On Geometrical Proportion. 420 



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portion when j = - ■., and any number of quantities, a> I, c, 



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d, e,f, &c, have the same proportion when j = 7 = -7. &c. 



j 



Hence we have a criterion by which proportional quan- 

 tities may easily be distinguished, viz. an equality of ratios ; 

 and this being understood, the whole doctrine of proportion 

 flows immediately from the above obvious principles. 



Four proportional quantities are commonly expressed by 

 saying that a is to b as c to d, and they are usually written 

 thus, a: b : : c : d; where Z>and a are called the mean terms, 

 and a and d the extremes ; also a and c are called antece- 

 dents, and b and d their consequents. The subject is further 

 illustrated in the following articles : 



Article I. — When four quantities are proportional, the 

 product of the two means is equal to the product of the two 



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extremes. For since, by hypothesis, -,-= — , multiply both 



sides of the equation by bd, and we have -j- =z-j } orad—bc. 



Also, conversely, if the product of any two quantities be 

 equal to the product of two others, the four quantities are 

 proportional. For, since ad = be, divide by bd, and we 

 . ad be a c 



havc ld= M' or T=7 ; ^^a:b::c:d. 



AriicleW. — If four quantities are proportional when taken 



directly, they will be proportional when taken inversely ; 



that is, \f a : b : : c : d, then will b : a : : d : c. 



a, c 

 For when a : b : : c : d, we have 7- = -j, and dividing 



unity by each of these ratios, or inverting them, we ge| 



h d i 1 



- = -, or b : a : : a : c. 



a c 



Article III. — When four quantities are directly propor- 

 tional, they will also be proportional when taken alternately ; 

 that is, if a : b : : c : d, then will a : c : : b : a. 



For, because y = -,-, multiply both sides by — , and we 



la Ic a e . . . . 



have , = v> or -«= 3 : that is, a ; c : : b : d, 

 be dc re d ' 





