mixture the proportions of the remaining articles thereto are found thus: If, instead, it is desired to mix a given quantity, say 100 gallons, and proportioned, say as in first example, the quantity to be taken of each is ascertained by the following RULE. As the sum of the relative quantities is to the quantity required, so is each relative quantity to the quantity required of it respectively. The sum of the relative quantities alluded to is 6 + 30+ 50+10 =96; then, INVOLUTION Consists in involving, that is, in multiplying a number one or more times into itself. The number so involved is called the root, and the product arising from such involution, its power. The second power, or square, of the root, is obtained by multiplying the root once into itself, as 4 × 4=16; 4 being the root and 16 its square. The third power, or cube, of a number, is obtained by multiplying the number twice into itself, as 4 X 4 X 4-64; and so on for any power whatever. When a number is to be involved into itself, a small figure called the index or exponent is placed at its right, indicating the number of times it is to be so involved, or the power to which it is to be raised. Thus, 34 = 3 × 3 × 3 × 3=81; and 43 = 4 × 4 × 4 — 64. EVOLUTION. EVOLUTION is the opposite of Involution. It consists in finding a root of a given number, instead of a power of a given root. When the root of a number is required or indicated, the number is written with the before it: and the character or denomination of the root, if it be other than the square root, is defined by an index figure placed over the sign. When the square root of a number is required, the sign (✔) is placed before the number, but the index (2) is usually omitted. Thus, 25, shows that the square root of 25 is required, or to be taken; and 25 shows that the cube root is required. The operation is usually called extracting the root. TO EXTRACT THE SQUARE ROOT. RULE- 1. Separate the given number into periods of two figures each, by placing a point over the first figure, third, fifth, &c., counting from right to left — the root will consist of as many figures as there are periods. 2. Find the greatest square in the left hand period, and place its root in the quotient; subtract the square of the root from the left hand period, and to the remainder bring down the next period for a dividend. 3. Multiply the root so far found - the figure in the quotient - by 2, for a divisor; see how many times the divisor is contained in the dividend, except the right hand figure, and place the result (the number of times it is contained) in the quotient, to the right of the figure already there, and also to the right of the divisor; multiply the divisor, thus increased, by the last figure in the quotient, and subtract the product from the dividend, and to the remainder bring down the next period for a dividend. 4. Multiply the quotient - the root so far found (now consisting of two figures) by 2, as before, and take the product for a divisor; see how many times the divisor is contained in the dividend, except the right hand figure, and place the result in the quotient, and to the right of the divisor, as before; multiply the divisor, as it now stands, by the figure last placed in the quotient, and subtract the product from the dividend, and to the remainder bring down the next period for a dividend, as before. 5. Multiply the quotient (now consisting of 3 figures) by 2, as before, and take the product for a divisor, and in all respects proceed as when seeking for the last two figures in the quotient. The quotient, when all the periods have been brought down and divided, will be the root sought. NOTE. 1. If there is a remainder after finding the integer of a root, annex periods of ciphers thereto, and proceed as when seeking for the integer. The quotient figures will be the decimal portion of the root. 2. If the given number is a decimal, or consists of a whole number and decimal, point off the decimal from left to right, by placing the point over the second, fourth, sixth, &c., figures therein, and fill the last period, if incomplete, by annexing a cipher. 3. If the dividend does not contain the divisor, a cipher must be placed in the quotient, and also at the right of the divisor, and the next period brought down; then the dividend must be divided by the divisor as increased. 4. If the quotient figure, obtained by dividing by the double of the root, is too large, as will sometimes be the case, (see 2d Example) it must be dropped, and a less — one which is the true measure taken in its stead. EXAMPLE.Required the square root of 123456.432. 123456.4320 (351.3636+. Ans. EXAMPLE. 9 65) 334 701) 956 7023) 25543 70266) 447420 702723) 2582400 2108169 7027266) 47423100 42163596 5259504 Required the square root of 10621. Also, of 28561 RULE 1. Separate the given number into periods of three figures each, by placing a point over the first, fourth, seventh, &c., counting from right to left the root will consist of as many figures as there are periods. 2. Find the greatest cube in the left hand period, and place its root in the quotient; subtract the cube of the root from the left hand period, and to the remainder bring down the next period for a dividend. 3. Multiply the square of the quotient by 300, for a divisor; see how many times the divisor is contained in the dividend, and place the result (except that the remainder is large, diminished by one or two units) in the quotient. 4. Multiply the divisor by the figure last placed in the quotient, and to the product add the square of the same figure, multiplied by the other figure, or figures, in the quotient, and by 30; and add also thereto the cube of the same figure, and take the sum for the subtrahend; subtract the subtrahend from the dividend, and to the remainder bring down the next period for a dividend, with which proceed as with the preceding, so continuing until the whole is completed. NOTE-1. Decimals must be pointed from left to right, by placing a point over the third, sixth, &c., figures in that direction. 2. If the divisor is not contained by the dividend, place a cipher in the quotient, and annex two ciphers to the divisor, and bring down the next period for a dividend, and use the divisor, as thus increased, for finding the next quotient figure. 3. If there is a remainder after finding the integer of the root, annex a period of three ciphers thereto, and proceed for the decimal of the root as if seeking for the integer, annexing a period of three ciphers to each remainder until the decimal is carried to as many places of figures as desired. EXAMPLE. - Required the cube root of 47421875.6324. 47421875.632400 (361.959+. Ans. EXAMPLE.Required the cube root of 32768. Also, of 8489664. General Rule for extracting the roots of all powers, or for finding any proposed root of a given number. 1. Point off the given number into periods of as many figures each, counting from right to left, as correspond with the denomination of the root required; that is, if the cube root be required, into periods of three figures, if the fourth root, into periods of four figures, &c. 2. Find the first figure of the root by inspection or trial, and place it at the right of the number, in the form of a quotient; raise this quotient figure to a power corresponding with the denomination of the root sought, and subtract that power from the left hand period, and to the remainder bring down the first figure of the next period, for a dividend. 3. Raise the root thus far found (the quotient figure) to a power next inferior in denomination to that of the root required, multiply this power by the number or index figure of the root required, and take the product for a divisor; find the number of times the divisor is contained in the dividend, and place the result (except that the remainder is large, diminished by one or two units) in the quotient, for the second figure of the root. 4. Raise the root thus far found (now consisting of two figures) to a power corresponding in denomination with the root required, and subtract that power from the two left hand periods, and to the remainder bring down the first figure of the third period, for a dividend; find a new divisor, as before, and so proceed until the whole root is extracted. EXAMPLE. - Required the fifth root of 45435424. |
