...successively equal to each other. Example : 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. Let To Prove Proof bdfh b+d+f+hf lj (2i H ^3> a) From (1). 7=7 ^4) H (5) a/= be (6) ] cf=de (7) «/ = /«...

...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)...

...A'B' = BC : B'C' = CD : C'D', etc. § 351 .-. AB + BC + etc. : A'B' + B'C' + etc. = AB : A'B', § 335 (in a series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent). That is, ' = AB:A'B'. QED Ex. 252. If the line joining the middle points of the bases of a trapezoid...

...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...

...antecedents and the denominators the consequents. The general truth was shown in that article, that in a series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent. AREAS OF POLYGONS 36. Definitions. — The area of a surface is the superficial space included within...