Editing Blaise Pascal (section)

===''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''&nbsp;+&nbsp;1)th row and (''n''&nbsp;+&nbsp;1)th column ''t''<sub>''mn''</sub>. Then ''t''<sub>''mn''</sub>&nbsp;=&nbsp;''t''<sub>''m''–1,''n''</sub>&nbsp;+&nbsp;''t''<sub>''m'',''n''–1</sub>, for ''m''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;... and ''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;... The boundary conditions are ''t''<sub>''m'',−1</sub>&nbsp;=&nbsp;0, ''t''<sub>−1,''n''</sub>&nbsp;=&nbsp;0 for ''m''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;... and ''n''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;... The generator ''t''<sub>00</sub>&nbsp;=&nbsp;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.
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