=viewDocument& nodeId=2907). Gray 1995 (p. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number. [43] Cantor's construction starts with the set of transcendentals T and removes a countable subset {tn} (for example, tn = e / n). Call this set T’. Then T = T’ ∪ {tn} = T’ ∪ {t2n-1} ∪ {t2n}. The set of reals R = T ∪ {an} = T’ ∪ {tn} ∪ {an} where an is the sequence of real algebraic numbers. So both T and R are the union of three disjoint sets: T’ and two countable sets. A one-to-one correspondence between T and R is

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Georg Cantor given by the function: f(t) = t if t ∈ T’, f(t2n-1) = tn, and f(t2n) = an. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (Cantor 1932, p. 142). [44] Dauben 1977, p. 89. [45] The English translation is Cantor 1955. [46] Wallace 2003, p. 259. [47] Dauben 1979, p. 69; 324 63n. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it. [48] Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is W. Hugh Woodin. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2. [49] Dauben 1979, pp. 240–270; see especially pp. 241, 259. [50] Hallett 1986. [51] Weir 1998, p. 766: "...it may well be seriously mistaken to think of Cantor's Mengenlehre [set theory] as naive..." [52] Dauben 1979, p. 295. [53] Dauben, 1979, p. 120. [54] Hallett 1986, p. 13. Compare to the writings of Thomas Aquinas. [55] Dauben 1979, p. 225 [56] Snapper 1979, p. 3 [57] Davenport 1997, p.3 [58] Dauben, 1977, p. 85. [59] Cantor 1932, p. 404. Translation in Dauben 1977, p. 95. [60] Dauben 1979, p. 296. [61] Dauben, 1979, p. 144. [62] Dauben 1977 pp. 91–93. [63] On Cantor, Husserl, and Gottlob Frege, see Hill and Rosado Haddock (2000). [64] Dauben 1979, p. 96. [65] Purkert and Ilgauds 1987, p. 15. [66] E.g., Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement. [67] For example, Jewish Encyclopedia, art. "Cantor, Georg"; Jewish Year Book 1896–97, "List of Jewish Celebrities of the Nineteenth Century", p. 119; this list has a star against people with one Jewish parent, but Cantor is not starred. [68] For more information, see: Dauben 1979, p. 1 and notes; Grattan-Guinness 1971, pp. 350–352 and notes; Purkert and Ilgauds 1985; the letter is from Aczel 2000, pp. 93–94, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included. [69] Paul Tannery, Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, 1934, p. 306 [70] Georg Cantor: his mathematics and philosophy of the infinite, Princeton University Press, 1990, By Joseph Warren Dauben, page 274 [71] Modern Jews and their musical agendas, Ezra Mendelsohn, Oxford University Press, 1993, page 9 [72] Ismerjükoket?: zsidó származású nevezetes magyarok arcképcsarnoka, István Reményi Gyenes Ex Libris, (Budapest 1997), pages 132–133 [73] Bertrand Russell, Autobiography, vol. I, p. 229. In English in the original; italics also as in the original. [74] Grattan-Guinness 1971, p. 350. [75] Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p. 1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.) [76] Georg Cantor, Joseph Dauben, Harvard University Press (1979). Reprinted as a paperback by Princeton University (1990).

References Older sources on Cantor's life should be treated with caution. See Historiography section above. Primary literature in English • Cantor, Georg (1955) [1915], Philip Jourdain, ed., Contributions to the Founding of the Theory of Transfinite Numbers (http://www.archive.org/details/contributionstot003626mbp), New York: Dover, ISBN 978-0-486-60045-1. • Ewald, William B., ed. (1996), From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, New York: Oxford University Press, ISBN 978-0-19-853271-2. Primary literature in German

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Georg Cantor • Cantor, Georg (1874), "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen" (http://bolyai. cs.elte.hu/~badam/matbsc/11o/cantor1874.pdf), Journal für die Reine und Angewandte Mathematik 77: 258–262. • Cantor, Georg (1932), Ernst Zermelo, ed. (PDF), Gesammelte Abhandlungen mathematischen und philosophischen inhalts (http://philosophons.free.fr/philosophes/cantor1932.pdf). Almost everything that Cantor wrote. • Hilbert, David (1926), "Über das Unendliche" (http://www.digizeitschriften.de/main/dms/img/ ?PPN=GDZPPN002270641), Mathematische Annalen 95: 161–190, doi:10.1007/BF01206605. Secondary literature • Aczel, Amir D. (2000), The mystery of the Aleph: Mathematics, the Kabbala, and the Human Mind, New York: Four Walls Eight Windows Publishing. ISBN 0-7607-7778-0. A popular treatment of infinity, in which Cantor is frequently mentioned. • Dauben, Joseph W. (1977), "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite", Journal of the History of Ideas 38 (1): 85–108. • Dauben, Joseph W. (1979), Georg Cantor: his mathematics and philosophy of the infinite, Boston: Harvard University Press. The definitive biography to date. ISBN 978-0-691-02447-9 • Dauben, Joseph W. (June 1983), "Georg Cantor and the Origins of Transfinite Set Theory", Scientific American 248 (6): 122–131 • Dauben, Joseph (1993, 2004), " Georg Cantor and the Battle for Transfinite Set Theory (http://www. acmsonline.org/Dauben93.htm)", Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA), pp. 1–22. Internet version published in Journal of the ACMS 2004. • Davenport, Anne A. (1997), "The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century", Isis 88 (2): 263–295. • Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought, Basel, Switzerland: Birkhäuser. ISBN 3-7643-8349-6 Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory. • Grattan-Guinness, Ivor (1971), "Towards a Biography of Georg Cantor", Annals of Science 27: 345–391. • Grattan-Guinness, Ivor (2000), The Search for Mathematical Roots: 1870–1940, Princeton University Press. ISBN 978-0-691-05858-0 • Gray, Robert (1994), " Georg Cantor and Transcendental Numbers (http://mathdl.maa.org/mathDL/22/ ?pa=content&sa=viewDocument&nodeId=2907)", American Mathematical Monthly 101: 819–832. • Hallett, Michael (1986), Cantorian Set Theory and Limitation of Size, New York: Oxford University Press. ISBN 0-19-853283-0 • Halmos, Paul (1998, 1960), Naive Set Theory, New York & Berlin: Springer. ISBN 3-540-90092-6 • Hill, C. O.; Rosado Haddock, G. E. (2000), Husserl or Frege? Meaning, Objectivity, and Mathematics, Chicago: Open Court. ISBN 0-8126-9538-0 Three chapters and 18 index entries on Cantor. • Johnson, Phillip E. (1972), "The Genesis and Development of Set Theory", The Two-Year College Mathematics Journal 3 (1): 55–62. • Meschkowski, Herbert (1983), Georg Cantor, Leben, Werk und Wirkung (George Cantor, Life, Work and Influence, in German), Wieveg, Braunschweig • Moore, A.W. (April 1995), "A brief history of infinity", Scientific American 272 (4): 112–116. • Penrose, Roger (2004), The Road to Reality, Alfred A. Knopf. ISBN 0-679-77631-1 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist. • Purkert, Walter; Ilgauds, Hans Joachim (1985), Georg Cantor: 1845–1918, Birkhäuser. ISBN 0-8176-1770-1 • Reid, Constance (1996), Hilbert, New York: Springer-Verlag. ISBN 0-387-04999-1 • Rucker, Rudy (2005, 1982), Infinity and the Mind, Princeton University Press. ISBN 0-553-25531-2 Deals with similar topics to Aczel, but in more depth.

