## A Primer of Infinitesimal AnalysisOne of the most remarkable recent occurrences in mathematics is the re-founding, on a rigorous basis, the idea of infinitesimal quantity, a notion which played an important role in the early development of the calculus and mathematical analysis. In this new and updated edition, basic calculus, together with some of its applications to simple physical problems, are presented through the use of a straightforward, rigorous, axiomatically formulated concept of 'zero-square', or 'nilpotent' infinitesimal - that is, a quantity so small that its square and all higher powers can be set, to zero. The systematic employment of these infinitesimals reduces the differential calculus to simple algebra and, at the same time, restores to use the "infinitesimal" methods figuring in traditional applications of the calculus to physical problems - a number of which are discussed in this book. This edition also contains an expanded historical and philosophical introduction. |

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### Contents

2 | 24 |

3 | 35 |

c | 39 |

4 | 49 |

48 The KeplerNewton areal law of motion under a central | 67 |

5 | 69 |

6 | 89 |

7 | 96 |

8 | 102 |

### Common terms and phrases

2-microvector abscissa arbitrary assertion assume axioms base point beam BSIA calculation called cancelling centre of mass centroid Chapter circle classical closed interval coincides cone consider Constancy Principle constant coordinates cos2 cross-section curve parameter deﬁned deﬁnite derivative determine differential calculus differential geometry distance equation exercise ﬁeld ﬁgure ﬁnd ﬁrst ﬁxed follows given gives Hence identical implies inertia intersection intuitionistic intuitionistic logic kth-order length logic Microcancellation microlinear microneighbourhood micropolynomiality microquantities microsegment Microstraightness moment of inertia natural number nonstandard analysis obtain pair plane polynomial Principle of Microafﬁneness proof quantiﬁer radius real numbers rotating rule satisﬁes segment Show sin2 smooth inﬁnitesimal analysis smooth worlds space spacetime stationary point straight line suppose surface surface of revolution tangent bundle tangent space tangent vector Taylor's theorem theorem unique variables vector field volume write x-axis y-coordinate yields zero