a (b + x)2 (6 + x)3 Hence if x + + - be a proper fraction, the integral quo10 a 3(10 a) 3 tient arising from the division of N — (10 a)3 by 3 (10 a)”, will be the second digit. But as x may be very nearly equal to 1, and b may be as large as 9, while a may be as small as 1, the value of the above expression may be very nearly as large as 143 ; or there may be an excess of 14 in the quotient above the second digit, and even if a be 9, there may be an excess of 2. But as the ratio is diminished or increased, the possible error is diminished or increased. And if b be less than 3, the error cannot exceed 1. No Rule however can be formed for determining the exact amount of error in every particular case. Suppose then the second digit found; we have now to see whether the number obtained be the exact, or nearest, cube root. To do this, we must subtract the cube of the root from the given number. But since (10 a +b)3 = (10 a)3 + 3 (10 a)2 6+3 (10 a)b2 +63, and in forming the first remainder we subtracted (10 a)3, therefore we need only to subtract 3 (10 a)2 b + 3 (10 a) 62 +63 from this remainder. Now this expression may be put into the form ( 3 (10 a)2 + {3(10 a) +0}6} 6, so that if to the divisor 3 (10 a)? there be added, the product by the second part of the root of the number, formed by adding the second part to three times the first part, and this whole sum be multiplied by the second part, the subtrahend will be formed. If there be no remainder, the number found will be the exact root, but if there be a remainder, either there is no exact root, or the second digit in that found is too small. To determine whether this latter be the case, we observe that the addition of 1 to any number increases its cube by 1 more than three times the product of the original, by the increased number; therefore the remainder must not be greater than this product, if the second digit be correct. Now let the given number contain more than 6 and not more than 9 digits; its nearest cube root therefore will contain 3 digits. Let the number be pointed according to the Rule; then it may be shewn as before that the whole number of tens in the root is the greatest number, whose cube does not exceed the whole number of thousands in the given number, that is the first two periods. Hence we have to find the nearest cube root of the number composed of the first two periods, which, containing two digits, may be found as already explained. Let a stand for this number, a of course being less than 100. Then the whole root will be expressed by 10 a +b + x, 6 being the number of units, & a fraction less than 1. In precisely the same way as before it may be shewn, that the digit b may be found by dividing the remainder, after the subtraction of (10 a)s by 3 (10 a). Only in this case, as a is not less than 10, the greatest possible excess of this quotient over b will be 2. Having found the digit b, we have to determine, as before, whether the exact or nearest root has been obtained, by subtracting the cube of the root found from the given number; or by subtracting from the last reinainder a subtrahend formed in the same manner as the former one. The method of forming the second trial-divisor without the labour of squaring the root may be shewn thus. Let a, b, be the first two digits in the root, then the number of hundreds in the next trial-divisor will be 3 (10 a + b)2, or 3 (10 a)2 +2{3(10 a +6} 6+6o, or is the sum of b the square of the last figure in the root. In the same way it may be shewn how to extract the cube root of any number whatever. And the process is seen to be that described in the Rule, unnecessary ciphers being omitted. If the given number be a decimal, the number of decimal places must be made some multiple of three, (Prop. 72), and the cube root of the number, considered as integral, being effected, the root of the decimal is found by marking off as decimals one-third as many as there are in the given number. Prop. 76.-If the cube root of a number contain 2 n + 2 digits, and n + 2 have been found by the ordinary Rule, the remaining n may be found by dividing the remainder by the corresponding trial-divisor. a a For if a, b, be the two parts of the root, the one a containing n +2 significant digits, followed by n ciphers, and the other b containing n digits, then the remainder, after a has been found, and its cube subtracted, will be 3 ao b + 3 a 62 +63, which, being divided by the trial-divisor 3 a, gives 62 63 62 63 a quotient bt-t: differing from b by the quantity - + If 3 aa 3a2 this quantity be less than 1, b will be correctly found by the division. Now b containing n digits is less than 10n, and a containing in all 2 n + 2 digits 62 63 102 n 103n is not less than 102n +l; therefore + is less than -+ 3 a2 102n+1 3X104n+2 1 1 1 1 i.e. is less than + -inuch more therefore, is less than + 10 3X10n +2 10 100 11 or Hence b is accurately found by dividing the remainder by the 100 trial-divisor. a Prop. 77.--In any Arithmetic Series the sum of any two terms, equidistant from the extremes is always the same; and, when the number of terms is odd, twice the middle term is equal to the sum of the extremes. For every Arithmetic series, whose terins increase by a common difference, may by inverting the order of the terms, be written as a decreasing series, and any term of the increasing series will be as much greater than the first term as the corresponding term of the decreasing series is less than its first term: that is if a and I be the first and last terms of the series, and 6 and c two terms equidistant from a and b, the excess of 6 above a will be equal to the defect of c from l, or bma=l-c: hence b+c=a+l. Also if m be the middle term, then m - a=i- m, or 2 m = a +1. Prop. 78.