Scenes from the History of Real Functions by Professor Dr. Fyodor A. Medvedev (auth.)

By Professor Dr. Fyodor A. Medvedev (auth.)

To try to assemble a comparatively whole bibliography of the idea of capabilities of a true variable with the needful bibliographical information, to enumer­ ate the names of the mathematicians who've studied this topic, express their basic effects, and in addition comprise the main crucial biographical info approximately them, to behavior a list of the thoughts and strategies which were and stay utilized within the conception of features of a true variable ... briefly, to hold out somebody of those initiatives with acceptable completeness will require a separate ebook related to a corresponding volume of labor. accordingly the note essays happens within the identify of the current paintings, permitting a few freedom within the number of fabric. In justification of this option, it's moderate to aim to signify to some extent the topic to whose heritage those essays are dedicated. the reality of the problem is this is a hopeless firm if one calls for this type of characterization to be exhaustively whole and concise. No residing topic may be given a last definition with out upsetting a few objections, often critical ones. but when we make no such claims, a characterization is feasible; and if the 1st essay of the current e-book seems unconvincing to a person, the reason being the non-public fault of the writer, and never the target necessity of the attempt.

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Additional resources for Scenes from the History of Real Functions

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These values had to be described somehow. The descriptions given by Euler are interesting in two respects. First, "a variable quantity is an indefinite or universal quantity containing within itself all possible values" [1, p. 4]. In other words for Euler a variable quantity is not a flowing, changing quantity, as it was, for example, for Maclaurin [1, pp. 52-57, and many other places], but rather some real structure: "just as the concepts of species and genera are formed from the concepts of individuals, so a variable quantity is a genus in which all the definite quantities are contained" [1, p.

15J and d'Alembert made the same assertion (Natucci [1, p. 91]). Enest rom-a rather scrupulous historian of mathematics--considered this assertion unfounded, although it may be that someone did indeed use the word function in this sense. 48 Chapter 2 The history of the concept of a f1lllction from the latter" [1, T. 1, p. 34 Lacroix not only introduced this definition into his basic course [1] but also stated it in some more elementary textbooks on mathematical analysis, as well as in the second edition of the treatise under discussion.

Having precisely John Bernoulli's definition of a function in mind (with his own amendments to it), Euler expressed the fundamental idea that "all of infinitesimal analysis revolves about variable quantities and their functions" [1, p. VIII] and he adopted the Bernoulli definition as the point of departure. But since the term variable magnitude or variable quantity occurs in the Bernoulli definition, Euler preceded the definition of the concept of a function with an explanation of this term. The Euler explanation deserves attention for the following reasons.

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