204 



THE POPULAR EDUCATOE. 



4. Subtracting the last two terms from the first two, 



a -c :b - d: . a :b, or a- c :b d . : c . d, etc. 



5. Subtracting the consequents from the antecedents, 



a b:b::Cd:d,OTa-.a b::C:Cd, etc. 

 The alteration expressed by the last of these forms is called 

 conversion. 



6. Subtracting the antecedents from the consequents, 



b a:a::dc-.c, or b :b a . . d : d c, etc. 



7. Adding and subtracting, a + b : a b -. -. c + d -. c d ; 

 that is, the sum of the first two terms is to their difference as 

 the sum of the last two to their difference. 



Co>\ If any compound quantities, arranged as in the pre- 

 ceding examples, are proportional, the simple quantities of 

 which they are compounded are proportional also. 



Thus, if a -}- b :b -. . c + d : d, then a : b -. : c -. d. This is called 

 dii-ision. (Euclid V. 17.) 



CASE V. COMPOUNDING PROPORTIONS. 

 If the corresponding terms of two or more ranks of propor- 

 tional quantities be multiplied together, the products will be 

 proportional. 



This process is called compounding proportions. It is the 

 same as compounding ratios. It should be distinguished from 

 what is called composition, which is an addition of the terms of 

 a ratio. 



If a-.b::c:d 12:4::6:2 



And h-.l::m-.n 10 : 5 : : 8 : 4 



Then ah : bl : : cm -. dn. 120 : 20 : :48 :8. 



For, from the nature of proportion, the two ratios in the first 

 rank are equal, and also the ratios in the second rank. And 

 multiplying the corresponding terms is multiplying the ratios 

 that is, multiplying equals by equals, so that the ratios will still 

 be equal, and therefore the four products must be proportional. 

 The same proof is applicable to any number of proportions. 



(a: b : : C : d} 

 If < h : I : : m -. n > then ahp :blq:: cmx : dny. 



(p:q::x:y) 



From this it is evident that if the terms of a proportion be 

 multiplied each into itself, that is, if they be raised to any power, 

 they will still be proportional. 



If a-.b::C:d 2:4:6:12 



a : b : : c : d 2:4:6:12 



Then a- : b- : -. <? : d-. 4 : 16 : : 36 : 144. 



Proportionals will also be obtained by reversing this process, 

 that is, by extracting the roots of the terms. 



If a : b : : c : d, then Va :/&:: v/c : \'d. 



For taking the products of the extremes and means, ad = be. 

 And extracting the root of both sides, Vad = Vbc. 

 That is, Va :</&:: Vc -. Vd. 



CASE VI. INVOLUTION AND EVOLUTION OF THE TERMS. 

 If several quantities are proportional, their like powers or 

 like roots are proportional. 



If a : b : : C : d, 

 Then a : b -. -. c" : d", and " Va -."Vb: : " Vc : * Vd. 



And " A/O. : m Vb" : : A/C : m Vd ; that is, aT : b m : : c m : d. ' 



It must not be inferred from this, that quantities have the 

 same ratio as their like powers or like roots. 



If the terms in one rank of proportionals be divided by the 

 corresponding terms in another rank, the quotients will be pro- 

 portional. 



This is sometimes called the resolution of ratios. 



If a : b : : C : d 12 : 6 : : 18 : 9 



And h : I : : m : n 6 : 2 : : 9 : 3 



rp, a b c d 12 6 18 9 



Ihen :-::-:- - : - : : : - 



h I m n 6293 



This is merely reversing the process in Case V., and may be 

 demonstrated in a similar manner. 



N.B. This should be distinguished from what geometricians 

 call division, which is a subtraction of the terms of a ratio. 



When proportions are compounded by multiplication, it wil 



often be the case that the same factor will be found in two 

 analogous or two homologous terms. 



Thus, if a . b . : c : d 



And m : a : : n : c 



am : ab : : en : cd 



Here a is in the first two terms, and c in the last two. 

