## An elementary essay on the computation of logarithms |

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An Elementary Essay on the Computation of Logarithms John Radford Young Keine Leseprobe verfügbar - 2010 |

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2log 3d order 50th logarithm 9 pie abscissas asymptotes base become infinitely Borda's formula Borda's theorem calculated chapter ciphers coefficients commence COMPUTATION OF LOGARITHMS compute the logarithm computing logarithms constant fourth difference convergency curve decimal places deduce determining the logarithm diffe Double log equal equation fourth order given number greater number hence hyperbolic increasing interpolation Lagrange last rem log.p loga logarithms of numbers method METHOD OF DIFFERENCES modulus Napierian logarithm Napierian system negative number number is infinitely number of decimals number of terms numbers from 45000 obtain the logarithm obviously odd number order of differences ordinates original series places of decimals positive preceding prime numbers proposed number proposed series quadruple quotients rences resulting logarithms rithms root second difference significant figure small number substituting Subtract log suppose the fourth table of logarithms three first prime tion true

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Seite 45 - ... computation is carried. The chief part of the working consists in merely setting down a number under itself removed one or more places to the right, and subtracting, and repeating this operation ; and consequently is very little liable to mistake. Moreover, from the commodious manner in which the work stands, it may be revised with extreme rapidity. It may be performed after a few minutes instruction by any one who is competent to subtract.

Seite 45 - I trust, rhe following method will supply, while at the same time it is peculiarly easy of application, requiring no division, multiplication, or extraction of roots, and has its relative advantages highly increased by increasing the number of decimal places to which the computation is carried.

Seite 64 - FIND the sum of the m* powers of the roots of an equation in terms of the coefficients and the sums of the inferior powers.

Seite 48 - Jind the hyperbolic Logarithm of any Number not exceeding 2. RULE. Set the number under itself, to be subtracted from itself, but removed so many places to the right as shall be necessary to make the remainder greater than 1 ; subtract. Proceed in the same manner with the remainder, and so on till the remainder becomes i followed by £ as many cyphers as the number of decimal places you work to ; suppose at the end of the operation you find that you have removed one place to the right and subtracted...

Seite 48 - ... under itself, to be subtracted from itself, but removed so many places to the right as shall be necessary to make the remainder greater than 1 ; subtract. Proceed in the same manner with the remainder, and so on till the remainder becomes i followed by |- as many cyphers as the number of decimal places you work to ; suppose at the end of the operation you find that you have removed one place to the right and subtracted b times ; two places, c times ; three places, d times, &c. ; then bx h. 1....

Seite 55 - Therefore the log. of the original number = bx log. - + cx log. — + dx log. — + &c. + log. of last remainder. Now the last remainder being unity followed by a certain number of decimal cyphers, its correct hyp. log., as far as twice that number of places, is (as is well known ) the decimal part itself of that remainder. Hence the rule is manifest. A similar method, by addition only, by means of the ready computed logarithms of li, -j-|i, -ff^, &c.

Seite 52 - RULE. Reduce the given number ( if necessary ) to a whole, or mixt number less than 2, by setting the decimal point after the first significant figure, or if the given number be 10 or a power of 10 after the first o; and then dividing by 2 (if necessary) till the integral part is i.* Find the h. 1. of this reduced number by the last rule, and add to it or subtract from it as many times the h.

Seite 52 - ... decimal point after the first, significant figure, or if the given number be 10 or a power of 10 after the first o ; and then dividing by. 2 (.if necessary) till the integral part is i.* Find the h. 1. of this reduced number by the lastrufc, and; add to it or subtract from it as many times the h.

Seite 46 - ... rapidity. It may be performed after a few minutes instruction by any one who is competent to subtract. It is as easy for large numbers as for small ; and on an average about 27 subtractions will furnish a logarithm acccurately to to places of decimals. In general 9x^i^ subtractions will be accurate to a

Seite 2 - ... from which we obtain this general expression for determining the logarithm of any number, p+1, when we know the logarithm of the preceding number, p, viz.