The Evolution of Mathematics

Elementary forms of mathematics have probably been with man throughout his evolution. As human societies advanced, so too did mathematics. From the 1500s to the present, a long lineage of mathematicians have revolutionized the field. These men were often of European origin. Only in the last century has the United States and Japan emerged as dominant mathematical forces. At present, either of these nations could lead the field into the future.

The first systems of numeration were invented by the Greeks and the Romans (Struik, 1987, p. 8081). Later, the Western merchant, Leonardo of Pisa, introduced the HinduArabic system of numeration into Western Europe. Europeans came to accept these ten symbols gradually. For many centuries, the Greek system of numeration remained popular. Computation was usually performed on the ancient abacus. Roman numerals were then used to register the result. Resistance to HinduArabic numerals resulted from the fact that they made the ledgers difficult to read. Throughout the Middle Ages, Roman numerals can be found in merchants' ledgers. As trade increased, interest in mathematics increased also. Initially, this interest was a matter of practical necessity. Arithmetic and algebra were taught outside the universities by selfmade, Rechenmeisters ("reckon masters," arithmeticians). This knowledge was needed primarily for bookkeeping and navigation (Str

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e., a formula or statement) about some integer "n." This proposition is then shown to be true for all positive integers "n," or at least for all integers greater than or equal to some specific integer n(0). With the eighteenth century, mathematics primarily focused on calculus and its application to mechanics (Struik, 1987, p. 117). Important figures during this time were the Bernoulli brothers, Jakob and Johann, as well as Leonhard Euler, PierreSimon Laplace, and JosephLuis Lagrange (Struik, 1987, p. 117139). Due to their contribution to the problem of brachistochrone, Jakob and Johann Bernoulli are considered the inventors of the calculus of variations. This work basically involves the curve of quickest descent for a mass point moving between two points in a gravitational field. It was Johann's son, Daniel, who concentrated on applications of mathematics and eventually established Bernoulli's law on hydraulic pressure. A student of the Bernoullis, Leonhard Euler, then approached the problem of the vibrating string. His investigations led to the theory of partial differential equations. Although Euler became totally blind in 1766, he continued his research (Struik, 1987, p. 120). Eventually, Euler made contribu
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