## Foundations of Differential CalculusWhat differential calculus, and, in general, analysis ofthe infinite, might be can hardly be explainedto those innocent ofany knowledge ofit. 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 courseofthe development ofthe 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 distinc tion 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. |

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### Contents

On Finite Differences | 1 |

On the Use of Differences in the Theory of Series | 25 |

On the Infinite and the Infinitely Small | 47 |

On the Nature of Differentials of Each Order | 63 |

On the Differentiation of Algebraic Functions of One Variable | 77 |

On the Differentiation of Transcendental Functions | 99 |

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2x dx a/dx algebraic function algebraic quantities arithmetic progression becomes clear constant differential constant quantity denominator differential calculus differential dx differential equation differential is assumed differential is equal dx dx dx dy dx is constant dxdy dy d2x easily equal to zero example exponent find the differential finite difference finite equation finite quantity follows formula fraction function of x Furthermore geometric ratio given equation given expression given function given series Hence higher differentials higher order homogeneous homogeneous function infinite number infinite series infinitely divisible infinitely large quantity infinitely small quantity infinity integral calculus let dy logarithm method obtain paragraph partial sum pd2x polynomial powers of dx Q dy qdx2 quotient reason relationship rule second and higher second difference second differential set constant similar sine subtracted term third differences three variables vanishing increments x2dy