History of Probability

The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers.[4]
According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances. However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.

Aside from elementary work by Girolamo Cardano in the 16th century, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics. See Ian Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of the early development of the very concept of mathematical probability.

The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.
Pierre-Simon Laplace (1774) first tried to deduce a rule for combining observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve y = φ(x), x being any error and y its probability, and laid down three properties of this curve:

It is symmetric as to the y-axis;
The x-axis is an asymptote, the probability of the error being 0;
The area enclosed is 1, it being certain that an error exists.
He also provided, in 1781, a formula for the law of facility of error (a term Lagrange used in 1774), but it led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
Adrien-Marie Legendre (1805) developed the method of least squares, and introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error,

h being a constant depending on precision of observation, and c a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W. F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for r, the probable error of a single observation, is well known.

In the nineteenth century authors on the general theory included Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion, and Karl Pearson. Augustus De Morgan and George Boole improved the exposition of the theory.

Andrey Markov introduced the notion of Markov chains (1906), which played an important role in stochastic processes theory and its applications. The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov (1931).
On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).



Note: This isn't my work, this is copied from wikipedia and some other sources and are edited by me. I am not trying to plagiarise.

Sofia Kovalevskaya

Sofia Kovalevsky

Sofia Kovalevsky, born in Moscow, displayed unusual mathematical ability from childhood. Desperate to escape the strictures of gentry womanhood, at eighteen she contracted a "fictive" marriage with the paleontologist and social activist Vladimir Kovalevsky, who took her to the Western Europe to study. In 1874 Kovalevskaya, mentored by the eminent German mathematician Karl Weierstrass, received a doctorate from Göttingen University. Afterward, the Kovalevskys, now married in fact, returned to St. Petersburg, where their daughter was born in 1878. In 1883, faced with worsening mood swings and the possibility of being prosecuted for his role in a stock swindle, Vladimir committed suicide.

Unable to find suitable teaching work in Russia, Kovalevskaya, at the urging of Weierstrass and the Swedish mathematician Gustav Mittag-Leffler, accepted a professorship in the newly established Stockholm University, becoming the first woman in modern Europe to hold such a post. In Sweden the homesick Kovalevskaya wrote her vivid reminiscences of girlhood; a novella based on a true incident, “The Nihilist Girl”; two plays written in Swedish with writer Anna Charlotte Leffler under the title “Struggle for Happiness”, concerning the contrast between real and ideal fates in life; and some journalistic articles. In 1888 Kovalevskaya received the prestigious French Prix Bordin for mathematics. Death from pneumonia in 1891 cut short Kovalevskaya's dual careers as mature scientist and budding author. In the early twentieth century her story served as inspiration for science-minded girls throughout Europe. Her mathematics - in particular, equations describing the motions of rotating solids over time ("Kovalevsky's top") - has particular relevance in the space age.

Stephen Smale

Stephen Smale was born in Flint, Michigan in 1930. At the age of five he lived on a farm while his father worked in the city for General Motors. For eight years Stephen attended elementary school and afterwards, he studied at high school where his favourite subject was chemistry.When he entered in the University of Michigan he was interested in physics, but after failing a course, he changed to mathematics. Smale finally earned his Ph.D. in 1957.

His career began as an instructor in the University of Chicago from 1956 until 1958. In 1958, he firstly astounded the world with a proof of a sphere aversion. In that year, he also learnt about Pontryagin's work on structurally stable vector fields and he began to apply topological methods to study the these problems.

Between 1958 and 1960 he worked at the Institute for Advanced Study at Princeton on a National Science Foundation Postdoctoral Fellowship. He was allowed to study chaotic phenomena.

In 1960 Smale was appointed an associate professor of mathematics at the University of California at Berkeley, moving to a professorship at Columbia University the following year. In 1964 he returned to a professorship at the University of California at Berkeley where he has spent the main part of his career. He retired from Berkeley in 1995 and took up a post as professor at the City University of Hong Kong.

Smale's impressive results was his work on the generalised Poincaré conjecture, which is one of the famous problems of 20^th-century mathematics, but another areas in which Smale has contributed enormously is in Morse theory which he has applied to multiple integral problems, and in strange attractors.