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Georg Cantor • Rodych, Victor (2007), " Wittgenstein's Philosophy of Mathematics (http://plato.stanford.edu/entries/ wittgenstein-mathematics/)", in Edward N. Zalta, The Stanford Encyclopedia of Philosophy. • Snapper, Ernst (1979), " The Three Crises in Mathematics: Logicism, Intuitionism and Formalism (http://math. boisestate.edu/~tconklin/MATH547/Main/Exhibits/Three Crises in Math A.pdf)", Mathematics Magazine 524: 207–216. • Suppes, Patrick (1972, 1960), Axiomatic Set Theory, New York: Dover. ISBN 0-486-61630-4 Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics. • Wallace, David Foster (2003), Everything and More: A Compact History of Infinity, New York: W.W. Norton and Company. ISBN 0-393-00338-8 • Weir, Alan (1998), "Naive Set Theory is Innocent!", Mind 107 (428): 763–798.

External links • O'Connor, John J.; Robertson, Edmund F., "Georg Cantor" (http://www-history.mcs.st-andrews.ac.uk/ Biographies/Cantor.html), MacTutor History of Mathematics archive, University of St Andrews. • O'Connor, John J.; Robertson, Edmund F., "A history of set theory" (http://www-history.mcs.st-andrews.ac. uk/HistTopics/Beginnings_of_set_theory.html), MacTutor History of Mathematics archive, University of St Andrews. Mainly devoted to Cantor's accomplishment. • Georg Cantor (http://genealogy.math.ndsu.nodak.edu/id.php?id=29561) at the Mathematics Genealogy Project • Stanford Encyclopedia of Philosophy: Set theory (http://plato.stanford.edu/entries/set-theory/) by Thomas Jech. • Grammar school Georg-Cantor Halle (Saale): Georg-Cantor-Gynmasium Halle (http://www.cantor-gymnasium. de)

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384

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Article Sources and Contributors Singing Badger, The Thing That Should Not Be, The last goodnight, The undertow, TheSun, Theda, Thedjatclubrock, Thehelpfulone, Thejerm, Theresa knott, Thetwistedblue, ThiagoRuiz, Thingg, This user has left wikipedia, Thomas Gilling, ThomasK, Tiddly Tom, Tide rolls, TigerShark, Timboslicej23, Timur lenk, Timwi, Tiramisoo, Tirral, TitaniumDreads, Tkuvho, Tmichael, Tom harrison, Tomasboij, Tomaxer, Tomcat7, Tommy2010, Tomos, Tomwsulcer, Tonyfaull, Top.Squark, Topazinutah, Trekker114, Tresoldi, Trust Is All You Need, Truthanado, Tsop, Tucapl, Tunheim, TurtleTurtle, Tuxedo junction, Twice25, Twipley, TwoTwoHello, Twsx, Tycho, Tyrol5, Uglinessman, Ukexpat, Ulric1313, Uncle Dick, Uncle G, Unitanode, Unixer, Unschool, Uriah923, Utcursch, Vambiant, Vanished User 1004, Vanished User 8a9b4725f8376, Vanished user 39948282, Vary, Versus22, Vikaszt, VirtualDelight, Vulcanparty, Vwilmot, Wahabijaz, Wangi, Washburnmav, Wegetthepussy, WestJohnson, Wester, Where, Whisky drinker, Whoistheroach, Whosyourjudas, Wik, Wiki alf, WikiDao, WikiFew, WikiPedant, Wikidea, Wikididact, Wikieditor06, Wikilolli9, WikipedianMarlith, Wikipelli, Wikiwikifast, Will Beback, Will Beback Auto, Wimt, Windsok, WireAlbatross, Wknight94, Wolfdog, Womble, Woohookitty, Workinggirl007, XJamRastafire, Xaosflux, Xastic, Xcrime94, Xenfreak, Xipirho, Y control, Yamamoto Ichiro, Yawdamper, YellowMonkey, Yurik, Zachlipton, Zaffes, Zawersh, ZeroEgo, Zlatno Pile, Zonath, Zsinj, Zzyzx11, Александър, 2271 ,‫ לערי ריינהארט‬anonymous edits Pierre de Fermat  Source: http://en.wikipedia.org/w/index.php?oldid=531609185  Contributors: *drew, 15turnsm, 28421u2232nfenfcenc, 65.68.87.xxx, 9258fahsflkh917fas, ACEOREVIVED, Abductive, Abrech, Acroterion, AdamSmithee, Adashiel, Akasseb, Albedo, Alberto da Calvairate, Alexander101, Allens, Allstarecho, Alphachimp, Ammubhave, Andre Engels, Antandrus, Anterior1, Antonio Lopez, Arthena, Asyndeton, Atomician, Avatar09, B424, Ben Tillman, Ben Webber, Ben morphett, Ben-Zin, Benjaminmakjiaming, Big Bird, Billymac00, Bkell, Blanchardb, Bob247, Bobo192, Bonadea, Bongwarrior, Bookmite, Bporopat, Brunislav, C.Fred, CBOrgatrope, CanadianLinuxUser, CapnPrep, Catslash, Cey0005, Chick Bowen, Commander Shepard, Conversion script, Courcelles, Cradel, Curps, D6, DORA, Da Joe, Daggerbox, Dajes13, Dckilgore, Delldot, Den fjättrade ankan, Denisarona, Deor, DerHexer, Dfrg.msc, Dissident, Doomedtx, Dr pda, Drengor, Eclecticology, ElNuevoEinstein, Elockid, EmilJ, Emilio Juanatey, Epbr123, Equendil, Etale, Europe22, Everyking, Exol, Explicit, Fangz, FannyCrumble, Farosdaughter, Favonian, FlyingToaster, Foxj, Fyyer, Gadfium, Gary King, Gauss, Gciriani, Geni, Giftlite, Good Olfactory, Graham87, Green caterpillar, Gregbard, Grokmoo, Gurt McRusski, Guðsþegn, Halfdan, Happysailor, Hardouin, Harland1, Hewhoamareismyself, Hkpawn, Hqb, Iceager, Iitisgood, J.delanoy, JDP90, JackofOz, Jaerik, Jaraalbe, Jaredwf, Jauhienij, Jennavecia, Jsc83, Jumbuck, JuneGloom07, Jusdafax, Karl-Henner, Katieh5584, KeithB, Kingvashy, Kkm010, KnowledgeOfSelf, Koogunmo, Korg, Kriggens, Kubigula, Kwamikagami, LDH, LateToTheGame, Leszek Jańczuk, LiDaobing, Lir, Little Mountain 5, Loadmaster, Lord.lucan, Lradrama, LukeTH, Madhero88, Magnus Manske, Mandarax, Martin451, Mastersrabbitry2, MathMartin, Mathilda, Mathypolic, MattKingston, Meaningful Username, Mel Etitis, Mentifisto, Mercury McKinnon, Michael Hardy, Minthellen, Modeha, Monegasque, Mormegil, Mouse Nightshirt, Mschlindwein, Msh210, Myasuda, Nairebis, NeoUrfahraner, Nev1, Newone, Nguyen Thanh Quang, Nicola.fragnito, Nthitz, OAC, Oftopladb, Olivier, Omnipaedista, Oracleofottawa, Oxymoron83, Party, Paul August, Paul Drye, Paul Niquette, Pax:Vobiscum, Penwhale, PeteSF, Phatalbert, Philip Trueman, Phlembowper99, Pizza Puzzle, Portalian, Pseudomonas, Psychonaut, Publius3, Razorflame, Rdanneskjold, Richard L. 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Jensen, MarkSutton, Martarius, Martian.knight, Martinp23, Materialscientist, Mathiasrex, Mathman712, Matt.T, Mav, Mavs41, Maxamegalon2000, Maxim, Maxis ftw, Mayhap, Mayumashu, Mboverload, Mdebets, Media lib, Meelar, Mel Etitis, Melodychan210, Mentifisto, Merlion444, Merovingian, Methusulah, Miaow Miaow, Michael Devore, Michael Hardy, Mifter Public, Mikeo, Miquonranger03, Miranda, Monegasque, Moonriddengirl, Mr Gullible, Mrh30, Mschneblin, Muhandes, Mycreations, Mygerardromance, NYArtsnWords, Nagy, Nagytibi, NawlinWiki, Neddyseagoon, Nedim Ardoğa, Nescio, Netkinetic, Neurolysis, Neutrality, NewEnglandYankee, Nfm, Nicastpj, Nick, Nickin, NightCrawler, Nik froud, Nikkimaria, Niluap, Ninarosa, Nishkid64, Njamesdingman, No Guru, Noctibus, NortyNort, NorwegianBlue, Nsaa, Oaktwig, Ocee, Oda Mari, Ohconfucius, Olivier, Omcnew, Omegastar, Omicronpersei8, Omnipaedista, Orange Suede Sofa, Orphan Wiki, OwenX, Oxnard27, Oxymoron83, PGWG, PL290, Paine, Paine Ellsworth, Panairjdde, Paradiso, Pascal.Tesson, Paul A, Paul August, Peak Freak, Pedant17, Pedro, Per Ardua, Perceval, Pgk, Phantomsteve, Phil1988, Philip Trueman, PhilipC, Piano non troppo, Picaroon, Pimpc, Poetaris, Poindexter Propellerhead, Polylerus, Popsracer, Postdlf, Ppanzini, Prashanthns, Protohiro, PseudoOne, Pufacz, PurplePiers, Python eggs, Quadell, Quintote, Qxz, R.e.b., RDBury, RJASE1, RTC, Radioflux, Randomblue, Raul654, Rdanneskjold, Rdsmith4, Reconsider the static, Reddi, Regibox, RexNL, Rholton, Rich Farmbrough, Richaraj, Ricky81682, Rjhatl, Rjwilmsi, Rklawton, RobertG, Robertgreer, Robocoder, Rohedin, Room429, Roundhouse0, Roxychik126, Rrburke, Rursus, Ryan