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How did Joseph-Louis Lagrange try to give calculus an algebraic foundation?

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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.

Joseph-Louis Lagrange sought to provide calculus with an algebraic foundation by removing reliance on geometric constructions, infinitesimals, and the concept of vanishing limits, which he viewed as logically problematic.12

In his 1797 treatise, Théorie des fonctions analytiques, Lagrange aimed to reduce the subject to the "algebraic analysis of finite quantities". Building on Leonhard Euler’s definition of a function as an analytic expression, Lagrange defined the "derived function" (or derivative) $f^{'} (x)$ as the coefficient of $i$ in the Taylor series expansion of $f (x + i)$ . By assuming that such power series expansions were generally possible, he attempted to build a complete theory of differential and integral calculus based entirely on algebraic operations.2

While this approach represented the pinnacle of the 18th-century analytic conception of calculus, it was eventually superseded in the 19th century by the work of mathematicians like Augustin-Louis Cauchy, who established the field on a more rigorous arithmetic basis using the concept of limit.12