Let the given power, or number be repre- G. sented by } A. the index, or exponent, in the question by X. the assumed power, by the assumed root, by and the required root by Q. Then X+1×A+X—1×G : X+1×G+X—1×A :: Q: R That is, as the sum of X+1 times A and X1 times G, is to the sum of X+1 times G and X-1 times A, so is the assumed root, Q, to the required root, R,-nearly; and the operation may be repeated as many times, as we chuse, by using always the root last found for the assumed root, and this, involved according to the given index, for the assumed power.* EXAMPLES. 1. Required the Cube root of 789. "This is a very general approximating rule," says Dr. Hutton, "of which that for the cube root is a particular case, and is the best adapted for practice and for memory, of any that I have yet seen. It was first discovered in this form by myseif, and the investigation and use of it were given at large in my Tracts page 45 &c." = Here G-789, X=3, Q=9, A=93 -729, X+1 4 and X-1=2. And 4×729-2916 4×789-3156 2×789=1578 2×729=1458 In the foregoing example the answer is strictly correct in its integral part and also in the three first decimal places; but if more decimals were wanted, and if their exactness were likewise requisite, the present answer might be taken for the assumed root, and the whole operation should be repeated. 2. Required the biquadrate root of 2.0743. Here G=2.0743, Q=1.2, A=1.2* =2.0736, X=4, X+1=5, and X--1=3. And 5×2.0736=10.3680 5×2.0743=10.3715 3×2.0743= 6.2229 3×2.0736= 6.2208 Then 16.5909 : 16.5923 [:: 1.2 1.2001+Anş. Required the fifth root of 21035.8 Ans.-7.3213+ Required the seventh root of 21035.8 SECTION HI. OF LOGARITHMS. LOGARITHMS are a series of numbers, so contrived, that by them the work of mutiplication may be performed by addition; and the operation of division may be done by subtraction. Or, Logarithms are the indices, or series of numbers in arithmetical progression, corresponding to another series of numbers in geometrical progression. Thus, 0, 1, 2, 3, 4, 5, 6, &c. Indices or Logarithms. 1, 2, 4, 8, 16, 32, 64, &c. Geometrical progression. 1, 3, 9, 27, 81, 243, 729, &c. Geometrical Series. 0, 1, 2, 3, Or 5, 6,&c.I.orL. 1, 10, 100, 1000, 10000, 100000, 1000000, &c. Geometrical series,-where the same indices serve equally for any Geometrical series, or progression. Hence it appears that there may be as many kinds of indices, or logarithms, as there can be taken kinds of geometrical series. But the Logarithms most convenient for common uses are those adapted to a geometrical series increasing in a tenfold progression, as in the last of the foregoing examples. In the geometrical series 1, 10, 100, 1000, &c. if between the terms 1 and 10, the numbers 2, 3, 4, 5, 6, 7, 8, 9 were interposed, indices might also be adapted to them in an arithmetical progression, suited to the terms interposed between 1 and 10, considered as a geometrical progression. Moreover, proper indices may be found to all the numbers, that can be interposed between any two terms of the Geometrical series. But it is evident that all the indices to the numbers under 10, must be less than 1; that is, they must be fractions. Those to the numbers between 10 and 100, must fall between 1 and 2; that is, they are mixed numbers, consisting of 1 and some fraction. Likewise the indices to the numbers between 100 and 1000, will fall between 2 and 3; that is, they are mixed numbers, consisting of 2 and some fraction; and so of the other indices. Hereafter the integral part only of these indices will be called the Index; and the fractional part will be called the Logarithm. The computation of these fractional parts, is called making Logarithms; and the most troublesome part of this work is to make the Logarithms of Prime Numbers, or those which cannot be divided by any other numbers than themselves and unity. RULE For Computing the Logarithms of Numbers. Let the sum of its proposed number and the next less number be called A. Divide 0.8685889638׆ The number 0.8685889638+ is the quotient of 2 divided by 2.302585093, which is the logarithm of 10, according to the first by A, and reserve the quotient. Divide the reserved quotient by the square of A, and reserve this quotient Divide the last reserved quotient by the square of A, reserving the quotient still; and thus proceed as long as division can be made. Write the reserved quotients orderly under one another, the first being uppermost. Divide these quotients respectively by the odd numbers 1, 3, 5, 7, 9, 11, &c.; that is, divide the first reserved quotient by 1, the second by 3, the third by 5, the fourth by 7, &c. and let these quotients be written orderly under one another; add them together and their sum will be a logarithm. To this logarithm add the logarithm of the next less number, and the sum will be the logarithm of the number proposed. form of Lord Napier, the inventor of logarithms. The manner in which Napier's logarithm of 10 is found, may be seen in most books of Algebra, but it is here omitted, because students of Surveying are too generally unacquainted with the principles of that science, and the subject is too extensive for the present treatise. Those, however, who have not an opportunity for entering thoroughly into this subject, may with more propriety grant the truth of one sumber, and thereby be enabled to try the correctness of any logarithm in the tables, than receive those tables, as truly computed, without any means of examining their accuracy. E... |