RATIO AND PROPORTION.. DEFINITIONS. (It is necessary to understand the elementary principles of ratio and proportion before entering upon the Books that are to follow. It is therefore introduced here, but not numbered as one of the Books of Geometry, as it belongs properly to Algebra. Reference to the propositions in ratio and proportion will be made by the abbreviation Pn., with the number of the article annexed.) 1. Ratio is the relation of one quantity to another of the same kind; or it is the quotient which arises from dividing one quantity by another of the same kind. Ratio is indicated by writing the two quantities after one another with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to b is written, α a: b, or ; read, a is to b, or a divided by b. b 2. The Terms of a ratio are the quantities compared, whether simple or compound. The first term of a ratio is called the antecedent, the other the consequent; the two terms together are called a couplet. 3. An Inverse or Reciprocal Ratio of any two quantities is the ratio of their reciprocals. Thus, the direct ratio of a to b a is ab, that is, the inverse ratio of a to b is 1 1 b - or b: a. a 4. Proportion is an equality of ratios. Four quantities are in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth. 2 The equality of two ratios is indicated by the sign of equality (=), or by four dots (: :). Thus, a : b cd, or ab::c:d, or α b с - read a to b equals c to d, or a is to b as c is to d, or a divided by b equals c divided by d. 5. In a proportion the antecedents and consequents of the two ratios are respectively the antecedents and consequents of the proportion. The first and fourth terms are called the extremes, and the second and third the means. 6. When three quantities are in proportion, e. g. a : b = b : c, the second is called a mean proportional between the other two; and the third, a third proportional to the first and second. 7. A proportion is transformed by Alternation when antecedent is compared with antecedent, and consequent with consequent. 8. A proportion is transformed by Inversion when the antecedents are made consequents, and the consequents antecedents. 9. A proportion is transformed by Composition when in each couplet the sum of the antecedent and consequent is compared with the antecedent or with the consequent. 10. A proportion is transformed by Division when in each couplet the difference of the antecedent and consequent is compared with the antecedent or with the consequent. 11. Axiom. Two ratios respectively equal to a third are equal to each other. THEOREM I. 12. In a proportion the product of the extremes is equal to the product of the means. 13. Scholium. A proportion is an equation; and making the product of the extremes equal to the product of the means is merely clearing the equation of fractions. THEOREM II. 14. If the product of two quantities is equal to the product of two others, the factors of either product may be made the extremes, and the factors of the other the means of a proportion. 15. If four quantities are in proportion, they will be in proportion by alternation. Let By (12) By (14) a: bc: d ad bc a:cb: d THEOREM IV. 16. If four quantities are in proportion, they will be in pro 17. If four quantities are in proportion, they will be in proportion by composition. 18. If four quantities are in proportion, they will be in proportion by division. 19. Corollary. From (17) and (18), by means of (15) and (11), If then abc: d a+bab=c+d:c-d THEOREM VII. 20. Equimultiples of two quantities have the same ratio as the quantities themselves. 21. Corollary. It follows that either couplet of a proportion may be multiplied or divided by any quantity, and the resulting quantities will be in proportion. And since by (15), if abma: mb, a: m a bmb or ma: amb: b, it follows that both consequents, or both antecedents, may be multiplied or divided by any quantity, and the resulting quantities will be in proportion. THEOREM VIII. 22. If four quantities are in proportion, like powers or like roots of these quantities will be in proportion. Since n may be either integral or fractional, the theorem is proved. |