Reich, RyanCross, RyanGerbil10, Ryulong, SE7, Sadalmelik, Salix alba, Sam Korn, Samsam9292, Sandius, SandyGeorgia, Sanfranman59, Sango123, Sankalpdravid, Sanxiyn, Schlier22, Scieberking, Scientizzle, Scohoust, Sergio1, Sethmahoney, Shanel, Shanenenenene, Shell Kinney, Shoeofdeath, Shortskimmer, Shoy, Sierrasurfer3, Sigma 7, Silver hr, Simonides, Sinn, Sjakkalle, Skew-t, Skizzik, Skomorokh, Skunkboy74, Skyezx, Slakr, SlaveToTheWage, Sleternel, Slon02, Smeira, Snigbrook, Snowolf, Snoyes, Solitude, Some jerk on the Internet, Soporaeternus, Sorarail, Sozo316, SpK, Spangineer, Sparkzy, Spirals31, Spitfire, Splash, SpuriousQ, StAnselm, Stan Shunpike, StaticGull, StaticVision, Stephan202, Stephen Burnett, Stephenb, Steven J. Anderson, Stevenkopelke, Stevenmitchell, Stl193, StoatBringer, Stout gat, Studerby, StuffOfInterest, Supasheep, Supernino, Symane, T. Anthony, T.J.V., Tabletop, Tamas, Tangent747, Tangotango, Tariqabjotu, Tbhotch, Tedickey, Tempodivalse, Texture, Tfl, The Anonymous One, The Rambling Man, The Thing That Should Not Be, TheMadBaron, TheSimkin, TheTrueSora, Thefourdotelipsis, Theoneintraining, Thingg, Tiddly Tom, Tide rolls, Timothy Titus, Titoxd, Tmoiynmwg, Toddst1, Tohd8BohaithuGh1, Tom harrison, Tommy2010, Tothebarricades.tk, Tpbradbury, Trangdoannguyen, Transity, Trevor Parscal, Triwbe, Tuffley, Tuplanolla, Twinkler4, U608854, Ulysse2000, Uncle Dick, UpstateNYer, Useight, Utcursch, UtherSRG, Utorak-sedamdeset, Vaceituno, Vanished user psdfiwnef3niurunfiuh234ruhfwdb7, Varith, Vary, Vegaswikian, View full catalog record, Vincentz, VivaEmilyDavies, Wahabijaz, Warfvinge, WatermelonPotion, Wavelength, Wayward, Weregerbil, Whatever (again), Whitfield Larrabee, Whosyourjudas, WikHead, Wiki alf, WikiFan04, WikiPedant, Wikiklrsc, Wikipelli, Wikismile, William Avery, Windchaser, Wingjob, Wmahan, WolfmanSF, World, Woudloper, Writ Keeper, Wysprgr2005, XJamRastafire, Xamuel, Xanchester, YDZ, Yamaguchi先 生, Yamamoto Ichiro, Yanksox, Yansa, YnnusOiramo, Ytrottier, Yuckfoo, Yugsdrawkcabeht, Zanimum, Zundark, Évangéline, రవిచంద్ర, 2098 anonymous edits Isaac Newton  Source: http://en.wikipedia.org/w/index.php?oldid=531627839  Contributors: (Didie), .marc., 0dd1, 123Steben456, 1929Depression, 1exec1, 1pezguy, 213.253.39.xxx, 2fort5r, 345Kai, 478jjjz, A bit iffy, A. B., A. di M., A.C. Norman, AJR, AKGhetto, AMCKen, AQUIMISMO, ASSHOLE, AVIosad, Abarzani, Abdallak, Acc60, Acetic Acid, Acjelen, Ackatsis, Ackees, Adam Grenberg, Adam1213, AdamSmithee, Adamk, Adamrce, Adashiel, Adib Khaled, AdilKarabora, Aecis, Aesopos, Aetheling, Afg96, Afuhz, AgentPeppermint, Agile, Agutie, Ahmed91981, Ahoerstemeier, Akajune, Akamad, Akrabbim, Aksi great, Aladue, Alan Canon, Alan Liefting, Aldis90, Aldy, Alexthe5th, Alfvaen, Algorithme, Ali K, Ali1986, Alibinmustafa, Alison, Alistair032, All 4 Jesus is 1, All Is One, Allen Jesus, Allen234, Allthingstoallpeople, Alphachimp, Alrasheedan, Amaraiel, AmiDaniel, An Siarach, Ancheta Wis, Andie142105, Andre Engels,

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Article Sources and Contributors Andreita933, Andrejj, Andrew Gray, Andrew Levine, AndrewHowse, AndrewTJ31, Andrewpmk, Andy Marchbanks, Andycjp, Anglius, Angmering, Angrymansr, Ankitsingh83, Anlace, AnnaFrance, Ansell, Antandrus, Anteriorlobe, Anthon.Eff, Anthony, Antidote, Antonrojo, Apothecia, Aprock, Apyule, Aquillion, Arcadie, Ariel., Aris Katsaris, Aristophanes68, Aristotle16, Arkachatterjea, Arne List, Arthursimms, Arturo 7, Arzel, Asav, Aschnack, Ashmoo, Ashnard, Askiser, Aslapnatickle, Aston martin-jacko, AstroNomer, Astrochemist, Atlant, Atrystan, AtticusX, Auréola, Autodidactyl, Autonova, Avalokitesvara, Avicennasis, AxelBoldt, AxiomShell, Ayla, Azeira, Azucar, BSTemple, Baby123gurl, Backburner001, BadgerBadger, Balloonthesixth, Ballz4kidz, Banes, Barbara Shack, Baronjonas, Barrettmagic, Basharh, Basho, Basilicofresco, Bastique, Batman2005, Bbatsell, Bbrown8370, Bcartolo, Bcrowell, Bejnar, Beland, Belovedfreak, Ben davison, Ben-Zin, Benbest, Bencoland, Bender235, Benhealy, 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Pichler, AMorozov, Abdullais4u, Achim1999, Almit39, Alves, Andre Engels, Arcandam, Arcfrk, Archimedes100, AxelBoldt, Babylon56, Baccyak4H, Baghdadian, Battamer, Beetstra, Bender235, Berland, Betacommand, BigNum, Bikasuishin, Binary TSO, Brasiliero, C S, CRGreathouse, Catapult, Cbigorgne, Chad.brewbaker, Charles Matthews, Chtito, CiaPan, Ckruschke, CmdrSunshine, Conversion script, Cvfgdw, Dantheox, DavidCW, Dfeldmann, Djun Kim, Dmcq, Dmharvey, Dominus, Drbreznjev, Drew R. Smith, Druseltal2005, Dsp13, Ed Poor, Elroch, Fredrik, Galoa2804, Gandalf61, Gauge, Gene Ward Smith, Georg Muntingh, Gfis, Giftlite, H.ehsaan, Haklo, Hari, Harvester, Headbomb, Herbee, Hu12, J. Finkelstein, J. Van Meter, JackSchmidt, Jagged 85, Jbergquist, Jitse Niesen, Jmk, Josilot, Jsondow, Jtc1313, Julian Birdbath, Kaldari, Kilom691, KittySaturn, Kletka, Lambiam, Lightmouse, LilHelpa, Linas, Lipedia, Loisel, Lupin, Magioladitis, Mathdata, Maxal, McKay, Melchoir, Merovingian, Michael Hardy, Mindmatrix, Monkeyfox, Nbarth, Nixdorf, Norm mit, Oleg Alexandrov, Osman-pasha, Paul Curtz, Paulbeeb, PeterStJohn, Plouffe, Pmanderson, Policron, Polluxian, PrimeHunter, Pt, R.e.b., RDBury, Rar, Rcingham, Rjwilmsi, Roadrunner, Rob Hooft, RobHar, Robert The Rebuilder, Ruzulo, SDC, Saxbryn, Scientific data, Selket, Silly rabbit, Simetrical, Small potato, SoWhy, Spacepotato, Spiffy sperry, Stikonas, Sumitkumarjha75, Sun Creator, TakuyaMurata, The new math, The-tenth-zdog, Thenub314, Toolnut, Uker, Ulner, Vahid alpha, Weialawaga, Wikiman211, Wirkstoff, Wmroche, XJamRastafire, Xnn, Yahord, Zstk, 220 anonymous edits Bernoulli polynomials  Source: http://en.wikipedia.org/w/index.php?oldid=524763902  Contributors: A. Pichler, Alberto da Calvairate, Bender235, CRGreathouse, Cuzkatzimhut, Dominus, Druseltal2005, Eric Kvaalen, Francos, Fredrik, Giftlite, Headbomb, Jsondow, Kiensvay, Kilom691, LeadSongDog, Lemontea, Linas, MarkSweep, Michael Hardy, Michael Ross, Moink, OMERZEN, Oleg Alexandrov, R.e.b., Rjwilmsi, RobHar, Silly rabbit, Slawekb, Vanish2, Vyznev Xnebara, William Ackerman, Wirkstoff, Xeno onex, ZeroOne, Zundark, 34 anonymous edits Bernoulli process  Source: http://en.wikipedia.org/w/index.php?oldid=523971847  Contributors: ABCD, Ap, AxelBoldt, Charles Matthews, Chassain, ClaesWallin, Dpbert, Espertus, FF2010, False vacuum, First Harmonic, Giftlite, Isnow, JHunterJ, Jamelan, Jmkim dot com, Linas, MathsPoetry, Meaning of Lif, Melcombe, Michael Hardy, MrOllie, Nbarth, P64, Panicpgh, Schmock, Skittleys, Tabletop, Talgalili, Weialawaga, Youandme, 19 anonymous edits Bernoulli trial  Source: http://en.wikipedia.org/w/index.php?oldid=528957373  Contributors: 16@r, Andrzej5335, AxelBoldt, Ayda D, BenMcLean, Benwing, Bgeelhoed, Bkell, Calvin 1998, Charles Matthews, Creidieki, Darrel francis, Dekart, Eramesan, False vacuum, Flavio Guitian, Fredrik, Gengizkhan, Giftlite, Huntnb, Jason Quinn, JeanM, Jiejunkong, Jugander, Kondormari, LimoWreck, Linas, Mackseem, Matt Kovacs, Melcombe, Michael Hardy, Michael miceli, Policron, RDBury, Ram einstein, Robinh, Shishir0610, Sigma0 1, Stijn Vermeeren, TedPavlic, Weialawaga, 57 anonymous edits Bernoulli's principle  Source: http://en.wikipedia.org/w/index.php?oldid=532037446  Contributors: !jim, 124Nick, 6birc, A little insignificant, A3RO, ABF, AHands, AbaCal, Abce2, Acather96, AdjustShift, Against the current, Ahoerstemeier, Ajayfermi, Alexmikesell, Allens, Allstarecho, Alphachimp, Amaury, AmiDaniel, Andonic, Andre Engels, Antonrojo, Aquinex, Aristotle1990,

