On Geometrical Proportion. 431 



For, since y m — , we have ac = b 2 : multiply by <z, and 



a a* 



o?c = a b 1 : or, = - bv Art. I., therefore a : c : : a z : b 1 . 



C V 1 J 



This proportion is often used in Dynamics. 



Article VIII. — When any number of quantities are pro- 

 portional, as one antecedent is to its consequent, so is the 

 sum of all the antecedents to the sum of ail the consequents ; 



cl c e 

 that is, if a : b : : c : d : : e : f &c, or -.- = -.- = >-, &c, 



then will a : b : : a + c + e : b + d + f 



For, since r- = — = — , &c, we have ad = be, also 



af=be, and by adding equals to equals, ad -J- af = 



be -f- b e; add a b to both sides, and a b + ad -f- af = b a 



-\- b c + b e y or a x (b + d -\-f) = b x (a + c + e); that is, 



/ a t x a aJ r c + e 7 i t 



(Art. I.) j = f±j±j> or a : b \ : a -f- c + e : b + d +f. 



Article IX. — When four quantities are proportional, any 

 like powers or roots of these quantities will be proportional : 

 that is/ if a : h ? i c : d, then will a m : b m : : c m : d m ; where 

 m may be either a whole number or a fraction. 



„ . a c , _ a m c m _ 



For, since -.- = -7, therefore ;— = -7^, or a m : b m : : c m : a 13 . 



Article X. — If the corresponding terms of two ranks of 

 proportional quantities be multiplied together, their products 

 will be proportional; that is, if a : b : : c : d, 



and e:f::g:h, 

 then will ae : If: : eg : dh. 



CL C 6 S 



For, since , = -., and --. -= y, multiply equalsby equals, 



and „= S, or ae : bf : : eg : dh ; and the same is true for 



any number of ranks of proportionals. 



Article XI. — When four quantities are proportional, if 

 the antecedents or consequents be multiplied or divided by 

 . any quantity, the products or quotients will be proportional; 

 that is, if a : b : : c : d, then will ma : mb : : ns : nd. 



For. 



