When the antecedent is equal to the consequent, it is called a ratio of equality ; thus the ratio of 6 to 6 = = 1. But if the antecedent be larger than the consequent, it is a ratio of greater inequality ; and if the antecedent be less than the consequent, it is a ratio of less inequality. The antecedent and consequent are called the terms of the ratio ; and the quotient of the two terms is the index or exponent of the ratio. Compound ratio is made up of two or more ratios, by multiplying their terms and exponents together. The ratio of 8 to 6 and of 4 to 2 may be compounded; thus, 8 to 6=&; 4 to 2 = ; 8 x 4 to 6 x2=6*4; 32 to 12 = 14. If a ratio be compounded of two equal ratios, it is called a duplicate ratio ; of three ratios, it is called a triplicate ratio, &c. Thus, if the ratio of 4 to 2 be 2, and the ratio of 6 to 3 be 2, the ratio of 4 x 6 to 2 X 3 will be 2 X 2, that is, the ratio of 24 to 6 will be 22, &c. If the terms of a ratio be prime to each other, no quantities can be found in the same ratio but what would be multiples thereof. Numbers that are prime to each other are the least of all numbers in the same ratio. If, therefore, we wish to ascertain whether the ratio of 3 to 7 is greater or less than the ratio of 4 to 9, since these ratios are represented by the fractions and $, we reduce them to a common denominator, 27 and 2 ; and, since the latter of these is greater than the former, it is evident that the ratio of 3 to 7 is less than the ratio of 4 to 9. If we have the terms of a ratio given in large numbers, that are prime to each other, and we wish to find a ratio nearly equivalent, whose terms are expressed by smaller numbers, we adopt the following RULE. — Divide the greater term by the less, and that divisor by the remainder, as in Sect. XVI., Case I., of Vulgar Fractions. Then, if the antecedent be greater than the consequent, the first quotient divided by 1 gives the first ratio ; if less, a unit divided by the first quotient will express the first ratio. Multiply the terms of the first ratio by the second quotient, and add a unit to the numerator or denominator, according as the antecedent of the original terms is greater or less than its consequent, and we have the second ratio. Then, as a general principle, we multiply the terms of the ratio last found by the next succeeding quotient, and to the product we add the corresponding terms of the preceding ratio, and we have the next succeeding ratio; and thus we proceed until there is no remainder, or until we have arrived at a sufficient approximation. 1. Let it be required to find a series of ratios in' less numbers, constantly approaching to the ratio of 314159 to 100000, which is nearly the ratio of the circumference of a circle to its diameter. OPERATION, 100000)314159(3 300000 99113 13305 854 33, &c. 3= , the first ratio. 1x7 7=4, the second ratio, being the approximation of Ar [chimedes. 22x1513 =, the third ratio. 333x1 +22 =ff, the fourth ratio, the approximation of Metius. PROPORTION is the likeness or equalities of ratios. Thus, because 5 has the same relation or ratio to 10 that 8 has to 16, we say such numbers are in proportion to each other, and are therefore called proportionals. If any four numbers whatever be taken, the first is said to have the same ratio or relation to the second, that the third has to the fourth, when the first number or term contains the second. as many times as the third contains the fourth, or when the second contains the first as many times as the fourth does the third. Thus, 8 has the same ratio to 4 that 12 has to 6, be. cause 8 contains 4 as many times as 12 does 6. And 3 has the same relation to 9 that 4 has to 12, because 9 contains 3 as many times as 12 does 4. Ratios are represented by colons, and the equalities of ratios by double colons. 3:9::8:24 is read thus:- 3 has the same ratio or relation to 9 as 8 to 24. The first and third numbers of a proportion are called antecedents, and the second and fourth are called consequents; also, the first and fourth are called extremes, and the second and third are called means. Whatever four numbers are proportionals, if their antece. dents or consequents be multiplied or divided by the same numbers, they are still proportionals; and if the terms of one proportion be multiplied or divided by the corresponding term of another proportion, their products and quotients are still proportionals. This will appear evident from the various changes that the following example admits. 4:8::3:6 Directly. 4:3 ::8:6 By permutation. 8-4:8::6- 3:6 By division. :8:: : By division. That the product of the extremes is equal to that of the means is evident from the following consideration. Let the following proportionals be taken. 