Abstract
Let \(y=f(x)\) always be a function of the independent variable x, i an infinitely small quantity and h a finite quantity. If we set \( i = \alpha h, \) \( \alpha \) will also be an infinitely small quantity, and we will have identically
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Notes
- 1.
This requirement is necessary to ensure the function is well defined for negative values of x. Cauchy must also be assuming \(x \ne 0\) for the function \(\frac{a}{x}.\) He views this location as a solution of continuity (a point of discontinuity).
- 2.
When used in this manner, the term arc is an old trigonometric term referring to the signed length of the curve measured on a unit circle generated by a given angle at the center; therefore, at the same time it is the signed value of the angle.
- 3.
Cauchy uses the phrase ligne trigonométrique, translated here as trigonometric line, which is another old trigonometric term used to denote the signed lengths of the right triangle line segments which are represented by the various trigonometric functions. It is used to indicate circular function values, today generally referred to as the values of the six trigonometric functions.
- 4.
One of our modern differentiation rules – the derivative of a constant is zero.
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Cates, D.M. (2019). DIFFERENTIALS OF FUNCTIONS OF A SINGLE VARIABLE.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_4
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DOI: https://doi.org/10.1007/978-3-030-11036-9_4
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