THE SCHOLAR'S ARITHMETIC. SECTION III. Rules occafionally useful to men in particular callings and purfuits of life. § 1. Involution. INVOLUTION, or the railing of powers is the multiplying of any given number into itself continually, a certain number of times. this way produced, are called powers of the given number. 4x4-16 is the 2d. power, or square of 4. 4×4464 is the 3d. power, or cube of 4. The quantities in Thus, -42 43 4× 4×4×4256 is the 4th power, or biquadrate of 4. —4+ The given number, (4) is called the firft power; and the fmall 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 whofe continual multiplication into itfelf produces the power, and is denominated the fquare, cube, biquadrate, or 2d, 3d, 4th, root, &c. accordingly as it is, when raifed to the 2d, 3d, 4th, &c. power, equal to that power. Thus 4 is the fquare root of 16, because 4×4 16. 4 alfo is the cube root of 64, because 4 x 4x464; and 3 is the fquare root of 9, and 12 is the fquare root of 144, and the cube root of 1728, because 12x12x12 1728, and so on. To every number there is a root, although there are numbers, the precise foots of which can never be obtained. But, by the help of decimals, we can approximate towards thofe roots, to any neceffary degree of exactness. Such roots are called Surd Roots, in distinction from thofe, perfectly accurate, which are called Rational Roots. The square root is denoted by this character placed before the power; the other roots by the fame character, with the index of the root placed over it. Thus the fquare root of 16 is expreffed✔ 16, and the cube root of 27 is √27, &c. 3 When the power is expreffed by feveral numbers with the fign+ or between them, a line is drawn from the top of the fign over all the parts of it; thus, the second root of 21-5 is √ 21-5, and the 3d. root of 56+8 3 is 56+8, &c. The fecond, third, fourth, and fifth powers of the nine digits may be seen in the following or 3d. Powers. 8 27 64 125 216 343 512 729 Cubes, § 3. Extraction of the Square Hoot. To extract the fquare root of any number, is to find another number which multiplied by, or into itself, will produce the given number; and after the root is found, fuch a multiplication is a proof of the work. RULE. 1. "Diftinguifh 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 fhew the number of figures the root will confift of. 2. "Find the greatest fquare 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 divifion) for the first figure of the root, and the fquare number, under the period, and fubtract 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 divifor. 4. "Seek how often the divifor is contained in the dividend (except the right hand figure) and place the anfwer in the root for the fecond figure of it, and likewife on the right hand of the divifor; multiply the divifor with the figure last annexed by the figure last placed in the root, and fubtract the product from the dividend: To the remainder join the next period for a new dividend. 5. "Double the figure already found in the root, for a new divifor, (or bring down your laft divifor for a new one, doubling the right hand figure of it) and from these, find the next figure in the root as laft directed, and continue the operation in the fame 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 fhould 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 units If, when there are integers, the first period in the decimals be deficient, it may be completed by annexing fo many cyphers as the power requires : And the root must be made to confift of fo 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 at pleasure by annexing cyphers." EXAMPLES. 1. What is the fquare root of 729 ? OPERATION. 729(27 the root. 4 47(329 329 000 PROOF. 27 27 189 54 729 The given number being diftinguifhed into periods, I feek the greatest fquare 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 fquare number (4) fubtracted 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 divifor, I fay how often 4 in 32? (excepting 9 the right hand figure) the anfwer is 7, which I place in the root for the fecond figure of it, and alfo to the right hand of the divifor; then multiplying the divifor thus increased by the figure (7) last obtained in the root, I place the product underneath the dividend, and fubtract it therefrom, and the work is done. DEMONSTRATION. Of the reafon and nature of the various steps in the extraction of the SQUARE ROOT. The fuperficial content of any thing, that is, the number of fquare feet, yards, or inches, &c. contained in the furface 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 fquare, then the meafure of one of the fides as of a room, is the root, of which the fuperficial content in the floor of that room, is the second power. So that having the fuperiicial contents of the floor of a fquare room, if we extract the fquare root, we hall have the length of one fide of that room. On the other hand, having the length of one fide of a fquare room, if we multiply that number into itfelf, that is to raife it to the fecond power, we fhall then have the fuperficial contents of the floor of that room. The extraction of the fquare root, therefore has this operation on numbers, to arrange the number of which we extract the root into a fquare form. As if a man fhould have 625 yards of carpeting, 1 yard wide, if he extract the fquare root of that number (625) he will then have the length of one fide of a fquare room, the floor of which, 625 yards, will be just fufficient to cover.