Pascal's triangle was being used long before Pascal published it.
The triangle was known to the Chinese as early as the twelfth century, which was about five centuries before the time of Pascal.
Ji Xian is attributed with writing the triangle out to the 6th row, and identifying the method we know today of generating it:
a given element of the triangle is found by adding together the two values above it.
Yang Hui provided the first known presentation of Pascal's triangle in 1261, written out to the 6th row, and it was later depicted
by Chu Shi-Kie in 1303 written out to the 8th row, as shown below(Humphries & Scott, 2003).
In 1654, Pascal began his work on probability theory and his own arithmetic triangle.
Among the people that Pascal was associated with, gambling was a popular distraction.
Two of the men in this circle sought an answer to the question of what to do with money already
wagered if a game has to be interrupted. The question was "how should it be divided according to what each player
might reasonably expect to win (or lose) if the play were continued?"(Davidson, 1983, pg. 14). Pascal and his close associate
Fermat saw this as a mathematical problem, and in an exchange of letters from July to October of 1654 they were able to work out a solution.
This is when Pascal first started studying the triangle and how it could be used (Davidson, 1983).
Pascal stared constructing his own triangle by studying how Chu Shi-Kie constructed his triangle.
He started with two lines that formed a right angle. He then divided it into equal lengths and connected the resulting points to
make a triangle composed of boxes similar to the picture shown below. He noticed that numbers can now be seen with
reference to other numbers in a context of systematic and repeating relationships. The numbers on the outside
(on the very top and very left of the triangle) were simply there to organize it and are not actually included in the triangle itself.
Once we get into the actual triangle we can see that any number (x) turns out to be the sum of the number in the box directly
to the left of (x) plus the number in the square directly above (x) (Davidson, 1983).
Today we use a triangle that looks more like the following picture.
We describe the triangle in terms of rows rather than parallel rank and perpendicular rank.
To construct today's triangle we start with the "zero" row, which is a 1.
To construct the rest of the rows, into infinity, we simply start and end each row with a 1.
To form the other numbers in each row we would just have to add the two numbers in the row immediately above it.
For example, in row 4 we have 1 4 6 4 1. We can see that we start and end the row with a one.
Then for the first 4 in the row we would need to add the 3 and the 1 that are immediately above it in order to get the sum of 4.
Once again, to get the 6 we would add the 3 and the 3 immediately above it. This pattern continues until each desired row is complete.
