Newton's "De analysi per aequationes numero terminorum infinitas" presents the foundational principles of calculus, specifically focusing on methods for calculating areas and tangents. Its central thesis is that quantities can be represented and manipulated using infinite series, enabling the solution of problems previously intractable with finite methods. The book demonstrates techniques for approximating areas under curves and finding instantaneous rates of change by treating these quantities as sums of infinitely many infinitesimal parts.
Readers gain an understanding of how infinite series can be used as a powerful analytical tool. Key ideas include the concept of approximating functions with polynomials and the derivation of derivatives and integrals from these series. The work establishes the analytical power of approaching continuous change through discrete, infinitely divisible components, laying the groundwork for differential and integral calculus.
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Key concepts
- Infinite Series — A sequence of numbers that continues indefinitely, used here to represent continuous quantities.
- Method of Fluxions — Newton's term for calculus, focusing on the rate of change (fluxions) of continuously varying quantities.
- Quadrature — The process of finding the area under a curve, achieved by summing infinitely many infinitesimally thin rectangles.
- Tangents — The method for finding the slope of a curve at any given point, representing its instantaneous rate of change.