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Image Sources, Licenses and Contributors

Image Sources, Licenses and Contributors File:Zeno of Elea Tibaldi or Carducci Escorial.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Zeno_of_Elea_Tibaldi_or_Carducci_Escorial.jpg  License: Public Domain  Contributors: Bartolomeo Carducci or Pellegrino Tibaldi Image:wikisource-logo.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wikisource-logo.svg  License: logo  Contributors: Guillom, Jarekt, MichaelMaggs, NielsF, Rei-artur, Rocket000 File:Domenico-Fetti Archimedes 1620.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Domenico-Fetti_Archimedes_1620.jpg  License: Public Domain  Contributors: A. Wagner, Andreagrossmann, AndreasPraefcke, Boo-Boo Baroo, Bukk, Christophe.Finot, FranzK, Gene.arboit, Ianmacm, Karel K., Kilom691, Kramer Associates, Luestling, Mattes, Plindenbaum, Serge Lachinov, Shakko, Wst, 3 anonymous edits File:Gerhard Thieme Archimedes.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Gerhard_Thieme_Archimedes.jpg  License: GNU Free Documentation License  Contributors: Editor at Large, Gerardus, Ianmacm, Kilom691, Lobo, Lotse, SpreeTom, Srittau, Wst, 3 anonymous edits File:Archimedes sphere and cylinder.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Archimedes_sphere_and_cylinder.svg  License: Creative Commons Attribution-Sharealike 2.5  Contributors: derivative work: Pbroks13 (talk) Archimedes_sphere_and_cylinder.png: André Karwath aka Aka File:Archimedes water balance.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Archimedes_water_balance.gif  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Tonyle File:Archimedes-screw one-screw-threads with-ball 3D-view animated small.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Archimedes-screw_one-screw-threads_with-ball_3D-view_animated_small.gif  License: Creative Commons Attribution-Sharealike 2.5  Contributors: Silberwolf (size changed by: Jahobr) File:Archimedes Heat Ray conceptual diagram.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Archimedes_Heat_Ray_conceptual_diagram.svg  License: GNU Free Documentation License  Contributors: Finnrind (original); Pbroks13 (talk) (redraw) File:PiArchimede4.svg  Source: http://en.wikipedia.org/w/index.php?title=File:PiArchimede4.svg  License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0  Contributors: HB File:Parabolic segment and inscribed triangle.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Parabolic_segment_and_inscribed_triangle.svg  License: Public Domain  Contributors: User:Vladislav Pogorelov, derivative of works by Pbroks13 and Jim.belk File:Archimedes lever (Small).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Archimedes_lever_(Small).jpg  License: Public Domain  Contributors: Mechanics Magazine File:Stomachion.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Stomachion.JPG  License: Public Domain  Contributors: Ianmacm at en.wikipedia File:FieldsMedalFront.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:FieldsMedalFront.jpg  License: Public Domain  Contributors: Stefan Zachow of the International Mathematical Union; retouched by King of Hearts File:Works of Archimedes Lemmas.PNG  Source: http://en.wikipedia.org/w/index.php?title=File:Works_of_Archimedes_Lemmas.PNG  License: Public Domain  Contributors: Archimedes; Thomas Heath File:Arbelos.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Arbelos.svg  License: unknown  Contributors: Grafite, Tintazul File:Salinon shaded.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Salinon_shaded.svg  License: Public Domain  Contributors: Pbroks13 Image:Stomachion.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Stomachion.JPG  License: Public Domain  Contributors: Ianmacm at en.wikipedia File:ArPalimTypPage.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:ArPalimTypPage.jpg  License: Creative Commons Attribution 3.0  Contributors: The Walters Museum File:ArPalimTyp2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:ArPalimTyp2.jpg  License: Creative Commons Attribution 3.0  Contributors: The Walters Museum File:Frans Hals - Portret van René Descartes.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Frans_Hals_-_Portret_van_René_Descartes.jpg  License: Public Domain  Contributors: Beria, Bohème, Dedden, Ecummenic, Kigsz, Kilom691, Mcke, Miniwark, Serge Lachinov, Shakko, Vincent Steenberg, 1 anonymous edits File:Firma Descartes.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Firma_Descartes.svg  License: GNU Free Documentation License  Contributors: Firma_Descartes.jpg: René Descartes derivative work: Addicted04 (talk) Image:DescartesGraduationRegistry.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:DescartesGraduationRegistry.JPG  License: Public Domain  Contributors: Ayack, Emijrp, Jan Arkesteijn, World Imaging, 1 anonymous edits File:René Descartes i samtal med Sveriges drottning, Kristina.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:René_Descartes_i_samtal_med_Sveriges_drottning,_Kristina.jpg  License: Public Domain  Contributors: Pierre Louis Dumesnil (1698-1781); 1884 copy by Nils Forsberg Image:DescartesAshes.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:DescartesAshes.jpg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: PHGCOM Image:Descartes3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Descartes3.jpg  License: Public Domain  Contributors: Emijrp, Tomisti File:Handwritten letter by Descartes December 1638.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Handwritten_letter_by_Descartes_December_1638.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: PHGCOM File:Pierre de Fermat.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Pierre_de_Fermat.jpg  License: Public Domain  Contributors: File:France Toulouse Capitole Fermat 072007.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:France_Toulouse_Capitole_Fermat_072007.jpg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Pinpin File:Pierre de Fermat.png  Source: http://en.wikipedia.org/w/index.php?title=File:Pierre_de_Fermat.png  License: Public Domain  Contributors: Bohème, Guérin Nicolas, Sir Gawain, Thuresson File:Fermat burial plaque.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Fermat_burial_plaque.jpg  License: Public Domain  Contributors: Original uploader was Gciriani at fr.wikipedia File:Fermats will.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Fermats_will.jpg  License: Public Domain  Contributors: Giorgiomonteforti, Samariay File:Blaise_pascal.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Blaise_pascal.jpg  License: Public Domain  Contributors: Anarkman, Crux, Deadstar, GianniG46, Killiondude File:Arts et Metiers Pascaline dsc03869.