14^ ARITHIvIETlC. 



5. A icuncr melts lolb. of gold of 20 carp.ts fine with 

 loib. of 18 carats fine ; how much alloy must he put to 

 it to make it 22 carats fine ? 



Ans. It is not fine enough by 3^ carats, so that no al- 

 io v must be put to it, but more gold. 



ALLIGATION ALTERNATE. 



Alligation alternate is the method of nnding \vhat quan- 

 tity of any number of simples, whose rates are given, will 

 compose a mixture of a given rate ; so that it is the re- 

 verse of alligation medial, and may be proved by it. 



1. Write the rates of the simples in a column under 

 each other. 



2. Connect 



* Demonstration. By connecting the less rate to the 

 greater, and placing the differences between them and the mean 

 rate alternately, the quantities resulting are such, that there is pre- 

 cisely as much gained by one quantity as is lost by the other, and 

 therefore the gain and loss upon the whole are equal, and are ex- 

 actly the proposed rate : and the same will be true of any other- 

 two simples, managed according to the rule. 



In like manner, let the number of simples be what it may, and 

 with how many soever each is linked, since it is always a lisss with 

 a greater than the mean price, there will be an equal balance of 

 loss and gain between every two, and consequently an equal bal- 

 ance on the whole. Q^ E. D. 



It is obvious from the- rule, that questions of this sort admit 

 of a great variety of answers ; for, having found one answer, we 

 may find as many more as we please, by only multiplying or di- 

 viding each of tlie quantities found by 2, 3, or 4, &c. the reason 

 of which is evident ; for, if two quantities of two simples make a 

 balance of loss and gain, with respect to the mean price, so must 



also 



