## Principles of the Differential and Integral Calculus ... |

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2ada abscissa acº algebraic expression approach area increasing axis becomes equal circle circumference common logarithms compound expression constant quantity cosecant coversine cube curve cycloid cylinder denotes determine diameter difference Differential Calculus differential coefficient diminish divided dºu dºw ellipse equation example EXERCISES ferential find the differential Find the integral formly fraction function geometrical given Hence hyperbola illustrate inch per second increase uni increment indefinitely small independent variable infinite Integral Calculus learner limit move uniformly multiplied Naperian logarithm number of terms ordinate parabola perpendicular pupil quotient radius of curvature rate of increase rate of variation rectangle Required the developement right angles rule side becomes sine ſº solidity square inch straight line Substituting these values subtangent supposed surface tangent Taylor's Theorem triangle uniform rate variable quantity whilst

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Page vii - ... processes, without the slightest traces of logical reasoning to exercise and improve the intellect; we should bear in mind that the simple execution of analytical operations, acquired by dint of practice and experience, is a mere common species of labour, often merely mechanical ; whilst a distinct apprehension of the specific object and meaning of the operations, and a contemplation of the clearness and beauty of the various arguments employed, constitute the intellectual lore that gratifies...

Page iii - PRINCIPLES of GEOMETRY, familiarly Illustrated, and applied to a variety of useful purposes. Designed for the Instruction of Young Persons. Second Edition, revised...

Page 53 - When is this possible? 4. Divide a given straight line into two parts, so that the rectangle contained by the parts may be equal to a given rectangle.

Page 177 - ... as a line by the motion of a point ; a surface by the motion of a line ; and a solid by the motion of a surface.

Page 2 - ... to the area of the circle than that of the first. By continuing to double the number of sides, the area of the polygon will approach nearer and nearer to that of the circle, and- may be made to differ from it by a quantity less than any finite quantity. Hence the circle is said to be the limit of all its inscribed polygons.

Page 16 - ... of the curve in their immediate vicinity, we can easily trace the remainder of the curve, by assigning to x and y arbitrary values at pleasure. INTEGRAL CALCULUS. SECTION I, INTEGRATION OF MONOMIAL DIFFERENTIALS — OF BINOMIAL DIFFERENTIALS — OF THE DIFFERENTIALS OF CIRCULAR ARCS. ARTICLE (291.) THE Integral Calculus is the reverse of the Differential Calculus, its object being to determine the expression or function from which a given differential has been derived. Thus we have found that...

Page 96 - Y, and x + y is the sum of their logarithms; from which it follows that the sum of the logarithms of two numbers is equal to the logarithm of their product. Hence...

Page 57 - The solidity of a cylinder is equal to the area of its base multiplied by its altitude.

Page 88 - TV inch per second, at what rate is its solidity increasing when the diameter of the base becomes 10 inches, the height being constantly one foot ? Ans.

Page vii - It is to be regretted that most of our academical treatises on this as well as other subjects, abound so much with complex algebraical processes, without the slightest traces of logical reasoning to exercise and improve the intellect; we should bear in mind that the simple execution of analytical operations, acquired by dint of practice and experience, is a mere common species of labour, often merely mechanical ; whilst...