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Arts_et_Metiers_Pascaline_dsc03869.jpg  License: Creative Commons Attribution-Sharealike 2.0  Contributors: © 2005 David Monniaux, User:David.Monniaux File:PascalTriangleAnimated2.gif  Source: http://en.wikipedia.org/w/index.php?title=File:PascalTriangleAnimated2.gif  License: Creative Commons Attribution-Share Alike  Contributors: Hersfold on the English Wikipedia File:Blaise Pascal 2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Blaise_Pascal_2.jpg  License: Public Domain  Contributors: Deadstar, Phrood File:Pascal's Barrel.png  Source: http://en.wikipedia.org/w/index.php?title=File:Pascal's_Barrel.png  License: Public Domain  Contributors: 84user, Arbitrarily0 File:Pascal Pajou Louvre RF2981.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Pascal_Pajou_Louvre_RF2981.jpg  License: Public Domain  Contributors: User:Jastrow File:Epitaph Blaise Pascal Saint-Etienne.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Epitaph_Blaise_Pascal_Saint-Etienne.jpg  License: Public Domain  Contributors: Jastrow File:001Paskal.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:001Paskal.JPG  License: Public Domain  Contributors: Original uploader was Mary trump at en.wikipedia File:wikisource-logo.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wikisource-logo.svg  License: logo  Contributors: Guillom, Jarekt, MichaelMaggs, NielsF, Rei-artur, Rocket000 File:GodfreyKneller-IsaacNewton-1689.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:GodfreyKneller-IsaacNewton-1689.jpg  License: Public Domain  Contributors: Algorithme, Beyond My Ken, Bjankuloski06en, Grenavitar, Infrogmation, Kelson, Kilom691, Porao, Saperaud, Semnoz, Siebrand, Sparkit, Thomas Gun, Vonvon, Wikiklaas, Wknight94, Wst, Zaphod, 7 anonymous edits File:Isaac Newton signature.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Isaac_Newton_signature.svg  License: Public Domain  Contributors: derivative work: Pbroks13 (talk) Isaac_Newton_signature.png: Isaac Newton File:Sir Isaac Newton by Sir Godfrey Kneller, Bt.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Sir_Isaac_Newton_by_Sir_Godfrey_Kneller,_Bt.jpg  License: Public Domain  Contributors: User:Dcoetzee File:Bolton-newton.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Bolton-newton.jpg  License: Public Domain  Contributors: David.Monniaux, Jacklee, Red devil 666, Saperaud, Wst

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Image Sources, Licenses and Contributors File:NewtonsTelescopeReplica.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:NewtonsTelescopeReplica.jpg  License: Creative Commons Attribution-Share Alike 2.0 Generic  Contributors: Basilicofresco, DrJunge, Eoghanacht, Fabian Commons, HHahn, JMCC1, Jahobr, MartinPoulter, Plindenbaum, Rimshot, Sanao, Solipsist, Thomas Gun, Thuresson, Wst, Xocoyote, Zahn, 15 anonymous edits File:Dispersive Prism Illustration.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Dispersive_Prism_Illustration.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Dispersive_Prism_Illustration_by_Spigget.jpg: Spigget derivative work: Cepheiden (talk) Image:Newton-letter-to-briggs 03.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Newton-letter-to-briggs_03.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: User:Xtrasystole File:NewtonsPrincipia.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:NewtonsPrincipia.jpg  License: Creative Commons Attribution-Share Alike 2.0 Generic  Contributors: Acc60, Anarkman, Aristeas, Aunuki, Daniele Pugliesi, Duesentrieb, Ephraim33, JackyR, Mdd, Piero, Solipsist, Svdmolen, Wst, 3 anonymous edits File:Newton 25.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Newton_25.jpg  License: Public Domain  Contributors: User:Shizhao File:ENG COA Newton.svg  Source: http://en.wikipedia.org/w/index.php?title=File:ENG_COA_Newton.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Avalokitesvara File:Isaac Newton statue.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Isaac_Newton_statue.jpg  License: GNU Free Documentation License  Contributors: Andrew Gray File:Newton by Eduardo Paolozzi 2003-03-10.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Newton_by_Eduardo_Paolozzi_2003-03-10.jpg  License: Creative Commons Attribution 3.0  Contributors: John McCullough File:Isaac Newton grave in Westminster Abbey.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Isaac_Newton_grave_in_Westminster_Abbey.jpg  License: Public Domain  Contributors: Klaus-Dieter Keller, Germany File:Newton-WilliamBlake.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Newton-WilliamBlake.jpg  License: Public Domain  Contributors: Barosaul, Jeremiahpatrick, Lamiot, Lombroso, Madmedea, Matanya (usurped), Mattes, Petropoxy (Lithoderm Proxy), Plindenbaum, Waldir Image:Newton's tree, Botanic Gardens, Cambridge.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Newton's_tree,_Botanic_Gardens,_Cambridge.JPG  License: Public Domain  Contributors: User:Azeira Image:Newtons apple.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Newtons_apple.jpg  License: GNU Free Documentation License  Contributors: Original uploader was Bcartolo at en.wikipedia Image:Newtons laws in latin.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Newtons_laws_in_latin.jpg  License: Public Domain  Contributors: Bestiasonica, JdH, Man vyi, Tttrung, Wst, 4 anonymous edits File:Sir Isaac Newton (1643-1727).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Sir_Isaac_Newton_(1643-1727).jpg  License: Public Domain  Contributors: David.Monniaux, Hystrix, Kilom691, Luestling, Thomas Gun, Umherirrender, Wst, 4 anonymous edits Image:Skaters showing newtons third law.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Skaters_showing_newtons_third_law.svg  License: GNU Free Documentation License  Contributors: Benjamin Crowell (Wikipedia user bcrowell) Image:NewtonsPrincipia.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:NewtonsPrincipia.jpg  License: Creative Commons Attribution-Share Alike 2.0 Generic  Contributors: Acc60, Anarkman, Aristeas, Aunuki, Daniele Pugliesi, Duesentrieb, Ephraim33, JackyR, Mdd, Piero, Solipsist, Svdmolen, Wst, 3 anonymous edits Image:Principia Page 1726.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Principia_Page_1726.jpg  License: Public Domain  Contributors: Duesentrieb, FSII, Kjetil r, Svdmolen File:Gottfried Wilhelm von Leibniz.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Gottfried_Wilhelm_von_Leibniz.jpg  License: Public Domain  Contributors: AndreasPraefcke, Auntof6, Beria, Beyond My Ken, Boo-Boo Baroo, Cirt, Davidlud, Ecummenic, Eusebius, Factumquintus, FalconL, Gabor, Luestling, Mattes, Schaengel89, Shakko, Svencb, Tomisti, 5 anonymous edits File:Leibnitz signature.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Leibnitz_signature.svg  License: Public Domain  Contributors: AndreasPraefcke, Connormah, Florentyna, McSush, Shaddim File:Leibnitzrechenmaschine.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Leibnitzrechenmaschine.jpg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: User:Kolossos File:Korespondencja Gottfrieda Leibniza.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Korespondencja_Gottfrieda_Leibniza.jpg  License: Public Domain  Contributors: AndreasPraefcke, BurgererSF File:Jakob_Bernoulli.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Jakob_Bernoulli.jpg  License: Public Domain  Contributors: Funck77, Kilom691, Malo, Materialscientist, Xgoni Image:Basler Muenster Bernoulli.