GEOMETRY. BOOK III. OF RATIOS AND PROPORTIONS. DEFINITIONS. 1. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if the numbers 3 and 6 have the same unit, the ratio of 3 to 6 will be expressed by 6 2. And in general, if A and B represent quantities of the same kind, the ratio of A to B will be expressed by B 2. If there be four numbers, 2, 4, 8, 16, having such values that the second divided by the first is equal to the fourth di vided by the third, the numbers are said to be in proportion. And in general, if there be four quantities, A, B, C, and D, having such values that B D then, A is said to have the same ratio to B, that C has to D, or, the ratio of A to B is equal to the ratio of C to D When Of Ratios and Proportions. four quantities have this relation to each other, they are said to be in proportion. Hence, the proportion of four quantities. results from an equality of their ratios taken two and two. To express that the ratio of A to B is equal to the ratio of C to D, we write the quantities thus: and read, A : B :: C : D; A is to B, as C to D. The quantities which are compared together are called the terms of the proportion. The first and last terms are called the extremes, and the second and third terms, the means. Thus, A and D are the extremes, and B and C the means. 3. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order. Thus, in the last proportion, A and C are the antecedents, and B and D the consequents. 4. Three quantities are in proportion when the first has the same ratio to the second, that the second has to the third; and then the middle term is said to be a mean proportional between the other two. For example, and 6 is a mean proportional between 3 and 12. 5. Quantities are said to be in proportion by inversion, or inversely, when the consequents are made the antsedents and the antecedents the consequents. Thus, if we have the proportion 3 : 6 :: 8 the inverse proportion would be 16. 6: 3 :: 16 : 8. Of Ratios and Proportions. 6. Quantities are said to be in proportion by alternation, or alternately, when antecedent is compared with antecedent and consequent with consequent. Thus, if we have the proportion 3 : 6 :: 8 the alternate proportion would be : 16, 3 : 8 :: 6 : 16. 7. Quantities are said to be in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent. Thus, if we have the proportion 8. Quantities are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with the antecedent or consequent. Thus, if we have the proportion 3 : 9 :: 12 : 36, 9-3 : 9 :: 36-12 : 36; the proportion by division will be that is, 6 : 9 :: 24 : 36. 9. Equimultiples of two or more quantities are the products which arise from multiplying the quantities by the same number. Thus, if we have any two numbers, as 6 and 5, and multiply Of Ratios and Proportions. them both by any number, as 9, the equimultiples will be 54 and 45; for Also, mxA and mx B are equimultiples of A and B, the common multiplier being m. 10. Two variable quantities, A and B, are said to be reciprocally proportional, or inversely proportional, when one increases in the same ratio as the other diminishes. When this relation exists, either of them is equal to a constant quantity divided by the other. Thus, if we had any two numbers, as 2 and 4, so related to each other that if we divided one by any number we must multiply the other by the same number, one would increase in the same ratio as the other would diminish, and their product would not be changed.f THEOREM I. If four quantities are in proportion, the product of the two extremes will be equal to the product of the two means. If we have the proportion A : B :: C: D we have, by Def. 2, B D and by clearing the equation of fractions, we have BC-AD Sch. The general principle is verified in the proportion between the numbers 2 : 10 :: 12 : 60 which gives 2×60=10x12=120 Of Ratios and Proportions. THEOREM II. If four quantities are so related to each other, that the product of two of them is equal to the product of the other two; then, two of them may be made the means, and the other two the extremes of a proportion. Let A, B, C, and D, have such values that BX C=AXD Divide both sides of the equation by A, and we have Then divide both sides of the last equation by C, and we have hence, by Def. 2, we have B D A : B :: C : D. Sch. The general truth may be verified by the numbers which give 2x18=9x4 2: 4 :: 9 : 18 THEOREM III. If three quantities are in proportion, the product of the tu.. extremes will be equal to the square of the middle term. and by clearing the equation of its fractions, we have |