shall find the several results to agree with the respective limits established by the formula. Thus, 5,37824, which is the fifth power of 1,4, is within 3 units; 6,864, &c., which is the fifth power of 1,47, is within; 6,9816, &c., which is the fifth power of 1,475, is within 80, that is, within; and so on for the rest. Having found the figures 1,4757, we suppress one figure of the divisor, and venture to find, by division, one more figure, which is 7, and, without continuing the operation farther, we raise 1,47577 to the fifth power, and have 6,9999250187372420137021657, which not only differs from 7 by less than a thousandth, as was proposed, but by less than a ten-thousandth of a unit. = 819 4./281950621875 = 195 7. 1/4850333677308998938550332032 =9018 8. Required the fifth root of, so that the fifth power of the root may differ from by less than a thousandth of a unit. 7 Ans. ,91356, the fifth power of which differs from by less than a ten-thousandth of a unit. 9. Required the fifth root of, so that the fifth power of the root may differ from by less than a ten-millionth of a unit. Ans. ,922107911, the difference between the fifth power of which and is,000000017414005424186469844153340045905861156, which is evidently less than a ten-millionth of a unit. 10. Required the seventh root of 13, so that the seventh power of the root may differ from 13 by less than a thousandth of a unit. Ans. 1,44256. The seventh power of this root being 12,99981583631852561343253184235175936, is evidently within the required limit. SECTION XXI. OF AN EQUATION-GEOMETRICAL PROPORTION-COMPOUND PROPORTION. Of an Equation. 397. WHEN two quantities, each consisting of one or more terms, have the sign between them, the whole expression is called an equation; thus, each of the expressions ab, ad a+b=c; =c, a―c=b, ad bc, 1+2 - }=}, ad=bc, b is an equation. = Each quan The expression on the left of the sign is called the first member; that on the right, the second member. tity contained in either member is called a term. Now, it is plain that, as the members of an equation are equal, we may add to or subtract from each the same or equal quantities: also, that we can multiply or divide each by the same or by equal quantities without disturbing the equation; that is to say, the results will, in each case, be equal. 398. Any term, in either member, may be removed (transposed) to the other member, by merely changing its sign when so removed. For if it is a positive quantity, it is, when removed, evidently subtracted from the member from which it is taken; and, in placing it in the other member, its subtraction from that member is signified by placing before it the negative sign. Again, if it is a negative quantity, we must, in removing it, consider that we have added a quantity equal to it to that member from which it is removed, because such addition (95) would only destroy it; and, in placing it in the other member with the sign+, we signify its addition to that member also. Wherefore, as equals added to equals, or subtracted from equals, must give results which are equal, the equation is not disturbed. Thus, in the equation +38 , if we remove the negative quantity to the second member, and change its sign, we have + 3 = √ + }· Again, if we remove to the first member, and change its sign, we have + } } = {, or +. Hence also it is plain that all the signs of both members may be changed without disturbing the equation. 399. An equation is cleared of fractions by multiplying both members by each denominator successively, or by the least common multiple of all the denominators of those fractions. Thus, in the above equation, 1+1=§, if we multiply both sides by 6, we have 3 + 2 400. When the data of a mathematical question—that is to say, the quantities expressed or implied in its enunciation—are put in equation, the solution of the equation is often the solution of the question. An equation containing one simple unknown quantity is solved by bringing all the terms involving the unknown quantity to one side of the equation, and the known quantities to the other; at the same time, by the above methods, reducing the equation to its simplest form. The unknown quantity being expressed in terms of quantities which are known, thus becomes known. It is not the province of common arithmetic to treat of this subject at large, but of Universal Arithmetic, or Algebra, to which the student is referred for its full development; it is merely introduced in this place to enable the student more clearly to comprehend some important parts of arithmetic. Examples. 