How did Euler make the concept of a function central to analysis?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
Leonhard Euler transformed mathematical analysis by systematically separating calculus from its origins in the geometry of curves, establishing the function as the central organizing concept of the field.1
In his seminal 1748 work, Introductio in Analysin Infinitorum, Euler defined a function as an analytical expression composed of variables and constants. By shifting the focus from geometric diagrams to these analytic expressions, he allowed mathematicians to study relationships between quantities independently of geometric constructions.12
Euler’s approach is exemplified by his treatment of trigonometric functions. Rather than viewing sines and cosines as lengths of lines within a circle, he redefined them as numerical ratios, enabling their study through algebraic and analytic methods. Furthermore, he introduced standardized notation, including the use of the letter f and parentheses to denote a function, which remains the standard today. By prioritizing this analytic framework, Euler provided a unified basis for calculus that moved beyond disparate geometric techniques, setting a precedent for modern mathematical practice.12