First Principles of the Differential and Integral Calculus, Or, the Doctrines of Fluxions: Intended as an Introduction to the Physico-mathematical Sciences |
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abscisses ACRN algebraical angle arithmetical progression axis becomes binomial calculation circle coefficients consequently considered constant quantity curve denominator determine difference differential equation divided dx a² dx and dy dx dy dx² dy dx dy dy dy² ellipse equa equal to zero exact differential example exponent expression factors fraction function geometrical progression gives half segment hyperbola increasing latitude indeterminate integral KPLM logarithm manner method method of exhaustions multiplied ordinates parabola parallel perpendicular point of inflexion positive whole number proposed differential radius ratio reduced similar triangles sine solidity straight line substituting subtangent surface tang tangent tion variable wherefore x d x x2 dx x²)³ y d x
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Page 180 - Now, since the areas of similar polygons are to each other as the squares of their homologous sides (B.
Page 4 - It contains the rules necessary to calculate quantities of any definite magnitude whatever. But quantities are sometimes considered as varying in magnitude, or as having arrived at a given state of magnitude by successive variations. This gives rise to the higher analysis, which is of the greatest use in the physico-mathematical sciences. Two objects are here proposed : First, to descend from quantities to their elements. The method of effecting this is called the differential calculus.
Page 3 - The area of a regular polygon is equal to the product of its perimeter, by half of the perpendicular let fall from the centre upon one of the sides.
Page 13 - ... to the wants of those who have but little time to devote to such studies. The easier and more important parts are distinguished from those which are more difficult or of less frequent use, by being printed in a larger character. In the Introduction, taken from Carnot's Reflexions sur la Metaphysique du Calcul Infinitesimal, a few examples are given to show the truth of the infinitesimal method, independently of its technical form. Moreover in the 4th of the notes, subjoined at the end, some account...
Page 13 - His works were, therefore, at this time, rather old, but his calculus was selected in preference to others " ou account of the plain and perspicuous manner for which the author is so well known, as also on account of its brevity and adaptation in other respects to the wants of those who have but little time to devote to such studies.
Page 4 - But the sum of the terms of such a progression is found by multiplying the sum of the first and last terms by half the number of terms.
Page 181 - Though few things more ingenious than this method have been devised, and though nothing could be more conclusive than the demonstrations resulting from it, yet it laboured under two very considerable defects. In the first place, the process by which the demonstration was obtained was long and difficult; and, in the second place, it was indirect, giving no insight into the principle on which...
Page 3 - If two lines are drawn through the same point across a circle, the products of the two distances on each line from this point to the circumference are equal to each other.
Page 72 - RB\]r, and therefore d± 1 ds ~R' ie the curvature of a circle is measured by the reciprocal of its radius. Hence, if p be the radius of the circle which has the same curvature as the given curve at the point P, we have A circle of this radius, having the same tangent at P, and its concavity turned the same way, as in the given curve, is called the 'circle of curvature,' its radius is called the 'radius of curvature,' and its centre the 'centre of curvature.
Page 53 - From the point D, with a radius equal to AB, describe an arc ; and from the point B as a centre, with a radius equal to AD, describe another arc cutting the former in the point C. Draw the straight lines CD, CB; and the parallelogram ABCD will be the one required.