nificant figure below the fourth place from the right. We therefore reject the 6, separating it by an accent, and divide 15,(15000), by 6,(600), twice the hundreds of the root. The quotient is 2, (20), which we put as the second figure or tens of the root; we also place it at the right of the divisor. The divisor thus increased is 62, (620), and answers to 2a+b, which we multiply by 6 or b, and have 124, (12400), answering to (2 a+b) b. We subtract 124 from the dividend 156, and have for a remainder 32, to the right of which we annex the last period 25, and regard the result 3225 as a new dividend. This dividend corresponds to 2 (a + b) c + c2, or [2 (a+b)+c] c; that is, it contains twice the sum of the hundreds and tens multiplied by the units, plus the second power of the units. To find the units of the root, therefore, we must divide by twice the sum of the hundreds and tens, that is, by twice the whole root, so far as it has been already ascertained. But since hundreds and tens, multiplied by units, must have one zero at the right, this product can have no significant figure below the second from the right; we therefore reject the figure 5, separating it by an accent. Twice the hundreds and tens make 64, (640), corresponding to 2(a+b), which is contained in 322 (3220) five times. We put 5, which is the presumed value of c, as the next figure of the root, also at the right of the divisor. The divisor thus increased answers to 2 (a+b)+c, which multiplied by 5, or c, gives 3225, corresponding to [2 (a + b)+c] c; this subtracted from the dividend leaves no remainder. Hence 325 is the root required. Of the two formulæ, already given, of the second power of a+b+c, viz., a2+2ab+2(a+b) c + c2, and a2+ (2 a+b)b+[2 (a+b)+c] c, the former shows, that, after the first figure has been found, each succeeding figure is to be sought by dividing by twice the whole of the root previously found; and the latter shows that, in each case, the quotient is to be placed at the right of the divisor, and that the divisor, thus increased, is to be multiplied by the quotient. Moreover, from the rank of the figures, it is evident that twice the root already found can produce no significant figure below the second from the right in each dividend. Formulæ might be given for the second power of four or more figures; but from what has been already shown, the mode of proceeding, in all cases, will be sufficiently manifest. We exhibit below the process of extracting a root consisting of five figures. Operation. 28/11/22/64/41 (53021. Root. 25 31'1 (103 309 2226'4 ( 10602 10604'1 (106041 0. In this example we find that the second divisor 106 is not contained in the dividend 222, the right-hand figure being rejected; this shows that there are no hundreds in required root. In such a case, we place a zero in the root, also at the right of the divisor, and annex the suc ceeding period to form a dividend. ART. 85. From the foregoing examples and explanations we deduce the following RULE FOR EXTRACTING THE SECOND ROOTS OF NUMBERS. 1. Begin at the right, and, by means of accents or other marks, separate the number into periods of two figures each. The left-hand period may contain one or two figures. 2. Find the greatest second power in the left-hand period, place its root at the right of the proposed number, separating it by a line, and subtract the second power from the left-hand period. 3. To the right of the remainder bring down the next period to form a dividend. Double the root already found for a divisor. Seek how many times the divisor is contained in the dividend, the right-hand figure being rejected. Place the quotient in the root, at the right of the figure previously found, and also at the right of the divisor. Multiply the divisor, thus increased, by the last figure of the root, and subtract the product from the whole dividend. 4. Repeat the process in part third of the rule, until all the periods have been brought down. Remark 1st. If the dividend will not contain the divisor, the right-hand figure of the former being rejected, place a zero in the root, also at the right of the divisor, and bring down the next period. Remark 2d. We may observe, that, if the last figure of the preceding divisor be doubled, the root will be doubled; for that divisor contains twice the whole root found, with the exception of the figure last obtained. Remark 3d. To find the root of a product, as will be shown hereafter, we take the root of each factor and multiply these roots together. Thus, the root of 81.225 is 9.15135. second powers, and the roots of such as are not perfect powers cannot be found exactly, either in whole numbers or fractions. Thus, the second root of 5 is between 2 and 3; but no number can be obtained, which, multiplied by itself, will produce exactly 5. Such a root may, however, be approximated to any degree of accuracy. All whole numbers and all definite fractions are called commensurable or rational, because they have a common measure with unity, or their ratio to unity can be exactly obtained. But the root of a number which is not a perfect power, is called incommensurable or irrational, because it has no common measure with unity, or its ratio to unity cannot be exactly found. Such roots, or rather expressions of them, are called also surds. The second root of a quantity is denoted either by the exponent, or by the sign, called the radical sign Thus, 43 or √ 4 = 2, and 3 or √3 means the second root of 3. The second root of a negative quantity is called imaginary, because no quantity, either positive or negative, can, when multiplied by itself, produce a negative quantity. Thus, (-4) or 4 is an imaginary quantity. An imaginary result to a problem generally indicates absurdity in the conditions of that problem. SECTION XXVIII. SECOND ROOTS OF FRACTIONS, AND EXTRACTION OF SECOND ROOTS BY APPROXIMATION. ART. 87. A fraction is raised to the second power by raising both numerator and denominator to that power, this being equivalent to multiplying the fraction by itself. 2 2 4 2 m m Thus, ()'=.=; and (m)'= ".; 3 3 n n Consequently, the second root of a fraction is found by extracting the root of both numerator and denominator. AṚT. 88. But if either the numerator or denominator is not an exact second power, we can find only an approximate root of the fraction. Thus, the root of 15 is between and, or 1, the latter being nearer the true root than the former. We can always make the denominator of a fraction a perfect second power, by multiplying both numerator and denominator by the denominator. This does not change the value, but only the form of the fraction. For example, Remark. The sign +, placed after an approximate |