3. Annex the quotient figure to the first co-efficient, and the sum will be the first number under the said co-ethicient : each of the two remaining numbers will be found by increasing the number above it by the quotient figure. 4. Multiply each of the two upper members of the first column, in succession, by the quotient figure, and the opposite number in the second column will be found by adding the product to the number above it. 5. Multiply the first member of the second column by the quotient figure, and the opposite pumber in the third column will be found by subtracting it from the absolute number. Then, if the product be less than the absolute number, the quotient figure is the second figure of the root to be extracted; but if not, the work must be repeated. Now, considering the third number in the first column as a frist co-efficient, the second number in the second column as a second coefficient, and the remainder as an absolute number, the next or third figure of the root will be found in the same manner as the second ; and so on. Er. Extract the cube root of the number 13. Here the nearest cube to 13 is 8, the root of which is 2; therefore the co-efficients of the first step are 6 and 12, and the remainder, or absolute number is 5. Now 5 will be found to contain 1, which is the second co-efficient without the last figure five times : but this will be found not to succeed ; therefore try 3 in the operation. 6 12 5.. (3 63 1389 833 66 1587 69 Proceed with the above columns of numbers according to the 2d, dg,4th, and 5th parts of the rule. Then, since 3 succeeds, divide 883 by 158, which is the co-efficient without the last figure, and the quotient 5 is the next figure of the root, which must now succeed ; therefore proceed with the next step. 69 1587 833.. (5 695 162175 22125 700 165675 705 Again, divide 22125 by 16567, and the quotient 1 is the next figure of the root; therefore proceed with the next step. 165675 22125.. (1 7051 16574551 5550449 7052 0081003 703 And so on, so that the root is 2351. 705 This process being sufficiently understood, the learner may then work the whole of the steps in one continued operation : thus, 6 5...(2.351 833.. 1587 : 7053 In this operation we may observe, that the multiplications and additions, as also the multiplications and subtractions, are performed in one line, as may be found in some of our best systenis of arithmetic. It may here be remarked, that wherever we stop in the operation, as many figures, except one, may be found as the number of figures in the root already obtained, by dividing the last remainder by the second co-efficient, wanting as many of its last figures as the number of figures to be found : thus, in the present instance, 5550449 divided by 16581 gives 334, which, annexed to the part 2.351 of the root already found, gives 2.351334, which is true to the last figure. Method of Proof. This operation will admit of a proof at every step, which may be done by the following Rule. Consider the co-efficients and remainder from which the step to be proved is found as whole numbers, and the figure of the root as a decimal in the place of tenths; then add into one sum the cube of the new figure, the product of the first co-efficient, and the square of the new figure the product of the second co-efficient, and the new figure itself together with the last remainder ; then, if the work is right, the sum will be equal to the preceding remainder or absolute number, Er. The co-efficients and absolute number by which the third figure of the root 5, in the example given, are 69,1587 and 833 considered as whole numbers ; then (5)'= 125 69 (5)= 17:25 1587 X (-5) =793.5 22:125 = 22:125 833.000 OF RATIOS, OR PROPORTION. DEFINITIONS. (96.) Ratio, is the relation which one quantity bears to another, with respect to magnitude. This relation exists only between quantities of a similar kind; thus, a number must be compared with a number ; a line with a line ; &c. &c. for it would be absurd to compare a certain number of feet with: a certain number of pounds ; &c. &c. (97.) The magnitude of quantities may be compared in two ways. First, with regard to their difference; and then the question asked, is, “ How much one quantity is greater or less than another.” This relation of quantities to each other, is called their Arithmetical ratio. The second way in which they may be compared, is, by inquiring “ How often one quantity is contained in the other." This relation between quantities is called their Geometrical ratio. Obs. The term ratio, when simply applied, is generally understood in the latter sense ; and it is in this sense that the word will be made use of in the present Article. (98.) In examining how often one quantity is contained in nother, the natural process is to divide the one by the other. Thus, in comparing the number 12 with the numbers 4 and 3, we find that 4 is contained in 12 three times, and that 3 is contained in 12 four times; from which we infer, the ratio of 12 :3 is greater than the ratio of 12 to 4, the magnitude of a ratio being measured by the number of times one quantity is contained in another. For the same reason, we may say, that the ratio of 11 : 7 is less than the ratio of 1 : 5. When a ratio is thus expressed, the first term of it is called the antecedent, the last term the consequent, of that ratio. Nobe.