...Example: ci c e _ g — — — — — —. GLC* 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. Let b~d''f~h' « + c + e + g_e b+d+f+h~f a c e _ g To Prove Proof ? = 4 <2> § = ^ ( 3 ) 7=7 w 1=7 (5)...

...equal to each other. Example : acea BOOK III 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. H=H- m To Prove «±i+£±4-«. Proof Add (6), (7), (8), and (9), and factor. /(a + c + e + g)=e(b...

...? with the ratio of any antecedent to its consequent ? PRINCIPLE 13. — In any multiple proportion the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Principle 13 may be established as follows : Let a :b = с : d = e :/= g : h. It is to be proved that...

...two terms is to the second term as the difference of the last two terms is to the fourth term. 335. In a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. 338. Like powers of the terms of a proportion are in proportion. 342. If a line is drawn through two...

...this in any particular case, it will be found sufficient to substitute ra for b and re for d. 245. In a series of equal ratios, the sum of the antecedents...the sum of the consequents as any antecedent is to ifs consequent. we may put r fur each of these ratios. ,,., « xeg I hen, . . = r, — = r, = r, -...

...particular case, it will be found sufficient to substitute ra for b and re for d. 245. In a serie» of equal ratios, the sum of the antecedents is to...consequents as any antecedent is to its consequent. For,« 6 dfh we may put r for each of these ratios. H а-* г- И.-. a = br, c = dr, e=fr, g = kr .:...

...? with the ratio of any antecedent to its consequent ? PRINCIPLE 13. — In any multiple proportion the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Principle 13 may be established as follows : Let a:b = c:d = e:f=g:h. It is to be proved that a + c...

...division.) (By composition.) a + b c + d PROPOSITION LXXXIV. THEOREM In a series of equal ratios, the sinn of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Let a:b = c:d — e:f To prove that a + с + e:b + d +/: : a : 6 a6 = 6a (The product of two quantities...

...Division. а — с : с : : Ь — d: d. 295. In a series of equal ratios, the sum of the antecedente is to the sum of the consequents as any antecedent is to us consequent. H-;-f r may be pat for each of these ratios. i---S--7"-i--- .-. a =- br, с=- dr, e...

...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...