Blaise Pascal Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ==Mathematics== ===Probability=== Pascal's development of [[probability theory]] was his most influential contribution to mathematics. 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."<ref>{{cite journal|last1=Ross|first1=John F.|year=2004|title=Pascal's legacy|journal=EMBO Reports|volume=5|issue=Suppl 1|pages=S7–S10|doi=10.1038/sj.embor.7400229|pmc=1299210|pmid=15459727}}</ref> However, Pascal and Fermat, though doing important early work in probability theory, did not 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]]. In 1654, prompted by his friend the [[Chevalier de Méré]], he corresponded with [[Pierre de Fermat]] on the subject of gambling problems, and from that collaboration was born the mathematical theory of [[probability|probabilities]].{{sfn|Devlin|p=24}} 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 [[division of the stakes|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 [[Gottfried Wilhelm Leibniz|Leibniz]]' formulation of the [[calculus]].<ref>{{cite web|title=The Mathematical Leibniz|url=http://www.math.rutgers.edu/courses/436/Honors02/leibniz.html|access-date=16 August 2009|publisher=Math.rutgers.edu|archive-date=3 February 2017|archive-url=https://web.archive.org/web/20170203084344/http://www.math.rutgers.edu/courses/436/Honors02/leibniz.html|url-status=live}}</ref> ===''Treatise on the Arithmetical Triangle''=== {{Main|Pascal's triangle}} [[File:PascalTriangleAnimated2.gif|thumb|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.]] Pascal's ''Traité du triangle arithmétique'', written in 1654 but published posthumously in 1665, described a convenient tabular presentation for [[binomial coefficient]]s which he called the arithmetical triangle, but is now called [[Pascal's triangle]].<ref name=":1">{{Cite book|last=Katz|first=Victor|title=A History of Mathematics: An Introduction|publisher=Addison-Wesley|year=2009|isbn=978-0-321-38700-4|pages=491|chapter=14.3: Elementary Probability}}</ref><ref>{{cite book| url = http://www.bookrags.com/research/pascals-triangle-wom/| title = Pascal's triangle {{!}} World of Mathematics Summary| access-date = 4 December 2020| archive-date = 4 March 2016| archive-url = https://web.archive.org/web/20160304065153/http://www.bookrags.com/research/pascals-triangle-wom/| url-status = live}}</ref> The triangle can also be represented: {| class="wikitable" |- ! style="width:20px;" | ! style="width:20px;" |0 ! style="width:20px;" |1 ! style="width:20px;" |2 ! style="width:20px;" |3 ! style="width:20px;" |4 ! style="width:20px;" |5 ! style="width:20px;" |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 defined the numbers in the triangle by [[recursion]]: Call the number in the (''m'' + 1)th row and (''n'' + 1)th column ''t''<sub>''mn''</sub>. Then ''t''<sub>''mn''</sub> = ''t''<sub>''m''–1,''n''</sub> + ''t''<sub>''m'',''n''–1</sub>, for ''m'' = 0, 1, 2, ... and ''n'' = 0, 1, 2, ... The boundary conditions are ''t''<sub>''m'',−1</sub> = 0, ''t''<sub>−1,''n''</sub> = 0 for ''m'' = 1, 2, 3, ... and ''n'' = 1, 2, 3, ... The generator ''t''<sub>00</sub> = 1. Pascal concluded with the proof, :<math>t_{mn} = \frac{(m+n)(m+n-1)\cdots(m+1)}{n(n-1)\cdots 1}.</math> In the same treatise, Pascal gave an explicit statement of the principle of [[mathematical induction]].<ref name=":1" /> In 1654, he proved [[Faulhaber's formula|''Pascal's identity'']] relating the sums of the ''p''-th powers of the first ''n'' positive integers for ''p'' = 0, 1, 2, ..., ''k''.<ref>{{cite journal|author=Kieren MacMillan, Jonathan Sondow|title=Proofs of power sum and binomial coefficient congruences via Pascal's identity |journal=[[American Mathematical Monthly]] |year=2011 |volume=118 |issue=6 |pages=549–551 |doi=10.4169/amer.math.monthly.118.06.549|arxiv=1011.0076|s2cid=207521003 }}</ref> That same year, Pascal had a religious experience, and mostly gave up work in mathematics. {{Clear}} ===Cycloid=== [[File:Pascal Pajou Louvre RF2981.jpg|thumb|upright|Pascal studying the [[cycloid]], by [[Augustin Pajou]], 1785, [[Louvre]]|alt=]] In 1658, Pascal, while suffering from a toothache, began considering several problems concerning the cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay<ref name="Ball_1960">{{cite book |last=Ball |first= W. W. Rouse |date=2010-09-16 |title=A Short Account of the History of Mathematics |url=https://www.gutenberg.org/files/31246/31246-pdf.pdf |location=New York, NY, USA |publisher=Dover Publications, Inc |page=234 |isbn=978-0486206301}}</ref> and, to publicize the results, proposed a contest.<ref name="Ferroli_1935">{{cite journal |last1=Ferroli |first1=D. |date=April 1935 |title=A Note on Blaise Pascal (1623-1662). A Forerunner of Leibnitz and Newton in the Discovery of the Calculus |url=https://www.jstor.org/stable/24221628 |journal=Current Science |volume=3 |issue=10 |pages=459 |access-date=2024-03-02}}</ref> Pascal proposed three questions relating to the [[Center of mass|center of gravity]], area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish [[doubloon]]s. Pascal, [[Gilles de Roberval]] and [[Pierre de Carcavi]] were the judges, and neither of the two submissions (by [[John Wallis]] and [[Antoine de Lalouvère]]) were judged to be adequate.<ref>{{citation | last=Conner | first=James A. | title=Pascal's Wager: The Man Who Played Dice with God | pages=[https://archive.org/details/pascalswagermanw00conn/page/224 224] | isbn=9780060766917 | edition=1st | year=2006 | publisher=HarperCollins | url-access=registration | url=https://archive.org/details/pascalswagermanw00conn/page/224 }}</ref> While the contest was ongoing, [[Christopher Wren]] sent Pascal a proposal for a proof of the [[arc length|rectification]] of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis's ''Tractus Duo'', giving Wren priority for the first published proof. {{Clear}} Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page