On prime and ultimate Ratios. 187 



The part b of the ratio is called the antecedent) and a is 

 its consequent. 



The antecedent may be equal to the .consequent, and 

 then the ratio is called a ratio of equality ; though it would 

 be more proper to say, the terms of the ratio are equal :— 

 when the terms of a ratio are equal, its measure is always 

 equal to unity. 



If the terms of a ratio vary, the measure of the ratio 

 may have any magnitude whatever ; and if one term re- 

 main constant while the olher\ varies, the measure of the 

 ratio will vary with the varying term. 



Let — represent any ratio, and let a remain constant, 



while b is variable; it is obvious that if b decrease, the 

 measure of the ratio will increase ; and, when b is become 

 indefinitely small, the measure of the ratio is then indefi- 

 nitely near to a; and when b entirely vanishes, the mea- 

 sure of the ratio is exactly equal to a. 



On the contrary, when b increases, the measure of the 

 ratio decreases. Again, let b remain constant while a is 

 variable: then, as a increases the measure of the ratio in- 

 creases, but it decreases as a decreases, and when a entirely 



vanishes the measure of the ratio is equal to —. 



As another example, suppose we have the ratio > 



where x is variable and a constant ; the measure of this 

 ratio may vary through all possible degrees of magnitude, as 

 in the preceding example. 



1. Let x continually increase ; then, the measure of the 



ratio — - will decrease; and when x is indefinitely great, it 



will become nearly a constant ratio, or a ratio of equality ; 

 that is, the terms of the ratio will be nearly equal : be- 

 cause the addition of a to a quantity x which is indefi- 

 nitely great, will alter the measure of the ratio only in an 

 indefinitely small degree: hence it continually verges to a 

 ratio of equality as a limit. 



2. Let x decrease ; then, the measure of the ratio r 



x 

 will increase; and when x is indefinitely small, the measure 

 or the ratio is indefinitely .near to a:' when x vanishes, the 

 ratio is equal to a, exactly. 



From the above illustrations it is exceedingly obvious, 

 that a ratio in which the terms continually vary, or where 

 one is variable and the other constant, or where part of one 

 term is constant, as in this latter example, — it is obvious, 



I sa\> 



