6 INTRODUCTION. 



If a : h-nc : d^aistob ascto dfOra l b'.'.c I d. 



28. Theorem. Of four proportionals, the product of 

 the extremes is equal to that of the means. 



Since a I hz^c ', d, a l b. bdzzc I d. bd. (17), or adzzcb. 



29. Theorem. If the product of the extremes of 

 four numbers is equal to that of the means, the numbers 

 are proportional. 



If adzz.cb, ad ; bdzzcb ', W(18), and a ', biz.c \ d; also ad I cdzzcb I 

 cd, and a '. crzb '. d. 



30. Theorem. Four proportionals are proportional 

 alternately. 



I( a :b::c : d, ad-=zbc (28), therefore a ', c\\b : d (29). 



31. Theorem. Four proportionals are proportional 

 by inversion. 



If a I &t ;c I rf, adzzbc, ad ', aczzbc '. ac, and d I czzh '. a. 



32. Theorem. Four proportionals are proportional 

 by composition. 



If « : 6: ',c : d, a-\-b : 5: ',c+d : c?; for since adzzbc, ad+bdzzbc-^bd 

 (15), or (a+J). dzzic+d). b, therefore a+b : b: :c+d : d{29). 



33. Theorem. Four proportionals are proportional 

 by division. 



If a l b'.'.c l d, a — b '. b'. '.c — d '. d; for since ad:z:bc, ad — bdzzbc — 

 bd (16), (a— 6). dn{c~d). b, and a—b : b: '.c—d ; d (29). 



34. Theorem. If any number of quantities are pro- 

 portional, the sum of the antecedents is in the same ratio 

 to the sum of the consequents. 



If a : i: '.c I d, a I b: ',a-\-c '. 6+rf; for since adzz.bc, ab-\-adzz.ab-\- 

 be, a. {b-\-d) zzb. (a+c), and a :b::a-\-c : b+d (29). 



35. Theorem. If any number of antecedents and any 

 number of consequents be added together, the ratio of the 

 sums will be less than the greatest of the single ratios, 

 when those ratios are unequal. 



