Elements of the Differential and Integral Calculus: By a New Method, Founded on the True System of Sir Isaac Newton, Without the Use of Infinitesimals Or Limits |
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algebraic angle asymptote axes axis of abscissas base becomes zero binomial calculus circumference concave cone convex coordinates corresponding cosine cube cycloid d2 u dx2 decreasing diameter differential calculus direction distance divided dx d2 dx² dy dx ellipse envelope equa equal to zero evolute exponent ferential find the differential finite formula found Art function given curve gives Hence the curve hyperbola infinite infinity length logarithm logarithmic spiral measuring circle method multiplied obtain origin osculatory circle parabola parallel pass perpendicular point of tangency polar curve positive value PROPOSITION radius of curvature radius vector rate of change rate of increase ratio rectangle reduce second differential coefficient side subnormal Substituting these values Substituting this value subtangent suppose surface symbol tangent line tends to move tion triangle versed sine vertex whence
Popular passages
Page 133 - MRS^ at a point on the indifference curve we can do so by drawing tangent at the point on the indifference curve and then measuring the slope by estimating the value of the tangent of the angle which the tangent line makes with the X-axis.
Page xxiii - But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases.
Page 315 - Ttie area of a circle is equal to the square of the radius multiplied by тт.
Page xxxiii - So far that letter. And these last words relate to a treatise I composed on that subject in the year 1671. The foundation of that general method is contained in the preceding Lemma.
Page xxiii - And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish.
Page 229 - The cycloid is the curve described by a point in the circumference of a circle, as it rolls along a straight line. Let OX be the straight line. As the circle NPT, with radius a, rolls along this line, the point P describes the cycloid OBO'.
Page xxiii - Perhaps it may be objected that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alleged that a body arriving at a certain place, and there stopping, has no ultimate velocity: because the velocity » before the body comes to the place, is not its ultimate velocity; when it has arrived, is none.
Page 69 - NUMERATOR AND DENOMINATOR IS THE DIFFERENTIAL OF THE NUMERATOR MULTIPLIED BY THE DENOMINATOR, MINUS THE DIFFERENTIAL OF THE DENOMINATOR MULTIPLIED BY THE NUMERATOR, DIVIDED BY THE SQUARE OF THE DENOMINATOR.
Page 163 - ... the differential of a surface of revolution is equal to the circumference of a circle perpendicular to the axis, into the differential of the arc of the meridian curve.