PROPORTION. 103 



8, 16, taken thus, 2, 4, 8, 16, or thus, 4, 2, 16, 8, are ge- 

 ometrical proportionals.* 



Proportion is distinguished into continued and discontinued^ 



If, of several couplets of proportionals written down in 

 a serieS) the difference or ratio of each consequent and the 

 antecedent of the next following couplet be the same as 

 the common difference or ratio of the couplets, the pro- 

 portion is said to be continued^ and the numbers themselves 

 a series of continued arithmetical or geometrical proportio7Ja!s> 

 So 2, 4, 6, 8, form an arithmetical progression *, for 4 — 2 

 = 6 — 4=28 — 6 = 2 ; and 2, 4, 8, 16, a geometrical pro- 

 gression -, for 4- = ^= '^ - 2. 



But if the difference or ratio of the consequent of one 

 couplet and the antecedent of the next couplet be not the 

 same as the common difference or ratio of the couplets, 

 the proportion is said to be discontinued. So 4, 2, 8, 6, are 

 in discontinued arithmetical proportion ; for 4-— 2=8 — 6=2, 

 but 8 — 2 = 6 j also 4, 2, 1 6, 8, are in discontinued geomet'^ 

 rical proportion ; for ^= ^ = 2, but '-/ = 8. 



Four numbers are directly proportional y when the ratio of 

 the first to the second is the same as that of the third to 

 the fourth. As 2 : 4 : : 3 : 6. Four numbers are said 

 to be reciprocally or inversely proportional, when the first is to 

 the second as the fourth is to the third, and vice versa. 

 Thus, 2, 6, 9 and 3, are reciprocal proportionals ; 

 2:6:13:9. 



Three or four numbers are said to be in harmonical pro- 

 portion, v/hen, in the former case, the difference of the first 



and 



* In geometrical proportionals a colon Is placed between the 

 terms of each couplet, and a double colon between the couplets ; 

 in arithmetical proportionals a colon may be tmned horizontally 

 between the terms of each couplet, and two colons written be- 

 tween the couplets. Thus the above geometrical proportionals are 

 written thus, 2 : 4 : : 8 : 16, and 4:2:; 16:8.: the arithmet- 

 ical, 2 •• 4 : : 6-8, and 4 •• 2 : : 8 •• 6. 



