ABSTRACT

In 1807, a meeting of the French Academy of Science was the scene of one of the most electrifying moments in mathematical history: the presentation of a paper by J. Fourier in which he asserted that any function could be described as a sum of sine and cosine functions. It was soon demonstrated that Fourier's claim was too all encompassing to be totally accurate, but nevertheless he was close enough to the truth that a vast area of mathematics has evolved from his germinal observation. Brook Taylor published in the early eighteenth century a formula for expanding functions as infinite series; it appears however that Gregory had essentially discovered this expression some 40 years previously, but as is often the case in mathematics, the glory was to remain with another. Both Taylor polynomials and the Taylor series expansions will be dealt with presently; they represent two of the crowning early achievements of mathematical analysis.