...solving equations in the fractional form, as will he seen in the solution of Ex. 4, Art. 408. 408. In a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. •Ei •£ « I' e for, if - = = = •••, b 'I- j we may put each of these equal ratios equal...

...other. Example : - = - = — = -, etc. bdf ti PROPOSITION X. THEOREM 443. In a continued proportion the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. (1) Let To Prove Proof bdfh o_±-2_±A+_fl! = ?. b+d+f+hf T = ~ (2) From (1). 5 = 7 (3) df - = - (4)...

...Divide, - r = - v a — bc — d Or a + b : a — b = c + d : c — d. QED PROPOSITION IX. THEOREM. 335. In a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. Let a:b = c:d = e:f = g:h. To prove that a + c + e + g:b + d+f+h = a:b. Let r = 2 = £==f = f. bdfh...

...first (or the second) as the sum of the third and fourth is to the third (or the fourth). PRINCIPLE 6. In a series of equal ratios the sum of the antecedents is to the sum of tl<e consequents as any antecedent is to its own consequent. Thus, if a : b = c : d = e :f=g : A= •••...

...first (or the second) as the sum of the third and fourth is to the third (or the fourth}. PRINCIPLE 6. In a series of equal ratios the sum of the antecedents is to tlie sum of tJie consequents as any antecedent is to its own consequent. Thus, if a: b = c: d = e :f=g:...

...And O2 = «c. §195. Hence, ¿>=j/ac. Thus, in the proportion 9: 18=18: 36, we have 18=1/8 x 36. 205. In a, series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent. ampx Suppose that _=_=£.=_== onqy Ti a Let Ъ=г' mpx . . Then — =r. -=r. -—r, etc. Axiom 7. n...

...(1) bd raising each member of (1) to the nth power, whether n is integral or fractional, 6"~d"' XI. In a series of equal ratios the sum of the antecedents...sum of the consequents as any antecedent is to its own consequent. If a_c._m_x si*. , . a с mx /г,-. let - = r, - = r, - = r, -=r, (2) bdny clearing...

...each member of (1) to the nth power, whether n is integral or fractional, an _ cn b'1 ~ cF XI. _Zn д series of equal ratios the sum of the antecedents...sum of the consequents as any antecedent is to its own consequent. К а с т х ,л^ T=J=— = -•> t1) о dn у let f.* «-,,»_,, S.,, (2) о а...

...is said to be derived from the given proportion by compositivn and division.) XI. 2. In any number of equal ratios, the sum of the antecedents is to...consequents as any antecedent is to its consequent. Hyp. If a: 6 : : c : e : :/: g : : etc., Cone.: then a + c+/+ ...: b + e + g+ ... : : a: 6 : : c: e...

...composition 7:2 = 21:0; by division, 3:2 = 0:0; and by composition and division 7 :3 = 21 : 9. 216. In a series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent. aceg For, let - = j = - = f • bd / h Denote the value of each of these ratios by r. acea Then - =...