Pascal's Treatise on the Arithmetical Triangle: Mathematical Induction, Combinations, the Binomial Theorem and Fermat's Theorem

Abstract

Introduction Blaise Pascal (1623–1662) was born in Clermont-Ferrand in central France. Even as a teenager his father introduced him to meetings for mathematical discussion in Paris run by Marin Mersenne, who served as a primary conduit for transmitting mathematical ideas widely at that time, before the existence of any research journals. He quickly became involved in the development of projective geometry, the first in a sequence of highly creative mathematical and scientific episodes in his life, punctuated by periods of religious fervor. Around age twenty-one he spent several years developing a mechanical addition and subtraction machine, in part to help his father in tax computations as a local administrator. It was the first of its kind ever to be marketed. Then for several years he was at the center of efforts to understand vacuum, which led to an understanding of barometric pressure. In fact the scientific unit of pressure is named the pascal . He is also known for Pascal's Law on the behavior of fluid pressure. Around 1654 Pascal conducted his studies on the Arithmetical Triangle (“Pascal's Triangle”) and its relationship to probabilities. His correspondence with Pierre de Fermat (1601–1665) in that year marks the beginning of probability theory. Several years later, Pascal refined his ideas on area problems via the method of indivisibles already being developed by others, and solved various problems of areas, volumes, centers of gravity, and lengths of curves.