Leonhard Euler and Differentials

Leonhard Euler and Differentials

Euler was the most prolific and one of the most influential mathematicians who ever lived. He made major contributions to both pure and applied mathematics and his collected works amount to over 70 volumes. So strong was his influence that historians like Boyer and Edwards refer to the eighteenth century as the Age of Euler.

Euler made the function concept fundamental in analysis. He saw a function as both any quantity depending on variables and also as any algebraic combination of constants and variables (including infinite sums or products). This is obviously not a modern definition of a function. Still, Euler used his function concept to maximal advantage. As we examine some of Euler’s computations, keep in mind the immense insight and unity he achieved with the function approach-a point of view we now take for granted.

In his Introducio in analysin infinitorum (1748), one sees the first systematic interpretation of logarithms as exponents. Prior to Euler, logarithms were typically viewed as terms of an arithmetic series in one-to-one correspondence with terms of a geometric series. Euler viewed trigonometric functions as numerical ratios rather than as ratios of line segments. He also studied properties of the elementary transcendental functions by the frequent use of their infinite series expansions. Euler often used infinite series indiscriminately, without regard to questions of convergence.

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