Summary
Newton's "De analysi per aequationes numero terminorum infinitas" (On the Analysis by Infinite Series) establishes the foundational principles of calculus, specifically introducing the method of infinite series to solve problems previously intractable. Its central thesis is that continuous quantities can be represented and manipulated using infinite series, enabling the calculation of areas, tangents, and rates of change for a wide range of functions. The book demonstrates how to expand functions into these series and how to perform operations like differentiation and integration on them.
The work provides concrete methods for deriving the coefficients of these series and shows their application in solving differential equations and determining the areas under curves. Readers gain a direct understanding of how infinite series serve as a powerful tool for approximating and precisely determining values related to continuous functions, marking a pivotal shift in mathematical analysis.
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Key concepts
- Infinite Series — A sum of an infinite sequence of numbers, used by Newton to represent functions.
- Method of Fluxions — Newton's term for calculus, focusing on rates of change (fluxions) and their integrals.
- Tangent Method — A procedure for finding the slope of a curve at any point using infinite series.
- Quadrature Method — A technique for finding the area under a curve by integrating its infinite series representation.
- Binomial Theorem — Implicitly used and demonstrated for expanding expressions like $(a+b)^n$ into infinite series.