Pascal's Triangle

History

Although Pascal’s Triangle is named after seventeenth century mathematician, Blaise Pascal, several other mathematicians knew about and applied their knowledge of the triangle hundreds of years before the birth of Pascal in 1623. As of today, the triangle appears to have been discovered independently by both the Persians and the Chinese during the eleventh century. Although no longer in existence, the work of Chinese mathematician Chia Hsien (ca. 1050) showed that he “was using the triangle to extract square and cube roots of numbers” (Clawson 133). Also having a method of extracting roots of numbers, Omar Khayyam (1048? – 1113?), a Persian mathematician, seemed to have had knowledge of the so-called Pascal’s Triangle. In China, after Chia Hsien’s discovery of the relationship between extracting roots and the binomial coefficients of the triangle, work continued on this topic by several Chinese algebraists to solve higher than cubic equations (Calinger 189). One of these mathematicians, Yang Hui (ca. 1261-75), has provided for current researchers the earliest display of Pascal’s Triangle. Chinese mathematician, Zhu Shijie, once again, revealed a visual representation of the triangle in 1303. It was in his work that Shijie spoke of the triangle as being ancient in his time (Eves 159). Thus, Pascal’s Triangle and the use of its binomial coefficients were known long before the mathematical minds of Pascal and Newton came into this world.

Years after it first appeared in Persia and China, the triangle came to be known as Pascal’s Triangle with Blaise Pascal’s completion of Traité du triangle arithmétique in 1654. Making use of the already known array of binomial coefficients, French mathematician Pascal developed many of the triangle’s properties and applications within these writings. Although Pascal is best known for his work with the arithmetic triangle, he made many other contributions to mathematics during his lifetime. Throughout his thirty-nine years, Pascal also discovered an important theorem in geometry, worked with cycloids, invented a calculating machine, laid the foundations of probability, and planted the seeds of calculus (Eves 242-6). Pascal’s contributions to mathematics, especially of ‘his’ triangle, were unquestionably brought forth from the mind of a highly intelligent man.