27.6047. for the assumed root, it will give 27.60491. the root still nearer, 2. Required the cube root of 3214? Ans. 14.757583. Required the cube root of 2? Ans. 1.25992. 4. Required the cube root of 256? Ans. 6.349. A GENERAL RULE For extracting any Root whatever. Find by trial a number, which when involved, to the power denoted by the index of the requir ed root, shall come nearest to the given number, whether greater or less; and let that number be called the assumed root, and when thus involved, the assumed power. Let the given power, or number be repre- G. sented by the index, or exponent, in the question by X. the assumed power by the assumed root by and the required root by A. Q. R. Then X+1xA+X—1×G:X+1×G+X—T×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 time 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.* "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 EXAMPLES. 1. Required the Cube root of 789. Here G=789, X=3, Q=9, A=93=729, X+1 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. form by myself, and the investigation and use of it were given at large in my Tracts-page 45, &c." The algebraic form of Is this: As P2A:A+2P::r: R. Or, As P+2A: PA::r: Rur; When P is the given number, A the assumed nearest cube, r the cube root of A, and R the root of P sought. Here G=2.0743, Q=1.2,A=1.2=2.0736, X = 4, X+1=5, and X-1=3, And 5 x 2.0736 = 10.3680 5 × 2.0743 = 10.3715 6.2229 3 X 2.0736 = 6.2208 3 x 2.0743= 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 Required the cube root of 2 Ans. =3.3322+ 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, 0,1,2,3, 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. 0*. 1, 2, 3, Or, 5, 6,&c.I.orE. 1, 10. 100, 1000, 10000, 100000, 1000000, &c. Geometrical series,-where the same indices serve equally for any Geometrical series, or progres sion. 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 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 progres sion, suited to the terms interposed between I 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. * 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 xoa, then by squaring x o = a2 hence a2a, and consequently by division a=1, from whence it is evident, that the log. of 1 is always o, in any system; for more on this subject, and the algebraical form of the rule for computing Logarithms, see Bonnycastle's Al gebra, page 200, New-York Edition. 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 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 CĂFE—1. In CA take AB and Ab, on each side, or 0.1 And, erecting the per pendiculars BD, bd; half the sum of the spaces AD and Ad will be 0.001 0.00001 0.0000001 + + + 3 and the half diff. &c. Which reduced will stand thus, = 0.1 0.1000000000000,0.0050000000000 the sum of these-0.1053605156577—Ad 3333333333 20000000 142857 1111 9 -0.0953101798043- AD 1666666 In like manner, putting AB and A¿. 250000000 And the diff. 12500 each 0.2 there is obtained 0.1003353477310, 0.0050251679267 Having thus the hyperbolic Logarithms of the four decimal numbers 0.8, 0.9, 1.1, and 1.2; and since ·X. 1.2 0.8 1.2 0.9 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, be 10, we have 2.3025850929933, the Log. of 10. Hence by one addition 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 |