2. To extract the Cube Root by another Method.* 1. By trials find the nearest rational cube to the given number, whether it be greater or less, and call it the assumed cube. 2. Then say, by the Rule of Three, as the sum of the given number and double the assumed cube is to the sum of the assumed, and double the given number, so is the root of the assumed cube to the root required, nearly. Or, as the first sum is to the difference of the given and assumed cube, so is the assumed root to the diffefence of the roots, nearly. 3. By using, in like manner, the cube of the root last found as a new assumed cube, another root will be obtained still nearer. And so on as far as we please; using always the cube of the last found root for the assumed cube. EXAMPLES. 1. To find the cube root of 21035.8. Here the root therefore 27, its Then, Taking is soon found between 27 and 28. As 60401.8: 61754.6: 27: 27.6047, is equal to 20+5 cubed; but 20+5 cubed is equivalent to 8000+300 × 4x5+30×2×5×5+125, or to 203+(300×4+30×2×5+5×5)×5= 20×20×20+3×5×20×20+3×20×25+125=3d power or, 8000+300x4×5+30×2×25+125. Here the rule is evident. In the same manner, the operation may be illustrated in every case. For a demonstration of this rule in general terms, the reader is referred to the Editor's "Treatise on Algebra, Theoretical and Practical."-ED. *This rule is found in Hutton's Mathematics. There have been different rules given for extracting the cube root, among which this, and another rule given in Pike's Arithmetic (by approximation), are very expeditious. Therefore 27.6047 is the root nearly. Again, by repeating the operation, and taking 27.6047 for the assumed root, it will give 27.60491 the root still nearer. 2. Required the cube root of 3214? Ans. 14.75758. 3. Required the cube root of 2? Ans. 1.25992. 4. Required the cube root of 256? Ans. 6.349. 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, (0, 1, 2, 3, 4, 5, 6, &c. indices or logarithms. (0, 1, 2, 3, 4, 5, 6, &c. ind. or log. 1, 3, 9, 27, 81, 243, 729, &c. geometrical series. S0", 1, 2, 3, 1, 10, 100, 1000, 10000, 100000, 1000000, &c. geometrical series, where the same indices servé 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 arith * In any system of logarithms the log. of 1 is 0; for logarithms may be considered as the exponents of the powers to which a given or invariable number must be raised, in order to produce all the common or natural numbers, therefore by assuming xa, then by squaring xoaa hence a2=a, and consequently by division a=1, from whence it is evident that the log. of 1 is always 0, in any system; for more on this subject, and the algebraical form of the rule for computing logarithms, see Bonnycastle's Algebra, page 200, New-York edition; or my Treatise on Algebra, page 332, second edition.-ED. = metical 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 one 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+ by A, and reserve * The number 0.8685889638+ is twice the reciprocal of the hyperbolic log. 2.302585093, which is the log. of 10, according to the first form of Lord Napier, the inventor of logarithms; which log. according to the excellent Sir I. Newton's method is calculated thus; let DFD (Pl. 14, fig. 1) be an hyperbola whose centre is C, vertex F, and interposed square CAFE=1. In CA take AB and Ab, on each side, or 0.1; and, erecting the perpendiculars BD, bd, half the sum of the spaces AD and Ad will be 0.1+ and the half diff. + + 0.001 0.00001 0.0000001 3 5 7 0.01 0.0001 0.000001 Which reduced will stand thus, &c. 0.00000001 &c. 8 , 0.1000000000000,0.0050000000000 Sum of these=0.1053605156577=Ad 3333333333 20000000 142857 1111 9 250000000 And the diff. -0.0953101798043 AD 1666666 In like manner putting AB and Ab 100 Ad=0.2231435513142, and 0.1003353477310,0.0050251679267 1.2 1.2. Having thus the hyperbolic logarithms of the four decimal numbers 0.8, 0.9, 1.1, and 1.2; and since 2, and 0.8 and 0.9 are less than unity, adding their logarithms to double the log. of 1.2, we have 0.6931471805597, the hyperbolic log. of 2. To the triple of this adding the log. of 0.8, because 10, we have 2.3025850929933, the log. of 10. Hence by one addition 21212 0,8 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. EXAMPLE 1. Required the logarithm of the number 2. Here the next less number is 1, and 2+1=3=A, and A or 329; then 3)0.868588964 9)0.289529654÷ 1=0.289529654 9)0.032169962÷ 3=0.010723321 9)0.003574440÷ 5=0.000714888 9)0.000397160÷ 7=0.000056737 9)0.000044129÷ 9=0.000004903 9)0.000004903-11 0.000000446 9)0.000000545÷13 0.000000042 0.00000006115 0.000000004 are found the logarithms of 9 and 11: And thus the logarithms of all the prime numbers are prepared, that is, 2, 3, 5, 11, &c. Moreover, by only depressing the numbers above computed, lower in the decimal places, and adding, are obtained the logarithms of the decimals 0.98, 0.99, 1.01, 1.02; as also of these, 0.998, 0.999, 1.001, 1.002. And hence, by addition and subtraction, will arise the logarithms of the primes 7, 13, 17, 37, &c. All which logarithms being divided by 2.3025850929933 (the hyperbolic log. of 10), or multiplied by its reciprocal, .4342944819, give the common logarithms to be inserted in the table. Note.-For further illustration on this subject, the reader is referred to Hutton's Tables. To this logarithm 0.301029995 add the logarithm of 1=0.000000000 Their sum=0.301029995=log. of 2. The manner in which the division is here carried on may be readily perceived by dividing, in the first place, the given decimal by A, and the succeeding quotients by A2; then letting these quotients remain in their situation, as seen in the example, divide them respectively by the odd numbers, and place the new quotients in a column by themselves. By employing this process, the operation is considerably abbreviated. EXAMPLE 2. Required the logarithm of the number 3. Here the next less number is 2; and 3+2=5=A, and A2=25. 5)0.868588964 25)0.173717793 1 0.173717793 25)0.006948712÷3=0.002316237 25)0.0002779485 0.000055590 25)0.000011118÷7=0.000001588 25)0.0000004459 0.000000049 0.000000018 11 0.000000002 To this logarithm 0.176091259 add the logarithm of 2=0.301029995 Their sum=0.477121254=log. of 3. Then, because the sum of the logarithms of numbers gives the logarithm of their product; and the difference of the logarithms gives the logarithm of the quotient of the numbers: from the two preceding logarithms, and the logarithm of 10, which is 1, a great many logarithms can be easily made, as in the following examples. Example 3. Required the logarithm of 4. Since 4=2×2, then to the logarithm of 2=0.301029995 add the logarithm of 2=0.301029995 The sum logarithm of 4=0.602059990 |