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Reflexions on the Metaphysical Principles of the Infinitesimal Analysis
No preview available - 2015
absciss absolutely zero amongst arbitrary quantities Assigned Quantities auxiliary quantities Calculus of Variations called infinitely small circle common algebra consequently coordinates Corollary curve differ as little Differential Calculus differing infinitely little difficult eliminate employed equal to zero equations errors example exist expression find the ratio finite quantities functions geometricians hence hypothesis indeterminate quantities infinite number infinitely small quantities Infinitesimal Analysis Infinitesimal Calculus Infinitesimal quantities Integral Calculus John Landen Lagrange latter law of continuity Leibnitz less Maxima and Minima means method of exhaustion method of fluxions method of Indeterminate method of limits neglecting Newton notation number of sides object operations ordinate perfectly exact polygon prime and ultimate proposed quantities quan question real quantities reduced regard result shew simplified subtangent supposed as small tangent theory tion tities triangle ultimate ratios unassigned vanishing quantities variable variation velocity whilst
Page 137 - This book is a preservation photocopy. It is made in compliance with copyright law and produced on acid-free archival 60# book weight paper which meets the requirements of ANSI/NISO Z39.48-1992 (permanence of paper) Preservation photocopying and binding by Acme Bookbinding Charlestown, Massachusetts 2002 2044 022 705
Page 44 - For example, by regarding a curve as a polygon, with an infinite number of sides each infinitely small, and which when produced is the tangent of the curve, it is clear that we make
Page 3 - when it is impossible to find the exact solution of a question, it is natural to endeavour to approach to it, as nearly as possible, by neglecting quantities which embarrass the combinations, if it be foreseen that these quantities which have been
Page 73 - its radius the mean proportional between the side of the cone and the radius of the circle of the base
Page 7 - is equal to the product of its perimeter into the half of the perpendicular drawn from the centre upon one of its sides ; hence the circle being considered as a polygon of a great number of sides, its area ought to equal the product of the circumference into half the radius;
Page 114 - convinced of the exactness of its results by the geometrical method of prime and ultimate ratios, or by the
Page 71 - gave to this operation the name of " the method of Exhaustion." As these polygons terminated by straight lines were known figures, their continual approach to the curve gave an idea of it more and more precise, and the law of continuity serving as a guide,
Page 71 - could eventually arrive at the exact knowledge of its properties. But it was not sufficient for geometricians to have observed, and as it were guessed, at these properties : it was necessary to verify them in an unexceptionable way; and this they did by proving, that every supposition contrary to the existence of these properties would necessarily lead to some contradiction
Page 59 - by a constant quantity. For the same reason, if any two quantities whatever differ in an infinitely small degree from each other, their differentials will also differ from one another infinitely little: and reciprocally, if two differential quantities differ infinitely little from one another, their integrals, putting aside the constant, can also differ but infinitely little one from the other.