## First Principles of the Differential and Integral Calculus: Or The Doctrine of Fluxions |

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abscissas algebraical altitude angle arithmetical progression asymptote axis calculation called circle circumference coefficients conic section consequently considered constant quantity curve line cycloid denominator determine difference differential equation divided dy dx ellipse ellipsoid equal to zero exact differential example exponent expression factors finite fraction function geometrical progression given gives hyperbola indeterminate infinite number infinitely small quantities infinitesimal analysis integral KPLM logarithm manner method method of exhaustions multiple point multiplied obtain ordinates parabola parallel parallelepiped perpendicular point of inflexion polygon positive whole number proposed differential radius radius of curvature reduced represent rule second differential segment similar triangles sine solidity square straight line substituting subtangent supposition surface tang tangent tion variable whence we deduce wherefore x d x y d x

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Page 182 - Now, since the areas of similar polygons are to each other as the squares of their homologous sides (B.

Page 7 - 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 183 - ... zone is equal to the circumference of a great circle multiplied by the altitude of the zone.

Page 3 - Since the area of a great circle is equal to the product of its circumference by half the radius, or by one-fourth of the diameter (Bk.

Page 5 - The sum of a series of quantities in arithmetical progression is found by multiplying the sum of the first and last terms by half the number of terms.

Page 183 - ... itself. The method of Exhaustions was the name given to the indirect demonstrations thus formed. 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 the investigation...

Page iii - ... University at Cambridge, New England. Boston, 1824. Pp. 195. This book forms a part of Farrar's Cambridge Mathematics. It is the first -work published in this country employing the notation of Leibnitz and the infinitesimal method.

Page iii - 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 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.