Proof. Let the values of the respective ratios A : B, Divide the first equation by p and the second by n, (1) Divide this equation by i and the last of (1) by nq, 374. Def. When the ratio of the first antecedent to the last consequent is formed by multiplying a series of intermediate ratios, the ratio thus obtained is said to be compounded of these intermediate ratios. CHAPTER II. LINEAR PROPORTIONS. Definitions. B 375. Def. Similar figures are those of which the angles taken in the same order are equal, and of which the sides between the equal angles are proportional. EXAMPLE. The figure ABCD is similar to A'B'C'D' when A B AB: BC: CD: DA :: A'B' : B'C' : C'D' : D'A'. 376. Def. In two or more similar figures any side of the one is said to be homologous to the corresponding side of the other. EXAMPLE. In the above figure, the sides AB and A'B' are homologous, 377. Def. When a finite straight line, as AB, is internally at P, and the two parts AP and BP are called segments. 378. Def. If the straight line AB is produced, and cut at a point outside of A and B, it is said to be divided externally at Q, and the lines AQ and BQ are called segments. Corollary 1. A line cut internally is equal to the sum of its segments. Cor. 2. A line cut externally is equal to the difference of its segments. 379. Def. Two straight lines are said to be similarly divided when the different segments of the one have the same ratios as the corresponding segments of the other. A N M B EXAMPLE 1. If the line AB is divided at M and the line cd at N in such manner that ly divided at M and N. EXAMPLE 2. If the lines AB and CD are divided at M, P, N, A and Q in such wise that AM: MP: PB :: CN: NQ : QD, they are similarly divided. 380. Def. If three straight lines, a, b, c, are so related that a: bbc, the line b is said to be a mean proportional between a and c. THEOREM I. a 381. If two straight lines are similarly divided, each part of the first has the same ratio to the corresponding part of the second that the whole of the first has to the whole of the second. AP: PQ: QB :: A'P' : P'Q' : Q'B'. or, expressed as a multiple proportion, AP: PQ: QB : AB :: A'P' : P'Q' : Q'B' : A'B'. Proof. In the proportion of the hypothesis the sum of the antecedents is AB, and the sum of the consequents A'B'. Therefore (§371) AB: A'B': AP: A'P' :: PQ: P'Q', etc. Q.E.D. THEOREM II. 382. A line cannot be divided at two different points, both internal or both external, into segments having the same ratio to each other. Hypothesis I. A line, AB, divided internally at the points P and Q. Ꭺ P Q Conclusion. The ratio AP: PB will be different from the ratio AQ QB. m Proof. Let the ratio AP: PB be Then AP will con n tain m parts, and PB n equal parts. Because AQ is greater than AP, it will contain more than m parts; and because QB is less than PB, it will contain less than n parts. Therefore the numerator of the ratio AQ: QB will be greater than m, and its denominator less than n, whence it and cannot be equal to it. m must be greater than n Therefore there is no other point of division than P for which the ratio of the segments will be the same as AP: PB. Hypothesis II. A line, AB, divided externally at the points P and Q. Conclusion. A The ratio B AP BP will be different from the ratio AQ: BQ. P Q Proof. Let m be the number of equal parts in AP; n, the number in BP; and s, the number in PQ. Then If we reduce these fractions to a common denominator and take their difference, we find it to be Because m and n are necessarily different, this fraction cannot be zero, and the ratio AP: BP is different from AQ BQ. Q.E.D. 383. Corollary 1. When the point of division P is nearer to B than to A, AP is greater than BP, and the ratio AP BP is greater than unity. When it is nearer to A than to B, AP is less than BP, the ratio is less than unity. 384. Cor. 2. If we suppose the point P to move from A toward B, the ratio AP: PB will be equal to zero as P starts from A, will be unity when P is half way between A and B, and will increase without limit as P approaches B. 385. Cor. 3. A line cannot be divided externally into segments having the ratio unity. 386. Cor. 4. Two different points may be found, the one internal and the other external, which shall divide a line into segments having the same given ratio. EXERCISES. 1. Draw a line, AB, and cut it internally in several points so that the ratios of the segments shall be 16, 25, 3:4, 4:3, 5:2, 6: 1. 2. Cut the same line externally in the ratios 3. A line 7 inches long is to be divided into segments having the ratio 4: 5. How long are the segments? 4. A line 6 inches long is to segments having the ratio 58. division from each end of the line. be divided externally into How far is the point of 5. If the line AB (§377) is 3 centimetres in length, and the points P and Q divide it both internally and externally into segments having the same ratio 1 : 2 (§ 386), find the. lengths AP, AB, and AQ. |