1. Suppose that into an empty cistern, the capacity of which is 1800 gallons, the water runs during a heavy shower for some time at the rate of 30 gallons per minute; and that, by the increase of the shower, it then runs into it at the rate of 55 gals. per minute, until the cistern is full. During how many minutes must it run at the rate of 55 gals. per minute, that the cistern may be filled in 50 minutes from the commencement of the shower, and what part of the cistern is filled in each interval of time? Let = the required number of minutes; then 50 - x the number of minutes during which the water runs at the rate of 30 gals. per minute. Also 55x the number of gallons received by the cistern after the increase of the shower, and (50-x)30 the number of gals. previously received; then, 55x+(50 — x)30: 1800, or 55x + 1500 30x or 25x 1800 1500 = 300 300; wherefore, x= Hence the water runs, during 12 minutes, at the rate of 55 = 1800, = 12. gals. per minute, and consequently during 38 minutes at the rate of 30 gals. per minute; then we have 55X12: 660 gals. filled after the increase of the shower; also 30X381140 before the increase Proof. += 38=1, or the whole cistern. 2. Suppose that a man and his son can together cut 24 cords of wood in 6 days, and that the man alone can do it in 10 days; in how many days can the son alone perform it, and how much wood does each cut per day? Ans. The son can perform it in 15 days; also, the man cuts 2 cords, and his son 1g cords per day. 3. What number is that, the square root of which is double its cube root? Ans. 64. 4. What number is that, of which half the cube root is equal to of its square root? Ans. 11,390625. Prove this result. Geometrical Proportion-Rule of Three, or Golden Rule. 401. The ratio of two quantities is often expressed by two points, one above the other, placed between them. Thus, 6 3 signifies the ratio of 6 to 3, and is read 6 is to 3; a: b, signifies the ratio of a to b, and is read a is to b. The scholar already knows that the value of the first of these is § =2, and that the value of the last is expressed by α The two homogeneous quantities put in ratio are called the terms of the ratio; the first term is called the antecedent, and the last the consequent. Thus 6 and a are the antecedents, and 3 and 6, the consequents of the above ratios. 402. A proportion is the expression of the relation existing between the terms of two equal ratios. Thus, 8: 4 and 6: 3 form a proportion, each ratio being equal to 2. Four points are placed between the ratios thus-8: 4 :: 6 : 3, and the proportion is read 8 is to 4 as 6 is to 3. Now it is plain, that as the value of each ratio is 2, the first term bears the same rela tion to the second that the third does to the fourth. Whenever, therefore, it may be said of four quantities, that the first is to the second as the third to the fourth, they are in proportion. When this cannot be said, they are not in proportion, the ratios being unequal. The first and last terms of a proportion are called the extremes, and the two middle terms the means. a b * 403. Let ab::c:d be any proportion whatever; then as if we multiply both fractions by b, the denominator of с d the first fraction, which evidently will not disturb the equality, bc d' we have a== Again, if we multiply both sides of the equation by d, we have ad- be; that is to say, in every proportion the product of the extremes is equal to the product of the means. This may therefore be called the fundamental property of a proportion. 404. We have seen (108) that the product of any two abstract numbers may be considered as representing the content or surface of a rectangular parallelogram. The product of the extremes and the product of the means may therefore be considered as representing two equal rectangles; hence, doubtless, the name geometrical has been applied to this kind of proportion, as well as to the ratio upon which it is based. If any object that, in applying this kind of proportion to the solution of mercantile questions, there would be evident impropriety in supposing a quantity of tobacco to be one side of a parallelogram, and the price of another quantity of tobacco the other; we should remember that this application does not change the fundamental property in the abstract, and that it is by taking the given numbers in an abstract sense that we are able to apply the principle to the solution of such questions. 405. When three terms of a proportion are given, the fourth is, by the fundamental property explained above, easily found. For the product of two numbers, divided by either of them, gives the other; and as the product of the extremes is the same as that of the means, the product of the extremes, divided by either mean, gives the other mean; also, the product of the means, divided by either extreme, gives the other extreme. Now, as in all mercantile questions, which are solved by means *Les moyens, (French,) middle terms. |