-In expressing the ratio of two quantities, the word "19" is generally supplied by two dots. thus, the ratio of “ a to 6” is expressed by “u:b. Corol. It appears, from this mode of estimating the magnitude of a ratio, that when the consequent of a ratio is not an aliquot part of the antecedent, the value of the ratio must be expressed by a fraction, the numerator of which is the antecedent, and denominator the consequent of that ratio. Thus, the magnitude of the ratio of 15 : 7 is expressed by the 4. fraction and that of the ratio 4 : 13 by the fraction 7 13' (99.) When the antecedent of a ratio is greater than the consequent, it is called a ratio of greater inequality ; when the antecedent is less than the consequent, a ratio of lesser inequulity; and if the two terms of a ratio be the same, then it is said to be a ratio of equality. (100.) But the foregoing definitions apply to those instances only, in which the consequent of a ratio is contained a certain number of times in the antecedent, or in which the magnitude of the ratio may be expressed by some definite fraction. It 15 does not, therefore, comprehend such ratios as ✓ 2 : 5; V3: 37; 4: %10, &c. &c. in which the values of the quan tities ✓2, 13, 37, &c. can only be expressed in decimal frac tions that do not terminate. The ratio that exists between quantities of this latter kind, when the radical quantity is expressed by a decimal fraction, is called their approximate ratio. (101.) Proportion may be defined the equality of ratios ; for, since 4 is contained in 12, the same number of times that 6 is in 18, the ratio of 12 : 4 is said to be equal to the ratio of 18 : 6, or, in other words, that 12:4 :: 18:6. The first and last terms of every proportion are called the extremes, and the second and third the means of that proportion. (102.) A continued proportion, is that in which a set of quantities are related to each other in the following manner, viz. a:b:: b:C::C:d::d:e, &c. where the consequent of every preceding ratio is the antecedent of the following one ; and if only three quantities be concerned, as in the proportion a: b:: b:c, then b is called a mean proportional between the two extremes a and c. Nole.-In s'ating a proportion, the words " is to," and " to," are generally supplied by two dors, and the words “so is,” by four dots ; thus, the pro. portion is to-b, so is c to d," is expressed by a:b::c:d." (103.) Since the proportion a : b::c:d denotes the equality of the ratios a :b and c:d; and since the magnitude of the ratio a :b is measured by the fraction and that of the ratio ā c:d by the fraction, it follows that that is, when four bod quantities are proportional, the quotient of the first divided by the second, is equal to the quotient of the third divided by the fourth ; and rice rersa, if there be four quantities a, b, c, d, such, that bd =, then those four quantities are proportional, or a : b::c:d. с On the Comparison and Composition of Ratios. (104.) On the comparison of Ratios. 1. Since the ratio of a : b may be expressed by the fraction if both the numerator and denominator of this fraction be D multiplied by any quantity m (m being either integral or frac ina vonal), then mbb and, consequently, ihe ratio of ma : mh is the same with the ratio of a :b; from which we infer, that if the terms of a ratio be multiplied or divided by the same quantity, it does not alter the value of the ratio. From whence, also, it appears, that a ratio is reduced to its lowest terms by dividing both its antecedent and consequent by their greatest common measure. 2. Ratios are compared together by reducing to a common denominator the fractions by which their values are respectively represented. 8 Thus, we represent the ratio of 8 : 5, by the fraction and the ratio of 9 : 6 by the fraction ; ; reduce these fractions to others of the same value, having a common denominator, and they become 48 45 48 and 45 respectively, and because is greater than the 30 30 30 30' ratio 8 : 5 is greater than the ratio of 9:6. (105.) A ratio of greater inequality is diminished, and a ratio of lesser inequality is increased, by adding the same quantity to both its terms. Let a+b: a represent a ratio of greater inequality, and let Z be added to each of its terms, and it then becomes the ratio of a +6+x:a+r. But the ratio of 2+b a+b+r Q+ =. atx Now reduce these fractions to others of the same value, having qo+ab+ar+lr a common denominator, and they become and a(a + x) a + altar respectively; and since a2 + ab + ax + bx, is evidently a(a +r) greater than a' + ab + ar, the ratio of atb: a is greater than the ratio of a +6+x : a+x. In other words, the ratio of a +b: a has been diminished by adding x to each of its terms. Again, let a-6: a represent a ratio of lesser inequality ; then pro a-b+x ceeding with the fractions and as in the former instance, ats aabtax-2 amabtar the resulting fractions are and and since a(a + x) a(a + x) a |