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Pascal's triangle is named after the
French mathematician and philosopher Blaise Pascal (1623-62),
who wrote a *Treatise on the Arithmetical Triangle* describing
it. But Pascal was not the first to draw out this triangle or
to notice its amazing properties!

Long before
Pascal, 10th century Indian mathematicians described this array
of numbers as useful for representing the number of combinations
of short and long sounds in poetic meters. The triangle also appears
in the writings of Omar
Khayyam, the great eleventh-century astronomer, poet, philosopher,
and mathematician, who lived in what is modern-day Iran.

The Chinese mathematician Chu
Shih Chieh depicted the triangle and indicated its use in
providing coefficients for the binomial expansion of in
his 1303 treatise *The Precious Mirror of the Four Elements*.
Below is a reproduction of the triangle from Chu Shih Chieh, in
Chinese numerals

and in our arabic numerals:

(Both illustrations from Georges Ifrah, *The Universal History
of Numbers from Prehistory to the Invention of the Computer*.
New York: John Wiley & Sons, 1981, 1998.)

Pascal's work on the triangle
stemmed from the popularity of gambling. A French nobleman had
approached him with a question about gambling with dice. Pascal
shared the question with another famous mathematician, Fermat,
and Pascal's *Arithmetical Triangle* was the result.

Using Pascal's triangle, one can in fact find the number of ways of choosing
*k* items from a set of *n *items simply by looking
at the *k*th entry on the *n*th row of the triangle.
So, to see how many different trios you could form using the 45
members of your jazz band, you would look at the 3nd entry on
the 45th row. (The "1"* *at the top of the triangle
is considered the "0"th row, and the first entry on
each row is labeled the "0"th entry on the row.)

Since Pascal's time, mathematicians have found numerous patterns in Pascal's triangle. Some of the most interesting patterns are obtained by coloring in multiples of various numbers in Pascal's triangle; the results form endlessly repeating patterns called fractals.

**Blaise Pascal.**

(Source: http://www-groups.dcs.st-andrews.ac.uk/~history/PictDisplay/Pascal.html)

Pascal made several other
important contributions to the history of mathematics, including
the first digital calculator, which he designed to help his father,
who was a tax collector. Adding French currency was difficult,
because the currency consisted of livres, sols, and deniers, with
12 deniers in a sol and 20 sols in a livre.

Pascal's
machine, called the *Pascaline*, never was a great success,
however. Fifty prototypes were manufactured, but the machine did
not sell well--perhaps because the only arithmetical function
it could perform was addition!

(Illustration source: http://starform.infj.ulst.ac.uk/Billsweb/PGCert/intranets/Graham/Assignment/GIFS/Pascul.htm)

We arrive at truth, not by reason only, but also by the heart.

*Pensees* (1670)

It is not certain that everything is uncertain.

*Pensees* (1670)

The excitement that a gambler feels when making a bet is equal
to the amount he might win times the probability of winning it.

N Rose *Mathematical Maxims and Minims* (Raleigh N C 1988).

Let us weigh the gain and the loss in wagering that God is.
Let us consider the two possibilities. If you gain, you gain all;
if you lose, you lose nothing. Hesitate not, then, to wager that
He is.

The last thing one knows when writing a book is what to put
first.

*Pensees* (1670)

The more I see of men, the better I like my dog.

H Eves *Return to Mathematical Circles* (Boston 1988).

I have made this letter longer than usual, because I lack the time to make it short.

(Source for quotations: http://www-groups.dcs.st-andrews.ac.uk/~history/Quotations/Pascal.html)

*Fun links:*

Patterns in Pascal's triangle, with an applet to allow you
virtually to color entries divisible by *x*: http://www.mat.bham.ac.uk/funfair/java/pascal/

Probability Plaza, with applets for calculating permutations and combinations and a Pascal's triangle row generator: http://library.thinkquest.org/C006087/english/interactive.shtml

More coloring: http://ted.educ.sfu.ca/people/students/jane/act1.html

And: http://www.cs.washington.edu/homes/jbaer/classes/blaise/bigblaise.html

And: http://www.cecm.sfu.ca/cgi-bin/organics/pascalform

Pascal's triangle generator; you specify how many rows: http://forum.swarthmore.edu/~ken/pascal.cgi

A poem on Pascal's triangle: http://www.contrib.andrew.cmu.edu/~dr4b/writing/newstuff/sestina

Generate your own Pascal's triangle and check it on line: http://www.educ.sfu.ca/people/students/jane/

Use Pascal's triangle to compute the number of presents received in the 12 days of Christmas: http://dimacs.rutgers.edu/~judyann/LP/lessons/12.days.pascal.html

*For more information:*

http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Pascal.html

http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/pascal_bla.html

http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html

http://www.britannica.com/eb/article?eu=114515

http://www.math.sfu.ca/histmath/Europe/17thCenturyAD/Blaise.htm

http://www.maa.org/mathland/mathland_2_10.html

http://www.math.usouthal.edu/~brick/teaching/math110/pascal-tri.html

Pascal's triangle and probability: http://www.peddie.org/Resources/MathJournal/binomial.htm

And: http://ted.educ.sfu.ca/people/students/jane/prob.html

On the Pascaline (or Pascal's calculating maching): http://starform.infj.ulst.ac.uk/Billsweb/PGCert/intranets/Graham/Assignment/History3.htm

And: http://www.maxmon.com/1640ad.htm

By Laura Smoller, UALR Department of History.

March 2001.