How did Leonhard Euler help separate calculus from geometry?
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.
In the early 18th century, calculus was heavily rooted in the geometry of curves, which served as the primary object of study. Leonhard Euler was instrumental in shifting the field toward a purely analytic basis by systematically removing this reliance on geometric diagrams and constructions.1
Euler achieved this separation primarily through his work in the mid-18th century, most notably in his Introduction to the Analysis of the Infinite (1748). In this text, he established the function as the central organizing concept of analysis, defining it as an analytical expression composed of variables and constants.1
A key example of this transition was his treatment of trigonometry. While earlier mathematicians interpreted sine and cosine functions geometrically through lines and angles within a circle, Euler introduced them as quantities whose relationships could be studied independently of any geometric figure. By generalizing these concepts and focusing on algebraic expressions, Euler helped transform calculus into a discipline defined by analytic methods rather than spatial intuition.12