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Foundations of Differential Calculus by Euler John D Blanton



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Foundations of Differential Calculus by Euler , Translated by John D Blanton.
What differential calculus, and, in general, analysis of the infinite, might be can hardly be explained to those innocent of any knowledge of it. Nor can we here offer a definition at the beginning of this dissertation as is sometimes done in other disciplines. It is not that there is no clear definition of this calculus; rather, the fact is that in order to understand the definition there are concepts that must first be understood. Besides those ideas in common usage, there are also others from finite analysis that are much less common and are usually explained in the course of the development of the differential calculus. For this reason, it is not possible to understand a definition before its principles are sufficiently clearly seen.
In the first place, this calculus is concerned with variable quantities. Although every quantity can naturally be increased or decreased without limit, still, since calculus is directed to a certain purpose, we think of some quantities as being constantly the same magnitude, while others change through all the stages of increasing and decreasing. We note this distinction and call the former constant quantities and the latter variables. This characteristic difference is not required by the nature of things, but rather because of the special question addressed by the calculus.

Foundations of Differential Calculus by Euler cover the following topics.

  • 1. On Finite Difference
    2. On the Use of Difference in the Theory of Series
    3. On the finite and the Infinitly Small
    4. On the Nature of Differentials of Each Order
    5. On the Differentiation of Algebraic Functions on One Variable
    6. On the Differentiation of Transcendental Functions
    7. On the Differentiation of Functions of Two or More Variable
    8. On the Higher Differentiation of Differential Formulas
    9. On Differential Equations

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