reduced to the single proportion 2A-B: A:: B: C, or to A, B, C are numbers (or are regarded as such), we shall Hence, if the second of three harmonicals is multiplied by the ratio of the first to twice the first diminished by the second, the product will give the third term. If the harmonicals are numbers, the equation C= being written in the form 1 1 1 1 AB 2A-B' 1 C AB B A' gives Α' and if B, C, D, E, etc., are also harmon Hence, since the numbers A, B, C, D, etc., on the suppositions here made, are said to be in harmonical progression, it follows that their reciprocals, 1, 1 1 1 1 etc., are in A'B'C' D'E' arithmetical progression. Conversely, it is easy to show that the reciprocals of a series of numbers in arithmetical progression will be a series of numbers in harmonical progression. Thus, by taking the reciprocals of the arithmeticals 2, 5, 8, we get the harmonicals or their equiva 40 16 10 lents 80' 80' 80 1 1 1 2' 5' 8' ; consequently, since we have 40:10::40 -16 1610, it follows that the reciprocals of the arithmeticals are in harmonical progression. Again, resuming the proportion 2A-B: A:: B: C, and by alternation by (13), writing it in the form 2A – B: B:: A: C; then by composition, according to (19), we shall get the proportion 2A : B:: A+ C: C, or the equivalent equa2A A+ C tion = B C From the preceding equation we easily get B = 2A × C A+ C ; which, if A, B, C are numbers, may be written in Hence, the mean of three harmonicals equals twice the first multiplied by the ratio of the third to the sum of the first and third. 2. Four numbers or quantities of the same kind are in harmonical proportion when the first is to the fourth as the difference of the first and second is to that of the third and fourth; where it is to be noticed that the fourth term is called a fourth harmonical proportional to the first three terms. Thus, if A, B, C, D are in harmonical proportion, then we must either have A: D::A B: CD, or A:D::BA: D C. Proceeding as in the case of three harmonicals, these proportions are easily reduced to the single proportion 2A-B: A::C: D, or to the equation DC x which may be written in the form D = AC 2A B harmonicals are numbers, or are regarded as such. A 2A-B' when the Hence, the fourth proportional to three given harmonicals equals the third term multiplied by the ratio of the first to twice the first less the second term. To illustrate what has been said concerning ratios and proportions. 1. To find the ratios of 45 to 9 and 40 to 8. metrical Proportion, we have the proportion 45 : 9 :: 40: 8. 2. To compare the ratios of 17 to 3 and 23 to 4. 17 Because the ratio of 17 to 3 is expressed by 3 23 and that of 23 to 4 by 4 5.75, it follows that 17 has a less ratio to 3 than 23 has to 4, which we may (according to custom) express by the form 17: 3 <23: 4. 3. To compare the ratios of 25 to 6 and 29 to 7. Because the ratios are expressed by 29 25 1 = and = 7 it follows that 25 has a greater ratio to 6 than 29 has to 7, which may be expressed by the form 25: 6 > 29: 7. 4. Supposing A, B, and C to be three numbers or quantities of the same kind, then it is proposed to compare the A A+ C ratios and B 1st. Because = it is clear that the ra tios will be equal, and each equal to unity, when A = B; noticing that such ratios, in which each antecedent equals its consequent, are called ratios of equality. 2d. If A is less than B, then A, B, and C, being positive, 3d. Similarly, if A is greater than B, it is manifest that A+ C the ratio B+C A is less than the ratio B Remarks. When the antecedent is greater than the consequent, the ratio is sometimes called a ratio of greater inequality; and when the antecedent equals the consequent, it is called a ratio of equality; also, when the antecedent is less than the consequent, the ratio is called a ratio of lesser inequality. 5. To find a fourth proportional to 6, 12, and 15; also, a third proportional to 4 and 6. 12 From (31), we shall have 15 × = 15 × 2 = 30= the 6 6 fourth proportional; and from (32) we shall have 6 × 6 x 1.59 the third proportional. = = 4 6. To find a mean proportional between 2 and 8, and between 4 and 9. From (32) it follows that the sought mean proportionals are 7. If 25 yards of a certain kind of cloth cost £30 108., it is proposed to find how much 60 yards of the same kind of cloth is worth. If we denote the required value by x, then the question, properly stated, gives the proportion 25 yds.: 60 yds. :: £30 108.:. Hence, since 60 yds. ÷ 25 yds. = 2 = 2.4, we shall get x = £30 10s. × 2.4 = £73 48. If 25 yds. and 60 yds. are treated as numbers, our propor£30 10s. x 60 tion becomes 25: 60 :: £30 10s. : x = 25 £73 48., which mode of proceeding is in conformity to the common method of working the statement of a question in the Rule of Three. 8. If 36 men eat a certain quantity of provisions in 6 weeks, then how long will it take 48 men to eat the same quantity? Because 48 men will clearly eat the provisions in less time than 36 men, or (which is the same), since more requires less, this question clearly falls under what is commonly called the Rule of Three Inverse. If a stands for the required time, the question gives the proportion 36 m. : 48 m. : : 6 w.; which, by inverting the terms (according to (12)), becomes 48 m. : 36 m. :: 6 w. : x; consequently, since 36 m. 48 m. = get x = 6 w. x 0.75 4.5 weeks. = 36 3 = 48 4 = 0.75, we shall If 48 m. and 36 m. are regarded as numbers, our propor tion becomes 48: 36 :: 6 w. : x, which gives x = 36 x 6 w. 48 9. Given, the proportion a +x: b+x:: 3 : 2, to find x. Dividing according to (20), we get a b: b+x: 1: 2, a x a a b b or we shall get x = (2 × (a - b) b) = 2a2b-b2a-3b, as required. If we put 2a 36 for x in the given proportion, it becomes 3a3b2a26: 3:2; which is equivalent to the identical proportion 3: 2 :: 3 : 2, as it clearly ought to be. 10. To find a third harmonical proportional to 12 and 8. 6 the third harmonical; and it is easy to perceive that 12, 8, and 6 are in harmonical proportion or progression. 11. To find a harmonical mean between 3 and 6. 12. To find a fourth harmonical proportional to 5, 2, and 16. From (43) we get 16 x 5 = 10, for the required pro portional, and it is easy to perceive that 5, 2, 16, and 10 are in harmonical proportion. (45.) LIMITS OF VARIABLE RATIOS. If the antecedent and consequent of a variable ratio are such that (without actually changing the signs of any of their terms) they can be varied, so that the ratio can be made to differ from a certain fixed ratio, by a difference which is less than any given difference, then we shall call the fixed ratio a limit of the variable ratio. Thus, if we take the ratio a + x and bare constant or invariable, while a is variable, then it is clear that the ratio will be variable. Supposing a to diminish until it differs from naught (0) by a difference less |