--To find the sum of an Arithmetic Series. Let the order of the terms of the series be inverted, and another series formed; and let the corresponding terms of these two series be added together, the result will (Prop. 77) be a series of terms, each equal to the sum of the first and last terms of the original series, and the number being the same as in that series. Also the sum of these terms, being the sum of two identical series, is twice the sum of one of them. Hence twice the sum of the series is equal to the sum of the first and last terms multiplied by the number of terms: or the sum of an Arithmetic series is the sum of the first and last terms multiplied by half the number of terms. Prop. 79:-— To find any required term of an Arithmetic Series. Since every term is greater than the preceding or following by the common difference, therefore the second differs from the first by once the common difference, the third differs from the first by twice the difference, and so on. So it appears that to form any term from the first and the common difference, we must increase or diminish the first term by the product of the common difference multiplied by a number less by one than the number of the term. Cor. 1. Hence the last term is equal to the first term increased or diminished by the product of the common difference multiplied by a number less by one than the number of terms. Cor. 2. Hence if the difference between the first and last terms of an Arithmetic series be divided by a number less by one than the number of terms, the result will be the common difference. This divisor being greater by one than the number of means, the common difference may be found by dividing the difference between the first and last terms by the number of means increased by 1. Cor. 3. Hence is apparent the method of inserting Arithmetic means between two given numbers; for the common difference being found as above, the successive terms may be readily formed from the first by the Prop. Prop. 80.--To find any required term in a Geometric series from the first term, and the common ratio. In a Geometric series the ratio of any term to the preceding is the same, so that any term may be formed from the preceding by multiplying by the same number, which is called the common ratio. Hence in the second term the common ratio will enter as a factor once; in the third term twice ; in the fourth term three times ; and so on. Whence it appears that the common ratio will appear as a factor of any term a number of times less by one than the number of the term. Cor. 1. Hence the last term is equal to the product of the first term multiplied by a power of the common ratio of a degree less by one than the number of terms. Cor. 2. Hence if the first and last terms be given of a Geometric series, the quotient of the first by the last will be a power of the common ratio of a degree less by one than the uumber of terms; whence the common ratio may be found. Cor. 3. Hence if it be required to insert any number of Geometric means between two numbers, the common ratio being determined, the terms themselves may be found by the Prop. Prop. 81.- To find the sum of a Geometric series. Let the terms of the series be multiplied by the common ratio; the result will be a series containing all the terms of the former except the first, and another term, which is equal to the product of the last term by the common ratio; also it is manifest, that the sum of this series is equal to the sum of the former multiplied by the common ratio. Hence the difference between these series is equal to the difference between the first term, and the last multiplied by the common ratio; but it is also equal to the difference between the sum of the first and the same sum multiplied by the common ratio, i.e. to the product of the sum of the first multiplied by the difference between 1 and the common ratio. Hence the sum may be obtained by R dividing the difference between the first term and the last multiplied by the common ratio, by the difference between 1 and the common ratio. Prop. 82.-The powers of a number, greater than unity, in crease, while those of a number, less than unity, decrease continually without limit. Any number repeated more than once is greater than itself; but if a number greater than unity be multiplied by itself, it will be repeated more than once; therefore the square of such a number is greater than the first power; the cube is greater than the square, &c. The powers therefore of a number greater than unity increase continually. And this increase has no limit. For if a number however small be continually added to another, there will in time result a number greater than any other, which may be assigned. Much more therefore, when the numbers successively added are continually increased, will this be the case. But in forming the cube from the square, a greater number is added than in forming the square from the first power, since in the former case a number greater than the square is added; and in the latter, a number less than the square. Similarly it may be shown, that in forming all the successive powers of a number greater than unity, numbers are added which continually increase. Hence by raising the number to some power, a number may he formed greater than any, which may be assigned. Again : a part, or parts of a number less than the whole number of parts in it, is less than the whole. But if a number be multiplied by a proper fraction, it is divided into a number of parts equal to the denominator, and a nu of these parts are taken equal to the numerator, and therefore less than the denominator. Therefore the product is less than the multiplicand. Hence the square, cube, fourth powers, &c. of a proper fraction, are respectively less than the first, second, third, &c. powers, or the powers of a proper fraction continually decrease. And this decrease has no limit. For a proper fraction is equal to unity divided by the same fraction inverted, which is greater than unity; and which therefore can be made as large as we please, by raising it to some power. By raising the proper fraction therefore to some power, a result may be obtained equal to unity divided by a number as large as we please, i.e. such a part of unity, as that any number of these parts we please are required to form a quantity as large as unity, that is, as small a part of unity as we please. Hence the powers of a proper fraction may be made as small as we please, or decrease continually without limit. Prop. 83.—To find the limit of the sum of an infinite de creasing Geometric series. |