A short view of the first principles of the differential calculus

by Arthur Browne

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1824 edition. ...p(x + h) may be represented by a series of a finite number of terms. Cor. 6. Hence, if j (x) be any rational function of (x), in which the highest power of (x) is less than (n), and if A, B, C, A' be the values of d.b(x) dp(x) dn-lp(x).W' dx ' dx dxn-1 respectively, when x = a, then 1.2 (n-i)' and, dividing this equation by (x-a)n, we obtain the equation (A). This formula will afterwards be useful in integration. k SECT. III. ON LOGARITHMS. If () be any constant quantity, and (x) and (y) two other numbers, whose relation is expressed by the equation y = '; then (x) is termed the logarithm of (y) to the base (a). It has been shown, (Wood's Algebra, Arts. 383, 384, &c.) that the sum of the logarithms of two numbers is the logarithm of their product, that the difference of their logarithms, is the logarithm of their quotient; and that if the logarithm of a number be multiplied or divided by (n), the result will be the logarithm of the (nth) power or (nth) root of that number. The calculation of logarithms must, at first, have been a laborious task. They have, however, been calculated and arranged in tables, and the utility of these tables in arithmetical operations is manifest. It would scarcely have been possible to have ascertained the value of such quantities as 71 or 1.05, without the assistance of logarithmic tables. Now, however, that these tables have been constructed, we can find the value of such quantities as (7)7'13 by the following more simple process. Find, from the tables, the logarithm of (7); multiply the logarithm thus found by 7-13, and the result will be the logarithm of the number required; and, looking again into the tables for the number which answers to this logarithm, we find the value ofY'. The common...

32 pages, Paperback

Published September 13, 2013

About the author

Arthur Browne is the recently retired editor and publisher of the New York Daily News, where he spent over four decades reporting on and shaping the story of New York City. Throughout his career, he chronicled the administrations of six mayors, from Abe Beame to Bill de Blasio, and coauthored I, Koch, a biography of Mayor Ed Koch. As editorial page editor, he led the Daily News team that won the 2007 Pulitzer Prize for a powerful series of editorials exposing the health crisis faced by 9/11 rescue and recovery workers.