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Basler_Muenster_Bernoulli.jpg  License: GNU Free Documentation License  Contributors: Wladyslaw Sojka, Uploaded to Commons by Modulo File:Johann_Bernoulli2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Johann_Bernoulli2.jpg  License: Public Domain  Contributors: Boo-Boo Baroo, Common Good, Funck77, Gill110951, Leyo, Materialscientist, 1 anonymous edits Image:BernoulliNumbersByZetaLowRes.png  Source: http://en.wikipedia.org/w/index.php?title=File:BernoulliNumbersByZetaLowRes.png  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Wirkstoff Image:SCWoonTree.png  Source: http://en.wikipedia.org/w/index.php?title=File:SCWoonTree.png  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Wirkstoff Image:SeidelAlgorithmForTn.png  Source: http://en.wikipedia.org/w/index.php?title=File:SeidelAlgorithmForTn.png  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Wirkstoff Image:Seki Kowa Katsuyo Sampo Bernoulli numbers.png  Source: http://en.wikipedia.org/w/index.php?title=File:Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png  License: Public Domain  Contributors: Seki Kowa Original uploader was Dominus at en.wikipedia Image:JakobBernoulliSummaePotestatum.png  Source: http://en.wikipedia.org/w/index.php?title=File:JakobBernoulliSummaePotestatum.png  License: Public Domain  Contributors: Creator:Wirkstoff Image:Bernoulli polynomials.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Bernoulli_polynomials.svg  License: GNU Free Documentation License  Contributors: User:Linas File:VenturiFlow.png  Source: http://en.wikipedia.org/w/index.php?title=File:VenturiFlow.png  License: GNU Free Documentation License  Contributors: user:ComputerGeezer and Geof. Original uploader was ComputerGeezer at en.wikipedia File:BernoullisLawDerivationDiagram.svg  Source: http://en.wikipedia.org/w/index.php?title=File:BernoullisLawDerivationDiagram.svg  License: GNU Free Documentation License  Contributors: MannyMax (original) File:Cloud over A340 wing.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Cloud_over_A340_wing.JPG  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Mgw89 File:Leonhard Euler 2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Leonhard_Euler_2.jpg  License: Public Domain  Contributors: Haham hanuka, Herbythyme, Plindenbaum, Rd232, Serge Lachinov, Shakko, Soerfm, 6 anonymous edits File:Euler's signature.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Euler's_signature.svg  License: Public Domain  Contributors: derivative work: Pbroks13 (talk) Euler's_signature.png: Leonhard Euler Image:Euler-10 Swiss Franc banknote (front).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Euler-10_Swiss_Franc_banknote_(front).jpg  License: Public Domain  Contributors: Pascal.Tesson, Soerfm, Szajci, 1 anonymous edits Image:Euler-USSR-1957-stamp.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Euler-USSR-1957-stamp.jpg  License: Public Domain  Contributors: Butko, Pascal.Tesson, 2 anonymous edits Image:Euler GDR stamp.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Euler_GDR_stamp.jpg  License: Public Domain  Contributors: Deutsche Post der DDR Image:Leonhard Euler.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Leonhard_Euler.jpg  License: Public Domain  Contributors: User:Wars Image:Euler Grave at Alexander Nevsky Monastry.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Euler_Grave_at_Alexander_Nevsky_Monastry.jpg  License: Public Domain  Contributors: Berteun, IgorMagic, Kaganer, Knakts, Man vyi, Mikkalai, Soerfm, Verica Atrebatum Image:Euler's formula.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Euler's_formula.svg  License: GNU Free Documentation License  Contributors: User:Wereon, User:lasindi, user:gunther Image:Konigsberg bridges.png  Source: http://en.wikipedia.org/w/index.php?title=File:Konigsberg_bridges.png  License: GNU Free Documentation License  Contributors: Bogdan Giuşcă

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Image Sources, Licenses and Contributors Image:Methodus inveniendi - Leonhard Euler - 1744.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Methodus_inveniendi_-_Leonhard_Euler_-_1744.jpg  License: Public Domain  Contributors: Unknown, assumed that Euler himself didn't draw this. File:Langrange portrait.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Langrange_portrait.jpg  License: Public Domain  Contributors: 84user, Breeze, Ellywa, G.dallorto, Gabor, Mu File:Lagrange's tomb at the Pantheon.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Lagrange's_tomb_at_the_Pantheon.jpg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Bohème, Carcharoth (Commons), G.dallorto, Kilom691 File:Pierre-Simon Laplace.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Pierre-Simon_Laplace.jpg  License: Public Domain  Contributors: Ashill, Ecummenic, Elcobbola, Gene.arboit, Jimmy44, Olivier2, 霧 木 諒 二, 1 anonymous edits File:Pierre-Simon Laplace signature.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Pierre-Simon_Laplace_signature.svg  License: Public Domain  Contributors: Pierre-Simon Laplace Created in vector format by Scewing File:Rotating spherical harmonics.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Rotating_spherical_harmonics.gif  License: GNU Free Documentation License  Contributors: User:Cyp File:Laplace house Arcueil.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Laplace_house_Arcueil.jpg  License: Public Domain  Contributors: User:cutler File:Pierre-Simon-Laplace (1749-1827).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Pierre-Simon-Laplace_(1749-1827).jpg  License: Public Domain  Contributors: Gabor, Luestling, Olivier2, Umherirrender File:Gaspard monge litho delpech.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Gaspard_monge_litho_delpech.jpg  License: Public Domain  Contributors: Kelson, Kilom691, Mu, Romary, Ylebru File:Perelachaise-Monge-p1000360.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Perelachaise-Monge-p1000360.jpg  License: unknown  Contributors: User:Rama File:Fourier2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Fourier2.jpg  License: Public domain  Contributors: Original uploader was User:Bunzil at en.wikipedia File:Legendre and Fourier (1820).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Legendre_and_Fourier_(1820).jpg  License: Public Domain  Contributors: Original uploader was Libb Thims at en.wikipedia File:Joseph Fourier (circa 1820).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Joseph_Fourier_(circa_1820).jpg  License: Public Domain  Contributors: Original uploader was Libb Thims at en.wikipedia File:Grenoble - ancien évêché - Joseph Fourier.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Grenoble_-_ancien_évêché_-_Joseph_Fourier.JPG  License: Creative Commons Attribution 3.0  Contributors: Coyau, Eusebius, Hydrel, Jean-Frédéric, Kilom691 File:Perelachaise-p1000361.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Perelachaise-p1000361.jpg  License: unknown  Contributors: User:Rama File:Jean-Victor Poncelet.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Jean-Victor_Poncelet.jpg  License: Public Domain  Contributors: Badzil, Gdr, Kilom691, Verbex File:Ponce sig.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Ponce_sig.jpg  License: unknown  Contributors: Poncelet (it is his signature) File:Cloître_sainte_Constance_(lycée_Fabert_de_Metz).JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Cloître_sainte_Constance_(lycée_Fabert_de_Metz).JPG  License: Creative Commons Attribution 3.0  Contributors: Graoully Image:Ponceletsteiner.PNG  Source: http://en.wikipedia.org/w/index.php?title=File:Ponceletsteiner.PNG  License: Public Domain  Contributors: Me File:Carl Friedrich Gauss.