Pascal

“Man's Greatness Lies in the Power of Thought”
-Blaise Pascal

Jessica: Hello, we are very pleased to have scientist, philosopher and mathematician, Blaise Pascal with us today. Mr. Pascal, thank you for joining us, would you like to start off by telling us a little bit about your childhood and early years?

Pascal: Well, I was born at Clermont on June 19th, 1623. My father was a local judge. When I was young, my parents had me stay at home because they wanted me to become learned in the area of linguistics.

Interestingly enough, I was not originally supposed to learn mathematics. However, the fact that I was not suppose to learn about mathematics intrigued me and I gave up my playtime to study geometry. When my father saw the aptitude I had for mathematics, he gave me a copy of Euclid's Elements. That book was one of the best presents that I ever got. It seemed to open up a whole new world to me, and I couldn't wait to explore it. When I was fourteen years old I was admitted to the weekly meetings of French geometriticians, such as Roberval, Mersenne, and Mydorge. As you can imagine, this was quite an honor, especially at my age. When I was eighteen years old, I constructed my first arithmetical machine, which I later improved on when I was twenty-six. At this time, I was mainly interested in analytical geometry and physics. I even repeated some of Torricelli's experiments that had to do with the pressure of the atmosphere.

Jessica: From what I understand, around 1650, you suddenly abandoned your studies on mathematics and the sciences and instead chose to study religion. What was the cause of this sudden shift in focus?

Pascal: Yes, at this time in my life, I felt a calling to contemplate the greatness and the misery of man; I chose to do this by studying religion. I even helped to persuade my youngest sister to also become involved in the more religious aspects of life.

Jessica: So what took you away from these contemplative studies and brought you back to your old life?

Pascal: I was brought back to my old life by necessity. In 1653, I was summoned to administer my father's estates. This summons also brought me back to my old studies and I experimented with the pressures that are exerted by liquids and gases. Around this time, I also invented the arithmetical triangle and created the calculus of probabilities in collaboration with Fermat.

Jessica: What would you consider to be one of your major accomplishments in the world of mathematics?

Pascal: Well, when I was in my early twenties I finally came out with a final standard workable model for a calculating device, which did bring me some fame. My work with the arithmetical triangle was a major accomplishment. The triangle was in existence for over a hundred years before I started working with it. The triangle became one of the major objects of my study. I wrote “A Treatise on the Arithmetic Triangle,” in 1653 and I found many new properties that had not been discovered yet. In the triangle, each horizontal line being formed forms the one above it. In this way, every number is equal to the sum of those above and to the left of it in the row above it. So, for example, 20 is equal to 10 +10, the numbers above and to the left of it.

My triangle can also be used for binomial expansion, the number in the diagonals give the coefficients of the expansion of a binomial. For example, if you look in the fifth diagonal, the numbers are 1,4,6,4,1, which are the coefficients of the binomial expansion of (a+b)4.

(a+b)^4=1a^4+4a^3b+6a^2b^2+4ab^3+1b^4

Jessica: Mr. Pascal, thank you for joining us today and sharing some of your personal experiences as well as explaining this small part of the work you did with the arithmetic triangle, that is now known as Pascal's Triangle.

Here are some examples of Binomial Expansion in Pascal's Triangle. The bold numbers are called binomial coefficients, they are the numbers that appear in my triangle.

(a+b)^0=1

(a+b)^1=1a + 1b

(a+b)^2=1a^2+2ab+1b^2

(a+b)^3=1a^3+3a^2b+1b^3

(a+b)^4=1a^4+4a^3b+6a^2b^2+4ab^3+1b^4

Pascal's Triangle




Works Cited

Ball, W.W. Rouse. “Blaise Pascal (1623-1662).” A Short Account of the History of Mathematics. 1908.

This was a very helpful source, it helped me to understand some of the mathematical concepts.

Green, Thomas and Charles Hamberg. Pascal's Triangle. Dale Seymour Publications, 1986.

This source was very helpful, it helped me to understand more about the way that the triangle worked and it did a very good job of explaining some of the patterns and rules within Pascal's triangle.

“Life of Blaise Pascal.” Top Biography. http://top-biography.com/9012 -Blaise%20Pascal/Life1.htm.

This source was moderately useful, it gave a good overview of Pascal's life.

O'Conner, JJ and E F Robertson. “Blaise Pascal.” School of Mathematics and Statistics University of St. Andrews, Scotland. December 1996.

This resource was pretty helpful for an overview of Pascal's life and a few of his major accomplishments.

Oppy, Graham. “On Rescher On Pascal's Wager.” International Journal for Philosophy Of Religion. 1990.

This resource was interesting, but not especially useful for this particular project.

Seymour, Dale. Visual Patterns in Pascal's Triangle. Dale Seymour Publications, 1986.

This book was very useful for learning about binomial expansion.