--To proceed then to the demonstration. EXAMPLE 2. Suppofing a man has 625 yards of carpeting, 1 yard wide, what will be the length of one fide of a fquare room, the floor of which his carpeting will cover? The first step is to point off the number into periods of two figures each. This determines the number of figures of which the root will confift, and is done on this principle, that the product of any two numbers can have at mojì but se many places of figures as there are places in both the factors, and at least, but one lefs, of which any perfon may fatisfy himfelf at pleafure. OPERATION. d. a. 625(20 4. 225 FIG. I. A. 20 20 20 400 20 The number being pointed off, as the rule directs, we find we have two periods; confequently, the root will confist of two figures. The greatest fquare number in the left hand period (6) is 4, of which two is the root; there fore, 2 is the first figure of the root, and as it is certain we have one figure more to find in the root, we may for the prefent fupply the place of that figure by a cypher, (20) then 20 will exprefs the just value of that part of the root now obtained. But it must be remembered, that a root is the fide of a fquare of equal fides. Let us then form a fquare, A. Fig. I. each fide of which fhall be fuppofed 20 yards. Now the fide a b of this fquare, or either of the fides, fhews the root, 20, which we have obtained. To proceed then by the rule, " place the fquare number underneath the period fubtract, and to the remainder bring down the next period." Now the fquare number (4) is the fuperficial content of the fquare A-made evident thus-each fide of the fquare A, measures 20 yards, which number multiplied into itself, produces 400, the fuperficial contents of the fquare A; alfo the fquare number, or the fquare of the figure 2 already found in the root, is 4, which placed under the period (6) as it falls in the place of hundreds, is in reality 400, as might be feen alfo by filling the places to the right hand with cyphers, then 4 fubtracted from 6 and to the remainder (2) the next period (25) being brought down, it is plain, the fum 225 has been diminished by the deduction of 400, a number equal to the fuperficial contents of the fquare A. Hence, Fig. 1. exhibits the exact progrefs of the operation. By the operation, 400 yards of the carpeting have been difpofed of, and by the figure is feen the difpofition made of them. Now the fquare A, is to be enlarged by the addition of the 225 yards which remain, and this addition must be so made that the figure, at the fame time, fhall continue to be a complete and perfect fquare. If the addition be made to one fide only, the figure would lofe its fquare form; it must be made to two fides; for this reafon the rule directs," place the double of the root already found on the left hand of the dividend for a divifor." The double of the root is juft equal to two fides b c and c d of the square, A, as may be seen by what fclows. The fquare A C D b D 55 =400 yds. Cef100 g b=100 25 25 C 20 5 100 Proof 635 yds. 5 h Again, by the rule, "Seek how often the divifor is contained in the divi dend (except the right hand figure) and place the answer in the root, for the fecond figure of it, and on the right hand of the divifor." Now if the fides bc and cd of the fquare A, Fig. II. is the length to which the remaining 225 yards are to be added, and the divifor (4 tens) is the fum of thefe two fides, it is then evident, that 225 divided by the length of the two fides, that is by the divifor (4 tens) will give the breadth of this new addition of the 225 yards to the fides b and c d of the fquare, A. except But we are directed to the right hand figure," and alio to "place the quotient figure on the right band of the divifor;" the reafon of which is that the addition, Cef and C g h to the fides be and cd of the fquare, A, do not leave the figure a complete iquare, but there is a deficiency, D, at the corner. Therefore, in dividing, the right hand figure is expected, to leave something of the dividend, for this deficiency; and as the deficiency, D, is limited by the additions, Cef and Cg h, and as the quotient figure (5) is the width of these additions, confequently equal to one fide of the fquare, D; therefore, the quotient figure (5) placed to the right hand of the divifor (4 tens) and multiplied into itfelf, gives the contents of the fquare, D, and the 4 tens to the sum of the fides, be and cd of the addition Cef and Cgh, multiplied by the quotient figure, (5) the width of thofe additions, give the contents Cef and Cg h, which together fubtracted from the dividend, and there being no remainder, fhew that the 225 yards are dif posed in the new additions Ce f, Cg h, and D, and the figure is feen to be continued a complete fquare. Confequently, fig. II. fhews the dimenfions of a fquare room, 25 yards on a fide, the floor of which, 625 yards of carpeting, 1 yard wide will be fufficient to cover. The proof is feen by adding together the different parts of the figure. Such are the principles, on which the operation of extracting the fquare root is grounded. |