Principles of the Differential and Integral Calculus ... (Google eBook)

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Taylor and Walton, 1847 - Calculus - 193 pages
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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...

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