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Ulrich Felgner

    Sudhoffs Archiv
    |April 19, 2018
    PubMed
    Summary
    This summary is machine-generated.

    Fermat's method of adequality, crucial for calculus, is clarified. The study shows it relies on a "less-than" relation and polynomial analysis, not infinitesimals, making Fermat's mathematics clear and correct.

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    Area of Science:

    • History of Mathematics
    • Calculus Development
    • Number Theory

    Background:

    • Fermat's method of adequality is a foundational concept in early calculus, essential for determining maxima, minima, and tangents.
    • The precise mathematical underpinnings of Fermat's adequality have remained a subject of historical debate, often misinterpreted as requiring infinitesimals.
    • The 'parísotē s' from Diophantus is identified as a key source influencing Fermat's approach.

    Purpose of the Study:

    • To provide a clear and accurate interpretation of Fermat's method of adequality.
    • To demonstrate that Fermat's method does not rely on infinitesimal calculus.
    • To analyze the historical origins and mathematical techniques employed by Fermat.

    Main Methods:

    • Analysis of Fermat's use of the 'less-than' relation in optimization problems.

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  • Examination of Fermat's construction of auxiliary polynomials of the form hψ(h).
  • Demonstration of how these polynomials establish strict positivity near zero, leading to a double root at h=0.

    • Main Results:

    • Fermat's adequality is shown to be based on a rigorous comparison of values using inequalities, not infinitesimals.
    • The method involves analyzing auxiliary polynomials to deduce that the constant term of ψ(h) must be zero.
    • This interpretation resolves ambiguities and validates Fermat's mathematical techniques within his historical context.

    Conclusions:

    • Fermat's method of adequality is mathematically sound and does not require concepts beyond those available in his time.
    • The interpretation clarifies Fermat's transition from 'adequality' (inequality) to 'equality' through polynomial root analysis.
    • The study offers a historically accurate understanding of a pivotal method in the development of calculus.