N. B. The following questions are left without any answers, that the Scholar may operate and prove each question. 19. What will 11 yards of flannel, at 2s. 6d. per yard come to ? 20. What will 13lb. of cotton cost at 3s. 4d. per lb. ? 21. What will 183 yards of ribbon come to at 8d. per yard! THE SCHOLAR'S ARITHMETIC SECTION III. RULES OCCASIONALLY USEFUL TO MEN IN PARTICULAR CALLINGS AND PURSUITS OF LIFE. § 1. INVOLUTION. INVOLUTION, or the raising of powers, is the multiplying of any given number into itself continually, a certain number of times. The quantities in this way produced, are called powers of the given number. Thus, 4X4 16 is the second power or square of 4. 4X4X4 64 is the 3d power, or cube of 4. 4X4X4X4=256 is the 4th power or biquadrate of 4. :42 =43 :44 The given number, (4) is called the first power; and the small figure, which points out the order of the power, is called the Index or the Exponent. 2. EVOLUTION. EVOLUTION, or the extraction of roots, is the operation by which we find any root of any given number. The root is a number whose continual multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th, root, &c. accordingly as it is, when raised to the 2d, 3d, 4th, &c power, equal to that power. Thus, 4 is the square root of 16, because 4X4 16. 4 also is the cube root of 64, because 4X4X4-64; and 3 is the square root of 9, and 12 is the square root of 144, and the cube root of 1728, because 12x12x12=1728, and so on. 1 To every number there is a root, although there are numbers, the precise roots of which can never be obtained. But by the help of decimals, we can approximate towards those roots, to any necessary degree of exactness. Such roots are called Surd Roots, in distinction from those perfectly accurate, which are called Rational Roots. The square root is denoted by this character ✔placed before the power; the other roots by the same character, with the index of the root placed over it. Thus the square root of 16 is expressed ✔ 16, and the cube root 3 of 27 is 27, &c. When the power is expressed by several numbers, with the sign + or— between them, a line is drawn from the top of the sign over all the parts of it; thus the second power of 21-5 is 21-5, and the 3d power of 3 56+8 is 56+8, &c. The second, third, fourth and fifth powers of the nine digits may be seen in the following Roots or 4th Powers 1|16| 81 256 625 1296 | 2401 | 4096 | or 5th Powers | 1 | 32 | 243 | 1024 | 3125 | 7776 | 16807 | 32768 | 59049 41. 5 16 | 64 6561 3. EXTRACTION OF THE SQUARE ROOT. To extract the square root of any number, is to find another number which multiplied by or into itself, would produce the given number; and after the root is found, such a multiplication is a proof of the work. RULE. 1. "Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on, which points shew the number of figures the root will consist of. 2. "Find the greatest square number in the first, or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number, under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend. 3. Place the double of the root, already found, on the left hand of the dividend for a divisor. 4. "Seek how often the divisor is contained in the dividend, (except the right hand figure) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor; multiply the divisor with the figure last annexed by the figure last placed in the root, and subtract the product from the dividend; to the remainder join the next period for new dividend. 5. "Double the figures already found in the root for a new divisor, (or bring down your last divisor for a new one, doubling the right hand figure of it) and from these find the next figure in the root, as last directed, and continue the operation in the same manner till you have brought down all the periods." "Note 1. If, when the given power is pointed off as the power requires, the left hand period should be deficient, it must nevertheless stand as the first period." "Note 2. If there be decimals in the given number it must be pointed both ways from the place of us; If, when there are integers, the first period in the decimals be debent, it may be completed by annexing so many cyphers as the power requires: And the root must be made to consist of so many whole numbers and decimals as there are periods belonging to each; and when the periods belonging to the given number are exhausted, the operation may be continued a pleasure by annexing cyphers." OPERATION. 729(27 the root. 4 47)329 329 000 PROOF. 1 The given number being distinguished into periods, I seek the greatest square number in the left hand period (7) which is 4, of which the root (2) being placed to the right hand of the given number, after the manner of a quotient, and the square number (4) subtracted from the period (7) to the remainder (3) I bring down the next period (29) making for a dividend, 329. Then the double of the root (4) being placed to the left hand for a divisor, I say how often 4 in 32 ? (excepting 9 the right hand figure) the answer is 7, which I place in the root for the second figure of it, and also to the right hand of the divisor; then multiplying the divisor thus increased by the figure (7) last obtained in the root, I place the product underneath the dividend, and subtract it therefrom, and the work is done. 27 27 189 54 729 DEMONSTRATION Of the reason and nature of the various steps in the extraction of the SQUARE ROOT. The superficial content of any thing, that is the number of square feet, yards or inches, &c. contained in the surface of a thing, as of a table or floor, a picture, a field, &c. is found by multiplying the length into the breadth. If the length and breadth be equal, it is a square, then the measure of one of the sides as of a room, is the root, of which the superficial content in the floor of that room is the second power. So that having the superficial contents of the floor of a square room, if we extract the square root, we shall have the length of one side of that room. On the other hand, having the length of one side of a square room, if we multiply that number into itself, that is, raise it to the second power, we shall then have the super ficial contents of the floor of that room. The extraction of the square root therefore has this operation on numbers, to arrange the numbers of which we extract the root into a square |