430 On Geometrical Proportion. 



Note. — Here, unless the four quantities are all of ttie 

 same kind, the alternation cannot take place. 



Article IV. — When four quantities are proportional, the 

 first together with the second is, to the second, as the third 

 together with the fourth is to the fourth; that is, when 

 a : b : : c : d, a -f- b : b : : c -f d : d. 



For, since = -j> add unity to each side, and -- -f- 1 = 



c t » t • n . a + b 



-7 + 1 3 and reducing each to an improper fraction, — . — 



c + d 

 = ~j—> that is, a -f b : b : : c -f d : d. 



Article V. — When four quantities are proportional, the 

 excess of the first above the second is, to the second, as the 

 excess of the third above the fourth is to the fourth ; that 

 is, when a : b : : c : d, a — b : b : : c — d : d. 



— , . a c . a c a — h 



ror. since -.- = -,-, we have -.- — 1 = — — i, or — \ — 

 b a' b d b 



c — d 



= — — , that is, a — b : b : : c — d : d. 



Article VT. — When four quantities are proportional, the 

 sum of the first and second is, to their difference, as the 

 sum of the third and fourth to their difference ; that is, when 

 a : b : : c : d, then will a -{• b : a — b : : c + d : c — d. 



■ For, by Art. IV., ^±* = c -±£ and by Art. V., £j2 = 



£ d 4 



; multiply both equations by b d, and d x (a + b) ss 



b x (c + d), also d x (a — h) = b X {c — d), and if we 



divide equals by equals, the quotients will be equal; therefore 



dx(ai-l) bx(c+d) , . a+b c+d 



— j yr as ; that is, 7 = ■ - , or a + b : 



dx(a—/r) bx{c — d)' 3 a—b c — d ' 



a — i : : c + d : c - c?. 



Article VI F. — When three quantities are proportional, the 



first is, to the third, as the square of the first to the square 



. a b 

 of the second ; that is, if -.- as , or a : b : : I : c, then will 

 is c J 



a : c : : a 1 : b\ 



For, 