 Dividing by these, the proportion becomes 

 m . b : : n : d. Hence, 



In compounding proportions, equal factors or divisors in two 

 analogous or homologous terms may be rejected. 



a : b : : c: d 12:4::9:3 



b :h : : d : I 4:8::3:6 



h :m: : I :n 8 : 20 : : 6 : 15 



If 



Then a : m : -. c : n 12 : 20 : : 9 : 15 



This rule may be applied to the cases to which the terms 

 "ex cequo" and "ex cequo perturbata" refer. One of the methods, 

 may serve to verify the other. 



When four quantities are proportional, if the first be greater 

 than the second, the third will be greater than tho fourth ; if 

 equal , equal ; if less, less. 



( a = b, c = d. 



Suppose a : b : : c : d, then if < a > b, c > d. 

 ( a<b, c< d. 



If four quantities are proportional, their reciprocals are pro- 

 portional, and vice versa. 



Ifa:b::c:d, then l : - : : I : -. 

 a b c d 



For in each of these proportions, we have, by reduction, 

 ad = be. 



PROBLEMS IN GEOMETRICAL PROPORTION. 

 EXAMPLE. Divide the number 49 into two such parts, that- 

 the greater increased by 6 may be to the less diminished by 1 1 

 as 9 to 2. 



Let x = the greater, and 49 - x = the less. 

 By the conditions proposed, x + G : 38 - x -. . 9 : 2. 



Adding terms, tc+6:44::9:ll. 



Dividing the consequents, x + 6 : 4 : : 9 : 1. 



Multiplying the extremes and means, x + 6 = 36 ; and x = 30 r 

 the greater part, and 49 - x = 49 30 = 19, the lesser part. 



EXERCISE 70. 



1. What number is that, to which if 3, 8, and 17 be severally added, 

 the first sum shall be to the second as the second to the third ? 



2. Find two numbers, the greater of which shall be to the less as 

 their sum to 42, and as their difference to 6. 



3. Divide the number 13 into two such parts, that the squares of 

 those parts may be in the ratio of 25 to 16. 



4. Divide the number 14 into two such parts, that the quotient 

 of the greater divided by the less shall be to the quotient of the less- 

 divided by the greater as 16 to 9. 



5. If the number 20 be divided into two parts, which are to eacb 

 other in the duplicate ratio of 3 to 1, what number is a mean propor- 

 tional between those parts ? 



6. There are two numbers whose product is 24, and the difference 

 of their cubes is to the cube of their difference as 19 to 1. What are 

 the numbers ? 



7. There are two numbers in the proportion of 5 : 6 ; the first being- 

 increased by 4 and the last by 6, the proportion will be as 4 : 5. What 

 are the numbers ? 



8. A farmer has a quantity of corn in his granary, and sells a certain 

 number of bushels, which is to the number of bushels remaining as 

 4 : 5. He then feeds out 10 bushels, which is to the number sold as 

 1 : 2. How many bushels had he at first, and how many did he sell ? 



9. There are two numbers whose product is 135, and the difference- 

 of their squares is to the square of their difference as 4 to 1. What 

 are the numbers ? 



10. What two numbers are those, whose difference, sum, and pro- 

 duct are as the numbers 2_ 3, and 5 respectively ? 



11. Divide the number 24 into two such parts, that their product 

 shall be to the sum of their squares as 3 to 10. 



12. In a mixture of rum and brandy, the difference between the 

 quantities of each is to the quantity of brandy as 100 is to the number 

 of gallons of rum ; and the same difference is to the quantity of rum 

 as 4 to the number of gallons of brandy. How many gallons are there 

 of each ? 



13. There are two numbers which are to each other as 3 to 2 ; if 

 6 be added to the greater and subtracted from the less, the sum and 

 remainder will be to each other as 3 to 1. What are the numbers ? 



14. There are two numbers whose product is 320 ; and the difference 



