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How did Newton’s approach to calculus differ from Leibniz’s approach?

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transformation of a circular region into an approximately rectangular region — 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.

Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus in the late 17th century, but their approaches differed significantly in methodology, notation, and conceptual foundation.12

Newton’s Approach: Fluxions and Series

Newton viewed calculus primarily through the lens of infinite series and kinematics. He conceptualized variables as "fluents"—magnitudes that flow over time—and their rates of change as "fluxions". His work focused on algebraic manipulation, using power series to differentiate and integrate functions. In his Principia, he later moved toward a geometric theory of limits, known as the calculus of first and last ratios, to avoid analytic processes.23

Leibniz’s Approach: Infinitesimals and Notation

Leibniz approached calculus with a focus on logic and notation, treating continuous quantities as if they were discrete. He developed the now-standard symbols $d$ for differentials and $\int$ for integration. By defining the derivative as a quotient of infinitesimals ( $d f / d x$ ), he created a powerful, general algorithm that was more easily applied to a wide range of problems than Newton’s geometric methods.23

While both men faced criticism regarding the logical rigor of their use of infinitesimals, Leibniz’s superior notation and his ability to foster a community of researchers ensured his methods became the standard for modern analysis.23