Explanation of the "Theory of the Calculus"

Lee and Shepard, 1870 - Differential calculus - 12 pages

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Page 59 - The volume of a spherical pyramid is equal to the area of its base multiplied by one-third of the radius of the sphere.
Page 84 - Napier's system. 427. The logarithm of a number in any system is equal to the modulus of that system multiplied by the Naperian logarithm of the number. If we designate Naperian logarithms by Nap. log., and logarithms in any other system by log., then, since the modulus of Napier's system is unity, we have log. (l+m)^M(m-— +— -, etc.), noo2 '779^ Tv-r ^ '/H v ' Nap. log. (l + m)=m — O
Page 50 - When the exponent of the variable without the parenthesis, increased by unity, is exactly divisible by the exponent of the variable within the parenthesis.
Page 22 - ... of any number of variables is equal to the sum of the partial differentials.
Page 82 - The logarithms of the same number in different systems are to each other as the moduli of those systems. This is evident from the general logarithmic series. Thus the logarithm of 1 + x in a system whose modulus is m...
Page 32 - T2 which was to be proved ; and which, being freed from all consideration of infinite, is necessarily and rigorously exact. 3. To determine in what manner to divide a quantity, a, into two parts, in such a manner that the product of these parts shall be the greatest possible. Let x be one of the parts, the other will be a — x, and the product will be ax — x2.
Page 64 - If also (pf'(x') = \[/'(x) for the same value of x, the equation for h has three roots zero and the curves cut in three ultimately coincident points at P. There are now two contiguous tangents common, and the contact is said to be of the second order; and so on. Similarly for curves given by their polar equations, if...
Page 49 - That is, the derivative with respect to x of the sum of any number of functions of x is equal to the sum of their derivatives. Proof of V. Consider first the case of two factors. »-\ \ lim ruh"+(v+h")hr\ A=o[" h 475 As h approaches the limit 0, h" also approaches 0, and therefore the limiting value of v + h
Page 90 - 1.2.3.4.5 ' " for the value of the sine of an arc in terms of the arc itself.
Page 15 - The derived function of the product of a variable by a constant is equal to the derived function of that variable multiplied by that constant.