a 404. DEFINITIONS. = C BOOK III A proportion is the equality of ratios. is a proportion, and expresses the fact that the ratio of b d a to b is equal to the ratio of c to d. The proportion may also be written a : b c d and a: b:: c In the proportion a b = с ď ::c: d. α с b d the first and fourth terms (a and d) are called the extremes, and the second and third terms (b and c) are called the means. The first and third terms (a and c) are the antecedents, and the second and fourth terms (b and d) are the consequents. If the means of a proportion are equal, either mean is a mean proportional or a geometrical mean between the extremes. α b Thus in the proportion = b is a mean proportional between " b с a and c. In this same proportion, c is called a third proportional to a and b. PROPOSITION I. THEOREM 405. In a proportion, the product of the extremes is equal to the product of the means. Proof. [Clear fractions in (1) by multiplying both members 406. COROLLARY. The mean proportional between two quan tities is equal to the square root of their product. Proof. [Apply § 405 to (1), and extract the square root of both members.] 407. EXERCISE. Find x in 408. EXERCISE. between 9 and 4? 409. EXERCISE. One of them is 16. 410. EXERCISE. and a2 - 2 ab + b2. Q.E.D. What is the geometrical mean or mean proportional 12 is the geometrical mean between two numbers. What is the other? Find the mean proportional between a2 + 2 ab + b2 PROPOSITION II. THEOREM. (CONVERSE OF PROP. I.) 411. If the product of two factors is equal to the product of two other factors, the factors of either product may be made the means, and the factors of the other product the extremes of a proportion. Proof. [Divide both members of (1) by bd.] 412. EXERCISE. Q.E.D. From the equation ad = bc, derive the following Let To Prove 415. EXERCISE. Form a proportion from a3 + b3 = x2 — y2. 416. DEFINITION. A proportion is arranged by alternation when antecedent is compared with antecedent and consequent with consequent. 417. If four quantities are in proportion, they are in proportion by alternation. 419. DEFINITION. A proportion is arranged by inversion when the antecedents are made consequents, and the consequents are made antecedents. a If the proportion is arranged by inversion, it be b d с b d comes α C PROPOSITION IV. THEOREM 420. If four quantities are in proportion, they are in proportion by inversion. Let To Prove 81001 α с (1) (2) Q.E.D. 421. DEFINITION. A proportion is arranged by composition when the sum of antecedent and consequent is compared with either antecedent or consequent. 422. If four quantities are in proportion, they are in proportion by composition. 423. NOTE. The student may discover for himself the steps of the solution of this and the succeeding propositions by studying the analysis of the theorem. In the analysis we assume the conclusion (the part to be proved) to be a true equation. Working upon this conclusion by algebraic transformations, we produce the hypothesis. The solution of the theorem begins with the last step of the analysis and reverses the work, step by step, until the first step or conclusion is reached. Let the student show that the solution of Prop. V. as given on the preceding page may be obtained by reversing the steps of this analysis. 427. DEFINITION. A proportion is arranged by division when the difference between antecedent and consequent is compared with either antecedent or consequent. |