He has won Fields Medal, National Medal of Science and Wolf Prize in addition to other prizes and honorary degrees.

Ludovica Russo and Paula Cascante

Archimedes of Siracuse

Archimedes of Syracuse (287 BC-212 BC) was a Greek mathematician, inventor and astronomer. He is famous for using the method of exhaustion to compute the area of a shape by filling the circle with a polygon of a greater and greater number of sides. He was killed during the second Punic war by a roman soldier which didn't known who he was. Cicero wrote about his visit to Archimedes' tomb, which was surmounted by a sphere inscribed within a cylinder.

Muhammad ibn Musa al- Khwarizmi Biography

Al-Khwarizmi was one of the most brilliant muslim scholars in the early Islamic culture. He was an astronomer, geographer, astrologer, and most importantly a mathematician.
Thanks to his devotion to math, we have the term "Algebra" from Hisab al-Jabr wa-al-Muqabala
His algebra book was the first book that had the systematic solution of linear and quadratic equations, hence, he is considered the "FATHER OF ALGEBRA" He was also on of the first first to use zero as a place holder in positional base notation.




>. Statue of Muhammad ibn Mūsā al-Khwārizmī, sitting in front of Khiva, north west of Uzbekistan.
Al-Khwārizmī was born in Khiva and lived in Persia in the year 800s.

Photo from:http://www.flickr.com/photos/michelv/1678696282/

EUCLID

Euclid was a Greek mathematician, often referred to as the "Father of geometry".

He was born in Athens but he was active in Alexandria during the reign of Ptolemy I.

His Elements is one of the most influential works in the history of mathematics. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.

Euclid is the anglicized version of the Greek name Εὐκλείδης, meaning "Good Glory"

Leonardo Fibonacci

Leonardo Fibonacci (c.1170 - c.1250) was an Italian mathematician best known to the modern world for the spreading of Hindu-Arabic numeral system in Europe with is booko "Liber Abaci" and for the number sequence named as him.
The Fibonacci number, very used in Algebra, is a sequence of numbers, each of one is the sum of the previous two numbers and it starts with 0: 0,1,1,2,3,5,8,13,21, 34, 55, 89, 144, 233, 377, 610, 987, ... .

Archimedes of Syracuse

Archimedes of Syracuse (c. 287BC- c. 212 BC) - was a Greek mathematician, physicist, engineer, inventor, and astronomer. He was one of the greatest of mathematicians.Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia.

The Archimedes Screw

The Archimedes screw can raise water efficiently.

A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer described how King Hieron II commissioned Archimedes to design a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a naval warship. It was capable of carrying 600 people and included garden decorations, a gymnasium and a temple. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes screw was 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.

Archimedes' Principle:

A gold crown for 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. The weight of the crown was known,but it had irregular shape.Archimedes thought about this problem every day.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. According to the legend Archimedes ran at the street naked and he was crying "Eureka!".The test was conducted successfully, proving that silver had indeed been mixed in.

Euclid of Alexandria

There isn't clear information about Euclid's life. where/when he was born and dead unknown. No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's imagination
the few historical references about Euclid written centuries by Proclus and Pappus  of Alexandria after he died. Euclid wrote a book which called Elements. When King Ptolemy asked if there was a shorter path to learning geometry than Euclid's Elements, Euclid answered there isn't royal road to geometry. Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.
There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are "from the edition of Theon" or the "lectures of Theon",while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.
Although best-known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization the Euclidean algorithm for finding the greatest common divisor of two numbers.
The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

Homework for the week commencing 11 July 2011

The topic of this week is Past, Present and Future which culminates in a visit to Science Museum on Friday.

The web-site MathsICSsummer201.blogspot.com contains a list of famous mathematicians and Noble prize winners (right side of the from page). Please, choose one, research on his/her biography and mathematics and blog it.

No copying and pasting, please. Pictures of respective mathematicians are welcome and invitations has been duly sent to your email.

Please, NOTE, that if you do NOT have google.mail account, you will be guided through steps to create one, once you have received the invite.

Good luck!

Welcome!

This page is created as a resource form students attending ICS in the summer 2011.
Location: Hyde Park