Blaise Pascal is best known for what is known as Pascal's Triangle (figure above). Pascal's Triangle is created by starting with 1 at the top, then to find each number you add the sum of the two numbers above it (it will always be 1 if it only has one number above it i.e. if it is on a side). Thus in the 6^th row (the row with 6 as it's second number), 15 is formed by adding 5 and 10. This process never ends and will continue well past the 7^th row (as the picture shows). Although this triangle is very simple to create it actually has many complex mathematical properties hidden in it. For instance if you add up the numbers in the nth row the sum is 2ⁿ. An example is the sum of the numbers in the 4^th row is 1 + 4 + 6 + 4 + 1=16 and 16 = 2⁴. Although that property is very interesting to mathematicians, a more useful property is that Pascal's Triangle can be used to find binomial expansions. This can be demonstrated by looking at (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ where the coefficients of the expression are 1, 4, 6, 4, 1; the 4^th row of Pascal's Triangle. Many more properties of the triangle are known and Pascal did discover several more results, but none that are as interesting and easily understood as the triangle and some of it's properties.

by Josiah Thornton

        The 17^th Century was full of mathematicians working on unsolved problems and creating new fields of mathematics. Some of these mathematicians are Sirsaac Newton and Gottfried Wilhelm Leibnitz (both credited with the discovery of modern calculus) and the Bernoullis. These mathematicians studied in locations throughout Europe, but in a time of nationalism and prestige France produced several important mathematicians. Three of these mathematicians are René Descartes, Blaise Pascal, and Pierre de Fermat.
         René Descartes major contribution to mathematics was in the field known as analytical geometry. Most of the results that he obtained are found in his La Géométrie. The advance that is generally noted as Descartes' greatest is the use of what is known as the Cartesian Coordinate System. His discovery involving this coordinate system is that any point in the plane (flat 2D surface) can be denoted by the distances from the point to two orthogonal lines. In other words a point two units to the left of the origin and 3 units above the origin can be denoted by (-2, 3).
         Descartes' La Géométrie is a collection of three books. In the first book, Descartes discussed a problem that had been posed by mathematicians during the time of the Greeks. To solve this problem he developed analytical geometry. In the second book, Descartes studied curves, by separating them into two different classes, geometrical -where dy/dx is algebraic- and mechanical - where dy/dx is transcendental- and discussing the implications that can be made by which class a curve falls into. He also used circles to determine the tangents to curves and given points (a task that was made simpler by the development of calculus 40 years later). His third and final book deals mostly with an analysis of algebra. One of the largest results found in this book is that he developed a method to find the number of positive and the number of negative roots of a function. Also, the use of letters at the beginning of the alphabet for known values and the letters at the end of the alphabet for unknown values originated from this book.