ON RATIOS AND PROPORTION. 95. By the term RATIO, is meant the relation which one quantity bears to another, with respect to magnitude. It is obvious, that this relation can exist only between quantities of the same kind, as of a line to a line, a surface to a surface, &c. 96. One quantity may be compared with another, either with a view of finding out their difference, or of finding how often the one contains the other. The relation which one quantity bears to another, with respect to their difference, is called their Arithmetical ratio. If the object of the comparison be to discover what multiple, part, or parts, the one is of the other, this relation is called their Geometrical ratio. In this latter sense, the term ratio is in the mean time to be used. 97. In order to discover the ratio subsisting between two quantities, the natural process is to divide the one by the other; in which case, the quotient will evidently express the fraction or multiple which the one is of the other. Thus, in comparing 2 with 6, we observe that the latter contains 6 the former 3 times, being 3. Hence, then, 3 expresses 2 = the relative magnitude of 2 and 6, meaning that 6 is 3 times as great as 2; or, which amounts to the same thing, that 2 is part of 6. In the same manner, the fractions which express the ratio or relative magnitude of 4 and 6, 5 2 5 8 4 and 6, and 8 and 10, are and = and respec6'6 10 3'6 5 tively. Or, in general, the ratio of a to b is expressed by α 4 5 the fraction The ratio of one quantity to another, as of a to b, is expressed by placing two points between them; thus, ab; where the former, from the circumstance of its standing first, is called the antecedent of the ratio, while the latter, from its following the other, is denominated the consequent. 98. If the terms of a ratio be multiplied or divided by the same quantity, the ratio is not altered. Thus, the ratio of a to b is the same with that of ma to mb, because ma (No. 42. Arith.) Hence, a ratio will be reduced to its mb' lowest terms, by dividing both its terms by their greatest common measure. 99. That ratio is the greater, whose antecedent is the greater multiple part or parts of its consequent. Thus, the 3 ratio of 3: 4 is greater than the ratio of 2:3; because 4 100. A ratio is called a ratio of less inequality, of equality, or of greater inequality, according as the antecedent is less than, equal to, or greater than the consequent. Thus, 3: 4 is a ratio of less inequality; 4: 4 of equality; and 5: 3 of greater inequality. 101. A ratio of less inequality is increased, and of greater inequality diminished, by adding any quantity to both its terms. Thus, if to the terms of the ratio 3:5, 2 be added, it becomes the ratio of 5: 7, which is a greater ratio 5 3 than the former, because is greater than Or, to give a 7 5 general demonstration, let a: a+b represent any ratio of less inequality; let c be added to both terms, then the ratio becomes that of a+c: a+b+c. Now, the ratio of a: a+b, α is otherwise expressed by ; and that of a+c: a+b+c by a+b a a+c Let now the two fractions and a+b. a+b+c a+c a+b+c be reduced to a common denominator, and they become a2+ab+ac a2+2ab+b2+ac+bc a2+ab+ac+bc and a2+2ab+b+ac+bc* Now, since the numerator of the latter fraction is greater than that of the a+c: a+b+c is a greater ratio than that of a: a+b, and hence that the ratio of a: a+b has been increased by adding the same quantity to both its terms. In the same manner, it might be shown, that any ratio of greater inequality, as that of a+b: a, is diminished by adding the same quantity c to both its terms; that a ratio of less inequality is diminished, while a ratio of greater inequality is increased, by subtracting the same quantity from both its terms. 