...±2,0 : D ±^D -t qy T f which was to be proved. PEOPOSITION XI. THEOREM. In any continued proportion, the sum of the antecedents is to the sum of the consequents, as any antecedent to its corresponding consequent. From the definition of a continued proportion (D. 3), Adding and factoring,...

...2 _ ! = 1 _ i bld L' that is, _, bd or. a — b : b : : с — d : d. a. ED PROPOSITION VIII. 266. 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. We are to prove a + e+e + ¿r:6 + Denote each ratio by r. [-~,',...

...Then 2 _ ! = f _ lf that is, _, bd or, a — 6 : 6 : : с — d : d. QED PROPOSITION VIII. 266. 1n a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. Let a : b = с : d ! = e : f = g : h. We are to prove a + с + e + g : b + d + f+ h : : a : b. Denote...

...of similar polygons are proportional). .-. AB + BC, etc. : A'B' + B'C', etc. : : AB : A'B', § 266 (in a series of equal ratios the sum of the antecedents...the sum of the consequents as any antecedent is to it That is P : P' A П : A' B'. GEOMETRY. BOOK III. PROPOSITION XVII. THEOREM. 296. The homologous...

...of similar polygons are proportional). .-.A£ + BC, etc. : A'B' + B'C', etc. : : AB : A'B', § 266 (in a series of equal ratios the sum of the antecedents is to the sum of the consequents ns any antecedent, is to its consequent). That is P : P' : : A£ : A'B'. GEOMETRY. BOOK III. PROPOSITION...

...proportional). .-.AB+ BC, etc. : A'B' + B'C', etc. : : AB : A'B', § 266 (in a series of equal ratins the sum of the antecedents is to the sum of the consequents an any an1ecedent is to its consequent). That is P : P' : : A Б : A'В'. GЕOMETRY. BOOK III. PROPOSITION...

...oblique cone with a circular base. 5 66 The same proportion is true for every other element ; and since the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent, we have I. Ad' : I. dc :: AD : DC; in which the symbol 2* is used...

...their difference as the sum of the third and fourth is to their difference. COR. 3. — In any number of equal ratios, the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent. riH a + e + e + &c. a „ . . , , 5 -2 — - — = r = r = &c....

...circle as the conjugate semi-axis is to the transverse semi-axis. VIII. HIGHER ALGEBRA. 1. Prove that in a series of equal ratios, the sum of the antecedents is to the sums of the consequents as any one antecedent is to its consequent. 2. Insert three arithmetical, three...

...obtained by: VI. Composition. a + c: c : : b -\- d : d. VII. Division. a — c:c::b — d:d. ' 350. In a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. r may be put for each of these ratios. Then " = r, l=r, l=r, ?-=r, oa / ft /. a = 6r, c = dr, e =/>•,...