, sented by the index, or exponent, in the question by X. the assumed power, by A. the assumed root, by Q. and the required root by R. Then X+1xA+X-1XG:X+1xG+X-1RA ::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, 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 myself, 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=95=729, X+1 34 and X-1=2. And 4729=2916 4X789=3156 2 x 7891578 2x729=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. 3 x 2.0743= 6.2229 3x2.0736= 6.2208 Then 16.5909 16.5923 [:: 1.2: 1.2001+Ans. Required the fifth root of 21035.8 Ans. = 7.3213+ Required the sixth root of 21035,8 Ans. =5.2540+ Required the cube root of 999 Ans. =9.9966+ Required the fourth root of 97.41 Ans. =3.1416 Required the cube root of .037 Ans. =.33322+ Required the cube root of 2 Ans. = 1.2599+ Required the seventh root of 21035.8 Answer= [4,1454. SECTION III. OF LOGARITHMS. LOGARITHMS are a series of numbers, so contrived, that by them the work of multiplication 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, 10,1,2,3, 4, 5, 6, &c. Indices or Logarithms. ( 1,2,4,8,16,32,64, &c. Geometrical progression. Or, 0, 1, 2, 3, 4, 5, 6, &c. Ind. or Log. | 1, 3, 9, 27, 81, 243,729, &c. Geometrical Series. Or, 2, 0, 1, 3, 4, 5, 6,&c.I.orL. 1, 10, 100, 1000, 10000, 100000, 1000000, &c. Geometrical series-where the same indices serye 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 cominon uses are those adapted to a geometrical series increasing in a ten-fold 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*t † The number 0.8685889638+ is the quotient of 2 divided by 2.302585093, which is the logarithm of 10, according to the first |