12 : 3 ::8:2. From the definition of proportion, the first term contains the second as many times as the third does the fourth ; therefore, 12 = ' ; but 12 = 24, and i2*; and if 24, the numerator of the first fraction, which is a substitute for the first term, be multiplied by 6, the denominator of the second fraction, and a substitute for the fourth term, the product will be the same as if 6, the denominator of the first fraction, and a substitute for the second term, be multiplied by 24, the numerator of the second fraction, and a substitute for the third term. Thus 24 x 6=6 x 24. Therefore the product of the extremes is, in all cases, equal to that of the means. If, then, one of the extremes be wanting, divide the product of the means by the extreme given ; or, if one of the means be wanting, divide the product of the extremes by the means given, and the result will be the term sought. To apply this, we will take the following question. If 5 yards of cloth cost $15, what will 7 yards cost ? It is evident that twice the quantity of cloth would cost twice the sum, and that three times the quantity, three times the sum, &c.; that is, the price will be in proportion to the quantity purchased. We then have three terms of a proportion given, one of the extremes and the two means, to find the other extreme. Thus, 5:7:: 15. Therefore, to find the other extreme by the rule above stated, we multiply the two means, 7 and 15, and divide their product by the extreme given, and the quotient is the extreme required.7 x 15 = 105. 105 = 5= 21 dollars, the answer required. To perform this question by analysis, we reason thus. If 5 yards cost 15 dollars, 1 yard will cost one fifth as much, which is 3 dollars; and if 1 yard cost 3 dollars, 7 yards will cost 7 times as much, which is 21 dollars. Rule.* — State the question by making that number which is of the same name or quality of the answer required the third term; then, if the answer required is to be greater than the third term, make the second term greater than the first ; but if the answer is to be less than the third term, make the second less than the first. Reduce the first and second terms to the lowest denomination mentioned in either, and the third term to the lowest denomination mentioned in it. Multiply the second and third terms together, and divide their product * This rule was formerly divided into the Rule of Three Direct, and the Rule of Three Inverse. The Rule of Three Direct included those questions where more required more and less required less ; thus, — If 516. of coffee cost 60 cents, what would be the value of 10lb.? would be a question in the Rule of Three Direct, because the more coffee there was the more money it would take to purchase it. But if the question were thus:- If 4 men can mow a certain field in 12 days, how long would it take 8 men ?- it would be in Inverse, because the more men the less would be the time to perform the labor, that is, more would require less. The method for stating questions was this : — To make that number which is the demand of the question the third term, that which is of the same name the first, and that which is of the same name as the answer required, the second term. If the question was direct, the second and third terms must be multiplied together, and their product divided by the first ; but if it was inverse, the first and second terms must be multiplied together, and their product divided by the third. by the first, and the quotient is the answer, in the same denomination to which the third is reduced. If any thing remains after division, reduce it to the next lower denomination, and divide as before. If either of the terms consists of fractions, slate the question as in whole numbers, and reduce the mixed numbers to improper fractions, compound fractions to simple ones, and invert the first term, and then multiply the three terms continually together, and the product is the answer to the question. Or the fractions may be reduced to a common denominator ; and their numerators may be used as whole numbers. For when fractions are reduced to a common denominator, their relative value is as their numerators. Note 1. - In the Rule of Three, the second term is the quantity whose price is wanted; the third term is the value of the first term; when, therefore, the second term is multiplied by the third, the answer is as much more than it should be, as the first term is greater than unity; therefore, by dividing by the first term, we have the value of the quantity required. Or, multiplying the third by the number of times which the second contains the first will produce the answer. Note 2. - The pupil should perform every question by analysis, previous to his performing it by Proportion. EXAMPLES. m. 24 1. If a man travel 243 miles in 9 days, how far will he travel in 24 days? Ans. 648 miles. da. da. As the answer to the question must be in miles, we make the third term miles (243); and from the nature of 972 the question we know that he will travel farther in 24 days than in 9 days; 9)5832 we therefore place 24, as the larger of Ans. 648 miles. the two remaining terms, in the second place, and the remaining number, 9 27 24 x 248 P = 648 miles, Ans. To perform this question by analysis, we proceed thus:If he travel 243 miles in 9 days, he will in one day travel of 243 miles, which is 27 miles; then if he travel 27 miles in |