therefore, the co-efficient 3 of the first term is 300, and the whole term means 300 times the square of a, that is, of the figure already found. The second term contains the first power of a, which is tens, affecting both the co-efficient 3 and the other factor b; therefore the second term is 30 times the product of the figure already found, and the figure which we are finding. At this stage of the operation, therefore, the value of b must be determined by trial; and it may take two or three trials of a beginner before the right one is discovered, because we are not yet in possession of the whole divisor. When the value of bis found, it must be multiplied by a and then by 30, and made the second line of the divisor, properly arranged under 300 times the square of a, the first figure, which is the first line of the divisor. The third term of the divisor is the square of b; and in respect of the other lines of the divisor, the right hand figure of it is units, because b is units, considered in relation to a as tens, and therefore the right hand figure of any power of it must be units, because the powers of quantities must be of the same kind with the quantities themselves. Thus the complete divisor for finding the second figure of a cube root consists of three lines. First, 300 times the square of the figure already found; secondly, 30 times the product of the figure already found and the figure which we are finding ; and thirdly, the square of the figure which we are finding. These three lines are to be added together, and their product by the figure which we are finding, subtracted from the dividend; and the figure is the second one of the root; while the remainder with the next period annexed makes a new dividend, out of which to find the third figure. At the third and every subsequent step of the operation, all the part of the root found, which always stands in the relation of tens to the next figure as units, is to be used as a, and the figure we are finding as b; and with this understanding every step of the operation, however long it may be, is an application of the same formula. If the root does not terminate in integers, so as to leave no remainder, it will not terminate in decimals; but it may be carried to any approximate degree of accuracy that may be required, by annexing three Os to every remainder, and continuing the operation as before. If the number is wholly decimal, we must begin with three figures immediately after the decimal point; and as often as three Os occur between the decimal point and the significant figures, one 0 must be placed after the decimal point and before the other figures in the root. The cube roots of fractions may be obtained by taking those of the numerator and denominator; and those of numbers expressed by factors, may be taken in the cube roots of the factors, and multiplied together for the general root. But as cubes occur much less frequently in the natural order of the numbers than squares, it is seldom that any advantage can be gained by this method. It is sometimes, however, of advantage to take the cube root of one factor of a number, and express that of the other by the radical sign. Thus, the cube root of 54, that is 37 54, may be expressed by 32; for 54 is the product of 27 and 2, the first of which is the cube of 3. We shall now give one short example, as illustrative of the application to numbers : Let it be required to find the cube root of 12812904? Beginning at the right, and dividing into periods of three figures, because the exponent is 3, we have 12'812 904. The nearest cube to 12, the first period, is 8, the root of which is 2; therefore 2 is the first figure of the root, and its cube subtracted from the given number leaves 4'812'90 4. We have now subtracted the cube of a, and there remains 4812, out of which to find the second figure. a and b stand to each other in the relation of a tens and b units, therefore the divisor, as found by the general operation, is 300 a2+30 a b+b2; and we must apply this to our number, and determine b by trials. 300 a2 is the largest part of our divisor, so that we may compare it with the dividend, a=2 and 300 a2 = 1200 | 4'812. We would get 1200 four times in 4800; but the remainder 12 would not be equal to the cube of 4, and we want another term which is 30 a, or 60 times the number we are seeking, therefore we must try 3 for the value of b, our second figure; and as it is probable that 3 may answer, we may complete our operation. We have now found 2 and 3 for the first and second figures of the root, 2 considered as tens, and 3 as units; we have subtracted the cube of their sum, which is 23, from the first and second periods of our number, and there remains 645. If we next annex the next period of our number to this, we shall have the following number for a new dividend, 645'904. a is now = 23, but with this exception our divisor is exactly the same as before; therefore, using the numbers in place of the letters, we have 300 x 23o 158700 the trial divisor. Comparing this with the dividend, we have 158700 | 645'904. = Comparing 15 with 64, which are the corresponding figures of the divisor and dividend, there being an equal number of figures to the right hand of them in each, we find it can be got 4 times, so that we may try 4 as our next figure; and the complete operation, using these numeral values in place of letters, will be 300 x 232 = Multiply by. Remains 30 x 23 x 4 = 158700 *645'904 Consequently, 234 is the cube root of the given number 12812904; and the cube root of any number might be found in a similar manner; only when there are decimals in the number, this must be pointed from the left, while the integers are pointed from the right. Any one who first examines the investigation of the formula with sufficient care, then goes over the steps of this operation, and bears in mind that the divisor is 300 a2+30 ab+b2, and also that a means the figure or figures of the root already found, and b the figure which one is finding, can have no difficulty in extracting the cube root of any number whatever. Formulæ for all other roots may be found by taking the corresponding power of a+b, omitting the first term, diminishing the exponent of b by unity, or 1, in each remaining term, and bearing in mind that there are always as many Os on the right of the numeral co-efficient in each term as there are terms to the right of that one in the formula. SECTION XV. ARITHMETIC OF EXPONENTS-LOGARITHMS. In the case of every root or power, that is, of every quantity considered either as a root or as a power-and any quantity whatever, when spoken of and expressed generally, may be considered in this manner-there are three distinct subjects, any one of which may be the object of our inquiry. First, there is the root; and if the root is given, the power is found by multiplication, as already explained. Secondly, if the power is given, the root may be found by the methods explained in the latter part of last section. But thirdly, there is the exponent, and it also may become that which is sought; or, a and b being any quantities or positive numbers whatsoever, and " any exponent whatsoever, an answer may be demanded to the question, what power a is of b, or, in other words, to find such a value of " as that bra. |