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August 17, 2011

Pierre de Fermat (Born Too Lose)

**** * * Pierre de Fermat * * * *

Born August 17, 1601 Beaumont-de-Lomagne, France Died January 12, 1665 (aged 63) Castres, France Residence France Nationality French Fields Mathematics and Law Known for Number theory Analytic geometry Fermat's principle Probability Fermat's Last Theorem Influences François Viète

* * ***** * Pierre de Fermat * * ***** * French pronunciation: [pjɛːʁ dəfɛʁˈma]; 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 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 then unknown differential calculus, 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. Contents 1 Life and work 1.1 Work 1.2 Death 2 Assessment of his work 3 See also 4 Notes 4.1 Books referenced 5 Further reading 6 External links 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. Bust in the Salle des Illustres in Capitole de Toulouse 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 the title of councillor at the High Court of Judicature in 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[citation needed], 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. Pierre de Fermat 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.[citation needed] 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 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, 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 him 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 term Fermat functional was named in recognition of this role.[11] Death Plaque at the place of burial of Pierre de Fermat 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 He 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. Assessment of his work Holographic will handwritten by Fermat on 4 March 1660 — kept at the Departmental Archives of Haute-Garonne, in Toulouse 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 Weil wrote that "... what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for the modern theory of 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. See also Diagonal form Euler's theorem Fermat cubic Fermat Prize Fermat pseudoprime Fermat quotient Fermat's spiral Fermat's theorem (stationary points) dogmeat

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Quick Tip: Ever Thought About Using @Font-face for Icons?

Wayne Helman on Apr 23rd 2010 with 116 comments

Tutorial Details

Technology: CSS
Estimated Completion Time: 15 Minutes
Difficulty: Beginner

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This entry is part 11 of 16 in the CSS3 Mastery Session - Show All
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The evolution of Internet technologies never ceases to amaze. Seemingly daily, new concepts and techniques are being thought up by creative and talented people. With modern browsers being adopted at a greater rate, systems like CSS3 are becoming more and more viable for use on projects of all sizes. Clearly, this can be seen by looking at new services sprouting on-line like TypeKit. Conceptually, if we deconstruct a font down to it’s basic elements, we can make use of this technology for things other than type, icons.

The Need for Speed

For a short period of time, developers began producing websites with little regard for bandwidth consumption. HTML and CSS where restrictive and Adobe Flash was an open canvas for designers and developers to stuff animations and complex layouts into. This resulted in some extremely bandwidth heavy sites—we all remember a few. Those were the days before the proliferation of mobile smart phones.

With smart phones accessing the Internet more frequently, bandwidth and page load speeds have suddenly returned to the forefront. Thankfully, advances in HTML, CSS, and JavaScript have made that all possible. Central to webpage speed and responsiveness is the number of HTTP requests a page load must make. Modern browsers limit the number of requests to a single server. The W3C HTTP 1.1 specification reads

“A single-user client SHOULD NOT maintain more than 2 connections with any server or proxy. A proxy SHOULD use up to 2*N connections to another server or proxy, where N is the number of simultaneously active users. These guidelines are intended to improve HTTP response times and avoid congestion.”

One technique that has become increasingly popular is the use of CSS sprites. CSS sprites are designed to reduce the number of HTTP requests to the web server by combining many smaller images into a single larger image and defining a block level CSS element to only show a defined portion of the larger image. The technique is simple, but ingenious.

Deconstructing the Font

Fonts at their most basic molecular level are a series of vector glyphs packaged up into a single “glyph archive”.

CSS3 has introduced to the web development world the ability to embed fonts with the @face-face declaration. Without question, this advancement in Internet technologies is one of the most exciting and important stages in our brief history. With developers able to embed fonts of their choice, designers can produce layouts that will render far more consistently from platform to platform bringing the art of interactive layout closer to it’s print cousin.

If we take a closer look at the technology behind a font, we can gain a far better understanding of how they can be used and deployed. Fonts at their most basic molecular level are a series of vector glyphs packaged up into a single “glyph archive”. We can then reference each glyph by its corresponding character code. Theoretically, it’s very similar to the way in which we reference an array in almost any programming language—through a key/value pair.

With this in mind, the glyphs we reference can really be any vector-based single color image. This is nothing new—we’ve all seen Dingbats and Webdings. They are two examples of non-type fonts, that is, a series of vector based images compiled into a single font archive.

Abstracting and Expanding @font-face

With the advent of font embedding and the realization that fonts are essentially a series of simple vector glyphs, I began to experiment on how to use this format to my advantage. Conceptually, if I placed all required icons for a particular site into a custom font, I would then be able to use those icons anywhere on the site with the ability to change size and color, add backgrounds, shadows and rotation, and just about anything else CSS will allow for text. The added advantage being a single CSS sprite-like HTTP request.

To illustrate, I’ve compiled a new font with a few of the great icons from Brightmix.

I’ve used the lower case slots for plain icons, and the uppercase slots for the same icon in a circular treatment.

To use my new Icon Pack, I’ll first have to export my font set as a number of different font files (.eot, .woff, .ttf, .svg) to be compatible with all browsers. The topic of font embedding and file format converting is covered elsewhere, so I will avoid a detailed explanation here. However, the CSS would look something like this.

view plain copy to clipboard print ?

@font-face {
font-family: 'IconPack';
src: url('iconpack.eot');
src: local('IconPack'),
url('iconpack.woff') format('woff'),
url('iconpack.ttf') format('truetype'),
url('iconpack.svg#IconPack') format('svg');
}

@font-face { font-family: 'IconPack'; src: url('iconpack.eot'); src: local('IconPack'), url('iconpack.woff') format('woff'), url('iconpack.ttf') format('truetype'), url('iconpack.svg#IconPack') format('svg'); }

Once embedded, I now have a complete icon set in vector format to reference. To reference an icon I simply need a style that includes the font-family of “IconPack”.

image alt tag

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