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Carl_Friedrich_Gauss.jpg  License: Public Domain  Contributors: Gottlieb BiermannA. Wittmann (photo) File:Carl Friedrich Gauß signature.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Carl_Friedrich_Gauß_signature.svg  License: Public Domain  Contributors: derivative work: Pbroks13 (talk) Carl_Friedrich_Gauß,_Namenszug_von_1794.jpg: Carl Friedrich Gauß (1777-1855) File:Speaker Icon.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Speaker_Icon.svg  License: Public Domain  Contributors: Blast, G.Hagedorn, Mobius, Tehdog, 2 anonymous edits File:Statue-of-Gauss-in-Braunschweig.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Statue-of-Gauss-in-Braunschweig.jpg  License: Creative Commons Attribution-Sharealike 2.5  Contributors: Mascdman File:Disqvisitiones-800.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Disqvisitiones-800.jpg  License: Public Domain  Contributors: Achird, Aristeas, Gveret Tered, Juiced lemon, Maksim, Toobaz, Ufudu, Wst File:Bendixen - Carl Friedrich Gauß, 1828.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Bendixen_-_Carl_Friedrich_Gauß,_1828.jpg  License: Public Domain  Contributors: Siegfried Detlev Bendixen File:Normal distribution pdf.png  Source: http://en.wikipedia.org/w/index.php?title=File:Normal_distribution_pdf.png  License: GNU General Public License  Contributors: Ardonik, Gerbrant, Grendelkhan, Inductiveload, Juiced lemon, MarkSweep, Wikiwide, 10 anonymous edits File:Carl Friedrich Gauss on his Deathbed, 1855.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Carl_Friedrich_Gauss_on_his_Deathbed,_1855.jpg  License: Public Domain  Contributors: Connormah, Ephraim33, Nicolas Perrault III, Tamba52 File:Göttingen-Grave.of.Gauß.06.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Göttingen-Grave.of.Gauß.06.jpg  License: Creative Commons Sharealike 1.0  Contributors: Jonathan Groß, Kresspahl, Longbow4u, Martin H. File:Therese Gauss.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Therese_Gauss.jpg  License: Public Domain  Contributors: Churchh, Shakko, Skraemer File:10 DM Serie4 Vorderseite.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:10_DM_Serie4_Vorderseite.jpg  License: unknown  Contributors: Deutsche Bundesbank, Frankfurt am Main, Germany File:Stamps of Germany (DDR) 1977, MiNr 2215.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Stamps_of_Germany_(DDR)_1977,_MiNr_2215.jpg  License: Public Domain  Contributors: Hochgeladen von --Nightflyer (talk) 18:24, 21 November 2009 (UTC) File:Augustin-Louis Cauchy 1901.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Augustin-Louis_Cauchy_1901.jpg  License: Public Domain  Contributors: Public domain File:Augustin-Louis Cauchy.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Augustin-Louis_Cauchy.jpg  License: Public Domain  Contributors: Charles H. Reutlinger Image:Cauchy.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Cauchy.jpg  License: Public Domain  Contributors: Augustin-Louis Cauchy (1789-1857) File:Lobachevsky.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Lobachevsky.jpg  License: Public Domain  Contributors: $toic File:Celebrating the birthday of Lobachevsky.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Celebrating_the_birthday_of_Lobachevsky.JPG  License: Creative Commons Attribution-Sharealike 3.0  Contributors: User:Жабыш File:Niels Henrik Abel.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Niels_Henrik_Abel.jpg  License: Public Domain  Contributors: Gene.arboit, Kelson, Kåre-Olav, Magnus Manske, Materialscientist, Verdlanco, Wst File:Niels_Henrik_Abel_signature.png  Source: http://en.wikipedia.org/w/index.php?title=File:Niels_Henrik_Abel_signature.png  License: Public Domain  Contributors: Kåre-Olav Image:Søren_Georg_Abel.png  Source: http://en.wikipedia.org/w/index.php?title=File:Søren_Georg_Abel.png  License: Public Domain  Contributors: Kåre-Olav Image:Anne_Marie_Abel.png  Source: http://en.wikipedia.org/w/index.php?title=File:Anne_Marie_Abel.png  License: Public Domain  Contributors: Kåre-Olav File:Gjerstad kirke or Risør.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Gjerstad_kirke_or_Risør.jpg  License: Public Domain  Contributors: Peder Bentsen Melaas Image:Abel-kladd.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Abel-kladd.jpg  License: Public Domain  Contributors: Niels Henrik Abel Image:AbelStatue.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:AbelStatue.jpg  License: GNU Free Documentation License  Contributors: Raflmoe Image:Holmengard-125.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Holmengard-125.JPG  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Torgrim111 Image:Holmengard-119.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Holmengard-119.JPG  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Torgrim111 Image:Stamps of Norway, 1929-Niels Henrik Abel1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Stamps_of_Norway,_1929-Niels_Henrik_Abel1.jpg  License: Public Domain  Contributors: Cekli829 Image:Stamps of Norway, 1929-Niels Henrik Abel2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Stamps_of_Norway,_1929-Niels_Henrik_Abel2.jpg  License: Public Domain  Contributors: Cekli829 Image:Stamps of Norway, 1929-Niels Henrik Abel3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Stamps_of_Norway,_1929-Niels_Henrik_Abel3.jpg  License: Public Domain  Contributors: Cekli829 Image:Stamps of Norway, 1929-Niels Henrik Abel4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Stamps_of_Norway,_1929-Niels_Henrik_Abel4.jpg  License: Public Domain  Contributors: Cekli829

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Image Sources, Licenses and Contributors File:Carl Jacobi.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Carl_Jacobi.jpg  License: Public Domain  Contributors: Gabor, Mehlauge, Sebastian Wallroth, Siebrand, Stern, 1 anonymous edits File:Carl Jacobi2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Carl_Jacobi2.jpg  License: Public Domain  Contributors: Kristof vt File:William Rowan Hamilton portrait oval combined.png  Source: http://en.wikipedia.org/w/index.php?title=File:William_Rowan_Hamilton_portrait_oval_combined.png  License: Public Domain  Contributors: William_Rowan_Hamilton_portrait_oval.png: William_Rowan_Hamilton_portrait_oval_2.png: derivative work: Quibik (talk) Image:William Rowan Hamilton Plaque - geograph.org.uk - 347941.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:William_Rowan_Hamilton_Plaque_-_geograph.org.uk_-_347941.jpg  License: Creative Commons Attribution-Share Alike 2.0 Generic  Contributors: Anne Bauval, Silverblaster File:Evariste galois.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Evariste_galois.jpg  License: Public Domain  Contributors: Anarkman, FSII, Knakts, Mikhail Ryazanov, Mu, Tohma File:Galois-Signature.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Galois-Signature.svg  License: Public Domain  Contributors: Galois-1830.JPG: Évariste Galois derivative work: EIDETIV (talk) File:Tombe galois.