102. Ratios are compounded together, by multiplying their antecedents together for a new antecedent, and their consequents together for a new consequent. Thus, if the ratios of a: b and c d be compounded together, the resulting ratio is that of ac: bd. If the ratios of 3: 4, 5: 6, and 7:10, be compounded together, the resulting ratio is that of 3x5×7:4x6x10 105: 240 = (dividing both by 15) 7:16, (No. 98.) = 103. If any ratio a: b be compounded with itself once, twice, thrice, &c., the resulting ratios are those of a2: b2, a3: b3, a1: b1, &c. The ratio of a b2, is called the duplicate of a:b; a3: b3, the triplicate; a: b, the quadruplicate, &c. Hence the duplicate ratio of two quantities is the same as that of their squares; the triplicate, as that of their cubes, &c. &c. These ratios evidently receive their names from the indices of a and b; in this way, the ratio of √a:√b is called the sub-duplicate of a:b; Va: √b, the sub-triplicate, &c. &c. 104. If any number of ratios, of which the consequent of the preceding ratio is the same with the antecedent of the succeeding one, be compounded together, the resulting ratio is that of the first antecedent to the last consequent. Thus, if the ratios a: b, b: c, c: d, &c. be compounded together, the resulting ratio is that of abc &c. bed &c. or (dividing both by bc) that of a: d, : 105. A ratio of less inequality, compounded with another, diminishes it, while a ratio of greater inequality, compounded with another, increases it. Thus, let y be com pounded with the ratio of a: b, the resulting ratio is that of ax: by, which is less or greater than ab, according as àx is less or greater than that is, according as a is less or by greater than y. 106. If the difference between the antecedent and consequent of a ratio be small, when compared with either of them, the ratio of their squares will be nearly obtained by doubling this difference. Let a+x: a be a ratio where x is small when compared with a; then a2+2ax+x2: a2 is the true ratio of their squares; but as x, by supposition, is small in comparison with a, 2 will be much smaller when compared with a2+ 2ax, it may therefore be left out without much error; and hence a2+2ax: a2, or (dividing by a) a+2x: a, will very nearly express the ratio of the squares of a+ and a. Thus, an approximation to the duplicate ratio of 1001 : 1000, is that of 1002: 1000. In the same manner, it might be shown that the ratios of a+3x: a, a+ 4x: a, or, in general, of a+ma: a, are nearly equal to the ratios of (a+x)3 : a3, (a+x)1 ; a1, and (a+x): am, if x be small in comparison with a. ON PROPORTION. 107. Proportion consists in the equality of ratios. If the ratio of a b be the same with that of c: d, or, which comes α C to the same thing, if = then these four quantities are b ď called proportionals, and are commonly written thus, a : b= c:d, or more generally thus, ab:: cd. This is expressed in language, by saying, that a is to b, as c to d.The first and last terms a and d are called the extremes, and the two middle terms, b and c, the means. 108. If four quantities be proportionals, the product of the extremes is equal to the product of the means. For, if a:b::c:d, then (No. 107.), and hence, by multiplying both sides of the equation by bd, we have ad= bc. From this, it is manifest, that if any three terms of a proportion or analogy (as it is often called) be given, the fourth may be ad found. For, since ad = bc, a = bc bc d=- b== and c= 109. From the preceding theorem, it follows, that if three quantities be continually proportional, that is, if the first be to the second, as the second to the third, the product of the extremes is equal to the square of the mean. For, if a:bb:c, then (No. 108.) ac = b2. Hence, a mean proportional between two quantities is equal to the square root of their product. For, let x be a mean proportional between a and c, so that a:x:x:c, then is a2 = ac, and, therefore, x= Jac. 110. The converse of each of the preceding theorems is also true, that is, If the product of two quantities be equal to the product of two others, or if the product of two be equal to the square of one, then the FOUR, or the THREE quantities will constitute an an analogy, provided the terms of the one product be made the extremes, and the terms of the other the means. a b For, if ad bc, then, dividing both by bd, we shall have In like manner, if ac = b2, then, dividing both by bc, we have a b b = and hence, as before, a:b::b:c. 111. Ratios that are equal to the same ratio, are equal to one another, that is, if a:b::c: d, and e: d::e:, then shall 112. If four quantities be proportionals, they are also proportionals when taken alternately, that is the first: third: : second fourth. : |