RATIO. 



pect to 3, which it contains twice ; again, 

 in comparing iv with 2, we see that it has 

 a different relative magnitude, for it con- 

 tains 2 three times, or it is greater when 

 compared with 2 than it is when compar- 

 ed with 3. The ratio of a 10 b is usually 

 expressed by two points placed between 

 them, thus, a \ b and the former is called 

 the antecedent of the ratio, the latter the 

 consequent. When one antecedent is the 

 same multiple part, or parts, of its conse- 

 quent, that another antecedent is of its 

 consequent, the ratios are equal. Thus, 

 the ratio of 4 : 6 is equal to the ratio of 

 2 : 3, i. e 4 has the same magnitude when 

 compared with 6, that -2 has when com- 



4 2 



pared with 3, since = the ratio of a : 

 b o 



b is equal to the ratio of c : d, if- 



d' 



because and -r, represent the multiple, 



part, or parts, that a is of b, and c ofd. 



If the terms of a ratio be multiplied or 

 divided by the same quantity, the ratio is 



a ma 

 not altered. For 7- = 7-. 



b mo 



That ratio is greater than -aiother, 

 whose antecedent is the greater multiple, 

 part, or parts of its consequent. Thus, 

 the ratio of 7 : 4 is greater than the ratio 



7 35 . 



of 8 : 5 ; because - or ^-~ is greater than 

 4 20 



-or ~- These conclusiens follow im- 



5 20 



mediately from our idea of ratio. 



" A ratio is called a ratio of greater in- 

 equality, of less inequality, or of equality, 

 according as the antecedent is greater, 

 less than, or equal to the consequent." 



" A ratio of greater inequality is dimi- 

 nished, and of less inequality increased, 

 by adding any quantity to both its terms. 

 If to the terms of the ratio 7 : 4, 1 be add- 

 ed, it becomes the ratio of 8 : 5, which is 

 less than the former. And in general, 

 let x be added to the terms of the ratio 

 a : A, and it becomes a -j- x : b -f- x , 

 which is greater, or less than the for- 



a -f x . 

 mer, according as -~r is greater or 



less than y ; or by reducing then\ to a 



common denominator, as 



b.b 



greater or less than . ; that is, as 



b .b -f x 



b is greater or less than a. Hence, a ra- 

 tio of greater inequality is increased, and 



of ess inequality diminished, by taking 1 

 from the terms a quantity less than either 

 of them. 



If the antecedents of any ratios be mul- 

 tiplied together, and also the 

 quents, a new ratio results, which is said 

 to be compounded of ihe former. Tr.us, 

 a c . b d is said to be compounded of the 

 two a : b and c : d. It is also son: 

 called the sum of the ratios; ami -vhen 

 the ratio a b is compounded with itself, 

 the resulting ratio, a 1 : fr, is called ihe 

 double of the ratio of a : b\ and if three of 

 these ratios be compounded together, 

 the result a3 : b3. is called the triple of 

 the first, &,c. Also, the ratio of a b is 

 said to be one third of the ratio of a3 : fa ; 



1^ _! 



and a m : b*n is said to be an 7n th part of 

 the ratio of a : b. 



Let the first ratio be a : 1 ; then J : 1, 

 fl3 : 1, ....an : 1, are twice three times, ....n 

 times the first ratio ; where n the index of 

 a, shows what multiple, or part, of the 

 ratio an -. 1, the first ratio, a : I, is. On 

 this account, the indices, 1, 2, 3, ...n, are 

 called measures of the ratios a 1 : 1, a* : 1, 

 a3 : 1, an : 1. 



" If the consequent of the preceding 

 ratio be the antecedent of the succeeding 

 one, and any number of such ratios be ta- 

 ken, the ratio which arises from their 

 composition is that of their first antece- 

 dent to the last consequent." Let a : b, 

 b . c, c : d, &c. be the ratios, the com- 

 pound ratio is a X^Xc:6x^X^; 

 or dividing by b X <-', : d. 



" A ratio of greater inequality, com- 

 pounded with another, increases it ; and 

 a ratio of less inequality diminishes it." 

 Let the ratio of x : y ?>e compounded with 

 the ratio of a : 6, and the resulting ratio 

 ax : by is greater or less than the ratio 



a : b, according as - is greater or less 



than i. e. according as x is greater or 



less than y. 



"If the difference between the ante- 

 cedent and consequent of a ratio be 

 small when compared with either of them, 

 the double of the ratio, or the ratio of 

 their squares^ is nearly obtained by doub- 

 ling this difference." 



Let a -f- x : a be the proposed ratio, 

 where xis small when compared with a ; 

 then a 1 + 2 ax -+ x* : a 1 is the ratio of the 

 squares of the antecedent and conse- 

 quent ; but since x is small when compar- 

 ed with a, x* or x X x 1S small when, 

 compared with 2 a X x t and much small- 



- 



