## 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 becomes calculation called circle coefficients consequently considered constant quantity curve line denominator determine difference differential equation divided dy-H ellipse 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 parallelopipeds perpendicular point of inflexion polygon positive whole number proposed differential radius ratio reduced rule second differential segment similar triangles sine ſº solidity solids of revolution square straight line substituting subtangent supposition surface tang tangent tion variable wherefore

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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 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.