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Tombe_galois.JPG  License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0  Contributors: Beachboy68 File:James Joseph Sylvester.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:James_Joseph_Sylvester.jpg  License: Public Domain  Contributors: Avraham, Dabomb87, MesserWoland, PMG, Sobi3ch File:Karl Weierstrass.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Karl_Weierstrass.jpg  License: Public Domain  Contributors: AndreasPraefcke, ArtMechanic, Katpatuka File:Arthur Cayley.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Arthur_Cayley.jpg  License: Public Domain  Contributors: Aliman5040, Asturius, Kilom691, Scewing, Zuirdj, 1 anonymous edits File:Sofja_Wassiljewna_Kowalewskaja_1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Sofja_Wassiljewna_Kowalewskaja_1.jpg  License: Public Domain  Contributors: INeverCry, Skraemer, Túrelio, Vonvon, 1 anonymous edits File:Sofya Vasilyevna Kovalevskaya Bust.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Sofya_Vasilyevna_Kovalevskaya_Bust.jpg  License: Public Domain  Contributors: Published by Macmillan, NY, 1895 File:RR5110-0034R.gif  Source: http://en.wikipedia.org/w/index.php?title=File:RR5110-0034R.gif  License: Public Domain  Contributors: Банк России.. Original uploader was Permjak at ru.wikipedia File:Stamp of USSR 1635g.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Stamp_of_USSR_1635g.jpg  License: Public Domain  Contributors: Post of USSR File:George Boole.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:George_Boole.jpg  License: Public Domain  Contributors: Haks File:3 Pottergate - geograph.org.uk - 657140.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:3_Pottergate_-_geograph.org.uk_-_657140.jpg  License: Creative Commons Attribution-Share Alike 2.0 Generic  Contributors: Ardfern, Charles Matthews, Fæ File:BoolePlacque.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:BoolePlacque.jpg  License: Public Domain  Contributors: Logicus File:BooleWindow(bottom third).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:BooleWindow(bottom_third).jpg  License: Public Domain  Contributors: Logicus File:BoolePlaque2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:BoolePlaque2.jpg  License: Public Domain  Contributors: Logicus File:Boole_House_Cork.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Boole_House_Cork.jpg  License: Creative Commons Zero  Contributors: SandStone File:2010-05-26 at 18-05-02.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:2010-05-26_at_18-05-02.jpg  License: Creative Commons Attribution 3.0  Contributors: Marcovanhogan File:Hasse2Free.png  Source: http://en.wikipedia.org/w/index.php?title=File:Hasse2Free.png  License: Public Domain  Contributors: CommonsDelinker, Darapti, Juiced lemon, Kilom691, Lipedia, Maksim, Mdd, 2 anonymous edits File:Charles Hermite circa 1901 edit.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Charles_Hermite_circa_1901_edit.jpg  License: Public Domain  Contributors: Charles_Hermite_circa_1901.jpg: Unknown derivative work: Quibik (talk) File:Charles Hermite circa 1887.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Charles_Hermite_circa_1887.jpg  License: Public Domain  Contributors: Pirou File:Leopold Kronecker.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Leopold_Kronecker.jpg  License: Public Domain  Contributors: Uploaded on En by http://en.wikipedia.org/wiki/User:SuperGirl File:KroneckerGrab.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:KroneckerGrab.jpg  License: GNU Free Documentation License  Contributors: Furfur File:Georg Friedrich Bernhard Riemann.jpeg  Source: http://en.wikipedia.org/w/index.php?title=File:Georg_Friedrich_Bernhard_Riemann.jpeg  License: Public Domain  Contributors: Ariadacapo, Bdk, Red Rooster, Ævar Arnfjörð Bjarmason, 2 anonymous edits File:Bernhard Riemann signature.png  Source: http://en.wikipedia.org/w/index.php?title=File:Bernhard_Riemann_signature.png  License: Public Domain  Contributors: Ariadacapo File:ErnstKummer.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:ErnstKummer.jpg  License: Public Domain  Contributors: PDH File:Dedekind.jpeg  Source: http://en.wikipedia.org/w/index.php?title=File:Dedekind.jpeg  License: Public Domain  Contributors: not found File:Stamps of Germany (DDR) 1981, MiNr 2605.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Stamps_of_Germany_(DDR)_1981,_MiNr_2605.jpg  License: Public Domain  Contributors: Hochgeladen von --Nightflyer (talk) 18:38, 21 November 2009 (UTC) File:Henri_Poincaré-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Henri_Poincaré-2.jpg  License: Public Domain  Contributors: unknown File:Henri Poincaré Signature.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Henri_Poincaré_Signature.svg  License: Public Domain  Contributors: Henri Poincaré Image:Young Poincare.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Young_Poincare.jpg  License: Public Domain  Contributors: Badzil, Christian1985, Clindberg, Hohum, Homonihilis, Materialscientist, Soerfm File:Poincaré gravestone.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Poincaré_gravestone.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: EmilJ Image:Curie and Poincare 1911 Solvay.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Curie_and_Poincare_1911_Solvay.jpg  License: Public Domain  Contributors: Donarreiskoffer, Fastfission, JdH, Mu, 1 anonymous edits Image:Mug and Torus morph.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Mug_and_Torus_morph.gif  License: Public Domain  Contributors: Abnormaal, Durova, Howcheng, Kieff, Kri, Manco Capac, Maximaximax, Rovnet, SharkD, Takabeg, 16 anonymous edits File: Phase Portrait Sadle.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Phase_Portrait_Sadle.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Podshumok File: Phase Portrait Stable Focus.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Phase_Portrait_Stable_Focus.svg  License: Public Domain  Contributors: Podshumok File: Phase portrait center.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Phase_portrait_center.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Podshumok File: Phase Portrait Stable Node.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Phase_Portrait_Stable_Node.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Podshumok File:Henri Poincaré by H Manuel.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Henri_Poincaré_by_H_Manuel.jpg  License: unknown  Contributors: Henri Manuel (1874-1947) File:Georg Cantor2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Georg_Cantor2.jpg  License: Public Domain  Contributors: Unknown File:Georg Cantor3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Georg_Cantor3.jpg  License: Public Domain  Contributors: unknown File:Diagonal argument 2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Diagonal_argument_2.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Cronholm144 File:Bijection.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Bijection.svg  License: Public Domain  Contributors: en:User:Schapel File:Blackboard Georg Cantor (11-line V O building 24).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Blackboard_Georg_Cantor_(11-line_V_O_building_24).jpg  License: Public Domain  Contributors: AKA MBG

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