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abscissas algebraical angle arithmetical progression axis becomes calculation circle coefficients conic section consequently considered constant quantity curve line cycloid d(AP denominator determine difference differential equation divided dºr dy-H dz-H ellipse equal to zero exact differential example exponent expression factors function geometrical progression given gives hyperbola indeterminate infinite number infinitely small quantities infinitesimal analysis KPLM logarithm manner method method of exhaustions multiplied number of sides obtain ordinates parabola parallel perpendicular point of inflexion polygon positive whole number proposed differential radius ratio reduced rule second differential segment similar triangles sine ſº solidity straight line substituting subtangent supposition surface tang tangent tion variable whence we deduce wherefore
Page 172 - Now, since the areas of similar polygons are to each other as the squares of their homologous sides (B.
Page 5 - 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 173 - ... zone is equal to the circumference of a great circle multiplied by the altitude of the zone.
Page 1 - 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 3 - 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 173 - ... 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 1 - 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 62 - 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.